TSINGHUA SCIENCE AND TECHNOLOGY ISSNll1007-0214ll02/08llpp125-136 Volume 18, Number 2, April 2013
Methods to Identify Individual Eddy Structures in Turbulent Flow
Shengwen Wang, Jack Goldfeather, Ellen K. Longmire, and Victoria Interrante
Abstract: Turbulent flows are intrinsic to many processes in science and engineering, and efforts to elucidate the physics of turbulence are of critical importance to many fields. However, ongoing efforts to achieve a fundamental understanding of the mechanisms of turbulent flow are hindered by the difficulty of quantifying the complex, non- linear interactions between individual eddies in these flows. The difficulty of this task is compounded by the lack of robust methods for accurately identifying individual eddy structures and characterizing their dynamic evolution and organization across multiple scales. In this paper we address this problem by proposing several novel approaches for more accurately segmenting individual eddy structures in turbulent flows.
Key words: flow visualization; segmentation; vortices; turbulent flow
organization and performance in practical devices. While many methods have been proposed for 1 Introduction defining and visualizing eddies in 3-D flow data[3], Three-dimensional (3-D), time-varying turbulent flows little attention has been devoted to the problem of have been a subject of intense research for many robustly ensuring that the identified regions accurately years because of their critical importance in many correspond to individual eddy structures, as opposed applications. A long-term goal of the ongoing research to containing entangled clusters of closely spaced and in the turbulence community has been to develop an potentially intertwined vortices. However, the success improved understanding of the dynamically-evolving of efforts to effectively analyze the flow dynamics at eddy structure and organization in various canonical the level of individual structures critically depends on flows, including turbulent boundary layers[1, 2]. Such this ability. improved understanding is critical to the development We begin this paper by briefly reviewing current and validation of accurate numerical models of these methods for extracting features and defining vortices flows, which are needed for predictive purposes in 3-D turbulent flows, and explaining their limitations in engineering and environmental applications. This for our purposes. We then outline two novel approaches improved understanding also has the potential to enable that we have begun to pursue in our efforts to achieve the development of effective strategies to control eddy a robust segmentation of a 3-D flow dataset into individual vortex regions. The first approach is based on Shengwen Wang and Victoria Interrante are with the Department of Computer Science and Engineering, University the intuition that while any single scalar measure may of Minnesota, Minneapolis, MN 55455, USA. E-mail: not provide enough information by itself to robustly [email protected]; [email protected]. define an appropriate segmentation of a composite Jack Goldfeather is with the Mathematics and Computer structure into individual eddies, it is possible that we Science Department, Carleton College, Northfield, Minnesota, can do better by considering multiple, non-redundant, MN 55057, USA. E-mail: [email protected]. local scalar, and vector measures in combination. To Ellen K. Longmire is with the Department of Aerospace this end we present a careful mathematical analysis Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA. E-mail: [email protected]. of the interrelationships between swirl and vorticity, To whom correspondence should be addressed. and show how these measures can be used together Manuscript received: 2013-02-21; accepted: 2013-03-10 to resolve ambiguities in structure identification that 126 Tsinghua Science and Technology, April 2013, 18(2): 125-136 cannot be as successfully determined using either scalar quantity such as swirl strength, inevitably fail measure alone. In a second, different, approach to the to differentiate individual structures located in close problem, we introduce a novel method for automatically proximity. The three images in Fig. 1 illustrate how detecting potential compound structures in an initial reducing the threshold is insufficient to ensure the segmentation of a flow and coherently partitioning the separation of the individual eddies that otherwise appear constituent components. This method is based on the connected. combined use of vortex core lines and