The influence of control on proper orthogonal decomposition of wall-bounded turbulent flows

Abstract

This paper explores the effects of several wall-based, turbulence control strategies on the structure of the basis functions determined using the proper orthogonal decomposition (POD). This research is motivated by the observation that the POD basis functions are only optimal for the flow for which they were created. Under the action of control, the POD basis may be significantly altered so that the common assumption that effective reduced-order models for predictive control can be constructed from the POD basis of an uncontrolled flow may be suspect. This issue is explored for plane, incompressible, turbulent channel flow at Reynolds number, Reτ=180. Based on well- resolved large eddy simulations, POD bases are constructed for three flows: no control; opposition control, which achieves a 25% drag reduction; and optimal control, which gives a 40% drag reduction. Both controlled flows use wall transpiration as the control mechanism and only differ in the technique used to predict the control. For both controlled flows, the POD basis is altered from that of the no-control flow by the introduction of a localized shear layer near the walls and a nearly impenetrable virtual wall that hinders momentum transfer in the wall-normal direction thereby leading to drag reduction. A major difference between the two controlled flows is that the shear layer and associated virtual wall are located farther away from the physical wall when using optimal compared to opposition control. From this investigation, it is concluded that a no-control POD basis used as a low-dimensional model will not capture the key features of these controlled flows. In particular, it is shown that such an approximation leads to grossly underpredicted Reynolds stresses. These results indicate that a no-control POD basis should be supplemented with features of a controlled flow before using it as a low-dimensional approximation for predictive control.