CN111695307B - Water hammer finite volume simulation method considering dynamic friction resistance explicitly - Google Patents

Water hammer finite volume simulation method considering dynamic friction resistance explicitly Download PDF

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CN111695307B
CN111695307B CN202010434805.2A CN202010434805A CN111695307B CN 111695307 B CN111695307 B CN 111695307B CN 202010434805 A CN202010434805 A CN 202010434805A CN 111695307 B CN111695307 B CN 111695307B
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周领
曹云
刘静
李赟杰
方浩宇
薛子剑
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Hohai University HHU
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Abstract

The invention discloses a water hammer finite volume simulation method considering dynamic friction resistance explicitly. Firstly, a model is discretized through a finite volume method, a solver of Riemann is used for solving discrete flux, and meanwhile, a convection term is introduced into calculation. And then reconstructing by adopting a MUSCL-Hancock method to obtain a Godunov format with second-order precision, and avoiding introducing a MINMOD slope limiter into false oscillation by adopting an explicit solving method and taking a dynamic friction item into consideration. And finally, realizing the unified calculation of all nodes and boundaries of the calculation region by using a virtual boundary processing method. The method adopts a second-order finite volume method Godunov format, explicitly considers the influence of dynamic friction resistance, and accurately and efficiently performs numerical simulation on the pipeline water hammer process, thereby providing reference for accurately and efficiently predicting and monitoring hydraulic transient which may occur in the pipeline in real time.

Description

Water hammer finite volume simulation method considering dynamic friction resistance explicitly
Technical Field
The invention relates to simulation of a hydraulic transient process, in particular to a finite volume simulation method of a water hammer with explicit consideration of dynamic friction resistance.
Background
In the process of energy conversion, a pumped storage power station pipeline system often generates complex hydraulic transient state in order to meet the dynamic change of the load of a power grid. Dangerous water hammer events due to some improper operation of the turbine/valve in the water system can create abnormally high pressures, causing pipe breaks and other hydraulic damage. The method can accurately and effectively simulate the water hammer events, and can provide reference for reasonable design and safe operation of the pumped storage power station pipeline system. The existing simulation methods mainly include a Method Of Characteristics (MOC), a Finite Volume Method (FVM), and a Computational Fluid Dynamics (CFD) Method.
The characteristic line Method (MOC) is simple, efficient and accurate, and is widely applied to solving the problem of transient flow of the pipeline. However, since the actual water pipe system of the pumped storage power station often includes different pipe sections of different materials and different lengths, it is impossible to make the barntt number in all the pipes of a complex pipe system completely equal to 1, so that when the MOC of the fixed grid is used for calculation and analysis, the barntt number condition may need to be satisfied by changing the wave speed or the grid length, resulting in increased complexity of calculation and easy occurrence of large errors.
When the Finite Volume Method (FVM) is used for the problems of gas dynamics and shallow water equations, a reasonable solution scheme can be provided for the discontinuous problems under the condition of ensuring the conservation of mass and energy, and false oscillation can be effectively avoided. The FVM-based Godunov format (GTS) was introduced to simulate simple transient pipe flow. For a simple water hammer event, FVM GTS can achieve the same result as fixed grid MOC when the countt number is 1. However, for more complex hydraulic transient problems, including dynamic friction, there is a drawback of poor simulation accuracy.
Computational Fluid Dynamics (CFD) methods, when applied to the implementation of three-dimensional (3D) flow simulations of the entire system of upstream and downstream reservoirs, can accurately predict transient pressure and energy conversion, vividly reflecting physical processes. However, due to the large amount of calculation and long time consumption, it is difficult to realize real-time dynamic simulation and monitoring by using the three-dimensional CFD method.
Disclosure of Invention
The purpose of the invention is as follows: the application aims to provide a water hammer finite volume simulation method considering dynamic friction resistance explicitly, and the method overcomes the defects of low simulation calculation precision and calculation speed in the water hammer process.
