In this thesis we investigate the phenomena of shocks, focusing on the structure of the shock and its amplitude and thickness. We begin by using Burgers’
model for waves of small but finite pressure amplitude, restricting attention to one-dimensional propagation. We begin by considering the physical effects of nonlinearity attenuation and viscous dissipation, studying how the solution changes as it progresses. This investigation advances to study the nonlinear propagation in a viscous relaxing gas. In the second part of the thesis we
consider the augmented Burgers’ equation accompanied with a set of relaxation equations for the study of the finite-amplitude nonlinear propagation in a relaxing medium.
In the second chapter, we look at Burger’s equation in the plane for the nonlinear finite amplitude flow propagating in thermoviscous medium. Each section is devoted to a different technique used to find solution of Burgers’
equation.
We begin in section §2.2 by neglecting the viscous effects on the shock wave and Burger’s equation is then reduced to the inviscid version. We apply the method of characteristics to look for the exact solutions of the inviscid Burgers’
equation. Such solutions resemble wave propagation in a nondissipative medium. Depending on the initial conditions, the method of characteristics may predict the appearance of multivalued solutions as a result of the intersection of characteristic curves. Thus, using the weak shock theory, we consider a weak solution that is associated with these characteristic curves to recover the single-valued solutions by fitting a discontinuity in the waveform which propagates as time advances [7]. This discontinuity that appears after a certain time is called a shock, where the characteristic curves on both sides intersect for the first time with the shock curve. We focus on investigating the dynamics of shock waves such as shock formation, shock location and the interaction of multiple shocks. The Rankine-Hugoniot relation which includes the particle velocity on both sides of the shock can be applied to ascertain the shock location and amplitude (see for instance, Pierce [72] section 11-3). The theory is concluded with an alternative method referred to as equal area rule to fit the shock.
We then include in the second section §2.3 a very small viscous effect in Burgers’ equation. In this case we implemented a perturbation method to look for the analytical approximation of the solution. The asymptotic analysis follows the perturbation method proposed by Crighton and Scott [25], and the
shock profile is analysed. An outer solution is deduced form the characteristic solution and for the inner solution a Taylor viscous shock is obtained. The inner and outer expansions are matched to second order and an expression is obtained for the location of the thermoviscous shock. We end the section with results on the shock width which is associated with shock wave asymptotic solutions.
In section §2.4 we use the well known Cole-Hopf transformation to determine an analytical solution of the viscid Burgers’ model. We then derive a detailed explanation for the case of significantly small viscosity using the stationary phase method.
Finally in §2.5 we conclude this chapter with a numerical implementation to approximate the solution of the non-linear Burgers’ equation. We choose a pseudo-spectral method for spatial discretization using truncated Fourier modes, and the fourth-order Runge-Kutta for time marching. A rectangular unit pulse is defined as the initial representative waveform of the nonlinear problem. The Fourier pseudo-spectral scheme is modified to improve the numerical stability of the solution. We validate the outcomes of the numerical scheme by comparisons with the asymptotic and Cole-Hopf solutions.
In the third chapter, the augmented Burgers’ equation is stated for the propagation in viscous and relaxing gases. Each relaxation process is characterised by two parameters, and these parameters are accommodated in a governing equation ( Pierce [72] Ch 10-7). Travelling waves in dimensionless form are considered in §3.4 for a single relaxation mode. The asymptotic behaviour of the shock solution is analysed and conditions depending on the order of magnitude of the relaxation parameters are obtained for different classes of shock structure. The analysis clarified that for particular values of the relaxation parameters, the relaxation mode does not fully control the shock and a finer sub-shock controlled by viscosity arises, otherwise the shock is totally governed by the relaxation process. Lighthill [55] referred to the former shock
as partially dispersed and fully-dispersed for the latter shock. We then seek the inner and outer expansions, which are matched to second order to obtain an expression for the shock position. The asymptotic analysis is summed up with a calculations of the shock width by looking at the maximum values of the velocity gradient.
In §3.4.7 we generate a numerical approximation of the travelling wave solution to validate the asymptotic solutions. In §3.5 we seek a numerical solution for the augmented Burgers’ equation by applying the numerical approach described earlier in §2.5, with a simple modification to include the relaxation terms. Thus, we use the Fourier pseudo-spectral (FPS) numerical method for discretizing the problem and we then apply fourth-order Runge-Kutta for time marching. We compare the outcomes of this scheme with both numerical and asymptotic solutions of the travelling wave.
The chapter is summed up in §3.5.2 with comparisons between the delicate asymptotic measurements of the maximum slope and the numerically computed results of the FPS method.
In the fourth chapter, we look at one-dimensional small amplitude sound wave propagation subject to thermoviscous dissipation and two vibrational relaxation modes. Since for propagation through the atmosphere, it is believed that the vibrational states of N2 and O2 molecules are significant. This chapter therefore is an in-depth analysis where we obtain analytical and numerical solutions in the presence of two relaxation modes. In this chapter, once again, we follow the analysis presented earlier in the third chapter to bring asymptotic and numerical solutions for the travelling wave in the nondimensional form in
§4.2. We examine how the shock structure is affected by this combination of thermoviscosity, nonlinearity and the two relaxation processes. In the presence of thermoviscous dissipation and two molecular relaxation modes, in this case the shock may consist of three regimes [44] depending on the relaxation parameters. In §4.2.2 the sensitive shock width with this multiple layers is studied by looking at highest value of the gradient. In §4.3 the Fourier
pseudo-spectral numerical (FPS) method proposed earlier in §3.5 is used with the inclusion of the second relaxation mode. The analysis of augmented Burgers’ equation coupled with two relaxation modes ends in §4.3.3 with comparison results of the asymptotic suggestions of the maximum gradient together with the FPS outputs.
In the fifth chapter, we look at the parametric example proposed by Pierce and Kang [73] to estimate the shock waveform of sonic boom shocks on the ground.
We use the atmospheric measured values in Pierce and Kang’s example to compare our predictions of the pressure profile and shock width with their results. The thesis is then summed up with some conclusion remarks.