Computational Aeroacoustics: Overview and Numerical Methods

Transcription

1 Computational Aeroacoustics: Overview and Numerical Methods Contents Tim Colonius California Institute of Technology Pasadena, CA USA 1 Overview of CAA Goals of CAA Direct Numerical Simulation Turbulence and acoustic source modeling Numerical Methods Finite-difference schemes Dispersion and dissipation Optimized schemes Spurious waves, artificial viscosity, and filtering Boundary closures Time marching schemes Computational efficiency Other issues Boundary Conditions Outlook Abstract In these lecture notes, we discuss the goals of computational aeroacoustics (CAA) and the numerical techniques that have been developed to achieve them. We first survey the scientific and engineering issues that have motivated computational approaches to aeroacoustic problems over the past few decades, and define choices of flow model and numerical algorithms that are appropriate to the differing goals and applications. Next we examine numerical algorithms for computation of aerodynamic sound in detail, with the aim of acquainting the student with ssues that drive the design of algorithms. In order to keep these notes as brief as possible, no attempt is made to survey the extensive literature on CAA. The student may consult recent reviews of CAA [1, 2] for a more complete summary of the field and complete references to the archival literature. 1

2 1 Overview of CAA 1.1 Goals of CAA An overall goal of computational aeroacoustics is to predict the sound radiated by turbulent flow, and perhaps more importantly, to investigate strategies by which noise could be reduced. The challenge is the enormous range of applications and their complexity. Difficulties arise because (i) the flows of interest are usually turbulent and involve a range of length and time scales that are difficult to resolve in a computation, (ii) the flows of interest stem from complex engineering systems (e.g. the complete exhaust system of an aircraft engine), (iii) the physics are complicated when additional features such as shock waves, multiphase flow, chemical reactions, and so on are present, and (iv) the fact that all of these complexities can occur in the same application! Like all difficult engineering problems, this complexity must be broken down into simpler unit problems, many of which are in themselves quite challenging. One can put these unit problems into different but overlapping categories: The development of accurate and robust computational methods for solving flow equations (either Navier-Stokes or modeled equations) The use of computations (in conjunction with experiments) to investigate fundamental mechanisms of sound generation The use of computation (in conjunction with experiments and theory) to derive simpler models of turbulence or sound generation processes The integration of theory, models, and computation into predictive tools that can be used for engineering design, optimization, and noise reduction strategies. An important aspect of these categories is that only one of them (the first) only involves computation and even there, there are issues that cannot be resolved without recourse to the physics of the underlying fluid dynamic processes. It is important to recognize that progress in CAA requires progress in experiments, theory, and modeling. Computational approaches to complex problems that ignore (or wish away) the physical challenges are doomed to produce nothing but pretty pictures. On the other hand, computational approaches to unit problems can provide insights that drive the modeling and theory to ever more practical and complex predictions. Prandtl is often quoted as saying There is nothing more practical than a good theory. 1.2 Direct Numerical Simulation Sound generation, propagation, scattering, etc. are all described in the continuum limit (comprising the vast majority of relevant applications) by the compressible equations for conservation of mass, momentum, and energy, together with constitutive models (e.g. Newtonian Fluid, Fourier s law), equations of state, chemical reactions, etc. These relations must 2

3 be closed by appropriate boundary conditions (solid surfaces, fluid interfaces) and, in addition, initial or boundary data must often be supplied. We will not dwell on these equations since the objective of these notes is to introduce the student to the overall issues, and they can be found in standard texts of fluid dynamics. Once the equations are assembled, we have exhausted the principles that physics has supplied to us in order to solve the problem. Can we then turn the mathematical crank and solve them? The answer is yes and no. We can certainly use numerical analysis to discretize the equations and then solve them by computer. However, as we discuss here, accurate solution would in most cases require computer resources far in excess of what will exist in our lifetimes Turbulence scales and DNS By Direct Numerical Simulation (DNS), we imply that all relevant scales of the true motion of the fluid are adequately resolved in a computation. Only first principles (continuum mechanics) are used to derive the equations, and, in turn, numerical analysis is used to guarantee that the approximate discretized solution is close enough to the true continuous solution. This latter aspect is know as verification of the numerical solution. In addition, we must ensure that any auxiliary data (e.g initial and boundary conditions), simplifications, etc. are adequate for the problem at hand; this is usually accomplished by comparing the result to a related experiment and is termed validation. Turbulence is characterized by disorganized motion over a range of length and timescales. The concept of an energy cascade wherein an equilibrium exists between drawing energy from mean flow gradients (production) and dissipating that energy at the smallest scales allows one to estimate the range of scales for a given flow. The smallest Kolmogorov length scale, η, is a motion whose length scale and related velocity fluctuation are such they form a Reynolds number of order unity so that they are rapidly diffused. The largest scales are essentially set by the extent of the turbulent flow, and are often characterized in terms of an integral scale, L, that is a measure of the extent over which the motion is coherent, as gauged by two-point correlations of the velocity, and the magnitude of velocity fluctuations, u. In turbulent thin shear layers (e.g. mixing layers and jets) it is proportional to the thickness of the layer, δ, and u is proportional to the mean flow velocity or velocity difference, U. Equating production and dissipation results in an estimate: η L = 1 (1) Re 3 4 L where Re = u L Uδ. ν ν In a DNS, we must resolve length scales as small as η and as large as L. Thus the grid spacing, x must be small like η and the extent of the domain large like L, so that in any coordinate: N L η Re 3 4 L 3

