A finite volume method for calculating the average water surface slope of river network branches

Abstract

The present invention belongs to the field of one-dimensional river network hydrodynamic model, and provides a method for calculating the average water surface slope of a river network fork point using a finite volume method, including: calculating the water level at the connection between the river channel and the fork point grid, calculating the coordinates of the center point of the interface, obtaining the spatial coordinate sequence of the center point of the interface, and solving the average water surface slope of the fork point. The method of the present invention makes full use of the effective information of each sub-river section and the fork point, fits the best fork point plane to obtain the average water surface slope of the fork point, can solve the problem of difficult processing of the water surface slope of the fork point where multiple river sections meet, and has the advantages of physical rationality, reliable accuracy, stable algorithm and efficient calculation.

Description

Technical Field

[0001] The invention belongs to the field of one-dimensional river network hydrodynamic models, and particularly relates to a method for calculating the average water surface slope of river network branch points using a finite volume method. Background Art

[0002] The one-dimensional river network hydrodynamic model based on the Godunov-type finite volume method needs to solve the water surface slope term of the fork grid during model calculation. Since the fork grid is formed by the intersection of multiple sub-river sections, the water levels of the sub-river sections are different, which makes it difficult to calculate the water surface slope of the fork. Therefore, a method for calculating the average water surface slope of the fork needs to be proposed.

[0003] In the existing research literature, there is no report on the calculation of the water surface slope of the fork grid. This technical field is blank and a new calculation method is urgently needed. In addition, the water surface slope of the fork grid involves multiple rivers that converge to the fork. The obtained water surface slope has a great influence on the stability and accuracy of the one-dimensional hydrodynamic model calculation, and is a key link in the one-dimensional hydrodynamic model calculation. Summary of the invention

[0004] The present invention is based on a one-dimensional river network hydrodynamic model of a Godunov-type format finite volume method, and proposes a method for calculating the average water surface slope of a river network fork using a finite volume method. The method makes full use of the effective information of each sub-river section and fork, fits the optimal fork plane to obtain the average water surface slope of the fork, and can be used to solve the problem that the water surface slope of a fork at the intersection of multiple rivers is difficult to calculate.

[0005] The object of the present invention is achieved in that:

[0006] A method for calculating the average water surface slope of a river network branch point using a finite volume method comprises the following steps:

[0007] Step 1: Calculate the water level at the junction of the river channel and the branch point grid:

[0008] Calculate the water level at the junction of each river channel and the branch point grid in the study area, first perform non-negative hydrostatic reconstruction, and then calculate the interface water level;

[0009] Step 2: Calculate the coordinates of the center point of the interface:

[0010] According to the centroid coordinates of the branch point grid and the centroid coordinates of each river channel grid, the coordinates of the center point of the interface between each river channel and the branch point are calculated by arithmetic average method;

[0011] Step 3: Get the spatial coordinate sequence of the center point of the interface:

[0012] According to the calculation results obtained in step 1 and step 2, the three-dimensional spatial coordinates of the center point of the interface between the river channel and the branch point are formed, and the spatial coordinates of the center point of the interface between each river channel and the branch point are collected to form a spatial coordinate sequence of the center point of the interface;

[0013] Step 4: Calculate the average slope of the water surface at the branch point:

[0014] Using the plane equation Ax+By+Cz+D=0 and the spatial coordinate sequence of the center point of the interface obtained in step 3, the average slope of the water surface at the branch point is solved.

[0015] Furthermore, in step 1, the specific method for calculating the water level at the junction of each river channel and the branch point grid in the study area is:

[0016] First, non-negative hydrostatic reconstruction is performed: the water level and grid bottom elevation that are lower than the adjacent grid bottom elevation are raised to the adjacent grid bottom elevation;

[0017] Then, calculate the interface water level: the interface water level is equal to the arithmetic mean of the water levels on both sides of the interface.

[0018] Furthermore, in step 2, the coordinates of the center point of the interface between the river channel and the branch point are (x i ,y i ), and its calculation formula is:

[0019]

[0020]

[0021] In the formula, (X JUNC ,Y JUNC ) is the centroid coordinate of the branch point grid; (X i ,Y i ) are the centroid coordinates of the river grid; i = 1, 2, ..., n, where n is the number of rivers converging into the branch point.

[0022] Furthermore, in step 4, the specific method for solving the average slope of the branch point water surface is:

[0023] Substitute the spatial coordinate sequence of the center point of the interface obtained in step 3 into the plane equation Ax+By+Cz+D=0 to obtain a set of linear equations. The least squares method is used to solve the equations and the plane normal vector to calculate the average slope of the water surface at the branch point.

