Incident Shock Mach Number Research Articles

Numerical Investigation of Reflected Shock/Vortex Interaction near an Open-Ended Duct

The problem of planar reflected-shock/spiral-vortex interactions near an open-ended duct is considered. The reflected shock is formed after exiting the duct and impinging on a vertical wall that is located downstream of the duct. Two cases associated with the shock diffraction around the duct exit are investigated for understanding the detailed flow structure and acoustic waves due to the reflected shock/spiral-vortex interaction. One is a 90-deg diffraction; the other is a 180-deg diffraction. An Euler solver with a high-resolution scheme of weighted essential nonoscillation is used to study these complicated flow problems. The Euler solver is validated to be reasonably accurate by comparing the computed solution with experimental data. For the study of reflected-shock/spiral-vortex interactions, the techniques of computational shadowgraph, computational schlieren image, and computational interferometry are used. Detailed flow structures of reflected shock/vortex interactions are reported for different incident shock Mach numbers

Normal velocity freeze-out of the Richtmyer-Meshkov instability when a shock is reflected.

It is known that for some values of the initial parameters that define the Richtmyer-Meshkov instability, the normal velocity at the contact surface vanishes asymptotically in time. This phenomenon, called freeze-out, is studied here with an exact analytic model. The instability freeze-out, already considered by previous authors [K.O. Mikaelian, Phys. Fluids 6, 356 (1994), Y. Yang, Q. Zhang, and D.H. Sharp, Phys. Fluids 6, 1856 (1994), and A.L. Velikovich, Phys. Fluids 8, 1666 (1996)], is the result of a subtle interaction between the unstable surface and the corrugated shock fronts. In particular, it is seen that the transmitted shock at the contact surface plays a key role in determining the asymptotic behavior of the normal velocity at the contact surface. By properly tuning the fluids compressibilities, the density jump, and the incident shock Mach number, the value of the initial circulation deposited by the reflected and transmitted shocks at the material interface can be adjusted in such a way that the normal growth at the contact surface will vanish for large times. The conditions for this to happen are calculated exactly, by expressing the initial density ratio as a function of the other parameters of the problem: fluids compressibilities and incident shock Mach number. This is done by means of a linear theory model developed in a previous work [J.G. Wouchuk, Phys. Rev. E. 63, 056303 (2001)]. A general and qualitative criterion to decide the conditions for freezing-out is derived, and the evolution of different cases (freeze-out and non-freeze-out) are studied with some detail. A comparison with previous works is also presented.

A study of the impulse wave discharged from the exits of two parallel tubes

The twin impulse wave leads to very complicated flow fields, such as Mach stem, spherical waves, and vortex ring. The twin impulse wave discharged from the exits of the two tubes placed in parallel is investigated to understand the detailed flow physics associated with the twin impulse wave, compared with those in a single impulse wave. In the current study, the merging phenomena and propagation characteristics of the impulse waves are investigated using a shock tube experiment and by numerical computations. The Harten-Yee’s total variation diminishing (TVD) scheme is used to solve the unsteady two-dimensional compressible Euler equations. The Mach number M s of incident shock wave is changed below 1.5 and the distance between two-parallel tubes, L/d, is changed from 1.2 to 4.0. In the shock tube experiment, the twin impulse waves are visualized by a Schlieren optical system for the purpose of validation of computational work. The results obtained show that on the symmetric axis between two-parallel tubes, the peak pressure produced by the twin impulse waves and its location strongly depend upon the distance between two-parallel tubes, L/d and the incident shock Mach number, M s. The predicted Schlieren images represent the measured twin-impulse wave with a good accuracy.

Interferometric CT measurement of three-dimensional flow phenomena on shock waves and vortices discharged from open ends

Three-dimensional flow phenomena are observed in a shock tube experiment for shock waves and vortices discharged from a square open end and a pair of circular open ends by using an interferometric CT (Computed Tomography) technique. To take multidirectional finite-fringe interferograms for a three-dimensional flow field, we introduce a rotating duct model in the test section of the shock tube. The experiments are performed for incident shock Mach numbers 1.50 and 1.30 in nitrogen gas of 98 kPa initial pressure. Good quality CT images of the density distribution are obtained by carefully selecting the projection images for a reproducible flow within a prescribed accuracy. The three-dimensional flow features are clearly visualized for a vortex ring, a secondary shock wave, shock-vortex interaction, and shock-shock interaction through the pseudo-color image of density distribution and the isopycnic surface. Their meanings in gas dynamics and shock dynamics are discussed. We demonstrate that studies of various three-dimensional problems in shock dynamics are possible using the present CT technique.

