CN109165402A - A method of solving general power function shape open channel optimal hydraulic cross-section - Google Patents

A method of solving general power function shape open channel optimal hydraulic cross-section Download PDF

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CN109165402A
CN109165402A CN201810756137.8A CN201810756137A CN109165402A CN 109165402 A CN109165402 A CN 109165402A CN 201810756137 A CN201810756137 A CN 201810756137A CN 109165402 A CN109165402 A CN 109165402A
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open channel
power function
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optimal hydraulic
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韩延成
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University of Jinan
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    • GPHYSICS
    • G06COMPUTING; CALCULATING; COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
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    • G06F30/20Design optimisation, verification or simulation
    • GPHYSICS
    • G06COMPUTING; CALCULATING; COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00Computer-aided design [CAD]
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    • GPHYSICS
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    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00Details relating to CAD techniques
    • G06F2111/10Numerical modelling

Abstract

The invention discloses a kind of methods for solving general power function shape open channel optimal hydraulic cross-section, it indicates the section configuration equation of general power function shape open channel the following steps are included: step 1;Step 2, the hydraulic elements of open channel section are solved;Step 3, the optimal hydraulic cross-section model of open channel is established;Step 4, the wetted perimeter of open channel section is described using Gaussian hypergeometric function expression formula;Step 5, the optimal solution of optimal hydraulic cross-section model is solved.The present invention states the wetted perimeter of general power function shape open channel using Gaussian hypergeometric function, the optimal solution of general power function shape open channel wetted perimeter is derived using method of Lagrange multipliers, the accurate solution of power function shape open channel optimal hydraulic cross-section when can not only quickly obtain aleatory variable, and significantly improve solving speed and precision.

Description

A method of solving general power function shape open channel optimal hydraulic cross-section
Technical field
The present invention relates to a kind of methods for solving general power function shape open channel optimal hydraulic cross-section, belong to irrigated area canal for water conveyance Planning and designing technical field.
Background technique
It is well known that canal cross section has a major impact conveyance capacity, construction cost, therefore canal cross section optimization design is One of important content of channel designing.Optimal hydraulic cross-section be area of passage or wetted perimeter it is certain in the case where, pass through maximum flow Section (or in the case that flow is certain, area of passage or the smallest section of wetted perimeter), conveyance capacity can not only be made maximum, also Construction cost can be made to tend to be minimum, be widely used in channel designing, be the basis of channel designing.Known trapezoidal channel Optimal hydraulic cross-section be(wherein b is bottom width, and h is the depth of water, and m is slope coefficient).
Scholars generally believe power function shape canal cross section (y=a | x |k, k is variable) and it has the following advantages: (1) power function Shape section is the common version of various parabolic-shaped sections;(2) power function shape section can be fitted various natures or artificial canal Shape.However since power function shape section k is variable, complexity is solved, therefore, existing research generally assumes that k takes specified value (k=1,1.5,2,3) then establish a set of model solution optimal section respectively.In terms of presently disclosed research achievement, power function There is also following defects for wetted perimeter and the normal depth of flow calculating of the optimal hydraulic fracture of shape section:
(1) according to mathematical knowledge, these parabolic-shaped sections (k=1,1.5,2,3) are one kind of power function shape section Special shape.The common version of power function shape section is y=a | x |k(x is abscissa, and y is ordinate, and a is form factor, and k is The index of power function).It is traditional square parabolic-shaped section as the index k=2 of power function, is half cube of throwing as k=1.5 The linear section of object.Current research is to establish a set of solution waterpower respectively for specific several index k (k=1,2,3) Method of optimal section, such as Wei Wen gift etc. (2006) establish optimal models with extremum method, obtain semi-cubical parabola shape section Optimal hydraulic cross-section, Han (2016) with ellptic integral function method solve wetted perimeter, and then using method of Lagrange multipliers establish most Excellent model, and obtain the optimal hydraulic cross-section of cubic parabola shape section.These methods are all not suitable for any k value (such as k= 1.8,2.2 etc.).How a set of unified mathematical model is established, the water of power function shape canal cross section when can solve any k value Power optimal solution is a problem.If optimal hydraulic when a set of simple and practical method solves any k value can be established to break Face, so that it may so that problem is simplified.
