Asymptotic behavior of localized perturbations in free shear layers

A study is made in the linear approximation, within the scope of the ideal fluid, of the asymptotic behavior of three-dimensional localized perturbations of the parameters of a shear flow which over considerable periods of time turn into growing and propagating wave packets. The behavior of the packets is studied in every possible system of coordinates moving with constant velocity parallel to the plane of the velocity shear. Mathematically, the problem reduces to using the method of steepest descent to study the asymptotic behavior of double Fourier integrals which depend parametrically on these velocities. The saddle points which determine this asymptotic behavior are found numerically. A region is indicated in a plane of flow parallel to the velocity shear which is moving and expanding linearly with time, and in which growing perturbations are found over long periods of time. The results obtained enabled us to write down the criteria for absolute and convective instability. This problem has been considered previously for flows of an ideal fluid with a shear discontinuity in the velocity [1, 2] and for flows in a wake [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, 8–14, March–April, 1987.The author wishes to express his sincere gratitude to A. G. Kulikovskii for formulating the problem and for advice on numerous occasions.     

Development of perturbations at the interface between two liquids

The asymptotic behavior is studied in the case of large times of initially localized, one-dimensional, small perturbations of the interface between two liquids in the presence of a tangential velocity discontinuity, taking account of surface tension and the force of gravity. The asymptotic behavior of the perturbed region is found; i.e., on the plane x, t a sector is shown with vertex at the origin of the coordinates, inside of which the perturbations tend to infinity with increase of t, and outside of which the perturbations tend to zero, and the velocities of motion of the boundaries of the perturbed region are calculated. The conditions are shown for which the instability of the tangential discontinuity will not be absolute; i.e., when they are fulfilled, flows with a tangential velocity discontinuity can occur. For the case where the effect of the force of gravity can be neglected, these conditions are independent of the magnitude of the surface tension.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 46–49, September–October, 1977.     

Development of two-dimensional perturbations on the surface of a tangential discontinuity

A study is made of the asymptotic behavior at long times of initially localized small two-dimensional perturbations of the interface of two fluids in the presence of a tangential discontinuity of the velocity; surface tension is taken into account. The development of one-dimensional perturbations was considered earlier in [1]. The asymptotic behavior of the perturbed region is found, i.e., in the xyt space there is found a cone with apex at the origin such that perturbations tend to infinity with increasing t along rays within the cone, while perturbations tend to zero along the remaining rays. Conditions are found under which the instability of the tangential discontinuity is not absolute, i.e., when these conditions are satisfied, flows with tangential discontinuity of the velocity can take place. These conditions, like the shape of the cone, do not depend on the magnitude of the surface tension.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 12–16, May–June, 1979.     

The motion of snow avalanches in the conditions of limiting friction

We investigate the equations of motion of large snow avalanches, and in contrast with [1–3] we take into account the fact that the dry friction can reach a critical value above which the snow in the avalanche or the underlaying material cannot sustain the friction. We find asymptotic solutions for long times after the beginning of motion. These solutions describe the avalanche motion in which a part of the snow moves in the conditions of limiting friction over a tilted plane with a uniform layer of snow. The equations which are used to find these asymptotic solutions have the property that for certain depths the flow velocity of small perturbations decreases with increasing depth. This is related to a number of unusual features (from the hydraulic point of view) of the solutions. In particular, on relatively gentle slopes two zones are formed in the avalanche: the forward part, with a large velocity and thickness of the moving layer, and the rear part, which is significantly slower and thinner. The two parts are separated by a narrow region characterized by a sharp decline in velocity and thickness of the moving layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 30–37, September–October, 1977.     

Locality properties of the problem of hydrodynamic stability

Locality properties are formulated for short-wave length disturbances in the problem of hydrodynamic stability, which together with global flow stability enable us to study the stability of particular sections of the stream, e.g., the flow core or the zone next to the wall. The locality properties are illustrated in the spectrum of small perturbations of plane Poiseuille flow and flows which are obtained by deforming a small section of the Poiseuille parabola. Such a deformation produces points of inflection which lead to the appearance of growing perturbations with wavelength of the order of the deformation zone. It is shown that discontinuities in the velocity profile leads to the loss of stability for high enough Reynolds' numbers.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 2, pp. 56–61, March–April, 1970.     

