On the Arnold Stability of a Solid in a Plane Steady Flow of an Ideal Incompressible Fluid

We study the stability of a rigid body in a steady rotational flow of an inviscid incompressible fluid. We consider the two-dimensional problem: a body is an infinite cylinder with arbitrary cross section moving perpendicularly to its axis, a flow is two-dimensional, i.e., it does not depend on the coordinate along the axis of a cylinder; both body and fluid are in a two-dimensional bounded domain with an arbitrary smooth boundary. Arnold's method is exploited to obtain sufficient conditions for linear stability of an equilibrium of a body in a steady rotational flow. We first establish a new energy-type variational principle which is a natural generalization of the well-known Arnold's result (1965a, 1966) to the system “body + fluid.” Then, by Arnold's technique, a general sufficient condition for linear stability is obtained. Received 21 February 1997 and accepted 23 June 1997     

Integrals of two-dimensional motions of a perfect incompressible fluid of nonuniform density

Integrals of motion are obtained which remain the same by virtue of the linearized equations of two-dimensional motions of a perfect incompressible fluid stratified with respect to the density and which are quadratic forms of the perturbation fields. The properties of these integrals and the variational principles associated with them are discussed. A number of conclusions concerning the problem of stability of stratified flows are drawn.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 16–20, May–June, 1987.     

Dynamics of an elliptic vortex

The two-dimensional flow of an ideal incompressible fluid is considered. Within an ellipse, the flow has a constant vorticity, while outside the ellipse there is an irrotational flow with circulation. Such motion can be described by Lagrange equations for a dynamical system with an infinite number of degrees of freedom. For a system with four degrees of freedom a new steady solution is obtained and its stability investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 55–60, July–August, 1983.We thank L. I. Sedov for his interest in the work.     

Chains of axisymmetric vortices

Chains of coaxial vortex formations of the “vortex breakdown“ type in axisymmetric swirling incompressible viscous and ideal fluid flows are represented in analytic form. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 174–176, March–April, 1998.     

Formation of Symmetric Structures in the Dynamics of Repelling Particles

We consider Hamiltonian systems corresponding to the motions of a system of N repelling particles evolving in space under the action deriving from a very long range potential energy; the asymptotic behavior of the system is analysed for the cases U=− ln r and . Only special “asymptotic?shapes” are reached, which may present quite interesting symmetries and correspond to the critical points of a gradient system. The relationships between the original Hamiltonian and the asymptotic gradient system are discussed. Accepted: May 25, 1999     

Poroelasticity-I: Governing Equations of the Mechanics of Fluid-Saturated Porous Materials

We present here the approach to the theory of fluid-filled poroelastics based on consideration of poroelastics as a continuum of “macropoints” (representative elementary volumes), which “internal” states can be described by as a set of internal parameters, such as local relative velocity of fluid and solid, density of fluid, internal strain tensor, specific area, and position of the center of mass of porous space. We use the generalized Cauchy–Born hypothesis and suggest that there is a system of (structural) relationships between external parameters, describing the deformation of the continuum and internal parameters, characterizing the state of representative elementary volumes. We show that in nonhomogenous (and, particularly, nonlinear) poroelastics, an interaction force between solid and fluid appears. Because this force is proportional to the gradient of porosity, absent in homogeneous poroelastics, and one can neglect with dynamics of internal degrees of freedom, this force is equivalent to the interaction force, introduced earlier by Nikolaevskiy from phenomenological reasons. At last, we show that developed theory naturally incorporates three mechanisms of energy absorption: visco-inertial Darcy mechanism, “squirt flow” attenuation, and skeleton attenuation.     

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     

On the Generation of Discontinuous Shearing Motions of a Non-Newtonian Fluid

The system under study models unsteady, one-dimensional shear flow of a highly elastic and viscous incompressible non-Newtonian fluid with fading memory under isothermal conditions. The flow, in a channel, is driven by a constant pressure gradient, is symmetric about the center line, and satisfies a no-slip boundary condition at the wall. The non-Newtonian contribution to the stress is assumed to obey a differential constitutive law (due to Oldroyd, Johnson & Segalman), the key feature of which is a non-monotone relation between the total steady shear stress and strain rate. In a regime in which the Reynolds number is much smaller than the Deborah (or Weissenberg) number, one obtains a degenerate, singularly perturbed system of nonlinear reaction-diffusion equations. It is shown that if the driving pressure gradient exceeds a critical value (the local shear stress maximum of the steady stress vs. strain rate relation), then the solution to the governing system, starting from rest at , tends as to a particular discontinuous steady state solution (the “top-jumping” steady state), except in a small neighborhood of the discontinuity. This discontinuous steady state is shown to be nonlinearly stable in a precise sense with respect to perturbations yielding smooth initial data. Such discontinuous steady states have been proposed to explain “spurting” flows, which exhibit a large increase in mean flow rate when the driving pressure is raised above a critical value. (Accepted April 22, 1996)     

