1 Introduction

Cook et al. and Hassanzadeh et al. stated that, 52% of the destruction of bridges is due to scouring in the area of their piers. Studies have shown that the local scouring around the bridge pier is the major cause of the bridge failure [8, 11]. Therefore, due to the effect of scouring on the stability of bridges, the bridge pier scour depth should be carefully calculated and taken into account in the design of bridges. With climate change, every day we see severe and frequent floods all over the world.

Large floods can cause the bridge deck to be submerged and the flow under it to be pressurized. The pressurized flow pattern starts when the flow area decreases due to the submerged bridge deck, resulting in increasing flow velocity. This increasing flow velocity can further cause additional shear stress on the channel bed, thus increasing the bed scour in the contraction region. Under these conditions, the scouring of bridge piers is intensified due to increased shear stress and other factors in the bridge pier area. Since the bed materials is transported out of the contraction region, the cross-sectional area gradually increases, and the velocity decreases. Ultimately, the bed shear stress reduces to below the threshold condition, the scouring of the bed is reduced [13]. In comparison with open channel flow, pressure flow significantly increases the erosion potential because scouring is one of the ways energy is dissipated in flood conditions [10].

In addition to the pressure flow, the bridge pier scour is also affected by factors concerning flow patterns including cross-section shape parameter, floodplain vegetation density and bridge pier etc. Until now, many researchers have proposed empirical equations to estimate the bridge pier scour considering these factors.

Kouchakzadeh and Townsend, using a laboratory study, investigated the effects of the lateral momentum of the flow intercepted by abutment. They found that the flow related to the width of the channel at the end of the abutment is an important parameter for calculating the lateral momentum and should be included in the scour depth estimation equations. They proposed a relationship to predict the maximum local scour depth in the bridge abutment under conditions of flow ratio, Froude number and critical Froude number related to sediment size [12].

Ettema et al. [9] also stated that velocity is not a determining parameter for estimating the maximum scour depth. Generally, scouring consists of three main components of (1) river bed degradation in the long term, (2) contraction scouring in bridges, and (3) local scouring at the bridge pier. For example, Laursen [14] suggested Eq. (1) for clear-water contraction scour.

$${\text{z}}_{{{\text{max}}}} \, = \,\left[ {\frac{{{\text{Q}}^{{\text{2}}} }}{{40{\text{D}}_{{\text{m}}}^{{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}}} {\text{b}}_{{\text{c}}}^{2} }}} \right]$$
(1)

where Zmax is the maximum scour depth, Q is the flow discharge, bc is the Bottom width of the contracted section less pier widths, Dm is Diameter of the bed material (1.25 d50) in the contracted section and d50 is the average diameter of sediment particles [15]. Colorado State University (CSU) proposed Eq. (2) for bridge pier scour in moving bed and clear water conditions [18].

$$\begin{array}{*{20}l} {\frac{{Z_{{\max }} }}{{h_{a} }}\, = \,2.0K_{1} K_{2} \left( {\frac{{d_{p} }}{{h_{a} }}} \right)^{{0.65}} } & {Fr^{{0.43}} } \\ \end{array}$$
(2)

Based on experimental data performed in clear-water conditions, Abed stated that the maximum scour depth around a bridge pier in pressurized flow is 2.3–10 times the amount of scouring in free flow conditions [2]. This underscores the significant effect of pressurized flow conditions on bridge pier scour.

Arneson defined critical velocity Uc with incipient sediment motion based on ha as [3]:

$${\text{U}}_{{\text{c}}} {\text{ = 1}}{\text{.52}}\sqrt {{\text{g}}\left( {{\text{s}} - 1} \right){\text{d}}_{{{\text{50}}}} } \left( {\frac{{{\text{h}}_{{\text{a}}} }}{{{\text{d}}_{{{\text{50}}}} }}} \right)^{{{\text{1}}/{\text{6}}}}$$
(3)

where s is the specific gravity of sediment and g is the gravitational acceleration.

Umbrell et al. performed experiments in clear water, without pier, and for both submerged and non-submerged deck conditions. They presented Eq. (4) to calculate the scouring in pressurized conditions [28].

$$\frac{{Z_{{\max }} \, + \,h_{a} }}{{h_{a} }}\, = \,\left\{ {1.102\left[ {\left( {1 - \frac{{W_{e} }}{{h_{a} }}} \right)\frac{U}{{U_{c} }}} \right]^{{0.603}} } \right\}$$
(4)

where hb is the distance between the original bottom and the lower deck edge of the bridge, U is the average flow velocity upstream of the bridge and we is the water level at the top edge of the bridge deck (Fig. 1).

