One-dimensional Dam-Break Flood Simulation Method Based on ROE Format

Abstract

The one-dimensional dam-break flood simulation method based on the ROE format disclosed by the present invention reads the basic data of the calculation area in a computer and processes the data; discretizes the Saint-Venant equations by using the finite volume method in Godunov format, performs time integration on the Saint-Venant equations by using the predictor-corrector method, and calculates hydraulic elements such as grid-averaged water depth, flow velocity, and discharge at the half-time level; obtains the discharges and momenta at the left and right interfaces of each grid by using the numerical flux function in ROE format; advances the grid-averaged hydraulic elements to the next complete time step by using a corrector; and finally outputs the result file simulated by the model. The one-dimensional dam-break flood simulation method based on the ROE format of the present invention has accurate calculation, short time consumption, can simulate complex dam-break flows, and realizes large-scale dam-break flow simulation.

Description

Technical Field

[0001] The present invention belongs to the technical field of numerical simulation, and particularly relates to a one-dimensional dam-break flood simulation method based on the ROE format. Background Art

[0002] Dam-break flood refers to the flood formed by the instantaneous breach of a dam or other water retaining structures, resulting in the sudden release of water. The causes of its formation can generally be divided into two major factors: natural and human. For example, dam breaches caused by floods exceeding the design standards, ice jams, earthquakes, etc. belong to natural factors; dam breaches caused by poor design, construction, management, war damage, etc. belong to human factors. In recent years, with the continuous change of the global climate, the frequency of extreme rainfall and other events has been increasing, and the threat of floods exceeding the standards caused by this to dams has also been increasing accordingly. Accurate and timely calculation of dam-break floods is an effective means to quantitatively estimate the impact of the failure of reservoirs and levees, and to reasonably determine the flood control design standards and risk avoidance measures for reservoirs or dams.

[0003] With the development of computer technology, the use of numerical models to simulate the process of dam-break floods has gradually come into view. Through numerical models, the results under different initial dam-break conditions can be constructed and calculated conveniently in a short time, and the physical process of the water flow evolving downstream can be described more precisely, making numerical models more widely used in the simulation of dam-break floods. The main content of dam-break flood calculation is to obtain the flow rate, water depth, flow velocity, wave front and the arrival time of the flood peak at various points along the downstream flood evolution process based on the flow rate and water level process line at the dam-break site. However, due to the very complex composition of dam-break water flow, usually including shock waves, subcritical flow, supercritical flow and other flow patterns, numerical oscillation and instability are likely to occur during its numerical simulation. In addition, due to the large scale of the downstream evolution process of dam-break floods, the calculation time of using two-dimensional models is relatively long, which is difficult to meet the need for timely assessment of flood risks. Summary of the Invention

[0004] The purpose of the present invention is to provide a one-dimensional dam-break flood simulation method based on the ROE format to achieve accurate and efficient simulation of the dam-break flood evolution process.

[0005] The technical solution adopted by the present invention is as follows: The one-dimensional dam-break flood simulation method based on the ROE format is specifically implemented according to the following steps:

[0006] Step 1: Read historical data in a computer. The historical data includes river cross-section data, Manning coefficient, dam-break flow process, calculation duration, and divide the calculation area into grids of different sizes according to the river cross-section spacing, and then process the river cross-section data to obtain the water depth-flow area and water depth-wetted perimeter relationships of each grid.

[0007] Step 2: Discretize the Saint-Venant equations using the finite volume method in Godunov format to obtain the flux term, bottom slope source term, friction source term, and time stepping;

[0008] Step 3: Perform time integration on the discretized Saint-Venant equations using the predictor-corrector method to calculate the hydraulic elements on each grid in Step 1. The hydraulic elements include the average water depth and flow velocity. Then, obtain the corresponding cross-sectional area according to the water depth-cross-sectional area relationship obtained in Step 1; Use the Superbee limiter to limit the water depth, flow velocity, and cross-sectional area to avoid abnormal gradients. Then, use the MUSCL-type format to extrapolate and construct the corresponding water depth, flow velocity, and cross-sectional area on the left and right interfaces of each grid, so that the solution has second-order spatial accuracy;

[0009] Step 4: According to the average water depth, flow velocity, and cross-sectional area on each grid obtained in Step 3, as well as the corresponding water depth, flow velocity, and cross-sectional area on the left and right interfaces of the grid, use the numerical flux function in ROE format to calculate the flow rate and momentum on the left and right interfaces of each grid;

[0010] Step 5: Update and correct the average water depth, flow velocity, and cross-sectional area on each grid obtained in Step 3 using the semi-implicit calculation method and advance them to the next time step;

[0011] Step 6: Update and output the flow rate, momentum, average water depth, flow velocity, and cross-sectional area calculated in Steps 4 and 5 according to the time step to obtain the hydraulic element values of each grid cell at each moment.

