Shock Waves In Gas Dynamics Research Articles

On the Accuracy of Discontinuous Galerkin Method Calculating Gas-Dynamic Shock Waves

On the Accuracy of Discontinuous Galerkin Method Calculating Gas-Dynamic Shock Waves

ON THE ACCURACY OF DISCONTINUOUS GALERKIN METHOD CALCULATING GAS-DYNAMIC SHOCK WAVES

The results of a numerical calculation of gas-dynamic shock waves that arise when solving the Cauchy problem with smooth periodic initial data are presented using three variants of the DG (Discontinuous Galerkin) method, in which the solution is sought in the form of a piecewise linear discontinuous function. It is shown that the methods DG1A1 and DG1A2, for which the Cockburn limiter with parameters A1 = 1 and A2 = 2 are used for monotonization, have approximately the same accuracy in the influence areas of shocks (arising as a result of gradient catastrophes within the computational domain), while the nonmonotonic DG1 method, in which this limiter is not used, has a significantly higher accuracy in these areas, despite noticeable non-physical oscillations on shocks. With this in mind, the combined scheme obtained by the joint application of the DG1 and DG1A1 methods monotonously localizes the shocks and maintains increased accuracy in the areas of their influence.

The effect of initial amplitude and convergence ratio on instability development and deposited fluctuating kinetic energy in the single-mode Richtmyer–Meshkov instability in spherical implosions

This paper investigates the growth of perturbations on the inner surface of a dense imploding spherical shell due to hydrodynamic instabilities. The perturbations change in amplitude due to Richtmyer–Meshkov instability and Rayleigh-Taylor instability, geometric convergence, and compressibility. Two mode numbers (ℓ=5,50) and three different perturbation amplitudes (a0=0.1λ,0.01λ,0.001λ) are applied to the surface of the dense inner shell. Two independent codes were used to perform simulations with these six perturbation profiles at four convergence ratios, ranging from 3.6 to 7.3, for a total of 48 cases. The mixing layer amplitudes show good agreement between the two simulation codes across the range of convergence ratios, mode numbers, and initial amplitudes. The growth of the mixing layer is employed to validate an extension of a recently proposed Bell-Plesset model, showing good agreement across the range of convergence ratios. Persistent and substantial shock-deposited fluctuating kinetic energy is observed within the light core, away from the perturbed interface. Temporal evolution of fluctuating kinetic energy indicates a time delay between the peak radial and theta directions, consistent with a build-up of vortical motion. A “jet” like phenomena is observed at the peaks and troughs of the initial perturbations in both simulation codes across a variety of cases. It is postulated that these occur due to the shape singularity shown to develop in spatially periodic perturbed planar shock waves in ideal gas dynamics. Significant anisotropy of kinetic energy components is present at all times.

An assessment of theories modeling vortex breakdown as a transition between cylindrical flow states

This study assesses the validity of two theories proposed to explain vortex breakdown occurring in swirling flows in pipes [Benjamin, J. Fluid Mech. 14, 593–629 (1962) and Wang and Rusak, J. Fluid Mech. 340, 177–223 (1997)]. Both model vortex breakdown as a steady, inviscid, streamwise transition between axisymmetric, cylindrical (streamwise-invariant) flow states, with the downstream “conjugate” state predicted differently by each based on the upstream inflow state. In this study, these conjugate solutions are computed for three distinct swirling inflow profiles by solving the Bragg–Hawthorne equation based on the inflow conditions. It is first shown that the “adjacent” conjugate solution proposed by Benjamin exhibits stronger flow reversal when the inflow swirl strength is decreased. This is in direct contradiction to trends observed in experiments, indicating that this aspect of the theory is invalid. Following this, the “global minimizer” conjugate solution proposed by Wang and Rusak is examined. In addition to numerical computations, an analytic expression for this conjugate solution is derived for the case of a Rankine vortex as the inflow. For various inflow profiles, it is shown that these conjugate solutions exhibit many trends similar to those observed in experiments. However, the results also indicate that the stagnation zone associated with these solutions expands radially in an unbounded fashion in the absence of confinement effects, implying that viscous effects might play a crucial role in limiting the radial expansion of the flow. Finally, based on these results and the inverse relationship between the swirl parameter and Mach number, it is argued that modeling vortex breakdown as directly analogous to a gasdynamic normal shock wave is erroneous.

