IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. X, NO. Y, MONTH 2007 1

Visualization of Vorticity and Vortices inWall-Bounded Turbulent Flows

Anders Helgeland, Member, IEEE, B. Anders Pettersson Reif, Øyvind Andreassen, and Carl Erik Wasberg

Abstract— This study was initiated by the scientifically inter-esting prospect of applying advanced visualization techniques togain further insight into various spatio-temporal characteristicsof turbulent flows. The ability to study complex kinematical anddynamical features of turbulence provides means of extractingthe underlying physics of turbulent fluid motion. The objectiveis to analyze the use of a vorticity field line approach to studynumerically generated incompressible turbulent flows. In orderto study the vorticity field, we present a field line animationtechnique which uses a specialized particle advection and seed-ing strategy. Efficient analysis is achieved by decoupling therendering stage from the preceding stages of the visualizationmethod. This allows interactive exploration of multiple fieldssimultaneously, which sets the stage for a more complete analysisof the flow field. Multifield visualizations are obtained using aflexible volume rendering framework which is presented in thispaper. Vorticity field lines have been employed as indicators toprovide a means to identify “ejection” and “sweep” regions; twoparticularly important spatio-temporal events in wall-boundedturbulent flows. Their relation to the rate of turbulent kineticenergy production and viscous dissipation, respectively, have beenidentified.

Index Terms— 3D vector field visualization, unsteady flowvisualization, time-varying volume data, features in volume datasets, multifield visualization, fluid dynamics, turbulence

I. INTRODUCTION

Turbulent flows, bounded by impermeable surfaces, proba-bly constitute the most frequently occurring flow configurationin practice. Examples are external boundary layers on cars,ships and aeroplanes, and internal flows in turbines, pumps andpipes, to only mention a few. The large-scale structure of tur-bulent flows near rigid boundaries is affected in several ways:by strong mean shear; by kinematic blocking of turbulentfluctuations; by fluctuating pressure reflections; and by movinginternal shear layers as produced by the large-scale structuresthemselves. Elongated streamwise vortices are formed withlength scales comparable to, or larger than, the boundary layerthickness. These structures vigorously mix momentum – highmomentum fluid in the outer part is transported toward thesurface and, conversely, low momentum fluid is transportedfrom the near-wall region toward the outer part of the bound-ary layer. The complexity of these structures, and many ofthe characteristic phenomena associated with them, are stillnot well understood. Advances of our understanding of thephysics of fluid turbulence is however of crucial scientific and

A. Helgeland is with the University of Oslo and the University GraduateCenter, Kjeller, Norway. E-mail: andershe@ifi.uio.no

B. A. Pettersson Reif, Ø. Andreassen and C. E. Wasberg is with theNorwegian Defence Research Establishment, FFI, Kjeller, Norway. E-mail:{Bjorn.Reif,Oyvind.Andreassen,Carl-Erik.Wasberg}@ffi.no

practical importance. An excellent introduction to the theoryof turbulent shear flows can be found in the text by Townsend[1].

Carefully conducted direct numerical simulations (DNS) en-able a deterministic approach to study the seemingly stochasticturbulent motion – DNS provides a pointwise solution tothe Navier-Stokes equations both in time and space. Thisstudy was motivated by the interesting prospect of applyingadvanced visualization techniques to highly accurate DNS datain order to gain further insight into various spatio-temporalcharacteristics of turbulent flows. The study is based on DNSof fully developed plane turbulent channel flow using anadvanced high-order spectral element code.

The primary objective is to employ a vorticity field line ap-proach to study the spatio-temporal behavior of wall-boundedincompressible turbulent flows. The very nature of turbulentmotion makes the utilization of flow vorticity the most con-venient approach to characterize the flow field. The premiseof the work is that the kinematical and dynamical evolutionof an incompressible fluid can equivalently be expressed interms of the velocity and the vorticity fields. As such, noinformation about the numerically simulated flow field is lostby considering the latter. The cascade of turbulent kineticenergy is characterized by vorticity processes like strainingand connection/reconnection, all of which can be quantifiedpointwise in a DNS field. Since vorticity is linked to flowtopology, the time evolution of the vorticity field expresseschanges in flow topology. Such processes can be identifiedthrough visualization. Of particular interest in the presentpaper, is the study of spatio-temporal kinematical “ejection”and “sweep” events and their relation to the rate of productionand viscous dissipation of turbulent kinetic energy.

A notable feature of the vorticity field in an inviscidfluid, is that the corresponding field lines are equivalent tomaterial lines of the fluid. Material lines, formed by injectingtracers, are used to visualize the fluid motion in physicalexperimental settings with the objective to reveal dominatingstructures/patterns of the flow field. Material lines “per se”do not reveal dynamically important features as vorticity fieldlines are able to. There exist, however, no flow visualizationtechniques in physical fluid experiments that capture the 3Dvorticity field. In numerical experiments, on the other hand,the vorticity field is easy to derive. Nevertheless, viscousdissipation is always present in turbulent flows, and one of thegoals of the present study is to provide a quantitative measureof the error of interpreting the vorticity field lines as materiallines.

In this paper, we discuss visualization and animation of

2 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. X, NO. Y, MONTH 2007

x

y

z

U(z)

Fig. 1. The computational domain and the corresponding sub-domain usedfor visualization purposes. Lower wall is colored blue.

the vorticity field through field lines, and through “fuzzyobjects” referred to as “vortices”. The depiction of the lattercan be based on a number of existing criteria that includes thevorticity field – although no universally accepted definition ofa vortex in turbulence presently exists [2]. The vorticity fieldline animation is done by injecting a collection of particles intothe domain. These particles are then tracked along their pathlines. At each time step, the particles are used as seed pointsto generate vorticity field lines. In this way, the animationshows the advection of particles, while each frame in theanimation shows the instantaneous vorticity field. To improverendering performance, we decouple the rendering stage fromthe preceding stages of the visualization method. This allowsinteractive exploration of multiple fields simultaneously, whichsets the stage for a more complete analysis of the flow field.To facilitate the analysis of multiple data fields, we presenta flexible volume rendering framework which is capable ofproducing effective visualizations of multiple volumetricalfields.