Unsteady convective surface heat flux measurements on cylinder for CFD code validation

The surface convective heat transfer rates to a cylinder have been measured using platinum thin film gauges in a shock tube and the results have been used to validate two numerical codes. The investigations have been carried out at different incident shock Mach numbers and flow Reynolds numbers. The measured and simulated results give an insight into the transient flow fields around the model in the shock tube and are a valuable means of complementing the numerical and experimental techniques used in this study.

Mach number effects on shock-bubble interaction

An investigation of Mach number effects on the interaction of a shock wave with a cylindrical bubble, is presented. We have conducted simulations with the Euler equations for various incident shock Mach numbers (\(M_S\)) in the range of \( 1.22 \le M_S \le 6\), using high-resolution Godunov-type methods and an implicit solver. Our results are found in a very good agreement with previous investigations and further reveal additional gasdynamic features with increasing the Mach number. At higher Mach numbers larger deformations of the bubble occur and a secondary-reflected shock wave arises upstream of the bubble. Negative vorticity forms at all Mach numbers, but the “c-shaped” vortical structure appeared at \(M_S=1.22\) gives its place to a circular-shaped structure at higher Mach numbers. The computations reveal that the (instantaneous) displacements of the upstream, downstream and jet interfaces are not significantly affected by the incident Mach number for values (approximately) greater than \(M_S=2.5\). With increasing the incident Mach number, the speed of the jet (arising from the centre of the bubble during the interaction) also increases.

Growth rate of the Richtmyer–Meshkov instability when a rarefaction is reflected

A model is presented that calculates the asymptotic growth rate of the linear Richtmyer–Meshkov instability when a rarefaction is reflected at the contact surface. The result is valid for any value of the incident shock Mach number and initial fluids parameters. There is very good agreement with previous numerical simulations and experiments done at high compressions. The technique developed in the model is seen to be highly accurate and allows us a fast evaluation of the asymptotic normal velocity at the interface.

Analytical Predictions of Strong Pseudo-Steady Mach Reflections for the Polyatomic Gas: SF6

ABSTRACTStrong pseudo-steady Mach reflections in sulfur hexafluoride (SF6) are analyzed using the three-shock and local three-shock theories, where both the vibrationally-frozen (γ = 1.333) and -equilibrated (γ = 1.093) perfect-gas models are used to compare with existing experiments. The ranges of the incident shock Mach number and reflecting wedge angle studied are 1.49 ≤ Ms ≤ 5.95 and 10° ≤ θω ≤ 42°, respectively. It is found that predicted angles between the incident and reflected shocks from the local three-shock theory using the vibrationally-equilibrated fictitious perfect-gas model (i.e., γ = 1.093) agree closely with those, currently available in literature, measured experimentally; while these predicted angles obtained using the vibrationally-frozen perfect-gas model (i.e., γ = 1.333) differ significantly from the existing experiments. Taking the convex Mach stem curvature at the triple point into consideration, it is shown that both the triple point trajectory angle and the angle between the incident and reflected shocks of strong pseudo-steady Mach reflections in SF6 can be more accurately determined for wide ranges of Ms and θω from the three-shock theory using the vibrationally-equilibrated fictitious perfect-gas model than those without considering this effect.

Non-self-similar behavior of the von Neumann reflection

The purpose of the present paper is to consider the von Neumann reflection (vNR) which takes place for a small incident shock Mach number and a small wedge angle. A series of experiments has been performed with ordinary smooth straight wedges and step-like wedges that simulate the former. The reflection configuration over the step-like wedge has suggested unsteady characteristics of the vNR. Contrary to the established notion of shock reflection phenomena over straight wedges, while the triple point trajectory was approximately a straight line through the apex of the wedge, the vNR showed unsteadiness in the relation between angles of incidence and reflection (ωi,ωr), and thus the flow-field proved to be non-self-similar near the triple point. Based on the measured values, the incident angle for the reflected wave and the Mach number of the flow ahead of the reflected wave were estimated. These values show that the reflected wave is not a Mach wave, but it moves on the (ωi,ωr)-plane almost along a trivial solution. In particular, the flow Mach number ahead of the reflected wave approaches unity, which leads to analytical formulas for angles of incidence and reflection as functions of the incident shock Mach number only. The reflected wave degenerates to a normal Mach wave as the incident shock proceeds. In the case of the vNR, the von Neumann paradox takes place only at an early stage of reflection, and the paradox is resolved later, since the flow properties such as angles of incidence and reflection are given by the trivial solution which is a particular solution for the three-shock theory.

