Vortical flow analysis

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331

Journal of Mechanics in Medicine and BiologyVol. 10, No. 2 (2010) 191–212c© World Scientific Publishing CompanyDOI: 10.1142/S0219519410003319

VORTICAL FLOW ANALYSIS

KELVIN K. L. WONG∗,†, JIYUAN TU†and RICHARD M. KELSO‡

†School of Aerospace, Mechanical and Manufacturing Engineering,and Health Innovations Research Institute (HIRi),RMIT University, Bundoora VIC 3083, Australia

‡School of Mechanical EngineeringUniversity of Adelaide, SA 5005, Australia

∗[email protected]

Received 8 June 2009Accepted 1 September 2009

The minimization of errors due to random noise in statistical properties such as the meanand root mean square of a vorticity distribution is investigated. We map two-dimensional(2D) vorticity fields and present the distribution of vorticity with an intensity map. Then,we quantify the reliability of the visualization configuration using statistical propertiesof this distribution. To validate our methodology, we devise vortical flow fields based onthe analytical Lamb–Oseen vortex. The reliability of the vorticity measurements fromour results shows that the size of the flow vector sampling and random noise in thedata affect the fidelity of the vorticity maps. Based on this analysis, we establish anoptimised configuration that minimises the influence of noise on the computed vorticityhistograms. The concept outlined in this study may be used to reduce the effects of noisein a vorticity calculation, and thereby improve the effectiveness of flow visualization.

Keywords: Keywords: vorticity; flow visualization; reliability; histogram; Lamb–Oseenvortex.

1. Introduction

Flow visualization is the technique of examining fluid flow to obtain qualitativeinformation about other flow parameters within the flow field. Two-dimensional(2D) flow imaging such as particle image velocimetry,1, 2 phase contrast magneticresonance velocimetry3, 4 or fluid motion estimating5 generate vector fields of fluidflow. However, imperfect measurement conditions and equipment noise fluctuationsoften generate a certain percentage of error in measurement.

In many instances, post-processing of flow data, such as the identification ofinconsistent or outlying vectors using a median test,2 followed by replacement withinterpolated data, can be employed to improve the flow measurement accuracy.Derivative estimation based on the processed data is typically performed for amore detailed flow analysis. Despite efforts to reduce noise in the data and to

∗Corresponding author.

191

http://dx.doi.org/10.1142/S0219519410003319

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331

192 K. K. L. Wong, J. Tu & R. M. Kelso

improve accuracy of the derivative estimation,6 the computed vorticity will oftenbe biased and noisy due to the presence of noise in the flow field. Recent studieshave quantified the error of measurement based on different vorticity measurementschemes.7 The vorticity calculation will be affected by noise based on a minimallysized interrogation window (for example, of a 3 × 3 grid point array) and such aconfiguration cannot provide sufficient averaging and inherent smoothing to delivera representative vorticity field.

From first principles, we have developed a set of equations that explain how toderive flow derivatives based on multiple layers of vector contours around a regionof interrogation. Using more than one layer of contours, which are encapsulatedby a larger sampling window size, we intrinsically average a larger group of theflow data while providing an indication of the fluid vorticity at a point of interest.However, the sampling window configuration of the calculation needs to be selectedoptimally to produce vorticity maps that relate to that of an analytically correctvortex to the maximum extent. Oversizing the window attenuates the critical flowfeatures excessively, while an undersized window is more susceptible to noise andso gives a poor global representation of the vorticity map of a vortex. It is thereforeof interest to develop a framework for analyzing the reliability of vorticity samplingand calculation to best represent vortical flow mapping and visualization. To solvethis problem, we developed a framework that can test the reliability of the vorticitycomputation based on a specific sampling window configuration. This paper studiesthe effect of image resolution, sampling window size and addition of noise, eithersingly or combined, on the vorticity measurement. The framework may be appliedto improve visualization of vortices in cardiovascular flow, which can be enhancedby a statistical quantification of the vorticity field.4, 8 From a clinical perspective,such vorticity information can potentially be utilized to investigate hemodynamicsand may also be used to characterise the cardiac health of a patient.

A background on the Lamb–Oseen vortex formulation is provided since thesynthetic flow fields used in our experiments are constructed using the definingequations. We also examine the different methods for the visualization of vorticitytransport in unsteady flow as a form of literature review of existing techniques.Vorticity quantification and visualization are described in the Theory section. Next,techniques for the flow setup and synthetic vector field generation are described.The results and discussion section analyses the statistical behavior of vorticity fieldsand conceptualises the framework for optimization of vorticity visualization. Finally,a summary of the implemented methodology and vorticity visualization results isgiven.

2. Background

2.1. Analytical formulation of vortex

The Lamb–Oseen vortex9–11 is a theoretical model of a real vortex which analyti-cally defines its velocity, vorticity, and circulation. The mathematical presentation

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331

Vortical Flow Analysis 193

of the vortex by Meunier et al.10 is such that Γ is the circulation and a is thecharacteristic core radius. The angular velocity ω(r) is given by Eq. (1) as

ω(r) =Γ

πa2

(e−

r2

a2

). (1)

The tangential velocity vθ(r) is a function of r in Eq. (2) such that:

vθ(r) =Γ

2πr

(1 − e−

r2

a2

). (2)

We digitise the analytic velocity field over a data grid with each coordinatedenoted by (x, y), and with velocity interrogation spacing ∆ to produce a velocityvector flow field that is given by (u, v). This allows the effect of velocity field res-olution based on the analytical data produced for our computational approach tobe quantified. The velocity profiles generated by these equations are plotted as afunction of r in Fig. 1. The angular and tangential velocities vary with respect tothe radius from the vortex centre and their magnitudes can be represented usinggray-scale intensity. Note that the tangential velocity at the core is zero despitehaving a finite vorticity. The velocities ω and vθ vary from 0 to maximum values ofωmax and vθmax, respectively.

