Large-eddy simulation of jet mixing in a supersonic turbulent crossflow

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1 Center for Turbulence Research Annual Research Briefs 8 9 Large-eddy simulation of jet mixing in a supersonic turbulent crossflow By S. Kawai AN S. K. Lele. Motivation and objectives Important recent research efforts have focused on the application of large-eddy simulation (LES) to compressible turbulent flows. The engineering motivation for compressible LES is to provide a realistic turbulent flowfield and to elucidate unsteady phenomena of such interest, mixing, combustion, heat-transfer, sound-generation, unsteady load, etc. Jet mixing in a supersonic crossflow (JISC) is a type of flow where compressible LES can play an important role in understanding the detailed physics of the turbulent mixing mechanisms and predicting the flowfield. Inside a supersonic combustor, due to the limited flow residence time, the enhancement of supersonic turbulent mixing of jet fuel and crossflow air is a critical issue in developing supersonic air-breathing engines. Typical flow structures resulting from a sonic under-expanded transverse jet injection into a supersonic crossflow are illustrated in Fig. (Ben-Yakar et al. 6; Gruber et al. 995). An under-expanded jet expands through a Prandtl-Meyer fan at the lip of the jet orifice before the jet flow is compressed by a barrel shock and Mach disk. Thus the flowfield involves complex - unsteady shocks, contact surfaces, turbulence and their interactions. Accurately simulating the flows involving these interactions is a significant challenge because numerical algorithms need to satisfy two contradictory requirements: the scheme needs to capture different types of discontinuities and also simultaneously resolve the scales of turbulence. LES and detached-eddy simulation have been performed and showed some large-scale vortex structures (von Lavante et al. ; Peterson et al. 6), though turbulent eddies are under-resolved. This is primarily because of the conventional low-order upwinding finite volume schemes employed in the simulations. These schemes work well in the sense of discontinuity capturing but are too dissipative for LES to properly resolve the scales of turbulence. It is important for LES not to damp turbulence artificially. Clearly, LES of the supersonic jet mixing presents challenges for simultaneously capturing flows with complex unsteady shocks and contact surfaces and resolving the broadband turbulent eddying motions present in high-reynolds number flows. In this paper, high-order compact differencing/filtering schemes (Lele 99; Gaitonde & Visbal ) are coupled with recently developed localized artificial diffusivity methodology (Kawai & Lele 8a) in the context of LES to obtain insights into the physics of an under-expanded sonic jet injection into a supersonic turbulent crossflow. JISC with a laminar boundary layer is first discussed. Key physics of the jet mixing in a supersonic crossflow are highlighted from the observations of instantaneous flowfields. Then, the effects of approaching turbulent boundary layer on the mixing mechanisms are discussed by comparing with the laminar and turbulent crossflow cases. The flow condition examined here is based on the experiment of Santiago & utton (997) and the numerical results are compared with the experimental data. A series of mesh refinement studies is also performed to verify the numerical scheme.

2 4 S. Kawai and S. K. Lele Figure. Schematics of the transverse injection of an under-expanded jet into a supersonic crossflow (Ben-Yakar etal. 6; Gruber etal. 995).. Mathematical models The compressible Navier-Stokes equations for an ideal non-reactive gas are: ρu t ρ + (ρu) =, (.) t + (ρuu + pδ τ ) =, (.) E + [Eu + τ ) u κ T] =, t (pδ (.) ρy k + (ρuy k ) (ρ k Y k ) =, t (.4) E = p γ + ρu u, p = ρrt, (.5) where ρ is the density, u is the velocity vector, p is the static pressure, E is the total energy, T is the temperature, γ (=.4: air) is the ratio of specific heats, R is the gas constant, κ is the thermal conductivity, δ is the unit tensor. Equation (.4) is the transport equation for passive scalar of jet fluid Y k to understand the mixing mechanisms between the jet and crossflow where k is the species diffusion coefficient. The viscous stress tensor τ is = µ(s ) + (β τ µ)( u)δ, (.6) where µ is the dynamic (shear) viscosity, β is the bulk viscosity, and S is the strain rate tensor, S = ( u + ( u)t )... Numerical schemes Spatially filtered Navier-Stokes equations in conservative form are solved in generalized curvilinear coordinates, where spatial derivatives for convective terms, viscous terms, metrics and Jacobian are evaluated by a sixth-order compact differencing scheme (Lele 99). Because of their spectral-like resolution and non-dissipative characteristics, highorder compact differencing schemes are an attractive choice for LES to properly resolve the scales of turbulence. At boundary points and and correspondingly at imax- and imax, second- and fourth-order one-sided formulas are utilized that retain the tridiagonal form of the equation set (Gaitonde & Visbal 998). Symmetric Gauss-Seidel alternate directional implicit factorization scheme with three multiple subiterations (Newton-Raphson iteration) is used for time integration. The im-

