Alluvial channel hydrodynamics around tandem piers with downward seepage

Abstract

In this paper, we report the turbulent flow structures and the scour geometry around two piers with different diameters. An experiment was conducted on a non-uniform sand bed with two types of tandem arrangements, namely, pier (T1) with a 75 mm front and 90 mm rear, and pier (T2) with a 90 mm front and 75 mm rear, with and without-seepage flows, respectively. A strong wake region was observed behind the piers, but the vortex strength diminished with downward seepage. Streamwise velocity was found to be maximum near the bed downstream of the piers and at the edge of the scour hole upstream of the piers. Quadrant analysis was used to recognize the susceptible region for sediment entrainment and deposition. Upstream of the piers near the bed, the moments, turbulent kinetic energy (TKE), and TKE fluxes were found to decrease with downward seepage, in contrast to those in a plane mobile bed without piers. The reduction percentages of scour depth at the rear pier compared with the front one were approximately 40% for T1 and 60% for T2. Downward seepage also resulted in restrained growth of scouring with time.

Keywords

scour / seepage / Strouhal number / tandem arrangement / turbulent characteristics

Introduction

In alluvial channels, complicated flow structures around bridge piers result in scouring, which causes structural collapse. Many researchers have discovered that the vortex systems developed around bridge piers are responsible for the scouring around piers [¹–⁶]. Melville [¹] and Chiew [⁴] stated that a vortex system that consists of a horseshoe vortex and wake vortices is the primary cause of scouring around piers. The horseshoe vortex increases the velocity near the bed, and the wake vortices maintain the sediment particle in suspension. Qadar [³] reported that flow separates at a pier and rolls up near the bed region, thereby causing scouring around the piers. According to Melville and Coleman [⁷], a flow field can be characterized by a downflow at the upstream face of the pier, horseshoe vortex near the bed, and surface roller and wake vortices behind the pier. Izadinia et al. [⁸] investigated the flow characteristics around a single pier under a fixed bed condition. Previously published research has shown that the turbulent flow characteristics around a single bridge pier are complex. Thus, these characteristics can become increasingly complex around a group of piers.

Currently, groups of piers are extensively used in bridge design for geotechnical and economic reasons [⁹]. In crowded cities, constructing bridges with multiple lanes or new bridges is mandatory. In practice, many bridges are wide and comprise several piers aligned in the flow direction to support the structure. In the present study, we investigate piers’ tandem arrangement, which is a simple arrangement of groups of piers. The scouring phenomenon varies more around a single pier than around a group of piers [¹⁰]. Many researchers have conducted laboratory experiments on the tandem arrangement of piers to investigate the effect of spacing between piers on scouring [¹¹,¹²]. Mahjoub Said et al. [¹³] conducted an experimental investigation using two arrangements (i.e., single and tandem piers) under a submerged condition and stated that the flow characteristics vary for tandem piers in accordance with the distance between piers. Palau-Salvador et al. [¹⁴] reported that a second pier disturbs the wake vortices behind the first pier. Ataie-Ashtiani and Aslani-Kordkandi [¹⁵] conducted experiments and investigated the variation in flow statistics resulting from single and tandem piers. Several researchers have conducted experiments on tandem piers in wind tunnels [¹⁶–¹⁸].

A vital hydraulic feature of alluvial channels is their generation of lateral flow as seepage. Natural streams generally comprise permeable boundaries, through which water continuously seeps into (upward seepage) or out from (downward seepage) the channel bed or banks, depending on the level of the surrounding groundwater table. Sharma and Chawla [¹⁹] estimated the seepage losses of the water supplied at the head of a channel to be 45%. Krishnamurthy and Rao [²⁰] reported seepage losses of 2.2 m³ per day per length in the Ganga canal. The seepage losses in a canal in New York vary from 12% to 20% [²¹]; Tanji and Kielen [²²] determined 20%–50% seepage losses in an unlined canal in a semi-arid region. Kinzli et al. [²³] and Martin and Gates [²⁴] estimated the seepage losses to be 40% and 15%, respectively. In addition to losses, downward seepage leads to changes in hydraulic features and the bed geometry of the channel [²⁵,²⁶]. Many researchers have stated that a lateral flow through boundaries may modify the channel geometry by separating the particles from the channel boundaries, thereby increasing the sediment transport rate [²⁷]. Francalanci et al. [²⁸] conducted various experiments and observed that downward seepage results in scouring, whereas an upward seepage produces deposition. Qi et al. [²⁹] conducted experiments to understand the effect of seepage on scour depth around a circular pier and found that the scour depth decreases when 2% downward seepage is applied.

