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Interaction of a current with a circular cylinder near a rigid bed
ARTICLE IN PRESS Ocean Engineering 35 (2008) 1492– 1504
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Interaction of a current with a circular cylinder near a rigid bed A. Alper Oner a, M. Salih Kirkgoz b,, M. Sami Akoz b a b
Civil Engineering Department, Aksaray University, 68100 Aksaray, Turkey Civil Engineering Department, Cukurova University, 01330 Adana, Turkey
a r t i c l e in f o
a b s t r a c t
Article history: Received 18 December 2007 Accepted 13 June 2008 Available online 19 June 2008
The interaction of fluid flow with a circular cylinder is important in many design situations. In the study of fluid–body interaction problems, it is essential that one is familiar with the affected flow structure in terms of its kinematics and related properties. Existing experimental data show that the vertical gap, G, between a horizontal cylinder and a plane boundary is the major parameter that affects the flow structure around the cylinder. In this study, the PIV technique is used to measure the velocity field in a steady, two-dimensional, turbulent flow around a horizontal circular cylinder near a plane boundary. The resulting velocity profiles, streamlines and isovorticity contours are presented for the gap ratios, G/D ¼ 0.0, 0.1, 0.2, 0.3, 0.6, 1.0 and 2.0. Experiments for the seven different G/D ratios are repeated for Reynolds numbers ReD ¼ 840, 4150 and 9500. Using the measured velocity fields, the effect of gap ratio on the wall boundary–layer separation, the positions of the stagnation and separation points on the cylinder, and the Strouhal number are investigated. Present experimental results indicate that the changes in the flow structure become very slow when G/DX0.3, and the wall proximity effect on the flow around the cylinder becomes insignificant when G/DX1.0. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Steady flow Circular cylinder Particle image velocimetry Velocity field Wall proximity Vortex shedding
1. Introduction The interaction of fluid flow with two-dimensional cylindrical structures may be encountered in a variety of engineering applications. Cylinders with circular cross-sections are the most common submerged structural elements in civil, mechanical, coastal and ocean engineering practices. In coastal and ocean engineering, the design of submarine pipelines is one of the most important practical examples. When a pipeline is initially placed on an erodible bed, interaction of the disturbed flow field with the sea floor may create erosion problems. Scour may occur below the pipe and, as a result, the pipeline may become a freespanning structure. Pipelines, either resting on the seabed or suspended above the bed with a small gap, may be subjected to transversely flowing steady currents in deep, intermediate and shallow water conditions. The dynamic forces exerted by the fluid flow on submerged cylindrical elements and their response to these forces are the primary concerns in engineering design. Fig. 1 illustrates four typical regions in the disturbed flow field around a cylinder (Sumer and Fredsoe, 1997; Zdravkovich, 1997). These are: (a) the upstream retarded flow region; (b) the boundary layers that develop along the cylinder surface; (c) the
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E-mail address: [email protected] (M. Salih Kirkgoz). 0029-8018/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2008.06.005
curvilinear, accelerated sidewise flow regions; and (d) the downstream separated flow region called ‘‘the wake’’. The wake region is the most complicated part of the disturbed flow field. At the same time, it is the most influential in the interaction between the fluid flow and the cylindrical body. Flow structures affected by cylindrical bodies near solid boundaries have been the subject to numerous experimental and theoretical investigations. They have mostly been directed toward determining wall proximity effects on various flow properties such as the wall and cylinder boundary–layer separation points, the position of the front stagnation point, the pressure distribution on the cylinder surface, the drag and lift coefficients, the Strouhal number for vortex shedding frequency, and the vortical structure and turbulence characteristics of the wake behind the cylinder (Bearman and Zdravkovich, 1978; Grass et al., 1984; Zdravkovich, 1985; Lei et al., 1999; Choi and Lee, 2000; Price et al., 2002). In the earlier experimental studies, different flow measurement techniques were employed to determine the flow behavior around the cylinder, including hot-wire anemometry (Bearman and Zdravkovich, 1978; Choi and Lee, 2000), hot-film anemometry (Price et al., 2002), particle image velocimetry (PIV; Price et al., 2002; Akilli et al., 2005) and flow visualization (Choi and Lee, 2000; Hatipoglu and Avci, 2003). Different experimental conditions have been used, ranging from low Reynolds numbers of ReD ¼ 40 (Williamson, 1989) and 170 (Taneda, 1965) to high Reynolds numbers of ReD ¼ 7.1 106 (Schewe, 1983) and 107 (Roshko, 1961).
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Fig. 1. Definition sketch for the variables related to the steady flow field around a horizontal circular cylinder near plane boundary.
In these earlier experiments, it was observed that the freestream velocity, u0, the depth, h, and the boundary layer thickness, d, of the flow, the kinematic viscosity of the fluid, n, the cylinder diameter, D, and the gap between the cylinder and the wall, G, are the principal quantities that affect the flow past a horizontal circular cylinder, which is either placed on the bottom or is mounted above it with a small gap (see Figs. 1 and 2, for symbols). The dimensionless numbers governing the characteristic features of the flow were found to be the Reynolds number, ReD ¼ u0D/n, the gap ratio, G/D, and the ratio of the free-stream boundary layer thickness to the cylinder diameter, d/D. Much experimental information is available in the literature on the flow field around submerged cylindrical structures. However, further experimental effort is needed either to confirm the earlier findings or to provide new information for different experimental conditions. In this study, PIV is used to measure the timedependent and -averaged velocity fields in steady, two-dimensional, turbulent flows around a horizontal smooth cylinder with different gaps from a plane boundary for flow conditions with ReD ¼ 840, 4150 and 9500. Some characteristic features are then described.
2. Experiments 2.1. Experimental setup The experiments were performed in a transparent plexiglaswalled laboratory channel that was 1 m wide, 0.75 m deep and 14 m long. The water depth in the channel was kept constant at 0.60 m during the experiments. As may be seen in Fig. 2, a smooth flat plate made from plexiglas was mounted in the central part of the channel. The plate was 2 m long and spanned the channel width, and it had a leading sharp edge to avoid flow disturbances. The plate was positioned 0.28 m above the channel bed, and the flow depth on the plate was h ¼ 0.32 m. A smooth circular plexiglas test cylinder with a diameter D ¼ 50 mm was mounted horizontally 1.5 m ( ¼ 30D) downstream from the leading edge of the plate. Tests were undertaken with ReD ¼ 840, 4150 and 9500. The details of the experimental conditions are given in Table 1. The free-stream velocities, u0, and the boundary layer thicknesses, d, were obtained from the velocity profiles measured at the location of the cylinder in the channel without the cylinder in place. The ratios d/D at the test location were 1.6, 1.5 and 1.4 for the given ReD values, respectively. The flow regime around the cylinder is
classified as subcritical when the Reynolds number is in the range of 300oReDo3 105 (Sumer and Fredsoe, 1997). According to this criterion, the three flow cases used in the present tests are categorized as subcritical, which has the characteristic features of a completely turbulent wake and laminar boundary layer separation. For each of the flow conditions listed in Table 1, the experiments were repeated with seven different gaps G ¼ 0, 5, 10, 15, 30, 50, and 100 mm corresponding to gap ratios G/D ¼ 0.0, 0.1, 0.2, 0.3, 0.6, 1.0 and 2.0.
2.2. Velocity measurements using PIV technique PIV is an optical method used to measure the velocities and related properties in fluid flows. The fluid is seeded with particles (tracers) that are assumed to follow the flow. The motion of the seeding particles is used to calculate the velocity field. The PIV is a non-intrusive technique capable of measuring an entire twodimensional geometry of the flow field simultaneously. Compared to other techniques such as hot-film anemometry and hot-wire anemometry, the PIV technique has the advantage of determining the time-dependent vortical formation simultaneously in the twodimensional flow field around a circular cylinder. The instantaneous velocities in the flow field around the cylinder at the mid-span of the channel were measured using the PIV system and Flow Manager Software. A schematic view of the PIV system is seen in Fig. 2. The water in the channel was seeded with neutrally buoyant silver-coated spherical particles of 14 mm in diameter. The flow was illuminated via a 2-mm-thick laser sheet of argon-ion green light with a wavelength of 532 nm from a pair of double-pulsed Nd:YAG laser units. Flow map processor that controls the timing of the data acquisition was used to synchronize the image taking and the laser units. Within the two-dimensional measuring area of the flow, the instantaneous velocity vectors were determined by recording the displacements of the seeding particles between the two locations during a specified time interval of the two pulses, which was 1.5 ms throughout the experiments. The movements of the particles were recorded by a CCD camera with a resolution of 1024 1024 pixels, equipped with a 60 mm focal length lens. Each pixel covered a square of 0.065 0.065 mm in the field of measurement. From the recorded particle image data, the displacement vectors were obtained using digital PIV software employing a frame-to-frame cross-correlation technique. In the image processing, 32 32 pixel rectangular effective interrogation windows were used and an overlap of 50% was employed, yielding a total of approximately
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Fig. 2. Experimental arrangement for the circular cylinder in the channel and the schematic diagram of the PIV system.
