# Advancing Turbulence Insight: The Power of Non-Linear SBSL-EARSM Model

## Key Takeaways

- The Explicit Algebraic Reynolds Stress Model (EARSM) simplifies complex physical processes from Reynolds Stress Models into a more manageable two-equation framework.
- Non-linear eddy viscosity models (EVMs), with their quadratic terms defined as a function of strain and vorticity, are able to overcome the limitations of linear models. This allows them to accurately predict the high Reynolds stress anisotropy that results from complex flow physics.
- Non-linear EVMs have received significant interest because they offer the potential to return better predictions than linear EVMs but with only a moderate increase in required computing resources.

## Key Techniques for Implementing the S-BSL-EARSM Turbulence Model

EARSM builds on the standard two-equation turbulent models. Derived from the Reynolds Stress Transport Equation, EARSM establishes a non-linear relationship between the Reynolds stresses, the mean strain rate, and the vorticity tensor. Consequently, EARSM is much less computationally demanding than the Reynolds stress model (RSM) while capturing important turbulence features such as anisotropy in normal stresses and secondary flows, which linear eddy viscosity turbulence models cannot.

The EARSM model implemented in Fidelity Open is the simplified baseline explicit algebraic Reynolds stress model (S-BSL-EARSM) proposed by Menter et al. (2009), which is based on the BSL k-ω model of Menter (1994) and allows the inclusion of anisotropic effects into the turbulence model.

In the S-BSL-EARSM model, the Reynolds-stress tensor is expressed using an effective eddy-viscosity formulation, including a corrective extra-anisotropy tensor :

where the effective turbulence eddy viscosity is defined as: , and the extra anisotropy terms are defined as:

All the terms and coefficients of the effective turbulence eddy viscosity and the extra anisotropy tensor are defined as follows:

The turbulence time scale,

The equations for the turbulence model are as follows:

where the production term is: and the turbulent kinematic viscosity in the equations is defined as:

The constants , denoted with the generic symbol are defined as:

, with the coefficients:

Inner model constants:

Outer model constants :

The auxiliary blending function , designed to blend the model coefficients of the original model in boundary layer zones with the transformed model in free-shear layer and free-stream zones, is defined as:

## Wall Functions for S-BSL-EARSM

The value of in the first inner cell is obtained from the tabulated values already used for the model. in the first inner cell can be imposed based on the value of the wall friction velocity .

In the intermediate region, is approximated with the form of interpolation between the viscous and log layer values:

**References**

Menter F., 1994, "Two-equation eddy viscosity turbulence models for engineering applications", AIAA Journal, vol. 32:1299-1310.

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.

Assumptions and Insights of k-epsilon Low Re Yang-Shih Turbulence Model

Theoretical Foundations of k-ω Menter-Shear Stress Transport Turbulence Model