Hysteresis phenomenon in supersonic flow past a system of cylinders

The interaction between the bow shocks ahead of a system of bodies in a supersonic flow is numerically investigated. Flow past cylinders with parallel axes lying in the plane perpendicular to the flow direction is calculated. Three different shock interaction patterns are obtained in modeling flow past an infinite periodic lattice, namely, the regular and Mach-type regimes and a regime with flow choking. In the case of two bodies a collective bow shock is formed. Transition between the flow patterns with variation in the freestream Mach number and the spacing between the cylinders is studied. Regular-to-Mach-configuration transition and vice versa occurs at particular angles of inclination of the bow shock at the interaction point determined by the criterion of a maximum flow deflection angle and the von Neumann criterion, respectively.     

Collective Bow Shock Ahead of a Transverse System of Spheres in a Supersonic Flow Behind a Moving Shock Wave

The results of investigating the dynamics and physical conditions of formation of a collective bow shock ahead of a system of spheres with the line of centers normal to the supersonic flow behind a traveling shock wave are presented. Two types of shock-wave patterns that necessarily precede the formation of the collective shock wave and correspond to regular and Mach interaction of the bow shocks were detected experimentally. On the basis of a local gasdynamic-discontinuity interference theory, quantitative criteria of the existence of these regimes and of the formation of a common shock wave are determined. These criteria are confirmed in a series of experiments for the transitional regimes.     

Study of unsteady flow past bodies behind shock waves

Several theoretical and experimental studies have been devoted to the problem of the nonstationary action of the stream behind a shock wave on bodies of varied shape. In particular, in [1], the pressure and density are calculated for flow about bodies of the more typical shapes in the initial stage of the process. The basic relations which accompany the interaction of shock waves are considered in [2, 3]. The analysis of the phenomena of diffraction of shock waves on the sphere, cylinder, and cone is presented in [4]. Problems of unsteady flow about a wing are examined in [5, 6]. A detailed review of the foreign studies on unsteady flow is given in [7]. Of great practical interest is the question of the time for flow formation and the magnitudes of the unsteady loads during this period. Experimental investigations have been made recently [8, 9] in which some criteria are presented for estimating the bow shock formation time for supersonic flow about the sphere and cylinder with flat blunting. However the question of the formation time of the stationary pressure on the body surface is not referred to in these studies and no relationship is shown between the transient position of the reflected wave and the corresponding unsteady pressure on the surface. Moreover, in [8] the dimensionless time criterion is determined very approximately, independently of the Mach number of the shock wave. The present study was undertaken with the object of determining the basic criteria which characterize unsteady flow about bodies behind a plane shock wave which has time-independent parameters, and clarification of the shock wave reflected from the body and the pressure on the surface of the body during the transient period. The most typical body shapes were studied: 1) a cylinder with flat face aligned with the stream; 2) a spherically-blunted cylinder; and 3) a cylinder transverse to the stream. The experiments were conducted in a conventional shock tube using the single-diaphragm scheme. The measurements of the pressure on the models and the velocity of the incident shock wave were made using the technique analogous to that of [10, 11]. A highspeed movie camera was used to record the pattern of the wave diffraction on the body. The Mach number of the incident shock wave varied in the range from M=1.5 to M≈6.0, which corresponded to a range of Mach numbers M_∞ of the stream behind the shock wave from 0.6 to 2.1. The calculations of the required gas dynamic parameters for high temperatures were made with account for equilibrium dissociation of the air on the basis of the data of [10, 12, 13]. The magnitude of the relative maximal shock wave standoff Δ at the stagnation point obtained in the present experiments was compared with the values of Δ from other studies. In the case of the flat-blunted cylinder it was in good agreement with the results of [8–14], and in the case of the spherically-blunted cylinder and the transverse cylinder it was in agreement with the results of [15].     

Unsteady reflection of a shock wave from a body with a cylindrical recess

In a number of cases of supersonic flow past bodies with recesses pulsations in the flow arise [1–3]. Experiments [4, 5] indicate that stabilization of the steady supersonic flow past the body with a recess on which a shock wave is incident takes place after a series of oscillations of the bow wave. Numerical calculation of the interaction of a supersonic jet with a cylindrical cavity [6] reveals that damped pressure pulsations arise inside the cavity if the jet is homogeneous, and undamped pulsations it is inhomogeneous. The authors explain the damping of the pulsations by the influence of artificial viscosity. This paper investigates experimentally and theoretically (by numerical methods) the oscillations of the bow shock wave and the parameters of the flow behind it in the case of unsteady reflection of a shock wave from a body with a cylindrical recess turned towards the flow. The problem is posed as follows. A plane shock wave with constant parameters impinges on a cylinder with a cavity. The unsteady flow originating from this interaction is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 199–202, September–October, 1984.     