Table 1 Experimental conditions Flow case
h (m)
u0 (mm/s)
d (mm)
ReD ( ¼ u0D/n)
d/D
1 2 3
0.32 0.32 0.32
17.3 86 197
80 75 70
840 4150 9500
1.6 1.5 1.4
3844 velocity vectors. The size of the velocity measurement field viewed by the laser sheet was taken as 80 80 mm upstream and downstream of the cylinder, and 50 50 mm above and below it. For each measurement, a total of 490 patterns of instantaneous particle images were taken and recorded on a computer during a period of 49 s (i.e. at a rate of 10 Hz). By post-processing of the measured instantaneous velocity field, the time-averaged velocity vector field and the corresponding streamlines and isovorticity contour lines were determined. The time-averaged velocity for each interrogation window was obtained by taking the arithmetic mean of 490 instantaneous velocities measured sequentially for the seeding particles in that window. The measurement uncertainty in velocities was judged to be within 72%. More detail of the experimental procedure is given by Oner (2007).
around the cylinder for gap ratios G/D ¼ 0.0, 0.2, 0.3 and 1.0 at ReD ¼ 9500 are shown in Figs. 3–6. The boundaries of development of boundary layers on both side of the cylinder surface and upper and lower free-shear layers behind the cylinder are shown in dashed lines. The four typical disturbed regions of the flow around the cylinder, defined earlier, may be detected from the velocity profiles provided in the above plots. The reformation of the wall boundary layer through the gap is indicated with a dotted line for G/D ¼ 0.2 and 0.3 in Figs. 4 and 5. The points where the inner boundary lines of the shear layers detach from the cylinder show the approximate time-averaged positions of the upper and lower separation points on the cylinder surface. From the velocity distributions for G/D ¼ 1.0 in Fig. 6, it is seen that the disturbed velocity field around the cylinder is nearly symmetrical about its horizontal axis. That means the influence of the solid bed on the velocity field around the cylinder becomes weak when the gap ratio increases to G/D ¼ 1.0. The streamlines and isovorticity contour lines calculated from the measured time-averaged velocity data upstream and downstream of the cylinder are provided in Figs. 7–9 for gap ratios G/D ¼ 0.0, 0.1, 0.2, 0.3 and 1.0 from the three flows with ReD ¼ 840, 4150 and 9500. The z-component of the vorticity vector of the two-dimensional flow field in the xy plane is 2oz ¼
3. Experimental results 3.1. Velocity distributions, streamlines and isovorticity contour lines The vertical distributions of the time-averaged horizontal velocity components, u, at different locations, x/D, of the flow
qv qu . qx qy
(1)
In Eq. (1), u and v are the horizontal and vertical velocity components in the x- and y-directions, respectively. The local vorticities in the xy plane were determined as the differential circulation per unit area using a finite difference scheme. The solid lines in Figs. 7–9 represent positive vorticities (counterclockwise), and the incremental values of the vorticity contours are taken
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Fig. 3. Vertical distributions of horizontal velocity component at different locations of flow around the cylinder for G/D ¼ 0.0 at ReD ¼ 9500.
Fig. 4. Vertical distributions of horizontal velocity component at different locations of flow around the cylinder for G/D ¼ 0.2 at ReD ¼ 9500.
Fig. 5. Vertical distributions of horizontal velocity component at different locations of flow around the cylinder for G/D ¼ 0.3 at ReD ¼ 9500.
equal to the minimum values that are given in the figures. As can be seen, the isovorticity contours are concentrated mostly in the boundary layers of the cylinder and wall and also in the freeshear layers downstream of the cylinder: that is, in the regions
where high-gradient velocity distributions exist, as observed in Figs. 3–6. From the streamline patterns, it is possible to determine the positions of the front stagnation point on the cylinder. The effects
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Fig. 6. Vertical distributions of horizontal velocity component at different locations of flow around the cylinder for G/D ¼ 1.0 at ReD ¼ 9500.
Fig. 7. Time-averaged streamlines and isovorticity contour lines upstream and downstream of the cylinder for different G/D at ReD ¼ 840. Solid lines represent positive vorticities. The minimum and incremental values of vorticity are 1 s1.
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Fig. 8. Time-averaged streamlines and isovorticity contour lines upstream and downstream of the cylinder for different G/D at ReD ¼ 4150. Solid lines represent positive vorticities. The minimum and incremental values of vorticity are 5 s5.
of wall proximity on the time-averaged flow structures are clearly displayed in the figures. It is seen that the streamline patterns and isovorticity contour lines tend to become symmetrical as the gap ratio increases toward G/D ¼ 1.0. Almost symmetrical velocity distributions, streamline patterns and isovorticity contours about the horizontal axis of the cylinder for G/D ¼ 1.0 suggest that the wall proximity effect on the flow becomes negligible when G/D increases to 1.0. However, the minor influence of the channel boundary layer on the flow field should not be disregarded, as the values of d/D at the test location are all greater than G/D ¼ 1.0. Thus, the cylinder is partly buried within the upper portion of the channel boundary layer and is not yet completely free from the wall boundary layer effect. A typical record of instantaneous streamlines, measured at Dt ¼ 0.1 s intervals, is shown in Fig. 10 for conditions downstream of the cylinder with a gap ratio G/D ¼ 0.3 at ReD ¼ 9500. Alternate vortex shedding and changes in the vortex formation downstream
of the cylinder are clearly displayed in the figure. The frequency of vortex shedding for this particular case may be estimated by determining the recurrence interval of any one of the instantaneous streamline patterns. Visual examination of the instantaneous streamline records shows that streamline pattern in Fig. 10(a) is more or less repeated in Fig. 10(n), although they are not completely identical. The time elapse between Fig. 10(a) and (n) is 1.3 s, which corresponds to a vortex shedding frequency of approximately 0.77 Hz for this case. Similar repetitions are observed in the subsequent images: that is in Figs. 10(b) and (o), and in Figs. 10(c) and (p). However, the preceding analysis for shedding frequency, using the instantaneous streamlines pertains to one shedding period only and its precision is affected by the time interval of the recordings. However, the frequency of the vortex shedding determined in this way can be used to roughly confirm the finding of a more reliable quantitative evaluation given later.
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Fig. 9. The-averaged streamlines and isovorticity contour lines upstream and downstream of the cylinder for different G/D at ReD ¼ 9500. Solid lines represent positive vorticities. The minimum and incremental values of vorticity are 5 s1.
3.2. Upstream and downstream wall boundary layer separations Price et al. (2002) reported that for Reynolds numbers in the range 1200pReDp1400, wall boundary layer separation upstream of the cylinder is observed for G/Dp0.75 and no wall separation occurs when G/DX1.0. From the examination of the present experimental time-averaged streamlines, it is possible to determine the flow conditions where the wall boundary layer separations occur. For the present test conditions shown in Figs. 7(b)–(d), 8(b)–(d), and 9(b), it can be seen that the flow separates from the bed upstream of the cylinder and then reattaches as it passes through the gap. For the flow conditions in Figs. 7(b) and (c), and 8(b), the downstream wall boundary layer is forced to separate again as the streamlines in the inner shear layer are deflected away from the wall. From the timeaveraged streamlines obtained in the present experiments, the ranges of G/D for which the upstream and downstream wall separations occur are given in Table 2. The values in the table show that the upper limit of the gap ratio where upstream or
downstream wall separation occurs reduces as ReD increases. Upstream wall separation occurs for G/Dp0.6, 0.3 and 0.1 at ReD ¼ 840, 4150 and 9500, respectively; and the downstream wall separation exists when G/Dp0.2 and 0.1 at ReD ¼ 840 and 4150, respectively. The streamline patterns at ReD ¼ 9500 do not seem to display any downstream wall separations for the present gap ratios. Price et al. (2002) stated that for G/D between 0.25 and 1.0, vortex shedding from the lower side of the cylinder affected the downstream wall boundary layer, causing it to separate at the same frequency as that of the vortex shedding from the cylinder. This sort of vortex pair occurrence between the inner shear layer and downstream separated wall boundary layer is also observed in some cases of the present tests. The instantaneous streamline patterns demonstrate the changes in the vortical structure of the flow downstream of the cylinder. It is seen that during a vortexshedding period on the lower cylinder side, the instantaneous streamline patterns in Figs. 10(g)–(k) display a temporary local wall separation about one cylinder diameter downstream of the cylinder. As pointed out by Price et al. (2002), and confirmed by
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Fig. 10. Instantaneous streamlines patterns with Dt ¼ 0.1 s intervals downstream of the cylinder for G/D ¼ 0.3 at ReD ¼ 9500.
