Average Mach Number Research Articles

Supersonic motions of galaxies in clusters

We study motions of galaxies in galaxy clusters formed in the concordance Λ cold dark matter cosmology. We use high-resolution cosmological simulations that follow the dynamics of dark matter and gas and include various physical processes critical for galaxy formation: gas cooling, heating and star formation. Analysing the motions of galaxies and the properties of intracluster gas in a sample of eight simulated clusters at z= 0, we study the velocity dispersion profiles of the dark matter, gas and galaxies. We measure the mean velocity of galaxy motions and gas sound speed as a function of radius and calculate the average Mach number of galaxy motions. The simulations show that galaxies, on average, move supersonically with the average Mach number of ≈1.4, approximately independent of the cluster-centric radius. The supersonic motions of galaxies may potentially provide an important source of heating for the intracluster gas by driving weak shocks and via dynamical friction, although these heating processes appear to be inefficient in our simulations. We also find that galaxies move slightly faster than the dark matter particles. The magnitude of the velocity bias, bv≈ 1.1, is, however, smaller than the bias estimated for subhaloes in dissipationless simulations. Interestingly, we find velocity bias in the tangential component of the velocity dispersion, but not in the radial component. Finally, we find significant random bulk motions of gas. The typical gas velocities are of order ≈20–30 per cent of the gas sound speed. These random motions provide about 10 per cent of the total pressure support in our simulated clusters. The non-thermal pressure support, if neglected, will bias measurements of the total mass in the hydrostatic analyses of the X-ray cluster observations.

Forebody compressibility research of hypersonic vehicle

Three kinds of forebody model of hypersonic vehicles were studied with numerical simulation method. It shows that the two-order compressive ramp model is the best selection among the three for its good evaluative parameters value at the cowl of the inlet. This model can provide higher value of flux coefficient and total pressure recovery coefficient and lower average Mach number compared with those of the other two models. Simultaneously different compressive angles may have different effects. The configuration which the first-order of compressive angle is 4° and the second 5° is the optimum combination. Furthermore factors such as attack angle were concerned. Better result may be obtained with a range of attack angles. Based on the work above the integrated design for forebody/inlet of a hypersonic vehicle was performed. The numerical result shows that this integrated model provides good flow field quality for inlet and engine work.

Influence of Mach Number on Tornado Corner Flow Dynamics

Abstract Since maximum Mach numbers in an F5 tornado are expected to, at least, reach ∼0.4, a study was undertaken to determine the major potential compressibility effects in such flows. Results from compressible large-eddy simulations of tornado corner flow dynamics are summarized. Comparison with previous incompressible simulations indicates that Mach number effects tend to be modest and may be estimated by an isentropic approximation. As the average maximum Mach number M within the tornado increases above one-half, the largest changes occur for “low-swirl” corner flows, those exhibiting a central vertical jet off the surface capped by a vortex breakdown, with the vortex breakdown found to occur at significantly greater heights as M increases. It also appears that the highest Mach numbers are most likely to occur during rapid transients in near-surface intensification that can sometimes occur during a tornado's evolution.

Three‐dimensional Simulations of Jets from Keplerian Disks: Self‐regulatory Stability

We present the extension of previous two-dimensional simulations of the time-dependent evolution of nonrelativistic outflows from the surface of Keplerian accretion disks to three dimensions. As in the previous work, we investigate the outflow that arises from a magnetized accretion disk that is initially in hydrostatic balance with its surrounding cold corona. The accretion disk itself is taken to provide a set of fixed boundary conditions for the problem. We find that the mechanism of jet acceleration is identical to what was established from the previous two-dimensional simulations. The three-dimensional results are consistent with the theory of steady, axisymmetric, centrifugally driven disk winds up to the Alfven surface of the outflow. Beyond the Alfven surface, however, the jet in three dimensions becomes unstable to nonaxisymmetric, Kelvin-Helmholtz instabilities. The most important result of our work is that while the jet is unstable at super-Alfvenic speeds, it survives the onset of unstable modes that appear in this physical regime. We show that jets maintain their long-term stability through a self-limiting process wherein the average Alfvenic Mach number within the jet is maintained to the order of unity. This is accomplished in at least two ways. First, the poloidal magnetic field is concentrated along the central axis of the jet forming a backbone in which the Alfven speed is sufficiently high to reduce the average jet Alfvenic Mach number to unity. Second, the onset of higher order Kelvin-Helmholtz flute modes (m ≥ 2) reduces the efficiency with which the jet material is accelerated and transfers kinetic energy of the outflow into the stretched, poloidal field lines of the distorted jet. This too has the effect of increasing the Alfven speed and thereby reducing the Alfvenic Mach number. The jet is able to survive the onset of the more destructive m = 1 mode in this way. Our simulations also show that jets can acquire corkscrew or wobbling types of geometries in this relatively stable end state depending on the nature of the perturbations on them. Finally, we suggest that jets go into alternating periods of low and high activity since the disappearance of unstable modes in the sub-Alfvenic regime enables another cycle of acceleration to super-Alfvenic speeds.

