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International Journal of Computational Fluid Dynamics
ISSN: 1061-8562 (Print) 1029-0257 (Online) Journal homepage: https://www.tandfonline.com/loi/gcfd20
A simple technique for the visualisation of eddykinematics in turbulent flows
Geoffrey Lee , Carlo Scalo & Ugo Piomelli
To cite this article: Geoffrey Lee , Carlo Scalo & Ugo Piomelli (2012) A simple technique for thevisualisation of eddy kinematics in turbulent flows, International Journal of Computational FluidDynamics, 26:4, 263-274, DOI: 10.1080/10618562.2012.693605
To link to this article: https://doi.org/10.1080/10618562.2012.693605
Published online: 11 Jul 2012.
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A simple technique for the visualisation of eddy kinematics in turbulent flows
Geoffrey Lee, Carlo Scalo* and Ugo Piomelli
Department of Mechanical and Materials Engineering, Queen’s University, Kingston, ON, K7L 3N6, Canada
(Received 13 January 2012; final version received 7 May 2012)
We developed and tested a simple technique to predict, for flow visualisation purposes only, the evolution ofcoherent structures in between two given realisations of a turbulent flow. Classic coherent-structure eductionmethods are adopted, such as the Q-criterion, pressure fluctuations and contours of velocity fluctuations. Thekinematics of the evolving structures are reconstructed by means of an advection-based reconstruction techniqueand captured in a movie. The resulting quality of the animations has been assessed via the Structural SimilarityIndex (SSIM). The sensitivity to increasing spacing in time of the available flow realisations has been tested andseveral improvements implemented. The abrupt transition from reconstructed frames of the animation to theavailable realisations results in a noticeable lack of smoothness. The replacement of the available realisations with asimilar advection-based average increases the perceived smoothness of the movies. This is confirmed by the reducedtotal variation of the SSIM index. The residual minor periodic variations of accuracy have been reduced byintroducing a stochastic weighting function. The sensitivity of the results to changes in Reynolds number, resolutionand structure representation methods has been tested.
Keywords: flow visualisation; large-eddy simulation; turbulent structures kinematics; Structural Similarity Index;advective interpolation; movie accuracy and smoothness; human visual system
1. Introduction
In turbulent flows, coherent motions (known as‘eddies’) play a crucial role in the transport of mass,momentum and energy (Robinson 1991). Hence, asignificant effort has concentrated, in the past, on theeduction, quantification and visualisation of coherenteddies.
Coherent structures were first discovered with theaid of experimental flow visualisation methods (Liep-mann 1979). Early flow visualisations relied mostly oneither line-of-sight (optical interferometry, schlierenand shadowgraphy) or light-scattering (smoke or dyevisualisation) methods. Among the first examples ofsuch discoveries were the near-wall streaks (Kline et al.1967) and the spanwise rollers in mixing layers (Brownand Roshko 1974). Numerical simulations of turbulentflows (and, in particular, large-eddy and directsimulations) have given a major contribution to thissubject. From a flow visualisation standpoint, theypresent significant advantages, compared to experi-mental results. First, since three-dimensional (3D)pressure and velocity fields are available, one canconsider a variety of identifiers, and choose the onemost appropriate to the flow under investigation. Also,the extraction of the information is local, instanta-neous and non-intrusive. Only recently, experimental
techniques that yield 3D, instantaneous flow fields suchas, for instance, tomographic and time-resolvedParticle-Image Velocimetry (Elsinga et al. 2006) havebecome available.
While early experimental studies only yieldedparticle paths, streamlines or streaklines, the avail-ability of instantaneous data has made it possible tovisualise a variety of quantities, highlighting particularaspects of the turbulent flow field. Among the mostcommon choices are vorticity and pressure-relatedquantities and various invariants of the velocitygradient tensors. Most commonly, two-dimensional(2D) contours are shown in planes of interest;isosurfaces of various quantities are also very oftenshown, to highlight the 3D structure of the turbulenteddies. Comparisons of the advantages and disadvan-tages of the various methods can be found in Robinsonet al. (1989) and Dubief and Delcayre (2000). Variousvortex-identification methods have also been proposed(Kida and Miura 1998, Haller 2005).
In addition to the instantaneous snapshots, moviesor animations highlighting the kinematic developmentof these eddies have become widespread. Early movies,obtained from experiments, typically used smoke ordye injection and proved to be particularly effective inanalysing mechanisms of interaction between coherent
*Corresponding author. Email: [email protected]
International Journal of Computational Fluid Dynamics
Vol. 26, No. 4, April–May 2012, 263–274
ISSN 1061-8562 print/ISSN 1029-0257 online
� 2012 Taylor & Francis
http://dx.doi.org/10.1080/10618562.2012.693605
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motions. For example, Head (1969) successfullycaptured the time evolution of the intermittent(turbulent/non-turbulent) edge of a smoke-filled tur-bulent boundary layer on a 16-mm film. With the samesetup, Head and Graham (1969) looked at theinteraction of trailing edge vortices and near-wallcoherent eddies. A more detailed discussion on theimplications of this structure visualisation method(with particular focus on hairpin/horseshoe vortices) ispresent in Head and Bandyopadhyay (1981). Recently,the data obtained from direct and large-eddy simula-tions are increasingly used to generate such movies,which are often part of refereed article submissions andform an integral part of scientific contributions.
