The symmetries of flow structures are often prescribed by their mechanical instability and geometry. Here, as an example, we present the homotopy of the rotating 3-fold spiral state that is robust in a spherical Couette flow towards the hybrid system with a thermal stratification effect. It has not yet been confirmed that the rotating wave state smoothly connects to the thermal stratification system. Through continuation, the most dangerous mode at a purely spherical Couette flow of $m=4$ modes of spherical harmonics is replaced by $l=4$ and $m=3$ in a purely thermal convective system. For the state obtained at the limit under only the thermal effect, the residual quantities of both the torque to the outer sphere and meridional circulation are discussed in detail.
SCF experiments were conducted according to the work of Egbers and Rath [Acta Mech. 111 pp. 125--140 (1995)]. Through visualization using aluminium flakes drifting on a horizontal plane illuminated by a laser sheet, the flow was identified as a spiral wave with azimuthal wavenumber $m=3$, using the experimentally obtained and numerically deduced comparison between phase velocities. By solving the equation of motion for the infinitesimal planar particles advecting in the flow field of the spiral wave, a visual distribution of reflected light was reproduced virtually, which is in good agreement …
An investigation of the effect of a destabilizing cross-stream temperature gradient on the transient growth phenomenon of plane Poiseuille flow and plane Couette flow is presented. Only the streamwise-uniform and nearly streamwise-uniform disturbances are highly influenced by the Rayleigh number Ra and Prandtl number Pr. The maximum optimal transient growth G max of streamwise-uniform disturbances increases slowly with increasing Ra and decreasing Pr. For all Ra and Pr, at moderately large Reynolds numbersRe, the supremum of G max is always attained for streamwise-uniform perturbations (or nea…
From Sheared Annular Centrifugal Rayleigh-Bénard Convection to Radially Heated Taylor-Couette Flow: Exploring the Impact of Buoyancy and Shear on Heat Transfer and Flow Structure
This study numerically investigates the bifurcation aspect of the wide-gap spherical Couette flow (SCF), with an emphasis on the competition among polygonal coherence with different wave numbers observed over transitional Reynolds numbers. Focusing on a representative case, the half-radius ratio, we confirm that the axisymmetric state becomes unstable over the first transitional Reynolds number at which the 4-fold spiral state bifurcates, using the continuation method based on the Newton-Raphson algorithm. The Galerkin-spectral method was employed to numerically solve the governing equations. …
Rayleigh-Benard convection and Taylor-Couette flow are two canonical flows that have many properties in common. We here compare the two flows in detail for parameter values where the Nusselt numbers, i.e. the thermal transport and the angular momentum transport normalized by the corresponding laminar values, coincide. We study turbulent Rayleigh-Benard convection in air at Rayleigh number Ra=1e7 and Taylor-Couette flow at shear Reynolds number Re_S=2e4 for two different mean rotation rates but the same Nusselt numbers. For individual pairwise related fields and convective currents, we compare …
Numerical calculations of vortex flows in Taylor-Couette systems with counter rotating cylinders are presented. The full, time dependent Navier-Stokes equations are solved with a combination of a finite difference and a Galerkin method. Annular gaps of radius ratio $η=0.5$ and of several heights are simulated. They are closed by nonrotating lids that produce localized Ekman vortices in their vicinity and that prevent axial phase propagation of spiral vortices. Existence and spatio temporal properties of rotating defects, of modulated Ekman vortices, and of the spiral vortex structures in the b…
We generalize an analogy between rotating and stratified shear flows. This analogy is summarized in Table 1. We use this analogy in the unstable case (centrifugally unstable flow v.s. convection) to compute the torque in Taylor-Couette configuration, as a function of the Reynolds number. At low Reynolds numbers, when most of the dissipation comes from the mean flow, we predict that the non-dimensional torque $G=T/ν^2L$, where $L$ is the cylinder length, scales with Reynolds number $R$ and gap width $η$, $G=1.46 η^{3/2} (1-η)^{-7/4}R^{3/2}$. At larger Reynolds number, velocity fluctuations beco…
We investigate the local self-sustained process underlying spiral turbulence in counter-rotating Taylor-Couette flow using a periodic annular domain, shaped as a parallelogram, two of whose sides are aligned with the cylindrical helix described by the spiral pattern. The primary focus of the study is placed on the emergence of drifting-rotating waves ({\sc drw}) that capture, in a relatively small domain, the main features of coherent structures typically observed in developed turbulence. The transitional dynamics of the subcritical region, far below the first instability of the laminar circul…
