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.     

On the Perturbation of Hamel Flow Due to Unevenness of the Channel Walls

A method of analyzing the receptivity of longitudinally inhomogeneous flows is proposed. The process of excitation of natural oscillations is studied with reference to the simplest inhomogeneous flow: the two-dimensional flow of a viscous incompressible fluid in a channel with plane nonparallel walls. As physical factors generating perturbations, the cases of a stationary irregularity and localized vibration of the channel walls are considered. By changing the independent variables and unknown functions of the perturbed flow, the problem of the generation of stationary perturbations above an irregularity is reduced to a longitudinally homogeneous boundary-value problem which is solved using a Fourier transform in the longitudinal variable. The same problem is investigated using another method based on representing the required solution in the form of a superposition of solutions of the homogeneous problem and a forced solution calculated in the locally homogeneous approximation. As a result, the problem of calculating the longitudinal distributions of the amplitudes of the normal modes is reduced to the solution of an infinite-dimensional inhomogeneous system of ordinary differential equations. The numerical solution obtained using this method is tested by comparison with an exact calculation based on the Fourier method. Using the method proposed, the problem of flow receptivity to harmonic oscillations of parts of the channel walls is analyzed. The calculations performed show that the method is promising for investigating the receptivity of longitudinally inhomogeneous flow in a laminar boundary layer.     

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.     

Unsteady effects in weakly perturbed pulsatile channel flows

Pulsatile incompressible laminar flow in a plane channel with slight asymmetric deformation of the walls is considered on the basis of an asymptotic approach to the solution of the Navier-Stokes equations at large Reynolds numbers. It is shown that for extended deformations the change in the direction of the undisturbed flow in the wall regions is accompanied by a sharp increase in the amplitude of the perturbations; this is an essentially unsteady process. For flows with a small positive friction stress two classes of eigensolutions are obtained in the quasisteady approximation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 96–104, March–April, 1987.The author is grateful to V. V. Sychev and A. I. Ruban for a number of useful consultations in the course of the work.     

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.     

Rarefied gas flow into a vacuum from a plane long channel closed at one end

The time-dependent problem of rarefied gas flow into a vacuum from a plane long channel closed at one end is solved on the basis of the kinetic S-model. The effect of diffuse molecular reflection from the channel walls on the flow velocity and the process of channel cavity vacuumization is studied as a function of the channel length and the extent of gas rarefaction under the condition that the wall temperature is maintained to be constant. The kinetic equation is solved numerically using a conservative finite-difference method of the second order of accuracy in spatial coordinates. The possibility of simplification of the problem for long times by means of reduction to the diffusion process is considered.     

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.     

Stability of the plane Couette flow of a disperse medium with a finite volume fraction of the particles

The linear hydrodynamic stability of the plane Couette flow of a suspension with a finite volume fraction of the particles is considered. The two-phase medium flow is described within the framework of the model of mutually penetrating continua which allows for the finiteness of the volume occupied by the particles. In the main flow the phase velocities are the same, while gravity is not taken into account. The stability of disperse flows with both uniform and nonuniform particle distributions is studied. The linearized system of the equations of suspension motion with the no-slip boundary conditions imposed on solid walls is reduced to the eigenvalue problem for an ordinary differential fourth-order equation in the stream function. The eigenvalues are sought using the orthogonolization method. The parametric investigation of the stability characteristics of the disperse flow is performed. It is shown that in the case of the uniform spatial distribution of the particles in the main flow, the presence of an admixture in the flow leads to a slight variation in the wave decay rates, while the flow remains stable for any permissible combinations of the dimensionless governing parameters. In the case of nonuniform distribution of inclusions the flow loses stability already for low Reynolds numbers on a wide range of the dimensionless governing parameters.     

Algorithm for transient growth of perturbations in channel Poiseuille flow

This study develops a direct optimal growth algorithm for three-dimensional transient growth analysis of perturbations in channel flows which are globally stable but locally unstable. Different from traditional non-modal methods based on the OrrSommerfeld and Squire(OSS) equations that assume simple base flows, this algorithm can be applied to arbitrarily complex base flows. In the proposed algorithm, a reorthogonalization Arnoldi method is used to improve orthogonality of the orthogonal basis of the Krylov subspace generated by solving the linearized forward and adjoint Navier-Stokes(N-S) equations. The linearized adjoint N-S equations with the specific boundary conditions for the channel are derived, and a new convergence criterion is proposed. The algorithm is then applied to a one-dimensional base flow(the plane Poiseuille flow) and a two-dimensional base flow(the plane Poiseuille flow with a low-speed streak)in a channel. For one-dimensional cases, the effects of the spanwise width of the channel and the Reynolds number on the transient growth of perturbations are studied. For two-dimensional cases, the effect of strength of initial low-speed streak is discussed. The presence of the streak in the plane Poiseuille flow leads to a larger and quicker growth of the perturbations than that in the one-dimensional case. For both cases, the results show that an optimal flow field leading to the largest growth of perturbations is characterized by high-and low-speed streaks and the corresponding streamwise vortical structures.The lift-up mechanism that induces the transient growth of perturbations is discussed.The performance of the re-orthogonalization Arnoldi technique in the algorithm for both one-and two-dimensional base flows is demonstrated, and the algorithm is validated by comparing the results with those obtained from the OSS equations method and the crosscheck method.     