Transition between the regular and Mach shock interaction regimes initiated by a temperature disturbance

The results of searching numerically temperature disturbances which make it possible to convert flow from the steady-state Mach to regular shock interaction when both regimes may coexist simultaneously are given. The flow sensitivity to the temperature variations is calculated by solving the adjoint equations. The control perturbation is sought using gradient methods. The numerical experiments show that transition from the regular to Mach regime can be realized over the Mach number range from 3.45 to 6 by means of an increase in the temperature. Transition from the Mach to regular regime requires to decrease the temperature, it is possible for moderate Mach numbers and cannot be realized for the higher Mach numbers.     

Investigation into the interaction of a shock wave with an acute cone

A numerical investigation has been made into the nonstationary axisymmetric flow which arises from the interaction of a shock wave with a fixed acute cone. A solution has been obtained in self-similar coordinates for regimes in which a shock wave attached to the point of the cone is formed in the gas flow. The region of existence of these regimes has been established and the method of stabilization used to calculate the gas-dynamic functions in the disturbed region. The main attention is devoted to studying the transition from regular reflection to Mach reflection, a study of the features of double Mach reflection, and comparison of the results for the axisynuaetric and two-dimensional cases. The pressure distributions along the surface of the cone are given in a form which makes it possible to obtain dependences valid in a wide range of variation of the intensities of the incident shock wave and for different values of the adiabatlc exponent of the gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 98–104, May–June, 1980.     

Transition between regular and Mach reflection of shock waves: new numerical and experimental results

New numerical and experimental results on the transition between regular and Mach reflections of steady shock waves are presented. The influence of flow three-dimensionality on transition between steady regular and Mach reflection has been studied in detail both numerically and experimentally. Characteristic features of 3D shock wave configuration, such as peripheral Mach reflection, non-monotonous Mach stem variation in transverse direction, the existence of combined Mach-regular-peripheral Mach shock wave configuration, have been found in the numerical simulations. The application of laser sheet imaging technique in streamwise direction allowed us to confirm all the details of shock wave configuration in the experiments. Close agreement of the numerical and experimental data on Mach stem heights is shown. Received 23 November 2000 / Accepted 25 April 2001     

Vortical regime of the flow behind the bow shock wave

A new nonstationary regime of the flow around a step and a cylinder was found to exist at high free-stream Mach numbers for gas specific heat ratios below 1.2. The main features of the flow are strong vortices in the shock-compressed region with supersonic reversal velocities at the body face. The bow shock wave takes on a complicated shape, fluctuating in time. The vortical regimes can result from local heterogeneities in the free stream. The case of the heterogeneity is studied in this paper in the form of a thin thermal layer of limited length. The vortical regime remains in existence after the source of disturbances is removed. The results have been obtained through computer simulations through the use of Eulerian hydrodynamic equations and by way of several numerical methods: FLIC, Godunov's scheme, TVD, and PPM. The influence of viscosity on the development of the vortical regime has been studied by computer solving the Navier–Stokes equations. Received 21 August 1998 / Accepted 6 June 2001     

Effects of a Single-pulse Energy Deposition on Steady Shock Wave Reflection

The effects of energy deposition in the free stream on steady regular and Mach shock wave reflections are studied numerically. A short-duration laser pulse is focused upstream of the incident shock waves. It causes formation of the expanding blast wave and the residual hot-spot interacting in a complex way with the steady shock wave reflection. It was found that the laser energy addition in the free stream may force the transition from regular to Mach reflection in the dual solution domain. In contrast to previously reported numerical results, the transition from Mach to regular reflection has not been reproduced in our refined computations since the Mach reflection is restored after the flow perturbation.     

