KR100593822B1 - One-Dimensional Numerical Analysis of Fluid Flows by the Upstream McCorem Method - Google Patents

Abstract

The present invention relates to a one-dimensional numerical analysis of fluid flows by the upstream McCorem technique, in particular discontinuous fluid flows due to various types of flows occurring in rivers, that is, numerical vibrations occurring near discontinuous points. The present invention relates to a one-dimensional numerical analysis method of fluid flow that can accurately simulate various types of fluid flows generated under various conditions by improving the McCormak technique that cannot be applied. In the one-dimensional numerical analysis method of the fluid flow according to the present invention, the distance between the water level, the flow rate, and the flow term and the generation term, which constitute the variable term of the one-dimensional governing equation of the fluid flow with respect to the reference time by inputting initial conditions Calculating a value for the grid; Calculating boundary conditions of an upstream end and a downstream end of the distance grating with respect to the reference time using the initial condition; Calculating a normalized Jacobian in an upstream direction of the flow term for distance grids except upstream and downstream ends of the distance grid for the reference time; Applying the normal Jacobian in the upstream direction to the corresponding distance grating of the flow term to the difference value and the generation term of the adjacent distance grid of the flow term calculated in the first step, the water level, the flow rate and the upstream end and A fourth step of calculating an upstream flow rate and level for each distance grid in the next temporal grid using boundary conditions at the downstream end; Using the result calculated in step 4, a value for each distance grating is calculated for the variables of the flow term and the generation term of the one-dimensional governing equation of the fluid flow, the upstream end of the distance grating for the reference time, and A fifth step of calculating a normal Jacobian in the downstream direction of the flow term for a distance grid excluding a downstream end; The difference value of the adjacent distance grid of the flow term calculated in step 5 and the generated Jacobian in the downstream direction with respect to the corresponding distance lattice of the flow term are applied to the generation term, and the water level, the flow rate and the upstream end constituting the variable term. And a sixth step of calculating the flow rate and water level in the downstream direction for each distance grid in the next temporal grid by using the boundary condition of the downstream stage; The flow rate and level in the upstream direction for each distance grid calculated in the fourth step and the flow rate and level in the downstream direction for each distance grid calculated in the sixth step are averaged for each distance grid in the next time grid. A seventh step of calculating the water level and the flow rate; And returning to the second step to transfer to the next time grid to continuously calculate the water level and the flow rate for each distance grid at each time grid.

Description

ONE DIMENSIONAL NUMERICAL ANALYSIS METHOD BY UPWIND MCCORMACK SCHEME}

1 is a flowchart illustrating a numerical analysis method according to an embodiment of the present invention.

Figure 2 is a graph comparing the analysis results and the theoretical solution according to the numerical analysis method according to an embodiment of the present invention for the dam collapse flow.

3 to 6 is a graph comparing the analysis results and the theoretical solution according to the numerical analysis method according to an embodiment of the present invention for the steady flow occurring in the beam.

7 to 10 is a graph comparing the analysis results and the theoretical solution according to the numerical analysis method according to an embodiment of the present invention for the steady flow in the case of non-uniform lower width.

Figure 11 is a graph comparing the analysis results and actual water level according to the numerical analysis method according to an embodiment of the present invention for the dam collapse test of Bellos et al.

The present invention relates to a one-dimensional numerical analysis of fluid flows by the upstream McCorem technique, in particular discontinuous fluid flows due to various types of flows occurring in rivers, that is, numerical vibrations occurring near the discontinuities. The present invention relates to a one-dimensional numerical analysis method of fluid flow that can accurately simulate various types of fluid flows generated under various conditions by improving the McCormak technique that cannot be applied.

The McCormack scheme, which is used for the numerical analysis of fluid flow, is widely used in the second-order accuracy-understanding central difference method because it has a second-order accuracy and a simpler form than other techniques. The McCormak technique is based on the Lax-Wendroff technique. The basic concept of the Lax-Wendroff technique is to solve the instability of the central difference method and to secure second-order accuracy in time and space.

