Experiments on two immiscible fluids in spherical Couette flow

This paper deals with the experimental instabilities analysis of spherical Couette flow. We consider the flow of two immiscible fluids superimposed between concentric spheres when the outer sphere is fixed and the inner one rotates. The working fluids have rather different viscosities and thus different Reynolds numbers. The obtained results are compared with a reference case of filled gap using one fluid (Γ _max = 20). Experiments are performed for different aspect ratio values, and Laser photometric technique is used for visualization. Our analysis is mainly focused on the type of instabilities and their relationship with the laminar-turbulent transition regime. We intend to explore the combined effects of the aspect ratio and the interaction between the two superposed fluids on the appearance of different instability evolutions. The evolution of the phase velocity for different aspect ratio of heavy fluid Γ _HF = H _HF/d is presented. The immiscible fluids are separated by a liquid–liquid interface (water–oil). In order to control instability occurrence, Taylor number variation is presented versus aspect ratio. Instability phenomena are found to be the same as for the nominal case for large heavy fluid aspect ratios. The first equatorial symmetry breaking of the flow is observed for a critical value Γ _c  = 13 where the Taylor vortex flow is introduced with three stationary cells. For the same aspect ratio, the interaction of the immiscible fluids leads to the appearance of gravitational waves near the equatorial zone. A surface cell, starting before the appearance of Taylor vortices, is detected in the light fluid for low aspect ratios. This cell of Ekman type has not been observed before, to our best knowledge, in spherical Couette flow.     

Numerical and experimental study on the flow history effects of axial flow on the Couette–Taylor flow

In this research, experimental and numerical techniques are used to study the flow history effects of axial flow on the Couette–Taylor flow. For the experimental investigation, the flow is visualized using the PIV technique with reflective particles with a density of 1.62 g/cm³. Dispersed in a solution, the particles have a strong refraction index equal to 1.85. In this study, two protocols are adopted to study the effect of an axial flow superimposed on a Couette–Taylor flow, and of the history of the flow. The first one, the direct protocol, consists of imposing an azimuthal flow to the inner cylinder. In this case, when the regime is established, the axial flow is superimposed. The second protocol, the inverse protocol, consists of imposing first the axial flow in the gap of the system, after which an azimuthal flow is conveyed. The Couette–Taylor flow with axial flow is strongly dependent on the flow history (the protocol). Thus, the flow structures and development for different protocols are studied and analyzed here experimentally and numerically. In addition, from the numerical results, mathematical models for the two protocols are presented. For the direct protocol, a new relation between the axial Reynolds number, which stabilizes the Couette–Taylor flow, and the Taylor number is presented; for the inverse protocol, a new mathematical model for the critical Taylor number is developed as a function of the axial Reynolds number and also the first critical Taylor number without axial flow.     

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.     

Nonlinear stability of convective flow between heated vertical planes

A weakly nonlinear stability analysis is described for the buoyancy-driven flow between infinite vertical planes that are subject to a constant vertical temperature gradient and a constant horizontal temperature difference. For sufficiently high values of a Rayleigh number based on the vertical temperature gradient and gap width, and for large Prandtl numbers, the critical two-dimensional mode of instability occurs as stationary convection. A nonlinear amplitude equation governing the spatial and temporal development near the critical point is determined.     

Thermocapillary instability of a viscoelastic liquid layer

Summary The stability of a layer of viscoelastic fluid heated from below is studied in the case that the instability is driven by surface-tension gradients at the upper surface. The operative parameter for instability is the Marangoni number and the critical value of this parameter is calculated as a function of Prandtl number, heat-transfer coefficient and elasticity parameters. It is shown that when the elasticity is very small instability sets in as steady convection, as for a Newtonian fluid, but at larger elasticities oscillatory convection is the first mode of instability to appear.With 5 Figures     