4 We must also integrate the equations over time. Generally in compressible flow there is a Courant number that limits the maximum time step for stable and accurate results: Cmax = a t x O(1) where a is the speed of sound. We must integrate for a total time of several turnover times of the largest scales, estimated by L with time step t. We therefore obtain an estimate for u the total computational work to resolve a region of turbulence: number of operations N 4 Re3 L M where M = u is the relevant Mach number. Thus at high Re and/or low M, the number of a operations required is staggering. We note that computing a portion of any radiated acoustic field requires even more operations; this is discussed in the next subsection. There is, of course, a minimum Re L for which turbulence persists and by the early 1980 s, computers were large and fast enough to allow barely turbulent flows to be computed. For channel flow, as an example, this amounts to a Reynolds number of several thousand based on the gap width and average velocity. Flows relevant to CAA have Reynolds numbers at least 1000 times higher. Computers have become exponentially faster in the intervening years according to Moore s law, and by now we can achieve Reynolds numbers about 10 times what was possible in the early 80 s. At that pace a turbulent jet with a Reynolds number of 10 6 would be feasible in about 60 to 80 years 1. This indicates a real need for models that permit some features of a flow to be computed without fully resolving all the scales. Such turbulence models are discussed in the next subsection, and, for CAA, the presence of the any model for the turbulence also has ramifications for the sound generation process. That is to say, we will often also need to model the acoustic sources since they are no longer resolved directly by the numerics. Our estimates here are crude, and do not take account of a constant of proportionality in our scaling laws. In real flows like turbulent jets, this constant may itself be a large number, especially for flows that are geometrically complex such as the exhaust system of a jet aircraft. Even the jet itself, downstream of the nozzle, consists of 4 distinct regions the near nozzle turbulent boundary layers, the annular mixing layer surrounding the potential core, and the fully developed jet further downstream (see Figure 1) Implications of acoustic inefficiency and wavelength Some features of sound generation, namely acoustic inefficiency and the inherent wavelength mismatch between flow and acoustic scales at low Mach number, require additional consideration beyond resolving the range of turbulence scales. A principle contribution of the Lighthill theory is to highlight the role of compact sources wherein retarded-time variations 1 Provided of course the Moore s law can be extrapolated that far, which is perhaps unlikely without a paradigm shift in processor design 4

5 Mixing layer Fully developed jet U j θ o Potential core U c (x) δ(x) x D j Figure 1: Schematic of a turbulent jet. Reprinted from [2] with permission from Elsevier. over the turbulent source region can be neglected in estimating the far field sound, and the delicate cancellation in acoustic sources lead to a tiny fraction of the flow energy being radiated in the form of sound. Acoustic inefficiency places demands on the numerical method above and beyond the requirements for resolution of turbulent scales discussed above. Since sound waves are small, we must be careful to use sufficiently accurate discretizations so that the acoustic waves generated by the flow are resolved and allowed to propagate to the far field (or computational boundary). If equivalent acoustic sources are to be computed, we must also ensure, for example, that sufficient accuracy is achieved in the computation so that errors in source calculation do not overwhelm the acoustic field. These constraints have led to some specialized high-order-accurate, low dissipation numerical methods used in CAA and these are the topic of section 2. The compact source approximation at low Mach number implies that the time-scale (or frequency) of the turbulence is preserved in the acoustic far field, but that the acoustic field has a length-scale (wavelength, λ) that is bigger than the turbulence scale (e.g. Crighton [3]): λ L 1 M Thus if our computational domain is to include about a wavelength of the radiated sound, then our estimate for total computational resources is increased to: number of operations N 4 Re3 L M 4 and, like high Re, it is simply not feasible to compute any significant portion of the acoustic field for small M (e.g. underwater acoustics). Such a compact source description is not always appropriate at high subsonic to supersonic flow conditions, but more detailed analysis [2] reveals that the acoustic wavelength is 5

