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Searching for Order in Turbulent Flow

Björn Hof, Nazmi Burak Budanur

2017
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Physics
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M ost of us associate turbulence with a bumpy ride on a plane. Yet turbulent flow is practically everywhere, even if we aren't aware of it. The air we exhale, the blood pumped through our bodies' ascending aorta, and the steam rising from a hot cup of coffee are all in turbulent motion. Even if we could see these flows, it would be difficult to discern any order in the whirling currents that they contain. A group led by Michael Schatz at the Georgia Institute of Technology in Atlanta has now
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... wn with experiments and simulations that flow patterns with a surprising amount of order can underlie turbulent flow [1], at least when the turbulence is weak. Having identified these ordered structures in the lab, they were able to predict the evolution of a turbulent flow, which had previously been difficult to do. There is no short answer to the question "What is turbulence?" In contrast to the well-ordered laminar flow that prevails at low flow speeds, turbulence is disordered and fluctuating, and it contains whirls of many different sizes that exchange energy with one another. As such, turbulence has proven extremely difficult to describe from first principles. Although the Navier-Stokes equations governing fluid flow are well known, their nonlinear nature means that they can only be solved in exceptional cases (such as laminar pipe flow). Thus there is little hope to obtain an expression that would encompass the complex spatiotemporal dynamics of turbulence. Researchers have therefore largely described turbulence in statistical terms, focusing on deriving averaged quantities and scaling relationships, such as that between the amount of energy in an eddy and its size. A different tack that has become increasingly popular [2] [3] [4] [5] , and the one that Schatz and co-workers follow, is to view fluid flow as a dynamical system, whose possible states lie in a high-dimensional space [6] . In this framework, the time evolution of a flow's velocity field is represented by a trajectory through state space, in which each point corresponds to a solution to the Navier-Stokes equations. Here, laminar flow is a fixed point (an equilibrium solution) in Figure 1: The time evolution of a turbulent flow's velocity field can be represented by a trajectory through a state space, in which each point corresponds to a solution to the Navier-Stokes equations. The panels indicate the flow's velocity field at several points along the trajectory (colors indicate the vorticity of the flow). The trajectory will be attracted to an unstable equilibrium point (grey and red spheres), but will eventually exit along the unstable manifold (the red curve indicates the dominant part of the unstable manifold). Schatz and colleagues showed that once they had identified an unstable equilibrium point and its associated manifolds, they could simulate the fluid's evolution. (B. Suri et al., Phys. Rev. Lett. (2017)) state space, whereas turbulent flow is a chaotic trajectory. This chaotic motion takes place in a region of state space that accommodates infinitely many unstable periodic orbits and unstable equilibrium points, collectively known as exact coherent structures (ECS). Each of these structures is flanked by stable and unstable manifolds: bundles of trajectories that approach and leave, respectively, the vicinity of an ECS. A turbulent trajectory continuously bounces from one unstable solution to another (Fig. 1) : it is attracted by the stable manifold of an equilibrium point, where the fluid's dynamics slows down and its velocity field resembles that of the equilibrium state; it then leaves this region along the unstable manifold and heads to another ECS, and so forth. The physics.aps.org

doi:10.1103/physics.10.25
fatcat:xwr3iqtpnfam3g5azhsiqo723y