CN106503396B - The multidimensional Hydraulic Power System transition analogy method coupled based on finite difference calculus with finite volume method - Google Patents
The multidimensional Hydraulic Power System transition analogy method coupled based on finite difference calculus with finite volume method Download PDFInfo
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Abstract
A kind of multidimensional Hydraulic Power System transition analogy method coupled based on finite difference calculus with finite volume method, belongs to pipe network system Hydraulic Transient numerical simulation field.The present invention utilizes one-dimensional pressure conduit unsteady flow governing equation equation, one-dimensional open channel shallow water equation, two-dimentional open channel shallow water equation, Three-dimensional Flow equation, and the thought coupled based on finite difference calculus with finite volume method, using identical coupling process, by the non-constant finite difference calculus of one-dimensional pressure conduit, respectively with one-dimensional pressure conduit unsteady flow finite volume method, one-dimensional unsteady flow in open finite volume method, two-dimentional unsteady flow in open finite volume method, the coupling of three-dimension complex flow finite volume method.On the one hand calculating speed of the One Dimensional Finite calculus of finite differences in terms of Complex hydraulic system simulation is utilized in the coupling model fast, the simple advantage of boundary condition treatment, on the other hand multidimensional finite volume method feature with high accuracy when Complex Flows calculate is utilized, realizes efficiency and precision optimum organization.
Description
Technical field
The invention belongs to pipe network system Hydraulic Transient numerical simulation field, in particular to a kind of finite difference calculus and limited body
The analogy method of the non-constant process of Hydraulic Power System of area method coupling.
Technical background
Pipeline pressure flow is dynamic and channel Shallow-water Flow is two kinds of most common flowings in hydraulic system, using one
Dimension, the method for two and three dimensions can simulate it.For pressure conduit unsteady flow, one-dimensional method is mostly used to simulate,
The method of characteristic curves and finite difference calculus are easy due to programming in ONE-DIMENSIONAL METHOD, and boundary condition treatment is simple, can satisfy computational accuracy
And it is widely used in the simulation of pipe-line system water hammer pressure.And the relatively fewer of one-dimensional water hammer is simulated using finite volume method, and
And it is rarely used in engineering practice.
Simulation for open channel shallow water equation, mostly uses finite volume method method to be simulated, and is on the one hand due to channel
It is influenced by orographic condition, it is often necessary to non-orthogonal mesh is used, and the grid of finite difference calculus is mostly orthogonal grid, another party
Face is such as dam-break water flow, and the simulation of the Shallow-water Flows such as river flood often refers to complicated flowing, for example exists simultaneously urgency
Stream and unhurried current and the mutual conversion between them, also relate to drylyly and the conversion of Wetland boundary condition, using finite difference
Method is difficult to simulate these complicated flowings, needs to handle using finite volume method.
The dynamic governing equation of one peacekeeping Wave of Two-Dimension Shallow Water is nonlinear Hyperbolic Partial Differential Equations, and finite difference calculus needs
Orthogonal grid and it is not easy the violent flowing of analog variation, such as torrent and critical flow.However the region of required calculating is past
Past is irregular, therefore the simulation for shallow water equation, and do not transported finite difference calculus extensively as finite volume method
With.Finite volume method based on Godunov format is widely used for solving nonlinear hyperbolic equation, the key of Godunov
It is the solution to Riemannian problem, it can be using accurate solution or the method for approximate solution.Such as based on conservation equation it is dull windward in
Heart difference scheme, HLL method, and the Roe approximate solution being widely used, can immediately arrive at Godunov flux expression formula simultaneously
Solution suitable for a variety of physical problems.
Either dynamic for pressure flow or Open Channels, when simulated object local turbulence phenomenon is obvious, such as water power
The bright full mixed flow stood in generating system by piloting water, needs to be related to Two-phase flow's separation, and such as hydraulic internal flow
Simulation need to be related to fluid structure interaction mode etc., using it is one-dimensional or it is two-dimentional it cannot all be simulated well, at this time
It needs to calculate it using threedimensional model, if simulated pressure oscillation in movable propeller turbine, analyzes surge-chamber head
Loss etc..
Hydraulic transient process is related to the coupling of three aspect of hydraulic-mechanical-power grid, and one-dimensional model has simply,
The fast advantage of calculating speed, result can describe the macroscopic view variation of pressure and flow velocity in such as long pipeline and channel, and energy
Enough reflect the dynamic characteristic of system.On the other hand, two and three dimensions model can capture Complex Flows in regional area
Information of flow, such as water inlet and the indoor vortex of pressure regulation.More accurate computation model, such as turbulence model, multiphase flow mould
Type and fluid structure interaction mode can apply in three-dimensional calculation method.However compared with one-dimensional model, three-dimensional computations are more multiple
It is miscellaneous and generally require the longer calculating time, therefore can not will include governor, transformer, other hydraulics such as power grid
The whole system of equipment is all made of three-dimensional method and is simulated.In this case, one-dimensional and two-dimentional or one-dimensional and three-dimensional coupling
The model of conjunction can play ONE-DIMENSIONAL METHOD processing boundary condition it is simple, the fast advantage of calculating speed, but can using two dimension or three-dimensional
The accurate feature of model result.
