Ekman vortices and the centrifugal instability in counter-rotating cylindrical Couette flow

The finite length of a Taylor–Couette cell introduces endwall effects that interact with the centrifugal instability. We investigate the interaction between the endwall Ekman boundary layers and the vortical structures in a finite-length cavity with counter-rotating cylinders via direct numerical simulation using a three-dimensional spectral method. To analyze the nature of the interaction between the vortices and the endwall layers we consider four endwall boundary conditions: fixed endwalls, endwalls rotating with the outer cylinder, endwalls rotating with the inner cylinder, and stress-free endwalls. The vortical structure of the flow depends on the endwall conditions. The waviness of the vortices is suppressed only very near the endwall, primarily due to zero axial velocity at the endwall rather than viscous effects. In spite of their waviness and random behavior, the vortices generally stay inside of the v=0 isosurface by adjusting quickly to the radial transport of azimuthal momentum. The thickness and strength of the Ekman layer at the endwall match with that predicted from a simple theoretical approach.     

Three-dimensional instability analysis of boundary layers perturbed by streamwise vortices

A parametric study is presented for the incompressible, zero-pressure-gradient flat-plate boundary layer perturbed by streamwise vortices. The vortices are placed near the leading edge and model the vortices induced by miniature vortex generators (MVGs), which consist in a spanwise-periodic array of small winglet pairs. The introduction of MVGs has been experimentally proved to be a successful passive flow control strategy for delaying laminar-turbulent transition caused by Tollmien–Schlichting (TS) waves. The counter-rotating vortex pairs induce non-modal, transient growth that leads to a streaky boundary layer flow. The initial intensity of the vortices and their wall-normal distances to the plate wall are varied with the aim of finding the most effective location for streak generation and the effect on the instability characteristics of the perturbed flow. The study includes the solution of the three-dimensional, stationary, streaky boundary layer flows by using the boundary region equations, and the three-dimensional instability analysis of the resulting basic flows by using the plane-marching parabolized stability equations. Depending on the initial circulation and positioning of the vortices, planar TS waves are stabilized by the presence of the streaks, resulting in a reduction in the region of instability and shrink of the neutral stability curve. For a fixed maximum streak amplitude below the threshold for secondary instability (SI), the most effective wall-normal distance for the formation of the streaks is found to also offer the most stabilization of TS waves. By setting a maximum streak amplitude above the threshold for SI, sinuous shear layer modes become unstable, as well as another instability mode that is amplified in a narrow region near the vortex inlet position.     

Two- and three-dimensional flows in nearly rectangular cavities driven by collinear motion of two facing walls

Two- and three-dimensional flows in nearly cuboidal cavities are investigated experimentally. A tight cavity is formed in the gap between two long and parallel cylinders of large radii by adding rigid top, bottom, and end walls. The cross-section perpendicular to the axes of the cylinders is nearly rectangular with aspect ratio Γ. The axial aspect ratio Λ > 10 is large to suppress end-wall effects. The fluid motion is driven by independent and steady rotation of the cylinders about their axes which defines two Reynolds numbers Re _1,2. Stability boundaries of the nearly two-dimensional steady flow have been determined as functions of Re _1,2 for Γ = 0.76 and Γ = 1. Up to six different three-dimensional supercritical modes have been identified. The critical thresholds for the onset of most of the three-dimensional modes, three of which have been observed for the first time, agree well with corresponding linear-stability calculations. Particular attention is paid to the flow for Γ = 1 under symmetric and parallel wall motion. In that case the basic flow consists of two mirror symmetric counter-rotating parallel vortices. They become modulated in span-wise direction as the driving increases. Detailed LDV measurements of the supercritical three-dimensional velocity field and the bifurcation show an excellent agreement with numerical simulations.