Because of these advantages, the McCormak technique has been widely used since the 1970s. However, this technique, like other second-accuracy central difference method, has a limitation that cannot be applied to discontinuous flow due to the problem of numerical vibration occurring near the point of discontinuity. The proposed method to overcome this limitation is to use artificial viscous terms, which also have the disadvantage of testing the variables for each target flow.

Another way to solve the numerical vibration problem is to use a controller such as Total Variation Diminishing (TVD). The TVD controller is a method that can suppress abnormal numerical vibration and instability without testing additional variables, but it is complicated and time-consuming compared to the general technique. It is also unclear at this time whether such a TVD controller can be applied in the case of a natural river having a very strong term. Because of this, the existing McCormack technique is not actually used in various fields despite its advantages.

Therefore, there is a need to improve the existing McCormak technique, which cannot be applied to discontinuous fluid flow due to the problem of numerical vibration.

Disclosure of the Invention An object of the present invention is to improve the McCormak technique, which is not applicable to discontinuous fluid flow due to various types of flows generated in rivers, that is, numerical vibration occurring near discontinuous points, and thus, various types of fluids generated under various conditions. It is to provide a one-dimensional numerical analysis method of the fluid flow that can accurately simulate the flow.

The object of the present invention is to input an initial condition and set a value for each distance grid for the variables of the water level and the flow rate and the flow term and the generation term that constitute the variable term of the one-dimensional governing equation of the fluid flow with respect to the reference time. A first step of computing; Calculating boundary conditions of an upstream end and a downstream end of the distance grating with respect to the reference time using the initial condition; Calculating a normalized Jacobian in an upstream direction of the flow term for distance grids except upstream and downstream ends of the distance grid for the reference time; Applying the normal Jacobian in the upstream direction to the corresponding distance grating of the flow term to the difference value and the generation term of the adjacent distance grid of the flow term calculated in the first step, the water level, the flow rate and the upstream end and A fourth step of calculating an upstream flow rate and level for each distance grid in the next temporal grid using boundary conditions at the downstream end; Using the result calculated in step 4, the values for each distance grating are calculated for the flow terms and the generation terms of the one-dimensional governing equations of the fluid flow, and the upstream end of the distance grating for the reference time and A fifth step of calculating a normal Jacobian in the downstream direction of the flow term for a distance grid excluding a downstream end; The difference value of the adjacent distance grid of the flow term calculated in step 5 and the generated Jacobian in the downstream direction with respect to the corresponding distance lattice of the flow term are applied to the generation term, and the water level, the flow rate and the upstream end constituting the variable term. And a sixth step of calculating the flow rate and water level in the downstream direction for each distance grid in the next temporal grid by using the boundary condition of the downstream stage; The flow rate and level in the upstream direction for each distance grid calculated in the fourth step and the flow rate and level in the downstream direction for each distance grid calculated in the sixth step are averaged for each distance grid in the next time grid. A seventh step of calculating the water level and the flow rate; And a eighth step of returning to the second step to transfer to the next time grid and continuously calculating the water level and the flow rate for each distance grid at each time grid.

DESCRIPTION OF THE PREFERRED EMBODIMENTS Preferred embodiments will now be described in detail with reference to the accompanying drawings as examples only.

One-dimensional governing equations in rivers are shown in Equation 1 below.

Where U is a variable term, F is a flow term, and S is a generation term, as defined by Equations 2 to 4 below.

In Equation 2, A represents the effective area, Q (x, t) is the flow rate, g is the gravitational acceleration, I ₁ is the hydrostatic pressure of the effective area represented by Equation 5 below, S ₀ And S _f each represent a bottom slope and a friction slope.

In Equation 5, h is the depth and β is the surface width when the depth η is defined as in Equation 6 below.

In Equation 4 representing the generation term, I ₂ is a term representing a force generated by the expansion and contraction of the lower cross section, and is defined as in Equation 7 below.

