Reasoning about Fluid Motion I: Finding Structures

Abstract :With the increasing role of high performance com-

puting in attacking complex physical problems, thereis an urgent need for the development of advancedcomputational technology to provide scientists withhigh-level assistance in the analysis, interpretation,and modeling of a massive amount of quantitativedata. A critical area where this need is quite evident isthe problem of turbulence . The overall research goalis to develop a computational environment to helpscientists efficiently make observations, test hypothe-ses, and make conceptual models of turbulence datasets . This paper presents the intellectual challengesand progress of this project. My approach is based ontwo key ideas: (1) Representing fluid behavior as localinteractions amongsemi-persistent flexible objects likevortices enables automatic high-level qualitative inter-pretation of turbulence data, and (2) Abstracting fromthe particular features of fluid dynamcal reasoning, Ipropose five core operations - aggregation, classifica-tion, re-description, spatial inference, and configura-tion change - as part of a general theory of imagisticreasoning. A new vortex-finding algorithm is also pre-sented .

"Feymann said . . . when Einstein stopped cre-ating it was because he stopped thinking in con-crete physical images and became a manipula-tor of equations . "

1 Introduction

Genius , James Gleick, p244, 1992 .

It is commonly believed that there are two styles ofsci-entific thinking : analytical, a logical chain of symbolicreasoning from premises to conclusions, and visual,the holding of imagistic, analogue representations of aproblem in one's mind so that perceptual and symbolicoperations can be brought to bear to make deductions .Neither style is to be preferred a priori over the other.However, for problems whose complexity precludes a

' The research was funded in part by NSF NYI AwardECS-9357773.

Kenneth Man-kam YipDepartment of Computer Science

Yale UniversityP.O . Box 208285, Yale StationNew Haven, CT 06520-8285 .

(yip-ken@cs .yale.edu)

direct analytical approach, a certain amount of quali-tative and visual imagination is needed to provide thenecessary "feel" or "understanding" of the physicalphenomena. Once the picture is clear, the analyticalmathematics can take over and lead more efficientlyto logical conclusions .This "feel and physical understanding" is rather in-

formal, imprecise, and apparently unteachable, butnecessary for scientists . My research goal is to formal-ize the visual style of thinking as computer programs,and to demonstrate the power of these programs bytheir ability to reason about the structure and motionof turbulent fluid flows.The choice of fluid flows as a domain may seem arbi-

trary and turbulence may add unnecessary difficultyto the project. However I believe the subject matteris fascinating, ripe for attack, more constrained thana commonsense theory of liquids, and, most impor-tantly, is worthwhile because new tools for advancingturbulence research can have enormous scientific val-ues.Turbulent flows have been intensely studied for

many decades . But only recently is the capacity toperform direct numerical simulation (DNS) of turbu-lent flows at moderate Reynolds numbers with enoughaccuracy realized . 1 So for the first time detailed so-lution fields are available to scientists . Some of theimportant questions facing scientists are: How doesone make observations from the data? How does onemake theories based on new observations? How aredata used to test theories?Currently there is a large effort in Scientific Visu-

alization whose goal is to develop computer graphicsto facilitate the presentation of large datasets . How-ever the process of discovering interesting structuresin the datasets and extracting physics from them isup to the human experts. Even with the help of mod-ern visualization software (such as AVS and Explorer),most human experts find the task time-consuming andprone to human visualization error. 2In collaboration with experts in fluid dynamics from

1The terminology is explained in sections 2 and 3.2Further discussion of this issue and current approaches in

the visualization literature can be found in [13] .

Yale and MIT, I am building computer tools that canreduce the post-processing time of CFD data by ordersof magnitude. My approach is based on the followingkey idea : Active Visualization and Abstraction(AVA) . s By "active" I mean the reasoning processhas three properties :

1 . Autonomous - the computer holds pictures in itsmind so that perceptual operations can be used tomake deductions which are otherwise too difficultto make by analytical methods alone.

2 . Purposeful - the visualization is done in the con-text of making observations and testing theories .

3 . Generative - the results of visualization guide thecomputer to make further observations .

