The model assumes that particles in the sedimentary bed and transition layer are completely elastic spheres, so the energy loss caused by the impact and collision between particles due to plastic deformation can be ignored, the particle transport is limited in a two-dimensional plane composed of gravity direction and flow direction, and the influence of random disturbance is not considered.

Based on the assumptions above, as shown in Figure 1, the collision motion can be simplified as particle A in the transition layer colliding with the stationary particle B on the sedimentary bed at the collision angle α (the angle between the particle movement direction and flow direction), and the contact angle between A and B is β (the angle between the particle center line and the flow direction). Mt ₀ is the momentum when particle A collides with B. It can be decomposed into Mt _0h and Mt _0v along the flow and gravity directions, where Mt _0v is the momentum accumulation of particle A along the gravity direction before collision. Without considering the influence of parabolic motion in the transition layer on the change of particle momentum, if the transportation of particle A maintains continuity, the momentum of A in the gravity direction after collision should be greater than Mt _0v . Further, Mt ₀ can be decomposed into Mt _L and Mt ₁ along the line of the particle center line and its vertical direction, where Mt _L is the momentum component that will be transferred to particle B after collision, and Mt ₁ is the initial momentum of particle A after collision. Therefore, when the component Mt _1v of Mt ₁ along the gravity direction is greater than Mt _0v , the continuity of particle transport can be maintained. Although the two decompositions above have the same dynamic significance, they express the momentum distribution state before and after particle collision with different emphases. Based on the above analysis, it can be concluded that the essence of particle collision lies in the momentum exchange and redistribution between particles. The key to maintaining the continuity of particle motion lies in the relative sizes of Mt _1v and Mt _0v . Based on the geometric relationships between the components, Δ M t 1 v M t 0 h = M t 1 v − M t 0 v M t 0 h = sin ( β − α ) cos ⁡ β cos ⁡ α − tan ⁡ α (2)

According to Eq. 2, under a given collision angle α, the relative sizes of Mt _0v and Mt _1v are only related to the contact angle β, but the value of β is constrained by the arrangement of particles on the sedimentary bed. When there is no gap between particles on the bed surface, the minimum contact angle β_min is obtained when the particles are tangent to the adjacent particles C along the direction of transport velocity. The maximum contact angle β_max is obtained when the particle is tangent to the static particle B along the velocity direction. According to the geometric relationship between the particles, it can be concluded that β max = 90 ∘ + α (3) β min = arcsin ( 1 − sin ⁡ α ) + α (4)

When there is a gap between particles, β_min decreases with an increase in the gap distance. Under the condition that the moving and static particles are spheres of the same size and the collision angle α is given, the probability of the contact angle β between the moving particles and the static particles on the bed will vary with the size of the impact area. When β = α, the impact area of the moving particles is the largest, and it corresponds to the large circle of the static particles. According to the geometric relationship, the collision area and β, α satisfy the following relationship: C = 2 π R ⁡ cos ( β − α ) (5) where C is the collision area, and R is the radius of the particle circle. Correspondingly, under the condition of a given collision angle α, the larger the value of β is, the lower the probability of its occurrence.

It can be observed from Figure 1 that when the moving particle A collides with the stationary particle B on the bed surface, momentum exchange Mt _L is generated along the direction of the line connecting the A and B centers. The size of the residual momentum Mt ₁ determines whether the particles can move sustainably, and its direction determines the direction of transport speed. According to the geometric relations, the direction α′ of Mt ₁ satisfies α ′ = β − 90 ∘ (6)

