Development of finite-amplitude two-dimensional and three-dimensional disturbances in jet flows

The laminar-turbulent transition zone is investigated for a broad class of jet flows. The problem is considered in terms of the inviscid model. The solution of the initial-boundary value problem for three-dimensional unsteady Euler equations is found by the Bubnov-Galerkin method using the generalized Rayleigh approach [1–4]. The occurrence, subsequent nonlinear evolution and interaction of two-dimensional wave disturbances are studied, together with their secondary instability with respect to three-dimensional disturbances.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 8–19, September–October, 1985.     

Structure of artificial disturbances induced by an external acoustic field in a supersonic boundary layer

The wave structure of the artificial disturbances generated by an external acoustic field in a supersonic boundary layer is investigated. The disturbances are classified with respect to phase velocity. Disturbances whose phase velocity in the direction of flow is greater than unity and waves located at the boundary of the discrete and continuous spectra are detected.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 82–86, May–June, 1989.     

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.     

Traveling disturbances in rotating-disk flow

The stability curves for traveling disturbances in rotating-disk flow are computed using the sixth-order system of incompressible linear stability equations. It is found that the neutral curve has two minima for disturbances with positive frequencies as found earlier by Malik (1986) for stationary disturbances. The upper branch minimum occurs at =–2.9, R=283.6 while the lower branch minimum occurs at =7.9, R=64.46 where R is the Reynolds number. There exists a critical angle of approximately –35.34° (which is about 15° from the direction of maximum wall shear) below which all the waves are linearly damped. The results also show that at high frequencies the wave number for lower branch neutral disturbances varies with Reynolds number like R ^–1 while for stationary waves it behaves like R ^–1/2. The eigenfunction distribution suggests that the structure of the nonstationary high-frequency lower branch neutral disturbances are different from the structure of the viscous stationary disturbances.This work was sponsored under NASA Contract NAS1-18240.     

Interaction between a supersonic boundary layer and acoustic disturbances

The development of disturbances in a boundary layer that have been induced by an external acoustic field are investigated. The problem is considered in the linear formulation. It is shown that the oscillations inside the supersonic boundary layer can have several times the intensity of the external disturbances. The susceptibility of the boundary layer to the acoustic disturbances increases with increasing Mach number. Cooling of the surface leads to a small decrease in the intensity of the longitudinal velocity oscillations in the layer. The effect of the parameters of the acoustic wave is considered, i.e., the effect of the frequency and phase velocity on the development of the disturbances.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 51–56, November–December, 1977.     

Propagation of wave packets in the boundary layer on a curved surface

The development of wave packets excited in a boundary layer by means of a local deformation of the surface in the longitudinal-transverse interaction regime is considered. A solution of the linearized system of equations of interaction theory is constructed using a Laplace transformation with respect to time and a Fourier transformation with respect to the space variables. Two problems are separately examined. In the first, the disturbances are induced by a surface deformation sinusoidal in the transverse direction. It is shown that the center of the wave packet with the greatest oscillation amplitude moves in a direction opposite to that of the flow in the boundary layer. At the same time the wave packet expands, so that in the course of time any fixed point will enter the region of growing oscillations. In the second problem the source of the disturbances is isolated. In this case the wave packet acquires a horseshoe shape. Expanding, it carries the disturbances away from the source in all directions, including upstream relative to the flow in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 59–68, March–April, 1990.     

Natural oscillations arising from loss of stability in parallel flows of a viscous liquid under long-wavelength periodic disturbances

It was shown in [1] that a parallel flow with an arbitrary nonconstant velocity profile is unstable for long-wavelength spatially periodic disturbances along the flow. The present paper shows that this instability leads to a supercritical natural oscillation mode of the simple wave type. This mode is calculated using the Lyapunov-Schmidt method in the form given in [2], along with the asymptotic curve of the wavelengths [1]. If the long wavelength disturbances are the most dangerous (this occurs, for example, when there is a sinusoidal velocity profile), then the natural oscillation mode is stable for spatially periodic disturbances having the same wavelength.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 32–35, January–February, 1973.     

Instability and structure of axisymmetric rotating flow

The development of instability in an axisymmetric flow has been experimentally investigated. The evolution of disturbances ranging from high-frequency fluctuations to large-scale ordered structures with wave numbersn=2, 3, 4, 5, 6, and 7 is traced. An experimental diagram of the regions of stable existence of various types of disturbances is constructed. The periodic flop-over cycles of a hydrodynamical system with respect ton are obtained experimentally for constant external conditions in the absence of forcing.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 60–67, September–October, 1993.     

