Transition to turbulence in the wake of a fixed sphere in mixed convection
Résumé
The thermal effect on axisymmetry breaking and transition to turbulence in the wake of a fixed heated sphere is investigated in the mixed convection configurations commonly known as 'assisting'and 'opposing' flows in which the buoyancy tends, respectively, to accelerate and decelerate the flow. The study is carried out in the Ri –Re parameter plane (Ri being the mixed convection parameter – the Richardson number) for two values of Prandtl number – 0.72 (≈air and many gases) and 7 (≈water). We show that convection affects considerably the transition (as compared to that observed in the wake of an unheated sphere) even at moderate Richardson numbers. The latter are taken to be positive in assisting flow and negative in opposing one. In this notation, it can be said that convection shifts the primary-instability threshold to higher Reynolds numbers with increasing Richardson number. In assisting flow, the primary bifurcation is always regular, but at Ri > 0.6 it appears in azimuthal subspaces associated with higher azimuthal wavenumbers m > 1. The transition scenario is characterized by a large variety of regimes explainable by nonlinear interactions between different azimuthal subspaces. On the side of higher (positive) Richardson numbers the axisymmetric flow is found stable up to Re = 1400 at Pr = 0.72 and Ri = 0.7. In opposing flow, the m = 1 subspace is always the most unstable, but the regular bifurcation gives way to a Hopf one at Ri < −0.1. Close to the junction of both bifurcations a similar variety of regimes precedes the transition to chaos as in assisting flow. On the side of negative Richardson numbers the primary (Hopf) bifurcation threshold is found as low as Re = 100 at Ri = −0.25 and at both investigated Prandtl numbers. After a primary periodic regime characterized by vortex shedding with a symmetry plane, the transition proceeds via a series of increasingly irregular helical regimes.