One of the methods related to the finite difference method, which is used to analyze the one-dimensional governing equation of the fluid flow represented by Equation 1, is the McCormak technique mentioned above. This conventional McCormack technique is represented by Equations 8 to 10 below.

,

는 각각 거리 격자의 간격 및 시간 격자의 간격을 나타내며, n은 임의의 기준 시간 격자, n+1은 다음 시간 격자를 나타내는 것으로 n시간에 대한 값은 기지이며, n+1시간에 대한 값이 미지수가 된다. 또한, i는 임의의 기준 거리 격자, i-1은 i의 직상류 거리 격자, i+1은 i의 직하류 거리 격자를 나타낸다.here,

,

Denotes the interval of the distance grid and the interval of the time grid, respectively, n represents an arbitrary reference time grid, n + 1 represents the next time grid, the value for n hours is known, and the value for n + 1 hours is unknown. Becomes In addition, i represents an arbitrary reference distance grating, i-1 represents an upstream distance grating of i, and i + 1 represents an upstream distance grating of i.

As described above, the conventional McCormak technique represented by Equations 8 to 10 is as follows.

를 계산하여 다음 시간에 대한 상류방향의 유량과 수위를 계산하게 된다. Equation (8) calculates the flow term F and the generation term S for the upstream flow direction, and consists of a variable term consisting of the water level and the flow rate at the point.

Calculate the upstream flow rate and level for the next time.

와 생성항

를 계산하고, 해당지점의 수위와 유량으로 구성되는 변수항

를 계산하여 다음 시간에 대한 하류방향의 수위와 유량을 계산하게 된다.Equation 9 is a flow term for the downstream flow direction based on the upstream flow rate and level for the next time calculated by Equation 8

And generating term

Is a variable term consisting of the water level and the flow rate at the point.

We calculate the water level and the flow rate in the downstream direction for next time.

를 구성하는 수위와 유량을 계산하게 된다.Equation 10 is an average of the variable terms calculated by Equation 8 and Equation 9 described above, and finally is a variable term corresponding to each distance grid of the next time.

It will calculate the water level and the flow rate constituting.

However, the conventional McCormack technique, as mentioned above, has a fundamental limitation that cannot be applied to discontinuous fluid flow due to the problem of numerical vibration occurring near the discontinuous point.

The inventors of the present invention have the advantages of secondary accuracy and simplicity, but in order to improve the conventional McCormack technique, which has the problems of numerical vibration described above, the upstream transfer type is incorporated by combining the conventional McCormack technique with the upstream transfer technique. McCormack technique was created.

The upstream McCormak technique created by the inventor of the present invention is represented by Equations 11 to 13 below.

는 수학식 1의 흐름항

의 상류방향 및 하류방향의 정규 야코비안(normalized Jacobian)을 나타내며, 아래의 수학식 14에 의해 정의된다.In Equation 11 and Equation 12

Is the flow term of Equation 1

The normalized Jacobian in the upstream and downstream directions of is defined by Equation 14 below.

는 유사전환행렬,

은

의 역행렬을 나타내며,

는 정규화된 고유값으로 아래의 수학식 15에 의해 정의된다.here,

Is the variation matrix,

silver

Represents the inverse of,

Is a normalized eigenvalue and is defined by Equation 15 below.

는 아래의 수학식 16에 의해 정의되는 수학식 1의 흐름항

의 야코비안의 고유값이다.In Equation 15, sgn represents a sign function, and

Is the flow term of Equation 1 defined by Equation 16 below

Eigenvalue of Jacobian of.

,

,

의 관계가 성립한다.In Equation 16, u represents a flow rate, c represents a wave velocity,

The relationship is established.

