AIAA th AIAA Aerospace Sciences Meeting & Exhibit 8-11 January 2001 / Reno, NV

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1 AIAA A Comparison of Turbulence Models for a Supersonic Jet in Transonic Crossflow Jeffrey L. Payne, Christopher J. Roy, and Steven J. Beresh Sandia National Laboratories Albuquerque, NM 39th AIAA Aerospace Sciences Meeting & Exhibit 8-11 January 21 / Reno, NV

2 A Comparison of Turbulence Models for a Supersonic Jet in Transonic Crossflow Jeffrey L. Payne, Christopher J. Roy, and Steven J. Beresh Sandia National Laboratories P. O. Box 58 Albuquerque, NM Abstract Numerical simulations of a supersonic jet in a subsonic compressible crossflow are conducted using three turbulence models. The numerical results are compared to existing experimental data. The comparisons with the experiment include separation points, reattachment points, and surface pressures in the near jet region as well as mean total pressure and velocity measurements at 1 and 4 diameters downstream of the jet. The simulations employ a finite volume Navier-Stokes code on structured multi-block grids. The turbulence models considered in this study include the Spalart-Allmaras one-equation model, a low Reynolds number k-ε model and the Wilcox k-ω model. The Spalart-Allmaras model gave poor results in the near-field of the jet, but showed excellent agreement with the downstream total pressure and vorticity data. The two-equations models, while giving good results in the jet near-field, tended to over predict both the strength and the lift-off height of the jetinduced vortex pair. Nomenclature γ ratio of specific heats C p pressure coefficient = (p-p c )/(γp c M 2 c / 2) D turbulence diffusion term, nozzle exit diameter (D=3 mm) Principal Member of Technical Staff, Mail Stop 825, E- mail: jlpayne@sandia.gov, Member AIAA Senior Member of Technical Staff, Mail Stop 835, cjroy@sandia.gov, Member AIAA Senior Member of Technical Staff, Mail Stop 825, sjberes@sandia.gov, Member AIAA * Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC4-94AL85. This paper is declared a work of the U. S. Government and is not subject to copyright protection in the United States. d distance to solid surface, m k specific turbulent kinetic energy, m 2 /s 2 M Mach number p pressure, N/m 2 γ γ 1 p t total pressure p M 2 ( γ 1) =, N/m 2 γ Ω vorticity = V, 1/s Ω x X axis component of vorticity, 1/s S P turbulence production source term S D turbulence destruction source term S ij strain rate tensor, 1/s T temperature, K Tu percent freestream turbulence intensity t time, s X,Y,Z cartesian coordinates, m U velocity magnitude, m/s u i velocity in the i th direction, m/s δ ij Kronecker delta function (= 1 when i=j) ε specific turbulent dissipation rate, m 2 /s 3 µ absolute viscosity, N s/m 2 ν kinematic viscosity, m 2 /s νˆ Spalart-Allmaras working variable, m 2 /s ρ density, kg/m 3 τ turbulent stress tensor, m 2 /s 2 ij ϕ generic turbulent transport quantity Ω ij rotation tensor, 1/s ω specific turbulence frequency, 1/s Subscripts c crossflow value eff effective value (turbulent + laminar) i,j indices for tensor notation j jet value t turbulent quantity or stagnation value freestream value 2

