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NASA NTRS2003

Magnetohydrodynamic Jump Conditions for Oblique Relativistic Shocks with Gyrotropic Pressure

Shock jump conditions, i.e., the specification of the downstream parameters of the gas in terms of the upstream parameters, are obtained for steady-state, plane shocks with oblique magnetic fields and arbitrary flow speeds. This is done by combining the continuity of particle number flux and the electromagnetic boundary conditions at the shock with the magnetohydrodynamic conservation laws derived from the stress-energy tensor. For ultrarelativistic and nonrelativistic shocks, the jump conditions may be solved analytically. For mildly relativistic shocks, analytic solutions are obtained for isotropic pressure using an approximation for the adiabatic index that is valid in high sonic Mach number cases. Examples assuming isotropic pressure illustrate how the shock compression ratio depends on the shock speed and obliquity. In the more general case of gyrotropic pressure, the jump conditions cannot be solved analytically with- out additional assumptions, and the effects of gyrotropic pressure are investigated by parameterizing the distribution of pressure parallel and perpendicular to the magnetic field. Our numerical solutions reveal that relatively small departures from isotropy (e.g., approximately 20%) produce significant changes in the shock compression ratio, r , at all shock Lorentz factors, including ultrarelativistic ones, where an analytic solution with gyrotropic pressure is obtained. In particular, either dynamically important fields or significant pressure anisotropies can incur marked departures from the canonical gas dynamic value of r = 3 for a shocked ultrarelativistic flow and this may impact models of particle acceleration in gamma-ray bursts and other environments where relativistic shocks are inferred. The jump conditions presented apply directly to test-particle acceleration, and will facilitate future self-consistent numerical modeling of particle acceleration at oblique, relativistic shocks; such models include the modification of the fluid velocity profile due to the contribution of energetic particles to the momentum and energy fluxes.

Double, Glen P.↗
NASA NTRS1998

The Application of Nonstandard Analysis to the Study of Inviscid Shock Wave Jump Conditions

The use of conservation laws in nonconservative form for deriving shock jump conditions by Schwartz distribution theory leads to ambiguous products of generalized functions. Nonstandard analysis is used to define a class of Heaviside functions where the jump from zero to one occurs on an infinitesimal interval. These Heaviside functions differ by their microstructure near x = 0, i.e., by the nature of the rise within the infinitesimal interval it is shown that the conservation laws in nonconservative form can relate the different Heaviside functions used to define jumps in different flow parameters. There are no mathematical or logical ambiguities in the derivation of the jump conditions. An important result is that the microstructure of the Heaviside function of the jump in entropy has a positive peak greater than one within the infinitesimal interval where the jump occurs. This phenomena is known from more sophisticated studies of the structure of shock waves using viscous fluid assumption. However, the present analysis is simpler and more direct.

Farassat, F.↗
NASA NTRS2007

Nonstandard Analysis and Shock Wave Jump Conditions in a One-Dimensional Compressible Gas

Nonstandard analysis is a relatively new area of mathematics in which infinitesimal numbers can be defined and manipulated rigorously like real numbers. This report presents a fairly comprehensive tutorial on nonstandard analysis for physicists and engineers with many examples applicable to generalized functions. To demonstrate the power of the subject, the problem of shock wave jump conditions is studied for a one-dimensional compressible gas. It is assumed that the shock thickness occurs on an infinitesimal interval and the jump functions in the thermodynamic and fluid dynamic parameters occur smoothly across this interval. To use conservations laws, smooth pre-distributions of the Dirac delta measure are applied whose supports are contained within the shock thickness. Furthermore, smooth pre-distributions of the Heaviside function are applied which vary from zero to one across the shock wave. It is shown that if the equations of motion are expressed in nonconservative form then the relationships between the jump functions for the flow parameters may be found unambiguously. The analysis yields the classical Rankine-Hugoniot jump conditions for an inviscid shock wave. Moreover, non-monotonic entropy jump conditions are obtained for both inviscid and viscous flows. The report shows that products of generalized functions may be defined consistently using nonstandard analysis; however, physically meaningful products of generalized functions must be determined from the physics of the problem and not the mathematical form of the governing equations.

