Formation of the three-dimensional structure of the subharmonic transition regime in Blasius flow

The formation of the angles of the three-dimensional structure associated with subharmonic-type transition in a Blasius boundary layer is investigated. It is shown that the observed structure is determined by the selection of a spatially symmetric pair of Tollmien-Schlichting waves at parametric resonance of the background perturbations in the presence of induced oscillation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 77–81, May–June, 1989.     

Evolution of resonance disturbances in a hartmann flow

A rapid increase of energy of fluctuation motion is observed after a severe loss of stability of laminar regimes. This phenomenon does not find explanation in the scope of the linear theory of stability, which, though it predicts an exponential increase of disturbances in the supercritical region, gives quite small values of the increments. The explosionlike turbulence is due to a nonlinear mechanism. The simplest collective interaction of disturbances is illustrated by a set of three harmonic oscillations whose parameters are associated by resonance relations. Such triplets, being an elementary but sufficiently meaningful model of the nonlinear theory of hydrodynamic stability, have become in recent years the object of interesting investigations [1–4]. In [5–7] branching of stationary triplets of small amplitude from laminar regimes was investigated and it was shown that, beginning with certain Reynolds numbers, the triplet can be composed of neutral waves and Tolman-Schlichting waves increasing according to the linear theory. It is shown in the article that a quite rich example in this case is Hartmann flow, where the existence of triplets of disturbances having a different symmetry relative to the axis of the channel is admitted. The evolution of triplets is studied for near-critical values of the parameters in the framework of amplitude equations obtained on the basis of the Galerkin method with the use of eigenfunctions of the linear theory of stability as the basis [8]. Regimes stationary in the mean are calculated in the supercritical region: limiting cycles and strange attractors; in the latter case a spectral analysis is carried out.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 33–39, September–October, 1978.The authors thank M. A. Gol'dshtik and M. I. Rabinovich for discussing the 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.     

Perturbations of finite amplitude in a boundary layer

The nonlinear development of Tollmien-Schlichting waves in a boundary layer is studied taking account of the nonparallel character of the mean flow. The investigation is made using a modified Stuart-Watson method [1–3]. A comparison is made with results obtained with the usual quasiparallel approach.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 150–153, January–February, 1976.     

Excitation of Tollmien-Schlichting waves py a vibrating section of a surface with flow round it

In the context of the problem of describing the transition of a laminar boundary layer to a turbulent, great interest attaches to the study of susceptibility, i.e., of the reaction of the flow to various external influences, such as acoustic perturbations, surface roughness, vibration of the wall, turbulence of the unperturbed flow, etc. A general property of the effect of the factors mentioned above on the flow in a laminar boundary layer was discovered in experimental and numerical studies and is noted in [1]: in all cases an external forcing perturbation leads to the excitation of normal modes of oscillation in the boundary layer which propagate downstream, namely, Tollmien-Schlichting waves. There is an analytical calculation in [2, 3] of the amplitude of a wave excited by harmonic oscillations of a narrow band on the surface of a plane plate, the Reynolds number having been assumed to be infinitely large, and the frequency of the vibrator corresponding to the neighborhood of the lower branch of the neutral cuirve [4], In [5] the amplitude of the wave of instability generated is calculated by the method of expansion of the solution in a biorthogonal system of eigenfunctions. The amplitudes of the Tollmien-Schlichting waves are calculated below by means of a generalization of the method of [2] for the whole range of Reynolds numbers and frequencies of the vibrator corresponding to the region of instability: for moderate Reynolds numbers the problem is solved numerically, while for large Reynolds numbers an asymptotic solution is constructed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 46–51, July–August, 1987.The author is grateful to M. N. Kogan and V. V. Mikhailov for useful discussions of the results of the study.     

Laminarization of boundary layer flow on a vibrating wing

A new type of Tollmien-Schlichting wave excitation, experimentally detected in [6] in investigating the unsteady perturbation field downstream from roughness on the surface of a vibrating wing, is studied. It is shown that the generation mechanism consists in the nonlinear interaction between the unsteady disturbance produced by the vibrations of the smooth wall and the steady nonuniformity of the boundary layer above the roughness.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 26–34, March–April, 1993.     

Method of mean-mass parameters for a three-dimensional boundary layer in a flow with vorticity

When a gas flows with hypersonic velocity over a slender blunt body, the bow shock induces large entropy gradients and vorticity near the wall in the disturbed flow region (in the high-entropy layer) [1]. The boundary layer on the body develops in an essentially inhomogeneous inviscid flow, so that it is necessary to take into account the difference between the values of the gas parameters on the outer edge of the boundary layer and their values on the wall in the inviscid flow. This vortex interaction is usually accompanied by a growth in the frictional stress and heat flux at the wall [2, 3]. In three-dimensional flows in which the spreading of the gas on the windward sections of the body causes the high-entropy layer to become narrower, the vortex interaction can be expected to be particularly important. The first investigations in this direction [4–6] studied the attachment lines of a three-dimensional boundary layer. The method proposed in the present paper for calculating the heat transfer generalizes the approach realized in [5] for the attachment lines and makes it possible to take into account this effect on the complete surface of a blunt body for three-dimensional laminar, transition, or turbulent flow regime in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 80–87, January–February, 1981.     

