Validation of High-Speed Turbulent Boundary Layer and Shock …lyrintzi/AIAA-2006-894... · 2006-01-19 · Validation of High-Speed Turbulent Boundary Layer and Shock-Boundary Layer

Validation of High-Speed Turbulent Boundary Layer

and Shock-Boundary Layer Interaction Computations

with the OVERFLOW Code

A. B. Oliver∗, R. P. Lillard†, G. A. Blaisdell‡, and A. S. Lyrintzis§

School of Aeronautics and Astronautics

Purdue University

West Lafayette, IN 47907

The capability of the OVERFLOW code to accurately compute high-speed turbulentboundary layers and turbulent shock-boundary layer interactions is evaluated. Config-urations investigated include a Mach 6 flat plate to compare experimental surface heattransfer, a direct numerical simulation (DNS) at Mach 2.25 for turbulent quantities, anda Mach 2.87 flat plate for boundary layer properties and growth. Additionally, severalMach 3 compression ramps are used to compare computations of shock-boundary layerinteractions to experimental laser doppler velocimetry (LDV) data, pitot-probe measure-ments, skin friction and surface pressure measurements. The present paper describes thestudy and presents preliminary results for two of the flat plate cases and two small-anglecompression corner test cases.

Nomenclature

Cf = Local skin friction coefficientLsep = Separation lengthM = Local Mach numberNSt,Ref = Stanton number, NSt,Ref = q̇

ρRef URef Cp(Taw−Tw)

Ps = Static pressurer = Recovery factor, r = 0.89Rex,Ref = Reynolds number, Rex,e = Vex/νe

s = Arc lengthT = TemperatureTaw = Adiabatic wall temperature, Taw = Te

(1 + r

(γ−1

2

)M2

e

)u′v′ = Reynolds shear stressV = Velocity magnitudex = Streamwise coordinate (relative to corner location)y = Crossflow coordinatez = Vertical coordinate (normal to flat plate)δ = Boundary layer thickness (99%Ve)δ∗ = Boundary layer displacement thickness, δ∗ =

∫ 1.2δ

01− ρV

ρeVeds

θ = Boundary layer momentum thickness, θ =∫ 1.2δ

0ρV

ρeVe

(1− V

Ve

)ds

ν = Kinematic viscosity

∗Graduate Research Assistant, Student Member AIAA.†PhD Candidate, Student Member AIAA; Aerospace Engineer, NASA/JSC, Aerosciences and CFD Branch‡Associate Professor, Senior Member AIAA.§Professor, Associate Fellow AIAA.

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American Institute of Aeronautics and Astronautics

44th AIAA Aerospace Sciences Meeting and Exhibit9 - 12 January 2006, Reno, Nevada

AIAA 2006-894

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

Subscriptse = Edge conditions (at s = 1.2δ)Ref = Reference value for data reductionw = Wall condition0 = Stagnation condition∞ = Freestream condition

I. Introduction

Auseful Computational Fluid Dynamics (CFD) tool for the Space Shuttle and future space vehicles needsto accurately predict the surface heat transfer over the entire acreage of the vehicle, including any

localized effects due to protuberances or shock-boundary layer interactions. The acreage flow is generallyattached, so for these regions, the CFD code will be required to accurately model turbulent boundary layers.Perhaps more important is the accurate prediction of any shock-boundary layer interactions, as the heatingrates in these localized regions can be several times larger than the acreage heating levels.

Advances in computer speeds have given rise to advances in the modeling of turbulent flows using DirectNumerical Simulations (DNS) and Large Eddy Simulations (LES). For complex configurations such as theSpace Shuttle, DNS or even LES are still impractical because of the grid spacing requirements and the needfor time-accurate solutions. Although DNS and LES employ a more physics based representation of the fluiddynamics, the Reynolds Averaged Navier Stokes (RANS) Equations, along with a turbulence model, are stilla valuable tool for aerodynamic analysis. The averaging process leaves variables that need to be modeledbecause no explicit relationship between the new quantities and the mean flowfield averaged quantities aregiven (the closure problem). The main quantities that need modeling are the Reynolds stress tensor (RST)and the turbulent heat flux vector. These turbulence models typically use the Boussinesq approximationfor this closure, relating mean strain rate to the RST in a linear fashion. Despite the speed advantages, theperformance of turbulence models are often less than desirable. The models can not capture all the relevantphysics of the problem and are prone to inaccuracies in adverse pressure gradient regions, especially thosewith shock wave-boundary layer interactions that occur in many compressible flows.

