Applications and Analysis

Transcription

1 SPH and α-sph: Applications and Analysis John Anthony Mansour B.Sc.(Hons) Thesis sumitted for the degree of Doctor of Philosophy School of Mathematical Sciences Monash University Octoer 2007

2 for my parents

3 Contents Acknowledgements iii Declarations iv Summary v 1 Introduction 1 2 Smoothed Particle Hydrodynamics The SPH discrete approximation The approximation function The first derivative The second derivative The kernel SPH applied to fluid dynamics The continuity equation The momentum equation The energy equations Viscosity Equation of state Integrals of motion Implementation Timestepping Variale Resolution Implementations Neighouring particle list Boundaries Summary The α-sph turulence model LANS-α α-sph: equations of motion The filtered velocity The momentum equation Integrals of motion Implementation Timestepping Iteration for filtered velocity Summary i

4 4 One-Dimensional Tests Burgers equation Colliding shocks The steepening shock front One-dimensional Navier-Stokes The Euler system Forced Navier-Stokes simulations Summary and Conclusions The Kelvin-Helmholtz instaility Constant velocity fluids in relative motion Linear results Computational configuration Determination of mode growth rates Results The hyperolic tangent velocity profile Linear results Computational configuration Results Conclusion Two-Dimensional Turulence Computational configuration Forcing Viscosity Scales Equation of state Intermediate scale forcing Large scale forcing Quasi-steady solutions Steady solutions Steady solutions incorporating α-sph Conclusion Conclusion 134 A α-sph: variale-h terms 138 B α-sph: Resulting differential equations 142 C The spectral method 145 C.1 Non-linear terms C.2 Iteration for filtered velocity C.3 Timestepping C.4 Normal modes of linearised energy D Fourier mode construction using particle data 151 ii

5 Acknowledgements iii

6 Declarations This thesis contains no material which has een accepted for the award of any other degree or diploma in any university or other institution. To the est of my knowledge, this thesis contains no material previously pulished or written y another person, except where due reference is made within the text of the thesis. iv

7 Summary In this thesis a study of the Smoothed Particle Hydrodynamics (SPH) method is undertaken. Furthermore, a recent modification to SPH known as α-sph is also considered. This variant is designed for application to turulent fluid dynamics. Both SPH and α-sph are applied to test prolems, with solutions analysed within physical and spectral space. In Chapter 2 an extensive review of the SPH method is undertaken. First the details of SPH as a general numerical method are considered, followed y specifics of the SPH application to fluid dynamics, including derivation of equations of motion within a Lagrangian framework. Likewise in Chapter 3, the particulars of α-sph are considered, including derivation of the α-sph equations and discussion of conservation properties. An extensive array of one-dimensional tests are performed in Chapter 4. Where availale, results are compared with analytic solutions. Elsewhere, highly accurate spectral solutions form enchmarks for SPH and α-sph simulations. Numerous oservations are made y considering SPH solutions in spectral space. In particular, the importance of a variale smoothing length implementation to correct non-linear energy cascades and the influence of secondary SPH pressure gradient terms. Simulations of α-sph demonstrate it to e successful in inducing closure of Euler dynamics, though results for small values of the turulence parameter are found to e unsatisfactory. Linear regime studies of the Kelvin-Helmholtz simulations are presented in Chapter 5. Expected growth rates from linear staility theory are recovered for SPH simulations where sufficient resolution is utilised. For poorly resolved Kelvin-Helmholtz perturations, incorrect growth rates are shown to e directly related to deficiencies of the SPH pressure gradient. We demonstrate that the α-sph scheme is successful in reducing growth rates. Random forcing is used to induce two-dimensional turulence in Chapter 6. Large scale dynamics are found to compare favouraly with theoretical expectations, though a numer of difficulties are encountered. At the shortest scales, numerical artifacts related to insufficient resolution are apparent, though these do not appear to significantly influence large scales. Results for α-sph method are given, though simulations are restricted to small values for the turulence parameter, and findings are inconclusive. v

8 Chapter 1 Introduction The advent of digital computers in the twentieth century heralded an era of strong proliferation in the mathematical sciences. A new avenue of investigation was opened, and previously intractale prolems were now ale to e tackled using the rute numerical force afforded y the digital calculator. Indeed the potential application of computational techniques has increased in line with the ever increasing processing power of the computer. Despite recent pessimism, technological developments continue to yield advancements in line with Moore s law, which states that computational speed doules approximately every two years. Even so, for many classes of prolems, solution via pure numerical force is still very much a distant dream, and in some cases a fundamental impossiility. Perhaps the most prominent example of such a prolem is that of fluid turulence. The solution to prolems of turulence defies numerical calculation y virtue of the huge range of scales which must e resolved for accurate integration. Indeed the required processing power can e many orders of magnitude greater than what is currently availale. It is perhaps ironic then that most flows encountered in prolems of engineering interest are turulent in nature. While the straightforward approach of resolving all relevant dynamics is usually not a possiility, numerous techniques have een developed wherey approximations are introduced to reduce the computational costs to practical levels. Indeed the field of turulence research is extremely active, with an extensive lirary of pulications appearing in the literature. Researchers are drawn y the self-perpetuating complexity of turulence, the challenge of conquering an open prolem, or simply the visual allure and eauty of turulent flows. Many of the greatest scientists of our time have put their minds to prolems of turulence, though not all have met with success. Indeed Horace Lam, in speech to the British Association for the Advancement of Science, is reputed to have quipped, I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turulent motion of fluids. And aout the former I am rather optimistic. With such luminaries struggling to gain insight to the mysteries of turulence, it seems there is little hope of developing a full understanding of the physical phenomena at play, nor of eing ale to accurately simulate highly turulent flows. However turulence is a field as road as it is complex, and piece y piece, progress is made. In this thesis we consider the solution to various dynamical prolems of a tur- 1

9 2 ulent nature. To this end, we utilise the numerical method known as Smoothed Particle Hydrodynamics (SPH) (Lucy, 1977; Gingold and Monaghan, 1977). As inferred y it s name, SPH divides a flow into a set of discrete particles, with the evolution of each particle determined y the SPH equations of motion. One enefit of SPH is that it does not require a regular grid, instead only relying on the radial direction and displacement etween particles to construct the required quantaties. This endows SPH with a geometric generality which allows for it to e trivially applied to a wide range of prolems. In Chapter 2, a comprehensive overview of the SPH method is given. We consider the various developments and modifications to the method, oth from the perspective of numerical methods, and also from the view of SPH in application to fluid dynamics. SPH has seen widespread application in many fields, perhaps none more so than astrophysics and fluid dynamics. While for most prolems in these fields we can expect turulent dynamics to play some role, the significance of turulence is often overlooked or ignored. Indeed the roust nature of SPH can e a doule-edged sword, allowing simulations to proceed without giving indication of poorly-resolved dynamics. Alternatively, and as is usually the case, the cost of resolving all physically relevant scales simply puts direct simulation out of the question. In light of these limitations, a new modification of the standard SPH algorithm has een devised. The new implementation, known as α-sph (Monaghan, 2002), purports to provide a more mathematically rigorous handling of short scales than what might e encounter for standard SPH. Based on the Lagrangian-averaged Navier-Stokes differential equations of Chen et al. (1998), α-sph may e considered a minimal model of turulence, the asic premise eing that energy propagation to short scales is inhiited, while simultaneously maintaining the conservation of key physical quantaties. The details of α-sph are explored in Chapter 3, and various numerical tests in oth one and two dimensions are performed throughout the thesis to evaluate the effectiveness of this new model. However this thesis is only partially concerned with issues of turulence modelling. Perhaps the more prominent theme is the ehavior of the SPH algorithm itself, afterall we cannot make a judgement on the effectiveness of any modelling regime without first knowing the accuracy of the standard algorithm. While modern technology allows the use of large SPH particle populations, with simulations utilising of order one million particles realistic on a current desktop machine, important turulent dynamics occur at all scales, and may easily saturate the availale andwidth, potentially introducing error. In this thesis, we attempted to develop an understanding of the ehavior of SPH where dynamics are marginally resolved, oth through direct comparison with highly accurate spectral calculations, and with comparison with expected theoretical results. Presented within is the first thorough investigation of SPH dynamical ehavior in spectral space. A novel new method is utilised to take generally distriuted particle data, and recast this data in terms of trigonometric functions. In Chapter 4, this allows us to make a very direct comparison etween a one-dimensional SPH algorithm and equivalent spectral algorithm. Whereas previous investigation primarily consider physical space quantities and properties, here we are also ale to quantify SPH dynamics across the entire spectrum of modes. Insight into the significance of secondary terms orn of the SPH approximation is gained, along with the influence of the SPH smoothing length

10 3 Introduction parameter, and the cost and enefits of a Lagrangian derivation. A similar investigate is performed in two dimensions for SPH simulations of the Kelvin- Helmholtz instaility (Chapter 5). Here the interface of two counter-streaming fluids is tracked to accurately determine the growth of perturations in the linear regime. Application of varying wavelength perturations allows us to map the change in growth rates as resolution limits are approached. The aove one and two-dimensional tests are also performed for the α-sph algorithm, with results compared and contrasted with standard SPH. While one-dimensional results give some indication of the performance of SPH and α-sph, we do not expect results to necessarily generalise to higher dimensions given the geometric simplicity and asence of transverse waves. Likewise, linear staility results only hint at what might e found for road spectrum turulent simulations. Bearing this in mind, fully turulent two-dimensional simulations have een performed in Chapter 6, with forcing applied in a periodic ox to induce a Kraichnan (1967) regime. Through the use of the spectral recomposition method, comparison is ale to e made with theoretical predictions and to findings in the literature, the vast majority of which is presented in the spectral domain. This is the first such simulation using the SPH method, and indeed one of a small numer of Kraichnan turulence simulations where spectral methods have not een applied. This regime provides a challenging vehicle y which the intricacies of SPH may e studied, along with a truly turulent foundation for the evaluation of the α-sph modification.

11 Chapter 2 Smoothed Particle Hydrodynamics Recent advancements in computer technology has rought the practical aility to perform numerics on a very large scale. This has allowed for the analysis of differential equations through the use of discretisation techniques, and many such techniques exists. One such method, and the suject of the current thesis, is the method of Smoothed Particle Hydrodynamics (SPH) (Lucy, 1977; Gingold and Monaghan, 1977). Here, a physical domain is decomposed into a numer of nodes, or particles, which then interact with other nodes according to discrete approximations to prolem dynamics. While the quality of the decomposition will to an extent e determined y the arrangement of nodes, SPH does not place any particular requirements on node configuration. Further, where we use SPH techniques to simulate hydrodynamics, nodes are advected with the fluid flow. We now make the analogy of nodes as physical particles, carrying with them physical attriutes such as mass and temperature. In this sense, SPH forms a native approximation to Lagrangian fluid dynamics, and allows for construction of simulations in an intuitive fashion. Unless otherwise explicitly stated, we discuss SPH with respect to its application to fluid dynamics. SPH is one of a family of techniques known as methless methods. As the name suggests, these methods do not rely on a mesh or grid to construct function approximations, or derivative approximations y which differential equations may e calculated. This relieves simulations of the inherent geometry found in other methods such as finite differences. Such geometric deficiencies can lead to difficulty or errors in a numer of situations, such as simulations involving discontinuities not aligned with the numerical grid, with excessive numerical diffusion potentially resulting. Mesh ased methods also do not lend themselves naturally to simulations involving complex geometries, with special measures eing required, such as the construction of prolem specific grids. In contrast, SPH calculations only rely on the radial interactions of particles and as such show minimal preference to particular geometry configurations. Furthermore, algorithms may e easily constructed for prolems involving interacting ojects, such as moving walls or floating odies, and free surfaces are also handled with minimal difficulty. For techniques requiring grids, such simulations certainly present great difficulty and increased complexity. 4

12 5 Smoothed Particle Hydrodynamics SPH naturally leads to solutions with a resolution which varies in space and time. This results from the Lagrangian nature of SPH, where our numerical nodes are considered as particles carrying mass, and therefore concentrate resolution in regions of high density. Indeed, on account of this density, such regions often hold relatively large proportions of energy, and it follows that accurate calculations of their dynamics should take precedence. Though grid refinement methods are certainly capale of achieving similar results, this comes at additional cost and refinement implementations are often neither straightforward nor intuitive. SPH provides such capaility natively, though certain simulations, such as those of nearly incompressile fluids, may not enefit from this inherent variale resolution. We also find that SPH algorithms prove roust in simulations where turulent dynamics are under-resolved, tending to redistriute energy away from poorly represented scales in a non-dissipative fashion (see Chapter 4). This ehavior is similar to that required of turulence modelling schemes, and indeed we may consider it as implicit turulence modelling. This ehavior of SPH allows integrations to continue though under-resolved, and mean dynamics are often still found to e reproduced correctly (see Chapter 4). Analogue simulations using spectral methods lead to energy accumulating at the resolution limit, and solution corruption usually follows. It should e noted that this may e advantageous at times, indicating insufficient resolution. Furthermore, correct dynamical ehavior cannot e expected where we rely on implicit SPH turulence modelling, though an aim of this thesis is to investigate a quantifiale and physical approach to turulence modelling which may minimise reliance on this implicit ehavior. The earliest inceptions of smooth particle hydrodynamics (SPH) date to the work of Lucy (1977) and Gingold and Monaghan (1977) which rings SPH into its thirtieth year of development. In this time, it has seen application to many areas of numerical modelling, most notaly perhaps eing the vast array of pulications concerning SPH applications to astrophysics: interstellar gas dynamics and star formation (Bromm et al., 1999; Lattanzio et al., 1985), magnetohydrodynamics (Price and Monaghan, 2004; Phillips and Monaghan, 1985) and cosmology (Springel and Hernquist, 2002; Thacker et al., 2000) to name ut a few (see Monaghan (1992) for further references). Application has also een found in areas such as fracture dynamics ((Benz and Asphaug, 1995; Bonet and Kulasegaram, 2005), elastic (Gray et al., 2001) and viscoelastic (Ellero et al., 2002) flows, free surface and incompressile flows (Monaghan, 1994; Morris et al., 1997; Cummins et al., 1997). Recently SPH has also found use in the animation industry owing to its roust nature and straightforward application to complex geometries. This chapter presents an overview of the theory, derivation, and implementation of SPH, much of which is found in the review articles of Monaghan (1992, 2005). It is structured as follows. In the first section we consider SPH from a numerical perspective, with formulations of approximation functions and derivatives outlined, along with respective errors, and appropriate forms for our approximation kernel. The second section is concerned with application of SPH to fluid dynamics. Here we outline a variational derivation of our constitutive equations of continuity, momentum and energy. Different forms of equation of state and viscosity are presented, along with consideration of integrals of motion. Finally we turn to the practical considerations of computing SPH approximations, such as timestepping and staility, as well as the self-consistent

13 2.1 The SPH discrete approximation 6 resolution formulation and oundary implementations. 2.1 The SPH discrete approximation The approximation function The foundation of the SPH scheme is the weighted average integral, which leads to an approximation A h (r) to some function A(r): A h (r) = A(r )W (r r, h) dr, (2.1) R where W is our kernel (or weight function), r is a position vector, and dr is an element of the spatial domain. The action of parameter h will depend on the choice of kernel, ut generally it will determine the domain over which the averaging occurs. Throughout this work it will e referred to as the smoothing length parameter. To simplify notation, we will not explicitly state the dependence of the kernel on h for the time eing, nor state the integration domain R. A Taylor expansion of A(r ) aout r gives A h (r) = A(r) W (r r ) dr + W (r r ) ((r r) r ) A(r ) dr (2.2) r =r + 1 [ ] W (r r ) ((r r) r ) 2 A(r ) + O((r r) 3 ) dr. 2 r =r An example in two dimensions is given to clarify notation: ((r r) r ) A(r ) = r =r (x x) A x (r) + (y y) A y (r), for r = (x, y). Consistency requires W (r r ) dr = 1, (2.3) and where the kernel is symmetric aout its argument, the second term in (2.2) disappears and we can write A h = A + O(h 2 ). We also note that the aove expansions require that our function A e smooth on some scale r = r r. The leap to SPH egins with the discretisation of equation (2.1). In one dimension this can e achieved y simple application of the trapezoidal rule. For a set of nodes at positions r with respective nodal values A we have A h S(r) = A W (r r ) r A h (r) with r = 1 2 (r R r L ). Here r L and r R are the nodes respectively to the left and right of the node at r. This sum is performed over all nodes, however since kernels with compact

14 7 Smoothed Particle Hydrodynamics support are normally chosen, typically a small suset of nodes (those within close proximity to position r) are the only non-zero contriutions. Stepping up to higher dimensions, this decomposition of the domain into finite elements ceases to ecome trivial. For prolems of fluid dynamics, the method of SPH deals with this y assigning a mass to each node. Rewriting equation (2.1) as A(r A h ) (r) = ρ(r ) ρ(r )W (r r ) dr, (2.4) where ρ is a material density, we can now make the following interpretation: ρ(r )dr = dm(r ) where dm(r ) is an element of mass. Nodes can now carry with them a certain mass, and as such form a good analogy to physical particles. Numerical quadrature is thus achieved, and we can write the following approximation A h S(r) = m ρ A W (r r ) A h (r) (2.5) where m and ρ are the mass and density associated with particle, while A = A(r ). This can e recast using the notation of finite elements, where we can now define a shape function as and the approximation is then written φ = m ρ W (r r ) A h S(r) = φ (r)a. The discretisations outlined aove form the foundation upon which all SPH schemes can e constructed. Naturally, this secondary step of approximation rings with it some added degree of error. Given that there is no prescription on the arrangement of nodes, quantifying this additional error is no trivial task and will e different for each simulation. Considering Taylor expansions can yield some insight however. Equation (2.2) is rewritten with summations replacing integrals: A h S(r) A(r) V W (r r ) + V W (r r ) + 1 V W (r r ) 2 [ ] ((r r) r ) A(r ) r =r ] [ ((r r) r ) 2 A(r ) r =r (2.6) for some volume element V associated with node. Inspection of equation (2.6) reveals that our normalisation (2.3) is now only approximate, and first order terms no longer vanish exactly. As such, we do not expect to reproduce constant functions exactly. Analysis of Monaghan (2005) for one-dimensional equispaced data indicates that a Gaussian kernel (see section 2.1.4) will lead

15 2.1 The SPH discrete approximation 8 to errors which diminish exponentially in (h/ r) 2, for particle spacing r. So reasonale accuracy can perhaps e expected where data is approximately equispaced, and for appropriately chosen smoothing lengths. A correction which restores exact constant reproduction is attriuted to Shepard (1968). It can e derived easily. We rewrite equation (2.5): A h S(r) = V A W (r r ), and now for constant A = k we have k = A h S (r) V W (r r ), so we can now define a new corrected scheme: A h shep(r) = A h S (r) V W (r r ), which is exact where A is constant. The extra computation involved in calculating the aove come at negligile cost. Further corrections to the summation approximation can e constructed to ensure linear completeness, such as the moving least squares method (Krongauz and Belytschko, 1996) or the reproducing kernel method (Liu et al., 1995). These methods introduce significant additional operation count to calculations, though for certain simulations this extra cost may prove worthwhile The first derivative We may otain an approximation to a first derivative y exact differentiation of the approximation integral (2.1) ( A) h (r) = A(r )W (r r ) dr. (2.7) Integration y parts then yields ( A) h (r) = A(r )W (r r ) ds R R A(r ) r W (r r ) dr. (2.8) Here ds is an element along the domain oundary. In application, we usually have the first term on the left of (2.8) disappearing, either exactly or approximately. We then write ( A) h (r) = A(r ) r W (r r ) dr = A(r ) r W (r r ) dr = A h (r). (2.9) We now take Taylor series expansions of A(r ) aout r to find A h (r) = A(r) r W (r r ) dr + r W (r r ) ((r r) r ) A(r ) dr (2.10) r =r + 1 [ ] r W (r r ) ((r r) r ) 2 A(r ) + O((r r) 3 ) dr. 2 r =r

16 9 Smoothed Particle Hydrodynamics The first term in the aove equation will vanish as long as the kernel is chosen to e symmetric aout its argument. Via integration y parts, the second term will reduce to the required gradient. This however relies on the kernel eing normalised correctly (2.3). We are left with A h (r) = A(r) (2.11) [ ] + ((r r) r ) 2 A(r ) + O((r r) 3 ) r W (r r ) dr. r =r In SPH, we approximated equation (2.9) with a summation: A h S(r) = V A(r ) r W (r r ) A h (r). (2.12) Unfortunately, in making this approximation, we only approximately find the aove simplifications. Equation (2.10) is written, with truncation of high order terms, A h S(r) A(r) + V r W (r r ) V ((r r) r A(r)) r W (r r ) (2.13) As the first term in (2.13) is not identically zero, this approximation does not yield correct derivatives for constants. We say that it lacks zeroth-order completeness. For most calculations, we are usually only concerned with evaluating functions at the nodes. Equation (2.12) ecomes A h S(r a ) = V A(r ) r W (r r ) r=ra A(r a ) + V r W (r r ) r=ra (2.14) V ( (r r) r A(r) ) r W (r r ) r=ra. We can ensure that the first term on the right hand side of equation (2.14) vanishes for constant functions y writing equations in symmetric form (Monaghan, 1988): A h S(r a ) = V ( A(r ) A(r a ) ) r W (r r ) r=ra. (2.15) However, this correction is only applicale when the function is eing evaluated at a node. A generalised route to all forms of this correction is given in Monaghan (2005), which suggests we write, for any differentiale function Φ, Our SPH summation (2.14) then yields A h S(r a ) = A = 1 Φ( (AΦ) A (Φ) ). (2.16) 1 ( V Φ A(r ) A(r a ) ) r W (r r ) Φ r=ra a

17 2.1 The SPH discrete approximation 10 We find equation (2.15) y setting Φ = 1. A correction leading to equivalent order accuracy may e had y taking derivatives of the Shepard function (2.1.1). This has the added advantage of eing applicale anywhere in the domain. Further corrections to (2.14) can e had y renormalising to remove the coefficient that will appear in front of the gradient term (this coefficient is precisely unity for the integral approximation (2.11), ut only approximates this for the summation (2.14)). Modifications to restore linear completeness at all points in the domain have also een suggested. Johnson and Beissel (1996) provided corrections for calculations performed on axisymmetric fields, while Krongauz and Belytschko (1997) presented results for general fields. These modifications tend to e computationally cumersome however, and also give unreliale conservation of key physical quantities due to asymmetry (see section 2.2.6). All calculations presented here only make use of the symmetrisation correction (2.15). SPH forms for other first derivative operators can e derived in a similar fashion (Monaghan, 1992) The second derivative The second derivative may e otained in a similar fashion to the first derivative. In analogy to equation (2.7), we write ( 2 A) h (r) = 2 A(r )W (r r ) dr. (2.17) Now we apply integration y parts twice, again assuming that surface terms vanish, to find 2 A h (r) = A(r ) 2 rw (r r ) dr. (2.18) As previously, we can now approximate this with summations. However, it turns out that approximations to second derivatives found in this way are excessively sensitive to the node configurations. Because of this, a numer of approaches exist in the literature wherey the second derivative is determined indirectly, usually requiring only the first derivative of the kernel. One such approach egins with an integral approximation to the second derivative (Brookshaw, 1985; Cleary and Monaghan, 1999; Español and Revenga, 2003). Perhaps the most general of these approximation equations is given y Español and Revenga with (A(r) A(r ) ) F ( (r r ) ) [5 (r r ) α (r r ) β (r r ) 2 δ αβ] dr F ( (r r ) ) = (r r ) r W (r r ) (r r ) 2 = 2 A(r) r α r β + O(h2 ) (2.19) and where Greek indices indicate Cartesian component of suject. The aove applies to integrations performed in three dimensions and assumes kernels which

18 11 Smoothed Particle Hydrodynamics exhiit spherically symmetry. An SPH equivalent can then e written ( 2 ) A(r) r α r β ( V A(ra ) A(r ) ) F ( (r a r ) ) [5 (r a r ) α (r a r ) β a (r a r ) 2 δ αβ]. (2.20) We will e concerned with constructing the Laplacian in later sections. Using equation (2.19), 2 A(r) = 2 A x A y A z 2 2 (A(r) A(r ) ) F ( (r r ) ) dr, (2.21) with (r 1, r 2, r 3 ) = (x, y, z). This rings us to a similar integral approximant to that given in Brookshaw (1985) and Cleary and Monaghan (1999). That this indeed leads to the Laplacian of A(r) is easily verified using Taylor expansions of A(r ). The SPH equivalent then takes the simpler form ( 2 A(r) ) 2 ( V a A(ra ) A(r ) ) F ( (r a r ) ). (2.22) Other approaches which utilises only first order kernel derivatives are given in Flee et al. (1994) and Watkins et al. (1996). Though sutly different, oth methods largely consist of performing SPH differentiation twice to arrive at a second derivative. To calculate 2 A, we can write F (r a ) = ( A(r)) a = V ( A(r ) A(r a ) ) r W (r r ) r=ra. Then taking the divergence of the aove, we have ( 2 A(r) ) = ( F (r)) a a = ( V F (r ) F (r a ) ) r W (r r ) r=ra. Watkins performed tests comparing the aove methods under a numer of node and vector field configurations. These results suggest that calculating second derivatives using recursive first derivatives leads to superior results, especially where nodes are arranged randomly, though testing was far from exhaustive. In general, SPH simulations of hydrodynamics prolems will not lead to such degree of disorder (Monaghan, 2005). It follows that the significance of Watkins results for such simulations is questionale. Furthermore, any potential improvements in accuracy come at a cost, as further summations are now required to e calculated separately The kernel Thus far, our definition of the SPH method has not specified the properties the smoothing kernel W should exhiit, except that it is required to meet the normalisation defined y equation (2.3). Further, the kernel should tend towards the Dirac delta function in the limit of diminishing smoothing length: lim W (r h 0 r, h) = δ(r r ). (2.23)

19 2.1 The SPH discrete approximation 12 Infinitely many kernels may e constructed to satisfy these properties, and throughout the literature a large range have een used. While not strictly necessary, it is usually desirale for any given kernel to e a function of radial distance alone. This ensures the required even symmetry for our approximation function (2.1) to exhiit second order accuracy in h. Furthermore, this symmetry is required for the conservation of key physical quantities such as momentum (see section 2.2.6). A general form for these kernels can e written W ( r, h) = κ f(q), (2.24) hν where r = r r and q = r /h. The parameter ν is the kernel s dimensionality, while κ is a constant, determined such that (2.24) satisfies equation (2.3). In the early SPH work of Gingold and Monaghan (1977) the Gaussian kernel was employed. It takes the form W ( r, h) = κ h ν exp( q2 ). (2.25) For normalisation, we require the constant κ takes the values 1/π 1/2, 1/π and 1/π 3/2 for one, two and three dimensions respectively. An alternate kernel, constructed of cuic splines, is found in Monaghan and Lattanzio (1985): W ( r, h) = κ h ν q q3 0 q (2 q)3 1 q q. (2.26) Here κ takes the values 2/3, 10/7π and 1/π for one, two and three dimensions respectively. Both the Gaussian and the cuic spline kernel form appropriate kernels for SPH summations. The Gaussian kernel has the added enefit of eing infinitely differentiale, which leads to improved staility qualities (Price and Monaghan, 2004). This however comes at the cost of computational efficiency, as the Gaussian kernel does not have compact support. Therefore, all particles contriute to the summation in equation (2.5), and computations scale as N 2, where N is the numer of particles/nodes. In practise though, contriutions from particles eyond a few smoothing lengths are negligile, and can e ignored. This truncation allows for efficient computations, ut is a trade off with accuracy. It is preferale to choose kernels with finite supports, such as the cuic spline kernel. No truncations are required for such kernels; summation contriutions are zero eyond their domain of influence. For example the cuic spline falls to zero for distances eyond 2h, allowing for computations which scale as N. The cuic spline kernel is used for all calculations found in this work. Owing to its construction, the cuic spline kernel only exhiits smooth first order derivatives. This may lead to inferior staility traits (Morris, 1996) in comparison with smoother kernels such as the Gaussian, even though our calculations do not use kernel second derivatives (see section 2.1.3). Higher order spline kernels may certainly e constructed to address this issue, such as the

