10TH INTERNATIONAL SYMPOSIUM ON PARTICLE IMAGE VELOCIMETRY - PIV13 Delft, The Netherlands, July 1-3, 2013
Cross-Plane Stereo PIV Measurements of a Turbulent Boundary Layer Overlying
Irregular Roughness
Julio M. Barros and Kenneth T. Christensen
Department of Mechanical and Science Engineering, University of Illinois at Urbana-Champaign, USA jmbarros@illinois.edu, ktc@illinois.edu
ABSTRACT
The structural attributes of turbulent flow over a complex roughness topography are explored with both low and high frame-rate stereo particle-image velocimetry (sPIV) measurements in a wall-normal–spanwise measurement plane (y − z). Particular attention is paid on the structures of the outer layer and well as within the roughness sublayer. The roughness under consideration was replicated from a turbine blade damaged by deposition of foreign materials and contains a broad range of topographical scales arranged in a highly irregular manner. This roughness was reproduced over a long streamwise fetch in a boundary-layer wind tunnel and measurements were performed at a momentum thickness Reynolds number of 14,000. Instantaneous velocity fields from the low-frame-rate measurements revealed structural attributes qualitatively consistent with smooth-wall flow structure, particularly patterns consistent with large-scale motions termed hairpin vortex packets. However, single-point turbulence statistics revealed significant statistical heterogeneity in the form of low- and high-momentum flow pathways marked by enhanced Reynolds stresses and turbulent kinetic energy. The low-momentum flow pathways were also marked by intense vortical activity along their spanwise boundaries, indicating that these pathways could represent preferential “channeling” of large-scale motions due to the roughness below or the generation of trains of vortical structures shed from the roughness that advect downstream along a common path. In addition, some of these flow pathways were found to extend well into the outer layer of the flow. The high-frame-rate sPIV measurements in the spanwise–wall-normal plane were conducted at a lower momentum thickness Reynolds number of 4500 and revealed the dynamical nature of the flow both in the outer region and in the roughness sublayer.
1. Introduction
The spatial characteristics of large-scale motions in wall turbulence have been the subject of intense study, particularly with recent evidence suggesting that these scales modulate the smaller-scale motions in the near-wall region [1]. The streamwise–wall-normal (x − y) plane two-dimensional particle-image velocimetry (PIV) measurements of Adrian et al. [2] provided a direct visualization of the coherent ordering of hairpin-like structures into larger-scale structural entities termed hairpin vortex packets. In particular, the streamwise alignment of individual hairpin-like structures into larger-scale packets observed by Adrian et al. [2] across the boundary layer in a hierarchy of scales is marked by an inclined interface formed by the spanwise-oriented heads of each structure beneath which a region of streamwise momentum deficit is apparent due to the collectively-induced ejection events generated by each of the vortices in a packet. Thus, these large-scale packets induce low-momentum regions (LMRs) previously identified in streamwise–spanwise (x − z) plane PIV measurements that are bounded by wall-normal vortex cores likely associated with the legs/necks of the individual vortices of hairpin packets [3, 4, 5] and within which intense ejections of low-speed fluid are generated [3, 5].
Instantaneous PIV fields in the x − z plane within the log layer also reveal the existence of high-momentum regions (HMRs) adjacent to LMRs within which strong sweep events are observed. This spanwise-alternating behavior of LMRs and HMRs is consistent with the spanwise-alternating sign of the two-point correlation of streamwise velocity in the x − z plane [6, 5]. More recently, hot-wire measurements indicate that the LMRs observed in δ-scale PIV studies (where δ is the boundary-layer thickness) can actually extend several δ in the streamwise direction [7]. These ‘superstructures’ can meander significantly in the spanwise direction [7] and can embody a significant fraction of the turbulent kinetic energy and Reynolds shear stress [8]. It is these motions that appear to amplitude modulate the smaller scales in the near-wall region of the flow [1]. Leveraging these amplitude-modulation observations, Marusic et al. [9] and Mathis et al. [10] proposed a predictive inner–outer model for the streamwise turbulence statistics in smooth-wall turbulence at high Re. While providing significant information about the structural characteristics of the flow, measurements at fixed wall-normal locations (i.e., fixed x − z PIV planes) unfortunately do not provide details as to the wall-normal dependence of the dominant spanwise scales of the flow.