Two types of costs are associated with thegeneration of animations: CPU time and storage.While generating 2D contours is reasonably inexpen-sive, the rendering and the encoding of the animationof the evolution of a turbulent field through iso-surfaces, at a frame rate comparable to the timeresolution of the computation, can take longer thanthe simulation itself. For example, one Navier–Stokessolver step for a Large Eddy Simulation (LES) with1.6 million points requires, with the code used in thiswork, 1.83 s on a 2.8 GHz AMD Opteron1; a call tothe Matlab1 isosurface routine for the same flow fieldand on the same machine takes (including renderingtime) 3.2 s for one frame; its cost scales linearly withthe total number of grid points. Storage can alsobecome an important issue. If in a numerical simula-tion only the three velocity components and thepressure field are stored (in double precision), one ofthe largest direct numerical simulations (DNS) com-putations of a boundary layer (by Schlatter and Orlu2010) would require 96 GB for every instantaneousfield. Possible alternatives to mitigate these issues areto decrease the spatial resolution of the data set, theframe rate or the temporal accuracy. The firstapproach leads to undesirable loss in spatial accuracyand unphysical smoothing of the data. The secondapproach can result in a poor kinematic description ofthe coherent motions and, also, in a visually unat-tractive movie. The third is the most commonly usedone, and the one we will be considered here.
For a flow visualisation to be effective in terms ofthe accuracy with which the data are represented theinterval between successive frames must be a fractionof the integral scale of the structure being observed.Taking a turbulent channel flow as an example, at aReynolds number accessible by DNS, the integral timescale of eddies in the buffer layer is Tin ¼ o(y/ut)(where y is the distance from the wall and ut ¼ (tw/r )1/2 is the friction velocity, where tw is the wall-shearstress and r is the fluid density), while in the outerreaches of the logarithmic layer Tout ¼ o(d/Ub) where
d is the channel half-width and Ub is the averageflow velocity. The time-step, which is limited by theCourant-Friedrichs-Levy (CFL) condition in simula-tions of this type, is of the order of Dt ¼ o(Ub/Dx). Fora DNS of channel flow at Ret ¼ 400 (Moser et al.1999), assuming a (maximum) interval between framesof one-tenth of the Tout, the number of time framesrequired to reproduce accurately the structure evolu-tion in one flow-through time, Tfthr ¼ 2pd/Ub, wouldbe 10 Tfthr/Tout. The number of complete flow fields tobe stored would be approximately 63, resulting inapproximately 128 GB of storage.
The need for visual smoothness (the differencebetween successive frames must be small enough not tobe detectable by the human eye) may increase thisrequirement. Zooming on a small region of the flow,for instance, would amplify the relative motion of thestructures, and require that more finely spaced flowrealisations be stored.
CPU and storage issues are both addressed in thework by Jobard et al. (2002) who propose a method foradvecting textures with an assigned 2D flow field.Images are seeded with particles that are then trackedat the pixel level with a hybrid Lagrangian andEulerian scheme. The velocity field is contaminatedby a low percentage of noise in order to compensatefor pixel duplication and spurious flow divergenceeffects. Noise has two distinct roles: it is advected (inparallel with the textures) and re-inserted in theLagrangian coordinate integration step and, just forpost-processing purposes, is used in the blending offrames in order to achieve high temporal correlations,therefore, visual pleasantness of the animations. Theissues with rendering time are not present sinceadvection is carried out at the pixel level. This methodresults in savings of both computational resources(since the rendering is only carried out on the coarselyspaced fields) and storage, however, its effectiveness inturbulent flows is unknown.
In this article, we propose an advection-basedmethod inspired by the work of Jobard et al. (2002).We choose, however, to advect physical quantities (thevariables to be plotted) rather than the pixels. Thus,our approach does not decrease the CPU time requiredby a visualisation, but only the storage required. Wehope, however, that the physical knowledge on thestructure of turbulence that can be used with ourapproach will compensate the higher cost of ourmethod. Our goal is to develop a method that allowsthe generation of visually pleasing animations, usingtime-coarsened data sets. Given a series of flowrealisations that are sufficiently closely spaced (intime) to represent accurately the flow-field evolution,intermediate fields can be reconstructed that result in avisually smooth evolution of the structures. We
264 G. Lee et al.
emphasise again that the coarsening can only becarried to the level that allows the accurate descriptionof the flow physics; the aim of this method is toseparate this requirement from that of visualsmoothness.
Some difficulties may arise when trying to assess theperceptual quality, rather than the physical accuracy,of animations capturing the evolution of turbulentstructures. An appropriate metric is needed to addressthe aforementioned quality issues. Simple point-to-point comparison between a reference (distortion-free)field and its approximate reconstruction can bemisleading as shown in Figure 1. The original frame(a) has been distorted in five different ways (Figure1b–f), yet achieving the same root-mean square (rms)difference (of the 8-bit greyscale field) with respect tothe original image. Images (b) and (c) clearly have ahigher perceived quality than others. The loss ofinformation in the compressed image (d) and theblurring in image (e) diminishes the perceived quality.An average quality score should be given to image (f),which was perturbed with scattered black and whitepixels, but still shows a good structural integrity.Image defects, therefore, have to be weighted based ontheir visibility to the human visual system (HSV) in theglobal quality assessment criteria.