By abstraction, I mean the process of reducing theencoding length of the datasets . A moderate-sizedinstantaneous flow field might consist of 0(107) gridpoints each with a 4-vector (three components for ve-locity and one for pressure) associated with it . It ishard to imagine four numbers or even one at eachpoint of space. Could we re-represent the flow fieldby semi-persistent objects that are much more com-pact and easy to visualize? That the physical pictureof moving and deforming objects, called vortices, is auseful level of abstraction for reasoning about fluids isan important claim of AVA .What does AVA consist of? I conjecture that AVA

consists of five core operations : 41 . Aggregation2 . Classification3 . Re-description4. Spatial inference5. Configuration change

Aggregation is the grouping of primitive objects ac-cording to some measure of similarity (e .g ., closeness,continuity, symmetry) . For a familiar example, con-sider a vector field . Points in the vector field can begrouped into integral curves or orbits . Classificationis the assignment of labels to the aggregate objects.Each label denotes a bundle of characteristic proper-ties . Re-description gives a more concise representa-tion of the aggregate . These aggregates might possesscharacteristic shape and might touch or overlap. De-duction of these geometric relations is the province ofspatial inference. The characteristic properties of anobject might constrain what other objects can exist

3The concept is similar in spirit to the idea of "active vi-sion" in the vision literature, which means the active controlof camera movements and focus of attention to improve therobustness and stability of vision . I apply the concept to ahigher-level cognitive process.

4These core operations are distilled from a rethinking of howphase space analysis programs - like KAM [15] or MAPS (16] -work. I suspect much of the work in mechanism analysis usingconfigurationspace (e .g. (9]) can be cast in the same framework.

or change or deform in its neighborhood . Finally, thewhole cycle of core operations can be reapplied at amore abstract level .A word about related works. Commonsense reason-

ing about fluids is a central problem in naive physics[7, 6] . The problem is hard because fluids do not con-veniently divide into discrete pieces that can be easilycombined . Ken Forbus and his group have done im-portant work in extending and partially implementingPat Hayes' ideas for representing fluids using both thecontained-liquid and piece-of-stuff ontologies [5, 4] .The work described here is closer to the scientific endof the formalization spectrum . As a consequence thetheory is less general than Hayes' theory, but morerelevant to the research in the turbulence community.I expect the two lines of research to have fruitful in-teractions .

2

Properties of fluid motionA fluid is either a gas or a liquid ; it is not a solid.The behavior of fluid is interesting and often surpris-ing. One can get a good understanding of fluids bythinking about the properties of water. Solids resistdeformation; fluids yield to any shear stress . Solidscare how far they deform; fluid how fast they deform .The measure of the ease of deformation is viscosity.A more viscous fluid like honey deforms more slowlythan a less viscous one like water . If the fluid is notmoving, the only force it exerts is pressure, which al-ways acts in the direction normal to any surface in thefluid. If the fluid is moving, a number of new forcescome into play. Viscosity, an internal friction so tospeak, causes a shear stress to develop between layersof fluid moving in different velocity. A fluid movingacross a solid surface tends to drag the surface with it .This property is known as the no-slip condition, i.e .,the velocity of the fluid is exactly zero at the solid sur-face . The frictional forces always act in the oppositedirection of motion . A fluid moving over an asymmet-rical solid surface can create lift, a force perpendicularto the direction of motion .Real fluids are quite complicated, so I will make

some assumptions about the properties of the flow tosimplify the discussion . The assumptions are conve-nient fictions, but in many interesting cases they arerather good approximations :

1. Fluids are incompressible . This is really the con-servation of mass. The volumetric flow rate is thesame at every point in the fluid; no fluid can ac-cumulate anywhere .

2. Fluids are Newtonian. This gives a particularlysimple linear relationship between the appliedshear stress and the rate of deformation.

3 . No temperature variation, Coriolis forces, nor elec-tromagnetic forces exist that might affect the fluid .

The Newtonian, incompressible fluid is alreadycomplicated enough .