A simple physical theory of weak Mach reflection over plane surfaces

It is shown that simple physical principles coupled with the inviscid shock jump relations can be applied to the problem of weak Mach reflection to the extent that the triple point path can be predicted from the incident shock Mach number \(M_{\rm i}\), gas specific heat ratio \(\gamma\) and the inclination angle \(\theta_{\rm w}\) of the reflecting surface to the shock normal. Comparison with the Euler code data and with experiments show close agreement for conditions both far and close to transition and that the general shape of the reflected and Mach stem shocks follow simple curves except in the neighbourhood of the triple point. The conflict at the triple point in matching the flow deflection angles and pressures across the contact discontinuity remains. It is shown however that the simple model presented here gives a close match to the cfd and experimental overall shock and contact surface shapes although it cannot predict these or the flow properties in any detail.

Numerical investigation of axisymmetric shock wave focusing over paraboloidal reflectors

The problem of a plane shock wave incident to a paraboloidal reflector is numerically investigated. The numerical solver used is developed by an improved, implicit, upwind total variation diminishing scheme in a finite-volume approach. The real-gas effect is taken into account if high temperature occurs. The solver is validated on four test problems. The complicated flow fields of axisymmetric shock wave focusing for different-depth reflectors at various incident shock Mach numbers are studied. An interesting result of a maximum pressure happening at the reflector center is found. This is due to the occurrence of an implosion phenomenon. A maximum temperature might occur at the reflector center or at other locations, depending on the incident shock Mach number and the reflector depth. Moreover, vortical flows induced by shock wave focusing and their formation mechanism are explored. It was found that the vortices near the reflector are caused by a ring-shaped shock/slipline interaction. Owing to the slipline on the symmetry axis, a jet flow is induced, resulting in the formation of vortices near the symmetry axis.

Self-similar hypervelocity shock interactions with oblique contact discontinuities

The two-dimensional inviscid refraction of a shock wave at an oblique contact discontinuity is self-similar; i.e. it depends only upon the variables \(\xi\equiv x/t\), \(\eta\equiv y/t\). We transform the compressible Euler equations into the self-similar (\(\xi,\eta\)) coordinates and solve the resulting boundary-value problem by an implicit equilibrium flux method. We present results for strong shock (\(M\ge10\) where M is the incident shock Mach number) interactions with an oblique contact discontinuity separating an inert gas from a gas which exhibits high-temperature gas chemistry effects. To model high-temperature effects we employ Lighthill's ideal dissociating gas (IDG) model. Comparison between the frozen and equilibrium limits, both of which are self-similar, indicate large changes in peak density and temperatures. Significant differences in the overall flow pattern between the frozen and equilibrium limits are observed for interfaces with low negative Atwood ratio and high positive Atwood ratio. Results from a local analysis are presented when the shock refraction is regular. The critical angle at which the transition from regular to irregular refraction occurs is slightly larger for the equilibrium chemistry case.

Numerical Study of the Shock Wave Propagating through the Plenum Chamber with Borda's Mouthpiece.

In this paper, the shock wave propagating through the plenum chamber with Borda's mouthpiece was studied by numerical simulation. Computations were carried out by solving the two-dimensional compressible Navier-Stokes equations by using the total variation diminishing (TVD) scheme. Computations were performed for five types of the plenum chamber with Borda's mouthpiece [coaxial mouthpiece, off axis mouthpiece, oblique edge mouthpiece, no mouthpiece (coaxis, off axis)] and two incident shock Mach numbers (Ms=1.5, 2.0). The flow fields were numerically visualized by the pressure and vorticity contours, velocity vectors and sonic lines. The pressure distributions and time histories of the pressure contours in the wall and the axis of the duct were also studied, and the attenuating process of the shock front and the diminishing process of the nonuniformity of the shock front were explored.

Numerical Study of Shock Waves Propagating in Bmanched Ducts. The Generating and Diminishing Process of the Nonuniformity of the Shock Front.

In this paper, the generating process and the diminishing process of the nonuniformity regarding the strength and the shape of the shock wave front propagating through branched ducts were explored by numerical simulation. Computations were carried out by solving the two-dimensional compressible Navier-Stokes equations by using the total variation diminishing (TVD) scheme. Computations were performed for six types of branch (90° branch, 1/2 size 90° branch, T-shape (1/1, 1/2, 1/3 size) branch, 45° branch) and two incident shock Mach numbers (Ms=1.3, 2.0). The flow field were numerically visualized by the pressure and vorticity contours, velocity vectors, pressure distributions on the walls and time histories of the pressure contours on the walls. The mechanism of the generating process and the diminishing process of the nonuniformity of propagating shock front was clarified.

