KR20110072551A - Method for analyzing shallow water flow using the two-dimensional river flow model with tensor-type eddy viscosity - Google Patents

Abstract

PURPOSE: The eddy viscosity coefficient having 4 directionalities in the horizontal 2D plane is introduced and 2D finite element model based on the tensor type eddy viscosity coefficient is applied. The complex flux structure is reconstructed accurately in the various flow conditions. CONSTITUTION: The eddy viscosity coefficient showing the turbulence formation grade to the direction which does not accord with the cartesian coordinate system and the eddy viscosity coefficient showing distance to the river bed, the roughness coefficient, and about turbulence generation to the x/y-direction at the cartesian coordinate system and the gravitational acceleration are inputted with the input unit to the main memory. A CPU applies the finite element method to input data and equations. The depth of water at the arbitrary time and location, and the longitudinal speed of a current and traverse flux are produced.

Description

Method for analyzing shallow water flow using the two-dimensional river flow model with tensor-type eddy viscosity}

The present invention relates to a method for analyzing shallow water flow using a two-dimensional stream flow model having a tensor type eddy viscosity, and more specifically, to introduce a eddy viscosity having four directional viscosities in a horizontal two-dimensional plane and tensor. By applying the two-dimensional finite element model based on the vortex eddy viscosity coefficient, it is possible to more accurately reproduce the complex flow velocity structure under various flow conditions. Using a two-dimensional stream flow model with tensor-type eddy viscosity, which improves accuracy, applicability, and reliability by enabling lateral flow velocity distribution, partial water level analysis and detailed analysis of sediment and erosion by centrifugal force It is about how to interpret the water flow.

In general rivers, estuaries and coastal regions, the width is very wide compared to the depth, so the shallow water approximation is widely used to analyze the flow at the average velocity in the longitudinal direction and the transverse velocity in the depth direction.

The shallow water equation can be obtained by integrating three-dimensional continuous equations and momentum conservation equations with respect to time and depth, which include incompressible fluid, hydrostatic and bushesines approximation, introduction of eddy viscosity coefficients, and kinematic free water surface. Various assumptions and approximations are included, including boundary conditions (KFSBC), inertness at the bottom, net pressure conditions ignoring density stratification, and shallow water flow conditions where the horizontal dimension is much larger than the vertical direction.

The shallow water equations, which represent the shallow water flow characteristics, are composed of mass conservation equations and momentum conservation equations, and are numerically modeled from three partial differential equations, and the depth, longitudinal velocity, and transverse velocity are determined by inputting topographic data, boundary conditions, and parameters. The solution of the shallow water equation is used to interpret hydraulic behavior in the design and design of hydraulic structures such as weirs, dams and bridges, and to predict erosion and sedimentation due to meandering, shoal and cattle and turbulent distributions. It can be used for. In addition, the obtained velocity field is used as an input data for analyzing pollution diffusion and similar transport problems, and the ecological model is also used to provide flow velocity and depth information, which are basic data for estimating the suitability of fish habitat, and It can also be used as a basic analysis engine.

Here, the eddy viscosity term of the momentum conservation equation is divided into general type and tensor type according to the isotropy and anisotropy of the viscosity coefficient. The general type eddy viscosity coefficient is based on the concept of turbulent viscosity, and the Reynolds shear stress is proportional to the average velocity gradient. In the process of approximation, the proportional constant is set to a single value, so that the flow characteristics in the actual stream are ignored except that the main flow direction coincides with the Cartesian coordinate system, ignoring the direction due to the density of the fluid, the magnitude of the flow velocity, and the flow conditions. It is difficult to predict accurately. Existing domestic and foreign two-dimensional surface water analysis commercial model ignores the anisotropy of eddy viscosity, and analyzes the flow of fluid with general eddy viscosity, so that the flow direction is changed when the main flow direction is changed or the direction is not coincident with the rectangular coordinate system Because they are not properly simulated, they represent a significant difference from the flow behavior in natural rivers.

SUMMARY OF THE INVENTION The present invention has been made to solve the above problems, and an object of the present invention is to introduce a eddy viscosity having four directional viscosities in a horizontal two-dimensional plane and to develop a two-dimensional finite element model based on a tensor type eddy viscosity. It is possible to more accurately reproduce complex flow structures under various flow conditions.In particular, analysis of main flow distribution in rivers with meandering characteristics and lateral flow velocity distribution and single-level phenomena due to centrifugal force in the channel where the main flow direction changes rapidly. And a method for analyzing shallow water flow using a two-dimensional stream flow model with a tensor-type eddy viscosity, which improves accuracy, applicability, and reliability by enabling detailed analysis of sediment and erosion parts. .

