Entitled Assessment of Turbulence Models and Corrections with Application to the Orion Launch Abort System

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1 Graduate School ETD Form 9 (Revised 12/07) PURDUE UNIVERSITY GRADUATE SCHOOL Thesis/Dissertation Acceptance This is to certify that the thesis/dissertation prepared By Nico Gross Entitled Assessment of Turbulence Models and Corrections with Application to the Orion Launch Abort System For the degree of Master of Science in Engineering Is approved by the final examining committee: Gregory A. Blaisdell Chair Anastasios S. Lyrintzis Charles L. Merkle To the best of my knowledge and as understood by the student in the Research Integrity and Copyright Disclaimer (Graduate School Form 20), this thesis/dissertation adheres to the provisions of Purdue University s Policy on Integrity in Research and the use of copyrighted material. Approved by Major Professor(s): Gregory A. Blaisdell Anastasios S. Lyrintzis Approved by: Anastasios S. Lyrintzis 4/15/10 Head of the Graduate Program Date

2 Graduate School Form 20 (Revised 1/10) PURDUE UNIVERSITY GRADUATE SCHOOL Research Integrity and Copyright Disclaimer Title of Thesis/Dissertation: Assessment of Turbulence Models and Corrections with Application to the Orion Launch Abort System For the degree of Master of Science in Engineering I certify that in the preparation of this thesis, I have observed the provisions of Purdue University Teaching, Research, and Outreach Policy on Research Misconduct (VIII.3.1), October 1, 2008.* Further, I certify that this work is free of plagiarism and all materials appearing in this thesis/dissertation have been properly quoted and attributed. I certify that all copyrighted material incorporated into this thesis/dissertation is in compliance with the United States copyright law and that I have received written permission from the copyright owners for my use of their work, which is beyond the scope of the law. I agree to indemnify and save harmless Purdue University from any and all claims that may be asserted or that may arise from any copyright violation. Nico Gross Printed Name and Signature of Candidate 4/15/10 Date (month/day/year) *Located at

3 ASSESSMENT OF TURBULENCE MODELS AND CORRECTIONS WITH APPLICATION TO THE ORION LAUNCH ABORT SYSTEM A Thesis Submitted to the Faculty of Purdue University by Nico Gross In Partial Fulfillment of the Requirements for the Degree of Master of Science in Engineering May 2010 Purdue University West Lafayette, Indiana

4 For Brittany and Mingo ii

5 iii ACKNOWLEDGMENTS This work was supported by NASA Johnson Space Center under grant number NNX09AN06G. I would like to thank Brandon Oliver, Randy Lillard, Alan Schwing, and Darby Vicker in the Aerosciences and CFD Branch at JSC for all their help in using OVERFLOW. Also, Robert Childs and Andrea Nelson at NASA Ames Research Center were very helpful for questions related to turbulence modeling and the Glenn experiment. My advisers, Professors Greg Blaisdell and Tasos Lyrintzsis, spent a lot of time discussing my research, offering advice, and providing guidance. Finally, Matt Churchfield helped me get started with using OVERFLOW.

6 iv TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES ABSTRACT Page 1 INTRODUCTION Background Physics of Jet Flows Discussion of Previous Work Experimental Investigations Computational Analyses Objectives Organization of Thesis TEST CASES Eggers Jet Wishart Jet Glenn Jet Other Test Cases COMPUTATIONAL METHODS Code Description Turbulence Models Spalart-Allmaras Model Baldwin-Barth Model k-ω Model SST Model Turbulence Model Corrections Compressible Turbulence Modeling Sarkar Compressibility Correction Zeman Compressibility Correction Wilcox Compressibility Correction Pressure Dilatation Abdol-Hamid Temperature Correction Spalart-Shur Rotation/Curvature Correction Grid Generation Axisymmetric Jets vii viii xiii

7 v Page D Jet in Crossflow Boundary Layer Mixing Layer Boundary Conditions Axisymmetric Jets D Jet in Crossflow Flat Plate Mixing Layer Solution Method Numerical Solution Convergence Computer Resources RESULTS FOR AXISYMMETRIC JETS Grid Convergence Eggers Jet Turbulence Model Assessment Compressibility Corrections Pressure Dilatation Correction Heated and Perfectly Expanded Wishart Jet Isothermal and Underexpanded Wishart Jet Heated and Underexpanded Wishart Jet High Pressure and Temperature Parametric Study RESULTS FOR 3-D JET IN CROSSFLOW Grid Convergence Without Capsule Turbulence Models Turbulence Model Corrections With Capsule Impact on Flight Mechanics Experiment Uncertainties RESULTS FOR COMPRESSIBILITY CORRECTION ANALYSIS Background Formulation of New Corrections Application to an Axisymmetric Jet Application to a 3-D Jet in Crossflow Application to Compressible Boundary Layers Application to Compressible Mixing Layers SUMMARY Conclusions Axisymmetric Jets

8 vi Page D Jet in Crossflow Compressibility Correction Analysis Future Work LIST OF REFERENCES APPENDIX VITA

9 vii Table LIST OF TABLES Page 2.1 Reference Conditions for Test Cases Considered Number of Grid Points for Test Cases Changes in Aerodynamic Coefficients Caused by Compressibility Correction Maximum Turbulence Mach Number for Flat Plate Conditions for Compressible Mixing Layer

10 viii Figure LIST OF FIGURES Page 1.1 Diagram of Launch Abort System showing Abort Motors (AM), Jettison Motors (JM), and Attitude Control Motors (ACM). Courtesy of NASA [2] Diagram of jet flow. Used with permission from Wishart [3] Diagram of vortex structure for jet in crossflow. Used with permission from Fric and Roshko [4] Diagram of Glenn test experimental setup Diagram of Glenn test experimental setup showing PIV planes downstream of nozzle Schematic showing locations of pressure transducers in streamwise and circumferential directions (a) Schematic of overall grid system for axisymmetric jets (showing every fourth grid line). (b) Detail view of the nozzle lip region (a) Schematic of center (Y=0) cut of grid system for Glenn jet without capsule. (b) Schematic showing grid on tunnel centerbody (green), nozzle body (red), and nozzle (blue) walls, and center (Y=0) cut of plume grids (orange) Schematic of grid around capsule for Glenn jet Grids for conical nozzle and velocity vectors specified in plenum for Glenn jet Grid topology for generic boundary layer (showing every third grid line) Grid topology for generic mixing layer (showing every fifth grid line) Typical residual behavior for Glenn jet Force and moment monitoring on capsule surface for Glenn jet Grid refinement for Wishart underexpanded jet showing centerline Mach profile Grid refinement for Wishart underexpanded jet showing downstream Mach profiles

11 ix Figure Page 4.3 Grid refinement for Wishart underexpanded jet showing downstream mean temperature profiles Normalized centerline velocity for Eggers jet Inverse velocity for Eggers jet with the x-axis shifted using the Witze correlation Normalized radial velocity profiles for Eggers jet Normalized half radius for Eggers jet (a) Normalized centerline velocity and (b) normalized half radius for Eggers jet showing effect of compressibility corrections (a) Normalized centerline velocity and (b) normalized half radius for Eggers jet showing effect of including pressure dilatation Centerline quantities for heated and perfectly expanded Wishart jet. (a) Mach number. (b) Normalized temperature Radial Mach number profiles for heated and perfectly expanded Wishart jet Centerline quantities for isothermal and underexpanded Wishart jet. (a) Mach number. (b) Normalized temperature Radial Mach number profiles for isothermal and underexpanded Wishart jet Contours of Mach number showing shock cell structure for heated and underexpanded Wishart jet for cases CC=0, TC=0 (top), CC=1, TC=0 (middle), and CC=1, TC=1 (bottom) Centerline quantities for heated and underexpanded Wishart jet. (a) Mach number. (b) Normalized temperature Radial Mach number profiles for heated and underexpanded Wishart jet Normalized half radius for underexpanded jet at elevated conditions. (a) Increasing pressure ratio comparing CC. (b) Increasing temperature ratio comparing TC Grid refinement for Glenn jet showing normalized velocity magnitude profiles along center streamline Grid refinement for Glenn jet showing downstream normalized velocity magnitude profiles

12 x Figure Page 5.3 Downstream velocity magnitude contours for Glenn jet showing effect of baseline turbulence model Downstream velocity magnitude contours for Glenn jet showing effect of compressibility corrections Normalized downstream velocity magnitude profiles for Glenn jet showing effect of compressibility corrections Downstream vorticity magnitude contours for Glenn jet (a) Jet position and (b) normalized velocity along center streamline for Glenn jet Normalized downstream velocity magnitude profiles for Glenn jet showing effect of including pressure dilatation Contours of Mach number and surface pressure coefficient on capsule Downstream velocity magnitude contours for Glenn jet with capsule Normalized downstream velocity magnitude profiles for Glenn jet with capsule Surface pressure coefficient for streamwise measurements for Glenn jet with capsule Surface pressure coefficient for radial measurements for Glenn jet with capsule Delta C p contours on capsule surface Normalized mixing layer growth rate data compared with previous efforts and recent experiments my Rossman. Used with permission from Rossman [36] Dissipation ratio as a function of turbulence Mach number for compressibility corrections (a) on log-log scale, along with DNS data by Blaisdell el al. [22], and (b) with regular scale in range of interest Contours of turbulence Mach number for Eggers jet (a) Normalized centerline velocity and (b) normalized half radius for Eggers jet using modified compressibility corrections Contours of turbulence Mach number for Glenn jet Downstream velocity magnitude contours for Glenn jet using modified compressibility corrections

13 xi Figure Page 6.7 Normalized downstream velocity magnitude profiles for Glenn jet using modified compressibility corrections Normalized downstream velocity magnitude profiles for Glenn jet with capsule using modified compressibility corrections Skin friction predictions as a function of Reynolds number based on momentum thickness, compared with Van Driest II correlation, for flat plate. Computations use standard compressibility correction implementation Skin friction predictions as a function of Reynolds number based on momentum thickness, compared with Van Driest II correlation, for flat plate. Computations use modified compressibility correction implementation Mixing layer growth as function of convective Mach number

14 xii Appendix Figure Page A.1 Centerline quantities for isothermal and perfectly expanded Wishart jet. (a) Mach number. (b) Normalized temperature A.2 Radial Mach number profiles for isothermal and perfectly expanded Wishart jet A.3 Centerline quantities for isothermal and overexpended Wishart jet. (a) Mach number. (b) Normalized temperature A.4 Radial Mach number profiles for isothermal and overexpanded Wishart jet A.5 Centerline quantities for heated and overexpended Wishart jet. (a) Mach number. (b) Normalized temperature A.6 Radial Mach number profiles for heated and overexpanded Wishart jet. 113