A non-primitive boundary element technique for modeling flow through non-deformable porous medium using Brinkman equation

In this study, we develop a non-primitive boundary integral equation (BIE) method for steady two-dimensional flows of an incompressible Newtonian fluids through porous medium. We assume that the porous medium is isotropic and homogeneous, and use Brinkman equation to model the fluid flow. First, we present BIE method for 2D Brinkman equation in terms of the non-primitive variables namely, stream-function and vorticity variables. Subsequently, a test problem namely, the lid-driven porous cavity over a unit square domain is presented to assert the accuracy of our BEM code. Finally, we discuss an application of our proposed method to flows through porous wavy channel, which is a problem of significant interest in the micro-fluidics, biological domains and groundwater flows. We observe that the rate of convergence (\(R_{c}\)) increases with increasing Darcy number. For low Darcy number streamlines follow the curvature of the wavy-walled channel and no circulation occurs irrespective of the wave–amplitude, while for high Darcy number the flow circulation occurs near the crest of the wavy-walled channel, when the wave–amplitude is large enough.     

Propagation of acoustic perturbations along a supersonic jet flowing out into the atmosphere

The propagation of acoustic perturbations (specified in the outlet cross section of a particular channel) along a supersonic jet flowing out of the channel is considered; also considered is acoustic emission from the surface of the jet into the atmosphere. The solution of these problems is obtained by a numerical method on the linear approximation. The laws governing the propagation of the perturbations as a function of the perturbation frequency and other determining parameters are investigated; these parameters include the velocity and temperature of the jet, the velocity of the subsonic accompanying flow in the external medium, and the character of the perturbation in the initial cross section of the jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 92–99, March–April, 1977.     

Approximate calculation of channel flows under force and energy actions

Gasdynamic channel flows under force and energy actions are considered. An approximate method is proposed for solving the gasdynamic equations that describe these flows. Themethod includes the separation of an “active” flow volume in which the integral electric force and applied power, whose densities are assumed to be uniform, are concentrated and the numerical integration of the system of hydrodynamic equations over the entire channel (in laminar and turbulent variants) with the piecewise constant force and energy sources obtained. The results of experimental investigation are presented for the flow that arises after two accessories mounted on the opposite walls of the vertical rectangular channel of constant cross-section, which create a dielectric barrier discharge (DBD actuators). This flow is numerically simulated using the method developed. On the basis of the method proposed the flow characteristics are determined for a model subsonic diffuser on whose lower wall, immediately in front of the separation zone, the DBD actuator is mounted. The efficiency of this accessory in reducing the gasdynamic losses is demonstrated.     

Numerical Analysis of Laminar-Turbulent Transition in a Circular Pipe with Periodic Inflow Perturbations

The problem of the spatio-temporal evolution of perturbations introduced into the inlet cross-section of a circular pipe is solved numerically. The case of time-periodic inflow perturbations is considered for Re = 4000. It is shown that for relatively small inflow perturbations periodic flow regimes and for greater perturbations chaotic regimes are established.Periodic regimes the flow is a superposition of steady flow and a damped wave propagating downstream. The velocity profile of the steady component differs essentially from both the parabolic Poiseuille and developed turbulent flows and is strongly inhomogeneous in the angular direction. The angular distortion of the velocity profile is caused by longitudinal vortices developing as a result of the nonlinear interaction of inflow perturbations.Chaotic flow regimes develop when the amplitude of the inflow perturbations exceeds a certain threshold level. Stochastic high-frequency pulsations appear after the formation of longitudinal vortices in the regions of maximum angular gradient of the axial velocity. In the downstream part of the flow, remote from the transition region, the developed turbulent regime is formed. The distributions of all the statistical moments along the pipe level off and approach the values measured experimentally and calculated numerically for developed turbulent flows.     

The supercritical regime of nonlinear interaction of subsonic and supersonic inviscid jets in a two-dimensional channel

The nonlinear problem of the supercritical regime of interaction between sub- and supersonic inviscid jets flowing in a two-dimensional channel is investigated. The propagation of small pressure perturbations is considered. A numerical calculation is made of the shape of the streamline which separates the two jets, whose nonlinear perturbations have an oscillatory nature. The dependence is obtained of the amplitude of the oscillations on the similarity parameter representing the integrated characteristic of the profile of the unperturbed flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 156–160, September–October, 1985.     

