10TH INTERNATIONAL SYMPOSIUM ON PARTICLE IMAGE VELOCIMETRY – PIV13 Delft, The Netherlands, July 1-3, 2013
Characteristics of discrete vortices near the wall in a turbulent channel
flow with a rectangular cylinder
M. Senda, K. Inaba and K. Inaoka
Department of Mechanical Engineering, Doshisha University, Kyoto, Japan msenda@mail.doshisha.ac.jp
ABSTRACT
PIV measurement was made to clarify the characteristics of discrete vortices near the wall in a turbulent channel flow with a rectangular cylinder. Phase-averaged velocity statistics of the discrete vortex were described by comparison with the Karman vortex shed from the cylinder. The near-wall shear layer separates in the region of adverse pressure gradient behind the rectangular cylinder and develops into the periodic formation of the discrete vortices which are eventually caught up by the Karman vortex and entrained into it. The vorticity of the discrete vortices is negative (clockwise rotating) and about the same order of magnitude as the Karman vortices. The coherent normal stresses of the Karman vortices are much larger than those of the discrete vortices, while the incoherent ones are not all that different with each other.
1. INTRODUCTION
Separated flows past a bluff body with various geometries have been studied extensively in relation to the heat transfer enhancement in heat exchangers. In turbulent channel flow with a rectangular cylinder, one of the main features is the periodic and alternating vortex (Karman vortex) shedding from the cylinder. The Strouhal number and the flow pattern in the near wake of the rectangular cylinder are well known to change with its aspect ratio [1].
We investigated experimentally the flow and thermal characteristics of this flow configuration by the simultaneous measurement of the flow visualization and the unsteady heat flux at the wall [2]. It was revealed that the heat transfer was enhanced in the extensive downstream region behind the cylinder by the wall-ward flow induced between the counter-rotating Karman vortices (vortex pair) shed alternately from the upper and lower edges of the cylinder. Characteristics of the velocity field involved the Karman vortices have been studied by many researchers [3]-[6]. On the other hand, Yao et al.[7] made the flow visualization and the heat transfer measurement for a turbulent channel flow obstructed with an inserted square rod, in which the space between the square rod and the duct wall was changed. They reported the periodical formation of the secondary vortices (discrete vortices) near the channel wall, which were developed by an interaction of the near-wall flow with the Karman vortices, and suggested that the washing action [8] of discrete vortices was an important key mechanism to produce the enhancement of heat transfer, especially when the square rod was mounted in asymmetric positions. However, the characteristics of the velocity field for the discrete vortices and their interaction with the Karman vortices have not been studied in detail.
In this paper, we present the phase-averaged velocity statistics of the discrete vortices obtained by PIV measurement in a turbulent channel flow with a rectangular cylinder and discuss their characteristics by comparison with the Karman vortices.
2. EXPERIMENTAL APPARATUS AND PROCEDURE
Experiment was carried out in a closed water channel in which the rectangular cylinder was located symmetrically. Figure 1 shows the schematic of test section and the definitions of several geometric parameters and the coordinate system are described. The cross section of the channel was 50mm☓350mm. The cylinder height was fixed as h=10mm, giving the blockage ratio equal to 20%, and its width b was changed so that the aspect ratios were b/h=0.5, 1.0, 2.0 and 3.0. The velocity profile of approaching flow was uniform at the inlet of the test section and the Reynolds number based on the channel height H and the mean velocity Um at the inlet was Re=13,000.
PIV measurement was carried out in the x-y mid plane (in line with the flow direction). A Nd:YLF laser delivering 10mJ light pulses was employed to make a light sheet of 1mm in thickness and the flow was seeded with the nylon tracer particles of 50μm in diameter. PIV images were recorded with a CMOS camera (1024☓768 pixels) at a flame rate of 1000 fps and processed by a multi-pass iterative scheme starting with a interrogation area of 32☓32 pixels and a final pass at 16☓16 pixels with 50% overlap between adjacent correlation regions. The spatial resolution of the camera was 0.05mm per pixel.
0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] x*/h y/ h 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] x*/h y/ h x*/h y/h 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 x*/h y/h 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 x*/h y/ h 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 x*/h y/ h 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 x*/h y/h 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 x*/h y/h 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5
Figure 1 Schematic of the test section.
