Investigation of the far and near fields in the problem of exponentially stratified flow over a bottom irregularity

The three-dimensional problem of finite-depth stratified flow over a small bottom irregularity is considered in mixed Euler-Lagrange variables. The Brunt-Väisälä frequency is assumed to be constant and small, and the free surface condition is replaced by the rigid roof condition. Investigation of the far field showed that the principal wave perturbations lie within an angle which for large values of the internal Froude number is much less than theKelvin angle, while the wave amplitude at infinity is of the order of l/r, where r is the polar radius. The ring perturbations are exponentially damped. As distinct from point source models, the model in question does not lead to divergence of the integrals on the flow axis [1-3]. Appproximate expressions for the radial and ring waves in terms of certain universai functions were obtained for investigating the near and far fields when the bottom irregularity is hemispherical. For the radial waves a law of similarity was obtained for which the characteristic dimension in the direction of the flow axis is the ratio of the flow velocity to the Brunt-Väisälä frequency, and the characteristic dimension in a direction perpendicular to the flow axis the depth of the fluid. In the first approximation the ring perturbations do not depend on the Brunt-Väisälä frequency. It is shown that in the near field the zone of intense wave perturbations is of the order of the fluid depth and not of the dimensions of the obstacle as for Kelvin ship waves on the surface of a homogeneous fluid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 86–94, September–October, 1987.     

Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media

An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow-through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite-dimensional manifold. An abstract theory is developed for gradient flows in length spaces, which shows how degenerate convexity (or even non-convexity) — if uniformly controlled — will quantify contractivity (limit expansivity) of the flow.     

Improved bounds on the energy-minimizing strains in martensitic polycrystals

This paper is concerned with the theoretical prediction of the energy-minimizing (or recoverable) strains in martensitic polycrystals, considering a nonlinear elasticity model of phase transformation at finite strains. The main results are some rigorous upper bounds on the set of energy-minimizing strains. Those bounds depend on the polycrystalline texture through the volume fractions of the different orientations. The simplest form of the bounds presented is obtained by combining recent results for single crystals with a homogenization approach proposed previously for martensitic polycrystals. However, the polycrystalline bound delivered by that procedure may fail to recover the monocrystalline bound in the homogeneous limit, as is demonstrated in this paper by considering an example related to tetragonal martensite. This motivates the development of a more detailed analysis, leading to improved polycrystalline bounds that are notably consistent with results for single crystals in the homogeneous limit. A two-orientation polycrystal of tetragonal martensite is studied as an illustration. In that case, analytical expressions of the upper bounds are derived and the results are compared with lower bounds obtained by considering laminate textures.     

Instability of the flow around an impacting sphere

Results from an experimental and numerical study of the flow generated by a sphere immersed in fluid, impacting normally without rebound on a solid wall, are presented. The parameters are the running distance before impact and the sphere Reynolds number. For running lengths less than 7.5 diameters, the sphere wake before impact is axisymmetric in the form of an attached vortex ring. After impact, this ring overtakes the sphere and spreads out along the wall. For Reynolds numbers below 1000, the flow remains axisymmetric at all times. For higher values, perturbations of azimuthal wavenumbers 20–25 are observed on the vortex ring, leading to a breakdown of the flow. We analyse different hypotheses concerning the origin of this instability, with the conclusion that a centrifugal instability mechanism is likely to be acting in this flow. Comparisons are made with the flow involving an isolated vortex ring approaching a wall. Numerical simulations of this case have revealed that two distinct instability mechanisms are operating, one of which appears to be similar to the centrifugal instability observed for the sphere impact.     

On the Perturbation of Hamel Flow Due to Unevenness of the Channel Walls

A method of analyzing the receptivity of longitudinally inhomogeneous flows is proposed. The process of excitation of natural oscillations is studied with reference to the simplest inhomogeneous flow: the two-dimensional flow of a viscous incompressible fluid in a channel with plane nonparallel walls. As physical factors generating perturbations, the cases of a stationary irregularity and localized vibration of the channel walls are considered. By changing the independent variables and unknown functions of the perturbed flow, the problem of the generation of stationary perturbations above an irregularity is reduced to a longitudinally homogeneous boundary-value problem which is solved using a Fourier transform in the longitudinal variable. The same problem is investigated using another method based on representing the required solution in the form of a superposition of solutions of the homogeneous problem and a forced solution calculated in the locally homogeneous approximation. As a result, the problem of calculating the longitudinal distributions of the amplitudes of the normal modes is reduced to the solution of an infinite-dimensional inhomogeneous system of ordinary differential equations. The numerical solution obtained using this method is tested by comparison with an exact calculation based on the Fourier method. Using the method proposed, the problem of flow receptivity to harmonic oscillations of parts of the channel walls is analyzed. The calculations performed show that the method is promising for investigating the receptivity of longitudinally inhomogeneous flow in a laminar boundary layer.     

