Ideal Incompressible Fluid Research Articles

Lagrangian averages, averaged Lagrangians, and the mean effects of fluctuations in fluid dynamics.

We begin by placing the generalized Lagrangian mean (GLM) equations for a compressible adiabatic fluid into the Euler-Poincare (EP) variational framework of fluid dynamics, for an averaged Lagrangian. This is the Lagrangian averaged Euler-Poincare (LAEP) theorem. Next, we derive a set of approximate small amplitude GLM equations (glm equations) at second order in the fluctuating displacement of a Lagrangian trajectory from its mean position. These equations express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The derivation of the glm equations uses the linearized relations between Eulerian and Lagrangian fluctuations, in the tradition of Lagrangian stability analysis for fluids. The glm derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction. Next, the new glm EP motion equations for incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM (or glm) fluid theory with a Taylor hypothesis closure. Such closures are based on the linearized fluctuation relations that determine the dynamics of the Lagrangian statistical quantities in the Euler-alpha equations. Thus, by using the LAEP theorem, we bridge between the GLM equations and the Euler-alpha closure equations, through the small-amplitude glm approximation in the EP variational framework. We conclude by highlighting a new application of the GLM, glm, and alpha-model results for Lagrangian averaged ideal magnetohydrodynamics. (c) 2002 American Institute of Physics.

On a Motion of a Perfect Fluid in a Domain with Sources and Sinks

We prove the global weak solvability of the problem on a flow of an ideal incompressible fluid in a domain containing various types of sources and sinks. In the first part of the paper the case of point sources and sinks is considered. This situation is characterized by singularities of the velocity vector field, whose second powers are not integrable. We suggest a new generalized formulation of the problem, where the singularities are described by corresponding measures. As a development of this technique, we solve in the second part of the paper the problem on a flow of an ideal fluid through a given domain.

Structure of isobaric flows in a gas and in an ideal incompressible fluid

Structure of isobaric flows in a gas and in an ideal incompressible fluid

Nonlinear stationary surface waves above an underwater ridge.

Given an ideal incompressible heavy irrotational fluid, we consider the exact statement of the problem on gravitational-capillary surface waves of small amplitude travelling along an underwater ridge. We show that, under some requirements on the shape of the bottom and the surface tension, the equations of an ideal incompressible fluid have smooth solutions periodic in the variable directed along the underwater ridge and decreasing exponentially with a small positive exponent in the perpendicular direction.

Exact solutions of problems of the three-dimensional flow of ideal viscous incompressible fluids near cylindrical surfaces

Three-dimensional flows of an incompressible fluid, the parameters of which depend on two coordinates and time, are considered. The stream surfaces of such flows are cylindrical. The equations of continuity and the Navier-Stokes equations can be transformed to relations, one of which is the equation for the stream function the other is the integral of the equations relating the pressure and the stream function, and the third is a linear equation for the projection of the velocity vector onto the axis parallel to the generatrix of the cylindrical surfaces. The problems of modelling the flows are considered on the basis of the exact solutions of the Navier-Stokes equations and Euler's equations using examples. Relations for the distribution of the flow parameters in the channel created by hyperbolical cylinders are derived for the case of unsteady inviscid flow. The streamlines of these flows are situated on the side surfaces of the hyperbolical cylinders and intercept the generatrices of the cylinders at certain indirect angles. The flow around a circular cylinder and the flow of fluid inside an elliptic cylinder are considered in the case of steady inviscid flow. The streamlines on the circular cylinder are arranged transverse to the cylinder (the projection of the velocity vector onto the coordinate axis, parallel to the generatrix of the cylinder, is equal to zero). Far from the cylinder the streamlines are also situated on a cylindrical surfaces, but not transverse to the cylinder, making certain indirect angles with the generatrix. Viscous three-dimensional flows, possessing a certain symmetry, are considered. In the case of radial symmetry the streamlines are helical lines. The non-planar Couette flow between parallel moving planes is characterized by the fact that the velocity vectors, being situated in the same plane, are collinear, while the velocity vectors in parallel planes are not collinear. Relations for viscous steady three-dimensional flows, using well-known relations, obtained for the stream function of two-dimensional flows, are given.

