A Discontinuous Galerkin Conservative Level Set Scheme for Simulating Turbulent Primary Atomization

ILASS-Americas 3rd Annual Conference on Liquid Atomization and Spray Systems, Ventura, CA, May 011 A Discontinuous Galerkin Conservative Level Set Scheme for Simulating Turbulent Primary Atomization M. F. Czajkowski and O. Desjardins Department of Mechanical Engineering University of Colorado, Boulder, CO 80309-047 Abstract In an effort to improve the numerical schemes available for simulating of multiphase flows, a novel numerical method is developed to track the interface location. This method combines the excellent mass conservation properties of the accurate conservative level set (ACLS) method of Desjardins et al. [J. Comp. Physics 7 (18), 8395 8416] with a discontinuous Galerkin (DG) discretization. DG provides an arbitrarily high order representation of the level set function without requiring a large computational stencil, resulting in a highly accurate and parallelizable method. A conservative reinitialization equation, discretized with DG as well, is used to reset the level set function to a hyperbolic tangent profile, ensuring the good conservation properties. Finally, several curvature computation strategies are explored in the context of a high order polynomial representation of the level set function. The performance of the method is demonstrated for a variety of test cases. Combined with a new discretization of the momentum convection term in the Navier-Stokes equations, and an immersed boundary methodology that allows to use non body-fitted grids while conserving both mass and momentum discretely, this scheme provide both accuracy, scalability, and robustness even in the presence of turbulence, high shear, and large density ratios. This new methodology is then employed in a detailed study of the turbulent airblast atomization of n-dodecane. Numerical predictions are compared to experimental measurements, showing the satisfactory behavior of the proposed level set scheme. In particular, the onset of break-up, most unstable wavelength, and drop size and velocity distributions are in good agreement, suggesting that the fundamental physics of air-blast atomization are well captured by the simulations. Corresponding Author: olivier.desjardins@colorado.edu

Introduction Optimizing the process by which a liquid fuel is injected and atomized in a combustion chamber requires a detailed understanding of the physics of turbulent multiphase flows. Typically, experimental investigations of turbulent atomizing liquids have remained limited in the amount of information that could be gathered at the onset of break-up: by their very nature, these flows tend to generate a large number of optically opaque fuel droplets, and therefore detailed optical diagnostics of atomizing liquids are sparse. Considering these challenges, Computational Fluid Dynamics (CFD) has the potential to provide much needed information on the nature of the atomization process, leading the way to the development of practical engineering atomization models that could be used in the industry sector. However, simulating turbulent liquid atomization remains a challenging endeavor for several reasons. The phase-interface needs to be tracked accurately, with minimal volume conservation errors, and the interfacial curvature must converges under mesh refinement. In addition, a robust method must be employed to ensure stability in the presence of a large density difference between the liquid and the gas. Both the singular nature of the surface tension force and the discontinuity in viscosity require a special treatment. The development of numerical methods that are accurate enough to satisfactorily capture the key physics of turbulent two-phase flows is the topic of this paper. Two key numerical methods have been developed and added to an advanced CFD code to allow for the simulation of n-dodecane air-blast atomization. The base code, NGA [1], uses the ghostfluid method (GFM) [] to handle the pressure and density discontinuities. It has been extended with capabilities to handle high density ratios and more accurately transport the phase interface using a discontinuous Galerkin conservative level set method. Using this combination of numerical methods, simulations of primary atomization at realistic injection parameters become feasible. The paper is organized