CN114330152A - Surface production confluence calculation method based on smooth particle fluid dynamics - Google Patents
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Abstract
The invention provides a surface production confluence calculation method based on smooth particle fluid dynamics, which comprises the following steps: initializing calculation conditions, describing surface information by bottom particles, and describing surface production confluence by fluid particles; calculating acceleration according to the speed and the position of the fluid particles at the current moment; calculating the speed and position of the particle at the next moment according to the acceleration and position of the fluid particle; calculating the density and smooth length of the particle after moving to the new position; calculating the mass of the fluid particles according to the rainfall and the infiltration water amount, and performing momentum correction on the fluid particles; adding fluid particles in an initial state to the upstream region without particle coverage; and calculating each time step, and outputting a calculation result of the surface production convergence. The method provided by the invention does not need to establish a grid, and the modeling process is more efficient; the rainfall and infiltration calculation is supported, and the earth surface production confluence process can be simulated more completely.
Description
Technical Field
The invention relates to a method for calculating surface production confluence, in particular to a method for calculating surface production confluence based on smooth particle fluid dynamics.
Background
In recent years, inland water logging disasters have become the most common and serious problems in cities. When heavy rainstorm or short-term heavy rainfall occurs, the low-lying part of the urban terrain cannot drain water in time, and water is accumulated in the low-lying part of the urban terrain easily to form waterlogging. Alleviating waterlogging can be started from two aspects. Firstly, the source is controlled, and the peak flow of surface runoff is reduced. Secondly, the urban drainage system is optimized, and the risk of waterlogging is reduced by increasing the collection and discharge capacity of drainage facilities. In the implementation process of the two measures, the calculation of the surface production convergence is a basis, and is directly related to the early design and the later operation management.
The calculation of surface production convergence has been a difficult problem and is difficult to obtain through simple calculation. To obtain more reliable computational results, researchers computer model the process of production convergence, which models are based on physical-based models, using detailed, rigorous physical process expressions, based on the laws of mass, momentum, and energy conservation. The models can be divided into a grid method and a non-grid method according to whether a numerical method adopted in solving the nonlinear partial differential equation system needs to divide grids or not.
1) Grid method
Most of the existing distributed convergence physical models adopt the method, such as European hydrological System model (SHE), UK's hydrological institute distributed model (IHDM), Australia has THALES model and CSIRO TOPOG-dynamic model, United states ' comprehensive hydrological model of VanderKwaak and Loague and Donner et al's gridded surface/underground hydrological analysis model (GSSHA). A method and system for hierarchical modeling of surface grids (CN110532641B) provides a method for hierarchical modeling of surface grids, which divides a surface structure into a plurality of layers of grids, respectively performs rainfall runoff generation and in-layer confluence calculation inside each layer of grid, and then performs interlayer confluence calculation on each layer of grid with confluence relation, thereby realizing real simulation of the physical process of complicated multilayer structure runoff; a watershed surface runoff confluence simulation method (CN107590354B) based on a stable water flow field provides a stable water flow field concept, an N-S equation under a grid mode is established, a water flow mixing simulation link is added in runoff simulation, and the establishment of the watershed stable water flow field under the grid mode is realized.
The advantages and disadvantages are as follows: these models have been developed for a long time and are well studied to achieve satisfactory analysis results, but the grid has limited their application to many problems. In the grid numerical method, a prerequisite for numerical simulation is the generation of a grid in the calculation area. For euler mesh methods, such as finite difference methods, it is difficult to construct regular meshes on complex geometries and often requires additional mathematical transformations, which are sometimes even more cumbersome than solving the problem itself. It is also a difficult task to accurately determine the position of the free surface, the deformation boundary, the moving interface and the inhomogeneous material on a fixed euler grid. For lagrange mesh methods, such as the finite volume method, a mesh must be built on the study object before simulation, and this operation often takes a large amount of computational effort. In the method based on the lagrangian mesh, the mesh generally needs to be divided again when large deformation is processed, and the subdivision of the mesh is very complicated and time-consuming, and further brings solving errors.