(2) current research is just for specific several index k.Have result of study is that a few is parabola shaped Open channel optimal hydraulic cross-section includes k=1.5,2.0,3.0, not can solve the optimal section problem of arbitrary characteristics k.Because of canal When road designs, suitable section configuration not only needs to consider conveyance capacity, it is also necessary to consider hydrogeologic condition, construction cost etc. Factor.Such as k=1.8 (y=ax1.8) parabolic-shaped section may be more suitable for the hydrogeologic condition in a region, and k= 2.2 (y=ax2.2) parabolic-shaped section may be more suitable for the hydrogeologic condition in another region.But there is presently no k= Research achievement when 1.8 and k=2.2, also without k=2.1,2.2,2.3,2.6,2.7,2.8,2.9,3.1 ... when it is optimal Section.
(3) difficult point that power function shape optimal hydraulic cross-section solves is wetted perimeter.Conventional method is numerical integrating, needs to compile Processing procedure sequence, difficulty are big.It is difficult to promote in engineering design or management practice, it is also difficult to quickly estimate wetted perimeter.Therefore compel to be essential Easy algorithm is wanted, the algorithm of result can be especially obtained by hand computation, it can be to avoid calculating hypergeometric function or numerical value product Point, it is provided conveniently for engineer application.
(4) it is well known that being non-linear relation between the open channel uniform flow depth of water and flow, when according to the flow rate calculation depth of water, Generally to be obtained by drawing or solution nonlinear equation.To power function shape open channel, drawing is not only needed through integration method meter Calculate wetted perimeter, and low precision, it is difficult to meet engineering practice.The method of solution nonlinear equation is also required to through numerical integration method meter Wetted perimeter is calculated, the calculating for solving normal depth of flow is extremely difficult, is not easy to engineer application.Therefore, there is an urgent need to not have to integral, without Nonlinear equation is solved, the algorithm that can be obtained by normal depth of flow by artificial hand computation is simple, to meet engineering practice needs.
Summary of the invention
In view of the deficiencies of the prior art, the invention proposes a kind of solution general power function shape (y=a | x |k, k is variable) The method of open channel optimal hydraulic cross-section can quickly obtain power function shape open channel optimal hydraulic cross-section when k is aleatory variable Accurate solution, hence it is evident that improve solving speed and precision.
The present invention solves its technical problem and adopts the technical scheme that:
It is provided in an embodiment of the present invention it is a kind of solve general power function shape open channel optimal hydraulic cross-section method, it include with Lower step:
Step 1, the section configuration equation of general power function shape open channel is indicated;
Step 2, the hydraulic elements of open channel section are solved;
Step 3, the optimal hydraulic cross-section model of open channel is established;
Step 4, the wetted perimeter of open channel section is described using Gaussian hypergeometric function expression formula;
Step 5, the optimal solution of optimal hydraulic cross-section model is solved;
Step 6, a kind of algorithm of rapid solving power function shape open channel optimal hydraulic cross-section is provided;
Step 5 includes following procedure:
It (1) is the partial differential equation for solving power function optimal section by optimal hydraulic cross-section model conversation;
(2) one is converted by partial differential equation to refer to about breadth depth ratio η (ratio of water surface width and the depth of water) and power function The universal equation of number k, obtains the optimal hydraulic cross-section parameter (optimal breadth depth ratio η) when any given k;
(3) parameter of other optimal hydraulic cross-sections is solved.
As a kind of possible implementation of the present embodiment, in step 1, the section shape of the general power function shape open channel Shape equation is indicated using following formula:
Y=a | x |k (1)
In formula, a is the form factor of open channel section, and x is abscissa (m), and y is ordinate (m), and k is index, and k >=1.
As a kind of possible implementation of the present embodiment, the detailed process of the step 2 are as follows:
If when x=B/2, y=h can then obtain the relationship of water surface width B and form factor a according to formula (1):
In formula, h is the depth of water (m), and B is water surface width (m);
The grade of side slope at the water surface can be obtained according to formula (1) are as follows:
In formula, Z is the grade of side slope at the water surface;
The area A of the cross-section of river is obtained according to formula (1) and general power function shape open channel section structure are as follows:
The wetted perimeter P integral representation of general power function shape open channel section are as follows:
As a kind of possible implementation of the present embodiment, in step 3, the optimal hydraulic cross-section model of the open channel Are as follows:
Objective function is that area of passage is minimum, it may be assumed that
It minimizes:
Constraint condition meets Manning formula between uniform flow condition down-off and cross dimensions:
In formula, it is the function about h, B, k that Φ, which is constraint condition function,.Q is flow (m3/ s), n is roughness, and i is canal bottom Longitudinal slope (m/m).
