CFD数学模型的线性化方法及其应用

计算流体力学(CFD)方法不仅仅起到数值模拟的作用,它本身是一个复杂的非线性系统。在流动稳定性分析、气动弹性分析、优化设计以及流动控制等领域,从系统的角度出发,对CFD数学模型线性化后,可以对模型的系统矩阵进行定量分析获得更多的系统特性。但是CFD数学模型往往非常复杂且阶数很高,因此其线性化系统矩阵的获得比较困难。鉴于此,采用人工编程和自动微分相结合,构造有限体积法并行CFD模型的线性化系统矩阵。其中自动微分只被用来得到每个界面通量的局部雅可比矩阵,而采用人工编程方法来实现并行环境下的稀疏雅可比矩阵的组装。线性化系统的并行求解采用了块雅可比预处理的广义最小残量法,每个并行进程内部则采用零填充不完全LU分解预处理。为了验证这种线性化方法,上述方法被用于:① NACA 0012翼型的非定常绕流线性系统构造与求解;② NACA 0012翼型稳态流动的伴随方程构造与求解;③ AGARD wing 445.6机翼颤振问题降阶建模。上述三个算例的结果与CFD模拟的吻合一致。

Method of linearizing computational fluid dynamics model and its applications

Computational fluid dynamics (CFD) is not just a simulation method, but a kind of complicated mathematical model for fluid flows. In the fields like flow stability analysis, aeroelastic analysis, aerodynamic optimization and flow control, from the viewpoint of dynamics system, the system matrix of a CFD model can be constructed for quantitative analysis, obtaining more systematic information about the CFD model. However, CFD model is a complicated high order nonlinear system. It is difficult to construct the system matrix directly. In this paper, automatic differential method is cooperated with manual coding to construct the Jacobian of a parallel finite volume CFD solver based on multiblock structured grid. Automatic differential is applied to obtaining the local Jacobian of the flux across each interface. And by means of manual coding, each local Jacobian is assembled into the global distributed sparse Jacobian. In order to solve the linearized system, preconditioned GMRES method is adopted. In the parallel environment, the block Jacobi preconditioner is used while ILU(0) preconditioner is applied to each parallel thread. In the numerical tests, this procedure is applied to ① constructing and solving the linear system of an unsteady flow around NACA0012 airfoil; ② sensitivity analysis based on the adjoint equation for a steady flow of NACA0012 airfoil; ③ reduced order modeling for the aeroelastic problem of AGARD wing 445.6. The results agree excellently with the data of CFD simulations.