Accepted for J. Hydraulic Research 2014 Experimental study on the role of spanwise vorticity and vortex filaments in the outer region of open-channel flow QIGANG CHEN, PhD Student, Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China Email: cqg09@mails.tsinghua.edu.cn RONALD J. ADRIAN, Regents' Professor, School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ 85287, USA Email: rjadrian@asu.edu (Corresponding Author) QIANG ZHONG, PhD Student, Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China Email: zhongq08@mails.tsinghua.edu.cn DANXUN LI, Professor, Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China Email: lidx@tsinghua.edu.cn XINGKUI WANG, Professor, Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China Email: wangxk@tsinghua.edu.cn 1 Accepted for J. Hydraulic Research 2014 Experimental study on the role of spanwise vorticity and vortex filaments in the outer region of open-channel flow ABSTRACT The dynamic importance of spanwise vorticity and vortex filaments has been assessed in steady, uniform open-channel flows by means of particle image velocimetry (PIV). By expressing the net force due to Reynolds’ turbulent shear stress, ∂ (−uv) ∂y , in terms of two velocity-vorticity correlations, vωz and wω y , the results show that both spanwise vorticity ω z and the portion of it that is due to spanwise filaments make important contributions to the net force and hence the shape of the mean flow profile. Using the swirling strength to identify spanwise vortex filaments, it is found that they account for about 45% of vω z , the remainder coming from non-filamentary spanwise vorticity, i.e. shear. The mechanism underlying this contribution is the movement of vortex filaments away from the wall. The contribution of spanwise vortex filaments to the Reynolds stress is small because they occupy a small fraction of the flow. The contribution of the induced motion of the spanwise vortex filaments is significant. Keywords: Hairpin vortex; net force; open-channel flow; Reynolds shear stress; spanwise vortex filament; turbulence Running Head: Spanwise vorticity in open-channel flow 1. Introduction Since the pioneering work of Theodorsen (1952), hairpin vortices have been hypothesized to be important coherent elements of wall turbulence. A wide body of evidence based on visualization experiments (Head and Bandyopadhyay 1981), modelling (Perry and Chong 1982, Perry et al. 1986, Perry and Marusic 1995, Marusic 2001), low Reynolds number direct numerical simulations (DNS) (Robinson 1991, Zhou et al.1996, 1999, Adrian and Liu 2002) and quantitative experiments using of particle image velocimetry (PIV) (Liu et al. 1991, 2 Accepted for J. Hydraulic Research 2014 Meinhart and Adrian 1995, Adrian et al. 2000b, Christensen and Adrian 2001, Ganapathisubramani et al. 2003, Tomkins and Adrian 2003, Camussi and Di Felice 2006) in turbulent boundary layers and channel flows provided further evidence that hairpin vortex packets are important coherent structures in wall turbulence (Adrian 2007, Adrian and Marusic 2012). More recently, well-resolved DNS datasets in developing boundary layers (Wu and Moin 2009, Schlatter and Orlu 2012) and three-dimensional PIV experiments in boundary layers (Elsinga et al. 2010, Dennis and Nickels 2011) provided three-dimensional evidence of the abundance of hairpins and hairpin packets. The theoretical model of Perry and Chong (1982) was based entirely on the hairpin vortex paradigm. In their model, the turbulent boundary layer was conceived as a random forest of self-similar hairpin-shaped vortices. These vortices originated and commenced their growth from the wall and contained all the vorticity in the flow. Later refinements of the model by Perry et al. (1986), Perry and Marusic (1995) and Marusic (2001) successfully reproduced many statistical quantities including mean velocity, Reynolds stress and spectra. The Reynolds stress is especially interesting as it relates the mean flow to the turbulence, and it is the "closure" term in Reynolds-averaged Navier-Stokes (RANS) equations. Hairpin vortices have been used to qualitatively explain intense events, such as ejections and sweeps that produce most of the Reynolds shear stress (Robinson 1991, Smith et al. 1991). More recently, velocity fields measured by PIV made it possible to quantitatively assess the contribution of hairpins to Reynolds shear stress. Ganapathisubramani et al. (2003) reported the first PIV experiments to measure their contribution in a turbulent boundary layer. By measuring the velocity fields in streamwise–spanwise planes, groups of legs of hairpin vortices and associated patches of induced-motion were identified. Their results showed that the induced motions contribute about 25% to the total Reynolds shear stress while occupying only 4% of the total area. In another experiment, Wu and Christensen (2006) measured the velocity fields in streamwise–wall-normal planes of turbulent boundary layer and channel flow. They reported that the induced motion of spanwise vortex filaments contribute significantly to the total shear stress. 