Shock Jumps Research Articles

A spectrum improved peridynamic model for shock wave problems

Most previous peridynamics (PD) based shock wave modeling focuses on the field discontinuity problem but neglects the temporal discontinuous feature. Considering it is of vital importance to reproduce the shock jumps, the spectral properties of PD are studied and improved in this paper. A systematic theoretical analysis is firstly conducted and reveals the overly-dispersive behavior of PD model, including the oscillatory dispersion relation and zero energy modes. To suppress the consequent spurious oscillation, the dissipation effect is nonlocally introduced by viscosity as a minimum variation of conventional PD. Rather than applying the existing viscosities directly, the spectral balance is considered for the good control of both the oscillation and shock width, leading to the proposal of polynomial and force-limited viscous force states. The proposed model is then verified by several impact and explosion cases. The quantification of spectral balance is established and is employed for parameter calibration. The results suggest that the model shows good performance in shock wave modeling, with which both the suppression of oscillation and control of shock width near the shock jump are well achieved.

Reconnection layer bounded by switch‐off shocks: Dayside magnetopause crossing by THEMIS D

AbstractWe discuss observations of reconnection, obtained by Time History of Events and Macroscale Interactions during Substorms (THEMIS) D during an outward bound traversal of the low‐latitude dayside magnetopause. The reconnection signatures include high magnetic shear, a southward directed Alfvénic jet, bounded by slow‐mode shocks near the switch‐off limit (as in the symmetric Petschek geometry), a small, sunward directed normal magnetic field and plasma inflow into the jet from both sides. We conclude that cold, unmeasured ionospheric ions helped establish the symmetry. The effective ion mass, estimated from the switch‐off condition, was 2.39 amu on the magnetospheric side, where the number density was inferred from the spacecraft potential, and 1.09 amu on the magnetosheath side. After a modest pressure correction in the magnetospheric shock, the MHD jump conditions for density, pressure, temperature, and entropy were well satisfied. The shock jumps were much larger on the magnetosphere side than on the magnetosheath side; we show this to be a plasma β effect. The main dissipation mechanism appears to be irreversible transfer between thermal motion parallel and perpendicular to the field, such that both shocks bring about approximate downstream temperature isotropy. Hall currents and electric fields were present, albeit in a strongly asymmetric configuration. The magnetospheric shock had longer duration than the magnetosheath one, possibly as a result of a nonconstant magnetopause speed. We infer an average earthward magnetopause speed (14 km/s), corresponding nominal shock thicknesses (12 and 6 λi), dimensionless reconnection rates (0.061–0.085), and reconnection wedge angles (5° between shocks; 13° between separatrices).

Shocking features in the merging galaxy cluster RXJ0334.2−0111

We present a 66 ksec $\textit{Chandra}$ X-ray observation of the galaxy cluster RXJ0334.2-0111. This deep observation revealed a unique bow shock system associated with a wide angle tail (WAT) radio galaxy and several intriguing substructures. The temperature across the bow shock jumps by a factor of $\sim$ 1.5 (from 4.1 keV to 6.2 keV), and is consistent with the Mach number $M = 1.6_{-0.3}^{+0.5}$. A second inner surface brightness edge is a cold front that marks the border between infalling subcluster cool core and the ICM of the main cluster. The temperature across the cold front increases from $1.3_{-0.8}^{+0.3}$ keV to $6.2_{-0.6}^{+0.6}$ keV. We find an overpressurized region $\sim$ 250 kpc east of the cold front that is named "the eastern extension (EE)". The EE may be a part of the third subcluster in the ongoing merger. We also find a tail shaped feature that originates near the bow shock and may extend up to a distance of $\sim$ 1 Mpc. This feature is also likely overpressurized. The luminous FR-I radio galaxy, 3C89, appears to be the cD galaxy of the infalling subcluster. We estimated 3C89's jet power from jet bending and the possible interaction between the X-ray gas and the radio lobes. A comparison between the shock stand-off distance and the Mach number for all known shock front/cold front combinations suggests that the core is continuously shrinking in size by stripping.

Shock model description of the interaction radiation pulse in nested wire array z-pinches

Bow shock structures are observed in a nested wire array z-pinch as ablation streams from the outer array pass the inner array. The jump in plasma conditions across these shocks results in an enhancement of snowplow emission from the imploding plasma piston. Results from a snowplow model modified to account for the shock jumps are discussed and compared to experimental data from MAGPIE. Magnetohydrodynamic simulations indicate that this is the primary heating mechanism responsible for the interaction pulse recorded on the Z generator, which is required for pulse shaping for inertial confinement fusion.

Computation of compaction in compressible granular material

Equations modeling compaction in a mixture of granular high explosive and interstitial gas are solved numerically. Both phases are modeled as compressible, viscous fluids. This overcomes well known difficulties associated with computing shock jumps in the inviscid version of the equations, which cannot be posed in a fully conservative form. One-dimensional shock tube and piston-driven compaction solutions compare favorably with experiment and known analytic solutions. A simple two-dimensional extension is presented.