3 LES of jet mixing in a supersonic crossflow 4 plicit scheme uses the idea of four-factored symmetric Gauss-Seidel (Fujii 999) that is based on alternate directional implicit factorization with the sweep procedure as a kind of symmetric Gauss-Seidel relaxation. The implicit portion of the algorithm uses secondorder accurate three-point backward differencing for the derivative. In the present study, the computational time step is approximately t= sec ( t c s /=.6) at which the maximum Courant-Friedrichs-Lewy (CFL) number is less than., where c s is the speed of sound of the freestream, and is the diameter of the nozzle exit... Localized artificial diffusivity When central differencing schemes, such as high-order compact differencing schemes, are applied to solve flows that involve steep gradients, such as shock waves, contact surfaces or material discontinuities, non-physical spurious oscillations that make the simulation unstable are generated. A key issue here is how to properly remove the non-physical spurious oscillations without damping the resolved scales of turbulence. Our discontinuity-capturing scheme, localized artificial diffusivity scheme, is based on adding grid-dependent artificial fluid transport coefficients to the coefficients appearing in Eqs. (.), (.4) and (.6) proposed by Cook (7), µ = µ f + µ, β = β f + β, κ = κ f + κ, k = f,k + k, (.7) where the f subscripts and asterisks denote fluid and artificial transport coefficients. The artificial fluid transport coefficients are designed to automatically vanish in smooth well-resolved regions and provide damping in non-smooth unresolved regions to capture different types of discontinuity. We model the localized artificial diffusivity on a multi-dimensional generalized coordinate system defined by Kawai & Lele (8a): µ = C µ ρ l= k = C c s l= r S ξ r l r Y k ξ r l, β = C β ρ r S ξ r ξl r l l= l, ρc s κ = C r e κ T ξ r ξl r l l= l, ξl r l + C Y c s [Y k ]H(Y k ) Y k [ H(Y k )] Yk, (.8) ξ r l l where C µ, C β, C κ, C and C Y are user-specified constants, S is the magnitude of the strain rate tensor, c s is sound speed, e is internal energy defined by e = p γ ρ, and H is the Heaviside function. If r is sufficiently high, the high-wavenumber biased (k r ) artificial diffusivity is only activated in the region where needed, to capture discontinuities using high-order compact differencing schemes. The overbar denotes an approximate truncated- Gaussian filter (Cook & Cabot 4). In the limit of l, Eq. (.8). Therefore, the governing equations are consistent with the original Navier-Stokes equations. In the present study r=4 is adopted in Eq. (.8). The fourth derivatives, 4 S, 4 e ξ 4 ξ 4 l l and 4 Y k, are evaluated by the fourth-order explicit scheme at interior points. The userspecified constants are set to C µ =., C β =, C κ =., C =. and C Y = (Kawai ξ 4 l & Lele 8a). These constants allows the scheme to capture discontinuities and not to introduce excessive dissipation. Although the constants work well for a wide range of test cases when r = 4 in Eq. (.8) and sixth-order compact/eighth-order filtering schemes are used, the constants will possibly need to be adjusted if different value of r or other numerical schemes are adopted. The method captures the discontinuities on curvilinear and anisotropic meshes with