,

는 각각 거리 격자의 간격 및 시간 격자의 간격을 나타내며,

은 초기조건에 의해 연산되는 기준 시간 격자에서의 해당 거리 격자의 수위와 유량을 나타내며,

및

는 각각 초기조건 에 의해 연산되는 기준 시간 격자에서의 해당 거리 격자와 기준 시간 격자에서의 해당 거리 격자의 직상류 거리 격자에서의 흐름항을 나타내며,

는 초기조건에 의해 연산되는 해당 거리 격자에서의 생성항을 나타내며,

는 기준 시간에서의 해당 거리 격자에서의 상류방향의 정규 야코비안을 나타낸다.In Equations 11 to 13,

,

Represents the spacing of the distance grid and the spacing of the time grid, respectively.

Represents the level and flow rate of the distance grid in the reference time grid computed by the initial conditions,

And

Denotes the flow term in the distance grids upstream of the distance grids in the reference time grid and the distance grids in the reference time grid, respectively, computed by the initial conditions,

Denotes the generation term in the corresponding distance grid computed by the initial conditions,

Denotes the normal Jacobian upstream in the distance grid at the reference time.

및

는 각각 하류방향의 수위와 유량을 기초로 연산되는 기준 시간 격자의 다음 시간 격자에서의 해당 거리 격자의 직하류 거리 격자와 기준 시간 격자의 다음 시간 격자에서의 해당 거리 격자의 흐름항을 나타내며,

는 하류방향의 수위와 유량을 기초로 연산되는 해당 거리 격자에서의 생성항을 나타내며,

는 기준 시간에서의 해당 거리 격자에서의 하류방향의 정규 야코비안을 나타낸다.Also,

And

Represents the flow term of the distance grid directly downstream of the corresponding distance grid at the next time grid of the reference time grid and the corresponding distance grid at the next time grid of the reference time grid, respectively calculated based on the water level and flow rate in the downstream direction,

Represents the product term in the distance grid calculated based on the downstream water level and flow rate,

Denotes the normal Jacobian downstream of the corresponding distance grid at the reference time.

는 기준 시간 격자의 다음 시간 격자에서의 해당 거리 격자에 대한 상류방향의 수위와 유량을 나타내며, 수학식 12의

는 기준 시간 격자의 다음 시간 격자에서의 해당 거리 격자에 대한 하류방향의 수위와 유량을 나타내고, 수학식 13의

은 기준 시간 격자의 다음 시간 격자에서의 해당 거리 격자에 대한 상류방향의 수위 및 유량과, 기준 시간 격자의 다음 시간 격자에서의 해당 거리 격자에 대한 하류방향의 수위와 유량의 평균에 의해 연산되는 최종적인 기준 시간 격자의 다음 시간 격자에서의 해당 거리 격자에 대한 수위 및 유량을 나타낸다.Of Equation 11

Is the upstream water level and flow rate for the distance grid in the next time grid of the reference time grid,

Is the downstream water level and flow rate for the distance grid in the next time grid of the reference time grid,

Is calculated by the average of the upstream water level and flow rate for that distance grid in the next time grid of the reference time grid, and the downstream water level and flow rate for the corresponding distance grid in the next time grid of the reference time grid. Level and flow rate for the distance grid in the next time grid of a typical reference time grid.

Referring now to Figures 1 and 11 to 13, the one-dimensional numerical analysis method of the fluid flow according to the present invention will be described.

)과 흐름항과 생성항의 변수(

,

)에 대해 각 거리 격자에 대한 값을 연산한다.In step 10 shown in FIG. 1, initial conditions such as initial data, boundary data, roughness coefficient, stream section, and time interval are inputted to form a water level and a flow rate constituting a variable term of a fluid flow with respect to a reference time.

) And the variables in the flow and creation terms (

,

For each distance grid.

Here, the initial data means the known value of the reference time, and the boundary data means the data at both ends of the next time to be calculated. The roughness coefficient represents the roughness of the stream, and the river cross section represents the physical shape of the river.

In step 11, the boundary conditions of the upstream and downstream ends of the distance grid with respect to the reference time are calculated using the above initial conditions.

)을 연산한다.Then, in step 12, using the equations (14) to (16), the normal Jacobian in the upstream direction of the flow term for each distance grid except for the upstream and downstream ends of the distance grid for the reference time (

) Is calculated.

