Similarity solutions for two-dimensional weak shock waves

The general forms of self-similar solutions for two-dimensional weak shock waves in fluid dynamics are obtained. The functional form describing the area under the initial pulse is characterized under which the general system of PDEs admits similarity solutions. It is shown how one can construct new solutions with shock discontinuity from these self-similar solutions. In particular, a plane-wave solution is joined with a self-similar solution across a non-trivial shock. Furthermore, a new class of non-trivial simple wave solutions is presented.     

Approximate solutions of some problems of scattering of surface water waves by vertical barriers

A class of mixed boundary value problems (bvps), occurring in the study of scattering of surface water waves by thin vertical rigid barriers placed in water of finite depth, is examined for their approximate solutions. Two different placings of vertical barriers are analyzed, namely, (i) a partially immersed barrier and (ii) a bottom standing barrier. The solutions of the bvps are obtained by utilizing the eigenfunction expansion method, leading to a mathematical problem of solving over-determined systems of linear algebraic equations. The methods of analytical least-square approximation as well as algebraic least-square approximation are employed to solve the corresponding over-determined system of linear algebraic equations and thereby evaluate the physical quantities, namely, the reflection and transmission coefficients. Further, the absolute values of the reflection coefficients are compared to the known results obtained by utilizing a Galerkin type of approximate method after reducing the bvps to integral equations whose complete solutions are difficult to be determined. Various combinations of discretization of the resulting dual series relations obtained in the present analysis are employed to determine the least-square solution.     

Approximate solutions for non-linear random vibration problems

Two methods are described for obtaining approximate solutions to non-linear random vibration problems. The first method approximates the non-linear system with the linear system whose corresponding probability density function best solves the Fokker-Planck equation associated with the non-linear system. In the second method, a class of non-linear systems with known solutions to the Fokker-Planck equation is used to best approximate the non-linear system of interest. Two illustrative examples are presented and the results are compared with existing methods.     

Approximate transonic profile flow with shock

Summary The present work gives a procedure for computing approximately steady inviscid transonic profile flow with shock. Using an analysis similar to that adopted byHosokawa in [11] it extends the shock-free transonic solution ofNiyogi andMitra [9] to the case of flow with shock. Supercritical flow past parabolic are profiles and a NACA profile are computed and compared with theoretical results ofOswatitsch andZierep and finite difference solution ofMurman andCole [13] and with experimental results. The agreements are satisfactory.

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Eine Näherungslösung der schallnahen Strömung um Profile unter Berücksichtigung von Stoßwellen

Zusammenfassung In dieser Arbeit wird eine Methode zur näherungsweisen Berechnung der stationären, reibungsfreien, schallnahen Strömung um Profile unter Berücksichtigung von Stoßwellen angegeben. Unter Verwendung einer der vonHosokawa [11] angewendeten ähnlichen Vorgangsweise wird die vonNiyogi undMitra [9] ermittelte stoßwellenfrei schallnahe Lösung auf den Fall mit Stoßwellen erweitert. Die überkritischen Strömungen längs eines Parabelbogen- und eines NACA-Profiles werden berechnet und mit theoretischen Ergebnissen vonOswatitsch undZierep, mit solchen nach dem Differenzenverfahren vonMurman undCole [13] ermittelten und mit experimentellen Ergebnissen verglichen. Die Übereinstimmung ist zufriedenstellend.

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With 5 Figures     

Approximate Riemann solutions of the two-dimensional shallow-water equations

A finite-difference scheme based on flux difference splitting is presented for the solution of the two-dimensional shallow-water equations of ideal fluid flow. A linearised problem, analogous to that of Riemann for gasdynamics, is defined and a scheme, based on numerical characteristic decomposition, is presented for obtaining approximate solutions to the linearised problem. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second-order scheme which avoids non-physical, spurious oscillations. An extension to the two-dimensional equations with source terms, is included. The scheme is applied to a dam-break problem with cylindrical symmetry.     

