Life-cycle of streaks in the buffer layer of wall-bounded turbulence
aa r X i v : . [ phy s i c s . f l u - dyn ] F e b Life-cycle of streaks in the buffer layer of wall-bounded turbulence
H. Jane Bae ∗ School of Engineering and Applied Sciences,Harvard University, Cambridge, MA 02139, USA andGraduate Aerospace Laboratories, CaliforniaInstitute of Technology, Pasadena, CA 91125, USA
Myoungkyu Lee † Combustion Research Facility, Sandia National Laboratories, Livermore, CA 94550, USA (Dated: February 9, 2021)
Abstract
Streaks in the buffer layer of wall-bounded turbulence are tracked in time to study their life-cycle.Spatially and temporally resolved direct numerical simulation data is used to analyze the strongwall-parallel movements conditioned to low-speed streamwise flow. The analysis of the streaksshows that there is a clear distinction between wall-attached and detached streaks, and that theformer can be further categorized into streaks that are contained in the buffer layer and the onesthat reach the outer region. The results reveal that streaks are born in the buffer layer, coalescingwith each other to create larger streaks that are still attached to the wall. Once the streak becomeslarge enough, it starts to meander due to the large streamwise to the wall-normal aspect ratio, andconsequently the elongation in the streamwise direction, which makes it more difficult for the streakto be oriented strictly in the streamwise direction. While the continuous interaction of the streaksallows the super-structure to span extremely long temporal scales and streamwise dimensions,individual streak components are relatively small and short-lived. Tall-attached streaks eventuallysplit into wall-attached and wall-detached components. These wall-detached streaks have a strongwall-normal velocity away from the wall, similar to ejections or bursts observed in the literature.Conditional averaging of flow fields to these split events show that the detached streak has notonly a larger wall-normal velocity compared to the wall-attached counterpart, it also has a larger(less negative) streamwise velocity, similar to the velocity field at the tip of a vortex cluster. ∗ [email protected] † [email protected] . INTRODUCTION Among the many organized structures observed in near-wall turbulent flows, streaks,defined as regions of slowly moving fluid elongated in the direction of the mean flow, areconsidered to be of major importance for their role in the regeneration of turbulent energy.In the flow-visualization study by Kline et al. [1], it was established that important charac-teristics of the near-wall region of wall-bounded turbulent flows are the streak formation inthe viscous sublayer and the subsequent ejection of the low-velocity fluid to the outer regionof the flow.Flow visualization employing dye, particles, bubbles, and smoke has played a majorrole in the study of turbulent coherent motions. Quantitative analyses of flow-visualizationstudies by Corino & Brodkey [2], Kim et al. [3], and Grass [4] indicated that the ejectionof low-velocity fluid from the wall region was associated with a major part of the Reynoldsstress and turbulent energy production. According to these studies, the low-velocity streakslowly lifts away from the wall, at which time the streak filament begins to oscillate in boththe spanwise and wall-normal directions. The bursting process continues as the loops of thestreak filaments eject away from the wall. Finally, the ejected streak filaments eventuallybreak up in a chaotic process.Since most of the turbulence production in the near-wall region occurs during thosebursts, many following studies have been performed using probes to measure the velocityand pressure fields associated with bursts. Examples of works using conditional samplingtechniques to identify the structures involved in the bursting process using probe measure-ments can be found in the uv -quadrant method [5, 6], u -level detection method [7], the VITA(variable-interval time average) method [8], and the VISA (variable-interval space average)method [9]. Key aspects of the conditional sampling methods for turbulence structures arereviewed by Bogard & Tiederman [10], among others.The advent of particle-image velocimetry (PIV) provided two-dimensional instantaneousflow fields, which allowed a more in-depth analysis of instantaneous flow fields comparedto probe measurements. PIV experiments led to studies linking groups of ejections to low-momentum streaks [11, 12]. Tomkins & Adrian [13] and Ganapathisubramani et al. [14]used PIV data on wall-parallel planes in a turbulent boundary layer to quantify the con-tribution to the Reynolds stresses by packets of hairpins. Kevin et al. [15, 16] used PIV2easurements from spanwise homogeneous and heterogeneous boundary layers to identifystreak meandering. Three-dimensional measurements using high-speed PIV coupled withTaylor’s hypothesis [17, 18] has been utilized to track vortices and long structures in turbu-lent boundary layers.Simultaneously, the rise of computational power has allowed access to fully resolved three-dimensional data sets, which led to the study of instantaneous three-dimensional coherentstructures extracted from direct numerical simulations (DNS) of wall-bounded flow [19],such as the characterization of clusters