Development of New Numerical Methods Based on Nonlocally Exact Differencing and Numerical Investigation on Naturally Convected Heat Transfer

1)提出了一种关于非局部精确数值差分概念的数值方法。该方法的实质在于确定差分格式的系数，使在控制体积表面插值数值通量的结果方程满足带吸收项和源项的输运方程的解析解。基于四区域模型，考虑输运方程系数在相邻三个控制体之间的空间分布，该模型在相邻两个控制体之间的边界上施加解的连续条件。2)采用特征多项式分析方法对该方案的非振荡特性进行了分析。结果表明，该格式对含吸收的稳态对流扩散方程具有非振荡性质。通过数值实验对该格式进行了验证，并给出了具有吸收项和源项的输运方程的较好解。3)对具有强空间依赖系数的输运方程的数值实验表明，SDCLENS格式的解明显优于局部精确格式LENS和QUICK格式。4)采用SDCLENS格式对质量、动量和能量方程进行离散化，编制了二维热工分析程序。5)通过剪切驱动空腔流动的数值实验验证了上述计算机程序的正确性。6)利用本程序对方形腔体中具有高瑞利数Ra 10^5-10^9的自然循环流动进行了数值模拟，克服了传统数值方法在多维场中由于热液耦合而容易产生较大数值扩散和不稳定解的问题。在Ra = 10^9范围内得到稳定的层流解。然而，当Ra = 10^8时，解不是稳定的，而是时间相关的。对壁面施加温差后涡的形成发展和涡的混沌结构进行了数值研究。此外，根据Ra = 10^9时的统计分析结果，计算了雷诺应力和波动能谱。结果表明，在紊流充分发展的情况下，能量谱大致为1/f谱，与Kolmogorov功率谱（-5/3）不同。少

1) A numerical method on the concept of nonlocally exact numerical differencing is presented. The essence of this method consists in determining the coefficients of the difference scheme such that the resulting equation interpolating numerical fluxes at the control volume surface satisfies the analytical solution of transport equations with absorption and source terms. The spatial distribution of the coefficients of transport equations is taken into consideration based on a four- region model among adjacent three control volumes, in which continuous conditions for solutions are imposed on the boundary between adjacent two regions.2) The analysis on nonoscillation properties of the present scheme was performed using the characteristic polynomial analysis method. It was found that the present scheme possesses potential of nonoscillation properties for stationary convection- diffusion equations with absorption. The present scheme is examined through numerical experiments, and shows good s … More olutions for transport equations with absorption and source terms.3) Numerical experiments for transport equations with strongly spatially dependent coefficients show that the present SDCLENS scheme gives solutions distinctly better than solutions with the locally exact scheme LENS and the QUICK scheme.4) The mass, momentum and energy equations were discretized using the SDCLENS scheme and a 2- D thermal hydraulics analysis program has been developed.5) The above computer program was validated through numerical experiments for the shear-driven cavity flow with the exact solution.6) The numerical simulation using the present computer program was performed for naturally circulating flows with high Rayleigh numbers Ra 10^5-10^9 in a square cavity, where conventional numerical methods tend to suffer from large numerical diffusions and unstable solutions because of thermohydraulically coupling in multidimensional fields. Steady laminar solutions were obtained up to Ra = 10^9. However, in case of Ra = 10^8the solutions were not steady but temporally dependent. Developing of vortex formation and the chaotic structure of vortex after the temperature difference was imposed on the walls are made numerically investigated. Further, as results of statistical analysis in case of Ra = 10^9, a Reynolds stress and a fluctuation energy spectrum were evaluated. It was shown that the energy spectrum is roughly 1/f spectrum, which is different from the Kolmogorov's (-5/3) power spectrum in case of fully developed turbulent flows. Less