Experimental and numerical investigation of local scour around a submerged vertical circular cylinder in steady currents

Coastal Engineering 57 (2010) 709–721

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Coastal Engineering j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c o a s t a l e n g

Experimental and numerical investigation of local scour around a submerged vertical circular cylinder in steady currents Ming Zhao ⁎, Liang Cheng, Zhipeng Zang School of Civil and Resource Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

a r t i c l e

i n f o

Article history: Received 5 May 2009 Received in revised form 24 February 2010 Accepted 5 March 2010 Available online 5 April 2010 Keywords: Local scour Current Circular cylinder Horseshoe vortex Numerical model

a b s t r a c t Local scour around a submerged vertical circular cylinder in steady currents was studied both experimentally and numerically. The physical experiments were conducted for two different cylinder diameters with a range of cylinder height-to-diameter ratios. Transient scour depth at the stagnation point (upstream edge) of the cylinder was measured using the so-called conductivity scour probes. Three-dimensional (3D) seabed topography around each model cylinder was measured using a laser profiler. The effect of the height-todiameter ratio on the scour depth was investigated. The experimental results show that the scour depth at the stagnation point is independent on cylinder height-to-diameter ratio when the later is smaller than 2. The increase rate of equilibrium scour depth with cylinder height increases with an increase in Shields parameter. A three-dimensional finite element numerical model is developed for simulating local scour around submerged vertical cylinders. Steady flow around the cylinder is simulated by solving the ReynoldsAveraged Navier–Stokes (RANS) equations with a k–ω turbulence closure. Both suspended load and bed load sediment transport rates were included in the model. Efforts are made to reduce the computational time associated with three-dimensional morphological modelling including the use of wall-function to avoid resolving the near-wall boundary layers, large morphological time step and parallel computing techniques. Scour around a submerged vertical circular cylinder founded on the seabed is simulated and the numerical results are validated against the test data. The scour mechanisms around the submerged cylinder are investigated using the numerical model. © 2010 Elsevier B.V. All rights reserved.

1. Introduction When a structure is placed on an erodible bed, it causes an increase in local sediment transport capacity and this consequently leads to scour in the vicinity of the structure. This is referred to as local scour in literature. Local scour has been identified as one of the key factors that cause failures of structures in bridge engineering, coastal and offshore engineering. Numerous studies on local scour around vertical piles in steady currents have been reported in the past. Most of these studies deal with laboratory model studies. Reviews of the subject can be found in Breusers et al. (1977), Breusers and Raudkivi (1991), Richardson and Davies (1995), Dey (1997a,b), Hoffmans and Verheij (1997), Raudkivi (1998), Whitehouse (1998), Melville and Coleman (2000), Sumer and Fredsøe (2002) and Melville (2008). It has been understood that the horseshoe vortex in front of the cylinder, the vortex shedding flow in the wake of the cylinder and the streamline contraction at the two sides of the cylinder are the main flow mechanisms that are responsible for scour in steady currents (Sumer

⁎ Corresponding author. Tel.: +61 8 6488 7356; fax: +61 8 6488 1018. E-mail address: [email protected] (M. Zhao). 0378-3839/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2010.03.002

and Fredsøe, 2002). Dey and Raikar (2006) reported strong turbulence around scour holes in the vicinity of cylindrical piers through experimental measurement. Despite several decades of research on bridge pier scour, the scale effect inherent in the small scale laboratory tests is still one of the vexing problems (Ettema et al., 1998; Lee and Sturm, 2008). Studies showed that laboratory-based formulae appear to overestimate the scour depths obtained from field measurements (Lee and Sturm, 2008). Sheppard et al. (2004) investigated clear water local pier scour at large model scales. In their tests, they found that the concentration of suspended fine sediment (wash load) has significant effect on scour depths. Ettema et al. (2006) found that similitude of large-scale turbulence is an important consideration in model design as it influences the equilibrium depth of local scour around cylinders. A direct correlation between equilibrium scour depth (normalized with cylinder diameter) and the intensity and frequency of large-scale turbulence shed from each cylinder was identified. It was found that values of scour depth normalized by cylinder diameter increases as the cylinder diameter decreases. Numerical studies about scour around a vertical circular cylinder are not as well documented as experimental investigations, mostly due to the limitation of available computer capacities to the scour

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research community. The advantage of numerical method is that it is free from scaling effect as it can be run in the prototype size. However, numerical models do need to use empirical formulae for sediments transport rates which were mostly derived from laboratory tests. The earliest three-dimensional (3D) numerical study on local scour was reported by Olsen and Melaaen (1993), where a steady state Navier– Stokes solver coupled with a sediment transport algorithm was employed to simulate the growth of a scour hole at the base of a circular pier. Since the transient terms were neglected in the numerical model, the validity of numerical model in simulating local scour was questioned. Olsen and Melaaen (1993) stated in their paper that “This study does not verify that the numerical model will be able to handle all cases of scour around obstacles. Nor is it verified that the model will calculate maximum depth in the scour hole correctly.” However, their study shows that three-dimensional numerical models may be able to calculate the scour around an obstacle in a general complex geometry. Olsen and Kjellesvig (1998) extended the work reported in Olsen and Melaaen (1993), by using unsteady Navier–Stokes equations and simulating the entire scour process. Richardson and Panchang (1998) used a fully three-dimensional hydrodynamic model to simulate the flow at the base of a bridge pier within a scour hole. They compared their numerical results of velocity and fluid particle trajectories with the test results by Melville and Raudkivi (1977) and obtained a good agreement. Roulund et al. (2005) simulated local scour around a vertical pile numerically. In their numerical model, RANS equations with a k–ω turbulence closure were solved to simulate the turbulent flow and a bed load sediment transport was considered in the scour simulation. Their numerical results of scour development agreed well with their experimental data. They found that the size of the horseshoe vortex and the bed shear stress were affected by the boundary-layer-thickness-to-pilediameter ratio (δ/D) and the Reynolds number (Re). They also found that the influence of the bed roughness on the horseshoe vortex was not very extensive. In offshore oil and gas engineering, sub-sea structures such as subsea caissons, gravity anchors, platform foundations etc. are commonly treated as short vertical cylindrical structures when local scour depth is assessed (see Fig. 1). If the cylinder height-to-diameter ratio is smaller than a critical value, the size of the horseshoe vortex in front of the cylinder will decrease with a decrease in cylinder height (Baker, 1985; Sumer and Fredsøe, 2002). Accordingly, a decrease in cylinder height will lead to a reduced scour depth in front of the cylinder. The vortex shedding behind the cylinder will also be weakened if the cylinder height decreases, leading to a reduced scour depth behind the cylinder. Although the mechanism of the scour around a vertical cylinder has been understood, little has been done to quantify the effects of the cylinder height on the scour depth. In this study, local scour around a vertical submerged circular cylinder installed on a sandy bed was investigated through both