The payoff of these assumptions is large. The onlyrelevant forces on the fluid, besides pressure, are in-ertial and viscous. The inertial forces keep the fluidgoing; the viscous try to stop it . The ratio betweenthe two forces, a dimensionless number, is called theReynolds number, denoted by the symbol Re .One reason why the Reynolds number is an impor-

tant parameter is that the character of fluid motionis strongly dependent on it . A very high Reynoldsnumber flow (Re » 1000) is dominated by inertialforces and tends to favor turbulence in which individ-ual "fluid particles" move in a random unpredictablefashion even when the fluid as a whole is moving in adefinite direction. Large viscous forces such as thosein a flow with very low Reynolds number should dampturbulence and maintain a laminarflow in which fluidparticles move more or less parallel to each other . Inbetween the very high and very low Reynolds numberflows are the transition flows whose character dependson the circumstances .To get an idea of the different regimes of a real fluid

flow, consider a steady incompressible flow past asym-metric "bluff body" (a non-streamlined body such asa circular cylinder or a sphere) . See Fig. 1 . At verylow Reynolds number, when the flow is slow, the flowpattern consists of smooth symmetrical streamlines,the trajectories of the fluid particles. As the Reynoldsnumber is increased, a pair of symmetrical eddies orvortices appears in the rear of the cylinder where fluidparticles curl around . If the flow speed continues toincrease, the vortices become elongated and at somepoint break off, traveling downstream with the fluid .The phenomenon is known as flow separation . Newvortices begin to form near the behind of the cylin-der and shed alternatively into the cylinder wake. Atsufficiently high Reynolds number, turbulence sets inand the wake begins to oscillate; the flow is unsteadyand highly irregular. Vortices of many different lengthscales are also visible .What really happens inside a turbulent flow? Is the

motion of fluid particles purely random? Or is thereorganized motion within a background of smaller scalerandom fluctuation? Experimental evidence seems tofavor the latter alternative . Nobody really knows forsure . But why do we care anyway? The reason is thatalthough we do not understand turbulence, its effectsare highly significant . Turbulence increases drag, mix-ing of materials, and transport of heat . In designingan air transport, we might want to suppress turbu-lence in order to reduce drag ; in a combustion engineon the other hand we might want to enhance turbu-lence to increase mixing rate . If large-scale structuresdo exist inside a turbulent flow and are found to be re-sponsible for the enhanced mixing and transport prop-erties, then it might be possible to control turbulence

(a) Re < 10

creeping flow

(b) 10 < Re < 40 Attached vortices

(d) Re > 200 .000

Fully turbulent wake

(c) 40 < Re < 200 .000

von Karman street

Figure 1 : Schematic flow patterns around a circularcylinder . (a) Symmetrical streamlines and absence ofvortices . (b) A pair of vortices in the rear of cylin-der . (c) Alternating vortices shed downstream . (d) Aturbulent wake with vortices of many length scales .

by direct interference with these structures .

3

Reasoning tasksGiven a numerical solution field of a turbulent flow, aspectrum of reasoning tasks can be defined. The fol-lowing list is roughly in the order of increasing com-plexity:

1 . Making observations . Find out structures, if any,that exist in the solution field. Are there vortices?What are their shapes and sizes? How are theydistributed? How are they created? How do theyevolve and interact?

2 . Making correlations . Determine how the shapeand distribution of structures correlate with fluidvelocities, pressure, dissipation, and other statis-tical properties of the flow .

3 . Incremental analysis . Givenan instantaneous con-figuration of structures, predict the possible short-time behaviors .

4 . Causal analysis . Explain and summarize the evo-lution of structures by a set of elementary inter-action rules.

5 . Testing theories . Given a hypothesis about struc-ture formation or interaction, gather evidence tosupport or disprove the hypothesis .

