On solving the problem of an ideal incompressible fluid flow around large-aspect-ratio axisymmetric bodies

Based on Babenko’s fundamental mathematical ideas, principally new (unsaturated) algorithms are developed for the numerical solution of problems of a potential axisymmetric ideal fluid flow around bodies of revolution, in particular, an ellipsoid of revolution with an aspect ratio equal to 1000. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 56–67, September–October, 2006.     

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.     

Finite element solutions of axisymmetric Euler equations for an incompressible and inviscid fluid

In this paper we present a finite element method for the numerical solution of axisymmetric flows. The governing equations of the flow are the axisymmetric Euler equations. We use a streamfunction angular velocity and vorticity formulation of these equations, and we consider the non-stationary and the stationary problems. For industrial applications we have developed a general model which computes the flow past an annular aerofoil and a duct propeller. It is able to take into account jumps of angular velocity and vorticiy in order to model the flow in the presence of a propeller. Moreover, we compute the complete flow around the after-body of a ship and the interaction between a ducted propeller and the stern. In the stationary case we have developed a simple and efficient version of the characteristics/finite element method. Numerical tests have shown that this last method leads to a very fast solver for the Euler equations. The numerical results are in good agreement with experimental data.     

Exact solutions of equations of rotationally symmetric motion of an ideal incompressible liquid

A partially invariant solution of the Euler equations is considered, where the vertical component of velocity is a function of the vertical coordinate and time, whereas the remaining components of velocity and pressure are independent of the polar angle in a cylindrical coordinate system. Using the classification of equations obtained by analysis of an overdetermined system, we consider two hyperbolic systems: the first one describes the motion of a cylindrical layer of an ideal incompressible liquid under a punch, and the second system allows obtaining solutions in a halfcylinder with singularities at the axis of symmetry. A class of new exact solutions is obtained, which describe vortex motion of an ideal incompressible liquid, including the motion with singularities (sources of vortices) located along the axis of symmetry.     

Kinematic problem of an ideal incompressible fluid flow around an arbitrarily moving airfoil

An unsteady kinematic problem for arbitrary two-dimensional motion of an airfoil in an ideal incompressible fluid with formation of one and two vortex wakes is solved. The problem is solved by the method of conformal mapping of the flow domain onto a circle exterior; solution singularities in the vicinity of a sharp edge are analyzed, and the initial asymptotics of the solution is taken into account. The calculated results are found to be in good agreement with available experimental data on visualization of the flow pattern. The necessity of correct modeling of the initial stage of vortex-wake formation is demonstrated. A regular flow pattern is found to form after three and more periods of oscillations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 120–128, March-April, 2009.     

Plane linear problem of the immersion of an elastic plate in an ideal incompressible fluid

A plane unsteady-state linear problem of the immersion of an elastic plate of finite length in an ideal incompressible weightless fluid is considered. The deflection of the plate and the velocity of its points are known at the initial moment of time. The fluid occupies the lower halfplane, and its boundary outside the plate is free. The plate which is the bottom of a structure immersed in the fluid with a constant velocity is modeled by an Euler beam. At the initial stage of immersion, when the displacement of the structure is much smaller than the length of the plate, the plate deflection and the distribution of bending stresses in it are determined. The model used allows one to estimate the maximum stresses occurring in the elastic plate during its impact on water and to predict the moment and site of its occurrence. Calculations are performed under the conditions of the experiment carried out in MARINTEX (Norway). Qualitative agreement between the numerical and experimental results is shown. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 150–160, May–June, 1999.     