A number of time-periodic flows have been found to have a property called RZIF: when a linear stability analysis is carried out about the temporal mean (rather than the usual steady state), an eigenvalue is obtained whose Real part is Zero and whose Imaginary part is the nonlinear Frequency. For two-dimensional thermosolutal convection, a Hopf bifurcation leads to traveling waves which satisfy the RZIF property and standing waves which do not. We have investigated this property numerically for counter-rotating Couette-Taylor flow, in which a Hopf bifurcation gives rise to branches of upwards a…
Investigations of counter-rotating Taylor-Couette flow (TCF) in the narrow gap limit are conducted in a very large aspect ratio apparatus. The phase diagram is presented and compared to that obtained by Andereck et al. The spiral turbulence regime is studied by varying both internal and external Reynolds numbers. Spiral turbulence is shown to emerge from the fully turbulent regime via a continuous transition appearing first as a modulated turbulent state, which eventually relaxes locally to the laminar flow. The connection with the intermittent regimes of the plane Couette flow (pCf) is discus…
We investigate the linear stability of a thin, suspended, annular film of conducting fluid with a voltage difference applied between its inner and outer edges. For a sufficiently large voltage, such a film is unstable to radially-driven electroconvection due to charges which develop on its free surfaces. The film can also be subjected to a Couette shear by rotating its inner edge. This combination is experimentally realized using films of smectic A liquid crystals. In the absence of shear, the convective flow consists of a stationary, azimuthally one-dimensional pattern of symmetric, counter-r…
Rayleigh-Bénard convection (RBC) and Taylor-Couette Flow (TCF) are two paradigmatic fluid dynamical systems frequently discussed together because of their many similarities despite their different geometries and forcing. Often these analogies require approximations, but in the limit of large radii where TCF becomes rotating plane Couette flow (RPC) exact relations can be established. When the flows are restricted to two spatial degrees of freedom there is an exact specification that maps the three velocity components in RPC to the two velocity components and one temperature field in RBC. Using…
This paper numerically investigates the shear flow between double concentric spherical boundaries rotating differentially, so-called spherical Couette flow, under unstable thermal stratification, focusing on the boundary of the axisymmetric/non-axisymmetric transition in wide gap cases where the inner radius is comparable to the clearance width. While the transition of SCF has been confirmed experimentally in cases without thermal factor, insufficient knowledge on SCF subject to thermal instability, related to geophysical problems especially in wide gap cases, has been accumulated mainly based…
Taylor-Couette (TC) flow, the flow between two independently rotating and co-axial cylinders is commonly used as a canonical model for shear flows. Unlike plane Couette, pinned secondary flows can be found in TC flow. These are known as Taylor rolls and drastically affect the flow behaviour. We study the possibility of modifying these secondary structures using patterns of stress-free and no-slip boundary conditions on the inner cylinder. For this, we perform direct numerical simulations of narrow-gap TC flow with pure inner cylinder rotation at four different shear Reynolds numbers up to $Re_…
The transition to turbulence in Taylor-Couette flow often occurs via a sequence of supercritical bifurcations to progressively more complex, yet stable, flows. We describe a subcritical laminar-turbulent transition in the counter-rotating regime mediated by an unstable intermediate state in a system with an axial aspect ratio of $Γ=5.26$ and a radius ratio of $η=0.905$. In this regime, flow visualization experiments and numerical simulations indicate the intermediate state corresponds to an aperiodic flow featuring interpenetrating spirals. Furthermore, the reverse transition out of turbulence…
When the Rayleigh number is low, Rayleigh-Bénard convection in a nonrotating spherical shell with central gravity has symmetric solutions in terms of three-dimensional discrete rotation. All the known patterns with the regular polyhedral symmetries accompany reflection symmetry. We found a new type of steady convection in a nonrotating spherical shell by computer simulations. The pattern has the discrete rotational symmetry of a regular tetrahedron with no reflection symmetry. The convection consists of six pairs of spiral rolls placed on 12 faces of a spherical dodecahedron. Depending on the …
We consider the effect of radial fluid injection and suction on Taylor-Couette flow. Injection at the outer cylinder and suction at the inner cylinder generally results in a linearly unstable steady spiralling flow, even for cylindrical shears that are linearly stable in the absence of a radial flux. We study nonlinear aspects of the unstable motions with the energy stability method. Our results, though specialized, may have implications for drag reduction by suction, accretion in astrophysical disks, and perhaps even in the flow in the earth's polar vortex.
