Tollmien-Schlichting/vortex interactions in compressible boundary-layer flows

A weakly nonlinear interaction of oblique Tollmien-Schlichtingwaves and longitudinal vortices in compressible, high Reynoldsnumber, boundary-layer flow over a flat plate is consideredfor all ranges of the Mach number. The interaction equationsconsist of equations for the vortex which is indirectly forcedby the waves via a boundary condition, whereas a vortex termappears in the amplitude equation for the wave pressure. Thedownstream solution properties of interaction equations arefound to depend on the sign of an interaction coefficient. Thisparticular type of weakly nonlinear interaction was first proposedby Hall & Smith (1989), who considered incompressible flows;however, there are some errors in their formulation. Correctedresults for the incompressible regime are presented for comparisonwith those calculated for compressible flows. Compressibilityis found to have a significant effect on the interaction properties,principally through its impact on the waves and their governingmechanism, the ‘triple-deck’ structure. It is foundthat, in general, the flow quantities will grow slowly withincreasing downstream coordinate. However, for flows with Machnumber values below 2, there exists a small band of wave obliquenessangles for which the solutions terminate in abrupt, finite-distance‘break-ups’.     

Rotating flows with acceleration and compaction in the model of hard spheres

We construct a model of the interaction between two flows in a gas of hard spheres, each of which has the structure of an accelerating and compacting rotating flow. We obtain several conditions sufficient for the uniform integral or the “mixed” norm of the difference between the left- and right-hand sides of the Boltzmann equation to be arbitrarily small.     

Numerical analysis of the Navier–Stokes/Darcy coupling

We consider a differential system based on the coupling of the Navier–Stokes and Darcy equations for modeling the interaction between surface and porous-media flows. We formulate the problem as an interface equation, we analyze the associated (nonlinear) Steklov–Poincaré operators, and we prove its well-posedness. We propose and analyze iterative methods to solve a conforming finite element approximation of the coupled problem.     

A Survey on dynamical transport distances

In this paper, we review some transport models based on the continuity equation, starting with the so-called Benamou − Brenier formula, which is nothing but a fluid mechanics reformulation of the Monge − Kantorovich problem with cost c(x, y) = |x − y|². We discuss some of its applications (gradient flows, sharp functional inequalities, etc.), as well as some variants and generalizations to dynamical transport problems, where interaction effects among mass particles are considered. Bibliography: 43 titles.     

Investigation of surface destruction of construction materials under intense periodic heat transfer

The study simulates the surface destruction of structural steel elements in power installations as a result of interaction with periodic high-temperature chemically active flows. The system of equations of thermoelasticity with two spatial observations is used. The existence of high tangential and axial compressive stresses is established in a thin layer near the metal-gas interface. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996. pp. 115–118.     

On geometric properties of stochastic flows related to the Lyapunov spectrum

We study the geometric properties of two stochastic flows on spheres in Euclidean space. The underlying one-point motion in both cases is Brownian. Both flows arise from the action of a Lie group valued Brownian motion on a quotient. For both flows the curvature of a curve moving under the flow is shown to be a diffusion, null recurrent in one case and transient in the other. Received: 30 April 1999 / Revised version: 4 October 1999 / Published online: 8 August 2000     

On the stability of jetlike magnetohydrodynamic flows

The linear stability problem is under study for steady axisymmetric translational flows of a density-homogeneous nonviscous incompressible ideal conducting fluid with free surface and “frozen-in” poloidal magnetic field. By the direct Lyapunov method, some sufficient conditions are obtained for the stability of these flows under small long-wave perturbations with the same symmetry. These stability conditions have partial converses; and, for unstable stationary flows, an a priori exponential lower bound is constructed on the growth of small perturbations under consideration, while the increment of the appearing exponent serves as an arbitrary positive parameter. An illustrative analytical example is given of steady flows with superimposed small long-wave axisymmetric perturbations growing in time in accordance with the estimate.     

Variation of drag coefficient with wind wave development

Turbulent air flows over developing wind waves in the air-sea boundary layer are numerically simulated without considering wave breaking. Influences of wind waves on air flows are considered using a model of significant wave and surface roughness, with a formula proposed for calculating the surface roughness. κ-ε model is adopted to simulate turbulent flows. The results of the drag coefficient and turbulence characteristics agree well with the observations. Project supported by the National Natural Science Foundation of China (Grant No. 19332010).     

On upper bounds for the growth rate in the extended Taylor-Goldstein problem of hydrodynamic stability

For the extended Taylor-Goldstein problem of hydrodynamic stability governing the stability of shear flows of an inviscid, incompressible but density stratified fluid in sea straits of arbitrary cross-section a new estimate for the growth rate of an arbitrary unstable normal mode is given for a class of basic flows. Furthermore the Howard’s conjecture, namely, the growth rate kc _i → 0 as the wave number k → ∞ is proved for two classes of basic flows.     

Multicomponent generalization of the hierarchy of the Landau-Lifshitz equation

We construct a second-order 2N-component integrable system (with arbitrary N) whose spectral parameter lies on a curve of genus g=1+(N-3)2^N−2. The odd-order flows admit N-component reductions, which for N=3 coincide with the odd-order flows of the hierarchy of the Landau-Lifshitz equation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 1, pp. 62–71, July, 2000.     

Acceleration of heavy-current ionic flows

The process of acceleration of heavy-current ionic flows in a magnetic plasma emitter is studied. Results of interaction of ionic flows under their motion in the emitter are obtained. In view of the small mobility of electrons in acceleration directions it was possible to involve ions of the whole volume of emitter in the acceleration process. The interaction of the ionic flow with the electrode of the accelerator determined the work of the ionic accelerator. With the help of the energy of an electric field a forced cooling of the plasma emitter and acceleration of 700 ampere ionic flows were realized.Translated from Dinamicheskie Sistemy, No. 8, pp. 107–113, 1989.     

