# Numerical Simulation of Supersonic Turbulent Separated Flows Based on k–ω Turbulence Models with Different Compressibility Corrections

## Abstract

**:**

## 1. Introduction

_{T}, will have difficulty in accurately estimating the effects of compressibility and thus is not suitable for simulations of supersonic and hypersonic turbulent flows. Sarkar [3] and Zeman [4] have devised very elegant models of compressibility corrections for the simulation of compressible mixing layers. Their work is very innovative and useful, although their models work best only for strained homogeneous flows. Based on previous work, Wilcox [5,6] postulated a similar model, and it shows encouraging applicability to the simulation of wall-bounded flows. Suzen and Hoffmann [7] made compressibility corrections for Menter’s turbulence model and obtained improved predictions for supersonic jet exhaust flows. It should be noted that a certain compressibility correction method that has been proposed has been proven to be effective for a certain class of flows, but its universality for complex compressible flows still needs further verification and confirmation.

## 2. Numerical Methods

^{2}norm of the residual of all the variables (mean flow and turbulence equation) drops by nine orders of magnitude.

_{i}, U

_{j}, and U

_{k}are the flow velocities. As for the BSL turbulence model [18,19], the k-equation is formally identical to the original k–ω model. The difference is that a cross-diffusion term is added in the ω-equation, and it is written as [18,19]

_{1}is the blending function designed by Menter, and its specific expression is given by [18,19]

_{1}k, with a

_{1}being a model constant. In the SST model, the transport equations of k and ω are exactly the same as for the BSL model, and the eddy viscosity, μ

_{t}, is determined by the following equation [18,19]

_{2}is another blending function, and its specific expression is given by [18,19]

_{1}and F

_{2}are equal to one inside the boundary layer and switch over to zero away from the wall surface. The model constants for the above three different versions of the k–ω model can be found in the literature [17,18,19].

_{∞}is the free-stream speed of sound. From the above equation, it follows that the free-stream turbulent viscosity ratio, μ

_{t}/μ

_{∞}= 0.009. The boundary condition for k and ω at a solid surface is

_{1}= 0.075, Δy

_{1}is the distance to the next grid point away from the wall, and ν

_{1}is the fluid kinematic viscosity at the grid point location.

#### 2.1. Rapid Compression Fix

#### 2.2. Wilcox’s Compressibility Correction

_{T0}= 0.25, and $\mathcal{H}$ (·) is the Heaviside function. It can be seen from Equation (11) that, with an increase in the turbulence Mach number, the destruction term of the k-equation increases, while the destruction term of the ω-equation decreases. Overall, the modified turbulence model can obtain smaller k and larger ω values compared to the original model, thus reducing the turbulent viscosity. The introduction of M

_{T0}is to ensure that the compressibility correction is only valid for the compressible free shear layer and does not work in the near-wall boundary layer, where the turbulence Mach number is relatively low. For turbulent boundary layers, when the incoming Mach number is greater than five or the flow compressibility effect is strong, the turbulence Mach number within the boundary layer can easily exceed a value of 0.25, which makes the wall skin coefficient and heat flux predicted by the modified model significantly low. Brown [25] introduced the F

_{1}function designed by Menter to protect the boundary layer near the wall to avoid the adverse effects of the correction, and the specific expression is written as follows [25]

#### 2.3. Suzen and Hoffmann’s Compressibility Correction

_{1}= 1.0, α

_{2}= 0.4, and α

_{3}= 0.2, are determined [7] based on DNS results.

## 3. Results and Discussion

#### 3.1. Supersonic Ramped Cavity Flow

^{−4}cm, which ensures y

^{+}≈ 1 for the wall turbulence simulation. A mesh refinement study using three distinct mesh resolutions (coarse, medium, and fine) is also conducted based on the original SST turbulence model, and the comparison results are presented in Figure 3. As the number of grid points increases, the present CFD results hardly change. For RANS calculations, the baseline grid is already fine enough, and continuing to refine the mesh has a minimal-to-negligible effect on the computational results.