To the best of our knowledge, no work has determined the effect of downward seepage on the tandem arrangement with different pier diameters, although several researchers have studied the tandem arrangement of piers. The study of the flow statistics around piers and their linkage to scouring and bed morphology around tandem piers is inadequate. Therefore, this study focuses on the influence of downward seepage on the flow field and the consequences of scouring around tandem piers with different diameters.

Experiment

In the present study, experiments were conducted in a 17.2 m long, 1 m wide, and 0.72 m deep tilting flume (Fig. 1). The slope of the flume was 0.05% for all the experiments. The basic details of the experimental setup can be obtained from the research of Chavan et al. [³⁰], who conducted experiments using the same setup. The entire length of the channel, except for a 2 m length at the upstream limit, was made porous by covering a fine mesh (0.1 mm × 0.1 mm aperture), which was supported with the help of a steel structure with a height of 0.22 m, placed at the bottom of the channel below the sand bed. The sand was placed on the mesh to prevent its entrance into the bottom chamber. The seepage chamber, with a length 15.2 m from the downstream end of the flume, a width of 1 m, and a depth of 0.22 m, was provided beneath the main channel, as shown in Fig. 1. The seepage chamber was used to extract water from the channel bed (uniformly) in the form of downward seepage. The amount of downward seepage was controlled by valves connected to the seepage chamber.

Materials

Two piers of Perspex material, with a height of 15 cm, were used in this study. To avoid a wall effect, the diameters of the piers should be less than 1/10 the channel width [33]. Thus, 7.5 cm (P1) and 9 cm (P2) were selected as the diameters of the piers. In this study, the piers were placed in tandem arrangements. Riverbeds comprise non-uniform sediments [³¹]; therefore, we conducted the experiments in non-uniform sand beds using two types of sands. Through sieve analysis (Fig. 2), the d₅₀ of sand was found to be 0.5 and 0.395 mm, and the standard deviations (s_g) were 1.65 and 1.85, respectively (greater than 1.4), thereby confirming the non-uniformity of sand [³²]. The permeability and void ratio for both 0.5 and 0.395 sands are 1.8 × 10⁻⁴ and 2.1 × 10⁻⁴ m/s and 0.584 and 0.635, respectively. The ratio of pier diameter to median sand diameter has been maintained at greater than 50 to prevent the influence of gradation on scour depth [³³].

Methodology

Initially, the flume was filled with sand up to a height of 17 cm. Before each run, the plane sand bed was prepared, after which the piers were installed. The water could enter the flume slowly, such that the flow did not disturb the bed, by controlling a valve attached to the overhead tank until the desired discharge was obtained. Two types of tandem arrangements (T1 and T2) were used, with P1 in front and P2 at the rear and with P2 in front and P1 at the rear, respectively. A test section 5–10 m from the downstream end of the flume was selected. In each arrangement, the front pier was placed 7.5 m from the downstream end of the flume, which was in the test section of 5–10 m, to obtain a fully developed flow condition. For arrangements T1 and T2, the center-to-center distance between the piers was three times the diameter of the large pier. This spacing was maintained to avoid the intrusion effect of the piers [¹²]. The experiment was performed using five discharges on both types of sands. For seepage experiments, 10% and 15% seepages were applied to all five discharges. The experiments were conducted for 24 h. Figure 3 shows a snapshot of the scoured bed after 24 h of the experimental run. Table 1 displays the details of the experiment. The 3D effect was avoided because the aspect ratio was greater than 6 [³⁴].

Morphological measurements were performed by an ultrasonic ranging system (SeaTek®) with respect to time. Four pairs of transducers were attached to an automated trolley. The wavelength of 5-MHz sound waves in water was 0.3 mm, with a resolution of 0.1 mm. Turbulent measurements were obtained by an acoustic Doppler velocimeter (ADV) around the piers at various depths. The measurements were captured at various points around the piers. However, Sections A, B, and C are critical and were used for further analysis (Fig. 1). The ADV collects data in a remote sampling volume located 5 cm below its central transmitter, with an acoustic frequency of 10 MHz and \pm \mathrm{0.1} mm precision. The velocity samples were collected within 5 min at a sampling rate of 200 Hz.

Using the ADV, the velocity data were obtained as follows:

(1)\begin{array}{l}\overline{u}={\displaystyle \frac{\mathrm{1}}{N}}{\displaystyle \sum }_{i=\mathrm{1}}^N{U}_i,\\ \overline{w}={\displaystyle \frac{\mathrm{1}}{N}}{\displaystyle \sum }_{i=\mathrm{1}}^N{W}_i,\end{array}

where \bar{u} and \bar{w} are the time-averaged velocities, and U_i and W_i are the instantaneous velocities in the streamwise and vertical directions, respectively. N is the number of instantaneous velocity samples. Reynolds shear stresses (RSSs) were calculated as

(2)\begin{array}{l}{\tau }_{\mathrm{u}\mathrm{w}}=\rho \overline{u\text{'}w\text{'}},\\ u\text{'}=U_i-\overline{u},\\ w\text{'}=W_i-\overline{w},\end{array}

where r is the density of the fluid, and u^{\prime } and w^{\prime } are the fluctuating components in the streamwise and vertical directions, respectively.