The human eye is highly adapted to extractstructural information from the viewing field and theStructural Similarity Index (SSIM) has been designed
to serve specifically for this purpose (Wang et al. 2004).To improve the relevance of our quality assessment ofthe generated animation, the first step is to focusthe analysis on the 2D bitmap image representation ofthe 3D data, via the SSIM index, rather than thereconstructed 3D data itself.
In the following, we will describe the numericalmethod used to generate the data, and the metrics usedto evaluate image quality and smoothness. We willthen apply the proposed method to turbulent channelflow and discuss the results, and draw concludingremarks.
2. Problem formulation
2.1. Data set description
The available set of instantaneous velocity fields istaken from a large-eddy simulation of a turbulentchannel flow at ReB ¼ UB d/n ¼ 5000 and 11,000 basedon the channel half-width d and the streamwise averagevelocity UB, corresponding to Ret ¼ utd/n ¼ 300 and600, respectively. The filtered equations of conservationof mass and momentum are:
@ui@xi¼ 0 ð1Þ
@uj@tþ @uiuj
@xi¼ � @p
@xj� @tij@xiþ 1
ReB
@2uj@xi @xi
ð2Þ
Figure 1. Comparison of ‘Boat’ images with different types of distortions, all with MSE ¼ 210. (a) Original image (256 6 256resolution at 8 bits/pixel), (b) contrast-stretched image, SSIM ¼ 0.9168, (c) mean-shifted image, MSSIM ¼ 0.9900, (d) JPEG-compressed image, SSIM ¼ 0.6949, (e) blurred image, SSIM ¼ 0.7052, (f) salt–pepper impulsive noise contaminated image,SSIM ¼ 0.7748. Reproduced with permission from Wang et al. (2004).
International Journal of Computational Fluid Dynamics 265
where x1, x2 and x3 (or x, y and z) are, respectively, thestreamwise, wall-normal and spanwise directions, and uithe filtered velocity components in those directions. Theequations are solved in discretised form using a well-validated second-order-accurate finite difference code(Keating et al. 2004), based on a staggered gridarrangement. The code is parallelised using the MessagePassing Interface (MPI) protocol. The sub-grid scalemodel used is the classic Plane-Averaged DynamicEddy-Viscosity model (Germano et al. 1991, Lilly1992). Periodic boundary conditions are applied to allquantities in the streamwise and spanwise direction. Thevelocity field obeys no-slip conditions at the walls.
At ReB ¼ 5000, two-grid resolutions are used, with64664664 and 1286966128 points, respectively, inthe streamwise, normal-to-wall and spanwise directions.Only the finer mesh is used at the higher ReB. The reso-lution in wall units of the fine ReB ¼ 5000 calculation isDx ¼ 14þ, Dz ¼ 7þ and Dymin ¼ 0.7þ; the coarse ReB5000 and ReB ¼ 11,000 calculations share the sameresolution of Dx ¼ 28þ, Dz ¼ 14þ and Dymin ¼ 1.4þ.
The equations of motion were integrated in timeuntil a statistical steady state is reached; statistics arethen accumulated for approximately 10 large-eddyturnover times (LETOTs) d/ut. The mean velocity andReynolds stresses were found to be in good agreementwith the reference data (Moser et al. 1999), withmaximum differences of 6%. Realisations of the flowfield were then stored, spaced by 0.1 d/UB, equal to0.005 LETOTs, or 1:5 n=u2t (viscous time units) atReB ¼ 5000 and 3 n=u2t at ReB ¼ 11,000.
2.2. Advective-based interpolation technique
We intend to reproduce the evolution of turbulentstructures using only a subset of all of the availableclosely spaced realisations with T ¼ 0.1 d/UB. We willrefer generally to f as any scalar field used forvisualisation purposes. Each of these quantities calcu-lated from the ith realisation is indicated with fi ¼f(x,ti), where ti ¼ (i 7 1)T. We will coarsen the set ofrealisations by an (integer) factor n so that, in thecoarsened space, the interval between two consecutiverealisations is tiþn7ti ¼ nT, and the correspondingscalar fields are fi and fiþn. For n ¼ 6, the spacing intime is equal to 1/30th of an LETOT.
One way to reconstruct the evolution of f is anadvection-based interpolation method:
@f@tþ Vt � rf ¼
DtfDt¼ 0 ð3Þ
where Vt is an advection velocity that needs to bespecified (numerical experiments have shown that thebest choice for Vt is the local mean velocity). The basic
physical assumption underlying Equation (3) is thatbetween one realisation and the other the evolutionof f is governed only by mean flow transport,neglecting, among other things, transport by turbulentfluctuations and diffusion.
It is now possible to predict f at time ti 5 t 5tiþn, based on the available information at times ti andtiþn, by advancing Equation (3) in time first forward,from ti to t, then backwards, from tiþn to t, and takinga weighted average of the two solutions:
f�ðx; tÞ ¼ ð1�WiðtÞÞfþ þWiðtÞf� ð4Þ
where
WiðtÞ ¼t� tinT
ð5Þ
is a weighting function. The forward and backwardsolutions of the advection equations
fþ ¼Z t
ti
DtfDt
dtþ fi; f� ¼Z t
tiþn
DtfDt
dtþ fiþn
ð6Þ
can be approximated, to first order, by
fþ ’ fiðx� ðt� tiÞVtÞ ð7Þ
f� ’ fiþnðxþ ðtiþn � tÞVtÞ ð8Þ
where the right-hand-sides of Equations (7) and (8) canbe conveniently computed numerically using tri-linearinterpolation.