Measurements in the wall-normal–spanwise (y − z) plane overcome such limitations; however, PIV measurements in this cross-flow plane are extremely challenging, as the bulk flow direction is normal to the lasersheet. Nevertheless, a few studies have successfully employed PIV to study wall turbulence in the cross-stream plane [6, 11, 12]. In particular, Hutchins et al. [11] and Ganapathisubramani et al. [6] used stereo PIV in cross-stream planes inclined at 45◦ and 135◦to the streamwise direction in a replication of the original flow-visualization imaging planes of Head and Bandyopadhyay [13]. These measurements revealed inclined vortical structures bounding LMRs that are consistent with the hairpin vortex packet model of wall turbulence. Spanwise-adjacent HMRs were also observed in the instantaneous fields, with both LMR and HMR events extending well into the outer layer of the flow. Analysis of spatial correlations of velocity in these inclined cross-stream planes also uncovered imprints consistent with hairpin vortex packets. Hutchins
and Marusic [7] used channel flow DNS fields to compute the conditionally-averaged velocity field associated with an LMR in the wall-normal–spanwise plane at low Reynolds number (Re). This field was characterized by an LMR bounded on either spanwise side by an HMR, between which streamwise vortices resided. Similar conditional average results for the large scales were reported by Chung and McKeon [14] from large-eddy simulations (LES) of turbulent channel flow at friction Reynolds numbers (Re) of 2,000 and 200,000.
The impact of roughness on this structural skeleton of smooth-wall flow is not yet fully understood. While some efforts indicate that roughness alters the structural and/or statistical attributes of the flow throughout the entire boundary layer [15, 16, 17, 18], other studies [19, 20, 21, 22, 5, 23] indicate that the effect of roughness is confined within the immediate vicinity of the roughness–the so-called roughness sublayer ( 3-5k, where k is a measure of the characteristic roughness height). This latter notion is consistent with Townsend’s wall similarity hypothesis [24], extended to rough-wall turbulence by Raupach et al. [20], which states that at high Re, surface conditions set the wall shear stress and the boundary-layer thickness, δ, while the turbulence outside the roughness sublayer adjusts itself to these conditions in an universal manner. A necessary condition for this similarity to exist is a broad scale separation between k and the outer length scale of the flow (typically taken as δ). Previous efforts indicate δ/k must exceed 40–50 for this similarity to exist [25, 26]. The geometrical details of the roughness can also play a critical role as to the existence of outer-layer similarity, with flow over three-dimensional (3D) roughness topographies often displaying such similarity in contrast to flow over two-dimensional (2D) topographies wherein the large spanwise extent of the roughness generates large-scale flow structures that grow well into the outer layer [27, 28, 29].
From a structural viewpoint, the PIV measurements of Nakagawa and Hanratty [30] in the x − y plane of turbulent channel flow with a wavy bottom wall revealed the spatial coherence of this flow to be quite similar to that of smooth-wall flow in the outer region. This observation is interesting given that the wavy wall under consideration was 2D in nature. Volino et al. [21] observed the spatial signatures of hairpin vortex packets in instantaneous PIV velocity fields in x − y and x − z measurement planes for a turbulent boundary layer (TBL) over woven wire mesh (3D roughness). Two-point correlations indicated a slight reduction in the streamwise spatial coherence close to the wall, compared to smooth-wall flow, that quickly diminished with increasing wall-normal position. Finally, Wu and Christensen [22] reported outer-layer similarity for flow over highly-irregular roughness replicated from a turbine blade damaged by deposition of foreign materials based on PIV measurements in the x − y plane. In a follow-up effort, Wu and Christensen [5] reported that this irregular roughness altered the characteristic streamwise and, to a lesser extent, the spanwise length scales of the flow based on stereo PIV measurements in a streamwise–spanwise plane near the outer edge of the roughness sublayer (y ≈ 0.2δ relative to the mean elevation of the roughness). Nevertheless, the rough-wall flow was still found to embody many of the structural attributes of hairpin vortex packets, including elongated LMRs bounded by wall-normal vortex cores interpreted as slices through the legs/necks of hairpin vortices.
