A uniformly valid solution for the hypersonic flow past blunted bodies

1968 ◽  
Vol 31 (2) ◽  
pp. 397-415 ◽  
Author(s):  
W. Schneider
Keyword(s):  
Flow Field ◽  
Hypersonic Flow ◽  
Local Thermodynamic Equilibrium ◽  
Density Ratio ◽  
Intersection Point ◽  
The Body ◽  
Boundary Values ◽  
Stagnation Region ◽  
Flow Past ◽  
Valid Solution

The plane and axisymmetric hypersonic flow past blunted bodies is investigated as an inverse problem (shock shape given). The fluid may behave as a real gas in local thermodynamic equilibrium. Viscosity and heat conduction are neglected. An analytical solution uniformly valid in the whole flow field (from the stagnation region up to large distances from the body nose) is given. The solution is based on two main assumptions: (i) the density ratio ε across the shock is very small, (ii) the pressure at a pointPof the disturbed flow field isnotvery small compared with the pressure immediately behind the shock in the intersection point of the shock surface with its normal throughP.TermsO(ε) are neglected in comparison with 1, but it is not necessary for the shock layer to be thin. The change of velocity along streamlines is taken into account. In order to calculate the flow quantities one has to evaluate only two integrals (equations (49) and (53) together with the boundary values (5) and (10)). The application of the solution is illustrated and the accuracy is tested in some examples.

Density Distribution in Stagnation Region of Saffman Equation for Dusty Gas

2002 ◽  
Author(s):  
Kazuhiro Tsuboi
Keyword(s):  
Flow Field ◽  
Particle Density ◽  
Stokes Number ◽  
The Body ◽  
Particle Flow ◽  
Particle Phase ◽  
Primary Interest ◽  
Stagnation Region ◽  
Two Fluid ◽  
Good Agreement

We investigate the behaviour of flow field around an obstacle placed in uniform particle flow based on two-fluid Saffman equation. Particle density in the vicinity of the front stagnation point is, in particular, the primary interest in the present study. In the case of small Stokes number, in which particle impingement does not occur, there exists the exact solution of the flow field of particle phase is obtained. Perturbed solution is also obtained in the reciprocal of Stokes number when Stokes number is large enough. Comparison between numerical results and these solutions shows good agreement and the peak of particle density appears near the threshold of partide impingement to the body surface.

Hypersonic Swirling Flow past Blunt Bodies

1973 ◽  
Vol 24 (4) ◽  
pp. 241-251 ◽  
Author(s):  
Roger Smith
Keyword(s):  
High Speed ◽  
Swirling Flow ◽  
Pressure Coefficient ◽  
Density Ratio ◽  
Perturbation Expansion ◽  
The Body ◽  
Constant Density ◽  
Flow Past ◽  
Speed Flow ◽  
Blunt Bodies

SummaryThe effect of swirl on the high speed flow past blunt bodies is analysed by assuming constant density in the region between the shock wave and the body. For small swirl the stand-off distance is only slightly affected, but it is shown that there is a critical value of the swirl parameter which, if exceeded, will cause a jump in the position of the shock. This is demonstrated by solving the full constant-density equations for the flow past a sphere and by a perturbation expansion in powers of the density ratio across the shock for a more general body shape. The perturbation solution shows that the pressure coefficient on the body is constant at the critical swirl number.

scholarly journals The steady oblique path of buoyancy-driven disks and spheres

2012 ◽  
Vol 707 ◽  
pp. 24-36 ◽  
Author(s):  
David Fabre ◽  
Joël Tchoufag ◽  
Jacques Magnaudet
Keyword(s):  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Density Ratio ◽  
Steady Motion ◽  
The Body ◽  
Body Motion ◽  
Angle Of Incidence ◽  
Weakly Nonlinear ◽  
Numerical Studies ◽  
Flow Past

AbstractWe consider the steady motion of disks of various thicknesses in a weakly viscous flow, in the case where the angle of incidence $\ensuremath{\alpha} $ (defined as that between the disk axis and its velocity) is small. We derive the structure of the steady flow past the body and the associated hydrodynamic force and torque through a weakly nonlinear expansion of the flow with respect to $\ensuremath{\alpha} $. When buoyancy drives the body motion, we obtain a solution corresponding to an oblique path with a non-zero incidence by requiring the torque to vanish and the hydrodynamic and net buoyancy forces to balance each other. This oblique solution is shown to arise through a bifurcation at a critical Reynolds number ${\mathit{Re}}^{\mathit{SO}} $ which does not depend upon the body-to-fluid density ratio and is distinct from the critical Reynolds number ${\mathit{Re}}^{\mathit{SS}} $ corresponding to the steady bifurcation of the flow past the body held fixed with $\ensuremath{\alpha} = 0$. We then apply the same approach to the related problem of a sphere that weakly rotates about an axis perpendicular to its path and show that an oblique path sets in at a critical Reynolds number ${\mathit{Re}}^{\mathit{SO}} $ slightly lower than ${\mathit{Re}}^{\mathit{SS}} $, in agreement with available numerical studies.

