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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

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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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  3. Xin Wen
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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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