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

## Key Takeaways

- Widely recognized in turbulence modeling, the k-ω (M-SST) model uses a two-equation framework with additional transport equations to capture fluids' turbulence characteristics, considering historical effects like convection and diffusion.
- The k-ω (M-SST) model excels in adverse pressure gradient and separating flow but tends to overestimate turbulence in areas with high normal strain.
- The k-ω formulation in the inner boundary layer allows the k-ω (M-SST) model to be used directly on the wall. It functions as a low Reynolds number turbulence model without extra damping functions.

Turbulence is a complex and chaotic state of fluid flow characterized by irregular fluctuations in velocity, pressure, and other properties. Unlike laminar flow, where fluid moves in smooth and orderly layers, turbulent flow is marked by eddies, vortices, and rapid variations over various scales. This chaotic nature of turbulence makes it inherently difficult to predict and model, but it is essential for understanding many natural and engineering systems, from weather patterns to aircraft design.

Turbulent flow is typically analyzed using statistical methods and turbulence models because the direct simulation of all the scales of turbulence (known as Direct Numerical Simulation or DNS) is computationally infeasible for most practical applications. Turbulence models, such as the Reynolds-Averaged Navier-Stokes (RANS) equations, approximate turbulence's effects by averaging the Navier-Stokes equations and solving for the mean flow properties.

The k-ω (M-SST) model is a widely used turbulence model designed to predict turbulent flows more accurately. It combines the strengths of the k-omega model in the near-wall region with the k-epsilon model in the far field. This combination makes the SST model particularly effective for dealing with complex boundary layers, adverse pressure gradients, and separating flows, which are common in many engineering applications.

## Exploring the k-ω (M-SST) Turbulence Model

Before the introduction of the k-ω M-SST model, the Wilcox k-ω turbulence model existed, which is a prominent two-equation model due to its superior accuracy and robustness near the solid wall. However, it falls short of accurately predicting the asymptotic behavior of turbulence eddy viscosity. Another limitation of this model is its inability to accurately replicate the DNS profiles of turbulent kinetic energy k and dissipation rate ɛ in the boundary layer. The most significant drawback of this model is its high sensitivity to the small free stream value of ω in the free shear layer, and adverse pressure gradient boundary layer flows. To address this issue, Menter (1994) proposed a blended model that incorporates the Wilcox k-ω model near the solid wall and the standard k-ɛ model in a k-ω formulation in the free stream.

To blend the k-ω and k-ɛ models, the latter is transformed into the k-ω formulation. The differences between this formulation and the original k-ω model are the inclusion of an additional cross-diffusion term in the ω equation and different modeling constants. The original k-ω model is then multiplied by a function F1 and the transformed k-ε model by the function 1-F1, and the corresponding equations of each model are added together. The function F1 is designed to be a value of one in the half inner part of the boundary layer (where the model behaves like the original model) and decreases to vanish far from the wall.

## Steps for Effective Implementation of the k-ω (M-SST) Turbulence Model

In this model, the turbulent kinematic viscosity is defined as a function of the turbulent kinetic energy, k, and the specific dissipation rate of the turbulent frequency, ω, as follows:

The two transport model equations for the k and ω scalar turbulence scales are defined below.

Where

Notice that this is a modification recently brought to the model by Menter et al. (2003), who replaced vorticity with the strain rate in the eddy-viscosity expression. The limiter on the eddy viscosity was added for robustness issues, but it can be removed.

The standard coefficients for the k and ω by Menter (1994):

The constant for the k and ω by Menter (2003):

All other constant values are the same as Menter (1994) described.

## Extended Wall Function for k-ω (M-SST) Model

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

Where , and is the VonKarman constant (default value 0.41)

with

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

## Estimating Turbulent Kinetic Energy and Turbulent Dissipation

If no (experimental) data is available for the turbulence intensity or the wall shear stress, the inlet boundary conditions for turbulence can be estimated in the following way.

First, an estimation of the ratio of the turbulent viscosity to the laminar viscosity is needed:

For internal flows (such as turbomachinery flow), typical values are

For external flows (in aerodynamics computations), typical values are

The value of the turbulent kinetic energy, k, is defined as:

Where U is the streamwise velocity, L_ref is the characteristic length of the domain, ν the kinematic viscosity of the fluid, and C_μ is 0.09.

The value of ɛ, the turbulent dissipation, is:

## Key Features of K-Omega M-SST Model:

- Two-Equation Model: It solves two separate transport equations, one for the turbulent kinetic energy (k) and one for the specific dissipation rate (ω).
- Inner Boundary Layer Handling: The k-ω formulation is used in the inner part of the boundary layer, allowing the model to be applied directly to the wall through the viscous sublayer.
- Outer Layer Performance: In the outer part of the boundary layer and free shear flows, the SST model behaves like the K-ɛ model, ensuring better performance for various flow conditions.
- Low Reynolds Number Capability: The model can be used as a low Reynolds number turbulence model without requiring additional damping functions, making it versatile and robust.

By capturing near-wall and far-field turbulent effects, the K-ω (M-SST) model offers improved accuracy and reliability in simulating turbulent flows, especially in scenarios involving complex boundary layer interactions and strong adverse pressure gradients. This makes it a preferred choice for many computational fluid dynamics applications, including aerospace, automotive, and turbomachinery design.

## References

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

Menter, F. R., Kuntz, M., and Langtry, R., "Ten Years of Industrial Experience with the SST Turbulence Model," Turbulence, Heat and Mass Transfer 4, ed: K. Hanjalic, Y. Nagano, and M. Tummers, Begell House, Inc., 2003, pp. 625 - 632.