Shock Reflection Research Articles

Aerodynamic interference between high-speed slender bodies

Under supersonic flow conditions, slender bodies in close proximity induce aerodynamic interference effects. This paper aims to quantify the magnitude of the resulting interference loads and to understand the underlying flow-physics mechanisms that cause them. A pair of identical slender bodies are investigated through a series of wind-tunnel experiments and supporting computational fluid dynamics (CFD) predictions. The bodies induce a complex interference flowfield, which tends to be bespoke to each configuration. The flow features include impinging shock and expansion waves, conical shock reflections, strong skewing of the boundary-layer flows and shock diffraction. The effects of axial stagger, lateral separation and the strength of the primary disturbance flow field are evaluated. The interference loads are found to be most sensitive to the initial location of the primary disturbance but are also affected by its strength. In addition, maximum interference loads which equate to an effective incidence of up to 6° are observed. Finally, very good agreement is found between the measurements and the CFD predicted normal force and pitching moment.

Generation of radiative knots in a randomly pulsed protostellar jet

HH objects are characterized by a complex knotty morphology detected mainly along the axis of protostellar jets in a wide range of bands. Evidence of interactions between knots formed in different epochs have been found, suggesting that jets may result from the ejection of plasma blobs from the source. We aim at investigating the physical mechanism leading to the irregular knotty structure observed in jets in different bands and the complex interactions occurring among blobs of plasma ejected from the stellar source. We perform 2D axisymmetric HD simulations of a randomly ejected pulsed jet. The jet consists of a train of blobs which ram with supersonic speed into the ambient medium. The initial random velocity of each blob follows an exponential distribution. We explore the ejection rate parameter to derive constraints on the physical properties of protostellar jets by comparison of model results with observations. Our model takes into account radiative losses and thermal conduction. We find that the mutual interactions of blobs ejected at different epochs and with different speed lead to a variety of plasma components not described by current models. The main features characterizing the random pulsed jet scenario are: single high speed knots, showing a measurable proper motion in nice agreement with observations; irregular chains of knots aligned along the jet axis and possibly interacting with each other; reverse shocks interacting with outgoing knots; oblique shocks produced by the reflection of shocks at the jet cocoon. All these structures concur to determine the morphology of the jet in different bands. We also find that the thermal conduction plays a crucial role in damping out HD instabilities that would develop within the cocoon and that contribute to the jet breaking.

SELF-SIMILAR SOLUTIONS FOR THE 2-D BURGERS SYSTEM IN INFINITE SUBSONIC CHANNELS

We establish the existence of weak solutions in an infinite subsonic channel in the self-similar plane to the two-dimensional Burgers system. We consider a boundary value problem in a fixed domain such that a part of the domain is degenerate, and the system becomes a second order elliptic equation in the channel. The problem is motivated by the study of the weak shock reflection problem and 2-D Riemann problems. The two-dimensional Burgers system is obtained through an asymptotic reduction of the 2-D full Euler equations to study weak shock reflection by a ramp.

Experimental Measurements of Starting Loads and Model Behaviors in the Indraft Supersonic Wind Tunnel

Measurements of starting load in the indraft supersonic wind tunnel of Muroran Institute of Technology were conducted for Mach 2, 3 and 4 conditions with the AGARD-B model. The high-speed photographs covering the behaviors of the wind tunnel model from the start to end of the operation were taken. Those photographs make clear that the oscillations of the model coincide with the measured starting load oscillation and starting loads were caused by two shock waves. The first shock wave is the reflection shock, generated at the nozzle throat by expansion wave reflection. The second wave is comprised asymmetric oblique shock waves (AOS) coming from upstream. AOS can generate asymmetric conical shock (ACS) around the nose cone of the model, which would have directly caused the starting loads on the wind tunnel model. Based on these observations, propose a conical shock theory, as an alternative starting load prediction theory instead of the normal shock theory.

