Mixing is an omnipresent process in a wide range of industrial applications, which supports scientific efforts to devise techniques for optimizing mixing processes under time and energy constraints. In this endeavour, we present a computational framework based on nonlinear direct-adjoint looping for the enhancement of mixing efficiency in a binary fluid system. The governing equations consist of the nonlinear Navier–Stokes equations, complemented by an evolution equation for a passive scalar. Immersed and moving stirrers are treated by a Brinkman penalization technique, and the full system of equations is solved using a Fourier-based pseudospectral approach. The adjoint equations provide gradient and sensitivity information which is in turn used to improve an initial mixing strategy, based on shape, rotational and path modifications. We utilize a Fourier-based approach for parameterizing and optimizing the embedded stirrers and consider a variety of geometries to achieve enhanced mixing efficiency. We consider a restricted optimization space by limiting the time for mixing and the rotational velocities of all stirrers. In all cases, non-intuitive shapes are found which produce significantly enhanced mixing efficiency.

中文翻译：

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用于混合二元流体的搅拌棒的形状优化

混合是广泛工业应用中无所不在的过程，它支持科学的努力来设计出在时间和能量限制下优化混合过程的技术。为此，我们提出了一种基于非线性直接伴随循环的计算框架，用于提高二元流体系统中的混合效率。控制方程由非线性Navier–Stokes方程组成，并辅以无源标量的演化方程。沉入式和移动式搅拌器采用Brinkman惩罚技术处理，整个方程组使用基于傅立叶的伪谱方法求解。伴随方程式提供了坡度和灵敏度信息，这些信息又用于根据形状，旋转和路径修改来改善初始混合策略。我们利用基于傅立叶的方法对嵌入式搅拌器进行参数化和优化，并考虑了各种几何形状以提高混合效率。我们通过限制混合时间和所有搅拌器的旋转速度来考虑有限的优化空间。在所有情况下，都发现了非直觉的形状，可显着提高混合效率。