15 xiii ABSTRACT Gross, Nico. M.S.E., Purdue University, May Assessment of Turbulence Models and Corrections with Application to the Orion Launch Abort System. Major Professors: Gregory A. Blaisdell and Anastasios S. Lyrintzis. The aim of this research is to assess turbulence modeling issues for rocket plumes with application to the Orion Launch Abort System (LAS). Turbulence models, used with the Reynolds-averaged Navier-Stokes (RANS) equations, are critically important for an accurate flow field prediction. NASA s OVERFLOW code is used for computations of supersonic jets. First, baseline turbulence performance is assessed. Then, corrections for compressibility, temperature, and rotation/curvature effects are evaluated based on comparisons with experimental data. Test cases progressively increase in complexity and include axisymmetric jets, 3-D jets with crossflow, and a full LAS-like configuration. For the axisymmetric jets, the test cases are isothermal and heated, perfectly expanded and underexpanded axisymmetric jets with design Mach numbers around two. The 3-D jet in crossflow is based on a NASA Glenn experiment and includes elevated conditions (higher Mach numbers and temperatures) more similar to the actual LAS. The SST turbulence model shows the best performance and is chosen for the flows examined in this study. For the axisymmetric jet, results demonstrate that using a compressibility correction is important for a good prediction of the jet development. The temperature correction shows improved results for the heated cases, but its effect is not as significant. For the 3-D jet in crossflow, conclusions from using corrections are not as decisive. While use of the compressibility correction improves the vorticity contours shape, the velocity decay is slightly underpredicted. The temperature and rotation/curvature corrections show little improvement. The overall recommendation

16 xiv is that the corrections must be used with caution, especially when approaching high temperature and high Mach number regimes. Finally, two modified compressibility correction are introduced based on the notion that the growth rate in compressible free shear flows levels off at a certain turbulence Mach number. The modified corrections do not disturb the beneficial behavior of the standard corrections observed for the axisymmetric jets at the given conditions. For the 3-D jet in crossflow, the modified corrections show significant improvement over the Sarkar and uncorrected models. The compressible boundary layer test case shows that there is still an underprediction of the wall skin friction coefficient, but this can be solved by using a simple modification in the turbulence model. Finally, analysis of the compressible mixing layer shows that the modified corrections exhibit the correct trend in limiting the growth rate, but do not match exactly with the experimentally-obtained results.

17 1 1. INTRODUCTION 1.1 Background The modeling of exhaust plumes has been a problem of interest since the earliest development of rocket and jet engines. Sutton et al. [1] explain some of the motivations for understanding rocket plumes. For a propulsion system at given operating conditions, it is important to predict the plume dimensions and temperature and pressure profiles. The aerodynamic interaction of the plume with the freestream around the vehicle can cause significant changes in drag and aerodynamic coefficients. It is also important to understand the heat transfer to other components of the vehicle, test facility, propulsion system or launcher, and prevent damage by design changes. There are various other motivations for understanding jet flows. For commercial airplanes, the study of the noise generated by the aircraft engine exhaust jets is very important. This is frequently related to the mixing of the coaxial flows of a turbofan engine. In internal combustion engines, a jet of fuel is injected into the chamber to mix with the oxidizer and combust. Jets also have film cooling applications on turbine blades, where the goal is to keep the jet attached to the wall to shield the blades from the high temperatures of the combustion products. For this study, understanding the interaction between the rocket exhaust jet plume and the capsule is an important issue in design and analysis for the Crew Exploration Vehicle (CEV) Launch Abort System (LAS), currently under the development as part of NASA s Project Constellation. The CEV, also known as Orion, will be the new capsule used to return humans to orbit, the Moon, and eventually to Mars. The LAS separates the capsule from the main rocket in the case of an emergency. The various solid rocket motors of the LAS are shown in Fig Most of this study is focused on analysis of a single Abort Motor (AM). If the plume impinges on the capsule surface,

18 2 large localized pressure and temperature peaks may be created due to surface, shock, and freestream interactions. This could be particularly problematic for heat transfer predictions. When the jet is not directly impinging on the surface, the shock cells present near the capsule may cause large changes in the pressure coefficients on the capsule. This can have significant effects on the flight mechanics, such as the pitching moment, of the entire vehicle. Figure 1.1. Diagram of Launch Abort System showing Abort Motors (AM), Jettison Motors (JM), and Attitude Control Motors (ACM). Courtesy of NASA [2]. 1.2 Physics of Jet Flows In fluid mechanics, jets represent the viscous and turbulent interaction between a coherent stream exiting a nozzle and the surrounding freestream. A diagram of an axisymmetric jet flow is shown in Fig The shear layers, which form at the nozzle boundary layer, mix with the surrounding fluid. The center of the jet, known as the potential core, remains essentially inviscid until the shear layers meet. After this point, the jet becomes fully developed and shows a constant decay in velocity. The velocity profile in the fully developed region is known to be self similar. Parameters of interest when studying jets include the extent of the potential core, the spread rate, and the velocity decay rate. When the jet is no longer axisymmetric, three-dimensional effects must be considered and the jet structure becomes more complex, as shown in Fig For a

19 3 Figure 1.2. Diagram of jet flow. Used with permission from Wishart [3]. jet in crossflow, the interaction with the freestream causes various vortex formations. For example, the freestream moving around the jet causes the formation of counter-rotating vortex pairs in the main jet stream, and wake vortex interactions downstream. Figure 1.3. Diagram of vortex structure for jet in crossflow. Used with permission from Fric and Roshko [4]. There are a number of physical processes that affect the structure of the plumes, all of which need to be modeled sufficiently to allow for good computational fluid dynamics (CFD) solutions. These processes include, but are not limited to, compressibility,

20 4 high temperature, variable specific heats, vortex stretching, and curvature effects. In addition, turbulence and distortion from the combustion process may be important. Various nonlinear interactions between all these phenomena are expected [5]. Choosing suitable computational models is key to a good prediction of the flowfield. For example, the turbulence model determines parameters such as the shock locations, the jet spread rate, and the rate of diffusion of the plume, all of which have significant impact on the overall design and performance of the vehicle. 1.3 Discussion of Previous Work Experimental Investigations A significant amount of work has been done to study supersonic jet flows. Experimental investigations into jet flows received attention with the development of more advanced aircraft and launch vehicles in the 1960 s. Understanding the behavior of rocket or jet exhaust plumes became significantly more important when dealing with flows at higher Mach numbers. Eggers [6] performed measurements of velocity profiles and eddy viscosity distributions of a Mach 2.22 jet exiting into ambient air. These provided some of the first detailed measurements of the near- and far-field of a supersonic jet. Birch and Eggers [7] compiled many of the early experiments performed for free shear flows into the Langley curve, which is commonly used to calibrate turbulence models. It was found that for mixing layers, the spread rate decreases with increasing convective Mach number. Lau [8] examined subsonic and supersonic round jets at elevated temperature ratios. He noticed that the jet spreading rate falls with increasing temperature for subsonic flows and rises for supersonic flows. Seiner et al. [9] also examined heated supersonic jets. They found that an increased jet total temperature gradient enhanced the mixing in the jet, which is consistent with the findings by Lau. They reasoned that the enhanced density fluctuations in the flow cause additional instabilities and lead

21 5 to a more rapid diffusion into the freestream. More recently, Wishart [3] completed similar study including heated and off-design conditions Computational Analyses Advances in CFD have allowed numerical analysis of jet flows. Most of the viscous CFD codes in use for engineering analysis and design work solve the Reynoldsaveraged Navier-Stokes (RANS) equations. The RANS equations are obtained by time-averaging the Navier-Stokes equations which govern fluid flow. The additional term produced by the averaging process, consisting of the Reynolds stresses, can not be exactly modeled based on the mean flow variables. As a result, a turbulence model is necessary to provide closure to the system of equations. Most popular models in use today linearly relate the Reynolds stresses to the strain rate via the Boussinesq approximation. Through these assumptions, many inaccuracies are inherent in the modeling of turbulence. Recent improvements in computing power have increased the popularity of DNS, LES, and hybrid RANS/LES approaches for modeling or simulating turbulent flows. In direct numerical simulation (DNS), all spatial and temporal scales of the turbulence are resolved directly. This requires a very fine computational mesh that captures everything from the smallest scales, known as Kolmogorov length scales, to the larger integral length scales. In practice, DNS has only proven useful for very simple flows. When using DNS for jet flows, an axisymmetric analysis is not allowed because the full 3-D effects of the turbulence must be simulated. In order to have sufficient grid resolution for all length scales, the increase in computational cost is proportional to Re 3, where Re is the Reynolds number. Because of this cost, the DNS of a jet has only been performed up to a Reynolds number of about 5,000, such as the simulations of Muppidi and Mahesh [10]. In large eddy simulation (LES), only the larger turbulence length scales, which are presumed to have the more dominant effect on the mean flow, are simulated,

22 6 while the finest scales are filtered out and modeled. A number of LES computations for jet flows have been performed. For example, Uzun [11] looked at compressible turbulent jets at various Reynolds numbers up to 400,000 and compared mean flow properties, turbulence statistics, and aeroacoustics data. While LES has proven useful for simplified jets, its application for complex configurations, such as the LAS, is still impractical due to the computational cost. At present, the development of improved RANS modeling capabilities for these flows is of interest. This research focuses on the application of the RANS models to jet flows. Many computational analyses intended to evaluate turbulence models have made use of the before-mentioned experiments for baseline validation. Georgiadis et al. [12] evaluated modified two-equation turbulence models specifically developed for jet flows. They found that all models showed a deficiency in the initial jet mixing rate and turbulence kinetic energy predictions. Slight improvements in the modified models over the standard models showed better axial velocities, but not better turbulence kinetic energy predictions. Abdol-Hamid et al. [13] applied two-equation turbulence models to a heated jet, comparing to the experiment by Seiner et al [9]. They determined that an additional convective heat transfer model is needed to accurately predict jets with a temperature gradient. Goto et al. [14] examined supersonic underexpanded jets impinging on an inclined flat plate and compared RANS and implicit LES (ILES) results. This allowed examination of the causes of the localized pressure peaks and their unsteady behavior. They found that the key factors leading to the pressure peaks were strong shock waves in the upstream area, stagnation points in the upstream area, and interaction between the shock and the boundary layer. 1.4 Objectives The overall purpose of this research work is to improve jet plume modeling capabilities for the NASA CEV Aerosciences Project (CAP) team. The main objectives of this research are listed below:

23 7 Assess the capability of the OVERFLOW code to model jet flows. Evaluate the performance of turbulence models for baseline jet flows. Determine the effects of turbulence model corrections for heated and supersonic jets. Analyze a three dimensional jet in crossflow at conditions similar to those of the LAS. Assess effects of jet plume modeling on the CEV capsule surface and determine the impact on vehicle flight mechanics. Devise, implement, and test a new compressibility correction which improves predictions for jet flows. 1.5 Organization of Thesis The rest of the thesis is structured as follows. First, the various experiments which are used for comparison to CFD results are presented. Then, a background of the computational methods used for the subsequent analysis is discussed. This includes the choice of turbulence models and corrections, grid generation, and the solution method. Next, the results for all the test cases are presented and discussed. The first set of analyses is performed for axisymmetric, perfectly expanded, and isothermal jets. Next, compressibility, temperature, and three-dimensionality effects are analyzed. The analysis is then extended to a 3-D jet in crossflow. The final test case is a LASlike configuration, which includes the effects of the turbulent plume on the capsule. Finally, two modified compressibility corrections are formulated and evaluated using several test cases. From the analysis, recommendations for the modeling of LAS jet flows are made.

