Hostname: page-component-5d59c44645-lfgmx Total loading time: 0 Render date: 2024-02-22T10:02:03.268Z Has data issue: false hasContentIssue false

Shape optimisation for a stochastic two-dimensional cylinder wake using ensemble variation

Published online by Cambridge University Press:  16 March 2023

Javier Lorente-Macias
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
Yacine Bengana
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
Yongyun Hwang*
Department of Aeronautics, Imperial College London, London SW7 2AZ, UK
Email address for correspondence:


In the present study, the shape of a two-dimensional cylinder is optimised to minimise the mean drag in laminar unsteady flow under a noisy environment. A small inline stochastic oscillation in the free-stream velocity, which follows the Ornstein–Uhlenbeck process, is considered for the noise. The small noise is found to yield a large random fluctuation in instantaneous drag of the cylinder due to the effect of added mass. Subject to the strong random fluctuation of drag, the shape optimisation is performed using an ensemble-variation-based method (EnVar), as the conventional adjoint-based optimisation is not applicable to such a flow environment with unknown free-stream noise. The optimised cylinder geometry is found to be a nearly-symmetric slender oval at a low Reynolds number. As the Reynolds number is increased, two optimal shapes emerge: one is identical to the oval obtained at the low Reynolds number, and the other is an asymmetric oval, the rear side of which is more slender than the front side, reminiscent of an aerofoil. Despite the large random fluctuation in the instantaneous drag, the optimal cylinder shapes obtained for different levels of the upstream noise are found to be almost identical. It is shown that the robust nature of the optimal cylinder shape originates from the limited influence of the small upstream noise on the mean flow properties of the cylinder wake. Finally, the optimised cylinder primarily reduces the pressure component of the drag, associated mainly with vortex shedding in the wake, and this is achieved by marginally increasing the viscous drag through the shape change.

