Nonlinear Strong Shock Interactions: A Shock-Fitted Approach

This paper addresses nonlinear effects which result from the interaction of shock waves with vortices. A series of experiments are carried out, which involve the interaction of a strong shock wave with a single plane vorticity wave and a randomly distributed wave system. These experiments are first conducted in the linear regime to obtain a mutual verification of theory and computation. They are subsequently extended into the nonlinear regime. A systematic study of the interaction of a plane shock wave and a single vortex is then conducted. Specifically, we investigate the conditions under which nonlinear effects become important, both as a function of shock Mach number, M ₁, and incident vortex strength (characterized by its circulation Γ). The shock Mach number is varied from 2 to 8, while the circulation of the vortex is varied from infinitesimally small values (linear theory) to unity. Budgets of vorticity, dilatation, and pressure are obtained. They indicate that nonlinear effects become more significant as both the shock Mach number and the circulation increase. For Mach numbers equal to 5 and above, the dilatation in the vortex core grows quadratically with circulation. An acoustic wave propagates radially outward from the vortex center. As circulation increases, its upstream-facing front steepens at low Mach numbers, and its downstream-facing front steepens at high Mach numbers. A high Mach number asymptotic expansion of the Rankine--Hugoniot conditions reveals that nonlinear effects dominate both the shock motion and the downstream flow for ΓM ₁ > 1. Received 28 June 1997 and accepted 25 November 1997     

Behavior of the vorticity vector in supersonic axisymmetric swirl flows behind a detonation wave

The behavior of the vorticity vector on a discontinuity surface arising in a supersonic nonuniform combustible gas flow with the formation of a shock or detonation wave is studied. In the general case, it is a vortex flow with prescribed distributions of parameters. It is demonstrated that the ratio of the tangential component of vorticity to density remains continuous in passing through the discontinuity surface, while the quantities proper become discontinuous. Results calculated for flow vorticity behind a steady-state detonation wave in an axisymmetric supersonic flow of a combustible mixture of gases are presented. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 15–21, November–December, 2007     

Ideal incompressible flow around a wedge tip

A planar analog of conical flows is considered: an inviscid incompressible fluid flow around a wedge tip. A class of conical flows is found where vorticity is transported along streamlines by the potential component of velocity. Problems of a wave “locked” in the corner region and of a flow accelerating along the rib of a dihedral angle are considered. By analogy with an axisymmetric quasi-conical flow, a planar quasi-conical flow of the fluid is determined, namely, the flow inside and outside the region bounded by tangent curves described by a power law. Conditions are found where vorticity and swirl produce a significant effect. An approximate solution of the problem of the fluid flow inside a “zero” angle is obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 57–65, March–April, 2007.     

Internal waves induced by potential vortices

A linearized equation of the internal waves developing in an ideal stratified gas under the action of potential vortices concentrated in a vertical cylinder is obtained. The Cauchy problem for the internal wave equation with right side depending on the vortex intensity is solved by the integral transform method. In the case of a vortex filament the exact solution is found. Approximate formulas are obtained on the basis of the steady-phase method when the vorticity is exponentially stratified along the vertical. Expressions for the phase velocity and amplitude of the radial wave traveling away from the cylindrical vortex are found. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 118–123, January–February, 1998. The work was carried out with the support of the Russian Foundation for Fundamental Research (project No. 96-01-04599).     

Equations of the shallow water model on a rotating attracting sphere. 1. Derivation and general properties

A shallow water model on a rotating attracting sphere is proposed to describe large-scale motions of the gas in planetary atmospheres and of the liquid in the world ocean. The equations of the model coincide with the equations of gas-dynamic of a polytropic gas in the case of spherical gas motions on the surface of a rotating sphere. The range of applicability of the model is discussed, and the conservation of potential vorticity along the trajectories is proved. The equations of stationary shallow water motions are presented in the form of Bernoulli and potential vorticity integrals, which relate the liquid depth to the stream function. The simplest stationary solutions that describe the equilibrium state differing from the spherically symmetric state and the zonal flows along the parallels are found. It is demonstrated that the stationary equations of the model admit the infinitely dimensional Lie group of equivalence. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 24–36, March–April, 2009.     

