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Acceleration feature points of unsteady shear flows Jens Kasten Leipzig University, PF 100 920, 04009 Leipzig, Germany∗ Jan Reininghaus IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria Ingrid Hotz DLR Braunschweig, Lilienthalplatz 7, 38108 Braunschweig Hans-Christian Hege 4 Zuse Institute Berlin (ZIB), Takustr. 7, 14195 Berlin, Germany 1 0 Bernd R. Noack 2 Institut P’, CNRS – Université de Poitiers – ENSMA, n UPR 3346, Départment Fluides, Thermique, Combustion, CEAT, a 43 rue de l’Aérodrome, F-86036 POITIERS Cedex, France J 0 Guillaume Daviller 1 CERFACS, 42 Avenue Gaspard Coriolis, 31057 Toulouse Cedex 01, France ] Marek Morzyn´ski n y Poznan´ University of Technology, Institute of Combustion Engines d and Transportation, ul. Piotrowo 3, PL-60-965 Poznan´, Poland - u In this paper, we propose a novel framework to extract features such as vortex cores and sad- l dle points in two-dimensional unsteady flows. This feature extraction strategy generalizes critical f . points of snapshot topology in a Galilean-invariant manner, allows to prioritize features according s c totheirstrengthandlongevity,enablestotrackthetemporalevolutionoffeatures,isrobustagainst i noise and has no subjective parameters. These characteristics are realized via several constitutive s elements. First,accelerationisemployedasafeatureidentifierfollowingGotoandVassilicos(2006), y h thus ensuring Galilean invariance. Second, the acceleration magnitude is used as basis for a math- p ematically well-developed scalar field topology. The minima of this field are called acceleration [ featurepoints,asupersetoftheaccelerationzeros. Thesepointsarediscriminatedintovorticesand saddle points depending the spectral properties of the velocity Jacobian. Third, all operations are 1 based on discrete topology for the scalar field with combinatorial algorithms. This parameter-free v foundationallows(1)tousepersistenceasaphysicallymeaningfulimportancemeasuretoprioritize 2 featurepoints,(2)ensuresrobustnesssincenodifferentiationandinterpolationneedtobeperformed 6 with the data, and (3) enables a natural and robust tracking algorithm for the temporal feature 4 evolution. In particular, we can track vortex merging events in an unsupervised manner. Data 2 basedanalysesarepresentedforanincompressibleperiodiccylinderwake,anincompressibleplanar . 1 mixing layer and a weakly compressible planar jet. They demonstrate the power of the tracking 0 approach, which provides a spatiotemporal hierarchy of the minima. 4 1 : I. INTRODUCTION mix — just to name a few applications. Velocity snap- v i shot topology provides invaluable insights into laminar X or time-averaged flows [1–5], or, in general, into velocity r Computational fluid dynamics and particle image ve- fields with a distinguished frame of reference and a low a locimetry can provide highly resolved flow data in space feature density. and time. One challenge is to quickly extract the im- Suchatopologyisalwaysbasedonthezerosoftheve- portant kinematic features from these data. Topological locity field and thus is intrinsically Galilean-variant, i.e., methodsappliedtosnapshotsareoneofthefirstchoices. dependsonthechosenframeofreference. Inanunsteady Flow topology may provide information about the size flow, a zero or critical point at one instant is generally of separation bubbles and vortices, about the length of a notazeroatanotherinstant. Thequestionwhatcritical dead-water region, and about flow regions, which do not point, ’connector’ and other topological elements physically mean for an unsteady situation immediately arises and its answer is far from being clear. ∗Electronicaddress: [email protected] In some cases, e.g., the flow over an obstacle, a nat- 2 urally preferred body-fixed frame of reference is given. tribute to the distillation of vortex cores in three major Here, Galilean invariance of the topology appears to be points: (1)arobustextractionofthefeaturepointsinthe a purely academic requirement. In many cases, however, presence of noise; (2) an efficient tracking of them over the proper frame of reference is far less obvious. In a time; (3) a filtering strategy that is based on a hierarchy wake or mixing layer, for instance, topology may resolve of the vortex cores and trajectories. The extraction and the initial vortex formation in a body-fixed frame of ref- trackingisbasedonacombinatorialframework [10,11]. erence, but the convecting vortices do not give rise to The resulting explicit representation of the vortex core velocity zeros as they convect downstream. Now, the lines enables a detailed analysis of the interacting struc- choice of the ’right’ frame of reference is subject to per- tures in a flow field. In principle, an analogous feature sonal preferences. extraction can be effected for saddles. A second challenge is that critical points are associ- This paper is structured as follows: In Sec. II, key ated to the smallest structures on the flow. In a fully elements of the analysis are motivated for simple ana- turbulent flow, the average distance of fixed points is of lytically defined flows. In Sec. III, a feature extraction the