In order to achieve the above object, the present invention provides a gravitational acceleration (g), distance from the baseline (H), roughness coefficient (n), the degree of turbulence in the x direction and the y direction in the Cartesian coordinate system. The eddy viscosity coefficients (ν _xx , ν _yy ) and the eddy viscosity coefficients (ν _xy , ν _yx ) representing the degree of turbulence formation in a direction not coincident with the rectangular coordinate system are input to the main memory by the input device as input data, The central processing unit applies the following equations, i.e., using a program in the main memory which is numerically modeled by applying the finite element method to the input data and the following equations.

,(a)

,

및(b)

And

을 만족하는 임의의 시간과 위치에서의 수심(h)과 종방향 유속(u) 및 횡방향 유속(v)을 산출하는 것을 그 기본 특징으로 한다.(c)

The basic feature is to calculate the water depth h, the longitudinal flow rate u and the transverse flow rate v at any time and position satisfying.

As described above, the method of analyzing the shallow water flow using the two-dimensional stream flow model having the tensor type eddy viscosity coefficient according to the present invention introduces a eddy viscosity coefficient having four directional viscous coefficients in a horizontal two-dimensional plane and the tensor type eddy viscosity coefficient. By applying the two-dimensional finite element model based on, the complex flow velocity structure can be more accurately reproduced under various flow conditions. Improves accuracy, applicability, and reliability by directional flow rate distribution, partial level analysis, and detailed analysis of sediment and erosion, and improves the habitat environment of fish and aquatic ecosystems by accurately predicting streams and cattle. It has an effect.

When described in detail with reference to the accompanying drawings a preferred embodiment of the present invention configured as described above are as follows.

1 is a diagram showing a plan view of a meandering channel experimental apparatus and a topographic grid for numerical simulation, FIG. 2 is a diagram showing a kerosene velocity and a velocity vector diagram of a meandering channel according to the eddy viscosity coefficient shape, and FIG. 3 is a meandering channel 4 is a view showing a flow rate comparison at the curved vertex of Fig. 4 is a view showing the flow rate and streamline distribution according to the eddy viscosity coefficient in the rectangular channel, Figure 5 is a view showing a flow rate comparison in the diamond channel, 6 is a diagram illustrating a flow rate comparison in a circular channel.

Except when there is a vertical stratification at the top or bottom of a water body due to a density difference or a temperature difference, or when there is a circulation or residual circulation occurring in a bay where the depth is rapidly deepened. In rivers, estuaries and coastal regions, the width is very wide compared to the depth, so shallow water approximation is widely used to analyze the flow at the average velocity in the longitudinal direction and the transverse velocity in the depth direction.

The shallow water equation, which expresses the characteristics of the shallow water flow, is composed of the mass conservation equation of Equation 1 and the momentum conservation equation of Equation 2 and Equation 3. The depth, longitudinal velocity and transverse velocity at and are found.

In Equations 1 to 3, h is depth, u and v are longitudinal depth mean velocity, g is gravity acceleration, n is roughness coefficient indicating roughness of bed and wall, H is distance from baseline to bed , ν _T represents the eddy viscosity coefficient. Equations 2 and 3 are represented by tensors as follows.

Here, the subscripts i and j have the values of 1 and 2, respectively, and mean the sum of x ₁ and x ₂ when the same subscripts are continuously located.

The eddy viscosity (ν _T ) included in Equation 4 is an isotropic property defined in the same way, ignoring the directionality existing in the horizontal two-dimensional plane, and thus has an anisotropy having four directionalities as shown in Equation 5 below. (anisotropy) Tensor type eddy viscosity (ν _ij ) is introduced.

Substituting the values of 1 and 2 into the subscripts i and j of Equation 5 to solve the problem, Equation 6 and Equation 7 are as follows.

In the equations (6) and (7), ν _xx and ν _yy components of the tensor-type eddy viscosity coefficients are coefficients representing the degree of turbulence in the x and y directions in the rectangular coordinate system, and ν _xy and ν _yx are the rectangular coordinate system. Coefficient indicating the degree of turbulence formation in a direction that does not coincide with. According to the tensor theory, since ν _xy and ν _yx have the same value, the present invention expresses the tensor type eddy viscosity by three vortex viscosity coefficients ν _xx , ν _xy and ν _yy .