For simple one-dimensional, two dimension or three-dimensional calculating, computation model also relative maturity, therefore for different dimensions
Coupling calculation, key is two aspects, one is the transmitting of data, the i.e. processing of coupled boundary condition, the second is
The transition of dimension, i.e., one-dimensional point data are exchanged with multidimensional face data.For second point, simplest mode is to put down multidimensional face
, depression of order is one-dimensional data, then is docked with ONE-DIMENSIONAL METHOD.Therefore the essence of coupling is the processing of interface boundary condition.
The utilization of coupling process is up to the simulation of one peacekeeping two dimension coupling process of shallow water equation at present, applies to river lake
The simulation of pool system flood, the method that peacekeeping two-dimensional portion therein is all made of finite volume method, the master of different coupling process
It is distinguished as the processing of interface boundary condition, such as the method based on coupled interface pressure iteration, a peacekeeping two on interface
Dimension mutually explicitly provides the method etc. of boundary condition with last moment value.
It is one-dimensional also usually to apply to actual engineering problem with the three-dimensional method coupled, one-dimensional with the three-dimensional boundary coupled
On face, the parameter of 3D region usually uses the method depression of order being averaged for dimension parameter, then again with one-dimensional regional edge
Parameter in boundary carries out coupling transmitting.
In coupling process mentioned above, the calculation method of different dimensions mostly uses greatly same difference scheme, therefore
The problem of coupling, is converted to the problem of interface flux calculates, and between different difference scheme coupling (such as finite difference calculus with
The coupling of finite volume method), relevant achievement is also less, and main innovation point of the invention is exactly by One Dimensional Finite calculus of finite differences method
Respectively with it is one-dimensional, two dimension, Three-D limited volumetric method method coupling, using unified coupling process, and apply to respectively different
In problem.Wherein, one-dimensional to couple the coupling including pressure conduit and pressure conduit, the coupling of pressure conduit and open channel with one-dimensional
It closes, provides basis for dimension and multi-dimensional Coupling, it is one-dimensional to couple mainly one-dimensional pressure conduit and two-dimensional shallow water stream with two-dimensional
Dynamic coupling, one-dimensional is mainly one-dimensional to have pressure and three-dimensional to have pressure, the coupling between three-dimensional free pressure flow with three-dimensional coupling.
Summary of the invention
Existing insufficient during simulation Complex hydraulic system is non-constant for the prior art, the present invention has using one-dimensional
Pressure pipeline unsteady flow governing equation, one-dimensional open channel shallow water equation, two-dimentional open channel shallow water equation, Three-dimensional Flow equation propose one
The multidimensional Hydraulic Power System transition analogy method that kind is coupled based on finite difference calculus with finite volume method.
The present invention is based on the multidimensional Hydraulic Power System transition analogy method that finite difference calculus is coupled with finite volume method, features
Be the following steps are included:
Step 1, non-constant using finite difference calculus-discrete one-dimensional pressure conduit of 4, the space Pressimann implicit schemes
Flow control equation；
Step 2, there is one-dimensional pressure conduit unsteady flow governing equation using finite volume method-Godunov format is discrete；
Step 3, governing equation is fluctuated using the discrete peacekeeping two dimension open channel shallow water of finite volume method-Godunov format；
Step 4, Three-dimensional Flow equation is solved based on the solver of OpenFOAM using finite volume method-；
Step 5, the boundary condition equation for deriving pressure conduit flowing and Open Channels respectively as finite difference calculus and has
Limit the boundary condition of volumetric method coupling；
Step 6, based on the boundary condition equation of derivation, one-dimensional pressure conduit finite difference calculus and one-dimensional pressure conduit are coupled
Finite volume method, or couple one-dimensional pressure conduit finite difference calculus and one-dimensional open channel finite volume method, or coupling is one-dimensional pressure pipe
Road finite difference calculus and two-dimentional open channel finite volume method, or the one-dimensional pressure conduit finite difference calculus of coupling and the limited body of Complex Flows
Area method.
The step 5, further, the characteristic value of demand solution pressure conduit unsteady flow governing equation and corresponding
Right feature vector lists the differential relationship between equation variable and right feature vector, along characteristic curve integral differential equation, obtains
Boundary face calculates the linear relation known to moment variable and grid element center between moment variable, as pressure conduit limited bulk
The relational expression of head and flow velocity, is denoted as the boundary condition side of pressure conduit inlet and outlet herein in method inlet and outlet boundary face
Journey；Using same method, the pass of head and flow velocity in open channel shallow water fluctuation finite volume method inlet and outlet boundary face is acquired
It is formula, is denoted as the boundary condition of open channel inlet and outlet herein.
