«Characteristics of vortex packets in turbulent boundary layers By B H A R A T H R A M G A N A P A T H I S U B R A M A N I, E L L E N K. L O N G M I R ...»
J. Fluid Mech. In Press.
Characteristics of vortex packets in
turbulent boundary layers
By B H A R A T H R A M G A N A P A T H I S U B R A M A N I,
E L L E N K. L O N G M I R E A N D I V A N M A R U S I C
Department of Aerospace Engineering and Mechanics, University of Minnesota,
107 Akerman Hall, 110 Union Street SE, Minneapolis, MN 55455, USA
e-mail : email@example.com
(Received 07 August 2002 and in revised form 25 October 2002) Stereoscopic PIV was used to measure all three instantaneous components of the velocity ﬁeld in streamwise-spanwise planes of a turbulent boundary layer at Reτ = 1060 (Reθ = 2500). Datasets were obtained in the logarithmic layer and beyond. The vector ﬁelds in the log layer (z + = 92 and 150) revealed signatures of vortex packets similar to those proposed by Adrian and co-workers in their PIV experiments. Groups of legs of hairpin vortices appeared to be coherently arranged along the streamwise direction. These regions also generated substantial Reynolds shear stress, sometimes as high as 40 times −uw. A feature extraction algorithm was developed to automate the identiﬁcation and characterisation of these packets of hairpin vortices. Identiﬁed patches contributed 28% to −uw while occupying only 4% of the total area at z + = 92. At z + = 150, these patches occupied 4.5% of the total area while contributing 25% to −uw. Beyond the log layer (z + = 198 and 530), the spatial organisation into packets is seen to break down.
1. Introduction The structure of the turbulent boundary layer has been the subject of much research over the past 60 years because of its importance in practical applications. Some of the current studies are aimed at developing models to represent the boundary layer eﬃciently at higher Reynolds numbers. Various models have been proposed for a dominant structure including long quasi-streamwise vortices in the viscous buﬀer region (see Lyons, Hanratty & McLaughlin 1989 and Heist, Hanratty & Na 2000). However, a growing consensus is emerging for a model based on individual hairpin vortices (Theodorsen 1952) which is probably the simplest structural model that explains most of the features observed in wall turbulence. The existence of asymmetric hairpins in the region above the viscous layer was documented by Robinson (1991) in his analysis of early DNS results. Earlier, Oﬀen & Kline (1975) used a hairpin model to explain the existence of ejections (Q2 events, u 0, w 0), volumes of low speed ﬂuid that are pushed away from the wall and sweeps (Q4 events, u 0, w 0), volumes of faster ﬂuid pushed towards the wall, in the logarithmic part of the boundary layer. In this paper u, v and w are deﬁned as the ﬂuctuating velocity components in the streamwise (x), spanwise (y) and wall-normal (z) directions respectively. Willmarth & Lu (1971) and Blackwelder & Kaplan (1976), among others, documented that ejections and sweeps make a substantial contribution to the Reynolds shear stress and therefore to the drag associated with the bounding surface.
Bandyopadhyay (1980) and Head & Bandyopadhyay (1981) proposed that hairpin vortices travel in groups after studying the ﬂow visualisation experiments they performed 2 B. Ganapathisubramani, E. K. Longmire and I. Marusic on a zero pressure gradient boundary layer. Theoretical studies by Smith et al. (1991) and the study of DNS datasets at low Reynolds number (Reτ = δUτ /ν = 180, where δ is the boundary layer thickness, Uτ is the skin friction velocity and ν is the kinematic viscosity) performed by Zhou et al. (1999) demonstrated that a single vortex of suﬃcient circulation could spawn a trailing group of hairpins that convected at the same speed as the leading structure. More recently, Adrian, Meinhart & Tomkins (2000b), who performed particle image velocimetry (PIV) experiments in x−z planes of a zero pressure gradient boundary layer over a Reynolds number range of 355 Reτ 2000, identiﬁed packets of hairpin vortex heads that appeared regularly in the near-wall and logarithmic layers. Groups of 5hairpins, which extended over a length of 2δ, were determined to convect at a uniform streamwise velocity. The authors noted also that the presence of vortex packets explained the multiple ejections associated with individual bursts observed by Bogard & Tiederman (1986). The elongated zones of uniform streamwise velocity associated with the packets also help explain the long tails in two-point correlations of streamwise velocity found in Kovasznay, Kibens & Blackwelder (1970), Townsend (1976) and Brown & Thomas (1977).
