Analyzing three-dimensional wake vortex dynamics...

17th International Symposium on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2014

- 1 -

Analyzing three-dimensional wake vortex dynamics using time-resolved

planar PIV

Chris Morton*, Serhiy Yarusevych

Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, Canada

* correspondent author: [email protected] Abstract The present study investigates the three-dimensional wake development of complex cylindrical geometries using time-resolved, planar, two-component Particle Image Velocimetry data. A new technique is proposed for reconstructing salient aspects of wake topology based on planar data collected in multiple planes. The technique produces phase-average reconstructions based on Proper Orthogonal Decomposition analysis of velocity fields. It is validated on the turbulent wake of a low aspect ratio dual step cylinder geometry, consisting of a large diameter cylinder (D) with small aspect ratio (L/D) attached to the mid-span of a small diameter cylinder (d). Experiments were performed in a water flume facility for ReD = 2100, D/d = 2, and L/D = 1. The results show that the proposed technique successfully reconstructs the dominant periodic wake vortex interactions and can be extended to a wide range of flows.

1. Introduction Vortex shedding from cylindrical structures occurs in many engineering applications, for example, aircraft wing-body junctions, landing gears in cross-flow, and marine structures, such as off-shore risers exposed to ocean currents. The bluff body geometries and flow conditions involved in such applications are often far more complex compared to the canonical case of a uniform circular cylinder in uniform cross-flow, which has been investigated extensively over the last several decades (Roshko, 1954; Oertel Jr., 1990; Williamson, 1996). It is well known that, for non-uniform flows and complex bluff body geometries, vortex shedding is often highly three-dimensional (e.g., Zdravkovich et al., 1989; Williamson, 1992; Morton & Yarusevych, 2014a). Moreover, under turbulent vortex shedding conditions, which are ubiquitous in engineering applications, there are often spatial and temporal variations in vortex strength, coherence, and frequency (e.g., Zdravkovich et al., 1989). In order to investigate such flows experimentally, quantitative measurements of the three-dimensional flow field can be completed using tomographic Particle Image Velocimetry (PIV), three-dimensional PTV, multi-plane scanning PIV, and holography (Arroyo & Hinsch, 2008). However, these techniques are often restricted by spatial and temporal resolution limits, and are not yet common research tools in experimental facilities world-wide, many of which employ more conventional techniques, such as 2D PIV. Over the last few decades, several techniques of 2D PIV data analysis have proved useful for studying vortical structures in turbulent flows, including: (i) phase-averaging of dominant periodic structures, (ii) vortical structure identification through Proper Orthogonal Decomposition (POD), (iii) linear stochastic estimation, (iv) pattern recognition methods, and combinations of these techniques. The present investigation presents a new approach for investigating vortex dynamics in turbulent flows by linking POD and phase-averaging approaches. The goal is to reconstruct the essential features of three-dimensional vortex topology using two-dimensional PIV measurements in multiple planes. Traditional phase-averaging of PIV measurements involves obtaining a reference signal for resolving the phase variation of dominant flow structures. This typically requires a time-resolved quantitative measurement of the flow velocity or surface pressure. Once phase variation is obtained in the measurement signal, the PIV data can be grouped according to specific phase angles and then averaged. This type of procedure provides a statistical description of the dominant structures, and has been successfully applied in several investigations (e.g., Konstantinidis et al., 2005; Perrin et al., 2007a; Sattari et al., 2012). The POD was recommended by Lumley (1967) as a tool for identifying coherent structures in turbulent flows. If the flow is strictly periodic, the basis functions of the POD become equivalent to Fourier modes, and the POD reduces to a harmonic orthogonal decomposition (Berkooz et al., 1993). As a consequence, in the wake of a bluff body, a relatively small number of modes is generally sufficient to characterize the dominant vortical structures in the flow field due to the strong periodicity associated with vortex shedding phenomenon. The