Analysis of velocity equation of steady flow of a viscous incompressible fluid in channel with porous walls

Steady flow of a viscous incompressible fluid in a channel, driven by suction or injection of the fluid through the channel walls, is investigated. The velocity equation of this problem is reduced to nonlinear ordinary differential equation with two boundary conditions by appropriate transformation and convert the two‐point boundary‐value problem for the similarity function into an initial‐value problem in which the position of the upper channel. Then obtained differential equation is solved analytically using differential transformation method and compare with He's variational iteration method and numerical solution. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and sciences. Copyright © 2009 John Wiley & Sons, Ltd.     

Simulation of the Action of a Permeable Barrier on an Ideal Incompressible Channel Flow

Plane ideal incompressible flow in a rectangular channel partitioned by a thin permeable barrier (lattice) is considered. In flowing through the lattice the stream suddenly (jumpwise) changes direction and loses energy. The flow is assumed to be vortical; the vorticity is discontinuous on the lattice. A mathematical formulation of the problem for the stream function is proposed in the form of a nonlinear elliptic equation with coefficients discontinuous on the lattice line. A numerical solution is constructed using the finite-element iteration method. The results of the numerical simulation show how the flow velocity profile in the channel can be controlled by means of permeable barriers.     

Stabilité linéaire d'écoulements symétriques presque parallèles en canalLinear stability of nearly parallel symmetric flows in a channel.

In this Note, we present a temporal linear stability analysis of symmetric developing flows slightly perturbed from Poiseuille flow. The Chebyshev spectral collocation method is used to resolve the Orr–Sommerfeld equation. For the main flow, the solution considered is analytic. The results of the stability study depend essentially on the shape and amplitude of the velocity profiles imposed at the channel entry. To cite this article: A. Hifdi et al., C. R. Mecanique 332 (2004).     

Hydrodynamic flows and heat transfer in slightly noncylindrical channels

Viscous incompressible laminar flow and heat transfer in channels with a small arbitrary deviation from a cylindrical surface are examined. A linear system of equations and boundary conditions for the disturbed dynamic and thermal fields, obtained by linearizing the complete system of Navier-Stokes equations with respect to the solution for developed flows in cylindrical tubes of arbitrary cross section, is presented. In the important practical case in which the perturbations of the channel surface are concentrated on an interval of finite length it is shown that the integral dynamic and thermal characteristics of the channel can be found without solving the three-dimensional equations by going over to effective two-dimensional boundary-value problems which are fundamentally no more difficult to solve than those for developed flows. Extensions of the theory to flows with low-efficiency power sources are given. Applications to plane channels and circular tubes with deformed surfaces are considered. Among the numerous applications requiring information about the integral characteristics of flows in channels whose initially cylindrical surface is slighty deformed, we note the problem of heat transfer intensification by slightly deforming the tube surface with careful estimation of the accompanying increase in resistance [1] and the calculation of the resistance of capillaries and biological transport systems in the form of tubes and channels when the walls are deformed [2]. Below we consider laminar flow in channels with deformed walls. Whereas for the first problem this class of flows is only one of those possible (in general it is necessary to analyze the transition, turbulence and flow separation effects), in the second case, which is characterized by low Reynolds numbers, the laminar flow model is perfectly adequate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 21–30, March–April, 1990.The authors are grateful to A. Yu. Klimenko for useful discussions.     

Optimal disturbances of a dusty-gas plane-channel flow with a nonuniform distribution of particles

The classical stability theory for multiphase flows, based on an analysis of one (most unstable) mode, is generalized. A method for studying an algebraic (non-modal) instability of a disperse medium, which consists in examining the energy of linear combinations of three-dimensional modes with given wave vectors, is proposed. An algebraic instability of a dusty-gas flow in a plane channel with a nonuniform particle distribution in the form of two layers arranged symmetrically with respect to the flow axis is investigated. For all possible values of governing parameters, the optimal disturbances of the disperse flow have zero wavenumber in the flow direction, which indicates their banded structure (“streaks”). The presence of dispersed particles in the flow increases the algebraic instability, since the energy of optimal disturbances in the disperse medium exceeds that for the pure-fluid flow. It is found that for a homogeneous particle distribution the increase in the energy of optimal perturbations is proportional to the square of the sum of unity and the particle mass concentration and is almost independent of particle inertia. For a non-uniform distribution of the dispersed phase, the largest increase in the initial energy of disturbances is achieved in the case when the dust layers are located in the middle between the center line of the flow and the walls.