20 13 Smoothed Particle Hydrodynamics quartic spline kernel (Schoenerg, 1946): W ( r, h) = κ h ν ( 5 2 q)4 5( 3 2 q)4 + 10( 1 2 q)4 0 q 1 2 ( 5 2 q)4 5( 3 2 q)4 1 2 q 3 2 ( 5 2 q)4 3 2 q q. (2.27) In practise, we do not expect such kernels to impart significant additional accuracy, despite improved staility qualities. Most importantly, though many forms are possile which satisfy the conditions aove, est accuracy is found when the kernel takes a symmetric ell type profile (Fulk and Quinn, 1996; Hongin and Xin, 2005). Accurate approximations are found where we take h > x for node seperation x. For the cuic spline, smoothing length usually takes the value h = 1.3 x. Larger smoothing lengths give etter discretisations of (2.1.1), owing to the larger numer of nodes representing it. Though the effective resolution of the approximation will e reduced (approximation is now smoother ), and computational cost will e increased significiantly. Expected superior accuracy of large scales is generally outweighed y increased computational cost, and inferior representation of short scales. For multidimensional modelling, anisotropic kernels may e costructed with the symmetry W (r) = W ( r), such as those resulting from tensor products of one-dimensional kernels. This may e desirale where the distriution is largely anisotropic (Shapiro et al., 1996), resulting in insufficient summation contriution in certain directions, and correspondingly inferior surface representations. Such kernels will not affect linear momentum conservation, and where anisotropy is defined y local particle configuration, angular momentum can also e conserved. 2.2 SPH applied to fluid dynamics We now turn to the application of SPH to prolems of hydrodynamics, from which the method of SPH was orn. While in the previous section we considered SPH from a purely numerical point of view, we now introduce physics to the prolem. We make the interpretation of nodes as physical particles, each carrying with them attriutes such as mass and velocity. This analogy proves useful in developing a sense of intuition when working with SPH The continuity equation SPH algorithms may e constructed to satisfy continuity via a numer of routes. The simplest is perhaps a straightforward application of equation (2.5) to reveal ρ a = m W a. (2.28) We have made the following notation simplifications in the aove: ρ a = ρ(r a ), W a = W (r a r ). The explicit dependence of the kernel on smoothing length is omitted for the time eing. Alternatively, we can write an equation for the

21 2.2 SPH applied to fluid dynamics 14 time rate of change density. Beginning with the continuity equation written for the Lagrangian frame, dρ = ρ v, (2.29) dt which we recast in SPH summation form: dρ a dt = ρ m a v a a W a. (2.30) ρ Here the notation a specifies the gradient with respect to coordinate r a, while v a = v a v. The particular choice of continuity formulation will largely depend on the dynamical prolem at hand, though perhaps use of the density time derivative is usually preferale. In particular, for simulations involving fluids of different densities, the summation (2.28) will result in false density gradients at fluid-fluid interfaces. Similarly, at free surfaces, summations will e incorrectly normalised, leading to lower than expected near surface densities (see Bonet and Rodriguez-Paz (2005) for a formulation which accounts for this). Equation (2.30) can also take a numer of forms. For instance, taking a direct Lagrangian derivative of (2.28) gives us dρ a dt = m v a a W a. (2.31) In analogy to equation (2.16), Price (2004) writes a general form from which all such equations may e found. For some scalar function φ, the divergence (2.29) is given as [ dρ dt = φ v from which we may now write ( ) ρ φ ( )] ρv φ (2.32) dρ a dt = φ m a v a a W a. (2.33) φ If we take φ = ρ 2 σ, setting σ = 1 leads to equation (2.30), while for σ = 2 we yield (2.31). We note that different forms for the continuity equation will in general lead to different momentum equations where the momentum equation is derived through a variational framework (see section 2.2.2). From this perspective, some authors have given versions of the momentum equation which are inconsistent with their choice of continuity equation. The enefits of strict consistency are not necessarily ovious, though variational derivations may assure conservation of various integrals of motion, such as linear and angular momentum. Key differences etween continuity implementation are often to e found at the interface etween fluids of differing density. It is common practice in SPH to integrate such systems as a single fluid with different mass particles, though in reality the two fluids should e considered separately, with oundary forces acting on each at the interface. So we then wish to determine which method gives the est approximation to such configurations. Differences arise in the ways with which equations (2.30) and (2.31) calculate the divergence v (Monaghan, 2005). Calculations of divergence near an interface using

22 15 Smoothed Particle Hydrodynamics (2.31) result in divergences with dependence on particle mass. However, v is not a function of mass. Equation (2.30) on the other hand includes the term m /ρ within the summation, which we interpret as the volume element occupied y particle, ie m /ρ = V. Calculations of divergence via (2.30) give no dependence on particle mass, as we require. Improved accuracy is found for multi-fluid simulations using (2.30) (Colagrossi, 2004). Monaghan (2005) asserts that either (2.30) or (2.31) provide sufficient accuracy where density ratios are less than 2 : 1. As discussed previously, using mass summations (2.28) leads to density gradients across the interface. This is highly undesirale as such gradients will cause unphysical pressure forces. This may e realised as surface oscillations in simulations involved free surfaces (Monaghan, 1992). Several authors (Ott and Schnetter, 2003; Tartakovsky and Meakin, 2005) have suggested an alternate summation yielding numer densities can e used to avoid density gradients at multi-fluid interfaces: n a = W a, (2.34) where n a is the numer density at particle a. Tartakovsky and Meakin used the aove with a modified momentum equation to perform Rayleigh-Taylor simulations for miscile flows, with particle masses evolved according to a diffusion equation. Summation equation (2.34) proves advantageous in implementations where particle mass may vary in time, with equations of state written in terms of numer densities. However such summations will not correct erroneous normalisation at free surfaces, and false density gradients ensue The momentum equation Neglecting viscosity, our acceleration equation reduces to the Euler equation, dv dt = 1 P, (2.35) ρ for a pressure force P. Application of equation (2.15) gives dv a dt = 1 ρ a m ρ (P P a ) a W a. (2.36) While the aove returns zero acceleration for constant pressure fields, it fails to conserve linear or angular momentum exactly. We instead prefer to construct our momentum equation such that it exhiits symmetry leading to line of sight forcings etween particle pairs. A natural path to such formulations egins with the Lagrangian ( ) 1 L = ρv v u(ρ, s) dr, (2.37) 2 for thermal energy per unit mass u, as function of density ρ and entropy s (Eckart, 1960). We now write a discrete approximation to this: L = ( ) 1 m 2 v v u(ρ, s ). (2.38)

23 2.2 SPH applied to fluid dynamics 16 The equation of motion for particle a is then found through an application of the Euler-Lagrange equation: ( ) d L dt va i L ra i = 0, (2.39) where the superscript i refers to a vector component. Where we take (r, v) as our canonical variales, equation (2.39) ecomes dva i m a dt + ( ) u ρ m ρ s ra i = 0. (2.40) An application of the first law of thermodynamics yields ( ) u = P ρ ρ 2. We now differentiate our summation density (2.28) to find s ρ ra i = c m c W c r i a (δ a δ ac ) and putting all this together we have m a dv i a dt = = = c c c P W c m m c ρ 2 ra i (δ a δ ac ) P W c m m c ρ 2 δ a ra i + m a m c P a ρ 2 a W ac r i a + c m m a P ρ 2 P W c m m c ρ 2 δ ac ra i W a ra i. Where our kernel is radially symmetric we finally have (returning to vector notation), dv a = ( Pa m dt ρ 2 + P ) a ρ 2 a W a. (2.41) An alternate route to this equation is demonstrated in Monaghan (1992), where the term on the right hand side of (2.35) is rewritten ( ) P P ρ = + P ρ ρ 2 ρ, and then recast using the SPH summation (2.12) to find equation (2.41). Many symmetric variants are possile, though if variational consistency is desired, care should e taken in selecting the appropriate form for the continuity equation. Indeed, Bonet and Lok (1999) demonstrate that calculations using mis-matched continuity and momentum equations lead to inferior results. Where continuity is achieved through the use of rate of change equations (such as (2.30)), a slightly different approach is required for the variational derivation of momentum. We egin y writing our action (Price and Monaghan, 2004) S = t2 t 1 L(r 1,..., r N, v 1,..., v N )dt,

24 17 Smoothed Particle Hydrodynamics where we have N particles. We require the action to e stationary for any small perturation to the path etween the given limits. Taking a small deviation δr a to coordinates of particle a, we have to first order δs = t2 t 1 ( δr a L + dδr a L r a dt v a ) dt, and integration y parts on the second term gives t2 ( L δs = δr a + d ) L dt. t 1 r a dt v a We note here that the secondary term (which arises due to integration y parts) disappears, since our coordinate deviation goes to zero at the limits. Still to first order, we can now write ( ) δs = t2 t 1 δr a m a dv a dt + m u ρ δρ r a where δρ represents the co-moving variation in ρ, which will depend on our choice of continuity equation. As we require the first order change in action to e zero for any such variale perturation, we then have dt, m a dv a dt = m u ρ δρ r a. (2.42) As an example, we consider the general form of continuity (equation (2.33)) given in Price (2004): m c δρ = φ (δr δr c ) W c. φ c c Inserting this into (2.42), and making use of the first law of thermodynamics, yields dv a = ( Pa φ a m dt ρ 2 + P ) φ a φ ρ 2 a W a, (2.43) φ a where we have again exploited the kernel s symmetry. For density equation (2.30), we set φ = ρ, so we then have dv a dt = m ( Pa + P ρ a ρ ) a W a, (2.44) while letting φ = 1 leads to momentum equation (2.41) found earlier via the density summation (2.28) The energy equations The equation for rate of change of thermal energy can e easily determined starting with the first law of thermodynamics, du = T ds P dv = T ds + P ρ 2,

25 2.2 SPH applied to fluid dynamics 18 which then for constant entropy, leads to du dt = P dρ ρ 2 dt. From here we may apply our particular choice of SPH continuity equation. Using equation (2.30), we have du a dt = P a ρ a m ρ v a a W a. (2.45) For variationally consistent derivations, the total energy of the system will e equal to the total thermal and kinetic energy, which we may write ê = m ( 1 2 v v + u ), (2.46) and then taking the co-moving derivative: dê dt = ( m v dv dt + du ). dt Now using equations (2.44) and (2.45), we arrive at dê dt = m c { Pc m c v + P } v c ρ ρ c ρ ρ c W c, and we can now infer that the change of total energy of particle can e written de dt = { Pc m c v + P } v c W c. (2.47) ρ c ρ c ρ ρ c Viscosity As with most aspects of SPH, there exists many different approaches y which viscosity may e implemented. Again, the choice will e largely determined y the particular prolem in consideration. Many methodologies are geared towards evolving astrophysical dynamics, often where high Mach numer velocities, and correspondingly shocks, may e found. Viscosity is often then added only as a tool; a method to tame numerical artifacts orn of shocks and discontinuities. Monaghan (1992) gives the following such form for viscosity term α c a ζ a + βζa 2 v Π a = a r a < 0 ρ a (2.48) 0 v a r a 0 where ζ a = hv a r a r a r a + η 2. (2.49) This viscosity enters into our SPH algorithm via the momentum equation, where we write dv a = m (F (P a, P, ρ a, ρ ) + Π a ) a W a. (2.50) dt

26 19 Smoothed Particle Hydrodynamics Here F is some pressure term, taking various forms as outlined earlier. For the viscous term (2.48), we have that ρ a and c a represent average density and sound speeds of particles a and 1, and also that r a = r a r. The parameters α and β are usually set to α = 1 and β = 2 for astrophysical simulations, though resulting integrations are not overly sensitive to the precise value used (Monaghan, 1992). The term associated with α results in Navier-Stokes type ulk and shear viscosity, as Taylor expansions will reveal. Further, we have the term linked to our β parameter which mimics a von Neumann-Richtmyer type viscosity. It will largely come into effect with high Mach numer shocks, helping to prevent particle penetration. Indeed, this artificial viscosity is constructed explicitly to handle shocks and is not required elsewhere in the domain, hence the action of (2.48) wherey the viscosity is set to zero where fluid elements are moving apart. Also, parameter η is usually taken to e some small fraction of smoothing length h, and is included simply to prevent singularities. For this thesis we are concerned with constructing approximations to Navier- Stokes type dynamics. Mainly considered are fluids of a largely incompressile nature (see section 2.2.5), where we do not expect shocks to present given low Mach numers. We as such require a more appropriate formulation of viscosity than that given aove, and a numer of possiilities are suggested in the literature. The straightforward approach of Flee et al. (1994) and Watkins et al. (1996) is to directly compute all viscous terms of the Navier-Stokes equations. Both authors similarly use recursive application of the first order derivatives outlined in section to arrive at higher order derivatives (see section 2.1.3). An implementation applied to low Reynolds numers is given y Morris et al. (1997) using an alternate formulation for the SPH velocity derivatives. Cleary (1998) outlines a viscosity using derivatives found in a similar fashion to term (2.49), and allowing for simulation of fluids of differing densities and viscosities: Π a = ξ ( ) 4µ a µ va r a ρ a ρ µ a + µ r a r a + η 2. (2.51) Here we have the dynamic viscosities µ a and µ for particles a and. The factor ξ is a type of normalisation constant determined such that our dynamic viscosity µ coincides with that of the Navier-Stokes equations. It will depend on the dimensionality of the prolem, as well as the particular choice of kernel and smoothing length h. It may e determined either empirically with numerical experiment, or via Taylor expansions. The viscosity formulation used for most work presented here is written Π a = c ( ) aα va r a (2.52) ρ a r a (Monaghan, 1997), where we modulate viscosity strength using the α parameter. Taylor expansions of (2.52) lead us to a relation etween the α parameter and a kinematic viscosity: α = ν κ c a h, (2.53) with κ eing similar to the ξ of Cleary s viscosity. It s value will also e determined y the choice of kernel and dimensionality. For example, in onedimension where the cuic spline kernel is used, we have κ = 14/15, while in 1 For these averaged values, we have utilised the harmonic mean.

27 2.2 SPH applied to fluid dynamics 20 two-dimensions the value κ = 15/224 is appropriate. It is also worth noting the effect of the density term ρ a in (2.52). We find that our viscosity in the one-dimensional continuum limit leads to terms of the form A 1 ρ a x ( ρ v x for some constant A. Now replacing ρ a with ρ, we instead find ( 2 ) v A x 2 So somewhat sutle changes to our viscosity can lead to large changes in dynamics. The difference etween the two aove terms will e minimal for nearly incompressile flows, where density will e almost constant. Where large density gradients exist, significant departure is found. This is demonstrated in section where Burgers equation is considered. Averaged terms such as ρ a are often chosen to maintain the symmetry from which conservation stems. However at times compromise is required and sacrifice of conservation may e preferale to achieve other goals. For instance, Watkins et al. (1996) required a precise specification of the ulk and shear viscosities, ut this comes at the cost of angular momentum conservation. Likewise, Morris et al. (1997) also sacrifice exact angular momentum for an implementation they state performs etter for low Reynolds numers. The viscosity used here, (2.52) conserves oth linear and angular momentum, while also having the additional desirale attriutes of eing Galilean invariant, and vanishing for solid ody rotation. The addition of viscous dynamics necessitates modification of the thermal energy equation (2.45) to account for energy eing leached from kinetic to thermal. To determine our modified thermal energy equation, we note that given conservation of energy, we can write de k dt ). a = de T dt, where E K is a total kinetic energy, while E T is total thermal energy. We now write de k = d ( ) 1 dt dt 2 m av a v a = a = a = a a m a v a dv a dt m a v a m a { P a ρ a m ( Pa + P ρ a ρ, a ) + Π a a W a m ρ v a a W a } m Π a v a a W a, and so we conclude du a dt = P a ρ a m ρ v a a W a m Π a v a a W a. (2.54)

28 21 Smoothed Particle Hydrodynamics This is equivalent to equation (2.45) with the addition of a term due to viscosity, which can e shown (Monaghan, 1997) always acts to increase thermal energy Equation of state The SPH method natively leads to approximations of the compressile Navier- Stokes equations. As such, we need to complement the equations outlined aove with a further equation which will relate our state variales of pressure and density. The equation of state may take a numer of forms which depend on the physics of the medium we are modelling. Most of the work undertaken in this thesis considers almost incompressile fluid regimes where thermal effects are unimportant, so we consider pressure a function of density only. In nature, we may find such regimes in fluids such as water, and quiet often the gaseous dynamics of the earth s atmosphere. Most of the literature on fluid dynamics makes an approximation to these systems, considering them to e fully incompressile and thus allowing simplification of the Navier-Stokes equations. We are required to make a similar approximation, though we instead construct models which are more compressile than the actual system. Fluids such as water have very large sound speeds; this results in very short timesteps (see section 2.3.1), and expensive computations. We hence make approximations to these fluids using smaller sounds speeds. While this allows for numerically tractale simulations, it leads us to the forementioned increased compressiility. We may modulate sound speed through the equation of state. One common choice is the equation of state given in Batchelor (1967): {( ) γ ρ P = B 1} (2.55) ρ 0 where we normally take γ = 7, which results in large variations in pressure for small density changes. We also have the reference density ρ 0, and the factor B which will e used to determine sound speed. It can e shown that for sound speed c s, we have c 2 s = P ρ = γb ( ) γ 1 ρ, (2.56) ρ 0 ρ 0 so that the unpertured fluid takes the sound speed c 2 s = γb/ρ 0. A scale analysis of the Euler equation (2.35) indicates the following relation: V 2 max c 2 s = δρ ρ, for some characterising maximum velocity V max of our fluid. We wish to restrict density fluctuations to mimic incompressile regimes. We set our sound speed to c s = 10V max, resulting in density fluctuations of order one percent. The parameter B is then determined according to equation (2.56). Where we consider flows with fluids of different reference densities, it may e desirale for them to have equivalent sound speeds. We therefore write equation (2.55) as {( ) γ ρ P = ρ 0 B 1}. (2.57) ρ 0

29 2.2 SPH applied to fluid dynamics Integrals of motion The derivation given aove maintains various symmetries found in the physical system. This leads to conservation of important quantities such as momentum, which therefore imposes constraints on our modelled dynamics which reflect those of the physical system. Conservation of linear momentum follows from homogeneity of space, wherey parallel translations leave closed mechanical systems unchanged (Landau and Lifshitz, 1976). It follows that for such homogeneity, we require that the Lagrangian e translation invariant. This can e easily shown y taking a new co-ordinate r = r + δr, for some constant deviation δr. Clearly we have v = dr /dt = dr/dt = v, so velocity is unchanged y this transformation. Likewise for density ρ a = = = m W (r a r ) m W ((r a δr) (r δr)) m W (r a r ) = ρ a. Our kernel is a function of the magnitude of it s argument, which is invariant to translations δr, so we have r a r = r a r, and we now have that for Lagrangian (2.38), L = L (assuming constant entropy), and so we have translation invariance. As such, if our equations of motion are variationally consistent, then they must conserve momentum. That this is the case can e verified y consideration of the momentum equation (either (2.44) or (2.41)). We first note that we can write a W a = r a F a where we assume that W = W ( r a ) and that therefore F = F ( r a ). This is an assumption made to reach momentum equations (2.44) or (2.41) via the variational framework, so it s use here is warranted. Now simply noting that d dt ( a m a v a ) = ( a = a = 0 m a dv a dt ) m a m ( Pa + P ρ a ρ ) r a F a where momentum equation (2.44) has een used and we have exploited symmetry. It can also e seen that we have particle pairs applying equal and opposite forces upon each other, from which momentum conservation follows. Similarly, conservation of angular momentum follows from isotropy of space, which is realised through the Lagrangian s invariance to rotations. This invariance is again trivial. Density invariance follows y the same arguments as aove, and the square of the velocity will clearly not e effected y rotation, so it follows that the Lagrangian is not changed. Now considering the momentum

30 23 Smoothed Particle Hydrodynamics equation, as aove, we find: d dt ( a m a r a v a ) = ( a = a = a = 0 m a r a dv a dt ) m a m ( Pa + P ρ a ρ m a m ( Pa + P ρ a ρ ) r a r a F a ) r a r F a where we again exploit symmetries in the summations. While we have not included viscosity in the aove calculations, where viscosity takes a symmetric form such as (2.52), we expect momentum conservation will not e violated. A further integral of motion is that of total energy, as given y equation (2.46). Homogeneity in time is the symmetry from which this stems. So given a time independent Lagrangian, such as will e used in this thesis, total energy conservation follows. 2.3 Implementation Timestepping The discritisation of spatial derivatives via SPH leaves us with a set of ordinary differential equations for which we seek solution. Where we use a summation density, they can e represented as dr a = v a dt (2.58a) dv a = g a (r 1,..., r n, v 1,..., v n ), dt (2.58) for some vector function g a, with the suscript denoting evaluated at coordinate r a. Where viscous dissipation is not included, our formulation can e derived in a variationally consistent manner (as in section 2.2.2). It is then desirale to use integrators which reflect the symmetries of the Hamiltonian system. We use the second-order Verlet integration scheme, which for system (2.58) is written r 1/2 a = r 0 a t v0 a (2.59a) v 1 a = v 0 a + t g 1/2 a (2.59) r 1 a = r 1/2 a t v1 a, (2.59c) for fixed timestep t. Superscripts here determine relative time coordinate of particles. This iteration scheme is part of a class of schemes known as geometric integrators which are designed to reproduce various geometric traits of the true system. The Verlet scheme reproduces a numer of key features of the Hamiltonian system, such as reversiility and symplecticity, which leads to excellent energy conservation qualities, as well as conservation of angular momentum (McLachlan and Quispel, 1999; Leimkuhler et al., 1996). Very long time integrations are possile using symplectic methods, where standard methods may not maintain energy conservation to sufficient degree.

31 2.3 Implementation 24 For the timestep t, we simply use a Courant condition, which we may write ( ) h t = 0.5 min v sig for smoothing length h and signal speed v sig, which is usually taken to e (2.60) v sig = c a + c (2.61) for interacting particles a and. However, where symplecticity is to e preserved, we require that our timestep remain constant for the entire integration. This requirement stems from artifacts that are introduced at the intermediate steps of equations (2.59). Where the timestep is constant, these artifacts are cancelled at proceeding steps, ut for variale timestep size, symmetry is lost and errors do not cancel (Leimkuhler et al., 1996). It is still possile to construct reversile integrators for variale timesteps (Monaghan, 2005). However, though reversile methods retain some of the desirale features of geometric integrators, energy conservation suffers and can e expected to drift from its initial value (McLachlan and Perlmutter, 2004). Where density is evolved via continuity equation (2.31), instead of (2.58) we have dr a = v a dt (2.62a) dρ a dt = k a(r 1,..., r n, v 1,..., v n ) (2.62) dv a = g a (r 1,..., r n, v 1,..., v n ). dt (2.62c) Here the function k a is evaluated at the coordinates of particle a, and the equivalent Verlet scheme to (2.59) is r 1/2 a = r 0 a t v0 a (2.63a) ρ 1/2 a = ρ 0 a t k0 a (2.63) v 1 a = v 0 a + t g 1/2 a (2.63c) r 1 a = r 1/2 a t v1 a (2.63d) ρ 1 a = ρ 1/2 a t k1 a. (2.63e) The aove maintains all the desirale attriutes of the equivalent for the summation density. Where continuity equation (2.31) is used, the function k a ecomes dependent on particle densities. The final density step (2.63e) is now implicit and iteration is required. We note that equations (2.59) also ecome implicit where viscosity is included, as the force function g a will require knowledge of v 1/2 for the velocity step. In practise, convergence may e found with a small numer of point iterations, at least for simulations encountered here. However, as our system is no longer Hamiltonian, there is perhaps little reason to use the Verlet integrator, and other choices such as a modified Euler may e preferale.