The intent of the present contribution is to further explore the structural attributes of a TBL in the presence of the highly-irregular roughness topography employed in our previous efforts [22, 5, 23]. However, in contrast to these initial PIV measurements in the streamwise–wall-normal plane and a streamwise–spanwise plane at the outer edge of the roughness sublayer, the stereo PIV measurements reported herein were conducted in the wall-normal–spanwise plane to provide a simultaneous assessment of the flow’s spanwise spatial characteristics as well as their coherence in the wall-normal direction.
2. Experimental Setup
The TBL experiments were conducted in an open-circuit Eiffel-type, boundary-layer wind tunnel. The test section of the tunnel is 6 m long, 45.7 cm tall and 91.4 cm wide, and all boundary layers were formed on a smooth boundary-layer plate suspended above the bottom wall of the tunnel. This plate consists of two 3-m long and 91.4-cm wide smooth-wall sections smoothly joined at the streamwise center. Zero-pressure-gradient conditions were achieved via an adjustable ceiling in the test section.
The rough surface used was the same as that originally fabricated and studied by Wu and Christensen [22, 5]. This surface is a scaled version of a profilometric surface scan of a turbine blade damaged by deposition of foreign materials, which was first reported by Bons et al. [31]. Figure 1(a) presents a topographical map of the rough surface, which is marked by a broad range of topographical scales arranged in an irregular manner. The average peak-to-valley roughness height of this surface is k = 4.25 mm while the root-mean square (RMS) roughness height, krms, is 1.0 mm. As described in Wu and Christensen [5], a 3-m long replica of this topography was achieved by mirroring it in both the streamwise and spanwise directions and fabricated with a powder-deposition printer. This roughness was mounted on cast aluminum plates and placed along the downstream half of the boundary-layer plate by adjusting its height above the bottom wall of the tunnel such that the mean elevation of the roughness was coincident with the upstream smooth-wall conditions. Thus, the boundary layers under study were allowed to initially develop over the first 3 m of the smooth boundary-layer plate followed by an additional 3 m of development over the roughness. In all cases the flow was tripped with a cylindrical rod near the upstream end of the boundary-layer plate and all measurements were conducted approximately 2.3 m downstream of the leading edge of the roughness. Wu and Christensen [22] previously reported this rough-wall flow to have achieved self-similar conditions at this measurement location. Figure 1(c) presents a zoomed-in photo of a portion of the roughness replica in the wind tunnel. This photo highlights the complex, multi-scale nature of the topography that, despite being replicated from a damaged turbine blade. As in the present experiments, the features of these natural topographies typically protrude into the outer (logarithmic) region of the flow but are an order of magnitude smaller than the characteristic flow depth (δ).
Two sets of stereo PIV experiments were conducted: low frame-rate and high frame-rate measurements. Figure 2(a) presents a schematic of the stereo PIV arrangement for the low-frame rate experiments. The system employed consisted of two 4k × 2.75k pixel, 12-bit, frame-straddle CCD cameras (TSI 11MP) and a 190 mJ/pulse, dual-cavity pulsed Nd:YAG laser (Quantel). A 1.0 mm thick laser lightsheet was formed by three cylindrical lenses and directed into the tunnel’s test section in the y − z plane. The cameras viewed the y − z-oriented lightsheet from upstream through optical-grade glass side-walls of the wind tunnel at angles of ±45◦from
Figure 1: (a) Topographical map of the roughness. (b) pdf of roughness height about the mean elevation. (c) Photo of the replicated roughness in the wind tunnel along the flow direction.
