Unsteady Behavior of Necklace Vortex Produced by a Square Plate Protrusion

In this study, the unsteady behavior of necklace vortices formed in front of a square flat plate was experimentally investigated by flow visualization and PIV analysis by using a water channel. As a result, the necklace vortices started to oscillate when the Reynolds number increased beyond approximately 2000. Then, an amalgamation behavior of the necklace vortices took place when the Reynolds number increased beyond approximately 2650. Furthermore, in the same Reynolds number range, a breakaway behavior appeared when the relative height of the square plate was beyond approximately h/δ = 4.0. The features of the necklace vortices behaviors in the oscillation, amalgamation and breakaway states were explained by observing the time-series image of path lines and by analyzing the frequency of velocity fluctuation.


Introduction
It is well known that the necklace vortex system is formed around a three-dimensional bluff body.Regarding the steady necklace vortex, a circular cylinder, a rectangular cylinder or a flat plate was mainly selected as a three-dimensional bluff body.The steady necklace vortex around a circular cylinder was experimentally studied by Baker [1].The rectangular cylinder was dealt with by Seal et al. [2].The flat plate was dealt with by Lin et al. [3] and Nakahara et al. [4].In particular, the characteristics of the unsteady necklace vortex Open Journal of Fluid Dynamics measuring the velocity distribution [5].Visbal showed streamlines of the steady necklace vortices and the unsteady processes by using numerical calculations [6].Seal et al. studied a breakaway behavior of the necklace vortices formed by a long rectangular cylinder which protrudes through the water surface [7].Tsuruno et al. and Matsuguchi et al. studied the transition process of an unsteady necklace vortex system in front of a short square cylinder with a width-to-height ratio of 2 [8] [9] [10].Lin et al. investigated the oscillation and breakaway processes of a necklace vortex upstream from a vertical rectangular plate (h/w = 0.5 -4.0) and a short square cylinder (h/w = 1.08) on a ground wall.They explained that the boundary layer thickness is an important length scale controlling the cycle in the oscillation and breakaway processes of the necklace vortex [3] [11].
In studies on the necklace vortex, various three-dimensional bluff body shapes have been used, as mentioned above.Among them, a thin plate can be regarded as a typical bluff body, because there is no reattachment to itself downstream of the separation on the plate edge.However, there are few reports on unsteady necklace vortices produced by the thin square flat plate, except the reports of Lin et al.Lin et al. did not distinguish between flat plates having different aspect ratios, w/h, in his experiment results.The necklace vortex produced by the thin square flat plate should be researched in more details, because it is thought to be the most basic shape among vertical thin plates.
Therefore, we focused on the unsteady necklace vortices produced in front of the thin square flat plate protrusion perpendicular to the ground wall on which the laminar boundary layer exists.The purpose of this paper is to clarify the influence of the Reynolds number R e and the reference laminar boundary layer thickness δ on the behavior of the unsteady necklace vortices produced by the thin square plate on the ground wall.On this account, flow visualization and PIV analysis were carried out while an unsteady necklace vortex state appeared in the Reynolds numbers between 2000 and 3250.

Experimental Apparatus and Procedure
Figure 1 shows an open test section of a water channel used in this experiment.
The test section is made of transparent acrylic walls, and has a width of 300 mm, a height of 300 mm and a length of 1100 mm.The coordinate system and nomenclature also are shown in the same figure.The contraction area ratio of the inlet nozzle is 0.5.A perforated plate, a honeycomb and four woven wire screens are installed upstream of the inlet nozzle.A ground wall, which let a square plate protrusion stand, is an acrylic plate with a width of 300 mm, a length of 1000 mm and a thickness of 5 mm.The ground wall, whose leading-edge shape is one half the major axis of an ellipse, was submerged in a position 50 mm above the test section floor.Therefore, a new laminar boundary layer was formed on the ground wall.
The unsteady necklace vortices were produced by the protrusion of a thin square plate with a thickness of 1.5 mm standing perpendicularly to the ground Open Journal of Fluid Dynamics wall on which the laminar boundary layer exists.The sizes h or w of the square plate and the blockage ratio are shown in Table 1.The blockage effect of the square plate on this flow field will not need to be considered, because the maximum blockage ratio is 4.38%.The square plate was stood at position 100, 150, 320, 500 or 800 mm from the front edge of ground wall.
In the experimental procedure, fine nylon particles, with an average diameter of 50 μm (specific gravity about 1.03), were mixed in the water channel.The particle-images in the flow field were visualized by a metal-halide light irradiating from a slit of about 2 mm, and were successively taken by a digital camera (1280 × 1024 pixels) at a frame rate of 100 or 200 fps.Each path lines-image was created by mutually superposing 50 particle-images.The resolution of the pixel was set in the range of 0.07 to 0.09 mm/pixel.
The velocity profile upstream of the square plate protrusion was obtained by using a time-series PIV.The uncertainty of velocity measurement by PIV was estimated to be approximately 6.2%, by comparing the Blasius' boundary layer velocity profile with the velocity profile measured by PIV at 320 mm position downstream from the front edge of the ground wall.On this account, the original boundary layer can be determined by the Blasius' solution.Therefore, the reference laminar boundary layer thickness δ (=δ 99 ) at the position assumed to stand the square plate was also obtained by the Blasius' solution.The flow visualization and PIV analysis were carried out by varying the relative plate height h/δ between approximately 2.0 and 5.0, and by varying the Reynolds numbers R e ( = U 0 •h/ν) between approximately 2000 and 3250.