II. TURBULENT CHANNEL FLOW AND DIRECT NUMERICALSIMULATIONS

The present study considers fully developed turbulent flowin a plane straight channel bounded by infinite plates. Fig. 1displays a schematic of the channel flow configuration. Thisparticular configuration constitutes a well defined case that issuited for the purpose of this study. Although the mean flowfield (i.e. the ensemble averaged flow field) is unidirectionaland steady, the turbulence field is highly three-dimensionaland time-varying, and as such extremely complex. The presentDNS has been performed at frictional Reynolds number Reτ ≡uτ h/ν = 180, where uτ ≡

√

τwall/ρ , h, and ν ≡ µ/ρ denotethe friction velocity, channel half height, and kinematic viscos-ity of the fluid, respectively. µ and ρ are the dynamic viscosityand fluid density, whereas τwall ≡ µ(dU/dz)wall represents thewall shear stress. Physical parameters throughout the paperhave been made nondimensional (referred to as ’plus units’ or’viscous units’) by using the velocity scale uτ and kinematicviscosity ν .

Mean flow kinetic energy is transferred to turbulent kineticenergy by the action of mean flow gradients at the largestscales of motion. The energy is on average transferred fromthe largest turbulent scale to the smallest scales through theturbulent cascade process, where it subsequently is dissipatedinto heat by the action of viscosity. Since the channel flowfield is fully developed, i.e. statistically steady, it reaches a

statistical equilibrium between the rate of production and rateof viscous dissipation of turbulent kinetic energy, and is assuch also independent of inflow conditions.

The direct numerical simulation was conducted using aspectral element method, where the computational domainis divided into elements, and the solution is representedby high order polynomials on each element. For details onthe implementation, see [3], [4]. The simulation was carriedout on a computational domain of size (L+

x , L+y , L+

z ) =(1440, 720, 360) viscous units (Fig. 1). The total numberof nodal points equals 128 × 128 × 129 in the streamwise(x), spanwise (y), and wall-normal (z) directions, respectively.The solution was advanced in time with a time-step corre-sponding to 0.18 viscous time-units (t+ = ν/u2

τ ), and with50% polynomial filtering [5] on each time-step. In order toensure sufficiently converged statistics the flow was evolvedapproximately 54 flow-through times along the computationalbox. This is important in order to achieve accurate fluctuatingvelocity and pressure fields used in the visualization (these areobtained by subtracting the mean flow from the instantaneousfield). No-slip boundary conditions are applied at the solidwalls, whereas periodicity are imposed in the streamwise (x)and spanwise (y) directions.

III. VORTICITY AND VORTICES IN TURBULENT FLOWS

In the early 1990’s, numerical realizations of turbulentflows at low to moderate Reynolds numbers revealed thatvorticity was in part concentrated in “tubes” with characteristicthickness in the order of a few dissipation scales [6], [7].Although these “tube-like” structures often are referred to asvortices, there has not yet been given any universally accepteddefinition of a “vortex”. Several authors have proposed anumber of different criteria that can be used to identify andclassify a vortex within complex fluid topologies, cf. e.g. [8],[9]. The concept of tube-like vortices and the recognition oftheir dynamical importance in turbulent flows has spawned anumber of research efforts with the objective to develop sim-pler models of near-wall turbulent motion. In the mid 1990’s,Banks and Singer [10], [11] (e.g.) proposed a technique forvortex tube identification, representation, and reconstructionin turbulent flows through a predictor-corrector techniquewhere vortices were visualized and used as an exploratorytool. In [12], a predictor-corrector scheme is used to studyvortex structures in a wall-bounded flow during the transitionphase from laminar to turbulent motion. A comprehensivecomparison between various vortex identification schemes canbe found for instance in [2] and [13], [14].

As alluded to above, the majority of previous studiesfocused on the identification of vortex topology as compacttube-like objects occupying local regions. Due to the natureof such methods, vortex identification schemes are, in general,not able to represent the kinematic behavior related to internalvelocity shear. Velocity shear spawn regions of vorticity thatare characterized by sheet-like structures. These structures arefor some applications, such as wall-bounded flows, dynami-cally very important. For example in the immediate vicinity ofan impermeable wall, the kinematic blocking of wall-normal

HELGELAND ET AL.: VISUALIZATION OF VORTICITY AND VORTICES IN WALL-BOUNDED TURBULENT FLOWS 3

velocity fluctuations causes the flow field to be more sheet-likethan tube-like, even if the flow is fully turbulent. It is withinthis narrow region turbulence is produced and subsequentlydissipated into heat by the action of the fluid viscosity; inevery respect this is a fundamentally crucial portion of wall-bounded turbulent flows. It is therefore highly desirable to notonly consider tube-like vortices but also to characterize thetopology of the vorticity field in general. This requirementcalls specifically on the ability to visualize vector fields, andin particular vorticity field lines. While vorticity field lineswere visualized in [15] to reveal the flow topology in anengineering application, the present study is chiefly concernedwith developing and assessing faithful methodologies that canassist our interpretation of the physics of turbulent flows.

In this paper, we present an animation technique that canbe used to visualize temporally evolving vector fields. Wediscuss the quantification of the error of interpreting vorticityfield lines as material lines. This is of interest when evolvingvorticity lines are used to infer the physical characteristics oftemporally evolving turbulent flow fields. It should be noted,however, that irrespectively of the error, the presented fieldline animation technique will provide qualitative informationabout the flow evolution.

The vorticity transport equation as derived from the incom-pressible Navier-Stokes equations states that the local timechange of vorticity at a fixed point in space can be written

∂ωωω∂ t = ∇× (u×ωωω)+ν∇2ωωω, (1)

where ωωω ≡ ∇× u is the vorticity vector and u denotes thevelocity vector. Let us now consider the flux of the vorticityfield integrated over a moving contour attached to a fluidelement. Let dA be an element of the surface outlined bythe contour and let ds be a line element along the contour.Then the total rate of change in the vorticity flux (φ ), can bewritten using Stokes’ theorem as

dφdt =

∫∫ ∂ωωω∂ t ·dA+

∮

(u×ds) ·ωωω

=∫∫

(

∂ωωω∂ t −∇× (u×ωωω)

)

·dA

= ν∫∫

∇2ωωω ·dA, (2)

which vanishes if viscous effects can be neglected; the flux ofvorticity through a contour that follows the fluid particles isthen constant.

