Computational simulations of microscale shock–vortex interaction using a mixed discontinuous Galerkin method

Extensively investigations of the net vorticity in micro-scale shock/vortex interaction were conducted by validated direct simulation of Monte Carlo (DSMC). The major characteristic is found that the strong shock/vortex interaction in micro-scale can cause an increase in enstrophy, which differs from the previous DSMC studies. Investigations also show that the increase of the enstrophy causes an increase dissipation rate in during the strong interaction. The shock Mach number, vortex size and vortex Mach number turn out to play a critical role in the strength of interaction, which in turn govern the change in the dissipation rate. The ratio, of shock thickness to vortex core radius, is founded to be an important parameter which determines the tendency of enstrophy. When the ratio is more than about 0.5, the increasing of enstrophy disappear, which is different from the previous studies of macro-scale shock/vortex interaction. When the shock wave is strong enough to make the ratio less than 0.5, the macro ...

Investigate the shock focusing under a single vortex disturbance using 2D Saint-Venant equations with a shock-capturing scheme

Context. We revisit the study of shocks interacting with molecular clouds, incorporating coupled gas-dust dynamics. Aims. We study the effect of different parameters on the shock-cloud interaction, such as the dust-to-gas ratio or the Mach number of the impinging shock. By solving self-consistently for drag-coupled gas and dust evolutions, we can assess the frequently made assumption that the dust is locked to the dynamics of the gas so that dust observations would result in direct information on the gas distribution. Methods. We used a multi-fluid model where the dust is represented by grain-size specific pressureless fluids. The dust and gas interact through a drag force, and we used four dust species with weighted representative sizes between 1 and 500 nm. We use the open source code MPI-AMRVAC for a parametric study of the effect of the gas-dust ratio and the Mach number of the shock. By using the radiative transfer code SKIRT, we create synthetic millimeter wavelength maps to connect to observations. Results. We find that the presence of dust does not significantly affect the dynamics of the gas for realistic dust-gas ratios, and this is the case throughout the range of Mach numbers explored (1.5−10). For high Mach numbers, we find a significant discrepancy between the distribution of the dust and gas after the cloud-shock interaction with the larger dust species clearly lagging the heavily mixed and accelerated gas (re)distribution. Conclusions. We conclude that observational studies of dusty environments may need to account for clearly separated spatial distributions of dust and gas, especially those studies that are representative of molecular clouds that have been interacting with high Mach number shock fronts.

Analysis of multipolar vortices in the interaction of a shock with a strong moving vortex

In this paper, we introduce and analyze a mixed discontinuous Galerkin method for the Helmholtz equation. The mixed discontinuous Galerkin method is designed by using a discontinuous Pp+1−1−Pp−1 finite element pair for the flux variable and the scattered field with p≥0. We can get optimal order convergence for the flux variable in both Hdiv-like norm and L2 norm and the scattered field in L2 norm numerically. Moreover, we conduct the numerical experiments on the Helmholtz equation with perturbation and the rectangular waveguide, which also demonstrate the good performance of the mixed discontinuous Galerkin method.

Deformation and breakup of bubbles interacting with single vortex rings

Whitham's approximation for handling shock wave propagation in area changes (reductions) in a duct was checked in comparison with a numerical solution. Also the Whitham approximation for shock wave propagation from a constant cross-sectional duct to a duct of a smaller cross-sectional area was studied and compared with a numerical solution. It was found that for modest incident shock Mach numbers and modest area reductions the Whitham approximation provided a fair solution for the shock Mach number and for the post-shock pressure. For higher shock Mach numbers and/or area reductions, large discrepancies exit between the approximate and exact solutions. A wider range of applicability of the Whitham approximation is found for the monotonical area reduction case; it is quite narrow for the passage of a shock wave from a wider to a narrower duct case. In addition, the effect of the extent of the area change region on the time required for reaching a quasi-steady flow was studied. It was shown that the longer the area change segment is, the longer it takes to reach a quasi-steady flow.