Figure 2: Schematic of the experimental arrangements in the wall-normal–spanwise (y − z) measurement plane. (a) Low-frame-rate stereo PIV; (b) High-frame-rate stereo PIV.
the streamwise (x) direction. In the measurement plane, the angle between each lens and camera CCD array was adjusted to satisfy the Scheimpflug condition ensuring uniform focus across the field of view. The flow was seeded with 1 µm olive-oil droplets generated by a Laskin nozzle and timing of the cameras, lasers and image acquisition was controlled with a timing unit with 1 ns resolution. Figure 2(b) presents a schematic of the stereo PIV arrangement for the high-frame-rate experiments. The system consisted of two 1k × 1k pixel, 10-bit, CMOS cameras (Fastcam APX-RS Photron) and a 30 mJ/pulse at 1 kHz, dual-cavity pulsed Nd:YLF laser (Litron). A 1.0 mm thick laser lightsheet was formed by three cylindrical lenses and directed into the tunnel’s test section in the y − z plane. The cameras viewed the y − z-oriented lightsheet from a forward-scattering perspective to maximize the intensity of the scattered light imaged by the cameras, with one camera upstream to the laser lightsheet and the other downstream of it, through optical-grade glass side-walls of the wind tunnel at angles of ±45◦from the streamwise (x) direction. In the measurement plane, the angle between each lens and camera CMOS array was adjusted to satisfy the Scheimpflug condition which ensured uniform focus across the field of view. The flow was again seeded with 1 µm olive-oil droplets generated by a Laskin nozzle and timing of the cameras, lasers and image acquisition was controlled with the same timing unit as mentioned above.
Accurate stereo PIV measurements required careful calibration of the imaging system to properly map the image coordinate system to the object plane defined by the laser lightsheet. A single-plane target consisting of dots spaced at 2.5 mm in both the horizontal and vertical directions was utilized in the cross-stream experiments. The front face of this target was carefully aligned with the center of the lightsheet. Images of this target were acquired by both cameras at this position as well as with the target translated ±250 µm upstream and downstream of lightsheet center. The resulting calibration images for each measurement plane were used to generate calibration mapping functions to map the two, 2-D image planes to the 3-D space defined by the laser lightsheet using the least-squares method of Soloff et al. [32]. Thus, the out-of-plane fluid motion was discerned from the distinct views of the tracer-particle motion within the
Table 1: Summary of the experimental parameters for the turbulent boundary layer experiments over rough-wall Surface U∞(m/s) Reθ δ (mm) k(mm) δ/k Field of View Acq. Freq (Hz) # of Realizations
Rough 17.5 13700 94.1 4.25 22.1 1.5δ × 3.0δ 2 10000
Rough 6.0 4500 90.0 4.25 21.2 0.8δ × 1.3δ 1500 15 runs of 3000
Figure 3: Ensemble-averaged, single-point statistics in the wall-normal–spanwise (y − z) plane. (a) Mean streamwise velocity, U /Ue; (b) Turbulent kinetic energy, T KE/Ue2; (c) Reynolds shear stresse, −hu0v0i/Ue2; (d) Ensemble-averaged signed swirling strength (δ/Ue)hΛcii. Solid and dashed lines in (d) represent positive and negative contour levels, respectively.
laser lightsheet as imaged by the two cameras for each of the stereo PIV experiments.
For the low-frame rate experiments, ten thousand statistically-independent planar, three-component velocity fields were acquired in the cross-flow measurement plane over the roughness at Reθ' 14000. Each three-component velocity field was derived from two, 2-D displacement fields generated from the time-delayed pairs of images acquired by each camera. These pairs of time-delayed images were interrogated using a recursive, two-frame cross-correlation methodology. The first-pass interrogation was performed with a bulk window offset to minimize loss of particle pairs, while the final-pass interrogation was performed with square interrogation spots of size 162pixels with 50% overlap to satisfy the Nyquist sampling criterion, and the second window was locally offset by an integer pixel displacement determined during the first-pass interrogation. Statistical validation tools were employed between passes to identify and replace erroneous vectors as well as after the final interrogation pass was completed, including Rohaly–Hart [33] replacement with displacements assessed from alternate correlation peaks identified during the interrogation process. All fields were then low-pass filtered with a narrow Gaussian filter to remove high-frequency noise. Each pair of 2D displacement fields was then recombined using the aforementioned mapping function to reconcile all three instantaneous velocity components on the measurement plane defined by the laser lightsheet. The field of view was 1.5δ × 3.0δ (wall-normal by spanwise), resulting in a vector grid spacing of 520 µm in both spatial directions.
