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.     

气液两相流动与固壁相互作用耦合求解的研究

气液两相流动与固壁相互作用的研究是液滴撞击壁面运动研究的重要基础.以结合了VOF和Level Set两种方法优点的用于气液相界面追踪的复合Level Set-VOF方法和利用唯象分析方法建立的能够反映接触角滞后性及壁面性质对润湿过程影响的壁面润湿模型为基础,提出了气液两相流动与固壁相互作用耦合求解流程,给出了气液两相流动与固壁相互作用耦合求解过程中接触线速度的计算方法及边界条件的确定方法.通过与已有实验结果的对比,对提出的气液两相流动与固壁相互作用耦合求解方法的有效性进行了验证.     

Rotationally symmetric flow over a rotating disk

The rotationally symmetric flow over a rotating disk in an incompressible viscous fluid is analyzed by a new method when the fluid at infinity is in a state of rigid rotation (in the same or in the opposite sense) about the same axis as that of the disk. Asymptotic expansions for the velocity field over the entire flow field are obtained for the general class of one-parameter rotationally symmetric flows. This method is further extended to the case when a uniform suction or injection is assumed at the rotating disk. Fluid motion induced by oscillatory suction of small amplitude at the rotating disk is also discussed.An initial-value analysis reveals that resonance is possible only when the angular velocity of the rotating fluid is greater than that of the rotating disk.     

Numerical modelling of rotating tangential layers (jets) in shells under strong uniform magnetic field

A numerical study of tangential layers in steady‐state magnetohydrodynamic rotating flows is presented using CFD to solve the inductionless governing equations. The analysis considers two basic flow configurations. In the first, a fluid is enclosed in a cylinder with electrically perfect conducting walls and the flow is driven by a small rotating, conducting disk. In the second, a flow is considered in a spherical shell with an inner rotating sphere. The fluid in both cases is subjected to an external axial uniform magnetic field. The results show that these flows exhibit two different types of flow cores separated from each other by a tangential layer parallel to the axis of rotation. The inner core follows a solid‐body rotation while the outer is quasistagnant. A counter‐rotating jet is developed in the tangential layer between the cores. The characteristics of the tangential layer and the properties of the meridional motion are determined for a wide range of Hartmann numbers. Distributions of angular velocity of circumferential flow and electric potential are obtained and the results are compared with those of analytic methods. Copyright © 2009 John Wiley & Sons, Ltd.     

Computation of Viscoelastic flow in a cylindrical tank with a rotating lid

An orthogonal collocation method is used to compute steady flows of viscoelastic fluids in a cylindrical tank covered by a rotating disk. The stability of these flows with respect to small disturbances is also analyzed. The Criminale-Ericksen-Filbey constitutive equation is used, since the primary flow is viscometric and the secondary flow is small. The solutions satisfy the equation of continuity and the boundary conditions exactly, including the velocity discontinuity at the edge of the disk. The computed flows for aqueous solutions of Separan 30 exhibit single or double vortices, according to the concentration and the rotation speed. Reasonable agreement is found with the data of Hill [7,8].     

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.     

Stability of flows of the Hamel type in channels with permeable walls

The stability of nonparallel flows of a viscous incompressible fluid in an expanding channel with permeable walls is studied. The fluid is supplied to the channel through the walls with a constant velocity v₀ and through the entrance cross section, where a Hamel velocity profile is assigned. The resulting flow in the channel depends on the ratio of flow rates of the mixing streams. This flow was studied through the solution of the Navier—Stokes equations by the finite-difference method. It is shown that for strong enough injection of fluid through the permeable walls and at a distance from the initial cross section of the channel the flow approaches the vortical flow of an ideal fluid studied in [1]. The steady-state solutions obtained were studied for stability in a linear approximation using a modified Orr—Sommerfeld equation in which the nonparallel nature of the flow and of the channel walls were taken into account. Such an approach to the study of the stability of nonparallel flows was used in [2] for self-similar Berman flow in a channel and in [3] for non-self-similar flows obtained through a numerical solution of the Navier—Stokes equations. The critical parameters *, R*, and Cr* at the point of loss of stability are presented as functions of the Reynolds number R₀, characterizing the injection of fluid through the walls, and the parameter , characterizing the type of Hamel flow. A comparison is made with the results of [4] on the stability of Hamel flows with R₀ = 0.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 125–129, November–December, 1977.The author thanks G.I. Petrov for a discussion of the results of the work at a seminar at the Institute of Mechanics of Moscow State University.     

