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.

The effect of spanwise wall oscillation on turbulent pipe flow structures resulting in drag reduction

“…In the realm of control, KL methods have been used to produce drag reduction in a turbulent channel flow by phase randomization of the structures 21 and to understand the effect of drag reduction by controlled wall normal suction and blowing. 22 In the present study, the KL framework is used to examine the differences in the turbulent structures and dynamics between turbulent pipe flow with and without spanwise wall oscillation.…”

Low-order models. Optimal sampling and linearized control strategies

“…The POD basis gives an optimal representation, in terms of kinetic energy, of the database of snapshots used to build the basis itself and generated by the system. However, when the input parameters vary, the basis becomes inaccurate, as it is the case in control problems (see [18,3]). The focus of this section is to improve the representation capabilities of a POD basis of a given flow when the Reynolds number (input parameter of the system) varies in a given range, so as to provide a single ROM that is efficient for the considered range.…”

Enablers for robust POD models

“…The main drawback for flow control is that the POD basis is only able to give an optimal representation of the snapshots set from which it was derived. The approximation properties of the basis can be greatly degraded under variation of some input system parameters values, as control parameters [15][16][17] . For flow control purposes, some special care has to be taken to build the POD basis functions.…”

“…Indeed, for flow control purpose, it was demonstrated [15][16][17] that POD basis functions built from a flow database generated with a given set of control parameters is not able to represent the main features of a flow generated with another set of control parameters. To overcome this problem, we propose to derive methods allowing to adapt the POD basis functions at low numerical costs.…”