Flows in annuli with longitudinal grooves

Analysis of pressure losses in laminar flows through annuli fitted with longitudinal grooves has been carried out. The additional pressure gradient required in order to maintain the same flow rate in the grooved annuli, as well as in the reference smooth annuli, is used as a measure of the loss. The groove-induced changes can be represented as a superposition of a pressure drop due to a change in the average position of the bounding cylinders and a pressure drop due to flow modulations induced by the shape of the grooves. The former effect can be evaluated analytically while the latter requires explicit computations. It has been demonstrated that a reduced-order model is an effective tool for extraction of the features of groove geometry that lead to flow modulations relevant to drag generation. One Fourier mode from the Fourier expansion representing the annulus geometry is sufficient to predict pressure losses with an accuracy sufficient for most applications in the case of equal-depth grooves. It is shown that the presence of the grooves may lead to a reduction of pressure loss in spite of an increase of the surface wetted area. The drag-decreasing grooves are characterized by the groove wavenumber $M/ {R}_{1} $ being smaller than a certain critical value, where $M$ denotes the number of grooves and ${R}_{1} $ stands for the radius of the annulus. This number marginally depends on the groove amplitude and does not depend on the flow Reynolds number. It is shown that the drag reduction mechanism relies on the re-arrangement of the bulk flow that leads to the largest mass flow taking place in the area of the largest annulus opening. The form of the optimal grooves from the point of view of the maximum drag reduction has been determined. This form depends on the type of constraints imposed. In general, the optimal shape can be described using the reduced-order model involving only a few Fourier modes. It is shown that in the case of equal-depth grooves, the optimal shape can be approximated using a special form of trapezoid. In the case of unequal-depth grooves, where the groove depth needs to be determined as part of the optimization procedure, the optimal geometry, consisting of the optimal depth and the corresponding optimal shape, can be approximated using a delta function. The maximum possible drag reduction, corresponding to the optimal geometry, has been determined.