of vortices [20–23], the generalized three-dimensionalquadrant analysis [24], and the study of intense regions of individual velocity components[25]. Temporally resolved data sets allow an additional dimension in the coherent structureanalysis, providing a full picture of how the structures evolve in space and time [26].These observations led to structural models that explain the dynamics of wall-boundedturbulence. The most established models describe motions in the buffer layer; examplesinclude the papers by Jim´enez & Moin [27], Jim´enez & Pinelli [28], Schoppa & Hussain [29],and Kawahara et al. [30], among others. While a lot of effort has been devoted to studyingthe dynamics of the buffer layer, most of them have been in idealized conditions with simpli-fied dynamics, with most focus given in identifying the dynamics of a single isolated streak.Furthermore, studies of coherent structures mentioned throughout this introduction havemainly focused on vortical structures and Reynolds stress, and with good reason. Reynoldsstress is strongly related to the turbulent kinetic energy production – the understanding ofthe mechanism of Reynolds stress generation is central to predicting the effects of turbu-lence in a wide variety of natural settings and engineering applications. Quadrant analysisclassifies the Reynolds shear stress into four categories based on the sign of the streamwise( u ) and wall-normal fluctuations ( v ). Q2 ( u < v >
0) and Q4 ( u > v <
0) events playimportant roles in most of the structural models explaining how turbulent kinetic energyand momentum are redistributed. These models are loosely based on the attached-eddyhypothesis by Townsend [31] and involve wall-attached vortical loops growing from the wallinto the outer region [32]. The study of these structures led to further understanding ofnear-wall turbulence, especially with the characterization in space and time [26], but re-search focused on the kinematics and the dynamics of the temporally and spatially resolvedinteraction of the streaks in the buffer layer is still incomplete. A broader view of the fulllife-cycle of streaks, involving ejections and bursts, could complement the ongoing study of3eynolds stresses and vortical clusters.The goal of this paper is to study the life-cycle of streaks, classified in terms of the stream-wise and spanwise fluctuations, and to study the time evolution of the size and meanderingof the streaks. We classify the streaks into tall-attached, detached, and buffer layer struc-tures and study how the detached streaks correlate with the Q2 or bursting events. For thispurpose, we track streaks in spatially and temporally resolved flow fields of low-Reynolds-number turbulent channel flow. The remainder of this paper is organized as follows. Thedetails of the numerical simulations are given in § II. The methodology used to identify andtrack individual streak structures are introduced in § III. Static and temporal analysis ofstreaks are given in § IV and § V, respectively. Finally, the summary of the work and theconclusions are offered in § VI.
II. NUMERICAL EXPERIMENT
A DNS of a channel flow at friction Reynolds number Re τ = u τ δ/ν ≈
186 is performed,where ν is the kinematic viscosity, δ is the channel half-height, and u τ is the friction velocityat the wall. Throughout the paper, x , y , and z denote the streamwise, wall-normal, andspanwise directions, respectively. The corresponding fluctuating velocity components are u , v , and w . The root-mean-squared (r.m.s.) intensities are given by u ′ , v ′ , and w ′ , respectively.The only non-zero mean velocity is in the streamwise component and denoted as U ( y ).The simulations are computed with a staggered second-order finite difference [33] and afractional-step method [34] with a third-order Runge-Kutta time-advancing scheme [35].Periodic boundary conditions are imposed in the streamwise and spanwise directions andthe no-slip and no-penetration boundary conditions are used at the top and bottom walls.The code has been validated in previous studies in turbulent channel flows [36–39] andflat-plate boundary layers [40].The numerical domain is 8 πδ × δ × πδ in the streamwise, wall-normal, and spanwisedirections, respectively. While this domain is not long enough to capture the longest ofnear-wall streaks that are of the order of 10 − wall units long, these streaks are known tobe formed by coalescence of several shorter ones [41], and thus for our purposes, the currentdomain is adequate for tracking individual streak structures. The domain is discretized using768 × ×
288 grid points in each direction. This corresponds to uniform grid spacings in the4treamwise and spanwise directions of ∆ x + = 6, ∆ z + = 3 . y + ) = 0 .
16 and max(∆ y + ) = 7 . ν and u τ . The simulationswere run for 100 eddy turnover times (defined as δ/u τ ) after transients to compute the meanquantities. The analysis of the temporal evolution of the flow requires storing approximately2 × snapshots spaced ∆ t + ≈ . δ/u τ , whereas the longest lifetimes Reynolds stress structures are less than 2 δ/u τ [26].While individual streaks have longer lifetimes than those of the Reynolds stress structures,our analysis in § V shows that the current temporal range is enough to capture the longestlife-cycle of streaks as well.