6 still much longer than the turbulence integral scale at high subsonic Mach number. For a laboratory scale turbulent jet with M = 0.9 (unheated) for example (Figure 1), it can be estimated that the acoustic wavelength corresponding to the peak radiated frequency at 30 degrees to the jet axis is about 20 times the integral scale two diameters downstream of the nozzle. For sound radiated to 90 degrees, the wavelength to integral scale ratio is about 10 times the integral scale. The integral scale is approximately equal to the shear layer thickness at this streamwise position, and thus even at M = 0.9, there is still a large mismatch that requires an extensive computation domain to resolve a portion of the radiated sound. Just as the range of length scales associated with turbulence drives us to consider turbulence models, the acoustic inefficiency and wavelength mismatch drive us to consider whether we can somehow compute the turbulence with DNS at low Mach number (perhaps even with incompressible flow equations) and then use equivalent sources from acoustic analog theories (e.g. Lighthill) to compute, independently, the far-field sound. In many flows including jets, the acoustic waves themselves have no significant effect on the flow itself, and such a split seems reasonable. This is the basis then of so-called hybrid methods for CAA, which are discussed in more detail in subsection Boundary conditions (BC) and inflow forcing Another complication with directly computing a flow together with its radiated sound is the need to impose BC that accurately reflect the physics of the flow. Often, only a portion of a relevant flow can be computed, for example the portion of a turbulent jet downstream of the nozzle lip, but only extending to perhaps 12 diameters downstream in the fully developed portion of the jet. Outflow BC are then required at the downstream boundary that allow turbulent flow structures to leave the domain. It is essential (and nontrivial) that this occur with producing acoustic reflections that could be larger than the flow generated sound. Similarly, at upstream and normal boundaries, we must allow the acoustic waves in the domain to cleanly exit without producing sizable reflections that would contaminate the computation. Such nonreflecting and outflow BC were in fact a pacing item in CAA for many years, and even though several accurate techniques are now available, they can be difficult to use in practice. The entire issue of BC is therefore discussed in greater detail in another lecture in this series. An additional issue arises at the inflow boundary in the above example. There are either turbulent or laminar boundary layers developing inside the nozzle upstream of the computation. For the former case, we would need to specify some fluctuations at the inflow boundary that suitably approximate the true turbulent fluctuations. Of course, if it were a simple matter to make up accurate turbulence fluctuations than the whole topic of turbulence modeling would a simpler matter (which it is not!). Indeed, the issue is generic to DNS of turbulent flows in general, and a variety of techniques have been developed. While it is beyond the scope of these notes to describe these, we note that they involve forcing the flow with disturbances through the inflow boundary, or perhaps in a small region just upstream of the physical portion of the computational domain (e.g. [4]). In either case, it is important that these fluctuations are specified in a manner such that they do 6

7 not act as spurious sources of acoustic waves. For a region, one can require the imposed fluctuations to be divergence free, and this greatly lessens any sound produced directly by the forcing. It should also be noted that there will exist a development length over which the inlet forcing relaxes to a more realistic turbulent flow. This portion of the computation needs to be sacrificed in the sense that the flow will not be physical in this region. A somewhat more computationally intensive method of specifying inflow disturbances is to feed results from another turbulence simulation through the inflow boundary. For example, Freund [5] computed turbulence in a periodic annular mixing region to obtain accurate turbulence fluctuations with a set boundary layer thickness to generate a database that could be used to excite the inflow of a corresponding spatially evolving jet. For laminar incoming boundary layers (and these can exist even at high Reynolds numbers when the nozzle contraction ratio is large), the specification of inflow is considerably simpler. There is a caveat, however. Computations tend to be quieter than experiments in the sense of external disturbances. If laminar upstream boundary layers are not seeded with (small) disturbances, they can obtain unnaturally long laminar runs (with low flow spreading rate) before transitioning to turbulence. For the jet, for example, this results in an unrealistically long potential core. Thus inlet perturbations are typically added at or near the inflow boundary in order to excite natural instabilities in the initially laminar shear layer. Often it is useful to use eigenfunctions from linear stability theory as the form of these seed disturbances Domain extensions for far-field sound At progressively larger distances from the near-field unsteady flow region, fluctuation levels are decaying and at a certain distance nonlinear effects become negligible (except for radiated weak shock waves). At this point, pressure fluctuations are generally governed by a linear wave equation (provided that there is at most uniform flow or something close to it outside the source region). A corollary of the wave equation is that in this far-field region, the acoustic waves at any point may be expressed in terms of an integral of the the time history of the pressure on a surface surrounding all sources. Such a Kirchhoff Surface therefore allows the flow computation to be truncated at a point where fluctuations become linear 2. The methodology can be carried out in two ways. Linearized equations (linearized Euler equations or their reduction to a wave equation) can be discretized on a grid surrounding the flow computation grid, with data from the flow computation specified along the common (Kirchhoff) surface. More traditionally, an integral formulation of the wave equation is used and evaluated with numerical quadrature. Brentner [6] discusses some of the common sources of error in such quadrature. Sometimes it is not possible to define a closed surface around all the sources, for example when a computational domain ends with unsteady flow passing through an outflow BC. Errors will be incurred if the surface is drawn through the flow (which obviously does not satisfy the wave equation) [7, 8], or if it left open [9]. An alternative is to use an integral formulation due to Ffowcs Williams and Hawkings (Ff-H) [10, 11] that has been shown 2 This assumes that an accurate BC for the flow solver can be posed at this surface, as discussed in section 2.9 7

