Topological considerations in support of PIV vector field analyses

10TH INTERNATIONAL SYMPOSIUM ON PARTICLE IMAGE VELOCIMETRY - PIV13 Delft, The Netherlands, July 1-3, 2013

Topological considerations in support of PIV vector field analyses

1_{Department of Mechanical Engineering, Michigan State University, East Lansing, Michigan, USA}

foss@egr.msu.edu

2_{LaVision Inc., Ypsilanti, Michigan, USA}

ABSTRACT

Stereo and planar PIV methods provide a continuous vector field overlaid on a planar surface. A subset of the more general topological considerations are ideally suited for the evaluation of such vector fields. Namely, these methods can: i) identify the singular points and assign their correct index values, and ii) determine if the identified singular points satisfy the Poincare-Hopf theorem. Hence, once a specific vector field is established, the methods presented in this paper can be used to evaluate the kinematic attributes of the vector field and to ensure that the identified singular points represent a topologically valid vector field.

1. Introduction

1.1 Purpose and Objectives

Topological considerations can provide substantive contributions to the evaluations of the inferred singular points in PIV images. In addition to local uncertainty evaluations, these topological considerations provide a whole-field evaluation of the image. Paraphrasing A.E. Perry, the flows singular points serve as the skeleton upon which the kinematics of the flow are supported. The purpose of this paper is to introduce or reinforce the readers understanding of the appropriate topological considerations and to offer a systematic analysis structure by which these considerations can be applied. The objectives are to enable the maximum extraction of information from PIV images and to add these considerations to the developing methods of uncertainty analyses for PIV images.

1.2 Vector Field

The topological considerations refer to the isolated1singular points of a continuous vector field that is overlaid on a surface. (The vectors are in the plane of the surface). The desired vector field is most reliably established experimentally using Stereo PIV since it most accurately represents the projection of a three-component vector onto the surface of the planar image (with its two components of the 3-component vector).

Tracing a streamline, which is understood to be tangent to the instantaneous vector field at all locations, to an isolated singular (or critical) point leads to a condition of ~Vsur f ace= 0. The issue of how the vectors at a physical surface are used in these analyses in spite of the no-slip condition is dealt with in Foss [2]. That point will, however, have an index. That value: ±1, is defined by using the

operation shown in §2.0 with the result that a node2is designated as+1 and a saddle is designated as −1. (The operation to identify

the index is also quite often useful in identifying the presence of an isolated singular point in an otherwise uncertain vector field).

1.3 Poincare-Hopf Theorem3

The Euler characteristic:χ, for a surface is equal to the sum of the indices of the isolated singular points on the surface. The following specific example of this general rule is referred to as the hairy sphere theorem.

Combing the hairs on a sphere into the simplest pattern results in two nodes. The hairy sphere theorem follows:χsphere= 2. If a hole penetrates the surface of the sphere and/or if a handle is added to the sphere, the Euler characteristic is changed as

χsur f ace= 2 −Σholes− 2Σhandles. (1)

1_{To clarify the term isolated, consider an annular jet that is directed onto a surface with a circular hole in the impact plate at the centerline of the jet.} The resulting flow field would have a continuous ring of stagnation points - but none that are isolated.

2_{There are numerous styles of nodes; see Perry and Chong [10] albeit each has an index equal to+1.}

3_{The background information presented in this paper is abbreviated from that of Foss [2]. The intent of this foreshortened description is to make} plausible the basic considerations for the topological considerations and to reserve space for the applications.

It is important to recognize that the vector field must be uniformly directed onto or away-from the surface at a hole. It is useful to consider (1) to represent the apriori Euler characteristic. That is, theχvalue of (1) is known once the analyst has selected the domain to be interrogated.

1.4 Planar Surfaces

Given that a PIV image is represented by a planar surface, it is appropriate to identify two techniques of obtaining such a surface having started with a sphere.

1.4.1 A Sphere Plus One Hole

Penetrate a spheres surface with a hole (say at the South Pole) and let the remaining surface shrink nearly to the north pole. The resulting surface can then be stretched to cover the PIV image as a disk. Realizing that the perimeter of the disk constitutes a hole, the surface vectors must be oriented either uniformly onto or away from the surface perimeter. If the disk is placed perpendicular to an oncoming flow, the resulting flow field is easily visualized: a centered stagnation point (a node) and a radially outward flow at the perimeter. Sinceχ= 2 − 1 = +1, the centered stagnation point satisfies the topological constraint and the outward flow at the perimeter

satisfies the constraint at a hole.