JFM Papers
© The Author(s), 2023. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Abergel, F. & Temam, R. 1990 On some control problems in fluid mechanics. Theor. Comput. Fluid Dyn. 1 (6), 303325.CrossRefGoogle Scholar
Arcas, D. & Redekopp, L. 2004 Aspects of wake vortex control through base blowing/suction. Phys. Fluids 16, 452456.Google Scholar
Baek, S. & Sung, H.J. 1998 Numerical simulation of the flow behind a rotary oscillating circular cylinder. Phys. Fluids 10, 869876.CrossRefGoogle Scholar
Barkley, D. & Henderson, R.D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215241.CrossRefGoogle Scholar
Bearman, P.W. 1965 Investigation of the flow behind a two-dimensional model with a blunt trailing edge and fitted with splitter plates. J. Fluid Mech. 21, 241255.CrossRefGoogle Scholar
Bearman, P.W. 1967 The effect of base bleed on the flow behind a two-dimensional model with a blunt trailing edge. Aeronaut. Q. 18, 207224.CrossRefGoogle Scholar
Bearman, P.W. & Owen, J.C. 1998 Reduction of bluff-body drag and suppression of vortex shedding by the introduction of wavy separation lines. J. Fluids Struct. 12, 123130.CrossRefGoogle Scholar
Bewley, T.R., Moin, P. & Temam, R. 2001 DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms. J. Fluid Mech. 447, 179225.CrossRefGoogle Scholar
Blackburn, H. & Henderson, R. 1999 A study of two-dimensional flow past an oscillating cylinder. J. Fluid Mech. 385, 255286.CrossRefGoogle Scholar
Blanchard, A., Bergman, L.A. & Vakakis, A.F. 2020 Vortex-induced vibration of a linearly sprung cylinder with an internal rotational nonlinear energy sink in turbulent flow. Nonlinear Dyn. 99 (1), 593609.CrossRefGoogle Scholar
Brewster, J. & Juniper, M. 2020 Shape sensitivity of eigenvalues in hydrodynamic stability, with physical interpretation for the flow around a cylinder. Eur. J. Mech. (B/Fluids) 80, 8091.Google Scholar
Choi, H., Jeon, W.-P. & Kim, J. 2008 Control of flow over a bluff body. Annu. Rev. Fluid Mech. 40, 113139.CrossRefGoogle Scholar
Choi, S., Choi, H. & Kang, S. 2002 Characteristics of flow over a rotationally oscillating cylinder at low Reynolds number. Phys. Fluids 140, 27672777.CrossRefGoogle Scholar
Chomaz, J.M. 2005 Global instabilities in spatially developing flows: nonnormality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357392.CrossRefGoogle Scholar
Colburn, C.H., Cessna, J.B. & Bewley, T.R. 2011 State estimation in wall-bounded flow systems. Part 3. The ensemble Kalman filter. J. Fluid Mech. 682, 289303.CrossRefGoogle Scholar
Darekar, R.M. & Sherwin, S.J. 2001 Flow past a square-section cylinder with a wavy stagnation face. J. Fluid Mech. 426, 263295.CrossRefGoogle Scholar
Evensen, G. 2009 Data Assimilation: The Ensemble Kalman Filter. Springer.CrossRefGoogle Scholar
Giannetti, F. & Luchini, F. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167197.CrossRefGoogle Scholar
Hecht, F. 2012 New development in FreeFem++. J. Numer. Maths 20 (3–4), 251265.Google Scholar
Huerre, P. & Monkewitz, P.A. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151168.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P.A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473537.CrossRefGoogle Scholar
Jahanbakhshi, R. & Zaki, T.A. 2019 Nonlinearly most dangerous disturbance for high-speed boundary-layer transition. J. Fluid Mech. 876, 87121.CrossRefGoogle Scholar
Kato, H., Yoshizawa, A., Ueno, G. & Obayashi, S. 2015 A data assimilation methodology for reconstructing turbulent flows around aircraft. J. Comput. Phys. 283, 559581.CrossRefGoogle Scholar
Kwon, K. & Choi, H. 1996 Control of laminar vortex shedding behind a circular cylinder using splitter plates. Phys. Fluids 8, 478496.CrossRefGoogle Scholar
Lasagna, D. 2018 Sensitivity analysis of chaotic systems using unstable periodic orbits. SIAM J. Appl. Dyn. Sys. 17 (1), 547580.CrossRefGoogle Scholar
Lasagna, D., Sharma, A. & Meyers, J. 2019 Periodic shadowing sensitivity analysis of chaotic systems. J. Comput. Phys. 391, 119141.CrossRefGoogle Scholar
Lewis, J.M., Lakshmivarahan, S. & Dhall, S.K. 2006 Dynamic Data Assimilation: A Least Squares Approach. Cambridge University Press.CrossRefGoogle Scholar
Lim, S. & Choi, H. 2004 Optimal shape design of a two-dimensional asymmetric diffuser in turbulent flow. AIAA J. 42 (6), 11541169.Google Scholar
Mao, X., Blackburn, H.M. & Sherwin, S.J. 2015 Nonlinear optimal suppression of vortex shedding from a circular cylinder. J. Fluid Mech. 775, 241265.CrossRefGoogle Scholar
Marensi, E., Willis, A.P. & Kerswell, R.R. 2020 Designing a minimal baffle to destabilise turbulence in pipe flows. J. Fluid Mech. 900, A31.CrossRefGoogle Scholar
Marquet, O., Sipp, D. & Jacquin, L. 2008 Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221252.CrossRefGoogle Scholar
Min, C. & Choi, H. 1999 Suboptimal feedback control of vortex shedding at low Reynolds numbers. J. Fluid Mech. 401, 123156.Google Scholar
Mohammadi, B. & Pironneau, O. 2009 Applied Shape Optimization for Fluids. Oxford Scholarship Online.CrossRefGoogle Scholar
Mohan, P., Fitzsimmons, N. & Moser, R.D. 2017 Scaling of Lyapunov exponents in homogeneous isotropic turbulence. Phys. Rev. Fluids 2 (11), 114606.CrossRefGoogle Scholar
Mons, V., Chassaing, J.-C., Gomez, T. & Sagaut, P. 2016 Reconstruction of unsteady viscous flows using data assimilation schemes. J. Comput. Phys. 316, 255280.CrossRefGoogle Scholar
Mons, V., Du, Y. & Zaki, T.A. 2021 Ensemble-variational assimilation of statistical data in large-eddy simulation. Phys. Rev. Fluids 6, 104607.CrossRefGoogle Scholar
Ni, A., Wang, Q., Fernandez, P. & Talnikar, C. 2019 Sensitivity analysis on chaotic dynamical systems by finite difference non-intrusive least squares shadowing (FD-NILSS). J. Comput. Phys. 394, 615631.CrossRefGoogle Scholar
Nocedal, J. & Wright, S.J. 2006 Numerical Optimization, 2nd edn. Springer.Google Scholar
Park, H., Jeon, W.-P., Choi, H. & Yoo, J.Y. 2006 Drag reduction in flow over a two-dimensional bluff body with a blunt trailing edge using a new passive device. J. Fluid Mech. 563, 389414.CrossRefGoogle Scholar
Pironneau, O. 1974 On optimum design in fluid mechanics. J. Fluid Mech. 64, 97110.CrossRefGoogle Scholar
Roshko, A. 1955 On the wake and drag of bluff bodies. J. Aeronaut. Sci. 22, 12.Google Scholar
Spall, J.C. 2003 Introduction to Stochastic Search and Optimization: Estimation, Simulation and Control. Wiley.CrossRefGoogle Scholar
Strykowski, P.J. & Sreenivasan, K.R. 1990 On the formation and suppression of vortex ‘shedding’ at low Reynolds number. J. Fluid Mech. 218, 71107.CrossRefGoogle Scholar
Suzuki, T. 2012 Reduced-order Kalman-filtered hybrid simulation combining particle tracking velocimetry and direct numerical simulation. J. Fluid Mech. 705, 249288.CrossRefGoogle Scholar
Tokumaru, P.T. & Dimotakis, P.E. 1991 Rotary oscillation control of a cylinder wake. J. Fluid Mech. 224, 7790.CrossRefGoogle Scholar
Tombazis, N. & Bearman, P.W. 1997 A study of three dimensional aspects of vortex shedding from a bluff body with a mild geometric disturbance. J. Fluid Mech. 330, 85112.CrossRefGoogle Scholar
Wang, Q., Hu, R. & Blonigan, P. 2014 Least squares shadowing sensitivity analysis of chaotic limit cycle oscillations. J. Comput. Phys. 267, 210224.CrossRefGoogle Scholar
Williamson, C.H.K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28, 477539.CrossRefGoogle Scholar
Wood, C.J. 1967 The effect of base bleed on a periodic wake. J. Aeronaut. Soc. 68, 477482.CrossRefGoogle Scholar
Wright, S. & Nocedal, J. 1999 Numerical optimization. Springer Sci. 35 (67–68), 7.Google Scholar
Yang, Y., Robinson, C., Heitz, D. & Mémin, E. 2015 Enhanced ensemble-based 4DVar scheme for data assimilation. Comput. Fluids 115, 201210.CrossRefGoogle Scholar
Zhou, K., Doyle, J.C. & Glover, K. 1996 Robust and Optimal Control. Prentice Hall.Google Scholar