JP7102741B2 - Fluid analyzer, fluid analysis method, and fluid analysis program - Google Patents

Description

The present invention relates to a fluid analyzer, a fluid analysis method, and a fluid analysis program.

Conventionally, a technique using computational fluid dynamics (CFD) for numerically analyzing a fluid flow by a computer has been known. For example, a technique for obtaining the shear stress of a fluid model in analyzing a fluid flow using a solid model and a fluid model modeled with finite elements has been proposed (see, for example, Patent Document 1). In this technique, the motion of a fluid model is represented by the Navier-Stokes equation based on the Immersed particle (IB) method, which is an example of the orthogonal lattice method. Then, the shear stress of the fluid model acting on the boundary between the solid model and the fluid model is obtained. When calculating the shear stress on this boundary, the velocity is interpolated from the point on the boundary to the point on the perpendicular line extending from the point on the boundary using the information of the center of gravity of the element in the fluid region, and the value is used. The virtual velocity is extrapolated to the calculation grid inside the boundary (inside the object), and the flow field is calculated by referring to the value.

In addition, in order to analyze the hydrodynamic effect of the shielding plate on an object such as an automobile or a railroad vehicle, a virtual value such that the velocity difference between the fluid and the boundary becomes 0 using the solid volume ratio occupied by the object in the calculation grid. A technique has been proposed in which a fluid force is obtained and an object shape is expressed by using the obtained virtual fluid force (see, for example, Patent Document 2). In this technique, the velocity is interpolated in the calculation grid near the boundary.

Further, a technique for simply performing a coupled simulation of a fluid and a rigid body has been proposed (see, for example, Patent Document 3). In this technique, the solid volume ratio included in the calculation grid is calculated based on the shape data of the rigid body, and the velocity of the rigid body motion is added to the calculation grid in the rigid body region based on the calculated solid volume ratio value, and the flow Calculate the field.

Japanese Unexamined Patent Publication No. 2014-102574 Japanese Unexamined Patent Publication No. 2016-164706 Japanese Unexamined Patent Publication No. 2004-21883

However, in the technique of Patent Document 1, the flow velocity is interpolated in the calculation grid inside the boundary, and the equation is discreteized in the calculation grid adjacent to the boundary in the same manner as the calculation grid away from the boundary. Higher-order equations must be used, which complicates the extrapolation algorithm and increases the computational load. In addition, the velocity gradient on the boundary using the flow field information is used to calculate the shear stress, but the obtained velocity gradient is only the gradient in the normal direction, and it is not possible to obtain the velocity gradient in the three-dimensional direction on the boundary. Have difficulty. Since the technique of Patent Document 2 calculates the virtual fluid force using the solid volume ratio occupied by the object in the calculation grid, it does not deal with the flow field near the actual boundary, and the calculation accuracy of the flow field near the boundary is high. descend. The technique of Patent Document 3 also determines the calculation grid of the rigid body region from the solid volume ratio and adds the velocity obtained from the rigid body motion to the calculation grid. Therefore, it does not deal with the flow field near the actual boundary, but near the boundary. The calculation accuracy of the flow field is reduced. Therefore, there is room for improvement in providing a fluid analysis device that analyzes a fluid flow with high accuracy with a simple configuration while suppressing a calculation load.

The present invention has been made in consideration of the above facts, and is a fluid analyzer, a fluid analysis method, and a fluid analysis program capable of analyzing a fluid flow with high accuracy with a simple configuration while suppressing a calculation load. The purpose is to provide.

In order to achieve the above object, the fluid analyzer of the present invention
A space including an object and a fluid that flows around the object and whose viscosity changes spatially is composed of a plurality of elements by a plurality of lattice lines that do not coincide with the boundary between the object and the fluid and are orthogonal to each other . The model definition part that defines the divided numerical calculation target model, and
It is related to the flow of fluid in the predetermined direction of the boundary based on the information in each of the plurality of elements including the boundary continuous in the predetermined direction using the numerical calculation target model defined in the model definition unit. A first calculation unit that calculates a first physical quantity indicating the velocity gradient of the fluid using the distance from the center point or the center of gravity point of the grid composed of the plurality of grid lines to the boundary .
Based on the first physical quantity calculated by the first calculation unit, a second physical quantity indicating the velocity gradient of the fluid related to the flow of the fluid in the direction intersecting the predetermined direction of the boundary is obtained . A second calculation unit that calculates using the first physical quantity and a relational expression relating to a predetermined velocity gradient that holds at the boundary .
Based on the first physical quantity calculated by the first calculation unit and the second physical quantity calculated by the second calculation unit, the behavior of the fluid flowing in a predetermined region of interest including the boundary is described. The derivation part to be derived and
It has.

According to the fluid analysis apparatus of the present invention, the first calculation unit includes a boundary between an object and a fluid that are continuous in a predetermined direction using a numerical calculation target model defined by dividing a space by a plurality of elements. Based on the information in each of the plurality of elements, the first physical quantity related to the flow of the fluid in the predetermined direction of the boundary is calculated. The second calculation unit calculates a second physical quantity related to the flow of the fluid in the direction intersecting the predetermined direction of the boundary based on the first physical quantity calculated by the first calculation unit. The derivation unit derives the behavior of the fluid flowing in the predetermined region of interest including the boundary based on the first physical quantity and the second physical quantity. In this way, since the second physical quantity related to the fluid flow in the intersecting direction is calculated based on the first physical quantity related to the fluid flow in the predetermined direction of the boundary, it is easy while suppressing the calculation load. With the configuration, the fluid flow can be analyzed with high accuracy.

