Simulation Study on Local Scour Characteristics of Tandem Bridge Piers in a Straight River under a Changing Environment

Abstract

Hydrodynamics is a common manifestation that causes natural scouring of riverbeds, and it is one of the factors that exacerbate the natural disasters of local scouring of bridge piers, causing sustainability of environmental changes in the water. The evolution pattern and scour characteristics of the bed surface around the submerged structures under different scouring conditions vary greatly. In order to investigate the scour mechanism, the reformed group (RNG) turbulence model in computational fluid dynamics (CFD) simulation software (v11.2) was used to simulate the scour under the clear-water scour and live-bed scour environments, and different scour morphology characteristics around the tandem piers under the clear-water scour and live-bed scour environments were obtained in the final simulation. By capturing the cross-sectional vortex and bed shear stress during the scouring process, the characteristic pattern of scouring topography around the pier and the relationship between the scour hole structure scale were analyzed, and the relationship equation between the development of scour depth and time scale was established. The study shows that: under the clear-water scouring environment, the sediment transport rate lags behind, but the contribution time is superior; under the live-bed scouring environment, by the shading and reinforcement influence of the upstream piers, the extent and development of the downstream pier surrounding the scour hole is small; the development trend of the maximum sediment transport rate of the scour hole and the great value of the shear stress is more synergistic, and the peri-pier eddy is positively correlated with the bed shear stress; through the regression equation to compare the relevant test and simulation results, the two are in good agreement, indicating that the simulated local scour evolution law is consistent with the actual law.

1. Introduction

The problem of local scouring of submarine structures is characterized by complex evolutionary patterns of sediment transport and water flow structures at three-dimensional spatial and temporal scales; therefore, numerical simulation studies have attracted much attention [1,2,3], aiming to deeply resolve the scouring mechanism and improve the accuracy of scour depth prediction. Combined with the trigger mechanism and transient mechanism of foundation scour, the mathematical model of local scour involves fluid motion, sediment transport and streambed deformation, and the process of sediment transport is complicated, including the criteria for discriminating between suspended and bed load transition modes and the critical starting elements of particles, etc., which may lead to bias in the numerical simulation results of the local scour if missing.

Direct simulation and large eddy simulation of a transient analytical solution of the natural riverbed surface turbulence model need to dissipate many resources pooling, so the practical design mostly uses a Reynolds time-averaged equation (RANS) for simulation study [4,5]. The turbulence characteristic term of the RANS equation is based on an eddy viscosity model, and various models are derived for eddy structure, which provides a selection strategy to deal with complex eddies. A large number of scholars [6,7,8] have used different turbulence models to explore the effect of each model on the characteristic physical quantities such as near-wall flow field, return region and vortex structure. In the reformed group (RNG) model, the three-dimensional transient vortex street is taken into account, which is superior for the simulation of vortex evolution characteristics [9].

Considering the spatial and temporal evolution characteristics of the pier surrounding the scour, the pier foundation scour [10] is divided into two states: clear-water scour and live-bed scour, due to the interaction between the bypass flow field and sand bed scour; the research [11,12,13] extensively focuses on the dispersion of turbulent flow characteristics and clear-water scouring patterns on the pier side, with typical indoor model experiments as the main means. The existing research results mainly focus on the clear-water scouring process around piers, and the research on the water–sand coupling law, scouring transient mechanism and time accumulation effect under the muddy water condition needs to be explored continuously. Burkow and Griebel [14] selected the bed load transport equation for clear-water scour tests; Roulund et al. [15] simulated the velocity field around a cylindrical pier, capturing the key parameters, but only explored the role of bed load; Baykal et al. [16] added a suspended mass parameter term to the sediment scour module based on the premise of Roulund et al.’s [15] conclusion, and the related results showed that the contribution of suspended mass to local scour was large; Sumer et al. [17] described the water–sand movement in the live bed by the LES large eddy model, adjusted the starting shear stress to estimate the sediment migration intensity and compared with the results of Williams et al. [18] to arrive at basically the same conclusion, and the influence of upstream incoming sand transport on pier circumferential scouring was analyzed.

This paper addresses the foundation scouring problem of bridge pier structures in natural river sections. Taking the Yellow River Special Bridge in Hohhot, Inner Mongolia (see Figure 1), as the prototype pier, model experimental research on the scour of bridge piers with tandem equal diameter in series was carried out using model flow analysis software under clear-water scour and live-bed scour conditions. By capturing the cross-sectional eddies and bed shear stresses during the scouring process, the characteristic pattern of scouring topography around the pier and the spatial and temporal evolution characteristics of the scour hole were analyzed, and the relationship equation between the development of scour depth and time scale was established, and the accuracy of numerical calculation was verified against the physical test data, in order to analyze the physical mechanism of the river’s water–sand interaction. At present, the research on bridge pier scouring is mostly limited to indoor clear-water scouring research, but this paper simulates the live-bed scouring with reference to the field site bridge pier, which makes a breakthrough for solving the bridge pier scouring issue.