2) Gridless method
It is some of the limitations of the mesh that the meshless method is becoming more popular in fluid dynamics research, with the Smooth Particle Hydrodynamics (SPH) method being the most widely used. The method has the main idea that: an accurate and stable numerical solution is obtained by solving an integral equation or a partial differential equation system having various boundary conditions using any series of distributed particles. The smooth particle hydrodynamic method (SPH) finds application in many fields, such as: CN 103729555B-a method and a device for simulating the action of blood flow and blood vessel wall, & lt CN 107633123B-a method for simulating bleeding and treating acceleration by smooth particle hydrodynamics, & lt CN 106096215B-a realistic fluid simulation method relating to heat conduction and dynamic viscosity, & lt 106777662B & gt an aircraft fuel tank oil-mixing characteristic optimization method based on smooth particle hydrodynamics, & lt 112069689A & gt an aircraft engine fuel oil atomization characteristic simulation method and system, & lt 110569541A & gt a pipeline water hammer analysis method, & lt 106570308A & gt a method for analyzing non-grid particles of instantaneous flow of a pipeline containing trapped air mass.
The advantages and disadvantages are as follows: SPH has significant advantages in modeling compared to traditional mesh-based approaches, such as: (1) no numerical oscillation is caused because there is no nonlinear term in the control equation and the mass is completely conserved; (2) can effectively process mixed flow state containing subcritical, supercritical and supercritical flow; (3) complex surface variations can be easily represented using the bottom particles and no additional processing is required for the wet-dry interface. However, there is currently no solution that applies SPH to the surface production convergence process and supports infiltration calculations, so it is not possible to simulate a complete surface production convergence process.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a surface production convergence calculation method based on smooth particle fluid dynamics, which does not need to establish a grid and has more efficient modeling process; the rainfall and infiltration calculation is supported, and the earth surface production confluence process can be simulated more completely.
The technical scheme of the invention is as follows:
a surface production confluence calculation method based on smooth particle fluid dynamics comprises the following steps:
s1: initializing calculation conditions: the spatial dimension is set to two dimensions; collecting earth surface information data, and describing the earth surface information data by using bottom particles; describing a flow field of surface production confluence by using fluid particles, and setting initial state data of the fluid particles; the initial state data of the fluid particles comprises mass, smooth length, position and velocity; setting an analog time length T; setting a simulation step length delta t;
s2: calculating the acceleration of the fluid particles according to the state data of the particles at the time t; calculating the speed and position of the particle at the t + delta t moment according to the state data and the acceleration of the particle at the t moment;
s3: calculating the density and the smooth length of the fluid particles after the fluid particles move to the position at the time t + delta t;
s4: calculating the mass of the fluid particles at the t + delta t moment according to the rainfall and the infiltration water amount, and performing momentum correction on the fluid particles;
s5: partitioning the calculation area, checking whether each partition is covered by particles, and adding initial particles to an upstream area without particle coverage; the initial particles refer to the fluid particles in the initial state in step S1;
s6: checking the calculated total time length, if the calculated total time length is greater than the simulation time length T, finishing the calculation, and outputting a calculation result; if the time is less than the simulation time period T, the process returns to S2, and the time T +2 × Δ T is calculated.
Further, the step S3 includes the following steps:
s31: equation 1 listing the density of the fluid particles:
wherein m isjIs the mass of the fluid particle j, Wi(xj,hi) Is a formula of kernel function, hiIs the smooth length, x, of the fluid particle ijIs the distance between the fluid particles i and j;
s32: equation 2 listing the smooth length of the fluid particle:
wherein V0,iAnd ViRespectively the initial area of the particle i and the area at the current moment, dmIs that the spatial dimension takes the value of 2, h0,iIs the initial smooth length, h, of the fluid particle iiIs the smooth length of the fluid particle i at the current moment;
s33: and (3) combining equation 1 and equation 2, and solving the density and the smooth length of the fluid particles by a numerical calculation method.
Further, the numerical calculation method is a (Newton-Raphson) Newton-Raphson iteration method.
Further, the step S4 includes the following steps:
s41: obtaining the infiltration quantity d at the t + delta t moment according to an infiltration modeli；
S42: obtaining the rainfall d at the t + delta t moment according to the rainfall datar；
S43: according to the quantity d of rainfalliAnd the amount of the lower seepage drCalculating the height d 'of the fluid particles at the time t + delta t't+Δt：
d′t+Δt＝dt+Δt+dr-di
Wherein d ist+ΔtThe particle height at the time of t + delta t without considering rainfall and infiltration; drIs the particle height of increased rainfall; diIs the reduced particle height due to infiltration;
s44: particle height d 'calculated according to step S43't+ΔtAnd correcting data such as mass, volume, speed and the like of the fluid particles.
