Near Wake Structure of a Blunt Trailing Edge Profiled Body

LAKSHMANA DODDIPATLA1, ARASH NAGHIB-LAHOUTI2 and HORIA HANGAN3

Wind Engineering, Energy, and Environment Research Institute, The University of Western Ontario, London, Ontario, N6A 5B9

lakshmana.doddipatla@fmglobal.com, alahouti@utias.utoronto.ca, hmh@blwtl.uwo.ca Abstract: - Wake flows behind two dimensional bodies are mainly dominated by two coherent structures, namely the Karman-Benard vortices and the streamwise vortices, also referred to as rolls and ribs respectively. The three-dimensional wake instabilities lead to distinct instability modes (mode-A, mode-B and mode-C or mode S) depending on the flow Reynolds number and geometric shape. The present investigation explores the mechanism by which the flow transitions to three dimensionality in the near wake of a profiled leading edge and blunt trailing edge body. A combination of Planar Laser Induced Fluorescence visualizations and Particle Image Velocimetry measurements are conducted for Reynolds numbers ranging from 250 to 2300. The results indicate that three instability modes (mode-A, mode-B and mode-C) appear in the wake transition to three dimensionality, and their order of appearance does not follow the traditional sequence as observed in circular cylinder flows. It is found that mode-C instability with a spanwise spacing of 2.4D (D being the trailing edge thickness) dominates the near wake development.

Key-Words: - Blunt trailing-edge-profiled body, Vortex shedding, Near wake instabilities, Particle Image Velocimetry (PIV), Planar Laser Induced Fluorescence (PLIF), Proper Orthogonal Decomposition (POD)

1 Introduction Wake flows behind nominally two dimensional bodies are mainly dominated by two coherent structures, namely the von Karman-Benard spanwise vortices and the streamwise vortices, also referred to as rolls and ribs respectively. It has been established that ribs wrap around rolls and are interconnected.1 The development of both two- and three-dimensional instabilities and their sensitivity to various actuation techniques are found to play a dominant role in control strategies to mitigate vortex shedding and minimize lift fluctuations and drag.2,3,4,5,6 Identifying the most unstable secondary wake instability and triggering it would be the most efficient way to develop a wake control mechanism. This work focuses on identifying the spanwise spacing of the most unstable streamwise wake instability using PLIF (Planar Laser Induced Fluorescence) and PIV (Particle Image Velocimetry) measurements in the near wake, and by application of Proper Orthogonal Decomposition (POD) on the PIV data.

Over the last two decades extensive research has been conducted to understand the three-dimensional near wake flow topology of various wake generators, such as circular cylinders (Karniadakis and Triantafullou,7 Bay-Muchmore and Ahmed,8 Mansy et al.,9 Zhang et al.,10 Brede et al.,11 Williamson,12 Barkley and Henderson13), square cylinders (Robichaux et al.,14 Luo et al.,15,16 Dobre and Hangan,17 Sheard et al.18), thin flat plates (Meiburg and Lashares,19 Julien et al.,6), blunt trailing edge 1 Currently Senior Research Scientist at F M Global, 1151 Boston Providence Turnpike, Norwood, MA, 02062

2 Currently Post Doctoral Research Assistant at University of Toronto Institute for Aerospace Studies, 4925 Dufferin Street, Toronto, Ontario, Canada M3H 5T6

profiled body (Ryan et al.20), bluff rings (Sheard et al.21), blunt trailing edge airfoil (El-Gammal and Hangan22).

For circular cylinder flows, Williamson12 demonstrated that two distinct three-dimensional instability modes occur, i.e. mode-A and mode-B, depending on the flow Reynolds numbers. Mode-A occurs at ReD>180 and is gradually replaced by mode-B at ReD>230. The spanwise spacing between streamwise vortices of Mode-A is scattered between 3D to 5D,12 but for Mode-B it was consistently found to be around 1D12 for various Reynolds numbers. The spatio-temporal structure of mode-A and mode-B are shown in Figure 1. For mode-A, the sign of streamwise vorticity, and thereby direction of the secondary vortices, alternates twice every shedding cycle. For mode-B the streamwise vorticity pattern shows spanwise periodicity. Yet in contrast to mode-A, the sense of rotation does not change with time for a given spanwise location.