The Cauchy-Poisson problem for forced waves of finite amplitude

Using the Linshtedt-Poincaré method, Sretenskii gave an approximate solution of the Cauchy-Poisson problem for free waves of finite amplitude constructed so as to be free of secular terms [1]. In [2] the Cauchy-Poisson problem was solved by the same method, but for somewhat modified initial conditions. It would appear reasonable to generalize the results of [1] to include the case of forced waves of finite amplitude and to describe their development with time. In the present paper, in order to solve this problem the Krylov-Bogolyubov method is employed and the principal and subharmonic resonances are investigated.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 116–121, July–August, 1995.     

Three-wave resonance interaction of disturbances in a boundary layer

The three-frequency resonance of Tolman-Schlichting waves, one of which propagates along the stream while the other two propagate at adjacent angles to it, is investigated as a function of the spectrum and initial intensity in incompressible flows of the boundary-layer type within the framework of a weakly nonlinear theory. In the parallel-flow approximation such an interaction leads to the formation of unstable self-oscillations. The spatial evolution of the associated disturbances is studied with allowance for the self-similar deformation of the velocity profile of the main flow. It is shown that such development can lead to a sharp amplification of the oscillations, primarily of those propagating at an angle to the flow. The role of the effects under consideration in the transitional process and the connection with experimental data are discussed. As experiments [1, 2] show, in the process of a transition from a laminar boundary layer to a turbulent region, well described by the linear theory of hydrodynamic stability, there first comes a section of the excitation of harmonics of a Tolman-Schlichting wave, the appearance of three-dimensional structures, and a rapid growth in the intensity of low-frequency oscillations. There is no doubt that in this section the phenomena are dependent on the nonlinear character of the development with disturbances. The resonance interaction of wave triads can play an important role in this. For small enough amplitudes such an interaction is described by a first-order theory [3, 4], and in the general case the nonlinear effects associated with them should occur sooner than others. The importance of resonance triads for the explanation of the development of three-dimensional structures in a layer and the generation of intense pulsations has already been emphasized in [5, 6]. The clarification of the properties of the evolution of resonantly interacting disturbances therefore is important for an understanding of this transitional process.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 78–84, September–October, 1978.The authors thank V. Ya. Levchenko for a discussion of the work.     

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.     

Interaction of a laminar boundary layer with external turbulence

Transition in the boundary layer on a flat plate in a turbulent flow is investigated experimentally and theoretically. It is established that over a broad range of flow conditions (variation of the intensity and scale of the external turbulence, the angle of attack, the shape of the leading edge, etc.) transition takes place without the formation of Tollmien-Schlichting waves, and its initial stages, including the amplification of disturbances, are described by the linearized unsteady three-dimensional boundary layer equations without a pressure gradient.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 55–65, September–October, 1989.The authors are grateful to N. F. Polyakov, V. S. Kosorygin, and O. S. Ryzhov for useful discussions and to N. N. Bychkov and O. N. Konstantinovskii for assisting with the experiments.     

Nonlinear Mechanisms of the Initial Stage of the Laminar–Turbulent Transition at Hypersonic Velocities

Weakly nonlinear development of waves in an axisymmetric hypersonic boundary layer is studied by the method of bispectral analysis. The type of nonlinear interaction that was not observed previously in such flows is found. The possibility of subharmonic resonance of the second mode at the nonlinear stage of transition is demonstrated. The previously discovered nonlinear generation of the harmonic of the fundamental wave of the second mode of disturbances is observed.     

Flow processes with change of regime in fractured reservoirs

Experimental and industrial observations indicate a strong nonlinear dependence of the parameters of the flow processes in a fractured reservoir on its state of stress. Two problems with change of boundary condition at the well — pressure recovery and transition from constant flow to fixed bottom pressure — are analyzed for such a reservoir. The latter problem may be formulated, for example, so as not to permit closure of the fractures in the bottom zone. For comparison, the cases of linear [1] and nonlinear [2] fractured porous media and a fractured medium [3] are considered, and solutions are obtained in a unified manner using the integral method described in [1]. Nonlinear elastic flow regimes were previously considered in [3–6], where the pressure recovery process was investigated in the linearized formulation. Problems involving a change of well operating regime were examined for a porous reservoir in [7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 67–73, May–June, 1991.     

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.     

An experimental study of receptivity of acoustic waves in laminar boundary layers

The receptivity of a laminar boundary layer developing on a flat plate was studied with two- and three-dimensional roughness elements. The layer was subjected to acoustic waves from speakers orientated perpendicular to the surface of the plate. Visualization of the transition patterns were obtained by heating temperature sensitive liquid crystals on the plate and observing the cooling patterns associated with the different flow regimes. Hot-wire data showed that the most amplified Tollmien-Schlichting waves dominated the downstream growth when the roughnesses were placed within the linearly unstable regime. The receptivity depended upon the size and aspect ratio of the three-dimensional roughness as predicted by Choudhari and Kerschen 1990. This research was partially funded by the Office of Naval Research under Contract N00014-89-J-1400. Their support is gratefully acknowledged. We also thank one of the reviewers for his helpful comments.     

