Stability of a laminar boundary layer of a power-law no n-newtonian liquid

The stability of a laminar boundary layer of a power-law non-Newtonian fluid is studied. The validity of the Squire theorem on the possibility of reducing the flow stability problem for a power-law fluid relative to three-dimensional disturbances to a problem with two-dimensional disturbances is demonstrated. A numerical method of integrating the generalized Orr-Sommerfeld equation is constructed on the basis of previously proposed [1] transformations. Stability characteristics of the boundary layer on a longitudinally streamlined semiinfinite plate are considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 101–106, January–February, 1976.     

Linear response of a plasma in a magnetic field to localized initial vorticity

It is established that for three-dimensional disturbances the long-range effect is observed even in the absence of boundaries. The problem of the evolution of the electrodynamic and gas dynamic disturbances created by a localized vorticity source is considered. It is shown that acoustic disturbances of a nonlocal nature are formed. The spatial structure of the electric potential and the nonlocal electric field created by localized initial vorticity at a finite value of the Hall parameter is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 109–114, March–April, 1990.     

Energy analysis of the stability of MHD flows

The analog of Orr's problem is formulated for MHD flows. Arbitrary three-dimensional disturbances satisfying the continuity equations are considered. It is established that direct interaction of the disturbances of the magnetic field and the velocity field cannot increase the energy estimate of the critical Reynolds number. Numerical calculations for Hartmann flow and modified Couette flows are made for the particular case of small magnetic Reynolds numbers, The minimum value of R is attained for disturbances with a wave vector perpendicular to the velocity vector of the main flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–9, July–August, 1971.The authors thank M. A. Gol'dshtik for his interest in their work.     

Experimental investigation of Richtmayer-Meshkov instability development on a contact discontinuity with three-dimensional disturbances

The results of an experimental investigation of Richtmayer-Meshkov instability on a contact discontinuity with three-dimensional disturbances are presented and compared with previously obtained results for contact discontinuities with two-dimensional disturbances.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 111–117, November–December, 1995.     

Stability of Hartmann flows

The stability of Hartmann flows for arbitrary magnetic Reynolds numbers is investigated in the framework of linear theory. The initial three-dimensional problem reduces to the equivalent two-dimensional problem. Perturbation theory is used to find asymptotic expressions for the eigenvalues. Distinguishing two types of disturbances — magnetic and hydrodynamic — is shown to be advantageous in a number of cases. Simple features of the stability are considered for particular cases. The well-know Lundquist result is generalized. An energy approach is applied to the problem of stability. The results of simulations involving the solution of the linear stability problem are described. A distinctive picture of stability is developed. There are several types of instability and they can develop simultaneously. The hydrodynamic and magnetic phenomena interact with each other in a very complex fashion. The magnetic field can either enhance flow stability or reduce it.Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 6, pp. 17–31, November–December, 1972.     

Stable nonlinear waves in a poiseuille flow

Nonlinear Tolmin-Schlichting waves are studied [1–8]. The investigation is carried out by means of a modified Stuart-Watson method [1–3]. In the case of a rigid regime of excitation terms to the fifth order are taken into account in expansions with respect to the amplitude of self-excited oscillations. The stability of self-excited oscillations with respect to two- and three-dimensional disturbances is examined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 40–45, September–October, 1978.The author thanks S. Ya. Gertsenshtein for attention to the work and discussion of the results.     

Asymptotic structure of inviscid disturbances in a thin shock layer

The WKB method, used in [4] to analyze the short-wave instability of a supersonic mixing layer, is employed to investigate various types of inviscid three-dimensional short-wave disturbances in a thin shock layer of perfect gas with arbitrary velocity and temperature distributions across the layer. Simple analytic expressions for the dispersion relations are obtained for neutral disturbances. The results of an asymptotic analysis are compared with direct numerical calculations for a simple model of the shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 72–79, November–December, 1988.     

Allowance for the unsteadiness of the flow in determining the critical cavitation number for marine propellers

A method of estimating the critical cavitation number for marine propeller blades is proposed. This method is based on the reduction of the three-dimensional unsteady problem to the three-dimensional steady problem and a series of two-dimensional unsteady problems.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 78–85, January–February, 1993.The authors are grateful to S. V. Kaprantsev for assisting with the experiments.     

