Penetrative turbulent Rayleigh–Bénard convection in two and three dimensions

Abstract

Penetrative turbulent Rayleigh-Bénard convection which depends on the density maximum of water near 4 • C is studied using two-dimensional (2D) and three-dimensional (3D) direct numerical simulations (DNS). The working fluid is water near 4 • C with Prandtl number P r = 11.57. The considered Rayleigh numbers Ra range from 10 7 to 10 10 . The density inversion parameter θ m varies from 0 to 0.9. It is found that the ratio of the top and bottom thermal boundary-layer thickness (F λ = λ θ t /λ θ b ) increases with increasing θ m , and the relationship between F λ and θ m seems to be independent of Ra. The centre temperature θ c is enhanced compared to that of Oberbeck-Boussinesq (OB) cases, as θ c is related to F λ with 1/θ c = 1/F λ + 1, θ c is also found to have a universal relationship with θ m which is independent of Ra. Both the Nusselt number N u and the Reynolds number Re decrease with increasing θ m , the normalized Nusselt number N u(θ m )/N u(0) and Reynolds number Re(θ m )/Re(0) also have universal relationships with θ m which seem to be independent of both Ra and the aspect ratio Γ . The scaling exponents of N u ∼ Ra α and Re ∼ Ra β are found to be insensitive to θ m despite of the remarkable change of the flow organizations.

From zonal flow to convection rolls in Rayleigh–Bénard convection with free-slip plates

“…For zonal flow γ Nu is much smaller, namely only 0.17. The effective scaling exponent γ Re in Re ∼ Ra γ Re is approximately 0.6 for zonal flow and approximately 0.67 for the convection roll state, which is also close to the GL predication of 2/3 for the I < ∞ and III ∞ regimes (Grossmann & Lohse 2001;Shishkina et al 2017) for no-slip cases, while it is larger than 0.6 that has been reported for 2-D RB convection with no-slip plates (Sugiyama et al 2009;Zhang et al 2017;Wang et al 2019c).…”

“…For the 2-D horizontally periodic cases with no-slip plates, Nu ∼ Ra 0.29 is found with Pr = 1, Ra ≤ 10 10 (Johnston & Doering 2009;Zhu et al 2018a). For 2-D RB convection with no-slip plates and sidewalls with unit aspect ratio, several studies have shown that Nu ∼ Ra 0.3 and Re ∼ Ra 0.6 (Sugiyama et al 2009;Zhang, Zhou & Sun 2017;Wang et al 2019c). However, how these effective scaling relations will change for free-slip plates has not been Figure 8(a) reveals that the heat transfer in the convection roll state is much higher than that of the zonal flow state.…”

“…2009; Zhang, Zhou & Sun 2017; Wang et al. 2019 c ). However, how these effective scaling relations will change for free-slip plates has not been explored, especially not for convection roll states, which are only present in a large enough domain size.…”

“…2017; Wang et al. 2019 c ).

Figure 8.The ( a ) and ( b ) as functions of for different convection roll states (see legend in panel ( d )) for , and the zonal flow state (see orange stars on solid orange line) for , .

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Shape and size of large-scale vortices: A generic fluid pattern in geophysical fluid dynamics

“…As a result, the fluid self organizes into a well-mixed convective layer adjacent to a stably stratified layer. The use of a nonlinear equation of state with a density maximum is motivated by the similar behavior of water near its density maximum of 4 • C [37][38][39] and allows to study the coupled dynamics of two-layer mixed turbulent and stably stratified geophysical and astrophysical fluids. Here, β(T ) is piecewise-constant and is positive for T T inv = 0 and negative for T < T inv = 0 (i.e., we use T inv as reference temperature).…”