## 1. Introduction

_{t}. However, in strongly separated flows, the actual dependence of the modeled turbulent shear stresses to the mean strain is non-linear. For alleviating this problem, various non-linear corrections have been devised. In general, non-linear models perform better than linear ones. Still, in many practical aerospace configurations, the accuracy of Reynolds Averaged Navier–Stokes (RANS) simulation results is not satisfactory. Higher order schemes, which do not require a turbulence model for closing the equations, have been developed. The present day computing power is sufficient for the application of Large Eddy Simulation (LES) to simple configurations, like those examined in the present study, although the simulated Reynolds numbers are still rather low. Guidance to modeling of LES is provided by the more computing power demanding Direct Numerical Simulation (DNS). Nevertheless, RANS calculations will continue for many years to support the aerospace industry. Even when LES reach the level of application in aerospace components or complete configurations, it will be more economic to apply RANS in an optimization procedure and subsequently to simulate the optimum configuration by LES.

**Figure 1.**Formation of extensive crossflow separation: (

**a**) Delta wing at high-incidence flow. Picture taken by H. Werle (ONERA) in a water-tank; (

**b**) Swept shock/boundary layer interaction in a fin/plate configuration. The quasi-conical separation vortex is visualized by the contours of the eigenvalues of the velocity gradient field [1].

Simulation accuracy is good for mild interactions, marginal for strong ones. For wall heat transfer rate, the deviation of the calculated results from experiment ranges from 40% to 150%. Calculations predict “more turbulent” flows, compared to experiments.

**Figure 2.**Structure of the separated flow in a swept-shock/turbulent boundary-layer interaction. Flow conditions: M

_{∞}= 4.0, α = 16° [5].

## 2. Description of Codes and Turbulence Models

_{t}. For the calculation of the turbulent viscosity coefficient, various turbulence models have been developed, in which the Reynolds stresses and other terms of turbulent fluctuation parameters are related to mean values of the flow: ${\overline{u}}_{i},\overline{T},\overline{\rho}$. The Boussinesq equation is,

_{ij}is the strain rate tensor given by,

_{t}, is related to mean and turbulent quantities by,

_{ij}is the rotation rate tensor,

_{1}is obtained by a solution of a cubic equation. The non-linear relation of Rumsey and Gatski [11] is coupled with the regular k–ω equations of Wilcox [10].

_{max}is the maximum value of the moment of vorticity:

_{max}is the value of η at which F(η), Equation (12), is maximum. D is the van Driest damping factor (see [12]).

_{dif}is the difference between maximum and minimum velocity in the velocity profile. The thickness of the boundary layer is defined by: δ = η

_{max}/C

_{kleb}. The constants appearing in the previous relations are: C

_{wk}= 0.25, C

_{Kleb}= 0.3.

_{t}is smaller than the value predicted by the regular equations of the Baldwin–Lomax model. The developed model was tested both in high-angle of attack flows and shock wave turbulent boundary layer interactions (Panaras [7,13]). The agreement with the experimental evidence was very good. In its initial form, the Panaras modeling is difficult to use, since it requires a preliminary run, so that the user will define “reference conditions” for the wake function of the Baldwin–Lomax model. Also, the prediction of the line of separation for the application of the new relation is required. In this paper, a new version is presented, in which the Baldwin–Lomax model keeps its original formulation, but one of the basic equations for the estimation of the turbulent viscosity coefficient is changed, to follow the structure of turbulent separated flows. More particularly, the η

_{max}in Equation (10) is replaced by a reference value, η

_{ref}. The value of η

_{ref}differs from flow to flow and it is based on existing semi-empirical relations that define the boundary layer parameters of an equivalent flat plate flow of the same Reynolds number.

_{ref}along a flat-plate flow, the semi-empirical analysis of Falkner [14] is used, which is valid for a Reynolds number between 10

^{5}and 10

^{10}. According to Falkner [14], the boundary layer growth along a flat plate is,

_{max}/C

_{Kleb}, then,

_{ref}at a characteristic separation length of each examined configuration, L

_{sep}, the final calculation scheme is,

_{ref}has been non-dimensionalized by the length of the body, L.

_{sep}= L. But if there is extensive crossflow separation, as in the case of slender bodies at incidence, more accurate results are obtained if the characteristic length is equal to a crossflow length (L

_{sep}= d, for axisymmetric bodies). This assumption is reasonable, since in high-alpha flows, the flowfield is dominated by the separated crossflow.