hierarchical In our first efforts to robustly identify individual region identification. The datasets used in this paper vortex structures in turbulent flow, we investigate the are extracted from a direct numerical simulation of extent to which it is possible to achieve a superior turbulent channel flow in Ref. [4]. segmentation of a flow dataset by using multiple scalar and/or vector flow features in combination rather than 2 Related Work relying on a single scalar feature alone. The first step Over the years, significant attention has been devoted in this process is to identify promising complementary to the problem of developing robust methods for flow features indicative of the presence of a vortex. identifying and effectively visualizing vortical We begin by observing that although Q, , and structures in general 3-D flows. Historically, these 2 represent different scalar measures, the regions efforts have been complicated by the lack of a robust, comprehensive, precise mathematical definition of what a vortex is. The most common method of vortex identification is based on the demarcation of regions of swirling flow by threshold levels of a scalar quantity indicative of swirling motion, typically derived from the velocity gradient tensor V . Among the suggested [5]r [6] criteria are: the Q criterion , the criterion , the 2 criterion[7], and swirl strength[8]. One complication in using this approach for automatic vortex identification (a) is the need to define an appropriate threshold level to demarcate the region boundary; the spatial extent of the vortex regions identified, as well as the extent to which weak vortices are captured, strongly depends on the threshold level chosen. The use of a scale-space approach, as suggested by Bauer and Peikert[9], offers one possibility to achieve increased robustness; an alternative is to focus on the identification of vortex core lines[10-14]. Other promising approaches are the use (b) of a predictor-corrector method[15-17] and methods that use Lyapunov exponents to define Lagrangian-coherent structures in time-varying flows[18-20]. 3 Our First Method In turbulent flows, the vortices that occur near the boundary often occur in densely intertwined, tightly packed clumps (see Fig. 1, which was obtained from a three-dimensional, high Reynolds number (Re = (c) 934), direct numerical simulation of turbulent channel Fig. 1 Swirling regions are identified at three different [4] flow ). In this data, we have found that methods theshold levels increasing from top to bottom at 3.5, 5.0, which identify regions of swirling flow, according and 8.0. Bottom of plotted domain is the bottom bounding to any single globally-defined threshold level of a surface. Flow is from left to right. Shengwen Wang et al.: Methods to Identify Individual Eddy Structures in Turbulent Flow 127 identified by isosurfaces of each of these quantities with s 2 t 2 1 be the associated eigenvectors of jNj C jNj D are somewhat similar in form in that they are long the Jacobian V of a velocity vector field V . Let be r narrow tubes[21]. Hence we investigate the potential of the real eigenvalue with associated real eigenvector r. N using vector valued features to differentiate neighboring Theorem 1 c n b, so that the angle  between N ı N D structures. c and n is given by N N b 3.1 Definition of vorticity and swirl cos  : D c n There are multiple choices for vectors that describe An immediate corollary toj Theorem Njj Nj 1 is as follows. rotation. For example, the vorticity of a velocity vector Corollary 1 The swirl b is always less than half of field V (also called the curl of V ) is defined as c N D the magnitude of the vorticity. V . The direction of this vector c determines the r N Hence regions with high swirl must also have high local axis around which rotation occurs as well as the vorticity. sense of the rotation (using a right-hand rule). The Theorem 2 Let 0 1 0 1 swirl of a velocity vector field is defined when two s .a /sTr bt Tr N N N N N of the eigenvalues of V are complex. This measure B Bt C ; w BbsTr .a /t TrC ; r D @ NA N D @ N N C N NA is designed to differentiate swirling motion about an r 0 axis from rotation due to a simple shear, both of N then which are identified by vorticity. In this case, there is 1 r c B w; a rotation in the plane determined by the two complex N N D N so that the angle ˛ between c and r is given by eigenvectors. There are two different ways to think of N N B 1w the axis of this rotation. On the one hand, it is common sin ˛ j N j: to think of the normal, n, to this plane