The technical scheme is as follows: the invention provides a water hammer finite volume simulation method considering dynamic friction resistance explicitly, which adopts a second-order Godunov format based on a finite volume method to simulate a pipeline hydraulic transient process by a method considering dynamic friction resistance explicitly through calculation, and comprises the following steps:
(1) dividing a computational grid according to a finite volume method, establishing a control equation aiming at divided control units, and determining an initial condition and a boundary condition based on an engineering example;
(2) calculating the numerical flux of a water body discrete equation at the interface of the control unit in a second-order Godunov format;
(3) adding a dynamic friction resistance term, and utilizing a Runge-Kutta method to carry out explicit solution on a numerical integral term of a water body discrete equation;
(4) performing virtual boundary processing on the boundary of the water body;
(5) determining a maximum time step according to the CFL condition and the source item stability constraint met by the Godunov format of the second-order explicit finite volume method;
(6) and solving the hydraulic transient process.
Further, the control equation established in step (1) is expressed as:
Figure BDA0002501845290000021
wherein the content of the first and second substances,
Figure BDA0002501845290000022
wherein i is the number of the control unit in the water body, 1<i<N, N is the total number of the control units; f. ofi-1/2Is the i-1/2 interface numerical flux; f. ofi+1/2Is the i +1/2 interface number flux; x is the distance along the tube axis; t is the time; h is a piezometer tube water head; v is the flow velocity of water flow in the pipeline; a is the wave velocity; g is the acceleration of gravity; j. the design is a squareQAnd JUHead losses caused by quasi-constant friction and non-constant friction in the pipeline flow, respectively; d is the diameter of the pipeline; Δ t is the time step; Δ x is the length of each control unit; the superscript n represents time t; the superscript n +1 denotes the time t + Δ t.
Further, the step (2) comprises:
(21) the flux at the interface of the internal control unit of the water body in the first order Godunov format was calculated:
based on the riemann problem, according to the Godunov format, for any control unit i in the water body, the flux at the interface i +1/2 is as follows:
Figure BDA0002501845290000023
wherein the content of the first and second substances,
Figure BDA0002501845290000031
the average value of time u to the left of the i +1/2 interface at time n,
Figure BDA0002501845290000032
the average value of u at the right side of the i +1/2 interface at time n;
(22) using MUSCL-Hancock format pair fi+1/2Linear interpolation reconstruction is carried out to obtain the second-order Godunov format under t epsilon [ t [ [ t ]n,tn+1]The flux value at all control unit interfaces i +1/2 at time instant.
Further, in step (22), the MUSCL-Hancock format is adopted for fi+1/2When linear interpolation reconstruction is carried out, a slope limiter function MINMOD is introduced to avoid false oscillation in the second-order Godunov format during reconstruction.
Further, in the step (3), the dynamic friction term is included in the source term s, the time step of n +1 is obtained through explicit solution, and the flux of the control unit i
Figure BDA0002501845290000033
The form of the solution is represented as:
Figure BDA0002501845290000034
wherein the content of the first and second substances,
Figure BDA0002501845290000035
in the time step of n +1, the control unit i flows the flux of the variable u in the pure convection;
Figure BDA0002501845290000036
for first updating by time-splittingFlux.
Further, the virtual boundary processing at the water body boundary in the step (4) includes: two virtual control units-1 and 0, and N +1 and N +2 are respectively constructed on the upstream side of the starting control unit 1 and the downstream side of the end point control unit N, and the flow of the water body at the virtual units is consistent with the boundary so as to solve the boundary Riemann problem.
Further, the step (5) includes obtaining a maximum time step according to the CFL condition and the source item stability constraint.