The small scale pipe plays an important role in the heat and mass transfer of industrial application. For the study of the flow characteristics in small scale pipe, a modal mixed Discontinuous Galerkin (DG) method was proposed for solving axial-symmetrical Navier-Stokes (NS) equations with solid slip boundary conditions. A new limiter of modal DG was proposed. And, the special treatments of velocity slip and temperature jump in wall solid boundary were given in the numerical scheme. Extensive numerical studies for the flow characteristics of small scale pipe were conducted. For validation, the comparison between the present study and experiment data were performed. The heat transfer coefficients and frictional factor are found to be increased with the diameter decreasing. Furthermore, it is observed that the frictional factor of small scale pipe is larger than that of macro scale pipe. And, the proposed DG-NS method provides a new way for the investigation of the flow characteristics in small scale flow.

• A numerical strategy for capturing shock-vortex interactions with Discontinuous Galerkin schemes, including modal DG and nodal DGSEM, is presented • The two schemes are evaluated and compared for an inviscid Mach 3 forward step case, a viscous shock-wall interaction and the supersonic Taylor-Green vortex • In all cases, it is shown that both schemes provide good shock and vortex capturing properties, considering coarse grids, unstructured meshes and high polynomial degrees • DG and DGSEM provide similar solutions on equivalent grids, while DGSEM displays better performance than DG, in particular when considering higher polynomial degrees This paper presents the assessment and comparison of high-order nodal and modal Discontinuous Galerkin schemes for unsteady simulations of freely-developing and wall-bounded scale-resolved shock-turbulence and shock-vortex interaction problems. A numerical strategy that is suitable for both schemes, combining a positivity-preserving limiter, a localized artificial viscosity operator and a turbulence sensor, is presented. This choice of numerical ingredients aims specifically at regularizing shocks while providing a high-accuracy for the representation of vortices and turbulent scales. The accuracy, robustness and performance of the proposed methodologies are assessed in the context of scale-resolved shock-turbulence and shock-vortex interactions considering various resolutions, polynomial degrees and mesh element topology. For all cases considered, it is shown that both schemes provide good shock and vortex capturing properties, even when considering coarse grids, unstructured meshes and high polynomial degrees.

Shocks are ubiquitous in astrophysical environments and cosmic-rays (CRs) are known to be accelerated at collisionless shocks via diffusive shock acceleration. It is believed that the CR pressure is important in the evolution of the interstellar medium of our galaxy and most of galactic CRs with energies up to eV are accelerated by supernova remnant shocks. In this contribution we have studied the CR acceleration at shocks through numerical simulation of 1D, quasi-parallel shocks for a wide range of shock Mach numbers and shock speeds. We show that CR modified shocks evolve to time-asymptotic states by the time injected particles are accelerated to moderately relativistic energies, and that two shocks with the same Mach number, but with different shock speeds, evolve qualitatively similarly when the results are presented in terms of a characteristic diffusion length and diffusion time. We find that of the particles passed through the shock are accelerated to form the CR population, and the injection rate is higher for shocks with higher Mach number. The time asymptotic value for the CR acceleration efficiency is controlled mainly by shock Mach number, and high Mach number shocks all evolve towards efficiencies , regardless of the injection rate and upstream CR pressure. We conclude that the injection rates in strong quasi-parallel shocks are sufficient to lead to significant nonlinear modifications to the shock structures, implying the importance of the CR acceleration at astrophysical shocks.

