## 1. Introduction

Turbulence modeling is one of the most challenging problems in numerical flow simulation. It is common experience that different turbulence models result in different predictions, when applied to a particular complex flow. It is doubtful whether a universally valid turbulence model, capable of describing all complex flows, could be devised. Most of the turbulence models are based on the Boussinesq hypothesis, according to which the apparent turbulent shear stresses are related linearly to the rate of mean strain through an apparent scalar turbulent or “eddy” viscosity coefficient, μ_{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.

Object of the present study is the turbulence modeling of flows with extensive crossflow separation,

i.e., three-dimensional flows in which the separated boundary layer rolls up into longitudinal vortices. This type of separation appears in many practical aerospace configurations; for example, subsonic/supersonic flow about slender bodies or delta wings at high incidence or in components of supersonic/hypersonic air vehicles when swept shock waves interact with boundary layers (

Figure 1).

**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].

**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].

In the early 1950s, H. Werle of ONERA did pioneer work in visualizing high-angle of attack flows, like the delta wing shown in

Figure 1a, by using a water tunnel and injecting colors from small holes at the expected regions of generation of the separation longitudinal vortices. The appearance of separation quasi-conical vortices of flattened shape in swept shock/boundary layer interactions, like that shown in

Figure 1b for a fin/plate configuration, was hypothesized in the 1970s, but it was proved much later. Indeed, this early period oil-flow visualization revealed the existence on the surface of the plate, below the separation bubble, of a separation and an attachment line, which away from the apex of the configuration are straight and intersect upstream of the apex and close to it. The trace of the inviscid shock also passes through this intersection. It has been proposed that the separation bubble is actually a conical flat vortex. More than 20 years of experimental and computational research were required for proving these early hypotheses (see [

2,

3] for details).

Returning to turbulence modeling, it is mentioned that according to published evidence, the accuracy of prediction of flows with extensive crossflow separation is marginal. Particularly for shock wave/turbulent boundary layer interactions (SBLIs), Knight and Degrez [

4] summarizing results of comparisons of several contributions (organized by Advisory Group for Aerospace Research and Development (AGARD)), using the RANS equations with a wide range of turbulence model from zero equations to full Reynolds Stress Equation formulations, state:

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.

To explain this condition, the present author [

5] studied the effect of the longitudinal separation vortices on the turbulence of the flow. It is reminded that a characteristic feature of vortices is their strong swirling motion, allowing them to promote large-scale mixing of fluids with possibly different momentum and energy. Panaras [

5] hypothesized that the longitudinal vortices generated in the types of flows shown in

Figure 1, transfer external inviscid air into the lower turbulent part of the separated flow, decreasing its turbulence. To prove his hypothesis, Panaras [

5] studied numerically the structure of the separation vortex, which appears in a strong swept shock wave/turbulent boundary layer interaction generated in a fin/plate configuration. He studied the flowfield by means of stream surfaces starting at the inflow plane within the undisturbed boundary layer, which is initially parallel to the plate. Each of these surfaces was represented by a number of streamlines. Calculation of the spatial evolution of selected stream surfaces revealed that the inner layers of the undisturbed boundary layer, where the eddy viscosity is high, wind around the core of the separation vortex. However, the outer layers, which have low turbulence, rotate over the separation vortex and penetrate into the separation bubble at the attachment region, forming a low-turbulence tongue, which lies along the plate, underneath the vortex (

Figure 2). The intermittency of the fluid that constitutes the tongue is very small, that is, the flow is almost laminar there. At the initial stage of development, the conical separation vortex is completely composed of turbulent fluid, but as it grows linearly downstream the low-turbulence tongue is formed.

The analysis of Panaras [

5] leads to the conclusion that laminarization of the initially turbulent flow appears in case of extensive crossflow separation. Hence, the Boussinesq equation is not adequate for the estimation of the Reynolds stresses. With reference to

Figure 2, it is mentioned that although the mean strain is very high in the near wall reversed flow, the flow there is almost laminar. Evidently, application of Boussinesq’s equation in this region predicts higher Reynolds stresses than the actual ones. Related examples will be given in

Section 3.1.

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

M_{∞} = 4.0,

α = 16° [

5].

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

M_{∞} = 4.0,

α = 16° [

5].

