ON THE STABILITY OF DISTORTED LAMINAR FLOW(Ⅱ)——THE LINEAR STABILITY ANALYSIS OF DISTORTED PARALLEL SHEAR FLOW

Based on the hydrodynamic stability theory of distorted laminar flow and the kind of distortion profiles on the mean velocity in parallel shear flow given in paper [1], this paper investigates the linear stability behaviour of parallel shear flow, presents unstable results of plane Couette flow and pipe Poiseuille flow to two-dimensional or axisymmetric disturbances for the first time, and obtains neutral curves of these two motions under certain definition.     

ON THE STABILITY OF DISTORTED LAMINAR FLOW (I)—BASIC IDEAS AND THEORY

This paper suggests a hydrodynamic stability theory of distorted laminar flow, and presents a kind of distortion profile of mean velocity in parallel shear flow. With such distortion profiles, the new theory can be used to investigate the stability behaviour of parallel shear flow, and thus suggests a new possible approach to instability. The Project Supported by National Natural Science Foundation of China.     

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.     

Nonparallel stability of the flow in an axially rotating pipe

The linear stability of the developing flow in an axially rotating pipe is analyzed using parabolized stability equations (PSE). The results are compared with those obtained from a near-parallel stability approximation that only takes into account the axial variation of the basic flow. Though the PSE results obviously coincide with the near-parallel ones far downstream, when the flow has reached a Hagen-Poiseuille axial velocity profile with superimposed solid-body rotation, they differ significantly in the developing region. Therefore, the onset of instability strongly depends on the axial evolution of the perturbations. The PSE results are also compared with experimental data from Imao et al. [Exp. Fluids 12 (1992) 277], showing a good agreement in the frequencies and wavelengths of the unstable disturbances, that take the form of spiral waves. Finally, a simple method for detecting one of the conditions to characterize the onset of absolute instability using PSE is given.     

Hard excitation of auto-oscillations in a Hagen-Poiseuille flow

Sufficiently powerful perturbations of the flow of a liquid moving in circular pipes results in turbulence, starting with Reynolds numbers of the order of 2200–2300 [1]. It has been established theoretically [2, 3] that the flow of a viscous incompressible liquid in a pipe of circular section (Hagen-Poiseuille flow) is stable with respect to infinitesimally small perturbations for all Reynolds numbers. Attempts to obtain finite-amplitude flow instability by considering only two-dimensional perturbations [4, 5] were also unsuccessful. This paper shows that the considered flow is unstable with respect to three-dimensional perturbations of finite amplitude.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 181–183, September–October, 1984.The author wishes to express sincere gratitude to G. I. Petrov and S. Ya. Gertsenshtein for their interest in his work.     

Analytical solution for laminar flow through leaky tube

The laminar flow through a leaky tube is investigated, and the momentum and conservation of energy equations are solved analytically. By using the Hagen-Poiseuille velocity profile and defining unknown functions for the axial and radial velocity components, the pressure and mass transfer equations are obtained, and their profiles are plotted according to different parameters. The results indicate that the axial velocity, the radial velocity, the mass transfer parameter, and the pressure in the tube decrease as the fluid moves along the tube.     

Mass and heat transfer during solid sphere dissolution in a non-uniform fluid flow

We studied a nonisothermal dissolution of a solvable solid spherical particle in an axisymmetric non-uniform fluid flow when the concentration level of the solute in the solvent is finite (finite dilution of solute approximation). It is shown that simultaneous heat and mass transfer during solid sphere dissolution in a uniform fluid flow, axisymmetric shear flow, shear-translational flow and flow with a parabolic velocity profile can be described by a system of generalized equations of convective diffusion and energy. Solutions of diffusion and energy equations are obtained in an exact analytical form. Using a general solution the asymptotic solutions for heat and mass transfer problem during spherical solid particle dissolution in a uniform fluid flow, axisymmetric shear flow, shear-translational flow and flow with parabolic velocity profile are derived. Theoretical results are in compliance with the available experimental data on falling urea particles dissolution in water and for solid sphere dissolution in a shear flow.     

The effect of slip in the flow of a branched PP melt: experiments and simulations

In this paper visualisation and direct velocity profile measurement experiments for a branched polypropylene melt in a 10:1 axisymmetric contraction demonstrate the onset of wall slip. Video processing of the flow shows the formation of vortices and their diminution with increasing flow rate. Numerical simulations using a multimode K-BKZ viscoelastic and a purely viscous (Cross) model—both of them incorporating a nonlinear slip law—were used to predict the flow kinematics and dynamics as well as to deduce the slip velocity function by performing fitting to the velocity profiles. It was found that the numerical predictions agree well with the experimental results for the velocity profiles, and vortex formation, growth and reduction. It is suggested that such experiments (visualisation of entrance flow and direct velocity profile measurement) can be useful in evaluating the validity of constitutive equations and slip laws in the flow of polymer melts through processing equipment.     

