Circulation characteristics of horseshoe vortex in scour region around circular piers

Subhasish DAS*, Rajib DAS, Asis MAZUMDAR

School of Water Resources Engineering, Jadavpur University, Kolkata 700032, India

Circulation characteristics of horseshoe vortex in scour region around circular piers

Subhasish DAS*, Rajib DAS, Asis MAZUMDAR

School of Water Resources Engineering, Jadavpur University, Kolkata 700032, India

This paper presents an experimental investigation of the circulation of the horseshoe vortex system within the equilibrium scour hole at a circular pier, with the data measured by an acoustic Doppler velocimeter (ADV). Velocity vector plots and vorticity contours of the flow field on the upstream plane of symmetry (y = 0 cm) and on the planes ±3 cm away from the plane of symmetry (y = ±3 cm) are presented. The vorticity and circulation of the horseshoe vortices were determined using the forward difference technique and Stokes theorem, respectively. The results show that the magnitudes of circulations are similar on the planes y = 3 cm and y = –3 cm, which are less than those on the plane y = 0 cm. The circulation decreases with the increase of flow shallowness, and increases with the densimetric Froude number. It also increases with the pier Reynolds number at a constant densimetric Froude number, or at a constant flow shallowness. The relative vortex strength (dimensionless circulation) decreases with the increase of the pier Reynolds number. Some empirical equations are proposed based on the results. The predicted circulation values with these equations match the measured data, which indicates that these equations can be used to estimate the circulation in future studies.

experimental investigation; open channel turbulent flow; scour; horseshoe vortex; circulation; circular pier; forward difference technique; Stokes theorem

1 Introduction

Vorticity, or circulation per unit area, reflects the tendency for fluid elements to spin. It is important to know the magnitude of circulation as it implies the strength of the vortex. Circulation or vortex strength (Γ) will increase if the Reynolds number increases and if the viscous effect is negligible. The vortex strength is related to the occurrence of scour around the pier. For this reason it is essential to study the vortex strength around the pier and, moreover, a thorough study of the flow field around the pier is very important for gaining a better understanding of occurrence of scour. Numerous studies have been carried out with the purpose of predicting the scour depth, and various equations have been developed by many researchers, including Laursen and Toch (1956), Liu et al. (1961), Shen et al. (1969), Breusers et al. (1977), Jain and Fischer (1979), Froehlich (1989), Melville (1992), Abed and Gasser (1993), Richardson and Richardson (1994), Barbhuiya and Dey (2004), and Khwairakpam et al. (2012).

Raudkivi and Ettema (1983) derived an equation for estimating the maximum depth of local scour at circular piers based on laboratory experiments for cohesionless bed sediment. They concluded that the equilibrium depth of local scour (dse) decreased as the geometric standard deviation of sediment (σg) increased (an exception occurs when σg＜1.5). A similar phenomenon, that the scour depth decreased with the increase of σg(1.17＜σg＜2.77), was also observed by Pagliara (2007). In the case of a non-uniform material, i.e., σg＞1.3, the scour is less compared with that of a uniform material with the same median particle size (d50). In another work, Pagliara et. al. (2008) considered sediment material with σgup to 1.3 to be uniform. The pier diameter (b) relative to the median particle size (d50) is known as the sediment coarseness (b d50). The equilibrium scour depth decreases with the decreasing sediment coarseness for values less than about 20. It also decreases at a greater rate with the decreasing flow depth for smaller values of the flow shallowness or relative inflow depth (h b, where h is the approaching flow depth), which is one of the main parameters influencing the local scour.

However, these studies mainly focused on the estimation of the maximum scour depth at piers and abutments. Therefore, it is very important to study the horseshoe vortex to gain a clear understanding of scour around circular piers. For a better understanding of horseshoe vortex characteristics, some researchers have focused on the flow field around circular piers.