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Three-dimensional viscoelastic flows through a rectangular channel with a cavity

Three-dimensional viscoelastic flows in a rectangular channel with a cavity were studied both numerically and experimentally. In the numerical study, computations were carried out using the Phan-Thien–Tanner (PTT) model as a constitutive equation. A finite volume method (FVM) using colocated grids was applied. A three-dimensional structure in the cavity was observed even when three-dimensional behavior was not remarkable in main flows. At high Weissenberg number, the flow in the cavity spirals to move towards the center plane of the channel. In the experiments, the flow of polymer solutions was visualized to observe three-dimensional flow behavior near the cavity part. It was confirmed that the spiraling flow moving towards the center plane emerged in the cavity.     

Global flow instability in a lid‐driven cavity

The stability of flow in a lid‐driven cavity is investigated using an accurate numerical technique based on a hybrid scheme with spectral collocation and high‐order finite differences. A global stability analysis is carried out and critical parameters are identified for various aspect ratios. It is found that while there is reasonable agreement with the literature for the critical parameters leading to loss of stability for the square cavity, there are significant discrepancies for cavities of aspect ratios 1.5 and 2. Simulations of the linearized unsteady equations confirm the results from the global stability analysis for aspect ratios A = 1, 1.5 and A = 2. Copyright © 2009 John Wiley & Sons, Ltd.     

Effect of a permeable partition on convective instability in a rectangular cavity

The equilibrium stability of a fluid, heated from below, in a rectangular cavity with a vertical permeable partition is investigated. The small perturbation problem is solved by the Galerkin-Kantorovich method. The relations obtained for the dependence of the critical Rayleigh numbers on the partition parameters and the cavity dimensions make it possible to identify regions in which either even or odd perturbations, sensitive to only the normal or only the tangential resistance of the partition, respectively, are responsible for equilibrium crisis. The effect of a permeable partition on the convective instability of a horizontal layer of fluid under various heating conditions was considered in [1–3], where a significant dependence of the critical Rayleigh numbers on the properties of the partition was established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 6–10, May–June, 1989.     

Natural convection in a rectangular cavity with wall temperature decreasing at a uniform rate

The transient natural convection in a fluid contained in a rectangular enclosure, the wall of which is maintained at a uniform temperature which changes at a steady rate, is approached by a numerical method. Numerical solutions are obtained forPr=0.73, 7.3 and 73 and a range of Rayleigh numbersRa=10² ～ 10⁸. At relatively low Rayleigh numbers the flow is characterized by the development of double cells with flow up the center and down the sidewalk However it was found that an increase of the Rayleigh number leads to the development of strong secondary circulation on the axis of symmetry of the cavity near the top wall. Thus, as the Rayleigh number is increased the secondary cells grow in size. The effects of the secondary cells on the temperature field and heat transfer coefficients are discussed. Most results are obtained for the case of a square cavity (E=2) but the influence of the aspect ratio of the cavity is also studied forE=1 and 4.     

三相驱动下含两质量块刚体的平面运动

本文研究了一类含两个质量块的振动驱动系统在粘性摩擦下的平面运动,其中两个内部质量在相互垂直的水平槽道上作三相运动。利用第二类拉格朗日方程,建立了系统的动力学方程;其次,利用系统直线运动时的理论解,验证速度Verlet积分法的可靠性;利用这种算法分析了系统的平面运动,得到了内部驱动参数与系统运动轨迹、运动速度的关系;然后,基于数值分析结果,通过调节内部驱动参数,得到了系统在不同轨迹间的六种转换形式;最后,通过组合不同的转换形式,得到了曲率连续变化的平面运动路径。     

Horizontal convection in a rectangular enclosure driven by a linear temperature profile

The horizontal convection in a square enclosure driven by a linear temperature profile along the bottom boundary is investigated numerically by using a finite difference method. The Prandtl number is fixed at 4.38, and the Rayleigh number Ra ranges from107 to 1011. The convective flow is steady at a relatively low Rayleigh number, and no thermal plume is observed, whereas it transits to be unsteady when the Rayleigh number increases beyond the critical value. The scaling law for the Nusselt number Nu changes from Rossby's scaling Nu ～ Ra~(1/5) in a steady regime to Nu ～ Ra~(1/4) in an unsteady regime, which agrees well with the theoretically predicted results. Accordingly,the Reynolds number Re scaling varies from Re ～ Ra~(3/11) to Re ～ Ra~(2/5). The investigation on the mean flows shows that the thermal and kinetic boundary layer thickness and the mean temperature in the bulk zone decrease with the increasing Ra. The intensity of fluctuating velocity increases with the increasing Ra.     