2.2. Methods for visualization of vortex

Vorticity ω or curl of velocity u is defined as ω = ∇ × u and represents bothorientation and angular velocity of local fluid rotation. When computed from 2Dsectional velocity field data, the vorticity vector is oriented normal to the plane ofthe velocity data. The display of a 2D vorticity map of a single vortex based onits in-plane velocity field can be used to indicate the location of its vortex core,its overall shape and its direction of circulation. By analysing the distribution and

(a) Angular velocity (b) Tangential velocity

Fig. 1. Velocity characteristics of a Lamb–Oseen vortex. Variation of profiles with respect tovortex radius r is shown for the angular and tangential velocities labeled as ω and vθ in (a) and(b), respectively. The velocity profile for the vortex angular and tangential components can beillustrated using varying gray-scale intensity. We plot the variation of the presented vortex basedon analytical formulations (Eqs. (1) and (2)).

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


gradient of vorticity around its core, the strength and length scale of the vortex canbe characterized.

Other than the examination of vorticity gradient, we can also visualize vorticesin space using volume reconstruction of surfaces of uniform vorticity magnitude.12

Their isovolume components can then be used to locate vortex structures and theirevolution over time. Other techniques include tracing of vortex cores. There are aremany ways to define and locate the vortex core. For example, points of maximumenstrophy (one-half the square of the relative vorticity) are defined to exist alongthe core.13 Other definitions include the local maxima of normalised helicity,14

where helicity is defined as the extent to which fluid travels in a helical motion.A study by Banks15 uses a vorticity-predictor pressure-corrector scheme to followvortex cores, where low pressure and a large magnitude of vorticity indicate thepresence of a vortex. The surface reconstruction of vortices using regions that followthe vortex cores can be performed. Instantaneous streamline mapping is anothermethod to indirectly indicate the presence of vorticity. Streamline tracing on a planenormal to the vortex core can be performed to exhibit the pattern of lines over thevortex region.16–18 However, this method is not always conclusive as vorticity canbe present in shear flow without displaying any form of swirling motion in thestreamline pattern.19

3. Theory

From the fluid mechanical perspective, the velocity field alone is not sufficient toestablish a complete description of a flow. For example, relative motion of theobserver with respect to the flow features will influence the observed motion. On theother hand, differential quantities such as vorticity and strain rate are independentof the relative velocity of the observer, and so provide an unambiguous description ofthe flow dynamics. In this way, examination of the vorticity field is able to enhancethe description of the fluid dynamics and provide a deeper insight.

3.1. First order finite differentiation

A pre-requisite to examining the suggested flow parameters is the requirementto develop a finite element differentiation technique using velocity data sampledwithin an interrogation window. Numerical first-order differential operators basedon single-level differential operators2 have been described.

Assume that f(x) is sampled at discrete locations such that fi = f(xi). A first-order differential implementation for data spaced at uniform ∆x intervals along thex-axis can be given by a central difference (CD) scheme such that:

f′i (2∆x) ≈ fi+1 − fi−1

2∆x. (3)

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


We sample multiple discrete locations such that [fi−N , . . . , fi, . . . , fi+N ] areobtained at [xi−N , . . . , xi, . . . , xi+N ]. The distance between each pair of sampledpoints, xi−n and xi+n about xi is given by 2n∆x.

By averaging of f′i (2n∆x) for n = 1 to N , a first-order differential at point xi

becomes: (δf

δx

)i

=1N

N∑n=1

f′i (2n∆x) ≈ 1

N

N∑n=1

fi+n − fi−n

2n∆x. (4)

The proposed multi-step gradient averaging also has the advantage of smoothingout local discontinuities and/or noise within the function. It also results in highererrors at points of inflexion. Using this derived equation, we will be able to calculatethe approximate differential quantities of the flow by numerical differentiation ofthe flow field in the x and y directions.

Graphs with inflexion points exist for f(x) = xk given that k assumes odd num-ber integer values. The analytical differentiation of f(x) results in f

′(x) = kxk−1

where, at x=0, this gradient is calculated to be zero. Using the finite differentiationmethod and knowing that fi−n = −fi+n, the gradient at each interval step of ∆x

becomes a finite non-zero value of fi+n/∆x since fi+n > 0 at x > 0. We arrive atthe conclusion that: (

δf

δx

)i

= kxk−1i �= 1

N

N∑n=1

(xki+n)

n∆x, (5)

given the condition xi = 0 and xi+n �= 0. Note that as n increases when moresampling points are taken, the rate of change of fi+n becomes higher in value,contributing to an increment of error in the numerical results. Let us revisit first-order differentiation from a 2D perspective. Based on the velocity field, with its x

and y components as Vx(i, j) and Vy(i, j) respectively, at a point of interest locatedat (i, j), N represents the number of layers of the contour within the interrogationwindow or sampling frame and ∆x and ∆y represent the horizontal and verticalintervals between neighboring measurement points. The vorticity distribution canthen be determined based on numerical differentiation when deriving the differentialquantities in Eq. (6).

