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Hypersonic Turbulent Flow Predictions using CFD++

Metacomp Technologies, Inc., Agoura Hills, CA, USA.

Corresponding author: Uriel Goldberg, ucg@metacomptech.com Hypersonic turbulent wall-bounded flows pose a difficult challenge to CFD due to severe velocity and temperature gradients adjacent to solid surfaces, the presence of laminar to turbulent flow transition and strong shock/boundary-layer interactions with attendant massive flow separation. Due to the high Mach numbers involved, very few reliable sets of experimental data exist, leaving engineers with little choice but to rely on CFD methodology. Predicting hypersonic turbulent flows efficiently and with high level of confidence requires a combination of robust yet accurate numerical methods and a powerful turbulence modeling capability. CFD++ [1] possesses these attributes and is well qualified to predict hypersonic turbulent flows with a high confidence level. A number of hypersonic flow examples are described in this paper. Results using several turbulence models are compared with experimental data. All computations were done using CFD++, a Navier-Stokes solver for either compressible or incompressible fluid flows. It features a second order Total Variation Diminishing (TVD) discretisation scheme based on a multi-dimensional interpolation framework. For the results presented here, an HLLC (Harten, Lax, van Leer, with Contact wave) Riemann solver [2] was used to define the (limited) upwind fluxes. This Riemann solver is particularly suitable for hypersonic flow applications since, unlike classical linear solvers such as Roe's scheme, it automatically enforces entropy and positivity conditions. Flow examples 1. Hypersonic Flow over a Curved Ramp This example concerns hypersonic flow over a curved compression surface, with experimental data by Holden [3]. The compression creates an oblique shock which induces a large increase in heat transfer to the cooled wall. Some flow conditions are: M=8.03, Re=16.31X10^6/ft,

∞T =90.6 R, ∞TTw / =5.89. The experimental data indicate laminar to turbulent flow transition approximately between x=5 and x=12 inches. The calculations, using a k-ε model [4,5], were done twice: with and without a transition model sensitised to abrupt boundary layer thickening. The computations were carried out on a 39,000 cell grid

with at least four cells inside the viscous sublayer and ≤+y 0.75 at the first layer of grid centroids away from the wall. Figure 1 shows geometry and main flow features. Figure 2 is a Mach contour plot, showing the ramp-induced shock, and Figure 3 compares results of heat transfer prediction with experimental data. The calculations were done with the following initial/free-stream conditions: free-stream turbulence level T'=0.1% and turbulence

AIAA/CIRA 13th International Space Planes and Hypersonics Systems and Technologies AIAA 2005-3214

Copyright © 2005 by Metacomp Technologies, Inc. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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length-scale ∞l =0.1 mm, corresponding to an eddy to molecular

viscosity ratio µµ /t =0.69. The k-ε model induces laminar to turbulent flow by-pass transition, the location of which is shifted slightly upstream when the transition model is switched on. Predictions with and without the transition model straddle the experimental data.

Fig. 1 Curved ramp schematic view (not to scale)

Fig. 2 Curved ramp Mach contours

Fig. 3 Curved ramp wall heat transfer

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2. Hypersonic Flow in a Double-Wedge Inlet In recent years there has been renewed interest in hypersonic flight vehicles. The engine inlets of these vehicles typically involve compression ramps which, through a series of shocks, reduce the engine inflow Mach number to supersonic levels to enable supersonic combustion. Such a shock system imposes, however, a severe penalty in terms of surface heating, requiring careful attention to the choice of materials and/or cooling devices to avoid the possibility of local melting of the vehicle's skin. It is important, therefore, to be able to predict hypersonic flow over ramp and wedge configurations, including surface heating characteristics, with the aim of using this capability for analysis and design purposes of vehicle components such as engine inlets. The ability to predict turbulent hypersonic flows with high level of confidence carries much broader benefits, namely entire vehicle external/internal flow prediction capability for preliminary design and, later, for various analysis purposes. Kussoy et al. [6] performed extensive experimental measurements on a Mach 8.3 flow in a wedge inlet configuration, with 0/TTw =0.27. To predict this complex 3D flow-field, involving crossing shock/boundary-layer interactions, the CFD solver was used on a mesh consisting of approximately 250,000 cells. First centroidal locations away from

walls were at ≈+y 60 to avoid a much larger grid size. A wall function which accounts for compressibility, heat transfer and pressure gradient effects was employed. This wall function uses the van Driest transformed velocity (see White [7]) in conjunction with a modified version of the Launder-Spalding Law-of-the-Wall [8] which is based on

