Rayleigh–Taylor and Richtmyer-Meshkov instability induced flow, turbulence, and mixing. II

This paper refers to the work of Moeckel (1952) on the interaction of an oblique shock wave with a shear layer in steady supersonic flow and the work of Chester (1955) and Chisnell (1957) on the propagation of a shock wave down a non-uniform tube. It is shown that their basic results can be obtained by the application of the following simple rule. The relevant equations of motion are first written in characteristic form. Then the rule is to apply the differential relation which must be satisfied by the flow quantities along a characteristic to the flow quantities just behind the shock wave. Together with the shock relations this rule determines the motion of the shock wave. The accuracy of the results for a wide range of problems and for all shock strengths is truly surprising.The results are exactly the same as were found by the authors cited above. The derivation given here is simpler to perfom (although the original methods were by no means involved) and of somewhat wider application, but the main reason for presenting this discussion is to try to throw further light on these remarkable results.In discussing the underlying reasons for this rule, it is convenient to use the propagation in a non-uniform tube as a typical example, but applications to a number of problems are given later. A list of some of these appears at the beginning of the introductory section.

On the propagation of shock waves through regions of non-uniform area or flow

This paper gives the extension of the approximate theory developed in Part 1 (Whitham 1957) to three-dimensional problems. The basic equations are derived in §1, using the original assumption of a functional relation between the strength of the shock wave at any point and the area of the ray tube. An analogy with steady supersonic flow is found. For the diffraction of a plane shock wave by an obstacle, the equations and boundary conditions are exactly the same as those for steady supersonic potential flow past that obstacle, with a special choice of the density-speed relation. The successive positions of the shock wave are the equipotential surfaces of the supersonic flow. The 'shock-shocks’ introduced in Part 1, i.e. discontinuities in the slope and Mach number of the shock wave, correspond to the steady oblique shock waves in the supersonic flow problem. They arise when Mach reflexion occurs.In §2 the theory is applied in detail to the diffraction of a plane shock wave by a cone. Then, in §3, a small perturbation theory is applied to the two typical problems of (i) diffraction by a slender axi-symmetrical body of general shape, and (ii) the stability of a plane shock. Many further applications would be possible and some brief comments on these are made in §4.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S002211205900026X

A new approach to problems of shock dynamics Part 2. Three-dimensional problems

The development, form, and engineering applications of the transient equation of motion of rigid particles, bubbles, and droplets are presented. Some of the early work on the equation of motion, as well as recent advances, are exposed. Particular emphasis is placed on the semiempirical forms of the equation, which are widely used in engineering practice. The creeping flow assumption, on which most of the known applications are based, is critically examined and its limitations are pointed out. Recent results on particle flow, which include the effect of the advection of a downstream wake and are applicable to finite (but small) Reynolds numbers are also presented. The form of the history (Basset) term is discussed, in the light of recent work and its effect on the integrated results of the equation of motion is examined. Recommendations are given on the appearance, importance, and significance of the history and added mass terms for those who may use the semiempirical form of the transient equation of spheres in a differential or integrated form.

Review—The Transient Equation of Motion for Particles, Bubbles, and Droplets

Recent advances on the analytical form of the hydrodynamic force and heat/mass transfer from a particle, bubble or drop are examined critically. Also some of the recent computational studies, which help strengthen or clarify our knowledge of the complex velocity and temperature fields associated with the momentum and heat/mass transfer processes are also mentioned in a succinct way. Whenever possible, the processes of energy mass exchange and of momentum exchange from spheres and spheroids are examined simultaneously and any common results and possible analogies between these processes are pointed out. This approach results in a better comprehension of the transport processes, which are very similar in nature, as well as in the better understanding of the theoretical expressions that are currently used to model these processes. Of the various terms that appear in the transient equations, emphasis is given to the history terms, which are lesser known and more difficult to calculate. The origin, form, and method of computation of the history terms are pointed out as well as the effects of various parameters on them

Hydrodynamic Force and Heat/Mass Transfer From Particles, Bubbles, and Drops—The Freeman Scholar Lecture

A first-order relationship between changes in area and shock strength is derived for the case of a shock moving through a small area change in a channel. By integration of this relationship the area of the channel is obtained as a function of the shock strength in closed form. This result is interpreted as giving the average strength of a shock at a given time as it moves along a channel of arbitrary shape.By suitable choices of the shape of the channel, descriptions of converging cylindrical and spherical shocks are obtained. These descriptions are found to be in close agreement with the similarity solutions valid near the points of collapse of the shocks. The reason for such good agreement is examined.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112057000130

The motion of a shock wave in a channel, with applications to cylindrical and spherical shock waves

The similarity solution describing the motion of converging spherical and cylindrical shocks is governed by a set of three ordinary differential equations. Previous descriptions of the shock motion have been based on numerical solutions of these differential equations. In the present paper a study of the singular points of the differential equations leads to an analytic description of the flow and a determination of the similarity exponent which is in excellent agreement with the earlier numerical values. Limiting values of the ratio of specific heats are considered. It is shown that as the ratio tends to unity the shock becomes ‘freely propagating’ and the first terms in a power series for the similarity exponent are obtained. Large values of the ratio of specific heats are briefly considered and provide a further check on the analytic description of this paper. Finally in the Appendix the condition for the pressure to have a maximum is clarified and the location of the maximum provides further strong evidence of the high accuracy of the analytic approach of this paper.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112097007775

An analytic description of converging shock waves

On obtient le champ d'ecoulement instationnaire a partir des equations de Stokes applicables aux ecoulements a faible nombre de Reynolds. On montre qu'il n'y a pas de variation de la contrainte normale a la surface de la goutte. On presente les resultats numeriques pour une goutte d'eau se deplacant a travers l'air

The unsteady motion of a drop moving vertically under gravity

A note on Whitham's rule

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112082002882

R. F. Chisnell

Papers

The motion of a shock wave in a channel, with applications to cylindrical and spherical shock waves

An analytic description of converging shock waves

The unsteady motion of a drop moving vertically under gravity

A note on Whitham's rule

The effect of the overtaking disturbance on a shock wave moving in a non-uniform medium