The velocity and turbulence data obtained using the ADV contain spikes, given the interference between the communicated and obtained signals. A spike removal technique [³⁵] was used to satisfy Kolmogorov’s – 5/3 scaling law in the inertial subrange, with acceleration threshold values that range from 1 to 1.5 [³⁶]. The correlation coefficient among the communicated and obtained signals was 70, and a signal-to-noise ratio in the range from 10 to 15 was maintained in all the experiments. Figure 4 depicts the velocity power spectra F_uu(f) at z = 0.01 m for the no-seepage and seepage conditions. In addition, the filtered data are consistent with Kolmogorov’s – 5/3 law in the inertial subrange.

Results and discussions

In the vertical plane, the flow reaches the front pier and separates in the form of a downflow at the face of the pier and side circulations, thus further resulting in the development of wake vortices behind the pier. The flow between both piers is noticeably complex, given the wake behind the first pier and the hindrance of the flow by the second pier. The wake vortices form behind the rear pier and result in a reversal flow near the free surface. However, to understand the effect of downward seepage on the flow field and its influence on the scour geometry around the piers, the velocity along the flow depth was measured around the piers. The two arrangements show the same trend of turbulent statistics.

Velocity

The instantaneous velocities at Sections A, B, and C, as exhibited in Fig. 1, were obtained using the ADV. The time-averaged velocities were derived by decomposing the time instantaneous velocity samples into the time-averaged velocity and velocity fluctuations [³⁷]. For the T1 and T2 arrangements, Sections A and C are upstream of the front pier and downstream of the rear pier, 6 cm away from the respective pier walls. Both sections were measured 6 cm away from the wall of the respective piers. Section B is exactly in the middle of the gap between the piers. The trend of velocity profiles is the same for both arrangements. Figure 5 displays the velocity profiles with respect to the normalized flow depth z/h at all three sections for the with- and without-seepage runs, where z is the depth from the bed at which velocity is measured and h is the flow depth.

In upstream Section A, the streamwise velocity is negative near the bed region. However, moving away from the bed, the streamwise velocity becomes positive. The maximum velocity is obtained nearly at the edge of the scour hole, after which it becomes nearly constant moving toward the free surface. In Section C, the velocity is negative and moving toward the bed near the free surface, and the velocity becomes positive and attains a high value near the bed (z/h≈ 0.1). In Sections A and C, the negative value of the streamwise velocity indicates the presence of reverse flow in this region. In between the piers, i.e., in Section B, the streamwise component of velocity is positive but small near the free surface, and it increases moving away from the free surface. The small positive value of streamwise velocity near the free surface indicates that the wake vortices behind the front pier are weakened by the presence of the rear pier. The velocity of reverse flow around the piers decreases with an increase in seepage. However, with the increase in the seepage ratio, the velocity increases moving away from the bed and near the bed in Sections A and C, respectively, where no reversal flow exists. The downflow at the upstream face of the pier caused by flow separation hits the bed. Within the developing scour hole, the flow moves along the slanting slope of the scour hole in the inverse direction to that of the approaching flow. The negative velocity near the bed upstream of the piers confirms the reversal flow, which is primarily responsible for scouring. Downstream of the piers, a negative velocity is obtained, given the wake vortices. However, the downward seepage counters the reverse flow and diminishes its strength, thereby eroding the bed material.

Reynolds shear stress (RSS)

The flow separation at the piers enhances the Reynolds stresses near the piers, thereby leading to scouring of the bed material. Figure 6 presents the RSS profile around the piers for all the runs. The RSS is negative in the regions where reversal flow occurred. In Sections B and C, the Reynolds stresses fluctuate significantly, given the dominance of wake vortices. Figure 6 illustrates that, in Section A, the RSS is small at the free surface, reaches the highest value at the edge of the scour hole, and attains a negative value moving toward the bed. In Section C, negative Reynolds stresses become positive while moving toward the bed. The negative Reynolds stresses confirm the reversal flow in that region. Figure 6 also shows that the negative Reynolds stresses in upstream Section A and downstream Section C decrease with downward seepage, thereby revealing that downward seepage reduces the strength of the reversal flow. The decreasing value of Reynolds stresses upon applying seepage signifies the reduced erosive capacity of the flow.