32 25 Smoothed Particle Hydrodynamics Variale Resolution Implementations Where particles remain largely equispaced throughout integration, such as for incompressile fluid flows simulations, constant-h techniques generally perform adequately. However where we expect a large variation in particle numer densities over short scales, constant-h implementations suffer on a numer of fronts. Possily the foremost limitation of constant-h SPH is that of resolution. By construction, the SPH approximation will only resolve to length scales of order h. That this is the case can e demonstrated with a simple example. We consider the approximation integral (2.1), which effectively takes some average of the suject function over a sudomain (determined y the kernel s asis). For simplicity, we choose our kernel to e the top-hat function, which we define in one dimension as W T H ( x x, h) = 1 2h { 1 0 x x h 0 h < x x, (2.64) and consider the function A(x) = sin(kx), for which the approximation integral now gives sin(kx) sin(kx) = sin(kx )W T H (x x, h) dx R x+h 1 = x h 2h sin(kx ) dx [ ] sin(kh) = sin(kx) = sin(kx)f (kh). (2.65) kh We see that the approximation integral reproduces our suject function sin(kx) along with a factor F (kh). The factor F depends on kh = 2πh/λ, for wavelength λ, and attenuates the function sin(kx) with strength depending on the ratio of smoothing length to wavelength. We see that in the limit h/λ 0, we have F 1, and so (2.65) exactly recovers sin(kr), as we expect. For the values h = λ/5, λ/10, λ/100 we have respectfully F 0.76, 0.94, So for reasonale accuracy, we perhaps require a smoothing length one tenth the size of the shortest length scale to e reproduced. Of course here we have used a top-hat kernel which we expect leads to greater smoothing than a Gaussian ased kernel. So we wish to reduce h and therey improve reproduction of short length scale dynamics. The next question is perhaps how far we can reduce h? To answer this we must rememer that the approximation integral (2.1) is discretised in the SPH technique, and so we must retain sufficient nodes (ie. particles) to reproduce it accurately. This rings us to another deficiency of constant-h SPH, which may e encountered for example in simulations involving expanding gasses. In this situation, we may use insufficient particles to accurately approximate (2.1). Monaghan (2005) shows that for equispaced particles approximating linear functions, errors are small where h >, for particle spacing. Using variale-h SPH, we may therefore ensure that our smoothing length is sufficiently large such that the h > requirement is met. At the other end of the spectrum we may have that h. With regard to computational cost, we have the undesirale situation where a large numer of particles fall under a kernel s umrella. This leads to inefficient calculations

33 2.3 Implementation 26 tending towards operation counts of N 2 for an SPH population of N particles. Our scheme may now also suffer loss of accuracy. Firstly, we have intrinsic resolution limitation imposed y the approximate function (2.1) as outlined aove. This leads to our second and more critical point of failure. Given that the approximation function now smoothes all scales elow h, we may fail to capture the true density profile appropriate to certain dense particle configurations. However, our calculations rely on density to determine our numerical quadrature, and we now have the situation where our volume element V may not correspond to m /ρ. SPH relies on this quadrature to accurately approximate the integral (2.1). So we now make the requirement m a ρ a = V a = ( r a ) ν (2.66) where r gives an indication of typical particle separations at particle a, and ν is the dimensionality. For h a = σ r a we then have h a = σ ( ma ρ a ) 1 ν. (2.67) The value σ will depend on the form of the kernel used. For the cuic spline, 1.2 < σ < 2 is usually appropriate. We note that the aove leads to an implicit form for summation density (2.28). We restate it here: ρ a = = m W (r a r, h a ) (2.68) m W (r a r, σ(m a /ρ a ) 1 ν ) (2.69) This represents a self-consistent formulation for ρ a which only relies on parameter σ for which it is not overly sensitive (as long as σ is chosen within an appropriate range). The aove density equation is highly non-linear, the details of which depend on the form of kernel used. In practise, we solve equations (2.67) and (2.68) iteratively until some convergence criterion is met. This approach yields values for the density and smoothing length which are consistent. A more computationally efficient method is to simply use density values from the previous timestep, thus circumventing the requirement for iteration, though sacrificing precise consistency. Bonet and Rodriguez-Paz (2005) report that inferior results ensue, and additional costs of iterative approach may possily e offset with lower particle numers. Another possiility is to evolve the smoothing length in time dh a dt = h a dρ a νρ a dt though this will also not guarantee consistency of smoothing length and density. To maintain symmetry and the various conserved qualities that follow, its then desirale to write our kernels as either an average W (r a, h a, h ) = 1 2( W (ra, h a ) + W (r a, h ) ) (2.70) or to take averages of smoothing lengths W (r a, h a, h ) = W ( r a, 1 2 (h a + h ) ). (2.71)

34 27 Smoothed Particle Hydrodynamics We recall our variational derivation of the momentum equation, and in particular the transition from a continuum Lagrangian (2.37) to the discrete form (2.38). The accuracy of this discretisation will depend on our specification of density, or rather the accuracy thereof. The most accurate specification for density can e expected to follow from the implicit form (2.69), though our use of this density in the Lagrangian (2.38) leads to an alternate momentum equation. We follow the derivation of Monaghan (2002) starting from (2.40), which we reproduce here: m a dv i a dt + Our density derivative now ecomes and we have ρ W a (h ) Ω ra i = m a ra i ( ) u ρ m ρ s ra i = 0. (2.72) + δ a c c W ac (h a ) m c ra i, (2.73) Ω = 1 h W c (h ) m c (2.74) ρ h where we use equation (2.67) to determine the smoothing length derivative. Putting all this together, we arrive at our momentum equation for variale-h SPH: dv a dt = ( Pa m Ω a ρ 2 a W a (h a ) + a P Ω ρ 2 ) a W a (h ). (2.75) As for the constant-h version, the aove conserves angular and linear momentum, and total energy. An equivalent formulation has een given y Springel and Hernquist (2002), though derived in a slightly different manner. The earlier momentum equation (2.41) is recovered for constant smoothing length, as we would expect. It should e noted that additional terms now present in momentum equation (2.75) result from a new density specification. Such terms are often dued the h terms, and while not incorrect, this view is misleading in that it implies an intrinsic reliance of dynamics on the smoothing length h. A more natural perspective is found y forgetting smoothing length, noting that all kernels may instead e written as functions of density: W (r a r, f(ρ a ), f(ρ )). The new algorithm still only requires a single numerical parameter (the related parameter σ). Where particle distriutions are anisotropic, prolems may e encountered due to a lack of interacting particles in certain directions. In this case, perhaps radial kernels are not ideal and isotropic kernels may e a etter choice (see section 2.1.4). Another option may e to use remeshing strategies Neighouring particle list Where gloal kernels, such as the Gaussian, are used, summations involve contriutions from all particles (though small for distant particles). As such, compuations scale as N 2 for a SPH population of N particles; highly expensive computations follow. This motivates choice of kernels with compact support,

35 2.3 Implementation 28 which restrict the numer of particle-particle interactions. Though this leads to sustantial savings, there are some minor extra costs incurred. Namedly, we must construct list of particles such that our algorithms know which particles are required for summations. For constant smoothing length (ie constant kernel support size), the procedure is straightforward. For an interaction radius R i (for the cuic spline, we have R i = 2h), we divide the domain into cells of width R i. For each cell, we then create a list of particles contained within that cell. For two-dimensional simulations, summations for any said particle in cell C i,j, then only consider particles within the same cell (C i,j ), as well as the eight cells surrounding cells (C i 1,j+1, C i,j+1, C i+1,j+1, C i 1,j, C i+1,j, C i 1,j 1, C i,j 1, C i+1,j 1 ). Particleparticle interaction symmetries are also exploited to improve efficiency. We may use this method for variale smoothing length implementations, where the largest particle interaction radius must e used to construct cells. However efficiency declines as our variation in h increases, and some other neighour finding algorithm such as rank listing may e preferale Boundaries For fluid dynamics prolems, we usually require some form of oundary condition. Boundaries may serve the simple purpose of confining the fluid within a prescried domain, or perhaps may form a more integral part of dynamics, such as flow past a cylinder. Various types of oundary conditions may e required, such as free slip, no slip, periodic or open. Such conditions may e included in SPH computations through a numer of methods. Perhaps the most straightforward implementation for impenetrale oundary conditions is to simply use fixed SPH fluid particles. We calculate density for these oundary particles in the same way as for fluid particles, with respective pressure forces yielding the desired condition. For a free slip requirement, we simply exclude oundary particles from viscosity calculations, else we have a no-slip oundary. There are a numer of advantages to using this approach. As symmetry is not altered, we retain conservation of momentum (where forces on oundary are not disregarded). Initial settling periods are also not required (on account of oundaries), as oundaries apply equivalent forces to fluid particles. The disadvantages to this approach are as follows. Firstly, additional costs are incurred as we require numerous oundary particle layers to prevent particle penetration. The numer of layers will depend on the kernel support, with oundary particles eing required to extend eyond fluid particle interaction radii. Secondly, such oundaries present non-constant constituitive forces, as felt y particles moving along the oundary. This may cause difficulty in simulations which are sensitive to such noise, such as Poiseuille flow. Boundary force noise issues are increasingly prolematic for insufficiently resolved curved oundaries. An alternate oundary methodology instead gives oundary particles some explicitly defined repulsive force to prevent particle penetration. This force acts along the line of centre for any fluid/oundary particle interaction. A typical choice is the Lennard-Jones force, though many possiilities exist (Monaghan, 1994). The noise issues discussed aove may now e addressed y interpolating oundary forces (Monaghan et al., 2003), and in a similar fashion we may construct smooth curved oundaries (Monaghan, 1995). We only require one

36 29 Smoothed Particle Hydrodynamics line of particles for oundaries, though we must now use the time evolution forms for continuity, as summations will lead to errors near the oundaries. Bonet and Rodriguez-Paz (2005) propose a renormalisation correction for summations near oundaries, which they also include in their variationally derived momentum equation. Another option is the use of mirror particles, where particles are reflected through the oundary plane using temporary ghost particles. This yields a low noise oundary condition, though is only really applicale for flat oundaries. We may alternatively wish to implement periodic oundary conditions. For our simulations, the linked list data structures fascilitate periodicity. For example, particles near the right oundary, ie those in the rightmost linked list cell, directly interact with particles in the leftmost linked list cell, though the righthand particles percieve a transposed version of the lefthand particles. An equivalent (if slightly more costly) situation sees the lefthand particles copied temporarilty eyond the right oundary. We may also implement other symmetries in a similar fashion, such as point symmetry aout the origin. 2.4 Summary An outline of the SPH numerical method has een given. Initial attention was directed towards analytic considerations of the SPH approximation. In section we found that where symmetric kernels are used, the smoothing function yields O(h 2 ) approximations, and for equispaced data, the discrete smoothing function also tends towards O(h 2 ) accuracy. Next the first and second derivatives were considered (sections and 2.1.3). It was shown for the first derivative that improved accuracy is found where we write summations in symmetric form, while second derivatives are found via an integral approximation. Kernel requirements were discussed in section Next the application of SPH to Navier-Stokes equations were considered. Various forms of the continuity equation were given, their appropriateness to various simulations discussed (section 2.2.1). The various ensuing forms for momentum were determined via a variational framework (section 2.2.2), along with corresponding energy terms (section 2.2.3). Viscosity formulations are given in section The equation of state, and coefficients required to determine sound speed and compressiility, are discussed in section We finally estalish the conservation properties for our governing discrete equations (section 2.2.6). Issues related to computational time integration of forementioned equations are then discussed. Various qualities of the Verlet timestepping algorithm are discussed, along with the Courant timestep requirement (section 2.3.1). Section deals with the different implementations for oundary conditions. A selfconsistent variale resolution formulation is discussed in section 2.3.2, along with the shortcomings of fixed smoothing length algorithms.

37 Chapter 3 The α-sph turulence model Turulent flow regimes are characterised y irregular fluid motion acting over a very road spectrum of length scales in a seemingly chaotic nature. The mathematical realisation of these flows contain many degrees of freedom, which computationally translate to expensive simulations. We may characterise our flow y a Reynolds numer, which we define as Re = UL/ν, for length and velocity scales L and U and kinematic viscosity ν. Typically, turulent regimes occur for medium to high Reynolds numers (Re 10 3 ). Such systems will tend to exhiit energy cascade phenomona, where we have large eddies reaking into smaller eddies, which themselves follow on to seed even smaller eddies, and so forth. Cascades also act in the opposite direction, with small eddies comining to spawn larger eddies, as is the predominant ehavior in two-dimensional turulence. This phenomena is a defining characteristic of turulent flows, and continues down to a scale where viscosity ecomes dominant. This length scale is known as the Kolmogorov scale η K, and it can e shown that in three dimensions it varies with Reynolds numers as η K L = Re 3/4 (3.1) (Kolmogorov, 1941). For a three-dimensional Navier-Stokes flow, the numer of degrees of freedom then scales as Re 9/4 and so ecomes large for turulent Reynolds numers. Indeed douling the Reynolds numer will result in five times the numer of degrees of freedom, and for a spectral code will increase the computational work y an order of magnitude. Techniques such as direct numerical simulation (DNS), where all energy containing modes are resolved, remain completely intractale using today s processing technology. Furthermore, at the rate at which computational power is increasing, a full DNS of most turulent flows will remain eyond reach for the foreseeale future. Generally, numerical turulence methodologies use a model to account for length scales eyond resolution limit, while directly simulating ulk dynamics. One such method, known as Reynolds averaged Navier-Stokes (RANS), involves averaging out fluctuation from our variales, with the flow split into a mean and fluctuating component. Differential equations for the mean components are then derived from the Navier-Stokes equations, and a model is used to account for 30

38 31 The α-sph turulence model the effect of fluctuating components on the mean. Given that all fluctuation is removed from direct computation, the outcome relies heavily on the model used. Experiments are generally required to calirate the sugrid model, and final results are accepted as approximations. A method which uses modeling less aggressively is large eddy simulation (LES). Here the flow is resolved down to computationally realistic length scales, and a model used to account for scales eyond this point. To this end, velocity is filtered to remove fluctuations elow the required length scale, with the required differential equations eing determined through filtration of the Navier-Stokes equations. As with RANS, the filtered equations posses a dependence on sugrid terms for which a model is required. There are many different models used for the sugrid dependence, such as the Smargorinsky model and the scale similarity model. An alternate method involves averaging at the variational principle level. Fluid elements are moved with a local filtered velocity, and a new Lagrangian is defined from which the so called Lagrangian-averaged Navier-Stokes equations (LANS) are otained. This derivation naturally yields sugrid stress terms, and the resulting modified acceleration equation ensures that circulation is conserved for non viscous flows. Furthermore, a quadratic invariant related to the kinetic energy is conserved. Analysis due to Monaghan (2002) led to an SPH equivalent of LANS known as α-sph, where particles are moved with a filtered velocity under action of a modified acceleration equation. Originally, a version of SPH known as XSPH was devised to help prevent particle interpenetration (Monaghan, 1999). In similarity to α-sph, XSPH moved particles with a local averaged velocity, though energy conservation was compromised. Conservation is recovered y deriving our SPH equations from a discrete averaged Lagrangian, yielding the α-sph formulation. The derivation of α-sph is given here, along with discussion of it s conserved quantities and time integration. 3.1 LANS-α A relatively recent technique devised to simulate turulent fluid dynamics is the Lagrangian Averaged Navier-Stokes alpha (LANS-α) method. The key distinguishing feature of this method is that an averaging is performed at the Lagrangian level, and following this averaging we derive the required equations. This contrasts methods such as large eddy simulation, and Reynolds average Navier-Stokes, where an averaging (or filtering) procedure is performed on the actual equations of motion. LANS-α originates from the one-dimensional Camassa-Holm equations (Camassa and Holm, 1993), which deals with non-linear shallow water wave dynamics. Generalisation of the Camassa-Holm equations to three dimensions yields the Lagrangian averaged Euler alpha equations (LAE-α) (Holm et al., 1998a,). These equations provide a closed realisation of Euler dynamics, and we egin to see hints of the methods potential for modeling turulent fluid regimes. The inclusion of viscous dynamics y Chen et al. (1998,1999a) rings us to the full LANS-α scheme, with application to turulence given impetus y favourale results for turulent pipe flow. Here analytic solutions for steady mean velocity profiles reproduced those found in experiments at moderate to high Reynolds numers. Direct numerical simulations of LANS-α were performed for three-

39 3.1 LANS-α 32 dimensional isotropic turulence in Chen et al. (1999). Key results indicated that large scale Navier-Stokes dynamics were reproducile using LANS-α, while the latter only requires a reduced andwidth for complete spectral representation. The LANS-α equations can e written (Holm, 1999) with and v t + u v + ut v + P = ν v (3.2) v = (1 α 2 )u (3.3) u = 0. (3.4) We note that two different velocities are required, v and u. They are related y equation (3.3). The parameter α gives a length scale which will e discussed shortly. The vector u represents the velocity with which the fluid is transported. It is determined via inversion of equation (3.3), and as such takes values which are weighted averages of velocity v, with averaging performed over the length scale α. Velocity u is therefore a smoother realisation of velocity v. Where the aove equations are derived of an averaged Lagrangian, the velocity v can e interpreted as the momentum per unit mass of the Lagrangian averaged motion (Holm, 1999). A new length scale α is introduced to dynamics in the LANS-α system. Technically α is defined as the distance of typical deviations of a Lagrangian path from its time-averaged trajectory (Holm et al., 2005). It may also e interpreted as the scale elow which fluctuations are passively advected y the flow, and do not participate in cascades, nor affect their own advection. Typically it is desirale to make maximum utilisation of numerical resources, and so α may then e chosen to e some fixed multiple of the numerical resolution length. In this way we maximise the andwidth of Navier-Stokes like dynamics, i.e., those which occur at lengths scales aove α. Care must e taken however to ensure that the LANS-α dynamics are fully resolved, or that numerical schemes ehaves agreealy for marginally resolved dynamics. Comparative studies of LANS-α with other closure methods such as LES have een performed in Domaradzki and Holm (2001) and Geurts and Holm (2002), and it is found that α modeling techniques generally perform at least as well the est LES methods.. While similarities exist etween such methods, LANS-α methods modify the nonlinear advection of the Navier-Stokes system, where LES methods introduce energy diffusion or dispersion. Indeed LES can often e overly dissipative when compared to LANS-α techniques which conserve circulation (for inviscid flow regions) and as such are ale to produce greater variaility. It should e mentioned that this increased degree of small scale dynamics may not always e desirale however, with Geurts and Holm (2002) reporting overly large growth rates, and instaility. Of course the purpose of all sugrid schemes is to reduce the resolution requirements of turulence calculations, and savings naturally depend on how aggressively modeling is used. For LANS-α, dimensional arguments indicate that energy scales as k 3 for length scales elow α (Foias et al., 2001; Monaghan, 2002). It follows that resolution requirements for fully resolved LANS-α scale as

40 33 The α-sph turulence model Re 2, which presents a considerale saving compared with Re 3 requirements for full Navier-Stokes dynamics, though overheads involved in calculating additional quantities must also e considered. Further information on the LANS-α model can e found in Foias et al. (2001) and references therein. 3.2 α-sph: equations of motion We seek an SPH equivalent to the continuum LANS-α equations. The derivation given in Monaghan (2002) is outlined. We start with the discrete Lagrangian L = [ 1 ] m 2 ˆv v u(ρ, s ), (3.5) where in addition to the previously defined quantities, we have a new velocity ˆv. This velocity represents an averaged or smoothed realisation of the velocity v. This is the velocity with which particles are advected. We consider it s particulars now The filtered velocity The precise specification of ˆv may take a numer of forms. For the calculations found herein, we use an implicit definition given y ˆv a = v a + ɛ m ρ a (ˆv ˆv a ) W a, (3.6) where we have the parameter ɛ which modulates the degree of smoothing. The kernel W a is an averaged kernel, as given in equation (2.70). Choice of kernel is an open question, though the cuic spline has een used here (see 2.1.4) for simplicity. Furthermore, the asis size of our kernel (with respect to particle spacing) will affect the filtering qualities of (3.6). An alternate specification for the filtered velocity is ˆv a = v a + ɛ m ρ a (v v a ) W a. (3.7) This is the form used for XSPH implementations. The implicit version, equation (3.6), is preferred for our current simulations for a numer of reasons, which we will come to shortly. To gain further understanding of the ehavior of equation (3.6), we considering the limiting situation as smoothing length goes to zero. Taylor expansions reveal with ˆv j = v j + A2 ρ (ρ ˆv j) ( ρ ˆv j ) h A 2 ρ h (3.8) A 2 = ɛ (x x) 2 W (r r, h(r)) dr. (3.9) 2 Where we assume variations in density (and therefore smoothing length) can e neglected, equation (3.8) reduces to ˆv j = v j + A 2 2ˆv j, (3.10)

41 3.2 α-sph: equations of motion 34 which may e compared with the LANS-α equivalent (3.3). In the aove, superscript j denotes dimensional component. We find that our smoothed velocity is equivalent to that of LANS-α under constant density conditions, where we now write α 2 = A 2 = ɛ (x x) 2 W (r r, h(r)) dr. (3.11) 2 In one dimension, for the cuic spline kernel with smoothing length h, we have α 2 = 1 6 h2 ɛ. (3.12) So, at least for the limiting constant density one-dimensional situation, we can alter ɛ or smoothing length to modulate α, and therefore the degree of smoothing. The effect of douling smoothing length will e identical to that of quadrupling ɛ. The action of (3.10) in filtering components of v is est understood in spectral space. For simplicity, we restrict dynamics to one dimension, and expand our velocity fields in trigonometric functions: v = m ˆv = m v m exp(ik m x) (3.13) ˆv m exp(ik m x), (3.14) for imaginary numer i and wavenumer k m = 2πm/L. Inserting (3.13) and (3.14) into equation (3.10) rings us to ˆv m = (k m α) 2 v m = F i (k m, α) v m (3.15) The factor F i (k m, α) acts to attenuate mode amplitudes as we push into higher wavenumers. Using (3.12), and setting g = h/λ for wavelength λ, we recast F i : 1 F i = 1 + (k m α) 2 = ɛ(2πg) 2. (3.16) This attenuation factor is displayed in Figure 3.1 for the values of ɛ = 0.1, 0.25, 0.5 and 1.0. We see that the smoothing operation acts as a low pass filter, with the expected result of greater mode damping for larger values of ɛ. We turn our attention riefly to the explicit smooth velocity (3.7). For the constant density situation, we find the following continuum equivalent ˆv j = v j + α 2 2 v j, (3.17) which is equivalent to (3.10) ut with the Laplacian operating on the standard velocity v. Mode coefficients are determined y the following: ˆv m = ( 1 (k m α) 2) v m = F e (k m, α) v m. (3.18) We plot F e in Figure 3.2. It is found that this form of smoothing may lead to mode phase changes, and more critically in amplification of certain modes. These issues present where values of ɛ > 0.5 are taken. Application of this form of smoothing is therefore limited, as we may not e ale to alter frequencies at

42 35 The α-sph turulence model 1 ε = 0.10 ε = 0.25 ε = 0.50 ε = F i g = h / λ = hk/2 Figure 3.1: Implicit filtered velocity attenuation profile at various values of ɛ. the lower part of the spectrum without causing instailities for high frequencies. In practise, these issues are not found to e as severe as Figure 3.2 may suggest. This is possily owing to the fact that modes where staility prolems are expected are marginally represent on the SPH grid, and as such may e numerically damped. Other possiilities for filtered velocity include using higher order Laplacian operators (or their SPH equivalents), leaving a larger portion of the spectrum unchanged elow some cut off frequency, while strongly damping modes eyond. Resulting dynamics have not een investigated The momentum equation Equip with filtered velocity (3.6), we return to the Lagrangian. Lagrange equations for this system read: Our Euler- ( ) d L dt ˆv a i L ra i = 0, (3.19) and we note that the advection velocity ˆv is used for our generalised velocity coordinate. Canonical momentum is considered first: L ˆv a i { = [ 1 ˆv a i m 2 ˆv v + u(ρ, s )] }.

43 3.2 α-sph: equations of motion 36 1 ε = 0.10 ε = 0.25 ε = 0.50 ε = F e g = h / λ = hk/2 Figure 3.2: Explicit filtered velocity attenuation profile at various values of ɛ. Note that large values of ɛ may cause excitation of high order wavenumers, leading to instaility. The thermal energy term in the aove disappears. Using (3.6) we rewrite the kinetic energy term: 1 2 m ˆv v = 1 m ˆv ˆv ɛ 2 2 c = 1 m ˆv ˆv + ɛ 2 4 The canonical momentum now ecomes L ˆv a i { = 1 ˆv a i 2 = ˆv i a { 1 2 } m ˆv v m ˆv ˆv + ɛ 4 c c i (ˆv ˆv a i ) W a m m c ˆv (ˆv c ˆv ) W c ρ c m m c ρ c (ˆv c ˆv ) 2 W c. (3.20) m m c ρ c (ˆv c ˆv ) 2 W c = m aˆv a i + ɛ m a m ρ a = m a va. i (3.21) So we find that the canonical momenta for the current system reduces to the standard momentum for the Navier-Stokes regime. We turn to the second term }

44 37 The α-sph turulence model of equation (3.19): L r i a { = [ 1 ra i m 2 ˆv v + u(ρ, s )] }. (3.22) Thermal energy terms in the aove yield pressure terms in the momentum equation, as per standard SPH (see Section 2.2.2): r i a { } ( Pa m u(ρ, s ) = m a m ρ 2 a + P ) ρ 2 a W a. (3.23) For clarity and simplicity, we neglect terms which may arise owing to nonconstant smoothing lengths. Letting ˆv 2 c = (ˆv c ˆv ) 2, We consider the velocity terms in (3.22): 1 2 r i a { } m ˆv v = r i a { ɛ 4 = ɛ m m cˆv 2 c 4 c = ɛ { m m cˆv c 2 4 c c m m c ˆv ρ cw 2 c c r i a + ɛ m m cˆv c 2 4 c } { } 1 W c ρ c 1 W c ra i ρ c } { 1 ρ c W c r i a } (3.24) Now working with the first term: ɛ m m cˆv 2 4 cw c c c 1 ra i ρ c = ɛ m m cˆv 2 4 cw c r i a ( ρ ), 2 ρ c

45 3.2 α-sph: equations of motion 38 symmetry then allows us to write = ɛ m m cˆv cw c r i c a ρ = ɛ m m cˆv 2 1 ρ 4 cw c ρ 2 c ra i = ɛ m m cˆv cw c ρ 2 c ra i m d W d d = ɛ m m cˆv 2 1 W d 4 cw c ρ 2 m d c r i d a = ɛ m m cˆv 2 1 W d 4 cw c ρ 2 m d c r i (δ a δ ad ) d = ɛ 4 m a m c m dˆv 2 1 W ad acw ac ρ 2 c d a ra i ɛ 4 m a m m cˆv 2 1 W a cw c ρ 2 c ra i = ɛ 4 m a d = ɛ 4 m a d m d 1 ρ 2 a c m cˆv acw 2 W ad ac ra i ζ a W ad m d ρ 2 a ra i ɛ 4 m a = ɛ 4 m ( ζa a m ρ 2 + ζ a ρ 2 where we designate ζ a = c ) Wa r i a ɛ 4 m a ζ W a m ρ 2 ra i m 1 ρ 2 c m cˆv 2 cw c W a r i a, (3.25) m cˆv 2 acw ac. (3.26) Now for the second term of (3.24): ɛ 4 c 1 m m c ˆv 2 W c c ρ c ra i = ɛ 4 c = ɛ 4 c = ɛ 2 m a 1 m m c ˆv 2 W c c ρ c r i (δ a δ ac ) m a m c 1 ρ ac ˆv 2 ac W ac r i a 1 m ˆv 2 W a a ρ a ra i + ɛ 4 1 m a m ˆv 2 W a a ρ a ra i (3.27) We put all this together (equations (3.21), (3.23), (3.25) and (3.27)) to arrive at our α-sph momentum equation: )} a W a. (3.28) dv a dt = m { Pa ρ 2 a + P ρ 2 ɛ ( ˆv 2 a ζ a 2 ρ a 2ρ 2 ζ a 2ρ 2 We note that the co-moving derivative is defined using the advection velocity ˆv: d dt = + ˆv. (3.29) t

46 39 The α-sph turulence model Insight to the ehavior of the α-sph momentum equation is gained y considering it s continuum limit. Letting particle spacing tend to zero and performing Taylor series expansions, we find (see Appendix B): dv dt = P ρ [ + α2 ˆv l( (ρ ˆv l ) ) + ρ ρ 2 ( ˆv l ˆv l)] (3.30) with α (defined y (3.11)) held constant as smoothing length goes to zero. Superscripts define spatial components and summation convention is used. Where we assume an incompressile flow field, (3.30) reduces to v i t + ˆv vi + v j ˆvj x i + ( P x i ρ 1 2 ˆvj ˆv j α2 ˆv j ˆv j ) 2 x k x k = 0 together with velocity filter (3.10). This is equivalent to LANS-α momentum equation given in Holm (1999), and so our system can e interpreted as a particle discretisation of LANS-α. Where a variale smoothing length has een considered, the required equation is (see Appendix A): with dv a dt = [{ Pa m Ω a ρ 2 a ɛ ( ˆv 2 a + 1 h a ν a 2ζ )} a a W a (h a ) 4 ρ a Ω a ρ a Ω a { P + Ω ρ 2 ɛ ( ˆv 2 a + 1 h ν 2ζ 4 ρ a Ω ρ Ω )} ] a W a (h ), (3.31) ν k = c ζ k = c m c ˆv 2 W kc kc (h k ) ρ kc h k m c A kc ρ 2 ˆv kcw 2 kc kc A kl = ρ kl ρ k Ω k = 1 h k W kc m c (h k ), ρ k h k and where smoothing length h may e determined according to (2.67). Viscous dynamics may now e included in the same manner as undertaken for standard SPH (see Section 2.2.4). We introduce viscosity term Π a (2.52) to our momentum equation, though it now depends on smooth velocity ˆv: Π a = c aα ρ a c ( ) ˆva r a. (3.32) r a The use of the smoothed velocity is desirale to ensure that the addition of viscosity leads to an increase in thermal energy (see Monaghan (2002) for further