Taking into consideration that for incompressible fluids thevelocity field is solenoidal i.e. ∂uk/∂xk = ∂u/∂x + ∂v/∂y +∂w/∂ z = 0, equation (1) can be rewritten as

dωidt = si jω j +ν

∂ 2ωi∂xk∂xk

, (3)

where si j ≡12 (∂ui/∂x j + ∂u j/∂xi) denotes the components

of the rate-of-strain tensor. Mathematically, in order for thevorticity field lines to be perfectly advected by the fluid, thedynamic influence of the viscous term (ν∂ 2ωi/∂xk∂xk) inequation (3) must vanish. If the viscous term is small comparedto the strain term, the idea of a perfectly advected field line is

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 20 40 60 80 100 120 140 160 180z+

Rate of strain termViscous term

Fig. 2. Instantaneous distribution over horizontally averages planes (xy) ofthe terms in equation (3) across half the channel (nondimensionalized by uτand ν).

at best approximative. This constraint seems to be fulfilled inthe present case only within a quite narrow region close to thewall (5 / z+ / 60). This is demonstrated in Fig. 2, wherethe norm of the averaged rate-of-strain term (〈‖si jω j‖〉xy)is compared with the norm of the average viscous term(〈‖ν∂ 2ωi/∂xk∂xk)‖〉xy) plotted against the distance from theboundary. Here, <>xy means the average is calculated overan xy plane parallel to the bounding wall. The no-slip bound-ary condition, u = 0, implies that the rate-of-strain sourceof vorticity si jω j vanishes at the boundaries. Although theresults shown in Fig. 2 would suggest approximative match ofmaterial lines and vorticity field lines within the narrow regionclose to the wall, the analysis presented in section VII showsthat this picture is too simple and that the vorticity field linesand material lines exhibit significant spatial separation after arelatively short time interval.

IV. VORTICITY FIELD ANIMATION

A technique for animating three-dimensional time-dependent vector fields was developed in [16]. The methodis a hybrid solution based on both the use of path lines andfield lines, similar to the ideas used in DLIC [17] and UFAC[18]. In principle, the field lines can be based on a differentvector field than the velocity field, but only the velocity fieldwas studied in [16]. In this paper, we develop the previouswork further to animate the field lines of the vorticity field incombination with the underlying flow field.

The vorticity field line animation is done by injecting acollection of “evenly distributed particles” throughout thephysical domain (Section V-A). These particles are thentracked along the time-dependent velocity field by calculatingtheir path lines. At each time step, the particles are used asseed points to generate field lines using the vorticity field. Inthis way, the animation shows the advection of particles, whileeach frame in the animation shows the instantaneous vorticityfield. To obtain a global representation of the flow, particlesare injected and removed in areas with too low or too highdensity, respectively.

The particle placement is also dependent on a seconddistribution criterion, which has the effect of producing fieldlines that are approximately evenly spaced at each time step.Such a solution gives still images (volumes) that are very easy

4 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. X, NO. Y, MONTH 2007

Inflow Treatment

Particle Advection

Initial Seed Point Placement

Output TextureMaintaining Particle Density

Next Time Step

Fig. 3. Flowchart of the particle advection algorithm.

to interpret and is nearly optimal for diminishing clutteringissues.

The time-dependent vector field visualization algorithm isdivided into three parts:

1) Particle Advection – pre-process (Section V),2) Field Line Generation – pre-process (Section VI),3) Volume Rendering (Section VIII).To improve rendering performance, the rendering stage is

decoupled from the rest of the visualization pipeline. Hence,the final rendering of the flow data will achieve an increasein performance [16] compared to the combined advection andvolume rendering algorithms. A compact description of thealgorithm is given in the next few sections. More details canbe found in [16].

V. THE PARTICLE ADVECTION STRATEGY

One of the main challenges encountered when visualiz-ing three-dimensional vector fields is to find a good visualrepresentation of the vector field due to perceptual issuessuch as occlusion and cluttering. This is especially truefor dense texture-based representations of 3D flows such asthe Lagrangian-Eulerian Advection (LEA) [19], [20] and theImage Based Flow Visualization (IBFV) [21], [22] methods.Visual perception can be improved by replacing the completelydense input noise texture by a sparse input texture [23], [24].The sparsely collected points will act as seed points with theresult of rendering a collection of densely placed field linesinstead of a more or less solid object as is the case for a denserepresentation.

The fundamental problem with the sparse texture approachis to choose appropriate seed points for the particle tracing ofthe flow. Such a particle advection method needs to addressa number of issues: It must be able to depict all importantfeatures of a flow, maintain a certain particle density both intime and space, properly take into account certain boundaryeffects such as inflow and outflow, and make sure that thefield lines traced from these seed points are separated bya minimum distance. All these issues are addressed in thealgorithm presented in this paper. A flowchart of the algorithmis shown in Fig. 3.

A. Initial Seed Point PlacementThe purpose of the seed point placement algorithm is to

obtain a collection of evenly distributed seed points throughout

the domain, while meeting two criteria. First, the chosen seedpoints should be separated in such a way that all field linestraced from the points are separated by a minimum distance.Second, the distribution of points should be somewhat randomin order to avoid a completely uniform distribution, whichproduces visual artifacts.

The initial seeding algorithm consists of inserting one seedpoint at a time randomly into the domain. Then, a field line iscomputed in positive and negative direction for a user-definedlength. If any sample point along the computed field lineis closer to any already inserted field line than a prescribedminimum separating distance, the particle is rejected. If not,the particle is inserted into the domain and all cells or voxelscovered by the field line are marked. A single sample pointis checked for validation by checking neighboring voxels formarked values. The sparse particle texture is defined by settingthe voxels at the chosen seed points to 255 (which is themaximum value for a byte texture of type unsigned char),while the remaining voxels are set to zero. This texture willat a later stage of the algorithm be treated as input to atexture-based method for representing directional informationvia patterns of correlation in a texture. This is covered inSection VI.