3 AIAA 99-xxxx Superscripts + quantity in wall coordinates ~ denotes Favre (density-weighted) averaging denotes Reynolds (time-based) averaging denotes Favre fluctuating quantity Introduction The introduction of a transverse supersonic jet into the flow over the surface of a transonic flight vehicle generates a number of complex fluid structures. A region of separated flow is created by the pressure rise as the crossflow stagnates at the front of the jet and a low pressure separated region forms in the jet s wake. Shock waves are also present in the form of a Mach disk within the jet plume and, depending upon the freestream Mach number, a separation shock upstream of the jet and critical shocks along its sides. These latter shocks induce the additional complications associated with shock/boundary layer interactions. The far-field of the jet is dominated by a counter-rotating vortex pair created in its wake, which is carried along just beneath the jet and retains its strength even as the jet itself dissipates. Turbulent mixing and entrainment issues accompany the presence of the dissipating jet and the vortex pair. One of the major difficulties in simulating these complex flowfields is the accurate modeling of the effects of the turbulent fluctuations. The proper modeling of these turbulent fluctuations with Reynolds-Averaged Navier- Stokes (RANS) codes has been an area of intense research in the field of fluid dynamics. This research has resulted in an array of turbulence models with varying degrees of complexity. Often, these models are accurate only for certain kinds of flows, such as attached boundary layers or free shear layers. The flowfield generated by a supersonic jet in a subsonic crossflow, however, contains several of these complex fluid structures, each of which interact with the others. An assessment of the ability of turbulence models to predict such complex flowfields in the transonic regime has been hampered by a dearth of experimental data. While most of the compressible experiments conducted in the past have involved supersonic jets exhausting into supersonic crossflow, a few programs have examined a supersonic jet in subsonic compressible crossflow. One of the most useful is reported by Chocinski et al. 1 and Chocinski, 2 who provide a variety of surface and flowfield measurements that can be compared with numerical results. Other studies have provided more limited measurements of similar flowfields 3-5 and a few analytical efforts also have been published. 6-8 Some additional studies have involved sonic jets exhausting into a subsonic compressible freestream The experimental results reported in Refs. 1 and 2 are used in the current investigation to assess the ability of a number of widely-used turbulence models to predict the complex features found in transonic jet-in-crossflow situations. The present study focuses on predicting the location of the counter-rotating vortex pair at various stations downstream of the jet. The issues of grid convergence and iterative convergence are also addressed. al Description A detailed discussion of the experimental setup and facilities can be found in Refs. 1 and 2. The experiment was performed in a solid-wall blow-down transonic tunnel at CEAT. Figure 1: (reproduced from Ref. 1) shows the experimental set-up. The jet nozzle was a Laval nozzle with a 1 half angle downstream of the throat and a 3 mm exit diameter. It exhausted transversely from one side wall of the wind tunnel. Instrumentation included mean surface pressure taps both in the vicinity of the nozzle and on the opposite wall, and a five-hole probe that was used to interrogate the flowfield for Mach number and total pressure coefficient. The wind tunnel conditions reported by Chocinski et al. are given in Table 1. These conditions produced a static jet-to-crossflow pressure ratio of 1.5 and a jet-to-crossflow momentum ratio of 16.7, and the jet is underexpanded.. Table 1 Flow Conditions Crossflow Computational Method The SACCARA Code The distributed massively parallel CFD code currently under development at Sandia National Laboratories as part of the Department of Energy Accelerated Strategic Computing Initiative program is called SACCARA (Sandia Advanced Code for Compressible Aerothermodynamics Research and Analysis). The SACCARA code was developed from PINCA, 14,15 a distributed parallel version of the commercial, finite volume, Navier- Jet Mach Number Stagnation Pressure (Pa) Stagnation Temperature (K) Reynolds Number (based on jet exit diameter)

4 AIAA 99-xxxx Stokes code INCA, 16 from Amtec, Inc. The SACCARA code solves the 2-D/axisymmetric and full 3-D Navier- Stokes equations for laminar and turbulent flows. The SACCARA code is applicable to flows from subsonic through hypersonic speeds and has options for perfect gas, thermochemical equilibrium, and thermochemical nonequilibrium. Standard zero-, one-, and two-equation turbulence models are also available. The code employs multi-block structured grids with point-to-point match up at the block interfaces. The solution is driven to a steady state using the lower-upper symmetric Gauss- Seidel (LU-SGS) or diagonal implicit solution advancement scheme based on a combination of the work of Yoon et al. 17,18 and Peery and Imlay. 19 The inviscid fluxes are evaluated using the flux vector splitting of Steger and Warming 2 or the symmetric TVD flux function of Yee. 21 The latter is employed in this paper and is nominally second-order accurate in smooth regions of the flow. The viscous terms are discretized using a standard central differencing scheme. Turbulence Models Nearly every turbulence model was initially developed for incompressible flows, and the extension to compressible flows is not always straightforward. The general form of the turbulence transport equation for compressible flow may be written as ( ρϕ) + ( ρũ t x j ϕ) = D+ S P S D j (1) where ϕ is the transported quantity (k, ε, etc.), D is the diffusion term, and S P and S D represent source terms for production and destruction, respectively. The general form for the compressible diffusion term is D ϕ = µeff x j x j (2) where the effective viscosity µ eff is some combination of the molecular (or laminar) viscosity µ and the turbulent viscosity µ t. The following development utilizes Favre (overtilde) and Reynolds (overbar) averaging. See Ref. 22 for notation and details on these averaging procedures. Spalart-Allmaras Model A transport equation for determining the eddy viscosity with near-wall effects included has been developed by Spalart and Allmaras. 23,24 The governing equation was originally developed for incompressible flows and was formulated in substantial derivative form. When transformed into compressible form (i.e., Eq. (1)), the working variable and source terms take the following form: ϕ = νˆ, S P = c b1 ρνˆ Ŝ S D = c w1 f w ρ -- νˆd 2 (3) where the eddy viscosity is defined as µ t = ρνˆ f v1, and the following functions are used: χ 3 χ f v1 = , χ 3 f 3 v2 = c 1 + χ f v1 v1 f w g 1 c 6 + w3 1 6 = g 6, χ = 6 + c w3 g = r + c w2 ( r 6 r), r = νˆ f v2 Ŝ = 2Ω ij Ω ij κ 2 d 2 The original formulation for the diffusion term can be written as D where the effective viscosity is defined as (4) ν+ νˆ (5) σ When Eq. (4) is transformed into the compressible form shown in Eq. (2), an additional term appears which contains a density gradient: This density gradient term (shown in brackets) has been found to cause numerical stability problems for highspeed flows. Menter has shown that when an eddy viscosity transport equation is developed from the compressible form of the k-ε equations, this density gradient term does not appear. 27,28 This term is therefore omitted from the current formulation, 25,26 and the diffusion term is now defined as: νˆ -- ν νˆ Ŝκ 2 d 2 ρ νˆ c ν x eff b2 ρ νˆ νˆ = j x j σ x j x j ν eff µ eff = = ρ D νˆ µ x eff ρ νˆ = ν j x j eff x j x j c b2 ρ νˆ νˆ σ x j x j 4

