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Optimal sensor placement for variational data assimilation of unsteady flows past a rotationally oscillating cylinder

Published online by Cambridge University Press:  16 June 2017

Vincent Mons Open the ORCID record for Vincent Mons [Opens in a new window] ,
Jean-Camille Chassaing  and
Pierre Sagaut
Vincent Mons*
Affiliation:
Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France
Jean-Camille Chassaing
Affiliation:
Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France
Pierre Sagaut
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, UMR 7340, M2P2, 13451 Marseille CEDEX 13, France
*
†Email address for correspondence: [email protected]
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Abstract

An optimal sensor placement procedure is proposed within the framework of variational data assimilation (DA) for unsteady flows, with the aim of maximizing the efficiency of the DA procedure. It is dedicated to the a priori design of a sensor network, and relies on a first-order adjoint approach. The proposed methodology first consists in identifying, via optimal control, the locations in the flow that have the greatest sensitivity with respect to a change in the initial condition, boundary conditions or model parameters. In a second step, sensors are placed at these locations for DA purposes. The use of this optimal sensor placement procedure does not require extra development in the case where a variational DA suite is available. The proposed methodology is applied to the reconstruction of unsteady bidimensional flows past a rotationally oscillating cylinder. More precisely, the possibilities of reconstructing the rotational speed of the cylinder and the initial flow, which here encompasses upstream conditions, from various types of observations are investigated via variational DA. Then, the observation optimization procedure is employed to identify optimal locations for placing velocity sensors downstream of the cylinder. Both reduction in the computational cost and improvement in the quality of the reconstructed flow are achieved through optimal sensor placement, encouraging the application of the proposed methodology to more complex and realistic flows.

JFM classification

Mathematical Foundations: Variational methods Vortex Flows: Vortex shedding Wakes/Jets: Wakes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 823 , 25 July 2017 , pp. 230 - 277
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Copyright
© 2017 Cambridge University Press 