as the axis of D r c N jNjj Nj rotation. On the other hand, the real eigenvector, r, Theorem 3 If any two of c, n, and r are in the same N N N N better defines the direction of the swirl isosurface tube direction, so is the third. (see Fig. 2 and Ref. [8]). We explore using c, r, and n These theorems suggest that we may be able to N N N to segment the vortices. gain insight into how turbulence evolves over time by exploring how the various rotation directions change 3.2 Relationship between vorticity and swirl relative to each other. Although c, n, and r all tell us something about N N N 3.3 Segmenting compound structures using rotation direction in 3-space, in general they each point vorticity and swirl in a different direction. In order to better understand how vortices change over time and to help us devise Figure 3 shows a series of images in which structures disentaglement algorithms, we need to know how have been segmented solely on the basis of scalar these vectors are related mathematically. The following values of swirl strength. The segmentation is defined facts can be established (proofs are provided in the as follows. We begin by choosing an aribtrary point appendix). whose swirl strength is above the threshold level. We then define the structure to which that point belongs by Let 1 a bi and 2 a bi with b > 0 be the D C D implementing a recursive flood fill algorithm, adding complex eigenvalues and v1 s ti and v2 s ti N DN C N N DN N a neighboring point to the structure if, and only if, its swirl strength is also above the threshold level. When we find that no additional points can be added to a structure, we repeat the process by choosing a new seed point and continue until all points in the volume, whose swirl values are above the threshold, have been tagged as belonging to one structure or another. The structures in each image are each displayed in a different color to clearly label the segmentation. We can see that the lower threshold yields smaller connected regions (e.g., Fig. 2 Swirl rotation axes. the dark green region in the larger circle in Fig. 3a), but it still remains in a compound form even after the 128 Tsinghua Science and Technology, April 2013, 18(2): 125-136
(a) (a)
(b) (b) Fig. 3 Individual vortices segmented according to a Fig. 4 Individual vortices defined by similar threshold levels threshold level of swirl strength (only), for two different of swirl strength, and by threshold differences between thresholds. Compound form in the bottom image still the swirl and vorticity directions at neighboring points. In remains by varying swirl strength only. comparison with Fig. 3, one can see fewer compound structures at each threshold level of swirl strength. threshold has been reduced. It also causes the loss of weaker eddy structures (such as the bright magenta consistency criteria does not result in the spurious eddy surrounded in the smaller circle). subdivision of coherent structures into fragments, even Figure 4 shows a series of images in which in cases where the vortices are quite bent in their overall structures have been successfully segmented into proper shape. constituent pieces both on the basis of swirl strength This suggests that there is inherently more promises and on the basis of the angles between the directions in an approach that uses both swirl strength and of c, n, and r at neighboring voxels. Specifically, the the consistency of swirl and vorticity directions N N N same flood fill approach is used to define a connected between neighboring points to identify individual region, but in this case a region is expanded to include vortex structures than in an approach that considers only a neighboring voxel only when the swirl and vorticity the level of swirl relative to a threshold. The problem directions at that voxel are within a threshold degree of automatically defining appropriate thresholds of both of consistency with the swirl and vorticity directions at the scalar quantities and vector differences to use in its neighboring points that have already been identified this process is a separate question, which we intend to as belonging to the region. It can be clearly seen that, address in future work. regardless of the particular threshold values chosen, 3.4 Our second method superior segmentation results are achieved using this combination of scalar and vector flow features than In this section, we introduce a novel method are achieved using swirl strength alone. It can also for automatically identifying individual (potentially be seen, by inspection, that