陳炳煇Chen, Ping-Hei臺灣大學:機械工程學研究所郭龍生Kuo, Long-ShengLong-ShengKuo2010-06-302018-06-282010-06-302018-06-282009U0001-2107200916492800http://ntur.lib.ntu.edu.tw//handle/246246/187249This work studies the effects of hydrodynamic and thermal slip boundaries on Rayleigh-B′enard convection (RB convection) using lattice Boltzmann method (LBM).irstly, the theoretical relations of the critical Rayleigh number (Rc) and corresponding wavenumber (ac) of RB convection with partially slippery boundary conditions on infinite horizontal plates are derived using the linear stability analysis. The results make previous studies by others under both slip and nonslip boundary conditions as special cases of the general relations.econdly, before numerical study of the effects of various slip side walls on RB convection in a 2D box, a new implementation of partial slip boundary conditions in LBM is developed by using the parameter of tangential momentum accommodation coefficient (TMAC,σ). This new implementation utilizes the native expressions of velocity gradients in LBM and then eliminates the need for information of neighbor nodes to estimate the velocity and temperature gradients, which is inevitable for conventional numerical technique like finite-difference method.hirdly, the numerical simulations show that the vertical slip side walls have similar impacts as the horizontal slip plates on the determination of Rc and the relations for infinite horizontal plates can be used as guidelines for the cases with side walls. The observations of pattern selection in the box with aspect ratio equal to 2 show that when σh (σ of horizontal plates) is less than 0.02, the preferred pattern is the one-roll mode. When σh ≥ 0.02, the fluid prefers the two-roll mode in which the fluid moves upwards in the center of the box if σv ≤ 0.1 (σ of vertical side walls), while the fluid switches the rotation directions if σv ≥ 0.2. The investigation of initial disturbance indicates the existence of the threshold of the initial amplitude for the desired mode. This result reveals the importance of initial conditions to RB convection.Acknowledgment . . .. . .. . .. . .. . .. . .. . .. . .iibstract . . .. . .. . .. . .. . .. . .. . .. . .. . .iv Introduction . . .. . .. . .. . .. . .. . .. . .. . .1 Rayleigh-B′enard Convection . . .. . .. . .. . .. . .6.1 Governing equations . . . . . . . . . . . .. . . . 6.2 Linear stability analysis . . . . . . . . . . . . 8 Lattice Boltzmann Method . . .. . .. . .. . .. . .. 10.1 Discretization and lattice Boltzmann equation . . 10.2 Lattice BGK model . . . . . . . . . . . . . . . . 13.3 External force field . . . . . . . . . . . . . . 16.4 Derivation of hydrodynamic equations . . . . . . 18.4.1 Multi-scale perturbation for LBE . . . . . . . 18.4.2 Solutions of f(0)i and f(1)i . . . . . . . . . 19.4.3 Conservation of mass . . . . . . . . . . . . . 20.4.4 Conservation of momentum . . . . . . . . . . . 20.5 Two-distribution lattice Boltzmann method (TLBM) 22.5.1 Distribution for temperature field . . . . . . . 23.5.2 Equilibrium distributions . . . . . . . . . . . 24.5.3 Derivation of temperature equation in continuous space . . . 25.5.4 Derivation of temperature equation in discrete space . . . . . 28 Slip Boundary Conditions in RB Convection . . .. . .30.1 The principle of the exchange of stabilities . . 30.2 The variational principle . . . . . . . . . . . . 32.3 Exact solutions of the characteristic value problem . 36.4 The variational solutions . . . . . . . . . . . . 38.5 Numerical results and discussions . . . . . . . . 41.5.1 Hydrodynamic slip only (ζ = 0) . . . . . . . . 41.5.2 Thermal slip (ζ>0) . . . . . . . . . . . . . . 41 Slip Boundary Conditions in LBM . . .. . .. . .. . .45.1 Slip phenomena and tangential momentum accommodation coefficient . . .. . .. . .. . .. . .. . .. . .. . .. 46.2 Slip boundary conditions in LBM . . . . . . . . . 48.2.1 D2Q9 Lattice . . . . . . . . . . . . . . . . . 50.2.2 D3Q15 Lattice . . . . . . . . . . . . . . . . . 52.3 Numerical Results . . . . . . . . . . . . . . . . 56.3.1 Couette flows . . . . . . . . . . . . . . . . . 57.3.2 Planar Poiseuille flows . . . . . . . . . . . . 58 Thermal Boundary Conditions in TLBM . . .. . .. . ..62.1 Lattice Boltzmann Models . . . . . . . . . . . . 62.2 Thermal boundary conditions . . . . . . . . . . . 63.2.1 Adiabatic boundary conditions . . . . . . . . . 64.2.2 Isothermal boundary conditions . . . . . . . .. 65.2.3 Prescribed-flux boundary conditions . . . . . . 67.3 Simulation results . . . . . . . . . . . . . . . 69.3.1 1D transient heat diffusion problem . . . . . . 69.3.2 2D Rayleigh-B′enard convection . . . . . . . . 72 Effects of Side Walls and Various Boundary Conditions 77.1 Code verification . . . . . . . . . . . . . . . . . 78.1.1 Slip horizontal plates . . . . . . . . . . . . . 79.1.2 Nonslip horizontal plates . . . . . . . . . . . 79.1.3 Grid dependence test . . . . . . . . . . . . . 80.2 Effect of Bi on Critical Rayleigh number and Nusselt number . . . . . 81.3 Effects of slip sidewalls . . . . . . . . . . . . . 82.3.1 Critical Rayleigh number . . . . . . .. . . . . . 83.3.2 Pattern selection . . . . . . . . . . . . . . . . 85.3.3 Pattern transition . . . . . . . . . . . . . . . 90 Conclusions . . .. . .. . .. . .. . .. . .. . .. . .100ibliography. .. .. .. .. .. .. .. .. .. .. .. .. .. .103 TLBM in D2Q9 and D3Q15 lattices . . .. . .. . .. . .109.1 D2Q9 lattice . . . . . . . . . . . . . . . . . . . . . . . 110.2 D3Q15 lattice . . . . . . . . . . . . . . . . . . . . . . . 1101722988 bytesapplication/pdfen-US雷利-本那 熱對流滑動邊界晶格波茲曼方法切線動量調節係數樣式變換Rayleigh-Benard convectionslip boundarylattice Boltzmann methodtangential momentum accommodation coefficientpattern transitionthesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/187249/1/ntu-98-D94522017-1.pdf