## Parametrically forced stably stratified cavity flow: complicated nonlinear dynamics near the onset of instability.(English)Zbl 1419.76203

Summary: 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.

### MSC:

 76D50 Stratification effects in viscous fluids 76E17 Interfacial stability and instability in hydrodynamic stability 76E20 Stability and instability of geophysical and astrophysical flows

### Keywords:

parametric instability; stratified flows
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### References:

 [1] Abshagen, J.; Lopez, J. M.; Marques, F.; Pfister, G., Mode competition of rotating waves in reflection-symmetric Taylor-Couette flow, J. Fluid Mech., 540, 269-299, (2005) · Zbl 1082.76041 [2] Benielli, D.; Sommeria, J., Excitation and breaking of internal gravity waves by parametric instability, J. Fluid Mech., 374, 117-144, (1998) · Zbl 0941.76514 [3] Benjamin, T. B.; Ursell, F., The stability of the plane free surface of a liquid in vertical periodic motion, Proc. R. Soc. Lond. A, 225, 505-515, (1954) · Zbl 0057.18801 [4] Bouruet-Aubertot, P.; Sommeria, J.; Staquet, C., Breaking of standing internal gravity waves through two-dimensional instabilities, J. Fluid Mech., 285, 265-301, (1995) · Zbl 0848.76017 [5] Broer, H.; Simó, C.; Vitolo, R., Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: analysis of a resonance bubble, Physica D, 237, 1773-1799, (2008) · Zbl 1165.37017 [6] Dauxois, T.; Joubaud, S.; Odier, P.; Venaille, A., Instabilities of internal gravity wave beams, Annu. Rev. Fluid Mech., 50, 1-28, (2018) [7] Drazin, P. G., On the instability of an internal gravity wave, Proc. R. Soc. Lond. A, 356, 411-432, (1977) · Zbl 0367.76043 [8] Fauve, S.; Kumar, K.; Laroche, C.; Beysens, D.; Garrabos, Y., Parametric instability of a liquid-vapour interface close to the critical point, Phys. Rev. Lett., 68, 3160-3163, (1992) [9] Feigenbaum, M. J., Quantitative universality for a class of nonlinear transformations, J. Stat. Phys., 19, 25-52, (1978) · Zbl 0509.58037 [10] Gaspard, P., Measurement of the instability rate of a far-from-equilibrium steady state at an infinite period bifurcation, J. Phys. Chem., 94, 1-3, (1990) [11] Glendinning, P., Bifurcations near homoclinic orbits with symmetry, Phys. Lett., 103A, 163-166, (1984) [12] Kumar, K.; Tuckerman, L. S., Parametric instability of the interface between two fluids, J. Fluid Mech., 279, 49-68, (1994) · Zbl 0823.76026 [13] Kuznetsov, Y. A., Elements of Applied Bifurcation Theory, (2004), Springer · Zbl 1082.37002 [14] Lopez, J. M.; Marques, F., Dynamics of 3-tori in a periodically forced Navier-Stokes flow, Phys. Rev. Lett., 85, 972-975, (2000) [15] Lopez, J. M.; Marques, F.; Shen, J., Complex dynamics in a short annular cylinder with rotating bottom and inner cylinder, J. Fluid Mech., 501, 327-354, (2004) · Zbl 1071.76064 [16] Lopez, J. M.; Welfert, B. D.; Wu, K.; Yalim, J., Transition to complex dynamics in the cubic lid-driven cavity, Phys. Rev. Fluids, 2, (2017) [17] Marques, F.; Lopez, J. M.; Shen, J., A periodically forced flow displaying symmetry breaking via a three-tori gluing bifurcation and two-tori resonances, Physica D, 156, 81-97, (2001) · Zbl 1028.76010 [18] May, R. M., Simple mathematical models with very complicated dynamics, Nature, 261, 459-467, (1976) · Zbl 1369.37088 [19] McEwan, A. D., Degeneration of resonantly-excited standing internal gravity waves, J. Fluid Mech., 50, 431-448, (1971) · Zbl 0239.76018 [20] McEwan, A. D., The kinematics of stratified mixing through internal wavebreaking, J. Fluid Mech., 128, 47-57, (1983) [21] McEwan, A. D.; Mander, D. W.; Smith, R. K., Forced resonant second-order interaction between damped internal waves, J. Fluid Mech., 55, 589-608, (1972) · Zbl 0253.76020 [22] McEwan, A. D.; Robinson, R. M., Parametric instability of internal gravity waves, J. Fluid Mech., 67, 667-687, (1975) · Zbl 0303.76022 [23] Mercader, I.; Batiste, O.; Alonso, A., An efficient spectral code for incompressible flows in cylindrical geometries, Comput. Fluids, 39, 215-224, (2010) · Zbl 1242.76221 [24] Miles, J.; Henderson, D. M., Parametrically forced surface-waves, Annu. Rev. Fluid Mech., 22, 143-165, (1990) [25] Oldeman, B. E.; Krauskopf, B.; Champneys, A. R., Death of period-doublings: locating the homoclinic-doubling cascade, Physica D, 146, 100-120, (2000) · Zbl 1011.34036 [26] Orlanski, I., On the breaking of standing internal gravity waves, J. Fluid Mech., 54, 577-598, (1972) · Zbl 0247.76018 [27] Orlanski, I., Trapeze instability as a source of internal gravity waves. Part I, J. Atmos. Sci., 30, 1007-1016, (1973) [28] Sherman, F. S.; Imberger, J.; Corcos, G. M., Turbulence and mixing in stably stratified waters, Annu. Rev. Fluid Mech., 10, 267-288, (1978) · Zbl 0405.76023 [29] Smale, S., Differentiable dynamical systems, Bull. Am. Math. Soc., 73, 747-817, (1967) · Zbl 0202.55202 [30] Staquet, C., Gravity and inertia-gravity waves: breaking processes and induced mixing, Surv. Geophys., 25, 281-314, (2004) [31] Thorpe, S. A., On standing internal gravity waves of finite amplitude, J. Fluid Mech., 32, 489-528, (1968) · Zbl 0155.56402 [32] Thorpe, S. A., Observations of parametric instability and breaking waves in an oscillating tilted tube, J. Fluid Mech., 261, 33-45, (1994) [33] Wu, K.; Welfert, B. D.; Lopez, J. M., Complex dynamics in a stratified lid-driven square cavity flow, J. Fluid Mech., 855, 43-66, (2018) · Zbl 1415.76216 [34] Yalim, J.; Lopez, J. M.; Welfert, B. D., Vertically forced stably stratified cavity flow: instabilities of the basic state, J. Fluid Mech., 851, R6, (2018) · Zbl 1415.76113 [35] Yalim, J.; Welfert, B.; Lopez, J.; Wu, K., Fluid flow in a vertically oscillating, stably stratified cubic cavity, 70th Annual Meeting of the APS Division of Fluid Dynamics, (2017) [36] Yalim, J.; Welfert, B.; Lopez, J.; Wu, K., V0066: Resonant collapse in a harmonically forced stratified cavity, 70th Annual Meeting of the APS Division of Fluid Dynamics, (2017) [37] Yih, C.-S., Gravity waves in a stratified fluid, J. Fluid Mech., 8, 481-508, (1960) · Zbl 0094.41003
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