The open source CFD toolbox
epsilonWallFunction

# Properties

• 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.

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


# Model equations

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 stepwise switch (discontinuous) method

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 switch (discontinuous) method

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 binomial blending (continuous) method

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 exponential blending (continuous) method

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 $$G$$

# Usage

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)


## 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.
• 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.

# Further information

Tutorial

Source code

History

• Introduced in version 1.6