Scale Model Experiment on Local Scour around Submarine Pipelines under Bidirectional Tidal Currents

: In nearshore regions, bidirectional tidal ﬂow is the main hydrodynamic factor, which induces local scour around submarine pipelines. So far, most studies on scour around submarine pipelines only consider the action of unidirectional, steady currents and little attention has been paid to the situation of bidirectional tidal currents. To deeply understand scour characteristics and produce a more accurate prediction method in bidirectional tidal currents for engineering application, a series of laboratory scale experiments were conducted in a bidirectional current ﬂume. The experiments were carried out at a length scale of 1:20 and the tidal currents were scaled with ﬁeld measurements from Cezhen pipeline in Hangzhou Bay, China. The experimental results showed that under bidirectional tidal currents, the scour depth increased signiﬁcantly during the ﬁrst half of the tidal cycle and it only increased slightly when the ﬂow of the tidal velocity was near maximum ﬂood or ebb in the following tidal cycle. Compared with scour under a unidirectional steady current, the scour proﬁle under a bidirectional tidal current was more symmetrical, and the scour depth in a bidirectional tidal current was on average 80% of that under a unidirectional, steady current based on maximum peak velocity. Based on previous research and the present experimental data, a more accurate ﬁtted equation to predict the tidally induced live-bed scour depth around submarine pipelines was proposed and has been veriﬁed using ﬁeld data from the Cezhen pipeline. a bidirectional current was predicted based on a new ﬁtted equation for live-bed scour depth in unidirectional current coupling with a reduction coefﬁcient of 0.8. This equation was used to predict scour depth in practical engineering and showed good agreement with ﬁeld data, indicating that the present new equation could accurately predict live-bed scour depth around submarine pipelines under bidirectional currents in the ﬁeld.


