CN111767684B - Modeling method of optimized friction source item implicit format two-dimensional shallow water equation - Google Patents
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Abstract
The invention discloses an optimized friction source item implicit format two-dimensional shallow water equation modeling method, which comprises the steps of firstly dispersing terrain data of a river basin by adopting rectangular grids in a structural grid, and assigning a value to each parameter according to boundary conditions; adopting a central-format space discrete two-dimensional shallow water equation by a finite volume method; dividing a solving area into a plurality of small misaligned subareas based on a limited volume method of Godunov format, adopting an approximation solving Riemann problem method proposed by Harten, lax and Leer, namely an HLLC Riemann operator, and controlling flux calculation on a unit interface; and then calculating a source item and a stability condition, and finally updating the single-width flow, so that the modeling method of the optimized friction source item implicit format two-dimensional shallow water equation is completed. The invention improves the simulation efficiency and the calculation precision of the surface water dynamic process and provides effective technical support for the prediction of the associated process based on the surface water.
Description
Technical Field
The invention belongs to the technical field of surface water numerical simulation methods, and particularly relates to an optimized friction source term implicit format two-dimensional shallow water equation modeling method.
Background
The two-dimensional shallow water equation numerical model based on dynamic waves can simulate surface runoff and associated processes thereof, such as urban rainfall flood process, sub-flood process, pollutant diffusion process and sediment transport process. The method can provide effective technical support for prediction and early warning of urban flood disasters, river flood control, non-point source pollution and water and soil loss.
The high-efficiency accurate simulation calculation result is a basis for guaranteeing the accuracy and timeliness of prediction early warning, the high-resolution terrain data can truly represent the terrain, and the model simulation precision is improved. Meanwhile, the high-resolution topographic data restricts the operation efficiency of the model, and one of the restriction factors is the calculation of friction source items in the model operation. Proper treatment of friction source terms is one of the important conditions to ensure the computational stability of the model at shallow waters. Currently, the active and effective friction source treatment is based on an implicit method. However, the iteration of implicit calculations greatly increases the computational burden of the model, especially for high resolution terrain over large scale areas. Therefore, development of a new scheme for treating friction source item with positive and efficient is needed. Provides technical support for disaster prevention, disaster reduction, water pollution prevention and ecological civilization construction.
Disclosure of Invention
The invention aims to provide an optimized friction source item implicit format two-dimensional shallow water equation modeling method, which improves the simulation efficiency and calculation accuracy of a surface water power process and provides effective technical support for prediction of a surface water-based associated process.
The technical scheme adopted by the invention is that an optimized friction source item implicit format two-dimensional shallow water equation modeling method is implemented according to the following steps:
step 1, dispersing terrain data of a river basin by adopting rectangular grids in a structural grid, and assigning a value to each parameter according to boundary conditions;
step 2, adopting a central-format space discrete two-dimensional shallow water equation by a finite volume method;
Step 3, dividing the solving area into a plurality of small non-coincident subareas based on a limited volume method of Godunov format, adopting a method for solving the Riemann problem by approximation proposed by Harten, lax and Leer, namely an HLLC Riemann operator, and controlling flux calculation on a unit interface;
Step 4, source item calculation: the source item part comprises a bottom slope item and a friction item, the bottom slope source item processing adopts integration of decomposing integration on a cell into sub-units, and the bottom slope source item of the grid is converted into flux on the surface of the grid by assuming linear change of the height and the water level of the bed bottom in the grid; the friction source item treatment adopts a friction source item calculation method;
step 5, calculating stability conditions: the stability condition is satisfied by the time step and the space step;
And 6, updating the single wide flow, and thus, completing the modeling method of the two-dimensional shallow water equation with the implicit format of the optimized friction source item.
