Y+ Calculator

Estimate the wall distance for your CFD analysis using the Y+ calculator.

First cell height at your target y⁺17.8283 µmfor y⁺ = 1, that is 17.8283 µm — and the CENTROID of the first cell is what y⁺ refers to, so a wall-normal cell of twice this height puts its centre in the right place

Friction velocity u_τ0.83014 m/s√(τ_w/ρ) — this, and NOT the free-stream velocity, is what sets the wall spacing. It is 4.151% of the free stream

Wall shear stress844.1904 mPa

Skin friction coefficient0.0034460.058 Re⁻⁰·² for a turbulent flat plate

Reynolds number1.3514e+6

Viscous length scale ν/u_τ17.8283 µmone wall unit — y⁺ counts these, so the first cell height is simply this times your target

Total height of that first cell35.6566 µmtwice the centroid distance

Which wall treatment this suitswall-resolved, y⁺ ≲ 5resolving the viscous sublayer directly, where u⁺ = y⁺

Cell height for y⁺ = 30534.8485 µmthe wall-function alternative, thirty times coarser and far cheaper

Why not the bulk Reynolds numberit is the wrong velocityy⁺ is built from the FRICTION velocity, which here is only 4.15% of the free stream. Substituting the bulk velocity inflates the answer by a factor of about 24.1, which is why meshes built that way are hopelessly wrong at the wall

The formula

u_τ = √(τ_w/ρ); y⁺ = ρu_τy/µ; C_f = 0.058Re⁻⁰·²

y⁺ is built on the friction velocity

The wall coordinate y⁺ = ρu_τy/µ uses the friction velocity u_τ = √(τ_w/ρ), which comes from the wall shear stress. It is typically three to five percent of the free-stream velocity, so substituting the bulk velocity — a tempting shortcut, since a bulk Reynolds number is easy to hand — inflates the answer by a factor of twenty or more and produces a mesh that resolves nothing near the wall.

Getting there needs the skin friction coefficient first, which is why this calculator asks for the geometry: the correlation for a flat plate is not the correlation for a pipe.

Stay out of the buffer layer

Below about y⁺ = 5 the viscous sublayer dominates and u⁺ = y⁺ holds directly, so a mesh fine enough to sit there resolves the wall properly. Above about y⁺ = 30 the logarithmic law holds and a wall function can bridge the gap cheaply. Between the two lies the buffer layer, where neither description is valid and both approaches are wrong.

A first cell landing at y⁺ = 15 is therefore worse than one at either 1 or 50. Note also that y⁺ refers to the CENTROID of the first cell, not its outer face, so the cell itself is twice the height this calculator reports.