III. STREAK IDENTIFICATION AND TRACKING
In the present work, we identify streaks as a connected set of points where the streamwisevelocity fluctuation is less than zero and the wall-parallel velocity magnitude exceeds a giventhreshold, i.e., n ( x, y, z ) : u ( x, y, z ) < p u ( x, y, z ) + w ( x, y, z ) > αu τ o , (1)where α is a threshold value. This is similar to the traditional definition of streaks, butwith additional contribution from the spanwise fluctuations. The spanwise fluctuationswere included to account for the strong meandering events of streaks, where the spanwisevelocity component would become dominant. Disregarding the streamwise component, inthis case, would result in these meandering events being classified as streaks separating fromone another, which is undesirable. Connectivity in space is defined in terms of six orthogonalneighbors in the Cartesian mesh of the DNS.The threshold α can be obtained from a percolation analysis [21, 23, 24] (see Figure 1a).Percolation theory describes the behavior of a network when nodes or links are removed.Here, percolation analysis is applied to the variations of the volume of the connected objectsextracted by Eq. (1) with the threshold parameter α . When α is very large, the identificationonly yields a few small streaks. Decreasing α introduces new streaks while the previouslyidentified ones grow in size. At first, the ratio of the volume of the largest streak to thevolume of all identified streaks, V max /V tot decreases, but it increases rapidly as streaks merge5 -1 (a) (b) (c) FIG. 1. (a) Percolation diagram for the identification of streaks. Curves indicate ratio of thevolume of the largest object to the volume of all identified objects, V max /V tot (solid line) and ratioof the number of identified objects to the maximum number of objects, N/ max( N ), (dashed line).The vertical dotted line indicates the chosen threshold, α = 3 .
4. (b) Streamwise (solid), spanwise(dot-dashed) and wall-normal (dashed) r.m.s. intensity contribution from the full channel (red) andonly from the identified streaks (black). (c) Streaks identified in a single snapshot using threshold α = 3 . and form larger streaks. This percolation crisis occurs around 1 . α .
4. The value of α = 3 .
4, which lies within these bounds was chosen to maximize the number of streaksidentified ( N ). Small changes to the parameter α do not change the conclusions of thecurrent study (not shown), and thus we only focus on results given by α = 3 .
4. This6hreshold corresponds to about 1 . Re τ = 186 case. This ensures that the majority of the streaks identified are located in thebuffer layer, with only very strong events identified in the outer region, which is in line withour study of buffer layer streaks and their connection to these strong outer layer streaks.Once the streaks satisfying the threshold are identified, streaks with volume less than 30 wall units are discarded to reduce noise in terms of identifying streak-to-streak interactions.Figure 1(b) shows the contribution of the turbulent intensities from the streaks comparedto the full channel for each velocity component. The streaks carry more than 60% of thestreamwise turbulence intensity in the peak of the buffer region. Also, the streaks areresponsible for 20% of the total kinetic energy of the entire domain while only taking up 2%of the volume. The identified streaks for a single snapshot are shown in Figure 1(c), whichshows that indeed the majority of the streaks lie in the buffer region, with some streaksidentified in the outer region.For each streak identified, a few key quantities are computed to characterize the streak,namely size, volume, meandering, and orientation of the streak. First, the size of the streakis given by the dimensions of the bounding box (∆ x × ∆ y × ∆ z ), which is defined asthe smallest box that can encapsulate the streak (Figure 2a). The volume of the streakis computed as the volume the streak occupies in the domain, not by the volume of thebounding box. The spine of the streak, Σ = { ( x s , y s , z s ) } , is computed as the geometriccenter in the yz -plane for each streamwise location the streak occupies. The spine is thenfitted into a line Λ s = { ( x, y, z ) : ( x − ¯ x s ) /a x = ( y − ¯ y s ) /a y = ( z − ¯ z s ) /a z } that minimizesthe L -norm, as shown in Figure 2(a). Here, (¯ x s , ¯ y s , ¯ z s ) is the mean x , y , z coordinate ofthe spine. The meandering of the streak is quantified as the deviation of the spine to thelinear fit, i.e., the mean-squared error defined as kE k ≡ R ~ x s ∈ Σ min ~ x ∈ Λ s ( ~ x s − ~ x ) d S ∆ x s ∆ y s ∆ z s , (2)where ~ x = ( x, y, z ), ~ x s = ( x s , y s , z s ), and ∆ x s , ∆ y s , and ∆ z s is the streamwise, wall-normal, and spanwise length of the spine, respectively [16]. This definition of meanderingensures that resizing of the streaks (multiplication by a constant factor in all dimensions)would not affect the meandering coefficient kE k . Using the spine and its linear fit to quantifymeandering ensures that the width of the streak and the linear orientation of the spines doesnot play a role. The orientation of the streak is quantified by the azimuth and elevation7 a)(b) FIG. 2. (a) Spine (black solid line) and its linear fit (red solid line) for a single identified streak.Dotted lines indicate the bounding box of the streak. (b) Diagram depicting the azimuth ( φ ) andelevation ( θ ) angles of the linear fit for the case shown in (a). angles (see Figure 2b), which are computed as φ = tan − ( a z /a x ) and θ = tan − ( a y /a x ),respectively. The azimuth angle indicates how