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The finite-difference method which Peter D. Lax developed for treating unsteady inviscid flow fields is used to study the transient flow in the shock layer of a sphere that has been struck by a normal shock wave. Transient flow of this sort is encountered when a shock tube is used as a supersonic wind tunnel. Time histories of the shock detachment distance and the stagnation-point pressure and tangential velocity gradient are presented for ranges of the incident-shock Mach number and the perfect-gas specific-heat ratio. These results show that the stagnation-point pressure approaches the steady value much more rapidly than the shock detachment distance. In general, the stagnation-point pressure but more rapidly than the shock detachment distance. As the specific-heat ratio is decreased and the incident-shock Mach number is increased, the variation of the velocity gradient with respect to the shock detachment distance becomes more nearly linear.
This book provides an elementary introduction to one-dimensional fluid flow problems involving shock waves in air. The differential equations of fluid flow are approximated by finite difference equations and these in turn are numerically integrated in a stepwise manner, with artificial viscosity introduced into the numerical calculations in order to deal with shocks. This treatment of the subject is focused on the finite-difference approach to solve the coupled differential equations of fluid flow and presents the results arising from the numerical solution using Mathcad programming. Both plane and spherical shock waves are discussed with particular emphasis on very strong explosive shocks in air. This expanded second edition features substantial new material on sound wave parameters, Riemann's method for numerical integration of the equations of motion, approximate analytical expressions for weak shock waves, short duration piston motion, numerical results for shock wave interactions, and new appendices on the piston withdrawal problem and numerical results for a closed shock tube. This text will appeal to students, researchers, and professionals in shock wave research and related fields. Students in particular will appreciate the benefits of numerical methods in fluid mechanics and the level of presentation.
The accuracy of the result obtained in a fundamental paper by Kantrowitz (NACA TN 1225) that a small short-time lowering of the back pressure in steady, shock-free, transonic diffuser flow causes a stationary or trapped shock to form near the critical sonic channel throat is investigated by considering the contribution of a higher-order term in the short-time calculations which was neglected in Kantrowitz's paper. In this higher approximation to the short-time effects, the shock is no longer stationary or trapped unless it is supported by a negative steady-flow back pressure; the result thus is no long in disagreement with steady-flow solutions for stationary shocks.
Because the chemical reaction rates needed to predict the dependence of degree of dissociation on distance behind the shock are not known, order-of-magnitude estimates of their values have been used in a numerical example, the purpose of which is to illustrate the use of reaction-rate equations to predict relaxation time and distance behind the shock front.
This book presents two distinct aspects of wave dynamics – wave propagation and diffraction – with a focus on wave diffraction. The authors apply different mathematical methods to the solution of typical problems in the theory of wave propagation and diffraction and analyze the obtained results. The rigorous diffraction theory distinguishes three approaches: the method of surface currents, where the diffracted field is represented as a superposition of secondary spherical waves emitted by each element (the Huygens–Fresnel principle); the Fourier method; and the separation of variables and Wiener–Hopf transformation method. Chapter 1 presents mathematical methods related to studying the problems of wave diffraction theory, while Chapter 2 deals with spectral methods in the theory of wave propagation, focusing mainly on the Fourier methods to study the Stokes (gravity) waves on the surface of inviscid fluid. Chapter 3 then presents some results of modeling the refraction of surf ace gravity waves on the basis of the ray method, which originates from geometrical optics. Chapter 4 is devoted to the diffraction of surface gravity waves and the final two chapters discuss the diffraction of waves by semi-infinite domains on the basis of method of images and present some results on the problem of propagation of tsunami waves. Lastly, it provides insights into directions for further developing the wave diffraction theory.
This monograph presents the mathematical description and numerical computation of the high-frequency diffracted wave by an immersed elastic wave with normal incidence. The mathematical analysis is based on the explicit description of the principal symbol of the pseudo-differential operator connected with the coupled linear problem elasticity/fluid by the wedge interface. This description is subsequently used to derive an accurate numerical computation of diffraction diagrams for different incoming waves in the fluid, and for different wedge angles. The method can be applied to any problem of coupled waves by a wedge interface. This work is of interest for any researcher concerned with high frequency wave scattering, especially mathematicians, acousticians, engineers.