The findings from this thesis have shown that balanced models of high-dimensional globally unstable flows can be obtained with projection-free BPOD and ERA at a low computational cost and enable the robust stabilisation of these systems. It would now be of great interest to explore how well these conclusions carry over to more complex and industrially relevant flows. In particular, high Reynolds number flows or flows with many unstable modes may be expected to be particularly challenging. The second natural extension of the present work would be to extend it to ex- perimental setups. Using ERA, the response to a small impulse may be sufficient to obtain an accurate model of the input-output dynamics. However, several issues need to be overcome in experiments. First, noise, disturbances, and nonlinearities are likely to deteriorate the quality of the impulse response signal. To mitigate this, the experiment can be repeated a number of times in order to average the res- ults. Alternatively, the OKID algorithm may also be applicable in such cases. The second issue concerns the ability to initialise the flow in the unstable equilibrium state, which is unlikely to be exactly possible experimentally. If the flow can be reliably initialised in a state “close” to the base flow, a method was proposed to improve the model accuracy, by removing the contribution of the output from this imperfect initial condition.
In the second part of this thesis, it was shown that it is possible to apply the sensitivity reduction approach to a fully three-dimensional wake, in order to obtain a predictable attenuation of the wake fluctuations and an associated increase in the time-averaged base pressure. A first point that may benefit from further analysis is the effect that the control has on the flow. A number of modal decomposition techniques (e.g. POD, DMD, OMD) exist and may provide some valuable insight into the modification of the flow dynamics. In particular, it would be instructive to apply this type of analysis to work that is currently under-way [56, 67], where the technique is applied to simplified square-back road vehicle geometries, both at laminar and turbulent Reynolds numbers.
Another unanswered question at this stage is how the different terms in the mean drag coefficient equation (5.3) are affected by the control. Specifically, for highly nonlinear flows, it can be expected that a significant attenuation of wake fluctuations terms will also result in non-negligible changes in time-averaged quantities. An associated critical issue to investigate is the extent to which the control affects the linearised dynamics themselves. In some cases, further drag reduction may be
Finally, the sensitivity reduction approach relies on the fact that the fluctuations measured by the sensor give an accurate estimation of the losses due to unsteady structures in the flow. In order to avoid suboptimal performance due to observability issues, testing several actuator-sensor combinations for any given application may prove to be critical, especially if the flow dynamics are complex and not particularly well understood.
A number of more general recommendations can be made regarding different approaches to model and control of bluff body flows. First, only linear techniques were considered here. For both approaches, they were shown to be adequate choices, leading to the desired control goals. The linear approach provides a powerful set of modelling and control tools, but must be applied with caution. For instance, in the balanced model approach, the model is only accurate near the unstable equilibrium and in the sensitivity reduction approach, the model does not attempt to predict the behaviour of the full flow, which may be strongly nonlinear.
Regarding controller design, many studies have focused on optimisation-based methods, whereas in the present work, only loop-shaping techniques were used (al-
though H∞ loop-shaping does also include an optimisation stage). Loop-shaping
techniques have associated downsides: they require a deeper understanding of con- trol theory, they involve a manual (and hence inherently suboptimal) design stage, and comparing the performance of different controllers quantitatively is difficult. However, with loop-shaping methods, specific performance specifications can be en- forced. This feature is central to both approaches discussed in the present work. It can also result in robust closed-loop systems, an arguably essential aspect for controllers designed from low-order linear models and applied to high-dimensional complex, nonlinear flows.
[1] Achenbach, E. The effects of surface roughness and tunnel blockage on the flow past spheres. J. Fluid Mech. 65 (1974), 133–125.
[2] Agostini, L., Touber, E., and Leschziner, M. A. Spanwise oscillatory wall motion in channel flow: drag-reduction mechanisms inferred from DNS- predicted phase-wise property variations at. J. Fluid Mech. 743 (2014), 606– 635.
[3] Ahmed, S. R., Ramm, G., and Faltin, G. Some salient features of the time-averaged ground vehicle wake. SAE Tech. Pap., 840300 (1984).
[4] Ahuja, S. Reduction methods for feedback stabilization of fluid flows. PhD thesis, Princeton University, 2009.
[5] Ahuja, S., and Rowley, C. W. Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators. J. Fluid Mech. 645 (2010), 447–478.
[6] Aider, J.-L., Beaudoin, J.-F., and Wesfreid, J.-E. Drag and lift re- duction of a 3D bluff-body using active vortex generators. Exp. Fluids 48 (2009), 771–789.