The remainder of this paper is organized as follows: Section 2 introduces the underlying theory and feature scales required for model construction and validates the model accuracy in Section 3; Section 4 and Section 5 present the feature quantities and arithmetic analysis of pier circumferential scouring; and the last section concludes.

2. Theoretical Model Construction

The whole calculation procedure of the model is an iterative solution process of the flow solid-phase environment, based on the hydrodynamic and riverbed deformation module, to simulate the water and sand transport around the pier, and the specific calculation process is shown in Figure 2.

2.1. Flow Movement

2.1.1. Flow Foundation Equation

The continuity equation and the equation of motion of the incompressible fluid constrain the water flow motion. Given the software-specific mesh technique [19], the continuity equation and Navier–Stokes (N-S) equation, which are also the control equations, are used to simulate the water flow motion. The control equations in the Cartesian coordinate system [20] are as follows:

Continuity equation:

$$\begin{array}{c}\partial \left(A_i{u}_i\right)/\partial x_i=0\end{array}$$

(1)

N-S equation of motion:

$$\frac{\partial u_i}{\partial t}+\frac{1}{V_F}\left(u_j{A}_j\frac{\partial u_i}{\partial x_j}\right)=-\frac{1}{\rho }\frac{\partial P}{\partial x_i}+G_i+F_i$$

(2)

where A_i is the area parameter of flow in the direction i; u_i is the flow velocity vector in the direction i; x_i is the coordinate component of the Cartesian coordinate system; t is the time; V_F is the volume parameter of flowable; ρ is the fluid density; P is the fluid pressure; G_i is the body acceleration in the direction i; and F_i is the viscous force acceleration in the direction i.

2.1.2. Turbulence Control Equations

Incompressible fluids are divided into two types of motion: turbulent and laminar, both of which are distinguished by the presence of viscous shear stress caused by the relative motion between the flow layers. The fluid particles between the layers of turbulent flow in the process of motion will continue to diffuse and mix with the local flow velocity and pressure adjustment to form a vortex, with three-dimensional characteristics of probability, instability and vorticity.

The operational efficiency and accuracy errors of numerical calculations in computational fluid dynamics (CFD) are closely related to the accuracy of the turbulence model selection. Six turbulence models are built into the simulation software [21]: the Plante mixed length model, the single equation turbulent energy k model, the dual equation k-ε model, the dual equation k-w model, the reformed group (RNG) model and the large eddy simulation (LES) model.

To more thoroughly address the intricate details of the complicated vortex system structure, this paper selects the RNG model with dynamic calculation, which can shorten the calculation time to improve efficiency and accurately capture the vortex structure around the pier with wider applicability compared to the large vortex model [22]. Its control equation [23] is as follows:

$$\frac{\partial k}{\partial t}+u_j\frac{\partial k}{\partial x_i}=T_{ij}\frac{\partial u_i}{\partial x_j}+\frac{\partial }{\partial x_j}\left[\frac{1}{\rho }\left(\mu +\frac{{\mu }_t}{{\sigma }_k}\right)\frac{\partial k}{\partial x_j}\right]-\epsilon$$

(3)

$$\frac{\partial \epsilon }{\partial t}+u_j\frac{\partial \epsilon }{\partial x_i}=C_{1\epsilon }\frac{\epsilon }{k}{T}_{ij}\frac{\partial u_i}{\partial x_j}+\frac{\partial }{\partial x_j}\left[\frac{1}{\rho }\left(\mu +\frac{{\mu }_t}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_j}\right]-C_{2\epsilon }\rho \frac{{\epsilon }^2}{k}$$

(4)

where k is the Reynolds average kinetic energy, $k=\frac{1}{2}\overline{u_i^{\prime }{u}_j^{\prime }}$; ε is the turbulent kinetic energy dissipation rate, $\epsilon =\nu \left(\frac{\partial U_i^{\prime }}{\partial x_k}\right)\left(\frac{\partial u_j^{\prime }}{\partial x_k}\right)$; μ_t is the turbulent vortex viscosity, ${\mu }_t=\frac{c_{\mu }\rho k^2}{\epsilon }$; T_ij is the strain rate tensor, $T_{ij}=\frac{{\mu }_t}{\rho }\left(\frac{\partial U_i}{\partial x_j}+\frac{\partial u_j}{\partial x_j}\right)-\frac{2}{3}\rho k{\delta }_{ij}$; and C_1ε, C_2ε, σ_ε and c_μ are the model coefficients.

2.2. Bed Scouring

2.2.1. Determination of Starting Shear Stress

The dimensionless representation of shear stress is the Shields number, defined [24] as follows:

$${\theta }_n=\frac{\tau }{g{d}_n\left({\rho }_n-{\rho }_f\right)}$$

(5)

where g is the acceleration of gravity; ρ_n is the sediment density; ρ_f is the fluid density; and d_n is the sediment particle size.

The bed shear stress is calculated by considering the bed roughness, assuming that the Nicolaides rough height k_s of the bed is equivalent to the median grain size of the bed sand:

$$k_s=c_r{d}_{50}$$

(6)

where c_r is the scaling factor, which takes the value of 2.5 in this paper.

