CN109299569B - Large vortex simulation method of incompressible viscous fluid based on coherent structure - Google Patents

Large vortex simulation method of incompressible viscous fluid based on coherent structure Download PDF

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CN109299569B
CN109299569B CN201811241675.XA CN201811241675A CN109299569B CN 109299569 B CN109299569 B CN 109299569B CN 201811241675 A CN201811241675 A CN 201811241675A CN 109299569 B CN109299569 B CN 109299569B
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陈宇明
傅秀聪
吴池力
欧宝星
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Guangzhou HKUST Fok Ying Tung Research Institute
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Abstract

The invention discloses a coherent structure-based large vortex simulation method for incompressible viscous fluid, which comprises the steps of establishing a lattice boltzmann method model; in the conventional Smagorinsky model, the vortex viscosity v t From the filtered strain rate tensor S αβ Filter scale Δ x and Smagorinsky constant C S Determining; in a lattice Boltzmann method-large vortex simulation method model, momentum flux Q is established αβ Strain rate tensor S αβ Smigorinsky constant C S A relation equation of relaxation time and average density of the system; galileo invariants Q introduced into sub-lattice flow fields LES Define the Galileo invariant Q LES And strain rate tensor S αβ Tensor W of rotation rate αβ The relational equation of (a); obtaining the relation of relaxation time under different time steps in a multi-relaxation-time lattice Boltzmann method model; obtaining the vortex viscosity in the large vortex simulation; the invention has the beneficial effects that: galileo invariant Q introduced into sub-lattice flow field LES The coherent structure function based on the Smagorinsky model is constructed, the vortex viscosity of the LES is dynamically calculated, the calculation of the relaxation time is simple, and the accurate value is easily obtained.

Description

Large vortex simulation method of incompressible viscous fluid based on coherent structure
Technical Field
The invention belongs to a large vortex simulation method, and particularly relates to a large vortex simulation method of incompressible viscous fluid based on a coherent structure.
Background
Turbulence research has been a significant challenge to scientists and engineers for over a hundred years, despite much effort. At this stage, the most efficient tool to understand this physical phenomenon and to guide the theoretical development is the Computational Fluid Dynamics (CFD) method. Real turbulence is usually three-dimensional and the time and economic costs of existing computer power to directly simulate these flows are prohibitive. In order to reduce the time consumption of direct numerical simulation while maintaining high calculation accuracy, researchers have proposed a Large Eddy Simulation (LES) method. In large vortex simulation, large unsteady turbulent motion is directly expressed, and the effect of small scale motion is modeled. Since large-scale unsteady flow is explicitly expressed, large-vortex simulation has higher accuracy and is more reliable than a Reynolds stress model in large-scale effect-dominated flow simulation (such as blunt body streaming).
The smagonrinsky vortex-stick model is the most common of the large vortex simulation methods. It assumes that the energy transfer from resolvable scale to non-resolvable scale pulsation is equal to turbulent kinetic energy dissipation. This method does not require averaging of the model parameters, i.e. the model parameters are determined locally and always positive, so that the numerical calculation is very stable. In addition, a dynamic Smagorinsky model is provided, and model parameters are derived by a Germano formula by taking the Smagorinsky vortex-viscous model as a reference. The model parameters can be dynamically adjusted according to the flow type, and the asymptotic behavior of the wall vortex viscosity can be correctly reproduced without determining the wall damping function. In laminar flow, the model parameters will automatically change to zero, so the dynamic Smagorinsky model is also suitable for simulating laminar flow. Since the model parameters are determined by local physical quantities, either positive or negative, numerical calculations can be made divergent; even if the minimum error method of d.k.lilly is used to ensure that the model parameters are positive, it requires averaging over the entire flow field; the stress tensor component under the test grid also needs to be calculated, and more calculation time is consumed.
In the last three decades, the Lattice Boltzmann Method (LBM) has developed into a mature CFD Method. The LBM procedure is much simpler to implement than the conventional CFD method. Since the relaxation process of LBM is local and its communication mode is unidirectional, making it easy to parallelize, the computation performance increases almost linearly with the number of computation cores.