Melville (1975) was the pioneer who measured the turbulent flow field within a scour hole at a circular pier using a hot-film anemometer. He measured the flow field along the upstream axis of symmetry and the near-bed turbulence intensity for the case of a flat bed, intermediate scour, and an equilibrium scour hole. Dey et al. (1995) investigated the vortex flow field in clear-water quasi-equilibrium scour holes around circular piers. They measured velocity vectors on the planes with azimuthal angles of 0°, 15°, 30°, 45°, 60°, and 75° with a five-hole Pitot probe. They also presented the variation of circulation with the pier Reynolds number, Rp(equal to Ubν, where U is the depth-averaged approaching flow velocity and ν is the kinematic viscosity), on a 0° plane for all eighteen tests. The study showed satisfactory agreement with the observations of Melville (1975). Ahmed and Rajaratnam (1998) attempted to describe the velocity distributions along the upstream axis of symmetry within a scour hole at a circular pier using a Clauser-type defect method. Melville and Coleman (2000) explained that the strength of the horseshoe vortex depended on Rpand h b. Thus, the circulation of the horseshoe vortex was also a function of Rpand h b. Graf and Istiarto (2002) experimentally investigated the three-dimensional flow field in an equilibrium scour hole. They used an acoustic Doppler velocity profiler (ADVP) to measure the three components of the velocities on the vertical symmetry (stagnation) plane of the flow before and after the circular pier. They also calculated the turbulence intensities, Reynolds stresses, bed-shear stresses, and vorticities of the flow field on different azimuthal planes within the equilibrium scour hole. Results of the study showed that a vortex system was established in front of the circular pier and a trailing wake-vortex system of strong turbulence was formed at the rear of thecircular pier.

However, until now observations by these researchers on the variations of circulations of the horseshoe vortices at circular piers with respect to the flow shallowness and pier Reynolds number are particularly scanty for 10 000≤Rp≤35 000. Based on that, an initiative has been taken in this study to measure the turbulent flow field at circular piers of different sizes within a clear-water equilibrium scour hole. The time-averaged velocity vectors and vorticity contours are presented on the plane y = 0 cm (that is, on the upstream plane of symmetry) and the planes y = ±3 cm (3 cm away from the plane of symmetry). On the vertical planes, the planes y = ±3 cm were chosen to observe the nature of circulation for two cases: case 1 in which these planes were not obstructed by the pier with a diameter of b = 5 cm, and case 2 in which these planes were obstructed by the pier with a diameter of b = 7.5 cm or 10 cm. The obtained comprehensive data set demonstrates some important relations between the flow shallowness, circulation, densimetric Froude number, and pier Reynolds number. In addition, some comparative studies were carried out in non-dimensional forms.

2 Experimental setup

In this study, experimental investigation of the scour depth and velocity around a circular pier was carried out with an ADV. The experimental setup and conditions are shown in Fig. 1. All the experiments were conducted in a re-circulating tilting flume with a length of 11 m, a width of 0.81 m, and a depth of 0.60 m in the Fluvial Hydraulics Laboratory of the School of Water Resources Engineering at Jadavpur University in Kolkata, India. The working section ofthe flume was filled with sand to a uniform thickness of 0.20 m, the length of the sand bed being 3 m, and the width being 0.81 m. The sand bed was located 2.9 m upstream from the flume inlet. The re-circulating flow system was served by a 10 hp variable-speed centrifugal pump located at the upstream end of the tilting flume. The pump had a rotational speed of 1 430 r/min, a power capacity of 7.5 kW, and a maximum discharge of 25.5 L/s. The water discharge was measured with a flow meter connected to the upstream pipe at the inlet of the flume. Water ran directly into the flume through a 0.2 m-diameter pipe line. A vernier point gauge with an accuracy of 0.1 mm, fixed with a movable trolley, was placed on the flume to measure the water level, initial bed level, and scour depth. A Cartesian coordinate system (Fig. 1) for all the experiments is used to represent the turbulence flow fields where the time-averaged velocity components in thex,y, andzdirections are represented byu,v, andw, respectively. In Fig. 1,i,j, andkdenote the direction indices in thex,y, andzdirections, respectively, andx?is the dynamic angle of response. The ADV readings were taken along several vertical planes (y= 0, andy= ±3 cm), with the lowest longitudinal, transverse, and vertical resolution, i.e. Δx, Δy, and Δzbeing 1.5 cm, 3 cm, and 2 mm, respectively. Fig. 2 shows the horizontal planes for ADV measurements for different pier diameters: 5 cm, 7.5 cm, and 10 cm.