Fig. 1
figure 1

Longitudinal profile of the channel bed with pressurized flow conditions [13]

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Based on Arneson and Abt [4] orifice flow data, Richardson and Davis [19] presented Eq. (5) to estimate the maximum scour depth.

$$\frac{{\text{z}}_{\text{max}}}{{\text{h}}_{\text{a}}}\text{=}-\text{5.08+1.27}\left(\frac{{\text{h}}_{\text{a}}}{{\text{h}}_{\text{b}}}\right)\text{+4.44}{\left(\frac{{\text{h}}_{\text{a}}}{{\text{h}}_{\text{b}}}\right)}^{-1}\text{+0.19}\left(\frac{{\text{U}}_{\text{b}}}{{\text{U}}_{\text{c}}}\right)$$
(5)

Cardoso and Bettess research have been done to investigate the local scouring in the bridge abutment. The results of this research show that the temporal development of local scouring is in good agreement with Ettema and Franzetti theories [6]. Also, experiments are described to investigate for Time and Channel Geometry on scour at bridge abutments. For example, Melville and Chiew [16] showed that the scour depth after 10% of the time to equilibrium is between about 50% and 80% of the equilibrium scour depth, depending on the approach flow velocity.

total scouring includes the sum of components of contraction scouring in bridges and local scouring at bridge piers [19]. Therefore, to calculate the total amount of scouring of the bridge pier in the contraction range under clear water conditions, both Eqs. (1) and (2) were used:

$$Z_{{\max }} = 2.0({\text{K}}_{1} {\text{K}}_{{\text{2}}} )\left( {\frac{{{\text{d}}_{{\text{p}}} }}{{{\text{h}}_{{\text{a}}} }}} \right)^{{0.65}} {\text{Fr}}^{{0.43}} {\text{h}}_{{\text{a}}} + \left[ {\frac{{{\text{Q}}^{2} }}{{40{\text{D}}_{{\text{m}}}^{{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}}} {\text{b}}_{{\text{c}}}^{{\text{2}}} }}} \right]^{{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 7}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$7$}}}}$$
(6)

Sturm proposed a common relation for scour depth in clear water conditions for compound open channel. He considers the non-uniform distribution of average lateral velocity to be affected by backwater caused by the presence of bridge abutments [24].

Guo et al. [10] studied pressured flow under bridges with girders in clear-water conditions both analytically and experimentally. Their experiments revealed that the maximum scour position is located at a location 15.4% of the deck width from the downstream edge of the deck. Scour begins at approximately one deck width upstream of the bridge, and deposition starts at approximately 2.5 deck widths downstream.

Calappi et al. [5] by removing some empirical variables and approaching velocity, presented the following equation:

$$\begin{array}{*{20}l} {\frac{{Z_{{\max }} \,}}{{h_{a} }}\, = \,K_{1} K_{2} K_{3} K_{4} \left( {\frac{{d_{p} }}{{h_{a} }}} \right)^{{1.78}} } & {Fr^{0} } \\ \end{array} \, + \,0.28\,$$
(7)

where Zmax is the maximum scour depth around the bridge pier, K1 is the pier nose shape coefficient, K2 is the correction factor for an angle of attack of flow, K3 is the correction factor for bed condition (plane bed, dune, ripple), K4 is the armoring correction factor that related bed material size, dp is pier width, and Fr is Froude number at upstream of the pier. Melville [15] investigated scouring around hydraulic structures such as sluice gates, submerged bridges, and low weirs, proposing Eq. (8) to calculate the maximum scour depth:

$$\begin{gathered} \begin{array}{*{20}l} {\frac{{\left( {{\text{Z}}_{{{\text{max}}}} } \right)}}{{{\text{h}}_{{\text{a}}} }}{\text{ = 0}}{\text{.75}}\left( {\frac{{{\text{U}}_{{\text{b}}} }}{{{\text{U}}_{{\text{c}}} }} - 0.4} \right)} & {{\text{0}}{\text{.4}} \le \frac{{{\text{U}}_{{\text{b}}} }}{{{\text{U}}_{{\text{c}}} }}{\text{ < 1}}} \\ \end{array} \hfill \\ \begin{array}{*{20}l} {\frac{{\left( {{\text{Z}}_{{{\text{max}}}} } \right)}}{{{\text{h}}_{{\text{a}}} }}{\text{ = 0}}{\text{.45~}}} & {{\text{1}} \le \frac{{{\text{U}}_{{\text{b}}} }}{{{\text{U}}_{{\text{c}}} }}{\text{ < 2}}{\text{.5}}} \\ \end{array} \hfill \\ \end{gathered}$$
(8)

