Three-Dimensional Modeling Shock-Wave Interaction with a Fin at Mach 5

The three-dimensional single-fin configuration finds application in an intake geometry where the cowl-shock wave interacts with the side-wall boundary layer. Accurate numerical simulation of such three-dimensional shock/turbulent boundary-layer interaction flows, which are characterized by the appearance of strong crossflow separation, is a challenging task. Reynolds-averaged Navier–Stokes computations using the shock-unsteadiness modified Spalart–Allmaras model is carried out at Mach of 5 at large fin angle of 23∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$23^{\circ }$$\end{document}. The computed results using the modified model are compared to the standard Spalart–Allmaras model and validated against the experimental data. The focus of the work is to implement the modified model and to study the flow physics in detail in the complex region of swept-shock-wave turbulent boundary-layer interaction in terms of the shock structure, expansion fan, shear layer and the surface streamlines. The flow structure is correlated with the wall pressure and skin friction in detail. It is observed that the standard model predicts an initial pressure location downstream of the experiments. The modified model reduces the eddy viscosity at the shock and predicts close to the experiments. Overall, the surface pressure using modified model has predicted accurately at all the locations. The skin friction is under-predicted by both the models in the reattachment region and is attributed to the poor performance of turbulence models due to flow laminarization.


Introduction
The most fundamental three-dimensional shock/boundarylayer interaction is generated on a single-fin configuration. It consists of a flat plate with a sharp fin mounted perpendicular to it. The oblique shock wave generated by the fin interacts with the turbulent boundary layer on the plate and results in flow separation. The three-dimensional vortical flow thus generated alters the inviscid flow pattern. Additional shock waves, expansion regions and free shear layers are generated that result in a complex flow in the interaction region. Practical applications of single-fin configuration include scramjet inlets, where the oblique shock generated by the cowl interacts with the side wall boundary layer [1].
The single-fin shock boundary-layer interaction flows are characterized by localized regions of increased pressure, skin friction and heat transfer rate. Prediction of these surface properties is important in the design of scramjet inlets. In addition, the flow distortion caused by the interaction can degrade the performance of these inlets significantly. Therefore, the aerodynamic loads generated from these interactions play a significant role in the structural integrity of hypersonic vehicle [2,3]. Computational fluid dynamic approach is a useful tool to understand the complex three-dimensional flow pattern in these shock-wave/boundary-layer interactions and to predict its influence on the wall data. The direct numerical simulation and large eddy simulation demand a large number of grid points at high Reynolds number flows leading to large computational resources and computational times to capture the fine features of shock-wave/turbulent boundary-layer interaction cases [4][5][6][7]. As an engineering approach, Reynolds-averaged Navier-Stokes (RANS) method is applied in the present work along with oneequation turbulence models to compute these flowfields.
Experiments and computations were carried by many authors [8][9][10][11][12][13] for single-fin geometry. Kubota et al. [8] based on the experimental study, proposed a flow model of for single-fin geometry with deflection angle θ = 15 • and M ∞ = 2.3. They showed that for this weak shock/boundarylayer interaction, a strong and stiff vortex is formed attached to a fin and a weak stretched vortex is formed above it, attached to the flat plate. As the interaction is weak, no lambda shock was observed in this case. Alvi et al. [9] performed experiments for single fin with deflection angle θ = 16 • and M ∞ = 2.95. The proposed flowfield model depicted an inviscid shock which bifurcated to form a separation shock and a rear shock, representing a lambda structure in a cross plane perpendicular to the flow direction. The separated conical vortical flow region is formed underneath the lambda shock structure. A slip line is generated from the triple point of intersection of three shock waves, and a set of compression and expansion waves are formed between the shear layer and the slip line. Panaras [11,13] computed using RANS code for deflection angle of θ = 20 • and M ∞ = 3.0. The wall pressure data were improved using modified Baldwin-Lomax turbulence model, compared to the standard Baldwin-Lomax model. Edwards et al. [10] studied the performance of four different one-equation turbulence models at M ∞ = 8 and θ = 15 • . Among them, the standard Spalart-Allmaras model has shown to predict the surface properties close to the experiments. This model