Baty, Roy S.↗
NASA NTRS1991

A determination of relativistic shock jump conditions using Monte Carlo techniques

Monte Carlo techniques are used, assuming isotropic elastic scattering of all particles, to calculate jump conditions in parallel relativistic collisionless shocks in the absence of Fermi acceleration. The shock velocity and compression ratios are shown for arbitrary flow velocities and for any upstream temperature. Both single-component electron-positron plasma and two-component proton-electron plasmas are considered. It is shown that protons and electrons must share energy, directly or through the mediation of plasma waves, in order to satisfy the basic conservation conditions, and the electron and proton temperatures are determined for a particular microscopic, kinetic-theory model, namely, that protons always scatter elastically. The results are directly applicable to shocks in which waves of scattering superthermal particles are absent.

Ellison, Donald C.↗
NASA NTRS2008

Nonstandard Analysis and Jump Conditions for Converging Shock Waves

Nonstandard analysis is an area of modern mathematics which studies abstract number systems containing both infinitesimal and infinite numbers. This article applies nonstandard analysis to derive jump conditions for one-dimensional, converging shock waves in a compressible, inviscid, perfect gas. It is assumed that the shock thickness occurs on an infinitesimal interval and the jump functions in the thermodynamic and fluid dynamic parameters occur smoothly across this interval. Predistributions of the Heaviside function and the Dirac delta measure are introduced to model the flow parameters across a shock wave. The equations of motion expressed in nonconservative form are then applied to derive unambiguous relationships between the jump functions for the flow parameters.

Baty, Roy S.↗
NASA NTRS1977

Application of the wavenumber jump condition to the normal and oblique interaction of a plane acoustic wave and a plane shock

The kinematics of normal and oblique interactions between a plane acoustic wave and a plane shock wave are investigated separately using an approach whereby the shock is considered as a sharp discontinuity surface separating two half-spaces, so that the dispersion relation on either side of the shock and the wavenumber jump condition across a discontinuity surface completely specify the kinematics of the problem in the whole space independently of the acoustic-field dynamics. The normal interaction is analyzed for a stationary shock, and the spectral change of the incident wave is investigated. The normal interaction is then examined for the case of a shock wave traveling into an ambient region where an acoustic disturbance is propagating in the opposite direction. Detailed attention is given to the consequences of the existence of a critical shock speed above which the frequency of the transmitted wave becomes negative. Finally, the oblique interaction with a fixed shock is considered, and the existence and nature of the transmitted wave is investigated, particularly as a function of the angle of incidence.

Kleinstein, G. G.↗
NASA NTRS1995

Anisotropic shock jump conditions: Theory and observations

The MHD Rankine-Hugoniot (RH) relations for shock waves in a collisionless plasma with bi-Maxwellian distribution functions are considered. While by introducing the pressure anisotropy parameter xi in the RH relations, the number of unknowns -- B, V, n, p and xi (a total of 9) -- becomes one more than the total number of the conservation equations, it is possible to use the observed quantities on both sides of the shock to study the anisotropy changes across the shock. A simple relation for the anisotropy change across the shock is derived as a function of the ratio of magnetic fields m(= B'/B), the shock normal angle theta(sub Bn) and the plasma beta and beta' (primes are downstream values). Since m and theta(sub Bn) can be determined accurately in observation, the reliability of the anisotropy change deduced is mostly dependent on the accuracy of the measurements beta and beta'. We have applied the results to six low-beta quasi-perpendicular (Q perpendicular) laminar bow shock crossings with temperature anisotropy measured in the magnetosheath. In the six test cases, it is found that the predicted pressure anisotropies agree well with those observed in the magnetosheath.

