Dissociation effect on the hypersonic flow past a circular cylinder

The effects of dissociation of air on hypersonic flow past a circular cylinder at zero angle of incidence are considered under the assumptions that the shock wave is in the shape of a circular cylinder, the density ratio across the shock is constant, the flow behind the shock is at constant density and dissociation occurs only behind the shock wave. In the present paper, the velocity, pressure and drag coefficients, vorticity, shock detachment distance, stagnation point velocity gradient and sonic points on the shock and the surface have been obtained in the presence of dissociation. The results have been compared with the corresponding results obtained in the case when dissociation dose not occur and the corresponding results in the case of the sphere in the presence of dissociation.     

Dissociation effects in hypersonic flow past a circular cone at an angle of attack

The analytical solution to the steady, compressible, non-viscous, inviscid hypersonic flow past a circular cone at an angle of incidence, with an attached Shockwave, in the presence of dissociation of air in the shock layer, has been obtained here under the assumption of thermal equilibrium. Expression for the velocity, pressure, temperature, density, velocity of air, Mach number, pressure, drag and lift coefficients have been obtained both in the shocklayer outside the vortical layer and on the surface of the cone inside the vortical layer.     

Flow Structure behind a Shock Wave in a Channel with Periodically Arranged Obstacles

We study the propagation of a pressure wave in a rectangular channel with periodically arranged obstacles and show that a flow corresponding to a discontinuity structure may exist in such a channel. The discontinuity structure is a complex consisting of a leading shock wave and a zone in which pressure relaxation occurs. The pressure at the end of the relaxation zone can be much higher than the pressure immediately behind the gas-dynamic shock. We derive an approximate formula that relates the gas parameters behind the discontinuity structure to the average velocity of the structure. The calculations of the pressure, velocity, and density of the gas behind the structure that are based on the average velocity of the structure agree well with the results of gas-dynamic calculations. The approximate dependences obtained allow us to estimate the minimum pressure at which there exists a flow with a discontinuity structure. This estimate is confirmed by gas-dynamic calculations.     

The shapes of bodies with maximum lift-to-drag ratio in supersonic flow

Assuming that the pressure coefficient on the body surface is defined by the angle between the local normal to it and the velocity vector of the undisturbed flow, the problem of the shape of a body which possesses the maximum lift-to-drag ratio is solved. When the bottom section area and the constant coefficient of friction are given, the optimal body has a plane windward surface positioned at the angle of attack to the undisturbed flow. The leeward surface of the optimal body is parallel to the velocity vector of the undisturbed flow. The absolutely optimal body is a two-dimensional wedge. When additional constraints on the external dimensions of the body are specified, solutions of variational problems are obtained on the basis of which bodies which have the maximum lift-to-drag ratio in supersonic flow are designed.     

Rapid cylindrically and spherically symmetric strong compression of a perfect gas

The problem of the rapid cylindrically and spherically symmetric strong compression of a perfect (non-viscous and non-heat-conducting) gas is solved. The term “rapid” denotes that the compression time is much less than the run time of a sound wave across the initial cylindrical or spherical volume, while the term “strong” in this case means the simultaneous attainment of as large a density and temperature as desired. By definition, rapid compression must begin in a strong shock wave, which propagates to the axis or centre of symmetry. When the shock wave approaches the centre of symmetry this flow is described by the self-similar Guderley equation with an unbounded rise in temperature, pressure and velocity and a finite increase in the density at the centre of symmetry both behind the arriving and behind the reflected shock waves. To obtain as high an increase in the density as desired one must add on a centred compression wave with focus at the centre of symmetry to the overtaking shock wave at the instant it arrives at the centre of symmetry C⁻-characteristic. Outside a small neighbourhood of the focus one can calculate, by the method of characteristics, the centred wave and the trajectory of the piston which produces it. As for any centred wave, this calculation must be carried out from the centre of symmetry. Since some of the parameters at the focus (certainly the pressure, temperature and velocity of the gas) are unbounded, it is necessary to preface the calculation by the method of characteristics by constructing an analytic solution which holds in a small neighbourhood of the centre of symmetry. Below, after constructing the required solution, the centred waves corresponding to it and the trajectories of the piston producing them are calculated.     

The motion of a submerged sphere in a liquid under an ice sheet

The solution of the linear steady problem of the flow of an inviscid, incompressible and infinitely deep liquid around a sphere under an ice sheet, which is modelled by a thin elastic stressed plate of constant thickness is constructed. Special cases of this problem are the motion of a submerged sphere under broken ice, a membrane, and also under the free surface both in the presence and absence of capillary effects. The method of multipole expansions is used in the framework of the linear potential wave theory. The hydrodynamic loads (the wave drag and the buoyancy) acting on the body and also the distribution of the deflections of the ice sheet are calculated as a function of the body velocity, the ice thickness and the value of the compressing or stretching forces. It is shown that all the flow characteristics depend considerably on the ratio of the body velocity and the critical velocity of flexural-gravitational waves.     

