Spherical Couette Flow Research Articles

Spherical Couette flow in a dipolar magnetic field

We consider numerically the flow of an electrically conducting fluid in a differentially rotating spherical shell, in a dipolar magnetic field. For infinitesimal differential rotation the flow consists of a super-rotating region, concentrated on the particular field line C just touching the outer sphere, in agreement with previous results. Finite differential rotation suppresses this super-rotation, and pushes it inward, toward the equator of the inner sphere. For sufficiently strong differential rotation the outer boundary layer becomes unstable, yielding time-dependent solutions. Adding an overall rotation suppresses these instabilities again. The results are in qualitative agreement with the DTS liquid sodium experiment.

Pulse-train solutions of a spatially heterogeneous amplitude equation arising in the subcritical instability of narrow gap spherical Couette flow

We investigate some complex solutions a ( x , t ) of the heterogeneous complex-Ginzburg–Landau equation ∂ a / ∂ t = [ λ ( x ) + i x − | a | 2 ] a + ∂ 2 a / ∂ x 2 , in which the real driving coefficient λ ( x ) is either constant or the quadratic λ ( 0 ) − Υ ε 2 x 2 . This CGL equation arises in the weakly nonlinear theory of spherical Couette flow between two concentric spheres caused by rotating them both about a common axis with distinct angular velocities in the narrow gap (aspect ratio ε ) limit. Roughly square Taylor vortices are confined near the equator, where their amplitude modulation a ( x , t ) varies with a suitably ‘stretched’ latitude x . The value of Υ ε , which depends on sphere angular velocity ratio, generally tends to zero with ε . Though we report new solutions for Υ ε ≠ 0 , our main focus is the physically more interesting limit Υ ε = 0 . When λ = constant , uniformly bounded solutions of our CGL equation on − ∞ < x < ∞ have some remarkable related features, which occur at all values of λ . Firstly, the linearised equation has no non-trivial neutral modes a ¯ ( x ) exp ( i Ω t ) with any real frequency Ω including zero. Secondly, all evidence indicates that there are no steady solutions a ¯ ( x ) of the nonlinear equation either. Nevertheless, Bassom and Soward [A.P. Bassom, A.M. Soward, On finite amplitude subcritical instability in narrow-gap spherical Couette flow, J. Fluid. Mech. 499 (2004) 277–314. Referred to as BS] identified oscillatory finite amplitude solutions, a ( x , t ) = ∑ n ∈ Z a ¯ ( x − x n ) exp { i [ ( 2 n + 1 ) Ω t + ϕ n ] } , expressed in terms of the single complex amplitude a ¯ ( x ) , which is localised as a pulse on the length scale L P S = 2 Ω about x = 0 . Each pulse-amplitude a ¯ ( x − x n ) exp ( i ϕ n ) is identical up to the phase ϕ n = ( − 1 ) n π / 4 , is centred at x n = ( n + 1 2 ) L P S and oscillates at frequency ( 2 n + 1 ) Ω . The survival of the pulse-train depends upon the nonlinear mutual interaction of close neighbours; self-interaction is inadequate, as the absence of steady solutions shows. For given constant values of λ in excess of some threshold λ MIN ( > 0 ) , solutions with pulse-separation L P S were located on a finite range L min ( λ ) ≤ L P S ≤ L max ( λ ) . Here, we seek new pulse-train solutions, for which the product a ( x , t ) exp ( − i x t ) is spatially periodic on the length 2 L = N L P S , N ∈ N . The BS-mode at small λ has N = 2 , and on increasing λ it bifurcates to another symmetry-broken N = 2 solution. Other bifurcations to N = 6 were located. Solution branches with N odd, namely 3, 5, 7, were only found after solving initial value problems. Many of the large amplitude solutions are stable. Generally, the BS-mode is preferred at moderate λ , while that preference yields to the other symmetry-broken N = 2 solution at larger λ . Quasi-periodic solutions are also common. We conclude that finite amplitude solutions, not necessarily of BS-form, are robust in the sense that they persist and do not evaporate.

