A modeling and simulation method for gas-water transient flow in long-distance water delivery pipeline

Abstract

The application discloses a modeling simulation method for air-water transient flow of long-distance water delivery pipeline, and comprises the following steps: constructing control equations of air mass entrainment fast filling water body considering the step-like non-steady friction model; constructing a numerical format of air mass entrainment fast filling water body of the non-steady friction model; performing the boundary condition of the pressurized air and the water body, and obtaining simulation results through calculation. The application develops a stable numerical method for the air mass entrainment fast filling process, and the proposed Gudonov air-water format is more stable and accurate in reproducing the measured air mass pressure fluctuation, even in the case of the Coulomb number less than 1; the second-order Gudonov scheme adds the non-steady friction in the fast filling simulation, so that the energy consumption process can be simulated more accurately.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of numerical simulation of water power station (pumping station) hydraulics, and particularly relates to a modeling simulation method for air-water transient flow in long-distance water conveying pipeline. BACKGROUND

[0002] Air masses often appear at the high points and uneven cross sections of water supply pipe networks. Pressurized air masses in the pipeline system can cause significant harm, and even can cause intermittent spring eruption or system component failure. Therefore, it is necessary to accurately numerically simulate all such air-water transient events, which is crucial for providing safe operation procedures for the pipeline system and better understanding the related air-water mode mechanism.

[0003] Characteristic schemes have been widely used to simulate the pipeline transient flow induced by entrained air masses. The rigid column method for the fast filling process of entrained air masses was first proposed. In order to improve the accuracy of the method, other factors have been gradually added to the existing model, such as filling length variation, water body elasticity and unsteady friction. The elastic water body model considering water body elasticity is often used to simulate the fast filling process of entrained air masses. So far, most of the existing fast filling models are characteristic schemes, and are developed and verified in the case of single characteristic pipeline, in which the Coulomb number is adjusted to 1.

[0004] Real water pipe systems often have pipe sections of various lengths, diameters and even materials. In these complex pipeline systems, the Coulomb number cannot be completely equal to 1. Therefore, the characteristic scheme must use interpolation or wave speed adjustment in the pipeline, which can cause inaccuracy in numerical simulation. SUMMARY

[0005] The purpose of the application is to provide a modeling simulation method for air-water transient flow in long-distance water conveying pipeline, aiming at the numerical weakening problem of the characteristic scheme when the Coulomb number is less than 1, and to develop a stable numerical method for the fast filling process of entrained air masses. The proposed Gudonnu air-water scheme is more robust and accurate in reproducing the measured air mass pressure fluctuations, even when the Coulomb number is less than 1. The second-order Gudonnu scheme adds unsteady friction in the fast filling simulation to facilitate more accurate simulation of the energy consumption process.

[0006] Technical scheme: In order to achieve the above purpose, the application provides a modeling simulation method for air-water transient flow in long-distance water conveying pipeline, comprising the following steps:

[0007] S1: constructing control equations of the fast filling water body of entrained air masses considering the unsteady friction model of the step function;

[0008] S2: based on the constructed control equations, constructing a numerical scheme of the fast filling water body of entrained air masses of the unsteady friction model.

[0009] S3: Combining the constructed control equations and numerical schemes, the boundary conditions of the pressurized air and water body are calculated, and the simulation results are obtained.

[0010] Furthermore, the governing equations in step S1 include the non-steady water body governing equations considering unsteady friction, the pressurized gas governing equations, and the gas-water interface governing equations.

[0011] Furthermore, the expression for the governing equation of the unsteady water body considering unsteady friction is as follows:

[0012]

[0013]

[0014] In the formula, H is the piezometric head height; V is the average flow velocity; c is the speed of sound propagating in water; g is the gravitational acceleration; z is the distance along the pipeline direction; t is the time; f is the Darcy-Visbach friction coefficient; k is the Waddy friction coefficient; when the flow velocity is greater than or equal to zero, sign(V) = 1; when the flow velocity is less than zero, sign(V) = -1; D is the pipe diameter; the right side of equation (2) is an unsteady friction term.