much the streak deviates in the spanwisedirection whereas the elevation indicates how much it deviates in the wall-normal direction.For temporal analysis, we track the evolution of individual streaks following the methodin [26]. The streaks from every two consecutive snapshots are evaluated for overlaps betweenthem. When comparing consecutive snapshots for overlap, the preceding flow field is shiftedby U ( y )∆ t in the streamwise direction to account for the advection, which is mostly dueto the mean flow [42–44]. All of the streaks with some overlap are considered connected.The connections in time identified is then organized into graphs such that each streak (at8 ·· ··· (a) (b) (c)(d) (e)(f) FIG. 3. Examples of (a) buffer-layer, (b) tall-attached, and (c) detached streaks. Sketch of a (d)merger and (e) split event. (f) Graph associated with the evolution shown in (d,e). all time steps) is considered a vertex and the connection between consecutive times areconsidered an edge. This way, the temporal evolution of a single streak can be identifiedthrough a connected component within this graph. If the streak has more than one backwardconnection, i.e. more than one streak is connected to a single streak in the next time step, itis considered a merger event. Conversely, if a streak has more than one forward connection,it is considered a split event. See Figure 3(d–f) for a sketch of merging and splitting eventsand the corresponding graph. A single connected component of the graph signifies theevolution of all of the structures that interact with each other at some point in time andmay have multiple merger and split events, possibly making the structure more complex.For simplicity, each of these connected components will be simply referred to as a ‘graph’,and the superset graph containing all the connected components will be referred to as the‘supergraph’.Graphs are then organized into ‘branches’ representing individual structures within the9 · · · · · · · · · · ·········· ········· · · · · · ·
FIG. 4. Examples of (a) primary, (b) incoming, (c) outgoing, and (d) connector branches. graph. For each merger and split event, i.e., when a node has more than one forward orbackward edge, the weight ∆
V /V o of each of the edges is computed, where ∆ V is the volumedifference in the two streaks in the two end nodes of the edge and V o is the volume overlap.The edge with the smallest weight is considered the primary connection and the rest areconsidered secondary. The secondary connections in a merger event are considered the endof that branch that merged into a larger branch, and the secondary connections in a splitevent are newly created branches that split from the main branch. This way, the complexspatio-temporal interaction between various streaks can be broken down into individualstreaks. Note that a merger or split event is a single event in time and, thus, identified as anode in the graph whereas a branch is a series of connected nodes and identify the evolutionof a streak in time.Each branch is classified as ‘primary’, ‘incoming’, ‘outgoing’, and ‘connector’ dependingon how they are created and destroyed (see Figure 4). Primary branches have no forwardor backward connections and are created from and dissipate into the turbulent background.Outgoing and incoming branches either split from or merge into another streak, respectively.Connector branches start and end in another streak, created through a split event and dyingin a merger event, effectively connecting two streaks. See Lozano-Dur´an et al. [26] for a morecomprehensive explanation of the tracking algorithm.For the current data set, graphs that traverse the entirety of the temporal length ( T + ≈ T u τ /δ ≈ .
4) account for only two out of the 5,810 graphs identified. However, dueto the complexity of these graphs, the two graphs account for 73.9% of all streak structuresidentified. The branches that compose these graphs do not span a long time period, witha mean duration of 28 wall units. This indicates that the extremely long-lasting graph10 (a) (b) FIG. 5. (a) Premultiplied p.d.f. of y +min (solid line), y +max (dashed line) and centroid, y c , (dot-dashed line) of the streaks. (b) C.d.f. of y +max as a function of y +min . Solid line indicates the medianquantity and shaded region indicate 25 and 75 percentiles. structures are sustained from the continuous merging of individual branches. This is alsoevident in the breakdown of branch categories. The primary branches account for 9.2% ofall the branches, composing only a small fraction of all branches. Incoming, outgoing, andconnector branches account for 14.9, 37.2, and 38.7% of the branches, respectively. IV. STATIC ANALYSIS
First, we consider all the nodes of the supergraph as individual entities and perform astatic analysis. The goal is to study the overall statistics of streaks. Figure 5(a) showsthe probability density function (p.d.f.), denoted P ( · ), of the wall-normal locations of thebottom ( y min ), the top ( y max ), and the centroid ( y c ) of the streak structures. As expected,most of the streaks occur close to the wall ( y +min <
20) with a much smaller secondary peakin the p.d.f of y min centered around y + ≃
90. The p.d.f.s of y c and y max are centered around y + ≃
20 and y + ≃
70, respectively and have a wider spread than that of y min .The cumulative distribution function (c.d.f.) y max as a function of y min in Figure 5(b)shows a clear divide in streaks that are wall-attached versus wall-detached. Wall-detachedstreaks ( y +min >