5 AIAA 99-xxxx D = νˆ c µ x eff b2 ρ νˆ νˆ j x j σ x j x j (6) µ t µ eff,ε = µ σ ε The model constants used for the Spalart-Allmaras model are: c b1 =.1355, c b2 =.622, σ = 2 3 κ =.41, c w1 c b1 κ c b2 = σ c w2 =.3, c w3 = 2., c v1 = 7.1 Low Reynolds Number k-ε Model The original k-ε formulation of Jones and Launder 29 is appropriate for high Reynolds number flows only. In order to apply this model to wall-bounded flows, damping terms may be added to allow this model to be integrated to the wall. The low Reynolds number formulation of the k-ε model used in the current work is the Nagano and Hishida 3 formulation. The transported quantity and source terms for the turbulent kinetic energy equation in compressible form (see Eq. (1)) are: k ϕ k = k, D k = µeff x,k j x j S Pk k = ρp, S Dk = ρε 2µ x n The effective eddy viscosity is given by (7) The formulation of the k-ε equations currently implemented in the SACCARA code uses the following functions f 1 = 1, f 2 = 1.3e R t R t k 2 y = νε, f µ 1 exp = and the production term P employs the incompressible formulation, i.e., ũ i P τ ij x ν S = = ũ i t ij j x j The variable x n refers to the wall-normal direction and ũ t to the wall-tangent component of velocity as in standard boundary layer theory. The model constants for the current implementation are: c ε1 = 1.44, c ε2 = 1.92 c µ =.9, σ k = 1., σ ε = 1.3 Wilcox (1998) k-ω Model For the Wilcox (1998) k-ω model, 22 the terms in Eq. (1) are as follows: 2 µ t µ eff,k = µ σ k where the turbulent viscosity is defined as µ t = c µ f µ ρk 2 ε. (8) The compressible form of the specific dissipation rate equation is k ϕ k, = k, D k = µeff x,k j x j S Pk = ρp C, S Dk = β * ρkω ω ϕ ω, = ω, D ω = µeff x,ω j x j S Pω = ρ---- γ P ν C, S Dω = βρω 2 t (1) (11) ε ϕ ε = ε, D ε = µeff x,ε j x j S Pε = ε - c k ε1 f 1 ρp µν t ( 1 f µ ) 2 ũ t x2 n S Dε = c ε2 f 2 ρε where the effective eddy viscosity is now: (9) The effective eddy viscosity is given by µ eff,k = µ + σ k µ t, µ eff,ω = µ + σ ω µ t where the turbulent viscosity is defined as: µ t = ρk ω (12) The compressible form of the turbulence production 5

6 AIAA 99-xxxx term P C is used, and is given by ũ i 1 P C τ ij ν x t S ij -- ũ k δ 2 = = j 3 x ij --kδ k 3 ij ũ i x j The following relations and standard model constants are used for the Wilcox (1998) k-ω model: α 13 = -----, β = β 25 f β, β * * = β f * β 1 * 9 σ k = σ ω = --, β 2 = , χ k 2 1 k f * = χ β k, χ k , χ 2 k > ω ω x j x j 1 + 4χ k χ ω Ω β , f 125 β χ ij Ω jk S = =, ki 1+ 8χ ω ω * 3 ( β ω) Computational Mesh Three computational meshes were used in the calculations: a fine mesh, a medium mesh and a coarse mesh. The medium mesh was created from the fine mesh by removing every other mesh cell from each computational direction. In the same way, the coarse mesh was created by removing every other cell from the medium mesh. All the meshes consisted of seven structured blocks and the total number of mesh cells is given in Table 2. Ideally, the number of mesh cells should increase by a factor of 64 between the coarse and fine meshes and a factor of 8 between the medium and fine meshes. A number of grid lines had to be preserved through the flowfield to ensure point-to-point match-up on the block boundaries. However properly maintaining these block interfaces resulted in a factor slightly less than eight between the different mesh levels. The near-jet region of the fine mesh used for the calculation is shown in Figure 2:. The boundaries between the blocks have point-to-point match-up and are connected through the use of ghost cells. Each block has two ghost cells at the boundaries allowing the numerical solution method to maintain second order spatial accuracy across block interfaces. The mesh height of 85 mm and width of 65 mm matches the wind tunnel test section dimensions. 1 The origin of the axis system is located at centerline of the jet located on the surface of the plate. The mesh extends 5 jet diameters upstream and 2 jet diameters downstream. Only mean flow properties were measured experimentally and flowfield measurements were reported from an Y/D= (center of the jet) to Y/D=6 (i.e. half the jet). Based on the type and location of the experimental data a plane of symmetry is assumed along a plane that is parallel to the freestream velocity vector and intersects the center of the nozzle. Attention has been given to the wall spacing to ensure enough points exist in the boundary layer to allow accurate integration to the wall. Also, grid points were clustered in the near-jet region to help resolve the high gradients created by the jetfreestream interaction. The nozzle wall grid spacing for the fine mesh was mm with a total of 12, mesh points in the nozzle. The physical grid spacing on the wall of the flat plate is mm. This physical grid spacing on the flat plate resulted in a maximum y + value of.1 on the fine mesh. Mesh level Table 2 Flow Conditions Fine 3,192,864 Medium 437,447 Coarse 53,797 Mesh cells Boundary Conditions A uniform velocity profile was assumed at the inflow plane and the boundary layer was allowed to grow naturally. At the subsonic inflow boundary, total pressure (p t ) and total temperature (T t ) and the direction cosine of the velocity vectors are specified. The simulations used the total conditions shown in Table 1. An adiabatic noslip wall boundary condition was applied along the flat plate and the jet walls. The jet inflow boundary was assumed to start at the nozzle throat where a sonic boundary condition was applied. A symmetry boundary condition was applied along the centerline of the jet, parallel to the freestream velocity vector and a free-slip boundary condition was assumed along the remaining tunnel walls. The subsonic outflow boundary condition is defined by specifying a constant static pressure. The value of static pressure used at the outflow boundary was determined by varying its value and comparing the Mach number distribution on the wall directly opposite the jet with experimental data from Ref. 2. This procedure resulted in a static pressure value of 6.2 x 1 4 Pa. The comparison of the resulting Mach number distribution produced by the static pressure value used in the calculations is shown in Figure 3:. The results of simula- 6