For square cylinder, Robichaux et al.14 demonstrated that wake transition modes and sequence are similar to the ones observed in the case of circular cylinder, with slightly larger wavelengths. These observation were verified by different authors15,16,17,18 later both experimentally and numerically. Recently Ryan et al.20 numerically demonstrated that the sequence of near wake transition for a bluff body with profiled leading edge and blunt trailing edge is dependent on the body aspect ratio, summarized here is Table 1. It can be observed from Table 1 that the order of appearance of the near wake three dimensional secondary wake instabilities depends on the aspect ratio. They reported that mode-A wake instability dominates the near wake

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development when AR < 7.5, with a spanwise wavelength of 3.5D, and by a mode-B instability for

Figure 1. Description of Mode-A, Mode-B and Mode-C

structures in XY and XZ planes

AR >7.5 with a spanwise wavelength of 2.2D. However their study is limited by the capability of the linear stability analysis to predict the transition regimes for higher Reynolds number. To the authors’ knowledge there has been no prior experimental verification of these findings reported in the literature, and this forms the main objective of the current investigation. The scope of the current investigation is to • Perform well-controlled experiments using Planar

Laser Induced Fluorescence (PLIF) and Particle Image Velocimetry (PIV) to verify the findings of the numerical work by Ryan et al.20 for the same geometry with an aspect ratio of 12.5.

• Extract the dominant coherent structures by performing Proper Orthogonal Decomposition (POD) on the obtained PIV data, and relate the POD modes and associated eigenvalues to the dominant coherent structures of near wake

• Identify the spatio-temporal topologies of the near wake instabilities and their corresponding unstable wavelengths

• Extend the study up to a Reynolds number of 2300, investigate if the modes A, B and C coexist, and identify which instability mode dominates the near wake development.

Table 1. Summary of the wake instabilities for blunt trailing edge profiled body (Ryan et al.20)

Aspect Ratio (AR)

1st Instability mode

2nd Instability mode

3rd Instability mode

2.5 Mode A – 3.5D (Re = 400)

-- --

7.5 Mode A – 3.9D (Re ≈ 470)

Mode B – 2.2D (400<Re <500)

Mode C – 0.9D (Re > 500)

12.5 Mode B – 2.2D (Re ≈ 410)

Mode A – 3.5D (Re ≈ 600)

Mode C – 0.9D (Re > 600)

17.5 Mode B – 2.2D (Re ≈ 430)

Mode C – 0.7D (Re ≈ 690)

Mode A – 3.5D (Re > 700)

2 Experimental Setup

Transparent end plate

Flow direction

Elliptic leading edge

Imaging Planes

LX = 12.5D

Lz = 48D

D = 0.0127 m

Z X

Y

Dye injection slot

34D

Figure 2. Flat plate model used for this investigation, along with schematic of the region in the near wake of the model imaged by

the PIV system

All of the measurements for this investigation are performed in the 0.61 m x 0.305 m x 3 (m) cross section water tunnel facility at The Boundary Layer Wind Tunnel Laboratory of The University of Western Ontario. This is an open return, closed section water tunnel producing a maximum free-stream velocity of 0.2 m/s. The water introduced into the water tunnel is passed through a settling chamber consisting of honeycomb and screens to break down large scale non-uniformities. The mean velocity profile varies within 1% across the test section and the turbulence intensity is less than 1%, where the experiments are performed. The water tunnel was operated at free stream velocities of 0.02 to 0.2 m/s, equivalent to ReD varying from 250 to 2300. The model shown in Figure 2, has a thickness of D = 0.0127m, a chord of 0.1587m (12.5D), and a total span of 0.6m (48D). PLIF Visualizations and PIV measurements have been carried out over a 34D portion of the span, which is bounded by two transparent endplates. The endplates isolate the body from end effects. The fluorescent dye is introduced into the flow through a thin spanwise slot on the lower surface of the body, located 1D upstream of the trailing edge. A horizontal light sheet is generated using an Nd-YAG laser with 532nm wavelength. The dye, which is a Rhodamine 6G solution, emits a fluorescent light with 560nm wavelength when exposed to the laser. The 550nm color filter blocks the laser light reflections and allows the CCD camera to capture only the fluorescent light emitted by the dye. The images have been recorded at a capture rate of 15 frames per second using a camera resolution of 1200 x 1600 pixels. PLIF measurements were performed in XY, XZ and YZ planes. The same system is used for the PIV measurements. The wake velocity field is determined from raw images using cross correlation in 50% overlapping interrogation windows of 32x32 pixels, yielding 99 x 74 vectors with a spatial resolution of 1.83