On the receptivity and nonparallel stability of traveling disturbances in rotating-disk flow

The generation and evolution of small amplitude long wavelength traveling disturbances in rotating-disk flow is the subject of this paper. The steady rotational speed of the disk is perturbed so as to introduce high-frequency oscillations in the flow field. Secondly, we introduce surface imperfections on the disk such as roughness elements. The interaction of these two disturbances will generate the instability waves whose evolution is governed by parabolic partial differential equations which are solved numerically. It is found that, for the class of disturbances considered here (wavelength on the order of Reynolds number), eigensolutions exist which decay or grow algebraically in the radial direction. However, these solutions grow only for frequencies larger than 4.58 times the steady rotational speed of the disk. The computed receptivity coefficient shows that there is an optimum size of roughness for which these modes are preferentially excited. The width of these roughness elements in the radial direction is about 0.1r ₀ ^* where r ₀ ^* is the radial location of the roughness. It is also found that the receptivity coefficient is larger for a negative spanwise wave number than for a positive one. The cumulative wave pattern produced from the roughness site shows that the typical wave angles for these disturbances are about –26° with about seven waves around the circumference. This is in contrast with the wave angles of 10°–14° found for the 30 or so inviscid cross-flow vortices.This work was sponsored by NASA Langley Research Center under Contracts NAS1-18240 (P.B. and M.R.M.) and NAS1-18605 (P.H.).     

Experimental investigation of the development of harmonic disturbances in the boundary layer on a flat plate at mach number M=4

The development of three-dimensional wave packets artificially introduced into a boundary layer has been experimentally investigated. The measurements were made by the hot-wire anemometer method in the boundary layer on a flat plate at a Mach number M = 4. The artificial disturbances were introduced into the boundary layer by means of an electric discharge. A Fourier analysis of the data made it possible to obtain the wave characteristics of the plane waves. The composition of the disturbances was analyzed and those most dangerous from the instability standpoint were identified. The data obtained are compared with the results of experiments carried out at M = 2. The differences in the data are discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 54–58, November–December, 1990.     

Growth of artificially induced disturbances in a supersonic boundary layer

This work proposes a method of inducing artificial disturbances of adjustable amplitude in a supersonic boundary layer. Using the proposed method, an experimental study is made of the development of a three-dimensional wave packet of low intensity at a frequency of 20 kHz in the boundary layer of a flat plate at Mach number M = 2.0. The Fourier components of the wave packet are determined. The data obtained are compared with the results of calculating the linear stability of the supersonic boundary layer in a plane-parallel flow approximation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 37–43, September–October, 1984.     

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.     

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.     

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.     

Effect of fluid microstructure on stability of plane two-layer flows

The stability of plane two-layer Couette and Poiseuille flows, where the lower layer consists of a Grad-model fluid and the upper layer is a viscous Newtonian fluid, is investigated. The disturbances are assumed to be of the long-wave type, and the analysis involves expansion in wave numbers and is limited by two approximations. Numerical calculations are made for some values of the parameters. The calculations indicate that the rotational energy of the fluid in the lower layer has a destabilizing effect on the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 125–127, July–August, 1978.     

Spatial structure of two-dimensional wave disturbances in a boundary layer

In [1] on the basis of a numerical integration of the Navier-Stokes equations the authors investigated the nonlinear evolution of two-dimensional disturbances of the traveling wave type in the boundary layer on a flat plate. The process of interaction of two waves with different wave numbers and initial amplitudes was examined. In this article the study of these interactions is continued. Special attention is paid to the spatial structure of the disturbances with respect to the cross-flow coordinate (with respect to the longitudinal coordinate the disturbances are assumed to be periodic) at various moments of time. It is shown that if the initial amplitude of one of the waves is sufficiently large, i.e., exceeds a certain threshold value, an undamped quasisteady regime is established during the interaction process. At lower amplitudes the process degenerates and the waves develop independently. In these two cases the evolution of the spatial distribution of the perturbation amplitudes is qualitatively different. In the first case the shape of the amplitude distribution varies only slightly with time, while in the second it depends importantly on the parameters of the wave numbers and the Reynolds number. When the parameters are such that one of the finite-amplitude waves is damped, its amplitude distribution rapidly evolves into the form characteristic of disturbances of the continuous spectrum of the linear stability problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 19–24, September–October, 1990.     