[7] ˚Akervik, E., Brandt, L., Henningson, D. S., Hœpffner, J.,
Marxen, O., and Schlatter, P. Steady solutions of the Navier-Stokes equations by selective frequency damping. Phys. Fluids 18 (2006), 68102.
[8] ˚Akervik, E., Hœpffner, J., Ehrenstein, U., and Henningson, D. S.
Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes. J. Fluid Mech. 579 (2007), 305–314.
[9] Al-Saggaf, U. M. Model reduction for discrete unstable systems based on generalized normal representations. Int. J. Control 55 (1992), 431–443.
[10] Amitay, M., Smith, D. R., Kibens, V., Parekh, D. E., and Glezer, A. Aerodynamic Flow Control over an Unconventional Airfoil Using Synthetic Jet Actuators. AIAA J. 39, 3 (2001), 361–370.
[11] Anderson, E. A., and Szewczyk, A. A. Effects of a splitter plate on the near wake of a circular cylinder in 2 and 3-dimensional flow configurations. Exp. Fluids 23 (1997), 161–174.
[12] Antoulas, A. C. Approximation of large-scale dynamical systems. SIAM, Houston, Texas, 2005.
[13] Aubrun, S., McNally, J., Alvi, F. S., and Kourta, A. Separation flow control on a generic ground vehicle using steady microjet arrays. Exp. Fluids 51 (2011), 1177–1187.
[14] Aubry, N., Holmes, P., Lumley, J. L., and Stone, E. The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192 (1988), 115–173.
[15] Bagheri, S., ˚Akervik, E., Brandt, L., and Henningson, D. S. Matrix-
free methods for the stability and control of boundary layers. AIAA J. 47 (2009), 1057–1068.
[16] Bagheri, S., Brandt, L., and Henningson, D. S. Input-output analysis, model reduction and control of the flat-plate boundary layer. J. Fluid Mech. 620 (2009), 263–298.
[17] Bagheri, S., Henningson, D. S., Hœpffner, J., and Schmid, P. J. Input-output analysis and control design applied to a linear model of spatially developing flows. Appl. Mech. Rev. 62 (2009), 020803.
[18] Baker, C. J. Flow and dispersion in ground vehicle wakes. J. Fluid Struct. 15 (2001), 1031–1060.
[19] Barbagallo, A., Dergham, G., Sipp, D., Schmid, P. J., and Robinet, J.-C. Closed-loop control of unsteadiness over a rounded backward- facing step. J. Fluid Mech. 703 (2012), 326–362.
[20] Barbagallo, A., Sipp, D., and Schmid, P. J. Closed-loop control of an open cavity flow using reduced-order models. J. Fluid Mech. 641 (2009), 1–50.
[21] Barbagallo, A., Sipp, D., and Schmid, P. J. Input-output measures for model reduction and closed-loop control: application to global modes. J. Fluid Mech. 685 (2011), 23–53.
[22] Bearman, P. W. Investigation of the flow behind a two-dimensional model with a blunt trailing edge and fitted with splitter plates. J. Fluid Mech. 21 (1965), 241–255.
[23] Bearman, P. W. Near wake flows behind two- and three-dimensional bluff bodies. J. Wind Eng. Ind. Aerod. 69 (1997), 33–54.
[24] Beaudoin, J.-F., Cadot, O., Aider, J.-L., and Wesfreid, J.-E. Bluff- body drag reduction by extremum-seeking control. J. Fluid Struct. 22 (2006), 973–978.
[25] Beaudoin, J.-F., Cadot, O., Aider, J.-L., and Wesfreid, J.-E. Drag reduction of a bluff body using adaptive control methods. Phys. Fluids 18 (2006), 085107.
[26] Becker, R., Garwon, M., Gutknecht, C., Barwolff, G., and King, R. Robust control of separated shear flows in simulation and experiment. J. Process Contr. 15 (2005), 691–700.
[27] Becker, R., King, R., Petz, R., and Nitsche, W. Adaptive closed-loop separation control on a high-lift configuration using extremum seeking. AIAA J. 45 (2007), 1382–1392.
[28] Belson, B. A., Semeraro, O., Rowley, C. W., and Henningson, D. S. Feedback control of instabilities in the two-dimensional Blasius bound- ary layer: The role of sensors and actuators. Phys. Fluids 25 (2013), 054106. [29] Bergmann, M., and Cordier, L. Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models. J. Comput. Phys. 227 (2008), 7813–7840.