On the principle of minimal entropy production for Navier-Stokes-Fourier fluids

In this paper we discuss the principle of minimal entropy production, proposed by Prigogine [1], which affirms that the global entropy production approaches a minimum as a process becomes stationary. We point out in two particular cases that this principle produces field equations that do not agree with the equations of balance of mass, momentum and energy. The processes considered are: • heat conduction in a fluid at rest • shear flow and heat conduction in an incompressible fluid. Now is the appropriate time to review Prigogine's principle, since in recent years a new, and different principle of minimal entropy production has been proposed. This is the “minimax principle” postulated by Struchtrup & Weiss [2]. Within the context of extended thermodynamics this new principle shows great promise. Received April 1, 1999     

Regarding some thermoelastic models of “coating-substrate” system deformation

In present research, we investigate dynamic coupled thermoelasticity problem for a “coating-substrate” system. We present a number of models of thermoelastic deformation of the “coating-substrate” system with thermomechanical characteristics which may vary both continuously and discontinuously. To solve these problems, we use the variational principle of coupled thermoelasticity in the Laplace transforms space and hypotheses on a distribution of temperature and displacements transforms. The transforms inversion is realized according to the Durbin method. The calculations were carried out based on both proposed simplified models and FEM.     

Motion of a solidcontaininga spherical damper

For the purpose of modeling the motion of a solid with a cavity filled with a viscous fluid, M. A. Lavrent'ev [1] has proposed a model in the form of a solid with a spherical cavity in which another solid, spherical in shape, is enclosed. The sphere is separated from the cavity walls by a small, clearance in which viscous forces act (a lubricating film). This simple model with a finite number of degrees of freedom possesses certain mechanical properties of a solid with a cavity containing a viscous fluid. Study of this model is therefore of interest.The present paper examines certain properties of the model, which will be termed a solid with a damper. It is shown that for a highviscosity lubricant the motion of a solid with a damper can be described by the same equations which pertain to the motion of a solid with a spherical cavity filled with a high-viscosity fluid. Expressions relating the parameters of the systems are obtained. If these relations are fulfilled, the systems become mechanically equivalent.The steady motions of a free solid with a damper and their stability conditions are determined.These motions and stability conditions hold for a body with a cavity filled with a viscous fluid [2].     

Modeling of jet flows of a viscous fluid by the discrete vortex method

Results obtained previously by the discrete vortex method with a “viscous” correction are generalized. The boundaries of applicability of this method are determined. Previous results obtained for a flow past a flat plate are supplemented with solution convergence estimates. Exhaustion of a plane jet of a viscous incompressible fluid into the ambient space is modeled. The geometric parameters of the jet (its half-width, shapes of the streamwise velocity profiles, and intensity of oscillations) are analyzed. The calculated results are found to agree well with experimental data and with results calculated by other methods.     

On relative equilibria and integrable dynamics of point vortices in periodic domains

The motion of two point vortices defines an integrable Hamiltonian dynamical system in either singly or doubly periodic domains. The motion of three point vortices in these domains is also integrable when the net circulation is zero. The relative vortex motion in both domains can be reduced to advection of a passive particle by fixed vortices in an equivalent Hamiltonian system. A survey of the solutions for vortex motion in these systems is discussed. Some initial conditions lead to relative equilibria, or vortex configurations that move without change of shape or size. These configurations can be determined as stagnation points in the reduced problem or through explicit solution of the governing equations. These periodic point-vortex systems present a rich collection of interesting solutions despite the few degrees of freedom, and several questions on this subject remain open.     

Variational methods of solving dynamic problems for fluid-containing bodies

A variational approach to solving linear and nonlinear problems for a body with cavities partially filled with a perfect incompressible fluid is enunciated. The approach applies a nonclassical variational principle to describe the spatial motion of a finite fluid with a free surface and the classical variational principle, which is widely used in rigid body dynamics. These principles are used to formulate variational problems that are the basis of direct methods of solving nonlinear and linear dynamic problems for body-fluid systems. The approach allows us to derive an infinite system of nonlinear ordinary differential equations describing the joint motion of the rigid body and fluid and to develop an algorithm for determining the hydrodynamic coefficients. Linearized differential equations of motion of the mechanical system are presented and approximate methods are given to solve linear boundary-value problems and to determine the hydrodynamic coefficients.Translated from Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 37–77, October 2004.The study was partially sponsored by the German Research Fund (der Deutsche Forschungsgemeinschaft), Grant 436 UKR113/33/0-3.     