predicted flow separation downstream of the experiments and under-predicted the skin friction in the reattachment region. Thivet [12] computed three different single-fin configuration cases with M ∞ = 3, θ = 15 • , M ∞ = 4, θ = 20 • and M ∞ = 4, θ = 30.6 • , using the two-equation standard and modified k−ω turbulence models. The prediction of secondary vortex region was improved using modified k−ω model, hence improving the wall data in this region. The review articles [14][15][16][17][18] discusses different single-fin configurations flow physics and wall data in detail. The review articles discuss the experimental study for singlefin configurations with deflection angles of 8 • , 12 • , 15 • , 16 • , 18 • , 20 • , 24 • , 31 • and Mach numbers in the vicinity of 3, 4, 5 and 8. The Reynolds number based on the boundary-layer thickness upstream of the interaction region lies in the vicinity of 2 × 10 5 for most of the cases. Surface oil flow patterns, wall pressure and skin friction measurements have been studied. These configurations have been computed using RANS method using algebraic, one-and two-equation turbulence models for predicting the surface pressure and skin friction. The computations for 20 • at M ∞ = 4 showed that the k−ω model showed a delayed separation and higher values of wall pressure and C f in the plateau of conical recirculation region [18]. The standard k− and Spalart-Allmaras models show similar trend to k−ω model predictions [14,16]. Similar trend in wall data was observed by these turbulence models for the weaker interaction case of 15 • at Mach 3 [17]. The computations using RANS method for single-fin flows with an angle of α = 20 • and free stream Mach number of 3.0 predicted some of the features of crossflow vortices and shock waves [15]. The three-dimensional contour plots were plotted using eigen values of the velocity gradient field. The flow consists of the vortical structure with the elliptical cross section in the core of this cone. This vorticity sheet lifts up at separation line and forms the conical vortex.
Several modifications have been proposed in the literature [19], namely compressibility correction, realizability constraint, rapid-distortion correction and length-scale correction. The implementation of pressure gradient across shock waves to the Spalart-Allmaras model, modified wake function to the Baldwin-Lomax model and limitation of production of turbulent kinetic energy to k− model is also used to improve the wall pressure in shock/boundary-layer interaction flows [20][21][22]. Their performance varied from one test case to another.
The shock-unsteadiness modified Spalart-Allmaras model of Sinha et al. [23] has shown potential in improving separation bubble prediction in two-dimensional and axisymmetric flows [23,24]. The application of shock-unsteadiness modification by Sinha et al. [23] to Spalart-Allmaras model was limited to simple compression corner geometry at supersonic Mach number of 2.8 at deflection angle 24 • . The standard Spalart-Allmaras model predicted lower values of eddy vis-cosity and resulted in initial pressure location upstream of experiments. Therefore, a larger separation bubble size is predicted; consequently, the wall pressure and skin friction are not predicted accurately as compared to the experiments. The shock-unsteadiness modified Spalart-Allmaras model corrects the amplification of eddy viscosity across the shock. The higher values of eddy viscosity predicted by the modified model push the initial pressure location close to the experiments and predict accurate separation bubble size. Both the standard and modified Spalart-Allmaras models under-predicted the skin friction coefficient in the reattachment region as compared to the experiments. The modified Spalart-Allmaras model showed higher values of skin friction as compared to the standard Spalart-Allmaras model. Later, Pasha et al. [24] applied the shock-unsteadiness modified Spalart-Allmaras model to axisymmetric coneflare geometry at significantly higher Mach numbers of 11-13 with two flare angles, 36 • and 42 • . The higher pressure rise across flare-shock results in largely separated flows. In contrast, to simulate separated flows at supersonic Mach numbers, it is comparatively difficult to simulate such flows at hypersonic Mach flows. The computed results were compared to the experimental data. The standard Spalart-Allmaras turbulence model suppressed flow separation at the corner and therefore did not reproduce the correct shock structure in the interaction region due to a high level of turbulence predicted at the shock wave. By comparison, the shock-unsteadiness modified Spalart-Allmaras model dampens the turbulence amplification at the shock. It thereby reproduced the separation bubble size observed experimentally and matched the wall pressure measurements well. The shock-unsteadiness model also predicted the flow topology in the interaction region correctly. The modification had a negligible effect on weak interactions, where the standard Spalart-Allmaras model performed reasonably well.
