EVALUATION OF TWO-DIMENSIONAL HIGH-SPEED TURBULENT BOUNDARY LAYER AND SHOCK-BOUNDARY LAYER INTERACTION COMPUTATIONS WITH THE OVERFLOW CODE.

Transcription

1 EVALUATION OF TWO-DIMENSIONAL HIGH-SPEED TURBULENT BOUNDARY LAYER AND SHOCK-BOUNDARY LAYER INTERACTION COMPUTATIONS WITH THE OVERFLOW CODE A Thesis Submitted to the Faculty of Purdue University by Anthony Brandon Oliver In Partial Fulfillment of the Requirements for the Degree of Master of Science in Engineering August 2006 Purdue University West Lafayette, Indiana

2 For my parents, whose endless support and encouragement have made this possible. ii

3 iii ACKNOWLEDGMENTS First and foremost, I would like to thank Randy Lillard for his help and support throughout this work. I would also like to thank Tom Gatski, Dave Kuntz, and Lex Smits for providing us with documents and information on each of their experiments. A special thanks is also due to Megan and David, who had to put up with me and did wonders keeping me on track during the final months of this project. This work was supported by the NASA Johnson Space Center under Grant No. NNJ04HI12G.

4 iv TABLE OF CONTENTS Page LIST OF TABLES vii LIST OF FIGURES viii SYMBOLS xi ABBREVIATIONS xiii ABSTRACT xiv 1 Introduction Test Cases Flat Plates Cary Experiments Gatski DNS Compression Ramps Princeton Kuntz Cone-Flare Computational Methods Code Description Turbulence Models Spalart-Allmaras (SA) Model Menter s (SST) Model Olsen & Coakley s Lag Model Grid Generation Flat plates Compression ramps Cone-Flares

5 v Page 3.4 Boundary Conditions Numerical Solution Grid Convergence Data Reduction Integrated Boundary Layer Properties Reynolds Shear Stress Sutherland s Viscosity Law Formulation Results Grid Convergence Gatski Flat Plate DNS Cary Flat Plates Princeton Compression Ramps Kuntz Compression Ramps Holden Cone-Flares Zero Pressure-Gradient Boundary Layers Flat Plate Discussion Shock-Boundary Layer Interactions Surface Pressure Skin Friction & Heat Transfer Separation Length Velocity Profiles Reynolds Shear Stress Profiles SWBLI Discussion Summary Conclusions Future Work Comments on Future Aerothermal Heating Environment CFD Validation Experiments

6 vi Page LIST OF REFERENCES A 2-D Richardson Extrapolation Formulation B Qualitative Grid Convergence Plots

7 vii Table LIST OF TABLES Page 2.1 Freestream and inflow boundary layer data for compression ramp cases Grid dimensions used in presented computations Compression corner case-dependent geometric parameters OVERFLOW flow condition inputs used for all test cases Reference values used in data reduction Gatski DNS Results Cary flat plate: Grid convergence results Princeton Compression Corners: Grid convergence results Kuntz Compression Corners: Grid convergence results RMS change of skin friction due to shock grid, scaled by upstream skin friction Holden Cone-Flares: Grid convergence results Princeton Compression Ramps: 8 and 16 inflow boundary layer results 59

8 viii Figure LIST OF FIGURES Page 1.1 Illustration of a typical shock-boundary layer interaction. Note that the ramp angle and coordinate definitions are shown in red Grid topology for the Cary flat plate cases. (a) Entire domain, (b) Detail of plate leading edge colored by static pressure General grid topology for the compression ramp cases. (a) Entire domain, (b) Detail of corner region Static pressure contours over the Princeton 16 ramp: (a) Without shockgrid, (b) With shock-grid General grid topology for the cone-flare cases. (a) Entire domain, (b) Detail of tip region, (c) Detail of corner region, colored by static pressure Velocity profile for Gatski flat plate test case [19, 20] Reynolds shear stress profile for Gatski flat plate test case [19, 20] Heat transfer distribution on the Mach 4.9 Cary flat plates. (a) T w /T 0 = 0.6, (b) T w /T 0 = Heat transfer distribution on the Mach 6.0 Cary flat plate. T w /T 0 = Surface pressure distribution on 8 ramps. (a) Princeton, (b) Kuntz Surface pressure distribution on 16 ramps. (a) Princeton, (b) Kuntz Surface pressure distribution on 20 ramps. (a) Princeton, (b) Kuntz Surface pressure distribution on 24 ramps. (a) Princeton, (b) Kuntz Surface pressure distribution on cone-flares. (a) 36 flare, (b) 42 flare Skin friction distribution for Princeton dataset. (a) 8, (b) Skin friction distribution for Princeton dataset. (a) 20, (b) Heat transfer distribution for Princeton 16 ramp Heat transfer distribution on cone-flares. (a) 36 flare, (b) 42 flare Separation length vs. ramp angle: (a) Princeton dataset, (b) Kuntz dataset. 72

9 ix Figure Page 4.15 Normalized separation length for Princeton and Kuntz datasets vs. (a) Ramp angle, (b) Re δ Kuntz 8 ramp. (a) Contours of local Mach number, with profile locations indicated, (b) Velocity magnitude profiles, (c) Reynolds shear stress profiles Kuntz 16 ramp. (a) Contours of local Mach number, with profile locations indicated, (b) Velocity magnitude profiles, (c) Reynolds shear stress profiles Kuntz 20 ramp. (a) Contours of local Mach number, with profile locations indicated, (b) Velocity magnitude profiles, (c) Reynolds shear stress profiles Kuntz 24 ramp. (a) Contours of local Mach number, with profile locations indicated, (b) Velocity magnitude profiles, (c) Reynolds shear stress profiles Princeton 8 ramp. (a) Contours of local Mach number, with mean profile locations indicated, (b) Velocity magnitude profiles Princeton 16 ramp. (a) Contours of local Mach number, with mean profile locations indicated, (b) Velocity magnitude profiles Princeton 20 ramp. (a) Contours of local Mach number, with mean profile locations indicated, (b) Velocity magnitude profiles Princeton 24 ramp. (a) Contours of local Mach number, with mean profile locations indicated, (b) Velocity magnitude profiles Princeton 8 ramp. (a) Contours of local Mach number, with turbulent profile locations indicated, (b) Reynolds shear stress profiles Princeton 16 ramp. (a) Contours of local Mach number, with turbulent profile locations indicated, (b) Reynolds shear stress profiles Princeton 20 ramp. (a) Contours of local Mach number, with turbulent profile locations indicated, (b) Reynolds shear stress profiles Profile comparison at x/δ Ref 3 on the 20 ramps. (a) Mean velocity profiles - Princeton mean profile #8, Kuntz profile #4; (b) Reynolds shear stress profiles - Princeton RSS profile #6, Kuntz profile # A.1 (a) Coarse grid of the 42 cone-flare case. (b) Leading error terms (scaled by the local a 0 ) of the skin friction distribution B.1 Grid convergence of skin friction, Kuntz dataset (a) 8, (b) B.2 Grid convergence of skin friction, Kuntz dataset (a) 20, (b) B.3 Grid convergence of skin friction, Princeton dataset (a) 8, (b) B.4 Grid convergence of skin friction, Princeton dataset (a) 20, (b)

10 x Figure Page B.5 Grid convergence of skin friction, Evans 16 ramp B.6 Grid convergence of skin friction, cone-flare dataset (a) 36 flare, (b) 42 flare.105

11 xi SYMBOLS a C f C h k L c L sep M P q Re s T u u v v V x y y + z C f L sep x δ Local speed of sound Skin friction Stanton number Turbulent kinetic energy Separation length correlation Separation length (streamwise extent only) Local Mach number Pressure Dynamic pressure Reynolds number Arc length Temperature Velocity component in x-direction (streamwise) Reynolds shear stress Velocity component in z-direction (wall-normal) Velocity magnitude Streamwise coordinate Spanwise coordinate Non-dimensional turbulent wall spacing (wall-normal) Wall-normal coordinate Relative change in skin friction due to grid coarsening Relative change in separation length due to grid coarsening Grid spacing Boundary layer thickness

12 xii δ ε θ µ Viscosity ν t ρ ω Boundary layer displacement thickness Relative error Boundary layer momentum thickness Eddy viscosity Density Turbulent specific dissipation rate Subscripts 0 Stagnation condition e Ref s Boundary layer edge condition Reference condition Static condition Freestream condition

13 xiii ABBREVIATIONS CFD DNS FP GCI LDV LES NS RANS RE RMS RSS RST SA SST SWBLI Computational fluid dynamics Direct numerical simulation Flat plate Grid Convergence Index Laser doppler velocimetry Large eddy simulation Navier-Stokes equations Reynolds-averaged Navier-Stokes equations Richardson extrapolation Root-mean-square Reynolds shear stress Reynolds stress tensor Spalart-Allmaras turbulence model Menter s shear stress transport turbulence model Shock-wave boundary layer interaction

14 xiv ABSTRACT Oliver, A. Brandon M.S.E., Purdue University, August, Evaluation of Two- Dimensional High-Speed Turbulent Boundary Layer and Shock-Boundary Layer Interaction Computations with the OVERFLOW Code. Major Professors: Anastasios S. Lyrintzis and Gregory A. Blaisdell. Accurate prediction of turbulent boundary layers and turbulent shock-boundary layer interactions is a critical capability required for a useful computational fluid dynamics (CFD) tool used to predict the aerodynamic and aerothermal loads on present and future space vehicles. The purpose of the present study is to provide information on the performance of baseline turbulence models, which will be useful in evaluating future turbulence model modifications. Two popular turbulence models, the Spalart-Allmaras model and Menter s SST model, and one relatively new model, Olsen & Coakley s Lag model, are evaluated using the OVERFLOW code. The zeropressure gradient turbulent boundary layer predictions are evaluated by comparison to: experimental heat transfer data from a Mach 6 flat plate experiment, flat plate direct numerical simulation (DNS) results for turbulent properties at Mach 2.25, and experimental measurements of flat plate boundary-layer growth at Mach Turbulent shock-boundary layer interaction predictions are evaluated with three different experimental datasets: a series of compression ramps at Mach 2.87, a series of compression ramps at Mach 2.94, and an axisymmetric cone-flare at Mach 11. Each of the experimental datasets include flows with no separation, moderate separation, and significant separation, and different experimental measurement methods, including laser doppler velocimetry (LDV), pitot-probe, inclined hot-wire probe, Preston tube skin friction, and surface pressure. It is shown that the zero pressure gradient predictions of boundary layer properties and velocity profiles are in reasonable agreement with the experimental data, although all models have a tendency to overpredict heat transfer

15 xv and miss the decay rate. The predictions of surface pressure, skin friction, and velocity profiles for weak shock-boundary layer interactions are reasonably accurate. For strong shock-boundary layer interactions, all of the turbulence models fail to predict the correct skin friction distribution and separation size. In nearly every case, surface pressure predictions show too much upstream influence and early separation. Heat transfer predictions show large variations between the three models. Profiles of mean velocity indicate too much flow deceleration near the wall for stronger interactions. Reynolds shear stress profiles show peaks too close to the wall and too rapid a recovery from the interaction. Overall, it seems that these models are capable of predicting the surface pressure on simple 2D compression-corner shock boundary layer interactions with reasonable accuracy (except for the initial pressure rise for separated flow), but much work remains before accurate predictions of thermal environments over a flight vehicle may be obtained.

16 1 1. INTRODUCTION A useful computational fluid dynamics (CFD) tool for the Space Shuttle and future space vehicles needs to accurately predict the surface heat transfer over the entire vehicle, including any localized effects due to protuberances or shock wave-boundary layer interactions (SWBLIs). The acreage flow (where the variations in geometry are relatively gradual) is generally attached, so for these regions, the CFD code will be required to accurately model turbulent boundary layers. Perhaps more importantly, however, the flow around protuberances generates shock-boundary layer interactions that can produce localized peaks in the heating rates that may be several times larger than the acreage heating levels. Figure 1.1 shows an illustration of a supersonic compression ramp SWBLI. From an understanding of inviscid compressible flow, it is known that a concave corner will generate an oblique shock wave, and that the static pressure will rise across the shock. When the viscous boundary layer is introduced in the corner, the behavior changes. The low-velocity fluid near the wall is subsonic, so the pressure downstream of the shock wave can feed upstream. The pressure rise appears to the boundary layer as an adverse pressure gradient. If the pressure rise is strong enough, the boundary layer will separate, and form a separation bubble, as shown in Figure 1.1. The recirculation zone appears to the inviscid flow as a small bridge across the corner, making it look like two small corners (one at the separation point, one at the reattachment point) rather than a single corner. Each corner generates compression waves that eventually merge into the single inviscid oblique shock, forming the so-called lambda shock system. This structure is typical of all separated compression-corner type shock-boundary layer interactions and is not a specific characteristic of laminar or turbulent boundary layers; although as would be expected, turbulent boundary layers are more resistant to separation than laminar boundary layers and the interaction tends to be unsteady.

