On velocity gradients in PIV interrogation
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This paper presents a generalization of the description of the displacement-correlation peak in particle image velocimetry (PIV) to include the effects due to local velocity gradients at the scale of the interrogation domain. A general expression is derived that describes the amplitude, location and width of the displacement-correlation peak in the presence of local velocity gradients. Simplified expressions are obtained for the peak centroid and peak width for simple non-uniform motions. The results confirm that local gradients can be ignored provided that the variation of the displacement within the interrogation domain does not exceed the (mean) particle-image diameter. An additional bias occurs for a spatially accelerating or decelerating fluid, which implies an artificial "particle inertia" even when the particles can be considered as ideal tracers.
KeywordsParticle Image Velocimetry Displacement Field Simple Shear Tracer Particle Interrogation Window
For uniform displacements the correlation peak detectability is proportional to N I F I F O, where N I is the image density and F I and F O are the in-plane and out-of-plane loss-of-correlation due to in-plane and out-of-plane motion of the tracer particles (Keane and Adrian 1990). This can be generalized to: N I F I F O F Δ, where the term F Δ accounts for the loss-of-correlation due to the local variation of the displacement field (Westerweel 2004; Hain and Kähler 2007).
The reduction of the displacement-correlation peak means that the peak detectability is reduced. This implies a higher occurrence of spurious vectors in regions with strong local gradients (Keane and Adrian 1992). If the gradient only occurs in the in-plane components of the displacement, then it is possible to (partially) compensate for the effects of the gradients in the displacement by means of local image deformation (Huang et al. 1993; Tokumaru and Dimotakis 1995; Fincham and Delerce 2000; Scarano 2002). However, in most turbulent flows the small-scale turbulence is (nearly) isotropic, which means that the in-plane variation of the displacements are of the same magnitude as the out-of-plane variation of the displacements when the light-sheet thickness is of the order of the in-plane dimension of the interrogation domain; then the out-of-plane gradients cannot be compensated by planar deformation methods, which means that the local velocity gradients irreversibly deteriorate the interrogation performance. In multi-frame PIV local velocity gradients can easily dominate the peak detectability when the temporal separation of the interrogation images is increased (Hain and Kähler 2007). For micro-PIV the depth-of-correlation (Olsen and Adrian 2000b) can be of the order of the flow domain (i.e., channel depth), which means that a large range of displacements can occur in a single interrogation domain. As mentioned before, Brownian motion of the tracer particles adds to the broadening of the correlation peak.
In this paper, the mathematical support for the empirical relation in Eq. (1) is derived. First the theoretical analysis for PIV interrogation in case of a uniform displacement field is summarized (Sect. 2). Then an expression for the local displacement distribution is derived (Sect. 3). This is used to generalize the existing theory for uniform displacements to include non-uniform displacement fields (Sect. 4). The analysis follows a more rigorous approach than used by Olsen and Adrian (2001). Based on the extended description several effects are described where local gradients affect the interrogation analysis. The last section (Sect. 5) summarizes the main results.
2 Interrogation by spatial cross-correlation
The point of departure for the analysis is the original theoretical description of the interrogation analysis given by Adrian (1988), Keane and Adrian (1990), Westerweel (1993), and Olsen and Adrian (2001), which is summarized here.
In general the bias error will be small, i.e., typically about 0.06 px for particle images with a diameter of 2 px in a 32 × 32-pixel interrogation domain (Westerweel 1997). This is small with respect to the random error (typically 0.1 pixel units) in instantaneous data, but can be significant when evaluating flow statistics, such as the mean flow velocity. The bias error can be eliminated in several ways. It is evident that the bias error vanishes when the gradient of F I is zero. For uniform interrogation windows this can be accomplished either by using two interrogation windows of different size (Keane and Adrian 1992), or by using offset interrogation regions (Westerweel et al. 1997), so that the displacement-correlation peak is located at the maximum of F I(s) (i.e., ∂F I/∂s = 0). Another approach is to divide the correlation values by F I(s) (Westerweel 1997).
3 The distribution function of a displacement field
4 The displacement-correlation peak for non-uniform displacements
4.1 Approximate expression
4.2 Simple flows
In this section the effect of spatial gradients in the displacement field on the location, height and shape of the displacement-correlation peak is investigated for the case of simple, one-dimensional displacement fields, such as simple shear and uniaxial strain. To reduce the complexity of the analysis, only uniform interrogation windows are considered. Analytical expressions for the peak centroid and the peak width are found by means of the approximate expression Eq. (41) that was derived in the preceding section. These are compared against numerical solutions of the exact expression Eq. (8).