Kumcu [13] developed another equation based on Arneson [3] and Umbrell et al. [28] experimental data:

$$\begin{gathered} \frac{{\left( {{\text{h}}_{{\text{b}}} {\text{ + Z}}_{{{\text{max}}}} } \right)}}{{{\text{h}}_{{\text{a}}} }}{\text{ = 0}}{\text{.65 + 0}}{\text{.5}}\left( {\frac{{{\text{U}}_{{\text{b}}} }}{{{\text{U}}_{{\text{c}}} }}} \right){\text{0}}{\text{.5}} \le \frac{{{\text{U}}_{{\text{b}}} }}{{{\text{U}}_{{\text{c}}} }}{\text{ < 1}} \hfill \\ \begin{array}{*{20}l} {\frac{{\left( {{\text{h}}_{{\text{b}}} {\text{ + Z}}_{{{\text{max}}}} } \right)}}{{{\text{h}}_{{\text{a}}} }}{\text{ = 1}}{\text{.025 + 0}}{\text{.125}}\left( {\frac{{{\text{U}}_{{\text{b}}} }}{{{\text{U}}_{{\text{c}}} }}} \right)} & {{\text{1}} \le \frac{{{\text{U}}_{{\text{b}}} }}{{{\text{U}}_{{\text{c}}} }} \le {\text{1}}{\text{.8}}} \\ \end{array} \hfill \\ \end{gathered}$$
(9)

Carnacina et al. [7] conducted several experiments to investigate the effects of flow characteristics under free and pressurized flow conditions on the bridge pier scour and its evolution. This study showed that the scouring process is non-linear, and the scour depth increases around the bridge pier in the pressurized flow conditions.

Sung-Uk Choi and Byungwoong Choi [25] proposed Eq. (10) that the predicts the time to equilibrium.

$$\frac{{\text{z}}_{\text{t}}}{{\text{Z}}_{\text{max}}}\text{=exp}\left\{\text{0.065}{\left[\frac{\text{U}}{{\text{U}}_{\text{c}}}\right]}^{\text{0.35}}{\left[\frac{{\text{h}}_{\text{a}}}{{\text{d}}_{\text{p}}}\right]}^{\text{0.19}}\text{ln}\left[\frac{\text{t}}{{\text{t}}_{\text{e}}}\right]\right\}$$
(10)

where Zt scour depth in each time t and te is the time to equilibrium scour depth.

Abdelaziz and Lim, using 7 abutment ratios for the bridge in the floodplain of compound open channel, concluded that the average width of the abutment is approximately 3.5 times the depth of the flow, and this result is different from the result recommended by the FHWA (Federal Highway Administration, 2009) [1].

The cross-section of many rivers in the vicinity of residential and agricultural areas is a compound channel. In most of these areas, the floodplains are covered by vegetation. Due to the difference in flow velocity in the main channel and floodplains in the compound section of the river, significant changes occur in the flow structure near the interface between the main channel and floodplains. These changes may also create vortices resulting in excess energy loss in the flow. Many studies have been undertaken to understand the hydraulic flow conditions in compound channels with and without vegetation on floodplains, including Shiono and Knight [22], Rameshwaran and Shiono [17], Zarrati et al. [29], Shan et al. [21], Tanino and Nepf[27], Tang et al. [26], Sonnenwald et al. [23] and Samadirahim et al. [20]. For example, Shiono and Knight [22] presented the application of the Navier-Stokes equation under constant and uniform conditions without vegetation:

$$\begin{gathered} \begin{array}{*{20}c} {\rho gHS_{{ox}} + } & {\frac{\partial }{{\partial y}}\left( {\rho \lambda \sqrt {\frac{f}{8}} ~~H^{2} U_{d} \frac{{\partial U_{d} }}{{\partial y}}} \right) - } & {\frac{{\rho \cdot f}}{8}~\sqrt {1 + S_{{oy}}^{2} } ~~U_{d}^{2} = \Gamma } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {{\text{U}}_{{\text{d}}} {\text{ = }}\frac{{\text{1}}}{{\text{H}}}\int\limits_{0}^{{\text{H}}} {\text{u}} \cdot {\text{dz,}}\,} & {{\text{u}}_{{\text{*}}} {\text{ = }}\sqrt {\frac{{\tau _{{\text{b}}} }}{\rho }} {\text{,}}\,} & {\Gamma = \frac{\partial }{{\partial y}}\left[ {{\text{H}}\left( {\rho {\text{UV}}} \right)_{{\text{d}}} } \right]} \\ \end{array} \hfill \\ \end{gathered}$$
(11)

where H is the flow depth in the main channel (equivalent with ha), Sox is the bed slope, Soy is the side slope of the channel, λ is dimensionless eddy viscosity coefficient, f is Darcy-Wisbach roughness coefficient, Г is secondary flow term, Ud is depth-averaged velocity, u* is shear velocity. In addition, x and y indicate the longitudinal and transverse directions of the flow, respectively, and U and V are the velocities in the x and y directions, respectively.

