CN110008526B - Source-item-containing shallow water flow simulation method based on large time step format - Google Patents

Abstract

The invention discloses a source item-containing shallow water flow simulation method based on a large time step format, which comprises the following steps: (10) acquiring initial parameters of the river channel: acquiring parameters of a river channel and water flow; (20) River channel unit division, namely dividing a river channel into a plurality of units along the length direction, and taking the initial state of river channel water flow as a state value in each river channel unit; (30) shallow water flow simulation of the river channel unit: calculating the inter-unit state according to the intra-unit state of the river channel, and updating the intra-unit state of the river channel according to the inter-unit state to obtain the shallow water flowing state of each river channel unit at the next moment; (40) convergence judgment: comparing the shallow water flowing state at the next moment with the shallow water flowing state at the previous moment, and finishing the simulation when the error meets the convergence threshold; otherwise, setting the shallow water flowing state at the next moment as the shallow water flowing state at the previous moment, and turning to (30) the inter-unit state calculating step. The shallow water flow determination method is high in calculation efficiency and good in result convergence.

Description

Simulation method of shallow water flow with source term based on large time step format

technical field

The invention belongs to the technical field of shallow water dynamics, and is a method for simulating shallow water flow with source items based on a large time step format with high calculation efficiency and good result convergence.

Background technique

Shallow water flow refers to the flow whose horizontal movement scale is much larger than the vertical movement scale. It is characterized by negligible vertical flow velocity and acceleration, so that the water pressure is close to the static pressure distribution, there is a free liquid surface, and gravity is the main driving force. Usually the water movement in the channel can be regarded as shallow water flow.

Shallow water equation is an important mathematical model to describe shallow water flow in hydraulics. Therefore, the simulation of the shallow water flow state is mainly realized by solving the shallow water equation.

The shallow water equation is a partial differential equation, which is difficult to solve directly. At present, the numerical simulation of this type of equation is to obtain the Riemann solution at the interface between adjacent units after discretization, and then update the two units on both sides of the interface. However, due to the limitation of CFL conditions, this method is computationally inefficient.

In order to improve the calculation efficiency, the industry proposes to consider the use of a large time step format for simulating nonlinear hyperbolic partial differential equations. By changing the original update method, the limitation of the number of CFLs is broken, and the calculation efficiency is greatly improved.

However, when this method uses a large time step format to update the units experienced by a single wave, in the large time step format, a single wave will pass through multiple units in one time step, and the last unit is not properly processed Violating entropy solutions will be obtained, and oscillations will occur at the platform, which makes it difficult for the shallow water equation with a large time step format to accurately describe the shallow water flow state, which limits its application in determining shallow water flow.

Contents of the invention

The purpose of the present invention is to provide a method for determining shallow water flow with a source term based on a shallow water equation with a large time step, which has high calculation efficiency and good convergence of results.

The technical solution that realizes the object of the present invention is:

A method for simulating shallow water flow with a source term based on a large time step format, comprising the following steps:

(10) Acquisition of initial parameters of the river course: acquisition of river course parameters, including the length and width of the river course, the initial state of the river flow, including the initial water depth, flow velocity, upstream inlet flow and downstream water level;

(20) River course unit division: the river course is divided into multiple units along the length direction, and the initial state of river flow is used as the state value in each river course unit, and the total length of the river course is divided by the number of river course units to obtain the length of each river course unit;

(30) River unit shallow water flow simulation: Calculate the inter-unit state according to the state in the river unit, update the state in the river unit according to the inter-unit state, and obtain the shallow water flow state of each river unit at the next moment;

(40) Convergence judgment: compare the shallow water flow state at the next moment with the shallow water flow state at the previous moment, and when the error meets the convergence threshold, the simulation ends; otherwise, set the shallow water flow state at the next moment as the shallow water flow at the previous moment state, go to (30) inter-unit state calculation step.

Compared with the prior art, the present invention has the remarkable advantages of:

1. The result has good convergence: due to the use of the fixed selection method, the numerical viscosity is increased, the oscillation is suppressed, and a better convergence effect is obtained.