Table 2 Experimental conditions for wall boundary layer separations ReD
Upstream wall boundary layer separation occurs
Downstream wall boundary layer separation occurs
Wall separation due to vortex pair downstream of gap flow
840 4150 9500
0.0oG/Dp0.6 0.0oG/Dp0.3 0.0oG/Dp0.1
0.0oG/Dp0.2 0.0oG/Dp0.1 –
G/D ¼ 0.3–0.6 G/D ¼ 0.2–0.6 G/D ¼ 0.2–0.6
the present instantaneous streamline patterns, this occurrence may be attributed to the existence of a vortex pair between the inner shear layer of the cylinder wake and the wall boundary layer. The test conditions in which this vortex pair occurs are given in Table 2.
3.3. Position of stagnation point on the cylinder Lei et al. (1999) conducted experiments on steady flows around a horizontal cylinder with different gap ratios for ReD between 13,000 and 14,500. In their experiments, the cylinder was
positioned at 2D, 3D and 11D distances from the leading edge of the bottom plate, along the developing plane boundary layer, where the relative boundary layer thicknesses were d/D ¼ 0.14, 0.25 and 0.48, respectively. Using the measured pressure distribution on the cylinder surface, they determined the front stagnation-point angle, ys. The stagnation point was found to be displaced upward rapidly with an increasing gap ratio up to G/D ¼ 1.2, and continued to be displaced until G/D ¼ 2.0. However, the change was negligible for G/D41.2. It was also observed that an increase of the boundary layer thickness, d/D, caused an upward displacement of the stagnation point on the cylinder.
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Fig. 11. Variations of stagnation-point angles with gap ratio at different ReD.
From the present experiments, the variations of measured ys with G/D at ReD ¼ 840, 4150 and 9500 are given in Fig. 11. The results of Lei et al. (1999) are also shown. The present values for ys were determined from the measured time-averaged streamline patterns, some of which are provided earlier in the paper. The stagnation point at ReD ¼ 840 lies on the upper portion of the cylinder surface for G/Do0.6, while it is on the lower part for all gap ratios at ReD ¼ 4150 and 9500. The present results show that ys changes from 101, 12.31, and 19.51 for G/D ¼ 0.0 at ReD ¼ 840, 4150 and 9500, to almost the same (average) value of 1.61 for G/D ¼ 1.0 at all three values of ReD. For ReD ¼ 4150 and 9500, ys changes sharply between G/D ¼ 0.0 and 0.3; but for G/D40.3, the variation of ys is very slow and it only changes from about 2.81 to 0.91 between G/D ¼ 0.3 and 2.0. The variations in the stagnation-point angle with G/D indicate that the streamline pattern around the cylinder becomes nearly symmetrical about its horizontal axis when G/DX1.0. The experimental results for ys at ReD ¼ 4150 and 9500 and those of Lei et al. (1999) display similar behavior with respect to the gap ratio. However, in terms of the absolute values of ys, the present results are considerably smaller. The Reynolds number conditions in the two studies may partly account for the differences. But, it is more likely that the differences are attributable to the effects of the different boundary layer thicknesses, d/D, measured at the cylinder locations.
3.4. Separation points on the cylinder The instantaneous streamline patterns discussed earlier show that the flow in the separation zone is of unsteady character and, therefore, the boundaries of the separation zone and the points of separation on the cylinder change with time. However, using the time-averaged velocity field, it is possible to define a timeaveraged separation zone behind the cylinder. Past experimental results show that the time-averaged angular positions, 7ysep, of upper and lower separation points on a horizontal cylinder depend on ReD. For isolated cylinders, Batchelor (1974) gave the separation angles as ysep ¼ 7801 and 71201 at ReD ¼ 105 and 7 105, respectively. Sumer and Fredsoe (1997), in their experiments at ReD ¼ 6000, found ysep ¼ 801 and 1101 for G/D ¼ 0.1, and ysep ¼ 901 and 931 for G/D ¼ 1.0 at the free-stream side and wall side, respectively.
In this study, the approximate angles of the separation points on the cylinder were determined using Fig. 12, which shows the measured time-averaged streamlines above and below the cylinder for G/D ¼ 0.0, 0.2, 0.3 and 1.0 of the three flows at ReD ¼ 840, 4150 and 9500. Because there is no flow below the cylinder for G ¼ 0, the time-averaged streamlines are given for the upper half of the cylinder only for the cases of G/D ¼ 0.0. The streamline patterns in the figure clearly show that the wake region becomes gradually symmetrical about the horizontal axis of the cylinder as the gap ratio increases from G/D ¼ 0.2 to 1.0. The points where the streamlines for the boundaries of the timeaveraged separation zone detach from the cylinder may be assumed to represent the approximate time-averaged positions of the flow separations on the cylinder surface. Some examples for the inner and outer time-averaged separation-point angles obtained from the streamline patterns are given in Table 3. The values of 7ysep in the table show that the flow separation points on the cylinder shift in a clockwise direction as the cylinder moves away from the bottom. The results for G/D ¼ 1.0 indicate some asymmetry for the angular positions of the separation points, amounting to a difference of 51 to 81 between the lower and upper separation-point angles.
3.5. Vortex shedding and Strouhal number The instantaneous streamline patterns show that the flow behavior in the separation zone is time-dependent and, as a result, vortices grow and then are shed alternately from the upper and lower sides of the cylinder in a periodic manner. The time scale and the period of the oscillating flow are related by the Strouhal number, which is defined as follows: St ¼
fD . u0
(2)
Visualization of the vortex-shedding behavior in the cylinder wake can simply be achieved by injecting dye into the flow. Images, recorded using a digital camera, can then be used to examine the wake vortex formation. The onset of the vortex shedding and its frequency of occurrence can be quantitatively observed and evaluated by using the technique of power spectral analysis of either the lift force coefficient for the cylinder or the flow velocities in the wake of the cylinder. Lei et al. (1999) studied the vortex-shedding problem using the power spectrum of the lift
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Fig. 12. Time-averaged streamlines above and below the cylinder for different G/D at (a) ReD ¼ 840, (b) ReD ¼ 4150, and (c) ReD ¼ 9500.