In this paper, an attempt is made to extend the shockunsteadiness modified Spalart-Allmaras to three-dimensional single-fin configuration flows and study the flowfield in detail. First, the test case is described, which is followed by the simulation methodology. Next, the computed flowfield using modified Spalart-Allmaras model is explained for the strongest shock strength case with fin deflection angle of 23 • and M ∞ = 5. Next, the computed wall pressure and skin friction using modified model [23] and the standard model [25] are compared with the experimental results [26].

Test Case
The schematic of the single-fin configuration used in the experiments of Schulein [26] is shown in Fig. 1 Three-dimensional single-fin configuration with a fin mounted on the flat plate. The surface measurements [26] were taken along the dashed lines K and stagnation pressure P 0 = 2.12 MPa were taken in the reservoir of the Ludweig tube experimental facility. The free stream temperature and pressure corresponding to their stagnation values are T ∞ = 68.3 K and p ∞ = 4008.5 N/m 2 . A fin of height = 100 mm normal to the flat plate is taken. The fin tip is placed at a distance of 286 mm downstream of the flat plate edge. Both the flat plate and fin wall are maintained under the isothermal condition of 300 K. The flow is turbulent, upstream of the interaction region with a unit Reynolds number of Re 1∞ = 37 × 10 6 m −1 . Wall data like pressure, skin friction and heat transfer rates were measured at different cross sections in the shock-wave/turbulent boundary-layer interaction region (see Fig. 1). The undisturbed turbulent boundary-layer properties were measured on the flat plate at a distance of 20 mm upstream of the fin tip. The boundary-layer thickness δ of 3.8 mm, displacement thickness = 1.6 mm, momentum thickness = 0.16 mm and skin friction C f = 1.35 × 10 −3 is measured.

Simulation Methodology
The three-dimensional Reynolds-averaged Navier-Stokes equations [27] are used in the numerical simulations. The turbulence model equations are fully coupled to the mean flow equations. The shock-unsteadiness modified Spalart-Allmaras model of Sinha et al. [23] and its standard version [25] are used for calculating the eddy viscosity. A compressible correction to standard Spalart-Allmaras model has been proposed by Catris et al. [28] by modifying the diffusion laws in the turbulence model, but this strategy complicates the numerical implementation for three-dimensional flows [29]. Therefore, the compressibility correction is not used in the present work. The governing equations are discretized in a finite-volume formulation where the inviscid fluxes are computed using a modified low-dissipation form of the Steger-Warming flux-splitting approach. The method is second-order accurate both in stream-wise and wall-normal directions. The viscous fluxes and the turbulent source terms are evaluated using central difference method. More details of the numerical method are given in Ref. [30]. The code is capable of running on parallel machines and has been used successfully in several supersonic and hypersonic applications [23,24,31,32].