The step 6 specifically includes following method:
(1) simultaneous has one-dimensional pressure conduit outlet difference equation and one-dimensional pressure conduit inlet boundary conditional equation, or
One-dimensional pressure conduit export boundary condition equation and one-dimensional pressure conduit import difference equation, solution obtain the flow velocity in boundary face
And pressure further finds out finite difference calculus as the boundary condition in finite difference calculus and finite volume method coupling interface
Zoning, each node in finite volume method zoning inside and the intracorporal head of control and flow velocity, realize one-dimensional pressure conduit
The coupling of finite difference calculus and one-dimensional pressure conduit finite volume method；
(2) simultaneous has one-dimensional pressure conduit outlet difference equation and one-dimensional open channel inlet boundary conditional equation or one-dimensional
Open channel export boundary condition equation and one-dimensional pressure conduit import difference equation, solution obtain the flow velocity and the depth of water in boundary face,
As the boundary condition in finite difference calculus and finite volume method coupling interface, further finds out finite difference calculus and calculate area
Domain, each node in finite volume method zoning inside and the intracorporal depth of water of control and flow velocity, realize one-dimensional pressure conduit finite difference
The coupling of point-score and one-dimensional open channel finite volume method；
(3) simultaneous has one-dimensional pressure conduit outlet difference equation and two-dimentional open channel inlet boundary conditional equation, or two dimension
Open channel export boundary condition equation and one-dimensional pressure conduit import difference equation, solution obtain the flow velocity and the depth of water in boundary face,
As the boundary condition in finite difference calculus and finite volume method coupling interface, further finds out finite difference calculus and calculate area
Domain, each node in finite volume method zoning inside and the intracorporal flow velocity of control and the depth of water, realize one-dimensional pressure conduit finite difference
The coupling of point-score and two-dimentional open channel finite volume method；
(4) the one-dimensional pressure conduit outlet difference equation of simultaneous and three-dimension complex flow inlet boundary conditional equation, Huo Zheyi
Tie up pressure conduit import difference equation and three-dimension complex flow export boundary condition equation, solve obtain flow velocity in boundary face and
Then pressure finds out finite difference calculus zoning, finite volume method meter according to the flow velocity and pressure that find out in boundary face respectively
It calculates each node inside region and controls intracorporal flow velocity and pressure, realize one-dimensional pressure conduit finite difference calculus and Three Dimensional Finite Body
The coupling of area method.
The step 1 is non-constant using the discrete one-dimensional pressure conduit of 4, the space Pressimann implicit finite difference format
Flow control equation obtains the linear relationship expression formula between adjacent sections flow and head；In conjunction with inlets or outlets perimeter strip
Part, the linear relationship expression formula between simultaneous adjacent sections flow and head, recursion exported or admission section on flow
Linear relationship expression formula between head has been denoted as pressure pipe as one of the absorbing boundary equation coupled with finite volume method herein
Road inlet and outlet differential relationship formula.
Nonlinear terms in one-dimensional pressure conduit unsteady flow governing equation are done approximate processing by the step 2, will be non-
Conservation scheme is converted into conservation scheme, acquires the one-dimensional pressure conduit unsteady flow controlling party of conservation form using Godunov format
The accurate solution of journey solves the expression formula of the interface numerical flux of finite volume method according to accurate solution.
The step 3 specifically comprises the following steps:
(1) nonlinear terms in one-dimensional open channel diving equation are done into approximate processing, converts conservation lattice for Conservative Schemes
Formula is accurately solved using the One-dimensional Shallow Water Equations that Godunov format acquires conservation form, solves finite volume method according to accurate solution
The expression formula of interface numerical flux；
(2) nonlinear terms in two-dimentional open channel diving equation are done into approximate processing, converts conservation lattice for Conservative Schemes
Formula is accurately solved using the two-dimensional shallow water equation that Godunov format acquires conservation form, solves finite volume method according to accurate solution
The expression formula of interface numerical flux；
The step 4 further comprises following steps:
(1) corresponding OpenFOAM solver is selected according to the type of flowing, no matter pressure flow is moved or open channel is shallow
The Complex Flows such as water flowing, the coupling process proposed can be applicable in；
(2) corresponding boundary condition is set for Three-D limited volumetric method, if setting boundary condition be flow velocity at any time
Know, then pressure boundary is set as gradient, and pressure gradient is obtained from the grid of the finite difference calculus adjacent with coupling interface；Such as
Fruit setting boundary condition is that pressure changes over time it is known that then setting gradient for the flow velocity boundary on variable cross-section, gradient value is set
It is set to 0.
The present invention is based on the thoughts that finite difference calculus is coupled with finite volume method will be one-dimensional using identical coupling process
The non-constant finite difference calculus of pressure conduit, respectively with one-dimensional pressure conduit unsteady flow finite volume method, one-dimensional open channel is non-constant
Flow finite volume method, two-dimentional unsteady flow in open finite volume method, the coupling of three-dimension complex flow finite volume method.The coupling model
On the one hand it is utilized that calculating speed of the One Dimensional Finite calculus of finite differences in terms of Complex hydraulic system simulation is fast, and boundary condition treatment is simple
Advantage, multidimensional finite volume method feature with high accuracy when Complex Flows calculate on the other hand is utilized, realize efficiency with
Precision optimizing combination.
