From target y⁺ to a real dimension
You do not mesh in y⁺ — you mesh in millimeters. First cell height is the physical thickness of the first cell off the wall, and you back it out from a y⁺ target using a flat-plate estimate of the wall shear stress.
The standard chain looks like this:
- Reynolds number:
Re_x = U · x / ν - Skin-friction coefficient (one common turbulent flat-plate form):
C_f ≈ 0.026 / Re_x^(1/7) - Wall shear stress:
τ_w = C_f · ½ · ρ · U² - Friction velocity:
u_τ = √(τ_w / ρ) - First cell height:
Δy = (y⁺_target · ν) / u_τ
It is only an estimate — real τ_w varies over the body — so treat the result as a starting point and confirm the realized y⁺ after the first solve.
Watch the cell-centroid nuance
Many solvers evaluate y⁺ at the first cell centroid, not at the wall. For a cell-centered finite-volume mesh, the wall-adjacent cell is then about twice the centroid distance, so the Δy from the formula is the centroid height. Build the cell accordingly rather than doubling by accident.
Prism layers and growth ratio
A single thin cell is not enough; you stack prism (inflation) layers to march the fine near-wall spacing out to the edge of the boundary layer.
- Growth ratio: keep it gentle, usually 1.1–1.2. Ratios above ~1.3 jump cell size too fast and hurt accuracy.
- Layer count: enough total thickness to cover the boundary layer, often 10–30 layers.
- Transition: the last prism should roughly match the neighboring volume cell to avoid a sudden size jump.
How AutoMesh-Geo helps
The formula gives you one number; the hard part is holding that spacing and a clean growth stack over curved, pinched, real-world geometry. AutoMesh-Geo grows conforming near-wall cells that follow the surface, so the first cell height you designed for is what the solver actually sees.