order of the Taylor scale [6, 7]. Under these condi- strategyisdescribed. Resultsarepresentedforthreepla- tions, critical points loose their meaning as ’markers’ of nar shear flows with increasing level of complexity (Sec. large-scale coherent structures. IV). Finally (Sec. V), the paper concludes with a sum- Third, every measured or simulated data naturally mary, the relation to other topological analyses and an containsasmallamountofnoise. Thisnoisecomplicates identification of further research questions. theextractionoffeaturepointssuchaszeros. Therefore, important physical structures may be missed. II. ANALYTIC ILLUSTRATING EXAMPLES To address the first challenge, Goto & Vassilicos [8] used the acceleration to define a set of feature points. Theyproposetousezerosoftheaccelerationvectorfield In this section, two 2D incompressible flows are con- (zero acceleration points (ZAPs))fortheanalysisoftwo- sidered: the Stuart solution of the inviscid mixing layer dimensional flows. The motivation for the definition of (Sec. IIA) and the Oseen vortex pair (Sec. IIB). These ZAPswastofindaframemovingwithvortices,suchthat analytical examples show that local minima of the total the persistence of streamlines is maximized. However, acceleration magnitude are good indicators of vortices also the extraction of physically meaningful zeros of the and saddles. These results motivate the definition of ac- accelerationisacomplextask–especiallyinthepresence celeration feature points as key elements of the feature of noise. extraction strategy elaborated in the next section. We In this paper, we investigate a time-dependent coun- includeasimplethree-dimensionalflow(Sec.IIC)inad- terpart of the fixed points of the velocity field topology. dition to the two-dimensional examples. Ourdefinitionisbasedonthreerequirements,namely(1) choosing a Lagrangian viewpoint, (2) requiring Galilean invariance and (3) having standard velocity topology as A. Stuart solution of the mixing layer limiting case for steady flows. It is shown that the minima of the acceleration magnitude, called acceleration AnincompressiblemixinglayerisdescribedinaCarte- feature points, fulfill these criteria. These points are sian coordinate system x = (x,y), where x and y repre- Galilean-invariant and their physical meaning is inferred sent the streamwise and transverse coordinate, respec- from the Jacobian. They form a superset of the afore- tively. The origin 0 is placed in one saddle. The velocity mentionedzeroaccelerationpointsbyGoto&Vassilicos. is denoted by u = (u,v), where u and v represent its x In contrast to their interesting work, our concept can be and y components, respectively. All quantities are nor- generalized to three dimensional flows, in particular to malized with half of the relative velocity difference and one-dimensional features. half of the vorticity thickness. The usage of minima enables us to use the powerful Targeting a simple analytical example, we consider a concept of scalar field topology and associated combina- streamwiseperiodicmixinglayerwithconstantwidth,as torialextractionmethods,whicharerobustagainstlarge described by the inviscid Stuart solution [12]: noiselevelsinthedata. Theapplicationofthesemethods sinh(y) enablestheusageofpersistenthomology[9]. Itserves(a) u = u + , (1a) c as a filter for the robust extraction in the first step and cosh(y) 0.25cos(x uct) − − (b) as a spatial importance measure for the acceleration sin(x t) v = 0.25 − , (1b) feature points. − · cosh(y) 0.25cos(x u t) c Asubsetoftheaccelerationfeaturepointscanbeinter- − − preted as vortex cores. Within our combinatorial frame- where u represents the convection velocity. c work, we track these points over time. The combination The Stuart vortices are depicted in Fig. 1 as stream- ofpersistencewiththelifetimeofthevortices,weareable linesusingplanarlineintegralconvolution(LIC)[13,14]. to discriminate short-living unimportant features from ThetoppicturerepresentsEqs.(1)andshowsthefamous long-living and dominant vortices. We therefore con- cat eyes in a periodic sequence of centers (vortices) and 3 y The saddles and centers of a Stuart solution are not 6 only zeros of the velocity field but also zeros of the material acceleration field a=D u=∂ u+u u. (2) t t 0 ·∇ Here, ∂ represents the partial derivative with respect t to time, the nabla operator and the dot the tensor ∇ · contraction. The acceleration zeros are derived from a -6 Galilean-invariantfieldanddonotdependonthechosen 0 1 u x y inertial frame of reference. Figure 2 illustrates the accel- 6 erationfieldasheightfield. Thezerosoftheacceleration 0 y x -6 0 1 u x y ||a|| 6 0 FIG. 2: The acceleration field of convecting Stuart vortices. The coloring at the bottom is determined by the vorticity. -6 The height field shows the acceleration magnitude and the 0 1 u -2π 0 2π x curves depict integral lines of the acceleration vector field. Theyellowsphereshighlighttheaccelerationminima,orange spheres vorticity maxima. Note that center-like acceleration FIG. 1: Stuart vortices in various convecting frames. The minima and vorticity maxima coincide (orange spheres hide meanvelocityprofileisshownattheleft. TheStuartvortices