ashfordeOU Boundary Layer Theory (aerodynamics/boundary-layer/boundary-layer-theory)
Use when the task is flat-plate boundary-layer estimation: thickness, displacement and momentum thickness, skin friction, and the laminar to turbulent transition.
Domain quick reference
- The boundary layer is the thin viscous region next to a surface where the velocity rises from zero at the wall (no-slip) to the edge value U_e. The local Reynolds number Re_x = rho * U * x / mu = U * x / nu sets the flow regime at station x.
- Laminar flat plate (Blasius similarity solution, 1908): 99-percent thickness delta = 5.0 * x / sqrt(Re_x), displacement thickness delta* = 1.7208 * x / sqrt(Re_x), momentum thickness theta = 0.664 * x / sqrt(Re_x), shape factor H = delta* / theta = 2.5916, local skin friction Cf = 0.664 / sqrt(Re_x), average Cf over one side = 1.328 / sqrt(Re_x).
- Turbulent flat plate (1/7 power law, Re_x up to about 1e7): delta = 0.37 * x / Re_x^(1/5), delta* = delta / 8, theta = 7 * delta / 72, H = 9 / 7 = 1.286, local Cf = 0.0592 / Re_x^(1/5), average Cf = 0.074 / Re_x^(1/5). Above about 1e7 prefer the fully turbulent log-law correlation Cf = 0.455 / (log10 Re_x)^2.58.
- Displacement thickness delta* = integral_0^inf (1 - u / U_e) dy is the mass deficit of the layer; the outer flow behaves as if the body were thickened by delta*. Momentum thickness theta = integral_0^inf (u / U_e) * (1 - u / U_e) dy is the momentum deficit.
- The von Karman momentum integral d(theta)/dx + (H + 2) * (theta / U_e) * dU_e/dx = Cf / 2 relates the thickness growth to the edge velocity gradient for pressure-gradient layers.
- Transition: the laminar layer destabilizes through Tollmien- Schlichting waves. On a smooth flat plate with low free-stream turbulence the transition Reynolds number is near Re_x = 5e5; values range from about 3e5 (rough surface, high turbulence) to 3e6 (very quiet flow). Favorable pressure gradients delay transition, adverse gradients advance it.
- Reynolds-number regimes (flat plate): laminar below the transition Reynolds number, turbulent above; a fully laminar boundary layer over a whole chord is rare at flight Reynolds numbers (typically 1e6 to 1e8).
Workflow
- Establish the condition: speed U, density rho, dynamic viscosity mu, and the station x (or chord position).
- Compute Re_x with reynolds_number; use kinematic_viscosity when only nu = mu / rho is available.
- Pick the regime with classify_regime (default transition at Re_x = 5e5) and use the laminar or turbulent functions.
- Estimate delta, delta*, theta with the Blasius or 1/7 power-law thickness functions; form the shape factor with shape_factor.
- Evaluate local and average skin friction with the cf functions; switch to cf_turbulent_log_law above Re_x ~ 1e7.
- State the transition assumption explicitly; recheck when the surface is rough or the free-stream turbulence is high.
Pitfalls
- Using the Blasius laminar values beyond transition: the turbulent layer is several times thicker and draggier at the same Re_x.
- Quoting delta where delta* was meant: the displacement thickness is about one third of delta for laminar flow and one eighth for turbulent flow.
- Mixing the local Cf with the average (total-drag) Cf: the laminar pair differs by a factor of two (0.664 vs 1.328), the turbulent pair by about 25 percent (0.0592 vs 0.074).
- Using the freestream speed where the edge velocity U_e differs (wing suction side, curved bodies): the correlations then misstate the thicknesses.
- Treating the transition point as fixed: roughness, free-stream turbulence, and pressure gradient move it by orders of magnitude.
- Applying the 1/7 power law near the wall or at very high Re_x: the profile has an infinite wall gradient; use the log-law correlation for friction above Re_x ~ 1e7.
- Forgetting the two-sided factor when converting skin friction to total surface drag: delta and Cf are per side.
- Mixing the length scale: the delta correlations are keyed to the local Re_x at station x, not a global chord Reynolds number.
Behavior contract (gate 3)
The boundary-layer logic is exercised by the gate 3 contract test: scripts/test_boundary_layer_theory.py against scripts/boundary_layer_theory_logic.py (stdlib unittest, offline). Run: python3 scripts/test_boundary_layer_theory.py
Compliance
- The Blasius solution and the 1/7 power-law and log-law correlations are classical physics results (public-domain knowledge), paraphrased here. NACA Report 824 is cited as reference only for the aerodynamics validation context; no proprietary or copyrighted text is reproduced.
- compliance: STANDARDS-REF, gated: false.