Falkner-Skan Equations

The Falkner-Skan (FS) equations are a obtained by reducing the 2D compressible boundary layer equations to an ODE system. All assumptions from the boundary layer equations carry over.

Inherited assumptions

Steady, two-dimensional, laminar flow
Calorically perfect gas (\(\gamma\), \(c_p\), \(\mathrm{Pr}\) constant)
Temperature-dependent viscosity \(\mu = \mu(T)\)
Thin boundary layer (\(\delta \ll L\))

Similarity Ansatz

\[u_e = C x^m, \qquad \psi = \sqrt{2\xi}\,f(\eta), \qquad u = u_e\,f'(\eta)\]

Outer Flow

\[-\frac{dp}{dx} = \rho_e u_e \frac{du_e}{dx}\]

Definitions

Stream function ²³:

\[\rho u = \frac{\partial\psi}{\partial y}, \qquad \rho v = -\frac{\partial\psi}{\partial x}\]

Levy-Lees similarity coordinates ¹²:

\[\xi = \int_0^x \rho_e \mu_e u_e\,dx', \qquad \eta = \frac{u_e}{\sqrt{2\xi}}\int_0^y \rho\,dy'\]

Hartree parameter, Chapman-Rubesin factor, temperature ratio:

\[\beta_H = \frac{2m}{m+1}, \qquad C = \frac{\rho\mu}{\rho_e\mu_e}, \qquad \tau = \frac{T}{T_e}\]

Edge Mach number, Prandtl number:

\[M_e = \frac{u_e}{\sqrt{\gamma R T_e}}, \qquad \mathrm{Pr} = \frac{\mu c_p}{k}\]

ODE System

The 2D compressible BL equations reduce to ODEs in \(\eta\) (see derivation below).

x-momentum:

\[(Cf'')' + ff'' + \beta_H(\tau - f'^2) = 0\]

Energy:

\[\left(\frac{C}{\mathrm{Pr}}\tau'\right)' + f\tau' + (\gamma-1)M_e^2\!\left[Cf''^2 - \beta_H\tau f'\right] = 0\]

Boundary Conditions

Wall (\(\eta = 0\)):

\[f(0) = 0, \qquad f'(0) = 0\]

Isothermal: \(\tau(0) = T_w/T_e\) (prescribed)
Adiabatic: \(\tau'(0) = 0\)

Edge (\(\eta \to \infty\)):

\[f' = 1, \qquad \tau = 1\]

Cohen, C. B. & Reshotko, E. (1955). Similar solutions for the compressible laminar boundary layer with heat transfer and pressure gradient. NACA TN 1293. PDF ↩
White, F. M. (2006). Viscous Fluid Flow, 3rd ed. McGraw-Hill, New York. ↩↩
Schlichting, H. & Gersten, K. (2017). Boundary Layer Theory, 9th ed. Springer. DOI: 10.1007/978-3-662-52919-5 ↩