波包在后掠翼三维边界层中的演化特征

研究了点源产生的孤立波包在后掠平板三维边界层中的演化特征．理论计算的波包增长路径和ＮｉｔＳＣｈｋｅ－Ｋｏｗｓｔｙ所做流场显示得到的条纹结构一致,证明条纹是波包的等相位线;波包方程的渐近解表明在三维边界层中ｅＮ方法应沿着为实数的方向积分;ｅＮ方法过高预测了扰动幅值的增长率,与Ｄｅｒｈｌｅ的实验结论一致．     

Localized disturbances in parallel shear flows

The development of localized disturbances in parallel shear flows is reviewed. The inviscid case is considered, first for a general velocity profile and then in the special case of plane Couette flow so as to bring out the key asymptotic results in an explicit form. In this context, the distinctive differences between the wave-packet associated with the asymptotic behavior of eigenmodes and the non-dispersive (inviscid) continuous spectrum is highlighted. The largest growth is found for three-dimensional disturbances and occurs in the normal vorticity component. It is due to an algebraic instability associated with the lift-up effect. Comparison is also made between the analytical results and some numerical calculations.Next the viscous case is treated, where the complete solution to the initial value problem is presented for bounded flows using eigenfunction expansions. The asymptotic, wave-packet type behaviour is analyzed using the method of steepest descent and kinematic wave theory. For short times, on the other hand, transient growth can be large, particularly for three-dimensional disturbances. This growth is associated with cancelation of non-orthogonal modes and is the viscous equivalent of the algebraic instability. The maximum transient growth possible to obtain from this mechanism is also presented, the so called optimal growth.Lastly the application of the dynamics of three dimensional disturbances in modeling of coherent structures in turbulent flows is discussed.     

Flow and instability of capillary jets interacting with the surrounding medium

The flow of an axially symmetric capillary jet of a viscous incompressible liquid in the space occupied by another liquid is investigated. The problem of stationary flow in the jet and in the surrounding medium under the action of viscosity, capillary forces, and gravity was obtained numerically. The instability problem of this flow to small perturbations in the form of running waves is stated and solved numerically. The values of the dimensionless Reynolds, Weber, and Froude numbers are explained, as well as the effect of the initial velocity profile in the jet, its instability, and subsequent jet decay into drops.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 50–59, November–December, 1978.     

Certain types of vibrationally convective instability of a two-dimensional fluid layer in zero gravity

The stability of a new equilibrium configuration possible in a two-dimensional layer of nonisothermal fluid executing high-frequency vibrations in zero gravity is investigated in the framework of the linear theory. A study is made on the basis of the equations of vibrational convection. Instability with respect to one-dimensional and two-dimensional perturbations is studied. An elementary exact solution is obtained for the one-dimensional perturbations. Vibrationally connective instability of a fluid in zero gravity has been studied in a number of papers [1-3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 4–7, September–October, 1987.The author expresses his gratitude to G. Z. Gershuni for his constant interest in my work.     

Signal velocity for transient waves in linear dissipative media

The aim of this paper is to discuss the concept of signal velocity for transient sinusoidal waves in linear dissipative media. The method of the steepest descent path, first introduced by Brillouin in this respect, is used to show whether the signal velocity may equal the group velocity or the phase velocity.     

Construction of the solution of the linear problem of shock wave stability

Within the framework of the linear theory a solution is obtained in explicit form for a solitary plane shock using Fourier and Laplace transforms and assuming only the finiteness of the small perturbations. In the case of three-dimensional flows the small deformations of the shock wave surface are represented in the form of integral functionals, with Poisson kernels, of the initial perturbations of both the shape of the shock wave and the parameters of the flow field beyond it. The solution for plane flows is then constructed by the method of descent. From the equations obtained it follows that: for the region of stability and the intermediate region the solution has a finite domain of dependence on the initial perturbations; despite the fact that the structure of the domain of dependence in these regions is different, at large times the damping of the perturbations proceeds in accordance with a single law at a rate that depends on the dimensionality of the shock front.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. A, pp. 130–138, July–August, 1988.     

Role of two-dimensional acoustic wave cutoff in the instability of subsonic MHD flows

One of the important factors affecting the structure of the natural vibrations and the conditions under which they build up in an inhomogeneous subsonic flow may be the cutoff of non-one-dimensional sound waves expressed in the strong reflection of such waves from the critical sections (caustics). In this study the case of natural two-dimensional acoustic perturbations in an inhomogeneous subsonic conducting gas flow in the presence of critical sections is subjected to an asymptotic analysis. Special attention is paid to the conditions of growth of the two-dimensional acoustic perturbations in the internal resonator formed by two critical sections and the walls of an MHD channel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 26–36, March–April, 1988.The authors are grateful to seminar participants L. M. Biberman and G. A. Lyubimov for useful discussions.     

Asymptotic analysis of a pulsating flow of viscous fluid in a symmetric deformed channel

The time-periodic flow of a viscous incompressible fluid in a two-dimensional symmetric channel with slightly deformed walls is considered. The solution of the Navier-Stokes equations is constructed by means of the method of matched asymptotic expansions [1] at large characteristic Reynolds numbers. It is shown that in an unsteady flow a region of nonlinear perturbations surrounds the line of zero velocity inside the fluid. The formation and development of such nonlinear zones with respect to time is considered. An alternation of the topological features of the streamline pattern in the nonlinear perturbation zone is discovered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 17–23, July–August, 1987.The author is deeply grateful to V. V. Sychev for his formulation of the problem and his attentive attitude to my work.     