47 3.3 Integrals of motion 40 details). We may now write our momentum equation dv a dt = [{ Pa m Ω a ρ 2 a ɛ ( ˆv 2 a + 1 h a ν a 2ζ ) a + 1 } 4 ρ a Ω a ρ a Ω a 2 Π a a W a (h a ) { P + Ω ρ 2 ɛ ( ˆv 2 a + 1 h ν 2ζ 4 ρ a Ω ρ Ω 3.3 Integrals of motion ) + 1 } ] 2 Π a a W a (h ). (3.33) A key quality of α-sph/lans-α modeling methodologies is their conservation of certain integrals of motion. The derivation of these systems through variational methods provides a natural pathway to such conservation, whereas other closure techniques usually do not provide roust assurances. As discussed in Section for standard SPH, conservation may e determined through inspection of the Lagrangian. Conservation of momentum follows from homogeneity of space, which requires linear translation invariance of the Lagrangian (3.5). The invariance of thermal energy term u(ρ, s) has een estalished for standard SPH in Section and is unchanged for the current system. Where we take canonical variales r and ˆv, we now have a kinetic energy term which is a function of variale r. We consider equation (3.20). The first term is not dependent on r, while the second term is translation invariant owing to the kernel s invariance (see Section 2.2.6), and so we have estalished momentum conservation for the current system. The conserved momentum is the same as for the standard Navier-Stokes system (see equation (3.21)). Conservation of angular momentum is determined in a similar fashion, following from the Lagrangian s invariance to rotation. We note from equation (3.20) that only velocity magnitudes enter the Lagrangian, and these are unchanged y rotations. Density terms also remain constant for rotations of the coordinate system, completing the kinetic energy s invariance, while also assuring invariance for the thermal energy. We thus have conservation of the angular momentum: d m r v a = 0. dt As Lagrangian (3.5) has no explicit dependence on time, we expect to conserve a final additive integral of motion, that of energy. Manipulation of the Euler-Lagrange equations reveal: E = = = ˆv L L ˆv [ 1 ] m 2 ˆv v + u(ρ, s ) m [ 1 2 ˆv ˆv + ɛ 4 c m ] c (ˆv c ˆv ) 2 W c + u(ρ, s ), ρ c where we have used (3.20). The SPH summation leads to the following contin-

48 41 The α-sph turulence model uum equivalent where density variations have een neglected: m c (ˆv c ˆv ) 2 W c = 1 ρ c 2 α ˆvi ˆv. i c Where parameter α is set to zero, we recover conservation of the same energy as that of the Navier-Stokes system. Furthermore, where high order velocity modes are zero or small, the conserved energy is expected to e close to that of the Navier-Stokes system. So in summary, we have a conserved energy, quadratic in velocities, which constrains the flow in a similar fashion to the Navier-Stokes energy, and only diverges from Navier-Stokes conservation where energy occupies typically under-resolved velocity modes. A further quantity conserved for the α-sph system is a discrete equivalent of circulation. Monaghan (2002) gives an outline of this invariant using a socalled necklace transformation, which is followed here. We imagine a closed path within the domain defined y a necklace of particles. Now each particle in the loop is translated to the location of its neighour (with all particles moving in the same sense), and given it s neighour s velocity ˆv. Where we assume all particles have identical mass and entropy, we can write: δl = j ( L r i j δrj i + L ˆv j i δˆv j i ) = 0, with suscript j giving consecutive loop particle laels, δrj i = ri j+1 ri j and δˆv j i = ˆvi j+1 ˆvi j. We apply the Euler-Lagrange equation to the aove: 0 = ( ) L r i δrj i + L j j ˆv j i δˆv j i = ( ( ) ) d L dt ˆv i δrj i + L j j ˆv j i δˆv j i = ( ) d L dt ˆv i δrj i j j = d L dt ˆv j i δrj, i and using (3.21), j 0 = d v i dt jδrj. i Our turulence closure therefore does not impinge upon the conservation of fluid circulation. This owes to the fact that α models in effect enslaves the circulation present at short length scales to the dynamics of the larger scales, whereas other turulence closures result in a diffusion of short scale circulation (Holm et al., 2005). 3.4 Implementation A numer of implementation issues arise resulting from the aove modifications to standard SPH. Firstly we have an acceleration equation which explic- j

49 3.4 Implementation 42 itly depends on particle velocities, and so scheme (2.59) requires modification. Secondly, advection velocity ˆv is determined as an implicit solution of equation (3.6), and therefore an iterative scheme is required. We investigate these matters here Timestepping For the current system, the set of ordinary differential equations for which we seek solution are: dr a dt dv a dt = ˆv a (3.34a) = g a (r 1,..., r n, ˆv 1,..., ˆv n ), (3.34) along with ˆv a = f a (r 1,..., r n, v a, ˆv 1,..., ˆv n ) (3.34c) for functions f a and g a determined respectively y (3.6) and (3.33), with f a solved interactively. The Verlet scheme may now e written r 1/2 a = r 0 a t ˆv0 a (3.35a) v 1/2 a = v 0 a t g0 a (3.35) va 1/2 = va t g1/2 a (3.35c) va 1 = va 1/ t g1/2 a (3.35d) ra 1 = ra 1/ t ˆv1 a, (3.35e) For the aove system, iteration is required at a numer of points. Most significant is the requirement of iteration at equation (3.35c), which requires the calculation of summation (3.33), along with solution to (3.6), which itself requires iteration. In practise, convergence is usually attained with only a few iterations, though this will depend on the value of parameter α, with larger values requiring larger iteration counts. As particles coordinates do not change during this iteration, it is possile during the first pass to create lists which determine exact particle interactions. These lists may e used at future iterates for improved efficiency. The construction of a smoothed velocity at the final time point (ˆv 1 ) also presents difficulty, as particles are advected to their final position (r 1 ) using this velocity, though the final particle positions are required in its construction. So we again require point iteration, now aout ˆv 1 and r 1. In practise, we often instead construct the filtered velocity using the approximation ˆv 1 = 2ˆv 1/2 ˆv 0. This approximation suffices where we have smoothing parameter ɛ less than unity. For larger values, it is found that an instaility develops in the filtered velocity field where such approximations are used, and instead we are required to perform iteration aout the final step. Nested iterations lead to sustantially increased computational cost. Explicit timestepping schemes may e constructed with equivalent order of accuracy, though such schemes will not respect geometric properties of our Lagrangian system. As such, the additional cost of the Verlet scheme may prove

50 43 The α-sph turulence model worthwhile for the maintenance of reversiility, and where timestep is held constant, symplecticity. Also, though the variational properties of our system are violated where viscosity is introduced, use of the geometric scheme may still e desirale given that many particle-particle interactions occur with minimal viscous forcing. Such issues warrant further investigation Iteration for filtered velocity We require solution to implicit equation (3.6). The simplest approach is to use point iteration until convergence is found, with unfiltered velocity v used as an initial guess. Faster convergence may e found through use of a Jacoi iterative type scheme. We write: ˆv a = v a + ɛ = v a + ɛ ( v a + ɛ = m (ˆv ˆv a ) W a ρ a m m ˆv W a ɛˆv a W a ρ a ρ a m )( ˆv W a 1 + ɛ ρ a m ρ a W a ) 1. Further improvement may e found y utilising the latest iterated velocity values as they are determined in a Gauss-Seidel approach. Again, as particles do not move within iterations, we may construct particle interaction lists to improve computational speed. 3.5 Summary A new methodology due to Monaghan (2002), the SPH-α model, has een outlined. This new scheme is constructed for the calculation of turulent fluid regimes, and it has een shown that it is analogous to the LANS-α turulence model. Integrals of motion have een discussed, with the equivalence of particle versions estalished. Issues relating to implementation have also een covered.

51 Chapter 4 One-Dimensional Tests We start our investigation of SPH and α-sph y considering one-dimensional test prolems for which non-linear advection plays a central role. Restricting dynamics to one-dimension allows for a thorough exploration of parameter space while still retaining the key physics of higher dimensional systems. Furthermore, a greater degree of analytic tractaility is often found, as is the case for Burgers equation which is consider first. Following this, we turn to simulations of the one-dimensional Euler system, and a forced compressile Navier-Stokes system. 4.1 Burgers equation The difficulties of simulating and comprehending the full Navier-Stokes equations arise due to their complex geometric nature and nonlinearity, which give rise to turulent energy cascades and shock formation. Burgers equation provides a simplified framework for studying such phenomenon. It is the simplest equation which leads to the competition etween convection and diffusion. As such, Burgers equation is often first considered when investigating new models for hydrodynamics. It is most commonly written u t + u u x = ν 2 u x 2, (4.1) which first appeared in a paper due to Bateman (1915). Burgers (1948) investigated a recast version of this equation as a toy model to Navier-Stokes turulence, demonstrating the cascade of energy through different length scales consequent of non-linearity, along with the arrest of this cascade due to diffusion. This early work led to equation (4.1) coming to e known as the Burgers equation. Analytic solutions to Burgers equation are availale due to Cole (1951), who derived a transformation which leads to a general solution for known initial and simple oundary conditions. Application has een found to a wide range of disciplines, and so it is perhaps no surprise that many distinct and significant solutions have appeared in the literature. A collation in taular form of many such solutions may e found in Benton and Platzman (1972). Another aspect of Burgers dynamics which make it attractive as a numerical testing ground is the potential for shock formation. Many natural phenomena are suject to shocks, and the aility to model them accurately, or at the least 44

52 45 One-Dimensional Tests prevent corruption due to their presence, is often of prime concern. A common approach is to add a von Neumann-Richtmyer type artificial viscosity to help prevent the steepening of shock fronts, though such methodologies may prove overly dissipative, oth at the shock and throughout the domain. We investigate here an alternate route to regularised shocks, where dispersion is utilised to moilise energy away from short length-scales, as opposed to removing this energy via viscous diffusion. To this end, averaged Lagrangian techniques provide the required framework. We consider standard SPH, the α-sph model, and a spectral method equivalent (see Appendix C), with influence alpha modifications impart on the dynamics eing of concern. Two initial conditions are utilised: firstly, that which results in two approaching shock fronts which eventually merge; secondly, the classical sine wave initial condition Colliding shocks In this section we consider the situation where initially two shock fronts move towards each other, and eventually merging. Here the analytic solution to Burgers equation is given y sinh(βx) u(x, t) = 2βν cosh βx + exp ( β 2 νt), (4.2) (Benton and Platzman, 1972), where β is a parameter which determines our length scale and ν is a kinematic viscosity. This solution may e found in Figure 4.1 for various times. Our velocity scale, as determined y equation (4.2), is U βν, while we define a length-scale of L 1/β. Using these definitions to determine a Reynolds numer Re = UL/ν, results in a constant Reynolds numer, and so non-linear terms scale with viscous terms. While this limits the potential insight to e gained, solution (4.2) still provides a worthwhile preliminary to more complex dynamics. Parameters are determined y first selecting a value for SPH viscosity parameter α, then using (4.4) to determine ν. We fix U max = 1, from which β follows. Initial time was chosen to give a clear initial separation of the approaching wave fronts. To meet inflow oundary conditions, particles were added as required at the oundaries (±L max ) and given initial velocities of u(±l max, t) = U max, L max eing chosen sufficiently large that u(±l max, t) u(±, t). For the presented integrations we set α = 1, and initially 126 particles have een used to span the domain, though due to the inflow condition, this grows to 260 particles y the end of the simulation. Most of this resolution is in effect not utilised, and many particles could e removed if efficiency were a concern. Acceleration equation (3.31) is used where pressure terms are removed to produce the required dynamic. We use a version of the SPH viscosity term written with corresponding kinematic viscosity given y Π a = α ( ) va r a. (4.3) ρ r a α = 7 ν 15 h. (4.4)

53 4.1 Burgers equation 46 1 time = time = v time = time = v Figure 4.1: Solution where a compressile SPH viscosity term has een used. Density gradients result in increased viscosity at fronts. Solid curve corresponds to analytic solution (4.2). This yields dissipation terms of the form found in (4.1). Where an average density is used in the denominator of (4.3), we yield viscous terms involving density gradients (see Section 2.2.4). This results in departure from the Burgers dynamics of equation (4.1), and also therefore solution (4.2). We compare Figures 4.1 and 4.2 where respectively a compressile and incompressile viscosity term have een used. The solution flow field represented in Figure 4.2 clearly reproduces the analytic solution with greater accuracy. While these are simply questions of consistency, they are worth noting and often overlooked y SPH practitioners. Sutle change to the SPH equations can yield significant changes in dynamics. Often a symmetric form for the SPH equations is deemed the highest priority due to the conservation properties which follow. As demonstrated here however, it may e worth sacrificing conservation for strict consistency. This will naturally depended on the relative significance of the different dynamical terms eing modelled. Returning to Figure 4.2, it is noted that in the final frame, minor discrepancies develop at the front crests, with numerical solutions tending to overshoot the correct result, leading to a steeper shock front than required. An expansion on this region is given in Figure 4.3, where we also display profiles resulting from variations to our algorithm, which we now discuss. Thus far we have utilised an SPH implementation for which smoothing length h is held constant. We introduce a variale smoothing length together with self consistent density evaluation (see Section for details). Increasing densi-

54 47 One-Dimensional Tests 1 time = time = v time = time = v Figure 4.2: Solution where incompressile viscosity term (4.3) has een used. As expected, numerical solutions correspond with greater accuracy to exact results. ties, owing to particle accumulation at shock front centres, leads to diminishing smoothing length, and correspondingly small timesteps. We therefore limit the minimum size for smoothing length y applying the following rule for it s determination: h a = σ m a β ( ) β ρ a, (4.5) where for the cuic spline we use σ = 1.3, and the parameter β determines a minimum limit for for smoothing length. For results presented, the value β = 10 has een taken, which allows the smoothing length to take a minimum value of one tenth it s initial setting. We must now also modify our viscosity term, as kinematic viscosity will vary with smoothing length h, according to equation (4.4). Instead of (4.3) we write Π a = α ρ ( va r a r a ) h0 h a. (4.6) In comparing Figures 4.3a and 4.3 we see that the largest differences occurs near the origin. Here the fixed smoothing length routines result in dissipation acting over a larger area than expected, owing to large particle numer densities, and corresponding increased contriutions to SPH viscosity summations. Furthermore, density summations tend to give underestimates in such regions, and particle contriutions may then e overestimated due to poor quadrature. We recognise that smoothing length defines an implicit resolution limit for simulations. Allowing for a variale smoothing length addresses these issues, and

55 4.1 Burgers equation v (a) () v, ˆv (c) (d) v, ˆv (e) (f) Figure 4.3: Comparison of velocity fields for algorithm variations: (a) Standard SPH with a constant smoothing length; () variale smoothing length; (c) constant smoothing length and filtered velocity; (d) full α-sph algorithm again using constant smoothing length; (e) equivalent to (c) ut using incompressile filtered velocity (4.7); (f) equivalent to (d) ut using momentum (4.8) in addition to filtered velocity (4.7). We have symol (+) for standard velocity, and ( ) for filtered velocity. hence Figure 4.3 exhiits reduced spurious diffusion. We next introduce the filtered velocity to investigate it s influence on dynamics, again using the fixed smoothing length algorithm. Firstly the filtered (3.6) is used along with the standard acceleration equation (2.41). A smoothing parameter of ɛ = 1 is used for these calculations, which results in a frequency cutoff parameter given y α 2 = 1 6 h2. The two velocity fields v and ˆv are given in Figure 4.3c. The filtered field ˆv produces the desired result of a smoother velocity field. The standard velocity v appears to have suffered less diffusion that that of standard SPH (Figure 4.3a), owing to reduced energy in short length scales and therefore reduced viscosity (note that the filtered velocity ˆv is used for the viscosity calculation). Figure 4.3d shows the outcome of using the full α-sph algorithm. Results correspond well with analytic solution, and we note that velocity v is not pulled down with smooth velocity ˆv, as in Figure 4.3c. This ehavior may possily e attriuted to the energy conservation found for the full scheme. Filtered velocities have een given y equation (3.6), however such summations lead to dependencies on density gradients, viz equation (3.8). We may

56 49 One-Dimensional Tests instead construct filtered velocities using the following: ˆv a = v a + ɛ m ρ (ˆv ˆv a ) W a, (4.7) which results in a continuum equivalent given y equation (3.10). We also modify our additional acceleration equation terms to suit, writing equation (3.28) as dv a = { ( ɛ ˆv 2 m a ζ a ζ )} a W a. (4.8) dt 2 ρ 2ρ a 2ρ together with ζ a = c m c ρ c ˆv 2 acw ac. It is perhaps not clear which form of filtered velocity and acceleration is preferale. On one hand, our Burgers equation (4.1) has no dependence on density and perhaps we should expect our discretisation to follow suit, with density only acting as a numerical tool to determine quadrature. However, the compressile version is derived of a variational framework, and as such exhiits etter symmetry properties which may e desirale, though we note that the addition of viscosity perhaps voids any enefits. For the colliding waves prolem, there is little to separate the formulations, as a comparison of Figures 4.3c, 4.3d, 4.3e and 4.3f reveals. In summary, there is little to separate the methods considered under this configuration, unless we expand upon regions where difficulties tend to present, though the significance of differences is questionale. Largest improvement is possily attriuted to allowing for a variale smoothing length, though this comes at greatly increased cost due reduced timestep. We now consider the classic Burgers realisation of a flow field initiated with a single sinusoidal mode The steepening shock front Our Burgers simulation is now initiated using u 0 (x) = sin(2πx), (4.9) which is defined in the periodic domain 0 x < 1. The parameter which determines ehavior of the system is the Reynolds numer which we now define as Re = UL (4.10) ν where U is a velocity scale, L is a length scale and ν is the kinematic viscosity. For the sine wave initial perturation, we take L 1 and U 1. The value the kinematic viscosity coefficient takes will depend on the SPH viscosity coefficient α and the form of the SPH viscosity term used. For the lower Reynolds numer regimes, energy largely resides in large lengths scales, eing diffused away efore nonlinear advection takes it into the short scales. Velocity fields are correspondingly smooth, and tend to e represented with greater easy via discrete representations. For larger Reynolds numers, viscosity is less ale to arrest energy propagation to short length scales. For any discrete method then, there exists a regime where energy is carried

57 4.1 Burgers equation 50 1 time = 0.1 time = v time = 0.3 time = v Figure 4.4: Multivalued solutions resulting from insufficient resolution of viscous length scales. This simulation utilised a fixed smoothing length with one-hundred particles, Re = eyond resolution allowances, and ensuing evolution will depend on the details of the discretisation. While exact analytic solutions exist for the current configuration (Cole, 1951), where Reynolds numers are high (Re > 100), evaluation ecomes impractical due to slow convergence of series solutions. We rely on comparison with other numerical results, though there appears to e some inconsistencies in the literature. For simulations with Re 10 5, Zhang et al. (1997) have compared simulations with those of Kakuda and Tosaka (1990). They found significant differences in early time evolution, where velocity fields were still relatively smooth, though late time velocity profiles appeared to agree well. They have also compared results with Varoglu and Finn (1980), with good correspondence at all points in the domain with the exception of near the discontinuity, where specific characteristics of the particular numerical method may e of influence. More recently Wei and Gu (2002) have given a method y which they are ale to determine numerical solutions up to the inviscid regime, while Xu and Duan (2001) have also calculated results using Reynolds numers of up to Re = 10 5, though some artifacts are apparent. SPH simulations will e compared with these author s results. For large Reynolds numers, we correspondingly have small dissipative length scales which must e resolved y the numerical method. Where insufficient resolution is used, SPH simulations yield solutions which have multivalued velocity fields (see Figure 4.4), though Burgers equation (4.1) cannot admit such solu-

58 51 One-Dimensional Tests 1 time = 0.1 time = v time = 0.5 time = v Figure 4.5: Burgers solution at various times. Re = 10 2, 100 particles, β = 10. tions. In this situations, the SPH realisation of viscosity is in effect watered down due summations over large numer of particles with few contriuting significant viscous effect. We also note that the assumption of local smoothness used in the deriving continuum viscosity equivalents (see Section 2.2.4) is no longer valid. To correct this, it is desirale to reduce smoothing length. We may increase the particle population while using a constant-h implementation, though as we only require increased resolution at the shock, it is more efficient to simply allow smoothing length to vary. Indeed SPH lends itself very well to such simulations, concentrating resolutions where most required. We require the use of limited smoothing length rule (4.5) to maintain a timestep which allows for reasonale time evolution. All calculations are performed using one-hundred particles, though we allow smoothing length parameter β to vary such that higher Reynolds numer regimes are effectively calculated at higher resolution. Regimes considered are Re = 10 2, 10 3 and 10 5, with respectively β = 10, 50 and 600 (Figures ). All particle cross-streaming has een prevented through use of increased resolution. We also note that the representations given do not convey the near shock velocity profile well. For instance, the Re = 100 simulation, Figure 4.5, at time t = 1.0, shows a velocity profile which appears to have discontinuous derivative near the shock, where in actual fact particles follow a smooth path past the peak, and into the centre of the domain. Comparison with results of Varoglu and Finn (1980) (Figure 4.7) reveal only minor discrepancy in Re = 10 5 simulations, lending support to their solutions over those of Kakuda and Tosaka (1990). Results of Wei and Gu (2002) are also shown in Figure 4.7 and are in excellent correspondence with

59 4.1 Burgers equation 52 1 time = 0.1 time = v time = 0.5 time = v Figure 4.6: Burgers solution at various times. Re = 10 3, 100 particles, β = 50. The solid curve represents solutions found via a spectral method using 256 trigonometric modes. SPH results. Solutions have also een calculated using a spectral method (Re = 10 3 ), and excellent agreement with SPH solutions is found (Figure 4.6). Energy spectrums corresponding to given times are to e found in Figure 4.8. Specific definitions for energy spectrum e(k) are to e found in Appendix C, with only kinetic energy components eing considered here. The energy cascade process can e clearly identified, with all energy (originally residing in the fundamental mode) eventually propagating through the spectrum down to the dissipative length scale. Once the turulence has developed, an inertial surange also ecomes evident. For Burgers equation, we expect this range to exhiit a k 2 scaling (Burgers, 1948), which is indeed what is found. Burgers also predicted an exponential decay within the dissipation range which can e seen. As time progresses energy can e seen to decay under viscous forces, with the dissipative range growing as peak velocity falls. Finally, we note the minor artifact at the spectrum tail which results from energy reaching our highest wavenumer mode. The cascade process must finish at this point, and so energy accumulates until viscous dissipation is sufficiently activated. For the spectral simulation, the Re = 10 3 regime may e considered largely resolved, so energy accumulation at short scales does not present an issue, with viscosity dissipating energy efore amplitudes ecome significant. Had a larger Reynold s regime een simulated however, this accumulation would ecome large, and spurious oscillations throughout the physical domain would

60 53 One-Dimensional Tests v time = 0.1 time = v time = 0.5 time = Figure 4.7: Burgers solution at various times. Re = 10 5, 100 particles, β = 600. Blue curve corresponds to Varoglu and Finn (1980), while green curve is solution due to Wei and Gu (2002). ensue. We note that the ehavior of solutions under such under-resolved circumstances will rest upon the details of the particular numerical scheme, with spectral methods yielding unphysical high order oscillations throughout the domain, while SPH techniques result in the related issue of multivalued solutions. Of course, these issues may e addressed y improving resolution. In SPH we may increase particle numers or allow smoothing length to vary, or oth. For spectral techniques we introduce further modes. While this achieves the desired result, largely increased computational cost is incurred due to reduced timestep, which may e compounded y increased operation count per step where more modes or particles are required. In analogy to computational Navier-Stokes turulence, we may wish to instead provide closure to the Burgers regime at some scale larger than that required of viscous closure. It seems that the α-sph model may provide means of this end, though unfortunately it is found that the model proves to e unstale under the current configuration. Though a formal staility analysis has not een performed, the instaility occurs and similar in nature for oth SPH and spectral simulations. From this we may possily conclude that the instaility is intrinsic to the differential dynamics and not a product of the particular discretisation taken. We also note that the instaility is only apparent where turulence terms are included in our momentum equation, and does not present where only the filtered advection velocity is used. Geurts and Holm (2002) also makes mention of this Burgers instaility, attriuting it to excessive energy ack-scatter resulting from antidiffusive turulence terms. We

61 4.2 One-dimensional Navier-Stokes time = 0.1 time = 0.3 E(k) time = 0.5 time = E(k) k/π Figure 4.8: Energy spectrum at various times for the Re = 10 3 spectral solution given in Figure 4.6. The dashed line gives a reference for the k 2 energy spectrum scaling expected for the Burgers inertial range. k/π finally note that the instaility occurs regardless of which momentum equation is utilised, whether it e the compressile form given y momentum equation (3.31) and velocity filter (3.6), or the incompressile version defined y momentum equation (4.8) and velocity (4.7). Some examples of unstale solutions may e found in Figure One-dimensional Navier-Stokes As we are unale to perform simulations of Burgers regime incorporating alpha turulence terms, we instead wish to consider an alternate system which most importantly presents energy cascade phenomena y which we may investigate the action of oth standard SPH and α-sph. Various one-dimensional models have een proposed y numerous authors which are designed to mimic ehavior of multidimensional turulent flows, or rather to recreate various energy scalings found in higher dimensional systems. Bartello and Warn (1988) have given a model where in effect severe mode truncation is performed in all ut one dimension, which has proven successful in reproducing two-dimensional energy scalings of Kraichnan (1967) and Batchelor (1969), ut not as successful for the three-dimensional Kolmogorov (1941) scaling (Bartello, 1992). An alternative model due to Qian (1984) modifies the one-dimensional advection term and introduces an artificial pressure term such that Kolmogorov s k 5/3 energy scaling is reproduced. Furthermore, a model which is ased on a postulated

62 55 One-Dimensional Tests time = 0.10 time = ˆv time = 0.20 time = ˆv Figure 4.9: Solution demonstrating instaility of alpha model to the steepening shock Burgers configuration. Solution computed using incompressile version of alpha model. SPH solution represented with + symols, Re = , 100 particles, β = 10, ɛ 0 = 1, ɛ a = ɛ 0h 2 0/h 2 a. Solid curve solution is computed using spectral methods with 1024 modes. stochastic advection is given y Kerstein (1999). However the ehavior of alpha turulence terms under such models are not necessarily expected to mimic those of higher dimensional systems, and contruction of such models within the SPH framework presents difficulty. We instead move on to consider a related system to that of Burgers equation which simply incorporates a pressure gradient term, hence yielding the one-dimensional compressile Navier-Stokes system. This provides the necessary ingredients of conservation of total energy, along with the non-linear terms required for energy cascades. Naturally we still do not expect dynamics imparted y turulence terms to necessarily represent those found in higher dimensions, owing to the sacrificed physics in restricting dynamics to one-dimension. We approach results with the view of gaining perspective to the mathematics of cascade processes realised in standard and α-sph simulations. Our onedimensional system is then (see Appendix B for details): v v + ˆv t x = 1 P ρ ˆv = v + α2 ρ ( ( x + α2 ˆv ρ x ( ρ ˆv x x ) 2 ρ ˆv 2ˆv ) + 2ρ x x x 2 + ν ( ρ ˆv ) (4.11) ρ x x ), (4.12)

63 4.2 One-dimensional Navier-Stokes 56 with continuity equation ρ ρ ˆv + ˆv = ρ t x x, (4.13) and equation of state P = ρ γ. (4.14) Pressure perturations then move at the sound speed C s, where C 2 s = P/ ρ. We initiate all flows with velocity perturation v 0 = v(x, 0) = 0.05 C s sin(2πx), (4.15) where the mach numer has een taken to ensure density perturations are within five percent of the ase density. Integrations are performed over a periodic domain 0 x < 1, and so we find that the fundamental mode perturation will oscillate with period T = 1/C s. We define the nondimensional time t = t /T for dimensional time t The Euler system First considered is the dynamics of the Euler system, and a modified equivalent where alpha terms have een included. In the standard guise, the Euler equations present a open system in the sense that infinite spectral components will ecome active given sufficient time. The multidimensional physical realisation of this is an infinite inertial energy cascade range, where large eddies spawn smaller eddies, which themselves spawn further eddies onto infinity. In one dimension we have a related situation where as with the Burgers simulation we find a shockfront which steepens with time, though now the shockfront oscillates. This contrasts the full Navier-Stokes regime where eventually energy will e acted upon y viscosity, halting any further cascade. Numerically, turulent Euler simulations only represent continuum dynamics up until the point where resolution limitations prevent or slow further cascades. The modified Euler algorithm introduces the length scale α eyond which it is expected that energy cascades will e inhiited, thus providing an alternate route to closure. We present here simulations for the modified algorithm, along with it s standard counterpart. Both SPH and spectral techniques are utilised and compared. Standard Euler First considered is the integration of Euler s equations using the spectral algorithm. Figure 4.10 gives the velocity and density profiles for a simulation utilising 256 trigonometric modes, with Figure 4.11 showing closer views at certain time points. The initial sinusoidal perturation evolves to forms two shock fronts which oscillate ack and forth within the domain. Eventually this steepening process cannot continue as the spectral decomposition is unale to represent shorter length scales, with resulting dynamics given in the latter frames of Figure 4.10.

