×

Reconstruction of unsteady viscous flows using data assimilation schemes. (English) Zbl 1349.76763

Summary: This paper investigates the use of various data assimilation (DA) approaches for the reconstruction of the unsteady flow past a cylinder in the presence of incident coherent gusts. Variational, ensemble Kalman filter-based and ensemble-based variational DA techniques are deployed along with a 2D compressible Navier-Stokes flow solver, which is also used to generate synthetic observations of a reference flow. The performance of these DA schemes is thoroughly analyzed for various types of observations ranging from the global aerodynamic coefficients of the cylinder to the full 2D flow field. Moreover, different reconstruction scenarios are investigated in order to assess the robustness of these methods for large scale DA problems with up to \(10^{5}\) control variables. In particular, we show how an iterative procedure can be used within the framework of ensemble-based methods to deal with both non-uniform unsteady boundary conditions and initial field reconstruction. The different methodologies developed and assessed in this work give a review of what can be done with DA schemes in computational fluid dynamics (CFD) paradigm. In the same time, this work also provides useful information which can also turn out to be rational arguments in the DA scheme choice dedicated to a specific CFD application.

MSC:

76M35 Stochastic analysis applied to problems in fluid mechanics
65J22 Numerical solution to inverse problems in abstract spaces
62M20 Inference from stochastic processes and prediction
65C60 Computational problems in statistics (MSC2010)
76N15 Gas dynamics (general theory)

Software:

L-BFGS; EnKF
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lewis, J. M.; Lakshmivarahan, S.; Dhall, S. K., Dynamic Data Assimilation: A Least Squares Approach, Encyclopedia of Mathematics and Its Applications, vol. 104 (2006), Cambridge University Press · Zbl 1268.62003
[2] Evensen, G., Data Assimilation: The Ensemble Kalman Filter (2009), Springer-Verlag · Zbl 1395.93534
[3] Gronskis, A.; Heitz, D.; Mémin, E., Inflow and initial conditions for direct numerical simulation based on adjoint data assimilation, J. Comput. Phys., 242, 480-497 (2013) · Zbl 1299.76099
[4] Kato, H.; Yoshizawa, A.; Ueno, G.; Obayashi, S., A data assimilation methodology for reconstructing turbulent flows around aircraft, J. Comput. Phys., 283, 559-581 (2015) · Zbl 1351.76036
[5] Suzuki, T., Reduced-order Kalman-filtered hybrid simulation combining particle tracking velocimetry and direct numerical simulation, J. Fluid Mech., 709, 249-288 (2012) · Zbl 1275.76028
[6] Mons, V.; Chassaing, J.-C.; Gomez, T.; Sagaut, P., 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 (2014)
[7] Le Dimet, F. X.; Talagrand, O., Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects, Tellus, 38A, 97-110 (1986)
[8] Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations (1971), Springer-Verlag · Zbl 0203.09001
[9] Kalman, R. E., A new approach to linear filtering and prediction problems, J. Basic Eng., 82, 35-45 (1960)
[10] van Leeuwen, P. J.; Evensen, G., Data assimilation and inverse methods in terms of a probabilistic formulation, Mon. Weather Rev., 124, 2898-2913 (1996)
[11] Wikle, C. K.; Berliner, L. M., A Bayesian tutorial for data assimilation, Physica D, 230, 1-16 (2007) · Zbl 1113.62032
[12] D’Adamo, J.; Papadakis, N.; Mémin, E.; Artana, G., Variational assimilation of POD low-order dynamical systems, J. Turbul., 8, 1-22 (2007) · Zbl 1273.76133
[13] Artana, G.; Cammilleri, A.; Carlier, J.; Mémin, E., Strong and weak constraint variational assimilations for reduced order fluid flow modeling, J. Comput. Phys., 231, 3264-3288 (2012) · Zbl 1401.76060
[14] Foures, D. P.G.; Dovetta, N.; Sipp, D.; Schmid, P. J., A data-assimilation method for Reynolds-averaged Navier-Stokes-driven mean flow reconstruction, J. Fluid Mech., 759, 404-431 (2014) · Zbl 1446.76121
[15] Evensen, G., Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, J. Geophys. Res., 99, 10143-10162 (1994)
[16] Colburn, C. H.; Cessna, J. B.; Bewley, T. R., State estimation in wall-bounded flow systems. Part 3. The ensemble Kalman filter, J. Fluid Mech., 682, 289-303 (2011) · Zbl 1241.76303
[17] Kato, H.; Obayashi, S., Approach for uncertainty of turbulence modeling based on data assimilation technique, Comput. Fluids, 85, 2-7 (2013) · Zbl 1290.76042
[18] Peter, J. E.V.; Dwight, R. P., Numerical sensitivity analysis for aerodynamic optimization: a survey of approaches, Comput. Fluids, 39, 373-391 (2010) · Zbl 1242.76301
[19] Houtekamer, P. L.; Mitchell, H. L., A sequential ensemble Kalman filter for atmospheric data assimilation, Mon. Weather Rev., 129, 123-137 (2001)
[20] Anderson, J. L.; Anderson, S. L., A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts, Mon. Weather Rev., 127, 2741-2758 (1999)
[21] Liu, C.; Xiao, Q.; Wang, B., An ensemble-based four-dimensional variational data assimilation scheme. Part I: technical formulation and preliminary test, Mon. Weather Rev., 136, 3363-3373 (2008)
[22] Liu, C.; Xiao, Q.; Wang, B., An ensemble-based four-dimensional variational data assimilation scheme. part II: observing system simulation experiments with Advanced Research WRF (ARW), Mon. Weather Rev., 137, 1687-1704 (2009)
[23] Liu, C.; Xiao, Q., An ensemble-based four-dimensional variational data assimilation scheme. Part III: antarctic applications with advanced research WRF using real data, Mon. Weather Rev., 141, 2721-2739 (2013)
[24] Yang, Y.; Robinson, C.; Heitz, D.; Mémin, E., Enhanced ensemble-based 4DVar scheme for data assimilation, Comput. Fluids, 115, 201-210 (2015) · Zbl 06893440
[25] Afgan, I.; Benhamadouche, S.; Han, X.; Sagaut, P.; Laurence, D., Flow over a flat plate with uniform inlet and incident coherent gusts, J. Fluid Mech., 720, 457-485 (2013) · Zbl 1284.76188
[26] Lorenc, A. C., Analysis methods for numerical weather prediction, Q. J. R. Meteorol. Soc., 112, 1177-1194 (1986)
[27] Courtier, P.; Thépaut, J.-N.; Hollingsworth, A., A strategy for operational implementation of 4D-Var, using an incremental approach, Q. J. R. Meteorol. Soc., 120, 1367-1387 (1994)
[28] Evensen, G.; van Leeuwen, P. J., An ensemble Kalman smoother for nonlinear dynamics, Mon. Weather Rev., 128, 1852-1867 (2000)
[29] Cosme, E.; Verron, J.; Brasseur, P.; Blum, J.; Auroux, D., Smoothing problems in a Bayesian framework and their linear Gaussian solutions, Mon. Weather Rev., 140, 683-695 (2012)
[30] Burgers, G.; van Leeuwen, P. J.; Evensen, G., Analysis scheme in the ensemble Kalman filter, Mon. Weather Rev., 126, 1719-1724 (1998)
[31] Buehner, M.; Houtekamer, P. L.; Charette, C.; Mitchell, H. L.; He, B., Intercomparison of variational data assimilation and the ensemble Kalman filter for global deterministic NWP. Part I: description and single-observation experiments, Mon. Weather Rev., 138, 1550-1566 (2010)
[32] Bocquet, M.; Sakov, P., An iterative ensemble Kalman smoother, Q. J. R. Meteorol. Soc., 140, 1521-1535 (2014)
[33] Nadarajah, S. K.; Jameson, A., Studies of continuous and discrete adjoint approaches to viscous automatic aerodynamic shape optimization (2001), AIAA Paper No. 2001-2530
[34] Carpentieri, G.; Koren, B.; van Tooren, M. J.L., Adjoint-based aerodynamic shape optimization on unstructured meshes, J. Comput. Phys., 224, 267-287 (2007) · Zbl 1117.76056
[35] Nocedal, J., Updating quasi-Newton matrices with limited storage, Math. Comput., 35, 773-782 (1980) · Zbl 0464.65037
[36] Armijo, L., Minimization of functions having Lipschitz continuous first partial derivatives, Pac. J. Math., 16, 1-3 (1966) · Zbl 0202.46105
[37] Nerger, L.; Schulte, S.; Bunse-Gerstner, A., On the influence of model nonlinearity and localization on ensemble Kalman smoothing, Q. J. R. Meteorol. Soc., 140, 2249-2259 (2014)
[38] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357-372 (1981) · Zbl 0474.65066
[39] Jawahar, P.; Kamath, H., A high-resolution procedure for Euler and Navier-Stokes computations on unstructured grids, J. Comput. Phys., 164, 165-203 (2000) · Zbl 0992.76063
[40] Jameson, A., Time-dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings (1991), AIAA Paper No. 91-1596
[41] Sharov, D.; Nakahashi, K., Reordering of 3-D hybrid unstructured grids for vectorized LU-SGS Navier-Stokes computations (1997), AIAA Paper No. 97-2102
[42] Luo, H.; Baum, J. D.; Löhner, R., An accurate, fast, matrix-free implicit method for computing unsteady flows on unstructured grids, Comput. Fluids, 30, 137-159 (2001) · Zbl 0983.76056
[43] Liang, C.; Premasuthan, S.; Jameson, A., High-order accurate simulation of low-Mach laminar flow past two side-by-side cylinders using spectral difference method, Comput. Struct., 87, 812-827 (2009)
[44] Bierbooms, W.; Cheng, P.-W., Stochastic gust model for design calculations of wind turbines, J. Wind Eng. Ind. Aerodyn., 90, 1237-1251 (2002)
[45] Bierbooms, W., A gust model for wind turbine design, JSME Int. J. Ser. B Fluids Therm. Eng., 47, 378-386 (2004)
[46] Cadot, O.; Desai, A.; Mittal, S.; Saxena, S.; Chandra, B., Statistics and dynamics of the boundary layer reattachments during the drag crisis transitions of a circular cylinder, Phys. Fluids, 27, Article 014101 pp. (2015)
[47] Bannister, R. N., A review of forecast error covariance statistics in atmospheric variational data assimilation. I: characteristics and measurements of forecast error covariances, Q. J. R. Meteorol. Soc., 134, 1951-1970 (2008)
[48] Bannister, R. N., A review of forecast error covariance statistics in atmospheric variational data assimilation. II: modelling the forecast error covariance statistics, Q. J. R. Meteorol. Soc., 134, 1971-1996 (2008)
[49] Bennett, A. F.; Leslie, L. M.; Hagelberg, C. R.; Powers, P. E., Tropical cyclone prediction using a barotropic model initialized by a generalized inverse method, Mon. Weather Rev., 121, 1714-1729 (1993)
[50] Ménard, R.; Chang, L.-P., Assimilation of stratospheric chemical tracer observations using a Kalman filter. Part II: \( \chi^2\)-validated results and analysis of variance and correlation dynamics, Mon. Weather Rev., 128, 2672-2686 (2000)
[51] Tarantola, A., Inverse Problem Theory and Methods for Model Parameter Estimation (2005), SIAM · Zbl 1074.65013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.