yellow spheres). are depicted by visualizing the instantaneous velocity field using line integral convolution. The coloring is determined field and the local minima of the acceleration magnitude by vorticity; more intense red corresponds to higher vorticity. Thecriticalpointsofstandardvelocityfieldtopologyare (yellowspheres)coincideinthisexample. Ingeneral,the displayed as red (centers) and green (saddles) spheres. The latterquantityisasupersetofthefirst. Theacceleration maxima of the vorticity are added as orange spheres. minima, however, enable to identify vortices and saddles in case of a non-uniform convection velocity. saddles for a vortex-fixed frame of reference (uc = 0). B. Oseen vortex pair The middle picture depicts the same structures but in a frame of reference moving to the left with the lower In this section, a pair of equal Oseen vortices in ambi- streamatvelocity( 0.7,0),or,equivalently,thevortices ent flow is considered. They rotate around the origin at − movingtotherightatuc =0.7. Thecentersandsaddles constantdistanceR=1/√2withuniformangularveloc- aredisplacedtowardstheslowerstream. Thebottompic- ityΩ. Letx =(x ,y ), i=1,2bethecentersofthetwo i i i ture illustrates the same flow with a frame of reference vortices. Then, moving at velocity ( 1.2,0), i.e. u = 1.2 in Eq. (1). c − Now, no zeros are observed. These pictures recall the x = R cosΩt, y =R sinΩt, (3a) 1 1 well-known fact, emphasized in many textbooks in fluid x = R cosΩt, y = R sinΩt. (3b) 2 2 mechanics, that velocity field topology strongly depends − − on the frame of reference, i.e. is not Galilean-invariant. The induced velocity uθ of a single Oseen vortex in the In case of the Stuart solution, one might argue that the circumferential direction θ is given by frame of reference convecting with the structures is the Γ (cid:16) (cid:17) most natural one. However, the convection velocity of a u (r)= 1 e−(r/rc)2 , (4) θ 2πr − jet and many other flows depend on the streamwise position, i.e., generally no single natural frame of reference where r is the distance from the center of the vortex, exists for topological considerations. r determines the core radius and Γ is the circulation of c 4 example would violate the Biot-Savart law, i.e. is not y physically realizable. C. Simple 3D flow r c 0 1 r c � 0 rc 0 rc x � 1 -1 0 0 FIG. 3: Depiction of two co-rotating Oseen vortices by their 1.5 instantaneousstreamlinesusinglineintegralconvolution. The color-coding is determined by the acceleration magnitude; 3 more intense red encodes higher values. The minima of the acceleration magnitude are marked by yellow spheres. For comparison,thecriticalpointsofstandardvelocityfieldtopol- FIG. 4: Depiction of a simple three-dimensional stationary ogy (green and red spheres) and the maxima of the vorticity flow field. The particle trajectories are indicated by the il- (orange spheres) are added. luminated streamlines. In the center, the vortex core line is extractedasminimallineoftheaccelerationmagnitude. Note thattheparticlesalongthevortexcorelinehavenon-zeroac- celeration. the vortex. In our example, r is chosen as 0.5 and Γ as c 2π. The velocity of each vortex, cf. Eq. (3), corresponds Following Wu et al. [15], we consider a simple linear to the induced velocity, cf. Eq. (4), i.e. RΩ = u (2R). θ velocity field with swirl and strain. The velocity field is The diffusion of vorticity is ignored to keep this analyti- given by calexamplesimple. Figure3illustratestheOseenvortex pair as described by Eq. (3) and Eq. (4). Here, the in- u = 0.75x 100y, stantaneous streamlines can be inferred from the planar − − LIC. v = 0.75y+100x, (5) − Redandgreenspheresmarkthecorrespondingcritical w = 1.5z. points of the velocity field. Regions of large acceleration are marked by red. The minima (zeros) of the accelera- The material acceleration a=(au,av,aw) of this field is tionfieldaredenotedbyyellowcircles. Itshouldbenoted given by that the maxima of the vorticity (indicated by orange a = 9999.4375x+150y, spheres) are much closer to the minima (zeros) of the u − acceleration than to the zeros of the velocities. This dif- a = 9999.4375y 150x, (6) v − − ference is insignificant for ’frozen’ vortices but increases a = 2.25z. w withincreasingangularrotation. Hence,thedifferenceis correlated with the radial acceleration of the vortex mo- In Fig. 4, some streamlines of this field are randomly tion. Only a non-inertial co-rotating frame of reference placed in the volume. One streamline is emphasized minimizesthedifferencebetweenvorticitymaximumand showing swirling motion around the vortex core line (z- the center of the velocity field. axis). The points on the core line are minima of the It may be noted that any definition of feature points acceleration magnitude of the flow field with respect to is based on the concept of an ’idealized template’, like a a two-dimensional cross-section. Using the terminology slowly accelerating saddle or vortex — as compared to of three-dimensional scalar field topology, such lines are the surrounding acceleration maximum. And any defini- called valley lines or minimal lines. They can be ex- tioncanbechallengedbyconstructedlimitingexamples. tracted