Cosmological shocks form as an inevitable consequence of gravitational collapse during the large scale structure formation and cosmic-rays (CRs) are known to be accelerated at collisionless shocks via diffusive shock acceleration (DSA). We have calculated the evolution of CR modified shocks for a wide range of shock Mach numbers and shock speeds through numerical simulations of DSA in 1D quasi-parallel plane shocks. The simulations include thermal leakage injection of seed CRs, as well as pre-existing, upstream CR populations. Bohm-like diffusion is assumed. We show that CR modified shocks evolve to time-asymptotic states by the time injected particles are accelerated to moderately relativistic energies (p/mc <TEX>$\ge$</TEX> 1), and that two shocks with the same Mach number, but with different shock speeds, evolve qualitatively similarly when the results are presented in terms of a characteristic diffusion length and diffusion time. We find that <TEX>$10^{-4} - 10^{-3}$</TEX> of the particles passed through the shock are accelerated to form the CR population, and the injection rate is higher for shocks with higher Mach number. The CR acceleration efficiency increases with shock Mach number, but it asymptotes to <TEX>${\~}50\%$</TEX> in high Mach number shocks, regardless of the injection rate and upstream CR pressure. On the other hand, in moderate strength shocks (<TEX>$M_s {\le} 5$</TEX>), the pre-existing CRs increase the overall CR energy. We conclude that the CR acceleration at cosmological shocks is efficient enough to lead to significant nonlinear modifications to the shock structures.

A stabilized mixed discontinuous Galerkin formulation for double porosity/permeability model

A two-dimensional, linearized treatment, including real gas effects, of shock curvature in a shock tube is presented. An expression for shock shape as a function of shock Mach number and initial pressure of the test gas is presented. The results are compared with the available experimental data obtained in argon at low shock strengths and in air at high shock Mach numbers. Within the scatter of the data in the latter experiments, there is relatively good agreement with theory, while theory falls approximately 30 ± 10% above the data in argon. Some of this disagreement is attributed to application of the two-dimensional theoretical result to axisymmetric shock tubes of finite dimensions used in the experiments.

An exact analytical model for the interaction between an isolated shock wave and an isotropic turbulent vorticity field is presented. The interaction with a single-mode two-dimensional (2D) divergence-free vorticity field is analyzed in detail, giving the time and space evolutions of the perturbed quantities downstream. The results are generalized to study the interaction of a planar shock wave with an isotropic three-dimensional (3D) or 2D preshock vorticity field. This field is decomposed into Fourier modes, and each mode is assumed to interact independently with the shock front. Averages of the downstream quantities are made by integrating over the angles that define the orientation of the upstream velocity field. The ratio of downstream/upstream kinetic energies is in good agreement with existing numerical and experimental results for both 3D and 2D preshock vorticity fields. The generation of sound and the sonic energy flux radiated downstream from the shock front is also discussed in detail, as well as the amplification of transverse vorticity across the shock front. The anisotropy is calculated for the far downstream fields of both velocity and vorticity. All the quantities characteristic of the shock-turbulence interaction are reduced to closed-form exact analytical expressions. They are presented as explicit functions of the two parameters that govern the dynamics of the interaction: the adiabatic exponent gamma and the shock Mach number M1 . These formulas are further reduced to simpler exact asymptotic expressions in the limits of weak and strong shock waves (M_{1}-11, M_{1}1) and high shock compressibility of the gas (gamma-->1) .

This article presents molecular-level analysis of a reactive, near-continuum, Mach 21 nitrogen flow over a blunt wedge using the direct molecular simulation (DMS) method. The flow conditions lead to internal energy excitation and dissociation in the flow field, resulting in thermal and chemical nonequilibrium in the flow. Thermal nonequilibrium in the vibrational mode is observed to extend to the molecular level, where the vibrational energy distributions at various points in the flow field are observed to be non-Boltzmann. Furthermore, this is the first reactive DMS calculation where the wall is assumed to be isothermal and full momentum accommodation of the particles is enforced, hence incorporating viscous wall effects. Since the DMS method uses a quantum mechanically generated interaction potential as its only modeling input, all thermochemical and transport properties of the flow field can directly be attributed to the ab initio potential energy surface. Using the DMS solution as a benchmark, this article assesses the performance of Navier–Stokes computational fluid dynamics solutions using lower fidelity two-temperature models. Two models are chosen as points of comparison: the well-known Park two-temperature model and the recently developed modified Marrone and Treanor model.