Numerical Simulation Studies Of Heat Transfer And Temperature Profiles In Turbulent Thermal Convection With Slippery Surfaces

Turbulent thermal convection with different slippery boundary conditions is a major process of heat transfer in many natural systems,such as in ocean circulation,Earth’s mantle motion and planetary atmospheric motion,where heat is exchanged at the fluidfluid interface.In many engineering applications,slippery boundary conditions also play an important role in system heat transfer,for example,by using superhydrophobic surfaces,internal heating and injecting additives into the system to reduce the resistance at the solid-liquid interface.Therefore,studying the effect of slip boundaries on heat transfer can help to understand heat transport phenomena in nature as well as can provide guidance for solving heat transfer problems in engineering applications.The main research of this paper is as follows.First of all,we numerically report the flow characteristics and heat transfer characteristics of turbulent Rayleigh-Benard thermal convection systems with different slip boundary conditions.We also discuss the mean temperature Θ and temperature variance Ωboundary layer profile.In 2D and 3D closed RB thermal convection systems,the Rayleigh number(Ra)is in the range 106 ≤Ra≤1010,the aspect ratio Γ=W/H(W and H are the width and height of the cell,respectively)ranges from 1<Γ<10 and the Prandtl number Pr=4.3.The slip length b is used to characterize the slip boundary from the no-slip boundary condition(b=0)to the free slip boundary condition(b→∞).The aspect ratio of the convection system is set to Γ=1/4 in the three-dimensional RB convection.The calculation results show the heat transfer Nu of the convective system for free-slip plates conditions increases by～80%in the two-dimensional case,compared with the no-slip boundary condition.When plotted b/λ0 as the abscissa,the heat transport Nu/Nu0 for different Ra numbers and different slip boundary conditions b could be collapsed onto a single curve Nu/Nu0＝N0 × tanh(b/λ0)+1,where Nu0 and λ0 are the heat transport and temperature boundary layer thickness for the no-slip boundary case,respectively,the parameter No=NuF/Nu0-1 and NuF is the heat transport for the free-slip case.Next,introducing the turbulent fluctuation,we established the mean temperature and temperature variance boundary layer equations.Both the turbulent diffusion coefficients Kt and κf can be approximated by～ξp with ξ=z/λ,where λ denotes the thermal boundary layer thickness.The parameter p decreases monotonically from p=3 for the no-slip boundary conditions to p=2 for the free-slip boundary conditions as the slip length b/λ0 increases.In this paper,we derived the mean temperature Θ and temperature pulsation Ω boundary layer equations for different slip boundary conditions,and found that the temperature profiles obtained from direct numerical simulation are in agreement with the predicted equation for different scale ratios Γ,different Ra numbers and different slip boundary conditions b in two-dimensional and three-dimensional closed RB thermal convection systems.The mean temperature and temperature variance boundary layer equations can reveal well the physical mechanism of the enhancement of heat transport in the closed RB convection system with the slippery surfaces.Furthermore,we also calculated the two-dimensional periodic RB thermal convection with horizontally periodic boundary conditions and slippery surface by using direct numerical simulations.When the flow state is the large scale circulation state,the heat transport of the system increases with increasing slip length b/Ao.When the slip length b/λ0(?)20,the flow state of the fluid changes from a large-scale circulation state to a zonal flow state and the heat transport of the system decreases sharply.The slip boundary condition affects the flow state and the heat transport of the system,and also affects the scaling law of heat transfer and Rayleigh number Nu～RaγNu.The scaling law exponent is γNu=0.31 when the flow state is a large-scale circulation state,while it is yNu=0.16 when the flow state is a zonal flow,which is much smaller than the scaling law exponent for large-scale circulation.Finally,we numerically study the effect of the geometrical confinement on the flow dynamics and heat transport,as well as on the onset of the large-scale circulation(LSC)in horizontal cylinder with no-slip boundary conditions.The numerical study is conducted with the aspect ratio Γ=W/H(width-to-height)varied in the range 1/64 ≤Γ≤1/2 and the Rayleigh number(Ra)varied in the range 103 ≤Ra ≤1010 at a fixed Prandtl number Pr=4.4.Our results show that the heat transfer data(Nu-1)/Ra1/3 could be collapsed onto a single curve for varying Γ and Ra,once plotted versus Rayleigh number Ral≈Ra(1+0.25Γ-2)-3/2.The heat transfer is optimal at the Rayleigh number Ral≈9 × 105 for an optimal aspect ratio Γ at a fixed Rayleigh number Ra.Based on the Fourier modal analysis method,we extracted the flow pattern for different Rayleigh numbers and different aspect ratios Γ.It can be found that the normalized Rayleigh number Ral of the formation of large-scale circulation is also Ral≈9 × 105,which indicates that Ral can also reveal the formation mechanism of large-scale circulation at the optimal Γ.