On the theory of hypersonic flow past plane and axially symmetric bluff bodies

1956 ◽  
Vol 1 (4) ◽  
pp. 366-387 ◽  
Author(s):  
N. C. Freeman
Keyword(s):  
Hypersonic Flow ◽  
The Body ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Radius Of Curvature ◽  
Bluff Bodies ◽  
Axially Symmetric ◽  
Nose Radius ◽  
Flow Past ◽  
Past Plane

The ‘Newtonian-plus-centrifugal’ approximate solution (Busemann (1933) and Ivey (1948)) for hypersonic flow past plane and axially symmetric bluff bodies in gases with the ratio of the specific heats λ constant and equal to unity is rederived using ‘boundary layer’ techniques together with the von Mises variables x and ψ. A method of successive approximations then gives a closer approximation to this solution for ε (λ − 1)/(λ + 1) small and the free-strea Mach number infinite. Formulae for the streamlines, shock shape and pressure distribution are determined to this approximation. These formulae are valid for any plane or axially symmetric shape, giving the ‘stand-off’ distance of the shock wave from the body as ½εlog(4|3ε) and ε times the nose radius of curvature for plane and axially-symmetric flows respectively. Particular results are computed for a number of special shapes. For certain shapes, the theory has a singular point where the first approximation to the pressure vanishes (θ = 60° for a sphere). Actually, the theory is not applicable where the pressure becomes too small. The corresponding theory for gases of general thermodynamic properties is deduced, the approximation being valid provided the total energy of the gas is large compared with the energy contained in the translational modes of the gas molecules.

On the viscous flow in the nose region of a symmetric blunt body in hypersonic flow

1966 ◽  
Vol 26 (4) ◽  
pp. 829-839 ◽  
Author(s):  
B. S. H. Rarity
Keyword(s):  
Hypersonic Flow ◽  
Blunt Body ◽  
Density Ratio ◽  
The Body ◽  
Entire Region ◽  
Nose Radius ◽  
Stagnation Line ◽  
Total Enthalpy ◽  
Post Shock ◽  
The Given

The flow field in the nose region of a blunt body in hypersonic flow is studied by considering the transport of vorticity and enthalpy. The entire region between body and shock is considered to be viscous, not necessarily thin in comparison with the nose radius of the body and to be of slowly varying density. The (given) post-shock vorticity need not be small and the density ratio ρ∞/ρs may either be small or near unity, the analysis being valid asymptotically at both limits.It is found that the vorticity equation may be uncoupled from the total enthalpy equation if μ√ρ is constant. While the equations are not expected to be necessarily restricted to the immediate vicinity of the stagnation line, only there can the solution be written down explicitly; elsewhere, numerical integration is required.

Computational analysis of the flow field structure of a non-reacting hypersonic flow over forward-facing steps

2014 ◽  
Vol 763 ◽  
pp. 460-499 ◽  
Author(s):  
P. H. M. Leite ◽  
W. F. N. Santos
Keyword(s):  
Heat Transfer ◽  
Flow Field ◽  
Hypersonic Flow ◽  
Field Structure ◽  
Recirculation Region ◽  
Transition Flow ◽  
Face Surface ◽  
Face Height ◽  
Flow Past ◽  
Forward Facing Step

AbstractThis work is a computational study of a rarefied non-reacting hypersonic flow past a forward-facing step at zero-degree angle of attack in thermal non-equilibrium. Effects on the flow field structure and on the aerodynamic surface quantities due to changes in step frontal-face height are investigated by employing the direct simulation Monte Carlo method. The work focuses the attention of designers of hypersonic configurations on the fundamental parameter of surface discontinuity, which can have an important impact on even initial design. The results presented highlight the sensitivity of the primary flow field properties, velocity, density, pressure and temperature, to changes in the step frontal-face height. In addition, the behaviour of heat transfer, pressure and skin friction coefficients with variation of the step frontal-face height is detailed. The analysis shows that hypersonic flow past a forward-facing step in the transition flow regime is characterized by a strong compression ahead of the frontal face, which influences the aerodynamic surface properties upstream and adjacent to the frontal face. The analysis also shows that the extension of the upstream disturbance depends on the step frontal-face height. It was found that the recirculation region ahead of the step is also a function of the frontal-face height. A sequence of Moffatt eddies of decreasing size and intensity is observed in the concave step corner. Locally high heating and pressure loads were observed at three locations along the surface, i.e. on the lower surface, on the frontal face and on the upper surface. The results showed that both loads rely on the frontal-face height. The peak values for the heat transfer coefficient on the frontal-face surface were at least one order of magnitude larger than the maximum value observed for a smooth surface, i.e. a flat plate without a step. A comparison of the present simulation results with numerical and experimental data showed close agreement concerning the wall pressure acting on the step surface.