1.4.2 A Collapsed Sphere4

A planar surface can also (and more generally) be established by collapsing a sphere such that it forms a two-sided planar surface. The constraint in this case is that the vector field at the perimeter of the collapsed sphere must be tangent to the perimeter.

The Rule is unchanged; however, because of the two sides, it is stated as

χsur f ace= 2ΣN+ΣN′− 2ΣS−ΣS′ (2)

for the planar surface. N and S represent full nodes and saddles; they appear on the image inside the perimeter. N′and S′represent half-nodes and half-saddles. They appear at the perimeter. The designations: full and one-half were introduced by Hunt and Graham [4]. Equation 2 is the form that is most useful for PIV applications, which provide a purely planar vector field.

2. Evaluation of the Singular Points’ Index

Consider a streamline pattern that suggests, but does not clearly show, a singular point. It will be characteristically true that the surrounding streamlines will be adequate for the following evaluation.

Prepare a circle of diameter D (where D is small with respect to the length scales of the surrounding flow field) around the candidate location and, using a pointed marker, translate the markers base clockwise around the circle with the base on the circle and the marker pointed in the velocity direction at each azimuthal location of the circle. A node has been encircled if the pointer rotates clockwise, a saddle has been encircled if the pointer rotates counterclockwise and no net singular point is encircled if the marker does not complete a full rotation.

Figure 1 shows these operations for a node and a saddle. If a combination of both singular points were inside the circle the marker would not make a complete rotation; the reader is invited to confirm this.

3. Exemplar Flow Fields

Four selected flow fields are presented in this section to demonstrate the application of Equation 2. An investigation of the use of the Topological Rule in the analysis of a PIV images is conducted and described in detail.

It is useful to identify two statements of the Euler characteristic for a given flow field. The first is theχvalue that follows from the a’priori representation of the surface as designated by (1), and is noted asχA. The second,χE, is defined by the experimentally observed nodes and saddles; that is, Equation 2. The Topological Rule is satisfied ifχA=χE.

3.1 A Low Reynolds Number Single Stream Shear Layer

Bade and Foss [1] obtained and interpreted PIV images of a relatively low Reynolds number single stream shear layer, Re_θ= 6118

whereθ(x) = 20.2 mm at the center of the image plane. Two images, similar to those of the publication, are shown in Figures 2 and 3.

The streamlines of the flow are shown in Figure 2a. The same data are used to show the streamlines in a convected frame: (U0/2), in

Figure 3a.

4_{Alternatively, represent the vector field on a planar surface and stitch an identical image and surface such that their back sides are aligned. The} vector fields must be tangent at the perimeter except for the hole locations.

Evaluation Circle Node Saddle 2π counter-clockwise rotation of marker 2π clockwise rotation of marker marker, first position at 3:00 marker, first position at 3:00 3:00 3:00 6:00 6:00 9:00 9:00 12:00 12:00 3:00 6:00 9:00 12:00 3:00 6:00 9:00 12:00

Figure 1: An operational definition to specify the value of the index (±1) for an isolated singular point.

A collapsed sphere, denoted by (1,2,3,4), is shown in Figure 2b. The perimeter is marked by 1-2 and 3-4; the outflow hole: (2-3) and the inflow hole: (4-1) clearly satisfy the uniformly directed vector field direction at a hole constraint. The Euler characteristic (χA) is zero (two holes) and the evident node and saddle satisfy the Rule:χA=χE.

Figure 3b identifies the topological features of the flow in the convected reference frame. The rhs perimeter (1-2) is unchanged as is the lhs perimeter (3-4). However the full span holes: (2-3) and (4-1) are now replaced by holes (2-a) and (b-3), and similarly, (4-c) and (d-1). The perimeter is expanded by (a-b) and (c-d) resulting in two additional holes; hence,χA= 2 − 4 = −2. There is one evident node in Figure 3b; with this one singular point,χ6=χE.

