Shock Structures Using the OBurnett Equations in Combination with the Holian Conjecture

In the present work, we study the normal shock wave flow problem using a combination of the OBurnett equations and the Holian conjecture. The numerical results of the OBurnett equations for normal shocks established several fundamental aspects of the equations such as the thermodynamic consistency of the equations, and the existence of the heteroclinic trajectory and smooth shock structures at all Mach numbers. The shock profiles for the hydrodynamic field variables were found to be in quantitative agreement with the direct simulation Monte Carlo (DSMC) results in the upstream region, whereas further improvement was desirable in the downstream region of the shock. For the discrepancy in the downstream region, we conjecture that the viscosity–temperature relation (μ∝Tφ) needs to be modified in order to achieve increased dissipation and thereby achieve better agreement with the benchmark results in the downstream region. In this respect, we examine the Holian conjecture (HC), wherein transport coefficients (absolute viscosity and thermal conductivity) are evaluated using the temperature in the direction of shock propagation rather than the average temperature. The results of the modified theory (OBurnett + HC) are compared against the benchmark results and we find that the modified theory improves upon the OBurnett results, especially in the case of the heat flux shock profile. We find that the accuracy gain is marginal at lower Mach numbers, while the shock profiles are described better using the modified theory for the case of strong shocks.

Linear stability of shock profiles for a quasilinear Benney system in ℝ2 × ℝ+

Following Dias et al. [Vanishing viscosity with short wave-long wave interactions for multi-D scalar conservation laws, J. Differential Equations 251 (2007) 555–563], we study the linearized stability of a pair [Formula: see text], where [Formula: see text] is a shock profile for a family of quasilinear hyperbolic conservation laws in [Formula: see text] coupled with a semilinear Schrödinger equation.

A deep insight into the ion foreshock with the help of test particle two-dimensional simulations

Two-dimensional (2D) test particle simulations based on shock profiles issued from 2D full particle-in-cell (PIC) simulations are used in order to analyze the formation processes of ions back streaming within the upstream region after interacting with a quasi-perpendicular curved shock front. Two different types of simulations have been performed based on (i) a fully consistent expansion (FCE) model, which includes all self-consistent shock profiles at different times, and (ii) a homothetic expansion (HE) model in which shock profiles are fixed at certain times and artificially expanded in space. The comparison of both configurations allows one to analyze the impact of the front nonstationarity on the back-streaming population. Moreover, the role of the space charge electric field El is analyzed by either including or canceling the El component in the simulations. A detailed comparison of these last two different configurations allows one to show that this El component plays a key role in the ion reflection process within the whole quasi-perpendicular propagation range. Simulations provide evidence that the different field-aligned beam (FAB) and gyro-phase bunched (GPB) populations observed in situ are essentially formed by a Et×B drift in the velocity space involving the convective electric field Et. Simultaneously, the study emphasizes (i) the essential action of the magnetic field component on the GPB population (i.e., mirror reflection) and (ii) the leading role of the convective field Et in the FAB energy gain. In addition, the electrostatic field component El is essential for reflecting ions at high θBn angles and, in particular, at the edge of the ion foreshock around 70∘. Moreover, the HE model shows that the rate BI% of back-streaming ions is strongly dependent on the shock front profile, which varies because of the shock front nonstationarity. In particular, reflected ions appear to escape periodically from the shock front as bursts with an occurrence time period associated to the self-reformation of the shock front.

A deep insight into the Ion Foreshock with the help of Test-particles Two-dimensional simulations

Two dimensional test-particles simulations based on shock profiles issued from 2D full PIC simulations are used in order to analyze the formation processes of ions backstreaming within the upstream region after these interact with a quasi-perpendicular curved shock front. Two different types of simulations have been performed based on (i) a FCE (Full Consistent Expansion) model which includes all self-consistent shock profiles at different times, and (ii) a HE (Homothetic Expansion) model where shock profiles are fixed at certain times and artificially expanded in space. The comparison of both configurations allows to analyze the impact of the front non stationarity on the backstreaming population. Moreover, the role of the space charge electric field is analyzed by switching it in/off in the simulations. A detailled comparison of these two last different configurations allows to show that the electric field component plays a key role in the ion reflection process within the whole quasi-perpendicular propagation range. Simulations evidence that the different populations observed in-situ namely the FAB (Field-Aligned Beam) and GBP (Gyro-Phase Bunch) populations are essentially formed by a E→t × B→ drift involving the convective electric field E→t. Simultaneously, the study emphasizes the leading role of the electrostatic (longitudinal) field E→l built up within the shock front in the acceleration process in addition to the magnetic mirror reflection (Fast Fermi). This electrostatic field component appears as essential to form backstreaming ions at high θBn angles and in particular at the edge of the ion foreshock around 70°. Moreover, the HE model shows that the rate BI% of reflected ions is strongly dependent on the shock front profile which varies because of the shock front non stationarity. In particular, reflected ions appear to escape periodically from the shock front as bursts with an occurrence time period associated to the self-reformation of the shock front.

Boltzmann–Curtiss Description for Flows Under Translational Nonequilibrium

Continuum-based theories, such as Navier–Stokes (NS) equations, have been considered inappropriate for flows under nonequilibrium conditions. In part, it is due to the lack of rotational degrees-of-freedom in the Maxwell–Boltzmann distribution. The Boltzmann–Curtiss formulation describes gases allowing both rotational and translational degrees-of-freedom and forms morphing continuum theory (MCT). The first-order solution to Boltzmann–Curtiss equation yields a stress tensor that contains a coupling coefficient that is dependent on the particles number density, the temperature, and the total relaxation time. A new bulk viscosity model derived from the Boltzmann–Curtiss distribution is employed for shock structure and temperature profile under translational and rotational nonequilibrium. Numerical simulations of argon and nitrogen shock profiles are performed in the Mach number range of 1.2–9. The current study, when comparing with experimental measurements and direct simulation Monte Carlo (DSMC) method, shows a significant improvement in the density profile, normal stresses, and shock thickness at nonequilibrium conditions than NS equations. The results indicate that equations derived from the Boltzmann–Curtiss distribution are valid for a wide range of nonequilibrium conditions than those from the Maxwell–Boltzmann distribution.

Stability of planar rarefaction wave to the 3D bipolar Vlasov–Poisson–Boltzmann system

We investigate the time-asymptotic stability of planar rarefaction wave for the 3D bipolar Vlasov–Poisson Boltzmann (VPB) system, based on the micro–macro decompositions introduced in [T. P. Liu and S. H. Yu, Boltzmann equation: Micro–macro decompositions and positivity of shock profiles, Comm. Math. Phys. 246 (2004) 133–179; Energy method for the Boltzmann equation, Physica D 188 (2004) 178–192] and our new observations on the underlying wave structures of the equation to overcome the difficulties due to the wave propagation along the transverse directions and its interactions with the planar rarefaction wave. Note that this is the first stability result of basic wave patterns for bipolar VPB system in three dimensions.