Fig. 1. Definition sketch of flow around a submerged vertical cylinder of height of a height of hc and a diameter of D.

laboratory experiments and numerical simulations. The dependence of scour depth on the cylinder height was quantified. The effect of the model scale on the scour was studied by conducting physical experiments with two different cylinder diameters. A three-dimensional finite element numerical model was also established for simulating local scour around submerged cylinders. The ReynoldsAveraged Navier–Stokes equations are coupled with the bed morphological model to simulate the scour process. Both suspended load and bed load sediment transport rates are taken into account in the model. The suspended sediment concentration is determined through a convection–diffusion equation with proper sediment concentration boundary conditions on the bed boundary. The bed load sediment transport rate is calculated using an existing empirical formula. Efforts are made to reduce the computational time associated with threedimensional morphological modelling including the use of wallfunction, large morphological time step and parallel computing techniques. Petrov–Galerkin FEM and the sand slide model are employed to stabilize the numerical scheme. Scour around a submerged wall mounted vertical circular cylinder is simulated and the numerical results are validated against the test data. The scour mechanism around a submerged cylinder is investigated using the present numerical model. 2. Laboratory tests 2.1. Experimental setup Experiments were conducted in a water flume of 4 m wide, 2.5 m deep and 45 m long. The flume is equipped with a pump of 1 m3/s capacity. Fig. 2 shows the overall view of the flume. The water temperature in the flume was around 10 to 15 degrees centigrade. A concrete sand pit of 4 m long, 4 m wide and 0.25 m deep was built in the test section, as shown in Fig. 2(a). The depth of the sand pit was determined based on the consideration that it would not be exceeded

Fig. 2. Test setup (unit: m).

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by the scour depth. Only one type of sands with a median particle size pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of d50 = 0.40 mm and a uniformity parameter of d85 = d15 = 1:42 was used in the experiments. The specific grain density of the sediment (ratio of sediment grain density to the water density) is 2.69 and angle of repose of the sand is ϕs = 35°. The water depth in the test section was maintained at 0.5 m for all the tests. Fig. 2(b) is the plan view of the model arrangement in the sand pit. In each of the test runs, three model cylinders of different heights were installed side by side as shown in Fig. 2(b) in order to reduce the total number of test runs in the testing program. The cylinder-to-cylinder distance (centre to centre) was 1.35 m. Trial tests have been conducted to examine potential interference effects of such an arrangement on local scour. It was found that the interference effects were negligible at this cylinder-to-cylinder distance. In some of the tests, time-dependence of scour depth at the stagnation point of the cylinder were measured using a type of conductivity scour probes, which were previously by Yeow and Cheng (2004) and Rambabu et al. (2003) for measuring the scour depth development. The conductivity probe used in this study is identical to those used by Yeow and Cheng (2004). After each test, the water in the flume was drained out and three-dimensional (3D) seabed topography around the model cylinders was measured using a laser profiler developed at University of Western Australia. The laser profiler is comprised of a 2D actuator and a laser sensor. The 2D actuator is the Parker linear actuators HLE-RB50 Belt drive and covers a plan area of 1.00 m by 0.50 m with a travel accuracy of ±0.2 mm at a speed up to 5 m/s. A SICK DME 3000-111P laser sensor is installed on the profiler. Table 1 shows the parameters used in the tests. The tests were conducted for two model cylinder diameters, namely D = 60 mm and 100 mm. Two incoming flow velocities were employed in the tests. The cylinder height-to-diameter ratio hc/D ranges from 0.25 to 5.0 for D = 100 mm and 0.25 to 8.33 for D = 60 mm. The top of the cylinders were actually above the free surface if the cylinder height is same as the water depth.