Table 3 Cylinder boundary layer separation angles G/D
0.0 0.2 1.0
Cylinder separation angle, 7ysep (1) ReD ¼ 840
ReD ¼ 4150
ReD ¼ 9500
94, – 98, 112 100, 105
90, – 92, 110 96, 104
88, – 90, 109 97, 104
coefficient. Using 20 pressure tappings on the surface of the cylinder at its mid-span, they measured the pressure histories and, based on the pressure data, they calculated the power spectra of the lift coefficient for different G/D values to determine the
occurrence of the dominant frequency, which represents the frequency of the alternate vortex shedding. Other investigators have determined the vortex-shedding frequency by using the power spectra of the velocity histories measured at a point in the cylinder wake; different experimental techniques such as hotwire anemometry (Choi and Lee, 2000), hot-film anemometry (Price et al., 2002) and PIV (Akilli et al., 2005) have been used to measure the flow velocities in the wake. Analysis of vortex shedding using the power spectral density (PSD) distribution of the wake velocities may not be as accurate as that of the lift coefficients, since the strength and appearance of the spectrum is somewhat dependent on the position of the velocity measuring point in the wake. However, since the velocity field has to follow the periodic behavior of the flow in the cylinder wake, the dominant frequency of the PSD distribution obtained from the velocities measured at a point in the shear layers of the near-wake
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region should adequately represent the vortex-shedding frequency. As the present experimental program did not involve pressure measurements, determination of the vortex-shedding properties was based on the PIV measurements of the velocity field. Bearman and Zdravkovich (1978) reported that vortex shedding was suppressed for G/Do0.3; and St for G/DX0.3 was remarkably constant at about 0.20 for flow conditions having ReD ¼ 25,000 and 45,000. Grass et al. (1984) obtained a critical value of G/D ¼ 0.3 below which vortex shedding was suppressed for ReD ¼ 1785 and 3570. Lei et al. (1999) found that for ReD ¼ 13,000–14,500, no regular vortex shedding occurred downstream of the cylinder for G/Dp0.2–0.3, depending on the boundary layer thickness. In their experiments, St did not change significantly with either G/D or Du/uc and remained constant at about 0.20 for G/DX0.3 (Du is the velocity difference between the top and bottom levels of the cylinder and uc is the velocity at the center of the cylinder). Choi and Lee (2000), using spectral analyses of the velocity signals at ReD ¼ 14,000, found that when G/DX0.3, a dominant peak at the vortex-shedding frequency appeared in the power spectra and St gradually increased from 0.185 to 0.196, as G/D increased. St was almost independent of the boundary layer thickness. For flows with ReD ¼ 1200–4960, Price et al. (2002) found that regular vortex shedding began when G/DX0.5–0.75, depending on the boundary layer thickness; and changes in the vortex-shedding behavior were negligible for G/DX1.0. For G/Dp0.25, the wake
Strouhal number was associated with periodicities in the outer shear layer and, for G/Dp0.125, no regular vortex shedding occurred downstream of the cylinder. For ReDp2600, St was much more sensitive to G/D, varying in the range of 0.27–0.24 and 0.41–0.21 for G/D of 1.0 and 0.25, respectively. For ReD ¼ 2600–4000, St varied in the range of 0.25–0.24 and 0.21–0.20 for G/D of 1.0 and 0.25, respectively, with St decreasing with increasing ReD. At ReDX4000, St was found to vary between 0.24 and 0.20 as G/D decreased from 1.0 to 0.25. St was independent of d/D, which varied between 0.2 and 0.5. In order to determine the periodicity of vortex shedding in the present study, the power spectral density (PSD) distributions of the horizontal velocity components were calculated at the upper and lower edges of the cylinder wake at about x/D ¼ 1.0 and y/D ¼ 70.5. The results for different gap ratios and Reynolds numbers are given in Fig. 13. It is seen that the PSD distributions do not present any clear peaks for G/D ¼ 0.1 for any Reynolds number, which means that the vortex shedding is suppressed for these conditions. For cases when G/DX0.2, there are dominant peaks in the PSD distributions, which indicate the occurrence of regular vortex shedding. However, the PSD distributions at ReD ¼ 840 in Fig. 13(a) contain secondary peaks as well. It is seen that for ReD ¼ 840, St is much more sensitive to the changes in G/D, and the range of upper and lower wake frequencies varies between 0.11 and 0.17 for G/D ¼ 0.2 and 0.23 and 0.27 for G/D ¼ 1.0. On the other hand, the Strouhal numbers obtained at the upper and lower sides of the cylinder wake in Figs. 13(b) and (c)
Fig. 13. Power spectral density (PSD) distributions of horizontal wake velocities at the upper and lower edges of the cylinder for different G/D at: (a) ReD ¼ 840, (b) ReD ¼ 4150, and (c) ReD ¼ 9500.
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Fig. 14. Variations of Strouhal number with gap ratio at ReD ¼ 4150 and 9500.
are almost identical for ReD ¼ 4150 and 9500. From the PSD distributions, it is seen that the strength of the vortices weakens as G/D decreases for all three Reynolds numbers. At low Reynolds numbers, the occurrence of a secondary peak in the spectrum was also reported by Price et al. (2002). However, according to their results, as ReD was increased from 1440, the two peaks appeared to merge into one broad peak, and a further increase in ReD produced a wake with a very distinct frequency at ReD ¼ 2840 for G/D ¼ 0.25–0.5. The existence of the secondary peak in the PSD distribution may be attributed to instabilities in the vortex formation caused by the transition from laminar to turbulent behavior in the wake, which takes place during the early stages of the subcritical regime characterized by low ReD conditions. The variations of St with G/D at ReD ¼ 4150 and 9500, obtained from the PSD distributions, are given in Fig. 14. The results for ReD ¼ 840 are not included in Fig. 14 because, as mentioned above, there are uncertainties in determining single values for St to represent upper and lower vortex-shedding frequencies for these flow cases. As may be seen, when G/DX0.6, St becomes nearly constant at about 0.24 and 0.21 for ReD ¼ 4150 and 9500, respectively. These values for St are in agreement with the results found by Price et al. (2002) at ReDX4000. It is seen that for G/Do0.6, St decreases with a decrease in G/D to a value of 0.19 for G/D ¼ 0.2 at both Reynolds number conditions. As mentioned earlier, St may roughly be calculated using the approximate vortex-shedding frequency, f, from the instantaneous streamline patterns. When the value f ¼ 0.77 Hz for G/D ¼ 0.3 at ReD ¼ 9500, found from Fig. 10, is used in Eq. (2), St may be estimated as 0.195 that confirms the value St ¼ 0.20 obtained from the PSD distribution.
4. Conclusions Using the PIV technique, experiments were conducted to measure the velocity fields for steady, two-dimensional, turbulent flows around a horizontal smooth circular cylinder near a plane wall. Using seven different gap ratios, G/D ¼ 0.0, 0.1, 0.2, 0.3, 0.6, 1.0 and 2.0, experiments were repeated for the flow conditions at ReD ¼ 840, 4150 and 9500. From the measured velocity fields, the effects of wall proximity on various flow properties were investigated.
Results show that upstream wall separation occurs for G/Dp0.6, 0.3 and 0.1 for ReD ¼ 840, 4150 and 9500, respectively; and downstream wall separation exists for G/Dp0.2 and 0.1 at ReD ¼ 840 and 4150, respectively. Wall separation due to the vortex pair downstream of the cylinder appears for conditions between G/D ¼ 0.3 and 0.6 for ReD ¼ 840, and between G/D ¼ 0.2 and 0.6 for ReD ¼ 4150 and 9500. The stagnation-point angles for G/D ¼ 0.0 are ys ¼ 101, 12.31, and 19.51 for ReD ¼ 840, 4150 and 9500, respectively; and the stagnation-point angle for G/D ¼ 1.0 is ys ¼ 1.61 at all three Reynolds numbers. For ReD ¼ 4150 and 9500, ys changes sharply between G/D ¼ 0.0 and 0.3, but for G/DX0.3, the variation of ys becomes very slow: it only changes from about 2.81 to 0.91 between G/D ¼ 0.3 and 2.0. The power spectral density distributions of the instantaneous wake velocities indicate that the vortex shedding behind the cylinder is suppressed for G/D ¼ 0.1 and that regular vortex shedding occurs for G/DX0.2 at all three Reynolds numbers. When the gap ratio G/DX0.6, St remains constant at about 0.24 and 0.21 at ReD ¼ 4150 and 9500, respectively; and for G/Do0.6, St decreases with decreasing gap ratio to a minimum value of 0.19 for G/D ¼ 0.2. Finally, the measured time-averaged velocity distributions, streamline patterns and the isovorticity contours show that the effect of wall proximity on the flow field around the cylinder becomes insignificant when the gap ratio G/DX1.0.
Acknowledgments The work was partly supported by the Cukurova University Research Fund under project no: MMF2004D4. This support is gratefully acknowledged. Suggestions of Eur Ing Terry S. Hedges at the University of Liverpool are also acknowledged. References Akilli, H., Sahin, B., Tumen, N.F., 2005. Suppression of vortex shedding of circular cylinder in shallow water by a splitter plate. Flow Measurement and Instrumentation 16, 211–219. Batchelor, G.K., 1974. An Introduction to Fluid Dynamics. Cambridge University Press. Bearman, P.W., Zdravkovich, M.M., 1978. Flow around a circular cylinder near a plane boundary. Journal of Fluid Mechanics 89, 33–47.
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Choi, J.H., Lee, S.J., 2000. Ground effect of flow around an elliptic cylinder in a turbulent boundary layer. Journal of Fluids and Structures 14, 697–709. Grass, A.J., Raven, P.W.J., Stuart, R.J., Bray, J.A., 1984. The influence of boundary layer velocity gradients and bed proximity on vortex shedding from free spanning pipelines. Journal of Energy Resources Technology 106, 70–78. Hatipoglu, F., Avci, I., 2003. Flow around a partly buried cylinder in a steady current. Ocean Engineering 30, 239–249. Lei, C., Cheng, L., Kavanagh, K., 1999. Re-examination of the effect of a plane boundary on force and vortex shedding of a circular cylinder. Journal of Wind Engineering and Industrial Aerodynamics 80, 263–286. Oner, A., 2007. Experimental investigation of flow around horizontal circular cylinders. Ph.D. Thesis, Cukurova University, Adana, Turkey. Price, S.J., Sumner, D., Smith, J.G., Leong, K., Paidoussis, M.P., 2002. Flow visualization around a circular cylinder near to a plane wall. Journal of Fluids and Structures 16 (2), 175–191.
