Mangler Transformation

Mangler transformation, also known as Mangler-Stepanov transformation (Stepanov 1947, Mangler 1948, Schlichting 1955), reduces the axisymmetric boundary layer equations to the plane boundary layer equations.

The transformation transforms the equations of axisymmetric boundary layer with external velocity $U$ in terms of original variables $x,y,u,v$ into the equations of plane boundary layer with external velocity ${\bar {U}}$ in terms of the new variables ${\bar {x}},{\bar {y}},{\bar {u}},{\bar {v}}$. The transformation is given by the formulas

${\begin{array}{l}\displaystyle {\bar {x}}={\frac {1}{L^{2}}}\int \limits _{0}^{x}r^{2}(x)dx,\quad {\bar {y}}={\frac {r(x)}{L}}y,\\\displaystyle {\bar {u}}=u,\quad {\bar {v}}={\frac {L}{r}}\left(v+{\frac {r'}{r}}yu\right),\\\displaystyle {\bar {U}}=U,\end{array}}$

where $L$ is a constant length, $r(x)$ is the distance from the point on the wall to the axis.

References

• Stepanov, E. I. (1947), "Integration of laminar boundary-layer equations for motions with a symmetry axis", Prikl. Mat. Mekh., 9 (1).
• Mangler, W. (1948), "Zusammenhang zwischen ebenen und rotationssymmetrischen Grenzschichten in kompressiblen Flüssigkeiten", Journal of Applied Mathematics and Mechanics, 28 (4): 97–103, Bibcode:1948ZaMM...28...97M, doi:10.1002/zamm.19480280401.
• Schlichting, H. (1955), Boundary-layer theory, trans. by J. Kestin, London: Pergamon Press