3.2. Vorticity

Vorticity is calculated by examining the magnitude of rotation of fluid about aspecific examined point or region in a 2D velocity field. This is demonstrated byschematic elements in Fig. 2. The circulation around a fluid region, or interroga-tion window, is obtained by the summation of the tangential velocity componentsmultiplied by the finite interval distance in the counter-clockwise (CCW) directionaround a closed circuit bounding this region. The vorticity ω is then obtained bydividing the circulation by the area of the interrogation window.

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


Fig. 2. Vorticity computation using finite elements. Display of vorticity and calculations based onvelocity vectors along a closed contour around a node in the vector map. Vorticity computationis based on the curl of the velocity about a point of interest.

Mathematically, we have:

ω =(

∂Vy

∂x− ∂Vx

∂y

)i,j

=1

N2

N∑m=1

N∑n=1

[Vy(i + m, j + n) − Vy(i − m, j − n)

2m∆x

− Vx(i + m, j + n) − Vx(i + m, j − n)2n∆y

]. (6)

Note that in order to evaluate such integrals, an appropriate circuit or integra-tion window must be chosen, and these often enclose the vortex core.7, 20 From theformulation, positive values signify CCW rotation, whereas negative values repre-sent clockwise (CW) motion of the fluid. Therefore, the magnitudes of these valuesgive an indication of the rate of rotation and its polarity signifies the direction ofthe rotation. These may be represented by a color scale with maximum CCW andCW absolute values corresponding to red and blue, respectively.

3.3. Statistics of differential flow map

We perform a statistical quantification of the vorticity distribution, producing ahistogram of the vorticity magnitudes in the map. Such a histogram can be repre-sented using a bar plot in Fig. 3(a).

The histogram that is produced based on the percentage of map area versusthe vorticity values ω(s−1) throughout the entire flow map is featured in Fig. 3(b).A high resolution of the histogram bins causes the histogram bar width to be overly

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


s-1

Num

ber

of p

ixel

s

ω0

s -1ω0

−veValues

+veValues

CCW VorticityCW Vorticity

Are

a A

(%)

(a) Number of data points versus vorticity ω(s−1) (b) Area A (%) versus vorticity ω(s−1)

Fig. 3. Histogram representation of vorticity distribution in a flow map. A histogram depictingthe distribution of vorticity values within the region of analysis gives an indication of the generalcharacter and overall sense of rotation of the fluid flow. In this example, the mean µ and median m,shown by the blue striaght and dash lines respectively, are on the left side of the zero-value centerline shown in gray. Therefore, we can deduce that the flow is substantially CW since vorticitymean and median values are negative.

small, and we propose the display of a frequency graph using a line that joins theheight of each bar. In addition, we smooth the lines using spline interpolation toprovide a more moderate estimation of the true frequency from each discrete barheight.

The histograms pertaining to the vorticity field plots can be computed based onthis statistical framework. We are able to calculate the mean (µ) and median (m)from each of these histograms. Percentiles provide an indication of how the datavalues are spread over the interval from the smallest value to the largest value. Themedian is a robust measure of central tendency in the presence of outlier values.A quantification of average vorticity value is computed by taking the mean, ωµ,or median, ωm of the frequency histograms generated from vorticity maps. Themagntiude of these parameters is represented as the blue solid and dash lines, whilethe center zero ω line is superimposed onto the frequency graph in gray.

The vorticity standard deviations from the flow map measure the distributionaround the mean and median respectively. Standard deviation σ with respect to µ

can be computed by considering the variation about the mean and is denoted asσµ. Based on a similar mode of computation, calculating the degree of variationabout the median will give σm.

Standardisation of the histogram is performed for a set of vorticity maps toattain a more controlled comparison by setting the total number of grid pointsmeasured to a constant. If the velocity field map demonstrated that most regionsof fluid are stationary, the mean vorticity should be close to zero. In the caseof a single vortex, it is likely that the mean vorticity will shift away from the

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


zero flow value, and broaden the distribution of the histogram. Such an analyticalmethodology can help in assessing the relative intensity of flow property of thefluid. If the same number and strength of CW and CCW vortices co-exist, then themean will approximate zero. The standard deviation of the histogram can indicatethe size of the region with rotational flow.

4. Vortex Visualization Using Theoretical Formulation

4.1. Formulation of vortex flow field

We create analytical data using a theoretical vortex formulation to test the robust-ness of our vorticity measurement. The flow fields are generated based on twodifferent configurations of the vortices. In general, we have developed flow fieldsof the single vortex and double vortices pertaining to their two specific sizesrespectively.

