Parametrically forced stably stratified cavity flow: complicated nonlinear dynamics near the onset of instability

Abstract

The dynamics of a fluid-filled square cavity with stable thermal stratification subjected to harmonic vertical oscillations is investigated numerically. The nonlinear responses to this parametric excitation are studied over a comprehensive range of forcing frequencies up to two and a half times the buoyancy frequency. The nonlinear results are in general agreement with the Floquet analysis, indicating the presence of nested resonance tongues corresponding to the intrinsic $m:n$ eigenmodes of the stratified cavity. For the lowest-order subharmonic $1:1$ tongue, the responses are analysed in great detail, with complex dynamics identified near onset, most of which involves interactions with unstable saddle states of a homoclinic or heteroclinic nature.

Parametrically forced stably stratified flow in a three-dimensional rectangular container

“…Space is discretized via spectral collocation with Chebyshev polynomials of degree in barycentric form, and time evolution uses the fractional-step improved projection scheme of Mercader, Batiste & Alonso (2010). We have implemented this method previously in related problems (Lopez et al 2017; Wu, Welfert & Lopez 2018; Yalim et al 2018, 2019 b ). Most of the results presented in this study were obtained with Chebyshev polynomials of degree in the , , and directions and up to 200 time steps per forcing period.…”

“…They had (, , ) and a Schmidt number of order 700 (corresponding to salt in water), which are quite challenging for an extensive numerical parametric study, even in two dimensions. In our 2-D numerical study (Yalim et al 2019 b ), we used and for an extensive study in the forcing frequency and amplitude, considering and . In the present 3-D numerical study, we use the same geometry as Benielli & Sommeria (1998), with , and also consider the response in the 1 : 0 : 1 subharmonic resonance tongue, with and .…”

“…Figure 2 presents bifurcation diagrams from simulations in two dimensions (reproduced from Yalim et al (2019 b )) and in three dimensions with , both at , and . The state measure in both is the variance in the velocity at a point, and .…”

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Figure 2.Bifurcation diagrams at , and for the 2-D study and the 3-D study with . In both panels, the primary subharmonic limit cycles are denoted in red, the symmetric-conjugate limit cycles in blue, quasi-periodic states of two or three frequencies in yellow, and a myriad of complex states, including chaos, in green.

Figure 3.Snapshots of subharmonic limit cycle response flows for , and forcing frequency at forcing amplitude as indicated: ( a ) spanwise vorticity and isotherms of 2-D flows from Yalim et al (2019 b ); ( b ) and the isotherms of 3-D flows in the spanwise midplane , but at different as indicated. The isotherms and are shown a half-forcing-period apart.

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“…The Floquet analysis, however, does not provide any indication as to what the response flow is when the equilibrium is unstable. Yalim, Welfert & Lopez (2019 b ) addressed this by studying the fully nonlinear 2-D problem, uncovering a wealth of complex dynamics. Restricting the simulations to two dimensions allowed for a very detailed exploration of the nonlinear dynamics, particularly near the tip and centre of the broadest resonance tongue, the subharmonic 1 : 1 tongue.…”