K-epsilon

The k-epsilon (k-\(\epsilon\)) model for turbulence is the most common to simulate the mean flow characteristics for turbulent flow conditions. This is a 2 equation model which gives a general description of turbulence by means of two transport equations (PDEs), which accounts for the history effects like convection and diffusion of turbulent energy. The 2 transported variables are turbulent kinetic energy k, which determine the energy in turbulence, and turbulent dissipation \(\epsilon\), which determines the rate of dissipation of the turbulent kinetic energy. The k-\(\epsilon\) model is shown to be applicable for free-shear flows, such as the ones with relatively small pressure gradients [1], but might not be the best model for problems involving large adverse pressure gradients [2]. Hence this model might not be suitable for inlets and compressors.

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, \(\epsilon\)) 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 for estimating 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 turbulent dissipation rate can be calcaulted using the following formula

\[\epsilon ={ { C }_{ \mu } }^{ \frac { 3 }{ 4 } }\frac { { k }^{ \frac { 3 }{ 2 } } }{ l }\]

where \({C}_{\mu}\) is the turbulence model constant which usually takes the value 0.09, 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] Bardina, J.E., Huang, P.G., Coakley, T.J. (1997), “Turbulence Modeling Validation, Testing, and Development”, NASA Technical Memorandum 110446

[2] Wilcox, David C (1998). “Turbulence Modeling for CFD”. Second edition. Anaheim: DCW Industries, 1998. pp. 174

[3] http://www.cfd-online.com/Wiki/Turbulence_free-stream_boundary_conditions