Power-law Bodies Research Articles

Truncated Power-Law Bodies as a Result of an Approximate Solution to the Newton Problem on a Body Having the Minimum Drag

Truncated Power-Law Bodies as a Result of an Approximate Solution to the Newton Problem on a Body Having the Minimum Drag

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

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.

Effect of Fineness Ratio on Minimum-Drag Shapes in Hypersonic Flows

The importance of fineness ratio in determining the aerodynamic shapes of minimum-drag, zero-lift axisymmetric bodies in inviscid hypersonic flows is investigated using a shape optimization framework. The framework employs modified Newtonian theory for surface pressure computation, Bezier curves for geometric parameterization and a steepest-descent approach for minimization of the wave-drag coefficient. Studies are performed on axisymmetric bodies of a given length for fineness ratios varying from 1 to 6 and Mach numbers ranging from 6 to 12. It is shown that the optimal axisymmetric body for a given freestream Mach number is blunt-nosed for smaller fineness ratios but becomes sharp-nosed at larger fineness ratios. The fineness ratio at which the optimal body of revolution transitions from blunt-nosed to sharp-nosed is found to be nearly constant at hypersonic speeds and is around 3. Results indicate that the optimal bodies derived from the framework are superior in terms of wave drag to the von Kármán ogive but comparable to the 0.7 power-law body. High-resolution Euler simulations of these optimal forebodies demonstrate that, while the modified Newtonian theory underpredicts the wave drag for lower values of the similarity parameter, the derived configurations remain “near-optimal” for the entire range of the similarity parameter in the hypersonic regime. The study hints at the possibility of multiple near-optimal configurations in inviscid hypersonic flows and affirms the utility of the present framework as a low-fidelity approach for the design of minimum-drag configurations.

Retransmission Delays With Bounded Packets: Power-Law Body and Exponential Tail

Retransmissions serve as the basic building block that communication protocols use to achieve reliable data transfer. Until recently, the number of retransmissions was thought to follow a geometric (light-tailed) distribution. However, recent work shows that when the distribution of the packet sizes have infinite support, retransmission-based protocols may result in heavy-tailed delays and possibly zero throughput even when the aforementioned distribution is light-tailed. In reality, however, packet sizes are often bounded by the maximum transmission unit (MTU), and thus the aforementioned result merits a deeper investigation. To that end, in this paper, we allow the distribution of the packet size $L$ to have finite support. Under mild conditions, we show that the transmission duration distribution exhibits a transition from a power-law main body to an exponential tail. The timescale to observe the power-law main body is roughly equal to the average transmission duration of the longest packet. The power-law main body, if significant, may cause the channel throughput to be very close to zero. These theoretical findings provide an understanding on why some empirical measurements suggest heavy tails. We use these results to further highlight the engineering implications of distributions with power-law main bodies and light tails by analyzing two cases: 1) the throughput of on–off channels with retransmissions, where we show that even when packet sizes have small means and bounded support the variability in their sizes can greatly impact system performance; 2) the distribution of the number of jobs in an $M/M/\infty$ queue with server failures. Here, we show that retransmissions can cause long-range dependence and quantify the impact of the maximum job sizes on the long-range dependence.

Controlling the level of the sonic boom generated by a flying vehicle by means of cryogenic forcing. 2. Distributed injection of a supercooled gas from the vehicle surface

The possibility of improving the efficiency of cryogenic forcing on the parameters of the hanging shock determining the length of the region of minimization of the sonic boom (middle zone) generated by a modified power-law body is studied. The effect of distributed injection of the coolant from the body surface on the formation of a perturbed flow near the body and at large distances from the body is considered. The scheme of distributed injection and the regime of coolant exhaustion are demonstrated to exert a significant effect on the length of the middle zone of the sonic boom. A scheme of cryogenic forcing is determined, which ensures reduction of bow shock wave intensity by more than 40% at distances corresponding to 7000 body diameters. The mechanism of cryogenic forcing on the flow structure near the body is discussed.