Further, the infiltration model is a Horton infiltration model.
Further, the step S5 includes the following steps:
s51: dividing the calculation area into several areas, and making grid division on the area to be researched, and the length and width of grid are all
Wherein A is0Represents the area of the fluid particles in the initial state;
s52: after the mesh division, the position occupied by each particle is marked, and the length and the width of each particle are assumed to be
Wherein A isiRepresenting the area of the fluid particle in the current state;
s53: an eight-direction method is introduced to distinguish upstream and downstream of the grid, and a parameter D8 (c) is adoptedi) To indicate the order of cells in the runoff for labeling;
s54: find the upstream mesh without fluid particles, add the initial particles.
Further, the eight-direction method comprises the following calculation steps:
s61: divide the calculation area into length and width
In the cell of (A), wherein0Represents the area of the fluid particles in the initial state;
s62: the flow direction of each unit cell is distributed to one of eight adjacent unit cells, wherein the flow direction can be up, down, left and right, or diagonal, and the flow direction is the direction with the steepest gradient;
s63: using the parameter D8 (c)i) To represent the order of the cells in the runoff, the number represents how many cells including the current cell water will flow into the current cell, and the calculation formula is as follows:
wherein, ciIs a cell in the calculation region, N is the flow direction ciAdjacent cells of (a).
The invention has the beneficial effects that:
(1) according to the method, the smooth particle fluid dynamics model is used, the surface runoff convergence calculation is realized, the method is suitable for various complex terrain scenes, and more accurate surface runoff data can be provided for the drainage pipe network design;
(2) the method provided by the invention supports the infiltration calculation, and can more completely simulate the surface production confluence process;
(3) the method provided by the invention is based on a numerical method of smooth particle fluid dynamics, and due to the non-grid property, the grid does not need to be established during calculation and simulation, and the method has the characteristics of low modeling cost and high practicability;
(4) the method for adding the initial particles in the particle-free area is provided by the invention, the original particle reconstruction method is replaced, the initial particles can be effectively supplemented, and the support is provided for the subsequent rainfall runoff calculation. Compared with the original method, the method has the advantages that higher calculation accuracy is realized by using fewer particles, the calculation efficiency and accuracy are improved, and the obtained surface runoff data can participate in operation management of a drainage pipe network in time.
Drawings
FIG. 1 is a flow diagram of the present invention;
FIG. 2 is a calculated area topography of an embodiment;
FIG. 3 shows t of an embodiment0Fluid particle profile at time (initial state);
FIG. 4 is a schematic illustration of an embodiment of integrated rainfall and infiltration;
FIG. 5 shows t of an embodiment0Schematic diagram of the height and area change of the fluid particles to time t;
FIG. 6 is a fluid particle distribution plot at time t + Δ t for an embodiment;
FIG. 7 is a study area grid division diagram of an embodiment;
FIG. 8 is a schematic view of a region occupied by a marker particle of an embodiment;
FIG. 9 is a schematic representation of the eight-way method of the embodiment computationally downstream;
FIG. 10 is a fluid particle distribution plot after the addition of initial particles at time t + Δ t for an example;
FIG. 11 is a graph of the calculation results of the example.
Detailed Description
The invention will now be described in detail by way of an example with reference to the accompanying figures 1 to 11.
The calculation steps are as follows:
s1: the calculation conditions are initialized. This step can be subdivided into 4 steps:
(1) the spatial dimension being set to two dimensions, i.e. dm＝2；
(2) And collecting the earth surface information data, and describing the earth surface information data by using the bottom particles. The topographic map of the calculated area is shown in figure 2 and is a natural terrain of 200m multiplied by 100 m; the terrain is high on the right and low on the left; the water depth at the initial surface is 0, i.e. dry surface conditions; water can only enter the calculation area in a rainfall mode, and an open boundary is arranged at a left downstream outlet and is free to flow out.