[0015] Furthermore, the expression for the governing equation of the pressurized gas is:

[0016]

[0017] In the formula, P a V is the pressure of the gas after it has been compressed; a P is the velocity of the gas under pressure; a0 and V a0 It is P a and V a The initial value of ; m is the gas state index, which can be taken as a constant when heat transfer is not considered.

[0018] Furthermore, the governing equation at the air-water interface is expressed as follows:

[0019]

[0020] In the formula, L W It is the length of the water body, L W0 It is L W The initial value of the gas can be obtained by integrating the velocity of the gas under pressure over time to obtain the length of the water body after a period of time. By solving equations (3) and (4), a method for capturing the pressure and velocity at the interface between the moving gas mass and water is established.

[0021] Further, the numerical format in the step S2 is a non-constant water body numerical format considering non-constant friction, which is specifically as follows:

[0022]

[0023] In the formula,

[0024] w, S is a matrix converted into a Gudon method; f is a flux of a solving process; the water body can be divided into N unit bodies, and the integral of the formula (3) along the interface of each control body i can obtain a result;

[0025] For the i-th control volume, the integral between the control surfaces i-1 / 2 and i+1 / 2 can be expressed as:

[0026]

[0027] In the formula, Δt is a time interval, Δx is a space step, the superscripts n and n+1 represent the time steps t and t+Δt respectively, W is the average value of w at the interface [i-1 / 2, i+1 / 2] of the control body, And is the unit flux inside the i-th unit cell.

[0028] Further, the step S3 is specifically as follows:

[0029] When the boundary unit of the water body and the bulk air mass is calculated, the Gudon method and the characteristic scheme are combined:

[0030]

[0031]

[0032]

[0033]

[0034] In the formula, H A , V A is the value of W at the N-1 / 2 interface; H C , V C is the pressure and velocity at the interface of the water body and the bulk air mass; H P , V P is the pressure and velocity obtained by combining the formula (7) and (8); H E , V E is the pressure and velocity of the last time of H P , V P ; H G , V G is the pressure and velocity of H P , V, V P current time and the pressure and velocity interpolated from the previous time;H D , V D pressure and velocity obtained at the gas-water interface of the water body; ΔL w a small water body segment with the length constantly changing with the expansion of the compressed gas;W N+1 , W N+2 a virtual unit value.

[0035] The application provides a second-order finite volume Godunov scheme to simulate the fast filling flow process of a long-distance water pipeline entraining large-volume air, and the scheme has not been used to solve the problem of impacting large-volume air. In a complex pipeline system, when the Coulomb number is less than 1, the Godunov scheme performs better than the characteristic scheme used in the past in the simulation of transient pressure due to the advantages of the numerical format, and less numerical format attenuation is generated.

[0036] Advantages: Compared with the prior art, the application has the following advantages:

[0037] 1. The Godunov format is used to simulate the process of entraining air mass fast filling, and the numerical format of the fast filling process of the air mass simulated by the previous characteristic scheme is improved. When the Coulomb number is less than 1, the Godunov numerical format has the advantage of generating less numerical format attenuation, and the Godunov performs better than the characteristic scheme used in the past in the simulation of transient pressure.

[0038] 2. The step-like non-constant friction term is added to the control equation and the improved Godunov numerical format. Unlike the constant friction term, the constant friction is determined by the constant pressure gradient along the pipeline axis. In this case, the wall shear stress or the head loss is usually expressed as a function of the average velocity. However, in the transient process containing air pockets, the pressure gradient of the filling water column changes with time, and therefore the velocity distribution can be very complex. In the boundary layer region, the friction force is dominant, and in the vicinity of the pipeline center, the inertial force is dominant. Therefore, the wall shear stress is not expressed as a function of the average velocity, and the addition of the step-like non-constant friction term can more accurately simulate the energy change process of the entrained air mass.