Fig. 1Schematic diagram of grid points for ADV measurements

Fig. 2Horizontal planes for ADV measurements for different pier diameters (Unit: cm)

Table 1 Experimental conditions for all tests

3 Methodology

The critical condition for bed material movement was checked before each test using the following steps:

(1) The depth-averaged approaching flow velocity (U) was calculated using Manning’s equation and Strickler’s formula. Considering steady uniform flow in a rectangular flume, thebed shear stress (τ0) can be expressed as τ0=ρfgRsin α, where ρfis the mass density of fluid, g is the gravitational acceleration, R is the hydraulic radius, and α is the angle between the longitudinal sloping bed and the horizontal direction.

(2) The critical bed shear stress (τ0c) was determined using the expression τ0c=ΘcΔρfgd50, where Δ=s?1, and Θcis the critical Shields parameter and is calculated using the following van Rijn’s empirical equations for the Shields curve (van Rijn 1984):

When a negligible difference (1 mm or less) of scour depth was observed at an interval of 2 hours after the experiment lasted for 60 hours, it was assumed that an equilibrium stage of the scour hole had been attained. The total duration of each experiment of 67 hours was adequate for achieving the equilibrium scour (Dey and Raikar 2007). After the run was stopped, the maximum equilibrium scour depth, observed at the upstream base of the pier, was carefully measured by the vernier point gauge. After carefully draining out the water from the scour hole, when the bed was reasonably dry, a synthetic resin mixed with water (1:3 by volume) was sprayed uniformly over the scoured bed to stabilize and freeze it. The sand bed was sufficiently filled with the resin when it was left to set for a period of 48 hours. Having dried further for up to 72 hour, the scoured bed profile became rock-hard, facilitating the ADV measurements.

A three-beam 5-cm down-looking ADV (16 MHz MicroADV Lab Model), manufactured by Sontek, was used to measure the instantaneous three-dimensional velocity components. Asampling rate of 50 Hz and cylindrical sampling volume of 0.09 cm3, having a 2 to 5 mm sampling height (Δz), were set for measurements. Sampling heights of 5 mm and 2 mm were used for measurement of the velocity components above and within the interfacial sub-layer, respectively. Sampling durations varied from 120 to 300 seconds to achieve a statistically time-independent average velocity. The sampling durations were relatively long near the bed. It is impossible to measure the flow field with the ADV probe within the range from 0 to 4.5 mm above the sand bed, because the ADV needs a measuring volume of 0.09 cm3. The output data from the ADV were filtered using the software WinADV32 version 2.027, developed by Wahl (2003). It is important to point out that the ADV sensor had an outer radius of 2.5 cm, and three receiving transducers mounted on short arms around the transmitting transducer at 120° azimuth intervals, which made it possible to measure the flow as close as 2 cm from the pier boundary.

4 Results and discussion

The literature review revealed that, for the ripple-forming sediment having d50＜0.7 mm , it is rarely possible to maintain a plane bed (Breusers and Raudkivi 1991). Ripple formation was also described by Raudkivi and Ettema (1983) for non-cohesive alluvial sediments with the particle size of 0.05 to 0.7 mm, which form distinctive small ripples when bed shear stresses are slightly greater than the threshold value. For flow with uniform ripple-forming sediments, the scour depth is less than that with non-ripple-forming sediments. The reason is that it is impossible to maintain a flat sand bed under the near-threshold conditions. Thus, ripples develop, and a small amount of sediment transport takes place, replenishing some of the sand scoured at the pier. The sand used in this study had a median grain size of 0.825 mm. Thus, the true clear-water scour conditions could be maintained experimentally in this study.

Fig. 3 shows the experimental results plotted on the Shields diagram (Shields 1936). The threshold of sediment motion occurs when Θ＞Θc, or τ0＞τ0c, or u*＞u*c, where Θ is the Shields parameter and u*is the shear velocity. The flow is hydraulically laminar or turbulent when the particle Reynolds number is less than 2 or more than 500, respectively. The region below the solid line in Fig. 3 indicates that no sediment motion occurs in these experimental conditions. Fig. 3 shows that the discharge during each test was lower than the minimum discharge required for the threshold conditions of the bed particles. Therefore, it can be said that all the experiments were carried out under the clear-water scour conditions.