Resonance properties of two-dimensional wave disturbances in a boundary layer

The nonlinear development of disturbances of the traveling wave type in the boundary layer on a flat plate is examined. The investigation is restricted to two-dimensional disturbances periodic with respect to the longitudinal space coordinate and evolving in time. Attention is concentrated on the interactions of two waves of finite amplitude with multiple wave numbers. The problem is solved by numerically integrating the Navier-Stokes equations for an incompressible fluid. The pseudospectral method used in the calculations is an extension to the multidimensional case of a method previously developed by the authors [1, 2] in connection with the study of nonlinear wave processes in one-dimensional systems. Its use makes it possible to obtain reliable results even at very large amplitudes of the velocity perturbations (up to 20% of the free-stream velocity). The time dependence of the amplitudes of the disturbances and their phase velocities is determined. It is shown that for a fairly large amplitude of the harmonic and a particular choice of wave number and Reynolds number the interacting waves are synchronized. In this case the amplitude of the subharmonic grows strongly and quickly reaches a value comparable with that for the harmonic. As distinct from the resonance effects reported in [3, 4], which are typical only of the three-dimensional problem, the effect described is essentially two-dimensional.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 37–44, March–April, 1990.     

Instability of two-layer film flow along an inclined surface

In [1–3] the method of expansion in a small wave number is used to investigate stability of two-layer flows; the results are valid for the neutral curves and in their neighborhood. Here, the eigenvalue problem is solved numerically, the wave disturbances are considered over the entire region of instability and the effect of the governing parameters on the characteristics of the most unstable disturbances is established.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 10–18, March–April, 1992.     

Three-dimensional problem of exponentially stratified flow over bottom roughness in the case of finite depth

The three-dimensional problem of the flow of an exponentially stratified fluid of finite depth over bottom roughness is considered in the rigid roof approximation and in the presence of a free surface. In the rigid roof approximation the solution is obtained in the form of a Fourier series in the vertical Lagrangian coordinate, and the series coefficients are expressed in terms of single integrals outside a horizontal strip whose sides are parallel to the flow axis and tangential to the projection of the support of the function describing the bottom roughness. This makes it possible to investigate the near field in regions not considered in [1, 2]. The presence of a small parameter in the boundary condition at the free surface makes it possible to find, in the first approximation, the wave motions and nonwave disturbances at the free surface in the near and far fields. In the near field the width of the wave zone is of the order of the flow depth, expands with distance from the bottom and is broadest at the free surface. As distinct from the annular disturbances within the fluid, the pattern of the nonwave disturbances at the free surface depends on the polar angle. The law of similarity for the diverging waves at the free surface is also obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 101–111, May–June, 1990.The authors are grateful to É. V. Teodorovich for discussing the formulation of the problem.     

Stability of a nonstationary round jet of an ideal incompressible fluid

The stability of transient flow in a cylinder of an ideal incompressible fluid with a free boundary is studied. There are 20 different cases of the behavior of small disturbances as a function of the parameters of the problem. In particular, if surface tension is not taken into account a round jet is stable with respect to axially symmetrical disturbances, but the introduction of capillary forces leads to a strong instability.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 80–84, July–August, 1972.In conclusion the author thanks V. V. Pukhnachev for formulation of the problem and valuable advice.     

Quasi-three-dimensional calculation of flow in blade passages of turbines

The flow in turbomachines is currently calculated either on the basis of a single successive solution of an axisymmetric problem (see, for example, [1-A]) and the problem of flow past cascades of blades in a layer of variable thickness [1, 5], or by solution of a quasi-three-dimensional problem [6–8], or on the basis of three-dimensional models of the motion [9–11]. In this paper, we derive equations of a three-dimensional model of the flow of an ideal incompressible fluid for an arbitrary curvilinear system of coordinates based on averaging the equations of motion in the Gromek–Lamb form in the azimuthal direction; the pulsation terms are taken into account in the equations of the quasi-three-dimensional motion. An algorithm for numerical solution of the problem is described. The results of calculations are given and compared with experimental data for flows in the blade passages of an axial pump and a rotating-blade turbine. The obtained results are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 69–76, March–April, 1991.I thank A. I. Kuzin and A. V. Gol'din for supplying the results of the experimental investigations.     

