Shock jump conditions for an anisotropic plasma.
Finite discontinuity jump conditions for plasma in strong magnetic field determined by approximation and Maxwell equation, noting possible application to satellite data
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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.↗
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.↗
A study of the effects of Reynolds number and Mach number on constant pressure coefficient jump for shock-induced trailing-edge separation
The influence of Mach and Reynolds numbers as well as airfoil and planform geometry on the phenomenon of constant shock jump pressure coefficient for conditions of shock induced trailing edge separation (SITES) was studied. It was demonstrated that the phenomenon does exist for a wide variety of two and three dimensional flow cases and that the influence of free stream Mach number was not significant. The influence of Reynolds number was found to be important but was not strong. Airfoil and planform geometric characteristics were found to be very important where the pressure coefficient jump was shown to vary with the sum of: (1) airfoil curvature at the upper surface crest, and (2) camber surface slope at the trailing edge. It was also determined that the onset of SITES could be defined as a function of airfoil geometric parameters and Mach number normal to the leading edge. This onset prediction was shown to predict the angle of onset to within + or - 1 deg accuracy or better for about 90% of the cases studied.
Cunningham, Atlee M., Jr.↗
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.↗
Entropy jump across an inviscid shock wave
The shock jump conditions for the Euler equations in their primitive form are derived by using generalized functions. The shock profiles for specific volume, speed, and pressure are shown to be the same, however density has a different shock profile. Careful study of the equations that govern the entropy shows that the inviscid entropy profile has a local maximum within the shock layer. We demonstrate that because of this phenomenon, the entropy, propagation equation cannot be used as a conservation law.
Salas, Manuel D.↗
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.↗
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.↗
MHD intermediate shocks in coronal mass ejections
A simplified model of coronal mass ejections is considered in which at least a portion of the interaction with the background corona involves a shock wave, and the allowable shock solutions and their compressive signatures are examined. The MHD shock-jump equations have a maximum of three possible types of solutions with an entropy rise for fixed values of the physical variables (slow, intermediate, and fast shocks). However, one of the three solution classes (the intermediate shock) is widely believed to not occur in nature and is regarded as nonevolutionary or extraneous. Without the intermediate shock, there is no multiplicity of solutions in that only one shock (or none) can occur for given physical values. All three potential shock types are considered, and it is shown solely on the basis of the shock-jump equations, that intermediate shocks must exist along some segment of the shock front for certain parametric regimes and for conditions that probably occur in some coronal mass ejections.
Steinolfson, R. S.↗
Quantum-cascade-laser-absorption-spectroscopy diagnostic for temperature, pressure, and NO X 2 Π 1/2 at 500 kHz in shock-heated air at elevated pressures
The design, validation, and application of a quantum-cascade-laser-absorption-spectroscopy diagnostic for measuring gas temperature, pressure, and nitric oxide (NO) in high-temperature air are presented. A distributed-feedback quantum-cascade laser (QCL) centered near 1976 c m − <#comment/> 1 was used to scan across two transitions of NO in its ground electronic state ( X 2 Π <#comment/> 1 / 2 ). A measurement rate of 500 kHz was achieved using a single QCL by: (1) performing current modulation through a bias-tee, and (2) targeting closely spaced transitions with a large difference in lower-state energy. The diagnostic was validated in a mixture of 95% argon and 5% NO, which was shock-heated to ≈ <#comment/> 2000 to 3700 K. The average mean percent differences between laser-absorption-spectroscopy (LAS) measurements and predictions from shock-jump relations for temperature, pressure, and NO mole fraction were 3.1%, 4.1%, and 6.5%, respectively. The diagnostic was then applied to characterize shock-heated air at high temperatures (up to ≈ <#comment/> 5500 K ) and high pressures (up to 12 atm) behind either incident or reflected shocks. The LAS measurements were compared to theoretical predictions from shock-jump relations, pressure sensors mounted in the wall of the shock tube, and equilibrium values of the NO mole fraction. The average mean percent differences between LAS measurements and their aforementioned reference values were 3.2%, 10.8%, and 10.4% for temperature, pressure, and NO mole fraction, respectively. Last, a comparison between a measured NO mole fraction time history and a time-stepped homogeneous reactor simulation performed using two different chemical kinetics mechanisms is presented.
Gilvey, Jonathan J. (ORCID:0000000317664383)↗
Unsteady aerodynamic flow field analysis of the space shuttle configuration. Part 2: Launch vehicle aeroelastic analysis
An exploratory analysis has been made of the aeroelastic stability of the Space Shuttle Launch Configuration, with the objective of defining critical flow phenomena with adverse aeroelastic effects and developing simple analytic means of describing the time-dependent flow-interference effects so that they can be incorporated into a computer program to predict the aeroelastic stability of all free-free modes of the shuttle launch configuration. Three critical flow phenomana have been identified: (1) discontinuous jump of orbiter wing shock, (2) inlet flow between orbiter and booster, and (3) H.O. tank base flow. All involve highly nonlinear and often discontinuous aerodynamics which cause limit cycle oscillations of certain critical modes. Given the appropriate static data, the dynamic effects of the wing shock jump and the HO tank bulbous base effect can be analyzed using the developed quasi-steady techniques. However, further analytic and experimental efforts are required before the dynamic effects of the inlet flow phenomenon can be predicted for the shuttle launch configuration.
