### Abstract

A new approach for the interpolation of a filtered turbulence velocity field given random point samples of unfiltered turbulence velocity data is described. In this optimal interpolation method, the best possible value of the interpolated filtered field is obtained as a stochastic estimate of a conditional average, which minimizes the mean square error between the interpolated filtered velocity field and the true filtered velocity field. Besides its origins in approximation theory, the optimal interpolation method also has other advantages over more commonly used ad hoc interpolation methods (like the 'adaptive Gaussian window'). The optimal estimate of the filtered velocity field can be guaranteed to preserve the solenoidal nature of the filtered velocity field and also the underlying correlation structure of both the filtered and the unfiltered velocity fields. The a posteriori performance of the optimal interpolation method is evaluated using data obtained from high-resolution direct numerical simulation of isotropic turbulence. Our results show that for a given sample data density, there exists an optimal choice of the characteristic width of cut-off filter that gives the least possible relative mean square error between the true filtered velocity and the interpolated filtered velocity. The width of this 'optimal' filter and the corresponding minimum relative error appear to decrease with increase in sample data density. Errors due to the optimal interpolation method are observed to be quite low for appropriate choices of the data density and the characteristic width of the filter. The optimal interpolation method is also seen to outperform the 'adaptive Gaussian window', in representing the interpolated field given the data at random sample locations. The overall a posteriori performance of the optimal interpolation method was found to be quite good and hence makes a potential candidate for use in interpolation of PTV and super-resolution PIV data.

Original language | English (US) |
---|---|

Pages (from-to) | 213-221 |

Number of pages | 9 |

Journal | Experiments in Fluids |

Volume | 39 |

Issue number | 2 |

DOIs | |

State | Published - Aug 2005 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mechanical Engineering
- Fluid Flow and Transfer Processes
- Mechanics of Materials
- Computational Mechanics

### Cite this

**Optimal solenoidal interpolation of turbulent vector fields : Application to PTV and super-resolution PIV.** / Vedula, Prakash; Adrian, Ronald.

Research output: Contribution to journal › Article

*Experiments in Fluids*, vol. 39, no. 2, pp. 213-221. https://doi.org/10.1007/s00348-005-1020-6

}

TY - JOUR

T1 - Optimal solenoidal interpolation of turbulent vector fields

T2 - Application to PTV and super-resolution PIV

AU - Vedula, Prakash

AU - Adrian, Ronald

PY - 2005/8

Y1 - 2005/8

N2 - A new approach for the interpolation of a filtered turbulence velocity field given random point samples of unfiltered turbulence velocity data is described. In this optimal interpolation method, the best possible value of the interpolated filtered field is obtained as a stochastic estimate of a conditional average, which minimizes the mean square error between the interpolated filtered velocity field and the true filtered velocity field. Besides its origins in approximation theory, the optimal interpolation method also has other advantages over more commonly used ad hoc interpolation methods (like the 'adaptive Gaussian window'). The optimal estimate of the filtered velocity field can be guaranteed to preserve the solenoidal nature of the filtered velocity field and also the underlying correlation structure of both the filtered and the unfiltered velocity fields. The a posteriori performance of the optimal interpolation method is evaluated using data obtained from high-resolution direct numerical simulation of isotropic turbulence. Our results show that for a given sample data density, there exists an optimal choice of the characteristic width of cut-off filter that gives the least possible relative mean square error between the true filtered velocity and the interpolated filtered velocity. The width of this 'optimal' filter and the corresponding minimum relative error appear to decrease with increase in sample data density. Errors due to the optimal interpolation method are observed to be quite low for appropriate choices of the data density and the characteristic width of the filter. The optimal interpolation method is also seen to outperform the 'adaptive Gaussian window', in representing the interpolated field given the data at random sample locations. The overall a posteriori performance of the optimal interpolation method was found to be quite good and hence makes a potential candidate for use in interpolation of PTV and super-resolution PIV data.

AB - A new approach for the interpolation of a filtered turbulence velocity field given random point samples of unfiltered turbulence velocity data is described. In this optimal interpolation method, the best possible value of the interpolated filtered field is obtained as a stochastic estimate of a conditional average, which minimizes the mean square error between the interpolated filtered velocity field and the true filtered velocity field. Besides its origins in approximation theory, the optimal interpolation method also has other advantages over more commonly used ad hoc interpolation methods (like the 'adaptive Gaussian window'). The optimal estimate of the filtered velocity field can be guaranteed to preserve the solenoidal nature of the filtered velocity field and also the underlying correlation structure of both the filtered and the unfiltered velocity fields. The a posteriori performance of the optimal interpolation method is evaluated using data obtained from high-resolution direct numerical simulation of isotropic turbulence. Our results show that for a given sample data density, there exists an optimal choice of the characteristic width of cut-off filter that gives the least possible relative mean square error between the true filtered velocity and the interpolated filtered velocity. The width of this 'optimal' filter and the corresponding minimum relative error appear to decrease with increase in sample data density. Errors due to the optimal interpolation method are observed to be quite low for appropriate choices of the data density and the characteristic width of the filter. The optimal interpolation method is also seen to outperform the 'adaptive Gaussian window', in representing the interpolated field given the data at random sample locations. The overall a posteriori performance of the optimal interpolation method was found to be quite good and hence makes a potential candidate for use in interpolation of PTV and super-resolution PIV data.

UR - http://www.scopus.com/inward/record.url?scp=24144498033&partnerID=8YFLogxK

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U2 - 10.1007/s00348-005-1020-6

DO - 10.1007/s00348-005-1020-6

M3 - Article

AN - SCOPUS:24144498033

VL - 39

SP - 213

EP - 221

JO - Experiments in Fluids

JF - Experiments in Fluids

SN - 0723-4864

IS - 2

ER -