Three-Dimensional Simulations of Jets from Keplerian Disks: Stability Issues

Three-dimensional simulations of the time-dependent evolution of non-relativistic outflows from the surface of Keplerian accretion disks are presented. We investigate the outflow that arises from a magnetized accretion disk that is initially in hydrostatic balance with its surrounding cold corona. Our simulations show that jets maintain their long-term stability through a self-limiting process wherein the average Alfvenic Mach number within the jet is maintained to order unity. This is accomplished in at least two ways. First, poloidal magnetic field is concentrated along the central axis of the jet forming a ‘backbone’ in which the Alfven speed is sufficiently high to reduce the average jet Alfvenic Mach number to unity. Second, the onset of higher order Kelvin-Helmholtz ‘flute’ modes (m ≥ 2) reduce the efficiency with which the jet material is accelerated, and transfer kinetic energy of the outflow into the stretched, poloidal field lines of the distorted jet. This too has the effect of increasing the Alfven speed and thereby reducing the Alfvenic Mach number. The jet is able to survive the onset of the more destructive m = 1 mode in this way.

Hydrodynamical simulations of the decay of high-speed molecular turbulence — I. Dense molecular regions

We present the results from three-dimensional hydrodynamical simulations of decaying high-speed turbulence in dense molecular clouds. We compare our results, which include a detailed cooling function, molecular hydrogen chemistry and a limited C and O chemistry, with those previously obtained for decaying isothermal turbulence. After an initial phase of shock formation, power-law decay regimes are uncovered, as in the isothermal case. We find that the turbulence decays faster than in the isothermal case because the average Mach number remains higher, owing to the radiative cooling. The total thermal energy, initially raised by the introduction of turbulence, decays only a little more slowly than the kinetic energy. We discover that molecule reformation, as the fast turbulence decays, is several times faster than that predicted for a non-turbulent medium. This is caused by moderate speed shocks which sweep through a large fraction of the volume, compressing the gas and dust. Through reformation, the molecular density and molecular column appear as complex patterns of filaments, clumps and some diffuse structure. In contrast, the molecular fraction has a wider distribution of highly distorted clumps and copious diffuse structure, so that density and molecular density are almost identically distributed during the reformation phase. We conclude that molecules form in swept-up clumps but effectively mix throughout via subsequent expansions and compressions.

Sound Speed in a Random Flow and Turbulent Shifts of the Solar Eigenfrequencies

Perturbation theory is used to calculate the frequency shift of acoustic modes of a homogeneous turbulent fluid and the frequency shifts of solar modes due to turbulent convection. For sound waves in a random flow, the fractional frequency shift is +(11/30)M2 in the long-wavelength limit, and the shift at short wavelengths is -2/3M2, where M is the average Mach number of the flow. In our model of solar convection, the low-degree f-mode shift is positive and unobservably small, whereas the fractional frequency shift at high degrees (l 500) is -M2, where M is the Mach number of convection near the solar surface.

Longwave approximation model for gas shear flow in a channel of varying area

An approximate system of equations that describe unsteady flow of an inviscid non-heat-conducting gas in a narrow channel of varying area is derived. Generalized characteristics and hyperbolicity conditions are obtained for this system of equations. In connection with characteristics theory, the average Mach number and the flow criticality condition are introduced. Exact solutions that describe steady transonic channel flows are investigated.

Evolution of large eddies in compressible shear layers

The evolution of large turbulent eddies has been investigated in seven supersonic shear layers with average convective Mach numbers Mc¯ ranging from 0.22 to 0.86 and with large variation in density and velocity ratios. A two-laser, single-detector planar laser-induced fluorescence technique was used to visualize the flow and its evolution. Two-dimensional pattern matching yielded the convective velocity of the eddies. For Mc¯>0.3, fast and slow modes of eddy propagation were detected in supersonic–subsonic and supersonic–supersonic combinations, respectively. An empirical model for the convective velocity is proposed. Plan views reveal coexistence of two- and three-dimensional disturbances. Interaction among eddies appears significantly suppressed. The findings have direct impact on supersonic jet noise and are very relevant to supersonic combustion.