as follows. A mathematical description of governing equations is given first, followed by a description of the numerical methods, along with a few test cases. The simulation of a n- dodecane air-blast injector is then presented, including comparison to experiments. Finally, conclusions are provided. Mathematical Formulation The gas-liquid flow of interest can be described using the continuity and Navier-Stokes equations. Assuming that both phases are incompressible, i.e., u = 0, the continuity equation is written Dρ Dt = ρ + u ρ = 0, (1) t where u is the velocity field and ρ is the density. The Navier-Stokes equations are written ρu t + (ρu u) = p+ (µ [ u + u T]) +ρg, () where p is the pressure, g is the gravitational acceleration, and µ is the dynamic viscosity. The material properties are considered to be constant in each phase. The subscript l is used to describe the density and the viscosity in the liquid, respectively ρ l and µ l. Similarly, the subscript g corresponds to the density and the viscosity in the gas, respectively ρ g and µ g. It is convenient to introduce the jump of these quantities across the phase-interface Γ, defined by [ρ] Γ = ρ l ρ g and [µ] Γ = µ l µ g. In the absence of mass transfer between the two phases, the velocity field is continuous across Γ, i.e., [u] Γ = 0. In contrast, the existence of surface tension forces will lead to a discontinuity in the normal stresses at the gas-liquid interface. This translates into a pressure jump that can be expressed as [p] Γ = σκ + [µ] Γ n T u n, (3) where σ is the surface tension coefficient, κ is the curvature of the phase-interface, and n is the phaseinterface normal. Numerical Methodology The numerical methods used in this work are based on arbitrarily high-order accurate fully conservative finite difference schemes implemented in the NGA code [1]. These schemes are both robust and accurate, and are therefore ideally suited to the numerical simulation of turbulence. In this work, the second-order version of these schemes is used, since it greatly simplifies the implementation of the multiphase numerics. Equation 3 highlights the need for a specific strategy regarding discontinuities. In this work, the discontinuities in density and pressure that arise in the pressure gradient term of Eq. are treated using the ghost-fluid method (GFM) of Fedkiw et al. []. This approach provides a sharp and robust discretization of the discontinuities across the phaseinterface. However, it presents specific challenges when considering the viscous term discontinuities, and for this reason this term is discretized using a height function strategy, as presented for example

by Sussman et al. [3]. For more details on the numerical discretization for multiphase flow problems in NGA, the reader is referred to prior work [1, 4]. Two new developments have been introduced here, namely a quadrature-free discontinuous Galerkin conservative level set method for interface transport, and a novel discretization of the momentum convection term that allows for robust high density ratio simulations. These two methods are presented briefly below, along with numerical tests. Discontinuous Galerkin Conservative Level Set Turbulent primary atomization is a process that includes significant interface dynamics that result in a large quantity of small droplets. Hence, an interface scheme must combine excellent accuracy with excellent mass conservation properties in order to be successfully used to simulate primary atomization. We present here a novel methodology that is based on the Accurate Conservative Level Set (ACLS) technique [5], but relies on a quadrature-free discontinuous Galerkin discretization for improved accuracy. This approach has the following properties: (1) it is very accurate since it uses a polynomial representation of the level set function to an arbitrarily high order in each cell, () it has excellent mass conservation properties since it uses only discretely conserving schemes, and (3) it has the smallest possible numerical stencil size, which leads to excellent parallel scalability. Accurate Conservative Level Set Approach According to the level set methodology, the interface is defined implicitly as an iso-surface of a smooth function ψ, and is transported by solving ψ t + (uψ) = 0. (4) Note that since the velocity field is solenoidal, this equation implies that