KR20120012249A - Method for Calculating Conservation Variable and Numerical Analysis Method of 2-dimentional Fluid Flow using the Same - Google Patents

Abstract

PURPOSE: A conservation variables constructing method in a finite volume method and a 2D flux flow analysis method are provided to reconstruct a conservation variable in a boundary surface of a calculation lattice. CONSTITUTION: A conservation variable from a time n to n+1 is calculated. The conservation variable of a time n+1 reconstructs the conservation variable based on an area weight, and is obtained by using the conservation variable, an x-direction flux rate and a y-direction flux rate in a boundary surface of a calculation lattice, and a generating term on the calculation lattice(S40). The generating term includes a down slope term and a frictional slope term.

Description

Method for constructing conserved variables by finite volume method and method for analyzing two-dimensional fluid flow using the method {Method for Calculating Conservation Variable and Numerical Analysis Method of 2-dimentional Fluid Flow using the Same}

The present invention relates to the construction of a conserved variable of the finite volume method and a method for analyzing two-dimensional fluid flow using the same. It is about an interpretation method.

Two-dimensional numerical modeling techniques for flow, similarity, and contaminant analysis require numerically stable and consistent algorithms. In particular, new techniques have been studied that extend the finite volume technique for discontinuous flow analysis.

The application of the MUSCL technique to improve the accuracy of two-dimensional finite volume models has been studied at home and abroad. However, most of the MUSCL techniques applied up to now have reconstructed the conservation variables by allocating the conservation variables calculated in consideration of the conservation variables of the computational grid and the neighboring grid to the interface or by equally distribution based on the average width of the computational grid. .

However, these conventional methods may be a method of implementing an accurate reconstruction on the orthopedic lattice, but there is a limit to the application of the atypical lattice.

One object of an embodiment of the present invention for solving the above problems is to provide a method for constructing a conserved variable by a finite volume method applicable to an atypical lattice and a method for analyzing a two-dimensional fluid flow using the same.

Another object of the present invention is to provide a method of constructing a conserved variable by a finite volume method capable of reconstructing a conserved variable at a boundary of a calculated lattice using area weights and a method of analyzing a two-dimensional fluid flow using the same. It is.

In addition, an object of the present invention is to provide a method for constructing a conserved variable by a finite volume method and a two-dimensional fluid flow analysis method using the same, which can be used for dam collapse analysis.

According to a preferred embodiment of the present invention for achieving the above objects, a method for constructing a conserved variable in a two-dimensional fluid flow analysis by a finite volume method according to an embodiment of the present invention, to obtain the center of the calculation grid, The triangular small lattice may be divided based on, to obtain area weights, and the conservation variable at the interface may be obtained using the area weights.

Preferably, the area weight can be obtained by the following equation.

Where ω is the area weight value in the first triangle, a ₁ is the area of the first triangle, and a ₃ is the area of the third triangle located on the opposite side from the center of the city center.

In the two-dimensional fluid flow analysis method by the finite volume method according to an embodiment of the present invention, the storage variable from n hours to n + 1 hours is calculated, the storage variable of n + 1 hours, using the area weight Reconstructing the conserved variable, obtaining an x-direction flow rate and a y-direction flow rate at the boundary of the calculation grid, obtaining a generation term from the calculation grid, and the retention variable, an x-direction flow rate, and a y-direction flow Obtaining a conservation variable of n + 1 hours using the rate and generation term.

The storage variable at n + 1 hour can be obtained by the following equation.

Where A _l , N _l are the area of the computational grid l, the number of interfaces, F _{l , k} , G _{l , k} are the flow rates in the x and y directions at the k th interface, and n _{l , k} and L _{l , k} are It represents the outward unit vector of the kth boundary surface of the calculation grid l and the length of the boundary surface.

The generation term may include a bottom diameter matter and a friction diameter matter.

Preferably, the generation term can be obtained by the following equation.

Here, 1, 2, 3, and 4 represent the order of the nodes constituting the computational grid.

Preferably, the lattice structure may be an amorphous lattice.

According to the method of constructing the conserved variable by the finite volume method and the two-dimensional fluid flow analysis method using the same according to an embodiment of the present invention, there is an effect that can be applied to both orthopedic lattice and atypical lattice.

In addition, according to the method of constructing the storage variable by the finite volume method and the two-dimensional fluid flow analysis method using the same according to an embodiment of the present invention, the effect of reconstructing the storage variable at the boundary of the computational grid using the area weight value There is.

In addition, according to the method of constructing the storage variable by the finite volume method and the two-dimensional fluid flow analysis method using the same according to an embodiment of the present invention, there is an effect that can be efficiently used for dam collapse analysis.