Low-Reynolds-number motion of a deformable drop between two parallel plane walls

The motion of a three-dimensional deformable drop between two parallel plane walls in a low-Reynolds-number Poiseuille flow is examined using a boundary-integral algorithm that employs the Green’s function for the domain between two infinite plane walls, which incorporates the wall effects without discretization of the walls. We have developed an economical calculation scheme that allows long-time dynamical simulations, so that both transient and steady-state shapes and velocities are obtained. Results are presented for neutrally buoyant drops having various viscosity, size, deformability, and channel position. For nearly spherical drops, the decrease in translational velocity relative to the undisturbed fluid velocity at the drop center increases with drop size, proximity of the drop to one or both walls, and drop-to-medium viscosity ratio. When deformable drops are initially placed off the centerline of flow, lateral migration towards the channel center is observed, where the drops obtain steady shapes and translational velocities for subcritical capillary numbers. With increasing capillary number, the drops become more deformed and have larger steady velocities due to larger drop-to-wall clearances. Non-monotonic behavior for the lateral migration velocities with increasing viscosity ratio is observed. Simulation results for large drops with non-deformed spherical diameters exceeding the channel height are also presented.     

Impulsive motion of a non-Newtonian fluid between two oscillating parallel plates

An exact solution for the flow of an incompressible viscoelastic fluid between two infinitely extended parallel plates, due to the harmonic oscillations of the upper plate and the impulsively started harmonic oscillations of the lower plate from rest, in the respective planes of the plates, has been obtained. The momentum transfer towards the central region and the skin friction of the lower plate are found to be greater for the viscoelastic fluid than that for viscous fluid. The effect of out-of-phase oscillations of the plates with different amplitudes on the flow characteristics has also been investigated.     

Vibrational thermal convection in a rectangular cavity

Vibrational thermal convection in a rectangular cavity under conditions of weightlessness is studied. Some equilibrium configurations were obtained in earlier papers of two of the authors [1, 2] and their linear stability investigated. In the present paper, a numerical investigation is made of the developed vibrational convection which arises under conditions when equilibrium is impossible. The structure of the average vibrational-convective flows and the characteristics of the heat transfer are determined. The change of regimes and the connection with the stability problem are discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 94–99, July–August, 1982.     

Combined radiation and natural convection in a rectangular cavity with a transparent wall and containing a non-participating fluid

This paper describes a numerical method for the study of combined natural convection and radiation in a rectangular, two-dimensional cavity containing a non-participating (i.e. transparent) fluid. One wall of the cavity is isothermal, being heated either by solar radiation or independently. The opposite wall is partially transparent, permitting radiation exchanges between the cavity and its surroundings and/or the Sun; that wall also exchanges heat by convection from its external surface to the surroundings. The other two walls are adiabatic: convection and radiation there are balanced, so that there is no heat transfer through those walls. The equations of motion and energy are solved by finite difference methods. Coupled to these equations are the radiative flux boundary conditions which are used to determine the temperature distribution along the non-isothermal walls. A two-band radiation model has been employed. Results are presented for a square cavity with a vertical hot wall at 150 °C, the ambient at 20 °C and 10⁴ ? Ra ? 3 × 10⁵, in the absence of direct insolation. The effects on the flow and heat transfer in the cavity of radiation and external convection have been examined. More extensive results will be presented in subsequent papers.     