As a kind of possible implementation of the present embodiment, in step 4, the Gaussian hypergeometric function of open channel section wetted perimeter Expression formula are as follows:
In formula, G1It is the Gaussian hypergeometric function about parameter B, k and h, G1Concrete form indicate are as follows:
If dimensionless group η=B/h, then form factor a, discharge section area A and wetted perimeter P are respectively indicated are as follows:
A=2kη-kh1-k (11)
In formula,
As a kind of possible implementation of the present embodiment, the detailed process of the step 5 the following steps are included:
(1) partial differential equation of power function optimal section derive
According to the objective function peace treaty for the optimal hydraulic cross-section model for optimizing method of Lagrange multipliers theory and open channel Beam condition constructs a new LagrangianL:
It minimizes L=A (η, h, k)+λ Φ (η, h, k) (15)
In formula, L is Lagrangian, is the function about h, B, k;λ is Lagrange multiplier;
According to method of Lagrange multipliers, formula (15) are indicated are as follows:
It is obtained by formula (16) λ that disappears:
Formula (18), (19) substitution formula (17) is available
Due to A (h, η, k) > 0, feasible solution only has
Formula (21) is to solve the partial differential equation of power function optimal hydraulic cross-section.
(2) derivation of power function shape section optimal hydraulic cross-section universal equation
Derivative of the A (h, η, k) about η and h are as follows:
Partial derivative of the P (h, η, k) (formula (13)) about η and h are as follows:
In formula, G1,G2For Gaussian hypergeometric function, they are expressed as
Formula (22), (23), (24) and (25) substitution formula (21) is obtained:
Formula (28) be solve power function open channel section (y=a | x |k) optimal hydraulic cross-section of k when being arbitrary value accurately solve General formula.Method particularly includes: any k is given, formula (28) is solved and obtains the accurate solution of optimal hydraulic cross-section breadth depth ratio η.Such as k When=1.5, k=1.5 is substituted into formula (28), obtains equation after simplifyingIt solves equation and obtains water Parameter η=B/h=2.0186 of power optimal section.The result of k=1.0~6.0 is shown in embodiment table 1;
(3) parameter of other optimal hydraulic cross-sections is solved
η substitution formula (11) is obtained into optimum shape coefficient a, by η and k substitute into formula (12) and (13) obtain area of passage A with Wetted perimeter P.Specific method is shown in embodiment.
Compared with prior art, the present invention regards k as variable, and complicated power function open channel optimal hydraulic cross-section problem is final The unified equation (formula (28)) for solving power function optimal hydraulic cross-section at one has been derived, this equation has been solved and can be obtained by and appoint Optimal hydraulic cross-section when given k value of anticipating, makes the solution of power function open channel optimal hydraulic cross-section become more simple.It avoids pair Any k requires to repeat the complex process that step 1 establishes Optimized model and derivation equation to step 5.
Further, the present invention provides a kind of explicit algorithm of rapid solving power function shape open channel optimal hydraulic cross-section.
According to formula (28), according to formula (28), can be obtained corresponding optimal hydraulic cross-section when k=1.0~6.0 parameter η (see Embodiment table 1), obtaining the relationship between k and η using optimization of curve fitting method is
Compared with prior art, invention further provides an explicit approximate formula (29), to any given k value, It substitutes into formula (29), optimal breadth depth ratio η can be readily available, further simplify the optimal hydraulic fracture solution side of power function Method keeps solution more simple, using calculator with regard to achievable.Compared with solving equation method (formula (28)), the phase of formula (29) To error between 0.0003% and 0.087%, illustrate that formula (29) has good precision.