For the high-frame-rate measurements, 15 runs of three thousand stereo PIV fields were in the same cross-flow plane over the roughness at Reθ' 4500 at a rate of 1500 vector fields/s. The same calibration and interrogation methodology described above was used for these measurements. The final field of view was 0.8δ × 1.3δ (wall-normal by spanwise), resulting in a grid spacing of 680 µm in both spatial directions. Table 1 summarizes the relevant experimental parameters for these measurements. For reference, the boundary-layer thickness, δ, was taken as the wall-normal location where the mean streamwise velocity, U , was 99% of the free-stream velocity, Ue. 3. Results
3.1 Single-Point Statistics
Figure 3 presents ensemble-averaged, single-point statistics in the y − z measurement plane in perspective view with the upstream roughness topography included as well. These statistics were computed by averaging over the ensemble of 10,000 statistically independent velocity realizations acquired in this measurement plane. No spatial averaging was performed in order to retain the streamwise and spanwise dependence of these statistics. The mean, outer-scaled streamwise velocity (U /Ue; Figure 3a) shows strong heterogeneity in the form of a spanwise-localized low-momentum pathway (LMP) bounded in the spanwise direction by
high-momentum pathways (HMPs). These heterogeneities are quite distinct from the character of smooth-wall turbulence, which reflects the expected uniform (homogeneous) U . This result suggests that the roughness under consideration induces a “channeling” effect in the flow or the possible existence of persistent wakes generated by dominant roughness features in the case of the LMP. These low- and high-momentum pathways are distinct from the LMRs and HMRs identified in the instantaneous realization of Figure 4a (and in the aforementioned studies of smooth-wall turbulence) because they appear in the ensemble-averaged field of the streamwise velocity. In particular, the LMPs and HMPs can be interpreted as preferential pathways for these low- and high-momentum motions in contrast to the instantaneous LMRs and HMRs that, in the case of smooth-wall flow, occur randomly in space and have clear spatial coherence. Thus, the roughness under consideration appears to generate large-scale heterogeneity in the mean streamwise velocity in the form of preferred spatial pathways for low- and high-momentum events. While preferential flow paths have been observed, and are likely expected, in the near-wall region of flow over periodic surface features [34], this flow channeling phenomenon persists despite the complex nature of the present surface topography [Figure 1(c)].
Figures 3b and 3c present contour plots of outer-scaled mean turbulent kinetic energy (TKE; 1/2hu02+ v02+ w02i/U2
e) and RSS (−hu0v0i/U2
e) in the y − z plane. Both of these single-point turbulence statistics display significant heterogeneity in the spanwise direction, particularly enhanced regions of both TKE and RSS spatially coincident with the identified LMP in the mean streamwise velocity (Figure 3a). Similar regions of enhanced behavior are also notable in the other RSS components (hu0w0i and hv0w0i which are quite weak in smooth-wall flow; not shown for brevity) spatially coincident with the identified LMP. Since the Reynolds shear stresses play a defining role in the production of TKE from the mean flow, these results suggest that the roughness under consideration may promote generation of TKE in preferential regions within the roughness sublayer. In addition, the unique view afforded by the y − z measurement plane reveals that these regions of enhanced TKE and RSS are not confined to the near-wall region but rather extend quite far from the wall (y ≈ 0.6δ).
