Stratified Taylor–Couette flow: nonlinear dynamics

Abstract

A series of experiments on stratified Taylor–Couette flow in short aspect ratio wide-gap annuli found an intriguing and not-well-understood dynamics: nonlinear coherent structures appearing and disappearing periodically, along with density layering reminiscent of staircase profiles. A detailed numerical study is presented of the nonlinear dynamics near onset of instability in this setting, which explains most of the characteristics found in the experiments. The simulations show that centrifugal instability of the boundary layer on the inner rotating cylinder produces jets of angular momentum forming Taylor cells that are compressed axially due to the strong stratification. These cells are not axisymmetric from the onset, but are in fact two sets of Taylor cells displaced axially that meet in localized azimuthal defect regions where the cells are patched together; the whole structure is a rotating wave with azimuthal wavenumber $m=1$ . The presence of endwalls in this short aspect ratio annulus is critical for the understanding of the dynamics. Their impact cannot be accounted for in idealized axially periodic models. Another key ingredient is the role played by the symmetries of the system. Although the axial reflection symmetry is weakly broken by centrifugal buoyancy effects, following instability there are various branches of solutions corresponding to the different ways the system's symmetries may be broken.

Stably stratified Taylor–Couette flows

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Figure 3( a ) Stratified Taylor–Couette flow with outer cylinder at rest and η∼0.4; experimental shadowgraph and its schematic interpretation (reproduced from [6] with permission of the publisher). ( b ) DNS at Re=800, Fr=0.53, η=0.417 and γ=3 showing contours of axial temperature gradient in an false(θ,zfalse) cylindrical plane at 1% gap distance from the inner cylinder (reproduced from [11] with permission of the publisher). ( c ) Temperature profile of ( b ) at the same radial distance from the inner cylinder and θ=π/2.

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“…The experiments [14–16] on linearly stratified Taylor–Couette flows used salt as the stratifying agent, and parameter values η=0.417 and γ=3. These experiments motivated a DNS study [11] that reproduced these results and shed light on the instability mechanisms and flow structure. The governing equations for the DNS study are (1.1) and (1.2), and the parameter values such as η=0.417, γ=3, Ar=1010 and Fr=0.53 were fixed.…”

“…The Ekman vortices at the endwalls change size to accommodate the different number of cells that exist in the two regions separated by the defect regions, and play a subordinate role in the dynamics.

Figure 5( a – h ) Rotating wave at Re=800, showing contours of angular momentum Γ=rv, axial temperature gradient ∂zT, streamfunction ψ and velocity vectors projected onto a meridional plane for the angles θ=π/2 and 3π/2 in figure 3 b , where the outgoing radial jets are strongest (reproduced from [11] with permission of the publisher). (Online version in colour.

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“…( a – h ) Rotating wave at Re=800, showing contours of angular momentum Γ=rv, axial temperature gradient ∂zT, streamfunction ψ and velocity vectors projected onto a meridional plane for the angles θ=π/2 and 3π/2 in figure 3 b , where the outgoing radial jets are strongest (reproduced from [11] with permission of the publisher). (Online version in colour.…”

“…Another startling phenomena not yet fully understood that occurs in sheared stratified flows is the formation of staircase horizontal layers in the vertical density profile; these are widely observed in many situations [47]. They have been observed in stratified Taylor–Couette flow in the medium gap case [6,11] (figure 5). These structures also appear in stratified plane Couette flows.…”