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Data assimilation of mean velocity from 2D PIV measurements of flow over an idealized airfoil

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Data assimilation can be used to combine experimental and numerical realizations of the same flow to produce hybrid flow fields. These have the advantages of less noise contamination and higher resolution while simultaneously reproducing the main physical features of the measured flow. This study investigates data assimilation of the mean flow around an idealized airfoil (Re = 13,500) obtained from time-averaged two-dimensional particle image velocimetry (PIV) data. The experimental data, which constitute a low-dimensional representation of the full flow field due to resolution and field-of-view limitations, are incorporated into a simulation governed by the two-dimensional, incompressible Reynolds-averaged Navier–Stokes (RANS) equations with an unknown momentum forcing. This forcing, which corresponds to the divergence of the Reynolds stress tensor, is calculated from a direct-adjoint optimization procedure to match the experimental and numerical mean velocity fields. The simulation is projected onto the low-dimensional subspace of the experiment to calculate the discrepancy and a smoothing procedure is used to recover adjoint solutions on the higher dimensional subspace of the simulation. The study quantifies how well data assimilation can reconstruct the mean flow and the minimum experimental measurements needed by altering the resolution and domain size of the time-averaged PIV.

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  • Adrian RJ (1991) Particle-imaging techniques for experimental fluid mechanics. Annu Rev Fluid Mech 23:261–304

    Article  Google Scholar 

  • Adrian RJ, Westerweel J (2011) Particle image velocimetry. Cambridge University, New York

    MATH  Google Scholar 

  • Bewley TR, Moin P, Temam R (2001) DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms. J Fluid Mech 447:179–225

    Article  MATH  MathSciNet  Google Scholar 

  • Bukshtynov V, Volkov O, Protas B (2011) On optimal reconstruction of constitutive relations. Physica D 240(16):1228–1244

    Article  MATH  MathSciNet  Google Scholar 

  • Flemming J (2011) Generalized Tikhonov regularization: Basic theory and comprehensive results on convergence rates. PhD thesis, Techn Univ Chemnitz

  • Foures DPG, Dovetta N, Sipp D, Schmid PJ (2014) A data-assimilation method for Reynolds-averaged Navier-Stokes-driven mean flow reconstruction. J Fluid Mech 759:404–431

    Article  MATH  Google Scholar 

  • Gharib M (1983) The effect of flow oscillations on cavity drag, and a technique for their control. PhD thesis, California Institute of Technology

  • Giannetti F, Luchini P (2007) Structural sensitivity analysis of the first instability of the cylinder wake. J Fluid Mech 581:167–197

    Article  MATH  MathSciNet  Google Scholar 

  • Gonzalez C, Bruhat JF, Wainfan B, McKeon BJ (2010) Control of laminar separation on an idealized airfoil using periodic dynamic roughness actuation. In 63rd Meeting of the American Physical Society Division of Fluid Dynamics. Gallery of Fluid Motion, Poster 26

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • Kim J, Bewley TR (2007) A linear systems approach to flow control. Annu Rev Fluid Mech 39:383–417

    Article  MATH  MathSciNet  Google Scholar 

  • Kompenhans J, Raffel M, Willert C (2007) Particle image velocimetry—a pratical guide. Springer, Berlin

    Google Scholar 

  • Lewis JM, Lakshmivarahan S, Dhall S (2006) Dynamic data assimilation: a least squares approach. In: Encyclopedia of mathematics and its application 104, vol 13. Cambridge University Press

  • Marquet O, Sipp D, Jacquin L (2008) Sensitivity analysis and passive control of cylinder flow. J Fluid Mech 516:221–252

    Article  MATH  MathSciNet  Google Scholar 

  • Meliga P, Pujals G, Serre E (2012) Sensitivity of 2-D turbulent flow past a D-shaped cylinder using global stability. Phys Fluids 24(6):61701

