Exploring Non-Linear Consecutive Relations in Turbulence: SSC-EARSM

Key Takeaways

• The separation-sensitive corrected explicit algebraic Reynolds stress model (SSC-EARSM) is designed to better predict separated flows compared to the Simplified Baseline Explicit Algebraic Reynolds Stress Model (S-BSL-EARSM).
• The SSC-EARSM introduces three key updates: (1) an EARSM upgrade to handle near-wall anisotropy, (2) Shear Stress Enhancement (SSE) to increase turbulent mixing in shear layers, and (3) Scale-Adaptive Simulation (SAS) to boost further mixing in separated flow regions.

Key Techniques for Implementing the SSC-EARSM Turbulence Model

The SSC-EARSM is a modification of the S-BSL-EARSM derived by Menter et al. [2009]. The following three upgrades have been introduced to the SSC-EARSM. The first correction enhances the near-wall behavior of the anisotropy, which is related to the EARSM. The other two corrections, SSE and SAS, have been applied to increase the turbulent mixing in the shear layers between the separated flow and the free stream.

Low Reynolds Correction

Skote et al. [2016] proposed a modification to the EARSM to account for low Reynolds number effects using a Van Driest-type damping function. This function, which is active only near the wall, is similar to eddy viscosity damping. Unlike traditional damping functions based on  , this approach uses , which has been shown to be more suitable for separated flows, as demonstrated by Skote et al. [2016]. The damping function is defined as:

Where,

The of the S-BSL_EARSM are modified as follows:

A-posteriori have shown that eddy viscosity damping was insufficient to obtain the correct turbulence energy peak for planar flows. Although the production of turbulence energy is accurately predicted, the dissipation rate in the buffer region (i.e., region of maximum turbulence intensity) is too large. This leads to the excessive dissipation of turbulence energy. Therefore, reducing the dissipation rate in the buffer region is necessary for accurate turbulence modeling. To achieve this, the dissipation rate in the k-equation is damped using the damping function proposed by Abe et al. [2003]. This function is based on the ratio between the turbulent and laminar viscosity and is active only in the vicinity of the wall. This damping function reads:

Where, 

The k-equation is modified as follows:

The dissipation rate in the very near-wall region is inaccurate. However, the weak turbulence and dominant viscous diffusion balance the unrealistic dissipation, so this limitation does not significantly impact the overall flow.

Shear Stress Enhancement

The second correction to the S-BSL-EARSM is a modification of the turbulence energy production to enhance the shear stress in the separated region of the shear layer. In a specific flow region, the turbulence energy production in the k-equation is increased and is modified as follows:

Where  is the SSE function.

When the flow separates, the streamwise velocity exhibits an inflexion point. Mathematically, it is defined as the point where the second derivative of the streamwise velocity goes to zero. Thus, this point is the location of the maximum shear region of the flow, i.e., the separated bubble shear layer. The Von Karman length scale is an excellent means for identifying the enhancement region. It is defined as:

The Von Karman length scale approaches infinity near the inflection point. Thus, the ratio of the integral length scale, , to the Von Karman length scale goes to zero near the wall around the inflexion point. It is then possible to formulate an  , such that the function has a unitary value around the inflexion point and quickly approaches zero elsewhere. To avoid the correction affecting the near-wall region, the function is multiplied by the damping function defined previously for the low-Reynolds number correction.This function is deactivated in the boundary layer using the  function of Spalart [2006] to minimize the impact of the S-BSL-EARSM when computing attached boundary layers. 

The  function is used in hybrid models where it has a unitary value in LES regions and is zero elsewhere. Upon conducting further research, was defined as follows:

The 1/10 constant in the  function controls the extent of the correction. Furthermore, the function depends on the local Reynolds number. For low Reynolds numbers, the amount of the correction should be increased. The coefficient  in the  function is also a function of the local Reynolds number. The constant values for high and low local Reynolds numbers are blended using  . Note that  is a damping function similar to , but to a greater degree. The values of this constant result from the best compromise obtained in predicting various low and high Reynolds test cases.

Sensitization to the Separation Correction Term

The SAS term of Maduta et al. [2015] . It consists of a recalibrated version of Menter's SAS formulation for the Jakirlíc and Hanjalíc [2002] Reynold Stress Model. It is given by:

As for the SSE term, the value of   is blended for high and low Reynolds numbers. The coefficient's value is derived from the optimal balance achieved across different low and high Reynolds test scenarios. In practical applications, the SAS term activates in the narrow separation zone,  increasing the production of the specific dissipation rate, which leads to an increase in turbulent viscosity. This rise in eddy viscosity eases the separation.

References

Menter F., Garbaruk A.V., Egorov Y., 2009, "Explicit algebraic Reynolds stress models for anisotropic wall-bounded flows," EUCASS ‐ 3rd European Conference for Aero-Space Sciences, Versailles.

Skote M., and Wallin., 2016, "Near-wall damping in model predictions of separated flows," Int. J. Comput. Fluid Dyn., vol. 30:218-230.

Abe K., Jang Y.-J., and Leschziner M. A.,2003, "An investigation of wall-anisotropy expressions and length-scale equations for non-linear eddy-viscosity models," Int. J. Heat Fluid Flow, vol. 24:181‐198.

Spalart P. R., Deck S., Shur M. L., Squires K. D., Strelets M. K., and Travin A., 2006, "A New Version of Detached-eddy Simulation, Resistant to Ambiguous Grid Densities," Theor. Comput. Fluid Dyn., vol. 20:181‐195.

Maduta R. and Jakirlic S., 2015, "Extending the bounds of steady RANS closures: Toward an instability-sensitive Reynolds stress model," Int. J. Heat Fluid Flow, vol. 51:175‐194.

Jakirlíc S. and Hanjalíc K., 2003, "A new approach to modeling near-wall turbulence energy and stress dissipation," J. Fluid Mech., vol. 459:139‐166.

Monté S., Temmerman L., Tartinville B., Léonard B., and Hirsch C., 2019, “Towards a Separation Sensitive Explicit Algebraic Reynolds Stress RANS Model,” ERCOFTAC Bulletin 121, P-61.