Highly Efficient Lattice Boltzmann Model for Compressible Fluids: Two-Dimensional Case

We present a highly efficient lattice Boltzmann model for simulating compressible flows. This model is based on the combination of an appropriate finite difference scheme, a 16-discrete-velocity model [Kataoka and Tsutahara, Phys. Rev. E 69 (2004) 035701(R)] and reasonable dispersion and dissipation terms. The dispersion term effectively reduces the oscillation at the discontinuity and enhances numerical precision. The dissipation term makes the new model more easily meet with the von Neumann stability condition. This model works for both high-speed and low-speed flows with arbitrary specific-heat-ratio. With the new model simulation results for the well-known benchmark problems get a high accuracy compared with the analytic or experimental ones. The used benchmark tests include (i) Shock tubes such as the Sod, Lax, Sjogreen, Colella explosion wave, and collision of two strong shocks, (ii) Regular and Mach shock reflections, and (iii) Shock wave reaction on cylindrical bubble problems. With a more realistic equation of state or free-energy functional, the new model has the potential tostudy the complex procedure of shock wave reaction on porous materials.

Hyperbolic Conservation Laws

Hyperbolic Conservation Laws

ADMISSIBILITY OF SELF-SIMILAR WEAK SOLUTIONS OF SYSTEMS OF CONSERVATION LAWS IN TWO SPACE VARIABLES AND TIME

By means of an example, we postulate that the familiar local entropy conditions on discontinuities are far from sufficient to distinguish admissible weak solutions of systems of conservation laws in two space dimensions and time. We consider the familiar problem of the reflection of an incident shock by a wedge for the "nonlinear wave system", finding a plethora of self-similar weak solutions (or possibly approximate weak solutions) satisfying the usual entropy condition for this system. The multiplicity of such solutions arises from unneeded freedom in the algorithm for constructing solutions, and is directly related to a modest reduction in the assumed regularity of the solution in comparison to that assumed in previous work on this problem. We conclude that if the limit of vanishing viscosity indeed exists for such problems, an additional physical principle is needed to characterize admissible weak solutions.

Analysis of shock wave reflection from fixed and moving boundaries using a stabilized particle method

In the present paper, the efficiency of an enhanced formulation of the stabilized corrective smoothed particle method (CSPM) for simulation of shock wave propagation and reflection from fixed and moving solid boundaries in compressible fluids is investigated. The Lagrangian nature and its accuracy for imposing the boundary conditions are the two main reasons for adoption of CSPM. The governing equations are further modified for imposition of moving solid boundary conditions. In addition to the traditional artificial viscosity, which can remove numerically induced abnormal jumps in the field values, a velocity field smoothing technique is introduced as an efficient method for stabilizing the solution. The method has been implemented for one- and two-dimensional shock wave propagation and reflection from fixed and moving boundaries and the results have been compared with other available solutions. The method has also been adopted for simulation of shock wave propagation and reflection from infinite and finite solid boundaries.

Evolution, Propagation, Reflection, and Interactions of 1D-normal Shocks in Air and FC70 Using hpk Finite Element Computational Framework

In this paper, the mathematical models based on Navier-Stokes equations incorporating physical mechanism of viscosity and conductivity are utilized in h, p, k finite element computational framework to investigate 1-D normal shocks in air and FC70. In the case of air, shock evolution, propagation, repeated reflection and interactions are simulated using ideal gas law with constant viscosity and thermal conductivity. For FC70, numerical studies are presented for initial conditions inside the BZT zone as well as for those away from the BZT zone to investigate the possibility of compression as well as rarefaction shocks. The Van der Waals equation of state with constant transport properties is used for FC70. In all cases, “the rate of production of entropy per unit volume” is used to establish: (i) evolution of shock; (ii) sustained propagation; (iii) repeated reflections and subsequent sustained propagation; and (iv) interactions and sustained propagation of reflected shocks. In all numerical studies care is taken in choosing h, p, k computational parameters to ensure that the computational processes are free of measurable numerical dispersion and that the non-discretized form of the governing differential equations are satisfied accurately in the point wise sense and hence, ensuring time accuracy of the evolutions. 1-D Riemann shock tube is used as a model problem.