from the acceleration magnitude field without In the example of an Oseen vortex pair, the limit of in- prior knowledge of the corresponding cross-sections. In finitely fast rotation of the vortices around their center this example, the values of the acceleration are non-zero constitutes such an example. However, this kinematic everywhere along the line. While for two-dimensional 5 fields, the centripetal acceleration of a vortex always in- duces a zero point, this characterization is not trans- ferable to the three-dimensional case. Now, minimal lines provide a meaningful generalization of the two- dimensional feature points to the three-dimensional setting. Velocity III. FEATURE EXTRACTION Our feature extraction pipeline consists of three steps: (1) spatial feature extraction, resulting in isolated feature points; (2) computation of the temporal evolution of these points; and (3) spatiotemporal filtering of the Material Acceleration Magnitude resulting structures, cf. Fig.5. First (Sec. IIIA), distinguished feature points are defined. One important implication of this definition for the pressure field is described in Sec. IIIB. Finally, we describe a mathematical model for the extraction and tracking algorithm (Sec. IIIC). Minima = Lagrangian Equilibrium Points A. Acceleration feature points In this section, the definition of the considered feature is introduced. Starting point is a critical review of the velocitysnapshottopology. Topologicalanalysisofveloc- ityfieldshasbeensuccessfullyappliedforexaminationof flow fields with a distinguished frame of reference. How- Tracked and Filtered LEPs ever, its applicability is limited, as location and number of critical points depend on the frame of reference. The goal of the current study is a definition of an alternative featureconcept,whichgeneralizesthesnapshottopology FIG. 5: Pipeline of the proposed approach: After computing in a local sense and overcomes the above-mentioned lim- theaccelerationmagnitudefieldfromthevelocity,itsminima itations. Thefeaturepointdefinitionismotivatedbythe are extracted, which are referred to as Lagrangian Equilib- observations in the previous examples and the following rium Points (LEPs). These LEPs are tracked over time and three requirements: prioritizedbyaspatiotemporalimportancemeasure. Theim- portancemeasurecombinesaspatialstrengthandthelifetime of the feature. (R1) Correspondence to velocity topology: A flow field iscalledsteady,ifthereexistsadistinguishedframe of reference for which the vector field is stationary, i.e., it does not change in time. Such flow fields (R2) Galilean invariance: A Galilean-invariant feature consist of frozen convective structures. They sat- identifier reveals the same structures when chang- isfy Taylor’s hypothesis [16]. With respect to this ing the frame of reference. An example illustrating distinguished frame of reference, critical points of the influence of a Galilean transformation on the the velocity field correspond to the position of vor- velocity field and its feature points is illustrated tex cores and saddles. This concept is not applica- in Fig. 6. Let v(x,t) be a time-dependent vec- ble to unsteady flow fields, since there is no such tor field with a single Oseen vortex, Eq. 4 with distinguishedframeofreference. Aimingforagen- r = 2, Γ = 8π, convecting with constant velocity c eralization of velocity topology, the newly defined (x (t)=v t,y (t)=0) from left to right. The in- 0 0 0 featurepointsshouldcoincideforsteadyflowfields stantaneous velocity field for one point in time is withthezerosofthevelocityfield. Thisalsomeans shown in Fig. 6 (a) including the maximum of the that the classification of the points as saddles or vorticity (orange) and the fixed point (red). The centers is preserved. Note that this requirement is two points do not coincide. Since this is a steady not fulfilled by Haller’s definition of an objective flow field, there exists a distinguished frame of ref- vortex [17]. Rotational invariant features cannot erencemovingwiththecenter,thatcanbereached distinguish saddles and centers. by a Galilean transformation. The resulting veloc- 6 of acceleration zeros and these zeros are a superset of the critical points of the velocity field. Hence, acceleration feature points can be considered to be a generalization of the critical points of the velocity fields. Accelera- tionisaGalilean-invariantquantity. Itiscomputedfrom the velocity using the material or Lagrangian derivative thatlinkstheEuleriantotheLagrangianviewpoint[18]. Moreover, acceleration feature points satisfy all requirements R1 to R3. The acceleration feature points can exhibit vortex- as well as saddle-like behavior, depending on the eigenval- (a)Convectingcenter (b)Distinguishedframeof uesofthevelocityJacobian. Realeigenvaluescorrespond reference to saddles while a complex-conjugate eigenvalue pair in- dicates a vortex. Alternative synonymous discriminants FIG. 6: Convecting Oseen vortex displayed with respect to two different reference frames. A dense streamline visual- have been proposed for 2D flows: Goto & Vassilicos [8] ization builds the background texture for both images. The show that saddles