# Nonuniqueness of Triple-deck Solutions for Axisymmetric Supersonic Flow with Separation

• Ph. Gittler
• A. Kluwick
Conference paper
Part of the International Union of Theoretical and Applied Mechanics book series (IUTAM)

## Abstract

The triple-deck equations (e.g.[1]) for axisymmetric supersonic laminar boundary layers,[2]
$$\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0,u\frac{{\partial u}}{{\partial x}} + v\frac{{\partial v}}{{\partial y}} = - \frac{{dy}}{{dx}} + \frac{{{\partial ^2}u}}{{\partial {y^2}}}$$
$$u=v=0fory=F\left(x\right),u=yforx\to-\infty,u=y+ A\left(x\right)fory\to\infty$$
$$p\left( x \right) = - A'\left( x \right) + \frac{1}{a}\int\limits_{ - \infty }^x W \left( {\frac{{x - \xi }}{a}} \right)A'\left( \xi \right)d\xi ,W\left( z \right) = \int\limits_0^\infty {\frac{{\exp \left( { - \lambda z} \right)}}{{K_1^2\left( \lambda \right) + {\pi ^2}I_1^2\left( \lambda \right)}}} \frac{{d\lambda }}{\lambda },$$
where
$$F(x) = \left\{ {\begin{array}{*{20}{c}} 0 \\ a \\ {ax} \end{array}} \right.\left( {\frac{{{x^2}}}{{4\rho }} + \frac{x}{2} + \frac{\rho }{4}} \right)\begin{array}{*{20}{c}} {forx < - \rho } \\ {for - \rho \underline{\underline < } x\underline{\underline < } \rho } \\ {forx > \rho } \end{array}$$
describes a smoothed cylinder-cone configuration, have been solved for a=1., ρ=1. and moderately large values of the scaled flare angle α. Here a denotes the scaled radius of the cylinder.

## Keywords

Supersonic Flow Cone Angle Separation Point Pressure Plateau Planar Flow
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Smith, F.T.: On the High Reynoldsnumber Theory of Laminar Flows, IMA J. Appl. Math. 28 (1982), 207–281.
2. 2.
Gittler, Ph. and Kluwick, A.: Triple-Deck Solutions for Supersonic Flows past Flared Cylinders, J. Fluid Mech. (1987), in press.Google Scholar