Further, the explicit algorithm of a kind of easy power function shape open channel section wetted perimeter provided by the invention:
To formula (6), using Gauss Legendre's integration method, the three point scheme of available power function open channel section wetted perimeter Approximate algorithm:
In formula, the precision that B can obtain formula (30) by formula (2) is examined by following methods: taking a= 0.4, k=1.5-3.0, step-length 0.5, h=0.5-4.0.Theoretical value is calculated by (6) formula or (9) formula, and the two result is identical.It is right Than the result shows that, the relative error of formula (30) is between 0.0003% and 0.087%.Show that formula (30) has good essence Degree.Its advantage is that can be by hand computation, it can be to avoid hypergeometric function or numerical integration be calculated, for the engineer application side of providing Just.
Further, the iterative algorithm of a kind of power function normal depth of flow provided by the invention:
It is non-linear relation (formula (8)) between the open channel uniform flow depth of water and flow, when according to the flow rate calculation depth of water, generally wants It is obtained by drawing or solution nonlinear equation.Drawing not only needs to calculate wetted perimeter, but also low precision by integration method, it is difficult to Meet engineering practice.The method of solution nonlinear equation is also required to calculate wetted perimeter by numerical integration method, solves normal depth of flow It calculates extremely difficult, is not easy to engineer application.When solution of the invention: formula (30) and (5) being substituted into formula (8), constructed A kind of iterative algorithm of normal depth of flow are as follows:
Wherein,
hj+1For+1 iterative value (m) of normal depth of flow jth, h is iteration j value (m).
What the technical solution of the embodiment of the present invention can have has the beneficial effect that:
To overcome the problems, such as that power function shape open channel optimal hydraulic cross-section solves difficult and low precision, technology of the embodiment of the present invention Scheme derives general power using method of Lagrange multipliers using the wetted perimeter of Gaussian hypergeometric function statement general power function shape open channel The optimal solution of function shape open channel wetted perimeter is proposed a kind of combined using Gaussian hypergeometric function and method of Lagrange multipliers and quickly asked The method that accurately solves of solution power function, solve existing method low precision, discontinuous while being 1 (such as water surface width and depth ratio) and Complicated problem is calculated, the accurate solution of power function shape open channel optimal hydraulic cross-section when can not only quickly obtain aleatory variable, And significantly improve solving speed and precision.
Existing research method is to regard k value as constant, for different k values, establishes one respectively using different methods Method for solving is covered, and the present invention provides a kind of unified method for solving.The present invention regards k as variable, by complicated power function Open channel optimal hydraulic cross-section problem has finally derived the unified equation that power function optimal hydraulic cross-section is solved at one, solves this Equation can be obtained by optimal hydraulic cross-section when any k value, become the solution of power function open channel optimal hydraulic cross-section more Simply, the complex process that complex model and solution are all established to any k is avoided.According to the unification of power function optimal hydraulic cross-section The explicit algorithm for the rapid solving power function shape open channel optimal hydraulic cross-section that equation and curve-fitting method obtain, power function waterpower is most The solution of excellent section is further simplified, and optimal breadth depth ratio just can be obtained using calculator.
Detailed description of the invention
Fig. 1 is a kind of side for solving general power function shape open channel optimal hydraulic cross-section shown according to an exemplary embodiment The flow chart of method;
Fig. 2 is the section configuration schematic diagram of power function shape open channel shown according to an exemplary embodiment;
The section configuration of power function shape open channel when Fig. 3 is different index k shown according to an exemplary embodiment is illustrated Figure;
Fig. 4 is a kind of relation curve of the optimal breadth depth ratio shown according to an exemplary embodiment with power function index variation Figure.
Specific embodiment
In order to clarify the technical characteristics of the invention, below by specific embodiment, and its attached drawing is combined, to this hair It is bright to be described in detail.Following disclosure provides many different embodiments or example is used to realize different knots of the invention Structure.In order to simplify disclosure of the invention, hereinafter the component of specific examples and setting are described.In addition, the present invention can be with Repeat reference numerals and/or letter in different examples.This repetition is that for purposes of simplicity and clarity, itself is not indicated Relationship between various embodiments and/or setting is discussed.It should be noted that illustrated component is not necessarily to scale in the accompanying drawings It draws.Present invention omits the descriptions to known assemblies and treatment technology and process to avoid the present invention is unnecessarily limiting.