[30] Bewley, T. R., and Liu, S. Optimal and robust control and estimation of linear paths to transition. J. Fluid Mech. 365 (1998), 305–349.
[31] Bewley, T. R., Moin, P., and Temam, R. DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms. J. Fluid Mech. 447 (2001), 179–225.
[32] Bewley, T. R., Temam, R., and Ziane, M. A general framework for robust control in fluid mechanics. Physica D 138 (2000), 360–392.
[33] Bhattacharjee, S., Scheelke, B., and Troutt, T. R. Modification of vortex interactions in a reattaching separated flow. AIAA J. 24 (1986), 623–629.
[34] Bradley, R. Technology roadmap for the 21st century truck program. Tech. rep., Washington DC, 2000.
[35] Brunton, S. L., Dawson, S. T. M., and Rowley, C. W. State-space model identification and feedback control of unsteady aerodynamic forces. J. Fluid Struct. 50 (2014), 253–270.
[36] Brunton, S. L., and Noack, B. R. Closed-loop turbulence control: pro- gress and challenges. Appl. Mech. Rev. 67 (2015), 050801.
[37] Brunton, S. L., Rowley, C. W., and Williams, D. R. Reduced-order unsteady aerodynamic models at low Reynolds numbers. J. Fluid Mech. 724 (2013), 203–233.
[38] Cabitza, S. Active control of the wake from a rectangular-sectioned body. PhD thesis, Imperial College London, 2014.
[39] Cadot, O., Evrard, A., and Pastur, L. Imperfect supercritical bifurca- tion in a three-dimensional turbulent wake. Phys. Rev. E 91 (2015), 063005. [40] Castro, I. P., and Robins, A. G. The flow around a surface-mounted
cube in uniform and turbulent streams. J. Fluid Mech. 79 (1977), 307–335. [41] Cattafesta, L. N., Song, Q., Williams, D. R., Rowley, C. W., and
Alvi, F. S. Active control of flow-induced cavity oscillations. Prog. Aerosp. Sci. 44 (2008), 479–502.
[42] Chen, K. K., and Rowley, C. W. H2 optimal actuator and sensor place- ment in the linearised complex Ginzburg–Landau system. J. Fluid Mech. 681 (2011), 241–260.
[43] Chiu, T.-Y. Model reduction by the low-frequency approximation balancing method for unstable systems. IEEE T. Automat. Contr. 41 (1996), 995–997.
[44] Choi, H., Jeon, W.-P., and Kim, J. Control of flow over a bluff body. Annu. Rev. Fluid Mech. 40 (2008), 113–139.
[45] Choi, H., Lee, J., and Park, H. Aerodynamics of heavy vehicles. Annu. Rev. Fluid Mech. 46 (2014), 441–468.
[46] Choi, J., Colonius, T., and Williams, D. R. Surging and plunging oscillations of an airfoil at low Reynolds number. J. Fluid Mech. 763 (2015), 237–253.
[47] Choi, J., Jeon, W.-P., and Choi, H. Mechanism of drag reduction by dimples on a sphere. Phys. Fluids 18 (2006), 041702.
[48] Chomaz, J.-M. Global instabilities in spatially developing flows: non-
normality and nonlinearity. Annu. Rev. Fluid Mech. 37 (2005), 357–392. [49] Chun, K. B., and Sung, H. J. Control of turbulent separated flow over a
backward-facing step by local forcing. Exp. Fluids 21 (1996), 417–426. [50] Cohen, K., Siegel, S. G., McLaughlin, T., Gillies, E. A., and My-
att, J. H. Closed-loop approaches to control of a wake flow modeled by the Ginzburg–Landau equation. Comput. Fluids 34 (2005), 927–949.
[51] Collis, S. S., Joslin, R. D., Seifert, A., and Theofilis, V. Issues in active flow control: Theory, control, simulation, and experiment. Prog. Aerosp. Sci. 40 (2004), 237–289.
[52] Colonius, T., and Taira, K. A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions. Comput. Method Appl. M. 197 (2008), 2131–2146.
[53] Dadfar, R., Semeraro, O., Hanifi, A., and Henningson, D. S. Out- put feedback control of Blasius flow with leading edge using plasma actuator. AIAA J. 51 (2013), 2192–2207.
[54] Dahan, J. A. Linear feedback control for form drag reduction on bluff bodies with a blunt trailing edge. PhD thesis, Imperial College London, 2013.