Compared to the prior art, the invention has the advantages that and the utility model has the advantages that
1, global macroscopic properties and localized micro characteristic are considered simultaneously in Complex hydraulic system transition numerical simulation, in conjunction with
One-dimensional simulation method calculating speed is fast, saves and calculates time, two and three dimensions analogy method accurate advantage；
2, the coupling process proposed is versatile, and method is simple, for explicit coupling, does not need iteration, can couple pipe
Road pressure flow, channel Shallow-water Flow and the apparent turbulent flow of three-dimensional character.
3, the coupling process proposed is applied widely, suitable for the coupling of multidimentional system, is suitable for One Dimensional Finite difference
The coupling of method and one-dimensional, two-dimentional Three-D limited volumetric method；
4, the coupling process proposed is convenient for and other commercial or open source software interfaces, is suitable for such as Fluent commercialization
Software and such as OpenFOAM open source software.
Detailed description of the invention
Fig. 1 finite volume method grid schematic diagram.
Fig. 2 pressure conduit One Dimensional Finite calculus of finite differences couples schematic diagram with pressure conduit One Dimensional Finite volumetric method.
Fig. 3 pressure conduit One Dimensional Finite calculus of finite differences couples schematic diagram with open channel One Dimensional Finite volumetric method.
Fig. 4 pressure conduit One Dimensional Finite calculus of finite differences couples schematic diagram with open channel two-dimensional finite volumetric method.
Fig. 5 pressure conduit One Dimensional Finite calculus of finite differences couples schematic diagram with Three-D limited volumetric method.
Fig. 6 uses power station of the example-containing settling pit.
Fig. 7 uses example-downstream river course water-level fluctuation power station.
Fig. 8 has the power station of annular pressure measuring unit with example-.
Fig. 9 has the power station of complicated bifurcated pipe with example-.
Specific embodiment
Embodiment 1: technical solution of the present invention is described further below in conjunction with attached drawing.
Specific step is as follows for the method for the present invention:
Step 1, using the discrete one-dimensional pressure conduit unsteady flow of 4, the space finite difference calculus-Pressimann implicit schemes
Governing equation.Ignore the one-dimensional prism transient pipe flow of pipeline on slope governing equation be continuity equation formula (1) and
Momentum equation (2):
Wherein, x and t is room and time coordinate, and V is the mean flow rate of section, and H is piezometric head, and A is the disconnected of pipeline
Face area, f are the Darcy-Weisbach coefficient of friction resistances, and α is velocity of wave, and g is acceleration of gravity, and D is the diameter of pipeline.
Using 4, the space the Preissmann implicit finite difference format of a weight coefficient, one-dimensional have with this format is discrete
Pressure pipeline unsteady flow governing equation (1) and formula (2), with flow replace flow velocity, can be obtained shown in formula (3) and formula (4) from
Dissipate equation:
A1·Hi+1+B1·Qi+1=C1Hi+D1·Qi+F1 (3)
A2·Hi+1+B2·Qi+1=C2Hi+D2·Qi+F2 (4)
Formula (3) and formula (4) express the relational expression of pressure and flow between two adjacent nodes, if it is known that inlet boundary item
Part (5):
f(Q1, H1)=0 (5)
After boundary condition (5) linearisation, in conjunction with (3) and (4) formula, pressure and flow in the available outlet border of recursion
Linear representation (6):
QN=EEN·HN+FFN (6)
If it is known that export boundary condition, using same method, the line of pressure and flow on available inlet boundary
Sexual intercourse formula (7):
Q1=EE1·H1+FF1 (7)
In above formula, coefficient A1 (A2), B1 (B2), C1 (C2), D1 (D2), F1 (F2), EE and FF are given value.Joint
(6) formula and downstream boundary condition, or joint (7) formula and upstream boundary condition, can solve exported or import section on
Pressure and flow in conjunction with (3) formula and (4) formula can solve the pressure and flow that obtain cross sections.
Step 2, there is one-dimensional pressure conduit unsteady flow governing equation using finite volume method-Godunov format is discrete.
The finite volume method method of solution water hammer based on Godunov format, basic idea is from Riemann
The numerical flux expression formula that accurate solution is back-calculated to obtain interface can immediately arrive at accurate bound on solutions face numerical value for linear equation
Flux, steps are as follows:
Equation (1) and (2) are indicated in the form of matrix are as follows:
Wherein:
The method for converting conservation scheme for nonlinear terms in Hyperbolic type Systems, equation (8) can with approximately equivalent in:
Formula (9) is the water hammer of conservation form, in which:
It is the average value of V, whenWhen, as ignore the convective term in equation (1) and (2).Using arithmetic mean of instantaneous value
EstimationThat is:
For control volume i, the continuity equation and the equation of momentum of integrated form be can be written as:
UiAnd SiFor the average value in control volume i, then equation (10) indicates are as follows:
The solution procedure that Godunov format solves hyperbolic equation is as follows, for the difference equation as shown in formula (12):
Wherein:
For n moment U left side grid average value,For the average value of n moment U grid on the right.