The present work is carried out to continue the development of the capability to model high-speed com-pressible flows over complex vehicles. The goal of the current project is to evaluate the performance of theturbulence models in the OVERFLOW code1,2 for high-speed, non-reacting flow. The idea of using OVER-FLOW as an aerothermal analysis tool has received some criticism based on the fact that it was designedto compute aerodynamic forces in transonic and low supersonic flows. Previous work3,4 has indicated thatthe code is indeed capable of accurately modeling laminar hypersonic flows and does an adequate job ofcapturing heat transfer at those speeds. The present study is now directing attention at validating the use ofOVERFLOW for high-speed turbulent flows. Three turbulence models in OVERFLOW will be consideredin this paper: the Spalart-Allmaras model,5 the SST model,6 and the Lag model.7 Results from each modelwill be compared to several different experiments and direct numerical simulation (DNS) computations inorder to benchmark their behavior.

A. Code Description

The OVERFLOW code is an overset (chimera) grid Navier-Stokes solver, which makes it ideal for computingthe flow fields over complex geometries.8 It uses a finite-difference formulation with flow quantities storedat the grid nodes. OVERFLOW has central- and Roe upwind-difference options, and uses a diagonalized,implicit approximate factorization scheme for time advancement. Local timestepping, multigrid techniquesand grid sequencing are all used to accelerate convergence to a steady state solution. In the present study,the Swanson/Turkel matrix dissipation model9 was utilized. With the correct parameters,3 this allows thecentral-differenced scheme to mimic a Total Variation Diminishing (TVD) upwind biased scheme in regionsof flow discontinuities. This method takes advantage of the speed of central differencing while maintainingthe robustness of the upwind TVD scheme. For a complete discussion of the scheme, see Olsen et al.3

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B. Turbulence Models

OVERFLOW has numerous turbulence models coded in, but this study will focus on three models, theSpalart-Allmaras (SA) model,5 the SST model,6 and the Lag model.7 The turbulence models included inthe code all make use of the Boussinesq approximation, meaning that the effects of the Reynolds stressterms are included in the Navier-Stokes equations through an eddy viscosity. The SA model is a popularmodel for aerodynamic flows that solves a single partial differential equation directly for the eddy viscosity.The SST model is a two-equation model which is a hybrid of the k-ε and k-ω turbulence models and solvesequations for the turbulent kinetic energy and the specific dissipation rate. The eddy viscosity is then derivedfrom an algebraic relation that chooses the minimum value between the standard formulation based on theturbulence kinetic energy and the specific dissipation rate and a formulation based on an assumed value ofthe normalized Reynolds shear stress. The Lag model is a recent extension of the k-ω model that accountsfor non-equilibrium effects by carrying an additional differential equation that relaxes the eddy viscosity tothe equilibrium value. This third equation accounts for the time required for the turbulence to respond tochanges in the mean flowfield. Each of the test cases is computed with all three turbulence models to allowcomparisons of the models.

C. Study Objectives

The objective of the current study is to benchmark the performance of these turbulence models, as imple-mented within OVERFLOW, for the high-supersonic to hypersonic regimes experienced by the space shuttleduring ascent. Surface properties, such as surface pressure and heat transfer, are the primary variables ofinterest; however, to aid in the development and implementation of improved models and model corrections,profiles of flow variables and turbulent quantities are also being generated. This paper presents the surfaceproperties and select profiles for the test cases that have been completed.

II. Test Cases

In order to validate the code’s ability to capture a high-speed zero pressure gradient boundary layer,several flat plate cases will be run. Two-dimensional compression ramp experiments have been selectedto analyze the shock/boundary layer interaction behavior, namely the size of the separation region, theprofiles of turbulent quantities downstream of the shock, and the quality of the prediction in the recoveringboundary layer. Since experiments often have to make some compromises in terms of what measurementscan be made, multiple experiments have been chosen for both geometries to capture specific features of theturbulent boundary layers. The experiments have been chosen based on perceived quality of the data andMach number of interest.