8 in certain cases to retain accuracy even when the boundary is drawn through a nonlinear flow region. Note that the Ff-H surface is also used in connection with hybrid methods as discussed below DNS Examples Before turning to modeling issues in the next subsection, it is useful to present a brief resume of those flows that have and are being computed with DNS discussed in this section. Here we include only those computations where the noise was directly captured in the simulation, and for which no turbulence or noise source models were used (those issues are discussed in section 1.3. These flows fall into two categories, depending on the level of physical realism. In the first category are model problems, often two-dimensional, which may be physically unrealizable but which can be used to address fundamental issues of aeroacoustic theory, or to provide benchmark problems for development of future algorithms. In the second are actual turbulent flows computed by DNS which can be directly validated against experiment and used as databases for modeling studies. This list is meant to be illustrative, not exhaustive. Model problems: 1. Interaction of vortices, including co-rotating vortices [12], collisions of vortex rings [13]. 2. Vortex pairing in mixing layers [14, 15] and jets [16] and with mixing layer with adjoint-based control [17]. 3. Mach wave radiation from supersonic jets [5, 18, 19, 20]. 4. Shock vortex interaction [21, 22, 23, 24, 25, 26]. 5. Jet screech and shock-associated noise [27, 28] 6. Flow over open cavity [29, 30, 31, 32] Turbulent flows: 1. Jet [5, 4] 2. Vortex ring [33] 1.3 Turbulence and acoustic source modeling Turbulence modeling As discussed in the last section, it is presently impossible to simulate flow at high Reynolds number using DNS. Alternatives to DNS include Large Eddy Simulation (LES) and Reynolds Averaged Navier-Stokes equations RANS. LES takes the approach of filtering out scales below a cutoff,, in equations of motion, whereas RANS takes the approach of averaging velocities and other flow quantities over a long time, or over a large ensemble of realizations. Both LES and RANS result in an unclosed term in the resulting equations that cannot even 8

9 in principle be exactly recovered from the remaining filtered or averaged variables. A wide range of models are available that provide this Reynolds or Leonard stress, in the case of RANS and LES, respectively. It should be noted that when the LES equations are solved numerically, the scale cutoff must be at a minimum the grid spacing, x. However, the dynamics of these smallest computed scales is not correct owing to truncation error. Whether or not is made larger than x depends on the particular model employed, but regardless of this, it should be assumed that the smallest motions represented on the grid (and in particular their sound radiation) are effected by discretization errors. There are obviously complex issues and trade offs involved in choosing a particular closure. The main objective of this section is to acquaint the student with those CAA issues that arise above and beyond the normal requirements and capabilities of the models. These have principally to do with three inter-related issues: 1. Whether any of the sound generation process is directly captured by the modeled equations 2. Whether an additional model (above and beyond the closure) can be supplied to model the missing acoustic sources 3. Whether the closure model leads to unphysical acoustic radiation RANS in its simplest form seeks the time-averaged flow field and thus resolves no acoustic generation or radiation, except possibly stationary Mach waves in supersonic flow. There the issue turns completely to modeling of the acoustic source in terms of the mean flow and whatever statistics are represented by the model, usually the turbulence kinetic energy, k and the dissipation, ɛ. LES, on the other hand, resolves the energetic (large) scales of motion and presumably captures with that some portion of the radiated sound. Sound radiation from small scales is missing, although without further analysis it is unclear what portion of the total acoustic spectrum this might represent. Further details about this are given below. The final issue about whether the closure leads to nonphysical noise sources has received comparably little attention in the literature. It can be shown that they do represent equivalent sound sources in a traditional acoustic analogy framework (Lighthill) [34], and some estimates for acoustic output of the SGS terms have been made for isotropic turbulence [35, 36]. On the other hand, several existing LES models seem to show good correspondence between the resolved portion of the acoustic spectrum and experiment or model spectra. This agreement suggests that at least with the particular models employed, direct radiation from the SGS models is not a major contributor to the overall sound. For the case of a turbulent jet it is possible to make estimates for what portion of the acoustic spectrum would be resolved for a particular LES grid resolution [2]. Using an empirical model of the three-dimensional energy spectrum, Lele estimates that over the inertial range, the typical eddy frequency scales like: f f o ( ) 2 Lo 3 L 9

10 where f is the frequency, L is the length scale of the eddy, and L o and f o are the length scale and frequency of the integral scale (peak in energy spectrum). For example if LES places 32 points per integral scale, then we can resolve frequencies relative integral scale to the up to With 128 points, this goes up to around 25. However, this assumes that the eddies are well-represented all the way to the grid-scale, which as noted above is not the case in an actual computation. Better estimates reduce 10 and 25 to 5 and 13, respectively, when high-order-accurate numerical methods are used [2]. Note also that the estimate is for a single integral scale. A typical computation would involve a flow region many times the integral scale in extent. A nice illustration of the analysis is provided in the work of Bodony and Lele [37] and is reproduced in figure 2. The impact of using finer resolution in LES is to fill in a portion of the missing spectra This higher-frequency sound that is difficult to resolve can be important in SPL St Figure 2: Predicted far-field noise spectrum from LES of an unheated, M j = 0.9 jet by Bodony & Lele [37]; LES with 10 6 points; LES with 10 5 points; Empirical jet noise spectra from Tam. Figure reproduced from [2] with permission from Elsevier applications, particular in jet noise where metrics like Perceived noise level are used that penalize the high frequencies that people find most annoying. Thus even with LES, it may be necessary in some applications to supply a model for the missing sound generated by the smallest scales. Efforts toward that goal are being made by several groups [38, 39, 40, 41] Noise source modeling Aside from noise source models for SGS terms in LES, the development of more general models characterizing the acoustic sources in turbulent flows remains a important goal, especially in the context of employing RANS to supply mean velocity, turbulence kinetic energy and dissipation as inputs to the models. Available codes include the JeNo (for Jet Noise) and MGBK software developed at NASA Glenn Research Center [42]. These models generally seek to represent the two-point, two-time correlation functions needed to statistically model the Lighthill source. Without delving into the details, it is worth mentioning an important application of DNS (and perhaps LES) results in this context. That is in providing a detailed database of both the turbulent fluctuations and the far-field sound that can be used to examine statistical noise source models. The statistics required for the noise sources are quite difficult to measure 10