- 2 -

POD has more recently been adapted to perform statistical phase-averaging of the dominant flow structures in the wake of a circular cylinder and square cylinder in a single two-dimensional PIV measurement plane (van Oudheusden et al., 2005; Perrin et al., 2007b). This POD-based phase-averaging methodology has been shown to be an improvement over traditional phase-averaging techniques due to a reduction in phase-jitter effects (Perrin et al., 2007b). The present study employs POD-based phase-averaging of planar, two-component PIV measurements in multiple planes for a test model of a dual step cylinder in uniform flow. As shown in Fig. 1, a dual step cylinder consists of two different diameter cylinders, with the larger diameter cylinder attached coaxially to the mid-span of the smaller diameter cylinder. Previous experimental studies completed on dual step cylinders have shown that the flow development is influenced by the Reynolds number (ReD=UoD/ν, where Uo is the free stream velocity, D is the large cylinder diameter, and ν is the kinematic viscosity of the fluid), large cylinder aspect ratio (L/D, where L is the length of the large cylinder), and diameter ratio (D/d, where d is the diameter of the small cylinder) (Williamson, 1992; Morton & Yarusevych, 2012; Morton & Yarusevych, 2014a). Williamson (1992) investigated the wake development of a dual step cylinder in the laminar vortex shedding regime for Red ≤ 200, 1.1 ≤ D/d ≤ 2.0, and L/d = 0.5. His results showed that periodic vortex shedding occurs in the wake of each small diameter cylinders, and complex vortex interactions and vortex dislocations occur in the wake of the large cylinder. The vortex dislocations occur periodically, and are linked to two dominant frequencies detectable in the large cylinder wake. The flow visualization images and sketches by Williamson (1992) suggest that the essential features of the wake vortex dynamics may be described by these two frequencies. Morton & Yarusevych (2012, 2014a) investigated the dual step cylinder wake topology in the turbulent shedding regime for 1050 ≤ ReD ≤ 2100, D/d = 2, and 0.2 ≤ L/D ≤ 17. Similar to the results of Williamson (1992), at low aspect ratios less than about L/D = 0.5, vortex dislocations between small cylinder vortices occur periodically in the wake of the large cylinder. On the other hand, for larger L/D, up to four different wake topologies can occur: (i) vortex shedding at a frequency lower than that expected from a uniform cylinder at the same Reynolds number, (ii) irregular vortex shedding, (iii) vortex shedding at a frequency higher than that expected for a uniform cylinder, and (iv) vortex shedding in multiple constant frequency cells along the span of the large cylinder. The wide range of vortex wake topologies observed for a dual step cylinder makes this geometry a suitable test case for investigating complex, three-dimensional, turbulent flows which contain some underlying periodicity. The present study examines the three-dimensional wake topology of a dual step cylinders for ReD = 2100, D/d = 2, and L/D = 1. 2. Experimental Methodology A water flume facility at the University of Waterloo was used for all experiments. The dual step cylinder

Fig. 1 Dual step cylinder model.


- 3 -

models were placed in a uniform flow and mounted between circular endplates, as shown in Fig. 2. Qualitative visualization of the wake development was completed with hydrogen bubble technique. For these visualizations, a DC voltage is applied to a 0.085 mm diameter stainless steel wire placed approximately 0.7D upstream of the model axis and along its entire span. Additional details of the hydrogen bubble system and its implementation can be found in Morton & Yarusevych (2014b). Two-dimensional velocity measurements were completed using a LaVision PIV system comprised of a Photonics Nd-YLF pulsed laser and two Photron SA4 high speed cameras. The flow was seeded with 10 μm hollow glass spheres. The PIV system was arranged in order to acquire measurements in vertical (x-z) and horizontal (x-y) planes simultaneously, as shown in Fig. 2. For these measurements, the vertical plane remained stationary at y/D = -0.5, while eighteen horizontal measurement planes were investigated in the range -3.2 ≤ z/D ≤ 0. Considering the relatively low free-stream velocity (U ≈ 85 mm/s), the PIV system was operated in single-frame mode. A total of 5456 particle images were acquired at a fixed rate of 100Hz for each measurement plane and processed using LaVision DaVis 8 software. The data acquisition rate employed is over fifty times the vortex shedding frequencies of interest. Cross-correlation was performed using an iterative, multi-grid approach with window deformation (Scarano & Riethmuller, 2000). The final interrogation window size was 16x16 pixels with a 75% overlap applied between adjacent windows, resulting in a vector spacing of approximately 0.03D. The direction and displacement of particles within each window was approximately constant and the number of particles within each window was maintained at approximately ten, as per the recommendations by Keane & Adrian (1992). The flow development was investigated using POD analysis of both vertical and horizontal PIV measurements. For the horizontal planes, it is well-established that, in periodic wake flows, the first pair of POD modes combined with the mean flow gives a low order approximation of the dominant wake structures (e.g., van Oudheusden et al., 2005). As shown in Fig. 3, a cross-plot of the first pair of temporal coefficients lies on a circle given by Equation 1.

