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).

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.

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.

scholarly journals Path oscillations and enhanced drag of light rising spheres

2018 ◽  
Vol 841 ◽  
pp. 228-266 ◽  
Author(s):  
Franck Auguste ◽  
Jacques Magnaudet
Keyword(s):  
Density Ratio ◽  
Reynolds Numbers ◽  
Relative Magnitude ◽  
The Body ◽  
Sphere Diameter ◽  
Viscous Effects ◽  
Flow Parameters ◽  
Route To Chaos ◽  
Set Up ◽  
Equivalent Conditions

The dynamics of light spheres rising freely under buoyancy in a large expanse of viscous fluid at rest at infinity is investigated numerically. For this purpose, the computational approach developed by Mougin & Magnaudet (Intl J. Multiphase Flow, vol. 28, 2002, pp. 1837–1851) is improved to account for the instantaneous viscous loads induced by the translational and rotational sphere accelerations, which play a crucial role in the dynamics of very light spheres. A comprehensive map of the rise regimes encountered up to Reynolds numbers (based on the sphere diameter and mean rise velocity) of the order of $10^{3}$ is set up by varying independently the body-to-fluid density ratio and the relative magnitude of inertial and viscous effects in approximately 250 distinct combinations. These computations confirm or reveal the presence of several distinct periodic regions on the route to chaos, most of which only exist within a finite range of the sphere relative density and Reynolds number. The wake structure is analysed in these various regimes, evidencing the existence of markedly different shedding modes according to the style of path. The variation of the drag force with the flow parameters is also examined, revealing that only one of the styles of path specific to very light spheres yields a non-standard drag behaviour, with drag coefficients significantly larger than those measured on a fixed sphere under equivalent conditions. The outcomes of this investigation are compared with available experimental and numerical results, evidencing points of consensus and disagreement.

scholarly journals A Functional Form for Fine Sediment Interception in Vegetated Environments

Geosciences ◽  
2021 ◽  
Vol 11 (4) ◽  
pp. 157
Author(s):  
Samuel Stein ◽  
Jordan Wingenroth ◽  
Laurel Larsen
Keyword(s):  
Functional Relationship ◽  
Functional Form ◽  
Density Ratio ◽  
Diameter Ratio ◽  
The Body ◽  
Aquatic Environments ◽  
Vegetation Canopy ◽  
Flume Experiments ◽  
Canopy Density ◽  
Total Sedimentation

The body of literature seeking to evaluate particle interception in vegetated, aquatic environments is growing; however, comparing the results of these studies is difficult due to large variation in flow regime, particle size, vegetation canopy density, and stem configuration. In this work, we synthesize data from these studies and develop a functional form of particle interception efficiency (η) as a function of stem Reynolds number (Rec), stem diameter, vegetation frontal area, particle–collector diameter ratio, flow velocity, and kinematic viscosity. We develop this functional relationship based on a dimensional analysis and hypothesize that the coefficients would exhibit regimes within different Rec ranges. We test this hypothesis by synthesizing data from 80 flume experiments reported in the literature and in-house flume experiments. Contrary to our hypothesis, data from different Rec ranges follow a single functional form for particle interception. In this form, η varies strongly with collector density and particle–collector diameter ratio, and weakly with Rec and particle–fluid density ratio. This work enables more accurate modeling of the flux terms in sedimentation budgets, which can inform ongoing modeling and management efforts in marsh environments. For example, we show that by integrating the new functional form of particle interception into established models of marsh elevation change, interception may account for up to 60% of total sedimentation in a typical silt-dominated marsh ecosystem with emergent vegetation.

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.