2.2. Undisturbed flows Before the scour tests were conducted, the undisturbed velocity profiles above the sandy bed at three locations where the model cylinders were installed later on (as shown in Fig. 2(b)), were measured. The velocity profiles were measured using a SonTek/YSI 16-MHz Micro ADV velocimeter. The velocities were recorded at a rate of 25 samples per second. The mean velocity profiles were obtained by averaging the recorded velocity samples over 1 min. Fig. 3 shows the measured velocity profiles at the locations of the model cylinders. The measured velocity profiles are correlated to the logarithmic law of the velocity profile: uðzÞ =

Test Cylinder DepthShields no. diameter averaged parameter D (m) velocity θ V (m/s)

Cylinder Duration Scour Skin height (h) depth friction hc/D S/D Shields parameter θs

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.044 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.06 0.06 0.06 0.06 0.06 0.06 0.06

0.362 0.362 0.362 0.362 0.362 0.362 0.362 0.362 0.362 0.362 0.362 0.362 0.362 0.362 0.441 0.441 0.441 0.441 0.441 0.441 0.441 0.441 0.441 0.441 0.441 0.441 0.441 0.441

0.148 0.148 0.148 0.148 0.148 0.148 0.148 0.148 0.148 0.148 0.148 0.148 0.148 0.148 0.352 0.352 0.352 0.352 0.352 0.352 0.352 0.352 0.352 0.352 0.352 0.352 0.352 0.352

0.25 0.50 1.00 1.50 2.50 3.50 5.00 0.25 0.50 1.00 1.50 2.50 3.50 8.33 0.25 0.50 1.00 1.50 2.50 3.50 5.00 0.25 0.50 1.00 1.50 2.50 3.50 8.33

4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 2.35 2.35 2.35 2.35 2.35 2.35 2.35 2.35 2.35 2.35 2.35 2.35 2.35 2.35

0.57 0.63 0.95 1.00 1.13 1.18 1.17 0.48 0.69 0.76 0.96 0.98 1.04 1.06 0.58 0.75 0.93 0.97 0.99 1.02 1.03 0.62 0.66 0.90 0.97 0.99 1.01 1.04

  uf z ln 30 κ ks

ð1Þ

where z is the vertical distance from the seabed, uf is the friction velocity, ks is the Nikurase roughness and κ (=0.41) is the von Kármán constant. In some of the tests, sand ripples were observed at the bed surface. The sand ripple height and length was found to increase with the increase in flow velocity. The maximum ripple height was observed to be about 1 cm when depth-averaged velocity V = 36.2 cm/s and 5 cm when velocity V = 44.1 cm/s. The correlation of ripple length with flow velocity was not clear because ripples become very irregular and are not two dimensional when velocity is 44.1 cm/s. In all the tests of different velocities the ripple length was between 10 cm and 20 cm. The origin of the vertical coordinate z was defined at the mean bed level if ripples exist. The uf and ks in Eq. (1) are obtained by the least square method based on the measured velocities. It can be seen that the velocity distributions in the vertical direction follow logarithmic distribution in the whole water depth. Table 2 shows the flow parameters for two different velocities, which were used in the scour tests later on. The Shields parameter θ is the non-dimensional bed shear stress, which is defined by θ=

Table 1 Test parameters.

711

τb ρðs−1Þgd50

ð2Þ

where τb =ρu2f is the bed shear stress in the horizontal direction, ρ is the water density, g is the gravitational acceleration, s=ρs / ρ is the specific gravity of the sediment grain, ρs is the density of the sediment. τb in Eq. (2) is the total shear stress which includes the effects of skin friction and the shear stress due to the bed forms such as sand ripples. For sediment transport purpose, the skin friction is responsible for bed load transport and entrainment of sand from the bed (Soulsby, 1997). The bed shear stress and the Shields parameter are calculated from the depthaveraged velocity based on logarithmic velocity distribution. According to logarithmic profile of velocity, the bed shear stress due to skin friction is (Soulsby, 1997) τbs = ρCDsU ̅2, where CDs = {κ / [ln(h / z0s) − 1]}2, z0s =kss /30 and skin Nikuradse roughness kss is related to the sediment grain size by kss =2.5d50. Due to the existence of sand ripples, the roughness determined from velocity measurements are much larger than the skin friction roughness. This leads to a total Shields parameter larger than the skin friction Shields parameter. The sediment transport occurs if the skin friction Shields parameter exceeds the critical Shields parameter. One of the widely used empirical formulae for calculating the critical Shields parameter is (Soulsby, 1997) θcr =

0:30 + 0:055½1− expð−0:020D* Þ 1 + 1:2D*

ð3Þ

where D⁎ = [g(s − 1) / ν2]1/3d50 is the non-dimensional grain size, ν is the kinematic viscosity of the fluid. Eq. (3) was derived by curve fitting the Shields diagram for the critical Shields parameter (see Soulsby, 1997, p. 105 for the Shields diagram). The critical Shields parameter for sand size d50 = 0.40 mm is estimated to be 0.033 based on Eq. (3).

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Fig. 3. Undisturbed mean velocity profile in the vertical direction (y-locations of model cylinders are marked in Fig. 2).

In the experiments, sediments were observed to be mobile at V = 36.2 cm/s, where V is the depth average velocity. This suggests that tests were conducted under live-bed conditions. The skin friction Shields parameter θs for V = 36.2 cm/s is 0.044. Although it is larger than the critical Shields parameter, the undisturbed sediment transport rate corresponding to V = 36.2 cm/s was observed to be rather small. However, at the front edge of the cylinder, large quantity of the sediment can be clearly be seen being stirred up by the horseshoe vortex and drift downstream at the cylinder sides. It is believed that sediment suspension is one of main courses of scour. When the velocity was increased to V = 44.1 cm/s, it was observed that large quantity of sediment was transported along the bed. Water in the near-bed region became murky due to the suspension of sediment shortly after test was started. At the very early stage the sediment suspension below horseshoe vortex can still be seen and is much stronger than that in the undisturbed area. 2.3. Time-development of the scour depth Fig. 4 shows the time developments of the scour depth (S) at the upstream stagnation point of the cylinder (D = 0.1) for different cylinder heights. It can be seen that the scour depths reached equilibrium almost for all tests. It was deemed that the scour reached equilibrium when the mean bed level change rate (averaged over half hour) at the front edge of the cylinder is less than 0.03D per hour. From Fig. 4 it was found that the scour depth for Tests 2, 3 and 5, in which the Shields parameter is small, reaches equilibrium after about 4.5 h of scour. After 4.5 h the scour depth changes little with time. For Tests 16 and 19, in which the Shields parameter is large, considerable