20) are positively correlated with the corresponding y +max with y +max ∼ ( y +min ) . . Thus, the height of the streaks grows only sublinearly with distance to the wall,11ith the streak height becoming smaller beyond a certain threshold. However, with wall-attached structures, more structures end with y +max >
70, forming ‘tall’ attached streaks.While larger Reynolds number simulations will have a fully developed logarithmic layer [45],the current Reynolds number is too small to identify a distinctive logarithmic layer. Thus,it is difficult to provide a clear cut definition of ‘tall’ structures whose y max outside thebuffer layer; however, we choose the cutoff for ‘tall’ structures to be y +cutoff = 70 as the twopeaks of the bimodal distribution of y c in Figure 5(a) are separated at y + ≈
70, providinga natural divide. The y + = 70 cutoff also coincides with the location where the dominantproduction mechanism chanages [46]. We classify the streaks as wall-detached if y +min > y +min <
20 and y +max >
70, and buffer layer if y +min <
20 and y +max <
70 (seeFigure 3a–c). The definitions of the wall-attached/detached streaks follow similar analysisfor Reynolds stresses and clusters [23, 24, 26]. Based on these studies, the attached streaksshould behave differently compared to the detached ones [31, 47] with the detached streaksbeing more isotropic and the attached structures in the log-layer forming self-similar familieswith approximately constant geometric aspect ratios [48, 49]. The current domain lacks awell-defined log layer, and will not contain a distinct layer of self-similar streaks; however,we expect self-similar streaks to be dominant in higher Reynolds number flows. While thisthreshold for ‘tall’ structures is arbitrary, changing the limit to y +cutoff = 60 or 80 does notchange the conclusions of this paper, and thus, the results are resilient to the choice of thisthreshold.We study the distribution of key statistics of the streaks for the tall-attached, detached,and buffer layer streaks. Of all the identified streaks, the majority are categorized as bufferlayer streaks (73.6%). The tall-attached and detached streaks account for 12.3% and 14% ofthe streaks, respectively. In Figure 6, the p.d.f. of volume, meandering coefficient, azimuth,and elevation angle of the spine, and aspect ratios of the bounding box are shown for thethree classes of streaks. The three categories have distinct properties. For example, thetall-attached streaks tend to be larger in volume, meander more, are aligned along thestreamwise direction with φ ≃
0, and are more elongated. This shows that even if the tall-attached streaks meander more, their larger dimensions allow them to align their spine inthe streamwise direction, as depicted by their azimuth angle. On the other hand, detachedstreaks are smaller and more isotropic, but have varying orientations, as shown by thewide range of azimuth and elevation angles. Even with varying orientations, the size of12 (a) (b) -1 -0.5 0 0.5 1012345678 (c) -1 -0.5 0 0.5 102468101214 (d) (e) (f) FIG. 6. (a) Premultiplied p.d.f. of the volume and the p.d.f. of (b) meandering quantity, (c)azimuth and (d) elevation angle of the spine, and (e) streamwise/wall-normal and (f) spanwise/wall-normal aspect ratio of the bounding box. Lines indicate tall attached streaks (black solid line),detached streaks (red dashed line) and buffer layer streaks (blue dot-dashed line). Dotted lines are(c) φ = 0, (d) θ = 0, (e) ∆ x/ ∆ y = 2 and (f) ∆ x/ ∆ z = 1. φ ≈ θ >
0, indicating, on average, a positive tilt.On average, the size of the bounding box is 5 . δ × . δ × . δ for the tall attachedstreaks, 0 . δ × . δ × . δ for the detached streaks, and 1 . δ × . δ × . δ (or300 ν/u τ × ν/u τ × ν/u τ ) for the buffer layer. The tall-attached streaks and the buffer layerstreaks have similar aspect ratios (Figure 7a), although the trend deviates at high values of∆ x/ ∆ y . These aspect ratios of the attached streaks are similar to what other people observefor buffer layer streaks [25]. The streamwise and wall-normal dimensions of the boundingbox for tall-attached flows follows ∆ x ∼ ∆ y and ∆ x ∼ ∆ z (Figure 7b), which indicates thestreaks grow in the streamwise and spanwise direction at a faster rate than the wall-normaldirection. Detached streaks are more isotropic in size. They are also less aligned with thestreamwise direction and might be the reason that in the outer region, oblique features be-come more prominent [16]. The volume of the tall-attached streaks follow V ∼ ∆ x / , ∆ y (Figure 7c), showing that the volume of the bounding box (∆ x ∆ y ∆ z ∼ ∆ x / , y ) is not agood representation of the volume of the streak, most likely due to a large amount of mean-dering, especially for large streaks. We also see that meandering intensifies with increasingstreak volume, albeit weakly, following kE k ∼ (∆ x/ ∆ y ) / . This indicates that meanderingis a byproduct of the elongating streak, which makes the streak less likely to orient in thesame direction throughout the entirety of its structure. V. TEMPORAL ANALYSIS