Helical Outflow through a Circular Hole in the Bottom of a Semi-Infinite Cylinder (Modification of a Slezkin Problem)

The steady axisymmetric helical flow of an incompressible ideal fluid in a semi-infinite cylinder due to the presence of a circular hole in its bottom is analyzed. At an infinite distance from the bottom, in contrast to the similar problem considered by N. A. Slezkin, constant axial and angular components of the quasi-rigid rotation are maintained and the hole-induced flow is uniformly helical in the Zhukovsky sense, that is, the vorticity vector of the absolute motion is collinear to the relative velocity. In the coordinate system rotating with the fluid, this flow can be represented as the superposition of a straight-line translational flow in the direction of the bottom and a uniformly helical Gromeka-Beltrami flow. The problem is solved using the generalized stream function. The limiting cases of a helical sink in the bottom of a semi-infinite cylinder and helical flow out of a half-space through a circular hole on the boundary are considered. The results are compared with those for potential flow.

Vertical Impact on a Rigid Body Floating on the Surface of an Ideal Incompressible Fluid in a Bounded Basin of Arbitrary Shape

In the case of basin walls located at a large distance from the body, the asymptotics of the main impact parameters are found. As concrete examples, impact problems for regions with spherical boundaries are considered.

Hydrodynamics with quadratic pressure. 2. Examples

Exact solutions of Euler equations that describe the motion of an ideal incompressible fluid with quadratic pressure are studied. The solutions are described by explicit formulas and can be physically interpreted. The dynamics of a spherical fluid volume is studied for specified initial velocity fields. It is shown that under certain initial conditions, the spherical volume can evolve into a torus‐shaped body, thereby changing the connectivity of the region occupied by the fluid.

A new approach to studying the dynamics of a thin curved vortex

The paper studies the dynamics of a thin curved vortex in a potential flow of an ideal incompressible fluid. The flow is specified by a number of geometrical restrictions and does not satisfy the Biot–Savart law. The form of the derived equation of the vortex dynamics coincides with the form of the well‐known equation of local induction for self‐induced vortex motion. The parameters of the new equation are simultaneously flow parameters, and in this sense, they do not show uncertainty typical of classical equations. The coefficient of the new equation can take any specified values (not necessarily much greater than unity, as required according to the concept of local induction) and generally is a function of a natural filament parameter.

Numerical simulation of the interaction of a solitary wave with a partially immersed body

- In this paper we give the results on the calculation of the wave motion of a fluid on adaptive grids when a solitary wave runs up at a stationary rectangular body which is partially immersed in a fluid and is not in contact with the bottom. We study the effect of the length and immersion of a body and the amplitude of an incident wave on a maximum wave run-up, the wave pattern in front of and behind a body, force action on a body. Numerical algorithms are based on a one-dimensional nonlinear shallow-water model and a nonlinear model of potential plane flows of an ideal incompressible fluid. We compare the computational results obtained by various models with the experimental data in [9].

Ideal fluid flow past obstacles in an arbitrary channel: comparison of numerical and experimental results

This paper analyses a numerical model of a flow around an obstacle bounded in a channel with arbitrary walls. The model is based on the usual two-dimensional model of ideal incompressible weightless fluid. It is validated here by comparison with analytical findings and experimental data, which consisted of the velocity field determined by particle image velocimetry (PIV). Despite the simplicity of our wake model, the numerical data are usually in good agreement with the analytical and experimental data.

The asymptotic stability of a steady flow of an ideal incompressible fluid through a permeable domain

are the given functions ( = 0 for all t ≥ 0 ), and the inlet and the outlet are the parts of the boundary S through which the fluid inflows into domain D and,respectively, outflows from it at the time moment t ≥ 0 .Therefore, by definition, and It will be recalled that Kochin was the first, who hasanalyzed the general problem of unsteady fluid flow [1].In his study, the boundary conditions (2) were pro-posed. Yudovich [2] established that a two-dimensionalproblem defined by Eq. (1) and boundary conditions (2)is globally solvable. In what follows, we consider onlythis two-dimensional problem and refer it to as theproblem Y.∂

Dispersive–Dissipative Eddy Parameterization in a Barotropic Model

Recently a new class of coarse-grained equations, known as a models, have been proposed for the mean motion of an ideal incompressible fluid. The use of one such model to represent the time-mean component of a turbulent b-plane circulation characterized by potential vorticity mixing is considered. In particular, the focus is on the wind-driven circulation in a shallow ocean basin, a problem well studied as a prototype of more realistic ocean dynamics. The authors demonstrate the ability of an a model to reproduce qualitatively the structure of a four-gyre circulation that forms (in the time mean) when the barotropic vorticity equation is driven by a symmetric, double-gyre wind forcing, and when the dissipation is weak. This is offered as a first step in assessing the utility of the a-model approach to simulating more complex geophysical flows.

