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Three-dimensional instability of axisymmetric flow in a rotating lid–cylinder enclosure
Abstract and Figures
The axisymmetry-breaking three-dimensional instability of the axisymmetric flow
between a rotating lid and a stationary cylinder is analysed. The flow is governed by
two parameters – the Reynolds number Re and the aspect ratio γ (=height/radius).
Published experimental results indicate that in different ranges of γ axisymmetric
or non-axisymmetric instabilities can be observed. Previous analyses considered only
axisymmetric instability. The present analysis is devoted to the linear stability of the
basic axisymmetric flow with respect to the non-axisymmetric perturbations. After
the linearization the stability problem separates into a family of quasi-axisymmetric
subproblems for discrete values of the azimuthal wavenumber k. The computations
are done using the global Galerkin method. The stability analysis is carried out at
various densely distributed values of γ in the range 1 < γ < 3.5. It is shown that the
axisymmetric perturbations are dominant in the range 1.63 < γ < 2.76. Outside this
range, for γ < 1.63 and for γ > 2.76, the instability is three-dimensional and sets in with
k = 2 and k = 3 or 4, respectively. The azimuthal periodicity, patterns, characteristic
frequencies and phase velocities of the dominant perturbations are discussed.
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... [5] and Gelfgat et al. [6]. The flow behaviours were also studied using computational techniques. ...
... Swirling flows or rotating flow in an enclosure replicates many natural phenomena such as tornadoes, water circulation in oceans, meteorological movements, as mentioned by Childs [1] in his book called "Rotating flow". The swirling flow in a cylindrical cavity has been studied for more than 50 years by Escudier [2], Lopez et al. [3], Westergaard et al. [4], Sørensen et al. [5], Gelfgat et al. [6] and others. The cylinder with the rotating lid being the most straightforward configuration has been extensively used for understanding the transitions and instability in the rotating flow, both experimentally and numerically. ...
... As many authors have described, the rotating cavity flow chiefly depends on two factors: (a) the aspect ratio and (b) the Reynolds number. The flow within the cylindrical cavity can be categorized as follows: (a) steady, laminar, axisymmetric without separation (Peres et al. [41,42]) (b) steady laminar, axisymmetric with vortex breakdown composed of two bubbles in the axis of symmetry (Gelfgat et al. [6]) (c) periodic laminar unsteady axisymmetric and non-axisymmetric flow with several azimuthal waves (fluctuations) depending on the aspect ratio and Reynolds number (Sørensen et al. [5], Peres et al. [41]) (d) transitional unsteady flow (Sørensen et al. [5]) and (e) fully turbulent flow (Sørensen et al. [5], Gelfgat et al. [6]). ...
In this paper, we analyse the flow in a cylindrical cavity of aspect ratio H / R = 2, with a top lid rotating with a sinusoidal time-varying speed. The Reynolds numbers, based on the time-averaged rotating speed and the radius of the cavity (Re = $$ \Omega _0 R^2/\nu $$ Ω 0 R 2 / ν ) are 2800, 3500 and 4100. Particle Image velocimetry measurements and numerical simulations are performed to determine the effect on the flow of different imposed frequencies on the rotating lid. The acceleration and deceleration of the lid produce a marked pulsatile flow behaviour. It is found that at low Reynolds numbers the frequencies measured in the flow coincide with the frequency imposed on the rotating lid, while at the largest Re considered the flow frequencies are significantly lower, especially when the frequency of the disk is large. The fluctuations induced in the vertical velocity component are about 100% of the instantaneous values while for the angular component the fluctuations are of the same order as the instantaneous values.
... General three-dimensional perturbations to the axisymmetric, steady basic state have been examined in the linear stability analysis of Gelfgat et al . [4] for the range 1 < Λ < 4. For low aspect ratios, 1 < Λ < 1.63, the analysis predicts symmetry breaking via a Hopf bifurcation to a RW state with azimuthal wavenumber k = 2. For high aspect ratios, Λ > 2.76, the analysis again predicts a symmetry-breaking Hopf to a RW with k = 4. ...