The critical Shields number [24] is used to define the critical streambed shear stress at which both sediment hostage and bedload transport begin sediment movement:

$${\theta }_{cr,n}=\frac{{\tau }_{cr,n}}{g{d}_n\left({\rho }_n-{\rho }_f\right)}$$

(7)

The critical Shields number can be determined by calculating the Soulsby–Whitehouse equation [25], and the resulting curve is consistent with the trend of the Shields curve, which is widely used [26]:

$${\theta }_{cr,n}=\frac{0.3}{1+1.2d_{\ast ,n}}+0.055\left[1-\exp \left(-0.02d_{\ast ,n}\right)\right]$$

(8)

where d_∗,n is the dimensionless particle size, ${\mathrm{d}}_{\ast ,n}=d_n{\left[\frac{g\left(s_n-1\right)}{v_f^2}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$; $s_n={\rho }_n/{\rho }_f$; and v_f is the kinematic viscosity coefficient of the fluid.

At the inclined bed interface, the sediment particles are more unstable, so gravity exerts a tangential force component to stabilize the bed according to the flow direction. If the fluid rises in the slope, the critical shear stress will increase, and vice versa, it will decrease. Given the effect of slope on the critical starting of sediment particles, the above equation is modified [25] to

$${\theta }_{cr,n}^{\prime }={\theta }_{cr,n}\frac{\cos \psi \sin \beta +\sqrt{{\cos }^2\beta {\tan }^2{\varphi }_n-{\sin }^2\psi {\sin }^2\beta }}{\tan {\varphi }_n}$$

(9)

where β is the bed inclination angle; ϕ_n is the silt rest angle, defined as the steepest slope angle before the sand slides on its own; and ψ is the angle between the direction of fluid flow near the sloping bed and the upslope direction, ranging from 0° to 180°, where 0° is the upslope flow and 180° is the downslope flow.

Sediment transport causes bed deformation and is divided into suspended and bed load transport with reference to the difference in sediment transport forms; the effect of both is considered by sediment grain size and flow velocity. Under the natural channel flow condition, the actual sediment size is defined by the critical size D_c [23], where the critical size is:

$$D_c=U^2/360g$$

(10)

where U is the instantaneous velocity; and g is the acceleration of free fall.

2.2.2. Scour Module Calculation

The sediment scouring module of the CFD software can simulate and calculate the structurally complex particle erosion motion; it mainly interprets the coupled motion of sediment and water flow by predicting the three-dimensional characteristics of sediment erosion, transport and deposition.

1.

Sand carrying and deposition module

For the sediment particle entrainment velocity, the velocity of particles leaving the sand bed is classified as the improved velocity and is calculated based on Winterwerp et al. [27]:

$$u_{lift,n}={\alpha }_n{n}_b{d}_{\ast ,n}^{0.3}{\left({\theta }_n-{\theta }_{cr,n}\right)}^{1.5}\sqrt{g{d}_n\left(s_n-1\right)}$$

(11)

where α_n is the entrainment coefficient (takes the value of 0.018); $s_n={\rho }_n/{\rho }_f$; and n_b is the outer normal direction of the bed load surface.

Sedimentation is the process by which sediment particles settle out of suspension onto a filled bed or remain in bed load transport due to their weight. In the sediment particle deposition process, the settling velocity of Soulsby and Damgaard [25] was used

$$u_{settle,n}=\frac{v_f}{d_n}\left[{\left({10.36}^2+1.049d_{\ast ,n}^3\right)}^{0.5}-10.36\right]$$

(12)

where v_f is the coefficient of viscosity of fluid movement.

2.

Bed Load Module

The key to the bed load movement of sediment is determined by calculating the bed load transport rate, and the modal flow analysis software considers three equations, namely the Meyer-Peter and Muller, Nielsen [28] and van Rijn [29] equations. In this paper, the Meyer-Peter bed load sand transport rate equation is used:

$${\mathsf{\Phi }}_n=B_n{\left({\theta }_n-{\theta }_{cr,n}\right)}^{1.5}{C}_{b,n}$$

(13)

where B_n is the bed load factor, and the range of values is about 5.0–5.7 for low transport and about 8.0–13.0 for medium and high transport.

The dimensionless form of the volume bed load transport rate per unit width of the bed surface is defined by the grid method [30] as follows:

$$q_{b,n}={\mathsf{\Phi }}_n{\left[g\left(S_n-1\right)d_n^3\right]}^{0.5}$$

(14)

The bottom load layer thickness is calculated by the van Rijn [29] equation as follows:

$${\delta }_n=0.3d_n{d}_{\ast ,n}^{0.7}{\left[\left({\theta }_n-{\theta }_{cr,n}\right)/{\theta }_{cr,n}\right]}^{0.5}$$

(15)

To solve for the sediment particle motion at a single grid, the following equation is used to convert the single-width volumetric sand transport rate $q_{b,n}$ into the bed load flow rate $u_{b,n}$:

$$u_{b,n}=q_{b,n}/{\delta }_n{f}_b{c}_{b,n}$$

(16)

3. Verification of Geometric Modeling

3.1. Physical Model Conditions

The numerical simulations in this article were verified by indoor bridge pier scour test data. The test was conducted in a multi-functional variable slope flume, and the hydraulic model experimental device is shown in Figure 3. The maximum variable slope of the flume was 0–1%. The flow cross-section was rectangular, and the flume inlet was equipped with rectification equipment to eliminate the sudden change in water flow caused by external disturbance; the outlet was a movable tailgate with a controlled opening to achieve stable water flow conditions.