The time-averaged bed shear stress for the flat bed without a pier was also estimated using the distribution of Reynolds stresses, as was done by Dey and Barbhuiya (2005):

where τxand τyare the bed shear stresses in the x and y directions, respectively, and u′, v′, and w′ are the fluctuations of u, v, and w, respectively.

Fig. 3Experimental data plotted on Shields diagram

The maximum value ofτ0was obtained for Test 15 using Eq. (2), and was equal to 0.348 5 N/m2. The time-averaged critical bed shear stressτ0con the sloping bed was also estimated using the method proposed by Dey (2003a, 2003b), and was equal to 0.401 8 N/m2. This also indicates that a clear-water condition occurred during all the tests.

As an example, the contour lines of the equilibrium scour holes at the circular piers, plotted with the Golden software Surfer 8, for Test 11 and Test 12 are shown in Figs. 4(a) and 4(b), respectively. Here both scours are slightly asymmetric due to some local effects. Table 2 shows the equilibrium scour depthdse, equilibrium scour lengthlse, and equilibrium scour widthwsefor all tests.

Fig. 4Contours of equilibrium scour holes around piers (Unit: cm)

Table 2Equilibrium scour depths, lengths, and widths for all tests cm

Fig. 5Scour-affected zones around piers

Fig. 6Velocity vectors onxzplanes for equilibrium scour hole for Test 11

Figs. 8 and 9 show the vorticity contours at equilibrium scour holes on thexzplanes (y= –3 cm,y= 0 cm, andy= 3 cm) for Test 11 and Test 12, respectively. Theycomponent of vorticity,ω, which is equal to?u?z??w?x, was computed for each test by converting the partial differential equation into a finite difference equation with the help of the forward difference technique of computational hydrodynamics. Theycomponent of vorticity at the grid point (i,j,k) can be expressed as

Fig. 7Velocity vectors onxzplanes for equilibrium scour hole for Test 12

Fig. 8Vorticity contours at equilibrium scour hole onxzplanes for Test 11 (Unit: s-1)

The circulation value (Γ) of the vortex was estimated from the vorticity contours for different Cartesian planes using the following equation:

whereVis the velocity vector,sis the displacement vector along a closed curvec, andAis the enclosed area.

The detailed methodology for computingΓwas also described by Dey and Raikar (2007). The anticlockwise direction, by convention, was considered positive for circulation. From Table 3,we can see that forh= 8 cm on the planey= 3 cm, the magnitudes ofΓfor the piers with diameters of 7.5 cm and 10 cm are approximately 1.5 to 2.5 and 2.7 to 5.1 times those for the pier with a diameter of 5 cm, respectively. Similarly, on the planey= –3 cm, these values are 2.3 to 3.1 and 4.0 to 4.8 times, respectively, whereas, on the planey= 0 cm, these values are only 1.3 to 1.9 and 1.6 to 2.2 times, respectively. Therefore, it is clear that the circulationsincrease rapidly on the planesy= 3 cm andy= –3 cm compared with those on the planey= 0 cm if the pier diameter increases from 5 to 10 cm. It is noticeable from these observations that the circulations increase rapidly on the planes (such as the planesy= 3 cm andy= –3 cm for Tests 4 through 9) which are obstructed by the pier with a diameter of 7.5 or 10 cm, compared with those on the planes (such as the planesy= 3 cm andy= –3 cm for Tests 1 through 3) which are not obstructed by the pier with a diameter of 5 cm.