introducing the directional compound) structures in a flow, then identifying Shengwen Wang et al.: Methods to Identify Individual Eddy Structures in Turbulent Flow 129 individual constituent eddies in each of these structures by leveraging fundamental known characteristics about the physics of flows. Specifically, we make use of the basic intuition that vortices do not branch. Each individual vortex in what appears to be a composite clump will generally be characterized by a separate core line. If we can robustly determine the core line locations, we should be able to use that information to accurately segment each cluster into its true consituent individual eddies as shown in an example of Fig. 5. Essentially, our method works by combining vortex core line detection with hierarchical region identification in Fig. 6. We begin by identifying a potentially compound structure defined by a connected set of voxels that correspond to locations in the flow where the swirl strength is higher than a specified threshold level. Examples of such structures are shown in red in Fig. 7 above, in the context of other surrounding structures (drawn in grey) within a small region of the flow; the same structure of the image in Fig. 7b is shown in the image in Fig. 8a, in a closer view without the Fig. 6 A flowchart describing our algorithm. surrounding structures. This initial structure is defined using a simple flood fill algorithm. After the subsequent steps are applied to this structure, another structure is selected, and the process repeats until all voxels in the volume have been classified as belonging to one eddy or another. Once we have identified a single connected region, shown in Fig. 5a, we begin the process of determining
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Fig. 5 An example shows the main steps of our Fig. 7 Vortices defined by a threshold level of swirl algorithm. (a) The identified primary region as an interesting strength. (a) A compound structure that is a region marked vortex. (b) The identified subregions as candidate sub- in red and is defined by a connected set of neighboring voxels vortices within the primary region. (c) Sub-vortices all with supra-threshold values. (b) Another compound correlated by extracted vortex cores. (d) Two identified structure that is an extremely complex connected region. segments shown as blue and red regions. 130 Tsinghua Science and Technology, April 2013, 18(2): 125-136
that the vortex core line does not extend beyond the boundaries of the domain. Second, we stop tracing a core line when the cross sectional area of the portion of the structure through which the core line is currently passing decreases to zero. Third, as a safety precaution, we terminate the growth of a core line if it reaches a total length that is more than twice as long as the longest side of the domain. After this, we successively consider subregions, within our initial structure, that are defined by (a) decreasing threshold levels of swirl strength, beginning from the maximum value and continuing down to the value that was used to define the extent of the initial structure. At each step in this hierarchical progression, as new subregions develop, we determine whether or not they should be classified as an extension (possibly disconnected) of the existing subregions or as an independent subregion, belonging to a different component vortex in the compound cluster. We make this determination on the basis of the paths taken by the vortex cores through each subregion. If the vortex (b) cores emanating from each region intersect the other Fig. 8 (a) A connected collection of voxels delimited by region, we consider the two sub-regions as belonging an isosurface of swirl strength. (b) The segmentation of to the same vortex as in Fig. 5c. Because the vortex this composite clump into individual eddy structures that is automatically produced by our method. core line paths can be noisy, we also need to consider additional circumstances. If one of the vortex core whether and how it should be subdivided. We start lines passes very close to but just misses intersecting by identifing sub-regions of higher thresholds in the the subregion from which the other emanates, and the vortex as shown in Fig. 5b. Then, we incorporate second subregion is very small, we also consider the the vortex line extraction by identifying a point in a two subregions to belong to the same structure. A region where the swirl strength assumes a maximal simple example in Fig. 5 shows steps of one iteration value. We take this