epsilonWallFunction

Table of Contents

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.

Required fields: 

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