Nonlinear motion equations for a Non-Newtonian incompressible fluid in an orthogonal coordinate system

Summary Starting with an assumed relationship between the stress tensor, the non-Newtonian viscosity, and the strain rate tensor, the nonlinear equations of motion are developed for use in any orthogonal coordinate system. The resulting equations are written in terms of the scalar velocities, the non-Newtonian viscosity, the metric coefficients, and their derivatives.The non-Newtonian viscosity is assumed to be a scalar function of the strain rate tensor, and so depends upon the invariants of the strain rate tensor. For convenience, the necessary invariants are written out in complete form for use in any orthogonal coordinate system, in terms of the scalar velocities, the metric coefficients, and their derivatives.Using the resulting motion equations and a model of this type of viscosity, theOstwald-de Waele model, an example of time dependent flow is solved using a continuous time-discrete space method programmed on an analog computer. e _ij strain rate tensor - body force density, dynes/cm³ - F ₁,F ₂,F ₃ components of body force density, dynes/cm³ - g acceleration of gravity - H function of time - h ₁,h ₂,h ₃ metric coefficients - I ₁,I ₂,I ₃ invariants - m constant - P pressure, dynes/cm² - r radius, cm - t time, sec - velocity vector, cm/sec - v ₁,v ₂,v ₃ velocities in thex ₁,x ₂ andx ₃ directions, respectively, cm/sec - v _n (t) velocity of thenth node, cm/sec - x ₁,x ₂,x ₃ coordinate directions - z coordinate, cm - unit tensor - _ij Kronecker delta - _ij 2e _ij - nabla - _ijk alternating unit tensor - non-Newtonian viscosity, dynes/cm² - ₀, ₁ constant viscosities, dynes sec/cm², dynes sec ^m /cm² - angle, radians - v ₀,v ₁ constant kinematic viscosities, cm²/sec, cm² sec ^m-2 - density, g/cm³ - _ij stress tensor - fluid dilation With 3 figures     

Nonlinear motion equations for a non-Newtonian incompressible fluid in an orthogonal coordinate system

This paper describes the work carried out to investigate the pore pressures occurring in secondary consolidation. A theoretical approach and an experimental technique was developed in order to conduct the study. By considering compression to occur only due to water leaving the soil it was possible to derive an expression for the dissipation of pore pressure in the secondary phase. By further simplified assumptions which are based on experimental observations, the above general solution was reduced to a simple formula which predicted the observed behaviour of pore water pressures during secondary consolidation.     

The dual Least Action Problem for an ideal,incompressible fluid

Communicated by J. M. Ball     

Numerical solution of a nonlinear problem of the motion of a plane hydrofoil beneath the free surface of an ideal incompressible fluid

A nonlinear problem of the motion of a hydrofoil of infinite span beneath the free surface of an ideal incompressible fluid with gravity is studied. The stream function is used as the dependent variable. Iterative algorithms for small and large Froude numbers based upon solving a linear boundary value problem in each step with subsequent updating of the shape of the free boundary are proposed. Typical predictions are given for a symmetric profile at different values of the submersion depth, the Froude number and the angle of attack. The free surface and streamlines shapes are shown. The dependence of the lift on the submersion depth for motion through fluid layers of different thickness is presented.Dnepropetrovsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 100–107, July–August, 1995.     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID(Ⅱ)

The solvability of the Euler equations about incompressible inviscid fluid based on the stratification theory is discussed. And the conditions for the existence of formal solutions and the methods are presented for calculating all kinds of ill-posed initial value problems. Two examples are given as the evidences that the initial problems at the hyper surface does not exist any unique solution. Foundation item: the National Natural Science Foundation of China (19971054) Biography: Shen Zhen (1977−)     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID(Ⅰ)

The ill posed initial value problem of the Euler equations and the formal solvability of ill posed problem based on stratification theory are discussed. For some ill posed initial value problems, the existence conditions of formal solutions and the methods of how to construct a formal solution are given. Finally, an example is given to discuss the ill posedness of the initial value problem on hyper plane {t=0} in R⁴, and explain that the problem has more than one solution. Foundation item: the National Natural Science Foundation of China (19971054) Biography: Shen Zhen (1977−)     

Effect of initial strains on the propagation of waves in an incompressible cylinder located in an ideal fluid

S. P. Timoshenko Institute of Mechanics, Academy of Sciences of the Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 8, pp. 31–34, August, 1994.