Wall Functions

Wall functions provide algebraic approximations of the main quantities—velocity, temperature, turbulence quantities—in the inner layer of a turbulent boundary layer, independent of the turbulence model formulation or the wall-treatment. Wall functions are in that sense universal under the conditions for which they are derived.

The following types of wall function are available in Simcenter STAR-CCM+:

Standard Wall Functions
Standard wall functions are functions that are specifically defined for the viscous sublayer or the log layer. No standard wall functions are available for the buffer layer.
Blended Wall Functions
Blended wall functions are continuous functions that cover all three sublayers. They represent the buffer layer by appropriately blending the viscous sublayer and the log layer.

Wall functions are defined in terms of non-dimensional quantities, which makes the algebraic approximations independent of the Reynolds number of the flow. The following table lists the required quantities and the corresponding non-dimensional definitions:


Quantity	Non-dimensional Definition
Wall distance $y$	Figure 1. EQUATION_DISPLAY $y^{+} = \frac{y {ρ u}_{*}}{μ}$ (1584)
Wall-tangential velocity component $u$ of the velocity vector	Figure 2. EQUATION_DISPLAY $u^{+} = \frac{u}{u_{*}}$ (1585)
Temperature $T$	Figure 3. EQUATION_DISPLAY $T^{+} = ρ C_{p} u_{*} \frac{T - T_{w}}{\dot{q}''}$ (1586)
Turbulent eddy viscosity $μ_{t}$	Figure 4. EQUATION_DISPLAY $μ_{t}^{+} = \frac{μ_{t}}{μ}$ (1587)
Modified diffusivity $\tilde{v}$	Figure 5. EQUATION_DISPLAY ${\tilde{v}}^{+} = \frac{ρ \tilde{v}}{μ}$ (1588)
Production of modified diffusivity $P_{\tilde{v}}$	Figure 6. EQUATION_DISPLAY $P_{\tilde{v}}^{+} = \frac{P_{\tilde{v}}}{u_{*}^{2}}$ (1589)
Turbulent kinetic energy $k$	Figure 7. EQUATION_DISPLAY $k^{+} = \frac{k}{u_{*}^{2}}$ (1590)
Production of turbulent kinetic energy $P_{k}$	Figure 8. EQUATION_DISPLAY $P_{k}^{+} = \frac{P_{k} μ}{ρ^{2} u_{*}^{4}}$ (1591)
Dissipation rate $ε$	Figure 9. EQUATION_DISPLAY $ε^{+} = \frac{ε μ}{ρ u_{*}^{4}}$ (1592)
Specific dissipation rate $ω$	Figure 10. EQUATION_DISPLAY $ω^{+} = \frac{ω μ}{ρ u_{*}^{2}}$ (1593)

where:

$μ$ is the dynamic viscosity.
$ρ$ is the density.
$C_{p}$ is the specific heat.
$\dot{q}''$ is the wall heat flux.

Velocity Scale

The velocity scale $u_{*}$ is a representative of the flow velocity in the near-wall region and can be approximated iteratively or non-iteratively.

Non-Iterative Method

u_{*}

depends on the type of wall function as:


Sublayer	Standard Wall Function	Blended Wall Functions
Viscous sublayer	Figure 11. EQUATION_DISPLAY $u_{*} = \sqrt{\frac{μ \| v_{tangential} \|}{ρ y}}$ (1594)	Figure 12. EQUATION_DISPLAY $u_{*} = γ \sqrt{\frac{μ \| v_{tangential} \|}{ρ y}} + (1 - γ) C_{μ}^{1 / 4} k^{1 / 2}$ (1595)
Buffer layer	-
Log layer	Figure 13. EQUATION_DISPLAY $u_{*} = C_{μ}^{1 / 4} k^{1 / 2}$ (1596)

where:

$v_{tangential}$ is the wall-tangential velocity vector.
$C_{μ}$ is a turbulence model coefficient, see Turbulence.

The blending function $γ$ is given as:

Figure 14. EQUATION_DISPLAY

γ = \exp (- \frac{{Re}_{d}}{11})

(1597)

where ${Re}_{d}$ is the wall-distance Reynolds number given by Eqn. (1134).

Iterative Method

$u_{*}$ is calculated iteratively by equating the computed value of $u^{+}$ given by Eqn. (1585) with the corresponding wall function definition for $u^{+}$ (see Wall Functions for Velocity).