In order to rectify the apparent imbalance of theχvalues, it is necessary to visualize the nature of the three-dimensional sphere that has been collapsed to form the surface of Figure 3b with the overlaid vectors on its surface. As an aid in that visualization process, begin with a ”pillowcase” (where the opening has been sewn shut) and the material of the four corners has been cut off. Overlay the pillowcase on the flow field of Figure 3b. The vectors, obtained from the PIV image are understood to be overlaid on both sides of the pillowcase and if the pillowcase is the made flat along the line 2-b, Figure 4 would be resolved. With this visualization, it is apparent that the vector field at a is different from that at 2. Specifically, if the pillowcase is opened up and flattened, there would be a half-saddle at a, but not at ”2”. Similarly, there are half saddles at b, c and d. Hence, Equation 2 for Figure 3b becomes:

χE= 2ΣN+ΣN′− 2ΣS−ΣS′= 2(1) + 0 − 2(0) − 4 = −2, (3)

and the Rule is satisfied.

3.2 A Simulated Urban Environment

Monnier and Wark [5] have investigated the flow fields of a simulated urban environment with PIV techniques. Their recent PIV results [6] have been shared with the present authors as instructive images that demonstrate the principles under consideration.

A schematic representation of their wind tunnel configuration is shown in Figure 5. The upwind blocks are to thicken the boundary layer. The buildings are modeled with rectangular parallelepipeds of base 25x25 mm2_{and height 49 mm. Using the base length, b, as}

the experiments length scale, the buildings height is 1.96 b. The buildings are separated by 1.5 b in the streamwise direction.

Fig. 6 identifies the location of the light sheet. The camera image extends from nominally 1mm above the floor to 66 mm above the floor (or 0.04 b to 2.64 b). The light sheet was positioned 0.14 b from the buildings centerline in order to capture the highly three-dimensional flow field in this region. The stereo cameras view the light sheet from below the glass floor of the tunnel.

Fig. 7a presents the streamlines that were added to the discrete vector field from Monnier and Ward [6] by DRN. This figure provides a representative example of the 10 images that were made available to the present authors. As shown, these data offer an inviting set of flow field features for topological analysis. The indicated streamline pattern of the complete image in this figure can readily be segregated into two regions: I and II; as noted in Figures 7b and 7c.

y (mm) x (m m ) -150 -100 -50 0 50 100 450 500 550 600 650 700

(a) Streamlines in a low Reθ single stream shear layer; laboratory reference frame.

y (mm) x (m m ) -150 -100 -50 0 50 100 450 500 550 600 650 700 2 4 3 1 S N

(b) Streamlines of Figure 2a with the added topological features: i) seams (1-2) and (3-4), plus ii) singular points (N and S)

Figure 2

3.2.1 Region I

In Region I of Figures 7b and 7c, the large node at x/b≈-0.47 and y/b≈1.72 is balanced by the saddle at x/b≈-1.44 and y/b≈1.9. There

are a total of six holes in Region I including two on the lhs, one long hole and three additional ones on the rhs. These six holes establish χA= 2 − 6 = −4. TheχE values follows from the notations on Figure ; viz,

χE= 2ΣN+ΣN′− 2ΣS−ΣS′= 2(1) + 0 − 2(2) − 2 = −4. (4)

The Topological Rule is, therefore, satisfied for Region I.

It is noted that the two S′designations near the rhs border follow from the pillowcase argument in the discussion of the Bade/Foss flow field in §3.1.

3.2.2 Region II

Figure 7c provides a focus on Region II with the understanding that a portion of Figure 7b will be required to complete the interpretation of Region II. The upper left portion provides the inflow to Region II with two prominent seams forming the boundaries. The terminating line, at the bottom of Region II, is broken at four locations to allow (reading from left to right) the outflow: a-b, the inflow: c-d, and the outflow: e-f. The seams connecting b-c and d-e can either be extended to enclose or to exclude the nodes. As shown in Figure 7c, the lower two seams exclude the nodes from being a part of Region II and the four holes lead to aχAvalue of -2. Region II, in Figure 7c, has one saddle such thatχEis also -2 and the Rule is satisfied.

Continuing the evaluation Region II, if the lower seams are drawn as shown in 7b, then the two nodes are included in Region II. As discussed for the convected reference frame results of §3.1, the connecting seams below the nodes must be terminated with half-saddles at points b,c,d, and e; as shown in Figure 7b. TheχEvalue is represented asχE= 2(2) + 0 − 2(1) − 4 = −2 which (again) agrees with χAand the Rule is satisfied.