3 Accepted for J. Hydraulic Research 2014 Much of the evidence for hairpin vortices derives from velocity vector fields and spanwise (z) vorticity fields measured on a planar slice of the flow, usually in the streamwise (x)– wall-normal (y) plane, by PIV. Roughly circular, concentrated regions of spanwise vorticity in the x-y measurement plane are interpreted to be the cross-sections of vortex filaments in the heads of the hairpins (Adrian et al. 2000b). The vorticity-containing filament, also called a vortex core (Chakraborty et al. 2005), is surrounded by inviscid, induced flow whose streamlines in the cross-sectional plane of the vortex filament are roughly circular when viewed in a coordinate system moving with its centre. It is notable that although vortex is widely taken to mean vortex filament in turbulence (Marusic and Adrian 2013), the commonly defined vortex also contains the essential characteristics of the flow induced by a vortex filament (Chakraborty et al. 2005). The long, thin vortex filaments are one type of vortex in a picture that also contains vortex shear layers and vortex blobs. The flows around these filaments are known to be important to the mean structure of turbulent flow. However, while filaments and patterns consistent with hairpin vortices are widely observed in two-dimensional PIV experiments, many investigators using three-dimensional DNS report difficulty in seeing the hairpin shaped filaments in three-dimensional simulations, causing some to question the significance and even the existence of hairpins (Del Alamo et al. 2006). The three-dimensional simulation of a boundary layer starting from laminar flow and passing through transition by Wu and Moin (2009) went far to dispel questions about hairpin existence, for their results exhibited an exceptionally clear forest of hairpins. But, questions have since been raised concerning the fate of the hairpins at larger downstream distances where the Reynolds number is larger and DNS researchers report difficulty in observing hairpins (Schlatter et al. 2012). Del Alamo et al. (2006) report only “clusters of vortices”, not hairpin packets, in channel flow, and Schlatter et al. (2012) see vortex legs sticking up from the wall region, but they do not see the hairpin heads connecting the legs in the downstream regions of the boundary layer, well beyond transition. They also see that the legs often appear to be one sided, i.e., a left but no right, as shown earlier by Guezennec et al. (1989). The failures to see hairpin heads or missing legs may be artefacts of three-dimensional visualization of turbulent fields caused by the levels of vorticity being lower in the heads than 4 Accepted for J. Hydraulic Research 2014 the legs, or in one leg versus the other, and it is conceded that visualization of well-formed packets at high Reynolds number is difficult. Even Baltzer et al. (2013) reported difficulty. On the other hand, there is three-dimensional visualization evidence for hairpins and hairpin packets in the work of Elsinga et al. (2010) and Dennis and Nickels (2011), and if one interprets the regions of compact transverse vorticity and swirling strength to be consequences of hairpins intersecting the x-y plane, then there is ample evidence for the existence of hairpins. Thus, the resolution of these conflicting observations is important to the progress of turbulence research, and it is desirable to determine the existence and significance of filamentary vortex heads using means other than qualitative visualization. Much is known about these filamentary vortices. For example, the region just above the buffer layer is found to be densely populated with young, first generation hairpins, and their number decreases across the logarithmic layer and into the wake region (Carlier and Stanislas 2005, Wu and Christensen 2006, Pirozzoli et al. 2008, Stanislas et