NUMERICAL STUDY OF TURBULENCE TRANSITION MODELS KSHITIJ D. NEROORKAR GARY CHENG, COMMITTEE CHAIR ROBERT NICHOLS ROY P. KOOMULLIL A THESIS

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1 NUMERICAL STUDY OF TURBULENCE TRANSITION MODELS by KSHITIJ D. NEROORKAR GARY CHENG, COMMITTEE CHAIR ROBERT NICHOLS ROY P. KOOMULLIL A THESIS Submitted to the graduate faculty of The University of Alabama at Birmingham, in partial fulfillment of the requirements for the degree of Master of Science BIRMINGHAM, ALABAMA 2007

2 NUMERICAL STUDY OF TURBULENCE TRANSITION MODELS KSHITIJ D. NEROORKAR MASTER OF SCIENCE IN MECHANICAL ENGINEERING ABSTRACT The transition of an attached boundary layer from laminar to turbulent greatly affects the performance of aerospace and turbomachinery devices. In the past, a wide range of methods have been employed for modeling this physical phenomenon, including the Stability theory, Direct Numerical Simulation, and numerous proposed modifications to the existing Reynolds-Averaged Navier-Stokes turbulence models. It is believed that the Reynolds-Averaged Navier-Stokes based models, of all the existing models, are easiest for coupling with production Computational Fluid Dynamics codes in order to achieve rapid, satisfactorily accurate estimates. There exists a huge body of literature on predicting turbulence transition using models based on Reynolds-Averaged Navier-Stokes, however, due to the complexity of the underlying physics involved in the transition process, these models are usually tuned to handle only certain types of flows and may not be applicable to different flow conditions and geometries. Therefore, the first objective of this thesis is to select, from the available literature, Reynolds-Averaged Navier-Stokes based transition predicting models that have the most generalized formulation. The second goal is to implement these models into a Computational Fluid Dynamics flow solver. The third goal is to validate them on some standard benchmark test cases. This thesis aims at achieving a neutral evaluation of the state-of-the-art turbulence transition models. ii

3 ACKNOWLEDGMENTS I would like to thank my mentor, Dr. Gary Cheng, for his constant encouragement and guidance throughout this work. A special thanks to Dr. Robert Nichols, I have benefited greatly from his experience in the field of turbulence and transition modeling. I am very thankful to Dr. Roy Koomullil for serving on my thesis committee and providing valuable feedback and suggestions. I thank the Department of Defense (project number CFD-KY7-002) and Dr. Cheng for providing funding for this research. I would like to acknowledge the support given by my parents, my sister Samiksha, my fiancée Sneha, and all my friends here at the University of Alabama at Birmingham and back home in India. Finally,a very special thanks to Nitin Bhagat, Deepak Kini, Amit Yadav, Bappa Patankar, Neelesh Bhopatkar, Deepesh Dimble, Balaji Shankar Venkatachari, and Sandeep Kulathu. iii

4 TABLE OF CONTENTS Page ABSTRACT... ii ACKNOWLEDGMENTS... iii LIST OF FIGURES...vi NOMENCLATURE... viii CHAPTER 1. INTRODUCTION Importance of Transition Modeling Overall Goal Tasks Task 1. Literature Survey Task 2. Implementation of the Shear Stress Transport model (SST) Task 3. Implementing the Transition Models LITERATURE REVIEW AND BACKGROUND INFORMATION The Physics of Transition The Linear Stability Theory for Transition Literature Study for Transition Models Models Based On Stability Theory Models Not Based On Stability Theory Literature Study for Experimental Data Conclusion from the Literature Review iv

5 3. MODEL FORMULATION Model of Menter et al Shear Stress Transport Model Transition Model Walters-Leylek model MODEL VALIDATION Shear Stress Transport Model Turbulent Flow Past A Flatplate Case Transonic Flow Over An Axisymmetric Bump Transition Models Flat Plate Test Cases Hypersonic Cone Test Cases CONCLUSION Summary Future Direction for Model Development LIST OF REFERENCES v

6 LIST OF FIGURES Page Fig. 1 Fig. 2 Growth of a turbulent spot originated at A, in a flat plate laminar boundary layer. (The flat plate is in the ZX plane.)... 5 A typical stability diagram showing the frequency variation with streamwise distance Fig. 3 Computational grid on the flat plate test case Fig. 4 Comparison of flat plate simulation results with analytical solution Fig. 5 Geometry of the circular bump Fig. 6 Computational grid on the bump Fig. 7 Convergence history for U velocity of point on the bump Fig. 8 Mach number contour for flow over bump Fig. 9 Comparison of velocity plots for transonic flow over the bump Fig. 10 Computational grid used for T3-series of simulations Fig. 11 Skin friction variation for T3AM case Fig. 12 U+ versus Y+ for T3AM test case in the transition region for three different grids using the Walters-Leylek model Fig. 13 U+ versus Y+ for T3AM test case in the transition region for three different grids using the LCTM model Fig. 14 Comparison of the experimental freestream turbulence intensity with numerical results for T3AM case Fig. 15 Skin friction variation for T3A case Fig. 16 U+ versus Y+ for T3A test case in the transition region for three different grids using the Walters-Leylek model vi

7 Fig. 17 U+ versus Y+ for T3A test case in the transition region for four different grids using the LCTM model Fig. 18 Comparison of the experimental freestream turbulence intensity with numerical results for T3A case Fig. 19 Skin friction variation for T3B case Fig. 20 U+ versus Y+ for T3B test case in the transition region for three different grids using the Walters-Leylek model Fig. 21 U+ versus Y+ for T3B test case in the transition region for three different grids using the LCTM model Fig. 22 Comparison of the experimental freestream turbulence intensity with numerical results for T3B case Fig. 23 Skin friction variation for T3A test case for four different grids using the LCTM Fig. 24 Computational grid for hypersonic flow over a cone Fig. 25 Stanton number variation for hypersonic cone using Walters- Leylek model Fig. 26 Stanton number variation for hypersonic case using LCTM and SST model vii

8 NOMENCLATURE = skin friction coefficient, = turbulent kinetic energy = laminar turbulent kinetic energy = Reynolds number, = momentum thickness Reynolds number, = transition onset momentum thickness, = vorticity Reynolds number S = absolute value of strain rate, = strain rate tensor, Tu = turbulence intensity = local velocity = local freestream velocity = inlet freestream velocity U+ = nondimensional velocity, u = streamvise component of velocity = friction velocity Y+ = distance in wall normal coordinates, y = normal distance to nearest wall = turbulence dissipation rate viii

9 = momentum thickness = molecular viscosity = eddy viscosity = molecular kinematic viscosity = eddy kinematic viscosity = density = intermittency = wall shear stress = magnitude of vorticity, = vorticity tensor, = specific turbulence dissipation rate Subscripts t = transition onset s = small scale l = large scale ix

