omegaWallFunction

Properties

  • 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]

Model equations

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 stepwise switch (discontinuous) method

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 switch (discontinuous) method

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 binomial2 blending (continuous) method

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 binomial blending (continuous) method

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 exponential blending (continuous) method

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 tanh blending (continuous) method

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

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

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

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

Usage

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)

Notes on entries

  • 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).

Further information

Tutorial

Source code