point as the seed point from which when one threshold value is considered, and two a vortex core line is grown, in both directions. We segments are identified in Fig. 5d. We continue this define the vortex core line using a variation of the process until all predefined threshold values of swirl predictor-corrector method described by Banks and strength have been considered. [15] Singer that relies on vorticity and swirl strength The groups for each individual vortex core and each rather than vorticity and pressure for the prediction and of the remaining sub-vortices are all new candidate correction steps, respectively. This strategy is similar individual vortex cores. The expansion of the regions [16] to the approach by Stegmaier et al. that replaced from grid cells of individual candidate vortex cores pressure of the predictor-corrector method with 2 for is conducted by a recursive flood fill algorithm that more reliability and applicability. The swirl strength is applied to all sub-vortices simultaneously, and in our line-based method is the quantity that was also is also similar to the flood approach in our first used for hierarchical vortex region method. Thus, the method. Then, the grid cells within the compound identified core line by vortex core detection based on vortex are categorized into different groups, such that swirl strength benefits the overall procedure of our each group of grid cells belongs to one of the individual individual vortex identificiton. segmented structures. The criteria we use to determine when to terminate The previous stage produces a temporary the extension of a vortex core line in the procedure of segmentation result for a compound vortex. This vortex core detection are as follows. First, we ensure result has to be stored so that the program can start over Shengwen Wang et al.: Methods to Identify Individual Eddy Structures in Turbulent Flow 131 on another examination with similar processes. We can store these data in memory because the sub-domain we are examining is relatively small and the information we retrieve can be efficiently compacted . We successively consider subregions, within our initial structure, that are defined by decreasing threshold levels of swirl strength, beginning from the maximum value and continuing down to the one that was used to define the extent of the initial structure. We continue this process until all predefined threshold values of swirl srength have been examined. Our algorithm examines multiple candidate levels, one at a time, to insure that all potential individual vortex cores can be identified. When all levels have been examined, our algorithm proceeds to the next stage. (a) The final segmentation depends on the segmentations produced at all of the considered threshold levels. Our algorithm examines the results from each iteration and determines the individual vortices across each result. Sub-structures from all candidate levels with the same or closed vortex core lines are considered to belong to the same group. The final segmentation is determined by expanding regions from different final candidate sub-structures. The flowchart in Fig. 6 shows the complete steps of our algorithm. As a last step, we render each separate group of grid cells using a different color; all primitives are painted according to the color of their group. Our hierarchical region detection method (b) successfully segments a variety of compound structures into individual vortices, and paints each vortex using a different color. Hierarchical level regions provide premature hints for the candidate segmentation, and the corresponding core lines are used to connect different individual sub-elements together, as shown in Fig. 8. All possible levels are examined to determine the proper categorized regions. Figures 9-11 show more results. Figure 9 shows a clear success case of our algorithm; a formerly compound structure is successfully segmented into four component pieces. The three images illustrate two intermediate results at different hierarchical levels and a final image. In Fig. 9a, we can see that three groups of sub-regions are (c) identified by using multiple core lines grown from Fig. 9 A success case of our algorithm. The top and middle seed points located within the long blue, small cyan, images show the different segments that are identified at each and tiny red sub-regions. Figure 9b shows that at a intermediate level, and the bottom image shows the successful different threshold level, four groups of sub-regions are segmentation after multiple iterations. identified; note the additional tiny cyan blob. The 132 Tsinghua Science and Technology, April 2013, 18(2): 125-136