)과 생성항(

)에 단계 12에서 연산한 해당 거리격자에 대한 상류방향의 정규 야코비안(

)을 적용하고, 단계 10에서 연산한 기준 시간 격자의 해당 거리 격자에서의 수위 및 유량(

)과 단계 11에서 연산한 상류단 및 하류단의 경계조건을 이용하여 다음 시간 격자에서의 각 거리 격자에 대한 상류방형의 유량과 수위(

)를 연산한다.In step 13, the flow rate and level in the upstream direction for each distance grid in the next time grid are calculated using Equation 11. Specifically, the difference value of the adjacent distance grid of the flow term calculated in step 10 (

) And the creation term (

), The upstream normal Jacobian (

) And the water level and flow rate at the corresponding distance grid of the reference time grid calculated in step 10.

) And the upstream flow and water level for each distance grid in the next temporal grid, using the boundary conditions of the upstream and downstream ends computed in step 11).

) Is calculated.

Referring to FIG. 1, a person skilled in the art will understand that the above description repeats the routines of steps 12 and 13 by incrementing by one step the i value representing the distance grid.

)를 이용하여 기준 시간 격자의 다음 시간 격자에서의 흐름항과 생성항의 변수(

,

)에 대해 각 거리 격자에 대한 값을 연산하고, 수학식 14 내지 수학식 16을 참조하여 기준 시간에 대한 거리 격자의 상류단 및 하류단을 제외한 거리 격자에 대해 흐름항의 하류방향의 정규 야코비안(

)을 연산한다.In step 14, the result calculated in step 13

), The variables in the flow term and the generation term in the

,

For each distance grid, calculate a value for each distance grid, and refer to Equations 14 to 16, and for the distance grids excluding the upstream and downstream ends of the distance grid for the reference time,

) Is calculated.

)과 생성항(

)에 단계 14에서 연산한 해당 거리격자에 대한 하류방향의 정규 야코비안(

)을 적용하고, 단계 10에서 연산한 기준 시간 격자의 해당 거리 격자에서의 수위 및 유량(

)과 단계 11에서 연산한 상류단 및 하류단의 경계조건을 이용하여 다음 시간 격자에서의 각 거리 격자에 대한 하류방향의 유량과 수위(

)를 연산한다.In step 15, the flow rate and level in the downstream direction for each distance grid in the next time grid of the reference time grid are calculated using Equation 12. Specifically, the difference value of the adjacent distance grid of the flow term calculated in step 14 (

) And the creation term (

), The normal Jacobian downstream of the corresponding distance grating computed in step 14

) And the water level and flow rate at the corresponding distance grid of the reference time grid calculated in step 10.

) And the downstream flow and water level for each distance grid in the next temporal grid, using the boundary conditions of the upstream and downstream ends computed in step 11).

) Is calculated.

Further, referring to FIG. 1, a person skilled in the art will understand that the above description repeats the routines of steps 14 and 15 by incrementing by one step the i value representing the distance grid.

)와 단계 15에서 연산한 각 거리 격자에 대한 하류방향의 유량 및 수위(

)를 평균하여 기준 시간 격자의 다음 시간 격자에서의 각 거리 격자에 대한 수위와 유량(

)을 연산한다.In steps 16 and 17, equation (13) is applied to the flow rate and water level in the upstream direction for each distance grid calculated in step 13 (

) And the downstream flow and level for each distance grid computed in step 15 (

) And the water level and flow rate (for each distance grid in the next time grid of the reference time grid)

) Is calculated.

In addition, after step 17, the flow returns to step 11, transfers to the next time grid, and repeats the routine of steps 11 to 17 to calculate the water level and the flow rate for each distance grid in each time grid.

When all the calculation processes are completed, the routine exits the steps 11 to 17, and all operations are completed in step 18.

The inventor of the present invention has applied the numerical method according to the present invention to various fluid flows in order to verify the efficiency of the numerical method according to the present invention.