An improved CE/SE scheme for numerical simulation of gaseous and two-phase detonations

A new structure of solution elements and conservation elements based on rectangular mesh was proposed and an improved space-time conservation element and solution element (CE/SE) scheme with second-order accuracy was constructed. Furthermore, the application of improved CE/SE scheme was extended to detonation simulation. Three models were used for chemical reaction in gaseous detonation. And a two-fluid model was used for two-phase (gas–droplet) detonation. Shock reflections were simulated by the improved CE/SE scheme and the numerical results were compared with those obtained by other different numerical schemes. Gaseous and gas–droplet planar detonations were simulated and the numerical results were carefully compared with the experimental data and theoretical results based on C–J theory. Mach reflection of a cellular detonation was also simulated, and the numerical cellular patterns were compared with experimental ones. Comparisons show that the improved CE/SE scheme is clear in physical concept, easy to be implemented and high accurate for above-mentioned problems.

Surface pressure effects from shock wave impact on inclined and curved clothing

Previous studies of the impact of a blast or shock wave on a layer of textile positioned a short distance in front of a surface have shown that significant pressure amplification can occur. These studies have all dealt with the case of head-on impact on plane textile surfaces. This study extends such work to plane textile specimens inclined to the wave propagation direction, both for the textile freely suspended in order to examine the transmitted wave and with it positioned a short distance in front of a solid surface. Tests are also conducted with the textile suspended in a cylindrical shape positioned in front of a curved surface to approximate a more realistic practical loading case on a torso. Three textile types were used, varying significantly in permeability. In the oblique impact case it is found that the pressure amplification decreases as the wall angle decreases. The behaviour of the wave system in the gap between the textile layer and the body is clarified, showing a succession of oblique shock reflections and transmissions back through the textile layer. Tests on the cylindrical body surface at low Mach number show no significant amplification at the most forward position and steadily decreasing pressure around the sides of the surface. The concern regarding very large increases in pressure which arose from tests on head-on impact on plane surfaces appear to be ameliorated when the body is curved. Computer simulations are used to confirm the wave pattern for the cylindrical case by modeling a curved porous surface a short distance in front of a rigid wall.

Numerical study on spontaneous ignition of pressurized hydrogen release through a length of tube

The issue of spontaneous ignition of highly pressurized hydrogen release is of important safety concern, e.g. in the assessment of risk and design of safety measures. This paper reports on recent numerical investigation of this phenomenon through releases via a length of tube. This mimics a potential accidental scenario involving release through instrument line. The implicit large eddy simulation (ILES) approach was used with the 5th-order weighted essentially non-oscillatory (WENO) scheme. A mixture-averaged multi-component approach was used for accurate calculation of molecular transport. The thin flame was resolved with fine grid resolution and the autoignition and combustion chemistry were accounted for using a 21-step kinetic scheme. The numerical study revealed that the finite rupture process of the initial pressure boundary plays an important role in the spontaneous ignition. The rupture process induces significant turbulent mixing at the contact region via shock reflections and interactions. The predicted leading shock velocity inside the tube increases during the early stages of the release and then stabilizes at a nearly constant value which is higher than that predicted by one-dimensional analysis. The air behind the leading shock is shock-heated and mixes with the released hydrogen in the contact region. Ignition is firstly initiated inside the tube and then a partially premixed flame is developed. Significant amount of shock-heated air and well developed partially premixed flames are two major factors providing potential energy to overcome the strong under-expansion and flow divergence following spouting from the tube. Parametric studies were also conducted to investigate the effect of rupture time, release pressure, tube length and diameter on the likelihood of spontaneous ignition. It was found that a slower rupture time and a lower release pressure will lead to increases in ignition delay time and hence reduces the likelihood of spontaneous ignition. If the tube length is smaller than a certain value, even though ignition could take place inside the tube, the flame is unlikely to be sufficiently strong to overcome under-expansion and flow divergence after spouting from the tube and hence is likely to be quenched.

Diffuse holographic interferometric observation of shock wave reflection from a skewed wedge

The pattern of shock wave reflection over a wedge is, in general, either a regular reflection or a Mach reflection, depending on wedge angles, shock wave Mach numbers, and specific heat ratios of gases. However, regular and Mach reflections can coexist, in particular, over a three-dimensional wedge surface, whose inclination angles locally vary normal to the direction of shock propagation. This paper reports a result of diffuse double exposure holographic interferometric observations of shock wave reflections over a skewed wedge surface placed in a 100 × 180 mm shock tube. The wedge consists of a straight generating line whose local inclination angle varies continuously from 30° to 60°. Painting its surface with fluorescent spray paint and irradiating its surface with a collimated object beam at a time interval of a few microseconds, we succeeded in visualizing three-dimensional shock reflection over the skewed wedge surface. Experiments were performed at shock Mach numbers, 1.55, 2.02, and 2.53 in air. From reconstructed holographic images, we estimated critical transition angles at these shock wave Mach numbers and found that these were very close to those over straight wedges. This is attributable to the flow three-dimensionality.