[0012] The features of the present invention also lie in that,

[0013] In Step 1, the specific method for processing the river cross-section data is as follows:

[0014] Divide any river cross-section into n narrow strips along the river width direction. The width of the narrow strip is dx, where the starting distance corresponding to the left boundary of the i-th narrow strip is x i , and the corresponding river bottom elevation is z i , and the starting distance corresponding to the right boundary is x i+1 , and the corresponding river bottom elevation is z i+1 ; Assume that the water depth of the cross-section at any moment is h, and the area of the i-th narrow strip is S i , then the calculation formulas for the water depth h and S i in the i-th grid are as follows:

[0015] (1) When h > max(z i , z i+1 )

[0016]

[0017] (2) When zi >h>z i+1 When

[0018]

[0019] (3) When z i <h<z i+1 When

[0020]

[0021] After cycling through n narrow strips, the corresponding relationship between the water depth and the flow area of the cross-section can be obtained:

[0022]

[0023] In step 2, the Saint-Venant equations are specifically:

[0024]

[0025] In the formula, t is time; Ω is the integration interval, which is the grid length in the one-dimensional model; A is the cross-sectional area of the water flow; V is the average flow velocity of the cross-sectional area; g is the acceleration due to gravity; S O is the bottom slope source term; S f is the friction source term;

[0026] Among them, S O and S f are specifically

[0027]

[0028]

[0029] In the formula, z is the water level; n is the Manning coefficient; R is the hydraulic radius.

[0030] In step 3, the solution processes for the average water depth, flow velocity, and flow area on each grid are respectively:

[0031]

[0032]

[0033] In the formula, A is the flow area, V is the flow velocity, the subscript j represents the j-th computational grid, the superscript k represents the k-th computational time step, and k + 1 / 2 represents half a time step after the k-th moment; and are both obtained after being limited by the Superbee limiter;

[0034] The Superbee limiter is specifically

[0035]

[0036] and specifically

[0037]

[0038]

[0039]

[0040] In the formula, h is the water depth, respectively represent the water depth gradient, flow velocity gradient and flow area gradient of each grid.

[0041] In step 3, the MUSCL-type format extrapolation is used to construct the corresponding water depth, flow velocity and flow area on the left and right interfaces of each grid:

[0042]

[0043]

[0044] In the formula, the subscripts L and R represent the left and right interfaces of the grid respectively; the flow area A L and A R are obtained by interpolation according to the water depth-flow area relationship obtained in step 1.

[0045] In step 4, the numerical flux function of the ROE format is used to calculate the flow rate and momentum on the left and right interfaces of each grid, specifically:

[0046]

[0047] In the formula, F L and F R are respectively the fluxes calculated according to the hydraulic elements on the left and right of the grid extrapolated by the formula (5) in step 3 and the MUSCL extrapolation; and ΔU are given by the following formulas respectively:

[0048]

[0049]

[0050] In the formula, specifically:

[0051]

[0052] In the formula, and specifically:

[0053]

[0054]

[0055]

[0056] Among them, g is the acceleration due to gravity.

[0057] The formula specifically used for correcting the calculation result in step 5 is as follows:

[0058]

[0059] In the formula, Δt is the calculation time step; k + 1 / 2 represents half a time step after the k-th moment, and k + 1 represents one time step after the k-th moment; j ± 1 / 2 respectively represent the left and right interfaces of the j-th grid.

[0060] The beneficial effects of the present invention are as follows: The present invention solves the problems of numerical oscillation and instability that are prone to occur in traditional simulation of dam-break flows. The Saint-Venant equations are solved using the finite volume method with the Godunov format, and the Riemann solver with the ROE format is used to solve the grid interface fluxes, which can accurately simulate complex dam-break flow processes including shock waves, subcritical flows, supercritical flows, etc.

