Gas-dynamic colliders: Numerical simulations

A collision of supersonic flows of gas mixtures with disparate molecular weights, which are limited in their cross-sectional size, in vacuum leads to formation of a cloud with an elevated concentration and elevated temperature of the heavy gas. Under certain conditions, the governing factor is the collision of molecules of the heavy gas being compressed at the center of the collision of the flows. The generator of such a flow can be called a collider. Results of studying the flows in jet-type, cylindrical, and mixed two-stage colliders are described. The main attention is paid to separation of gases in terms of energy and composition. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 142–151, May–June, 2007.     

Wall Effect on the Convective–Absolute Boundary for the Compressible Shear Layer

The linear stability of inviscid compressible shear layers is studied. When the layer develops at the vicinity of a wall, the two parallel flows can have a velocity of the same sign or of opposite signs. This situation is examined in order to obtain first hints on the stability of separated flows in the compressible regime. The shear layer is described by a hyperbolic tangent profile for the velocity component and the Crocco relation for the temperature profile. Gravity effects and the superficial tension are neglected. By examining the temporal growth rate at the saddle point in the wave-number space, the flow is characterized as being either absolutely unstable or convectively unstable. This study principally shows the effect of the wall on the convective–absolute transition in compressible shear flow. Results are presented, showing the amount of the backflow necessary to have this type of transition for a range of primary flow Mach numbers M ₁ up to 3.0. The boundary of the convective–absolute transition is defined as a function of the velocity ratio, the temperature ratio and the Mach number. Unstable solutions are calculated for both streamwise and oblique disturbances in the shear layer. Received 9 May 2001 and accepted 21 August 2001     

Subcritical and supercritical shear flows above an uneven bottom

The concepts of subcritical and supercritical flows are introduced for the long-wave approximation model describing stationary free-boundary rotational flows of an ideal incompressible fluid. Shear flows of a fluid layer above an uneven bottom are analyzed. Exact solutions describing different flow regimes are constructed, and the flow properties are studied as a function of the flow regime. Flows with backward streamlines are considered. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 26–38, July–August, 2006. An erratum to this article is available at .     

Local subsonic zones in supersonic jet flows

The results of the experimental investigation of supersonic turbulent jets with local subsonic zones of forward and reverse flow exhausting into the ambient atmosphere or an outer stream, either parallel or transverse to the jet, are presented. Some gasdynamic features of the flows containing these zones, which have not been adequately addressed in the literature, are analyzed. Thus, supersonic flows with back pressure, e.g., highly overexpanded and underexpanded jet flows, and those upstream and downstream of a jet on the leeward side of a cone in a supersonic gas stream, are studied. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 143–150, January–February, 1998.     

Breaking of waves of limiting amplitude over an obstacle

Homogeneous heavy fluid flows over an uneven bottom are studied in a long-wave approximation. A mathematical model is proposed that takes into account both the dispersion effects and the formation of a turbulent upper layer due to the breaking of surface gravity waves. The asymptotic behavior of nonlinear perturbations at the wave front is studied, and the conditions of transition from smooth flows to breaking waves are obtained for steady-state supercritical flow over a local obstacle. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 3–11, May–June, 2006.     

Calculation of spatial flow past an inclined flat plate with a detached shock wave

The results of a numerical analysis of a supersonic underexpanded jet impinging on an inclined flat plate are presented. The effects of the angle between the plate and the jet symmetry axis, the distance from the nozzle exit section, the exit Mach number, and the off-design conditions on the distribution of the gasdynamic parameters in the jet flowfield and on the plate surface are demonstrated. Specific features of the compressed layer and obstacle surface flows are revealed. The three-dimensional flow is simulated using the large particle method on the basis of the nonstationary Euler equations written in the cylindrical coordinate system. The calculated results are compared with experimental data. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 31–35, January–February, 1997.     