Sonnenwald et al. [23] proposed Eq. 12 to estimate the drag coefficient (CD) by adding a new parameter based on element diameter and vegetation density.

$$\begin{array}{*{20}c} {{\text{C}}_{{\text{D}}} {\text{ = 2}}\left( {\frac{{{\text{6475d + 32}}}}{{{\text{Re}}_{{{\text{rod}}}} }}{\text{ + 17d + 3}}{\text{.2}}\phi {\text{ + 0}}{\text{.50}}} \right),} & {\phi \, = \,\frac{{{\text{N}}_{{\text{v}}} \pi {\text{d}}^{{\text{2}}} }}{{\text{4}}},} & {{\text{Re}}_{{rod}} {\text{ = }}\,\frac{{{\text{U}}_{{\text{p}}} {\text{d}}}}{\upsilon }} \\ \end{array}$$
(12)

where Rerod is the Reynolds number of the rod, ϕ is obstruction volume fraction, d is rod diameter, Nv number of rods per unit area, ν is kinematic viscosity, and Up is average vertical velocity near the rods.

Tanino and Nepf [27] showed that the drag coefficient decreases with increasing Reynolds number for ϕ < 0.09, and Rerod > 1000, it is in the range of 1-1.05. In previous studies, the rate of bridge pier scouring under pressure flow conditions in simple rectangular sections and compound channels has not been investigated thoroughly. Due to the hydraulic complexity of the flow in the compound channels, particularly with floodplains covered with vegetation, the flow field can be completely different from that in free flow conditions, further impacting on the bridge pier scour.

In previous studies, the effects of vegetation density, compound channels, bridge pier and pressurized flow conditions on scouring rate have been studied separately, but their combined effects have not been investigated yet. Therefore, this paper aims to experimentally investigate the combined effects of vegetation density, and pressurized flow under the bridge deck with different relative depths in a compound channel with three different width ratios on the bridge pier scour depth. Moreover, an optimized prediction model for the bridge scour depth considering these combined effects was proposed in the present study.

In the 2 section, laboratory methods, equipment and facilities are introduced and experiments with different conditions are explained. Then, the hydraulic analysis of the flow approaching the bridge base and the relevant parameters will be done. Also, similar to the existing equations for calculating the scouring of the bridge pier in free flow conditions, a new equation has been presented to estimate the amount of scouring pier under pressurized condition and in compound channels with vegetated.

2 Experimental set-up and methodology

The experiments were performed in a 10 m long and 1.5 m wide flume with a fixed bed slope of 0.001 (Fig. 2). This channel consists of the main channel of a trapezoidal cross-section with a bottom width of 0.3 m and a side slope of 1:1, the bank-full depth of 0.15 m, and two symmetrical floodplains with each being a width of 0.45 m. The deck of the rectangular bridge is 1.5 m long and 0.3 m wide on a cylindrical pier with a diameter of 0.05 m at a distance of 6 m upstream and the middle of the main channel.

Fig. 2
figure 2

General view of a compound open channel

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The dimensionless parameters in this research are obtained by using П-method as follows.

$$\frac{{{\text{Z}}_{{{\text{max}}}} }}{{{\text{h}}_{{\text{a}}} }}{\text{ = f}}\left( {\frac{{\text{U}}}{{{\text{U}}_{{\text{C}}} }} \cdot \frac{{{\text{h}}_{{\text{a}}} }}{{{\text{h}}_{{\text{b}}} }} \cdot \frac{{\text{B}}}{{\text{b}}} \cdot \frac{{{\text{y}}_{{\text{f}}} }}{{{\text{h}}_{{\text{a}}} }} \cdot {\text{T}}} \right){\text{ = f}}\left( {\xi \cdot \psi \cdot \chi \cdot {\text{Dr}} \cdot {\text{T}}} \right)$$
(13)

where ξ is relative velocity (U/UC), ψ is ratio of ha/hb, χ is width ratios of compound sections(B/b), B is Half the width of the flume, b is Half the width of the main channel, yf is the flow depth in the floodplain, Dr is relative depths (yf /ha) and T vegetation densities.