The shock-unsteadiness modified Spalart-Allmaras model of Sinha et al. [23] accounts for the effect of unsteady shock motion in a steady mean flow. The shock-unsteadiness correction is achieved by adding a source term of the form −c b 1ρν S ii in the transport equation forρν, whereρ is mean density,ν is modified turbulent kinematic viscosity, S ii is mean dilatation and shock-unsteadiness parameter Note that this additional term is effective only in regions of strong dilatation and therefore does not alter the standard Spalart-Allmaras model elsewhere. Here, b 1 is model parameter and c 1 = 1.25+0.2(M 1n −1). The model parameter b 1 represents the coupling between the unsteady shock motion and the upstream velocity fluctuations, and is given by, It is zero for subsonic flows without shock waves, and it reaches an asymptotic value of 0.4 for high Mach numbers. The detailed formulation of shock-unsteadiness modified Spalart-Allmaras model and its implementation for twodimensional compression corner and axisymmetric coneflare flows at supersonic and hypersonic Mach numbers is presented in Refs. [23,24]. In the current work, the modification is applied to the three-dimensional flows which are more difficult to simulate as compared to their counterpart two-dimensional flows. The computational domain and boundary conditions for three-dimensional single-fin configuration are identified in A single-block grid is generated using a code, and a careful grid refinement study is performed by systematically varying the number of grid points in each direction, as well as refining the cell size in critical regions. The origin is taken at a tip of the single-fin, and the grid is stretched exponentially in the upstream and downstream directions of origin with initial grid size of 1 × 10 −4 m. A structured mesh with exponential stretching normal to the plate and fin walls are used to span the computational domain. Wall pressure and skin friction coefficient are found to be sensitive to the computational mesh and are used to identify a grid converged solution. First, the number of points in the wall parallel direction is refined and it is observed that among the three grids, the 100 × 110 × 110, 140 × 110 × 110, and 200 × 110 × 110, the last two grids match. Next, 140 × 110 × 110 grid is taken and only the distance of the first cell center from the wall is successively reduced by halve-times from its initial value of 2 × 10 −6 m. Wall pressure and C f variation indicate that 1 × 10 −6 m is sufficient for an accurate solution. The z + 2 varies between 0.06 in the undisturbed boundary layer to a maximum value of z + 2 < 0.6 in the shock/boundary-layer interaction region. This value of the z + 2 is sufficient to capture the high gradients of the mean and turbulent variables in the boundary layer in the wall-normal direction, especially at the reattachment region. In the next step, using 140 × 110 × 110 grid, the number of points in the transverse direction is increased. It is observed that variation in wall pressure and skin friction for 140 × 160 × 110 and 140 × 200 × 110 is less

Separation shock
Flat-plate than 3%. Finally, the number of points in the flat plate normal direction is increased to arrive at the 140 × 160 × 160 grid. Surface properties variation shows that further refinement to 140 × 160 × 200 grid points yields less than 2% variation in wall pressure and skin friction. A converged grid of 140 × 160 × 160 is obtained with 140 points along the stream-wise direction, 160 points transverse to the flow and 160 points normal to the plate. In the present computations, a CFL of 0.05 is used at the beginning and it is gradually increased to 10 in the first 3500 iterations. It is further increased to 100 at 6000 iterations and to 1000 at 8000 iterations. A maximum CFL of 5000 is used after 10,000 iterations. The corresponding time steps vary from 2 × 10 −12 s for the initial iterations to 2 × 10 −5 s at steady state solution. It takes 190 cpu-hours to reach the steady state solution in 30,000 iterations. The time step was calculated using the relationship τ = CFL z min /(u + a), where z min is the minimum cell size and, u and a refer to the local velocity and speed of sound [33].

Flow Physics
In this section, the computed results using modified Spalart-Allmaras model are discussed. An isometric view of the flow solution for the fin is shown in Fig. 3. The pressure contours are taken at two x-sections of 92 and 182 mm to identify the shock structure. The flow pattern is indicated by the streamlines taken in the cell adjacent to the flat plate and behave similarly to the skin friction lines. A planar shock wave generated by the fin interacts with the turbulent boundary-layer on the plate. It separates the boundary-layer and results in the formation of the separa- tion shock. The flow separates at the primary separation line S1 and attaches at the primary reattachment line R1 near the fin wall. The streamlines converge at separation line S1, and the fluid moves upwards normal to the plate and then turns in a counter clockwise direction to form a helical flow as shown in Fig. 4. Two stream surfaces originating at z/δ 0 = 0.8 and z/δ 0 = 0.25 are shown, where z is the normal distance from the plate and δ 0 is the undisturbed boundary-layer thickness. The fluid impinges on the plate at the reattachment line from the top, making the streamlines diverge in either direction. The inviscid shock wave in Fig. 4 bifurcates into a lambda structure and encloses the vortex region.