Incompressible Viscous Fluid Flows in a Thin Spherical Shell

Linearized stability of incompressible viscous fluid flows in a thin spherical shell is studied by using the two-dimensional Navier–Stokes equations on a sphere. The stationary flow on the sphere has two singularities (a sink and a source) at the North and South poles of the sphere. We prove analytically for the linearized Navier–Stokes equations that the stationary flow is asymptotically stable. When the spherical layer is truncated between two symmetrical rings, we study eigenvalues of the linearized equations numerically by using power series solutions and show that the stationary flow remains asymptotically stable for all Reynolds numbers.      

Suspended load and bed-load transport of particle-laden gravity currents: the role of particle–bed interaction

The development of particle-enriched regions (bed-load) at the base of particle-laden gravity currents has been widely observed, yet the controls and relative partitioning of material into the bed-load is poorly understood. We examine particle-laden gravity currents whose initial mixture (particle and fluid) density is greater than the ambient fluid, but whose interstitial fluid density is less than the ambient fluid (such as occurs in pyroclastic flows produced during volcanic eruptions or when sediment-enriched river discharge enters the ocean, generating hyperpycnal turbidity currents). A multifluid numerical approach is employed to assess suspended load and bed-load transport in particle-laden gravity currents under varying boundary conditions. Particle-laden flows that traverse denser fluid (such as pyroclastic flows crossing water) have leaky boundaries that provide the conceptual framework to study suspended load in isolation from bed-load transport. We develop leaky and saltation boundary conditions to study the influence of flow substrate on the development of bed-load. Flows with saltating boundaries develop particle–enriched basal layers (bed-load) where momentum transfer is primarily a result of particle–particle collisions. The grain size distribution is more homogeneous in the bed-load and the saltation boundaries increase the run-out distance and residence time of particles in the flow by as much as 25% over leaky boundary conditions. Transport over a leaky substrate removes particles that reach the bottom boundary and only the suspended load remains. Particle transport to the boundary is proportional to the settling velocity of particles, and flow dilution results in shear and buoyancy instabilities at the upper interface of these flows. These instabilities entrain ambient fluid, and the continued dilution ultimately results in these currents becoming less dense than the ambient fluid. A unifying concept is energy dissipation due to particle–boundary interaction: leaky boundaries dissipate energy more efficiently at the boundary than their saltating counterparts and have smaller run-out distance.

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Some features of the flow around rapidly rotating bodies made of cellular-porous materials

Two types of gas flows arising near a rapidly rotating cellular-porous disk are studied numerically and experimentally. Steady-state limits for the flow around a disk rotating in free space and the type and scenario of the loss of stability are determined. Transitional flows are characterized by formation of a vortex sheet at the boundary of the exhausting jet. Numerical simulations of the flow around a cellular-porous disk rotating near a flat screen show that it is possible to form a closed swirl flow responsible for redistribution of swirl in the gap between the disk and the flat screen. The computed results offer an explanation for the experimentally observed excess of tangential velocity of the flow in the gap over the velocity of disk rotation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 86–96, January–February, 2007.     

“Non-free” interaction of plane shock waves and the boundary layer in the neighborhood of the leading edge of a plate with slip

The interaction between a normally impinging shock wave and the boundary layer on a plate with slip is studied in the neighborhood of the leading edge using various experimental methods, including special laser technology, to visualize the supersonic conical gas flows. It is found that in the “non-free” interaction, when the leading edge impedes the propagation of the boundary layer separation line upstream, the structure of the disturbed flow is largely identical to that in the developed “free” interaction, but with higher parameter values and gradients in the leading part of the separation zone. The fundamental property of developed separation flows, namely, coincidence of the values of the pressure “plateau” in the separation zone and the pressure behind the oblique shock above the separation zone of the turbulent boundary layer, is conserved. Moscow. e-mail: ostap@inmech.msu.su. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 57–69, May–June, 2000. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 97-01-00099).     