4 4 S. Kawai and S. K. Lele minimum impact on the smooth flow regions. The comparisons between the proposed method and high-order shock-capturing schemes illustrate the advantage of the method for the simulation of flows involving shocks, turbulence and their interactions. A complete discussion of the localized artificial diffusivity scheme may be found in Kawai & Lele (8a). Since the LES of a sonic jet in a supersonic crossflow involves a wall-bounded viscous flow simulation, the following van riest wall damping function multiplies the grid spacing ( l ) in the localized artificial diffusivity scheme to force the artificial viscosity to vanish in the near wall portion of the boundary layers, f vd = exp y+ A +, µ = C µ ρ r S ξl r l= ξ r l l f vd, β = C β ρ r S ξl r l= ξ r l l f vd,(.9) where the van riest constant A + is 6. In the present study no explicit subgrid models are used. The localized artificial diffusivity and filtering act to suppress near grid scale fluctuations. The performance of such an implicit approach to LES is the subject of further investigation... Flow configuration The flow condition examined in this study is based on the experiments of Santiago & utton (997) in order to validate the present LES. The computation uses M =.6, Re =.4 4. The density and pressure ratios between the nozzle chamber and crossflow are ρ j /ρ =5.55 and p oj /p =8.4. Based on these flow conditions, the resulting jet-to-crossflow momentum flux ratio is J=.7. Only the Reynolds number differs from the experiment; the value is six times lower than in the experiment to maintain the LES resolution requirement under currently acceptable computational costs. Although the Reynolds number is not matched, the upstream boundary layer thickness, δ 99 / =.775 (. mm), is matched at = -5. Figure shows the grid geometry of the computational domain employed in the present study. Every fifth grid point on the coarse mesh is presented in the figure. Overset grids consist of three structured grids, background, circular nozzle and nozzle axis grids. The nozzle axis grid covers the singular line in the nozzle grid. The geometry of the nozzle matches the experiment (Santiago & utton 997). The computational test section of the background mesh extends from the center of the nozzle exit to 5 upstream and downstream in the streamwise direction (=-5 to 5), on both sides in the spanwise direction (z/=- to ) and. in the wall-normal direction (y/= to.). In the focused region, a uniformly spaced grid is adopted in streamwise and spanwise directions. In the wall-normal direction, the grid is clustered near the wall in the region y/= to. and then a uniformly spaced grid is used for y/=. to.. Sponge layers with the lengths of, and are placed at the outlet, both sides and the upper boundaries. Three levels of mesh refinement by the factor of in each direction are conducted on the background mesh. Table summarizes the number of grid points and the grid resolutions for the background meshes. The grid resolution in wall units is based on the wall friction coefficient at =-5 measured in the experiment and for the reduced Reynolds number used in this study. The number of grid points for the nozzle mesh and nozzle axis mesh are and in the ξ, η, ζ direction, respectively.

5 LES of jet mixing in a supersonic crossflow 4 Grid N ξ N η N ζ + in ξ + in η + in ζ Fine Medium Coarse Table. Computational grids for background mesh. 5 5 η ξ ζ. ξ. grid points inside wall η Figure. Computational grids (every fifth grid point on coarse mesh). Side view at z/= on left, close-up view near nozzle exit in the middle, and top view on right. ξ The grid resolution for the circular nozzle and nozzle axis meshes is designed based on the fine background mesh. Communication between the grids is handled through a two-point fringe at boundaries by using sixth-order Lagrangian interpolation (Sherer & Visbal ). The bottom boundary of the nozzle is set to nozzle chamber conditions. The solid wall boundary condition is treated as a non-slip adiabatic wall. A characteristic boundary condition is applied to the upper, side and outlet boundaries. Large sponge layers are introduced around these boundaries to remove turbulent fluctuations and their reflection from the boundaries.. Numerical results... Statistics.. JISC with laminar boundary layer In this JISC with laminar incoming boundary layer case, mean physical properties at the crossflow inlet boundary at =-5 on the background mesh are fixed at the profiles obtained from a - laminar computation. Although the boundary thickness at the station matches that of the experiment (.775:. mm), the experimental velocity measurement at =-5 possesses a turbulent boundary layer. Simulated time-averaged flowfield shows a pair of counter-rotating vortices as often discussed in the literature and most of the jet fluid passes through the barrel shock and Mach disk and then turns downstream (not shown here). Further discussions regarding the time-averaged flowfield may be found in Kawai & Lele (8b). Comparisons of streamwise and wall-normal velocities between the LES with three levels of mesh refinement and experimental data at midline plane z/= and downstream stations = and 4 are shown in Fig.. A qualitative comparison can be made by comparing the LES with the experiment, although precise quantitative agreement with the experiment cannot be expected because of the lower Reynolds number used in the LES and uncertainty involved in the experiment. There are discrepancies between the