Chao, J. K.↗
NASA NTRS1998

The Local ISM and its Interaction with the Winds of Nearby Late-type Stars

We present new Goddard High-Resolution Spectrograph (GHRS) observations of the Ly-alpha and Mg II absorption lines seen toward the nearby stars 61 Cyg A and 40 Eri A. We use these data to measure interstellar properties along these lines of sight and to search for evidence of circumstellar hydrogen walls, which are produced by collisions between the stellar winds and the Local InterStellar Medium (LISM). We were able to model the Ly-alpha lines of both stars without hydrogen-wall absorption components, but for 61 Cyg A the fit required a stellar Ly-alpha, line profile with an improbably deep self-reversal, and for 40 Eri A the fit required a very low deuterium-to-hydrogen ratio that is inconsistent with previous GHRS measurements. Since these problems could be rectified simply by including stellar hydrogen-wall components with reasonable attributes, our preferred fits to the data include these components. We have explored several ways in which the hydrogen-wall properties measured here and in previous work can be used to study stellar winds and the LISM. We argue that the existence of a hydrogen wall around 40 Eri A and a low H I column density along that line of sight imply that either the interstellar density must decrease toward 40 Eri A or the hydrogen ionization fraction (chi) must increase. We find that hydrogen-wall temperatures are larger for stars with faster velocities through the LISM. The observed temperature-velocity relation is consistent with the predictions of hydromagnetic shock jump conditions. More precise comparison of the data and the jump conditions suggests crude upper limits for both chi and the ratio of magnetic to thermal pressure in the LISM (alpha): chi less than 0.6 and alpha less than 2. The latter upper limit corresponds to a limit on the LISM magnetic field of B less than 5 micro G. These results imply that the plasma Mach number of the interstellar wind flowing into the heliosphere is M(sub A) greater than 1.3, which indicates that the collision is supersonic and that there should therefore be a bow shock outside the heliopause in the upwind direction. Finally, we estimate stellar wind pressures (P sub wind) from the measured hydrogen-wall column densities. These estimates represent the first empirical measurements of wind properties for late-type main-sequence stars. The wind pressures appear to be correlated with stellar X-ray surface fluxes, F(x), in a manner consistent with the relation P(wind) varies as F(x)(exp -1/2), a relation that is also consistent with the variations of P(sub wind) and F(sub x) observed during the solar activity cycle. If this relation can in fact be generalized to solar-like stars, as is suggested by our data, then it is possible to estimate stellar wind properties simply by measuring stellar X-rays. One implication of this is that stellar wind pressures and mass-loss rates are then predicted to increase with time, since F(sub x) is known to decrease with stellar age.

Wood, Brian E.↗
NASA NTRS1987

Shock structure measured in a transonic fan using laser anemometry

Shock structure measurements acquired in a low aspect ratio transonic fan rotor are presented and analyzed. The rotor aspect ratio is 1.56 and the design tip relative Mach number is 1.38. The rotor flowfield was surveyed at near maximum efficiency and near stall operating conditions. Intra-blade velocity measurements acquired with a laser fringe anemometer on blade-to-blade planes in the supersonic region from 10 to 60 percent span are presented. The three-dimensional shock surface determined from the velocity measurments is used to determine the shock surface normal Mach number in order to properly calculate the ideal shock jump conditions. The ideal jump conditions are calculated based upon the Mach numbers measured on a surface of revolution and based upon the normal Mach number to indicate the importance of accounting for shock three dimensionality in turbomachinery design. Comparison of the shock locations with those predicted by a 3-D Euler code showed very good agreement and indicated the usefulness of integrating computational and experimental work to enhance the understanding of the flow physics occurring in transonic turbomachinery passages.