Comparing Equations 6 and 7, which are momentum conservation equations having a tensor type eddy viscosity, and Equations 2 and 3, which are momentum conservation equations having a general vortex viscosity coefficient, have one vortex viscosity (ν _T ) Increased to 4 (ν _xx , ν _xy , ν _yx , ν _yy ).

In actual natural phenomena, the magnitude and directionality of the eddy viscosity is very different depending on the density of the fluid, the magnitude of the flow rate, and the flow conditions. Therefore, the momentum conservation equation with the general eddy viscosity, ignoring the directionality, is limited in applicability. It is known that there is.

On the other hand, the shallow water equation is a partial differential equation that expresses the horizontal flow of natural rivers, but it has an analysis solution under certain boundary conditions and flow conditions, but it considers only one direction velocity and made many assumptions in deriving the solution through Laplace transformation. Therefore, the analysis of flow behavior using analysis solution in rivers with complex topography is very limited. Therefore, various numerical techniques exist to obtain approximate solutions. The finite difference method, which has the longest history, is very effective and easy to apply to structured grids, but it has to be applied to orthogonal curve grids with complex boundaries such as rivers. However, it takes a lot of time and effort to generate it, and the accuracy of numerical calculation is also lowered when a grid network lacking orthogonality is generated. Also, because dependent variables are defined at grid points, it is difficult to physically match the preservation of the flow rate through each side of the test volume. The finite volume method has advantages in preserving continuity of fluid transfer and handling transition flow analysis or dry / wetting phenomena, but it is not practical in areas with many lattice numbers due to the long calculation time. The finite element method, on the other hand, has a relative advantage in streams with complicated flow boundaries because it is free to adopt an unstructured grid that combines triangles or squares. In addition, it is possible to optimize the size of the grid, the number of nodes, and the accuracy of the solution by using an adaptive mesh. In addition, it is very direct and easy to apply to various Neumann boundary conditions, and since it is a numerical method based on a solid mathematical basis, it has the advantage of being able to perform mathematical analysis on error, convergence and solution accuracy.

In other words, the finite element method is based on elements, not nodes, and various interpolation functions can be employed to obtain accurate numerical solutions for nodes within elements. Therefore, in the present invention, in order to accurately and effectively analyze the flow of natural water generated from natural bodies such as rivers and coasts, the finite element model is applied to a motion equation having a tensor-type eddy viscosity coefficient.

Next, the present invention numerically models the momentum conservation equation and the continuous equation having a tensor type eddy viscosity describing the water flow through the process of Equations 8 to 17 below. Representing the main unknown flow velocity and depth as u can be approximated by Equation 8 according to the time (t) and space (x, y) variables.

는 수학적으로 근사된 해를 의미한다.Here, the subscript j means the position of the grid point, N _j is a shape function that is determined according to the shape and the order of the element at the j point,

Is the mathematically approximate solution.

Substituting Equation 8 into Equation 1, Equation 6, and Equation 7, which are governing equations, is an approximated value, and thus a non-zero residual R ( u ^h ) (Equation 9) can be obtained.

When the inner product of Equation 9 is taken using the weighted residual method by the Galerkin method, the integral in the element inner region of the equation multiplied by the shape function is zero as shown in Equation 10 below.

Here, Ω means the inner region of the element.

Since the momentum conservation equation is a quadratic partial differential equation, it is transformed into weak form using Green's theorem in the eddy viscosity term, and expressed in the form of an integral equation for the lattice points of each element. The momentum conservation equations are represented by Equations 11 to 13, respectively.

,

및

은 수심, 종방향 유속 및 횡방향 유속의 시간미분을 의미하며, Ω^e 는 개별 요소의 영역을 나타내고, n 은 각 요소에 포함되어 있는 격자점의 갯수이다.In Equations 11 to 13

,

And

Is the time differential of depth, longitudinal and lateral flow, Ω ^e represents the area of the individual elements, and n is the number of lattice points contained in each element.

Equations 11 to 13 may be simply expressed as in Equation 14 below.

은 유속 및 수심의 시간미분벡터를 의미한다.Where M is the mass matrix, K is the stiffness matrix,

Is the time differential vector of velocity and depth.

The x direction component n _x of the normal vector at any lattice point (i), as shown in Equation 15, in order to subtract the temporal derivative of Equation 14 using the Theta method and reflect the irregular terrain boundary A rotation matrix T (rotation matrix) consisting of and the y direction component n _y is introduced.