_{T}) function, the low-speed flow regions close to the wall are still subject to the effect of the compressibility correction. For both corrections (Wilcox’s correction and Suzen and Hoffmann’s correction), the action area and intensity of the compressibility corrections can be illustrated by the contours of functions F(M

_{T})(1 − F

_{1}) and (M

_{T})

^{2}(1 − F

_{1}), respectively. With the introduction of the blending function, F

_{1}, Wilcox’s compressibility correction is limited to high-speed flow regions, such as the separated shear layer and the subsequent reattached boundary layer far away from the wall, as shown in Figure 10a. In the case of Suzen and Hoffmann’s compressibility correction, the correction area is roughly the same as that for Wilcox’s correction, but the range is slightly enlarged and the correction intensity is slightly increased, as shown in Figure 10b. As for Suzen and Hoffmann’s correction, the pressure dilatation term also acts almost in the same regions as the dilatation dissipation term. In the action regions, the pressure dilatation term is negative, which, like the effect of the dilatation dissipation term, reduces the turbulent kinetic energy, k, and increases the value of ω, thus reducing the turbulent viscosity of the model. As for the rapid compression fix (not used in the SST model), the velocity divergence, ∇·

**V**, can be used to indicate the action area, as shown in Figure 10c. In Equation (10), ∇·

**V**< 0 or ∇·

**V**> 0 will increase or decrease the production term in the ω-equation of RANS models. In the compression region (near the oblique shock wave), the value of the velocity divergence is less than zero, which directly leads to an increase in w. From Equation (4), the increase in w is an important reason for the decrease in turbulent viscosity. In the expansion region (near the step), the value of the velocity divergence is larger than zero, which directly leads to a decrease in w. Nevertheless, the expansion is weak and has a very limited effect on w. Although these three compressibility corrections are very different, they cause the same effect on the original RANS models, that is, an overall reduction in the turbulent viscosity of the model. The original RANS models always produce too high levels of turbulent viscosity in the non-equilibrium region after separation and overestimate the initial spreading rate of the free shear layer. An adaptive reduction in the turbulent viscosity through compressibility corrections slows down this spreading rate. The overall lower levels of turbulent viscosity in the correction “C2”, when compared to the correction “C1”, are responsible for the better predictions.

#### 3.2. Supersonic Compression Corner Flow

^{−5}δ, resulting in y

^{+}< 1 upstream of both the corners. A mesh refinement study using three distinct mesh resolutions (coarse, medium, and fine) is also performed based on the SST model, and the refinement has little effect on the numerical results, as shown in Figure 13.

_{1}, Wilcox’s correction and Suzen and Hoffmann’s correction, which are based on the turbulence Mach number, M

_{T}, hardly work for the RANS models. Instead, the rapid compression fix plays a major role in the corrections. At the 24° corner angle, it can be seen from Figure 14b that the turbulence Mach number is higher than the threshold, M

_{T}

_{0}= 0.25, after flowing through the shock wave. The maximum turbulence Mach number is about 0.45 in the recirculation region at the corner. Even near the wall region, the turbulence Mach number has a large value. Under the shielding effect of the blending function F

_{1}, the action region of dilatation dissipation and pressure dilatation in the compressibility corrections is constrained away from the wall, which significantly minimizes the detrimental effect of corrections on the prediction of low-speed flows near the wall.

**V**, is used to indicate the action area of the rapid compression fix, as shown in Figure 15. As mentioned previously, this compressibility correction is near the oblique shock wave and will increase the separation bubble size predicated by the original k–ω or BSL model. The contours of the functions F(M

_{T})(1 − F

_{1}) and (M

_{T})

^{2}(1 − F

_{1}) are used to illustrate the action area and the intensity of both Wilcox’s correction and Suzen and Hoffmann’s correction, respectively. For the 24° compression corner flow, a local function value of F(M

_{T})(1 − F

_{1}) and (M

_{T})

^{2}(1 − F

_{1}) greater than zero indicates that the compressibility correction plays a role in this region, and the larger the function value, the more pronounced the correction. As can be seen in Figure 16, the local function values are overall less than 0.08 with Wilcox’s correction, whereas with Suzen and Hoffmann’s correction, there exists a large region of local function values of about 0.1. Suzen and Hoffmann’s correction acts on a larger region than Wilcox’s correction, and the correction is more intense.