But do not
The existing lattice boltzmann method-large eddy simulation method (LBM-LES): most of control equations are based on Navier-Stokes (NS) equations of original variables, and other equations such as vorticity transport and the like are used as control equations in many projects, but the calculation of relaxation time is complex and accurate values are difficult to obtain, and meanwhile, as LBM adopts a rectangular grid, accurate strain rate tensors are difficult to obtain when curve or curved surface boundaries are calculated.
Disclosure of Invention
The invention aims to overcome the defects of the prior art, provides a coherent structure-based large vortex simulation method for incompressible viscous fluid, and solves the technical problems that the calculation of relaxation time is complex and an accurate value is difficult to obtain in a lattice boltzmann method-large vortex simulation method.
In order to solve the problems, the invention is realized according to the following technical scheme:
a large vortex simulation method of incompressible viscous fluid based on a coherent structure comprises the following steps:
establishing a lattice Boltzmann method model to obtain a calculation formula of fluid viscosity v, wherein the fluid viscosity v comprises molecular viscosity v 0 And a vortex viscosity v t ；
In the conventional Smigorinsky model, the vortical viscosity v t From the filtered strain rate tensor S αβ Filter scale Δ x and Smagorinsky constant C S Determining;
in a lattice Boltzmann method-large vortex simulation method model, a filtering scale delta x is taken as a unit length, and a momentum flux Q is established αβ The strain rate tensor S αβ Smigorinsky constant C S A relation equation of relaxation time and average density of the system;
galileo invariants Q introduced into sub-lattice flow fields LES Define the Galileo invariant Q LES And the strain rate tensor S αβ Tensor W of rotation rate αβ Defining model parameters C to obtain the model parameters C and the Smagorinsky constant C S The relationship of (a);
in a multi-relaxation-time lattice Boltzmann method model, obtaining the relationship of relaxation time under different time step lengths according to the relationship among relaxation time, vortex viscosity and momentum flux;
the vortical viscosity was determined in a large vortex simulation.
More preferably, the Galileo invariant Q LES And the strain rate tensor S αβ Tensor W of rotation rate αβ The relational equation of (a) is:
Figure GDA0004091516990000021
more preferably, the model parameters C are defined as:
C＝C CSM |F CSM | 3/2 F Ω
wherein C is a model parameter, C CSM To self-define constants, F CSM As a function of coherent structure, F Ω Is an energy decay suppression function.
4. The method for modeling large vortices in an incompressible viscous fluid according to claim 3 wherein the model parameters C define:
Figure GDA0004091516990000031
Figure GDA0004091516990000032
F Ω ＝1-F CSM ，
Figure GDA0004091516990000033
where E is the velocity gradient tensor.
Preferably, the model parameter C and the Smagorinsky constant C are obtained according to the definition of the model parameter C S The relationship of (1):
Figure GDA0004091516990000034
preferably, in the conventional Smagorinsky model, the vortical viscosity v t And the filtered strain rate tensor S αβ Filter scale Δ x and Smagorinsky constant C S The relation of (A) is as follows:
Figure GDA0004091516990000035
wherein the content of the first and second substances,
Figure GDA0004091516990000036
preferably, in a lattice boltzmann method-large vortex simulation method model, the filtering scale delta x is taken as the unit length, and the momentum flux Q is established αβ The strain rate tensor S αβ Smigorinsky constant C S Relaxation time τ, mean density of the system ρ 0 Equation of the relation (c):
Figure GDA0004091516990000037
preferably, in the multi-relaxation-time lattice boltzmann method model, according to the relationship among relaxation time, vortex viscosity and momentum flux, the relationship among relaxation time under different time steps is obtained:
Figure GDA0004091516990000038
wherein, tau 0 、τ t All are relaxation times.
The invention has the beneficial effects that:
galileo invariants Q introduced into sub-lattice flow fields LES A coherent structure function based on a Smagorinsky model is constructed, the vortex viscosity of the LES is dynamically calculated, the calculation of the relaxation time is simple, and an accurate value is easily obtained;
the model parameters are always positive, and the numerical calculation is stable;
the coherent structure function only comprises a strain rate tensor and a rotation rate tensor, does not need a filtering function, is simple to calculate and is convenient for engineering application;
the model is constructed based on a coherent structure representing turbulence, the coherent structure is shielded, and the model can also be suitable for laminar flow, so that the model has universality;
the model parameters consist of a fixed model parameter and a coherent structure function, the coherent structure function is a Galileo invariant normalized by the magnitude of the velocity gradient tensor, and has the function of wall surface damping without using a wall surface function;
the strain rate tensor of the curve boundary is implicitly obtained in the model, and the calculation precision is high;
can be used for rotating homogeneous turbulence and channel turbulence;
the programming is simple, the parallelization is easy, and the calculation efficiency is improved.
