Whitham's Rule Research Articles

Shock-Formation Down a Non-Uniform Tube in Two Phase Flow

In present paper an attempt is made to study the one dimensional formulation of flow in a tube of varying cross-sectional area for two phase flow when equilibrium is established eventually and particle volume fraction is taken as one of the additional variable. Conservation laws and shock condition are obtained and using Whitham rule (1974) of characteristic a relation between cross- sectional area and particle volume fraction is obtained and result is discussed for different values of Mach number.

IMPLODING CYLINDRICAL AND SPHERICAL SHOCK WAVES IN A NON-IDEAL MEDIUM

Converging shock waves in an almost ideal medium are considered. The kinematics of one-dimensional motion have been applied to construct an evolution equation for strong cylindrical and spherical shock waves propagating into a low density gas at rest. The approximate value of the similarity parameter obtained from there is compared with those derived from Whitham's Rule and the exact similarity solution at the instant of collapse of the shock wave. The above computation is carried out for different values of the parameter α, which depends on the internal volume of the gas molecules.

Asymptotic solution of a non-linear wave motion in an electron plasma

The theory of high-frequency waves has been used to calculate first and second-order asymptotic solutions for the propagation of non-linear waves in a cylindrical symmetric flow of an electron plasma. The behaviour of acceleration waves and weak shock waves has been analysed through these solutions and Whitham's rule for a weak shock wave on any wavelet has been confirmed through the first-order solution. The appearance of a weak shock wave on any wavelet has been determined and its strength, the location, and the speed of propagation have been found from the asymptotic solution presented in this paper. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

Focusing of Reflected Shock Waves Analyzed by Geometrical Shock Dynamics.

In the present work, a numerical scheme for the unsteady motion of shock waves in gases is developed based on Whitham's rule of geometrical shock dynamics (GSD). To investigate the effect of the initial shock strength on the shock reflection and diffraction on a wall, the two-dimensional focusing of cylindrical blast waves reflected from a concave elliptic wall is analyzed under various initial conditions. The results are compared with the finite difference solutions based on the piecewise linear method (PLM) as well as the experimental results. It is found that the geometrical shock dynamics is a practical and economical method to analyze the shock focusing process of the blast waves produced by exploding wires. Pressure change along the major axis of the ellipse shows good agreement when the present computations are compared with the experimental results as well as with PLM solutions.

Self-similar solutions of isothermal flows behind a strong MHD shock

Self-similar solutions for isothermal flows headed by a cylindrical magneto-hydrodynamic shock wave have been obtained. It is assumed that a strong MHD shock wave is propagating into an inhomogeneous atmosphere at rest with density varying according to a power law of distance from the axis of symmetry. The gaseous atmosphere is assumed to be electrically conducting with infinite electrical conductivity. The azimuthal magnetic field is permeated due to a constant line current along the axis of symmetry. Under these assumptions an analytical and numerical treatment has been presented and the flow distributions in a disturbed region headed by a blast wave have been determined. The law of propagation has been determined by using Whitham's rule. Numerical solutions of self-similar flows have been obtained and graphed.

Cumulative nonlinear distortion of an acoustic wave propagating through non-uniform flow

In this paper we examine how the unsteady flow field radiated from an oscillating body is altered from the result of acoustic theory as the direct consequence of disturbances propagating through the non-uniform flow produced by the presence of the body. Taking the specific example of an oscillating airfoil placed in supersonic flow and having the contour of a parabolic arc, we derive a closed-form representation for the unsteady flow field in terms of the confluent hypergeometric function. The analytical expression reveals explicitly that, though the body shape has a negligible effect in the near field, it inextricably affects the unsteady flow at a large distance, both in its amplitude and phase, and substantially modifies the results of acoustic theory. In addition, we display the relation of this solution to the ‘fundamental solution’ and the other salient physical features connected with disturbances propagating through non-uniform flow. The present results recover Whitham's rule in the limit of zero frequency of oscillation and also include, as another special case, the unsteady solution for a wedge obtained by Carrier and Van Dyke.

The decay of a plane shock wave

An analytic solution of the non-isentropic equations of gas-dynamics, for the one-dimensional motion of a non-viscous and non-conductive medium, is derived in this paper for the first time. This is a particular solution which contains only one arbitrary function. On the basis of this solution, the interaction of a centred simple wave with a shock of moderate strength is analyzed; and it is shown that, for a weak shock, this analysis is compatible with Friedrichs's theory. Furthermore, in the light of this analysis, it is explained why the empirical methods employed by the shock-expansion theory, including Whitham's rule for determining the shock path, work.