are associated with sources of the ma- maximaofthevorticityaremarkedbyorangespheres. They terialaccelerationfieldwhilevorticescorrespondtosinks. coincidewiththeaccelerationminimawhicharethereforenot One advantage of their definition is that it relies purely shown in both images. The fixed-points of the velocity field on the acceleration without reference to the velocity Ja- are represented as a red sphere. The vorticity maximum re- cobian. Basdevant & Philipovitch [19] critically assess vealsavortexcoremovingalongthex-axis. Inimage(a),this the use of the Weiss criterion as discriminant. pointdoesnotcoincidewiththecenterofthestreamlines. Ap- plying a Galilean transformation such that the resulting flow issteadyleadstoaconvectedframeofreference(b)revealing a different picture. Now the fixed point coincides with the B. Implications of acceleration feature points maximum of vorticity and the minimum of the acceleration. Another perspective onto the acceleration minima is given by their relation to the pressure gradient via the ity field is shown in Fig. 6 (b). While the flow be- incompressible Navier-Stokes equation: havior itself is not changed by the transformation, the visual output is different. In particular, both 1 feature points now coincide. a(x,t)= p(x,t)+ν∆u(x,t), −ρ ∇ (7) (R3) Lagrangian viewpoint: To guarantee a physically 0= u(x,t), ∇· sensible feature identifier, we focus on particle motion. This Lagrangian viewpoint implies the focus where p is the pressure of the flow field, ρ and ν are the on Galilean-invariant properties of fluid particles, kinematic viscosity and density of the fluid, respectively, but it does not include tracking finite-time fluid and ∆ is the spatial Laplacian operator. For ideal flows, particle motion. This restricted Lagrangian view- the equations reduce to the Euler equation: point is consistent with the general notion of ’La- grangian coherent structures.’ 1 a(x,t)= p(x,t), −ρ ∇ (8) These requirements and the observations from Sec. II 0= u(x,t). suggest to relate feature points to the material accelera- ∇· tion field. The particle acceleration a is the total deriva- Then, local extrema of the pressure field, which are zero tive of the flow field u. In other words, the acceleration points of the pressure gradient coincide with zeros of the in a space-time point (x,t) is given by Eq. (2). accelerationfield. Inthiscase,theabovedefinedacceler- Definition: Aminimumofthemagnitudeofthemate- ation feature points form a superset of local extrema of rial acceleration a is called acceleration feature point. (cid:107) (cid:107) the pressure field, the minima of which are often associ- Such points can be classified on the basis of the Jaco- ated with vortices. bian of the velocity field, u. A feature point is called ∇ saddle-like if its eigenvalues are real and center-like if its eigenvalues are complex. A feature trajectory is defined C. Feature Point Extraction Strategy by the temporal evolution of a minimum in the acceleration field. In the following, this definition is shown to satisfy re- A second task, besides the definition of physically ap- quirements R1 to R3. Let x be a zero of the velocity propriate feature points, is the selection of a mathemat- 0 field u(x ,t) 0. This implies that the material acceler- ical model that serves as basis for an algorithmic imple- 0 ≡ ationa =(∂ u+u u) =0vanishesatx . Thus, mentation. The selection of a mathematical framework |x0 t ·∇ |x0 0 the minima of the acceleration magnitude are a superset is guided by the following criteria: 7 (C1)Itshouldfacilitatearobustandefficientextraction high feature density are statistical methods. Another without subjective parameters to enable an unsu- way, pursued in this work, is to facilitate an importance pervised extraction of the structures. measure to build a feature hierarchy. A commonly used importance measure for critical points in context of (C2) It should allow to generate a feature hierarchy scalar field topology is persistent homology [9]. based on an intrinsic filtering mechanism. This Persistence measures the stability of critical points eases the interpretation of the results and becomes with respect to changes in the data as introduced by necessary as soon as one moves on from simple noise, for instance. To do so, persistence analyzes the showcases or when the data exhibit high feature topological changes of the sublevel sets of a scalar func- densities. tion. At critical points, the topology of the sublevel sets (C3) It should allow for tracking of features over time, changes by increasing the sublevel parameter. In two di- based on neighborhood relations. mensions, there are four events possible: First, a new connected component in the sub-level set can be born. From a mathematical point of view, extrema and ridges This happens at a minimum. Furthermore, two con- of scalar fields can be subsumed under the framework nected components can merge, which occurs when the of scalar field topology. This provides access to many parameter t passes the value of a saddle. Third, a new powerful algorithmic tools developed for extraction and hole can also be born at a saddle. Last, a hole in a con- tracking