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Recovering turbulent flow field from local quantity measurement: turbulence modeling using ensemble-Kalman-filter-based data assimilation

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Abstract

This paper is focused on the recovery of the global flow field through data assimilation of local flow quantity measurement and Reynolds-averaged Navier–Stokes (RANS) modeling. Particular attention is given to the optimization of various RANS model constants using the ensemble Kalman filter (EnKF) approach. To this end, a free round jet at Reynolds number Re = 6000 is experimentally measured using particle image velocimetry (PIV), serving as the observation data and validation purpose. A total of four different RANS models are separately employed as system models in the data assimilation, i.e., the Spalart–Allmaras, \(k - \varepsilon\), \(k - \omega\), and shear stress transport models. The results convincingly demonstrate that all models with EnKF augmentation are considerably improved compared with their original counterparts. Among all models, the \(k - \varepsilon\) model with EnKF augmentation showed the best performance in predicating the time-averaged flow quantities. Subsequently, the \(k - \varepsilon\) model with EnKF augmentation is examined to demonstrate its robustness and sensitivity for different observational data. Three different selection strategies of observational data are documented here: the velocity distributions in a region, along a line, and at a single point. For all of these selections, the observational data in the jet transition region are shown to be the best candidate for flow field recovery.

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Abbreviations

d 0 :

Depth of glass tank

D :

Diameter of the jet (mm)

N :

Ensemble size number

N m :

Maximum number of iteration steps

H :

Observation operator

K :

Kalman gain matrix

L :

Length of the round pipe [mm]

P :

Ensemble covariance matrix of systematic samples

Re :

Reynolds number (based on the diameter of jet)

R :

Covariance matrix of measurement perturbations

u :

Non-dimensional velocity in system model

\(\tilde{u}\) :

Non-dimensional velocity in observation model

\(U_{0}\) :

Bulk velocity of the jet (m s−1)

\(v\) :

System noise

w :

Synthetic experiment noise

\(x_{f}^{i}\) :

Forecasted state variables of each ensemble member (\(i = 1, \ldots N\))

\(x_{a}^{i}\) :

Analysis state variables of each ensemble member (\(i = 1, \ldots N\))

\(\bar{x}_{f}\) :

Mean state variables of \(x_{f}^{i}\)

\(\bar{x}_{a}\) :

Mean state variables of \(x_{a}^{i}\)

\(X_{t}\) :

N-dimensional matrix of \(x_{t}^{(i)}\)

\(y_{\exp }\) :

Experiment data

\(\overline{y}_{f}\) :

Mean of the ensemble prediction matrix

\( C_{v1} , \, C_{b1} , \, C_{b2}, \, C_{w2} ,C_{w3} , \, \sigma \) :

Constants in the SA model

\( C_{\mu } , \, C_{1\varepsilon } , \, C_{2\varepsilon } , \, \sigma_{k} ,\sigma_{\varepsilon } \, \) :

Constants in the \(k - \varepsilon\) model

\(\alpha , \, \beta *, \, \beta_{i} , \sigma_{k} , \, \sigma_{w} \) :

Constants in the \(k - \omega\) model

\( \alpha*, \,\beta*,\,\beta_{i,1}, \, \beta_{i,2}, \,\sigma_{k,1}, \, \sigma_{k,2}, \sigma_{w,1} , \, \sigma_{w,2} ,a_{1} \) :

Constants in the SST model

\(\beta\) :

Relaxation factor

\(\theta\) :

Constants of the RANS model

\(\xi\) :

Non-dimensional relative error of velocity

CFD:

Computational fluid dynamics

DNS:

Direct numerical simulation

DA:

Data assimilation

EnKF:

Ensemble Kalman filter

EnKS:

Ensemble Kalman smoother

LES:

Large eddy simulation

PIV:

Particle image velocimetry

RANS:

Reynolds-averaged Navier–Stokes

SA:

Spalart–Allmaras

SST:

Shear stress transport

References

  • Dimet FL, Talagrand O (1986) Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus Ser A Dyn Meteorol Oceanogr 38A(2):97–110

    Article  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 Oceans 99(C5):10143–10162

    Article  Google Scholar 

  • Evensen G (2006) Data assimilation: the ensemble Kalman filter. Springer, New York Inc

    MATH  Google Scholar 

  • Gao X, Wang Y, Overton N, Zupanski M, Tu X (2017) Data-assimilated computational fluid dynamics modeling of convection-diffusion-reaction problems. J Comput Sci 21:38–59

    Article  MathSciNet  Google Scholar 

  • He C, Liu Y (2017) Proper orthogonal decomposition-based spatial refinement of TR-PIV realizations using high-resolution non-TR-PIV measurements. Exp Fluids 58(7):86

    Article  Google Scholar 

  • He C, Liu Y, Savas Y (2018) Large-eddy simulation of circular jet mixing: lip-and inner-ribbed nozzles. Comput Fluids 168:245–268

    Article  MathSciNet  Google Scholar 

  • Kato H, Obayashi S (2012) Statistical approach for determining parameters of a turbulence model. In: International conference on information fusion

  • Kato H, Obayashi S (2013) Data assimilation for turbulent flows. In: Aiaa non-deterministic approaches conference

  • Kato H, Yoshizawa A, Ueno G, Obayashi S (2015) A data assimilation methodology for reconstructing turbulent flows around aircraft. J Comput Phys 283(C):559–581

    Article  MathSciNet  Google Scholar 

  • Launder BE, Spalding DB (1983) The numerical computation of turbulent flows. Numerical prediction of flow, heat transfer, turbulence and combustion. Elsevier, Amsterdam, pp 96–116

    Book  Google Scholar 

  • Law KJH, Stuart AM, Zygalakis KC (2015) Data assimilation: a mathematical introduction. Rev Bras Meteorol 26(3):433–442

    MATH  Google Scholar 

  • Mckay MD, Beckman RJ, Conover WJ (2000) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 42(1):55–61

    Article  Google Scholar 

  • Menter FR (1994) Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J 32(8):1598–1605

    Article  Google Scholar 

  • Moghaddam AA, Sadaghiyani A (2018) A deep learning framework for turbulence modeling using data assimilation and feature extraction. arXiv preprint arXiv:1802.06106

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

    Article  MathSciNet  Google Scholar 

  • Spalart P, Allmaras S (1994) A one-equation turbulence model for aerodynamic flows. Rech Aerosp 1(1):5–21

    Google Scholar 

  • Wilcox DC (1988) Multiscale model for turbulent flows. AIAA J 1(11):1311–1320

    Article  MathSciNet  Google Scholar 

  • Wilcox DC (1998) Turbulence modeling for CFD (Vol. 2): DCW industries La Canada, CA, pp 73-92

  • Zhang X, Su G, Yuan H, Chen J, Huang Q (2014) Modified ensemble Kalman filter for nuclear accident atmospheric dispersion: prediction improved and source estimated. J Hazard Mater 280:143–155

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge financial support for this study from the National Natural Science Foundation of China (11725209).

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Authors and Affiliations

  1. Key Lab of Education Ministry for Power Machinery and Engineering, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China

    Zhiwen Deng, Chuangxin He, Xin Wen & Yingzheng Liu

  2. Gas Turbine Research Institute, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China

    Zhiwen Deng, Chuangxin He, Xin Wen & Yingzheng Liu

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  1. Zhiwen Deng
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  2. Chuangxin He
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  4. Yingzheng Liu
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Correspondence to Yingzheng Liu.

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Deng, Z., He, C., Wen, X. et al. Recovering turbulent flow field from local quantity measurement: turbulence modeling using ensemble-Kalman-filter-based data assimilation. J Vis 21, 1043–1063 (2018). https://doi.org/10.1007/s12650-018-0508-0

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