5 results
Liquid inertia versus bubble cloud buoyancy in circular plunging jet experiments
- Narendra Dev, J. John Soundar Jerome, Hélène Scolan, Jean-Philippe Matas
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- Journal:
- Journal of Fluid Mechanics / Volume 978 / 10 January 2024
- Published online by Cambridge University Press:
- 05 January 2024, A23
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When a liquid jet plunges into a pool, it can generate a bubble-laden jet flow underneath the surface. This common and simple phenomenon is investigated experimentally for circular jets to illustrate and quantify the role played by the net gas/liquid void fraction on the maximum bubble penetration depth. It is first shown that an increase in either the impact diameter or the jet fall height to diameter ratio at constant impact momentum leads to a reduction in the bubble cloud size. By measuring systematically the local void fraction using optical probes in the biphasic jet, it is then demonstrated that this effect is a direct consequence of the increase in air content within the cloud. A simple momentum balance model, including only inertia and the buoyancy force, is shown to predict the bubble cloud depth without any fitting parameters. Finally, a Froude number based on the bubble terminal velocity, the cloud depth and also the net void fraction is introduced to propose a simple criterion for the threshold between the inertia-dominated and buoyancy-dominated regimes.
Bubble rise in a Hele-Shaw cell: bridging the gap between viscous and inertial regimes
- Benjamin Monnet, Christopher Madec, Valérie Vidal, Sylvain Joubaud, J. John Soundar Jerome
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- Journal:
- Journal of Fluid Mechanics / Volume 942 / 10 July 2022
- Published online by Cambridge University Press:
- 18 May 2022, R3
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The rise of a single bubble confined between two vertical plates is investigated over a wide range of Reynolds numbers. In particular, we focus on the evolution of the bubble speed, aspect ratio and drag coefficient during the transition from the viscous to the inertial regime. For sufficiently large bubbles, a simple model based on power balance captures the transition for the bubble velocity and matches all the experimental data despite strong time variations of bubble aspect ratio at large Reynolds numbers. Surprisingly, bubbles in the viscous regime systematically exhibit an ellipse elongated along its direction of motion while bubbles in the inertia-dominated regime are always flattened perpendicularly to it.
Inertial drag-out problem: sheets and films on a rotating disc
- J. John Soundar Jerome, Sébastien Thevenin, Mickaël Bourgoin, Jean-Philippe Matas
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- Journal:
- Journal of Fluid Mechanics / Volume 908 / 10 February 2021
- Published online by Cambridge University Press:
- 03 December 2020, A7
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The so-called Landau–Levich–Deryaguin problem treats the coating flow dynamics of a thin viscous liquid film entrained by a moving solid surface. In this context, we use a simple experimental set-up consisting of a partially immersed rotating disc in a liquid tank to study the role of inertia, and also curvature, on liquid entrainment. Using water and UCON$^{\textrm{TM} }$ mixtures, we point out a rich phenomenology in the presence of strong inertia: ejection of multiple liquid sheets on the emerging side of the disc, sheet fragmentation, ligament formation and atomization of the liquid flux entrained over the disc's rim. We focus our study on a single liquid sheet and the related average liquid flow rate entrained over a thin disc for various depth-to-radius ratio $h/R < 1$. We show that the liquid sheet is created via a ballistic mechanism as liquid is lifted out of the pool by the rotating disc. We then show that the flow rate in the entrained liquid film is controlled by both viscous and surface tension forces as in the classical Landau–Levich–Deryaguin problem, despite the three-dimensional, non-uniform and unsteady nature of the flow, and also despite the large values of the film thickness based flow Reynolds number. When the characteristic Froude and Weber numbers become significant, strong inertial effects influence the entrained liquid flux over the disc at large radius-to-immersion-depth ratio, namely via entrainment by the disc's lateral walls and via a contribution to the flow rate extracted from the three-dimensional liquid sheet itself, respectively.
