TY - JOUR
T1 - Scaling relations in large-Prandtl-number natural thermal convection
AU - Shishkina, Olga
AU - Emran, Mohammad S.
AU - Grossmann, Siegfried
AU - Lohse, Detlef
PY - 2017/10/1
Y1 - 2017/10/1
N2 - In this study, we follow Grossmann and Lohse [Phys. Rev. Lett. 86, 3316 (2001)10.1103/PhysRevLett.86.3316], who derived various scalings regimes for the dependence of the Nusselt number Nu and the Reynolds number Re on the Rayleigh number Ra and the Prandtl number Pr. We focus on theoretical arguments as well as on numerical simulations for the case of large-Pr natural thermal convection. Based on an analysis of self-similarity of the boundary layer equations, we derive that in this case the limiting large-Pr boundary-layer dominated regime is I<, introduced and defined by Grossmann and Lohse [Phys. Rev. Lett. 86, 3316 (2001)10.1103/PhysRevLett.86.3316], with the scaling relations Nu∼Pr0Ra1/3 and Re∼Pr-1Ra2/3. Our direct numerical simulations for Ra from 104 to 109 and Pr from 0.1 to 200 show that the regime I< is almost indistinguishable from the regime III, where the kinetic dissipation is bulk-dominated. With increasing Ra, the scaling relations undergo a transition to those in IVu of Grossmann and Lohse [Phys. Rev. Lett. 86, 3316 (2001)10.1103/PhysRevLett.86.3316], where the thermal dissipation is determined by its bulk contribution.
AB - In this study, we follow Grossmann and Lohse [Phys. Rev. Lett. 86, 3316 (2001)10.1103/PhysRevLett.86.3316], who derived various scalings regimes for the dependence of the Nusselt number Nu and the Reynolds number Re on the Rayleigh number Ra and the Prandtl number Pr. We focus on theoretical arguments as well as on numerical simulations for the case of large-Pr natural thermal convection. Based on an analysis of self-similarity of the boundary layer equations, we derive that in this case the limiting large-Pr boundary-layer dominated regime is I<, introduced and defined by Grossmann and Lohse [Phys. Rev. Lett. 86, 3316 (2001)10.1103/PhysRevLett.86.3316], with the scaling relations Nu∼Pr0Ra1/3 and Re∼Pr-1Ra2/3. Our direct numerical simulations for Ra from 104 to 109 and Pr from 0.1 to 200 show that the regime I< is almost indistinguishable from the regime III, where the kinetic dissipation is bulk-dominated. With increasing Ra, the scaling relations undergo a transition to those in IVu of Grossmann and Lohse [Phys. Rev. Lett. 86, 3316 (2001)10.1103/PhysRevLett.86.3316], where the thermal dissipation is determined by its bulk contribution.
UR - http://www.scopus.com/inward/record.url?scp=85036512967&partnerID=8YFLogxK
U2 - 10.1103/PhysRevFluids.2.103502
DO - 10.1103/PhysRevFluids.2.103502
M3 - Article
AN - SCOPUS:85036512967
VL - 2
JO - Physical review fluids
JF - Physical review fluids
SN - 2469-990X
IS - 10
M1 - 103502
ER -