We propose a new flow condition for conducting DNS of internal flows: Instead of keeping either a flow rate or a pressure gradient constant the new simulation approach is to maintain the total power input, at a constant value. This new condition is closer to a practical situation where the power input, i.e. the product of flow rate and pressure drop (and not flow rate or pressure drop independently), is given by a pump for example. From the flow control point of view, the constant power input approach is the most suitable approach for an energetic analyses of flows with/without control. BACKGROUND & OBJECTIVES Typically direct numerical simulations (DNS) of turbulent wall flows are carried out while keeping constant in time either the flow rate (CFR) or, less often, the pressure gradient (CPG), under the generally acknowledged assumption that the statistics of the flow are insensitive to this arbitrary choice. However, this apparently uninfluential choice becomes critical when drag-redution techniques are applied to turbulent channel or pipe flows. Under the CFR condition, a successful drag reducing technique reduces friction drag, which immediately translates into a reduction of the pumping power. One important drawback of the CFR constraint is that the wall shear stress, which is a dominant factor in near-wall turbulence dynamics, is changed due to the applied control, so that it is difficult to extract the essential effects of a control input itself owing to superimposed Reynolds number effects. On the other hand, when the CPG condition is used, friction drag is unchanged by design, and 'drag reduction' manifests itself through an increase of the flow rate, which implies an increase in the power required to drive the flow. Recently, we introduced a novel way of assessing the energetic performance of flow control techniques by explicitly ac- counting for the total power spent for pumping and control (Frohnapfel et al. (2012)). The presented energy-convenience plane allows the designer to weight the energetic cost of the applied control against the increased performance, that is observed through an increased flow rate. This evaluation plane graphically emphasizes that besides CFR and CPG, many different routes link an uncontrolled turbulent state to the laminar flow state, which is the ultimate goal in drag reduction control. Among these various routes, the one of constant power input (CPI) is an alternative that puts the energetic advan- tage of laminarizing the flow into an interesting perspective: at a Reynolds number of 75000 relaminarisation corresponds to the total energy savings of 86% for CPI while stunning 50 times larger flow rate will be reached for CPG. In addition to this novel evaluation perspective that might be useful in judging the potential economical benefit of (active) flow control techniques, the CPI approach provides an interesting tool for the fundamental analysis of turbulent flows and their control. If we assume that an analysis of the energy flow through the system and its modifications in flow control can teach us something about the physics of turbulence itself and possibly allow generalized statements on how to efficiently control a flow, we need to be able to compare different flow control scenarios at the same energy flux and dissipation rate. This is basically impossible to realize with CFR or CPG but straightforward to implement with the CPI approach. The aim of the present study is thus to introduce CPI as a physically sound strategy to carry out numerical experiments. We show how the CPI approach naturally leads to identifying a characteristic velocity scale, based on the power consumption, and thus a related Reynolds number. In this way it becomes natural and easy to carry out DNS under the CPI condition. First results for CPI in controlled flows are presented.

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Quadrio Maurizio

Papers

Direct numerical simulation of turbulent wall flows at constant power input