We now consider each branch (individual structures within each graph) as one entity,providing temporal analysis of the life-cycle of streaks. The p.d.f. of the lifetimes, T s , forthe different branch types are given in Figure 8(a). All branches have a long tail, with thelongest lifetime being T + s ≈ ν/u τ , withthe mean lifetime being longest for the main branches (32.1 ν/u τ ) and shortest for outgoingbranches (21.5 ν/u τ ). This shows that the current temporal domain is long enough to analyze14 (a) (b) (c) (d) FIG. 7. (a) Joint p.d.f. of the aspect ratios ∆ x/ ∆ y and ∆ z/ ∆ y for tall attached streaks (blacksolid line), detached streaks (red dashed line) and buffer layer streaks (blue dot-dashed line). (b)Joint p.d.f. of the logarithms of dimensions of the bounding box ∆ x with ∆ y (solid lines) and ∆ z (dashed lines) for tall-attached streaks. Dotted line indicates ∆ x ∼ ∆ y and ∆ x ∼ ∆ z . (c) Jointp.d.f. of the logarithms of volume with ∆ x (solid lines) and ∆ y (dashed lines) for tall-attachedstreaks. Dotted lines are V ∼ ∆ x / and V ∼ ∆ y . (d) Joint p.d.f. of the logarithms of meanderingquantity and ∆ x/ ∆ y for tall attached streaks. Dotted line is kEk ∼ (∆ x/ ∆ y ) / . Contour levelsare 50, 70, 90% of maximum value. (a) (b) FIG. 8. (a) Premultiplied p.d.f. of lifetimes of streaks for primary (yellow), incoming (blue),outgoing (red), and connector (green) branches. (b) Percentage of branches starting from (first)and ending in (last) each streak category. the full life-cycle of streaks, including the longest ones observed.We then analyze how the streaks change within each branch. The evolution of a branchdefined by the classification of its first and last streak (node) in time is summarized inFigure 8(b). Majority of the branches stay in the same category they were created in –either as tall-attached, detached, or buffer layer streaks. Only a small percentage (8%) ofthe branches created as wall-attached streaks, either tall or buffer layer, detach from thewall with time. On the contrary, a large portion (22%) of the branches ending as wall-detached streaks were initially wall-attached. Less than 0.1% of the streaks are created aswall-detached and then attach to the wall. This indicates that the branches that start aswall attached can, albeit rarely, detach and form detached streaks, but not the other wayaround. The most frequent change in classification is between tall-attached and buffer layerstreaks, which occur in similar numbers in either direction; however, this is expected as the y + = 70 cutoff for ‘tall’ structures are arbitrary and fluctuations of the streak height aboutthis cutoff will result in a change in classification.We also study the classification of the first and last streak (node) in a branch for eachbranch type (Table I). Branches created from a turbulent background (primary and incomingbranches) rarely contain tall-attached streaks because there is not enough time for the16 ABLE I. Percent distribution of the first and last streak category for each branch type.tall-attached detached buffer layerFirst streak incoming branch 0.7 2.1 97.2outgoing branch 13.3 55.6 31.0connector branch 34.9 10.4 54.7primary branch 1.0 9.6 89.3Last streak incoming branch 8.7 2.3 89.0outgoing branch 2.9 69.5 29.6connector branch 33.2 11.8 55.0primary branch 3.2 16.7 80.1 streaks generated this way to grow into tall-attached structures before they merge intoanother streak or disappear. The majority of the branches created this way start as buffer-layer streaks, accounting for 95% of these cases. Similarly, branches disintegrating to theturbulent background rarely contain tall-attached streaks, mostly because the streaks breakdown into smaller streaks before dying. Near the end of the streak life-cycle (outgoingbranches and ends of primary branches), the majority of these streaks are detached streaks,with 57% being detached and 43% being buffer layer streaks. Tall-attached streaks appearmostly in the connector branches.The interaction of various branches can be studied by observing the composition of theprimary and secondary structures of each split and merger event, where the primary structureis a node of the main branch in the merger and split events. For each split event, the twostreaks with the lowest weight, ∆
V /V o , emerging from the same streak are analyzed inFigure 9(a). In most cases, the primary structure is a tall-attached streak with similarprobabilities for tall-attached, detached, or buffer layer streak splitting off as the secondarystructure. Buffer layer streaks splitting into two buffer layers streaks are the next common.Detached streaks rarely are the primary structures in splitting events. In the case of mergerevents, the two streaks with the lowest weight merging into the same streak are analyzed inFigure 9(b). The buffer layer streak merging into a tall-attached streak accounts for 42%of all merger events. The tall-attached streaks merging into tall-attached streaks and bufferlayer streaks merging into buffer layer streaks also play a significant role. Detached streaks17 a) (b) FIG. 9. Percentage of primary and secondary streaks in (a) split and (b) merger events. Thesecondary streaks split off the primary streaks in a split event, and the secondary streaks mergeinto the primary streaks in a merger event. only occasionally merge into tall-attached streaks.The observation provided above shows that the majority of the streaks start as buffer layerstreaks and stay as buffer layer streaks. These buffer layer streaks then either die or mergeinto other existing streaks, which in turn may become tall-attached streaks. Tall-attachedstreaks are maintained through a continuous coalescing of other buffer-layer or tall-attachedstreaks. Near the end of their life-time, these streaks disintegrate into detached and buffer-layer streaks, which then dissolves into the turbulent background. The disintegration intodetached and buffer-layer streaks can be thought of as the bursting phenomena, where theproduction of turbulence in the boundary layer via violent outward eruptions of near-wallfluid. A more detailed analysis of detached streaks and their connection to bursting is givenlater in this section.The lifetimes, T