Free surface effects on the flow between conical cylinders

Summary An experimental study is presented dealing with free surface effects on the flow between two truncated conical cylinders, the inner being rotated and the outer at rest. For different filling ratios and angular velocities different flow modes are observed. The basic flow and the occurring vortices in the present arrangement show similarities to those observed for the case of a completely liquid-filled annulus. In both systems upwards travelling toroidal cells, steady Taylor vortices and downwards moving helical vortice appear. The variation of the wavelength of toroidal vortices for small filling ratios as well as the occurrence of a helical upward motion are, however, new effects which are solely related to the free surface. In a partially liquid-filled annulus both helical systems may coexist, whereby the helical vortices with opposite inclinations of the vortex axes propagate in opposite directions. The characteristics of the various flow modes are discussed for different Reynolds numbers, filling ratios and initial conditions. A comparison with the case of a completely liquid-filled annulus and with the case of circular cylinders is made.     

Arterial geometry,flow pattern,wall shear and mass transport: potential physiological significance

We have studied numerically steady and unsteady flow in a straight and a helically stented common carotid artery, in order to model porcine experimental results that show reduced intimal hyperplasia (IH) in the helical case. The combination of flow pulsatility and three-dimensionality generates a sweeping motion of the Dean vortices, which overall reduced extremes of both oxygen flux to the vessel wall and wall shear stress (WSS). Since IH and atherosclerosis affect preferentially low WSS regions, these findings imply that vessel three-dimensionality and flow pulsatility can play important protective roles in respect of these diseases. The amplitude and frequency of the velocity waveform are important parameters of the system. Increase in amplitude increases WSS and oxygen flux to the vessel wall. Increase in frequency has a small effect; it increases WSS but has no effect on the oxygen flux to the vessel wall.     

Finite element analysis of incompressible viscous flow in a helical pipe

This paper presents an accurate finite element procedure to deal with steady state, fully developed and incompressible viscous flow in helical pipes with arbitrary curvatures and torsions. The full Navier-Stokes equations and continuity equation have been explicitly derived using a non-orthogonal helical coordinate system. To obtain the final simultaneous non-linear algebraic equations, a pressure-velocity finite element formulation is formulated based on the Galerkin Method.The combined influence of finite curvature and finite torsion on the helical flow is studied. The secondary flow patterns and contours of axial velocity of helical flows show the significant distinction with those of toroidal flows. Further, the effect of torsion on flow rates can be neglected.Several numerical examples are presented. Excellent correlations between the computed results and available referenced solutions can be drawn.     

Symmetric and asymmetric Taylor vortex flow in spherical gaps

Summary The rotationally symmetric flow between two concentric rotating spheres is investigated both theoretically and experimentally. The non-uniqueness of the supercritical flow exhibits three different modes with zero, one and two Taylor vortices in each hemisphere. These modes are realized by different accelerations of the inner sphere from the state at rest. A initial value code, based on a finite difference method, is used for the numerical simulation. The existence regions of the different supercritical flows are connected with symmetric and asymmetric transitions. It is found, that a steady state can exist asymmetric with respect to the equator. The flow is analyzed by plotting the size of the Taylor vortices, the depending variables , ,V, the velocity distributions and the torque. A comparison between theory and experiments for the observed modes of flow is given.     

Onset of Taylor-Görtler instability in accelerated circular Couette flow

Summary The onset of Taylor-G?rtler vortices in a developing Couette flow induced by the inner cylinder rotating with time-dependent manner is analyzed using linear theory. It is well known that there is a critical Taylor number Ta _c at which Taylor vortices set in between two concentric cylinders. For Ta > Ta _c Taylor-G?rtler vortices are detected experimentally at a certain elapsed time. In the present study the critical time t _c to represent the onset of a fastest growing instability, which then grows as toroidal vortices, is analyzed using the propagation theory. Available experimental data indicate that for large Ta secondary motion is detected starting from a certain time t _m ≈ 4t _c. This means that the growth period of initiated instabilities is needed for secondary motion to be detected experimentally. The new measures to represent the onset of a fastest growing instability in the primary time-dependent Couette flow are suggested.     