The first physical quantity and the second physical quantity are velocity gradients of the fluid . In order to correctly grasp the behavior of the fluid , it is preferable to specify the velocity of the fluid based on the equation of motion of the fluid even in the vicinity of the boundary. For that purpose, it is preferable to use the velocity gradient of the fluid on the boundary when determining the velocity of the fluid near the boundary.

で表される式を用いて演算することができる。
また、前記第２演算部は、前記第２の物理量を、
予め定めた方向と交差する方向をｙ方向とｚ方向とし、前記ｙ方向と前記ｚ方向の流体の速度勾配を（∂ｕ／∂ｙ） _ｂ と（∂ｕ／∂ｚ） _ｂ とし、前記境界における接線方向の単位ベクトルをｓとｔ、前記物体が運動する際の回転角速度ベクトルをωとするとき、

で表される式を用いて演算することができる。
The numerical calculation target model is defined by being divided into a plurality of elements by a plurality of orthogonal grid lines . It is preferable that the model to be numerically calculated can be easily analyzed by numerical calculation and accurately represents the regions of an object and a fluid. Therefore, each region can be easily handled by defining the model to be calculated numerically by dividing it into a plurality of elements by a plurality of orthogonal grid lines.
The first calculation unit uses the first physical quantity.
(I, J) is the position of the element divided by the plurality of grid lines in the predetermined direction and the position in the direction intersecting the predetermined direction, the predetermined direction is the x direction, and b is (I). , J) Information indicating that it is the intersection of the straight line in the predetermined direction passing through the center point or the center of gravity point and the boundary, and the velocity gradient of the fluid in the x direction is (∂u _i / ). When ∂x) _b , the x component of the flow velocity is u, Δx is the width of the element in the x direction, and ε _x is the information obtained by normalizing the width from the center point or center of gravity of the lattice to the boundary with Δx. ,

It can be calculated using the formula represented by.
In addition, the second calculation unit calculates the second physical quantity.
The directions that intersect the predetermined directions are the y and z directions, and the velocity gradients of the fluids in the y and z directions are (∂u / ∂y) _b and (∂u / ∂z) _b , and the boundary is defined. When the unit vector in the tangential direction in is s and t, and the rotation angle velocity vector when the object moves is ω,

It can be calculated using the formula represented by.

The directions that intersect the predetermined directions are two directions that are orthogonal to each other and are orthogonal to the predetermined directions.
In order to grasp the flow of the fluid, it is preferable to specify the velocity component in the three-dimensional direction regarding the flow of the fluid. Therefore, in addition to the velocity gradients in the predetermined directions, the velocity components in the three-dimensional directions of the fluid flow are obtained by using the velocity gradients in the two directions orthogonal to each other and orthogonal to the predetermined directions. It can be grasped accurately.

The fluid analysis method of the present invention
The computer
A space containing an object and a fluid that flows around the object and whose viscosity changes spatially is incompatible with the boundary between the object and the fluid, and is composed of a plurality of elements by a plurality of orthogonal grid lines . The fluid related to the flow of the fluid in the predetermined direction of the boundary based on the information in each of the plurality of elements including the boundary continuous in the predetermined direction using the divided and defined numerical calculation target model. The first physical quantity indicating the velocity gradient of is calculated by using the distance from the center point or the center of gravity point of the lattice composed of the plurality of lattice lines to the boundary .
Based on the calculated first physical quantity, the second physical quantity indicating the velocity gradient of the fluid related to the flow of the fluid in the direction intersecting the predetermined direction of the boundary is referred to as the first physical quantity. Calculated using the relational expression related to the predetermined velocity gradient that holds at the boundary .
Based on the calculated first physical quantity and the calculated second physical quantity, the behavior of the fluid flowing in the predetermined region of interest including the boundary is derived.

The fluid analysis program of the present invention causes the computer to function as the fluid analysis device according to any one of claims 1 to 4.

As described above, according to the present invention, it is possible to obtain the effect that the fluid flow can be analyzed with high accuracy with a simple configuration while suppressing the calculation load.

It is a block diagram which shows an example of the structure of the fluid analysis apparatus which concerns on embodiment. It is an image diagram which shows an example which divided the region including the boundary between an object region and a fluid region using an orthogonal grid. It is an image diagram which shows an example of a two-dimensional object which moves rigidly. It is an image diagram which shows an example of the case which has a boundary in two directions. It is explanatory drawing of the extrapolation method of a comparative example. It is a flowchart which shows an example of the flow of the process for executing the fluid analysis method by the fluid analysis apparatus by the computer which concerns on embodiment. FIG. 5 is an image diagram showing an example of a schematic structure of an object through which a fluid flows, which is the object of verification in the first verification for verifying a fluid flow with the fluid analyzer according to the embodiment. It is an image diagram which shows the verification result of the 1st verification. It is an image diagram which shows the verification result of the 1st verification including a comparative method. FIG. 5 is an image diagram showing an example of a schematic structure of an object through which a fluid flows, which is the object of verification in the second verification for verifying the flow of fluid with the fluid analyzer according to the embodiment.