[0039] 3. The method combines the Gudon method and the characteristic scheme to track the moving air-water interface at the boundary, which improves the calculation accuracy of the Gudon method while solving the assignment problem at the boundary by combining the characteristic scheme. Due to the compression and expansion of the air column, the length of the short water column ΔL w changes over time. These results show that the variable W of the Nth control volume cannot be obtained by integrating the control surface i-1 / 2 to i+1 / 2 once. In order to update W from the current time to the next time, detailed boundary processing shows that Δt / (ΔL w / a) integrations are required. However, starting from the second integration, the values of the 4 control volumes adjacent to the Nth element are no longer updated due to the lack of a suitable method. The difficulty in updating the control volume determines the need for a new method to solve this problem, and the characteristic line method is verified to be a feasible method for assigning values to the 4 control volumes adjacent to the Nth element. BRIEF DESCRIPTION OF DRAWINGS

[0040] Figure 1 is the implementation flowchart of the present application;

[0041] Figure 2 is the Gudon grid diagram of the water body during the entrainment of the air mass in the rapid charging process of the present application;

[0042] Figure 3 is a schematic diagram of the boundary conditions during the entrainment of the air mass in the rapid charging process of the present application;

[0043] Figure 4 is a comparison diagram of the air mass pressure between the experimental values and the simulation of the second-order Gudon format considering the non-steady friction of the step;

[0044] Figure 5 is a comparison diagram of the air mass pressure between the characteristic scheme and the simulation of the second-order Gudon format considering the non-steady friction of the step. DETAILED DESCRIPTION

[0045] The present application will be further illustrated below in conjunction with the drawings and specific embodiments, and it should be understood that these embodiments are only used to illustrate the present application and do not limit the scope of the present application, and various equivalent modifications of the present application made by those skilled in the art after reading the present application fall within the scope defined by the appended claims.

[0046] The present application provides a modeling and simulation method for air-water transient flow in a long-distance water conveying pipeline, as shown in Figure 1 , comprising the following steps:

[0047] S1: constructing a control equation of an air mass entrainment water body considering a non-steady friction model of a step;

[0048] S2: based on the constructed control equation, constructing a numerical format of an air mass entrainment water body of a non-steady friction model;

[0049] S3: Combine the constructed control equation and numerical format, and calculate the simulation results by considering the boundary conditions of the pressurized air and water body.

[0050] In this embodiment, the above method is applied as an example, and the specific process is as follows:

[0051] In order to verify and analyze the simulation effect of the Gudon method considering the non-steady friction of the entrained air mass fast filling process, this embodiment selects the pipeline hydraulic transient experiment device system designed and built by Zhou in 2018 to verify the effectiveness of the transient flow simulation method in the fast filling pipeline with the air bag interlayer. Embodiment 1: The pipeline is 8.862 m long, the diameter is 0.04 m, the upstream water tank has a constant water head of 0.08 MPa, the wave speed is 850 m / s, the initial state is static, the initial air mass length is 0.3 m, the initial air mass pressure is 9.794 m, and the valve is opened instantaneously. Embodiment 2: The pipeline is 101 m long, the diameter is 0.04 m, the upstream water tank has a constant water head of 1.35 MPa, the wave speed is 850 m / s, the initial state is static, the initial air mass length is 1 m, the initial air mass pressure is 9.794 m, and the valve is opened instantaneously.

[0052] The specific implementation steps are as follows:

[0053] S1: Construct the control equation of the entrained air mass fast filling water body considering the non-steady friction model:

[0054] The control equation includes a non-constant water body control equation considering non-steady friction, a pressurized gas control equation, and an air mass-water interface control equation.

[0055] The expression of the non-constant water body control equation considering non-steady friction is:

[0056]

[0057]

[0058] In the formula, H is the water head height of the pressure measuring pipe; V is the average flow velocity; c is the sound wave speed propagating in water; g is the gravitational acceleration; z is the distance along the pipeline direction; t is the time; f is the Darcy-Weisbach friction coefficient; k is the Wadis friction coefficient; sign(V) = 1 when the flow velocity is greater than or equal to zero; sign(V) = -1 when the flow velocity is less than zero; D is the pipeline diameter; and the right side of equation (2) is the non-steady friction term.