Hypersonic flow over spherically blunted cone capsules for atmospheric entry. Part 1. The sharp cone and the sphere

2019 ◽  
Vol 871 ◽  
pp. 1097-1116 ◽  
Author(s):  
H. G. Hornung ◽  
Jan Martinez Schramm ◽  
Klaus Hannemann
Keyword(s):  
Hypersonic Flow ◽  
Functional Form ◽  
Cone Angle ◽  
Density Ratio ◽  
The Body ◽  
Normal Shock ◽  
Viscous Effects ◽  
Atmospheric Entry ◽  
Shock Density ◽  
Sharp Cone

Depending on the cone half-angle and the inverse normal-shock density ratio $\unicode[STIX]{x1D700}$, hypersonic flow over a spherically blunted cone exhibits two regimes separated by an almost discontinuous jump of the body end of the sonic line from a point on the spherical nose to the shoulder of the cone, here called sphere behaviour and cone behaviour. The inflection point of the shock wave in sphere behaviour is explained. In Part 1 we explore the two elements of the capsule shape, the sphere and the sharp cone with detached shock, theoretically and computationally, in order to put the treatment of the full capsule shape on a sound basis. Starting from the analytical expression for the shock detachment angle of a cone given by Hayes & Probstein (Hypersonic Flow Theory, 1959, Academic Press) we make a hypothesis for the sharp cone, about the functional form of the dependence of dimensionless quantities on $\unicode[STIX]{x1D700}$ and a cone angle parameter, $\unicode[STIX]{x1D702}$. In the critical part of atmospheric entry the shock shape and drag of the capsule are insensitive to viscous effects, so that much can be learned from inviscid studies. Accordingly, the hypothesis is tested by making a large number of Euler computations to cover the parameter space: Mach number, specific heat ratio and cone angle. The results confirm the hypothesis in the case of the dimensionless shock stand-off distance as well as for the drag coefficient, yielding accurate analytical functions for both. This reduces the number of independent parameters of the problem from three to two. A functional form of the shock stand-off distance is found for the transition from the $90^{\circ }$ cone to the sphere. Although the analysis assumes a calorically perfect gas, the results may be carried over to the high-enthalpy real-gas situation if the normal-shock density ratio is replaced by the density ratio based on the average density along the stagnation streamline (see e.g. Stulov, Izv. AN SSSR Mech. Zhidk. Gaza, vol. 4, 1969, pp. 142–146; Hornung, J. Fluid Mech., vol. 53, 1972, pp. 149–176; Wen & Hornung, J. Fluid Mech., vol. 299, 1995, pp. 389–405).

Some Special Aspects of Hypersonic Flow Fields

1959 ◽  
Vol 63 (585) ◽  
pp. 508-512 ◽  
Author(s):  
K. W. Mangler
Keyword(s):  
Hypersonic Flow ◽  
High Speed ◽  
Flow Fields ◽  
The Body ◽  
Lower Velocity ◽  
Bow Wave ◽  
Total Head ◽  
Bow Shocks ◽  
Lower Entropy ◽  
Subsonic Region

When a body moves through air at very high speed at such a height that the air can be considered as a continuum, the distinction between sharp and blunt noses with their attached or detached bow shocks loses its significance, since, in practical cases, the bow wave is always detached and fairly strong. In practice, all bodies behave as blunt shapes with a smaller or larger subsonic region near the nose where the entropy and the corresponding loss of total head change from streamline to streamline due to the curvature of the bow shock. These entropy gradients determine the behaviour of the hypersonic flow fields to a large extent. Even in regions where viscosity effects are small they give rise to gradients of the velocity and shear layers with a lower velocity and a higher entropy near the surface than would occur in their absence. Thus one can expect to gain some relief in the heating problems arising on the surface of the body. On the other hand, one would lose farther downstream on long slender shapes as more and more air of lower entropy is entrained into the boundary layer so that the heat transfer to the surface goes up again. Both these flow regions will be discussed here for the simple case of a body of axial symmetry at zero incidence. Finally, some remarks on the flow field past a lifting body will be made. Recently, a great deal of information on these subjects has appeared in a number of reviewing papers so that little can be added. The numerical results on the subsonic flow regions in Section 2 have not been published before.

The Supersonic Flow Past a Slender Ducted Body of Revolution with an Annular Intake

1950 ◽  
Vol 1 (4) ◽  
pp. 305-318
Author(s):  
G. N. Ward
Keyword(s):  
Supersonic Flow ◽  
Velocity Potential ◽  
Operational Calculus ◽  
The Body ◽  
Body Of Revolution ◽  
Internal Flows ◽  
Flow Past ◽  
External Forces ◽  
Internal Pressures ◽  
Slender Body Of Revolution

SummaryThe approximate supersonic flow past a slender ducted body of revolution having an annular intake is determined by using the Heaviside operational calculus applied to the linearised equation for the velocity potential. It is assumed that the external and internal flows are independent. The pressures on the body are integrated to find the drag, lift and moment coefficients of the external forces. The lift and moment coefficients have the same values as for a slender body of revolution without an intake, but the formula for the drag has extra terms given in equations (32) and (56). Under extra assumptions, the lift force due to the internal pressures is estimated. The results are applicable to propulsive ducts working under the specified condition of no “ spill-over “ at the intake.

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