Asymmetric, Helical, and Mirror-Symmetric Traveling Waves in Pipe Flow

Chris C. T. Pringle and Rich R. Kerswell

Phys. Rev. Lett. 99, 074502 – Published 16 August 2007

Abstract

New families of three-dimensional nonlinear traveling waves are discovered in pipe flow. In contrast with known waves [H. Faisst and B. Eckhardt, Phys. Rev. Lett. 91, 224502 (2003); H. Wedin and R. R. Kerswell, J. Fluid Mech. 508, 333 (2004)], they possess no discrete rotational symmetry and exist at a significantly lower Reynolds numbers (Re). First to appear is a mirror-symmetric traveling wave which is born in a saddle node bifurcation at $Re = 773$ . As Re increases, “asymmetric” modes arise through a symmetry-breaking bifurcation. These look to be a minimal coherent unit consisting of one slow streak sandwiched between two fast streaks located preferentially to one side of the pipe. Helical and nonhelical rotating waves are also found, emphasizing the richness of phase space even at these very low Reynolds numbers.

Received 20 March 2007

DOI:https://doi.org/10.1103/PhysRevLett.99.074502

Authors & Affiliations

Chris C. T. Pringle^* and Rich R. Kerswell^†

Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom

^*Chris.Pringle@bris.ac.uk
^†R.R.Kerswell@bris.ac.uk

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Issue

Vol. 99, Iss. 7 — 17 August 2007

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Images

Figure 1
Velocity fields for the asymmetric mode at $Re = 2900$ (top) and the mirror-symmetric mode at $Re = 1344$ (bottom) (both at $α = 0.75$ ). An instantaneous state is shown on the left and a streamwise-averaged state on the right. The coloring indicates the downstream velocity relative to the parabolic laminar profile: red (dark) through white (light) represents slow through fast (with zero corresponding to the shading outside the pipe). In-plane velocity components are shown by vectors. The maximum and minimum streamwise velocities (with the laminar flow subtracted) and maximum in-plane speed for the asymmetric mode are 0.33, $- 0.42$ , and 0.03, respectively, while for the mirror-symmetric mode they are 0.31, $- 0.43$ , and 0.08 (all in units of $U$ ).Reuse & Permissions
Figure 2
Phase velocity $C$ in units of $U$ as a function of $α$ for the mirror-symmetric modes (solid lines) and asymmetric modes (dashed) at 4 values of Re near the saddle node bifurcation at $Re = 773$ .Reuse & Permissions
Figure 3
A schematic picture of how all the new traveling wave branches fit together in ( $β$ , Re, $C$ ) space (at $α = 0.75$ ). The main parabolic curve in the $β = 0$ plane is the mirror-symmetric branch off which the asymmetric branch bifurcates (uppermost line). Helical branches bulge out of the $β = 0$ plane and connect upper and lower parts of the mirror symmetric. Across a finite range of Re, these helical modes perforate the $β = 0$ plane in between the mirror-symmetric branches, creating an isola of nonhelical rotating TWs (closed dashed-dotted loop). Helical waves also connect the asymmetric branch and the helical solutions which originate from the mirror-symmetric solutions. [solid (dashed) lines indicate confirmed (inferred) behavior].Reuse & Permissions
Figure 4
Two velocity slices across a helical mode taken at the same instant of time but $25 D$ apart with $α = 0.75$ , $β = 0.019$ , $ω = - 0.0011$ at $Re = 1344$ . The velocity representation is as in Fig. 1.Reuse & Permissions
Figure 5
The four fast streaks of the helical mode shown in Fig. 4 plotted over one $β$ wavelength $\approx 170 D$ .Reuse & Permissions
Figure 6
Friction factor $Λ$ against Re for the various TW families. The lower dashed line indicates the laminar value $Λ_{lam} = 64 / Re$ and the upper dash-dotted line indicates the log-law parametrization of experimental data $1 / \sqrt{Λ} = 2.0 \log (Re \sqrt{Λ})$ . The labels are $m$ values for the rotational symmetry $R_{m}$ of the different TW families all drawn at the wave number which leads to the lowest saddle node bifurcation. The new TWs shown—mirror-symmetric modes (solid) and asymmetric modes (dashed)—correspond to $m = 1$ and $α = 1.44$ . The bifurcation point where the asymmetric waves are born is marked with a dot. The inset shows the phase velocity $C$ (in units of $U$ ) versus Re for all the TWs (upper branch TWs have smaller phase speeds than lower branch TWs).Reuse & Permissions

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