SPH-fem coupled simulation method for introducing water jet turbulent flow velocity distribution

Abstract

The application relates to the technical field of tunnel and underground engineering construction, and discloses an SPH-FEM coupling simulation method introducing water jet turbulent flow velocity distribution, which comprises the following steps: determining key parameters of a scouring system, calculating turbulent jet cross-section velocity distribution, assigning SPH particle velocity, constructing an SPH-FEM coupling numerical model, setting a contact algorithm, executing explicit dynamics calculation of a scouring process, and extracting and analyzing scouring results; based on the characteristics of the SPH method, the turbulent jet theory is rewritten and discretized, is introduced into the SPH-FEM coupling framework, a refined model capable of accurately describing the non-uniform velocity distribution of the jet is constructed, and the SPH jet inflow boundary condition is reconstructed; the interaction dynamics process of the jet and the mud cake is simulated truly; the peeling and crushing mechanism of the mud cake under the influence of multiple parameters is revealed; and solid theoretical basis and advanced technical means are provided for the optimization design and intelligent control of a shield cutterhead scouring system.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of tunnel and underground engineering construction, and particularly relates to an SPH-FEM coupling simulation method introducing water jet turbulent flow velocity distribution. BACKGROUND

[0002] As the mainstream method of tunnel and underground engineering construction, shield method faces the serious challenge of cutterhead mud cake when crossing high viscosity strata. The formation of mud cake can significantly reduce the shield tunneling efficiency, aggravate cutter wear, and even cause equipment downtime, which seriously restricts construction safety and engineering progress. High-pressure water jet scouring is a key technology for preventing cutterhead mud cake, and its scouring effect is directly related to the continuous and efficient tunneling of the shield machine. Therefore, accurately revealing the action mechanism of water jet scouring mud cake and establishing a reliable prediction and optimization method are of great significance to guide the site construction.

[0003] In the prior art, numerical simulation is an important means to study the water jet scouring process. The SPH-FEM coupling algorithm combining the Smoothed Particle Hydrodynamics (SPH) method and the Finite Element Method (FEM) is widely used due to its unique advantages in handling large deformation fluid-structure coupling problems. However, when using the SPH-FEM method to simulate water jet scouring, there is a key defect: the jet inlet boundary condition is usually simplified as a uniform flow velocity distribution. This simplified assumption is seriously inconsistent with the significant turbulent characteristics of high-pressure water jet in engineering practice, resulting in serious distortion of the simulated jet energy distribution, velocity decay law, and mud cake damage form, which cannot accurately reproduce the real peeling process and damage form of the jet on the mud cake. In addition, existing researches focus on macroscopic flow field analysis, and lack systematic quantitative analysis of the scouring mechanism under the coupling of multiple parameters such as scouring distance and mud cake density, making it difficult to establish an accurate scouring effect prediction model and parameter optimization method.

[0004] Therefore, it is necessary to provide a new SPH-FEM coupling simulation method to solve the above technical problems. SUMMARY

[0005] The main purpose of the present application is to provide an SPH-FEM coupling simulation method introducing water jet turbulent flow velocity distribution. In the construction of the water jet model, the radial non-uniformly distributed flow velocity field is calculated by introducing the turbulent jet theory, and is accurately assigned to the SPH particle set as the initial boundary condition, aiming to solve the problem of low simulation accuracy caused by the simplification of the jet inlet boundary condition to uniform flow velocity distribution in the existing method.

[0006] To achieve the above purpose, the present application provides an SPH-FEM coupling simulation method introducing water jet turbulent flow velocity distribution, comprising the following steps:

[0007] Step one, obtaining a parameter set, the parameter set comprising a nozzle diameter of a shield cutter flushing system, a jet initial velocity and a target flushing distance, a density, an elastic modulus and a compressive strength of a mud cake material;

[0008] Step two, rewriting and discretization of the turbulent jet formula, specifically: introducing the SPH particle coordinate parameters to convert the turbulent jet formula and rewriting it into a discrete form in the cylindrical coordinate system to obtain a new turbulent jet formula;

[0009] Step three, calculating the turbulent jet cross-sectional velocity distribution, specifically: based on the nozzle diameter of the shield cutter flushing system, the jet initial velocity, the target flushing distance and the new turbulent jet formula obtained in step two, calculating the radial velocity distribution of the SPH particles on the jet cross section at the specified flushing distance;