Substituting variables multiplied by a T in the momentum conservation equations in order to change the above equation 11 to equation (13) in the rotated coordinate system by T u = ^T u _T is expressed by the following equation (16) of.

Here, T ^T is the transpose matrix of the rotation matrix represented by Equation 15, θ is a value for adjusting the degree of sound solution introduced in the process of differential time derivative, u _{T , n} and u _{T , n + 1} Are the flow velocity and depth values at the current time and next time, respectively.

Since the stiffness matrix K of Equation 16 includes a nonlinear term based on the transfer term, a Jacobian matrix obtained by Newton-Raphson method is linearized as shown in Equation 17 below. This is made up. As a result, since it is difficult to analytically determine the depth, longitudinal velocity, and transverse velocity at any time and position from Equations 1, 6, and 7, the finite element method is used as in Equations 8 to 17. The numerical model is applied to obtain a numerical solution. The numerical model is a program in which an algorithm is directly coded through a programming language (Fortran, etc.) to be finally executed by a computer. The computer's central processing unit is stored in main memory and entered into main memory by the input unit (parameters including terrain data, boundary conditions and eddy viscosity) and at any time and location using the program. The depth, longitudinal flow rate and lateral flow rate of are calculated.

는 다음 시간에서의 유속 및 수심을 구하기 위해 가정 및 계산된 임시적 변수이며, Δu_T , Δv_T 및 Δh 는 유속 및 수심변수의 변화량이다.Where Δt _n is the time calculation interval,

Are temporary variables that are assumed and calculated to find the velocity and depth at the next time, and Δu _T , Δv _T and Δh are variations in the velocity and depth variables.

The components of the matrix of Equation 17 are as shown in Equations 18 to 28 below.

는 형상함수의 행 방향 성분의 x방향 편미분값,

는 형상함수의 행 방향 성분의 y방향 편미분값,

는 형상함수의 열 방향 성분의 x방향 편미분값,

는 형상함수의 열 방향 성분의 y방향 편미분값,

는 하천바닥의 x방향 편미분값,

는 하천바닥의 y방향 편미분값, u^k 는 다음 시간에서의 x방향 유속을 구하기 위한 임시적 변수, v^k 는 다음 시간에서의 y방향 유속을 구하기 위한 임시적 변수, h^k 는 다음 시간에서의 수심을 구하기 위한 임시적 변수이다.Where N _i is the row direction component of the shape function, N _j is the column direction component of the shape function,

Is the partial differential value in the x direction of the row component of the shape function,

Is the y-direction partial derivative of the row-direction component of the shape function,

Is the partial differential value in the x direction of the column component of the shape function,

Is the partial differential value in the y direction of the column component of the shape function,

Is the partial differential value in the x direction of the river bed,

Is the partial derivative of the y direction of the river bed, u ^k Is A temporary variable for obtaining the velocity in the x direction at the next time, v ^k is a temporary variable for obtaining the velocity in the y direction at the next time, and h ^k is a temporary variable for determining the depth at the next time.

In the numerical model developed in the present invention, the integration of Equations 18 to 28 is performed by using a Gaussian quadrature method, and in the case of n = 3, the triangular element network corresponds to the quadrangular element network. Write code to use a combination of triangle and square networks depending on the terrain.

In addition, the algebraic equation solution of Equation 17 is obtained by using the frontal method. On the other hand, when the general eddy viscosity is applied, the eddy viscosity of Equations 19 and 20 corresponding to the eddy viscosity is expressed as ν _T , which is not one of ν _xx , ν _xy , ν _yx , ν _yy . .

Furthermore, the numerical model developed in the present invention is applied to a rectangular cross-section meandering channel having two curved portions having a width of 1 m, a center angle of 150 °, and a meandering angle of 1.52 as shown in FIG. In the case of the hydraulic test cases measured using the micro-ADV device capable of measuring 50 flow rates per second, a specific case having the most noticeable lateral flow difference in the inside and the outside of the curved part was selected to apply the general vortex viscosity coefficient and the tensor type eddy viscosity coefficient. The model is verified by comparing the applied cases with the hydraulic test results as follows.

In order to derive the result value at each lattice point by dividing the hydraulic experimental apparatus of FIG. 1 (a) into a finite number of elements and a lattice, a rectangular finite element network of 448 elements and 513 grids as shown in FIG. And set the flow rate of 60 liter / sec and the depth of 30 cm as input conditions as shown in the numerical simulation conditions shown in Table 1 below, and the roughness coefficient of roughness in the case of painted smooth iron as presented in Chow (1973). Let it be 0.013 which is degree.