4.1.1. Single vortex

Using mathematically generated velocity information based on the Lamb–Oseenvortex, we have examined the flow with different vorticity distributions. This allowsus to simulate the principal features of a planar velocity field measurement of avortical flow. We assume that the length scale corresponding to one standard devi-ation of the Gaussian vorticity distribution, L is 1 mm. The computational domainis (−5L ≤ x ≤ 5L) and (−5L ≤ y ≤ 5L), which is equivalent to (−5 ≤ x ≤ 5)mmand (−5 ≤ y ≤ 5)mm, whereby (x, y) represents the Cartesian coordinates of theflow field, and the vortex has been scaled such that its maximum tangential veloc-ity is 10 mms−1. An array of 160 by 160 data points represents the velocity flowfield of this vortex. Figure 4 describes the distribution of the velocity magnitudesusing flow maps depicting the (a) absolute, (b) horizontal, and (c) vertical velocitycomponents.

4.1.2. Double vortices

We now position two Lamb–Oseen vortices with core centers at a distance of 5 mmapart from each other. The vortices have different polarities in rotation. This sim-ulates flow consisting of two adjacent vortices, with some vorticity cancellationbetween. Each vortex is constructed in such a way that we computationally set theflow field to span 10mm by 10mm in space, and its maximum velocity magnitudeto 10mms−1. Therefore, the maximum absolute velocity in the flow field is lessthan 20mms−1. Note that the velocity field of this flow is represented by an arrayof 160 × 240 (width × height) data points. Figure 5 describes the layout of thevelocity magnitudes using flow maps depicting the (a) absolute, (b) horizontal, and(c) vertical velocity components.

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


SINGLE VORTEX WITH 0% NOISE

ABSOLUTE HORIZONTAL VERTICAL

(a) Absolute velocity v (b) x-Component velocity vx (c) y-Component velocity vy







Fig. 4. Artificially generated single Lamb–Oseen vortex velocity flow field maps. The Lamb–Oseenvortex has a non-uniform distribution of velocity magnitudes which follows a profile defined by aset of equations. The figure shows the absolute, x-component and y-component velocity magnitudedistributions, represented by a color map. Vector field pertubation at 10% and 20% is carried outto simulate the effect of noise in flow measurement.

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


DOUBLE VORTEX WITH 0% NOISE









Fig. 5. Artificially generated double Lamb–Oseen vortex velocity flow field maps. Two Lamb–Oseen vortices in opposite directions of rotation are generated based on the analytical formulations.The figure shows the absolute, x-component and y-component velocity magnitude distributions,represented by a color map. Flow fields with perturbation at 10% and 20% are displayed.

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


4.1.3. Perturbation of flow field Vectors

The analytical flow fields are now perturbed by random noise. This is performedusing a matrix of random values of range 0 to 1, which is created using a randomassignment of values based on the Gaussian distribution at specific percentages.The matrix is multiplied onto the x- and y-components of the flow field to modifyvector orientations, thereby adding noise to the flow field. The flow field is thenre-scaled to a range of 0 to 10mms−1.

In this study, Gaussian noise is added at percentages of 0% to 25% onto theanalytical velocity field to examine the effect of varying sampling window size onvorticity measurements in the presence of different levels of random noise. The effectof increasing the vorticity sampling area will be analyzed using these data sets. Toillustrate the effect of noise on the variation of velocity and vorticity, only the flowfield maps for 20% noise addition will be presented as an example.

4.2. Configuration for vorticity measurement

In the preceding section, we discussed the scheme used to calculate the vorticitydistribution. Using the analytically-generated velocity fields, the size of the inter-rogration window is varied in order to test effects of sampling window size on thefidelity of the measured vorticity distribution. Note that regions where the samplingwindows are exposed partially outside the frame at the edge of the data grid willbe padded with zero velocity vectors.

From the results describing single and double vortices in the flow field, we areable to visually observe their cores and strength using vorticity flow maps. In addi-tion, from the histogram pertaining to each flow map, we can extract some use-ful statistical information about the flow, including the presence of vorticity withopposing sign.

4.2.1. Single vortex

We then calculate the vorticity field and represent this at each data point using col-ors ranging from blue to red for the maximum CW to maximum CCW values respec-tively. Various vorticity sampling frame sizes are implemented incrementally toobserve their effect in smoothing the calculated vorticity values. Histograms of thevorticity maps are devised, based on a bin resolution of 1 s−1. Spline-interpolatedresults are displayed for ω range of −20 to 20 s−1. Statistical properties of the maps,such as means, medians, and standard deviations, are stated below each set of mapsand histograms.

In this study, we apply sampling mask sizes for vorticity computations rangingfrom (3 × 3) to (67 × 67) grid points. Each frame has a dimension of 1/16mm.Therefore we are looking at sampling frame sizes of (0.189 × 0.189) to (4.19 ×4.19)mm. Note that the vorticity map spans 10 by 10mm. The analysis is performed

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


for flow fields perturbed by various levels of noise. However, only the 20% noise mapsas shown in Fig. 6 will be displayed.

4.2.2. Double vortices

For the double vortices, we apply the same sampling mask sizes for vorticity com-putations ranging from (3 × 3) to (67 × 67) grid points, whereby each frame has adimension of 1

16 mm. This is a similar vorticity measurement configuration as theone for single vortex. The vorticity map for these two vortices spans 160 by 240grid points. The mean and median of histograms for these flow maps should ideallybe zero since the there are two vortices with opposite directions of rotation.

The same configuration for histograms of the maps applies to both the singleand double vortices flow field. The bin resolution is set at 1 s−1. Range of ω is −20to 20 s−1. Statistical properties of the map such as means, medians, and standarddeviations are computed based on the vorticity distribution in the map. The pro-cedure described above is carried out for flow fields for various noise additions aswell. However, only the results for 20% noise addition is shown in Fig. 7.