122 2

22

1

21

aa

(1)

where a1 and a2 are the first two temporal coefficients, and λ1 and λ2 are the first two eigenvalues obtained from the POD. The phase angle of the vortex shedding is given by:

Fig. 2 PIV measurement setup. PIV measurements are conducted simultaneously in the x‐z plane at y/D =

‐0.5 and x‐y plane, with multiple x‐y planes investigated. For illustrative purposes, velocity fields are

shown in place of the raw particle image pairs.


- 4 -

ta

ta

1

2

2

1arctan

(2)

Since the data points in Fig. 3 correspond to individual velocity fields, the plot can be used for phase-averaging by separating the data into discrete bins associated with a specified phase angle, as illustrated in the figure. In the present study, a bin size of 20 degrees was employed. This bin size was selected as a compromise between phase resolution and the desired statistically significant number of velocity fields in each bin. In contrast with the numerous studies investigating phase-averaging techniques in bluff body flows with a single dominant frequency, few have considered flows with multiple dominant frequencies, e.g., the wake of a dual-step cylinder. One approach to investigating such flows is to obtain multiple reference signals simultaneously, with each signal containing a distinct phase reference for one of the dominant frequencies. If the phase of each dominant frequency is known for each vector field, a conditional phase-averaging approach can be employed. For a complex three-dimensional geometry, such as a dual step cylinder, vortical structures of varying size, shape, and frequency are being shed from the body. Hence, a single point reference signal will not capture the dynamics in the entire flow field. Although an array of sensors may lead to reasonable results, the performance is expected to depend dramatically on the placement of the probes so as to ensure that the wake structures are being properly detected. In contrast, as will be shown in this study, no external reference signal is required for phase-averaging planar PIV data, as the POD can be used for conditional phase-averaging and three-dimensional reconstruction of the wake development. 3. Results A conditional phase averaging technique which allows reconstruction of complex three-dimensional wake topology from planar PIV measurements is presented here. A dual step cylinder is used as a test model and is investigated for ReD = 2100, D/d = 2, and L/D = 1. The results are divided into three sections. First, the characteristics of the wake development for a low aspect ratio dual step cylinder are presented using flow visualization. Following this, a conditional phase-averaging technique is developed and implemented.

3.1 Dual Step Cylinder Wake Development The main features of the flow development past a dual step cylinder are illustrated in the sequence of hydrogen bubble flow visualization images presented in Fig. 4. The hydrogen bubbles highlight the vortex cores in the wake and show that both the large and small cylinder shed vortices periodically, but at different frequencies. Analysis of video records revealed that the dominant frequencies in the large and small cylinder wake are fLD/Uo = 0.26, and fSD/Uo = 0.40, respectively, as was later confirmed from PIV measurements. Morton & Yarusevych (2014a) showed that, under the investigated conditions, small and large cylinder vortices move in and out of phase periodically at the beat frequency f* = fS - fL, and each 2π phase variation between the large

Fig. 3 Cross plot of normalized temporal coefficients a1 and a2 from POD analysis of x‐y planar PIV data in

the wake of a dual step cylinder. A unit circle is drawn with a red dashed line for reference.

