# On velocity gradients in PIV interrogation

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## Abstract

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.

## Keywords

Particle Image Velocimetry Displacement Field Simple Shear Tracer Particle Interrogation Window## 1 Introduction

*d*

_{τ}(Adrian 1988; Westerweel 2000b). In a simulation study it was shown that the velocity gradients can be ignored when the variation

*a*of the local particle-image displacement is small with respect to

*d*

_{τ}(Keane and Adrian 1992), i.e.,

*M*is the image magnification, Δ

*t*is the exposure time delay, and Δ

*u*represents the local variation of the velocity field, i.e.,

*L*is a typical dimension of the interrogation volume, e.g., the thickness of the light sheet Δ

*z*

_{0}or the equivalent in-plane dimension of the interrogation region

*D*

_{I}/

*M*. The effect on the appearance of the correlation as a function of an increasing variation

*a*of the displacement within the interrogation volume is shown in Fig. 1. In many practical situations the ratio

*d*

_{τ}/

*D*

_{I}is very small, and typically should not exceed 3–5% in order to preserve a well-defined correlation peak (Keane and Adrian 1992). This means that the local gradients have to be small in order to comply with this requirement. For increasing gradients the correlation peak amplitude decreases, while the width of the correlation peak increases in proportion to the variation of the displacement. Such a broadening of the displacement-correlation peak also occurs in micro-PIV as the result of Brownian motion of the tracer particles (Olsen and Adrian 2000a), which can be used to estimate the local temperature-dependent viscosity (Hohreiter et al. 2002).

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.

*I*

_{1}and

*I*

_{2}, which represent the two image frames recorded with a time delay Δ

*t*=

*t*

_{2}−

*t*

_{1}(Olsen and Adrian 2001):

*W*

_{1}and

*W*

_{2}are the weighting functions that define the interrogation windows,

*I*

_{01}and

*I*

_{02}define the illumination pulses, τ

_{0}is the particle-image intensity per unit illumination,

*M*the image magnification,

^{1}

*and*

**X***are the coordinates in the image domain and flow field domain, respectively, and*

**x**

**x**_{ i }(

*t*) at time

*t*. It is common to split

*g*into mean and fluctuating parts, i.e.,

*g*= 〈

*g*〉 +

*g*′, with 〈

*g*〉 =

*C*and 〈

*g*′〉 = 0, where

*C*is the mean number density of tracer particles (Adrian 1988).

*R*(

*) for*

**s***continuous*image fields is defined by

*R*(

*) with the self-correlation of the spatial pixel sensitivity and sampling the result at discrete separations, as described by Westerweel (1993, 1997). For sufficiently large particle images, i.e.,*

**s***d*

_{τ}/

*d*

_{ r }≪1 (where

*d*

_{ r }is the pixel size),

*R*(

*) closely approximates the spatial correlation for digital PIV images (Westerweel 2000a, b).*

**s***R*(

*) can be written as (Keane and Adrian 1992)*

**s***R*

_{ C }is the correlation of the mean image intensities,

*R*

_{ F }the correlation of the mean image intensity of

*I*

_{1}with the fluctuating part of

*I*

_{2}and vice versa,

^{2}and

*R*

_{ D }the correlation of the fluctuating parts of

*I*

_{1}and

*I*

_{2}. The terms

*R*

_{ C }and

*R*

_{ F }vanish when the mean image intensity is subtracted from

*I*

_{1}and

*I*

_{2}. The remaining term

*R*

_{ D }can be split into mean and fluctuating parts, where the averaging is taken over an ensemble of tracer patterns for a given (fixed) velocity field

*(*

**u***,*

**X***t*) (Adrian 1988; Westerweel 1993):

*R*

_{ D }(

*)|*

**s***〉 is commonly referred to as the*

**u***displacement-correlation peak*and

*R*′

_{D}(

*) as the*

**s***random correlation term*(Westerweel 2000b).

^{3}(2) that the light sheet intensity distribution is only a function of the out-of-plane coordinate

^{4}(here denoted as

*z*); (3) that the optical axis is normal to the light-sheet plane; and (4) that the two exposures of the light sheet occur in the same plane. The ensemble mean of the spatial correlation can then be written as (Adrian 1988; Westerweel 1993):

*g*′

_{1}(

*′)*

**x***g*′

_{2}(

*′′)|*

**x***〉 is the conditional two-point ensemble cross-correlation over all possible tracer pattern fluctuations for a given flow field*