Introduction
With the development and exploitation of offshore oil and gas resources, submarine pipelines are widely used to transport oil and gas from offshore to onshore areas. However, in a harsh ocean environment, suspension of submarine pipelines caused by local scour often occurs, leading to instability and breakage of the pipelines.
Local scour around submarine pipelines is a typical scour phenomenon in a marine environment. In past decades, extensive studies have been carried out to investigate the scour mechanism, scouring process and scour protection around submarine pipelines [1][2][3][4][5][6][7][8]. Mao (1986) [9] studied the scour characteristics under a steady current in an experimental flume and discussed the effect of vortices on the onset of scour. Chiew (1990) [10] carried out laboratory experiments to investigate the onset of scour and found that scour around a pipeline could be prevented by placing a plate on the upstream side of the pipeline. Some researchers [11][12][13] developed various clear-water empirical formulas to predict equilibrium scour depth based on their own experimental data under a steady current, respectively. However previous studies mostly focused only on the clear-water scour conditions (i.e., u < u c , where u is flow velocity and u c is the critical velocity for sediment incipient motion). For the live bed conditions (u > u c ) (Figure 1), the sandy bed outside the scour hole is also eroded due to high flow intensity and sediment is transported into the The scour around submarine pipeline under waves, combined waves and currents were also paid attention to. Under wave conditions, wake structure occurred on both sides of the pipeline and induced lee-wake erosion on both sides of the pipe [16,17]. The scour depth under the wave action was mainly influenced by the wave period and the maximum velocity. Sumer and Fredsøe (1990) [18] experimentally studied the scour around pipelines under wave action with different Keulegan-Carpenter numbers (KC = uwmaxTw/D, uwmax,Tw which are the wave peak velocity and period, respectively, and D is the pipeline diameter) and found that the relative scour depth dsm/D (dsm is the maximum scour depth) is a function of KC k (k = 0.5). Several other relative empirical formulas about KC were also obtained to predict the scour depth caused by waves [19][20][21][22][23]. For wave plus current, scour development has also been investigated by , Zhang et al. (2017) [24] and Li et al. (2020) [25].
In real nearshore regions, the tidal currents induced by the tidal force of celestial bodies are unsteady, with time varying velocities and flow directions [26,27]. Figure 2 shows a typical time process of tidal average vertical velocities and tidal level from field data from the Cezhen pipeline in Hangzhou Bay, China. It was seen that the flow velocity varied as sinusoids and the flow direction changed bidirectionally, which was similar to waves. However, the tidal period was nearly 12 h, which was much longer than common wave time periods (T < 100 s). Time-varying velocity with longer time periods made the tidal current different from steady currents and waves [28]; hence, the scour induced by tidal current was also different from steady currents and waves. However, at present, there is no study on the scour around submarine pipelines under bidirectional tidal currents. In practical engineering, the scour depth under bidirectional tidal flow is predicted by equations under steady currents or waves, which overestimates [29] the scour depth. The objective of this study was to gain further insights into the scour characteristics around submarine pipelines under bidirectional tidal currents. A more accurate equation was also developed to predict the maximum scour depth beneath submarine pipelines for engineering applications. The scour around submarine pipeline under waves, combined waves and currents were also paid attention to. Under wave conditions, wake structure occurred on both sides of the pipeline and induced lee-wake erosion on both sides of the pipe [16,17]. The scour depth under the wave action was mainly influenced by the wave period and the maximum velocity. Sumer and Fredsøe (1990) [18] experimentally studied the scour around pipelines under wave action with different Keulegan-Carpenter numbers (KC = u wmax T w /D, u wmax , T w which are the wave peak velocity and period, respectively, and D is the pipeline diameter) and found that the relative scour depth d sm /D (d sm is the maximum scour depth) is a function of KC k (k = 0.5). Several other relative empirical formulas about KC were also obtained to predict the scour depth caused by waves [19][20][21][22][23]. For wave plus current, scour development has also been investigated by , Zhang et al. (2017) [24] and Li et al. (2020) [25].
In real nearshore regions, the tidal currents induced by the tidal force of celestial bodies are unsteady, with time varying velocities and flow directions [26,27]. Figure 2 shows a typical time process of tidal average vertical velocities and tidal level from field data from the Cezhen pipeline in Hangzhou Bay, China. It was seen that the flow velocity varied as sinusoids and the flow direction changed bidirectionally, which was similar to waves. However, the tidal period was nearly 12 h, which was much longer than common wave time periods (T < 100 s). Time-varying velocity with longer time periods made the tidal current different from steady currents and waves [28]; hence, the scour induced by tidal current was also different from steady currents and waves. However, at present, there is no study on the scour around submarine pipelines under bidirectional tidal currents. In practical engineering, the scour depth under bidirectional tidal flow is predicted by equations under steady currents or waves, which overestimates [29] the scour depth. The objective of this study was to gain further insights into the scour characteristics around submarine pipelines under bidirectional tidal currents. A more accurate equation was also developed to predict the maximum scour depth beneath submarine pipelines for engineering applications.

Experiment Setup
The laboratory scale model experiments for the scour around submarine pipelines under tidal current were conducted in a flume in the Key Laboratory of Estuary and Coast of Zhejiang Province in China. Figure 3 shows the experimental setup with top and side views. The flume was 50 m in length, 4 m in width and 0.5 m in depth. There were eight reversal pumps at both sides of the flume and the bidirectional tidal current could be generated using a multi-pump frequency control system. There was a sand basin (4.0 m × 4.0 m × 0.2 m) placed in the middle of the flume (see Figure 3). The sand basin was filled with sand with a median diameter d50 = 0.15 mm, density ρs = 2650 kg/m 3 and angle of repose 32°. According to the critical Shields parameter of sediment and the logarithmic form of the velocity profile, the critical velocity for sediment incipient motion was calculated uc = 0.28 m/s. The diameter (D) of the modeled pipeline was 0.045 m, which was scaled down from the Cezhen pipeline (D = 0.9 m) in Hangzhou Bay, China, with a scaling factor of 1:20. In addition, another pipeline with D = 0.075 m was also used. A Cartesian coordinate system was established with the origin being the contact point of the pipeline and sand bed at the central line. An ADV (acoustic Doppler velocimetry) was used to measure the flow velocity upstream of pipeline. Additionally, there was a camera recording the pictures of the scour profile at regular intervals and the data of the scour profiles were acquired from the pictures using image recognition. This method was also compared with traditional depth probes and showed an accuracy of 1 mm.

