zbMATH — the first resource for mathematics

Complex dynamics in a stratified lid-driven square cavity flow. (English) Zbl 1415.76216
Summary: The dynamic response to shear of a fluid-filled square cavity with stable temperature stratification is investigated numerically. The shear is imposed by the constant translation of the top lid, and is quantified by the associated Reynolds number. The stratification, quantified by a Richardson number, is imposed by maintaining the temperature of the top lid at a higher constant temperature than that of the bottom, and the side walls are insulating. The Navier-Stokes equations under the Boussinesq approximation are solved, using a pseudospectral approximation, over a wide range of Reynolds and Richardson numbers. Particular attention is paid to the dynamical mechanisms associated with the onset of instability of steady state solutions, and to the complex and rich dynamics occurring beyond.

76D50 Stratification effects in viscous fluids
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
76D33 Waves for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI
[1] Ansong, J. K.; Sutherland, B. R., Internal gravity waves generated by convective plumes, J. Fluid Mech., 648, 405-434, (2010) · Zbl 1189.76012
[2] Auteri, F.; Parolini, N.; Quartapelle, L., Numerical investigation of the stability of singular driven cavity flow, J. Comput. Phys., 183, 1-25, (2002) · Zbl 1021.76040
[3] Blanchette, F.; Peacock, T.; Cousin, R., Stability of a stratified fluid with a vertically moving sidewall, J. Fluid Mech., 609, 305-317, (2008) · Zbl 1147.76032
[4] Boyland, P. L., Bifurcations of circle maps: Arnol’d tongues, bistability and rotation intervals, Commun. Math. Phys., 106, 353-381, (1986) · Zbl 0612.58032
[5] Brezillon, A.; Girault, G.; Cadou, J. M., A numerical algorithm coupling a bifurcating indicator and a direct method for the computation of Hopf bifurcation points in fluid mechanics, Comput. Fluids, 39, 1226-1240, (2010) · Zbl 1242.76113
[6] Cheng, T. S.; Liu, W.-H., Effect of temperature gradient orientation on the characteristics of mixed convection flow in a lid-driven square cavity, Comput. Fluids, 39, 965-978, (2010) · Zbl 1242.76313
[7] Cohen, N.; Eidelman, A.; Elperin, T.; Kleerin, N.; Rogachevskii, I., Sheared stably stratified turbulence and large-scale waves in a lid driven cavity, Phys. Fluids, 26, (2014)
[8] Dohan, K.; Sutherland, B. R., Internal waves generated from a turbulent mixed region, Phys. Fluids, 15, 488-498, (2003) · Zbl 1185.76112
[9] Fernando, H. J. S., Turbulent mixing in stratified fluids, Annu. Rev. Fluid Mech., 23, 455-493, (1991)
[10] Fortin, A.; Jardak, M.; Gervais, J. J.; Pierre, R., Localization of Hopf bifurcations in fluid flow problems, Intl J. Numer. Meth. Fluids, 24, 1185-1210, (1997) · Zbl 0886.76042
[11] Hanratty, T. J.; Rosen, E. M.; Kabel, R. L., Effect of heat transfer on flow field at low Reynolds numbers in vertical tubes, Ind. Engng Chem., 50, 815-818, (1958)
[12] Hugues, S.; Randriamampianina, A., An improved projection scheme applied to pseudospectral methods for the incompressible Navier-Stokes equations, Intl J. Numer. Meth. Fluids, 28, 501-521, (1998) · Zbl 0932.76065
[13] Ivey, G. N.; Winters, K. B.; Koseff, J. R., Density stratification, turbulence, but how much mixing?, Annu. Rev. Fluid Mech., 40, 169-184, (2008) · Zbl 1136.76026
[14] Iwatsu, R.; Hyun, J. H.; Kuwahara, K., Mixed convection in a driven cavity with a stable vertical temperature gradient, Intl J. Heat Mass Transfer, 36, 1601-1608, (1993)
[15] Ji, T. H.; Kim, S. Y.; Hyun, J. M., Transient mixed convection in an enclosure driven by a sliding lid, Heat Mass Transfer, 43, 629-638, (2007)
[16] Knobloch, E.; Proctor, M. R. E., The double Hopf bifurcation with 2 : 1 resonance, Proc. R. Soc. Lond. A, 415, 61-90, (1988) · Zbl 0646.34042
[17] Koseff, J. R.; Street, R. L., Circulation structure in a stratified cavity flow, J. Hydraul. Engng, 111, 334-354, (1985)
[18] Kuznetsov, Y. A., Elements of Applied Bifurcation Theory, (2004), Springer · Zbl 1082.37002
[19] Lopez, J. M.; Marques, F., Centrifugal effects in rotating convection: nonlinear dynamics, J. Fluid Mech., 628, 269-297, (2009) · Zbl 1181.76064
[20] 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)
[21] Maiti, M. K.; Gupta, A. S.; Bhattacharyya, S., Stable/unstable stratification in thermosolutal convection in a square cavity, Trans. ASME J. Heat Transfer, 130, (2008)
[22] Majda, A. J.; Shefter, M. G., Elementary stratified flows with instability at large Richardson number, J. Fluid Mech., 376, 319-350, (1998) · Zbl 0936.76024
[23] Majda, A. J.; Shefter, M. G., Nonlinear instability of elementary stratified flows at large Richardson number, Chaos, 10, 3-27, (2000) · Zbl 0983.76022
[24] Michaelian, M. E.; Maxworthy, T.; Redekopp, L. G., The coupling between turbulent, penetrative convection and internal waves, Eur. J. Mech. (B/Fluids), 21, 1-28, (2002) · Zbl 0995.76502
[25] Munroe, J. R.; Sutherland, B. R., Internal wave energy radiated from a turbulent mixed layer, Phys. Fluids, 26, (2014)
[26] Nicolás, A.; Bermúdez, B., 2D thermal/isothermal incompressible viscous flows, Intl J. Numer. Meth. Fluids, 48, 349-366, (2005) · Zbl 1068.76064
[27] Peltier, W. R.; Caulfield, C. P., Mixing efficiency in stratified shear flows, Annu. Rev. Fluid Mech., 35, 135-167, (2003) · Zbl 1041.76024
[28] Pham, H. T.; Sarkar, S.; Brucker, K. A., Dynamics of a stratified shear layer above a region of uniform stratification, J. Fluid Mech., 630, 191-223, (2009) · Zbl 1181.76079
[29] Poliashenko, M.; Aidun, C. K., A direct method for computation of simple bifurcations, J. Comput. Phys., 121, 246-260, (1995) · Zbl 0840.76076
[30] 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
[31] Stevens, C.; Imberger, J., The initial response of a stratified lake to a surface shear stress, J. Fluid Mech., 312, 39-66, (1996)
[32] Turner, J. S., Buoyancy Effects in Fluids, (1979), Cambridge University Press · Zbl 0443.76091
[33] Vanel, J.-M.; Peyret, R.; Bontoux, P.; Morton, K. W.; Baines, M. J., A pseudo-spectral solution of vorticity-stream function equations using the influence matrix technique, Numerical Methods for Fluid Dynamics II, 463-475, (1986), Clarendon Press · Zbl 0606.76030
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.