0-0.5-1.0-1.5 1.51.00.5

1.5

1.0

0.5

0

-0.5

-1.0

-1.5

a2/√2λ2

a1/√2λ1

θ


- 5 -

and small cylinder vortices involves vortex splitting of large cylinder vortices and substantial deformations of vortex filaments in the wake of the small cylinder, as can be seen at x/D = 2 and -0.5 ≤ z/D ≤ -1.5 in Fig. 4c. Moreover, smaller-scale deformations of the spanwise vortex filaments result in the formation of streamwise vortices which connect between the spanwise vortex cores, e.g., at x/D = 2.0 and z/D = 1.75 in Fig. 4b. 3.2 Development of a POD‐based phase averaging methodology A POD-based phase-averaging approach is developed in this section using experimental data obtained in the wake of a dual step cylinder. The methodology involves determining if the temporal coefficients obtained from POD in each horizontal PIV plane can be used to extract the phase variation of the small cylinder and large cylinder vortices simultaneously. Figure 5 shows the mode energy distribution and cumulative mode energy distribution for the first fifty POD modes. The results indicate that, in the wake of the large diameter cylinder as well as sufficiently far away from the step (z/D < -1.8), the first two modes capture over 50% of the wake energy content (Fig. 5b). However, between these regions, where vortex interactions take place (Fig. 4), the energy content in the first two modes is considerably lower. Figure 6 presents the first two spatial modes obtained from the POD of x-y planar PIV data for three selected spanwise locations, z/D = 0, -1.0, and -3.2. The first two modes at each spanwise location are qualitatively

(a) t* = 0 (b) t* = 0.44 (c) t* = 1.14

Fig. 4 Sequence of hydrogen bubble flow visualization images in the wake of a dual step cylinder. t* is the

number of large cylinder vortex shedding periods. The images presented are negatives of the raw hydrogen

bubble flow visualization images. The hydrogen bubbles trace vortex cores on one side of the wake only.

2.5

2.0

1.5

1.0

0.5

z/D

-0.5

-1.0

-1.5

-2.0

-2.5

x/D0.5 1.0 1.5 2.0 2.5 3.0 0 3.5 4.0

2.5

2.0

1.5

1.0

0.5

z/D

-0.5

-1.0

-1.5

-2.0

-2.5

x/D0.5 1.0 1.5 2.0 2.5 3.0 0 3.5 4.0

2.5

2.0

1.5

1.0

0.5

z/D

-0.5

-1.0

-1.5

-2.0

-2.5

x/D0.5 1.0 1.5 2.0 2.5 3.0 0 3.5 4.0

(a) (b)

Fig. 5 Energy distribution over the POD modes.


- 6 -

similar to those found for a uniform circular cylinder (e.g., Perrin et al., 2007), indicating strong periodicity in the flow. Figure 7 shows spectra for the first temporal coefficient at z/D = 0, -1.0, and -3.2. Note that the results for the second temporal coefficient feature the same spectral content and are not included for brevity. At z/D = 0 (Fig. 7a), the majority of the energy content is centered on the large cylinder vortex shedding frequency, fD/Uo ≈ 0.26. At z/D = -3.2 (Fig. 7c), the energy content of the first temporal coefficient signal is centered on the small cylinder vortex shedding frequency, fD/Uo ≈ 0.40. Spectral analysis of other temporal coefficients associated with lower energy modes at z/D = 0 and -3.2 showed no evidence of a secondary dominant frequency. Hence, at these spanwise locations, a single dominant vortex shedding phenomenon is present, i.e., the vortex shedding from the large cylinder at z/D = 0, and the vortex shedding from the small cylinder at z/D = -3.2. In

(a) z/D = 0, φ1 (b) z/D = 0, φ2

(c) z/D = ‐1.0, φ1 (d) z/D = ‐1.0, φ2

(e) z/D = ‐3.2, φ1 (f) z/D = ‐3.2, φ2

Fig. 6 Spatial eigenmodes obtained from POD of x‐y planar PIV data acquired in the wake of a dual step

cylinder.

