非定常流函数涡量方程的一种数值解法的研究

对非定常流函数涡量方程的数值求解方法进行了改进，其中流函数一阶导数即速度项采用四阶精度的Hermitian公式，对流项由一般二阶精度的中心差分提高到四阶精度离散差分，包含温度方程在内的离散方程组采用ADI迭代方法求得定常解．以无内热体及有一内热体的封闭方腔内自然对流为例，进行了不同瑞利数（Ra）条件下的数值研究．结果表明，该方法推导简单，求解精度高且计算稳定，适用于封闭腔内高瑞利数复杂混合对流的数值模拟．

STUDY OF ONE NUMERICAL METHOD FOR SOLVING THE UNSTEADY EQUATIONS OF STREAM AND VORTICITY FUNCTIONS

The method for solving the unsteady equations of stream and vorticity functions has been improved. The Hermitian formulas of forth-order accuracy are adopted for first partial derivatives of stream function (i.e. velocities). The forth-order accuracy finite difference is used for the convective terms instead of the second-order central difference. The ADI successive method is used for solving the equations which contain the temperature equation to obtain the steady solutions.Second-order central difference is used for the convective terms of the equations. The method is used for the numerical simulations of the natural convection in an enclosure. When the Rayleigh number (Ra) is not so high, the numerical results of the method are consistent with the results of other numerical methods.When the Rayleigh number is higher than 105, the numerical results of the method are different from the results of other numerical methods. The calculations are not stable and steady results are not obtained. In the paper a new method is put forward to try to improve the above situations. The finite different accuracy for the convective terms of the equations could be raised from second-order to forth-order. The new method is also used for the numerical simulations of the natural convection in an enclosure. When the Rayleigh numbers are from 103 to 106, steady solutions could be obtained by using the improved method and the results are consistent with the results of the other numerical methods.A numerical method of forth-order accuracy is chosen and compared with the improved method in the paper. The comparison between the two methods shows that the numerical accuracy of the improved method could also be forth-order and the formulas are derived more easily than that of the chosen method.The natural convection in an enclosure with a heat-conducting body is complex mixed convection and under the condition of high Rayleigh numbers the numerical simulations for it are much more strickly checked for the numerical stability of the solution method. In the paper by using the improved method heat transfers are simulated for Ra=105 and the heat-conducting body on the centre of the cavity. The numerical results are consistent with the results of the SIMPLER method. It shows that the improved method is desirable for the numerical calculations of complex mixed convection in a closed cavity and is stable for the calculations. In the paper the steady solutions are obtained by using the improved numerical method for the unsteady equations of stream, vorticity and temperature functions. The examples show that the improved method is forth-order accuracy and is stalble for calculations. It is suited to the numerical simulations of high Rayleigh number complex mixed convection in a closed cavity. The numerical study shows that raising the difference accuracy of the convective terms of the equations is important for the raising of the numerical accuracy and the numerical stability.