11 experimentally, especially in high speed flows. Indeed, Freund [43] has used his simulations of the M = 0.9, Re=3600 turbulent jet to examine in detail statistics associated with the correlation functions. Even though the Reynolds numbers reached in simulation are low, some correlations derived from the data may hold at much higher Reynolds number [43] Hybrid methods Especially for low Mach number flows, it is advantageous to develop methods that do not directly capture the radiated sound but instead rely on a second calculation, or post-processing step, to predict the noise. In some cases this step may be carried out concurrently with the flow simulation, but generally by hybrid methods we mean ones for which the flow field itself is evolved independently of the acoustic radiation. Two related approaches have been developed and are summarized in figure 3. Hybrid Methods (II) Flow Computation Compressible/Incompressible DNS/LES Acoustic Analogy Equivalent Source Secondary Acoustic Computation Equation Splitting (singular Perturbation) Green s Function solution of wave equation Direct solution on overlapping FD mesh Tim Colonius Caltech Figure 3: A guide to hybrid methods VKI Lecture Series 44 Acoustic-analogy-based methods The equivalent sources for an acoustic analogy are computed from data from DNS or LES simulations of the turbulent near-field region. Any acoustic analogy can in principle be used, including Lighthill [44], Lilley [45], recent generalized acoustic analogies by Goldstein [46]. The wave operator (for example in Lighthill s equation) is then inverted either in integral form with an appropriate Green s function, or by direct finite-difference solution. For low Mach number, it is usually appropriate to make compact source assumptions in evaluating the Green s function integrals, and this can be important numerically since small quadrature errors could in principle prevent appropriate quadrupole cancellations. While these methods are most effective for low Mach number flows, where the acoustic Courant constraint requires tiny time steps, they can in principle be applied to higher Mach numbers, provided of course that compact source assumptions are relaxed. However, one must evaluate whether it makes sense to do so, since a direct computation that resolves the sound generation at moderate subsonic Mach number may be less expensive and prone to 11

12 error. No general analysis has been done to determine at which Mach number one approach becomes less expensive than the other. Incompressible solvers typically involve iterations on a Poisson equation to determine the pressure (i.e. to satisfy the divergence-free constraint), which must be balanced against the larger number of variables and smaller time-steps of a compressible solver. For example, a fully compressible flow solver at M=0.5 together with a highly stretched mesh near the far-field boundary would probably be competitive with an incompressible flow solver in terms of total computational time required. As a practical matter it is necessary to ensure that the equivalent source terms decay sufficiently towards the computational boundaries. In cases where outflow of organized disturbances from the flow domain elevates the source-terms to a significant level at the boundary, it is necessary to adequately treat this aspect, since frozen convection of a disturbance at subsonic speed either into or out of the computational domain should not be a source of sound [47]. The choice of Green s function depends on the particular left-hand-side linear operator, examples: 1. The standard wave equation (e.g. Lighthill). The flow outside the domain must be quiescent or uniform flow (with a change in reference frame). Refractions effects in the near field are lumped in with the acoustic sources. 2. The third-order wave equation for parallel shear flows (e.g. Lilley). The base flow is a parallel shear flow and the wave operator includes effects of refraction consistent with that flow. Any refraction due to sound interaction with more general velocity gradients is lumped in with the acoustic source. If the parallel flow is inflectional, then unbounded instability waves (i.e. Kelvin-Helmholtz) will also be a solution. These can lead to numerical difficulties or ambiguities in evaluating the sound and a variety of methods are being developed to suppress the instability-wave solution [48, 49, 50, 51]. 3. Generalizations to more general (non-parallel) base states that result in the Linearized Euler Equations (LEE) [48, 49] or other operators [46, 52] on the left-hand-side of the acoustic analogy. If mean flow spreading is included in the operator, the instabilitywave solution is bounded and can be taken as a particular solution driven by the sources [53]. The Green s function is obtained by replacing the right-hand-side source with a delta function and finding a solution to the left-hand-side operator. The Green s function is then convolved with the right-hand-side source. Beyond this, there are additional issues and choices that arise in finding the Green s function: 1. If there are solid surfaces, then a Green s function that satisfies appropriate acoustic BC on these surfaces can be used. In this case the convolution will be an integral over the volume including the sources. 2. A free-space Green s function may be used in which case convolution will be an integral over the volume and apparent sources on the solid surfaces. 12