6 44 S. Kawai and S. K. Lele y / y / y / y / y / y / y / y / - (a) Fine mesh (b) Medium mesh - (c) Coarse mesh (d) Experiment y/.5 y/.5 y/.5 y/ U/U V/U (e) U/U and V/U at =.5 U/U V/U (f) U/U and V/U at =4 Figure. Comparisons of streamwise (top) and wall-normal (middle) velocities between LES and experiment (Santiago & utton 997) at midline plane z/= and downstream stations = and 4. Blue line, fine mesh; red line, medium mesh; green line, coarse mesh;, experiment. LES and experiment in the recirculating regions upstream and downstream of the jet. The LES shows larger recirculating regions than the experiment. One possible reason for the disagreement between the LES results and experiment is the presence of a laminar boundary layer upstream of jet injection in LES (discussed in Sec..). However, overall the locations of the shock structures and jet development downstream agree reasonably well with the experiment. Three velocity components at cross-view planes = and 5 also show good agreements with the experiment (although not shown here). The three levels of mesh refinement show the reasonable convergence of the time-averaged flow quantities. Turbulent kinetic energy (TKE) distributions, ( u u + v v + w w )/U, and instantaneous snapshots of density gradient magnitude at midplane obtained by the three levels of mesh refinement are shown in Fig. 4. Three high TKE regions in the midline plane correspond to the regions where the turbulent structures are observed at the windward and leeward boundaries of the jet and under the leeward jet boundary. The jet fluid is progressively diluted in the regions where high T KE is observed. Although overall T KE distributions are not changed between the three meshes, the higher intensity of T KE is obtained by refining the mesh. This is primarily because the turbulent structures are highly well-resolved by the finer mesh as shown in the instantaneous snapshots of density gradient magnitude.

7 LES of jet mixing in a supersonic crossflow 45 y / y / y / y / y / y / (a) Fine mesh (b) Medium mesh (c) Coarse mesh Figure 4. Turbulent kinetic energy distributions (top) and instantaneous snapshots of density gradient magnitude (bottom) with three levels of mesh refinement at midline. y / y / z / z / Figure 5. Instantaneous snapshots of density gradient magnitude (left) and jet fluid (right) at midline plane z/= (top) and wall-parallel plane y/= (bottom) obtained by fine mesh.... Instantaneous flowfields Figure 5 shows instantaneous snapshots of density gradient magnitude on the left and passive scalar Y k of jet fluid on the right obtained by the fine mesh. Side-view and top-view planes are at z/= and y/= planes. The high-order compact differencing scheme with localized artificial diffusivity methodology captures the - unsteady front bow shock, barrel shock, Mach disk and contact surface without spurious wiggles and also simultaneously resolve the turbulent structures. The progressive mesh refinement and the high-resolution scheme allow for proper resolution of these turbulent eddies. Most of the jet fluid passes through the barrel shock and Mach disk; jet mixing progressively occurs after the jet fluid passes through the shocks. The vortex structures that create hairpin-like structures in the windward and leeward jet boundaries break down to finer turbulent structures in the downstream. The turbulent structures play an important role in determining the behavior of jet fluid stirring and subsequent mixing. Figure 6 shows time-series pressure data inside the upstream recirculation region and