Explanation of constant mean angular momentum in high-Reynolds-number Taylor--Couette turbulence in terms of history effects
Department of Physics & Research and Education Center for Natural Sciences, Hiyoshi Campus, Keio University（庆应义塾大学日吉校区物理系及自然科学研究教育中心）；Department of Mechanical Engineering, Doshisha University（立命馆大学机械工程系）；Laboratory of Complex Flows, Faculty of Science and Engineering, Kindai University（近畿大学理工学部复杂流动实验室）
We study the nonlinear mode competition of various spiral instabilities in magnetised Taylor-Couette flow. The resulting finite-amplitude mixed-mode solution branches are tracked using the annular-parallelogram periodic domain approach developed by Deguchi & Altmeyer (2013). Mode competition phenomena are studied in both the anti-cyclonic and cyclonic Rayleigh-stable regimes. In the anti-cyclonic sub-rotation regime, with the inner cylinder rotating faster than the outer, Hollerbach, Teeluck & Rudiger (2010) found competing axisymmetric and non-axisymmetric magneto-rotational linearly unstable…
The single-arm spiral roll state in the system of Boussinesq fluid confined between non-rotating double concentric spherical boundaries with an opposing temperature gradient previously reported by Zhang et al.(Phys. Rev. E, v66, 055203, 2002) has been numerically investigated in detail. It is presented that the state can exist even in a relatively thicker gap rather than that used in our previous reports as a travelling wave solution with a finite constant angular velocity.
This paper is the first part of a two-fold study of mixing, i.e. the formation of layers and upwelling of buoyancy, in axially stratified Taylor--Couette flow, with fixed outer cylinder. Using linear analysis and direct numerical simulation, we show the critical role played by non-axisymmetric instability modes, despite the fact that the flow is centrifugally unstable in the sense of Rayleigh's criterion. Interactions between helical modes of opposite handedness leads to the formation of nonlinear coherent structures: (mixed)-ribbons and (mixed)-cross-spirals. These give birth to complex densi…
Large-scale patterns, which are well-known from the spiral defect chaos regime of thermal convection at Rayleigh numbers $Ra < 10^4$, continue to exist in three-dimensional numerical simulations of turbulent Rayleigh-Bénard convection in extended cylindrical cells with an aspect ratio $Γ=50$ and $Ra>10^5$. They are uncovered when the turbulent fields are averaged in time and turbulent fluctuations are thus removed. We apply the Boussinesq closure to estimate turbulent viscosities and diffusivities, respectively. The resulting turbulent Rayleigh number $Ra_{\ast}$, that describes the convection…
Rotating spiral patterns in Rayleigh-Bénard convection are known to induce azimuthal flows, which raises the question of how different neighboring spirals interact with each other in spiral chaos, and the role of hydrodynamics in this regime. Far from the core, we show that spiral rotations lead to an azimuthal body force that is irrotational and of magnitude proportional to the topological index of the spiral and its angular frequency. The force, although irrotational, cannot be included in the pressure field as it would lead to a nonphysical, multivalued pressure. We calculate the asymptotic…
We study the stability and dynamics of non-Boussinesq convection in pure gases (CO$_2$ and SF$_6$) with Prandtl numbers near $Pr\simeq 1$ and in a H$_2$-Xe mixture with $Pr=0.17$. Focusing on the strongly nonlinear regime we employ Galerkin stability analyses and direct numerical simulations of the Navier-Stokes equations. For $Pr \simeq 1$ and intermediate non-Boussinesq effects we find reentrance of stable hexagons as the Rayleigh number is increased. For stronger non-Boussinesq effects the hexagons do not exhibit any amplitude instability to rolls. Seemingly, this result contradicts the exp…