10 1. INTRODUCTION 1.1 Importance of Transition Modeling Prediction of flow transition is extremely important in the design of turbomachinery and aerospace devices in which wall shear stress or wall heat transfer is critical. In turbomachinery, early transition on the blade can prevent separation of the suction side boundary layer, thereby leading to a reduction in total pressure loss. As a result, the proper prediction of transition can be used to minimize the number of blades and stages and thereby reduce the cost. At low Reynolds numbers with low freestream turbulence, the boundary layer on the airfoil surface has a tendency to remain laminar and hence to separate before it becomes turbulent. This reduces efficiency and increases fuel consumption. In high speed aerospace applications, the onset of transition increases the heat transfer rate, skin friction, and surface ablation rate of the thermal protection system (TPS). This makes the prediction of transition very critical in designing the TPS for hypersonic reentry vehicles [1]. Reda [2] stated that roughness elements are formed on the surface of the carbonaceous nosetip materials during reentry. These roughness elements create disturbances in the laminar boundary layer, eventually leading to transition. Scaling wind tunnel data to actual flight conditions is still a black art, because there is no reliable means of accounting for transition. 1

11 For vehicles that are powered by air breathing propulsion systems, most of the atmospheric hypersonic flight is found to take place at conditions where the flow will be transitional [3]. In addition, hypersonic vehicles like the National Aerospace Plane [NASP] have been known to encounter flows that may be transitional over a large portion of the vehicle [4]. Information about transition mechanisms is also important in drag reduction on supersonic aircrafts and for development of applications that benefit from extensive regions of laminar flow, for example, laminar flow aircraft. All these examples indicate that the performance, weight, and cost associated with turbomachines and aerospace devices are affected by turbulence transition and, therefore, its proper modeling is of utmost important for the efficient design of such applications. 1.2 Overall Goal The main aims of this thesis are as follows: 1. Conduct a literature review of both, the existing turbulence transition models and the available benchmark test data. 2. Select some appropriate transition models. 3. Implement these models into a Computational Fluid Dynamics (CFD) flow solver so as to assess their performance with some benchmark test cases. 2

12 1.3 Tasks To achieve these goals, the following tasks were performed Task 1. Literature Survey A literature survey was conducted to get information about past modeling efforts and available experimental data Task 2. Implementation of the Shear Stress Transport Model (SST) The Local Correlation based Transition Model (LCTM) of Menter et al. [5, 6], and the Walters-Leylek [7] model were chosen from the literature. The LCTM is based on the Shear Stress Transport (SST) model [8], which was not available in the employed CFD solver (Finite Difference Navier-Stokes solver (FDNS) [9]) and hence the first task was to implement and validate this turbulence model. The test cases chosen to validate the implemented SST model were the flow over a flat plate and the flow over a 2-D bump [10] Task 3. Implementing the Transition Models The next task was to implement the LCTM and the Walters-Leylek model into the FDNS code and to simulate the T3AM, T3A, and the T3B test cases [11] and the hypersonic flow over a cone case [12]. 3

13 2. LITERATURE REVIEW AND BACKGROUND INFORMATION This chapter is divided into four sections. The first section reviews the physics involved in the transition process, the second section reviews the theoretical explanation of the transition process, the third section contains a literature survey of the past modeling efforts in this field, and the final section reviews the test data available for benchmarking turbulence transition codes. 2.1 The Physics of Transition The laminar boundary layer is susceptible to disturbances from both the freestream and the body surface. This is defined as the receptivity of the boundary layer. The freestream disturbances include acoustic waves, particles in the flow, and pressure fluctuations. Surface disturbances include the roughness of the body as well as any vibrations the body may have. Many of these disturbances eventually get damped. The transition process starts when a disturbance in the boundary layer is no longer damped but gets amplified. The result is the formation of the two-dimensional(2d) Tollmien- Schlichting (TS) waves, which are the first mode instability of the flow. While these waves travel downstream, three-dimensional(3d) waves and vortices begin to develop. At certain points in the boundary layer, there occur small, irregularly shaped turbulent spots, which then travel in a wedge-shaped region as shown in Fig.1. 4

14 These spots, known as Emmons spots, appear at random locations on the plates at irregular time intervals (Schlichting [13]). As these spots move downstream, they grow and eventually fuse with each other to encompass the entire boundary layer, leading to fully developed turbulent flow. Fig. 1 Growth of a turbulent spot originated at A, in a flat plate laminar boundary layer. (The flat plate is in the ZX plane.) This transition mechanism is known as natural transition; however, flows with strong disturbances, especially from the freestream turbulence, by-pass the mechanism of the formation of TS waves, and turbulent spots are directly formed. This is called bypass turbulence (Morkovin [14]). For compressible flows, most notably flows with a Mach number of 2.2 or greater, Mack [15, 16] showed that multiple modes of instability exist. For flows with Mach numbers between 4.5 and 2.2, the first mode of the instability (known as the viscous instability) is the most unstable. 5

15 As the Mach number is increased beyond 4.5, the second mode (known as the inviscid instability) becomes the most unstable mode. This second mode includes acoustical disturbances that are characterized by very large fluctuations in pressure and temperature. 2.2 The Linear Stability Theory for Transition The theoretical investigations performed on transition flows evolve from the concept that some of the disturbances in the laminar boundary layer amplify and eventually lead to transition. Most of the literature presented in this section has been obtained from Schlichting [13] and White [17]. The theory of stability was formulated in order to find the critical Reynolds number for a flow, i.e., the Reynolds number at which the laminar flow breaks down. According to this theory, the fluid motion is decomposed into two components i.e., a mean flow and a disturbance superimposed on it. Therefore, if U,V, W, and P are the x, y, z velocities and pressure of the mean (steady) flow, if the corresponding unsteady disturbances are, and, respectively, then the resultant velocity and pressure components are (1) The analysis on such a flow may be carried out using two different methods, the energy method and the method of small disturbances. The method of small disturbances has been proven successful in this investigation (Schlichting [13]) and is explained as follows. 6

16 Consider the 2D incompressible Navier-Stokes equations for simplicity and assume that the mean flow is defined by velocity U = U(y), V=W=0, and P=P(x, y). The superimposed disturbances in this mean flow may be given as (2) The resultant flow, according to Eq.(1), may be described as (3) The mean flow expressed by U, V, and P will satisfy the Navier-Stokes equations, and so it is required that the resultant flow must also satisfy the equations. Thereby, substituting the expressions given in Eq.(3) into the Navier-Stokes equations, subtracting the original equations represented by U, V, and P and neglecting the nonlinear terms, we get (4) (5) (6) These are the three equations for the disturbances and Each disturbance is assumed to take the form of a traveling wave whose amplitude varies in y only (parallel flow assumption) and which is propagated in the x direction. These waves are called the TS waves, which are the first indications of instability in the laminar flow. Therefore the disturbances may be specified as Where is the wave number and is the frequency. (7) 7

17 Depending upon the nature of and, two theories are possible. 1) Temporal amplification: In this theory, is a real quantity and is complex In this case, is the circular frequency of the partial oscillation, whereas is the (8) temporal amplification factor, so that if <0, the disturbances are damped, and if >0, the disturbances get amplified. 2) Spatial amplification theory: In this theory, is complex and is real, so that now the disturbances are damped if >0 and amplified if <0. Substituting the expressions from Eq.(7) into Eqs.(4) (6), and by eliminating the pressure, we get the following fourth-order, differential equation (9) Where is the propagation speed of the disturbances. This is the fundamental differential equation for the disturbance, which is known as the stability equation or Orr- Sommerfeld equation. The terms on the left-hand side represent the inertia terms, and the terms on the right-hand side represent the viscous terms from the N-S equations. The three parameters, in Eqn.(9),, c, and (which may be also expressed as ). For any specified value of Re we get values for and c. By plotting the locus of the points at which for the temporal stability and for the spatial stability, we get a curve that represents the neutral disturbances (disturbances that neither amplify nor damp out). The plot showing this neutral curve is called the stability diagram. 8