the phase-interface undergoes material transport, in accordance with Eq. 1. In ACLS [5], the level set function is defined as an hyperbolic tangent profile, written ψ (x, t) = 1 ( ( ) ) φ (x, t) tanh + 1, (5) ε where ε is a parameter that sets the thickness of the profile. We set ε to half the characteristic cell size in all simulations in this paper. In the previous equation, φ corresponds to a standard signed distance function, i.e., φ (x, t) = x x Γ, (6) where x Γ corresponds to the closest point on the interface from x, and φ (x, t) > 0 on one side of the interface, and φ (x, t) < 0 on the other side. Finally, a reinitialization equation is introduced in order to preserve the hyperbolic tangent profile. This equation is written ψ τ + (ψ (1 ψ) n) = (ε ( ψ n) n), (7) and is solved in pseudo-time τ at each time-step. This approach is chosen because it combines the accuracy of level set methods with excellent conservation properties. The ACLS methodology, in particular its coupling with the GFM, is described in more details by Desjardins et al.[5]. Discontinuous Galerkin Discretization Discontinuous Galerkin (DG) schemes offer a variety of desirable properties when applied to the conservative level set method. DG allows for arbitrarily high order of accuracy without the necessity of a large stencil resulting in a robust, accurate, and highly scalable scheme. The novel discretization introduced here is presented in two parts: the first deals with the transport of the conservative level set, the second considers the reinitialization equation. The formulation for both equations is derived in a quadrature-free form, where all surface and volume integrals that appear in the derivation are computed a priori in order to reduce computational cost. The hyperbolic tangent level set is conservatively transported with a quadrature-free DG strategy. The main idea of DG is to represent the level set function as a linear combination of basis functions φ n, such as ψ(x, t) = N g n (t)φ n (x), (8) n=0 where N +1 is the number of degrees of freedom and φ n is the basis function with associated weight g n. Although almost any set of basis functions will work, Legendre polynomials have the beneficial property of being orthogonal to each other and are therefore used in this work. To obtain the quadrature-free DG version of the transport equation, we multiply Eq. 4 by each basis function, and integrate by parts over each computational cell, leading to g n t V φ n φ t dv g i u j + (u j ) face (g i ) up N j V S φ t x j φ i dv (φ i ) up φ t ds = 0, (9) using Einstein s summation notation and (g i ) up defined as the i th weight upwinded based on the ve- 3

locity field. ( ) face is defined as quantity ( ) interpolated to the face. It should be noted that an assumption of u j being constant within each cell allows for it to be removed from the integral in the previous equation. This assumption is reasonable for us since we make use of a second order accurate velocity solver. In addition, the surface normal N j is a constant for flat faces found in regular grids. Removing u j and N j from the integrals allows us to write the conservative transport equation in quadrature-free form, i.e. where all the integrals are only a function of the basis functions and not a function of time. As a result, they can be precomputed for a chosen set of basis functions, and then Eq. 9 can easily be updated without evaluating any integrals. Additionally, the equation uses only data from the cell of interest and the faces of neighboring cells. Such a small stencil leads to excellent parallelization properties. The resulting level set representation is illustrated in Fig. 1 for a circle resolved by 5 cells, using second order polynomials. Figure 1. Discontinuous polynomial representation of the conservative level set field. Reinitialization is used to maintain the thickness of the hyperbolic tangent profile and control mass loss. The reinitialization equation presented above, Eq. 7 is solved using DG applied in a similar fashion as described in the previous section. However, the reinitialization equation contains a diffusion term that requires special treatment, and is handled here following the ideas of Lou et al. [6]. This method involves constructing a function over the two cells that contain the face, i.e. R = N r n φn, (10) n=0 where φ n is a scaled basis