7 AIAA 99-xxxx tions on the fine mesh for all three turbulence models are compared to the experimental data. There is an initial increase in the Mach number due to the presence of the jet. All three turbulence models track the rise in Mach number with the k-ω model producing a slightly more accurate prediction of the Mach number rise. After X/D = 2 the simulations predict a continued rise in the Mach number while the experimental data indicates constant or slightly decreasing Mach number. It is believed that the prediction of the increase in Mach number is due to a reduction in the effective area of the test section. The area reduction is caused by the presence of the jet, the vortices and the boundary growth on the wall. The wind tunnel test section is given as a rectangular geometry, 1 but the Mach number distribution given by the experimental data suggests that some method was employed to maintain the effective area of the test section. This discrepancy leads to additional uncertainty in how well the experiment is being characterized in the current simulations. Boundary Conditions for Turbulent Quantities Spalart-Allmaras model The Spalart-Allmaras turbulence model requires the specification of the turbulent viscosity at the tunnel and jet inflow boundaries. For this case the freestream turbulent viscosity is set equal to the freestream laminar viscosity, i.e., In the above equation, the first equality will always be satisfied when Eq. (16) is used. The second equality will only be satisfied if all three meshes have been sufficiently refined so as to be in the second order asymptotic range. The normalized errors from Eq. (18) are given in Figcrossflow: µ t = µ = Ns m jet: µ t = µ = Ns m Low Reynolds Number k-ε Model The turbulent quantities at the inflow boundaries were estimated by assuming a turbulence intensity T u of 3% for both the crossflow and the jet. Since the turbulence intensity levels in the wind tunnel were not reported, a value of 3% was assumed to ensure a turbulent flowfield. The turbulent kinetic energy is then determined by Eq. (13): k 1.2 Tu = U (13) By again assuming that the freestream turbulent viscosity is equal to the freestream laminar viscosity, the initial turbulent dissipation rate can be estimated using Eq. (14): where C µ =.9. k 2 ε = ρc µ ---- µ t (14) Wilcox (1998) k-ω Model The initial values for the k-ω model can be obtained from Eq. (13) and Eq. (14) using the additional relationship given below: ω = ε C µ k (15) Numerical Accuracy Spatial convergence has been judged from the steadystate solutions on three meshes: 1, 2, and 3 (from finest to coarsest). The Richardson extrapolation procedure 31 has been used to obtain a more accurate result from the relation f RE = f 1 + ( f 1 f 2 ) 3. (16) The above relation assumes that the numerical scheme is second-order, that both mesh levels are in the asymptotic grid convergence regime and a mesh refinement factor of two (i.e., grid halving). The accuracy of the solutions on the three meshes has been estimated with the exact solution approximated with f RE which gives the solution error as % Error of f k = 1( f k f RE ) f RE (17) where k=1, 2, 3 is the mesh level. If the mesh has been refined sufficiently such that the solution displays a second-order behavior, then the errors on the three meshes will obey the following relationship: % Error of f 2 % Error of f 1 = % Error of f 3 = (18) 7

