KR20100055556A - Computer simulation method by thermal lattice boltzmann method - Google Patents

Computer simulation method by thermal lattice boltzmann method Download PDF

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KR20100055556A
KR20100055556A KR1020080114342A KR20080114342A KR20100055556A KR 20100055556 A KR20100055556 A KR 20100055556A KR 1020080114342 A KR1020080114342 A KR 1020080114342A KR 20080114342 A KR20080114342 A KR 20080114342A KR 20100055556 A KR20100055556 A KR 20100055556A
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equation
boltzmann
lattice boltzmann
discrete
temperature
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심재완
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    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
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Abstract

The present invention relates to a method for predicting the physical quantity of an object by a computer simulation method. Herein, the object includes all known properties such as solid, liquid, gas, and plasma, and the physical quantity refers to temperature, density, pressure, speed, and the like.

One of the outstanding methods proposed in the related art is the Lattice Boltzmann Method. However, to date, the lattice Boltzmann method is limited to isothermal fluids, and the thermal lattice Boltzmann method, which can handle nonisothermal fluids, has been proposed, but there are many problems with stability. . In order to solve this problem, the present invention proposes a thermal lattice Boltzmann method different from the conventional method.

The present invention is characterized by including not only the velocity vector variable but also the temperature variable in the discretization of the Boltzmann equation and the BGK collision term as the Taylor expansion variable. In the process of determining constants for discretization, the constants are determined by comparing the original and discrete equations as well as the Taylor expansion equation.

The present invention can set the temperature reference point as the best point at the time of Taylor development, and thus can perform more stable computer simulation, and its predictive power is superior to the previous method.

Description

Two-dimensional and three-dimensional computer simulation using thermal lattice Boltzmann method {Computer simulation method by thermal lattice Boltzmann method}

The present invention relates to a method for predicting the physical quantity of an object by computer simulation. Herein, the object includes all known properties such as solid, liquid, gas, and plasma, and the physical quantity refers to temperature, density, pressure, speed, and the like.

For example, when designing a submarine or aircraft, it is essential to predict the physical quantity of fluid flowing around it. The distribution of physical quantities (pressure, velocity, temperature, etc.) we want to predict can be obtained by solving a governing equation, such as the Navier-Stokes equation, which is very difficult Approximated solution is obtained using a method such as finite element method. However, this method requires a continuum hypothesis of the object, which gives the correct answer for a rare gas or fluid that flows around an object of the size of micro or nano units. can not do it. Conventional methods also include complex fluid systems, bubble or droplet dynamics, hydrophilic or hydrophobic properties on solid surfaces, interfacial slip, There is no suitable method for predicting fluid flow through porous materials, fluid flow in blood vessels, and the like. Therefore, in order to solve this problem, it is possible to solve the equation of motion for each particle at the molecular level by considering the fluid as a set of molecules constituting it, not as a continuum, but it is very inefficient. One of the proposed methods to solve this problem is the lattice Boltzmann method. This can be achieved by discretizing or discretizing the Boltzmann equation and the Batnagar-Gross-Krook or BGK collision term. However, to date, the lattice Boltzmann method is limited to isothermal fluids, and the thermal lattice Boltzmann method, which can handle nonisothermal fluids, has been proposed, but there are many problems with stability. . The column grid modifier "columns" in the Boltzmann method is for the following reasons: Since the lattice Boltzmann method was initially applicable only to isothermal fluids, the term “thermal” was added to mean that it can be applied to nonisothermal fluids as well as isothermal fluids.

Literature Information of the Prior Art

[1] X. He and L.-S. Luo, Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation, Physical Review E, volume 56, number 6, 1997

[Reference 2] F.J. Alexander, S. Chen and J.D. Sterling, Lattice Boltzmann thermohydrodynamics, Physical Review E, volume 47, number 4, 1993

It is an object of the present invention to provide a method for efficiently and stably performing nonisothermal two-dimensional and three-dimensional computer simulations by proposing a thermal lattice Boltzmann method having better stability than the previously proposed method.

The present invention is characterized by including not only the velocity vector variable but also the temperature variable in the discretization of the Boltzmann equation and the BGK collision term as the Taylor expansion variable. In the process of determining constants for discretization, the constants are determined by comparing the original and discrete equations as well as the Taylor expansion equation.

The present invention can set the temperature reference point as the best point at the time of Taylor development, and thus can perform more stable computer simulation, and its predictive power is superior to the previous method.