[0061] In addition, when calculating in a one-dimensional model, it is very important to calculate the relationship between the water depth at any cross-section and other hydraulic elements. In this method, the river cross-section is discretized into many narrow strips, the element values of each narrow strip are calculated separately, and finally accumulated. The number of narrow strips can be appropriately increased according to the cross-section width to ensure the accuracy of the calculation result.

[0062] By using a one-dimensional grid for simulation, the key hydraulic elements at any downstream cross-section can be provided quickly and accurately. Compared with a two-dimensional model, the simulation speed is significantly improved when simulating large-scale dam-break flows. A one-dimensional dam-break flood simulation method based on the ROE format of the present invention has accurate calculations, short calculation time, can simulate complex dam-break flows, and can achieve large-scale dam-break flow simulations. Brief Description of the Drawings

[0063] Figure 1 is a schematic diagram of cross-section discretization of a one-dimensional dam-break flood simulation method based on the ROE format of the present invention;

[0064] Figure 2 is a process diagram of solving a one-dimensional dam-break flood simulation method based on the ROE format of the present invention;

[0065] Figure 3 is a comparison diagram of the along-channel Froude number at a certain moment when the downstream wet grid is for a dam-break example with a triangular cross-section in a one-dimensional dam-break flood simulation method based on the ROE format of the present invention;

[0066] Figure 4

[0066] is a triangular cross-section dam-break example of a one-dimensional dam-break flood simulation method based on the ROE format of the present invention, and a comparison diagram of the flow velocity along the downstream dry grid at a certain moment; Detailed implementation manner

[0067] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0068] The one-dimensional dam-break flood simulation method based on the ROE format of the present invention, as Figure 2 shown, is specifically implemented according to the following steps:

[0069] Step 1: Read basic data in the computer, including river cross-section data, Manning coefficient, dam-break flow process, calculation duration, etc., divide the calculation area into grids of different sizes according to the section spacing, and then process the model input section file to obtain the water depth-overflow area and water depth-wetted perimeter relationships of each grid. The calculation schematic diagram is as Figure 1 shown;

[0070] Step 2: Discretize the Saint-Venant equation by using the finite volume method in Godunov format to obtain the flux term, bottom slope source term, friction source term, and time step that can be solved by the computer;

[0071] Step 3: Perform time integration on the discretized Saint-Venant equation by using the predictor-corrector method to calculate the average water depth, flow velocity and other hydraulic elements on each grid in Step 1, and then obtain the corresponding overflow area according to the water depth-overflow area relationship obtained in Step 1; use the Superbee limiter to limit the water depth to avoid abnormal gradients, and then use the MUSCL-type format to extrapolate and construct the corresponding water depth, flow velocity, overflow area and other hydraulic elements on the left and right interfaces of each grid, so that the solution has second-order spatial accuracy. The calculation flow chart of this method is as Figure 2 shown, and the part within the dashed box is the core solution part of the model;

[0072] Step 4: Since it is difficult to accurately calculate the interface flux by solving the Riemann problem, the general method is to solve the locally linearized Riemann problem through a numerical flux function. In this method, according to the grid average values obtained in Step 3 and the corresponding values on the interface, the numerical flux function in ROE format is used to solve the flow rate and momentum passing through each grid interface;

[0073] Step 5: In the correction step, the new grid average hydraulic elements are advanced to the next complete time step by using the semi-implicit calculation method based on the grid average hydraulic elements obtained in Step 3;

[0074] Step 6: Update and calculate the hydraulic elements such as water depth, flow velocity, flow area, and discharge for each grid cell;

[0075] Step 7: Output the hydraulic elements of each grid cell in Step 6 to obtain the hydraulic element values of each grid cell at each moment.

[0076] The specific method for processing the model input cross-section file in Step 1 to obtain the corresponding relationship between water depth and other elements is as follows:

[0077] Divide any river cross-section into n narrow strips along the river width direction, with the width of the narrow strip being dx. The starting distance corresponding to the left boundary of the i-th narrow strip is x i , and the corresponding river bottom elevation is z i , and the starting distance corresponding to the right boundary is x i+1 , and the corresponding river bottom elevation is z i+1 . Let the water depth of the cross-section at any moment be h, and the area of the i-th narrow strip be S i , then the calculation formula for the water depth h and S i in the i-th grid is as follows:

[0078] (1) When h > max(z i , z i+1 )

[0079]

[0080] (2) When z i > h > z i+1

[0081]

[0082] (3) When z i < h < z i+1

[0083]