24 8 2. TEST CASES In order to assess the ability of the OVERFLOW code to model supersonic jets, simple test cases are considered. A number of experiments are selected to show the variation of several parameters of interest. Various axisymmetric jet experiments are selected to analyze the effects of compressibility and temperature. Then, a 3-D jet in an angled crossflow is used to assess further parameters of interest which could not be studied using the axisymmetric jets. To analyze these jets, off-body mean flow data, including velocities and temperatures, are of primary interest. From this, the jet is evaluated using parameters such as the potential core length, the jet spread rate, and the temperature rise and decay. The experiments are chosen based on the quality of the data available and the conditions of interest. For axisymmetric supersonic jet flows, the experiments by Eggers [6], Lau [15] [8], and Seiner et al. [9] are the baseline test cases widely used for turbulence model assessment. All these tests include perfectly expanded supersonic jets with a exit Mach number around two. For this analysis, the Eggers jet is used for all baseline analysis and calibration. In order to assess jets operating at off-design conditions, a test case by Wishart [3] is included. The 3-D jet in crossflow provides higher pressure ratios and three-dimensional effects of plume and freestream interactions. Experimental data measured using particle image velocimetry and pressure transducers in the flow field are available for comparison to CFD results. Key conditions for the test cases considered are summarized in Table Eggers Jet The Eggers [6] jet refers to a 1966 experiment characterizing supersonic turbulent jets. A Mach 2.22 round, isothermal jet with nozzle exit diameter of 1.01 was

25 9 Table 2.1 Reference Conditions for Test Cases Considered Test Case Classification M nozzle P 0 /P 0, T 0 /T 0, Eggers perfectly expanded, isothermal Wishart overexpanded, isothermal Wishart overexpanded, heated Wishart perfectly expanded, isothermal Wishart perfectly expanded, heated Wishart underexpanded, isothermal Wishart underexpanded, heated Glenn underexpanded, heated exhausted into ambient air, with a pressure ratio of 11. Isothermal does not imply that the static temperature remains constant in the jet, rather that the stagnation temperature is the same between the nozzle inlet and the ambient. The nozzle was designed using the method of characteristics to ensure purely axial flow at the exit. The experiment supplies velocity profiles and eddy viscosity distributions in the jet up to 150 nozzle radii downstream. This test case provides a standard test case for compressible jet validation. 2.2 Wishart Jet The Wishart [3] experiment, performed at Florida State University, is a similar study to that of Eggers, but with conditions more closely resembling those of the LAS. For this test case, emphasis was placed on the near-field and data is not available as far downstream as for the Eggers case. The resolution of the data downstream is not adequate, so the data is limited to 17 exit diameters behind the nozzle. This range is of primary interest because it corresponds to the relative location of the capsule with respect to the nozzle. Also, the data is digitized from the manuscript, so due to practicality, only selected data sets are used.

26 10 The jet used a nozzle designed for Mach 2, a nozzle exit diameter of 1.15, and was operated at ideal and off-design conditions. A pressure ratio corresponding to each underexpanded, perfectly expanded, and overexpanded flow conditions was selected. All cases were run isothermal and heated. For the heated cases, the the nozzle stagnation temperature is larger than the ambient temperature. These conditions allow examination of the shock-cell structure and temperature effects on the development of the jet. Various test conditions are included for completeness, and the results are shown in the Appendix. However, the focus of our study is on heated and underexpanded jets. These effects are first analyzed individually and then together. 2.3 Glenn Jet The Glenn jet experiment (Wernet et al. [16]), performed at NASA Glenn Research Center, was used to acquire data for a hot supersonic jet in subsonic crossflow. It was performed as part of the aerodynamics analysis and design process for the LAS. The initial request for the experiment arose because of numerous significant discrepancies which were observed in LAS aerodynamic force coefficient predictions due to turbulence model selection. Namely, the SST two-equation and Lag three-equation turbulence models showed a deviation of over 100% in the axial force coefficient. Thus, further analysis was needed to validate the turbulence model selection. Actual flight conditions of the LAS, 2,000 psi stagnation pressure and 5,500 R stagnation temperature, were not possible in any wind tunnel. The jet conditions used here were at the highest possible pressures and temperatures attainable in the test facility, which was stagnation pressure and temperature of 410 psi and 1,350 R, respectively. It was assumed that these conditions are sufficiently high to start examining compressibility and temperature effects on turbulence models [5]. A survey of available literature showed that no other experiments had been performed at similar conditions.

27 11 Fig. 2.1 shows a diagram of the experimental setup. The objects were placed in a wind tunnel with cold freestream air at Mach 0.3. In the centerbody, air was heated using a natural gas combustor and fed through the nozzle body. The convergingdiverging nozzle, with a conical diverging section and exit diameter of 0.81, had a design exit Mach number of about 3.0, but because the flow was underexpanded, Mach numbers exceeding four were seen shortly after the nozzle exit. The conical shape closely resembles an actual rocket engine than the perfectly expanded shape used for the previous jets. While momentum losses are created because the flow is not purely axial, the reduction in weight is typically of increased advantage when designing a rocket engine. The nozzle exhausted air at an angle of 25 degrees relative to the tunnel freestream. For the first case, the capsule was removed and the experiment was simply a jet in an angled cross flow. For the second case, the capsule was put in place behind the nozzle, with a similar relative spacing as seen with the actual launch abort system in Fig The jet exiting the nozzle showed significant interaction with the capsule but did not impinge directly on it. Experimental data was obtained with Stereo Particle Image Velocimetry (PIV) at various locations downstream of the nozzle exit and normal to the freestream, producing two dimensional velocity vector fields, as depicted in Fig The PIV was taken mainly where the flow affects the capsule, and not further downstream. Pressure taps were also placed on the capsule surface at various axial and radial locations, as shown in Fig Other Test Cases In addition to the supersonic jets, test cases are also computed for a generic boundary layer and mixing layer. These cases are not based on particular experiments.

28 12 Figure 2.1. Diagram of Glenn test experimental setup. Figure 2.2. Diagram of Glenn test experimental setup showing PIV planes downstream of nozzle.

29 Figure 2.3. Schematic showing locations of pressure transducers in streamwise and circumferential directions. 13

30 Code Description 3. COMPUTATIONAL METHODS OVERFLOW version 2.1z is used as the solver for all computations presented here. The OVERFLOW code [17] is an overset grid solver developed by NASA. It is currently used for aerodynamics and aerothermodynamics analysis of the Space Shuttle and Orion vehicles. It solves the time-dependent, RANS equations for compressible flows. Its overset, or Chimera, grid capability makes it useful for computing flow fields involving complex geometries. The code is formulated using a finite difference method and has various central and upwind schemes for spatial discretization built in. A diagonalized, implicit approximate factorization (ADI) scheme is used for time advancement. Scalar transport equations are used for multi-species calculations. Other capabilities, such as local time-stepping and grid sequencing, are also available to accelerate convergence. 3.2 Turbulence Models Choosing the appropriate turbulence model is of vast importance in the prediction of jet flows. The model determines important parameters in a jet (e.g., the location of shocks) and thus can have a significant impact on the overall design of the vehicle. The OVERFLOW code contains various algebraic, one-, and two-equation turbulence models. For this study, four of the most popular turbulence models are used for comparison: the Spalart-Allmaras (SA), Baldwin-Barth (BB), k-ω, and shear stress transport (SST) models.

31 Spalart-Allmaras Model The Spalart-Allmaras [18] model is a one-equation turbulence model developed for aerodynamic flows. The model computes one partial differential equation to compute the eddy viscosity. It does not solve for the turbulence kinetic energy. The simplicity and good predictions for a variety of flows have made the SA one of the most popular models in use today Baldwin-Barth Model The Baldwin-Barth [19] model is another one-equation model. It was initially developed for wall-bounded flows, but has shown some promising results for turbulent jet computations [5] k-ω Model Wilcox s [20] k-ω model is a two-equation eddy-viscosity model. The first differential equation is solved for the production of turbulence kinetic energy, k. The second equation solves for the specific dissipation rate, ω. The eddy viscosity is then derived from an algebraic relation of k and ω. There are three versions of this model (1988, 1998, 2006). The implementation in OVERFLOW used in this study is the 1988 version of this model. The k-ω is one of the most widely-used turbulence models that performs particularly well for wall-bounded flows SST Model Menter s [21] SST model is another two-equation eddy-viscosity model that has recently become very popular for application to engineering flows. It is essentially a hybrid between the k-ǫ and k-ω and is formulated in terms of the turbulence kinetic energy, k, and the specific dissipation rate, ω. The eddy viscosity is computed from the standard k-ω formulation of the normalized Reynolds shear stress. The model

32 16 uses a hyperbolic tangent blending function based on the distance to the nearest wall and behaves like the k-ǫ model in regions of free shear, while retaining the favorable behavior of the k-ω model near the walls. 3.3 Turbulence Model Corrections Current turbulence models have been designed for low-speed, isothermal flows. While developing a new model that is general for all types of flows should be the longterm goal of research efforts, the current approach is to modify current turbulence models for more complicated flows. Most corrections have been devised to work with two-equation turbulence models. For this study, all corrections are applied to the SST model, but they could be implemented for other two-equation models, such as the k-ω model, as well. The intent of this work is to assess the various corrections for turbulence models that are applicable to the flow of interest, namely compressibility, high temperature and rotation/curvature. Compressibility effects clearly become important for jets at high Mach numbers where the velocity is no longer divergence free. Temperature effects must also be considered due to the high temperatures experienced in the rocket nozzle exhaust. Lastly, rotation and curvature become important because of the curvature of the jet in the freestream and the resulting vortex interactions Compressible Turbulence Modeling When deriving the RANS equations for compressible flows using Favre averaging, additional terms are introduced due to density fluctuations which would be zero for the incompressible case. Blaisdell et al. [22] explain the meaning of the additional terms in the mass-averaged turbulence kinetic energy equation and assess the relative importance of the terms to turbulence modeling: 1) The solenoidal dissipation rate is the same as the incompressible dissipation rate and does not change for compressible flows. The term acts to decrease the turbulence growth. 2) The dilatation dissipa-

33 17 tion occurs only for compressible flows. It is proportional to the divergence of the velocity and inhibits the turbulence growth. 3) The pressure dilatation represents the exchange between internal and kinetic energy in the flow. This term may act to increase or decrease the turbulence growth Sarkar Compressibility Correction The compressibility correction is devised to deal with additional effects seen for higher Mach number flows, specifically, the effects of compressibility on the dissipation rate of the turbulent kinetic energy. For free shear flows, this is exhibited as the growth rate in the mixing layer [7]. Current turbulence models do not account for this Mach number dependence, and thus a compressibility correction is used. Sarkar et al. [23] first showed that for compressible flows, an extra dilatation dissipation term, ε d, can be introduced in the turbulence kinetic energy and specific dissipation rate equations for two-equation models. This term is included in addition to the solenoidal, or incompressible, dissipation. The effect thus is that the growth rate in turbulence equation, k, is inhibited when the correction is active. The ratio of the t dilatation dissipation to the solenoidal dissipation, ε d /ε s, is modeled as a function of the turbulence Mach number, M t, defined as M t = 2k a (3.1) For Sarkar s model, ε d ε s = F(M t ) = M 2 t (3.2) All compressibility corrections use this basic formulation, but the details vary. In OVERFLOW, the compressibility correction is only implemented for the k-ǫ part of the SST turbulence model. Wilcox [20] showed that compressibility corrections can produce unwanted effects in the near-wall regions, such as the tendency