On the problem of the stability of convective two-phase medium flows under high-frequency vibration

The principles and methods of constructing a model of vibrational convection in a medium consisting of a liquid (gas) and a solid admixture are discussed. A closed system of averaged equations is first obtained. The system admits passage to the limits of the equations of both vibrational convection in a homogeneous fluid and convection in a dusty medium in the static case. As a measure of the difference with respect to the homogeneous fluid, in addition to the sedimentation parameter, which also manifests itself in the absence of vibrational accelerations, it is possible to take the inhomogeneity parameter introduced in this study and responsible for the pulsatory transport of the average fields. The problem of the stability of plane parallel flow in a vertical layer of a two-phase medium under horizontal longitudinal vibration with respect to infinitesimal perturbations is considered. It is shown that the introduction of particles into the flow leads to qualitatively novel effects which cannot be predicted within the framework of the homogeneous fluid model.     

Spatial Evolution of Nonstationary Perturbations in Hamel Flow

The propagation of small perturbations in longitudinally inhomogeneous flows is investigated. The evolution of the perturbations is studied with reference to the radial flow of a viscous incompressible fluid between plane nonparallel walls, the simplest inhomogeneous flow. Using a generalized method of variation of constants, the corresponding boundary-value problem is reduced to an infinite-dimensional evolutionary system of ordinary differential equations for the complex amplitudes of the eigensolutions of a locally homogeneous problem. Physically, the method can be interpreted as a representation of the perturbation evolution process via two concomitant processes: the independent amplification (attenuation) of normal modes of the locally homogeneous problem and the rescattering of these modes into each other on local inhomogeneities of the base flow. The calculations show that reduced versions of the method are promising for describing the linear stage of laminar-turbulent transition in a boundary layer.     

On asymptotic theory of separated flows past bodies

Symmetric two-dimensional steady flow past a body in a homogeneous incompressible fluid stream at high Reynolds numbers is considered. A slow motion in the reverse flow zone is investigated and the solution for the flow in the external region is obtained in the second approximation. Additional considerations of the fact that the flow in the closure region of the separation zone and in the wake behind this zone is turbulent are presented. The laminar-turbulent transition in the mixing layer is analyzed and an analogy between this process and the propagation of perturbations upstream of the boundary layer interaction regions is revealed.     

Simulation of turbulent dispersion using Weierstrass modes

A simple fractal model is proposed for the dispersion of passive scalars in an incompressible homogeneous turbulent flow field. The dispersion process is based on a three-dimensional velocity field which is assumed to be a linear superposition of Taylor-Green vortices with wave numbers and amplitudes as those in a Weierstrass function. The chosen velocity field satisfies the continuity equation as well as Kolmogorov's inertial range power law, and has a fractal dimension D between upper and lower length-scale bounds. A cloud of tracer particles is relased into the flow field and dispersed by the fluid motion, the trajectories are numerically integrated from the velocity field. The puffs which result from this process are used to examine some aspects of turbulent dispersion, through comparisons with integrated concentration wind-tunnel measurements. The agreement between numerical and experimental results indicates the significance of the proposed simulation model.     

Long-wave instability of a vortex ring

The instability of barrel-shaped vibrations of a vortex ring in an ideal fluid is investigated. These vibrations, stable for a vortex ring with a piecewise-uniform vorticity profile, appear to be unstable for a vortex ring with a smooth vorticity profile. The instability growth rate is found on the basis of the energy balance equation determining the energy transport from perturbations with negative energy in the critical layer to perturbations with positive energy in the rest of the flow. The curvature of the vortex ring, by virtue of which the perturbations with energies of different signs appear to be connected, plays a prominent role in the mechanism under consideration.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 72–78, November–December, 1995.     

Onset of convection in a multicomponent fluid layer in the presence of a uniform magnetic field

The principle of the exchange of stabilities for magnetohydrodynamic multicomponent convection is established. If this sufficient condition holds and there are perturbations, oscillatory motions of neutral or growing amplitude can exist in the fluid. The upper bounds for the complex growth rate of such motions when at least one of the boundaries is rigid are obtained.     

