KR20130022800A - Numerical analysis method using 2-dimensional finite volume model applicable to mixed meshes - Google Patents

Abstract

PURPOSE: A fluid flow analysis method of a 2D finite volume method, capable of applying a mixed mesh is provided to find an adjacent mesh of a computational mesh and to compute a flow ratio at a mesh boundary side. CONSTITUTION: Fluid flow is analyzed by using a conservation variable of computational meshes. The conservation variable is obtained by a flow ratio and a generating function in boundary sides of a computational mesh and an adjacent mesh. The computational mesh and the adjacent mesh are distinguished through a property which shares two breakpoints(S150). The computational mesh is formed with a triangular mesh at a point which change of a flow velocity property and a flow property is large. The rest is formed with a rectangular mesh. [Reference numerals] (S110) Step of finding an angle between a central coordinate of a computational mesh and nodal points; (S120) Step of determining a sequence of the nodal point; (S130) Step of reading a nodal point number; (S140) Step of repeating the same steps for residual meshes; (S150) Determining a mesh with two same nodal points as an adjacent mesh

Description

Numerical Analysis Method using 2-dimensional Finite Volume Model applicable to Mixed Meshes

The present invention relates to a fluid flow analysis method using a two-dimensional finite volume method that can be applied to the mixed lattice, and more specifically, the application of the mixed lattice, which can calculate the flow rate at the lattice boundary by finding adjacent lattice of the calculated lattice A fluid flow analysis method by two-dimensional finite volume method is possible.

Most flows in nature are very complex and difficult to interpret in consideration of all flow characteristics. Therefore, a simplification process for complex flow phenomena is required. For this purpose, the flow characteristics can be used in a number in which the horizontal size of the water body is much larger than the vertical size. Nonlinear shallow water equations derived from these characteristics are widely used as numerical model governing equations in order to analyze the flows of free water channels such as tides, tsunamis, flood waves, and dam collapse waves.

Recently, many studies on finite volume techniques have been conducted to interpret shallow water equations. In Korea, most of the studies using two-dimensional finite volume technique for channel flow analysis have been mainly composed of the calculation region of the rectangular grid for the experimental road and the analysis using the first-order accuracy method or the second-order accuracy method. In order to solve the problem of the balance between the generation term and the flow rate, a high order accuracy was applied. In addition, although the research on the second-order accuracy finite volume model using the triangular lattice has been made, the study on the mixed lattice has not been carried out. Overseas, we constructed triangular and rectangular grids for river basins and applied the first-order accuracy of the Roe technique. We developed a finite volume model that can be applied to triangular and square grids using HLLC-WAF technique. However, the application of the mixed lattice was not shown. In addition, we propose a UFF (upstream flux-splitting finite-volume) technique to calculate the numerical flow rate, and apply a HRM (hydraulic reconstruction method) to balance the flow rate and generation term. It was developed and applied to river flow analysis, but it did not show applicability in flood analysis.

As such, the research on the two-dimensional finite volume model to which the mixed lattice can be applied has not been studied so far at home and abroad.

One object of an embodiment of the present invention for solving the above problems is to provide a fluid flow analysis method using a two-dimensional finite volume model that can be applied to a triangular and square grating, and a mixed grid of two gratings combined have.

Another object of the present invention is to efficiently produce a two-dimensional grid for natural rivers and floodplains with irregular topography, confluences of main streams and tributaries of rivers, neighborhoods of hand structures, and urban areas including buildings and roads. To provide a fluid flow analysis method that can be configured.

According to a preferred aspect of the present invention for achieving the above objects, the fluid flow analysis method, the fluid flow analysis using the storage parameters in each calculation grid, the storage variable is the interface between the calculation grid and the adjacent grid The flow rate at the interface is calculated based on the discrimination between the computational lattice and the adjacent lattice, and the computational lattice and the adjacent lattice are characterized by the characteristics of sharing two nodes. Determine.

Preferably, the determination of the adjacent lattice, the step of obtaining the angle between the coordinates of the center coordinates and the nodes of the calculation grid, the order of the nodes in the order of the smallest angle size, two low order in the calculation grid Reading the node number from the node, repeating the number of nodes in the same manner as the calculation lattice for the remaining grids except the calculation lattice, until it matches the two node numbers of the calculation lattice; and The method may include determining a grid having a computational grid and two shared nodes as an adjacent grid.

Preferably, the generation term may include the bottom surface matter and the friction diameter matter.

Preferably, the riverbed matter may be calculated by distinguishing the triangular lattice and the rectangular lattice.