34 18 to produce too little wall skin friction, which is investigated in a later chapter, and the wrong boundary layer separation length in a shock-boundary layer interaction. The OVERFLOW implementation is initially presented in the paper by Suzen and Hoffmann [24]. The SST model is formulated, in Cartesian tensor notation, as follows: ρk t + ρku j x j = P k ρωβ k [1 + α 1 F(M t )(1 F 1 )] + (1 F 1 )p d + [ (µ + σ k µ t ) k ] x i x i (3.3) ρω t + ρωu j x j = γp k ν t (1 F 1 ) p d ν t β ρω 2 + (1 F 1 )β α 1 F(M t )ρω ρσ ω2 ω (1 F 1) k ωx j x j + x i [ (µ + σ k µ t ) k ] x i (3.4) where P k is the production term. The first equation is solved for the turbulence kinetic energy and the second equation is solved for the specific dissipation rate. In both equations, a compressibility correction, F(M t ), is added to the dissipation term. In addition, a term for the pressure dilatation, p d, is added, and is modeled separately as shown below. Both of the correction terms are multiplied by the SST blending function, F 1, which effectively disables the corrections in the near wall regions Zeman Compressibility Correction Zeman [25] also devised a correction based on an effort to match mixing layer growth rates. His model is given by F(M t ) = { 1 exp [ 12 ]} (γ + 1)(M t M t0 ) 2 /Λ 2 H(M t M t0 ) (3.5) where M t0 = /(γ + 1). The function is multiplied by a constant, ξ, which controls the magnitude of the shift. ξ is 3/4 for this model.

35 Wilcox Compressibility Correction As mentioned before, Sarkar s model can produce unwanted effects in boundary layers. Thus, Wilcox [20] suggested a similar model that uses a Heaviside step function to activate and deactivate this correction. The model constant, M t0, is set to disable the correction below M t of The constant ξ is set to 2. The model is formulated as F(M t ) = (M 2 t M2 t0 )H(M t M t0 ) (3.6) Pressure Dilatation When looking at the mass-averaged RANS equations, a term known as the pressuredilatation must also be modeled. Sometimes, this term is simply ignored when solving the equations. Sarkar [26] proposed a term for the pressure-dilatation mean product. The formulation is based on simulations of simple compressible flows. The model is implemented as follows: p d = α 2 P k M 2 t + α 3ρεM 2 t (3.7) The coefficients α 2 and α 3 are set to 0.15 and 0.2, respectively. This term may either increase or decrease the turbulence levels in the flow, depending on the relative strength of production and dissipation. More recent DNS data has shown that these terms are relatively insignificant for inhomogeneous flows such as mixing layers and channel flows [20]. Please note that unless otherwise noted, the pressure dilatation is enabled, as is default in OVERFLOW.

36 Abdol-Hamid Temperature Correction Abdol-Hamid et al. [13] suggested a turbulence model correction to account for large temperature fluctuations, which can also have significant impact on the turbulence levels in a flow. The correction is formulated as follows. First, the total temperature gradient is calculated as T g = (T t)(k 3/2 /ǫ) T t (3.8) where (T t ) = ( T t x i ) 2. The total temperature is chosen over density fluctuations as the determining parameter because it is not dependent on Mach number and not influenced by flow features such as the compressions and expansions seen in shock cells. Seiner et al. [9] showed that these shock cells only change the axial and radial components of turbulence intensity but have no increase in the overall turbulence kinetic energy. The modified eddy viscosity, C µ, is calculated from the total temperature gradient and a compressibility factor, F(M t ), [ T 3 ] g C µ = F(M t ) (3.9) F(M t ) is formulated the same as in Eq. 3.6 above, except the model constant M t0 is 0.1. Based on this formulation, a large total temperature gradient would cause an increase in the eddy viscosity, thus increasing the diffusion of the jet. The correction also considers compressibility effects using the F(M t ) term. For large turbulence Mach numbers, the eddy viscosity would be decreased and suppress mixing. The compressibility effects in the temperature correction are calculated separately from those of an implicit compressibility correction such as Sarkar s. However, for Abdol- Hamid s supersonic jet test case [13], all analysis is completed while also using Sarkar s compressibility correction.

37 Spalart-Shur Rotation/Curvature Correction Spalart and Shur [27] devised a turbulence model correction that deals with streamline curvature and/or system rotation. These effects become important in flows such as U-turn ducts and curved mixing layers. For our work, the problem of interest is a jet in crossflow. The curvature of the jet due to the crossflow, as well as the vorticity developed by vortex systems (such as counter-rotating vortices), all need to be accounted for. Mani et al. [28] showed an improvement using the Spalart- Allmaras rotation curvature/correction (SARC) for U-turn flows and a high-speed ground/jet interaction. Churchfield and Blaisdell [29] also saw a positive effect of the correction for near-field wingtip vortex flows. The correction tracks the direction of the principal axes of the strain rate tensor, making it Galilean invariant. The rotation function term, f r1, is computed from second order derivatives of the flowfield. Then, it is implemented into the turbulence model by multiplying the source term in the production equation. The production term is cut off at 0 and limited to a maximum of This correction can be applied to the production term in the one-equation SA model (for which it was devised) as well as a two-equation model like the SST model. 3.4 Grid Generation All grids are generated using the Chimera Grid Tools package [30] which allows for the use of overset grids. The SRAP tool is used to define grid distribution along curves. Then, surfaces are generated using the SURGRD tool. The HYPGEN tool is used to generate the off-body grids using a hyperbolic tangent stretching function. For axisymmetric grids, the GRIDED tool is used to create an axisymmetric input from the 2-D grids. Tools like DIAGNOS are used to check for adequate grid quality, such as cell stretching ratios. The grids are generated in inches and processed with double-precision floating point variables. Once completed, the grids are run through the domain connectivity software PEGASUS [31] to obtain the overlap region in-

38 22 terpolation stencils. The Overset grid generation guidelines [32] are followed for the generation and modification of grids. Table 3.1 summarizes the number of grid points used for each test case. Test Case Table 3.1 Number of Grid Points for Test Cases Number of Grid Points Eggers Jet (axisymmetric) Wishart Jet (axisymmetric) Glenn Jet without Capsule (3-D) Glenn Jet with Capsule (3-D) Flat Plate (2-D) Mixing Layer (2-D) Axisymmetric Jets Grids for the Eggers and Wishart jets are very similar. The nozzle geometry is extracted from the initial experiments. The grid is shown in Fig. 3.1a, and a detailed view of the lip region is shown in Fig. 3.1b. Four zones are used for the nozzle, top, aft, and the nozzle lip regions. The X-axis represents the jet s axis of symmetry. The collar grid around the nozzle lip is used to better resolve the high flow gradients present in this region. The grid spacing along the inner nozzle boundary is designed to keep the wall y + below one in order to adequately resolve the viscous sublayer near the wall. The grid is particularly fine in the aft region off the nozzle lip to resolve the shear layer that determines the jet development. In order to sufficiently model the freestream and eliminate upper and aft boundary effects, the domain of the plume extends at least 30 diameters in the radial and 80 diameters in the axial directions, measured from the nozzle exit plane. The final Eggers and Wishart grid systems contain and grid points, respectively.

39 D Jet in Crossflow For the Glenn jet, grid scripts were obtained from researchers at NASA Ames Research Center. The model includes the entire wind tunnel. The grid scripts are modified to minimize orphans, eliminate interpolation problems, and improve resolution in the mixing region. Similar to the grids for the axisymmetric jets, particular attention is paid to ensure sufficient resolution along the nozzle lip and the mixing region. For the final grid, the plume region contained nine million grid points, leading to a total of 29 million grid points for the grid without capsule, and 31 million for the grid with capsule. Fig. 3.2 and 3.3 show the grids for the case without and with the capsule, respectively. The normal spacing off the wall for viscous grids is at most inches, and inches everywhere on the capsule. The wall y + is less than 0.2 everywhere on the capsule surface. The Walldist code, an in-house tool at NASA JSC, is used to create a file which contains the distance to the nearest wall for all points in the grid. This is important information for turbulence models, especially the SST, which switches formulations based on the distance to the nearest wall Boundary Layer The grid for the boundary layer is a single zone grid with particular refinement near the wall to capture all three layers of the boundary layer (viscous sublayer, log layer, and defect layer). The grid is shown in Fig Mixing Layer The grid for the mixing layer, shown in Fig. 3.6, is also a single zone grid refined in the mixing region. The top and bottom boundaries extend out sufficiently far to avoid any interference. The domain also extends far enough downstream to ensure the development of a self-similar flowfield.

40 Boundary Conditions Note: Abbreviations in parentheses ( ) refer to the flag name of the setting in the OVERFLOW code Axisymmetric Jets Freestream boundary conditions (IBTYP=40) are imposed at the top of the nozzle, with Mach 0.01 and standard pressure and temperature. The walls are specified as viscous and adiabatic (IBTYP=5). The nozzle conditions are set as a ratio of stagnation pressure and temperature from the nozzle plenum to the freestream (IB- TYP=41). The top and aft domain boundaries are specified as pressure outflow boundaries (IBTYP=32) with a given value of static pressure D Jet in Crossflow The entrance of the tunnel is modeled using a free stream boundary condition (IBTYP=40). Here, values are calculated using a specified freestream Mach number of 0.3 and temperature of 540 R, the specific heats ratio (1.4), and zero degrees angle of attack. A freestream Reynolds number, based on the freestream velocity and the grid length unit, is calculated as 165,000. The freestream and nozzle plenum turbulence (ratio of turbulent to laminar eddy viscosity) and turbulent kinetic energy are specified as 0.1 and , respectively. The outflow boundary conditions are modeled as a specified pressure outflow (IBTYP=32), due to the setup of the tunnel. All surfaces are specified as viscous, adiabatic walls (IBTYP=5). For the nozzle, a prescribed q file (containing the conserved variables) is specified at the inflow plane upstream of the throat (IBTYP=43). Conditions are calculated from measured experiment conditions (pressure, stagnation temperature and Mach number), specific heats ratio, and molecular weight. Nozzle conditions at the inflow, calculated from the experiment, have a Mach number of 0.3, stagnation pressure ratio

41 25 of 28.5, and total temperature ratio of 2.3, calculated with respect to the free stream. The specified q file is used with a characteristic condition (IBTYP=31). The nozzle inflow file also uses a slow start over the first 500 iterations. The nozzle grid and velocity vector are shown in Fig Because a natural gas combustor is used to heat the air exiting the jet, the hot exhaust gas is approximated using a specific heats ratio of 1.32 and a gas constant of ft lb f lb m. The gas properties are calculated with a program that uses cu- R bic spline fits for air and stoichiometric products of combustion with fuels having a hydrogen-to-carbon mass ratio of Of course, the combustor is not always running stoichiometrically, but this approach provides better values than using cold air. A sensitivity study of the use of variable specific heats ratio shows that its use had little effect compared with the other variables in the problem Flat Plate Boundary conditions for the two-dimensional flat plate test case include a viscous adiabatic wall (IBTYP=5), standard free stream inflow conditions (IBTYP=40) and outflow (IBTYP=30). The freestream Reynolds number, which is based on the grid length unit, is set to The freestream temperature is set to 180 R and the freestream Mach number is varied from 2 to Mixing Layer The final test case is a generic two-dimensional mixing layer. The velocity is fast on one side and slow the other, and there is a uniform temperature of 460 R across the inflow. The inflow is modeled as two streams at different velocities. A prescribed q file (IBTYP=42) with the hyperbolic tangent velocity profile, along with a characteristic condition (IBTYP=31), is specified at the inlet. The far-fields on top and bottom, as well as the aft boundary, are specified with a supersonic characteristic condition (IBTYP=31).