Controlling the level of the sonic boom generated by a flying vehicle by means of cryogenic forcing. 1. Cooling of the vehicle surface

The possibility of controlling the sonic boom level by means of cooling the surface of a flying vehicle is discussed. The effect of surface cooling on the formation of the perturbed flow structure at large distances from the vehicle is demonstrated by an example of a modified power-law body of revolution. The intensity of the intermediate shock wave and the perturbed pressure pulse near the body are seen to decrease, which expands the altitude range of the region where the sonic boom is reduced (down to 50%). At larger distances from the body, cryogenic forcing ensures a 12% decrease in the bow shock wave intensity. The possibility of controlling the process of formation of wave structures near the surface, such as barrel shock waves, is demonstrated. An explanation of the cryogenic forcing mechanism is offered.

Optimization of the shape of a body with respect to two criteria: Radiation heat flux and wave drag

The numerical solution of the two-criteria variational problem of the body contour with minimum radiation heat flux and wave drag is obtained in the class of axisymmetric and plane slender bodies in hypersonic flow. Solutions obtained using the Pareto, ideal point and minimax methods are compared. It is shown that in the class of axisymmetric slender bodies the optimum body gives a decrease in the radiation heat flux as compared with a cone of up to 15% for the Pareto method, up to 13% for the ideal point method, and up to 5% for the minimax method. A solution is also obtained in the subclass of power-law slender bodies and it is shown that the optimum power-law bodies are inferior, as compared with the optimum bodies from the general class of such bodies, in reducing both radiation heating and resistance.

Analytical solution in the problem of constructing axisymmetric noses with minimum wave drag

An analytical procedure for determining the axisymmetric nose shapes that ensure minimum wave drag is developed. The solution is constructed as an improving variation on the conical shape. The target function is built up in an approximate form on the basis of the assumption that the relationship between the geometric and gasdynamic parameters is local in nature. It is shown that the optimal bodies are truncated power-law bodies with an exponent equal to 2/3. The bodies thus obtained are compared with those constructed in accordance with an exact formulation of the problem.

Leading-edge bluntness effects on aerodynamic heating and drag of power law body in low-density hypersonic flow

A numerical study is reported on power law shaped leading edges situated in a rarefied hypersonic flow. The sensitivity of the heat flux and drag coefficient to shape variations of such leading edges is calculated by using a Direct Simulation Monte Carlo method. Calculations show that the stagnation point heating on power law leading edges with finite radius of curvature follows the same relation for classical blunt body in continuum flow; it scales inversely with the square root of the curvature radius at the nose. Furthermore, for those leading edges with zero or infinity radii of curvature, the heat transfer behavior is in surprising agreement with that for classical blunt body far from the nose of the leading edge.

Calculation of Shock Wave Structure over Power Law Bodies in Hypersonic Flow

A numerical study is conducted of power law shaped leading edges situated in a rarefied hypersonic flow. The motivation is interest in investigating the flowfield structure of power law shaped leading edges as possible candidates for blunting geometries of hypersonic leading edges. The primary aim is to examine in detail the effect of geometry of such leading edges on the shock wave structure. The sensitivity of the shock wave shape, standoff distance, and shock wave thickness to shape variations of the power law class of leading edges is calculated by using the direct simulation Monte Carlo method. Calculations show that the shape of the shock wave is in agreement with that predicted by the analytic hypersonic small disturbance theory for the flow conditions considered.

Composite models (shock and photoionization) for the interpretation of AGN spectra

AGN spectra are investigated by comparing model calculations with observations. Line intensity ratios show that the emitting gas is heated by shocks. Therefore, composite models which consistently account for both shocks, accompanying the radial motion of the gaseous clouds, and a photoionizing radiation flux are used. Liners are ambiguous objects between AGN and Starburst galaxies, therefore, power-law or black body radiations are considered in the models, depending on the characteristics of the line spectra. The SUMA code is used.