Eq. 6 shows that α′ depends on contact angle β. When β is less than 90°, the component Mt _1v of Mt ₁ is opposite to gravity, and particles can jump again after collision when Mt _1v is sufficient to overcome Mt _0v . When β is greater than 90°, Mt _1v is in the same direction as gravity, which means that the transported particle A will continue to collide with the stationary particle D after contacting the stationary particle B. As the initial momentum Mt ₀ causes momentum loss Mt _L along the direction of the central line of A and B, the initial momentum of the secondary collision is Mt ₁ . According to the geometric relationship, the collision and contact angles corresponding to the second collision can be obtained as follows: α ′ = β − 90 ∘ (7) β ′ = arcsin ( 1 − sin ⁡ α ′ ) + α ′ (8) where α′ and β′ are the collision and contact angles of the second collision, respectively; whether the particles can jump again depends on the relative size of Mt _2V and Mt _3V . According to Eq. 2, the following can be concluded: Δ M t 1 v M t 0 h = sin ( β − α ) cos ⁡ α [ sin ( β ′ − α ′ ) cos ⁡ β ′ − sin ⁡ α ′ ] (9)

After completing the collision motion between particles and momentum exchange, the parabolic motion in the transition layer begins. As shown in Figure 1, the parabolic motion of the particles is divided into two stages: upward and downward. In the upward stage, the particles decelerate along the gravity direction under the action of gravity G and fluid resistance F _Dy . When they reach the highest point P, the corresponding vertical velocity v _↑ = 0; in the downward stage, the directions of G and F _Dy are opposite, and the maximum vertical velocity is reached when the particles are about to collide with the static particles in the sedimentary bed. It can be observed that the parabolic motion also contains frequent stress interactions and momentum exchange between the liquid and solid. In this section, we analyze the stress state of the particles in the transition layer, the dynamic model of the particle parabolic motion, and the corresponding transfer probability matrix. On this basis, the trend and convergence of momentum in the parabolic motion of particle swarms are discussed.

As shown in Figure 5, during the upward stage of the particles, the changes in velocity satisfy m d v ↑ d t = − m g − 1 2 ρ f A C D v ↑ 2 (47) where ρ _f is the density of seawater, A is the particle cross-sectional area, v is the particle movement speed, and C _D is the drag coefficient with dimension 1, which is a function of the Reynolds number, Re, of the particle movement.

The momentum of the particle swarm gradually converges to a stable state as the collision progresses, and the corresponding Re should also converge under the given conditions of flow velocity and particle diameter. As shown in Figure 4, the Re probability distribution of the particle swarm at the steady state can be obtained according to Figure 4, and combined with the change curve of the drag coefficient C _D with Re (Syamlal and O’Brien, 1987). As shown in Figure 4, when Re is small, the transport medium is in laminar flow state. At this time, the particle settlement resistance is mainly viscous force, and C _D has a linear relationship with Re; With the increase of Re value, the movement of transport medium is gradually transformed into turbulence. At this time, the viscous force and turbulent resistance received by particle settlement can not be ignored, and C _D has a curve relationship with Re; When the value of Re increases to a certain extent, the transport medium is in the turbulent region. At this time, the viscous resistance of particle settlement can be ignored, and C _D remains constant with the change of Re. The fluctuation range of the value of the particle swarm C _D is negligible. Therefore, it can be considered that C _D remains constant during the parabolic motion, and its size can be measured by the terminal velocity of the particle settling: C D = 2 m g ρ f A v t 2 (48) where v _t is the terminal velocity. Substituting Eq. 48 into Eq. 47, we obtain − 1 1 + ( v ↑ / v t ) 2 d ( v ↑ v t ) = d ( t v t / g ) (49)

Integrating both sides of Eq. 49, and taking the particle motion state at the highest point P as the boundary condition, that is, t = 0 and v _↑ = 0 are satisfied, the motion equation of the particle upward stage can be obtained: v ↑ v t = tan ( − t v t / g ) (50)

The motion equation of particles in the downward stage is as follows: v ↓ v t = 1 − 2 1 + exp ( 2 t v t / g ) (51)