Table 1 shows the range of changes of the studied parameters.

Table 1 Domain of changes of the studied parameters
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The experiments included three χ = B/b = 2.5, 2.17, and 1.83, three Dr = 0.3, 0.4, 0.5, three vegetation densities (T1, T2, T3) and without vegetation experiments (T0). All experiments of this research were performed in clear water conditions. To simulate vegetation, cylindrical plastic elements with a diameter of 0.01 m have been used. The vegetation density is calculated by Table 2 summarizes the specifications of three types of emergent vegetation densities. Table 3 also shows the general details of the experiments.

Table 2 Characteristics of different emergent vegetation densities
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Table 3 General details of experimental programs in this research
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To evaluate the flow pressure under the bridge deck, several piezometers was installed on the deck in three rows in the center and at the ends of the main channel. The cross-section was selected in the middle of the bridge deck. Flow velocity was measured at transverse and vertical interval distances of 0.05 m and 0.025 m, respectively and they were measured by a two-dimensional velocimetry made by Delft Hydraulic Company. The bottom of the main channel was covered with non-cohesive erodible materials with uniform sediment sand of d50 = 1.3 mm. in this material uniformity coefficient (Cu), median size (Dg) and gradation coefficient(σg) equal 1.7, 1.16, 1.5, respectively.

The scour depth and longitudinal profile of the bed around the bridge pier were measured automatically using a bed digitizing machine. Due to the symmetry of the channel, the velocity measurements were performed for only half of the cross-section. To establish constant flow discharge in the laboratory channel by considering three B/b ratios, the corresponding stage-discharge curves were plotted with and without vegetation (four types) on the floodplain, as shown in Fig. 3.

Fig. 3
figure 3

Stage-discharge curve in expriments

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To determine the effective dimensionless parameters on the depth of the maximum bridge pier scour in pressurized flow conditions and based on the uncertainty analysis of parameters, the percentage of coefficient of variation for the effective parameters of ξ = U/Uc, ψ = ha/hb ,χ = B/b, Dr and T were 12.62, 20.41, 84.52, 12.96, and 8.45%, respectively.

Based on the equation provided by Shan et al. [21] in smooth and vegetated compound channels, the relationship between the bed shear stress and the velocity in the vicinity of the bed (Ub) in each section can be expressed as follows.

$$\tau _{{\text{b}}} {\text{ = A}}_{{\text{i}}} {\text{U}}_{{\text{b}}}^{{\text{2}}} {\text{~,}}\,{\text{A}}_{{\text{i}}} {\text{ = }}\,\rho \left( {{\raise0.7ex\hbox{${{\text{f}}_{{\text{i}}} }$} \!\mathord{\left/ {\vphantom {{{\text{f}}_{{\text{i}}} } {\text{8}}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\text{8}}$}}} \right){\text{K}}_{{\text{i}}}^{{\text{2}}} {\text{,}}\,{\text{K}}_{{\text{i}}} {\text{ = }}\frac{{{\text{U}}_{{\text{d}}} }}{{{\text{U}}_{{\text{b}}} }}$$
(14)

where Ai is the dimensionless parameter and subscript i is number of each section. In this method, based on the width of the interaction area (δ) and three different values ​​of Ai for the three areas of floodplain, the main channel and the distance between the main channel and floodplain are determined. The friction factor (fi) can be calculated using the Colebrook-White equation:

$${\text{f}}_{{\text{i}}} = \left[ { - 2\log \left( {\frac{{3.02\nu }}{{\sqrt {128{\text{gH}}^{3} {\text{S}}_{{{\text{ox}}}} } }} + \frac{{{\text{k}}_{{\text{s}}} }}{{\Delta {\text{H}}}}} \right)} \right]^{{ - 2}}$$
(15)

where ks is the equivalent roughness of sand (ks value is similar to Nikuradse’s sand grain dimension), Δ equal to 12.3 and 1.2 for smooth and vegetated beds, respectively.

In steady conditions, the apparent shear stress between the main channel and floodplains has been calculated based on the balance of effective forces at the junction of the main channel and floodplains (Fig. 4).