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References

Akhtar, I., Borggaard, J., Burns, J. A., Imtiaz, H. & Zietsman, L. 2015 Using functional gains for effective sensor location in flow control: a reduced-order modelling approach. J. Fluid Mech. 781, 622–656.Google Scholar
Akhtar, I., Borggaard, J., Stoyanov, M. & Zietsman, L.2010 On commutation of reduction and control: linear feedback control of a von Kármán street. AIAA Paper 2010-4832.Google Scholar
Anderson, J. L. & Anderson, S. L. 1999 A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Weath. Rev. 127, 2741–2758.Google Scholar
Armijo, L. 1966 Minimization of functions having Lipschitz continuous first partial derivatives. Pac. J. Maths 16, 1–3.Google Scholar
Artana, G., Cammilleri, A., Carlier, J. & Mémin, E. 2012 Strong and weak constraint variational assimilations for reduced order fluid flow modeling. J. Comput. Phys. 231, 3264–3288.Google Scholar
Baek, S.-J. & Sung, H. J. 2000 Quasi-periodicity in the wake of a rotationally oscillating cylinder. J. Fluid Mech. 408, 275–300.Google Scholar
Baker, N. L. & Daley, R. 2000 Observation and background adjoint sensitivity in the adaptive observation-targeting problem. Q. J. R. Meteorol. Soc. 126, 1431–1454.CrossRefGoogle Scholar
Belson, B. A., Semeraro, O., Rowley, C. W. & Henningson, D. S. 2013 Feedback control of instabilities in the two-dimensional Blasius boundary layer: the role of sensors and actuators. Phys. Fluids 25, 054106.Google Scholar
Bergmann, M., Cordier, L. & Brancher, J.-P. 2005 Optimal rotary control of the cylinder wake using proper orthogonal decomposition reduced-order model. Phys. Fluids 17, 097101.Google Scholar
Bewley, T. R. & Protas, B. 2004 Skin friction and pressure: the footprints of turbulence. Physica D 196, 28–44.Google Scholar
Borggaard, J., Stoyanov, M. & Zietsman, L. 2010 Linear feedback control of a von Kármán street by cylinder rotation. In Proceedings of the 2010 American Control Conference, pp. 5674–5681. IEEE.CrossRefGoogle Scholar
Carpentieri, G., Koren, B. & van Tooren, M. J. L. 2007 Adjoint-based aerodynamic shape optimization on unstructured meshes. J. Comput. Phys. 224, 267–287.Google Scholar
Chen, K. K. & Rowley, C. W. 2011 H2 optimal actuator and sensor placement in the linearised complex Ginzburg–Landau system. J. Fluid Mech. 681, 241–260.Google Scholar
Chevalier, M., Hoepffner, J., Bewley, T. R. & Henningson, D. 2006 State estimation in wall-bounded flow systems. Part 2. Turbulent flows. J. Fluid Mech. 552, 167–187.Google Scholar
Choi, H., Jeon, W.-P. & Kim, J. 2008 Control of flow over a bluff body. Annu. Rev. Fluid Mech. 40, 113–139.Google Scholar
Choi, S., Choi, H. & Kang, S. 2002 Characteristics of flow over a rotationally oscillating cylinder at low Reynolds number. Phys. Fluids 14, 2767–2777.CrossRefGoogle Scholar
Cioaca, A. & Sandu, A. 2014 An optimization framework to improve 4D-var data assimilation system performance. J. Comput. Phys. 275, 377–389.Google Scholar
Cohen, K., Siegel, S. & McLaughlin, T. 2006 A heuristic approach to effective sensor placement for modeling of a cylinder wake. Comput. Fluids 35, 103–120.Google 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, 289–303.Google Scholar
Daescu, D. N. 2008 On the sensitivity equations of four-dimensional variational (4D-Var) data assimilation. Mon. Weath. Rev. 136, 3050–3065.Google Scholar
Ding, H., Shu, C., Yeo, K. S. & Xu, D. 2007 Numerical simulation of flows around two circular cylinders by mesh-free least square-based finite difference methods. Intl J. Numer. Meth. Fluids 53, 305–332.Google Scholar
Evensen, G. 1994 Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res. 99, 10143–10162.CrossRefGoogle Scholar
Flinois, T. L. B. & Colonius, T. 2015 Optimal control of circular cylinder wakes using long control horizons. Phys. Fluids 27, 087105.Google Scholar
Foures, D. P. G., Dovetta, N., Sipp, D. & Schmid, P. J. 2014 A data-assimilation method for Reynolds-averaged Navier–Stokes-driven mean flow reconstruction. J. Fluid Mech. 759, 404–431.Google Scholar
Green, L. L., Newman, P. A. & Haigler, K. J. 1996 Sensitivity derivatives for advanced CFD algorithm and viscous modeling parameters via automatic differentiation. J. Comput. Phys. 125, 313–324.Google Scholar
Gronskis, A., Heitz, D. & Mémin, E. 2013 Inflow and initial conditions for direct numerical simulation based on adjoint data assimilation. J. Comput. Phys. 242, 480–497.CrossRefGoogle Scholar
Hayase, T. 2015 Numerical simulation of real-world flows. Fluid Dyn. Res. 47, 051201.Google Scholar
He, J.-W., Glowinski, R., Metcalfe, R., Nordlander, A. & Periaux, J. 2000 Active control and drag optimization for flow past a circular cylinder: I. Oscillatory cylinder rotation. J. Comput. Phys. 163, 83–117.Google Scholar
Heitz, D., Mémin, E. & Schnörr, C. 2010 Variational fluid flow measurements from image sequences: synopsis and perspectives. Exp. Fluids 48, 369–393.Google Scholar
Hoepffner, J., Chevalier, M., Bewley, T. R. & Henningson, D. 2005 State estimation in wall-bounded flow systems. Part 1. Perturbed laminar flows. J. Fluid Mech. 534, 263–294.Google Scholar
Homescu, C., Navon, I. M. & Li, Z. 2002 Suppression of vortex shedding for flow around a circular cylinder using optimal control. Intl J. Numer. Meth. Fluids 38, 43–69.CrossRefGoogle Scholar
Houtekamer, P. L. & Mitchell, H. L. 2001 A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Weath. Rev. 129, 123–137.Google Scholar
Jameson, A.1991 Time-dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings. AIAA Paper 91-1596.Google Scholar
Jawahar, P. & Kamath, H. 2000 A high-resolution procedure for Euler and Navier–Stokes computations on unstructured grids. J. Comput. Phys. 164, 165–203.Google Scholar