Hereinafter, embodiments according to the present invention will be described with reference to the drawings.
This embodiment describes an example of analyzing the flow of a fluid acting by the movement of a rigid body.

FIG. 1 shows an example of a configuration in which the fluid analyzer 10 according to the present embodiment is realized by a computer.
As shown in FIG. 1, the computer operating as the fluid analysis device 10 includes a device main body 12 including a CPU (Central Processing Unit) 12A, a RAM (Random Access Memory) 12B, and a ROM (Read Only Memory) 12C. It is configured. The ROM 12C includes a fluid analysis program 12P for realizing various functions for simulating the behavior of a fluid acting by the movement of a rigid body. The apparatus main body 12 includes an input / output interface (I / O) 12D, and the CPU 12A, the RAM 12B, the ROM 12C, and the I / O 12D are connected via the bus 12E so that commands and data can be exchanged. The I / O 12D includes an input unit 12F that enables numerical input, character input, and command input of a keyboard and mouse, a display unit 12G that enables image display, a communication unit 12H that communicates with an external device, and various types. A storage unit 12M for storing data is connected.

The apparatus main body 12 operates as the fluid analysis apparatus 10 when the fluid analysis program 12P is read from the ROM 12C and expanded into the RAM 12B, and the fluid analysis program 12P expanded in the RAM 12B is executed by the CPU 12A. The fluid analysis program 12P includes a process for realizing various functions for analyzing the flow of the fluid (details will be described later).

In the present embodiment, a non-Newtonian fluid in which the viscosity coefficient changes spatially will be described as an example of the fluid according to the present embodiment.

First, in order to enable the fluid analysis device 10 according to the present embodiment to numerically analyze the flow of the fluid, a method of decentralizing the flow field and deriving a velocity gradient at the boundary between the object and the fluid will be described.

When analyzing a fluid flow, the behavior of the fluid is expressed using a basic equation of the flow field or the like. The following equations (1) and (2) show an example of the basic equation of the flow field used in this embodiment. Equation (1) shows the continuity equation of the incompressible fluid, and equation (2) shows the equation of Navier-Stokes.

However, the symbols in the equations (1) and (2) indicate the following. _ui is the flow velocity, p is the pressure, and ρ is the density. τ _ij indicates a viscous stress tensor. This viscous stress tensor τ _ij can be expressed by the following equation (3) using the viscosity η and the strain rate tensor D _ij .

In the following description, the divergence of each of the first term on the right side and the second term on the right side of the above equation (3) will be referred to as a viscosity term 1 and a viscosity term 2, respectively.

In this embodiment, an analysis method for calculating a fluid flow using a calculation grid that does not match the boundary will be described. In this embodiment, the fluid flow is analyzed using an orthogonal grid. An example of a technique for analyzing a fluid flow using an orthogonal lattice is Document 1 (Sato et al., Direct dispersal method near the object boundary in the orthogonal lattice method (high accuracy considering the consistency between velocity field and pressure field). ), The technique shown in Proceedings of the Japan Society of Mechanical Engineers B, 79 (2013) 605-621).

Further, in the present embodiment, the collocation lattice method is used for spatial discretization. Then, the basic equation is discretized for the calculation grid adjacent to the boundary. In the following description, the discretization of the viscosity term related to this embodiment will be described.

FIG. 2 shows a region of interest 1 including a boundary 4 between the object region 2 and the fluid region 3 in which the object region 2 and the fluid region 3 are discretized using an orthogonal grid. The region of interest 1 shown in FIG. 2 shows a part of an object region 2 and a fluid region 3 in a two-dimensional plane passing through the X-axis and the Y-axis.

As shown in FIG. 2, the region of interest 1 is defined by a plurality of elements using a calculation grid that does not fit the boundary 4 between the object region 2 and the fluid region 3. A case where the object region 2 and the fluid region 3 are divided by using an orthogonal grid, that is, defined by a plurality of calculation grids 6 (9 in 3x3 in the vertical and horizontal directions in the example of FIG. 2) divided by orthogonal grid lines 5 Shown. In each of the calculation grids 6, calculation points 7 for calculating the velocity, pressure, temperature, etc. of the fluid when analyzing the flow of the fluid are defined. The calculation point 7 may be defined at any position in the calculation grid 6, but since it functions as a representative point representing the calculation grid 6, it is preferable to set the center point or the center of gravity point of the calculation grid 6. In the present embodiment, the case where the calculation point 7 is set as the center point of the calculation grid 6 will be described.

In the following description, when each of the plurality of calculation grids 6 is described separately, it will be described as a calculation grid (I, J) with an identification code indicating a position.

Further, in the following description, as shown in FIG. 2, the discretization of the viscosity term will be described for a two-dimensional calculation grid 6 having a boundary in the x direction. That is, the calculation grid (I, J), which is a specific calculation grid 6, has a boundary 4 only in the calculation grids (I-1, J) in the x direction among the adjacent calculation grids 6, and the boundary 4 in the y direction. Does not have.