function that spans the two cells and r n is the weight associated with the scaled basis function. The weights are calculated by imposing the following constraints in a least-squares sense, g i φ φ i t dv = r i φi φt dv, (11a) V V g + i φ φ i t dv = r i φi φt dv. (11b) V + V + Using this approach a unique flux can be determined at each face. For brevity, the details of the derivation of the algorithm are omitted here, and only the final discretized quadrature-free equation is shown: g n τ φ t φ n φ t dv g i n j φ i dv V V x j + (g i ) up (n j ) face N j (φ i ) up φ t ds S φ t + g i g k n j φ i φ k dv V x j (g i g k ) up (n j ) face N j (φ i φ k ) up φ t ds = εg i n j n k V S φ i x j φ t x k dv + εr i (n j n k ) face N k S φ i x j φ t ds. (1) Curvature Computation An accurate curvature scheme is key component of any level set approach. For pde-based level set schemes that rely on a signed distance function, curvature can typically be obtained easily using finite differencing to compute the divergence of the normal vector. However, because the proposed level set is a hyperbolic tangent function, it is significantly more curved in its normal direction than in its tangential directions, making finite differencing very inaccurate. This problem is similar to the challenge faced by volume of fluid (VOF) users, for which it has been shown that using a height function was an appropriate strategy [7]. As a consequence, a height function approach was implemented and tested. It consists of integrating the level set function in its normal direction in order to form a height, which is then fitted using a second order polynomial, leading to a curvature. The convergence of this approach is assessed on a circle discretized with an increasingly finer mesh. The L and L norm of the curvature error are shown in Fig., where it can be observed that the curvature converges with a rate between second and third order accuracy. Scalability The DG conservative level set method relies on a fourth order Runge-Kutta temporal integration. 4

Curvature error 10 0 10-1 10-10 -3 ~ x 3 ~ x 10 100 Mesh size Figure. Convergence of the curvature error on a circle: L norm (dashed line) and L norm (solid line). Because of the combination of an explicit temporal integration scheme and a minimal stencil size, excellent scaling properties are obtained for this method, as illustrated in Fig. 3 for a Zalesak disk test case. Figure 3. Weak scaling test for level set transport and re-initialization. High Density Ratio Correction Transport inconsistencies between density and velocity give rise to numerical difficulties, as spurious errors in momentum can appear, leading to spurious variations in kinetic energy. This problem is exacerbated by the presence of strong shear at the interface and high density ratios, and ultimately causes numerical stability issues. Note that it can affect both level set [8] and VOF methods [9], since these approaches rely on very specific strategies for transporting the phase-interface, which are likely to differ from the way momentum is convected. In the context of VOF, Rudman [10] suggested using VOF density fluxes when calculating the momentum convection term, thereby forcing a discrete compatibility between density and momentum transport. This strategy for dealing with high density ratio flows was then adapted to level set methods by Raessi [11] and Raessi & Pitsch [1], although their work was limited to one- and two-dimensional problems. In this work, we make use of a novel approach for extending this correction to level set methods. This new high-density scheme performs well regardless of the density ratio and shear rate, making it ideal for airblast fuel injection problems. The Navier-Stokes equations are advanced by solving where u un t P n+ 1 k = 1 C n+ 1 = V n+ 1 = ρ n+ 1 1 ρ n+ 1 = P n+ 1 k C n+ 1 V n+ 1, (13) ( u n+ 1 p n+ 1 k, 1 un+ ( µ n+ 1 ), and [ u n+ 1 + 1 un+ T]) are the pressure, convective, and viscous terms, respectively. In order to improve the coupling between momentum convection and level set transport, the convective term is re-cast in a form that includes density, which will be carefully constructed from the level set. The new convection term is obtained by writing the discrete Navier-Stokes equation in terms of the momentum instead of the velocity, giving Ĉ n+ 1 = 1 ( ) ˆρ n+1 ˆρ n+ 1 n+ u 1 1 un+ + 1 ˆρ n+1 u n ˆρ n u n ˆρ n+1. (14) t In the previous expression, the density can vary by several orders of magnitude, which requires special attention. We start by writing ( ) ˆρ n i = ρ g + [ρ] Γ h φ n i, φ n 1 i+, (15) 1 where the height function h is defined by 1 if φ n 0 and φ n 0 i ( ) 1 i+ 1 h φ n i, φ n 0 if φ 1 i+ = n < 0 and φ n < 0 1 i 1 i+ 1 φ n+ i 1 +φ n+ i+ 1 φ n i 1 + φ n i+ 1 otherwise, (16) where a + = max (a, 0). Next, ˆρ n+ 1 u n+ 1 is treated as a density flux for the continuity equation, allowing 5