The Rayleigh–Taylor instability for inviscid and viscous fluids

The classical Rayleigh–Taylor instability occurs when two inviscid fluids, with a sharp interface separating them, lie in two horizontal layers with the heavier fluid above the lighter one. A small sinusoidal disturbance on the interface grows rapidly in time in this unstable situation, as the heavier upper fluid begins to move downwards through the lighter lower fluid. This paper presents a novel numerical method for computing the growth of the interface. The technique is based on a spectral representation of the solution. The results are accurate right up to the time when a curvature singularity forms at the interface and the inviscid model loses its validity. A spectral method is then presented to study the same instability in a viscous Boussinesq fluid. The results are shown to agree closely with the inviscid calculations for small to moderate times. However, the high interface curvatures that develop in the inviscid model are prevented from occurring in viscous fluid by the growth of regions of high vorticity at precisely these singular points. This leads to over-turning of the interface, to form mushroom-shaped profiles. It is shown that different initial interface configurations can lead to very different geometrical outcomes, as a result of the flow instability. These can include situations when detached bubbles form in the fluid.     

Taylor instability of superposed fluids for wave numbers near linear cut-off wave number

Summary A solution of the hydrodynamic equations for the case of (Taylor) instability of the interface of two fluids of different but constant densities when accelerated from the heavier fluid to the lighter one near the linear cut-off wave number is sought. Since the PLK method does not lead to valid results near this wave number, we apply the method of multiple scales. It was shown that the interface grows even at the linear critical wave number and the density ratio plays a role in determining the nonlinear critical wave number.

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Taylor Instabilität überlagerter Fluide für Wellenzahlen nahe der linearen Grenzwellenzahl

Zusammenfassung Es wird eine Lösung der hydrodynamischen Gleichungen für den Fall der (Taylor-) Instabilität der Trennfläche zweier Strömungen mit unterschiedlichen, aber konstanten Dichten gesucht, wenn in Richtung vom schwereren Fluid zum leichteren nahe der linearen Grenzwellenzahl beschleunigt wird. Da die PLK-Methode nahe diesen Wellenzahlen keine gültigen Resultate liefert, wird hier die Methode der variierten Maßstäbe angewendet. Es wird gezeigt, daß die Trennfläche sogar bei der linearen, kritischen Wellenzahl größer wird und daß das Dichteverhältnis eine wichtige Rolle für die Bestimmung der nichtlinearen, kritischen Wellenzahl spielt.

     

薄膜结构气弹动力稳定性研究

将扁壳的无矩理论和流体的理想势流理论结合起来对薄膜结构的气弹动力稳定性进行了研究,提出了结构失稳的判别准则,确定了结构失稳临界风速。首先应用扁壳的无矩理论建立了薄膜结构的动力平衡方程。然后假设来流为均匀的理想势流,考虑流固耦合作用,对风向沿结构拱向和垂向时分别采用不同的气弹模型确定了作用于薄膜表面的气动力,得到了两种情况下薄膜结构的气弹动力耦合作用方程。利用Bubnov-Galerkin方法将此耦合作用方程转化为一常系数二阶微分方程,并根据Routh-Hurwitz稳定性准则确定了薄膜的失稳临界风速。最后通过对临界风速的影响因素进行分析,得到了一些重要结论,并提出了防止薄膜结构气弹失稳的一些基本措施。     

Environmental dispersion in a two-layer wetland: Analytical solution by method of concentration moments

For the most typical two-layer wetland with free-water-surface-effect, analytically explored in this paper is the environmental dispersion in terms of the longitudinal evolution of the depth-averaged concentration. Taylor’s classical analysis for soluble matter dispersion in a single phase fluid flow is rigorously generalized for the two-layer case of a wetland flow to develop the dispersion model. The velocity profile of the fully developed flow through the wetland is derived, with that for the single-layer wetland flow included as a special case. Aris’s method of concentration moments is applied to determine the dispersivity with an asymptotic time variation. For typical contaminant constituents of chemical oxygen demand, biochemical oxygen demand, total phosphorus, total nitrogen and heavy metal, the process of dispersion is shown characteristic of a hierarchical structure in the critical length and duration of the contaminant cloud.     

Dispersion in a two-zone packed tube: An extended Taylor’s analysis

Taylor’s classical analysis for scalar dispersion in a single phase fluid flow is rigorously generalized for an instantaneous release of scalar substances into a fully developed flow through a long tube of two zones distinctively packed with porous media. A formulation for the dispersion is presented by cross-sectionally averaging the concentration transport equations and introducing a closure model for the concentration deviation terms produced in the averaging procedure. The velocity distribution of the flow through the tube is derived, with existing solution for a single zone tube flow included as a special case. Corresponding dispersivity is analytically determined, and Taylor’s well-known result for a single phase flow in a single-zone tube is recovered by setting corresponding parameters as unity. The effects of relevant parameters on both velocity profile and Taylor dispersivity are illustrated.     