    Article  Google Scholar 

  • Mettot C, Sipp D (2014) Quasi-laminar stability and sensitivity analyses for turbulent flows: Prediction of low-frequency unsteadiness and passive control. Phys Fluids 26(4):45112

    Article  Google Scholar 

  • Mons V, Chassaing JC, Gomez T, Sagaut P (2016) Reconstruction of unsteady viscous flows using data assimilation schemes. J Comput Phys 316:255–280

    Article  MATH  MathSciNet  Google Scholar 

  • Nisugi K, Hayase T, Shirai A (2004) Fundamental study of hybrid wind tunnel integrating numerical simulation and experiment in analysis of flow field. JSME Int J 47:593–604

    Article  Google Scholar 

  • Polak E, Ribière G (1969) Note sur la convergence de méthodes de directions conjuguées. Revue française d’informatique et de recherche opérationnelle, série rouge 3(1):35–43

    Article  MATH  Google Scholar 

  • Prasad A, Williamson CHK (1997) The instability of the shear layer separating from a bluff body. J Fluid Mech 333:375–402

    Article  Google Scholar 

  • Sciacchitano A, Wieneke B (2016) PIV uncertainty propagation. Meas Sci Technol 27(8):1–16

    Article  Google Scholar 

  • Sipp D, Marquet O, Meliga P, Barbagallo A (2010) Dynamics and control of global instabilities in open-flows: a linearized approach. Appl Mech Rev 63(3):030801

    Article  Google Scholar 

  • Strykowski P, Sreenivasan K (1990) On the formation and suppression of vortex ‘shedding’ at low Reynolds numbers. J Fluid Mech 218:71–107

    Article  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

    Article  MATH  MathSciNet  Google Scholar 

  • Suzuki T, Ji H, Yamamoto F (2009a) Unsteady PTV velocity field past an airfoil solved with DNS: Part 1. Algorithm of hybrid simulation and hybrid velocity field at \({R}e \approx 10^3\). Exp Fluids 47:957–976

    Article  Google Scholar 

  • Suzuki T, Sanse A, Mizushima T, Yamamoto F (2009b) Unsteady PTV velocity field past an airfoil solved with DNS: Part 2. Validation and application at Reynolds numbers up to \({R}e \le 10^4\). Exp Fluids 47:977–994

    Article  Google Scholar 

  • Wallace RD, McKeon BJ (2012) Laminar separation bubble manipulation with dynamic roughness. In: Proceedings of the 6th AIAA flow control conference. AIAA 2012–2680

  • Westerweel J (2008) On velocity gradients in PIV interrogation. Exp Fluids 44:831–842

    Article  Google Scholar 

  • Westerweel J, Scarano F (2005) Universal outlier detection for PIV data. Exp Fluids 39(6):1096–1100

    Article  Google Scholar 

  • Wieneke B (2015) PIV uncertainty quantification from correlation statistics. Meas Sci Technol 26(7):1–10

    Article  Google Scholar 

  • Wilson BM, Smith BL (2013) Uncertainty on PIV mean and fluctuating velocity due to bias and random errors. Meas Sci Technol 24(3):1–15

    Google Scholar 

  • Zielinska BJA, Goujon-Durand S, Dusek J, Wesfried JE (1997) Strongly nonlinear effect in unstable wakes. Phys Rev Lett 79:3893–3896

    Article  Google Scholar 

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Support from a National Science Foundation Graduate Fellowship (S.S.) is gratefully acknowledged. The authors would also like to thank Professors Tim Colonius and Anthony Leonard for discussions about the direct-adjoint optimization framework as well as Kevin Rosenberg for his assistance in implementing the optimization procedure on a cluster. Finally, the authors wish to thank the referees for their suggestions which improved the clarity of the paper.

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Correspondence to Sean Symon.

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Symon, S., Dovetta, N., McKeon, B.J. et al. Data assimilation of mean velocity from 2D PIV measurements of flow over an idealized airfoil. Exp Fluids 58, 61 (2017).

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