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Mangler Transformation

The Mangler transformation 1 maps the axisymmetric BL equations to an equivalent 2D boundary layer, allowing the Falkner-Skan similarity solver to be applied directly to bodies of revolution such as cones, ogives, and cylinders.

Warning

The Mangler transformation applies to 2D (Falkner-Skan) flows only. Extension to swept axisymmetric bodies (Falkner-Skan-Cooke) is not standard and is not covered here.

Coordinate Transformation

Introduce transformed coordinates

\[ \tilde{x} = \frac{1}{L^2}\int_0^x r_0(\xi)^2\,d\xi, \qquad \tilde{y} = \frac{r_0(x)}{L}\,y \]

where \(r_0(x)\) is the local body radius, \(y\) is the physical wall-normal coordinate, and \(L\) is an arbitrary reference length. Under this map the axisymmetric continuity equation takes the standard 2D form in \((\tilde{x}, \tilde{y})\), so the momentum and energy equations in tilde space are identical to those of a 2D flat-plate or wedge flow.

Inverse Transform

After solving the 2D similarity problem in tilde space (obtaining \(\tilde{y}(\eta)\) via the Levy-Lees inverse), the physical wall-normal coordinate follows from inverting the second relation:

\[ y(x, \eta) = \frac{L}{r_0(x)}\,\tilde{y}(\eta) \]

The streamwise direction requires inverting \(\tilde{x}(x)\) for the body geometry of interest; \(y\) does not depend on this inversion.

Composition with the Existing eta2y Transforms

The Mangler map wraps the standard Levy-Lees eta2y transform in three steps:

  1. Preprocess — compute the Mangler-transformed station \(\tilde{x}\):

$\(\tilde{x} = \frac{1}{L^2}\int_0^x r_0(\xi)^2\,d\xi\)$

  1. Call eta2y with \(\tilde{x}\) in place of \(x\) to obtain \(\tilde{y}(\eta)\).

  2. Postprocess — scale back to physical \(y\):

$\(y = \frac{L}{r_0(x)}\,\tilde{y}\)$

The eta2y transform itself is unchanged; Mangler enters only as a coordinate substitution before and a radial rescaling after.

Common Body Geometries

Body \(r_0(x)\) \(\tilde{x}\)
Sharp cone (half-angle \(\theta\)) \(x\sin\theta\) \(\dfrac{\sin^2\theta}{3L^2}x^3\)
Cylinder (radius \(R\)) \(R\) \(\dfrac{R^2}{L^2}x\)
Ogive / arbitrary body tabulated \(r_0(x)\) numerical quadrature

For a sharp cone the Mangler \(\tilde{x}\) grows as \(x^3\), which compresses the effective streamwise coordinate and produces a thinner equivalent 2D BL relative to the physical arc length.

Reference Length Choice

\(L\) cancels in the final physical coordinate \(y\) because \(\tilde{y} \propto 1/L\) while the postprocessing scale is \(L/r_0\). The choice of \(L\) therefore affects only the numerical magnitude of \(\tilde{x}\) and \(\tilde{y}\) individually, not the computed \(y\). A convenient default is \(L = 1\,\text{m}\) (or \(L = r_\text{nose}\) for ogive bodies) to keep \(\tilde{x}\) of order unity.

  1. Mangler, W. (1948). Zusammenhang zwischen ebenen und rotationssymmetrischen Grenzschichten in kompressiblen Flüssigkeiten. Zeitschrift für Angewandte Mathematik und Mechanik, 28(4), 97–103. https://doi.org/10.1002/zamm.19480280401 ↩