64 57 One-Dimensional Tests Figure 4.10: Velocity and density profiles for compressile Euler simulation using spectral algorithm. Velocity and density are represented with red and gray curves respectively. Frames are given in 0.4 T increments. Density scale shows values within five percent of static density. Another perspective is found in considering the energy spectrum of the flow, with energy modes defined in Appendix C. We oserve in Figure 4.12 the cascade of energy down from the initial fundamental mode perturation, with what appears to e a tendency towards k 2 energy scaling. Naturally no developed turulent state can e reached given the infinite spectral modes required. We see that once energy reaches the final mode, it is unale to e passed down any further, and we have so-called spectral locking where energy is continually dumped into this final mode. Preceding modes soon also struggle to pass down energy, owing to excited state of the final mode, so they also egin to grow. Eventually the solution is dominated y this ehavior. For this calculation, total system energy is conserved to within less than one percent of the initial pertured energy up to time t = 10. We now turn to SPH simulations of the compressile Euler configuration. First attempted are calculations using a fixed smoothing length implementation. Unfortunately, these are not met with success. While qualitatively we have similar ehavior to that found using our spectral algorithm, there are a numer of significant departures.

65 4.2 One-dimensional Navier-Stokes 58 1 time = 2.3 time = v 0 1 ρ -1 1 time = 5.9 time = v 0 1 ρ Figure 4.11: Velocity and density profiles for compressile Euler simulation using spectral algorithm. Accumulation of energy at the spectral limit ecomes evident at later time points, physically realised as oscillations throughout the domain. Velocity is represent with red curve, while density is given y grey curve time = 2.3 time = E(k) time = 5.9 time = E(k) k/2π k/2π Figure 4.12: Energy spectrum for flow configurations found in Figure The dashed line gives a reference for k 2 energy spectrum scaling.

66 59 One-Dimensional Tests E(k) spectral h = 1.3 x h = 2.0 x h = 2.5 x k/2π Figure 4.13: Energy spectrum for different implementations at time t = 2.5. We first perform simulations using one-thousand particles, with smoothing length set at h = 1.3 x for initial particle spacing x. It is found that perturations do no not move at the expected velocity, with results lagging ehind the spectral method equivalent. Further to this, it appears the nature of the nonlinearity is modified, with energy cascades processes taking longer to reach an equivalent state. We note that even if we rescale the integration time, our simulations still do not coincide, with cascade processes eing slowed to a greater extent than sound speed. Simulations appear convergent, with increased particle numers and shortened timestep leaving results unchanged. Increasing smoothing length to h = 2 x leads to more favourale results. Pressure perturations now in effect travel with only slightly larger than expect speeds, and energy cascades proceed at a comparale rate to those of spectral calculations. If we now increase smoothing length further still to h = 2.5 x, sound speed is again found to fall short of the expected figure. Now however, the energy cascade is enhanced and energy propagates through the spectrum faster than found in the spectral integration. These results are demonstrated in Figure 4.14 where velocity profiles in time are given, and Figure 4.13 which shows energy spectrums. Details for the determination of spectral mode coefficients corresponding to SPH particle simulations are to e found in Appendix D.

67 4.2 One-dimensional Navier-Stokes 60 Figure 4.14: Velocity profiles in time increments of 0.1 T. The lack dashed line represents results for the spectral algorithm; the green, lue and magenta curves correspond to simulations using h = 1.3 x, h = 2.0 x and h = 2.5 x respectively.

68 61 One-Dimensional Tests Figure 4.15: Velocity profiles for variale-h SPH simulation (lue) alongside results otained using spectral algorithm (red). Frames are given in 0.4 T increments. While est results appear to correspond to h = 2.0 x, we are left unsure as to whether this choice of smoothing length parameter will est suit all configurations, or even all regimes within a simulation (such as transition to turulence, or developed turulence). We instead seek solutions using a variale-h implementation, which are found to correspond closer to spectral solutions. Figure 4.15 gives ensuing velocity profiles, where we find that results are nearly identical up until the point where resolution limits are met. In spectral space (Figure 4.16), a few differences ecome apparent. Firstly we notice that mode growth rates fall slightly short of those found for the spectral method, though the difference is relatively minor. More significant is the attenuation of high order modes at later times. A divergence first occurs when energy reaches modes of order one-hundred, at approximately time t = 3. From this time on a second spectral range forms with a steeper energy scaling, as can e seen in the third frame of Figure We have now in effect reached a point where solutions will egin to diverge from strict Euler dynamics, with the higher order details of the SPH technique ecoming significant. Energy however still attempts to filter down the spectrum from aove, with Euler dynamics tending towards a k 2 energy scaling. This results in accumulation of energy at particular modes, with more energy entering than is ale to leave, and the spectrum then tending to uckle forming kinks at certain modes (final frame Figure 4.16). This is not dissimilar to the spectral locking oserved for the spectral algorithm. Here however, overly excited modes are eventually ale to pass energy down, with kinks then moving through the spectrum. SPH thus provides a less arupt cascade cutoff, though this occurs at a much larger

69 4.2 One-dimensional Navier-Stokes time = 2.0 time = E(k) time = 5.3 time = E(k) k/2π k/2π Figure 4.16: Energy spectrum for variale smoothing length SPH implementation (lue), alongside spectrum for spectral algorithm (red). The dashed line gives a reference for k 2 energy spectrum scaling. wavelength than found for the spectral technique. The variale smoothing length is defined according to equation (2.67), for which the parameter σ determines the numer of particles which fall over the kernel s support domain, and therefore modifies the smoothing length. Some examples demonstrating the influence of varying σ (and therefore smoothing length) are to e found in Figure We can clearly see modulation of the cascade cutoff point as smoothing length is increased, with cutoff point moving to larger scales for increased smoothing lengths. Similarly, we may instead vary particle numers, maintaining a smoothing length parameter σ = 1.3 (Figure 4.18). We are still here in effect varying smoothing length, and so it is not surprising that we find similar results to those of Figure 4.17, with the consequence again eing an alteration of cutoff frequency. A similar change in cutoff point is oserved for equivalent changes in smoothing length (i.e. n = or σ = ). To test the convergence of the SPH summation interpolant (2.5), we wish to increase σ while keeping the asolute ase smoothing length fixed, so therefore must increase particle numers (Figure 4.19). What is found is a slight increase in cutoff point, indicating that σ = 1.3 is not sufficient for true convergence of the summation interpolant. We note that the difference is relatively minor, and increased costs incurred may perhaps e etter utilised simply increasing particle numers while using σ = 1.3. Finally, we consider the influence of the omega terms (2.74) found in the momentum equation (2.75). These terms

70 63 One-Dimensional Tests E(k) σ = 1.3 σ = 2.0 σ = 2.6 spectral k/2π Figure 4.17: Energy spectrum for different smoothing lengths with fixed particles numers. Results are taken at time t = E(k) n = 500 n = 1000 n = 2000 n = 4000 spectral k/2π Figure 4.18: Energy spectrum for different particle populations using smoothing length parameter σ = 1.3. Results are taken at time t = 5.0.

71 4.2 One-dimensional Navier-Stokes E(k) n = 1000 n = 2000 Ω = 1 spectral k/2π Figure 4.19: Energy spectrum keeping ase smoothing length fixed at h 0 = 1.3 x 0 with x 0 = Results are taken at time t = are related to the gradient of the smoothing length, and are required for a variationally consistent SPH derivation. We can see in Figure 4.19 that in setting Ω = 1 we find medium-scale modes suffer significantly less damping. However, the sacrificed variational consistency appears to result in phase speed inaccuracies, and in physical space results are found to step away from the spectral solution, though not as severely as found where smoothing length is kept fixed. So despite the increased mode damping, we choose to retain our calculation of Ω according to equation (2.74), though the differences found in setting Ω = 1 are worth noting. In conclusion, we can say that while the SPH technique is ale to produce the ulk dynamics of Euler flow, mode attenuation ecomes significant from length scales of order ten times larger than smoothing length. This imposes a significant restriction on the simulation of medium to short scale Euler dynamics which can only e resolved through use of very large particle populations. While in one-dimension this is perhaps on option, even stepping up to two-dimensions we will find computer resources quickly overwhelmed. On the other hand, we may interpret this as an implicit turulence type ehavior, with large scales eing calculated as desired and short scales eing inhiited. For a simulation such as Navier-Stokes flow, where dissipation will provide eventual closure, this may indeed prove useful in preventing spurious short-scale ehavior corrupting ulk dynamics. Further investigation of this implicit turulence modelling ehavior is certainly warranted, though for now we turn to the explicit turulence modelling scheme outlined in Chapter 3.

72 65 One-Dimensional Tests E(k) time = 2.3 time = E(k) time = 5.9 time = k/2π k/2π Figure 4.20: Energy spectrum for standard Euler (red) and modified Euler (lue) simulations. 256 modes are utilised with k α 596. Dashed line gives a reference for k 2 energy spectrum scaling. Modified Euler Our modified Euler systems is defined as the one-dimensional version of the alpha continuum model, given y equations (3.30) and (3.8): [ ( v v + ˆv t x = 1 ) 2 P ρ x + α2 ˆv ρ ˆv 2ˆv ] + 2ρ ρ x x x x 2 ( ˆv = v + α2 ρ ˆv ), ρ x x with continuity equation ρ ρ ˆv + ˆv = ρ t x x. It is expected that the additional terms will act to slow and then halt energy cascade processes for modes of order kα 1, therefore closing the Euler system at some wavenumer where kα 1. We consider results for the spectral algorithm first. Figure 4.20 gives the progression of energy modes for the modified scheme alongside the results found aove for standard Euler. For this calculation 256 modes are used, and the effective alpha cutoff wavenumer is k α 596, where we define k α = 1/α. This choice is taken to coincide with values used for SPH simulations.

73 4.2 One-dimensional Navier-Stokes 66 Figure 4.21: Velocity and density profiles for compressile Euler simulation using spectral algorithm with alpha parameter k α 596. Velocity and density are represented with red and gray curves respectively. Frames are given in 0.4 T increments. Density scale shows values within five percent of static density. We oserve that oth simulations proceed identically (first frame Figure 4.20) until the point where modes egin to feel the affects of alpha turulence terms, which under the current parameters occurs from mode m = 30. The energy spectrum is then turned down with energy propagation inhiited (second frame Figure 4.20). As seen previously in SPH simulations, we then find kinks developing in the energy spectrum as energy mode net fluxes ecome non-zero (third frame Figure 4.20), and a train of such kinks eventually develop further along the energy spectrum (fourth frame Figure 4.20). The spectrum has now reached a largely statistically steady state, and so we may conclude that closure has een attained. We note that for large length scales, the spectrum appears ale to maintain the expected k 2 energy scaling, whereas where turulence terms have not een included eventually noise overwhelms all wavenumer modes. The simulation in physical space is shown in Figure 4.21 for density and velocity, and may e compared with results without turulence terms found in Figure The most ovious change is the reduction in short scale noise, which for standard Euler simulations appears to overwhelm the solution. This is especially evident in the final frame of It may e said that solutions are

74 67 One-Dimensional Tests E(k) k α = k α = 5958 k α = 2665 k α = 1884 k α = 843 k α = 596 k α = k/2π Figure 4.22: Spectral algorithm energy spectrum for different alpha turulence parameters at time t = 5. regularised y the addition of alpha turulence terms. We find in Figures 4.22 and 4.23 results for different values of parameter k α. At the earlier time (Figure 4.22), we see that all simulations inhiit energy propagation with strength relative to alpha cutoff parameter k α, as expected. For higher wavenumer cutoff parameters, energy is still ale to reach the highest integrated wavenumer (n = 256), though at this simulation time minimal energy accumulation appears to e occurring. Turning to the later time results (4.23) it is found that simulations with higher alpha cutoff wavenumers (such as k α 2000) are ale to produce improved large scale dynamics despite requiring greater resolution for true closure. Lower wavenumer cutoff parameters, while significantly reducing noise at short scales, tend to result in large scales which deviate from Euler dynamics. Instead, an overaundance of energy is found due to inhiited propagation, with incorrect scaling caused y neary energy kinks. In general it is desirale to resolve most if not all of the alpha dispersive energy surange, though computational cost may rule this out. Bearing these points in mind, for closest approximation to Euler dynamics the alpha cutoff wavenumer should e set at some value short of the maximum wavenumer which allows for reasonale integration of the dispersive surange. Returning to Figure 4.23, we find that parameter k α = 843 meets these requirements, perhaps giving the est compromise etween short scale noise and large scale accuracy. In turning to α-sph simulations, a few points need to e considered. Firstly, where we use variale resolution, our filtered velocity cutoff length scale will depend on smoothing length according to equation (3.11). However, we instead wish to have a fixed cutoff frequency for comparison with the spectral algorithm.

75 4.2 One-dimensional Navier-Stokes E(k) k α = k α = 5958 k α = 2665 k α = 1884 k α = 843 k α = 596 k α = k/2π Figure 4.23: Energy spectrum for different alpha turulence parameters at time t = 10. This may e achieved y allowing parameter ɛ to e vary with smoothing length: ɛ a = h2 0 h 2 ɛ 0, (4.16) a for initial values h o and ɛ 0. In application, the est approach (fixed or variale cutoff frequency) is unclear. Using a variale cutoff allows us to slow energy cascading as they approach our resolution scale, which practically may e desirale. For the current simulation, density variations are small and so we do not expect a large difference etween approaches though for consistency with the spectral algorithm we make use of equation (4.16). Secondly, as oserved aove for the SPH Euler simulations, there exists a secondary dynamic which acts to disperse energy away from short length scales. Taylor series expansions reveal the following for the SPH pressure gradient term: m { Pa ρ 2 a + P } ρ 2 a W a ρ 1 P + β 2 ( ρ 2 ρ P +P f + 3P f + 3P f + P f) where primes denote derivatives with respect to coordinate (example P = P x ), we set f = 1/ρ, and in analogy to the definition of α (equation (3.11)) we define the following: β 2 = 1 (x x) 2 W (r r, h(r)) dr. 2

76 69 One-Dimensional Tests Where we use equation of state P = ρ γ, the aove ecomes m { Pa ρ 2 a + P } ρ 2 a W a γρ γ 2 ρ + β 2( γρ γ 2 ρ + 3ρ γ 3 ρ ρ (1 γ)(2 γ) ) ρ γ 4 (ρ ) 3 (1 γ)(2 γ)(3 γ). For the cuic spline kernel, the parameter β takes the value β 2 = h 2 /6. We postulate that these terms are responsile for the dispersive ehavior oserved for the earlier SPH Euler simulations. If we wish to investigate the effects of alpha terms introduced in α-sph, we must set α β to minimise the influence of eta terms. Unfortunately this leads to expensive computations, as we required ɛ 1 which necessitates many iterations for filtered velocity convergence. Bearing the aove in mind, we proceed to consider the α-sph simulations, and in Figure 4.24 we find results where one-thousand particles are utilised and different values of ɛ are taken. Interestingly, it is found that for small values of ɛ, the additional turulence terms tend to allow increased propagation of energy into short scales. The exact mechanisms y which this is ale to occur is unclear, though it appears that turulence terms perhaps reduce the strength of the secondary terms discussed aove. In comparing with the equivalent spectral algorithm results found in Figure 4.22, we find that reasonale agreement is found for values ɛ greater than unity, with improved correspondence for increasing values of ɛ, as predicted aove. Results at a later time are to e found in Figure 4.25, and can e compared with Figure As for results at the earlier time, est correspondence is found for largest values of ɛ. Simulations are performed to investigate dynamics where the relative strength of α and β terms are varied (Figure 4.26). To this end, a fixed value of k α = 1884 is taken, with the value of ɛ is varied with values of smoothing length chosen to suit. For the value ɛ = 1, results appear to e in good agreement on average with the spectral results, though stepping up to ɛ = 2 we see that short scale energy is olstered. Simulations for ɛ = 5 and ɛ = 10 closely correspond to those found via spectral methods. We also note that for all simulations modes eyond n = 300 are inactive, with exception of results for ɛ = 2 where modes eyond n = 380 are inactive. This indicates that improved results are not directly consequent of increased SPH grid resolution, ut rather due to the diminished significance of the SPH modified differential equation terms (i.e. the eta terms). To investigate this further, simulations for fixed asolute smoothing length and fixed parameter ɛ are performed while the particle spacing is decreased (or equivalently particle populations increased). Therefore, summations involve larger numer of particles (so smoothing length parameter σ is increased), and it is expected that such summations will form etter approximations to the integral approximant. Turning to Figure 4.27, we indeed find significant differences at all ut the largest scales. While results where σ = 1.3 on average replicate the spectral result, the spectrum form appears to e generally incorrect. Convergence appears to e attained for h x, though consideraly increased computational cost is incurred. These results echo those found earlier for standard SPH (Figure 4.19). Parameter configurations will in general require some

77 4.2 One-dimensional Navier-Stokes E(k) k α =, k α = 5958, k α = 2665, k α = 1884, k α = 843, k α = 596, k α = 377, ɛ = 0 ɛ = 0.1 ɛ = 0.5 ɛ = 1.0 ɛ = 5.0 ɛ = 10 ɛ = k/2π Figure 4.24: Energy spectrum for different alpha turulence parameters at time t = 5. Results for α-sph simulations, using one-thousand particles. To e compared with the equivalent spectral results found in Figure E(k) k α =, k α = 5958, k α = 2665, k α = 1884, k α = 843, k α = 596, k α = 377, ɛ = 0 ɛ = 0.1 ɛ = 0.5 ɛ = 1.0 ɛ = 5.0 ɛ = 10 ɛ = k/2π Figure 4.25: Energy spectrum for different alpha turulence parameters. Results for α-sph simulations, using one-thousand particles. To e compared with the equivalent spectral results found in Figure Time t = 10.

78 71 One-Dimensional Tests E(k) spectral ɛ = 1 ɛ = 2 ɛ = 5 ɛ = k/2π Figure 4.26: Energy spectrum where ɛ is varied with fixed parameter k α = Particle populations for ɛ = 1, 2, 5 and 10 are respectively n = 1000, 1414, 2236 and Time t = E(k) spectral n = 1000, σ = 1.3 n = 1308, σ = 1.7 n = 1424, σ = 1.85 n = 1500, σ = 1.95 n = 1538, σ = 2.0 n = 2308, σ = k/2π Figure 4.27: Energy spectrum for varied particle densities, with ase smoothing length kept at the same value of h = for all simulations. For all simulations we use ɛ = 1. Results are at time t = 5 with k α = 1884.

79 4.2 One-dimensional Navier-Stokes Energy time Figure 4.28: Time progression of energy in modes and total energy. Uppermost curve gives total energy, with curves elow showing energy in the first, second, third and fourth mode respectively. Spectral algorithm utilising 256 modes, with Re = degree of tuning for an optimal alance of accuracy and efficiency, though it appears that values σ 2 are desirale Forced Navier-Stokes simulations Integrations are performed for the full one-dimensional Navier-Stokes equations. The inclusion of viscosity introduces a length scale where energy dissipation will eventually outweight non-linear cascades, and hence ring aout closure. For real multi-dimensional turulent flows, this process usually occurs at scales which fall well eyond the limits of computational resolution. For onedimensional simulations, full direct numerical simulations of governing equations is a realistic prospect however, which allows for useful comparison with the modified dynamics of LANS, despite the incurred simplification of physics. The forcing regime essentially maintains the initial energy of the fundamental mode. In effect, we force the fundamental mode velocity amplitude such that it always accelerates towards the inviscid linear solution (where total fundamental energy remains constant in time). Density perturations follow suit. Specifically, for the spectral algorithm we apply the follow for velocity forcing v 1 t = f AB 1 + C( v L (t) v 1 ) (4.17) with f AB eing the forcing from our Adam-Bashforth scheme, and v L (t) eing the linear solution. The constant C adjusts the aggression with which forcing

80 73 One-Dimensional Tests 10-3 time = 2 time = E(k) time = 6 time = E(k) k/2π k/2π Figure 4.29: Energy spectrum for Re = 1000 simulation, with 256 modes. The dashed line gives a reference for k 2 energy spectrum scaling. is applied, and typically is set such that forcing acceleration is one tenth the strength of maximum linear mode acceleration. Total energy is found to increase until the point where modes of short scale ecome sufficiently active for viscous dissipation to ecome appreciale. Eventually a alance is found where the forced energy input is equal to the viscous energy dissipation (Figure 4.28). We also oserve a relatively small degree of energy oscillation which may e attriuted to the oscillation of the fundamental mode, consequent of the forcing scheme. Alternatively we may have chosen to explicitly drive the fundamental velocity and density modes according to the linear regime, though an equivalent SPH simulation would have een difficult. Simulations are performed using the spectral scheme with 256 modes. We define a Reynolds numer as previously: Re = UL ν (4.18) for length scale L, velocity scale U and kinematic viscosity ν. The length and velocity scales are respectively defined as the domain size and initial velocity. The energy spectrum at various times is displayed in Figure Since viscosity predominantly acts upon short scales, only once these scales ecome sufficiently active do we have appreciale deviation from the inviscid simulation. Energy in short scales is dissipated under the action of viscosity, and therefore tapers off within the dissipative surange. By the final time frame, a statistically steady state has een achieved, and we expect the energy spectrum to remain stationary. This may e confirmed with reference to the energy progression of Figure 4.28, and also y inspection of the solution in the physical domain

81 4.2 One-dimensional Navier-Stokes 74 Figure 4.30: Velocity and density profiles for Navier-Stokes simulation using spectral algorithm. Velocity and density are represented with red and gray curves respectively. Frames are given in 0.4 T increments. Density scale shows values within five percent of static density. (Figure 4.30). The physical space solution is oserved to evolved from the sinusoidal initial condition to eventually form two shock fronts which oscillate ack and forth. We note that the viscous dissipation range appears to not e completely resolved, with a small degree of spectral locking to e found at the tail of the spectrum. Only the slightest hint of this is found in the physical space solution, with very slight oscillation at shock fronts. Resolution is sufficient such that viscosity ensures that energy does not accumulate. The inertial surange scaling of k 2 may e clearly seen. Forced spectral simulation is performed for various viscosity levels, with steady state results found in Figure Larger Reynolds numers push the dissipation range to higher and higher wavenumers, with the result eing that larger amounts of energy reach the numerical limit. Naturally, a larger degree of energy accumulation occurs at these short scales, with the physical space realisation eing short scale noise overlayed on solutions. Indeed, since a smaller portion of the viscosity range is simulated, the short scales must increase in energy to effect equivalent viscous dissipation. It is not until we step up to Re = 10 6 that viscosity is insufficient to ensure staility, and the entire spectral

82 75 One-Dimensional Tests E(k) Re = 250 Re = 500 Re = 1000 Re = 2000 Re = 4000 Re = k/2π Figure 4.31: Energy spectrum at various Reynolds numers. All results are taken at time t = 10, where a steady state has een reached. range ecomes corrupted, with results similar to those of Figure In this case forcing leads to a continual increase in total energy which we expect to eventually cause simulation failure. Forced Navier-Stokes simulations are also performed using the SPH algorithm. The forcing scheme is equivalent to that of the spectral algorithm. We use the following equation for particle velocity forcing: v a t = f SP H a + C( v L (t) v 1 ) sin(2πx a ). (4.19) Here fa SP H is the standard SPH summation force, v a is the particle velocity, and x a is the particle position. Function v L (t) is as prior, the linear mode solution. The value v 1 is the fundamental mode coefficient, and is determined using methods outlined in Appendix D. Figure 4.32 shows total energy progression, and is in excellent agreement with the equivalent spectral result (Figure 4.28), though a slight increase in total energy oscillation is oserved. The SPH energy spectrum is given in Figure As insufficient resolution is utilised, neither the SPH nor spectral methods give correct ehavior at short scales. The SPH solution however appears to suffer to a larger extent owing to eta terms which slow energy cascades, resulting in energy accumulation points in the spectrum. SPH solutions also exhiit a degree of oscillation in the high wavenumer section of the spectrum, though statistically solutions are steady. Steady state solutions are compared for oth the spectral and SPH code at different resolutions in Figure The spectral method s superiority is evident, with the coarser simulation replicating the higher resolution result for all ut it s highest wavenumers. It appears that eta terms implicit in SPH

83 4.2 One-dimensional Navier-Stokes Energy time Figure 4.32: Time progression of energy in modes and total energy for SPH simulation using 1000 with Re = Uppermost curve gives total energy, with curves elow showing energy in the first, second, third and fourth mode respectively. integrations, and the resulting impedance of energy cascades, leads to unphysical energy accumulation in the coarse SPH simulation (at n 40). For the higher resolution simulation, the viscosity range is sufficiently resolved to prevent such accumulation, though we still do not find the expected exponential decay. This suggests that even where two-thousand particles are used, SPH simulations are still under-resolved for Re = Results for SPH simulations at various Reynolds numers are given in Figure Results appear to e sufficiently resolved for the Re = 100 and Re = 250 simulations. The Re = 500 simulations egins to show evidence that the poorly simulated high wavenumers are having influence, while results for larger Reynolds numer are clearly suject to these inadequacies. We proceed to perform calculations for the alpha modified forced Navier- Stokes regime. Solutions at varying alpha parameter are carried out for a Reynolds numer of Re = Results are presented in Figure Similarities are to e found with previous modified Euler (and SPH) simulations with the formation of kinks in the spectrum where energy propagation is slowed. All simulations (with the exception of that with smallest alpha parameter) are successful in reducing energy at the shortest scales, though spectrums in the intermediate range now certainly deviate largely from the Navier-Stokes solution. We note that for one-dimensional simulations, viscous dissipation ensures solutions remain regular even for very large Reynolds numers (as demonstrated for Re = 10000), despite spectral locking. It appears there is no practical application for the alpha turulence methodologies in one-dimension, which despite

84 77 One-Dimensional Tests 10-3 time = 2 time = E(k) time = 6 time = E(k) k/2π k/2π Figure 4.33: Energy spectrum for Re = 1000 simulation. SPH simulations given y lue points, while spectral results are given in red. The dashed line gives a reference for k 2 energy spectrum scaling. reducing short scale energy, yield large deviates at all ut the smallest wavenumers. Regardless, the alpha turulence dynamics are worth investigating in their own right. Though we do not expect to produced any improvement to solutions, the alpha modifications to our SPH algorithm are also explored. A Reynolds numer of Re = 2500 is taken while the parameter ɛ (or conversely α) is varied (Figure 4.37). Qualitatively results are again very similar to those found previous, oth where alpha terms have een included, and for standard SPH. Similarities are of course due to the slowing of energy cascades at high wavenumers with energy at larger scales trying to flow down at a higher rate. Results echo those found for the modified Euler SPH simulations, where low values of ɛ result in a oost in energy at short scales. As we reach higher values of ɛ however, increased impedance of cascade processes results, leading to reduced energy propagation into high wavenumers. We note that for oth the spectral and SPH simulations, the decrease in energy at short scales reduces the aility of viscosity to effect energy dissipation. This results in a increased transition period to reach a steady state, and also an increased system total energy.