A study on the mechanism of high-lift generation by an airfoil in unsteady motion at low reynolds number

The aerodynamic force and flow structure of NACA 0012 airfoil performing an unsteady motion at low Reynolds number (Re=100) are calculated by solving Navier-Stokes equations. The motion consists of three parts: the first translation, rotation and the second translation in the direction opposite to the first. The rotation and the second translation in this motion are expected to represent the rotation and translation of the wing-section of a hovering insect. The flow structure is used in combination with the theory of vorticity dynamics to explain the generation of unsteady aerodynamic force in the motion. During the rotation, due to the creation of strong vortices in short time, large aerodynamic force is produced and the force is almost normal to the airfoil chord. During the second translation, large lift coefficient can be maintained for certain time period and , the lift coefficient averaged over four chord lengths of travel, is larger than 2 (the corresponding steady-state lift coefficient is only 0.9). The large lift coefficient is due to two effects. The first is the delayed shedding of the stall vortex. The second is that the vortices created during the airfoil rotation and in the near wake left by previous translation form a short “vortex street” in front of the airfoil and the “vortex street” induces a “wind”; against this “wind” the airfoil translates, increasing its relative speed. The above results provide insights to the understanding of the mechanism of high-lift generation by a hovering insect. The project supported by the National Natural Science Foundation of China (19725210)     

Frictional passage of fluid through inhomogeneous porous system: a variational principle

A Fermat-like principle of minimum time is formulated for nonlinear steady paths of fluid flow in inhomogeneous isotropic porous media where fluid streamlines are curved by a location dependent hydraulic conductivity. The principle describes an optimal nature of nonlinear paths in steady Darcy’s flows of fluids. An expression for the total path resistance leads to a basic analytical formula for an optimal shape of a steady trajectory. In the physical space an optimal curved path ensures the maximum flux or shortest transition time of the fluid through the porous medium. A sort of “law of bending” holds for the frictional fluid flux in Lagrange coordinates. This law shows that—by minimizing the total resistance—a ray spanned between two given points takes the shape assuring that a relatively large part of it resides in the region of lower flow resistance (a ‘rarer’ region of the medium).     

Two methods of solving dynamic response by discretization of Gurtin variational principle

Based on the Gurtin variational principle and the finite element method two recurrence formulae of solving dynamic response are derived in this paper. The first recurrence formula which is conditionally stable is obtained by adopting “dual temporal finite element” to discretize the Gurtin variational principle and then making use of convolution, while the second, an unconditionally stable one derived by discritizing the variational principle in each time step and taking approprlate integration parameter θ. The paper had been accepted by the XVIth International Congress of IUTAM, Lyngby, Denmark, August, 1984.     

A Cartesian cut cell based two-way strong fluid–solid coupling algorithm for 2D floating bodies

In a recent paper Kelly et al. (2015) [PICIN: A Particle-In-Cell solver for incompressible free surface flows with two-way fluid–solid coupling. SIAM Journal on Scientific Computing 37 (3), B403–24.] detailed the PICIN full particle Particle-In-Cell (PIC) solver for incompressible free-surface flows. The model described in that paper employed a tailored version of the Distributed Lagrange Multiplier (DLM) method for the strong coupling of fluid–solid interaction. In this paper we propose an alternative strong fluid–solid coupling algorithm based on a modification to the cut cell methodology that is informed by the variational approach. The solid velocity flux/integral on the boundary is expressed purely in terms of pressure leading to a revised pressure Poisson equation that is discretised in a finite volume sense. This approach allows the PICIN model to simulate the motion of floating bodies of arbitrary configuration. 2D test cases involving floating bodies with one or more degrees of freedom (DoF) are used to validate the modified PICIN model. The results presented show that the modified PICIN model is able to both efficiently and robustly predict the motions of surface-piercing floating structures under either regular or extreme wave action.     

Impulse generated during unsteady maneuvering of swimming fish

The relationship between the maneuvering kinematics of a Giant Danio (Danio aequipinnatus) and the resulting vortical wake is investigated for a rapid, ‘C’-start maneuver using fully time-resolved (500 Hz) particle image velocimetry (PIV). PIV illuminates the two distinct vortices formed during the turn. The fish body rotation is facilitated by the initial, or “maneuvering” vortex formation, and the final fish velocity is augmented by the strength of the second, “propulsive” vortex. Results confirm that the axisymmetric vortex ring model is reasonable to use in calculating the hydrodynamic impulse acting on the fish. The total linear momentum change of the fish from its initial swimming trajectory to its final swimming trajectory is balanced by the vector sum of the impulses of both vortex rings. The timing of vortex formation is uniquely synchronized with the fish motion, and the choreography of the maneuver is addressed in the context of the resulting hydrodynamic forces.