Time-varying velocity in the near wake flow past a cylindrical object includes a periodic component due to a large-scale organized motion and the instantaneous velocity signal ui(t) was analyzed in terms of the triple decomposition [9] as ui(t) = Ui + uip + uit, where uip is the periodic or coherent component and uit stands for the turbulent or incoherent one. Phase-averaged velocity statistics were obtained by using the transverse velocity at a position on the centerline in the near wake as a reference signal. Phase-averaged statistics are denoted below by the symbol < >.
3. RESULTS AND DISCUSSION
Figure 2 shows the phase-averaged coherent velocity vectors, <up> and <vp>, at several phases for the case of the aspect ratio b/h=1.0. A total of 20 phase bins was used for a cycle period and the ensemble averages were made with about 50 samples per phase bin. The abscissa of the figure is the non-dimensional streamwise distance x*/h from the
(a) Phase 12 (b) Phase 16
(c) Phase 0
(d) Phase 4 (e) Phase 8
Figure 2 Phase-averaged coherent velocity vectors, <up> and <vp>, for the case of b/h=1.0.
∼∼ ∼∼ b x* x y 0.6H 10H H h ∼∼ ∼∼ b x* x y 0.6H 10H H h 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] x*/h y/h 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] x*/h y/h
rear surface of the cylinder and the lower half of the flow passage is indicated here, because the flow is symmetrical about the centerline of y/h=0. Near the centerline, the Karman vortices, whose spatial scales are as large as the channel height, are distinctly seen to be shed alternately from the upper and lower corners of the cylinder and to be convected downstream. It is also observed that there is a strong wall-ward flow induced between these counter-rotating Karman vortices.
At Phase 12 the flow separates around x*/h=1.0 and there appears a small secondary vortex (discrete vortex) with negative spanwise vorticity at x*/h=2.0 near the channel wall. This discrete vortex moves downward along the wall, interacting with the Karman vortex, as the phase progresses. Since the convection velocity of the discrete vortex (0.56Um) is smaller than that of the Karman vortex (0.96Um) as is observed from the figure, the discrete vortex is caught up by the Karman vortex with positive vorticity and disappears eventually in the region of around x*/h=6.5. Suzuki et al.[8] carried out a numerical study for the same configuration of the flow with a square rod, but in low Reynolds number range (laminar flow). They found that the high vorticity near-wall layer interacts with the Karman vortices shed from the rod, intermittently separating and development into the periodical formation of discrete vortices. The discrete vortex with negative spanwise vorticity (clockwise rotating vortex) entrains fresh or cooler fluid from its downstream side to the space between itself and the channel wall. This fluid entrainment has the effect of making the temperature gradient at the wall large and, therefore, the effect of augmenting the wall heat transfer. This is called the washing action by Suzuki and Suzuki [8]. The present result suggests that the washing action of discrete vortices, which is related to the heat transfer enhancement, can hold also for turbulent flows [7].
Similarly, the discrete vortices are observed for the other aspect ratios except for the case of b/h=0.5 near the wall just behind the rectangular cylinder. Characteristics such as the Strouhal number and the convection velocities of the Karman and the discrete vortices are summarized in Table 1.
Figure 3 shows the streamwise distributions of the static pressure coefficient Cp (=2∆p/ρUm2), where ∆p is the static pressure difference with that at the outlet of the test section. Here, x/H is the streamwise distance from the inlet of the test section normalized with the channel height. In the lower part of the figure are shown the regions where the discrete vortices exist. The region for appearance of the discrete vortices corresponds well to that of the adverse pressure gradient behind the rectangular cylinder. It is then considered that the near-wall shear layer separates from the wall due to the adverse pressure gradient and develops into the periodic formation of discrete vortex with negative spanwise vorticity by an interaction with the Karman vortex with positive vorticity. In the present study, the discrete vortices were not observed for the case of b/h=0.5. This is because a noticeable adverse pressure gradient does not exist behind the cylinder as is indicated in Fig. 3.