Further, the maximum time step is obtained by:
(51) according to the CFL condition, obtaining the maximum time step length of the CFL; the CFL conditions were:
Figure BDA0002501845290000037
the maximum time step of the CFL is:
Figure BDA0002501845290000038
(52) obtaining a source item constraint maximum time step according to the source item stability constraint, wherein the source item constraint conditions are as follows:
Figure BDA0002501845290000039
the source constraint maximum time step is:
Figure BDA0002501845290000041
(53) taking the smaller value of the CFL maximum time step and the source item constraint maximum time step as the maximum time step delta tmaxNamely:
Δtmax＝min(Δtmax,CFL,Δtmax,s)。
has the advantages that: compared with the prior art, the method has the advantages that the Godunov based on the finite volume method is utilized, the hydraulic transient problem is solved simply and is easy to realize; the dynamic friction resistance is explicitly considered and taken into a source item, and the existing various dynamic friction resistance models are applied to the finite volume method-based water hammer solving, so that the water head loss can be more accurately considered, and the calculation precision is further improved. In addition, the nonlinear convection term is added into the solution, so that the application range of the model can be expanded to the problem of large Mach number, and the accuracy is improved.
Drawings
FIG. 1 is a schematic flow diagram of a simulation method of the present application;
FIG. 2 is a schematic diagram of meshing in the method of the present application;
FIG. 3 is a schematic structural diagram of an experimental apparatus used in an effect evaluation experiment of the simulation method of the present application;
FIG. 4 is a hydraulic transient process pressure change diagram under the laminar flow state combined with different dynamic friction models for effect evaluation of the simulation method of the present application;
FIG. 5 is a hydraulic transient process pressure change diagram under a turbulent flow state combined with different dynamic friction models for effect evaluation of the simulation method.
Detailed Description
The invention is further described below with reference to the following figures and examples:
the invention provides a water hammer finite volume simulation method considering dynamic friction resistance explicitly, which adopts a second-order Godunov format based on a finite volume method to simulate a pipeline hydraulic transient process by a method considering dynamic friction resistance explicitly through calculation, as shown in figure 1, and comprises the following steps:
s101, dividing a computational grid according to a finite volume method, establishing a control equation aiming at the divided control units, and determining an initial condition and a boundary condition based on an engineering example. Wherein, according to the concrete working condition example, the initial condition mainly comprises an upstream water head, a pipeline length, a pipeline inner diameter and the like.
Specifically, the motion equation and continuity equation describing the state of the pipe water flow are written in the form of a matrix:
Figure BDA0002501845290000051
wherein the content of the first and second substances,
Figure BDA0002501845290000052
wherein x is the distance along the tube axis; t is the time; h is a piezometer tube water head; v is the flow velocity of water flow in the pipeline; a is the wave velocity; g is the acceleration of gravity; j. the design is a squareQAnd JURespectively refer to head loss caused by quasi-constant friction resistance and non-constant friction resistance in the pipeline flow; d is the diameter of the pipeline.
The computational grid is divided according to the finite volume method, and as shown in fig. 2, the numbers of the upstream and downstream interfaces of the control unit i are defined as i-1/2 and i +1/2 respectively.
For the control unit i, an integral equation of the flow variable u is established, namely, the integral equation of the formula (1) is integrated, and the control equation of the control unit i is obtained as follows:
Figure BDA0002501845290000053
wherein the content of the first and second substances,
Figure BDA0002501845290000054
wherein i is the number of the control unit in the water body, 1<i<N, N is the total number of the control units; f. ofi-1/2Is the i-1/2 interface numerical flux; f. ofi+1/2Is the i +1/2 interface number flux; Δ t is the time step; Δ x is the length of each control unit; the superscript n represents time t; the superscript n +1 denotes the time t + Δ t.
S102, calculating the numerical flux of the water body discrete equation at the interface of the control unit in the second-order Godunov format.
Specifically, the calculation is performed by the following steps:
calculating the flux at an interface i +1/2 for any control unit i in the water body according to Godunov format based on Riemann problem:
according to the Godunov method, the Riemann question is the following initial value question, which is specifically as follows:
Figure BDA0002501845290000055
Figure BDA0002501845290000056
in the formula (I), the compound is shown in the specification,
Figure BDA0002501845290000057
the average value of time u to the left of the i +1/2 interface at time n,
Figure BDA0002501845290000058
the average value of the time u at n on the right side of the i +1/2 interface is obtained, and t e [ t ] is obtainedn,tn+1]Flux value at time control unit i interface i + 1/2:
Figure BDA0002501845290000059
wherein the content of the first and second substances,
Figure BDA0002501845290000061
namely, the value of the variable average distribution in the unit is adopted to replace the value of the left and right of the interface, so that the format has only first-order precision.