1 is a flowchart showing a method for obtaining a conservation variable at n + 1 time in a two-dimensional fluid flow analysis method by a finite volume method according to the present invention;
2 shows a computational grid divided into triangular small lattice by the method according to the invention;
3 is a numerical simulation graph of an experiment taking into account the influence of a building on dam collapse;
4 is a graph comparing the measured depth and the depth calculated by the present invention for the case where the water level difference between the upstream and downstream portions of the dam is 4.9 cm;
5 is a graph comparing the measured depth and the depth calculated by the present invention for the case where the water level difference between the upstream and downstream portions of the dam is 19.9 cm.

Hereinafter, preferred embodiments of the present invention will be described in detail with reference to the accompanying drawings, but the present invention is not limited or limited by 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.

A two-dimensional shallow water equation consisting of conserved variables is expressed in [Equation 1].

[Equation 1]

Where U is the conserved variable vector, F (U), G (U) is the flow rate in the x- and y-directions, and S (U) is the generation term, which are shown in Equation 2 below.

[Equation 2]

Where u and v are the flow velocities in the x and y directions, g is the gravitational acceleration, and h is the depth. S _o is a bed slope and S _f represents a friction slope.

Integrating [Equation 1] with respect to the computational lattice (l) having an area A ₁ and applying Green's theorem, a basic equation as shown in [Equation 3] can be obtained.

&Quot; (3) "

Where N _l and L _{l , k} are the number of boundaries of the computational lattice (l) and the k-th boundary length, F _{l , k} and G _{l, k} are the x and y flow rates at the k-th boundary of the lattice (l), φ _k is the counterclockwise angle between the x-axis and the normal vector of the k-th interface and S _l represents the generation term.

For the calculation of the flow rates F and G in [Equation 3], the flow rate of mass and momentum at the boundary of the finite volume is calculated using the HLLC method, which is an approximate Riemann solution. The structure of the HLLC Riemann solution is divided into four sections according to the wave velocity S _L , S _R and S _* , and the numerical flow rate is calculated using Equation 4.

&Quot; (4) "

The MUSCL technique reconstructs the conserved variable at the boundary of the computational grid before calculating the flow rate. It uses the value of the adjacent lattice data to reconstruct the conserved variable and has a second-order accuracy in space. By predicting the value, we have a second order accuracy over time.

The two-dimensional fluid flow analysis method using the finite volume method according to the present invention uses the MUSCL method of reconstructing the area weighted value in order to enable physical reconstruction of the atypical lattice to which the MUSCL method is applied.

1 is a flowchart of a method for analyzing a two-dimensional fluid flow by the finite volume method according to the present invention, and FIG. 2 illustrates a computational grid divided into triangular small lattice.

Two-dimensional fluid flow analysis by the finite volume method according to the present invention reconstructs the conservation variable using the area weight. (S110)

Referring to FIG. 2, in order to reconstruct by area weights, the center grid G of the computational grid is obtained and the rectangular grid is divided into four triangular small grids based on the center grid G. In the present embodiment, a rectangular grid is described as an example, but is not necessarily limited thereto. For example, the computational lattice may be an orthopedic lattice, such as a pentagon or hexagon.

The areas a ₂ and a ₄ of the triangles DELTA 2G3 and DELTA 1G4 in the X-axis direction are calculated, and the areas a ₁ and a ₃ of the triangles DELTA 1G2 and DELTA 3G4 in the y-axis direction are calculated.

Using Equation 5, area weights for each triangle are calculated.

[Equation 5]

For reconstructing the conserved variables _, we apply ωa _{l , k} with area weights, where a _{l, k} represents the area of the k th boundary of the computational grid l. The reconstructed conservation variable is shown in Equation 6 below.

&Quot; (6) "

Here, m = 2 in the x direction and m = 3 y = 1 in the n = 4 y direction.

The x-direction flow rate and the y-direction flow rate are calculated at the computational grid boundary (S120), and the generation term is processed (S130).

Conservation variables between n + 1 can be obtained by Equation 7 (S140).

[Equation 7]

Where A _l and N _l are the area of the computational grid l and the number of interfaces, respectively, and F _{l , k} and G _{l , k} are the x and y flow rates at the k th boundary of the grid l, where k = 2 , 4 and in the y direction k = 1, 3. And n _{l , k} and L _{l , k} represent the outward unit vector and the boundary length of the k th boundary of the grid l, respectively.

As the generating term, the bottom slope and the friction inclination can be considered, and the generating term can be treated as shown in [Equation 8].

[Equation 8]

Here, subscripts 1, 2, 3, and 4 indicate the order of nodes constituting the computational grid.

In the boundary grid of the computational area, the flow rate with the adjacent grid is required, so a virtual grid is required at the boundary. Therefore, boundary conditions can be imparted by performing appropriate numerical processing on the virtual lattice adjacent to the boundary. In the present invention, the closed boundary condition in which the flow is blocked by the wall and the flow cannot exist in the direction perpendicular to the boundary surface is expressed by Equation 9, and the free outflow boundary condition is a condition in which conditions outside the boundary surface do not affect the adjacent flow. Equation 10 applies.

[Equation 9]

,

,

,

,

[Equation 10]

,

,

,

,

Here, h _* , u _*, and v _* represent the water depth in a virtual lattice, and the flow velocity of x and y directions.