A Comparison Principle for Hamilton–Jacobi Equations Related to Controlled Gradient Flows in Infinite Dimensions

We develop new comparison principles for viscosity solutions of Hamilton–Jacobi equations associated with controlled gradient flows in function spaces as well as the space of probability measures. Our examples are optimal control of Ginzburg–Landau and Fokker–Planck equations. They arise in limit considerations of externally forced non-equilibrium statistical mechanics models, or through the large deviation principle for interacting particle systems. Our approach is based on two key ingredients: an appropriate choice of geometric structure defining the gradient flow, and a free energy inequality resulting from such gradient flow structure. The approach allows us to handle Hamiltonians with singular state dependency in the nonlinear term, as well as Hamiltonians with a state space which does not satisfy the Radon–Nikodym property. In the case where the state space is a Hilbert space, the method simplifies existing theories by avoiding the perturbed optimization principle.     

On the Solution of a Boltzmann System for Gas Mixtures

We study the Boltzmann equation for a mixture of two gases in one space dimension with initial condition of one gas near vacuum and the other near a Maxwellian equilibrium state. A qualitative-quantitative mathematical analysis is developed to study this mass diffusion problem based on the Green’s function of the Boltzmann equation for the single species hard sphere collision model in Liu andYu (Commun Pure Appl Math 57:1543–1608, 2004). The cross-species resonance of the mass diffusion and the diffusion-sound wave is investigated. An exponentially sharp global solution is obtained.     

Combined Forced and Free Convective Flow in a Vertical Porous Channel: The Effects of Viscous Dissipation and Pressure Work

Buoyant flow is analysed for a vertical fluid saturated porous layer bounded by an isothermal plane and an isoflux plane in the case of a fully developed flow with a parallel velocity field. The effects of viscous dissipation and pressure work are taken into account in the framework of the Oberbeck–Boussinesq approximation scheme and of the Darcy flow model. Momentum and energy balances are combined in a dimensionless nonlinear ordinary differential equation solved numerically by a Runge–Kutta method. Both cases of upward pressure force (upward driven flows) and of downward pressure force (downward driven flows) are examined. The thermal behaviour for upward driven flows and downward driven flows is quite different. For upward driven flows, the combined effects of viscous dissipation and pressure work may produce a net cooling of the fluid even in the case of a positive heat input from the isoflux wall. For downward driven flows, viscous dissipation and pressure work yield a net heating of the fluid. A general reflection on the roles played by the effects of viscous dissipation and pressure work with respect to the Oberbeck–Boussinesq approximation is proposed.     

Heat and mass transfer analogy for condensation of humid air in a vertical channel

This study examines energy transport associated with liquid film condensation in natural convection flows driven by differences in density due to temperature and concentration gradients. The condensation problem is based on the thin-film assumptions. The most common compositional gradient, which is encountered in humid air at ambient temperature is considered. A steady laminar Boussinesq flow of an ideal gas–vapor mixture is studied for the case of a vertical parallel plate channel. New correlations for the latent and sensible Nusselt numbers are established, and the heat and mass transfer analogy between the sensible Nusselt number and Sherwood number is demonstrated. Received on 15 November 1999     

A mixing layer in a homogeneous fluid

A mathematical model for the evolution of a mixing layer in shear flows is constructed. The problem of a mixing layer with pressure gradient is solved: in particular, the distributions of the velocity and basic characteristics of turbulent flow in the mixing layer are obtained. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 81–92, July–August, 2000.     

Conjugate flows and smooth bores in a weakly stratified fluid

The problem of steady-state flows in a layer of a continuously stratified fluid is considered. The sufficient condition of existence of families of shear flows that are consistent with the meaning of the laws of conservation of mass, momentum, and energy with a uniform flow is given. Approximate solutions of the smooth-bore type, which describe the wave transitions for pairs of conjugate flows of the first spectral mode, are obtained. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 69–78. March–April, 1999.     

