The legendre condition in optimum problems of supersonic gasdynamics : PMM vol. 39, n 6, 1975, pp. 1032–1042

The necessary Legendre condition for problems of optimum (in the sense of minimum wave drag) supersonic flow past bodies is obtained. Plane and axisymmetric flows are considered on the assumption of imposition of isoperimetric constraints of a general form. Shock-free flows and flows with attached shock waves are investigated. The method here proposed is used for deriving the second order condition in the particular case when it is possible to pass to the reference contour, and which has been earlier obtained by Shmyglevskii [1] and then by Guderley and others [2].     

The Evolution of Large-Horizontal-Scale Disturbances in Marginally Stable,Inviscid, Shear Flows. II. Solutions of the Boussinesq Equation

In part I the Boussinesq equation was found to describe large-horizontal-scale disturbances in certain marginally stable flows. In this paper the properties of the solutions of this equation are explored. Solutions that are bounded for all time and those that blow up in finite time are observed analytically and numerically. Sufficient conditions for both boundedness and blowup are given.     

Numerical Simulation of Incompressible Flows Within Simple Boundaries. I. Galerkin (Spectral) Representations

Galerkin (spectral) methods are explored for the numerical simulation of incompressible flows within simple boundaries. A major part of the paper is devoted to the development of transform methods for efficient simulation of flows in box geometries with periodic and free-slip boundary conditions. Techniques for incorporating known symmetries and invariances into transform methods are illustrated for the Taylor-Green vortex. Galerkin methods for accurate and efficient representation of rigid no-slip boundary conditions are also explained. A class of pseudospectral approximations is introduced in order to handle more complicated dynamical interactions in more complicated geometries. Later papers in this series will demonstrate the important advantages of spectral methods over finite-difference methods for simulation of many of the flows of current interest and will present specific numerical results for various transition and turbulent flows.     

On the “equivalence” rule for flows of perfect gas : PMM vol. 38, n 6, 1974, pp. 1004–1014

An extension of the equivalence of “area” rule [1, 2] is presented. The rule was initially derived for stationary flows of perfect (inviscid and non-heat-conducting) gas past slender fine pointed bodies (or blunted bodies in the hypersonic flow case) whose transverse dimensions are small in comparison with their length. According to that rule the wave drag of a three-dimensional body is equal to the wave drag of an axisymmetric body with the same distribution of cross-sectional areas along the axis. The rule is extended here to stationary and nonstationary flows past nonslender bodies and to internal flows, using the procedure of averaging with respect to the angular variable of a cylindrical system of coordinates. That procedure is, strictly speaking, valid for nearly axisymmetric bodies. However the numerical solutions obtained by the authors for a fairly wide range of external and internal problems show that the generalized equivalence rule is applicable to substantially nonaxisymmetric configurations (^*) (see next page).     

Supersonic flow past triangular wings with ribs on their surface PMM vol. 31, no. 6, 1967, pp. 1050–1056

Various types of partitions are a common feature of lifting surfaces. These partitions can take the form of stiffening ribs, deflectors for preventing secondary flows or flow separation, etc. The presence of partitions has a marked effect on the character of flow and on the values of the aerodynamic parameters. Flow past such wings cannot be computed in the general case. Wings of a special type are amenable to simple solution, however, and this will be considered below. One special case of interaction between a partition and an infinite wing is also considered in [1].     

Dissipation,topography, and statistical theories for large-scale coherent structure