Transition from regular reflection to mach reflection when a shock wave interacts with a wall in a two-phase gas—liquid medium

A study is made of the transition from regular reflection to Mach reflection when a plane moderately strong or weak shock wave interacts with a wall in a two-phase gas—liquid medium. An equilibrium model that differs from the model of Parkin et al. [1] by the introduction of the adiabatic velocity of sound is used to investigate shock wave reflection in the complete range of gas concentrations. For the reflection of weak shock waves, nonlinear asymptotic expansions [2] are used. In the limiting cases, the results agree with those already known for single-phase media [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 190–192, September–October, 1983.     

Propagation of modulated nonlinear waves in a relaxing gas-liquid mixture

The propagation of nonstationary weak shock waves in a chemically active medium is essentially dispersive and dissipative. The equations for short-wavelength waves for such media were obtained and investigated in [1–4]. It is of interest to study quasimonochromatic waves with slowly varying amplitude and phase. A general method for obtaining the equations for modulated oscillations in nonlinear dispersive media without dissipation was proposed in [5–8]. In the present paper, for a dispersive, weakly nonlinear and weakly dissipative medium we derive in the three-dimensional formulation equations for waves of short wavelength and a Schrödinger equation, which describes slow modulations of the amplitude and phase of an arbitrary wave. The coefficients of the equations are particularized for the considered gas-liquid mixture. Solutions are obtained for narrow beams in a given defocusing medium as well as linear and nonlinear solutions in the neighborhood of a diffraction beam. A solution near a caustic for quasimonochromatic waves was found in [9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 133–143, January–February, 1980.     

Traveling gravity water waves in two and three dimensions

This paper discusses the bifurcation theory for the equations for traveling surface water waves, based on the formulation of Zakharov [58] and of Craig and Sulem [15] in terms of integro-differential equations on the free surface. This theory recovers the well-known picture of bifurcation curves of Stokes progressive wavetrains in two-dimensions, with the bifurcation parameter being the phase velocity of the solution. In three dimensions the phase velocity is a two-dimensional vector, and the resulting bifurcation equations describe two-dimensional bifurcation surfaces, with multiple intersections at simple bifurcation points. The integro-differential formulation on the free surface is posed in terms of the Dirichlet–Neumann operator for the fluid domain. This lends itself naturally to numerical computations through the fast Fourier transform and surface spectral methods, which has been implemented in Nicholls [32]. We present a perturbation analysis of the resulting bifurcation surfaces for the three-dimensional problem, some analytic results for these bifurcation problems, and numerical solutions of the surface water waves problem, based on a numerical continuation method which uses the spectral formulation of the problem in surface variables. Our numerical results address the problem in both two and three dimensions, and for both the shallow and deep water cases. In particular we describe the formation of steep hexagonal traveling wave patterns in the three-dimensional shallow water regime, and their transition to rolling waves, on high aspect ratio rectangular patterns as the depth increases to infinity.     

Bifurcation of developed flow in rectangular channels rotating about the transverse axis

The linear problem of the stability of Poiseuille flow between two infinite plates rotating about an axis parallel to the plates and normal to the flow direction was studied in [1, 2]. It was established that the flow is least stable with respect to disturbances in the form of standing waves known as Taylor eddies. The experimental data of [1, 3] and the results [4] of a numerical integration of the Navier-Stokes equations for channels with cross sections highly elongated in the direction of the axis of rotation are in good agreement with the conclusions of the linear theory. In the case of channels with a side ratio of the order of unity, which is of greater practical interest, the primary flow becomes essentially three-dimensional and evolves with variation of the governing criteria: the Reynolds and Rossby numbers. Obviously, this seriously complicates the use of the methods of the linear theory. The effect of the ratio of the sides of the cross section on the stability of the primary flow regime was studied experimentally in [3]. The present article describes the results of an investigation of the problem based on a numerical method of integrating the nonlinear Navier-Stokes equations. Moreover, an asymptotic estimate of the stability limit of the primary regime, based on a local condition of inviscid instability of rotating flows, is presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 27–32, September–October, 1985.     

Unsteady gravity-elastic and gravity-capillary ship waves

The development of three-dimensional waves generated by a region of pressures moving uniformly and rectilinearly over the surface of a thin elastic isotropic plate covering an ideal fluid layer of finite depth is investigated. The pressures act starting at a certain instant. A qualitative similarity between the waves occurring and gravity-capillary waves is noted. The calculations are made for an ice cover. This model problem permits examining a number of properties of the oscillations of the ice cover occurring when hauling freight over ice roads, landing and takeoff of aircraft from ice fields, etc. [1]. The development of ship waves in a fluid of finite depth in the absence of a floating plate was investigated in [2, 3] and gravity-capillary waves were studied in [4–6]. Certain properties of steady three-dimensional waves occurring during movement of a load over the surface of a floating elastic plate were established in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 26–32, September–October, 1978.