Experiment Setup
The laboratory scale model experiments for the scour around submarine pipelines under tidal current were conducted in a flume in the Key Laboratory of Estuary and Coast of Zhejiang Province in China. Figure 3 shows the experimental setup with top and side views. The flume was 50 m in length, 4 m in width and 0.5 m in depth. There were eight reversal pumps at both sides of the flume and the bidirectional tidal current could be generated using a multi-pump frequency control system. There was a sand basin (4.0 m × 4.0 m × 0.2 m) placed in the middle of the flume (see Figure 3). The sand basin was filled with sand with a median diameter d 50 = 0.15 mm, density ρ s = 2650 kg/m 3 and angle of repose 32 • . According to the critical Shields parameter of sediment and the logarithmic form of the velocity profile, the critical velocity for sediment incipient motion was calculated u c = 0.28 m/s. The diameter (D) of the modeled pipeline was 0.045 m, which was scaled down from the Cezhen pipeline (D = 0.9 m) in Hangzhou Bay, China, with a scaling factor of 1:20. In addition, another pipeline with D = 0.075 m was also used. A Cartesian coordinate system was established with the origin being the contact point of the pipeline and sand bed at the central line. An ADV (acoustic Doppler velocimetry) was used to measure the flow velocity upstream of pipeline. Additionally, there was a camera recording the pictures of the scour profile at regular intervals and the data of the scour profiles were acquired from the pictures using image recognition. This method was also compared with traditional depth probes and showed an accuracy of 1 mm.  Experimental runs were carried out under bidirectional tidal current and unidirectional current conditions, respectively. Considering the characteristics of the stronger flow in Hangzhou Bay, the live-bed unidirectional current conditions (u > uc) were studied in the experiment. Details of the experimental flow conditions are listed in Table 1. According to Froude's law, the length scale between the model experiment and prototype was Experimental runs were carried out under bidirectional tidal current and unidirectional current conditions, respectively. Considering the characteristics of the stronger flow in Hangzhou Bay, the live-bed unidirectional current conditions (u > u c ) were studied in the experiment. Details of the experimental flow conditions are listed in Table 1. According to Froude's law, the length scale between the model experiment and prototype was 1:20, and the time scale between the laboratory experiment and the prototype was 1:4.47. Therefore, the period of tidal current in the laboratory experiments was 2.68 h, corresponding to the period of 12 h in the prototype. The tidal current in the experiment was simplified as a regular sinusoid tide: where u t is tidal velocity at time t, u max is the maximum tidal velocity during the flood and ebb phases and T is tidal period. Considering the varying tidal velocity, root mean square velocity u r was also calculated as effective flow intensity, and for the sinusoidal form of the tidal velocity, u r could be estimated by u r = 2u max /π.  Figure 4 shows the time history of the simulated flow velocity generated by a multipump frequency control system and the theoretical velocity calculated by Equation (1) for u max = 0.4 m/s. The time history of the experimentally generated tidal velocity was in good agreement with the theoretical value, indicating that the multi-pump frequency control system could generate an excellent bidirectional tidal current.
For the unidirectional current, the flow was kept at constant velocity and depth. The unidirectional velocity u 0 was equal to the maximum peak velocity u max in a tide period and the water depth was also equal to the value of the bidirectional tidal current (see Table 1 for details of the experimental runs).