... As noted above, the first bifurcation to occur in this flow (for Λ = 2.5) as Reynolds numbers are increased is a supercritical Hopf that preserves axisymmetry; this occurs at Re = 2707 [4,6]. Thereafter, while further solution branches appear, it is the underlying axisymmetric flow that dominates the behaviour in each case, up to the maximum Reynolds number for this study, Re = 4300. ...
Rotating waves are a generic instability mode of flows that possess a rotation symmetry, such as Taylor-Couette flow. In the enclosed swirling flow that is generated in a closed cylindrical container by a rotating end-wall, the initial bifurcation to unsteadiness produced by increasing the endwall rotation rate can be either to rotating waves or unsteady axisymmetric flow, depending on the aspect ratio of the cylinder. We examine a case where the initial bifurcation is to a periodic axisymmetric state, and follow subsequent bifurcations to other states where the flow breaks axisymmetry as well. These resulting states possess rotating waves that are modulated by the underlying axisymmetric behaviour. Although this flow also has axial vortex breakdowns, they appear to play no dynamical role in the symmetry breaking, remaining essentially axisymmetric.
... The validation focused on the behavior of axisymmetric and non-oscillatory flow structures which limits the comparison to 2600 Re [20]. Fig.2 illustrates the effect of injecting a fluid slightly heavier than the ambient one, i.e. ρa > ρj. ...
... We recall that outside the prescribed range of Re, the flow becomes time dependent (bubble oscillations evidenced by 3D simulations along the cavity axis [20]). ...
Thermal buoyancy, induced by injection or by differential heating of a tiny rod is explored to control breakdown in the core of a helical flow driven by the lid rotation of a cylinder. Three main parameters are required to characterize numerically the flow behavior; namely, the rotational Reynolds number Re, the cavity aspect ratio and the Richardson number Ri. Warm injection/rod, Ri > 0, is shown to prevent on-axis flow stagnation while breakdown enhancement is evidenced when Ri < 0. Results revealed that a bubble vortex evolves into a ring type structure which may remain robust, as observed in prior related experiments or, in contrast, disappear over a given range of parameters (Λh, Re, Ri > 0). Besides, the emergence of such a toroidal mode was not found to occur under thermal stratification induced by a differentially heated rod. Moreover, three state diagrams were established which provide detailed flow characteristics under the distinct and combined effects of buoyancy strength, viscous effects and cavity aspect ratio.
... Various studies have been performed to control the vortex breakdown bubble [26,27]. The azimuthal perturbation in the 3D instability analysis was observed by Lugt & Abboud [28] and Gelfgat et al [29]. In a recent work [30], the presence of Taylor-like vortices is reported in VE flows, which corresponds to the chaotic state of the flow. ...
The effect of compressibility for flow inside the cylinder with the top rotating lid (Vogel-Escudier flow) is examined. Three-dimensional Navier–Stokes equations for compressible flow in Cartesian coordinates are used to simulate the flow using open-source OpenFOAM software. The Mach number (Ma) of the flow is varied from 0.1 to 0.3, and the Reynolds number (Re) is varied from 1000 to 5000 for a fixed aspect ratio (Γ = 2.5) of the cylinder. The flow is found to have a transition from a steady axisymmetric state to a non-axisymmetric state exhibiting multiple azimuthal waves as the Mach number and the Reynolds number are varied. The flow field changes significantly with an increase in Ma for unsteady flow at higher Re. An increase in Ma increases the side wall azimuthal instability, as found in the perturbation contour plots and time series analysis. Further, we reconstruct phase portraits to show the dynamics of the flow finally becoming chaotic as the Reynolds number is increased to 5000. Finally, we support the argument with Lyapunov exponents for the higher Re samples. The Lyapunov Exponent is found to increase with Ma.
... This map shows the number of vortex breakdown bubbles along with the steady and unsteady regimes of the flow for a wide range of Reynolds numbers and aspect ratios of the cylinder. There is a considerable discussion on the mechanism of the vortex breakdown in the VE flow, which can be found in the literature [6,[13][14][15][16][17]. Early three-dimensional studies (both computational and experimental) of the flow have revealed the existence of the azimuthal rotating waves in the flow [10,11,[18][19][20] as well as the effect of perturbations on the vortex breakdown bubble [7,21,22]. ...