The computational lattice may be any one of a triangular lattice or a rectangular lattice, and in particular, the computational lattice may be composed of a triangular lattice and the remaining portion may be formed of a rectangular lattice at a point where a change in flow velocity and flow characteristics is large. In addition, the rectangular grid can be used for two-dimensional flow and flood analysis in places where terrain is difficult to express (confluence of main streams and tributaries, neighboring parts of hand structures, and surrounding buildings in urban areas) It is possible to shorten the calculation time than using triangular lattice. In flood and flood analysis, rivers can be composed of square grids, floodplains are composed of triangular grids, and in urban flooding analysis, roads and buildings can be composed of square grids and the remaining flood zones can be composed of triangular grids.

According to a preferred aspect of the present invention, the method for determining the adjacent lattice of the mixed lattice in the fluid flow analysis by the two-dimensional finite volume method, the angle formed by the coordinates of the center coordinates of the calculation lattice and the nodes constituting the calculation lattice Determine the number of nodes in the order of the smallest angle, read the node number from two nodes of low order in the grid, and read the number of nodes for the remaining grids except for the grid Repeating until the two node numbers of the computational grid coincide with each other, the grid having the computational grid and the two shared nodes can be determined as the neighboring grid.

According to the fluid flow analysis method using the two-dimensional finite volume method that can be applied to the mixed lattice according to an embodiment of the present invention, there is an effect that can be applied to the triangular and square lattice, and the mixed lattice combined with the two lattice.

In addition, according to the fluid flow analysis method using the two-dimensional finite volume method that can be applied to the mixed lattice according to an embodiment of the present invention, the confluence of the natural stream and floodplain, the main stream and the tributaries of the irregular terrain, the hand structure It is possible to apply mixed grids that can construct two-dimensional grids efficiently for neighborhoods and urban areas including buildings and roads, so that they can have all the advantages of using square and triangular grids. It can compensate for the shortcomings of the use, the difficulties appearing in the use of triangular and square lattice has an effect that can be overcome.

1 shows the structure of the HLLC approximation Riemann solution.
2 is a flowchart for determining an adjacent grid according to the present invention.
3 shows a method of calculating the angle between the center point and the node of each grating for finding adjacent lattice of triangular grating and rectangular grating.
4 shows a dam collapse test road connecting a rectangular reservoir to an L-shaped channel.
5 is a finite volume model using a mixed grid according to the present invention.
FIG. 6 shows the flow velocity of dam collapse waves propagating to wet sewage at 1 and 8 seconds after dam collapse.
FIG. 7 shows an aspect of flow rate of dam collapse waves propagating to dry sewage at 1 and 8 seconds after dam collapse.
FIG. 8 shows the comparison between the water level calculated by the square grid and the water level calculated by the mixed lattice, and the measured water level and the results of Petaccia.
FIG. 9 shows the comparison between the water level calculated by the square grid and the water level calculated by the mixed lattice, the measured water level, and the results of Petaccia.
10 shows the irradiation points of Malpasset dam collapse.
FIG. 11 shows triangular and rectangular mixed grids for topographic elevation and dam collapse simulation in the downstream area of the Malpasset Dam.
FIG. 12 shows a comparison between the calculated results of the model, the highest trace level investigated in the field, and the measured flood level measured in the laboratory.
13 and 14 show two- and three-dimensional flood depths and patterns at 1, 5, 20, and 35 minutes after dam collapse, respectively.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Hereinafter, preferred embodiments of the present invention will be described in detail with reference to the accompanying drawings. However, the present invention is not limited to the embodiments. For reference, in the present description, the same numbers refer to substantially the same grid, and may be described by quoting the contents described in other drawings under the above rules, and the contents repeated or deemed apparent to those skilled in the art may be omitted.

The conserved form of the two-dimensional shallow water equation can be expressed as Equation 1 below.

[Equation 1]

Where U is the conservation variable, F (U) and G (U) are the flow rates in the x and y directions, respectively, and S (U) is the bottom term for the x and y directions as the product term, S _0x , S _oy and friction Slope _fx , S _fy . H represents the depth and u and v represent the flow velocity in the x and y directions, respectively.

By integrating [Equation 1] for any computational lattice, the basic equation of the finite volume technique as in [Equation 2] can be obtained.

&Quot; (2) "

Where Ω is the boundary of the lattice, dΩ is the length of the boundary, and E = [F, G] ^T is the flow rate vector, and n is the outward unit vector perpendicular to the boundary Ω.

Assuming that the conserved variable U has a constant value in the computational lattice, the basic equation of the finite volume technique of [Equation 2] can be discretized as shown in [Equation 3].