The propagation of weak shocks in non-uniform flows

A new theory of the propagation of weak shocks into non-uniform, two-dimensional flows is introduced. The theory is based on a description of shock propagation in terms of a manifold equation together with compatibility conditions for shock strength and its normal derivatives behind the shock. This approach was developed by Ravindran & Prasad (1993) for shocks of arbitrary strength propagating into a medium at rest and is extended here to non-uniform media and restricted to moderately weak shocks. The theory is tested against known analytical solutions for cylindrical and plane shocks, and against a full direct numerical simulation (DNS) of a shock propagating into a sinusoidal shear flow. The test against DNS shows that the present theory accurately predicts the evolution of a moderately weak shock front, including the formation of shock-shocks due to shock focusing. The theory is then applied to the focusing of an initially parabolic shock, and to the propagation of an initially straight shock into a variety of simple flows (sinusoidal shear, vortex array, point-vortex array) exhibiting some fundamental properties of turbulent flows. A number of relations are deduced for the variation of shock quantities with initial shock strength MS0 and the Mach number of the flow ahead of the shock MU (e.g. separation of shock-shocks and maximum shock strength at a focus). It is found that shock-shocks are likely to form in turbulent flows with Mt/M1N > 0.14–0.25, where Mt is the average Mach number of the turbulence and M1N is the Mach number of the shock in a flow at rest. The shock moves up to 1.5% faster in a two-dimensional vortex array than in uniform flow.

The influence of wall cooling on hypersonic boundary-layer separation and stability

The effect of wall cooling on hypersonic boundary-layer separation near a compression ramp is considered. Two cases are identified corresponding to the value of the average Mach number $\overline{M}$ across the upstream boundary layer approaching the compression ramp. The flow is referred to as supercritical for $\overline > 1$ and subcritical for $\overline{M} < 1$. The interaction is described by triple-deck theory, and numerical results are given for both cases for various ramp angles and levels of wall cooling. The effect of wall cooling on the absolute instability described recently by Cassel, Ruban & Walker (1995) for an uncooled wall is of particular interest; a stabilizing effect is observed for supercritical boundary layers, but a strong destabilizing influence occurs in the subcritical case. Wall cooling also influences the location and size of the separated region. For supercritical flow, progressive wall cooling reduces the size of the recirculating-flow region, the separation point moves downstream, and upstream influence is diminished. In contrast for the subcritical case downstream influence is reduced with increased cooling. In either situation, a sufficient level of wall cooling eliminates separation altogether for the ramp angles considered. The present numerical results closely confirm the strong wall cooling theory of Kerimbekov, Ruban & Walker (1994).

Preliminary assessment of tunnel wall interference in the NDA cryogenic wind tunnel

As the second stage of a preliminary cryogenic airfoil testing program, the test section interference of the National Defense Academy cryogenic tunnel was evaluated. A R4 airfoil model, which has the chord length of 12 cm and the aspect ratio of 0.5, was tested in the range of Mach number 0.5 to 0.75, and that of Reynolds number 7 x 106 to about 1.1 x 10 7. The experimental results were corrected with empirical wall interference correction methods, the Barnwell-Sewall method for sidewall boundary layers, and the Blackwell method for the top and bottom walls. This preliminary evaluation showed that the sidewall boundary layers dominate the tunnel wall interference of the present cryogenic tunnel, and there may be some possibility of utilizing the tunnel for performing two-dimensional airfoil tests if more precise wall interference parameters are obtained. Nomenclature b = tunnel width Cp = pressure coefficient Cp = equivalent pressure coefficient c = chord length of the airfoil cn = section normal force coefficient H = shape factor M - local Mach number Mc = corrected freestream Mach number ME = averaged experimental Mach number at the wall over the airfoil MT = averaged theoretical Mach number at the wall over the airfoil Mx = tunnel freestream Mach number M^, = equivalent Mach number a = angle of attack AMB = blockage Mach number correction AMy = sequential Mach number correction AMw = sidewall Mach number correction 8* = sidewall displacement thickness Subscripts ambient condition

The relationship between the magnetic field in the Martian magnetotail and upstream solar wind parameters

