Wall treatment

How the turbulence model meets the wall. Wall function (default) bridges the near-wall region with Spalding's law of the wall and suits a first cell anywhere up to $y^+ = 300$ ; low-Re resolves the viscous sublayer and needs a first cell at $y^+ \approx 1$ . The choice sets the wall boundary conditions and, on auto, the target $y^+$ , the number of layers and their expansion ratio.

Last updated 2026-11-03

Theory

The turbulence model is $k$ - $\omega$ SST (Menter, 1994). At a wall the program writes, for every wall and both sides of every internal wall:

Treatment	Turbulent viscosity $\nu_t$	$k$	$\omega$
Wall function	`nutUSpaldingWallFunction`	`kqRWallFunction`	`omegaWallFunction`
Low-Re	`nutLowReWallFunction` (zero)	`kLowReWallFunction`	`omegaWallFunction`

A wall with a roughness height $k_s > 0$ (Boundary Conditions step) uses nutkRoughWallFunction instead of Spalding's law in the wall-function treatment.

With the wall function, the wall shear stress follows Spalding's single law of the wall, equation (1) (Spalding, 1961): it joins the linear sublayer $u^+ = y^+$ and the logarithmic law, equation (2), in one curve, so the first cell may sit in the sublayer, the buffer layer or the logarithmic region. The constants are $\kappa = 0.41$ and $B = 5.2$ , i.e. $E = e^{\kappa B} = 8.43$ , equation (3); OpenFOAM's own default $E = 9.8$ corresponds to $B = 5.57$ . Both values of $B$ are within the scatter of measured log laws; the program uses 5.2 so that its results can be compared directly with the reference models it is validated against.

With low-Re, $\nu_t \to 0$ at the wall and the velocity gradient is resolved by the mesh itself, which requires the first cell centre at $y^+ \approx 1$ and many thin layers.

The choice also moves three auto settings together: target $y^+$ 30 or 1, the number of layers (6 with curvature sizing or 3 with uniform sizing; 12 with low-Re) and the expansion ratio (1.2 or 1.15). With low-Re the layer stack is defined by its outermost layer (0.1 of the wall cell); with the wall function and curvature sizing by its first layer, computed from the target $y^+$ (see Target y+).

Equations

y^+ = u^+ + \frac{1}{E}\left[e^{\kappa u^+} - 1 - \kappa u^+ - \frac{(\kappa u^+)^2}{2} - \frac{(\kappa u^+)^3}{6}\right]

(1)

u^+ = \frac{1}{\kappa}\ln\!\left(E\,y^+\right) = \frac{1}{\kappa}\ln y^+ + B

(2)

E = e^{\kappa B} = e^{0.41 \times 5.2} = 8.43

(3)

where	meaning	unit
$y^+$	Wall distance of the first cell centre in wall units, $y\,u_\tau/\nu$	-
$u^+$	Velocity in wall units, $U/u_\tau$	-
$u_\tau$	Friction velocity, $\sqrt{\tau_w/\rho}$	m/s
$\kappa$	von Kármán constant, 0.41	-
$B$	Log-law intercept, 5.2	-
$E$	Log-law constant written to the boundary condition	-

Default and rule

Default

wall function

Rule

Wall function. It is valid for any $y^+$ up to the gate limit, so it tolerates the wide spread of $y^+$ a muffler produces (fast jets next to nearly still chambers), at a far lower cell count than a resolved wall.

Source

Spalding's law (Spalding, 1961); equation (1) is given in its commonly quoted form. The constants $\kappa = 0.41$ and $B = 5.2$ are the program's fixed values.

Quality gate 2 sets no lower bound on $y^+$ for the wall function. The plain log-law convention $30 \le y^+ \le 300$ would reject meshes that Spalding's law handles correctly, for example a validation run with an average $y^+$ of 24.5. The 12 layers of low-Re are the low end of the 12-15 layers commonly recommended for a resolved sublayer; no reference is cited for this guidance.

Valid range wall function | low-Re (resolved wall)

Wall function

Gate 2 passes for an average wall $y^+ \le 300$ , with no lower bound. An average between 5 and 30 (buffer layer) is reported as advisory: the wall shear is least accurate there.

Low-Re

Gate 2 fails above an average $y^+$ of 5. A typed target above 5 or fewer than 12 layers gets a warning on the Mesh step.

Cell count

The low-Re layer estimate of the cell-count model has not been checked against a real 12-layer mesh.

Theory

Equations

Default and rule

Valid range wall function | low-Re (resolved wall)

Related 4

References 2