Energy analysis of development of kinematic perturbations in weakly inhomogeneous viscous fluids

The deformation stability relative to small perturbations is analyzed for weakly inhomogeneous viscous media on the assumption that both the main flow and perturbation field are three-dimensional. To test the damping or growth of initial perturbations, sufficient estimates based on the use of variational inequalities in different function spaces (energy estimates) are obtained. The choice of function space determines the measures of the parameter deviations, which may be different for the initial and current parameters. The unperturbed process chosen is a fairly arbitrary unsteady flow of homogeneous incompressible viscous fluid in a three-dimensional region of Eulerian space. At the initial instant, not only the kinematics of the motion but also the density and viscosity of the fluid are disturbed and the medium is therefore called weakly inhomogeneous. On the basis of the integral relation methods developed in recent years, sufficient integral estimates are obtained for lack of perturbation growth in the mean-square sense (in theL ₂ space measure). The rate of growth or damping of the kinematic perturbations depends linearly on the initial variations of the kinematics, density and viscosity. Illustrations of the general result are given. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 56–67, March–April, 2000. The work was supported by the Russian Foundation for Basic Research (projects No. 99-01-00125 and No. 99-01-00250) and by the Federal Special “Integration” Program (project No. 426).     

The Onset of Double-Diffusive Convection in a Superposed Fluid and Porous Layer Under High-Frequency and Small-Amplitude Vibrations

We numerically simulate the initiation of an average convective flow in a system composed of a horizontal binary fluid layer overlying a homogeneous porous layer saturated with the same fluid under gravitational field and vibration. In the layers, fixed equilibrium temperature and concentration gradients are set. The layers execute high-frequency oscillations in the vertical direction. The vibration period is small compared with characteristic timescales of the problem. The averaging method is applied to obtain vibrational convection equations. Using for computation the shooting method, a numerical investigation is carried out for an aqueous ammonium chloride solution and packed glass spheres saturated with the solution. The instability threshold is determined under two heating conditions—on heating from below and from above. When the solution is heated from below, the instability character changes abruptly with increasing solutal Rayleigh number, i.e., there is a jump-wise transition from the most dangerous shortwave perturbations localized in the fluid layer to the long-wave perturbations covering both layers. The perturbation wavelength increases by almost 10 times. Vibrations significantly stabilize the fluid equilibrium state and lead to an increase in the wavelength of its perturbations. When the fluid with the stabilizing concentration gradient is heated from below, convection can occur not only in a monotonous manner but also in an oscillatory manner. The frequency of critical oscillatory perturbations decreases by 10 times, when the long-wave instability replaces the shortwave instability. When the fluid is heated from above, only stationary convection is excited over the entire range of the examined parameters. A lower monotonic instability level is associated with the development of perturbations with longer wavelength even at a relatively large fluid layer thickness. Vibrations speed up the stationary convection onset and lead to a decrease in the wavelength of most dangerous perturbations of the motionless equilibrium state. In this case, high enough amplitudes of vibration are needed for a remarkable change in the stability threshold. The results of numerical simulation show good agreement with the data of earlier works in the limiting case of zero fluid layer thickness.     

Stability of advective flow in an inclined plane fluid layer bounded by solid planes with a longitudinal temperature gradient. 1. Unstable stratification

The stability of steady convective flow in an inclined plane fluid layer bounded by ideally heat conducting solid planes is studied in the presence of a homogeneous longitudinal temperature gradient under unstable stratification conditions where the layer is inclined so that the temperature is higher in the lower part than in the upper part. It is shown that the inclination leads to the transition from critical perturbations to long-wavelength helical perturbations. Flow stability maps are given for the entire range of Prandtl numbers and inclination angles corresponding to unstable stratification.     

Stability of Rheometric Viscoelastic Fluid Flow with Respect to Shear Perturbations

Flow between two plates is considered for a fluid obeying the DeWitt rheological equation of state with the Jaumann derivative. It is found analytically that the steady-state Couette flow is stable or unstable with respect to plane shear perturbations when the Weissenberg numbers are less or greater than unity, respectively. The flow acceleration stage is studied analytically and numerically, a comparison with the case of an Oldroyd fluid is carried out, and the neutral stability curves are constructed. The fundamental role of perturbations of the type considered among the set of instability types which can act on the fluid in such a flow is noted.     