8 AIAA 99-xxxx ure 4: for the near-jet region using the Spalart-Allmaras model. The fine grid errors are within 3% upstream of the jet and within 5% downstream of the jet. The fact that the normalized coarse grid errors in the near-jet regions do not match with the normalized errors on the fine and medium grids indicates that the three mesh solutions are not all within the asymptotic grid convergence range. Away from the jet, the errors in the surface pressure are below 1% and appear to be grid resolved. The normalized errors for the k-ε model are presented in Figure 5:. The results in the near-jet region are not fully grid resolved on all three meshes; however the downstream near-jet regions appears to be better resolved than in the Spalart-Allmaras prediction. The fine grid errors in surface pressure near the upstream separation region are within 4%, while the errors at the downstream reattachment region are below 3%. Away from the jet, the errors in surface pressure are again below 1% and are grid resolved. Normalized errors for the surface pressure are shown in Figure 6: for the Wilcox k-ω model. These results appear to be the least grid resolved downstream of the jet, both in the near-jet region and farther downstream. The normalized errors near the reattachment point are greater than those seen for the other two turbulence models, indicating the need for further grid refinement with this model. The upstream region does appear to be fairly well grid resolved, with maximum fine grid errors of less than 1.5%. In the downstream region, the maximum fine grid errors are near 1%; however, the poor agreement between the normalized errors in this region means that there is significant uncertainty in this error estimate. It should be noted that obtaining grid-resolved solutions for complex, three-dimensional problems is extremely challenging due to the difficulty of obtaining adequate grid quality and resolution for these flows. Results Figure 7: illustrates some of the flowfield features captured in the simulation. The white line segments represent the core of the vortices. Streamlines have been inserted in the region of the vortices as well as in the center of the jet. A pair of counter-rotating vortices are clearly evident behind the jet, although only one vortex is shown since symmetry is assumed. An additional pair of vortices can be seen near the surface of the plate. This horse-shoe vortical flow was also observed in the experiment. 1 Critical Points In Figure 8: an iso-surface of the axial component of velocity (u=) is shown. The iso-surface is colored by Mach number. This surface shows the region where the presence of the jet forced the axial component of velocity to change sign and thus bounds the recirculation region. Points 1 and 2, labeled on the figure, represent critical points that have been measured experimentally in Ref. 1. Point 1 is a saddle point of separation located in front of the jet. This region of separation is caused by the large adverse pressure gradient generated as the crossflow is forced to slow and negotiate the jet. The crossflow then accelerates around the jet creating a low pressure separated region behind the jet. Point 2 is the reattachment point of the separated region behind the jet. Table 3 shows a comparison between the experimentally measured and the calculated critical points as well as the percent error between the calculations and the experiment. Using Richardson Extrapolation the critical points predicted by the calculations have been extrapolated from the medium and fines meshes to provide a more accurate estimate of their location. The Spalart-Allmaras model prediction shows the best agreement with the experimental measured location of the saddle point. The k-ε model does a good job of predicting the location of point 1. This result is somewhat surprising since the k-ε model has been shown to inaccurately predict the location of separated regions generated by adverse pressure gradients. 22 The k-ω model also overpredicted the size of this separated region. All three models underpredict the location of point 2. Rather than using the Richardson Extrapolation values to compare with the experimental critical points, these extrapolated values may instead be used to estimate the error in the fine grid predictions. If the extrapolated values for point 2 are used to calculate the percent error of the fine mesh predictions, then the fine mesh errors are 2.1%, 1.7% and -6.% for the Spalart-Allmaras, k-ε, and k-ω models, respectively. The large error for k-ω model is an indication that an additional mesh refinement may be required to ensure an accurate extrapolation of this flow feature. Near the surface of the plate the jet behaves as a solid obstruction to the oncoming fluid. Predicting these types of reattachment points is difficult. The predictions of the critical point locations may also be sensitive to the freestream turbulence quantities and the boundary conditions. These sensitivities have not been explored in this work. Separation Point Table 3 Critical Separation Points Point 1 (X/D) Percent Error Point 2 (X/D) Percent Error S-Α model % % 8