Roshko, A., 1961. Experiments on the flow past a circular cylinder at very high Reynolds number. Journal of Fluid Mechanics 10, 345–356. Schewe, G., 1983. On the force fluctuations acting on a circular cylinder in crossflow from subcritical up to transcritical Reynolds numbers. Journal of Fluid Mechanics 133, 265–285. Sumer, B.M., Fredsoe, J., 1997. Hydrodynamics Around Cylindrical Structures. World Scientific, Singapore. Taneda, S., 1965. Experimental investigation of vortex streets. Journal of the Physical Society of Japan 20, 1714–1721. Williamson, C.H.K., 1989. Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds number. Journal of Fluid Mechanics 206, 579–627. Zdravkovich, M.M., 1985. Forces on a circular cylinder near a plane wall. Applied Ocean Research 7, 197–201. Zdravkovich, M.M., 1997. Flow Around Circular Cylinders, Vol. 1: Fundamentals. Oxford University Press.
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Interaction of a current with a circular cylinder near a rigid bed A. Alper Oner a, M. Salih Kirkgoz b,, M. Sami Akoz b a b
Civil Engineering Department, Aksaray University, 68100 Aksaray, Turkey Civil Engineering Department, Cukurova University, 01330 Adana, Turkey
a r t i c l e in f o
a b s t r a c t
Article history: Received 18 December 2007 Accepted 13 June 2008 Available online 19 June 2008
The interaction of fluid flow with a circular cylinder is important in many design situations. In the study of fluid–body interaction problems, it is essential that one is familiar with the affected flow structure in terms of its kinematics and related properties. Existing experimental data show that the vertical gap, G, between a horizontal cylinder and a plane boundary is the major parameter that affects the flow structure around the cylinder. In this study, the PIV technique is used to measure the velocity field in a steady, two-dimensional, turbulent flow around a horizontal circular cylinder near a plane boundary. The resulting velocity profiles, streamlines and isovorticity contours are presented for the gap ratios, G/D ¼ 0.0, 0.1, 0.2, 0.3, 0.6, 1.0 and 2.0. Experiments for the seven different G/D ratios are repeated for Reynolds numbers ReD ¼ 840, 4150 and 9500. Using the measured velocity fields, the effect of gap ratio on the wall boundary–layer separation, the positions of the stagnation and separation points on the cylinder, and the Strouhal number are investigated. Present experimental results indicate that the changes in the flow structure become very slow when G/DX0.3, and the wall proximity effect on the flow around the cylinder becomes insignificant when G/DX1.0. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Steady flow Circular cylinder Particle image velocimetry Velocity field Wall proximity Vortex shedding
1. Introduction The interaction of fluid flow with two-dimensional cylindrical structures may be encountered in a variety of engineering applications. Cylinders with circular cross-sections are the most common submerged structural elements in civil, mechanical, coastal and ocean engineering practices. In coastal and ocean engineering, the design of submarine pipelines is one of the most important practical examples. When a pipeline is initially placed on an erodible bed, interaction of the disturbed flow field with the sea floor may create erosion problems. Scour may occur below the pipe and, as a result, the pipeline may become a freespanning structure. Pipelines, either resting on the seabed or suspended above the bed with a small gap, may be subjected to transversely flowing steady currents in deep, intermediate and shallow water conditions. The dynamic forces exerted by the fluid flow on submerged cylindrical elements and their response to these forces are the primary concerns in engineering design. Fig. 1 illustrates four typical regions in the disturbed flow field around a cylinder (Sumer and Fredsoe, 1997; Zdravkovich, 1997). These are: (a) the upstream retarded flow region; (b) the boundary layers that develop along the cylinder surface; (c) the
Corresponding author. Tel.: +90 322 3386702; fax: +90 322 3386702.
E-mail address: [email protected] (M. Salih Kirkgoz). 0029-8018/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2008.06.005
curvilinear, accelerated sidewise flow regions; and (d) the downstream separated flow region called ‘‘the wake’’. The wake region is the most complicated part of the disturbed flow field. At the same time, it is the most influential in the interaction between the fluid flow and the cylindrical body. Flow structures affected by cylindrical bodies near solid boundaries have been the subject to numerous experimental and theoretical investigations. They have mostly been directed toward determining wall proximity effects on various flow properties such as the wall and cylinder boundary–layer separation points, the position of the front stagnation point, the pressure distribution on the cylinder surface, the drag and lift coefficients, the Strouhal number for vortex shedding frequency, and the vortical structure and turbulence characteristics of the wake behind the cylinder (Bearman and Zdravkovich, 1978; Grass et al., 1984; Zdravkovich, 1985; Lei et al., 1999; Choi and Lee, 2000; Price et al., 2002). In the earlier experimental studies, different flow measurement techniques were employed to determine the flow behavior around the cylinder, including hot-wire anemometry (Bearman and Zdravkovich, 1978; Choi and Lee, 2000), hot-film anemometry (Price et al., 2002), particle image velocimetry (PIV; Price et al., 2002; Akilli et al., 2005) and flow visualization (Choi and Lee, 2000; Hatipoglu and Avci, 2003). Different experimental conditions have been used, ranging from low Reynolds numbers of ReD ¼ 40 (Williamson, 1989) and 170 (Taneda, 1965) to high Reynolds numbers of ReD ¼ 7.1 106 (Schewe, 1983) and 107 (Roshko, 1961).
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Fig. 1. Definition sketch for the variables related to the steady flow field around a horizontal circular cylinder near plane boundary.
In these earlier experiments, it was observed that the freestream velocity, u0, the depth, h, and the boundary layer thickness, d, of the flow, the kinematic viscosity of the fluid, n, the cylinder diameter, D, and the gap between the cylinder and the wall, G, are the principal quantities that affect the flow past a horizontal circular cylinder, which is either placed on the bottom or is mounted above it with a small gap (see Figs. 1 and 2, for symbols). The dimensionless numbers governing the characteristic features of the flow were found to be the Reynolds number, ReD ¼ u0D/n, the gap ratio, G/D, and the ratio of the free-stream boundary layer thickness to the cylinder diameter, d/D. Much experimental information is available in the literature on the flow field around submerged cylindrical structures. However, further experimental effort is needed either to confirm the earlier findings or to provide new information for different experimental conditions. In this study, PIV is used to measure the timedependent and -averaged velocity fields in steady, two-dimensional, turbulent flows around a horizontal smooth cylinder with different gaps from a plane boundary for flow conditions with ReD ¼ 840, 4150 and 9500. Some characteristic features are then described.
2. Experiments 2.1. Experimental setup The experiments were performed in a transparent plexiglaswalled laboratory channel that was 1 m wide, 0.75 m deep and 14 m long. The water depth in the channel was kept constant at 0.60 m during the experiments. As may be seen in Fig. 2, a smooth flat plate made from plexiglas was mounted in the central part of the channel. The plate was 2 m long and spanned the channel width, and it had a leading sharp edge to avoid flow disturbances. The plate was positioned 0.28 m above the channel bed, and the flow depth on the plate was h ¼ 0.32 m. A smooth circular plexiglas test cylinder with a diameter D ¼ 50 mm was mounted horizontally 1.5 m ( ¼ 30D) downstream from the leading edge of the plate. Tests were undertaken with ReD ¼ 840, 4150 and 9500. The details of the experimental conditions are given in Table 1. The free-stream velocities, u0, and the boundary layer thicknesses, d, were obtained from the velocity profiles measured at the location of the cylinder in the channel without the cylinder in place. The ratios d/D at the test location were 1.6, 1.5 and 1.4 for the given ReD values, respectively. The flow regime around the cylinder is
classified as subcritical when the Reynolds number is in the range of 300oReDo3 105 (Sumer and Fredsoe, 1997). According to this criterion, the three flow cases used in the present tests are categorized as subcritical, which has the characteristic features of a completely turbulent wake and laminar boundary layer separation. For each of the flow conditions listed in Table 1, the experiments were repeated with seven different gaps G ¼ 0, 5, 10, 15, 30, 50, and 100 mm corresponding to gap ratios G/D ¼ 0.0, 0.1, 0.2, 0.3, 0.6, 1.0 and 2.0.