4

Domain TheoryThe entire theory of incompressible flow is containedin the Navier-Stokes equations :

V-u = 0

at + (u _0)u

-

-Op+ vO2u

The first equation expresses the conservation ofmass ; it is the incompressibility assumption . It saysthe velocity u, which has three components, has zerodivergence. The name divergence is well-chosen for0 - u is a measure of how much u spreads out (di-verges) . If it is positive (negative) at a point P, thenP is asource (sink) . Zero divergence means no sourceor sink inside the flow .The second equation comes from Newton's second

law of motion ; it equates fluid acceleration with ap-plied forces . The acceleration terms, the left hand sideof the second equation, represent the change of veloc-ity in time and space. On the right hand side, thefirst term -Vp is the pressure gradient . The secondterm, v is the kinematic viscosity which is the ratio ofthe viscosity and density of the fluid; the product of vand the second spatial derivatives of the velocity Vuis the force due to viscosity.Since we can associate with every point (x, y, z)

in space and instance t with a velocity u, we callu(x, y, z, t) a velocity field. A field is any physicalquantity that takes on different values at different po-sitions and time . u(x, y, z, t), like a magnetic field, isa vector field because at each space-time point, we as-sociate a vector with three components . The pressurefield p(x, y, z, t) like temperature is a scalar field - onenumber for each point .While useful for some specialized flows, the veloc-

ity field and its topology are not a particularly goodrepresentation of turbulent flows. The velocity fieldnot only is not Galilean invariant (what an observersees depends on how fast she is moving), but also canchange quite unpredictably from one instance to an-other . As a consequence, we cannot visualize what ishappening - not easily . Just as some scientists preferto think in terms of interaction of charges and mag-nets rather than the resulting magnetic field, manyfluid dynamicists prefer to think in terms of a vectorquantity called vorticity, denoted by the symbol w,which is the curl of the velocity, w= Vx u. The curlof u at a point P measures how much the vector fieldu curls around P.Suppose you float a paddlewheel on a bathtub. If it

starts to turn, then the point it is placed has a non-zero curl . A region with a large curl is an eddy, awhirlpool . The curl of u is a vector ; its direction isassigned by the right hand rule : if the water surface isthe xy-plane and the paddlewheel turns counterclock-wise, the curl points to the upward z-direction.

The reason we introduce the vorticity fieldw(x, y, z, t) is that it simplifies the description of fluidmotion and gives ontological primitives which are eas-ier to reason with . A vortex line is an integral curveof the vorticity field. The set of vortex lines pass-ing through a simple closed curve in space is said toform the boundary of a vortex tube . Vortex lines andvortex tubes have nice invariant properties . In par-ticular, they can be treated as material objects . Thetruth of this last statement follows from the so-calledHelmholtz laws of vortex motion . For incompressible,inviscid (i .e ., v = 0) flow, Helmholtz proved the fol-lowing theorems [2] :

1 . The vorticity field has zero divergence (becausethe divergence of a curl is always zero) .

2 . Vortex lines move with the fluid, i.e ., fluid parti-cles that at any time lie on a vortex line continueto lie on it .

3 . The strength I' of a vortex tube, defined as cir-culation I' = fs w - ndS is the same for all cross-sections S of the vortex tube and is constant intime .

The first and second theorems explain why vortexlines or tubes can be treated as material bodies. Thethird theorem is just an expression of the conservationof angular momentum . The skater's spin is a nice il-lustration of the third theorem at work . By bringing'in her arms and thereby shrinking the cross-section,the skater spins faster because the total angular mo-mentum is conserved.Turbulent flows are not inviscid, but at very high

Reynolds number the viscous effect is very small ex-cept in a thin region close to the solid boundaries . Soit is reasonable to expect the Helmholtz laws to beapproximately correct .Although the exact analysis of the motion of a con-

figuration of vortex lines and vortex tubes can be verycomplicated, the simplest situations can be under-stood by a few qualitative principles :

Qualitative Rules of Vortex Motion

1 . A vortex line accelerates velocity on one side andslows down on another. A lift force perpendicularto the vortex line is generated due to the Bernoullieffect .

2 . A vortex line is convected by the fluid which exertsa drag force to push the vortex line in the directionof fluid motion .

3 . A vortex tube stretched (compressed) in one direc-tion increases (decreases) the velocity componentsin the other two directions .