18 This curve separates the region in which the flow is stable from the region in which it is unstable. The smallest Reynolds number on this curve is called the critical Reynolds number for the flow under consideration. This indicates the smallest Reynolds number at which disturbances can be amplified. It may be noted that this theoretical Re is called the point of instability, and the experimental value, which is further downstream, is called the point of transition. 2.3 Literature Study for Transition Models This section gives an overview of past transition modeling efforts. The models have been broadly categorized in two sections: a) Models based on stability theory. b) Models not based on stability theory Models Based On Stability Theory The e n method proposed by Smith and Gamberoni [18] and Van Ingen [19] is one of the most popular methods available for transition prediction. There are three steps in the application of the e n. In the first step, the laminar velocity and temperature profiles at different streamwise locations for the given flow are calculated. In the second step, the amplification rates of the most unstable waves are calculated for each profile by using the e n method. In the third step, these amplification rates are used to calculate the transition location. 9

19 The e n method is based on the linear stability theory. As mentioned in the previous section, a stability diagram for a given flow can be computed, showing the frequencies of the disturbances as functions of the distance from the leading edge. An example of such a curve is shown in Fig.2. Fig. 2 A typical stability diagram showing the frequency variation with streamwise distance. Now if we consider a wave that propagates downstream with a constant frequency f, then, it is damped before point x0. Beyond x0 it gets amplified up to x1 and then gets damped again after x1. 10

20 At any given station x (between x0 and x1) the total amplification rate is defined as (10) Where A is the wave amplitude as it evolves downstream, and A o is the amplitude at the onset position Xo of instability. In the case of the temporal theory, is calculated by dividing the complex frequency by the group velocity. However, the spatial amplification theory is preferred since it gives the amplitude change directly with the streamwise length. The n factor is then defined as the maximum of the amplitude ratios at each streamwise location. (11) By correlations with experiments, transition has been found to occur at a value of n between 7 and 9. Transition will occur when the most unstable frequency is amplified by a factor of e 7 to e 9. One of the major criticisms that the e n method has received is that it assumes that the flow is locally parallel and, the value of the n factor for transition is not universal and needs experimental data. The linear Parabolized Stability Equations (PSE) method represents the nonparallel effects neglected in the linear stability theory and assumes that the mean flow, amplitude functions, and wave number are dependent upon the streamwise distance (x). A further development of the linear PSE known as the nonlinear PSE incorporates the nonlinear effects that have been neglected in the linear stability theory. 11

21 Methods based on the stability theory have a number of drawbacks: 1. They need to track the growth of the disturbance amplitude ratio along the streamline. This limitation poses a significant problem for 3D flows where the streamline direction is not aligned with the grid. 2. These methods require the steady state solution, which may not be available in certain cases. 3. Coupling of such methods with CFD codes requires an unrealistically high grid density to yield the boundary layer data with the required level of accuracy. The main advantage of these methods is that they give the correct treatment of the surface curvature. Some different techniques have been employed to use these stability-based methods more efficiently. One of them is to generate a database of the solution of the linear stability equation for different velocity profiles in advance. The local flow stability can then be determined quickly, based on the local velocity calculated from CFD codes. However, the validity of these models is still limited to the range of velocity profiles available in the database. 12

22 2.3.2 Models Not Based On Stability Theory Models with known transition onset All of the transition region models mentioned in this section are unable to predict the location of the transition and require empirical data or data from an e n computation so as to get the location of the transition onset. In this case, transition region modeling is done by modifying existing turbulence models. McKeel [20] implemented six transition models into the Navier-Stokes code GASP. These six models were the Baldwin-Lomax model [21], the Wilcox model [22, 23]; the Schmidt and Patankar low-re model [24] which had a production term modified for modeling transition; the Warren, Harris and Hasan one-equation model [4]; the algebraic transition model developed at ONERA/CERT [25, 1]; and the linear combination transition model developed by Dey and Narasimha [26]. These models were used to simulate hypersonic experimental cases that included transition on a cone at Mach 6 [27], a compression ramp at Mach [28], and five flared-cone test cases at Mach 7.93 [29, 12]. Of the five flared cones used, there were two with favorable pressure gradients, two with adverse gradients, and one with a zero pressure gradient. Baldwin-Lomax Algebraic Turbulence model [21]. This is an algebraic turbulence model so no differential equations need to be solved. In [20], it was used to predict the transition region by setting the eddy viscosity to zero ahead of the given location of transition. The turbulence model was turned off for the laminar region by setting the eddy viscosity equal to zero and then turned on at the transition point. 13

23 It was found that, in most of the cases, this model adequately predicted the peak heat transfer, but under- predicted the transition length [20]. Warren, Harris, and Hassan (WHH) One-Equation model [4]. This model attempts to include the effect of second-mode disturbances in addition to the first mode (TS waves). The transitional stress, incorporating both modes, is calculated by using the following formula for the eddy viscosity length scale ( ) Where and are the length scale contributions from the first and second modes, (12) respectively, and are calculated from experimental correlations. is the turbulent length scale, and is the intermittency factor. The intermittency of the flow is defined as the fraction of the time that the flow is turbulent. The expression for intermittency used here was developed by Dhawan and Narasimha [30] and is given by (13) Where x t is the location of transition onset, is the streamwise distance between points where is 0.25 and Warren et al.[4] used this model to simulate cases in which the first mode disturbances dominate the transition process (M < 4) and also cases in which the second modes are dominant (M > 4). In all cases the model performed satisfactorily. This model was later implemented by McKeel [20] and again was found to be quite accurate. 14

24 Wilcox Turbulence model [22, 23]. The low-re, model developed by Wilcox was used to predict the transition region by McKeel [20]. The prediction of the transition region was obtained by tripping the boundary layer at a given point by decreasing the value of the dissipation of turbulence kinetic energy (TKE) so as to destabilize the boundary layer and cause a transition. This is similar to the use of roughness strips in wind tunnels to cause transition. The application of this model by McKeel [20] showed that values for the height and length of the roughness strip proposed by Wilcox [22, 23] did not produce transition behavior at the required location. The results obtained for the experimental cases in [20] showed that the model predicted a short transition length and that sometimes the heat transfer peak predicted was too high. Schmidt and Patankar Production Term Modifications [24]. Schmidt and Patankar have developed modifications to the production term in the TKE equation of the Lam and Bremhorst [31] model. These modifications limited the production of the kinetic energy. For the use of this model, a trial and error method was needed to make the transition to occur at the desired position by varying the inlet conditions. The results in the experimental cases in [20] were not very satisfactory, and the method was found to be very sensitive to the grid spacing near the wall. Due to the defects in this model, a few modifications were suggested in [20]. Since it was found that it was difficult to trigger transition, a spot with high TKE was introduced into the boundary layer. This spot then grew and caused transition to take place. 15