Self-similar problems of propagation of an extended hydrofracture in a permeable medium

The propagation of an extended hydrofracture in a permeable elastic medium under the influence of an injected viscous fluid is considered within the framework of the model proposed in [1, 2]. It is assumed that the motion of the fluid in the fracture is turbulent. The flow of the fluid in the porous medium is described by the filtration equation. In the quasisteady approximation and for locally one-dimensional leakage [3] new self-similarity solutions of the problem of the hydraulic fracture of a permeable reservoir with an exponential self-similar variable are obtained for plane and axial symmetry. The solution of this two-dimensional evolution problem is reduced to the integration of a one-dimensional integral equation. The asymptotic behavior of the solution near the well and the tip of the fracture is analyzed. The difficulties of using the quasisteady approximation for solving problems of the hydraulic fracture of permeable reservoirs are discussed. Other similarity solutions of the problem of the propagation of plane hydrofractures in the locally one-dimensional leakage approximation were considered in [3, 4] and for leakage constant along the surface of the fracture in [5–7].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 91–101, March–April, 1992.     

Two methods for calculating the load on the surface of a slender body executing axisymmetric vibrations in a sonic gas flow

Low-frequency axisymmetric vibrations of the surface of a slender body in a sonic flow are considered. The distribution of the stationary longitudinal velocity on the body is assumed to be linear. The linear equation with variable coefficients for the nonstationary part of the velocity potential is solved by two methods: by separation of the variables, as was done in [1] for a two-dimensional flow, and by the method of superposition of sources. Particular solutions with the required singularity are obtained.Translated from Izvestiya Akaderaii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 151–154, March–April, 1980.     

Investigation of instability in vertical liquid films as an initial value problem

The instability of plane-parallel vertical viscous layer downflow is investigated. We solve not the classical eigenvalue problem for the Orr-Sommerfeld equation but a Cauchy problem with respect to time and a boundary-value problem with respect to the spatial variable for a linearized system of equations. The problem is solved by means of a Laplace transformation in time and a Fourier transformation in the spatial variable. Subsequently, using the residue theorem and the method of steepest descent makes it possible to predict asymptotically the perturbation behavior as time t → ∞. The system is convectively unstable and a localized perturbation spreads out at the velocities of the trailing and leading fronts. The packet behavior is investigated over a wide range of the flow parameters.     

Generation of secondary instability modes in the interaction of a Tollmien-Schlichting wave with isolated roughness

The infinite plane channel flow arising from the impingement of a plane instability wave of finite amplitude on isolated three-dimensional wall roughness is considered. The problem of the transformation of perturbations developing on the roughness in growing modes of secondary instability is solved. This problem describes the development of perturbations leading to the occurrence of a turbulent wedge. Simple relations describing the flow at large distances from the roughness are obtained. From these relations it follows that the angle at the vertex of the turbulent wedge is determined by the amplitude of the impinging wave, while the value of the perturbations generated is proportional to the roughness volume.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 28–38, May–June, 1995.The work was carried out with the financial support of the Russian Foundation for Fundamental Research (project No. 93-013-17613).     

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.     

Boundary layer in the neighborhood of the intersection of a concave cylindrical surface and a plane

In studies devoted to the theoretical and experimental investigation of longitudinal flow of a viscous fluid past corner regions, a corner formed by the intersection of two planes is usually considered [1–3]. In contrast, the present paper is concerned with the flow in the neighborhood of the line of intersection of a plane and a concave cylindrical surface (see Fig. 1). The asymptotic behavior of the Navier-Stokes equations at large Re is investigated for such a flow. Estimates are obtained for the velocity and characteristic scales of the flow. It is shown that curvature of one of the surfaces qualitatively changes the pattern of the longitudinal flow of a viscous fluid past a corner. The development of a three-dimensional boundary layer on a plane in the domain of influence of a concave cylindrical surface is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 160–165, March–April, 1981.     

Stability of plane magnetohydrodynamic couette flow with asymmetrical velocity profile

The hydrodynamic stability of plane magnetohydrodynamic Couette flow with asymmetrical velocity profile formed by a transverse magnetic field is investigated within the framework of the linear theory. The complete spectrum of the small perturbations is studied for the characteristic Hartmann numbers. The perturbations are classified in accordance with their phase velocity at large wave numbers. It is established that the stability of the flow is controlled by only one type of perturbations. The critical parameters of the problem are determined. The instability in question recalls the instability of Hartmann flow against asymmetrical perturbations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 12–18, May–June, 1971.The author thanks M. A. Gol'dshtik for interest in the work and V. A. Sapozhnikov and V. N. Shtern for useful discussions.