The specific parameter values are as follows:
i. initial fluid particles
ii. Closed boundary particle
Particle arrangement interval: 0.5m
Initial smooth length: 0.6m
Area of single particle: 0.25m2
iii, bottom particles
iv, Horton infiltration parameters
Initial permeability f0：1.977×10-4m/s
Soil stability infiltration rate fc：3.272×10-5m/s
Infiltration attenuation coefficient k: 2.43X 10-3s-1
v, rainfall conditions
Total rainfall: 75000m3
And (3) rainfall for a period of time: 250min
(3) The flow field representing the flow of the surface production confluence is described by fluid particles, and initial state data (including mass, smooth length, position and velocity) of the fluid particles are set. The distribution diagram of the fluid particles in the initial state is shown in FIG. 3;
(4) the simulation time period T is set to 350 minutes and the simulation step Δ T is set to 1 minute.
S2: according to the position of the fluid particles, finding out the correspondingThe height and roughness condition of the bottom particles, and the acceleration a of the fluid particles at the time t is calculatedi：
Wherein t isi＝Ti/mi，Sf,iIs a friction source term and can be calculated by the following formula under the condition that the Manning coefficient is known:
wherein n isiThe Manning coefficient of the particle i can be obtained by adopting SPH interpolation calculation through each time step:
wherein
Means that the jth bottom particle is located at
The manning coefficient of (c). (a)iThe calculation method of (2) is from the literature: vacond R, Rogers BD, Stansby PK, Stansby P K. accurate particulate for smooth particulate hydrodynamics in show water with shot capturing [ J].International Journal for Numerical Methods in Fluids,2012,69(8):1377-1410)
According to the acceleration aiAnd the position x of the fluid particle at time tiCalculating the velocity v of the fluid particles at time t + Δ tiAnd position xi。
S3: the density and the smooth length of the fluid particles after moving to the position at time t + Δ t are calculated. This step can be subdivided into 3 steps:
(1) equation 1 listing the density of the fluid particles:
wherein m isjIs the mass of the fluid particle j, Wi(xj,hi) Is a formula of kernel function, hiIs the smooth length, x, of the fluid particle ijIs the distance between the fluid particles i and j;
(2) equation 2 listing the smooth length of the fluid particle:
wherein V0,iAnd ViRespectively the initial area of the particle i and the area at the current moment, dmIs a spatial dimension with a value dm＝2，h0,iIs the initial smooth length, h, of the fluid particle iiIs the smooth length of the fluid particle i at the current moment;
(3) by combining equation 1 and equation 2, the density and the smooth length of the fluid particles at the time t + delta t are obtained by a (Newton-Raphson) Newton-Raphson iteration method.
S4: and calculating the mass of the fluid particles at the t + delta t moment according to the rainfall and the infiltration water amount, and performing momentum correction on the fluid particles. This step can be subdivided into 4 steps:
(1) obtaining the infiltration quantity d at the t + delta t moment according to a Horton infiltration modeli。
(2) Obtaining the rainfall d at the t + delta t moment according to the rainfall datar。
(3) According to the quantity d of rainfalliAnd the amount of the lower seepage drThe height of the fluid particles at time t + Δ t is calculated. As shown in FIG. 4, the height of the particles is added with the amount of water increased by rainfall and the amount of infiltration is subtracted, drIndicated water columns and by diThe indicated water columns represent the amount of increased rainfall and decreased infiltration, respectively. After considering rainfall, the particle height can be calculated by the following equation:
d′t+Δt＝dt+Δt+dr-di
wherein d ist+ΔtThe particle height at the time of t + delta t without considering rainfall and infiltration; drIs the particle height of increased rainfall; diIs the reduced particle height due to infiltration;
(4) according to new particle height d't+ΔtAnd correcting data such as mass, volume, speed and the like of the fluid particles. For the case of only infiltration, there is no need to correct the flow rate, but if there is rainfall, because the raindrops fall down and become flowing water, the flow rate of each fluid particle will change due to momentum conservation, and the flow rates of fluid particles of different masses need to be corrected by the following formula:
(m+Δmi-Δmf)·v′＝(m-Δmf)·v
where v' is the corrected flow velocity, Δ miIs the increase in particle mass, Δ m, caused by rainfallfIs the reduction in particle mass due to infiltration.
S5: the calculation region is partitioned, whether each partition has particle coverage is checked, and initial particles (i.e., fluid particles in an initial state in step S1) are added to the upstream region having no particle coverage. Due to the characteristics of the smooth particle hydrodynamics method, the particles can be set to be uniformly distributed at the beginning (time t 0) (as shown in fig. 3), and at this time, the research area is considered to be covered by the particles, so the rainfall at this time can be well expressed by the mass change of the particles.