25 In order to improve the prediction of the length of the transition region, an exponential function was used for the maximum allowable production of TKE. Though the results obtained from this modification were improved for some cases, they deteriorated for other cases. Algebraic Transition model [25, 1]. The algebraic transition region model was developed at ONERA/CERT and is described in Arnal [25, 1]. The form of this model in Singer et al. [32, 33] was implemented in [20]. This model predicts transition by multiplying the eddy viscosity by a transition function before adding it to the molecular viscosity. This function was found to be related to the momentum thickness growth. As a result, in test cases with severe adverse pressure gradients, where the momentum thickness decreases, the model did not undergo transition. Theoretically, this model should be compatible with any turbulence model; however, it was found that this model did not perform well with two-equation models, so in [20], the model was used with the Baldwin-Lomax model. In [20], a few corrections to the calibration of the transition function are suggested in order to better simulate high-speed flows. The new model predicted the cases tested in [20] better than the original model. Linear Combination Transition model [26]. This model was developed by Dey and Narasimha and is based on the concept that the transition flow is a combination of laminar and turbulent flow fields. This model requires that a complete laminar simulation be run first. This is followed by a turbulent one, with the turbulent boundary layer starting at the point of transition. 16

26 The model then uses these two solutions to generate the transitional solution. For example, the mean velocity (U) and skin friction (C f ) will be calculated as follows: (14) (15) In this equation, the subscripts L and T stand for values in the laminar and turbulent boundary layers, respectively, and is the intermittency factor developed by Dhawan and Narasimha [30] and described earlier. The peak heat transfer was not predicted in the test cases run McKeel [20]. The main difficulty in getting accurate results with this model was that one of the modeling constants needed to be modified from case to case to obtain good results. Many researchers, including Abid [34], have used the intermittency function from the linear combination model as an algebraic transition region function to proportion the amount of eddy viscosity added to the molecular viscosity. The results using this method were found to be very similar to the linear combination model mentioned previously, but there are some noticeable differences (McKeel [20]). The transition length was always predicted shorter. For the cases with no pressure gradient and adverse pressure gradients, the heat transfer predicted at the end of transition and through the turbulent region was significantly higher. Roy and Blottner [35] evaluated the performance of the Spalart-Allmaras [36] and the Baldwin-Barth [37] one-equation models, and three two-equation models for simulating hypersonic transition were evaluated. 17

27 The two-equation models assessed included a low-re model with the modifications of Nagano and Hishida [38], the hybrid model of Menter [8], and the Wilcox model [39]. The Sandia Advanced Code for Compressible Aerothermodynamics Research and Analysis (SACCARA) was used by Roy and Blottner [35]. Two flow cases were used to evaluate these models. The first case was the flow over a flat plate at Mach 8 with flow conditions corresponding to an altitude of 15 km, where a perfect gas model is appropriate. The second flow case considered was the flow over a Reentry F flight vehicle at Mach 20 and an altitude of 24.4 km [40], where real gas effects need to be taken into account. The method employed by Roy and Blottner [35] to specify transition from laminar to turbulent flow is described as follows. The turbulence transport equations were solved over the entire domain, with a transition plane specified by the user. Upstream of this plane, the effective viscosity was simply the laminar value, whereas downstream the effective viscosity was the sum of the molecular and turbulent viscosities. An advantage of this approach was that the turbulence transport equations were solved over the whole domain, thus promoting turbulent behavior downstream of the transition plane. On the other hand, if the turbulence source terms were simply turned on after the transition plane, the turbulence model might not have transitioned to turbulent flow until further downstream, depending on the freestream turbulence values. A disadvantage of this approach was that a discontinuity in the total viscosity (laminar plus turbulent) could occur at the transition plane. 18

28 Spalart-Allmaras One-Equation model [36]. The Spalart-Allmaras one-equation model was employed by Roy and Blottner [35]. The diffusion term of the model used was modified to incorporate compressibility effects. It was found that, for high speed flows, it was difficult to control the onset of transition using the trip terms that are a part of the original model. There was also no control over the length of transition, so three different methods were suggested to control transition onset and length. Of these the most reasonable solution involved the variation of the coefficient of the source terms in the transition region. Baldwin Barth One-Equation model [37]. The Baldwin Barth one-equation model was also tested by Roy and Blottner [35], however, it has been found that this model does not constitute a well-posed set of governing equations (Menter [41]). In boundary layer and shear flows, it was observed that the method does not converge to a unique solution as the mesh is refined. Wilcox model [39]. The Wilcox model was modified in 1998 to improve its accuracy in simulation of shear flows. Roy and Blottner [35] employed this modified model for modeling hypersonic transition. The compressible form of the production term of the turbulent transport equations was used. Low-Re model of Nagano and Hishida [38]. This model was evaluated by Roy and Blottner [35] for modeling hypersonic transition flows. This model was developed for incompressible flows and was employed without any compressibility corrections. 19

29 Menter s SST model [8]. This is a hybrid model that uses a blending function to combine the advantages of the and the turbulence models. In the near-wall region a model is used, and in the outer edges of the boundary layers as well as in free shear flows the model transforms into a model. This transformation is accomplished through a blending function. The production term used for the turbulence transport equations included the compressibility. All models except the Spalart Allmaras and the low-re model predicted the transition at the correct location for the flat-plate case at Mach 8. For these two models, the freestream turbulence values needed to be increased to get transition at the correct location. All of the models tested in [35] provided the correct skin friction levels for this case. For the Reentry F flight vehicle [40], the wall heat flux predicted by the Spalart Allmaras model, Menter s SST model, and the Wilcox model were found to be in reasonable agreement with the experimental data. The Baldwin-Barth and the low-re model greatly overpredicted the heat flux in the turbulent region. Models with onset point unknown The models listed in this section have a built-in capability not only to predict the onset of transition but also to simulate the characteristics of the transition region. The Turbulence/Transition model [42]. In [42], a turbulence model was used to study the effect of high-disturbance environments (HIDE) on the turbulence transition in conventional hypersonic facilities. 20

30 Since HIDE cannot be described by the linear stability theory, a minimum heat flux criterion was used to determine the onset of transition. This is done by assuming initial transition onset location. After running a few iterations, the minimum heat flux criterion is employed to find the locations where the wall heat flux is lowest. The solution is independent of the initial guess as long as the initial transition points are upstream of the actual locations. The approach was similar to the WHH model. In this case, the eddy viscosity was modified using the formula (16) Where is the intermittency factor calculated using the algebraic formula of Dhawan and Narasimha [30] and is the eddy viscosity resulting from the nonturbulent fluctuations. This is calculated by the formula (17) Where is the nonturbulent time scale. In addition, the dissipation time scale in the turbulent kinetic energy equation was also chosen as the combination of time scales of turbulent and nonturbulent fluctuations. The time scale for calculating the eddy viscosity and the dissipation time scale were derived for three different transition mechanisms i.e., crossflow instabilities, second mode instabilities, and HIDE. These three mechanisms were selected because they were believed to be responsible for transition over 3D bodies in conventional hypersonic tunnels (Mokovin [43]). The simulations were performed on an elliptic cone at a mach number of 7.93 and were compared with experimental results. 21