The test piers are equal-diameter tandem piers, pier width was 4.8 cm, and the model piers were field-measured tandem combination piers made by scaling 3D printing, as shown in Figure 4. Dynamic bed section laid two kinds of non-homogeneous sand simulated riverbeds, d₅₀ values were 0.63 mm and 0.51 mm and geometric standard deviation [2] ${\sigma }_g={\left(d_{84}/d_{16}\right)}^{0.5}>1.20$; the test simulated two kinds of scour conditions, respectively: when the profundity and scope of the scour pit on the pier side are no longer developed, the pier tail pile morphology height is no longer developed, and the prediction of local scouring reaches a quasi-equilibrium state.

3.2. Numerical Model Parameters

According to the sediment evolution characteristics of the riverbed, the relevant water sediment coupling model was established, and the specific geometric specifications and hydraulic parameters are shown in

Table 1. Geometric specifications and hydraulic parameters of the model.

Numerical Model	Scouring Environment	Pier Diameter	Water Depth	V/Vc	Fr	Re
A	Clear water	4.8 cm	34 cm	0.82	0.14	11,808
B	Live bed	4.8 cm	34 cm	1.21	0.20	17,424

.

The dimensions of the bridge pier numerical model correspond to the indoor hydraulic model and were divided into three computational domains from top to bottom (see Figure 5). Whether it is a transient flow-around model or a sediment erosion model, the size of the pier diameter D of the bridge pier constrains the size of the computational domain; the computational domain established in this study has a total transverse width of 13D, and the axial center of the bridge pier coincides with the center of the domain, which can ignore the effect of the contraction of the flume sidewall on the central region [2]; the total longitudinal length was 29D, and the axial center point of the front pier was 8D from the inlet boundary, and the axial center point of the rear pier was 16D from the outlet boundary, so that the front of the pier and the area behind the pier are not disturbed by flow and allow the full development of water flow [31]. To stabilize the water flow and prevent the bed material from eroding, solid baffles of the same width and height (blue area) were installed in the upstream and downstream areas of the moving bed section, respectively.

The quality of the grid division plays a crucial role in the simulation accuracy of deepening vortex core identification and scouring terrain. In this current study, a hexahedral mesh was employed to develop the spatial discrete solution, and the grid division was combined with nested grids to encrypt the grid to refine the pier surroundings (see Figure 6), which compresses the grid volume and improves the convergence speed while ensuring the accuracy. The overall area was symmetrical according to the downstream flow direction of the test section, and the grid scale was 1:2 from inside to outside, and the area with the inner edge length of 21D × 8D × 9D was partially encrypted, so that the horizontal grid size of the water–sand coupling area around the pier converges to 0.08D. After the pre-processing calculation, the amount of local nested meshes was 1.35 million.

To ensure the accuracy and efficiency of the numerical results, the value of the iteration time step was always taken to satisfy the Courant number in the range of 0.5–1.0. The whole process was obtained by Taylor series expansion [32].

$${\mathsf{\Phi }}_n={\mathsf{\Phi }}_{n+1}-\frac{\Delta x}{1!}{\left(\frac{\partial \mathsf{\Phi }}{\partial x}\right)}_2+\frac{1}{2!}{\left(\Delta x\right)}^2{\left(\frac{{\partial }^2\mathsf{\Phi }}{\partial x^2}\right)}_2-\dots$$

(17)

$${\mathsf{\Phi }}_{n+2}={\mathsf{\Phi }}_{n+1}+\frac{\Delta x}{1!}{\left(\frac{\partial \mathsf{\Phi }}{\partial x}\right)}_2+\frac{1}{2!}{\left(\Delta x\right)}^2{\left(\frac{{\partial }^2\mathsf{\Phi }}{\partial x^2}\right)}_2+\dots$$

(18)

Independence analysis of grid size and time step is indispensable, this paper uses different sizes of grids and time steps to validate the dynamic bed scour example, and the results are shown in

Table 2. Comparison of results for different grid sizes and time steps.

Model	Grid Size (m)	Time Step (s)	Iteration Residual
1	0.020	0.010	10−4
2	0.010	0.010	10−6
3	0.005	0.010	10−5
4	0.010	0.005	10−6

, in which the simulation convergence criterion was set as the residuals of the pressure iteration, which were less than 1 × 10⁴. According to the results in Table 2, from the point of view of the simulation efficiency, this paper selects the combination of model 2 to carry out the numerical simulation.

