Surface streaming on nonbreaking wind waves

<p>The energetic, coherent vortical motions in the aqueous surface layer beneath the wind waves dominate the liquid-phase controlled transport processes across the air-water interface. Through interacting with the interface, these coherent vortices manifest themselves by forming quasi-streamwise, high-speed streaks on the wind waves. The density of these streamwise streaks, which can be quantified by the transverse spacing of streaks, thus characterizes the interfacial transfer contributed by the coherent vortices. The formation of surface streaming on the wind waves is geometrically similar to the low-speed streaks observed in the turbulent wall layers. It is generally accepted that the mean spanwise spacing between these low-speed streaks, when scaled by the viscous length, would exhibit a universal value of 100. Observations in wind-wave flumes, however, show that the transverse scale between high-speed streaming on nonbreaking wind waves is narrower than that between low-speed streaks next to no-slip wall. Comparative numerical simulations of shear flow bounded by flat and wavy surfaces are conducted to explain the variation. Analysis of the vorticity transport in the simulated flows bounded by a wavy surface reveals that the presence of surface waves enhances the production of streamwise enstrophy and, consequently, intensifies the generation of quasi-streamwise vortices that form the elongated streaks.<br>This work is supported by the Taiwan Ministry of Science and Technology (107-2611-M-002 -014 -MY3 and 110-2923-M-002 -014 -MY3).</p>

Self-organisation of morphology and sediment transport in alluvial rivers

<div> <div> <div> <p>The coupling of sediment transport with the flow that drives it shapes the bed of alluvial rivers. The channel steers the flow, which in turns deforms the bed through erosion and sedimentation. To investigate this process, we produce a small river in a laboratory experiment by pouring a viscous fluid on a layer of plastic sediment. This laminar river gradually reaches its equilibrium shape. In the absence of sediment transport, the combination of gravity and flow-induced stress maintains the bed surface at the threshold of motion (Seizilles et al., 2013). If we impose a sediment discharge, the river widens and shallows to accommodate this input. Particle tracking reveals that the grains entrained by the flow behave as random walkers. Accordingly, they diffuse towards the less active areas of the bed (Seizilles et al., 2014). The river then adjusts its shape to maintain the balance between this diffusive flux, which pushes the grains towards the banks, and gravity, which pulls them towards the center of the channel. This dynamical equilibrium results in a peculiar Boltzmann distribution, in which the local sediment flux decreases exponentially with the elevation of the bed (Abramian et al., 2019). As the sediment discharge increases, the channel gets wider and shallower. Eventually, it destabilizes into multiple channels. A linear stability analysis suggests that it is diffusion that causes this instability, which could explain the formation of braided rivers (Abramian, Devauchelle, and Lajeunesse, 2019).</p> </div> </div> </div><p> </p><p>References:</p><ul><li>Abramian, A., Devauchelle, O., and Lajeunesse, E., “Streamwise streaks induced by bedload diffusion,” Journal of Fluid Mechanics 863, 601–619 (2019).</li> <li>Abramian, A., Devauchelle, O., Seizilles, G., and Lajeunesse, E., “Boltzmann distribution of sediment transport,” Physical review letters 123, 014501 (2019).</li> <li>Seizilles, G., Devauchelle, O., Lajeunesse, E., and M ́etivier, F., “Width of laminar laboratory rivers,” Phys. Rev. E. 87, 052204 (2013).</li> <li> <p>Seizilles, G., Lajeunesse, E., Devauchelle, O., and Bak, M., “Cross-stream diffusion in bedload transport,” Phys. of Fluids 26, 013302 (2014).</p> </li> </ul>

Streamwise streaks induced by bedload diffusion

A fluid flowing over a granular bed can move its superficial grains, and eventually deform it by erosion and deposition. This coupling generates a beautiful variety of patterns, such as ripples, bars and streamwise streaks. Here, we investigate the latter, sometimes called ‘sand ridges’ or ‘sand ribbons’. We perturb a sediment bed with sinusoidal streaks, the crests of which are aligned with the flow. We find that, when their wavelength is much larger than the flow depth, bedload diffusion brings mobile grains from troughs, where they are more numerous, to crests. Surprisingly, gravity can only counter this destabilising mechanism when sediment transport is intense enough. Relaxing the long-wavelength approximation, we find that the cross-stream diffusion of momentum mitigates the influence of the bed perturbation on the flow, and even reverses it for short wavelengths. Viscosity thus opposes the diffusion of entrained grains to select the most unstable wavelength. This instability might turn single-thread alluvial rivers into braided channels.

Transient energy growth of a swirling jet with vortex breakdown

We investigate the existence of short-time, local transient growth in the helical modes of a rapidly swirling, high-speed jet that has transitioned into an axisymmetric bubble breakdown state. The time-averaged flow consisting of the bubble and its wake downstream constitute the base state, which we show to exhibit strong transient amplification owing to the non-modal behaviour of the continuous eigenspectrum. A pseudospectrum analysis mathematically identifies the so-called potential modes within this continuous spectrum and the resultant non-orthogonality between these modes and the existing discrete stable modes is shown to be the main contributor to such growth. As the swirling flow develops post the collapsed bubble, the potential spectrum moves further toward the unstable half-plane, which along with the concurrent weakening of exponential growth from the discrete unstable modes, increases the dynamic importance of transient growth inside the wake region. The transient amplifications calculated at several locations inside the bubble and wake confirm this, where strong growths inside the wake far outstrip the corresponding modal growths (if available) at shorter times, but especially at the higher helical orders and smaller streamwise wavenumbers. The corresponding optimal perturbations at initial times consist of streamwise streaks of azimuthal velocity, which if concentrated inside the core vortical region, unfold via the classical Orr mechanism to yield structures resembling core (or viscous) Kelvin waves of the corresponding Lamb–Oseen vortex. However, in contrast to that in Lamb–Oseen vortex flow, where critical-layer waves are associated with higher transient gains, here, such core Kelvin modes with the more compact spiral structure at the vortex core are seen to yield the maximum transient amplifications.

