用于模拟粘性流体流动的格子Boltzmann方法
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摘要
本文主要利用格子Boltzmann方法计算粘性不可压缩流体的流动。首先对格子Boltzmann方法的基本思想和发展历史及国内外的研究现状进行综述,介绍了该方法的统计机理、格子Boltzmann方法的基本模型及和连续Boltzmann方程的关系,应用Chapman-Enskog展开法推导了BGK模型对应的不可压缩Navier-Stokes方程,介绍了格子Boltzmann方法的稳定性、渐近性及误差分析等理论。本文主要利用格子Boltzmann方法数值计算了以下几个粘性流体流动问题:两侧面和四侧面驱动方腔流的数值模拟;壁面驱动半圆形空腔流的数值模拟;非光滑表面减阻的数值模拟;旋转皮带驱动的圆腔流体搅动的数值模拟。数值计算结果合理、可信,对工程实践有重要的参考价值。
The lattice Boltzmann method (LBM) is a meso-scale method for computational fluid dynamics, in which a discretizaion of space and velocity is employed. The basic idea of LBM is to use a very simple microscopic model of gas, which is nevertheless capable of correctly describing the macroscopic flow behavior. The macroscopic dynamics of fluid is the result of the collective behavior of many microscopic particles in the system. The main features of LBM are: the velocity set of the lattice Boltzmann equation is limited, the space is divided into the regular lattices. At each node, the particles are allowed to move along a few directions to its neighbour nodes.
The microscopic approach of LBM has its origins in the theory of lattice gas automata and is closely related to discrete velocity models of the Boltzmann equation. In its standard form, LBM is an explicit finite difference approximation of a discrete- velocity Boltzmann equation with a collision operator of relaxation type. Unlike conventional numerical schemes based on macroscopic equations, LBM can be considered as a physical model for describing flow or a discretization scheme of macroscopic continuum equations.
The Boltzmann equation is an integro-differential equation for the single particle distribution function: here x = q1, v = p1 m, m is the particle mass, a = K m is the acceleration induced by the body force, Q ( f ,f ) be the collision integral.
When dealing with the Boltzmann equation, a particular simple linearized version of the collision operator is used, which is called usually the BGK approximation.
The appropriate discreted velocity model and its corresponding to equilibrium distribution function must be choosed to obtain the correct macroscopic equations. Although f eq is an explicit expression of time, it is also a function of the macroscopic quantitiesρ, u ,T. So it is very critical to evaluateρ, u ,Tbefore discreting the lattice Boltzmann equation.
Macroscopic quantities, such as mass, momentum, and energy are defined as follows: The discretized equilibrium distribution function can be expressed as: The coefficient is ( )
D2Q9 discretized velocity model is applied. A discrete velocity model of the Boltzmann equation is given by
Many fluid dynamical phenomena can be described by solutions of the Navier-Stokes equation. It is of practical interest to derive the Navier-Stokes equation from the Boltzmann equation. The expansion parameter of Chapman-Enskog is the Knudsen number Kn, i.e. the ratio between the mean free lengthλand the characteristic space scale of the system. Usually, the time scales t 0,t 1 =εt0,t 2 =ε2t0and the space scales x0 , x1 =εx0 are used and the equilibrium distribution function can be expansed as follows:
Combining with the different time scale equations, for incompressible flow,ρ=ρ0 = const, and by taking the limit of small Mach number and considering the two time-scale, the resulting Navier-Stokes equations are recoved:
For practical simulations, the implementation of boundary conditions for the lattice Boltzmann method is very important, and has a great effect on the accuracy, the stability and the efficiency of the method. The particle distribution functions are unknown on the boundary. So we must construct schemes to give the boundary conditions on the microscopic level. Some usual implementations of boundary conditions are outlined, such as bounce-back scheme, hydrodynamic boundary schemes, extrapolation schemes, and the treatment of complex boundary. An improved curved boundary treatment scheme is applied in this paper.
The lattice Boltzmann scheme consists of two computational steps: collision:streaming: We can see that the collision process take place only near the grid and the corresponding distribution function is transferred to neighbor grid along velocity direction during the streaming process.
To finish the streaming step In order to evaluate fα( x b,t), a fictitious equilibrium distribution function is given by here u bf is a fictitious velocity.
A linear interpolation constructed by fα( x f, t) and ( )fα* x b,t is as follow For D2Q9 model, the equilibrium distribution function is given by By calculating, we have q is defined by The treatment of boundary condition can lead to a second-order accurate.