Stability of Couette flow and Dean flow in micropolar fluids

In this article, we have investigated the onset of instability in the flow of the micropolar fluid between two rotating concentric cylinders and the flow in a curved channel, under small gap approximation. The solution of the eigenvalue problems, governing the onset of instability, are obtained by using the Shooting Method. It is seen that the flows are stable when the micropolar elements are present in the fluid.     

Thermal relaxation effect on magnetohydrodynamic instability in a rotating micropolar fluid layer heated from below

Summary. The effect of a vertical magnetic field on the onset of convective instability in a conducting micropolar fluid (Oldroyd fluid) layer heated from below confined between two horizontal planes under the simultaneous action of the rotation of the system and a vertical temperature gradient is considered. Linear stability theory and normal mode analysis are used to derive an eigenvalue system of order twelve, and an exact eigenvalue equation for natural instability is obtained. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. The effects of magnetic field, the relaxation time and micropolar parameters on the critical Rayleigh number and critical wave number are discussed and presented graphically. The analysis presented in this paper is more general than any previous investigation.     

The onset of liquid entrainment from a stratified two-phase region through small branches

A new criterion has been developed to predict the onset of liquid (heavier fluid) entrainment from a stratified two-phase region. The criterion was developed based on the local instability of the interface between two fluids due to the suction effect associated with the discharging of the lighter fluid. To validate the proposed criterion, comparisons were conducted between the measured critical height at the onset and those predicted using a three-dimensional analysis of the flow through two configurations: (1) a single branch mounted on an inclined wall with an inclination angle ranging between ?90° and + 60° and (2) dual branches mounted on a vertical wall with the plane passing through the branch centerlines and inclined with an angle α ranging between 0° and 60°. Comparisons demonstrate a very good agreement between the predicted and the measured values for both single and dual branches. This verifies that the onset of liquid entrainment mechanism occurs due to local flow instability of the interface, analogous to Rayleigh–Taylor instability.     

A particle method for two‐phase flows with large density difference

A new numerical approach for solving incompressible two‐phase flows is presented in the framework of the recently developed Consistent Particle Method (CPM). In the context of the Lagrangian particle formulation, the CPM computes spatial derivatives based on the generalized finite difference scheme and produces good results for single‐phase flow problems. Nevertheless, for two‐phase flows, the method cannot be directly applied near the fluid interface because of the abrupt discontinuity of fluid density resulting in large change in pressure gradient. This problem is resolved by dealing with the pressure gradient normalized by density, leading to a two‐phase CPM of which the original singlephase CPM is a special case. In addition, a new adaptive particle selection scheme is proposed to overcome the problem of ill‐conditioned coefficient matrix of pressure Poisson equation when particles are sparse and non‐uniformly spaced. Numerical examples of Rayleigh–Taylor instability, gravity current flow, water‐air sloshing and dam break are presented to demonstrate the accuracy of the proposed method in wave profile and pressure solution. Copyright © 2015 John Wiley & Sons, Ltd.     

Stochastic-Deterministic Modeling of the Development of Hydrodynamic Instability in Nonisothermal Filtration

Consideration is given to the stochastic-deterministic approach to the modeling of the Saffman–Taylor hydrodynamic instability in nonisothermal filtration of miscible fluids in a porous medium. The numerical model is employed to investigate the dynamics of development of the instability of the case of displacement of a more viscous cold fluid by a less viscous hot fluid in uniformly and nonuniformly permeable media.     

Stability of magnetohydrodynamic stratified shear flows

Summary A study is made of the stability of a stratified shear flow in a perfectly conducting fluid in the presence of an external magnetic field aligned with the flow. A semi-circle theorem for the present hydromagnetic case is proved. The magnetic field is found to have a stabilizing effect on the flow. The Rayleigh-Taylor instability problem in a stratified conducting fluid is discussed. Finally, a study is made of the absorption of wave energy by the mean flow in the hydromagnetic case by considering a shear flow with an anti-symmetric velocity profile given byU=tanhz. Unlike the hydrodynamic case, it is found that, in the critical layer atU=0, the transfer of the wave energy to the mean flow occurs for any value of the Richardson number. This result implies again the stabilizing effect of the magnetic field on the shear flow.     