A. Flat Plate

In order to adequately address each relevant feature of a high-speed zero pressure gradient boundary layer,three datasets have been selected for comparison. Experimental studies by Cary10 at Mach 6.0 (Tw/T0 = 0.6)and Mach 4.9 (Tw/T0 = 0.3, & 0.6) have been chosen for surface heating data. A DNS simulation byGatski11,12 provides two detailed profiles of mean and turbulent properties. Finally, experiments conductedby Smits, et al.13 at Mach 2.87 have profiles at multiple locations upstream of 8◦ and 16◦ compression corners,and will be used for comparison of high Reynolds number experimental velocity profiles and boundary layergrowth measurements.

B. Compression Ramp

Two supersonic compression ramp datasets have been chosen for comparison. The Mach 2.9 experiments ofKuntz14,15 provide two component laser doppler velocimetry (LDV) measurements of turbulent boundarylayers on ramps of 8◦, 12◦, 16◦, 20◦, and 24◦, and the Mach 2.9 series of experiments 13,16 conducted atthe Princeton Gas Dynamics Laboratory will be used to compare turbulence quantities downstream of theshock and for skin friction and surface pressure distributions on ramps of 8◦, 16◦, 20◦, and 24◦. Settles andDodson report in their 1994 shock-boundary layer interaction database17 that these two datasets disagreeon the magnitude of the Reynolds stresses by as much as a factor of 4. To date, the authors are not aware of

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an explanation of this discrepancy, so following the suggestion of Settles and Dodson, these values are takento be reasonable bounds of the actual Reynolds stresses.

III. Solution Methods

A. Grid Generation and Boundary Conditions

The Gatski flat plate test case uses a simple single-zone grid that does not model the radius of the plateleading edge (the viscous wall extends all the way to the inflow plane). The Cary flat plate cases use single-zone grids that model a finite-radius leading edge, with a radius of 0.0015 inches (pictured in Figure 1). Tomatch the experimental conditions, the Mach 4.9 cases were run using the Mach 6.0 freestream conditionswith the flat plate inclined 8.1◦ to the freestream.

Figure 1. Grid topology for the Cary flat plate cases.

The compression ramp cases use a system similar to that depicted in Figure 2. The long flat plate regionupstream of the corner is used to develop a turbulent boundary layer that matches the momentum thicknessof the experimentally provided inflow profile. The OVERFLOW cases are run fully turbulent, so the lengthof this plate must be adjusted for each case to match the experimental momentum thickness. The cornerand ramp regions are defined by three grids: two high resolution near-wall grids (split for load-balancing onmultiple processors) and a grid in the freestream above the shock wave. It was found that the hyperbolictangent stretching function used for the wall-normal direction did not place the desired number of grid pointsin the freestream above the shock wave, thus a separate zone was created. The final grid is a low resolutionsponge grid to contain the Mach wave generated at the upstream boundary. Sponge grids are generally usedin LES simulations to prevent disturbances from reflecting off of the boundary back into the domain. Inthis case, it was found that the reflected Mach wave interfered with the boundary layer development, sothe sponge grid was used to prevent this interference. An additional grid to better resolve the compressionshock wave has been used, but as will be discussed later in the paper, this grid did not change the computedsurface properties and was not used for the presented results. Table 1 shows the approximate number ofgrid points used in each grid discussed herein.

The boundary conditions for the Gatski and compression corner grid systems are freestream at the inflowboundary, characteristic freestream on the boundary opposite the viscous wall, and extrapolation at theoutflow. The viscous wall is either adiabatic or isothermal, based on the specific dataset being compared.The tops of the compression ramps were included in the simulation so that the expansion fan would insulate

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Table 1. Approximate number of grid points used in presented computations

Case Grid PointsGatski Flat Plate 33.0× 103

Cary Flat Plate 24.9× 103

Smits 8◦ Ramp 85.5× 103

Smits 16◦ Ramp 84.6× 103

the compression corner and ramp face from upstream influences due to the 0th order extrapolation outflowboundary condition and for accuracy of the computations. The Cary cases use characteristic freestream onthe boundary opposite the isothermal viscous wall and outflow at the remaining boundaries.