4.2.1 Simple shear
4.2.2 Uniaxial strain
So, for an accelerating fluid (a > 0) the measured displacement is smaller than the true local mean displacement, whereas for a decelerating fluid (a < 0) the measured displacement is larger. It is as if the tracer particles have some inertia, even if the tracer particles themselves are ideal.
5 Discussion and conclusion
The previous sections describe the effect of local gradients at the scale of the interrogation domain on the shape of the displacement-correlation peak. This is a generalization of the existing theoretical expression for the displacement-correlation peak. An approximate expression is derived, which accurately predicts the bias error and peak width for simple flows. It is confirmed that the broadening and splitting of the displacement-correlation peak due to local variations of the displacement does not occur as long as the variation a of the displacements over the interrogation volume does not exceed the particle image diameter d τ, as stated in Eq. (1).
determine the local distribution function F Δ(s) of the displacement field;
convolve F Δ(s) with the particle-image self-correlation F τ(s);
multiply the result with the in-plane loss-of-pairs F I(s) and out-of-plane loss-of-pairs F O(Δz).
In the case of a uniaxial strain, i.e., a spatially accelerating or decelerating fluid, the centroid of the displacement-correlation peak yields an additional bias that can be interpreted as an artificial ‘inertia’ of the particles. This even occurs when the particles can be considered as ideal tracers, and is the result of the finite dimensions of the interrogation domain.
Also, the analysis shows that a simple sinusoidal fluid motion with a wavelength equal or less than the dimension of the interrogation domain leads to the appearance of two correlation peaks (see Fig. 3) when the displacement amplitude becomes larger than d τ. This is an example of the so-called peak splitting. In a practical situation, i.e., with a finite number of particle images, it is likely that the interrogation analysis just finds only one of the two peaks (ignoring the other peak as a possible random-correlation peak). Then the measured displacement would correspond to the local minimum or maximum displacement. 7 This means that the measured displacement for the case of a sinusoidal displacement field is not proportional to the locally averaged displacement; this invalidates the commonly accepted assumption that the measured displacement is equal to the locally averaged displacement (Willert and Gharib 1991; Olsen and Adrian 2001; Hart 2000; Nogueira et al. 1999).
An application where the local gradients can become dominant is micro-PIV, in particular for measurements where the interrogation domain extends over a substantial part along the observation direction. For pressure-driven Stokes flow in a channel geometry the velocity profile along the optical axis has a parabolic shape. This shape is approximated by the sinusoidal distribution in Fig. 3 for: 0 ≤ x ≤ 1; hence, F Δ(s) is approximately given by Eq. (32) for s ≥ 0. This particular shape of the displacement-correlation peak is reported by Wereley and Whitacre (2006). In this particular situation the PIV measurement yields the maximum velocity in the measurement domain, rather than the mean displacement.
In order to absorb larger local variations of the displacement, one could increase the particle-image diameter d τ. In the case of diffraction-limited particle images, d τ ≈d s , with d s = 2.44(M + 1)f #λ, where f # is the aperture number of the lens and λ the light wavelength. So, d τ can be increased by increasing f #, i.e., by reducing the lens aperture. Unfortunately, this also reduces the collected amount of light scattered by the tracer particles, and—in the case of micro-PIV—implies an unfavorable increase of the correlation depth (Olsen and Adrian 2000b). An increase of d τ also implies a proportional increase of the random error, given in Eq. (56), and a general deterioration of the overall performance of the PIV system (Adrian 1997). One could possibly determine an optimum between increasing d τ to improve peak detectability, while accepting a (small) increase in the random error. The expressions given in this paper can be used as a guideline.
Note that M is only a scalar in the case of paraxial imaging and a thin light sheet; in general M will be a projection matrix that maps x onto X; see, e.g., Prasad (2000).
In older texts the term R F is mistakenly referred to as the random correlation term, but this part of the signal is actually included in R D ; (see Westerweel 2000b).
This assumption is generally satisfied for diffraction-limited imaging of small tracer particles; see Adrian (1984). These particle images can have different intensities based on their position within the light sheet.
This condition is generally satisfied at the local scale of the equivalent interrogation volume in the flow.
Note that M is a projection of the volumetric flow domain onto the planar image domain.
Provided that the particle image diameter is at least two pixel units in a digital PIV image.
As the result of the bias toward s = 0, it is more likely that the peak closest to the origin of the correlation domain is detected.
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