17 2 The primary region of interest for high-speed vehicle design is near the reattachment region. At this point, the boundary layer has become very thin, and the boundary layer begins to recover back to a zero-pressure gradient boundary layer. The skin friction and surface heating in the recovering boundary layer are typically extremely large, and can be orders of magnitude larger than flat-plate boundary layer values at high Mach numbers. The location of the reattachment point is largely dependent on the separation location. As it turns out, predicting the separation point is difficult, so accurately predicting the reattachment point is very hard. Thus, obtaining accurate estimates of maximum heating and pressure loads in a compression corner is a very challenging task. A significant amount of work has been done to study turbulent shock-separated compression corner flows. Smits and Dussauge [1] provided and excellent discussion of compressible turbulence phenomena and dedicate a chapter to a discussion of SWBLI experimental and DNS results. Dolling [2] discusses the history and current issues regarding turbulent SWBLI research, with particular emphasis on shock unsteadiness issues. In 1991 (and updated in 1994), Settles & Dodson [3,4] performed a thorough review of turbulent SWBLI experiments in an effort to provide a database of highquality experiments suitable for CFD turbulence model validation. All of the shockboundary layer interaction test cases chosen for the present study were labeled as acceptable for CFD validation in this database. Most of the Navier-Stokes (NS) computational codes that are used for design work solve the Reynolds Averaged Navier-Stokes (RANS) equations. These equations are based on the NS equations, except that the flow variables are replaced by the sum of mean and fluctuating components and the equations are averaged. These equations can be rearranged such that the equations of the mean flow terms match the original NS equations, with the addition of one more term: x j ρu i u j. The ρu i u j is known as the Reynolds stress tensor (RST) and is a function of the turbulence in the flow. This term cannot be exactly computed based on the mean flow variables alone (the closure problem). Turbulence models are necessary to estimate this term. Most turbulence

18 3 models use the Boussinesq approximation to relate the RST to the mean strain rate in a linear fashion. These models can not capture all the relevant physics of the problem and are prone to inaccuracies in adverse pressure gradient regions, especially those with shock wave-boundary layer interactions that occur in many compressible flows. A thorough review of previously conducted RANS studies of shock-boundary layer interactions was not performed; however, two reviews have been published by Wilcox [5] and Knight et al. [6] that summarize common trends in RANS computations of SWBLI flows. Wilcox [5] gives a list of four traits that are typical of RANS SWBLI computations: Too little upstream influence (computed wall pressure begins rising downstream of measured start of adverse pressure gradient) Surface pressure too high in the separated region Skin friction and heat transfer too high downstream of reattachment Velocity profiles downstream of interaction show too much flow deceleration near the wall Wilcox describes several model modifications that have been tried, but none have been successful at improving all four points. Many of the corrections, however, have questionable physical basis (corrections using wall functions in the separated region, for instance). In 2003, Knight et al. [6] conducted a survey of recent SWBLI CFD studies and discuss the capabilities and limitations of the computations. In general, they state that RANS methods are reasonably accurate for weak interactions, but show significant discrepancies with experiment for strong interactions. RANS computations of strong interactions typically have a difficult time predicting surface pressure distributions and the separation lengths, with the separation point being particularly difficult to predict. They also place strong emphasis on the fact that RANS methods fail to predict the shock unsteadiness discussed by Dolling [2], and note that the

19 4 discrepancies of the pressure predictions are possibly a result of this shortcoming. Similarly, RANS calculations do not provide an RMS fluctuating pressure in the separated region an important parameter for aerodynamic design. Knight et al. finally note that several ad-hoc modifications have been tried, with varying degrees of success; however the corrections are not typically general in nature and not extensible to generic flows. The results of the present study will be discussed in light of the points mentioned by Wilcox and Knight et al. One feature of turbulent SWBLIs is that turbulence is amplified by the shock wave. The Boussinesq approximation is believed [7] to amplify turbulence too greatly. Sinha et al. [8] have produced a modified verison of the Spalart-Allmaras model that reduces the production of turbulence due to the shock wave. This model is applied to a coneflare experimental dataset, and it is shown that the model significantly underpredicts the surface pressures and heat transfer in the Mach 11 flow. This study will consider this experimental dataset as well and will be used to breifly compare the OVERFLOW results to the results of another CFD code [9]. Advances in computer speeds have given rise to advances in high accuracy methods of simulating and modeling turbulent flows. Direct numerical simulations (DNS) do not model any turbulence effects, and directly solve for the time-dependent motion of turbulent structures. However, grid requirements typically restrict DNS to low Reynolds numbers. Large eddy simulations (LES) are similar to DNS in that they solve for the time-dependent motion of the large scale turbulent structures; however, the fine-scales are filtered out and modeled with a sub-grid-scale turbulence model. For complex configurations such as the Space Shuttle, grid requirements and the need for time accurate solutions make DNS and LES impractical. Although DNS and LES employ a more physics based representation of the fluid dynamics, the RANS equations, along with a turbulence model, are still a valuable tool for aerodynamic analysis. The present work continues the development of modeling high-speed compressible flows over complex vehicles. The goal of the current project is to evaluate the per-

20 5 formance of a set of baseline turbulence models using the OVERFLOW code [10, 11] for high-speed, non-reacting flow. The idea of using OVERFLOW as an aerothermal analysis tool has received some criticism based on the fact that it was designed to compute aerodynamic forces in transonic and low supersonic flows. Previous work [12,13] has indicated that the code is indeed capable of accurately modeling laminar hypersonic flows and does an adequate job of capturing heat transfer at those speeds in some simple cases. The present study is now directing attention at validating the use of OVERFLOW for high-speed turbulent flows. As this study is one of the first steps of many in the development of the aeroheating capability of the OVERFLOW code, only simple two-dimensional canonical flow configurations are considered and no chemistry effects are included. The geometries considered in this study are flat plates for zero pressure gradient boundary layers and compression corners for SWBLIs. It should be noted that the compressioncorner type SWBLI is only one of several different types of SWBLIs. This type was chosen for comparison because of the availability of data in the open literature and the relative simplicity of the flow. This study focuses only on cold-flow conditions (i.e. negligible chemistry effects), so these conditions would more closely represent conditions experienced by the Space Shuttle on ascent rather than atmospheric reentry. These simplifications are significant as many real-world computations will require much more complicated geometries and flow patterns, as well as some degree of chemistry effects. Three turbulence models in OVERFLOW will be considered in this paper: the Spalart-Allmaras model [14], the SST model [15], and the lag model [16]. Results from each model will be compared to several different experiments and DNS computations in order to benchmark their behavior with canonical flows. Surface properties, such as surface pressure, skin friction, and heat transfer, are the primary variables of interest; however, to aid in the development and implementation of improved models and model corrections, profiles of flow variables and turbulent quantities will also be

21 6 presented. Based on the results of this work, future studies will consider modifications to these turbulence models in order to improve their accuracy. Chapter 2 of this document outlines the experimental and DNS test cases used to benchmark the turbulence model performance. Chapter 3 gives a detailed description of the numerical methods, grid generation, and boundary conditions used. Chapter 4 presents representative results of the CFD solutions, and Chapter 5 summarizes and concludes the work. The appendices contain grid refinement data results not fully discussed in Chapters 3 and 4. Throughout this document, an effort has been made to retain the original units used in each of the test cases considered. That makes this document a little more difficult to read, but it is done to reduce the likelihood of introducing errors. Preliminary results can be found in reference [17]. Fig Illustration of a typical shock-boundary layer interaction. Note that the ramp angle and coordinate definitions are shown in red.

22 7 2. TEST CASES In order to assess the code s ability to capture a high-speed zero pressure gradient boundary layer, several flat plate cases have been run. Also, two-dimensional compression ramp experiments and axisymmetric cone-flare experiments were selected to analyze the shock/boundary layer interaction behavior, namely the size of the separated region, the profiles of mean velocity and turbulence quantities downstream of the shock, and the quality of the prediction of surface pressure, skin friction and heat transfer in the recovering boundary layer. Since experiments often have to make some compromises in terms of what measurements can be made, multiple experiments have been chosen for these geometries to capture specific features of the turbulent boundary layers. The experiments have been chosen based on available experimental data, perceived quality of the data, and Mach number of interest. There have been many experiments performed that could relate to this study, however many (if not most) of the experiments are not documented well enough or of a high enough quality to be used for meaningful code assessment and validation. For instance, in their shockboundary layer interaction database, Settles and Dodson [3] examined 105 papers, but only list 12 experiments as being of acceptable quality for CFD validation. None of the chosen datasets contain all the data that we would prefer for the assessment and all of the datasets are rather old (the latest experiment being published in 1991). Many of the cases considered are supersonic instead of hypersonic, simply because there is not enough useful data in the hypersonic regime to achieve the turbulence modeling goals. There are several aspects of each test case that qualifies the conclusions that may be drawn. These qualifications are noted in the following description of each dataset.

23 8 2.1 Flat Plates In order to address several relevant features of a high-speed zero pressure gradient boundary layers, three datasets have been selected for comparison. Experimental studies by Cary [18] at Mach 6.0 and Mach 4.9 have been chosen for surface heating data. A DNS simulation by Gatski [19, 20] provides detailed profiles of mean and turbulent properties. Finally, experiments conducted at the Princeton Gas Dynamics Lab [21, 22] 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 layer growth measurements Cary Experiments Cary [18] conducted several experiments on a cooled flat plate model in Mach 6.0 flow for his 1969 masters thesis. He measured static pressure and heat transfer on the surface of the model in order to study transition to turbulence at high speeds, where the wall temperature to stagnation temperature ratio was a varied parameter (the wall temperature was varied). In addition, he presents data for Mach 4.9 flow obtained by inclining the plate 8.1 with respect to the freestream. The Mach 6.0 (T w /T 0 = 0.6) and Mach 4.9 (T w /T 0 = 0.3 & 0.6) runs have been chosen for surface heating comparisons in the present study. This dataset was identified and selected prior to the author s joining the study, and is reportedly [23] one of the few datasets available with heat transfer on a flat plate model. This data was generated primarily to study transition for varying Mach numbers and wall-temperature ratios. Since the heat transfer values were not the objective (the trends were), the scatter in the heat transfer measurements is rather large and the Reynolds numbers achieved are somewhat lower than would be desired (fully turbulent boundary layers were not of interest in the experiment).

24 Gatski DNS Gatski [19, 20] generated a DNS solution of a spatially evolving flat plate boundary layer for use as the inflow conditions for an impinging SWBLI simulation. The freestream Mach number was 2.25 and the freestream Reynolds number was in 1. The numerical scheme used an upwind biased, implicit, finite difference scheme with 2nd order temporal and 4th order spatial accuracy. The flow variables at the inflow boundary was specified by a laminar similarity solution, and transition was forced by wall suction and blowing. The wall temperature was held constant at the theoretical adiabatic wall temperature. Although the freestream boundary opposite the wall was specified as a symmetry plane (meaning that the simulation is a channel flow simulation rather than a flat plate simulation), flat plate behavior can be assumed due to the large ratio of momentum thickness to channel width (about 300) and because grid stretching was used to attenuate the disturbances near the boundary. Power spectra and two point correlations were used to verify the domain size and the solution quality. Gatski has provided us with the mean skin friction distribution on the plate and a profile of mean and fluctuating flow variables at Re x = (Re θ = ). This DNS provides a very detailed profile with mean and turbulence data; however, the method used a non-physical method to generate the fully turbulent boundary layer (similar to most DNS studies), so assumptions had to be made about the leading edge conditions to compare to the data. In the present study, the leading edge is not explicitly modeled (see Chapter 3) and comparisons are made where the CFD momentum thickness matches the DNS momentum thickness. This, in effect, limits the comparison to a comparison of the boundary layer shape factor. The profiles do still provide a valuable comparison, however.

25 Compression Ramps Two supersonic compression ramp datasets with freestream Mach numbers near 2.9 have been chosen for comparison. The experiments of Kuntz [24 26] provide two component laser doppler velocimetry (LDV) measurements of turbulent boundary layers on ramps of 8, 12, 16, 20, and 24, and the series of experiments [21, 22] conducted at the Princeton Gas Dynamics Laboratory will be used to compare turbulence quantities downstream of the shock and for skin friction and surface pressure distributions on ramps of 8, 16, 20, and 24. There are a few common problems with these datasets that should be explicitly noted. First of all, it should be noted that the two-dimensionality assumption is a rather large assumption, and it does not necessarily hold for the the larger ramp angles. Additionally, both of the datasets use ramp models attached to the floor of the wind tunnel, so the turbulent boundary layers had previously experienced a strong distortion through the nozzle. The effect this had on the turbulence is unknown. Finally, Settles and Dodson report in their 1994 shock-boundary layer interaction database [4] that these two datasets disagree on the magnitude of the Reynolds stresses by as much as a factor of 4. To date, the authors are not aware of an explanation for this discrepancy, so following the suggestion of Settles and Dodson, these values are taken to be reasonable bounds of the actual Reynolds stresses. Despite these problems (and a few ones particular to each dataset described below), these two datasets appear to be the most rigorous experiments to cover a large range of interaction strengths in the same experiment Princeton Several researchers conducted a series of compression corner experiments in the Princeton 20 cm 20 cm high Reynolds number supersonic wind tunnel. A significant number of parameters were studied in this series of experiments; however, only a

26 11 subset of this data will be considered here. A description of these selected experiments follows. The ramp angles considered (8, 16, 20, and 24 ) provide flows that range from unseparated to significantly separated. The 6 inch wide compression ramp models were mounted on the wind tunnel floor with one inch of clearance on either side to permit the passage of the sidewall boundary layers. The ramp model station was located far enough downstream of the nozzle that the turbulence in the boundary layer appeared to have fully recovered from distortion in the nozzle. The boundary layer was nominally 26 mm thick (θ = 1.3 mm) upstream of the ramp corner location, although the specific values varied for each case. The freestream Reynolds number was approximately m 1, which gives Re θ = , two inches upstream of the ramp corner. The temperature of the tunnel walls and ramp models was not controlled; however, the temperature was observed to be approximately 1.05 times the stagnation temperature (hot-wall conditions). A summary of the case specific conditions is given in Table 2.1. Settles et al. [21] analyzed the mean flowfields using several methods. Pitot and static pressure probes were used to make measurements of mean velocity in the boundary layers upstream, inside, and downstream of the interaction. Static pressure measurements were made on the surfaces of all models, and Preston tubes were used to measure skin friction. Surface streak and schlieren photography were used to observe separation behavior. Two dimensionality of the flow was verified with surface streak measurements (aerodynamic fences were necessary for the higher ramp angles to achieve two dimensionality). Interference effects due to the various probes were assessed with schlieren photos, surface pressure measurements, and surface streak observations. Smits et al. [22] revisited the Settles et al. [21] dataset and added to it by using hot-wires to measure fluctuating quantities. It does not appear that Smits et al. repeated any new mean flow measurements (likely because the flow conditions were not significantly different from those used by Settles et al.) and used the Settles et al.