Equations 50, 51 indicate that when the particle motion velocity v and the corresponding time t are normalized by the end-settlement velocity v _t , the change in the dimensionless velocity v / v t with dimensionless time t v t / g is determined by the above equation. Taking the combined Eq. 48, v _t is only related to the particle size under a constant C _D . Therefore, the motion law of the particles is uniquely determined under the given distribution of particle size. Figure 5 shows the variation of v / v t with t v t / g , where the velocity of the particles rapidly drops to 0 during the ascending process, and then slowly rises to v _t during the descending process. When the particle moves at the initial velocity v 0 / v t = 3 (point A in Figure 5), the corresponding time t 0 v t / g can be obtained from Eq. 50, and the path lengths of the particles rising and falling are equal, indicating that the shadow area in the corresponding figure is equal. Combining Eqs 50, 51, the corresponding time when the particles fall to the bed (point B in Figure 5) can be obtained as follows: ∫ A 0 v ↑ v t d ( t v t / g ) = ∫ 0 B v ↓ v t d ( t v t / g ) (52)

Equation 52 is substituted into Eq. 51 to obtain the velocity v 1 / v t when the particles fall to the bed, and then the next parabolic motion starts with v 1 / v t as the initial velocity (point C in Figure 5) and cycles in turns. It can be concluded from the path of gain and loss of particle momentum shown in Figure 5 that although the variation of particle momentum Δ v A B / v t and Δ v C D / v t decreases with the increase in motion time, the momentum gain during parabolic motion is negative. This means that without considering the momentum gain brought by the collision process, the particle momentum will gradually converge to zero, and the continuity of particle transport will no longer be satisfied.

In this section, the particle swarm velocity distribution profile is obtained by analyzing the change law of the momentum of the particles during the transport process. Moreover, we considered the interaction between the fluid in the transition layer and the particles on the bed, and the mass of the particles flowing into the transition layer per unit time was obtained. Finally, on this basis, the disturbance law of particle transport in the flow field of the transition layer is discussed.

The transport of particles essentially alternates between the collision motion and the parabolic motion, that is, the particles begin the parabolic motion in the transition layer after the collision motion with the bed surface, and the change law of particle swarm momentum should satisfy X 1 = X 0 PR (53) where P and R are the transfer probability matrix of the momentum change in the collision motion and the parabolic motion of the particle swarm, and the matrix S = PR obtained by the multiplication of the two is also the transfer probability matrix according to Eq. 16, which corresponds to the change in momentum of the particle swarm in the transport process.

When the particle moves to the highest point during the parabolic motion, the velocity along the gravity direction reaches the minimum value, which indicates that the particles stay at this position for the longest time. The velocity v _→ of particles in the flow direction at this position can then be taken as the value of the velocity profile in the transport process of the particle swarm, which can be obtained by traversing the particle swarm momentum sample space, as shown in Figure 6. The particle swarm velocity increases with increasing elevation, but the increasing speed gradually slows down. This is because with an increase in v _→ the fluid resistance of particles along the flow direction decreases gradually, that is, the velocity of particles remains constant until v _→ increases to the flow velocity. When the transition layer fluid accelerates the particles, it will also be subject to the reaction of particles, which causes the original flow field to evolve with the transport process. According to the conservation of the fluid mass in the transition layer, the following can be obtained: v ˜ ( y ) = 1 1 + I P F v f ( y ) + I P F 1 + I P F v → ( y ) (54) where v ˜ ( y ) is the liquid–solid mixing velocity of the fluid at the corresponding y position in the transition layer, and IPF is the activation probability function of the static particles in the sedimentary bed, which is defined as the mass of the particles in the sedimentary bed that can be activated by the unit mass of the transition layer fluid velocity distribution v _f . This is related to the shear force of the transition layer fluid τ _a , τ _w , and the dimension is 1. In Eq. 57, both v f ( y ) and v → ( y ) are known quantities; therefore, when the probability function IPF is obtained, the variation law of the transition layer flow field distribution can be obtained.