B. Particle AdvectionAfter the initial placement of the seed points, these points

are treated as particles and are advected along the velocityfield for a short time period. Both path lines and field linesare computed using a fourth-order Runge-Kutta method. Toachieve correct discrete Runge-Kutta integration along pathlines, intermediate steps are calculated from linearly interpo-lated vector field values. To avoid cluttering of field lines,all particles that come too close to any other particle areremoved after each time step. When particles are insertedinto the domain, the minimum separating distance betweenadjacent particles apply for all sample points along the fieldlines traced from these particles. However when removingparticles, the separating distance is just computed betweenthe particles. The reason for this weaker constraint duringremoval of particles, is that we wish to preserve the life span ofeach individual particle as long as possible. This is motivatedboth for physical reasons as well as for producing smoothand coherent animations. For instance, the formation processof vortex structures leads to a concentration of vorticity fieldlines. Such a flow property could be suppressed if it was notfor this weaker constraint when removing particles. As theproposed field line animation technique allows individual fieldlines to fall closer to each other locally over time, significantflow information such as concentration of vorticity field linescan still be seen from the animation even though we suppresssome flow information by allowing particles to be removed.

C. Inflow TreatmentWhile all particles leaving the physical domain are removed

naturally in the particle advection step, special attention has tobe given to particles entering the domain. After each advectionstep, every cell or voxel on the boundary is checked in

HELGELAND ET AL.: VISUALIZATION OF VORTICITY AND VORTICES IN WALL-BOUNDED TURBULENT FLOWS 5

random order for inflow. New particles are then inserted at theboundary using the same criteria as were used for the initialseed point placement.

D. Maintaining Particle Density in Time and SpaceAs time evolves and particles start to cluster, some areas

of the domain will have lower particle density than the initialdistribution. To maintain an approximately even distributionin space, particles are injected in areas with low density. Thisensures that all parts of the flow are represented at all times.By doing this, we are in fact emulating a field descriptionrather than a true particle description of the flow. However,since efforts are made not to distort the interpretation of thephysics, some clustering of particles is allowed to happen. Asa result, the separating distance between adjacent field lineswill vary during the animation. The injected points are chosenaccording to the same criteria as was used for the initial seedpoint placement (see Section V-A).

In order to maintain the avarage density of particles in time,a maximum number of seed points allowed in the domainis computed by the initial seed point placement algorithmas the number of starting particles. Only when particles areremoved, either due to outflow or clustering, new particlescan be injected into the domain.

VI. FIELD LINE GENERATION

After each time step in the particle advection algorithm,the resulting sparse particle texture can be used as input toa texture-based algorithm for visualizing field lines. Severaldifferent approaches for generating field lines were discussedin [16], including Seed LIC [23] and anisotropic diffusion[25]. Due to significantly shorter running times, we here onlyemploy the direct approach described in [16]. In this field linegeneration method, all voxel values are set directly during thefield line integration step. This means that the final ’particleand field line’ (PFL) volume can be generated directly duringthe particle advection algorithm presented in Section V.

To incorporate orientational as well as directional informa-tion in each output 3D texture, each voxel value along the fieldline in the negative direction is set with decreasing intensityvalues. This will have the same effect as the OLIC methodproposed by Wegenkittl et al. [26]. While all voxels alongfield lines in the positive direction are set to 255, the voxelintensity, I, in the negative direction at xi = σσσ(si) is computedby the formula

I(xi) =(

(M− i)/M)

·255.

Here, M denotes the max number of samples along the fieldline in each direction. The curve σσσ(s) is parameterized by thearc-length s.

To convey the 3D shape and depth relations among the fieldlines we employ the limb darkening technique used in [23].Limb darkening creates a halo effect around each field line andis obtained by manipulating transfer functions (TFs). Sinceall TFs are handled by the graphics card and only requirea 1D texture stored in memory, this is an efficient methodfor shading. Limb darkening is achieved by assigning darker

values and decreasing opacity near the edges of a volumefeature. To emphasize the halo effect, the output texturesare convolved with an isotropic 3 × 3 × 3 filter [23]. Thisleads to a smearing of the field lines, resulting in a smootherrepresentation.

VII. PHYSICAL INTERPRETATION

The animation technique used in this work employs a hybridsolution based on both the use of path and field lines. Thephysical interpretation of the animation is not straightforwardand therefore deserves additional comments. Path lines areused to track individual particles over time, while field linesare used to convey the instantaneous topology of a vector fieldat each instant of time. The field lines in our solution are tobe regarded as advanced glyphs attached to each individualparticle. During the animation, these instantaneous curves orglyphs are tracked along the time-dependent velocity field.This is done by tracking the center point of the curve alongits path line. This means that particle motion is only observedby watching a sequence of images.

The main objective with this particular visualization tech-nique was to clearly convey the instantaneous topology oftime-dependent vector fields and their time evolution. How-ever, when used to visualize inviscid flows, the animationof vorticity field lines has additional physical meaning. Thereason is that in such cases they can be considered to bematerial lines (Section III). For such flows, the presentedtechnique can also be used to track individual vorticity fieldlines over time. As the results shown in Fig. 2 suggest areasonably good match of material lines and vorticity fieldlines, it is of interest to quantify this difference.

A. Analysis - Vorticity Field Lines VS Material LinesThe spatial separation (i.e. the “error”) between material

lines and vorticity field lines is calculated in the followingway. First, a section of a given length of a vorticity field lineis computed in both directions from the seed point. Then a setof particles is inserted along the vorticity field line at evenlydistributed sample points. All particles, including the seedpoint, are subsequently advected along the velocity field for agiven time period. A new vorticity field line is then computedfrom the new seed point position, and finally the error at eachsample point is computed by calculating the distance betweenthe sample points along the new vorticity field line and theadvected particles (Fig. 4). For simplicity, the local errors arecalculated from points that constitute original neighbors, andnot from the closest point along the vorticity field line.

To obtain sufficient statistics, vorticity field lines are inserteddensely into the domain using the initial seed point placementalgorithm presented in Section V-A. In order to obtain errorinformation at different z+ positions, all local errors foundat each sample point are averaged in xy-planes. From Figs.5 and 6 it is clear that the vorticity field lines and materiallines disperse after a relatively short time interval. The errorsshown are given in percentage of the distance traveled (i.e.advected length). For instance, an error of 5% means that whena particle has traveled 20 units, the error between the advected

6 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. X, NO. Y, MONTH 2007

Vorticityfield line

field line

Advectedseed point

Seed point

Advected particles Particles

New vorticity

Error

Fig. 4. A sketch of how error measurements between the vorticity field lineand the material line are obtained.