1. Dam collapse flow

The numerical analysis method according to the present invention was applied to the analysis of dam collapse flow occurring in a horizontal unloaded frictionless road. The total channel length was 2000m and the dam was set in the middle. The initial level was set at 10m upstream and 5m downstream. The distance grating spacing was set to 6.25 m and the time grating spacing was set to 0.4 seconds.

Figure 2 is a graphical analysis of the analysis (analytical solution) based on the above set values, the conventional McCormack technique, a comparison between the numerical analysis method according to the present invention. The thick solid line (Analytical solution) represents the analysis solution, the thin solid line (upwind McCormack) is the result using the method of the present invention, the dotted line (McCormack) is the result using the existing method. As shown in FIG. 2, the conventional McCormak technique shows a strong numerical vibration in the vicinity of the discontinuous point, but in the case of the method according to the present invention, it can be seen that it is approximated to the analysis solution without the numerical vibration.

2. Steady Flow in Beams

In order to simulate the steady flow occurring in the beam, the beam is set in the form of Equation 17.

These plots are presented in the Dam Collapse Wave Worksheet (Goutal and Maurel, 1997) and are commonly used to verify numerical models because they can be calculated using energy equations. The distance interval for the calculation was set to 0.25m, the flow rate of 0.18m ³ / sec for the upstream boundary conditions and the water level of 0.33m for the downstream boundary conditions. Initial conditions were given at a flow rate of 0.18 m ³ / sec and a water level of 0.5 m in all sections.

Figure 3 is a comparison of the water level at 13 seconds after the start of the calculation, while the conventional McCormak technique shows a large numerical oscillation with a dotted line, while the numerical analysis method of the present invention indicated by "o" converged stably You can see that it is going. In the case of the conventional McCormak technique, it is impossible to converge to an analysis solution because it is impossible to calculate further due to such numerical vibration, but the numerical analysis method according to the present invention converges well to the analysis solution without numerical vibration through FIG. 4. You can check it.

5 and 6 is a comparison of the conventional McCormack technique and the numerical analysis method according to the present invention with respect to the flow rate. Figure 5 compares the flow rate at 13 seconds after the start of the calculation, Figure 6 shows the final converged flow rate. The flow rate also shows the same result as the water level. That is, the conventional McCormak technique cannot calculate due to the problem of numerical vibration, but it can be seen that the numerical method according to the present invention converges well to the analysis solution.

3. Steady flow when having uneven lower width

In order to verify the discontinuous flow occurring in the lower road where the width is changed, a lower road of 500m length is assumed as shown in Equation 18 below.

The initial condition was set to 2m water level and 20m ³ / sec flow rate, the upstream boundary condition was set to 20m ³ / sec flow rate and the downstream boundary condition was set to 1.775m water level.

7 shows the water level at 320 seconds after the start of the calculation, while the conventional McCormack technique, which is represented by a dotted line, shows a large numerical vibration, while the numerical analysis method according to the present invention indicated by "o" is stably converged. It can be seen that. 8 shows the final convergent water level, the conventional McCormak technique does not converge itself by numerical vibration, but it can be seen that the numerical analysis method according to the present invention is almost identical to the analytical solution.

9 and 10 show the results of the flow rate, FIG. 9 shows the change of the flow rate when 320 seconds after the start of the calculation, and FIG. 10 shows the final converged result.

4. Damos collapse experiments such as Bellos

Bellos et al conducted a dam collapse test in 1992 on a rectangular sewer with varying width. The target underpass is a horizontal underpass of 22m in length with a lower width in the middle. The initial water level was given as 0.25m upstream of the dam and 0.101m downstream of the dam, and the roughness coefficient was given as 0.016. The distance grid spacing is 0.5m and the time grid spacing is 0.001 seconds.

11 shows the calculated water level and the measured water level for a specific point downstream of the dam. Although the conventional McCormak technique shows a large numerical vibration, the result of the numerical analysis method according to the present invention shown in solid line is shown. It can be seen that is almost identical to the measured water level indicated by "o".