27 12 measurements in their data reduction process (see note on page 64 of reference [3]). Single normal hot-wires were used to measure the normal Reynolds stresses and longitudinal mass flux fluctuations; inclined wires were used to measure Reynolds shear stresses. Evans et al. [27] instrumented the 16 ramp with thin-film gages and measured heat transfer from the surface of the ramp in the recovering boundary layer. These runs deviate from the Settles et al. flow conditions more than the Smits et al. runs, as the wall temperature increased by 13 K (5%). While not explicitly stated in the literature, this change could be the result of taking measurements at the start of the run when the tunnel wall was at room temperature instead of later in the run after the walls have had some time to cool down closer to the flow recovery temperature. Of course, it is also possible that the ambient conditions in the facilities were colder when Settles was doing his experiments than they were for Evans. The Settles et al. [21] dataset and the Smits et al. [22] datasets are both listed as acceptable experiments in the Settles and Dodson database [3, 4], and the Settles et al. [21] dataset is included in the AFOSR-HTTM-Stanford Conference on Complex Turbulent Flows [28] Kuntz Kuntz et al. [24 26] conducted a series of compression corner SWBLI experiments using a non-intrusive two-component LDV system to measure flow velocity and turbulence properties. The range of ramp angles were chosen to span the full range of possibilities: flows with no separation, incipient separation, and significant separation. The ramp models were located on the floor of a 10.2 cm 10.2 cm blow-down wind tunnel, and spanned the entire width of the tunnel test section. The Mach number in the test section was determined to be 2.94 based on surface static pressure measurements, and two-dimensionality was determined via observation of surface-streak patterns. Three-dimensionality was observed in the 16, 20, and 24 cases, but the reattachment line was reportedly straight for the inner 7 cm of the 10.2 cm test sec-

28 13 tion width. The stagnation pressure was approximately 483 kpa (70 psia), and the total temperature varied from 21 C to 29 C (the room temperature of the facilities). Given the short run times (90 seconds max) and the thick aluminum construction, the walls likely approximated isothermal walls near the stagnation temperature [29]. Incoming boundary layer properties were measured using the LDV system in the tunnel without a ramp model in place. The no-ramp setup was also used to determine the freestream conditions. These runs determined that the boundary layer at the corner location was 8.27 mm thick (based on 99% of freestream velocity), and numerical integration of the compressible profiles (assuming adiabatic behavior of the boundary layer to determine density) yielded displacement and momentum thicknesses of 3.11 mm and 0.57 mm, respectively. The Reynolds number based on boundary layer thickness was Re δ0 = (Re θ = ). The profiles were least-squares fit to find the wake strength parameter, Π, and the skin friction, C f, using the transformed wall-wake law presented by Maise and McDonald [30]. This fit found Π = 0.98 and C f = for the undisturbed boundary layer. Kuntz [26] says that the value of the wake strength parameter is rather large, but agrees with several other experiments. Table 2.1 provides the detailed flow conditions reported for this series of experiments. Schlieren photography was used to ensure the entire flow was supersonic for each model, and some unsteadiness in the shock was observed. Separation was observed in the 16, 20, & 24 corners with both Schlieren photography and surface streak measurements. It was observed that lengths of all but the 24 ramp were long enough to permit full pressure recovery. However, the ramp lengths were not long enough to permit the turbulence to return to an equilibrium state after passing through the shock. Settles and Dodson reported in [3] that the boundary conditions were not clear and that the LDV data may be inconsistent. This paper only references the AIAA journal article [24]. The experiment was tagged unacceptable. However, in Reference [4], they change their minds about this dataset and label it as acceptable. They note, however,

29 14 that the magnitudes of the Reynolds stresses measured with the LDV system are 2 to 4 times larger than the values reported by Smits et al. [22] using hot-wires. Kuntz et al. [26] speculate that difficulties calibrating inclined hot-wires are responsible for the discrepancy. However, the actual source of the discrepancy is not known. Settles and Dodson [4] decided to include both datasets to provide an idea of the reasonable limits of Reynolds shear stress in this class of flow. 2.3 Cone-Flare One major drawback to the two-dimensional plane compression ramp is that the flow is not exactly two dimensional; end effects are always present and their influence on the interaction is likely very complicated. An axisymmetric cone-flare model at zero angle of attack alleviates this problem. The dataset obtained by Holden et al. [31] reports heat transfer and surface pressure measurements on two cone-flare models in Mach 11 flow in the Calspan 96-inch shock tunnel. These models used a 9 foot (4.12 m) long 6 cone to generate a turbulent boundary layer with natural transition. Flares with angles of 36 and 42 (measured from the axis of symmetry) attached to the end of the cone result in compressioncorner SWBLIs that are unseparated and separated, respectively. The freestream Reynolds number outside the conical shock wave is Re = in 1. There is a fair amount of uncertainty surrounding this dataset at the time of this writing. Only one paper has been found that describes this dataset, although the data also appears in electronic format in the Settles and Dodson database (the database description outlines the cone-flare experiments; however, the references it cites describe three-dimensional swept shock boundary layer interaction experiments). The experimental boundary layer thickness is not explicitly stated in the paper (although it appears as a note in one of the plots). The wall temperature is not stated, so room temperature was assumed. Likewise, the exact model geometry is not provided; however, the geometry is given in the Settles and Dodson description of the experiment.

30 15 Finally, the freestream stagnation conditions are not explicitly stated and had to be backed out from the stagnation values prior to the shock heating (the paper describes how to do this). Sinha [9] considered this test case, but appears to have used flare angles 6 too small due to the unclear definition of the flare geometry. Despite these issues, this test case is still briefly considered in this study. The heat transfer data shows significant heating at the reattachment point, a common and important feature of SWBLI s that is not addressed by the Mach 3 compression corner experiments. We are also interested in comparing the OVERFLOW results to the results of Sinha [9], who showed that the SA model significantly underpredicts the pressure and heat transfer on the cone-flare models.

31 16 Table 2.1 Freestream and inflow boundary layer data for compression ramp cases Princeton Ramps M T 0 [K] P 0 [Pa] 6.90E E E E+05 δ [mm] δ [mm] θ [mm] x upstream [mm] Re [m 1 ] Re δ Kuntz Ramps M T 0 [K] P 0 [Pa] 4.83E E E E+05 δ [mm] δ [mm] θ [mm] x upstream [mm] Re [m 1 ] Re δ

32 Code Description 3. COMPUTATIONAL METHODS The OVERFLOW code is an overset grid (chimera) Navier-Stokes solver, which makes it ideal for computing the flow fields over complex geometries [32]. It uses a finite-difference formulation with flow quantities stored at the grid nodes. OVER- FLOW has central- and Roe upwind-difference options, and uses a diagonalized, implicit approximate factorization scheme for time advancement. Local timestepping, multigrid techniques and grid sequencing are all used to accelerate convergence to a steady state solution. In the present study, the Swanson/Turkel matrix dissipation model [33] was utilized. With the appropriate parameters [12], this allows the central-differenced scheme to mimic a Total Variation Diminishing (TVD) upwind biased scheme in regions of flow discontinuities. This method takes advantage of the speed of central differencing while maintaining the robustness of the upwind TVD scheme. For a complete discussion of the scheme, see Olsen et al. [12]. 3.2 Turbulence Models OVERFLOW has numerous turbulence models coded in, but this study focuses on the two most popular turbulence models in the OVERFLOW code: the Spalart- Allmaras (SA) model and the SST model, and also on a promising new model: the lag model. All three of these turbulence models make use of the Boussinesq approximation, meaning that the effects of the Reynolds stress terms are included in the Navier-Stokes equations through an eddy viscosity. Each of the test cases is computed with all three turbulence models to allow comparisons of the models.

33 Spalart-Allmaras (SA) Model The SA model [14] is a one-equation turbulence model developed specifically for aerodynamic applications. Being a one-equation turbulence model, it only needs to solve one partial differential equation to find the eddy viscosity. It is popular for design use because of its simplicity, low cost (low grid resolution requirements near wall), and robustness for aerodynamic flows. It has been shown [34] to be capable of providing accurate results in the hypersonic regime and is one of the most commonly used turbulence models in OVERFLOW. The specific implementation of the SA model in the OVERFLOW code makes use of loose coupling and first order accurate spatial derivatives Menter s (SST) Model The SST model [15] is a two-equation model which is a hybrid of the k-ɛ and k-ω turbulence models and solves partial differential equations for the turbulent kinetic energy and the specific dissipation rate. The eddy viscosity is then derived from an algebraic relation that chooses the minimum value between the standard formulation (based on the turbulence kinetic energy and the specific dissipation rate) and a formulation based on an assumed value of the normalized Reynolds shear stress. This hybrid formulation is popular for aerodynamic design and analysis because it retains the accuracy of the k-ω near-wall behavior and the k-ɛ wake region behavior Olsen & Coakley s Lag Model The Lag model [16, 35] is a recent extension of the k-ω model [5] that accounts for non-equilibrium effects by carrying an additional differential equation that relaxes the eddy viscosity to the equilibrium value. This third equation accounts for the time required for the turbulence to respond to changes in the mean flowfield. This model

34 19 is based on the 1998 k ω model, except that the eddy viscosity, ν t, is given by the field PDE ρν t t instead of k/ω as in the k ω model defined by ρk t + x i + x i (ρu i ν t ) = a 0 ρω (ν te ν t ), (3.1) ( ρu i k (µ + σ k ρν t ) k x i ) = P k ε k (3.2) and where: with parameters: ρω t + ( ρu i ω (µ + σ ω ρν t ) ω ) = P ω ε ω, (3.3) x i x i ν te = k ω s ij = 1 2 ( u i x j P k = τ ij s ij P ω = αρs 2 ε k = β ρkω ε ω = βρω 2 S = 2 (s ij s ij s 2 kk /3) τ ij = ρ ( 2 kδ ( 3 ij ν t 2sij 2s )) 3 kkδ ij ) + u j x i α = 5/9 a 0 = 0.35 β = β = 0.09 σ k = 1.5 σ e = 0.5. The lag equation alleviates the sensitivity to freestream turbulence levels that the 1998 version of the k ω model is known to have. The 2005 formulation [35] of the lag model (shown below) is computationally simple and requires less CPU time per iteration [16] than the SA and SST models despite requiring the storage of an additional field variable. Information about the wall distance is not required for the lag model as it is for the SA and SST models. The OVERFLOW implementation of the lag model requires 2 nd order upwind spatial discretizations to reduce numerical dissipation and grid density requirements. This turbulence model is still in the development stage and the present study is used to aid in the model verification and validation.

35 Grid Generation All of the grids were generated using the Chimera Grid Tools package [36] and simple in-house grid generation codes. For all cases, scripts were written to automate the process and help ensure consistency between similar cases. Surface definitions were generated by an in-house code that simply took a list of points in space and wrote them to a Plot3D grid file. SRAP was used to define the surface grid distribution, SETZETA was used to define the off-body grid marching distance, and HYPGEN was used to hyperbolically generate the off-body grids. For the Gatski flat plates and for the compression corner feeder grids, an in-house code to build box-grids with hyperbolic tangent stretching functions (with the same formulation as used in HYPGEN) was used to generate the grids. The hyperbolic tangent stretching function was used for all grids. All grids were generated in inches and were processed with double-precision floating point variables. Double fringes were used in all overlap region interpolation stencils, and grids split simply for load balancing had 5 points of perfect overlap. Table 3.1 summarizes the number of grid points used in each case Flat plates The Gatski flat plate test case uses a simple single-zone 129x256 grid that does not model the radius of the plate leading edge (i.e. the viscous wall extends all the way to the inflow plane). Grid points are clustered near the viscous wall, to resolve the boundary layer, and towards the leading edge, to contain the disturbances that result by specifying non-physical freestream conditions at the grid point immediately next to a viscous wall at the inflow plane. The wall spacing is constant inches along the entire grid, although the spacing was specified to be inches in the input file (due to the formulation of the hyperbolic tangent stretching function used in HYPGEN, the wall spacing in the resulting grid is not necessarily the same as the specified wall spacing). This wall spacing corresponds to a y + value of approximately 0.22 in the region of interest.