B. Solution Method

Each of the solutions is initialized to freestream conditions and iterated on a sequence of progressively finergrids until the finest grid level is obtained. These grids are created internally in OVERFLOW by specifyinga number of grid levels (three grid levels would be the original fine grid, a medium grid with every othergrid point used, and a coarse grid with every fourth grid point used). For this study, it was found that threegrid levels and between 500 and 1,000 iterations on the coarse and medium grid levels reliably started upthe solutions. Multigrid sequencing was utilized to accelerate convergence for most of the cases.

In order to rapidly obtain a high quality steady-state solution, the dissipation parameters for the matrixdissipation scheme9 are ramped down to their optimum level as the solution converges. The four parameterscontrolled are the 2nd and 4th-order dissipation coefficients, κ2 and κ4, and the eigenvalue limiters Vεη

andVε`

(DIS2, DIS4, VEPSN, and VEPSL in OVERFLOW, respectively). The presented solutions were startedwith these values set to (10, 0.2, 1.0, 1.0) and were ramped down to (2, 0.1, 0.3, 0.3). For more informationon this process, see Lillard et al.4

The solutions are considered to be iteratively converged with the optimal dissipation settings when: (1)the residuals have dropped by more than three orders of magnitude and, (2) the skin friction at severalrepresentative locations has converged to at least six significant digits. The degree of grid convergence wasdetermined using Richardson extrapolation similar to the method of Roache.18 To quantify the grid conver-gence, the skin friction distributions computed on the fine grids (Table 1) and slightly coarser grids (withgrid spacings approximately 25% greater than spacings in the fine grids) were combined with Richardsonextrapolation to obtain a higher-order prediction of skin friction. Average and maximum deviations of thefine-grid skin friction from the extrapolated skin friction for each case are presented in Table 2 (’RMS ∆Cf ’and ’Max ∆Cf ’ respectively). Additionally, the percent change in length of the separation bubble (∆LSep),determined by negative skin friction, was tabulated as well to determine the convergence of the separationsize. It should be noted that the results presented below are not the extrapolated results, but rather the finegrid solutions.

IV. Results

Work on this project is continuing; however, several computations have been completed and are presentedbelow. The given solutions include the Gatski flat plate, the Cary flat plates, and the mean flow portion ofthe 8◦ and 16◦ compression corners from the Princeton series of experiments. The reference values in Table3 are used to scale the following data.

A. Flat Plate

Gatski DNS

The data available from the Gatski dataset includes skin friction and a velocity profile in the fully turbulentregion of the flat plate (indicated in reference11,12 as the x = 8.8in profile). The streamwise location of theprofiles presented in this section were determined by choosing the location in the fully-turbulent OVERFLOWcomputation that matched the momentum thickness of the available DNS profile.

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Figure 2. Grid topology for 16◦ ramp with every other grid line shown. (a) Overall grid system, (b) Detailview of corner region.

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Table 2. Grid convergence results

Case RMS ∆Cf Max ∆Cf ∆LSep

Gatski Flat Plate (SST) 0.6% 0.8% -Gatski Flat Plate (SA) 0.7% 0.9% -Gatski Flat Plate (Lag) 0.7% 1.1% -Cary Flat Plate: Mach 4.9, Tw/T0 = 0.3 (SST) 5.0% 6.5% -Cary Flat Plate: Mach 4.9, Tw/T0 = 0.3 (SA) 3.2% 4.1% -Cary Flat Plate: Mach 4.9, Tw/T0 = 0.3 (Lag) 6.0% 7.8% -Cary Flat Plate: Mach 4.9, Tw/T0 = 0.6 (SST) 0.3% 0.5% -Cary Flat Plate: Mach 4.9, Tw/T0 = 0.6 (SA) 0.9% 1.0% -Cary Flat Plate: Mach 4.9, Tw/T0 = 0.6 (Lag) 0.4% 0.7% -Cary Flat Plate: Mach 6.0, Tw/T0 = 0.6 (SST) 1.1% 3.2% -Cary Flat Plate: Mach 6.0, Tw/T0 = 0.6 (SA) 0.6% 0.7% -Cary Flat Plate: Mach 6.0, Tw/T0 = 0.6 (Lag) 0.4% 0.7% -Smits 8◦ Ramp (SST) 0.8% 2.0% 2.1%Smits 8◦ Ramp (SA) 1.9% 8.5% 7.9%Smits 8◦ Ramp (Lag) 1.7% 3.7% 3.5%Smits 16◦ Ramp (SST) 2.2% 17.8% 2.6%Smits 16◦ Ramp (SA) 2.0% 8.8% 0.6%Smits 16◦ Ramp (Lag) 2.1% 5.5% 1.1%