of topological structures in scalar fields. The nected component can die. This occurs at a maximum. algorithm to extract minima of scalar fields used in this paperisbasedondiscrete Morse theory [20]. Itispurely The persistence value of the critical points is determined in the following way: A new-born connected com- combinatorial and guarantees topological consistency ponent is labeled with the associated minimum. At a of the extracted structures [10]. The robustness of saddle,twoconnectedcomponentsmergethatarelabeled the algorithm and lack of any algorithmic parameter with two different minima. The persistence value for the allow an unsupervised extraction of structures. This saddle and the minimum with the higher scalar value guaranteestheapplicabilityofthemethodstolargedata is defined as their scalar value difference. The merged sets. In the following, we will briefly describe the used component is labeled with the remaining minimum. A methods and summarize the concept of discrete Morse maximumispairedwiththesaddlewiththehighestfunc- theory. tion value and, again, their persistence value is defined Combinatorial extraction of two-dimensional scalar as their scalar value difference. field topology – Typically, scalar field topology is ex- Algorithmically, persistence can be computed in two tracted by analyzing the gradient of the scalar function. dimensionsbyanapproachthatcanalsobeusedtocom- This approach utilizes derivatives that amplify noise pute the matching, see Ref. [10]. For two-dimensional when computed in a discrete setting. In contrast, fields of reasonable resolution, the implementation com- combinatorial methods do not rely on interpolation or putespersistenceinafewseconds. Thisenablesustoan- derivatives. Therefore, we decided to use a combinato- alyze time-dependent fields with many time slices. The rial setting, here, following the ideas of Reininghaus et advantage of this algorithm is that it not only computes al. [10]. Note that our data is given on a polygonal grid the persistence values for the critical points but it is also for each time slice. able to simplify the combinatorial gradient field follow- In the following, we briefly describe how to extract ing the hierarchy introduced by persistence. This can be scalar field topology using the aforementioned approach: usedtoreducenoiseandthereforefacilitatesthetracking The grid that holds the data is transferred to a polyg- of the critical points in the next step. onal graph. In this graph, each node represents a Notethatcriticalpointsthatarepairedbypersistence d-dimensional face (point, edge, polygon) and each link are not necessarily adjacent. For a simple example representstheconnectionbetweentwoadjacentfaces. In illustrating the concept of persistent homology, we refer accordance to Forman [20], the combinatorial gradient the reader to Figure 7(a). field is given as a matching on this graph. The critical points are represented by the unmatched nodes and Temporal feature development – To get an under- the streamlines are alternating paths with respect to standing of the temporal evolution of acceleration the matching. An optimization algorithm enforces the feature points, the minima are tracked over time. In the matching to represent the actual analytic gradient field context of discrete Morse theory, one can make use of best. Loosely speaking, it has to be assured that a com- combinatorial feature flow fields (CFFF), as proposed binatorial stream line corresponds best to the analytical in Ref. [11]. The basic idea of this tracking algorithm stream lines. A couple of algorithmic implementations is to construct a discrete gradient field in space-time, of this idea have been proposed; we follow the approach describing the development of the acceleration minima, presented by Robins et al. [21]. such that tracking of those minima results in an integration of the discrete gradient field. The approach Generating a feature hierarchy for acceleration fea- constructs two fields: A forward and a backward field ture points – One way to approach the problem of a representingthedestinationofacriticalpointinforward 8 or backward direction, respectively. Two minima of layer and center of the orifice for the jet. The velocity two adjacent time slices are uniquely connected to each u = (u,v) is expressed in the same system, u and v be- other, if they are connected in the forward as well as in ing the x- and y-components of the velocity. The time is the backward field. Intuitively, these two minima are denoted by t, the pressure by p and the material acceler- uniquely connected if they fall into each other’s basin, ation by a. All quantities are non-dimensionalized with see Figure 7(b). The result of this tracking is a set of a characteristic length-scale L, a characteristic velocity temporal feature lines without mergers and splits. By U and the density of the fluid ρ. L denotes the cylin- furtherincorporatingtheaforementionedspatiotemporal der diameter for the wake, the vorticity thickness for the gradient fields, we were able to extract the mergers that mixing layer, and the width of the origin for the planar occur for vortex core lines in a