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Altmeyer, Sebastian Andreas

Hof, Björn

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Invariant solutions organizing pipe flow turbulence

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Invariant solutions organizing pipe flow turbulenceSebastian Altmeyer ∗, Bjo¨rn Hof †Invariant unstable solutions such as (relative) periodic orbits (RPOs) and travellingwaves (TWs) have been suggested to act as building blocks of turbulence in basicshear flows. A large number of such invariant solutions have been determined inrecent years, yet most observed states typically possess spatial symmetry (e.g. torotation, reflection or translation) due to artificial symmetry restrictions. In contrastturbulence does not have any of such symmetry in general. Commonly used recurrencemethods are unlikely to capture orbits in full space due to their complexity and theshort visiting times of turbulent trajectories.Nevertheless, looking for periodic modulations instead of full recurrences we havebeen able to extract asymmetric invariant solutions, RPOs and TWs, dynamicallyembedded in turbulence. Compared to other invariant solutions in subspaces theisosurfaces look less smooth indicating various different length scales within thesestructures. This is a typical observation in turbulent flows which also show strongfluctuations, e.g. in the internal arrangement of high and low velocity streaks. Thecomplexity of the underlying manifold results in closed curves either in Re and kdefining parameters with up to four solutions for a single RPO.(a) (b) (c)Figure 1: (a) Variation of Ekin with Re for RPOs (index numbers represent the period time)and TWs as indicated. Flow visualization of RPO6.511 (Re = 2550) and TW (Re = 2450);Shown are streamwise velocity uz = ±0.4U (red is positive and blue is negative; flow fromleft to right) and space-time averaged cross section perpendicular to the pipe axis. Thedownstream velocity relative to the parabolic laminar profile is shown in color ranging formred (fast) to blue (slow). In-plane velocity components are indicated by vectors.∗Castelldefels School of Telecom and Aerospace Engineering (EETAC), Universitat Polytecnicade Catalunya (UPC)†Institute of Science and Technology (IST) Austria

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Invariant solutions organizing pipe flow turbulence

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