2.3. Performance metrics
Given that the focus of this article is on the flowvisualisation, the quality assessment (accuracy andsmoothness evaluation) of the resulting animationsmust be based on the comparison between the originaland reconstructed images (respectively, indicated asunstarred and starred quantities), rather than actual3D data. We will indicate the projection operation ofthe 3D original, f, and reconstructed, f*, fields onto a2D image as follows:
fðx; tÞ ! IðtÞ; f�ðx; tÞ ! I�ðtÞ ð9Þ
where by cursive letters, such as IðtÞ, we will conciselyindicate the collection of red, green and blue (RGB)values for each pixel (three monochromatic images)composing the final colour frame of the animation attime t. The accuracy of the animations will, therefore,be determined by comparing the reconstructed imageI�ðtÞ with the corresponding frame from the original
266 G. Lee et al.
data IðtÞ, that is to say, by evaluating some appro-priate metric
A ¼ AðI�ðtÞ; IðtÞÞ: ð10Þ
The smoothness of the animations, on the otherhand, is a self-referenced quality measurement, and itcan be evaluated by comparing two successive frames,at t and t þ Dt, of the reconstructed animations
S ¼ SðI�ðtÞ; I�ðtþ DtÞÞ: ð11Þ
As mentioned in Section 1, the development ofappropriate quality measurements is critical, since thepixel-to-pixel difference may be misleading. In thiswork, the norms in Equations (10) and (11) areevaluated by a locally weighted and image-averagedversion of the SSIM (Wang et al. 2004). The SSIM wasdeveloped to assess the quality of an image, withrespect to a reference one, based on the degradation ofstructural information. Pixels can exhibit strongcorrelations, especially when spatially proximate.Apart from detecting simple differences between twoimages, such as different levels of luminance andcontrast, the SSIM index compares normalised dis-tribution patterns of pixel intensities in order to extractpurely structural information. This parameter is veryuseful in our case because we want to focus onstructural information change in the image rather thana simple norm of a pixel-to-pixel difference. Thismakes the SSIM index ideal for assessing both theperceived quality and the accuracy of the reproducedanimations of coherent structures.
The SSIM index can be defined by considering twomonochromatic images, X and Y of the same size. Thesymbols represent the arrays of pixels each with itsown luminance intensity
X ¼ fx1;x2; . . . ; xNg; Y ¼ fy1; y2; . . . ; yNg ð12Þ
where N is the total number of pixels in the images.For example, for 8-bit greyscale images, values of xk,yk range from 0 to 255. The statistical (unbiased)estimates of the mean intensity, the signal contrast andthe correlation between the two images X and Y are,respectively
mx ¼1
N
XNk¼1
xk ð13Þ
sx ¼1
N� 1
XNk¼1
xk � mxð Þ2" #1=2
ð14Þ
sxy ¼1
N� 1
XNk¼1
xk � mxð Þ yk � my� �
: ð15Þ
Taking into account the differences in luminance,contrast and image structure (the former measured bythe normalised correlation sxy), we obtain the finalform of the SSIM index (Wang et al. 2004)
SSIMðX ;YÞ ¼2mxmy þ C1
� �2sxy þ C2
� �s2xs2y þ C1
� �m2xm2y þ C2
� � ð16Þ
where C1 and C2 are stabilising constants (to avoiddivisions of zero by zero) taken as C1 ¼ (0.01L)2 andC2 ¼ (0.03L)2, where L is the dynamic range of thepixel values (255 for 8-bit greyscale images). Forimage-quality assessment, it is useful to apply theSSIM index locally rather than globally by adopting awindowing approach across the whole image. Theimage statistics Equations (13), (14) and (15) arelocally estimated using a 11 6 11 circular-symmetricGaussian weighting function instead of simple arith-metic average. This leads to the definition of a meanSSIM index (MSSIM) used to evaluate the overallimage quality (Wang et al. 2004)
MSSIMðX ;YÞ ¼ 1
M
XMj
SSIM fXgj; fYgj� �
ð17Þ
where fXgj and fYgj are the image contents at thelocal j-th window of the reference and the distortedimage, respectively, with Gaussian weighting. Forcolour images, it will be implied that such mean indexis averaged over the three RGB channels, separatelytreated as monochromatic images. The accuracy andsmoothness indices (10) and (11) will, therefore, becalculated as
A ¼MSSIMðI�ðtÞ; IðtÞÞ ð18Þ
S ¼MSSIMðI�ðtÞ; I�ðtþ DtÞÞ ð19Þ
3. Results
We will now perform a quantitative assessment of theaccuracy and smoothness of the reconstructed anima-tions; two additional corrections are found to benecessary to improve the animation quality; they willbe presented and their effects are discussed. The effectsof changes in Reynolds number, spatial resolution andvisualisation methods are discussed as well as resultsfrom a brief cost analysis.
