scholarly journals Approximate Analytic Solution of a Self-Similar Piston Moving in an Inhomogeneous Medium

2020 ◽  
pp. 1-9
Author(s):  
B.N. Prasad
Keyword(s):  
Spherical Symmetry ◽  
Analytic Solution ◽  
Density Ratio ◽  
Strong Shock ◽  
Power Laws ◽  
Ideal Gas ◽  
Approximate Analytic Solution ◽  
Piston Problem ◽  
Self Similar ◽  
Flow Variables

Self-similar motion for the flow between a piston and strong shock propagating in a non uniform ideal gas at rest has been studied. The solution to the problem is similar to that of hypersonic flows past the power law bodies. The gas ahead of the shock is assumed to be uniform and at rest. This is considered as a particular case of radiative piston problem. The shock is assumed to be very strong and propagating in a medium at rest in which density obeys power laws. This problem with spherical symmetry has got importance in astrophysics. To solve the gas dynamics problem, Chernyii’s expansion techniques have been used in which flow variables are expanded in a series of powers of ε, the density ratio across the strong shock. The approximate analytic solution has been obtained in closed form to the zeroth approximation. The problem discussed belongs to the self-similar motion of the first kind. The resulting analytic solution gives the flow variables distribution for plane, cylindrical, and spherical symmetry for different cases which satisfy the similarity conditions with accurate trend and values.

Nearly spherical constant-power detonation waves as driven by focused radiation

1973 ◽  
Vol 61 (3) ◽  
pp. 481-498 ◽  
Author(s):  
Y. H. George ◽  
F. K. Moore
Keyword(s):  
Spherical Symmetry ◽  
Three Dimensional ◽  
Tangential Velocity ◽  
Polar Angle ◽  
Strong Shock ◽  
Order System ◽  
Detonation Waves ◽  
Self Similarity ◽  
Self Similar ◽  
Pressure Perturbations

An analysis is made of the flow within a three-dimensional explosion, or spark, created in a gas absorbing energy from a steady conical beam of radiation with nearly spherical symmetry. The radiation, typically from an array of lasers with a common focus, is assumed to be very intense, and absorbed immediately behind an outwardly advancing strong shock. The resulting self-similar flow has previously been studied for spherical symmetry; somewhat improved calculations for that case are presented here.Departures of the laser power from spherical uniformity, which would result from practical problems of arrangement, are conveniently represented by an ascending series of Legendre polynomials in the polar angle. For non-uniformities of small amplitude, first-order perturbations of the flow field are analysed in detail. Self-similarity is shown to be retained, for zero counter-pressure and power constant with time.For the first five harmonics in power distortion, the resulting fourth-order system of equations is solved numerically for profiles of velocity components, density and pressure, and for shock shape. Results are presented graphically. These solutions are singular near the focus, but are nevertheless fully determined. In the limit of large wavenumber, the core of the flow has vanishing tangential velocity and pressure perturbations, and hence the governing equations are only of second order, except presumably in a boundary layer appearing near the shock.Study of the nonlinear case of large wavenumber along the axis of symmetry shows that the singularity at the focus reflects the existence of a ‘forbidden zone’ whose extent depends on the degree of asymmetry. It is argued that this zone is one within which diffusional processes must dominate.

scholarly journals Similarity Solutions of Strong Shock Waves for Isothermal Flow in an Ideal Gas

2019 ◽  
Vol 4 (5) ◽  
pp. 1094-1107
Author(s):  
Astha Chauhan ◽  
Rajan Arora
Keyword(s):  
Strong Shock ◽  
Unsteady Motion ◽  
Similarity Solutions ◽  
Ideal Gas ◽  
Isothermal Flow ◽  
Linear Partial Differential Equations ◽  
The One ◽  
Similar Solutions ◽  
Flow Variables ◽  
Strong Shock Waves

Self-similar solutions of the system of non-linear partial differential equations are obtained using the Lie group of invariance technique. The system of equations governs the one dimensional and unsteady motion for the isothermal flow of an ideal gas. The medium has been taken the uniform. From the expressions of infinitesimal generators involving arbitrary constants, different cases arise as per the choice of the arbitrary constants. In this paper, the case of a collapse of an implosion of a cylindrical shock wave is shown in detail along with the comparison between the similarity exponent obtained by Guderley's method and by Crammer's rule. Also, the effects of the adiabatic index and the ambient density exponent on the flow variables are illustrated through the figures. The flow variables are computed behind the leading shock and are shown graphically.

Self-similar piston problem in non-ideal gas

1976 ◽  
Vol 14 (1) ◽  
pp. 91-97 ◽  
Author(s):  
Melam P. Ranga Rao ◽  
Niranjan K. Purohit
Keyword(s):  
Ideal Gas ◽  
Piston Problem ◽  
Self Similar

Analytical solution for unsteady flow behind ionizing shock wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field

2021 ◽  
Vol 76 (3) ◽  
pp. 265-283
Author(s):  
G. Nath
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Analytical Solution ◽  
Axial Magnetic Field ◽  
Order Approximation ◽  
Ideal Gas ◽  
Field Region ◽  
First Order ◽  
First Order Approximation ◽  
Flow Variables

Abstract The approximate analytical solution for the propagation of gas ionizing cylindrical blast (shock) wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field is investigated. The axial and azimuthal components of fluid velocity are taken into consideration and these flow variables, magnetic field in the ambient medium are assumed to be varying according to the power laws with distance from the axis of symmetry. The shock is supposed to be strong one for the ratio C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ to be a negligible small quantity, where C 0 is the sound velocity in undisturbed fluid and V S is the shock velocity. In the undisturbed medium the density is assumed to be constant to obtain the similarity solution. The flow variables in power series of C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ are expanded to obtain the approximate analytical solutions. The first order and second order approximations to the solutions are discussed with the help of power series expansion. For the first order approximation the analytical solutions are derived. In the flow-field region behind the blast wave the distribution of the flow variables in the case of first order approximation is shown in graphs. It is observed that in the flow field region the quantity J 0 increases with an increase in the value of gas non-idealness parameter or Alfven-Mach number or rotational parameter. Hence, the non-idealness of the gas and the presence of rotation or magnetic field have decaying effect on shock wave.

scholarly journals An approximate analytic solution of a particular boundary value problem

2001 ◽  
Vol 27 (8) ◽  
pp. 513-520
Author(s):  
Ugur Tanriver ◽  
Aravinda Kar
Keyword(s):  
Boundary Value Problem ◽  
Heat Conduction Equation ◽  
Analytic Solution ◽  
Boundary Value ◽  
Three Dimensional ◽  
Welding Process ◽  
Approximate Analytic Solution ◽  
Analytic Expression ◽  
Steady State Heat ◽  
Laser Welding Process

This note is concerned with the three-dimensional quasi-steady-state heat conduction equation subject to certain boundary conditions in the wholex′y′-plane and finite inz′-direction. This type of boundary value problem arises in laser welding process. The solution to this problem can be represented by an integral using Fourier analysis. This integral is approximated to obtain a simple analytic expression for the temperature distribution.

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