Reding, J. P.↗
Acceleration of positrons in supernova shocks
During this project we investigated the acceleration of leptons (electrons and positrons) in collisionless shock waves. In particular, we were interested in how leptons are accelerated in the blast waves existing in the remnants of supernova explosions. Supernova remnants (SNRs) have long been considered as the most likely source of galactic cosmic rays but no definite connection between SNRs and the cosmic rays seen at earth can be made. Only by understanding lepton acceleration in shocks can the rich SNR data base be properly used to understand cosmic ray origins. Our project was directed at the neglected aspects of lepton acceleration. We showed that the efficiency of lepton acceleration depended critically on the lepton injection energy. We showed that, even when infection effects are not important, that proton and lepton distribution functions produced by shocks are quite different in the critical energy range for producing the observed synchrotron emission. We also showed that transrelativistic effects produced proton spectra that were not in agreement with standard results from radio observations, but that the lepton spectra were, in fact, consistent with observations. We performed simulations of relativistic shocks (shocks where the flow speed is a sizable fraction of the speed of light) and discovered some interesting effects. We first demonstrated the power of the Monte Carlo technique by determining the shock jump conditions in relativistic shocks. We then proceeded to determine how relativistic shocks accelerate particles. We found that nonlinear relativistic shocks treat protons and leptons even more differently than nonrelativistic shocks. The transrelativistic effects on the shock structure from the heavy ion component reduces the lepton acceleration to a tiny fraction of the ion acceleration. This effect is dramatic even if high energy leptons (many times thermal energy) are injected, and was totally unexpected. Our results have important consequences for astrophysical environments expected to harbor relativistic flows such as extra-galactic radio sources and accretion onto compact objects.
Ellison, Donald C.↗
Numerical computation of three-dimensional blunt body flow fields with an impinging shock
A time-marching finite-difference method was used to solve the compressible Navier-Stokes equations for the three-dimensional wing-leading-edge shock impingement problem. The bow shock was treated as a discontinuity across which the exact shock jump conditions were applied. All interior shock layer detail such as shear layers, shock waves, jets, and the wall boundary layer were automatically captured in the solution. The impinging shock was introduced by discontinuously changing the freestream conditions across the intersection line at the bow shock. A special storage-saving procedure for sweeping through the finite-difference mesh was developed which reduces the required amount of computer storage by at least a factor of two without sacrificing the execution time. Numerical results are presented for infinite cylinder blunt body cases as well as the three-dimensional shock impingement case. The numerical results are compared with existing experimental and theoretical results.
Holst, T. L.↗
Electron heating and the potential jump across fast mode shocks
Two different methods were applied to determine the cross-shock potential jump in the de Hoffmann-Teller reference frame, using a data set that represented 66 crossings of the terrestrial bow shock and 14 interplanetary shocks observed by various ISEE spacecraft, and one crossing each of the Jovian bow shock and the Uranian bow shock made by the Voyager spacecraft. Results for estimates of the electrostatic potential based on an estimate of the jump in electron enthalpy correlated well with estimates based on Liouville's theorem, although the Liouville-determined values were systematically the higher of the two, suggesting that significant irreversible processes contribute to the shape of the downstream distribution. The potential jump corresponds to approximately 12-15 percent of the incident ion ram kinetic energy, and was found not to be controlled by the Mach number, plasma beta, shock geometry, or electron to ion temperature ratios.
Schwartz, Steven J.↗
The special relativistic shock tube
The shock-tube problem has served as a popular test for numerical hydrodynamics codes. The development of relativistic hydrodynamics codes has created a need for a similar test problem in relativistic hydrodynamics. The analytical solution to the special relativistic shock-tube problem is presented here. The relativistic shock-jump conditions and rarefaction solution which make up the shock tube are derived. The Newtonian limit of the calculations is given throughout.
Thompson, Kevin W.↗
Two-temperature radiative shocks with electron thermal conduction
The influence of electron thermal conduction on radiative shock structure is studied for both one- and two-temperature plasmas. The dimensionless ratio of the conductive length to the cooling length determines whether or not conduction is important, and shock jump conditions with conduction are established for a collisionless shock front. Approximate solutions are obtained, with the assumptions that the ionization state of the gas is constant and the cooling rate is a function of temperature alone. In the absence of magnetic fields, these solutions indicate that conduction noticeably influences normal-abundance interstellar shocks with velocities 50-100 km/s and dramatically affects metal-dominated shocks over a wide range of shock velocities.
Borkowski, Kazimierz J.↗
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.↗
Gas-grain energy transfer in solar nebula shock waves: Implications for the origin of chondrules
Meteoritic chondrules provide evidence for the occurrence of rapid transient heating events in the protoplanetary nebula. Astronomical evidence suggests that gas dynamic shock waves are likely to be excited in protostellar accretion disks by processes such as protosolar mass ejections, nonaxisymmetric structures in an evolving disk, and impact on the nebula surface of infalling 'clumps' of circumstellar gas. Previous detailed calculations of gas-grain energy and momentum transfer have supported the possibility that such shock waves could have melted pre-existing chondrule-sized grains. The main requirement for grains to reach melting temperatures in shock waves with plausibly low Mach numbers is that grains existed in dust-rich zones (optical depth greater than 1) where radiative cooling of a given grain can be nearly balanced by radiation from surrounding grains. Localized dust-rich zones also provide a means of explaining the apparent small spatial scale of heating events. For example, the scale size of at least some optically thick dust-rich zones must have been relatively small (less than 10 kilometers) to be consistent with petrologic evidence for accretion of hot material onto cold chondrules. The implied number density of mm-sized grains for these zones would be greater than 30 m(exp -3). In this paper, we make several improvements of our earlier calculations to include radiation self-consistently in the shock jump conditions, and we include heating of grains due to radiation from the shocked gas. In addition, we estimate the importance of momentum feedback of dust concentrations onto the shocked gas which would tend to reduce the efficiency of gas dynamic heating of grains in the center of the dust cloud.
Hood, L. L.↗