Experimental Results and Discussions
When the Reynolds number was larger than approximately 2000, the waveform of velocity fluctuation became wavy and the turbulent intensity also increased in  the vicinity of the necklace vortices.In addition, the dominant frequency appeared in FFT analysis of the velocity fluctuation.However, the necklace vortices were not always completely steady in R e < 2000, because the turbulent intensity in the vortex region increased gradually as the Reynolds number approached 2000.Therefore, the unsteady state in the necklace vortices may be induced by the turbulence inherent in the vortices.Blue, yellow and gray lines indicate the x-positions of each vortex center against the time of t = 8.5 s in order to investigate a vortex movement.The velocity measurement positions P 1 , P 2 and P 3 were also placed in a slightly upper position on V 1 , V 2 and V 3 at the same time.Four clockwise necklace vortices (first vortex V 1 , second vortex V 2 , third vortex V 3 and fourth vortex V 4 ) oscillated upstream and downstream, as shown in Figure 2. When the first vortex V 1 moved downstream, the second vortex V 2 also moved downstream belatedly.Following this, the third vortex V 3 also moved downstream by following V 2 .

Oscillation Behavior
Figure 3(a) shows the waveforms of y-direction velocity v measured simultaneously at measurement positions P 1 , P 2 , and P 3 .The waveforms had a phase shift to one another as mentioned above.The positive velocity in the waveform at each measurement position means that each corresponding vortex moves downstream, and negative velocity means vortex moving upstream.The waveforms at P 1 and P 2 were almost opposite phases, and the waveform at P 3 was later about 1/4 the period than that at P 2 .Therefore, it is assumed that vortex V 1 and vortex V 2 approach each other in the time zone of A, as shown in Figure 3(a).In addition, V 2 and V 3 approach each other in the time zone of B. Figure 3(b) shows a frequency analysis of the y-direction velocity component measured in a time length of 41 s at the positions P 1 , P 2 , and P 3 .The dominant frequency was f D = 0.195 Hz at three measurement positions, and corresponds to the Strouhal number S t (= f•h /U 0 ) of 0.15.This means that the necklace vortices V 1 , V 2 and V 3 oscillate at the same period of T = 5.1 s. Figure 4(a) shows the waveforms of x-direction velocity u measured at the measurement positions P 1 , P 2 and P 3 .The phase of the waveforms almost corresponded with the phase of the y-direction velocity waveforms at P 1 , P 2 and P 3 , respectively.The x-direction velocity waveforms also indicated f D = 0.195 Hz, as shown in Figure 4(b).The second vortex V 2 lately started to move downstream beyond t = 4 s.After the third vortex V 3 amalgamated the fourth vortex V 4 in the time of t = 4 -5 s, V 3 lately started to move downstream together with V 2 .For that reason, V 2 was enclosed in the narrow region between V 1 and V 3 in the time of t = 5.5 -6.5 s.When V 2 closely approached V 1 , a part of V 2 was amalgamated by V 1 in the time of t = 5.5 -6 s.Successively, the remaining part of V 2 was amalgamated by V 3 in the time of t = 6.5 -7 s.After V 3 amalgamated the remaining part of V 2 , V 3 and V 4 moved downstream until the original positions of V 2 and V 3 , respectively.They became newly V 2 and V 3 at the end time of period.

FLOW (a)
Square plate front Figure 6(a) shows the waveforms of y-direction velocity v measured simultaneously at the measurement positions P 1 , P 2 and P 3 seen in Figure 5.Each waveform indicated evidently periodic patterns.The measurement positions P 1 , P 2 and P 3 were placed in a slightly upper position on V 1 , V 2 and V 3 at the time of t = 0 s.When the velocity is negative at P 1 and positive at P 2 , it is expected that the vortices V 1 and V 2 approach each other.In fact, V 1 and V 2 amalgamated in the time zone A, written in Figure 6(a).In similar manner, V 2 and V 3 amalgamated in the time zone B when the velocity was negative at P 2 and positive at P 3 .was f D = 0.11 Hz.The period is T = 9.1 s.The waveform at the positions P 2 was distorted due to V 2 being periodically amalgamated by V 1 and V 3 , as mentioned above.On this account, the dominant frequency at the position P 2 became f D = 0.11 and 0.22 Hz, which was equal with the dominant frequency f D at the position P 3 .The Strouhal number for these dominant frequencies were S t = 0.11 and 0.23.first vortex V 1 was pushed downstream by the second vortex V 2 coming downstream at first, and was amalgamated in the time of t = 13 s.Soon after that, the coming vortex V 2 became a new first vortex V 1 in approximately a time of t = 14 s in the upstream position of P 1 line.

