Similarity solutions for unsteady flow behind an exponential shock in an axisymmetric rotating non-ideal gas

Abstract

One-dimensional self-similar unsteady isothermal and adiabatic flows behind a strong exponential shock wave driven out by a cylindrical piston moving with time according to an exponential law in a rotational axisymmetric non-ideal gas is investigated. The medium is assumed to be non-ideal gas rotating about the axis of symmetry. The fluid velocities in the ambient medium are assumed to be varying with time according to an exponential law. Similarity solutions exist only when the surrounding medium is of constant density. Solutions are obtained, in both the cases, when the flow between the shock and the piston is isothermal or adiabatic by taking into account components of vorticity vector. It is found that the assumption of zero temperature gradient brings a profound change in the density and compressibility distributions as compared to that of the adiabatic case. The effect of an increase in the value of the parameter of the non-idealness of the gas is investigated. Also, a comparison between the solutions in the cases of isothermal and adiabatic flows is made. Further, it is shown that the consideration of zero temperature gradient and the effect of variation of the parameter of non-idealness of the gas decrease the shock strength and widens the disturbed region between the shock and piston. The shock waves in non-ideal gas can be important for description of shocks in supernova explosions, in the study of a flare produced shock in solar wind, central part of star burst galaxies, nuclear explosion, rupture of pressurized vessel, in the analysis of data from exploding wire experiments, and cylindrically symmetric hypersonic flow problems associated with meteors or reentry vehicles, etc. The findings of the present work provided a clear picture of whether and how the non-idealness of the gas and consideration of zero temperature gradient affect the propagation of shock and the flow behind it.

Appendix

From the basic equations of continuity, momentum and energy in Eulearian co-ordinates for the rotational axisymmetric flow of non-ideal gas with the similarity transformations

$$ \eta = \frac{r}{R}, $$

(61)

$$ u = \frac{r}{t}U(\eta ),v = \frac{r}{t}\phi (\eta ),w = \frac{r}{t}W(\eta ),\rho = \rho_{a} D\left( \eta \right),p = \frac{{r^{2} }}{{t^{2} }}\rho_{a} P(\eta ), $$

(62)

where the variable η assumes the value ‘1’ at the shock front and η _p on the piston surface, such that the piston radius r _p  = η _p R, R(∼ t ⁿ⁺¹)being the shock radius, we obtain

$$ \left[ {U - (n + 1)} \right]\frac{dD}{d\eta } + D\frac{dU}{d\eta } + \frac{2DU}{\eta } = 0 $$

(63)

$$ \left[ {U - (n + 1)} \right]\frac{dU}{d\eta } + \frac{1}{D}\frac{dP}{d\eta } + \frac{U(U - 1)}{\eta } + \frac{{(2P - \phi^{2} D)}}{D\eta } = 0 $$

(64)

$$ \left[ {U - (n + 1)} \right]\frac{d\phi }{d\eta } + \frac{(2U - 1)\phi }{\eta } = 0 $$

(65)

$$ \left[ {U - (n + 1)} \right]\frac{dW}{d\eta } + \frac{(U - 1)W}{\eta } = 0 $$

(66)

$$ \frac{dP}{d\eta } - \frac{P\,\gamma }{{D(1 - \bar{b}D)}}\frac{dD}{d\eta } + \frac{2(U - 1)P}{{\eta \left[ {U - (n + 1)} \right]}} = 0 $$

(67)

The boundary conditions for a strong shock in the rotating non-ideal gas at η = 1 are given by

$$ U\left( 1\right) = \left( { 1 - \beta } \right)\left( {n + 1} \right),\;D(1) = \frac{1}{\beta },\;P\left( 1\right) = \left( { 1 - \beta } \right)\left( {n + 1} \right)^{ 2} , $$

$$ \phi (1) = C^{*} \left( C \right)^{{ - \frac{1}{n + 1}}} ,W(1) = E\left( C \right)^{{ - \frac{1}{n + 1}}} $$

(68)

where it was necessary to use $\alpha = \delta = \frac{n}{n + 1}$ for existence of similarity solutions. In addition to the shock conditions (68), the kinematic condition U(η _p ) = (n + 1) at the piston surface must be satisfied. From Eqs. (63) and (67), we obtain the relation

$$ \frac{{D^{{\gamma + \,(\frac{n}{n + 1})}} \,}}{{(1 - \bar{b}D)^{\gamma } }} = \frac{{P\,\,\,\eta^{{(\frac{2}{n + 1})}} }}{{C_{1} }}\,\,[U - (n + 1)]^{{ - (\frac{n}{n + 1})}} $$

(69)

where C ₁ is a constant to be determined from (68).