Experiments on Guderley Mach reflection

Experiments have been conducted in a large shock tube to examine the four-wave shock reflection pattern, now known as Guderley reflection (GR). The fourth wave, an expansion, is clearly identified, as is the supersonic patch behind the reflected wave. A shocklet terminating the supersonic patch behind the reflected wave is identified, which forms a second triple point further down the Mach stem. Evidence is presented showing the presence of more than one expansion wave and more than one shocklet, thus indicating the existence of more than one supersonic patch. In order to distinguish between cases with a single patch without the shocklet as originally proposed by Guderley and found in some computations, and the indications of a multi-patch geometry found here, and also in other computations, this latter case is designated Guderley Mach reflection (GMR). Multi-exposure images of the shock propagation superimposed on a single image frame enable estimates to be made of the strength of the major waves, and it is shown that the reflected wave is very weak.

Numerical simulation of a single bubble by compressible two‐phase fluids

AbstractThe present work deals with the numerical investigation of a collapsing bubble in a liquid–gas fluid, which is modeled as a single compressible medium. The medium is characterized by the stiffened gas law using different material parameters for the two phases. For the discretization of the stiffened gas model, the approach of Saurel and Abgrall is employed where the flow equations, here the Euler equations, for the conserved quantities are approximated by a finite volume scheme, and an upwind discretization is used for the non‐conservative transport equations of the pressure law coefficients. The original first‐order discretization is extended to higher order applying second‐order ENO reconstruction to the primitive variables. The derivation of the non‐conservative upwind discretization for the phase indicator, here the gas fraction, is presented for arbitrary unstructured grids. The efficiency of the numerical scheme is significantly improved by employing local grid adaptation. For this purpose, multiscale‐based grid adaptation is used in combination with a multilevel time stepping strategy to avoid small time steps for coarse cells. The resulting numerical scheme is then applied to the numerical investigation of the 2‐D axisymmetric collapse of a gas bubble in a free flow field and near to a rigid wall. The numerical investigation predicts physical features such as bubble collapse, bubble splitting and the formation of a liquid jet that can be observed in experiments with laser‐induced cavitation bubbles. Opposite to the experiments, the computations reveal insight to the state inside the bubble clearly indicating that these features are caused by the acceleration of the gas due to shock wave focusing and reflection as well as wave interaction processes. While incompressible models have been used to provide useful predictions on the change of the bubble shape of a collapsing bubble near a solid boundary, we wish to study the effects of shock wave emissions into the ambient liquid on the bubble collapse, a phenomenon that may not be captured using an incompressible fluid model. Copyright © 2009 John Wiley & Sons, Ltd.

Shock and wave dynamics in cavitating compressible liquid flows in injection nozzles

Due to the exceptional high inlet pressures up to 2,000 bar flow dynamics and efficiency of modern injection systems are controlled by high frequency wave dynamics of the compressible liquid flow. Corresponding to alternating shock and expansion waves the liquid fluid evaporates and recondenses instantaneously. Here we present CFD simulations of the time accurate evolution of cavitating flows in 2-D plane and in six-hole injection nozzles with focus on the wave dynamics just after initialisation of the flow and within the time scale Δt ≤ 10−4 s of pilot and multi-point injection. Due to shock reflections at the bottom of the sack hole the instantaneous maximum pressure increases more than three times higher as compared with the prescribed pressure at the nozzle inlet. For instance, in case of an inlet pressure of 600 bar the maximum pressure in the sack and therefore ahead of the nozzle bore holes reaches about 2,100 bar. It is quite reasonable that this amplification of the pressure affects the evolution of the convective flow and therefore the mass flow through the nozzle bore holes.

