Home HomeD.G. Crighton, Cambridge University
This talk will report work, part of a joint investigation with G. I. Barenblatt, on shock wave propagation when small-amplitude time-dependent fluctuations are created behind the shock. The analysis is carried out in the framework of Burgers' equation, and we seek a self-consistent theory in a two-parameter plane (perturbation amplitude, Reynolds number) showing how the perturbations are convected towards the shock, and how they significantly distort the shock profile while being absorbed by the shock. A closed equation is found, within a self-consistent framework, for the mean shock profile, and this is solved by singular perturbation techniques to show that in certain regions of parameter space it is possible for small perturbations to cause a great broadening of the shock width, and a splitting of the shock itself into three regions. Two of these are conventional Taylor shocks, in which convection and thermoviscous diffusion balance; and in the region between them there is a much less rapid transition, deep within the shock, controlled by the fluctuations themselves.
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John K. Hunter, University of California at Davis (U.S.A.)
Moysey Brio, University of Arizona (U.S.A.)
The two-dimensional Burgers equation, or unsteady transonic small disturbance equation, is the simplest model equation that describes shock wave propagation in several space dimensions. It provides a quantitative, asymptotic description of the reflection, diffraction, and focusing of weak shock waves in gas dynamics and other physical applications. We will present a number of numerical solutions of the two-dimensional Burgers equation which illustrate these phenomena. In particular, high resolution numerical solutions of the equation for irregular weak shock reflection provide evidence of a small supersonic bubble behind the triple point. Based on these numerical results, we will present a theoretically consistent structure of irregular weak shock wave reflection in which there is a centered expansion fan at the triple point and in which a nonuniform supersonic wave behind the triple point is generated by the reflection of incoming characteristics off an embedded sonic line.
sv970563.pdf (Scanned) TOPGabi Ben-Dor, Ben-Gurion University of the Negev (Israel)
The state-of-the-art regarding the hysteresis phenomenon in the regular-to-Mach reflection transition in steady flows as it has been established in the past decade is summarized.
sv970561.pdf (Scanned) TOPN. Kevlahan, LMD-CNRS (France)
A new theory of the propagation of weak shocks into non-uniform, two-dimensional flows is introduced. The theory is based on a description of shock propagation in terms of a manifold equation together with compatibility conditions for shock strength and its normal derivatives behind the shock. This approach was developed by Ravindran and Prasad (1993) for shocks of arbitrary strength propagating into a medium at rest and is extended here to non-uniform media and restricted to moderately weak shocks. The theory is tested against known analytical solutions for cylindrical and plane shocks, and against a full direct numerical simulation (DNS) of a shock propagating into a sinusoidal shear flow. The test against DNS shows that the present theory accurately predicts the evolution of a moderately weak shock front, including the formation of shock-shocks due to shock focusing. The theory is then applied to the focusing of an initially parabolic shock, and to the propagation of an initially straight shock into a variety of simple flows (sinusoidal shear, vortex array, point-vortex array) exhibiting some fundamental properties of turbulent flows. A number of relations are deduced for the variation of shock quantities with initial shock strength M_S0 and the Mach number of the flow ahead of the shock M_U (e.g. separation of shock-shocks and maximum shock strength at a focus). It is found that shock-shocks are likely to form in turbulent flows with M_t/M_1N > 0.14-0.25, where M_t is the average Mach number of the turbulence and M_1N is the Mach number of the shock in a flow at rest. The shock moves up to 1.5% faster in a two-dimensional vortex array than in uniform flow.
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