Multiplicity of steady solutions in a two-sided lid-driven cavity with different aspect ratios

This study presents a continuation method to calculate flow bifurcation in a two-sided lid-driven cavity with different aspect ratios for anti-parallel motion. In anti-parallel motion, the top and bottom walls of the cavity move in opposite directions simultaneously, while the two walls both moving to the right give parallel motion at the same speed. Comprehensive bifurcation diagrams of the cavity flows with different aspect ratios of the cavities are derived via Keller’s continuation method, and linear- stability analysis is used to identify the nature of the various flow solutions. The Reynolds number (1 ≤ Re ≤ 1,200) is used as the continuation parameter to trace the solution curves. In anti-parallel motion, the evolution of the bifurcation diagrams in cases with different aspect ratios (1 ≤ AR ≤ 2.5) is illustrated. Two stable symmetric flows and one stable asymmetric flow are identified, and the existent regions of the stable flows in the aspect ratios and Reynolds numbers are distinguished. The newly found asymmetric flow state can be obtained at a high aspect ratio and a low Reynolds number.     

Some features of the flow around rapidly rotating bodies made of cellular-porous materials

Two types of gas flows arising near a rapidly rotating cellular-porous disk are studied numerically and experimentally. Steady-state limits for the flow around a disk rotating in free space and the type and scenario of the loss of stability are determined. Transitional flows are characterized by formation of a vortex sheet at the boundary of the exhausting jet. Numerical simulations of the flow around a cellular-porous disk rotating near a flat screen show that it is possible to form a closed swirl flow responsible for redistribution of swirl in the gap between the disk and the flat screen. The computed results offer an explanation for the experimentally observed excess of tangential velocity of the flow in the gap over the velocity of disk rotation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 86–96, January–February, 2007.     

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.     

Transient flow near a rotating disk

Summary The solution of the time dependent flow due to the impulsive starting of a single infinite disk from rest is obtained numerically for the entire history of the transient. The primary tangential velocity exhibits a single overshoot of its steady value while the growth of the secondary flows is monotonic. The overshoot is seen to be a direct consequence of the lag in the development of the secondary flows. An analytical solution is obtained for a related linearized problem: The angular velocity of an infinite disk, initially rotating with an infinite environment, is perturbed. The oscillatory decays to the steady state, which occur in both unbounded and bounded linearized analyses, are discussed in relation to the overshoot in the impulsively started disk problem.     

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.     

Three-dimensional rotational MHD flows in bounded volumes

Three-dimensional laminar flows of a viscous conducting gas in the neighborhood of a rotating disk are considered. The simultaneous impact of an external magnetic field, suction from the disk surface, and the axial temperature gradient as well as the action of the external axial magnetic field on three-dimensional flows in the neighborhood of rigid permeable surfaces are first studied. An exact analytic solution of the system of the boundary layer equations is obtained. It is found that the direction of the radial flow initiated in the boundary layer can be varied by changing the temperature ratio in the external flow and on the disk for various Prandtl numbers Pr. An approximate solution of the problem of flow in the rotating cylinder in the presence of a retarding cover is constructed on the basis of the approach developed for extended disks.     

Turbulent time-events in channel with rough walls

This brief communication quantifies the time-events that contribute to the dynamics of wall-bounded flows with rough walls. Lumley’s Proper Orthogonal Decomposition (POD) methodology has been used to extract the energetic modes of the flow. We have used the concept of entropy, a representation of lack of organization in the flow, to represent the extent of spread of turbulent kinetic energy to higher modes. The rough-wall dynamics is dominated by fast activity (short time period) propagating modes and slow activity (long time period) roll modes. A single dominant timescale has been captured for all the propagating modes in flows over smooth walls; multiple dominant timescales representing various vortex shedding events are captured for rough walls. Variable-interval time averaging technique has been used to obtain the bursting frequency. The bursting frequency of rough-wall turbulence is higher compared to smooth-wall turbulence, suggesting that roughness enhances turbulence production activity. Another insightful observation for rough walls revealed by our study is that the vortex shedding frequency of roughness elements is much higher compared to the bursting frequency of rough-wall turbulence. POD provides a straightforward method to extract the natural frequency of shed vortices due to roughness, an important dynamical activity in rough-wall turbulent boundary layers.     

Flows around rotating disks with and without rim-shroud gap

Experimental and numerical approaches have been used to study the effect of the radial rim-shroud gap on the flow structures found around a rotating disk in a finite cylindrical casing. When the radius of the disk and the inner radius of the casing are comparable and there is no radial gap, instabilities bring spiral rolls with a positive front angle in the Bödewadt layer on the end wall of the stationary casing. When the disk radius is smaller than the inner radius of the casing, vortex flows appear within the radial gap between the disk rim and the side wall of the casing. If the disk is thin, but not too thin, disturbances generated by these vortex flows proceed inward and the spiral rolls with a negative front angle appear in the Bödewadt layer. In the case of a thick disk, wavy Taylor vortex-like flow appears in the radial gap. The disturbances formed by the vortex flow do not well propagate into the inner region, and a flow pattern of bead-like vortices or a chain of vortices consisting of a series of small vortices are found around the disk in the visualized figure parallel to the disk.     