Fig. 1 is a kind of side for solving general power function shape open channel optimal hydraulic cross-section shown according to an exemplary embodiment The flow chart of method.As shown in Figure 1, a kind of method of solution general power function shape open channel optimal hydraulic cross-section of the present embodiment, it It may comprise steps of:
Step 1, the section configuration equation of general power function shape open channel is indicated.
Fig. 2 is the section configuration schematic diagram of power function shape open channel shown according to an exemplary embodiment.The general power The section configuration equation of function shape open channel can be indicated using following formula:
Y=a | xk| (1)
In formula, a is the form factor of open channel section, and x is abscissa (m), and k is index, and k >=1, y are ordinate (m).
Section configuration when k is different value is as shown in Figure 3.When k=1, section configuration is common triangular section, k It is common square parabolic-shaped section when=2.Power function shape section is the common version of various parabolic-shaped sections, and k value can Be it is any be greater than 1 value, therefore can produce countless form of fracture.In addition power function shape section can be fitted it is various natural or Artificial canal's shape.
Step 2, the hydraulic elements of open channel section are solved.
According to formula (1) and Fig. 2 it is found that as x=B/2, y=h can then obtain the relationship of water surface width B and form factor a:
In formula, h is the depth of water (m), and B is water surface width (m);
The grade of side slope at the water surface can be obtained according to formula (1) are as follows:
Grade of side slope in formula, at the Z=water surface;
The area A of the cross-section of river is obtained according to formula (1) and general power function shape open channel section structure shown in Fig. 2 are as follows:
The wetted perimeter P integral representation of general power function shape open channel section are as follows:
Step 3, the optimal hydraulic cross-section model of open channel is established.
In hydraulic engineering, optimal hydraulic cross-section be defined as area of passage or wetted perimeter it is certain in the case where, pass through maximum flow Section or flow it is certain in the case where, area of passage or the smallest section of wetted perimeter.The present embodiment uses the latter.It needs to illustrate , it is the same that two kinds of definition, which solve obtained final result,.
The optimal hydraulic cross-section model of the open channel are as follows:
Objective function is that area of passage is minimum, it may be assumed that
It minimizes:
Constraint condition meets Manning formula between uniform flow condition down-off and cross dimensions:
In formula, Φ is constraint condition function, and Q is flow (m3/ s), n is roughness, and i is canal bottom longitudinal slope (m/m).
Step 4, the wetted perimeter of open channel section is described using Gaussian hypergeometric function expression formula.
The Gaussian hypergeometric function expression formula of open channel section wetted perimeter are as follows:
In formula, G1It is the Gaussian hypergeometric function about parameter k, B and h, G1Concrete form indicate are as follows:
If dimensionless group η=B/h, then form factor a, discharge section area A and wetted perimeter P are respectively indicated are as follows:
A=2kη-kh1-k (11)
In formula,
Step 5, the optimal solution of optimal hydraulic cross-section model is solved.
(1) partial differential equation of power function optimal section derive
According to the optimal hydraulic cross-section for optimizing open channel shown in method of Lagrange multipliers theory and formula (7) and formula (8) The objective function and constraint condition of model construct a new LagrangianL:
It minimizes L=A (η, h, k)+λ Φ (η, h, k) (15)
In formula, L is Lagrangian, is the function about h, B, k;λ is Lagrange multiplier;
According to method of Lagrange multipliers, formula (15) are indicated are as follows:
It is obtained by formula (16) λ that disappears:
Formula (18), (19) substitution formula (17) is available
Due to A (h, η, k) > 0, feasible solution only has
Formula (21) is to solve the partial differential equation of power function optimal hydraulic cross-section.
(2) derivation of power function shape section optimal hydraulic cross-section universal equation
Derivative of the A (h, η, k) about η and h are as follows:
Partial derivative of the P (h, η, k) (formula (13)) about η and h are as follows:
In formula, G1,G2For Gaussian hypergeometric function, they are expressed as
Formula (22), (23), (24) and (25) substitution formula (21) is obtained:
Formula (28) be solve power function open channel section (y=a | x |k) optimal hydraulic cross-section of k when being arbitrary value accurately solve General formula.
