Boundary layers and y⁺

First-cell spacing

$$y^+=\frac{u_\tau y}{\nu}, \qquad u_\tau=\sqrt{\frac{|\tau_w|}{\rho}}.$$

$y$ is the distance from the wall to the relevant first solution point, $u_\tau$ is friction velocity and $\tau_w$ is wall shear stress. Check your solver’s definition of that point. The full height of a first prism and its center-to-wall distance differ.

For a wall-integrated RANS model, $y^+\lesssim 1$ is a common starting target, subject to the model and the accuracy needed. For wall functions, use the implementation’s target range and assumptions. A low value near a separation line can reflect weak wall shear, so $y^+$ alone cannot establish adequate resolution.

Design the layer stack

Set first spacing, total thickness, number of layers and growth together. Resolve the expected boundary layer, and transition smoothly into the surrounding cells. Inspect junctions, tips and narrow gaps for collapsed layers and abrupt changes. Plot the solved $y^+$ distribution over the whole surface.

For LES and hybrid RANS–LES

Wall-resolved LES also needs adequate wall-parallel resolution, often expressed as $\Delta x^+$ and $\Delta z^+$. WMLES follows its own sampling and spacing guidance. In DES-family methods, the grid scale can directly affect model behavior, so a refinement can change which flow scales are modeled and resolved.

Sources

• NASA TMR · flat-plate verification grids and wall-spacing study
• NASA LAVA · high-lift wall-modeled LES
• Spalart et al. · DDES and ambiguous grid densities