The onset of transient Marangoni convection in a liquid layer subjected to rotation about a vertical axis

In view of the interesting possibilities of controlling surface tension-driven convection, anticipated in space experiments involving fluid interfaces, the problem of the stability of a thin horizontal fluid layer subjected to rotation about a vertical axis, when the thermal (or concentration) gradient is not uniform is examined by linear stability analysis. Attention is focussed on the situation where the critical Marangoni number is greater than that for the case of uniform thermal gradient and the convection is not, in general, maintained. The case of adiabatic boundary condition is examined because it brings out the effect of surface tension at the free surfaces and allows a simple application of the Galerkin technique, which gives useful results. Numerical results are obtained for special cases and some general conclusions about the destabilizing effects of various basic temperature profiles and the stabilizing effect of coriolis force are presented. The results indicate that the most destabilizing temperature gradient is one for which the temperature gradient is a step function of the depth. Increase in Taylor number and the inverted parabolic basic temperature profile suppress the onset of convection.     

Asymptotic Laws of Thermovibrational Convecton in a Horizontal Fluid Layer

Theoretical study of convective instability is applied to a horizontal layer of incompressible single-component fluid subjected to the uniform steady gravity, longitudinal vibrations of arbitrary frequency and initial temperature difference. The mathematical model of thermovibrational convection has the form of initial boundary value problem for the Oberbeck-Boussinesq system of equations. The problems are solved using different simulation strategies, like the method of averaging, method of multiple scales, Galerkin approach, Wentzel-Kramers-Brillouin method and Floquet technique. The numerical analysis has shown that the effect of vibrations on the stability threshold is complex: vibrations can either stabilize or destabilize the basic state depending on values of the parameters. The influence of the Prandtl number on the instability thresholds is investigated. The asymptotic behaviour of critical values of the parameters is studied in two limiting cases: (i) small amplitude and (ii) low frequency of vibration. In case (i), the instability is due to the influence of thermovibrational mechanism on the classical Rayleigh-Benard convective instability. In case (ii), the nature of the instability is related to the instability of oscillating counter-streams with a cubic profile.     

液氢加注系统竖直管道内Taylor气泡的行为特性

针对液氢加注系统竖直管道内气液两相流实验化困难的问题,运用建模仿真的方法建立了竖直管道内Taylor气泡的运动模型,对Taylor气泡的形成过程、大小以及充分发展的Taylor气泡上升速度进行了研究.采用VOF方法对气液两相的交界面进行追踪,并引入CSF模型对两相间的表面张力进行计算.仿真结果表明:Taylor气泡是由...     

Current-induced instabilities of hydromagnetic flow in a small gap between rotating conducting cylinders

Summary.  In considering the stability of an electrically conducting fluid between rotating perfectly conducting cylinders with a current-induced pressure gradient acting in the azimuthal direction and with an applied axial magnetic field, the assumption of small-gap approximation is made and the governing equations with respect to both axisymmetric and non-axisymmetric disturbances are solved by a direct numerical procedure. A parametric study covering wide ranges of Q, the Hartmann number which represents the strength of the axial magnetic field, and β, a parameter characterizing the ratio of current-induced and rotation velocities, is conducted for the situation where the outer cylinder is stationary and the inner cylinder is rotating. It is found that the stability characteristics are thoroughly different from those of the case of weakly conducting walls. The variation of the onset mode is shown in the (β, Q)-plane, and the transition of the corresponding neutral curves is discussed in detail. Results for the critical Taylor number and wavenumber pertaining to the critical disturbances are presented. The critical values of radial current density required for the onset of instability are also determined. Received May 8, 2002; revised November 11, 2002 Published online: May 8, 2003 The financial support for this work from National Science Council of Taiwan, ROC, through Grant No. NSC 89-2212-E-132-006 is gratefully acknowledged.