5. Discussion

New parameters are defined for testing reliability of measurement as well as compar-ing vorticities in order to determine the fidelity of the derived vorticity distribution.The reliability of vorticity measurements from our results shows that the size of thesampling array and noise in the flow field affect the fidelity of the computed vortic-ity fields maps. Based on the statistics of these field maps represented by reliabilityand comparison graphs in this section, we are able to establish an optimised sam-pling window size that computes vorticity fields to approximate statistically theideal vortex.

5.1. Reliability of vorticity measurement

A study on the influence of sampling window size and perturbation of flow fieldvectors is conducted using ideal flow maps and flow maps with noise addition (suchas the ones shown in Figs. 6 and 7). From the visualization and statistics of thevorticity maps computed from noisy flow fields, a sampling window based on asmaller number of grid points results in a relatively higher spatial variation ofvorticity magnitudes, as compared to the ideal flow maps. However, as the samplingwindow size increases, the variability in the vorticity map decreases. This is due tothe intrinsic 2D smoothing of the vorticity map, thereby decreasing the distinctionbetween positive and negative vorticity values. In the presence of noise and for smallsampling windows, the flow details cannot be captured sufficiently and ideally foran accurate vorticity mapping. This can be explained by the inability of a smallsampling window to encapsulate enough data to smooth the random noise. However,

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331



CONTOUR MAP VECTOR MAP HISTOGRAM

(i) Vorticity sampling window (9 × 9) grid points,

»ωσ

–µ

=

»1.678.22

–s−1,

»ωσ

–m

=

»1.088.24

–s−1.

(ii) Vorticity sampling window (17 × 17) grid points,

»ωσ

–µ

=

»1.673.85

–s−1,

»ωσ

–m

=

»0.933.92

–s−1.

(iii) Vorticity sampling window (25 × 25) grid points,

»ωσ

–µ

=

»1.663.03

–s−1,

»ωσ

–m

=

»0.773.15

–s−1.

Fig. 6. Flow visualization of Lamb–Oseen vortex (20% noise). Flow visualization is performed onthe Lamb–Oseen vortex flow field with 20% noise added. The vorticity contour maps are presentedfor sampling window sizes in the range of (3 × 3), (17 × 17), and (25 × 25) grid points size or1.87 × 10−2 to 1.56 × 10−1 times relative to the image of 160 by 160 grid points. Histogramspertaining to each vorticity map are presented in the last column.

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331



CONTOUR MAP VECTOR MAP HISTOGRAM

(i) Vorticity sampling window (3 × 3) grid points,

»ωσ

–µ

=

»0.0110.67

–s−1,

»ωσ

–m

=

»0.0010.67

–s−1.

(ii) Vorticity sampling window (17 × 17) grid points,

»ωσ

–µ

=

»0.025.39

–s−1,

»ωσ

–m

=

»0.005.39

–s−1.

(iii) Vorticity sampling window (25 × 25) grid points,

»ωσ

–µ

=

»0.034.45

–s−1,

»ωσ

–m

=

»0.004.45

–s−1.

Fig. 7. Flow visualization of double Lamb–Oseen vortices (20% noise). Flow visualization is per-formed on the flow field of two Lamb–Oseen vortices with 20% noise added. The vorticity contourmaps are presented for sampling window sizes in the range of (3×3), (17×17), and (25×25) gridpoints size or 1.87 × 10−2 to 1.56 × 10−1 times relative to the image of 160 by 240 grid points.Histograms pertaining to each vorticity map are presented in the last column.

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


it is important to note that a large sampling window, on the contrary, may create anoverly smoothed vorticity field such that some genuine flow features are attenuated.It is good practice to vary the sampling size until the variance of the vorticity valuesis sufficiently small for a clear visualization of the flow field, and yet large enoughto avoid losing significant flow details.

It is useful to establish a measure for reliability of the system to accuratelydevelop vorticity maps of the flow using a given technique. The reliability of thevorticity measurement ρ can be defined as the ratio of the true vorticity varianceto total variance that comprises the true and error components21 as

ρ =σ2

True

σ2True + σ2

Error

, (7)

where σ2True is the variance pertaining to the vorticity map of the ideal vortex, which

is measured using a (3× 3) points sampling window size, whereas that of the mea-sured variance given by vortical flow field with noise is equivalent to σ2

True + σ2Error.

To obtain the error map of a measured vorticity field, based on a given samplingwindow size, grid-point by grid-point differencing between the ideal vorticity mapand that of the measured one is conducted. The computation of the error variancein our measurement is based on the standard deviation of this error map. Relia-bility curves based on 5%–25% noise are plotted as a function of sampling windowsizes and shown in Figs. 8 and 9. These figures also plot the normalized samplingwindow size, which is the ratio of the window size to the overall width of the datagrid.

Note that the reliability of the vorticity measurement improves sharply withincreasing sampling window size, reaching a broad plateau, and then decreasinggradually. The optimal reliability is weakly dependent on the level of noise addedto the velocity flow field. From the graphs, the optimal sampling window to use forvorticity maps based on a single vortical flow with 5%, 10%, 15%, 20%, and 25%noise is (35×35) grid points. This corresponds to a normalized sampling window sizeof 0.22, or approximately 2.2 vortex length scales (where the length scale is definedas one standard deviation of the Gaussian vorticity distribution, as described in theprevious section). The same results hold for the double-vortex flow field.