Given that the software does not provide a fully developed turbulent inlet boundary, the simulations in this paper first run the pierless flow-around model separately to obtain a fully developed boundary layer; the simulation results based on the fixed-bed flow-around model were embedded in the sediment model for the live-bed scour sequential simulation, in which the inlet boundary was a grid-covered boundary, aiming to assign a constant turbulent flux to the upstream inlet boundary layer of the sequential model; both the outlet and top surface boundaries were assumed as pressure boundaries, and both sides were assumed to be wall boundaries (see

Table 3. Boundary conditions for local scour models.

Software	Upstream Boundary	Downstream Boundary	Free Surface Boundary	Floor Boundary	Lateral Boundaries
Flow3d	Grid overlay	Specified pressure	Specified pressure	Wall	Wall

).

3.3. Verification of Scour Results

For the verification of the results of numerical calculation and experimental model under clear-water conditions, comparing to Figure 7, it can be found visually that the numerical results and the experimental results of the pier surrounding the scour hole structure morphology were almost the same, the pier surrounding the scour hole through the adjacent pier-surrounding area and pier-end dune accumulation were significantly “butterfly-shaped”, and there was siltation between the front and rear piers to prove that the shading and the consolidation effect of the anterior pier were significant. Overall, the distribution of numerical simulation and experimental measurement basically fits, and the simulation results can represent the actual measurement results approximately.

Figure 8 shows the comparison between the numerical results and the experimental results of the scour topography contours around the pier at 30 min, and the results show that the scour topography of the two is generally consistent. Only in front of the upstream pier are there differences, the simulated results of the waterfront scour hole morphology are not fully developed, and it is also a challenging issue for the present numerical simulation to address [33,34]. The highest scouring position of the adjacent pier perimeter area at this moment is at point b on both sides of the front pier, the simulated scouring depth value is 4.9 cm, and the measurement error of the test scouring depth value of 5.4 cm is about 9.3%; although the numerical simulation of the scour depth is smaller than the model test, it can still reflect the change pattern of topographic erosion and siltation near the pier column.

The maximum scour depth of pier circumference developed with time is defined as dt, and the maximum scour depth at 60 min of stable scour depth is defined as d_t60. Then, the evolution trend of d_t/d_t60 developed with time in the first 60 min is shown in Figure 9. The relative scour depth simulation value in the initial stage is slower than the test value growth rate, and the scour depth value between the two in the equilibrium stage is similar, and the scour development ephemeral curve shows that the numerical model prediction results agree well with the test results.

4. Local Field Characteristics

There are complex turbulent vortices near the bottom bed of the bridge pier, and the sand initiation around the pier is influenced by the vortex motion; many scholars [35,36,37] have conducted a lot of investigations on the structural characteristics of the vortex system around the pier, and this paper focuses on the correlation between the vortex coherent structure, bed shear stress and sediment transport on the pier side under different scouring environments.

4.1. Vortex Distribution

Based on the vortex kernel feature identification criterion, the vortex structure is predicted quantitatively by extracting the vortex kernel lines to discriminate the vortex size and then evaluate its contribution to sediment transport. The categories of vortex kernel identification criteria include Q, λ₂, Δ and Ω criteria, all of which are Eulerian and have Galilean invariance [38].

In this paper, we use the widely used Q criterion for eddy recognition research to quantitatively evaluate the influence of vorticity on the flushing area in conjunction with the maximum vortex value:

$$Q=\frac{1}{2}\left[tr{\left(\overline{D}\right)}^2-tr\left({\overline{D}}^2\right)\right]=\frac{1}{2}\left({{\omega }_{ij}}^2-{s_{ij}}^2\right)$$

(19)

where ${\omega }_{ij}$ is the vortex; $s_{ij}$ is the tension tensor; $\overline{D}$ is the local velocity gradient tensor; tr is the trace of the matrix; as usually Q > 0, characterizing the vortex plays a dominant role.

The vortex volume can describe the characteristic region of the eddy channel architecture, and the characteristic equation for each axial direction can be defined [39] as follows:

$${\omega }_i=\frac{\partial u_j}{\partial x_k}-\frac{\partial u_k}{\partial x_j}$$

(20)

For the changing characteristics of the transient flow field, this paper captures the cross-sectional vortex distribution at the cross-section of the upstream pier column (refer to Figure 6, 1–1 section front view) at four characteristic moments in the course of both classes of scouring (refer to Figure 10 and Figure 11). In the initial stage t₀, significant eddies appear at the bottom of the pier perimeter in both types of scouring; in the pre-development stage, the sand transport rate of live-bed scouring is larger in Figure 11b, and the depth of scouring reaches Figure 10b in a short time; in the later development stage, the sand transport rate slows down in both Figure 10c and Figure 11c, and the depth of scouring approaches the maximum; in the equilibrium stage, the scour depths in Figure 10d and Figure 11d are slightly higher than those in the development stage, but the overall clear-water scour depths are greater than those in the live bed. There are two types of eddies causing scouring at the scour hole on the pier side, which are the main eddy and the secondary eddy [40]; the sand transport rate of the main eddy is larger, and its motion makes up a small part of the eddy kinetic energy transfer to the secondary eddy, while the majority of the eddy kinetic energy moves to the tail eddy with sand, forming a deeper scour hole.

