KR101562863B1 - Method for simulating fluid flow by using the lattice Boltzmann theory and recording medium for performing the method - Google Patents

Abstract

More particularly, the present invention relates to a fluid flow simulation method using a Lattice Boltzmann theory or a Lattice Boltzmann method and a recording medium for carrying out the method. The Lattice Boltzmann theory of simulating actual fluid flow in such a way that fictitious particles move across normalized lattice points can be used, for example, to solve the Navier-Stokes equations. The virtual particles cross the lattice points at unit time intervals. Likewise, the rule for determining the method of rearrangement of the particle velocities before and after the collision, assuming collision of particles at every unit time interval, is the core of the Lattice Boltzmann method. In order to improve the conventional unstable rearrangement rule, a rearrangement rule having the discrete velocity as a parameter is obtained, and a parameter is set to maintain a non-negative value with respect to a change in the macroscopic velocity and temperature required in the simulation. Invented a wide relocation rule.

Description

TECHNICAL FIELD The present invention relates to a fluid flow simulation method using a Lattice Boltzmann theory and a recording medium for realizing the method.

More particularly, the present invention relates to a fluid flow simulation method using a Lattice Boltzmann theory or a Lattice Boltzmann method and a recording medium for carrying out the method.

The physical quantity of fluid can be described and predicted by mathematical equations. Predicting physical quantities such as density, pressure, velocity, and temperature of a fluid at any given time and location is required in many scientific and engineering fields. Some examples include weather forecasts, analysis of flows around automobiles, ships and aircraft, and many others. The differential equation using the physical quantity as a variable is a Navier-Stokes equation. The Lattice Boltzmann theory of simulating actual fluid flow in such a way that fictitious particles move across normalized lattice points can also be used to solve the above equations. Considering that the macroscopic physical quantities including the physical quantities are expressed by averaging the microscopic movements of each of the molecules constituting the fluid, the solution of the equation obtained by the Lattice Boltzmann method is very accurate and efficient according to the detailed conditions applied to the solution . Let's look at this method in detail. In a highly efficient standard model, virtual particles travel in a grid-wise fashion and can only exist on lattice points, where the velocity-vector quantity that any single particle can have, and the speed of the wind in the ordinary sense, There are differences in the accuracy and efficiency of the simulation results, depending on how many are limited to the same macroscopic speed and micro speed. Also, the probability that a particle with a given velocity exists at any given time and position is expressed as a floating point. Particles cross the lattice points at every unit time interval. Likewise, the rearrangement rule that assumes the collision of particles at every unit time interval and re-determines the probability distribution of the particle velocity after each collision considering the pre-collision distribution is the core of the Lattice Boltzmann method can do. If the movement is continued without a rearrangement step after the collision, the virtual fluid becomes infinitely viscous. The value of the relocation rule is a probability related value, and it is difficult to expect the correct simulation result when the definition of the Lattice Boltzmann equation and relocation rule to be described later is negative. The direction and size of any virtual particle in the Lattice Boltzmann method is limited to maximize efficiency, so that given the expected macroscopic velocity and temperature range of the fluid to be simulated, the mass, It is tricky to make relocation rules that have nonnegative values while maintaining the momentum and energy conservation. Therefore, finding a relocation rule that can be applied to a wider range is a challenge.

Thus, it is possible to provide a rearrangement rule applicable at a wider macroscopic speed and temperature range.

According to another aspect of the present invention, there is provided a fluid flow simulation method comprising: discretizing a space in which a fluid flows into a grid at regular intervals; Assuming that the fluid particles repeatedly move and collide on the grating; Assuming that the discrete particle velocity set that the particle may have is a finite set; Obtaining a relocation rule having the discrete rate as a parameter; Determining a parameter to maintain a non-negative value for changes in the macroscopic speed and temperature required in the simulation; And deriving a Lattice Boltzmann model using this.

In an embodiment of the present invention, the method may further comprise measuring at least one of a physical quantity including temperature, density, pressure, and velocity of the fluid over time based on the Lattice Boltzmann model.

In an embodiment of the present invention, the regular spacing of the gratings may correspond to a mean free path.

In an embodiment of the present invention, the particles of the fluid may be present only at the lattice points on the lattice.

In an embodiment of the present invention, the probability distribution of the relocation rule may further comprise rounding off the decimal place to determine the approximate value.