Various facets of the equilibrium statistical theories for large-scale coherent structures for two-dimensional flows with and without topography are studied here. The classical few-constraint statistical theories involving energy-enstrophy principles or point vortices are shown to be statistically sharp in the more recent statistical theories with an infinite number of constraints; in other words, at the macrostates of the few-constraint theories, the many-constraint theory provides no additional statistical information. These results are established through a general link between these statistical theories, generalized “selective decay” principles, and statistical sharpness. Through an asymptotic procedure, the many-constraint statistical theories for flows with topography and small-potential vorticity are shown to yield the simpler energy-enstrophy macroscopic states at leading order with systematic higher-order corrections involving a renormalized topography that includes higher moments of the microscopic potential vorticity distribution. For nonequilibrium flows with and without topography, the utility of crude approximate dynamics based on “adiabatic approximation” through the macrostates of few-constraint statistical theory is developed here. It is established that for nonequilibrium decaying flows with viscous dissipation, the crude dynamics based on macrostates involving statistical point vortices yields an excellent approximation; the role of “selective decay” principles is also clarified and compared quantitatively in this context through both mathematical theory and numerical experiments. Surprisingly, these approximate dynamics yield a much poorer approximation with moderate Ekman drag as the dissipative mechanism, and a simple analytical explanation is provided here. Finally, all of these issues are pursued more briefly for damped and driven flows with topography. © 1997 John Wiley & Sons, Inc.     

Flows op a reacting mixture in laval nozzles under conditions op a quasi-frozen process : PMM vol. 39, n 6, 1975, pp. 1068–1072

Flows of a chemically active gas mixture are considered in a small region of a La val nozzle, where their mode changes from subsonic to supersonic (the frozen speed of sound is considered) are analyzed. Continuous solutions and solutions with shock waves are derived. Conditions of shock-free flows are obtained.     

The projective translation equation and rational plane flows. II. Corrections and additions

In this second part of the work, we correct the flaw which was left in the proof of the main Theorem in the first part. This affects only a small part of the text in this first part and two consecutive papers. Yet, some additional arguments are needed to claim the validity of the classification results. With these new results, algebraic and rational flows can be much more easily and transparently classified. It also turns out that the notion of an algebraic projective flow is a very natural one. For example, we give an inductive (on dimension) method to build algebraic projective flows with rational vector fields, and ask whether these account for all such flows. Further, we expand on results concerning rational flows in dimension 2. Previously we found all such flows symmetric with respect to a linear involution \(i_{0}(x,y)=(y,x)\). Here we find all rational flows symmetric with respect to a non-linear 1-homogeneous involution \(i(x,y)=(\frac{y^2}{x},y)\). We also find all solenoidal rational flows. Up to linear conjugation, there appears to be exactly two non-trivial examples.     

On Wave Interactions III. Miscellaneous Results

In this paper, a collection of results pertaining to resonant triads in fluid flow systems is presented. The main topic is the extension of the analysis by Mahoney for wind-driven shear flows to general, layered flows, that is to say, flows which have piecewise continuous velocity and density profiles. It is seen how the results for calculating the triad interaction coefficients for two-layer flows generalize and carry over for multilayer flows.     

Geometric Structures, Lobe Dynamics, and Lagrangian Transport in Flows with Aperiodic Time-Dependence, with Applications to Rossby Wave Flow

Summary. In this paper we develop the mathematical framework for studying transport in two-dimensional flows with aperiodic time dependence from the geometrical point of view of dynamical systems theory. We show how the notion of a hyperbolic fixed point, or periodic trajectory, and its stable and unstable manifolds generalize to the aperiodically time-dependent setting. We show how these stable and unstable manifolds act as mediators of transport, and we extend the technique of lobe dynamics to this context. We discuss Melnikov's method for two classes of systems having aperiodic time dependence. We develop a numerical method for computing the stable and unstable manifolds of hyperbolic trajectories in two-dimensional flows with aperiodic time dependence. The theory and the numerical techniques are applied to study the transport in a kinematic model of Rossby wave flow studied earlier by Pierrehumbert [1991a]. He considered flows with periodic time dependence, and we continue his study by considering flows having quasi-periodic, wave-packet, and purely aperiodic time dependencies. These numerical simulations exhibit a variety of new transport phenomena mediated by the stable and unstable manifolds of hyperbolic trajectories that are unique to the case of aperiodic time dependence. Received April 10, 1997; revision received August 1, 1997 and accepted October 7, 1997     