[0059] The expression of the pressurized gas control equation is:

[0060]

[0061] In the formula, Pa P is the pressure of the gas; V is the velocity of the gas a P is the pressure of the gas; V is the velocity of the gas a0 P is the pressure of the gas; V is the velocity of the gas a0 P is the pressure of the gas; V is the velocity of the gas a P is the pressure of the gas; V is the velocity of the gas a P is the pressure of the gas; V is the velocity of the gas; m is the state index of the gas, which can be taken as a constant when the heat transfer is not considered.

[0062] The expression of the control equation of the air-water interface is:

[0063]

[0064] In the formula, L W is the length of the water body, L W0 is the initial value of L W , and the change in the length of the water body after a period of time can be obtained by integrating the moving velocity of the gas over time. By solving equations (3) and (4), a method for capturing the pressure and velocity of the moving air-water interface is established.

[0065] S2: Based on the constructed control equation, a numerical format for the entrainment of large air masses into the fast-filling water body of the unsteady friction model is constructed:

[0066]

[0067] In the formula,

[0068] w, S is the matrix converted into the Gauss-Seidel method; f is the flux in the solving process; the water body can be divided into N unit bodies, and the result can be obtained by integrating formula (5) along the interface of each control body i.

[0069] For the i-th control volume, the integral between the control surfaces i-1 / 2 and i+1 / 2 can be expressed as:

[0070]

[0071] In the formula, Δt is the time interval, Δx is the spatial step, the superscripts n and n+1 represent the time steps t and t+Δt respectively, W is the average value of w at the interface [i-1 / 2, i+1 / 2] of the control body, and is the unit flux inside the i-th unit cell.

[0072] S3: Based on the constructed control equation and numerical format, the boundary conditions of the pressurized air and water body are calculated to obtain the simulation results:

[0073] Referring to Figure 3 , when calculating the boundary unit of the water body and the large-volume air mass interface, the Gauss-Seidel method and the characteristic scheme are combined:

[0074]

[0075]

[0076]

[0077]

[0078] where H A , V A are the values of the N-1 / 2 interface W; H C , V C are the pressure and velocity at the water-air interface; H P , V P are the pressure and velocity from the combination of equations (7) and (8); H E , V E are the pressure and velocity from the combination of equations (7) and (8); H P , V P are the pressure and velocity from the previous time step; H G , V G are the pressure and velocity from the previous time step; H P , V P are the pressure and velocity from the previous time step; H D , V D are the pressure and velocity from the previous time step; ΔL w is the small water segment whose length is constantly changing with the expansion and contraction of the air; W N+1 , W N+2 are the values of the virtual cell.

[0079] The moving air-water interface is tracked at the boundary by combining the Godunov method and the characteristic scheme. The Godunov method is used to improve the calculation accuracy, and the characteristic scheme is used to solve the assignment problem at the boundary. Due to the compression and expansion of the air column, the length of the short water column ΔL w changes constantly over time. These results show that the variable W of the Nth control volume cannot be obtained by integrating once through the control surface i-1 / 2 to i+1 / 2. In order to update W from the current time to the next time, the detailed boundary treatment as shown in Figure 2 indicates that Δt / (ΔL w / a) integrations are needed. However, starting from the second integration, the values of the 4 control volumes adjacent to the Nth cell are no longer updated due to the lack of a suitable method. The difficulty in updating the control volume determines the need for a new method to solve this problem. Assigning values to the 4 control volumes adjacent to the Nth cell by combining the characteristic line method is a feasible method that has been verified.

[0080] The calculation results of the Guderly method of the entrained air mass fast charging process considering unsteady friction are compared with the experimental results of the embodiment, and the pressure curve is shown in Figure 4 The calculation results of the Guderly method of the entrained air mass fast charging process considering mixed format boundary processing are compared with the experimental results of the embodiment, and the pressure curve is shown in Figure 5 .

[0081] As shown in Figure 4 , compared with the model without considering unsteady friction, the unsteady friction model provided by the application can accurately describe the transient pipe pressure amplitude and attenuation.

