Wall Treatment for Rough Walls

Roughness effects are a critical issue in various engineering applications. Even small surface imperfections can lead to significant disturbances in the velocity field and change the flow behavior and thus the performance of a product. In most cases, roughness structures are too small to be resolved by the mesh. Therefore, wall roughness models are employed.

In general, the effect of wall roughness is modeled by moving the log layer of the inner boundary layer closer to the wall. Simcenter STAR-CCM+ provides two methods for incorporating this effect— Rough and Rough Displaced Origin.

Rough

The Rough method uses data correlation to calculate a roughness function $f$ that reduces the log law offset $E$ in the Wall Functions for Velocity and the Wall Functions for Temperature.

The roughness function $f$ is a function of the roughness parameter $R^{+}$ , which is defined as:

Figure 1. EQUATION_DISPLAY

R^{+} = \frac{r ρ u_{*}}{μ}

(1639)

where:

$r$ is the equivalent sand-grain roughness height and a Model Coefficient.
$ρ$ is the density.
$u_{*}$ is the velocity scale.
$μ$ is the dynamic viscosity.

The roughness function $f$ is based on the expression given in [391] and is defined as:

Figure 2. EQUATION_DISPLAY

f = {\begin{matrix} 1 & {; R}^{+} \leq R_{smooth}^{+} \\ {[B (\frac{R^{+} - R_{smooth}^{+}}{R_{rough}^{+} - R_{smooth}^{+}}) + C R^{+}]}^{a} & {; R}_{smooth}^{+} < R^{+} < R_{rough}^{+} \\ B + C R^{+} & {; R}^{+} > R_{rough}^{+} \end{matrix}

(1640)

with:

Figure 3. EQUATION_DISPLAY

a = \sin [\frac{π}{2} \frac{\log (R^{+} / R_{smooth}^{+})}{\log (R_{rough}^{+} / R_{smooth}^{+})}]

(1641)

where:

$B$ and $C$ are Model Coefficients.
$R_{smooth}^{+}$ and $R_{rough}^{+}$ are the roughness parameters that correspond to a fully smooth and fully rough surface, respectively, and are Model Coefficients.

For large values of $R^{+}$ , the logarithmic profile can stop intersecting with the assumed linear profile in the sublayer, as shown is the following figure:

If this event occurs in the fully rough regime ( $R^{+} > R_{rough}^{+}$ ), there is no problem, since the sublayer is irrelevant. If these curves do not intersect in the transitional roughness regime ( $R_{smooth}^{+} < R^{+} < R_{rough}^{+}$ ), Simcenter STAR-CCM+ uses the logarithmic profile, with the proviso that $u^{+}$ must never be less than zero.

For all turbulence models (as long as the default wall-treatment is used), this modelling approach is not physically meaningful if the roughness height is larger than the wall distance of the wall-adjacent cell ( $R^{+} > y^{+}$ ). In this case, Simcenter STAR-CCM+ limits the local roughness height and sets it equal to the wall distance of the wall-adjacent cell ( $R^{+} = y^{+}$ ).

Rough Displaced Origin

The Rough Displaced Origin method uses a model called displacement of origin ([389], [393]). This model displaces the origin of the boundary layer above the roughness height and uses specific wall functions for velocity, temperature, turbulent dissipation rate, and specific dissipation rate.

Velocity

For the standard wall function, the non-dimensional velocity

u^{+}

is described as:

Figure 4. EQUATION_DISPLAY

u^{+} = \frac{1}{κ} \log (\frac{y_{e f f}}{y_{0}}) = \frac{1}{κ} \log (\frac{y^{+} + y_{0}^{+}}{y_{0}^{+}})

(1642)

with:

Figure 5. EQUATION_DISPLAY

y_{0}^{+} = {\begin{matrix} \begin{matrix} 0.56 {(R^{+} / R_{s m o o t h}^{+})}^{2.5} \\ 0.63 ζ (R^{+}) + 0.028 R^{+} \end{matrix} & \begin{matrix} {; R}^{+} < R_{s m o o t h}^{+} \\ ; R_{s m o o t h}^{+} \leq R^{+} \leq R_{r o u g h}^{+} \end{matrix} \\ 0.031 R^{+} - 0.27 & ; R^{+} > R_{r o u g h}^{+} \end{matrix}

(1643)

Figure 6. EQUATION_DISPLAY

ζ (R^{+}) = \sin [π {(\frac{R^{+} - 20}{70})}^{0.9}]

(1644)

where:

$κ$ is the von Karman constant.
$y_{e f f}$ is the effective origin of the boundary layer.
$y_{0}$ is the displaced origin of the boundary layer.
$y^{+}$ is given by Eqn. (1584).
$y_{m}^{+}$ is the point at which the viscous sublayer meets the log-layer solution and is computed iteratively using a Newton-Raphson method.