Fig. 9Vorticity contours at equilibrium scour hole onxzplanes for Test 12 (Unit: s-1)

Table 3Magnitudes of circulation for all tests

In Fig. 10, the circulation is plotted against the pier Reynolds number at different pier diameters on the planesy= 0 cm,y= 3 cm, andy= –3 cm. The differences between the values measured in the present study at different pier diameters (b= 5 cm, 7.5 cm, 10 cm, and 11 cm) and the values obtained by Dey et al. (1995) on the planey= 0 cm are shown in Fig. 10. The increasing trend of the circulation of the present study on the planey= 0 cm corresponds closely with the results of Dey et al. (1995). It is observed that the circulation increases with the pier Reynolds number on the planesy= 3 cm andy= –3 cm. Exponential trendlines of circulations on the planesy= 0 cm,y= 3 cm, andy= –3 cm are also shown in Fig. 10 with solid lines, and can be respectively expressed as

The correlation coefficients (r) between Eq. (5), Eq. (6), and Eq. (7) and their corresponding observations are 0.954, 0.975, and 0.970, respectively, which also imply an almost perfect positive correlation.

Fig. 11 shows a comparison of observed and predicted values of circulation on the planey= 0 cm. The predicted values of circulation were calculated with Eq. (5). It is clear from Fig. 11 that the predicted data match the measured data with a deviation ranging from –25% to 25%.

Fig. 10Variation ofΓwithRpon planesy= 0 cm,y= 3 cm, andy= –3 cm

Fig. 11Comparison of observed and predicted values ofΓon planey= 0 cm

The most important observation is that 72% of the measured data of Dey et al. (1995) are also within this range when 10 000≤Rp≤35 000. The only remaining five numbers (28%) ofmeasured data are lying just below the line with a deviation of –25%. This may occur because of a change in the median particle size of sand (d50). In the present study,d50was considered to be 0.825 mm, whereas it was considered only 0.26 and 0.58 mm by Dey et al. (1995). It is well known that circulation increases as the scour increases and an increase ofd50implies an increase of scour. Eq. (5) also corresponds closely with the result of Melville (1975).

Figs. 10 and 11 show that the results of the present study on they= 0 cm agree with the observations of Dey et al. (1995) and Melville (1975). Based on that similarity, an attempt was also made to introduce empirical Eqs. (6) and (7) for prediction of circulation when 10 000≤Rp≤35 000 on the planesy= 3 cm andy= –3 cm, respectively, as shown in Fig. 10. The magnitudes ofΓon the planesy= 3 cm andy= –3 cm were always found to be lower than those on the planey= 0 cm for 10 000≤Rp≤35 000. This may be due to a decrease of the scour area on the planesy= ±3 cm, compared with the scour area on the planey= 0 cm.

Fig. 12 shows the variation of circulation with the pier Reynolds number on the planesy= 3 cm andy= –3 cm. The magnitudes of circulation should be similar on the planesy= 3 cm andy= –3 cm, as the two planes are symmetric. Therefore, based on the experimental data on the planesy= 3 cm andy= –3 cm, a single exponential trendline, as shown in Fig. 12, was introduced and expressed as

Here, the correlation coefficientsrbetween Eq. (8) and the observations is 0.959, which implies an almost perfect positive correlation. The predicted values of circulation were calculated from Eq. (8). Fig. 13 shows a comparison of observed and predicted values of circulation for the data considered on the planesy= ±3 cm for 10 000≤Rp≤35 000. The ±25% deviation intervals are added as dashed lines. It can be seen from Fig. 13 that the deviation between the predicted and measured data is mostly in a range of –25% to 25%.

Fig. 12Variation ofΓwithRpon planesy= ±3 cm

Fig. 13Comparison of observed and predicted values ofΓon planesy= ±3 cm

The non-dimensional circulations (Γn) on the planesy= 0 cm,y= 3 cm, andy= –3 cm are plotted against the flow shallowness (h/b) with different pier Reynolds numbers in Fig. 14.Fig. 14(a) shows thatΓnranges between 0.3 and 0.5. Figs. 14(b) and 14(c) indicate that the magnitudes ofΓnon the planesy= 3 cm andy= –3 cm are 0.6 to 0.8 times that on the planey= 0.