Flow stability of a flat plastic ring with free boundaries

The problem of two-dimensional unstable flow of an ideally plastic ring acted upon by internal pressure is formulated. The determination of the law of motion for the boundaries and of the time change of pressure is reduced to an ordinary nonlinear differential equation of the second order. For this equation a particular solution of the Cauchy problem is determined; this corresponds to a widening of the ring boundaries with a negative acceleration. For the field of initial velocities an estimate from above is available, expressed in terms of the original parameters. The very particular unstable flow obtained for an ideally plastic ring is also investigated with respect to stability to small harmonic perturbations of the velocity vector, the pressure, or the boundaries of the ring. It is shown that the fundamental flow is stable irrespective of the wave number. This result has been obtained by assuming that the inertial forces in the perturbed flow are small compared to the lasting ones.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 94–101, March–April, 1975.     

Rayleigh stability of shear flow in relaxing media

The stability of steady-state flow is considered in a medium with a nonlocal coupling between pressure and density. The equations for perturbations in such a medium are derived in the linear approximation. The results of numerical integration are given for shear motion. The stability of parallel layered flow in an inviscid homogeneous fluid has been studied for a hundred years. The mathematics for investigating an inviscid instability has been developed, and it has been given a physical interpretation. The first important results in flow stability of an incompressible fluid were obtained in the papers of Helmholtz, Rayleigh, and Kelvin [1] in the last century. Heisenberg [2] worked on this problem in the 1920's, and a series of interesting papers by Tollmien [3] appeared subsequently. Apparently one of the first problems in the stability of a compressible fluid was solved by Landau [4]. The first investigations on the boundary-layer stability of an ideal gas were carried out by Lees and Lin [5], and Dunn and Lin [6]. Mention should be made of a series of papers which have appeared quite recently [7–9]. In all the papers mentioned flow stability is investigated in the framework of classical single-phase hydrodynamics. Meanwhile, in recent years, the processes by which perturbations propagate in media with relaxation have been intensively studied [10–12].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 87–93, May–June, 1976.     

Simple Shear Flow of Suspensions of Elastic Capsules

The simple shear flow of homogeneous suspensions of two-dimensional capsules enclosed by elastic membranes is studied in the limit of vanishing Reynolds number, in the special case where the viscosity of the fluid enclosed by the capsules is equal to the viscosity of the ambient fluid. The deformation of capsules with circular, elliptical, and biconcave unstressed shapes, and the rheological and statistical properties of their infinitely dilute and moderately dense suspensions are investigated by dynamical simulation using the method of interfacial dynamics for Stokes flow. In a preliminary investigation, the behavior of solitary capsules suspended in an infinite fluid is studied as a function of the dimensionless membrane elasticity number expressing the capsule deformability or the strength of the shear flow. It is found that a critical elasticity number above which a capsule exhibits continued elongation does not exist, and an equilibrium configuration is reached no matter how large the shear rate, in agreement with previous results for three-dimensional flow. A correspondence is established between the elasticity numbers for two- and three-dimensional flow at which the capsules undergo the same degree of deformation. Simulations of pairwise capsule interceptions reveal behavior similar to that exhibited by liquid drops with uniform surface tension. Because of strong hydrodynamic interactions in two-dimensional Stokes flow, the concept of hydrodynamic diffusivity in the limit of infinite dilution is ill-defined in the absence of fluid inertia. Dynamical simulations of doubly periodic monodisperse suspensions with up to 50 capsules distributed in each periodic cell at areal fractions of 0.25 and 0.40 provide information on the effective rheological properties of the suspension and on the nature of the statistical properties of the particle motion. The character of the flow is found to be intermediate between that of liquid drops and rigid particles, and this is attributed to the membrane deformability and to the ability of the interfaces to perform tank-treading motion. The results are compared with rheological measurements of blood flow with good agreement. Received 26 April 1999 and accepted 5 October 1999     

Bounds on the drag for creeping flow of an ellis fluid past an assemblage of spheres

Upper and lower bounds for the creeping flow of an Ellis fluid past an assemblage of solid spheres are obtained using a combination of Happel's free surface model and variational principles. The arithmetic mean of the bounds agrees closely with the experimental data on flow through porous media. For Ellis numbers approaching infinity, the analysis also predicts the bounds for a power law fluid.