al. 2008, Herpin et al. 2010). These vortex filaments propagate on average more slowly than the mean velocity (Pirozzoli et al. 2008, Gao et al. 2011). The size of the filamentary vortices increases slowly with wall-normal position (Carlier and Stanislas 2005, Pirozzoli et al. 2008, Stanislas et al. 2008). When scaled with the local Kolmogorov length-scale, η, the mean diameter is about 8-12η (Tanahashi et al. 2004, Pirozzoli et al. 2008, Stanislas et al. 2008, Tanahashi et al. 2008, Gao et al. 2011, Marusic and Adrian 2013). However, the dynamic importance of vortex filaments in wall turbulence is not fully understood. While most turbulent flow research focuses on the Reynolds shear stress, the net force per unit volume or more rigorously, the mean effect of turbulent inertia (Morrill-Winter and Klewicki 2013), is equally important. It is defined as the divergence of the Reynolds stress tensor, (net force)i = ∂ (− ρ ui u j ) ∂x j (1) where ρ is the fluid density, and xi, ui, i=1,2,3 are the components of position and turbulent velocity fluctuation (i.e., deviation from the mean value). Eq. (1) offers a more direct way to understanding the importance of various structural elements. The net force appears in any 5 Accepted for J. Hydraulic Research 2014 Reynolds averaged form of the mean momentum equation, where it represents the Reynolds’ mean fictitious force per unit volume exerted by turbulent momentum transport on the mean flow. In the case of fully developed, parallel wall flows net force in x-direction = ∂ (− ρ uv) ∂y (2) where (x, y, z)=(x1, x2, x3), and (u, v, w)=(u1, u2, u3) represent the streamwise, wall-normal and spanwise components of position and velocity. Klewicki (1989) expressed the net force in this case in terms of two velocity-vorticity correlations, ∂ (−uv) = vωz − wω y ∂y (3) where ωy and ωz are fluctuating vorticity components in the wall-normal and spanwise directions, and the overbar denotes time averaging. Equation (3) is derived directly from the well-known vorticity form of the non-linear term in Navier-Stokes equation 1 u ⋅ ∇u = ∇ ⎛⎜ ui ui ⎞⎟ − u × ω ⎝2 ⎠ (4) under the conditions that the streamwise and spanwise gradients of the mean turbulent kinetic energy vanish in wide, fully developed wall flows, or they can be neglected (a standard approximation for turbulent boundary layers). The relationship in Eq. (3) shows unequivocally that spanwise vorticity and wall-normal vorticity are essential to the creation of net force. Klewicki’s (1989) experiments indicated that the two velocity-vorticity correlations are roughly equal in turbulent boundary layers. Later, Klewicki et al. (1994) found that features of the velocity-vorticity correlations coincide well with the vortex models proposed by Falco (1991) and Smith et al. (1991) in the near wall region. Recently, by measuring the spectra of velocity, vorticity and their co-spectra, Priyadarshana et al. (2007) further established that the velocity-vorticity interactions responsible for the "scale selection" feature of the co-spectra could be interpreted properly by the hairpin vortex packets proposed by Adrian et al. (2000b). These authors implied a tight connection between the net force and vortex filaments in wall turbulent flows, especially the filaments that constitute hairpins and hairpin packets. But, to make this conclusion 6 Accepted for J. Hydraulic Research 2014 well-grounded, it must be established that the spanwise vorticity is indeed due to filaments in order to make the association with hairpin heads. This paper describes an investigation of the contribution of spanwise vortex filaments to net force and Reynolds stress in open-channel flow. It is intended primarily to assess the role that spanwise vorticity and its filamentary regions play in determining the net force and thereby shaping the mean velocity profile. Although filamentary vortices in open-channel flows have been observed by many PIV experiments (Roussinova et al. 2009, Dwivedi et al. 2011, Sanjou and Nezu 2011), the similarities and differences between vortices in open-channel flow and other well-documented wall turbulent flows are still unclear. Therefore, mean properties of spanwise vortex filaments in open-channel flow are also investigated and compared with results in channel flow and turbulent boundary layer flow. The organization of this paper is as follows. In section 2, the open-channel flow facility and the PIV system used are described. The properties of the net force and its connection with spanwise vortex filaments are investigated in section 3. Contributions of spanwise vortex filaments to the Reynolds stress are analysed in section 4, and section 5 presents mean properties of spanwise vortex filaments in open-channel flow. Finally, main results and conclusions are given in section 6. 