Taylor–Couette flow is a canonical flow to study Taylor–Görtler (TG) instability or centrifugal instability and the associated vortices. TG instability has been traditionally associated with flow over curved surfaces or geometries. In the computational study, we confirm the presence of TG-like near-wall vortical structures in two lid-driven flow systems, the Vogel–Escudier (VE) and the lid-driven cavity (LDC) flows. The VE flow is generated inside a circular cylinder by a rotating lid (top lid in the present study), while the LDC flow is generated inside a square or rectangular cavity by the linear movement of the lid. We look at the emergence of these vortical structures through reconstructed phase space diagrams and find that the TG-like vortices are seen in the chaotic regimes in both flows. In the VE flow, these vortices are seen when the side-wall boundary layer instability sets in at large R e . The VE flow is observed to go to a chaotic state in a sequence of events from a steady state at low R e . In contrast to VE flows, in the LDC flow with no curved boundaries, TG-like vortices are seen at the emergence of unsteadiness when the flow exhibits a limit cycle. The LDC flow is observed to have transitioned to chaos from the steady state through a periodic oscillatory state. Various aspect ratio cavities are examined in both flows for the presence of TG-like vortices.
This article is part of the theme issue ‘Taylor–Couette and related flows on the centennial of Taylor’s seminal Philosophical transactions paper (Part 2)’.
... His work was followed by dozens of experimental and numerical studies, which found more and more interesting features in this closed swirling flow. In particular, in numerical linear analysis of the propagation of axisymmetric and azimuthal infinitesimal perturbations in the cavity, exponentially growing marginal and critical modes were found for cylinders of aspect ratios up to 5.5 for a wide range of Reynolds numbers [34][35] . The existence of marginal and critical modes was revealed in numerical calculations [32][33] and confirmed by experiments 36 , but with a background level of perturbations, which, due to the natural roughness of the cavity walls, that is smaller than that of the nano-grass shown in this work. ...
Experiments were carried out in a water-filled elongated cup of a “kitchen scale,” where motion was created by a rotating disk with various micro- and nano-roughness in the top of the cup. The obtained results have shown that for some patterns of nanostructures, there is a noticeable growth of a vortex, generated by the disk, while other roughnesses do not make visible changes in the flow structure. The results are of interest in assessing the efficiency of surfaces with nanoscale roughnesses. Indeed, the first type of nano-roughness may become useful for enhancing soft mixing in chemical and bio-reactors, including in the preparation of special food delicacies. On the other hand, the use of nanostructured surfaces that do not affect the main flow can help to solve some industrial problems of water and ice erosion, for example, in wind turbines or any other objects where disturbances of the main flow are undesirable.
... The first mode of instability from these studies were identified to rotating wave. The azimuthal perturbation modes were later detected in 3D instability analysis 17,18 . A recent numerical study 19 investigates the characteristic features of stream tubes for an axisymmetric flow and shows that the momentum flux of axial vortex flows is the most deciding factor for the occurrence of a breakdown bubble. ...
This paper examines the effect of unstable thermal stratification on vortex breakdown in Vogel–Escudier flow. A three-dimensional direct numerical simulation of Navier–Stokes and energy equations are used to simulate a flow inside a cylindrical container generated by rotating the top lid. The top and bottom are kept at two constant temperatures such that unstable stratification is maintained. The rotation speed is related to the Reynolds number (Re), and buoyancy is linked to the Rayleigh number (Ra). The streamline and vertical velocity contour plots indicate different regimes of the flow depending on the Re and Ra. The convection dominated (CD) regime has a characteristic large-scale circulation similar to the Rayleigh–Bénard convection, and the rotation dominated (RD) regime has a central axial vortex and breakdowns. A transitional regime between RD and CD regimes is also identified from energy consideration. The influence of Ra on a vortex breakdown bubble and its relation to azimuthal vorticity is investigated in detail. Consistent with the literature on Vogel–Escudier flow, the azimuthal vorticity is shown to be essential for the breakdown in the presence of buoyancy as well. In the low Re limits, the energy of flow tends to be associated with the r–z plane velocity field, while at large Re, the energy is associated with the out-of-the-plane velocity field. Thermal plumes align along the axis for large rotations and are affected by the vortex breakdown bubble. The velocity perturbation structures and plumes show a remarkable distinction between rotation and convection-dominated regimes in the topology.