Turbulence intensity

Turbulence intensity can be calculated using the square root of the second moment of velocity. It is defined as

(3)\begin{array}{l}{\sigma }_{\mathrm{u}}=\sqrt{\overline{u{\text{'}}^{\mathrm{2}}}}=\sqrt{{\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \sum _{i=\mathrm{1}}^n{\left(U_i-\overline{u}\right)}^{\mathrm{2}}}},\\ {\sigma }_{\mathrm{w}}=\sqrt{\overline{w{\text{'}}^{\mathrm{2}}}}=\sqrt{{\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \sum _{i=\mathrm{1}}^n{\left(W_i-\overline{w}\right)}^{\mathrm{2}}}},\end{array}

The vertical profiles of s_u and s_w in Sections A, B, and C are plotted for all experimental conditions, as depicted in Fig. 7. In upstream Section A, s_u decreases moving toward the bed with an increase in seepage, whereas in the plane bed without piers, the turbulent intensities increase with seepage. The vertical turbulence intensities are smaller in magnitude than the longitudinal turbulence intensities, thereby implying that the fluctuating components of velocity in the streamwise direction are more dominant in Section A. In Section B, the longitudinal and vertical turbulent intensities near the free surface are nearly the same in magnitude. However, moving away from the free surface, the longitudinal turbulent intensities increase with decreasing vertical turbulent intensities because the wake vortices diminish in strength while moving toward the bed. In Section C, the vertical turbulence intensities increase. High vertical turbulence intensities are evidence of wake formation behind the piers, thereby enhancing the vertical turbulence intensities in Section C. The longitudinal turbulence intensities are slightly smaller than those in upstream Section A. In downstream Section C, the turbulence intensities decrease near the free surface, where they increase near the bed in seepage runs. The decrease in turbulence intensities near the free surface behind the piers signifies the retardation of wake vortices as a result of the application of downward seepage.

Quadrant analysis

The conditional statistics of the deviations of velocity fluctuations u' and w' and the correlation of the turbulent bursting events with sediment transport were investigated using quadrant analysis. The distribution of velocity fluctuations u' and w' in the u'w' plane [³⁸] is used to characterize the bursting events. Quadrant analysis is conducted to study the contribution of different bursting events to the RSS at a given point. The plane is divided into four quadrants, which denote four events, as follows: outward interaction, Q1,(u\text{'}>\mathrm{0},w\text{'}>\mathrm{0}); ejection, Q2, ; inward interaction, Q3, ; and sweep, Q4, (u'>0, w'<0). At any point in a flow, the quadrant analysis can be calculated as follows:

(4)u\prime w{\prime }_{\mathrm{q},\mathrm{H}}=\underset{T\to \infty }{\lim }{\displaystyle \frac{\mathrm{1}}{T}}{\int }_{\mathrm{0}}^{T}u\prime \left(t\right)w\prime \left(t\right)I_{\mathrm{q},\mathrm{H}}\left(z,t\right)\mathrm{d}t,

where T is the sampling time and I_q,H is the indicator function, which is defined as

(5)I_{q,H}[u\text{'}(\mathrm{t})w\text{'}(\mathrm{t})]=\{\begin{array}{l}\mathrm{1},\hspace{0.17em}\hspace{0.17em}\mathrm{if}(u\text{'},w\text{'})\hspace{0.17em}\mathrm{i}\mathrm{s}\hspace{0.17em}\mathrm{i}\mathrm{n}\hspace{0.17em}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{t}\hspace{0.17em}\mathrm{q}\hspace{0.17em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace{0.17em}\left|u\text{'}w\text{'}\right|\ge H{\left(\overline{u{\text{'}}^{\mathrm{2}}}\right)}^{\mathrm{0.5}}{\left(\overline{w{\text{'}}^{\mathrm{2}}}\right)}^{\mathrm{0.5}}\\ \mathrm{0},\hspace{0.17em}\mathrm{otherwise}\end{array}

where H is the hyperbolic hole size defined by the following curve:

(6)\left|u\text{'}w\text{'}\right|=H\sqrt{\overline{u{\text{'}}^{\mathrm{2}}}}\sqrt{\overline{w{\text{'}}^{\mathrm{2}}}}.

The hole size, H= 0, denotes that all u\text{'} and the corresponding w^{\prime } data are included in the analysis. The fractional contribution from any event to the total RSS is defined as

(7)S_{q,\mathrm{H}}={\displaystyle \frac{u\text{'}w{\text{'}}_{q,\mathrm{H}}}{\overline{u\text{'}w\text{'}}}}.