Table 3. Reference values used in data reduction

Case Gatski Flat Plate Cary Flat Plates 8◦ Ramp 16◦ RampMRef 2.25 6.0 2.87 2.85

VRef (m/s) 220.5 969.8 590 580δRef (mm) 1.936 - 26 26PRef (kPa) - - 22.52 23.56

ρRef (kg/m3) - 0.1205 - -TRef (K) - 65.0 - -T0 (K) - 533 - -

Table 4 shows that the predicted displacement thickness is reasonably close to the DNS value. The SAmodel performs the best and the Lag model the worst of the three, but all do a decent job predicting theshape factor of a high-speed zero pressure gradient boundary layer. Table 4 also shows the skin friction atthe profile location for each turbulence model against the DNS value. Again, all three models perform well,with SA the best and SST the worst.

Figure 3 shows the velocity profiles for the three turbulence models against the DNS data. Again, allthree models show similar predictions, although the data and other models predict fuller velocity profilesthan the SST model. The DNS velocity profile is fuller than all three computed profiles near the wall.Figure 4 shows the Reynolds shear stress (u′v′) profiles at the same location, and again, the differences aresmall. A larger discrepancy is apparent between the DNS and RANS computations for the Reynolds shearstress than for the velocity profiles. The RANS computations appear to diffuse too much of the Reynoldsshear stress to the outer portion of the boundary layer.

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Figure 3. Velocity profile for Gatski flat plate test case.11,12

Figure 4. Reynolds shear stress profile for Gatski flat plate test case.11,12

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Table 4. Inflow matching results

Case θ [mm]† (% Diff.) δ∗ [mm] (% Diff.) Cf (% Diff.)Gatski Flat Plate (DNS) 0.161 0.545 2.25× 10−3

Gatski Flat Plate (SST) 0.161 (0.0%) 0.569 (4.4%) 2.13× 10−3 (5.1%)Gatski Flat Plate (SA) 0.161 (0.0%) 0.567 (3.9%) 2.21× 10−3 (1.5%)Gatski Flat Plate (Lag) 0.161 (0.0%) 0.573 (5.1%) 2.20× 10−3 (2.1%)Smits 8◦ Ramp (Experiment) 1.3 6.7 1.00× 10−3

Smits 8◦ Ramp (SST) 1.30 (0.1%) 6.71 (0.2%) 1.09× 10−3 (8.7%)Smits 8◦ Ramp (SA) 1.29 (0.4%) 6.72 (0.3%) 1.13× 10−3 (12.3%)Smits 8◦ Ramp (Lag) 1.32 (1.5%) 6.82 (1.8%) 1.11× 10−3 (10.2%)Smits 16◦ Ramp (Experiment) 1.3 6.3 9.84× 10−4

Smits 16◦ Ramp (SST) 1.30 (0.1%) 6.68 (6.1%) 1.08× 10−3 (9.4%)Smits 16◦ Ramp (SA) 1.30 (0.1%) 6.72 (6.7%) 1.11× 10−3 (13.1%)Smits 16◦ Ramp (Lag) 1.32 (1.3%) 6.79 (7.8%) 1.09× 10−3 (11.0%)

†: Note that the grid geometry is varied to match θ as closely as possible, δ∗ and Cf are not controlled andtaken to be a predicted results.