The definition of a symbol used in an equation is defined only once at the first occurrence of the letter. Therefore, when the same symbol comes after it, the first definition is followed. Boltzmann's equation is a nonlinear calculus equation. The equation includes a collision term, some of which have been suggested as a way of expressing how the physical quantity of a particle changes before and after a collision. Assuming that the BGK collision term is relatively simple and there is no external force, Boltzmann's equation is given by Equation 1. The definition of the symbol letter used in Equation 1 is as follows. Distribution function

Figure 112008079281391-PAT00001
Is
Figure 112008079281391-PAT00002
You can define indirectly by defining
Figure 112008079281391-PAT00003
Is an infinite phase space at time t
Figure 112008079281391-PAT00004
It shows the number of particles contained in. only
Figure 112008079281391-PAT00005
Is a position vector
Figure 112008079281391-PAT00006
Is the micro velocity vector. The microscopic velocity vector is distinguished from the macroscopic velocity vector in which the latter corresponds to the velocity of the fluid in general, and the former refers to the microscopic velocity of each particle in the fluid. a constant
Figure 112008079281391-PAT00007
Denotes the relaxation time, which means the speed of approaching the Maxwell-Boltzmann distribution by collisions between particles.

Distribution function

Figure 112008079281391-PAT00008
The macrophysical quantities can be obtained from Equations 2, 3, and 4. Previously described distribution function
Figure 112008079281391-PAT00009
The symbol letter n referred to in Equation 2, Equation 3, and Equation 4 by the definition of is a number density,
Figure 112008079281391-PAT00010
Is the macroscopic velocity and e is the energy per unit mass. The symbol letter e has a relationship between temperature and equation (5). only
Figure 112008079281391-PAT00011
Is the degree of freedom of the particle, k is the Boltzmann constant, T is the temperature and m is the mass of the particle. Distribution function
Figure 112008079281391-PAT00012
Maxwell-Boltzmann distribution as a special case of
Figure 112008079281391-PAT00013
Equation describes the equilibrium state most likely to occur under given conditions, and its form is shown in Equation 6. Where D is the dimension of space,
Figure 112008079281391-PAT00014
Denotes a parameter defined by Equation 7. In other words, the temperature T can be defined as a dimensionless temperature by scaling the constant k and m . therefore
Figure 112008079281391-PAT00015
Was introduced for the convenience of calculation, and in the future
Figure 112008079281391-PAT00016
Taylor unfolding for is no different than Taylor unfolding for temperature T. For reference, Equation 6
Figure 112008079281391-PAT00017
To
Figure 112008079281391-PAT00018
,
Figure 112008079281391-PAT00019
, explicitly indicates that it is a function with t as a variable. It is obvious that the reason by Equation 2, Equation 3, Equation 4 n,
Figure 112008079281391-PAT00020
,
Figure 112008079281391-PAT00021
Is a distribution function
Figure 112008079281391-PAT00022
Because it is calculated from. Of equation (6)
Figure 112008079281391-PAT00023
Generally satisfies Equations 2, 3, and 4. Equation 2, Equation 3, Equation 4
Figure 112008079281391-PAT00024
on
Figure 112008079281391-PAT00025
By substituting and integrating by Gaussian method, the result is shown in Equation 8, Equation 9 and Equation 10.

Discretizing Equation 1 with respect to time yields Equation 12 (see Literature 1). When discretizing space and velocity in Equation 12, the entire discretization work, that is, discretization of time, space, and velocity is completed. The biggest difficulty here is the Maxwell-Boltzmann distribution

Figure 112008079281391-PAT00026
Is about dioxide rate. Let's take a look at this from now on.

First, we will look at the case of two-dimensional space. Macro Velocity of Equation 6

Figure 112008079281391-PAT00027
Wow
Figure 112008079281391-PAT00028
In the second Taylor development of (ie, temperature T )
Figure 112008079281391-PAT00029
With = (0,0)
Figure 112008079281391-PAT00030
The result is as shown in Equation 13. only
Figure 112008079281391-PAT00031
,
Figure 112008079281391-PAT00032
,
Figure 112008079281391-PAT00033
,
Figure 112008079281391-PAT00034
The error term (or error term or Lagrange remainder) following Taylor expansion is omitted. here
Figure 112008079281391-PAT00035
Is
Figure 112008079281391-PAT00036
Of Taylor
Figure 112008079281391-PAT00037
Is the equation
Figure 112008079281391-PAT00038
Wow
Figure 112008079281391-PAT00039
Each on
Figure 112008079281391-PAT00040
Wow
Figure 112008079281391-PAT00041
It means that is substituted. We are
Figure 112008079281391-PAT00042
The optimal value can be chosen, resulting in a more stable and efficient thermal lattice Boltzmann method. For convenience, Equations 14 and 15 are defined by dividing the right side of Equation 13 into two parts.

Next, let's look at the three-dimensional space. The result at this time is the two-dimensional space and the part defined in Equation 15

Figure 112008079281391-PAT00043
The only difference is that in the case of 3D space,
Figure 112008079281391-PAT00044
The subscript "D3" used in the above is to distinguish it from the case in the two-dimensional space. Forward
Figure 112008079281391-PAT00045
Equation that indicates that there is a subscript "D3" in the three-dimensional space even if the subscript "D3" is not specified.