5.2. Comparison of vorticity measurement

Each of the histograms that accompanies a flow map in the illustrated figures isable to provide some information about the vortex. For flow fields containing asingle vortex, the counts pertaining to positive values of vorticity ω shown by thehistogram of the vorticity map based on Fig. 6 signifies the CCW rotation. Themeans of histograms show consistently small increments in magnitude as samplingis applied on a larger frame. A positive ω mean signifies overall positive rotationalflow.

From the histograms of the double vortices illustrated in Fig. 7, we are ableto observe a decrease in standard deviations as the sampling size increases, while

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sampling Window Size (grid point)

Rel

iabi

lity

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalised Sampling Window Sizeρ

5% Noise10% Noise15% Noise20% Noise25% Noise

0.02 0.04 0.07 0.09 0.12 0.14 0.17 0.19 0.22 0.24 0.27 0.29 0.32 0.34 0.37 0.39 0.42

Fig. 8. Reliability test for single vortex flow fields. The reliability for the flow field ρ is based onvalues σµ and σm that are the standard deviations with respect to the mean and median of theflow maps respectively. Both sets of reliability graphs are approximately the same, and so only oneset based on σµ is presented. The variation of the vorticity sampling window size can affect thereliability of computational measurement of the vorticity. As the sampling window size increases,the reliability of the vorticity map measurement improves. Beyond a specific sampling windowsize, the reliability starts to drop slightly.

the means and medians remain at zero regardless of the vorticity sampling. Theirhistograms are almost symmetrical about the zero center line, as there is an equaldistribution of negative and positive rotational flow generated by the vorticies intwo different polarities. We also note that larger vorticity sampling windows reducethe spatial vorticity variation due to random noise, and allow us to distinguish thetwo vortices with better clarity.

For analytical and measured vorticity maps with the same vorticity samplingwindow applied, the comparison of their histogram variances γ can be defined asthe ratio of the histogram variance from the analytically determined vorticity mapto the one for the measured vorticity map such that:

γ =σ2

True

σ2Measure

. (8)

We discuss the two factors that will cause the comparison parameter γ toincrease. The first one is the size of the sampling window used for vorticity cal-culation at every frame. Increments in the vorticity sampling size have the effectof reducing the vorticity variance due to the increased size of the sampling windowcompared with the underlying vorticity field. The second variable is noise within

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sampling Window Size (pixel)

Rel

iabi

lity

0.02 0.04 0.07 0.09 0.12 0.14 0.17 0.19 0.22 0.24 0.27 0.29 0.32 0.34 0.37 0.39 0.42

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalised Sampling Window Size

ρ5% Noise10% Noise15% Noise20% Noise25% Noise

Fig. 9. Reliability test for double vortex flow fields. The reliability for the flow field ρ based on σµ

is presented for flow fields with various degrees of noise addition. If one were to plot the reliabilitybased on σm, the graphs will be almost identical. The reliability curves based on different levelsof added noise follow the same variation as those based on the single vortex flow. The optimalsampling size for vorticity measurement may differ for every unique flow field.

the measured flow field. The addition of noise to the velocity field has the effectof increasing the vorticity variance, due to the increased variation in the velocitygradients within the data field.

If the variance pertaining to the vorticity map of the ideal vortex is σ2True,

and given that for a perfect flow field the measured variance given by σ2Measure is

equivalent to σ2True, then the measured vorticity map is identical to that of the true

flow information, and reliability becomes one. It is important to note that becauseof the implications of sampling window size and noise and an existing relationshipbetween the two variables, σ2

Error is not the same as the variance of noise componentσ2

Noise. This also implies that σ2Measure is not equivalent to the sum of contributing

variable σ2Noise and σ2

True. Therefore, the incorporation of the noise and adjustmentof sampling window size such that σ2

Measured is equivalent to σ2True gives the critical

point of equilibrium for approximating the measured histogram to the one basedon idealized flow condition.

In our case studies, using various sampling windows for vorticity measurement,the histograms of the flow fields with noise addition at 5–25% are compared withthe histograms of the analytical flow field with no noise. The results of the varianceratio curves with respect to the sampling window size from (3×3) to (41×41) gridpoints are shown in Figs. 10 and 11 for single and double vortex flow fields.

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Sampling Window Size (grid point)

Var

ianc

e R

atio

0.02 0.03 0.04 0.06 0.07 0.08 0.09 0.11 0.12 0.13 0.14 0.16 0.17 0.18 0.19 0.21 0.22 0.23 0.24 0.26

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Normalised Sampling Window Sizeγ


Fig. 10. Comparison of histograms for single vortex flow fields. Comparisons of histograms basedon the analytical and measured flow fields are made by taking the ratio of their variances γ. Thevariances σ2

µ and σ2m are based on standard deviations with respect to the means and medians

of the flow maps respectively. Based on our results, both sets of standard deviations are verysimilar. The variation of vorticity sampling window size shows that as the sampling window sizeincreases, the histogram variance from the measured flow approaches that of the analytical onefor the same vorticity measurement configuration. Beyond a specific sampling window size, themeasured histogram variance becomes smaller than the analytical one and γ exceeds one.

