K-omega SST

K-omega (k-\(\omega\)) model is one of the most commonly used models. This include 2 additional transport equations to represent turbulent properties of flow - to account for history effects like convection and diffusion of turbulent energy. The transport variable k determines the energy in turbulence and \(\omega\) determines the scale of turbulence [1].

The basic k-\(\omega\) model can be used for boundary layer problems, where the formulation works from the inner part throught the viscous sub-layer, till the walls - hence the k-\(\omega\) SST model can be used as a low reynolds flow applications without extra damping functions. SST stands for Shear Stress Transport. The SST formulation also switches to a k-\(\epsilon\) behaviour in the free-stream, which avoids that the k-\(\omega\) problem that the model is very sensitive to the inlet free-stream turbulence properties. The k-\(\omega\) SST model also accounts for its good behaviour in adverse pressure gradients and separating flow [1]. The k-\(\omega\) SST model does produce 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. The SST model has the ability to account for the transport of the principal shear stress in adverse pressure gradient boundary-layers [2].

The turbulent energy k is given by

\[k=\frac { 3 }{ 2 } { \left( UI \right) }^{ 2 }\]

where U is the mean flow velocity and I is the turbulence intensity. The turbulence internsity gives the level of turbulence and can be defined as follows

\[I \equiv \frac { u' }{ U }\]

where u’ is the root-mean-square of the turbulent velocity fluctuations and U is the mean velocity. The root-mean-square of the turbulent velocity fluctuations u’ is given as

\[u' \equiv \sqrt { \frac { 1 }{ 3 } \left( { { u' }_{ x } }^{ 2 } + { { u' }_{ y } }^{ 2 } + { { u' }_{ z } }^{ 2 } \right) } =\sqrt { \frac { 2 }{ 3 } k }\]

The mean velocity U can be calculated as follows

\[U \equiv \sqrt { { { U }_{ x } }^{ 2 }+{ { U }_{ y } }^{ 2 }+{ { U }_{ z } }^{ 2 }}\]

By default the SimScale platform takes up the default values of turbulence (like \(k\), \(\omega\)) depending on the domain of the problem. This is considered not just during the initial condition, but also for the specifications of various boundaries. If the problem requires distinct values of turbulence in a boundary condition then the Custom boundary condition option is defined. It is important to find the turbulence intensity at the inlets, which requires some form of measurements or previous experience to estimate. Here are a few examples of common estimations of the incoming turbulence intensity:

  • High-turbulence case (between 5% and 20%): Cases with high velocity flow inside complex geometries. Examples: Heat-exchangers, flow in rotating machienry etc.
  • Medium-turbulence case (between 1% and 5%): Flow in not-so-complex geometries or low speed flows. Examples: Flow in large pipes, ventilation flows etc.
  • Low-turbulence case (well below 1%): Cases with fluid that stands still or very high-quality wind-tunnels (low turbulence levels). Examples: External flow across cars, submarines, aircraft etc.

The turbulent intensity at the core of a pipe for a fully developed pipe flow can be estimated as follows.

\[I=0.16 { { Re }_{ { d }_{ h } } }^{ -\frac { 1 }{ 8 } }\]

where \({ Re }_{ { d }_{ h } }\) is the reynolds number for a pipe of hydraulic diameter \({ d }_{ h }\). The specific turbulent dissipation rate can be obtained as follows.

\[\omega ={ { C }_{ \mu } }^{ -\frac { 1 }{ 4 } }\frac { \sqrt { k } }{ l }\]

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. For a fully developed pipe flow this can be given as follows

\[l=0.07 { d }_{ h }\]

where \({d}_{h}\) is the hydraulic diameter.

References

[1] <a class=”reference external” target=”_blank” ref=”nofollow” href=http://www.cfd-online.com/Wiki/Turbulence_free-stream_boundary_conditions>http://www.cfd-online.com/Wiki/Turbulence_free-stream_boundary_conditions</a>

[2] Menter, Florian R. “Improved two-equation k-omega turbulence models for aerodynamic flows.” (1992).