[55] Dahan, J. A., Morgans, A. S., and Lardeau, S. Feedback control for form-drag reduction on a bluff body with a blunt trailing edge. J. Fluid Mech. 704 (2012), 360–387.
[56] Dalla Longa, L., and Morgans, A. S. Simulation, modeling and feed- back control of the flow around a square-back bluff body. In 68th Annual Meeting of the APS Division of Fluid Dynamics (Boston, MA, 2015).
[57] Dergham, G., Sipp, D., Robinet, J.-C., and Barbagallo, A. Model
reduction for fluids using frequential snapshots. Phys. Fluids 23 (2011),
064101.
[58] D’Hondt, M., Gilli´eron, P., and Devinant, P. Experimental invest-
igation on the flow around a simplified geometry of automotive engine com- partment. Exp. Fluids 50 (2011), 1317–1334.
[59] Dowling, A. P., and Morgans, A. S. Feedback control of combustion oscillations. Annu. Rev. Fluid Mech. 37 (2005), 151–182.
[60] Doyle, J. C. Guaranteed margins for LQG regulators. IEEE T. Automat. Contr. AC-23 (1978), 756–757.
[61] Duell, E. G., and George, A. R. Measurements in the unsteady near wakes of ground vehicle bodies. SAE Tech. Pap., 930298 (1993).
[62] Duell, E. G., and George, A. R. Experimental study of a ground vehicle body unsteady near wake. SAE Tech. Pap., 0812 (1999).
[63] Eaton, J. K., and Johnston, J. P. A review of research on subsonic turbulent flow reattachment. AIAA J. 19 (1981), 1093–1100.
[64] Ehrenstein, U., Passaggia, P.-Y., and Gallaire, F. Control of a sep- arated boundary layer: reduced-order modeling using global modes revisited. Theor. Comp. Fluid Dyn. 25 (2011), 195–207.
[65] Englar, R. J. Development of pneumatic aerodynamic devices to improve the performance , economics , and safety of heavy behicles. SAE Tech. Pap., 2208 (2000).
[66] Enns, D. F. Model reduction for control system design. PhD thesis, Stanford University, 1985.
[67] Evstafyeva, O., and Morgans, A. S. Feedback control of a square-back ahmed body flow for form-drag reduction. In 68th Annual Meeting of the APS Division of Fluid Dynamics (Boston, MA, 2015).
[68] Fabbiane, N., Semeraro, O., Bagheri, S., and Henningson, D. S. Adaptive and model-based control theory applied to convectively unstable flows. Appl. Mech. Rev. 66 (2014), 060801.
[69] Fabbiane, N., Simon, B., Fischer, F., Grundmann, S., Bagheri, S., and Henningson, D. S. On the role of adaptivity for robust laminar flow control. J. Fluid Mech. 767 (2015), 1–12.
[70] Fabris, D., Auguste, F., and Magnaudet, J. Bifurcations and sym- metry breaking in the wake of axisymmetric bodies. Phys. Fluids 20 (2008), 051702.
[71] Ffowcs Williams, J. E., and Zhao, B. C. The active control of vortex shedding. J. Fluid Struct. 3 (1989), 115–122.
[72] Flinois, T. L. B. Feedback control on three-dimensional blunt body wakes for road-vehicle form drag reduction. Tech. rep., Imperial College London, 2012.
[73] Flinois, T. L. B., and Colonius, T. Optimal control of circular cylinder wakes using long control horizons. Phys. Fluids 27 (2015), 087105.
[74] Flinois, T. L. B., and Morgans, A. S. Feedback control of unstable flows: a direct modelling approach using the eigensystem realisation algorithm. J. Fluid Mech. (in press) (2016).
[75] Flinois, T. L. B., Morgans, A. S., and Schmid, P. J. Projection-free approximate balanced truncation of large unstable systems. Phys. Rev. E 92 (2015), 023012.
[76] Fourri´e, G., Keirsbulck, L., Labraga, L., and Gilli´eron, P. Bluff-
body drag reduction using a deflector. Exp. Fluids 50 (2010), 385–395. [77] Gad-el Hak, M. Flow control: passive, active and reactive flow management.
Cambridge University Press, New York, NY, 2000.
[78] Garc´ıa, C. E., Prett, D. M., and Morari, M. Model predictive control: theory and practice—A survey. Automatica 25 (1989), 335–348.
