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Inflation Layer Calculator

Estimates the first-cell height, the boundary-layer thickness and the maximum growth ratio for inflation layers on a smooth wall, starting from a flat-plate skin-friction correlation.

Inputs

Flow conditions over the surface you are meshing. Defaults are air at room conditions over a 1 m plate at 10 m/s.

Air at 15 C is about 1.225.
Freestream speed over the surface.
Chord, plate length or similar reference.
Air at 15 C is about 1.81e-5.
1 for wall-resolved, 30 to 100 with wall functions.
Integer, typically 10 to 20 for RANS.

Results

First-cell height yH
-
Enter this as the first-layer thickness in your mesher.
Reynolds number
-
Skin-friction coefficient cf
-
Friction velocity
-
Boundary-layer thickness δ99
-
Maximum growth ratio r
-
Use a growth ratio below this maximum so the layers comfortably cover the boundary layer.
Final layer thickness
-
Compare with the core mesh size to judge the volume jump.

How it works

The calculator sizes the near-wall mesh for a turbulent boundary layer on a smooth wall. It starts from an empirical flat-plate skin-friction correlation to estimate the wall shear, converts the target y+ into a physical cell height, then uses the geometric-series formula for inflation layers to find the largest growth ratio that still covers the estimated boundary layer.

  1. Compute the Reynolds number from density, velocity, reference length and viscosity.
  2. Estimate the skin-friction coefficient with the Schlichting correlation for a smooth flat plate, valid below Re = 10^9.
  3. Get the wall shear stress, then the friction velocity.
  4. Rearrange the y+ definition to find the first-cell centroid height, then double it: mesh generators work with the full cell height.
  5. Estimate the boundary-layer thickness with the laminar Blasius relation or the turbulent power-law relation, depending on Re.
  6. Set the geometric-series thickness of N layers equal to the boundary-layer thickness and solve for the growth ratio r with a Newton iteration, falling back to bisection.
  7. Report the final layer thickness so you can judge the size jump into the core mesh.
Reynolds number
Re = ρ U L / μ
Skin-friction coefficient (Schlichting)
cf = [2 log10(Re) - 0.65]^-2.3
Wall shear stress and friction velocity
τw = (1/2) ρ U^2 cf   ;   uτ = √(τw / ρ)
First-cell height from target y+
yp = y+ μ / (ρ uτ)   ;   yH = 2 yp
Boundary-layer thickness
δ99 = 4.91 L / √Re  (Re < 5x10^5)   ;   δ99 = 0.38 L / Re^(1/5)  (turbulent)
Growth ratio from the geometric series
yT = yH (1 - r^N) / (1 - r) = δ99  →  f(r) = r^N - q r + q - 1 = 0,  q = δ99 / yH
Final layer thickness
yfinal = yH r^(N-1)
Re
Reynolds number based on the reference length
cf
Skin-friction coefficient
τw
Wall shear stress, Pa
uτ
Friction velocity, m/s
y+
Dimensionless wall distance, target value
yp, yH
First-cell centroid height and full first-cell height, m
δ99
Boundary-layer thickness, m
N, r
Number of layers and growth ratio between consecutive layers
q
Ratio of boundary-layer thickness to first-cell height
yfinal
Thickness of the last inflation layer, m

References

  • Schlichting, H. and Gersten, K., Boundary-Layer Theory, 7th ed., McGraw-Hill, 1979.
  • Cengel, Y. and Cimbala, J., Fluid Mechanics: Fundamentals and Applications, McGraw-Hill, 2006.