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Blasius limit

At \(M_e \to 0\), adiabatic wall, \(\beta = 0\) (flat plate), the compressible Falkner-Skan system reduces to the classical incompressible Blasius equation:

\[ f''' + \tfrac{1}{2} f f'' = 0 \]

The expected wall shear in the Levy-Lees non-dimensionalization is \(f''(0) = 0.4696\).

Why is the Levy-Lees value 0.4696, not the Blasius value 0.3321?

The Levy-Lees similarity coordinates are defined as:

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

For a flat plate with uniform edge conditions \(\xi = \rho_e \mu_e u_e x\), so \(\sqrt{2\xi} = \sqrt{2\rho_e \mu_e u_e x}\) and the wall-normal coordinate becomes:

\[ \eta_{LL} = \frac{u_e}{\sqrt{2\rho_e \mu_e u_e x}} \int_0^y \rho\,dy' \]

At \(M_e \to 0\), \(\rho \to \rho_e\), so the integral reduces to \(\rho_e y\):

\[ \eta_{LL} = \frac{u_e \rho_e}{\sqrt{2\rho_e \mu_e u_e x}}\,y = \sqrt{\frac{\rho_e u_e}{2\mu_e x}}\,y = \frac{1}{\sqrt{2}}\underbrace{\sqrt{\frac{u_e}{\nu x}}\,y}_{\eta_{Blasius}} \]

where \(\nu = \mu_e/\rho_e\). So:

\[ \eta_{LL} = \frac{1}{\sqrt{2}}\,\eta_{Blasius} \qquad \Longleftrightarrow \qquad \eta_{Blasius} = \sqrt{2}\,\eta_{LL} \]

Both systems describe the same physical velocity ratio \(u/u_e = f'\), so the two stream functions are related by:

\[ f'_{LL}(\eta_{LL}) = f'_{Blasius}(\eta_{Blasius}) = f'_{Blasius}(\sqrt{2}\,\eta_{LL}) \]

Differentiating both sides with respect to \(\eta_{LL}\) via the chain rule:

\[ f''_{LL}(\eta_{LL}) = \sqrt{2}\,f''_{Blasius}(\sqrt{2}\,\eta_{LL}) \]

Evaluating at the wall (\(\eta_{LL} = 0\)):

\[ f''_{LL}(0) = \sqrt{2}\,f''_{Blasius}(0) = \sqrt{2} \times 0.33206 \approx 0.4696 \]

At \(M_e \to 0\), adiabatic wall, \(\tau = T/T_e \approx 1\) everywhere.

Results

Profiles of \(f'(\eta)\), \(f''(\eta)\), and \((\tau - 1) \times 10^n\) (scaled so the deviation is \(O(1)\)). Dashed lines mark \(f''(0)_\text{ref} = 0.4696\) and \(\tau - 1 = 0\).

Me=0.1

Me=0.01

Me=0.001

Me=0.0001

Both \(|f''(0) - f''(0)_\text{ref}|\) and \(\max|\tau - 1|\) converge to zero as \(M_e \to 0\):

Convergence

Both errors converge monotonically to zero as \(M_e \to 0\), confirming that the solver correctly recovers the Blasius solution in the incompressible limit.

Run

The verification script is vnv/verification/falkner_skan/blasius/verification_blasius.py.

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python verification_blasius.py