Magnetic field data measured by the MAGMA instrument in the Martian magnetotail lobes are compared with the ram pressure of the upstream solar wind observed by the TAUS instrument in the circular orbits of the Phobos 2 spacecraft. High correlation was found between the magnetic field intensity in the Martian magnetotail lobes and the solar wind ram pressure. From this relationship the average flaring angle of the Martian magnetotail was determined as ∼13°, and the average magnetosonic Mach number was estimated as ∼5. The observed relationship between the Martian magnetotail magnetic field intensity and the solar wind magnetic field reflects the correlation of the solar wind magnetic field to the ram pressure providing a value of ∼7 for the average Alfvenic Mach number. The flaring angle obtained for the Martian magnetotail was found to be an intermediate value between the flaring angle of the magnetotail of the Earth and that of Venus at comparable distances.

Supersonic flow in the second-throat ejector-diffuser system

A computational analysis of the two-dimensional supersonic inviscid flowfield in a second-throat ejector-diffuser (STED) system is presented. A second-order high-resolution scheme for solving the new Lagrangian Euler equations is employed to accurately resolve the complicated shock patterns and associated slip lines and their interactions. A parametric study covering a variety of Xst and Ost is implemented to investigate their effects on the flow structure in STED as well as its performance. Results suggest that the averaged Mach number along the entrance plane of the second throat is a suitable criterion for the justification of the performance of STED. With this criterion, an optimal design insuring the largest pressure recovery can be achieved.

Evolution of the Orszag–Tang vortex system in a compressible medium. II. Supersonic flow

The numerical investigation of Orszag–Tang vortex system in compressible magnetofluids continues, this time using initial conditions with embedded supersonic regions. The simulations have initial average Mach numbers M=1.0 and 1.5 and β=10/3 with Lundquist numbers S=50, 100, or 200. Depending on the particular set of parameters, the numerical grid contains 2562 or 5122 collocation points. The behavior of the system differs significantly from that found previously for the incompressible and subsonic analogs. Shocks form at the downstream boundaries of the embedded supersonic regions outside the central magnetic X point and produce strong local current sheets that dissipate appreciable magnetic energy. Reconnection at the central X point, which dominates the incompressible and subsonic systems, peaks later and has a smaller impact as M increases from 0.6 to 1.5. Reconnection becomes significant only after shocks reach the central region, compressing the weak current sheet there. Similarly, the correlation between the momentum and magnetic field begins significant growth later than in subsonic and incompressible flows. The shocks bound large compression regions, which dominate the wave-number spectra of autocorrelations in mass density, velocity, and magnetic field. The normalized spectral amplitude of the cross helicity is almost zero over the middle and upper portions of the wave-number domain, unlike the incompressible and subsonic flows. The thermal and magnetic pressures are anticorrelated over a wide wave-number range during the earlier portion of the calculations, consistent with the presence of quasistationary structures bounded by shocks.

Inverse cascades in two-dimensional compressible turbulence. I. Incompressible forcing at low Mach number

A fundamental property of forced, dissipative, two-dimensional incompressible Navier–Stokes (NS) systems is the dominance at long times of the longest wavelengths available to the flow. This dominance, attributed to an inverse cascade of energy with respect to enstrophy [Kraichnan, Phys. Fluids 10, 1417 (1967)], has been observed in spectrally accurate numerical simulations (see Refs. 5–7) of the incompressible NS equations. The numerical investigation of this behavior is extended to the weakly compressible regime by means of the fully compressible Fourier collocation code combox with a solenoidal forcing function that imparts no net momentum and stirs the fluid in a wave-number band in the neighborhood of kf=11. A comparison of spectral results from combox simulations with an average Mach number of 0.22 with those from identically forced incompressible simulations, at Reynolds numbers ≤700, indicates (1) the compressible and incompressible wave-number dependences in both the energy cascading and enstrophy cascading regions are nearly identical; (2) in the compressible calculation, a dual power law is also observed in density and pressure fluctuations; and (3) continued (kf=11) forcing leads to overall continued growth in the longest accessible velocity field wavelength, in both the incompressible and compressible cases.

