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摘要:
跨声速定常流场的隐式求解相当于使用牛顿迭代法求解一个非线性方程组。为满足牛顿迭代收敛性的要求，通常需要对所求解问题进行全局化处理。在同伦延拓的框架内，提出了一种基于拉普拉斯算子的方程延拓方法，提高了定常流场隐式求解收敛速度。针对定常流场通常初始化为均匀来流的特点，一方面利用拉普拉斯算子的椭圆性加快边界条件信息向流场内部的传播，另一方面利用拉普拉斯算子的线性和正定性改善延拓问题的正则性，综合两者增加拟牛顿算法的稳定性，提高可用CFL数，最终达到提高流场求解效率的目的。由于流场问题的复杂性和非线性，难以通过理论分析得出先验的最优非线性求解策略。因此，通过无黏NACA0012翼型、湍流RAE2822翼型和三维ONERA M6机翼等算例的数值实验，研究了拉普拉斯项参数对收敛效率的影响，给出了效率较优的参数组合，验证了本文方法在跨声速情况下相对于经典伪时间推进法可以节约20%以上的CPU计算时间。
Abstract:The implicit solving approach of steady transonic flow field equals a Newton iteration for a nonlinear equation system. Globalization of Newton iteration is usually necessary in practice in order to fulfill the convergence requirement. In the framework of homogenous continuation, a Laplace operator based function continuation method which accelerates convergence of implicit solving of steady flow field is proposed. Considering that the steady flow field is usually initialized as uniform freestream condition, the Laplace operator is employed to speed up information propagation from wall boundary to internal flow field due to its ellipticity and to improve regularity of the problem due to its linearity and symmetric positive definite property. Thus the stability of Newton's method is improved then larger CFL number could be employed and finally the flow field solving efficiency is improved. Due to the complexity and nonlinearity of the flow field problem, a priori optimal nonlinear solving strategy is impossible to be obtained through theoretical analysis. Thus, the effect of Laplacian coefficient on convergence efficiency is investigated through numerical experiments on inviscid NACA0012 airfoil, turbulent RAE2822 airfoil and ONERA M6 3D wing test cases. Generally pragmatic combination of iteration parameters are also given and the proposed method is proved to gain over 20% saving in CPU computing time compared with the classic pseudo time marching method under transonic condition.
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Key words:
- nonlinear equations /
- implicit scheme /
- Newton's method /
- aerodynamics /
- transonic /
- steady flow
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图 1 无黏NACA0012翼型算例的计算网格
Figure 1. Computational grid of invicid NACA0012 airfoil test case
图 2 无黏NACA0012翼型算例升力系数、阻力系数及残差收敛曲线
Figure 2. Lift coefficient, drag coefficient and residual convergence history of invicid NACA0012 airfoil test case
图 3 无黏NACA0012翼型算例的表面压力系数分布
Figure 3. Surface pressure coefficient distribution of invicid NACA0012 airfoil test case
图 4 湍流RAE2822翼型算例的计算网格
Figure 4. Computational grid of turbulent RAE2822 airfoil test case
图 5 湍流RAE2822翼型算例升力系数、阻力系数及残差收敛曲线
Figure 5. Lift coefficient, drag coefficient and residual convergence history of turbulent RAE2822 airfoil test case
图 6 湍流RAE2822翼型算例的表面压力系数分布
Figure 6. Surface pressure coefficient distribution of turbulent RAE2822 airfoil test case
图 8 三维ONERA M6机翼算例升力系数、阻力系数及残差收敛曲线
Figure 8. Lift coefficient, drag coefficient and residual convergence history of 3D ONERA M6 wing test case
图 9 三维ONERA M6机翼算例表面压力系数分布
Figure 9. Surface pressure coefficient distribution of 3D ONERA M6 wing test case
表 1 无黏NACA0012翼型算例的延拓参数和收敛效率
Table 1. Continuation parameters and convergence efficiency of invicid NACA0012 airfoil test case
算例 CFL0 cLP0 n t/s 相对时间节约/% PTM1 6 152 38.79 -28.2 PTM2 8 109 30.25 0 LPTM1 20 5×10-2 67 25.77 14.8 LPTM2 20 5×10-3 66 22.07 27.0 LPTM3 20 1×10-3 64 22.50 25.6 LPTM4 20 5×10-4 62 21.37 29.4 LPTM5 10 5×10-5 92 25.56 15.5 
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表 2 湍流RAE2822翼型算例的延拓参数和收敛效率
Table 2. Continuation parameters and convergence efficiency for turbulent RAE2822 airfoil test case
算例 CFL0 cLP0 n t/s 相对时间节约/% PTM1 3 94 51.00 -20.4 PTM2 4 78 42.37 0 LPTM1 8 5×10-4 55 36.86 13.0 LPTM2 8 5×10-5 52 31.43 25.8 LPTM3 8 1×10-5 56 33.25 21.5 LPTM4 8 5×10-6 57 32.80 22.6 LPTM5 6 5×10-7 65 36.52 13.8
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表 3 ONERA M6机翼算例的延拓参数和收敛效率
Table 3. Continuation parameters and convergence efficiency for 3D ONERA M6 wing test case
算例 CFL0 cLP0 n t/s 相对时间节约/% PTM1 3 80 8 864.73 -12.5 PTM2 4 66 7 877.23 0 LPTM1 10 5×10-4 48 7 167.50 9.0 LPTM2 10 5×10-5 42 6 231.59 20.9 LPTM3 10 1×10-5 43 6 074.64 22.9 LPTM4 10 5×10-6 45 6 154.65 21.9 LPTM5 5 5×10-7 58 7 190.52 8.7
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