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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.

Many natural and industrial turbulent flows are subjected to time-dependent boundary conditions. Despite being ubiquitous, the influence of temporal modulations (with frequency f) on global transport properties has hardly been studied. Here, we perform numerical simulations of Rayleigh-Bénard convection with time periodic modulation in the temperature boundary condition and report how this modulation can lead to a significant heat flux (Nusselt number Nu) enhancement. Using the concept of Stokes thermal boundary layer, we can explain the onset frequency of the Nu enhancement and the optimal frequency at which Nu is maximal, and how they depend on the Rayleigh number Ra and Prandtl number Pr. From this, we construct a phase diagram in the 3D parameter space (f, Ra, Pr) and identify the following: (i) a regime where the modulation is too fast to affect Nu; (ii) a moderate modulation regime, where Nu increases with decreasing f, and (iii) slow modulation regime, where Nu decreases with further decreasing f. Our findings provide a framework to study other types of turbulent flows with timedependent forcing.

Thermal convection in a two-dimensional tilted cell with aspect ratio (Γ = width/height) 0.5 is studied using direct numerical simulations. The considered tilt angle β ranges from 0° to 90°. The Prandtl number Pr dependence is first studied in the range of 0.01 ≤ Pr ≤ 100 for a fixed Rayleigh number Ra = 107. The Ra dependence is also investigated in the range of 106 ≤ Ra ≤ 109 for a fixed Pr = 0.71. Different flow states are identified over the β − Pr parameter space. It is found that the flow tends to organize in stable vertically-stacked double-roll state (DRS) for small Pr and small β, while this DRS becomes unstable and flow reversals happen with the increase of β. This finding complements our previous study of flow reversals in tilted cells with Γ = 1 and 2 [Wang et al., J. Fluid Mech. 849, 355–372 (2018)]. For relatively larger Pr, the flow gives way to a stable triple-roll state or an unstable triple-roll state for small β. Moreover, multiple states in the turbulent regime are found for Ra ≥ 108, between which the flow can or cannot switch. In the latter case, the Nu are different for the two states with the same number of convection rolls, but different orientations. It is found that the Nu(β)/Nu(0) and Re(β)/Re(0) dependence is strongly influenced by a combination of Ra and Pr. In the present system, we interestingly find that the earlier conclusion that Nu decreases with increasing β close to β = 90° for Γ = 1 does not hold for the present Γ = 0.5 case with small Pr.

Abstract

Abstract

We offer a unifying theory for turbulent, purely internally heated convection, generalizing the unifying theories of Grossmann and Lohse (2000, https://doi.org/10.1017/S0022112099007545; 2001, https://doi.org/10.1103/PhysRevLett.86.3316) for Rayleigh‐Bénard turbulence and of Shishkina et al. (2016, https://doi.org/10.1002/2015GL067003) for turbulent horizontal convection, which are both based on the splitting of the kinetic and thermal dissipation rates in respective boundary and bulk contributions. We obtain the mean temperature of the system and the Reynolds number (which are the response parameters) as function of the control parameters, namely the internal thermal driving strength (called, when nondimensionalized, the Rayleigh‐Roberts number) and the Prandtl number. The results of the theory are consistent with our direct numerical simulations of the Boussinesq equations.