0

2

4

6

8

10

12

14

0 20 40 60 80 100 120 140 160 180

erro

r (%

)

z+

L+=28

L+=56

L+=112

t+=5.4t+=10.8t+=16.2

Fig. 5. Error measurements between the vorticity field lines and materiallines as a function of wall distance. L+ denotes the length of the vorticityfield line.

vorticity field line and the material line could be as much asone unit. A visualization of the error is shown in Fig. 7 ina plane parallel to the wall for a few field lines inserted atz+ = 40 and advected for t+ = 9.72 viscous time units. Thisimplies, according to the mean flow, that each field line inaverage has been advected about 142 x+ units. The length ofeach vorticity field line is L+ = 56 units.

VIII. VOLUME RENDERING

During the stages of particle advection and field line gen-eration, all vector data are converted to a series of byte scalardata sets carrying information of all the advected particlesand their resulting field lines. This leads to a reduction instorage requirements by a factor of 12, assuming the originalthree component vector data was represented as floats, andthe resulting particle and field line volume is of the same

0 1 2 3 4 5 6 7 8 9

10

0 20 40 60 80 100 120 140 160 180

erro

r (%

)

z+

L+=56

xyz

Fig. 6. Error measurements (shown in the three coordinate directions)between the vorticity field lines and material lines as a function of walldistance. L+ denotes the length of the vorticity field line.

Fig. 7. Visualization of the error between vorticity field lines (red) andmaterial lines (blue) in a horizontal plane parallel to the wall at z+ = 40. Theseed points are shown in yellow.

resolution as the original data. (i.e. 3 floats = 12 bytes → 1byte). Once these first two stages of the visualization pipelineare finished, the output, which is a time series of 3D textures,is sent to our rendering framework. Since the rendering stageis completely decoupled from the first two stages, it willensure a faster rendering compared to algorithms that haveto compute texture advection in addition to volume renderingon the graphics card [16]. These texture advection algorithmsalso have to handle much more data including the vectordata, all of which need to be stored in texture memory. Thepresented algorithm is therefore capable of handling largervolumes interactively than the combined advection and volumerendering algorithms.

The volumetric data sets are rendered using a standard 3Dtexture-based direct rendering approach [27], [28]. A stack ofview-aligned slicing polygons, serving as proxy geometries,are used to sample the volume and blended together in a back-to-front order to create the final image.

A. Multifield VisualizationEven though most CFD simulations involve the computation

of a multiple set of related data fields, much of the previousvisualization research have focused on methods and techniquesfor visualizing a single field variable only [29]. While single-variable visualizations can satisfy the needs of the user inmany applications, it is clear that for some areas, such asin fluid mechanics research, it would be extremely useful tobe able to effectively visualize multiple fields simultaneouslyand to visualize interactions between them. However, due toperceptual issues such as clutter and occlusion it can be verychallenging to produce an effective visualization of multiplevolumetrical fields.

Despite the less received attention, there are several tech-niques available that are capable of visualizing multiple vol-ume data sets simultaneously. The frameworks presented in[23], [30] use a similar slice-based multifield approach. Here,each volume in a scene is rendered using a separate setof slice planes. To achieve correct blending, the slices areintermixed and rendered in the correct geometrical order.Another approach is presented by Grimm et al. [31]. Here,scenes containing multiple volumetric data objects are repre-sented using a flexible data structure called V-Objects. Other

HELGELAND ET AL.: VISUALIZATION OF VORTICITY AND VORTICES IN WALL-BOUNDED TURBULENT FLOWS 7

methods involving the representation of multiple data sets areapproaches using multi-dimensional transfer functions [32],[33] and flexible focus+context visualization techniques basedon interactive feature specification [34], [35].

Our Multifield Approach: Here, we present a flexible vol-ume rendering framework for the rendering and analysis ofmultiple data fields. The framework was loosely presented in[16], however, in this paper we present a more formal anddetailed description.

The framework basically consists of data objects, volumeobjects, and a scene. First, the selected data fields are storedas data objects together with metainformation including datarange and time step. Then, volume objects are created from thedata sets and put in the scene. Each volume object is definedby an uniform 3D scalar field and stored as a separate 3Dbyte texture on the graphics card. For data objects stored asfloats or doubles, normalized byte data are generated from theoriginal data. The actual rendering of the scene depends onthe precise order and type of the volume objects.

The volume rendering framework currently supports threekinds of volume objects, which are luminance (L), alpha (oropacity) (A) and mask (M).

Luminance This is the standard type of volume object.Each value in the data set is used to defineboth the color and opacity in each voxel.

Alpha The data set values will replace the alpha valuesof the preceding luminance object.

Mask The alpha values of all preceding luminanceobjects are multiplied by the values associatedwith the mask object.

Using these three types of volume objects, we can create anumber of different visualization scenes. To facilitate interac-tive displays, all supported scenes are written as various shaderprograms. This could either be as a fragment program or viahigh-level shading languages such as Cg [36] and GLSL [37].We have chosen to write all of the shaders in a high-levelshading language, which is more human-readable and easierto maintain.

In order to identify the different scenes, we use a systemwhere each type of volume object is identified by a letter(see above). So, if we want to visualize the ’particle andfield line’ (PFL) volume together with another property fieldas two separate luminance objects, the shader program usedwould be identified as LL. A scene where both of these fieldsare masked by a third volume, would be identified as LLM.Whenever multiple luminance objects are used to create avisualization scene they are blended together. This is done inthe following way. First, every scalar value associated with aspecific volume object is mapped to a color and opacity valuethrough a transfer function. Then, the blended color (C) andopacity value (A) at each voxel or fragment are derived usingthe following formulas

A = 1−n

∏i=1

(1−Ai) (4)

C = (n∑i=1

AiCi)/n∑i=1

Ai, (5)

where n denotes the number of luminance objects in thescene and a single luminance object L after lookup, at eachvoxel, is given as Li = (Ci,Ai). Whenever a masking object isused, all alpha values of the “left hand side” volume objectsof a mask field are multiplied by the mask values. Thismeans that for instance in an LMLL scene, the two rightmostluminance objects will remain unaffected by the mask field.Both operations associated with the alpha and mask objectsare evaluated before the blending of the individual luminanceobjects. All volume objects used in a scene are associatedwith a separate transfer function, enabling the user to create avarious number of visualization effects.

