epsilonWallFunction

- The
`epsilonWallFunction`

boundary condition provides a wall constraint on the turbulent kinetic energy dissipation rate, i.e.`epsilon`

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

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

condition inherits the traits of the fixedValue boundary condition.

Required fields:

epsilon | Turbulent kinetic energy dissipation rate [m2/s3]

The model expressions:

\[ \epsilon_{vis} = 2 w k \frac{\nu_w}{y^2} \]

\[ \epsilon_{log} = w C_\mu \frac{k^{3/2}}{\nu_{t_w} y} \]

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

where

\( \epsilon \) | = | Turbulent kinetic energy dissipation rate [m2/s3] |

\( \epsilon_{vis} \) | = | \(\epsilon\) computed by the viscous sublayer assumptions [m2/s3] |

\( \epsilon_{log} \) | = | \(\epsilon\) computed by the inertial sublayer assumptions [m2/s3] |

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

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

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

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

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

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

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

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

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

The `epsilon`

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

stepwise | Stepwise switch (discontinuous) max | Maximum value switch (discontinuous) binomial | Binomial blending (smooth) exponential | Exponential 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 `epsilon`

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

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

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

where

\( \epsilon \) | = | \(\epsilon\) 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 `epsilon`

is set as the `epsilon`

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

\[ \epsilon = \max(\epsilon_{vis}, \epsilon_{log}) \]

The `epsilon`

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

\[ \epsilon = ((\epsilon_{vis})^n + (\epsilon_{log})^n)^{1/n} \]

where

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

The `epsilon`

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

\[ \epsilon = \epsilon_{vis} \exp[-\Gamma] +\epsilon_{log} \exp[-1/\Gamma] \]

where ([62], p. 193)

\( \Gamma_\epsilon \) | = | \(\Gamma = 0.001 (y^+)^4 / (1.0 + y^+)\) |

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

\( \Gamma_\epsilon \) | = | Blending expression for \(\epsilon\) |

\( \Gamma_G \) | = | Blending expression for \(G\) |

Example of the boundary condition specification:

<patchName> { // Mandatory entries (unmodifiable) type epsilonWallFunction; // Optional entries (unmodifiable) lowReCorrection false; blending stepwise; n 2.0; // Optional (inherited) entries ... }

where the entries mean:

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

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

lowReCorrection | Flag: apply low-Re correction | bool | no | false |

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) binomial | Binomial blending (smooth) exponential | Exponential 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. `lowReCorrection`

operates with only`stepwise`

blending treatment to ensure the backward compatibility.- If
`lowReCorrection`

is`on`

,`stepwise`

blending treatment is fully active. - If
`lowReCorrection`

is`off`

, only the inertial sublayer prediction is used in the wall function, hence high-Re mode operation.

Tutorial

Source code

History

- Introduced in version 1.6