31 It was concluded that the HIDE had a higher impact on transition than the other two mechanisms; however, it is impossible to ignore the fact that there may be nonlinear interactions playing some part in the transition process. The main disadvantages of this model are that it does not solve the nonturbulent fluctuations using transport equations and it uses an algebraic formula for the intermittency, and which limits the flexibility of this method. Papp and Dash model [44]. A concept that was analogous to that of WHH model was used. In this case, the model of So et al. [45] was employed with compressibility corrections for hypersonic flows. The only addition was that a separate transport equation was solved for the nonturbulent fluctuations. The nonturbulent fluctuations included first and second-mode mechanisms. This equation is similar to the turbulent kinetic energy equation of the model of So et al. [45]. In this case, the transition onset was determined by a condition formulated by Warren and Hassan [46] that is an alternative to the e n method mentioned earlier. This method uses a minimum skin friction criterion to locate the transition onset. The location of the onset of transition was said to be the minimum distance along the surface for which, where (18) Where is the turbulent viscosity coefficient with a value of 0.09, is the fluid kinematic viscosity, and is the eddy viscosity due to the nonturbulent fluctuations, which can be calculated as follows: 22

32 (19) Where is the viscosity time scale obtained for different transition mechanisms by empirical correlations, and is the laminar TKE, which is obtained by solving a transport equation in the Papp and Dash model [44] This model was incorporated into CFD flow solvers by multiplying the turbulent eddy viscosity with the intermittency before adding to the molecular viscosity. In all cases simulated, the transition onset was properly obtained; however in some cases the heat transfer peak was not reproduced correctly. This has been attributed to the algebraic nature of the intermittency function used, which is the biggest disadvantage of the model. Suzen and Huang model [47]. This model uses a transport equation for the intermittency factor. This equation not only reproduces the intermittency distribution of Dhawan and Narasimha [30], but also gives a realistic variation of the intermittency in the crossstream direction. The intermittency transport equation included source terms from two different models viz. the Steelant and Dick model [48] and the Cho and Chung model [49]. The model is incorporated into the Navier-Stokes solvers by simply multiplying the eddy viscosity obtained from the turbulence model by the intermittency factor. The Menter s SST model was used to calculate the turbulent quantities. The onset of transition was determined by comparing the local Reynolds number with a transition onset Reynolds number ( ) calculated using the correlation of Huang and Xiong [50], where is a function of freestream turbulent intensity and an acceleration parameter, which is calculated as follows: 23

33 Where U is the freestream velocity at the point of transition, s is the streamline coordinate defined as: (20) (21) This model was tested for flows with zero and variable pressure gradients and with different freestream turbulence intensities. The model showed good agreements with the T3 series experiment results [11]. The main disadvantage of this model is that it is not single point, because the calculation of the local Reynolds number requires integration, and this depends on the implementation of a search algorithm. Walters-Leylek model [7]. This model is a low-re model and is based on the concept that bypass transition is caused by very high amplitude streamwise fluctuations. These fluctuations are very different from turbulent fluctuations. Mayle and Schulz [51] proposed a second kinetic energy equation to describe these fluctuations. This kinetic energy was called laminar kinetic energy (k L ). In the near-wall region, the TKE (k T ) was split into small-scale energy and large-scale energy. The small-scale energy (k T,s ) contributes directly to the turbulence production, and the large-scale energy (k T,l ) contributes to the production of laminar kinetic energy. The eddy viscosities based on both scales are calculated from the respective-scale kinetic energies.for the onset of transition, a parameter is calculated from k T, the kinematic viscosity and the wall distance. When this parameter crosses a certain threshold, transition is assumed to start. 24

34 The onset of transition is associated with the reduction of k L and the consequent increase of k T, indicating the breakdown of laminar fluctuations into turbulence. This model was incorporated into a Reynolds-Averaged Navier-Stokes (RANS) flow solver for the calculation of the total eddy viscosity and eddy thermal diffusivity to account for contributions from the small-scale as well as large-scale turbulent kinetic energies. For all test cases simulated, the model responded correctly to increases in the freestream intensity, i.e., it predicted transition earlier for higher freestream intensity. It yielded reasonable results for cases with high pressure gradients and streamline curvatures. This method has the advantage of being easily implemented into CFD codes, since it is based on a RANS framework. This is a single-point transition model, meaning that it requires only local information, which makes this method easily applicable to unstructured and parallel computations. The main disadvantage is that the low-re turbulence models are typically not calibrated for transition prediction, but give the transition location as a by-product of their viscous sublayer formulation. Since this transition model is developed based on the low- Re model, the embedded viscous sublayer formulation coupled with the added transition prediction capability cannot be calibrated independently. Hence, a change in the transition formulation would affect the solution in the fully turbulent region. As a result, it is generally observed that these models are not flexible enough to sufficiently cover a wide range of the transition mechanisms observed in reality [5, 6] 25

35 Local Correlation Based Transition Model ( model or LCTM.) [5, 6]. This model is based on the SST model [8]. In this model the vorticity Reynolds number is an extremely important parameter and, being a local property, can be easily calculated in CFD codes. The maximum value of the vorticity Reynolds number in a boundary layer is directly proportional to the momentum thickness Reynolds number (. In this model, is used in triggering transition instead of directly using. This model solves a transport equation for intermittency and a transport equation for the Reynolds number based on the momentum thickness at transition onset. The first transport equation includes two terms that control production. These are, a parameter that controls the length of the transition zone, and, which is the momentum thickness Reynolds number at the point where the intermittency starts to increase in the boundary layer. These two variables are calculated from empirical functions of the local transition momentum thickness Reynolds number (. A second transport equation is required to solve for, and to include the nonlocal influence of the turbulence intensity, which varies with the freestream turbulence kinetic energy and the freestream velocity. In the case of flows with boundary layer separation, this transition model is modified so that the intermittency is allowed to exceed unity when the boundary layer separates. This event results in larger production of kinetic energy, leading to a correct prediction of flow reattachment [6]. This model is applied by modifying the production and destruction terms of the original SST model using the intermittency. The model was validated on many complicated 2D and 3D configurations. In all cases, good agreement with experimental data was obtained. 26

36 This model offers two main advantages: 1) it is based on local variables; 2) it is very flexible and can be used for any mechanism as long as the empirical correlation can be formulated. However, currently the employed empirical correlations are proprietary and are not available in the literature. 2.4 Literature Study for Experimental Data Some experimental studies have been conducted to examine the turbulence transition effect on friction force and heat transfer. They range from low to high speeds and have geometries with and without curvature effects, as well as with and without pressure gradients. The T3 series of experiments by Coupland [11] is one of the most widely examined test cases for transition model validations. These test cases include seven flat plate test cases with different freestream turbulence intensities and with and without pressure gradients. The test data provided for these cases include the mean velocity and turbulence data such as the non-dimensionalized distance away from the wall (Y+), the non-dimensionalized streamwise velocity (U+), momentum thickness, and displacement thickness, at different axial locations. The test data also has the skin friction and freestream turbulence intensity distribution along the axial direction. Radomsky and Thole [52] performed experiments on a stator vane geometry with turbulence levels comparable to those in the actual gas turbines. These experiments were performed with the intention of providing code-validation quality data for turbulence models and CFD codes. 27