13 3. The Ffowcs Williams-Hawkings approach [10] may be used wherein a generalized Green s formula is used to express the solution to the wave equation in terms of surface integrals over the body-surfaces immersed in the flow, integrals over other surfaces needed to enclose the integration domain, and volume integrals over the volume source distributions, see general derivation in Crighton et al. [54]. A review of this method, particularly from a viewpoint of use with CAA, has been provided by Lyrintzis [7]. 4. An adjoint formulation (Tam & Aurialt [55]) can be used where the role of source and observer in the acoustic field are interchanged so that the Green s function is found by solution of a scattering problem rather than a source problem. When the left-hand-side operator is the linearized Euler equations, this can lead to a computational advantage as it reduces the number of Green s functions that must be found from 3 to 1 [55]. Incompressible/acoustic split Several groups (e.g. [56, 57]) have proposed computational methods for predicting the radiated noise without an explicit use of an acoustic analogy. The general idea of these methods is to compute the unsteady flow responsible for the noise with incompressible equations and then overlay a simplified set of compressible flow equations to predict the radiated noise. The derivation of these methods lacks full rigor since the singular perturbation of the compressible equations [58, 3], in the limit of small Mach numbers, is not recognized in the derivation. It is interesting that Goldstein s [46] generalized acoustic analogy may provide a framework useful in rationalizing methods that invoke a split between the flow and the acoustic variables Summary and research directions The computational approaches discussed above are summarized in graphical form in figure 4. All elements of this matrix of methods are currently being developed by researchers in CAA and represent important unit problems in attacking noise prediction generally. The only completely unambiguous way to predict sound radiated by turbulent flow is by DNS, which as discussed here is restricted to sufficiently low Reynolds number so that practical engineering problems cannot be solved. This motivates modeling approaches for both the turbulence (LES,RANS) and the acoustic sources, or that portion of them that is not directly resolved by the flow computation. The development of these models is likely to drive a majority of work in CAA in the years to come. However, we see a strong role for employing DNS and LES of simpler, lower Reynolds number flows both model problems and full-blown DNS of turbulence. A summary of conclusions drawn from recent DNS computations is provided by Colonius & Lele [2]. We summarize the sorts of issues that DNS and LES can help resolve here. 1. Study simple model flows to try to learn how flows generate sound 2. Study canonical turbulent flows in order to measure equivalent sources for acoustic analogy, perform a priori tests of modeled sources in order to validate assumptions and approximations therein 13

14 Flow parameters, geometry Inflow Excitation Model Turbulence Model? No Subgrid Model? No! Yes Yes Implicit sub-grid model RANS/ URANS Hybrid methods (DES, NLDE) LES Vortex Methods, Reduced-order Models DNS Empirical noise source models? Subgrid noise source models? No! Extract acoustic sources? No No Implicit sub-grid model Flow computation must be compressible Yes Yes Yes! Scaling Laws Acoustic Analogy Sources Lighthill (integral or differential) Lilley Generalized Solution Methods Green s function (wave eq.) Differential (wave eq. or LEE) Adjoint Green s Function (LEE) Domain Extension (Kirchhoff/Ff-H, direct far-field, equation set matching, etc) Problem Setup Turbulence Modeling Flow computation Acoustic Source Modeling Acoustic computation Noise Prediction Figure 4: A hierarchy of noise prediction methods. Reprinted from [2] with permission from Elsevier. 14

15 3. Provide data for turbulence correlations related to aeroacoustic theory 4. Provide benchmarks against which hybrid/modeled methods may be compared 2 Numerical Methods A variety of accurate and robust numerical methods have been developed for sound generation and propagation propagation problems. As discussed above, acoustic inefficiency gives rise to waves whose amplitudes are small compared to near-field fluctuations; these waves may also need to propagate over substantial distances within the flow. Two key features of most CAA discretization schemes are therefore (i) high-order-accurate and optimized schemes such that the solution is adequately resolved with as few grid points as possible, and (ii) low numerical dissipation (or artificial viscosity) such that waves are not adversely attenuated. These same features also fulfill the requirements necessary of a good discretization scheme for DNS and LES where a range of scales need to be resolved, it is advantageous to use high-order-accurate methods. The present notes are an attempt to give the basic rationale for high-order-accurate and optimized methods in CAA, and to discuss some of the issues that arise in implementing them. For a more detailed account of these issues, see [2]. The 4 workshops on benchmark problems in CAA that have been sponsored by NASA [59] also provide a wealth of information regarding current practices and issues in CAA. Other things being equal, the best choice of discretization for a given geometry and accuracy requirement is the one that is most computationally efficient, i.e. the one that requires the smallest computing time for a given error tolerance. Other factors that determine the best choice of method include ease of implementation (and especially imposition of BC), efficiency of parallelization, memory requirements, and the potential for straightforward implementation in different geometries and flow configurations. The trade-offs between these issues have generally favored finite difference (FD) methods (especially high-order-accurate and optimized methods), but a variety of other methods are also useful. In particular, spectral (or pseudo-spectral) schemes are useful for flows possessing one or more homogeneous directions, combined with a finite-difference or alternative discretization of the inhomogeneous direction. Round jets, for example, are naturally described in cylindrical coordinates and Fourier-spectral methods for the azimuthal direction is a good choice. Spectral methods are generally efficient (more so than the schemes discussed below) for these cases. However, their implementation in inhomogeneous flow directions with BC discussed in section 2.9 is not advised. Finite element and spectral element methods have been developed for the compressible Euler and Navier-Stokes equations, particularly the Discontinuous Galerkin (DG) method (e.g. [60]) has been examined by a number of investigators for CAA problems [61, 62, 63, 64, 65, 66, 67, 68]. Finite-volume schemes on grids where fluxes are staggered with respect to the conserved variables are also attractive schemes for CAA, and staggered schemes for spectral element [61] and finite difference [69] discretizations have recently been extended to compressible flows. Such schemes confer an advantage especially in LES calcu- 15

