First layer height from y+ estimate

Definition: $y^+=\dfrac{\rho u_\tau y}{\mu}$ with $u_\tau=\sqrt{\tau_w/\rho}$. The estimate is done before meshing using flat-plate correlations.

(a) Reynolds number

$$Re_L=\frac{\rho UL}{\mu}=\frac{1.2\times30\times1.5}{1.8\times10^{-5}}=3.0\times10^{6}$$

(turbulent over most of the plate).

(b) Skin friction coefficient

$$C_f=0.026\,(3\times10^6)^{-1/7}=0.026\times0.1188=3.09\times10^{-3}$$

(c) Wall shear stress and friction velocity

$$\tau_w=\tfrac12C_f\rho U^2=0.5\times3.09\times10^{-3}\times1.2\times30^2=1.67\ \text{Pa}$$ $$u_\tau=\sqrt{\frac{\tau_w}{\rho}}=\sqrt{\frac{1.667}{1.2}}=1.179\ \text{m/s}$$

(d) First cell height

Kinematic viscosity $\nu=\mu/\rho=1.5\times10^{-5}$ m²/s. For $y^+=1$ the cell centre is at

$$y=\frac{y^+\nu}{u_\tau}=\frac{1\times1.5\times10^{-5}}{1.179}=1.27\times10^{-5}\ \text{m}$$

The first cell has its centre at half its height, so the first cell height is

$$\Delta y_1=2y=2.55\times10^{-5}\ \text{m}\approx 0.025\ \text{mm}$$

(e) Number of layers

Boundary layer thickness:

$$\delta=0.37\,L\,Re_L^{-1/5}=0.37\times1.5\times(3\times10^6)^{-0.2}=0.0281\ \text{m}$$

For $n$ layers with growth ratio $g=1.2$, total thickness is $\Delta y_1\dfrac{g^n-1}{g-1}\ge\delta$:

$$g^n\ge1+\frac{\delta(g-1)}{\Delta y_1}=1+\frac{0.0281\times0.2}{2.55\times10^{-5}}=221.4$$ $$n\ge\frac{\ln221.4}{\ln1.2}=29.6\ \Rightarrow\ n=30\ \text{layers}$$

Check: with 30 layers the total is $2.55\times10^{-5}\times\dfrac{1.2^{30}-1}{0.2}=0.0301$ m, which is above $\delta$.

Answer: $Re_L=3\times10^6$; $C_f=3.09\times10^{-3}$; $\tau_w=1.67$ Pa; $u_\tau=1.18$ m/s; first cell height $\approx2.5\times10^{-5}$ m; about 30 prism layers.