As the simulation progresses, the fluid particles move to a region with a lower topography, and in the region with a lower topography, the particles approach each other, the area decreases, and the height increases (as shown in fig. 5). This is expected to simulate "confluence" in the "rainfall confluence" process, with the lower the terrain, the greater the water depth.
However, as the particles move, it is inevitable that at a certain moment, in some areas with high terrain where rainfall occurs, there will be no particles, as shown in fig. 6, in the lower right corner of the terrain, and there will be no particles in this area after a while. This in turn affects the simulation of the rainfall process, since the areas without particles (i.e. "bare areas") have no physical quantities to represent the rainfall of the rainfall.
In this case, Chang et al used a method of particle reconstruction, i.e. the fluid particle positions are reinitialized for each calculation time step and evenly distributed in the calculation region (returning the particles to the state of fig. 3). This method ensures that all positions are covered by fluid particles, but also results in computational efficiency and accuracy losses. To improve this, the present invention adds initial particles separately to the particle-free region to provide a subsequently calculated physical quantity. The method mainly comprises the following steps:
(1) the calculation area is partitioned. Firstly, the research area is divided into grids, and the length and the width of the grids are the square root of the area of initial particles
As shown in fig. 7.
(2) The area actually occupied by each particle is marked. After the meshing, the position occupied by each particle is marked as shown in fig. 8. It is assumed here that the length and width of each particle is the square root of the current area
(3) Distinguishing upstream and downstream regions of the terrain. An eight-direction method is introduced to distinguish upstream and downstream of the grid, and a parameter D8 (c) is adoptedi) To indicate the order in which the cells are labeled in runoff as shown in figure 9. The eight-direction method comprises the following calculation steps:
i. divide the calculation area into length and width
In the cell of (A), wherein0Represents the area of the fluid particles in the initial state;
ii. The flow direction of each unit cell is distributed to one of eight adjacent unit cells, wherein the flow direction can be up, down, left and right, or diagonal, and the flow direction is the direction with the steepest gradient;
iii, using the parameter D8 (c)i) To represent the order of the cells in runoff, the number represents how much water from the cells will flow into the current cell (including itself), and the calculation formula is as follows:
wherein, ciIs a cell in the calculation region, N is the flow direction ciAdjacent cells of (a).
Processed in the manner described above, each cell will have a corresponding D8 (c)i) The value of (c). Based on this, the upstream region can be specified by a threshold variable D8, e.g., D8-5, then the initial particle is only added to D8 (c)i) Less than or equal to 5 cells. By doing so, it is possible to add particles only to the specified upstream region, thereby avoiding calculation errors.
(4) Find the upstream mesh without fluid particles, add the initial particles. Through the previous two steps, specific grids without particles can be obtained, and then initial particles can be added at the center of the grids, wherein the area of the initial particles is A0. The results after addition of the initial particles are shown in FIG. 10.
Through the treatment, each area of the research area can be covered by particles to the maximum extent at the current moment, and the water quantity increased due to rainfall can be reflected in the model. If the fluid particles are replenished in this way at each time step of the calculation, the accuracy of the result can be guaranteed to the greatest extent. Because the method does not need particle reconstruction, the calculation efficiency and precision can be improved compared with the reconstruction method theoretically. It is also easy to find that the accuracy of the method depends on the area of the initial particles, and if the area is smaller, the meshing is smaller, the accuracy is higher, and vice versa.
S6: checking the calculated total time length, if the calculated total time length is greater than the simulation time length T, finishing the calculation, and outputting a calculation result; if the time is less than the simulation time period T, the process returns to S4, and the time T +2 × Δ T is calculated.
The final calculation results are shown in fig. 11. The abscissa is time and the ordinate is outlet flow and total radial flow.
While the embodiments of the present invention have been disclosed above, it is not limited to the applications listed in the description and embodiments, but is fully applicable to various fields suitable for the present invention, and it will be apparent to those skilled in the art that various changes, modifications, substitutions and alterations can be made in the embodiments without departing from the principle and spirit of the present invention, and therefore the present invention is not limited to the specific details without departing from the general concept defined in the claims and the scope of equivalents thereof.
