Future work

The point above suggests that it would be worth studying several actuator/sensor pairs to avoid observability issues. As the sensitivity reduction approach is designed to be applied to complex flows, where little is known about the dynamics in general, some actuator-sensor combinations may result in transfer functions that are difficult

to control. More specifically, it may not be possible to reduce the gain of the

sensitivity function in the desired frequency ranges without increasing it in other problematic frequency ranges. As an example, with the present setup, a spanwise antisymmetric sensor/actuator pair was also tested. It was found that a very low frequency right half-plane zero always appeared in the transfer function and this prohibited the design of effective controllers in this case (even with more advanced controller design techniques).

As mentioned earlier in this chapter, this study was mainly intended to provide some insight into the challenges that the sensitivity reduction approach might en-

counter when applied to a three-dimensional flow. Compromises were therefore

made with regards to the resolution of the grid and the inflow boundary conditions and it may be possible to improve these in future work. Additionally, it would be instructive to study the effect of several parameters such as the width, angle, and profile of the jet, and the controller design method. Further insight into the effect of the control may also be obtained using more advanced analysis techniques such as POD, DMD [193, 188], or OMD [223].

Finally, for this particular flow, it would be interesting to investigate the effect of adding inflow perturbations. This can be expected to significantly increase the drag as the shear-layer instability would amplify these disturbances. We could expect that in this case, the unforced flow would have a higher drag with a reduced recirculation region and larger wake fluctuations. It might also allow the sensor to measure the fluctuations related to the shear layer roll-up, giving Controller 2 (which targets the associated frequency range) a better chance of being effective, as long as the changes in the dynamics introduced by the disturbances remain small.

Conclusions

The purpose of this thesis has been to investigate modelling and feedback control methods that aim to reduce the drag and wake fluctuations experienced by bluff bodies. In particular, we focused on two promising strategies that rely on linear model-based feedback control.

In the first approach, drag is reduced by fully stabilising a steady state of the flow. As bluff body flows are usually globally unstable, obtaining an accurate and low-order linear model of the input-output dynamics can be challenging. Indeed, in- stabilities either result in the unbounded growth of the state (in a linear framework) or in the nonlinear saturation of the system. Additionally, designing a controller that is able to stabilise the flow from off-design (potentially nonlinear) initial conditions can also be a major hurdle. These questions were addressed in the first part of this thesis, where it was shown that, despite apparent obstacles, balanced models can be obtained at a low computational cost, even for high-dimensional unstable flows, and only using data generated with a standard nonlinear flow solver. Based on these

models, controllers were synthesised using H∞ loop-shaping, which was found to be

an attractive framework for robustly stabilising nonlinear bluff body flows.

In the second approach, it is not assumed that full flow stabilisation is possible. In this case, modelling is also challenging as the unforced flow is usually evolving in an unsteady and strongly nonlinear manner. Additionally, choosing a linear control strategy given a linear model of the input-output dynamics about this unsteady nonlinear “base flow” is not straightforward. The second part of the thesis thus focuses on the the “sensitivity reduction approach”, whereby feedback control is used to reduce the fluctuations measured by a body-mounted sensor, in order to suppress the associated unsteady structures in the wake. The procedure was shown to provide promising results when applied to the flow over a fully three-dimensional geometry,

chosen as an intermediate step between canonical two-dimensional geometries and more complex and industrially relevant flows.

6.1

Summary

Over the last decade, balanced models have become increasingly popular in flow con- trol. The algorithms that produce them are based on solid theoretical foundations and have enabled the design of successful controllers for many stable and unstable flows. For stable systems, these models can be obtained either using a Galerkin projection approach – the balanced proper orthogonal decomposition (BPOD) – or a system identification approach – the eigensystem realisation algorithm (ERA). For unstable systems, the models are often identified in closed-loop with an exist- ing stabilising controller. However, such a controller may not be available or easily designed. Alternatively, the unstable subspace can first be identified and projec- ted out in order to only model the stable subspace, but this procedure can become computationally intractable for large systems.

One of the contributions of this thesis is to show that the standard BPOD and ERA algorithms can in fact be applied directly to unstable systems. This was done first from a purely theoretical perspective, by proving that the algorithms yield ad- equate and converged balanced models for sufficiently long impulse responses. Using the complex linearised Ginzburg-Landau equation, we then showed that in practice, finite precision arithmetic limits the accuracy of the models, leading to a trade-off between the growth of the unstable modes and the convergence of the models. Nev- ertheless, for this one-dimensional system, models of similar accuracy to the ones generated with other existing methods were obtained. Two notable extensions were also suggested. First, for difficult systems where simple impulse responses do not result in sufficient model accuracy, a slightly more expensive procedure based on a projection of the most unstable modes was introduced. Note that no global modes need to be identified here and the unstable subspace is still balanced. Second, both for stable and unstable systems, a considerably lower number of snapshots may be required for a given ROM accuracy if the snapshots are recorded and weighed fol- lowing a piecewise Gaussian quadrature as opposed to Newton-Cotes quadrature. As a result, the computational cost of the procedure can be further reduced.

The approach was then applied to the unstable flow over a D-shaped body at low Reynolds numbers. A body-mounted actuator and two sensor configurations were considered: a velocity sensor located in the wake and a force sensor located

from the response of the nonlinear flow to a small impulse with both sensors. These

models enabled the design of stabilising controllers using H∞loop-shaping and were

able to accurately predict the response of the flow in closed-loop. With the velocity sensor configuration, the wake was forced more aggressively, resulting in faster flow stabilisation than with the body-mounted force sensor. In both cases, the flow was also successfully stabilised from the unmodelled, nonlinear limit cycle at the design Reynolds number. However, at slightly increased Reynolds numbers, the flow was

only fully stabilised when the wake velocity sensor was used. Overall, the H∞

loop-shaping framework was found to be a suitable approach to design stabilising controllers from low-order models: while the controllers can be shaped to respect specific performance requirements, they are also robust to noise, disturbances, and model uncertainties.

For flows where a steady state cannot be stabilised, the sensitivity reduction approach is a promising alternative. It has previously been applied to a D-shaped body and a backward-facing step and resulted in a considerable drag reduction in both cases. Here, the same approach was applied to the fully three-dimensional wake over a backward-facing step with side walls. In order to obtain a low-order linear model of the input-output dynamics about the unsteady, nonlinear unforced flow, harmonic open-loop forcing was first applied. It allowed the identification of the gain and phase of the system’s response at a series of discrete frequencies. A low- order rational transfer function was then fitted through the measured data in order to obtain a linear model. The open-loop forcing simulations also showed that actu- ation at a range of frequencies resulted in the enhancement of the Kelvin-Helmholtz instability and a corresponding significant increase in drag. On the other hand, very low frequency forcing lead to an improved pressure recovery. In all cases, harmonic open-loop forcing increased the amplitude of the base pressure fluctuations.

Next, three controllers were designed to reduce the fluctuations measured by a base pressure sensor. The frequency ranges corresponding to the dominant unsteady wake structures were specifically targeted. In all cases, pressure fluctuations were suppressed, as predicted by the linear model, confirming that the linear framework was adequate to control this unsteady three-dimensional wake. The control also lead to a time-averaged base pressure increase in all cases. All controllers were found to reduce drag through a similar mechanism, whereby the dominant pumping motion of the recirculation region is somewhat suppressed and the flapping motion developing in the far wake due to the shear layer roll up is weakened.