Evolution of the Orszag–Tang vortex system in a compressible medium. I. Initial average subsonic flow

In this paper the results of fully compressible, Fourier collocation, numerical simulations of the Orszag–Tang vortex system are presented. The initial conditions for this system consist of a nonrandom, periodic field in which the magnetic and velocity field contain X points but differ in modal structure along one spatial direction. The velocity field is initially solenoidal, with the total initial pressure field consisting of the superposition of the appropriate incompressible pressure distribution upon a flat pressure field corresponding to the initial, average Mach number of the flow. In these numerical simulations, this initial Mach number is varied from 0.2–0.6. These values correspond to average plasma beta values ranging from 30.0 to 3.3, respectively. It is found that compressible effects develop within one or two Alfvén transit times, as manifested in the spectra of compressible quantities such as the mass density and the nonsolenoidal flow field. These effects include (1) a retardation of growth of correlation between the magnetic field and the velocity field, (2) the emergence of compressible small-scale structure such as massive jets, and (3) bifurcation of eddies in the compressible flow field. Differences between the incompressible and compressible results tend to increase with increasing initial average Mach number.

Identification of longitudinal acoustic modes associated with pressure oscillations in ramjets

A one-dimensiona l, isentropic analytical method is presented for calculating the longitudinal acoustic modes in an idealized dump-type ramjet engine. The analysis accounts for the complex acoustic admittances of the choked exit nozzle and of the normal shock in the supercritical inlet. The method assumes that all mean flow properties such as inlet and combustor temperatures, densities, and average Mach numbers are known. The predicted longitudinal modes are compared with data acquired from three different types of experiments. The first experiment concerned a wind tunnel model of a missile which utilized a ramjet with two side-dump inlets. The wind tunnel tests were primarily intended to determine the behavior of the supersonic inlets in the presence of downstream pressure oscillations. A mechanical device was used to excite periodic pressure perturbations in the simulated combustor exit. The second experiment involved connected pipe combustion tests of a side dump combustor. These tests included systematic changes in combustor length and in the entrance and exit boundary conditions. The third experiment utilized connected pipe tests of a coaxial dump combustor. The geometric length of the combustor was varied in order to examine the effects on combustion instability.

Observations of low‐energy ions in the wake of a magnetospheric satellite

Measurements of low‐energy ions made by the retarding ion mass spectrometer (RIMS) onboard the Dynamics Explorer 1 (DE 1) satellite are used to study some aspects of “body‐plasma interactions” in the terrestrial plasmasphere. Preliminary experimental results are presented in an empirical, parametric manner, yielding the degree of ion (current) depletion in the wake of the DE 1 satellite in terms of (1) specific and average ion Mach numbers (S), (2) average ion mass, (3) body size normalized to ionic Debye length (RDi), and (4) body potential normalized to ion thermal energy (ϕN). Some results from the RIMS measurements are compared with relevant results from the Explorer 31 and the Atmosphere Explorer C ionospheric satellites. These provide semiquantitative comparisons of certain aspects of “body‐plasma” interactions in two different flow regimes, subsonic/transonic (0.4 ≦ S ≦ 2.3) and supersonic (S > 3.3). From the DE 1 measurements, it is found that the ion species act independently in expanding into the wake of the satellite, a result which is in accord with theoretical predictions. Comparisons of measurements from the magnetospheric and ionospheric satellites indicate that for a “small” body (RDi ≲ 12), the Wake depletion varies approximately linearly with RDi, while for a “large” body (RDi > 50), the variation is exponential. The DE 1/RIMS measurements were also used to examine the utility of a simple “neutral approximation” for the maximum rarefaction region (wake) of a satellite orbiting in the terrestrial plasmasphere. The results indicate that the use of such a simple approximation is much more applicable for the subsonic/transonic flow regime (e.g., the plasmasphere) than it is for a supersonic/hypersonic flow regime (e.g., the ionosphere).

Langley Facility for Tests at Mach 7 of Subscale, Hydrogen-Burning, Airframe-Integratable, Scramjet Models

Modifications to a 20-megawatt arc-heated facility for testing a hydrogen-burning, airframe-integratable, subscale, scramjet model are described. Arc-heated flow is mixed with unheated air to furnish a test flow duplicating Mach 7 flight. (Stagnation temperature is 2220 K.) Modifications to the commercially available heater to improve survivability and smoothness are described. Pitot profiles show uniform flow and a slightly thinner nozzle boundary layer than predicted. Comparison of the tunnel boundary layer, which will be ingested by the engine model, with the boundary layer that a flight engine might ingest from its vehicle forebody shows a difference in the density distribution through the boundary layer. Calculations of wall heating and transient wall temperatures of the engine model show that for a 30-sec burn, the heat sink model requires cooling at selected locations to avoid thermal-stress, cycle-life problems. Model performance predictions show that fuel equivalence ratio and nozzle exit area both have large effects on thrust. Average inlet entrance Mach number (as affected by boundary-layer ingestion) has little effect on thrust.