Fig. 4
figure 4

Perspective view of control volume in compound channels

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$$\underbrace {{ - \left( {F_{{{\text{P1mc}}}} {\text{ }} - {\text{ }}F_{{{\text{P2mc + back~water}}}} } \right)}}_{{(1)}} - \underbrace {{\rho {\text{g}}\overline{{\text{A}}} _{{{\text{mc}}}} {\text{LS}}_{{\text{o}}} }}_{{(2)}} + \underbrace {{\left( {2\overline{{{\text{SF}}_{3} }} {\text{ }} + {\text{ }}\overline{{{\text{SF}}_{4} }} } \right){\text{L}}}}_{{(3)}} + \underbrace {{\rho \left( {\beta _{{2{\text{mc}}}} U_{{2{\text{mc}}}} Q_{{2{\text{mc }} - {\text{ }}}} \beta _{{1{\text{mc}}}} U_{{1{\text{mc}}}} Q_{{1{\text{mc}}}} } \right)}}_{{(4)}} + \underbrace {{2{\text{ASF}}_{{\text{V}}} {\mkern 1mu} = {\mkern 1mu} 0}}_{{(5)}}$$
(16)

where Fp is the pressure force upstream and downstream of the control volume, Wmc water weight in the main channel, Umc mean velocity in the main channel, Qmc is discharge passing through the main channel, β is the momentum correction factor, and L is the length of control volume. SF3 and SF4 are the shear forces of the wall and the bottom of the main channel, respectively. Subscripts 1 and 2 refer to the upstream and downstream sections of the control volume, l and r indicate the left and right floodplains and mc indicate the main channel. where the terms 1 to 5 express the force due to the hydrostatic pressure, water weight, shear force of the wall and bottom of the main channel, momentum flux, and apparent shear force, respectively.

3 Results and discussion

3.1 The velocity distribution

In Fig. 5, the depth velocity distribution in the sections A, B and C with and without vegetation on the floodplains is shown. Comparing the results shows that: in all cases and all sections without vegetation, moving away from the bottom of the channel and approaching the water surface, the difference between the measured velocity values increases with the logarithmic velocity distribution. This is due to the existence of the deck bridge and as a result of the flow retardation. Also, Vegetation causes fluctuations in the depth distribution of flow velocity and the difference between the measured and logarithmic velocity reaches its maximum on upper interface and on the floodplain. This phenomenon increases the amount of apparent shear stress in the interface area. This result has been confirmed by Samadirahim et al. [20].

3.2 Bed shear stress

Figure 6 shows the bed shear stress along the bed of compound channel with different vegetation densities and floodplain widths in upstream of the bridge. In all cases, a sudden drop in the amount of the bed shear stress occurs at the interface, which is due to the momentum transfer between the main channel and the floodplains. Furthermore, it is expected that for a certain relative depth, compared to smooth floodplain, the shear stress will increase by increasing roughness on floodplain. In these experiments, with increasing vegetation density on floodplain, discharge and consequently flow velocity decreased and finally shear stress is reduced. Examining the scouring results showed that there is a direct relationship between the bed shear stress and the scouring depth of the bridge pier. An increase in shear stress causes an increase in scour depth and the reduction of shear stress reduces the scour depth.

Fig. 5
figure 5

Measured and logarithmic veritcal profiles time-averaged velocity in the lower, upper interface and middle of the floodplain

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Fig. 6
figure 6

Boundary shear stress distribution

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3.3 Apparent shear force

The vertical apparent shear force (ASF) between the main channel and floodplains is calculated based on the momentum equation and the balance of effective forces at the interface of the main channel and floodplains, given in Eq. (16). Figure 7 shows the changes of this parameter. The results show that: With increasing vegetation density, the difference in flow velocity in the main channel and the floodplain increases, and as a result, the amount of apparent shear force increases. Also, the apparent shear force tends to increase with decreasing χ(Β/b) from 2.5 to 1.83 for a certain relative depth.

Changes in the apparent shear force to the amount of vegetation per square meter at interface are defined as follows:

$$\frac{{\partial {\text{ASF}}}}{{\partial {\text{T}}}}{\text{ = }}\frac{{\left( {{\text{ASF}}_{{{\text{i + 1}}}} - {\text{ ASF}}_{{\text{i}}} } \right)}}{{\left( {{\text{T}}_{{{\text{i + 1}}}} - {\text{ T}}_{{\text{i}}} } \right)}}$$
(17)

The changes of this parameter versus the relative depth indicate the inverse relationship between the relative depth increase and the apparent shear force (Fig. 8). This issue is important in determining the amount of effective bed shear stress in the main channel and, as a result, the amount of bed erosion in the vicinity of the bridge pier.

Fig. 7
figure 7

Mean apparent shear force at the interface of the main channel and floodplain (Section A-B).

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Fig. 8
figure 8

changes of the apparent shear force geradieant versus relative depth at section A-B.

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3.4 Bridge pier scour depth

The maximum scouring speed has been measured in all the experiments of this research. In almost all tests, the maximum scouring depth occurs in the first two hours of the tests, and then this process decreases sharply. However, in this research, all hydraulic and geometric parameters were measured after six hours.