Structural stability of shock waves in 2D compressible elastodynamics

We study the two-dimensional structural stability of shock waves in a compressible isentropic inviscid elastic material in the sense of the local-in-time existence and uniqueness of discontinuous shock front solutions of the equations of compressible elastodynamics in two space dimensions. By the energy method based on a symmetrization of the wave equation and giving an a priori estimate without loss of derivatives for solutions of the constant coefficients linearized problem we find a condition sufficient for the uniform stability of rectilinear shock waves. Comparing this condition with that for the uniform stability of shock waves in isentropic gas dynamics, we make the conclusion that the elastic force plays stabilizing role. In particular, we show that, as in isentropic gas dynamics, all compressive shock waves are uniformly stable for convex equations of state. Moreover, for some particular deformations (and general equations of state), by the direct test of the uniform Kreiss–Lopatinski condition we show that the stability condition found by the energy method is not only sufficient but also necessary for uniform stability. As is known, uniform stability implies structural stability of corresponding curved shock waves.

Self-Similarity Solution of Cylindrical MGD Shock Waves- Effect of Monochromatic Radiation

By using self similar method, propagation of cylindrical magneto gas dynamic shock waves in existence of magnetic flux under the effect of monochromatic radiation with variable density has been investigated. The equations of motion have been solved by using self similar solutions and numerical solutions are obtained subject to the boundary conditions. It is found that existence of magnetic field and monochromatic radiation decay the shock wave. All the analytical results are represented graphically.

Shock Waves: From Gas Dynamics to Granular Flows

This short article briefly discusses some aspects in shock wave studies in recent years, in particular on the difference between gas dynamics and granular flow problems. It compares the relations of oblique shock waves, where weak, strong and detached shock waves can be observed in both gas dynamic and granular conditions. If the upstream Froude number of granular flow becomes infinitely large a granular shock wave would still remain attached and oblique around a wedge angle near 90°, however an attached gas dynamic shock wave is limited by a maximum wedge angle, say, of 30°. On the other hand, the shock standoff distance for a detached granular shock wave tends to become infinitely small with the increase of the upstream Froude number since it is associated with the flow height ratio across the shock wave.

A geometric singular perturbation approach for planar stationary shock waves

The non-linear non-equilibrium nature of shock waves in gas dynamics is investigated for the planar case. Along each streamline, the Euler equations with non-equilibrium pressure are reduced to a set of ordinary differential equations defining a slow–fast system, and geometric singular perturbation theory is applied. The proposed theory shows that an orbit on the slow manifold corresponds to the smooth part of the solution to the Euler equation, where non-equilibrium effects are negligible; and a relaxation motion from the unsteady to the steady branch of the slow manifold corresponds to a shock wave, where the flow relaxes from non-equilibrium to equilibrium. Recognizing the shock wave as a fast motion is found to be especially useful for shock wave detection when post-processing experimental measured or numerical calculated flow fields. Various existing shock detection methods can be derived from the proposed theory in a rigorous mathematical manner. The proposed theory provides a new shock detection method based on its non-linear non-equilibrium nature, and may also serve as the theoretical foundation for many popular shock wave detection techniques.

New Self Similar Solution of Spherical Radiative Gasdynamic Shock Wave With Increasing Density Under The Effect Of Axial And Azimuthal Magnetic Field

In the present chapter we investigated the self similar flow behind a spherical shock wave propagating in a medium with increasing density, in the presence of magnetic fields. The medium is assumed to be non gravitational due to the heavy nucleus at origin. The medium ahead and behind the shock front are assumed to be inviscid. The initial density of gas is assumed to vary as some power of distance. It is assumed that gas is grey and opaque. The assumption of optically thick grey gas is physically consistant with the neglect of radiation pressure and radiation energy. Total energy of the flow field behind the spherical shock is assumed to be increasing with time, where the gas ahead of the shock is assumed to be at rest. The results of numerical calculations were shown in the form of graphs. A complete study was made for axial and azimuthal magnetic field. Also the effect of variation of initial density behind the shock, shock velocity and respective magnetic fields were investigated.