Since the analytically determined vortex is suggested to be the true flow fielddata, any measured vortex will have elements of noise in its flow field. From thegraphs of comparison curves, we observed that variation of noise in the flow fieldhas an effect on their gradients. Addition of flow field noise has an influence on thevariance of the vorticity map, and results in an increasing γ as the sampling sizebecomes larger.

Vorticity fields created using small sampling windows give a poor definition ofthe vorticity gradient in the map. An increase in the number of vectors being sam-pled can intrinsically smooth the vorticity map sufficiently to approximate that ofthe ideal vortex. However, when the sampling size exceeds a specific threshold, thevariance of the measured vorticity map becomes smaller than that of the analyti-cal one, which results in both poor correspondence between their histograms andreduced reliability of the vorticity measurement system. This signifies that over-smoothing of field vectors has occurred, attenuating the genuine flow features andproducing a vorticity map that starts to differ from that of the ideal vortex.

A critical sampling size can be determined by comparing the curves for differentnoise levels in Figs. 10 and 11. The intersection of these curves occurs at a position

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Sampling Window Size (pixel)

Var

ianc

e R

atio

0.02 0.03 0.04 0.06 0.07 0.08 0.09 0.11 0.12 0.13 0.14 0.16 0.17 0.18 0.19 0.21 0.22 0.23 0.24 0.26

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Normalised Sampling Window Sizeγ


Fig. 11. Comparison of histograms for double-vortex flow fields. The comparison of the histogramvariances is based on the variance of the analytical flow field divided by variance of the measuredone. Both sets of variances, with respect to the means and medians of the vorticity maps, havethe same values. This ratio, labeled as γ, is presented for different sampling window sizes andon flow fields that are comprised of double vortices, and with various levels of noise addition.The variance ratio curves based on different noise addition follow the same variation as thosepertaining to single-vortex flow.

that also coincides approximately with the line of absolute similarity (γ = 1). Thiscritical point of intercection gives the ideal sampling window size that can producevorticity histograms corresponding as closely as possible to the one pertaining tothe analytical vortex. It is worthwhile mentioning that the intercection with theabsolute similarity line remains substantially unaffected by any amount of noiseadded to the flow field.

From the results plotted in Fig. 10, we can conclude that in order to achieve avariance ratio of 1 in the single-vortex field, a critical sampling window of (20×20)grid points is required. This corresponds to a normalized sampling window sizeof 0.125, or 1.25 vortex length scales. In the case of the double-vortex system,the results are similar, with a critical sampling window of (21×21) grid points,corresponding to a normalized sampling window size of 0.13, or 1.3 vortex lengthscales.

5.3. Effect of grid resolution on vorticity measurement

The presence of both bias and random error exists for grid matrix that has a lowerresolution. Note that we use a reference flow field Λref with an arbitrary size of

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


(160× 160) points at a 16-grid points per millimeter resolution. We do a point-by-point differencing of a grid Λs that has resolution s grid points per millimeter withthis reference grid to obtain an error matrix. The average of all the elements in thatmatrix is computed using the function mean in Eq. (9). Note that s ranges from1 to 16. The maximum error is the subtraction of grids based on resolutions of 1and 16 grid points per millimeter (i.e. Λ1−Λ16, respectively). Normalization of theerror function ∆ is performed by taking the ratio of absolute mean of an arbitraryerror grid to the maximum error mean:

∆(s) =mean(|Λs − Λref |)mean(|Λ1 − Λref |) for s = 1, . . . , 16, and Λref = Λ16. (9)

Based on this configuration, normalized error ∆ ranges from 0 to 1. The equationcan give an estimate of the bias error as the result of reduction in grid resolution.We can observe from Fig. 12 that the error decreases as the quality of the gridimproves. The reliability equation that examines the variability of the differencemaps can be used to determine the nature of variation in terms of error fluctuations.The reliability is shown to approach 1 for higher resolutions of grid. This has beenshown to be consistent for vorticity maps measured using sampling window sizes of(3 × 3) to (15 × 15) points.

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 1600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Image size (Points)

Rel

iabi

lity

ρ

ρ by 3 points sampling window sizeρ by 7 points sampling window sizeρ by 11 points sampling window sizeρ by 15 points sampling window size

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Grid resolution (Points per mm)

Nor

mal

ised

mea

n er

ror

∆

∆ by 3 points sampling window size∆ by 7 points sampling window size∆ by 11 points sampling window size∆ by 15 points sampling window size

Reliability

Error

Fig. 12. Reliability and error deviation for multi-resolutional single vortex flow fields. The relia-bility ρ of the vorticity measurement depends on the variability in the error fluctuation, whereasthe normalized mean error ∆ is a function of the bias error in the differencing of two vorticityfields of dissimilar resolutions. Both parameters are shown to vary in the same fashion based onfour sampling window sizes. The reliability is improved as the error is lower for higher resolutiongrids.

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


The reliability of the vorticity measurement is inversely proportional to theerror deviation of the calculated vorticity field with respect to the true field. Asthe resolution of the grid increases, the flow detail becomes amplified, and vorticitymeasurement based on a finer flow grid is able to encapsulate more field vectorsfor calculations of higher accuracy. The graph demonstrates the nature of thisrelationship effectively.