In Figure 2, three animation frames reconstructedwith the advection-based technique (Equation (7)) areshown for different values of the spacing n for the fineReB ¼ 5000 calculation. The frames are all at a time, tf,at the midpoint of the coarsened interval, whichextends from the tf7n/2 to tfþn/2. We consider
International Journal of Computational Fluid Dynamics 267
isosurfaces of the second invariant of the velocitygradient tensor,1
Q ¼ � 1
2
@ui@xj
@uj@xi
: ð20Þ
The sensitivity of the reconstruction to the realisa-tion spacing n is highlighted by the pixel-to-pixel imagedifference between the reconstructed and originalframes, which grows in complexity as n increases(more detail of the original frame is loss). The accuracyindex decreases from A ¼ 0.90 to A ¼ 0.70 for nincreasing from 2 to 20. Major differences are evidentmostly around the edges of structures distorted byturbulent straining (which is not included in themodel). The prediction of the location of larger
structures remains rather accurate even for large valuesof n, as they are less sensitive to turbulent diffusion,and are dissipated at a slower rate, which results inqualitative agreement between the reconstructed andreference frame. For all but the coarsest (n ¼ 20)reconstruction, however, the general agreement be-tween the fields is reasonable, and the first three imagesconvey very similar information. For visualisationpurposes, an accuracy index A 4 0.8 appears to beacceptable.
The movie accuracy and smoothness indices asfunction of time are shown in Figure 3. The accuracyindex returns to unity every n frames, when an actualflow realisation is used, and there is no differencebetween I and I�. The accuracy decreases forreconstructed frames that are more distant (in time)
Figure 2. Reconstructed frame (left) and corresponding pixel-to-pixel difference (right) with respect to original frame for highlyresolved ReB ¼ 5000 case and Q ¼ 3:0U2
B=d2. Top: reference image. Reconstructed image for (a) n ¼ 2 (A ¼ 0.90), (b) n ¼ 4
(A ¼ 0.85), (c) n ¼ 6 (A ¼ 0.80) (d) n ¼ 20 (A ¼ 0.70).
268 G. Lee et al.
from the reference ones, reaching the lowest value atthe centre of the time interval tiþn 7 ti. The overallaccuracy decreases with n as expected. As mentionedabove, for n � 6, the reconstructed frame contains atleast 80% of the information contained in the originalone. The regular and abrupt variations in the accuracyindex have a striking effect to an observer of thecorrespondent animation, as movie smoothness be-comes an issue. Note that, for n ¼ 6, the structuresvisualised here (which are typically located at yþ ¼507150) move by approximately one-tenth of thechannel length (or approximately 10–13 grid points)between two actual realisation. If the advectiveinterpolation were not used, the smoothness of themovie would be appreciably decreased, to a value ofapproximately S ¼ 0.68, 0.660, 0.655, 0.65 for n ¼ 2,4, 6 and 20, respectively. This results in an unpleasantmovie where the under-resolved evolution of structurescannot be tracked by the human eye. Also note that,even when a very short interval between frames is used,smoothness levels very close to 1 cannot be achieved.If, for instance, we consider two realisations separatedby one-tenth of the separation used here (i.e. 3 6 1073
LETOTs), the smoothness index remains at 0.88. This
rapid decrease is due to many factors, such as theinterpolation required to calculate velocity gradients atthe cell centres from their staggered locations, and thedetails of the rendering algorithm, which may beaffected by small changes in the underlying field. Froma visual inspection of the movies, the smoothnessindices greater than approximately 0.7 are desirable.
For given n, at times coinciding with one of thereference snapshots (at time ti or tiþn), the interpola-tion method (Equation 4) returns the original, non-interpolated frame. This results in an abrupt increaseof detail in the animation, given the presence of smallstructures being rendered. To remove this problem, weintroduced a frame-substitution method in which thetwo interpolated fields f*(x,ti 7 Dt) and f*(x,ti þDt) are used as reference frames to recalculate f*(x,ti)via Equation (4) resulting in f*(x,ti) 6¼ fi(x,ti). Thetotal variation of the accuracy history is, therefore,reduced (Figure 4).
The periodic rise in accuracy and smoothnessbetween the reference realisations can, however, stillbe seen in Figure 4, which may be appreciable by ahuman observer in the animations, especially for largen: the evolution of the isosurfaces appears to change
Figure 3. Reconstructed-movie accuracy (a) and smoothness (b). Highly resolved ReB ¼ 5000 case with isovalue ofQ ¼ 3:0U2
B=d2, for n ¼ 2 (——), n ¼ 4 (– – –), n ¼ 6 (. . . . . . ) and n ¼ 20 (– � –).
Figure 4. Reconstructed-movie accuracy (a) and smoothness (b). Highly resolved ReB ¼ 5000 case with isovalue ofQ ¼ 3:0U2
B=d2, for n ¼ 6 without frame-substitution and e ¼ 0 (——), with frame-substitution and e ¼ 0 (– – –) and with frame-
substitution and e ¼ 0.05 (. . . . . . ).
International Journal of Computational Fluid Dynamics 269
with a regular, periodic, pattern between the tworeference realisations. To compensate for this un-wanted effect, a stochastic component, similar to thenoise-blending technique devised by Jobard et al.(2002), was added to the weighting function
W�i ðtÞ ¼t�tinT þ eNð0; 1Þ
1þ eð21Þ
where Nð0; 1Þ is a normally distributed Gaussianvariable and e ¼ 0.050. This value was chosen as theoptimal value, since this noise amplitude level wasstrong enough to eliminate the spurious periodicity ofvariations in smoothness and accuracy but not to affectsignificantly the overall accuracy of the animation(Figure 4a). An observer of the correspondent anima-tions can appreciate the absence of unpleasant periodicvariations in the movie quality as well as the(inevitable) slight deterioration of the overall accuracydue to the introduction of noise. Determining theoptimal value for e results, once again, in a trade-offbetween accuracy and visual pleasantness of theanimation. Overall, the introduction of noise can beconsidered as a second-order adjustment with respectto the frame-substitution method.