4.2. Shear Stress Distribution

The bed shear stress reflects the coupling between water and sand and is a key parameter for quantitative analysis of local scour. The application of Plante momentum transfer introduces time averaging to calculate the turbulent shear stress acting on the sand bed around the pier by Reynolds stress [41]:

$${\tau }_b={\left.{\left({\tau }_{\theta }^2+{\tau }_r^2\right)}^{0.5}\right|}_{at bed}$$

(21)

where ${\tau }_{\theta }=-\rho \left(\overline{w^{\prime }{u}^{\prime }}+\overline{v^{\prime }{u}^{\prime }}\right)$; ${\tau }_r=-\rho \left(\overline{u^{\prime }{v}^{\prime }}+\overline{w^{\prime }{v}^{\prime }}\right)$; and ρ is the fluid density.

For the convenience of observing the rate of change in tangential stress on the bottom layer of the sand bed during the process of scouring holes, the dimensionless bed shear stress was chosen to represent:

$${\widehat{\tau }}_b=\frac{{\tau }_b}{{\tau }_0}=\frac{1}{{\tau }_0}{\left.{\left({\tau }_{\theta }^2+{\tau }_r^2\right)}^{0.5}\right|}_{at bed}$$

(22)

where ${\tau }_0=\rho u_{\ast }^2$; and $u_{\ast }$ is the frictional flow rate, with the same scale as the flow rate.

Considering the transient characteristics of the turbulent flow field, the changes in the shear stress on the sand bed surface during the two types of scouring courses are calculated in this paper (see Figure 12 and Figure 13). It can be observed that at the initial moment, the maximum shear stress is at the location of point b around the pier surrounding the waterward side of the front pier, and the highest tangential stress max is 4.3 and 5.1 Pa in Figure 12 and Figure 13, respectively; at the development stage, the maximum shear-stress area (red area) gradually shifts to develop behind the backwater side of the front pier in the process of fluid flow-around, in which the rate of live-bed scouring is obviously ahead of clear-water scouring, and the overall change in the sand transport trend between the two is consistent; with the increase in the action time, the live-bed scour tends toward dynamic equilibrium faster, while the sand transfer by clear-water scour decays slowly and continues to develop until scour equilibrium is reached.

The alternating changes in the maximum shear-stress area around the pier are related to the vortex kinetic energy of the section. Simulation calculations were performed for the two types of scouring separately to obtain the vertical vortex kinetic energy and shear stress at point b of the pier surrounding; they were analyzed, and the correlation between them is shown in Figure 14. Simulations of the maximum vortex quantity of the vertical line obtained in the clear-water scour and live-bed scour conditions are 43 and 32 s⁻¹, respectively, and the same moment under the live-bed scour conditions of the vertical line vortex quantity sees an increase of 34.4%, and the maximum shear stress sees an increase of 18.6%; for the convenience of analysis, assume that the ratio of the increase rate of the characteristic quantity of live-bed scour and clear-water scour k_L/k_C is the time-averaged increase ratio λ, that is:

$$\lambda =k_L/k_C$$

(23)

The time-averaged increase ratio of about 1.3 can be quantified from Equation (23), which indicates that the contribution of the live-bed scour conditions to the development rate of scour holes is large.

It is found that the change in vortex quantity around the pier during the flushing process of the pier foundation is positively correlated with the shear stress; this is due to the obstruction of the head-on surface in front of the pier, and the surge of the undercutting intensity of the water causes the fluid turbulence to be intense and acts on the vortex structure, and the superposition effect of the two impacts the sand bed around the pier; during this period, it causes a large amount of sediment transport in the scour hole, accompanied by the lateral spread of the horseshoe vortex, and the area with the maximum shear stress moves backward with it. The maximum shear-stress area corresponds to the deeper area of the scour hole, indicating that the interaction of the vortex and shear stress leads to the evolution of the scour hole structure around the pier.

5. Analysis of Scouring Results

5.1. Evolution of Live Bed Scour Hole

In aiming to qualitatively analyze the morphological characteristics of scour pits, the local scour and silt topography around the piers were selected as the study area, in order to explore the development law of water–sand coupling under the cumulative effect of time.

