Structures and dynamics in a periodically forced Lorenz system

Skin friction drag causes substantial energy loss in transport systems. A promising method to reduce drag is to oscillate the surfaces of the flow transversely to the flow direction, leading to drag reduction of up to 25% (Quadrio, 2011). The wall oscillation leads to drag reduction via altering the flow structure in the near-wall region. However, a detailed description and understanding of the effect of periodic forcing on these structures is missing. It is well known (see e.g. Kawahara et al., 2012) that exact coherent structures, such as equilibria and periodic orbits form a 'skeleton' about which turbulent dynamics organise; therefore the reaction of these exact coherent structures to periodic forcing is of interest. As a first step towards understanding how periodic forcing impacts means of turbulent motion, such as drag, we study how such forcing impacts other low-dimensional chaotic processes, starting with the Lorenz system (Lorenz, 1963). We demonstrate how periodic forcing impacts averages in this system and relate this to structural changes of its unstable periodic orbits. These orbits have a direct impact on the averages in the Lorenz system (Eckhardt & Ott, 1994) and we demonstrate to what extent this remains true in a periodically forced version by examining the effect on chaotic averages, dynamics, and unstable periodic orbits. The methodology is applicable to a wider range of chaotic systems including turbulence, so that optimised strategies for drag reduction may be designed.