boundary-layer

name	boundary-layer
description	Boundary layer theory — Blasius, displacement/momentum thickness, shape factor, turbulent BL, transition, skin friction, Falkner-Skan, pressure gradient effects, separation.
metadata	{"priority":7,"promptSignals":{"phrases":["boundary layer","Blasius","skin friction","momentum thickness","displacement thickness","turbulent boundary layer","transition"],"minScore":3}}

Boundary Layer Theory — Complete Skill

Laminar Boundary Layer (Blasius Solution)

Flat Plate, Zero Pressure Gradient

BL thickness: δ = 5.0 x / √Re_x = 5.0 √(νx/U_∞) Displacement thickness: δ* = 1.72 x / √Re_x = 0.344 δ Momentum thickness: θ = 0.664 x / √Re_x = 0.133 δ Shape factor: H = δ*/θ = 2.59 (laminar)

Local skin friction: C_f,x = 0.664 / √Re_x (τ_w = C_f × ½ρU²) Average skin friction: C̄_f = 1.328 / √Re_L (integrated over plate length L)

Re_x = U_∞ x / ν

Transition

Flat plate: Re_x,tr ≈ 5×10⁵ (typical; ranges 2×10⁵ to 4×10⁶ depending on freestream turbulence) Turbulence intensity TI = u'/U∞ effect: higher TI → earlier transition Clean tunnel (TI < 0.1%): Re_tr = 3-4×10⁶; wind/rain (TI ≈ 5%): Re_tr ≈ 10⁵

Turbulent Boundary Layer (Flat Plate)

Turbulent BL Correlations (Empirical)

δ = 0.37 x Re_x^(-1/5) (1/7 power law) δ* = δ/8 (for 1/7 power law profile) θ = 7δ/72 = δ × 7/72 H = δ*/θ = 1.29 (turbulent, lower than laminar)

Local skin friction: C_f,x = 0.0592 Re_x^(-1/5) (Schlichting, 1/5 power law) More accurate: C_f = 0.455 / [log(Re_x)]^2.58 (Schlichting-Prandtl)

Average (mixed BL, transition at Re_tr): C̄_f = 0.074 Re_L^(-1/5) - A/Re_L A = 1700 (Re_tr = 5×10⁵); A = 3300 (Re_tr = 3×10⁵)

Log-Law Profile

u/u_τ = (1/κ) ln(y u_τ/ν) + B (log law, 30 < y⁺ < ~500) κ = 0.41 (von Kármán constant), B = 5.0 Viscous sublayer: u/u_τ = y⁺ (y⁺ < 5) Buffer layer: 5 < y⁺ < 30

Falkner-Skan (Pressure Gradient)

Similar solutions for: U_e(x) = U_0 x^m η = y √(m+1)U_0 x^m/(2νx) (similarity variable) β = 2m/(m+1) (Hartree pressure gradient parameter)

β > 0 (m > 0): favorable pressure gradient (U_e increasing) — BL thins, stabilizes β = 0 (m = 0): flat plate (Blasius) β < 0 (m < 0): adverse pressure gradient — BL thickens, approaches separation β = -0.19884: separation (τ_w = 0); corresponds to flow past stagnation → separation point

von Kármán integral equation: dθ/dx = C_f/2 - (δ*/U_e)(dU_e/dx) × H₁

H₁ = (H+1) → shape factor evolution; dθ/dx increases with adverse dP/dx

Separation

Separation: τ_w = 0; reversed flow zone begins Turbulent BL separates MUCH later than laminar (higher momentum near wall from mixing) → Dimples on golf ball trip BL turbulent → delayed separation → less drag

Separation prediction (Head's method): Compute H(x) as BL develops; separation when H → 3.5 (laminar) or H → 2.5 (turbulent)

Thermal Boundary Layer

For Pr near unity: thermal BL thickness ≈ hydrodynamic BL thickness δ_T / δ = Pr^(-1/3) (Pr > 0.6, turbulent analogy) Stanton number: St = C_f/2 × Pr^(-2/3) (Reynolds analogy modified, turbulent) Nu_x = 0.0296 Re_x^(4/5) Pr^(1/3) (turbulent, flat plate, Dittus-Boelter analog)

Output

Provide: δ(x) [mm], δ*(x), θ(x), H (shape factor), C_f,x (local), C̄_f (average), transition location x_tr [m], skin friction drag F_D [N], separation location (if pressure gradient given), heat transfer Nu_x if thermal BL needed.