[79] Garc´ıa-Villalba, M., Li, N., Rodi, W., and Leschziner, M. A. Large-eddy simulation of separated flow over a three-dimensional axisymmet- ric hill. J. Fluid Mech. 626 (2009), 55–96.
[80] Gautier, N., and Aider, J.-L.
backward-facing step flow. J. Fluid Mech. 759 (2014), 181–196.
[81] Gautier, N., Duriez, T., Aider, J.-L., Noack, B. R., Segond, M., and Abel, M. W. Closed-loop separation control using machine learning. J. Fluid Mech. 770 (2015), 442–457.
[82] Ghiglieri, J., and Ulbrich, S. Optimal flow control based on POD and MPC and an application to the cancellation of Tollmien-Schlichting waves. Optim. Method. Softw. 29 (2014), 1042–1074.
[83] Giannetti, F., and Luchini, P. Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581 (2007), 167–197.
[84] Gilli´eron, P. Flow control applied to the car: State of the art. M´ec. Ind. 3 (2002), 515–524.
[85] Gilli´eron, P., and Kourta, A. Aerodynamic drag reduction by vertical
splitter plates. Exp. Fluids 48 (2010), 1–16.
[86] Gilli´eron, P., and Kourta, A. A´erodynamique automobile pour
l’environnement, le design et la s´ecurit´e, 2nd editio ed. Cepadues Editions, Toulouse, France, 2011.
[87] Gloerfelt, X. Compressible proper orthogonal decomposition/Galerkin reduced-order model of self-sustained oscillations in a cavity. Phys. Fluids 20 (2008), 115105.
[88] Goldin, N., King, R., P¨atzold, A., Nitsche, W., Haller, D., and
Woias, P. Laminar flow control with distributed surface actuation: damping Tollmien-Schlichting waves with active surface displacement. Exp. Fluids 54 (2013), 1478.
[89] Graham, W. R., Peraire, J., and Tang, K. Y. Optimal control of vortex shedding using low-order models. Part I - Open-loop model development. Int. J. Numer. Meth. Eng. 44 (1999), 945–972.
[90] Graham, W. R., Peraire, J., and Tang, K. Y. Optimal control of vortex shedding using low-order models. Part II - Model-based control. Int. J. Numer. Meth. Eng. 44 (1999), 973–990.
[91] Grandemange, M. Analysis of three-dimensional turbulent wakes: from axisymmetric bodies to road vehicles. PhD thesis, Ecole Polytechnique - EN- STA ParisTech, 2013.
[92] Grandemange, M., Cadot, O., Courbois, A., Herbert, V., Ricot, D., Ruiz, T., and Vigneron, R. A study of wake effects on the drag of Ahmed squareback model at the industrial scale. J. Wind Eng. Ind. Aerod. 145 (2015), 282–291.
[93] Grandemange, M., Cadot, O., and Gohlke, M. Reflectional symmetry breaking of the separated flow over three-dimensional bluff bodies. Phys. Rev. E 86 (2012), 035302.
[94] Grandemange, M., Gohlke, M., and Cadot, O. Bi-stability in the turbulent wake past parallelepiped bodies with various aspect ratios and wall effects. Phys. Fluids 25 (2013), 095103.
[95] Grandemange, M., Gohlke, M., and Cadot, O. Turbulent wake past a three-dimensional blunt body part 1: Global modes and bi-stability. J. Fluid Mech. 722 (2013), 51–84.
[96] Grandemange, M., Gohlke, M., and Cadot, O. Statistical axisym- metry of the turbulent sphere wake. Exp. Fluids 55 (2014), 1838.
[97] Grandemange, M., Gohlke, M., and Cadot, O. Turbulent wake past a three-dimensional blunt body. part 2. Experimental sensitivity analysis. J. Fluid Mech. 752 (2014), 439–461.
[98] Grandemange, M., Mary, A., Gohlke, M., and Cadot, O. Effect on drag of the flow orientation at the base separation of a simplified blunt road vehicle. Exp. Fluids 54 (2013), 1529.
[99] Hamidy, E. The structure of wakes of 3D bluff bodies in proximity to the ground. PhD thesis, Imperial College London, 1991.
[100] Hanford, P. M. Measurements and calculations in three dimensional spear- ated flow. PhD thesis, Imperial College London, 1986.
[101] Heenan, A. F., and Morrison, J. F. Passive control of pressure fluctu- ations generated by separated flow. AIAA J. 36 (1998), 1014–1022.
[102] Henning, L., Becker, R., Feuerbach, G., Muminovic, R., King, R.,