k rather than on the friction velocity, τu , to avoid problems in separation and reattachment zones where the latter vanishes. The wall function was used to determine momentum and energy fluxes at walls. Fig. (4) is a sketch of the topography and Fig. (5) is an overview of the flow in the region of the wedges, showing streamlines and pressure contours on one wedge surface. The flow in the mid-region of the wedge is observed to maintain an approximately two-dimensional flavour. The high-pressure region downstream of the shoulder is due to shock impingement from the other wedge. The streamlines at the wall/wedge juncture clearly show streamwise separation due to the adverse pressure gradient downstream of the wedge shoulder. Flow spillage at the upper end of the wedge, due to cross-stream pressure gradient, is also observed. Fig.(6)compares predicted wall pressure and heat transfer, along the symmetry line, with corresponding measurements. The 1-equation tR model [9], used in the present example, yields very good agreement with both pressure and heat transfer data. These results demonstrate the capability of modern CFD to predict complex 3D hypersonic turbulent flows of engineering interest.

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Fig. 4 Double wedge inlet geometry Fig. 5 Double wedge inlet flow details

Fig. 6 Double wedge inlet wall pressure & heat transfer 3. Hypersonic flow over ramp

Mach 9 flow over a o38 two-dimensional cooled ramp was measured by Coleman and Stollery [10]. An oblique shock impinges on the boundary layer downstream of the ramp corner, inducing flow separation with subsequent reattachment onto the slanted ramp surface. The reattachment point possesses stagnation-like behavior and a large peak in pressure and heat transfer occurs there. Fig. 7 shows geometry and main flow features. The flow conditions are as follows:

.37.1,295,1070,5.64,/1047.0Re,22.9 06 ====×== ∞∞∞ γKTKTKTcmM w

The boundary layer was allowed to develop over the ramp from freestream conditions with ( ) .5/ =∞µµt A 250×200 mesh was used to guarantee grid-independent solutions. There were at least 20 cells within the viscous sublayer with 1.01 ≈+y . Streamwise grid clustering was centered at the ramp corner. Figures 8 and 9 show, respectively, wall pressure and heat transfer profiles. The tR closure [9] is

observed to predict the separation bubble extent better than the k-ε model does, resulting in more accurate shock capturing and closer agreement with the data.

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Fig. 7 2D ramp basic flow features

Fig. 8 2D ramp wall pressure profile Fig. 9 2D ramp wall heat transfer profile

4. Hypersonic flow over a 70 deg. sphere-cone Heat transfer measurements for a Mach 10 flow over a

70 o sphere/cone/40 o cone-frustum afterbody were done by Hollis and Perkins [11]. This configuration (referred to as Case MP1) was mounted on a sting which was also instrumented. Recently Brown [12] reported comparisons between these data and axisymmetric computations using several turbulence models. The 40,176 quadrilateral mesh used by Brown was employed also for the current calculations. The fluid was air and the freestream conditions were: M

∞= 9.795, T

∞= 52.45 K,

∞ρ = 0.00868

kg/m 3 and T w = 300 K. Freestream turbulence intensity was set to 1% and the length-scale was 2.5 mm. The base radius was

bR =2.54 cm and the Reynolds number, based on the corresponding diameter was 92,000. Due to this low Reynolds number, transition to turbulence occurred only on the sting section. Fig. 10 shows the geometry and eddy viscosity contours, indicating that turbulent flow is maintained on the sting only. Fig. 11 is a Mach contour plot, indicating the main flow features. The flow on the forebody downstream of the detached shock is

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subsonic, the sonic line being located just upstream of the shoulder. This is followed by a strong expansion fan and a large flow separation over the afterbody. The recompression shock is located on the sting, starting at the reattachment point. Fig. 12 compares heat transfer data with corresponding predictions by the S-A [13], SST [14] and k-ε [4] turbulence models. The latter closure includes a freestream generation term equal to the corresponding dissipation rate level in order to leave the k and ε transport equations in equilibrium at the far-field. The SST closure is seen to impose turbulent flow as soon as a solid surface is encountered hence it over-predicts heat transfer levels both on the forebody and on the portion of the sting subjected to reversed flow. The k-ε model, on the other hand, maintains a low level of turbulence along the surface which subsequently develops into a more substantial turbulence field over the aft portion of the sting, enabling very good agreement with the data. The S-A model yields the best prediction for this flow case.

