CN111368485B - Method for calculating surface roughness based on Manning equation - Google Patents

Method for calculating surface roughness based on Manning equation Download PDF

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CN111368485B
CN111368485B CN202010138905.0A CN202010138905A CN111368485B CN 111368485 B CN111368485 B CN 111368485B CN 202010138905 A CN202010138905 A CN 202010138905A CN 111368485 B CN111368485 B CN 111368485B
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杨恒月
戴玮
史源
章少辉
白美健
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China Institute of Water Resources and Hydropower Research
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Abstract

The invention discloses a method for calculating surface roughness based on Manning equation, which comprises the steps of measuring the speed, potential energy and water depth values of any adjacent three points, constructing an on-way resistance calculation model, calculating a momentum balance equation of a full-hydrodynamic model, obtaining a calculation model of the surface roughness by adopting a central difference format dispersion, and solving the calculation model of the surface roughness by using measured data. According to the method, the applicability of the farmland irrigation process is improved by constructing the on-way resistance calculation model of the ground surface to water flow under the farmland irrigation condition, and the calculation accuracy of the roughness coefficient is further improved.

Description

Method for calculating surface roughness based on Manning equation
Technical Field
The invention belongs to the technical field of roughness coefficient calculation under the condition of furrow irrigation, and particularly relates to a method for calculating surface roughness based on Manning equation.
Background
Currently, there are many studies on surface roughness, but none of them exceed the classical description of surface roughness in hydraulics. Some documents mention that the roughness coefficient n varies with the flow regime, in which case it is clear that the manning formula is not applicable, since the roughness coefficient n, which measures the resistance of the surface to water flow, should be a constant, regardless of the flow regime, rather than a function of the water flow regime, i.e. n ≠ f (h, u). This implies two concepts, one is that for the area around a certain point (the size is larger than a certain feature size), the roughness coefficient n is constant, and for the area around different points, the spatial variability of the surface roughness causes a variation in the roughness coefficient.
In hydraulics, the relationship between on-way resistance and speed is,
Sf＝λu2 (1)
in the formula (1), λ is an on-way drag coefficient which is a function of Reynolds number Re,
λ＝f(Re) (2)
for round tubes, the Reynolds number Re is defined as,
Figure BDA0002398325470000011
in the formula, u is the water flow speed, R is the characteristic scale of the restriction water flow movement boundary, and v is the viscosity coefficient of the water flow.
When the Re number is in a certain range, the water flow is in a laminar flow state, and at the moment
Figure BDA0002398325470000012
Then the on-way resistance SfIs a linear function of the speed u and,
Sf＝Au (4)
wherein A is a proportionality constant.
When Re is greater than a certain value, the water flow is in a turbulent state, and the on-way resistance is in direct proportion to the square of the speed through tests,
Sf＝Au2 (5)
that is, the water flow is in the "square area of resistance", and the coefficient A contains the coefficient of roughness n.
The conclusion of the "resistance square area" comes from a round tube test, and the test conditions clearly show that the roughness of the inner surface of the round tube is measured by the relative roughness delta/R, wherein delta is the diameter of sand grains on the inner wall of the round tube, and R is the radius of the round tube. It can be seen that, no matter whether the inner wall of the circular tube is rough or not, the distribution of the sand particles is uniform based on the characteristic dimension R from the statistical viewpoint.
In the furrow irrigation water flow motion equation, the resistance of the ground surface to the water flow is generally calculated by adopting the formula (5), so that the furrow irrigation water flow is considered to be in a resistance square area. There is an implicit assumption that the Re of the furrow irrigation water flow is large enough that the water flow conditions during furrow irrigation are always turbulent. However, how to define the Re number in the furrow irrigation process, no uniform definition is given in the literature at present. For example, what the characteristic dimension R in the Re number is during bedding. Based on the resistance square area, in the furrow irrigation process, if the selected test scale is smaller than the characteristic scale, the obtained water flow resistance calculation formula is unstable, namely, the scale effect exists, and when the test scale is larger than the characteristic scale, the form of the formula can be stable.
Compared with the round pipe test condition for obtaining the conclusion of the resistance square area, due to the existence of space variability, the field surface of the actual ridge field is a curved surface with different roughness distribution, so when different characteristic scales are selected to average the physical quantity in the ridge irrigation process, the ridge irrigation water flow is not always in the resistance square area. When the field relief degree is relatively uniform within the selected feature scale, a conclusion of a "resistance square region" may be obtained, and when the scale is enlarged, a part of the water flow may be in a "linear region" within the feature scale.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a method for accurately calculating the surface roughness based on the Manning equation.