Minimum-drag axisymmetric bodies in the supersonic/hypersonic flow regimes

A study of minimum-drag body shapes was conducted over a Mach range from 3 to 12. Numerical results show that power-law bodies result in low-drag shapes, where the power n = 0.69 (l/d = 3) or n = 0.70 (l/d = 5) shapes have lower drag than theoretical minimum results (« = 0.75 or 0.66, depending on the particular form of the theory). To evaluate the results, a numerical analysis was made, including viscous effects and the effect of a gas model. None of these considerations altered the conclusions. The Hayes minimum-drag body was analyzed and had a higher drag than the optimum power-law body. d i n r T rref x, y, 0 Nomenclature = drag coefficient based on the maximum cross-sectional area = skin-friction coefficient = pressure coefficient = body diameter at the base = marching plane index = unified supersonic-hypersonic similarity parameter, = body length = freestream Mach number = power-law exponent = body radius = temperature at the body surface = freestream temperature z - physical coordinates = circumferential angle

Pressure gradient effect in natural convection boundary layers

The high Grashof number laminar natural convection flow around the lower stagnation point of a symmetric bowl-shaped heated body is analyzed. A region is identified where the direct effect on the flow of the component of the buoyancy force tangential to the body surface is comparable to the indirect effect of the component normal to the surface, which acts through the gradient of the nonuniform pressure that it induces in the boundary layer. Analysis of this region provides a description of the evolution of the flow from a pressure-gradient dominated regime to a buoyancy dominated regime. Numerical results are presented for the flows above and below heated power-law body shapes, and the upstream propagation of small perturbations to the stationary flow is discussed. An asymptotic analysis is carried out for the flow below nearly flat horizontal bodies, for which the change from pressure-gradient-driven to buoyancy-driven flow occurs very rapidly in a short region. The influence of body edges located in the region of interest is also discussed.

Analytical solutions for hypersonic flow past slender power-law bodies at small angle of attack

Hypersonic flows around axisymmetrical power-law slender bodies are calculated for high, but finite, Mach numbers and for low angles of attack. This is done by a small perturbation expansion of self-similar solutions using the equivalence principle. The solutions depend only on the exponent n of the power law defining the body and on the specific heat ratio γ of the gas, which is assumed to be perfect and inviscid

Rheological models and interpretation of postglacial uplift

Accepted 1989 February 1. Received 1989 January 28; in original form 1988 September 26 SUMMARY paper investigates four generalized Maxwebl bodies (GMBs) and the classical Maxwell body (CMB) with respect to their potential to explain postglacial uplift. The GMBs are the Lomnitz body, defined by a logarithmic creep law, the Q power-law body, defined by a power-law dependence of the quality factor on frequency, the Caputo body, defined by a special stress-strain relation with memory, and the Burgers body. The Burgers body is a 4-parameter model, the other GMBs have 3 and the CMB has only 2 parameters. Theoretical uplift curves are calculated for simple space-time models of the Canadian and Fennoscandian deglaciations, and hedgehog inversion yields the successful model parameters. The GMBs explain the data better than the CMB. The parameters of the Burgers body are only rather weakly constrained because of trade-offs among them. From the success of the GMBs we conclude that long time-scale transient creep is a possible rheological behaviour of mantle rocks. Viscosity extrapolation from the time constant of postglacial uplift, about 10 ka, to the time constant of mantle convection, roughly 100 Ma, is made with the effective viscosity and yields approximately ld2 Pas (Q power-law body), loz3 Pas (Caputo body) and loz5 Pas (Lomnitz body). Hence, postglacial uplift does not strongly constrain the long time-scale mantle viscosity. It provides only a lower bound, about loz1 Pas; this value follows from data interpretation with the CMB model in which steady-state creep is present alone. Rayleigh number estimates for whole-mantle convection are clearly supercritical for the CMB, the Q power-law body and the Caputo body, but only about critical or even subcritical for the Lomnitz body. Both postglacial uplift and mantle convection is satisfactorily explained by the CMB, the Q power-law body and the Caputo body.

