Guide
y⁺ is the dimensionless distance from a wall to the centre of your first mesh cell. It tells you whether that cell is thin enough for the wall treatment of the turbulence model you picked. The calculator above takes your flow conditions and a target y⁺ and returns the first cell thickness to use when building prism (inflation) layers.
Near a wall, a turbulent boundary layer has distinct regions. Its structure is set by the wall shear stress τw, which defines the friction velocity uτ = √(τw/ρ) and the wall-scaled distance:
Where y is the height of the cell centre above the wall. Because both ρ and μ appear, the same y⁺ needs a much thinner first cell in water than in air, and at high speed than at low speed.
| Target | Wall treatment | Typical models |
|---|---|---|
| y⁺ ≈ 1 (below 5) | Resolve the boundary layer down to the wall | k-ω SST, low-Re k-ε, Spalart-Allmaras (fine mesh), transition models |
| 30 ≤ y⁺ ≤ 300 | Wall functions bridge the wall region | Standard or realizable k-ε with wall functions |
| 5 < y⁺ < 30 | Avoid: the first cell sits in the buffer layer | — |
Most modern solvers offer blended or “all-y⁺” wall treatments that tolerate a range of y⁺, but the safest practice is still to aim clearly at one of the two ranges above. Choose y⁺ ≈ 1 when you need wall heat transfer, separation onset or skin friction. y⁺ ≈ 30 to 100 is a pragmatic choice for external aerodynamics and internal flows where the overall pressure drop matters more than the detailed near-wall physics.
The calculator estimates wall shear stress from a skin-friction correlation, then inverts the y⁺ definition:
Re = ρ U L / μ, with L the plate length (external flow) or the pipe/duct diameter (internal flow).Cf = [2 log10(Re) − 0.65]−2.3. Internal smooth pipe, turbulent (Blasius): Darcy friction factor f = 0.316 Re−0.25.τw = Cf · ½ρU² for external flow, and τw = (f/8) ρU² for internal flow.uτ = √(τw / ρ).y = y⁺ μ / (ρ uτ).The calculator shows every step with your numbers substituted, so you can check the arithmetic or copy the values into a report. It also warns you when Re falls outside the range where the chosen correlation is valid (for example, below Re ≈ 5×10⁵ for the flat-plate formula, or above Re ≈ 10⁵ for Blasius).
| External: flat plate in air | Internal: water in a pipe | |
|---|---|---|
| Conditions | Air 20 °C, U = 50 m/s, L = 1 m | Water 20 °C, U = 2 m/s, D = 50 mm |
| Fluid properties | ρ = 1.2047 kg/m³, μ = 1.8205×10⁻⁵ Pa·s | ρ = 998.2 kg/m³, μ = 1.0016×10⁻³ Pa·s |
| Reynolds number | 3.31×10⁶ | 9.97×10⁴ |
| Friction coefficient | Cf = 0.003062 | f = 0.01779 |
| Wall shear stress | 4.61 Pa | 8.88 Pa |
| Friction velocity | 1.956 m/s | 0.0943 m/s |
| First cell, y⁺ = 1 | 7.7 µm | 10.6 µm |
| First cell, y⁺ = 30 | 232 µm | 319 µm |
| First cell, y⁺ = 100 | 772 µm | 1.06 mm |
The cost of y⁺ ≈ 1 shows up in layer count. For the flat plate, the turbulent boundary-layer thickness at the trailing edge is roughly δ ≈ 0.37 L Re−1/5 ≈ 18 mm. Growing from a 7.7 µm first cell at a ratio of 1.2 takes about 34 layers to span it, whereas a y⁺ ≈ 30 first cell of 232 µm needs only about 16.
y is a physical distance (metres). y⁺ scales that distance by the local viscous length ν/uτ, so it compares the cell height with the thickness of the viscous region. A first cell of 0.1 mm can be y⁺ = 1 in one flow and y⁺ = 300 in another.
The formula gives the wall distance y at which y⁺ equals your target. Solvers evaluate y⁺ at the centre of the wall-adjacent cell, so if your mesher asks for the full first-layer height, that height is about twice the centre distance, and using the calculator’s value directly as the layer height would give roughly half your target y⁺ at the cell centre. Check what your mesher’s input means and, if in doubt, verify the achieved y⁺ in the solution.
Aim for y⁺ ≈ 1 (below 5) with at least 10 to 15 layers across the boundary layer. k-ω SST also has an automatic wall treatment in many codes that tolerates y⁺ up to about 30, but it is most accurate when the sublayer is resolved.
The skin-friction correlations are calibrated for turbulent flow. If Re is below the transition range, the estimate of τw can be inaccurate, and the first cell you get may be far from the y⁺ you intended. Use the laminar relations, or a detailed simulation, for low Re.
Yes. Switch the unit system in the calculator and the inputs and results convert automatically.
See also the AgniCycle article Choosing y⁺ for your CFD mesh.
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