In order to achieve the purpose of the invention, the invention adopts the technical scheme that:
a method for calculating surface roughness based on Manning's equation comprises the following steps:
s1, measuring the speed, potential energy and water depth of any adjacent three points in the furrow irrigation process;
s2, constructing an on-way resistance calculation model of the ground surface to water flow under the furrow irrigation condition;
s3, calculating the ground surface friction force by adopting a Manning equation based on the on-way resistance calculation model to obtain a momentum balance equation of the full-hydrodynamic model;
s4, dispersing the momentum balance equation of the full-hydrodynamic model by adopting a central difference format to obtain a calculation model of the surface roughness;
and S5, solving the calculation model of the surface roughness by using the measurement data of the step S1.
Further, in step S1, a measuring ruler and a pitot tube are respectively disposed at any adjacent three measuring points, a time series of the physical quantity of the speed, the potential energy and the water depth value at each measuring point is obtained by measuring within a set time, and the time series of the physical quantity is subjected to statistical average processing.
Further, the step S2 is specifically:
based on a resistance square region theory, according to the influence of different characteristic scales on the furrow irrigation water flow in the furrow irrigation process, an on-way resistance calculation model of the ground surface to the water flow under the furrow irrigation condition is constructed and expressed as follows:
Sf＝A(u+u2)
wherein S isfThe on-way resistance of the surface to water flow is shown as A, a proportionality coefficient containing a roughness coefficient n and u, a water flow speed.
Further, the momentum balance equation of the full hydrodynamic model in the step S3 is expressed as:
Figure BDA0002398325470000041
wherein h is the water depth, z is the distance between the water surface line and the horizontal line, t is the time, x is the coordinate direction, and g is the acceleration of gravity.
Further, the momentum balance equation of the full hydrodynamic model in step S4 is discretely expressed in a central difference format as:
Figure BDA0002398325470000042
wherein u isi-1,ui,ui+1Respectively the water flow velocity, xi at the measuring points i-1, i, i +1i+1,ξi-1Respectively the potential energy at the measuring point i +1, i-1, hiFor the depth of water at measurement point i, Δ x is the distance between adjacent measurement points.
Further, the calculation model of the surface roughness in step S4 is represented as:
Figure BDA0002398325470000043
the invention has the following beneficial effects:
(1) aiming at the influence of different characteristic scales on the furrow irrigation water flow in the furrow irrigation process, the method comprehensively considers the relation between the resistance of the field surface of the laminar flow state and turbulent flow state area to the water flow and the flow speed, and accurately describes the on-way resistance by constructing an on-way resistance calculation model of the ground surface to the water flow under the furrow irrigation condition, so that the applicability of the on-way resistance calculation model to the furrow irrigation process is improved;
(2) according to the method, the roughness coefficient under the furrow irrigation condition is calculated based on the full-hydraulic dynamic model and the Manning equation, so that the calculation accuracy of the roughness coefficient is improved.
Drawings
FIG. 1 is a flow chart of a method of calculating surface roughness based on the Manning equation of the present invention;
fig. 2 is a schematic view of a ridge irrigation process parameter measurement structure in the embodiment of the invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
As shown in fig. 1, an embodiment of the present invention provides a method for calculating a surface roughness based on the manning equation, including the following steps S1 to S5:
s1, measuring the speed, potential energy and water depth of any adjacent three points in the furrow irrigation process;
in the embodiment, as shown in fig. 2, the invention selects any adjacent three measuring points i-1, i, i +1 along the ridge length, and a measuring ruler and a pitot tube are respectively arranged at the three measuring points. Height z of water surface line from horizontal line by measuring two measuring points i-1 and i +1i-1And zi+1Value to calculate potential energy gradient
Figure BDA0002398325470000051
And measuring the water depth h of the measurement point i by using a measuring ruleriMeasuring the water velocity u at each point by a pitot tube at that pointi. And measuring the speed, potential energy and water depth value time series of each point within a set time, and performing statistical average processing on the time series of the physical quantity within the set time in order to prevent numerical errors caused by numerical pulsation.
S2, constructing an on-way resistance calculation model of the ground surface to water flow under the furrow irrigation condition;
in this embodiment, because the horizontal dimensions are more than one order of magnitude larger than the vertical dimensions in the furrow irrigation process, some regions are in the laminar state due to low flow rate, and other regions in the turbulent state are not diffused due to the overlarge horizontal dimensions, so that the regions are in the laminar state until the flow rate is increased to change the state, and at this time, the resistance of the field surface of the region to the water flow is in a linear relationship with the flow rate.