② to obtain the Godunov format of second-order precision, adopting MUSCL-Hancock format to pair fi+1/2Linear interpolation reconstruction is carried out to obtain the second-order Godunov format under t epsilon [ t [ [ t ]n,tn+1]The flux values at the i +1/2 interface of all control units at the moment are as follows:
(a) data reconstruction
In order to avoid the phenomenon of false oscillation when the second-order format is reconstructed in the linear interpolation, a slope limiter function is introduced
Figure BDA0002501845290000062
Figure BDA0002501845290000063
Figure BDA0002501845290000064
Wherein the upper corner mark L represents x → xi-1/2And x>xi-1/2(ii) a Superscript R denotes x → xi+1/2And x<xi+1/2；
Figure BDA0002501845290000065
(b) Calculation of propulsion time
Figure BDA0002501845290000066
Figure BDA0002501845290000067
(c) Approximate solution to the Riemann problem
Figure BDA0002501845290000068
Substituting formula (10) into formula (5) to obtain second-order Godunov format in t e [ t [ [ t ]n,tn+1]The flux value at interface i +1/2 for all control units i at that moment.
S103, adding a dynamic friction resistance term, and performing explicit solution on a numerical integration term of the water body discrete equation by using a Runge-Kutta method.
Specifically, after the left and right fluxes are calculated, in order to solve the solution from the time n to the time n +1, the equation (3) needs to be integrated to obtain a numerical integral term of the water body discrete equation:
Figure BDA0002501845290000069
the dynamic friction resistance term is included in a source term s (u), and then the numerical integral term of the discrete equation of the water body is solved explicitly based on the Runge-Kutta method:
the first step is as follows:
Figure BDA0002501845290000071
the second step is that:
Figure BDA0002501845290000072
the third step:
Figure BDA0002501845290000073
wherein the content of the first and second substances,
Figure BDA0002501845290000074
in the time step of n +1, the control unit i flows the flux of the variable u in the pure convection;
Figure BDA0002501845290000075
the flux after the first update using time-splitting.
In the embodiment of the present application, three types of dynamic friction models with the most extensive applications can be adopted, which are respectively: zielke model, Vary-Brown model and Brunone model. The Zielke model is suitable for laminar flow, the Vary-Brown model and the Brunone model are suitable for turbulent flow. The dynamic friction terms for each model are explained below:
the Zielke dynamic friction model comprises the following steps:
Figure BDA0002501845290000076
wherein ν is the kinematic viscosity coefficient, m2S; u is a time variable integrated between 0 and t; w is a function of dimensionless time tau, tau 4 vt/D2W is calculated as follows:
when tau is more than or equal to 0.02:
Figure BDA0002501845290000077
when τ < 0.02:
Figure BDA0002501845290000078
in the formula, mi，niIs a coefficient, mi＝{26.3744；70.8493；135.0198；218.9216；322.5544}；ni＝{0.282095；-1.25；1.057855；0.9375；0.396696；-0.351563}。
Vary-Brown dynamic friction model:
Figure BDA0002501845290000081
in the formula, A*、B*Are coefficients.
Figure BDA0002501845290000082
Laminar flow:
B*＝210
turbulent flow:
B*＝0.135Reκ (19)
Figure BDA0002501845290000083
wherein Re is a Reynolds number,
Figure BDA0002501845290000084
third Brunone dynamic friction model:
Figure BDA0002501845290000085
wherein when V.gtoreq.0, SGN (V) is 1; when V < 0, sgn (V) ═ -1; k is a radical of3For Brunone coefficients, the following are calculated:
Figure BDA0002501845290000086
under laminar flow conditions:
C*＝0.00476
under turbulent conditions:
Figure BDA0002501845290000087
Figure BDA0002501845290000088
and S104, performing virtual boundary processing on the water body boundary.