85 4.2 One-dimensional Navier-Stokes E(k) Spectral Spectral SPH SPH 256 modes 512 modes 1000 particles 2000 particles k/2π Figure 4.34: Energy spectrum for SPH and spectral codes at different resolutions. All results are taken at time t = 10, where a steady state has een reached with Re = E(k) Re = 100 Re = 250 Re = 500 Re = 1000 Re = k/2π Figure 4.35: Energy spectrum at various Reynolds numers using SPH method. All results are taken at time t = 10, where a steady state has een reached.

86 79 One-Dimensional Tests E(k) k α = k α = k α = 5027 k α = 2513 k α = 1257 k α = k/2π Figure 4.36: Energy spectrum for different values of parameter α. All results given are for the steady state solution using the spectral algorithm with 256 modes and at Re = E(k) k α = k α = 1826 k α = 1289 k α = 576 k α = 408 k α = 258 ɛ = 0 ɛ = 0.5 ɛ = 1 ɛ = 5 ɛ = 10 ɛ = k/2π Figure 4.37: Energy spectrum for different values of parameter α using SPH algorithm with 1000 particles and h 0 = 1.9 x. Results given are for steady state solutions at Re = 2500.

87 4.3 Summary and Conclusions Summary and Conclusions Numerous one-dimensional results have een presented, with comparison eing made etween SPH and spectral algorithms. The SPH technique s ehavior in the limit of marginally resolved dynamics is examined, with the influence of simulation parameters and methodologies considered. Further to this, the inclusion of alpha terms and dynamics therein have een investigated. Initially simulations of Burgers equation were tested. Where a variale resolution implementation has een used, standard SPH proved very capale of producing highly accurate results, assisted y the natural concentration of resolution where most required. For moderate Reynolds regimes (Re = 10 3 ), results have een verified through comparison with spectral integrations, for which excellent correspondence was found. For higher Reynolds numers (Re = 10 6 ), we have compared SPH simulations with the Burgers results of Wei and Gu (2002), among others. Again excellent agreement was found. Less fortune was found however in investigating the effects of alpha terms in Burgers regime simulations, with an instaility resulting, possily owing to excessive ack-scatter (Geurts and Holm, 2002). Next the SPH algorithm was recast to perform simulations of Euler dynamics. Here we integrate an open system, and it is expected that eventually resolution limits will e encountered. The ehavior of our schemes in approaching and reaching this limit has een considered. The spectral technique provided enchmark results y which the accuracy of SPH could e quantified. Naturally the spectral method also has resolution limits which eventually lead to solution corruption. Up to this point accuracy is generally very good, and comparison with larger mode numer simulations show that only a small portion at the high end of the spectrum are effected y resolution shortcomings. Turning to the SPH simulations of Euler dynamics some interesting results are oserved. Firsty, it is found that where a constant SPH smoothing length is used, pressure perturations do not travel at the speed determined y the linearised system. While we may tune smoothing length parameter such that correct sound speeds are otained (Figure 4.14), the perhaps more significant issue of incorrect cascade rates persists (Figure 4.13). The use of a variale smoothing length corrects oth of these issues, though we now encounter secondary effects which ecome dominant at length scales of order 10h for smoothing length h. These effects tend to slow energy cascades into shorter scales (Figure 4.16). We relate this to eta terms which appears in the SPH resulting modified differential equation. Also worth note is the non-dissipative nature of these terms which may e attriuted to the variational derivation of the SPH scheme. The possile function of eta terms as an implicit turulence modelling methodology perhaps undermines the usefulness of SPH alpha turulence terms. Spectral simulations incorporating alpha terms indeed reveal ehavior which is at least qualitatively very similar to that encountered for standard SPH (compare Figures 4.16 and 4.20). With regards to the spectral results, we note that where a large enough alpha parameter has een taken, turulence terms are indeed ale to ring aout closure, with the largest scales still producing Euler energy scaling and energy conservation maintained. As such, eventual solution corruption due to resolution limitations is averted, so solutions may e said to e regularised. Where alpha terms have een included in SPH simulations,

88 81 One-Dimensional Tests some interesting results are found. Large values of parameter epsilon lead to the expected result of greater energy inhiition. For small values of the epsilon, we have the counter-intuitive result of decreased inhiition with greater energy reaching high wavenumer energy modes. Beta and alpha terms are expected to e of similar order in this limit, so this ehavior is perhaps owing to an interplay etween terms. We may fix parameter alpha and take larger values of epsilon via modulation of smoothing length, hence increasing the prominence of alpha terms with respect to eta terms. Then for large values of epsilon, the α-sph simulations produce results in line with those otained using the spectral algorithm. Resulting SPH simulations are however computationally expensive with large iteration counts required for convergence. The inclusion of viscosity yields the full Navier-Stokes equation. A forcing regime has een implemented which ensures constant energy in the fundamental mode, with integrations eventually yielding a statistically steady state. Spectral simulations for underresolved conditions yield the expected energy accumulation at the highest mode. While higher Reynolds regimes exacerate this shortcoming, it is not until we go aove Re = 10 5 that this leads to entire solution corruption. Increase mode numer simulations show that at least for Re = 1000, solution error is restricted to the final twenty percent of modes (Figure 4.34). SPH simulations reveal similar ehavior to that found earlier and attriuted to the eta terms of the modified differential equation. So where viscosity is insufficient to prevent significant energy propagation to scales shorter than approximately 10h, we have a slowing of further cascades and characteristic formation of kinks in the spectrum. Again we find similar ehavior for spectral results inclusive of alpha turulence terms, with the slowing of energy cascades. We perhaps cannot say that an improvement in accuracy is realised in the use of these terms for the underresolved simulations, though as with Euler simulations, they may prevent noise at high wavenumer from permeating through the spectral domain. SPH results also reflect those found for Euler simulations, with a oost in short scale energy for small values of parameter ɛ and similar ehavior to spectral results for large values of ɛ where alpha terms dominate. With regards to the SPH numerical method, all results indicate that a variale smoothing length should always e chosen over a fixed smoothing length implementation. For Burgers equation, this requirement is orn of the need to implement viscosity summations terms over short scales. The reasons for the vastly improved accuracy realised in Navier-Stokes simulations is less clear. Results also indicate that where we define a smoothing length according to equation (2.67), est summation accuracy is otained for σ 2, though the additional cost incurred may warrant use of smaller values. In conclusion we emphasise that results in one-dimension will not necessary reflect those to e found in higher dimensions, with very different physics to e encountered in two and three dimensions. Perhaps most significant is the appearance of transverse waves, which are likely to influence the action of oth alpha and eta terms. At least for one-dimensional simulations we can say that there appears to e no enefit in using alpha terms where instead we may vary the significance of eta terms through modulation of parameter σ.

89 Chapter 5 The Kelvin-Helmholtz instaility We consider the linear instailities which may present within a fluid containing layers in relative motion, the so-called Kelvin-Helmholtz instaility. While original mention of this instaility is generally attriuted to Helmholtz (Helmholtz, 1868), it is Lord Kelvin who first presented a thorough mathematical investigation of it s properties under various configurations (Kelvin, 1871). A minimum requirement for the instaility of inviscid unidirectional two-dimensional flow is the presence of an inflection point in the lateral velocity profile (Rayleigh, 1880). Meeting this requirement is the configuration wherey a fluid containing two regions of different ut constant velocity is separated y a surface of discontinuity (see Figure 5.1). The instaility of this flow was estalished y Helmholtz (1868), and it is found that sinusoidal perturations of all wavelengths grow exponentially in time with growth rate proportional to wavenumer. Instaility may also occurs for flows where the lateral profile of the mean streamwise velocity is given y a hyperolic tangent function (see Figure 5.1). Such profiles correspond to free oundary layers that form where two fluids of different velocity ut same direction meet, as can e found for instance at the edge of jets and wakes, and at the trailing edge of an asymmetric aerofoil. Similarly, splitter plate experiments produce profiles approximated very well y the hyperolic tangent function (Dimotakis and Brown, 1976; Slessor et al., 1998). Unlike the constant shearing configuration discussed aove, the splitter plate regime only exhiits instaility over a finite andwidth of sinusoidal modes. As such, there exists a particular wavelength of fastest growth, and an initial white noise perturation will eventually e dominated y growth of this mode. The ensuing non-linear evolution yields coherent vorticies and the phenomenology of mixing layers. In this chapter the aove two regimes are considered, with growth rates found for SPH simulations compared with values from staility theory. A numer of limitations for this comparison must e noted. Firstly, the linear theory assumes an incompressile fluid, while SPH integrations approximate the compressile Navier-Stokes system. The convective Mach numer (Bogdanoff, 1983), defined as M = (U 1 U 2 )/(c 1 + c 2 ), for mean stream velocities U 1 and U 2 and corresponding soundspeeds U 1 and U 2, is the most appropriate nondimensional 82

90 83 The Kelvin-Helmholtz instaility U 1 U 2 U 0 tanh(z) Figure 5.1: Mean velocity profiles. numer to parametrise mixing layer compressile dynamics (Papamoschou and Roshko, 1988; Raga and Wu, 1989). To simulate a nearly compressile fluid, we seek a Mach numer M 0.05, and so for U = U 1 = U 2 and C s = C 1 = C 2, we set soundspeed as C s = 20U. This yields Mach numer M = 0.05, from which it may e shown that δρ/ρ M 2 = Density variations throughout simulations should therefore e minimal. Furthermore, numerous compressile calculations (Raga and Wu, 1989; Sandham and Reynolds, 1991; Sauvage and Kourta, 1999) have shown that dynamics are very similar to incompressile simulations for Mach numers M < 0.5, with growth rates reduced y less than five percent at M = 0.2 and approximately twenty percent at M = 0.5 (Vreman et al., 1996). Another potential source of deviation from linear theory occurs due to the finite domain within which SPH simulations are performed. Linear solution eigenfunctions decay exponential however, so we can expect velocity perpendicular to oundaries to e negligile for sufficiently large computational domain size. A clearer definition of sufficiently large will follow, though for now we simply state that a domain size some multiple of the pertured mode wavelength should suffice. For the presented linear regime results, viscosity is not included. As integration are only performed over a long enough time to capture the exponential mode growth, it is not expected that any significant excitation of overly short (with respect to grid resolution) modes will occur, and so Euler simulations are valid. We also note that while linear results for the constant shearing velocity configuration gives mode growth proportional to wavenumer, SPH simulations tend to attenuate growth as wavelengths approach resolution length scale, preventing these modes outgrowing pertured modes. Any addition of viscosity however will have a stailising effect on all wavelength modes, with reduced growth rates (Raga and Wu, 1989). 5.1 Constant velocity fluids in relative motion We consider the Kelvin-Helmholtz instailities that may arise at the interface etween two fluid odies of constant velocity in relative motion.

91 5.1 Constant velocity fluids in relative motion Linear results The onset of instailities may e precipitated with perturations of the form u(x, z, t) = u(z) exp(ikx + n k t) (5.1) w(x, z, t) = w(z) exp(ikx + n k t) (5.2) where k is a wavenumer, n is a growth exponent and i = 1. These perturations are superimposed over the counter-streaming flow given y ū(z) = { +U z > 0 U z < 0. (5.3) for a domain of infinite extent. The linearised inviscid Navier-Stokes equations then yields (see Chandrasekhar (1961) for details), u(z) = { B( nk iku) exp(+kz) z < 0 B( n k iku) exp( kz) z > 0 (5.4) and from continuity we have w(z) = { B( ink + ku) exp(+kz) z < 0 B( in k ku) exp( kz) z > 0. (5.5) The characteristic equation (Chandrasekhar, 1961) then requires that n k = ±ku (5.6) and for mode growth we take the positive exponent. Clearly mode growth is directly proportional to shearing velocity U, and also directly proportional to wavenumer Computational configuration SPH simulations are performed as follows. Particles are initially arranged on a regular square grid with separation x, within a domain of width Lx and height Lz (see Figure 5.2). The domain is periodic in the horizontal, and enclosed from aove and elow with four layers of oundary particles (see Section for an outline of oundary implementations). Particles (including oundaries) intially move with the mean velocities (5.3). To this mean velocity, the perturations defined y (5.1), (5.2), (5.4) and (5.5) are added, with perturation strength given as some fraction of mean velocity U. For the initial field of equispaced particles, we have density set to unity, with particle mass calculated accordingly. The equation of state used for calculations is {( ) γ } ρ P = P 0 1 (5.7) ρ 0 with γ = 7 taken to yield strong variations in pressure for small changes in density 1. The constant P 0 will determine the soundspeed and degree of compressiility. As stated aove, we set P 0 such that Mach numer M = 0.05 is 1 Other values for gamma have een tested and results do not appear to e sensitive to the particular choice (within an appropriate range). For tested perturations, values 1.4 γ 7 yielding identical growth exponents.

92 85 The Kelvin-Helmholtz instaility Lz Lx Figure 5.2: Initial SPH particle configuration. Red and lue particles represent oundary particles and fluid particles respectively. expected. In Figure 5.3 results for three simulations are given where the Mach numers M = 0.1, 0.05 and are taken. The expected trend of reduced mode growth with increased Mach numer is oserved, with calculated growth exponents of n k = 11.4, 11.8 and 11.9 for M = 0.1, 0.05 and respectively (linear result give n k = 4π). So we choose M = 0.05 as further reduced Mach numer increases computational cost while not yielding significant change in growth rate. Details of growth rate measurement are to e found elow. For all calculations, a domain of half-width Lx = 1 is taken. To determine an appropriate domain height, we may first consider the eigenfunctions (5.4) and (5.5). It is easily shown then that for a wavenumer of wavelength λ k, the perturation drops to one percent of it s maximum value (found at the interface) at a height of approximately z = 0.8λ k. For a domain of height Lz = λ k, we do not then expect significant perturation velocity at the oundary. However, this alone cannot assure us that oundary effects will not e significant, and the large soundspeeds used for calculations ensures that oundaries are known throughout the domain. It does appear however that interface perturation growth is relatively insensitive to vertical domain size, with fundamental mode tests using Lz = 0.25λ k, 0.5λ k and λ k showing minimal difference. In all three cases, mode saturation occurs at almost the same time and largely identical growth rates are oserved. The main requirement is that interface perturations are allowed room to grow, and so a minimal domain height of Lz = 20 x is used for all simulations. Given the aove concerns, we define domain height as Lz = max(20 x, 2λ k ). Simulations in this section utilise oth the self-consistent summation density (2.69) and a fixed smoothing length, along with momentum equation (2.75) and timestepping regime (3.35). It is noted that some acoustic activity is evident in these simulations (often of greater amplitude than the perturation), though mode growth appears unaffected.

93 5.1 Constant velocity fluids in relative motion M = 0.1 M = 0.05 M = A(t) time Figure 5.3: Pertured mode displacement amplitude growth A(t), k = 4π. We note the reduction in growth rate for increasing Mach numer Determination of mode growth rates To determine mode growth exponents, we do not directly measure the velocity growth of the perturation. Instead, the interface etween the two fluids is considered, with the progression of disturances oserved and measured. To understand the expected ehavior of the interface, we use (5.2) to write an equation for the interface: w I (x, t) = w I (x) exp(n k t), where w I represents the vertical velocity at some position x along the interface at time t. Integration with respect to time yields I(x, t) = 1 n k w I (x) exp(n k t) + D(x), for interface displacement I(x, t) with respect to original position, and aritrary function D(x). The interface is initially unpertured, so I(x, 0) = 0, and we have D(x) = 1 n k w I (x) and may then write I(x, t) = 1 n k w I (x) (exp(n k t) 1). Where w I (x) is given y a trigonometric function, such as in (5.2), the term (exp(n k t) 1) acts to modulate the amplitude of perturations. For sufficiently large values of time, the aove ecomes I(x, t) 1 n k w I (x) exp(n k t) (5.8) = 1 n k w I (x)a(t). (5.9)

94 87 The Kelvin-Helmholtz instaility A(t) time k = π k = 10π Figure 5.4: Pertured mode displacement amplitude A(t) for modes k = π and k = 10π. Measurements of growth rates are possile at earlier times for shorter wavelength mode. The value of A(t) may e determined at each timestep of the simulations, and we may then measure the value of n k y first taking the logarithm of A(t): log(a(t)) = n k t, (5.10) from which it is clear that the gradient of the curve of log(a(t)) plotted against time should e determined y the mode growth exponent n k. To get an indication of what may e considered a sufficiently large time for (5.8) (and therefore (5.10)) to e valid, we may insist that exp(n k t) 20, in which case we have t 3/n k. For the current regime, with expected growth rates given y equation (5.6), modes of shorter wavelength fulfill this requirement much quicker than those of long wavelength. Considering Figure 5.4, it can e seen that a curve gradient may e extracted confidently at early times (0.2 < t < 0.5) for the k = 10π mode, while for k = π, a later time range (0.8 < t < 1.3) must e utilised. For all simulations, growth curves such as those of Figure 5.4 are visually inspected to determine the est time range to consider, with a least squares routine then used to give a line of est fit from which a gradient (and thus growth rate) is found. Initial velocity perturations (equations (5.1), (5.2), (5.4) and (5.5)) take amplitudes of strength B = U. Such small perturation velocities allow sufficient integration time for growth rate extraction to e performed confidently. Larger perturations result in earlier transition to non-linear dynamics, and for long wavelength modes this may not allow sufficient integration time for a definite growth rate calculation. For high order modes, larger perturation strengths may e used, though growth rates are not changed. The final required detail is the calculation of interface perturation amplitude A(t). The technique used consists of tracking the interface, and performing a Fourier analysis to determine interface perturation modes and amplitudes. The details of this procedure are now outlined, and should e considered with reference to Figure 5.5. Particles are initially configured such that rows occurs

95 5.1 Constant velocity fluids in relative motion 88 Figure 5.5: Interface tracking for the determination of mode amplitude. Bold green circles represent the SPH particles which are tagged as interface particles, with linear interpolation etween these eing used to determine data at equispaced points. at z = ±0.5 x. The particles at z = +0.5 x are flagged as the interface particles, though the z = 0.5 x particles (or any row actually) could have een used. Linear interpolation is then used etween interface particles to construct a continuous curve from which interface heights at regular intervals may e determined. Fourier analysis is then applied to the evenly spaced interface heights to determine mode amplitudes. To avoid aliasing issues, evenly space points are taken at four times the frequency of SPH particles Results Simulations are performed for various SPH parameters to develop an understanding of the influence of SPH resolution on perturation growth regimes, oth with respect to particle populations and smoothing length. A summary of simulation parameters is given in Tale 5.1. The SPH development of instailities for a typical simulation are given in Figure 5.6, with the corresponding interface amplitude given in Figure 5.7. The expected ehavior of exponential growth is oserved. Owing to the small initial perturation, much of the early exponential growth is not distinguishale in Figure 5.6, though can e clearly seen for the logarithmic scale amplitude given in Figure 5.7. Exponential growth continues until the perturation reaches saturation (just efore time t = 1), after whichpoint the wave reaks and the interface ceases to e one to one (final frame 5.6). For the current Euler regime simulations, dynamics eyond this point are largely chaotic and non-physical. Where viscosity is utilised, the classic Kelvin-Helmholtz vorticities emerge. Resulting growth rates for simulations using single mode perturations are to e found in Figure 5.8. All simulations result in slowed growth rates as perturation wavelengths approach the resolution length scale. Certainly as less particles per wavelength are availale, we expect integration accuracy to suffer. The damping of poorly resolved modes is in ways a desirale artifact,

96 89 The Kelvin-Helmholtz instaility Tale 5.1: Parameters for Kelvin-Helmholtz simulations Parameter Value Description x 0.025, Initial particle separation h x, 1.9 x Initial SPH smoothing length α 0 Viscosity parameter U 1 Shearing velocity B U Perturation strength M 0.05 Mach numer Lx 1 Horizontal domain extent Lz Lz = max(20 x, 2λ k ) Vertical domain extent ɛ 0.0, 0.1, 0.5, 1.0 Turulence cutoff parameter minimising their influence. In Figure 5.8 we also note the difference in ehavior where smoothing lengths of h = 1.3 x and h = 1.9 x are taken. The larger smoothing length exhiits more accurate growth rates in nearly all cases, ut perhaps more importantly is the consistency of results, and predictale change in growth rate as perturation length scales approach smoothing length. Where h = 1.3 x is used, growth within the linear regime for short wavelength perturations do not track an exponential curve with the accuracy found for h = 1.9 x simulations (see Figure 5.8). Hence the erratic calculated growth rates found in Figure 5.8 are a reflection of oth the inaccurate exponential growth, and the ensuing difficulty in determining growth rate exponents. We conclude that a smoothing length of h = 1.3 does not appear to e sufficient for accurate discrete representation of the approximation integral 2.1. It is noted however that larger smoothing lengths result in increased neighouring particles withing respective interaction radii, and so computational costs per timestep are increased significantly. For the current simulations, this increase is of order fifty percent, though larger timesteps may e taken which negate additional cost. Regardless, for the current Kelvin-Helmholtz simulations, it appears additional cost are warranted. Indeed, the h = 1.9 x simulations give excellent clear exponential growth regimes for all perturations up to k = 35π (approximately five particles per perturation wavelength). The lower resolution simulations of Figure 5.8 exhiit a similar trend to those found at higher particle numers, with results scaling according to smoothing length. Figure 5.10 gives numerical growth rates ˆn k scaled against theoretical growth rates n k as a function of particles per perturation wavelength. Results scaled very well for h = 1.9 x, with at least ten particles per wavelength required for growth rates within twenty percent of predicted values. Where smoothing length h = 1.3 x is used, the inconsistency again ecomes evident, although short lengthscale growth rates are closer to the theoretical result. Interestingly, for oth high resolution simulations, a fundamental mode perturation yields growth rates up to twenty percent higher than expected. Results presented aove were almost identical for oth variale and constant smoothing length algorithms. However, only the linear regime has een considered, with one-dimensional simulations indicating that variale smoothing lengths give significant improvements where non-linearity is of importance.

97 5.1 Constant velocity fluids in relative motion 90 Figure 5.6: Development of Kelvin-Helmholtz instaility. Bold points mark out particles used to determine interface growth. For this particular simulation a k = 4π perturation has een used, with parameters x = and h = 1.9 x. Corresponding interface amplitude growth is to e found in Figure A(t) time Figure 5.7: Interface perturations amplitudes for solution displayed in Figure 5.6. Broken line gives solution over a logarithmic scale, while solid line shows linear scale solution. 10-8

98 91 The Kelvin-Helmholtz instaility x = , h = 1.9 x x = , h = 1.3 x x = 0.025, h = 1.9 x x = 0.025, h = 1.3 x ˆnk k/π Figure 5.8: Perturation growth rates for SPH simulations at different fixed smoothing lengths and particle separations of x = and x = The old line gives growth rates predicted y linear theory A(t) h = 1.9 x h = 1.3 x time Figure 5.9: Comparison of exponential growth for different smoothing lengths. Perturation amplitudes are shown on a logarithmic scale to highlight differences. Shown results correspond to a k = 20π perturation with x =

99 5.2 The hyperolic tangent velocity profile nk h = 1.9 x h = 1.3 x Figure 5.10: Scaled growth rates n k = ˆn k /n k against particles per perturation wavelength. Filled circles and crosses give results for x = and x = respectively. Simulations have een performed incorporating the α-sph turulence terms, with results using h 0 = 1.3 x given in Figure The general oserved trend over all simulations is a damping of mode growth, with damping increasing for larger wavenumers, and for larger values of parameter ɛ. Results for the simulations with ɛ = 0.1 and ɛ = 0.5 are almost indistinguishale. For ɛ = 1.0, mode damping is oserved for all modes greater than k = 4π. Here modes greater than k = 17π do not produce sufficiently clear growth for an accurate determination of exponents. Indeed, as with standard SPH at h 0 = 1.3 x, there is some uncertainty in exponent calculation (see Figure 5.9) for all tested modes, with the degree of uncertainty increasing with wavenumer. For this reason there is some scatter in the data presented in Figure For standard SPH, as discussed aove, a clearer exponential growth range is encountered where smoothing length parameter h 0 = 1.9 x is utilised. Unfortunately α-sph simulations with h 0 = 1.9 x under the current Kelvin-Helmholtz configuration results in an interface instaility which appears to e related to the discontinuous velocity profile. A constant vertical velocity is oserved to grow across the entire interface, and in the asence of viscosity this leads to horizontal layers of particles elow the interface eing transported upwards, and vice-versa, with layers shearing past each other. Therefore Kelvin-Helmholtz mode growth does not occur. We leave this configuration and instead consider the continuous interface provided y the hyperolic tangent mean velocity profile. 5.2 The hyperolic tangent velocity profile We now turn to the instailities which may arises where the mean velocity profile takes the hyperolic tangent form (5.1). Such profiles are found where two fluids of different velocity ut coincident direction meet, with a oundary layer then resulting in a continuous velocity across the interface Linear results The linear staility results for hyperolic tangent mean flow are given y Michalke (1964). These results have een used to provide the initial velocity disturances

100 93 The Kelvin-Helmholtz instaility ɛ = 0.0 ɛ = 0.1 ɛ = 0.5 ɛ = ˆnk k/π Figure 5.11: Perturation growth rates for α-sph simulations using h = 1.3 x. from which exponential growth follows, and we outline the required method here. The flow is decomposed into mean and pertured quantities according to u(x, z, t) = ũ(x, z, t) + U(z) v(x, z, t) = ṽ(x, z, t) (5.11a) (5.11) with mean velocity for which perturations defined y U(z) = tanh(z), (5.12) ũ(x, z, t) = B exp(kct) {φ r(z) cos(kx) φ i(z) sin(kx)} (5.13a) ṽ(x, z, t) = kb exp(kct) {φ i (z) cos(kx) + φ r (z) sin(kx)} (5.13) are appropriate (Lin, 1955), with primes denoting differentiation with respect to z. These perturations are derived of a stream function which guarantees the divergence-free condition. The function φ(z) = φ r (z)+iφ i (z) and it s derivatives define the perturation lateral profile and from here dependence on z will not e explicated. Disturance growth is dictated y the value n kc, with phase speed for all wavelength disturances eing zero as a result of chosen mean velocity (5.12) (Michalke, 1964). Perturation amplitude may e determined y the free parameter B. Inserting equations (5.11) and (5.13) into the Euler equations yields the Rayleigh staility equation [U c] [ φ k 2 φ ] U φ = 0, (5.14)

101 5.2 The hyperolic tangent velocity profile n Figure 5.12: Perturation growth rates n variation with wavenumer k for hyperolic tangent mean velocity configuration. k where we have assumed ũ U and ṽ U and terms second order in the perturations are discarded. For given wavenumer k, we then require corresponding eigenvalue c (and hence growth exponent n = kc), along with eigenfunction φ. Solution to equation (5.14) is sought over a domain periodic in x (which is fulfilled y our perturations (5.13) and mean velocity (5.12)) and infinite in the vertical direction. We require that perturations go to zero as the vertical extent ecomes infinite. To determine eigenvalues, we simplify equation (5.14) y setting ( z ) φ(z) = exp Φ(z )dz 0 from which the Riccati equation is otained: Φ = k 2 Φ 2 + U U c. (5.15) To reduce our domain to a finite interval, the transformation y = tanh(z) is introduced. The following equations are then otained for Φ(y) = Φ r (y) + iφ i (y): dφ r dy = k2 Φ 2 r + Φ 2 i 1 y 2 2y2 y 2 + c 2 (5.16) dφ i dy = 2Φ rφ i 1 y 2 2cy y 2 + c 2. (5.17) The aove equations have een simultaneously solved using a fourth-order Runge-Kutta scheme coupled with a shooting method to home in on the required eigenvalue for chosen wavenumer k. The procedure is as follows. The symmetry of equations (5.16) and (5.17) is exploited, with integrations performed from y = 1 to y = 0. Boundary conditions follow from the requirement that

102 95 The Kelvin-Helmholtz instaility φ r φ i Figure 5.13: Solution eigenfunctions. From top to ottom, functions correspond to wavenumers k = 0.1, 0.3, 0.5, 0.7 and 0.9. The dashed curve corresponds to u = tanh(x) mean velocity profile, which has een included to illustrate the scale of eigenfunctions. z eigenfunctions vanish at infinity: Φ r ( 1) = k Φ i ( 1) = 0 dφ r dy ( 1) = 2 (1 + c 2 ) (k + 1) dφ i dy ( 1) = 2c (1 + c 2 ) (k + 1). Integration is then performed for different eigenvalues (holding wavenumer constant), with the correct value eing found where the following conditions are met: Φ r (0) = 0 dφ i (0) = 0. dy Initial ounding values of c = 0 and c = 0.5 are taken, with Newton s method eing utilised to locate the correct eigenvalues. Growth rate results are displayed in Figure Maximal mode growth is found for wavenumers k 0.44, with growth tending to zero as perturation wavelength ecome infinite.