oscillations of scour depth with time were observed. It is believed that such oscillations were caused by the migration of the sand ripples in the flume. However, the time-averaged mean developing rates of S/D for these two cases were close zero after about 2 h of scour. The scour rates at the early stage of scour increased with an increase in Shields parameter. For a constant current velocity, the scour rate decreases with the reduction of cylinder height. The durations of scour tests for tests with identical flow velocities were set the same. To ensure equilibrium scour depths were reached in the tests, the test durations were 4.5 h and 3 h for V = 0.362 m/s and V = 0.441 m/s, respectively. 2.4. Scour bed profiles Fig. 5 shows the photos of the scoured bed patterns taken after the scour test for D = 0.1 m. In Fig. 5(a), where the Shields parameter was small, the bed surface upstream of the cylinder was almost flat and the scour pit is clearly recognisable. A ridge-shape sand dune, which aligns in the flow direction, was formed in the wake of the cylinder. The peak of the sand ridge was formed at a location where the mean flow reverses. A few sand ripples were found upstream of the cylinder in Fig. 5(a). The heights of these ripples were observed to be smaller than 0.5 cm. The height of the sand ripples was observed to be about 1–3 cm in Fig. 5(b) where the Shields parameter was larger than that in Fig. 5(a). In high Shields parameter experiments, sediments were observed to build up behind the cylinder only at the early stage of

Table 2 Measured undisturbed velocity parameters. Test

1–14

15–28

Water depth d (cm) Mean flow velocity V (cm/s) Froude number Fr = V/(gh)1/2 Grain size d50 (mm) Friction velocity uf (cm/s) Shields parameter θ Nikuradse equivalent roughness ks (mm) Nikuradse equivalent sand roughness kss (mm) Skin friction velocity ufs (cm/s) Skin Friction Shields parameter θs

50 36.2 0.150 0.40 3.13 0.148 3.13 0.1 1.7 0.044

50 44.1 0.181 0.40 4.83 0.352 13.2 0.1 2.1 0.066

Fig. 4. Time history of scour depth at the most upstream point of the cylinder surface (D = 0.1 m).

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Fig. 5. Photos of the bed patterns after scour (D = 0.1 m).

scour development. The sand ridge was washed to the downstream as the test progressed. Although there were sand ripples around the scour pit, the bed in the scour pit was always smooth, especially upstream of the cylinder. It was observed that the sand ripples migrated in the flow direction. When a crest of a sand ripple reached the front edge of the scour pit, it collapsed and slid into the bottom of the scour pit. Fig. 5(c) shows the photos for two neighbour cylinders. Two distinct scour pits can be clearly seen in Fig. 5(c) and there is a wide space between them, indicating the distance between two cylinders has little effect on the scour. After each test, the water in the flume was drained out and scour profiles were measured using the laser profiler. Fig. 6 shows the bed

Fig. 6. Contours of bed level (z/D) after 4.5 h of scour (D = 0.1 m, V = 36.2 cm/s, θs = 0.044).

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level contours after the scour tests for V = 36.2 cm/s and D = 0.1 m. Each bed profile is comprised of a sand pit around the circular cylinder and a sand dune in the wake of the cylinder. Sediments that were washed away from the sand pit were deposited directly behind the cylinder. The deeper the sand pit was, the more sediment was deposited behind the cylinder. It was found that the height of the sand dune increases with the cylinder height. It is clearly seen in Fig. 6 that the cylinder is almost located at the centre of the sand pit. The radius and depth of the sand pit increase with an increase in cylinder height. The sizes of bed ripples are found to be too small to be identified at the contour interval of 0.1. If the scour is clear water scour, the deposition behind the cylinder will not be washed away because the flow is weak. The Shields parameter for V = 36.2 cm/s is just slightly larger (30% larger) than the critical value. Sediments are observed to move very slowly in the undisturbed bed. If the sand dune is dynamic, its movement would be extremely slow. No decrease of the sand dune height was observed during the test period for V = 36.2 cm/s. In the tests for V = 44.1, where the critical shear stress was substantially exceeded, the sand deposited behind the cylinder in the early stage was found to be washed to downstream gradually. The bed level contours for V = 44.1 cm/s and D = 0.1 m are shown in Fig. 7. The scour pits around the cylinder were similar to those shown in the earlier figures. When hc/D was 0.25 and 0.50, the sand dune still existed behind the cylinder. In tests with small cylinder heights, it was observed that significant portion of bed boundarylayer flow was diverted to the cylinder top. This weakened the flow around the sides of the cylinder. The weakened side flow reduced the strength of the vortex shedding in the wake of the cylinder, which is responsible for the scour there. As the cylinder height hc/D increases to above 1.0, vortex shedding become strong behind the cylinder, causing increases in seabed shear stress and sediment transport capacity. In Fig. 7(c) and (d), it can be seen that the heights of the sand dune became very small due to the strong vortex shedding. When V = 44.1 cm/s sand ripples can be identified in the bed contours. Fig. 8 shows the bed profiles in the xz plane measured at the end of each test for D = 0.1 m. Sand dunes downstream the cylinder in Fig. 8(a) were clearly seen. The sand dune for hc/D = 0.25 was closest to the cylinder among others. With the increase in cylinder height, the sand dune moves further downstream, mainly because of the increased strength of vortex shedding. In Fig. 8(b), where the Shields parameter is larger, the peak of the sand dune was washed to the downstream in the range of x/D N 5, except when hc/D = 0.25. It is observed in the tests that, at the bottom of the scour pit (close to the cylinder), the horseshoe vortex picked up the sediment particles into suspension continuously. The scour of sediment at the bottom of scour hole increased the slope angle of the scour pit. When the slope angle reached to a certain value, the slope collapsed into the scour pit. It can be seen that, in front of the cylinder, the bed slopes in the sand pit for all the profiles are almost same and are close to the angle of repose of the sediment. Both scour depths in front of and behind the cylinder increase with an increase in model cylinder height. The size of horseshoe vortex was observed to increase with cylinder height due to the increased flow blockage. This in turn produces a larger sediment transport capacity. The bed level in front of the cylinder was always lower than that behind the cylinder for both cases of flow velocity. This difference was more obvious when the cylinder height was small. This can be explained by studying the vortex shedding in the wake of the cylinder. The initiation of the scour around a vertical cylinder is the combination of the horseshoe vortex and the vortex shedding. The scour in front of the cylinder is due to the horseshoe vortex and that behind the cylinder is due to the vortex shedding. When the cylinder height is small the vortex shedding behind the cylinder becomes weak because of the over-topping flow from the top of the cylinder. The weaker of the vortex shedding, the smaller scour depth behind the cylinder is. In the present study, it was observed that no scour happened at the downstream edge of the cylinder when its height was very small (hc/D = 0.25).