Air blast and heat radiation from fuel-rich mixture detonations

Large scale experiments were carried out to study the effect of fuel concentration on air blast parameters and heat radiation from gaseous detonations. Hemispheric plastic envelope (4 meters in radius) was used with propane-air mixtures containing from 4 to 7 vol. % of fuel. The expressions for overpressures and impulses were determined in Sachs variables. The effect of fuel concentration on blast parameters is shown to be insignificant for the same amount of oxygen in the mixture volume. Thus the blast wave parameters can be described as for stoichiometric mixtures using additional scaling for the explosion energy according to oxygen content (cloud volume). The results of large scale experiments with fuel spray clouds containing 0.16–100 tons of fuel with mean concentration from stoichiometric () up to are reconsidered. These results confirm the proposed scaling of air blast parameters for a wide range of fuel types, cloud volumes and fuel concentrations. Detonations of fuel rich gaseous mixtures result in a strong heat radiation. Heat radiation energy, time and size of the fireball formed are studied as a function of fuel concentration. Received March 10, 1995 / Accepted March 12, 1995     

Improved methods for thin, surface boundary layer investigations

 New techniques are developed to improve the velocity flow-field measurement capability within a free-surface boundary layer region on which progressive capillary-gravity waves are present. Due to the extremely thin but rather vortical characteristics of the aforementioned boundary layer, conventional particle image velocimetry (PIV) methods fail to estimate velocity (and vorticity) vectors at an acceptable detection rate. This failure is a direct consequence of optimal PIV parameters that are difficult to achieve in practice for such flow situations. A new technique, Sub-pattern PIV, is developed. This method has features similar to both the super-resolution PIV (Keane et al. 1995) and the particle image distortion (PID) technique (Huang et al. 1993), but is predicated upon a very differential philosophy. Another difficulty that arises in experiments to investigate surface boundary layer flows is that the oscillating and deforming air–water interface has a mirror-like behavior that affects the images, and generates very noisy data. An alternative experimental setup that utilizes the Brewster angle phenomenon is adopted and the specular effects of the free-surface are removed successfully. This Brewster angle imaging, along with the Sub-pattern PIV technique, is used for the target application – a free-surface boundary layer investigation. It proved to be very effective. The methodology of both techniques is discussed, and the modified PIV procedure is validated by numerical probabilistic simulations. Application to the capillary-gravity wave boundary layer is presented in a subsequent paper. Received: 31 July 1997/Accepted: 4 February 1998     

Particle image velocimetry study of the shock-induced single mode Richtmyer–Meshkov instability

Full field particle image velocimetry (PIV) measurements are obtained for the first time in Richtmyer–Meshkov instability shock tube experiments. The experiments are carried out in a vertical shock tube in which the light gas (air) and the heavy gas (SF₆) flow from opposite ends of the shock tube driven section and exit through narrow slots at the interface location. A sinusoidal perturbation is given to the interface by oscillating the shock tube in the horizontal direction. Richtmyer–Meshkov instability is then produced by the interaction with a weak shock wave (M _s  = 1.21). PIV measurements are obtained by seeding the flow with 0.30 μm polystyrene Latex spheres which are illuminated using a double-pulsed Nd:YAG laser. PIV measurements indicate the vorticity to be distributed in a sheet-like distribution on the interface immediately after shock interaction and that this distribution quickly rolls up into compact vortices. The integration of the vorticity distribution over one half wave length shows the circulation to increase with time in qualitative agreement with the numerical study of Peng et al. (Phys. Fluids, 15, 3730–3744, 2003).     

Strong explosion near a parallelepipedic structure

The purpose of this paper is to report the blast loading characteristics resulting from the detonation of a stoichiometric propane–oxygen mixture, and to validate the approach which relies on simulating TNT explosions at large scale by small-scale experiments of gaseous explosions. Several dimensionless correlations are obtained from experimental data. These relationships allow determination of the parameters of a blast wave interacting with a structure as a function of the positions of the explosive charge and the structure. Simulations carried out with the Autodyn code show good correlation with experimental results. The Hopkinson law is suggested to predict the blast wave’s parameters at large scale on the basis of small-scale experiments and simulations. This paper was based on work that was presented at the 20th International Colloquium on the Dynamics of Explosions and Reactive Systems, Montreal, Canada, July 31–August 5, 2005.     