Wood, Jerry R.↗
NASA NTRS1994

An introduction to generalized functions with some applications in aerodynamics and aeroacoustics

In this paper, we start with the definition of generalized functions as continuous linear functionals on the space of infinitely differentiable functions with compact support. The concept of generalization differentiation is introduced next. This is the most important concept in generalized function theory and the applications we present utilize mainly this concept. First, some of the results of classical analysis, such as Leibniz rule of differentiation under the integral sign and the divergence theorem, are derived using the generalized function theory. It is shown that the divergence theorem remains valid for discontinuous vector fields provided that the derivatives are all viewed as generalized derivatives. This implies that all conservation laws of fluid mechanics are valid as they stand for discontinuous fields with all derivatives treated as generalized deriatives. Once these derivatives are written as ordinary derivatives and jumps in the field parameters across discontinuities, the jump conditions can be easily found. For example, the unsteady shock jump conditions can be derived from mass and momentum conservation laws. By using a generalized function theory, this derivative becomes trivial. Other applications of the generalized function theory in aerodynamics discussed in this paper are derivation of general transport theorems for deriving governing equations of fluid mechanics, the interpretation of finite part of divergent integrals, derivation of Oswatiitsch integral equation of transonic flow, and analysis of velocity field discontinuities as sources of vorticity. Applications in aeroacoustics presented here include the derivation of the Kirchoff formula for moving surfaces,the noise from moving surfaces, and shock noise source strength based on the Ffowcs Williams-Hawkings equation.

Farassat, F.↗
NASA NTRS1973

Analysis of embedded shock waves calculated by relaxation methods.

The requirements for uniqueness of the calculated jump conditions across embedded shock waves are investigated for type-dependent difference systems used in transonic flow studies. A mathematical analysis shows that sufficient conditions are (1) the equations should be differenced in conservative form and (2) a special difference operator should be used when switching from a hyperbolic to an elliptic operator. The latter results in a consistency condition on the integral equations, rather than the differential, at these points. Calculated jump conditions for several embedded and detached shock waves are analyzed in the physical and hodograph planes. Comparisons are made with previous results, a time-dependent calculation, and data.

Murman, E. M.↗
NASA NTRS1985

Hypersonic merged-layer flow on a sphere

With the objective to achieve the desired reduction in velocity by aerodynamic forces, an aeroassisted orbital transfer vehicle (AOTV) is expected to fly in the higher region of the atmosphere for a sustained period of time. This aeroassist maneuver will occur in the transitional regime, if the vehicle is designed for a low ballistic coefficient. In this case, a merged-layer (ML) is formed around the space vehicle. The flow characteristics of the vehicle can be studied by making use of the full Navier-Stokes (NS) equations, taking into account slip and temperature jump conditions. The present investigation has the objective to obtain a two-term series solution of the full Navier-Stokes equations with surface slip and temperature jump conditions for the ML flow on a sphere. Attention is given to the mathematical formulation of the problem, the numerical method of integration, and the results.

Jain, A. C.↗
NASA NTRS2001

Numerical Modeling of Liquid-Vapor Phase Change

We implemented a two- and three-dimensional finite difference/front tracking technique to solve liquid-vapor phase change problems. The mathematical and the numerical features of the method were explained in great detail in our previous reports, Briefly, we used a single formula representation which incorporated jump conditions into the governing equations. The interfacial terms were distributed as singular terms using delta functions so that the governing equations would be the same as conventional conservation equations away from the interface and in the vicinity of the interface they would provide correct jump conditions. We used a fixed staggered grid to discretize these equations and an unstructured grid to explicitly track the front. While in two dimensions the front was simply a connection of small line segments, in three dimensions it was represented by a connection of small triangular elements. The equations were written in conservative forms and during the course of computations we used regriding to control the size of the elements of the unstructured grid. Moreover, we implemented a coalescence in two dimensions which allowed the merging of different fronts or two segments of the same front when they were sufficiently close. We used our code to study thermocapillary migration of bubbles, burst of bubbles at a free surface, buoyancy-driven interactions of bubbles, evaporation of drops, rapid evaporation of an interface, planar solidification of an undercooled melt, dendritic solidification, and a host of other problems cited in the reference.

Esmaeeli, Asghar↗