Figure 112008079281391-PAT00046
To
Figure 112008079281391-PAT00047
For
Figure 112008079281391-PAT00048
If we define the discrete distribution of
Figure 112008079281391-PAT00049
Has the form as shown in Equation 17. only
Figure 112008079281391-PAT00050
Is a weighting coeffcient factor. Equation 17 is expressed in Equation 13
Figure 112008079281391-PAT00051
Discrete speed
Figure 112008079281391-PAT00052
Replace with and weight the discretization
Figure 112008079281391-PAT00053
It is obtained by introducing.

When ignoring the error due to Taylor expansion, Equation 18 is obtained. To perform discretization of velocity

Figure 112008079281391-PAT00054
Should be obtained. For this purpose, a constraint such as Equation 19 is assumed. only
Figure 112008079281391-PAT00055
Speed vector
Figure 112008079281391-PAT00056
Two component variables
Figure 112008079281391-PAT00057
Wow
Figure 112008079281391-PAT00058
Is a polynomial of. As a result of equations (18) and (19), equations (20) are obtained. Equation 20 is composed of three terms, which are defined as left, middle, and right sides with the equal sign as the boundary from the far left. If the solution does not exist by Equation 19, a solution may be obtained by comparing the left and right sides of Equation 20.
Figure 112008079281391-PAT00059
1,
Figure 112008079281391-PAT00060
or
Figure 112008079281391-PAT00061
In this case, when Equation 20 is applied, constraints such as Equation 21, Equation 22, and Equation 23 are obtained. only
Figure 112008079281391-PAT00062
Has the same relationship as In Equation 21, 22, 23
Figure 112008079281391-PAT00063
,
Figure 112008079281391-PAT00064
And
Figure 112008079281391-PAT00065
Can be directly obtained by integrating the Maxwell-Boltzmann distribution as in Equations 8, 9 and 10.

As a special case, let's look at the discretization of the 2-dimensional hexagonal lattice model. Polar coordinates are introduced for convenience. Therefore, micro and macro speeds are expressed as in Equation (24) and (25). Equations 14 and 15 are also expressed as in Equations 26 and 27. In addition, the left side of Equation 19 may be expressed as Equation 28.

Here, the assumption of (29) is introduced. Only in this family

Figure 112008079281391-PAT00066
It is done.
Figure 112008079281391-PAT00067
Wow
Figure 112008079281391-PAT00068
Finding is our purpose. Applying equation (29) to equation (28) yields equation (30).
Figure 112008079281391-PAT00069
,
Figure 112008079281391-PAT00070
And
Figure 112008079281391-PAT00071
In this case, if Equation 30 is calculated and the results of Equation 21, Equation 22, and Equation 23 are applied, Equation 31, which is a system of equations, is obtained.

By the way, the discretization of θ is achieved by the equation (32). Equation (32) is satisfied when n + m ≤ 5 and n and m are zero or natural numbers. Since our range discussed here satisfies this condition, we can use (32). Therefore, Equation 30 is expressed as Equation 33.

As a complete example, we discretize the Maxwell-Boltzmann distribution of 13 discrete velocity hexagonal models in two-dimensional space. The 13 discrete velocity hexagonal models are defined as a model of moving the lattice space with the discrete velocity as equation 34 and using the lattice points at which discrete particles with such discrete velocity can be located at any discrete time. Since there are six grid points closest to a grid point around it, we call it a hexagonal model, where c is a constant that will be determined later. Equation 33 is k = 2

Figure 112008079281391-PAT00072
,
Figure 112008079281391-PAT00073
And
Figure 112008079281391-PAT00074
When, we match 13 discrete speed hexagonal models. At this time, if the solution of Equation 31 is solved, there are two sets, which are the same as Equation 35 and Equation 36. Applying Equations 35 and 36 to Equation 33 results in discretized Maxwell-Boltzmann distribution Equations 37 and 38. However, the constant c of Equation 34 is as follows. In Equation 37, c = 1 and Equation 38
Figure 112008079281391-PAT00075
to be. Again, equations 37 and 38 satisfy equations 21, 22, and 23. Therefore, Equation 12 may be expressed as Equation 39 by applying the discrete speeds of Equation 34. So far we have looked at the discretization of speed.

At the end of the discretization of space, all discretization is complete. Space can be discretized to any point where a particle can be located at any discrete time by the discrete velocity in equation (34). An example of the thermal lattice Boltzmann method proposed by the present invention is to calculate the distribution of particles moving through the lattice point and the lattice point with the discrete velocity in the discrete space obtained by the equation (39). Thus, as the definition of discrete velocities changes, specific examples of various thermal lattice Boltzmann methods can be calculated.