Numerical modelling of three-dimensional steady fluid flows on adaptive grids

- This paper deals with the development of the iterative finite difference method of the numerical solution to three-dimensional stationary problems of an ideal incompressible fluid flow through channels of complex geometry. We use curvilinear adaptive grids and introduce new dependent variables, viz. the vector potential ψ⃗ and the vorticity vector ω⃗.

Forces Exerted on a Body in an Unsteady Vortex Separation Flow of an Ideal Incompressible Fluid

A formula relating the forces exerted on a three-dimensional body to the motion of a vortex and source system simulating that body is derived for an unsteady vortex separation flow of an ideal incompressible fluid. The shape of the body can vary with time. In the case of steady-state homogeneous flow past an airfoil the formula obtained coincides with the Joukovski formula.

Numerical Simulation of the Fall of a Body with a Corrugated Bottom on Water

A numerical model is proposed for the potential flow of an ideal incompressible fluid produced by impact of a body with concave bottom on water. Compression of the entrapped air is taken into account. The algorithm is based on joint solution of the equations of motion for the body and the fluid by the finite difference method with approximation in time. At each time, the boundary‐value problem for the Laplace equation is solved by the boundary‐element method. Calculation results are given. The effects of the air layer, dimensions and shape of the corrugations, initial velocity, and other parameters on the impact process are shown.

The motion of a rigid body in a fluid under the action of central newtonian attractive forces

The structure of the forces and moments which arise when a rigid body moves in an unbounded volume of an ideal incompressible fluid under the action of central Newtonian attractive forces is discussed, the equations of motion are written in explicit form, their first integrals are indicated and the properties of these integrals are investigated. The problem of possible simplifications in the formulation of the problem is investigated, based on an analogy with the classical “satellite approximation” and which hold when the translational velocity of the body can be regarded as being independent of its rotational motion. The question of the existence and stability of steady motions, in this case relative equilibria, is investigated in the satellite approximation.

The steady motions of a fluid-filled spheroid on a plane with friction

The motion of a thin-walled spheroid completely filled with an ideal incompressible fluid executing uniform vortex motion (a Kelvin top) is investigated. It is assumed that the spheroid rests on a horizontal plane from which it is acted upon by a normal reaction and the force of viscous sliding friction. The equations of motion of a Kelvin top on a plane with friction are set up, and the conditions for them to be consistent are obtained. The steady and periodic motions of a Kelvin top are found, and problems of the stability and branching of these motions are investigated.

On steady irrotational flow of incompressible ideal fluid in a circular domain with free surface

On steady irrotational flow of incompressible ideal fluid in a circular domain with free surface

On the normal mode instability of modons and Wu-Verkley waves

The normal mode instability of steady Wu-Verkley (1993) wave and modons by Verkley (1984, 1987, 1990) and Neven (1992) is considered. All these flows are solutions to the vorticity equation governing the motion of an ideal incompressible fluid on a rotating sphere. A conservation law for infinitesimal perturbations to each solution is derived and used to obtain a necessary condition for its exponential instability. By these conditions, Fjörtoft's (1953) average spectral number of the amplitude of an unstable mode must be equal to a specific number that depends on the degree of the solution in its inner and outer regions as well as on spectral distribution of the mode energy in these regions. Some properties of the conditions for different types of modons are discussed. The maximum growth (and decay) rate of the modes is estimated, and the orthogonality of the amplitude of each unstable, decaying, or non-stationary mode to the basic solution is shown in the energy inner product. The new instability conditions confine the unstable disturbances of the WV wave and modon to a hypersurface in the perturbation space and allow interpretation of their energy structure. They are also useful both in estimating the maximum growth rate of unstable modes and in testing the numerical algorithms designed for the linear stability study.