Kinematical conservation laws and applications

Abstract

Study of wave motion is important in the field of fluid mechanics. The equations governing the motion of an inviscid compressible fluid form a hyperbolic system of quasi-linear equations and hence give rise to nonlinear waves which are quite difficult to analyse mathematically. Because of the availability of powerful computational facilities, numerical studies are more common in applications. However, in some practical applications, it is very expensive to go for full numerical solutions and hence, in order to minimize the computational difficulties, certain approximations of the actual governing system are needed depending on the nature of the application. In applications like calculation of the sonic boom signature, extracorporeal shock wave lithotripsy to treat kidney stone disease, tracing the shock front or wavefront is essential. In such applications, it is not necessary to get the full information of the flow between shock and the body producing the shock. Moreover, in the case of the calculation of the sonic boom signature, one needs to calculate the far-field solution, as the main aim in this problem is to find the pressure disturbance on the ground when a supersonic aircraft flies at a high distance from the ground level. Hence, in such cases, it is expensive and sometimes may not be possible to go for a full numerical solution of gas dynamics equations. To minimize the computational effort, we need to apply certain physically realistic approximations to simplify the system. Acoustic ray theory or linear ray theory were used to trace the wavefronts or shock fronts, which works well in certain cases like geometrical optics. Since this theory does not take the nonlinear diffraction of the rays into account, a caustic region occurs beyond which the wavefront or the shock front tends to fold and form a cusp type of singularity. Hence, the linear ray theory is not suitable for applications where nonlinear effects play a crucial role. Whitham proposed in 1957 a theory called geometrical shock dynamics (GSD) through which he showed that there are nonlinear waves propagating on the shock fronts that make the amplitude in the focal region finite and instead of folding of the shock front, we get a discontinuity in the slope of the shock front, which is a kink (called shock-shock by Whitham). This behaviour of a weak shock front in the focal region was experimentally studied by Sturtevant and Kulkarni (1976). Kink is basically a geometrical phenomenon (Prasad (1995)) which appears as the image in (x,y)-plane of a shock as a solution of kinematical conservation laws (KCL): (gsin??)t+(mcos??)x=0(g \sin \theta)_t + (m \cos \theta)_x = 0(gsin?)t+(mcos?)x=0 (gcos??)t?(msin??)x=0(g \cos \theta)_t - (m \sin \theta)_x = 0(gcos?)t?(msin?)x=0 in an appropriately defined ray coordinate system (?,t\xi,t?,t). Here ggg is the metric along the wavefront (or shock front) t=constantt=\text{constant}t=constant, ?\theta? is the angle between the normal to the wavefront and the x-axis, and mmm is the wavefront (or shock front) Mach number. KCL was first derived by Morton, Prasad and Ravindran (1993). Prasad and Sangeetha (1999) used KCL to study the formation and propagation of kinks on nonlinear wavefronts. In the weakly nonlinear ray theory (WNLRT), used by Prasad and Sangeetha, KCL is hyperbolic in nature when the wavefront Mach number mmm is greater than 1 and hence the numerical study of the problem becomes quite easy and efficient. Since KCL represents evolution of any propagating curve CtC_tCt, it can be used effectively in many problems in science and engineering which involve propagation of an interface or discontinuity. But KCL, which has two equations in three unknowns, is an underdetermined system. To make the system closed, we need dynamical conditions depending on the nature of fit i.e., in different applications of KCL, we need different additional equations or relations between the unknowns. When fit is a shock front, dynamical consideration leads not only to one but infinity of compatibility conditions. Hence, a mathematical analysis of KCL is essential to understand the nature of the solution, limitation of the usage of the theory, etc. This thesis first deals with a mathematical analysis of the KCL with a simple additional closure relation, which appears to be physically realistic for a class of problems and then goes to applications of KCL to some problems including the propagation of a curved shock where more general closure equations are required. The thesis consists of seven chapters. Chapter 1 starts with a motivation for the work. A brief introduction to the WNLRT and the derivation of the KCL has been presented for the sake of making the material self-contained. However, a detailed discussion of the subject is available in the book by Prasad (2001). As mentioned above, the KCL is hyperbolic when the wavefront Mach number mmm is greater than 1. Therefore, we quote some results and definitions regarding hyperbolic systems for the readers to appreciate the work done in this thesis, especially the works discussed in Chapter 2 and Chapter 3. Riemann problem and the interaction of elementary waves are the building blocks for the solution of a general initial value problem for any hyperbolic system of conservation laws. So, it is not only interesting but also important to study the solution of the Riemann. Their suitable conservation forms to study this system numerically. In Chapter 5, we have given a brief introduction to NTSD and the method of deriving the governing equation for the SRT. We have modified the conservation form of the two compatibility conditions in SRT used by Monica and Prasad (2001). Our conservation forms appear to be more natural and follow a pattern which is valid for each of the infinite set of compatibility conditions for a curved shock of arbitrary strength. In Chapter 6, we solve some two-dimensional curved piston problems showing that the SRT with two compatibility conditions gives shock positions which are very close to the solution of the same problem obtained by numerical solution of Euler equations (NSEE). The comparison of the results obtained by geometrical shock dynamics (GSD) of Whitham, NSEE, and the numerical solutions obtained from SRT and WNLRT has been carried over in the case of accelerating and decelerating curved pistons. It has been shown that the results obtained by SRT agree well with NSEE, which shows that the SRT takes the acceleration and deceleration of the piston into account more effectively. The aim of this work is not just this comparison but in investigating the role of nonlinearity in accelerating the process of evolution of a shock, produced by an explosion of a non-circular finite charge, into a circular shock front. We find that the nonlinear waves propagating on the shock front appreciably accelerate this process. We also study the propagation of shock fronts produced by a wedge-shaped piston and also a piston of periodic shape. Assume that an aerofoil is moving with a constant speed up to some time, say t=t0t = t_0t=t0, and from this time onwards, it starts accelerating. This problem is complicated as linear rays tend to converge to focus and form a caustic region beyond which the linear solution becomes singular. Plotkin (2002) reviewed some methods used to calculate the sonic boom signature, which consists in finding not only the leading shock but also the flow behind it. From the results obtained in this thesis, we hope that SRT combined with WNLRT can be effectively used to solve the unsteady sonic boom problem completely and also can be used to solve the sonic boom problem in the case of a manoeuvring aircraft. In Chapter 7, we give a note on sonic booms and formulate the physical problem in which we are interested. 7, we give a note on sonic booms and formulate the physical problem m which we are interested. In this thesis, some basic analysis has been done for the system of kinematical conservation laws which are essential for general mathematical analysis of the system we have applied the system to a few practical problems. Mathematical questions i e existence and uniqueness of the solution of the KCL will be pursued in future and the calculation of sonic boom signature for an accelerating and any manoeuvring aerofoil are our ultimate aim in future, whose basic tools are developed and presented in this thesis.