37 The study provides the friction coefficients, Stanton numbers, and velocity measurements in the freestream as well as within the boundary layer. The turbulence levels in the experiment are comparable to those exiting the combustion chamber of the actual gas turbine. Unfortunately, the geometry of the vane was not provided in the report. Ubaldi et al. [53] performed boundary layer measurements at low freestream turbulence intensities in a large-scale turbine vane cascade. The data available is the skin friction and the velocity profiles. Ottavy et al. [54] tried to study the transition effects over the suction side of an axial compressor blade in the presence of wakes. Experimental investigation was performed on a compressor-like flat plate subjected to wakes coming from upstream bars. In addition, the flat plate was enclosed in a converging-diverging chamber to introduce a pressure gradient. This test case included the essential characteristics of boundary layer development on the suction surface of a compressor blade. The experimental data provided are the shape factor, the skin friction, the pressure variation, the momentum and displacement thickness, and the velocity profile in the boundary layer. However, the geometry of the converging-diverging chamber was not included in the literature. Kimmel [29, 12] examined the effect of pressure gradients on turbulence transition in hypersonic flows. This experimental study consists of five different cones (one straight cone, and four flared cones) at a Mach number of The first part of the domain is a straight 7º half angle cone with a length of m. The second part of the domain is either straight (one case) or flared (four cases). The straight geometry generates flow with no pressure gradient. 28

38 Two convex flared geometries produce flow with adverse pressure gradients, while two concave flared geometries create flow with favorable pressure gradients. The data available for comparison are the wall pressure distributions and the Stanton number variation. The wall temperature and mean flow conditions of the freestream were provided, but the freestream turbulence level was not reported. The experiment by Holden and Chadwick [28] measured the heat transfer in the transitional region of a geometry that included a 5º ramp followed by curved compression surface going into a 10º ramp in a hypersonic flow at a Mach number of 10. The curved compression surface creates a very severe adverse pressure gradient. The pressure and heat transfer variations were reported for comparison. Though the freestream mean flow conditions are provided in the report, the turbulence level is not available. Swearingen and Blackwelder [55] performed experiments on the concave surface of a curved plate with a 3.2m radius of curvature in a subsonic flow. The test result provides the effect of curvature on turbulence transition. The skin friction data from this case is available in [32]. The freestream turbulence intensity of this test case is very low (around 0.07%), where flow undergoes natural transition instead of bypass transition. 2.5 Conclusion from the Literature Review It was concluded that the LCTM [5, 6] and the Walters-Leylek model [7] constitute the best formulations for production CFD codes because they are both singlepoint models and can be easily incorporated into these codes. 29

39 Both of these models provide estimates of the location of turbulence transition and enable the CFD codes to simulate the flow characteristics in the transition region. These two models have been found to produce transition locations that respond properly to changes in freestream turbulence intensity and pressure gradients. A further investigation of two other models, the Papp and Dash model [44] and the turbulence/transition model [42], was deemed not worthwhile due to the aforementioned disadvantages. The experimental data selected for validating these models were as follows: 1. The T3 series of experiments [11], which constitute the flow over a flatplate with different freestream turbulence intensities and are the most widely accepted transition data. 2. The hypersonic flow over a nonflared cone (Kimmel [12, 29]) to examine the behavior of these models in predicting compressible flows. 30

40 3. MODEL FORMULATION 3.1 Model of Menter et al. Based on the literature review, two transition models, the LCTM and the Walters- Leylek model were selected to be evaluated for their accuracy and range of applicability. To examine these two transition models, they were implemented into a CFD code (FDNS) as part of this research work. The mathematical formulation of these models is described in this section Shear Stress Transport Model The first step in the implementation of the LCTM [5, 6] was the implementation and validation of the SST model [8]. As described in the previous section, this model is a combination of the and the models. In the near wall region, the model is activated, whereas in the freestream the model is activated. The switching between the two models is achieved through a blending function F 1. The formulation of the model is mentioned in the following discussion. The k-equation is as follows: (22) 31

41 The equation is given as follows: (23) Where (24) The constants are a combination of the respective constants of the original and the models. These are calculated as Where represents any constant in the model; 2 represents the corresponding (25) constant in the model. These constants are given as Set 1 Set 2 The blending function is formulated such that it is unity in the viscous sublayer and the log layer and gradually switches to zero in the wake region. It is defined as follows: (26) 32

42 Where (27) Where y is the normal distance to the closest wall and is the positive part of the cross diffusion term in the equation, Eq.(23). (28) The eddy viscosity is given as (29) Where is the absolute value of vorticity; a 1 is F 2 is given by (30) Where (31) Transition Model The next step was to program the LCTM of Menter et al. [5, 6], which is based on the SST model. This section discusses the formulation of this transition model as presented by Menter et al [6]. As mentioned in the previous section, this model uses two extra transport equations in addition to the turbulence equations of Menter s SST model [8]. The first additional equation is for the intermittency factor, and is given as follows: 33

43 (32) Where is the density, and is the velocity vector, is the dynamic viscosity, is the eddy viscosity, and is the model constant. The transport equation contains two production terms and, and two destruction terms and. Out of these, and represent the transition source terms, and and represent the relaminarization terms. These are formulated as follows: (33) (34) Where is the vorticity magnitude. the transition region. The term are the model constants, S is the strain rate magnitude and is an empirical correlation that controls the length of controls the transition onset and is calculated from the following functions: (35) (36) (37), (38) Where y is the normal distance from the nearest wall, k is the turbulence kinetic energy, and is the specific turbulence dissipation rate. is the critical momentum thickness Reynolds number where nonzero intermittency values first appear in the boundary layer. 34

44 This occurs upstream of the transition momentum thickness Reynolds number. The empirical correlations for and are proprietary and are not given in the original papers [5, 6]. The second additional equation is designed to capture the nonlocal influence of the turbulence intensity. In this equation, the transition momentum thickness Reynolds number is treated as a transported scalar quantity, and then the free stream value of calculated from the empirical correlations, is allowed to diffuse into the boundary layer. This equation is formulated as follows: (39) Where is the model constant. The source term is designed to force the transported scalar to match the local value of calculated from the empirical correlation, Eqn. (46), outside the boundary layer. The source term is defined as follows: (40) (41) (42) Here (43) (44) 35

45 Where u, v, and w are the X, Y, and Z components of velocity, respectively. The model constants are given as (45) Menter et al. [5] proposed a correlation for the transition momentum thickness Reynolds number. This is expressed as follows: (46) (47) (48) (49) Where the functions in are defined as (50) (51) (52) 36

46 Where is the local turbulence intensity in percentage, and is the momentum thickness, du/ds is the acceleration in the streamline direction, which can be computed as follows: (53) (54) (55) Where U is the local velocity magnitude as given in Eq.(43). In order to obtain, an iterative procedure is required, since both and are unknown. The steps for this procedure are as follows: a) Guess the initial value of ( is recommended). b) Calculate the and from Eqs. (48) and (49), respectively. c) Calculate the. d) Calculate the from Eq.(46). e) Calculate the from the definition of, f) Use the new value of as a guess for the next iteration and go to step b. To avoid unstable computations, the following limits are recommended for the values of and. (56) This transition model has been calibrated for use with the SST turbulence model as follows: 37