## 3. Results

#### 3.1. Swept-Shock Boundary Layer Interactions

**Figure 4.**Wall pressure and skin friction variation: (

**a**,

**b**) SF4 test case and (

**c**,

**d**) SF6 test case [13].

#### 3.1.1. Fin/Plate Configuration

_{∞}= 4.0, α = 16°, Re

_{δo}= 2.1 × 10

^{5}. The test case SF6 differs in the fin angle (α) relative to the freestream (stronger interaction): M

_{∞}= 4.0, α = 20°, Re

_{δo}= 2.1 × 10

^{5}. These two test cases were simulated by various contributors using the baseline Baldwin–Lomax, Panaras [7] B–L modification, k–ε and Spalart–Allmaras–Edwards turbulence models. The surface pressure along an arc across the flowfield is shown in Figure 4a,c. According to Knight and Degrez [4]:

The skin friction coefficient is displayed in Figure 4b,d.The Baldwin–Lomax–Panaras and Spalart–Allmaras–Edwards models are the most accurate. Both models predict the surface pressure in the plateau region within 5% to 10%. Also, both models display a pressure trough at the region of appearance of secondary separation, in agreement with the experiment. Both models overestimate the peak pressure in the vicinity of the corner. The predictions of the Baldwin–Lomax and k–ε models exhibit the general trends of the experiment but are less accurate.

It is reminded that the Spalart–Allmaras turbulence model solves a modeled transport equation for the kinematic eddy viscosity. The model was designed specifically for aerospace applications involving wall-bounded flows and has been shown to give good results for boundary layers subjected to adverse pressure gradients. Edwards and Chandra [15] developed a form, which improves the near-wall numerical behavior of the model (i.e., the goal was to improve the convergence behavior).In the stronger interaction (Figure 4d) the Baldwin–Lomax–Panaras and Spalart–Allmaras–Edwards models predict a peak in the vicinity of the corner which is not evident in the experiment; in particular, their computed values at the experimental location β = 26.5° are substantially above the experiment. Additional measurements in the region 22° < β < 26° would be helpful in determining whether a peak appears. Elsewhere, all four models provide generally good agreement with the experiment.

_{∞}= 4, α = 30.6°, Re

_{δo}= 1.6 × 10

^{5}) resulted in a remarkably accurate simulation. Herein, we present in Figure 5 the results of Thivet et al. [6], as they have been given by Zheltovodov [17], who is co-author of the basic paper and provided the experimental data. As seen in Figure 5b, the nonlinear model (WI-DU, line 4) shows a significant improvement relative to the linear k–ω model (line 3) and standard Baldwin–Lomax model (B–L, line 2) and correctly predicts wall pressure distribution as well as secondary separation phenomena in a vicinity of the local normal shock wave in the supersonic turbulent cross near-wall “reversed” flow (Figure 5a). Also, in contrast to other standard turbulence models (B–L, k–ω, k–ε), the weakly nonlinear model of Thivet et al. [6] predicts well the wall skin friction in the interaction region (see [6]).

**Figure 5.**Single fin flow at α = 30.6°, M

_{∞}= 4, Re

_{δo}= 1.6 × 10

^{5}: (

**a**) flowfield schematic in a cross section and (

**b**) wall pressure distribution. Reprinted [17] by permission from the author.

_{∞}= 5, α = 30.6°, Re

_{o}= 37.0 × 10

^{6}/m). The performed LES has demonstrated very good agreement with available experimental data in terms of the mean flowfield structure, the surface pressure, as well as the surface flow pattern. However, significant under-prediction in the surface skin-friction peak in the vicinity of the attachment line was observed. According to Fang et al. [8], the reasons of disagreement will be investigated. The wall pressure distribution is shown in Figure 6. The good agreement with the experimental data is evident. In Section 4, flowfield data from this LES study will be presented, which support and enforce the observation, done earlier by Panaras [5], for the laminarization of the initially turbulent flow by the induction mechanism of the separation vortex.

**Figure 6.**LES of fin/plate configuration. Comparison of wall pressure distribution. Reprinted [8] by permission from the authors.

#### 3.1.2. Crossing Shock Interaction

_{∞}= 5.0 is examined. The two fins are symmetric (Figure 7). The stagnation conditions were P

_{0}= 2.2 MPa and T

_{0}= 427 K, resulting in a freestream unit Reynolds number of 36.5 × 10

^{6}/m. The incoming boundary layer was an equilibrium turbulent boundary layer with an isothermal wall (T

_{w}= 295 K). The boundary and momentum thicknesses were δ

_{0}= 3.8 mm and θ = 0.157 mm at a distance of 6δ

_{0}upstream of the fin leading edge. A Navier–Stokes solver based on an upwind total variation diminishing (TVD) scheme and developed by A. Panaras and B. Mueller at DLR is used. The baseline and the modified B–L models are tested. For the application of the turbulence model, the existence of two intersecting surfaces dictates the use of a “modified distance”, originally proposed by Hung and MacCormack [19] to account for the turbulent mixing length near the intersection of the surfaces. The DLR upwind code incorporates this feature. For this reason this code is used, instead of ISAAC.

_{0}upstream of the leading edge of the fins. The height of the computational field is 26.8δ

_{0}. For the simulation of the undisturbed boundary layer, 45 points are used.

_{wall}/T

_{adiabatic}= 4.15.

**Figure 10.**Wall pressure prediction of Schmisseur and Gaitonde [20].

#### 3.2. Supersonic Ogive Cylinder

_{L}= 1.44 × 10

^{6}and incidence α = 10° to 20°. The measurements included surface-pressure distributions at various cross-sections. For natural transition, the authors used acenaphtene coating visualization to verify that the boundary layer remained laminar over the entire body at α = 0 degree. Transition was triggered by means of a 5 mm wide carborundum strip located at x/d = 1.0, where d is the diameter of the body.