In this paper, we experimentally study the influence of large-scale Taylor rolls on the small-scale statistics and the flow organization in fully turbulent Taylor-Couette flow {for Reynolds numbers up to $\text{Re}_S=3\times 10^5$}. The velocity field in the gap confined by coaxial and independently rotating cylinders at a radius ratio of $η=0.714$ is measured using planar {particle image velocimetry} in horizontal planes at different cylinder heights. Flow regions with and without prominent Taylor vortices are compared. We show that the local angular momentum transport (expressed in terms of …
The research is devoted to the stability of convective flow in a nonuniformly rotating layer of an electrically conducting fluid in a spiral magnetic field. The stationary and oscillatory modes of magnetic convection are considered depending on the profile of the angular rotation velocity (Rossby number $\textrm{Ro}$) and on the profile of the external azimuthal magnetic field (magnetic Rossby number $\textrm{Rb}$). The nonlinear dynamic system of Lorentz type equations is obtained by using the Galerkin method. Numerical analysis of these equations has shown the presence of chaotic behavior of…
In this study we report numerical results of turbulent transport of heat $Nu$ and angular momentum $ν_{t}/ν$ in Taylor-Couette (TC) flows subjected to a radial temperature gradient. Direct numerical simulations are performed in a TC cell with a radius ratio $η=0.5$ and an aspect ratio $Γ=8$ for two Rayleigh numbers ($Ra=10^5$, $10^6$) and two Prandtl numbers ($Pr=0.7$, $4.38$), while the Reynolds number $Re$ varies in the range of $0 \le Re \le 15000$. With increasing $Re$, the flows undergo two distinct transitions: the first transition being from the convection-dominated regime to the transi…
We present numerical simulations of vortices that appear via primary bifurcations out of the unstructured circular Couette flow in the Taylor-Couette system with counter-rotating as well as with co-rotating cylinders. The full, time dependent Navier-Stokes equations are solved with a combination of a finite difference and a Galerkin method for a fixed axial periodicity length of the vortex patterns and for a finite system of aspect ratio 12 with rigid nonrotating ends in a setup with radius ratio eta=0.5. Differences in structure, dynamics, symmetry properties, bifurcation and stability behavi…
In shear flows turbulence first occurs in the form of localized structures (puffs/spots) surrounded by laminar fluid. We here investigate such spatially intermittent flows in a pipe experiment showing that turbulent puffs have a well defined interaction distance, which sets the minimum spacing of puffs as well as the maximum observable turbulent fraction. Two methodologies are employed here. Starting from a laminar flow puffs can be created by locally injecting a jet of fluid through the pipe wall. When the perturbation is applied periodically at low frequencies, as expected, a regular sequenc…
This paper is the second part of a two-fold study of mixing, i.e. the formation of layers and upwelling of buoyancy, in axially stratified Taylor--Couette flow, with fixed outer cylinder. In a first paper, we showed that the dynamics of the flow was dominated by coherent structures made of a superposition of nonlinear waves. (Mixed)-ribbons and (mixed)-cross-spirals are generated by interactions between a pair of linearly unstable helical modes of opposite `handedness', and appear to be responsible for the formation of well-mixed layers and sharp density interfaces. In this paper, we show that…
The energy gradient theory is used to study the instability of Taylor-Couette flow between concentric rotating cylinders. This theory has been proposed in our previous works. In our previous studies, the energy gradient theory was demonstrated to be applicable for wall-bounded parallel flows. It was found that the critical value of the energy gradient parameter Kmax at turbulent transition is about 370-389 for wall-bounded parallel flows (which include plane Poiseuille flow, pipe Poiseuille flow and plane Couette flow) below which no turbulence occurs. In this paper, the detailed derivation fo…
