Periodic forcing of a chaotic fluid system #

Filip A Jovanovic, Tom S Eaves

11:10 Wednesday in 4Q04.

Part of the Nonlinear dynamics and applications session.

Abstract #

A significant challenge we are facing is the ever-growing demand for environmentally friendly and energy efficient transport as a result of emissions released from fossil fuels. A variety of approaches can be taken to optimise energy efficiency of transport and vehicles, one of which is to look at fluid flow drag reduction; surface drag (skin friction) plays a key role in causing unnecessary energy loss in vehicles. One potential method of reducing drag is to oscillate the surfaces of the flow transversely to the flow direction; simulations and experiments using this method lead to a drag reduction of up to 25% (Quadrio, 2011), depending on the Reynolds number of the flow.

To better understand how such dramatic drag reductions can be made, we study the impact of the oscillatory forcing on simplified fluid systems using a dynamical systems approach. Specifically, we study the effect of periodic forcing on the long-term averages and structural properties of the famous Lorenz system which was originally derived as a simplified model of convection. It is well known that unstable periodic orbits play a key role in the dynamics of chaotic systems (Cvitanovic, 1991), forming the structure, or skeleton, of the attractor on which the chaotic trajectories evolve. As such, periodic orbits directly impact long-time averages in certain chaotic systems, and it is possible to compute average quantities of the chaotic motion using only properties of the orbits, including their period, stability, and average quantities measured on each orbit. This computation typically converges rapidly with the number of orbits used in a ‘periodic orbit expansion’ of averages quantities of a chaotic attractor (Cvitanovic & Eckhardt, 1991). 

This presentation will detail the effect of periodic forcing on an average quantity of the Lorenz system (specifically a scaled heat-flux) along with the effect of this forcing on the unstable periodic orbits embedded within the Lorenz attractor. The ability of periodic orbit theory to predict averages in this oscillating system will be discussed, along with potential analytical tools (such as linear response theory) that will be developed to predict this behaviour a priori and hence deduce the response of chaotic fluid systems to oscillatory forcing using only properties of the unforced system. This work is helping to establish a mathematical framework with which to describe and explain the observed drag reduction and sets the groundwork for ongoing research into the effect of oscillatory forcing on the structures contained within the full Navier–Stokes equations. The ultimate objective of the project is to develop novel forcing techniques for drag reduction which lead to even greater savings, by understanding how structures within the fluid motion react to such forcing.

 References

Cvitanovic, Predrag 1991 Periodic orbits as the skeleton of classical and quantum chaos. Physica D: Nonlinear Phenomena 51 (1-3), 138–151.

Cvitanovic, Predrag & Eckhardt, Bruno 1991 Periodic orbit expansions for classical smooth flows. Journal of Physics A: Mathematical and General 24 (5), L237.

Quadrio, M. 2011 Drag reduction in turbulent boundary layers by in-plane wall motion. Philos. T. R. Soc. A 369 (1940), 1428–1442.