Drawings
FIG. 1 is a schematic diagram of a cylindrical flow winding of a large vortex simulation method of an incompressible viscous fluid based on a coherent structure according to the present invention;
FIG. 2 is a variation of resistance coefficient of cylindrical streaming around with time according to a large vortex simulation method of an incompressible viscous fluid based on a coherent structure;
FIG. 3 is a time-dependent variation of lift coefficient of a cylindrical streaming based on a large vortex simulation method of an incompressible viscous fluid with a coherent structure.
Detailed Description
The preferred embodiments of the present invention will be described in conjunction with the accompanying drawings, and it will be understood that they are described herein for the purpose of illustration and explanation and not limitation.
The invention provides a coherent structure-based large vortex simulation method for incompressible viscous fluid, which comprises the following steps of:
establishing a lattice Boltzmann method model to obtain a calculation formula of fluid viscosity v, wherein the fluid viscosity v comprises molecular viscosity v 0 And vortical viscosity v t ；
In the conventional Smigorinsky model, the vortical viscosity v t By passingFiltered strain rate tensor S αβ Filter scale Δ x and Smagorinsky constant C S Determining;
in a lattice Boltzmann method-large vortex simulation method model, a filtering scale delta x is taken as a unit length, and a momentum flux Q is established αβ The strain rate tensor S αβ Smagorinsky constant C S A relation equation of relaxation time and average density of the system;
galileo invariants Q introduced into sub-lattice flow fields LES Define the Galileo invariant Q LES And the strain rate tensor S αβ Tensor W of rotation rate αβ Defining a model parameter C to obtain the model parameter C and the Smigorinsky constant C S The relationship of (a);
in a multi-relaxation-time lattice Boltzmann method model, obtaining the relationship of relaxation time under different time step lengths according to the relationship among relaxation time, vortex viscosity and momentum flux;
the whirling viscosity was obtained in a large vortex simulation.
Preferably, said Galileo invariant Q LES And the strain rate tensor S αβ Tensor W of rotation rate αβ The relational equation of (a) is:
Figure GDA0004091516990000051
preferably, the model parameters C are defined as:
C＝C CSM |F CSM | 3/2 F Ω
wherein C is a model parameter, C CSM Is a custom constant, F CSM As a function of coherent structure, F Ω Is an energy decay suppression function.
Preferably, the model parameters C define:
Figure GDA0004091516990000052
Figure GDA0004091516990000053
F Ω ＝1-F CSM ，
Figure GDA0004091516990000054
where E is the velocity gradient tensor.
Preferably, the model parameter C and the Smagorinsky constant C are obtained according to the definition of the model parameter C S The relationship of (1):
Figure GDA0004091516990000055
preferably, in said conventional Smagorinsky model, the vortical viscosity v t And the filtered strain rate tensor S αβ Filter scale Δ x and Smagorinsky constant C S The relation of (A) is as follows:
Figure GDA0004091516990000056
wherein, the first and the second end of the pipe are connected with each other,
Figure GDA0004091516990000057
preferably, in the lattice boltzmann method-large vortex simulation method model, the filtering scale delta x is taken as the unit length, and the momentum flux Q is established αβ The strain rate tensor S αβ Smagorinsky constant C S Relaxation time τ, mean density of the system ρ 0 The relational equation of (1):
Figure GDA0004091516990000061
preferably, in the multi-relaxation-time lattice boltzmann method model, the relationship of the relaxation time at different time steps is obtained according to the relationship among the relaxation time, the vortex viscosity and the momentum flux:
Figure GDA0004091516990000062
wherein, tau 0 、τ t All are relaxation times.