Finally, Figure 3d presents contours of outer-scaled, ensemble-averaged signed swirling strength, hΛcii(δ/Ue). Here, Λci= λciωx/|ωx| where λciis the imaginary portion of the complex-conjugate eigenvalue pair of the local velocity gradient tensor which has previously been shown to be an effective vortex marker [2]. In particular, λciis frame independent and does not incorrectly identify regions of local shear as vortices (as vorticity can in wall-bounded flows). Thus, λci6= 0 indicates the presence of local vortical motion and is marked with the sign of the in-plane streamwise vorticity (ωx) to retain the sense of the rotation within λci. Interestingly, Λciin Figure 3d is quite heterogeneous in space, with a region of hΛcii < 0 at the left spanwise boundary of the LMP identified in Figure 3a and a region of hΛcii > 0 at the right spanwise boundary of this LMP. A region of hΛcii ≈ 0 is noted within the LMP. This pattern is consistent with a preferential alignment of clockwise- and counter-clockwise-rotating swirling motions aligned along the left and right boundaries of the LMP, respectively. The overall rotational sense of this counter-rotating vortical activity is consistent with the ejection of low-speed fluid from the near-wall region into the outer region of the boundary layer. Therefore, these patterns in hΛcii may be indicative of a preferential alignment of vortical motions due to the roughness upstream or the generation of such structures directly by the roughness. Alternatively, this pattern could also be interpreted as a train of structures that are the result of unsteady shedding from dominant roughness elements either upstream or below the measurement plane. However, under this scenario it is not clear whether such “trains” of vortices exhibit the same large-scale coherence as vortex packets. Regardless of the origin, this evidence suggests that vortical structures are likely responsible for the LMP identified in the mean streamwise velocity (Figure 3a) as well as the TKE (Figure 3b) and RSS (Figure 3c) enhancements noted deep within the roughness sublayer.
3.2 Instantaneous Fields
Figure 4a presents a representative instantaneous fluctuating velocity field in the y − z measurement plane with the in-plane wall-normal (v0) and spanwise (w0) velocity fluctuations shown as vectors and the out-of-plane streamwise velocity fluctuations (u0) presented as background contours. Note that the positive streamwise (x) direction, and hence the mean streamwise flow, is into Figure 4a. The streamwise velocity fluctuations are marked by large-scale (δ-scale) regions of low and high streamwise momentum that appear to alternate in the spanwise direction with a spacing of ∼ 0.5δ. These patterns are interpreted as the cross-plane signatures of the LMRs and HMRs and can often extent to the edge of the boundary layer. Focusing upon the large-scale LMR near z = 0.25δ in Figure 4c, its left boundary is populated by counter-clockwise-rotating vortex cores (Λci< 0; blue) while its right boundary is populated by vortex cores with clockwise rotation (Λci> 0; red). Furthermore, rather intense, positive wall-normal velocity fluctuations (v0) are observed within this LMR, resulting in a large-scale region of low-speed fluid ejected away from the wall which contributes heavily to the mean RSS. This LMR is flanked on its spanwise boundaries by HMRs within which intense, negative v0create a large-scale sweep of high-speed fluid towards the wall which also contributes heavily to the mean RSS (Figure 4b). Apart from these δ-scale events, smaller LMRs and HMRs are visualized in the near-wall region that are often bounded by streamwise vortex cores. These smaller-scale regions appear to co-exist beneath the larger-scale LMRs and HMRs, supporting the contention that such structures occur in a hierarchy of scales across the flow. As proposed by Adrian et al. [2] for smooth-wall turbulence, packets of varying size would be expected throughout the wall-normal extent of the flow, with smaller, younger, slower packets residing closer to the wall where they are likely formed and successively larger, older packets populating the outer region of the flow while maintaining a near-wall footprint. Thus, despite the presence of a rough boundary, the overall structural attributes of the flow are quite consistent, at least qualitatively, with those of smooth-wall turbulence. This observation is in accordance with Townsend’s wall similarity hypothesis [24] whereby the roughness sets the wall shear stress and the boundary-layer thickness and the turbulence in the outer region simply adjusts itself to these constraints in a universal manner.
3.3 Spatial Coherence
While well-defined regions of low- and high-momentum pathways were identified in the y − z single-point statistics of Figure 3, it is not known whether these pathways represent spatially-correlated turbulent events or a concatenation of uncorrelated turbulent events that simply advect along the same streamwise path. To assess the spatial coherence of these motions, two-point inhomogeneous velocity correlation coefficients were computed with the reference location taken at the center of the LMP identified in the measurement plane.