9 AIAA 99-xxxx Separation Point Table 3 Critical Separation Points Point 1 (X/D) Percent Error Point 2 (X/D) Percent Error k-ε model % % k-ω model % % Surface Pressure The surface pressure coefficient (C p ) is plotted along a line that is parallel to the inflow velocity vector and intersects the center of the jet. Comparisons between the experimental data 2 and the fine mesh solutions for all three turbulence models are shown in Figure 9:. The center of the jet is located at X/D= and Y/D=. The calculations capture the pressure rise upstream of the jet. The k-ε and k-ω models have slightly higher peak pressures before the jet. Downstream of the jet, the k-ε and k-ω models capture the trends in the experimental data. Both the k-ε model and k-ω model predict a region of moderately increasing C p within the first two diameters downstream of the jet and capture the slope change that occurs at approximately two diameters downstream of the jet. The k-ω model compares extremely well with the data as the flow continues to recompress. The k-ε model sightly overpredicts the C p value as the flow is recompressing. The Spalart-Allmaras model predicts a pressure rise and then a decrease in pressure within the first two diameters downstream. The Spalart-Allmaras model overpredicts the slope of the recompression and the peak value of C p. Figure 1: shows Mach number contours for the Spalart-Allmaras model in the plane that intersects the center of the jet. A region of high Mach number is clearly visible within the first two diameters downstream of the jet. Also, the region of separated flow appears to have been lifted off of the surface. Both of these phenomena appear to be non-physical. Figure 11: shows the same contour plot for the k-ω model. The figure does not show the high Mach number region found in the Spalart-Allmaras prediction. The high Mach number region predicted by the Spalart-Allmaras model and its effect on the local flowfield may contribute to the behavior shown in Figure 9: that differs from the k-ε and k-ω models. Although not shown, the k-ε results are similar to those predicted by the k-ω model. Flowfield Features Figure 12: shows a number of the flowfield features including the core of the vortices and streamlines integrated from the nozzle. The figure also shows a plane at X/D=1, downstream of the center of the jet exit. This plane is colored by Mach number and represents one of the locations in the flowfield where experimental data is available. Additional data was taken at X/D=4; however this location is not shown in the figure. The flowfield comparisons will be made with total pressure (p t ) and the X component of vorticity ( Ω x ). The first set of flowfield comparisons are at X/D=1 and are shown in Figures 13 through 15. The figures show a comparison between the measured p t and the predicted p t for the Spalart-Allmaras model, the k-ε model and k-ω model. The jet will produce total pressures greater than the freestream value, therefore the maximum total pressure at X/D=1 (labeled in the figures) represents the core of the jet. The vortices create a total pressure deficit with the minimum labeled value indicating the vortex core location. Figure 13: shows total pressure contours for the experiment and the prediction for the Spalart-Allmaras model. The Spalart- Allmaras model shows very good agreement with the experimental data. The prediction matches the location and magnitude of both the total pressure deficit (vortices) and maximum total pressure (jet). Both the k-ε model (Figure 14:) and k-ω model (Figure 15:) predictions capture the maximum total pressure but overpredict the total pressure deficit. Jet and vortex locations are not as well matched as for the Spalart-Allmaras model. Both models predict a splitting of the jet core that was not observed in the experiment. The k-ε model and k-ω model produce nearly identical results for all of the measured flowfield quantities. The k-ω model shows slightly better agreement with the experimental data. Since the k-ε and k-ω models produce similar results, only those from the Spalart-Allmaras model and k-ω model will be discussed hereafter. Figures 16 and 17 show comparisons of the X component of vorticity at X/D=1. In Figure 16: the experimental results are compared to the prediction using the Spalart-Allmaras turbulence model. The model s prediction of the vorticity contour levels matches the experiment. The results from the k-ω model, shown in Figure 17:, overpredict the maximum vorticity in the core of the vortex. The location of the core is predicted to be higher than is shown in the experimental data. The k-ω model predicts that the core of the vortices is 82% stronger than was measured. The overprediction of the vortex core strength provides an explanation of the splitting of the jet observed in Figure 15:. Additional experimental data was taken at an X/ D=4. By this location, the jet has completely dissipated, therefore the total pressure plots will not be shown. Figure 18: and Figure 19: show the X vorticity at X/ D=4 for the Spalart-Allmaras model and the k-ω model, respectively. Consistent with the results at X/D=1, the Spalart-Allmaras model matches both the location of 9