By the way, assuming that the viscosity is spatially constant, only the first term of the viscosity term needs to be dealt with, but in the present embodiment, the second term of the viscosity term is also discrete in consideration of the spatial distribution of the viscosity. To make it. The discretization equation of the first viscosity term in the calculation grid (I, J) of FIG. 2 can be expressed by the following equation (4).

However, the symbols in Eq. (4) indicate the following. δ / δx _i indicates that it is a quadratic precision center difference, and the subscript b is the intersection of the x-axis straight line passing through the calculation point 7 of the calculation grid (I, J) and the boundary 4 (FIG. 2). Indicates the value at the position indicated by the triangle symbol (hereinafter referred to as boundary point 8).

The velocity gradient (∂u _i / ∂x) _b in the x direction at the boundary point 8 can be obtained by the one-sided three-point difference equation (spatial quadratic accuracy) shown in the following equation (5).

Next, the second viscosity term is discretized. The discretized second viscosity term can be expressed by the following equation (6). The x component of the flow velocity u _i is u, and the y component is v.

The first term on the right side of the above equation (6) includes a velocity gradient (∂u / ∂x) _b in the x direction and a velocity gradient (∂u / ∂y) _b in the y direction at the boundary point 8. Therefore, in order to discretize the second term of the viscosity term, in addition to the velocity gradient (∂u / ∂x) _b appearing in the discretization equation (Equation (4)) of the first term of the viscosity term, it is different from the x direction. A directional velocity gradient, i.e. velocity gradient (∂u / ∂y) _b , is required. However, since the calculation point 7 does not exist on the straight line extending in the y direction from the boundary point 8 for the velocity gradient in the y direction, the equation ((5)) used for the calculation of the velocity gradient (∂u / ∂x) _b . ) Cannot be calculated in the same way.
Therefore, in the present embodiment, the velocity gradient in the y direction at the boundary point 8 is derived using the relational expression of the velocity gradient on the boundary 4 and the calculated velocity gradient in the x direction. The derivation of the velocity gradient in the direction different from the velocity gradient in the specific direction having a boundary, that is, the velocity gradient in the direction different from the x direction will be described below.

Here, for the sake of simplicity, a case of deriving a velocity gradient for a rigid body moving object will be described. In a rigid-moving object, the following equations (7) and (8) hold at the boundary points on the surface of the object.

However, the symbols in the equations (7) and (8) indicate the following. ∇u is a velocity gradient tensor, s and t are tangential (bidirectional) unit vectors at boundary point 8, and ω is a rotation angular velocity vector of rigid body motion.

Here, the equation (7) will be described with reference to FIG.
FIG. 3 shows an example of a two-dimensional object that moves rigidly.

As shown in FIG. 3, the velocity vectors up and u ₀ at the point _P on the surface of the object and its neighboring points Q can be represented by the following equations (7A) and (7B), respectively.

u _G and ω _G are the velocity vector and the angular velocity vector of the rigid body motion, and r _P and r _Q are the position vectors from the rotation center point G of the rigid body to the points P and Q. s is a unit vector between PQs (tangential unit vector on the surface of the object), and h is a distance between PQs. According to the equations (7A) and (7B), the velocity gradient between PQs, that is, the velocity gradient in the s direction on the surface of the rigid body moving object can be expressed by the following equation (7C).

Further, the equation (7C) can be further expressed by the following equation (7D).

Since the same applies to the equation (8), the description thereof will be omitted.

Here, considering the component of the flow velocity u, the above equations (7) and (8) can be expressed by the equations (9) and (10), respectively.

However, the symbols s _x , s _v , s _z in the equations (9) and (10) and t _x , _tv , t _z indicate the components of s and t, respectively. Considering that the velocity gradients (∂u / ∂x) _b in the x direction of equations (9) and (10) are known by the extrapolation equation of equation (5), the velocity gradients (∂u / ∂u /) are other components. ∂y) _b and the velocity gradient (∂u / ∂z) _b can be expressed by the following equations (11) from the simultaneous equations of equations (9) and (10).

Similarly, the y and z components of the velocity gradients v and w in the directions orthogonal to the flow velocity u can also be expressed by equations (12) and (13), respectively.

As described above, for the boundary point 8, the velocity gradient in the x direction can be derived using the equation (5), and the velocity gradient in the y direction and the z direction can be derived using the equations (11) to (13). Can be derived.

In the example shown in FIG. 2, in the case of the motion (rotation angular velocity ω) in the two-dimensional (xy) plane, the equations (12) and (13) are the following equations (14) and (15). It can be simplified as shown in the formula.

When the object is stationary or the object motion is only translational motion, the rotational angular velocity ω is a zero vector, and the right side of equations (9) and (10) is 0.

In the above, for convenience of explanation, the case where the velocity gradient is provided in the x direction shown in FIG. 2 has been described, but it goes without saying that the calculation grid 6 having boundaries in the y and z directions can also be discretized in the same manner. be.