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mm (0.144D). A schematic of the location in the near wake, where PIV measurements are performed relative to the flat plate model is shown in Figure 2. A total of 3000 PIV images were acquired for each experiment. PIV measurements were carried out in XY and XZ (Y/D = 0: Shear layer) planes.

3 Proper Orthogonal Decomposition For this investigation, Proper Orthogonal Decomposition (POD) is used to study the near wake instabilities. POD analysis yields a set of Eigenvectors or POD modes ( iφ )

that are optimal orthogonal basis functions. In this investigation, these POD modes are optimized for the kinetic energy and the combination of these POD modes provide information about the spatial structures of the investigated flow field. The Eigenvalues ( iλ )

associated with each of these POD modes, represent the kinetic energy captured by the corresponding POD mode. Using these POD modes the instantaneous velocity field can be reconstructed22 using

∑=

=N

n

nn xtatxu1

)()(),( φ (1)

where )(tai is the time varying coefficient for the ith

POD mode ( iφ ) at time t. These time varying

coefficients are determined by projecting the instantaneous velocity field ),( txu onto the POD modes.

When the number of grid points is much grater than the number of snapshots, it is appropriate to use the computationally more efficient method formulated by Sirovich.23 The advantage of this formulation is that the dimension of the problem is given by the number of instantaneous fields or snapshots, where in the original formulation, the problem depends on the size of the physical grid times the number of components of the signal. For the present analysis, method of snapshots is implemented for U and V components combined in vector format.

4 Results and Discussion 4.1 Spanwise vortices Figure 3 shows the relative energy captured by the first twenty 2D POD modes in XY plane at ReD = 550. The relative energies indicate that the flow structure is dominated by the first two modes. These two modes represent the Karman vortices, which capture about 80% of the total relative energy in the flow. Figure 4(a) shows the Karman-Benard vortices captured by the PLIF experiments. Figure 4(b) shows the phase averaged streakline plot in the vertical plane, which captures the spanwise vortices as it would appear to an observer moving downstream at a velocity of 0.87U∞. The

phase velocities are reconstructed using the first two modes using equation 1. The streamwise distance (λX/D) of the spanwise vortices is found to be 4.8D at ReD = 550. Table 2 summarizes the spacing’s for different Reynolds numbers.

4.2 Streamwise vortices

In order to capture the field in the XZ plane, the dye was injected through a spanwise slot located at the lower part of the trailing edge (Figure 2). Hence, the structure formed by the bottom shear layer appear darker compared to the ones from the top shear layer. For the low Reynolds number (ReD = 250) parallel shedding of the primary vortices is observed. Primary vortices shed regularly with almost no three dimensionality or vortex loops, as shown in Figure 5(a). Nevertheless, local vortex dislocation are also observed intermittently, see Figure 5(b). These large scale vortex dislocations are two sided as described by Williamson.24 Similar types of observations were made by Zhang et al.10 in circular cylinder wake and Luo et al.15 for square cylinder wake. The dislocations persist over several shedding cycles before the flow is restored to parallel vortex shedding. As the Reynolds number increase between 350 to 400, gradual initiation, growth and decay of the streamwise vortices is observed. In this range they form intermittently and decay after a few shedding cycles.

Similar observations were made by Wu et al.26 for circular cylinders in Re range of 150-175 and for square cylinder by Luo et al.15 in the Re range of 150-175. As the Reynolds number is increased, between 400 to 500, the wake topology displays small scale streamwise vortices corresponding to mode-B structure (Figure 6).