2. Experiments 2.1. Flow facility Experiments are conducted in a closed-circuit tilting open-channel flume 20m long, 0.3m wide and 0.4m high. The flume has a glass bed and side-walls to facilitate velocity measurement with PIV. Five honeycombs with decreasing diameters are set at the entrance of the flume to remove large-scale structures and make the flow uniform in the transverse direction. The test section is 12m downstream of the entrance. Velocity fields are measured in a streamwise–wall-normal plane located midway between the side walls. The measurements are made under four steady, uniform flow conditions as summarized in Table1. Froude number of each case is set to be nearly the same to enable comparisons. The ratios of width to depth are at least 7.5, to ensure two-dimensionality in the central region of the flow (Nezu 7 Accepted for J. Hydraulic Research 2014 2005). The friction velocity uτ is determined based on the logarithmic law with von Karman constant κ=0.412 and additive constant A=5.29, as suggested by Nezu and Rodi (1986). A superscript + denotes quantities normalized by the friction velocity uτ and kinematic viscosity ν. The x-axis is oriented along the main flow, parallel to the flume bed, the y-axis is normal to the bed and pointing toward water surface, and the transverse z-axis is normal to the flume side wall with an origin at the centreline on the flume bed. Table 1 Open-channel flow characteristics: S, bed slope; h, water depth; B, channel width; Um, bulk mean velocity; F= Um gh  , Froude number; R= Um h 𝜐, Reynolds number; Rτ = uτ h 𝜐, friction Reynolds number. S h B/h Um uτ F R Rτ (-) (cm) (-) (cms-1) (cms-1) (-) (-) (-) OCF380 0.001 2.5 12.0 25.4 1.55 0.51 5388 382 OCF490 0.001 3.0 10.0 27.8 1.68 0.51 6746 489 OCF610 0.001 3.5 8.6 30.6 1.77 0.52 8510 609 OCF740 0.001 4.0 7.5 34.4 1.91 0.55 10615 740 Case 2.2. Velocity measurements PIV is used to measure instantaneous, two-dimensional velocity vectors in the streamwise– wall-normal plane. The flow is seeded with polyamide seeding particles with a density of 3 1.03×10 kg/m3 and a mean diameter of 5µm. The field of view is illuminated with a 1mm-thick light-sheet formed from a continuous wave laser (5W). Particle images are captured by a 2560×1920 pixel 8-bit CMOS camera with a Canon EF 50mm f/1.2 USM lens. The PIV experiment parameters are summarized in Table 2. The camera is externally triggered to sample one pair of images per second to achieve statistical independence between successive velocity fields (based on the field of view and bulk mean velocity). Within each image pair, the time interval between the first frame and the second frame is determined by the frame rate of the camera (the frame rate is adjustable when the camera is used in ROI 8 Accepted for J. Hydraulic Research 2014 mode). Before the acquisition of images in each experimental case, the frame rate of the camera is adjusted according to the water velocity so that the maximum displacement of particle images from the first frame to the second frame satisfies the one-quarter rule (Adrian 1991). For each image, the exposure time is fixed to be 150µs to minimize the influence of streaking while maintaining the light intensity of particle images. A total number of 5000 image pairs are sampled in each experimental case to guarantee statistical convergence. Particle images are analysed with a multi-pass, multi-grid window deformation method similar to that of Scarano (2002). During image interrogation, a Gaussian window-weighting function is applied automatically to avoid the effect of the edge of the interrogation window. The correlation peaks are interpolated using three-point one-dimensional Gaussian peak fitting in two dimensions. The resultant velocity fields of each iterative stage are validated using the normalized median test proposed by Westerweel and Scarano (2005), and outliers are replaced by Gaussian-kernel weighted interpolation (Adrian and Westerweel 2011). The validated velocity fields are then smoothed before the next iterative stage to prevent the possible unstable behaviour of the interrogation process (Kim and Sung 2006). The interrogation window