Hartree (1937) wall shear
Sweeps \(\beta \in \{0.3,\, 0.4,\, 0.6,\, 0.8,\, 1.2,\, 1.6\}\) at \(M_e = 0.01\), adiabatic wall, and compares \(f''(0)\) against the tabulated values from Hartree (1937)1.
Incompressible approximation
The compressible solver is run at \(M_e = 0.01\) with an adiabatic wall. At this Mach number \(T \approx T_e\) throughout the boundary layer, so \(\mu/\mu_e \to 1\) and the compressible equations reduce to the incompressible Falkner-Skan system solved by Hartree. The Blasius limit case shows that the solver converges toward the incompressible solution as \(M_e \to 0\).
| Parameter | Value |
|---|---|
| \(M_e\) | 0.01 |
| \(T_e\) | 300 K |
| Wall BC | adiabatic |
Results
\(f'(\eta)\) and \(f''(\eta)\) profiles at each \(\beta\). The red dot marks \(f''(0)\) at the wall.
| \(\beta\) | \(f''(0)\) Hartree (1937)1 | \(f''(0)\) solver | rel. err. (%) |
|---|---|---|---|
| 0.3 | 0.7745 | 0.77474 | 0.032 |
| 0.4 | 0.854 | 0.85440 | 0.047 |
| 0.6 | 0.995 | 0.99580 | 0.080 |
| 0.8 | 1.121 | 1.12021 | 0.071 |
| 1.2 | 1.335 | 1.33562 | 0.046 |
| 1.6 | 1.522 | 1.52137 | 0.041 |
Sensitivity to initial guess and \(\eta_{max}\)
Convergence can be sensitive to the initial guess and the domain height \(\eta_\text{max}\). This is demonstrated at \(\beta = 0.6\) by varying both parameters using two different initialization methods.
| Method | Description |
|---|---|
| Cold start | Default initial guess: \(f''(0) = 0.7\), \(g = 1\) |
| \(\beta\)-continuation seed | Solve at a smaller \(\beta\) (milder pressure gradient), then step toward the target \(\beta\), using each converged result as the initial guess for the next step |
For \(\eta_\text{max} \geq 8\) the cold start leads to a spurious profile that happens to fulfill the boundary conditions exactly but clearly does not reach an asymptotic freestream. The continuation seed continues to converge to the correct profile.
Run
The verification script is
vnv/verification/falkner_skan/hartree/verification_hartree.py.
The initial-guess sensitivity figures are generated by
vnv/verification/falkner_skan/hartree/initial_guess_sensitivity_beta_0pt6.py.