Theory

Near a wall the velocity of a turbulent flow follows a universal profile once the distance is scaled with the viscous length $\nu/u_\tau$ . The scaled distance $y^+$ says where the first cell centre sits: in the viscous sublayer ( $y^+ \lesssim 5$ ), the buffer layer ( $5 < y^+ < 30$ ) or the logarithmic region ( $y^+ > 30$ ).

The flow is not solved yet when the mesh is built, so the friction velocity $u_\tau$ is estimated from a smooth-pipe skin-friction correlation, equation (2), at the bulk velocity $U$ of the fastest port and the equivalent circular diameter $D$ of that port. The flow through the outlets is shared by area, because the split is only known after the solve. Equation (3) then gives the first-layer thickness whose cell centre lies at the target $y^+$ .

With curvature sizing and wall functions, $y_1$ is used as a fraction of the finest wall cell $h_w$ , limited to $0.05\,h_w \le y_1 \le 0.3\,h_w$ ; the number of layers on auto is reduced (never below 3) until the whole stack, equation (4), fits in $0.8\,h_w$ . The plan is written to logs/layer_plan.log. The y+ estimate on the Mesh step and the y+ the solver reports after the run both quote the cell centre, so the three numbers can be compared directly.

Equations

y^+ = \frac{(y_1/2)\,u_\tau}{\nu}, \qquad u_\tau = U\sqrt{\frac{C_f}{2}}

(1)

C_f = 0.079\,Re^{-1/4}\ \ (Re < 10^5), \qquad C_f = 0.046\,Re^{-1/5}\ \ (Re \ge 10^5), \qquad Re = \frac{U D}{\nu}, \qquad D = \sqrt{4A/\pi}

(2)

where	meaning	unit
$y^+$	Distance of the first cell centre from the wall, in wall units	-
$y_1$	First-layer (first-cell) thickness	m
$u_\tau$	Friction velocity	m/s
$\nu$	Kinematic viscosity of the gas	m²/s
$C_f$	Fanning skin-friction coefficient of a smooth pipe	-
$U$	Bulk velocity of the fastest port, $Q/A$	m/s
$D$	Equivalent circular diameter of that port	m
$A$	Area of the port	m²

y_1 = \frac{2\,y^+\,\nu}{u_\tau}

(3)

\delta = y_1\,\frac{r^{N} - 1}{r - 1} \le 0.8\,h_w

(4)

where	meaning	unit
$\delta$	Thickness of the whole prism-layer stack	m
$r$	Expansion ratio between neighbouring layers	-
$N$	Number of layers	-
$h_w$	Size of the finest wall cell	m

Default and rule

Default: 30 (wall function), 1 (low-Re)Computed by the program
Rule: On auto the target follows the wall treatment: $y^+ = 30$ with the wall function, which puts the first cell centre at the start of the logarithmic region, past the buffer layer; $y^+ = 1$ with the low-Re treatment, which resolves the viscous sublayer. A typed value is kept as typed and gets a warning when it does not suit the wall treatment.
Source: The wall function is Spalding's single law of the wall (Spalding, 1961), valid from the sublayer to the logarithmic region. The target of 30 is the program's default: it places the first cell centre with a resolution margin past the buffer layer. The skin-friction correlations of equation (2) are the smooth-pipe Blasius form and its high-Reynolds-number counterpart.

Valid range y+ ≤ 300 (wall function), ≤ 5 (low-Re)

Continuous wall function

The wall function is valid for any $y^+$ (Spalding, 1961). A first cell in the buffer layer does not stop the run; the report marks it as advisory, because the wall shear there is the least accurate.

Upper limit

After the run, quality gate 2 fails when the average wall $y^+$ exceeds 300 (wall function) or 5 (low-Re): such a first cell is too coarse for any wall treatment.

Estimate before the solve

$u_\tau$ comes from a smooth-pipe correlation at the bulk velocity. Separated, swirling or jet flow inside the muffler moves the achieved $y^+$ away from the target; compare it with the solved value on the Results step.

References 1

Spalding, D. B. (1961). A single formula for the law of the wall. Journal of Applied Mechanics, 28(3), 455-458. DOI 10.1115/1.3641728

Theory

Equations

Default and rule

Valid range y+ ≤ 300 (wall function), ≤ 5 (low-Re)

Related 4

References 1