We perform direct numerical simulations of wall sheared Rayleigh-Bénard (RB) convection for Rayleigh numbers up to $Ra=10^8$, Prandtl number unity, and wall shear Reynolds numbers up to $Re_w=10000$. Using the Monin-Obukhov length $L_{MO}$ we identify three different flow states, a buoyancy dominated regime ($L_{MO} \lesssim λ_θ$; with $λ_θ$ the thermal boundary layer thickness), a transitional regime ($0.5H \gtrsim L_{MO} \gtrsim λ_θ$; with $H$ the height of the domain), and a shear dominated regime ($L_{MO} \gtrsim 0.5H$). In the buoyancy dominated regime the flow dynamics are similar to tha…
In thermal convection for very large Rayleigh numbers ($Ra$), the thermal and viscous boundary layers (BL) undergo a transition from a classical state to an ultimate state. In the former state, the BL thicknesses follow a laminar-like Prandtl-Blasius-Polhausen scaling, whereas in the latter, the BLs are turbulent with log-corrections in the sense of Prandtl and von Kármán. Here, we report evidence of this transition via changes in the BL structure of vertical natural convection (VC), which is a buoyancy driven flow between differentially heated vertical walls. The dataset spans $Ra$-values fro…
The influence of an axial through-flow on the spatiotemporal growth behavior of different vortex structures in the Taylor-Couette system with radius ratio eta=0.5 is determined. The Navier Stokes equations (NSE) linearized around the basic Couette-Poiseuille flow are solved numerically with a shooting method in a wide range of through-flow strengths Re and different rates of co- and counterrotating cylinders for toroidally closed vortices with azimuthal wave number m=0 and for spiral vortex flow with m=+1 and m=-1. For each of these three different vortex varieties we have investigated (i) axi…
Despite the presence of strong fluctuations, many turbulent systems such as Rayleigh-Bénard convection and Taylor-Couette flow display self-organized large-scale flow patterns. How do small-scale turbulent fluctuations impact the emergence and stability of such large-scale flow patterns? Here, we approach this question conceptually by investigating a class of pattern forming systems in the presence of random advection by a Kraichnan-Kazantsev velocity field. Combining tools from pattern formation with statistical theory and simulations, we show that random advection shifts the onset and the wa…
We present results from a numerical study of particle dispersion in the weakly nonlinear regime of Rayleigh-Bénard convection of a fluid with Prandtl number around unity, where bi-stability between ideal straight convection rolls and weak turbulence in the form of Spiral Defect Chaos exists. While Lagrangian pair statistics has become a common tool for studying fully developed turbulent flows at high Reynolds numbers, we show that key characteristics of mass transport can also be found in convection systems that show no or weak turbulence. Specifically, for short times, we find Richardson-like…
We present a new hybrid discrete exterior calculus (DEC) and finite difference (FD) method to simulate fully three-dimensional Boussinesq convection in spherical shells subject to internal heating and basal heating, relevant in the planetary and stellar phenomenon. We employ DEC to compute the surface spherical flows, taking advantage of its unique features of structure preservation (e.g., conservation of secondary quantities like kinetic energy) and coordinate system independence, while we discretize the radial direction using FD method. The grid employed for this novel method is free of prob…
The recently introduced consistent lattice Boltzmann model with energy conservation [S. Ansumali, I.V. Karlin, Phys. Rev. Lett. 95, 260605 (2005)] is extended to the simulation of thermal flows on standard lattices. The two-dimensional thermal model on the standard square lattice with nine velocities is developed and validated in the thermal Couette and Rayleigh-Bénard natural convection problems.