Orienting properties of discotic nematic liquid crystals in Jeffrey-Hamel flows

The orienting properties of incompressible discotic nematic liquid crystals for creeping flows between converging and diverging planar walls (Jeffrey-Hamel) are analyzed using the Leslie-Ericksen theory. The dependence of director orientation on the reactive parameter and the flow kinematics is presented. Closed form stationary solutions for the director orientation are found when elastic effects are neglected. Stationary numerical solutions for the velocity and director fields using the full Leslie-Ericksen theory are presented. The director field in converging flow is characterized by azimuthal (radial) centerline orientation, by being asymmetric with respect to the azimuthal (radial) direction, and by having an allowed orientation range that spans two half-quadrants (full quadrants). In the limiting case of perfectly flat disk ( –) the flow-induced director orientation in converging flow is the azimuthal direction, while in diverging flow the director rotates by a full n radians. By reducing the vertex angle between the walls to vanishingly small values, converging flow solutions properly reduce to those of flow between parallel plates, but diverging flows are expected to lead to a new instability.     

UNSTEADY CONFINED VISCOUS FLOWS WITH OSCILLATING WALLS AND MULTIPLE SEPARATION REGIONS OVER A DOWNSTREAM-FACING STEP

This paper presents computational solutions for unsteady viscous flows in channels with a downstream-facing step, followed by an oscillating floor. These solutions of the unsteady Navier–Stokes equations are obtained with a time-integration method using artificial compressibility in a fixed computational domain, which is obtained via a time-dependent coordinate transformation from the fluid domain with moving boundaries. The computational method is first validated for steady viscous flows past a downstream-facing step by comparison with previous numerical solutions and experimental results. This method is then used to obtain solutions for unsteady viscous flows with multiple separation regions over a downstream-facing step with oscillating walls, for which there are no previously known solutions. Thus, the present results may be used as benchmark solutions for the unsteady viscous flows with multiple separation regions between fixed and oscillating walls.     

Effects of the side walls on starting flows in ducts

This article considers the effects of the side walls on the unsteady flow of an incompressible viscous fluid in a duct of uniform cross-section. In order to show the effects of the side walls, three illustrative examples are given. They are: the starting flow in a duct of semicircular cross-section, the starting flow in a duct of rectangular cross-section and the starting flow in a duct of circular cross-section. The velocity distributions and the volume fluxes obtained for these flows are compared and it is shown that the flow in a duct of semicircular cross-section reaches steady state earlier than those for the flow in a duct of circular cross-section and for the flow in a duct of square cross-section. It is found that there are remarkable effects of the side walls of a duct on the required time to attain the asymptotic values of flow properties.     

Specific features of the modeling of boiling-fluid flows

Experiments show that in low-and high-velocity flows the boiling process is fundamentally different: in the former, the fluid boils on the walls, and in the latter in the volume. In high-velocity flows, the boiling intensity is orders of magnitude greater. In modeling fast and slow flows, the number of bubbles, which is a free parameter of the model and must be specified, differs by orders of magnitude. When high-speed flows of different kinds are modeled (vessel depressurization, nozzle flows) the number of bubbles specified also differs by orders of magnitude. In this study, we formulate the hypothesis that in both kinds of flows the process of boiling starts similarly, namely, on the walls. However, in high-speed flows the number of bubbles increases by orders of magnitude due to bubble fragmentation. As a result of intense fragmentation, the system “forgets” the initial number of bubbles and the process becomes volume boiling. This approach makes it possible to construct a universal model of boiling. To test this hypothesis, we constructed a mathematical model which takes into account the possibility of bubble fragmentation due to the instability developing under the action of centrifugal accelerations of the bubble surface. This model was used to calculate the process of depressurization of a high-pressure vessel. The calculations demonstrated that, for any initial number of bubbles, 1 ms after depressurization the bubble number attains the same level. Bubble fragmentation takes place in “self-sustained detonation waves”. The stationary structure of detonation waves in a boiling fluid is investigated. A scheme of the wave structure according to which the wave consists of a shock wave and a relaxation zone is proposed. Calculations of a boiling-fluid flow through a Laval nozzle reveal the periodic appearance of detonation waves. Accordingly, nozzle flows should be accompanied by significant oscillations of the parameters.     

Mass transfer enhancement in a sinusoidal wavy channel for pulsatile flow

We report experimental work on mass transfer enhancement through pulsatile flow in an asymmetric channel under the laminar flow condition. Mass transfer experiments were carried out by the electrochemical method. The vortices dynamics were visualized both experimentally and numerically. The present results were compared with previous ones in the case of a symmetric channel with the same sinusoidal wavy walls. The mass transport enhancement factor for the asymmetric channel is found to be larger than that for the symmetric channel, for a wide range of flow parameters. This is a consequence of the specific fluid mixing induced by vortices dynamics. It is also confirmed that the Sherwood number for pulsatile flow can be expressed in terms of the Sherwood numbers for steady and oscillatory flows at large amplitude of fluid oscillation, for both channels.