8 46 S. Kawai and S. K. Lele p/p y / y / Time steps (x ) 4 A y / y / y / y / B - - Figure 6. Time-series pressure data inside upstream recirculation region and representative time-series snapshots of jet fluid (passive scalar Y k ) overlapped with divergence of velocity contours at midline plane z/= obtained by fine mesh. - - representative time-series snapshots of the jet fluid overlapped with divergence of velocity contours at midline plane. The divergence of velocity contours represent shock structures. The pressure history data are obtained at location A as shown in snapshot. These six snapshots are taken at the time corresponding to the numbered markers in the pressure history. The simulated unsteady flowfield illustrates that the pressure fluctuation inside the recirculation region induces the large-scale dynamics of barrel shock and bow shock deforming and accompanies vortex formation. The barrel shock shows a kink in the timeseries images. Corresponding to the pressure rise from time to in the pressure history, the degree of expansion at the windward side of the nozzle edge reduces in order to keep the pressure balance across the jet boundary. Because of the lower Mach number distribution along the jet boundary at time, the jet shear layer is able to support the rapid growth of instability waves and starts to fluctuate. By way of contrast the jet shear at time does not show such fluctuations. Once the jet shear starts to fluctuate, a local shock wave appears within the jet because of the blockage of the supersonic jet by the deflected shear layer. Then, the local shock grows and connects to the original barrel shock, making a kink in the barrel shock as shown in snapshot 4. Once the kink appears, it moves downstream with an accompanying large-scale vortex and the jet shear layer rapidly deflects along the shock. The large-scale vortex entrains the crossflow, which stirs the jet fluid and enhances subsequent jet mixing. uring the large-scale dynamics, the acoustic wave is generated and propagates upstream; it interacts with the bow shock as shown in snapshot 6 location B. The interaction causes large-scale unsteady oscillation of the bow shock. Although the processes of the large-scale dynamics are discussed with reference to a certain time window, these dynamics are repeated numerous times during the LES time history when the pressure inside the upstream recirculation region becomes high. Similar large-scale dynamics of the barrel shock, bow shock and accompanied largescale vortex are also observed by the experiment (VanLerberghe, Santiago, utton & Lucht ; Ben-Yakar et al. 6)... JISC with turbulent boundary layer In order to investigate the effects of the approaching turbulent boundary layer on the mixing mechanisms, LES of JISC with an incoming turbulent boundary layer is discussed

9 LES of jet mixing in a supersonic crossflow 47 Rescale-reintroduction Extract for JISC inflow 7 Figure 7. JISC with incoming supersonic turbulent boundary layer U V 5 u, v, w y y/δ Figure 8. Mean and fluctuation (upper curve, u rms/u τ; lower curve, v rms/v τ; middle curve, w rms/w τ;) velocities. Blue line, fine mesh; red line, coarse mesh;, U + V = y+ and U + V =.5 log(y+ ) + 5.5;, NS by Spalart (988). in this section by comparing with the laminar crossflow case. Figure 7 shows how we simulate the JISC with turbulent boundary layer. This LES is basically the coupled simulation between the LES of JISC and supersonic turbulent boundary layer (STBL). The inflow conditions for the LES of JISC are extracted from the concurrently simulated LES of STBL. A compressible extension of the rescaling-reintroducing method (Urbin & Night ) is used to generate the inflow conditions for the LES of STBL. The domain size for the LES of STBL is 7 and.4 in streamwise and wall-normal directions and the spanwise domain size is matched to the JISC case. The LES of STBL has been performed on the coarse ( + = 9) and fine ( + = 5) meshes based on the same mesh distributions used in JISC as shown in Table. Figure 8 shows the mean and fluctuating velocity profiles. The LES on the fine mesh is in good agreement with a fully developed turbulent boundary layer profile with a logarithmic region and with the NS data by Spalart (988), whereas the coarse mesh shows a shift in a logarithmic region and over-predicts a peak in u + as usually observed by an unresolved LES. Also, the result illustrates the capability of the current compact differencing/filtering schemes with localized artificial diffusivity methodology for simulating the supersonic turbulent boundary layer in the context of LES.... Statistics Comparisons of the streamwise velocity between the LES with turbulent/laminar crossflow and experimental data at midline plane z/= and downstream stations =,