[0010] Step four, SPH particle velocity assignment, specifically: based on the radial velocity distribution of the SPH particles on the jet cross section at the specified flushing distance obtained in step three, assigning the corresponding initial velocity to the SPH particle set;

[0011] Step five, constructing an SPH-FEM coupled numerical model, specifically: based on the SPH particle velocity after the assignment in step four, establishing an SPH fluid particle domain representing the high-pressure water jet and establishing an FEM structure domain representing the mud cake and the shield cutter;

[0012] Step six, on the interface between the SPH fluid domain and the FEM structure domain, using a contact algorithm based on the virtual particle method for calculation; simulating the dynamic whole process of the high-pressure water jet impacting the mud cake to obtain the time sequence data of the damage pattern of the mud cake;

[0013] Step seven, based on the time sequence data of the damage pattern of the mud cake obtained in step six, extracting the volume damage ratio of the mud cake, the maximum gradient of the flushing pit cross section and the maximum flushing depth.

[0014] Preferably, the new turbulent jet formula in step two is as follows:

[0015] ;

[0016] ;

[0017] ;

[0018] Wherein: is the jet initial velocity; is the nozzle diameter; is the radial coordinate of the th discrete SPH particle; is the exponential function; is the elevation coordinate of the th discrete SPH particle; This is a discrete function of the jet center velocity, the value of which is related to the elevation coordinates of the SPH particles; It is a discrete function of the jet characteristic half-width, and its value is related to the elevation coordinates of the SPH particles; The jet diffusion coefficient; This is an empirical coefficient; In the cylindrical coordinate system, the first... The velocity of each SPH particle.

[0019] Preferably, the assignment of SPH particle velocity in step four specifically includes:

[0020] Based on Python, LS-Dyna was further developed, and a subroutine was written. The subroutine reads the coordinate information of SPH particles and calls the radial velocity distribution of SPH particles on the jet cross section at a specified scouring distance obtained in step three, and assigns a corresponding initial velocity value to each SPH particle.

[0021] Preferably, in step five: the FEM model of the mud cake adopts the Mat-Plastic-Kinematic material constitutive model, and the FEM model of the shield cutter adopts the Mat-Rigid rigid body material constitutive model.

[0022] Preferably, the contact algorithm of the virtual particle method in step six includes: generating a layer of virtual SPH particles on the FEM boundary; setting the contact parameters between the virtual particles and the fluid particles; and realizing momentum transfer through the contact algorithm.

[0023] Preferably, the simulation of the dynamic process of high-pressure water jet impacting the mud cake in step six specifically involves: starting explicit dynamic calculation in the LS-Dyna solver to simulate the dynamic process of high-pressure water jet impacting the mud cake; the explicit dynamic calculation uses the central difference method for time integration, and the time step satisfies the CFL stability condition.

[0024] Preferably, the process also includes a step for evaluating the scouring effect, which specifically involves evaluating the scouring effect based on the volumetric destruction ratio of the mud cake obtained in step seven, the maximum gradient of the scouring pit cross section, and the maximum scouring depth.

[0025] The technical solution of this invention has the following technical effects:

[0026] The SPH-FEM coupling simulation method of the water jet turbulent flow velocity distribution in the application comprises determining key parameters of the scour system, calculating a turbulent jet cross-section velocity distribution, assigning SPH particle velocity, constructing an SPH-FEM coupling numerical model, setting a fluid-structure coupling contact algorithm, performing explicit dynamics calculation of the scour process, and extracting and analyzing the scour results. BRIEF DESCRIPTION OF DRAWINGS

[0027] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the drawings needed to be used in the embodiments or the prior art description will be briefly introduced. Obviously, the drawings in the following description are only some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor on the basis of the drawings shown.