42 Solution Method Numerical Solution The numerical schemes are chosen based on robustness and rate of convergence. For consistency, all test cases use the same inputs for the numerical methods. Due to the presence of shocks in the plume, the HLLC upwind scheme (IRHS=5) is chosen, where the left-hand side is solved with an enhanced symmetric successive overrelaxation (SSOR) algorithm (ILHS=6). The implementation and validation of these schemes in OVERFLOW is discussed by Nichols et al [33]. A van Albada limiter is used for the upwind Euler terms (ILIMIT=3). Start-up iterations include added smoothing for the limiter. All spatial differencing (Navier-Stokes, turbulence model, and species equations) employs third order accuracy (FSO=3.0). Local time-step scaling with a constant Courant-Freidrichs-Lewy (CFL) number is employed, where the time-step scaling is based on the local cell Reynolds number (ITIME=4). The solutions are initialized to specified freestream conditions, as described in the previous chapter. Cases are run with a sequence of separate input files while the CFL is ramped up. Three grid sequencing levels are performed: 1,000 iterations are run on the coarse and medium grids to start up the solution, with a CFL of 10. Then, 2,500 iterations are run on the fine grid using a CFL of 10, followed by 4,000 fine grid iterations using a CFL of Convergence Both residuals and forces are used to monitor convergence. Residuals are expected to drop four orders of magnitude. An example of the residual convergence histories, for the L 2 norm of the conserved variables, q, in various grid zones, is shown in Fig Residuals are shown for important regions in the grid system. The spikes at 1,000 and 2,000 iterations are due to restarts during grid sequencing. Residuals decrease consistently and level out below after 6,000 iterations. Cases run

43 27 with the compressibility correction take slightly longer to converge. However, all solutions are still run for the same amount of iterations, which is based on the longer time required to converge the flow with the compressibility correction. For all jets, forces are monitored at downstream locations to ensure that pressure and viscous forces achieve equilibrium. The FOMOCO [34] tool is used to integrate pressure coefficients over the overset surfaces to obtain the aerodynamic coefficients. For the Glenn jet cases without the capsule, forces are monitored on the nozzle body to ensure that a steady state solution is reached. For cases with the capsule, the axial and normal force coefficients, C A and C N, and the moment coefficient, C m, are monitored on the capsule surface. The same convergence criteria employed by the CAP team is used: C A, C N, and C m change less than ±0.01, ±0.004 and ±0.002, respectively. An example for the force monitoring on the capsule surface is shown in Fig Clearly, all forces are well below their convergence criteria Computer Resources All calculations for the axisymmetric and 2-D cases are run on a 128 processor cluster at NASA JSC consisting of GHz and GHz processors and 124 GB RAM. Runs are split between 20 processors and use an average of 10 CPU hours. For the 3-D jet, cases are run on the NASA Advanced Supercomputing (NAS) cluster Columbia using 64 CPUs in parallel. The average time is between 1,700 and 2,000 CPU hours, depending on the case, the turbulence model, corrections, and whether species transport is modeled.

44 28 (a) (b) Figure 3.1. (a) Schematic of overall grid system for axisymmetric jets (showing every fourth grid line). (b) Detail view of the nozzle lip region.

45 29 (a) (b) Figure 3.2. (a) Schematic of center (Y=0) cut of grid system for Glenn jet without capsule. (b) Schematic showing grid on tunnel centerbody (green), nozzle body (red), and nozzle (blue) walls, and center (Y=0) cut of plume grids (orange).

46 30 Figure 3.3. Schematic of grid around capsule for Glenn jet. Figure 3.4. Grids for conical nozzle and velocity vectors specified in plenum for Glenn jet.

47 31 Figure 3.5. Grid topology for generic boundary layer (showing every third grid line). Figure 3.6. Grid topology for generic mixing layer (showing every fifth grid line).

48 Figure 3.7. Typical residual behavior for Glenn jet. 32

49 33 (a) (b) Figure 3.8. Force and moment monitoring on capsule surface for Glenn jet.

50 34 4. RESULTS FOR AXISYMMETRIC JETS 4.1 Grid Convergence Free shear flows such as this jet flow are sensitive to the computational grid. A grid that is too coarse may exhibit overly diffusive behavior. Grid refinement is thus performed to ensure that the solution is grid independent. Results are shown only for the underexpanded Wishart jet as this case shows the most grid sensitivity. The grid is refined in the regions of interest: the exit of the nozzle and the plume where mixing with the freestream occurs. The number of grid points in each direction is doubled successively. The velocity and temperature profiles downstream are parameters which are sensitive to grid refinement and important to the analysis. Fig. 4.1 shows the centerline Mach number for three levels of grid refinement, with , , and grid points in the plume region. The most significant distinction is seen in the resolution of the shock cells. While all grids predict the length of the shock cells equally, the peak is smeared with the coarse grid, where the peak Mach number is only 2.58, compared with 2.75 and 2.87 for the medium and fine grids, respectively. After the potential core, there is very little difference in the results. Fig. 4.2 and Fig. 4.3 show profiles at four downstream locations for Mach number and temperature, respectively. It appears that the temperature is slightly more sensitive to grid refinement than the velocity. While a small change is observed in going from the coarse to the medium grid level, there is essentially no difference between the medium and the fine grid. It is concluded that the medium grid level resolves all parameters of interest sufficiently, and thus is used in all subsequent computations.

51 Eggers Jet The Eggers jet is used as a starting point for the analysis. First, the turbulence models presented in the previous chapter are examined. Then, various turbulence model corrections are evaluated Turbulence Model Assessment Fig. 4.4 shows the jet centerline velocity, normalized by the jet exit velocity, as a function of downstream distance, normalized by jet exit diameter, for various uncorrected turbulence models. The velocity remains constant through the jet potential core, and then decays in the fully developed region. The potential core is defined as the distance from the nozzle exit until the centerline velocity drops below 95% of its exit value. The SST, k-ω, and Spalart-Allmaras models predict too much turbulenet mixing, thus showing a potential core length that is shorter than that of the experiment. The Baldwin-Barth model, on the other hand, significantly underpredicts the mixing and exhibits an excessively long potential core. The SST model gives the best prediction, especially further downstream. Fig. 4.5 shows the inverse velocity as a function of the Witze shift, W. The Witze shift is given by W = k(x x c )/r 0, where k is a constant based on the jet density and Mach number, x c is the location of the end of the potential core, and r 0 is the jet exit radius. This aligns the end of the potential core for the various cases and allows for examination of the jet s centerline decay rate. From the graph, it is clear that the k-ω and Spalart-Allmaras models decay much too quickly. The SST model shows an improvement and the Baldwin-Barth model gives the overall best prediction. Fig. 4.6 shows downstream axial profiles of velocity for the Eggers jet. Fig. 4.7 shows the jet half radius. The half radius is defined as the radius at which the velocity is 50% of the local centerline velocity, computed at all downstream (X) locations. This gives a representation of the jet s spreading rate. It can be seen that because of the increased mixing seen for the SST, k-ω, and Spalart-Allmaras models, the spread rate

52 36 is also overpredicted. The Baldwin-Barth model does not show enough mixing and does not spread as quickly. The SST model matches the experimental spread rate most closely. Based on this analysis, the SST turbulence model is chosen to be used for all subsequent computations Compressibility Corrections This section compares the compressibility corrections by Sarkar, Zeman, and Wilcox, which are implemented in the SST model in the OVERFLOW source code. Fig. 4.8a compares the centerline velocity. All three corrections show noteworthy improvement over the uncorrected model. Wilcox s model diffuses slightly more quickly compared with the experimental data. There is no significant difference between the Sarkar and Zeman models. In Fig. 4.8b, the spread rate for the three compressibility corrections is presented. Analogous to the centerline velocity, the Sarkar and Zeman corrections show the best performance, and all corrections show a significant improvement over the uncorrected model. There is no significant difference between the Sarkar and Zeman compressibility corrections examined, and hence Sarkar s model is used for all further analysis Pressure Dilatation Correction The effect of the pressure dilatation correction is briefly examined, and results are presented in this section. It is important to note that the use of this correction is included by default in the OVERFLOW code when using the SST turbulence model. As explained previously in Chapter 3, the pressure dilatation term may act to increase or decrease the growth of turbulence. Fig. 4.9 shows that the jet decays and spreads at a slower rate when used with the pressure dilatation. This indicates that the pressure dilatation term provides additional dissipation of turbulence kinetic energy for this case. This agrees with previous jet flow simulations by Lakshmanan and Abdol-Hamid [35]. It is also important to note that the effect is not nearly as strong

53 37 as that of the compressibility correction. For this data set, the use of the pressure dilatation improves the predictions. For the subsequent analysis in this section, the pressure dilatation correction is always used in conjunction with the compressibility correction. 4.3 Heated and Perfectly Expanded Wishart Jet The next sections include comparisons with the jet by Wishart for selected conditions. Please refer to the Appendix for additional cases. The isothermal and perfectly expanded Wishart jet compares very well with the Eggers jet. The indication clearly is that a compressibility correction is needed for accurate predictions of jet development. The first effect examined for the Wishart jet is the heated case. As discussed by Lau [8] and Seiner et al. [9], the enhanced density fluctuations in the jet increased the mixing in the jet. Thus, the jet is expected to decay and spread faster than an isothermal jet. Fig. 4.10a presents the centerline Mach number for the heated Wishart jet. The plot shown contains runs with the uncorrected model, only Sarkar s compressibility correction, and both Sarkar s compressibility correction and Abdol-Hamid s temperature correction. The case with only the temperature correction is not included because the correction is intended to be used with a compressibility correction. As observed with the Eggers jet, the compressibility correction decreases mixing and shifts the curve to the right. The temperature correction increases mixing when used with the compressibility correction. It is important to note that the compressibility correction has the more pronounced effect on the solution. While the use of no corrections most closely matches the experimental potential core length, the downstream behavior is best predicted when using the compressibility correction. In Fig. 4.10b, the temperature profile, normalized by the ambient temperature, is plotted along the centerline. The increase in temperature is caused by the compres-