The Effect of Power-Law Body Forces on a Thermally Driven Flow between Concentric Rotating Spheres

A numerical study is conducted to determine the effect of power-law body forces on a thermally driven axisymmetric flow field confined between concentric co-rotating spheres. This study is motivated by Spacelab geophysical fluid-flow experiments, which use an electrostatic force on a dielectric fluid to simulate gravity; this force exhibits a (1/r)5 distribution. Meridional velocity is found to increase when the electrostatic body form is imposed, relative to when the body force is uniform. Correlation among flow fields with uniform, inverse-square, and inverse-quintic force holds is obtained using a modified Grashof number.

Features of flow in the region of the stagnation point in supersonic flow past power-law bodies

Bodies of power-law shape with a generator of the form y=xα, 0.5<α<1, have at the apex a vertical tangent (like blunt bodies) and infinite curvature (like sharp ones). Supersonic flow past axisymmetric bodies with power-law longitudinal profile has been studied by Radvogin in [1, 2], where on the basis of the analysis of a large quantity of numerical calculations certain empirical laws of similarity were formulated. It follows from these relations that the position and form of the subsonic sector of the shock wave are not determined by the singularity in the body's profile, but by the position on the body of the sonic point and its “bluntness” in relation to the cone of the “critical” opening angle. In O. V. Titov's work the results obtained in [1, 2] are confirmed analytically, but here it is assumed that the curvature of the shock wave and the second derivative of the curvature with respect to the longitudinal curvilinear coordinate are finite at the apex. This assumption imposes a limitation on the contour of the body; if it is satisfied, the curvature of the profile past which the flow takes place is also finite. Therefore it is natural to consider the case when this assumption is not made. In this paper we study the flow past bodies of power-law shape with shock waves of infinite curvature at the apex and finite curvature but an infinite second derivative of the curvature with respect to the longitudinal coordinate.

A "Similarity" Solution for Laminar Swirling Core Flows

Spurk, J. H. and Gerber, N., Dust Collection Efficiency for Power Law Bodies in Hypersonic Flight, AIAA Journal, Vol. 10, June 1972, pp. 755-761. Peddieson, J. and Lyu, C.-H., Supersonic Two-Phase Flow Past a Cone, Proceedings of the 10th Southeastern Seminar on Thermal Sciences, edited by Watts and Sogin, Tulane Univ., New Orleans, La., 1974, pp. 467-482. Barren, R. M. and Wiley, J. T., Remarks on Dusty Hypersonic Wedge Flow, AIAA Journal, Vol. 17, March 1979, pp. 317-318. Barren, R. M. and Wiley, J. T., Newtonian Flow Theory for Slender Bodies in a Dusty Gas, Journal of Fluid Mechanics, Vol. 108, July 1981, pp. 147-157.

Unsteady Newton-Busemann flow theory. II - Bodies of revolution

Newtonian flow theory for unsteady flow past oscillating bodies of revolution at very high Mach numbers is completed by adding a centrifugal force correction to the impact pressures. Exact formulas for the unsteady pressure and the stability derivatives are obtained in closed form and are applicable to bodies of revolution that have arbitrary shapes, arbitrary thicknesses, and either sharp or blunt noses. The centrifugal force correction arising from the curved trajectories followed by the fluid particles in unsteady flow cannot be neglected even for the case of a circular cone. With this correction, the present theory is in excellent agreement with experimental results for sharp cones and for cones with small nose bluntness; gives poor agreement with the results of experiments in air for bodies with moderate or large nose bluntness. The pitching motions of slender power-law bodies of revulution are shown to be always dynamically stable according to Newton-Busemann theory.

Hypersonic boundary-layer flow with mass transfer on power-law bodies

Hypersonic boundary-layer flow with mass transfer on power-law bodies