3.3 The Flow Past a Circular Cylinder

The flow past a circular cylinder is one of the canonical flows in fluid mechanics. PIV images of the wake region for an investigation at the DLR Gottingen are included here for their instructive value; see Figure 8. The website5provides the details of the facility that produced the images of Figure 8. The ReDfor this image was approximately 12,000. The flow field of Figure 8a is undistinguished; the streamlines are essentially parallel and velocity magnitudes (not shown) indicate the classic wake profile.

In contrast, Figure 8b are the same point-wise data with thehUi and hV i subtracted from each spatial location. hUi and hV i represent

the magnitudes

y (mm) x (m m ) -150 -100 -50 0 50 100 450 500 550 600 650 700

(a) Streamlines in a low Reθ single stream shear layer; convective reference frame with Uconvection= U0/2.

y (mm) x (m m ) -150 -100 -50 0 50 100 450 500 550 600 650 700 2 4 3 1 c a b d N S’ S’ S’ S’

(b) Streamlines of Figure 3a with the added topological features: i) seams (a-b) and (c-d), plus ii) the required S′ points

Figure 3

2

a

(a) Qualitative streamline pattern in the x-y plane

a

2

(b) Pattern of the x-y streamlines as viewed as 2 sides of a pillowcase, opened up, and looked at from above.

Figure 4: Topological justification for the half-saddle located at location ”a” but not at location ”2” as shown in Figure 3b.

hUi and hV i = 1 IJ I,J

∑

∑

where I is the number of columns and J is the number of rows of the measurement locations. That is, the global averages of (u,v) are subtracted from each measurement location to form [ˆiu+ ˆjv] as the velocity to use for the streamlines of Figure 8b.

One of us (JFF) carried out a hand count of the singular points in Figure 8b and used the schemes of Figure 1 to determine the index of each point; the final count was 15 nodes and 17 saddles. An examination of the end planes reveals that there are three holes on each end of the image. For the lhs, two holes allowed flow to pass onto the subject surface and one delivers flow to the upstream region. This pattern is inverted on the rhs. The correspondingχAvalue is

χ= 2 − 6 = −4. (6)

From Equation 2,

χE= 2ΣN+ΣN′− 2ΣS−ΣS′= 2(15) + 0 − 2(17) − 0 = −4. (7)

Hence,χA=χEand the Rule is satisfied.

It is pertinent to note that the holes in Figure 8 at the left and right ends of the image are bounded by edges for which there are no S′ contributions. The explanation for the absence of the S′designations was established in the prior two exemplar flows.

Figure 5: The experimental configuration of Monnier and Wark [6]. Note, the stereo PIV system is location beneath the glass floor of the wind tunnel.

z

x

y

Figure 6: An overview of the simulated urban environment for [6] including the laser light sheet that is viewed from below.

The absence of singular points in Figure 8a is worthy of note. A collapsed sphere, in the plane of Figure 8a extending left to contact the aft one-half of the cylinder and extend beyond the wake in the lateral direction, terminating 10 diameters downstream, as drawn schematically in Figure 8c, would be characterized by three holes. The Rule would be satisfied by one S′ on the cylinder’s aft surface where the boundary separates to form the wake. Any singular points on the interior of the collapsed sphere would have to be self-canceling with those on the cylinder surface (not including the half-saddle that was just identified). In that sense, it is not surprising that there are no singular points in Figure 8a.

3.4 Topological Considerations in a High Re SSSL

Figure 10a presents a PIV image from the intermittent region (low speed side) of a large single stream shear layer. (The acquisition of the discrete vectors for this large scale single stream shear layer is primarily due to the efforts of RJP.) Theθ(x) value at the center of

this image is 13.2 cm. With the free stream velocity of 7.4 m/sec, the corresponding Re_θis 65,120. Figure 9 (from the MS thesis of A. Hellum [3]) is provided to show the self-preserving u(η)/U0where

η= y− y1/2

θ(x) . (8)

x/b y /b -1 -1 -0.5 -0.5 0.5 0.5 1 1 1.5 1.5 2 2 (a) x/b y /b -1 -1 -0.5 -0.5 0.5 0.5 1 1 1.5 1.5 2 2

I

II

I

II

Figure 7: a) Streamlines added to the discrete vector field provided by Monnier and Ward [6] with b) a focus on Region I, and c) a

focus on Region II.

thickness is linear (as expected) and described by Morris [7] as

θ(x)

θ(0)= 0.035

x

θ(0)+ 0.86. (9)

With these reference conditions and with the known lateral location in lab coordinates, the distinctive saddle point (at y≈ 70 cm) is

located atη= −4.0. The self-preserving form of the mean velocity field; see Figure 9 from Hellum [3], shows thatη= −4.0 is well

beyond the range of appreciable mean velocity. This raises interesting questions regarding the nature of the velocity and vorticity fields in the far region of the shear layer.