From the above figure, the following features can be observed: (1) In the initial stage of scouring, the tip-shaped structure in front of the upstream pier plays the role of diversion, the scouring on the pier side is first developed on both sides and symmetrically distributed, the downstream pier is shaded and reinforced by the upstream pier, and the development of the scouring hole is later than the front pier. (2) The development stage of scouring is shown in Figure 15b and Figure 16b, both sides of the bridge pier are gradually brushing deeper and widening, and the end of lower pier has an angle of about ±35°–75° development of an “eight-shaped” dune due to sediment accumulation, and the evolution of scouring and siltation in this stage of live-bed scouring is significantly faster than that of clear-water scouring. (3) In the scour equilibrium stage, both types of scouring are fully developed, and the distribution patterns of dunes are almost similar between the two, but there are significant differences in the patterns of scour holes. The overall distribution of scour pits on the pier side in the dynamic-bed scour environment is steep, and the radius area of scour holes is “raindrop-shaped” (wide on the upstream pier side and narrow on the downstream pier side), while the radius of the pier-side flushing pit of both piers of clear-water scour is nearly equal. From Figure 15c and Figure 16c, it is possible to observe that the highest flushing position in front of both piers reached 80–90% of the maximum local scour depth within 20–30 min of the test, respectively.

In order to further quantitatively study the scour hole scale pattern, longitudinal interception along the pier centerline along the xz profile range and transverse interception upstream and downstream of the scour pit b-b’ profile and g-g’ profile (refer to Figure 8) were investigated to explore the change trend of two different scour conditions.

From Figure 17, the maximum scour depth of the upstream pier in the clear-water scour and live-bed scour is located at spot b of the scouring pit (refer to Figure 8), the relative equilibrium scour depths were 1.0D and 0.9D, and the maximum flushing pit radii were 2.4D and 3.1D, respectively; the maximum scour depth of the downstream pier in the clear-water scour and live-bed scour is located at spot g of the scouring pit (refer to Figure 8), the relative equilibrium scour depths were 0.6D and 0.5D, and the maximum flushing pit radii were 2.3D and 1.5D, respectively. The maximum scour depth of the clear-water scour in the upstream pier is 8.9% deeper than that of live-bed scour, which is similar to the findings of [42], indicating that the contribution of sediment-laden flow in the clear-water scour hole is larger due to the absence of replenishment of incoming sediment; the maximum scour hole radius around the pier is compressed by 23.2% compared with that of the live-bed scour, indicating that the main vortex in the clear-water scour hole is mainly developed vertically, and the secondary vortex is poorly developed laterally. In the equilibrium stage of live-bed scouring, the radius of the upstream pier surrounding the scour hole is about 2.1 times that of the downstream, and the maximum scour depth is about 1.8 times that of the downstream, which is caused by the decrease in the depth and width oscillation of the back pier flush pit due to the change in transport frequency of the sand ripple passing through the front pier scour hole.

5.2. Scale of Scouring Hole Side Walls

The high-quality sediment transport collapse model can completely present the collapse process of the scour hole wall [43,44], the bed scouring of the pier foundation belongs to the continuous large-scale deformation process, and the sediment of the sidewall of the punch pit is adjusted by intermittent slumping until the slope of the side slope is less than the repose angle of the submerged sediment. Therefore, the longitudinal xz profile along the pier centerline and the transverse bb’ profile at the moment of balanced scouring were selected to explore the changed law of scour hole with slope under the conditions of clear-water scouring and live-bed scouring.

Figure 18 shows the comparison of scour hole slope between two scouring states in the scouring equilibrium stage, both of them have relatively steep scour holes along the front of the pier, while the slope at the end of the pier is gentle and the tail area shows a dune accumulation pattern, in which under the clear-water scouring condition, the gradient of the flush pit of the front pier undercut by the water flow (straight line A₁B₁ inclination in the diagram) is about 29°, and the gradient of the flush pit on the pier side (straight line C₁D₁ inclination in the figure) is about 30°. The gradient of the flush pit of the front pier undercut by the water flow (straight line A₂B₂ inclination in the figure) is about 23°, and the gradient of the flush pit on the pier side (straight line C₂D₂ inclination in the figure) is 24°; by comparison, it was found that the slope is smaller than the repose angle of the submerged sediment of 32°, indicating that the quasi-equilibrium scour hole has reached a stable state [45].

Observing the values of the scour hole slope for both types of scouring, the results show that (1) the slope in front of the pier is slightly smaller than the slope on the shoulder side, which is attributed to the cumulative influence of water flow, weakening the brush depth of the punching pit in advance of the pier and gradually slowing down the slope of the wall; and (2) in the equilibrium stage, the depth and slope values of the scour hole in front of the upstream pier are greater in the clear-water scour state than in the live-bed scour state, which indicates that the deeper the scour hole is, the steeper the slope of the scour hole. In general, the trend of the scour hole structure is consistent with the results of [46].