[0028] Figure 1 is a flow chart of the SPH-FEM coupling simulation method of the water jet turbulent flow velocity distribution in the embodiments of the present application;

[0029] Figure 2 is a turbulent jet along different cross-section velocity distribution chart in the embodiments of the present application;

[0030] Figure 3 is an SPH flow velocity turbulent distribution schematic diagram in the embodiments of the present application;

[0031] Figure 4 is a scour pit morphology evolution process chart in the embodiments of the present application;

[0032] Figure 5 is a scour effect evaluation chart in the embodiments of the present application;

[0033] Figure 6 is a method, a prior art method, and a test effect comparison chart, wherein (a) is a test result; (b) is a simulation result of the method of the present application; and (c) is a simulation result of the prior art method.

[0034] The implementation of the object of the present application, the functional characteristics and the advantages will be further described with reference to the embodiments and the drawings. DETAILED DESCRIPTION

[0035] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative work fall within the protection scope of the present application.

[0036] Embodiment:

[0037] A SPH-FEM coupling simulation method for introducing a water jet turbulent flow velocity distribution is described in detail in Figure 1 , comprising the following steps:

[0038] Step one, obtaining a parameter set, the parameter set including the nozzle diameter of the shield cutter flushing system, the jet initial velocity and the target flushing distance, the density, the elastic modulus and the compressive strength of the mud cake material;

[0039] Step two, rewriting and discretization of the turbulent jet formula, specifically: introducing the SPH particle coordinate parameters to convert the turbulent jet formula and rewriting it into a discrete form in the cylindrical coordinate system to obtain a new turbulent jet formula;

[0040] Step three, calculating the turbulent jet cross-sectional velocity distribution, specifically: based on the nozzle diameter of the shield cutter flushing system, the jet initial velocity, the target flushing distance and the new turbulent jet formula obtained in step two, calculating the radial velocity distribution of the SPH particles on the jet cross section at the specified flushing distance;

[0041] Step four, SPH particle velocity assignment, specifically: based on the radial velocity distribution of the SPH particles on the jet cross section at the specified flushing distance obtained in step three, assigning the corresponding initial velocity to the SPH particle set;

[0042] Step five, constructing a SPH-FEM coupling numerical model, specifically: based on the SPH particle velocity after the assignment in step four, establishing the SPH fluid particle domain representing the high-pressure water jet and establishing the FEM structure domain representing the mud cake and the shield cutter; this means is a conventional technology.

[0043] Step six, on the interface between the SPH fluid domain and the FEM structure domain, a contact algorithm based on the virtual particle method is used for calculation; the dynamic whole process of the high-pressure water jet impacting the mud cake is simulated, and this means is a conventional technology. Time sequence data of the damage form of the mud cake is obtained;

[0044] Step seven, measuring the volume damage ratio of the mud cake, the maximum gradient of the flushing pit cross section and the maximum flushing depth based on the time sequence data of the damage form of the mud cake obtained in step six.

[0045] In this preferred embodiment, the new turbulent jet formula in step two is as follows:

[0046] ;

[0047] ;

[0048] ;

[0049] in: The initial velocity of the jet; Nozzle diameter; For the first Radial coordinates of discrete SPH particles; It is an exponential function; For the first Discrete SPH particle elevation coordinates; This is a discrete function of the jet center velocity, the value of which is related to the elevation coordinates of the SPH particles; It is a discrete function of the jet characteristic half-width, and its value is related to the elevation coordinates of the SPH particles; The jet diffusion coefficient; This is an empirical coefficient; In the cylindrical coordinate system, the first... The velocity of each SPH particle.

[0050] In this preferred embodiment, the assignment of SPH particle velocity in step four specifically includes:

[0051] Based on Python, LS-Dyna is further developed by reading the coordinate information of SPH particles and calling the radial velocity distribution of SPH particles at a specified scouring distance obtained in step three to assign a corresponding initial velocity value to each SPH particle. More preferably, the subroutine developed based on the programming language can achieve the following functions: reading the initial coordinate information of SPH particles; calculating the velocity distribution matrix of the SPH particle set based on step two and the initial coordinate information of the SPH particles, and assigning a corresponding initial velocity to each SPH particle; and importing the assigned velocity field into the LS-Dyna calculation model.

[0052] In this preferred embodiment, in step five: the FEM model of the mud cake adopts a Mat-Plastic-Kinematic material constitutive model, and the FEM model of the tunnel boring machine cutter adopts a Mat-Rigid rigid body material constitutive model. This method is a conventional technique.

