In 2015, I was invited by F. Moisy (FAST) to collaborate on a research project on orbital sloshing. Two interns, R. Kostenko and J. Bouvard, performed extensive experimental campaigns that gave rise to two articles, published in Phys. Rev. Fluids and Europhys. Lett. (see here and here for preprints)
Orbital sloshing is just a fancy name for what we do when we swirl a glass of wine during a tasting ceremony. We move around the glass in a circular translational motion and naturally find the right shaking frequency that allows a resonant amplification of a gravity wave. The rotating wave comes along with a mean flow, that can be split into an azimuthal swirling flow (toroidal) and a poloidal recirculation.
The poloidal mean flow controls how the free surface is being renewed and thus contributes to liberating the aromas in the surrounding air. This is also used in orbital sloshing devices, used in many biological laboratories to softly blend and oxygenize liquids containing cells or micro-organisms (used to massively cultivate anti-biotics)
In the first part of this research projet, we have characterized the intensity and spatial structure of the mean flow that comes along with the rotating wave flow. We use stroboscopic PIV experiments and propose a simple scaling law for the intensity of this mean flow (ubar), for forcing frequencies (Omega) up to the resonant frequency (omega_1)
Here epsilon=A/R, the non-dimensional radius of the circular translation imposed to the container. Stroboscopic PIV can only measure the Lagrangian mean flow, composed of the Stokes drift (see this page for an explanation) of purely kinematic origin and a steady streaming flow that has a dynamical origin. We find that the Stokes drift quite decently describes the azimuthal mean flow, but also that a non-zero steady streaming flow is required to explain the observations at high Reynolds number. Non-linearities in the boundary layer under the moving contact line are likely in control of the poloidal recirculation.
This problem learned me that detailed analytical calculations do not always provide the best insights. I literally spent a few months trying to calculate the steady streaming flow, which is a surprisingly difficult task here. The standard technique is to calculate the nonlinear interactions in the viscous boundary layers that will effectively force a mean flow in the bulk. This may seem straightforward, but there is one big problem: boundary layers under the moving contact line are really not trivial at all and not available in analytical form. If we assume ordinary Stokes layers up until the contact line, one can calculate a steady streaming flow, but it certainly is very different from what we observe in experiments and this made me abandon the race. If we can’t get the viscous wave flow correct near the contact line, then it makes no sense to try to evaluate non-linearities there-in. As wave motion is the most intense near the contact line, it is my guess that this complex contact line region has a profound impact on the wave and on the steady streaming.
Other people have continued this story. F. Viola & F. Gallaire (Phys. Rev. Fluids, 3, 094801, 2018) have used a convincing model for the contact line dynamics to study how this affects the wave. In that work, there is a clear vortical region near the contact line and this will certainly affect the steady streaming. O.M. Faltinsen & A.N. Timokha present an inviscid mechanism (J. Fluid Mech. 865, 2019 , 884-903) that could explain the azimuthal part of the steady streaming. This model ignores the contact line specificity and also violates a basic kinematical principle that mass is transported on average by the Lagrangian mean flow (steady streaming + Stokes drift). Nevertheless, it is interesting to observe that it does reproduce our experimental data very well. Most recently, Huang et al. (see their preprint on arXiv) have shown experimentally how the capillary mensicus (the contact line) has an influence on the observed streaming patterns.
ORBITAL SLOSHING, part 2
Together with F. Moisy and J. Bouvard, we went on to study how the mean flow in an orbitally sloshed fluids is affected by polluting the free surface with buyoant particles. For cohesive particles (bubbles, glass microbeads ), this had an extra-ordinary effect. Above a critical density of particles, we observed that the floating raft could reverse its sens of rotation and be in contra-rotation with the imposed orbtial motion. This is visualised in the following movie
We were able to model this transition from co-rotation to counter-rotation for an idealised circular raft of cohesive particles. Physically, the counter-rotation is similar to what we see in a planetary gear train
Imagine that we take the green gear weel in our hands and that we shake it orbitally in anti-clockwise direction. At the right shaking frequency and if the grey gear wheel is free to rotate, it will roll anti-clockwisely along the circular track shown by the dashed line, rotating in the clockwise, opposite sense.
In our fluid system, the raft of cohesive particles behaves as an elastic solid, very deformable in the vertical direction but rather stiffly held together in the horizontal plane. Each particle in the raft feels a friction force that is proportional to the relative speed with the underlying flow and this because of this, the raft will want to follow the fluid particle motion. This results in a precessing motion at first order and co-rotating motion at second order, for small rafts that remain far from the boundary. Large rafts will also precess but in this motion, they will penetrate into a boundary region where the particle or fluid motion is hindered. This will exert a negative torque on the raft that can caus the counter-rotation.
The study was published in EPL and entirely readable online. A preprint is still available on arXiv.
ORBITAL SLOSHING, part 3
In my work on metal pad roll instability, I also encountered gravity waves and calculated theoretical viscous damping rate formulas. With this result, G.M. Horstmann was able to derive a linear viscous model for orbital sloshing in cylindrical devices with one- or two fluid layers that compares quite well to experimental measurements. This work is has now been published in JFM.
