dorsal/arxiv
View SchemaThe dynamics of transition to turbulence in plane Couette flow
| Authors | D. Viswanath |
|---|---|
| Categories | |
| ArXiv ID | physics/0701337 |
| URL | https://arxiv.org/abs/physics/0701337 |
| Journal | Mathematics and Computation, a Contemporary View. The Abel Symposium 2006, 2008 p. 109-127 |
Abstract
In plane Couette flow, the incompressible fluid between two plane parallel walls is driven by the motion of those walls. The laminar solution, in which the streamwise velocity varies linearly in the wall-normal direction, is known to be linearly stable at all Reynolds numbers ($Re$). Yet, in both experiments and computations, turbulence is observed for $Re \gtrsim 360$. In this article, we show that for certain {\it threshold} perturbations of the laminar flow, the flow approaches either steady or traveling wave solutions. These solutions exhibit some aspects of turbulence but are not fully turbulent even at $Re=4000$. However, these solutions are linearly unstable and flows that evolve along their unstable directions become fully turbulent. The solution approached by a threshold perturbation could depend upon the nature of the perturbation. Surprisingly, the positive eigenvalue that corresponds to one family of solutions decreases in magnitude with increasing $Re$, with the rate of decrease given by $Re^{\alpha}$ with $\alpha \approx -0.46$.
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"abstract": "In plane Couette flow, the incompressible fluid between two plane parallel\nwalls is driven by the motion of those walls. The laminar solution, in which\nthe streamwise velocity varies linearly in the wall-normal direction, is known\nto be linearly stable at all Reynolds numbers ($Re$). Yet, in both experiments\nand computations, turbulence is observed for $Re \\gtrsim 360$.\n In this article, we show that for certain {\\it threshold} perturbations of\nthe laminar flow, the flow approaches either steady or traveling wave\nsolutions. These solutions exhibit some aspects of turbulence but are not fully\nturbulent even at $Re=4000$. However, these solutions are linearly unstable and\nflows that evolve along their unstable directions become fully turbulent. The\nsolution approached by a threshold perturbation could depend upon the nature of\nthe perturbation. Surprisingly, the positive eigenvalue that corresponds to one\nfamily of solutions decreases in magnitude with increasing $Re$, with the rate\nof decrease given by $Re^{\\alpha}$ with $\\alpha \\approx -0.46$.",
"arxiv_id": "physics/0701337",
"authors": [
"D. Viswanath"
],
"categories": [
"physics.flu-dyn",
"physics.comp-ph"
],
"journal_ref": "Mathematics and Computation, a Contemporary View. The Abel\n Symposium 2006, 2008 p. 109-127",
"title": "The dynamics of transition to turbulence in plane Couette flow",
"url": "https://arxiv.org/abs/physics/0701337"
},
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