The lifting force action on a contour in a plane homogeneous vortex flow of an incompressible ideal fluid

The plane problem of homogeneous vortex flows of an incompressible inviscid fluid around a contour is considered. A method is developed for contours having a center or an axis of symmetry for calculating the lifting force acting on the contour which reduces the problem of determining the lifting force to an ordinary problem of a potential flow of a fluid around the given contour.     

Gauge principle and variational formulation for flows of an ideal fluid

A gauge principle is applied to mass flows of an ideal compressible fluid subject to Galilei transformation. A free-field Lagrangian defined at the outset is invariant with respeet to global SO(3) gauge transformations as well as Galilei transformations. The action principle leads to the equation of potential flows under constraint of a continuity equation. However, the irrotational flow is not invariant with respect to local SO(3) gauge transformations. According to the gauge principle, a gauge-covariant derivative is defined by introducing a new gauge field. Galilei invariance of the derivative requires the gauge field to coincide with the vorticity, i.e. the curl of the velocity field. A full gauge-covariant variational formulation is proposed on the basis of the Hamilton‘‘s principle and an assoicated Lagrangian. By means of an isentropic material variation taking into account individual particle motion, the Euler‘‘s equation of motion is derived for isentropic flows by using the covariant derivative. Noether‘‘s law associated with global SO(3) gauge invariance leads to the conservation of total angular momentum. In addition, the Lagrangian has a local symmetry of particle permutation which results in local conservation law equivalent to the vorticity equation.     

Some classes of two-dimensional vortex flows of an ideal fluid

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 109–117, January–February, 1989.     

Stability of two-dimensional curvilinear flows of an ideal barotropic fluid

It was established by Arnol'd [1] that the conservation laws for the energy and vorticity can be used to establish sufficient conditions of stability of two-dimensional curvilinear flows of an ideal incompressible fluid in the exact nonlinear formulation. It is shown below that one can obtain similarly conditions of stability of two-dimensional curvilinear steady flows of an ideal barotropic fluid in the linear approximation. One of the conditions has a significance similar to Rayleigh's criterion and its generalization by Arnol'd [1]; the other is the condition of subsonic flow. In addition, a variational principle is established and an expression found for the second variation of the corresponding functional; these can be used to prove the stability of these flows in the exact nonlinear formulation.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 5, pp. 19–25, September–October, 1981.I am sincerely grateful to V. L. Berdichevskii and A. G. Kulikovskii for constructive advice.     

Interaction of an elastic shell with two-dimensional flows of an ideal fluid

In [1, 2], Kiselev and Rapoport investigated the flow of a jet over an elastic plate and shell. In the present paper, the problem of two-sided flow past an elastic shell is investigated in the exact nonlinear formulation. At a sufficiently high rigidity and small curvature of the shell in undeformed state it is shown that the problem has a unique solution, and a method is proposed for finding it. Some results of calculations are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 139–143, September–October, 1981.     

Some plane adiabatic ideal gas flows containing shocks

We obtain the solution describing adiabatic flows of an ideal gas characterized by the two parameters a and b such that [a]=L ^m+1 T ^–1, [b]=ML ^–2–2m where m is arbitrary (m > 0).h This solution permits the construction of flows containing shocks.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 10, No. 3, pp. 71–73, May–June, 1969.     

Interaction of shear flows of an ideal incompressible fluid in a channel

The problem of the decay of an arbitrary discontinuity for the equations describing plane-parallel shear flows of an ideal fluid in a narrow channel is considered. The class of particular solutions corresponding to fluid flows with piecewise constant vorticity is studied. In this class, the existence of self-similar solutions describing all possible unsteady wave configurations resulting from the nonlinear interaction of the specified shear flows is established. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 34–47, November–December, 2006.     

Strong discontinuities in spatial stationary long-wave flows of an ideal incompressible fluid

Spatial stationary flows over an even bottom of a heavy ideal fluid with a free surface are considered. Jump relations for flows with a strong discontinuity are studied. It is shown that the flow parameters behind the jump are defined by a certain curve which is an analog of the (θ, p) diagram in gas dynamics. A shock polar and examples of flows with a hydraulic jump are constructed for a particular class of solutions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 37–45, March–April, 2009.     

Mixed flow of an ideal fluid on a plane containing a recess

In the exact formulation, a study is made of the solution to the problem of the flow of an ideal incompressible fluid on a flat surface in a recess in the form of a half-cylinder in the direction at right angles to the flow. In the recess, the flow is assumed to have uniform vorticity, while in the exterior unbounded flow it is irrotational. On the separating streamline, the Bernoulli constant has a discontinuity of a given magnitude.     

Axisymmetric irrotational potential flows of an ideal fluid in doubly connected domains

Amethod of constructing the model of ideal incompressible unbounded flow past an axisymmetric tube is developed; the tube shape and dimensions are determined in the process of the problem solution. The problem is solved using the method of spatial annular sources and sinks. It is shown that under certain conditions the resulting flow corresponds to that past a round finite-length tube, whose wall has a finite thickness. It is established that near the tunnel entry the flow is considerably restructured.     

Motion of a variable-volume sphere in an ideal fluid near a plane surface

The motion of a spherical cavity in a fluid is investigated. The radius of the sphere varies under the action of a constant pressure at infinity. The problems of the collapse of a cavity moving in an unbounded fluid and of the collapse of a cavity near a plane are solved in the exact formulation. The occurrence of an initial translational velocity or the presence of a solid surface, by contrast with the collapse of a sphere at rest in an unbounded fluid [1], yields a limiting radius at which the process of collapse ceases. A sphere initially at rest near a plane always comes into contact with the plane as a result of collapse. The radius and velocities at which the sphere arrives the plane are calculated for various initial distances from the latter. The possible mechanism of the action of a cavitation bubble on a solid surface is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 94–103, September–October, 1971.     

Applications of contour dynamics to two layer quasigeostrophic flows

We determine stationary states and examine dynamic mergers of isolated piecewise constant regions of potential vorticity in a two-layer quasigeostrophic model. We focus on the behavior of the critical initial separation distance for merger, d_c, as a function of γ⁻¹ (inverse Rossby radius) and δ, the ration of layer depths.