The present invention is also characterized in that,
The two-dimensional shallow water equation in step 2 is expressed as follows:
Wherein: t represents a time step, in s; x represents a horizontal transverse coordinate, and y represents a longitudinal coordinate; q represents a conservation variable, q x represents a single-width flow in the x direction, m 2/s,qx＝uh;qy represents a single-width flow in the y direction, and m 2/s,qy =vh; h represents the water depth, and the unit is m; u represents the average flow velocity in the x direction in m/s; v represents the average flow velocity in the y direction in m/s; f represents the flux vector in the x-direction; g represents the flux vector in the y-direction; s represents a source item vector; s b represents a bottom slope source term; s f represents a friction source item; z b represents the height of the river bed, and the unit is m; c f represents the bed surface roughness coefficient, C f＝gn2/h1 /3; n represents a Manning coefficient, the unit is s/m -1/3, eta represents the water level, and the unit is m; η=h+z b; i represents static rainfall in mm/h, i=i r-ii;ir represents rainfall intensity in mm/h; i i represents the infiltration rate in mm/h.
The step 3 is specifically as follows:
Step 3.1, firstly, performing spatial second-order precision reconstruction on the interface variables to form a Riemann problem, performing numerical reconstruction on the variables at the left side and the right side of the unit interface by adopting a MUSCL state interpolation method, and simultaneously limiting gradients by combining a Min-Mod limiter to effectively inhibit numerical oscillation;
and 3.2, introducing a non-negative water reconstruction method to ensure harmony and conservation of mass of numerical calculation at a dry-wet interface.
The friction resistance source term calculation formula in the step4 is specifically as follows:
Wherein: x represents a horizontal transverse coordinate, and y represents a longitudinal coordinate; q x denotes the single-width flow in x direction, the unit is m 2/s,qx＝uh;qy denotes the single-width flow in y direction, the unit is m 2/s,qy =vh; h represents the water depth, and the unit is m; u represents the average flow velocity in the x direction in m/s; v represents the average flow velocity in the y direction in m/s; s fx represents a friction source item in the x direction; s fy represents a friction source term in the y direction; c f represents the bed surface roughness coefficient, C f＝gn2/h1/3; n represents a Manning coefficient, and the unit is s/m -1/3.
The step 5 is specifically as follows:
under the space step length deltax, the calculation formula of the value of the time step length deltat is as follows:
Wherein: Δx min represents the minimum distance from the center of the grid cell to the corresponding interface, in m; u is the flow rate in m/s; g is the gravity acceleration, and the value is 9.8m/s 2; h is the depth of water in m.
The step6 is specifically as follows:
Step 6.1, the single-wide traffic is updated at a new time step:
qn+1＝qn+Δq+ΔtSf (8)
Wherein: q n+1 represents the single wide flow of n+1, the unit is m 2/s,qn represents the single wide flow at the time of n, the unit is m 2/s;Δq＝F(qn,hn)+ΔtSb, and the flow term is calculated through HLLC; f represents interface flux; s b represents the bottom slope source term flux; s f represents friction source term flux; Δt represents the time step in s;
And 6.2, substituting the calculation formula of the friction resistance source term in the step 4 into a formula (8) to obtain the following two-dimensional form:
Wherein: For single wide flow in x direction at time n+1, the unit is m 2/s,/> For single wide flow in the x direction at time n, the unit is m 2/s;Δqx which is a flow term calculated by HLLC in the x direction; /(I)And/>The single wide flow in the x direction at the time of n+1 and n is calculated by HLLC in the y direction, wherein the unit is m 2/s;Δqy; c f represents the roughness coefficient of the bed surface; h n represents the water depth at time n, and the unit is m; Δt represents the time step in s.
X represents a horizontal transverse coordinate, and y represents a longitudinal coordinate; q x denotes the single-width flow in x direction, the unit is m 2/s,qx＝uh;qy denotes the single-width flow in y direction, the unit is m 2/s,qy =vh; h represents the water depth, and the unit is m; u represents the average flow velocity in the x direction in m/s; v represents the average flow velocity in the y direction in m/s; s fx represents a friction source item in the x direction; s fy represents a friction source term in the y direction; c f represents the bed surface roughness coefficient, C f＝gn2/h1/3; n represents a Manning coefficient, and the unit is s/m -1/3.
