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Fluid trajectory evaluation based on an ensemble-averaged cross-correlation in time-resolved PIV

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Abstract

A novel multi-frame particle image velocimetry (PIV) method, able to evaluate a fluid trajectory by means of an ensemble-averaged cross-correlation, is introduced. The method integrates the advantages of the state-of-art time-resolved PIV (TR-PIV) methods to further enhance both robustness and dynamic range. The fluid trajectory follows a polynomial model with a prescribed order. A set of polynomial coefficients, which maximizes the ensemble-averaged cross-correlation value across the frames, is regarded as the most appropriate solution. To achieve a convergence of the trajectory in terms of polynomial coefficients, an ensemble-averaged cross-correlation map is constructed by sampling cross-correlation values near the predictor trajectory with respect to an imposed change of each polynomial coefficient. A relation between the given change and corresponding cross-correlation maps, which could be calculated from the ordinary cross-correlation, is derived. A disagreement between computational domain and corresponding physical domain is compensated by introducing the Jacobian matrix based on the image deformation scheme in accordance with the trajectory. An increased cost of the convergence calculation, associated with the nonlinearity of the fluid trajectory, is moderated by means of a V-cycle iteration. To validate enhancements of the present method, quantitative comparisons with the state-of-arts TR-PIV methods, e.g., the adaptive temporal interval, the multi-frame pyramid correlation and the fluid trajectory correlation, were carried out by using synthetically generated particle image sequences. The performances of the tested methods are discussed in algorithmic terms. A high-rate TR-PIV experiment of a flow over an airfoil demonstrates the effectiveness of the present method. It is shown that the present method is capable of reducing random errors in both velocity and material acceleration while suppressing spurious temporal fluctuations due to measurement noise.

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Abbreviations

a :

Set of polynomial coefficients, a = {a 1, …, a P }

a k :

kth-order polynomial coefficient

Δa :

Set of corrections vectors for a, Δa = {Δa 1, …, Δa P }

Δa k :

Corrections vectors for a k

C n (Г(x, n)):

Cross-correlation value between two time steps 0 and n along a trajectory

C ens :

Ensemble-averaged cross-correlation value

c n :

Contribution of a temporal interval n to bias error

Du/Dt :

Material acceleration

I n (x):

Particle image plane at a time step n

I Г(x,n) n :

Deformed particle image by Г(x, n)

J(Г(x, n)):

Jacobian matrix of a deformed image scheme by Г(x, n)

M :

Number of images

N c :

Number of averaged cross-correlation maps

n :

Normalized times step, n = t/Δt

n opt :

Optimal separation

P :

Polynomial order of a modeled trajectory

R(Δa):

Cross-correlation map for an entire correction of a

R k (Δa k ):

Cross-correlation map for an individual correction of a k

R Г(x,n) n (Δx):

Cross-correlation map between I 0 and I Г(x,n) n

Δs :

Scale of sampling step for a discrete construction of R k (Δa k )

Δs cri :

Criterion for a scale of sampling step

T :

Normalized maximum temporal interval

Δt :

Temporal separation between two subsequent recordings

u :

Velocity

u :

Exact horizontal displacement

W :

Interrogation window

w :

Window size

x :

Position in a particle image plane

x p :

Position of fluid parcel

Δx :

Coordinate of a computed cross-correlation map

Δx phy :

Physical vector which corresponds to Δx

β, σ :

Bias and random errors

β 1, σ 1 :

Basis error profiles from numerical assessment

β n , σ n :

Analytic errors from two-frame cross-correlation with an imposed interval n, n > 1

β FTC, σ FTC :

Analytic errors of the FTC method

β +, σ + :

Error profiles based on forward deformation scheme

δx, δy :

Grid spacings in horizontal and vertical directions

γ :

Scale factor of V-cycle iteration

Г(x, n):

Relative fluid trajectory to x

Γ′:

Regulated trajectory field

\({\varvec{\Gamma}}_{ \pm \delta x}^{{\prime }} ,{\varvec{\Gamma}}_{ \pm \delta y}^{{\prime }}\) :

Regulated trajectories at neighboring grid points

ε :

Signed random error

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Acknowledgments

The current work has been conducted as part of the AFDAR project, Advanced Flow Diagnostics for Aeronautical research, funded by the European Commission program FP7, Grant No. 265695.

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  1. Institut P PRIME, UPR3346, CNRS – Université de Poitiers – ISAE-ENSMA, 11 Boulevard Marie et Pierre Curie, 86962, Futuroscope Cedex, France

    Young Jin Jeon, Ludovic Chatellier & Laurent David

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  1. Young Jin Jeon
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  2. Ludovic Chatellier
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  3. Laurent David
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Correspondence to Young Jin Jeon.

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Jeon, Y.J., Chatellier, L. & David, L. Fluid trajectory evaluation based on an ensemble-averaged cross-correlation in time-resolved PIV. Exp Fluids 55, 1766 (2014). https://doi.org/10.1007/s00348-014-1766-9

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  • DOI: https://doi.org/10.1007/s00348-014-1766-9

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