On the modified dispersion-controlled dissipative (DCD) scheme for computation of flow supercavitation

In the present study, a robust scheme initially proposed for capturing shock waves in gasdynamics, the dispersion-controlled dissipative (DCD) scheme is modified to simulate supercavitating flows in the framework of one-fluid model. Due to the stiffness of the equations of state (EOS), the Steger-Warming method becomes inapplicable and the Lax–Friedrichs method should be used for the numerical flux splitting. Following the formulation, the updated scheme is validated by a series of comparisons with theoretical and experimental data. The validation shows that the modified DCD scheme performs very well for supercavitating flows in the presence of steep gradients of both density and EOS. Several one-fluid cavitation models and speed of sound (SoS) models are compared for the numerical investigation of supercavitating flows.

Conditional Stability of Unstable Viscous Shock Waves in Compressible Gas Dynamics and MHD

Extending our previous work in the strictly parabolic case, we show that a linearly unstable Lax-type viscous shock solution of a general quasilinear hyperbolic–parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small L 1 ∩ H 3 perturbations, converging time asymptotically to a translate of the unperturbed wave. That is, for a shock with p unstable eigenvalues, we establish conditional stability on a codimension-p manifold of initial data, with sharp rates of decay in all L p . For p = 0, we recover the result of unconditional stability obtained by Mascia and Zumbrun. The main new difficulty in the hyperbolic–parabolic case is to construct an invariant manifold in the absence of parabolic smoothing.

Hopf Bifurcation of Viscous Shock Waves in Compressible Gas Dynamics and MHD

Extending our previous results for artificial viscosity systems, we show, under suitable spectral hypotheses, that shock wave solutions of compressible Navier–Stokes and magnetohydrodynamics equations undergo Hopf bifurcation to nearby time-periodic solutions. The main new difficulty associated with physical viscosity and the corresponding absence of parabolic smoothing is the need to show that the difference between nonlinear and linearized solution operators is quadratically small in H s for data in H s . We accomplish this by a novel energy estimate carried out in Lagrangian coordinates; interestingly, this estimate is false in Eulerian coordinates. At the same time, we greatly sharpen and simplify the analysis of the previous work.

D215 超音速二相流ノズル出口に発生する膨張波と衝撃波(混合)

Two-phase flow nozzles are used in the total flow system for geothermal power plants and in the ejector of the refrigeration cycle, etc. There exist shock waves or rarefaction waves at the outlet of a supersonic nozzle of the ejector in the case of non-best fitting expansion conditions when the operation conditions of the nozzle are widely chosen. The purpose of the present study is to elucidate the character of the rarefaction waves and the shock waves at the outlet of the supersonic two-phase flow nozzle. In particular, this research treats only shock waves. Two-dimensional basic equations for the compressible two-phase flow are introduced considering the inter-phase momentum transfer. Two-dimensional oblique shock waves occurred in the high speed two-phase flows are calculated by the CIP method. Although the basic equations are controlled by the frozen Mach number, shock waves appear even for a frozen Mach number less than one. Angle of the oblique shock wave of two-phase flow can be calculated from the gas dynamic oblique shock wave relation, if sound speeds of two-phase flow may be used instead of that of gases.

Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics

A Bose-Einstein condensate (BEC) is a quantum fluid that gives rise to interesting shock wave nonlinear dynamics. Experiments depict a BEC that exhibits behavior similar to that of a shock wave in a compressible gas, eg. traveling fronts with steep gradients. However, the governing Gross-Pitaevskii (GP) equation that describes the mean field of a BEC admits no dissipation hence classical dissipative shock solutions do not explain the phenomena. Instead, wave dynamics with small dispersion is considered and it is shown that this provides a mechanism for the generation of a dispersive shock wave (DSW). Computations with the GP equation are compared to experiment with excellent agreement. A comparison between a canonical 1D dissipative and dispersive shock problem shows significant differences in shock structure and shock front speed. Numerical results associated with the three dimensional experiment show that three and two dimensional approximations are in excellent agreement and one dimensional approximations are in good qualitative agreement. Using one dimensional DSW theory it is argued that the experimentally observed blast waves may be viewed as dispersive shock waves.

Construction of uniform in time asymptotics for interaction of shock waves in gas dynamics

We construct uniform in time asymptotics which describes the interaction of two isothermal shock waves.

