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A few techniques to improve data-driven reduced-order simulations for unsteady flows. (English) Zbl 1519.76285

Summary: A key step to improve data-driven reduced-order simulations is to compute a transfer function that predicts the time evolution of the reduced-order modes accurately. We demonstrate a couple of useful techniques to achieve this objective: One is to pre-process time-series of reduced-order modes with a low-pass filter, e.g. a polynomial filter and B-spline, and the other is to compute a data-driven transfer function from multiple past time-steps, corresponding to a high-order temporal scheme. These techniques are exercised with POD modes generated from time-resolved planar PIV data. A fully separated flow past the NACA0012 airfoil at the angle of attack of \(30^\circ\) and Re = 1000 is measured in a water tunnel, and non-periodic unsteady flow is analyzed in two dimensions. From the first 1000 frames, transfer functions are calculated based on minimization of different cost functions, which define least-squares errors in the predicted POD modes at the next time step; subsequently, their prediction capabilities are evaluated during the following 1000 frames based on the accuracy of the predicted POD modes at the next time steps. The multistep schemes can reduce the root-mean-square errors of the predicted mode coefficients by up to 10% without a low-pass filter. Combining a low-pass filter with the second-order temporal scheme can further reduce the errors by 7%, and introduction of an \(L_1\)-norm constraint for the mode coefficients decreases it by extra 2%. In contrast, nonlinear transfer functions with more degrees of freedom deteriorate the prediction during time duration outside of the sampling period even relative to the linear prediction.

MSC:

76M99 Basic methods in fluid mechanics

Software:

DPIV
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[1] Salwen, H.; Grosch, C. E., The continuous spectrum of the Orr-Sommerfeld equation. Part 2. eigenfunction expansions, J Fluid Mech, 104, 445-465 (1981) · Zbl 0467.76051
[2] Hill, D. C., Adjoint systems and their role in the receptivity problem for boundary layers, J Fluid Mech, 292, 183-204 (1995) · Zbl 0866.76029
[3] Tumin, A., Receptivity of pipe Poiseuille flow, J Fluid Mech, 315, 119-137 (1996) · Zbl 0869.76019
[4] Theofilis, V., Advances in global linear instability analysis of nonparallel and three-dimensional flows, Prog Aerosp Sci, 39, 249-315 (2003)
[5] Herbert, T., Parabolized stability equations, Annu Rev Fluid Mech, 29, 245-283 (1997)
[6] Lumley, J. L., Stochastic tool in turbulence (1970), Academic Press · Zbl 0273.76035
[7] Aubry, N.; Holmes, P.; Lumley, J. L.; Stone, E., The dynamics of coherent structures in the wall region of turbulent boundary layer, J Fluid Mech, 192, 115-173 (1988) · Zbl 0643.76066
[8] Rowley, C. W.; Colonius, T.; Murray, R. M., Model reduction for compressible flows using POD and Galerkin projection, Physica D, 189, 115-129 (2004) · Zbl 1098.76602
[9] Farrell, B. F.; Ioannou, P. J., Stochastic forcing of the linearized Navier-Stokes equations, Phys Fluids, 5, 11, 2600-2609 (1993) · Zbl 0809.76078
[10] Mckeon, B. J.; Sharma, A. S., A critical layer model for turbulent pipe flow, J Fluid Mech, 658, 336-382 (2010) · Zbl 1205.76138
[11] Adrian, R. J.; Moin, P., Stochastic estimation of organized turbulent structure: homogeneous shear flow, J Fluid Mech, 190, 531-559 (1988) · Zbl 0642.76070
[12] Hasselmann, K., PIPS and POPs: the reduction of complex dynamical systems using principal interaction and oscillation patterns, J Geophys Res, 93, D9, 11015-11021 (1988)
[13] Schmid, P. J., Dynamic mode decomposition of numerical and experimental data, J Fluid Mech, 656, 5-28 (2010) · Zbl 1197.76091
[14] Chen, K. K.; Tu, J. H.; Rowley, C. W., Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses, J Nonlinear Sci, 22, 6, 887-915 (2012) · Zbl 1259.35009
[15] Wynn, A.; Pearson, D. S.; Ganapathisubramani, B.; Goulart, P. J., Optimal mode decomposition for unsteady flows, J Fluid Mech, 733, 473-503 (2013) · Zbl 1294.76205
[16] Jovanovic, M. R.; Schmid, P. J.; Nichols, J. W., Sparsity-promoting dynamic mode decomposition, Phys Fluids, 26, 2, 024103 (2014)
[17] Williams, M. O.; Kevrekidis, I. G.; Rowley, C. W., A data-driven approximation of the Koopman operator: extending dynamic mode decomposition, J Nonlinear Sci, 25, 6, 1307-1346 (2015) · Zbl 1329.65310
[18] Kutz, J. N.; Fu, X.; Brunton, S. L., Multiresolution dynamic mode decomposition, SIAM J Appl Dyn Syst, 15, 2, 713-735 (2016) · Zbl 1338.37121
[19] Perret, L.; Collin, E.; Delville, J., Polynomial identification of POD based low-order dynamical system, J Turbulence, 7, 1-15 (2006) · Zbl 1273.76134
[20] Iollo, A.; Lanteri, S.; Désidéri, J. A., Stability properties of POD-Galerkin approximations for the compressible Navier-Stokes equations, Theor Comp Fluid Dyn, 13, 6, 377-396 (2000) · Zbl 0987.76077
[21] Rempfer, D., On low-dimensional Galerkin models for fluid flow, Theor Comp Fluid Dyn, 14, 2, 75-88 (2000) · Zbl 0984.76068
[22] Ma, X.; Karniadakis, G. E.; Park, H.; Gharib, M., DPIV-driven flow simulation: a new computational paradigm, Proc R Soc Lond A, 459, 547-565 (2003) · Zbl 1116.76413
[23] Samimy, M.; Debiasi, M.; Caraballo, E.; Serrani, A.; Yuan, X.; Little, J., Feedback control of subsonic cavity flows using reduced-order models, J Fluid Mech, 579, 315-346 (2007) · Zbl 1113.76008
[24] Barbagallo, A.; Sipp, D.; Schmid, P. J., Closed-loop control of an open cavity flow using reduced-order models, J Fluid Mech, 641, 1-50 (2009) · Zbl 1183.76701
[25] D’adamo, J.; Papadakis, N.; Mémin, E.; Artana, G., Variational assimilation of POD low-order dynamical systems, J Turbulence, 8, 9, 1-22 (2007) · Zbl 1273.76133
[26] Daescu, D. N.; Navon, I. M., Efficiency of a POD-based reduced second-order adjoint model in 4D-Var data assimilation, Num Meth Fluids, 53, 6, 985-1004 (2007) · Zbl 1370.76122
[27] Leroux, R.; Chatellier, L.; David, L., Maximum likelihood estimation of missing data applied to flow reconstruction around NACA profiles, Fluid Dyn Res, 47, 1-23 (2015)
[28] Suzuki, T.; Yamamoto, F., Hierarchy of hybrid unsteady-flow simulations integrating time-resolved PTV with DNS and their data-assimilation capabilities, Fluid Dyn Res, 47, 051407 (2015)
[29] Nonomura, T.; Shibata, H.; Takaki, R., Extended-Kalman-filter-based dynamic mode decomposition for simultaneous system identification and denoising, PLoS One, 14, 2, e0209836 (2019)
[30] Jeong, S. H.; Bienkiewicz, B., Application of autoregressive modeling in proper orthogonal decomposition of building wind pressure, J Wind Eng Ind Aero, 69-71, 685-695 (1997)
[31] Kunisch, K.; Volkwein, S., Galerkin proper orthogonal decomposition methods for parabolic problems, Numer Math, 90, 1, 117-148 (2001) · Zbl 1005.65112
[32] Sirisup, S.; Karniadakis, G. E.; Xiu, D.; Kevrekidis, I. G., Equation-free/Galerkin-free POD-assisted computation of incompressible flows, J Comput Phys, 207, 568-587 (2005) · Zbl 1213.76146
[33] Stankiewicz, W.; Morzyński, M.; Noack, B. R.; Tadmor, G., Reduced order Galerkin models of flow around NACA-0012 airfoil, Math Modell Anal, 13, 1, 113-122 (2008) · Zbl 1258.76114
[34] Leroux, R.; Chatellier, L.; David, L., Bayesian inference applied to spatio-temporal reconstruction of flows around a NACA0012 airfoil, Exp Fluids, 55, 1699 (2014)
[35] Nankai, K.; Ozawa, Y.; Nonomura, T.; Asai, K., Linear reduced-order model based on PIV data of flow field around airfoil, Trans Japan Soc Aero Space Sci, 62, 4, 227-235 (2019)
[36] Sirovich, L., Turbulence and the dynamics of coherent structures. I. coherent structures. II. symmetries and transformations. III. dynamics and scaling, Quart Appl Math, 45, 561-590 (1987) · Zbl 0676.76047
[37] Suzuki, T., POD-based reduced-order hybrid simulation using the data-driven transfer function with time-resolved PTV feedback, Exp Fluids, 55, 8, 1798 (2014)
[38] Huber, P. J., Robust statistics (1981), John Wiley & Sons · Zbl 0536.62025
[39] Brunton, S. L.; Proctor, J. L.; Kutz, J. N., Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci. USA, 15 (2016) · Zbl 1355.94013
[40] de Boor, C., A practical guide to splines (1978), Springer · Zbl 0406.41003
[41] Fick, L.; Maday, Y.; Patera, A. T.; Taddei, T., A stabilized POD model for turbulent flows over a range of Reynolds numbers: optimal parameter sampling and constrained projection, J Comput Phys, 371, 15, 214-243 (2018) · Zbl 1415.76387
[42] Kalman, R. E., A new approach to linear filtering and prediction problems, Trans ASME: J Basic Eng, 82, D, 35-45 (1960)
[43] 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
[44] Julier, S. J.; Uhlmann, J. K., A new extension of the Kalman filter to nonlinear systems, Proc. AeroSense. (SPIE) (1997)
[45] Suzuki, T.; Chatellier, L.; Jeon, Y. J.; David, L., Unsteady pressure estimation and compensation capabilities of the hybrid simulation combining PIV and DNS, Meas Sci Technol, 29, 12, 125305 (2018)
[46] Jeon, Y. J.; Chatellier, L.; David, L., Fluid trajectory evaluation based on an ensemble-averaged cross-correlation in time-resolved PIV, Exp Fluids, 55, 1-16 (2014)
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