Instability due to three-dimensional disturbances

In the linear theory of the stability of parallel flows of a viscous fluid, most attention is usually given to plane-wave disturbances. The reason is the validity in many cases of the Squire theorem, which states that the critical Reynolds number R is determined by two-dimensional disturbances [1]. It is shown in the present paper that for large R the region generating the turbulence in the initial stage of its development is formed by three-dimensional disturbances. This feature applies both to the generating range of wave numbers and the dimension of the wall layer, where the fluctuating energy is produced. The consequences of the Squire transformations for parallel flows are analyzed. The contribution of resonant nonlinear triad coupling to the rapid growth of fluctuating energy is studied for the case of an explosive instability in an extended laminar mode. It is shown that the rate of turbulent energy production is not governed by the small derivatives of linear theory, but by nonlinear triad coupling of neutral and growing disturbances, with their three-dimensional nature playing an important role.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 29–34, September–October, 1976.The author thanks M. A. Gol'dshtik for his interest in the work and for discussion of the results.     

Numerical simulation of instabilities and secondary regimes in spherical couette flow

In all previous numerical investigations of spherical Couette flow only axisymmetric regimes were considered. At the same time, in experiments [1–4] it was found that when both spheres rotate and the layer is thin centrifugal instability of the main flow leads to the appearance of nonaxisymmetric secondary flows of the azimuthal traveling wave type. The results of an initial numerical investigation of these flows are presented below. Solving the linear problem of the stability of the main flow and simulating the secondary flows on the basis of the complete nonlinear Navier-Stokes equations has made it possible to supplement and explain many of the results obtained experimentally. The type of bifurcation and the structure of the disturbances whose growth leads to the appearance of three-dimensional nonstationary flows are determined, and the transitions between different secondary regimes in the region of weak supercriticality are described.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–15, January–February, 1995.     

Stability of viscous flow between rotating and stationary disks

The linear problem of the stability of viscous flow between rotating and stationary parallel disks is solved in the locally homogeneous formulation using the method of normal modes. The main flow is assumed to be selfsimilar with respect to the radial coordinate. The system of sixth-order equations, derived for the amplitude functions of the disturbances, is integrated by a finite difference method. The stability characteristics with respect to disturbances of four types are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 79–87, November–December, 1991.     

Numerical simulation of the nonuniform breakup of capillary jets

The problem of the evolution of the surface of a jet up to the stage at which it breaks up into droplets is solved numerically for two initial wave disturbances. The wave number of one of these coincides with the wave number of the disturbance that grows most strongly according to the linear theory, while the wave number of the other is varied. The effect of the wave numbers and the amplitude ratio of the initial disturbances on the breakup time and the appearance of nonuniformity is investigated.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 12–17, March–April, 1993.     

Velocity disturbance structure in the nonlinear stage of laminar-turbulent transition

The Herbert theory [1, 2] of laminar turbulent transition is generalized by taking into account the development of the secondary disturbances with allowance for their higher harmonics. The results of calculations of the development of three-dimensional pulsations, based on this theory, are in a agreement with the Herbert results in the initial stage of development and with the results of direct numerical modeling, obtained in this paper, and with experiment [3] in the later nonlinear stages of transition.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 3–9, November–December, 1993.     

The nonlinear evolution of two-dimensional and three-dimensional waves in mixing layers

The authors consider problems connected with stability [1–3] and the nonlinear development of perturbations in a plane mixing layer [4–7]. Attention is principally given to the problem of the nonlinear interaction of two-dimensional and three-dimensional perturbations [6, 7], and also to developing the corresponding method of numerical analysis based on the application to problems in the theory of hydrodynamic stability of the Bubnov—Galerkin method [8–14].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhldkosti i Gaza, No. 1, pp. 10–18, January–February, 1985.     

Development of three-dimensional disturbances in a boundary layer with pressure gradient

The results of an experimental investigation of the three-dimensional stability of a boundary layer with a pressure gradient are presented. A low-turbulence subsonic wind tunnel was employed. The development of a three-dimensional wave packet of oscillations harmonic in time in the boundary layer on a model wing is studied. The amplitudephase distributions of the pulsations in the wave packet are subjected to a Fourier analysis. Spectral (with respect to the wave numbers) decomposition of the oscillations enables the flow stability with respect to plane waves with different directions of propagation to be examined. The results are compared with the corresponding data obtained in flat plate experiments. The effect of the pressure gradient on the development of the three-dimensional spectral components of the disturbances and the dispersion properties of the flow is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 85–91, May–June, 1988.