**u***. For a*

**u***uniform*displacement this can be expressed as (Adrian 1988)

*=*

**x***Δ*

**u***t*. Under the condition that the particle-image diameter is small with respect to the typical dimension of the interrogation region (

*d*

_{τ}≪

*D*

_{ I }) and for a uniform particle-image displacement

**s**_{ D }(=

*M*Δ

*),*

**x**^{5}the displacement-correlation peak can be expressed as (Adrian 1988)

_{00}

^{2}= ∫τ

_{0}(

*)*

**X**^{2}d

*and*

**X***I*

_{ zk }= ∫

*I*

_{0k }(

*z*) d

*z*for

*k*= 1,2. The

*image density*

*N*

_{I}represents the mean number of particle images in the interrogation domain, the terms

*F*

_{I}and

*F*

_{O}are denoted as the

*in-plane*and

*out-of-plane loss of correlation*, respectively, and

*F*

_{τ}represents the particle-image self-correlation. Under the assumptions stated above, the displacement-correlation peak is a single sharp peak located at

**s**_{ D }. The exact position of the peak can be determined from either the peak centroid or peak maximum. For symmetric particle images and a uniform displacement the peak centroid and peak maximum are identical in the limit

*d*

_{τ}/

*D*

_{I}→0.

*F*

_{τ}(

*) is symmetric with its centroid located at*

**s**

**s**_{ D }. The in-plane loss-of-correlation

*F*

_{I}(

*)*

**s***skews*

*F*

_{τ}(

*−*

**s**

**s**_{ D }), which leads to a bias error in the measured displacement (Keane and Adrian 1990; Westerweel 1997). The bias error is directed toward smaller displacements, which is related to the fact that particle images with larger displacements are more likely to leave the interrogation domain between the two light pulses.

*F*

_{τ}(

*) is much smaller than the width of*

**s***F*

_{I}(

*), i.e.,*

**s***d*

_{τ}≪

*D*

_{I}. Hence,

*F*

_{I}(

*) can be written as a Taylor series around*

**s**

**s**_{ D }:

*F*

_{τ}(

*). Given that*

**s***F*

_{I}(

*) and the second moment of*

**s***F*

_{τ}(

*) are both positive, and that ∂*

**s***F*

_{I}/∂

*is directed toward the origin, \({\varvec{\mu}}_D\) is usually biased toward*

**s***=*

**s****0**(Keane and Adrian 1990; Westerweel 1997).

*is considered. The spatial derivative of*

**s***F*

_{I}with respect to

*s*is then given by

**s**_{ D }= (

*s*

_{ D },

*t*

_{ D }). (A similar expression can be found for the direction perpendicular to the direction of

*s*). The second moment of

*F*

_{τ}(

*) for identical Gaussian particle images is \(\frac{1}{8} d_{\tau}^2,\) so that the expression for the expected correlation peak centroid defined in Eq. (18) becomes:*

**s***F*

_{I}defined in Eq. (19) in are zero, this expression is exact. Indeed, the centroid is biased toward

*s*= 0. For uniform interrogation regions the displacement bias error is practically constant over a considerable range in

**s**_{ D }(Westerweel 1997).

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}/∂

*= 0). Another approach is to divide the correlation values by*

**s***F*

_{I}(

*) (Westerweel 1997).*

**s**## 3 The distribution function of a displacement field

*g*′

_{1}

*g*′

_{2}|

*〉 is evaluated over a finite measurement volume, i.e., δ*

**u***V*(

*′) = Δ*

**x***z*

_{0}· (

*D*

_{I}/

*M*)

^{2}, which is depicted in Fig. 2. This volume is equivalent to the weight function

*W*(

*) defined by Olsen and Adrian (2001). Due to the spatial variations in the displacement over δ*

**x***V*(

*′), the single displacement value that is represented by the δ-function in Eq. (9) is replaced by a displacement*

**x***distribution*function

*F*

_{Δ}(

*), which eliminates the explicit dependence on*

**s***′ and leads to the following expression for the displacement-correlation peak:*

**x**

**s**_{ D }is no longer uniquely defined: it may now refer to the maximum of

*F*

_{Δ}(

*) (viz., the most probable displacement), or the first moment of the distribution (viz., the local mean displacement), or any other convenient parameter that characterizes*

**s***F*

_{Δ}(

*).*

**s***(*

**x***) over a finite measurement volume δ*

**x***V*is given by

*=*

**k****0**, i.e.: \(\int F_{\Updelta}({\user2{s}}) {\rm d}{\user2{s}} \equiv 1,\) so that

*F*

_{Δ}(

*) is indeed a distribution function for arbitrary Δ*

**s***(*

**x***).*

**x***L*≤

*y*≤

*L*, where

*a*is a constant. Consider only the

*x*-coordinate, which implies a reduction to a one-dimensional problem. Substitution of Eq. (26) in Eq. (24) yields

*reduce*the correlation peak amplitude and

*increase*the width of the correlation peak, in correspondence to what occurs in Fig. 1 for increasing

*a*. Consequently, the local variations of the displacement field also

*reduce*the peak detectability, and

*enhance*the displacement bias error.

