Propagation of Perturbations in Plane Poiseuille Flow between Walls of Nonuniform Compliance

The evolution of small perturbations in longitudinally nonuniform flows is studied with reference to the problem of the propagation of flow perturbations in a plane channel with walls of variable elasticity. Using the solution of the problem of the receptivity of the flow to local vibrations of the walls, the problem considered can be reduced to the solution of an integral equation for a single function, namely, the complex vibration amplitude of the walls. A numerical method for solving this equation on the basis of a piecewise-linear approximation of the unknown function is proposed. It is shown that the instability wave amplitude changes discontinuously at the junction of the rigid and elastic channel sections. A second method of investigating the process of propagation of perturbations in the flow considered is proposed. This method is based on laws of evolution of perturbations in nonuniform flows and an analytic solution of the problem of perturbation scattering on the junction of walls with different compliance. On the basis of this method the classical stability theory is generalized to include the case of nonuniform flows.     

Spatial Evolution of Nonstationary Perturbations in Hamel Flow

The propagation of small perturbations in longitudinally inhomogeneous flows is investigated. The evolution of the perturbations is studied with reference to the radial flow of a viscous incompressible fluid between plane nonparallel walls, the simplest inhomogeneous flow. Using a generalized method of variation of constants, the corresponding boundary-value problem is reduced to an infinite-dimensional evolutionary system of ordinary differential equations for the complex amplitudes of the eigensolutions of a locally homogeneous problem. Physically, the method can be interpreted as a representation of the perturbation evolution process via two concomitant processes: the independent amplification (attenuation) of normal modes of the locally homogeneous problem and the rescattering of these modes into each other on local inhomogeneities of the base flow. The calculations show that reduced versions of the method are promising for describing the linear stage of laminar-turbulent transition in a boundary layer.     

Propagation of a Tollmien-Schlichting wave over the junction between rigid and compliant surfaces

The propagation of an instability wave over the junction region between rigid and compliant panels is studied theoretically. The problem is investigated using three different methods with reference to flow in a plane channel containing sections with elastic walls. Within the framework of the first approach, using the solution of the problem of flow receptivity to local wall vibration, the problem considered is reduced to the solution of an integro-differential equation for the complex wall oscillation amplitude. It is shown that at the junction of rigid and elastic channel walls the instability-wave amplitude changes stepwise. For calculating the step value, another, analytical, method of investigating the perturbation propagation process, based on representing the solution as a superposition of modes of the locally homogeneous problem, is proposed. This method is also applied to calculating the flow stability characteristics in channels containing one or more elastic sections or consisting of periodically alternating rigid and compliant sections. The third method represents the unknown solution as the sum of a local forced solution and a superposition of orthogonal modes of flow in a channel with rigid walls. The latter method can be used for calculating the eigenvalues and eigenfunctions of the stability problem for flow in a channel with uniformly compliant walls.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 31–48. Original Russian Text Copyright © 2004 by Manuilovich.     

Investigation of Mechanisms of Excitation and Growth of Unstable Perturbations in Pulsating Flows

General laws of the processes of generation and amplification of secondary perturbations in oscillating viscous fluid flows are studied theoretically. The stability and receptivity are analyzed with reference to perturbations generated by fluctuations of the flow rate of Poiseuille flow induced by small two-dimensional roughnesses of the channel walls. It is shown that the presence of roughness leads to excitation in the flow of perturbations at all multiples of the main flow oscillation frequency. Using the Fourier transform along the streamwise coordinate, the problem of calculating the frequency harmonics is reduced to a system of equations of the Orr-Sommerfeld type interrelated via the oscillatory component of the main flow. On the basis of an investigation of the analytic properties of the Fourier-image it is shown that upstream and downstream of the roughness the perturbation can be represented in the form of a superposition of modes of the time-dependent Poiseuille flow. The modes are classified and their spectrum is calculated. The structure of the mean-square fluctuations generated by free perturbations is investigated. Examples of calculating the evolution of forced perturbations are given for cases in which the scattering of the oscillations of the main flow on the roughness leads to the generation of one or two modes growing downstream.     