The computed stagnation pressure contours in Fig. 5 indicate that an inviscid shock bifurcates and forms a sep-  aration shock and a rear shock, and results in the formation of lambda structure in a cross plane perpendicular to the freestream flow direction. The shear layer is formed over the vortex region, which emanates from the separation point. It interacts with the rear shock wave and then rolls up and turns back to form a tongue. An entropy layer is generated from the triple point of the intersection of three shock waves and a set of compression, and expansion waves are formed between the shear layer and the slip line (see Fig. 7). A secondary flow separation region was observed in the experiments [26], whereas in the present computation, it is not predicted. The computed pressure contours in Fig. 6 indicate that an expansion fan is generated when the rear shock wave reflects from the shear layer. The computed Mach contours in Fig. 7 shows that a type-IV shock-shock interaction [34] is observed at the triple point. An alternate increase and decrease in Mach contours in the region between shear layer and jet represents the weak compression and expansion waves. The flow remains supersonic in the separated region. Small subsonic pockets are observed   Fig. 6. A small fin-vortex is formed when the fluid coming from the inviscid region interacts with fin wall and turns in the clockwise direction, as viewed from the downstream direction. Similar, corner-vortex was observed in the vapor screen images of the experiments [8]. The schematic sketches of these flowfield features are explained in detail in Refs. [9,18]. Figures 8 and 9 show the computed pressure and speed contours, overlapped with the wall pressure and skin friction on the flat plate at x-section = 122 mm. The distance along the y-axis is normalized with the corresponding xsection distance measured from the fin tip. The surface data are taken along the dashed lines as shown in Fig. 1. The wall pressure remains constant in the undisturbed boundarylayer before the interaction region. The separation shock affects the upstream flow at the point of influence U, and the wall pressure rises effectively across the separation shock at primary separation point S1. It remains constant in the separated vortex flow and rises to peak values at primary reattachment R1. The wall pressure then decreases away from the reattachment region and rises near the finplate junction. The skin friction does not vary significantly in the unperturbed boundary layer before the region of influence U. The boundary layer is compressed across the separation shock wave, and hence it increases the velocity gradient and thereby increases skin friction. The skin fric-  [26] using standard Spalart-Allmaras model [25] and modified Spalart-Allmaras model [23] tion reaches a peak value at the reattachment point R1 due to the high values of the velocity gradient and hence shear stress. Figure 10 indicates that at x-section = 152 mm, experiments give an initial pressure rise location at y/x = 1.22, where as the standard Spalart-Allmaras model predicts a delayed separation at y/x = 1.14. A similar trend is observed by the standard model at all other x-sections as shown in Table 1. A vortex region is calculated based on the distance between S1 and R1. The standard Spalart-Allmaras model predicts a small vortex region of 74 mm as compared to the experimental value of 82 mm. There is a difference of 11% in the vortex region between the Spalart-Allmaras model and the experimental value. The skin friction coefficient predicted by the standard model in Fig. 11 matches close to the experiments, except in the secondary flow separation region. The model also under-predicts the peak values in the reattachment region.

Comparison of Computed Wall Data with Experiments
The shock-unsteadiness parameter c b 1 as described in Eq. 1 is a function of the upstream normal Mach number M 1n and needs to be evaluated at each shock wave so as to implement the shock-unsteadiness correction. For two-dimensional compression corner flows [23] and axisymmetric cone-flare flows [24], an average value of M 1n was calculated and based on it c b 1 was calculated and implemented. This is the drawback and limitation of the shock-unsteadiness Spalart-Allmaras model. The three-dimensional shock structure in the single-fin configuration is quite complex. It is not easy to find the orientation of the different shock waves and the inclination of the upstream flow at each shock. Therefore, it is a difficult task to calculate the mean value of M 1n . An alternate approach is to use different values of c b 1 in the current single-fin case to improve the flow predictions. A higher magnitude of c b 1 yields a larger separation, and c b 1 = − 0.2 is found to match the experimental separation location closely and therefore is used in the present simulations. A similar approach by Gaitonde et al. [22] was used for simulating three-dimensional double-fin shock/boundary-layer interaction. The turbulence model constants were varied by limiting the value of production term in standard k-turbulence model. A smaller value of model constant resulted in lower turbulent kinetic energy. Hence, the computed solution resulted in a larger separated region and matched well with the experimental flowfield and wall pressure.