It is to be expected that the reconstruction methodproposed here will depend greatly on the variable beingplotted, and on the resolution. Turbulent quantitiesthat are representative of the large-scale motions willbe represented more accurately by the simple advec-tion-based model. Figure 5 shows contours of u0 and n0
fluctuations and isosurfaces of Q ¼ 2 and Q ¼ 3 for
the same flow realisation. First, we observe the largerscale of the u0 fluctuations, compared to n0. In Figure5(c) and (d), on the other hand, we note how for lowerlevels of Q more eddies are observed, especially in thebuffer layer. The proposed model cannot be expectedto be as accurate when reconstructing near-wallstructures, as it is for outer-layer eddies; similarly,the smaller eddies present in the contours of n0 cannotbe reconstructed as well as the larger streaky struc-tures. A quantitative assessment of the sensitivity ofthe proposed method to the structure size is shown inFigures 6 and 7, in which we see that a reducedstructure size has a beneficial effect on both theaccuracy (Figure 6a) and the smoothness (Figure 7a).The same test has been repeated for pressure iso-surfaces (Figures 6b and 7b), which highlight biggerstructures. In this case, increasing the threshold alsoresults in a sensible reduction of the size of thevisualised structures. Also note that, in the case of Qisosurfaces, the proposed interpolation method willnaturally decrease the population of visualised struc-tures (Figure 2); therefore, in order to have areconstructed movie with a similar number of visua-lised turbulent structures, a lower level of the thresholdneed to be chosen with respect to the original movie.All of the considerations made above remain un-changed when introducing noise in the weightingfunction (Equation 21).
Other parameters that affect the quality of theproposed reconstruction are the grid resolution andReynolds number that directly determine the visualisedstructure size. The reconstruction of a flow animation
Figure 5. Highly resolved ReB ¼ 5000 case; isosurfaces of Q ¼ 3:0U2B=d
2 (a) and Q ¼ 2:0U2B=d
2 (b), contours of streamwise (c)and wall-normal (d) fluctuating velocity.
270 G. Lee et al.
Figure 7. Comparison of smoothness for Q-isosurfaces (a) and p0 -isosurfaces (b) of advection-based interpolated moviefor e ¼ 0 and n ¼ 4 and frame replacement active. In (a) Q ¼ 2:0U2
B=d2 (——), Q ¼ 3:0U2
B=d2 (– – –) and Q ¼ 4:0U2
B=d2
(. . . . . . ); in (b) p0 ¼ �0:010rU2B (——), p0 ¼ �0:015 rU2
B (– – –) and p0 ¼ �0:020rU2B (. . . . . . ).
Figure 6. Comparison of accuracy for Q-isosurfaces (a) and p0 -isosurfaces (b) of advection-based interpolated movie for e ¼ 0and n ¼ 4 and frame replacement active. In (a) Q ¼ 2:0U2
B=d2 (——), Q ¼ 3:0U2
B=d2 (– – –) and Q ¼ 4:0U2
B=d2 (. . . . . . ); in (b)
p0 ¼ �0:010 rU2B (——), p0 ¼ �0:015 rU2
B (– – –) and p0 ¼ �0:020rU2B (. . . . . . ).
Figure 8. Reconstructed frame (left) and corresponding pixel-to-pixel difference with respect to original frame (right) showingeffects of Reynolds number on image accuracy. Low resolution ReB ¼ 5000 calculation (a) A ¼ 0.842, n ¼ 4 for Q ¼ 1:0U2
B=d2
and e ¼ 0; ReB ¼ 11,000, A ¼ 0.845, n ¼ 2 (b) for Q ¼ 4:0U2B=d
2 and e ¼ 0.
International Journal of Computational Fluid Dynamics 271
at higher Reynolds numbers, or with higher spatialresolution, is more challenging since the evolution of awider range of structure sizes must be predicted all atonce. For example, in order for the ReB ¼ 11,000animation to have an accuracy comparable to theReB ¼ 5000 one (at the same resolution in wall units),the distance between the realisations must be reducedand/or, in case of Q-isosurfaces, the threshold for thehigher Reynolds number data has to be increased(reducing the population of structures). For example,in order for a reconstructed frame from the higherReynolds number case to retain the same accuracy asthe lower Reynolds number counterpart, the value of nhad to be halved and the value of Q increased from 1to 4 to reduce the population of structures (Figure 8).This shows that the method performs better with largerresolved eddies in the flow that, also, evolve overlonger time scales, therefore, exhibit more predictablekinematics.
Figure 9 quantifies the aforementioned effectsallowing an assessment of Reynolds number effects(for a fixed resolution) and resolution effects (for afixed Reynolds number). Increasing the grid resolutionresults in a decrease in the accuracy (Figure 9a), whichis expected because the domain becomes more denselypopulated. Increasing the Reynolds number, keepingthe resolution (in wall units) fixed, causes an evenlarger drop in accuracy. In both cases, the visualisedresolved structures are much finer and a high correla-tion between the reconstructed and reference images ismore difficult to obtain. The method fails to correctlyreproduce both the actual and perceived evolution ofthe smaller structures. The smoothness index exhibitsthe same qualitative dependency from Reynoldsnumber and resolution (Figure 9b). Lower values ofsmoothness for the ReB ¼ 11,000 case (or highlyresolved ReB ¼ 5000 case) are related to the higherdensity of structures that will inevitably decrease thecorrelation between two consecutive frames. Lowervalues of smoothness are consistent with the highertotal variation of the accuracy time history.