When H = 0,

(8)S_{\mathrm{10}}+S_{\mathrm{20}}+S_{\mathrm{30}}+S_{\mathrm{40}}=\mathrm{1.}

Figure 8 demonstrates that the contributions of the Q1 and Q3 events to the total RSS are higher upstream of the pier, z/h≤0.1, than those in Q2 and Q4. The dominance of Q1 and Q3 indicates the weak flow at the bed in Section A. The reversal flow erodes the bed material. However, given that the capacity flow is insufficient to transport the material along with it, the particles return to the bed. The bed material is removed from the inclining slant of the scour hole, considering the inverse flow, and is deposited again at the bed. This phenomenon has been observed visually during the experiment. In contrast, the ejection and sweep events near the free surface become stronger than the inward and outward interactions. At the scour hole edge, Q4 is the dominant event. Sweep events are responsible for the arrival of high-speed fluid particles, resulting in deepening of the scour hole. In Section C, z/h≤0.3, the dominant ejection events signify the arrival of low-speed fluid parcels, considering wake vortices. The bed material remains in suspension because of ejection. At z/h>0.3, Q2 and Q4 contribute equally. In Section C, the bursting events are strong, considering their high Reynolds stresses. In Section B, Q2 and Q4 events contribute more than do Q1 and Q3.

The lateral flow through the channel boundaries affects the bursting events. The contributions of all four quadrants (Q1, Q2, Q3, and Q4) increase in the seepage cases. Figure 8 demonstrates that an increase in seepage causes an inrush of low-speed fluid parcels because flow retardation signifies ejection events. However, outside the scouring region, ejection and sweep events are dominant over inward and outward interactions.

The probability (P_i,H) of occurrence of bursting events is expressed as follows [³⁹]:

(9)P_{i,\mathrm{H}}={\displaystyle \frac{{\displaystyle {\int }_{t=\mathrm{0}}^{t=T}{I}_{q,\mathrm{H}}\mathrm{d}t}}{{\displaystyle {\int }_{t=\mathrm{0}}^{t=T}[I_{\mathrm{1},\mathrm{H}}+I_{\mathrm{2},\mathrm{H}}+I_{\mathrm{3},\mathrm{H}}+I_{\mathrm{4},\mathrm{H}}]\mathrm{d}t}}},

where I_q,H is an indicator function (Eq. (5)). The variation in P_i,H in the seepage and no-seepage flows with z/h is plotted in Fig. 10. In this figure, in Section A, the probability of occurrence is greater in Q1 and Q3 than in Q2 and Q4 because Q2 and Q4 are the governing events outside the scour hole. The fluctuations of all probabilities are more modest outside the scour hole than within the scour hole. Upstream of the piers, the occurrence probabilities for Q4 are the highest, followed by those of the Q2, Q1, and Q3 events. A sweep event results in significant entrainment of sediment particles and causes sediment transport. Thus, the dominant sweep events in Section A result in the maximum scour depth. The dominant ejection events at the pier’s downstream result in the transport and deposition of the bed material in that region.

Moments

The third-order moments of velocity fluctuations can be expressed as M_{lm}=\overline{{\widehat{u}}^l{\widehat{w}}^m}, where l+ m = 3, \widehat{u}={u\text{'}/\left(\overline{u\text{'}u\text{'}}\right)}^{\mathrm{0.5}}, and \widehat{w}={w\text{'}/\left(\overline{w\text{'}w\text{'}}\right)}^{\mathrm{0.5}} [⁴⁰]. M₃₀ (\overline{{\widehat{u}}^{\mathrm{3}}}) signifies the streamwise flux of the streamwise Reynolds normal stress (RNS) \overline{u\text{'}u\text{'}}, whereas M₀₃ (\overline{{\widehat{w}}^{\mathrm{3}}}) defines the vertical flux of the vertical RNS\overline{w\text{'}w\text{'}}. M₁₂ (\overline{u^{\mathrm{1}}{w}^{\mathrm{2}}}) and M₂₁ (\overline{u^{\mathrm{2}}{w}^{\mathrm{1}}}) indicate the diffusion of the vertical and streamwise RNS in the x- and z-direction, respectively.

The vertical distribution of moments for all the runs is displayed in Fig. 10. This figure shows that in upstream Section A, M₃₀ and M₁₂ start with nonnegative values and become negative moving toward the free surface. The change in M₃₀ and M₁₂ from positive to negative values suggest initial \overline{u\text{'}u\text{'}}-flux and \overline{w\text{'}w\text{'}}-diffusion in the x-direction. In contrast, \overline{u\text{'}u\text{'}}-flux and \overline{w\text{'}w\text{'}}-diffusion propagate opposite to the stream direction with an increase in depth. In the seepage runs, positive M₃₀ and M₁₂ decrease in Section A. This decrease contradicts the case of the plane mobile bed because the lateral flow through the channel boundaries hinders the reversal flow. M₀₃ and M₂₁ start with non-positive values and become positive, thereby revealing that \overline{w\text{'}w\text{'}}-flux and \overline{u\text{'}u\text{'}}-diffusion propagate toward the bed. The negativity of M₀₃ and M₂₁ decreases with downward seepage. Negative M₀₃ and M₂₁ and positive M₃₀ and M₁₂ demonstrate that flow is moving toward the bed, then conveyed in the stream direction. In downstream Section C, M₃₀ and M₀₃ are negative and positive, respectively, near the bed and become positive and negative, respectively, with increasing flow depth.