&Quot; (3) "

Where A is the area of the computational lattice, m is the exponent representing the lattice boundary, L ^m is the length of the boundary m, and E ^m is the flow rate vector calculated between the computational lattice and the adjacent lattice sharing the boundary m.

1 shows the structure of the HLLC approximation Riemann solution. The HLLC Riemann solution requires an intermediate wave speed S * in addition to the estimated wave speeds S _L and S _R in the HLLC Riemann solution. When the governing equation is interpreted using the HLLC Riemann solution, the conserved variable U is divided into four sections according to the wave speeds S _L and S _R and S _* as shown in FIG. 1. It can be calculated using Equation 4.

&Quot; (4) "

Here, m is a denotes the interface between the grid F _L and F _R is and can be calculated as the initial condition, F _{* L,} and F _{* R} U _{* L} and U _{* R} required for the computation of the equation 5] Can be calculated

[Equation 5]

Where k = L, R. In order to calculate the correct flow rate using the HLLC technique, the wave velocity S _L and S _* and S _R must be applied correctly. In the present invention, Equations 6 to 8 are used to calculate S _L and S _R , and Equation 9 is applied to calculate the wave velocity S * in the intermediate region.

&Quot; (6) "

When a shock wave with h _* > h _k occurs

[Equation 7]

When a shock wave with h _* ≤ h _k occurs

[Equation 8]

When dry areas appear with h _k = 0

&Quot; (9) "

Here, q _k (k = L, R) can be obtained by Equation 10. And the depth and flow rate h _* and u _* of the intermediate region was calculated by applying [Equation 11].

&Quot; (10) "

&Quot; (11) "

Where u _L , u _R , h _L and h _R are the left and right initial values for the Riemann problem.

In order to calculate the conserved variables U = [h, hu, hv] ^T in each lattice using the HLLC Riemann approximation, first we need to calculate the flow rate at the lattice boundary. And in order to calculate the flow rate at the interface, it is necessary to know the information about the computational lattice and the adjacent lattices that share the boundary with the computational lattice. Accordingly, in the present invention, the computational lattice and the neighboring lattice find the neighboring lattice of the computational lattice by using a property that always shares two nodes regardless of the shape of the grid.

FIG. 2 shows a flowchart for finding adjacent grids, and FIG. 3 shows a process for finding adjacent grids of a triangular grid and a rectangular grid.

For all the grids, as shown in FIG. 3, the angle formed by the coordinates of the center of the calculation lattice and the nodes constituting the calculation lattice is calculated using Equation 12, and the calculated angles of the nodes are in descending order. Set the order. (S110, S120)

[Equation 12]

After reading the node number counterclockwise from the two nodes of lower order in the calculation grid, read the node number in the same way as the calculation grid for the remaining grids except the calculation grid. Repeat until it matches. (S130, S140)

The lattice with two shared nodes becomes the neighboring lattice of the computational lattice and the computational lattice, and there are four neighboring lattice when the lattice is rectangular lattice and three adjacent lattice. (S150)

Calculate the retention rate U at the next time step by calculating the flow rate at the boundary of the computational and adjoining lattice and substituting the calculated flow rate into [Equation 3].

In the present invention, the method of finding the adjacent lattice can be easily applied to the complex terrain such as the experimental road under simple terrain and the natural river and the floodplain without any limitation on the configuration of the grid.

Proper numerical processing of the generated terms in the finite volume model is very important for calculating accurate results. In the present invention, the bottom slope and the friction slope are considered as the terms of the shallow water equation. In particular, the riverbed slope was calculated by dividing the triangular and rectangular grids for the application of the mixed grid.

The bottom slope of the triangular grid is found by the vector equation of the plane because the three nodes constituting the grid exist on one plane. In other words, the equation of the plane passing through the three nodes of the triangular grid is formulated and calculated in the x and y directions. In the present invention, Equation 13, in which the vector equations of planes are arranged, is used to establish a plane equation consisting of three nodes.

&Quot; (13) "

If Equation 13 is summarized briefly, it becomes an equation of plane like Equation 14.

&Quot; (14) "

Computation of the x and y directions, respectively, can be expressed as in [Equation 15].

&Quot; (15) "

The bottom slope for the rectangular grid is calculated as shown in [Equation 16].

&Quot; (16) "

In general, the Manning roughness coefficient is about 10 ^-2 in the channel, so the friction slope has very little effect on the flow rate. However, at shallower depths, friction is a very dominant force. In the present invention, Equation 17 is used to consider the friction point.

&Quot; (17) "

In order for the HLLC Riemann solution applied in the present invention to provide a stable solution, the characteristic line must be in the region used for the HLLC flow rate calculation. Therefore, in the present invention, the calculation time interval is calculated using the characteristic velocity and the distance between the grids, and the smallest value is determined to secure the stability of the calculation.