Extinction Versus Persistence in Strong Oscillating Flows

In this paper, we give some conditions for finite-time extinction or persistence of the solutions of diffusion–advection equations in strong and oscillating flows under Dirichlet boundary conditions. The enhancement of the diffusion rate depends on the interplay between strong advection and time-homogenization, and in particular on the ratio between the strength of the flow and its frequency parameter. Quantitative estimates of this ratio, which depend on the geometry of the domain, are provided in the case of a uniform flow. In the general time–space dependent case, the finite-time behavior of the solutions is related to the existence of first integrals of the flow.     

Prediction of separation flows around a 6：1 prolate spheroid using RANS/LES hybrid approaches

This paper presents hybrid Reynolds-averaged Navier–Stokes (RANS) and large-eddy-simulation (LES) methods for the separated flows at high angles of attack around a 6:1 prolate spheroid. The RANS/LES hybrid methods studied in this work include the detached eddy simulation (DES) based on Spalart–Allmaras (S–A), Menter’s k–ω shear-stress-transport (SST) and k–ω with weakly nonlinear eddy viscosity formulation (Wilcox–Durbin+, WD+) models and the zonal-RANS/LES methods based on the SST and WD+ models. The switch from RANS near the wall to LES in the core flow region is smooth through the implementation of a flow-dependent blending function for the zonal hybrid method. All the hybrid methods are designed to have a RANS mode for the attached flows and have a LES behavior for the separated flows. The main objective of this paper is to apply the hybrid methods for the high Reynolds number separated flows around prolate spheroid at high-incidences. A fourth-order central scheme with fourth-order artificial viscosity is applied for spatial differencing. The fully implicit lower–upper symmetric-Gauss–Seidel with pseudo time sub-iteration is taken as the temporal differentiation. Comparisons with available measurements are carried out for pressure distribution, skin friction, and profiles of velocity, etc. Reasonable agreement with the experiments, accounting for the effect on grids and fundamental turbulence models, is obtained for the separation flows. The project supported by the National Natural Science Foundation of China (10502030 and 90505005).     

Dissipative instability of fluid flows with piecewise linear velocity profiles

The viscous dissipative instability of two flows with continuous spectrum of neutrally-stable perturbations in the absence of dissipation is investigated. Ranges of wave numbers in which viscosity leads to flow destabilization are determined for a shear discontinuity in a smoothly-stratified fluid. A shear flow with a velocity in the transition layer that depends linearly on the coordinate has a continuum of neutral modes even in the case of an unstratified fluid. When viscosity is present in one of the layers with constant velocity, one of the branches of the spectrum becomes unstable. When the viscosity is the same above and below the shear layer, dissipation only leads to the damping of the perturbations. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 14–19, November–December, 1986.     

Two- and Four-Way Coupled Euler–Lagrangian Large-Eddy Simulation of Turbulent Particle-Laden Channel Flow

Large-eddy simulations (LES) of a vertical turbulent channel flow laden with a very large number of solid particles are performed. The motivation for this research is to get insight into fundamental aspects of co-current turbulent gas-particle flows, as encountered in riser reactors. The particle volume fraction equals about 1.3%, which is relatively high in the context of modern LES of two-phase flows. The channel flow simulations are based on large-eddy approximations of the compressible Navier–Stokes equations in a porous medium. The Euler–Lagrangian method is adopted, which means that for each individual particle an equation of motion is solved. The method incorporates four-way coupling, i.e., both the particle-fluid and particle–particle interactions are taken into account. The results are compared to single-phase channel flow in order to investigate the effect of the particles on turbulent statistics. The present results show that due to particle–fluid interactions the mean fluid profile is flattened and the boundary layer is thinner. Compared to single-phase turbulent flow, the streamwise turbulence intensity of the gas phase is increased, while the normal and spanwise turbulence intensities are reduced. This finding is generally consistent with existing experimental data. The four-way coupled simulations are also compared with two-way coupled simulations, in which the inelastic collisions between particles are neglected. The latter comparison clearly demonstrates that the collisions have a large influence on the main statistics of both phases. In addition, the four-way coupled simulations contain stronger coherent particle structures. It is thus essential to include the particle–particle interactions in numerical simulations of two-phase flow with volume fractions around one percent.