Supersonic low-concentration dusty-gas flow past a plane cylinder in the presence of an oblique shock wave interacting with the bow shock

The motion of an inertial dispersed admixture near a plane cylinder immersed in a steady-state hypersonic dusty flow in the presence of an oblique shock wave interacting with the bow shock is considered. It is assumed that the free-stream particle mass concentration is small and the particles do not affect the carrier flow. The III and IV shock wave interaction regimes are considered. The gas flow parameters in the shock layer are calculated from the numerical solution of the full Navier-Stokes equations for the perfect gas. A TVD second-order finite-difference scheme constructed on the basis of a finite volume method is used. For calculating the dispersed-phase parameters, including the concentration, the full Lagrangian method is used. On a wide range of variation of the particle inertia parameters, the patterns of the particle trajectories, velocity, concentration, and temperature in the shock layer are studied. The possibility of aerodynamic focusing of the particles behind the shock wave intersection point and the formation of narrow beams with a high particle concentration is revealed. These beams impinge on the cylinder surface and result in a sharp increase in the local heat fluxes. The maximal possible increase in the heat fluxes caused by the particles colliding with the cylinder surface is estimated for the flows with and without the incident oblique shock wave.     

Shock wave diffraction in the presence of a supersonic co-flow jet

The interaction between a diffracting shock wave and a uniform jet is a case that so far has only been partially investigated. This interaction is extremely important for the control of noise generation and improvement of combustor performance. To fill this knowledge gap, three geometries of the diffracting corner, namely a straight ramp, a serrated ramp, and a rounded corner, have been tested experimentally to study the interaction of shock diffraction with a supersonic co-flow jet at incident Mach numbers of 1.31 and 1.59, with Reynolds numbers of \(1.08\times 10^{6}\) and \(1.68\times 10^{6}\), respectively. Schlieren photography was employed to analyse the evolution of the flow phenomena. The aim is to provide a qualitative understanding of the interaction between the diffracting shock wave and the uniform jet relevant to future high-speed transport. The results show that the flow field evolves more rapidly and develops stronger structures for a higher shock Mach number. The diffraction around a rounded splitter develops a periodical vortical structure which continues after the disturbance introduced by the passage of the shock wave is removed.     

Gas content factor in the interaction of shock waves of different intensity in a two-phase gas-liquid medium

The transition from regular to Mach interaction is investigated in connection with the interaction of two plane weak or moderate shock waves of different intensity in a two-phase gas-liquid medium over the entire range of gas contents. A nonmonotonic dependence of the transition limit and the flow parameters on the gas content is detected. The investigation extends the results of [1] corresponding to the reflection of a shock wave from a wall. At intermediate gas contents in the case of opposing shock waves, analogous to the normal reflection of a shock wave from a solid wall, the results are in agreement with [2]. In the case of weak shock waves non-linear asymptotic expansions [3] are employed. In the extreme cases of single-phase media the results coincide with the findings of [3, 4]. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 172–174, November–December, 1986.     

An Embedding Method for Bluff Body Flows: Interactions of Two Side-by-Side Cylinder Wakes

We introduce a simple method for the numerical simulation of bluff body flows where the solid object is represented by a distributed body force in the Navier–Stokes equations. The body force density is found at every time step to reduce the velocity within the computational cells occupied by the rigid body to a prescribed value. The method combines certain ideas from the immersed boundary method which was developed to treat biofluid mechanical flows and the volume-of-fluid method for simulating flows with fluid–fluid interfaces. The main advantage of this embedding method is that the computations can be effected on a regular Cartesian grid, without the need to fit the grid to the bluff body surfaces. Thus, flow past several complex bodies can be treated as easily as flow past a single body. The method is validated by reproducing well-established results for vortex shedding from a stationary cylinder. The flow past two side-by-side cylinders is then investigated. When the distance between the cylinders is small, they are seen to shed vortices in-phase, whereas for larger distances, the shedding occurs in anti-phase. For intermediate distances, various shedding patterns are observed, including quasi-periodic, asymmetric and chaotic regimes. Mean values and phase portraits associated with the cylinder lift and drag coefficients, as well as spectral analysis of the same data, are used to describe the flow. A transition diagram that can be compared with experiments or models outlines the various dynamical regimes as a function of the distance between the cylinders and the Reynolds number.     

Computational study of shock wave control by pulse energy deposition

We have developed a computational code based on the axisymmetric Navier–Stokes equations with thermochemical kinetics for assessing wave drag reduction and other effects in pulse-energy deposition ahead of a bow shock by means of full simulations from generation of a laser-induced blast wave to interaction with the bow shock. Thermochemical nonequilibrium computations can reproduce the process of blast wave formation with laser ray tracing, and the computed low-density core inside the blast wave has a teardrop-like shape, depending on the laser input condition. The flowfield interacting with a bow shock formed in Mach 5 flow was computed. The result suggests that the shape of the low-density core affects the resultant wave drag, and parameters of an incident laser beam should be taken into account in exploring the optimal condition of the proposed wave-drag scheme.     