54 38 sion as the gas decelerates in the shear layer, and then decreases to the ambient value through mixing with the freestream. All models show an identical magnitude in the temperature rise which matches the experimental data well. Similarly to the Mach number profiles, the start of the shear layer is better predicted without correction, whereas the downstream behavior agrees more closely with the compressibility correction. In both plots, it is indiscernible whether adding the temperature correction provides a better prediction. Fig shows radial Mach number profiles for four downstream locations. The difference in jet development prediction is clearly demonstrated here. There is no noteworthy distinction between the models for X/D of 3.5 and 7 as the jet has not had ample time to mix with the ambient fluid. The compressibility correction again inhibits mixing, thus showing a longer potential core. For R/D, the radial coordinate, larger than 0.8, all models show a similar prediction as the jet has dissipated into the ambient fluid. For the two later locations, the experimental data lies between the CFD results and there is no clear model that exhibits a better behavior. However, at increasing radial distance, the model with the compressibility corrections seems to show better agreement with the experimental data. 4.4 Isothermal and Underexpanded Wishart Jet Next, an underexpanded jet is examined. The contour plots of Mach number in Fig illustrate the behavior of an underexpanded jet. The jet exit pressure is higher than the back pressure and expansion fans are formed at the lip of the jet. Once the expansion waves reach the jet boundary, they reflect as compression waves to match the constant pressure condition at the boundary. When the pressure ratio is increased, the compression waves no longer meet at the axis but instead form a stronger normal shock known as the Mach disc. The convergence of the expansion fan generates a cylindrical shock, known as the barrel shock, which terminates at the Mach disc. This leads to a pattern of shock cells, frequently seen in the exhaust of a

55 39 jet or rocket engine. The wavelength of the expansion and compressions depends on the Mach number and pressure ratio. For this case, the maximum Mach numbers are approaching three, much higher than the nozzle design Mach number of two. Fig. 4.12a and b show the centerline properties for the isothermal and underexpanded jet. Only cases with and without the compressibility correction are shown since the temperature correction has no effect for an isothermal jet. The resolution of the shocks and expansion fans is very good. Clearly, the case without the compressibility correction predicts premature turbulent mixing. Fig shows the radial Mach number profiles for the underexpanded jet. The Mach number in the potential core is no longer uniform because the expansions and contractions change the flow direction. The compressibility-corrected model agrees very well with the experimental data at all locations. A possible cause for why the compressibility correction performs better for this case than for the perfectly expanded case is that compressibility effects become more significant because higher Mach numbers are seen due to expansions and contractions. 4.5 Heated and Underexpanded Wishart Jet For the third case presented, both underexpanded and heated jet effects will be present. Fig shows Mach number contours for this jet using the various corrections. It is evident that the turbulence model corrections significantly impact the development of the shock cells. Using exclusively the compressibility correction (b) shows seven distinguishable shock cells, compared with only four for the uncorrected model (a). The use of the temperature correction (c) does not change the number of shock cells present but does slightly decrease the length of the potential core. Fig. 4.15a and b show the centerline Mach number and temperature profiles, respectively, for the heated and underexpanded jet. It is noted that the sharp increases in Mach number (due to expansion fans) correspond to sharp decreases in temperature, as expected, and the opposite is true for the compressions. The difference in the

56 40 number of shock cells present is also visible between the corrections. For X/D less than seven, all models match the experimental data very well. The models without the compressibility correction then decay too rapidly. The SST model with the compressibility correction (and the temperature correction enabled or disabled), matches the data very well, except for the slightly excessive potential core length. As before, the temperature and Mach number profiles correspond, and the compressibility-corrected models give an excellent match with the experimental data. Again, the magnitude of the temperature peak is predicted equally well by all models. Using the temperature correction increases turbulent mixing. The radial Mach number profiles for the heated and underexpanded jet are shown in Fig The compressibility-corrected models show an excellent prediction for all downstream locations. The uncorrected models diffuse too quickly, both in the radial and axial directions. It is unclear whether the temperature model improves the prediction, although at R/D greater than 0.6, the temperature-corrected model lies somewhat closer to the experimental data. 4.6 High Pressure and Temperature Parametric Study A parametric study is performed using a generic axisymmetric jet setup. The goal is to extrapolate the data to higher pressure ratio (PR) and temperature ratios (TR), more closely resembling those experienced during actual flight conditions. The pressure ratio is increased from the perfectly expanded conditions (PR = 11) to underexpanded ratios of 15, 25, and 35. The temperature ratio is successively increased from isothermal to ratios of 5 and 10. The compressibility and temperature corrections are examined. A survey of literature showed that there is very little experimental data available for verification at these conditions. Fig. 4.17a shows the jet half radius for increasing pressure ratios with the compressibility correction disabled (solid line) and enabled (dashed line). Initially, the half radius is largest for the highest pressure ratio. This occurs because the higher

57 41 pressure ratio causes the flow to expand more immediately after nozzle lip, forming the first barrel shock. There is basically no distinction between the different pressure ratios without the compressibility correction after X/D of 20. This is inconsistent with the observation that growth rate should decrease at higher Mach numbers and highlights the deficiency of the base turbulence models. For the compressibility correction, as expected, the growth rate for the higher pressure ratio is decreased. This could imply that that the compressibility correction is required to match the flow physics. Fig. 4.17b illustrates the jet half radius for increasing temperature ratios when the temperature correction is disabled (solid line) and enabled (dashed line). In both cases, the compressibility correction remains enabled, as this is how the temperature correction is intended to be used. The increasing temperature ratios lead to an increase in turbulence kinetic energy. From Eq. 3.1, this increase causes a rise in the turbulence Mach number. In turn, the turbulence dissipation is increased and the mixing is decreased, resulting to the smaller spread rate seen in the figure. The use of the temperature correction essentially returns the spread rate to the initial rate, offsetting the effect of the compressibility correction. Unfortunately, there is no experimental data at these elevated test conditions, so it cannot be determined which correction accurately models the flow physics. It does, however, warrant that caution must be exercised when using the corrections at conditions for which they have not been validated.

58 Figure 4.1. Grid refinement for Wishart underexpanded jet showing centerline Mach profile. 42

59 43 (a) X/D = 3.5 (b) X/D = 7 (c) X/D = 11 (d) X/D = 17 Figure 4.2. Grid refinement for Wishart underexpanded jet showing downstream Mach profiles.

60 44 (a) X/D = 3.5 (b) X/D = 7 (c) X/D = 11 (d) X/D = 17 Figure 4.3. Grid refinement for Wishart underexpanded jet showing downstream mean temperature profiles.

61 45 Figure 4.4. Normalized centerline velocity for Eggers jet. Figure 4.5. Inverse velocity for Eggers jet with the x-axis shifted using the Witze correlation.

62 46 (a) X/D = 25 (b) X/D = 50 (c) X/D = 100 (d) X/D = 150 Figure 4.6. Normalized radial velocity profiles for Eggers jet.

63 Figure 4.7. Normalized half radius for Eggers jet. 47

64 48 (a) (b) Figure 4.8. (a) Normalized centerline velocity and (b) normalized half radius for Eggers jet showing effect of compressibility corrections.

65 49 (a) (b) Figure 4.9. (a) Normalized centerline velocity and (b) normalized half radius for Eggers jet showing effect of including pressure dilatation.

66 50 (a) (b) Figure Centerline quantities for heated and perfectly expanded Wishart jet. (a) Mach number. (b) Normalized temperature.

67 Figure Radial Mach number profiles for heated and perfectly expanded Wishart jet. 51

68 52 (a) (b) Figure Centerline quantities for isothermal and underexpanded Wishart jet. (a) Mach number. (b) Normalized temperature.

69 Figure Radial Mach number profiles for isothermal and underexpanded Wishart jet. 53

70 Figure Contours of Mach number showing shock cell structure for heated and underexpanded Wishart jet for cases CC=0, TC=0 (top), CC=1, TC=0 (middle), and CC=1, TC=1 (bottom). 54

71 55 (a) (b) Figure Centerline quantities for heated and underexpanded Wishart jet. (a) Mach number. (b) Normalized temperature.

72 Figure Radial Mach number profiles for heated and underexpanded Wishart jet. 56

73 57 (a) (b) Figure Normalized half radius for underexpanded jet at elevated conditions. (a) Increasing pressure ratio comparing CC. (b) Increasing temperature ratio comparing TC.

74 58 5. RESULTS FOR 3-D JET IN CROSSFLOW Having completed preliminary analysis for various axisymmetric jets, the next step involves running a 3-D jet in crossflow. This chapter is split up into three parts: grid convergence, analysis for the jet without the capsule, and analysis for the jet with the capsule. 5.1 Grid Convergence Grid refinement is performed on the grid for the Glenn jet. The grid is refined only in the regions of interest: the exit of the nozzle and plume where mixing with the freestream occurs. Other grids are not refined to save computational time and because the flow is not important in these areas. The number of grid points in each index is increased by a factor of 3 2, thus doubling the total number of grid points with each refinement. This resulted in , and grid points for the coarse, medium, and fine grid levels, respectively, in the refinement region. For this case, the main quantities of interest are off-body velocity profiles and surface pressures. The off-body velocity profiles are much more sensitive to grid resolution and thus are used to judge grid convergence. Fig. 5.1 shows the velocity profiles along a streamline passing through the center of the nozzle. This allows for a similar comparison as the centerline velocity profiles for the axisymmetric jets. The observed behavior is very similar. Increased grid refinement shows improved resolution of the expansions and contractions caused by the shock reflections. The length of the shock cells is predicted equally well by all grid levels. Also, downstream behavior is not significantly impacted by grid refinement. In Fig. 5.2, two downstream axial velocity profiles are plotted. The coarsest grid is slightly too diffusive. There is essentially no difference between the medium and

75 59 fine levels of grid refinement. As a result, the medium grid level is deemed to have sufficient resolution and is used for all subsequent analysis. 5.2 Without Capsule Turbulence Models Baseline comparisons are made to verify that turbulence model behavior agrees with that of the analysis in the previous chapter. Two one-equation models (BB and S-A) and two two-equation models (k-ω and SST) are compared, all uncorrected. Velocity magnitude contours are shown for comparison to the PIV data in Fig Three slices are taken at various axial locations perpendicular to the freestream. The distance is normalized by the jet exit diameter. Fig. 5.3 shows the velocity magnitude contours compared to the above-mentioned turbulence models (Section 3.2). Clearly, the Baldwin-Barth shows a high velocity at all locations while the Spalart-Allmaras and the k-ω models diffuse too quickly. The SST model gives the closest match to velocity contours. Another trend of interest is the shape of the jet. The horseshoe shape exhibited in the experiment is caused by two counter-rotating pairs of vortices rolling up as the freestream moves past the jet. The amount of vorticity produced by this vortex roll-up determines the shape of the jet. The BB model predicts too much vorticity because the shape curls up more than seen in the experiment. The SA and the k-ω models diffuse too quickly, thus the shape is not curled enough. The SST again provides the closest agreement with experiment, which is consistent with the previous axisymmetric jet study (Fig. 4.6). This analysis confirms that the SST model is the best choice for this type of flows, and hence this model will be exclusively used in the following computations. It is also important to note that this validates the CAP team s current assumptions of using the SST model for LAS computations.