Scouring Process and Scour Profile
For the bidirectional tidal current, the reversal flow made the scour process more complicated than that under the unidirectional tidal current. Figure 5 shows the camera images of the scour profiles at different time instants for Tide 01: umax = 0.3 m/s, D = 0.045 m. As seen from the figure, in the first tidal half cycle (t < 0.5 T) (Figure 5a,b), the scour hole below the pipeline developed rapidly and there was a dune downstream of the pipeline, which gradually moved downstream with time. This phenomenon was similar to that under a unidirectional current. However, during the second half tidal cycle (0.5 T < t

Scouring Process and Scour Profile
For the bidirectional tidal current, the reversal flow made the scour process more complicated than that under the unidirectional tidal current. Figure 5 shows the camera images of the scour profiles at different time instants for Tide 01: u max = 0.3 m/s, D = 0.045 m. As seen from the figure, in the first tidal half cycle (t < 0.5 T) (Figure 5a,b), the scour hole below the pipeline developed rapidly and there was a dune downstream of the pipeline, which gradually moved downstream with time. This phenomenon was similar to that under a unidirectional current. However, during the second half tidal cycle (0.5 T < t < 1. To make readers clearly understand the distinction of scour profiles in different currents, the final equilibrium scour profiles for u 0 = u max = 0.3 m/s, 0.4 m/s and 0.5 m/s are plotted in Figure 6, respectively. As detailed in Figure 6, there were two significant differences of scour profiles between the unidirectional and bidirectional currents. The first one is that there was only one dune downstream of the pipeline for the unidirectional steady current, whereas there were two dunes for the bidirectional tidal current and the scour profile was more symmetric. Moreover, the scour depth and width under the unidirectional current were both larger than those of the bidirectional tidal current. Additionally, with increasing velocity the size of the scour hole, including scour depth and width, also both increased.

Temporal Development of Scour Depth
In practical engineering, significant attention is paid to the maximum scour depth of the pipeline by engineers because it influences the submarine pipeline design and scour protection. Figure 7 shows the temporal development of the normalized scour depth (ds/D) at x = 0.0 D versus dimensionless time t/T (T is tidal period) for various tidal currents, and the time history of relative flow velocity u t /u max is also shown. The figure demonstrated that scour depth increased rapidly during the first half of the tidal cycle. After that, the scour depth only increased slightly when the flow was near maximum flood or ebb tidal velocity and there was no scour development when the flow was close to zero velocity. It could also be seen that the decrease in scour depth induced by sediment backfilling did not obviously occur, which may be ascribed to the fact that the flow was turbulent enough to keep the sediment in suspension longer, allowing it to be transported out of the scour hole.
J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 6 of 18 that the scour profile at t = 3.0 T could be recognized as a dynamic equilibrium scour profile.
To make readers clearly understand the distinction of scour profiles in different currents, the final equilibrium scour profiles for u0 = umax = 0.3 m/s, 0.4 m/s and 0.5 m/s are plotted in Figure 6, respectively. As detailed in Figure 6, there were two significant differences of scour profiles between the unidirectional and bidirectional currents. The first one is that there was only one dune downstream of the pipeline for the unidirectional steady current, whereas there were two dunes for the bidirectional tidal current and the scour profile was more symmetric. Moreover, the scour depth and width under the unidirectional current were both larger than those of the bidirectional tidal current. Additionally, with increasing velocity the size of the scour hole, including scour depth and width, also both increased.

Temporal Development of Scour Depth
In practical engineering, significant attention is paid to the maximum scour depth of the pipeline by engineers because it influences the submarine pipeline design and scour protection. Figure 7 shows the temporal development of the normalized scour depth (ds/D) at x = 0.0 D versus dimensionless time t/T (T is tidal period) for various tidal currents, and the time history of relative flow velocity ut/umax is also shown. The figure demonstrated that scour depth increased rapidly during the first half of the tidal cycle.