two-dimensional setting. jet. U correspondstotheoncomingflowforthewake, to To do so, we utilize the forward tracking field to connect the velocity of the upper (faster) stream for the mixing the end of a uniquely tracked minima line to another layer, and to the maximum velocity at the orifice for the minima line in space time. The general approach is jet. described in [22]. Generating a feature hierarchy for temporal feature A. Cylinder wake lines – A typical importance measure applied for time dependent features is the concept of feature lifetime. Starting point is a benchmark problem of the data While leading to much simpler results, such filter visualization community: periodic vortex shedding be- methods are purely based on temporal measures. They hind a circular cylinder. The Reynolds number is set ignore pronounced short-lived features, which can play a to 100, which is well above the critical value for vor- significant role for the flow. Therefore, the temporal im- tex shedding at 47 [23, 24] and well below the criti- portance measure, given by the feature lifetime, should cal value for transition-related instabilities around 180 be combined with a spatial feature strength. The CFFF [25, 26]. The flow is simulated with a finite-element approach allows for a straightforward incorporation method solver with third-order accuracy in space and of persistent homology as spatial importance measure time,likein[27]. Therectangularcomputationaldomain for feature lines. In this paper, the spatiotemporal (x,y) [ 10,30] [ 15,15]withoutthediskK (0)for 1/2 importance measure is defined by integrating persis- ∈ − × − thecylinderisdiscretizedby277,576triangularelements. tence along the feature line, e.g., by accumulating all The numerical time step for implicit time integration is persistence values along the line. Since the measure is 0.1,whichalsocorrespondstothesamplingfrequencyfor not normalized, the lifetime of the feature is inherently the snapshots. depicted by this measure – longer living features are Figure 8 shows five vorticity related quantities of a more important if all structures have the same spatial cylinder wake snapshot. The vorticity field depicts the strength. Using this importance measure, it is possible separating shear-layers rolling up in a staggered array to filter out short-living weak features. of alternating vortices. The yellow balls mark the extrema, revealing the known fact that the ratio between the transverse of vortex displacement and the wave- IV. RESULTS FOR FREE SHEAR FLOWS length slightly increases downstream with vortex diffusion. The second subfigure shows the Okubo-Weiss pa- Three free shear flows are investigated: the cylinder rameter Q = S− 2 S+ 2 marking the maxima with (cid:107) (cid:107) −(cid:107) (cid:107) wake (Sec. IVA), the mixing layer (Sec. IVB), and the balls. This parameter employs the velocity Jacobian u ∇ planarjet(Sec.IVC).Theseconfigurationsrepresentdif- and compares the norm of the symmetric shear tensor (cid:104) (cid:105) ferent levels of spatiotemporal complexity from the peri- S+=1 u+( u)t andwiththenormoftheantisym- 2 ∇ ∇ odicwaketothebroadbanddynamicsandvortexpairing (cid:104) (cid:105) of the mixing layer and jet. The first two flows share a metric one S−=21 ∇u−(∇u)t . In the center of a ra- pronounced uniform far-wake convection velocity, while dially symmetric vortex, Q= ω 2 >0, since S+ van- the jet structures move slower with streamwise distance. ishes and S− becomes the n(cid:107)or(cid:107)m of the vort(cid:107)icity(cid:107) ω . (cid:107) (cid:107) (cid:107) (cid:107) Focus is placed on the vortex skeleton as identified by At a saddle point Q = S+ 2 < 0. Hence, maxima −(cid:107) (cid:107) theaccelerationfeaturepoints. Parsprototo,weperform of Q can be associated with vortex centers and minima astatisticalanalysisforthewake(Sec.IVA),investigate with saddles. The third subfigure shows λ . Its extrema 2 the vortex merging of the mixing layer (Sec. IVB), and are marked by balls and indicate vortex centers. Q and employ the persistence-filter of LEPs for the jet (Sec. λ aregenerallyconsideredtoprovidesynonymousinfor- 2 IVC). mation. The absolute value of the imaginary part of the All flows are described in a Cartesian coordinate sys- Jacobian uisshowninthefourthsubfigure. Thisquan- ∇ tem x=(x,y), of which the abscissa x points in stream- tity characterizes the angular frequency of revolution of wise direction and y in transversal direction. The origin a neighboring particles. Hence, its maxima marked by islocatedinthesourceoftheshearflow,i.e.centerofthe yellow balls indicate vortex centers. Finally, the magni- cylinder for the wake, center of the inflow for the mixing tude of the material acceleration field is depicted. The 9 f(x) ‖a‖ t = t 1 f(x ) 4 f(x ) 2 x f(x ) 3 t = t +Δt 1 f(x ) 1 x x x x x x 1 2 3 4 (a) (b) FIG. 7: (a): The employed spatial feature importance is given by the persistent homology of the critical points. It measures howstrongaminimumis,comparedtoitsneighbors. Thisisachievedbycorrectlypairingcriticalpointsandquantifyingtheir height difference. The image shows such a pairing, e.g., the critical points at x and x . (b): To extract