[0053] Preferably in the embodiment, the contact algorithm of the virtual particle method used in step six includes: generating a layer of virtual SPH particles on the FEM boundary; setting the contact parameters of the virtual particles and the fluid particles; and realizing momentum transfer through the contact algorithm. Further preferably, the virtual particle contact algorithm includes the following steps:

[0054] ① Generating a layer of virtual SPH particles on the FEM boundary;

[0055] ② Setting the contact parameters of the virtual particles and the fluid particles;

[0056] ③ Determining whether contact occurs through the contact detection algorithm and realizing momentum transfer.

[0057] The contact detection algorithm is as follows:

[0058] ;

[0059] Wherein: is the contact gap; is the SPH particle coordinate; is the FEM node coordinate; is the contact surface normal; if then it indicates that contact occurs and contact force needs to be applied for correction.

[0060] Preferably in the embodiment, the step of simulating the dynamic whole process of the high-pressure water jet impacting the mud cake is: starting explicit dynamics calculation in the LS-Dyna solver to simulate the dynamic whole process of the high-pressure water jet impacting the mud cake; the explicit dynamics calculation adopts central difference method for time integration, and the time step satisfies the CFL stability condition. This method is a conventional technique.

[0061] Preferably in the embodiment, the method further includes a scouring effect evaluation step, specifically: evaluating the scouring effect through the volume damage ratio of the mud cake obtained in step seven, the maximum gradient of the scour pit cross section, and the maximum scouring depth. Preferably, the evaluation indexes are: ① whether the volume damage ratio exceeds 15%; ② whether the absolute value of the maximum gradient of the scour pit cross section is less than 1; and ③ whether the maximum scouring depth exceeds 1 / 4 of the diameter of the tool. If the volume damage ratio exceeds 15%, the scour pit cross section gradient is less than 1, and the maximum scouring depth exceeds 1 / 4 of the diameter of the tool, it can be considered that the scouring effect is good. If only any two of the three standards are met, it can be considered that the scouring effect is medium. If only any one of the three standards is met, it can be considered that the scouring effect is qualified. If none of the three standards is met, it can be considered that the scouring effect is poor.

[0062] The mud-water shield scouring system of a certain actual project is described and calculated as follows:

[0063] Step one, obtaining the parameter set, specifically: obtaining the nozzle outlet diameter (10 cm) of the shield cutterhead scouring system, the initial jet velocity (50 m / s) and the target scouring distance (120 cm); the basic parameters of the mud cake material: density (1.92×10 3 kg / m³), elastic modulus (3.0×10 9 Pa) and compressive strength (3.67×10 5 Pa).

[0064] Step two, rewriting and discretization of the turbulent jet formula, specifically: introducing the SPH particle coordinate parameters to convert the turbulent jet formula and rewriting it into a discrete form in the cylindrical coordinate system to obtain a new turbulent jet formula.

[0065] Step three, calculating the turbulent jet cross-sectional velocity distribution, specifically: using the new turbulent jet formula to calculate the jet radial velocity distribution of the SPH particles at the specified scouring distance. Assuming that the nozzle outlet diameter is 10 cm; the initial jet velocity is 50 m / s; the target scouring distance is 120 cm (the turbulent jet flow velocity distribution at different cross sections along the way is shown in Figure 2 , which presents a non-uniform distribution characteristic that the center region is high and the edge region gradually decays, and the center flow velocity decreases and the jet characteristic half-width increases with the increase of the scouring distance. The coordinates of the SPH particle set are (120, 0), (120, 2), (120, 4)…, and the radial velocity distribution of some SPH particles is shown in Table 1:

[0066] Table 1 Coordinate data of SPH particle set

[0067]

[0068] Step four, SPH particle velocity assignment. In this embodiment, the subprogram sph_velocity.py reads the SPH particle coordinate information and calls the jet radial velocity distribution of the SPH particles at the specified scouring distance obtained in step three to assign the corresponding initial velocity value to each SPH particle. The state diagram after SPH particle assignment is shown in Figure 3 , which presents a non-uniform distribution characteristic that the center region is high and the edge region gradually decays.

[0069] Step five, building an SPH-FEM coupled numerical model.