Axisymmetric resonant disturbances in a homogeneous wave guide

The initial boundary-value linear stability problem for small localised axisymmetric disturbances in a homogeneous elastic wave guide, with the free upper surface and the lower surface being rigidly attached to a half-space, is formally solved by applying the Laplace transform in time and the Hankel transforms of zero and first orders in space. An asymptotic evaluation of the solution, expressed as a sum of inverse Laplace-Hankel integrals, is carried out by using the approach of the mathematical formalism of absolute and convective instabilities. It is shown that the dispersion-relation function of the problem D₀ (κ, ω), where the Hankel parameter κ is substituted by a wave number (and the Fourier parameter) κ, coincides with the dispersion-relation function D₀ (k, ω) for two-dimensional (2-D) disturbances in a homogeneous wave guide, where ω is the frequency (and the Laplace parameter) in both cases. An analysis for localised 2-D disturbances in a homogeneous wave guide is then applied. We obtain asymptotic expressions for wave packets, triggered by axisymmetric perturbations localised in space and finite in time, as well as for responses to axisymmetric sources localised in space, with the time dependence satisfying e^−iω₀t + O(e^−εt) for t → ∞, where Im ω₀ = 0, ε > 0, and t denotes time, i.e. for signalling with frequency ω₀. We demonstrate that, for certain combinations of physical parameters, axisymmetric wave packets with an algebraic temporal decay and axisymmetric signalling with an algebraic temporal growth, as √t, i.e., axisymmetric temporal resonances, are present in a neutrally stable homogeneous wave guide. The set of physically relevant wave guides having axisymmetric resonances is shown to be fairly wide. Furthermore, since an axisymmetric part of any source is L²-orthogonal to its non-axisymmetric part, a 3-D signalling with a non-vanishing axisymmetric component at an axisymmetric resonant frequency will generally grow algebraically in time. These results support our hypothesis concerning a possible resonant triggering mechanism of certain earthquakes, see Brevdo, 1998, J. Elasticity, 49, 201–237.     

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.     

Calculation of autooscillations of the type of azimuthal waves occurring at loss of stability of flow of a viscous fluid between concentric cylinders rotating in opposite directions

Starting with the Navier-Stokes equation we use the Lyapunov-Schmidt method to investigate the nature of the loss of stability of Couette flow between cylinders as the Reynolds number passes through its critical value. We consider the rotation of the cylinders in opposite directions with the ratio of the angular velocities such that the role of the most dangerous disturbances passes over from rotationally symmetric to nonrotationally symmetric disturbances. Branching nonstationary secondary flows (autooscillations) are found in the form of azimuthal waves; the longitudinal wave number and the azimuthal wave number m are assumed given. The amplitude of autooscillations and the wave velocity are calculated for m = 1, and it is shown that depending on the value of both weak excitation of stable and strong excitation of unstable autooscillations are possible and the wave number for which the critical Reynolds number is a minimum corresponds to a stable wave regime in the supercritical region. The linear problem of the stability of the circular flow of a viscous fluid with respect to nonrotationally symmetric disturbances is discussed in [1–3]. Di Prima [1] solved the problem numerically by the Galerkin method when the gap is small and the cylinders rotate in the same direction. Di Prima's analysis is extended in [2] to cylinders rotating in opposite directions, and in [3] it is extended to gaps which are not small. The nonlinear stability problem is treated in [4], where for fixed = 3 and cylinders rotating in opposite directions the axisymmetric stationary secondary flow the Taylor vortex is calculated. The formation of azimuthal waves in the fluid between the cylinders was studied experimentally in detail by Coles [5].Translated from Zhurnal Prikladnoi Mekhanika i Tekhnicheskoi Fiziki, No. 2, pp. 68–75, March–April, 1976.     

Dynamical properties of heterogeneous surface layers. capillary wave scattering

The dynamics of a heterogeneous liquid surface constituting a two-dimensional disperse system is considered. One of the surface phases (the dispersed phase) forms circular regions of diameter comparable with the characteristic length of the mechanical disturbances within the continuous disperse medium. Inhomogeneous boundary conditions for the Navier-Stokes equations with a discontinuity on the surface phase contact line are formulated. Special attention is paid to the conditions on this line. An approximate method of solving the surface wave diffraction problem and the results for the case of transverse surface wave scattering are described. It is shown that if the wavelength is close to the dimensions of the two-dimensional dispersed particles and their concentration is sufficiently high, the energy of the scattered waves may exceed that dissipated in the vorticity layer. Thus, a new nonclassical mechanism of surfactant action on capillary wave damping is possible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 129–137, January–February, 1991.