(3) parameter of other optimal hydraulic cross-sections is solved
η substitution formula (11) is obtained into optimum shape coefficient a, by η and k substitute into formula (12) and (13) obtain area of passage A with Wetted perimeter P.Specific method is shown in embodiment.
Compared with prior art, the present invention regards k as variable, and complicated power function open channel optimal hydraulic cross-section problem is final The unified equation (formula (28)) for solving power function optimal hydraulic cross-section at one has been derived, this equation has been solved and can be obtained by and appoint Optimal hydraulic cross-section when given k value of anticipating, makes the solution of power function open channel optimal hydraulic cross-section become more simple.It avoids pair Any k requires to repeat the complex process that step 1 establishes Optimized model and derivation equation to step 5.
By taking k=1.5 as an example.K=1.5 is substituted into formula (28), obtains equation after simplifying
Solution formula obtains optimal breadth depth ratio when k=1.5
η=B/h=2.0186
This result is identical as using result obtained from the complex processes such as modeling, optimization.By η=B/h=2.0186 generation Enter the available explicit algorithm formula for solving optimum shape coefficient of formula (11) are as follows:
A=0.98621h1-k=0.98621h-0.5
η and k is substituted into formula (12) and (13) available explicit algorithm formula for solving area of passage and wetted perimeter are as follows:
A=1.21116h2, P=2.89272h
By A=1.21116h2Formula (8) are substituted into P=2.89272h, available optimal section calculates flow according to the depth of water Explicit algorithm formula be
Solve the explicit algorithm formula for obtaining solving normal depth of flow h are as follows:
H=1.15698 φ3/8
In formula,By h=1.15698 φ3/8Substitution formula (3), it is available that shape system is calculated according to flow Q The explicit algorithm formula of number a are as follows:
A=0.91686 φ-3/16
With same method, any k value (k=1.0 herein, 1.5,2.0,2.3,3.0,3.5,4.0) is substituted into formula (28), And then the optimal hydraulic fracture breadth depth ratio of power function and other parameters when available different value of K, as shown in table 1.
The calculated result of power function open channel optimal hydraulic cross-section when 1 different value of K of table
In table 1,μ=φ3/8
A kind of algorithm of rapid solving power function shape open channel optimal hydraulic cross-section provided by the invention:
The parameter η of corresponding optimal hydraulic cross-section when k=1.0~6.0 can be obtained, answer according to formula (28) according to formula (28) Obtaining the relationship between k and η with optimization of curve fitting method is
Compared with prior art, invention further provides an explicit approximate formula (29), to any given k value, It substitutes into formula (29), optimal breadth depth ratio η can be readily available, further simplify the optimal hydraulic fracture solution side of power function Method keeps solution more simple, using calculator with regard to achievable.Compared with solving equation method (formula (28)), the phase of formula (29) To error between 0.0003% and 0.087%, illustrate that formula (29) has good precision.
A kind of explicit algorithm of easy power function shape open channel section wetted perimeter provided by the invention:
To formula (6), using Gauss Legendre's integration method, the three point scheme of available power function open channel section wetted perimeter Approximate algorithm:
In formula, the precision that B can obtain formula (30) by formula (2) is examined by following methods: taking a= 0.4, k=1.5-3.0, step-length 0.5, h=0.5-4.0.Theoretical value is calculated by (6) formula or (9) formula, and the two result is identical.It is right Than the result shows that, the relative error of formula (30) is between 0.0003% and 0.087%.Show that formula (30) has good essence Degree.Its advantage is that can be by hand computation, it can be to avoid hypergeometric function or numerical integration be calculated, for the engineer application side of providing Just.
A kind of iterative algorithm of power function normal depth of flow provided by the invention:
It is non-linear relation (formula (8)) between the open channel uniform flow depth of water and flow, when according to the flow rate calculation depth of water, generally wants It is obtained by drawing or solution nonlinear equation.Drawing not only needs to calculate wetted perimeter, but also low precision by integration method, it is difficult to Meet engineering practice.The method of solution nonlinear equation is also required to calculate wetted perimeter by numerical integration method, solves normal depth of flow It calculates extremely difficult, is not easy to engineer application.When solution of the invention: formula (30) and (5) being substituted into formula (8), constructed A kind of iterative algorithm of normal depth of flow are as follows:
Wherein,
hj+1For+1 iterative value (m) of normal depth of flow jth, h is iteration j value (m).