Particles in the sedimentary bed begin to move under the action of the transition layer fluid, and the mass of the activated particles is related to the mass flow rate of the transition layer fluid. The activation probability function can be expressed as follows (Syamlal and O’Brien, 1987): M = ∫ 0 δ ρ v T d y ⋅ I P F (55) where T is the movement period of particles, M is the mass of particles invading the transition layer during T, and IPF is the activation probability function of the particle, which can be expressed as (Sauermann et al., 2001) I P F = n ( τ l 0 τ t ) − 1 (56) where τ _l0 is the liquid-phase shear force at the junction of the sedimentary bed and the transition layer. τ _t is the critical starting stress of material particles, and its calculation method can be referred to (Ahmed, 2001). When τ _l0 > τ _t , then n > 1, and when τ _l0 < τ _t , then n = 1. Equation 56 shows that when and only when τ _l0 > τ _t , particles will be activated in the sedimentary bed, and the activation efficiency of these particles depends on the trend of n. Based on the Taylor expansion and Eq. 55, the following equation can be obtained: M = ∫ 0 δ ρ v T d y ⋅ γ ( τ l 0 τ t − 1 ) (57) where γ = ∂ n ∂ ( τ l 0 / τ t ) | τ l 0 / τ t = 1 , and it can be considered as the contribution of factors other than the liquid shear stress to the activation of particles in the sedimentary bed, which characterizes the erosion intensity of the transition layer on the sedimentary bed and can be obtained through laboratory experiments or numerical simulation (Syamlal and O’Brien, 1987).

Taking the micro-element control volume in the transition layer as the research object, based on the law of conservation of mass, it can be concluded that ∂ ∂ t ( ∫ 0 δ ρ d y ) + ∂ ∂ x ( ∫ 0 δ ρ v d y ) = ∫ 0 δ ρ v l d y ⋅ γ ( τ l 0 τ t − 1 ) (58) where ρ is the fluid density of the transition layer, v is the fluid velocity, l is the average transport distance of the particles, and δ is the thickness of the transition layer. According to the stress state of the micro-element control volume, the expression of τ _l0 can be obtained using the momentum theorem as follows: τ l 0 = τ 0 − ∫ 0 δ ρ s v l ( y ) Δ v ( y ) d y (59)

The equation is based on the assumption that the flow velocity is uniform and time-invariant, where Δ v ( y ) is the difference between the horizontal component of the downward velocity and upward velocity during particle transport at position y, l ( y ) is the horizontal distance of particles at position y from upward to downward, and τ ₀ is the flow shear force of the fluid at the interface between the transition layer and the sedimentary bed. Based on the assumption that the slurry satisfies Newton’s shear law, the following conclusions can be drawn: τ 0 = μ d v d y | y = 0 (60) where μ is the viscosity of the slurry in the transition layer, which is affected by the concentration of solid particles (Einstein, 1906): μ = ( 1 + 2.5 C v | y = 0 ) μ 0 (61) where μ ₀ is the viscosity of the slurry without solid particles, and C _v is the volume concentration of the solid phase in the transition layer. Equation 58 shows that the change in fluid mass in the micro-element control body depends on the mass flow gradient in the flow direction and the invasion rate of particles in the sedimentary bed, and its trend is related to the distribution of the velocity field and solid concentration in the transition layer during transportation. Therefore, the characteristics of the evolution of the slurry properties and motion parameters depend on these two important parameters.

Based on the boundary conditions above, the velocity distribution in the transition layer can be assumed to be a polynomial distribution with five unknowns: v ( x , y , 0 ) = U [ 2 ( y δ ) − 2 ( y δ ) 3 + ( y δ ) 4 ] (62)

The concentration distribution of the solid phase can be expressed as follows: C v ( x , y , 0 ) = 0 (63)

The flow field and solid concentration distributions in the transition layer gradually evolve with the transportation process. Based on the principle of mass conservation, the evolution of the solid concentration can be expressed as follows: C v ( x , y , t + Δ t ) = C v ( x , y , t + Δ t ) + ∫ 0 δ ρ ( x , t ) v ( x , y , t ) d y ⋅ I P F ⋅ M R ( y ) ∫ 0 δ ρ p v ( x , y , t ) d y (64)