Displacement of origin is combined with the blended wall function in a modification of Reichhardt's law as:

Figure 7. EQUATION_DISPLAY

u^{+} = \frac{1}{κ} \log (1 + κ y^{+}) + C [1 - \exp (- \frac{y^{+}}{y_{m}^{+}}) - \frac{y^{+}}{y_{m}^{+}} \exp (- b y^{+})]

(1645)

with:

Figure 8. EQUATION_DISPLAY

C = \frac{1}{κ} \ln (\frac{1}{κ y_{0}^{+}})

(1646)

where $b$ is given by Eqn. (1601).

To ensure that the blended Reichardt's formula produces reasonable velocity profiles when

R_{r o u g h}^{+} < R^{+} < R_{s m o o t h}^{+}

, Simcenter STAR-CCM+ clips

u^{+}

as:

u^{+} = {\begin{matrix} \frac{1}{κ} \log (\frac{y^{+} + y_{0}^{+}}{y_{0}^{+}}) & {; y}_{0}^{+} > y_{m}^{+} \\ u_{s m o o t h}^{+} & {; \begin{matrix} R \end{matrix}}^{+} < R_{s m o o t h}^{+} \end{matrix}}

(1647)

Temperature

For both standard and blended wall functions, the distribution of the non-dimensional temperature

T^{+}

is defined as:

Figure 9. EQUATION_DISPLAY

T^{+} = \exp (- Γ^{'}) \Pr y^{+} + \exp (- \frac{1}{Γ^{'}}) \Pr_{t} [\frac{1}{κ} \ln (\frac{y^{+} + y_{0}^{+}}{y_{0}^{+}}) + P]

(1648)

where:

$\Pr$ is the Prandtl number given by Eqn. (1141).
$\Pr_{t}$ is the turbulent Prandtl number.
$P$ is given by Eqn. (1610).

$Γ^{'}$ is a blending function defined as:

Figure 10. EQUATION_DISPLAY

Γ^{'} = \frac{0.01 {[\Pr (y^{+} + y_{0}^{+})]}^{4}}{1 + 5 \Pr^{3} (y^{+} + y_{0}^{+})}

(1649)

Turbulent Dissipation Rate

For both standard and blended wall functions, the non-dimensional turbulent dissipation rate

ε^{+}

is set as:

Figure 11. EQUATION_DISPLAY

ε^{+} = γ \frac{2 k^{+}}{{(y^{+})}^{2}} Y_{e} + (1 - γ) \frac{1}{κ y^{+}}

(1650)

where:

Figure 12. EQUATION_DISPLAY

Y_{e} = \frac{y^{2}}{{(\frac{y}{2.236} + {(y_{0}^{+})}^{3} \frac{μ}{{ρ u}_{*}})}^{2}}

(1651)

and:

$γ$ is given by Eqn. (1597), with the additional requirement that $γ = 0$ if $R^{+} > R_{rough}^{+}$ , where $R_{rough}^{+}$ is the roughness parameter for a fully rough surface. A fully rough surface is a surface where roughness effects are so large that they leave no visible viscous flow behavior near the wall.
$k^{+}$ is given by Eqn. (1590).
$y$ is the distance to the wall.
$u_{*}$ is given by Eqn. (1595).

Specific Dissipation Rate

For both standard and blended wall functions, the formulation for the non-dimensional specific dissipation rate

ω^{+}

depends on the wall treatment and is calculated as a function of wall roughness as described by Wilcox [395]:


Wall Treatment	$ω^{+}$
Low- $y^{+}$	Figure 13. EQUATION_DISPLAY $ω^{+} = S_{r}$ (1652)
All- $y^{+}$	Figure 14. EQUATION_DISPLAY $ω^{+} = γ S_{r} + (1 - γ) \frac{1}{κ \sqrt{C_{μ}}}$ (1653)

with:

Figure 15. EQUATION_DISPLAY

S_{r} = {\begin{matrix} {(\frac{50}{R^{+}})}^{2} & ; 5 \leq R^{+} \leq 25 \\ \frac{100}{R^{+}} & {; R}^{+} \geq 25 \end{matrix}

(1654)

and $C_{μ}$ is a K-Omega Model—Model Coefficient.

Model Coefficients


	$r$	$B$	$C$	$R_{smooth}^{+}$	$R_{rough}^{+}$
Rough	0	0	0.253	2.25	90
Rough Displaced Origin	0	-	-	20	90