10 48 S. Kawai and S. K. Lele y / (a) Turbulent crossflow y / (b) Laminar crossflow y / (c) Experiment y/.5 y/.5 y/ U/U (d) =.5 U/U (e) =.5 U/U (f) =4 Figure 9. Comparisons of streamwise velocity between LES with turbulent/laminar crossflow and experiment (Santiago & utton 997) at midline plane z/= and downstream stations =, and 4. Blue line, turbulent crossflow; red line, laminar crossflow;, experiment. and 4 are shown in Fig. 9. The results discussed here for the LES of JISC with turbulent boundary layer are obtained on the coarse mesh ( + = 9). The LES with finer mesh will be conducted in the near future. As expected, the turbulent crossflow suppresses the upstream and downstream separating regions and smaller recirculations are obtained compared to the laminar crossflow case. The upstream separating region obtained by the turbulent crossflow shows reasonable agreement with the experiment. Quantitative comparisons of the velocity at the downstream stations =, and 4 also show better agreement with the experiment by applying the turbulent crossflow. In order to evaluate the effects of the approaching turbulent boundary layer on the jet mixing, the RMS of the jet fluid tracer fluctuations and mean jet fluid distributions obtained by the turbulent and laminar crossflow conditions are shown in Fig.. As clearly shown, the high-intensity region of the RMS of jet fluid fluctuations obtained by the turbulent crossflow is observed in the upper part of the windward jet boundary where the jet fluid is progressively diluted. On the other hand, the laminar crossflow case shows the high intensity in the lower part of the windward jet. Therefore there is a clear difference in the region of jet mixing between the turbulent and laminar crossflows. JISC with the turbulent boundary layer shows progressive jet mixing in the windward jet boundary compared with the laminar crossflow case.... Instantaneous flowfields Figure shows a comparison between the turbulent and laminar supersonic crossflow conditions in the instantaneous snapshots of density gradient magnitude and passive scalar Y k of jet fluid. In the turbulent crossflow case, upstream turbulent structures interact with the bow shock and windward jet boundary. These interactions enhance the instability of the windward jet shear layer and support a more rapid breakdown in the jet shear structure to the turbulent state. As a result, the jet fluid is stirred with the ambient fluid entrained into the flow and subsequently mixing is enhanced in the

11 LES of jet mixing in a supersonic crossflow 49 y / y / y / y / (a) Turbulent crossflow (b) Laminar crossflow y/.5 y/.5 y/.5 y/ Scalar rms Scalar (c) Y RMS and Y at = Scalar rms (d) Y RMS and Y at = Figure. Comparisons of RMS of jet fluid tracer fluctuations (top) and time-averaged jet fluid distributions (middle) between turbulent (right) and laminar (left) supersonic crossflow conditions at midline plane z/= and downstream stations = and 4. Blue line, turbulent crossflow; red line, laminar crossflow. Scalar y/ y/ y/ y/ Figure. Comparisons of turbulent (right) and laminar (left) supersonic crossflow conditions in instantaneous snapshots of density gradient magnitude (top) and jet fluid (bottom). windward jet boundary where high intensity of the jet fluid fluctuations and progressive jet dilution are observed. The results illustrate the importance of turbulent structures in the upstream boundary layer on the jet-mixing process.