3 is a graph showing a numerical simulation of the experiment in consideration of the influence of the building during the collapse of the dam.

With reference to FIG. 3, the measured value was compared with the numerical simulation result by Noel, and the result by the two-dimensional fluid flow analysis method by the finite volume method which concerns on this invention.

At G1 to G5, measured points located downstream of the dam, the time when the calculated water level initially increases is slightly later than the measured water level. However, the time difference is not large, and the difference is almost constant at all five observation points. As a cause of this constant time difference, Noel, who conducted the experiment, said that it was due to the abnormal initial time setting of the observation instrument. In G1 adjacent to the upper wall of the building and the left wall of the lower road, the tip of the dam collapse wave is reflected by the left wall inclined at the bottom, causing a sudden increase in the water level in the second generation. As it was not able to communicate smoothly and was congested, the wave coming back was well simulated by the phenomenon of square frequency due to the reflection of the left wall. In G2, the frequency of the building was found in the 13th century, but there was a slight difference from the measured frequency of occurrence, which is considered to be due to the inaccurate roughness coefficient and the initial water level. Noel simulated the change of the water level according to the change of the roughness coefficient and the initial level of the channel, and showed that the roughness coefficient and the initial level of the model can have a great influence on the accuracy of the calculation results. In G4 located on the right side of the building, there is a sudden increase in the water level due to the strong reflection wave caused by the building in the 3rd generation, and in the 4th generation, the water level increases slightly as the rectangular power generated by the right wall of the channel reaches G4. From 12 seconds later, the water level gradually decreased due to the congestion caused by the building and the right wall. In the G6 located upstream of the reservoir, the decrease of the water level as the water of the reservoir flows downstream shows almost the same as the measured water level.

FIG. 4 is a graph comparing the measured depth for the case where the water level difference between the upstream portion and the downstream portion of the dam is 4.9 cm and the depth calculated by the present invention, and FIG. 5 is the case where the water level difference between the upstream portion and the downstream portion of the dam is 19.9 cm. It is a graph comparing the measured depth and the depth calculated by the present invention.

4 and 5, after the dam collapse, the shock wave propagates downstream, the shock wave reaching the downstream is reflected back upstream due to the influence of the beam, the reflected wave reaching the wall of the upstream end is reflected back downstream It can be seen that the propagation process of is almost similarly reproduced by the present invention.

As described above, although described with reference to the preferred embodiment of the present invention, those skilled in the art various modifications and variations of the present invention without departing from the spirit and scope of the invention described in the claims below I can understand that you can.

Included in this specification.

ω: Area weight
A _l , N _l : the area of the grid lattice l, the number of interfaces,
F _{l , k} , G _{l , k} : flow rates in the X and y directions at the k th interface,
n _{l , k,} L _{l , k} : Outward unit vector of the kth boundary of the computational grid l and the length of the boundary

Claims (7)

In the method of constructing the conserved variable in the two-dimensional fluid flow analysis by the finite volume method,
Two-dimensional fluid flow analysis by finite volume method characterized by obtaining the centroid of the computational lattice, dividing the triangular small lattice on the basis of the centroid, and obtaining the conservation variable at the interface using the area weight. To configure retention variables in the database.
The method of claim 1,
The area weights are obtained by the following equation.

Where ω is the area weight value in the first triangle, a ₁ is the area of the first triangle, and a ₃ is the area of the third triangle located on the opposite side from the center of the city center.
In the two-dimensional fluid flow analysis method by the finite volume method,
Compute the conserved variable from n hours to n + 1 hours,
The n + 1 hour retention variable,
Reconstructing the conservation variable using the area weights;
Obtaining a flow rate in the x direction and a flow rate in the y direction at the boundary of the computational grid;
Obtaining a generation term from the calculation grid; And
Obtaining a conservation variable of n + 1 time using the conservation variable, the x-direction flow rate, the y-direction flow rate, and the generation term;
Including,
2D Fluid Flow Analysis by Finite Volume Method.
The method of claim 3,
The storage variable at n + 1 time is obtained by the following equation, the two-dimensional fluid flow analysis method by the finite volume method.

Where A _l , N _l are the area of the computational lattice l, the number of interfaces, F _{l , k} , G _{l , k} are the flow rates in the X and y directions at the k th interface, and n _{l , k} and L _{l , k} are It represents the outward unit vector of the kth boundary of the computational grid l and the length of the boundary.
The method of claim 3,
The generation term is a two-dimensional fluid flow analysis method by a finite volume method including a bed diameter matter and friction diameter matter.
The method of claim 5,
The generation term is a two-dimensional fluid flow analysis method by the finite volume method characterized in that obtained by the following equation.

Here, 1, 2, 3, and 4 represent the order of nodes constituting the computational grid.
The method of claim 3,
The computational lattice structure is a two-dimensional fluid flow analysis method by the finite volume method, characterized in that the amorphous grid.

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