Fig. 7. Contours of bed level (z/D) after 2.5 h of scour (D=0.1 m, V=44.1 cm/s, θs =0.066).

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conditions was about S/D = 1.3 and was not sensitive to the Shields parameter. Sumer and Fredsøe (2002) derived an empirical formula for estimating the scour depth in front of a submerged vertical cylinder, which was   S h = 1− exp −A c D S0

Fig. 8. Bed profiles in the xz plane for D = 0.1 m.

2.5. Variation of scour depth with cylinder height Fig. 9 shows the variation of the maximum scour depth in front of the cylinder with the cylinder height. The scour depth was measured at the upstream stagnation point of the cylinder surface (x/D = −0.5 and y/D = 0). When θs = 0.066, the equilibrium scour depth S/D for a cylinder with a height of hc/D = 3.5 was 1.306 for the D = 0.06 m and 1.261 for D = 0.1 m, respectively. Roulund et al. (2005) carried out experiments of local scour around a surface-piercing circular cylinder (top of cylinder is above water surface). The parameters in Rounlund's experiments were: water depth d = 0.4 m; V = 0.46 m/s; D = 0.1 mm and d50 = 0.26 mm. The measured maximum scour depth in front of the cylinder was about S/D = 1.25, which was close to the measured S/D in this study. Sumer et al. (1992) found that the scour depth under live-bed

ð4Þ

where S0 is the scour depth for a pile with infinite height, A = 0.55 is a fitted coefficient. The data used to determine the coefficient A were measured from experiments where the Shields parameter was rather close to but slightly smaller than the critical value corresponding to threshold of motion of sediments (Personal communication with Sumer 2008). It was not clear how many data points were employed to determine the coefficient A, although only two data points were apparently shown in the figure provided in the book by Sumer and Fredsøe (2002). Attempts were made to derive an empirical formula for scour depth based on the present experimental data. It was assumed that variation of the scour depth with the cylinder height follows the same formula as that given by Eq. (4). The coefficients S0 and A in Eq. (4) were determined by curve fitting the experimental data using the least square method. The fitted curves are shown in Fig. 9. It can be seen from Fig. 9 that, if the Shields parameter is kept constant, the non-dimensional scour depth S0/D increases slightly with a decrease in the model size. The increase rate of scour depth with cylinder height decreases with a reduction in Shields parameter, especially when hc/D is small. This can also be seen by comparing the values of parameter A, which increase with the increase in Shields parameter. It is expected that the increase rate of scour depth with cylinder height will further decrease if the Shields parameter is in the clear water scour regime. The coefficient A= 0.55 obtained by Sumer and Fredsøe (2002), which was derived from the clear water tests, is smaller than those obtained in this study. 3. Numerical model 3.1. Flow model The turbulent flow around a submerged cylinder is simulated by solving the Reynold-Averaged Navier–Stokes (RANS) equations with a k–ω turbulence model. The Arbitrary Lagrangian Eulerian (ALE)

Fig. 9. Variation of the scour depth in front of the cylinder with the cylinder height.

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scheme was employed for solving the RANS in a computational domain with a continuously developing bed boundary. In the ALE scheme the position of each mesh node moves and convection terms are modified in order to consider the effects of the mesh moving speed. The RANS are expressed as " #   ∂u ∂ui 1 ∂p ∂ ∂u i =− + ν i + τij + uj −ujp ρ ∂xi ∂xj ∂t ∂xj ∂xj