*L*≤

*x*≤

*L*. The corresponding Fourier transform of the displacement distribution is equal to a zeroth order Bessel function of the first kind, i.e.,

*F*

_{Δ}(

*s*) for the sinusoidal field with that of the shear field (see Fig. 3, middle), the corresponding displacement distributions are quite different: the distribution for the sinusoidal field has

*two*peaks. This means that the displacement-correlation peak will have two maxima (provided that the local variations of the displacement field are larger than the equivalent width of the particle images; see discussion below). This is an example of the so-called

*peak splitting*, which occurs often in regions with very strong local fluctuations of the displacement. This means that for a finite number of particle images the interrogation analysis will only detect one of the two peaks; in this case the measured displacement can

*not*be considered as a local mean value of the displacement field.

## 4 The displacement-correlation peak for non-uniform displacements

*non-uniform*displacement field is given by, cf. Eq. (9)

*g*′

_{1}(

*′)*

**x***g*′

_{2}(

*′′)|*

**x***〉 is a function of*

**u***=*

**s***′′ −*

**x***′ only, which leads to a rather simple expression that was derived in Sect. 2. However, for a non-unform displacement field Eq. (33) is a function of*

**x***′, which makes it difficult to carry out a straightforward evaluation of Eq. (8). The expression in Eq. (8) can be solved numerically, but this is a rather cumbersome procedure for a generalized analysis of spatial gradients. Instead, it is possible to reduce Eq. (8) to that of a much simpler approximate equation, by making some general assumptions. This procedure is described in this section.*

**x**### 4.1 Approximate expression

*x*-coordinate, i.e.,

*x*and

*y*only, so that the double integral over

*z*′ and

*z*′′ can be replaced by

*F*

_{ I }(

*) is split from the integrand:*

**s***F*

_{τ}(

*)*

**s***=*

**X***M*Δ

*. This expression is then further reduced by the integration of the two δ-functions that replace the τ*

**x**_{0}-functions, i.e.

*a*| =

*M*|Δ

*u*|Δ

*t*≪

*D*

_{I},

*F*

_{I}(

*′) is replaced by*

**s***F*

_{I}(

**s**_{ D }), so that

*F*

_{I}(

**s**_{ D })

*D*

_{ I }

^{2}represents the normalization constant for the displacement distribution function. Hence, the approximation for Eq. (8) reads

*F*

_{τ}(

*) and*

**s***F*

_{Δ}(

*); this justifies the expression given in Eq. (22).*

**s**Evidently, the result in Eqs. (41–42) needs to be validated against the numerical solutions of the exact expression in Eq. (8). This is done in the next section.

### 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

*X*

_{0}plus a simple shear motion

*Y*only, so that

*F*

_{Δ}(

*), defined in Eq. (42), is given by*

**s***= (*

**s***s*,

*t*)

^{ T }. This integral was solved in Sect. 3 for a uniform simple shear and a uniform interrogation window. Hence, the corresponding displacement distribution

*F*

_{Δ}(

*s*,

*t*) is uniform in

*s*:

*a*. In the Gaussian approximation for

*F*

_{τ}(

*) and*

**s***F*

_{Δ}(

*), the width*

**s***d*

_{ D }of

*F*

_{τ}*

*F*

_{Δ}(

*) is given by*

**s***increase*of the bias magnitude due to the shear is equal for all values of

*d*

_{τ}. This is reflected in Fig. 6, in which the width of the displacement-correlation peak relative to its width for uniform displacements (i.e., \(\sqrt{2}d_{\tau}\)) is plotted as a function of the gradient relative to the particle-image diameter. The solid line represents Eq. (46), whereas the symbols represent numerical solutions of Eq. (8). This graph indicates that PIV images with large

*d*

_{τ}are less sensitive to variations in the displacement than those images with small

*d*

_{τ}. So, according to Eq. (18) the displacement bias error is determined by the width

*d*

_{ D }of the displacement-correlation peak, given by Eq. (47). This is shown in Fig. 7, in which the displacement bias error is plotted as a function of

*d*

_{ D }.