[0024] The advantages and beneficial effects of the present invention are:

[0025] The present invention is based on a one-dimensional river network hydrodynamic model of the Godunov-type format finite volume method, and proposes a method for calculating the average water surface slope of the river network forks using the finite volume method, making full use of the effective information of each sub-river section and the forks, which is the key to solving the problem of the difficulty in calculating the water surface slope of the forks at the intersection of multiple rivers. The three-dimensional spatial coordinates of the center points of the interface between each sub-river section and the forks are collected to form a three-dimensional linear equation group, and the optimal fork plane is fitted by solving the linear equation group by the least squares method to obtain the average water surface slope of the forks. This is a new idea for calculating the water surface slope of the forks at the intersection of multiple rivers, which can solve the problem of the difficulty in processing the water surface slope of the forks at the intersection of multiple river sections, and has the advantages of physical rationality, reliable accuracy, stable algorithm and efficient calculation, and has important academic significance and engineering application significance. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0027] Figure 1 It is a schematic diagram of a flow chart of a method for calculating the average water surface slope of a river network branch point using the finite volume method of the present invention;

[0028] Figure 2 Schematic diagram of a river channel confluence point according to Embodiment 3 of the present invention;

[0029] Figure 3 is a schematic diagram of non-negative hydrostatic reconstruction according to an embodiment of the present invention;

[0030] Figure 4 is a schematic diagram of calculating the coordinates of the interface according to an embodiment of the present invention;

[0031] Figure 5 It is a schematic diagram of calculating the average water surface slope using the normal vector in an embodiment of the present invention. DETAILED DESCRIPTION

[0032] Example:

[0033] like Figure 1 As shown, based on the one-dimensional river network hydrodynamic model of the Godunov type format finite volume method, this embodiment provides a method for calculating the average water surface slope of the river network branch points using the finite volume method, including the following steps:

[0034] Step 1: Calculate the water level at the junction of the river channel and the branch point grid:

[0035] Calculate the water level at the junction of each river channel and branch point grid in the study area (i.e., the interface between the river channel grid and the branch point grid). Since the water level at each interface is not exactly the same, the interface water level is set to be z i , i = 1, 2, ..., n, n is the number of rivers that flow into the fork. For the convenience of description, Figure 2As shown, this embodiment takes three rivers as an example, that is, n=3.

[0036] To prevent negative water depth and unreasonable flow, firstly, non-negative hydrostatic reconstruction is performed:

[0037] That is, the water level (Z1) and grid bottom elevation (zb1) that are lower than the adjacent grid bottom elevation (zb2) are raised to the adjacent grid bottom elevation (zb2). Figure 3 As shown, the bottom elevation zb1 and water level Z1 of the left grid are raised to zb2. Among them, zb1 is the bottom elevation of the grid before reconstruction requiring non-negative hydrostatic reconstruction; zb1' is the bottom elevation of the grid after reconstruction requiring non-negative hydrostatic reconstruction (equal to zb2).

[0038] Then, calculate the interface water level: Interface water level z i It is equal to the arithmetic mean of the water levels on both sides of the interface, and the calculation formula is:

[0039]

[0040] Where Z1' is the water level of the grid after reconstruction that requires non-negative hydrostatic reconstruction (equal to zb2); Z2 is the water level of the grid on the other side (the bottom elevation of the grid is zb2).

[0041] Step 2: Calculate the coordinates of the center point of the interface:

[0042] like Figure 4 As shown in the figure, based on the centroid coordinates of the fork point grid and the centroid coordinates of each river channel grid, the coordinates of the center point of the interface between each river channel and fork point are calculated by arithmetic average method. The coordinates of the center point of the interface between the river channel and fork point are (x i ,y i ), and its calculation formula is:

[0043]

[0044]

[0045] In the formula, (X JUNC ,Y JUNC ) is the centroid coordinate of the branch point grid; (X i ,Y i ) are the centroid coordinates of the river grid; i = 1, 2, ..., n, where n is the number of rivers converging into the branch point.

[0046] Step 3: Get the spatial coordinate sequence of the center point of the interface:

[0047] According to the calculation results obtained in step 1 and step 2, the three-dimensional spatial coordinates of the center point of the interface between the river channel and the branch point are formed, and the spatial coordinates of the center point of the interface between each river channel and the branch point are collected to form the spatial coordinate sequence of the center point of the interface (x i,y i ,z i ).

[0048] Step 4: Calculate the average slope of the water surface at the branch point:

[0049] The spatial coordinate sequence (x i ,y i ,z i ), substitute the plane equation Ax+By+Cz+D=0, the average slope of the water surface at the branch point, where A, B, C, and D are all unknown coefficients. Specifically:

[0050] Since the water surface of a natural river flood branch is usually not perpendicular to the horizontal plane, C≠0, and the above plane equation can be converted to Ax / C+By / C+z+D / C=0. Substituting the spatial coordinate sequence of the center point of the interface into the equation, we can get a set of linear equations with coefficients A / C, B / C, and D / C as unknown quantities. Since the number of natural branch river channels is generally less than 10, the least squares rule can be used to efficiently solve the system of equations and obtain the unknown coefficient values ​​with the highest fit to obtain the plane equation. The plane normal vector can be obtained according to the plane equation, and the angle θ between the plane normal vector and the horizontal plane can be solved by using the vector inner product to calculate the average water surface slope tan(90°-θ) of the branch. The schematic diagram of the normal vector to calculate the average water surface slope is shown in the figure. Figure 5 shown.