On the effect of mantle conductivity on the super-rotating jets near the liquid core surface

We consider hydromagnetic Couette flows in planar and spherical geometries with strong magnetic field (large Hartmann number, M ≫ 1 ). The highly conducting bottom boundary is in steady motion that drives the flow. The top boundary is stationary and is either a highly conducting thin shell or a weakly conducting thick mantle. The magnetic field, B 0 + b , is a combination of the strong, force-free background B 0 and a perturbation b induced by the flow. This perturbation generates strong streamwise electromagnetic stress inside the fluid which, in some regions, forms a jet moving faster than the driving boundary. The super-velocity, in the spherical geometry called super-rotation, is particularly prominent in the region where the ‘grazing’ line of B 0 has a point of tangent contact with the top boundary and where the Hartmann layer is singular. This is a consequence of topological discontinuity across that special field line. We explain why the magnitude of super-rotation already present when the top wall is insulating [Dormy, E., Jault, D., Soward, A.M., 2002. A super-rotating shear layer in magnetohydrodynamic spherical Couette flow. J. Fluid Mech. 452, 263–291], considerably increases when that wall is even slightly conducting. The asymptotic theory is valid when either the thickness of the top wall is small, δ ∼ M − 1 and its conductivity is high, ɛ ∼ 1 or when δ ∼ 1 and ɛ ∼ M − 1 . The theory predicts the super-velocity enhancement of the order of δ M 3 / 4 in the first case and ɛ M 3 / 4 in the second case. We also numerically solve the planar problem outside the asymptotic regime, for ɛ = 1 and δ = 1 , and find that with the particular B 0 that we chose the peak super-velocity scales like M 0.33 . This scaling is different from M 0.6 found in spherical geometry [Hollerbach, R., Skinner, S., 2001. Instabilities of magnetically induced shear layers and jets. Proc. R. Soc. Lond. A 457, 785–802].

Driven inertial waves in spherical Couette flow

Driven inertial waves in spherical Couette flow

Gravitational Radiation from Nonaxisymmetric Spherical Couette Flow in a Neutron Star

The gravitational wave signal generated by global, nonaxisymmetric shear flows in a neutron star is calculated numerically by integrating the incompressible Navier--Stokes equation in a spherical, differentially rotating shell. At Reynolds numbers $\Rey \gsim 3 \times 10^{3}$, the laminar Stokes flow is unstable and helical, oscillating Taylor--G\"ortler vortices develop. The gravitational wave strain generated by the resulting kinetic-energy fluctuations is computed in both $+$ and $\times$ polarizations as a function of time. It is found that the signal-to-noise ratio for a coherent, $10^{8}$-{\rm s} integration with LIGO II scales as $ 6.5 (\Omega_*/10^{4} {\rm rad} {\rm s}^{-1})^{7/2}$ for a star at 1 {\rm kpc} with angular velocity $\Omega_*$. This should be regarded as a lower limit: it excludes pressure fluctuations, herringbone flows, Stuart vortices, and fully developed turbulence (for $\Rey \gsim 10^{6}$).

Spherical Couette flow of Oldroyd 8-constant model Part I. Solution up to the second-order approximation

The steady flow of an incompressible Oldroyd 8-constant fluid in the annular region between two spheres, or so-called spherical Couette flow, is investigated. The inner sphere rotates with anangular velocity Ω about the z-axis, which passes through the center of the spheres, while the outer sphere is kept at rest. The viscoelasticity of the fluid is assumed to dominate the inertia such that the latter can be neglected in the momentum equation. An analytical solution is obtained through the expansion of the dynamical variables in a power series of the dimensionless retardation time. The leading velocity term denotes the Newtonian rotation about the z-axis. The first-order term results in a secondary flow represented by the stream function that divides the flow region into four symmetric parts. The second-order term is the viscoelastic contribution to the primary viscous flow. The first-order approximation depends on the viscosity and four of the material time-constants of the fluid. The second-order approximation depends on the eight viscometric parameters of the fluid. The torque acting on the outer sphere has an additional term due to viscoelasticity that depends on all the material parameters of the fluid under consideration. For an Oldroyd-B fluid this contributed term enhances the primary torque but in the case of fluids with higher elasticity the torque components may be enhanced or diminished depending on the values of the viscometric parameters.PACS Nos.: 47.15.Rq