The characteristic value of matrix are as follows:
Corresponding right feature vector are as follows:
Using following equations intensity of wave
Solution obtains:
In single order Godunov format,WithFor explicit scheme, it may be assumed that
The interface numerical flux of system of linear equations (12) Exact Solutions are as follows:
Therefore, the unknown parameter of each grid element center can be found out by following formula:
In formula: n is the time, and i is grid number, and Δ t is time step, and Δ x and Δ y are the direction x and y sizing grid.
Source item is approximate using the central value of control unit, i.e. expression formula are as follows:
Step 3, governing equation is fluctuated using the discrete one-dimensional open channel shallow water of finite volume method-Godunov format.
(1) one-dimensional shallow water governing equation is following continuity equation and the equation of momentum:
Every expression formula are as follows:
τ=gn2u|u|/h1/3
Wherein, t is the time, and x and y are cartesian coordinate system, and u is the vector of flow field variable, and f is the flux vector in the direction x,
S is source item, and u is the speed in the direction x, and h is the depth of water, and q is discharge per unit width, and z is channel bottom elevation, and τ is the frictional resistance of channel bottom
Loss.
The above governing equation is written as with U (u1,u2) be dependent variable form:
Wherein:
The above governing equation is Nonlinear System of Equations, needs to pass through iterative solution.The present invention is used Nonlinear System of Equations
Local linearization is calculated the numerical flux at interface using Roe average magnitude, and solves subsequent time using following Godunov format
Unknown quantity.
Wherein, subscript n represents time step number, and following table i represents grid number.Δ t and Δ x respectively indicate time step and net
Lattice size.The F of boundary flux in formula (27)i-1/2And Fi+1/2It is found out using the approximation method that Roe is proposed, its step are as follows:
Formula (26) is rewritten as following form:
Wherein A is the Jacobin matrix of F (U) to U:
By the above Nonlinear System of Equations local linearization are as follows:
Wherein,For constant matrices, by known part ULAnd URIt finds out, according to the suggestion of Roe, constant matricesHave with A
There is identical form, and meets hyperbolicity, compatibility, three conditions of convergence of discontinuous solution, the average scale of use proposed by Roe
The constant matrices reachedExpression formula it is as follows:
Wherein:
Characteristic value are as follows:
Corresponding right feature vector are as follows:
Using following equations intensity of wave
Solution obtains:
Wherein: Δ h=hR-hL, Δ u=uR-uL。
The interface numerical flux of the above Nonlinear System of Equations Exact Solutions are as follows:
Therefore, the unknown parameter of each grid element center can be found out by following formula:
In formula: n is the time, and i is grid number, and Δ t is time step, and Δ x and Δ y are the direction x and y sizing grid, source
The expression formula of item is as follows:
(1) two-dimensional shallow water governing equation is as follows:
Every expression formula is as follows:
Wherein, t is the time, and x and y are cartesian coordinate system, and u is the vector of flow field variable, and f and g are respectively the direction x and y
Flux vector, s is source item, and u is the speed in the direction x, and v is the speed in the direction y, and h is the depth of water, and z is channel bottom elevation, τbx:
And τbyFor the friction loss of channel bottom.
(41) formula left-half is written as with U (u1, u2, u3) be independent variable equation, as shown in formula (42):
Formula (42) is Nonlinear System of Equations, is equivalent to (43) formula:
Wherein A is the Jacobian matrix of F (U), and wherein B is the Jacobian matrix of G (U), expression formula are as follows:
(43) formula local linear is turned into (44) formula:
WhereinWithFor constant matrices, by local ULAnd URIt finds out.According to the suggestion of Roe, constant matricesWithWith A and
B form having the same, and by following Roe average magnitude explicit representation:
Wherein:
The interface numerical flux expression formula of system of linear equations (44) formula Exact Solutions is (45) and (46) formula:
WhereinWithRespectivelyCharacteristic strength, characteristic value and corresponding right feature vector；With
RespectivelyCharacteristic strength, characteristic value and corresponding right feature vector, expression formula be as follows:
Δ h=hr-hlΔ u=ur-ulΔ v=vr-vl
Source item in formula (41) is handled using the mode as shown in following formula (47):
Therefore, the approximate solution of Nonlinear System of Equations (41) is shown in following (48):
In formula: n is the time, and i, j are grid number, and Δ t is time step, and Δ x and Δ y are the direction x and y sizing grid.
Step 4, Three-dimensional Flow equation is solved based on the solver of OpenFOAM using finite volume method-.
The object that is considered of the present invention is related to pressure-wave propagation, it is therefore necessary to which consider fluid in the three-dimensional model can
Compressibility.In the compressible fluid solver sonicLiquidFoam of OpenFOAM, continuity equation formula (49) and momentum side
Formula (50) is as follows:
The compressibility of fluid is realized by the linear relationship of such as lower density and pressure:
Wherein K is the bulk modulus of fluid, and above formula is linearized, the pressure-dependent relationship of density is obtained:
Wherein ρ0And p0For reference density and reference pressure, i.e. ρ (p0)=ρ0, the velocity of sound in liquid passes through Newton-
Laplace equation obtains, it may be assumed that
In sonicLiquidFoam solver, water body elasticity passes through coefficientIt realizes, it is fixed by UDF in Fluent
The attribute of adopted material realizes the relationship of pressure and density, and can be with customized velocity of wave.