Features of the electric current and field distributions in flows with narrow layers of highly conductive gas

A solution obtained by Fourier's method provides the basis for analyzing the influence of a narrow gas layer, of higher conductivity than the rest of the flow, on the Joule dissipation and current distribution in the terminal zone of a plane magnetohydrodynamic channel with nonconducting walls. The MHD interaction parameter, Reynolds magnetic number, and Hall parameter are assumed small. It is shown that a narrow, highly conductive layer can on occasions be replaced by a surface of discontinuity, on which well-defined relations between the electric quantities are satisfied. The presence of such a layer leads to an increase in the Joule dissipation and a reduction in the lengths of the current lines. A hopeful arrangement for a magnetohydrodynamic energy converter is one in which an inhomogeneous flow is used, consisting of a continuous series of alternating very hot and less hot zones [1,2]. For this arrangement, it is worth examining the influence of the stratified conductivity distribution of the working body on the Joule dissipation and the electric currents in the channel. Numerous papers have discussed the case of inhomogeneous conductivity in the context of MHD system electrical characteristics. A general solution was obtained in [3] for the stationary problem on the electric field in a plane MHD channel with nonconducting walls when the magnetic field and conductivity are arbitrary functions of the longitudinal coordinate. In [4], where the braking of undeformed conducting clusters was investigated, the Joule dissipation, linked with the appearance of closed eddy currents in the cluster as it enters and leaves the magnetic field, was evaluated. The relationships between the electrical quantities, on moving through a narrow layer of low-conductivity liquid, were considered in [5].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 1, pp. 39–43, January–February, 1970.In conclusion, the author thanks A. B. Vatazhin for valuable advice and discussion.     

Calculation of the longitudinal velocity in the linear problem of a vibrator in a subsonic boundary layer

The linear problem is considered of a localized vibrator mounted on a flat plate in a subsonic boundary layer. The plate and the vibrator are assumed to be heat-insulated, and the dimensions of the vibrator and the frequency of the oscillations are such that the flow may be described by means of the equations of a boundary layer with self-induced pressure. The amplitude of the oscillations of the vibrator and the perturbations of the flow parameters corresponding to it are assumed to be small, and this makes it possible to linearize these equations. Integral transformations are used to construct a solution for values of the time greatly exceeding the period of the oscillations of the vibrator. The profiles of the perturbations of the longitudinal velocity are calculated in dependence on the transverse coordinate for various values of the longitudinal coordinate. A comparison is made with the profiles of the perturbations of the longitudinal velocity which have been obtained experimentally.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 61–67, May–June, 1987.     

Three-dimensional receptivity of boundary layers

The paper presents a review of results of some recent (mainly experimental) studies devoted to a quantitative investigation of the problem of receptivity of the 2D and 3D boundary layers with respect to various 3D (in general) external perturbations. The paper concentrates on the mechanisms of excitation and development of stationary and travelling instability modes in a 3D boundary layer on a swept wing, as well as in 2D boundary layers including the Blasius flow and a self-similar boundary layer with an adverse pressure gradient. In particular, the following problems of the boundary-layer receptivity are discussed: (i) receptivity to localized 3D surface roughness, (ii) receptivity to localized 3D surface vibrations, (iii) acoustic receptivity in presence of 3D surface roughness, and (iv) acoustic receptivity in the presence of 3D surface vibrations. All experiments described in the paper were conducted using controlled disturbance conditions with the help of simulation of the stationary and non-stationary perturbations by means of several disturbance generators. This approach gives us the possibility to obtain quantitative results which are independent of any uncontrolled background perturbations of the flow and the experimental model. In contrast to the data obtained at “natural” environmental conditions these results can be directly compared with calculations without any significant assumptions about the physical nature of the disturbances under investigation. The complex (amplitude and phase) coefficients of the boundary-layer receptivity to external perturbations, obtained as functions of the disturbance frequency and the spanwise wavenumber (or the wave propagation angle), represent the main results of the experiments described. These results can be used for the evaluation of the initial amplitudes and phases of the instability modes generated by various external perturbations, as well as for quantitative verification of linear receptivity theories. Several examples of the comparison of experimental results with calculations are also presented in this paper. A brief analysis of the state-of-art in the field is performed and some general properties of different receptivity mechanisms are discussed.     

Application of the Kantorovich-Galerkin Method for Solving Boundary Value Problems with Conditions on Moving Borders

The approximate Kantorovich-Galerkin method is considered for solving problems describing the vibrations of viscoelastic objects with conditions on moving boundaries and analyzing the resonance properties of these objects. The method makes it possible to take into account the effect of forces of environmental resistance on the system, flexural rigidity, and also boundary conditions with weak nonstationarity. The mathematical formulation of the problem involves a partial differential equation with respect to the desired displacement function and inhomogeneous boundary conditions. The Kantorovich-Galerkin method makes it possible to take into account the initial conditions, but they do not affect the resonance properties of linear systems, so in this case they are not taken into account. By introducing a new function into the problem, the boundary conditions are reduced to homogeneous ones. The solution is carried out in dimensionless variables to within a second order of smallness with respect to small parameters characterizing the velocity of the boundary motion and viscoelasticity. Using the Kantorovich-Galerkin method, an approximate solution of high accuracy of the problem of forced longitudinal vibrations of a viscoelastic rope of variable length, one end of which is wound on a drum, and the second is rigidly fixed, is found. The results obtained for the amplitude of oscillations corresponding to the nth dynamical mode are presented. The phenomenon of steady resonance and passage through resonance is investigated using numerical methods. A graphical dependence of the maximum amplitude of the rope oscillations as it passes through the resonance, depending on the coefficient characterizing the viscoelasticity of the object based on the Voigtmodel, is presented. The accuracy of the Kantorovich-Galerkin method is estimated.     