Non-axisymmetric instabilities in basic state spherical Couette flow

We consider the flow induced in a spherical shell by fixing the outer sphere and rotating the inner one, with the aspect ratio β = ( r o - r i ) / r i ranging from 0.1 to 10. The basic state consists of a jet in the equatorial plane, carrying fluid from the inner sphere to the outer, and involving also an azimuthal component. The azimuthal component dominates for β ≲ 1 , the radial component for β ≳ 1 . The basic state is otherwise much the same over the entire range 0.1 ⩽ β ⩽ 10 . We next linearize the Navier–Stokes equation about this basic state, and compute the linear onset of non-axisymmetric instabilities. For 0.1 ⩽ β ⩽ 3.8 the instabilities have the opposite equatorial symmetry as the basic state, and consist of a series of waves on the initially flat radial-azimuthal jet. The azimuthal wavenumber decreases monotonically from m = 12 at β = 0.1 to m = 2 at β = 3.8 , but with a puzzling transition between β = 0.27 and 0.28, where one m = 6 mode is replaced by another, very similar one. For 3.8 ⩽ β ⩽ 10 we obtain an m = 2 mode having the same equatorial symmetry as the basic state. This instability is further differentiated from the previous ones in that it consists of a modulation in the strength of the return flow after the jet has reached the outer sphere, rather than an instability of the jet itself. Finally, we solve the fully three-dimensional Navier–Stokes equation, and consider the equilibration of some of these modes in the supercritical regime. For β = 0.5 we can achieve Re = 1.15 Re c , and obtain results in excellent agreement with experiments. For β = 0.8 , 1.5 and 2.5 we can achieve up to 1.9 Re c , and find that in all three cases secondary bifurcations occur, in which the solutions develop a time-dependence more complicated than a simple drift in φ . The precise nature of the bifurcation is different in the three cases.

Experimental study of super-rotation in a magnetostrophic spherical Couette flow

We report measurements of electric potentials at the surface of a spherical container of liquid sodium in which a magnetized inner core is differentially rotating. The azimuthal angular velocities inferred from these potentials reveal a strong super-rotation of the liquid sodium in the equatorial region, for small differential rotation. Super-rotation was observed in numerical simulations by Dormy et al. (Dormy, E., Cardin, P. and Jault, D., MHD flow in a slightly differentially rotating spherical shell, with conducting inner core, in a dipolar magnetic field, Earth Planet. Sci. Lett., 1998, 160, 15--30). We find that the latitudinal variation of the electric potentials in our experiments differs markedly from the predictions of a similar numerical model, suggesting that some of the assumptions used in the model – steadiness, equatorial symmetry, and linear treatment for the evolution of both the magnetic and velocity fields – are violated in the experiments. In addition, radial velocity measurements, using ultrasonic Doppler velocimetry, provide evidence of oscillatory motion near the outer sphere at low latitude: it is viewed as the signature of an instability of the super-rotating region.

Transition phenomena in a spherical Couette flow between independently rotating spheres

The quasistatic laminar-turbulent transition of spherical Couette flow between two independently rotating concentric spheres for a gap ratio (ratio of the gap between two spheres to the inner-sphere radius) of 0.14 for which the Taylor instability occurs in the case of the inner sphere rotating and the outer sphere fixed has been investigated by simultaneous spectral and flow-visualization measurements. Rotation frequencies (phase velocities), scaled by the mean rotation frequency of both spheres, of various periodic disturbances do not depend much on the ratio Ro*∕Ri* of the outer-sphere rotating Reynolds number Ro* to the inner-sphere rotating Reynolds number Ri*, although the phase velocity differs by the kind of disturbance. As disturbances unobserved in the case of the fixed outer sphere, traveling azimuthal waves only on an inflow boundary of toroidal Taylor-Görtler (TG) vortices and herringbone waves are observed in the case of co-rotating spheres, and spiral vortices and twists in toroidal TG vortices are observed in the case of counter-rotating spheres. Kinds and occurrence regions of flow regimes in the Ro*-Ri* plane considerably differ by the history of Ro* and Ri*.

Relaminarization phenomena and external-disturbance effects in spherical Couette flow

The relaminarization phenomenon in the laminar turbulent transition process of spherical Couette flow with the inner sphere rotating and the outer sphere fixed has been experimentally investigated for 0.06≤β≤0.206, where β is the ratio of the clearance to the inner-sphere radius. The relaminarization occurs for 0.13 < β < 0.17, and is observed as a reverse Hopf bifurcation from the limit cycle to the fixed point. The kinetic energy in the high-frequency region of the fluctuating azimuthal velocity component continues to increase with Reynolds number Re in the case without relaminarization, but in the case with relaminarization it first increases and then decreases with increasing Re and finally vanishes with the onset of the relaminarization. For β=0.14 with relaminarization, a small artificial disturbance introduced externally into the spherical Couette flow has no influence on the lowest critical Reynolds number of the first instability that can be defined universally and uniquely

Numerical simulation of spherical Couette flow

A spectral method is used to study numerically flow between two concentric rotating spheres. Analytical expressions of eigenfunctions of a Stokes operator in the spherical gap region are presented, numerical experiments are carried out by applying these expressions to the spectral approximation of spherical Couette flow, and numerical results are given.