Further, in the above, the case where the velocity gradient is provided only in the x direction (FIG. 2) has been described, but the same method as described above can be independently applied to each direction even when the boundary is provided in a plurality of directions. .. For example, as shown in FIG. 4, when the boundary 4 is provided in two directions, the x direction and the y direction, the same can be applied to each of the x and y directions as described above.

In this embodiment, at the boundary point 8 (position of the triangle symbol in FIG. 2) on the boundary 4, the velocity gradient in the direction in which the difference reference point (fluid calculation point) exists (in the case of FIG. 2, (∂u / ∂x)). _b ) should be obtained from the difference equation using the information of the fluid calculation point, and the velocity gradient in the other directions (in the case of FIG. 2, the velocity gradient in the y direction (∂u / ∂y) _b ) should be established on the boundary. Obtained using the relational expression of the velocity gradient. Therefore, the velocity gradient (∂u / ∂y) _b in the y direction is guaranteed to have the same spatial accuracy as the velocity gradient (∂u / ∂x) _b in the x direction.

In the above equations (9) and (10), the "velocity gradient (∂u / ∂x) _b in the direction in which the difference reference point exists" was obtained by the difference equation ((5)) with secondary accuracy. However, it is also possible to change the spatial accuracy. For example, in the case of first-order accuracy, the following equation (5A) is used instead of equation (5).

The velocity gradient obtained by this technology has a viscosity coefficient of space, such as a calculation target with non-Newtonity of fluid, a calculation target with temperature dependence of viscosity, and a calculation target using a turbulent flow model such as LES (Large Eddy Simulation). It can be widely used for calculation of fluid distribution. It can also be used when calculating the frictional stress on the surface of an object.

By the way, as a general method for calculating the velocity gradient (∂u / ∂y) _b at the boundary point, the velocity gradient in each coordinate axis direction is obtained at the calculation point of the fluid region near the boundary, and the velocity gradient is extrapolated to the boundary point. An insertion method (extrapolation method) can be mentioned.
Here, a method of calculating the velocity gradient at the boundary point by the extrapolation method will be described as a comparative example. For example, when an extrapolation equation with spatial quadratic accuracy is applied to the velocity gradient (∂u / ∂y) _b shown in FIG. 2, it can be expressed by equation (16).

The velocity gradient on the right side of the above equation (16) can be calculated from the following equations (17) to (19).

FIG. 5 shows a region of interest 1 including a boundary 4 between the object region 2 and the fluid region 3 when the extrapolation method of the comparative example is applied.

As shown in FIG. 5, when the extrapolation method of the comparative example is applied, a large number of calculation points are required as compared with the method of deriving the velocity gradient according to the present embodiment. That is, as shown in the above equations (17) to (19), in the extrapolation method of the comparative example, a total of seven items indicated by black circles in FIG. 5 are used to calculate the velocity gradient (∂u / ∂y) _b . Information on calculation points is required. On the other hand, in the method for deriving the velocity gradient according to the present embodiment, the velocity gradient (∂u / ∂y) _b can be derived only by the difference equation of Eq. (5) and the analytical relational expression. Only the information of a total of three calculation points indicated by the X symbol is required. Further, in the present embodiment, when the equation (5) is used, the same spatial quadratic accuracy as that of the quadratic extrapolation method can be obtained. That is, the method for deriving the velocity gradient according to the present embodiment can obtain the same accuracy with less information on calculation points than the extrapolation method. Further, by minimizing the interpolation operation, the stability of the calculation is greatly improved, and the calculation can be performed stably in a large time interval as compared with the secondary extrapolation method, and the calculation time can be shortened.

Next, based on the method for deriving the velocity gradient described above, the simulation of the behavior of the fluid acting by the movement of the rigid body, that is, the flow of the fluid acting by the movement of the rigid body, which is executed in the fluid analyzer 10, is performed. The fluid analysis method to be analyzed will be described. In the ROM 12C of the fluid analysis device 10 realized by a computer, a processing flow (process) for executing the fluid analysis method according to the present embodiment is stored as a fluid analysis program 12P.

FIG. 6 shows an example of a processing flow for executing the fluid analysis method according to the present embodiment. In the apparatus main body 12, the fluid analysis program 12P is read from the ROM 12C and expanded in the RAM 12B, and the CPU 12A executes the fluid analysis program 12P expanded in the RAM 12B.

First, in step S100, an acquisition process of information indicating the calculation grid 6, calculation conditions, and initial values is executed.

The information indicating the calculation grid 6 is information in which an orthogonal grid that divides the object region 2 and the fluid region 3, that is, the spacing between the orthogonal grid lines 5 is defined. Information about the calculation grid 6 is stored in the ROM 12C or the storage unit 12M. The information indicating the calculation conditions is the conditions for numerically analyzing the fluid flow in the present embodiment. An example of the calculation conditions is the position information of the object model with respect to the fluid model, information such as the angular velocity that defines the movement of the object model, and an example of the calculation conditions of the fluid model is the boundary condition of the fluid model and the velocity field calculation. Predetermined information such as convergence conditions and the number of iterations is included. The calculation conditions are stored in the ROM 12C or the storage unit 12M. Further, the information indicating the initial value is information indicating the first various states in the case of numerically analyzing the flow of the fluid in the present embodiment. Examples of initial values include information such as angular velocity that defines the initial motion of the object model analysis, and information such as the initial velocity, pressure, and temperature of the fluid.

