A simplified model is used to identify the diffuser shape that maximises pressure recovery for several classes of non-uniform inflow. We find that optimal diffuser shapes strike a balance between not widening too soon, as this accentuates the non-uniform flow, and not staying narrow for too long, which is detrimental for wall drag. Three classes of non-uniform inflow are considered, with the axial velocity varying across the width of the diffuser entrance. The first case has inner and outer streams of different speeds, with a velocity jump between them that evolves into a shear layer downstream. The second case is a limiting case when these streams are of similar speed. The third case is a pure shear profile with linear velocity variation between the centre and outer edge of the diffuser. We describe the evolution of the time-averaged flow profile using a reduced mathematical model that has been previously tested against experiments and computational fluid dynamics models. The model consists of integrated mass and momentum equations, where wall drag is treated with a friction factor parameterisation. The governing equations of this model form the dynamics of an optimal control problem where the control is the diffuser channel shape. A numerical optimisation approach is used to solve the optimal control problem and Pontryagin’s maximum principle is used to find analytical solutions in the second and third cases. We show that some of the optimal diffuser shapes can be well approximated by piecewise linear sections. This suggests a low-dimensional parameterisation of the shapes, providing a structure in which more detailed and computationally expensive turbulence models can be used to find optimal shapes for more realistic flow behaviour.

中文翻译：

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非均匀流动扩散器形状的最佳控制

一个简化的模型用于确定扩散器形状，该形状可最大限度地提高几类非均匀流入的压力恢复。我们发现最佳扩散器形状在不会过快加宽之间取得平衡，因为这会加剧不均匀的流动，并且不会长时间保持狭窄，这对壁阻力是有害的。考虑了三类非均匀流入，轴向速度在扩散器入口的宽度上变化。第一种情况具有不同速度的内部和外部流，它们之间的速度跳跃演变为下游的剪切层。当这些流具有相似的速度时，第二种情况是极限情况。第三种情况是在扩散器的中心和外边缘之间具有线性速度变化的纯剪切剖面。我们使用简化的数学模型来描述时间平均流量剖面的演变，该模型之前已针对实验和计算流体动力学模型进行过测试。该模型由集成的质量和动量方程组成，其中壁阻力用摩擦系数参数化处理。该模型的控制方程形成了最优控制问题的动力学，其中控制是扩散器通道形状。数值优化方法用于解决最优控制问题，庞特里亚金极大值原理用于寻找第二种和第三种情况的解析解。我们表明，一些最佳扩散器形状可以通过分段线性部分很好地近似。这表明形状的低维参数化，