Taylor vortices in annular spherical flow at large aspect ratios

Motivated by recent theoretical and experimental work, we numerically investigate spherical Couette flow with a view to obtaining for the first time Taylor vortices at large aspect ratios σ such as 0.38, 0.42, and 0.48. It is found that Taylor vortices can exist, stable or time-dependent, in a range of Reynolds numbers [Re1, Re2] and their formation depends on the aspect ratio, on the imposition of various rotationary conditions on the boundaries, on the history of the flow and on the rate at which energy is transferred into the fluid to its final value. With increasing σ the range [Re1, Re2] manifests a clear tendency to shorten.

On finite-amplitude subcritical instability in narrow-gap spherical Couette flow

We consider the finite-amplitude instability of incompressible spherical Couette flow between two concentric spheres of radii longer than the wave period. In view of the several length and time scales involved only qualitative comparison with experimental results is feasible. Nevertheless, the pulse-train structure is robust and likely to provide the building block of the ensuing complex dynamics.

Numerical study of multiple periodic flow states in spherical Couette flow

The supercritical flow states of the spherical Couette flow between two concentric spheres with the inner sphere rotating are investigated via direct numerical simulation using a three-dimensional finite difference method. For comparison with experiments of Nakabayashi et al. and Wimmer, a narrow gap and a medium gap with clearance ratio β=0.06 and 0.18 respectively are considered for the Reynolds number range covering the first Hopf bifurcation point. With adequate initial conditions and temporary imposition of small wave-type perturbation, multiple periodic flow states with three different pair numbers of spiral Taylor-Gortler (TG) vortices have been simulated successfully for β=0.06, of which the 1-pair and 2-pair of spiral TG vortices are newly obtained. Three different periodic flow states with shear waves, Stuart vortices or wavy outflow boundary, have been obtained for β=0.18. Analysis of the numerical results reveals these higher flow modes in terms of fundamental frequency, wave number and spatial structure.

A steady-state solver for the simulation of Taylor vortices in spherical annular flow

A numerical technique is described by which the solution of the equations of steady axisymmetric flow of a viscous incompressible fluid in any orthogonal system of coordinates can be obtained. In the present analysis the equations of motion and the boundary conditions are approximated by finite differences that are of the second-order accuracy with respect to the grid size. This technique is used for the numerical solution of the equations of steady motion in spherical Couette flow, in order to obtain Taylor vortices symmetric and asymmetric at a prescribed medium and large annular gap and given range of the Reynolds number of the flow. The computed results are compared to results available from previous experimental and theoretical studies.

Experimental investigation of intermittency modes in a spherical Couette flow

Experimental investigation of intermittency modes in a spherical Couette flow

Global bifurcation to travelling waves with application to narrow gap spherical Couette flow

In a previous paper [Physica D 137 (2000) 260], an inhomogeneous complex Landau equation was derived in the context of the amplitude modulation of Taylor vortices between two rapidly rotating concentric spheres, which bound a narrow gap and almost co-rotate about a common axis of symmetry. In this weakly nonlinear regime the latitudinal vortex width is comparable to the gap between the shells. The vortices are located close to the equator and are modulated on a latitudinal length scale large compared to the gap width but small compared to the shell radius. The system is characterised by two parameters: λ, which is proportional to the Taylor number, and κ, which provides a measure of phase mixing. Only when the inner and outer spheres almost co-rotate is κ of order unity; otherwise κ is large. In [Physica D 137 (2000) 260], it was shown that there is a finite amplitude steady solution branch at fixed κ that connects the first two bifurcation points λ 0 and λ 1. For sufficiently large κ, the branch lies on λ 0≤ λ≤ λ 1, but for smaller κ it extends beyond λ 1; there on λ 1< λ< λ N a large and small amplitude solution co-exist and coalesce at the nose of the branch, λ N. In this paper we investigate both analytically and numerically the stability of the steady solutions and their subsequent evolution. Two types of modes exist—one (SP) preserves the reflectional symmetry of the steady solutions with respect to the equatorial plane, while the other (SB) breaks it. Three SP-global bifurcation scenarios are identified. Each lead to limit cycles, which correspond to vortices drifting towards the equator from both sides. For small κ, a heteroclinic connection is made between the steady nose solution and its reversed flow state (opposite sign). The same occurs for moderate κ except that two oppositely signed small amplitude steady solutions are connected. For large κ a homoclinic cycle forms joining the undisturbed state to itself and this leads to a gluing bifurcation. This homoclinic cycle evolves from the vacillating wave limit cycle shed by an SP-Hopf bifurcation of the large amplitude solution. An SB-pitchfork bifurcation of the steady solutions leads to asymmetric drifting-phase solutions (travelling waves). Time-stepping reveals that they are generally unstable and evolve into larger amplitude periodic solutions, which for small and moderate κ are SP-states. For large κ, the SB-drifting-phase solutions are strongly subcritical. The realised larger amplitude periodic state, to which they evolve, may be either of SB- or SP-type depending on the value of λ. These complicated solutions consist of trains of stationary pulses each modulating travelling waves with distinct frequencies. The asymmetric SB-waves correspond to vortices drifting across the equator; yet far from it, where these vortices are very weak, they drift towards the equator as in the case of the SP-waves.

