Mangler Transformation

Mangler

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

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

where ${\displaystyle L}$ is a constant length, ${\displaystyle r(x)}$ 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",
  doi:10.1002/zamm.19480280401
  .
• Schlichting, H. (1955), Boundary-layer theory, trans. by J. Kestin, London: Pergamon Press