Juillet, F., Schmid, P. J. & Huerre, P. 2013 Control of amplifier flows using subspace identification techniques. J. Fluid Mech. 725, 522–565.Google Scholar
Kalman, R. E. 1960 A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Engng 82, 35–45.CrossRefGoogle Scholar
Kang, W. & Xu, L. 2012 Optimal placement of mobile sensors for data assimilations. Tellus A 64, 17133.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, 559–581.Google Scholar
Kumar, S., Lopez, C., Probst, O., Francisco, G., Askari, D. & Yang, Y. 2013 Flow past a rotationally oscillating cylinder. J. Fluid Mech. 735, 307–346.CrossRefGoogle Scholar
Langland, R. H. & Baker, N. L. 2004 Estimation of observation impact using the NRL atmospheric variational data assimilation adjoint system. Tellus A 56, 189–201.Google Scholar
Le Dimet, F.-X., Navon, I. M. & Daescu, D. N. 2002 Second-order information in data assimilation. Mon. Weath. Rev. 130, 629–648.Google Scholar
Le Dimet, F.-X., Ngodock, H.-E., Luong, B. & Verron, J. 1997 Sensitivity analysis in variational data assimilation. J. Met. Soc. Japan 75, 245–255.Google Scholar
Le Dimet, F.-X. & Talagrand, O. 1986 Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus A 38A, 97–110.Google Scholar
van Leeuwen, P. J. & Evensen, G. 1996 Data assimilation and inverse methods in terms of a probabilistic formulation. Mon. Weath. Rev. 124, 2898–2913.Google Scholar
Lewis, J. M., Lakshmivarahan, S. & Dhall, S. K. 2006 Dynamic Data Assimilation: A Least Squares Approach, Encyclopedia of Mathematics and its Applications, vol. 104. Cambridge University Press.Google Scholar
Lions, J. L. 1971 Optimal Control of Systems Governed by Partial Differential Equations. Springer.Google Scholar
Liu, C., Xiao, Q. & Wang, B. 2008 An ensemble-based four-dimensional variational data assimilation scheme. Part I: technical formulation and preliminary test. Mon. Weath. Rev. 136, 3363–3373.CrossRefGoogle Scholar
Liu, C., Zheng, X. & Sung, C. H. 1998 Preconditioned multigrid methods for unsteady incompressible flows. J. Comput. Phys. 139, 35–57.Google Scholar
Luo, H., Baum, J. D. & Löhner, R. 2001 An accurate, fast, matrix-free implicit method for computing unsteady flows on unstructured grids. Comput. Fluids 30, 137–159.Google Scholar
Mohammadi, B. & Pironneau, O. 2010 Applied Shape Optimization for Fluids, 2nd edn. Oxford University Press.Google Scholar
Mokhasi, P. & Rempfer, D. 2004 Optimized sensor placement for urban flow measurement. Phys. Fluids 16, 1758–1764.CrossRefGoogle Scholar
Mons, V., Chassaing, J.-C., Gomez, T. & Sagaut, P. 2014 Is isotropic turbulence decay governed by asymptotic behavior of large scales? An eddy-damped quasi-normal Markovian-based data assimilation study. Phys. Fluids 26, 115105.Google Scholar
Mons, V., Chassaing, J.-C., Gomez, T. & Sagaut, P. 2016 Reconstruction of unsteady viscous flows using data assimilation schemes. J. Comput. Phys. 316, 255–280.Google Scholar
Nadarajah, S. K. & Jameson, A.2001 Studies of continuous and discrete adjoint approaches to viscous automatic aerodynamic shape optimization. AIAA Paper 2001-2530.Google Scholar
Nocedal, J. 1980 Updating quasi-Newton matrices with limited storage. Maths Comput. 35, 773–782.Google Scholar
Papadakis, N. & Mémin, E. 2008 Variational assimilation of fluid motion from image sequence. SIAM J. Imaging Sci. 1, 343–363.Google Scholar
Peter, J. E. V. & Dwight, R. P. 2010 Numerical sensitivity analysis for aerodynamic optimization: a survey of approaches. Comput. Fluids 39, 373–391.Google Scholar
Posdziech, O. & Grudmann, R. 2007 A systematic approach to the numerical calculation of fundamental quantities of the two-dimensional flow over a circular cylinder. J. Fluids Struct. 23, 479–499.CrossRefGoogle Scholar
Protas, B. & Styczek, A. 2002 Optimal rotary control of the cylinder wake in the laminar regime. Phys. Fluids 14, 2073–2087.Google Scholar
Qu, L., Norberg, C., Davidson, L., Peng, S.-H. & Wang, F. 2013 Quantitative numerical analysis of flow past a circular cylinder at Reynolds number between 50 and 200. J. Fluids Struct. 39, 347–370.Google Scholar
Roe, P. L. 1981 Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372.Google Scholar
Sharov, D. & Nakahashi, K.1997 Reordering of 3-D hybrid unstructured grids for vectorized LU-SGS Navier–Stokes computations. AIAA Paper 97-2102.CrossRefGoogle Scholar
Stoyanov, M. K.2009 Reduced order methods for large scale Riccati equations. PhD thesis, Virginia Polytechnic Institute and State University.Google Scholar
Suzuki, T. 2012 Reduced-order Kalman-filtered hybrid simulation combining particle tracking velocimetry and direct numerical simulation. J. Fluid Mech. 709, 249–288.Google Scholar
Talagrand, O. 1997 Assimilation of observations, an introduction. J. Met. Soc. Japan 75, 191–209.Google Scholar
Thiria, B., Goujon-Durand, S. & Wesfreid, J. E. 2006 The wake of a cylinder performing rotary oscillations. J. Fluid Mech. 560, 123–147.Google Scholar
Thiria, B. & Wesfreid, J. E. 2007 Stability properties of forced wakes. J. Fluid Mech. 579, 137–161.Google Scholar
Tokumaru, P. & Dimotakis, P. E. 1991 Rotary oscillation control of cylinder wake. J. Fluid Mech. 224, 77–90.CrossRefGoogle Scholar
Wang, Z., Navon, I. M., Le Dimet, F.-X. & Zou, X. 1992 The second order adjoint analysis: theory and applications. Meteorol. Atmos. Phys. 50, 3–20.Google Scholar
Wikle, C. K. & Berliner, L. M. 2007 A Bayesian tutorial for data assimilation. Physica D 230, 1–16.Google Scholar
Willcox, K. 2006 Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition. Comput. Fluids 35, 208–226.Google Scholar
Williamson, C. H. K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28, 477–539.Google Scholar
Yildirim, B., Chryssostomidis, C. & Karniadakis, G. E. 2009 Efficient sensor placement for ocean measurements using low-dimensional concepts. Ocean Model. 27, 160–173.Google Scholar