Traveling hairpin-shaped fluid vortices in plane Couette flow

K. Deguchi and M. Nagata

Phys. Rev. E 82, 056325 – Published 29 November 2010

Abstract

Traveling-wave solutions are discovered in plane Couette flow. They are obtained when the so-called steady hairpin vortex state found recently by Gibson et al. [J. Fluid Mech. 638, 243 (2009)] and Itano and Generalis [Phys. Rev. Lett. 102, 114501 (2009)] is continued to sliding Couette flow geometry between two concentric cylinders by using the radius ratio as a homotopy parameter. It turns out that in the plane Couette flow geometry two traveling waves having the phase velocities with opposite signs are associated with their appearance from the steady hairpin vortex state, where the amplitude of the phase velocities increases gradually from zero as the Reynolds number is increased. The solutions obviously inherit the streaky structure of the hairpin vortex state, but shape preserving flow patterns propagate in the streamwise direction. Other striking features of the solution are asymmetric mean flow profiles and strong quasistreamwise vortices which occupy the vicinity of only the top or bottom moving boundary, depending on the sign of the phase velocity. Furthermore, we find that the pitchfork bifurcation associated with the appearance of the solution becomes imperfect when the flow is perturbed by a Poiseuille flow component.

Received 18 March 2010

DOI:https://doi.org/10.1103/PhysRevE.82.056325

Authors & Affiliations

K. Deguchi and M. Nagata

Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan

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Vol. 82, Iss. 5 — November 2010

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Images

Figure 1
The continuation of the solution under the radius ratio $η$ for wave numbers $(α, β) = (0.75, 1.37)$ and $R = 200$ . (a) The momentum transport $M$ and (b) the phase velocity $c$ . The two closed circles on $η = 1$ indicate the solutions on the upper and lower HVS branches. The open circle on $η = 1$ indicates the traveling hairpin vortex in PCF.Reuse & Permissions
Figure 2
The bifurcation of the traveling-wave solution with $(α, β) = (0.75, 1.37)$ in PCF. (a) The momentum transport $M$ and (b) the phase velocity $c$ . The solid and dashed curves indicate the solution branches of HVS and the traveling wave, respectively. The two closed circles and the open circle on $R = 200$ correspond to those in Fig. 1.Reuse & Permissions
Figure 3
(Color online) The imperfect bifurcation of the traveling-wave solution in PCF when a plane Poiseuille flow component, $Q$ , is added for wave numbers $(α, β) = (0.75, 1.37)$ . The measures of the nonlinearity are (a) the momentum transport $M$ and (b) the phase velocity. The curves in red (solid), green (dashed), and blue (dotted) correspond to $Q = 0$ , $Q = 0.1$ , and $Q = 1$ , respectively.Reuse & Permissions
Figure 4
(Color online) The real part of the growth rate of a perturbation with $(α, β) = (0.75, 1.37)$ . The three largest real parts for $Q = 0$ and $Q = 0.1$ are shown in (a) and (b), respectively. The next largest real parts (from the fourth to the tenth) for $Q = 0$ and $Q = 0.1$ are shown in (c) and (d), respectively. The figures include two growth rates $σ \equiv 0$ for all $R$ , which results from the invariance of the solution to infinitesimal translations. The solid or dashed curves imply that $σ$ is real or complex conjugate, respectively. For $Q = 0$ , the red (thick) and green (thin) curves correspond to HVS and traveling waves, respectively. For $Q = 0.1$ , the red (thick, $142.6 \leq R \leq 143.8$ ), blue (thick, $143.3 \leq R \leq 143.8$ ), and green (thin, $143.3 \leq R \leq 143.8$ ) curves correspond to traveling-wave solutions with negative, smaller positive, and larger positive phase velocities, respectively.Reuse & Permissions
Figure 5
The mean flow $\bar{u}$ of HVS and the traveling-wave solutions with wave numbers $(α, β) = (0.75, 1.37)$ at $R = 200$ . The solid curve: the upper HVS branch solution $u^{E Q +}$ . The dashed curve: the lower HVS branch solution $u^{E Q -}$ . The dashed-dotted curve: the traveling-wave solution $u^{T W +}$ . The dotted curve: the traveling-wave solution $u^{T W -}$ .Reuse & Permissions
Figure 6
(Color online) Isosurfaces of the streamwise velocity component $u$ (gray) and the streamwise vorticity component $ω_{x}$ [red (light gray): positive and blue (dark gray): negative] for wave numbers $(α, β) = (0.75, 1.37)$ at $R = 200$ . (a) $u = 120$ and $| ω_{x} | = 300$ for $u^{E Q +}$ , (b) $u = 120$ and $| ω_{x} | = 100$ for $u^{E Q -}$ , (c) $u = 120$ and $| ω_{x} | = 250$ for $u^{T W +}$ , and (d) $u = 120$ and $| ω_{x} | = 250$ for $u^{T W -}$ . The isosurface of the streamwise velocity component $u$ at $u = - 120$ in each case is located near the boundary $z = 1$ symmetrically to the isosurface at $u = 120$ but not displayed for clarity.Reuse & Permissions

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