In an embodiment of the present invention, it may further comprise the step of defining the discrete rate ( v _i ) of the virtual particles to be symmetric in defining the discrete rate ( v _i ). For example, the discrete velocity defined in one-dimensional space is symmetric about the origin as follows.

υ ₁ = 0, υ _{2 i} = - υ _{2 i + 1} . However, i = 1, 2, ....

A computer program for performing the fluid flow simulation method described above is recorded in a computer-readable storage medium according to an embodiment for realizing the above-mentioned other object of the present invention.

The relocation rules of the Lattice Boltzmann model provided by the conventional method are already determined on the basis of the characteristics of the induction method, and there is no room for the parameters to be adjusted. However, the relocation rules provided by the present invention can control parameters by different induction methods As a result, the applicable simulation area is wider than the existing method.

로 갖는 - 단, θ ₀는 상수이고

는 변수인 - 일차원 공간 모델의 재배치 규칙

이 음이 아닌 값을 갖는 구역을 음영으로 표현한 그래프이다. 가로축은

이고, 세로축은

이다. 단, u 는 거시적 속도이다.
특히

일 때, 재배치 규칙이 음이 아닌 값을 갖는

의 범위가 가장 크다. 또한, 재배치 규칙

가 0이 되는

의 값을 z _i 라고 할 때, 음영 부분의 경계를 이루는 세개의 곡선은 각각 z ₁ , z ₂ , z ₃에 해당한다.Figure 1 shows the discrete particle velocities

Having a - stage, θ ₀ is a constant and

Is a variable - a relocation rule for one-dimensional spatial models

This is a graph in which shaded regions have non-negative values. The horizontal axis

And the vertical axis indicates

to be. However, u is the macroscopic speed.
Especially

, The relocation rule has a non-negative value.

Is the largest. In addition,

Becomes zero

And z _i , the three curves forming the boundary of the shadow part correspond to z ₁ , z ₂ and z ₃ , respectively.

Preferred embodiments of the present invention will be described in more detail. The definition of symbolic characters used in the mathematical expressions, unless otherwise stated, quotes the definition of the preceding occurrence of the character.

The movement of the virtual particle follows Equation 1 using the rearrangement rule to be defined in the future.

[Number 1]

이며, 이 값은 시간 t와 위치 х에서 υ _i 를 갖는 가상 입자의 밀도 f _i (x,t)와 재배치 규칙 R _i (x,t)의 값으로부터 관계식

으로 얻어진다. 거시적 물리량은 f _i (x,t)로부터 구할 수 있는데, 예를 들면, 어떤 주어진 х, t에서 밀도 ρ(x,t)는

이고, 속도 u(x,t)는

이고, 온도 T(x,t)는

이다. 단, q는 가상 입자가 가질 수 있는 이산 입자 속도의 갯수이고, m은 입자 질량, k는 볼츠만 상수이다. 래티스 볼츠만 방법의 성능을 결정하는 중요한 요소 중 하나는 재배치 규칙 R _i (x,t)이다. 앞으로 혼동의 여지가 없는 한, 어떤 주어진 위치 х와 시간 t는 생략한다. 예를 들면, R _i (x,t)는 R _i 로 간략하게 쓴다.However, f _i (x, t) is multiplied by the quantity of density in the probability that the virtual particles dog i th discrete velocity υ _i at a given location х and time _{t, R i (x, t} ) is х and The relocation rule for υ _i at t , Δt is the time interval, and ω _i is a constant. If permanent, and the time t + △ t when position х + υ _i, the density of the virtual particles having υ _i is

And, the value of relation from the value of time t, and density of the virtual particles having υ _i in position х f _i (x, t) and the relocation rule R _i (x, t)

. The macroscopic physical quantity can be obtained from f _i ( x , t ), for example, density ρ ( x , t ) at any given х , t

, And the velocity u ( x , t ) is

, And the temperature T ( x , t ) is

to be. Where q is the number of discrete particle velocities that the hypothetical particle can have, m is the particle mass, and k is the Boltzmann constant. One of the important factors determining the performance of the Lattice Boltzmann method is the relocation rule R _i ( x , t ). As long as there is no confusion in the future, any given position х and time t are omitted. For example, R _i ( x , t ) is abbreviated as R _i .

The relocation rule proposed by the present invention can be obtained as an embodiment as follows. For the convenience of discussion, we limit it to one-dimensional space. The relocation rules in two-dimensional and three-dimensional spaces can be represented by the tensor product of the relocation rule obtained in the one-dimensional space, in addition to the method of direct retrieval. First, the relocation rule including the parameter is R _i satisfying the equation (2).