ð5Þ

∂ui =0 ∂xi

ð6Þ

where xi (i = 1, 2 and 3) is the Cartesian coordinates, ui is the velocity component in xi direction, p is the pressure, ν is the kinematic viscosity, t is time, ρ is the fluid density, ujp is the velocity of the computational mesh movement, τij is the Reynolds stress which is defined as τij = νt(∂ui/∂xj + ∂uj/∂xi) − (2/3)kδij, k is the turbulent energy, νt is the turbulence viscosity. The k–ω SST (shear–stress transport) model (Menter, 1994) is employed to close the RANS. Menter (1994) found this turbulence model gave better results in flows where strong adverse pressure gradient exists. The governing equations were solved using a finite element model (FEM) developed by Zhao et al. (2009). In this model a Petrov– Galerkin finite element scheme proposed by Brooks and Hughes (1982) was extended to three-dimensional flow simulation. In the Petrov–Galerkin method upwind is realized by modifying the weighting function to make its value upstream a computational node larger than that downstream the same node. Detail of the FEM model used in this study can be seen in Zhao et al. (2009). Simulations are carried out at the same physical scales as the laboratory tests. For each simulation, a cuboid computational domain of 60D in length, 20D in width and 20D in height was used. The cylinder was located at the centre of the domain. At the inlet boundary (left), the transverse and the vertical velocity components are set to be zero. For each inlet velocity, a separate calculation of flow in a straight channel of a length of 200D is carried out and the velocity, sediment concentration and turbulent quantities at the outlet boundary of the straight channel are used as the inlet boundary for scour simulation. At the outlet (right) boundary, the gradients of the velocity and the turbulent quantities in the flow direction are set to zero and a reference pressure of zero is given. At the two side boundaries and the top boundary, the velocity component and gradients of other dependent variables in normal direction are set zero. Non-slip boundary condition (zero-velocity) is applied at the surface of the cylinder. At the bed boundary, the standard wallfunction boundary condition is implemented (Liang et al., 2005; Zhao and Cheng, 2008). Use of wall functions reduces the demand on extreme fine mesh near-wall boundaries thus results in improved computational efficiency. At the water surface boundary, rigid lid boundary condition is applied where the surface is rigid and the velocity component in the vertical direction and gradient of other variables in the vertical direction is zero. The study of Meville and Sutherland (1988) indicates that the scour depth is independent on the water depth if the water depth to cylinder diameter ratio is greater than 4. In this study, the minimum water depth to cylinder diameter ratio is 5. The application of rigid lid boundary condition is not expected to have significant effect on the numerical results. Fig. 10 shows the typical computational mesh around the circular cylinder for hc/D = 2.5. The computational domain was discretized by eight-node hexahedron elements. Fine elements were used at the edge and the toe of the cylinder in order to simulate the large velocity gradients there. A total of 128 nodes are used in the circumferential direction of the cylinder surface at the top and bottom slices of the cylinder. The mesh size in the normal direction of the cylinder surface is 0.001D.

Fig. 10. Computational meshes around the circular cylinder for hc/D = 2.5.

3.2. Scour model Both bed load and suspended load sediment transport rates are considered in the scour model. In the numerical calculation, a reference level za is specified, below which the sediment movement is considered to be in the form of bed load and above which the sediment transport is considered to the suspended load. The za in present paper is set to be 2d50. The suspended sediment was evaluated by the volume concentration (c), which was computed by solving the convection–diffusion equation   ∂c ∂c ∂c ∂ −ws = + uj −ujp ∂t ∂xj ∂x3 ∂xj

σc νt

∂c ∂xj

! ð7Þ

where σc is a constant which is set to be 1 in this paper, ws is the fall velocity of the sediment particles in still water. The spatial discretized finite element formula for Eq. (7) is " ∫ W Ω

#   ∂c ∂c ∂c ∂c ∂W ∂c dΩ = ∫ σc νt −Wws + σc νt + W uj −ujp WdΓ ∂t ∂xj ∂x3 ∂xj ∂xj ∂n Γ

ð8Þ where W is the weighting function, Ω represents computational domain, Γ is the boundary of computational domain, n represents the normal direction of boundary pointing outward of the computational domain. The boundary conditions for Eq. (8) are: (a) The sediment concentration profile at the inlet boundary is determined by an equilibrium profiles obtained from a separate uniform-channel flow simulation. (b) No sediment particles pass through the cylinder surface, top and the two side boundaries. (c) At the reference level (za), the quantity of sediment particles settling down is determined by the sediment concentration and its falling velocity and the sediment stirred up by the flow is determined by the bed Shields parameter. In Eq. (8) the last term in the left hand side of the equation (σcνt∂c/ ∂n) represents sediment quantity that is entrained into suspension per unit time per unit area of seabed. Large numerical error will be introduced into the model if the quantity of suspension is calculated directly by this term. The reasons are: (a) The turbulent viscosity is highly dependent on the turbulent model; (b) The extremely high

M. Zhao et al. / Coastal Engineering 57 (2010) 709–721

near-bed vertical gradient of sediment concentration cannot be modelled accurately unless an extremely high density mesh is used; (c) The parameter σc is chosen arbitrarily in the numerical model. Under equilibrium conditions, the sediment deposition (wsca) equals to the sediment suspension (− σcνt∂c/∂z), where ca is the reference sediment concentration at level za. For a slope bed the term ∫σc νt ∂c = ∂nWdΓ can be replaced by ∫ −σc νt ∂c = ∂zWdΓxy , with Γxy Γ

Γxy

being the projection of Γ on xy-plane. In this study the capacity of the sediment suspension − σcνt∂c/∂z was replaced by wsca, which can be calculated by a empirical formula according to the bed shear stress (Wu et al., 2000). The reference concentration (ca) at the bed is calculated using the empirical formula proposed by Zyserman and Fredsøe (1990) ( ca =

1:75

0:331ðθs −0:045Þ

  1:75 ; = 1 + 0:72ðθs −0:045Þ

if θs N 0:045 if θs ≤0:045

0;

717

where Ds is the deposition rate defined as Ds = wscb, the erosion rate Es = wsca. In order to save computational time, the morphological time step is set larger than the flow time step (Brørs, 1999; Liang, et al., 2005; Zhao and Cheng, 2008). The method for choosing morphological time step Δtb is same as the one used by Zhao and Cheng (2008). The criteria for choosing Δtb are: (1) bed change within one morphological time step is less than 0.0005D and (2) Δtb ≤ 10Δt with Δt being the flow time step. Both criteria have to be satisfied. According the criteria, the morphological time step is less than 10Δt in the early stage of scour and equal to 10Δt at the later stage of scour. Parallel computational code developed and all the computations in this study are carried out on the WASP (Western Australian Supercomputer Program). The WASP has 164 Cray Xt3 processors. For each simulation, 64 grid nodes on the cluster were used. Considering the communication among the grid nodes, the computational speed can increase to at least 48 times of that of a single processor.