Fig. 9 shows an example of a complex visualization scenegenerated using the above rendering framework. Here, threedifferent fields are visualized using an LLAL-scene. In thisscene, the first luminance object holds the vorticity fieldthrough the PFL volume. This volume is then blended with theluminance/alpha object, where the vorticity field has been usedto define the opacity and the z component of the velocity fieldhas been used to define the color. While the first volume objectis used together with a transfer function to create shadingeffects through limb darkening, the latter volume object hasthe effect of coloring the individual vorticity field lines. Thelast luminance object is used to visualize the rate of productionof turbulent kinetic energy together with the other two volumeobjects.

Fig. 12(b) shows another example of a complex visualiza-tion scene. Here, three different fields are used to create anLMLL-scene. Similarly to the above scene, the first luminanceobject holds the vorticity field through the PFL volume. Thisvolume is then blended with the two fields enstrophy (‖ωωω‖)and λ2 [9], both revealing vorticity structures of the flow. Inorder to show the direction of the vorticity field only inside thestructures associated with the other two fields, a masking fieldhas been applied to the PFL volume. In Fig. 12(b), enstrophyhas been used as a masking field. All three luminance fieldshave been used together with transfer functions to createshading effects through limb darkening.

The blending procedure of multiple volume objects pre-sented here has similarities both with the slicing techniquespresented in [23], [30] and the V-Object approach [31].However, there are some notable differences. For instance, thenumber of slices used in our approach is independent of thenumber of volume objects. This is not the case for the otherslicing methods [23], [30], which use a separate set of sliceplanes for each volume object. Instead we consider, just asin the V-Object approach, the volumes to be like clouds ofparticles and take into account all volumes simultaneously ateach sample position.

Another difference lies in the compositing scheme. Similarto the other three techniques, our method employs the stan-dard over operator [38] when blending the individual slices.However, for the blending of multiple volumes at each voxel,we present an alternative formula which opposed to the overoperator is independent of the precise order of the volume ob-jects to be blended (equations (4) and (5)). This merge operatoris identical to the plus operator [38], with the exception thatwe use the standard compositing scheme when calculating the

8 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. X, NO. Y, MONTH 2007

(a) (b) (c)

Fig. 8. Rendering results for different blending schemes used to visualize atorus together with a sphere. The torus and sphere data are white everywhereexcept for a small region in red and blue colors, respectively. The alphavalue for each volume is A = 0.5. The rendering shows a single slice polygoninside the volume. (a) Blending formula presented in equations (4) and (5).(b) Sphere over torus. (c) Torus over sphere.

blended alpha values. This ensures that both the final colorand the alpha values in each voxel are in the range [0,1].Hence, a multifield visualization of several transparent datafields will continue to be transparent when using the mergeoperator. This is not the case when using the plus operatorwhich very quickly accumulates opacity values higher thanone. In many applications, the merge operator is preferablecompared to the over operator since this expression holds noprecedence in any area covered by multiple data fields (seeFig 8). For other visualization purposes, other compositingoperators could be the natural choice. The presented renderingframework can also easily be extended to support a wider set ofuseful operators such as the compositing operators presentedby Porter and Duff [38].

Our rendering framework also differs from the two otherslicing methods and the V-object approach, in that our frame-work supports more than a single type of volume object.By using luminance objects in combination with mask andalpha objects, we can create far more advanced scenes suchas the ones in Figs. 9 and 12(b). Since each volume object isassociated with a separate transfer function, our framework canalso be used to create focus+context visualizations [34], [35],[39]. In focus+context visualizations, some objects or parts ofthe data are shown in detail, while other objects or parts act asa context. While the data “in focus” often are displayed ratheropaque, the rest of the data can be shown rather transparent.

Since too much blending of different colors at a singlevoxel could lead to visualizations which can be difficult tointerpret, the presented rendering framework should not beused uncritically. Careful use of the framework, however,enables in-depth analysis of topologically very complex data.To facilitate the analysis, each volume object is accompaniedby a check box that can be used to quickly turn on and offindividual objects from the scene. This way, the user caninteractively investigate the effect and contribution made bythe individual data fields.

B. Animation - Interactive AnalysisOnce the desired data is selected and an appropriate visu-

alization scene is created, our rendering framework handlestwo types of navigations through the time-varying data set.

The data set can be explored by either dragging a timeslider or by using the animation utility. The time slider isvery useful for investigating the data at different time steps.Once a new time step is selected, the visualization sceneis automatically updated using the same transfer functions.The time slider feature also simplifies the process of findingcolor and opacity tables that are well-suited for the wholetime-series. Finding good transfer functions is often a tediousprocess that sometimes involves clipping of data value ranges.The animation utility allows a more continuous visualizationof the time-dependent data. Here, the user can choose the orderof the data sets to be loaded, the step size, as well as pre-generated user interactions such as rotation and zooming.

In addition to manipulating the time slider, our renderingframework supports an additional set of tools to facilitate theanalysis of a three-dimensional time-dependent flow field. Thisincludes manipulation of transfer functions, clip planes, data-subset selections, and other user functions at interactive rates.These are all tools that can be used to diminish the occlusioneffects, by for instance creating transparent visualizations andreducing the complexity of the scene by focusing on a regionof interest.

C. Visual PerceptionWhile geometric objects have reflective properties so that

the use of light sources emphasizes the 3D shape perception,a volume could be thought of as emitting light, where theemitted light expresses the data value of a particular voxel.Without the use of light sources, one could then anticipatethat volume rendering would result in “flat” images with littleinformation about the depth relations in the volume. However,“simple” nonphotorealistic rendering (NPR) techniques can beused to enhance spatial structures and to give the necessarythree-dimensional appearance. In our framework, we use theemission-absorption [40] model in combination with limbdarkening [23] to enhance interesting features.

