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Similarity Method

This method estimates skin friction directly from a Falkner-Skan similarity solution1.

It uses the simbl similarity solver2 to obtain the wall-shear quantity \(f''(0)\), where the normalized streamwise velocity is \(\overline{u} = u/u_e\). Thus,

\[ f''(0) = \left.\frac{\partial \overline{u}}{\partial \eta}\right|_w, \]
\[ \left.\frac{\partial u}{\partial \eta}\right|_w = u_e f''(0). \]

Convert to physical wall-normal gradient using the Falkner-Skan coordinate transformation (see Falkner-Skan Derivation):

\[ \left.\frac{\partial \eta}{\partial y}\right|_w = \left.\frac{\rho u_e}{\sqrt{2\xi}}\right|_w = \frac{\rho_w u_e}{\sqrt{2\xi}}. \]

Here \(\xi\) is the transformed streamwise coordinate,

\[ \xi = \int_0^x \rho_e\mu_e u_e\,dx', \]

so

\[ \left.\frac{\partial u}{\partial y}\right|_w = \left.\frac{\partial u}{\partial \eta}\right|_w \left.\frac{\partial \eta}{\partial y}\right|_w = f''(0)\frac{\rho_w u_e^2}{\sqrt{2\xi}}. \]

Substitute into the skin-friction definition from the Skin Friction and Stanton Number overview to get

\[ c_f(\xi) = \frac{2\mu_w}{\rho_e u_e^2}\overbrace{\left.\frac{\partial u}{\partial y}\right|_w}^{f''(0)\frac{\rho_w u_e^2}{\sqrt{2\xi}}} = \frac{\mu_w\rho_w}{\rho_e}\,\frac{\sqrt{2}}{\sqrt{\xi}}\,f''(0). \]

The Stanton number is computed with the Reynolds analogy:

\[ C_h = \frac{c_f}{2\,\mathrm{Pr}^{2/3}}. \]

Limits

Zero Pressure Gradient Flat Plate

For a zero-pressure-gradient flat plate,

\[ \xi = \rho_e\mu_e u_e x. \]

The edge quantities (subscript e) are freestream quantities (subscript \(\infty\)). Therefore,

\[ \xi = \rho_\infty u_\infty \mu_\infty x. \]

Using the unit Reynolds number,

\[ Re_1 = \frac{\rho_\infty u_\infty}{\mu_\infty}, \]

and the Reynolds number based on the streamwise coordinate,

\[ Re_x = Re_1 x, \]

so

\[ \xi = Re_1 \mu_\infty^2 x = Re_x \mu_\infty^2. \]

Substituting this into the expression above and recognizing \(\rho_e = \rho_\infty\) results in

\[ c_f = \frac{\mu_w\rho_w}{\rho_\infty} \frac{\sqrt{2}}{\sqrt{Re_x \mu_\infty^2}} f''(0) = \frac{\mu_w\rho_w}{\mu_\infty \rho_\infty} \sqrt{\frac{2}{Re_x}} f''(0). \]

Circular Cone

For axisymmetric flow over a circular cone, the corresponding similarity reduction follows through the Mangler transformation. This gives the cone analogue of the flat-plate limit while preserving the same underlying Falkner-Skan structure.

In this limit, the edge quantities do not reduce to freestream values. Instead, they are the local cone-edge quantities from the Taylor-Maccoll solution:

\[ \begin{aligned} \rho_e &= \rho_{e,\mathrm{TM}} = \text{const.}, \\ u_e &= u_{e,\mathrm{TM}} = \text{const.}, \\ \mu_e &= \mu_{e,\mathrm{TM}} = \text{const.} \end{aligned} \]

With \(x_\mathrm{cone}\) being the distance along the cone surface measured from the virtual cone origin, the Mangler-equivalent flat-plate coordinate is \(\tilde{x}_{\mathrm{cone}}\):

\[ \tilde{x}_{\mathrm{cone}} = \frac{1}{L^2}\int_0^{x_\mathrm{cone}} r_0(\xi)^2\,d\xi = \frac{\sin^2\theta_\mathrm{cone}}{3L^2}\,x_\mathrm{cone}^3. \]

The corresponding local Reynolds number is

\[ Re_{x,\mathrm{cone}} = \frac{\rho_{e,\mathrm{TM}}\,u_{e,\mathrm{TM}}\,\tilde{x}_{\mathrm{cone}}}{\mu_{e,\mathrm{TM}}}. \]

The skin-friction expression keeps the same similarity structure, with the correction due to the Mangler transform:

\[ c_{f,\mathrm{cone}} = \sqrt{3}\,\frac{\mu_w\rho_w}{\rho_{e,\mathrm{TM}}\,\mu_{e,\mathrm{TM}}} \sqrt{\frac{2}{Re_{x,\mathrm{cone}}}}\,f''(0). \]

References

  1. Falkner, V.M. and Skan, S.W. (1931). Solutions of the boundary-layer equations. Philosophical Magazine, 12(80), 865-896. ↩

  2. Hader, C. (2026). simbl: Similarity boundary layer solver (Version 0.2.0) [Software]. DOI: 10.5281/zenodo.20648457. Repository: uahypersonics/similarity-bl. ↩