Wall Functions for Turbulence

The wall functions for turbulence describe the distributions of the non-dimensional turbulent dissipation $ε^{+}$ and the non-dimensional specific turbulent dissipation $ω^{+}$ in the three sublayers of the inner turbulent boundary layer. No wall functions are available for the modified diffusivity ${\tilde{v}}^{+}$ and the turbulent kinetic energy $k^{+}$ as these quantities strongly depend on the Reynolds number of the flow. However, their production terms $P_{\tilde{v}}^{+}$ and $P_{k}^{+}$ show universal behavior and can be described with wall functions.

Turbulent Dissipation

The following wall functions describe the distribution of the non-dimensional turbulent dissipation $ε^{+}$ :


Sublayer	Standard Wall Function	Blended Wall Function
Viscous sublayer	Figure 1. EQUATION_DISPLAY $ε^{+} = \frac{2 k^{+}}{{(y^{+})}^{2}}$ (1611)	Figure 2. EQUATION_DISPLAY $ε^{+} = γ \frac{2 k^{+}}{{(y^{+})}^{2}} + (1 - γ) \frac{1}{κ y^{+}}$ (1612)
Buffer layer	-
Log layer	Figure 3. EQUATION_DISPLAY $ε^{+} = \frac{1}{κ y^{+}}$ (1613)

where:

$κ$ is the von Karman constant.
$γ$ is the blending function given by Eqn. (1597).

Specific Turbulent Dissipation

The following wall functions describe the distribution of the non-dimensional specific turbulent dissipation $ω^{+}$ :


Sublayer	Standard Wall Function	Blended Wall Function
Viscous sublayer	Figure 4. EQUATION_DISPLAY $ω^{+} = \frac{6}{β_{1} {(y^{+})}^{2}}$ (1614)	Figure 5. EQUATION_DISPLAY $ω^{+} = \sqrt{{[\frac{6}{β_{1} {(y^{+})}^{2}}]}^{2} + {[\frac{1}{\sqrt{β^{*}} κ y^{+}}]}^{2}}$ (1615)
Buffer layer	-
Log layer	Figure 6. EQUATION_DISPLAY $ω^{+} = \frac{1}{\sqrt{β^{*}} κ y^{+}}$ (1616)

where $β_{1}$ and $β^{*}$ are turbulence model coefficients, see Turbulence.

Production of Modified Diffusivity

The following blended wall function describes the distribution of the non-dimensional modified diffusivity production $P_{\tilde{v}}^{+}$ :

Figure 7. EQUATION_DISPLAY

P_{\tilde{v}}^{+} = - \frac{1 + C_{b 2}}{σ_{\tilde{v}}} \frac{{({\tilde{v}}^{+})}^{2}}{{(y^{+})}^{2}}

(1617)

where $C_{b 2}$ and $σ_{\tilde{v}}$ are turbulence model coefficients, see Turbulence.

Production of Turbulent Kinetic Energy

The following wall functions describe the distribution of the non-dimensional turbulent kinetic energy production $P_{k}^{+}$ :


Sublayer	Standard Wall Functions	Blended Wall Functions
Viscous sublayer	Figure 8. EQUATION_DISPLAY $P_{k}^{+} = μ_{t}^{+} \frac{\partial u^{+}}{\partial y^{+}}$ (1618)	Figure 9. EQUATION_DISPLAY $P_{k}^{+} = γ (μ_{t}^{+} \frac{\partial u^{+}}{\partial y^{+}}) + (1 - γ) \frac{1}{κ y^{+}}$ (1619)
Buffer layer	-
Log layer	Figure 10. EQUATION_DISPLAY $P_{k}^{+} = \frac{1}{κ y^{+}}$ (1620)