A horizontal flow of two immiscible fluid layers with different densities, viscosities and thicknesses, subject to vertical gravitational forces and with an insoluble surfactant present at the interface, is investigated. The base Couette flow is driven by the horizontal motion of the channel walls. Linear and nonlinear stages of the (inertialess) surfactant and gravity dependent long-wave instability are studied using the lubrication approximation, which leads to a system of coupled nonlinear evolution equations for the interface and surfactant disturbances. The linear stability is determined …
A novel flow state consisting of two oppositely travelling waves (TWs) with oscillating amplitudes has been found in the counterrotating Taylor-Couette system by full numerical simulations. This structure bifurcates out of axially standing waves that are nonlinear superpositions of left and right handed spiral vortex waves with equal time-independent amplitudes. Beyond a critical driving the two spiral TW modes start to oscillate in counterphase due to a Hopf bifurcation. The trigger for this bifurcation is provided by a nonlinearly excited mode of different symmetry than the spiral TWs. A thr…
The effect of an externally applied flow on symmetry degenerated waves propagating into opposite directions and standing waves that exchange stability with the traveling waves via mixed states is analyzed. Wave structures that consist of spiral vortices in the counter rotating Taylor-Couette system are investigated by full numerical simulations and explained quantitatively by amplitude equations containing quintic coupling terms. The latter are appropriate to describe the influence of inversion symmetry breaking perturbations on many oscillatory instabilities with O(2) symmetry.
This study considers the onset of stress-free Boussinesq thermal convection in rotating spherical shells with aspect ratio $η=r_i/r_o=0.9$ ($r_i$ and $r_o$ being the inner and outer radius), Prandtl numbers ${\rm Pr} \in[10^{-4},10^{-1}]$, and Taylor numbers ${\rm Ta}\in[10^{4},10^{12}]$. We are particularly interested in the form of the convective cell pattern that develops, and in its time scales, since this may have observational consequences. For a fixed ${\rm Ta}<10^{9}$ and by decreasing ${\rm Pr}$ from 0.1 to $10^{-4}$ a transition between spiralling columnar (SC) and equatorially-attac…
In this paper, a thermal cascaded lattice Boltzmann method (TCLBM) is developed in combination with the double-distribution-function (DDF) approach. A density distribution function relaxed by the cascaded scheme is employed to solve the flow field, and a total energy distribution function relaxed by the BGK scheme is used to solve temperature field, where two distribution functions are coupled naturally. The forcing terms are incorporated by means of central moments, which is consistent with the previous force scheme [Premnath \emph{et al.}, Phys. Rev. E \textbf{80}, 036702 (2009)] but the der…
In this paper, a coupling lattice Boltzmann (LB) model for simulating thermal flows on the standard D2Q9 lattice is developed in the framework of the double-distribution-function (DDF) approach in which the viscous heat dissipation and compression work are considered. In the model, a density distribution function is used to simulate the flow field, while a total energy distribution function is employed to simulate the temperature field. The discrete equilibrium density and total energy distribution functions are obtained from the Hermite expansions of the corresponding continuous equilibrium d…
The study focuses on the 3D electro-hydrodynamic (EHD) instability for flow between to parallel electrodes with unipolar charge injection with and without cross-flow. Lattice Boltzmann Method (LBM) with two-relaxation time (TRT) model is used to study flow pattern. In the absence of cross-flow, the base-state solution is hydrostatic, and the electric field is one-dimensional. With strong charge injection and high electrical Rayleigh number, the system exhibits electro-convective vortices. Disturbed by different perturbation patterns, such as rolling pattern, square pattern, and hexagon pattern…