2. High computational efficiency: the present invention uses a large time step format to simultaneously update multiple units on both sides of the interface, breaking through the limitation of CFL conditions and greatly improving computational efficiency.

The present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments.

Description of drawings

Fig. 1 is the main flow chart of the method for simulating shallow water flow with source term based on the large time step format of the present invention.

Figure 2 shows the initial water level and bottom elevation of the river.

Figure 3 is a comparison chart between the calculated value of the converged solution and the true solution.

Fig. 4 is a flow chart of the steps of simulating the shallow water flow of the channel unit in Fig. 1 .

FIG. 5 is a flow chart of the steps of updating the status within the unit in FIG. 4 .

Fig. 6 is a schematic diagram of a single wave passing through multiple units.

Fig. 7 is the result calculated according to the method of the present invention.

Detailed ways

As shown in Figure 1, the method for simulating shallow water flow with source items based on the large time step format of the present invention includes the following steps:

(10) Acquisition of initial parameters of the river course: acquisition of river course parameters, including the length and width of the river course, the initial state of the river flow, including the initial water depth, flow velocity, upstream inlet flow and downstream water level;

The embodiment selects a channel with a length of 25m, the vertical height of the river bottom (along the z direction) is equal, and the distribution along the x direction is shown in Figure 2, and its functional expression satisfies the following relationship:

At the initial moment, the water level is uniformly 0.33m, as shown in Figure 2. The flow rate at the upstream inlet is 0.18m ³ /s, and the water level at the downstream outlet is 0.33m. The water surface curve in Figure 2 will change, and finally it will become what it is in Figure 3.

(20) River course unit division: the river course is divided into multiple units along the length direction, and the initial state of river flow is used as the state value in each river course unit, and the total length of the river course is divided by the number of river course units to obtain the length of each river course unit;

Embodiment The river course is divided into 250 units, each unit length is 0.1m. Calculate the flow velocity u ₁ , u ₂ ..., u ₂₅₀ and water depth h ₁ , h ₂ , ..., h ₂₅₀ of each unit. From the formula (1), when x is less than 8 or greater than 12:

When x is between 8 and 12:

(30) River unit shallow water flow simulation: Calculate the inter-unit state according to the state in the river unit, update the state in the river unit according to the inter-unit state, and obtain the shallow water flow state of each river unit at the next moment;

As shown in Figure 4, the (30) channel unit shallow water flow simulation steps include:

(31) Calculation of the state of the interface between units: according to the state of the channel unit, the Roe average value of every two adjacent channel units is characterized to obtain the wave velocity and wave intensity of the interface between the channel units, and the time step is determined according to the CFL value;

界面，第i-1单元在左边，第i单元在右边，分别用L和R表示。那么对于

界面：(311) Do Roe average on the interface between each segment, such as in Figure 6

In the interface, the i-1th unit is on the left, and the i-th unit is on the right, denoted by L and R respectively. then for

interface:

(312) Calculate wave speed and wave intensity. Two waves are emitted from the interface, and their wave speeds are:

The wave strength is:

in:

(32) Update the state in the unit: according to the wave velocity and wave intensity of the interface between the units, the fixed selection method is used to update the state in the unit to obtain the shallow water flow state at the next moment;

As shown in Figure 5, the state update step in the (32) unit includes:

(321) Calculate the number of affected units: multiply the inter-unit interface wave velocity by the time step, and divide it by the unit length to obtain the value a, whose integer part is b;

界面，由公式(6)可知，这个界面会发出速度一大一小两个波，取其中一个大的波速为λ₂为例，假如算得：For example, in Figure 6, in

The interface, as can be seen from the formula (6), this interface will emit two waves with a velocity of one large and one small, and take the velocity of one of the large waves as _λ2 as an example, if it is calculated:

Then the value of a is 2.5 and the value of b is 2.