2.2. Velocity measurements using PIV technique PIV is an optical method used to measure the velocities and related properties in fluid flows. The fluid is seeded with particles (tracers) that are assumed to follow the flow. The motion of the seeding particles is used to calculate the velocity field. The PIV is a non-intrusive technique capable of measuring an entire twodimensional geometry of the flow field simultaneously. Compared to other techniques such as hot-film anemometry and hot-wire anemometry, the PIV technique has the advantage of determining the time-dependent vortical formation simultaneously in the twodimensional flow field around a circular cylinder. The instantaneous velocities in the flow field around the cylinder at the mid-span of the channel were measured using the PIV system and Flow Manager Software. A schematic view of the PIV system is seen in Fig. 2. The water in the channel was seeded with neutrally buoyant silver-coated spherical particles of 14 mm in diameter. The flow was illuminated via a 2-mm-thick laser sheet of argon-ion green light with a wavelength of 532 nm from a pair of double-pulsed Nd:YAG laser units. Flow map processor that controls the timing of the data acquisition was used to synchronize the image taking and the laser units. Within the two-dimensional measuring area of the flow, the instantaneous velocity vectors were determined by recording the displacements of the seeding particles between the two locations during a specified time interval of the two pulses, which was 1.5 ms throughout the experiments. The movements of the particles were recorded by a CCD camera with a resolution of 1024 1024 pixels, equipped with a 60 mm focal length lens. Each pixel covered a square of 0.065 0.065 mm in the field of measurement. From the recorded particle image data, the displacement vectors were obtained using digital PIV software employing a frame-to-frame cross-correlation technique. In the image processing, 32 32 pixel rectangular effective interrogation windows were used and an overlap of 50% was employed, yielding a total of approximately
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Fig. 2. Experimental arrangement for the circular cylinder in the channel and the schematic diagram of the PIV system.
Table 1 Experimental conditions Flow case
h (m)
u0 (mm/s)
d (mm)
ReD ( ¼ u0D/n)
d/D
1 2 3
0.32 0.32 0.32
17.3 86 197
80 75 70
840 4150 9500
1.6 1.5 1.4
3844 velocity vectors. The size of the velocity measurement field viewed by the laser sheet was taken as 80 80 mm upstream and downstream of the cylinder, and 50 50 mm above and below it. For each measurement, a total of 490 patterns of instantaneous particle images were taken and recorded on a computer during a period of 49 s (i.e. at a rate of 10 Hz). By post-processing of the measured instantaneous velocity field, the time-averaged velocity vector field and the corresponding streamlines and isovorticity contour lines were determined. The time-averaged velocity for each interrogation window was obtained by taking the arithmetic mean of 490 instantaneous velocities measured sequentially for the seeding particles in that window. The measurement uncertainty in velocities was judged to be within 72%. More detail of the experimental procedure is given by Oner (2007).
around the cylinder for gap ratios G/D ¼ 0.0, 0.2, 0.3 and 1.0 at ReD ¼ 9500 are shown in Figs. 3–6. The boundaries of development of boundary layers on both side of the cylinder surface and upper and lower free-shear layers behind the cylinder are shown in dashed lines. The four typical disturbed regions of the flow around the cylinder, defined earlier, may be detected from the velocity profiles provided in the above plots. The reformation of the wall boundary layer through the gap is indicated with a dotted line for G/D ¼ 0.2 and 0.3 in Figs. 4 and 5. The points where the inner boundary lines of the shear layers detach from the cylinder show the approximate time-averaged positions of the upper and lower separation points on the cylinder surface. From the velocity distributions for G/D ¼ 1.0 in Fig. 6, it is seen that the disturbed velocity field around the cylinder is nearly symmetrical about its horizontal axis. That means the influence of the solid bed on the velocity field around the cylinder becomes weak when the gap ratio increases to G/D ¼ 1.0. The streamlines and isovorticity contour lines calculated from the measured time-averaged velocity data upstream and downstream of the cylinder are provided in Figs. 7–9 for gap ratios G/D ¼ 0.0, 0.1, 0.2, 0.3 and 1.0 from the three flows with ReD ¼ 840, 4150 and 9500. The z-component of the vorticity vector of the two-dimensional flow field in the xy plane is 2oz ¼
3. Experimental results 3.1. Velocity distributions, streamlines and isovorticity contour lines The vertical distributions of the time-averaged horizontal velocity components, u, at different locations, x/D, of the flow
qv qu . qx qy
(1)
In Eq. (1), u and v are the horizontal and vertical velocity components in the x- and y-directions, respectively. The local vorticities in the xy plane were determined as the differential circulation per unit area using a finite difference scheme. The solid lines in Figs. 7–9 represent positive vorticities (counterclockwise), and the incremental values of the vorticity contours are taken
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Fig. 3. Vertical distributions of horizontal velocity component at different locations of flow around the cylinder for G/D ¼ 0.0 at ReD ¼ 9500.
Fig. 4. Vertical distributions of horizontal velocity component at different locations of flow around the cylinder for G/D ¼ 0.2 at ReD ¼ 9500.
Fig. 5. Vertical distributions of horizontal velocity component at different locations of flow around the cylinder for G/D ¼ 0.3 at ReD ¼ 9500.
equal to the minimum values that are given in the figures. As can be seen, the isovorticity contours are concentrated mostly in the boundary layers of the cylinder and wall and also in the freeshear layers downstream of the cylinder: that is, in the regions
where high-gradient velocity distributions exist, as observed in Figs. 3–6. From the streamline patterns, it is possible to determine the positions of the front stagnation point on the cylinder. The effects
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Fig. 6. Vertical distributions of horizontal velocity component at different locations of flow around the cylinder for G/D ¼ 1.0 at ReD ¼ 9500.
Fig. 7. Time-averaged streamlines and isovorticity contour lines upstream and downstream of the cylinder for different G/D at ReD ¼ 840. Solid lines represent positive vorticities. The minimum and incremental values of vorticity are 1 s1.
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Fig. 8. Time-averaged streamlines and isovorticity contour lines upstream and downstream of the cylinder for different G/D at ReD ¼ 4150. Solid lines represent positive vorticities. The minimum and incremental values of vorticity are 5 s5.
of wall proximity on the time-averaged flow structures are clearly displayed in the figures. It is seen that the streamline patterns and isovorticity contour lines tend to become symmetrical as the gap ratio increases toward G/D ¼ 1.0. Almost symmetrical velocity distributions, streamline patterns and isovorticity contours about the horizontal axis of the cylinder for G/D ¼ 1.0 suggest that the wall proximity effect on the flow becomes negligible when G/D increases to 1.0. However, the minor influence of the channel boundary layer on the flow field should not be disregarded, as the values of d/D at the test location are all greater than G/D ¼ 1.0. Thus, the cylinder is partly buried within the upper portion of the channel boundary layer and is not yet completely free from the wall boundary layer effect. A typical record of instantaneous streamlines, measured at Dt ¼ 0.1 s intervals, is shown in Fig. 10 for conditions downstream of the cylinder with a gap ratio G/D ¼ 0.3 at ReD ¼ 9500. Alternate vortex shedding and changes in the vortex formation downstream
of the cylinder are clearly displayed in the figure. The frequency of vortex shedding for this particular case may be estimated by determining the recurrence interval of any one of the instantaneous streamline patterns. Visual examination of the instantaneous streamline records shows that streamline pattern in Fig. 10(a) is more or less repeated in Fig. 10(n), although they are not completely identical. The time elapse between Fig. 10(a) and (n) is 1.3 s, which corresponds to a vortex shedding frequency of approximately 0.77 Hz for this case. Similar repetitions are observed in the subsequent images: that is in Figs. 10(b) and (o), and in Figs. 10(c) and (p). However, the preceding analysis for shedding frequency, using the instantaneous streamlines pertains to one shedding period only and its precision is affected by the time interval of the recordings. However, the frequency of the vortex shedding determined in this way can be used to roughly confirm the finding of a more reliable quantitative evaluation given later.
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Fig. 9. The-averaged streamlines and isovorticity contour lines upstream and downstream of the cylinder for different G/D at ReD ¼ 9500. Solid lines represent positive vorticities. The minimum and incremental values of vorticity are 5 s1.