Step 5, the boundary condition equation for deriving pressure conduit flowing and Open Channels respectively as finite difference calculus and has
Limit the boundary condition of volumetric method coupling.
(1) derivation of the boundary condition equation of pressure conduit flowing:
For the Quasilinear Hyperbolic type partial differential equations (53) of following m dimension
Wherein W=[w1,w2,···wm]T, with ith feature value λiCorresponding right feature vector KiAre as follows:
Corresponding to a certain characteristic curve, there are following ODEs:
For the water hammer as shown in formula (1) and formula (2)
Along λ1Characteristic curve exists:
Integration type (58), obtains:
Similarly, along λ2Characteristic curve exists:
Above formula is integrated, is obtained:
For the boundary condition of conduit entrance, with (61) formula, the flow velocity at n+1 moment, pressure on first grid left margin
There are following relational expressions for the flow velocity and pressure at power and first grid element center n moment:
If the flow velocity on upstream edge interfaceOr pressureIt is known that then simultaneous (62) formula can solve boundary face
On parameter, further solve boundary face on flux.
For downstream boundary, with (59) formula, flow velocity on the last one grid right margin, pressure and the last one grid
There are following relational expressions for the flow velocity and pressure at center n moment:
If the flow velocity on downstream side interfaceOr pressureIt is known that then simultaneous (63) formula can solve boundary
Parameter on face further solves the flux in boundary face.
(2) derivation of open channel Shallow-water Flow boundary condition equation:
Boundary condition similar with pressure conduit, being imported and exported using similar method in the hope of open channel.For channel into
Mouthful boundary condition, the flow velocity of the flow velocity at n+1 moment on first grid left margin, the depth of water and first grid element center n moment
There are following relational expressions with the depth of water:
If the flow velocity on upstream edge interfaceOr the depth of waterIt is known that then simultaneous (64) formula can solve in boundary face
Parameter, further solve boundary face on flux.
For channel outlet border, when flow velocity, the depth of water and the last one grid element center n on the last one grid right margin
There are following relational expressions for the flow velocity and the depth of water at quarter:
If the flow velocity on downstream side interfaceOr the depth of waterIt is known that then simultaneous (65) formula can solve boundary face
On parameter, further solve boundary face on flux.
Step 6, it is limited to couple one-dimensional pressure conduit according to requirement of engineering for the inlet and outlet boundary condition equation based on derivation
Calculus of finite differences and one-dimensional pressure conduit finite volume method, or couple one-dimensional pressure conduit finite difference calculus and one-dimensional open channel limited bulk
Method, or the one-dimensional pressure conduit finite difference calculus of coupling and two-dimentional open channel finite volume method, or the one-dimensional pressure conduit finite difference of coupling
Point-score and three-dimensional open channel or pressure conduit finite volume method.
It is described as follows:
(1) one-dimensional pressure conduit finite difference calculus and one-dimensional pressure conduit finite volume method are coupled
The coupling of pressure conduit water hammer is calculated, reservoir-pipeline-valve system as shown in Figure 2 works as upstream and downstream
Outlet difierence equation (6) when finite difference calculus and finite volume method simulation is respectively adopted in pipeline, in simultaneous finite difference calculus
With the inlet boundary condition equation (62) in finite volume method, next information for calculating the moment on available coupling surface.
When upstream line is finite volume method calculating, and downstream is that finite difference calculus calculates, simultaneous finite volume method export boundary condition
Equation (63) and pressure conduit finite difference calculus import difierence equation (7), on available interface when next calculating
The parameter at quarter, then finite difference calculus and finite volume method are in the variable for solving respective interior zone node respectively.
(2) coupling of one-dimensional pressure conduit finite difference calculus and one-dimensional open channel finite volume method
Pond pipe-line system as shown in Figure 3, upstream pond are simulated using One Dimensional Finite volumetric method, and downstream line, which uses, to be had
Calculus of finite differences simulation is limited, on the coupling edge interface of finite volume method and finite difference calculus, there are one-dimensional open channel finite volume methods to go out
Mouth boundary condition invariant equation (65) and one-dimensional pressure conduit import finite difference calculus difierence equation (7), joint type
(65) and formula (7) can solve to obtain the unknown parameter on coupling boundary, and then finite volume method and finite difference calculus are respectively
Seek the unknown parameter for respectively calculating interior zone.