Further, in step S100, an object model showing an object and a fluid model showing a fluid are also acquired.

An object model is a model that defines, for example, the structure of a rigid body (non-fluid object) as an object. The object model may be defined by modeling with a finite number of elements that can be handled by the numerical analysis method. In addition, numerical data such as coordinate values and material properties (for example, viscosity, density, material, etc.) are defined in the object model. These numerical values are stored in the ROM 12C or the storage unit 12M.

A fluid model is a model that defines the structure of the space in which a fluid flows outside an object. The fluid model may be defined by modeling with a finite number of elements that can be handled by the numerical analysis method. In addition, numerical data such as coordinate values and material properties (eg, viscosity, density, and material) are defined in the fluid model. These numerical values are stored in the ROM 12C or the storage unit 12M.

Further, in step S100, a process of defining the entire region to be analyzed including the object and the fluid is executed using the acquired information. That is, the entire region including the object region 2 by the object model, the fluid region 3 by the fluid model, and the boundary 4 between the object region 2 and the fluid region 3 is divided by using an orthogonal grid, that is, divided by orthogonal grid lines 5. The process defined by the plurality of calculation grids 6 is executed. The position of the calculation point 7 is also defined in each of the calculation grids 6.
Therefore, step S100 has a function of operating as a model definition unit for defining a numerical calculation target model in which a space including an object and a fluid flowing around the object is divided by a plurality of elements.

Next, in step S102, a process related to the movement of the boundary is executed. The process of step S102 is a process of defining the position of the boundary 4 that moves as the position of the object moves from moment to moment. Therefore, if the position of the object is fixed, step S102 is omitted. Further, at the beginning of execution of this processing routine, the initial value may be used in step S102.

In the next step S104, the extraction process of the boundary point 8 which is the intersection of the boundary 4 and the calculation point 7 of the calculation grid 6 is executed. In this step S104, an operation for extracting the boundary point 8 is performed for each of the calculation grids 6 including the boundary 4.

In the next step S106, a process of calculating a unit vector in the tangential direction on the boundary surface at each of the boundary points 8 extracted in step S104 is executed. That is, using the above equations (7) and (8), the unit vector in the tangential direction on the boundary surface at each of the boundary points 8 is calculated.

In the next step S108, a process of calculating the velocity gradient in one direction at each of the boundary points 8 is executed. That is, for example, the velocity gradient (∂u _i / ∂x) _b and v in the x direction of the u component at the boundary point 8 by the one-sided three-point difference equation (spatial quadratic accuracy) using the above equation (5). The velocity gradient (∂v _i / ∂x) _b of the component in the x direction and the velocity gradient (∂w _i / ∂x) _b of the w component in the x direction are calculated respectively.

In the next step S110, a process of calculating the velocity gradient in the residual direction at each of the boundary points 8 is executed. That is, using the above equations (11) to (13), the velocity gradient (∂u / ∂y) _b of the u component, the velocity gradient (∂u / ∂z) _b , and the velocity gradient of the v component. (∂v / ∂y) _b and velocity gradient (∂v / ∂z) _b and the velocity gradient (∂w / ∂y) _b and velocity gradient (∂w / ∂z) _b of the w component. Calculate each.

In the next step S112, a process of calculating the velocity field and the pressure field of the fluid is executed. That is, using the velocity gradient in one direction at each of the boundary points 8 obtained in step S108 and the velocity gradient in the other direction obtained in step S110, the fluid is prepared according to the above equations (1) to (3). Calculate the velocity field and pressure field.

In the next step S114, the end determination process is executed, the process shifts to step S116 in the case of an affirmative determination, and returns to step S102 in the case of a negative determination. The end of step S114 is determined based on predetermined information such as the convergence condition of the velocity field calculation, the number of repetitions of the processing of steps S102 to 112, and the elapsed time from the start of calculation.

In the next step S116, a process of outputting the calculation result of step S112 is executed. In step S116, the calculation results of the fluid velocity field and the pressure field obtained in step S112 may be output as they are, and the fluid flow is based on the fluid velocity field and the pressure field calculation results obtained in step S112. The physical quantity such as speed and pressure may be calculated and the calculation result may be output.

As described above, according to the present embodiment, the behavior of the fluid can be simulated by obtaining the velocity gradient of the fluid acting by the movement of the rigid body. That is, it is possible to analyze the fluid flow with high accuracy with less information on calculation points than the extrapolation method.

Here, the analysis of the fluid flow in the fluid analyzer 10 according to the present embodiment is verified.
In the first verification, the accuracy of the fluid flow in the fluid flowing between the coaxial double cylinders is verified for the non-Newtonian fluid whose viscosity coefficient changes spatially.

FIG. 7 shows an example of a schematic structure of an object (rigid body) through which a fluid flows, which was the subject of verification in the first verification.

