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.

Flare

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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