Numerical Solution of a Flow-Control Problem: Vorticity Reduction by Dynamic Boundary Action
Numerical Solution of a Flow-Control Problem: Vorticity Reduction by Dynamic Boundary Action
复制标题
流量控制问题的数值解：动态边界作用降低涡度
DOI：
10.1137/s1064827595294678
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发表时间：
1998
影响因子：
3.1
通讯作者：
M. Berggren
中科院分区：
文献类型：
--
作者：
M. Berggren
为了使非定常内部流层叠化，采用最小二乘方法使涡度场最小化。流动模型为粘性不可压缩流体的Navier—Stokes方程，流动由部分边界上的吸力和吹气控制。用拟牛顿法求解含有涡量和正则化项的二次目标函数的最小化问题。Navier—Stokes方程在时间上用有限差分格式近似，在空间上用有限元近似近似。为了获得满意的收敛速度，需要精确的离散目标函数梯度表达式。因此，导出了在完全离散情况下目标函数最小的一阶必要条件。本文讨论了一种存储设备，如果没有它，任何实际大小的问题，特别是在三维空间中，在计算上仍然是难以处理的。数值实验表明，在雷诺数高到足以引起非线性效应的情况下，矩形腔内二维流动的最优控制方法是可行的。
In order to laminarize an unsteady, internal flow, the vorticity field is minimized, in a least-squares sense, using an optimal-control approach. The flow model is the Navier--Stokes equation for a viscous incompressible fluid, and the flow is controlled by suction and blowing on a part of the boundary. A quasi-Newton method is used for the minimization of a quadratic objective function involving a measure of the vorticity and a regularization term. The Navier--Stokes equations are approximated using a finite-difference scheme in time and finite-element approximations in space. Accurate expressions for the gradient of the discrete objective function are needed to obtain a satisfactory convergence rate of the minimization algorithm. Therefore, first-order necessary conditions for a minimizer of the objective function are derived in the fully discrete case. A memory-saving device is discussed without which problems of any realistic size, especially in three space dimensions, would remain computationally intractable. The feasibility of the optimal-control approach for flow-control problems is demonstrated by numerical experiments for a two-dimensional flow in a rectangular cavity at a Reynolds number high enough for nonlinear effects to be important.