同轴圆柱间滑移旋转流动问题的同伦解析解

从理论上研究了具有延伸柱面的同轴圆柱间滑移流动问题.通过引入适当的相似变换将控制方程组转化为一类非线性边值问题, 利用同伦分析方法首次获得问题的近似解析解, 分析讨论滑移参数、雷诺数和内外筒半径比对流动的影响.结果表明: 滑移边界参数对剪切应力有较大的影响, 滑移边界小的流体对壁面和边界层内流场施加了更大的剪切力; 增大雷诺数Re, 能增大内筒的壁面剪切力和扭矩; 内外筒半径比对流动结构也有较大的影响, 内筒半径固定, 增大外筒半径能够减少内筒的纵向剪切力.      

Influence of the initial conditions on axisymmetric jets in a parallel flow

An experimental investigation was made of the initial-section flow of axisymmetric helium, air, and freon-12 jets in a parallel air flow for two different velocity profiles at the nozzle exit near the boundary of the jet. In one case, the velocity profile was determined by boundary layers on the nozzle walls; in the other case, it was produced artificially by means of a honeycomb of tubes of variable length. Measurements were made of the profiles of the mean and the pulsation velocity and the temperature. The flow was also photographed. The investigations showed that, depending on the initial conditions, the intensity of mixing of the jets in the initial section at Reynolds numbers Re 10⁴ (calculated using the jet diameter) can change from the level determined by molecular diffusion to the level characteristic of developed turbulent flow. The flow structure in the annular mixing layer also depends strongly on the initial conditions. The observed ordered structures in the mixing layer are related to a section of development of perturbations near the nozzle. The ordered structures are strongly influenced by the effect on the jet of acoustic vibrations from an external source. When the initial velocity profile is produced by the honeycomb, the transition to developed turbulence may be due to the development of long-wavelength perturbations or to the development of small-scale turbulence generated by the flow over the end of the honeycomb.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 18–24, July–August, 1980.We thank V. M. levlev and K. I. Artamonov for assistance and for discussing the work.     

Global stability analysis of axisymmetric boundary layer over a circular cylinder

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier–Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi’s iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.     

Contrôle des profils de vitesse par déformation de parois dans une conduiteControl of velocity profils by walls deformation in a deformed conduit

Starting from a study of perturbation equations for stationary Newtonian fluid flow in deformed axisymmetric tubes, we propose a semi-analytical method allowing to generate in a given section of the conduit, an arbitrary velocity profil previously defined. We get this velocity profile by geometrical modification of the duct walls on a given length. Doing this, we create a specific perturbation of the upstream profile wich allows to generate the chosen velocity profile. To cite this article: M. Sijelmassi, J. Khalid Naciri, C. R. Mecanique 330 (2002) 153–158.     

Three-dimensional axisymmetric flow in perfectly plastic metals

A method is presented for solving the three-dimensional axisymmetric field equations for a perfectly plastic material which obeys the von-Mises yield criterion and the Levy-Mises flow law. The method is used for the particular case in which a small axisymmetric perturbed flow is superposed on a uniform flow without flow reversal taking place. The method then leads to solving a fourth order differential equation for the velocity potential. The special case of a thick cylindrical shell under compressive flow is examined. The solution so obtained, being derived, from the three dimensional theory, includes a correct treatment of transverse shear distortion. A preferred mode of instability is identified having a wave-length in reasonable agreement with that obtained experimentally by other workers.     

The study of the induction of subsonic wind tunnels with an axisymmetric working part

For the case of small perturbations of the velocity on the boundary of the flow and arbitrary degree of perforation of the wall of the wind tunnel which is constant along its length, a solution has been found for the axisymmetric boundary-value problem of subsonic flow around a thin body of revolution in perforated boundaries. The boundary condition connects the tangential component of the perturbed velocity and the component normal to the wall, and has a general form for the whole boundary. From the solution obtained, the optimum degree of perforation of the wall is found for which distortion of the pressure coefficient on the surface of the model is a minimum in comparison with the unbounded flow round the body. The questions of the induction of the walls of the working part of the wind tunnel, the formulation of the boundary condition, and the determination of the corrections to the aerodynamic characteristics of the model are considered in a number of studies [1–5]. Lately a promising method was been worked out for reducing the influence of the walls directly in the process of experiment by regulating the parameters of the gas near the boundary in such a way that they correspond to the parameters in flow round the same body by an unbounded flow [3, 5]. However, the design properties of many working tunnels do not permit this method to be applied. The induction of such tunnels as these can be reduced only by the choice of an optimum degree of perforation of the walls. In the articles [6–8] there is a study of the effect of perforated boundaries on the flow round a profile, the degree of perforation in [7] being assumed to be small.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 150–154, January–February, 1985.The author is grateful to V. M. Neiland for his constant attention to the study and for discussion of the results.     

Prediction of bubble concentration profiles in vertical turbulent two-phase flow

In vertical bubbly flow, the bubbles are not distributed evenly across the flow section. Several investigators have observed a wall-skewed bubble concentration profile in a vertical upward flow. This paper presents an analysis that predicts this type of bubble distribution by incorporating into the equation of motion a lateral force due to the relative velocity of the two phases and the eddy diffusivity of the liquid. Comparison of analysis and experiment shows good agreement.     