ð9Þ The skin friction Shields parameter θs is calculated by θs = u2fs / [g(s − 1)d50] and the skin friction velocity ufs is obtained according to the logarithmic law by ufs = κuT / ln(Δ1 / z0s), where uT is the velocity at the mesh node next to the wall, Δ1 is the distance between the wall and the mesh node. Bed load transport rate qb = (qbx, qby) is calculated by the semi-empirical equation proposed by Engelund and Fredsøe (1976) qb =

πd350 PEF U 6 d250 b

ð10Þ

where Ub is the velocity of the sediment movement, PEF is the percentage of the particles that are moving on the bed, which is calculated by    −1 = 4 πμd =6 4 PEF = 1 + θ−θc

ð11Þ

When sediments move along a slope bed as shown in Fig. 11, the critical Shields parameter is modified as (Zhao and Teng, 2001; Roulund et al., 2005) 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 2 sin β sin α cosα cosβ 2 5 − θc = θc0 4 cos β− μs u2s

3.3. Model validation Experimental results by Dargahi (1989) for flow past a surfacepiercing (top of the pile is above water surface) vertical circular cylinder were used in this study to validate the flow model. The parameters in the experiment by Dargahi (1989) were: Reynolds number Re = 3.9 × 104, water depth to cylinder diameter ratio hc/ D = 1.33, the boundary-layer thickness equals to the water depth. Both the cylinder surface and the bed were smooth. Fig. 12 shows the comparison of pressure distributions along the stagnation line (the cylinder's most upstream edge) and that along the upstream part of xaxis. Cps in Fig. 12(a) is defined as Cps = p / pa with pa being the pressure at the top of the stagnation line; Cpb in Fig. 12(b) is defined as Cps = p / pb with pb being the pressure at the toe of the stagnation line. The pressure difference between the top and bottom of the cylinder (shown in Fig. 12(a)) drives the flow downward and forms the horseshoe vortex. In Fig. 12(b), the constant pressure distribution between − 0.9 b x / D b −0.68 is (referred as plateau by Dargahi, 1989)

ð12Þ

in which θc0 is the critical Shields parameter for a flat bed, μs = tan ϕs is the static friction coefficient, β is the bed slope angle, α is the angle between the down-slope direction and the bed shear stress. The sediment movement speed Ub in Eq. (10) is calculated according to the force equilibrium of the moving particles (Roulund et al., 2005). Details about calculating Ub can be found in Roulund et al. (2005). Bed evolution is modelled by solving the mass balance equation of the sediment (Brørs, 1999; Liu and Garcia, 2008) ∂zb 1 ½−∇⋅qb + Ds −Es  = 1−n ∂t

Fig. 11. Critical Shields parameter on a slope.

ð13Þ

Fig. 12. Pressure distribution upstream the pile.

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attributed to the presence of the horseshoe vortex. This plateau was well predicted by the present numerical model. Fig. 13 shows the bed time-averaged Shields parameter distribution along the symmetry line (x-axis) upstream of a circular cylinder over a vortex shedding period. The calculation is carried out under the same conditions as those in the experiments of Roulund et al. (2005). The cylinder is surface-piercing and the surfaces of both the bed and pile were smooth. The cylinder diameter was 53.6 cm. The water depth was 54 cm and the depth-averaged approach velocity is V = 32.6 cm/s. The present numerical results, as well as those by Roulund et al. (2005) shown in Fig. 13 agree with the experimental data well except the minimum Shields parameter. The negative bed shear stress corresponding to the location of the horseshoe vortex is well predicted by the numerical models. Both numerical models underestimated the Shields parameter in the zone between − 0.75 b x/ D b −0.5. 3.3.1. Scour below a submerged vertical cylinder The above numerical model is employed to simulate local scour around a submerged wall mounted vertical circular cylinder. In the numerical simulation the water depth and the sediment grain size are kept the same as their counterparts used in the tests in this study. A total of 64 computer processors were used for each computation. In the calculations, the scour calculation was not started until the flow calculation reached the equilibrium state. The cylinder diameter is 0.1 m and the flow velocity is V = 44.1 cm/s. Two cylinder heights are used in the calculations, e.g. hc/D = 0.5 and 2.5. For each simulation, 64 grid nodes on the cluster were used. Each scour calculations were carried for about physical scour duration of 3 h. Simulating 3 h of scour requires about 14 days computational time on 64 grid nodes. In the following discussion, the time of scour refer to the physical scour time unless otherwise specified. It is understood that the horseshoe vortex is one of the key mechanisms that lead to local scour around a surface-piercing pier (Sumer and Fredsøe, 2002; Roulund et al., 2005). Fig. 14 shows the computed streamlines around the cylinders 2 min after the scour initiation. The scour hole has been clearly seen surrounding the upstream part of cylinder. In both two cases of hc/D, the streamlines at the upstream surface of the cylinder bend towards to the bed and roll up, forming a horseshoe vortex, which can be clearly seen in Fig. 14. The horseshoe vortex induces high bed shear stress, which leads to the suspension of the sediment in front of the cylinder. The suspended sediment is washed away to the downstream of the cylinder by the flow. The streamlines tend to bend to the bed direction after they pass the top of the cylinder, forming a large vortex in the y-direction. If the cylinder is short, large portion of the flow will deviates to the top of the cylinder and the flow goes around the sides of the cylinder becomes weak. In case of hc/D = 2.5, the vortex shedding can be clearly seen in the wake of the cylinder, while no vortex shedding is observed when hc/D = 0.5. It is interesting to observe that the initial scour hole is exactly below the horseshoe vortex and its shape is similar to horseshoe too. The initial scour hole in the front edge of the circular is quite similar to what was observed in the laboratory tests.