y

x

y

x

y

x

y

x

y

x

y

x


- 7 -

such a case, the planar wake topology can be reconstructed using a traditional POD-based phase-averaging, as outlined in the experimental methodology section (Fig. 3). However, a different approach is required for reconstructing the wake topology in the entire field where two dominant frequency-centred activities interact. At z/D = -1.0, the temporal coefficient spectrum in Fig. 7b contains energy contributions from two frequencies, namely, the large and small cylinder vortex shedding frequencies. It is important to note that the presence of two frequency-centred activities in the wake is not clear from the examination of the corresponding spatial modes (Figs. 6c and 6d), and the analysis of time resolved temporal coefficient is required. The temporal variation of the energy content associated with the two dominant wake frequencies can be characterized by band-pass filtering the first two temporal coefficient signals from the POD results. Specifically, the filtering is performed using a Fourier decomposition of the two temporal coefficient signals (a1 & a2), followed by signal reconstruction with a limited number of Fourier modes centered on each of the two dominant frequencies. This is illustrated in Figs. 8a-d for signal a1. Cross-plots of the filtered and normalized temporal coefficient signals are presented in Figs. 8e and 8f. The same analysis can be applied to the entire data set, including all spanwise locations. Figure 9 depicts cross-plots of filtered temporal coefficients for a range of z/D locations spanning

(a)

(b)

(c)

Fig. 7 Spectra of the first temporal coefficient, a1 for: (a) z/D = 0, (b) z/D = ‐1.0, and (c) z/D = ‐3.2.

0.1 0.26 0.40 1.0fD/U

0.1 0.26 0.40 1.0fD/U

0.1 0.26 0.40 1.0fD/U


- 8 -

the flow region of interest. Within the large cylinder wake and in the adjacent part of the interaction region (-1.4 ≤ z/D ≤ 0), the temporal coefficients are filtered based on the large cylinder vortex shedding frequency (Figs. 9a-d); whereas, in the small cylinder wake and within the adjacent interaction region (-3.2 ≤ z/D ≤ -0.8), the filter is centered on the small cylinder vortex shedding frequency (Figs. 9e-h). As expected, the results show that there is a reduction in both the energy content (related to the magnitude of the temporal coefficients) and the periodicity (reflected in how well the data falls onto a circular fit in the cross plots) as the survey plane is moved away from the regions dominated by either of the two frequency-centred activities (Figs. 9d & 9h). The plots in Fig. 9 can be used for conditional averaging, which allows one to reconstruct planar wake topology corresponding to any relative phase alignment between the two dominant frequency-centred activities.

3.1.2 Extracting phase variation along the span using PIV measurements in the vertical plane Although horizontal plane POD analysis is sufficient to phase-average planar velocity measurements in each individual plane, the knowledge of the phase variation along the span is required for three-dimensional reconstruction. This is due to the fact that, in the energy-based decomposition, phase angles computed from

(a) (b)

(c) (d)

(e) (f)

Fig. 8 Analysis of temporal coefficients obtained from POD of x‐y planar PIV data for z/D = ‐1.0. (a)

Temporal coefficient spectrum of a1 and filtered spectrum pertaining to fD/Uo ≈ 0.4, (b) Temporal coefficient

spectrum of a1 and filtered spectrum pertaining to fD/Uo ≈ 0.26, (c) & (d) Sample segment of temporal

coefficient a1 and filtered temporal coefficient, (e) & (f) Cross‐plot of filtered temporal coefficients, and

. Filtered data are shown with red lines for (a)‐(d). A red dashed line illustrating a unit circle is shown in

(e)‐(f).