The maximum scour depth around the bridge pier under free and pressurized flow conditions is shown in Fig. 9. The bridge pier scour depth under pressurized flow conditions (except for case Dr1) is approximately two to three times that of the bridge pier scour depth under free-flow conditions. This result agrees with the previous study by Abed [2]. In pressurized flow conditions, due to the presence of bridges and backwater, the rate of bed erosion decreases by increasing the vegetation density and decreasing the flow velocity near the bed. In addition, at a given floodplain width and a constant vegetation density, increasing the relative depth increases the flow pressure under the bridge deck, resulting in an increase of the scour depth by an average of 45%. It can be seen that increasing vegetation density on average will lead to the decrease of the scour depth which can be related to the decreasing bed shear stress along the floodplain side. The results showed that at the same floodplain width and relative depth, 95% increase of the vegetation density can reduce the scour depth by 15%. However, in free flow conditions, increasing the vegetation density on average will increase the scour depth. Also, results showed that as the flow discharge is proportional to the cross-section of the channel and the energy slope, the changes in the floodplain width will not have much effect on the bridge pier scour. Consequently, the relative depth parameter has more dominant effects on the scour over the other two parameters of vegetation density and floodplain width.

Fig. 9
figure 9

Maximum scour depth around bridge pier under pressurized and free flow conditions

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3.4.1 Similarity of scour profiles

In each experiment, the ratio of scour depth in the longitudinal direction of the flow to the maximum scour depth (Z/Zmax) has been calculated.

Fig. 10
figure 10

Longitudinal profile of bed scour in Dr2,B,T.

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According to Fig. 10, the best equation is fitted for the scour profile of the bed upstream and downstream of the bridge pier. These equations are independent of relative depth and vegetation density. For the scour profiles in Fig. 10, the similarity profile for X/W < zero is arranged into equation follows:

$$\frac{{{\text{Z}}_{{{\text{max}}}} }}{{\text{Z}}}\,{\text{ = }}\, - {\text{0}}{\text{.2414}}\left( {{\raise0.7ex\hbox{${\text{X}}$} \!\mathord{\left/ {\vphantom {{\text{X}} {\text{W}}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\text{W}}$}}} \right)^{{\text{2}}} - {\text{0}}{\text{.9994}}\left( {{\raise0.7ex\hbox{${\text{X}}$} \!\mathord{\left/ {\vphantom {{\text{X}} {\text{W}}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\text{W}}$}}} \right) - {\text{1}}{\text{.0245}}$$
(18)

For X/W > zero, the profile in Fig. 10 can be approximated by equation follows:

$$\frac{{{\text{Z}}_{{{\text{max}}}} }}{{\text{Z}}}\,{\text{ = }}\, - {\text{0}}{\text{.0623}}\left( {{\raise0.7ex\hbox{${\text{X}}$} \!\mathord{\left/ {\vphantom {{\text{X}} {\text{W}}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\text{W}}$}}} \right)^{{\text{2}}} - {\text{0}}{\text{.591}}\left( {{\raise0.7ex\hbox{${\text{X}}$} \!\mathord{\left/ {\vphantom {{\text{X}} {\text{W}}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\text{W}}$}}} \right) - {\text{0}}{\text{.9468}}$$
(19)
Fig. 11
figure 11

Normalized scouring profile

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Figure 11 was used to define the initiation of pressure flow scour and deposition. For applications, Fig. 11 gave a normalized scour profile with highlighted positions of interest. For reference, the bridge deck is shown in this figure. To develop a suitable relationship for estimating the maximum scour depth of the bridge base in pressurized conditions in compound channels with vegetation on floodplains, different scenarios about the parameters of vegetation densities (T), different floodplain width ratios (χ) and relative depths (Dr) were examined in Eq. (6). In addition to the results of sensitivity analysis, the scope of application of the relative depth parameter has priority and superiority over the two parameters of vegetation densities and floodplain width. Therefore, the relative depth was selected to generalize the previous formulas for compound channels. The results showed that considering the relative depth in Eq. (6), a correlation coefficient of 92% can be obtained between laboratory data and Eq. (20).