Fig. 10 Hollis body eddy viscosity contours Fig. 11 Hollis body Mach contours

Fig. 12 Hollis body surface heat transfer profile

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5. Hypersonic flow over a flat plate Mach 10 flow calculations over a flat plate, with both adiabatic and cooled walls ( 3/0TTw = ), were carried out on a 122X85 grid with 1≤+y at the first off-wall centroids. The plate's leading-edge was preceded by a section of free-stream flow parallel to it. A free-stream turbulence level, corresponding to µµ /t =50, was imposed at the inflow. The White-Christoph (W-C) skin friction correlation [7], corrected for compressibility effects, was used to compare with the computational results. Fig. 13 compares predictions using CFD++’s k-ε-R closure with the W-C correlation, showing excellent agreement. Fig. 14 repeats the previous calculations using the q-L model [15] and also shows the performance of two compressibility correction methods: Sarkar’s approach and Metacomp’s scheme. The former is not adequate for wall-bounded flows and is seen to severely under-predict the correlation data. Metacomp’s approach, however, yields very good agreement, moving the base-line predictions (already quite good) even closer to the data. Note that the turbulence models predict leading edge laminar-to-turbulent flow by-pass transition.

Fig. 13 M=10 plate flow skin friction Fig. 14 M=10 plate flow compressibility effect

Conclusions Several hypersonic flow cases of engineering and scientific interest have been computed using various turbulence models available in the CFD++ flow solver. In all cases very good agreement between predictions and experimental data of pressure, heat transfer and skin friction were obtained. These results demonstrate the power of CFD++ in predicting hypersonic flows with a high level of confidence due to a synergy between turbulence models and numerical algorithms.

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References [1] O. Peroomian, S. Chakravarthy, S. Palaniswamy and U. Goldberg, “Convergence acceleration for unified-grid formulation using preconditioned implicit relaxation”, AIAA Paper 98-0116, Reno, NV, Jan. 1998. [2] P. Batten, M. A. Leschziner and U. C. Goldberg, “Average-state Jacobians and implicit methods for compressible viscous and turbulent flows,” J. Computational Physics, Vol. 137, pp. 38-78, 1997. [3] M. S. Holden, “Turbulent boundary layer development on curved compression surfaces," Calspan Report No. 7724-1, 1992. [4] U. Goldberg, O. Peroomian and S. Chakravarthy, “A wall-distance-free k-ε model with enhanced near-wall treatment,” ASME J. Fluids Eng., Vol. 120, pp. 457-462, 1998. [5] U. Goldberg, P. Batten, S. Palaniswamy, S. Chakravarthy and O. Peroomian, “Hypersonic flow predictions using linear and nonlinear turbulence closures,” AIAA J. of Aircraft, Vol. 37, pp. 671-675, 2000. [6] M. I. Kussoy, K. C. Horstman and C. C. Horstman, “Hypersonic crossing shock-wave/turbulent-boundary-layer interactions,” AIAA Journal, Vol. 31, pp. 2197-2203, 1993. [7] F. M. White, Viscous Fluid Flow, 1st ed., McGraw-Hill Book Company, 1974. [8] B. E. Launder, B.E. “On the computation of convective heat transfer in complex turbulent flows,” ASME Journal of Heat Transfer, Vol. 110, pp. 1112-1128, 1988. [9] U. Goldberg, "Turbulence closure with a topography-parameter-free single equation model," Int. J. of CFD, Vol. 17(1), pp. 27-38, 2003. [10] G. T. Coleman and J. L. Stollery, “Heat transfer from hypersonic turbulent flow at a wedge compression corner,” J. Fluid Mechanics, Vol. 56, pp. 741-752, 1972. [11] B. Hollis and J. Perkins, “Comparisons of experimental and computational aerothermodynamics of a 70-deg sphere-cone,” AIAA Paper 96-1867, June 1996. [12] J. L. Brown, “Turbulence model validation for hypersonic flows,” AIAA Paper 02-3308, June 2002. [13] P. R. Spalart and S. R. Allmaras, “A one-equation turbulence model for aerodynamic flows,” La Recherche Aerospatiale, Vol. 1, No. 5, 1994. [14] F. R. Menter, “Two-equation eddy-viscosity turbulence models for engineering applications,” AIAA journal, Vol. 32, No. 8, pp. 1598-1605, 1994. [15] U. Goldberg, P. Batten and S. Paslaniswamy, “The q-ℓ turbulence closure for wall-bounded and free shear flows,” AIAA Paper 2004-269, January 2004.