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Fig. 10 Another automatic segmentation produced by our algorithm. The top image shows one connected clump. The middle and bottom images show the segmentation result with eight individual vortices seen from different viewing angles. (c) green and cyan blobs in this figure were combined in Fig. 11 A failure case of our algorithm. Instability in the larger cyan region in Fig. 9a. The cyan structure the computation of the core lines leads to spurious in Fig. 9b is identified at the current swirl level but not divergences. This can be corrected for by locally comparing at the previous levels. A transparent layer in these two the vortex core line direction to the vorticity direction and terminating the core line following when the alignment images encompasses these candidate sub-regions and becomes poor. illustrates our interesting target clump. Different colors Shengwen Wang et al.: Methods to Identify Individual Eddy Structures in Turbulent Flow 133 of a transparent layer represent different individual of individual vortices in a complicated turbulent vortices correspoinding to the candidate sub-regions at flow. Our first method derives the underlying theoretical the current threshold level. Figure 9c shows the final relationships among different measures of rotation result of our algorithm. Our approach considers all direction in the flows. The possibility of using these possible sub-regions detected at each threshold level properties for distinguishing different structures of to make an overall determination of the final segments vortices inspires new directions for research. The that comprise a single coherent structure. combined use of these multiple measures enables the In Fig. 10, a compound structure identified in Fig. 7a superior segmentation of vortical structure compared has been automatically segmented into eight different with when only a single measure is used. Further subregions. Each of the component structures appears investigation of proper selection of thresholds and to be appropriately defined, with the possible exception advanced coherence of these measures with respect of the large red structure which may or may not remain to the fluid dynamics are left for future work. Our in a compound form. Figure 11 illustrates a failure second method is capable of successfully performing case of our algorithm that requires some further effort automatic segmentation on complex regions comprised to resolve. In Fig. 11b, we can see three core lines, of closely intertwined individual vortices that can colored red, yellow, and cyan, that were grown from not be distinguished by conventional types of vortex seed points located within the large red and yellow sub- identification methods. However, our method may regions and within a tiny cyan sub-region barely visible sometimes fail, particularly in situations where a near the upper end of the yellow colored segment. In continuous vortex core line can not be detected in the this view, the yellow core line suggests a vortex that discrete data. The practical applications of our methods extends from the upper left to the lower right of the to time-varying datasets will be addressed in future structure, and the red core line indicates a vortex that work. extends from the rear of the structure to the forward tip Acknowledgements at the lower left corner of the image. The cyan core line begins by closely following the approximate path taken This work was funded by the US National Science by the yellow core line, but then diverges (presumably Foundation (No. CBET-0324898). as a result of numerical error) and eventually joins Appendix up with the red core line. Although this divergence is We begin by stating some well-known facts from linear difficult to detect at a local level as it occurs, due to the algebra. Thinking of a vector v as a 3 1 matrix and its universally noisy nature of the core line, we believe that N transpose vT as a 1 3 matrix, the following can easily there is good potential to flag such occurrences during N be proved: a post process by double checking the correspondence Lemma 1 A. vTv v 2. between the direction indicated by the core line at N N D j Nj A. vTv v 2. any point and the directions indicated by the vorticity N N D j Nj B. vTw wTv v w. vectors at the surrounding points. If these directions N N DN N DN ı N C. u .v w/ v .w u/ w .u v/. are wildly discordant, we can not have