Maximum Scour Depth under Bidirectional Tidal Currents
As reported above, the developments of scour depth under unidirectional and bidirectional tidal currents were different, and hence the final maximum scour depths under different current conditions were also different. Figure 9 shows the dimensional maximum scour depths caused by the bidirectional tidal current and unidirectional steady current, respectively. According to Figure 9, the live-bed scour depth dsm increased with flow velocity, and also increased with pipeline diameter in the same flow conditions. In order to evaluate the difference of maximum scour depth between unidirectional and bidirectional tidal currents, the equilibrium scour depths were transformed to dimensionless values dsm/D ( Figure 10). As seen from Figure 10, the maximum scour depths dsm/D at the same flow conditions were nearly the same with different pipeline diameters. Moreover, it was also seen that the maximum scour depth under the bidirectional tidal current dsm/D was smaller than those of the unidirectional current for u0 = umax, and was on average 0.8 times of that of the unidirectional current.

Maximum Scour Depth under Bidirectional Tidal Currents
As reported above, the developments of scour depth under unidirectional and bidirectional tidal currents were different, and hence the final maximum scour depths under different current conditions were also different. Figure 9 shows the dimensional maximum scour depths caused by the bidirectional tidal current and unidirectional steady current, respectively. According to Figure 9, the live-bed scour depth dsm increased with flow velocity, and also increased with pipeline diameter in the same flow conditions. In order to evaluate the difference of maximum scour depth between unidirectional and bidirectional tidal currents, the equilibrium scour depths were transformed to dimensionless values dsm/D ( Figure 10). As seen from Figure 10, the maximum scour depths dsm/D at the same flow conditions were nearly the same with different pipeline diameters. Moreover, it was also seen that the maximum scour depth under the bidirectional tidal current dsm/D was smaller than those of the unidirectional current for u0 = umax, and was on average 0.8 times of that of the unidirectional current.

Maximum Scour Depth under Bidirectional Tidal Currents
As reported above, the developments of scour depth under unidirectional and bidirectional tidal currents were different, and hence the final maximum scour depths under different current conditions were also different. Figure 9 shows the dimensional maximum scour depths caused by the bidirectional tidal current and unidirectional steady current, respectively. According to Figure 9, the live-bed scour depth d sm increased with flow velocity, and also increased with pipeline diameter in the same flow conditions. In order to evaluate the difference of maximum scour depth between unidirectional and bidirectional tidal currents, the equilibrium scour depths were transformed to dimensionless values d sm /D ( Figure 10). As seen from Figure 10, the maximum scour depths d sm /D at the same flow conditions were nearly the same with different pipeline diameters. Moreover, it was also seen that the maximum scour depth under the bidirectional tidal current d sm /D was smaller than those of the unidirectional current for u 0 = u max , and was on average 0.8 times of that of the unidirectional current.

Prediction of Maximum Scour Depth
According to the present experimental conditions, if the tidal current was regarded as longer time period wave, the KC number of each bidirectional tidal current was larger than 300. Sumer and Fredsøe [18] developed a scour depth equation about KC under wave and pointed out that when KC was larger than 300, the relative scour depth dsm/D was a constant value, 1.73. However, the scour depth of each tidal current case from the present experiment was far less than 1.73, which implied that the scour under the bidirectional tidal current could not be regarded as that under wave.
For the live-bed scour depth around submarine pipelines under unidirectional steady currents, Ibraim and Nalluri [14] studied live-bed scour depths around pipelines based on experiments and established an equation to predict equilibrium scour depth: where Fr is Froude number, = / ℎ. [18] suggested that the maximum scour depth was approximately a constant value:

Prediction of Maximum Scour Depth
According to the present experimental conditions, if the tidal current was regarded as longer time period wave, the KC number of each bidirectional tidal current was larger than 300. Sumer and Fredsøe [18] developed a scour depth equation about KC under wave and pointed out that when KC was larger than 300, the relative scour depth dsm/D was a constant value, 1.73. However, the scour depth of each tidal current case from the present experiment was far less than 1.73, which implied that the scour under the bidirectional tidal current could not be regarded as that under wave.
For the live-bed scour depth around submarine pipelines under unidirectional steady currents, Ibraim and Nalluri [14] studied live-bed scour depths around pipelines based on experiments and established an equation to predict equilibrium scour depth: where Fr is Froude number, = / ℎ. [18] suggested that the maximum scour depth was approximately a constant value:

Prediction of Maximum Scour Depth
According to the present experimental conditions, if the tidal current was regarded as longer time period wave, the KC number of each bidirectional tidal current was larger than 300. Sumer and Fredsøe [18] developed a scour depth equation about KC under wave and pointed out that when KC was larger than 300, the relative scour depth d sm /D was a constant value, 1.73. However, the scour depth of each tidal current case from the present experiment was far less than 1.73, which implied that the scour under the bidirectional tidal current could not be regarded as that under wave.
For the live-bed scour depth around submarine pipelines under unidirectional steady currents, Ibraim and Nalluri [14] studied live-bed scour depths around pipelines based on experiments and established an equation to predict equilibrium scour depth: where Fr is Froude number, Fr = u/ gh. Sumer and Fredsøe [18] suggested that the maximum scour depth was approximately a constant value: Also in practical engineering, except for the above equations, some clear-water scour equations are often used to predict scour depth, even though it is a live-bed condition.
Here a widely used empirical equation by Bijker and Leeuwestein [12] (4) In order to evaluate the accuracy of the above three equations, a total of 46 laboratory experimental data from Mao (1986) [9], Moncada et al. (1999) [15] and the present study were collected. Figure 11 shows the comparisons of experimental measured values and predicted values using Equations (2) and (3). It was found that predicted values by Equation (2) were both greater than 1.33 and were much larger than measured values, which implied that Equation (2) overestimated the live-bed scour depth. Additionally, most of the measured scour depths were larger than 0.6 D which revealed that Equation (3) underestimated the scour depth. The comparisons indicated that the above two predicted equations both have larger errors.
were collected. Figure 11 shows the comparisons of experimental measured values and predicted values using Equations (2) and (3). It was found that predicted values by Equation (2) were both greater than 1.33 and were much larger than measured values, which implied that Equation (2) overestimated the live-bed scour depth. Additionally, most of the measured scour depths were larger than 0.6 D which revealed that Equation (3) underestimated the scour depth. The comparisons indicated that the above two predicted equations both have larger errors. Figure 12 shows the comparisons between measured scour depths and predicted values by Equation (4). Generally, the measured values were larger than the predicted values which implied that Equation (4) also underestimated the scour depth and had a relative error of 40%. Considering larger errors of the above three equations, it was necessary to develop a more accurate equation. Based on collected experimental data, a new fitted equation with Fr and u0/uc was proposed by nonlinear multiple regression (see Equation (5)). Figure 13 shows the comparisons of predicted values using Equation (5) with measured values. It could be seen that the predicted values using Equation (5) were in good agreement with the measured values and the relative error was within 20%. Compared to Equations (2)-(5), it was more suitable for predicting the maximum scour depth of a steady current in live-bed conditions.  (2) Equation (3) ─20% +20% Figure 11. Comparisons of measured and predicted scour depth using Equations (2) and (3). Figure 12 shows the comparisons between measured scour depths and predicted values by Equation (4). Generally, the measured values were larger than the predicted values which implied that Equation (4) also underestimated the scour depth and had a relative error of 40%. Considering larger errors of the above three equations, it was necessary to develop a more accurate equation. Based on collected experimental data, a new fitted equation with Fr and u 0 /u c was proposed by nonlinear multiple regression (see Equation (5)). Figure 13 shows the comparisons of predicted values using Equation (5) with measured values. It could be seen that the predicted values using Equation (5) were in good agreement with the measured values and the relative error was within 20%. Compared to Equations (2)-(5), it was more suitable for predicting the maximum scour depth of a steady current in live-bed conditions.   As discussed in 3.3, the maximum scour depth under the bidirectional tidal current was on average 80% of that under the unidirectional steady current. Moreover, considering that a more accurate predicted Equation (5) for scour depth under the steady current has been developed in this study, we can predict the scour depth under the bidirectional tidal current by adding a reduction coefficient (0.8) to Equation (5). Therefore, the maximum live-bed scour depth under the bidirectional tidal current could be calculated as follows: Seen from Figure 14, the predicted values using Equation (6) were also in good agreement with the measured values and the relative error was also within 20%.   [9] Experimental data from [15] ─20% +20% Figure 12. Comparisons of measured and predicted scour depth using Equation (4).  As discussed in 3.3, the maximum scour depth under the bidirectional tidal current was on average 80% of that under the unidirectional steady current. Moreover, considering that a more accurate predicted Equation (5) for scour depth under the steady current has been developed in this study, we can predict the scour depth under the bidirectional tidal current by adding a reduction coefficient (0.8) to Equation (5). Therefore, the maximum live-bed scour depth under the bidirectional tidal current could be calculated as follows: Seen from Figure 14, the predicted values using Equation (6) were also in good agreement with the measured values and the relative error was also within 20%.  [9] Experimental data from [15] ─20% +20% Figure 13. Comparisons of measured and predicted scour depth using Equation (5).
As discussed in 3.3, the maximum scour depth under the bidirectional tidal current was on average 80% of that under the unidirectional steady current. Moreover, considering that a more accurate predicted Equation (5) for scour depth under the steady current has been developed in this study, we can predict the scour depth under the bidirectional tidal current by adding a reduction coefficient (0.8) to Equation (5). Therefore, the maximum live-bed scour depth under the bidirectional tidal current could be calculated as follows: Seen from Figure 14, the predicted values using Equation (6) were also in good agreement with the measured values and the relative error was also within 20%.