A note on Stokes’ problem in dense granular media using the $\unicode[STIX]{x1D707}(I)$-rheology
- J. John Soundar Jerome, B. Di Pierro
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- Journal:
- Journal of Fluid Mechanics / Volume 847 / 25 July 2018
- Published online by Cambridge University Press:
- 23 May 2018, pp. 365-385
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The classical Stokes’ problem describing the fluid motion due to a steadily moving infinite wall is revisited in the context of dense granular flows of mono-dispersed beads using the recently proposed $\unicode[STIX]{x1D707}(I)$-rheology. In Newtonian fluids, molecular diffusion brings about a self-similar velocity profile and the boundary layer in which the fluid motion takes place increases indefinitely with time $t$ as $\sqrt{\unicode[STIX]{x1D708}t}$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity. For a dense granular viscoplastic liquid, it is shown that the local shear stress, when properly rescaled, exhibits self-similar behaviour at short time scales and it then rapidly evolves towards a steady-state solution. The resulting shear layer increases in thickness as $\sqrt{\unicode[STIX]{x1D708}_{g}t}$ analogous to a Newtonian fluid where $\unicode[STIX]{x1D708}_{g}$ is an equivalent granular kinematic viscosity depending not only on the intrinsic properties of the granular medium, such as grain diameter $d$, density $\unicode[STIX]{x1D70C}$ and friction coefficients, but also on the applied pressure $p_{w}$ at the moving wall and the solid fraction $\unicode[STIX]{x1D719}$ (constant). In addition, the $\unicode[STIX]{x1D707}(I)$-rheology indicates that this growth continues until reaching the steady-state boundary layer thickness $\unicode[STIX]{x1D6FF}_{s}=\unicode[STIX]{x1D6FD}_{w}(p_{w}/\unicode[STIX]{x1D719}\unicode[STIX]{x1D70C}g)$, independent of the grain size, at approximately a finite time proportional to $\unicode[STIX]{x1D6FD}_{w}^{2}(p_{w}/\unicode[STIX]{x1D70C}gd)^{3/2}\sqrt{d/g}$, where $g$ is the acceleration due to gravity and $\unicode[STIX]{x1D6FD}_{w}=(\unicode[STIX]{x1D70F}_{w}-\unicode[STIX]{x1D70F}_{s})/\unicode[STIX]{x1D70F}_{s}$ is the relative surplus of the steady-state wall shear stress $\unicode[STIX]{x1D70F}_{w}$ over the critical wall shear stress $\unicode[STIX]{x1D70F}_{s}$ (yield stress) that is needed to bring the granular medium into motion. For the case of Stokes’ first problem when the wall shear stress $\unicode[STIX]{x1D70F}_{w}$ is imposed externally, the $\unicode[STIX]{x1D707}(I)$-rheology suggests that the wall velocity simply grows as $\sqrt{t}$ before saturating to a constant value whereby the internal resistance of the granular medium balances out the applied stresses. In contrast, for the case with an externally imposed wall speed $u_{w}$, the dense granular medium near the wall initially maintains a shear stress very close to $\unicode[STIX]{x1D70F}_{d}$ which is the maximum internal resistance via grain–grain contact friction within the context of the $\unicode[STIX]{x1D707}(I)$-rheology. Then the wall shear stress $\unicode[STIX]{x1D70F}_{w}$ decreases as $1/\sqrt{t}$ until ultimately saturating to a constant value so that it gives precisely the same steady-state solution as for the imposed shear-stress case. Thereby, the steady-state wall velocity, wall shear stress and the applied wall pressure are related as $u_{w}\sim (g\unicode[STIX]{x1D6FF}_{s}^{2}/\unicode[STIX]{x1D708}_{g})f(\unicode[STIX]{x1D6FD}_{w})$ where $f(\unicode[STIX]{x1D6FD}_{w})$ is either $O(1)$ if $\unicode[STIX]{x1D70F}_{w}\sim \unicode[STIX]{x1D70F}_{s}$ or logarithmically large as $\unicode[STIX]{x1D70F}_{w}$ approaches $\unicode[STIX]{x1D70F}_{d}$.
Extended Squire’s transformation and its consequences for transient growth in a confined shear flow
- J. John Soundar Jerome, Jean-Marc Chomaz
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- Journal:
- Journal of Fluid Mechanics / Volume 744 / 10 April 2014
- Published online by Cambridge University Press:
- 13 March 2014, pp. 430-456
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The classical Squire transformation is extended to the entire eigenfunction structure of both Orr–Sommerfeld and Squire modes. For arbitrary Reynolds numbers $\mathit{Re}$, this transformation allows the solution of the initial-value problem for an arbitrary three-dimensional (3D) disturbance via a two-dimensional (2D) initial-value problem at a smaller Reynolds number $\mathit{Re}_{2D}$. Its implications for the transient growth of arbitrary 3D disturbances is studied. Using the Squire transformation, the general solution of the initial-value problem is shown to predict large-Reynolds-number scaling for the optimal gain at all optimization times $t$ with ${t}/{\mathit{Re}}$ finite or large. This result is an extension of the well-known scaling laws first obtained by Gustavsson (J. Fluid Mech., vol. 224, 1991, pp. 241–260) and Reddy & Henningson (J. Fluid Mech., vol. 252, 1993, pp. 209–238) for arbitrary $\alpha \mathit{Re}$, where $\alpha $ is the streamwise wavenumber. The Squire transformation is also extended to the adjoint problem and, hence, the adjoint Orr–Sommerfeld and Squire modes. It is, thus, demonstrated that the long-time optimal growth of 3D perturbations as given by the exponential growth (or decay) of the leading eigenmode times an extra gain representing its receptivity, may be decomposed as a product of the gains arising from purely 2D mechanisms and an analytical contribution representing 3D growth mechanisms equal to $1+ \left (\beta \mathit{Re}/\mathit{Re}_{2D}\right )^2 \mathcal{G}$, where $\beta $ is the spanwise wavenumber and $\mathcal{G}$ is a known expression. For example, when the leading eigenmode is an Orr–Sommerfeld mode, it is given by the product of respective gains from the 2D Orr mechanism and an analytical expression representing the 3D lift-up mechanism. Whereas if the leading eigenmode is a Squire mode, the extra gain is shown to be solely due to the 3D lift-up mechanism. Direct numerical solutions of the optimal gain for plane Poiseuille and plane Couette flow confirm the novel predictions of the Squire transformation extended to the initial-value problem. These results are also extended to confined shear flows in the presence of a temperature gradient.