| dc.description.abstract | Waves are ubiquitous in the physical world. The study of general laws that govern their propagation is, therefore, important. In particular, nonlinear hyperbolic waves are frequently encountered, for some of which the velocity of propagation depends on the amplitude of the wave. The distinguishing effect of this type of nonlinearity is its tendency to distort the initially smooth wave profile, leading to wave breaking and formation of shocks. Across a shock, the medium undergoes sudden and often considerable changes in various parameters like velocity, pressure, and temperature.
Some numerical results of curved nonlinear wavefront propagation, wherein the front develops shocks after some time, are available in Prasad and Sangeeta (1999). It is interesting to study the propagation of a shock front itself. A simple approximate theory for the propagation of curved shocks (Whitham, 1974) shows that there are nonlinear waves propagating on a shock front itself. As a result, kinks (called shock-shocks by Whitham) appear on the shock front (Prasad, 1995). The behaviour of kinks appearing on a weak shock front was experimentally studied by Sturtevant and Kulkarni (1976). Through their carefully designed experiments, they showed that nonlinear effects play a major role in the propagation of a weak shock front.
The questions which modern applied science asks of the area of nonlinear hyperbolic equations concern an analysis of the equations, a search for effective numerical methods, and an understanding of the solution. This thesis embodies the results of numerical investigations of the effect of nonlinearity on the propagation of arbitrarily curved weak shock fronts.
The thesis consists of four chapters. Chapter 1 is a brief introduction and describes the basic properties and methods peculiar to hyperbolic equations. It mainly includes the discussion of some of the ray methods for linear hyperbolic waves. These methods, in short-wavelength approximation (geometrical optics method), provide an adequate description of hyperbolic waves in terms of rays and propagation laws (transport equations) for the amplitude along the rays and have played an important role in finding the successive positions of the front. Linear geometrical optics, in the leading-order approximation, fails at caustics as the amplitude becomes infinite there. This necessitates the application of a nonlinear theory to the problem of propagation of a front.
Whitham’s theory of geometrical shock dynamics was the first one which came close to solving the problem with reasonably good results in some applications of the theory. This theory, however, does not take into account the effect of the flow behind the shock on the propagation of the shock front. The new theory of shock dynamics (NTSD) given by Ravindran and Prasad (1990) takes these effects into account and provides a powerful method for finding the successive positions of the shock front and formation and propagation of kinks on it. The theory is applied to derive the weak shock ray equations for the propagation of a shock in a polytropic gas.
To find the numerical solution of the weak shock ray equations, it is essential to put them in a conservation form. The conservation form facilitates the study of propagation of the shock front even after the kinks have appeared. There are various schemes available in the literature which can then be applied to solve the system. Chapter 2 of this thesis deals with writing the system of equations in the desired conservation form. The shock ray equations consist of four equations in four variables; namely, M, N, G, and ?. Here, M is the Mach number of the shock, N is proportional to the normal derivative of M, G is the metric associated with the transformation to ray coordinates, and ? is the angle that the normal to the shock front makes with the x-axis. The equation for N is not in strict conservation form and has a source-like term, but still the system can be subjected to numerical calculations.
Chapter 3 contains the results of extensive numerical computations done with the system of weak shock ray equations in the conservation form, for a propagating curved weak shock front. A total variation bounded finite difference scheme based on Lax-Friedrichs flux has been used to solve the system of conservation laws. Strang splitting was used to deal with the source term that appears in one of the equations. We start our investigations with an initially concave shock front, i.e., a parabola extended as straight lines on either side. This configuration of the initial front develops kinks after some critical time. Effects of changing the initial strength of the shock or that of its normal derivative, as also the effect of initial curvature on the formation, propagation, and separation of kinks have been studied. Then we go on to study an initially periodic sinusoidal front. In this case, it is seen that pairs of kinks are formed. The kinks from either side move closer to one another, interact, and give rise to another set of kinks which move apart, and the process continues. In fact, we could study more general configurations which are periodic in nature. In all the cases, the front propagation can be studied long after the kinks have appeared on the shock front.
The most interesting result is that for N ? 0, the Mach number M of the shock decreases continuously, no matter what initial configuration we start with. In fact, for weak curved shocks, the Mach number always decays to 1, N (the normal derivative of M) tends to zero, and the angle ? that the normal to the curve makes with the x-axis (which is the direction of propagation) also tends to zero as t ? ?.
The interesting results of Chapter 3 motivated us to do a comparative study of NTSD to Whitham’s theory of geometrical shock dynamics. In Chapter 4, we have presented analytical and numerical results comparing the two theories. Also, we have juxtaposed the results of the weakly nonlinear ray theory (Prasad and Sangeeta, 1999) in this context.
NTSD is a mathematically convincing theory to calculate the successive positions of a weak shock front. Our results show that the amplitude of an initially weak converging shock always remains small so that the small amplitude theory used here remains valid. Efforts are on to compare the results with the solution of full gas dynamic equations using Direct Numerical Simulation (DNS). | |