(322) Complete wave and unit wave velocity update: when the wave velocity of the inter-unit interface is greater than 0, it means that the wave propagates to the right, and a wave intensity is added to the b units on the right side of the unit interface; when the wave velocity of the inter-unit interface is less than 0, it means that the wave Propagate to the left, and add a wave intensity to the b units on the left side of the unit interface; in the example in Figure 6, the update method for the two units on the right side of the interface (unit i and unit i+1) is:

U ^n+l =U ⁿ +ΔU (13)

(323) Update the wave velocity of partially affected units: for the b+1th unit partially affected, set a value between 0 and 1, if |a-b| is less than this value, then the b+1th unit remains unchanged If |a-b| is greater than or equal to this value, a wave intensity is added to the unit.

In the example in Figure 6, the update method for the b+1th unit on the right side of the interface, which is the third unit (i+2 unit), is as follows:

|a-b| is 0.5, set a number between 0 and 1 (0.9 in this example), 0.5 is less than 0.9, so the third unit remains unchanged.

(33) Interface traversal check: judge whether the interface of the internal state updating unit is the last interface, if not, go to step (31), if yes, end this process.

(40) Convergence judgment: compare the shallow water flow state at the next moment with the shallow water flow state at the previous moment, and when the error meets the convergence threshold, the simulation ends; otherwise, set the shallow water flow state at the next moment as the shallow water flow at the previous moment state, go to (30) inter-unit state calculation step.

The calculation results are shown in Figure 7. It can be seen from the calculation results that due to the adoption of the fixed selection method, the numerical viscosity is increased and the oscillation is suppressed; at the same time, due to the use of a large time step format to update multiple units on both sides of the interface at the same time, it breaks through the CFL The limitation of conditions greatly improves the calculation efficiency.

Claims (1)

1. A method for simulating shallow water flow with a source term based on a large time step format, characterized in that it comprises the following steps: (10) Acquisition of initial parameters of the river course: acquisition of river course parameters, including the length and width of the river course, the initial state of the river flow, including the initial water depth, flow velocity, upstream inlet flow and downstream water level; (20) River course unit division: the river course is divided into multiple units along the length direction, and the initial state of river flow is used as the state value in each river course unit, and the total length of the river course is divided by the number of river course units to obtain the length of each river course unit; (30) River unit shallow water flow simulation: Calculate the inter-unit state according to the state in the river unit, update the state in the river unit according to the inter-unit state, and obtain the shallow water flow state of each river unit at the next moment; (40) Convergence judgment: compare the shallow water flow state at the next moment with the shallow water flow state at the previous moment, and when the error meets the convergence threshold, the simulation ends; otherwise, set the shallow water flow state at the next moment as the shallow water flow at the previous moment State, go to (30) inter-unit state calculation step; The (30) channel unit shallow water flow simulation steps include: (31) Calculation of the state of the interface between units: according to the state of the channel unit, the Roe average value of every two adjacent channel units is characterized to obtain the wave velocity and wave intensity of the interface between the channel units, and the time step is determined according to the CFL value; (32) Update the state in the unit: according to the wave velocity and wave intensity of the interface between the units, the fixed selection method is used to update the state in the unit to obtain the shallow water flow state at the next moment; (33) Interface traversal inspection: judge whether the interface of the internal state update unit is the last interface, if not, go to (31) step, if so, end this process; The (32) step of updating the state in the unit includes: (321) Calculate the number of affected units: multiply the inter-unit interface wave velocity by the time step, and divide it by the unit length to obtain the value a, whose integer part is b; (322) Complete wave and unit wave velocity update: when the wave velocity of the inter-unit interface is greater than 0, it means that the wave propagates to the right, and a wave intensity is added to the b units on the right side of the unit interface; when the wave velocity of the inter-unit interface is less than 0, it means that the wave Propagate to the left, add a wave intensity to the b units on the left side of the unit interface; (323) Update the wave velocity of partially affected units: for the b+1th unit partially affected, set a value between 0 and 1, if |a-b| is less than this value, then the b+1th unit remains unchanged If |a-b| is greater than or equal to this value, a wave intensity is added to the unit.

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