The experimental facility was carefully designed and fabricated to create a very low vortical disturbance level in the entrainment stream; see Morris [7] and Morris and Foss [8]. The present PIV image from this experiment is one of 1500 recorded images. A comprehensive manuscript, to capitalize on this extensive data base, will be forthcoming.

Considering the visualized streamlines of the entrainment stream (lhs) of the image, it is considered to be reasonable to represent these data as a collapsed sphere with two holes (as shown in Figure 10c). With that understanding, and with the quite evident saddle at (x,y) = (3870, -700) mm, one can infer that a single net node would be found in the domain; likely near (x,y) = (3750, -500) mm. That domain is boxed in Figure 10c and shown to a larger scale in Figure 10d.

Figure 10d identifies the N,S locations which were (laboriously) identified by KMB. It is noteworthy that these detailed considerations did reveal a net node to complement the evident saddle. Some of the more intricate extraction locations are shown in greater detail in Figure 12 for purposes of further analysis in §4.0.

The next section (§4.0, as created by KMB) explores further understandings regarding the extraction of information from such high Re PIV images.

4. Vector spacing consideration on topology

The densely located singular points demonstrated in the high Re SSSL data of §3.4 can benefit from an assessment of the accuracy of the streamlines, and singular points. One method to carry out this assessment is to artificially add an isolated erroneous velocity vector in the vector field. That is, such an addition can give the analyst a sense of how important is fine scale information that is contained in a high Reynolds number environment. In order to make this assessment, the flow field of Figure 10a has been investigated in detail in the vicinity of (x,y)=(-800, 4000) mm. In this region, the velocity vectors exist in an organized and nearly uniform direction, as shown

in Figure 11a. As expected, the streamlines in this region demonstrate a nominally parallel flow with no singular points.

In Figure 11a, a single velocity vector has been selected (noted at point A) to undergo an artificial change by setting this vector to the negative in both the u and v velocity components. The resulting streamline pattern is demonstrated in Figure 11b. This alteration of a single vector represents a worst-case scenario for an erroneous vector in a PIV vector field. While this type of isolated erroneous vector would likely be identified in in the PIV validation processes, it is possible that a less extreme erroneous vector may pass the validation processes. By definition, streamlines are tangent to the vector field at all points; thus, streamlines that pass through, or very near, this modified vector must accommodate its reversed orientation. The streamline algorithms accomplish this by allowing the streamlines to turn very sharply between the surrounding discrete vector locations. The resulting topology includes one node, and one saddle (a self-cancelling pair) positioned at approximately the discrete vector spacing; as demonstrated in Figure 11b. The resulting message from this exercise is that singular points in proximity roughly equal to the discrete vector spacing should not be interpreted as valid vectors. The extension of this evaluation process to multiple spurious vectors is not clear, thus, the spatial limit between cancelling singular points is taken as the discrete vector spacing.

This limiting reliable topology analysis may be employed in evaluations of the complex (many closely located singular points) flow field of Figure 10d. Figure 12 provides the flow field in the region near (x,y)=(-350,3780) mm in Figure 10d wherein a total of five nodes and five saddles were identified. It is clear that all node-saddle location have a minimum spacing of twice that of the discrete vector distances. Furthermore, the vector field demonstrates that the streamline pattern is supported by multiple vectors near each singular point. Thus, confidence is gained that this complicated flow region is accurately represented in terms of its identified singular points.

5. Summary

Topological methods, with specific focus on supporting PIV evaluations, have been presented. These concepts have been made explicit using four subject flow fields. The singular points so identified can be understood to represent the ”skeletons” of the kinematic field upon which the velocities are ”hung”... to paraphrase A.E. Perry. The indices (±1) of the singular points must satisfy the Poincase-Hopf

relationship (herein referred to as the Topological Rule) and the confirmation thatχA=χE indicates that the inferred singular points are not in violation of the Rule.χAis the a’priori Euler characteristic andχE is that derived from the experimental observations:

χA= 2 −Σholes− 2Σhandles, (10)

and

χE= 2ΣN+ΣN′− 2ΣS−ΣS′. (11)

REFERENCES

[1] Bade KM and Foss JF, Attributes of the large-scale coherent motions in a shear layer, Experiments in Fluids, 49, pp. 225-239, 2010.