The non-linear refraction of shock waves by upstream disturbances in steady supersonic flow

The general problem studied is the propagation of an oblique shock wave through a two-dimensional, steady, non-uniform oncoming flow. A higher-order theory is developed to treat the refraction of the incident oblique shock wave by irrotational or rotational disturbances of arbitrary amplitude provided the flow is supersonic behind the shock. A unique feature of the analysis is the formulation of the flow equations on the downstream side of the shock wave. It is shown that the cumulative effect of the downstream wave interactions on the propagation of the shock wave can be accounted for exactly by a single parameter Φ, the local ratio of the pressure gradients along the Mach wave characteristic directions at the rear of the shock front. The general shock refraction problem is then reduced to a single non-linear differential equation for the local shock turning angle θ as a function of upstream conditions and an unknown wave interaction parameter Φ. To lowest order in the expansion variable θΦ, this equation is equivalent to Whitham's (1958) approximate characteristic rule for the propagation of shock waves in non-uniform flow. While some further insight into the accuracy of Whitham's rule does emerge, the theory is not a selfcontained rational approach, since some knowledge of the wave interaction parameter Φ must be assumed. Analytical and numerical solutions to the basic shock refraction relation are presented for a broad range of flows in which the principal interaction occurs with disturbances generated upstream of the shock. These solutions include the passage of a weak oblique shock wave through: a supersonic shear layer, a converging or diverging flow, a pure pressure disturbance, Prandtl–-Meyer expansions of the same and opposite family, an isentropic non-simple wave region, and a constant pressure rotational flow. The comparison between analytic and numerical results is very satisfactory.

The Brinkley-Kirkwood theory and Whitham's rule

In dieser Arbeit wird die bekannte Brinkley-Kirkwood-Theorie (B. K.) mit der Whithamschen Regel und mit der Methode der Charakteristiken verglichen. Numerische Resultate wurden mit allen diesen Theorien errechnet fur einen Stoss, der sich in inhomogene Medien mit verschiedenen Dichteanderungen ausbreitet. Die Resultate der B.-K.-Theorie weichen im allgemeinen von denen, die durch die Charakteristiken-Methode erhalten wurden, um nicht mehr als 15% ab.

SIMILARITY SOLUTIONS FOR REACTIVE SHOCK WAVES

Possibilities for similarity solutions of the flow following the impact of a piston on a reactive γ-law gas are delineated for the cases where the transmitted shock is strong or of constant strength. Reaction-rate functions that can be used with power-law, exponential and logarithmic shock paths are obtained. A particular case is worked out, namely, plane reactive flow with the reaction rate proportional to the internal energy and the shock velocity varying exponentially with time. For certain values of the parameters, the solution is shown to contain an envelope of characteristics which corresponds to a singular point of the ordinary differential equation which determines the flow. Results are obtained regarding the behaviour of the flow variables, a special case where Whitham's rule holds exactly, and the possibility of accelerating a strong shock indefinitely, without introducing secondary shocks into the flow.

PROPAGATION OF A BORE PRODUCED BY THE SUDDEN BREAK OF A DAM

In this paper we study the progress of a bore, produced by the sudden break of a dam, when there is a flow of water ahead of the dam and the bed has a mild slope and offers resistance, employing Whitham's rule. We first derive certain interesting results from the general discussion of the differential equation, expressing the variation of the bore strength with the undisturbed Froude number, M0, ratio of bedslope to bed resistance, gα/R = a2 and the bore strength M(x)≡{h(x)/h0(x)} where h(x) and h0(x) are the bore height and the undisturbed height of the water immediately ahead of the bore, the horizontal distance x being measured from the dam. The parameters M0 and a2 combine to influence the bore strength in a very special way. We also examine the asymptotic cases when the bore strength M↑and M ↑ 1. The intermediate cases are investigated numerically to bring out the effects of the parameters, α, a2, M0 and the dam height on the strength of the bore, its velocity and the fluid velocity behind it.

The shock-expansion method and Whitham's rule

The assumptions of the shock-expansion method are re-examined. Although the shock-expansion method and Whitham's rule are used to treat two different classes of problems, certain similarities between these two methods are noted. It is suggested that a single procedure leads to solutions for the entire flow field for both classes of problems.

A note on Whitham's Rule

Whitham (1957, 1958) has developed a method for treating the propagation of shock waves into non-uniform media, which has been widely and successfully used. In shock diffraction problems, he considers the medium ahead of the shock to be at rest. The purpose of this note is to generalize Whitham's technique to problems having a uniform flow ahead of the shock, in which the inclination of the shock normal to the direction of flow ahead of the shock remains small. This extension was suggested to the author by Miles (1964) who considered the problem of diffraction of a strong shock by a thin supersonic wedge.

Paper 11: The Attenuation of Shock Waves in Rough Ducts

The rule formulated by Whitham (1)† is briefly described and the application of this rule to the attenuation of shock waves in rough ducts is mentioned. A review of previous work on shock wave attenuation shows that most investigators have used smooth-walled ducts for their experiments and a steady-flow, flat-plate boundary-layer analogy for their theory. An alternative approach based upon one-dimensional flow is examined. Previous work concerned with this alternative approach is outlined and the present problem is then explained in detail. The attenuation of moving, one-dimensional shock waves in rough ducts is considered using the theory of characteristics applied to one-dimensional, unsteady, compressible flow, including friction and heat transfer effects and using Whitham's rule. In order to evaluate this, steady-flow friction coefficients and the Reynolds analogy between pipe friction and heat transfer are used. Comparison of the present theory with previous experimental results shows good agreement with half of them. It is suggested that at least part of the difference between the present theory and the other half of the experiments can be attributed to the method used by the previous investigators when evaluating their experimental results.