A negative M₃₀ and a nonnegative M₀₃ suggest the \overline{u\text{'}u\text{'}}-flux opposite the flow direction and the \overline{w\text{'}w\text{'}}-flux in the z-direction, respectively. However, M₁₂ starts with a positive value and then becomes non-positive, and M₂₁ starts with a negative value and changes into a nonnegative value moving toward the free surface. This transformation denotes that the \overline{w\text{'}w\text{'}} and \overline{u\text{'}u\text{'}} diffusions in Section C near the bed propagate in the longitudinal and downward directions, respectively. A non-positive M₁₂ and a positive M₂₁ away from the bed infer that \overline{w\text{'}w\text{'}} and \overline{u\text{'}u\text{'}} diffusions occur in the reverse stream and upward directions, respectively. This phenomenon affirms the predominance of a secondary current caused by wake formation. In Section C, a negative M₃₀ and a positive M₀₃ reveal the arrival of slow-moving fluid parcels that are responsible for the ejection motion. In the seepage runs, the negativity of M₃₀ and M₂₁ and the positivity of M₀₃ and M₁₂ decrease. The moment trends in Section B are nearly the same as those observed in Section C.

Turbulent kinetic energy (TKE)

Figure 11 displays the vertical distribution of TKE {\displaystyle \frac{\rho }{\mathrm{2}}}(\overline{u{\text{'}}^{\mathrm{2}}}+\overline{v{\text{'}}^{\mathrm{2}}}+\overline{w{\text{'}}^{\mathrm{2}}}) around the piers for all runs. Upstream of the piers, the magnitude of the TKE is low above the scour hole. Because of the adverse pressure gradient and separation of flow, the TKE is at a maximum at the edge of the scoured region [⁴¹] and decreases again moving toward the bed. Scouring occurs around piers, given the high magnitude of the TKE. In Section C, the TKE decreases with increasing flow depth. In Section B, the TKE fluctuates intensely in a sporadic manner because of the complex flow pattern. The TKE profile shows significant changes with respect to increasing seepage percentage. Figure 10 illustrates that TKE decreases in seepage runs. In this investigation, unlike the plane bed, the level of turbulence is diminished when the lateral flow is allowed through the channel boundaries because it impedes the reversal flow.

TKE-flux components

The distributions of the streamwise (F_TKEu = f_TKEu/u*³) and vertical (F_TKEw = f_TKEw/u*³) TKE fluxes along the flow depth are presented in Fig. 12, where u* is the shear velocity. The streamwise and vertical fluxes of TKE can be evaluated as follows [⁴⁰]:

(10)\begin{array}{l}{f}_{TKEu}=\mathrm{0.75}\left(\overline{u\text{'}u\text{'}u\text{'}}+\overline{u\text{'}w\text{'}w\text{'}}\right),\\ f_{TKEw}=\mathrm{0.75}\left(\overline{u\text{'}u\text{'}w\text{'}}+\overline{w\text{'}w\text{'}w\text{'}}\right),\end{array}

In upstream Section A, F_TKEu and F_TKEw start with small positive and non-positive values, respectively. Moving toward the free surface, F_TKEu and F_TKEw become negative and positive values, respectively. The positive F_TKEu and non-positive F_TKEw reveal the streamwise and vertical TKE fluxes in the longitudinal and downward directions, respectively. However, with increasing flow depth, negative F_TKEu and nonnegative F_TKEw result in retardation, given the arrival of low-speed fluid parcels. In Fig. 12, F_TKEu and F_TKEw exhibit reduced positivity and negativity in the seepage runs near the bed.

Downstream of the piers, wake vortices result in prevailing secondary currents. In Section C, F_TKEu and F_TKEw are negative and positive near the bed, respectively. Moving toward the free surface, F_TKEu changes to positive, and F_TKEw changes to negative. Downstream of the piers, the retardation effect, given the wake vortices, can be confirmed from the negative F_TKEu and nonnegative F_TKEw. In contrast, positive F_TKEu and negative F_TKEw away from the bed resembles the streamwise and vertical fluxes of the TKE in the longitudinal and downward directions as consequences of the inrush of fluid parcels. The negativity of F_TKEu and the positivity of F_TKEw decrease near the bed with increasing seepage, indicating a low momentum transfer, considering the lateral flow through the channel boundaries. In Section B, the streamwise and vertical fluxes of the TKE fluctuate significantly, resulting in a pattern nearly similar to that in Section C.