103 5.2 The hyperolic tangent velocity profile 96 Tale 5.2: Parameters for hyperolic tangent mean velocity simulations Parameter Value Description x λ k /Nx Initial particle separation N x 20, 40, 60 Numer of particles per perturation h x Initial SPH smoothing length α 0 Viscosity parameter U 0 1 Hyperolic tangent amplitude B U 0 Perturation strength M 0.05 Mach numer Lx λ k Horizontal domain extent Lz From perturation Vertical domain extent ɛ 0.0, 0.5, 1.0, 5.0 Turulence cutoff parameter As wavelengths approach sizes of order of the oundary layer thickness (as determined y mean profile (5.12)), mode growth also goes to zero. Having determined eigenvalues, we may return to the Rayleigh staility equation to find corresponding eigenfunctions. Equation (5.14) is integrated numerically again using the fourth-order Runge-Kutta method. Symmetry of φ follows from symmetry of Φ, so we are only required to integrate for z 0. The conditions at z = 0 are: φ r (0) = 1, φ i (0) = 0 φ r(0) = 0, φ i(0) = Φ i (0). Resulting eigenfunctions may e found in Figure These functions, together with their derivatives (also otained during eigenfunction integration), complete the perturation specification given y equation (5.13). While results given in this section assume an infinite domain, solutions for a finite domain (IJzerman, 2000) are almost indistinguishale provided that a sufficient domain size has een taken. We define a sufficient domain with respect to any given eigenfunctions, and require that the eigenfunction is of negligile magnitude at the domain extent. Taking for example wavenumer k = 0.7 perturations (Figure 5.13), a domain of size Lz = 10 would result in negligile difference etween finite and infinite domain linear solutions. Otained eigenfunctions and eigenvalues compare favouraly with those of Michalke (1964) and IJzerman (2000) Computational configuration The numerical configuration used for the hyperolic tangent simulations is much the same as used earlier for the discontinuous velocity profile simulations. The key difference here is that for all simulations within a series, the numer of particles per disturance wavelength is held constant, so we write x = λ k /Nx, for perturations of wavelength λ k, and Nx particles spanning the domain horizontally. Simulation domain is then defined y the perturation wavelength and eigenfunctions, taking Lx = λ k and Lz chosen to sufficiently represent the required eigenfunction of Figure 5.13.

104 97 The Kelvin-Helmholtz instaility Nx = 60 Nx = 40 Nx = n k Figure 5.14: Measured growth rates for SPH simulations at different resolution. Perturation strength takes the value of B = U 0. Tests performed show negligile difference for perturation amplitudes in the range U 0 B 0.01U 0 (see Section for further details). All other configuration details are identical to those used earlier. A summary of parameters is given in Tale Results We perform all tests over the spectrum of wavenumers for which mode growth is expect. The particular wavenumers simulated are those from k = 0.1 to k = 0.9 in k = 0.1 increments. First considered is the change in mode growth where different particle numers are used to represent each perturation wavelength. Tests are performed for Nx = 20, 40 and 60 with results given in Figure While modes are simulated using equivalent numer of particles (for each series of simulations), we note that this gives different resolutions in the lateral direction with respect to the mean velocity profile. This however does not appear to e a limitation, with all simulations producing the required growth rates very well over the entire spectrum. It is also noted that in the limit of diminishing wavenumer, growth rate results reduce to those of the discontinuous velocity profile (equation (5.6)), in which case the coarsely represented vertical profile is not a limitation. For the simulations in Figure 5.14, est results are otained for Nx = 40 and Nx = 60, with each series showing improved accuracy at different wavenumers, ut neither clearly superior. The N x = 20 simulations, while still producing the required mode growth trend well, does exhiit a greater degree of scatter across the spectrum. The aove simulations have een performed for relatively weak perturation amplitude of B = U 0. While for any sufficiently small perturation the linear staility results presented aove are valid, simulations have een performed to determine any sensitivity to perturation strength. Fig-

105 5.2 The hyperolic tangent velocity profile B = 10 5 U 0 B = 10 4 U 0 B = 10 3 U 0 B = 10 2 U 0 B = 10 1 U n Figure 5.15: Growth rates oserved for different perturation strengths. For these simulations, parameter Nx = 40 has een used. k ure 5.15 gives measured growth rates for perturation strengths ranging ranging from B = U 0 to B = 0.1U 0. Results are visually identical for U 0 B 0.001U 0. For B = 0.01U 0 an almost indistinguishale reduction in mode growth is found at high wavenumers, and for B = 0.1U 0 a further reduction is oserved for medium to high wavenumers. Given that only minor differences are oserved over the range of perturation strengths tested, we are reassured of the validity of using small perturaton strengths. We turn to Kelvin-Helmholtz simulations which utilise the α-sph algorithm. Simulations are performed for turulence parameter ɛ = 0.5, 1.0 and 5.0, with resulting growth rates given in Figure 5.16 and growth rate change (with respect to standard SPH) given in Figure We see that for simulations using values of ɛ 1.0, growth rates are almost identical to simulations without turulence terms. No clear trend is apparent in the small deviations for these results, though generally a slight increase in growth rates is oserved. For the ɛ = 5.0 simulations, all modes exhiit a reduction in growth rates. The change in growth rates rought aout y the introduction of turulence term, n = n α n sph, is given in Figure 5.17, where we define n α and n sph as respectively the growth rates for α-sph and standard SPH. It is found that the reduction in growth exponent is approximately constant for the spectrum of modes tested. This is an expected result, as the effective turulence model cutoff lengthscale is proportional to the smoothing length, and so is held constant in relation to pertured wavenumers for given simulations. With reference to equation (3.11), we may write in two dimensions: α 2 = h2 ɛ. (5.18) So for ɛ = [0.5, 1.0, 5.0] we have α/λ k = [0.013, 0.019, 0.042]. Clearly the cutoff lengthscale α is much smaller than the pertured wavelength λ k, though in light

106 99 The Kelvin-Helmholtz instaility ɛ = 0.0 ɛ = 0.5 ɛ = 1.0 ɛ = n k Figure 5.16: Perturation growth rates for α-sph simulations n ɛ = 0.5 ɛ = 1.0 ɛ = Figure 5.17: Change in growth rates for α-sph simulations, n = n α n sph. k of the low order Helmholtz velocity filtering, we still expect a degree of mode attenuation (see Figure 3.1) at all tested turulence parameters. The results for the α-sph simulations may e contrasted with findings where only the filtered velocity has een used to advect particles, with turulence terms removed from the acceleration equation. Growth rates for simulations using equivalent turulence parameters to those of Figure 5.16 are given in Figure 5.18, with change in growth rate n displayed in Figure We oserve that for these simulations growth rates appear to e slowed proportionally to

107 5.2 The hyperolic tangent velocity profile ɛ = 0.0 ɛ = 0.5 ɛ = 1.0 ɛ = n k Figure 5.18: Perturation growth rates for standard SPH simulations with particles advected using filtered velocity n ɛ = 0.5 ɛ = 1.0 ɛ = Figure 5.19: Change in growth rates for filtered velocity SPH simulations, n = n α n sph. k

108 101 The Kelvin-Helmholtz instaility parameter ɛ. Comparison of Figures 5.17 and 5.19 reveals that mode attenuation is significantly larger where turulence terms have not een included. We may perhaps conclude that the additional constrains of energy and circulation conservation yields correct integration of coarse scale (with respect to α cutoff parameter) dynamics despite the damping effect of the filtered velocity at these length scales. However where a sufficiently large turulence parameter is utilised, it is certainly expected that even α-sph simulations will give reduced growth rates as the perturaton lengthscale approaches α. Sign of this is oserved for the ɛ = 5.0 simulation where reduced growth is oserved for α-sph, though to a lesser extent than what is found for filtered velocity advection alone. Indeed the value of turulence parameter ɛ (or α) determines which lengthscales are e considered as coarse scales, and which are to e consider as su-grid scales. Choice of appropriate turulence parameter will e dependent on computational resources, and the required minimum wavenumer for which dynamics will e expected to e governed y Navier-Stokes dynamics. This will e determined y the prolem eing considered, and y the ehavior of the standard SPH algorithm where resolution is limiting. With a filtered velocity alone, results indicate that mode growth attenuation acts over a roader scale than encountered for α-sph. Furthermore, we speculate that violation of energy conservation will lead to more striking differences for non-driven simulations 2, such as perhaps a travelling wave simulation, or decaying turulence, where the filtered velocity will act to dissipate any wave-like motion. 5.3 Conclusion The results presented here indicate that SPH is ale to correctly reproduce Kelvin-Helmholtz growth rates where sufficient particles are utilised to represent perturations. The growth rates are oserved to e attenuated as perturation lengthscale approaches SPH smoothing length scale. This slowing of growth rates may e linked to a weakness of the SPH pressure gradient operator in the limit as perturation wavelengths approach the SPH smoothing length. Furthermore, the noise oserved for simulations where we set h 0 = 1.3 x may also e traced to innacuracies in the pressure gradient calculation for this parameter. To quantify the performance of the pressure gradient, we define the quantity Q P (k): [ Q 2 sph P sph P ] k P (k) = [ ]. (5.19) P P Full details for the calculation of equation (5.19) may e found in Section For now suffice to say that it gives a measure in spectral space of the SPH pressure gradient (as determined y (2.41), and represented in the numerator of (5.19)), relative to an analytically calculated pressure gradient (represented in the denominator of (5.19)). Where the SPH pressure gradient produces results identical to the analytic pressure gradient, we expect the value Q P = 1, and if the SPH pressure gradient is weaker than the analytic equivalent, we expect the values 0 Q P 1. The function Q P (k) is displayed in Figure 5.20, 2 We consider the Kelvin-Helmholtz simulations driven in the sense that the mean flow provides a large kinetic energy potential from which perturation growth is driven. k

109 5.3 Conclusion n k Q P (k) Figure 5.20: Scaled growth rates n k = ˆn k /n k and pressure gradient quality factor against particles per wavelength. alongside the scaled growth rates of Figure Clearly there is a very strong correlation etween the weakness of the SPH gradient and the slowed growth rates of perturation where insufficient particles per wavelength are used. Note that the data used for calculation of Q P (k) is from the periodic turulence simulations of Chapter 6, though we expect the profile of Q P (k) will largely identical for the Kelvin-Helmholtz simulations. The second mean velocity profile considered was the hyperolic tangent, with the eigenvectors and eigenvalues determined as outlined in Section Theoretical growth rates were again found to e reproduced accurately y SPH simulations. Investigation of α-sph indicate that it is successful in reducing growth rates at short scales, though not as aggressively as found for simulations where only the filtered velocity is utilised. The difference etween the simulations lies in the additional acceleration terms of α-sph which act to restore the energy conservation violated y a filtered velocity, and appear to counter slowing of growth rates due to the velocity filtering. These findings give indication that α-sph may perform successfully as a turulence model, though linear regime simulations are certainly not challenging enough to draw strong conclusions aout the potential of the model. We now therefore consider full non-linear simulations of turulent two-dimensional flow.

110 Chapter 6 Two-Dimensional Turulence It is often contended that turulence cannot truly exist in two dimensions. Of course all physically realised flows must contain some degree of three dimensionality, ut at times motion may e largely constrained in a particular direction. This may e due to domain limitations, or other limiting forces, such as those which can arise due to rotation. These flows may then e categorised as twodimensional turulence in the sense that turulent dynamics (i.e. displaying significant variaility and irregularity) only occurs in two dimensions, or in the sense that they are approximated well mathematically y a two-dimensional truncation of the Navier-Stokes equations. This reduced system of equations in some ways presents a simplification of governing dynamics, with for instance vorticity stretching eing eliminated. It is perhaps ironic then that the two-dimensional Navier-Stokes system gives rise to seemingly counter-intuitive dynamics which differ significantly from the three-dimensional counterpart. In particular, it is oserved that contrasting the three-dimensional situation, energy in two-dimensional turulence tends to cascade towards smaller wavenumers (larger scales). The physical space realisation of the inverse cascade processes is the coalescence of similar sized vorticies (Frisch and Sulem, 1984). This process of self organisation culminates in the creation of large coherent structures which persist over long times, travelling under the action of almost inviscid advection. The most ovious example of this ehavior in nature may e found within the atmosphere 1 where we find that inverse cascade processes yield the phenomena of cyclones. Similarly, some aspects of ocean dynamics are approximated well y two-dimensional turulence. In the laoratory, a numer of approaches have een used to study two-dimensional turulence. Paret and Taeling (1997) have performed experiments using two thin layers of staly stratified fluid to create a quasi two-dimensional configuration, the upper layer in effect moving inviscidly. With turulence driven y magnetic fields, the formation of an inverse energy cascade was successfully oserved. Another common approach uses thin soap films as a fluid ase upon which excitations may e imposed (see for instance Martin et al. (1998)). A 1 Note that the vertical extent of the atmosphere is orders of magnitude smaller than the horizontal scale. 103

111 104 comprehensive review of related experimental results is given in Kellay and Goldurg (2002). The theory of two-dimensional turulence has een developed in the pioneering works of Kraichnan (1967) and Batchelor (1969). A numer of key conjectures were given in these pulications: 1. An inverse energy cascades exists, with kinetic energy moving from small to large scales. 2. A direct cascade of enstrophy (mean squared vorticity) exists, enstrophy eing transferred to shorter and shorter scales until it is acted upon y viscosity. 3. Under an appropriate scaling, the energy spectrum is self similar. This applies for a turulent two-dimensional homogeneous isotropic fluid of sufficiently high Reynolds numer such that an inertial range may form. Following Batchelor, the arguments leading to the aove conjectures egin with the evolution equation for vorticity in two dimensions. Taking the curl of the Navier-Stokes equations yields the required condition: dω dt = ν 2 ω, (6.1) where in two-dimensions we have ω = (0, 0, ω). It can e seen that in the inviscid limit, vorticity is then advected as a passive scalar. Most importantly, the mechanisms of vortex stretching are asent in equation (6.1). This key omission results in the conservation of kinetic energy in the limit as viscosity diminishes, the most significant departure from three-dimensional turulence where energy dissipation instead approaches a constant as viscosity disappears. Vortex stretching dynamics are integral to this finite dissipation in the threedimensional case, with any reduction in viscosity resulting in an intensification of vortex stretching, followed y amplification of small scale vorticity, and hence compensation of dissipation. In two dimensions, the kinetic energy evolution equation may e written 1 d 2 dt v v = ν ω2 (6.2) where spatial homogeneity has een assumed and overars denote averages. Further to this, the equation for the evolution of enstrophy is written 1 d 2 dt ω2 = ν ω ω, (6.3) from which it can e seen that enstrophy must decline monotonically. Equations (6.2) and (6.3) illustrates the previous contention that kinetic energy is approximately conserved for two-dimensional turulence in the limit of vanishing viscosity. We argue that since enstrophy is ound from aove via equation (6.3), energy dissipation therefore must vanish as viscosity goes to zero. As such, any significant cascade of energy from large to small scales (which would eventually lead to dissipation) must e precluded where viscosity is small, hinting at the possiility of an inverse cascade. In contrast to kinetic energy, as viscosity falls the dissipation of enstrophy need not vanish. We again note that vorticity will e advected as a passive

112 105 Two-Dimensional Turulence E(k) ɛ 2/3 k 5/3 β 2/3 k 3 k f k Figure 6.1: Log-log plot of the predicted energy spectrum E(k) where turulence is forced at wavenumer k f. scalar in the non-viscous limit (according to equation 6.1), and we expect vorticity within any closed susection of the domain to e continually twisted and wound up, with layers of differing vorticity eventually eing rought closer together, therefore driving vorticity gradients to ever higher values. Analogous to the three-dimensional result of energy dissipation tending to a finite value for diminishing viscosity, it may e shown that smaller viscous forcings are compensated y increased vorticity gradients, hence giving rise to constant dissipation of enstrophy as viscosity vanishes. This process gives an indication that we might expect a forward enstrophy cascade, facilitating finite enstrophy dissipation. Kraichnan (1967) considers the situation where an infinite two-dimensional fluid is excited y a small andwidth forcing centered on some wavenumer k f. We define the total energy per wavenumer E(k) such that for total kinetic energy E kin and mass m we have: E kin = 1 ρ v v dx = m E(k) dk (6.4) 2 R for the domain R. For the inertial section of the spectrum E(k), two similarity ranges are then identified. Where it is assumed that the spectrum depends only on wavenumer k and net energy transfer per unit mass ɛ, a scaling law equivalent to Kolmogorov s three-dimensional result is found: 0 E(k) = Cɛ 2/3 k 5/3. (6.5) Here C is some constant which is expected to e different to that of threedimensional turulence. As eluded to aove, we also expect the sign of the transfer rate ɛ to e reversed for two-dimensional turulence. We may instead

113 106 assume that spectrum E(k) depends only on k and the net enstrophy dissipation rate β, from which we find the scaling E(k) = D β 2/3 k 3, (6.6) for a constant D. Kraichnan gives weight to the consistency of relations (6.5) and (6.6) through use of Fourier mode triad interactions, showing that where (6.5) is valid, energy transfer ɛ is independent of k, and β is identically zero. Likewise, where (6.6) is valid, β is independent of k and the energy transfer rate ɛ is zero. So where energy is supplied at wavenumer k f, we may expect two inertial ranges to form. For wavenumers k < k f, a range of the form given y (6.5) is expected, with energy eing carried to lower wavenumers y the inverse energy range. For a finite periodic domain simulation, the inverse cascade continues until modes of wavelength similar to the domain size are excited. Energy will e continually injected at these large scales until such modes are sufficiently energetic for viscosity to act (Lesieur, 1990; Tran and Bowman, 2004). This process has een likened to Bose-Einstein condensation y Kraichnan. For wavenumers k > k f, an enstrophy cascade range may e expected, with spectrum given y (6.6), and enstrophy carried to higher wavenumers. Kraichnan also introduced a logarithmic correction to equation (6.6) in light of non-localness of interactions in spectral space, though there is little evidence to support this correction. For the enstrophy cascade, viscosity must eventually ecome significant, halting further cascades. The earliest simulations of forced two-dimensional turulence dates to work of Lilly (1972) who used a finite difference simulation forced at k f 8 along with a large scale friction and was ale to produce a k 5/3 inverse energy cascade range, along with the k 3 enstrophy cascade spectrum. Indeed the inverse energy cascade scaling appears to e roust with many authors oserving the predicted k 5/3 energy scaling in numerical simulations (Frisch and Sulem, 1984; Maltrud and Vallis, 1991; Boffetta et al., 1999; Tran and Bowman, 2004). Experimental evidence also suggests the existence of the k 5/3 range, including the electromagnetically driven shallow flows of Paret and Taeling (1997) and driven soap films of Rivera and Wu (2002). The enstrophy cascade range however appears to e a much more elusive phenomena, with most reporting energy scaling as k α with exponent α 3. While some have found the exponent α to e within the range 3 α 4 (for instance Maltrud and Vallis (1991) and Lindorg and Alvelius (2000)), others report values as large as α = 6 (Dahlurg et al., 1990; Basdevant et al., 1981). The experimental results of Martin et al. (1998) suggests an exponent α 3.3. Interestingly, the k 3 energy scaling has een oserved in the atmosphere, ut on the infrared side of the spectrum, with the inverse energy cascade k 5/3 instead found at shorter scales (Lindorg, 1999). A possile explanation is found in the energy spectrums of long time integrations y Tran and Bowman (2004). These show a striking similarity to the spectrum otained via wind data in Lindorg (1999), with a large scale k 3 spectrum eventually appearing as a result of long time Bose-Einstein condensation. In this chapter we consider oth SPH and α-sph simulations of forced twodimensional turulence. Forced simulations have een chosen to e investigated as they provide a asis for clear evaluation of SPH over a road spectrum of

114 107 Two-Dimensional Turulence lengthscales, yielding stationary spectrums which allow for direct comparisons etween different simulations. In this respect, the perhaps fleeting dynamics of decaying turulence are in ways more difficult to qualify in terms of the Batchelor theory, with spectrums which are transient in nature and therefore do not make for easy comparison etween simulations. The main purpose of performing these simulations is to evaluate the aility of the SPH and α-sph algorithms to yield results in line with theoretical expectations and with data presented in the literature. Given the geometric simplicity of the simulations, spectral methods are a etter choice for an investigation of two-dimensional turulence theory, and results here instead largely serve to illuminate difficulties the SPH numericist may encounter. The chapter is organised as follows. In Section 6.1 details of the computation configuration are considered. This is followed y simulation results and discussion for simulations utilising short scale forcing in Section 6.2, followed y results for large scale forcing in Section 6.3 where simulations including the α-sph model are also presented. Finally concluding comments are made in Section Computational configuration SPH simulations are carried out within a square domain periodic in oth the horizontal and vertical direction. Periodic oundaries are set at Lx = ±0.5 and Ly = ±0.5. As with previous simulations, density is initialised at ρ = 1, with particle masses determined accordingly. Particles are initialised on a regular grid of particle seperation x, with forcing then applied until an irregular configuration is achieved. Susequently, damping and viscosity are used to ring particles to rest. The initial SPH smoothing length is set at h = 1.9 x, and the variale smoothing length implementation is used Forcing To initiate and drive turulent hydrodynamics in the sense of Kraichnan (1967), a random forcing regime is required to inject energy at wavenumers falling within a thin annulus in wavenumer space. Many methods exist for implementing such a random forcing. Nadiga and Shkoller (2001) use a forcing which ensures a constant amplitude for all modes within a certain andwith. A variation to this instead ensures that total energy within any particular waveand is maintained at some predetermined level. A perhaps more physically realistic alternative proposed y Alvelius (1999) instead applies a forcing of constant power over a wavespace annulus, with the energy input expected to e eventually alanced y dissipation. These methods all apply trigonometric mode forcing, as is convenient where spectral techniques have een employed. Alternatively, forcing within physical space may e employed. Boffetta et al. (1999) have used a Gaussian forcing f(r, t) with correlation f(r, t)f(0, t ) = F 0 exp( (r/l f ) 2 )δ(t t ), such that forcing should rapidly decline for r l f. We note that this forcing cannot e used to investigate the direct enstrophy cascade of Batchelor (1969), and as most results presented in the literature use spectral forcing, it is appropriate for us to use similar techniques so that clear comparisons may e made. We consider the method of Alvelius in further detail.