Based on the resistance square region theory, according to the influence of different characteristic scales on the flow of the furrow irrigation in the furrow irrigation process, the invention constructs an on-way resistance calculation model of the ground surface to the flow under the furrow irrigation condition, which is expressed as follows:
Sf＝A(u+u2) (6)
wherein S isfThe on-way resistance of the surface to water flow is shown as A, a proportionality coefficient containing a roughness coefficient n and u, a water flow speed.
S3, calculating the ground surface friction force by adopting a Manning equation based on the on-way resistance calculation model to obtain a momentum balance equation of the full-hydrodynamic model;
in this embodiment, the invention calculates the surface friction force by using the manning equation based on the on-way resistance calculation model, and the momentum balance equation of the full-hydrodynamic model is expressed as:
Figure BDA0002398325470000061
wherein h is the water depth, z is the distance between the water surface line and the horizontal line, t is the time, x is the coordinate direction, and g is the acceleration of gravity.
The formula (7) is changed without considering the water flow reflux as follows:
Figure BDA0002398325470000062
in the furrow irrigation process, the water flow state is relatively stable at a certain position behind the propulsion peak, so that the change rate of the water flow velocity u along with the time is a high-order small quantity relative to other items, and only the spatial change rate of the velocity is considered, the change of the formula (8) is as follows:
Figure BDA0002398325470000063
and h + z is set to be xi, and xi is the water flow potential energy at the point. By introducing the water flow potential energy term xi, the formula (9) is changed into:
Figure BDA0002398325470000064
is provided with
K＝gn2h-4/3 (11)
Then equation (10) changes to:
Figure BDA0002398325470000071
since the same definition of velocity and flux is given for water flow per unit time and per unit area, equation (12) indicates that, in the water flow propulsion phase, the factors causing water flow flux of a certain cross section, regardless of the time rate of change of velocity, include potential energy gradient
Figure BDA0002398325470000072
And kinetic energy gradient (more precisely, transport of momentum gradient)
Figure BDA0002398325470000073
The coefficient K of equation (12) is a function of the coefficient of manning roughness n and the water depth h. Since the surface friction dissipates the energy of the water flow, the change in potential energy (i.e., potential energy gradient) and the change in kinetic energy (i.e., kinetic energy gradient) cannot be completely converted into flux of the water flow, then K should always be less than 1, and the measure of the surface friction dissipating the energy of the water flow is 1-K.
As can be seen from the definition of K, if the roughness coefficient above a certain scale in a field is a constant, then equation (11) is
K＝Ch-4/3 (13)
Wherein C is a constant and is a value for measuring the resistance of the earth's surface to water flow, since C is equal to gn2And is therefore a positive number.
From the above, the momentum balance equation of the full-hydrodynamic model can be interpreted as that the square of the water flow flux of any section is in direct proportion to the kinetic energy gradient and the potential energy gradient of the point, and the proportionality coefficient K is an exponential function of the water depth h.
If the zero inertia model is considered, equation (12) is changed to,
Figure BDA0002398325470000074
equation (14) can be interpreted as that the square of the flux causing the current at any section is proportional to the potential energy gradient at that point, and the proportionality coefficient K is an exponential function of the water depth h.
S4, dispersing the momentum balance equation of the full-hydrodynamic model by adopting a central difference format to obtain a calculation model of the surface roughness;
in this embodiment, the present invention discretizes equation (12) in a central difference format as follows:
Figure BDA0002398325470000081
wherein u isi-1,ui,ui+1Respectively the water flow velocity, xi at the measuring points i-1, i, i +1i+1,ξi-1Respectively the potential energy at the measuring point i +1, i-1, hiFor the depth of water at measurement point i, Δ x is the distance between adjacent measurement points. Equation (15) relates to the velocity, potential energy and water depth values at the three points. In the case where the measurement error is not considered, the calculation accuracy of equation (15) is second order, that is, the truncation error is calculated to be third order or more.
The equation (15) is transposed and gn is assumed2C is a constant and is represented by
Figure BDA0002398325470000082
The calculation model of the obtained surface roughness is represented as:
Figure BDA0002398325470000083
and S5, solving the calculation model of the surface roughness by using the measurement data of the step S1.
In this embodiment, in step S1, the method selects an appropriate Δ x, measures physical values of three adjacent points in the field, and substitutes the measured values into a model for calculating the surface roughness to calculate the surface roughness of the field accurately.
It will be appreciated by those of ordinary skill in the art that the embodiments described herein are intended to assist the reader in understanding the principles of the invention and are to be construed as being without limitation to such specifically recited embodiments and examples. Those skilled in the art can make various other specific changes and combinations based on the teachings of the present invention without departing from the spirit of the invention, and these changes and combinations are within the scope of the invention.