us to perform simple upwinding, leading to ˆρ n+ 1 ˆρ n i if u n+ 1 0 = i+ 1 i+ 1 otherwise. ˆρ n i+1 (17) Finally, we obtain ˆρ n+1 using ( ) ˆρ n+1 = ˆρ n t ˆρ n+ 1 n+ u 1. (18) This approach ensures a tight coupling between level set and momentum transport, since ˆρ n is defined directly from the level set at each time-step. While this scheme is first-order in both space and time, more accurate schemes can readily be used provided they maintain the boundedness of ˆρ n+1. Finally, note that the extension to three dimensions is straightforward. This momentum correction scheme is evaluated on a simple two-dimensional test case of droplet transport. A droplet of diameter D is placed in the middle of a periodic computational domain of size 5D 5D. Various meshes are considered, ranging from 3 to 18. The velocity field is initialized by giving the liquid a velocity u l = 1 and v l = 0, while the gas is initially at rest. The time-step size is chosen such that the convective CFL remains below 0.. Both gas and liquid viscosities, as well as the surface tension coefficient, are set to zero. Finally, the density ratio between the liquid and the gas is set to 10 6. Under these conditions, the droplet is expected to remain perfectly circular. However, the non-corrected scheme becomes unstable and fails in less than a third of a flow-through time regardless of the mesh refinement or time-step size. In comparison, the density-corrected scheme runs robustly with all meshes for any number of flow-through times. Some deformation of the drop is visible, although it does not appear to increase significantly with time. Figure 4 shows the phase-interface after one flowthrough time for the various meshes, compared to the exact solution. The drop shape converges satisfyingly toward the exact solution. Simulation of n-dodecane Air-Blast Injection An air-blast atomizer was designed after the one described by Marmottant and Villermaux [13]. The simple, externally mixed geometry is well-suited for numerical modeling and code validation. The injector geometry is detailed in Fig. 5 and consists of a straight jet of diameter d 1 surrounded by a co-flow of inner diameter d and thickness h. The length of the injector is not shown in the sketch but it is such that the flow leaving the nozzle is fully developed. n-dodecane was injected with a co-flow of nitrogen, their respective properties are shown in Table 1 Figure 4. Droplet shape after one flow-through time using the density-based momentum flux correction scheme. The arrow indicates mesh size increasing from 3 to 64 to 18, and the thick line is the exact solution. Figure 5. Air-blast injector dimensions. with the subscript l for liquid n-dodecane and g for nitrogen gas. ρ Density l 746 kg/m 3 ρ g 1.5 kg/m 3 Surface Tension σ.535 10 N/m µ Dynamic Viscosity l 1.34 10 3 kg/m s µ g 1.718 10 5 kg/m s Table 1. Properties of n-dodecane and nitrogen. The flow properties are characterized by nondimensional numbers that describe the relative importance of viscosity, inertia, and surface tension for both the gas and liquid phases. Four nondimensional numbers are calculated for this test 6

case: Re l = ρ lu l d 1 µ l = 1336, (19a) Re g = ρ gu g h µ g = 1453, (19b) We l = ρ lu l d 1 σ We g = ρ gu gd 1 σ = 17, (19c) = 31. (19d) the break-up mechanism is captured in the simulation. Both the flow of n-dodecane and the co-flow of N are laminar. The liquid is issued with a bulk velocity of 1.8 m/s, while the gas has a bulk velocity of 70 m/s. The atomization simulation was performed on 1,04 processors using a mesh of 51 56 56 grid cells. A CFL number below 0.9 was maintained throughout the simulation. Roughly 1.5 flow through times were used to allow the jet to reached a statistically steady state. Mass conservation errors are monitored during the jet development, until liquid starts exiting the domain (at