Insights into spatio-temporal dynamics during shock–droplet flame interaction

Context. Many problems in stellar astrophysics feature flows at low Mach numbers. Conventional compressible hydrodynamics schemes frequently used in the field have been developed for the transonic regime and exhibit excessive numerical dissipation for these flows. Aims. While schemes were proposed that solve hydrodynamics strictly in the low Mach regime and thus restrict their applicability, we aim at developing a scheme that correctly operates in a wide range of Mach numbers. Methods. Based on an analysis of the asymptotic behavior of the Euler equations in the low Mach limit we propose a novel scheme that is able to maintain a low Mach number flow setup while retaining all effects of compressibility. This is achieved by a suitable modification of the well-known Roe solver. Results. Numerical tests demonstrate the capability of this new scheme to reproduce slow flow structures even in moderate numerical resolution. Conclusions. Our scheme provides a promising approach to a consistent multidimensional hydrodynamical treatment of astrophysical low Mach number problems such as convection, instabilities, and mixing in stellar evolution.

A low diffusion version of the Harten–Lax–Leer convective pressure split (HLL-CPS) scheme for resolving the shear layers and the flow features at low Mach numbers is presented here. The low diffusion HLL-CPS scheme is obtained by reconstructing the velocities at the cell interface with the face normal Mach number and a pressure function. Asymptotic analysis of the modified scheme shows a correct scaling of the pressure at low Mach numbers and a significant reduction in numerical dissipation. The robustness of the HLL-CPS scheme for strong shock is improved by reducing the contribution of the contact wave in the vicinity of the shock. The improvement in robustness for strong shock is demonstrated analytically through linear perturbation and matrix stability analyses. A set of numerical test cases are solved to demonstrate the efficacy of the proposed scheme over a wide range of Mach numbers.

Features of the simulation of low-velocity inviscid and viscous compressible gas flows are considered, and a finite-volume discretization of gas dynamics equations at low Mach numbers on unstructured meshes is discussed. Preconditioning based on the use of physical variables is used to speed up the convergence of time marching to a steady state and to improve the accuracy of the steady-state solution. The structure of the preconditioning matrix and the diagonalization of the Jacobian of the preconditioned system of equations are discussed. The capabilities of this approach are demonstrated using model gasdynamic simulations ing a wide range of Mach numbers.

This work focuses on the study of the asymptotic preserving property of a semi-implicit method. The semi-implicit method, which is a pressure-based method, has been successfully used to simulate two-phase flows in numerous industrial applications. This method is used in our studies due to the fact that pressure-based methods generally perform well at low Mach numbers. The semi-implicit method is applied to the homogeneous equilibrium model (HEM) in this work to simulate two-phase flows. We show that the semi-implicit method is asymptotic preserving, i.e. the discretization for a compressible model tends to a consistent discretization for the related incompressible model at the low Mach number limit. Finally, test cases are performed to show that the numerical method is able to deal with low Mach number flows, as well as flows with a wide range of Mach numbers.

Entropy-based viscous regularization for the multi-dimensional Euler equations in low-Mach and transonic flows

Split form ALE discontinuous Galerkin methods with applications to under-resolved turbulent low-Mach number flows

Astrophysical shocks are often studied in the high Mach number limit but weakly compressive fast shocks can occur in magnetic reconnection outflows and are considered to be a site of particle energization in solar flares. Here we study the microphysics of such perpendicular, low Mach number collisionless shocks using two-dimensional particle-in-cell simulations with a reduced ion/electron mass ratio and employ a moving wall boundary method for initial generation of the shock. This moving wall method allows for more control of the shock speed, smaller simulation box sizes, and longer simulation times than the commonly used fixed wall, reflection method of shock formation. Our results, which are independent of the shock formation method, reveal the prevalence shock drift acceleration (SDA) of both electron and ions in a purely perpendicular shock with Alfvén Mach number MA=6.8 and ratio of thermal to magnetic pressure β=8. We determine the respective minimum energies required for electrons and ions to incur SDA. We derive a theoretical electron distribution via SDA that compares to the simulation results. We also show that a modified two-stream instability due to the incoming and reflecting ions in the shock transition region acts as the mechanism to generate collisionless plasma turbulence that sustains the shock.

Recent simulation results have revealed that energetic electron bursts are produced cyclically at the shock reformation period upstream of reforming shocks and are qualitatively very different from the continuous beam expected from time‐stationary shocks (Yuan et al., 2007a). This paper extends our previous studies by numerically investigating the dependence of electron burst events on shock parameters (the upstream plasma β and Mach number M A ). The test particle approximation is made for electrons, and the electron trajectories are traced exactly in the time‐dependent electromagnetic field profiles, generated by one‐dimensional hybrid simulation code. Simulation results indicate that the upstream incoming electrons can be reflected nonuniformly or continuously depending on the shock parameters. Bursty energetic electron events take place when the plasma beta is low ( β ≤ 0.4) and the shock Mach number is high ( M A ≥ 6). Time‐varying loss cone, beam, and ring beam features are observed in the upstream electron distribution functions. The beam density, speed, average kinetic energy, and speed spread cyclically change with time by factor of ∼2–4. In contrast, continuously reflected electrons are observed for low beta ( β ≤ 0.4), low Mach number ( M A ≤ 4) shocks, even when the shock is reforming because the changes in shock fields are relative small. The electron burst events disappear and the observed upstream electron distribution function contours are steady state. A continuous electron beam is formed, which is qualitative the same as the beam from steady‐state shocks. Increasing the plasma beta (providing the shock is still reforming) has minor effects on the upstream electron beam features.