Cary flat plates

Figure 5 and Figure 6 show the heat transfer results for the Mach 4.9 and Mach 6.0 Cary flat plates10

respectively. It appears that the experimental data for these cases are mostly transitional, and since transitionis not modeled, accurate predictions in this region are not expected. Above Rex = 1.0 × 107, however, thedistribution appears fully turbulent. By this point, all three turbulence models have developed a similardecay rate, and the only difference between the three is a change in magnitude. All of the cases overpredictthe heat transfer in the fully developed region, although the decay rate along the plate appears to be withinthe experimental scatter for each case. It should be noted that no adjustment was made for a virtual origin,since momentum boundary layer thickness data are not available for Cary’s experiment. The cold wallcase in Figure 5b shows the most disagreement with the experiment (note from Table 2 however, that theMe = 4.9, Tw/T0 = 0.3 case is not sufficiently grid converged). Each computational plot does not appearto decay smoothly, so it is quite possible that all of the solutions are not fully converged and/or not wellresolved. Work is continuing on these cases to ensure that the solutions are of high-quality.

Princeton compression corner inflow boundary layers

As seen from Table 4, the computed displacement thickness at the inflow location of the Smits13,16 8◦ rampis very close to the experimentally measured value. Analysis of the displacement and momentum thicknessat locations upstream and downstream of this point (data not shown) indicated that the displacement andmomentum thicknesses grow slowly, as expected. The range over which measurements were taken, how-ever, produced displacement and momentum thickness changes that were smaller than the experimentaluncertainty, so meaningful comparisons to computations could not be made. The 16◦ ramp displacementthicknesses are not quite as accurate as the 8◦ predictions, though they are still within reasonable expecta-tions. The skin friction shown in Table 4 is within the expected tolerance. Given the accuracy of the shapefactor and skin friction computations for both this dataset and the Gatski dataset, the turbulent boundarylayers produced over a flat plate in OVERFLOW are sufficient to provide the upstream boundary conditionof the compression corner test cases.

B. Compression Ramps

Freestream conditions for the compression ramp cases13,16 shown are given in Table 5. Figure 7 shows thestatic pressure in the flowfield and on the surface of the 8◦ ramp. For this case, all of the models do a decentjob representing the data. Figure 8 shows the pressure for the 16◦ ramp. The deviation from the experimentsis larger for this case. It appears that the SST model separates too early and the oversized separation bubble

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Figure 5. Heat transfer coefficient along flat plate for Cary test cases,10 Me = 4.9. (a)Tw/T0 = 0.6, (b)Tw/T0 = 0.3.

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Figure 6. Heat transfer coefficient along flat plate for Cary test cases,10 Me = 6.0, Tw/T0 = 0.6.

affects the pressure in the recovering boundary layer. The Lag model predicts the location of the pressurerise very well, but the pressure in the recovering boundary layer falls slightly below the measured pressure.The SA model, on the other hand, misses the pressure rise but gets the recovery pressure profile.

Table 5. Freestream and inflow conditions used in compression ramp computations

Ramp Angle 8◦ 16◦ 24◦

M∞ 2.87 2.85 2.84To (k) 280 268 262

Po (kPa) 680 690 690δ (mm) 26 26 23δ∗ (mm) 6.7 6.3 6.1θ (mm) 1.3 1.3 1.2

xupstream (mm) −25.4 −25.4 −63.5

Looking at the contour plots in Figures 7 and 8, it appears that the shock wave is rather thick, thecontour lines are somewhat wavy, and the pressure overshoots immediately behind the shock wave. This isan artifact of the numerical dissipation scheme used in OVERFLOW, which is known to be sensitive to thealignment of the grid and shock wave. To see whether or not the grid alignment affects the surface properties,a case was run with a special grid to capture and contain the shock wave. This grid, shown in Figure 9a,is aligned closely with the shock predicted from the 16◦ computation. Figure 9b shows contours of staticpressure, and it is apparent that the shock is thinner and less wavy. Figure 9c shows the skin friction for the16◦ ramp with and without the shock grid (using the SST turbulence model). The differences are negligible(less than 1% along the entire length), and similar differences were observed in the surface pressure (notshown), so the shock grids were not used. Shock grids will be used at a later date when profiles of mean andturbulent properties are analyzed in detail.