Table 1 Strouhal number St and convection velocities
(UK:Convection velocity of Karman vortex UD:Covection velocity of discrete vortex)
St=f h/Um UK/Um UD/Um b/h=1.0 0.16 0.96 0.56 b/h=2.0 0.24 0.84 0.43 b/h=3.0 0.25 0.93 0.45 St=f h/Um UK/Um UD/Um b/h=1.0 0.16 0.96 0.56 b/h=2.0 0.24 0.84 0.43 b/h=3.0 0.25 0.93 0.45
Figure 3 Static pressure coefficient and the region of discrete vortex. b/h=1.0 b/h=2.0 x*/h 0 10 20 30 40 b/h=3.0 0 10 20 30 x*/h x*/h 0 10 20 30 40
: Region of the discrete vortex near the wall
0
2
4
6
8
10
-0.4
0.0
0.4
0.8
Cp
x/H
without cylinder b/h=0.5 b/h=1.0 b/h=2.0 b/h=3.0 b/h=1.0 b/h=2.0 x*/h 0 10 20 30 40 b/h=3.0 0 10 20 30 x*/h x*/h 0 10 20 30 40: Region of the discrete vortex near the wall
0
2
4
6
8
10
-0.4
0.0
0.4
0.8
Cp
x/H
without cylinder b/h=0.5 b/h=1.0 b/h=2.0 b/h=3.0(a) b/h=1.0 (Phase 16) (b) b/h=2.0 (Phase 14) (c) b/h=3.0 (Phase 2)
Figure 4 Contour plot of spanwise vorticity.
Figure 4 shows the phase-averaged velocity vectors of <up> and <vp> together with the contour plot of spanwise vorticity <ωz>h/Um. It is noted that the vorticity of the discrete vortex is about the same order of magnitude as the Karman vortex. The flow pattern behind the cylinder is, however, different one another depending on the aspect ratio. In the case of b/h=1.0, separated flows at the leading edge of the cylinder are entrained alternately behind the cylinder without reattaching to the upper or lower side of the cylinder. On the other hand, the flow pattern in the near wake changes for b/h=2.0 and 3.0 due to the unsteady reattachment of separated flows to one or both sides of the cylinder [10]. In the case of b/h=3.0, separated flows reattach to one side of the cylinder and then re-separate at the trailing edge of the cylinder, while wide and narrow wakes appear intermittently for b/h=2.0 resulting from the reattachment to either side of the cylinder or from the simultaneous reattachment to the upper and lower surfaces. As a result of the mixed wakes, the velocity field of b/h=2.0 is seen to be less organized and the vorticity of discrete vortex is weaker compared with the other cases of the aspect ratio.
The normal stresses of coherent (periodic) velocity components, <up2> and <vp2>, and incoherent (turbulent) ones, <ut2> and <vt2>, are normalized with the mean velocity Um and shown in Fig.5 and Fig.6, respectively. The lower part of each figure indicates the region close to the wall on an enlarged scale, where the discrete vortices exist. A noteworthy feature for the Karman vortex is that the normal stresses of coherent velocities are large around the vortex core, while those of both incoherent components take large values in the vortex core. Furthermore, <vp2> that is large between the counter-rotating vortices dominates <up2>, which might be responsible for the enhancement of heat transfer due to the wall-ward flow induced by the Karman vortex pair [2]. The coherent normal stresses of the Karman vortices are seen to be much larger than those of the discrete vortices, while the incoherent normal stresses are not all that different with each other.
4. CONCLUDING REMARKS
PIV measurement was carried out to clarify the characteristics of the discrete vortices near the wall for a turbulent channel flow with a rectangular cylinder having the different aspect ratios. Phase averaged statistics of velocity field have been presented by comparison with the Karman vortices shed from the cylinder.
