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.


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


[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.; Sim√≥, 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.