(3) coupling of one-dimensional pressure conduit finite difference calculus and two-dimentional open channel finite volume method
The coupling of one-dimensional pressure conduit finite difference calculus and two-dimentional open channel finite volume method as shown in Figure 4, is handed in coupling
In junction, by two-dimentional open channel outlet finite volume method boundary condition equation formula (65) and One Dimensional Finite conduit entrance finite difference calculus
Differential relationship formula (7) obtains the unknown parameter on coupling edge interface, then uses two-dimensional finite volumetric method and One Dimensional Finite calculus of finite differences
The parameter of each node inside pond and pipeline is sought respectively.
(4) coupling of one-dimensional pressure conduit finite difference calculus and three-dimensional open channel or pressure conduit finite volume method
It is illustrated in figure 5 the schematic diagram of One Dimensional Finite calculus of finite differences and the coupling of Three-D limited volumetric method, in one-dimensional model and three
Dimension module couples on interface, and there are the linear difference equation formulas of pressure and flow on interface for one-dimensional model finite difference calculus
(6), and in terms of threedimensional model, there are finite volume method inlet boundary condition equation (62), joint is solved and be can be obtained
The current pressure and flow velocity calculated on moment interface, and with this as one-dimensional calculating section and three-dimensional computations portion boundary item
Part solves zoning internal process.
After connection and solution formula (6) and formula (62), using flow velocity as the velocity boundary conditions of three-dimensional computations regional export, simultaneously
Using the barometric gradient of one-dimensional first and second calculate node last moments of region as the pressure side of three-dimensional computations regional export
Boundary's condition solves the process inside 3D region.Boundary condition by pressure (or flow velocity) as one-dimensional zoning import, is asked
Solve one-dimensional region interior flow field.
Incorporation engineering example is described as follows above-mentioned coupling technique:
(1) the utilization example of One Dimensional Finite calculus of finite differences and One Dimensional Finite volumetric method coupling model
It is illustrated in figure 6 certain power station schematic diagram, is run-of-river power station, three units share conduit pipe, in upper water
An Open Channel is had between library and unit for doing settling pit.For upper pond to settling pit import, settling pit exports to downstream
Reservoir is simulated using one-dimensional pressure conduit finite difference calculus, and the water-level fluctuation in settling pit uses one-dimensional open channel finite volume method mould
It is quasi-, desilting position and shape parameter can be studied to great fluctuation process parameter during load rejection and unit normal course of operation
The influence of minor swing parameter, to rationally determine settling pit parameter in engineering design process.
(2) the utilization example of One Dimensional Finite calculus of finite differences and two-dimensional finite volumetric method coupling model
As shown in Fig. 7 certain the left bank power plant being made of 6 units, 3 Hydraulic units, every two units share
Tail water discharge, every three Hydraulic unit tail water hole outlets are connected with same river.
Therefore, load rejection or the operation of other change workings in a certain Hydraulic unit, and can be by downstream river course
Water-level fluctuation interfere the normal operations of other Hydraulic unit units.With bank three Hydraulic units must synchronously simulating could care for
And the dynamic characteristic of whole system, while needing to simulate the fluctuation of the open channel shallow water in downstream river course, from upper pond to draft tube
Outlet solves water hammer and other electromechanical net boundary conditions using One Dimensional Finite calculus of finite differences, and tail water river uses two-dimensional finite body
Area method solves shallow water wave equation, can analyze influence of the water-level fluctuation to system Extreme Parameters and stability in river.
(3) the utilization example 1 of One Dimensional Finite calculus of finite differences and Three-D limited volumetric method coupling model
As shown in figure 8, in power station, in order to measure the pressure in pipeline, usually annular pipe device electric pressure pulsation.
The reason is that the diameter of pipeline is generally large in power station, on the same section of vertical axis the pressure of difference there may be compared with
Big difference, be using ring device the pressure of a point is measured directly on tube wall due to there are pressure fluctuation to reduce and
Insecure influence obtains the average pressure of a section by annular connection pipe device approximation, improves the reliable of pressure measurement
Property.
Since the direction of ring device conduit axis is vertical with the direction of water flow in tested pipeline, one in ring device
As be not present or only exist the water flow flowing of very little, but since ring device from tested pipeline connects different points, if company
There are larger pressure differences for contact, then can cause the variation of water flow in ring pipe, and compare with tested pipeline, circulating line
Diameter wants very little, it is thus possible to cause biggish change in flow by changes in flow rate existing in pipeline, so that measuring device be drawn
The water hammer pressure risen is introduced into the result of measurement, is run counter to desire instead.Using One Dimensional Finite calculus of finite differences simulation pipe-line system and
The process of Load Rejection of Hydraulic Turbine, using Three-D limited volumetric method analog loop shape dress set flowing of interior water flow during removal of load and
The variation of pressure can inquire into the reliability for inquiring into annular pressure measuring unit.
(4) the utilization example 2 of One Dimensional Finite calculus of finite differences and Three-D limited volumetric method coupling model
In diversion as shown in Figure 9 or the longer power station of tailwater tunnel, usually by more units and pressed using bifurcated pipe
Advocate pipe or tail water pipeline is connected.On the one hand, at the time of identical, different units are likely to be at different operating statuses, than
If any unit operate normally, and other units are in removal of load process, cause to show complicated fluidised form in bifurcated pipe, another
Aspect due to the pipeline in bifurcated pipe there are variable cross-section, and branches off point so that the propagation of pressure shows complicated reflection, superposition
Phenomena such as, so in influence system other parameters variation.