Step 6.3, solving the steps (9) and (10), and calculating to obtain the single wide flow under the new time step as follows:
Wherein: For single wide flow in x direction at time n+1, the unit is m 2/s,/> For single wide flow in the x direction at time n, the unit is m 2/s;Δqx which is a flow term calculated by HLLC in the x direction; /(I)Is the single wide flow rate at the time of n+1 in the y direction, and the unit is m 2/s,/>The unit is m 2/s;Δqy, which is the single wide flow in the y direction at the time n, and is the flow term calculated by HLLC in the y direction; c f represents the roughness coefficient of the bed surface; h n represents the water depth at time n, and the unit is m; Δt represents the time step in s; θ represents a parameter introduced to avoid denominator of 0, in m 2/s,θ＝1.0e-12.
The invention has the beneficial effects that the modeling method of the optimized friction source item implicit format two-dimensional shallow water equation aims at solving the problem of unavoidable redundant iteration of the implicit scheme adopted by the existing numerical model in the aspect of processing friction source item calculation, and provides a new processing scheme for avoiding iteration, which simulates the surface runoff process based on dynamic waves under a structure limited volume numerical discrete frame, can accurately ensure the conservation of water quantity in the whole calculation process, and efficiently and stably simulate the motion process of the surface runoff under the conditions of large gradient and small water depth of a river basin.
Drawings
FIG. 1 is a schematic view of a "V" shaped basin;
FIG. 2 is a comparison of simulated and analyzed values of the "V" watershed slope flow process;
fig. 3 is a comparison of the simulated value and the analytical solution value of the "V" basin exit flow process.
Detailed Description
The invention will be described in detail below with reference to the drawings and the detailed description.
The invention discloses an optimized friction source item implicit format two-dimensional shallow water equation modeling method, which is implemented according to the following steps:
step 1, dispersing terrain data of a river basin by adopting rectangular grids in a structural grid, and assigning a value to each parameter according to boundary conditions;
step 2, adopting a central-format space discrete two-dimensional shallow water equation by a finite volume method;
the two-dimensional shallow water equation in step 2 is expressed as follows:
Wherein: t represents a time step, in s; x represents a horizontal transverse coordinate, and y represents a longitudinal coordinate; q represents a conservation variable, q x represents a single-width flow in the x direction, m 2/s,qx＝uh;qy represents a single-width flow in the y direction, and m 2/s,qy =vh; h represents the water depth, and the unit is m; u represents the average flow velocity in the x direction in m/s; v represents the average flow velocity in the y direction in m/s; f represents the flux vector in the x-direction; g represents the flux vector in the y-direction; s represents a source item vector; s b represents a bottom slope source term; s f represents a friction source item; z b represents the height of the river bed, and the unit is m; c f represents the bed surface roughness coefficient, C f＝gn2/h1 /3; n represents a Manning coefficient, the unit is s/m -1/3, eta represents the water level, and the unit is m; η=h+z b; i represents static rainfall in mm/h, i=i r-ii;ir represents rainfall intensity in mm/h; i i represents the infiltration rate in mm/h.
Step 3, dividing the solving area into a plurality of small non-coincident subareas based on a limited volume method of Godunov format, adopting a method for solving the Riemann problem by approximation proposed by Harten, lax and Leer, namely an HLLC Riemann operator, and controlling flux calculation on a unit interface;
the step 3 is specifically as follows:
step 3.1, firstly, performing spatial second-order precision reconstruction on the interface variables to form a Riemann problem, performing numerical reconstruction on the variables at the left side and the right side of the unit interface by adopting a MUSCL (Monotonic Upstream Scheme for Conservation Laws) state interpolation method, and simultaneously limiting gradients by combining a Min-Mod limiter to effectively inhibit numerical oscillation;
And 3.2, introducing a non-negative water reconstruction method in the process of calculating the boundary flux by using the HLLC Riemann operator so as to ensure harmony and conservation of mass of numerical calculation at a dry-wet interface.