Tu方程族的高阶双约束流的分离变量

本文给出了Tu方程族的高阶双约束流,其自由度为2N＋l。根据通常的办法,利用Lax矩阵仅能引入N＋l对标准分离变量和N＋l个分离变量方程。本文构造出另外N对分离变量及N个分离变量方程。此外,还建立了双约束流和Tu方程族的Jacobi反演问题。     

The output of a switch,or, effective bandwidths for networks

Consider a switch which queues traffic from many independent input flows. We show that in the large deviations limiting regime in which the number of inputs increases and the service rate and buffer size are increased in proportion, the statistical characteristics of a flow are essentially unchanged by passage through the switch. This significantly simplifies the analysis of networks of switches. It means that each traffic flow in a network can be assigned an effective bandwidth, independent of the other flows, and the behaviour of any switch in the network depends only on the effective bandwidths of the flows using it. This revised version was published online in June 2006 with corrections to the Cover Date.     

Guo族约束流的经典Poisson结构、Lax表示、r-矩阵和Liouville可积性

基于伴随表示,通过引入Ｊａｃｏｂｉ－Ｏｓｔｒｏｇｒａｄｓｋｙ坐标,获得了Ｇｕｏ族约束流的Ｌａｘ表示,Ｐｏｉｓｓｏｎ结构和ｒ－矩阵,最后,借助Ｐｏｉｓｓｏｎ结构和ｒ－矩阵,说明了Ｇｕｏ族约束 是Ｌｉｏｕｖｉｌｌｅ可积的。     

Symmetries for the Ablowitz–Ladik Hierarchy: Part II. Integrable Discrete Nonlinear Schrödinger Equations and Discrete AKNS Hierarchy

In the paper, we continue to consider symmetries related to the Ablowitz–Ladik hierarchy. We derive symmetries for the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. The integrable discrete nonlinear Schrödinger hierarchy is in scalar form and its two sets of symmetries are shown to form a Lie algebra. We also present discrete AKNS isospectral flows, non‐isospectral flows and their recursion operator. In continuous limit these flows go to the continuous AKNS flows and the recursion operator goes to the square of the AKNS recursion operator. These discrete AKNS flows form a Lie algebra that plays a key role in constructing symmetries and their algebraic structures for both the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. Structures of the obtained algebras are different structures from those in continuous cases which usually are centerless Kac–Moody–Virasoro type. These algebra deformations are explained through continuous limit and degree in terms of lattice spacing parameter h.     

Directed submodularity,ditroids and directed submodular flows

Set relations and operations such as inclusion, union and intersection are generalized to directed subsets whose elements are distinguished between forward and backward elements. The concepts of submodular functions, matroids and polymatroidal network flows are extended to the concepts of directed submodular functions, ditroids and directed submodular flows on directed subsets. Two unrelated matroids (submodular functions) can be embedded in one ditroid (directed submodular function). Total dual integrality is preserved in these generalizations and proved for very general set-function class-directed odd submodular functions.This work was partially supported by Chinese National Natural Science Fund.     

On the behavior of solutions of equations for double waves in the neighborhood of the quiescent region : PMM vol. 39, n 6, 1975, pp. 1043–1050