Figure 3. Relative energy captured by first twenty 2-D POD modes in XY plan at ReD = 550

(a) (b)

8

20 14

Y/D

X/D

4.8

4.8 Figure 4. (a) PLIF Visualization (b) streakline plot along with vorticity contours of the first two POD modes in XY plane at

ReD = 550

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The secondary vortices of the mode-B are simply connected over several shedding cycles and are not

associated with the severe distortion or reorientation of segments of Karman vortices. Williamson12 and Brede et al.11 suggest that the secondary vortices of mode-B may have the origin in the separated shear layer in the near wake of the wake generator which could undergo stretching in the braid region connecting the initially formed Karman vortices. The spanwise spacing of the streamwise vortices is found to vary between 1.0D to 1.6D.

Beyond Reynolds number 550, the wake topology displays two types of wake structures. In the near wake, the wake topology display a mode-C structure and in the intermediate wake (X/D>3), it display mode-A topology. Figures 7 (a) show the asymmetric pattern of the wavy structure indicative of mode-C instability. The spanwise spacing of the streamwise vortices is found to vary between 2.0D to 2.7D in the near wake. Figures 7 (b) shows the appearance of mode-A instability in the

intermediate wake region at the same Reynolds number of 550. The spanwise spacing of the streamwise vortices is found to vary between 2.8D to 3.6D beyond X/D>3. Figure 8(a) shows the vorticity contours by combining the different POD modes at Y/D = 0 for ReD = 550. The vorticity contours are constructed by expanding the data in time at X/D = 1.

From PLIF images, the von-Karman vortices are distorted considerably because of the streamwise vortices. Hence, it is expected to see the same effect in the POD modes. Figure 8(a) shows the vorticity contour by combining the first four POD modes. This clearly indicates that these modes represent the primary coherent structure i.e. the Karman-Benard vortices. Figure 8(b) shows the vorticity contours obtained by combining modes 5 and 6. Figure 8(b) represents the near wake (X/D < 3) structure of the mode-C topology, observed as sinusoidal pattern in the PLIF images Figure 7(a)). The spacing between the streamwise vortices is found to vary between 2.2D to 2.6D with an average wavelength of 2.4D. Figure 8(c) shows the vorticity contours obtained by combining modes 7 and 8. These represent the undulations caused by the streamwise vortices on the spanwise vortices in the intermediate wake (X/D >3) of the mode-A topology.

The spacing between the undulations is found to vary between 3.2D to 3.8D with an average wavelength of 3.5D. Figure 8(d) shows the vorticity contours obtained by combining modes 9 to 12. These modes display the small scale structure with a mode-C topology, with a spanwise wavelength of 1.0D. The spanwise spacing of this structure is similar to mode-B in circular cylinder flows; hence we denote this structure as mode-Cl. Figure 9 a, b & c show the vorticity contours combining modes 5 and 6, 7 and 8 and 9 to 12 in spatial coordinates. The

(a) (b)

Z/D

X/D 0

10

X/D 0

10

8

Z/D

Figure 5.. PLIF visualization at ReD = 250 in XZ plane (a)

parallel vortex shedding (b) vortex dislocation.

X/D X/D

10

0 8 0 8

10

Z/D

(a) (b)

Figure.6 PLIF Visualization in XZ plane at ReD = 400

Streamwise vortices

0

7

9 X/D

Z/D

X/D 0

7

9

(a) (b)

(c) (d)

Small scale vortices

Figure 7. PLIF Visualization in XZ plane at ReD = 550

(a) (b)

(c) (d)

Figure 8. ωy vorticity contours in XZ plane (Y/D = -0.5) at ReD = 550 by combing POD modes (a) 1 to 4, (b) 5 and 6, (c) 7 and 8, and (d) 9

to 12

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contours indicate that same spanwise spacing observed in the expansion of the data in time. Figure 9 (d) shows

the streamwise velocity component by adding the first 32 POD modes which correspond to 80% relative kinetic energy of the flow. Here the POD modes display the effect of the streamwise vortices, indicating considerable deformation by the streamwise vortices, ascertaining the results obtained from PLIF images at ReD = 550.

The spanwise wavelengths of the instabilities observed in the current flow configuration are inline with the results from Ryan et al.20 and Dobre and Hangan.17 Ryan et al.20 proposed that the dominant unstable wavelength that is associated with similar aspect ratio geometry that dominates the flow is 2.2D based on numerical simulations at a ReD of 600.