47 (57) Where and are the original production and destruction terms of the turbulent kinetic energy equation of the SST model, and are the modified production and (58) destruction terms, and is the effective intermittency obtained from (59) (60) (61) The SST model is modified by changing the blending function responsible for switching between the and models, as follows: (62) Where is the original blending function of the SST model. (63) The steps to be followed for implementing this model are as follows: a) Solve the mean flow equations viz. the momentum and continuity equations. b) Solve the SST model with the modified blending function and the modified turbulent kinetic energy equation. c) Calculate the new eddy viscosity. d) Solve the empirical correlation of the transition momentum thickness Reynolds number, Eq. (46). 38

48 e) Solve the and equations. The boundary conditions for the intermittency factor and at freestream, outlet, and walls have zero flux (64) Where is either the normal distance away from the wall or the distance in the streamwise direction. The inlet value recommended for is 1 and for is to be calculated from Eq. (46) by setting and setting Tu with the value of the freestream intensity at the inlet. The initial conditions for both and are set to their inlet values for the entire flow field. In the original LCTM [5, 6], the two parameters, and are calculated from empirical correlations and are both functions of ; however, both of these correlations are proprietary and unavailable in public literature. Correlations for these two parameters have been formulated by Suluksna and Juntasaro [56] based on numerical experiments. For, the same formula as (46) is used but with the freestream intensity instead of local intensity and ignoring the effect due to pressure gradients. The relations for and from [56] are as follows: (65) (66) Where is the freestream turbulence intensity at the leading edge. 39

49 The relation for is only applicable for zero-pressure gradient flows with freestream turbulence intensities in the range of of. Since and are no longer functions of, the value of obtained from the second transport equation (Eq. (39)) is not used in the transition model. This version of the transition model was implemented and tested in the current study. 3.2 Walters-Leylek model For the Walters-Leylek model [7], developed based on the kinetic energy in the near wall region is split into a small-scale. The turbulent (contributes to the turbulence production) and large-scale (contributes to the production of the laminar kinetic energy). The cut-off length scale is defined as Where y is the distance from the wall, is a modeling constant, and is the turbulent length scale given by: (67) (68) Where is the turbulent dissipation. The small-scale and large-scale kinetic energies are then calculated as follows: (69) (70) 40

50 The transport equation for the turbulent kinetic energy (, the laminar kinetic energy ( ), and the turbulent dissipation are given as: (71) (72) (73) Where and are modeling constants. The first term on the right-hand side of Eq. (71) is the production of turbulence (74) Where S is the strain rate, and the small-scale eddy viscosity is defined as: (75) Where is the turbulent viscosity coefficient and can be calculated as: (76) Where and are modeling constants, and and are damping functions calculated as: (77) (78) 41

51 Where and are modeling constants, and is the turbulent Reynolds number based on the small, turbulence producing scales, which is defined as: (79) Where and are the mean and effective time scales. (80) (81) The first term on the right-hand side of Eq.(72) is the production of the laminar kinetic energy by the large-scale turbulent fluctuations (82) The large-scale turbulent viscosity is modeled as: (83) Where is a modeling constant, and is a damping function, similar to (84) (85) Eq. (83) indicates that the only exists near the walls, since both vorticity ( and are zero away from the wall. 42

52 Both the and equations require an explicit calculation of the near-wall dissipation that arises due to the no-slip effect on the turbulent and laminar fluctuations. (86) (87) The term R appearing in Eqs. (71)-(73) represents the rate of production of turbulent kinetic energy by laminar kinetic energy during the bypass transition process. It appears with opposite signs in the Eqs.(71)-(72), resulting in no net change in total fluctuation energy. The term R is calculated as follows: (88) (89) Where is a modeling constant, and is the threshold function that controls the bypass transition process, which is calculated as follows: (90) (91) Where and are modeling constants. The model transitions are based on the concept that laminar fluctuations break down whenever the turbulent kinetic energy is greater than a certain threshold value, relative to the wall distance and fluid kinematic viscosity. 43

53 The coefficient used in Eq. (73) is calculated as: (92) The coefficient used in Eq. (73) is calculated as follows: (93) The turbulent scalar diffusivity in Eqs. (71)-(73) is given as: (94) Where is a modeling constant. The total eddy viscosity and eddy thermal diffusivity are then calculated as follows: (95) (96) Where and are modeling constants. Finally, the eddy viscosity and the eddy thermal diffusivity are used in the mean flow equations as follows: (97) Where is the kinematic Reynolds stress tensor, is the kinematic heat flux tensor, and T is the mean Reynolds averaged temperature. (98) The inlet boundary conditions for and are specified in a manner similar to any model. If the inlet is located in the freestream, then is set to zero. At solid walls, all three transport equations use zero flux conditions. 44

54 , (99) Where is the normal distance away from the wall. The model constants used in the model are as follows: ; ; ; ; ; ; ; ; ; ; (100) ; ; The value of the constant was not given in the original paper [7], so the value from [57] was used. 45

55 4. MODEL VALIDATION 4.1 Shear Stress Transport Model The LCTM is based on the SST turbulence model. Since the employed CFD code (FDNS [9]) was not equipped with the SST model, this turbulence model was first implemented into the code. Two cases were considered for the validation of the implemented SST model; the first case was a flatplate case, and the second was a transonic flow over a bump [10] Turbulent Flow Past A Flatplate Case The freestream condition for this test case has a Mach number of 0.04 and a Reynolds number of 1.00E6. The numerical mesh used for this purpose consisted of 139 X 86 points and is shown in Fig. 3. The grid was packed towards the wall. Two different near-wall grid spacings (achieved by using different packing parameters) were used for the grid sensitivity study. The numerical results of the SST model were compared with Spalding s law of the wall [17] and are plotted in Fig. 4. The result shows that though there is a minor difference between different grid spacings, both results agree very well with Spalding s law of the wall. 46

56 Fig. 3 Computational grid on the flat plate test case Fig. 4 Comparison of flat plate simulation results with analytical solution 47

57 4.1.2 Transonic Flow Over An Axisymmetric Bump This is a well-known benchmark test case for turbulence models and involves interaction between a shock wave and a turbulent boundary layer. The geometry, shown in Fig. 5, constitutes a circular arc bump fixed to a straight, circular cylinder that is aligned with the flow direction. Fig. 5 also shows the freestream conditions used. Fig. 5 Geometry of the circular bump The computational mesh used for this simulation has 227 X 170 gridpoints and is shown in Fig. 6. The simulation results were validated against the experimental data of Bachalo and Johnson taken from [10]. The convergence history of the axial velocity at a point near the bump surface is plotted in Fig

58 Fig. 8 shows the Mach number contour plot, and Fig. 9 shows the comparison between the experimental data, the implemented SST model, and the standard k and extended k [58] models originally embedded in the FDNS code. The result of the SST turbulence model is very close to that of the SST model implemented in the OVERFLOW code [59] and of the extended k model. Fig. 6 Computational grid on the bump 49

59 Fig. 7 Convergence history for U velocity of point on the bump Fig. 8 Mach number contour for flow over bump 50

60 X/C= X/C=0.563 X/C=0.625 X/C=0.688 X/C=1.00 X/C=1.375 Fig. 9 Comparison of velocity plots for transonic flow over the bump 51