Axial drive to nonlinear flow between rotating cylinders

Stability of pseudoplastic rotational flow between cylinders in presence of an independent axial component is investigated. The fluid is assumed to follow the Carreau model and mixed boundary conditions are imposed. The conservation of mass and momentum equations give rise to a four-dimensional low-order dynamical system, including additional nonlinear terms in the velocity components originated from the shear-dependent viscosity. In absence of the axial flow, as the pseudoplasticity effects increases, the purely-azimuthal base flow loses its stability to the vortex structure at a lower critical Taylor number. Emergence of the vortices corresponds to the onset of a supercritical bifurcation also present in the flow of a linear fluid. However, unlike the Newtonian case, pseudoplastic Taylor vortices lose their stability as the Taylor number reaches a second critical number corresponding to the onset of a Hopf bifurcation. Existence of an axial flow induced by a pressure gradient appears to further advance each critical point on the bifurcation diagram. In continuation, complete flow field together with viscosity maps is analyzed for different flow scenarios. Through evaluation of the Lyapunov exponent, flow stability and temporal behavior of the system for cases with and without axial flow are brought to attention.     

Experimental and computational validation of Hele-Shaw stagnation flow with varying shear stress

An in vitro flow model system with continuous variation of fluid shear stress can be used to test cell responses to a range of shear stresses. In this investigation, we validated such a flow system computationally for steady and unsteady flow conditions and experimentally for steady flow conditions. The unsteady flow validation is important for studying cells such as endothelial cells that experience unsteady flow conditions in their native environment. The system is capable of exposing cells in different regions of the chamber to steady or unsteady shear stress conditions with average values ranging linearly from 0 to 30 dyn/cm $^{2}$ . These tests and analyses demonstrate that the variable-width parallel plate flow system can be used to test the influence of a range of steady and unsteady fluid shear stress levels on cell activities.     

Transition of boundary layer flow past a stretching sheet due to a step change in applied constant mass flux

The problem of transition of boundary layer flow from the initial unsteady flow to the final steady flow past a permeable stretching sheet is considered in this paper. Two cases are considered: (1) Case-I deals with the transition of the unsteady flow due to a sudden application of constant suction/injection at the surface of the stretching sheet, from the initially prevailing steady flow; (2) Case-II deals with unsteady flow transition due to a sudden removal of constant suction/injection at the surface of the stretching sheet, from the initially prevailing steady flow. Numerical results are obtained using the implicit finite difference method of Crank-Nicholson type. The velocity and local skin friction for different values of injection/suction parameter are graphically presented and discussed. It is found that for the same magnitude of applied mass flux in Case-I and mass flux removal in Case-II, the time to reach steady-state flow is different.     

Vortex motion in the early stages of unsteady flow around a circular cylinder

In this paper, processes in the early stages of vortex motion and the development of flow structure behind an impulsively-started circular cylinder at high Reynolds number are investigated by combining the discrete vortex model with boundary layer theory, considering the separation of incoming flow boundary layer and rear shear layer in the recirculating flow region. The development of flow structure and vortex motion, particularly the formation and development of secondary vortex and a pair of secondary vortices and their effect on the flow field are calculated. The results clearly show that the flow structure and vortices motion went through a series of complicated processes before the symmetric main vortices change into asymmetric: development of main vortices induces secondary vortices; growth of the secondary vortices causes the main vortex sheets to break off and causes the symmetric main vortices to become “free” vortices, while a pair of secondary vortices is formed; then the vortex sheets, after breaking off, gradually extend downstream and the structure of a pair of secondary vortices becomes relaxed. These features of vortex motion look very much like the observed features in some available flow field visualizations. The action of the secondary vortices causes the main vortex sheets to break off and converts the main vortices into free vortices. This should be the immediate cause leading to the instability of the motion of the symmetric main vortices. The flow field structure such as the separation position of boundary layer and rear shear layer, the unsteady pressure distributions and the drag coefficient are calculated. Comparison with other results or experiments is also made. This work was presented at the First Asian Congress of Fluid Mechanics, Bangalore in December 1980.     