[0082] As shown in Figure 5 , the air mass pressure results simulated by the second-order Guderly method in the air mass fast charging process are consistent with the results obtained by the characteristic scheme, and when the Coulomb number is less than 1, numerical weakening will not occur, and the calculation results are more accurate and stable.

[0083] Therefore, the Guderly method calculation of the entrained air mass fast charging process considering unsteady friction can well solve the accuracy problem of the characteristic scheme in the entrained air mass fast charging process, and the scheme can better reproduce the air mass experimental pressure fluctuation and accurately predict the pressure damping of the air mass pipeline transient.

Claims

1. A modeling and simulation method for transient air-water flow in long-distance water transmission pipelines, characterized in that, Includes the following steps: S1: Construct the governing equations for the rapidly filling water body with entrained atmospheric masses, considering the unsteady friction model of Buruno; S2: Based on the constructed control equations, a numerical scheme for the unsteady friction model of rapidly filling water bodies with entrained atmospheric masses is constructed. S3: Combining the constructed control equations and numerical schemes, the boundary conditions between pressurized air and water are calculated to obtain simulation results; The governing equations in step S1 include the governing equations for unsteady water bodies that take into account unsteady friction, the governing equations for pressurized gas, and the governing equations for the gas-water interface. The numerical format in step S2 is a non-steady water body numerical format that considers unsteady friction, as detailed below: ； In the formula, ; ; . , , For the matrix to be converted to the Gadnav method; To solve for the flux of the process, the water body can be divided into N unit volumes. The result can be obtained by integrating formula (3) along the interface of each control volume i. For the i-th control volume, the integral between the control surfaces i – 1 / 2 and i + 1 / 2 can be expressed as: ； In the formula, It is a time interval. It is the spatial step size, and the superscripts n and n+1 represent respectively. and + Time Step for The average value at the control volume interface [i – 1 / 2, i + 1 / 2], and It is the internal cell flux of the i-th cell; Step S3 is as follows: When calculating the boundary elements at the interface between a water body and a large air mass, the Gadnaud method and characteristic scheme are combined: ； ； ； ； In the formula, , For the N-1 / 2th interface The value; , The pressure and velocity at the interface between the water body and the large air mass; , The pressure and velocity obtained by combining formulas (7) and (8); , for , The pressure and speed of the previous moment; , for , Pressure and velocity obtained by interpolation between the current moment and the previous moment; , The pressure and velocity at the gas-water interface were obtained. A tiny water segment whose length changes continuously as gas is compressed and expanded; , This is the value of the virtual unit.

2. The modeling and simulation method for transient gas-water flow in a long-distance water transmission pipeline according to claim 1, characterized in that, The expression for the governing equation of the unsteady water body considering unsteady friction is as follows: ； In the formula, H is the piezometric head; V is the average flow velocity; c is the speed of sound propagating in water; g is the acceleration due to gravity; z is the distance along the pipeline; t is time; f is the Darcy-Visbach friction coefficient; k is the Vadyl friction coefficient; when the flow velocity is greater than or equal to zero, (V) = 1; when the flow velocity is less than zero, (V) = −1; D is the pipe diameter; the right side of equation (2) is an unsteady friction term.

3. The modeling and simulation method for transient gas-water flow in a long-distance water transmission pipeline according to claim 1, characterized in that, The expression for the governing equation of the pressurized gas is: ； In the formula, This refers to the pressure of the gas after it has been compressed. This refers to the velocity of the gas under pressure. and yes and The initial value of ; m is the gas state index, which can be taken as a constant when heat transfer is not considered.

4. The modeling and simulation method for transient gas-water flow in a long-distance water transmission pipeline according to claim 1, characterized in that, The governing equation for the air-water interface is expressed as follows: ； In the formula, It is the length of the water body. yes The initial value of the gas can be obtained by integrating the velocity of the gas under pressure over time to obtain the length of the water body after a period of time. By solving equations (3) and (4), a method for capturing the pressure and velocity at the interface between the moving gas mass and water can be established.

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