16 lations, in that they can be arranged to admit a discrete integral conservation law similar to the continuous equations. Vortex particle methods, either together with acoustic analogy [70, 71], or with recent extensions to compressible flow [72] can also be used in aeroacoustic computations. Flows involving shock waves require special attention, and their spatial discretization will be discussed in a subsequent lecture (Pirozzoli) in this series. 2.1 Finite-difference schemes FD schemes have been used for a majority of problems in CAA, which stems from their flexibility and ease of use on structured grids, as well as their ability to extend to high-orderaccuracy with little additional complication. A centered FD scheme for the first derivative on a uniform mesh is written: N α j=1 ( ) α j f i+j + f i j + f i = h( 1 Na ( ) ) a j fi+j f i j + O(h n ). (2) The independent variable at the nodes is x i = h(i 1) for 1 i N, and the function values at the nodes are f i = f(x i ), and the values of the derivative are f i = f (x x i). The stencil refers to the maximal number of points required to compute the derivative at a point j (max(n a, N α )) and a centered scheme is one for which the stencil extends equally to the left and right of the point j. Biased or upwind schemes are similar but have a unique coefficient a j and α j for each N a < j < N a and N α < j < N α, respectively. If N α = 0 then the scheme is called explicit. Implicit schemes (also Padé or compact FD), by contrast, have N α 0 and require the solution of a system of equations to determine the derivatives of all nodes 1 i N simultaneously. Provided N α = 1 or 2, efficient schemes for finding the solution of the resulting tri- and penta-diagonal matrices can be used. Conventionally, the coefficients α j and a j are chosen to give the largest possible exponent, n, in the truncation error, for given stencil width (i.e. choice of N α and N a ). By Taylor series expansion of Equation (2), the maximum possible exponent is given by: j=1 n max = 2(N a + N α ), (3) provided that N a 1. Table 1 gives coefficients for several centered schemes that have been used in CAA studies. The leading order error term in Equation (2) is a measure of the actual error for asymptotically small h and also involves high-order derivatives of the underlying function. Other measures of the error can be more instructive, in particular wave propagation errors discussed A natural way to investigate the wave propagation characteristics is through a Fourier modified wavenumber analysis. Fourier analysis of numerical approximations to PDE dates back to the 1940s and the pioneering work of von Neumann. The analysis is strictly applicable only to systems with periodic BC, but this is not a serious limitation since we can usually interpret the results as applying locally in a given non-periodic problem. If a domain of length L is discretized with N uniformly spaced points on x [0, L) (with h = L ), the N 16

17 Scheme α 1 α 2 a 1 a 2 a 3 Order (n) E / E /3 1/12-4 E /4 3/20 1/60 6 C4 1/4-3/ C6 1/3-7/9 1/36-6 DRP - - (496 15π)/42c (1725π 5632)/84c (272 85π)/14c 4 LUI See caption 6 Table 1: Coefficients of explicit second, forth, and sixth-order schemes,compact forth and sixth order schemes, and the optimized schemes of Tam & Webb [73] (DRP; where c = 45π 128) and Lui & Lele [74] (LUI; where the constants are: α 1 = , α 2 = , a 1 = /2, a 2 = /4, a 3 = /6) discrete Fourier transform (DFT) of f (denoted by ˆf) is: ˆf j = 1 N f i e ik jx m, j = N/2,, N/2 1 (4) N m=1 where the wavenumber is k j = 2πj/L and x m = (m 1)h. The inverse transform is f m = N/2 1 j= N/2 ˆf j e ik jx m, m = 1,, N (5) It may be shown that the jth component of the DFT of f x, denoted ˆf j is simply ik j ˆfj. Taking the DFT of Equation (2) gives the approximate value of ˆf j, which we denote ( ˆf j) fd, as: ( ˆf j) fd = ik(k j h) ˆf j (6) where K(z) = Na n=1 2a n sin (nz) 1 + N α n=1 2α n cos (nz) is the so-called modified wavenumber. Note that k j h takes on values between π and π as j varies between N/2 and N/2. For k j h = π, the period of the wave is 2h (and, generally, the number of points per wavelength, N λ, is 2π k j ). Higher wavenumbers cannot be represented on the grid, and their energy is aliased onto the resolved wavenumbers. Modified wavenumber curves are plotted in Figure 5 for the FD schemes of Table 1. Exact differentiation would result in a modified wavenumber relation K(z) = z. The error between the exact and modified wavenumber, ɛ(z) = (K(z) z)/z is also plotted in Figure 5. For a given error, we can estimate the required number of grid points per wavelength required for a given scheme (k j h = 2πjh/L). As the desired error is reduced, low-order schemes require significantly larger numbers of grid points per wavelength. 17 (7)