the evolution of the 2 3 acceleration feature points in time, the minima are tracked. The applied method utilizes the concept of combinatorial feature flow fields (CFFF). In this setting, two minima of adjacent time slices are uniquely connected, if they lie in the basin of each other. minima(zeros)markbothvortexcentersandsaddles,i.e. pect. twiceasmanypointsinthevortexstreet. Thesetwofea- tures are distinguished based on the velocity Jacobian: two positive eigenvalues of the velocity Jacobian are as- B. Mixing layer sociated with a saddle, a complex conjugate pair with a vortex. The second investigated shear flow is a mixing-layer In the vortex street, all five vortex criteria provide with a velocity ratio between upper and lower stream of nearly identical locations. In the boundary-layer and in 3:1,followingearlierinvestigationsoftheauthors[28–30]. thenear-wakeofthecylindertherearepronounceddiffer- Theinflowisdescribedbyatanhprofilewithastochastic ences. Mathematically, the vortex criteria rely on quite perturbation. And the Reynolds number based on max- different formulae. They cannot be expected to exactly imum velocity and vorticity thickness is 500. The flow is coincide except for pronounced flow features, e.g. axis- computed with a compact finite-difference scheme of 6th symmetric vortices. In addition, the cylinder boundary order accuracy in space and 3rd oder accuracy in time. introduces a singular line u = 0, thus amplifying the The computational domain (x,y) [0,140] [ 28,28] is differences between the vortex criteria. ∈ × − discretized on a 960 384 grid. The sampling time for Figure 9 shows the spatial-temporal vortex evolution, the employed snapsho×ts is ∆t=0.05 corresponding 1/10 based on the tracked acceleration feature points. In of the computational time step. thefar-wake, auniformlyconvectingvonKármánvortex In contrast to the space- and time-periodic Stuart so- streetisobserved. Inthenear-wake,theconvectionspeed lution, the mixing layer generally shows several vortex is significantly slower. This aspect is highlighted in Fig. pairing events. In Fig. 11, the distance between vortex 10 (first subfigure). The streamwise velocity of each vor- acceleration feature points (marked by balls) are seen to texu isamonotonicallyincreasingfunctionfrom0.03to v increaseinstreamwisedirectionasresultofvortexmerg- about 0.85. The asymptote corresponds to the literature ing. Furthermore, the locations of the acceleration fea- value[26]. Thetransversespreadingofthevortexstreet, ture points nicely correlate with the local maxima of the noted in Fig. 8, is quantified in the following subfigure vorticity(top),thelocalminimaofthepressure(middle) with the transverse location y . v and the local minima of the magnitude of the material It should be noted that tracked acceleration feature acceleration (bottom). The correlation between vortic- points can be seen as markers of coherent structures. ity maxima and pressure minima in free shear flows is The acceleration-based framework provides a convenient well documented in the literature. The correlation be- means for determining convection velocities and evolu- tween pressure and acceleration magnitude minima may tion of spatial extensions. The following investigations be inferred from the non-dimensionalized Euler equation of the mixing layer and the jet flow emphasize this as- a = p, governing the predominantly inviscid dynam- −∇ 10 t y 2 Vorticity 0 30 -2 y x 2 Q 0 20 -2 y x 2 �2 0 10 -2 y x 2 Angular velocity 0 -2 0 10 20 x y x 2 Acceleration magnitude FIG. 9: Tracked vortices of the cylinder wake in an x-t-view. Red (blue) marks positive (negative) rotation the vortices. 0 u -2 v 0.85 0 4 8 12 16 x 0 y v x FIG. 8: Visualization of a cylinder wake snapshot. Five 1 vorticity-related quantities are depicted by color maps (red: positive values, blue: negative, gray: zero): (1) vorticity; (2)Okubo-Weissparameter; (3)λ ; (4)absolutevalueofthe 2 imaginary part of the eigenvalues of the velocity Jacobian – 0 correspondstotheangularvelocity;(5)materialacceleration 10 20 x magnitude. The yellow spheres depict the extremal points typically used as features for the respective quantity. -1 icsofthemixinglayer(seeSec.III).Apressureminimum (or maximum) implies p=0 and thus a=0. ∇ FIG.10: Plotsofthestreamwisevelocitycomponentuv (top) The vortex merging events are shown in Fig. 12. and the transverse displacement yv (bottom) along tracked vortices. Note that each figure contains the history of many Upstream, many Kelvin-Helmholtz vortices are formed. vortex evolutions from roll-up to convection out of the do- In streamwise direction numerous merging events can be main. Hence, several lines can be seen in each curve. identified, approximately 2 successive vortex mergers in the domain shown. Not all crossing of x,t-curves mark mergers since vortex pairs may rotate around their cen- C. Planar jet terbeforeeventualmerging. Thefigurestronglysuggests a nearly constant streamwise convection velocity, as expected from literature results and contrary to the cylin- Finally, the spatiotemporal evolution of the planar jet der wake dynamics. is investigated. Like the mixing layer, the jet shows