36 21 The Cary flat plate cases use a single-zone 387x257 grid that models a finiteradius leading edge with a radius of inches (pictured in Figure 3.1). The outer boundary is loosely fit to the shock wave attached to the leading edge. Again, grid points are clustered towards the viscous wall and the plate leading edge. The wall spacing is inches (the y + values that correspond to this are presented in the Results chapter). To match the experimental conditions, the Mach 4.9 cases were run using the Mach 6.0 freestream conditions with the flat plate inclined 8.1 to the freestream Compression ramps The compression ramp cases use a system similar to that depicted in Figure 3.2. The long flat plate region upstream of the corner is used to develop a turbulent boundary layer that matches the momentum thickness of the experimentally provided inflow profile. The OVERFLOW cases are run fully turbulent, so the length of this plate must be adjusted for each case to match the experimental momentum thickness. The corner and ramp regions are defined by three grids: two high resolution near-wall grids (split for load-balancing on multiple processors) and a grid in the freestream above the shock wave. It was found that the hyperbolic tangent stretching function used for the wall-normal direction did not place the desired number of grid points in the freestream above the shock wave, thus a separate zone was created. A low resolution sponge grid is used to contain the Mach wave generated at the upstream boundary. Sponge grids are generally used in LES simulations to prevent disturbances from reflecting off of the boundary back into the domain. In this case, it was found that without the sponge grid, the Mach wave reflects back into the domain and interferes with the boundary layer development (the sponge grid terminology may be misleading here as the goal is not to use extreme grid stretching to attenuate disturbances, but rather to cheaply move the freestream boundary away from the region of interest). The final grid zone is used to better resolve the shock wave gener-

37 22 ated in the corner. This zone was generated by first obtaining a solution without the shock-grid, then generating the shock-grid such that it was aligned with the inviscid shock (starting where the separation and reattachment shocks merge and ending at the end of the domain). As is shown later, the shock grid does not significantly alter the surface properties; however, it cleans up the solution in the flowfield behind the shock and gives a sharper shock wave (Figure 3.3). This grid was not used for the 24 ramps nor the Evans 16 ramp cases, as discussed in the results section. All of the Princeton cases use a wall spacing of in. ( in. actual) and the Kuntz cases use a wall spacing of in. ( in. actual). The feeder plate grid lengths varied for each case and turbulence model and are tabulated in Table 3.2. The tops of the compression ramps are included in the simulation so that the expansion fan will insulate the compression corner and ramp face from upstream influences due to the 0 th order extrapolation outflow boundary condition and for accuracy of the computations. Table 3.2 gives the specific ramp heights used for each case Cone-Flares The cone-flare cases use the single zone grids illustrated in Figure 3.4. The radius of the cone tip is assumed sharp, and an inviscid feeder block is used to permit the shock wave to develop as it may. The outer boundary is loosely fit to the shock wave and the wall spacing is in. As with the compression corner cases, the flare top (a cylindrical section at the end of the ramp) is included in the grid, although this geometric feature is not discussed in the literature (and probably was not present). The ramp length was determined based on maximum x-value shown in the data plots in reference [31].

38 Boundary Conditions The boundary conditions for the Gatski and compression corner grid systems are freestream (IBTYP=40) at the inflow boundary, characteristic freestream (IB- TYP=47) on the boundary opposite the viscous wall, and extrapolation (IBTYP=30) at the outflow. The viscous wall is adiabatic (IBTYP=5) for the Gatski cases and isothermal (IBTYP=7) for the compression ramps. The Cary cases use freestream (IBTYP=40) on the boundary opposite the isothermal viscous wall (IBTYP=7) and extrapolation (IBTYP=30) at both streamwise boundaries (since the grid wraps around the plate leading edge). The cone-flare cases use characteristic freestream (IB- TYP=47) opposite the isothermal viscous wall (IBTYP=8). Freestream (IBTYP=40) is specified on the inflow plane, the inviscid feeder block uses the inviscid adiabatic wall (IBTYP=1), and the outflow boundary uses extrapolation (IBTYP=30). 3.5 Numerical Solution Each of the solutions is initialized to freestream conditions (tabulated in Table 3.3) and iterated on a sequence of progressively finer grids until the finest grid level is obtained. These grids are created internally in OVERFLOW by specifying a number of grid levels (three grid levels would be the original fine grid, a medium grid with every other grid point used, and a coarse grid with every fourth grid point used). For this study, it was found that three grid levels and between 500 and 1,000 iterations on the coarse and medium grid levels reliably starts up the 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 matrix dissipation scheme [33] are ramped down to their optimum values as the solution converges. The four parameters controlled are the 2nd and 4th-order dissipation coefficients, κ 2 and κ 4, and the eigenvalue limiters V ɛη and V ɛl (DIS2, DIS4, VEPSN, and VEPSL in OVERFLOW, respectively). The presented

39 24 solutions were started with 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 information on this process, see Lillard et al. [13]. As with the dissipation settings, the timestep method and CFL numbers were varied throughout the course of the solution process. The process was started with ITIME=1 (where the timestep is computed based on grid metrics), then changed to ITIME=3 (constant CFL number; the timestep is based on the maximum eigenvalue and and specified CFLMAX). CFLMAX was varied throughout the solution process to keep the solutions from blowing up. After the dissipation parameters had been ramped down to their final values, CFLMAX was set to 5.0 for the compression ramp cases, which resulted in rapid convergence to a steady state solution. The final portion of the Cary flat plate solutions used CFLMAX=0.8, the Gatski flat plate solutions used CFLMAX=1.0, and the cone-flare cases used CFLMAX=0.5. 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 several representative locations has converged to at least five significant digits. Most of the solutions converged to greater than six significant digits; however, the Cary T w /T 0 = 0.3 case, the 8 Kuntz ramp, and the cone-flare cases were near this limit. Iterative convergence was typically attained within 45,000-60,000 iterations; however, this number could have been lower had the dissipation settings been ramped been more aggressively. 3.6 Grid Convergence Given that the primary interest in the present study is assessing the surface property predictions using OVERFLOW, the skin friction, C f = τ w /q Ref was chosen as the physical variable used to quantify the grid convergence. This variable was chosen because it has smooth behavior upstream of the shock-boundary layer interactions, it provides a measure of the separation bubble size (the size of the region with reversed flow at the wall), and has a distinct behavior in the recovery region. The skin friction

40 25 distribution along the model surface was used as opposed to an integrated viscous drag value in order to retain information about the degree of grid convergence in each region of the shock-boundary layer interaction. A considerable amount of effort went into determining a robust method for quantifying the grid convergence of the considered cases. Two methods considered were Richardson extrapolation and the grid convergence index (GCI). Richardson extrapolation (RE) takes two discrete solutions of the same problem with different grid spacings and, based on a Taylor series expansion of the solution, cancels terms in order to obtain an extrapolated solution with leading error terms one order higher than the method used to obtain the two initial solutions. For instance, using a solution obtained with a 2 nd order method on grids with spacings of x and 2 x, the extrapolated solution will theoretically be at least 3 rd order (4 th order for nonstretched grids and 2 nd order central differenced scheme). This 3 rd order solution could also be considered to be the best solution that can be obtained with a 2 nd order method, so it provides a reasonable way to quantify the relative error due to discrete derivatives of continuous partial differential equations (truncation error). The onedimensional formulation of the RE is given by where: f RE = f( x) + f = Discrete solution q = Grid refinement ratio f( x) f( x/q) q p 1 p = Order of numerical method (assumed 2 nd order) + O ( x p+1), (3.4) however, this formulation is also used for two-dimensional cases if the grid aspect ratio is held constant. The GCI of Roache [37] defines a relative error index that permits studies using different non-integer grid refinement ratios (i.e. studies that aren t restricted to using

41 26 grid doubling or halving) to be reasonably compared to one another. The relative error is scaled to give grid convergence index, ε = f coarse f fine f fine, (3.5) GCI = 3 ε q p 1. (3.6) This scaling term includes both the grid refinement ratio and the order of the method, and is derived from the RE. Roache shows [37] that the errors predicted with the GCI are of the same order as those predicted with RE. For grid doubling with a 2 nd order method (q = 2, p = 2), this term equals 1 so as to be consistent with previous studies which report relative solutions changes under grid doubling. Neither of these two methods proved to be robust and reliable measures of grid convergence for the present study. The primary weakness of these methods with regard to our solutions is that they rely on relative changes in the solution. Since the skin friction goes to zero at the separation and reattachment points, the grid convergence methods would either blow up or give unreasonable measures of the error in these regions. There were also questions regarding the precise order of the numerical scheme (especially in regions of flow discontinuities) and the effects of nonmonotonic grid convergence that remain unanswered. Given these difficulties, efforts to define a single, consistent value that contained information regarding all the grid convergence issues was not pursued. Instead, it was decided to use both methods where they applied and quantitatively look at specific points in the flow field as opposed to along the entire surface. Thus, grid convergence is reported on a case by case basis. The quantitative grid convergence measures chosen for the present study are the RMS (RMS C f ) and maximum relative differences (Max C f ) between the computed and the RE extrapolated skin friction. For the cases with flow separation, the relative difference between the computed and RE extrapolated separation bubble size ( L Sep ) is also considered. Due to the run-specific nature of the shock-grid generation

42 27 process, it would be difficult to perform a meaningful grid refinement study on these cases. Thus, the compression corner cases are refined to the point of grid convergence without the shock-grids, and the effect of the shock-grid is quantified by the RMS change in skin friction on the surface of the model. For all cases, the relative difference is reported as a percent change and as a GCI value. The grid convergence results and case specific implementation issues are presented along with the inflow matching results in the Results chapter. Finally, it should be noted that the results presented in this document are the fine grid results, not the extrapolated results. The investigation into the Richardson extrapolation did produce a 2D formulation of the RE which was used to help obtain adequately resolved grids. This 2D formulation provided information about the relative adequacy of the streamwise and wall-normal grid spacing to help the user decide how to refine a grid more efficiently than by just guessing. This method is described in detail in Appendix A. 3.7 Data Reduction Integrated Boundary Layer Properties In this study, the boundary layer displacement and momentum thickness values are defined by and θ = δ = δ 0 δ 0 ρv ρ e V e 1 ρv ρ e V e ds (3.7) (1 VVe ) ds, (3.8) respectively. Boundary layer profiles are generated by interpolating the CFD solutions to wall-normal profiles and integrating from the wall to the boundary layer edge. The boundary layer edge is defined in a similar manner as in the WIND code [38], with the edge of the boundary layer being defined as the point where dθ dδ = 0. This iterative method started at the second grid point off the wall and used the flow conditions at

43 28 that point as the edge conditions when integrating the momentum thickness up to that point. This was repeated for the next wall-normal grid point and the computed momentum thicknesses were compared. The process is repeated until the computed momentum thickness starts decreasing, and the point where dθ dδ = 0 is interpolated. This method proved to be more robust behind strong Mach waves and weak shock waves than just using the point where the velocity is 99% of the freestream velocity. With the exception of the Gatski profile, the experimental integrated boundary layer properties are used as reported without modification. For the Gatski profile, the above boundary layer edge definition is not used; δ 99 based on velocity is used instead Reynolds Shear Stress Each of the turbulence models used in the present study is formulated differently, and hence the Reynolds shear stress (RSS), u v computation is slightly different for each. In all cases, however, the RSS is backed out of the eddy viscosity using ( u u v = ν t y + v ). (3.9) x The SA eddy viscosity is computed and dimensionalized using the relations ν = q 7 ν χ = ν ν fv1 = χ 3 χ ν t = νfv1, (3.10) the SST eddy viscosity is computed and dimensionalized using ν t = k ω = q 7a L q 8, (3.11) the Lag model eddy viscosity is computed and dimensionalized using ν t = q 9 a L, (3.12)

44 29 where q n indicates the n th variable in the OVERFLOW solution (q) array. Here L is the conversion factor from the grid units to the desired units in the Reynolds shear stress (for this study, L = for grids in inches and resulting RSS in units of [m 2 /s 2 ]) Sutherland s Viscosity Law Formulation The specific formulation of Sutherland s viscosity law used in the present study is µ = bt 1.5 S + T, (3.13) with: b = [kg/m s K 0.5 ] S = [K]. It should be noted that some of the freestream temperatures used in this study are quite low (approx 65 K) and may exceed the range of validity of these model constants. No special attempts were made to tailor the viscosity formulation to one appropriate for the cold-flow coditions considered.

45 30 Table 3.1 Grid dimensions used in presented computations Grid Zone Total Points Gatski FP 129x E+3 Cary FP 387x E+3 Kuntz 8 172x x181 85x x53 221x39 120x E+3 Kuntz x x181 41x x53 221x39 75x E+3 Kuntz x x181 48x x53 221x39 30x E+3 Kuntz 24 72x x181 48x x53 221x E+3 Princeton 8 123x x x x28 250x39 120x E+3 Princeton x x x x28 240x39 120x E+3 Princeton x x x x35 221x39 45x E+3 Princeton x x x x35 221x E+3 Cone-Flare x E+3 Cone-Flare x E+3

46 31 Table 3.2 Compression corner case-dependent geometric parameters Feeder Length [in] Ramp Ramp Case SA SST Lag Height [in] Height/δ Ref Princeton Princeton Princeton 16 (Evans) Princeton Princeton Kuntz Kuntz Kuntz Kuntz

47 32 Table 3.3 OVERFLOW flow condition inputs used for all test cases Case FSMACH TINF REY TWALL ALPHA Gatski Cary Mach 4.9, T w /T 0 = Cary Mach 4.9, T w /T 0 = Cary Mach 6.0, T w /T 0 = Princeton 8 Ramp Princeton 16 Ramp Princeton 16 Ramp (Evans) Princeton 20 Ramp Princeton 24 Ramp Kuntz 8 Ramp Kuntz 16 Ramp Kuntz 20 Ramp Kuntz 24 Ramp Cone-flare : Note that a fudge factor of 0.01 was added to the reported freestream Mach number for the OVERFLOW runs. This was added to account for the drop in Mach number due to the leading edge Mach wave. This way, the Mach number over the ramp model matched that of the experiment.