Even though several other NPR techniques exist, includingwarm to cool shading, gradient enhancement, and depth en-hancement [41], [42], we feel that our rendering strategy incombination with interactivity provides enough spatial cues forthe purpose of data analysis. In this context, it is important toemphasize that interactivity is far more important than shading.Of course when producing illustrations of volume data, thereis much to gain on using more advanced NPR techniquesthan the ones presented in our framework. Our frameworkcan easily be extended to support a broader amount of variousNPR techniques as well as volume shading, as all visualizationscenes are written as separate shader programs.

One benefit with the presented rendering framework, is itsflexibility when rendering multiple data features in a singlescene. In order to communicate data field correlations withseveral data fields, it can sometimes be useful to focus onsome specific features using limb darkening while visualizingother properties more transparent without further feature en-hancements. Examples of such visualizations can be seen inFigs. 9 and 11(b). In Fig. 9, the field “in focus” is the vorticityfield. Here, correlation with high rate of turbulent production is

HELGELAND ET AL.: VISUALIZATION OF VORTICITY AND VORTICES IN WALL-BOUNDED TURBULENT FLOWS 9

shown through a more transparent visualization and throughthe use of a clip plane. In Fig. 11(b), the field in focus isenstrophy, whereas the regions of high energy dissipation areshown without any further feature enhancement.

IX. RESULTS

We have visualized various fields and combinations offields derived from the turbulent channel flow simulation. Inorder to reduce the amount of data, only 1/8 of the entirecomputational domain has been considered, cf. Fig. 1.

We employ vorticity field lines as indicators to providea means to identify “ejection” and “sweep” regions; twoparticularly important spatio-temporal events in wall-boundedturbulent flows. In sweep regions, high momentum fluid el-ements far from the walls are brought closer to the surface,thereby increasing the local momentum density in the vicinityof the wall. This results in increased internal shear close to thewalls, and subsequently also increased vorticity, and increasedkinetic energy dissipation rate. In ejection regions, on theother hand, low momentum fluid close to the wall is broughtoutwards whereby the resulting momentum density is reduced.This results in reduced internal shear, and consequently also alocally reduced vorticity and reduced kinetic energy dissipa-tion rate.

The outward motion in ejection regions creates vorticityloops with associated vortex stretching and subsequent vortexintensification. The rate of energy dissipation is also increasedin these loops. In Fig. 9, the fields are seen through a clipplane perpendicular to the streamwise direction. The vorticityfield lines are colored according to the vertical component ofthe velocity (w). They are colored green in the sweep regionswhere the fluid elements are approaching the wall, i.e. w < 0,and yellow in the ejection regions where w > 0. The field linesin the figure are distorted according to the fluid motion. Thevorticity field line topology shows that the ejection regions areassociated with a low vorticity field line density, whereas thefield line density is increased in the sweep regions. In the samescene the rate of instantaneous turbulent production P(x, t) =−uw ∂U(z)/∂ z is visualized. Positive values are assigned a redcolor. As expected from kinematical considerations, positiveproduction occurs in regions where the streamwise (u) andwall-normal (w) fluctuating velocity components have differentsigns. It can clearly be seen that both ejection and sweepregions are associated with P > 0; ejections and sweeps arethus characterized by streamwise velocity fluctuations u < 0and u > 0, respectively.

Positive turbulent energy production is associated witha transfer of mean flow energy to the turbulent motion.Mean flow energy is provided by the imposed streamwisepressure gradient. Fig. 10 shows the instantaneous rate ofturbulence production. Positive production is colored in redwhile negative production is colored in green. In regions withnegative turbulence production, kinetic energy is extractedfrom the turbulence and transfered to the mean flow field.Positive energy transfer dominates in an ensemble averagedsense, however, and this is also indicated in the instantaneousdistribution displayed in Fig. 10. The area and saturation of the

Fig. 9. Vorticity field lines colored to identify sweep (green) and ejection(yellow) regions. Regions with high rate of turbulent production are coloredred.

Fig. 10. Regions of positive (red) and negative (green) rate of turbulentproduction.

red patches are larger compared with the green ones, indicatinga net positive turbulent production rate.

In Fig. 11(a), the instantaneous turbulent energy dissipationrate ε(x, t) = 2ν(∂ui/∂x j)(∂ui/∂x j) is visualized togetherwith “vortices” through the λ2 criterion [9]. The rate ofdissipation is strongest close to the wall, where the velocityshear has its maximum. Here, the shear is manifested asstrong vorticity sheets consisting mainly of spanwise vorticity;vortices per se do not exist in this thin layer immediatelyabove the wall. The λ2 structures are shown in green and arevisualized as tubes with transparent cores in order to makethe dissipation rate visible inside them. The turbulent energydissipation rate is shown in red to yellow where the red colorindicates maximum values. Apparently the energy dissipationinside the core of the vortices is weak compared with thelevels observed within the sheet-like structures. The latter thusdominates.

In order to capture vorticity sheets in addition to “tube-

10 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. X, NO. Y, MONTH 2007

like” structures, enstrophy (‖ωωω‖) (colored in white to gray)is visualized together with turbulent energy dissipation rate,and the result is shown in Fig. 11(b). The highest values ofenstrophy are found close to the wall, through intense vorticitysheets. Further into the channel, the enstrophy is weaker andforms structures that are a mix of tubes and sheets. The opacitycorresponding to the interiors of the enstrophy structures aregiven low values so they are made transparent. Similarly toFig. 11(a), the energy dissipation is represented in colorsvarying from red to yellow with decreasing values. Apparentlythere is a close spatial correspondence between enstrophy andturbulent kinetic energy dissipation, as shown in Fig. 11(b),which in fact is loosely manifested through the exact relationε = 2ν(∂ui/∂x j)(∂ui/∂x j) = 2ν [ωiωi +(∂ui/∂x j)(∂u j/∂xi)],where the last term has a secondary effect.

Visualization of enstrophy together with λ2 structures showthat the vortices often have vorticity sheets wrapped aroundthem (Fig. 12(a)). To make the vortices more visible, theinteriors of the enstrophy structures are made completelytransparent. Vorticity sheets are formed in the strong shearregion close to the wall. They are distorted and advectedaround in the fluid and can be wrapped up around the vortices.The orientation of the vorticity field within the vorticity sheetsis generally inclined relative to the direction of the nearbyvortex cores, whereas the field lines inside the vortex coresare approximately oriented along the cores (Fig. 12(b)).