Claims (2)

1. A method for calculating surface roughness based on Manning's equation comprises the following steps:
s1, measuring the speed, potential energy and water depth of any adjacent three points in the furrow irrigation process;
s2, constructing an on-way resistance calculation model of the ground surface to water flow under the furrow irrigation condition, specifically:
based on a resistance square region theory, according to the influence of different characteristic scales on the furrow irrigation water flow in the furrow irrigation process, an on-way resistance calculation model of the ground surface to the water flow under the furrow irrigation condition is constructed and expressed as follows:
Sf＝A(u+u2)
wherein S isfThe on-way resistance of the earth's surface to water flow, A is a proportionality coefficient containing the roughness n of the earth's surface, and u isThe water flow rate;
s3, calculating the surface friction force by adopting a Manning equation based on the on-way resistance calculation model to obtain a momentum balance equation of the full-hydrodynamic model, wherein the momentum balance equation is expressed as:
Figure FDA0002690459220000011
wherein h is the water depth, z is the distance between the water surface line and the horizontal line, t is the time, x is the coordinate direction, and g is the gravity acceleration;
s4, dispersing the momentum balance equation of the full-hydrodynamic model by adopting a central difference format, and expressing as follows:
Figure FDA0002690459220000012
wherein u isi-1,ui,ui+1Respectively the water flow velocity, xi at the measuring points i-1, i, i +1i+1,ξi-1Respectively the potential energy at the measuring point i +1, i-1, hiAnd (3) obtaining a calculation model of the surface roughness by taking the water depth at the measuring point i and the deltax as the distance between adjacent measuring points, wherein the calculation model is expressed as:
Figure FDA0002690459220000013
and S5, solving the calculation model of the surface roughness by using the measurement data of the step S1.
2. The method for calculating surface roughness based on mannin' S equation as claimed in claim 1, wherein in step S1, a measuring ruler and a pitot tube are respectively arranged at any adjacent three measuring points, a time series of physical quantities of speed, potential energy and water depth values of each measuring point is measured within a set time, and the time series of physical quantities is subjected to statistical average processing.
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