48 33 (a) (b) Fig Grid topology for the Cary flat plate cases. (a) Entire domain, (b) Detail of plate leading edge colored by static pressure.

49 34 (a) (b) Fig General grid topology for the compression ramp cases. (a) Entire domain, (b) Detail of corner region.

50 35 (a) (b) Fig Static pressure contours over the Princeton 16 ramp: (a) Without shock-grid, (b) With shock-grid.

51 36 (a) (b) (c) Fig General grid topology for the cone-flare cases. (a) Entire domain, (b) Detail of tip region, (c) Detail of corner region, colored by static pressure.

52 37 4. RESULTS Results from the various test cases computed by OVERFLOW are presented in this chapter along with detailed discussion of the results. Where possible, each case uses the same non-dimensionalizing reference values used in the literature describing the dataset. These values are tabulated in Table Grid Convergence The results of the grid convergence study are presented in this section. Each case was run on a fine grid (with dimensions reported in Table 3.1) and then on a coarser grid. The coarsening ratio is 2 for the flat plate cases and 2 for the SWBLI cases. The ratio is reduced for the SWBLI cases to minimize non-monotonic grid convergence effects for the Richardson extrapolation. The relative change in the skin friction is reported, as discussed in Chapter Gatski Flat Plate DNS One fully turbulent profile is available from the Gatski flat plate DNS, so the grid convergence at that point is reported. Note that this location is different for each turbulence model, as the location in the OVERFLOW computations was determined by matching the momentum thickness, θ. Table 4.2 presents the non-dimensional wall spacing and grid convergence results for the Gatski flat plate DNS runs. On average, the skin friction at the point of interest is within 1% of the extrapolated 3 rd order solution (with GCIs less than 2%), so these cases appear to be sufficiently grid converged. The non-dimensional wall spacing values of approximately 0.22 for the SA case is interesting because it is much lower than the value y + = 1 commonly

53 38 believed to be adequate for the SA model. For the SST model, the y + value is less than 1, which agrees with Wilcox s recommendation [5] that the wall spacing should be y + < 1. It appears that the lag model grid requirements are indeed similar to those of the SA and SST models for a simple boundary layer Cary Flat Plates Table 4.3 shows the grid convergence and non-dimensional wall-spacing results for the Cary flat plate cases. Note that the reported values of the GCI use the integrated skin friction along a window from 5 in to 20 in from the leading edge (the same window used for the average and max C f computation). Not surprisingly, the cases with finer wall-spacing show better grid convergence, however all cases meet the minimum requirements for this study. Note that the non-dimensional wall spacing for the T w /T 0 = 0.3 case is larger than for the other cases. All of the Cary cases use the same grid, so the colder wall conditions require finer grids (dimensionally) than hotter walls. The larger non-dimensional wall spacing results in greater grid resolution uncertainties; however, the uncertainty level achieved with this grid is adequate for the present study Princeton Compression Ramps Table 4.4 shows the grid convergence results for the Princeton compression ramp cases. As indicated, the fine grid skin friction value is within 2% of the Richardson extrapolated skin friction at the point 2 inches upstream of the corner (where the inflow boundary layer properties are specified). The GCI s for the skin friction are all within 5%, which along with the RE differences, indicate that the inflow boundary layer is adequately resolve. The separation length convergence appears to show more variation than the upstream skin friction at the fine-grid resolution, however most of the separation bubbles are within 4% of the extrapolated values. Note that the 8 ramp has a very small separation bubble, so relative differences begin to amplify

54 39 errors, leading to the large reported differences. Plots of the skin friction distribution on the fine and coarse grids for all of the compression ramp cases are included in Appendix B for a qualitative measure of grid convergence along the entire ramp. Table 4.6 shows the average change in skin friction caused by the addition of the shock grid. It appears that the lag model is affected most by this change, however the influence is minimal. Note that these values are scaled by the upstream skin friction as local relative differences would not be meaningful near separation Kuntz Compression Ramps Table 4.5 shows the grid convergence results for the Kuntz compression ramp cases. The grid convergence of the upstream skin friction (1 inch upstream of the corner) and the separation length show similar trends to the Princeton ramps. It should be noted that the 24 SST separation point is very near the point where the skin friction point was taken, so the large error here is again a result of relative difference of very small values. Table 4.6 shows the average change in skin friction caused by the addition of the shock grids. Again, the shock grids show negligible influence on the final results. Note that all of the surface property distributions presented in this document were not produced on the grid systems containing the shock grids (those were only used for profiles) Holden Cone-Flares Table 4.7 shows the grid convergence results for the cone-flare test cases. This table shows that the upstream boundary layer is very well resolved at a point approximately 9 boundary layer thicknesses upstream of the corner. The separation lengths, however, do not show very good grid convergence. For the SA and lag models, the separation lengths are small, so the relative changes reported tend to exaggerate the errors. The SST model, however, shows significant separation so the large differences are not caused by dividing by a small number. This fact should be kept in mind when

55 40 analyzing the following results. The skin friction plots of the fine and coarse grids show that the recovery region is reasonably well converged; however, small differences are apparent. 4.2 Zero Pressure-Gradient Boundary Layers The data available from the Gatski dataset includes skin friction and a velocity profile in the fully turbulent region of the flat plate (indicated in reference [19, 20] as the x = 8.8 in profile). The streamwise location of the profiles presented in this section were determined by choosing the location in the fully-turbulent OVERFLOW computation that matched the momentum thickness of the available DNS profile. Table 4.2 shows that the predicted displacement thickness is reasonably close to the DNS value, although the SST model is off by 8.1%. The SA model performs the best at 5.6%. Table 4.2 also shows the skin friction at the profile location for each turbulence model against the DNS value. All three models perform well, with SA the best at 0.7% and SST the worst at 3.7%. Table 4.8 likewise shows the predicted skin friction and displacement thickness at the point where the momentum thickness matches the Princeton compression ramp experiments. It can be seen that the predicted displacement thickness for the 8 ramp is close to the experimental value; however, a larger discrepancy exists for the 16 ramp. The skin friction values for both ramps are somewhat overpredicted, with the SST model achieving the best accuracy and the SA model having the worst. Figure 4.1 shows the velocity profiles for the three turbulence models against the DNS data. All three models show excellent agreement, although the DNS velocity profile is slightly fuller than all three computed profiles near the wall. Figure 4.2 shows the Reynolds shear stress (RSS) profiles at the same location, and again, the differences between the models are small. A larger discrepancy is apparent between the DNS and RANS computations for the RSS than for the velocity profiles. The

56 41 RANS computations appear to diffuse too much of the Reynolds shear stress to the outer portion of the boundary layer and underpredict the peak RSS value by 5%-8%. Figures 4.3 and 4.4 show the heat transfer results for the Mach 4.9 and Mach 6.0 Cary flat plates, respectively. It appears that the experimental data for these cases are largely transitional. Transitional boundary layers display an overshoot in skin friction and heat transfer relative to fully-turbulent correlations; so, only the data at the high end of the Reynolds number range considered is meaningful. Above Re x = or Re x = , the distribution appears fully turbulent (when the heat transfer distribution shows a constant decay rate). By this point, all three turbulence models have developed a similar decay rate (similar to each other, not similar to the data), and the only difference between the three is a change in magnitude. All of the cases overpredict the heat transfer in the fully developed region. Deviations from experiment range from 15% at Mach 6.0 to 27% at Mach 4.9 within the Reynolds number range for which there is data. The decay rate along the plate appears to be significantly lower than experiment for all of the cases. The discrepancy is smallest for the Mach 4.9, T w /T 0 = 0.3 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. This could be a significant contribution to the observed discrepancy in the magnitude of the heat transfer. It is also possible that 3D relieving effects are present in the Mach 4.9 test cases that could be affecting the measured heat transfer data. Work is continuing to assess possible three-dimensionality in this case Flat Plate Discussion The results in this study show that the three turbulence models all do a decent job predicting the shape factor of a zero pressure gradient boundary layer and that the relation between the skin friction and momentum thickness is also adequate. The heat transfer predictions, on the other hand, could use some improvement. The

57 42 experimental data considered, however, is somewhat less comprehensive than would be desired for a reasonable validation. Insufficient data was available to adequately assess the boundary layer growth accuracy. Similarly, the heat transfer dataset did not achieve wide enough range of Reynolds numbers to provide sufficient data for the validation of the CFD. 4.3 Shock-Boundary Layer Interactions The shock-boundary layer interaction predictions are based on two plane-compression ramp experiments (with multiple ramp angles) and a cone-flare experiment (with two flare angles). Discussion of the turbulence model performance in prediction of surface pressure, skin friction, heat transfer, velocity profiles, and Reynolds shear stress profiles follows Surface Pressure Figures 4.5 through 4.8 show the surface pressure predicted by OVERFLOW for the Princeton and Kuntz compression ramps. The Kuntz dataset shows a greater tendency to separate, a likely effect of the lower Reynolds number. In general, for the 8 and 16 ramps, only small differences exist between the three turbulence models. Note that the apparent shift in the experimental pressures on the Kuntz 8 ramp is present in the experimental data. This offset is present for all ramp angles and no effort was made to shift this data; it was used as reported. This shift is discussed by Kuntz in reference [26]. Differences between the models start to become apparent at 16, and continue to grow as the ramp angle increases to 20 and 24. The SST model consistently shows the largest upstream influence, and the SA model shows the least. The SA model consistently predicts the most rapid pressure recovery of the three models, and the SST model shows the slowest. The lag model shows a tendency to rapidly recover once it reattaches, however the late reattachment appears to result in pressure predictions bounded by the SA and SST solutions. It is interesting

58 43 to note that as the ramp angle increases, no one model does the best job of representing the profile relative to the experimental data; for each ramp angle, the best model changes. In general, however, the SA model probably would be the best for conservative design work. The surface pressure distributions predicted on the cone-flare models are shown in Figure 4.9. As with the weak compression ramp interactions, the unseparated case (36 flare) shows only small differences between the three turbulence models. For the separated case, however, the SST model shows significantly larger upstream influence than the other two models, and appears to predict the upstream influence reasonably well for this ramp angle. The SST model also shows a significant overprediction of the peak heating on the flare. It is really interesting that the lag model mimics the SA model so closely for both geometries of this test case; the reason for this behavior is unknown. Wilcox [5] claims that RANS models typically show too little upstream influence and too large a wall pressure in the separated region. This observation appears to hold for the OVERFLOW solutions of the shallow ramp angles; however, it tends to predict too much upstream influence for the steeper ramps. It also appears that the OVERFLOW solutions predict relatively accurate plateau pressures, though the oversized separation bubbles at higher ramp angles do tend to result in an overprediction of the plateau pressure. As a broad generalization of the OVERFLOW results, the computed pressure plateaus tend to show an overly steep rise in pressure near the separation point. One possible reason for this discrepancy is shock-unsteadiness. In the experiments, the separation shock foot was observed to be unsteady. This means that the measured surface pressure near the separation point was actually a time-average, which would round-off the pressure rise due to the separation shock. The RANS models, by nature, cannot capture this unsteadiness and instead predict a single location for the separation shock foot and the resulting steep pressure rise.

59 44 Note that the drop in pressure at the right-hand side of most of the plots is evidence of the expansion fan at the top of the ramp. The computed surface pressure distribution for both 24 ramps appears to be influenced by this expansion fan, indicating the importance that the computed ramp height match the experimental geometry Skin Friction & Heat Transfer Figures 4.10 and 4.11 show the skin friction predicted by OVERFLOW for the Princeton compression ramps. For all of the compression ramps and all of the turbulence models, the computed skin friction profile shows a larger separation region compared to the experimental data. For all cases, the SST model predicts the largest separation bubble and the SA model predicts the smallest. In the recovery region, the lag model shows the most rapid recovery, typically more rapid than measured. The SA model recovers too slowly for all ramp angles. The SST model prediction is interesting in that it predicts the correct trend in skin friction recovery for all four ramp angles; however, it appears that the late reattachment results in the computed skin friction being lower in magnitude than in the experiment. In all of the lag model computations, little glitches can be seen in the skin friction distributions. These glitches appear to be due to a bug in the specific implementation used for these computations. The drop in skin friction corresponds to a significant increase in the specific dissipation on the wall. The grid cells affected by this typically occur in a region of the grid that has been split up for load balancing (internal to OVERFLOW 2.0), as they begin and end at reported MPI boundaries. These glitches appear to be local in nature and do not seem to adversely affect the solution downstream of them. These issues have been brought to the model developers attention. Figure 4.12 shows the heat transfer of the Princeton 16 ramp. The heat transfer predictions show large variations between the three turbulence models. The SST

60 45 model does a good job predicting the heat transfer early in the recovery, but drops below the heat transfer levels seen in the experiment towards the second-half of the ramp. The lag model overpredicts the increase in heat transfer near the corner, but also underpredicts the downstream levels. The SA model underpredicts the heat transfer along the entire length. It should be noted that the upstream heat transfer levels were not provided in reference [27], so the relative accuracy of the zero-pressure gradient boundary layer heat transfer computations is not available for this dataset. The heat transfer distributions on the cone-flares (Figure 4.13) show similar differences between the three turbulence models. For the unseparated case (36 flare), the SST and lag models appear to provide reasonable predictions of the heat transfer in the recovery region. The SA model, as before, does not show enough of a rise in heating. The same observation can be made for the separated case (42 flare); the SA model prediction does not appear to show a significant change in behavior due to the presence of separation. The SST model shows a significant overprediction of the peak heating, and the peak occurs farther downstream than observed. The delayed peak would seem to suggest late reattachment, however the SST model shows a (suprisingly) good prediction of the heating in the separation region (care should be taken with this observation, however, considering the degree of grid convergence for this case). The lag model recovery is too rapid, but it does a better job at estimating the peak heating rate. Wilcox [5] states that RANS models typically overpredict C f and C h downstream of a shock-boundary layer interaction. The computed results in the present study seem to show a general trend to underpredict the skin friction in the recovering boundary layer for flows with significant separation, although the lag model does occasionally show larger skin friction than the experiment. It is likely, however, that this underprediction is due primarily to the oversized separation bubble and the corresponding late reattachment point, as the 8 ramp (with only minor separation) shows too rapid a recovery for all three turbulence models. The heat transfer results in