Flow in rotating radial channels at small Rossby and Ekman numbers

The problem of flow of a viscous fluid in a rotating channel is considered in the region of very small Rossby and Ekman numbers and moderately large Reynolds numbers. Asymptotic expressions with respect to the Ekman number are found for the velocity components and the longitudinal pressure gradient by solving a system of linear differential equations using Fourier series. The stability limits of such flow are predicted. Attention is drawn to a similarity between the velocity profiles of these flows and flows of a magnetic fluid and a fluid executing longitudinal oscillations in a fixed channel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 11–15, January–February, 1984.     

Transient natural convection flow of a particulate suspension through a vertical channel

Continuum equations governing transient, laminar, fully-developed natural convection flow of a particulate suspension through an infinitely long vertical channel are developed. The equations account for particulate viscous effects which are absent from the original dusty-gas model. The walls of the channel are maintained at constant but different temperatures. No-slip boundary conditions are employed for the particle phase at the channel walls. The general transient problem is solved analytically using trigonometric Fourier series and the Laplace transform method. A parametric study of some physical parameters involved in the problem is performed to illustrate the influence of these parameters on the flow and thermal aspects of the problem.     

The Onset of Double-Diffusive Convection in a Superposed Fluid and Porous Layer Under High-Frequency and Small-Amplitude Vibrations

We numerically simulate the initiation of an average convective flow in a system composed of a horizontal binary fluid layer overlying a homogeneous porous layer saturated with the same fluid under gravitational field and vibration. In the layers, fixed equilibrium temperature and concentration gradients are set. The layers execute high-frequency oscillations in the vertical direction. The vibration period is small compared with characteristic timescales of the problem. The averaging method is applied to obtain vibrational convection equations. Using for computation the shooting method, a numerical investigation is carried out for an aqueous ammonium chloride solution and packed glass spheres saturated with the solution. The instability threshold is determined under two heating conditions—on heating from below and from above. When the solution is heated from below, the instability character changes abruptly with increasing solutal Rayleigh number, i.e., there is a jump-wise transition from the most dangerous shortwave perturbations localized in the fluid layer to the long-wave perturbations covering both layers. The perturbation wavelength increases by almost 10 times. Vibrations significantly stabilize the fluid equilibrium state and lead to an increase in the wavelength of its perturbations. When the fluid with the stabilizing concentration gradient is heated from below, convection can occur not only in a monotonous manner but also in an oscillatory manner. The frequency of critical oscillatory perturbations decreases by 10 times, when the long-wave instability replaces the shortwave instability. When the fluid is heated from above, only stationary convection is excited over the entire range of the examined parameters. A lower monotonic instability level is associated with the development of perturbations with longer wavelength even at a relatively large fluid layer thickness. Vibrations speed up the stationary convection onset and lead to a decrease in the wavelength of most dangerous perturbations of the motionless equilibrium state. In this case, high enough amplitudes of vibration are needed for a remarkable change in the stability threshold. The results of numerical simulation show good agreement with the data of earlier works in the limiting case of zero fluid layer thickness.     

Estimates of the evolution of small perturbations in the radial spread (drain) of a viscous ring

This paper studies the evolution of small perturbations in the kinematic and dynamic characteristics of the radial flow of a flat ring filled with a homogeneous Newtonian fluid or an ideal incompressible fluid. When the flow rate is specified as a function of time, the main motion is completely determined by the incompressibility condition regardless of the properties of the medium. A biparabolic equation for the stream function with four homogeneous boundary conditions which simulate adhesion to the expanding (contracting) walls of the ring is derived. Upper bounds for the perturbation are obtained using the method of integral relations for quadratic functionals. The case of an exponential decay of initial perturbations is considered in a finite or infinite time interval. The admissibility of the inviscid limit in this problem is proved, and upper and lower bounds for this limit are estimated.     

Example of an exact solution of the stationary problem of two-layer flows with evaporation at the interface

Stationary two-layer liquid and gas flows with fluid evaporation at the interface are studied. On the solid impermeable boundaries of the channel, no-slip conditions are satisfied and a linear temperature distribution along the longitudinal coordinate and a condition for the vapor concentration at the upper boundary are specified. On the thermocapillary interface, remaining undeformed, the following conditions are specified: kinematic and dynamic conditions, a condition for thermal flows with mass transfer, continuity conditions for the velocity, temperature, and mass balance, and a relation for the saturated vapor concentration. An exact solution of the stationary problem for a given gas flow rate is obtained. Examples of velocity profiles are given for stationary flows of the ethanol-nitrogen system under normal and reduced gravity are given. The effect of longitudinal temperature gradients specified at the boundaries of the channel on the flow pattern is investigated.     