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... For Γ = 2.5, azimuthal waves appear at Re ≈ 3400 with wavenumber k = 5. Figure 12a shows a breakdown bubble at an instant for Re = 3400, along with contours of u z in z = 0.1 plane showing nonaxisymmetry of the flow. Linear stability analysis of the flow 45 has also predicted that the wavenumber supported by the flow for Γ = 2.5 is k = 5. This azimuthal wave rotates in the same direction as the bulk flow. ...
The topology and the dynamics of Vogel–Escudier flow, which is the flow inside a circular cylinder with a top rotating lid, are presented in this paper. A three-dimensional direct numerical simulation of the Navier–Stokes equations in cylindrical coordinates is used to investigate the flow. Various combinations of Reynolds number and aspect ratio are studied and classified based on the flow topology. The flow is found to exhibit steady axisymmetric, unsteady axisymmetric, rotating azimuthal waves, and weak turbulence regimes. The perturbations found in the system are characteristically different for various flow regimes and are used for the classification of flow. The presence of several modes at high Reynolds number suggests a weak turbulence state, and a Taylor–Görtler type instability wave is found in the sidewall boundary layer.
In hydraulic turbines, either Francis or axial turbines, a high swirling flow is generated at PL conditions at the inlet of the draft tube resulting in the formation of a rotating vortex rope (RVR). This leads to pressure fluctuations which limit the operating range of single regulated turbines. In the present study, an axial hydraulic turbine has been numerically simulated by the k−ϵ and SST-based Scale-Adaptive Simulation model (SST SAS) turbulent models to capture helical RVR. Considering the formation of the RVR as the result of a global instability, linear global stability analysis of the time-averaged turbulent flow field has been conducted. For the first time in axial hydraulic turbines, how boundary conditions affect the unstable mode and which frequency, plunging or rotating, is related to the vortex rope instability have been studied. It is found that the flow inside the draft tube is sensitive to the asymmetrical disturbances with a frequency close to the rotational component.
The three-dimensional instability of an axisymmetric natural convection flow is investigated numericaUy using a global spectral Galerkin method. The linear stability problem separates for different azimuthal modes. This aUowsus to reduce the problem to a sequence of 2D-like problems. The formulation of the numerical approach and several test calculations are reported. The numerical results are successfully compared with an experiment on natural convection of water in a vertical cylinder, which shows an axisymmetry-breaking instability with a high azimuthal wavenumber.
The flow driven by a rotating end wall in a cylindrical container with aspect ratio H/R=2.5 is time dependent for Reynolds numbers Re=ΩR2/ν>2700. For Reynolds numbers up to 4000 three solution branches have been identified, and we examine a solution on each one. At Re=3000, the flow is axisymmetric and time periodic. At Re=3500, the flow is quasiperiodic with a low-frequency modulation and supports a modulated rotating wave with azimuthal wave number k=5. At Re=4000, the flow is time periodic with a qualitatively different mode of oscillation to that at Re=3500. It also supports a modulated rotating wave, with k=6. The peak kinetic energy of the nonaxisymmetric modes is associated with the jet-like azimuthal flow in the interior.
A comparison between the experimental visualization and numerical simulations of the occurrence of vortex breakdown in laminar swirling flows produced by a rotating endwall is presented. The experimental visualizations of Escudier (1984) were the first to detect the presence of multiple recirculation zones and the numerical model presented here, consisting of a numerical solution of the unsteady axisymmetric Navier-Stokes equations, faithfully reproduces these phenomena and all other observed characteristics of the flow. Further, the numerical calculations elucidate the onset of oscillatory flow, an aspect of the flow that was not clearly resolved by the flow visualization experiments. Part 2 of the paper examines the underlying physics of these vortex flows.