The distance from the wall is also an importantfactor affecting the visualised structure size andcharacteristic time scale, not only for the isosurfacesbut especially for contour plots. Accuracy can drop by7% from the channel’s centreline to the rms peaklocation for u0 and n0 animations (not shown). This isdue to the error made when approximating thestructure transporting velocity Vt with local averageone in Equation (3) in the regions of the flow where thevelocity variance is higher. Also, visualising smallerstructures, with respect to u0 contours, such ascontours of n0 or u0 n0 can cause accuracy to dropby 2 and 9%, respectively.
3.1. Cost analysis
The main advantage of the proposed frame-recon-struction technique is to reduce the data storagerequirements while preserving, within acceptable
Figure 9. Effect of Reynolds number and grid resolution on accuracy (a) and smoothness (b). Coarsely resolved ReB ¼ 5000(. . . . . . ), finely resolved ReB ¼ 5000 (– – –), low resolution ReB ¼ 11,000 (——) for e ¼ 0.0, n ¼ 4 and Q ¼ 3:0U2
B=d2.
Figure 10. Movie generation speed-up for coarsely resolvedReB ¼ 5000 (– – –) and finely resolved ReB ¼ 5000 (——)cases, Q ¼ 1:0U2
B=d2 (with circles), Q ¼ 3:0U2
B=d2 (without
circles), for frame substitution and stochastic weightingwith e ¼ 0.05. Speed-up is defined as the overall timenecessary to generate a non-interpolated movie over thetime required to generate the same movie by interpolatingevery n frames.
272 G. Lee et al.
levels, the visual pleasantness and accuracy of thegenerated animations. However, while reductions instorage requirements, in the proposed technique,increase linearly with the snapshot spacing n, the totalcomputational cost may increase or decrease depend-ing on the data size, the structure eduction method andother tuneable parameters of the technique.
The total movie-generation time is primarily takenup by the advection-based interpolation, image render-ing and I/O activity. Figure 10 shows the speed-up ingenerating Q-isosurface animations for the completeadvection-based frame-reconstruction technique, in-cluding noise and frame substitution. The latterintroduces a moderate computational overhead (notshown), vanishing with n, since one extra interpolationstep is needed every n frames, for visual pleasantnesspurposes. High values of Q yield animations with fewerisosurfaces and faster rendering times. In this case, thetotal movie generation time is taken up by theadvection-based interpolation time which increaseswith n, causing an overall loss in performance (speed-up is lower than 1). However, the computationaloverhead of the reconstruction method stays below20% for n 4 3 for the fine-grid case, being the onethat benefits the most from the reduced I/O activity.The generation time for the coarse data set animationat high Q is also limited by the 3D interpolation,having very little I/O activity, resulting in a significantperformance loss. In this case, however, very fewstructures are visualised.
On the other hand, low values of Q result inrendering-intensive movies (flow field is highly popu-lated with structures) which benefit from the methodprimarily due to the smoothing of the Q-field, causedby the interpolation, which decreases the number ofvisualised structures, and, therefore, the renderingtime. Speed-up finally becomes greater than onewhen I/O activity becomes important for larger datasets.
In all cases, the movie-generation speed-up in-creases with the data set size suggesting that theproposed method can be cost-effective for large-scalesimulations where I/O time becomes the limiting factorfor movie-generation time.
4. Conclusions
We have implemented and tested a simple technique topredict the evolution of coherent structures in betweentwo given realisations of a turbulent flow. The presentwork brings together, for the first time, framereconstruction methods used in texture advectiontechniques (extending them to 3D instantaneousturbulent flow fields) and image quality metrics, basedon the receptivity of the HSV, into one simple
visualisation technique designed with the aid of basicknowledge of turbulent flows. The overall objectivewas to generate high-quality animations of turbulentflows at a reduced cost especially for large scalecomputations.
The structures were identified as isosurfaces ofpositive values of Q, pressure isosurfaces and withflooded contour plots of velocity fluctuations. Theprediction method used is a modified linear interpola-tion technique based on a mean velocity advectionmodel. The accuracy of the resulting animations isassessed via the SSIM index by direct comparison withthe original data set. The degree of similarity ofadjacent frames was used as a quantitative measure ofthe perceived smoothness of the resulting animations.A frame-substitution method was implemented toavoid the abrupt transition between an interpolatedand non-interpolated frame, which compromises thesmoothness of the visualisation. The latter can beincreased by increasing the spacing between referencerealisations, n. This results, however, in an unrealisticstreamwise stretching of the structures occurringperiodically (with period n) in the animations. Torectify this, the weighting function was perturbed withrandom noise. Both adjustments result in a decrease ofaccuracy which is, on the other hand, beneficialtowards obtaining higher level of pleasantness of theanimations. The required amplitude of the randomnoise perturbation to the linear weighting was found tobe acceptable within the range of e ¼ 0.01 7 0.1.