Fig. 14Variation ofΓnwithh/bwith different values ofRpon planesy= 0 cm,y= 3 cm, andy= –3 cm

The circulations on the planesy= 0 cm,y= 3 cm, andy= –3 cm for 10 000≤Rp≤35 000 are plotted against pier Reynolds numbers with different values of flow shallowness or non-dimensional inflow depth (h/b) in Figs. 15(a), 15(b), and 15(c), respectively. All the figures clearly indicate that the circulation increases if the pier Reynolds number increases at a constant non-dimensional inflow depth. It is observed that both the circulation and pier Reynolds number decrease with the increase of the non-dimensional inflow depth. This implies that, at a constant approaching flow depth, if the pier diameter increases, then the circulation and pier Reynolds number will both increase, and vice versa.

Fig. 15Variation ofΓwithRpat different values ofh/bon planesy= 0 cm,y= 3 cm, andy= –3 cm

Fig. 16Variation ofΓwithRpat different values ofFdon planesy= 0 cm,y= 3 cm, andy= –3 cm

The circulations of the horseshoe vortex inside the scour hole are shown in Fig. 17. The trendline shown in Fig. 17 was proposed by Muzzammil and Gangadhariah (2003). Fig. 17 shows that the results of the present study agree with those of Melville and Raudkivi (1977) and Muzzammil and Gangadhariah (2003). The circulation or vortex strength in dimensionless form is plotted against the pier Reynolds number in log-log scale in Fig. 18. The variation ofΓnwithRpwas also compared with the results obtained by Dey et al. (1995), Baker (1979), Qadar (1981), Devenport and Simpson (1990), Eckerle and Awad (1991), Srivastava (1982), Muzzammil and Gangadhariah (2003), and Unger and Hager (2005), which shows a good agreement with the results of these researchers. However, as shown in Fig. 18, the trendline of the present study is more similar to the results of Dey et al. (1995), and Muzzammil and Gangadhariah (2003). Here, the value ofris 0.825, which implies a good positive correlation. An overall trend of the data considered herein indicates thatΓndecreases with the increase ofRp. Fig. 18 reveals that the dimensionless circulation or relative vortex strength depends on the pier Reynolds number and is almost inversely proportional to the pier Reynolds number for 10 000≤Rp≤35 000. Therefore, an increase in the pier Reynolds number causes a decrease of relative vortex strength or dimensionless circulation.

Fig. 17Characteristics of horseshoe vortex inside scour hole

Fig. 18Variation ofΓnwithRpon planey= 0 cm

5 Conclusions

Clear-water scour tests were performed on a single circular pier with varying inflow depths, pier Reynolds numbers, and densimetric Froude numbers. All sixteen experiments satisfy the clear-water scour conditions. The turbulent flow field was measured with an ADV. The time-averaged velocity vectors and vorticity contours on different Cartesian planes were presented. The vorticity was calculated using the forward difference technique of computational hydrodynamics. The strength of the horseshoe vortex, i.e., the circulation, was computed using the Stokes theorem. Some empirical equations are proposed based on the results. The main conclusions drawn from the present study for 10 000 ≤ Rp≤ 35 000 are summarized below:

(1) The flow is almost horizontal above the scour hole (z ＞ 0), but it is downward close to the pier. The velocity is reversed within the scour hole in the vertical direction, forming a horseshoe vortex.

(2) The predicted circulation values with the proposed empirical equations match the measured data, which indicates that these equations can be used to estimate the circulation in future studies.

(3) Magnitudes of circulations are similar on the planes y = 3 cm and y = –3 cm, which are less than those on the plane y = 0 cm.

(4) The circulation decreases with the increase of flow shallowness, and increases with the densimetric Froude number. It also increases with the pier Reynolds number at a constant densimetric Froude number, or at a constant flow shallowness. The relative vortex strength (dimensionless circulation) decreases with the increase of the pier Reynolds number.

Acknowledgements

The helpful suggestions from Professor (Dr.) Subhasish Dey, Brahmaputra Chair Professor for Water Resources of the Department of Civil Engineering, at the Indian Institute of Technology in Kharagpur, India are gratefully acknowledged. The authors also appreciate the help provided by Mr. Ranajit Midya and Mr. Ranadeep Ghosh, M.E. students of the School of Water Resources and Engineering, at Jadavpur University in Kolkata, India, during the investigation.

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(Edited by Yan LEI)

*Corresponding author (e-mail: subhasishju@gmail.com)

Received May 30, 2012; accepted Oct. 25, 2012