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K-Omega and K-Omega SST

SimScale allows different methods to model the turbulent effects appearing in a CFD simulation. This document sheds light on the popular k-omega and industry standard k-omega SST turbulence model.

Overview

The k-omega (\(k-\omega\)) turbulence model is one of the most commonly used models to capture the effect of turbulent flow conditions. It belongs to the Reynolds-averaged Navier-Stokes (RANS) family of turbulence models where all the effects of turbulence are modeled.

It is a two-equation model. That means in addition to the conservation equations, it solves two transport equations (PDEs), which account for the history effects like convection and diffusion of turbulent energy. The two transported variables are turbulent kinetic energy (\(k\)), which determines the energy in turbulence, and specific turbulent dissipation rate (\(\omega\)), which determines the rate of dissipation per unit turbulent kinetic energy. \(\omega\) is also referred to as the scale of turbulence.

The standard \(k-\omega\) model is a low \(Re\) model, i.e., it can be used for flows with low Reynolds number where the boundary layer is relatively thick and the viscous sublayer can be resolved.

Thus, the standard \(k-\omega\) model is best used for near-wall treatment. Other advantages include superior performance for complex boundary layer flows under adverse pressure gradients and separations (e.g., external aerodynamics and turbomachinery). On the contrary, this model is also shown to predict excessive and early separations.

There exist different variations of the k-omega model such as standard k-omega (discussed above), baseline k-omega, k-omega SST, etc., each with certain modifications to perform better under certain conditions of the fluid flow.

Understanding K-Omega SST

\(SST\) stands for shear stress transport. The \(SST\) formulation switches to a \(k-\epsilon\) behavior in the free-stream, which avoids the \(k-\omega\) problem of being sensitive to the inlet free-stream turbulence properties.

The \(k-\omega\ SST\) model provides a better prediction of flow separation than most RANS models and also accounts for its good behavior in adverse pressure gradients. It has the ability to account for the transport of the principal shear stress in adverse pressure gradient boundary layers. It is the most commonly used model in the industry given its high accuracy to expense ratio.

On the negative side, the \(SST\) model produces some large turbulence levels in regions with large normal strain, like stagnation regions and regions with strong acceleration. This effect is much less pronounced than with a normal k-epsilon model though.

Mathematical Representation

The turbulent energy \(k\) is given by:
$$k=\frac { 3 }{ 2 } { \left( UI \right) }^{ 2 }\tag{1}$$
where \(U\) is the mean flow velocity and \(I\) is the turbulence intensity.

The turbulence intensity gives the level of turbulence and can be defined as follows:
$$I = \frac { u’ }{ U }\tag{2}$$
where \(u’\) is the root-mean-square of the turbulent velocity fluctuations given as:

$$u’ = \sqrt { \frac { 1 }{ 3 } \left( { { u’ }_{ x } }^{ 2 } + { { u’ }_{ y } }^{ 2 } + { { u’ }_{ z } }^{ 2 } \right) } =\sqrt { \frac { 2 }{ 3 } k }\tag{3}$$

The mean velocity \(U\) can be calculated as follows:
$$U = \sqrt { { { U }_{ x } }^{ 2 }+{ { U }_{ y } }^{ 2 }+{ { U }_{ z } }^{ 2 }}\tag{4}$$

The specific turbulent dissipation rate can be calculted using the following formula:
$$\omega ={ { C }_{ \mu } }^{ \frac { 3 }{ 4 } }\frac { { k }^{ \frac { 1 }{ 2 } } }{ l }\tag{5}$$
where \({ { C }_{ \mu } }\) is the turbulence model constant which usually takes the value 0.009, \(k\) is the turbulent energy, \(l\) is the turbulent length scale.

The turbulence length scale describes the size of large energy-containing eddies in a turbulent flow.

The turbulent viscosity \(\nu_t\) is, thus, calculated as:

$$\nu_{t} = \frac{k}{\omega}\tag{6}$$

Inlet Turbulence

To realistically model a given problem, it is important to define the turbulence intensity at the inlets. Here are a few examples of common estimations of the incoming turbulence intensity:

  • High-turbulence (between 5% and 20%): Cases with high velocity flow inside complex geometries. Examples: heat exchangers, flow in rotating machinery like fans, engines, etc.
  • Medium-turbulence (between 1% and 5%): Flow in not-so-complex geometries or low speed flows. Examples: flow in large pipes, ventilation flows, etc.
  • Low-turbulence (well below 1%): Cases with fluids that stand still or highly viscous fluids, very high-quality wind tunnels. Examples: external flow across cars, submarines, aircraft, etc.

Did you know?

The turbulent intensity at the core of a pipe for a fully developed pipe flow can be estimated as follows: $$I=0.16 { { Re }_{ { d }_{ h } } }^{ -\frac { 1 }{ 8 } }\tag{7}$$ where \({ { Re }_{ { d }_{ h } } }\) is the Reynolds number for a pipe of hydraulic diameter \({ { d }_{ h } }\).

The turbulence length scale in this case is $$l=0.07{ d }_{ h }\tag{8}$$.

Applying K-Omega Model in SimScale

The k-omega SST turbulence model needs to be chosen at the beginning of the simulation setup inside the global settings panel. This is shown in the figure below:

selection of k omega sst turbulence model in simscale
Figure 1: Select k-omega SST under Turbulence model inside global settings by clicking on the dropdown symbol (circled).

By default, SimScale defines the initial values of turbulence variables \(k\) and \(\omega\) depending on the domain of the problem. If needed, they can be changed under Initial conditions. Further, if the user wants to specifically define the boundary conditions for these turbulence variables, then the Custom boundary condition can be used.

References

Last updated: September 16th, 2020

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