Figures 10 and 11 show local Mach contours, skin friction distributions, and velocity profiles for the 8◦

and 16◦ ramps, respectively. The skin friction on the 8◦ ramp shows a very small separated region where theexperiments do not observe one (although the resolution of the experiments may not be fine enough to capturea small separated region). The skin friction recovers too rapidly and levels off quicker for the computations as

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Figure 7. (a) Contours of PsPRef

and (b) surface pressure on 8◦ ramp.13,16

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Figure 8. (a) Contours of PsPRef

and (b) surface pressure on 16◦ ramp.13,16

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Figure 9. 16◦ ramp13,16 with shock grid using the SST turbulence model. (a) Grid system, (b) Contours ofPs

PRef, (c) Computed skin friction with and without shock grid.

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Figure 10. 8◦ ramp:13,16 (a) Contours of Mach number, (b) Skin friction distribution, (c) Profiles of velocity.

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Figure 11. 16◦ ramp:13,16 (a) Contours of Mach number, (b) Skin friction distribution, (c) Profiles of velocity.

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well. The 16◦ ramp shows similar behavior; however, the larger separation bubble changes the recovery ratefor each turbulence model. The SA model is interesting, in that it has the smallest separation bubble, butalso has the slowest recovery. The SST model shows the largest separated region, and misses the magnitudeof the skin friction in the recovery region as a result (the rate of recovery is good though). The Lag modelplot shows a smaller separation bubble than the SST and has the most rapid recovery of the three turbulencemodels. The Lag model also shows odd glitches upstream of the corner and more noticeably downstream ofthe corner. These glitches appear to be a bug in the implementation of the Lag model that occur when zonesare split up across multiple processors. We are working with the developer to resolve this issue, although forthe time being, it appears that the glitch is mostly cosmetic with only a local effect on the skin friction.

The profiles shown in Figures 10c and 11c are taken at the locations indicated in Figures 10a and 11a,respectively. For both of these ramps, the shock wave location is apparent, and a slight decrease in theboundary layer thickness is observed. For the 8◦ ramp, the shock-boundary layer interaction is weak enoughthat most of the differences between the three turbulence models are small. However, for the 16◦ ramp,the differences are larger and characteristic features of RANS predictions begin to appear. In each of theprofiles downstream of the shock, the flow near the wall slows down too much, and the effects of the oversizedseparation bubble are apparent in the profiles at stations 2 and 3.

It appears that the adverse pressure gradient created over the 16◦ ramp is strong enough to begin tohighlight differences in these three turbulence models, but more severe ramp angles need to be studied toanalyze sensitivity to pressure gradient. Work is presently underway to obtain steady solutions of the 20◦

and 24◦ compression corners; however, initial computations often do not converge to a steady solution. Oneoption currently being explored is running OVERFLOW in a time-accurate unsteady calculation, similar toOlsen et. al.19 This method looks promising and work is currently underway to ensure that the solutionsare well resolved.

V. Conclusions and Future Work

The primary objective of this project is to evaluate the capability of the OVERFLOW code to accuratelypredict heat transfer in turbulent hypersonic flows. Multiple flat plate and compression corner experimentshave been selected for the evaluation of specific features of turbulent high-speed boundary layers. Preliminaryresults seem to show that for weak shock-boundary layer interactions, the SA, SST, and Lag turbulencemodels behave similarly. However, differences begin to appear for a 16◦ ramp at Mach 3. All of the modelspredict separation not supported by the experiments, with the SST model being the worst at overpredictingthe separation size. The Lag model appears to do the best overall of the three models for the few casesshown in this paper. Current and future work in this study is focused on obtaining high-quality solutions withrigorous grid resolution studies for all test cases and identifying what aspects of the code need improvementto make OVERFLOW a useful aerothermal analysis tool.

Acknowledgments

This work is supported by the NASA Johnson Space Center under Grant No. NNJ04HI12G. We wouldalso like to thank Tom Gatski for providing information on his DNS simulations, Dave Kuntz and Lex Smitsfor providing us with documents and information on each of their experiments.

References

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7Olsen, M. E. and Coakley, T. J., “The Lag Model, a Turbulence Model for Non Equilibrium Flows,” AIAA Paper No.2001-2564, June 2001.

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9Swanson, R. and Turkel, E., “On Central-Difference and Upwind Schemes,” Tech. Rep. 182061, NASA Langley ResearchCenter, Hampton, VA, June 1993, NASA Contractor Report (ICASE).

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