6. Conclusion

A framework has been formulated to measure the vorticity distribution in a fluidflow. To validate this technique, we have tested the vorticity measurement schemeusing analytically-defined flow fields along with a range of sampling window sizesand several levels of random noise. The experiments show that the size of thesampling window used for vorticity computation plays a significant role in definingthe vortex accurately when there is noise in the flow measurement.

The results demonstrate that in order to achieve the best reliability, or minimizethe overall statistical variation from the ideal, the optimal sampling window sizeis approximately 2.2 vortex length scales. The same outcomes hold for the double-vortex flow field. By contrast, in order to achieve the correct variance value, ora variance ratio of 1, a sampling window of 1.25 or 1.3 vortex length scales isrequired for the single vortex and double-vortex cases respectively. These resultsare only weakly dependent on the level of noise imposed on the velocity field. Weconclude that, overall, the best results are achieved when the reliability is closestto 1. However, if the variance is the key parameter in an analysis, it is possibleto optimise the variance ratio. Furthermore, the result graphs suggest that thereliability is only slightly degraded when the variance ratio is optimised. Finally,an examination of the effects or grid resolution confirms that the highest reliabilityis achieved when the grid resolution is maximized.

This study supports the basis of vortical flow analysis, and can be applied tocharacterise the vortices present in physical situations such as cardiovascular flow.It can enhance the quality of vortical flow investigations by empirical modes ofmeasurement, including cardiac flow analysis based on phase contrast magneticresonance imaging.

References

1. Alahyari A, Longmire E, Particle image velocimetry in a variable densiy flow: appli-cation to a dynamically evolving microburst, Exp Fluids 17:434–440, 1994.

2. Raffel M, Willert C, Kompenhans J, Particle Image Velocimetry, Springer-Verlag,Berlin Heidelberg, Germany, 1998.

3. Raguin LG, Honecker SL, Georgiadis JG, MRI velocimetry in microchannel networks,3rd IEEE/EMBS Special Topic Conference on Microtechnology in Medicine and Biol-ogy, pp. 319–322, 2005.

June 29, 2010 7:35 WSPC/S0219-5194/170-JMMB 00331


4. Wong KKL, Kelso RM, Worthley SG, Sanders P, Mazumdar J, Abbott D, Cardiacflow analysis applied to phase contrast magnetic resonance imaging of the heart, AnnBiomed Eng 37(8):1495–1515, 2009.

5. Cuzol A, Hellier P, Memin E, A low dimensional fluid motion estimator, Int J ComputVision, 75(3):329–349, 2007.

6. Etebari A, Vlachos P, Improvements on the accuracy of derivative estimation fromDPIV velocity measurements, Exp Fluids, 39(6):1040–1050, 2004.

7. Fouras A, Soria J, Accuracy of out-of-plane vorticity measurements derived fromin-plane velocity field data, Exp Fluids 25:409–430, 1998.

8. Wong KKL, Tu JY, Kelso RM, Worthley SG, Sanders P, Mazumdar J, Abbott D,Cardiac flow component analysis, Med Eng Phys 32(2):174–188, 2010.

9. Saffman P, Vortex Dynamics, 1st ed., Cambridge University Press, 1992.10. Meunier P, Dizes SL, Leweke T, Physics of vortex merging, Comptes Rendus Physique,

6(4–5):431–450, 2005.11. Cariteau B, Flor JB, An experimental investigation on elliptical instability of a

strongly asymmetric vortex pair in a stable density stratification, Nonlinear ProcessesGeophys 13(6):641–649, 2006.

12. Silver D, Wang X, Volume tracking, Proceedings of the 7th Conference on Visualiza-tion ’96, p. 157, 1996.

13. Jiminez J, Wray AA, Saffman PG, Rogallo RS, Intense vorticity in isotropic turbu-lence, J Fluid Mech 255:65–90, 1993.

14. Yates LA, Chapman GT, Streamlines, vorticity lines, and vortices, AIAA Paper 91-0731, 1991.

15. Banks DC, A predictor-corrector technique for visualizing unsteady flow, IEEE TransVis Comput Graph 1(2):151–163, 1995.

16. Robinson SK, A review of vortex structures and associated coherent motions in tur-bulent boundary layers, Proceedings of Second IUTAM Symposium on Structure ofTurbulence and Drag Reduction, Federal Institute of Technology, Zurich, Switzerland,pp. 25–28, 1989.

17. Robinson SK, Coherent motions in the turbulent boundary layer, Ann Rev Fluid Mech23:601, 1991.

18. Robinson SK, Kline SJ, Spalart PR, A review of quasi-coherent structures in a numer-ically simulated boundary layer, NASA Technical Memorandum 102191, 1989.

19. Sadlo F, Peikert R, Sick M, Visualization tools for vorticity transport analysis inincompressible flow, IEEE Trans Vis Comput Graph 12(5):949–956, 2006.

20. Weigand A, Gharib M, On the evolution of laminar vortex rings, Exp Fluids,22(6):447–457, 1997.

21. Bohrnstedt G, Measurement, In PH Rossie et al. (eds.), Handbook of Survey Research,Academic Press, New York, 1983.

22. Ferziger JH, Peric M, Computational Methods for Fluid Dynamics, Springer, 2001.

Documents

Vortical flow analysis