As shown in FIG. 7, the region 20 to be verified includes an object region having a flow path 22 having an inner diameter _{ri = KR and an outer diameter r 0 =} R, and a fluid region flowing through the flow path 22. Specifically, when the radius on the outer peripheral side of the flow path 22 is R, it is 0.2R in the x direction and 2.4R in the y and z directions, respectively, and the inner diameter r is set in the center of the calculation region in the yz plane. The region including the flow path 22 in which the coaxial double-cylindrical flow path having _i = KR and the outer diameter r ₀ = R is formed is defined as the area 20 to be verified.

The radius ratio K of the inner diameter and the outer diameter is 0.6. The wall surface is in a non-slip condition, and the flow in the flow path 22 is driven by the average pressure gradient dp / dθ added in the circumferential direction. Periodic conditions are imposed on both end faces of the area 20 to be verified in the x direction. As the non-Newtonian model, the Ostwald-de Wayle model (power law model) represented by the following equation (20) is used.

The viscosity η changes spatially depending on the velocity gradient of the flow field, except in the case of n = 1 (Newtonian fluid).

There is an analytical solution for the behavior (flow) of the fluid flowing through the flow path 22 between the coaxial double cylinders, which is the verification target in the first verification. The analytical solution can be obtained by the following equation (21).

However, the symbol u ^* _θ in the equation (21) is a dimensionless circumferential flow velocity.

The above equation (21) can be derived as follows.
The Navier-Stokes equation for the coaxial double-cylindrical flow in the laminar watershed can be expressed by the following equation (21a) in the cylindrical coordinate system.

The viscous stress τ _rθ can be expressed by the following equation (21b) when a power law model is used.

However, the symbol γ _rθ in the equation (21b) is the shear rate, and the relationship shown in the following equation (21c) holds with the circumferential flow velocity _uθ .

By solving the equation (21c) from the above equation (21a), the analytical solution of the dimensionless circumferential flow velocity u ^* _θ can be derived as shown in the equation (21). An example of a technique for deriving this analytical solution is Reference 2 (N. Prasanth, UV Shenoy, “Poiseuille flow of a power-law fluid between coaxial cylinders”, Journal of Applied Polymer Science, Vol. 46 (1992), pp. . 1189-1194.).

The dimensionless circumferential flow velocity u ^* _θ in the above equation (21) and the dimensional flow velocity u _θ have the relationship shown in the following equation (22).

However, the symbol λ in the equation (21) is the radial distance at the position where the shear stress becomes 0. The continuity of the flow velocity distribution at r / R = λ can be obtained from the relationship of Eq. (23) shown below.

FIG. 8 shows the verification results obtained by performing the first verification using the fluid analyzer 10. In FIG. 8, the radial distribution of the circumferential flow velocity _uθ at each power index n is shown by dots (black circles shown in FIG. 8) under the condition of the lattice spacing 0.02R. In addition, the distribution of the analytical solution obtained by the above equation (21) is shown by a solid line.

As shown in FIG. 8, the velocity distribution predicted by this method, which is a fluid analysis method for analyzing the flow of a non-Newtonian fluid using the fluid analyzer 10, can be obtained at any position (any n) in the radial direction. Conforms with the analytical solution.

FIG. 9 shows the verification result of the first verification using the fluid analyzer 10 and the verification result of the comparative method for analyzing the fluid flow by extrapolation. FIG. 9 shows the radial distribution of the circumferential flow velocity u _θ under the conditions of n = 0.5 and the grid spacing 0.08R (low grid resolution).

The comparative method for analyzing the fluid flow by extrapolation used in the verification result shown in FIG. 9 is, as an example of a general analysis method, a method for analyzing the fluid flow by the Immersed boundary (IB) method based on linear interpolation. It is used as a comparison method. Examples of the technique of this comparative method include the technique of Patent Document 1 and Document 3 (R. Mittal et al., A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries, J. Comput. Phys. 227 (2008). ) 4827-4825.). In the technique of Document 3, the virtual flow velocity is extrapolated to the calculation grid inside the boundary (inside the object) by using the flow velocity on the boundary and the flow velocity of the surrounding fluid calculation points, and they are used in the calculation grid adjacent to the boundary. As a result, the basic equation of the fluid is discreteized and calculated in the same way as the computational grid away from the boundary.

As shown in FIG. 9, in the comparative method, the flow velocity distribution deviates from the analytical solution, whereas in this method, a result suitable for the analytical solution is obtained.

Next, a second verification that verifies the analysis of the fluid flow in the fluid analyzer 10 according to the present embodiment will be described.
In the second verification, the flow field around the rigidly moving object was verified as the fluid flow.

FIG. 10 shows an example of a schematic structure of an object (rigid body) through which a fluid flows, which was the subject of verification in the second verification.

As shown in FIG. 10, the region 30 to be verified includes an object region including a rigid body that rotates constantly in the same direction in the cylinder 32, and a fluid region flowing through the cylinder 32. Specifically, the object that moves rigidly in the cylinder 32 is a screw that rotates constantly in the same direction (an object in a region shown by a diagonal line in FIG. 10), and the surrounding fluid flows due to the rotation of the screw. The fluid had a viscosity characteristic equivalent to that of the molten resin, and the non-Newtonian property of the resin was taken into consideration in the Cross model. The calculation grids were orthogonally evenly spaced and were set to about 0.06 mm. The major axis of the screw is about 15 mm, the rotation speed is 200 rpm, and the maximum peripheral speed at the major axis end at this time is about 0.157 m / s.