4 . A bent vortex line, conceptualized as a spacecurve, exerts aself-induced motion along its binor-

Figure 2: Motion of vortex lines can be understoodby afew qualitative principles : Schematic pictures fortwo types of vortex line reconnection . (a) Two an-tiparallel vortex lines come together and reconnect toget rid of the region of opposing vorticity. (b) A sim-ilar reconnection occurs when a vortex line touches afree-slip boundary .

mal and the effect is largest at points of maximumcurvature [I]-

5. Vorticity can only be created at velocity disconti-nuities and solid boundaries .

6. When two antiparallel vortex lines are broughtclose together, they can break apart and recon-nect . (See Fig. 2)

7. When a vortex line meets a free-slip boundary, itcan reconnect.

The first four rules are consequences of Helmholtzlaws ; they are good approximations even when ap-plied to vortex tubes with finite core size as long asthe core is relatively thin . The fifth is a kinematicconsequence of the no-slip condition and the conser-vation of vorticity. The last two rules describe howthe topology of vortex lines can change. The detail ofthe reconnection mechanism is still an open issue, butreconnection appears to happen experimentally .The significance of these qualitative rules is that

they allow the synthesis of the velocity field - at leastthe rotational component of the velocity. Main qual-itative features of the flow field can often be deducedwithout complicated numerics . The use of these rulesin incremental motion analysis will be the subject ofa sequel paper.

5 Automatic Extraction of VortexStructures

5.1

Aggregating vortex linesA vortex line is the basic building block of a vortexstructure. Because of the divergence-free property, avortex line, like a magnetic field line, does not start

or stop in the interior of the fluid; it tends to run ina closed loop. However, a real turbulent flow alwayshas a background of randomly fluctuating vorticity.So it is reasonable to assume that only the relativelyhigh-intensity vortex structures remain coherent fromone time instance to another. To distinguish a "co-herent" vortex from amathematical vortex, I proposethe following definition for aggregation:

Definition A coherent vortex is a compact bundle ofadjacent, high-intensity vortex lines that aregeometrically similar.

There is some degree of arbitrariness in any defini-tion of coherent structure because the notion of co-herence is informal . I believe the definition here ismore faithful to the mathematical definition of a vor-tex tube . A coherent vortex can be tube-like or sheet-like depending on the shape of its cross-section . Unlikea mathematical vortex, a coherent vortex can start orend in the interior of the fluid.Aggregating vortex lines to form coherent structures

is not straightforward. Previous researchers [10, 12]have found that vortex lines are sensitive to initialconditions . Nearby vortex lines can diverge rapidly.If the initial conditions are not chosen carefully, theresulting vortex lines are likely to resemble badly tan-gled spaghetti wandering over the whole flow field,making the identification of organized structure ex-tremely difficult . This might explain why vortex lineshave not been widely used for structure identification .I believe the search algorithm below is the first suc-cessful structure identification based on vortex lines .Given a numerical vorticity field, the search algo-

rithm finds all coherent vortices . The key idea in thealgorithm is the adaptive control of the cross-sectionof the vortex tube : the cross-section is shrunk (ex-panded) when the vortex lines on the boundary ofthe cross-section are converging (diverging) . The al-gorithm has the following steps:

l . Find all grid points that are local extrema of vor-ticity magnitude and greater than a threshold .These are the seed points .

2 . On the plane normal to the largest vorticity vectorcomponent at the seed point, find an isocontourcentered at the point . The contour, discretizedinto points, represents the initial cross section ofthe surface.

3 . Use the advancing front method (explained later)to interleave the advancement of the cross sectionby integration and the tiling of the surface.

4. Use the geometry of the tiles to decide shrinkingor expanding the cross section locally.

5. The wavefront is periodically adjusted globally bycomputing its convex hull, and the diameter andwidth of the hull .

6. The forward integration terminates when the cir-culation on the cross section falls below certainthreshold .

7. Reconstruct the surface by integrating the lastcross section backwards until it reaches the initialcross section. No wavefront adjustment is neededin this step .

