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Falkner-Skan-Cooke Equations

The Falkner-Skan-Cooke (FSC) equations obtained by reducing the quasi-3D compressible boundary layer equations to an ODE system 1.

Inherited assumptions
  • Steady, laminar flow with three velocity components \((u, v, w)\)
  • Homogeneous flow in the spanwise direction (\(\partial/\partial z = 0\))
  • Calorically perfect gas (\(\gamma\), \(c_p\), \(\mathrm{Pr}\) constant)
  • Dynamic viscosity via Sutherland's law
  • Thin boundary layer (\(\delta \ll L\))

Similarity Ansatz

The edge velocity follows a power law in the transformed coordinate:

\[U_e(\tilde{\xi}) = C_1\cdot\tilde{\xi}^m\]

The Falkner-Skan similarity variable and stream function are:

\[\eta = \bar{\eta}\sqrt{\frac{m+1}{2}\frac{U_e}{\nu_{e0}\tilde{\xi}}}, \qquad \psi = f(\eta)\sqrt{\frac{2\nu_{e0}U_e\tilde{\xi}}{m+1}}\]

where \(\nu_{e0} = \mu_{e0}/\rho_{e0}\) is the kinematic viscosity at the reference point. The normalized velocity components are 1:

\[f'(\eta) = \frac{U}{U_e} = \frac{u}{u_e}, \qquad g(\eta) = \frac{w}{w_e}, \qquad \tau(\eta) = \frac{T}{T_e}\]

Illingworth-Stewartson Transformation

The IS transformation removes density from the governing equations 1:

\[d\tilde{\xi} = \frac{\mu_e a_e \rho_e}{\mu_{e0} a_{e0} \rho_{e0}}\,dx, \qquad d\bar{\eta} = \frac{a_e \rho}{a_{e0} \rho_{e0}}\,dy\]

where \(a\) is the local sound speed and subscript \(e0\) denotes conditions at the reference edge point. The stream function \(\psi\) in the transformed space is defined as:

\[\frac{\partial\psi}{\partial\bar{\eta}} = U, \qquad \frac{\partial\psi}{\partial\tilde{\xi}} = -V\]

Definitions

Hartree parameter:

\[\beta_H = \frac{2m}{m+1}\]

Chapman-Rubesin factor:

\[ C \equiv \frac{\rho\mu}{\rho_e\mu_e} \]

Compressibility parameters:

\[ K = \frac{1 + \dfrac{\gamma-1}{2}M_e^2}{1 + \dfrac{\gamma-1}{2}M_e^2\cos^2\!\Lambda}, \qquad S = 1 + \frac{\gamma-1}{2}M_e^2\cos^2\!\Lambda \]

where \(\Lambda\) is the local swept angle and \(M_e\) the local edge Mach number.

Deviation from Liu (2021)

Liu 1 defines \(K\) and \(S\) in terms of a reference Mach number \(Ma_{e,\mathrm{ref}}\) and a streamwise parameter \(\chi = (\tilde{\xi}/\tilde{\xi}_\mathrm{ref})^m\) that tracks the variation of edge conditions along the surface. Here we adopt a locally self-similar formulation, which sets \(\chi = 1\) and consequently \(Ma_{e,\mathrm{ref}} = M_e\). The parameters \(\beta_H\), \(K\), \(S\), and \(T_e\) are then known inputs evaluated from the local edge conditions at each station.

ODE System

x-momentum 1:

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

The Falkner-Skan x-momentum is:

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

z-momentum (crossflow) 1:

\[(Cg')' + fg' = 0\]

Energy 1:

\[\begin{aligned} &\left(\frac{C}{\mathrm{Pr}}\tau'\right)' + (S-1)\left(C(f'^2)'\right)' + (K-1)S\left(C(g^2)'\right)' \\ &\quad + f\!\left[\tau' + (S-1)(f'^2)' + (K-1)S(g^2)'\right] = 0 \end{aligned}\]
Compare to Falkner-Skan

The Falkner-Skan energy equation is:

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

Setting \(\Lambda = 0\) (no sweep angle) gives \(K = 1\), so the crossflow kinetic energy terms \((K-1)S(\cdot)\) vanish and \(g \equiv 0\). The FSC energy equation reduces to:

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

This is a total-temperature (stagnation-enthalpy) form with \(S = (\gamma-1)M_e^2\). The Falkner-Skan equation above is the equivalent static-temperature form — the two are related by substituting the x-momentum ODE into the energy equation.

Boundary Conditions

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

\[f = 0, \qquad f' = 0, \qquad g = 0\]
  • Isothermal: \(\tau(0) = T_w/T_e\) (prescribed)
  • Adiabatic: \(\tau'(0) = 0\)

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

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

  1. Liu, Z. (2021). Compressible Falkner–Skan–Cooke boundary layer on a flat plate. Physics of Fluids, 33(12). DOI: 10.1063/5.0075233 ↩↩↩↩↩↩↩