Claims (7)
1. A surface production confluence calculation method based on smooth particle fluid dynamics is characterized by comprising the following steps:
s1: initializing calculation conditions: the spatial dimension is set to two dimensions; collecting earth surface information data, and describing the earth surface information data by using bottom particles; describing a flow field of surface production confluence by using fluid particles, and setting initial state data of the fluid particles; the initial state data of the fluid particles comprises mass, smooth length, position and velocity; setting an analog time length T; setting a simulation step length delta t;
s2: calculating the acceleration of the fluid particles according to the state data of the particles at the time t; calculating the speed and position of the particle at the t + delta t moment according to the state data and the acceleration of the particle at the t moment;
s3: calculating the density and the smooth length of the fluid particles after the fluid particles move to the position at the time t + delta t;
s4: calculating the mass of the fluid particles at the t + delta t moment according to the rainfall and the infiltration water amount, and performing momentum correction on the fluid particles;
s5: partitioning the calculation area, checking whether each partition is covered by particles, and adding initial particles to an upstream area without particle coverage; the initial particles refer to the fluid particles in the initial state in step S1;
s6: checking the calculated total time length, if the calculated total time length is greater than the simulation time length T, finishing the calculation, and outputting a calculation result; if the time is less than the simulation time period T, the process returns to S2, and the time T +2 × Δ T is calculated.
2. The method of claim 1, wherein the step S3 comprises the steps of:
s31: equation 1 listing the density of the fluid particles:
wherein m isjIs the mass of the fluid particle j, Wi(xj,hi) Is a formula of kernel function, hiIs the smooth length, x, of the fluid particle ijIs the distance between the fluid particles i and j;
s32: equation 2 listing the smooth length of the fluid particle:
wherein V0,iAnd ViRespectively the initial area of the particle i and the area at the current moment, dmIs that the spatial dimension takes the value of 2, h0,iIs the initial smooth length, h, of the fluid particle iiIs the smooth length of the fluid particle i at the current moment;
s33: and (3) combining equation 1 and equation 2, and solving the density and the smooth length of the fluid particles by a numerical calculation method.
3. The smooth particle hydrodynamics-based surface production convergence calculation method of claim 2, wherein: the numerical calculation method is a (Newton-Raphson) Newton-Raphson iteration method.
4. The method of claim 1, wherein the step S4 comprises the steps of:
s41: obtaining the infiltration quantity d at the t + delta t moment according to an infiltration modeli；
S42: obtaining the rainfall d at the t + delta t moment according to the rainfall datar；
S43: according to the quantity d of rainfalliAnd the amount of the lower seepage drCalculating the height d 'of the fluid particles at the time t + delta t't+Δt：
d′t+Δt＝dt+Δt+dr-di
Wherein d ist+ΔtThe particle height at the time of t + delta t without considering rainfall and infiltration; drIs the particle height of increased rainfall; diIs the reduced particle height due to infiltration;
s44: particle height d 'calculated according to step S43't+ΔtAnd correcting data such as mass, volume, speed and the like of the fluid particles.
5. The smooth particle hydrodynamics-based surface production confluence calculation method of claim 4, wherein: the infiltration model is a Horton infiltration model.
6. The method of claim 1, wherein the step S5 comprises the steps of:
s51: dividing the calculation area into several areas, and making grid division on the area to be researched, and the length and width of grid are all
Wherein A is0Represents the area of the fluid particles in the initial state;
s52: after the mesh division, the position occupied by each particle is marked, and the length and the width of each particle are assumed to be
Wherein A isiRepresenting the area of the fluid particle in the current state;
s53: an eight-direction method is introduced to distinguish upstream and downstream of the grid, and a parameter D8 (c) is adoptedi) To indicate the order of cells in the runoff for labeling;
s54: find the upstream mesh without fluid particles, add the initial particles.
7. The smooth particle hydrodynamics-based surface production confluence calculation method of claim 6, wherein the eight-direction method comprises the following calculation steps:
s61: divide the calculation area into length and width
In the cell of (A), wherein0Represents the area of the fluid particles in the initial state;
s62: the flow direction of each unit cell is distributed to one of eight adjacent unit cells, wherein the flow direction can be up, down, left and right, or diagonal, and the flow direction is the direction with the steepest gradient;
s63: using the parameter D8 (c)i) To represent the order of the cells in the runoff, the number represents how many cells including the current cell water will flow into the current cell, and the calculation formula is as follows:
wherein, ciIs a cell in the calculation region, N is the flow direction ciAdjacent cells of (a).
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