Theory

OpenFOAM's nutkRoughWallFunction (OpenFOAM user guide, 2026), written by the program when Wall roughness height is above 0, uses $C_s$ in its roughness function. In the fully rough regime the downward shift of the logarithmic velocity profile is equation (1): a larger $C_s$ makes the same $k_s$ act as a rougher wall. In the transitional regime ( $2.25 < k_s^+ < 90$ ) the function blends from smooth to this form. With $k_s = 0$ the wall is smooth and $C_s$ has no effect.

Equations

\Delta B = \frac{1}{\kappa}\ln\left(1 + C_s k_s^+\right) \quad (k_s^+ \ge 90), \qquad k_s^+ = \frac{u_\tau k_s}{\nu}

(1)

where	meaning	unit
$\Delta B$	Shift of the log law	-
$\kappa$	von Kármán constant, 0.41	-
$C_s$	Roughness constant (this setting)	-
$k_s^+$	Roughness Reynolds number	-
$u_\tau$	Friction velocity	m/s
$\nu$	Kinematic viscosity	m²/s

Default and rule

Default: 0.5
Rule: 0.5.
Source: OpenFOAM's rough-wall function and its default (OpenFOAM user guide, 2026); the regime limits 2.25 and 90 and the meaning of 0.5 as uniform sand grains follow the usual rough-wall formulation.

Valid range 0.5-1 (uniform sand grains: 0.5); used only when k_s > 0

Values

0.5 for uniform sand grains; values up to about 1 are used for non-uniform roughness.

Only with roughness

No effect while the roughness height is 0.

References 1

OpenCFD Ltd. (2026). OpenFOAM user guide. openfoam.com. Web page

Theory

Equations

Default and rule

Valid range 0.5-1 (uniform sand grains: 0.5); used only when k_s > 0

Related 3

References 1