10 AIAA 99-xxxx the vortex core and the magnitude of the X vorticity. Figure 19: shows that the k-ω model has overpredicted the core vortex strength by 12%. Coupled with the overprediction of the core strength is the overprediction of the core location. The k-ω model s prediction of stronger vortices indicates a larger induced velocity. The interaction of this induced velocity between the counter-rotating vortices tends to create greater vortex lift-off and hence increase their height above the tunnel floor. The turbulent viscosity shown in Figures 2 and 21 provides insight into the k-ω model s over-prediction of the vortex strength. Figure 2: shows contours of the turbulent viscosity for the k-ω model. In Figure 2: the maximum contour level shown on the plot is representative the peak value of turbulent viscosity. Figure 21: show the same plot for the Spalart-Allmaras model with the same contour levels used in Figure 2:. The Spalart- Allmaras predicts a large region of turbulent viscosity with values greater than the peak value predicted by the k-ω model. The peak turbulent viscosity values present in the Spalart-Allmaras prediction were 3.4 times greater than those predicted by the k-ω model. The lower turbulent viscosity of the k-ω model allows the counterrotating vortex pair to retain its strength and leads to a greater vortex lift-off height. Conclusions Steady state RANS simulations of a supersonic jet in a subsonic compressible crossflow were completed using three different turbulence models. The simulations used three grid levels for each turbulence model. The grids were used to estimate the spatial accuracy of the fine grid solutions. The iterative convergence on each of the mesh levels was also examined. The simulation results were compared to existing experimental data. The location of the separation and reattachment points, the surface pressure and the X-axis vorticity were compared to experimental data. All of the models under-predicted the location of the reattachment point. The k-ω model and k-ε model accurately predicted the surface pressure. The one-equation Spalart-Allmaras model did not predict the surface pressure as accurately as the two-equation models. A high Mach number region was observed in the Spalart-Allmaras model near the jet, possibly a non-physical result. The two-equation models missed the location and the strength of the vortices. The Spalart-Allmaras model did an excellent job predicting both the strength and location of the vortices. The discrepancy between the two equation models and the experimental data is believed to be caused by their underprediction of the turbulent viscosity relative to the Spalart-Allmaras model. More advanced turbulence models are probably needed to adequately simulate this complex flow. There are a number of issues associated with this study which warrant continued investigation: the amount of unsteadiness in flowfield, the effect of the symmetry plane boundary condition on the solutions and the effect of varying the amount of freestream turbulence. Rather than investigating the jet-in-crossflow problem using the current experimental data, the results from this study will be used to help design an ongoing set of experiments at Sandia National Laboratories to investigate the transonic jet-in-crossflow problem. Specific areas of interest are the near-jet region, the amount of unsteadiness in the flowfield and the measurement of flowfield quantities with nonintrusive methods. Acknowledgments The authors would like to thank Fred Blottner, Walter Wolfe, and Mary McWherter-Payne, of Sandia National Laboratories, for their assistance and insights. This work was supported by Sandia National Laboratories and the Department of Energy s Accelerated Strategic Computing Initiative. References 1. Chocinski, D., Leblanc, R., and Hachemin, J.-V., "al/computational Investigation of Supersonic Jet in Subsonic Compressible Crossflow," AIAA Paper , 35th Aerospace Science Meeting and Exhibit, Reno, NV, January 6-1, Chocinski, D., "Contribution to the Modeling of Interactions of a Supersonic Jet in a Subsonic Compressible Crossflow," Doctoral Thesis, University of Poitiers, France, Reichenau, D. E. A., "Interference Effects Produced by a Cold Jet Issuing Normal to the Airstream from a Flat Plate," AEDC TR-67-22, October Lee, E. E. Jr., and Willis, C. M., "Interaction Effects of a Control Jet Exhausting Radially from the Nose of an Ogive-Cylinder Body at Transonic Speeds," NASA TN D-3752, January Manela, J., and Seginer, A., "Interaction of Multiple Supersonic Jets with a Transonic Flowfield," AIAA Journal, Vol. 24, No. 3, pp , March Kuiper, R. A., "Control Jet Effectiveness in the Subsonic and Transonic Flight Regimes," Philco Aeronautic Division publication U-2932, December Cassel, L. A., Durando, N. A., Bullard, C. W., and Kelso, J. M., "Jet Interaction Control Effectiveness for Subsonic and Supersonic Flight," Report No. RD-TR , U. S. Army Missile Command, Redstone Arsenal, September

11 AIAA 99-xxxx 8. Manela, J., and Seginer, A., "Jet Penetration Height in Transonic Flow," AIAA Journal, Vol. 24, No. 1, 67-73, January Spring, D. J., "An al Investigation of the Interference Effects due to a Lateral Jet Issuing from a Body of Revolution over the Mach Number Range of.8 to 5," Report No. RD-TR-68-1, U.S. Army Missile Command, Redstone Arsenal, August Dahlke, C. W., "An al Investigation of Downstream Flow-Field Properties Behind a Sonic Jet Injected into Transonic Free Stream from a Body of Revolution," AEDC RD-TR-69-2, February Street, T. A., "An al Investigation of a Transverse Jet Ejecting from a Flat Plate into a Subsonic Free Stream," AEDC RD-TM-7-5, May Shaw, C. S., and Margason, R. J., "An al Investigation of a Highly Underexpanded Sonic Jet Ejecting from a Flat Plate into a Subsonic Crossflow," NASA TN D-7314, December Wang, K. C., Smith, O. I., and Karagozian, A. R., "In-Flight Imaging of Transverse Gas Jets Injected into Compressible Crossflows," AIAA Journal, Vol. 33, No. 12, December Wong, C. C., Soetrisno, M., Blottner, F. G., Imlay, S. T., and Payne, J. L., PINCA: A Scalable Parallel Program for Compressible Gas Dynamics with Nonequilibrium Chemistry, SAND , Sandia National Laboratories, Albuquerque, NM, Wong, C. C., Blottner, F. G., Payne, J. L., and Soetrisno, M., Implementation of a Parallel Algorithm for Thermo-Chemical Nonequilibrium Flow Solutions, AIAA Paper No , Jan INCA User s Manual, Version 2., Amtec Engineering, Inc., Bellevue, WA, Yoon, S. and Jameson, A., An LU-SSOR Scheme for the Euler and Navier-Stokes Equations, AIAA Paper No. 87-6, Jan Yoon, S. and Kwak, D., Artificial Dissipation Models for Hypersonic External Flow, AIAA Paper No , Peery, K. M. and Imlay, S. T., An Efficient Implicit Method for Solving Viscous Multi-Stream Nozzle/Afterbody Flow Fields, AIAA Paper No , June Steger, J. L. and Warming, R. F., Flux Vector Splitting of the Inviscid Gasdynamic Equations with Applications to Finite Difference Methods, Journal of Computational Physics, Vol. 4, 1981, pp Yee, H. C., Implicit and Symmetric Shock Capturing Schemes, NASA TM-89464, May Wilcox, D. C., Turbulence Modeling for CFD, 2nd Ed., DCW Industries, Inc Palm Drive, La Canada, CA 9111, Spalart, P. R. and Allmaras, S. R., A One-Equation Turbulence Model for Aerodynamic Flows, AIAA Paper , Spalart, P. R. and Allmaras, S. R., A One-Equation Turbulence Model for Aerodynamic Flows, La Recherche Aerospatiale, No. 1, 1994, pp Roy, C. J. and Blottner, F. G., Assessment of One-and Two-Equation Turbulence Models for Hypersonic Transitional Flows, AIAA Paper 2-132, January Roy, C. J., and Blottner, F. G., Assessment of One-and Two-Equation Turbulence Models for Hypersonic Transitional Flows, to appear in the Journal of Spacecraft and Rockets, October Menter, F. R., Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications, AIAA Journal, Vol. 32, No. 8, Aug. 1994, pp Menter, F. R., Grotjans, H., and Unger, F., Numerical Aspects of Turbulence Modeling for the Reynolds Averaged Navier-Stokes Equations, Computational FluidDynamics Lecture Series , von Karman Institute for Fluid Dynamics, March 3-7, Jones, W. L. and Launder, B. E., The Prediction of Laminarization with a Two-Equation Model of Turbulence, International Journal of Heat and Mass Transfer, Vol. 15, 1972, pp Nagano, Y. and Hishida, M., Improved Form of the k ε Model for Wall Turbulent Shear Flows. Journal of Fluids Engineering, Vol. 19, June 1987, pp P. Roache, Ch. 3: A Methodology for Accuracy Verification of Codes: the Method of Manufactured Solutions, Verification and Validation in Computational Science and Engineering, Hermosa Publishers, New Mexico, Kenwright, D. N. and Haimes, R., Automatic Vortex Core Detection, IEEE Computer Graphics and Applications, Vol. 18, No. 4, 1998, pp