The above steps are explained in detail as follows:
multi-relaxation time Lattice Boltzmann Method (Multiple-relaxation-time Lattice Boltzmann Method, MRT-LBM) model:
definition grid x i Set of distribution functions of (a): { f α |α＝0,1,…,Q-1} (1-1)，
Q in the formula (1-1) is the speed direction number;
the collision occurs in the momentum space M = R Q In, translation occurs in velocity space V = R Q In, Q is the number of speed directions;
in three-dimensional lattice space delta x Z D And a discrete time t n ∈δ t N 0 In the following, D is the spatial dimension and the evolution equation is:
Figure GDA0004091516990000063
in the formula (1-2), M is a Q multiplied by Q matrix,
Figure GDA0004091516990000064
is a relaxation matrix, m is a momentum vector;
the mapping of the velocity and momentum space is achieved by a linear transformation as follows:
m＝M·f (1-3)，
f＝M -1 ·m (1-4)，
in the formula (1-4), m is a momentum vector, and f is a distribution function vector;
the respective parts in the equations (1-2) are expressed as follows:
f(x i +eδ t ,t n +δ t )＝(f 0 (x i ,t n +δ t ),…,f Q (x i +e Q δ t ,t n +δ t )) T (1-5)，
f(x i ,t n )＝(f 0 (x i ,t n ),f 1 (x i ,t n ),…,f Q (x i ,t n )) T (1-6)，
m＝(m 0 (x i ,t n ),m 1 (x i ,t n ),…,m Q (x i ,t n )) T (1-7)，
Figure GDA0004091516990000071
taking the D3Q19 model as an example, the matrix M is:
Figure GDA0004091516990000072
in the D3Q19MRT-LBM model, 19 discrete velocities are:
Figure GDA0004091516990000073
the density and mass flow rate are calculated by the following formula:
Figure GDA0004091516990000074
Figure GDA0004091516990000075
ρ in the formulas (1-10) (1-11) is the fluid density, ρ u is the mass flow rate;
the 19 components of the momentum vector are given in the following order:
m 0 ＝δρ (1-12)
m 1 ＝e (1-13)
m 2 ＝ε (1-14)
m 3,5,7 ＝j x,y,z (1-15)
m 4,6,8 ＝q x,y,z (1-16)
m 9 ＝3p xx (1-17)
m 10 ＝3π xx (1-18)
m 11 ＝p ww (1-19)
m 12 ＝π ww (1-20)
m 13,14,15 ＝p xy,yz,zx (1-21)
m 16,17,18 ＝m x,y,z (1-22)
delta in equations (1-12) is a numerical operator, e in equations (1-13) is a quantity related to energy, epsilon in equations (1-14) is a quantity related to the square of energy,
j＝(j x ,j y ,j z )＝ρ 0 u (1-23)
in the formula (1-23), j is the momentum, ρ 0 Is the average density of the system, generally fixed at 1; j in the formulae (1 to 15) x,y,z Is a component of j in equations (1-23);
q＝(q x ,q y ,q z ) (1-24)
in the formulas (1-24), q represents heat flux; q in the formula (1-16) x,y,z Are related to the components of q for equations (1-24);
equations (1-17) (1-19) (1-21) are all related to components of the symmetric, unscented strain rate tensor;
the formulas (1-18) (1-20) are four orders of momentum;
the formula (1-22) is third-order momentum;
the equilibrium momentum is a function of the conservation-type momentum, wherein the non-conservation-type momentum is:
Figure GDA0004091516990000081
Figure GDA0004091516990000082
Figure GDA0004091516990000083
Figure GDA0004091516990000084
Figure GDA0004091516990000091
Figure GDA0004091516990000092
Figure GDA0004091516990000093
Figure GDA0004091516990000094
Figure GDA0004091516990000095
Figure GDA0004091516990000096
Figure GDA0004091516990000097
relaxation matrix
Figure GDA0004091516990000098
For a diagonal matrix, the following is shown:
Figure GDA0004091516990000099
in the formulae (1 to 36) s e ＝1.19，s ε ＝s π ＝1.4，s q ＝1.2，s m =1.98, wherein s v Is the relaxation frequency (or relaxation frequency) associated with the viscosity of the fluid;
when the model lattice unit delta x ＝δ t When =1, the sound velocity is
Figure GDA00040915169900000910
The fluid viscosity at this time was: />
Figure GDA00040915169900000911
Formula (1-37) v is the fluid viscosity,
Figure GDA00040915169900000912
among the LES are:
ν＝ν 0 +ν t (1-38)
v in the formula (1-38) 0 Is the molecular viscosity, v t Vortex viscosity;
in the conventional Smigorinsky model, the vortical viscosity is represented by the filtered strain rate tensor S αβ Filter scale Δ x, smagorinsky constant C S Co-determinationThe following relationship:
Figure GDA00040915169900000913
in equations (1-39)
Figure GDA0004091516990000101
In LBM-LES, a matrix grid is typically used, generally given by:
Δ x ＝δ x ＝1 (1-40)
strain rate tensor S αβ From the unbalanced term of the density distribution function:
Figure GDA0004091516990000102
wherein momentum flux Q αβ The following relation:
Figure GDA0004091516990000103
the following relationships are obtained from equations (1-41) (1-42):
Figure GDA0004091516990000104
τ in the formulas (1-43) is the relaxation time;
galileo invariants Q introduced into sub-lattice flow fields LES Defined as the formula:
Figure GDA0004091516990000105
in the formulae (1-44)
Figure GDA0004091516990000106
Formula (II)(1-45)W αβ Is the rotation rate tensor;
the purpose of introducing the Galileo invariant is to determine the Smigorinsky constant C S The model parameters are defined as follows:
C＝C CSM |F CSM | 3/2 F Ω (1-46)
Figure GDA0004091516990000107
Figure GDA0004091516990000108
F Ω ＝1-F CSM (1-49)
Figure GDA0004091516990000109
c in the formulas (1-46) (1-47) (1-48) (1-49) (1-50) is a model parameter, C CSM Is a custom constant, F CSM As a function of coherent structure, F Ω Is an energy decay inhibition function, E is a velocity gradient tensor;
galileo invariant Q LES The tensor E and the velocity gradient can be obtained by a central difference method:
Figure GDA0004091516990000111
Figure GDA0004091516990000112
due to W αβ Regardless of the material frame, and with respect to the galileo invariant, if the flow occurs under a rotating frame, the following transformations are required:
Figure GDA0004091516990000113
Figure GDA0004091516990000114
in the equations (1-53) (1-54), the physical quantity in the rotating frame, ε, is shown by superscript αβγ In order to transform the tensor,
Figure GDA0004091516990000115
is a rotational velocity vector;
defining model parameters C and Smagorinsky constant C S The relation of (1):
Figure GDA0004091516990000116
in MRT-LBM, the relaxation rate S ν From relaxation time τ 0 、τ t A joint decision, defined as:
Figure GDA0004091516990000117
Figure GDA0004091516990000118
τ t ＝3ν t (1-58)
will also
Figure GDA0004091516990000119
Substituting equations (1-43) into (1-59) (1-55) yields the following:
Figure GDA00040915169900001110
combining (1-56), (1-57), (1-58) and (1-60) to obtain:
Figure GDA0004091516990000121
obtaining the vortical viscosity v in the simulation of the large vortex according to (1-58) t 。
For example, the following steps are carried out:
as shown in fig. 1, the coefficient of resistance of the cylindrical streaming is calculated:
the first step is as follows: constructing grids, constructing rectangular grids:
N x ×N y ×N z ＝704×133×133，δ x ＝δ y ＝δ z ＝1，δ t ＝1；
the second step: initializing a flow field:
Figure GDA0004091516990000122
V＝W＝0，ρ 0 =1; wherein U is the speed in the x direction, V, W is the speed in the y and z directions, respectively, H is the calculated domain length in the y and z directions, U max Maximum velocity at the entrance;
the collision process and the migration process are as follows:
Figure GDA0004091516990000123
the third step: loading boundary conditions, and processing curve boundaries by using a Yu method;
the fourth step: calculating the coefficient of resistance C D And coefficient of lift C L ：
Figure GDA0004091516990000124
Figure GDA0004091516990000125
The fourth step: calculating the change of the macroscopic physical quantity U, V, W;
the fifth step: calculating a model parameter C;
and a sixth step: updating the local vortex viscosity of the flow field;
the seventh step: judging convergence, and using a technicot postprocessing result, as shown in fig. 2 and 3, the abscissa of fig. 2 represents a time step, and the ordinate represents a resistance coefficient; the abscissa of fig. 3 represents the time step and the ordinate represents the lift coefficient.