2 shows the kerosene velocity and velocity vector diagram in a meandering channel according to the shape of the eddy viscosity.

In the case of applying general eddy viscosity, the eddy viscosity is assumed to be isotropic, so ν _T = 8 × 10 ^-3 ㎡ / s, considering that it is a rectified experimental channel at room temperature, and apply tensor eddy viscosity. In this case, the experimental channel is meandering and the angle between the main flow direction and the Cartesian coordinate system changes continuously, so ν _xx = 8 × 10 ^-3 , ν _xy = 10 ^-3 , ν _yy = 8 × 10 ⁻³ m 2 / s. The eddy viscosity coefficient input here should be selected by repeatedly adjusting to express well flow.

In the case of FIG. 2 (b) applying the tensor type eddy viscosity, the inner flow velocity is distributed about 40% faster than the outer one at the first and second bends. In the case of FIG. 2 (a) to which the general eddy viscosity coefficient is applied, it can be seen that the flow velocity before approaching the curved portion is rapidly distributed and the flow velocity distribution is asymmetric.

The numerical values of the lateral flow velocity distributions simulated at the first curved portion S4 and the second curved portion S9 are shown in FIG. 3 in comparison with the measured data. Where b is the width of the channel and y is the coordinate system defined transversely from the left eye to the right eye of the channel. In the case of using the tensor type eddy viscosity, the lateral velocity distribution is very similar to the measured value, and the maximum and minimum velocity ranges agree well. However, it can be seen that in case of using the general vortex viscosity coefficient, the flow velocity difference between the inside and the outside of the curved portion is greatly distorted, and only the average flow velocity at the point of y / b = 0.5 is shown to be consistent.

In addition, the two-dimensional finite element model based on the momentum conservation equations with the tensor-type eddy viscosity coefficients verified above is applied to the sectional enlargement coefficient, the diamond channel and the circular channel compared with the general eddy viscosity coefficient as follows. same.

Figure 4 shows the flow rate and streamline distribution according to the eddy viscosity by inputting the numerical simulation conditions of the rectangular channel in which the cross section of Table 1 is rapidly expanded.

Divide the upstream boundary into three and input the flow velocity only at the center, and change the eddy viscosity coefficients in the xy and yx directions to 10 ^-4 ㎡ / s, 10 ^-5 ㎡ / s and 10 ^-6 ㎡ / s, respectively. The streamline connecting the tangent of the velocity vector at each point of the fluid is shown. In the case of Fig. 4 (b), very weak vortices are observed near the outermost streamline and the input flow rate is zero, with the largest eddy viscosity coefficient in the xy and yx directions of 10 ⁻⁴ m 2 / s. However, in FIG. 4 (a) in which the eddy viscosity coefficients in the xy and yx directions are reduced by 10 ⁻¹ m 2 / s, the outermost streamline is rotated in the direction in which the input flow rate is 0 and compared to FIG. 4 (b) near the rotating part. Very large vortices occurred. In the case of FIG. 4 (c), the rotation of the outermost streamline was more prominent with the smallest viscous coefficient ν _xy = 10 ⁻⁶ m 2 / s. The strength and degree of vortex occur in a wide variety of forms depending on the density of the fluid, the longitudinal flow velocity gradient, and the flow conditions due to the water temperature. Therefore, it can be confirmed that such a turbulence can be more accurately reproduced by the tensor type eddy viscosity coefficient. have.

FIG. 5 shows a flow rate distribution in a case where a general vortex viscosity coefficient is applied and a tensor type eddy viscosity coefficient is applied in a diamond channel.

As shown in FIG. 5 (a), the terrain is constructed using 600 square element networks and 726 grids, and a flow rate of 0.2 m 3 / s and a depth of 1.0 m are input as boundary conditions. Fig. 5 (b) is a flow field obtained by inputting a general vortex viscosity coefficient of ν _T = 10 ^-5 m 2 / s into the momentum conservation equation, and the inner flow rate is at an angle of 135 ° past an initial straight line. Although it appears faster than the outside, the minimum flow rate occurs at the outer vertex of the diamond shape, and the maximum flow rate occurs before and after the vertex, so that it does not reflect the actual physical phenomenon well. On the other hand, in the case of FIG. 5 (c) to which the tensor type eddy viscosity coefficient is applied, a small input of the xy component and the yx component of the eddy viscosity coefficient is used to simulate the flow separation at the portion where the direction of the channel is changed. It shows the accurate flow velocity distribution considering the effect of centrifugal force in the diamond-shaped portion of the angle of 45 ° and it reproduces well that the inner velocity is distributed faster than the outer portion at the vertex of the diamond.