Experimental and numerical studies on interactions of a spherical flame with incident and reflected shocks

Observations are presented from experiments and calculations where a laminar spherical CH4/air flame is perturbed successively by incident and reflected shock waves. The experiments are performed in a standard shock tube arrangement, in which a high-speed shadowgraph imaging system is used to record evolutions of the flame. Numerical simulations are conducted by using second-order wave propagation algorithms, based on two-dimensional axisymmetric Navier-Stokes equations with detailed chemical reactions. Qualitative agreements are obtained between the experimental and numerical results. Under actions of incident shock waves, Richtmyer-Meshkov instability responsible for the flame deformation is induced in the flame, and the distoned flame takes a barrel shape. Then, under subsequent actions of the shock wave reflected from a planar wall, the flame takes an inclined non-symmetrical kidney shape in a symmetric cross section, which means a mushroom-like shape of the flame comes finally into being. The vorticity direction in the ring cap has been altered by the reflected shock＇s action, which makes the head of the mushroom-like flame extend quickly to the side wall.     

On the Mutual Penetrations of Two Fluids Whose Interface is Accelerated by a Shock Wave

A shock tube experimental investigation and numerical simulations are undertaken to study the evolution of a perturbed interface of two different gases accelerated by a shock wave. The experimental method is based on a high-speed camera laser sheet diagnostic technique, and simulations are provided by our code CARBUR based on a finite volume discretization of Navier–Stokes’s equations. Two gas pairs are used to illustrate both the heavy/light (air/He) and the light/ heavy (air/SF₆) cases. Two simultaneous large initial perturbations, one positive and one negative, are tested for an incident shock wave Mach number in air of about 1.3. The thin membrane (less than 1 μ) which materializes the initial interface between the two test gases presents 2D perturbations whose wave number is close to 1 in order to rapidly reach the non-linear regime. The development of the perturbations is captured at a frequency of 10 kHz after the interface acceleration, and the experiments are complemented with a numerical simulation to validate the interface deformations. Results show an asymmetric mutual gas penetration increasing with the absolute value of the Atwood’s number. Furthermore, they confirm that the heavier gas penetrates the lighter as thin spikes and the lighter gas penetrates the heavier as large bubbles. Moreover, we show that the spike moves faster than the bubble in the heavy/light case and slightly faster in the light/heavy one. Finally, numerical and experimental results are in agreement.     

Experimental investigation of vortex rings impinging on inclined surfaces

Vortex–ring interactions with oblique boundaries were studied experimentally to determine the effects of plate angle on the generation of secondary vorticity, the evolution of the primary vorticity and secondary vorticity as they interact near the boundary, and the associated energy dissipation. Vortex rings were generated using a mechanical piston-cylinder vortex ring generator at jet Reynolds numbers 2,000–4,000 and stroke length to piston diameter ratios (L/D) in the range 0.75–2.0. The plate angle relative to the initial axis of the vortex ring ranged from 3 to 60°. Flow analysis was performed using planar laser-induced fluorescence (PLIF), digital particle image velocimetry (DPIV), and defocusing digital particle tracking velocimetry (DDPTV). Results showed the generation of secondary vorticity at the plate and its subsequent ejection into the fluid. The trajectories of the centers of circulation showed a maximum ejection angle of the secondary vorticity occurring for an angle of incidence of 10°. At lower incidence angles (<20°), the lower portion of the ring, which interacted with the plate first, played an important role in generation of the secondary vorticity and is a key reason for the maximum ejection angle for the secondary vorticity occurring at an incidence angle of 10°. Higher Reynolds number vortex rings resulted in more rapid destabilization of the flow. The three-dimensional DDPTV results showed an arc of secondary vorticity and secondary flow along the sides of the primary vortex ring as it collided with the boundary. Computation of the moments and products of kinetic energy and vorticity magnitude about the centroid of each vortex ring showed increasing asymmetry in the flow as the vortex interaction with the boundary evolved and more rapid dissipation of kinetic energy for higher incidence angles.     