The discovery of the laminarization of the initially turbulent flow, in flows with extensive crossflow separation led to the development of new ideas in turbulence modeling for shock wave/turbulent boundary layer interactions (SBLIs). These new ideas include realizability (weak nonlinearity) and specific physical models, which incorporate known flowfield behavior [

6,

7]. In a short period the simulation accuracy of SBLIs was improved. This is clearly evident if the content of the critical survey of Computational Fluid Dynamics (CFD) prediction capabilities for SBLIs prepared by Knight and Degrez [

4] and of the similar review of Knight

et al. [

3], which was published only five years later, are compared. In spite of the spectacular improvement shown by the new models, even presently, the predicted skin friction and heat transfer are marginally accurate in simulations of flows with extensive crossflow separation. The present review has been prepared in order to stimulate further research, now that LES methods enter the field. Published and new data will be presented for 3-D SBLIs and high-alpha flows. New evidence will be given regarding the almost-laminar nature of the near wall reverse flow in these types of flow, including data from the first publication of fin/plate simulation by using LES [

8].

## 2. Description of Codes and Turbulence Models

The CFD code ISAAC, developed by Morrison [

9], is used in this study. ISAAC is a second-order, upwind, finite-volume method where advection terms in the mean and turbulence equations are solved by using Roe’s approximate Riemann solver coupled with the MUSCL scheme. Viscous terms are calculated with a central difference approximation. Mean and turbulence equations are solved coupled by using an implicit spatially split, diagonalized approximate factorization solver. ISAAC has been developed to test a large range of turbulence models. Algebraic models, various

k–

ε and

k–

ω formulations, and Reynolds stress transport models are included in ISAAC.

In the present article, the examined flows are calculated with the

k–ω turbulence model of Wilcox [

10], the Explicit Algebraic Reynolds Stress Model (EASM) of Rumsey and Gatski [

11] and a modification of the algebraic turbulence model of Baldwin and Lomax [

12] done by the present author, in order to improve its accuracy in separated flows.

It is known that the functional form of the Navier–Stokes equations is the same for laminar and turbulent flows. In the latter case, however, time average has been applied to equations, since in turbulent flows, each flow or thermodynamic parameter has a mean value and a random turbulent fluctuation (for example:

${u}_{i}={\overline{u}}_{i}+{u}_{i}^{\prime}$). The time averaged flow equations are known as Reynolds Averaged Navier–Stokes (RANS) equations. In them, new apparent shear stresses, known as Reynolds stresses:

${\tau}_{ij}=(\overline{\rho {u}_{i}^{\prime}{u}_{j}^{\prime}})$, have appeared, which need to be calculated. The calculation of the Reynolds stresses is not easy; transport equation for each of them must be solved, increasing the total CPU cost. Alternatively, the Boussinesq hypothesis is applied, according to which the apparent turbulent shear stresses are related to the rate of mean strain, through an apparent scalar turbulent or “eddy” viscosity coefficient,

μ_{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,

where

k is the turbulent kinetic energy and

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

The

k–ω turbulence model of Wilcox [

10] is a linear eddy viscosity model, which includes Equation (1) in its formulation. The eddy viscosity,

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

where

ω is the specific dissipation rate.

The turbulent kinetic energy and specific dissipation rate are calculated by solving transport equations similar to those that express the flow parameters (see [

10]).

The explicit algebraic stress model (EASM) of Rumsey and Gatski [

11] replaces the linear Boussinesq approximation with a non-linear relationship of the strain rate and rotation rate tensors,

where

W_{ij} is the rotation rate tensor,

The eddy viscosity is given by,

and

α_{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].

The Baldwin–Lomax [

12] model is a two-layer algebraic zero-equation model. The eddy viscosity coefficient is given by algebraic equations. In the inner layer it follows the Prandlt–Van Driest formulation:

where

Ω is the absolute value of the vorticity and

η is the distance normal to the wall.

In the outer layer the following equations are used:

The quantity

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

The parameter

η_{max} is the value of

η at which

F(

η), Equation (12), is maximum.

D is the van Driest damping factor (see [

12]).

The quantity u_{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.

The present author [

7] considered the discovered laminarization of the flow for the derivation of a new equation for the calculation of the eddy-viscosity coefficient in the region of a separation vortex. He followed the Baldwin–Lomax turbulence model. According to his equation, in regions of high transverse velocity gradients the predicted value of

μ_{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.