Step 4, source item calculation: the source item part comprises a bottom slope item and a friction item, the bottom slope source item processing adopts integration of decomposing integration on a cell into sub-units, and the bottom slope source item of the grid is converted into flux on the surface of the grid by assuming linear change of the height and the water level of the bed bottom in the grid; the friction source item treatment adopts a friction source item calculation method;
the friction resistance source term calculation formula in the step4 is specifically as follows:
Wherein: x represents a horizontal transverse coordinate, and y represents a longitudinal coordinate; q x denotes the single-width flow in x direction, the unit is m 2/s,qx＝uh;qy denotes the single-width flow in y direction, the unit is m 2/s,qy =vh; h represents the water depth, and the unit is m; u represents the average flow velocity in the x direction in m/s; v represents the average flow velocity in the y direction in m/s; s fx represents a friction source item in the x direction; s fy represents a friction source term in the y direction; c f represents the bed surface roughness coefficient, C f＝gn2/h1/3; n represents a Manning coefficient, and the unit is s/m -1/3.
Step 5, calculating stability conditions: the stability condition is satisfied by the time step and the space step;
the step 5 is specifically as follows:
under the space step length deltax, the calculation formula of the value of the time step length deltat is as follows:
Wherein: Δx min represents the minimum distance from the center of the grid cell to the corresponding interface, in m; u is the flow rate in m/s; g is the gravity acceleration, and the value is 9.8m/s 2; h is the depth of water in m.
And 6, updating single wide flow, namely finishing the modeling method of the two-dimensional shallow water equation with the implicit format of the optimized friction resistance source item, wherein the method comprises the following steps of:
Step 6.1, the single-wide traffic is updated at a new time step:
qn+1＝qn+Δq+ΔtSf (8)
Wherein: q n+1 represents the single wide flow of n+1, the unit is m 2/s,qn represents the single wide flow at the time of n, the unit is m 2/s;Δq＝F(qn,hn)+ΔtSb, and the flow term is calculated through HLLC; f represents interface flux; s b represents the bottom slope source term flux; s f represents friction source term flux; Δt represents the time step in s;
And 6.2, substituting the calculation formula of the friction resistance source term in the step 4 into a formula (8) to obtain the following two-dimensional form:
Wherein: For single wide flow in x direction at time n+1, the unit is m 2/s,/> For single wide flow in the x direction at time n, the unit is m 2/s;Δqx which is a flow term calculated by HLLC in the x direction; /(I)And/>The single wide flow in the x direction at the time of n+1 and n is calculated by HLLC in the y direction, wherein the unit is m 2/s;Δqy; c f represents the roughness coefficient of the bed surface; h n represents the water depth at time n, and the unit is m; Δt represents the time step in s.
X represents a horizontal transverse coordinate, and y represents a longitudinal coordinate; q x denotes the single-width flow in x direction, the unit is m 2/s,qx＝uh;qy denotes the single-width flow in y direction, the unit is m 2/s,qy =vh; h represents the water depth, and the unit is m; u represents the average flow velocity in the x direction in m/s; v represents the average flow velocity in the y direction in m/s; s fx represents a friction source item in the x direction; s fy represents a friction source term in the y direction; c f represents the bed surface roughness coefficient, C f＝gn2/h1/3; n represents a Manning coefficient, and the unit is s/m -1/3.
Step 6.3, solving the steps (9) and (10), and calculating to obtain the single wide flow under the new time step as follows:
Wherein: For single wide flow in x direction at time n+1, the unit is m 2/s,/> For single wide flow in the x direction at time n, the unit is m 2/s;Δqx which is a flow term calculated by HLLC in the x direction; /(I)Is the single wide flow rate at the time of n+1 in the y direction, and the unit is m 2/s,/>The unit is m 2/s;Δqy, which is the single wide flow in the y direction at the time n, and is the flow term calculated by HLLC in the y direction; c f represents the roughness coefficient of the bed surface; h n represents the water depth at time n, and the unit is m; Δt represents the time step in s; θ represents a parameter introduced to avoid denominator of 0, in m 2/s,θ＝1.0e-12.