Comparison with Field Data
As noted above, the maximum scour depth around submarine pipelines had a relationship with the pipeline diameter D, flow velocity (umax), water depth (h) and sediment properties (uc). However, in practical engineering, the hydrodynamic conditions are larger than in laboratory flow conditions. To accurately predict the scour depth in the field, it is necessary to make sure which equation is more applicable in practical engineering. Consequently, in this study, field data from 2014 from the Cezhen pipeline in tide-dominated Hangzhou Bay were used to validate the predicted equations. According to the field data, there were 15 free span segments mainly located in three regions KP3, KP12 and KP16, and the maximum scour depths of each span segment are listed as follows: As seen from above, the maximum scour depths varied largely even with nearly the same conditions in the same region. This may be because some scour depths of the span segments did not develope to dynamic equilibrium scour status, especially the smaller values. Moreover, the maximum values in each region were much closer to the dynamic equilibrium scour status. Consequently, it was reasonable that the maximum values of each region in place of all scour depths of 15 span segments were chosen to validate the scour equation. The dimensionless maximum scour depths of a span segment in each region are shown in Table 2 and the flow conditions of each region are also shown in this table. In addition, the mean sediment size, d50, of the seabed near the pipeline was approximately 0.01 mm and the critical velocity for sediment incipient motion ranged from 0.8 to 1.2 m/s [30] (Table 2), based on a famous fine sediment incipient formula of Zhang et al. [31]. According to the flow and sediment parameters in the field, the maximum scour depths were predicted, respectively, by Equations (3)- (6). Figure 15 shows the predicted values using different equations with field data and Table 3 lists the absolute error (predicted value field data) of each equation. It was shown that the predicted maximum scour depth by Equation (6)