size in the final iterative step is 16×16 pixels with 50% overlap. Inner scaled vector grid spaces, Δx+ and Δy+, are listed in Table 2. Table 2 PIV experimental parameters. Δx+=inner scaled grid space in streamwise direction; Δy+=inner scaled grid space in wall-normal direction. Image size Exposure Frequency NO. of (pixel) time (µs) (Hz) realizations OCF380 1280×384 150 1 OCF490 1280×480 150 OCF610 1280×608 OCF740 1280×720 Case Δx+ Δy+ 5000 8.1 8.1 1 5000 8.2 8.2 150 1 5000 8.1 8.1 150 1 5000 8.3 8.3 9 Accepted for J. Hydraulic Research 2014 Figure 1 Profiles of (a) mean velocity, (b) turbulent intensities and Reynolds shear stress, and (c) root mean square of spanwise vorticity. □, OCF380; ▽, OCF740; ─, Del Alamo et al. (2003), Rτ=550; ▲, Adrian et al. (2000), Rθ=930; ●, Adrian et al. (2000), Rθ=6845; ▼, Carlier and Stanislas (2005), Rθ=7500; ■, Carlier and Stanislas (2005), Rθ=7500 Since the implementation of the current PIV system is different from the standard PIV system which uses double-pulsed laser as light source, the accuracy of the measurements are firstly validated by comparing the statistics from the present measurements to those from DNS and standard PIV systems. Figures 1(a) and 1(b) present the profiles of mean velocity 10 Accepted for J. Hydraulic Research 2014 and turbulent intensities/Reynolds stress, respectively. The measured results agree well with the simulated results except in the upper part of the flow where the free surface strongly influence the flow. In figure 1(c), the measured results in the near wall region are expected to be underestimated due to the inadequate wall-normal resolution of PIV measurements. But, the global agreement with the existing results is fairly good. In general, the current PIV system works quite well in resolving most of the motions especially in the outer region of the measured flows. 3. Net force and its connection with spanwise vortex filaments 3.1. Properties of net force To explore the relative importance of the correlation terms involving spanwise and wall-normal vorticity in Eq. (3), the budget of net force has been calculated for each of the studied scenarios. The results of case OCF610 are presented in Fig. 2 (results of other cases are similar). The spanwise velocity and the velocity gradient components of ωy are not measured in the present two-dimensional PIV experiments, so wω y is computed from wω y = u ∂v ∂y (5) Equation (5) can be derived by applying incompressible continuity plus the conditions of streamwise and spanwise homogeneity of the means. The reliability of Eq. (5) is verified in Fig. 2 by comparing net force computed from ∂⟨-uv⟩/∂y (solid line) to that computed from ⟨vωz⟩-­‐⟨u(∂v/∂y)⟩ (dash-dot line), where ⟨·⟩ has the same meaning as an overbar. The small disparity below y+=60 is caused by differentiation errors. 11 Accepted for J. Hydraulic Research 2014 Figure 2 Budget of net force in case OCF610 In Fig. 2, the region between the curves of ⟨vωz⟩ and ⟨wωy⟩ is shaded to indicate the magnitude of the net force. The net force changes signs at yp, corresponding to the location of the maximum of Reynolds stress. At yp, the profiles of ⟨vωz⟩ and ⟨wωy⟩ intersect, resulting in a zero net force. In the region below yp, the net force is positive, as ⟨wωy⟩ exceeds ⟨vωz⟩. Acceleration of the flow near the wall increases the mean velocity gradient at the wall and causes turbulent wall shear stress to exceed that of the laminar case. In the region above yp, ⟨wωy⟩  increases more quickly than ⟨vωz⟩, leading to negative net force that reduces the mean flow velocity in the core of the flow. The behaviour of the mean velocity depends upon the delicate difference between ⟨vωz⟩ and ⟨wωy⟩. It may be useful to note that drag reduction can be accomplished either by making ⟨vωz⟩  more negative, or ⟨wωy⟩ less negative in the region below yp. From Fig. 2 one can conclude that the features of the net force profile and the mean velocity profile would change dramatically if the contribution from ⟨vωz⟩  were neglected. Morrill-Winter and Klewicki (2013) further demonstrated that ⟨vωz⟩  is largely dominant over ⟨wωy⟩ on a domain that approaches the boundary layer thickness with the increasing of Reynolds number. Thus, spanwise vorticity is essential to turbulence as we know it. The wall-normal evolution of inner scaled ⟨vωz⟩+ and ⟨wωy⟩+ are presented as a function of y+ in Figs. 3(a) and 3(b), respectively. In the near-wall region, ⟨vωz⟩+ decreases sharply in the logarithmic