The fluid forces are evaluated by the moment-exchange
Finally, some numerical results by the lattice Boltzmann method are given, and the results are obtained:
1. Two-dimensional flow driven by moving boundaries inside a square cavity is a classical problem. The popularity of the problem is due to the simplicity of the geometry while retaining interesting phenomena such as vortex formation, hydrodynamic stability and flow bifurcations. Incompressible two-dimensional laminar flow inside a square cavity is numerically simulated. Two-sided and four-sided driven cavity flows are considered. For low Reynolds number flow, the resulting flow field is symmetric about one of the cavity diagonals for the two-sided driven cavity and about both cavity diagonals for the four-sided driven cavity.
2. A wall-driven flow within a semi-circular cavity is simulated. The curved boundary scheme is used. The forces that fluid acts on the wall are evaluated by the moment-exchange method. The bifurcation behaviors are captured. The results show that when the Reynolds number is small, the final steady state consists of one vortex only. As the Reynolds number increases, a secondary vortex and then a tertiay vortex arises. The size of the vortices depends on the Reynolds number too.
3. The flows over several non-smooth surfaces are studied and simulated numerically by using the lattice Boltzmann method. These surfaces are rough with circle concave surface, circle convex surface, triangular concave surface, triangular convex surface, regular sinusoidal wavy surface, upwind sinusoidal wavy surface, downwind sinusoidal wavy surface, and regular sinusoidal wavy surface with double wave amplitude, respectively. The resulting streamline patterns of the flow reveal the formation of vortices under certain conditions. The streamwise vortices settle inside the concave surface, which do not move with time developing. Similar vortices are also generated behind the convex surface. These vortices act like a kind of fluid roller bearings and may reduce viscous drag. Since flow velocity with reverse direction was produced in the cavity and the direction of viscous drag acted on wall was also opposed to streamwise direction, the negative drag is generated in the local region. As a whole, it is helpful to reduce total wall drag. The result may provide certain value for designing the non-smooth surfaces configuration with drag reduction in engineering, meanwhile may validate the lattice Boltzmann method.
4. The present investigation aims to extensively and systematically study the fluid stirrings in a circular cavity with a variety of boundary conditions by the LBM. The two belts are set symmetrically and asymmetrically with obvious lengths and positions and rotated at the same or opposite directions on the boundary to drive the liquid system. The curved boundary schemes are used on the boundary. The results show that flow feature is related to the rotation protocols of the belts, the geometry of the belts, including position and length. The combination of these factors affects the fluid stirrings in a complicated manner. Further, the circular cavity flows provide a powerful model to theoretically study the fluid stirrings, and provide, if possible, guidance for the design of fluid mixers in the areas of food processing and medicine.
The lattice Boltzmann method (LBM) is a meso-scale method for computational fluid dynamics, in which a discretizaion of space and velocity is employed. The basic idea of LBM is to use a very simple microscopic model of gas, which is nevertheless capable of correctly describing the macroscopic flow behavior. The macroscopic dynamics of fluid is the result of the collective behavior of many microscopic particles in the system. The main features of LBM are: the velocity set of the lattice Boltzmann equation is limited, the space is divided into the regular lattices. At each node, the particles are allowed to move along a few directions to its neighbour nodes.
The microscopic approach of LBM has its origins in the theory of lattice gas automata and is closely related to discrete velocity models of the Boltzmann equation. In its standard form, LBM is an explicit finite difference approximation of a discrete- velocity Boltzmann equation with a collision operator of relaxation type. Unlike conventional numerical schemes based on macroscopic equations, LBM can be considered as a physical model for describing flow or a discretization scheme of macroscopic continuum equations.
The Boltzmann equation is an integro-differential equation for the single particle distribution function: here x = q1, v = p1 m, m is the particle mass, a = K m is the acceleration induced by the body force, Q ( f ,f ) be the collision integral.
When dealing with the Boltzmann equation, a particular simple linearized version of the collision operator is used, which is called usually the BGK approximation.
The appropriate discreted velocity model and its corresponding to equilibrium distribution function must be choosed to obtain the correct macroscopic equations. Although f eq is an explicit expression of time, it is also a function of the macroscopic quantitiesρ, u ,T. So it is very critical to evaluateρ, u ,Tbefore discreting the lattice Boltzmann equation.
