Depinning of liquid droplets on substrates by flow of a surrounding immiscible fluid is central to applications such as cross-flow microemulsification, oil recovery and waste cleanup. Surface roughness, either natural or engineered, can cause droplet pinning, so it is of both fundamental and practical interest to determine the flow strength of the surrounding fluid required for droplet depinning on rough substrates. Here, we develop a lubrication-theory-based model for droplet depinning on a substrate with topographical defects by flow of a surrounding immiscible fluid. The droplet and surrounding fluid are in a rectangular channel, a pressure gradient is imposed to drive flow and the defects are modelled as Gaussian-shaped bumps. Using a precursor-film/disjoining-pressure approach to capture contact-line motion, a nonlinear evolution equation is derived describing the droplet thickness as a function of distance along the channel and time. Numerical solutions of the evolution equation are used to investigate how the critical pressure gradient for droplet depinning depends on the viscosity ratio, surface wettability and droplet volume. Simple analytical models are able to account for many of the features observed in the numerical simulations. The influence of defect height is also investigated, and it is found that, when the maximum defect slope is larger than the receding contact angle of the droplet, smaller residual droplets are left behind at the defect after the original droplet depins and slides away. The model presented here yields considerably more information than commonly used models based on simple force balances, and provides a framework that can readily be extended to study more complicated situations involving chemical heterogeneity and three-dimensional effects.

Understanding settling motion of coral grains is important in terms of protection of coral reef systems and resotoration of the associated ecosystems. In this paper, a series of laboratory experiments was conducted to investigate the settling motion, using optical microscopy to measure shape parameters of coral grains and the particle-filtering-based object tracking to reconstruct the three-dimensional trajectory. Three characteristic descent regimes, namely, tumbling, chaotic and fluttering, are classified based on the three-dimensional trajectory, the spiral radius variation and the velocity spectrum. It is demonstrated that if one randomly picks up one coral grain, then the probabilities of occurrence of the three regimes are approximately $26\,\%$, $42\,\%$ and $32\,\%$, respectively. We have shown that first, the dimensionless settling velocity generally increases with the non-dimensional diameter and Corey shape factor and second, the drag coefficient generally decreases with the Reynolds number and Corey shape factor. Based on this, the applicability of existing models on predicting settling velocity and drag coefficient for coral grains is demonstrated further. Finally, we have proposed extended models for predicting the settling velocity. This study contributes to better understanding of settling motion and improves our predictive capacity of settling velocity for coral grains with complex geometry.

We investigated the stability of the bottom boundary layer (BBL) beneath periodic internal solitary waves (ISWs) of depression over a flat bottom through two-dimensional direct numerical simulations. We explored the convective versus absolute/global nature of the BBL instability in response to changes in Reynolds number, and the sensitivity of the instability to seeding noise in the front of the ISW – spanning laboratory to geophysical scales. The BBL was laminar at $Re_{ISW}=90$ and convectively unstable at $Re_{ISW}=300$. At laboratory-scale $Re_{ISW}=300$, the convective wave packet was periodically amplified by each successive ISW, until vortex shedding occurred. The associated noise-amplification behaviour potentially explains the discrepancies of the critical $Re_{ISW}$ between the lock–release laboratory experiments and our Dubreil–Jacotin–Long-initialized numerical simulations as the result of the difference in background noise. Instability energy decreased under the front shoulder of the ISW, analogous to flow relaminarization under a favourable pressure gradient. At geophysical-scale $Re_{ISW}=900$, the BBL was initially convectively unstable, and then the instability tracked with the ISW, appearing phenomenologically similar to a global instability. The simulated initial convective instability at both $Re_{ISW}=300$ and $Re_{ISW}=900$ is in agreement with local linear stability analysis which predicts that the instability group speed is always lower than the ISW celerity. Increased free stream perturbations in front of the ISW and larger $Re_{ISW}$ shift the location of vortex shedding (and enhanced bed shear stress) beneath the wave, closer to the ISW trough, thereby potentially changing the location of maximum sediment resuspension, in agreement with field observations at higher $Re_{ISW}$.

Saturated flow film boiling on a sphere has been numerically studied in this work for both vertical and horizontal flow configurations. The simulations were performed using a numerical methodology developed by the authors for boiling flows on three-dimensional unstructured meshes. For interface capturing, the coupled level set and volume of fluid method is used. The interface evolution, vapour wake dynamics and heat transfer have been thoroughly investigated by varying the saturated liquid flow velocity, sphere diameter and wall superheat. The relative importance of both the buoyancy and the inertial forces is described in terms of the Froude number $(Fr)$. The vapour bubble evolves periodically at low $Fr$ values, while a stable vapour column develops at high $Fr$ values. The interface evolution pattern obtained in the present work is in good agreement with the results of experimental studies available in the literature. For all the values of $Fr$, a stable vapour column develops for a large-diameter sphere and releases vapour bubbles of varying sizes. Furthermore, for a large-diameter sphere, surface capillary waves are observed at the interface, similar to the observations of some of the experimental studies available in the literature. The flow in the liquid and vapour wakes appears to be strongly coupled. The heat transfer in the present work is estimated using the spatially and temporally averaged Nusselt numbers. Finally, an fast Fourier transform analysis of the space-averaged Nusselt number reveals a strong interaction among the different forces.

Wall temperature has a significant effect on shock wave/turbulent boundary layer interactions (STBLIs) and has become a non-negligible factor in the design process of hypersonic vehicles. In this paper, direct numerical simulations are conducted to investigate the wall temperature effects on STBLIs over a 34° compression ramp at Mach number 6. Three values of the wall-to-recovery-temperature ratio (0.50, 0.75 and 1.0) are considered in the simulations. The results show that the size of the separation bubble declines significantly as the wall temperature decreases. This is because the momentum profile of the boundary layer becomes fuller with wall cooling, which means the near-wall fluid has a greater momentum to suppress flow separation. An equation based on the free-interaction theory is proposed to predict the distributions of the wall pressure upstream of the corner at different wall temperatures. The prediction results are generally consistent with the simulation results (Reynolds number Reτ ranges from 160 to 675). In addition, the low-frequency unsteadiness is studied through the weighted power spectral density of the wall pressure and the correlation between the upstream and downstream. The results indicate that the low-frequency motion of the separation shock is mainly driven by the downstream mechanism and that wall cooling can significantly suppress the low-frequency unsteadiness, including the strength and streamwise range of the low-frequency motions.