76 Turbulence Model Corrections The next comparisons are made for various corrections using the SST turbulence model. The three corrections examined are 1) Sarkar s compressibility correction, 2) Abdol-Hamid s temperature correction, and 3) Spalart-Shur s Rotation/Curvature correction. As before, the temperature correction assumes that the compressibility correction is enabled simultaneously. Fig. 5.4 shows the velocity magnitude contours at four downstream locations, normalized by the jet exit diameter, and compared with PIV data. Fig. 5.5 shows cuts through the middle of the velocity contours to give velocity profiles. For the uncorrected model, the velocity magnitude and vortex roll-up diffuse too quickly when compared with experiment. As seen for the axisymmetric analysis, the compressibility correction shows the largest effect on the problem. The velocity magnitude is too high at downstream locations because turbulent mixing in the shear layer is suppressed for too long. The compressibility correction also shows increased vorticity and thus a shape that is skewed excessively by the vortex roll-up. From this test case, it appears that Sarkar s model overcorrects by not showing enough mixing with the freestream. A potential cause of this, when compared with the favorable behavior for the axisymmetric jet, is that the current experiment is performed at much higher turbulent Mach numbers than the previous cases. The temperature correction slightly increases the mixing, but does not have a significant effect on the solution. The rotation correction, which is meant to account for mean streamline curvature in the flow, possibly due to vortical flows and streamline curvature, also has very little effect. Again, the compressibility correction is dominant for this problem. This is expected because compressibility effects are large for this flow due to high turbulence Mach numbers in the jet shear layer. It was previously explained that jets in a crossflow form counter-rotating vortices as the freestream passes around the jet. From Fig. 5.4, it is clear that the prediction of the vortex roll-up is important to the overall jet development. Thus, the magnitude of the streamwise vorticity is evaluated. Vorticity is defined as the curl of the velocity

77 61 field and is related to the amount of rotation in a fluid. Fig. 5.6 shows vorticity in the streamwise direction; so at each constant-z plane, the vorticity is determined by the velocity components in the x and y directions. In this figure, the red region implies a positive vorticity, meaning that the flow is curling counter-clockwise, and vice verse. From the figure, one can observe that there is a pair of counter-rotating vortices that forms in the jet shear layer. There are also various other vortices formed. The potential core of the jet remains inviscid and thus there is essentially no vorticity in this region. Looking at the standard turbulence model corrections, there is clearly very little difference in using the rotation correction, except for a very small increase in the vorticity magnitude. The compressibility correction gives a good prediction for the shape of vorticity at all locations. However, it does slightly overpredict the magnitude. This explains the overly skewed shape observed in the velocity contours when using the compressibility correction. This observation is consistent with the analysis performed by researchers at NASA Ames for the same experiment [16]. The uncorrected and SARC corrected cases diffuse too rapidly. It can be concluded that the use of the compressibility correction more closely resembles the actual vorticity measured in the experiment. One set of PIV data was taken in the streamwise direction, allowing for additional comparisons. A streamline that passes through the middle of the nozzle exit plane was traced for comparison of jet position as the jet mixes with the freestream, shown in Fig. 5.7a. Clearly, all cases seem to match rather well with experimental data. The cases with compressibility correction delay mixing with freestream and thus does not exhibit as much curvature as other cases. Velocity magnitude is also extracted along the center streamline, shown in Fig. 5.7b. The inviscid region, including the expansions and contractions, is predicted equally well by all models. The uncorrected and rotation-corrected models show premature velocity decay, but seem to match the experiment closely further downstream. The compressibility and temperature corrections suppress mixing excessively and do not decay rapidly enough. It is also

78 62 discernible that the compressibility-corrected models show more shock cells compared with the uncorrected model. The behavior of the corrections is consistent with the axisymmetric jet studies (Fig. 4.15). The effect of the pressure dilatation terms is also evaluated in Fig The effect of enabling the pressure dilatation correction again acts to decrease the rate of turbulence growth. This is consistent with the axisymmetric jet analysis (Fig. 4.9). Again, the shift is not as pronounced. However, while previously the use of this correction led to better comparison with the experimental data, for the Glenn jet, the correction excessively suppresses the turbulence. 5.3 With Capsule After full analysis of the jet and freestream interaction, a 4% scale model of the capsule is inserted into the wind tunnel. The distance from nozzle exit to tapercapsule compression corner, about 10 exit diameters, is very similar to the actual geometry of the LAS. The vertical location was chosen so that there was significant interaction between the jet and capsule but no direct impingement. Again, this is similar to flight conditions. A representative solution for this case showing the Mach numbers and pressure coefficients is shown in Fig A Y = 0 slice of the Mach number contours are shown along with the pressure coefficient on the surface capsule. The flowfield shows a high pressure at the front stagnation point and by the tapercapsule compression corner. The high velocity jet in proximity to the front of the taper causes a low pressure region at this point. This can also be seen on the capsule surface. It is expected that the turbulence model s ability to predict the jet spread rate, shock cell structure, and curvature will be very important in determining the capsule pressure distribution. Analogous to the previous analysis, comparisons are made for velocity magnitude contours in Fig When compared with profiles for the case without capsule, Fig. 5.11, which is run at similar conditions, some changes are evident due to the

79 63 presence of the capsule. For all profiles downstream of the capsule (the nose is at X/D = 4), the plume exhibits less vorticity. The presence of the wall dampens out the vorticity because of the additional viscous effects generated in the boundary layer. The experiment also shows increased velocity, especially when looking further downstream. A possible cause is that less vortical flow leaves more velocity components in the downstream direction. Consequently it is obvious that the presence of the capsule wall effects changes the development of the flowfield. The trends seen for the various corrections are similar to the previous comparisons. The compressibility correction overpredicts the velocity and underpredicts streamline curvature. All cases not using the compressibility correction exhibit velocity that diffuses too quickly. The temperature correction slightly decreases the velocity and vorticity in the jet, and the rotation correction has very little effect. However, the experimental data lie between using no corrections and using the compressibility correction, which is different from before. When considering the shape due to vorticity, the compressibility correction overpredicts the initial vorticity but shows a good prediction further downstream. Not using the compressibility correction exhibits excessive diffusivity; the vortex roll up is unnoticeable at X/D = 18. Comparisons for pressure on the capsule in the streamwise direction are shown in Fig Please refer to Fig. 2.3 for the pressure transducer locations on the capsule. The initial drop is the location at which the jet is near the capsule. The differences in pressure drop are caused by the spread rate of the jet. Because the cases without the compressibility correction show more diffusivity and a higher spread rate, they entrain more of the surrounding, inviscid fluid at higher velocities. As a consequence of Bernoulli s equation, this results in larger pressure drops on the taper of the capsule. The compression region is predicted well by cases without the compressibility correction. The region up to the end of the shell (the last spike) shows some ambiguity due to the experimental data, but it appears that the cases without compressibility correction give a better solution.

80 64 Four circumferential cuts for pressure comparison on the capsule are shown in Fig The first cut is located on the taper, the second immediately after the tapercapsule compression corner, and the last two cuts are on the capsule. In the first cut, the compressibility correction shows the right trend (an increase in pressure) while the cases without show a drop in pressure, similar to what is seen for the streamwise comparison. For the other cuts, the cases without compressibility correction yield the best solutions. There is no clear indicator of whether the temperature correction shows an improvement Impact on Flight Mechanics The goal of this work is to improve turbulence modeling capabilities for the CAP team s flight vehicle computations. To understand the impact of the turbulence model corrections on the flight mechanics of the vehicle, the change in the capsule pressure distribution is examined. Fig shows the change in the capsule surface pressure coefficient when enabling Sarkar s compressibility correction. The largest change is an increase in pressure on the nozzle taper caused by a smaller jet spread rate. For the same reason, there is also an increase in pressure over the capsule surface. In addition, there is a decrease in pressure after the compression. The V-shape is due to the increased vorticity seen when using the compressibility correction. FOMOCO is used to perform force and moment coefficient computations on the capsule surface. The normal and axial force coefficients, as well as the moment coefficient (calculated about the scaled CEV moment reference center), are computed, and shown in Table 5.1. The tolerances are used to judge convergence and are shown for reference. All coefficients show a significant change due to the compressibility correction, caused by the change in pressure over the capsule surface. It must be noted that the effects are shown only for a single plume; four plumes in flight at zero angle of attack would balance out the normal forces and moments, but would increase the change in the axial force. Thus, the change in prediction of the axial

81 65 force coefficient could be as high as 40%. The use of the compressibility correction does have a significant impact in predictions of the aerodynamic coefficients. Table 5.1 Changes in Aerodynamic Coefficients Caused by Compressibility Correction C A C N C m No Correction Comp. Correction Coefficient Tolerance Experiment Uncertainties Several experimental uncertainties could have led to discrepancies in the experimental data. Unsteady behavior in the wake region after the capsule could have propagated forward and affected the flow, and the time-averaging used in the experiment could have failed to capture this sufficiently. This could have led to the asymmetry exhibited in some of the experimental data, most notably for the PIV and pressure readings on the capsule (Fig and 5.13). The flow is also affected by the sting used to attach the capsule and hold it in place, and thus any data past the capsule heat shield was disregarded. Finally, rough internal plumbing leading up to the nozzle could have developed some swirl. This velocity distortion could have led to momentum losses passing through the nozzle or an excessively rapid shear layer growth exiting the nozzle. Unfortunately, no experimental data (such as velocity profiles) is available for the internal flow, and thus these potential problems are viewed as difficult to assess.

82 Figure 5.1. Grid refinement for Glenn jet showing normalized velocity magnitude profiles along center streamline. 66

83 67 (a) X/D = 10 (b) X/D = 17 Figure 5.2. Grid refinement for Glenn jet showing downstream normalized velocity magnitude profiles.

84 Figure 5.3. Downstream velocity magnitude contours for Glenn jet showing effect of baseline turbulence model. 68

85 Figure 5.4. Downstream velocity magnitude contours for Glenn jet showing effect of compressibility corrections. 69

86 70 (a) X/D = 6 (b) X/D = 10.5 (c) X/D = 14 (d) X/D = 18 Figure 5.5. Normalized downstream velocity magnitude profiles for Glenn jet showing effect of compressibility corrections.

87 Figure 5.6. Downstream vorticity magnitude contours for Glenn jet. 71

88 72 (a) (b) Figure 5.7. (a) Jet position and (b) normalized velocity along center streamline for Glenn jet.

89 73 (a) X/D = 6 (b) X/D = 10.5 (c) X/D = 14 (d) X/D = 18 Figure 5.8. Normalized downstream velocity magnitude profiles for Glenn jet showing effect of including pressure dilatation.

90 Figure 5.9. Contours of Mach number and surface pressure coefficient on capsule. 74

91 Figure Downstream velocity magnitude contours for Glenn jet with capsule. 75

92 76 (a) X/D = 6 (b) X/D = 10.5 (c) X/D = 14 (d) X/D = 18 Figure Normalized downstream velocity magnitude profiles for Glenn jet with capsule.

93 Figure Surface pressure coefficient for streamwise measurements for Glenn jet with capsule. 77

94 78 (a) X/D = 6.5 (b) X/D = 11 (c) X/D = 14.5 (d) X/D = 17 Figure Surface pressure coefficient for radial measurements for Glenn jet with capsule.