[0084] After cycling through n narrow strips, the water depth - flow area corresponding relationship of this cross-section can be obtained:

[0085]

[0086] The specific Saint-Venant equations in Step 2 are as follows:

[0087]

[0088] In the formula, t is time, s; Ω is the integration interval, which is the grid length in the one-dimensional model, m; A is the cross-sectional area of flow, m 2 ; V is the average flow velocity of the cross-sectional area of flow, m / s; g is the acceleration due to gravity, m / s 2; S O is the bottom slope source term; S f is the friction source term;

[0089] Among them, S O and S f are specifically:

[0090]

[0091]

[0092] In the formula, z is the water level; n is the Manning coefficient; R is the hydraulic radius.

[0093] The solution process of calculating hydraulic elements such as the average water depth, flow velocity, and cross-sectional area of the computational grid in step 3 is as follows:

[0094]

[0095]

[0096] In the formula, the subscript j represents the jth computational grid, the superscript k represents the kth computational time step, and k + 1 / 2 represents half a time step after the kth moment. and are all obtained after being limited by the Superbee limiter.

[0097] The Superbee limiter is specifically:

[0098]

[0099] and Specifically

[0100]

[0101]

[0102]

[0103] In the formula respectively represent the water depth gradient, flow velocity gradient, and cross-sectional area gradient of each grid.

[0104] The specific method of extrapolating the grid interface values according to the grid average values using the MUSCL format in step 3 is:

[0105]

[0106]

[0107] In the formula, the subscripts L and R represent the left and right interfaces of the grid respectively; AL and A R Interpolated according to the water depth - area relationship obtained in Step 2.

[0108] In Step 4, the equations for solving the flow rate and momentum at the left and right interfaces of each grid using the ROE - format numerical flux function are specifically as follows:

[0109]

[0110] In the formula, F L and F R are respectively the fluxes (including flow rate and momentum) calculated based on the hydraulic elements on the left and right of the grid extrapolated by MUSCL and according to Equation (5) in Step 3; and ΔU are respectively given by the following formulas: and ΔU are respectively given by the following formulas:

[0111]

[0112]

[0113] In the formula, Specifically:

[0114]

[0115] In the formula, and Specifically:

[0116]

[0117]

[0118]

[0119] The formula for correcting the calculation results in Step 5 is specifically as follows:

[0120]

[0121] In the formula, Δt is the calculation time step; k + 1 / 2 represents half a time step after the k - th moment, and k + 1 represents one time step after the k - th moment; j ± 1 / 2 respectively represent the left and right interfaces of the j - th grid.

[0122] In Step 7, the hydraulic elements of each grid cell are output. The main output elements are the values of hydraulic elements such as the water depth, wetted area, flow velocity, Froude number, and flow rate of all grids at each moment, which can be output in.dat format and.vts format for different display requirements.

[0123] Figure 3This is an example of dam-break with a triangular cross-section. The figure shows the comparison of the along-channel Froude number with the analytical solution and the results of other literature at t = 112.9 s for the downstream wet grid. It can be seen that there are complex flow regimes such as subcritical flow and supercritical flow in this example. The proposed method can well simulate such complex flows, and the simulated values are close to the analytical solution.

[0124] Figure 4 This is an example of dam-break with a triangular cross-section. The figure shows the comparison of the along-channel velocity with the analytical solution and the results of other literature at t = 45.16 s for the downstream dry grid. It can be seen that the simulation effect is close to the literature, indicating that the model can better handle the dam-break flow problem with alternating dry and wet grids.

[0125] The beneficial effect of the present invention is that a one-dimensional dam-break flood simulation method based on the ROE format of the present invention has accurate calculations, short computing time, can simulate complex dam-break flows, and can achieve large-scale dam-break flow simulations.