The evolution of the flow field distribution is expressed as follows: v ( x , y , t + Δ t ) = ∫ 0 δ ρ ( x , t ) v ( x , y , t ) d y ⋅ I P F ⋅ M R ( y ) ⋅ v → ( y ) + ρ ( x , t ) v ( x , y , t ) ⋅ v ( x , y , t ) ∫ 0 δ ρ ( x , t ) v ( x , y , t ) d y ⋅ I P F ⋅ M R ( y ) + ρ ( x , t ) v ( x , y , t ) (65) where MR(y) is the mass proportion of particles at the y position in the transition layer to the total mass of activated particles in Δt, as shown in Figure 14. In conclusion, on the basis of solving Eqs. 64, 65 to obtain the slurry movement law in the transition layer, the variation law of fluid mass flow can be analyzed using the continuous Eq. 58.

The solution of the mass conservation equation of the transition layer fluid is directly related to the characteristics of the evolution of the property parameters and motion parameters of the slurry; therefore, the solution of Eq. 58 must be accompanied by the real-time solution of the flow field and solid concentration distributions of the transition layer. Equation 58 contains three unknowns: slurry density ρ, velocity v, and liquid phase shear force τ _l0 , and Eq. 59 describes the decisive factors of τ _l0 : T V P ∂ Q ∂ t + ∂ Q ∂ x = Q l s ⋅ ( 1 − Q Q s ) (66) where Q _s is the saturated mass flow rate: Q s = ( τ − τ t ) l 〈 Δ v 〉 (67)

TVP, 〈 Δ v 〉 , and l _s are process parameters: T V P = ∫ 0 δ ρ v d y ∫ 0 δ ρ d y , 〈 Δ v 〉 = ∫ 0 δ ρ p v ⋅ Δ v ( y ) d y ∫ 0 δ ρ v d y , l s = l τ t γ ( τ − τ t ) (68)

Equation 66 shows that the trend of the fluid mass flow rate in the transition layer is consistent with the trend of the solid-phase shear stress τ _s . The inflow of particles in the transition layer increases the mass flow rate, and the momentum exchange between the liquid and solid phases is more intense. However, it also leads to a decrease in the mass source term on the right side of Eq. 66 and a decrease in the liquid phase shear force τ _l , thus weakening the growth rate of the mass flow rate and the solid phase shear force τ _s . Q _S corresponds to the flow shear force τ, which describes the carrying capacity of solid particles of the transition layer from the perspective of mass exchange and momentum exchange, respectively, and the evolution of Q and τ _s describes the negative feedback mechanism of the mass change in the transition layer from the qualitative and quantitative perspectives, respectively.

As shown in Figure 7, in this experiment, a centrifugal pump injected water into the plexiglass open channel through an injection pipeline to form a strong water flow. The particles flowed homogeneously through the sand–water mixing tank at the injection point, and the flow speed and direction stability were controlled through the rectifier plate and guide grid group to simulate the stable flow situation in the process of trough transportation accurately. An acoustic Doppler velocimeter was set up in the open channel to monitor the variation in velocity at different positions during transportation in real time. The acoustic doppler velocimeter measures the three-dimensional velocity of X, Y and Z. Y is perpendicular to the X direction, and X and Y are in the same plane, Z is vertical upward. The three directions comply with the right-hand rule. The four “legs” of the flow meter are data acquisition points in four different directions (x, y, z) at the measurement position to measure the flow velocity and direction at the corresponding position, in which the red rectangle represents the x-axis. The evolution of the sedimentary bed was observed by a high-speed camera in real time.

The scale of real hydraulic transportation is large, so it is impossible to restore the experimental simulation of transportation process 1:1 in the laboratory. Therefore, the hydraulic transport simulation experiment must be carried out through the appropriate similarity criterion. As mentioned above, in the process of hydraulic transport, whether in the macro evolution of the geometry of the sedimentary bed and the flow field in the transition layer, or in the micro collision and parabola movement of the particle swarm, the gravity is always the key factor affecting its motion form. Therefore, the Froude number similarity criterion is selected here. According to the Froude similarity criterion, F r = v m 2 g m l m = v t 2 g t l t (79)

Where v _m and v _t represent the transport medium velocity under laboratory simulation conditions and real hydraulic conveying conditions respectively, m/s, l _m and l _t represent the scale size under laboratory experiment and real conditions respectively, m, g_m and g_t represent the gravitational acceleration under laboratory experiment and real conditions respectively. The parameters of the hydraulic transport simulation experiment were designed as Table 1.