layer while approaching the wall (see Fig. 3(a)). This tendency is also observed in the DNS results of Crawford and Karniadakis (1997). The Reynolds number 12 Accepted for J. Hydraulic Research 2014 dependency in the logarithmic region reported by Klewicki et al. (1994) and Priyadarshana et al. (2007) is not observed in the present study, possibly because the span of Reynolds number is narrow in the present study. In the outer region, ⟨vωz⟩+ increases slightly with wall-normal position, but remains negative. In Fig. 3(b), ⟨wωy⟩+ increases from a negative peak in the buffer layer and becomes slightly positive in the outer region. A similar result is also reported by Crawford and Karniadakis (1997). The Reynolds number dependence of ⟨wωy⟩+ is not readily observed, due to the narrow span of Reynolds number in the present study. Figure 3 Profiles of inner scaled (a) ⟨vωz⟩+ and (b) ⟨wωy⟩+ versus y+. □, OCF380; , OCF490; ○, OCF610; , OCF740 3.2. Relationship between spanwise vortex filaments and net force A spanwise vortex filaments is identified in the x-y plane as an extremum of the swirling strength field that is surrounded by at least three points in which the swirling strength satisfies Λci (x, y) ≥ 1.5Λcirms ( y) (Wu and Christensen 2006) in both x and y. The signed swirling strength Λci = λci ωz ωz is used to distinguish between prograde (ωz < 0) and retrograde (ωz > 0) spanwise vortex filaments (Tomkins and Adrian 2003), where Λcirms ( y) is the root-mean-square value of Λci at wall-normal position y and λci is the imaginary part of the complex eigenvalue of the two-dimensional velocity gradient tensor (Zhou et al. 1999, Adrian et al. 2000a). Similar methods were used by Natrajan et al. (2007) and Herpin et al. (2010) to identify spanwise vortex filaments from PIV velocity fields. An example of filamentary spanwise vortices in open-channel flow is shown in Fig. 4. Intense, concentrated regions of Λci indicate the spanwise vortex filaments, and the velocity 13 Accepted for J. Hydraulic Research 2014 vectors around these regions are plotted to confirm the swirling nature. One can observe firstly that large populations of spanwise vortex filaments exist in open-channel flow, with prograde filaments exceeding the retrograde ones in quantity. Secondly, there are more spanwise vortex filaments in the lower part of the water depth. The above features agree well with the PIV results of Wu and Christensen (2006) and Natrajan et al. (2007) in turbulent boundary layer and channel flow. Figure 4 Example of a vortex identification result in case OCF380. Vectors denote velocity fields, local Galilean decomposition is performed; contour lines denote ωz, and solid and dashed lines represent positive and negative values, respectively; colour contours denote Λci within vortex cores, blue and red colour indicating prograde and retrograde vortex filaments, respectively After spanwise vortex filaments are identified, one can see that the regions covered by spanwise vortex filaments are contained within the regions covered by spanwise vorticity. That is, spanwise vortex filaments crossing the x-y plane always have spanwise vorticity, but some portions of the regions containing spanwise vorticity do not qualify as filaments. Thus, spanwise vorticity is due to both spanwise vortex filaments and non-filamentary structures. Since spanwise vortex filaments are tightly associated with spanwise vorticity, their relationship with net force is investigated by calculating the contribution of prograde (retrograde) vortex filaments to the total mean value of ⟨vωz⟩, 14 Accepted for J. Hydraulic Research 2014 ∑ v ( x , y) ω ( x , y ) ⋅ I ( x , y ) F ( ) ( y) = ∑ v ( x , y) ω ( x , y ) i pr z i p(r) i =1 i z i , (6) i i =1 where i is summed over the number of streamwise grid points in each velocity realization and the number of realizations in the ensemble. The indicator function, Ip(r), is given as ⎧1 if (xi , y) is within prograde (retrograde) vortex filaments I p(r) (xi , y) = ⎨ ⎩0 otherwise (7) It distinguishes grid points that fall within the boundary of identified vortex filaments from all others. Figures 5(a) and 5(b) plot Fp, the contribution from prograde vortex filaments to ⟨vωz⟩, as functions of y+ and y/h, respectively. The contributions from the retrograde filaments Fr are plotted in Figs 5(b) and 5(d). The prograde contributions increase sharply from about zero to local maxima exceeding 60% at about y/h=0.35 and then decrease slightly within 0.35