Comparison with Field Data
As noted above, the maximum scour depth around submarine pipelines had a relationship with the pipeline diameter D, flow velocity (u max ), water depth (h) and sediment properties (u c ). However, in practical engineering, the hydrodynamic conditions are larger than in laboratory flow conditions. To accurately predict the scour depth in the field, it is necessary to make sure which equation is more applicable in practical engineering. Consequently, in this study, field data from 2014 from the Cezhen pipeline in tide-dominated Hangzhou Bay were used to validate the predicted equations. According to the field data, there were 15 free span segments mainly located in three regions KP3, KP12 and KP16, and the maximum scour depths of each span segment are listed as follows: As seen from above, the maximum scour depths varied largely even with nearly the same conditions in the same region. This may be because some scour depths of the span segments did not develope to dynamic equilibrium scour status, especially the smaller values. Moreover, the maximum values in each region were much closer to the dynamic equilibrium scour status. Consequently, it was reasonable that the maximum values of each region in place of all scour depths of 15 span segments were chosen to validate the scour equation. The dimensionless maximum scour depths of a span segment in each region are shown in Table 2 and the flow conditions of each region are also shown in this table. In addition, the mean sediment size, d 50 , of the seabed near the pipeline was approximately 0.01 mm and the critical velocity for sediment incipient motion ranged from 0.8 to 1.2 m/s [30] (Table 2), based on a famous fine sediment incipient formula of Zhang et al. [31]. According to the flow and sediment parameters in the field, the maximum scour depths were predicted, respectively, by Equations (3)- (6). Figure 15 shows the predicted values using different equations with field data and Table 3 lists the absolute error (predicted value field data) of each equation. It was shown that the predicted maximum scour depth by Equation (6) was much closer to the field data, with absolute error within 0.1 D, whereas absolute errors using Equations (3) and (4) were 0.04-0.29 D and 0.13-0.38 D, respectively. In addition, it was also found that Equation (4) overestimated the scour values in the field data, while it underestimated the scour values in the experimental data ( Figure 12). This may be because Equation (4) was highly sensitive to sediment size d 50 , especially smaller values. By evaluating the application of Equation (6) in the field and laboratory experiments, it was validated that Equation (6) was more accurate than the present equations, both in practical engineering and on the laboratory scale. whereas absolute errors using Equations (3) and (4) were 0.04-0.29 D and 0.13-0.38 D, respectively. In addition, it was also found that Equation (4) overestimated the scour values in the field data, while it underestimated the scour values in the experimental data ( Figure 12). This may be because Equation (4) was highly sensitive to sediment size d50, especially smaller values. By evaluating the application of Equation (6) in the field and laboratory experiments, it was validated that Equation (6) was more accurate than the present equations, both in practical engineering and on the laboratory scale.    Figure 15. Comparisons of field data and predicted scour depths using different equations.

Conclusions
Scour around submarine pipelines under bidirectional tidal currents was investigated by carrying out laboratory 1:20 scale model experiments. The tidal current was simulated as a regular sinusoid tide with various velocities. The temporal development of the scour depth and scour profile characteristics under a bidirectional tidal current are analyzed. The main conclusions can be summarized as follows:

1.
Under a bidirectional tidal current, the scour profile, with two dunes on both sides of submarine pipeline, was more symmetric than that subject to a unidirectional current. In addition, the scour width and depth under a bidirectional tidal current were both smaller than those under a unidirectional tidal current.

2.
For bidirectional tidal currents, the scour depth increased rapidly during the first half of the tidal cycle. After that, the scour depth only increased slightly when the flow was near maximum flood or ebb tidal velocity, and there was no scour development when flow was close to zero velocity. Additionally, the equilibrium scour depth under a bidirectional current was averagely 0.8 times of that under a unidirectional current whose flow velocity was equal to the maximum tidal velocity.

3.
An equation to predict live-bed scour depth around submarine pipelines under bidirectional tidal currents was developed. In this equation, the scour depth under a bidirectional current was predicted based on a new fitted equation for live-bed scour depth in unidirectional current coupling with a reduction coefficient of 0.8. This equation was used to predict scour depth in practical engineering and showed good agreement with field data, indicating that the present new equation could accurately predict live-bed scour depth around submarine pipelines under bidirectional currents in the field.