12 Figure 1: Schematic of the experimental set-up (reproduced from Ref. 1) Figure 2: Computational mesh 12

13 Figure 3: Near wall Mach number distribution Figure 4: Normalized errors in surface pressure distributions, Spalart-Allmaras model. 13

14 Figure 5: Normalized errors in surface pressure distributions, low Reynolds number k-ε model. Figure 6: Normalized errors in surface pressure distributions, Wilcox k-ω model. 14

15 Figure 7: Vortex locations Point 1 Point 2 Figure 8: Iso-surface of u =. m/s, colored by Mach number 15

16 Figure 9: Surface pressure coefficient comparison Figure 1: Jet centerline Mach number contours, Spalart-Allmaras model 16

17 Figure 11: Jet centerline Mach number contours, k-ω model Comparison Plane Jet Vortex Core Figure 12: Comparison plane at X/D = 1 with Flow Features 17

18 Figure 13: al /computational comparison of p t 1-5 Pa at X/D = 1, Spalart-Allmaras model Figure 14: al /computational comparison of p t 1-5 Pa at X/D = 1, k-ε model 18

19 Figure 15: al /computational comparison of p t 1-5 Pa at X/D = 1, k-ω model Figure 16: al /computational comparison of the x component of vorticity at X/D = 1, Spalart-Allmaras model 19

20 Figure 17: al /computational comparison of the x component of vorticity at X/D = 1, k-ω model Figure 18: al /computational comparison of the x component of vorticity at X/D = 4, Spalart-Allmaras model 2

21 Figure 19: al /computational comparison of the x component of vorticity at X/D = 4, k-ω model Figure 2: Jet centerline eddy viscosity contours, k-ω model 21

22 Figure 21: Jet centerline eddy viscosity contours, Spalart-Allmaras model 22

23 . Figure 22: al /computational comparison at X/D = 1, Spalart-Allmaras model (the experimental plot was reproduced from Ref. 7) Figure 23: al /computational comparison at X/D = 1, k-ε model 23

24 Figure 24: al /computational comparison at X/D = 1, k-ω model Figure 25: al /computational comparison at X/D = 4, Spalart-Allmaras model 24

25 Figure 26: al /computational comparison at X/D = 4, k-ε model Figure 27: al /computational comparison at X/D = 4, k-ω model 25

26 Figure 28: al /computational comparison of p t 1-5 Pa at X/D = 4, Spalart-Allmaras model Figure 29: al /computational comparison of p t 1-5 Pa at X/D = 4, k-ε model 26

27 Figure 3: al /computational comparison of p t 1-5 Pa at X/D = 4, k-ω model Figure 31: al /computational comparison at X/D = 1, k-ε model 27

28 AIAA 99-xxxx Figure 32: al /computational comparison at X/D = 4, k-ε model 28