Observe the process from particle return to stopping upward return. At 1 min, there is basically no rock debris return. At 2 min, the particle concentration increases, and then returns stably. A sharp fan-shaped sedimentary bed is formed at the bottom of the flume along the ocean current direction. The deposition positions of particles with different particle sizes are different, and the geometric shape of the deposition unit formed by material particles with corresponding particle sizes is in accordance with Equation 76 (77).

Driven by the water flow, the material particles on the surface of the sedimentary bed begin to move and invade the transition layer against the resistance and resistance torque. The amount of invasion is controlled by Eq. 57. The moving particles climb over the upper surface of the paved sand layer after parabolic motion (50) (51) and collision motion (15). The sedimentary bed begins to evolve slowly as a whole, and the evolution law conforms to Eq. 58.

After 10 min of particle upward return process, all transported particles have entered the water tank. Keep the water velocity constant, observe the migration and deposition process of particles, and record the results every 5 min. In order to clearly observe the range of particle migration and deposition, the top view of the experimental process is shown in Figure 8.

When the particles stop returning, the sediment is in the shape of a sharp fan. Under the action of water flow, the tip of the sedimentary bed slowly disappears, the sand layer is a small sand dune along the width of the flume, and the sedimentary bed is accumulated in a horseshoe shape as a whole.

Taking the simulation and experimental parameters listed in Table 1, the model calculation and experimental results are compared, as shown in Figure 9, where the color bar represents the height of the deposition bed. It can be observed that both of them show that the sedimentary bed presents, in general, a horseshoe-shaped accumulation, which indicates that the change in the morphological distribution of the sedimentary bed is affected by the transport of particles at the leading edge and the accumulation of particles at the tail, and the morphology evolution is that the particles in the front position are transported to the middle and rear positions of the sedimentary bed under the action of the liquid phase. It can be concluded that the intense parabolic and collision motions of the particles in the transition layer are accompanied by the synchronous evolution of the morphological distribution of the sedimentary bed, which ensures the continuity of particle transportation.

Table 2 presents the comparison between the experimental and model calculation data, and it can be observed that the maximum relative error is no more than 4%, which proves the rationality of the model. The velocity distribution of the transition layer is shown in Figure 10, where the continuous curve is the model calculation result of the velocity distribution in the transition layer, and the discrete points are the finite velocity points in the transition layer measured by acoustic Doppler velocimeter. It can be observed that the invasion of particles changes the original flow field distribution in the transition layer, and the change range of the velocity profile is different at different positions, but the influence area is mainly concentrated between positions A and B, which indicates that the particle transport is concentrated in this position. The continuous curve in Figure 10 represents the calculated data of the model, and the discrete points represent the measured data. It can be observed that the two have good consistency.

Figure 11 shows the evolution of the velocity distribution in the transition layer under different flow velocities. It can be observed that with the increase in flow velocity, the influence of particle invasion on the original velocity field gradually decreases, which indicates that the flow velocity can reflect the hydraulic energy of the transport medium to a certain extent. With the increase in flow velocity, the activation ability of the transition layer with respect to the particles in the sedimentary bed increases, and the thickness of the sedimentary bed decreases as a whole, which is consistent with the trend of the distribution of the sedimentary bed in Figure 12.

Figure 13 shows the evolution of the bed morphology distribution for different particle size distributions. As can be observed, the bed thickness increases with an increase in the average particle diameter. However, according to Figure 14, the velocity distribution of the transition layer does not respond to the change in particle size, which indicates that the morphological change of the sedimentary bed is simply caused by the change in the initial deposition position caused by the change in particle size distribution, and the change in particle size distribution has little effect on the transport of the transport medium.