Modelling and feedback control of vortex shedding for drag reduction of a turbulent bluff body wake
Introduction
The flow over three-dimensional bluff bodies is of particular interest due to its relevance to automotive vehicles. Such bodies experience large pressure drag due to the large region of separated flow in the wake. For automotive vehicles operating at motorway speeds, this pressure drag is responsible for a significant proportion of the fuel consumption, therefore its reduction is a topic of key interest.
Three-dimensional bluff body wakes exhibit a number of key coherent structures, two of which are the static asymmetry and the quasi-oscillatory vortex shedding. Both features arise initially at very low Reynolds numbers (Re) as respectively a reflectional symmetry breaking (SB) mode (Grandemange et al., 2012), and a temporal symmetry breaking, yet both are also observed at the much higher Reynolds numbers typical of road vehicles. In the flows over rectilinear bluff bodies, the SB mode is observed as an instantaneous asymmetry, mainly appearing in the recirculation region as displayed in Fig. 1(a). Under aligned flow conditions the wake flips randomly between two asymmetric states, each mirror symmetric with respect to the other, a phenomenon known as wake bistability. The instantaneous asymmetry of the wake leads to a lateral force and contributes to the pressure drag on the body. The feature is quite general, having been observed for a range of three-dimensional bluff bodies of widely varying geometries (Rigas et al., 2015). Superposed on this, the vortex shedding is seen as quasi-periodic oscillations of the wake, occurring in both the lateral and the vertical dimensions (Grandemange, Gohlke, Cadot, 2013b, Volpe, Devinant, Kourta, 2015). The relative importance of these two features with respect to the pressure drag on the body remains unclear.
The objective of this research is to develop efficient drag reduction methods for three-dimensional bluff bodies, without making large geometric modifications. A promising method for doing this involves suppression of coherent structures, including the SB mode and vortex shedding. Some passive methods such as a control cylinder in the wake (Cadot et al., 2015), base cavity (Evrard et al., 2015) or perimetric slit (García de la Cruz et al., 2017) have already shown promising results for both the suppression of unsteadiness and pressure drag reduction. However while effective, such passive methods may involve large geometric modifications and the exact requirements for suppression of the coherent structures remains unclear. This motivates a careful analysis of the physics of coherent structures in the wake, and the use of active feedback control for their suppression. In this work we develop a stochastic model to describe the vortex shedding in the wake before making use of the model in feedback control design.
In the recent work of Brackston et al. (2016) a stochastic modelling approach was developed for the bistability arising from the SB mode of the flow. This model was then used to develop a feedback control strategy that ultimately proved energetically efficient at drag reduction. Similar drag reduction has also been achieved in other studies (Li et al., 2016) targeting the asymmetry of the wake. Such studies aimed to reduce the quasi-static asymmetry associated with the SB mode, but have the knock-on effect of increasing unsteadiness at some frequencies. In this work we target the unsteadiness of the vortex shedding directly, and further seek to quantify the effect of fluctuations on the drag.
The control of vortex shedding has been pursued before, although this has generally tended to be at low Reynolds number and for predominantly two-dimensional flows. For example, early experimental work was conducted for control of the laminar flow over a circular cylinder by Ffowcs Williams and Zhao (1989) and Roussopoulos and Monkewitz (1996) and was found to be effective. Such studies tended to empirically obtain controllers that were effective at suppressing the shedding, a methodology continued in more recent numerical studies by Son et al. (e.g. 2011). Other numerical studies have applied a more model based approach in which low-order, linear models of the flow are obtained from numerical simulations (Illingworth, Naito, Fukagata, 2014, Illingworth, 2016, Flinois, Morgans, 2016). While the controllers resulting from these models prove effective, applying them experimentally and at high Reynolds numbers is impractical. The only high (turbulent) Re implementation that the authors are aware of is from Pastoor et al. (2008), who suppressed the shedding over a two-dimensional D-shaped body at . In this work, we attempt the controlled suppression of vortex shedding for a flow that is both fully three-dimensional and at high Reynolds number .
The paper is structured as follows. In Section 2 we describe the experiment consisting of an Ahmed body fitted with dynamic forcing flaps. We then go on to describe the stochastic modelling in Section 3, before detailing the ensuing control design and analysis methodology in Section 4. Conclusions are finally given in Section 5.
Section snippets
Experimental setup
Investigations and feedback control are implemented experimentally on a scaled down, flat-back Ahmed body (Ahmed et al., 1984) of the proportions used in many other studies (Grandemange, Cadot, Gohlke, 2012, Cadot, Evrard, Pastur, 2015), giving Re of . This experimental setup is shown in Fig. 1(b). A force balance measures the total force and moment acting on the body while 8 Endevco 8507C pressure transducers take fluctuating pressure measurements, and an ESP-DTC pressure scanner
Stochastic modelling of coherent structures
A first step in the understanding and control of coherent structures in a turbulent flow, is the development of low-order models for their dynamics. Such models may then give insight into the nature of feedback control strategies that may be applied. A promising approach is based upon the observation that these coherent structures in the turbulent flow are often the persistence of the bifurcations seen at very low Re (Rigas et al., 2014). This allows the development of phenomenological models
Feedback control
For the Ahmed body wake, feedback control may proceed along one of two approaches. Either the static asymmetry of the wake may be targeted, aiming to achieve a more symmetric wake on average, or the unsteadiness associated with vortex shedding may be targeted. The former of these strategies was applied in the lateral dimension in which the bistability is present, and was found to be effective at providing efficient drag reduction (Brackston et al., 2016). The latter strategy is that applied
Concluding remarks
We have presented a stochastic modelling approach suitable for some of the coherent structures observed in three-dimensional bluff body wakes, here focussing on the vortex shedding. This approach takes the equation describing the underlying bifurcation observed at low Re, and adds a stochastic term to model phenomenologically the effect of turbulent fluctuations on the large-scale coherent structures. For the case of the SB mode, this approach has already been shown to be effective at modelling
Acknowledgements
This work was supported by EPSRC grant no 1370746. We are also grateful to Dr Juan Marcos García de la Cruz Lopez and Dr Georgios Rigas for helpful discussions.
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