[0051] Calculation example:

[0052] like Figure 2 As shown in the figure, a river network branch point is connected to three rivers, among which:

[0053] (1) Coordinates of the centroid of the branch point grid (X JUNC ,Y JUNC )=(0,0), the bottom elevation of the centroid of the branch point zb JUNC =1, water level Z JUNC =2;

[0054] (2) The centroid coordinates of the grid at the junction of channel 1 and the branch point are (X1, Y1) = (-2, 1), the centroid bottom elevation zb1 = -1, and the water level Z1 = 0;

[0055] (3) The centroid coordinates of the grid at the junction of channel 2 and the branch point are (X2, Y2) = (2, 1), the centroid bottom elevation zb2 = 0, and the water level Z2 = 2;

[0056] (4) The centroid coordinates of the grid at the junction of channel 3 and the branch point are (X3, Y3) = (0, -2), the centroid bottom elevation zb3 = 0.5, and the water level Z3 = 2;

[0057] The method described in the embodiment of the present application is used to solve the average water surface slope θ of the above branch point:

[0058] The bottom elevation of the centroid of the grid at the junction of the river channel 1 and the fork is zb1, and the water level Z1 is lower than the bottom elevation of the fork. Therefore, it is necessary to perform non-negative hydrostatic reconstruction on the grid at the junction of the river channel 1 and the fork: Z1'=zb1'=zb JUNC =1.

[0059] Calculate the water level at the interface:

[0060] z1=(Z1'+Z JUNC ) / 2=1.5.

[0061] z2=(Z2+Z JUNC ) / 2=2.

[0062] z3=(Z3+Z JUNC ) / 2=2.

[0063] Calculate the coordinates of the center point of the interface:

[0064] x1=(X1+X JUNC ) / 2=-1; y1=(Y1+Y JUNC ) / 2=0.5.

[0065] x2=(X2+X JUNC ) / 2=1;y2=(Y2+Y JUNC ) / 2=0.5.

[0066] x3=(X3+X JUNC ) / 2=0; y3=(Y3+Y JUNC ) / 2=-1.

[0067] The spatial coordinate sequence of the center point of the interface is obtained:

[0068] The coordinate sequence is: (-1,0.5,1.5), (1,0.5,2), (0,-1,2).

[0069] Linear system of equations:

[0070]

[0071] Using the least squares method to solve the above linear equations, the optimal solution of the unknown coefficients is:

[0072]

[0073] The best fitting plane equation is -0.25x+0.1666y+z-1.8334=0, and the plane normal vector n1=(-0.25, 0.1666, 1).

[0074] Angle with horizontal plane z=0 That is, θ=16.72°.

[0075] Finally, it should be noted that the above is only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred arrangement scheme, a person skilled in the art should understand that the technical solution of the present invention (such as the use of various formulas, the sequence of steps, etc.) can be modified or replaced by equivalents without departing from the spirit and scope of the technical solution of the present invention.

Claims

1. A method for calculating the average water surface slope of a river network branch point using the finite volume method, characterized in that: The method comprises the following steps: Step 1: Calculate the water level at the junction of the river channel and the branch point grid: Calculate the water level at the junction of each river channel and branch point grid in the study area; First, non-negative hydrostatic reconstruction is performed: the water level and grid bottom elevation that are lower than the adjacent grid bottom elevation are raised to the adjacent grid bottom elevation; Then, calculate the interface water level: the interface water level is equal to the arithmetic mean of the water levels on both sides of the interface; Step 2: Calculate the coordinates of the center point of the interface: According to the centroid coordinates of the branch point grid and the centroid coordinates of each river channel grid, the coordinates of the center point of the interface between each river channel and the branch point are calculated by arithmetic average method; Step 3: Get the spatial coordinate sequence of the center point of the interface: According to the calculation results obtained in step 1 and step 2, the three-dimensional spatial coordinates of the center point of the interface between the river channel and the branch point are formed, and the spatial coordinates of the center point of the interface between each river channel and the branch point are collected to form a spatial coordinate sequence of the center point of the interface; Step 4: Calculate the average slope of the water surface at the branch point: Substitute the spatial coordinate sequence of the center point of the interface obtained in step 3 into the plane equation Ax+By+Cz+D=0 to obtain a set of linear equations. The least squares method is used to solve the equations and the plane normal vector to calculate the average slope of the water surface at the branch point.

2. The calculation method according to claim 1, characterized in that: In step 2, the coordinates of the center point of the interface between the river channel and the branch point are (x i ,y i ), and its calculation formula is: In the formula, (X JUNC ,Y JUNC ) is the centroid coordinate of the branch point grid; (X i ,Y i ) are the centroid coordinates of the river grid; i = 1, 2, ..., n, where n is the number of rivers converging into the branch point.

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