On the stability of distorted laminar flow(III)—The nonlinear stability analysis of distorted parallel shear flow

Based on the hydrodynamic stability theory of distorted laminar flow and the kind of distortion profiles on the mean velocity in parallel shear flow given in paper [1], this paper investigates the nonlinear stability behaviour of parallel shear flow, carries on stability calculation taking account of the perturbations of background turbulence noise under certain assumption, and obtains some results in accordance qualitatively with those of experiment for plane Poiseuille flow and pipe Poiseuille flow.The author thanks Prof. Zhou Heng sincerely for his kind offer of his computer program of the artificial neutrality method on the stability in subcritical range of plane Poiseuille flow.     

Linear stability of viscous incompressible flow in a circular viscoelastic tube

Flow stability in rigid tubes has been the subject of much research [1]. The overwhelming majority of authors of both theoretical and experimental studies now conclude that Poiseuille flow in a circular rigid tube is linearly stable. However, real tubes all possess elastic properties, the influence of which has not been investigated in such detail. For certain selected values of the parameters characterizing an elastic tube it has been shown that with respect to infinitesimal axisymmetric perturbations Poiseuille flow in the tube can be unstable [2]. In this case boundary conditions that did not take into account the fairly large velocity gradient of the undisturbed flow near the tube wall were used. The present paper reports the results of a numerical investigation of the linear stability of Poiseuille flow in a circular elastic tube with respect to three-dimensional perturbations in the form of traveling waves propagated along the system (azimuthal perturbation modes with numbers 0, 1, 2, 3, 4, and 5 are considered). It is shown that the elastic properties of the tube can have an important influence on the linear stability spectrum. In the case of axisymmetric perturbations it is possible to detect an instability which, at Reynolds numbers of more than 200, exists only for tubes whose modulus of elasticity is substantially less than that of materials in common use. The instability to perturbations of the second azimuthal mode is different in character, inasmuch as at Reynolds numbers greater than unity it occurs in stiffer tubes. Moreover, as the Reynolds number increases it can also occur in tubes of greater stiffness. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 126–134, November–December, 1986.     

Large-Eddy Simulations of Forced Three-Dimensional Impinging Jets

Large-eddy simulations of the flow field around twin three-dimensional impinging jets were carried out to simulate the near-ground hover configuration of a vertical takeoff and landing (VTOL) aircraft. Both the impinging jet and the upwash caused by the collision of the wall jets are modeled in this study. The evolution of the vortical structures in the impinging jet flow field, due to the introduction of axisymmetric and azimuthal perturbations at the jet exit, has been investigated. The vortical structures formed in the jet shear layer due to azimuthal forcing, show significant three-dimensional vortex stretching effects when compared to the structures formed during axisymmetric forcing. Breakdown of the large-scale structures into smaller vortices also occurs much earlier during azimuthal forcing. When compared to the upwash formed during axisymmetric forcing, the azimuthally perturbed jet forms an upwash that is less coherent and results in a weaker upload or lift-off force on the aircraft undersurface. Comparison with available experimental data indicates good agreement for the centerline velocity decay, the wall pressure variation and the phase speed of the vortical structures.     

Stability of motion of the plane boundary separating two fluids

The effect of viscosity on the disintegration of liquid jets can be considered in two ways. First, viscous forces alter the basic flow: they form a boundary layer whose presence necessarily alters wave formation. Second, viscous forces can have a direct effect on the development of perturbations for a given velocity profile of the basic flow. In this case the study of stability must be based on the Navier-Stokes equations instead of the equations of an ideal fluid. This complicates analysis considerably. Available data [1] indicate that this influence is very minor in the case of moderately viscous fluids. It appears, therefore, that the principal role is played by changes in the velocity profile alone, and that the behavior of the perturbations is described by the equations of an ideal fluid.In the present study we investigated the stability of motion and wave formation at the boundary between two fluids in order to determine the effect of viscosity on the drop formation mechanism. The simplest case of oscillation of the boundary is chosen in order to keep the analysis as simple as possible.     

Flow instability of nanofuilds in jet

The flow instability of nanofluids in a jet is studied numerically under various shape factors of the velocity profile, Reynolds numbers, nanoparticle mass loadings,Knudsen numbers, and Stokes numbers. The numerical results are compared with the available theoretical results for validation. The results show that the presence of nanoparticles enhances the flow stability, and there exists a critical particle mass loading beyond which the flow is stable. As the shape factor of the velocity profile and the Reynolds number increase, the flow becomes more unstable. However, the flow becomes more stable with the increase of the particle mass loading. The wavenumber corresponding to the maximum of wave amplification becomes large with the increase of the shape factor of the velocity profile, and with the decrease of the particle mass loading and the Reynolds number. The variations of wave amplification with the Stokes number and the Knudsen number are not monotonic increasing or decreasing, and there exists a critical Stokes number and a Knudsen number with which the flow is relatively stable and most unstable,respectively, when other parameters remain unchanged. The perturbation with the first azimuthal mode makes the flow unstable more easily than that with the axisymmetric azimuthal mode. The wavenumbers corresponding to the maximum of wave amplification are more concentrated for the perturbation with the axisymmetric azimuthal mode.