Spherical shock wave generated by a moving piston in mixture of a non-ideal gas and small solid particles under a gravitational field

Self-similar solutions are obtained for one-dimensional isothermal and adiabatic unsteady flows behind a strong spherical shock wave propagating in a dusty gas. The shock is assumed to be driven out by a moving piston and the dusty gas to be a mixture of a non-ideal (or perfect) gas and small solid particles, in which solid particles are continuously distributed. It is assumed that the equilibrium flow-conditions are maintained and variable energy input is continuously supplied by the piston. The medium is under the influence of the gravitational field due to a heavy nucleus at the origin (Roche model). The effects of an increase in the mass concentration of solid particles, the ratio of the density of the solid particles to the initial density of the gas, the gravitational parameter and the parameter of non-idealness of the gas in the mixture, are investigated. It is shown that due to presence of gravitational field the compressibility of the medium at any point in the flow-field behind the shock decreases and all other flow-variables and the shock strength increase. A comparison has also been made between the isothermal and adiabatic flows. It is investigated that the singularity in the density and compressibility distributions near the piston in the case of adiabatic flow are removed when the flow is isothermal.     

The effect of gravity on flow past a semi-circular cylinder with a constant pressure wake

The flow past a semi-cylinder with a trailing wake region is considered. In the absence of gravity the only known high Reynolds number solutions have tangential separation from the body and a cusped shape at the back of the wake. This flow can be a simple model for several situations, including the classical approximation of a constant pressure wake and the flow past an object with a region of trapped fluid of different density (or an air cavity) attached on the downstream side. Here we relax the assumption of high flow speeds to examine the effects of gravity. It is shown that there are situations in which a stagnation point can form either on the body or at the tail of the wake and that there is a minimum velocity beneath which a cavity will not form. Non-uniqueness in the parameter space is found in certain cases.     

Converging strong shock waves in magnetogasdynamics under isothermal condition

The similarity solutions to the problem of cylindrically symmetric strong shock waves in an ideal gas with a constant azimuthal magnetic field are presented. The flow behind the shock wave is assumed to spatially isothermal rather than adiabatic. We use the method of Lie group invariance to determine the possible class of self-similar solutions. Infinitesimal generators of Lie group transformations are determined by using the invariance surface conditions to the system and on the basis of arbitrary constants occurring in the expressions for the generators, four different possible cases of the solutions are reckoned and we observed that only two out of all possibilities hold self-similar solutions, one of which follows the power law and another follows the exponential law. To obtain the similarity exponents numerical calculations have been performed and comparison is made with the existing results in the literature. The flow patterns behind the shock are analyzed graphically.

     

Isothermal flow behind a shock wave propagating in the solar wind

A Parker-type blast wave, which is headed by a strong shock, driven out by a propelling contact surface, moving into an ambient solar wind having a strictly inverse square law radial decay in density, is studied. Assuming the self-similar flow behind the shock to be isothermal, approximate analytical and exact numerical solutions are obtained. There is a good agreement between the approximate analytical and exact numerical solutions. It is observed that the mathematical singularity in density at the contact surface is removed for the isothermal flow.     

Shock Interactions in Nonequilibrium Hypersonic Flow

A shock interaction problem is solved with finite difference methods for a hypersonic flow of air with chemical reactions. If a body has two concave corners, a secondary shock is formed in the shock layer and it meets the main shock later. As the two shocks meet, the flow becomes singular at the interaction point, and a new main shock, a contact discontinuity and an expansion wave appear as a result of interaction between the two shocks. Therefore, the problem is very complicated. Using proper combinations of implicit and explicit finite difference schemes according to the property of the equations and the boundary conditions, we compute the flow behind the interaction point successfully.     

Energy considerations of the flow behind a plane hydromagnetic shock

Similarity solutions describing the flow behind a plane hydromagnetic shock propagating with a constant velocity into a uniform ideal gas at rest in the presence of a transverse magnetic field are obtained. The gas is assumed to be infinitely electrically conducting, inviscid and non-heat conducting. The gain in the total energy of the flow between the shock and the inner expanding surface is assumed to be time-dependent. The variations of the percentages of the magnetic, internal and kinetic energies with the strength of the shock are studied. It is shown that there exists two values of the strength of the shock at which equipartition of the internal and kinetic energies of the flow between the shock and the inner expanding surface can occur.     

Analytic solution of self-similar strong shocks in an exponential medium for zero temperature gradient

The self-similar one-dimensional propagation of a strong shock wave in a medium with an exponentially decreasing density is studied. The flow behind the shock is assumed to be spatially isothermal rather than adiabatic to simulate the conditions of large radiative transfer behind the shock. The solution in closed form is obtained. An analytic expression for the similarity exponent has also been obtained.     