Claims

1. One-dimensional dam-break flood simulation method based on the ROE format, Characterized in that, Specifically includes the following steps: Step 1: Read historical data in the computer. The historical data includes river cross-section data, Manning coefficient, dam-break flow process, calculation duration, and divide the calculation area into grids of different sizes according to the river cross-section spacing. Then process the river cross-section data to obtain the water depth - flow area and water depth - wetted perimeter relationships of each grid; Step 2: Discretize the Saint-Venant equations using the finite volume method in Godunov format to obtain the flux term, bottom slope source term, friction source term, and time stepping; Step 3: Perform time integration on the discretized Saint-Venant equations using the predictor-corrector method to calculate the hydraulic elements on each grid in Step 1. The hydraulic elements include average water depth and flow velocity, and then obtain the corresponding flow area according to the water depth - flow area relationship obtained in Step 1; Use the Superbee limiter to limit the water depth, flow velocity, and flow area to avoid abnormal gradients, and then use the MUSCL-type format to extrapolate and construct the corresponding water depth, flow velocity, and flow area on the left and right interfaces of each grid to make the solution have second-order spatial accuracy; Step 4: According to the average water depth, flow velocity, flow area on each grid obtained in Step 3 and the corresponding water depth, flow velocity, and flow area on the left and right interfaces of the grid, use the numerical flux function in ROE format to calculate the flow rate and momentum on the left and right interfaces of each grid; Step 5: Update and correct the average water depth, flow velocity, and flow area on each grid obtained in Step 3 using the semi-implicit calculation method and advance it to the next time step; Step 6: Update and output the flow rate, momentum, average water depth, flow velocity, and flow area calculated in Steps 4 and 5 according to the time step to obtain the hydraulic element values of each grid cell at each moment.

2. The one-dimensional dam-break flood simulation method based on the ROE format according to claim 1, Characterized in that, In Step 1, the specific method for processing the river cross-section data is: Divide any river cross-section into n narrow strips along the river width direction, with the width of each narrow strip being dx. The starting distance corresponding to the left boundary of the i-th narrow strip is x i , and the corresponding river bottom elevation is z i , and the starting distance corresponding to the right boundary is x i+1 , and the corresponding river bottom elevation is z i+1 ; Let the water depth of this cross-section at any moment be h, and the area of the i-th narrow strip be S i , then the calculation formulas for the water depth h and S i in the i-th grid are as follows: (1) When h > max(z i , z i+1 ) (2) When z i > h > z i+1 at this time (3) When z i <h < z i+1 at that time After cycling through n narrow strips, the corresponding relationship between the water depth and flow area of this cross-section can be obtained:

3. The one-dimensional dam-break flood simulation method based on the ROE format according to claim 2, Characterized in that, In Step 2, the Saint-Venant equations are specifically: where t is time; Ω is the integration interval, which is the grid length in a one-dimensional model; A is the cross-sectional area of the water flow; V is the average velocity of the cross-sectional water flow; g is the acceleration due to gravity; S O is the bottom slope source term; S f is the friction source term; Among them, S O and S f Specifically In the formula, z is the water level; n is the Manning coefficient; R is the hydraulic radius.

4. The one-dimensional dam-break flood simulation method based on the ROE format according to claim 3, Characterized in that, In Step 3, the solution processes for the average water depth, flow velocity, and flow area on each grid are respectively: Where A is the flow-through area, V is the flow velocity, the subscript j represents the j-th computational grid, the superscript k represents the k-th computational time step, and k+1 / 2 represents half a time step after the k-th moment; and are obtained after being limited by the Superbee limiter; The Superbee limiter is specifically and specifically where h is the water depth, respectively represent the water depth gradient, flow velocity gradient, and flow area gradient of each grid.

5. The one-dimensional dam-break flood simulation method based on the ROE format according to claim 4, Characterized in that, In Step 3, use the MUSCL-type format to extrapolate and construct the corresponding water depth, flow velocity, and flow area on the left and right interfaces of each grid: In the formula, the subscripts L and R represent the left and right interfaces of the grid respectively; the flow-through area A L and A R is obtained by interpolating according to the water depth-flow-through area relationship obtained in Step 1.

6. The one-dimensional dam-break flood simulation method based on the ROE format according to claim 5, Characterized in that, In Step 4, use the numerical flux function in ROE format to calculate the flow rate and momentum on the left and right interfaces of each grid, specifically: where, F L and F R are fluxes calculated from the hydraulic elements on the left and right of the grid extrapolated according to formula (5) in step 3 and MUSCL, respectively; and ΔU are given by the following expressions, respectively: In the formula, Specifically: Wherein, and Specifically: where g is the acceleration due to gravity.

7. The one-dimensional dam-break flood simulation method based on the ROE format according to claim 6, characterized in that the formula specifically used for correcting the calculation result in step 5 is: In the formula, Δt is the calculation time step; k + 1 / 2 represents half a time step after the k-th moment, and k + 1 represents one time step after the k-th moment; j ± 1 / 2 respectively represent the left and right interfaces of the j-th grid.

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