The formation of a secondary shock wave behind a shock wave diffracting at a convex corner

This paper deals with the formation of a secondary shock wave behind the shock wave diffracting at a two-dimensional convex corner for incident shock Mach numbers ranging from 1.03 to 1.74 in air. Experiments were carried out using a 60 mm $\times$ 150 mm shock tube equipped with holographic interferometry. The threshold incident shock wave Mach number ( $M_s$ ) at which a secondary shock wave appeared was found to be $M_s$ = 1.32 at an 81° corner and $M_s$ = 1.33 at a 120° corner. These secondary shock waves are formed due to the existence of a locally supersonic flow behind the diffracting shock wave. Behind the diffracting shock wave, the subsonic flow is accelerated and eventually becomes locally supersonic. A simple unsteady flow analysis revealed that for gases with specific heats ratio $\gamma = 1.4$ the threshold shock wave Mach number was $M_s$ = 1.346. When the value of $M_s$ is less than this, the vortex is formed at the corner without any discontinuous waves accompanying above the slip line. The viscosity was found to be less effective on the threshold of the secondary shock wave, although it attenuated the pressure jump at the secondary shock wave. This is well understood by the consideration of the effect of the wall friction in one-dimensional duct flows. In order to interpret the experimental results a numerical simulation using a shock adaptive unstructured grid Eulerian solver was also carried out.

COMPUTATION OF NON-STATIONARY SHOCK-WAVE/CYLINDER INTERACTION USING ADAPTIVE-GRID METHODS

A numerical study of shock wave propagation over a cylinder is presented. Hybrid, structured-unstructured adaptive grids are employed for the numerical solution of the Euler and Navier-Stokes equations. A second-order Godunov-type scheme is employed for the discretization of the inviscid fluxes, while central differences are used for the viscous terms. The time integration is obtained by a second-order prediction-corrector scheme. Simulation of the unsteady gasdynamic phenomena around the cylinder is obtained at different incident-shock Mach numbers. The shock-wave diffraction over the cylinder is investigated by means of various contour plots, as well as pressure and skin friction distributions. The calculations reveal that the Euler solutions are very close to the Navier-Stokes ones in the first half of the cylinder, but large differences between the two solutions exist in the second half of the cylinder and the wake of the flow field, where strong viscous-inviscid interaction occurs.

Development of a Diaphragmless shock Tunnel Using a Butterfly Valve.

A diaphragmless shock tunnel is developed using a butterfly valve, which has generally lower energy loss than that of a piston valve. The valve and its driving unit can be made from commercially available parts. Incident shock Mach numbers and stagnation pressures at the barrel end were measured and compared with those obtained when an aluminum diaphragm was used. The faster the velocity of opening of the valve (or the diaphragm), the larger the incident shock Mach number and the stagnation pressure. Moreover, uniformity of hypersonic flow in the test section was ensured by Mach number distribution based on Pitot pressure measurement. Those results show the feasibility of the valve mechanism.

A holographic interferometric study of shock wave focusing in a circular reflector

A holographic interferometric study was made of the focusing of reflected shock waves from a circular reflector. A diaphragmless shock tube was used for incident shock Mach numbers ranging from 1.03 to 1.74. Hence, the process of reflected shock wave focusing was quantitatively observed. It is found that a converging shock wave along the curved wall undergoes an unsteady evolution of mach reflection and its focusing is, therefore, subject to the evolution of the process of shock wave reflections. The collision of triple points terminates the focusing process at the geometrical focus. In order to interprete quantitatively these interferograms, a numerical simulation using an Eulerian solver combined with adaptive unstructured grids was carried out. It is found numerically that the highest density appears immediately after the triple point collision. This implies that the final stage of focusing is mainly determined by the interaction between shock waves and vortices. The interaction of finite strength shock waves, hence, prevents a curved shock wave from creating the infinite increase of density or pressure at a focal point which is otherwise predicted by the linear acoustic theory.

224 干渉縞画像処理システムによる軸対称衝撃波の観測

In order to obtain density variations in axisymmetrical flow field, an image-processing system for analyzing interferograms is developed. The system consists of a Mach-Zehnder interferometer, a nitrogen pulse laser, and a CCD-camera. Images are processed by personal computers. Axisymmetrical flow fields are generated by shock waves, which are dischargedfrom an open end, and observed by using the present system of the image-processing. The gas is nitrogen, the intial pressures are 49 to 196 kPa, and the incident shock Mach numbers are 1.2 to 2.0. The observed density distributions are found to be in good agreements with the CFD ones.

Numerical simulation of shock wave focusing over parabolic reflectors

The problem of a plane shock wave that propagates in an air media and then is reflected from a parabolic concave reflector and focuses at some region is considered. The shock focusing can greatly magnify the pressure and the temperature. The purpose of this study is to numerically simulate the shock focusing process of the reflection of shock waves from the parabolic reflectors with different depths and to analyze their associated flow fields in detail. The present solver developed is to solve the Euler equations using an improved, implicit, upwind Total Variation Diminishing scheme in a finite-volume approach. The effects of reflectors with different depths and of the incident shock Mach numbers on shock focusing are investigated. The real-gas effect is taken into account through a proper correction of the specific heat ratio of air, when high temperature occurs due to shock focusing.