It was found that with the proposed method,visualising fewer and bigger structures is beneficialtowards the overall quality of the animations. Forexample, at higher Reynolds numbers, where thestructures being visualised are finer, n needs to bemuch smaller in order to retain the same accuracy as alower Reynolds number case. Changing the thresholdvalue for Q and p0 isosurfaces has the same effect onthe resulting animations. The contours of velocityfluctuations achieve, in general, higher values ofaccuracy than isosurfaces of pressure or Q. The formerare, however, sensitive to the distance from the wall.Reconstructed animations in the buffer layer havelower accuracy given the higher rms of the instanta-neous transporting velocity than in the centreline.Finally, the introduction of noise is effective inreducing the periodic variations of accuracy andsmoothness which may still be present with theframe-substitution method. However, its effects areof secondary importance and do not affect significantlythe overall performance of the proposed method.
The primary limitations of the adopted reconstruc-tion method lie in its sensitivity to the visualisedstructure size. Tracking the evolution of smallerstructures, either due to high-Reynolds-number data
International Journal of Computational Fluid Dynamics 273
or higher resolution, requires finer spacing in timebetween the reference snapshots, reducing the cost-effectiveness of the method. Also, testing is required inmore complex flows, with 2D and 3D mean velocityfields, where artificial diffusion effects are expected tobe stronger. However, even in simple flows, such as theone investigated, due to the inherent inhomogeneity inthe wall-normal direction, the accuracy of the recon-struction of the 3D Q field, for example, will vary withy. Therefore, improvement, such as making the noisyweighting function dependent of the local intensity ofthe turbulent activity is one option worth investigating.Finally, we have shown that the proposed technique isbeneficial both in memory and in computationalrequirements in the case of rendering-intensive moviesprocessed from large data sets. A computationaloverhead is, however, present when the data set size(I/O activity) is not a limiting factor.
Acknowledgements
The authors thank the High Performance Computing VirtualLaboratory (HPCVL), Queen’s University site, for thecomputational support. GL acknowledges the support ofNSERC under the USRA program. UP acknowledges thesupport of the Canada Research Chair program, and ofNSERC, under the Discovery Grant program.
Note
1. From now on, all quantities reported are normalised bythe constant density r, Ub and d, unless otherwise stated.
References
Brown, G.L. and Roshko, A., 1974. On density effects andlarge structure in turbulent mixing layers. Journal ofFluid Mechanics, 64 (04), 775–816.
Dubief, Y. and Delcayre, F., 2000. On coherent vortexidentification in turbulence. Journal of Turbulence, 1, 1–22.
Elsinga, G., et al., 2006. Tomographic particle imagevelocimetry. Experiments in Fluids, 41, 933–947. doi:10.1007/s00348-006-0212-z.
Germano, M., et al., 1991. A dynamic subgrid-scale eddyviscosity model. Physics of Fluids A, 3, 1760–1765.
Haller, G., 2005. An objective definition of a vortex. Journalof Fluid Mechanics, 525, 1–26.
Head,M.R., 1969. Flow visualization in the turbulent boundarylayer (16 mm cine film). Cambridge: Cambridge Uni-versity Engineering Department, Technical report.
Head, M.R. and Bandyopadhyay, P., 1981. New aspects ofturbulent boundary-layer structure. Journal of FluidMechanics, 107, 297–338.
Head, M.R. and Graham, J.M.R., 1969. Flow visualization oftransition (16 mm cine film). Cambridge: CambridgeUniversity Engineering Department, Technical report.
Jobard, B., Erlebacher, G., and Hussaini, M.Y., 2002.Lagrangian-Eulerian advection of noise and dye texturesfor unsteady flow visualization. IEEE Transactions onVisualization and Computer Graphics, 8 (3), 211–222.
Keating, A., et al., 2004. Large-eddy simulation of heattransfer downstream of a backward-facing step. Journalof Turbulence, 5 (20), 1–27.
Kida, S. and Miura, H., 1998. Swirl condition in low-pressure vortices. Journal of Physical Society of Japan, 67(7), 2166–2169.
Kline, S.J., et al., 1967. The structure of turbulent boundarylayers. Journal of Fluid Mechanics, 30 (4), 741–773.
Liepmann, H., 1979. The rise and fall of ideas in turbulence.American Scientist, 67, 221–228.
Lilly, D.K., 1992. A proposed modification of the Germanosubgrid-scale closure method. Physics of Fluids A, 4, 633–635.
Moser, R.D., Kim, J., and Mansour, N.N., 1999. Directnumerical simulation of turbulent channel flow up toRet ¼ 590. Physics of Fluids, 11 (4), 943–945.
Robinson, S.K., 1991. Coherent motions in the turbulentboundary layer. Annual Review of Fluid Mechanics, 23(1), 601–639.
Robinson, S.K., Kline, S.J., and Spalart, P.R., 1989. A reviewof quasi-coherent structures in a numerically simulatedturbulent boundary layer. Washington, D.C.: NASA, TM102191.
Schlatter, P. and Orlu, R., 2010. Assessment of directnumerical simulation data of turbulent boundary layers.Journal of Fluid Mechanics, 659, 116–126.
Wang, Z., et al., 2004. Image quality assessment: from errormeasurement to structural similarity. IEEE Transactionson Image Processing, 13 (4), 600–612.
274 G. Lee et al.
