This paper reports an experimental investigation, using shadowgraphs and pressure measurements, of the detailed behaviour of converging weak shock waves near three different kinds of focus. Shocks are brought to a focus by reflecting initially plane fronts from concave end walls in a large shock tube. The reflectors are shaped to generate perfect foci, aretes and caustics. It is found that, near the focus of a shock discontinuity, a complex wave field develops, which always has the same basic character, and which is always essentially nonlinear. A diffracted wave field forms behind the non-uniform converging shock; its compressive portions steepen to form diffraction shocks, while diffracted expansion waves overtake and weaken the diffraction shocks. The diffraction shocks participate in a Mach reflexion process near the focus, whose development is determined by competition between the convergence of the sides of the focusing front and acceleration of its central portion. In fact, depending on the aperture of the convergence and the strength of the initial wave, the three-shock intersections of the Mach reflexions either cross on a surface of symmetry or remain uncrossed. In the former case, which is observed if the shock wave is relatively weak, the wavefronts emerge from focus crossed and folded, in accordance with the predictions of geometrical acoustics theory. In the latter, the strong-shock case, the fronts beyond focus are uncrossed, as predicted by the theory of shock dynamics. It is emphasized that in both cases the behaviour at the focus is nonlinear. The overtaking of the diffraction shocks by the diffracted expansions limits the amplitude of the converging wave near focus, and is the mechanism by which the maximum amplification factor observed at focus is determined. In all cases, maximum pressures are limited to rather low values.

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The focusing of weak shock waves

A set of lubrication models for the thin film flow of incompressible fluids on solid substrates is derived and studied. The models are obtained as asymptotic limits of the Navier-Stokes equations with the Navier-slip boundary condition for different orders of magnitude for the slip-length parameter. Specifically, the influence of slip on the dewetting behavior of fluids on hydrophobic substrates is investigated here. Matched asymptotics are used to describe the dynamic profiles for dewetting films and comparison is given with computational simulations. The motion of the dewetting front shows transitions from being nearly linear in time for no-slip to t 2/3 as the slip is increased. For much larger slip lengths the front motion appears to become linear again. Correspondingly, the dewetting profiles undergo a transition from oscillatory to monotone decay into the uniform film layer for large slip. Increasing the slip further, to very large values, is associated with an increasing degree of asymmetry in the structure of the dewetting ridge profile.

Lubrication Models with Small to Large Slip Lengths

A generalized Riemann problem for quasi-one-dimensional gas flows

Based on fundamental theory developed through the 1950s and 1960s, sonic boom modeling has evolved into practical tools. Over the past decade, there have been requirements for design tools for an advanced supersonic transport, and for tools for environmental assessment of various military and aerospace activities. This has resulted in a number of advances in the understanding of the physics of sonic booms, including shock wave rise times, propagation through turbulence, and blending sonic boom theory with modern computational fluid dynamics (CFD) aerodynamic design methods. This article reviews the early fundamental theory, recent advances in theory, and the application of these advances to practical models.

State of the art of sonic boom modeling

The SI+-SI− pair and interplanetary forward-reverse shock ensembles

Investigations into the behaviour of the gas flowbehind spherical or cylindrical blasts have shown that secondary shocks arise within the original detonation gases. The secondary shock, at first weak, is carried outward with the expanding gases. Subsequently it strengthens and bends back toward the origin, arriving there with high intensity.By using some recently developed techniques in shock dynamics and extending them where necessary, a theory is developed by which the motion of the main shock wave, as well as the formation and subsequent motion of the secondary shock, are given by explicit formulae. In addition, a method for determining, also by explicit formulae, the location of the contact surface between the detonation gases and the outside atmosphere is given. The results of a specific problem, which has been solved by numerically integrating the total equations of motion, and has also been checked experimentally, are compared with the results of the present theory.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112061000810

A simplified analysis of spherical and cylindrical blast waves

In the present paper the problem of a shock propagating through a variable atmosphere is considered, and a rather complete treatment is presented Techniques are given which permit the calculation of shock strength and location as a function of its initial configuration, the atmosphere through which it has propagated, and the distance it traveled A specific application of this theory is made in considering the "sonic boom" caused by a supersonic aircraft Problems such as complete acoustic refraction and/or accelerating aircraft, which give rise to shock configurations that are concave to the direction of propagation, are discussed Attempts at solving these problems by acoustic techniques sometimes lead to physically unrealistic situations involving cusped shocks of high intensity It is shown that when an approximation, better than the acoustic one, is used these difficulties are resolved a A(s) B(s) c e E(s) G(s) Tf -, I(s) = I = L = m = M = M(s) =

Effects of Atmosphere and Aircraft Motion on the Location and Intensity of a Sonic Boom

Two-Dimensional and Axisymmetric Rotational Flows Past a Transonic Corner

Sonic boom theories initiated by Friedman, Kane, and Sigalla have been extended to include the effects of diving and climbing aircraft on curved flight paths. It is shown that flight paths that are concave downward can cause focusing of the wave energy giving rise to high boom overpressures. The present theory indicates that undesirable focusing effects might be avoided by pitting aircraft acceleration against flight-path curvature.

Manfred P. Friedman

Papers

A simplified analysis of spherical and cylindrical blast waves

Effects of Atmosphere and Aircraft Motion on the Location and Intensity of a Sonic Boom

Two-Dimensional and Axisymmetric Rotational Flows Past a Transonic Corner

A Method for Calculating the Effect of Aircraft Maneuvers on Sonic Booms