95 Figure Delta C p contours on capsule surface. 79

96 80 6. RESULTS FOR COMPRESSIBILITY CORRECTION 6.1 Background ANALYSIS The previous chapters showed the effects of using the Sarkar compressibility correction for jet flows. For the axisymmetric jet analysis, at the given conditions, the compressibility correction yielded significantly improved predictions for all cases. For the 3-D jet, however, the compressibility correction tended to excessively suppress the turbulence growth, leading to a jet that does not diffuse as quickly as it should. The main differences between these jets is that the 3-D jet experienced significantly higher Mach numbers and included a crossflow. This poses the question of whether the shear layer growth rate continually decreases with increasing compressibility effects or whether there is a regime in which this growth rate levels off. In 2001, Rossman [36] performed experimental analysis for mixing layers with very high convective Mach numbers, M c, defined as M c = U 1 U 2 a 1 + a 2 (6.1) where U 1 is the fast side velocity and U 2 is the slow side velocity, and a 1 and a 2 are the sonic velocities in the streams. Fig. 6.1 shows the mixing layer growth rate data by Rossman. Previous data at lower M c only show the initial growth rate decrease with increasing M c. However, Rossman s experimental data (the blue boxes) clearly demonstrates that the mixing layer growth rate levels off beyond M c of 1.0, and then remains flat for larger M c. A similar trend is also seen for homogeneous shear flow. Blaisdell et al. [22] examined the compressibility effects in turbulent homogeneous shear flow using DNS.

97 81 First, it was verified that the growth rate of turbulence in a homogeneous shear flow is reduced for the compressible case when compared with the incompressible case. This result is attributed to an increase in the dissipation rate and an increase in the energy transfer caused by the pressure-dilatation term. The solenoidal dissipation remained the same as for the incompressible case, but the dilatation dissipation term increased with increasing Mach number, M rms. The second key observation was that the ratio of the dilatational to solenoidal dissipation rate levels off beyond M rms of 0.3. This implies that the decrease in growth rate with increasing Mach number reaches a minimum value. 6.2 Formulation of New Corrections Compressibility corrections provide extra dissipation based on the turbulence Mach number, M t. Sarkar s model is presented in Eq Here, the dissipation ratio continues to increase with M t. The DNS by Blaisdell et al. [22] is used as the guideline to formulate the modified compressibility correction, which levels off the dilatation dissipation after a certain turbulence Mach number. The formulation for the first modification, CC-A, is given by Mt 2 if 0 M t 0.1 f(m t ) = 0.1 if M t 0.1 (6.2) The second formulation, CC-B, is given by 10Mt 4 if 0 M t 0.1 f(m t ) = 0.1 if M t 0.1 (6.3) The reason for the fourth power on the M t term is that this may improve predictions in the boundary layers. Ristorcelli [37] first suggested that the dilatation dissipation scales as M4 t Re. This formulation simply uses the M4 t dependence.

98 82 Fig. 6.2 shows the ratio of dilatation to solenoidal dissipation, as a function of turbulence Mach number, for the compressibility corrections by Sarkar, Zeman, and Wilcox. Also, the two new compressibility corrections are shown. The DNS data strongly suggests that there is a value (0.1) at which the dissipation levels off. Note that the values above 0.1 in the figure and some of the scatter in the data are due to transients as simulations develop from different sets of initial conditions. In the region before M t = , the Zeman and Sarkar corrections provide the most additional turbulence dissipation, while Wilcox s correction gives the least. To this point, i.e., M t = 0.1, the modified correction CC-A acts exactly the same as Sarkar s correction. For M t greater than 0.1, the standard corrections continue to provide additional dissipation, while the modified corrections level off this dissipation. This achieves the goal of matching the observed limit in the dissipation ratio shown by the DNS data. These models are implemented in the OVERFLOW code and evaluated using a range of test cases. 6.3 Application to an Axisymmetric Jet The modified corrections are first evaluated for axisymmetric jets to ensure the beneficial behavior for the cases presented in Chapter 4 is not disturbed. Fig. 6.3 shows a contour plot of turbulence Mach number for the Eggers jet (with Sarkar compressibility correction). The maximum M t is 0.32, and the changes in the new formulation go in effect above approximately 0.3, thus it is expected that the modified corrections will not have a significant effect for this test case. The centerline velocity and normalized half radius for the Eggers jet, using the modified corrections, are shown in Fig As expected, there is almost no change in using the new corrections. Since the M t for this case is mostly below 0.3, the correction CC-A acts essentially like Sarkar s and there is no change in the predictions. Also, there is a noticeable difference in the results from CC-B. The spread rate is decreased which happens as a result of the reduced dissipation rate for the CC-B model. The

99 83 CC-B model does, however, still show significant improvement over the uncorrected model. The jet by Wishart is very similar to that of Eggers. Thus, it is expected that changes resulting from using the modified compressibility correction are also small because of the lower turbulence Mach numbers. 6.4 Application to a 3-D Jet in Crossflow The Glenn jet provides a higher Mach number test case which highlights the problems in using Sarkar s compressibility correction. Turbulence Mach number contours (using Sarkar s compressibility correction) for this jet are shown in Fig When comparing with Fig. 6.3, it is evident that the values of M t, caused by elevated Mach numbers, are significantly larger. The maximum M t is 0.46, so it is expected that the limiter implemented in the modified corrections will have more impact. Fig. 6.6 presents the velocity magnitude contours for the modified compressibility corrections. Both modified corrections clearly improve the predictions. The shape of the vortex roll-up is predicted exceptionally well by both modified corrections, while it is overpredicted by Sarkar s correction and underpredicted by the uncorrected model. Also, the decay of velocity going downstream is predicted much better when compared with the other models. The velocity profiles are shown in Fig It is clear that, when compared with Sarkar s correction, both modifications do not inhibit turbulence growth as much. This is particularly clear for the later locations. The CC-B formulation is marginally more diffusive than the CC-A formulations. Overall, the predictions are greatly improved. Previously, the uncorrected turbulence model showed the best behavior. Now, using CC-A or CC-B is unquestionably the best solution. Fig. 6.8 shows velocity profiles for the Glenn jet with the capsule. The trends observed here agree with the case without the capsule. The agreement in the decay of velocity shows a significant improvement over both the Sarkar and the uncorrected models.

100 Application to Compressible Boundary Layers In order to evaluate the modified compressibility correction in the boundary layer, the SST turbulence model is modified. The OVERFLOW formulation, presented in Eq. 3.3 and 3.4, multiplies the compressibility correction by the SST blending function, (1 F 1 ), based on the distance to the nearest wall. This disables the correction in the boundary layer, and would not allow for an accurate assessment of the compressibility correction. Therefore, the multiplication by the (1 F 1 ) term is removed from the model formulation in the following analysis. A generic two-dimensional flat plate test case is used for computations. Results are compared to the Van Driest II with Karman-Schoenherr correlation [38] for Mach numbers of 2, 4, 6, and 8. In Fig. 6.9, the wall shear stress, C f, is plotted as a function of the Reynolds number based on momentum displacement thickness, Re θ = U θ ν. First, results are validated with those by Wilcox [20]. Wilcox found that at Mach 4 and Re θ of 10 4, the Sarkar correction underpredicts skin friction by 18%. This compares well with the current results which reflect a 20% change at the same conditions. The CC-A model behaves identical to the Sarkar model. This can be explained by examining the M t levels. Table 6.1 shows the maximum M t seen for the flat plate cases using Sarkar s compressibility correction. The second correction, CC-B, shows improvement over Sarkar s model because it provides less dissipation below M t of 0.3. However, the uncorrected model still gives the best prediction. This shows that the modified corrections still exhibit a boundary layer problem like Sarkar s correction, especially at higher Mach numbers. Wilcox [20] suggests that Sarkar s compressibility correction exhibits too little skin friction for a flat plate boundary layer, while an uncorrected k-ω model predicts the skin friction very accurately for higher Mach numbers. Wilcox s solution is to use a Heaviside step function to disable the correction in regions of lower M t (i.e., the boundary layer), which is shown in Eq This presents one mitigation for the compressibility correction problems in the boundary layer. However, as seen in

101 85 Table 6.1 Maximum Turbulence Mach Number for Flat Plate Mach Number M t,max Fig. 4.8, the result for the axisymmetric jet are not as good as when using Sarkar s correction. A second possible way to address this problem is by incorporating the compressibility correction terms only in the k-ǫ part of the SST model, and leaving the near-wall k-ω model undisturbed, as proposed by Suzen and Hoffmann [24], and presented previously in Eq. 3.3 and 3.4. The results for this formulation are presented in Fig The first modified correction, CC-A, is not included because Fig. 6.9 shows no difference between it and Sarkar s formulation. Both Sarkar s correction and CC-B show very good agreement with the Van Driest II correlation for most of the Re θ range. 6.6 Application to Compressible Mixing Layers The final test case assessed is the compressible mixing layer. This is the classic test case for compressible turbulence modeling. The experimental data used is from the study by Barone et al [39]. The data comes from various experiments which use velocity profiles to determine shear layer thickness. Cases for various convective Mach numbers are run, with a uniform temperature across the inflow. Table 6.2 summarizes the conditions used for this analysis. The shear layer thickness, δ, is defined as the difference in the locations of y, y, where (U U 2 ) 2 /(U 1 U 2 ) 2 is 9/10 and 1/10. The growth rate is measured at various locations in the fully-developed region and averaged. To scale the growth rate for different velocity ratios, the relation based on classic free shear scaling is used:

102 86 Table 6.2 Conditions for Compressible Mixing Layer M c U 2 /U 1 M t,max [ ] dδ dx = C U1 U 2 1 (U 1 + U 2 )/2 (6.4) Fig shows the growth rate scaled by the incompressible growth rate to give the compressibility factor, Φ. The uncorrected model, which exhibits no decrease in growth rate, is not shown. Results using Sarkar s correction (and the SST model) compare well with the results by Barone et al. when using the k-ǫ model. In the first region, the growth rate decreases faster than the experiment, meaning that the compressibility correction provides too much added dissipation initially. In the region beyond M c of 0.6, the growth rate does not decrease as quickly as it should, implying that more dissipation is needed. The modified correction, CC-A, matches Sarkar s until M c of around 0.8. Then, the modified correction levels off the growth rate. This trend is also seen in the experimental data. However, the location at which it levels off, where Φ is around 60%, is too high compared with the experimental data, which levels off around 45%. A possible solution to this problem is limiting the dissipation ratio at a later location to better match the experimental data. Another mitigation includes using a different formulation for the dissipation ratio below the cutoff, in order to provide less dissipation initially and then increased dissipation after M c of 0.7. According

103 87 to Barone et al., discrepancies are also seen based on the baseline turbulence model behavior. For their study, the corrections perform better when used with the k-ω model as opposed to the k-ǫ model. Various turbulence models could be tested to assess the baseline turbulence model dependence.

104 Figure 6.1. Normalized mixing layer growth rate data compared with previous efforts and recent experiments my Rossman. Used with permission from Rossman [36]. 88

105 89 (a) (b) Figure 6.2. Dissipation ratio as a function of turbulence Mach number for compressibility corrections (a) on log-log scale, along with DNS data by Blaisdell el al. [22], and (b) with regular scale in range of interest.

106 Figure 6.3. Contours of turbulence Mach number for Eggers jet. 90

107 91 (a) (b) Figure 6.4. (a) Normalized centerline velocity and (b) normalized half radius for Eggers jet using modified compressibility corrections.

108 Figure 6.5. Contours of turbulence Mach number for Glenn jet. 92