&Quot; (18) "

Here, C _cfl is a Courant number and must satisfy the range of 0 <C _cfl ≤1.

Figure 4 shows the location of the planar shape, specifications and measurement points for the experimental underflow of the dam collapse experiment by connecting the rectangular reservoir to the L-shaped channel.

As shown in Figure 4, the experimental underpass is composed of a rectangular reservoir and a 90-degree bend, the bottom elevation of the L-shaped underpass is 0.33m higher than the bottom elevation of the reservoir. In addition, the reservoir and the sewer are separated by a hydrologic gate, which causes the dam to open suddenly, forming a dam collapse wave, and the end of the sewer is open to allow the collapsed wave to flow out.

FIG. 5 shows a calculation region composed of a mixed lattice with respect to the experimental diagram shown in FIG. 4, and a portion determined to be severely changed in flow velocity or flow characteristics is constituted by triangular lattice by dividing the rectangular lattice.

FIG. 6 shows an aspect of the flow rate of the dam collapse wave propagating to the wet channel at 1 and 8 seconds after the collapse of the dam, and FIG. 7 is an aspect of the velocity of wave propagating to the dry channel for the same time.

As shown in Fig. 6 and 7, even in the 0.33m elevation difference close to the vertical of the reservoir bottom and the bottom of the bottom well express the flow rate at the reservoir outflow, the flow rate in the reservoir and the reservoir and the sewer over time after the dam collapse It can be seen that the flow velocity gradually decreases in the vicinity of the boundary. In addition, the flow rate of the inner portion is greater than the outer portion before passing through the curved portion of the curved portion of the triangular lattice, but after passing through the curved portion, the flow velocity of the outer portion is larger than the inner portion. The transverse water level and flow rate distribution are shown in Table 1 for the four points indicated in FIG. 5.

[Table 1]

Comparing the wet and dry conditions, it can be seen that in both 1 and 8 second time periods, wet waves are spreading faster than dry conditions. One second after the collapse of the dam, flood waves from wet sewers spread about 4.3 meters, while those from dry sewers spread about 4 meters. In the case of a wet channel at 8 seconds, the flood wave is already escaping out of the channel, but in the dry channel, the flood wave propagates late and the flood wave starts to flow out from the channel. It can be seen that is faster than the propagation speed inside.

8 and 9 show the comparison between the water level calculated by the square grid and the water level calculated by the mixed lattice, the measured water level, and the results of Petaccia, respectively, for wet and dry waterways.

As shown in FIG. 8 and FIG. 9, the arrival time and level of the flood wave calculated at most points were relatively well matched with the measured values. In G2, G3 and G4, there are two levels of water level increase under both conditions. The first is the increase of water level due to the collapse of the dam, and the second is the increase of the water level due to the stagnation and reflection of the dam collapse flow due to the curvature of the channel. The closer to the bend, the shorter the second water level increase time and the longer the farther it is, the better the reflection and congestion caused by the bend (G1 ~ G4).

In FIG. 8 and FIG. 9, the comparison of each point of the wet and dry channels shows that the flood wave propagates faster than that of the dry channel, and the result of applying the mixed lattice is square grid. The accuracy of the mixed lattice application method is also confirmed because the accuracy is similar to that of In addition, FIG. 9 shows an analysis result of a dry road, which is one of difficult mathematical problems, and this model shows a reasonable result without numerical divergence even when a dry road with a depth of 0.0m is applied.

The method according to the present invention was applied to the collapse of the Malpasset Dam, where the actual maximum trace level exists to verify the actual natural rivers and floodplains so that efficient mixed lattice can be applied to complex terrain treatment.

The highest level of flood wave propagated downstream due to the collapse of the dam was investigated by the police as the trace level of the right bank of the Reyran river. The surveyed points were represented by P1 to P17 of FIG. The hydraulic model test for the Malpasset dam collapse was carried out in 1964 with a reduced scale of 1: 400 at the National Institute of Hydraulics (EDF-LNH) in France, and the highest flood level was measured in the downstream through the reproduced Malpasset dam collapse test. The measured point in the laboratory is represented by S6 ~ S14 of FIG. The actual trace level measured in the field survey and the hydraulic model test were used to verify the development model.