0.1 026 0.40 1.0fD/U

0.1 0.260.40 1.0fD/U

1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500−2

−1

0

1

2

Snapshot #1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500

−2

−1

0

1

2

Snapshot #

0-0.5-1.0-1.5 1.51.00.5

1.5

1.0

0.5

0

-0.5

-1.0

-1.5

θ

a2

a1

0-0.5-1.0-1.5 1.51.00.5

1.5

1.0

0.5

0

-0.5

-1.0

-1.5

θ

a2

a1


- 9 -

the POD correspond to the same instant in the shedding cycle along the entire span of the model. In a 3D reconstruction, this artificially aligns the vortices to be parallel to the model axis. Thus, the spanwise variation of the phase angle of the vortex filaments needs to be determined and used to properly adjust the relative phase alignment in the adjacent horizontal planes. This can be achieved based on the analysis of the simultaneously acquired vertical PIV plane measurements. As shown in Fig. 10, the first four spatial eigenmodes obtained from the POD of the vertical plane measurements describe the spatial arrangement of the dominant coherent structures. Specifically, the spatial arrangement of the small cylinder and large cylinder vortex filaments is obtained. Hence, an adequate reduced order model of the vertical PIV planar fields can be constructed using the first four POD modes. To determine spanwise phase variation, velocity measurements from the reduced order model were analyzed along x/D = 3, located just downstream of the vortex formation region. Fourier decomposition was applied to all data in order to extract the fluctuations pertaining to each of the two dominant frequencies in the wake. Following this, a Hilbert transform was applied to the data in order to obtain the instantaneous phase of the signal. Figure 11 depicts the results for z/D = -1.0, where two dominant frequencies are present in the velocity signal. The results are in agreement with the analysis of temporal coefficients completed at the same z/D location in Fig. 8. In order to determine how the phase of the signal differs from the instantaneous phase along the entire model span, the data in Fig. 11b is compared with a global phase reference signal extracted from filtering the first two temporal coefficient signals of the vertical plane PIV data (at the large cylinder vortex shedding frequency), and the data in Fig. 11d is compared with a similar reference signal extracted from filtering the third and fourth temporal coefficient signals (at the small cylinder vortex shedding frequency). Following this, the mean and standard deviation in the phase difference between these signals is computed. This data analysis is repeated for all spanwise locations where fluctuations at each of the two vortex shedding frequencies are detectable, and the results are presented in Fig. 12. The obtained spanwise variation in the mean phase angle of the small and large cylinder dominant wake frequencies,φ,isshowninFig.12,alongwith95% confidence interval bands illustrating the expected variability in phase. The results illustrate that, the mean phase angle based on the large cylinder dominant frequency does not vary significantly (-10 < φ < 0), while the mean phase angle based on the small cylinder dominant frequency decreases with increasing z/D, where φ lies in the range -50 < φ < 0. As expected, for both frequency-centred activities, the most notable phase variation occurs within and near the vortex interaction region.

3.3 Three dimensional reconstruction of wake topology The flow visualization results (Fig. 4) show that small and large cylinder vortices continuously move in and

Fig. 9 Cross plots of filtered temporal coefficient signals: (a)‐(d) temporal coefficient signals filtered based

on the large cylinder vortex shedding frequency for (a) z/D = 0, (b) z/D = ‐0.4, (c) z/D = ‐0.8, (d) z/D = ‐1.2.

(e)‐(h) temporal coefficient signals filtered based on the small cylinder vortex shedding frequency for (e)

z/D = ‐2.4, (f) z/D = ‐1.6, (g) z/D = ‐1.2, (h) z/D = ‐0.8.

0-0.4-0.8 -1.2 1.20.80.4

1.2

0.8

0.4

0

-0.4

-0.8

-1.2

a2

a10-0.4-0.8 -1.2 1.20.80.4

1.2

0.8

0.4

0

-0.4

-0.8

-1.2

a2

a10-0.4-0.8 -1.2 1.20.80.4

1.2

0.8

0.4

0

-0.4

-0.8

-1.2

a2

a1

0-0.4-0.8 -1.2 1.20.80.4

1.2

0.8

0.4

0

-0.4

-0.8

-1.2

a2

a1

0-0.4-0.8 -1.2 1.20.80.4

1.2

0.8

0.4

0

-0.4

-0.8

-1.2

a2

a1

0-0.4-0.8 -1.2 1.20.80.4

1.2

0.8

0.4

0

-0.4

-0.8

-1.2

a2

a1

0-0.4-0.8 -1.2 1.20.80.4

1.2

0.8

0.4

0

-0.4

-0.8

-1.2

a2

a10-0.4-0.8-1.2 1.20.80.4

1.2

0.8

0.4

0

-0.4

-0.8

-1.2

a2

a1

(a) (b) (c)

(e) (f) (g)

(d)

(h)