61 46 this study do not really support or discredit Wilcox s claim as the differences between each of the models straddles the experimental data. The results presented by Sinha [9] tell pretty much the same story as the results in this document. He found that the SA model underpredicted both the peak heating and pressure, and did not really show much of a peak. Neither of his computations show signs of separation; however, this may be expected due to him running flares of 30 and 36 (instead of 36 and 42 relative to the cone axis) Separation Length Figures 4.14(a) and (b) show plots of the predicted and measured separation bubble size for the Princeton and Kuntz series, respectively, non-dimensionalized by the incoming boundary layer thickness. For these plots, the computed separation bubble size is determined by measuring regions of negative skin friction and the experimental size was measured from surface-streak methods. A couple of points can be made about the data in these figures. First of all, the computed results almost unanimously overpredict the separation size. The overprediction becomes larger with increasing ramp angle. These figures again show that in all cases, the SST model predicts the largest bubble size and the SA model predicts the smallest (except for the 20 Princeton ramp). Second, a significant difference in magnitude of the separation size is apparent between the two experimental datasets. One possible explanation for this difference is that the boundary layer thickness based Reynolds number is significantly different for the two datasets (Princeton: Re δ = ; Kuntz: Re δ = ). Zheltovodov [39] presented a correlation of many different separation length experimental studies versus boundary layer Reynolds number. The separation lengths are non-dimensionalized by L c = δ ( P2 M 3 P P l ) 3.1, (4.1)

62 47 where ( ) 1 P P l = P 2 M + 1 (4.2) is the plateau pressure and P 2 is the inviscid post-shock pressure. The experimental datapoints in Figure 4.15 appears to agree reasonably well with Zheltovodov s correlation which, unfortunately, is not depicted on the figure as the formulation has not been obtained yet (the references point to a paper written in Russian). Figures 7 and 9 in reference [6] show this correlation. Yan presents the results of several RANS compression ramp SWBLIs in a form similar to Zheltovodov (results reproduced in Figure 9 of reference [6]). The RANS results of the present study (Figure 4.15(b)) show scatter very similar to the previously conducted RANS computations. It appears that L c does not collapse the RANS data very well. These observations should be qualified by two points. First of all, it should be noted that surface streak methods will tend to show smaller than actual separation bubble sizes because of the difference in shear stress magnitude on either side of the separation point; the flow on the upstream side of the separation point has a higher shear stress and is more effective at moving the oil or kerosene on the model surface. Second, the two-dimensional assumption for the experimental plane compression ramp configuration is questionable. The separated region of the flow in the corner is subsonic, so the conditions at the end of the ramp influence the conditions at the centerline. It is unknown how much the centerline behavior is affected by the very complicated and three-dimensional behavior of the corner region [40]. The two-dimensionality assumption is justified in both experiments by how straight the separation and reattachment lines were, but slight curvature was observed for both datasets. The overall influence of the end effects for both datasets is unknown Velocity Profiles Figures 4.16 through 4.26 present the velocity profiles and Reynolds shear stress profiles for the Kuntz and Princeton datasets. Each page begins with a local Mach

63 48 number contour plot with lines indicating the spatial location of the profiles presented below. The Kuntz dataset provides the Reynolds shear stress profiles at the same locations as the mean velocity profiles, so the turbulent and mean profiles are presented on the same page. The researchers at Princeton used different locations for their mean and turbulent profiles, so these are presented separately, each with their own legend. Several interesting points can be observed from the mean velocity plots. First, for all of the test cases, the upstream profile shows good agreement between the computed and experimental velocity profiles. Differences between the models become apparent downstream of the interaction, with the differences becoming larger with increasing ramp angle. The SST model tends to produce a less-full profile than the other turbulence models, which makes sense considering the SST model is the most separation-prone of the three. The lag model tends to produce the fullest profile near the wall; however, the SA model does occasionally show a fuller profile for the large ramp angles because of its earlier reattachment location. The predicted shock location tends to be reasonably accurate, although at the higher ramp angles, the oversized separation bubble causes the SST profiles to show the separation shock farther away from the wall than the experiment observed. It is interesting to note the difference in the recovering boundary layer between the two experimental datasets. The OVERFLOW results underpredict the velocities measured by pitot probes in the Princeton experiments but overpredicts the velocities measured by the LDV in the Kuntz experiments. For the Princeton dataset, all three turbulence models appear to recover too slowly from the shock-boundary layer interaction. For the Kuntz dataset, it appears that the mean flow is not affected by the interaction as much in the computations as in the experiment. Wilcox [5] claims that typical RANS computations show too much mean flow deceleration near the wall. The OVERFLOW computations do show this trend with the Princeton dataset, and it appears that the deceleration gets worse with increasing

64 49 ramp angle. The Kuntz experiment does not provide data near the wall, so it is not apparent if the same effect is observed in this dataset Reynolds Shear Stress Profiles The profiles of Reynolds shear stress are shown in Figures and Figures for the Kuntz and Princeton datasets, respectively. Each of these plots show the RSS measured in the unperturbed upstream boundary layer, then several profiles on the ramp, moving downstream (the profiles are oriented and located as indicated in the legend for each). A few global observations may be made regarding the comparisons to both datasets. First of all, the measured peak RSS is further away from the wall than the computations predict, even for the undisturbed boundary layer. Second, the turbulence relaxes back towards undisturbed levels much faster in the computational models than in the experiments. The SST model seems to show the greatest level of turbulence amplification downstream of the interaction, and SA shows the least. As apparent in the velocity profiles, there is some unknown fundamental difference between the Kuntz and Princeton datasets that results in rather large differences in the measured profiles. The RSS levels measured by Kuntz, as indicated previously by Settles and Dodson [4], are significantly larger than the levels measured by Smits et al. [22]. As a result, the OVERFLOW computations show opposite trends for each set. The OVERFLOW computations overpredict the RSS levels near the interaction for the Princeton cases and the turbulence relaxes closer to the measured levels, whereas the computations underpredict the RSS levels for the Kuntz cases and the turbulence relaxes further away from the experimental measurements. The reason for the difference in RSS between the two experimental datasets is not known, although it has been speculated that the difference was due primarily to the different measurement techniques used. Kuntz speculated in his thesis [26] that for the Princeton dataset, the high turbulent intensities in the perturbed boundary layer

65 50 resulted in flow that exceeded the lower calibration limit of the slanted hot wire. Gadel-Hak and Bandyopadhyay [41] also lend some credence to the idea that the Princeton values may be in error by claiming that hot-wire turbulence measurements appear to be unreliable if the probe lengths are greater than the viscous sublayer thickness (the length of the Princeton hot-wires is not known at the time of this writing). The Kuntz measurements could have been influenced by particle lag, measurement volume size, and LDV bias effects, although he claims that these effects did not significantly affect the turbulence measurements. Settles and Dodson [4] note that the experimental data does not support the idea that the difference is simply due to shock-unsteadiness. Gad-el-Hak and Bandyopadhyay [41] also summarize possible Reynolds number effects on compressible turbulence in their AIAA technical note. They state that the peak u-turbulence intensity increases slightly with Reynolds number, and that the location of the peak RSS moves away from the wall with increasing Reynolds number. They also claim that most of the the turbulence properties do not scale with inner-variables, indicating a dependence on the external flow conditions. A thorough analysis of the conditions for each experimental test case was not conducted in light of these claims; however, a preliminary examination indicates that the difference in freestream Reynolds number could be a significant contributor to differences in the actual RSS between these two datasets. Another potential source of this difference is that the turbulence in the Kuntz inflow boundary layer was not in equilibrium (i.e. it had not yet recovered from the favorable pressure gradient in the tunnel nozzle). Other factors due to the different physical facilities, such as physical dimensions, freestream turbulence levels, etc. could also be contributing to the discrepancy. Figure 4.27 shows a detailed view of the Princeton and Kuntz 20 ramp velocity and RSS profiles at x/δ Ref 3. In this format, the differences mentioned in the previous section are more visible, and the differences in RSS magnitude of the two experimental cases becomes clear. Both the computations and experimental data show that the peak RSS occurs slightly farther away from the wall for the Kuntz

66 51 conditions than the Princeton conditions, although the profile orientations are likely partially responsible for this difference. It should be noted that the very large peaks visible in some profiles (most significantly the SST traces in Figure 4.25) are due to the shock wave. It is unknown why the SST model is particularly sensitive to the shock wave; however, the SA and lag models do show a peak in RSS when the shock wave is in the boundary layer (profiles #2 and #3 in Figure 4.25, for instance). It is possible that the source of this peak is in the process used to extract the RSS from the eddy viscosity (Eqn 3.9), as the strain rate is used in the computation SWBLI Discussion Knight et al. [6] states that RANS computations for compression corner shock boundary layer interactions are accurate for weak interactions, but significant discrepancies existed for strong interactions. While it is difficult to define the dividing line between weak and strong interactions, the present results agree with this assessment. The 8 ramp results tend to be reasonably accurate, but the accuracy decreases as the interactions are strengthened. Knight et al. [6] places a strong emphasis on the fact that the RANS computations fail to predict the low-frequency shock unsteadiness [2]; the OVERFLOW solutions likewise do not predict any unsteadiness. This should be expected, however, as recent findings [42] suggest that the low frequency oscillations are the result of large streamwise turbulent structures in the upstream boundary layer. Since the RANS models average out all of the turbulent fluctuations, including these large streamwise features, there is no physical source for this type of shock unsteadiness. In preliminary computations, the shock waves did occasionally show an oscillatory motion of the shock foot location that prevented convergence to a steady solution; however, efforts to run these solutions time-accurate failed to produce consistent and repeatable

67 52 results. This unsteadiness was considered to be a numerical problem and was fixed by reducing the grid aspect ratio near separation and by increasing the CFL number. One more thing is interesting to note about the presented results. On the Princeton 16 ramp, Figure 4.6(a) shows a rather accurate prediction of the surface pressure, but Figure 4.12 shows a very poor heat transfer prediction. This shows that the code can accurately predict the inviscid flow field (and hence, gets the inviscid pressure correct), but still get the viscous boundary layer wrong. Assessing the accuracy of a CFD simulation using pressure only does not indicate anything about the accuracy of skin friction or heating prediction.

68 53 Table 4.1 Reference values used in data reduction Case δref [m] MRef VRef [m/s] PRef [Pa] ρref [kg/m 3 ] TRef [K] T0 [K] Gatski DNS Cary FP Princeton Ramps: Kuntz Ramps: Cone-Flares:

69 54 Table 4.2 Gatski DNS Results Case δ [mm] (% Diff.) C f (% Diff.) y + RMS C f Max C f GCI DNS E SA (-5.6%) 2.23E-03 (0.7%) % 0.8% 1.9% SST 0.590(-8.1%) 2.16E-03 (3.7%) % 1.1% 1.8% Lag (-7.1%) 2.22E-03 (1.1%) % 0.6% 1.6% Table 4.3 Cary flat plate: Grid convergence results M=4.9, T w /T 0 = 0.3: y + Range = Case RMS C f Max C f GCI SA 0.8% 1.1% 1.5% SST 1.4% 1.8% 2.7% Lag 1.8% 2.4% 3.4% M=4.9, T w /T 0 = 0.6: y + Range = Case RMS C f Max C f GCI SA 0.4% 0.5% 1.0% SST 0.2% 0.2% 0.4% Lag 0.2% 0.2% 0.4% M=6.0, T w /T 0 = 0.6: y + Range = Case RMS C f Max C f GCI SA 0.2% 0.2% 1.1% SST 0.1% 0.2% 0.7% Lag 0.4% 0.7% 1.8%

70 55 Table 4.4 Princeton Compression Corners: Grid convergence results Princeton 8 C f (GCI) C f (RE) L sep (GCI) L sep (RE) SA 4% 1% 30% 9% SST 2% 1% 7% 2% Lag 2% 1% 7% 2% Princeton 16 C f (GCI) C f (RE) L sep (GCI) L sep (RE) SA 4% 0% 2% -1% SST 2% 1% 6% -2% Lag 3% 1% 3% 1% Princeton 16 (Evans) C f (GCI) C f (RE) L sep (GCI) L sep (RE) SA 4% 1% 2% -1% SST 3% 1% 7% -2% Lag 5% -2% 3% 1% Princeton 20 C f (GCI) C f (RE) L sep (GCI) L sep (RE) SA 3% 1% 0% 0% SST 1% 0% 5% -2% Lag 4% -1% 11% 4% Princeton 24 C f (GCI) C f (RE) L sep (GCI) L sep (RE) SA 2% 1% 2% -1% SST 3% -1% 5% -2% Lag 0% 0% 9% 3%