#### 4.2.2 Uniaxial strain

*X*≥ 0 for all positions inside the interrogation window. Substitution of Eq. (48) in Eq. (42) yields

*W*

_{1}and

*W*

_{2}now represent one-dimensional functions.) The integrand only makes a contribution to the total integral when

*s*= Δ

*X*(

*X*) on the interval where

*W*

_{1}(

*X*)

*W*

_{2}(

*X*+

*s*) ≠ 0. This implies that the finite dimensions of the interrogation windows limit the range of displacements that can be measured.

*W*

_{1}is uniform with a mean displacement Δ

*X*

_{0}and a width

*M*|Δ

*u*|Δ

*t*. However, for Δ

*x*> 0 the integral in Eq. (49) is non-zero only for

*F*

_{Δ}(

*s*) ≠ 0 for \(|s-s_{\Updelta}| < \frac{1}{2} d_{\Updelta},\) with (see also Fig. 8)

*positive*uniaxial strain (

*a*=

*M*Δ

*u*Δ

*t*> 0), i.e., a spatially accelerating fluid, Eq. (49) implies that the local displacement-distribution is truncated at the side of the largest displacements. Consequently, the mean of the

*observed*local displacement-distribution is then biased toward a smaller displacement in comparison with the actual mean of the local displacement distribution. On the other hand, for a

*negative*uniaxial strain (

*a*=

*M*Δ

*u*Δ

*t*< 0), i.e., a spatially decelerating fluid, the distribution is truncated at the side of

*smallest*displacements, so that the mean measured displacement is

*larger*than the local mean displacement. This bias adds to the displacement bias error that is the result of the skew of the displacement-correlation peak due to

*F*

_{I}(Fig. 9).

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*.

*F*

_{Δ}(

*s*) in Eq. (22). In Fig. 10 are shown the displacement-correlation peaks for uniaxial strain with different values of

*a*. Note that the displacement-correlation peak for uniaxial strain has a larger bias than for simple shear (for the same value of

*a*), which is the result of the truncation of the displacement distribution for large displacements.

*d*

_{ D }of the displacement-correlation peak for a uniaxial normal stress is given by (Gaussian approximation):

*plus*the bias error due to the increase of the width of the displacement-correlation peak, i.e.:

*X*

_{0}= 0, only compensates for part of the bias and increase in peak width (Fig. 12).

## 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).

- 1.
determine the local distribution function

*F*_{Δ}() of the displacement field;**s** - 2.
convolve

*F*_{Δ}() with the particle-image self-correlation**s***F*_{τ}();**s** - 3.
multiply the result with the in-plane loss-of-pairs

*F*_{I}() and out-of-plane loss-of-pairs**s***F*_{O}(Δ*z*).

*a*the local displacement field is well approximated by a uniform displacement

*plus*a shear and/or uniaxial strain. Specific results for these fluid motions are obtained in Sects. 4.2.1 and 4.2.2.

*d*

_{ D }of the correlation peak is given by Eq. (46). The conservation of total volume then implies that the correlation peak amplitude is given by

*a*/

*d*

_{τ}= 0.0, 0.5, 1.0 and 1.5, respectively; these values correspond well with the (instantaneous) peak amplitudes shown in Fig. 1 (see also Hain and Kähler 2007). The

*one-quarter rules*for the in-plane and out-of-plane displacement imply a maximum loss-of-correlation that reduces the correlation peak height to 75% of the maximum amplitude. A similar drop in amplitude corresponds to |

*a*|/

*d*

_{τ}< 0.66, i.e., a

*two-third rule*for the displacement variation.

_{ΔX }is proportional to the width

*d*

_{ D }of the displacement-correlation peak, and for a simple shear it is approximately given by

^{6}

*c*= 0.05-7 (Westerweel 2000b).

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.

## Footnotes

- 1.
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 mapsonto**x**; see, e.g., Prasad (2000).**X** - 2.
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). - 3.
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.

- 4.
This condition is generally satisfied at the local scale of the equivalent interrogation volume in the flow.

- 5.
Note that

*M*is a projection of the volumetric flow domain onto the planar image domain. - 6.
Provided that the particle image diameter is at least two pixel units in a digital PIV image.

- 7.
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.

## Notes

### Open Access

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