3.2. Upstream and downstream wall boundary layer separations Price et al. (2002) reported that for Reynolds numbers in the range 1200pReDp1400, wall boundary layer separation upstream of the cylinder is observed for G/Dp0.75 and no wall separation occurs when G/DX1.0. From the examination of the present experimental time-averaged streamlines, it is possible to determine the flow conditions where the wall boundary layer separations occur. For the present test conditions shown in Figs. 7(b)–(d), 8(b)–(d), and 9(b), it can be seen that the flow separates from the bed upstream of the cylinder and then reattaches as it passes through the gap. For the flow conditions in Figs. 7(b) and (c), and 8(b), the downstream wall boundary layer is forced to separate again as the streamlines in the inner shear layer are deflected away from the wall. From the timeaveraged streamlines obtained in the present experiments, the ranges of G/D for which the upstream and downstream wall separations occur are given in Table 2. The values in the table show that the upper limit of the gap ratio where upstream or
downstream wall separation occurs reduces as ReD increases. Upstream wall separation occurs for G/Dp0.6, 0.3 and 0.1 at ReD ¼ 840, 4150 and 9500, respectively; and the downstream wall separation exists when G/Dp0.2 and 0.1 at ReD ¼ 840 and 4150, respectively. The streamline patterns at ReD ¼ 9500 do not seem to display any downstream wall separations for the present gap ratios. Price et al. (2002) stated that for G/D between 0.25 and 1.0, vortex shedding from the lower side of the cylinder affected the downstream wall boundary layer, causing it to separate at the same frequency as that of the vortex shedding from the cylinder. This sort of vortex pair occurrence between the inner shear layer and downstream separated wall boundary layer is also observed in some cases of the present tests. The instantaneous streamline patterns demonstrate the changes in the vortical structure of the flow downstream of the cylinder. It is seen that during a vortexshedding period on the lower cylinder side, the instantaneous streamline patterns in Figs. 10(g)–(k) display a temporary local wall separation about one cylinder diameter downstream of the cylinder. As pointed out by Price et al. (2002), and confirmed by
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Fig. 10. Instantaneous streamlines patterns with Dt ¼ 0.1 s intervals downstream of the cylinder for G/D ¼ 0.3 at ReD ¼ 9500.
Table 2 Experimental conditions for wall boundary layer separations ReD
Upstream wall boundary layer separation occurs
Downstream wall boundary layer separation occurs
Wall separation due to vortex pair downstream of gap flow
840 4150 9500
0.0oG/Dp0.6 0.0oG/Dp0.3 0.0oG/Dp0.1
0.0oG/Dp0.2 0.0oG/Dp0.1 –
G/D ¼ 0.3–0.6 G/D ¼ 0.2–0.6 G/D ¼ 0.2–0.6
the present instantaneous streamline patterns, this occurrence may be attributed to the existence of a vortex pair between the inner shear layer of the cylinder wake and the wall boundary layer. The test conditions in which this vortex pair occurs are given in Table 2.
3.3. Position of stagnation point on the cylinder Lei et al. (1999) conducted experiments on steady flows around a horizontal cylinder with different gap ratios for ReD between 13,000 and 14,500. In their experiments, the cylinder was
positioned at 2D, 3D and 11D distances from the leading edge of the bottom plate, along the developing plane boundary layer, where the relative boundary layer thicknesses were d/D ¼ 0.14, 0.25 and 0.48, respectively. Using the measured pressure distribution on the cylinder surface, they determined the front stagnation-point angle, ys. The stagnation point was found to be displaced upward rapidly with an increasing gap ratio up to G/D ¼ 1.2, and continued to be displaced until G/D ¼ 2.0. However, the change was negligible for G/D41.2. It was also observed that an increase of the boundary layer thickness, d/D, caused an upward displacement of the stagnation point on the cylinder.
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Fig. 11. Variations of stagnation-point angles with gap ratio at different ReD.
From the present experiments, the variations of measured ys with G/D at ReD ¼ 840, 4150 and 9500 are given in Fig. 11. The results of Lei et al. (1999) are also shown. The present values for ys were determined from the measured time-averaged streamline patterns, some of which are provided earlier in the paper. The stagnation point at ReD ¼ 840 lies on the upper portion of the cylinder surface for G/Do0.6, while it is on the lower part for all gap ratios at ReD ¼ 4150 and 9500. The present results show that ys changes from 101, 12.31, and 19.51 for G/D ¼ 0.0 at ReD ¼ 840, 4150 and 9500, to almost the same (average) value of 1.61 for G/D ¼ 1.0 at all three values of ReD. For ReD ¼ 4150 and 9500, ys changes sharply between G/D ¼ 0.0 and 0.3; but for G/D40.3, the variation of ys is very slow and it only changes from about 2.81 to 0.91 between G/D ¼ 0.3 and 2.0. The variations in the stagnation-point angle with G/D indicate that the streamline pattern around the cylinder becomes nearly symmetrical about its horizontal axis when G/DX1.0. The experimental results for ys at ReD ¼ 4150 and 9500 and those of Lei et al. (1999) display similar behavior with respect to the gap ratio. However, in terms of the absolute values of ys, the present results are considerably smaller. The Reynolds number conditions in the two studies may partly account for the differences. But, it is more likely that the differences are attributable to the effects of the different boundary layer thicknesses, d/D, measured at the cylinder locations.
3.4. Separation points on the cylinder The instantaneous streamline patterns discussed earlier show that the flow in the separation zone is of unsteady character and, therefore, the boundaries of the separation zone and the points of separation on the cylinder change with time. However, using the time-averaged velocity field, it is possible to define a timeaveraged separation zone behind the cylinder. Past experimental results show that the time-averaged angular positions, 7ysep, of upper and lower separation points on a horizontal cylinder depend on ReD. For isolated cylinders, Batchelor (1974) gave the separation angles as ysep ¼ 7801 and 71201 at ReD ¼ 105 and 7 105, respectively. Sumer and Fredsoe (1997), in their experiments at ReD ¼ 6000, found ysep ¼ 801 and 1101 for G/D ¼ 0.1, and ysep ¼ 901 and 931 for G/D ¼ 1.0 at the free-stream side and wall side, respectively.
In this study, the approximate angles of the separation points on the cylinder were determined using Fig. 12, which shows the measured time-averaged streamlines above and below the cylinder for G/D ¼ 0.0, 0.2, 0.3 and 1.0 of the three flows at ReD ¼ 840, 4150 and 9500. Because there is no flow below the cylinder for G ¼ 0, the time-averaged streamlines are given for the upper half of the cylinder only for the cases of G/D ¼ 0.0. The streamline patterns in the figure clearly show that the wake region becomes gradually symmetrical about the horizontal axis of the cylinder as the gap ratio increases from G/D ¼ 0.2 to 1.0. The points where the streamlines for the boundaries of the timeaveraged separation zone detach from the cylinder may be assumed to represent the approximate time-averaged positions of the flow separations on the cylinder surface. Some examples for the inner and outer time-averaged separation-point angles obtained from the streamline patterns are given in Table 3. The values of 7ysep in the table show that the flow separation points on the cylinder shift in a clockwise direction as the cylinder moves away from the bottom. The results for G/D ¼ 1.0 indicate some asymmetry for the angular positions of the separation points, amounting to a difference of 51 to 81 between the lower and upper separation-point angles.
3.5. Vortex shedding and Strouhal number The instantaneous streamline patterns show that the flow behavior in the separation zone is time-dependent and, as a result, vortices grow and then are shed alternately from the upper and lower sides of the cylinder in a periodic manner. The time scale and the period of the oscillating flow are related by the Strouhal number, which is defined as follows: St ¼
fD . u0
(2)
Visualization of the vortex-shedding behavior in the cylinder wake can simply be achieved by injecting dye into the flow. Images, recorded using a digital camera, can then be used to examine the wake vortex formation. The onset of the vortex shedding and its frequency of occurrence can be quantitatively observed and evaluated by using the technique of power spectral analysis of either the lift force coefficient for the cylinder or the flow velocities in the wake of the cylinder. Lei et al. (1999) studied the vortex-shedding problem using the power spectrum of the lift
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Fig. 12. Time-averaged streamlines above and below the cylinder for different G/D at (a) ReD ¼ 840, (b) ReD ¼ 4150, and (c) ReD ¼ 9500.
