Topography optimisation of fluid flow between parallel plates of spatially-varying spacing: revisiting the origin of fluid flow topology optimisation

This paper presents a topography optimisation model for fluid flow with varying channel height. First, the origin of topology optimisation for fluid flow problems is revisited. Here, a Poiseuille-based frictional resistance term was first introduced by Borrvall and Petersson (Int J Numer Methods Fluids 41(1):77–107, 2003) to parametrise regions of solid and fluid. However, the traditional model only works for true topology optimisation, where it is used to approximate solid regions as areas with very small channel height and, thus, very high frictional resistance. Herein, it is shown that if the channel height is allowed to vary continuously, the minimum channel height is relatively large and/or meaning is attributed to intermediate design field values, then the predictions of the traditional model are wrong. To remedy this problem, this work introduces an augmentation of the mass conservation equation to allow for continuously varying channel heights. The proposed planar model describes fully developed flow between two plates of varying channel height. It allows for a significant reduction in the number of degrees of freedom, while generally ensuring a high accuracy for low-to-moderate Reynolds numbers in the laminar regime. The accuracy and limitations of both the traditional and proposed models are explored using in-depth parametric studies. The proposed model is used to optimise the height of the fluid channel between two parallel plates and, thus, the topography of the plate surfaces for a flow distribution problem. Lastly, it is shown that when introducing penalisation into the proposed height-based design parametrisation, the proposed model can produce designs of similar performance as the traditional resistance term interpolation. Thus, the proposed model bridges a gap between topography and topology optimisation of fluid flow, since it is able to perform both seamlessly with good accuracy.