Prediction of incidence effects on oscillating airfoil aerodynamics by a locally analytical method

A complete mathematical model is formulated to analyse the effects of mean flow incidence angle on the unsteady aerodynamics of an oscillating airfoil in an incompressible flow field. A velocity potential formulation is utilized. The steady flow is independent of the unsteady flow field. However, the unsteady flow is coupled to the steady flow field through the boundary conditions on the oscillating airfoil. The numerical solution technique for both the steady and unsteady flow fields is based on a locally analytical method. In this method, analytical solutions are incorporated into the numerical technique, with the discrete algebraic equations which represent the differential flow field equations obtained from analytic solutions in individual local computational grid elements. This flow model and locally analytic numerical solution method are then verified through the excellent correlation obtained with the Theodorsen oscillating flat plate and Sears transverse gust classical solutions. The effects of mean flow incidence on the steady and oscillating airfoil aerodynamics are then investigated.     

Analysis of rotating membrane emulsification performance for oil droplet production based on the Taylor vortices approach

Membrane emulsification processes generally employ a cross flow of the continuous phase in order to produce shear stress. Modification of membrane emulsification for the o/w emulsion using the rotating system has been introduced such as the use of stirred cells or a rotating tubular membrane in a stationary liquid. This paper presents an examination of membrane rotation speed on droplet characteristics. The performance of rotating membrane emulsification on o/w droplet size, the coefficient of variation, and size distribution was investigated. In addition, the operated flow regime in the rotating membrane emulsification is addressed. It has been found that overall the droplet size decreased with the increase of membrane rotation speed. The droplet size below the pore size could be produced when operating at a high rotation speed (1500?rpm). The decrease of droplet size was believed due to the action of Taylor vortices on droplet detachment. Analysis of membrane rotation speed proposed that action of Taylor vortices facilitates droplet detachment. Calculation of Taylor numbers (having a value of 0–629) confirmed that the rotating membrane emulsification produced laminar flow with vortices.     

Unsteady free-surface flow induced by a line sink

The unsteady flow of fluid from a deep reservoir through a line sink beneath a free surface with surface tension is considered. Two different initial conditions are discussed; the first effectively represents impulsive withdrawal from rest, and the second can be regarded as a disturbance to an existing steady flow. Small-time expansions and numerical methods are used to investigate both the movement to steady states and the critical drawdown of the free surface in the two situations. It is shown that there are several different critical values of flow parameters at which the flow changes its nature. In the zero-surface-tension case, the situation is not fully resolved, but the addition of surface tension clarifies the flow behaviour greatly, and drawdown or movement to a steady state becomes evident. For the second class of initial conditions, it appears that either movement to a steady state or drawdown are the only subcritical possibilities.     

BEM and FEM based numerical simulations for biomagnetic fluid flow

We investigate numerically the biomagnetic fluid flow between parallel plates imposed to a magnetic source placed below the lower plate. The biomagnetic fluid is assumed to be Newtonian, viscous, incompressible, electrically nonconducting, and has magnetization varying linearly with temperature and magnetic field intensity. Both steady and unsteady, laminar, two-dimensional biomagnetic fluid flow equations taking into care the heat transfer between the plates are solved using both finite element and dual reciprocity boundary element methods. Treatment of nonlinear terms by using only the fundamental solution of the Laplace equation, and discretization of only the boundary of the region are the advantages of dual reciprocity boundary element method giving small algebraic systems to be solved at a small expense. Finite element method is capable of giving very accurate results by discretizing the region affected by the magnetic source very finely, but it results in large sized algebraic systems requiring high computational cost. The results indicate that the flow is appreciably affected with the presence of magnetic source in terms of vortices at the magnetic source area. The lengths of the vortices, and temperature increase with an increase in the intensity of the magnetic field.     

Phase slip solutions in magnetically modulated Taylor–Couette flow

We numerically investigate Taylor–Couette flow in a wide-gap configuration, with \({r_i/r_o=1/2}\), the inner cylinder rotating, and the outer cylinder stationary. The fluid is taken to be electrically conducting, and a magnetic field of the form \({B_z\approx(1 + \cos(2\pi z/z_0))/2}\) is externally imposed, where the wavelength \({z_0=50(r_o-r_i)}\). Taylor vortices form where the field is weak, but not where it is strong. As the Reynolds number measuring the rotation rate is increased, the initial onset of vortices involves phase slip events, whereby pairs of Taylor vortices are periodically formed and then drift outward, away from the midplane where \({B_z=0}\). Subsequent bifurcations lead to a variety of other solutions, including ones both symmetric and asymmetric about the midplane. For even larger Reynolds numbers, a different type of phase slip arises, in which vortices form at the outer edges of the pattern and drift inward, disappearing abruptly at a certain point. These solutions can also be symmetric or asymmetric about the midplane and co-exist at the same Reynolds number. Many of the dynamics of these phase slip solutions are qualitatively similar to previous results in geometrically ramped Taylor–Couette flows.     