Recently, Christensen & Adrian (2001) with PIV data in x − z planes of channel ﬂow at Reτ = 547 and 1734, used linear stochastic estimation to estimate the conditionally averaged velocity ﬁeld associated with swirling motion. They concluded that the mean structure consists of a series of swirling motions along a line inclined at 12◦ − 13◦ with the wall which is consistent with the earlier observations of packets of hairpin vortices.
An attached eddy model applied by Marusic (2001) also demonstrated that packets of eddies were required in order to match measured boundary layer turbulence statistics near the wall.
In recent work similar to the present study (Tomkins & Adrian 2002, see also the thesis by Tomkins 2001), PIV measurements of streamwise-spanwise planes of a turbulent boundary layer at 430 Reτ 2270 (only in-plane velocity components were measured) showed long low streamwise momentum zones enveloped by positive and negative vortex cores representing packets of hairpin structures.
The objective of the current study was to apply the stereo PIV technique to examine x−y planes in the outer region of a zero pressure gradient boundary layer (z + = zUτ /ν
60) in order to identify the typical structures and to determine their contribution to the Reynolds shear stress, thereby investigating the role, if any, the packet structures play in the transport of momentum in the boundary layers. Based on the previous work described above, feature identiﬁcation algorithms were developed to search for individual hairpin vortices as well as packets of hairpins.
2. Facility and methods
2.1. Experimental facility, methods and qualiﬁcation The measurements were carried out in a suction wind tunnel with working section of
0.33 m height, 1.22 m width and 4.8 m length. The coordinate system used is depicted in ﬁgure 1. Measurement planes were located 3.3 m downstream of a trip wire in a zeropressure-gradient ﬂow with freestream velocity U∞ = 5.9 m s−1. Hot-wire measurements showed that the turbulence intensity in the freestream was less than 0.2%. The wall shear stress (τw ) was computed using the mean velocity proﬁle and the Clauser chart method.
All quantities measured and computed are normalised using the skin friction velocity Uτ (= τw /ρ, where ρ is the density of the ﬂuid) and ν and are denoted with a superscript +. The Reynolds number based on the momentum thickness Reθ was 2500, and Reτ was
1060. The value of δ in the region of the measurement planes was 69 mm.
Vortex packets in boundary layers 3
Figure 1. Experimental facility.
The ﬂow was seeded with olive oil droplets (size ∼ 1 µm) that were generated by eight Laskin nozzle units set up in parallel. The oil droplets were ingested into the intake of the wind tunnel upstream of honeycomb straighteners and screens used for ﬂow conditioning.
At the test section, glass side-walls and a glass bottom wall were installed in the wind tunnel to provide high-quality optical access. The seed particles were illuminated by pulsed sheets from two Nd:YAG lasers (Big Sky CFR200) directed through one side window and oriented parallel with the bottom wall of the tunnel. (See ﬁgure 1). The laser pulse energy was 120 mJ, and the thickness of each sheet was 0.3 mm. Sets of digital images were captured by two Kodak Megaplus CCD cameras (1024 × 1024 pixels) at z + = 92, 198 (z/δ = 0.2) and 530 (z/δ = 0.5) and by TSI Powerview 2048 × 2048 pixel resolution cameras at z + = 150. Nikon Micro Nikkor 60 mm f/2.8 lenses were used with both camera types. A TSI synchroniser box controlled the strobing and timing of the cameras and lasers. The dual frame acquisition rate was 15 Hz. The cameras were aligned in a plane parallel with the x − y ﬂow plane and inclined at angles of 15◦ with the z axis as shown in ﬁgure 1. Additional details of the calibration, experimental arrangement and vector reconstruction schemes are given in Ganapathisubramani, Longmire & Marusic (2002).
For all derived quantities, the interrogation spot size used was 16×16 pixels (∼ 20×20 wall units) with 50% overlap. Hence the spacing between adjacent vectors in either direction is 10 wall units (∼ 0.65 mm). The measurements had two main sources of uncertainty.
First, the Gaussian peak ﬁt in the cross-correlation algorithm generated an uncertainty of approximately 0.1 pixels or 0.016U (where U is the mean velocity in the streamwise direction). Second, a residual error arose due to the least-square curve-ﬁt in solving the four pixel-displacement equations in three unknowns (see Ganapathisubramani et al.