_{1}, may need to be recalibrated for this type of supersonic flow. Compressibility corrections to the SST model instead give worse results at a large corner angle, whereas corrections to the BSL model lead to improved predictions, which have also been confirmed by Forsythe et al. [15]. On the other hand, when the shock-unsteadiness modification is applied to the k–ε, k–ω, and Spalart–Allmaras turbulence models, improved predictions can also be obtained, as shown in the numerical comparisons performed by Sinha et al. [16]. However, the shock-unsteadiness modification is not based on the turbulence Mach number. This modification is not as elegant as Wilcox’s correction and Suzen and Hoffmann’s correction, both of which are based on the turbulence Mach number. Tu et al. [22] introduced Catris’ modification in the SST model equations and obtained improved predictions for the 34° compression corner flow at a Mach number of 9.22. Catris’ modification brings benefits for hypersonic compression corner flows, whereas for supersonic compression corner flows with Mach numbers less than 3, this modification may not have much effect on the computational results.

## 4. Conclusions

_{1}can protect the flow in the near-wall region from compressibility corrections, its role is after all limited. Thus, it is recommended that a more suitable near-wall shielding function be introduced in the corrections. Compared with the correction “C2”, the correction “C1” tends to be conservative and at least does not have significant negative effects. However, for the prediction of compressible free shear layers, the correction “C2” is recommended.

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Plots of surface pressure (

**left**) and skin friction coefficient (

**right**) for the ramped cavity.

**Figure 5.**Velocity profiles in shear layer for turbulence models with and without compressibility considerations.

**Figure 6.**Turbulent kinetic energy distribution for RANS models with and without compressibility corrections.

**Figure 7.**Specific dissipation rate distribution for RANS models with and without compressibility corrections.

**Figure 8.**Turbulent viscosity distribution for RANS models with and without compressibility corrections.

**Figure 10.**Comparison of the area where a compressibility correction works for the ramped cavity flow: (

**a**) Wilcox’s correction; (

**b**) Suzen and Hoffmann’s correction; (

**c**) rapid compression fix.

**Figure 14.**Distribution of turbulence Mach number for the compression corner flows: (

**a**) 16°; (

**b**) 24°.

**Figure 16.**Comparison of the area where a compressibility consideration works for the 24° compression corner flow: (

**a**) Wilcox’s correction; (

**b**) Suzen and Hoffmann’s correction.

**Figure 17.**Plots of surface pressure (

**left**) and skin friction coefficient (

**right**) for the 16° supersonic compression corner: (

**a**) original k-ω models with or without corrections; (

**b**) BSL models with or without corrections; (

**c**) SST models with or without corrections.

**Figure 18.**Plots of surface pressure (

**left**) and skin friction coefficient (

**right**) for the 24° supersonic compression corner: (

**a**) original k-ω models with or without corrections; (

**b**) BSL models with or without corrections; (

**c**) SST models with or without corrections.

M_{∞} | Re/m | δ, cm | P_{∞}, Pa | T_{∞}, K |
---|---|---|---|---|

2.92 | 6.7 × 10^{7} | 0.29 | 21,240 | 95.37 |

Princeton Ramps | M_{∞} | P_{0}, Pa | T_{0}, K | δ, mm | Re_{∞}/m | Re_{δ} |
---|---|---|---|---|---|---|

16° | 2.85 | 6.9 × 10^{5} | 268 | 26 | 6.56 × 10^{7} | 1.71 × 10^{6} |

24° | 2.84 | 6.9 × 10^{5} | 262 | 23 | 6.83 × 10^{7} | 1.57 × 10^{6} |

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**MDPI and ACS Style**

Luo, D.
Numerical Simulation of Supersonic Turbulent Separated Flows Based on *k–ω* Turbulence Models with Different Compressibility Corrections. *Aerospace* **2023**, *10*, 1014.
https://doi.org/10.3390/aerospace10121014

**AMA Style**

Luo D.
Numerical Simulation of Supersonic Turbulent Separated Flows Based on *k–ω* Turbulence Models with Different Compressibility Corrections. *Aerospace*. 2023; 10(12):1014.
https://doi.org/10.3390/aerospace10121014

**Chicago/Turabian Style**

Luo, Dahai.
2023. "Numerical Simulation of Supersonic Turbulent Separated Flows Based on *k–ω* Turbulence Models with Different Compressibility Corrections" *Aerospace* 10, no. 12: 1014.
https://doi.org/10.3390/aerospace10121014