- 10 -

out of phase, resulting in a complex arrangement of vortex interactions that is presumably dependent on their relative phase alignment. Using PIV data, it is possible to perform a conditional phase averaging to reconstruct wake topologies observed through various relative phase alignments of vortices in the large and small cylinder wakes. First, the results in Fig. 12 are used to adjust the relative phase alignment between the planar results. Next, using the conditioned cross-plots of the first pair of temporal POD coefficients (similar to those shown in Fig. 9), the conditional averaging can be carried out. The large cylinder vortex shedding frequency is selected as the reference frequency (i.e., the conditional phase averaging is done based on the phase of the large cylinder) and a phase angle bin size is set at 20 degrees. This gives 288 instantaneous velocity fields, on the average, in each bin. For a given phase of the large cylinder vortices, a relative arrangement of vortex filaments can be reconstructed for any phase angle of small cylinder vortices, i.e., a conditional phase averaging can be performed. This leaves approximately 16 instantaneous velocity fields to be averaged in each conditional bin at each spanwise location. Figure 13 shows six three-dimensional reconstructions of the vorticity field in the wake of a dual step cylinder. In these reconstructions, the phase position of the large cylinder vortices is held fixed while the relative phase of the small cylinder vortices is advanced through six different phase angles. The results in Fig. 13 illustrate the dominant periodic vortex interactions occurring between the large and small cylinders in the near wake. On the average, small cylinder vortices incline back towards the step discontinuities as they are shed (e.g., Fig. 13b & 13c). The small cylinder vortices consistently form vortex connections with large cylinder vortices in

(a) φ1 (b) φ2

(c) φ3 (d) φ4

Fig. 10 Vertical plane spatial eigenmodes obtained from POD of PIV data acquired in the wake of a dual

step cylinder at y/D = ‐0.5.


- 11 -

the near wake, though the arrangement of these connections is dependent upon their relative phase alignment.. The consistent ‘Y-type’ connections observed in the near wake agree with those observed in flow visualization results (Fig. 4). Farther downstream, vortex connections between large and small cylinder vortices are maintained when these vortices are aligned in phase (e.g., Fig. 13d-f). When the phase separation becomes significant, e.g., Fig. 13b, there is no evidence of direct connections being maintained. Based on the results of Morton & Yarusevych (2012, 2014a), in this case, consecutive small cylinder vortices form half-loop connections which cannot be properly resolved with spanwise vorticity isosurfaces. Moreover, more significant cycle to cycle variations in the spatial arrangement of vortex connections are expected as x/D increases, and, hence, the conditional average cannot resolve such non-periodic features of vortex interactions. 4. Conclusions A POD-based methodology for reconstructing three-dimensional wake topology from planar velocity measurements has been developed and evaluated experimentally. The technique is applicable primarily to strongly periodic flows which can consist of more than one dominant frequency. It relies on time-resolved planar velocity measurements in multiple two-dimensional planes acquired simultaneously with velocity measurements in an orthogonal plane. The orthogonal plane is configured to determine quantitative information on the spatial and temporal phase variation of the vortical structures. Each of the two-dimensional planes is conditionally phase-averaged using POD. The flow is reconstructed three-dimensionally for different relative phase positions of the dominant vortical structures. A dual step cylinder in cross flow at ReD = 2100, D/d = 2, and L/D = 1 was investigated to test the proposed

Fig. 11 Analysis of fluctuating wake velocity data from a reduced order model of vertical plane PIV data

for x/D = 3.0, z/D = ‐1.0. (a) Sample segment of velocity data and velocity data filtered based on a frequency

band centred on the small cylinder vortex shedding frequency, (b) phase variation of the filtered velocity

data in (a), (c) Sample segment of velocity data and velocity data filtered based on a frequency band centred

on the large cylinder vortex shedding frequency, (d) phase variation of the filtered velocity data in (c).

Filtered data are shown with red lines for (a) and (c).

100 110 120 130 140 150 160 170 180−0.03

0

0.03

tU/D(a)

u’

100 110 120 130 140 150 160 170 1800

180

360

tU/D(b)

θ

100 110 120 130 140 150 160 170 180−0.03

0

0.03

tU/D(c)

u’

100 110 120 130 140 150 160 170 1800

180

360

tU/D(d)

θ


- 12 -

methodology. The results show that the proposed conditional phase-averaging approach successfully reconstructs the dominant periodic vortex interactions between spanwise vortices in the wake of a dual step cylinder. This technique is a significant improvement over traditional phase averaging methods. The technique provides a new statistical tool for the analysis of vortex dynamics in turbulent flows.

References

Arroyo MP, Hinsch KD (2008) “Recent Developments of PIV towards 3D Measurements,” Particle Image Velocimetry: New Developments and Recent Applications, Topics in Applied Physics 112: 127-154.