Fig. 13. Time-averaged bed shear stress distribution along the symmetry line upstream of the pile over one vortex shedding period.

Fig. 14. Streamlines around the cylinder.

In the numerical simulation the scour started in front of and extended gradually to the rear of the cylinder. Fig. 15 shows the contours of the absolute value of the Shields parameter normalized by the Shields parameter at the inlet (θ0) before the scour. The shear stress is amplified in the vicinity of the cylinder because of the horseshoe vortex and the streamline contraction. When hc/D = 2.5, the vortex shedding happens and the Shields parameters in Fig. 15 were obtained by averaging θ over a simulated vortex shedding period. The maximum Shields parameter for both hc/D happens at the two shoulders of the cylinder, where both horseshoe vortex and streamline contraction exist. The bed shear stress reduces if the cylinder height decreases. The maximum value of

Fig. 15. Contour of bed shear stress around the cylinder (θ/θ0) for V = 44.1 cm/s.

M. Zhao et al. / Coastal Engineering 57 (2010) 709–721

Fig. 16. Simulated and measured scour depth development at the front edge of the cylinder.

Fig. 17. Contours of bed level z/D after 3 h of physical scour time.

719

θ/θ0 is 3.5 for hc/D = 2.5 and 2.5 for hc/D = 0.5. When hc/D = 2.5, the area where θ/θ0 N 1 is also larger than that when hc/D = 0.5. The larger Shields parameter in front of and at the two sides of the cylinder induces a horseshoe-like scoured bed shape (as shown in Fig. 4) at the initial stage of the scour. Fig. 16 shows the comparison of the calculated scour depth development at the most front edge of the cylinder (stagnation point) with the experimental data. Similar to the test results, the calculated scour rate is very large at the early stage of scour and reduces very quickly. The simulated scour rate after 3 h of scour has been very small (b0.05D per hour). In both cases of hc/D, the calculated results of scour depths and scour rate at the stagnation point are smaller than the experimental data, probably because of the underestimation of the shear stress amplification factor below the horseshoe vortex as shown in Fig. 13. The calculated S/D for hc/D = 0.5 is about 20% smaller than the test result and that for hc/D = 0.25 is 10% smaller. Fig. 17 shows the perspective view of the bed topography and the bed level contours. The scoured pit for hc/D = 2.5 is much deeper than that when hc/D = 0.5. The scour shape in front of the cylinder is quite similar to those observed in the tests. The bed outside the scour pit is not flat in the numerical simulation. However, the height of the sand ripples, was not as high as that observed in the tests. Fig. 18 shows the comparison of scour depth along x-axis. The numerical model underestimated the scour depth both in front and at the rear edge of the cylinder for about 10–20%. At the wake of the cylinder, the numerical results of the bed level is smaller than the measured in the zone of x/D N 1. Fig. 19 is the comparison of the scour depth distribution along the y-axis. The agreement between the numerical results and the test results along y-axis is as good as that along x-axis. It is found that the bed slope angles at the two side of the cylinder are almost same as the repose angle of the sediment. Fig. 20 shows the comparison of the scour depth distribution along the cylinder circumference. The maximum scour depth happened in the most upstream edge of the cylinder. It decreases gradually when the measured position moves to the downstream side. In case of hc/ D = 2.5, the calculated scour depth at the side of the cylinder is 8% larger than that in front of the cylinder.

Fig. 18. Scour depth distribution along x-axis.

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under the same condition as those in the experiments. The main conclusions of this study are summarized as follows:

Fig. 19. Scour depth distribution along y-axis.

1. The scour process is governed by the combination of horseshoe vortex and vortex shedding. The sediment near the upstream surface of the cylinder was entrained into suspension by horseshoe vortex and washed downstream by the flow. The scour beneath the horseshoe vortex steepens the bed slope in front of the cylinder and made the sediment at the front edge of the scour pit slide towards the cylinder. It was observed that the bed slope of the scour pit in front of the cylinder was always very close to the repose angle of the sediment. The vortex shedding plays an important role in the scour process in the wake of the cylinder. When the cylinder is short enough (hc/D b 0.5), no scour was observed in the wake of the cylinder due to the absence of vortex shedding. 2. A decrease in cylinder height weakened the horseshoe vortex and vortex shedding. It was observed that the scour depth decreases if the cylinder height was reduced. The change rate of the scour depth with cylinder height reduces exponentially. When cylinder height-to-diameter ratio exceeds 2 the scour depth is almost independent on hc. The increase rate of scour depth with cylinder height increases with an increase in Shields parameter. 3. The mechanisms of the scour such as the horseshoe vortex and vortex shedding were well predicted by the numerical model. The predicted scour depth along the cylinder circumference by the present model is about 10 to 20% smaller than those measured in experiments. The computational results demonstrate the potential of the numerical model in modelling scour around a sub-sea structure. Acknowledgements The authors would like to acknowledge the support from The University of Western Australia Research Grant Program and CSIRO Flagship Collaboration Fund Clusters on Subsea Pipelines. References

Fig. 20. Scour depth distribution along cylinder surface.

4. Conclusions Experiments of scour around a submerged vertical circular cylinder in steady current were conducted. The effect of the cylinder height on the scour depth is investigated. A finite element numerical model was established for simulating three-dimensional scour. RANS equations were solved for simulating the flow. The transport equation for suspended sediment concentration was solved and the bed load was calculated by an empirical formula. Calculations were carried out

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