71 56 Table 4.5 Kuntz Compression Corners: Grid convergence results Kuntz 8 C f (GCI) C f (RE) L sep (GCI) L sep (RE) SA 2% 1% 10% 3% SST 0% 0% 4% 1% Lag 1% 0% 4% 1% Kuntz 16 C f (GCI) C f (RE) L sep (GCI) L sep (RE) SA 2% 1% 0% 0% SST 0% 0% 3% -1% Lag 5% 2% 4% 1% Kuntz 20 C f (GCI) C f (RE) L sep (GCI) L sep (RE) SA 1% 0% 3% -1% SST 0% 0% 1% 0% Lag 1% 0% 7% 2% Kuntz 24 C f (GCI) C f (RE) L sep (GCI) L sep (RE) SA 2% 0% 1% 0% SST 13% 19% 3% -1% Lag 1% 0% 6% 2%

72 57 Table 4.6 RMS change of skin friction due to shock grid, scaled by upstream skin friction Model Kuntz 8 Kuntz 16 Kuntz 20 Princeton 8 Princeton 16 Princeton 20 SA 0.0% 0.1% 2.1% 0.0% 0.1% 0.7% SST 0.0% 0.1% 0.0% 0.0% 0.1% 1.0% Lag 1.3% 0.8% 0.6% 1.4% 2.5% 1.2%

73 58 Table 4.7 Holden Cone-Flares: Grid convergence results 36 Flare C f (GCI) C f (RE) L sep (GCI) L sep (RE) SA 1% 0% 38% 11% SST 1% 0% 112% 60% Lag 1% 0% 19% 7% 42 Flare C f (GCI) C f (RE) L sep (GCI) L sep (RE) SA 1% 0% 88% 41% SST 1% 0% 147% 33% Lag 1% 0% 20% 7%

74 59 Table 4.8 Princeton Compression Ramps: 8 and 16 inflow boundary layer results Princeton 8 ramp Experiment SA SST Lag Cf 1.00E E-03 (12.5%) 1.09E-03 (9.0%) 1.11E-03 (10.8%) δ [mm] (14.0%) 26.6 (2.1%) 28.6 (9.9%) δ [mm] (1.1%) 6.8 (1.0%) 6.8 (1.8%) θ [mm] (0.0%) 1.3 (0.0%) 1.3 (0.5%) Princeton 16 ramp Experiment SA SST Lag Cf 0.984E E-03 (13.6%) 1.08E-03 (9.9%) 1.12E-03 (13.6%) δ [mm] (14.5%) 26.6 (2.3%) 29.8 (14.5%) δ [mm] (7.5%) 6.7 (7.1%) 6.8 (7.5%) θ [mm] (0.2%) 1.3 (-0.1%) 1.3 (0.2%)

75 60 Fig Velocity profile for Gatski flat plate test case [19, 20]. Fig Reynolds shear stress profile for Gatski flat plate test case [19, 20].

76 61 (a) (b) Fig Heat transfer distribution on the Mach 4.9 Cary flat plates. (a) T w /T 0 = 0.6, (b) T w /T 0 = 0.3.

77 Fig Heat transfer distribution on the Mach 6.0 Cary flat plate. T w /T 0 =

78 63 (a) (b) Fig Surface pressure distribution on 8 ramps. (a) Princeton, (b) Kuntz.

79 64 (a) (b) Fig Surface pressure distribution on 16 ramps. (a) Princeton, (b) Kuntz.

80 65 (a) (b) Fig Surface pressure distribution on 20 ramps. (a) Princeton, (b) Kuntz.

81 66 (a) (b) Fig Surface pressure distribution on 24 ramps. (a) Princeton, (b) Kuntz.

82 67 (a) (b) Fig Surface pressure distribution on cone-flares. (a) 36 flare, (b) 42 flare.

83 68 (a) (b) Fig Skin friction distribution for Princeton dataset. (a) 8, (b) 16.

84 69 (a) (b) Fig Skin friction distribution for Princeton dataset. (a) 20, (b) 24.

85 Fig Heat transfer distribution for Princeton 16 ramp. 70

86 71 (a) (b) Fig Heat transfer distribution on cone-flares. (a) 36 flare, (b) 42 flare.

87 72 (a) (b) Fig Separation length vs. ramp angle: (a) Princeton dataset, (b) Kuntz dataset.

88 73 (a) (b) Fig Normalized separation length for Princeton and Kuntz datasets vs. (a) Ramp angle, (b) Re δ.

89 74 (a) (b) (c) Fig Kuntz 8 ramp. (a) Contours of local Mach number, with profile locations indicated, (b) Velocity magnitude profiles, (c) Reynolds shear stress profiles.

90 75 (a) (b) (c) Fig Kuntz 16 ramp. (a) Contours of local Mach number, with profile locations indicated, (b) Velocity magnitude profiles, (c) Reynolds shear stress profiles.

91 76 (a) (b) (c) Fig Kuntz 20 ramp. (a) Contours of local Mach number, with profile locations indicated, (b) Velocity magnitude profiles, (c) Reynolds shear stress profiles.

92 77 (a) (b) (c) Fig Kuntz 24 ramp. (a) Contours of local Mach number, with profile locations indicated, (b) Velocity magnitude profiles, (c) Reynolds shear stress profiles.

93 78 (a) (b) Fig Princeton 8 ramp. (a) Contours of local Mach number, with mean profile locations indicated, (b) Velocity magnitude profiles.

94 79 (a) (b) Fig Princeton 16 ramp. (a) Contours of local Mach number, with mean profile locations indicated, (b) Velocity magnitude profiles.

95 80 (a) (b) Fig Princeton 20 ramp. (a) Contours of local Mach number, with mean profile locations indicated, (b) Velocity magnitude profiles.

96 81 (a) (b) Fig Princeton 24 ramp. (a) Contours of local Mach number, with mean profile locations indicated, (b) Velocity magnitude profiles.

97 82 (a) (b) Fig Princeton 8 ramp. (a) Contours of local Mach number, with turbulent profile locations indicated, (b) Reynolds shear stress profiles.

98 83 (a) (b) Fig Princeton 16 ramp. (a) Contours of local Mach number, with turbulent profile locations indicated, (b) Reynolds shear stress profiles.

99 84 (a) (b) Fig Princeton 20 ramp. (a) Contours of local Mach number, with turbulent profile locations indicated, (b) Reynolds shear stress profiles.

100 85 (a) (b) Fig Profile comparison at x/δ Ref 3 on the 20 ramps. (a) Mean velocity profiles - Princeton mean profile #8, Kuntz profile #4; (b) Reynolds shear stress profiles - Princeton RSS profile #6, Kuntz profile #4.

101 86 5. SUMMARY 5.1 Conclusions In this study, the SA, SST, and lag turbulence models in the OVERFLOW code are applied to several high-speed experiments of canonical flows in order to evaluate their predictive performance. This work was performed to aid in the development of new models and model corrections. Strong emphasis was placed on ensuring that numerical errors remained low (by fine iterative and grid convergence criterion) and consistent (by similar grid systems) in an effort to isolate differences in the turbulence models. Three flat plate experiments are considered, including: a Mach 6 flat plate for heat transfer, a Mach 2.9 flat plate for boundary layer properties and growth, and a Mach 2.25 flat plate DNS for profiles of mean velocity and turbulent shear stress. The analysis of the OVERFLOW computations shows that all three turbulence models perform well at predicting the shape factor of zero pressure gradient boundary layers. The mean velocity and Reynolds shear stress profiles agree well with DNS, with relatively minor deviations between the three turbulence models. The predicted displacement thicknesses, δ are within 8% at the point where the momentum thickness, θ, matches the DNS. All three turbulence models overpredict heat transfer, with deviations ranging from 15% at Mach 6.0 to 27% at Mach 4.9 over the range of Reynolds numbers considered. The SA model consistently predicts the lowest heat transfer of the three models, and the lag model predicts the largest heat transfer. All three turbulence models underpredict the decay rate of the heat transfer with increasing Reynolds number. Two Mach 2.9 compression corner datasets and a Mach 11 cone-flare dataset are used to evaluate the performance of the three turbulence models in shock-boundary

102 87 layer interactions. The turning angles used in these experiments generate flows ranging from flows with no separation to flows with significant separation. For the lowest ramp angle (i.e. only slightly separated flows) the turbulence models perform reasonably well, but the comparisons deteriorate as the ramp angles increase. The surface pressure predictions show little difference between the models at shallow angles. As the turning angle increases, differences in the models begin to appear. The SST model shows the largest upstream influence (as measured by the location of the first rise in surface pressure) of the three models, and in almost every case, overpredicts the upstream influence. The SA model consistently shows the least upstream influence, which sometimes is larger and sometimes smaller than the experiment. Recovery predictions are consistent, with the SA model pressure recovery the greatest of the three models for the compression ramp cases, often reasonably close to the experimental observations. The SST model shows the slowest pressure recovery, and underpredicts the experimental values on the compression ramp but greatly overpredicts the peak pressure on the separated cone-flare. The lag model shows a tendency to begin recovering slowly but then start a more rapid recovery up around the SA predicted levels. In all but one case, the compression ramp computations show separation is larger than seen in the experiments. Increasing the ramp angle increases the discrepancy between the measured and predicted separation length. The SST model always predicts larger separation bubbles than the other models, and the SA model predicts the smallest. Differences in the skin friction recovery behavior of each model also become apparent by increasing the ramp angle, with the lag model recovering quickest (often greater than the experiment) and the SA the slowest. The SST model captures the correct trend in the skin friction recovery; however, it appears that the oversized separation bubble causes the magnitude of the skin friction to be lower than the experiment. The heat transfer predictions for both the cone-flare and the compression ramp cases show significant differences between the three turbulence models. For all of the

103 88 cases, the SA model underpredicts the heating downstream of the interaction. The SST model seems to do the best at predicting heat transfer, however it significantly overpredicts the peak heating on the separated cone-flare case. The lag model shows overly rapid recovery near the interaction. It shows a significant overprediction of the heating rate for the 16 compression ramp, although it does the best job predicting the peak heating level for the separated cone-flare. All three turbulence models appear to recover too quickly from the 16 compression ramp interaction. Analysis of computed and measured velocity profiles shows that the computations are reasonably accurate at lower ramp angles, but the predictions get progressively worse as the ramp angle is increased. The turbulence models consistently overpredict the deceleration near the wall due to the interaction. The visible shock locations agree well with experiment, with the exception of the 24 ramp (due to the separation shock being generated too far upstream). The profile behavior differs between the two experimental datasets, such that the computed velocity magnitudes are consistently lower than the experimentally measured velocities for one set but consistently higher than the measured values for the other. For the Princeton series, the SA model appears to give the best prediction of the three models, whereas the Kuntz profiles show an additional inflection point in the profiles that the SST model captures (although the magnitude of the velocity is overpredicted). The Reynolds shear stress (RSS) profiles consistently show that the predicted peak in RSS occurs closer to the wall than it does in the experiments for all turbulence models. The differences between the two experimental datasets is apparent again, with the magnitude of the RSS being much larger for the Kuntz datasets (as Settles and Dodson report). For the Princeton dataset, the computed magnitudes of the peak RSS are reasonably close to experiment. The computations appear to overpredict the RSS near the interaction region and become more accurate farther downstream. For the Kuntz cases, the computations likewise get more accurate moving downstream, except that the predictions near the interaction underpredict the magnitude of RSS.

104 89 Overall, it appears that the single biggest shortcoming of the RANS computations is the failure to predict the size of the separated region. The delayed reattachment appears to be a large source of the discrepancies with experimental data in the reattachment regions. Close analysis of the data shows that many of the curves that appear the best in this study (skin friction, for instance) would likely show a large overprediction if the reattachment point were close to the experimentally observed location. The overall failure to predict the Reynolds shear stress profiles with greater accuracy indicates that important physical mechanisms are not adequately addressed by the turbulence models. It is interesting that all three turbulence models show somewhat similar behavior in these two categories (relative to experimental data, at least). This would lead one to believe that the source of the discrepancy is rooted in an assumption made for all three models, perhaps the Boussinesq approximation. In general, the SST model could be characterized as very prone to separation. The SA model, very appropriately considering its simple formulation, appears to capture the least flow physics. For the aerodynamic computations for which it was designed (surface pressure), it appears to be the best for conservative design work. The lag model, in most cases, shows behavior between the SA and SST models. The shockboundary layer interactions in the present study do not appear to benefit greatly from the non-equilibrium capability of the lag model. The results of the present study indicate that for simple 2D compression corner shock boundary layer interactions, these three RANS models are capable of predicting surface pressures to a level of accuracy adequate for most engineering design work. However, much work remains before accurate predictions of the thermal environments over a flight vehicle may be obtained. 5.2 Future Work Several tasks remain to be completed before the present line of research will be concluded. First of all, the effect of the lag equation in the lag model will be directly

105 90 assessed by repeating a few key cases with the base k ω model to determine what differences are due to the lag equation and which are due to the k ω formulation. Second, several three-dimensional computations will be performed of the inclined flat plate and the compression corner models to estimate the overall influence of end effects. Next, some cases (including the 3-D compression ramp cases) will be run time accurate to observe if any inviscid unsteadiness is created that could be adversely affecting the predicted results. Finally, computations of heat transfer will be performed on the Space Shuttle launch configuration and compared to experimental wind tunnel data. Work is presently underway to improve the prediction capability for turbulent shock boundary layer interactions on two fronts. First of all, a new model, the LagRST model, is being implemented in OVERFLOW [43]. This model does not assume that the Reynolds shear tensor is aligned with the mean strain rate, and also allows for non-equilibrium effects in the same manner as the lag model considered in the present study. Second, the use of a hybrid method, such as detached eddy simulation (DES), will be investigated for canonical SWBLI flows. It is hoped that a more physical representation of the shock-turbulence interaction in the corner region will improve the capability to predict skin friction and heat transfer for these interactions. 5.3 Comments on Future Aerothermal Heating Environment CFD Validation Experiments For the present line of research, it would have been nice to have a zero pressure gradient experimental study that provided heat transfer, skin friction, and detailed boundary layer profiles at multiple stations. This would have permitted a much more thorough analysis (and borderline validation) of the zero pressure gradient predictions. For the shock boundary layer interactions, more cases with heat transfer (with less uncertainty in the freestream conditions between the different measurements) would be desirable. It would also be nice if the flow conditions more closely approximated