Berkooz G, Holmes P, Lumley JL (1993) “The proper orthogonal decomposition in the analysis of turbulent flows,” Annual Review of Fluid Mechanics 25: 539-575.

Keane RD, Adrian RJ (1992) “Theory of cross-correlation of PIV images. Applied Scientific Research 49:191-215.

Konstantinidis E, Balabani S, Yianneskis M (2005) “Conditional averaging of PIV plane wake data using a cross-correlation approach,” Experiments in Fluids 39: 38-47.

Lumley JL (1967) “The structure of inhomogeneous turbulence,” In Atmospheric Turbulence and Radio Wave Propagation, ed. A.M Yaglom, V.I. Tatarski, Nauka, Moscow, 166-178. Morton C, Yarusevych S (2012) “An experimental investigation of flow past a dual step cylinder,”

Experiments in Fluids 52:69-83. Morton C, Yarusevych, S (2014a) “On vortex shedding from low aspect ratio dual step cylinders”, Journal of

Fluids and Structures 44:251-269. Morton, C, Yarusevych, S (2014b) “Three-dimensional flow and surface visualization using hydrogen bubble

technique”, Accepted to Journal of Visualization, JOVI-D-14-00012. Oertel Jr., H (1990) “Wakes behind blunt bodies,” Annual Review of Fluid Mechanics 22: 539-564. Perrin R, Cid E, Cazin S, Sevrain A, Braza M, Moradei F, Harran G (2007a) “Phase-averaged measurements

of the turbulence properties in the near wake of a circular cylinder at high Reynolds number by 2C-PIV

Fig. 12 Variation in phase along the span of a dual step cylinder. Red solid and dashed curved lines

represent the mean phase variation, and 95% confidence interval in the phase variation of the velocity

signals filtered at the small cylinder vortex shedding frequency, respectively. Black solid and dashed

curved lines represent the same for the velocity signals filtered at the large cylinder vortex shedding. The

horizontal lines represent the spanwise extents of the vortex interaction region.

0.5

0

-0.5

-1.0

-1.5

-2.0

-2.5

-3.0

z/D

-150 -100 -50 0 50 100 150φ


- 13 -

and 3C PIV” Experiments in Fluids 42: 93-109. Perrin R, Braza M, Cid E, Cazin S, Barthet A, Sevrain A, Mockett C, Thiele F (2007b) “Obtaining phase

averaged turbulence properties in the near wake of a circular cylinder at high Reynolds number using POD” Experiments in Fluids 43:341-355.

Roshko, A. (1954) “On the development of turbulent wakes from vortex streets,” NACA Report No. 1191. Sattari P, Bourgeois JA, Martinuzzi RJ (2012) “On the vortex dynamics in the wake of a finite surface-mounted

square cylinder,” Experiments in Fluids 52:1149-1167. Scarano F., and Riethmuller M. L. (2000) “Advances in iterative multigrid PIV image processing,”

Experiments in Fluids 29(1), pp. S051-S060. van Oudheusden BW, Scarano F, van Hinsberg NP, Watt DW (2005) “Phase-resolved characterization of

vortex shedding in the near wake of a square-section cylinder at incidence” Experiments in Fluids 39:86-98. Williamson CHK (1996) “Vortex dynamics in the cylinder wake”, Annual Review of Fluid Mechanics

28:477-539. Zdravkovich MM, Brand VP, Mathew G, Weston A (1989) “Flow past short circular cylinders with two free

ends,” Journal of Fluid Mechanics 203:557-575.

(a) (b) (c)

(d) (e) (f)

Fig. 13 Reconstructed three‐dimensional wake topology of a dual step cylinder for a large cylinder phase

of zero degrees and a small cylinder phase of: (a) 30◦, (b) 120◦, (c) 180◦, (d) 250◦, (e) 290◦, (f) 330◦. Isosurfaces

of vorticity at ωU/D ≈ 0.1 are shown and colored by the vorticity sign. Note that the data has been mirrored

about the z/D = 0 plane for illustrative purposes.

Analyzing three-dimensional wake vortex dynamics...

Documents

Transcript of Analyzing three-dimensional wake vortex dynamics...