Table 3 Cylinder boundary layer separation angles G/D
0.0 0.2 1.0
Cylinder separation angle, 7ysep (1) ReD ¼ 840
ReD ¼ 4150
ReD ¼ 9500
94, – 98, 112 100, 105
90, – 92, 110 96, 104
88, – 90, 109 97, 104
coefficient. Using 20 pressure tappings on the surface of the cylinder at its mid-span, they measured the pressure histories and, based on the pressure data, they calculated the power spectra of the lift coefficient for different G/D values to determine the
occurrence of the dominant frequency, which represents the frequency of the alternate vortex shedding. Other investigators have determined the vortex-shedding frequency by using the power spectra of the velocity histories measured at a point in the cylinder wake; different experimental techniques such as hotwire anemometry (Choi and Lee, 2000), hot-film anemometry (Price et al., 2002) and PIV (Akilli et al., 2005) have been used to measure the flow velocities in the wake. Analysis of vortex shedding using the power spectral density (PSD) distribution of the wake velocities may not be as accurate as that of the lift coefficients, since the strength and appearance of the spectrum is somewhat dependent on the position of the velocity measuring point in the wake. However, since the velocity field has to follow the periodic behavior of the flow in the cylinder wake, the dominant frequency of the PSD distribution obtained from the velocities measured at a point in the shear layers of the near-wake
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region should adequately represent the vortex-shedding frequency. As the present experimental program did not involve pressure measurements, determination of the vortex-shedding properties was based on the PIV measurements of the velocity field. Bearman and Zdravkovich (1978) reported that vortex shedding was suppressed for G/Do0.3; and St for G/DX0.3 was remarkably constant at about 0.20 for flow conditions having ReD ¼ 25,000 and 45,000. Grass et al. (1984) obtained a critical value of G/D ¼ 0.3 below which vortex shedding was suppressed for ReD ¼ 1785 and 3570. Lei et al. (1999) found that for ReD ¼ 13,000–14,500, no regular vortex shedding occurred downstream of the cylinder for G/Dp0.2–0.3, depending on the boundary layer thickness. In their experiments, St did not change significantly with either G/D or Du/uc and remained constant at about 0.20 for G/DX0.3 (Du is the velocity difference between the top and bottom levels of the cylinder and uc is the velocity at the center of the cylinder). Choi and Lee (2000), using spectral analyses of the velocity signals at ReD ¼ 14,000, found that when G/DX0.3, a dominant peak at the vortex-shedding frequency appeared in the power spectra and St gradually increased from 0.185 to 0.196, as G/D increased. St was almost independent of the boundary layer thickness. For flows with ReD ¼ 1200–4960, Price et al. (2002) found that regular vortex shedding began when G/DX0.5–0.75, depending on the boundary layer thickness; and changes in the vortex-shedding behavior were negligible for G/DX1.0. For G/Dp0.25, the wake
Strouhal number was associated with periodicities in the outer shear layer and, for G/Dp0.125, no regular vortex shedding occurred downstream of the cylinder. For ReDp2600, St was much more sensitive to G/D, varying in the range of 0.27–0.24 and 0.41–0.21 for G/D of 1.0 and 0.25, respectively. For ReD ¼ 2600–4000, St varied in the range of 0.25–0.24 and 0.21–0.20 for G/D of 1.0 and 0.25, respectively, with St decreasing with increasing ReD. At ReDX4000, St was found to vary between 0.24 and 0.20 as G/D decreased from 1.0 to 0.25. St was independent of d/D, which varied between 0.2 and 0.5. In order to determine the periodicity of vortex shedding in the present study, the power spectral density (PSD) distributions of the horizontal velocity components were calculated at the upper and lower edges of the cylinder wake at about x/D ¼ 1.0 and y/D ¼ 70.5. The results for different gap ratios and Reynolds numbers are given in Fig. 13. It is seen that the PSD distributions do not present any clear peaks for G/D ¼ 0.1 for any Reynolds number, which means that the vortex shedding is suppressed for these conditions. For cases when G/DX0.2, there are dominant peaks in the PSD distributions, which indicate the occurrence of regular vortex shedding. However, the PSD distributions at ReD ¼ 840 in Fig. 13(a) contain secondary peaks as well. It is seen that for ReD ¼ 840, St is much more sensitive to the changes in G/D, and the range of upper and lower wake frequencies varies between 0.11 and 0.17 for G/D ¼ 0.2 and 0.23 and 0.27 for G/D ¼ 1.0. On the other hand, the Strouhal numbers obtained at the upper and lower sides of the cylinder wake in Figs. 13(b) and (c)
Fig. 13. Power spectral density (PSD) distributions of horizontal wake velocities at the upper and lower edges of the cylinder for different G/D at: (a) ReD ¼ 840, (b) ReD ¼ 4150, and (c) ReD ¼ 9500.
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1503
Fig. 14. Variations of Strouhal number with gap ratio at ReD ¼ 4150 and 9500.
are almost identical for ReD ¼ 4150 and 9500. From the PSD distributions, it is seen that the strength of the vortices weakens as G/D decreases for all three Reynolds numbers. At low Reynolds numbers, the occurrence of a secondary peak in the spectrum was also reported by Price et al. (2002). However, according to their results, as ReD was increased from 1440, the two peaks appeared to merge into one broad peak, and a further increase in ReD produced a wake with a very distinct frequency at ReD ¼ 2840 for G/D ¼ 0.25–0.5. The existence of the secondary peak in the PSD distribution may be attributed to instabilities in the vortex formation caused by the transition from laminar to turbulent behavior in the wake, which takes place during the early stages of the subcritical regime characterized by low ReD conditions. The variations of St with G/D at ReD ¼ 4150 and 9500, obtained from the PSD distributions, are given in Fig. 14. The results for ReD ¼ 840 are not included in Fig. 14 because, as mentioned above, there are uncertainties in determining single values for St to represent upper and lower vortex-shedding frequencies for these flow cases. As may be seen, when G/DX0.6, St becomes nearly constant at about 0.24 and 0.21 for ReD ¼ 4150 and 9500, respectively. These values for St are in agreement with the results found by Price et al. (2002) at ReDX4000. It is seen that for G/Do0.6, St decreases with a decrease in G/D to a value of 0.19 for G/D ¼ 0.2 at both Reynolds number conditions. As mentioned earlier, St may roughly be calculated using the approximate vortex-shedding frequency, f, from the instantaneous streamline patterns. When the value f ¼ 0.77 Hz for G/D ¼ 0.3 at ReD ¼ 9500, found from Fig. 10, is used in Eq. (2), St may be estimated as 0.195 that confirms the value St ¼ 0.20 obtained from the PSD distribution.
4. Conclusions Using the PIV technique, experiments were conducted to measure the velocity fields for steady, two-dimensional, turbulent flows around a horizontal smooth circular cylinder near a plane wall. Using seven different gap ratios, G/D ¼ 0.0, 0.1, 0.2, 0.3, 0.6, 1.0 and 2.0, experiments were repeated for the flow conditions at ReD ¼ 840, 4150 and 9500. From the measured velocity fields, the effects of wall proximity on various flow properties were investigated.
Results show that upstream wall separation occurs for G/Dp0.6, 0.3 and 0.1 for ReD ¼ 840, 4150 and 9500, respectively; and downstream wall separation exists for G/Dp0.2 and 0.1 at ReD ¼ 840 and 4150, respectively. Wall separation due to the vortex pair downstream of the cylinder appears for conditions between G/D ¼ 0.3 and 0.6 for ReD ¼ 840, and between G/D ¼ 0.2 and 0.6 for ReD ¼ 4150 and 9500. The stagnation-point angles for G/D ¼ 0.0 are ys ¼ 101, 12.31, and 19.51 for ReD ¼ 840, 4150 and 9500, respectively; and the stagnation-point angle for G/D ¼ 1.0 is ys ¼ 1.61 at all three Reynolds numbers. For ReD ¼ 4150 and 9500, ys changes sharply between G/D ¼ 0.0 and 0.3, but for G/DX0.3, the variation of ys becomes very slow: it only changes from about 2.81 to 0.91 between G/D ¼ 0.3 and 2.0. The power spectral density distributions of the instantaneous wake velocities indicate that the vortex shedding behind the cylinder is suppressed for G/D ¼ 0.1 and that regular vortex shedding occurs for G/DX0.2 at all three Reynolds numbers. When the gap ratio G/DX0.6, St remains constant at about 0.24 and 0.21 at ReD ¼ 4150 and 9500, respectively; and for G/Do0.6, St decreases with decreasing gap ratio to a minimum value of 0.19 for G/D ¼ 0.2. Finally, the measured time-averaged velocity distributions, streamline patterns and the isovorticity contours show that the effect of wall proximity on the flow field around the cylinder becomes insignificant when the gap ratio G/DX1.0.
Acknowledgments The work was partly supported by the Cukurova University Research Fund under project no: MMF2004D4. This support is gratefully acknowledged. Suggestions of Eur Ing Terry S. Hedges at the University of Liverpool are also acknowledged. References Akilli, H., Sahin, B., Tumen, N.F., 2005. Suppression of vortex shedding of circular cylinder in shallow water by a splitter plate. Flow Measurement and Instrumentation 16, 211–219. Batchelor, G.K., 1974. An Introduction to Fluid Dynamics. Cambridge University Press. Bearman, P.W., Zdravkovich, M.M., 1978. Flow around a circular cylinder near a plane boundary. Journal of Fluid Mechanics 89, 33–47.
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