The method coupled using One Dimensional Finite calculus of finite differences with Three-D limited volumetric method, calculate the power station containing bifurcated pipe due to
Hydraulic Transient process caused by load rejection, as shown in figure 9, bifurcated pipe is simulated using Three-D limited volumetric method in system, other
Pipeline, surge-chamber, unit are all made of the simulation of One Dimensional Finite calculus of finite differences.
Claims (1)
1. the multidimensional Hydraulic Power System transition analogy method coupled based on finite difference calculus with finite volume method, it is characterised in that including
Following steps:
Step 1, using the discrete one-dimensional non-constant flow control of pressure conduit of finite difference calculus -4, the space Pressimann implicit schemes
Equation processed；
Step 2, there is one-dimensional pressure conduit unsteady flow governing equation using finite volume method-Godunov format is discrete；
Step 3, governing equation is fluctuated using the discrete peacekeeping two dimension open channel shallow water of finite volume method-Godunov format；
Step 4, Three-dimensional Flow equation is solved based on the solver of OpenFOAM using finite volume method-；
Step 5, the boundary condition equation for deriving pressure conduit flowing and Open Channels respectively, as finite difference calculus and limited body
The boundary condition of area method coupling；Specifically:
The characteristic value and corresponding right feature vector of demand solution pressure conduit unsteady flow governing equation, list equation variable with
Differential relationship between right feature vector obtains boundary face and calculates moment variable and grid along characteristic curve integral differential equation
Linear relation known to center between moment variable, as head in pressure conduit finite volume method inlet and outlet boundary face
With the relational expression of flow velocity, it is denoted as pressure conduit inlet and outlet boundary condition equation relational expression herein；
Using same method, the boundary condition equation of Open Channels, demand obtain the fluctuation finite volume method import of open channel shallow water and
The relational expression of head and flow velocity on outlet border face is denoted as open channel inlet and outlet boundary condition equation herein；
Step 6, it based on the boundary condition equation of derivation, couples one-dimensional pressure conduit finite difference calculus and one-dimensional pressure conduit is limited
Volumetric method, or one-dimensional pressure conduit finite difference calculus and one-dimensional open channel finite volume method are coupled, or the one-dimensional pressure conduit of coupling has
Limit calculus of finite differences and two-dimentional open channel finite volume method, or the one-dimensional pressure conduit finite difference calculus of coupling and Complex Flows limited bulk
Method；Specifically include following method:
(1) simultaneous has one-dimensional pressure conduit outlet difference equation and one-dimensional pressure conduit inlet boundary conditional equation or one-dimensional
Pressure conduit export boundary condition equation and one-dimensional pressure conduit import difference equation, solution obtain the flow velocity and pressure in boundary face
Power further finds out finite difference calculus calculating as the boundary condition in finite difference calculus and finite volume method coupling interface
Region, each node in finite volume method zoning inside and the intracorporal head of control and flow velocity, realize that one-dimensional pressure conduit is limited
The coupling of calculus of finite differences and one-dimensional pressure conduit finite volume method；
(2) the one-dimensional pressure conduit outlet difference equation of simultaneous and one-dimensional open channel inlet boundary conditional equation or one-dimensional open channel go out
Mouthful boundary condition equation and one-dimensional pressure conduit import difference equation, solution obtain the flow velocity and the depth of water in boundary face, as having
The boundary condition in calculus of finite differences and finite volume method coupling interface is limited, finite difference calculus zoning is further found out, it is limited
Each node and the intracorporal depth of water of control and flow velocity, realize one-dimensional pressure conduit finite difference calculus and one inside volumetric method zoning
Tie up the coupling of open channel finite volume method；
(3) simultaneous has one-dimensional pressure conduit outlet difference equation and two-dimentional open channel inlet boundary conditional equation, or two-dimentional open channel
Export boundary condition equation and one-dimensional pressure conduit import difference equation, solution obtain the flow velocity and the depth of water in boundary face, as
Boundary condition in finite difference calculus and finite volume method coupling interface, further finds out finite difference calculus zoning, has
Limit each node inside volumetric method zoning and control intracorporal flow velocity and the depth of water, realize one-dimensional pressure conduit finite difference calculus and
The coupling of two-dimentional open channel finite volume method；
(4) simultaneous one-dimensional pressure conduit outlet difference equation and three-dimension complex flow inlet boundary conditional equation or one-dimensional have
Pressure pipeline import difference equation and three-dimension complex flow export boundary condition equation, solution obtain the flow velocity and pressure in boundary face
Power, then finds out finite difference calculus zoning according to the flow velocity and pressure that find out in boundary face respectively, and finite volume method calculates
Each node and the intracorporal flow velocity of control and pressure, realize one-dimensional pressure conduit finite difference calculus and Three-D limited volume inside region
The coupling of method.
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