In the present case, a shock-unsteadiness parameter of c b 1 = − 0.2 is chosen to yield a larger separation and is found to match the experimental initial pressure rise location closely as indicated in Fig. 10. The shock-unsteadiness correction reduces the turbulent eddy viscosity in the region of the separation shock. This causes an increase in the length of the separation shock and hence brings the separation point predictions close to the experiments. The same trend is observed in the axisymmetric flows over cone-flare cases at hypersonic Mach numbers in Ref. [24]. Figures 12 and 13 indicate that the modification predicts lower values of μ T /μ ∞ in the region of y/x 1.2, as compared to the standard Spalart-Allmaras model. Hence, the modified model improves the initial pressure rise location S1 (see Fig. 10) to around 6%. Also, the pressure distribution μ T /μ ∞ Fig. 13 Computed normalized eddy-viscosity contours at x-section = 122 mm using shock-unsteadiness modified Spalart-Allmaras model [23] is well predicted in the reattachment region and the corner region of the plate fin junction by the modified model as compared to the standard model. Figure 11 depicts that the modified model over predicts the skin friction between y/x = 0.65 and 0.72, whereas it underpredicts the skin friction by 42% at the reattachment region R1. Panaras [21] attributed this under-prediction of skin friction due to the poor performance of turbulence models. The models indicated lower values of turbulence inside the separation vortices, making the flow almost laminar in this region. Also, it is the incapability of one-equation model to capture the peak velocity gradients in the reattachment region, though you put more number of grid points in this region to capture it. Therefore, both the models underestimate the skin friction than the experimental results. Similar observation was made in the numerical simulations of Edwards et al. [10] where the standard Spalart-Allmaras model under-predicted the skin friction in the reattachment region for flow over single-fin configuration with Mach 8 and θ = 15 • . More advanced two-equation turbulence models [35][36][37] can be applied to capture the velocity gradients and hence predict the peak skin friction at the reattachment region accurately. Currently, this is beyond the scope of work. The modified Spalart-Allmaras model predicts vortex region length between S1 and R1 to be 79 mm with a difference of 5% to the experimental value. The computed locations of primary separation point S1 and reattachment point R1 are compared with the experimental data at different x-sections in Table 1. Overall, the modified Spalart-Allmaras model matches to the experimental data at all locations.

Conclusion
Reynolds-averaged Navier-Stokes based computations were carried out to investigate the three-dimensional shock/boundarylayer interaction in a single-fin configuration at Mach 5 with a large deflection angle of 23 • . The shock-unsteadiness modified Spalart-Allmaras model and its standard version are used in the computations. The inviscid shock generated by the fin interacts with the boundary layer on the adjacent flat plate and results in a formation of a complex region. The viscous effects cause the bifurcation of inviscid shock and result in the formation of a lambda shock structure, one edge being the separation shock and the other being the rear shock. Type-IV shock-shock interaction results from the interaction of these shock waves. The lambda shock encloses a cross flow coni-cal vortex. A shear layer emanates from the separation point and interacts with the rear shock wave and then rolls up and turns back to form a tongue. An entropy layer is generated at the triple point, and a set of compression and expansion waves are embedded in it and the shear layer. These flow features influence the wall pressure and skin friction and a correlation between them is explained. The standard Spalart-Allmaras model predicts initial pressure location downstream of experiments. The shock-unsteadiness correction leads to an improvement in prediction of the initial pressure rise location. This leads to an accurate prediction of vortex size and hence the shock structure. The skin friction is under-predicted at reattachment region in comparison with experiments by both the modified model and its standard version. This behavior is attributed to the poor performance of these models due to the prediction of laminar flow in this region. Further improvements to the present computations are envisaged by simulating the more advanced two-equation turbulence models.