Breakaway Behavior
Figure 8(a) shows the waveforms of y-direction velocity v at the measurement positions P 1 and P 2 which were placed in a slightly upper position on V 1 and V 2 at the time of t = 11 s.Though the waveform at the position P 1 seems to mean a oscillation of the same vortex V 1 , V 1 is exchanged with the new vortex V 1 in the middle of the period.the waveform at the position P 2 seems to indicate an oscillation of the same vortex V 2 , V 2 also is exchanged with a new vortex V 2 in the end of the period.Since V 2 was always moving downstream, V 1 was amalgamated by V 2 when V 1 started to move upstream of the original position P 1 , as indicated by the time zone A. because the dominant frequency at the positions P 1 and P 2 were f D = 0.27 Hz (S t = 0.29).The period is T = 3.7 s.

Classification of Unsteady Behavior
When the Reynolds number was smaller than 2000, the velocity measured in a slightly upper position on the first vortex V 1 did not provide such wavy waveform and dominant frequency as shown in Figure 3. On the other hand, the necklace vortices indicated the obvious oscillation state at R e = 2040, as shown in Figure 3. Therefore, the boundary in the Reynolds number between the steady and unsteady necklace vortex states is considered to be approximately 2000.
In the Reynolds number investigated in this experiment, the unsteady necklace vortex was classified to the oscillation state, the amalgamation state and the breakaway state.Figure 9 shows what kinds of unsteady necklace vortex appear in both conditions of the Reynolds number R e and the relative plate height h/δ.The necklace vortex indicated the oscillation behavior between R e = 2000 and 2650.

Feature of Unsteady Necklace Vortex
The Strouhal number was calculated from the dominant frequency of the velocity waveform measured in a slightly upper position on the first vortex V 1 against the various cases of R e and h/δ.frequencies appeared in the amalgamation state, as mentioned in Session 3.2.
The Strouhal number in the oscillation state was almost constant with each h/δ.

Conclusions
In this study, the unsteady behavior of necklace vortices formed in front of a

Figure 1 .
Figure 1.Schematic diagram of experimental apparatus and coordinate system.
t so u rc e d ev ic e

Figure 2 (
Figure 2(a)-(e) shows the path lines-images in about one period in the oscillation behavior of the necklace vortices in the case of R e = 2040 and h/δ = 2.0.Blue, yellow and gray lines indicate the x-positions of each vortex center against the time of t = 8.5 s in order to investigate a vortex movement.The velocity

Figures 5 (Figure 2 .
Figures 5(a)-(l) shows the path lines-images in about one period in the

Figure 3 .
Figure 3. Analysis of y-direction velocity v in each measurement position in oscillation behavior: (a) Waveforms of velocity fluctuation; (b) FFT analysis of velocity fluctuation.

Figure 6 ( 1 -Figure 4 .
Figure 6(b) shows a frequency analysis of the y-direction velocity component measured in a time length of 170 s at the positions P 1 , P 2 , and P 3 .The vortex V 1 would oscillate with 0.11 Hz, because the dominant frequency at the positions P 1

Figures 7 ( 3
Figures 7(a)-(e) shows the path lines-images in about one period in the breakaway behavior of the necklace vortices in the case of R e = 3250 and h/δ = 5.0.Blue and yellow lines indicate the x-positions of each vortex center against the time of t = 11 s.The feature of breakaway behavior is that the second vortex V 2 in the upstream position of the first vortex moves downstream with growing and amalgamates with the first vortex V 1 , as mentioned by Lin et al.The time-series path lines-images in Figure 7 also support the same feature.In addition, the vortex downstream moving occurred for all vortices except the first vortex.The

Figure 8 (Figure 6 .
Figure 6.Analysis of y-direction velocity v in each measurement position in amalgamation behavior: (a) Waveforms of velocity fluctuation; (b) FFT analysis of velocity fluctuation.

Figure 8 .
Figure 8. Analysis of y-direction velocity v in each measurement position in breakaway behavior: (a) Waveforms of velocity fluctuation; (b) FFT analysis of velocity fluctuation.

2
square flat plate protrusion was clarified based on the time-series image of path lines and the frequency analysis of velocity fluctuation.The following results were gained: 1) There exist three kinds of unsteady necklace vortices in the range of 2000 ≤ R e ≤ 3250, those are the oscillation state, the amalgamation state, and the breakaway state.Open Journal of Fluid Dynamics

Figure 9 .
Figure 9. Classification of unsteady necklace vortex state.Red broken lines show the transition boundary of the unsteady necklace vortex state.

Figure 10 . 2 ) 3 ) 4 )
Figure 10.Strouhal number in unsteady necklace vortex movement.Red broken lines show the transition boundary of the unsteady necklace vortex state.

Table 1 .
Square plate size and blockage ratio.
Then, the Strouhal number was approximately S t = 0.15 in the case of h/δ = 2.0, and approximately S t = 0.19 in h/δ = 3.0.The first Strouhal number in the amalgamation state gradually increased from S t = 0.15 with increasing the Reynolds number.The second Strouhal number was approximately half of the first Strouhal number.The Strouhal number in the breakaway state was larger than that in the amalgamation state.