This paper presents the Large Eddy Simulation (LES) of a Low-Pressure Turbine (LPT) Nozzle Guide Vane (NGV) for different Reynolds (Re) and Mach number (Ma). The analysis is based on a slice of the blade that may be representative of midspan flow where secondary flows, hub and shroud contributions are lower. In LPT, the variation of the Re during the mission of the gas turbine is a well-known effect since its value can vary of a factor four between take-off and cruise. This can induce performance variations due to various phenomena with among them suction side boundary layer separation on the aft portion of the blade due to an adverse pressure gradient and laminar boundary layer that can be maintained due to the relatively low Re in LPT. Similarly, the Ma in the LPT may vary depending on the thrust required from the gas turbine at the considered mission phase. The current paper investigates through numerical simulation the flow representative of a medium-sized LPT with three different Reynolds number Re = 175’000 (cruise), 280’000 (mid-level altitude) and 500’000 (take-off) keeping the same characteristic Mach number Ma = 0.2 and three different Mach number Ma = 0.2, 0.5 and 0.8 keeping the same Reynolds number Re = 280’000. The study focuses on different flow characteristics: pressure distribution around the blade, near-wall flow behavior and wake analysis. This includes the related generation of losses and the effect of Re and Ma on these different phenomena. A special emphasis is given to the generation of loss based on an entropy approach and the redistribution of mean kinetic energy towards turbulent kinetic energy. The results show that the increase of the Re has a destabilizing effect on potential separation while the increase of the Ma has a stabilizing effect. The peak in the TKE downstream of the blade is also moved upstream closer to the trailing edge when the Ma is increased.

To explore the effects of airfoil-probe tubes and its installment position on the flow field of the compressor cascade, and find out the mechanism that how the airfoil-probes affect the aerodynamic characteristics of the compressor cascade, this paper performed both numerical and experimental works on the same compressor cascade. The experiment mainly focused on the cases of low Mach number (Ma = 0.1), and cases with different Mach numbers (0.1, 0.3, 0.7) and different incidence angles (-5, 0, 5) are investigated by the numerical method. The case without the airfoil-probe tube was referenced as the baseline, and other three cases with the airfoil-probe tubes installed in different chordwise positions (30%, 50%, 70% of the chord length) were studied. The diameter of the airfoil-probe tube is 3mm, which is configured as 300% amplification of some particular airfoil-probe according to the geometrical similarity principle. The results show that the airfoil-probe tubes have a negative influence on the flow capacity of the cascade at all investigation points. The separations and the large scale streamwise vortices that induced by the airfoil-probe tube on the pressure side cause most the losses at the high Mach number. The influence of the airfoil-probe tube on the flow field in the vicinity of the pressure side surface is local separation at the low Mach number. The airfoil-probe tubes also have a clearly effect on the leakage flow. It decreases the mass flow of the leakage flow and weakens the intensity of the leakage vortex, but enlarges the influence area. The total pressure loss of the case that the tube is installed at the half chordwise position is generally lower than other cases especially at the high Mach number, it can even decrease the losses compared with the basic case.

: Heat-transfer distribution data obtained in the arc-driven tunnels (hotshot type) of the von Karman Gas Dynamics Facility, AEDC, are presented. Data were taken on a 9-deg half-angle spherically blunted cone and a hemisphere-cylinder at Mach numbers between 17 and 20 and Reynolds numbers per foot in the free-stream between 100,000 and 800,000. The test data are compared with theory and available shock tube and shock tunnel results. Good agreement between Lees' theory and measured heat-transfer rates demonstrates the capability of the VKF hotshot tunnels to obtain heat-transfer distribution data. The data indicate a strong dependence of the heat-transfer distribution on pressure distribution, Mach number, and cone half-angle. The heat-transfer data are correlated over a wide range of Mach numbers and cone angles. Correlation curves and formulas are presented for the pressure and heat-transfer distribution to spherically blunted cones at hypersonic conditions.

A similitude has been obtained for an oscillating two-dimensional wedge in pitch with attached shock at high angle of incidence in hypersonic flow. A strip theory, in which flow at infinite span wise location is two dimensional and independent of each other is being used. Further this unites the similitude with piston theory to give one dimensional piston theory with large flow deflection. Closed form solution is obtained for stiffness and damping derivatives in pitch. The present theory is valid when the shock wave is attached with the nose of the wedge. With the increase in semi vertex angle of the wedge the stiffness and the damping derivatives are found to increase progressively, and with the increase in the Mach number both stiffness and damping derivatives are found to decrease then with increase in Mach number it becomes independent of Mach number. From the amalgamation of theory developed, we can evaluate the stiffness and damping derivatives for in pitch for a wide range of Mach numbers, for various pivot positions, and angle of attack. Significantly the same results are being validated with the analytical results of Liu and Hui [4] and Lighthill’s [3] theory for some cases.