2002 for details). This residual error can play a major role in the uncertainty if concentration gradients in seeding occur within the ﬂow ﬁeld. In this experiment, such gradients were not signiﬁcant, and this error was on average about half of the Gaussian error. The vector ﬁelds from the images acquired with the Megaplus cameras covered an area of
1.2δ × 1.2δ. For the Powerview cameras, the ﬁeld size was approximately 2.4δ × 2.4δ making the spacing between vectors similar to that in the other planes. The larger area was acquired by moving the plane of the cameras away from the object plane.
Velocity gradients were computed from vector ﬁelds using a second order central difB. Ganapathisubramani, E. K. Longmire and I. Marusic
ference scheme wherever possible in the domain and a ﬁrst order forward or backward diﬀerence at the boundaries. To identify swirling motion caused by the eddies in the ﬂow ﬁeld, we used swirl strength λci (Zhou et al. 1999), which is deﬁned as the magnitude of the imaginary part of the eigenvalue of the local velocity gradient tensor. Since the PIV images were planar, we used only the in-plane gradients and formed a two dimensional form of the tensor. Vortices could be identiﬁed by extracting iso-regions of λci (Adrian, Christensen & Liu 2000a).
The values of U computed from the PIV data at each wall-normal location agreed well with the hot-wire measurements. Table 1 shows a summary of some ensemble averaged turbulence statistics. The values shown compare well with previous data in the literature (Balint, Wallace & Vukoslavcevic 1991; Naguib & Wark 1992; Adrian et al. 2000b).
2.2. Feature extraction Although a variety of coherent structures can be found in a turbulent boundary layer, we focus on the hairpin vortices discussed in the introduction. Since, the stereo PIV data are obtained in streamwise-spanwise planes, hairpin legs could appear as neighbouring cores of positive and negative wall-normal vorticity (ωz ) aligned in the spanwise direction.
If these hairpins were to travel in groups, one might also see multiple pairs of cores aligned approximately along the streamwise direction. Also, individual or groups of angled hairpins are expected to generate large values of instantaneous Reynolds shear stress (−uw+ = −uw/Uτ 2 ). Hence, an algorithm was developed to identify regions in which all of these events occur. In the algorithm description, the ﬂow is assumed to be from left
to right :
• Step 1: Regions of positive vorticity 1.5σωz + and negative vorticity −1.5σωz + are detected. At z + = 92, this value was ±0.065.
• Step 2: Zones of strong Reynolds shear stress (−uw+ ) greater than 2σ−uw + (where σ−uw + is the r.m.s of −uw+ ) are detected independently. No distinction is attached to sweep or ejection events. The values of σ−uw + at z + = 92 and 150 were 3.43 and 2.68 respectively.
• Step 3: All the zones found in step 2 are searched to mark regions of high Reynolds shear stress occurring between strips of positive and negative vorticity (found in step
1) such that positive vorticity lies beneath negative vorticity. This vorticity arrangement was chosen to ﬁt the pattern displayed by the legs of a hairpin vortex.
• Step 4: Each location of large −uw+ marked in step 3 is used as a seed point for a region growing algorithm. This algorithm, based on the streamwise velocity, identiﬁes a connected region of neighbouring points lying within local thresholds such that the ﬁnal coherent region is a patch of uniform momentum. The lower bound of the velocity was ﬁxed as the minimum velocity in the Reynolds shear stress zone and the upper bound was the average velocity in the two surrounding vorticity regions. If multiple seed zones are enveloped by the same vorticity regions, the thresholds are adjusted such that the Vortex packets in boundary layers 5 lower bound is the minimum velocity in all the seed points of −uw+ in that region. If the region growth from one location intersects the patches from other seed points, then they are merged and marked as one patch.
• Step 5: After detection of all patches in a given vector ﬁeld, patches that are adjacent to one another along the streamwise direction are merged provided that the streamwise distance between the patches is less than the average width of the patches being merged.
3. Results and discussion Please note that the ﬂow is from left to right in all vector plots, and the local mean U is subtracted from the vectors to clearly illustrate the slow and fast moving zones. The contour/vector plots shown in ﬁgure 2 are from individual realisations, but the patterns are representative of those found in many vector ﬁelds. The plots revealing the statistical features of the hairpin packets (ﬁgure 3) were obtained by investigating all vector ﬁelds at a given wall normal position.