The invention has the beneficial effects that:
galileo invariants Q introduced into sub-lattice flow fields LES A coherent structure function based on a Smagorinsky model is constructed, the vortex viscosity of the LES is dynamically calculated, the calculation of the relaxation time is simple, and an accurate value is easily obtained;
the model parameters are always positive, and the numerical calculation is stable;
the coherent structure function only comprises a strain rate tensor and a rotation rate tensor, does not need a filtering function, is simple to calculate and is convenient for engineering application;
the model is constructed based on a coherent structure representing turbulence, the coherent structure is shielded, and the model is also suitable for laminar flow, so that the model has universality;
the model parameters consist of a fixed model parameter and a coherent structure function, the coherent structure function is Galileo invariant normalized by the size of the velocity gradient tensor, and has the function of wall surface damping without using a wall surface function;
the strain rate tensor of the curve boundary is implicitly obtained in the model, and the calculation precision is high;
can be used for rotating homogeneous turbulence and channel turbulence;
the programming is simple, the parallelization is easy, and the calculation efficiency is improved.
The above description is only a preferred embodiment of the present invention, and is not intended to limit the present invention in any way, so that any modification, equivalent change and modification made to the above embodiment according to the technical spirit of the present invention are within the scope of the technical solution of the present invention.

Claims (4)

1. A large vortex simulation method of incompressible viscous fluid based on coherent structure is characterized by comprising the following steps:
establishing a lattice Boltzmann method model to obtain a calculation formula of fluid viscosity v, wherein the fluid viscosity v comprises molecular viscosity v 0 And a vortex viscosity v t ；
In the conventional Smagorinsky model, the vortical viscosity v t From the filtered strain rate tensor S αβ Filter scale Δ x and Smagorinsky constant C S Determining;
in a lattice Boltzmann method-large vortex simulation method model, a filtering scale delta x is taken as a unit length, and a momentum flux Q is established αβ The strain rate tensor S αβ Smigorinsky constant C S A relation equation of relaxation time and average density of the system;
galileo invariants Q introduced into sub-lattice flow fields LES Define Galileo invariant Q LES And the strain rate tensor S αβ Tensor W of rotation rate αβ Defining a model parameter C to obtain the model parameter C and the Smigorinsky constant C S The relationship of (1);
in a multi-relaxation-time lattice Boltzmann method model, obtaining the relationship of relaxation time under different time step lengths according to the relationship among relaxation time, vortex viscosity and momentum flux;
obtaining vortex viscosity in a large vortex simulation;
the Galileo invariant Q LES And the strain rate tensor S αβ Tensor W of rotation rate αβ The relational equation of (A) is as follows:
Figure FDA0004091516980000011
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0004091516980000012
is a speed component in the alpha direction>
Figure FDA0004091516980000013
For a speed component in the beta direction>
Figure FDA0004091516980000014
Is a displacement in the direction of beta and,
Figure FDA0004091516980000015
displacement in the alpha direction;
the model parameters C are defined as:
C＝C CSM |F CSM | 3/2 F Ω
wherein C is a model parameter, C CSM Is a custom constant, F CSM As a function of coherent structure, F Ω Is an energy decay suppression function;
in the model parameter C definition:
Figure FDA0004091516980000016
Figure FDA0004091516980000017
F Ω ＝1-F CSM ，
Figure FDA0004091516980000021
wherein E is a velocity gradient tensor;
according to the definition of the model parameter C, obtaining the model parameter C and the Smigorinsky constant C S The relationship of (c):
Figure FDA0004091516980000022
2. the method of modeling large vortices of incompressible viscous fluid according to claim 1, wherein the Smagorinsky model has a vortical viscosity v t And the filtered strain rate tensor S αβ Filter scale Δ x and Smagorinsky constant C S The relation of (A) is as follows:
Figure FDA0004091516980000023
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0004091516980000024
3. the method for modeling macrovortices in an incompressible viscous fluid having a coherent structure as claimed in claim 1, wherein the momentum flux Q is established by taking the filtering dimension Δ x as a unit length in a lattice boltzmann method-macrovortex modeling method model αβ The strain rate tensor S αβ Smigorinsky constant C S Relaxation time τ, average density of the system ρ 0 The relational equation of (1):
Figure FDA0004091516980000025
4. the large vortex simulation method for incompressible viscous fluids based on coherent structure of claim 1, wherein in the multi-relaxation-time lattice boltzmann method model, the relationship of relaxation time at different time steps is obtained according to the relationship among relaxation time, vortex viscosity and momentum flux:
Figure FDA0004091516980000026
wherein, tau 0 、τ t Are all relaxation times, p 0 Is the average density of the system.
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