[2] Foss JF, Surface Selections and Topological Constraint Evaluations for Flow Field Analyses, Experiments in Fluids, 37, pp. 883-898, 2004.

[3] Hellum A, Intermittency and the viscous super layer in a single stream shear layer, MS Thesis, Department of Mechanical Engineering, Michigan State University, 2006.

[4] Hunt JCR and Graham JMR, Kinematical Studies of the Flows Around Obstacles; Applying Topology to Flow Visualization, Journal of Fluid Mechanics, 86, pp. 179-200, 1978.

[5] Monnier B, Neiswander B, and Wark C, Stereoscopic Particle Image Velocimetry Measurements in an Urban-Type Boundary Layer: Insight into Flow Regimes and Incidence Angle Effect, Boundary-Layer Meteorology, 135:2, pp. 243-268, 2010.

[6] Monnier B and Wark C, Energetic Large-scale Structures and their Gust/Turbulence Characteristics in an Urban-type Boundary Layer, 2013.

[7] Morris SC, The velocity and vorticity fields of a single stream shear layer, PhD Dissertation, Department of Mechanical Engineering, Michigan State University, 2002.

[8] Morris SC and Foss JF, Turbulent boundary layer to single-stream shear layer: the transition region, Journal Fluid Mechanics, 494, pp. 187-221, 2003.

[9] Peabody J, Identification of the viscous super layer on the low speed side of a single stream shear layer, MS Thesis, Department of Mechanical Engineering, Michigan State University, 2010.

[10] Perry AE and Chong MS, A Description of Eddying Motions and Flow Patterns Using Critical-Point Concepts, Annual Review of Fluid Mechanics, 19, pp. 125-155, 1987.

[11] Perry AE and Chong MS, Topology of flow patterns in vortex motions and turbulence, Applied Scientific Research, 53, pp. 357-374, 1994.

x/D 20 25 30 35 40 45 10 15 5.0 0 -6 -4 -2 6 4 2 -8 y/D (a) N N N N N N N N N N N N N N N S S S S S S S S S S S S S S S S S x/D 20 25 30 35 40 45 10 15 5.0 0 -6 -4 -2 6 4 2 -8 y/D (b) S’ (c)

Figure 8: The wake of a circular cylinder as described in §3.3 where a) Shows the streamlines representing the velocity field in

laboratory coordinated, b) shows the streamlines in a convected frame using the mean flow field subtraction operations of §3.3, and c) presents a rationale for the absence of singular points in Figure 8a.

Figure 9: Characterization of the canonical nature of the single stream shear layer.

(a) The complete image captured by a two-component PIV system. (b) A detailed view of the clustered fine-scale motion in the region contained in the box in Figure 10c.

S

(c) The identification of the sub-region to be interrogated for singular points.

N N N S S S N S N S N S N N S N S N S N S N S N S

(d) Node and Saddle locations in the fine-scale motion region (Note, a net node exists in this region, which will balance with the saddle identified in Figure 10c.

y (mm) x (m m ) -804 -802 -800 -798 -796 3998 4000 4002 4004 0.2m/s

A

(a) The location (A) of the reversed vector (not yet modified)

y (mm) x (m m ) -804 -802 -800 -798 -796 3998 4000 4002 4004 0.2m/s

S

N

(b) The recovered streamlines via the utilization of TecPlot. Notes, i) the altered streamline matter was confirmed using the matlab streamline algorithm. ii) the grid lines show the locations of the discrete velocity vectors.

Figure 11: An valuation of the sensitivity of a PIV image streamline pattern to the imposition of a spurious vector.

y (mm) x (m m ) -360 -355 -350 -345 -340 -335 -330 3770 3775 3780 3785 3790 1m/s

Figure 12: The resolved PIV vectors in a region of intense small scale motions. Notes: i) the grid lines show the base locations for the discrete vectors. ii) The reader is invited to use the material of §2.0 to identify the singular points in this image.