| Summary: | 博士 === 國立成功大學 === 航空太空工程學系碩博士班 === 90 === A central difference (dissipation-free) scheme has to conserve kinetic energy to avoid numerical instability when a long-term time integration for incompressible flow such as large eddy simulation or direct numerical simulation is performed. A scheme is called fully conservative if it can simultaneously conserve mass, momentum, and kinetic energy in the discrete sense. A theoretical analysis is performed to extend the fully conservative schemes to non-uniform grid systems without sacrificing any conservative properties. The main step is to design a convective scheme which conserves momentum and kinetic energy simultaneously. An analysis is made for the second-order accurate scheme of Harlow and Welsh for staggered grid systems and it is found that the flux velocities need to be viewed as mass fluxes across control surfaces to conserve kinetic energy.
To extend the analysis to higher order schemes, it is necessary to work in computational space. The contravariant-Cartensian velocity formulation for the convection term in computational space has the similar structure for the proposed fully conservative second-order scheme in physical space. Using the velocity formulation, the higher order convective schemes of Morinishi are extended to non-uniform staggered grid systems for the advective, divergence and skew-symmetric forms. The higher order schemes for scalar variables which conserve the square of the scalar variables are also derived. Several numerical tests are used to validate the conservative properties, accuracy and performance of the proposed higher order schemes. A series of LESs of turbulent heat transfer in channel flow to study the contributions of SGS motions and the influences of grid number on turbulent statistics.
Large eddy simulations are performed to study fully developed turbulent mixed convection in a vertical plane channel, Re_b = 5600 and Pr = 0.71, with uniform heating or cooling from both walls. The main features of turbulent mixed convection are produced.
For aiding flow, a transition Gr_q number, Gr_q = 1.40x10^8, exists. Before the transition number, the turbulence is generated mostly by the shear force driven by the pressure gradient. The turbulent statistics are similar in shape to those for forced convection while the magnitudes reduce slightly in the near-wall region for all turbulent statistics and the friction coefficient and the Nusselt number also decrease gradually. The buoyancy production term in the budget of turbulent kinetic energy remain small and negative over the whole channel.
Around the transition Gr_q number, the regeneration process of near-wall structures are destroyed mostly. Second order statistics show the severest deterioration in the near-wall region and the turbulence generated by buoyancy becomes apparent on turbulent statistics away from the wall. The friction coefficient and the Nusselt number decline to 85% and 45%, respectively, of that at Gr_q=0. The point of
the maximum mean velocity begins to shift away from the channel center and the Reynolds shear stress and streamwise turbulent heat flux change sign nearly at the location of the maximum mean velocity. The buoyancy production term changes sign, and thus the term becomes a main producing term while y is larger than the zero point. The similarity between u' and theta' begins to deteriorate.
After the transition Gr_q number, turbulence generated by buoyancy gradually increases its influence on turbulent statistics. The magnitudes increase gradually in the near-wall region for all turbulent statistics, the friction coefficient and the Nusselt number with increasing Gr_q number. The dissimilarity between u' and theta' increases gradually and the thermal plumes become the main structures at highest simulated Gr_q.
For opposing flow, the contributions of the buoyant force and Reynolds shear stress are in the opposite direction, and thus the turbulence intensity increases as the buoyant force increases. The turbulent statistics are similar in shape to those for forced convection while the magnitudes increase in the near-wall region for all turbulent statistics except for the mean streamwise velocity and the friction coefficient, and the Nusselt number also increases gradually with increasing Gr_q number. The near-wall streaky structures are similar to those of Gr_q = 0, but the dissimilarity between u' and theta' is observed at the highest simulated Gr_q.
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