Stability of two-layer fluid flow

The problem of two-layer convective flow of viscous incompressible fluids in a horizontal channel with solid walls in the presence of evaporation is considered in the Oberbeck–Boussinesq approximation assuming that the interface is an undeformable thermocapillary surface and taking into account the Dufour effect in the upper layer which is a mixture of gas and liquid vapor. The effects of longitudinal temperature gradients at the boundaries of the channel and the thicknesses of the layer on the flow pattern and the evaporation rate are studied under conditions of specified gas flow and the absence of vapor flow on the upper boundary of the channel. It is shown that the long-wavelength asymptotics for the decrement is determined from the flow characteristics, the longwavelength perturbations occurring in the system decay monotonically, and the thermal instability mechanism is not potentially the most dangerous.     

Acoustic,entropy, and vorticity perturbations in a channel of variable cross section

A study is made of the problem of the propagation of infinitesimally small perturbations in a gas stream moving in a channel of variable cross section when the flow cannot be regarded as isentropic and irrotational. The solution is found in the framework of the linear theory of the flow of an ideal gas and the quasi-one-dimensional hydraulic approximation for the steady regime. For irrotational and isentropic perturbations in a nozzle, this problem was considered in [1–4]. In [1], the problem is generalized to take into account entropy perturbations in the nozzle for the case of longitudinal oscillations. The present paper treats arbitrary modes in a nozzle and takes into account not only entropy but also vorticity perturbations in the moving stream. For each of the three perturbation types — acoustic, entropy, and vorticity — the solutions are expanded in series in cylindrical functions. It is shown that in the considered approximation each oscillation mode can be analyzed independently of the others. In the special case of flow in a Laval nozzle, the concept of impedance (admittance), which is widely used in acoustics, is generalized to take into account entropy and vorticity perturbations. The contribution to the flow dynamics of the acoustic, entropy, and vorticity perturbations is estimated numerically for longitudinal and transverse modes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 91–98, January–February, 1982.     

Parametric Excitation of a Secondary Flow in a Vertical Layer of a Fluid in the Presence of Small Solid Particles

Thermal convection is studied in an inhomogeneous medium consisting of a fluid and a solid admixture under conditions of finite–frequency vibrations. Convection equations are derived within the framework of the generalized Boussinesq approximation, and the problem of flow stability in a vertical layer of a viscous fluid with horizontal oscillations along the layer to infinitely small perturbations is considered. A comparison with experimental data is made.     

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.     

Longitudinal oscillations of bodies of revolution and flow at supersonic velocities

The method of aerodynamic derivatives [1–3] can be used for the investigation of the flow around a body executing oscillations with a small amplitude. The characteristics of the flow are expressed in the form of functions which are determined from the solution of the linearized equations of gasdynamics and describe the flow pattern with adequate accuracy. The present article is devoted to the discussion of the results of solution of the general nonstationary problem in nonlinear formulation. Supersonic flows around a hemisphere and a cylindrical front end executing arbitrary harmonic oscillations along the axis of symmetry or experiencing the corresponding oscillations of the flow (turbulent atmosphere) are discussed as examples. The effect of the nonlinearity on the flow pattern is demonstrated for different Strouhal numbers. The results are compared with those of the linear theory and with the results obtained from the solution of the corresponding stationary problems. The solution is obtained by using the characteristic method in form [4].     

Hopf Bifurcation in Viscous Incompressible Flow Down an Inclined Plane

We provide the Hopf bifurcation theorem, which guarantees the existence of time periodic solution bifurcating from the stationary flow down an inclined plane under certain assumptions on the eigenvalues of the problem obtained by linearization around the stationary flow. Since we reduce the problem to the fixed domain, the inhomogeneous terms of reduced equations and reduced boundary conditions contain the highest derivatives. To deal with these we apply the Lyapunov–Schmidt decomposition directly.     

Stability of a vertical liquid film with consideration of the marangoni effect and heat exchange with the environment

The stability of a free vertical liquid film under the combined action of gravity and thermocapillary forces has been studied. An exact solution of the Navier-Stokes and thermal conductivity equations is obtained for the case of plane steady flow with constant film thickness. It is shown that if the free surfaces of the film are perfectly heat insulated, the liquid flow rate through the cross section of the layer is zero. It is found that to close the model with consideration of the heat exchange with the environment, it is necessary to specify the liquid flow rate and the derivative of the temperature with respect to the longitudinal coordinate or the flow rate and the film thickness. The stability of the solution with constant film thickness at small wave numbers is studied. A solution of the spectral problem for perturbations in the form of damped oscillations is obtained.