FIG. 11 shows a mixed grid of 13,850 triangular and square grids to simulate topographic elevation and dam collapse in the downstream region of the Malpasset Dam. As the boundary condition for the simulation, the reservoir inflow is very small compared with the dam collapse flow, so it is not considered as the boundary condition. Therefore, the upstream boundary condition was designated as a closed boundary condition with no inflow, and the downstream boundary condition was designated as a free outflow boundary condition in which the flood wave flows out to the coast. As the initial condition, the initial water level of the reservoir was 100m and the initial depth of the downstream part of the dam was 0.0m. That is, the flow rate of the reservoir with the initial water level of 100m without the flow into the reservoir was simulated to propagate downstream to the dry bottom without depth due to the collapse of the dam. Manning roughness coefficient was set to 0.033 in all areas.

FIG. 12 shows a comparison between the calculated results of the model, the highest trace level investigated in the field, and the measured flood level measured in the laboratory. 12 (a) and (b) show a comparison between the highest trace level at the survey points of the left and right banks and the calculation level of the present model, and FIG. 12 (c) shows the measurement obtained through the hydraulic model experiment of the EDF. Show the comparison between the water level and the calculated water level. As can be seen in Figure 12, the calculation result of the development model was relatively in good agreement with field survey data and experimental data.

13 and 14 show two- and three-dimensional flood depths and patterns at 1, 5, 20, and 35 minutes after dam collapse, respectively. Flood waves show flooding as they propagate around the main rivers, and as they move away from the main rivers, the depth of the water becomes shallower. As can be seen in FIGS. 13 (a) and (b) and 14 (a) and (b), the dam collapse waves propagate rapidly downstream through the narrow, steep valleys downstream of the dam. 13 (c) and (d) and 14 (c) and (d), however, the lower the floodplain, the slower the propagation time of the flood wave, the shallower the depth, and the wider the tip of the wave. Spreads throughout the area In general, flood waves show flooding as they propagate around the main rivers, and as the distance from the main rivers increases, the depth of the water becomes shallower. And as the water level in the reservoir, which was initially 100 meters, was propagated downstream due to the dam collapse, it can be seen that the water level in the reservoir dropped sharply for 35 minutes after the dam collapsed. 13 and 14 show that the results of the development model calculated by applying the mixed grid well reproduce the main characteristics of the flood due to the collapse of the Malpasset dam. And it shows that the technique according to the present invention can be applied simply and efficiently to irregular terrain such as natural rivers.

Although the present invention has been described with reference to the preferred embodiments thereof, it will be apparent to those skilled in the art that various modifications and variations can be made in the present invention without departing from the spirit and scope of the invention as defined in the following claims. It can be understood that

Claims (7)

In the fluid flow analysis method by the two-dimensional finite volume method,
The fluid flow is interpreted using the conserved variables in each computational grid,
The conservation variable is obtained by the flow rate and generation term at the boundary between the computational lattice and the adjacent lattice, and the calculation of the flow rate at the boundary is based on the discrimination of the computational lattice and the adjacent lattice, sharing two nodes The fluid flow analysis method for determining the calculation grid and the adjacent grid through the characteristic to.
The method of claim 1,
The determination of the adjacent grid,
Obtaining an angle formed between the center coordinate of the computational grid and the coordinates of the nodes;
Ordering nodes in order of decreasing magnitude of the angle;
Reading node numbers from two nodes of low order in the calculation grid;
Repeating the number of nodes in the same manner as the calculation lattice for the remaining grids except for the calculation lattice, until the two grid numbers coincide with the two nodal numbers of the lattice lattice; And
Determining a lattice having the shared grid and two shared nodes as an adjacent grid;
Containing, fluid flow analysis method.
The method of claim 1,
Wherein said generating term includes a bed diameter and a friction diameter.
The method of claim 3,
Wherein the riverbed matter is calculated by distinguishing when the triangular grid and the rectangular grid.
The method of claim 1,
The computational grid is any one of a triangular grid or a square grid, fluid flow analysis method.
The method of claim 5,
The computational grid may consist of a triangular grid at the point where the variation of flow velocity and flow characteristics is large, and the rest of the grid may consist of a rectangular grid, or in the flow and flood analysis, the stream may be a square grid and the floodplain may be a triangular grid. Or, in urban flooding analysis, buildings and roads may consist of a rectangular grid and the remaining submerged area may consist of a triangular grid.
In the method of determining the adjacent lattice of the mixed lattice in the fluid flow analysis by the two-dimensional finite volume method,
Obtain an angle formed by the center coordinates of the calculation grid and the coordinates of the nodes constituting the calculation grid, determine the number of the nodes in order of decreasing magnitude of the angle, and select the node number from the two nodes with the lowest order in the calculation grid. After reading, repeat the number of nodes for the remaining grids except the computational grid until the two nodes of the computational grid coincide with each other. Method of determining adjacent grid of mixed lattice.

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