Figure 6 shows the flow rate distribution in the case of applying a general eddy viscosity and a tensor eddy viscosity in a circular channel.

As shown in FIG. 6 (a), the terrain is constructed using 1,790 square element networks and 1,980 grids, and as shown in Table 1, the flow rate of 0.06 m 3 / s and the depth of 0.3 m are input as boundary conditions. If the roughness coefficient has a small value, the inner and outer flow speed difference does not occur greatly. Fig. 6 (b) shows a uniform flow without generating internal and external flow velocity differences throughout the circular channel with a flow velocity field obtained by inputting a viscous coefficient value of ν _T = 10 ^-6 m 2 / s, which is a general eddy viscosity coefficient. On the other hand, in the case of FIG. 6 (c) to which the tensor-type eddy viscosity coefficient is applied, the smaller the xy component and the yx component of the eddy viscosity coefficient, the better the separation of the flow in the Cartesian coordinate system and the inclined direction, and the main flow direction changes continuously. Reflecting the effect of centrifugal force in the circular channel, the inner flow velocity is about 30% faster than the outer channel.

1 is a diagram showing a topographic lattice network for meandering waterway experimental apparatus plan and numerical simulation;

2 is a diagram showing a kerosene velocity and a velocity vector diagram of a meandering channel according to the eddy viscosity coefficient form.

3 is a view showing a flow rate comparison at a curved vertex portion of a meandering channel.

4 is a graph showing the flow velocity and streamline distribution according to the eddy viscosity in a rectangular channel.

5 shows a flow rate comparison in a diamond channel.

6 shows a flow rate comparison in a circular channel.

Claims (2)

Gravitational acceleration (g), distance from baseline to bottom (H), roughness coefficient (n), and eddy viscosity coefficients (ν _xx , ν _yy ) indicating the degree of turbulence in the x- and y-directions of the Cartesian coordinate system, and Eddy viscous coefficients (ν _xy , ν _yx ) representing the degree of turbulence formation in a direction not coincident with the rectangular coordinate system are input to the main memory by the input device as input data, and the central processing unit and the following mathematics The finite element method is applied to the equations, and the following equations, i.e. (a)

, (b)

And (c)

Using a two-dimensional stream flow model with a tensor-type eddy viscosity, which calculates the depth (h), the longitudinal velocity (u) and the transverse velocity (v) at any time and location satisfying How to interpret shallow water flow. The method of claim 1, The numerically modeled program is represented by the following equation,

(here,

,

,

,

,

,

,

,

,

,

And

) Are stored in the main memory and inputted by the central processing unit into the main memory by the input device (terrain data (Ω ^e is the area of the individual element, N _i is the row direction component of the shape function, N _j is The thermal component of the shape function,

Is the partial differential value in the x direction of the row component of the shape function,

Is the y-direction partial derivative of the row-direction component of the shape function,

Is the partial differential value in the x direction of the column component of the shape function,

Is the partial differential value in the y direction of the column component of the shape function,

Is the partial differential value in the x direction of the river bed,

Is the y-direction partial derivative of the river bed), the parameter (ν _xx is the x-direction component of the eddy viscosity, ν _yy Is The y-direction component of the eddy viscosity, ν _xy is the xy-direction component of the eddy viscosity, ν _yx Is the yx component of the eddy viscosity, n ² Is the square of the roughness coefficient), the gravitational acceleration (g), the rotation matrix ( T ), the transpose matrix ( T ^T ), the calculation time interval (Δt _n ), the degree of negative resolution of the time differential term (θ), and Temporary variables that are assumed and calculated to find the velocity and depth of

), the amount of change in the x-direction flow rate (Δu _T ), the amount of change in the y-direction flow rate (Δv _T ), the amount of change in the depth of water (Δh), the flow rate and the depth vector ( u _{T , n} ) at the current time, and the x direction at the next time. Temporary variable (u ^k ) for obtaining the flow rate, temporary variable (v ^k ) for obtaining the flow velocity in the y direction at the next time, temporary variable (h ^k ) for obtaining the depth at the next time, and arbitrary using the program Analyzing shallow water flow using a two-dimensional river flow model with a tensor-type eddy viscosity, characterized by calculating depth (h), longitudinal velocity (u), and transverse velocity (v) at time and location How to.

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