The structure of solutions of gasdynamic equations is investigated in the case of unsteady double waves in the neighborhood of the quiescent region. A general concept of double waves is presented in the form of special series with logarithmic terms. Results of numerical computations are given.The problem of determining the flow of plane and three-dimensional waves separated from the quiescent region by a weak discontinuity was considered in [1–3], where approximate solutions were derived for that neighborhood, and the formulation of boundary value problems required for solving the equation for the analog of the velocity potential in the hodograph plane was investigated.The more general problem (without the assumption of the degeneration of motion) of arbitrary potential flows of polytropic gas adjacent to the quiescent region and separated by a weak discontinuity was considerd in [4–8]. Solution of that problem was obtained in the form of special series in powers of the mo dulus of the velocity vector r in the space of the time hodograph. The value r = 0 corresponds to the surface of weak discontinuity that separates the perturbed motion region from that at rest. Some applications of derived solutions to problems such as the motion of a convex piston and the propagation of weak shock waves were also investigated in those papers. Convergence in the small of obtained series was proved in [9]. However the attempts of constructing series in powers of r, which were used in [4–8] for the presentation of equations of double waves in the neighborhood of the quiescent region, proved to be unsuccessful.Although parts of expansions in series in powers of r (accurate to within 0 (r²)), were constructed in [1–3], it was found that the coefficient at r⁸ in equations for double waves cannot be determined owing to the insolvability of its equation. This is related to the fact that the surface r = 0in the case of equations for double waves is simultaneously a line of parabolic degeneration and a characteristic.The object of the present note is the formulation of solutions of equations for plane unsteady double waves in the neighborhood of the quiescent region. Parts of the derived series, which generally are nonanalytic functions of r, can be used for defining flows at small r in particular those downstream of two-dimensional normal detonation waves [10] or in problems of angular pistons [11]. The method used for the derivation of series can be also applied in investigations of threedimensional self-similar flows with variables x₁/x₃ and x₂/x₃ (steady flows) or x₁/t, x₂/t and x₃/t (unsteady flows). However it was not possible to obtain in such cases regular series in powers of r.     

On perturbations associated with the creation of lift acting on a body in a transonic stream of a dissipative gas PMM vol. 31, no. 6, 1967, pp. 1035–1049

The flow pattern of a viscous imcompressible fluid past a finite body is well known; an approximate solution of the related problem can, for example, be found in the book by Landau and Lifshits [1]. Finn [2] made a rigorous and exhaustive study of plane-parallel flows. No fundamental difficulties arise in passing from the motion of an incompressible fluid to a transonic flow of a compressible gas, however the velocity field is different, when the velocity of particles becomes critical at infinity.

The pattern of a sonic flow past a body of circular cross-section was investigated in paper [3]. This paper deals with perturbations associated with the creation of lift acting on an arbitrary body in a three-dimensional flow. When solving this problem it is necessary to consider not only the external stream, but also the laminar vortex trail because of the velocity vector transverse components becoming infinitely great, if functions defining these are formally extended into the trail area. This difficulty arises in investigations of three-dimensional flows only. The solution defining perturbation damping in an axisymmetric sonic stream of a dissipative gas has in its first approximation one singular point only, and does not contain any other singularities along the axis of symmetry [3].

The external stream pattern is essentially formed by the action of normal viscous stresses and the longitudinal component of the heat flux vector, while the distribution of gas parameters in the laminar trail is defined by tangential stresses. The conjunction of solutions valid for each of these areas makes the closure of the problem, and the determination of all necessary parameters possible.     

Dynamical states of bubbling in vertically vibrated granular materials. Part I: Collective processes

Granular materials deform plastically like a solid under weak shear and they flow like a fluid under high shear. These materials exhibit other unusual kinds of behavior, including pattern formation in shaking of granular materials for which the onset characteristics of the various patterns are not well understood. Vertically shaken granular materials undergo a transition to a convective motion which can result in the formation of bubbles. In Part I, a detailed overview is presented of collective processes in gas-particle flows useful for developing a simplified model for molecular dynamic type simulations of dense gas-particle flows. The large eddy simulation method (LES) has been employed for simulating fluid flows through irregular array of particles. The results obtained may lead to scale-dependent closures for quantities such as the drag, stresses and effective dispersion. These are of use for developing a continuum approach for describing the deformation and flow of dense gas-particle mixtures described in Part II.     

Comments on “Simple chaotic flows with a line equilibrium” [Chaos,solitons & fractals 57 (2013) 79–84]

In the above paper, the authors proposed a search algorithm for finding simple chaotic systems with quadratic nonlinearities which have the unusual feature of having a line equilibrium and introduced nine novel chaotic flows. However, the equations of 4th–9th chaotic flows were incorrectly given. In this letter, the correct equations are presented for the future usage of these line equilibrium chaotic flows.