Shock wave propagation into a surface depression

An aircraft travelling at supersonic speeds close to the ground generates a bow wave which is reflected off the ground surface. If a valley is traversed a complex reflection pattern will be generated. Similar patterns will evolve with a plane wave traversing a depression on a surface or structure. To simulate the process a planar shock wave, generated in a shock tube, is moved over several notched wedge configurations. Schlieren photographs were produced to assist in identifying the resulting complex three-dimensional wave structures and then verified and extended by three- dimensional computations. The valley geometries investigated are rectangular, triangular, parabolic and conical with a number of valley floor inclinations. The main features are extracted in surface models to demonstrate the complexity of the flow, and in particular in the case where the reflection is regular on the ground plane and Mach reflection in the valley.      

Three-dimensional supersonic internal flows

In order to examine the transition between regular and Mach reflection in a three-dimensional flow, a range of special geometry test pieces, and inlets, were designed. The concept is to have a geometry consisting of two plane wedges which results in regular reflection between the incident waves off the top and bottom of the inlet capped by two curved end sections causing Mach reflection. The merging of these two reflection patterns and the resulting downstream flow are studied using laser vapor screen and shadowgraph imaging supported by numerical simulation. An angled Mach disc is formed which merges with the line of regular reflection. A complex wave pattern results with the generation of a bridging shock connecting the reflected wave from the Mach reflection with the reflected waves from the regular reflection. In order to experimentally access the flow within the duct, a number of tests were conducted with one end cap removed. This resulted in a modified flow due to the expansive flow at the open end the influence of which was also studied in more detail.     

Submerged shocks in conical gas flows in the presence of a mach shock-wave configuration

The results of a numerical study of a new type of singularities in the Mach shock-wave structure realized in supersonic nonsymmetric conical flows over V-wings with a bow shock attached to the leading edges are presented. Within the framework of the ideal gas model we study the changes in the shock system on transition, with increase in the sweep angle, from the region of nonsymmetric Mach interaction of the shocks attached to the leading edges of the wing to the region of special flow patterns, where on the windward cantilever surface a rarefaction flow is realized rather than a flow with an internal shock. It is shown, in particular, that in the region with special wing flow patterns a Mach system of shocks with a submerged shock proceeding from the branch point above the windward cantilever may exist.     

Shock wave diffraction on multi-facetted and curved walls

The two-dimensional diffraction of a shock wave over a wall made up of a series of plane and/or curved sections is considered. The analysis is based on the theory presented by, for the interaction of an originally plane shock wave with a corner. A method is presented by which the shock profile may be determined for a wall of any shape and for any incident Mach number, in regions where the characteristics form a simple wave. Comparisons are made between experimental measurements and theoretical predictions for convex walls consisting of a number of facets, and for circular arcs, for a range of incident shock wave Mach numbers. It is shown that the theory gives a satisfactory prediction of the wave shape, which improves as the Mach number increases. Modifications in the flow field behind the shock, compared to that for a simple corner made up of two plane walls is discussed, particularly relating to flow separation. For circular arc concave walls a inverse Mach reflection results experimentally, leading to regular reflection, for which the theory is of no use. PACS 47.40.Nm     

STUDY ON THE FUEL AIR MIXING INDUCED BY A SHOCK WAVE PROPAGATING INTO A H_2-AIR INTERFACE

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Numerical investigations of transition between regular and Mach reflections caused by free-stream disturbances

Numerical simulations have been performed to study the influence of the free-stream disturbances on the alternation of the steady shock wave reflection configurations in the dual solution domain. Different types of disturbances have been considered. The analysis of interaction between disturbances and the incident shock wave can be substantially simplified for the localized density disturbances. It is shown that such disturbances can indeed cause the transition from regular reflection to Mach reflection and back, so that within a certain range of angles of incidence the shock wave reflection configuration can be considered as a bi-stable system. The threshold amplitude of the localized density disturbance, able to induce the transition, has been estimated theoretically. The results of numerical computations convince of higher stability of the Mach reflection in the dual solution domain compared to the regular reflection, which is in accordance with available experimental data. Received 10 May 2001 / Accepted 15 November 2001 Published online 8 July 2002