[0070] Step six, setting the fluid-structure coupling contact algorithm, specifically: using the virtual particle method coupling algorithm on the interface between the SPH fluid domain and the FEM structure domain, and the specific setting parameters are: contact stiffness coefficient 0.8, friction coefficient 0.2, and adaptive virtual particles are used in the jet impact area;

[0071] Explicit dynamic calculations of the scouring process were performed, specifically: time integration was conducted using the central difference method in the LS-Dyna solver, with a time step set to 1×10⁻⁶. -6 The simulation was performed to ensure the CFL stability condition was met. The total computation time was set to 20ms, with an output interval of 0.1ms, simulating the dynamic process of high-pressure water jet impacting the mud cake. This method is a conventional technique.

[0072] Step 7: Extract and analyze the scour results. Measure the volumetric damage percentage of the mud cake, the maximum gradient of the scour pit cross-section, and the maximum scour depth from the calculation results in Step 6. Quantitatively evaluate the scour effect. The morphological evolution process of the scour pit is as follows: Figure 4 As shown, the center of the mud cake was destroyed first, and the scour pit spread outwards. The scour effect was assessed as follows: Figure 5 As shown in the scouring results, the volumetric damage rate of the mud cake was 19.8%; the maximum gradient of the scouring pit cross-section was 0.257; and the maximum scouring depth was 0.314 times the cutter width. Therefore, the scouring effect can be determined to be good.

[0073] The effect of applying the technical solution of this embodiment is:

[0074] Based on turbulent jet theory, and considering the characteristics of SPH particles, the turbulent jet formula is rewritten and discretized to calculate the velocity distribution matrix of the jet cross-section at a specific scouring distance from the nozzle. Secondary development of LS-Dyna using a programming language accurately assigns the calculated non-uniform velocity field to the SPH particle set, reconstructing the initial inflow boundary conditions of the water jet. Geometric models of the SPH fluid domain and the FEM cake and cutterhead structural domains are established. An SPH-FEM coupled contact algorithm based on the virtual particle method is set to simulate the dynamic interaction between the water jet and the cake. Explicit dynamic calculations are performed to obtain the scouring failure morphology and evolution process of the cake. Based on the simulation results, the cross-sectional shape of the scouring pit and the proportion of cake volume failure are measured to evaluate the scouring effect. The jet cross-sectional velocity distribution is calculated based on the initial jet velocity and the target distance, using the attenuation formulas for the discretized jet core region and main body. SPH particle velocity assignment is achieved by traversing the particle coordinates and calling the velocity distribution matrix calculated by the rewritten turbulent jet formula. The contact algorithm, a conventional technique, generates virtual particles on the impacted FEM boundary to transfer fluid forces and prevent penetration. The method of this invention boasts strong physical realism and effectively overcomes the limitations of traditional models that assume uniform flow velocity (i.e.,...). Figure 6 This method addresses the distortion problem of existing methods and can accurately predict the cake removal effect under different combinations of scouring parameters. It significantly improves the physical realism and calculation accuracy of jet scouring simulation and provides a reliable numerical analysis tool for the optimized design of shield cutterhead scouring systems.

[0075] The simulation method proposed in this invention is verified through systematic parameterized simulation and experimental testing, such as...Figure 6 As shown, specifically: the method of the present invention ( Figure 6 The calculated scour results (as shown in Figure (b)) show a funnel-shaped failure section, which is consistent with the experimental results ( Figure 6 Similar to (a) shown in the middle. However, existing methods ( Figure 6 As shown in (c), the calculated result shows a rectangular damaged interface, which is inconsistent with the experimental results and is severely distorted. Therefore, it can be concluded that the method of this invention can accurately quantify the influence of the coupled effects of multiple parameters such as scour distance, scour time, and mud cake density on the scour effect. Compared with traditional methods that are mostly limited to macroscopic flow field analysis, this method can not only accurately predict the morphological evolution of scour pits, but also clarify the optimal scour parameter range, providing a reliable quantitative basis for the precise design and operational parameter optimization of the shield cutterhead scour system, significantly enhancing the engineering guidance value of the method.

[0076] The above description is merely a preferred embodiment of the present invention and does not limit the patent scope of the present invention. Any equivalent structural transformations made using the contents of the present invention's specification and drawings under the inventive concept of the present invention, or direct / indirect applications in other related technical fields, are included within the patent protection scope of the present invention.