which point the expected liquid mass in the domain is unknown), and are found to remain below 0.7%. Primary atomization under the flow parameters described above results in an initially smooth jet that rapidly breaks up into droplets. Figure 7 shows a side-by-side comparison of snapshots from the experiment and the simulation. Qualitatively, there is excellent agreement in the shape of the jet, the length-scales of instabilities, the onset of liquid break-up, and the distribution of large droplets. There appear to be smaller droplets in the experiment that are not found in the simulation result. It is postulated that such small droplets are formed when liquid accumulates on the outside of the nozzle and the jet interacts with this liquid. Drops were identified and characterized in experiments and simulations since the size of droplets produced by primary atomization is an important result for combustion-related applications. Simulations used a band-growth algorithm [14] to identify droplets and compute their size and velocity. The drop size pdf shown in Fig. 6(a) illustrates the excellent agreement in the size of droplets between simulations and experiments, suggesting that the simulations are capable of accurately predicting the breakup dynamics and could be used to predict drop sizes for design applications. In addition to drop size distributions, droplet velocity distributions were also calculated. Figure 6(b) shows probability density functions of droplet axial velocity. Again, excellent agreement is found between the experiment and simulation, indicating that the droplets are forming with the correct velocity, which suggests in turn that (a) Drop sizes. (b) Drop axial velocities. Figure 6. Comparison of probability density function of (a) drop size and (b) drop velocity bewteen experiments and simulations. Conclusions A discontinuous Galerkin conservative level set method and a high density ratio correction scheme have been developed and coupled with a Navier- Stokes solver, allowing for simulations of primary atomization at relevant flow conditions. The new level set scheme is an extension of the accurate conservative level set method [1] using a quadraturefree discontinuous Galerkin discretization leading to a highly accurate and scalable scheme, while ensuring excellent mass conservation properties. The high density ratio correction provides a simple way of modifying the Navier-Stokes equations to improve the consistency between momentum convection and level set transport, leading to a scheme that performs well at any density ratio or shear rate. Using these numerical methods, a simulation of air-blast fuel injection was performed and compared 7

to experiments. In addition to excellent qualitative agreement, probability density functions of drop size and drop velocity are found to agree, suggesting that the mechanisms leading to break-up are accurately captured in the simulation. References [1] O. Desjardins, G. Blanquart, G. Balarac, and H. Pitsch. J. Comp. Phys., 7(15):715 7159, 008. [] Ronald P. Fedkiw, Tariq Aslam, Barry Merriman, and Stanley Osher. J. Comput. Phys., 15():457 49, 1999. [3] M. Sussman, K. M. Smith, M. Y. Hussaini, M. Ohta, and R. Zhi-Wei. J. Comput. Phys., 1():469 505, 007. [4] Olivier Desjardins and Heinz Pitsch. J. Comput. Phys., 8(5):1658 1677, 009. [5] O. Desjardins, V. Moureau, and H. Pitsch. J. Comp. Phys., 7(18):8395 8416, 008. [6] Hong Luo, Luqing Luo, Robert Nourgaliev, Vincent A. Mousseau, and Nam Dinh. J. Comp. Phys., 9(19):6961 6978, September 010. [7] Marianne M. Francois, Sharen J. Cummins, Edward D. Dendy, Douglas B. Kothe, James M. Sicilian, and Matthew W. Williams. Journal of Computational Physics, 13(1):141 173, 006. [8] James Albert Sethian. Level set methods and fast marching methods: evolving interfaces in... Cambridge University Press, 1999. [9] Ruben Scardovelli and Stephane Zaleski. Annu. Rev. Fluid Mech., 31(1):567 603, 1999. [10] Murray Rudman. Int. J. Numer. Meth. Fl., 4(7):671 691, 1997. [11] M Raessi. CTR An. Res. Briefs, p. 467, 008. [1] M Raessi and P Pitsch. CTR An. Res. Briefs, p. 156, 009. [13] P. Marmottant and E. Villermaux. J. Fluid Mech., 498(-1):73 111, 004. [14] M. Herrmann. J. Comp. Phys., 9(3):745 759, February 010. 8

(a) Photo of experiment. (b) Rendering of simulation result. Figure 7. Comparison of jet from (a) experiment and (b) simulation. 9