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 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.     

The stability of two-dimensional linear flows of an Oldroyd-type fluid

A linear stability analysis is made for an Oldroyd-type fluid undergoing steady two-dimensional flows in which the velocity field is a linear function of position throughout an unbounded region. This class of basic flows is characterized by a parameter λ which ranges from λ = 0 for simple shear flow to λ = 1 for pure extensional flow. The time derivatives in the constitutive equation can be varied continuously from co-rotational to co-deformational as a parameter β varies from 0 to 1. The linearized disturbance equations are analyzed to determine the asymptotic behavior as time t → ∞ of a spatially periodic initial disturbance. It is found that unbounded flows in the range 0 < λ ? 1 are unconditionally unstable with respect to periodic initial disturbances which have lines of constant phase parallel to the inlet streamline in the plane of the basic flow. When the Weissenberg number is sufficiently small, only disturbances with sufficiently small wavenumber α₃ in the direction normal to the basic flow plane are unstable. However, for certain values of β, critical Weissenberg numbers are found above which flows are unstable for all values of the wavenumber α₃.     

Steady axisymmetric flow with vorticity of an inviscid fluid in multistage turbomachines

The direct problem of steady axisymmetric flow of a gas with vorticity through a multistage turbomachine is formulated precisely and a generalized solution is constructed by a variational-difference method. The turbomachine is represented schematically by an annular channel in which there are fixed (₁) and rotating (₂) three-dimensional cascades and channels free of them (₀).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–15, October–December, 1981.     

Formulation of the inverse axisymmetric problem of steady rotational flow of an ideal incompressible fluid in turbomachinery

The problem of the design of rotor blades within the framework of the hypothesis of an infinitely large number of blades reduces to the solution of an inverse axisymmetric problem. In the Bauersfeld-Voznesenskii formulation [1–3] this problem may be stated as follows: for given meridional flow and angular rotor velocity, construct the blade surface S ₂ (Fig. 1) passing through a given inlet ab (or exit cd) edge and a given line of intersection ad of the blade with one axisymmetric stream surface of the given meridional flow. Henceforth, S ₂ will be used to designate the median (or concave) surface of the blade, which, under certain conditions, coincides with the median interblade stream surface abcd. In [1–4], in solving this problem, use is made of the condition of coplanarity of the streamline elementd r=dr, rd,dz located on the blade surface S ₂ the relative velocity vector w and the absolute vorticity vector ×c In [5, 6] it is shown that this condition is valid only for irrotational flow incident on the rotor; consequently, its use in [1, 2] is completely legitimate, while in [3, 4] its use is inadmissible in principle, since the the equi-velocity meridional flow (i. e., that stream in which along each normal n to the meridional streamlines s the meridional component w_s of the velocity is constant (w _s/ _n=0)) assumed therein is essentially rotational [=(×c) _u 0] in the curved channel leading to the rotor.In [7] Gravalos presents the formulation and method of solution of the inverse axisymmetric problem for any arbitrary rotational meridional flow (and not just an equi-velocity flow), but does not take into account the constriction of the flow by the rotor blades, or take note of special cases of degeneration of the order and type of the equations at the boundaries and within the region; moreover, the method of solution employed assumes reduction of the quasilinear hyperbolic equation to the normal form to permit its solution by the Picard method of successive approximations.Below, we present the mathematical formulation of the inverse axisymmetric problem for any arbitrary rotational meridional flow in which account is taken of flow constriction. Cases of degeneration of the order and type of the equations are considered, the case with formation of a line of parabolic degeneration is examined, and important practical cases of the formulation of the boundary and initial conditions (Goursat problem and mixed problems), which determine the possible forms of the inlet and exit edges are studied. The problems formulated for the quasilinear hyperbolic equation can be solved with the aid of the method of characteristics, the method of finite differences, the method of straight lines, and other numerical methods.The discussion is directly applicable to radial and axial hydraulic turbines; however, it can be applied in essentially the same form to pump impellers, hydraulic converters, and also to stationary guide vanes (=0).In conclusion, the author wishes to thank G. Yu. Stepanov for discussing this work.     

A class of steady axisymmetric incompressible flows

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 28–34, November–December, 1990.     

The Giesekus fluid in ω–D form for steady two-dimensional flows

Using the fact that for simple fluids the most general constitutive equation in constant stretch history flows for the extra stress tensor τ is known in an explicit form, the Giesekus fluid model is cast into this (ω–D) form for two-dimensional flows. The three material functions needed to characterize τ are listed. The explicit results for simple shear and planar elongation reveal that the parameter α should be restricted to values less than 0.5. It is demonstrated that in this explicit form the constitutive equation is free from thermodynamic objections and can thus be used as a starting point for numerical calculations of general, but steady, two-dimensional flows. Received: 9 November 1998 Accepted: 20 May 1999     

Motion of a deformed contour in a flow of an ideal incompressible liquid with a constant vorticity

The article gives a solution to the plane problem of the motion of a deformed contour in a flow of an ideal incompressible liquid with a constant vorticity. An explicit expression is obtained for the hydrodynamic force when the velocity of the external flow depends linearly on the coordinates. In the case of a contour of small dimensions, this expression is valid also for an arbitrary external flow.     

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.