Numerical Analysis of Droplet Impingement on Pipe Inner Surface Using a Particle Method

We investigated single droplet impingement to a solid wall by the MPS-AS method. The MPS-AS method was a unified particle method based on the MPS method for both compressible and incompressible flows. The speed and the profile of the shock wave from the impact point agreed well with the experimental result by Field et al. The shock reflection from the droplet surface was also captured. The same tendency as Rochester's experimental approximation was obtained with respect to the peak pressure.

Detonation diffraction through different geometries

We performed the study of the diffraction of a self-sustained detonation from a cylindrical tube (of inner diameter d) through different geometric configurations in order to characterise the transmission processes and to quantify the transmission criteria to the reception chamber. For the diffraction from a tube to the open space the transmission criteria is expressed by d c = k c ·λ (with λ the detonation cell size and k c depending on the mixture and on the operture configuration, classically 13 for alkane mixtures with oxygen). The studied geometries are: (a) a sharp increase of diameter (D/d > 1) with and without a central obstacle in the diffracting section, (b) a conical divergent with a central obstacle in the diffracting section and (c) an inversed intermediate one end closed tube insuring a double reflection before a final diffraction between the initiator tube and the reception chamber. The results for case A show that the reinitiation process depends on the ratio d/λ. For ratios below k c the re-ignition takes place at the receptor tube wall and at a fixed distance from the step, i.e. closely after the diffracted shock reflection shows a Mach stem configuration. For ratios below a limit ratio k lim (which depends on D/d) the re-ignition distance increases with the decrease of d/λ. For both case A and B the introduction of a central obstacle (of blockage ratio BR = 0.5) at the exit of the initiator tube decreases the critical transmission ratio k c by 50%. The results in configuration C show that the re-ignition process depends both on d/λ and the geometric conditions. Optimal configuration is found that provides the transmission through the two successive reflections (from d = 26 mm to D ch = 200 mm) at as small d/λ as 2.2 whatever the intermediate diameter D is. This configuration provides a significant improvement in the detonation transmission conditions.

Counterexamples to the Sonic Criterion

We consider self-similar (pseudo-steady) shock reflection at an oblique wall. There are three parameters: wall corner angle, Mach number, angle of incident shock. Ever since Ernst Mach discovered the irregular reflection named after him, it has been an open problem to predict precisely for what parameters the reflection is regular. Three conflicting proposals, the detachment, sonic and von Neumann criteria, have been studied extensively without a clear result. We demonstrate that the sonic criterion is not correct. We consider polytropic potential flow and prove that there is an open nonempty set of parameters that admit a global regular reflection with a reflected shock that is \emph{transonic}. We also provide a clear physical reason: the flow type (sub- or supersonic) is not decisive; instead the reflected shock type (weak or strong) determines whether structural perturbations decay towards the reflection point.

Regularity of solutions to regular shock reflection for potential flow

The shock reflection problem is one of the most important problems in mathematical fluid dynamics, since this problem not only arises in many important physical situations but also is fundamental for the theory of multidimensional conservation laws. However, most of the fundamental issues for shock reflection have not been understood. Therefore, it is important to establish the regularity of solutions to shock reflection in order to understand fully the phenomena of shock reflection. On the other hand, for a regular reflection configuration, the potential flow governs the exact behavior of the solution in $C^{1,1}$ across the pseudo-sonic circle even starting from the full Euler flow, that is, both of the nonlinear systems are actually the same in an physically significant region near the pseudo-sonic circle; thus, it becomes essential to understand the optimal regularity of solutions for the potential flow across the pseudo-sonic circle and at the point where the pseudo-sonic circle meets the reflected shock. In this paper, we study the regularity of solutions to regular shock reflection for potential flow. In particular, we prove that the $C^{1,1}$-regularity is optimal for the solution across the pseudo-sonic circle and at the point where the pseudo-sonic circle meets the reflected shock. We also obtain the $C^{2,\alpha}$ regularity of the solution up to the pseudo-sonic circle in the pseudo-subsonic region. The problem involves two types of transonic flow: one is a continuous transition through the pseudo-sonic circle from the pseudo-supersonic region to the pseudo-subsonic region; the other a jump transition through the transonic shock as a free boundary from another pseudo-supersonic region to the pseudo-subsonic region.