Solutions to the mixed problem of viscous incompressible flows in a channel

We study regularity of viscous incompressible fluid flows in a 2D channel with “do nothing” outflow boundary condition on the output for the steady Stokes and Navier–Stokes equations.     

A topological classification of integrable geodesic flows on the two-dimensional sphere with an additional integral quadratic in the momenta

Summary The analytic expression for a Riemannian metric on a 2-sphere, having integrable geodesic flow with an additional integral quadratic in momenta, is given in [Ko1]. We give the topological classification, up to topological equivalence of Liouville foliations, of all such metrics. The classification is computable, and the formula for calculating the complexity of the flow is straightforward. We prove Fomenko's conjecture that, from the point of view of complexity, the integrable geodesic flows with an additional integral linear or quadratic in momenta exhaust “almost all” integrable geodesic flows on the 2-dimensional sphere.     

Stochastic flows associated to coalescent processes

 We study a class of stochastic flows connected to the coalescent processes that have been studied recently by M?hle, Pitman, Sagitov and Schweinsberg in connection with asymptotic models for the genealogy of populations with a large fixed size. We define a bridge to be a right-continuous process (B(r),r[0,1]) with nondecreasing paths and exchangeable increments, such that B(0)=0 and B(1)=1. We show that flows of bridges are in one-to-one correspondence with the so-called exchangeable coalescents. This yields an infinite-dimensional version of the classical Kingman representation for exchangeable partitions of ℕ. We then propose a Poissonian construction of a general class of flows of bridges and identify the associated coalescents. We also discuss an important auxiliary measure-valued process, which is closely related to the genealogical structure coded by the coalescent and can be viewed as a generalized Fleming-Viot process. Received: 26 November 2002 / Revised version: 10 February 2003 / Published online: 15 April 2003 Mathematics Subject Classification (2000): 60G09, 60J25, 92D30 Key words or phrases: Flow – Coalescence – Exchangeability – Bridge     

On the unbounded increase in vorticity and velocity circulation in stratified and homogenous fluid flows

We consider the Oberbeck-Boussinesq system without dissipation (ideal convection) in a horizontal layer and in a “barrel” with flat bottom and flat cover. It is shown that the velocity circulation along a fluid contour consisting of two fluid curves on the bottom and on the cover connected by two isothermic fluid curves can be calculated explicitly and is a linear function of time. The serre result stating that the azimuthal component of vorticity in rotationally symmetric ideal fluid flows between coaxial cylinders increases linearly is generalized to the case of stratified fluids. It is proved that all plane and axially symmetric isothermic flows in a layer or in a barrel are unstable with respect to nonisothermic perturbations and in the case of a homogenous fluid all axially symmetric flows between coaxial cylinders are unstable in the sense of Lyapunov with respect to perturbations of the azimuthal component of the velocity in any metric including the maximum of magnitude of vorticity. Translated fromMaternaticheskie Zametki, Vol. 68, No. 4, pp. 627–636, October, 2000.     

Uniform Shrinking and Expansion under Isotropic Brownian Flows

We study some finite time transport properties of isotropic Brownian flows. Under a certain nondegeneracy condition on the potential spectral measure, we prove that uniform shrinking or expansion of balls under the flow over some bounded time interval can happen with positive probability. We also provide a control theorem for isotropic Brownian flows with drift. Finally, we apply the above results to show that, under the nondegeneracy condition, the length of a rectifiable curve evolving in an isotropic Brownian flow with strictly negative top Lyapunov exponent converges to zero as t→∞ with positive probability. P. Baxendale’s research was supported in part by NSF Grant DMS-05-04853.     

The centralizer of a rank-one flow

In [5], King proved that the centralizer of a rank-1 transformation equals the “weak closure” of its (positive and negative) powers (see below for a definition of the weak topology). We define rank-1 flows, and then show that simple modifications of King's proof yield an analogous statement for rank-1 flows.     

Geometric evolution equations in critical dimensions

We make a qualitative comparison of phenomena occurring in two different geometric flows: the harmonic map heat flow in two space dimensions and the Yang–Mills heat flow in four space dimensions. Our results are a regularity result for the degree-2 equivariant harmonic map flow, and a blow-up result for an equivariant Yang–Mills-like flow. The results show that qualitatively differing behaviours observed in the two flows can be attributed to the degree of the equivariance.     

The contact angle in inviscid fluid mechanics

We show that in general, the specification of a contact angle condition at the contact line in inviscid fluid motions is incompatible with the classical field equations and boundary conditions generally applicable to them. The limited conditions under which such a specification is permissible are derived; however, these include cases where the static meniscus is not flat. In view of this situation, the status of the many ‘solutions’ in the literature which prescribe a contact angle in potential flows comes into question. We suggest that these solutions which attempt to incorporate a phenomenological, but incompatible, condition are in some, imprecise sense ‘weak-type solutions’; they satisfy or are likely to satisfy, at least in the limit, the governing equations and boundary conditions everywhere except in the neighbourhood of the contact line. We discuss the implications of the result for the analysis of inviscid flows with free surfaces.     

The central limit theorem for geodesic flows onn-dimensional manifolds of negative curvature

In this paper we prove a central limit theorem for special flows built over shifts which satisfy a uniform mixing of type , 0<γ<1, α>0. The function defining the special flow is assumed to be continuous with modulus of continuity of type , 0<ρ<1, β>0, andd is the natural metric on sequence space. Geodesic flows on compact manifolds of negative curvature are isomorphic to special flows of this kind.