Bulletin of the American Physical Society
71st Annual Meeting of the APS Division of Fluid Dynamics
Volume 63, Number 13
Sunday–Tuesday, November 18–20, 2018; Atlanta, Georgia
Session A27: Flow Instability: Transition to Turbulence I
8:00 AM–9:57 AM,
Sunday, November 18, 2018
Georgia World Congress Center
Room: B315
Chair: Nigel Goldenfeld, University of Illinois at Urbana-Champaign
Abstract ID: BAPS.2018.DFD.A27.3
Abstract: A27.00003 : Eigenfunctions of the Orr-Sommerfeld/Squire Operator for Channel Flow
8:26 AM–8:39 AM
Presenter:
Anthony Leonard
(Caltech)
Author:
Anthony Leonard
(Caltech)
here we determine these eigenfunctions and eigenvalues semi-analytically for plane channel flow using the WKB approximation. The eigenfunctions are given in terms of the Bessel functions $J_{1/3}, J_{-1/3}, J_1,$ and $Y_1$. For the inviscid case, we find that for all $c$ in the real interval $[0, U_{max}]$ there is an infinite sequence of eigenfunctions
$v_j(y,c)$ with eigenvalue $k_{\perp j}^2(c)$ . We then show how these solutions may be used to solve the initial value problem. In the viscous case, we find that the eigenfunctions are composed of two types of solutions. One type is essentially an inviscid solution with a phase speed $c$ shifted by a complex factor proportional to $1/\sqrt{k_x Re}$ and is a relatively slowly varying function of $y$. The other is highly oscillatory, varying as $[(k_x Re)^{1/3}(y - y_c)]^{3/2}$ away from the critical layer at $y = y_c$.
Applications are made to plane Poiseuille flow.
To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2018.DFD.A27.3
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