confidence in N ı N N DN ı N N DN ı N N D. v .w v/ 0. the robustness of the path followed by the core line, and N ı N N D More interesting is the following result. it should probably be truncated. Suppose the matrix A = V has one real eigenvalue r 4 Conclusions with real eigenvector r and a pair of complex conjugate N eigenvalues and 1 a bi and 2 a bi with b > In this paper, we have presented two algorithms D C D 0, with associated complex eigenvectors v1 s ti that can be used to identify individual vortex N DN C N and v2 s ti. We can assume the eigenvectors are of strutures in volumetrially-defined 3-D turbulent flow N DN N 2 2 2 unit length so 1 v1 s t . Then, we can datasets. These methods can be of use both to D j N j D jNj C jNj derive researchers who seek to quantify information about Av1 Av2 As N C N a flow at the level of individual structures, and N D 2 D also to those who want to be able to automatically .a bi/.s ti/ .a bi/.s ti/ C N C N C N N as bt .1/ track the evolution and interaction of large numbers 2 D N N 134 Tsinghua Science and Technology, April 2013, 18(2): 125-136
Similarly, we can show Finding the angle between c and r is a bit more N N At bs at: .2/ complicated. First we note that Ar r implies: N D N C N N D N 0 1 0 1 sTAr .sTr/; s1 t1 N N D N N Lemma 2 Let s Bs C, t Bt C, and y t TAr .t Tr/ .4/ N D @ 2A N D @ 2A N D N N D N N s3 t3 Also from Eqs. (1) and (2) we can write 0 1 u sTATr rTAs arTs brTt; Bv C. Then N N DN N D N N N N @ A t TATr rTAt brTs arTt .5/ w N N DN N D N N C N N 0 1 Subtracting Eqs. (4) from Eqs. (5) we get 0 w v T T T T T s .A A/r .a /s r bt r; t B w 0 u C s y .t s/: N N D N N N N N @ A N DN ı N N T T T T v u 0 t .A A/r bs r .a /t r: N N D N N C N N Proof Using Theorem 1 together with Lemma 1C and 0 1 0 w v Lemma 1D we obtain t T B w 0 u C s s .r c/ .a /sTr bt Tr; N @ A N D N ı N N D N N N N v u 0 t .r c/ bsTr .a /t Tr; N ı N N D N N C N N t3. s2u s1v/ t2.s3u s1w/ t1. s3v s2w/ r .r c/ 0 .6/ C C C C D N ı N N D u.s3t2 s2t3/ v.s3t1 s1t3/ w.s2t1 s1t2/ If we let C C D 0 1 y .t s/: s1 s2 s3 N ı N N B Bt t t C ; We apply these results using Eqs. (1) and D @ 1 2 3 A (2). Multiplying both side of Eq. (1) by t T and r1 r2 r3 N both side of Eq. (2) by sT we obtain x r c; N N DN N T T T 0 1 t As at s bt t; .a /sTr bt Tr N N D N N N N N N N N sTAt bsTs asTt: w BbsTr .a /t TrC .7/ N N D N N C N N N D @ N N C N NA Subtracting the first equation from the second we 0 obtain we can rewrite Eqs. (6) as T T 2 2 s At t As b. s t / .3/ Bx w: N N N N D jNj C jNj N DN Using Lemma 1B, we rewrite sTAt as t TATs. Hence Hence r c B 1w. Then the angle  between r and c N N N N N N D N N N the left side of Eq. (3) becomes can be found using r c r c sin Â. We summarize 0 1 jN Nj D jNjj Nj 0 Qx Py Rx Pz this as Theorem 2. T T B C Theorem 2 If B and w are as defined in Eqs. (7) t .A A/s t @Py Qx 0 Ry QzA s: N N N D N N then Pz Rx Qz Ry 0 1 r c B w: But this is precisely the left hand side of the equation N N D N in Lemma 2 with y c. We summarize this as We can now prove Theorem 3 which we separate into N DN Theorem 1. three parts. Theorem 1 Suppose 1 a bi and 2 a bi Theorem 3.1 If r and n have the same direction, D C D N N with b > 0 are the complex eigenvalues and v1 s ti then c is also in this direction. 2 2 N DN C N N and v2 s ti with s t 1 are the associated Proof Suppose r and n have the same N DN N jNj C jNj D N N eigenvectors of the Jacobian V of a velocity vector direction. Then r is in the same direction as n t s r N N D N N field V . Then c n c .t s/ b. s 2 t 2/ b, and hence r is perpendicular to both s and t. Then w N ı N DN ı N N D jNj C jNj D N N N N so that the angle  between c and n is given by in Theorem 3 is the 0 vector and so r c 0. But N N N N D N b r c 0 only if r and c are in the same direction. cos  : N N D N N N D c t s Theorem 3.2 If r and c have the same direction, j NjjN Nj N N Shengwen Wang et al.: Methods to Identify Individual Eddy Structures in Turbulent Flow 135 then n is also in this direction. c . .a /.s s/ b.t s// c .b b.s s/ .a /.t s//; 1 N 2 N ProofN If r and c have the same direction, then r NıN C ıN C NıN ıN N N N 0 c1.w1 t/ c2.w2 t/ c 0, so by Theorem 3, B 1w 0. Hence w D N ı N C N ı D N D N N D N N D B0 0. But using the definition of w in Eqs. (7) we c1. b .a /.s t/ b.t t// c2. b.s t/ .a /.t t/ N N NıN C NıN C NıN NıN can thenD write N .13/ ! ! ! a b sTr 0 We can write Eqs. (13) as NT N : ! ! b a t r D 0 c1 0 N N M ; The determinant of this matrix is .a /2 b2 0, c2 D 0 so it is invertible and hence both sTr 0 andCt Tr ¤ 0. where N But this means that r is