Note on the Assessment of Flow Disturbances at a Blunt Body Traveling at Supersonic Speeds Owing to Flow Disturbances in Free Stream

1960 ◽  
Vol 27 (2) ◽  
pp. 223-229 ◽  
Author(s):  
M. V. Morkovin
Keyword(s):  
Boundary Layer ◽  
Blunt Body ◽  
Free Stream ◽  
Pressure Fluctuations ◽  
The Body ◽  
Normal Shock ◽  
Pressure Waves ◽  
Flow Disturbances ◽  
High Mach ◽  
Blunt Nose

For the purposes of assessing the magnitude of flow disturbances which would affect conditions on a blunt nose of a body moving at supersonic speeds, the detached shock is approximated by a purely normal shock. The disturbances downstream of the shock are expressed in terms of the “free-stream” disturbances by considering sinusoidal fluctuations. Pressure fluctuations generated by interactions of entropy-temperature disturbances with the normal shock may be considerable at high Mach numbers, but their effect on the transition of a laminar boundary layer to a turbulent one is a matter of speculation. However, conjectures that reflections of such pressure waves between the body and the shock wave might lead to high resonant amplifications are definitely disproved.

Axisymmetric hypersonic flow with strong viscous interaction

1968 ◽  
Vol 34 (4) ◽  
pp. 687-703 ◽  
Author(s):  
John Webster Ellinwood ◽  
Harold Mirels
Keyword(s):  
Shock Wave ◽  
Hypersonic Flow ◽  
Power Law ◽  
Axisymmetric Flow ◽  
Strong Shock ◽  
The Body ◽  
Strong Shock Wave ◽  
Viscous Interaction ◽  
Slender Bodies ◽  
Prandtl Numbers

Stewartson's theory for axisymmetric hypersonic flow of a model gas over slender bodies with strong viscous interaction and strong shock wave is extended to power-law viscosity variation and Prandtl numbers other than one. Flow properties at the body surface and shock are obtained without recourse to numerical integration. Numerical computations are presented for axisymmetric flow over a three-quarter power-law body with strong shock wave and viscous interactions that range from weak to strong.

Ship Roll Motion Damping Using Bilge Keels – A Detailed CFD Investigation

2015 ◽  
Author(s):  
Joshua Counsil ◽  
Kevin McTaggart ◽  
Dominic Groulx ◽  
Kiari Boulama
Keyword(s):  
Experimental Studies ◽  
The Body ◽  
Navier Stokes ◽  
Roll Motion ◽  
Empirical Methods ◽  
Viscous Effects ◽  
Vortex Interactions ◽  
Ship Roll ◽  
Semi Empirical ◽  
Bilge Keel

A study has been undertaken to test the value of unsteady Reynolds-averaged Navier-Stokes (URANS) and traditional semi-empirical methods in the face of complex ship roll phenomena, and provide insight into the selection of bilge keel span for varying roll amplitudes. The computational fluid dynamics (CFD) code STAR-CCM+ is employed and two-dimensional submerged bodies undergoing forced roll motion are analyzed. The spatial resolution and timestepping scheme are validated by comparison with published numerical and experimental studies. The model is then applied to a fully-submerged circular cylinder with bilge keels of varying span and undergoing roll motion at varying angular amplitudes. Extracted hydrodynamic coefficients indicate that in general, increasing displacement amplitude and bilge keel span yields increased added mass and increased damping. The relationship is complex and highly dependent upon vortex interactions with each other and the body. The semi-empirical methods used for comparison yield good predictions for simple vortex interactions but fail where viscous effects are strong. Hence, URANS methods are shown to be necessary for friction-dominated flows while semi-empirical methods remain useful for initial design considerations.

A Comparison of Theoretical and Experimental Pressure Distributions on Bodies of Revolution

1980 ◽  
Vol 24 (01) ◽  
pp. 60-65
Author(s):  
A. J. Smits ◽  
S. P. Law ◽  
P. N. Joubert
Keyword(s):  
Calculation Method ◽  
The Body ◽  
Viscous Effects ◽  
Bodies Of Revolution ◽  
Pressure Distributions ◽  
Wide Range ◽  
The Right ◽  
Right Order ◽  
Axisymmetric Bodies ◽  
Experimental Pressure

A wide range of experimental pressure distributions along axisymmetric bodies was compared with the results of Landweber's potential flow calculation method. Apart from certain viscous effects, some discrepancies were found, and it is shown that blockage corrections are of the right order to account for these discrepancies. The calculation method was also used to show that the pressure distribution over the nose of the body is largely independent of the tail shape, and vice versa.

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.

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