omegaWallFunction

- The
`omegaWallFunction`

boundary condition provides a wall constraint on the specific dissipation rate, i.e.`omega`

, and the turbulent kinetic energy production contribution, i.e.`G`

, for low- and high-Reynolds number turbulence models. - The
`omegaWallFunction`

condition inherits the traits of the fixedValue boundary condition.

Required fields:

omega | Specific dissipation rate [1/s]

The model expressions:

\[ \omega_{vis} = \frac{6 \nu_w}{\beta_1 y^2} \]

\[ \omega_{log} = \frac{\sqrt{k}}{C_\mu \kappa y} \]

\[ G = w (\nu_{t_w} + \nu_w) |\vec{n} \cdot (\grad{\u})_f | C_\mu^{1/4} \frac{\sqrt{k}}{\kappa y} \qquad if \quad y^+ > y^+_{lam} \]

where

\( \omega \) | = | Specific dissipation rate [1/s] |

\( \omega_{vis} \) | = | \(\omega\) computed by the viscous sublayer assumptions [1/s] |

\( \omega_{log} \) | = | \(\omega\) computed by the inertial sublayer assumptions [1/s] |

\( w \) | = | Cell-corner weights [-] |

\( k \) | = | Turbulent kinetic energy [m2/s2] |

\( y \) | = | Wall-normal distance [m] |

\( C_\mu \) | = | Empirical model constant [-] |

\( \nu_w \) | = | Kinematic viscosity of fluid near wall [m2/s] |

\( \nu_{t_w} \) | = | Turbulent viscosity near wall [m2/s] |

\( \vec{n} \) | = | Face unit normal vector [-] |

\( \u \) | = | Velocity [m/s] |

\( \kappa \) | = | von Kármán constant [-] |

The `omega`

predictions for the viscous and inertial sublayers can be blended by six different methods:

stepwise | Stepwise switch (discontinuous) max | Maximum value switch (discontinuous) binomial2 | Binomial blending (smooth) n = 2 binomial | Binomial blending (smooth) exponential | Exponential blending (smooth) tanh | Tanh blending (smooth)

`G`

predictions for the viscous and inertial sublayers are always blended in a stepwise manner, and `G`

below \(y^+_{lam}\) (i.e. in the viscous sublayer) is presumed to be zero.

The viscous and inertial sublayer estimations of `omega`

are switched between each other depending on the \(y^+\) value of the point of interrogation.

\[ \omega = \omega_{vis} \qquad if \quad y^+ <= y^+_{lam} \]

\[ \omega = \omega_{log} \qquad if \quad y^+ > y^+_{lam} \]

where

\( \omega \) | = | \(\omega\) at \(y^+\) |

\( y^+ \) | = | Estimated wall-normal distance of the cell centre in wall units |

\( y^+_{lam} \) | = | Estimated intersection of the viscous and inertial sublayers in wall units |

The maximum value of the viscous and inertial sublayer estimations of `omega`

is set as the `omega`

estimation at \(y^+\) ([62], Eq. 27).

\[ \omega = \max(\omega_{vis}, \omega_{log}) \]

The `omega`

estimation at \(y^+\) is blended between the viscous and intertial sublayer estimations by using a binomial function ([51], Eq. 15). This method is a special case of the following `binomial`

method in order to ensure bitwise regression.

\[ \omega = ((\omega_{vis})^2 + (\omega_{log})^2)^{1/2} \]

The `omega`

estimation at \(y^+\) is blended between the viscous and intertial sublayer estimations by using a binomial function ([51], Eqs. 15-16).

\[ \omega = ((\omega_{vis})^n + (\omega_{log})^n)^{1/n} \]

where

\( n \) | = | Binomial blending exponent |

The `omega`

estimation at \(y^+\) is blended between the viscous and intertial sublayer estimations by using an exponential function ([62], Eq. 32).

\[ \omega = \omega_{vis} \exp[-\Gamma] +\omega_{log} \exp[-1/\Gamma] \]

where ([62], Eq. 31)

\( \Gamma \) | = | Blending expression |

\( \Gamma \) | = | \(0.01 (y^+)^4 / (1.0 + 5.0 y^+)\) |

The `omega`

estimation at \(y^+\) is blended between the viscous and intertial sublayer estimations by using a tanh function ([33], Eqs. 33-34).

\[ \omega = \phi \omega_1 + (1 - \phi) \omega_2 \]

with

\[ \phi = tanh \left( \left(\frac{y^+}{10}\right)^4 \right) \]

\[ \omega_1 = \omega_{vis} + \omega_{log} \]

\[ \omega_2 = \left( (\omega_{vis})^{1.2} + (\omega_{log})^{1.2} \right)^{1/1.2} \]

Example of the boundary condition specification:

<patchName> { // Mandatory entries (unmodifiable) type omegaWallFunction; // Optional entries (unmodifiable) beta1 0.075; blending binomial2; n 2.0; // Optional (inherited) entries ... }

where the entries mean:

Property | Description | Type | Required | Default |
---|---|---|---|---|

type | Type name: omegaWallFunction | word | yes | - |

beta1 | Model coefficient | scalar | no | 0.075 |

blending | Viscous/inertial sublayer blending method | word | no | stepwise |

n | Binomial blending exponent | scalar | no | 2.0 |

The inherited entries are elaborated in:

`fixedValueFvPatchField`

`nutWallFunctionFvPatchScalarField`

Options for the `blending`

entry:

stepwise | Stepwise switch (discontinuous) max | Maximum value switch (discontinuous) binomial2 | Binomial blending (smooth) n = 2 binomial | Binomial blending (smooth) exponential | Exponential blending (smooth) tanh | Tanh blending (smooth)

- The coefficients
`Cmu`

,`kappa`

, and`E`

are obtained from the specified`nutWallFunction`

in order to ensure that each patch possesses the same set of values for these coefficients. - The reason why
`binomial2`

and`binomial`

blending methods exist at the same time is to ensure the bitwise regression with the previous versions since`binomial2`

and`binomial`

with`n=2`

will yield slightly different output due to the miniscule differences in the implementation of the basic functions (i.e.`pow`

,`sqrt`

,`sqr`

).

Tutorial

Source code