Examples
Discrete and conventional parameter assignment of the drainage basin terrain grid: the watershed topography was discretized into a structured uniform grid with 5m cell side length for a total of 64,800 cells. The length and width of the slope are 1000m and 800m respectively, and the gradient is 0.05. The channel width was 20m, the riverbed slope was 0.02, and the sidewall design of the trench was high enough to prevent the effects of backwater, as shown in FIG. 1. Then, a rainfall event with constant rainfall intensity of 10.8mm/h and rainfall duration of 1.5h was simulated. The Manning coefficients of the slope and the channel were taken to be 0.015s/m 1/3 and 0.15s/m 1/3, respectively.
The numerical calculation effect is solved for comparing the optimized friction source item implicit format, in addition, the friction source item solving method of the explicit type and split point implicit format is respectively modeled in a numerical mode and is respectively simulated. The rainfall-runoff process of the V-shaped basin is simulated based on the model for processing the friction source item by the 3 different algorithms, wherein FIG. 2 is a comparison of a simulation value and an analytic solution value of the slope flow process of the V-shaped basin, and FIG. 3 is a comparison of a simulation value and an analytic solution value of the outlet flow process of the V-shaped basin. As is evident from a comparison of the flow process lines in fig. 2 and 3, the model for processing the friction source item based on the implicit and explicit algorithms of the split point cannot truly reflect the flow process of the slope and the drainage basin outlet. The algorithm for processing the friction source term by applying the new format of the implicit algorithm in the model can better simulate the flow process of the slope and the channel outlet, and the result is very close to the value of the analytic solution.
Claims (3)
1. An optimized friction source term implicit format two-dimensional shallow water equation modeling method is characterized by comprising the following steps:
step 1, dispersing terrain data of a river basin by adopting rectangular grids in a structural grid, and assigning a value to each parameter according to boundary conditions;
step 2, adopting a central-format space discrete two-dimensional shallow water equation by a finite volume method;
Step 3, dividing the solving area into a plurality of small non-coincident subareas based on a limited volume method of Godunov format, adopting a method for solving the Riemann problem by approximation proposed by Harten, lax and Leer, namely an HLLC Riemann operator, and controlling flux calculation on a unit interface;
Step 4, source item calculation: the source item part comprises a bottom slope item and a friction item, the bottom slope source item processing adopts integration of decomposing integration on a cell into sub-units, and the bottom slope source item of the grid is converted into flux on the surface of the grid by assuming linear change of the height and the water level of the bed bottom in the grid; the friction source item treatment adopts a friction source item calculation method;
the friction resistance source term calculation formula in the step 4 is specifically as follows:
Wherein: x represents a horizontal transverse coordinate, and y represents a longitudinal coordinate; q x denotes the single-width flow in x direction, the unit is m 2/s,qx＝uh;qy denotes the single-width flow in y direction, the unit is m 2/s,qy =vh; h represents the water depth, and the unit is m; u represents the average flow velocity in the x direction in m/s; v represents the average flow velocity in the y direction in m/s; s fx represents a friction source item in the x direction; s fy represents a friction source term in the y direction; c f represents the bed surface roughness coefficient, C f＝gn2/h1/3; n represents a Manning coefficient, and the unit is s/m -1/3;
step 5, calculating stability conditions: the stability condition is satisfied by the time step and the space step;
The step 5 specifically comprises the following steps:
under the space step length deltax, the calculation formula of the value of the time step length deltat is as follows:
Wherein: Δx min represents the minimum distance from the center of the grid cell to the corresponding interface, in m; u is the flow rate in m/s; g is the gravity acceleration, and the value is 9.8m/s 2; h is the depth of water, and the unit is m;
step 6, updating single wide flow, so far, completing the modeling method of the two-dimensional shallow water equation with the implicit format of the optimized friction resistance source item,
The step 6 specifically comprises the following steps:
Step 6.1, the single-wide traffic is updated at a new time step:
qn+1＝qn+Δq+ΔtSf (8)
Wherein: q n+1 represents the single wide flow of n+1, the unit is m 2/s,qn represents the single wide flow at the time of n, the unit is m 2/s;Δq＝F(qn,hn)+ΔtSb, and the flow term is calculated through HLLC; f represents interface flux; s b represents the bottom slope source term flux; s f represents friction source term flux; Δt represents the time step in s;
And 6.2, substituting the calculation formula of the friction resistance source term in the step 4 into a formula (8) to obtain the following two-dimensional form:
Wherein: For single wide flow in x direction at time n+1, the unit is m 2/s,/> For single wide flow in the x direction at time n, the unit is m 2/s;Δqx which is a flow term calculated by HLLC in the x direction; /(I)And/>The single wide flow in the x direction at the time of n+1 and n is calculated by HLLC in the y direction, wherein the unit is m 2/s;Δqy; c f represents the roughness coefficient of the bed surface; h n represents the water depth at time n, and the unit is m; Δt represents the time step in s;
x represents a horizontal transverse coordinate, and y represents a longitudinal coordinate; q x denotes the single-width flow in x direction, the unit is m 2/s,qx＝uh;qy denotes the single-width flow in y direction, the unit is m 2/s,qy =vh; h represents the water depth, and the unit is m; u represents the average flow velocity in the x direction in m/s; v represents the average flow velocity in the y direction in m/s; s fx represents a friction source item in the x direction; s fy represents a friction source term in the y direction; c f represents the bed surface roughness coefficient, C f＝gn2/h1/3; n represents a Manning coefficient, and the unit is s/m -1/3;
Step 6.3, solving the steps (9) and (10), and calculating to obtain the single wide flow under the new time step as follows:
Wherein: For single wide flow in x direction at time n+1, the unit is m 2/s,/> For single wide flow in the x direction at time n, the unit is m 2/s;Δqx which is a flow term calculated by HLLC in the x direction; /(I)Is the single wide flow rate at the time of n+1 in the y direction, and the unit is m 2/s,/>The unit is m 2/s;Δqy, which is the single wide flow in the y direction at the time n, and is the flow term calculated by HLLC in the y direction; c f represents the roughness coefficient of the bed surface; h n represents the water depth at time n, and the unit is m; Δt represents the time step in s; θ represents a parameter introduced to avoid denominator of 0, in m 2/s,θ＝1.0e-12.
2. The modeling method of an optimized friction source term implicit format two-dimensional shallow water equation according to claim 1, wherein the two-dimensional shallow water equation in step 2 is represented as follows:
Wherein: t represents a time step, in s; x represents a horizontal transverse coordinate, and y represents a longitudinal coordinate; q represents a conservation variable, q x represents a single-width flow in the x direction, m 2/s,qx＝uh;qy represents a single-width flow in the y direction, and m 2/s,qy =vh; h represents the water depth, and the unit is m; u represents the average flow velocity in the x direction in m/s; v represents the average flow velocity in the y direction in m/s; f represents the flux vector in the x-direction; g represents the flux vector in the y-direction; s represents a source item vector; s b represents a bottom slope source term; s f represents a friction source item; z b represents the height of the river bed, and the unit is m; c f represents the bed surface roughness coefficient, C f＝gn2/h1/3; n represents a Manning coefficient, the unit is s/m -1/3, eta represents the water level, and the unit is m; η=h+z b; i represents static rainfall in mm/h, i=i r-ii;ir represents rainfall intensity in mm/h; i i represents the infiltration rate in mm/h.
3. The modeling method of the two-dimensional shallow water equation with the implicit format of the friction drag source term according to claim 2, wherein the step 3 is specifically as follows:
Step 3.1, firstly, performing spatial second-order precision reconstruction on the interface variables to form a Riemann problem, performing numerical reconstruction on the variables at the left side and the right side of the unit interface by adopting a MUSCL state interpolation method, and simultaneously limiting gradients by combining a Min-Mod limiter to effectively inhibit numerical oscillation;
and 3.2, introducing a non-negative water reconstruction method to ensure harmony and conservation of mass of numerical calculation at a dry-wet interface.
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