On the accuracy of finite-difference solutions for nonlinear water waves

This paper considers the relative accuracy and efficiency of low- and high-order finite-difference discretisations of the exact potential-flow problem for nonlinear water waves. The method developed is an extension of that employed by Li and Fleming (Coastal Engng 30: 235–238, 1997) to allow arbitrary-order finite-difference schemes and a variable grid spacing. Time-integration is performed using a fourth-order Runge–Kutta scheme. The linear accuracy, stability and convergence properties of the method are analysed and high-order schemes with a stretched vertical grid are found to be advantageous relative to second-order schemes on an even grid. Comparison with highly accurate periodic solutions shows that these conclusions carry over to nonlinear problems and that the advantages of high-order schemes improve with both increasing nonlinearity and increasing accuracy tolerance. The combination of non-uniform grid spacing in the vertical and fourth-order schemes is suggested as optimal for engineering purposes.     

Analytical solution for shock waves in dusty gases

Summary The structure of one dimensional shock waves is investigated using the Navier-Stokes equations for the gas phase and the particle phase. The resulting system of four ordinary nonlinear differential equations is reduced to a system of two autonomous nonlinear differential equations which are solved analytically. This solution is obtained formally by neglecting the viscosity (=0).     

Approximate analytic solutions of adiabatic capillary tube

Approximate analytic solution of capillary tube is valuable for theoretical analysis and engineering calculation. In this work, two kinds of approximate analytic solutions of adiabatic capillary tube have been developed. One is the explicit function of capillary tube length. Another is the explicit function of refrigerant mass flow rate. In these solutions, the choked flow condition is taken into account without iterative calculations. The approximate predictions are found to agree reasonably well with experimental data in open literatures.     

Spall in non-one-dimensional shock waves

Consideration is given to non-stationary shock wave on spallation. Spall strength can be written as (i) pressure — particle velocity diagram in a one-dimensional case, and (ii) pressure — angle of turn flow diagram in a two-dimensional case. Experimental procedure involves (i) loading of the explosive by sliding detonation and (ii) orthogonal flash X-raying. In this way, some metals in particular low-melting ones and plastics as well as liquids have been studied based on the present results and those of other researchers. The dependences of spall strength — deformation rate have been obtained for substances studied in the form of power or linear functions. Some aspects of spallation and of accompanying effects on the base relations obtained are discussed.     

Interference of nonstationary oblique shock waves

Special features of calculation of the flow parameters behind a nonstationary oblique shock wave moving in a stream of absolutely nonviscous gas are considered. The wave intensity at which the stream behind the shock wave may exhibit singularities is determined. The problem of calculating a nonstationary shock wave configuration formed during the interaction of a supersonic jet with an obstacle is solved.     

Approximate solutions to a nonlinear diffusion equation

Approximate similarity solutions to the porous-medium equation, c ₁ = · (c ^m c), are obtained in one and two dimensions. The problems considered arise in the modelling of dopant diffusion in semiconductors, the two-dimentional problems corresponding to diffusion under a mask edge.     

Approximate analytical solutions of a nonlinear diffusion equation

Approximate analytical solutions of a nonlinear diffusion equation $\frac{{\partial c}}{{\partial t}} = \frac{\partial }{{\partial x}}\left( {D(c)\frac{{\partial c}}{{\partial x}}} \right)$ are obtained in the practically important case of constant boundary conditions corresponding to the diffusion in a homogeneously doped half-space at a zero surface concentration for D(c) = ac, ac ², and a√c (a > 0). The error of approximation for these D(c) dependences in the concentration interval (1–2) × 10^?3 < c < (0.92–0.99) does not exceed 1–2%.     

On the question of universality of imploding shock waves

For the case of initially infinitesimally weak spherically and cylindrically imploding shocks, Ponchaut et al. (J. Fluid Mech., 560:102–122, 2006) recently obtained universal solutions. We study the effect of starting the shock with an initially finite strength on the trajectory of the shock by performing numerical calculations for the incoming shock imploding spherically into a diatomic perfect gas. Deviations from the universal solution are extremely small. A solution for the initially infinitesimally weak shock obtained by using Whitham’s (Linear and nonlinear waves, Wiley, New York, 1974) Shock Dynamics is virtually indistinguishable from the Ponchaut solution.     

Converging shock waves in porous media

We have numerically solved several problems related to converging shock waves, including (i) one-dimensional spherical and cylindrical waves with cumulation limited to a ball or cylinder of small radius and (ii) shock-wave flow in a cone-shaped solid target. The passage from a continuous loaded substance to a porous medium in these problems leads to a significant increase in both temperature and pressure in the sample. This character of pressure variation depending on the porosity qualitatively differs from the case of plane waves of constant intensity, for which an increase in the sample porosity under otherwise equal conditions of loading always leads to a decrease in the pressure.