Ion velocity space distributions measured with Los Alamos/Garching instruments on ISEE 1 and 2 across the earth's bow shock have revealed that the nearly perpendicular shock reflects some fraction of the incident solar wind ions over a wide range of Mach numbers (MA ≃ 2.0–12.4). The relative density of the reflected ions varies from 1–3% of the solar wind density at MA ≃ 2 to 15–25% at MA = 8–12, in qualitative agreement with computer simulations and with predictions that indicate that such ions are needed to provide essential dissipation at the shock. These ions gyrate about B in the plasma rest frame with a speed of ∼2 υin, where υin is the normal component of the solar wind flow, and their density increases by ∼2 orders of magnitude at the forward end of the ’foot‘ of the shock profile ∼0.7 ion gyroradii upstream from the shock ramp. This is substantial evidence that the bulk of these ions have been reflected at the shock in a nearly specular fashion. Depending on the shock geometry, some low numbers of reflected‐gyrating ions are sometimes observed a few gyroradii ahead of the outer edge of the foot. Model calculations suggest that these additional ions are produced by non‐specular reflection of a much smaller fraction of the incident solar wind. Downstream of the shock ramp, the secondary ions gyrate with a slightly higher speed, υg ≃ 2.15 υin, in the downstream rest frame. Individual groups of gyrating ions rapidly disperse in velocity space to form a high‐energy torus centered at the bulk velocity of the distributions. Downstream ion toruses are most clearly seen, sometimes persisting for many minutes, when the shock Mach number is low. Downstream of highly supercritical shocks (M ≫ 3), the distributions are more turbulent. In either case, the distributions retain some non‐Maxwellian features for a long time downstream from the ramp, in accordance with earlier observations. Reflected‐gyrating ions have also been identified in a number of crossings with more oblique field orientations (θnB ≳ 45°, i.e., for quasi‐perpendicular shocks). Specularly reflected ions play an important role in producing down‐stream ion thermalization for all θnB ≳ 45°, being relatively more important the higher is the Mach number.

This paper discusses the sound generated when an inhomogeneity in density is convected in a low Mach number steady flow through a contraction in a duct of infinite extent, and also when the inhomogeneity exhausts through a nozzle into free space. The analyses of Candel (1972) and Marble (1973) for the case of duct flow were based on a frequency decomposition of the incident inhomogeneity and cannot adequately deal with sharp-fronted inhomogeneities and entropy spots. However, the practical difficulties of this earlier work can be avoided at low flow Mach numbers by conducting the analysis in terms of an approximate expression for the acoustic Green's function in the manner described by Howe (1975). This method also permits a considerable extension of the range of the earlier investigations to the determination of the sound generated when the inhomogeneity is swept out of a nozzle orifice into free space. It is shown that the acoustic pressure perturbations developed in a duct at a contraction are in general proportional to the fractional difference between the density of the inhomogeneity and that of the mean flow times a typical mean flow pressure level, and are due principally to the fluctuation in thrust accompanying the passage of the inhomogeneity through the region of variable pressure gradient. The pressure waves generated at a nozzle orifice and radiated into free space are O(M0) smaller, where M0 is a mean flow Mach number based on the speed of sound in the jet.

Astrophysical shocks are often collisionless shocks. An open question about collisionless shocks is whether electrons and ions each establish their own post-shock temperature, or whether they quickly equilibrate in the shock region. Here we provide simple relations for the minimal amount of equilibration to expect. The basic assumption is that the enthalpy-flux of the electrons is conserved separately, but that all particle species should undergo the same density jump across the the shock. This assumption results in an analytic treatment of electron-ion equilibration that agrees with observations of collisionless shocks: at low Mach numbers ($<2$) the electrons and ions are close to equilibration, whereas for Mach numbers above $M \sim 60$ the electron-ion temperature ratio scales with the particle masses $T_e/T_i = m_e/m_i$. In between these two extremes the electron-ion temperature ratio scales as $T_e/T_i \propto 1/M_s^2$. This relation also hold if adiabatic compression of the electrons is taken into account. For magnetised plasmas the compression is governed by the magnetosonic Mach number, whereas the electron-ion temperatures are governed by the sonic Mach number. The derived equations are in agreement with observational data at low Mach numbers, but for supernova remnants the relation requires that the inferred Mach numbers for the observations are over- estimated, perhaps as a result of upstream heating in the cosmic-ray precursor. In addition to predicting a minimal electron/ion temperature ratio, we also heuristically incorporate ion-electron heat exchange at the shock, quantified with a dimensionless parameter ${\xi}$. Comparing the model to existing observations in the solar system and supernova remnants suggests that the data are best described by ${\xi} \sim 5$ percent. (Abridged abstract.)