Claims

1. A SPH-FEM coupled simulation method incorporating water jet turbulent velocity distribution, characterized in that, Includes the following steps: Step 1: Obtain the parameter set, which includes the nozzle diameter, initial jet velocity and target scouring distance of the shield cutterhead scouring system, density, elastic modulus and compressive strength of the mud cake material; Step 2: Rewriting and discretizing the turbulent jet formula. Specifically, the SPH particle coordinate parameters are introduced to transform the turbulent jet formula and rewrite it into a discretized form in cylindrical coordinates to obtain a new turbulent jet formula. Step 3: Calculate the velocity distribution of the turbulent jet cross section. Specifically, based on the nozzle diameter of the shield cutterhead scouring system, the initial velocity of the jet, the target scouring distance, and the new turbulent jet formula obtained in Step 2, calculate the radial velocity distribution of SPH particles on the jet cross section at the specified scouring distance. Step 4: Assigning SPH particle velocity values, specifically: Assigning the corresponding initial velocity to the SPH particle set based on the radial velocity distribution of the SPH particles on the jet cross section at the specified scouring distance obtained in Step 3. Step 5: Construct the SPH-FEM coupled numerical model, specifically: based on the SPH particle velocity assigned in Step 4, establish the SPH fluid particle domain representing the high-pressure water jet and the FEM structural domain representing the mud cake and the tunnel boring machine cutter. Step 6: At the interface between the SPH fluid domain and the FEM structural domain, a contact algorithm based on the virtual particle method is used for calculation; the dynamic process of high-pressure water jet impacting the mud cake is simulated to obtain time-series data of the mud cake's failure morphology. Step 7: Based on the time-series data of the failure morphology of the mud cake obtained in Step 6, extract the volumetric failure ratio of the mud cake, the maximum gradient of the scour pit cross section, and the maximum scour depth. The new turbulent jet formula in step two is as follows: ； ； ； in: The initial velocity of the jet; Nozzle diameter; For the first Radial coordinates of discrete SPH particles; It is an exponential function; For the first Discrete SPH particle elevation coordinates; The velocity at the center of the jet is a discrete function; The characteristic half-width discrete function of the jet; The jet diffusion coefficient; This is an empirical coefficient; In the cylindrical coordinate system, the first... The velocity of each SPH particle.

2. The SPH-FEM coupled simulation method for introducing water jet turbulent velocity distribution as described in claim 1, characterized in that, The specific steps in step four, including assigning SPH particle velocity values, include: Based on Python, LS-Dyna was further developed, and a subroutine was written. The subroutine reads the coordinate information of SPH particles and calls the radial velocity distribution of SPH particles on the jet cross section at a specified scouring distance obtained in step three, and assigns a corresponding initial velocity value to each SPH particle.

3. The SPH-FEM coupled simulation method for introducing turbulent velocity distribution in water jets as described in claim 2, characterized in that, In step five: the FEM model of the mud cake adopts the Mat-Plastic-Kinematic material constitutive model, and the FEM model of the shield cutter adopts the Mat-Rigid rigid body material constitutive model.

4. The SPH-FEM coupled simulation method for introducing water jet turbulent velocity distribution as described in claim 3, characterized in that, The contact algorithm of the virtual particle method in step six includes: generating a layer of virtual SPH particles on the FEM boundary; setting the contact parameters between the virtual particles and the fluid particles; and realizing momentum transfer through the contact algorithm.

5. The SPH-FEM coupled simulation method for introducing water jet turbulent velocity distribution as described in claim 4, characterized in that, The specific steps in step six for simulating the dynamic process of high-pressure water jet impacting the mud cake are as follows: In the LS-Dyna solver, an explicit dynamic calculation is started to simulate the dynamic process of high-pressure water jet impacting the mud cake; the explicit dynamic calculation uses the central difference method for time integration, and the time step satisfies the CFL stability condition.

6. The SPH-FEM coupled simulation method for introducing water jet turbulent velocity distribution as described in claim 5, characterized in that, It also includes a step to evaluate the scouring effect, which is to evaluate the scouring effect by the volume destruction ratio of the mud cake obtained in step seven, the maximum gradient of the scouring pit cross section, and the maximum scouring depth.

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