8. Remove the weak vortex lines.

The advancing front method is implemented as fol-lows . (See [8] for details.) The vortex surface consistsof a list of ribbons . Each ribbon has two tracers : a leftand a right . As its tracers are advanced, the ribbonis tiled by triangular meshes in such a way to keepwavefront nearly perpendicular to the integration di-rection. The aspect ratio of the quadrilateral formedby the last pair of alternating left and right trianglesin each ribbon is used to control the local adjustmentof the wavefront.Vortex lines can twist and turn, so the cross sections

can get distorted quite a bit as the surface is devel-oped . The vortex-finding algorithm keeps track of thenumber of tracers. If the number exceeds a threshold,which is indicative of many highly distorted quadrilat-erals, the entire cross section is replaced by the convexhull of the projected wavefront on a plane normal tothe vorticity vector at the centroid of the cross sec-tion . The convex hull is useful for three purposes : (1)it reduces the number of tracers, (2) it re-orders thetracers into adjacent positions along the vertices ofan oriented polygon, and (3) important shape infor-mation of the cross section such as its diameter andwidth can be computed in linear time from its convexhull [11] .

5 .2

Aggregation resultsAs the test case, I use the DNS results of a free sur-face turbulence provided by Professor Dick Yue in theOcean Engineering Department of MIT . The turbu-lence is generated by a shear flow in a 1283 rectan-gular box with periodic boundary conditions in thex and y direction . A 4th order polynomial interpo-lation is used to compute the interpolated vorticityvector from the grid values, and an adaptive 4th or-der Runge-Kutta integrator to integrate vortex lines.Hundreds of structures have been constructed by thealgorithm . A typical result is shown in Fig. 3a . Thevortex is reconstructed from 10 vortex lines. The com-putation including rendering (done by AVS 5) takesabout 90 seconds real time on a Sparc 10/51 .The algorithm is not sensitive to the initial choice

of isocontour value : it is self-adjusting . Contrast thiswith an ordinary integrator . The vorticity lines ob-tained diverge and are tangled (Fig . 3b). Moreover,small changes in the initial isocontour can result indrastically different vortex line patterns .

Figure 3: (a) Upper: A vortex structure reconstructedby the advancing wavefront method with backwardintegration. Each line is obtained by approximately100 integration steps. (b) Lower: The vortex linesobtained by integration with no adaptive control ofcross sections .

Figure 4: A generalized cylinder representation of thereconstructed vortex shown in Fig. 3a . The tilingis chosen to minimize twist between adjacent cross-sections .

5.3

Classification and Re-description

The reconstructed vortex must be interpreted in orderto perform spatial inferences and incremental motionanalysis . Classification is the assignment of labels tovortices . The assignment is determined by the shapeof the vortex. A vortex can be tube-like or sheet-like . Tube-like vortices are further differentiated bythe curvature function along their axes . It is impor-tant to identify the local curvature extrema becausethese extrema cause a self-induced motion (section 4) .To obtain a concise representation of a bundle of

vortex lines, a vortex is re-described as a generalizedcylinder (GC) . A GC consists of a spline, a cross-section, and a sweeping rule [3] . The GC represen-tation of a vortex is computed by the following steps:

1 . Pick the vortex line with the highest integratedvorticity as the candidate spline .

2 . Compute a scale-space representation of thecurvature of the candidate spline [14] .

3. The stable local extrema are chosen as knotpoints .

4. At each knot point P, compute where the vortexlines intersect the plane at P with the planenormal equal to the vorticity vector at P.

5. Fit an ellipse to the intersection points to obtaina cross-section of the vortex.

6. The spline of the GC is obtained by spline-fitting the centers of the elliptical cross-sections .

7. Compute the intrinsic shape descriptions of thespline, i .e ., its curvature and torsion.

By varying the criteria for stable curvature extrema,one can obtain generalized cylinders of different reso-lutions. Fig. 4 shows a rather fine generalized cylinderrepresentation of the reconstructed vortex first shownin Fig. 3a .

6 ConclusionThis paper presents three novel ideas:

1 . A zeroth-order formalization of the visual style ofthinking as a cycle of five core operations: aggre-gation, classification, re-description, spatial infer-ence, and configuration change .

2. A formalization of a theory of fluid flow basedon new ontological primitives : vortex line, vortextube, and coherent vortex, and a list of qualitativeinteraction rules.

3. A new vortex-finding algorithm based on vortexlines.

The implementation of spatial inference (such as de-termining spatial relations among vortices) and config-uration change (such as incremental analysis of vortexmotion) will be the subject of a sequel paper.

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