Flow of a binary mixture of incompressible Newtonian fluids in a rectangular channel

The problems concerning some simple steady and unsteady flows of a mixture composed of two incompressible Newtonian fluids in an infinitely long channel of rectangular cross-section are examined. By means of finite Fourier sine transforms, the exact solutions of the field equations are obtained for the following four problems: (i) steady Couette flow in a rectangular channel, (ii) unsteady Couette flow in a rectangular channel, (iii) steady Poiseuille flow in a rectangular channel, (iv) unsteady Poiseuille flow in a rectangular channel.     

High frequency oscillating viscous flow over elliptic cylinder at incidence

Summary The problem of high frequency oscillating viscous flow over an elliptic cylinder at incidence is investigated. The method of matched inner-outer asymptotic expansion to second order is used to solve the governing equations. The steady and the unsteady modes of flow, related to the present work, are identified and separated. Both steady and unsteady drag and lift coefficients are presented and discussed. The effect of different parameters such as the Strouhal number, Reynolds number, focal length and angle of attack are explored.     

Stability of power law fluid flow between rotating cylinders

The centrifugal instability of power law fluid flow when the fluid fills the gap between two rotating concentric cylinders of infinite length is addressed. The instability of the circular Couette flow is determined via linear stability analysis. In the narrow gap case, the critical Taylor number is determined analytically, while in the general gap case, a finite-difference numerical method is employed. The results are shown to be in agreement with existing theoretical and experimental findings. The super-critical flow is investigated by means of a weakly non-linear analysis method. The Taylor-vortex flow (the secondary stable flow) is obtained. The instability of this flow is determined as we present the critical Taylor number when the Taylor vortices begin to exhibit a waviness in the azimuth     

Visualization Measurement of Streaming Flows Associated with a Single-Acoustic Levitator

The purpose of the study is to experimentally investigate flow fields generated by an acoustic levitator. This flow field has been observed using flow visualization, PIV method. In the absent of a drop, the flow field was strongly influenced by sound pressure level (SPL). In light of the interfacial stability of a levitated drop, SPL was set at 161–163 [dB] in our experiments. In the case of any levitated drop at a pressure node of a standing wave, the toroidal vortices were appeared around a drop and clearly observed the flow fields around the drop by PIV measurement. It is found that the toroidal vortices around a levitated drop were strongly affected by the viscosity of a drop. For more detailed research, experiments in the reduced gravity were conducted with aircraft parabolic flights. By comparison with experimental results in the earth and reduced gravity, it is also indicated that the configuration of the external flow field around a drop is most likely to be affected by a position of a drop as well.     

Parallel simulation of unsteady hovering rotor wakes

Numerical simulation using low diffusion schemes, for example free‐vortex or vorticity transport methods, and theoretical stability analyses have shown the wakes of rotors in hover to be unsteady. This has also been observed in experiments, although the instabilities are not always repeatable. Hovering rotor wake stability is considered here using a finite‐volume compressible CFD code. An implicit unsteady, multiblock, multigrid, upwind solver, and structured multiblock grid generator are presented, and applied to lifting rotors in hover. To allow the use of very fine meshes and, hence, better representation of the flow physics, a parallel version of the code has been developed, and parallel performance using upto 1024 CPUs is presented. A four‐bladed rotor is considered, and it is demonstrated that once the grid density is sufficient to capture enough turns of the tip vortices, hover exhibits oscillatory behaviour of the wake, even using a steady formulation. An unsteady simulation is then performed, and also shows an unsteady wake. Detailed analysis of the time‐accurate wake history shows that three dominant unsteady modes are captured, for this four‐bladed case, with frequencies of one, four, and eight times the rotational frequency. A comparison with theoretical stability analysis is also presented. Copyright © 2006 John Wiley & Sons, Ltd.