5.3. Time Characteristics of Scour Development

Based on the simulation outcome of scouring tandem equal-diameter piers, it is known that the factors affecting the development of pier-side scour pits in unidirectional orthogonal water flow mainly include the following characteristics: time-average flow velocity (V), pier diameter (D), flow depth (H), sediment parameters (d_s, ρ_s), fluid parameters (V_c, ρ_w), motion parameters (g, ν and μ) and duration (t_e). In this paper, the multivariate analysis method is used to obtain the following equation:

$$f\left(d_s,H,D,V,V_c,g,{\rho }_w,{\rho }_s,\nu ,\mu ,t_e\right)=0$$

(24)

According to the Buckingham π theory [47,48], defining V, D and ρ_w as the basic variables, the following equation (dimensionless parameter relations dominated by the diameter of the pier) can be obtained:

$$f\left(\frac{d_s}{D},\frac{H}{D},\frac{V}{V_c},\frac{{\rho }_s}{{\rho }_w},\frac{\nu }{VD},\frac{gH}{V^2},\frac{t}{t_e}\right)=0$$

(25)

Given that ρ_s, D and H are constants in the model, the water flow strength V/V_c is equal to 0.82 and 1.21 in the clear-water scour and live-bed scour, respectively, and the Reynolds number Re and the water flow Froude number Fr have minimal contribution to the scour depth in the fully developed state of turbulence [49]; the above equation can be simplified to

$$f\left(\frac{d_s}{D},\frac{t}{t_e}\right)=0$$

(26)

In the above equation, the time scale t_e reflects the significant effect of scouring time on the scour depth and takes the integration of the scour depth ephemeral curve [50] to obtain:

$$t_e={{\displaystyle \int }}_0^{t_{max}}\frac{{\mathrm{d}}_{se}-{\mathrm{d}}_s}{{\mathrm{d}}_s}\mathrm{d}t$$

(27)

Under the clear-water scour conditions, the evolution of the scour depth with time for the existing scour depth of the bridge pier test results in this paper uses an exponential function [51] to derive the final equilibrium scour depth:

$$\frac{{\mathrm{d}}_s}{D}=1-\exp \left[-0.134{\left(\frac{t}{t_e}\right)}^{1.104}\right]$$

(28)

Melville and Chiew [52] considered that for the bridge pier live-bed scour test, the process of scouring to reach equilibrium needs to last for several days, so this paper uses the logarithmic function proposed by Wu et al. [53] to describe the relationship between the depth of live-bed scouring and the time scale as follows:

$$\frac{{\mathrm{d}}_s}{{\mathrm{d}}_{se}}=0.122\ln \left(\frac{t}{t_e}\right)+0.456$$

(29)

To further verify the reliability of the formula, the measured values from the indoor tests were compared with the calculated values from the regression formula in this paper, as shown in Figure 19. The live-bed scour experiment data of Melville [10] and the clear-water scour experiment data of Yilmaz et al. [54] were also selected and compared with the live-bed scour and clear-water scour experiments simulated in this study, respectively. The results showed that the overall measured values of the test are slightly larger than the calculated values of the empirical formula, but the error range of both is limited to within ±20%, indicating that the regression formula has strong reliability and high accuracy; comparison with the data of previous scholars shows that the values of the simulation results in this study are slightly smaller, but the error range is confined within the envelope; for both classes of scouring environment, the agreement of the relevant data is high, which proves that the regression formula can evolve the development law of the scour depth with time very well.

6. Conclusions

The local scouring conditions around the piers in the natural riverbed are extremely complex. In this paper, a numerical simulation study of pier scouring under clear-water scouring and live-bed scouring conditions was carried out for tandem of piers of equal diameter, respectively, and the contribution of both classes of scour incoming flow environment in the vortex effect on the pier side and bed shear stress was analyzed, and the relationship between scour topography characteristics, scour hole structure and time characteristic scale was explored, and the main conclusions were as follows:

(1) The maximum sediment transport rate of the scour hole occurs in the pre-development stage, and compared with the clear-water scour, the live-bed scour has a faster sediment transport rate on the time scale; the area of maximum shear stress corresponds to the area of maximum scour depth around the pier, and the maximum value of shear stress of the clear-water scour is higher than that of the live-bed scour in the equilibrium stage; the change trends of the two are strongly linked, and the eddy quantity around the pier is positively correlated with the bed shear stress, indicating that the superposition effect of the sediment transport rate and the bed shear stress causes the local scouring around the pier.

(2) Through qualitative and quantitative analysis of scour topography and structural scales, it is concluded that the contribution of clear-water conditions to the total scouring around the piers is greater than that of live-bed conditions, in which the maximum scour depth of the clear-water scour of the upstream piers is 8.9% deeper than that of the live-bed scour, and the slope of the scour hole tends to slow down with the depth of scouring.

(3) Compared with the live-bed scour, the scour rate of clear water is slow but the contribution time is longer; under the live-bed condition, the downstream piers are shaded and reinforced by the upstream piers, and the extent and development of the scour holes around the piers are much smaller than the upstream piers, so the design of the scour reduction and protection structure for the upstream piers should be paid attention to in the project.

(4) For the development trend of live-bed scouring and clear-water scouring, the regression equation for the evolution of scour depth with time scale was established, and the reliability of the regression equation was proved by comparing the agreement of relevant data, which indicated that the simulated local scour evolution law was consistent with the reality.

(5) Changing environmental conditions can cause direct structural damage and indirect economic losses to underwater structures. Simulation studies of scouring damage to bridge piers induced by hydrodynamics can be used for example validation and analysis, which are important for refining bridge design, reducing cost-effectiveness and contribute to sustainable development of in-water structures by quantifying the environmental loads.