[Number 2]

, 즉, 입자 속도 υ에 대한 d차원 공간에서의 맥스웰-볼츠만(Maxwell-Boltzmann) 속도 분포이고, θ = kT/m, ρ는 밀도, n은 q보다 작거나 같은 자연수를 가지며(i = 1,2,...,q), 입자 속도 υ와 이산 입자 속도 υ _i 는 텐서량으로서 임의의 d차원 공간에서 직교 좌표계의 j번째 방향(j = 1,2,...,d)을 х _j 라고 하고 그 방향 속도 성분을 υ _хj 라고 하면,

이고 n = n ₁ + n ₂ + … +n _d 이며 n _j 는 음이 아닌 정수이다. 따라서, 위의 정의의 이해를 돕고자 예를 들면, 2차원 공간에서

의 세가지 경우를 모두 나타낸다.only,

Boltzmann velocity distribution in the d- dimensional space with respect to the particle velocity v , θ = kT / m , ρ is density, n is a natural number less than or equal to q ( i = 1, 2, ..., q), particle velocity υ and discrete particle velocity υ _i is the j-th direction of the Cartesian coordinate system at an arbitrary d-dimensional space as a tensor quantity (j = 1,2, ..., d ) to _j х Speaking of the its velocity component that υ _хj,

N = n ₁ + n ₂ + + n _d and n _j is a non-negative integer. Therefore, to help understand the definition above, for example, in a two-dimensional space

Of all three cases.

로 정의하면, 재배치 규칙은 수학식 3과 같다.As an example, when q = 3 in a one-dimensional space, considering the case of an isothermal fluid, the temperature is fixed to some constant θ ₀ , θ = θ ₀ , and υ _i

, The rearrangement rule is expressed by Equation (3).

[Number 3]

이다. 이 예에서와 달리 일반적으로는 비등온 유체의 경우에 사용 가능하다. 예를 들어, 이산 입자 속도 υ _i 가 υ ₁ = 0, υ ₂ = a, υ ₃ = -a, υ ₄ = b, υ ₅ = -b 같은, 단, a ≠ b, 경우이다.only,

to be. Unlike in this example, it is generally usable for non-isothermal fluids. For example, the discrete particle velocity υ _i is such that υ ₁ = 0, υ ₂ = a, υ ₃ = - a , υ ₄ = b , υ ₅ = - b , where a ≠ b .

를 정하는 하나의 방법이다. 편의를 위해서 무차원 변수

를 정의한다. 재배치 규칙을

에 대한 함수로 보고 매개 변수를

로 하면, 모든 i에 대하여

를 만족하는 영역에서

의 범위가 최대일 때의

값을 사용하여 재배치 규칙을 만드는 것이다. 도면 1에서 회색 음영에 해당 하는 부분이

인 영역이다. 살피건데,

일 때, 모든 i에 대하여

인

의 범위가 가장 넓음을 알 수 있다. 재배치 규칙

의 해를 z _i 라고 하자. 이를 구하면 z _i 는

에 대한 함수

로 표현되는데, 도면 1에서

는 각각 음영 부분의 경계를 이루는 얇은 실선(검정), 굵은 실선(파랑), 점선(빨강) 곡선에 해당한다. 결국, 매개 변수를

로 정하면 가장 넓은 범위의

에서 재배치 규칙은 음이 아닌 값을 갖고, 따라서 시뮬레이션 적용 가능한 범위가 확대되고, 안정적인 래티스 볼츠만 모델이 된다. 종래의 기술은, 예를 들면, 그 유도 방법의 특성상

으로 결정된 채로 도출된 재배치 규칙을 사용한다.Next, the parameters

. For convenience, dimensionless variables

. Relocation rules

See the function as a parameter

For all i,

Lt; RTI ID = 0.0 >

When the range of

Value to create a relocation rule. In Figure 1, the portion corresponding to the gray shade

Lt; / RTI > Look,

, For all i

sign

Is the largest. Relocation rule

Let z _i be the solution. If we obtain this, z _i

Functions for

1, < / RTI >

Corresponds to a thin solid line (black), a thick solid line (blue), and a dotted line (red) curve, which form the boundary of each shading portion. After all,

The largest range of

, The relocation rule has a nonnegative value, so that the applicable range of the simulation is expanded and becomes a stable Lattice Boltzmann model. The prior art is, for example,

And the reallocation rule derived from the decision.