Power spectra analysis

The strength of the wake vortices was determined by conducting power spectral analysis. The power spectrum at each point was calculated using a fast Fourier transform of the autocovariance function of the velocity time-series data. The resultant power spectra were obtained from S(f)={\{S_x{(f)}^{\mathrm{2}}+S_y{(f)}^{\mathrm{2}}+S_z{(f)}^{\mathrm{2}}\}}^{\mathrm{0.5}}, where S_x(f), S_y(f), and S_z(f)are the velocity spectrum components in the x, y, and z directions, respectively. The strength of the wake vortices can be determined through peak frequency because the power associated with it determines the vortex strength. The Strouhal number (S_t) is calculated as fD/U_{\mathrm{0}}, where f is the vortex-shedding frequency, D is the diameter of the pier, and U₀ is the depth average flow velocity. The values of S_t near the free surface and bed region are presented in Table 2.

Table 2 shows that S_t is low in the bed region with and without seepage. The low Strouhal number near the bed indicates a low vortex strength. However, it is dominant near the free surface. The Strouhal number decreases with increasing seepage percentage for T1 and T2. Given that seepage hinders the reversal flow, the vortex strength for scouring the bed material diminishes. The Strouhal number downstream is less in arrangement T2 than in arrangement T1 for all conditions. The reduced Strouhal number with arrangement T2 signifies more depletion in the scouring of the bed material with arrangement T2 than with arrangement T1.

Bed morphology around the piers

Figures 13 and 14 present the contour profiles of the scoured bed for arrangements T1 and T2, respectively, for the seepage and without-seepage runs. When two similar piers are in line, the scour depth at the second pier is approximately half of the scour depth at the first one [⁴²]. In the present study, two piers with different diameters are placed in line at T1 and T2, with a 75 mm front and 90 mm rear pier and a 90 mm front and 75 mm rear pier, respectively. Figure 15 illustrates the scour depths at both piers for arrangements T1 and T2 under all experimental conditions. Figure 15 demonstrates that the scour depth is reduced at the rear pier compared with that at the front pier in both arrangements. However, the percentage of reduction in scour depth is greater at the rear pier than at the front pier in arrangements T2 (nearly 60%) and T1 (nearly 40%). In Fig. 15, the scour depths at the front and rear piers in both arrangements decrease with increasing seepage discharge of the corresponding main channel flow. The decreased erosive limit of the reversal flow decreases the scour depth in the seepage cases (Fig. 6). In the present study, the reduction of scour depth with seepage is consistent with previous studies by Qi et al. [²⁹], Chavan et al. [³⁰], and Chavan and Kumar [⁴³].

Scouring at bridge piers is a time-dependent phenomenon. The scouring depth gradually increases with time. However, the erosive capacity of the flowing stream and the resistance offered by the bed material attain an equilibrium after a certain period. The definition of the equilibrium state for the scour depth at the bridge piers is complicated. Many researchers have presented various criteria for equilibrium time. According to Kumar et al. [⁴⁴], an experiment must be stopped when the scour depth has not exceeded 1 mm after 3 h. Mia and Nago [⁴⁵] suggested stopping the experiments when the increment in the scour depth was less than 1 mm over 1 h. In this experimental study, we found that the scour depth increased rapidly over a period of 8 h. Subsequently, the change in scour depth became insignificant. The bed material was dislodged from the vicinity of the piers and deposited behind the piers. Afterward, the deposited material was eroded by the flow and resulted in extending the neighborhood scour. Within 12 h, the increase rate in the scour depth decreased significantly, and the increase in scour depth did not exceed 1 mm. In both arrangements, the scour depth upstream of the front pier was measured by URS at various time intervals, as displayed in Figs. 16 and 17. In this figure, scour depth increased with time, with and without seepage. However, the increase rate in the scour depth decreased gradually, up to approximately 50% after a certain period. The downward seepage restricted the reversal flow upstream of the piers. Thus, for T1 and T2, the extent of increase in scour depth was low under the 10% seepage condition and decreased further for the 15% seepage runs. The eroded sediments were carried along with the side circulations and accumulated behind the pier along and at both sides of the centerline. Figure 18 displays the graph of the bed elevation along the centerline in the longitudinal direction. Figure 18 also shows that the bed elevation height downstream of the piers increased with increasing seepage percentage for T1 and T2. However, the length of deposition decreased with increasing seepage.

Discussions

In alluvial channels, the sediment particles that rest on the channel boundary are subjected to hydrodynamic forces that affect the stability of the bed material. Turbulent flows play an important role in sediment transport and morphodynamic changes in a channel in terms of the entrainment and deposition of sediment particles. In addition, the lateral flow through the alluvial boundaries in the form of seepage enhances the mass and momentum exchange across the interface of the fluid and boundary. This lateral flow has the potential to alter the flow properties and sediment motion. Scouring around bridge piers has been a major area of interest for hydraulic engineers and designers. The study of turbulent flow characteristics around piers can help to understand the complex phenomenon of scouring around the piers. Given its complex nature, river flow or the flow around piers is difficult to simulate in laboratory experiments. However, simplified laboratory experiments can provide comprehensive knowledge of the complex flow structure.