s , of the streaks are correlated with their maximum volume over time,as seen in Figure 10(a). In all types of branches, a larger maximum volume correlates withlarger lifetimes. In the case of incoming branches, the maximum volume occurs near theend of the life-cycle, when in merges into another branch, as seen in Figure 10(b), indicatingthe longer the streak is sustained, the larger it is able to grow. For outgoing branches,the maximum volume occurs near the beginning of its life-cycle, where it splits off from18 (a) (b) FIG. 10. (a) C.d.f. of the lifetime of a streak as a function of maximum volume of a branch. (b)C.d.f. of normalized volume as a function of the lifetime of a streak. Lines indicate median valuesof primary (yellow), incoming (blue), outgoing (red), and connector (green) branches. Transparentregions are 25 and 75 percentiles. another branch, indicating streaks with larger volumes take longer to disintegrate. Forprimary branches, the volume grows and decays within the streak’s lifetime, as expected.Finally, for connector branches, the maximum volume occurs equally likely throughout itslifetime, which explains the smallest correlation between lifetimes and volume for these typesof branches in Figure 10(a). Interestingly, even though meandering correlates with volume(see Figure 7d for indirect comparison with aspect ratio ∆ x/ ∆ y ), statistically, meanderingis not affected by the time evolution of the streaks regardless of branch type (not shown).Finally, we study the relative position of the streak as a function of its lifetime (Fig-ure 11a). All types of streaks move up during their life-cycle – this is expected as streaksare events with intense u < v avg , for each branch is given in Figure 11(b), whichshows a similar distribution between the primary, incoming and connector branches, but apositively shifted distribution of wall-normal velocities for the outgoing branches. Condi-tionally sampling outgoing branches that are composed primarily of detached streaks andthose composed primarily of buffer-layer streaks (Figure 11c), we see that the discrepancy19s due to the detached streaks, which clearly show a different distribution of wall-normalvelocities. The significantly higher positive wall-normal velocities associated with the de-tached outgoing branches coincide with our observation that detached streaks correlate withbursts or strong ejection events. The incoming and primary branches are mostly composedof buffer layer streaks, so it is difficult to see the p.d.f. breakdown of the average wall-normalvelocity conditioned to a particular streak type, but the connector branch is evenly dividedamong tall-attached and buffer layer streaks. The average wall-normal velocity of the con-nector branch conditioned to mostly tall-attached streaks and buffer-layer streaks are shownin Figure 11(d). While there is some discrepancy in the distribution, the difference is notas noticeable as in the case of the detached streaks and similar to the distributions for theincoming and primary branches. This indicates that the detached streaks in the outgoingbranch indeed have a significantly stronger wall-normal velocity. This corroborates the the-ory that these streaks can be identified as bursting events as well as strong Q2 events orejections.The averaged flow field conditioned to a split event when a detached streak splits froma tall-attached streak is shown in Figure 12. Conditional averages are computed in thereference frame ( r x , r y , r z ) = (( x, y, z ) − ( x m , y m , z m )) / ∆ y m , (3)where ( x m , y m , z m ) is the midpoint of the shortest line connecting the center of the detachedstreak and the spine of the tall-attached streak and ∆ y m is the distance from y m to thewall where the tall-attached streak is attached. The spanwise direction is chosen such that r z > r x ≈ . , r y ≈ .
42, whereas the center of the averaged structuresignifying tall-attached streaks is located at r x ≈ − . , r y ≈ − .
64. This shows that thedetached streak splits from the tall-attached streak at a greater (less negative) streamwiseand wall-normal velocity, consistent with the observations regarding wall-normal velocitiesof detached streaks in Figure 11(c). The average tall-attached streak shows an elongatedshape with aspect ratios similar to the one observed in Figure 7(a).The relative position of the detached streak with respect to the tall-attached streak placesthe detached streak on the tip of the vortex cluster placed between high-speed and low-speedstreaks, similar to the findings regarding Q2 events with respect to vortex clusters [23, 24].The relatively higher value of u (less negative) and v (more positive) of the detached streak20 (a) -2 -1 (b) -2 -1 (c) -2 -1 (d) FIG. 11. (a) C.d.f. of the normalized centroid of the streak as a function of the lifetime of a streak.Lines indicate median values of primary (yellow), incoming (blue), outgoing (red), and connector(green) branches. Transparent regions are 25 and 75 percentiles. (b) Premultiplied p.d.f. of theaverage wall-normal velocity of branches for primary (yellow), incoming (blue), outgoing (red),and connector (green) branches. (c) Premultiplied p.d.f. of the average wall-normal velocity ofoutgoing branches (solid), outgoing branches consisting mostly detached streaks (dotted), andoutgoing streaks consisting mostly buffer-layer streaks (dot-dashed). (d) Premultiplied p.d.f. theaverage wall-normal velocity of connector branches (solid), connector branches consisting mostlytall-attached streaks (dashed), and outgoing streaks consisting mostly buffer-layer streaks (dot-dashed).