The near-wall shear layer separates in the region of adverse pressure gradient behind the cylinder and develops into the periodic formation of discrete vortices with negative spanwise vorticity by an interaction with the Karman vortices with positive vorticity. Since the convection velocity of the discrete vortex is smaller than that of the Karman vortex, the discrete vortex moved along the wall is caught up by the Karman vortex and entrained eventually into it. The vorticity of the discrete vortices is about the same order of magnitude as the Karman vortices, while the flow pattern of
‐165 ‐135 ‐105 ‐75.0 ‐45.0 0 45.0 75.0 105 135 <ωz>h/Um 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] x*/h y/ h ‐165 ‐135 ‐105 ‐75.0 ‐45.0 0 45.0 75.0 105 135 ‐165 ‐135 ‐105 ‐75.0 ‐45.0 0 45.0 75.0 105 135 <ωz>h/Um 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] x*/h y/ h 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] x*/h y/ h <ωz>h/Um ‐100 ‐80.0 ‐60.0 ‐40.0 ‐20.0 0 20.0 40.0 60.0 80.0 100 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] y/ h x*/h <ωz>h/Um ‐100 ‐80.0 ‐60.0 ‐40.0 ‐20.0 0 20.0 40.0 60.0 80.0 100 <ωz>h/Um ‐100 ‐80.0 ‐60.0 ‐40.0 ‐20.0 0 20.0 40.0 60.0 80.0 100 ‐100 ‐80.0 ‐60.0 ‐40.0 ‐20.0 0 20.0 40.0 60.0 80.0 100 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] y/ h x*/h <ωｚ>h/Um ‐135 ‐105 ‐75.0 ‐45.0 ‐30.0 0 30.0 45.0 75.0 105 135 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] y/ h x*/h <ωｚ>h/Um ‐135 ‐105 ‐75.0 ‐45.0 ‐30.0 0 30.0 45.0 75.0 105 135 <ωｚ>h/Um ‐135 ‐105 ‐75.0 ‐45.0 ‐30.0 0 30.0 45.0 75.0 105 135 ‐135 ‐105 ‐75.0 ‐45.0 ‐30.0 0 30.0 45.0 75.0 105 135 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] y/ h x*/h 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] y/ h x*/h
0 0.020 0.030 0.040 0.050 0.060 0 0.080 0.15 0.22 0.30 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] 0 1 2 3 4 5 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 x*/h y/ h y/ h x*/h <up2>/U m2 0 0.020 0.030 0.040 0.050 0.060 0 0.080 0.15 0.22 0.30 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] 0 1 2 3 4 5 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 x*/h y/ h y/ h x*/h 0 0.020 0.030 0.040 0.050 0.060 0 0.080 0.15 0.22 0.30 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] 0 1 2 3 4 5 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 x*/h y/ h y/ h x*/h <up2>/U m2 0 0.002 0.004 0.006 0.008 0.01 0 0.20 0.40 0.60 0.80 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] 0 1 2 3 4 5 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 x*/h y/ h y/ h x*/h <vp2>/Um2 0 0.002 0.004 0.006 0.008 0.01 0 0.20 0.40 0.60 0.80 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] 0 1 2 3 4 5 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 x*/h y/ h y/ h x*/h 0 0.002 0.004 0.006 0.008 0.01 0 0.20 0.40 0.60 0.80 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] 0 1 2 3 4 5 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 x*/h y/ h y/ h x*/h <vp2>/Um2
the near wake is different with each other depending on the aspect ratio of the cylinder. The coherent normal stresses of the Karman vortices, which are predominant around the vortex core, are much larger than those of the discrete vortices. The incoherent normal stresses of the Karman vortices take large values in the vortex core, and they are not all that different with the discrete vortices.
(a) <up2>/Um2 (b) <vp2>/Um2
Figure 5 Normal stresses of coherent velocities (b/h=1.0).
(a) <ut2>/Um2 (b) <vt2>/Um2
Figure 6 Normal stresses of incoherent velocities (b/h=1.0). REFERENCES
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0 0.030 0.040 0.050 0.060 0 0.0500 0.0750 0.100 0.125 0.150 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] 0 1 2 3 4 5 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 x*/h y/ h y/ h x*/h <ut2>/Um2 0 0.030 0.040 0.050 0.060 0 0.0500 0.0750 0.100 0.125 0.150 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] 0 1 2 3 4 5 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 x*/h y/ h y/ h x*/h 0 0.030 0.040 0.050 0.060 0 0.0500 0.0750 0.100 0.125 0.150 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] 0 1 2 3 4 5 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 x*/h y/ h y/ h x*/h <ut2>/Um2 0 0.005 0.006 0.007 0.008 0 0.050 0.10 0.15 0.20 0.25 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] 0 1 2 3 4 5 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 x*/h y/ h y/ h x*/h <vt2>/Um2 0 0.005 0.006 0.007 0.008 0 0.050 0.10 0.15 0.20 0.25 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] 0 1 2 3 4 5 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 x*/h y/ h y/ h x*/h 0 0.005 0.006 0.007 0.008 0 0.050 0.10 0.15 0.20 0.25 0 1 2 3 4 5 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0.1[m/s] 0 1 2 3 4 5 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 x*/h y/ h y/ h x*/h <vt2>/Um2