Flow Bifurcations in a Thin Gap Between Two Rotating Spheres

This paper presents the use of a parameter continuation method and a test function to solve the steady, axisymmetric incompressible Navier–Stokes equations for spherical Couette flow in a thin gap between two concentric, differentially rotating spheres. The study focuses principally on the prediction of multiple steady flow patterns and the construction of bifurcation diagrams. Linear stability analysis is conducted to determine whether or not the computed steady flow solutions are stable. In the case of a rotating inner sphere and a stationary outer sphere, a new unstable solution branch with two asymmetric vortex pairs is identified near the point of a symmetry-breaking pitchfork bifurcation which occurs at a Reynolds number equal to 789. This solution transforms smoothly into an unstable asymmetric 1-vortex solution as the Reynolds number increases. Another new pair of unstable 2-vortex flow modes whose solution branches are unconnected to previously known branches is calculated by the present two-parameter continuation method. In the case of two rotating spheres, the range of existence in the (Re1, Re2) plane of the one and two vortex states, the vortex sizes as a function of both Reynolds numbers are identified. Bifurcation theory is used to discuss the origin of the calculated flow modes. Parameter continuation indicates that the stable states are accompanied by certain unstable states.

Characteristics of disturbances in the laminar–turbulent transition of spherical Couette flow. 2. New disturbances observed for a medium gap

Supercritical flow in spherical Couette system with a rotating inner sphere and a fixed outer sphere has been investigated for a medium gap of the clearance ratio 0.206 in terms of the structure, wave-number, fundamental, and rotational frequencies of disturbances. Consequently, ring vortices and traveling letter-x-like waves were found within the Ekman-type secondary flow as new disturbances, in addition to Stuart vortices and shear waves. Also, interior waves were found within toroidal Taylor–Görtler (TG) vortices as a new disturbance, in addition to twists. Further, the wavy outflow boundary at the equator of the turbulent basic flow was found. Each ring vortex occurring successively around the pole moves toward the pole for one wavelength in the meridian direction (θ=10°) while it rotates for one wavelength in the azimuthal direction (φ=90°), because it has four traveling waves. The rotational direction of Stuart vortices is opposite to that of the inner sphere. The wave speed (rotation frequency) of interior waves or twists is the same as that of oscillating inflow boundary between two toroidal TG vortex cells, and the wave speed of shear waves is the same as that of the oscillating outflow boundary between the toroidal TG vortex cell and the Ekman-type secondary flow. Also, the wave speed of the oscillating outflow boundary is about two times the oscillating inflow boundary.

Characteristics of disturbances in the laminar–turbulent transition of spherical Couette flow. 1. Spiral Taylor–Görtler vortices and traveling waves for narrow gaps

The fundamental frequencies of velocity fluctuations caused by disturbances (vortices and waves) in the laminar–turbulent transition of spherical Couette flow between two concentric spheres with only the inner sphere rotating are investigated by simultaneous and flow-visualization measurements for two small ratios, 0.06 and 0.024, of the clearance to inner-sphere radius for which spiral Taylor–Görtler (TG) vortices and traveling waves on TG vortices play an important role. As the Reynolds number Re is increased for both clearance ratios, first toroidal TG vortices occur at the equator, but the flow is steady. Second, spiral TG vortices (fundamental frequency fSi) occur next to the toroidal ones, so that the flow becomes singly periodic. Third, in addition to these spiral TG vortices, new spiral TG vortices occur between them at the different colatitude θ, so that the pair number of spiral TG vortices around the spherical annulus differs by θ. Therefore, plural fundamental frequencies fSi (i=1,2,…) of spiral TG vortices are detected, corresponding to the different pair numbers. However, the flow is not quasi-periodic, because ratios of the different frequencies fSi are in rational. Fourth, traveling waves (fundamental frequency fW) occur on TG vortices, so that the flow becomes quasi-periodic (fSi and fW). Fifth, the plural frequencies fSi change to a broadband frequency fB, which indicates the onset of chaos. Sixth, fW disappears, so that the flow becomes chaotic spiral TG vortex-flow with only fB. The rotation frequency (wave speed) of traveling waves is larger than that of spiral TG vortices, and the value of fB exists about between the two rotation frequencies.