Drag reduction and instabilities of flows in longitudinally grooved annuli

Abstract

The primary and secondary laminar flows in annuli with longitudinal grooves and driven by pressure gradients have been analysed. There exist geometric configurations reducing pressure losses in primary flows in spite of an increase of the wall wetted area. The parameter ranges when such flows exist have been determined using linear stability theory. Two types of secondary flows have been identified. The first type has the form of the classical travelling waves driven by shear and modified by the grooves. The axisymmetric waves dominate for sufficiently large radii of the annuli while different spiral waves dominate for small radii. The secondary flow topology is unique in the former case and has the form of axisymmetric rings propagating in the axial direction. Topologies in the latter case are not unique, as spiral waves with left and right twists can emerge under the same conditions, resulting in flow structures varying from spatial rings to rhombic forms. The most intense motion of this type occurs near the walls. The second type of secondary flow has the form of travelling waves driven by inertial effects with characteristics very distinct from the shear waves. Its critical Reynolds number increases proportionally to $S^{-2}$, where $S$ denotes the groove amplitude, while the amplification rates increase proportionally to $S^{2}$. These waves exist only if $S$ is above a well-defined minimum and their axisymmetric forms dominate, with the most intense motion occurring near the annulus mid-section. Geometries that give preference to the latter waves have been identified. It is shown that the drag-reducing topographies stabilize the classical travelling waves; these waves are driven by viscous shear, so reduction of this shear decreases their amplification. The same topographies destabilize the new waves; these waves are driven by an inviscid mechanism associated with the formation of circumferential inflection points, and an increase of the groove amplitude increases their amplification. The flow conditions when the presence of grooves can be ignored, i.e. the annuli can be treated as being hydraulically smooth, have been determined.

Poiseuille flow of a Bingham fluid in a channel with a superhydrophobic groovy wall

“…Before moving further, note that, for the sake of completeness of our analysis, the order of the inertial terms (including the neglected nonlinear terms) in (3.19) is evaluated in § 2 of the online supplementary material. Also note that the analysis and treatment of the inertial terms in our work are consistent with those in previous studies (Moradi & Floryan 2016, 2019; Jiao & Floryan 2021), which have solved their problem for the base flow at large Reynolds numbers and, similar to our work, they have obtained converged results.…”

“…In this flow, with periodic positioning of the rectangular trenches, the authors have used the Floquet–Bloch theory to be able to examine the disturbances with wavelengths larger than that of the trenches. In some other studies, the researchers have solved for the inertial flow in confined geometries with periodic groovy walls, followed by flow stability analysis (Floryan 2005; Szumbarski 2007; Moradi & Floryan 2016, 2019).…”

Modeling of Shear Flows over Superhydrophobic Surfaces: From Newtonian to Non-Newtonian Fluids

“…When fluid flows over a surface, the characteristics of the surface itself can play a significant role in the flow dynamics. − Even small-scale features on the surface can result in major alterations on the flow physics. For instance, inspired by properties of biosurfaces, e.g.…”

Slowing down convective instabilities in corrugated Couette–Poiseuille flow

“…While the number of possible surface patterns is limitless, here we focus on the regular wall roughness in the form of longitudinal grooves positioned such that ridges of the geometry run parallel to the flow direction, as schematically illustrated in figure 1. This type of grooves has been investigated as a means to manipulate flow dynamics in a doubly periodic grooved channel (Szumbarski 2007; Mohammadi & Floryan 2014; Mohammadi, Moradi & Floryan 2015; Yadav, Gepner & Szumbarski 2017; Gepner & Floryan 2020; Gepner, Yadav & Szumbarski 2020), singly periodic corrugated duct (Yadav, Gepner & Szumbarski 2018; Pushenko & Gepner 2021) and grooved, annular (Moradi & Floryan 2019; Moradi & Tavoularis 2019) configurations. It has been shown that properly shaped longitudinal grooves lead to a reduction of hydraulic drag (Szumbarski & Błoński 2011; Szumbarski, Blonski & Kowalewski 2011; Mohammadi & Floryan 2015; Ng, Jaiman & Lim 2018; Moradi & Floryan 2019).…”

“…This type of grooves has been investigated as a means to manipulate flow dynamics in a doubly periodic grooved channel (Szumbarski 2007; Mohammadi & Floryan 2014; Mohammadi, Moradi & Floryan 2015; Yadav, Gepner & Szumbarski 2017; Gepner & Floryan 2020; Gepner, Yadav & Szumbarski 2020), singly periodic corrugated duct (Yadav, Gepner & Szumbarski 2018; Pushenko & Gepner 2021) and grooved, annular (Moradi & Floryan 2019; Moradi & Tavoularis 2019) configurations. It has been shown that properly shaped longitudinal grooves lead to a reduction of hydraulic drag (Szumbarski & Błoński 2011; Szumbarski, Blonski & Kowalewski 2011; Mohammadi & Floryan 2015; Ng, Jaiman & Lim 2018; Moradi & Floryan 2019). Interestingly, there are indications, both experimental (Kim & Hidrovo 2012; Bolognesi, Cottin-Bizonne & Pirat 2014) and theoretical (Crowdy 2017), that drag reduction, attributed to the superhydrophobic effect, could, at least in some cases, be related to drag reduction reported for flows through longitudinally patterned geometries, such as those considered here.…”