Dobre and Hangan17 proposed 2.4D as the most unstable wavelength for a square cylinder body at a ReD of 22000. It is observed that near wake is dominated by Mode-C wake topology and in the intermediate wake by mode-A wake topology. Dobre and Hangan17 proposed that a variant of mode-A topology dominated the intermediate wake for the square cylinder flow; where as Ryan et al.20 proposed that a Mode-B structure dominates the flow. The wavelength associated with mode-A structure in the current flow configuration i.e. 3.5D matches with that proposed by Ryan et al.20 The small scale structure observed in the current flow configuration i.e. mode-Cl at ReD = 550 matches that of the mode-C structure

proposed by Ryan et al.20 at the similar Reynolds number.

4.3 Effect of Reynolds Numbers

(a) (b)

(c) (d)

(e) (f)

Figure. 10 PLIF Visualization and streamwsie velocity POD modes in XZ planes at (a, b) ReD = 1050, (c, d) ReD = 1550, (e, f) ReD = 2300 Figure 10 shows the PLIF images and streamwise velocity POD modes for (a, b) ReD = 1050 (c, d) ReD = 1550 and (e, f) ReD = 2300. The streamwise vortices are chaotic and very difficult to visualize from the PLIF images in XZ plane. For clarity to the reader, the images are marked with lines to identify the streamwise vortices. The spanwise spacing of the streamwise vortices is summarized in Table 2 for various Reynolds numbers. The streamwise vortices display a mode-A structure in every shedding period. However, combining POD modes 1 to 32 for ReD = 1050, ReD = 1550 and ReD = 2300, display the distortion in the spanwise vortices due to streamwise vortices as undulations (Figure 10). We observe that the wavelength in the near wake (X/D < 3) varies between 2.0 to 2.8D, which from our previous discussion represents mode-C structure. In the intermediate wake (X/D > 3) the spanwise wavelength of the streamwise vortices varies between 3.0 to 4.0 D, which represents the mode-A structure. Table 2 gives a summary of the streamwise spacing of the spanwise vortices for various Reynolds numbers.

(a) (b)

(c) (d)

Figure 9. ωy vorticity contours of (a) Combing POD modes 5 and 6 (b) Combining POD modes 7 and 8 (c) Combining POD modes 9 to 12 and (d) Streamwise velocity, combining POD modes 1 to

32 at Y/D = 0

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Table 2 Summary of streamwise and spanwise spacing’s for different Reynolds numbers

Streamwise spacing of spanwise vortices (λx/D)

Spanwise spacing of streamwise vortices (λz/D)

Reynolds Number (ReD) PLIF PIV PLIF PIV 550 4.8 4.8 2.0-3.8 2.0-4.0 1050 4.5 4.5 2.0-3.3 2.0-4.0 1550 4.3 4.3 2.0-3.0 2.0-4.0 2300 3.8 3.8 2.0-3.0 2.0-4.0

5 Conclusions Well controlled PLIF and PIV experiments were conducted for a range of Reynolds number (250 to 2300)

and demonstrates the transition of secondary wake instabilities in the near wake of a profiled leading edge and blunt trailing edge flat plate body. It can be confirmed that for a blunt trailing profiled body, the sequence of transition modes is different from that observed in circular cylinder flows, i.e. mode-B transitions first and followed by mode-A supporting the results of Ryan et al.20 Similar spanwise wavelength for mode-A and mode-Cl are indentified, but different wavelengths from mode-B are observed as compared to Ryan et al.20 The wavelengths of mode-B observed in our study are typical of three dimensional wake arrangements observed in other studies,12,14 however the transition Reynolds numbers are much higher compared to circular cylinder flows. POD analysis reveals the relation between POD modes and vortex structure, and reconfirms the observation of the PLIF experiments. Mode-A is the only mode responsible for von-Karman vortex dislocation. Hence, it can be concluded that the implicit assumption that wake of two dimensional bodies undergo transition to three dimensional flow and eventually turbulence, through the same sequence of transition as observed in the circular cylinder flows is not universal. Other instability modes can play crucial role in the near wake development and the route to turbulence, because wake vortex structure depends on the geometric shape of the body.

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