As suggested so far, the inventors of the present invention, as a result of applying the numerical method according to the present invention for various fluid flows, were able to confirm that the present invention has an excellent effect compared to the conventional method.

So far, a preferred embodiment of the present invention has been described. However, the preferred embodiments described so far should only be taken as examples. That is, those skilled in the art to which the present invention pertains will be able to derive various modifications with reference to the preferred embodiment of the present invention. Accordingly, the technical scope of the present invention should be interpreted only by the appended claims.

 The present invention improves the McCormak technique, which cannot be applied to discontinuous fluid flow due to various types of flows generated in rivers, that is, numerical vibration occurring near the discontinuous point, and thus, various types of fluid flows generated under various conditions. It is effective in providing a one-dimensional numerical analysis method of fluid flow which can be accurately simulated.

Claims (3)

As a one-dimensional numerical method of fluid flow, A first step of inputting an initial condition and calculating values for each distance grid for the variables of the water level and the flow rate and the flow term and the generating term that constitute the variable term of the one-dimensional governing equation of the fluid flow with respect to the reference time; A second step of calculating boundary conditions of an upstream end and a downstream end of the distance grating with respect to the reference time using the initial condition; Calculating a normalized Jacobian in an upstream direction of the flow term for each distance grid except for the upstream and downstream ends of the distance grid with respect to the reference time; Applying the normal Jacobian in the upstream direction to the corresponding distance grating of the flow term to the difference value and the generation term of the adjacent distance grid of the flow term calculated in the first step, the water level, the flow rate and the upstream end and A fourth step of calculating the upstream flow rate and water level for each distance grid in the next temporal grid by using the downstream boundary condition; Using the result calculated in step 4, a value for each distance grating is calculated for the variables of the flow term and the generation term of the one-dimensional governing equation of the fluid flow, the upstream end of the distance grating for the reference time, and A fifth step of calculating a normal Jacobian in the downstream direction of the flow term for a distance grid except for a downstream end, The difference value of the adjacent distance grid of the flow term calculated in step 5 and the generated Jacobian in the downstream direction with respect to the corresponding distance lattice of the flow term are applied to the generation term, and the water level, the flow rate and the upstream end constituting the variable term. And a sixth step of calculating the flow rate and the water level in the downstream direction for each distance grid in the next temporal grid by using the boundary condition of the downstream stage; The flow rate and level in the upstream direction for each distance grid calculated in the fourth step and the flow rate and level in the downstream direction for each distance grid calculated in the sixth step are averaged for each distance grid in the next time grid. A seventh step of calculating the water level and the flow rate, And an eighth step of returning to the second step to transfer to the next time grid to continuously calculate the water level and the flow rate for each distance grid at each time grid. One-dimensional numerical method of fluid flow. The method of claim 1, Flow rate and water level in the upstream direction in the fourth step

Is

Calculated in the step 7 and the water level and the flow rate in the downstream direction

silver

Is computed by here,

,

Represents the spacing of the distance grid and the spacing of the time grid, respectively.

Denotes the water level and the flow rate of the corresponding distance grid in the reference time grid calculated by the initial conditions,

And

Denote a flow term in the distance grid in the reference time grid and the upstream distance grid of the distance grid in the reference time grid respectively calculated by the initial conditions,

Represents the generation term in the distance grid calculated by the initial condition,

Denotes the normal Jacobian in the upstream direction in the distance grid at the reference time, Also,

And

Is a flow of the distance grid directly downstream of the corresponding distance grid in the next time grid of the reference time grid and the corresponding distance grid in the next time grid of the reference time grid, respectively calculated based on the downstream water level and flow rate. Term,

Denotes the generation term in the distance grid calculated based on the downstream water level and flow rate,

Is a normal Jacobian in the downstream direction in the distance grid at the reference time, One-dimensional numerical method of fluid flow. The water level and the flow rate for each distance grid in the next time grid of the reference time in the seventh step.

silver

The method of one-dimensional numerical analysis of fluid flow, characterized in that it is calculated by.

Images