120 11 0 -1 -2 1-3 0-4 -1
FIG. 12. Averaged flow field conditioned to a split event of detached streak from a tall-attachedstreak. The detached streak is colored in blue for clarity. Isosurfaces are given by √ u + w > . u τ and u < compared to the tall-attached streak coincides with the direction of the vorticity at the tipof the so-called ‘horseshoe’ or ‘hairpin’ vortices. It is worth mentioning that the smoothshape of the conditional averaged quantity is not representative of the individual streaks inthe flow, which are more complex and increase in complexity with higher Reynolds numbers.Regardless, the conditional field allows a structural assessment of averaged events relatedto these events. While the low-speed streaks were argued to be a consequence of the Q2events in Lozano-Dur´an et al. [26], it could also be argued that the flow field conditioned toQ2 events are indeed conditioned to these breakup of streaks and both are consequences ofeach other. VI. CONCLUSIONS
We study the kinematics and dynamics of buffer layer streaks using temporally resolvedDNS data from a turbulent channel at Re τ = 186. The temporal resolution was fine enoughand the duration of the simulation was long enough to track hundreds of thousands of streaksfrom creation to disintegration. We found that although the interacting streaks could createconnected structures that span long spatial domains and time periods, beyond the spatialand temporal domain of the current study, these large structures are composite streaks and22an be broken down into much smaller individual streaks that only last a small fraction ofthat time.We first divide the streaks into tall-attached streaks that span from the wall to the outerregion, detached streaks that are detached from the wall, and buffer-layer streaks that arewall-attached and stay within the buffer layer. The tall-attached streaks tend to be largerin volume, meander more, but are still aligned with the streamwise direction. The buffer-layer streaks show a similar aspect ratio as the tall-attached streaks but are smaller in size.The detached streaks are more isotropic and tend to be less aligned with the streamwisedirection, showing a wide range of azimuth and elevation angles. The distribution of thedimensions of the tall-attached streaks is such that ∆ x ∼ ∆ y and ∆ x ∼ ∆ z , and thevolume is proportional to ∆ x / . It was also observed that the meandering is correlated withthe ∆ x/ ∆ y , which increases with volume, indicating that larger streaks start to meandermore due to the substantial elongation in the streamwise direction resulting from the largervolume.The tracking process resulted in the organization of the streaks into branches, which aredivided into four categories: primary, incoming, outgoing, and connector. A large portionof the branches identified were connector branches, eluding to the complex merging andsplitting of streaks throughout their lifetime. Each branch tends to be composed of streaksthat do not change in category, e.g. branches starting as buffer-layer streaks tend to staybuffer-layer streaks during their lifetime. Splitting of branches tends to happen with tall-attached streaks breaking into a tall-attached streak and another streak. Merging eventshappen most frequently between buffer-layer streaks and tall-attached streaks. The resultsshow that streaks are born in the buffer layer, coalescing with each other to create largerstreaks that are still attached to the wall. Once the streak becomes large enough, thetall-attached streak eventually splits into wall-attached and wall-detached components.The lifetime of the branches depends on the volume of the largest streak within the branch.For incoming and primary branches, this is because the long lifetime allows the streak togrow before it merges into another branch or disintegrated into the turbulent background.For outgoing branches, the initial volume indicates how long the branch will sustain beforedissipating. All branches move away from the wall during their lifetime, consistent withthe observation that most strong u < u < v >
0) [50, 51]. Thestrongest average wall-normal velocity is observed in outgoing branches primarily composed23f detached streaks, which can be seen as strong ejection or bursting events.Averaging the velocity fields conditioned to an event where a detached streak splitsfrom a tall-attached streak, we observe that the detached streak splits with a less negativestreamwise and a larger positive wall-normal velocity compared to the tall-attached streak.The relative position of the detached streak relative to the tall-attached streak is equivalentto the relative positioning of Q2 events with respect to low-speed streak seen in Lozano-Dur´an et al. [24]. The relatively larger streamwise and wall-normal velocities of the detachedstreak could also be seen as part of a spanwise vortical structure on top of the vortex clusterresting between the low- and high-speed streaks. The observations allude to the fact that noone structure or event is a cause for all the other events but rather a connected set of eventsthat happen synchronously and can be observed in various ways. Bursting is simultaneouslya strong Q2 event as well as attached streaks splitting into detached streaks.
ACKNOWLEDGEMENTS
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