Specifically, two virtual control units-1 and 0 and N +1 and N +2 are respectively constructed on the upstream side of the starting control unit 1 and the downstream side of the end point control unit N, and the flow of the water body at the virtual units is consistent with the boundary so as to solve the boundary Riemann problem.
Specifically, in the embodiment of the present application, the upstream reservoir constant water level boundary:
upstream boundary:
Figure BDA0002501845290000091
according to the negative characteristic line and the Riemann invariant equation, the method comprises the following steps:
Figure BDA0002501845290000092
downstream boundary:
Figure BDA0002501845290000093
according to the forward characteristic line and the Riemann invariant equation, the method comprises the following steps:
Figure BDA0002501845290000094
s105, determining the maximum time step according to the CFL condition and the source item stability constraint which are met by the Godunov format of the second-order explicit finite volume method.
The method specifically comprises the following steps:
firstly, according to the CFL condition, obtaining the maximum time step length of the CFL; the CFL conditions were:
Figure BDA0002501845290000095
the maximum time step of the CFL is:
Figure BDA0002501845290000096
secondly, according to the source item stability constraint, obtaining the maximum time step of the source item constraint, wherein the source item constraint conditions are as follows:
Figure BDA0002501845290000097
the source constraint maximum time step is:
Figure BDA0002501845290000098
(53) taking the smaller value of the CFL maximum time step and the source item constraint maximum time step as the maximum time step delta tmaxNamely:
Δtmax＝min(Δtmax,CFL,Δtmax,s) (31)
and S106, solving the hydraulic transient process according to the parameters obtained by solving in the steps.
And (3) experimental evaluation:
in order to verify and analyze the simulation effect of the hydraulic transient Godunov simulation method explicitly considering dynamic friction resistance, the Godunov simulation method is combined with three commonly used dynamic friction resistance models, namely a Zielke model, a Vary-Brown model and a Brunene model, and the experiment models and pressure results of Bergant and the like are adopted for verification, and the schematic diagram of a simplified experiment device is shown in figure 3. The experimental device comprises an upstream constant pressure tank, an upward inclined straight pipeline (the gradient is 5.45 percent), a downstream ball valve and a downstream pressure tank. Basic parameters of the experimental system: upstream head of HRAt 32m, a valve was installed downstream and closed linearly at 0.009 s. The length L of the pipeline is 37.23m, the inner diameter D of the pipeline is 0.0221m, the wall thickness of the pipeline is 1.63mm, the water temperature T is 15.4 ℃, the wave propagation speed a is 1319m/s, the kinematic viscosity coefficient v is 1.184 x 10-6m2/s, and 2 experimental working conditions are taken: the initial flow state of the working condition 1 is laminar flow, and the flow velocity V00.1 m/s; the initial flow state of the working condition 2 is low Reynolds number turbulent flow with flow velocity V0＝0.2m/s。
According to the method, the hydraulic transient process is calculated in a programming mode, pressure head change curves of the hydraulic transient process in different initial flow states in the embodiment are shown in fig. 4 and 5, and meanwhile, the calculation result of the quasi-constant friction resistance model is also shown. The curve result shows that the amplitude and the time response of the pressure calculated by the method are well consistent with the experimental result, and the water hammer pressure fluctuation in the pipeline can be accurately predicted. Meanwhile, it can be seen that neglecting the influence of the dynamic friction term causes large calculation errors in both the pressure amplitude and the pressure period.
As will be appreciated by one skilled in the art, embodiments of the present application may be provided as a method, system, or computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
The present application is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the application. It will be understood that each flow and/or block of the flow diagrams and/or block diagrams, and combinations of flows and/or blocks in the flow diagrams and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.