로 표현되고, 3차원 모델의 해당 재배치 규칙은

과 같다. 2차원 모델 설명에서 속도 c _l의 아랫첨자 l이 재배치 규칙

아랫첨자 l에 상응하고, 영문자 c와

는 편의상 도입한 것이며, 일반적으로 차원에 관계없이 υ와

로 나타낼 수 있음을 강조하기 위하여 순서쌍을 단일 문자로 표현하기 위해 도입하였다.A model operating in two-dimensional and three-dimensional space can also be directly obtained through Equation (2), but can also be obtained by extending by using a tensor product of the model operating in a one-dimensional space. For example, the 1-D using a q = 3 model, look derive a two-dimensional q = 9 model, the discrete particle velocity is _{c 1 = (υ 1, υ} 1), c 2 = (υ 1, υ _{2), c 3 = (υ} 1, υ 3), c 4 = (υ 2, υ 1), c 5 = (υ 2, υ 2), c 6 = (υ 2, υ 3), c 7 = a _{(υ 3, υ 1),} c 8 = (υ 3, υ 2), c 9 = (υ 3, υ 3). In the same way, the three-dimensional discrete particle velocities q = 27 can be obtained in the same way as ( υ _i , υ _j , υ _k ). The corresponding relocation rule of the two-dimensional model is

And the corresponding relocation rule of the three-dimensional model is expressed as

Respectively. In the two-dimensional model description, the subscript l of the velocity c _l is replaced by the relocation rule

Corresponds to the subscript l, and the letters c and

Is generally introduced for convenience, and regardless of dimension, υ and

In order to emphasize that it can be expressed as a single character.

Claims (4)

Discretizing the space in which the fluid flows into a regularly spaced lattice;
Assuming that the fluid particles repeatedly move and collide on the grating;
Assuming that a set of discrete particle velocities that the particle may have is a finite set;
Obtaining a relocation rule having the discrete particle velocity as a parameter;
Determining a parameter to maintain a non-negative value as a real number for a range of changes in the macro speed and temperature required in the simulation; And
And deriving a Lattice Boltzmann model using the method. 2. The method of claim 1, wherein obtaining a relocation rule having a discrete particle velocity as a parameter comprises:
Further comprising the step of deriving a relocation rule using the following conditional expression,

The number of conditional expressions is determined by the number of discrete particle velocity set elements q and the number of spatial dimensions, where n is a natural number less than or equal to q ( i = 1, 2, ..., q ) υ and the discrete particle velocity υ _i are the tensor amounts in the arbitrary d- dimensional space and the j- th direction ( j = 1, 2, ..., d ) of the Cartesian coordinate system is х _j and the direction velocity component is υ _хj ,

N = n ₁ + n ₂ + + n _d where n _j is a nonnegative integer such that the number of conditional equations is less than or equal to the number of possible configurations of all n and n _j and R _i ( x , t ) is the element of the discrete particle velocity set υ _i Is used in the following equation as a rearrangement rule,

However, f _i (x, t) is the value dog virtual particles at a given location х and time t multiplied by the density of the probability that the i-th discrete particle velocity υ _i, R _i (x, t) is х and t relocating rules for υ _i in, △ t is a time interval, ω _i is a constant. At time t +? T and at position x + v _i , the density of virtual particles with v _i is

, And density of the virtual particles having at х υ _i and f _i t (x, t) and from the value of a relocation rule R _i (x, t), equation

, And further,

, That is, Maxwell in d-dimensional space for the particle velocity υ - is Boltzmann (Maxwell-Boltzmann) velocity profile, θ = kT / m, ρ (x, t) is a fluid, characterized in that the density of the х and t Flow simulation method. 2. The method of claim 1,
Relocation rule dividing density for convenience

As a function of the macroscopic velocity u and temperature θ of the fluid at any given position х and time t , by considering the discrete particle velocity υ _i as a parameter and by adjusting υ _i

The range of u and θ with non-negative values varies, and when this range falls into any given requirement, υ _i You can set the value as the parameter value of the relocation rule,

Υ _i when the range of u and θ with non-negative values has the widest range including a specific point Value is determined as the parameter value of the rearrangement rule. A computer-readable recording medium on which a computer program for performing the method according to claim 1 is recorded.

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