Figure 4: (a) Representative instantaneous fluctuating velocity field in the y − z plane. Contours of (b) instantaneous Reynolds shear stress, u0v0, and (c) signed swirling strength, Λcifor field in (a). Solid and dashed line contours in (b) and (c) demarcate boundaries of HMRs and LMRs, respectively.
Figure 5: Two-point velocity correlation coefficients in the wall-normal–spanwise (y − z) plane for a reference point positioned within the LMP identified in the figure 3 at (yref, zref) = (0.1δ, 0.15δ); (a) Autocorrelation of streamwise velocity, ρ11; (b) Autocorrelation of wall-normal velocity, ρ22; (c) Autocorrelation of spanwise velocity, ρ33; (d) Cross-correlation of streamwise and wall-normal velocities, ρ12.
These correlations are computed as
ρi j(y, z; x, yref, zref) =
hu0i(yref, zref)u0j(x, y, z)i σi(x, yref, zref)σj(x, y, z)
, (1)
in the y − z plane at fixed x where (yref, zref) defines the spatial location of the reference point in this measurement plane.
Figure 5 presents two-point velocity correlation coefficients in the y − z measurement plane for a reference point situated within the LMP identified in Figure 3 at (yref, zref) = (0.1δ, 0.15δ). The ρ11correlation (Figure 5a) is marked by a primary peak at (yref, zref) = (0.1δ, 0.15δ) that is bounded in the spanwise direction by correlation minima. The correlation maximum at the reference location indicates that the motions that travel along the identified LMP have significant wall-normal coherence (δ-scale). The alternating positive/negative nature of ρ11 in this y − z plane is again consistent with the occurrence of spanwise-alternating LMRs and HMRs in instantaneous velocity fields (Figure 4) and indicates the LMP identified in Figure 3 is often bounded in the spanwise direction by high-speed pathways. In contrast, ρ22(Figure 5b) is smaller in spanwise scale, though it is still characterized by a primary correlation peak at (yref, zref) = (0.1δ, 0.15δ) that is bounded by correlation minima in the spanwise direction. Similar behavior is noted in ρ12 (Figure 5d). The characteristics of ρ11, ρ22and ρ12in the y − z plane are consistent with the imprint of streamwise-oriented vortices, or a larger-scale collection of such structures, that pump low-speed fluid away from the wall and draw high-speed fluid towards the wall. Such motions result in the generation of intense RSS-producing events. Finally, ρ33(Figure 5c) displays a V-shaped region of positive correlation above which a wall-normal-elongated region of strong negative correlation resides. The symmetry of this correlation with respect to the location of the LMP suggests consistency with spanwise pairs of counter-rotating streamwise vortex cores. In other words, the configuration of ρ33, particularly its persistence into the outer region of the boundary layer, is consistent with the combined action of the legs of hairpin-like, or similarly oriented, vortical structures.
Thus, the spatial coherence observed in the rough-wall boundary layer through the two-point correlations of velocity fluctuations in the y − z planes centered along the spatial position of the LMP identified in Figure 3 indicates a high degree of spatial coherence along this preferential flow path. As such, these observations do not support the hypothesis that the identified LMP is simply a concatenation of uncorrelated turbulent events simply advecting along the same streamwise path. In contrast, the spatial characteristics exhibited by the mean flow, particularly the identified LMP, appear to be associated with larger-scale spatially-coherent motions traveling along this preferred flow path. This observation is consistent with at least two structural scenarios: 1) trains of statistically-correlated vortices that are shed from dominant roughness features that travel along the common flow path demarcated by the LMP or, 2) a “channeling” of existing large-scale motions (perhaps existing hairpin vortex packets, for example) along preferred paths over the roughness. It should be noted that two-point correlations computed in regions devoid of LMPs and HMPs more closely resemble those of smooth-wall flow. 3.4 High-Frame-Rate Stereo PIV Results
The cross-plane (y − z) high-frame-rate sPIV allows one to qualitatively visualize the 3D structures present in the rough-wall flow studied herein. The streamwise displacement of the bulk flow between consecutive instantaneous vector fields was maintained at half of the lightsheet thickness, which is consistent with the in-plane spatial resolution. Under this scenario, Taylor’s hypothesis was utilized
Figure 6: Representative quasi-instantaneous 3D reconstruction of the flow using Taylor’s hypothesis where iso-surfaces of the 3D swirling strength are shown with coloring representing the local streamwise velocity. (a) Perspective view; (b) Side view.