Its advantage is that not having to integral, without nonlinear equation is solved, normal depth of flow can be obtained by by artificial hand computation, Algorithm is simple, can preferably meet engineering practice.
2 specific examples of the invention are described below:
(1) one power function shape open channel, Q=11.0m3/ s, i=1/11000, n=0.014. is according to Analysis of Slope Stability k =1.8, which can satisfy requirement of engineering, substitutes into formula for k=1.8:
Available optimal breadth depth ratio η=2.0407 of theory.Substitute into the Simple calculating formula of formulaAvailable η=2.0408.It can be seen that the two error is very small.
(2) one power function shape open channel y=axk, k=2.1, Q=8.0m3/ s, i=1/9000, n=0.014. substitute into k public Formula Simple calculating formulaAvailable η=2.0627, and substitute into public Formula:
It can be with Obtain theoretical value η=2.0626.It can be seen that the two error is very small.
The present invention is according to equation power function optimal hydraulic cross-section equation, using Optimization Method for Fitting, has obtained according to any k value Calculate the explicit algorithm of optimal hydraulic fracture breadth depth ratio ηFurther The optimal hydraulic fracture method for solving of power function is simplified, keeps solution more simple, can be completed using calculator.
The present invention provides a kind of explicit algorithms of easy power function shape open channel section wetted perimeter, analyze the result shows that its phase To error between 0.0003% and 0.087%, there is good precision.Its advantage is that can be by hand computation, it can be to avoid meter Hypergeometric function or numerical integration are calculated, is provided convenience for engineer application
The present invention provides a kind of iterative algorithm of power function normal depth of flow, overcomes existing drawing and solves non-linear side The shortcomings that journey.Because drawing not only needs to calculate wetted perimeter, but also low precision by integration method to power function shape open channel, it is difficult to Meet engineering practice.The method of solution nonlinear equation is also required to calculate wetted perimeter by numerical integration method, solves normal depth of flow It calculates extremely difficult, is not easy to engineer application.This method have the advantage that not having to integral, without nonlinear equation is solved, pass through Artificial hand computation can be obtained by normal depth of flow, and algorithm is simple, can preferably meet engineering practice.
The above is the preferred embodiment of the present invention, for those skilled in the art, Without departing from the principles of the invention, several improvements and modifications can also be made, these improvements and modifications are also regarded as this hair Bright protection scope.

Claims (10)

1. a kind of method for solving general power function shape open channel optimal hydraulic cross-section, characterized in that the following steps are included:
Step 1, the section configuration equation of general power function shape open channel is indicated;
Step 2, the hydraulic elements of open channel section are solved;
Step 3, the optimal hydraulic cross-section model of open channel is established;
Step 4, the wetted perimeter of open channel section is described using Gaussian hypergeometric function expression formula;
Step 5, the optimal solution of optimal hydraulic cross-section model is solved.
2. a kind of method for solving general power function shape open channel optimal hydraulic cross-section as described in claim 1, characterized in that In step 1, the section configuration equation of the general power function shape open channel is indicated using following formula:
Y=a | x |k (1)
In formula, a is the form factor of open channel section, and x is abscissa, and y is ordinate, and k is index, and k >=1.
3. a kind of method for solving general power function shape open channel optimal hydraulic cross-section as claimed in claim 2, characterized in that institute State the detailed process of step 2 are as follows:
If when x=B/2, y=h can then obtain the relationship of water surface width B and form factor a according to formula (1):
In formula, h is the depth of water (m), and B is water surface width (m);
The grade of side slope at the water surface can be obtained according to formula (1) are as follows:
In formula, Z is the grade of side slope at the water surface;
The area A of the cross-section of river is obtained according to formula (1) and general power function shape open channel section structure are as follows:
The wetted perimeter P integral representation of general power function shape open channel section are as follows:
4. a kind of method for solving general power function shape open channel optimal hydraulic cross-section as claimed in claim 3, characterized in that In step 3, the optimal hydraulic cross-section model of the open channel are as follows:
Objective function is that area of passage is minimum, it may be assumed that
It minimizes:
Constraint condition meets Manning formula between uniform flow condition down-off and cross dimensions:
In formula, Φ is constraint condition function, is the function about h, B, k, and Q is flow, and n is roughness, and i is canal bottom longitudinal slope.