In the second verification, in the analysis by this method, which is a fluid analysis method for analyzing the flow of non-Newtonian fluid using the fluid analyzer 10, the upper limit of the calculation time step that can be stably calculated is the comparison method by extrapolation. The result was about 7 times that of the analysis using, and the calculation time was significantly shortened.

The fluid analysis device 10 according to the present embodiment is an example of the fluid analysis device of the disclosed technique. Further, the step S100 process shown in FIG. 6 is an example of a function executed in the model definition unit. Further, the process of step S108 is an example of a function executed by the first arithmetic unit, the process of step S110 is an example of a function executed by the second arithmetic unit, and the process of step S112 is an example of the function executed by the derivation unit. This is an example of the function executed by.

The device main body 12 included in the fluid analysis device 10 according to the present embodiment may be constructed by hardware such as an electronic circuit having each function described above, and each component may be constructed. At least a part of the components may be constructed so as to realize the function by a computer.

Moreover, although the present invention has been described using the embodiments, the technical scope of the present invention is not limited to the scope described in the above embodiments. Various changes or improvements can be made to the above embodiments without departing from the gist of the invention, and the modified or improved forms are also included in the technical scope of the present invention. In addition, all documents, patent applications and technical standards described herein are to the same extent as if the individual documents, patent applications and technical standards were specifically and individually stated to be incorporated by reference. Incorporated herein by reference.

10 Fluid analyzer 12 Device body 12A CPU
12B RAM
12C ROM
12P fluid analysis program

Claims (6)

A space including an object and a fluid that flows around the object and whose viscosity changes spatially is composed of a plurality of elements by a plurality of lattice lines that do not coincide with the boundary between the object and the fluid and are orthogonal to each other . The model definition part that defines the divided numerical calculation target model, and
It is related to the flow of fluid in the predetermined direction of the boundary based on the information in each of the plurality of elements including the boundary continuous in the predetermined direction using the numerical calculation target model defined in the model definition unit. A first calculation unit that calculates a first physical quantity indicating the velocity gradient of the fluid using the distance from the center point or the center of gravity point of the grid composed of the plurality of grid lines to the boundary .
Based on the first physical quantity calculated by the first calculation unit, a second physical quantity indicating the velocity gradient of the fluid related to the flow of the fluid in the direction intersecting the predetermined direction of the boundary is obtained . A second calculation unit that calculates using the first physical quantity and a relational expression relating to a predetermined velocity gradient that holds at the boundary .
Based on the first physical quantity calculated by the first calculation unit and the second physical quantity calculated by the second calculation unit, the behavior of the fluid flowing in a predetermined region of interest including the boundary is described. The derivation part to be derived and
Fluid analyzer equipped with. The first calculation unit calculates the first physical quantity.
(I, J) is the position of the element divided by the plurality of grid lines in the predetermined direction and the position in the direction intersecting the predetermined direction, the predetermined direction is the x direction, and b is (I). , J) Information indicating that it is the intersection of the straight line in the predetermined direction passing through the center point or the center of gravity point and the boundary, and the velocity gradient of the fluid in the x direction is (∂u _i / ). When ∂x) _b , the x component of the flow velocity is u, Δx is the width of the element in the x direction, and ε _x is the information obtained by normalizing the width from the center point or center of gravity of the lattice to the boundary with Δx. ,

The fluid analyzer according to claim 1, wherein the calculation is performed using the formula represented by . The second calculation unit calculates the second physical quantity.
The directions that intersect the predetermined directions are the y and z directions, and the velocity gradients of the fluids in the y and z directions are (∂u / ∂y) _b and (∂u / ∂z) _b , and the boundary is defined. When the unit vector in the tangential direction in is s and t, and the rotation angle velocity vector when the object moves is ω,

The fluid analyzer according to claim 2, wherein the calculation is performed using the formula represented by . The directions that intersect the predetermined directions are two directions that are orthogonal to each other and are orthogonal to the predetermined directions.
The fluid analyzer according to claim 3 . The computer
A space containing an object and a fluid that flows around the object and whose viscosity changes spatially is incompatible with the boundary between the object and the fluid, and is composed of a plurality of elements by a plurality of orthogonal grid lines . The fluid related to the flow of the fluid in the predetermined direction of the boundary based on the information in each of the plurality of elements including the boundary continuous in the predetermined direction using the divided and defined numerical calculation target model. The first physical quantity indicating the velocity gradient of is calculated by using the distance from the center point or the center of gravity point of the lattice composed of the plurality of lattice lines to the boundary .
Based on the calculated first physical quantity, the second physical quantity indicating the velocity gradient of the fluid related to the flow of the fluid in the direction intersecting the predetermined direction of the boundary is referred to as the first physical quantity. Calculated using the relational expression related to the predetermined velocity gradient that holds at the boundary .
A fluid analysis method for deriving the behavior of the fluid flowing in a predetermined region of interest including the boundary based on the calculated first physical quantity and the calculated second physical quantity. A fluid analysis program for causing a computer to function as the fluid analyzer according to any one of claims 1 to 4.

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