A study of ambient upstream material properties using perpendicular laser driven radiative blast waves in atomic cluster gases

We report on the characterisation of the upstream medium ahead of a radiative cylindrical blast wave launched in an argon cluster gas with a 1 J, 1 ps, 1054 nm Nd:Glass laser system. By launching two perpendicular blast waves and introducing a time delay between the heating beams it is possible to determine the extent of the cluster medium by observing the high energy absorption region associated with clusters, as apposed to the low energy deposition in monatomic gas. It was found that argon ions launched from the initial laser driven cluster ionisation created a ballistic ion wave which sweeps out ahead of the hydrodynamic blast wave at an initial velocity of 1000 kms⁻¹. This ballistic wave disassembles the clusters ahead of the blast wave into a neutral gas medium before the arrival of a radiative precursor. This observation gives us confidence that the dynamics of a radiative blast wave in cluster based experiments is determined primarily by the properties of an upstream atomic gas, and is not significantly influenced by cluster affects on energy transport or other material properties.     

Direct numerical simulation of a liquid sheet in a compressible gas stream in axisymmetric and planar configurations

A thin liquid sheet present in the shear layer of a compressible gas jet is investigated using an Eulerian approach with mixed-fluid treatment for the governing equations describing the gas–liquid two-phase flow system, where the gas is treated as fully compressible and the liquid as incompressible. The effects of different topological configurations, surface tension, gas pressure and liquid sheet thickness on the flow development of the gas–liquid two-phase flow system have been examined by direct solution of the compressible Navier–Stokes equations using highly accurate numerical schemes. The interface dynamics are captured using volume of fluid and continuum surface force models. The simulations show that the dispersion of the liquid sheet is dominated by vortical structures formed at the jet shear layer due to the Kelvin–Helmholtz instability. The axisymmetric case is less vortical than its planar counterpart that exhibits formation of larger vortical structures and larger liquid dispersion. It has been identified that the vorticity development and the liquid dispersion in a planar configuration are increased at the absence of surface tension, which when present, tends to oppose the development of the Kelvin–Helmholtz instability. An opposite trend was observed for an axisymmetric configuration where surface tension tends to promote the development of vorticity. An increase in vorticity development and liquid dispersion was observed for increased liquid sheet thickness, while a decreasing trend was observed for higher gas pressure. Therefore surface tension, liquid sheet thickness and gas pressure factors all affect the flow vorticity which consequently affects the dispersion of the liquid.      

On Nonlinear Baroclinic Waves and Adjustment of Pancake Dynamics

Three-dimensional nonhydrostatic Euler–Boussinesq equations are studied for Bu=O(1) flows as well as in the asymptotic regime of strong stratification and weak rotation. Reduced prognostic equations for ageostrophic components (divergent velocity potential and geostrophic departure/thermal wind imbalance) are analyzed. We describe classes of nonlinear anisotropic ageostrophic baroclinic waves which are generated by the strong nonlinear interactions between the quasi-geostrophic modes and inertio-gravity waves. In the asymptotic regime of strong stratification and weak rotation we show how switching on weak rotation triggers frontogenesis. The mechanism of the front formation is contraction in the horizontal dimension balanced by vertical shearing through coupling of large horizontal and small vertical scales by weak rotation. Vertical slanting of these fronts is proportional to μ^−1/2 where μ is the ratio of the Coriolis and Brunt–V?is?l? parameters. These fronts select slow baroclinic waves through nonlinear adjustment of the horizontal scale to the vertical scale by weak rotation, and are the envelope of inertio-gravity waves. Mathematically, this is generated by asymptotic hyperbolic systems describing the strong nonlinear interactions between waves and potential vorticity dynamics. This frontogenesis yields vertical “gluing” of pancake dynamics, in contrast to the independent dynamics of horizontal layers in strongly stratified turbulence without rotation. Received 8 April 1997 and accepted 29 March 1998     

Balanced Atmosphere–Ocean Dynamics, Generalized Lighthill Radiation, and the Slow Quasi-Manifold

Three of Lighthill's many interests, (a) wave propagation in moving media, (b) acoustic streaming and related phenomena, and (c), in a very subtle and fascinating way, aerodynamic sound generation, are all turning out to be fundamental to understanding our atmospheric environment. Among other things there is the global-scale circulation that shapes the ozone layer and controls the rate of destruction of man-made chlorofluorocarbons (CFCs). Recent progress in this field is sketched, including progress in understanding the abstract structure of Hamiltonian theories of balanced motion, the so-called slow “manifold”. Here a generic phenomenon, “velocity splitting”, turns out to be intimately related to aerodynamic sound generation, particularly its generalization describing the spontaneous emission of inertia–gravity waves from unsteady vortical motions in stratified rotating flow. Received 5 January 1997 and accepted 2 May 1997     