A combined experimental and numerical investigation is presented of the multiple
oscillatory states that exist in the flows produced in a completely filled, enclosed,
circular cylinder driven by the constant rotation of one of its endwalls. The flow
in a cylinder of height to radius ratio 2.5 is interrogated experimentally using flow
visualization and digitized images to extract quantitative temporal information. Numerical
solutions of the axisymmetric Navier–Stokes equations are used to study the
same flow over a range of Reynolds numbers where the flow is observed to remain
axisymmetric. Three oscillatory states have been identified, two of them are periodic
and the third is quasi-periodic with a modulation frequency much smaller than the
base frequency. The range of Reynolds numbers for which the quasi-periodic flow
exists brackets the switch between the two periodic states. The results from the combined
experimental and numerical study agree both qualitatively and quantitatively,
providing unambiguous evidence of the existence and robustness of these multiple
time-dependent states.
The onset of convection in a uniformly rotating vertical cylinder of height h and radius d heated from below is studied. For non-zero azimuthal wavenumber the instability is a Hopf bifurcation regardless of the Prandtl number of the fluid, and leads to precessing spiral patterns. The patterns typically precess counter to the rotation direction. Two types of modes are distinguished: the fast modes with relatively high precession velocity whose amplitude peaks near the sidewall, and the slow modes whose amplitude peaks near the centre. For aspect ratios τ ≡ d / h of order one or less the fast modes always set in first as the Rayleigh number increases; for larger aspect ratios the slow modes are preferred provided that the rotation rate is sufficiently slow. The precession velocity of the slow modes vanishes as τ → ∞. Thus it is these modes which provide the connection between the results for a finite-aspect-ratio System and the unbounded layer in which the instability is a steady-state one, except in low Prandtl number fluids.
The linear stability problem is solved for several different sets of boundary conditions, and the results compared with recent experiments. Results are presented for Prandtl numbers σ in the range 6.7 ≤ σ ≤ 7.0 as a function of both the rotation rate and the aspect ratio. The results for rigid walls, thermally conducting top and bottom and an insulating sidewall agree well with the measured critical Rayleigh numbers and precession frequencies for water in a τ = 1 cylinder. A conducting sidewall raises the critical Rayleigh number, while free-slip boundary conditions lower it. The difference between the critical Rayleigh numbers with no-slip and free-slip boundaries becomes small for dimensionless rotation rates Ω h ² / v ≥ 200, where v is the kinematic viscosity.
Buoyancy-driven instability of a monocomponent or binary fluid which is
completely contained in a vertical circular cylinder is investigated,
including the influence of the Soret effect for the binary mixture. The
Boussinesq approximation is used, and the resulting linear stability
problem is solved using Galerkin's technique. The analysis considers
various types of fluid mixtures, ranging from gases to liquid metals, in
cylinders with a variety of radius-to-height ratios. The flow structure
is found to depend strongly on both the cylinder aspect ratio and the
magnitude of the Soret effect. Comparisons are made with experiments and
other theories, and the predicted stability limits are shown to agree
closely with observations.
The stability of steady axisymmetric solutions of the Rayleigh-Benard problem to nonaxisymmetric disturbances is discussed. In the first part of the paper, the critical Rayleigh numbers for the nonaxisymmetric modes of convection in a cylinder are determined. These are found to be Iower than the critical Rayleigh number for unbounded convection. In the second part, the stability of finite amplitude axisymmetric solutions is discussed, with particular emphasis on the stability of the fiywheel mode. The significance of these results for low Prandtl number convection is briefiy discussed, and the connection with solar granulation, in particular the exploding granule phenomenon, is mentioned.
Steady and oscillatory convection in rigid vertical cylinders heated from below studied by means of a numerical solution of the three-dimensional, time-dependent Boussinesq equations. Both adiabatic and ideal conducting sidewalls are considered. The effect of the geometry of the container on the onset of convective instability and the structure and symmetry of the flow are analysed and compared with the results of linear stability theories. The nonlinear evolution and stability of convective flows at Rayleigh numbers beyond the critical number for the onset of convective motion are investigated for Prandtl numbers of 0.02 to 6.7. The limits of stable axisymmetric solutions are an important finding of this study. The onset and the frequency of oscillatory instability are calculated for the small Prandtl number 0.02 and compared with experimental data. Calculated stream patterns and velocity profiles illustrate the three-dimensional structure of steady convection and the time-dependent behaviour of oscillatory flows.