_{d}= 1.6 × 10

^{5}). These results are also shown in Figure 13 and indicate that this prediction is much closer to the data, than the regular results shown in Figure 12. As regards the turbulence structure of the two solutions, the distribution of turbulent kinetic energy (TKE) is shown in Figure 14 at the cross-sections x/d = 4.95 and 8.5, for the calculations employing the EASM model. In this figure, it is evident that in the solution based on Re

_{d}, the flow is practically laminar within the separation bubble.

**Figure 12.**Ogive-cylinder: comparison of wall pressure coefficient at various cross sections. Reynolds number based on body length.

**Figure 13.**Ogive-cylinder: comparison of wall pressure coefficient at various cross sections. Reynolds number based on diameter.

**Figure 14.**EASM calculations: Turbulent kinetic energy (TKE) distribution in two cross-sections. Calculations using Reynolds number based on length or diameter of body. Inset: experiment, flow visualization by total pressure at cross-section x/d = 4.95 [21].

## 4. Further Evidence of Laminarization of the Separated Flow in 3-D SBLIs

_{∞}= 3.0, α = 10°. In Figure 15a, it is seen that qualitatively this flow develops as described in Section 1. A primary vortex is formed, around which a part of the separated boundary layer folds and penetrates into the separation region. However, the vortex is very weak (its spiral core is small), and the layer folded around originates from the inner part of the boundary layer and not from the outer one. Hence, there is no formation of a low-turbulence tongue. Observation of the surface skin-friction lines (not included here) has indicated that no secondary vortex appears. In the test case shown in Figure 15b (M

_{∞}= 4.0, α = 16°), the y/δ

_{0}= 0.7 stream surface envelops the secondary vortex and a low turbulence tongue is formed. Also, the stream surface y/δ

_{0}= 1.0 folds around the vortex. At the other extreme is the flow examined in Figure 15c. This interaction is generated in a M

_{∞}= 4.0, α = 20° fin/plate flow, and the stream surface y/δ

_{0}= 1.0 penetrates deeply into the separation region and it is, actually, a part of the secondary vortex. Furthermore, the layers of the tongue adjacent to the plate (under the primary vortex) are practically laminar, because they are composed of air that originates outside the boundary layer (between y/δ

_{0}= 1.0 and 1.23).

**Figure 15.**Cross-sections of various flows: (

**a**) M

_{∞}= 3.0, α = 10°; (

**b**) M

_{∞}= 4.0, α = 16°; and (

**c**) M

_{∞}= 4.0, α = 20° [5].

_{∞}= 5, α = 30.6°) are compared in Figure 16 with the milder interaction results (M

_{∞}= 4.0, α = 16°) examined by Panaras [5]. Fang et al. [8] observe that the streamlines originating from the near-wall region (Figure 16a, y/δ

_{0}= 0.53) fold around the separation vortex core. With the increase of the origin position (Figure 16b, y/δ

_{0}= 1.05), the streamlines may directly enter the reversed flow, rather than through the vortex core. Equivalent conditions from the simulation of Panaras [5] are shown in Figure 16c,d. The data of Fang et al. [8], shown in figure 16, support the observation of Panaras [5] that in strong 3-D SBLIs the upper, less turbulent, part of the interacting boundary layer folds around the separation vortex and forms the reversed flow. Furthermore, Fang et al. [8] present the turbulent kinetic energy (TKE) distribution on a spherical section of the studied flowfield, close to the exit plane. Their data are reproduced here in Figure 17. In this figure, it is clear that the air, which folds around the separation vortex and forms its lower part near the wall, has very low turbulent kinetic energy, exactly as the analysis of Panaras [5] predicts. Amplification of turbulence is observed only in the near wall region downstream of the secondary separation (S

_{2}), caused by the adverse pressure gradient existing in the region of S

_{2}. The accurate LES calculations of Fang et al. [8] prove the hypothesis of laminarization of an originally turbulent flow in a swept shock/turbulent boundary layer interaction. We hope that Fang et al. [8] will include the distribution of TKE in more spherical sections in a future publication, so that the gradual laminarization along the separation vortex will be shown.

**Figure 17.**Turbulent kinetic energy (TKE) on the section at R = 226.3 mm, normalized with the square of the wall friction velocity at the inlet. Reprinted [8] with permission from the authors.

Such improvement is associated with significant reduction in the peak of turbulent kinetic energy in the flow which penetrates from an external part of the shear layer to the wall in the place of formation of the secondary separation line (Figure 18b) in contrast to the calculations with a standard k–ω model (WI, Figure 18a), which are characterized by high turbulence level in the near wall “reverse” flow”.

**Figure 18.**Turbulent kinetic energy in cross section (x

_{1}= 122.5 mm): (

**a**) The linear Wilcox’s k–ω

**-**model (WI); (

**b**) Weakly nonlinear k–ω

**-**model (WI–DU). Reprinted [17] by permission from the authors.

## 5. Conclusions

## Acknowledgments

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