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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.

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 $$y^+$$ (, 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 (, 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 (, 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 (, Eq. 32).

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

where (, 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 (, 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).

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