Claims (6)

1. A water hammer finite volume simulation method considering dynamic friction resistance explicitly is characterized in that a second-order Godunov format based on a finite volume method is adopted, and a method for considering dynamic friction resistance by explicitly calculating is used for simulating a pipeline hydraulic transient process, and comprises the following steps:
(1) dividing a computational grid according to a finite volume method, establishing a control equation aiming at divided control units, and determining an initial condition and a boundary condition based on an engineering example;
(2) calculating the numerical flux of a water body discrete equation at the interface of the control unit in a second-order Godunov format;
(3) adding a dynamic friction resistance term, and utilizing a Runge-Kutta method to carry out explicit solution on a numerical integral term of a water body discrete equation;
(4) performing virtual boundary processing on the boundary of the water body;
(5) determining a maximum time step according to the CFL condition and the source item stability constraint met by the Godunov format of the second-order explicit finite volume method;
(6) solving a hydraulic transient process;
the control equation established in step (1) is expressed as:
Figure FDA0003036894570000011
wherein the content of the first and second substances,
Figure FDA0003036894570000012
wherein i is the number of the control unit in the water body, 1<i<N, N is the total number of the control units; f. ofi-1/2Is the i-1/2 interface numerical flux; f. ofi+1/2Is the i +1/2 interface number flux; x is the distance along the tube axis; t is the time; h is a piezometer tube water head; v is the flow velocity of water flow in the pipeline; a is the wave velocity; g is the acceleration of gravity; j. the design is a squareQAnd JUHead losses caused by quasi-constant friction and non-constant friction in the pipeline flow, respectively; Δ t is the time step; Δ x is the length of each control unit; the superscript n represents time t; superscript n +1 represents time t + Δ t;
in the step (3), a dynamic friction resistance term is included in a source term s (u), and then a numerical integral term of a discrete equation of the water body is solved explicitly based on a Runge-Kutta method:
the first step is as follows:
Figure FDA0003036894570000013
the second step is that:
Figure FDA0003036894570000014
the third step:
Figure FDA0003036894570000021
wherein the content of the first and second substances,
Figure FDA0003036894570000022
in the time step of n +1, the control unit i flows the flux of the variable u in the pure convection;
Figure FDA0003036894570000023
the flux after the first update using time-splitting.
2. The method of claim 1, wherein step (2) comprises:
(21) the flux at the interface of the internal control unit of the water body in the first order Godunov format was calculated:
based on the riemann problem, according to the Godunov format, for any control unit i in the water body, the flux at the interface i +1/2 is as follows:
Figure FDA0003036894570000024
wherein the content of the first and second substances,
Figure FDA0003036894570000025
the average value of time u to the left of the i +1/2 interface at time n,
Figure FDA0003036894570000026
the average value of u at the right side of the i +1/2 interface at time n;
(22) adopting MUSCL-Hancock formatTo fi+1/2Linear interpolation reconstruction is carried out to obtain the second-order Godunov format under t epsilon [ t [ [ t ]n,tn+1]The flux value at all control unit interfaces i +1/2 at time instant.
3. The method of claim 2, wherein in step (22), the pair f is in a MUSCL-Hancock formati+1/2When linear interpolation reconstruction is carried out, a slope limiter function MINMOD is introduced to avoid false oscillation in the second-order Godunov format during reconstruction.
4. The method of claim 1, wherein the virtual boundary processing at the boundary of the water body in the step (4) comprises: two virtual control units-1 and 0, and N +1 and N +2 are respectively constructed on the upstream side of the starting control unit 1 and the downstream side of the end point control unit N, and the flow of the water body at the virtual units is consistent with the boundary so as to solve the boundary Riemann problem.
5. The method of claim 4, wherein step (5) comprises deriving a maximum time step based on the CFL condition and a source item stability constraint.
6. The method of claim 5, wherein the maximum time step is obtained by:
(51) obtaining the CFL maximum time step according to the CFL condition; the CFL conditions are as follows:
Figure FDA0003036894570000027
the maximum time step of the CFL is:
Figure FDA0003036894570000031
(52) obtaining a source item constraint maximum time step according to a source item stability constraint, wherein the source item constraint conditions are as follows:
Figure FDA0003036894570000032
the source constraint maximum time step is:
Figure FDA0003036894570000033
(53) taking the smaller value of the CFL maximum time step and the source item constraint maximum time step as the maximum time step delta tmaxNamely:
Δtmax＝min(Δtmax,CFL,Δtmax,s)。
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