Analytical investigations of heat transfer during the vertical impingement of an unsubmerged axisymmetric liquid jet on a horizontal plate have been limited to the regions ahead of the jump. This limitation is due to the complex flow physics in the jump region arising from sudden changes in the flow field. This is addressed in here by extending the averaging theory (AT) introduced by Bohr et al. (Phys. Rev. Lett., vol. 79, issue 6, 1997, pp. 1038–1041) which was further developed by Watanabe et al. (J. Fluid Mech., vol. 480, 2003, pp. 233–265), to describe the heat transfer problem in circular hydraulic jumps including separation. The applicability of the resulting theory to determine the temperature field in the jump region is evaluated using the data available in the literature and also by means of fully resolved numerical solutions. Good agreement is observed for moderate Prandtl numbers. However, for sufficiently high Prandtl numbers, deviations become notable. The reasons for the deviations according to their relevance are (i) monotonically decreasing temperature profile inherent to the AT, whereas the fully resolved numerical solutions exhibit a local maximum in the temperature profile away from the plate; and (ii) inapplicability of the concept of dividing the flow field into a region affected and a region unaffected by heat transfer according to the thermal boundary layer thickness. This concept leads to the overestimation of the temperature close to the wall and to the existence of a threshold Prandtl number, for which the thermal boundary layer thickness does not meet the free surface anymore. Around this threshold Prandtl number, the temperature field shows a discontinuous behaviour.

In this paper a fully coupled particle-size segregation model for granular flows (Barker et al., J. Fluid Mech., vol. 909, 2021, p. A22) is used to simulate the development of the patterns in a triangular rotating drum. The results are compared with the experimental patterns formed with bidisperse and tridisperse granular mixtures, and with varying compositions and fill heights. In all cases the agreement between the simulations and experiments is remarkably good. The experimental patterns are generated in a narrow gap between transparent front and back sidewalls. These prevent three-dimensional motion, but also impose friction on the flow, making it thinner and faster than it would otherwise be. This promotes segregation, as it simultaneously increases the shear rate and reduces the local pressure. To obtain the correct flow dynamics and segregation, width-averaged sidewall friction is incorporated into the two-dimensional simulations, which are performed in OpenFOAM$^{\circledR}$. The free-surface avalanche forms a boundary layer within which all the segregation occurs. Material in the lower reach of the avalanche is continuously deposited into an underlying solid body of grains, which rotates with the drum, and is eventually re-entrained into the avalanche along its upper reach. The changing geometry of the granular region (as the drum rotates) implies that the avalanche is constantly adjusting its length, position and depth. This generates a complex quasi-periodic flow, which when combined with particle-size segregation generates amazing patterns in the solid rotating granular body after only two drum rotations.

Gust response has consistently been a concern in engineering. Critical theories have been proposed in the past to predict the unsteady lift response of an airfoil experiencing vertical gusts by Atassi, and longitudinal gusts by Greenberg. However, their applicability for an airfoil with non-zero angles of attack still needs clarification. Thus, force measurements are conducted to examine these theories’ validity and quasi-steady corrections are applied to compensate potential disparities between the idealised and real flow conditions. Velocity measurements are performed to scrutinise the effect of gusts on the flow around the airfoil, and subsequently to reveal the underlying mechanism governing the airfoil's response to gust-induced perturbations. In the study, two pitching vanes are arranged upstream to generate periodic vertical and longitudinal gusts, whereas a downstream airfoil with angles of attack of 0–12° is subjected to two gust types. It is found that Greenberg's theory demonstrates superior predictive capability in pre-stall regimes, with the potential for its effectiveness to be expanded to post-stall regimes through theoretical refinements. In contrast, Atassi's theory exhibits significant deviations from experimental outcomes across the measured angles of attack. Nevertheless, a modified version of the theory aligns better with experimental results at small angles of attack, whereas substantial discrepancies persist as the angle of attack increases. In the pre-stall regime, the aerodynamic response of the airfoil to vertical gusts displays a linear correlation with the flow angle near the leading edge. In the post-stall regime, the vertical gust induces dynamic stall of the airfoil. The flow angle has an essential effect on the lift coefficient but it alone is inadequate to dictate the trend of the lift coefficient. The vorticity statistics show that negative vortex circulation strongly correlates with the lift coefficient. Thus, further correction of the theory or a new vortex model can be expected to predict the lift variation.

Filaments are ubiquitous within the microscopic world, occurring in biological and industrial environments and displaying a varied dynamics. Their wide range of applications has spurred the development of a branch of asymptotics focused on the behaviour of filaments, called slender-body theory (SBT). Slender-body theories are computationally efficient and focus on the mechanics of an isolated fibre that is slender and not too curved. However, SBTs that work beyond these limits are needed to explore complex systems. Recently, we developed tubular-body theory (TBT), an approach like SBT that allows the hydrodynamic traction on any isolated fibre in a viscous fluid to be determined exactly. This paper extends TBT to model fibres near plane interfaces by performing a similar expansion on the single-layer boundary integrals (BIs) for bodies by a plane interface. This provides a well-behaved SBT inspired approach for fibres by interfaces with a similar versatility to the BIs but without the singular kernels. The derivation of the new theory, called tubular-body theory for interfaces (TBTi), also establishes a criterion for the convergence of the TBTi series representation. The TBTi equations are solved numerically using a approach similar to boundary element methods (BEMs), called TBTi-BEM, to investigate the properties of TBTi empirically. The TBTi-BEM is found to compare favourably with an existing BEM and the lubrication singularity on a sphere, suggesting TBTi is valid for all separations. Finally, we simulate the hydrodynamics of helices beneath a free interface and a plane wall to demonstrate the applicability of the technique.

Flow between axially rotating concentric cylinders is well known to exhibit rich dynamics. Hence, Taylor instabilities have been studied, both experimentally and theoretically, for many years. Although usually studied in the abstract, such geometries arise in a range of practical situations including drilling, when a drilling fluid flow enters a well via a pipe that is the centre body and returns via the annulus between the pipe and the borehole wall. In drilling, the centre body rotates and the annular flow contains rock cuttings. Here, we report the development of an Eulerian-Eulerian solver, based on OpenFOAM, that solves for this cuttings transport problem in the presence of both gravity and Taylor vortices. To check the reliability of the solver, we conduct a set of experiments spanning a wide range of complex flow regimes. We show that the model successfully predicts, in all regimes, the observed complex redistribution of particulates. However, for suspension flows under viscously dominated conditions, high particle concentrations and in rectilinear flow, particle pressure and normal stress differences are sufficient to capture particle migration. Results show that in more complex flows exemplified by the Taylor–Couette flow studied here, more realistic predictions of non-Brownian particle migration require inclusion of forces arising through the relative velocity of the two phases including lift forces originating both from inertia and particle rotation.