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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 $\unicode[STIX]{x1D70F}_{d}$.
Space–time dynamics of optimal wavepackets for streaks in a channel entrance flow
- F. Alizard, A. Cadiou, L. Le Penven, B. Di Pierro, M. Buffat
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- Journal:
- Journal of Fluid Mechanics / Volume 844 / 10 June 2018
- Published online by Cambridge University Press:
- 06 April 2018, pp. 669-706
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The laminar–turbulent transition of a plane channel entrance flow is revisited using global linear optimization analyses and direct numerical simulations. The investigated case corresponds to uniform upstream velocity conditions and a moderate value of Reynolds number so that the two-dimensional developing flow is linearly stable under the parallel flow assumption. However, the boundary layers in the entry zone are capable of supporting the development of streaks, which may experience secondary instability and evolve to turbulence. In this study, global optimal linear perturbations are computed and studied in the nonlinear regime for different values of streak amplitude and optimization time. These optimal perturbations take the form of wavepackets having either varicose or sinuous symmetry. It is shown that, for short optimization times, varicose wavepackets grow through a combination of Orr and lift-up effects, whereas for longer target times, both sinuous and varicose wavepackets exhibit an instability mechanism driven by the presence of inflection points in the streaky flow. In addition, while the optimal varicose modes obtained for short optimization times are localized near the inlet, where the base flow is strongly three-dimensional, when the target time is increased, the sinuous and varicose optimal modes are displaced farther downstream, in the nearly parallel streaky flow. Finally, the optimal wavepackets are found to lead to turbulence for sufficiently high initial amplitudes. It is noticed that the resulting turbulent flows have the same wall-shear stress, whether the wavepackets have been obtained for short or for long time optimization.