For the estimation of

η_{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,

If this equation is combined with the relation:

δ =

η_{max}/

C_{Kleb}, then,

Defining

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

L_{sep}, the final calculation scheme is,

where

η_{ref} has been non-dimensionalized by the length of the body,

L.

Then Equation (10) is replaced by,

For aerospace configurations we propose the equality: 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.

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

To return to the theoretical analysis, we note that in the basic article of Panaras [

5], the effect of the strength of the interaction on the laminarization of the separation domain has been addressed. By studying various test cases of fin/plate flows, Panaras [

5] found that increase in the strength of the interaction results in the folding of higher inviscid layers around the vortex. In the other extreme of a weak interaction, no low-turbulence tongue is formed. Crossflow cuts of three studied flows are shown in

Figure 15. The cuts correspond to station (iii) of

Figure 1b and they have been arranged according to the interaction strength. The first cut belongs to a weak interaction,

M_{∞} = 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].

**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].

The spatial evolution of a selected number of stream surfaces originating within the upstream boundary layer and initially parallel to the plate, has also been examined by Fang

et al. [

8] in their very recent LES calculations. Results from their simulated very strong interaction (

M_{∞} = 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 16.**
Development of stream surfaces originating at the inflow plate, at constant distance from the flat plate: (

**a**,

**b**) Data of Fang

et al. [

8]; (

**c**,

**d**) Data of Panaras [

5].

**Figure 16.**
Development of stream surfaces originating at the inflow plate, at constant distance from the flat plate: (

**a**,

**b**) Data of Fang

et al. [

8]; (

**c**,

**d**) Data of Panaras [

5].

**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.

**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.

Finally, indirect evidence of laminarization of the separated flow in 3-D SBLIs is given by Zheltovodov [

17], who commenting on the success of the weakly nonlinear model of Thivet

et al. [

6] to simulate accurately this class of flows, mentions that:

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.

**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

The reasons for the difficulty in simulating accurately with classical turbulence models strong 3-D SBLIs and high-alpha flows, which are characterized by the appearance of strong crossflow separation, are investigated. In view of recent additional evidence, an earlier flow analysis [

5], which attributes the poor performance of classical turbulence models to the observed laminarization of the separation domain, is examined. According to this analysis, the longitudinal vortices into which the separated boundary layer rolls up, in this type of flow, transfer external inviscid air into the adjacent to the wall part of the separation, decreasing its turbulence. Increase in the strength of the interaction results in the folding of higher inviscid layers around the vortex. In the other extreme of a weak interaction, there is no laminarization. Strong evidence of laminarization of the flow in a very strong fin/plate SBLI is given in a recent LES simulation, by means of the predicted small value of the turbulent kinetic energy within the reverse flow in cross sections of the flow.

Existing linear turbulence models for closing the RANS equations relate the apparent turbulent shear stresses linearly to the rate of mean strain (Boussinesq’s equation). However, in the types of flows examined in the present article, although the mean strain is very high in the near wall reversed flow, the flow there is almost laminar. Evidently, application of Boussinesq’s equation in this region predicts higher Reynolds stresses than the actual ones. Published and new CFD results are presented which indicate that the accuracy of linear turbulence models is moderate in strong 3-D SBLIs. On the contrary, two-equation non-linear models and algebraic models that incorporate the previously described flowfield behavior, provide accurate results, regarding the flow structure, the wall pressure and skin friction distribution.

Since the high-alpha flow around wings and slender bodies is also characterized by the appearance of longitudinal separation vortices, the supersonic flow around an ogive cylinder at high-incidence has been simulated by employing the k–ω turbulence model, an Explicit Algebraic Reynolds Stress Model (EASM) and a modified Baldwin–Lomax model. All these turbulence models provided results of accepted accuracy, with the EASM model demonstrating better performance. Subsequently, the turbulent flow calculations were repeated with the arbitrary assumption that the Reynolds number in these new simulations was based on the diameter, not on the length of the body (scaling of the Reynolds number by the length of the body). Then, the predicted wall pressure distribution approached the data closer. Simultaneously, the turbulent kinetic energy in the reversed flow (EASM model) was reduced at very small levels. Of course, we do not suggest the arbitrary scaling of the Reynolds number for better accuracy. Our numerical test aimed at demonstrating that turbulent flow simulations which result in almost laminar flow conditions within the reversed flow of the separation domain, provide accurate results. A task of future efforts would be the development of turbulence models that predict flows with characteristics similar to those described in this article.