Effect of density extremum on the solidification of water on a vertical wall of a rectangular cavity

Experiments were performed in a two-dimensional rectangular cavity to study the transient flow in an initially isothermal and motionless fluid due to a step decrease in temperature on one of the two vertical end walls. In the experiments water was used as the phase-change medium, with the cold-wall temperature maintained below the freezing temperature. The opposite vertical wall was kept at the initial temperature, greater than the temperature where the density extremum occurs. The growth of ice and the transient flow in the cavity were visualized with the aid of a tracer technique to examine the effect of density inversion. The temperature field was continuously recorded by an array of thermocouples. It was found that the density inversion of water strongly influences both the growth of ice and the convective flow in the liquid region of the test cavity.     

Biomagnetic fluid flow in a driven cavity

In this study, the fundamental problem of the biomagnetic fluid flow in a lid driven cavity under the influence of a steady localized magnetic field is studied. The mathematical model used for the formulation of the problem is consistent with the principles of Ferrohydrodynamics (FHD) and Magnetohydrodynamics (MHD). The biomagnetic fluid is considered as a homogeneous Newtonian fluid and is treated as an electrically conducting magnetic fluid which also exhibits magnetization. A known biomagnetic fluid which exhibits such magnetic properties is blood. For the numerical solution of the problem, which is described by a coupled, non linear system of PDEs, with appropriate boundary conditions, the SIMPLE algorithm is used. The solution is obtained by the development of a numerical methodology using finite volumes on a staggered, properly stretched, grid. Results concerning the velocity indicate that the presence of the magnetic field influences considerably the flow field.     

Kolmogorov scales in a driven cavity flow

Turbulent flow in a three-dimensional driven cavity has been simulated directly by solving the Navier–Stokes equations. The results at Re=3200 and 10 000 compare well with the experimental data. Viscous dissipation rate has been calculated without making the assumption of isotropy. Near the top moving wall, the instantaneous dissipation rate is very high and also has high amplitude. Its frequency increases but amplitude decreases as one moves away from the wall and it becomes intermittent in the vortex core. The high Reynolds number assumption that dissipation is mainly due to the fluctuating velocity components is seen to be true in the present case except near the wall. The Kolmogorov length scale attains higher values in the core of the primary vortex due to low dissipation rate there. A value of 0.01 times the size of the cubic cavity is a good representative value at Re=10 000. Even though the present (84×84×84) grid cannot resolve this scale very well, it can resolve all the scales dynamically significant in the flow as seen from the velocity and dissipation spectra.     

Control of a submerged jet in a thin rectangular cavity

An innovative method is presented for control of an oscillatory turbulent jet in a thin rectangular cavity with a thickness to width ratio of 0.16. Jet flow control is achieved by mass injection of a secondary jet into the region above the submerged primary jet nozzle exit and perpendicular to the primary nozzle axis. An experimental model, a 2-D and a 3-D computational fluid dynamics (CFD) model are used to investigate the flow characteristics under various secondary injection mass flow rates and injection positions. Two-dimensional laser Doppler anemometry (LDA) measurements are compared with results from the CFD models, which incorporate a standard k–ε turbulence model or a 2-D and 3-D realisable k–ε model. Experimental results show deflection angles up to 23.3° for 24.6% of relative secondary mass flow are possible. The key to high jet control sensitivity is found to be lateral jet momentum with the optimum injection position at 12% of cavity width (31.6% of the primary nozzle length) above the primary nozzle exit. CFD results also show that a standard k–ε turbulence closure with nonequilibrium wall functions provides the best predictions of the flow.     

Temperature gradient driven flow experiments of two interacting bubbles on a hot wall

This publication deals with the thermocapillary convection of two bubbles in a close proximity under a heated wall. The resultant toroidal vortex rings of the bubbles interfere and cause a distinctive threedimensional flow pattern. Additionally we observed the penetration depth of this flow configuration in dependence on the bubble spacing. Liquid crystal tracer particles serve for simultaneous flow- and temperature monitoring. This novel method is described in some detail. It is a very helpful tool for analyzing heat and mass transfer in liquids. Received on 25 April 1997