18 K ( kh ) / π E2 E4 E6 DRP C4 C6 LUI ε (kh) E2 E4 E6 DRP C4 C6 LUI kh / π N λ Figure 5: The modified wavenumber (left) and the relative error between the exact and modified wavenumber (right). Reprinted from [2] with permission from Elsevier. 2.2 Dispersion and dissipation When used in linear hyperbolic PDE, FD schemes admit a simple interpretation in terms of modified wave propagation characteristics. A useful toy model that demonstrates most of the effects is the advection equation: u t + cu x = 0. (8) where c is a constant with units of speed, and the equation admits simple waves with u = u(x ct). When the one-dimensional Euler equations are linearized about a constant base flow, they can be decoupled into a system of equations of the form of Equation (8). On infinite or periodic domains, the solutions of Equation (8) may be decomposed into their Fourier components in both x and t. That is, we set u(x, t) = ûe ikx e iωt + c.c. Then nontrivial solutions to Equation (8) may only be obtained when the dispersion relation: ω = ck (9) is satisfied. Since all Fourier components of the solution travel with the same constant phase speed, ω = c, waveforms comprised of a superposition of modes retain their shape as the k propagate, and are therefore called non-dispersive. There is no physical viscosity and waves are not attenuated (dissipated). If a finite-difference scheme is used in Equation (8), we obtain a modified dispersion relation: ωh = K(kh) (10) c and the phase speed of disturbances is now given by: c p c = ω ck = K(kh) kh 18 (11)

19 Different Fourier components travel with different phase speeds and such a system is termed dispersive, since the waveform will be altered as the wave propagates [75, 76]. Groups of waves with Fourier components whose wavenumbers are near k propagate at the group velocity, c g c = 1 ω c k = K (kh) (12) where the prime denotes differentiation with respect to the argument, kh. Group velocities for FD schemes are shown in Figure 6. An important feature of the plot is that, for a given frequency, there are now two discrete waves. The one with the long wavelength corresponds to the physical or smooth solution in the limit of h 0. The one with short wavelength has no physical counterpart and is termed a spurious wave. In physical space, these waves appear as wiggles in the contours or sawtooth on a line plot. 1 0 c g / c E2 E4 E6 DRP C4 C6 LUI kh / π Figure 6: The group velocity for several FD schemes. Reprinted from [2] with permission from Elsevier. An important observation from Figure 6 is that the spurious waves propagate in an opposite sense to the physical waves. Upstream propagating spurious waves may readily be seen in supersonic flow computations even though all wave motion is (physically) downstream. We can define two useful quantities from the figure as well. The first is k c which is the wavenumber beyond which waves are spurious, and ω c which is the maximum frequency that can be propagated on the mesh. It is important to notice that k c h π. Amplitude at a particular wavenumber is dispersed, but with the centered FD, there is still no attenuation of a given Fourier component. Wave energy is conserved. By contrast, when biased FD schemes are used, the modified dispersion relation will result in complex frequencies, ω. If a scheme is appropriately upwinded, meaning the stencil is biased in the direction of positive c, then the imaginary part is negative and waves are attenuated or dissipated as they propagate. If the scheme is downwind biased, ω i is positive and the solution blows up. Thus only upwind FD schemes are permitted generally. In systems of PDE 19

20 that can have propagation velocities in different direction, this requires that the variables be approximately decoupled into upstream and downstream components before they can be differenced. Various upwind schemes have been considered for CAA applications (e.g. [77, 78]). A drawback of upwind schemes is that there is significant dissipation of the highest resolved wavenumbers that can only be reduced by increasing the stencil size. 2.3 Optimized schemes The coefficients of the FD scheme discussed above were chosen to give the lowest truncation error, asymptotically for small h, via a Taylor series expansion. For finite grid size, h, it is also possible to reduce the error in the modified wavenumber by optimizing the coefficients. Typically some of the coefficients are still chosen to give a particular order-ofaccuracy by Taylor series expansion, and the remaining coefficients to reduce error in the modified wavenumber. Lele [79] and Lui & Lele[74] derived optimized schemes by requiring the modified wavenumber to be equal to exact differentiation at certain wavenumbers. Tam and Webb [73] derived dispersion-relation-preserving (DRP) schemes by minimizing the integral of the error in modified wavenumber with respect to the remaining coefficients. Coefficients for the optimized schemes are given in Table 1 and modified wavenumbers and group velocities plotted in figures 5 and 6, respectively. Recently, Pirozzoli [80] has extended the idea of optimized schemes to consider optimally efficient schemes, in the sense of minimizing the computational cost of achieving a particular accuracy goal. These schemes are discussed below in the context of computational efficiency. 2.4 Spurious waves, artificial viscosity, and filtering As discussed above, centered FD schemes yield two solutions for the wavenumber for a given frequency: the small wavenumber smooth solution has propagation characteristics similar to the underlying PDE, while the high wavenumber (near π) spurious solution propagates with a group velocity of the wrong sign. In a computation, spurious waves may be produced in several ways: 1. Initial conditions (IC). Any energy at wavenumbers kh > k c h present in the initial condition will propagate as spurious waves. For this reason, it is preferable to start with a well-resolved initial condition. In practice, however, it may be difficult to construct and there will inevitably be unphysical initial transients as the specified IC relaxes to an approximate solution of the governing equations. For some simple flows like vortices, vortex rings, etc., an approximate steady laminar solution can often be found that minimizes such transients. In other cases, it is possible to use filtering or artificial viscosity at early stages to remove poorly resolved waves. 2. Nonlinear cascade to small scales in DNS and LES. The nonlinear terms in the governing equations naturally produce finer scales (higher wavenumbers). In a DNS calculation, physical viscosity should be the only mechanism that supplies dissipation 20