Figure 7: Wall-parallel x − z view of an LMR at y/δ = 0.15 demarcated with contours of negative streamwise velocity fluctuation from Taylor’s hypothesis reconstruction. A zoomed-in region is also presented with color contours of signed swirling strength demarcating the locations of wall-normal vortex cores.
to convert the temporal dimension to equivalent streamwise position assuming that the turbulence is frozen with respect to the advection in the streamwise direction. A single advection velocity was utilized when reconstructing the instantaneous structures based on the bulk velocity of the flow, giving x ' (t◦− t) ¯U[35, 36, 37].
Figure 6 presents a representative quasi-instantaneous 3D reconstruction of the flow using Taylor’s hypothesis from 1000 instantaneous fluctuating cross-plane velocity fields. Iso-surfaces of the 3D swirling strength are shown and computed using all 9 components of the local velocity gradient tensor with coloring representing the local streamwise velocity. This figure highlights the broad range of instantaneous vortical structures that populate this rough-wall flow, including arch-type, streamwise elongated vortices as well as structures that resemble hairpin-like vortices.
Finally, Figure 7 presents a wall-parallel x − z view of an LMR at y/δ = 0.15 demarcated with contours of negative streamwise velocity fluctuation. This streamwise-elongated planar field is a wall-parallel slice through the quasi-3D field in Figure 6. A zoomed-in region is also presented with color contours of signed swirling strength demarcating the locations of wall-normal vortex cores. Focusing upon the zoomed-in region in Figure 7, it is apparent that the streamwise-elongated LMRs are bounded on the spanwise edges by counter-rotating vortex patterns which is consistent with the hairpin packet model. In addition, this figure highlights the elongated streamwise extent of these superstructures, which appear to extend 5 − 6δ in the streamwise direction and are quite reminiscent of similar patterns reported by Hutchins and Marusic [7] from measurements using a spanwise array of hot-wire sensors in conjunction with Taylor’s hypothesis to reconstruct streamwise-elongated, wall-parallel fields of view. As already mentioned previously and also shown in recent work [22, 5], these LMRs are qualitatively similar to the structures found in smooth-wall flow. However, as highlighted in the ensemble-averaged
streamwise velocity, the roughness under consideration seems to provide a preferential path for LMRs and HMRs, which could affect the spanwise meandering of these δ-scale structures. A direct comparison with smooth-wall data at the same Re is needed to explore these possibilities.
4. Summary
Cross-plane sPIV experiments were conducted on a turbulent boundary layer flow over highly irregular roughness containing a broad range of topographical scales distributed in irregular manner. Results from instantaneous fluctuating velocity fields suggest that the many features found in smooth-wall flow, such as the hairpin vortex packets, are also present in the rough-wall flow. On the other hand, the turbulence statistics are quite distinct from smooth-wall results. The mean streamwise velocity field shows a high degree of heterogeneity in the spanwise direction, with low-momentum and high-momentum pathways evident. The LMPs are characterized by an intense region of enhanced TKE and Reynolds shear stress and the spanwise boundaries of the LMPs are marked by imprints of counter-rotating vortical motions. To evaluate whether these regions represent spatially-correlated turbulent events, two-point correlations were computed at the spatial location of an identified LMP. The results show a high degree of spatial coherence along what appears to be a preferential flow path. Quasi-instantaneous 3D reconstructions of the flow from high-frame-rate sPIV measurements in the cross-plane in conjunction with Taylor’s hypothesis reveal the presence of a broad range of instantaneous structures. A planar slice of this reconstruction parallel to the wall reveals streamwise-elongated regions of streamwise momentum deficit that extend multiple δ in x, consistent with previous smooth-wall observations [7].
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