Spatial structures and scaling in the Convective Boundary Layer

We performed an investigation on spatial features of the Convective Boundary Layer (CBL) of the atmosphere, which was simulated in a laboratory model and analyzed by means of image analysis techniques. This flow is dominated by large, anisotropic vortical structures, whose spatial organization affects the scalar transport and therefore the fluxes across the boundary layer. With the aim of investigating the spatial structure and scaling in the Convective Boundary Layer, two-dimensional velocity fields were measured, on a vertical plane, by means of a pyramidal Lucas–Kanade algorithm. The coherent structures characterizing the turbulent convection were educed by analyzing the Finite-Time Lyapunov Exponent fields, which also revealed interesting phenomenological features linked to the mixing processes occurring in the Convective Boundary Layer. Both velocity and vorticity fields were analyzed in a scale-invariance framework. Data analysis showed that normalized probability distribution functions for velocity differences are dependent on the scale and tend to become Gaussian for large separations. Extended Self Similarity holds true for velocity structure functions computed within the mixing layer, and their scaling exponents are interpreted well in the phenomenological framework of the Hierarchical Structure Model. Specifically, β parameter, which is related to the similarity between weak and strong vortices, reveals a higher degree of intermittency for the vertical velocity component with respect to the horizontal one. On the other hand, the analysis of circulation structure functions shows that scaling exponents are fairly constant in the lowest part of the mixed layer, and their values are in agreement with those reported in Benzi et al. (Phys Rev E 55:3739–3742, 1997) for shear turbulence. Moreover, the relationship between circulation and velocity scaling exponents is analyzed, and it is found to be linear in the bottom part of the mixing layer. The investigation of the CBL spatial features, which has seldom been studied experimentally, has important implications for the comprehension of the mixing dynamics, as well as in turbulence closure models.     

Experimental study and mathematical simulation of the characteristics of a turbulent flow in a straight circular pipe rotating about its longitudinal axis

The vorticity formed in the cross section of a turbulent flow in a straight circular pipe rotating about its longitudinal axis decreases the values of the turbulent stresses, turbulence energy, and dissipation rate along the pipe. The results of laboratory experiments and calculations by the second-order closure model of turbulent transfer are presented. On the whole, the model using a system of transport equations yields better agreement with experimental data than the models with algebraic relations for second-order moments. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 103–116, March–April, 1998.     

Thermocapillary migration of bubbles and drops at moderate to large Marangoni number and moderate Reynolds number in reduced gravity

 Results from experiments on the thermocapillary migration of air bubbles and Fluorinert drops in a Dow–Corning silicone oil aboard a NASA Space Shuttle mission are presented and discussed. The experiments cover a wider range of Marangoni and Reynolds numbers than that attained in a prior flight experiment. The data are consistent with earlier results, and are compared with theoretical predictions. Large air bubbles were found to deform slightly in shape to oblate spheroids while the deformation of even the largest drops was within the uncertainty of the size measurements. Received: 18 December 1997/Accepted: 30 May 1998     

Jet formation in shock-heavy gas bubble interaction

The influences of the acoustic impedance and shock strength on the jet formation in shock-heavy gas bubble interaction are numerically studied in this work. The process of a shock interacting with a krypton or a SF 6 bubble is studied by the numerical method VAS2D. As a validation, the experiments of a SF 6 bubble accelerated by a planar shock were performed. The results indicate that, due to the mismatch of acoustic impedance, the way of jet formation in heavy gas bubble with different species is diversified under the same initial condition. With respect to the same bubble, the manner of jet formation is also distinctly different under different shock strengths. The disparities of the acoustic impedance result in different effects of shock focusing in the bubble, and different behaviors of shock wave inside and outside the bubble. The analyses of the wave pattern and the pressure variation indicate that the jet formation is closely associated with the pressure perturbation. Moreover, the analysis of the vorticity deposition, and comparisons of circulation and baroclinic torque show that the baroclinic vorticity also contributes to the jet formation. It is concluded that the pressure perturbation and baroclinic vorticity deposition are the two dominant factors for the jet formation in shock-heavy gas bubble interaction.