$${\text{z}}_{{{\text{max}}}} \,{\text{ = }}\,\left( {{\text{2}}{\text{.0K}}_{{\text{1}}} {\text{K}}_{{\text{2}}} \left( {\frac{{{\text{d}}_{{\text{p}}} }}{{{\text{h}}_{{\text{a}}} }}} \right)^{{{\text{0}}{\text{.65}}}} {\text{Fr}}^{{{\text{0}}{\text{.43}}}} {\text{h}}_{{\text{a}}} {\text{ + }}\left[ {\frac{{{\text{Q}}^{{\text{2}}} }}{{{\text{40D}}_{{\text{m}}}^{{{\raise0.7ex\hbox{${\text{2}}$} \!\mathord{\left/ {\vphantom {{\text{2}} {\text{3}}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\text{3}}$}}}} {\text{b}}_{{\text{c}}}^{{\text{2}}} }}} \right]^{{{\raise0.7ex\hbox{${\text{3}}$} \!\mathord{\left/ {\vphantom {{\text{3}} {\text{7}}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\text{7}}$}}}} } \right){\text{Dr}}$$
(20)

Equation (20) takes into account the presence of the bridge pier (first term), the effects of pressurized flow (second term), and the effect of geometry in the compound channel. In the development of Eq. (6), the effect of different vegetation densities on floodplains was initially considered. However, the effect of vegetation on the floodplain was eliminated to simplify the equation as given in Eq. (20) because the removal of the parameter of vegetation density results in a difference of less than 10%. Due to the lack of other data available in the literature for the parameters of Eq. (20), it is difficult to compare this equation with the previous equations. Figure 12 show that compares the maximum bridge scour depth based on Eqs. 20 and 6.

Fig. 12
figure 12

Comparison between maximum bridge scour depth in experiments versus Eqs. 20 and 6

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3.5 The effect of vegetation density, relative depth and the ratio of channel width to floodplain width

The increase of vegetation, in the conditions of pressurized flow, causing an increase the length of the upstream flow backwater of the bridge and ultimately leads to the reduction of scouring of the bridge pier. The increase of relative depth, causing an increase discharge and ultimately leads to the reduction of scouring of the bridge pier. When the ratio of channel width to floodplain width increases, cross section increases. The increase in cross section, leading to decreases of the flow velocities. As a result, scour depth decreases with increasing the ratio of channel width to floodplain width.

4 Conclusions

Bridge pier scouring is a very complex phenomenon and it is difficult to predict, but by simplifying some complex parameters, it is possible to arrive at empirical equations with good accuracy to predict the amount of bridge pier scouring.

Pressurized flow conditions and compound channels increase the complexity of this problem. In this study, the CSU equation was chosen as the starting point for the development of this model because the current scouring model is approved by the Federal Highway Administration (FHWA) and is widely used. The experiments were performed to investigate the hydraulic characteristics of the flow approaching the bridge, and the bridge pier scour depth with different vegetation densities in different compound channels. Finally, using the previous equations and based on the dimensionless parameters affecting scouring, a new equation was obtained to quantify the maximum scour depth around the bridge pier in the channel with vegetation on the floodplain. The predicted value using this optimized model exhibits good agreement with the experimental value (R2 = 0.92). In consideration of the combined effects on scour depth, this estimated result is proved to be closer to the actual data when compared to the previous model.

Among the dimensionless parameters, relative depth was the best parameter that has the highest correlation coefficient with experimental data. The findings of the present study to predict the scour depth of a pressurized flow can be summarized as follows:

  • Scour around the bridge piers in pressurized flow becomes faster than the free flow conditions.

  • To estimate the maximum scour depth in compound channels, the relative depth parameter has priority and superiority over the two parameters of vegetation density and floodplain width.

  • More data are required to evaluate the effect of other independent parameters on the scour depth in the pressurized flow.

  • In free and pressurized flow, increasing relative depth, will increase the scour depth of the bridge pier.

  • The rate of change in scour depth decreases over time and becomes almost zero as the scour approaches equilibrium.

  • The scour morphology before reaching the maximum scour depth, due to the pressure flow conditions, is two-dimensional, however, after reaching the maximum scour depth, the scour morphology becomes three-dimensional, due to the effect of free flow.

  • In pressurized flow, the position of the maximum scour depth is rapidly approaching its equilibrium position near the lower edge of the bridge deck.

  • At the same floodplain width and relative depth, in pressurized flow, increasing vegetation density on average will reduce the scour depth by 15%. While in free flow conditions, increasing the vegetation density on average will increase the scour depth.

For the effectiveness of the recommended equations, it would be useful to further research and also consider the following:

scouring experiments in clear water conditions and sediments mixed with sand as bed materials.

Investigating scouring of a series of bridge piers under pressurized flow conditions.

Carrying out scouring experiments of the bridge pier series in the conditions of pressurized flow and live bed (sediment recirculation in order to satisfy with the principle of mass conservation in the balance of sediments mass).