12 5 S. Kawai and S. K. Lele 4. Conclusions and future work High-order compact differencing/filtering schemes have been coupled with recently developed localized artificial diffusivity methodology in the context of large-eddy simulation to obtain additional insights into the complex - flow physics of an under-expanded sonic jet injection into a supersonic turbulent crossflow. The interaction between the turbulent structures in the upstream boundary layer and jet shear layer induces the instability of the windward jet shear layer and enhance the jet mixing. Key physics of the jet mixing in a supersonic crossflow have been highlighted. Pressure fluctuations inside the recirculation region induce large-scale unsteady dynamics of the barrel shock and bow shock deformation and accompanies vortex formation that entrains the crossflow and enhances subsequent windward jet mixing. The present LES qualitatively reproduce the unsteady dynamics of both barrel shock and bow shock as observed in the experiment. Statistics obtained by the LES also show good agreement with the experiment. Three levels of progressive mesh refinement in the laminar crossflow case showed a convergence in mean flow quantities but didn t show a complete convergence in fluctuating flow quantities although the overall distributions have not been changed. Future work includes the further investigation of the effects of the approaching turbulent boundary layer on the mixing mechanisms including a series of mesh refinement study. The issue of the performance of implicit subgrid scale modeling used in this study is ongoing research. Beside the non-reactive case, progress toward the LES of reactive JISC is also the subject of further investigation. Acknowledgements This work is supported by the Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative (MURI) Program. The authors gratefully acknowledge Professor J. G. Santiago and Professor J. C. utton for providing extensive experimental data and r. Eric Johnsen for comments on the manuscript. Present code is based on an extension to the code FLI provided by r. M. R. Visbal, whom we thank for this. REFERENCES Ben-Yakar, A., Mungal, M. G. & Hanson, R. K. 6 Time evolution and mixing characteristics of hydrogen and ethylene transverse jets in supersonic crossflows. Physics of Fluids. 8 (), 6. Cook, A. W. & Cabot, W. H. 4 A high-wavenumber viscosity for high-resolution numerical method. J. of Comp. Phys. 95 (), Cook, A. W. 7 Artificial fluid properties for large-eddy simulation of compressible turbulent mixing. Phys. of Fluids 9 (5), 55. Fujii, K. 999 Efficiency improvement of unified implicit relaxation/time integration algorithms. AIAA J. 7 (), 5 8. Gaitonde,. V. & Visbal, M. R. Padé-type higher-order boundary filters for the Navier-Stokes equations. AIAA J. 8 (),. Gaitonde,. V. & Visbal, M. R. 998 High-order schemes for Navier Stokes equations: algorithm and implementation into FLI. AFRL-VA-WP-TR Gruber, M. R., Nejad, A. S., Chen, T. H. & utton, C. J. 995 Mixing and

13 LES of jet mixing in a supersonic crossflow 5 penetration studies of sonic jets in a Mach freestream. J. of Propul. Power. (), 5. Kawai, S. & Lele, S. K. 8a Localized artificial diffusivity scheme for discontinuity capturing on curvilinear meshes. J. of Comp. Phys. 7 (), Kawai, S. & Lele, S. K. 8b Mechanisms of jet mixing in a supersonic crossflow: a study using large-eddy simulation. AIAA Paper von Lavante, E., Zeitz,. & Kallenberg, M. Numerical simulation of supersonic airflow with transverse hydrogen injection. J. of Propul. Power. 7 (6), 9 6. Lele, S. K. 99 Compact finite difference schemes with spectral-like resolution. J. of Comp. Phys. (), 6 4. VanLerberghe, W. M., Santiago, J. G., utton, C. J. & Lucht, R. P. Mixing of a sonic transverse jet injected into a supersonic flow. AIAA J. 8 (), Peterson,. M., Subbareddy, P. K. & Candler, G. V. 6 Assessment of synthetic inflow generation for simulating injection into a supersonic crossflow. AIAA Paper Santiago, J. G. & utton, C. J. 997 Velocity measurements of a jet injected into a supersonic crossflow. J. of Propul. Power. (), Sherer, S. E. & Visbal, M. R. Computational study of acoustic scattering from multiple bodies using a high-order overset grid approach. AIAA Paper. Spalart, R. P. 988 irect simulation of a turbulent boundary layer up to R θ =4. J. Fluid Mech. 87, Urbin, G. & Knight,. Large-eddy simulation of a supersonic boundary layer using an unstructured grid. AIAA J. 9 (7),