Now let's look at the discretization of a two-dimensional rectangular model. Using the Hermite-Gauss Quadrature equation of equation 42 for the discrete velocity given by the definition of equation 40, a discrete Maxwell-Boltzmann distribution such as equation 41 is obtained. Here, Equation 12 may be expressed as Equation 39 by applying a discrete speed according to Equation 40.

Now let's look at the discretization of the three-dimensional cube model. Using the Hermit-Gauss quadrature equation of Eq. 42 for the discrete velocity given by the definition of Eq. 43, we obtain a discrete Maxwell-Boltzmann distribution such as Eq. Here, Equation 12 may be expressed as Equation 39 by applying a discrete speed according to Equation 43.

[Equation]

Figure 112008079281391-PAT00076

Figure 112008079281391-PAT00077

Figure 112008079281391-PAT00078

Figure 112008079281391-PAT00079

Figure 112008079281391-PAT00080

Figure 112008079281391-PAT00081

Figure 112008079281391-PAT00082

Figure 112008079281391-PAT00083

Figure 112008079281391-PAT00084

Figure 112008079281391-PAT00085

Figure 112008079281391-PAT00086

Figure 112008079281391-PAT00087

Figure 112008079281391-PAT00088

Figure 112008079281391-PAT00089

Figure 112008079281391-PAT00090

Figure 112008079281391-PAT00091

Figure 112008079281391-PAT00092

Figure 112008079281391-PAT00093

Figure 112008079281391-PAT00094

Figure 112008079281391-PAT00095

Figure 112008079281391-PAT00096

Figure 112008079281391-PAT00097

Figure 112008079281391-PAT00098

Figure 112008079281391-PAT00099

Figure 112008079281391-PAT00100

Figure 112008079281391-PAT00101

Figure 112008079281391-PAT00102

Figure 112008079281391-PAT00103

Figure 112008079281391-PAT00104

Figure 112008079281391-PAT00105

Figure 112008079281391-PAT00106

Figure 112008079281391-PAT00107

Figure 112008079281391-PAT00108

Figure 112008079281391-PAT00109

Figure 112008079281391-PAT00110

Figure 112008079281391-PAT00111

Figure 112008079281391-PAT00112

Figure 112008079281391-PAT00113

Figure 112008079281391-PAT00114

Figure 112008079281391-PAT00115

Figure 112008079281391-PAT00116

Figure 112008079281391-PAT00117
Figure 112008079281391-PAT00118
where

Figure 112008079281391-PAT00119
Figure 112008079281391-PAT00120
Figure 112008079281391-PAT00121
Figure 112008079281391-PAT00122
Figure 112008079281391-PAT00123
Figure 112008079281391-PAT00124
Figure 112008079281391-PAT00125
where,, and,,.

Figure 112008079281391-PAT00126

Figure 112008079281391-PAT00127
Figure 112008079281391-PAT00128
where

Claims (6)

When the dimension of space is D, macroscopic velocity
Figure 112008079281391-PAT00129
And the temperature on the D + l variables to refer to a T Maxwell-Taylor expansion of the Boltzmann distribution (Maxwell-Boltzmann distribution) open lattice Boltzmann method through (Taylor expansion in D + 1 variables of velocity vector and temperature) (thermal lattice Boltzmann method How to get The method of claim 1, wherein the discrete Maxwell-Boltzmann distribution is obtained by applying Equation 20. The method of claim 1, wherein the discrete Maxwell-Boltzmann distribution is obtained by applying Equation 20a or Equation 20b. 3. The thermal lattice Boltzmann method using discrete speed according to the definition of Equation 34 and Equation 37 or Equation 38 according to claim 2. The thermal lattice Boltzmann method of claim 2 using the discrete speed defined by equation (40) and equation (41). 3. The thermal lattice Boltzmann method using discrete speed and equation 44 according to claim 43.
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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
KR20190023794A (en) 2017-08-30 2019-03-08 (주)웰빙해피팜 A preparing method of tissue repair treatment composition using the two step cross-linking and the composition therefrom
CN112182991A (en) * 2020-09-29 2021-01-05 中南大学 Mesoscopic simulation method for gas-liquid phase change
WO2022067498A1 (en) * 2020-09-29 2022-04-07 中南大学 Mesoscopic simulation method for gas-liquid phase transition

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
KR20190023794A (en) 2017-08-30 2019-03-08 (주)웰빙해피팜 A preparing method of tissue repair treatment composition using the two step cross-linking and the composition therefrom
CN112182991A (en) * 2020-09-29 2021-01-05 中南大学 Mesoscopic simulation method for gas-liquid phase change
WO2022067498A1 (en) * 2020-09-29 2022-04-07 中南大学 Mesoscopic simulation method for gas-liquid phase transition
CN112182991B (en) * 2020-09-29 2023-11-10 中南大学 Mesoscopic simulation method for gas-liquid phase change

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