5. a kind of method for solving general power function shape open channel optimal hydraulic cross-section as claimed in claim 4, characterized in that In step 4, the Gaussian hypergeometric function expression formula of open channel section wetted perimeter are as follows:
In formula, G1It is the Gaussian hypergeometric function about parameter B, k and h, G1Concrete form indicate are as follows:
If dimensionless group η=B/h, then form factor a, discharge section area A and wetted perimeter P are respectively indicated are as follows:
A=2kη-kh1-k (11)
In formula,
6. a kind of method for solving general power function shape open channel optimal hydraulic cross-section as claimed in claim 5, characterized in that institute State the detailed process of step 5 the following steps are included:
It step 51, is the partial differential equation of a solution power function optimal section by optimal hydraulic cross-section model conversation;
Step 52, a universal equation about breadth depth ratio η and power function index k is converted by partial differential equation, obtained any Optimal hydraulic breadth depth ratio η when given k;
Step 53, the parameter of other optimal hydraulic cross-sections is solved.
7. a kind of method for solving general power function shape open channel optimal hydraulic cross-section as claimed in claim 6, characterized in that institute The process that the detailed process of step 51 derives for the partial differential equation of power function optimal section is stated, specific as follows:
According to the objective function for the optimal hydraulic cross-section model for optimizing method of Lagrange multipliers theory and open channel and constraint item Part constructs a new LagrangianL:
It minimizes L=A (η, h, k)+λ Φ (η, h, k) (15)
In formula, L is Lagrangian, is the function about h, B, k;λ is Lagrange multiplier;According to Lagrange multiplier Method indicates formula (15) are as follows:
It is obtained by formula (16) λ that disappears:
Formula (18), (19) substitution formula (17) is available
Due to A (h, η, k) > 0, feasible solution only has
Formula (21) is to solve the partial differential equation of power function optimal hydraulic cross-section.
8. a kind of method for solving general power function shape open channel optimal hydraulic cross-section as claimed in claim 7, characterized in that institute The detailed process for stating step 52 is the derivation process of power function shape section optimal hydraulic cross-section universal equation, specific as follows:
Derivative of the A (h, η, k) about η and h are as follows:
Partial derivative of the P (h, η, k) (formula (13)) about η and h are as follows:
In formula, G1,G2For Gaussian hypergeometric function, they are expressed as
Formula (22), (23), (24) and (25) substitution formula (21) is obtained:
Formula (28) be solve power function open channel section (y=a | x |k) k be arbitrary value when optimal hydraulic cross-section accurately solve it is general Formula, i.e. basis give any k solution formula (28) and obtain the accurate solution of optimal hydraulic cross-section breadth depth ratio η.
9. a kind of method for solving general power function shape open channel optimal hydraulic cross-section as claimed in claim 8, characterized in that institute The detailed process for stating step 53 is to solve the process of other optimal hydraulic cross-section parameters, specific as follows:
η substitution formula (11) is obtained into optimum shape coefficient a, η and k is substituted into formula (12) and (13) and obtains area of passage A and wetted perimeter P。
10. a kind of explicit algorithm of rapid solving power function shape open channel optimal hydraulic cross-section, characterized in that including following procedure
According to the formula (28) in claim 8, the parameter η of corresponding optimal hydraulic cross-section when k=1.0~6.0 can be obtained, use Optimization of curve fitting method constructs k and the relationship of η is
Using formula (29), gives any k value and just obtain optimal breadth depth ratio η.
CN201810756137.8A 2018-07-11 2018-07-11 A method of solving general power function shape open channel optimal hydraulic cross-section Pending CN109165402A (en)

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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN109632256A (en) * 2019-01-14 2019-04-16 中国科学院、水利部成都山地灾害与环境研究所 A kind of special-shaped section experimental tank design method and its application

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN109632256A (en) * 2019-01-14 2019-04-16 中国科学院、水利部成都山地灾害与环境研究所 A kind of special-shaped section experimental tank design method and its application
CN109632256B (en) * 2019-01-14 2020-06-02 中国科学院、水利部成都山地灾害与环境研究所 Design method and application of special-shaped section test water tank

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