115 6.1 Computational configuration 108 For the verlet timestepping scheme (2.59), the velocity timestep is written v 1 = v 0 + t (g 1/2 + f 1/2 ) (6.7) for some predetermined random forcing function f = f(x, t). Dominant terms contriuting to kinetic energy power input averaged over a timestep are given y P = t 2 f 1/2 f 1/2 + f 1/2 v 0 = P 1 + P 2. (6.8) Here overars represent volume averages over the entire spatial domain. We require that the force-force correlation function P 1 makes the most significant contriution so that power input may e controlled and constant. To remove the timestep size dependence of P 1, forcing will e written as a function of timestep (f ( t) 1/2 ). The secondary contriution P 2 should e much smaller, though where a small numer of forcing modes are utilised and the timestep is small, the forcing function will e large and P 2 may correspondingly ecome significant for any given step. Alvelius considers forced three-dimensional simulations where forcing is implemented at large scales only, and involves a relatively small numer of modes. In such situations and where precise energy input is required, measures to ensure the term P 2 is zero may e necessary. For the current simulations where we force at intermediate wavenumers, typically over a hundred contriuting modes are utilised, so we may e confident that P 2 averages to zero over short times (relative to total simulation time). Simulations elow for large scale forcing do exhiit significant variaility in the energy input, and where this is of concern we instead force to achieve constant total energy within the forcing wavespace annulus. Further details for this are given in Section 6.3. Additional terms also arise due to the Navier-Stokes forces and are neglected in equation (6.8). These terms are either zero on average, or tend to zero with shortening timestep (Alvelius, 1999). Where P 2 = 0, we write (dropping the timestep index) P = t 2 f f = t { fx (k x, k y ) 2 f x(k x, k y ) + f y (k x, k y ) f y (k x, k y )} k x k y for complex spectral forcing component f(k x, k y ) where f(x, y) = f(kx, k y ) exp(i(k x x + k y y)). k x k y (6.9) We define functions F (k) and G(k) which give respectively the total and average power input due to all components of (6.9) falling within a wavespace annulus of radius k = kx 2 + ky. 2 We may then write equation (6.9) as P = t 2 0 F (k)dk = t 2 0 k G(k)dk. (6.10) 2π To ensure that forcing does not introduce compressile moments to the velocity field, we impose the constraint k f(k) = 0

116 109 Two-Dimensional Turulence which must e met for any forcing mode f(k). This may e satisfied y writing f(k) = 1 k (k y A ran, k x A ran ) with complex random numer A ran. It is required that the power input for a mode of this form is given y the function G(k): which is satisfied y G(k) = A ran A ran = 2π k F (k), 2π A ran = exp(i θ) k F (k) with random numer θ = [0, 2π]. This yields a forcing where all modes within any particular shell have equal amplitude, ut random phase. It is noted that this forcing is not truly isotropic, as there is some dependence on the principal axes with which wavenumers are defined. A further random rotation could possily e applied to forcing component f(k), though this would not e compatile with the square periodic domain (which in itself presents a degree of anisotropy). New random numers are taken at each timestep. It remains to define the function F (k) which will determine the shape of the forcing spectrum. A common choice is the Gaussian profile, centered on some dominant forcing wavenumer k f : ( F (k) = A exp ( ) 2 k kf ) The constant A is determined such that the required power input is P. With reference to equation (6.10) we have: 0 ( A exp ( k kf c c ) 2 ) dk = 2P t. For k f sufficiently large, and c sufficiently small, we may write A = 2P c π t. For the SPH algorithm, we define the particle forcing f for particle as f = n m i=1 f i exp(i k i x ), (6.11) where we have a list of contriuting forcing coefficients f i with wavenumer k i, and n m total contriuting modes. This equation is added to the left hand side of the SPH momentum equation Viscosity Viscosity in numerical simulations of two-dimensional turulence is often relegated to a secondary role. The more important Euler dynamics usually takes

117 6.1 Computational configuration 110 centerplace with viscosity considered a tool to simply remove energy that approaches the spectral limit in a quasi-physical manner. Indeed the Newtonian viscosity is most often replaced with so-called hyperviscous terms, which are constructed of higher order Laplacian operators: f ν = ( 1) n+1 ν n n v. (6.12) Here the parameter choice n = 1 yields the standard Newtonian viscosity. Hyperviscosity is otained where higher values of n are taken, with values ranging up to n = 8 often found in the literature (see for instance Maltrud and Vallis (1991)). Large values of n effectively yield higher order velocity filtration, with modes less than some cutoff wavenumer largely unchanged, and modes eyond strongly attenuated. Hyperviscosity in effect acts over a shorter andwidth, with a larger portion of the availale spectrum then to e considered inviscid. The SPH viscosity operator may take a numer of forms, as outlined in Section Generally, it attempts to replicate a Newtonian viscosity. For simulations presented in this section we take a viscosity forcing akin to that of Morris et al. (1997): fν sph (r a ) = 2ν ( ) m 1 W a v a. (6.13) ρ r a r a Taylor series expansions of the aove give to leading order f ν with n = 1, so we expect the correct ehavior for sufficiently smooth velocity fields, though this leads to questions of what may e considered sufficiently smooth. We wish to understand the performance of (6.13) for short length-scale velocity fluctuations, which are expected to e of significant for turulent flows. Spectral recompositions of SPH particle fields provide a powerful means y which the ehavior of fν sph and f ν may e compared quantitatively. The SPH particle velocities v a and viscous dissipation fν sph (r a ) are first sph recomposed in trigonometric functions with component amplitudes ˆv and ˆf ν respectively (Appendix D gives details of the required technique). We may then consider the total viscous dissipation rates R v f ν sph dx and R v f νdx. The term f ν is first evaluated analytically: f ν (x) = ν k v(k) exp(ik x) = ν k k 2 v(k) exp(ik x), (6.14) where k = k x k y. For any two vector functions a and we may write a dx = ā(k a ) (k ) exp(i(k a + k ) x) dx R R k a k = ā(k a ) (k ) k a k and for a square domain of size length L we have = L 2 ā(k a ) (k ) δ ka, k k a k R exp(i(k a + k ) x) dx,

118 111 Two-Dimensional Turulence h = 1.3 x h = 1.7 x h = 1.9 x 0.85 Q ν(k) k/2π Figure 6.2: The function Q ν(k) (equation (6.15)) quantifies the integrity of the SPH Laplacian operator. The aove results are otained for a particle simulation, where identical velocity fields are used to otain the required quantities. The particle configuration and velocity field is otained from a typical turulence simulation, as given in the results section of this chapter. where δ is the Kronecker delta function. Therefore we have a dx = L 2 (ā ) = [ā ] k L2 k R k x k y k with the term [ ā ] representing the sum of all terms ā within an annulus k k 1 2 k k in wavenumer space. We consider the function ] sph [ v f ν k Q ν (k) = [ v f ]. (6.15) ν This function effectively gives a measure of the quality of the SPH Laplacian operator. It is given in Figure 6.2 with different values of smoothing length used for the kernel in (6.13). We first note the for all simulations, the SPH derivative gives values short of expectations as we move to higher wavenumers. Indeed where approximately five particles are utilised per wavelength (k/2π = 160), the SPH viscous dissipation falls approximately thirty percent short of the expected result. For smaller smoothing lengths, there is less fall in dissipation strength with wavenumer, though results now exhiit a large degree of noise throughout the spectrum. For a smoothing length of h = 1.7 x a good compromise is perhaps found, with only a slight increase in noise over the h = 1.9 x result, k

119 6.1 Computational configuration 112 though only minor improvements in dissipation strength is otained. Simulations presented in this chapter use h = 1.9 x, though the potential use of a shorter smoothing length for viscosity certainly warrants further investigation. Noise at the small wavenumer limit is simply a result of the denominator in equation (6.15) vanishing, with relative errors actually quiet small. Importantly, the sign of SPH viscous forcing modes is everywhere correct, and where smoothing length is sufficient, viscosity diminishes in strength regularly. Note that all calulations in this chapter utilise a variale smoothing length, with smoothing length values given here representing the initial values. These results highlight the deficiencies of the SPH viscosity. On one hand, we would like to minimise the viscosity such than an inertial range may form. However given the weak dissipation at short scales, we must select a sufficiently large kinematic viscosity to prevent energy accumulation at the spectral limit. This proves to e a significant limitation, with viscosity then required to act over a much roader spectrum than even standard Newtonian viscosity, and certainly much roader than the hyperviscous dissipation often utilised in spectral techniques Scales Integration times are normalised using an eddy-turnover time defined as Z 1/2 where Z is the total enstrophy which is written Z = ω 2 = 0 k 2 E(k)dk. (6.16) For simulations presented where only a quasi-steady state is reached, the inverse energy cascade results in energy accumulation at small wavenumers, and so the spectrum in this region is not stationary. For time scaling, integrations are performed until the spectrum is steady down to some wavenumer k l k f, at which time the integral (6.16) is calculated and lael Z 0. We note however that equation (6.16) converges to value Z 0 very early in the simulation. Unless otherwise stated, times are given in units of the eddy-turnover time Z 1/2 0. An appropriate Reynolds numer to characterise the flow may e written Re = L f Ek ν (6.17) where L f is the forcing length scale. For an assumed enstrophy cascade range scaling of E(k) = β 2/3 k 3, it may then e shown (Lesieur, 1990) that resolution requirements will e dictated y k d k f = Re (6.18) where k d is the viscous dissipation wavenumer. Therefore a douling of Reynolds numers in two-dimensional turulence leads to an increase in computational work of order 2 2, a more favourale result than the order 2 3 increase encounter in three-dimensional turulence.

120 113 Two-Dimensional Turulence Tale 6.1: Parameters for forced turulence simulation Parameter Description A B Run x Initial particle separation ν Kinematic viscosity P Forcing power input k f /2π Forcing wavenumer c Forcing andwidth M Maximum run Mach numer Re Maximum run Reynolds numer E k Maximum run kinetic energy Z Maximum run enstrophy time Total run time Equation of state Results presented in this section use a similar equation of state to that of earlier computations: P = ρ γ 1. (6.19) The value γ = 1.4 is used, though use of other values (1 γ 7) does not affect results significantly. Removal of the offset term on the right hand side of (6.19) does however cause a large change in dynamics with simulations exhiiting a tendency towards certain particle configurations which appears to overwhelm results. We wish to simulate flows which may e considered largely incompressile, and so require that Mach numers are kept to a small value. This is achieved simply y use of sufficiently small forcing amplitudes, with ensuing Mach numers monitored to ensure they remain small throughout the simulation. Results for weakly compressile Kraichnan forced turulence have een reported y Dahlurg et al. (1990), with Mach numers up to M = 0.3 utilised. Importantly, kinetic energy scalings were found to e almost identical for oth compressile and incompressile calculations. 6.2 Intermediate scale forcing The growth in time of kinetic energy and enstrophy for the run A parameters of Tale 6.1 are to e found in Figure 6.3, with the corresponding kinetic energy spectrum at various times found in Figure 6.4. Kinetic energy is oserved to initially grow at a rapid rate until a time of approximately t 10 at which point the viscous dissipation range ecomes sufficiently active to slow the energy growth. From here on, total energy continues to grow at a slowed rate, and corresponds to the inverse energy cascade carrying some percentage of the energy input to large scales. This is apparent in Figure 6.4 where the short scale region of the spectrum is largely stationary for times greater than t 20, while large scales continue to ecome more energetic. Energy continues to rise until the simulation is stopped, though growth rate is gradually diminishing with viscous dissipation eventually expected to alance energy input. The total enstrophy

121 6.2 Intermediate scale forcing 114 in Figure 6.3, as calculated using 6.16, rapidly rises to a value of Z 4.5 at approximately t = 10. As total enstrophy is largely dictated y the short scale which are stationary eyond this point in time, enstrophy correspondingly does not vary significantly despite the growth in energy at large scales. Turning to Figure 6.4, a numer of oservations are to e made. We first note that a peak forms at the forcing wavenumers, with energy within the forcing range approximately of four times greater magnitude than of the surrounding wavenumers. This is a result of the strong viscosity required to prevent an excessive accumulation of energy at short scales. Indeed even with a relatively strong viscous dissipation, a degree of energy accumulation at the resolution limit 2 occurs, as is evident in the upward turned tail of the energy spectrum in Figure 6.4. With respect to energy spectrum scaling, the inverse energy range appears to trend with the expected Kolmogorov k 5/3 law, though the spectrum is still not fully developed at the conclusion of the simulation. For this reason we also do not oserved the accumulation of energy in the fundamental mode which leads to the development of domain scale coherent structures. The predicted enstrophy cascade section of the spectrum (k > k f ) deviates greatly from the Kraichnan k 3 law, and instead energy scales as k 6 in this region. Reasons for this deviation will e explored shortly. Figure 6.5 gives the vorticity in physical space. Early times are clearly dominated y the random forcing function. By the third frame of Figure 6.5 (t = 23.5), we see the emergence of coherent eddies which appear to have lengthscales typically of order of the forcing lengthscale. For greater times very little change is oserved in the characteristics of the vorticity field, despite the growth in kinetic energy. This may e understood with reference to the corresponding enstrophy spectrums found in Figure 6.6. As the inverse energy spectrum follows a k 5/3 spectrum, the peak in enstrophy is expected to e found at the forcing wavenumer. This is only true while significant energy has not accumulated at the domain scale, which would turn the spectrum up at the large scales potentially causing a peak in the enstrophy spectrum. We also note the tails of the enstrophy spectrum turn at the shortest scales, an artifact of the marginal resolution use for simulations. Due to the excessively slow growth of energy at the large scales, the simulation energy input was increase, with all other parameters left unchanged (see parameters for run B in Tale 6.1), resulting in a larger simulation Reynolds numer. The late-time kinetic energy spectrum is given in Figure 6.7. A more developed inverse energy range is otained, though energy at the largest scales is still growing slowly at the time the simulation is stopped. As with the lower power simulations, a Kolmogorov scaling is oserved in this range. The enstrophy cascade section of the spectrum is largely unchanged in form from the previous run, with a scaling of approximately k 6 found within this range. Naturally energy across the spectrum is increased from the previous simulation, though notaly short scales are significantly more energetic as a result of the insufficient resolution (or viscosity) utilised. Results given aove demonstrate the it is possile to produce Kraichnan like 2 Note that the most appropriate choice for a resolution maximum wavenumer is defined y the SPH smoothing length, though there is no clear spectral cutoff. For the current simulations, the smoothing length implies a maximum wavenumer of approximately k/2π = 210. Consideration of the kinetic energy as determined using (6.4) however suggests the slightly smaller value of k/2π = 200.

122 115 Two-Dimensional Turulence E k Z time Figure 6.3: Evolution of total kinetic energy E k (solid green curve) and total enstrophy Z (roken green curve) in time for simulation A. 0 dynamics within the SPH framework. An inverse energy cascade is oserved with the correct spectrum scaling, with kinetic energy oserved to e growing despite a stationary energy spectrum for k > k f. Also oserved is the appearance of coherent structures which are of order of the forcing lengthscale, and survive for numerous eddy turnover times. These vorticies are commonly encountered in oth forced and decaying two-dimensional turulence (for instance Legras et al. (1988) and McWilliams (1990)), and their significant to the inertial range theory is still a suject of deate. The enstrophy cascade for the simulations presented has an approximate k 6 spectrum. While this is far from the predicted Kraichnan k 3 spectrum, it is not out of line with the literature, many authors reporting spectrums k α with 3 α 6. The presence of coherent vortices is often cited as the cause for enstrophy range spectra with exponents α > 3. These long lived vorticies introduce a spatial and temporal intermittency which inhiits nonlinear transfers leading to a steepening of the enstrophy spectra (Benzi et al., 1986). Basdevant et al. (1981) considered the spectral effects of this intermittency, concluding that it resulted in localness of spectral space interaction which tended to steepen spectra. However simulations have een performed in Maltrud and Vallis (1991) which exhiited no coherent structures yet still resulted in enstrophy range spectrum steeper than k 3. In these test various means were introduced to inhiit the formation of vorticies. Tran and Bowman (2004) offer an alternative explanation for the steep enstrophy spectra ased on gloal conservation of energy and enstrophy, along with the inclusion of viscosity. Their arguments are ased on defining a quantity r which determines the strength of the inverse cascade, with the cascade considered weak unless 1 r 1. It is conjectured that unless the inverse cascade is strong, the enstrophy range spectra must e steeper that k 5. Furthermore, the inverse cascade is shown to e extremely roust, persisting in simulations where approximately eighty percent of energy is dissipated y viscosity. For our simulations, we consider the quantity ɛ 0 = de k /dt which is the

123 6.2 Intermediate scale forcing 116 t = 2.3 t = 12.9 t = 23.5 t = 44.5 E(k) t = 65.8 t = k/2π Figure 6.4: Log-log plot of kinetic energy spectrum for run A. The peak in the spectrum corresponds to the forcing wavenumers k f /2π = 30. The green roken line gives a reference for the Kolmogorov k 5/3 scaling, while the lue roken line corresponds to an enstrophy cascade k 3 scaling.

124 117 Two-Dimensional Turulence t = 2.3 t = 12.9 t = 23.5 t = 44.5 t = 65.8 t = Figure 6.5: Fluid vorticity given at times corresponding to energy spectrums found in Figure 6.4.

125 6.2 Intermediate scale forcing 118 t = 2.3 t = 12.9 t = 23.5 t = 44.5 k 2 E(k) t = 65.8 t = k/2π Figure 6.6: Log-log plot of enstrophy spectrum for run A. The peak in the spectrum corresponds to the forcing wavenumers k f /2π = 30.

126 119 Two-Dimensional Turulence E(k) k/2π Figure 6.7: Log-log plot of kinetic energy spectrum for run B (red) alongside run A (lue). The peak in the spectrum corresponds to the forcing wavenumers k f /2π = 30. The green roken line gives a reference for the Kolmogorov k 5/3 scaling, while the lue roken line corresponds to an enstrophy cascade k 3 scaling ɛ ν ɛ 0/ɛ time Figure 6.8: Energy dissipation rate ɛ ν (green) and inverse cascade factor ɛ 0/ɛ (lue) in time. The upper graph corresponds to simulation parameters A, while the lower curve is for simulation parameters B. Energy input due to forcing is denoted y the red line

127 6.2 Intermediate scale forcing ˆ v f ν ɛ ν k k/2π Figure 6.9: Viscous dissipation spectrum normalised y total viscous dissipation. SPH viscosity is given y red plus symols, while analytic dissipation (as given y equation (6.14)) is represented y green cross symols. Data is for simulation A parameters at time t = 105. rate of energy injected into larger scales due to inverse cascades. In effect, this quantity is the gradient of the kinetic energy given in Figure 6.3 from the point where total enstrophy has stailised. Alternatively we may consider the total SPH viscous dissipation: ɛ ν = m v f sph ν (6.20) for a sum of all SPH particles in the domain and SPH viscous forcing fν sph. We assuming a constant energy input rate ɛ. This is a reasonale assumption given that we use a random white forcing of constant amplitude, and as discussed in Section the velocity-forcing correlation P 2 can e expected to e small since approximately five-hundred discrete modes are forced in the presented simulations. Therefore once the flow has evolved to a stationary enstrophy range spectrum, we can write ɛ 0 = ɛ ɛ ν, and hence define an inverse cascade strength r = ɛ 0 /ɛ. The evolution of viscous dissipation rate ɛ ν and cascade factor ɛ 0 /ɛ are found in Figure 6.8. We consider times eyond t = 20 afterwhich the short scale spectrum region is stationary. It can e seen for oth simulations that at early times the inverse cascade factor is approximately r = 0.06 and therefore only a very weak inverse cascade is present. As the inverse cascade gradually feeds long scale modes, these ecome more energetic and viscous dissipation correspondingly increases at these scales. Hence we see the inverse cascade factor drop to approximately r = 0.01 y the end of the run, which is reflected in the kinetic energy gradient falling in Figure 6.3. At this point the inverse cascade is largely exhausted with energy input eing alanced y viscous dissipation. We therefore do not expect the k 5/3 range to extend to the domain scale, and so the appearance of large scale coherent vorticies which follow from Bose-Einstein condensation are also not expected. The viscous dissipation spectrum (Figure 6.9) reveals that significant dissipation is to e found across the enstrophy cascade range. Approximately

128 121 Two-Dimensional Turulence Tale 6.2: Parameters for quasi-steady turulence simulation Parameter Description C Run x Initial particle separation ν Kinematic viscosity P Forcing power input k f /2π 10 Forcing wavenumer c 0.5 Forcing andwidth M 0.06 Maximum run Mach numer Re 1064 Maximum run Reynolds numer E k Maximum run kinetic energy Z 0.79 Maximum run enstrophy time 116 Total run time twenty percent is dissipated within the forcing and, and fifty percent of total dissipation occurs within the first fifty modes. As such, requirements of the Kraichnan theory are not satisfied. However the road spectrum nature of the SPH viscosity (as discussed in Section 6.1.2) makes it difficult to reduce viscosity sufficiently such that minimal dissipation occurs at long and intermediate scales. Indeed particle populations used for the aove simulations appears to e marginally sufficient, and any further reduction in viscosity would result in excessive unphysical energy accumulation at short scales. An analytic viscosity, as per equation (6.14), is also show. While for wavenumers less than k/2π = 100, the SPH viscosity approximates a Newtonian viscosity with good accuracy, at short scales the weakness of the SPH viscosity is evident. So for equivalent total viscous dissipation, the SPH viscosity must also e significantly stronger at intermediate scales. A quality spectrum may e determined for the SPH pressure gradient terms, as has een done for the viscosity term in Section Here the power spectrum [ P P ] is constructed. A similar trend as that of Figure 6.2 is k recovered (see Figure 5.20), and so the SPH pressure gradient term also suffers a similar weakness to the viscosity term. The significance of this with respect to the current simulations has not een investigated, though it is expected to contriute to short scale deficiencies. Shorter tests were performed at higher resolutions (one million particles) which exhiited reduced spurious energy at short scales (as compared with Figure 6.7 for instance), though results appeared otherwise unchanged. 6.3 Large scale forcing Quasi-steady solutions We turn to simulations utilising a larger scale forcing. Given that the forcing waveand will e sujected to less viscous dissipation than the intermediate and forcing used aove, a stronger inverse cascade should result. Relevant parameters are given in Tale 6.2. Here the wavespace annulus k f /2π = 10 ± 0.5 is sujected to a continual random forcing as discussed in Section

129 6.3 Large scale forcing E k Z time Figure 6.10: Large scale forcing simulation. Evolution of total kinetic energy E k (solid green curve) and total enstrophy Z (roken green curve) in time for simulation A. While on average over a large enough time the dominant energy contriution is going to e due to the force-force correlation P 1, ecause here only thirty discrete modes are forced, we cannot expect the force-velocity correlation P 2 to e negligile, though it s long time contriution should still e small. For the current simulation this is not of concern as we simply wish to present an SPH simulation using large scale forcing, with repeataility not required. The progression of total energy and enstrophy are given in Figure The early time energy and enstrophy variaility results from the random fluctuations in forcing power due to P 2, and is effectively determine only y the energy within the forcing and. By a time of t = 40, the total enstrophy has reach a largely steady value, signaling the convergence of the short scales (k > k f ) to a steady state spectrum. From this point forward, the total energy and enstrophy exhiits a roustness to variations in forcing owing to an active dissipation range. Unlike the previous simulations, with the inverse cascade eventually halted y viscosity, we find here a strong inverse cascade which presists throughout the run. This is evident in Figure 6.10, where the kinetic energy is increasing at a steady rate up until where the simulation is halted. At this point, the inverse cascade strength paramater is the relatively large r = 0.14, and approximately half all kinetic energy resides within modes k 3.5. The energy and enstrophy spectrums are displayed in Figure 6.11, along with the corresponding velocity and vorticity fields in Figure We first note that the small energy spike at the forcing wavenumer is asent in this simulation due to reduced viscosity in the forcing range.. Again the inverse cascade range trends according to a Kolmogorov k 5/3 scaling. The enstrophy range for this simulation follows approximately k 5, and so is shallower than the k 6 scaling encountered for intermediate scale forcing. This is most proaly on account of the larger inverse cascade strength, in line with the theory of Tran and Bowman (2004). The horizontal velocity field (Figure 6.12) exhiits predominatly large scale features, as dictated y the energy spectrum. Similarly, the structures

130 123 Two-Dimensional Turulence of the vorticity field appear to e of order of the forcing length scale, which corresponds to the peak in the enstrophy spectrum. Also evident are linear structures, or sheets, of constant vorticity, giving weight to the earlier qualitative arguement for an enstrophy cascade. Dark regions for oth the vorticity and energy field correspond to negative values, and we note that a reversed colour image is qualitatively identical. For domain scale vorticity structures, we expect that the enstrophy spectrum must peak at the smallest wavenumers, so we are required to continue the simulation until sufficient energy has accumulated in long wavelength modes. Time constraints did not permit the simulation to e continued unfortunately Steady solutions A series of simulations are performed to determine the ehavior of SPH as resolution is reduced. We also consider some preliminary simulations utilising the α-sph algorithm, as well as simulations where the filtered velocity is used with a standard acceleration equation. For clarity of comparison, we make two modifications to the turulence regime. Firstly, forcing is implemented to achieve some predetermined total energy Ef m within the forcing andwidth. This is implement y applying forcing where E f (t) < Ef m, and no forcing for E f (t) > Ef m, where total energy within the forcing and is E f (t). Secondly, we apply large scale dissipation to shut down the inverse energy cascade for modes k < 3.5, allowing the simulation to eventually achieve a statistically steady state. Large scale dissipation is used extensively in spectral two-dimensional turulence simulations (for instance Gotoh (1998) and Lindorg and Alvelius (2000)), and may e implemented natively within the spectral framework. A form for an analogous SPH large scale dissipation term, perhaps akin to the standard SPH viscosity, is not ovious. Instead, large scale dissipation is effected y explicitly removing the required velocity components from the velocity field. This is applied using the velocity stepping, v 1 a = v 0 a t k <3.5 ν L k v1/2 exp(i k x 1/2 a ) (6.21) with large scale dissipation parameter ν L. The coefficients v 1/2 are determined using the methods found in Appendix D. Simulations are executed until the kinetic energy reaches a statistically steady state. Parameters may e found in Tale 6.3. Changes in the steadystate spectrum as resolution is varyed are displayed in Figure For this figure, energy spectrum data is averaged over approximately ten eddy turnover times to compose the presented spectra. A small peak occurs at the forcing wavenumer on account of the lower Reynolds numers encountered. Short scales exhiit increasing energy accumulation as resolution is reduced. This result is expected given the larger velocity amplitudes required to effect an equivalent total viscous dissipation. Also worth noting is the reduced effectiveness of viscous dissipation as mode wavelengths approach the SPH smoothing length. This SPH shortcoming will e more pronounced for low resolution simulations, resulting in a further increase in short scale energy to compensate for deficiencies of the viscosity operator. The dimensional arguements in Section indicate a dissipative wavenumer of k d /2π 200. The maximum

131 6.3 Large scale forcing 124 E(k) k 2 E(k) (a) () k/2π Figure 6.11: Log-log plot of kinetic energy (a) and enstrophy () spectrum for run C. The peak in the spectrum corresponds to the forcing wavenumers k f /2π = 10. The green roken line gives a reference for the Kolmogorov k 5/3 scaling, while the lue roken line corresponds to an enstrophy cascade k 3 scaling. Data is taken at t = 85. (a) () Figure 6.12: Horizontal velocity (a) and fluid vorticity () corresponding to spectrums found in Figure 6.11.

132 125 Two-Dimensional Turulence Tale 6.3: Parameters for steady-state forced turulence simulation Parameter Description D Run x 0.002, , Initial particle separation ν Kinematic viscosity ν L Large scale dissipation parameter P Forcing power input Ef m Target forcing and energy k f /2π 10 Forcing wavenumer c 0.5 Forcing andwidth M Maximum run Mach numer Re 500 Maximum run Reynolds numer wavenumer of the highest resolution simulation is k/2π 130 (ased on the SPH smoothing length), which is somewhat shy of the dissipative wavenumer. While the validity of dissipative length scale definition 6.18 must e questioned for the current simulations given the asence of a k 3 enstrophy range scaling, the high resolution simulation does indeed appear to e insufficiently resolved in Figure A true dissipative lengthscale for the current regime is unclear, though energy at small wavenumers seems to e largely convergent for the two highest resolution simulations, which correspond well at all ut the shortest scales. We conclude that the N = simulation is sufficiently resolved to give accurate large scale dynamics. The lowest resolution spectrum deviates significantly from the higher resolution counterparts however, with energy at medium and large scales greatly reduced. This appears to e due to insufficient particles eing availale to resolve coherent structures. Instead, the vorticity field is largely dominated y the random forcing, with only weak vorticies presenting (see Figure 6.13). However, the coalescence of coherent structures is elieved to e integral to the inverse energy cascade (Frisch and Sulem, 1984), and so we postulate that the asence of strong vorticies weakens the SPH inverse cascade. With a total particle population of 62500, there are approximately 625 particles per vorticy (assuming structures are of order of the forcing lengthscale), which we conclude is insufficient to correctly produce the ehavior of interacting two-dimensional vorticies. For the range k > k f, we note with reference to Figure 6.4 that the enstrophy cascade range amplitude is not determined y the forcing mode amplitude, ut rather y the amplitude of the k < k f range. This is due to non-localness of interactions in spectral space, and may not e the case where a sufficiently wide forcing range is applied. Interestingly, Figure 6.13 indicates significant differences in the vorticity field etween the N = and N = simulation, which is perhaps not ovious in the energy spectra of Figure For N = , the vorticity field exhiits a certain roustness, with seemingly stronger features such as sheets of constant vorticity. A further simulation artifact ecomes evident where the squared density spectrum is considered (Figure 6.15). Details for the determination of spectrum [ρ 2 ] f are similar to those used in Section While we note that maximum density variations are within one percent of the mean value, it is apparant that

133 6.3 Large scale forcing 126 N = N = N = Figure 6.13: Vorticity field for steady-state simulations at different resolutions particles particles particles E(k) Figure 6.14: Log-log plot of kinetic energy for simulation series D. Presented data corresponds to averages taken over ten eddy turnover times.