Dissipative Symmetrizers of Hyperbolic Problems and Their Applications to Shock Waves and Characteristic Discontinuities

By introducing the notations of dissipative and strictly dissipative p-symmetrizers of initial-boundary-value problems for linear hyperbolic systems we formalize the dissipative integrals technique [A. Blokhin, Yu. Trakhinin, in Handbook of Mathematical Fluid Dynamics, Vol. 1, North-Holland, Amsterdam, 2002, pp. 545-652] applied earlier to shock waves and characteristic discontinuities for various concrete systems of conservation laws. This enables us to prove the local in time existence of shock-front solutions of an abstract symmetric system of hyperbolic conservation laws, provided that the corresponding constant coefficients linearized problem has a strictly dissipative p-symmetrizer. Our result does not, in particular, require the fulfillment of Majda's block structure condition. A p-symmetrizer is, in some sense, a "secondary" Friedrichs symmetrizer for the symmetric system for the vector of p-derivatives of unknown functions, and the structure of p-symmetrizer takes into account (if applicable) the set of divergent constraints for the original system. After applying a p-symmetrizer, which is in general a set of matrices and vectors, the boundary conditions for a resulting symmetric system are dissipative (or strictly dissipative). We give concrete examples of p-symmetrizers. Our main examples are strictly dissipative 2-symmetrizers for shock waves in gas dynamics and magnetohydrodynamics. A general procedure for constructing a p-symmetrizer does not however exist. But, if it was somehow constructed, then we do not need to test the Lopatinski condition that is often connected with insuperable technical difficulties. As an illustration, we refer to the author's recent result [Yu. Trakhinin, Arch. Ration. Mech. Anal., 177 (2005), pp. 331-366] for compressible current-vortex sheets for which the construction of a dissipative 0-symmetrizer has first enabled the finding of sufficient conditions for their weak linearized stability.

Interaction of shock waves in gas dynamics: Uniform in time asymptotics

We construct a uniform in time asymptotics describing the interaction of two isothermal shock waves with opposite directions of motion. We show that any smooth regularization of the problem implies the realization of the stable scenario of interaction.

Note on a paper by Robinet, Gressier, Casalis & Moschetta

In Robinet et al. (2000), an instability for shock waves in gas dynamics is exhibited. The fluid was assumed to obey the polytropic perfect gas pressure law. We show in this note that such an instability does not occur, as proved in the extensive work of Majda (1983). Two arguments are developed: we give first a mathematical result that is violated by the conclusions of Robinet et al. (2000). Then we detail the results of Robinet et al. (2000) and show why they do not yield any conclusions. Finally, we develop a general calculation and show that an instability cannot occur.

Experimental study of λ-phase transition induced by shock compression

The λ-phase transition induced by shock compression was experimentally investigated by using the superfluid shock tube facility. A compression shock wave is generated in He II by the impingement of a gas dynamic shock wave onto a He II free surface. The shock-compressed He II can be converted to He I crossing the λ-line in the case of initial He II temperature close to T λ. The resultant temperature variation has been measured with in-house-made superconductive temperature sensors. Furthermore, the visualization photographs were taken by a digital still camera with the Schlieren optical method. It is found that the magnitude of the entropy production Δ s induced by shock compression process highly depends on the temperature difference between the temperature in the shock compressed state and the lambda temperature as well as on the shock strength. In particular, Δ s becomes extremely large in the case where the λ-phase transition is induced.

Gasdynamic Analogies in Problems of the Oblique Interaction of MHD Shock Waves

Gasdynamic analogies are constructed for the oblique interaction of MHD shock waves (counter colliding or overtaking). These analogies fairly adequately describe the complex dependences of the gas dynamic parameters of the medium on the magnetic field strength and inclination. The complete gas dynamic analogy in which the MHD interaction is simulated by the interaction of two gas dynamic shock waves with Mach numbers calculated on the basis of the fast magnetosonic speeds adequately describe the state of the medium for weak and moderate magnetic fields. The “hybrid” model, in which the state behind the interacting shock wave is calculated from the MHD relations on discontinuities and the gas dynamic analogy is then used, gives satisfactory results in a stronger field.