Early azimuthal instability during drop impact

When a drop impacts on a liquid surface its bottom is deformed by lubrication pressure and it entraps a thin disc of air, thereby making contact along a ring at a finite distance from the centreline. The outer edge of this contact moves radially at high speed, governed by the impact velocity and bottom radius of the drop. Then at a certain radial location an ejecta sheet emerges from the neck connecting the two liquid masses. Herein, we show the formation of an azimuthal instability at the base of this ejecta, in the sharp corners at the two sides of the ejecta. They promote regular radial vorticity, thereby breaking the axisymmetry of the motions on the finest scales. The azimuthal wavenumber grows with the impact Weber number, based on the bottom curvature of the drop, reaching over 400 streamwise streaks around the periphery. This instability occurs first at Reynolds numbers ($Re$) of ${\sim}7000$, but for larger $Re$ is overtaken by the subsequent axisymmetric vortex shedding and their interactions can form intricate tangles, loops or chains.

Influence of optimally amplified streamwise streaks on the Kelvin–Helmholtz instability

The optimal energy amplifications of streamwise-uniform and spanwise-periodic perturbations of the hyperbolic-tangent mixing layer are computed and found to be very large, with maximum amplifications increasing with the Reynolds number and with the spanwise wavelength of the perturbations. The optimal initial conditions are streamwise vortices and the most amplified structures are streamwise streaks with sinuous symmetry in the cross-stream plane. The leading suboptimal perturbations have opposite (varicose) symmetry. When forced with finite amplitudes these perturbations modify the characteristics of the Kelvin–Helmholtz instability. Maximum temporal growth rates are reduced by optimal sinuous perturbations and are slightly increased by varicose suboptimal ones. In contrast, the onset of absolute instability is delayed by varicose suboptimal perturbations and is slightly promoted by sinuous optimal ones. We show that if, instead of the computed fully nonlinear basic-flow distortions, the stability analysis is based on a shape assumption for the flow distortions, then opposite effects on the flow stability are predicted in most of the considered cases. These strong differences are attributed to the spanwise-uniform component of the nonlinear basic-flow distortion which, we conclude, should be systematically included in sensitivity analyses of the stability of two-dimensional basic flows to three-dimensional basic-flow perturbations. We finally show that the leading-order quadratic sensitivity of the eigenvalues to the amplitude of the streaks is preserved if the effects of the mean flow distortion are included in the sensitivity analysis.

Edge state modulation by mean viscosity gradients

Motivated by the relevance of edge state solutions as mediators of transition, we use direct numerical simulations to study the effect of spatially non-uniform viscosity on their energy and stability in minimal channel flows. What we seek is a theoretical support rooted in a fully nonlinear framework that explains the modified threshold for transition to turbulence in flows with temperature-dependent viscosity. Consistently over a range of subcritical Reynolds numbers, we find that decreasing viscosity away from the walls weakens the streamwise streaks and the vortical structures responsible for their regeneration. The entire self-sustained cycle of the edge state is maintained on a lower kinetic energy level with a smaller driving force, compared to a flow with constant viscosity. Increasing viscosity away from the walls has the opposite effect. In both cases, the effect is proportional to the strength of the viscosity gradient. The results presented highlight a local shift in the state space of the position of the edge state relative to the laminar attractor with the consequent modulation of its basin of attraction in the proximity of the edge state and of the surrounding manifold. The implication is that the threshold for transition is reduced for perturbations evolving in the neighbourhood of the edge state in the case that viscosity decreases away from the walls, and vice versa.

Non-modal stability analysis of the boundary layer under solitary waves

In the present work, a stability analysis of the bottom boundary layer under solitary waves based on energy bounds and non-modal theory is performed. The instability mechanism of this flow consists of a competition between streamwise streaks and two-dimensional perturbations. For lower Reynolds numbers and early times, streamwise streaks display larger amplification due to their quadratic dependence on the Reynolds number, whereas two-dimensional perturbations become dominant for larger Reynolds numbers and later times in the deceleration region of this flow, as the maximum amplification of two-dimensional perturbations grows exponentially with the Reynolds number. By means of the present findings, we can give some indications on the physical mechanism and on the interpretation of the results by direct numerical simulation in Vittori & Blondeaux (J. Fluid Mech., vol. 615, 2008, pp. 433–443) and Özdemir et al. (J. Fluid Mech., vol. 731, 2013, pp. 545–578) and by experiments in Sumer et al. (J. Fluid Mech., vol. 646, 2010, pp. 207–231). In addition, three critical Reynolds numbers can be defined for which the stability properties of the flow change. In particular, it is shown that this boundary layer changes from a monotonically stable to a non-monotonically stable flow at a Reynolds number of $Re_{\unicode[STIX]{x1D6FF}}=18$.