Multi-level lattice Boltzmann model on square lattice for compressible flows

A compressible lattice Boltzmann model is established on a square lattice. The model allows large variations in the mean velocity by introducing a large particle-velocity set. To maintain tractability, the support set of the equilibrium distribution is chosen to include only four directions and three particle-velocity levels in which the third level is introduced to improve the stability of the model. This simple structure of the equilibrium distribution makes the model efficient for the simulation of flows over a wide range of Mach numbers and gives it the capability of capturing shock jumps. Unlike the standard lattice Boltzmann model, the formulation eliminated the fourth-order velocity tensors, which were the source of concerns over the homogeneity of square lattices. A modified collision invariant eliminates the second-order discretization error of the fluctuation velocity in the macroscopic conservation equation from which the Navier–Stokes equation and energy equation are recovered. The model is suitable for both viscous and inviscid compressible flows with or without shocks. Two-dimensional shock-wave propagations and boundary layer flows were successfully simulated. The model can be easily extended to three-dimensional cubic lattices.

Analytical Shock Jump Formulae for Cryogenic Homogeneous Two-Phase Nozzle Flow.

Choked flux rates and shock jump relations were calculated both analytically and computationally for a converging-diverging nozzle. High inlet stagnation pressure with a maximum of 6.45MPa was applied to the cryogenic vapor-liquid nitrogen flow. The total inlet-to-static exit pressure ratio range was between 3.5 and 15.3. The analytical shock jump expressions, which were derived, are exact extensions of the Rankine-Hugoniot single-phase gas dynamic relations. Accordingly, a consistent agreement was obtained between analytical and computational pressure, Mach number and void fraction shock jumps. The numerical mass flux values were also successfully compared with experimental data and two separate analytical results, both obtained from a modified Henry-Fauske model and a homogeneous mass flux equation.

Three-dimensional flow computations with shock fitting

The present paper provides a shock-fitting technique for solving inviscid transonic three-dimensional (3-D) flows. The continuous flow field is computed by means of an implicit fast Euler solver, which separately integrates compatibility conditions, written in terms of generalized Riemann variables along appropriate bicharacteristic lines. The continuous 3-D flow problem is thus reduced to a sequence of simple quasi 1-D problems. The shock wave is computed by means of a shock-fitting technique, which enforces the proper shock jumps by an explicit use of the Rankine-Hugoniot equations. The computed shock thus develops into a discontinuity as it is in reality. The merits of the present approach are demonstrated by means of a few simple applications and by comparison with corresponding results computed using a flux-difference splitting methodology.

Theory of compressible irrotational flows including heat conductivity and longitudinal viscosity

A new exact analytical model was derived for the irrotational flows of compressible fluids when the effects of heat conductivity and molecular viscosity are allowed. This new model satisfies conservation of mass, momentum and energy exactly. In addition, it satisfies physical irrotationality conditions. Compared to the classical small perturbation viscous-transonic (V-T) equation, the new physically dissipative potential (PDP) equation contains a number of additional terms that are highly nonlinear. The new model is derived in a general vector operator form and in a scalar canonical form. The PDP equation is able to produce shock waves of different strengths depending on the ratio of secondary and shear viscosity coefficients. The one-dimensional, steady flow version of the PDP equation was integrated using a Runge-Kutta scheme and different values of the ratio of the two viscosities. The computed shock structures are symmetric. Rankine-Hugoniot shock jumps were obtained when Stokes' hypothesis was used in the PDP equation and isentropic shock jumps were obtained when the longitudinal viscosity was negligible.

Damped Euler-equation method to compute transonic flow around wing-body combinations

Inviscid transonic flows containing either strong shock waves or complex vortex structure call for the Euler equations as a realistic model. Presented here is a computational procedure for solving the Euler equations for transonic flow around a wing and fuselage upon an O-O mesh generated by transfinite interpolation. An explicit time-marching finite-volume procedure solves the flow equations and features a nonreflecting far field boundary condition, an internal mechanism for temporal damping, and use of the local time step, all of which improve the convergence of the computation. Converged after several hundred iterations, results computed on the CYBER 203 vector processor are compared with experimental data and potential-flow computations. The Euler-equation model is found to predict the existence of a tip vortex created by inviscid flow separation in the downstream region of the tip of the M6 wing where the radius of curvature approaches zero. AST year a workshop 1 was held in order to assess the currently used computational procedures, one against the other, in several carefully specified two-dimensional transonic flow problems. While reasonable agreement among the results given by the various full potential and Euler-equation methods was obtained in the subcritical cases, a disparity between the results of these two models was found to grow with cases of increasing shock strength. For the NACA 0012 airfoil and conditions M- 0.80 and a. = 1.25 deg, for example, the lift coefficients CL given by all the Euler methods ranged between 0.30 and 0.38, whereas the range of CL for the potential method was 0.28-1.1. This discrepancy has led to a reconsideration of the validity of the potential model when strong shocks are present in the flow, not only locally in terms of the isentropic shock jumps but also, and perhaps even more importantly, with regard to the correct modeling of vortex phenomena throughout the entire flowfield. In the potential representation the vorticity that is bound to the airfoil is accounted for by a jump in the potential across a line originating at the trailing edge and lying a priori along some chosen coordinate direction downstream, the so-called Kutta condition. The Euler equations, in contrast, admit vorticity in the solution and the equivalent to the Kutta condition evidently does not need to be enforced explicitly. This situation is currently under study by researchers using a variety of numerical methods. In three dimensions the flow past a finite wing is even more complex and less is known. Vortices, for example, are shed continuously from the wing tips and the entire trailing edge. Whether a Kutta condition or its equivalent is necessary in this case has not been in- vestigated, but because vorticity is so crucial to the realism achieved by inviscid flow models, interest has been aroused in questions like this and in numerical methods that solve the Euler equations. A number of methods 2'4 exist to solve the Euler equations for three-dimensional flow, but they have been developed exclusively for and applied only to internal flows. Apparently, the solution of the Euler equations for