To analyze the flow characteristics and scouring phenomena, the internal flow structures must be observed carefully. Thus, instantaneous velocity measurement was obtained using the ADV at several sections around the piers. Careful observations of the turbulent statistics for the no-seepage and seepage conditions provide several insights. From the time-averaged velocity profiles, reversal flows near the bed upstream of the piers and near the free surface downstream of the piers have been observed. The reversal decreases with increasing seepage percentage. Similarly, the RSS, turbulent intensities, high-order moments, TKE, and TKE fluxes near the bed in Section A are reduced in magnitude in the seepage runs. The weak sweep and ejection events near the bed in Section A show the inadequate capacity of the sediment particles to be entrained in a flow. This phenomenon signifies that the lateral flow through the channel boundaries surmounts the erosive capacity of the reversal flow. However, in Sections B and C, the magnitude of various turbulent statistics increases near the bed in the presence of downward seepage, indicating an increased level of turbulence at the respective sections in the seepage runs. The linkage between turbulence statistics and sediment motion is important because it signifies the morphological changes in the channel through erosion and deposition of sediment particles. Geometrical measurements of the channel indicate that the scour depth at the piers decreases with time and increasing seepage percentage.

Conclusions

This study was conducted to elucidate the linkage between turbulence flow characteristics and the scouring phenomenon when two piers with different diameters are placed in a tandem arrangement in an alluvial channel, with or without seepage. Two piers, of 75 and 90 mm diameters, were used. The various results for display velocity, turbulent intensity, RSS, moments, TKE, and TKE-flux in Sections A, B, and C are presented for the main channel discharge and with seepage discharge. Spectral analysis was performed in Section C of the arrangements for both with- and without-seepage conditions. The corresponding changes in bed morphology with respect to flow characteristics were discussed for all the experimental conditions. The following conclusions can be drawn from this study.

1) The reversal flow in Section A, which is primarily responsible for the scouring of the bed material, and that near the free surface in Section C maintain the sediment particles in a suspension downstream of the piers and decreases with increasing seepage percentage. In Section B, the most complex flow pattern and reversal flow are observed, given the wake region behind the front pier and because of the rear pier, respectively. In section B, with downward seepage, the velocity near bed increases.

2) The reduction in RSS with increasing seepage near the bed’s free surface shows a low erosive capacity for reversal flow in Section A and weak wake vortices in Section C. In Section B, the RSS fluctuates heavily and increases with the seepage flow.

3) The magnitude of longitudinal turbulence intensity is smaller in downstream Section C than in upstream Section A. However, the vertical turbulence intensity is low in the upstream section and increases in the downstream sections, considering the increasing oscillations in the wake region of the piers. In contrast to a plane mobile bed without piers, the turbulence intensity near the bed’s free surface decreases with downward seepage in Sections A and C.

4) The quadrant analysis indicates that upstream of the pier inside the scour hole near the bed region, inward and outward interactions are dominant because of the reversal flow and result in sedimentation. In Section C, strong ejection events induce the sediments to remain in suspension.

5) In Section A, the positivity of M₃₀ and M₁₂ decreases with increasing seepage flow, thus applying minimal transport of \overline{u\text{'}u\text{'}}flux and \overline{w\text{'}w\text{'}}diffusion in the longitudinal direction. In Section C, the decreased negativity of M₃₀ and the positivity of M₀₃ with the application of downward seepage indicate the low arrival of slow-moving particles.

6) A high TKE at the edge of the scoured region implies a high erosive capacity of a flow that decreases by applying downward seepage.

7) In the seepage runs, the low fluxes of streamwise TKE in the streamwise direction and the vertical kinetic energy in the vertical direction indicate a low momentum exchange, thus minimizing the scouring of the bed material.

8) In Section C, the spectrum of velocity components shifts toward lower frequency in the seepage runs. The Strouhal number decreases with increasing seepage, thereby indicating the reduced strength of the wake vortices.

9) In terms of both arrangements, namely, T1 and T2, the scour depth is reduced at the rear pier compared with that at the front pier. The reduction in the scour depth at the rear pier for arrangement T1 is approximately 40%, and this value is 60% in the case of arrangement T2. In the seepage runs, the scour depths at both piers for these arrangements decrease. The rate of scour depth increase for arrangement T2 is greater; however, with both arrangements, the rate of increase in scour depth decreases with an increase in seepage. The eroded bed particles are deposited downstream of the piers.

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