Macroscopic quantities, such as mass, momentum, and energy are defined as follows: The discretized equilibrium distribution function can be expressed as: The coefficient is ( )
D2Q9 discretized velocity model is applied. A discrete velocity model of the Boltzmann equation is given by
Many fluid dynamical phenomena can be described by solutions of the Navier-Stokes equation. It is of practical interest to derive the Navier-Stokes equation from the Boltzmann equation. The expansion parameter of Chapman-Enskog is the Knudsen number Kn, i.e. the ratio between the mean free lengthλand the characteristic space scale of the system. Usually, the time scales t 0,t 1 =εt0,t 2 =ε2t0and the space scales x0 , x1 =εx0 are used and the equilibrium distribution function can be expansed as follows:
Combining with the different time scale equations, for incompressible flow,ρ=ρ0 = const, and by taking the limit of small Mach number and considering the two time-scale, the resulting Navier-Stokes equations are recoved:
For practical simulations, the implementation of boundary conditions for the lattice Boltzmann method is very important, and has a great effect on the accuracy, the stability and the efficiency of the method. The particle distribution functions are unknown on the boundary. So we must construct schemes to give the boundary conditions on the microscopic level. Some usual implementations of boundary conditions are outlined, such as bounce-back scheme, hydrodynamic boundary schemes, extrapolation schemes, and the treatment of complex boundary. An improved curved boundary treatment scheme is applied in this paper.
The lattice Boltzmann scheme consists of two computational steps: collision:streaming: We can see that the collision process take place only near the grid and the corresponding distribution function is transferred to neighbor grid along velocity direction during the streaming process.
To finish the streaming step In order to evaluate fα( x b,t), a fictitious equilibrium distribution function is given by here u bf is a fictitious velocity.
A linear interpolation constructed by fα( x f, t) and ( )fα* x b,t is as follow For D2Q9 model, the equilibrium distribution function is given by By calculating, we have q is defined by The treatment of boundary condition can lead to a second-order accurate.
The fluid forces are evaluated by the moment-exchange
Finally, some numerical results by the lattice Boltzmann method are given, and the results are obtained:
1. Two-dimensional flow driven by moving boundaries inside a square cavity is a classical problem. The popularity of the problem is due to the simplicity of the geometry while retaining interesting phenomena such as vortex formation, hydrodynamic stability and flow bifurcations. Incompressible two-dimensional laminar flow inside a square cavity is numerically simulated. Two-sided and four-sided driven cavity flows are considered. For low Reynolds number flow, the resulting flow field is symmetric about one of the cavity diagonals for the two-sided driven cavity and about both cavity diagonals for the four-sided driven cavity.
2. A wall-driven flow within a semi-circular cavity is simulated. The curved boundary scheme is used. The forces that fluid acts on the wall are evaluated by the moment-exchange method. The bifurcation behaviors are captured. The results show that when the Reynolds number is small, the final steady state consists of one vortex only. As the Reynolds number increases, a secondary vortex and then a tertiay vortex arises. The size of the vortices depends on the Reynolds number too.
3. The flows over several non-smooth surfaces are studied and simulated numerically by using the lattice Boltzmann method. These surfaces are rough with circle concave surface, circle convex surface, triangular concave surface, triangular convex surface, regular sinusoidal wavy surface, upwind sinusoidal wavy surface, downwind sinusoidal wavy surface, and regular sinusoidal wavy surface with double wave amplitude, respectively. The resulting streamline patterns of the flow reveal the formation of vortices under certain conditions. The streamwise vortices settle inside the concave surface, which do not move with time developing. Similar vortices are also generated behind the convex surface. These vortices act like a kind of fluid roller bearings and may reduce viscous drag. Since flow velocity with reverse direction was produced in the cavity and the direction of viscous drag acted on wall was also opposed to streamwise direction, the negative drag is generated in the local region. As a whole, it is helpful to reduce total wall drag. The result may provide certain value for designing the non-smooth surfaces configuration with drag reduction in engineering, meanwhile may validate the lattice Boltzmann method.
4. The present investigation aims to extensively and systematically study the fluid stirrings in a circular cavity with a variety of boundary conditions by the LBM. The two belts are set symmetrically and asymmetrically with obvious lengths and positions and rotated at the same or opposite directions on the boundary to drive the liquid system. The curved boundary schemes are used on the boundary. The results show that flow feature is related to the rotation protocols of the belts, the geometry of the belts, including position and length. The combination of these factors affects the fluid stirrings in a complicated manner. Further, the circular cavity flows provide a powerful model to theoretically study the fluid stirrings, and provide, if possible, guidance for the design of fluid mixers in the areas of food processing and medicine.
引文
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