Similarity solution for a spherical shock wave in a non‐ideal gas under the influence of gravitational field and monochromatic radiation with increasing energy

The propagation of a spherical shock wave in a non‐ideal gas with or without gravitational effects is investigated under the action of monochromatic radiation. Similarity solutions are obtained for adiabatic flow between the shock and the piston. The numerical solutions are obtained using the Runge‐Kutta method of the fourth order. The density of the gas is assumed to be constant. The total energy of the shock wave is non‐constant and varies with time. The effects of change in values of non‐idealness parameter, gravitational parameter, shock Mach number, radiation parameter, and adiabatic exponent of the gas on shock strength and flow variables are worked out in detail. It is investigated that the presence of gravitational field increases the compressibility of the medium, due to which it is compressed and, therefore, the distance between the inner contact surface and the shock surface is reduced. A comparison is also made between the solutions in the cases of the gravitating and the non‐gravitating media. It is manifested that the gravitational parameter and the radiation parameter have in general opposite behaviour on the flow variables and the shock strength.     

Finite element calculations of flow past a spherical bubble rising on the axis of a cylindrical tube

The velocity and pressure fields of a Newtonian fluid with homogeneous and constant physical properties flowing around a sphere on the axis of a cylindrical tube with no slip, free slip and partial slip at the sphere surface and no slip at the cylinder wall have been calculated by solving the Navier-Stokes equations and the continuity equation using the finite element technique with the penalty function method. Terminal rise velocities of spherical air bubbles in water have been calculated as function of the bubble radius and some conclusions have been drawn about the nature of the interface. Finally, the influence of the presence of a cylindrical wall on the drag force has been determined and a new empirical equation is derived for the wall correction factor for a sphere rising with free slip at its surface at low Reynolds number.     

The applicability of continuum models in the transitional regime of hypersonic flow over blunt bodies

Hypersonic rarefied gas flow over blunt bodies in the transitional flow regime (from continuum to free-molecule) is investigated. Asymptotically correct boundary conditions on the body surface are derived for the full and thin viscous shock layer models. The effect of taking into account the slip velocity and the temperature jump in the boundary condition along the surface on the extension of the limits of applicability of continuum models to high free-stream Knudsen numbers is investigated. Analytic relations are obtained, by an asymptotic method, for the heat transfer coefficient, the skin friction coefficient and the pressure as functions of the free-stream parameters and the geometry of the body in the flow field at low Reynolds number; the values of these coefficients approach their values in free-molecule flow (for unit accommodation coefficient) as the Reynolds number approaches zero. Numerical solutions of the thin viscous shock layer and full viscous shock layer equations, both with the no-slip boundary conditions and with boundary conditions taking into account the effects slip on the surface are obtained by the implicit finite-difference marching method of high accuracy of approximation. The asymptotic and numerical solutions are compared with the results of calculations by the Direct Simulation Monte Carlo method for flow over bodies of different shape and for the free-stream conditions corresponding to altitudes of 75–150 km of the trajectory of the Space Shuttle, and also with the known solutions for the free-molecule flow regine. The areas of applicability of the thin and full viscous shock layer models for calculating the pressure, skin friction and heat transfer on blunt bodies, in the hypersonic gas flow are estimated for various free-stream Knudsen numbers.     

Global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone

In this paper, we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Theoretically, as indicated in [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, 1948], it follows from the Rankine-Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface, which correspond to the supersonic shock and the transonic shock respectively. In the references [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47-84; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Polytropic case, preprint, 2006; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Isothermal case, Pacific J. Math. 233 (2) (2007) 257-289] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101-132], the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow. At present, for the supersonic symmetric incoming flow, we will study the global transonic shock problem when the pressure at infinity is appropriately large.     

Numerical simulation of sphere water entry problem using Eulerian–Lagrangian method

This paper looks at the hydrodynamic’s numerical simulation of a free-falling sphere impacting the free surface of water by using the coupled Eulerian–Lagrangian (CEL) formulation included in the commercial software ABAQUS. A 3D model of a sphere with an unsteady viscous transient flow condition is used for numerical simulation. The simulation is performed for sphere with different density. The simulation results are verified by showing the computed shape of the air cavity, displacement of sphere, pinch-off time and depth that agree well with experimental results.     

Approximate analytical solutions for strong shocks with variable energy

Approximate analytical solutions are obtained for self-similar flows behind strong shocks with variable energy deposition or withdrawal at the wavefront in a perfect gas at rest with constant initial density. Numerical solutions are also obtained and the approximate solutions agree with these solutions. The effect of the adiabatic index on the solutions is investigated. The dependence of shock density ratio on the parameter characterizing the energy of the flow is studied. It is observed that the rate of deposition of energy at the wavefront decreases with increase of the parameter that specifies the total energy of the flow.