perpendicularN N D to both sNandD t M N N N D ! and hence r is in the same direction as n t s. .a /.s s/ b.t s/ b b.s s/ .a /.t s/ N N D N N N ı N C N ı N N ı N N ı N : Theorem 3.3 If c and n have the same direction, b .a /.s t/ b.t t/ b.s t/ .a /.t t/ then r is also in this direction.N N N ı N C N ı N N ı N N ı N ProofN Suppose c and n have the same The determinant of M can be written as 2 2 2 direction. Then c N kn for someN scalar k, so by ..s s/.t t/ .s t/ /..a / b / N D N N ı N N ı N N ı N C Theorem 1 which is greater than .a /2 b2 by the Cauchy- C b c n kn n .8/ Schwarz inequality. Hence M is invertible so c1 DN ı N D N ı N D c2 0. This completes the proof. Replacing c in Eqs. (6) by kn and using Lemma 1 C we D obtain N N References r .kn s/ .a /sTr bt Tr; N ı N N D N N N N r .kn t/ bsTr .a /t Tr .9/ [1] R. J. Adrian, C. D. Meinhart, and C. D. Tomkins, Vortex N N N ı N D N N C N organization in the outer region of the turbulent boundary which can be rewritten as layer, J. Fluid Mech., vol. 422, pp. 1-54, 2000. 0 r .kn s .a /s bt/ r w1; [2] Q. Gao, C. Ortiz-Duenas, and E. K. Longmire, Analysis DN ı N N N C N DN ı N 0 r .kn t bs .a /t/ r w2 .10/ of vortex populations in turbulent wall-bounded flows, DN ı N N N N DN ı N J. Fluid Mech., vol. 678, pp. 87-123, 2011. Now kn s lies in the plane of s and t since N N N N N [3] W. Jiang, R. Machiraju, and D. Thompson, Detection and it is perpendicular to n so w1 also lies in this N N visualization of vortices, in The Visualization Handbook, plane. Similarly, w2 lies in the plane of s and t. If we C. Hansen and C. 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Cantwell, A general Also, since classification of three-dimensional flow fields, Physics of Fluids., vol. 2, no. 5, pp. 765-777, 1990. .kn s/ t kn .t s/ kn n b; N N ı N D N ı N N D N ı N D [7] J. Jeong and F. Hussain, On the identification of a vortex, .kn t/ s kn .t s/ kn n b; J. Fluid Mech., vol. 285, pp. 69-94, 1995. N N ı N D N ı N N D N ı N D [8] J. Zhou, R. J. Adrian, S. Balachandar, and T. M. Kendall, we obtain Mechanisms for generating coherent packets of hairpin w1 t b .a /.s t/ b.t t/; N ı N D N ı N C N ı N vortices in channel flow, J. Fluid Mech., vol. 387, pp. 353- w s b b.s s/ .a /.t s/ .12/ 396, 1999. 2 N N ı N D N ı N ı N [9] D. Bauer and R. Peikert, Vortex tracking in scale-space, To show linear independence we must show that in Proc. Joint Eurographics-IEEE TCVG Symposium on c1w1 c2w2 0 implies that c1 c2 0. But N C N D N D D Visualization, Barcelona, Spain, 2002, pp. 233-240. c1w1 c2w2 0 together with Eqs. (11) and (12) N C N D N [10] D. Kenwright and R. 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Shengwen Wang received his PhD in Ellen K. Longmire joined the Department the Department of Computer Science at of Aerospace Engineering and Mechanics the University of Minnesota in 2011, at the University of Minnesota in 1990. She where his research focuses on scientific received her MS (1985) and PhD (1991) visualization, in particular, 3-D turbulent degrees in mechanical engineering from flow visualization and high dimensional Stanford University, where she specialized data visualization. His recent interest also in experimental fluid mechanics. She uses includes development of bioinformatics experimentation and analysis to answer tools. He is currently an assistant researcher at the “National” fundamental questions in fluid dynamics that affect industrial, Center for High Performance Computing in Taiwan, China. biomedical, and environmental applications. Recently, her work has focused on single- and multi-phase turbulent flows, liquid/liquid mixtures with surface tension, microscale flows, and biomedical flows.
Jack Goldfeather is the William H. Laird Victoria Interrante is an associate Professor of Mathematics, Computer professor in the Department of Science, and the Liberal Arts in Carleton Computer Science in the University College, where he has been in the of Minnesota. Her recent research focuses department since 1977. He received his on the application of insights from MS (1971) and PhD (1975) degrees in perceptual psychology, art, and illustration mathematics at Purdue University. Over to the design of more effective techniques the years, he has done research in algebraic for visualizing 3-D data. She received her topology, chip design for rapid graphics display, tracking in PhD in computer science from the University of North Carolina virtual reality systems, and most recently in analyzing and at Chapel Hill in 1996. visualizing vortex behavior in turbulent fluid flow.