106 91 those experienced in flight (Reynolds numbers, temperature ratios, boundary layer thickness to protuberance height, etc.); however, it is understood that in many cases this can be difficult and very expensive. For an experiment to be useful for comparison to a CFD solution of a predicted high-speed aerothermal environment, attention must be paid to several key points. First of all, the boundary conditions need to be clearly defined. Freestream conditions need to be highly repeatable and thoroughly documented, detailed profiles upstream of the region of interest (including C f and C h ) should be documented to alleviate transitional uncertainties, and the wall temperature should be accurately reported. Detailed geometrical definitions are necessary, not only for the region of interest but also for nearby features (i.e. ramp height, ramp width, aerodynamic fence dimensions, distance to tunnel walls, potential sources of disturbances, probe dimensions, downstream geometry, etc.). For the flowfield and surface measurements, non-intrusive methods would be ideal, but a thorough analysis of interference effects (with the results reported, not just summarized) would suffice. The use of multiple measurement techniques would also be helpful in providing data measurement uncertainty. Repeated, accurate measurements of heat transfer are a must. Special care should be taken to thoroughly assess the two dimensionality of the flow, with both pressure and heat transfer measurements taken at several locations in the spanwise direction. High-quality schlieren photos would be helpful to examine shock locations and possible unsteadiness. Other measurements that would be helpful include freestream noise and freestream turbulence for turbulence model boundary conditions, and any trip elements, if used, should be well characterized.

107 LIST OF REFERENCES

108 92 LIST OF REFERENCES [1] A. J. Smits and J.-P. Dussauge, Turbulent Shear Layers in Supersonic Flow, 2nd Edition. New York, NY: Springer, [2] D. S. Dolling, 50 years of shock wave/boundary layer interaction - what next?. AIAA Paper No , June [3] G. S. Settles and L. J. Dodson, Hypersonic Shock/Boundary-Layer Interaction Database, Tech. Rep , Pennsylvania State University, University Park, PA, April NASA Contractor Report. [4] G. S. Settles and L. J. Dodson, Hypersonic Shock/Boundary-Layer Interaction Database: New and Corrected Data, Tech. Rep , Pennsylvania State University, University Park, PA, April NASA Contractor Report. [5] D. C. Wilcox, Turbulence Modelling for CFD. La Canada, CA: DCW Industries, [6] D. Knight, H. Yan, A. G. Panaras, and A. Zheltovodov, Advances in CFD prediction of shock wave turbulent boundary layer interactions, Progress in Aerospace Sciences, vol. 39, pp , [7] K. Sinha, K. Mahesh, and G. Candler, RANS modeling of shock/turbulence interactions using DNS. AIAA Paper No , January [8] K. Sinha and G. Candler, Modeling the effect of shock unsteadiness in shockwave/turbulent boundary layer interactions. AIAA Paper No , January [9] K. Sinha, Shock unsteadiness model applied to hypersonic shockwave/turbulent boundary layer interactions. AIAA Paper No , January [10] D. C. Jespersen, T. H. Pulliam, and P. G. Buning, Recent enhancements to OVERFLOW. AIAA Paper No , January [11] P. Buning, D. Jespersen, T. Pulliam, G. Klopfer, W. Chan, J. Slotnick, S. Krist, and K. Renze, OVERFLOW User s Manual, Version 1.8s. NASA Langley Research Center, [12] M. E. Olsen and D. K. Prabhu, Application of OVERFLOW to hypersonic perfect gas flowfields. AIAA Paper No , June [13] R. P. Lillard and K. M. Dries, Laminar heating validation of the OVERFLOW code. AIAA Paper No , January 2005.

109 93 [14] P. R. Spalart and S. R. Allmaras, A one-equation turbulence model for aerodynamic flows. AIAA Paper No , January [15] F. Menter, H. Grotjans, and F. Unger, Numerical aspects of turbulence modeling for the Reynolds averaged Navier-Stokes equations, VKI Lecture Series, pp. 1 45, March [16] M. E. Olsen and T. J. Coakley, The lag model, a turbulence model for nonequilibrium flows. AIAA Paper No , June [17] A. B. Oliver, R. P. Lillard, G. A. Blaisdell, and A. S. Lyrintzis, Validation of high-speed turbulent boundary layer and shock-boundary layer interaction computations with the OVERFLOW code. AIAA Paper No , January [18] A. M. Cary, Turbulent Boundary Layer Heat Transfer and Transition Measurements With Extreme Surface Cooling in Hypersonic Flows, MS Thesis, University of Virginia, Charlottesville, VA, August [19] T. B. Gatski and G. Erlebacher, Numerical Sumulation of a Spatially Evolving Supersonic Turbulent Boundary Layer, Tech. Rep. NASA TM , NASA Langley Research Center, Hampton, VA, September [20] S. Pirozzoli, F. Grasso, and T. B. Gatski, Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at m=2.25, Physics of Fluids, vol. 16, pp , March [21] G. S. Settles, T. J. Fitzpatrick, and S. M. Bogdonoff, Detailed study of attached and separated compression corner flowfields in high Reynolds number supersonic flow, AIAA Journal, vol. 17, pp , June [22] A. Smits and K. Muck, Experimental study of three shock wave/turbulent boundary layer interactions, Journal of Fluid Mechanics, vol. 182, pp , Sept [23] R. P. Lillard Private Communication, [24] D. Kuntz, V. A. Amatucci, and A. L. Addy, Turbulent boundary-layer properties downstream of the shock-wave/boundary-layer interaction, AIAA Journal, vol. 25, pp , May [25] D. Kuntz, V. A. Amatucci, and A. L. Addy, The turbulent boundary layer properties downstream of the shock-wave/boundary-layer interaction. AIAA Paper No , [26] D. Kuntz, An Experimental Investigation of the Shock Wave-Turbulent Boundary Layer Interaction. PhD Thesis, University of Illinois, Urbana-Champaign, IL, May [27] T. T. Evans and A. J. Smits, Measurements of the mean heat transfer in a shock wave - turbulent boundary layer interaction, Experimental Thermal and Fluid Science, no. 12, pp , [28] S. J. Kline, ed., The AFOSR-HTTM-Stanford conference on complex turbulent flows: Comparison of computation and experiment, Feb

110 94 [29] D. W. Kuntz Private Communication, [30] G. Maise and H. McDonald, Mixing length and kinematic eddy viscosity in a compressible boundary layer, AIAA Journal, vol. 6, pp , January [31] M. Holden, Studies of the mean and unsteady structure of turbulent boundary layer separation in hypersonic flow. AIAA Paper No , June [32] W. M. Chan, R. J. Gomez, S. E. Rogers, and P. G. Buning, Best practices in overset grid generation. AIAA Paper No , June [33] R. Swanson and E. Turkel, On Central-Difference and Upwind Schemes, Tech. Rep , NASA Langley Research Center, Hampton, VA, June NASA Contractor Report (ICASE). [34] R. Paciorri, W. Dieudonne, G. Degrez, J.-M. Charbonnier, and H. Deconinck, Validation of the Spalart-Allmaras turbulence model for application in hypersonic flows. AIAA Paper No , June [35] M. E. Olsen, R. P. Lillard, and T. J. Coakley, The lag model applied to high speed flows. AIAA Paper No , January [36] W. Chan, S. Rogers, S. Nash, P. Buning, and R. Meakin, User s manual for chimera grid tools. On the WWW, September URL rogers/cgt/doc/man.html. [37] P. J. Roache, Perspective: A method for uniform reporting of grid refinement studies, Journal of Fluids Engineering, vol. 116, no. 3, pp , [38] WIND online documentation - equations used by CFPOST. On the WWW, January URL [39] A. A. Zheltovodov and E. K. Shilein, Development of separation in the region where a shock interacts with a turbulent boundary layer perturbed by rarefaction waves, Journal of Applied Mechanics and Technical Physics, vol. 34, no. 3, pp , [40] P. Batcho and J. Sullivan, The 3-D flowfield in a supersonic shock boundary layer corner interaction. AIAA Paper No , January [41] M. Gad-el-Hak and P. R. Bandyopadhyay, Field versus laboratory turbulent boundary layers, AIAA Journal, vol. 33, no. 2, pp , [42] B. Ganapathisubramani, N. Clemens, and D. Dolling, Planar imaging measurements to study the effect of spanwise structure of upstream turbulent boundary layer on shock induced separation. AIAA Paper No , January [43] R. P. Lillard, Turbulence Modeling for Shock Wave/Turbulent Boundary Layer Interactions. PhD Preliminary Report, Purdue University, West Lafayette, IN, November 2005.

111 APPENDIX

112 95 A. 2-D RICHARDSON EXTRAPOLATION FORMULATION The conventional Richardson extrapolation (RE) takes two discrete solutions of the same partial differential equation, using different grid spacings, and cancels terms in the Taylor series expansions of the solutions to obtain an extrapolated solution with leading error terms at the next order higher than the method used to obtain the two initial solutions. The Taylor expansion for the discrete solution, f( ), obtained on a grid with spacing x is f( x) = a 0 + a p x p + a p+1 x p , (A.1) and the Taylor series expansion for the discrete solution f( x/q) obtained on a grid with x/q is f( x/q) = a 0 + a p x p q p + a p+1 x p+1 q (p+1) +..., (A.2) where p is the order of accuracy of the solution method used to obtain f and q is the grid coarsening ratio (q < 1). The term a 0 is the exact solution to the problem, and the a n terms are error terms caused by the discretization (note that not all a n terms are non-zero; for central difference schemes, the odd terms are zero). Combining these two equations and solving for a 0 yields a 0 = f( x) + f( x) f( x/q) q p 1 + O ( x p+1). (A.3) This expression is the familiar 1-D Richardson extrapolation. While it only considers 1-D grid spacing, it is common practice to use this in grid convergence studies while maintaining the aspect ratio of a multidimensional grid constant. This same idea can be applied in two dimensions, however. If a discrete solution is assumed to be a function of two variables, the Taylor series expansion becomes f = a 0 + a 1,x x + a 1,y y + a 2,x x 2 + a 2,y y 2 + a 2,xy x y +... (A.4)

113 96 Assuming the expansion starts from the solution order of accuracy, and using notation similar to the 1-D formulation, this simplifies to f( x, y) = a 0 + a p,x x p + a p,y y p + O( x p+1, x p y,..., x y p, y p+1 ). (A.5) Allowing the x and y direction grid coarsening ratios to vary independently (while the other direction remains constant) gives two more relations: f( x/q x, y) = a 0 + a p,x x p q p x + a p,y y p + O( x p+1, y p+1 ) (A.6) and f( x, y/q y ) = a 0 + a p,x x p + a p,y y p q p y + O( x p+1, y p+1 ). (A.7) Equations A.5-A.7 give a system of 3 equations and 3 unknowns (a 0, a p,x, & a p,y ) which permits the solution of the exact value and each leading error term. Assuming q x = q y = q, solving for these terms yields a 0 = f( x, y) + 2f( x, y) f( x/q, y) f( x, y/q) q p 1 (A.8) a p,x = a p,y = f( x, y) f( x/q, y) (1 q p ) x p (A.9) f( x, y) f( x, y/q) (1 q p ) y p. (A.10) By first obtaining a discrete solution on a grid, then independently varying the grid resolution in the x and y directions (3 solutions total), Equations A.5-A.7 yield information, not only about the overall degree of grid convergence, but also about how the grid should be modified to reduce the truncation error. As an example, a coarse Holden 6 cone 42 flare will be considered. The discrete solution, f, is the skin friction along the surface of the cone computed with the SA turbulence model. The original grid is , q x = q y = 1/ 2, and p is assumed to be 2. Figure A.1 shows that the streamwise (x) error term is two orders of magnitude smaller than the wall-normal (y) error term upstream of the corner (x = [in]). Furthermore, the wall-normal error term is contributing less than 1% error to the prediction of the upstream boundary layer. Depending on the degree of accuracy

114 97 sought, this could indicate that any further refinement of this region would simply be wasting computational effort, and that the efficiency could be improved by coarsening the streamwise grid spacing. The the relative errors used in the extrapolation blow up in the interaction region; however, downstream of the interaction, things settle down and it can be seen that both the steamwise and wall normal error terms are contributing roughly 10% error to the prediction. One significant simplification made in Figure A.1 is choosing to plot the absolute values. It is possible that the streamwise and wall-normal error terms could have opposite signs, and partially cancel each other out. If such a case were to occur, this method would indicate that the obtained solution was much less accurate (relative to a 0 ) than it actually is. The theoretical background for this method is rather crude and many significant terms were neglected, so this method should not be used for obtaining higher-order solutions to a discretized problem. It is intended only to aid in obtaining an efficient, well resolved grid system by putting grid points where they are needed.

115 98 (a) (b) Fig. A.1. (a) Coarse grid of the 42 cone-flare case. (b) Leading error terms (scaled by the local a 0 ) of the skin friction distribution.