Contrary to isotropic turbulence, the turbulence in thechannel is highly anisotropic, and the degree of anisotropy ishigher toward the walls of the channel than in the center. Thevorticity is created close to the wall by internal shear and isadvected away from the wall in the ejection regions. This leadsto local intensification of the wall-normal vorticity throughstraining. In sweep regions, the flow is directed toward thewall with a diverging motion close to the wall, again leadingto intensification of vorticity through straining. Although thechannel flow is stationary from a statistical point of view, itis surprising that also the instantaneous vorticity structuresdevelop relatively slowly in time. The animations show thatduring the time it takes for the structures to traverse thecomputational domain (we visualize only half of that), thestructures show a relatively stationary behavior. However,on the smallest scales, the temporal changes are substantial.Animation of the vorticity field together with the vorticitystructures provides qualitative information about the detailsof the evolution of the flow field.

X. CONCLUSION

We have visualized and animated vorticity fields in a tur-bulent channel flow by letting the seed points for the vorticityfield be material particles following particle paths. This givesqualitative information of the spatial and temporal evolutionof the vorticity field. The resulting vorticity fields show theevolution of the flow. Usually scalar fields like enstrophy orfields obtained by the λ2 criteria is used to visualize vorticityfields. We have demonstrated that these approaches give alimited view of the vorticity field. For example, the λ2 methodis only of limited interest close to the wall since there are

few “tube-like” structures there. The direction of the vorticityfield contains important information, which is also lost whenvisualizing the vorticity as a scalar field.

In the case of turbulent channel flow, due to the presence ofthe wall, the turbulence is highly anisotropic. This is seen inthe structure of the vorticity field. In regions close to the wall,vortices (tube-like structures of vorticity) do not dominate asin isotropic turbulence. In the channel flow there is an intricatemix of vortices and vorticity sheets. The latter dominatesclose to the wall, where they are produced at a constant rateby the strong velocity shear. The production of vorticity bystraining also peaks in a region close to the wall. The vorticitysheets are advected away from the wall by ejection events anddistorted through straining. In the sweep regions, the vorticitymagnitude is increased by straining, leading to preferentialspanwise vorticity close to the wall. The energy dissipation isstrongest in these regions.

The error involved by assuming that vorticity field linesand material lines are equivalent in wall-bounded turbulentflows has been quantified in the present study. It has beendemonstrated that the error is largest (and approximatelyequal) in the directions parallel to the wall, and that the errorgrows approximately linearly in time as the fluid elements areadvected downstream. Even though the vorticity field lines,in general, do not evolve as material lines, the proposedanimation technique still enable a trustworthy qualitative in-terpretation of the spatio-temporal behavior of the vorticityfield in turbulent flows. The technique cannot be used to trackindividual vorticity field lines over time, but is still able toshow the instantaneous topology of the vorticity field and itstime evolution. However, the time evolution of the vorticityfield is an indicator of the motion of the fluid particles.

We animated the vorticity field according to the abovementioned seeding and tracking strategy. The vorticity field isrendered in regions of non-neglectable enstrophy in the samescene as the λ2 structures. The animation is carried out overthe time interval it takes for the structures to traverse the wholechannel. To the best of our knowledge, this is the first time ananimation like this has been done in wall-bounded turbulentflows. A notable outcome of the animation is the observedstationarity of the vorticity structures (in a reference framemoving with the flow).

The rendering stage of the field line animation techniqueis decoupled from the preceding stages of the visualizationmethod, in order to increase the efficiency. In combinationwith the presented volume rendering framework, this allowsinteractive exploration of multiple fields simultaneously, andthus sets the stage for a more complete analysis of the flowfield.

ACKNOWLEDGMENT

The authors would like to thank Dr. Xing Cai and the anony-mous reviewers for their valuable comments and suggestions.

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HELGELAND ET AL.: VISUALIZATION OF VORTICITY AND VORTICES IN WALL-BOUNDED TURBULENT FLOWS 11

(a) (b)

Fig. 11. (a) Vortices defined by λ2 < 0 are colored green, while regions of energy dissipation are colored from yellow for moderate dissipation to red formaximum dissipation. Evidently the dissipation is weak within the vortices. (b) Energy dissipation shown in red to yellow is visualized together with enstrophyin white to gray. Areas with high energy dissipation is spatially well correlated with enstrophy.

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Anders Helgeland received BS and MS degreesin computer science from the University of Oslo,Norway, in 2000 and 2002. He is a PhD candidate incomputer science at the University Graduate Center,Norway. His current research interests include flowvisualization and volume visualization. He was arecipient of the best poster award at the Proceedingsof the IEEE Visualization Conference 2002. He isa member of the IEEE, the IEEE Computer Societyand ACM SIGGRAPH.

B. Anders Pettersson Reif received MS and Dr.Ing.degrees in Applied Mechanics at Lulea TechnicalUniversity in 1992, and at the Norwegian Universityof Science and Technology in 1997, respectively.Pettersson Reif is working as a principal scientistat the Norwegian Defence Research Establishment(FFI) and is appointed adjunct professor in turbu-lence modeling at Chalmers University of Technol-ogy. Research interests include turbulence physicsand modeling, and computational fluid dynamics.

Øyvind Andreassen studied astrophysics at the Uni-versity of Oslo, where he received a MS in 1981 anda PhD in physics in 1994. Andreassen is currentlychief scientist at the Norwegian Defence ResearchEstablishment (FFI) and adjunct professor in appliedmathematics at University Graduate Center at Kjellerand University of Oslo. His research interests in-clude scientific computing and visualization, wavephysics, computational fluid dynamics. turbulenceand flow noise.

Carl Erik Wasberg received a MS degree in Indus-trial Mathematics from the Norwegian Institute ofTechnology in 1986, and a PhD degree in AppliedMathematics from the University of Bergen in 1995.He is now a senior scientist at the NorwegianDefence Research Establishment (FFI), where heworks in computational fluid dynamics. His researchinterests include numerical solution of partial dif-ferential equations, large-scale computing, and fluidmechanics.