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Effect of Prandtl Number on the Dynamics and Stability of Natural Convective Flows inside a Cubical Cavity

Dolors Puigjaner, Fen˛mens de Transport, Departament d'Enginyeria InformÓtica i MatemÓtiques, Universitat Rovira i Virgili, Av. dels Pa´sos Catalans 26, Tarragona, 43007, Spain, Joan Herrero, Fen˛mens de Transport, Departament d'Enginyeria QuÝmica, Universitat Rovira i Virgili, ,Av. dels Pa´sos Catalans 26, Tarragona, 43007, Spain, Carles Simo, MatemÓica Aplicada i AnÓlisis, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007-Barcelona, Catalunya, Spain, and Francesc Giralt, Universitat Rovira i Virgili, Departament d'Enginyeria QuÝmica, Av. dels Pa´sos Catalans 26, Tarragona, 43007, Spain.

Natural convection phenomena of Newtonian fluids heated from below provide appropriate flow models to study the role of flow patterns arising from instabilities and the transition from laminar to turbulent flows. It is well known (1,2) that the onset of Rayleigh-BÚnard convection in a parallelepiped cavity occurs at the same critical Rayleigh number (Rac) independently of Prandtl number (Pr). Bifurcation diagrams of steady convective flow patterns inside a cubical cavity with adiabatic lateral walls have been recently determined (3-5) at two Prandtl numbers, Pr=0.71 (air) and Pr=130 (silicone oil), for both adiabatic and perfectly conducting lateral walls. These results show that several types of steady convective flow patterns develop as the Rayleigh number is increased beyond Rac, in good agreement with previous visualization results reported in the literature (6-7). It is remarkable that in all cases studied different stable steady flow patterns coexist for different ranges of Ra.

On the other hand, results show that bifurcation diagrams depend strongly not only on boundary conditions but also on Prandtl number. Hence, the evolution of flow patterns as the Rayleigh number increases, their stability and heat transfer properties are highly dependent on Pr, this dependence being stronger in the case of perfectly conducting lateral walls. In the current work the effect of Prandtl number on flow stability and pattern configuration was studied for two prototypical flow patterns within the ranges 7000≤Ra≤150000 and 0.71≤Pr≤130. The three dimensional diagrams in the (Ra, Pr, Nu)-space, where Nu denotes the Nusselt number were obtained by means of a parameter continuation algorithm, which allows to choose the continuation parameter to be Ra or Pr. The continuation procedure is based on a Galerkin spectral solver that uses a free-divergent set of basis functions and includes tools for the analysis of stability and bifurcations. Present results show that more than one realizations of a particular flow pattern may be possible over certain ranges of Ra and Pr due to the presence of turning points.

[1] S. H Davis (1967). "Convection in a box: linear theory", J. Fluid Mechanics, vol 30, pp 465-478.

[2] I. Catton (1972). "The efect of insulating vertical walls on the onset of motion in a fluid heated from below", Int. J. Heat Mass Transfer, vol 15, pp 665-672.

[3] D. Puigjaner, J. Herrero, F. Giralt and C. Simˇ (2004). "Stability analysis of the flow in a cubical cavity heated from below", Physics of Fluids, vol 16, pp 3639-3655.

[4] D. Puigjaner, J. Herrero, F. Giralt and C. Simˇ (2006). "Bifurcation analysis of multiple steady flow patterns for Rayleigh-BÚnard convection in a cubical cavity at Pr=130", Physical Review E, vol 73(4), 046304 pp 1-16.

[5] D. Puigjaner (2005), "Study of a natural convection problem inside a cubical cavity from the point of view of dynamical system" Ph.D. thesis,Facultat de Matematiques. Universitat de Barcelona, Barcelona, Spain.

[6] W.H. Leong, K.G.T. Hollands and A.P. Brunger (1999) "Experimental Nusselt numbers for a cubical-cavity benchmark problem in natural convection", Int. J. Heat Mass Transfer, vol 42, pp 1979-1989.

[7] J. PallarŔs, M.P. Arroyo, F.X. Grau and F. Giralt (2001), "Experimental laminar Rayleigh-Benard convection in a cubical cavity at moderate Rayleigh and Prandtl numbers", Experiments in Fluids, vol 31, pp 208-218.

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