61 In Fig. 9, X is the axial distance from the starting point of the circular bump; Y represents the transverse distance; C is the chord length of the bump; U stands for the axial component of velocity; and Uinf is the freestream value of U The results obtained from these two test cases confirm that the SST turbulence model was correctly implemented into the CFD code. 4.2 Transition Models The transition models were used to simulate four test cases, three flat plate cases and one hypersonic cone case Flat Plate Test Cases Three widely examined test cases, i.e., the T3AM, T3A, and the T3B [11], are considered. These three cases involve bypass transition on flat plates with three different freestream turbulence intensities and no pressure gradients. The models were incorporated into the FDNS code [9]. Computations were performed using the second order central difference scheme. The grid used for the computations has 265 X 100 points and is shown in Fig. 10. Different near-wall spacings were also employed to study grid sensitivity. A small slip wall was included upstream of the flat plate so that a uniform profile could be used for all of the dependent variables. Isotropic turbulence is assumed, and the freestream turbulence intensity (Tu) is calculated by the following formula: 52

62 (101) Where is the reference velocity, and k is the turbulence kinetic energy. Fig. 10 Computational grid used for T3-series of simulations T3AM test case This test case has a freestream velocity of m/s and a freestream turbulence intensity of 0.9%. Figs. 11, 12, 13, and 14 show the comparisons of skin friction distribution along the flat plate, the non-dimensionalized velocity (U+) profile, and the freestream turbulence intensity distributions, respectively. The U+ profile is plotted at X equal to 1095mm, which is the middle of the transition zone. The skin friction plot also shows the results using the low-re extended k [60] model that was available in FDNS. The freestream Tu decay for this and the two subsequent flat plate cases was obtained along a line that was located at distance of 700 mm from the wall. 53

63 Fig. 11 Skin friction variation for T3AM case From Fig. 11, it can be seen that the Walters-Leylek model predicts transition much earlier than the experimental data, whereas the LCTM predicts the transition at the correct location. In addition, the skin friction distribution of the fully turbulent flow predicted by the implemented SST model agrees very well with the test data, but the low- Re extended model underpredicts the skin friction. 54

64 Fig. 12 U+ versus Y+ for T3AM test case in the transition region for three different grids using the Walters-Leylek model. The U+ versus Y+ profile for the Walters-Leylek model (Fig. 12) shows that the maximum U+ obtained falls short of the experimental value. The early transition predicted by the Walters-Leylek model leads to this large discrepancy in the predicted U+ profile. Again, the result of the LCTM (Fig. 13) is much closer to the experimental data; however, the result of the LCTM using a near-wall grid spacing (Y+) equal to 3 is substantially different from the test data. 55

65 Fig. 13 U+ versus Y+ for T3AM test case in the transition region for three different grids using the LCTM model. From Fig. 14, it can be seen that both models are unable to properly predict the decay of the freestream turbulence intensity, but, the Walters-Leylek model predicts the slope of intensity decay better than the LCTM. 56

66 Fig. 14 Comparison of the experimental freestream turbulence intensity with numerical results for T3AM case T3A test case Compared to the previous test case, this case has a lower freestream velocity (5.4 m/s) and a higher freestream turbulence intensity (3.5%). Figs. 15, 16, 17, and 18 show the comparison of skin friction distribution along the flat plate, the non-dimensionalized velocity (U+) profile, and the freestream turbulence intensity distributions, respectively. The U+ profile is plotted at X equal to 595mm, which is the middle of the transition zone. 57

67 Fig. 15 Skin friction variation for T3A case It can be seen from Fig. 15 that both models accurately predict the transition location. The implemented SST model also accurately predicts the skin friction distribution of the fully turbulent flow. However, in this case too, the low-re model underpredicts the skin friction. 58

68 Fig. 16 U+ versus Y+ for T3A test case in the transition region for three different grids using the Walters-Leylek model. The U+ profile predicted by the Walters-Leylek model (Fig. 16) shows that the maximum U+ again falls short of the experimental value. The U+ profile for the LCTM (Fig. 17) shows that the model over- and underpredicts the maximum value of the U+ depending on the near wall spacing. In addition, it can be seen that the results of LCTM change greatly as Y+ increases. It should be noted that the U+ profiles predicted by the Walters-Leylek model with different near-wall grid spacings vary much less than those predicted by the LCTM. 59

69 Fig. 17 U+ versus Y+ for T3A test case in the transition region for four different grids using the LCTM model. From Fig. 18, it can be seen that both models are unable to accurately predict the proper decay of the freestream turbulence intensity, but, in this case, the LCTM predicts the slope of the decay slightly better than the Walters-Leylek model. 60

70 Fig. 18 Comparison of the experimental freestream turbulence intensity with numerical results for T3A case T3B test case This test case has a freestream velocity of 9.5 m/s and freestream turbulence intensity of 6%. Figs. 19, 20, 21, and 22 show the comparisons of skin friction distribution along the flat plate, the non-dimensionalized velocity (U+) profile, and the freestream turbulence intensity distributions, respectively. The U+ profile is plotted at X = 145mm, which is at the middle of the transition zone. 61

71 Fig. 19 Skin friction variation for T3B case From Fig. 19, it can be seen again that both models predict the correct onset of transition and satisfactorily capture the characteristics of the transition region. The implemented SST model again accurately predicts the skin friction distribution of the fully turbulent flow. The low-re extended model greatly underpredicts the skin friction. 62

72 Fig. 20 U+ versus Y+ for T3B test case in the transition region for three different grids using the Walters-Leylek model. Fig. 20 shows that the maximum U+ predicted by the Walters-Leylek model falls short of the experimental value but is much better in this case compared to the previous two. The LCTM (Fig. 21) again over- and underpredicts the maximum value of the U+, depending on the near-wall grid spacing. In addition, the result of this case also indicates that the LCTM is more sensitive to the near wall grid spacing than the Walters-Leylek model. 63

73 Fig. 21 U+ versus Y+ for T3B test case in the transition region for three different grids using the LCTM model. In the case of the freestream turbulence intensity (Fig. 22), it can be seen that again both models are unable to properly predict the decay of the freestream turbulence intensity. The results are much better than the previous two cases however, and the Walters-Leylek model gives a much better decay of freestream turbulence intensity. 64

74 Fig. 22 Comparison of the experimental freestream turbulence intensity with numerical results for T3B case To give an idea about the grid dependency of the LCTM model, the skin friction distribution for the T3A case, using different near-wall grid spacings, is shown in Fig. 23. It can be seen from Fig. 23 that the transition location moves upstream as the near-wall grid spacing is increased. 65

75 Fig. 23 Skin friction variation for T3A test case for four different grids using the LCTM Hypersonic Cone Test Case The final test case considered for evaluating the transition models was the hypersonic flow over a non-flared 7 o cone [12]. The freestream conditions for this case as follows: Mach number = 7.93, Reynolds number = 6.6E6 m -1, Total temperature= 722K and Wall temperature= K. Fig. 24 shows the computational grid used for this case. The grid had 109 X 75 gridpoints and a near-wall Y+ of

76 Fig. 24 Computational grid for hypersonic flow over a cone The experimental data used for comparison is the Stanton number, which is calculated as follows: (102) Where is the wall heat transfer rate, and are the freestream density and velocity, respectively, is the enthalpy at the wall, and is the freestream stagnation enthalpy upstream of the shock. For this case, the freestream turbulence information is not given and, a freestream intensity of 0.5% was assumed. The Fig. 25 shows the Stanton number variation obtained for the Walters-Leylek model and the low-re extended model. 67