CN111368487B - Flow field processing method for simulating periodic movement of particles based on lattice Boltzmann model - Google Patents
Flow field processing method for simulating periodic movement of particles based on lattice Boltzmann model Download PDFInfo
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Abstract
The invention discloses a flow field processing method for simulating periodic movement of particles based on a lattice Boltzmann model, which comprises the following steps: 1) initializing a flow field, 2) evolving the flow field, 3) carrying out boundary processing, 4) correcting by a momentum exchange method, and 5) calculating the macroscopic quantity. The method improves the existing simulation method that particles in the simulation flow field cannot directly flow out from one end of the flow field and flow in from the other end of the flow field, and compared with other fluid-solid coupling methods, the method adopts a momentum exchange method, and has the advantages of calculating the hydraulic power of the solid surface, being independent of the geometric shape of the particles, having simple form and high calculation precision; the modification and improvement of the momentum exchange can simulate a complex-shaped pipeline which cannot be simulated by the traditional momentum exchange method and other fluid-solid coupling methods, so that simulation scenes simulating different experimental modes such as particle sorting, aggregation and migration are increased, and the consumption of memory is greatly reduced.
Description
Technical Field
The invention relates to the technical fields of statistical physics, fluid mechanics and computer numerical simulation, in particular to a flow field processing method for simulating periodic movement of particles based on a lattice Boltzmann model.
Background
With the rapid development of computer technology, the numerical simulation method gradually becomes one of three scientific research methods parallel to experiments and theoretical analysis by utilizing the characteristic that computer high-speed operation is not tired. Among them, in the field of computational fluid dynamics, the Lattice Boltzmann Method (LBM) is a mainstream numerical simulation method. LBM has clear physical background, simple and efficient calculation, and can simulate complex fluid behavior at mesoscale, and has been successfully utilized in the simulation of complex systems such as turbulence, multiphase flow, microfluid, etc.
To study particle suspension problems, there are three main approaches to simulated flow-solid phase interactions: momentum exchange method, immersed boundary method and pressure tensor integration method; the latter two adopt to solve the calculus equation, the computational process is complex, inefficiency, especially tensor integration method can introduce the error, reduce the stability of calculating, calculate the load than momentum exchange method is big. Momentum Exchange (GME) is computationally compatible with LBM meshing features, and flow-solid interaction is obtained by summing momentum elements across boundaries, without extrapolation or interpolation. Compared with a method for solving a calculus equation, the GME can solve stress on the boundary of a complex shape, and the computational locality and parallelism of the LBM are fully maintained.
Boundary conditions are important factors for driving a flow field in LBM, and different implementation methods of the boundary conditions have great influence on the calculation accuracy, the numerical stability and the calculation efficiency of the LBM. LBM boundary formats are broadly divided into heuristic, kinetic and interpolation formats; again, the speed boundaries and pressure boundaries can be categorized by type. Because the memory capacity and the computing capacity of the computer equipment are limited, the computing simulation domain cannot be infinitely large or infinitely far away, people want to simulate a relatively long or relatively wide watershed in reality, and the effect of performing periodic simulation by utilizing the similarity of the local flow field can also be achieved completely according to the objective watershed size. The method solves the problem that particles cannot pass through an inlet and an outlet of a flow field in a current general grid moving mode, namely, after the particles move forwards for a small distance, the flow field is integrally moved backwards, and under the condition that the characteristic of an original flow field is changed in a negligible way, the particles always move stably in a small section of interval inside the flow field, and the method has the defect that only a straight pipeline with unchanged density gradient is suitable. Conventional periodic boundaries often employ a physical driving scheme that assumes that each fluid point in the flow field is spontaneously subjected to a directional force to drive the movement of the fluid. The driving mode has the defect that the driving mode of the objective world flow field cannot be comprehensively realized, for example, a test solution carrying particles in a bent pipeline moves under the pushing of pressure, the direction of the stress of each fluid point is difficult to determine by using physical driving, and thus, the motion of the fluid cannot be accurately simulated.
The periodic pressure boundary processing method can simulate a bent pipeline which cannot be accurately simulated originally, such as a snake-shaped pipeline and an annular pipeline, and the coupled momentum exchange method is used for researching the motion rule of particles in a complex pipeline, and can be applied to experiments for simulating cell transportation, aggregation, sorting and the like in the fields of microfluidics and microfluidics.
Disclosure of Invention
The invention aims to overcome the defects that a periodic physical force driving simulation driving mode is incomplete and a moving grid cannot accurately simulate a bending pipeline, and provides a flow field processing method for simulating periodic movement of particles based on a lattice Boltzmann model.
The technical scheme for realizing the aim of the invention is as follows:
a flow field processing method for simulating periodic movement of particles based on a lattice Boltzmann model comprises the following steps:
1) Initializing a flow field: according to a Navier-Stokes equation derived from a discretized grid gas model, a discrete grid model DnQm of n-dimensional discrete space and m-speed space is adopted, a simulated flow field simulating periodic movement of particles is established, a solving range of the flow field, boundary conditions and initial conditions are set, the boundary conditions are periodic pressure boundaries, and the initial conditions comprise a fluid distribution function, density, initial speed, node type, radius of the particles and rotational inertia of the particles;
2) And (3) flow field evolution: according to an LBM evolution equation, carrying out collision and flow of a distribution function on the fluid nodes in the whole simulation flow field established in the step 1), and completing simulation of most of non-boundary processing areas of the flow field in a time step delta t;
3) Boundary processing: processing the boundary conditions set in the step 1), wherein the boundary conditions comprise a closed boundary and a dynamic open boundary; in the same time step delta t, firstly, performing half-way rebound boundary processing on the upper and lower closed boundaries of the pipeline, and solving to obtain a distribution function of which the upper and lower rows are lost when the distribution function is flowing due to the fact that no boundary point exists; secondly, processing the curve boundary of the particle surface by using a quadratic interpolation method; finally, a periodic pressure boundary condition is adopted at the left and right dynamic open boundaries, and the distribution functions of the inlet inflow and the outlet outflow are processed so as to maintain the stable evolution of the flow field;
4) Momentum exchange method correction: in order to stabilize the calculation stress of the round particles crossing the inlet, correcting the momentum exchange of the round particles in the crossing state, and respectively correcting the distribution functions on the fluid-solid connecting lines at the two ends of the inlet and the outlet in the crossing state; when the round particles enter a crossing state at the next moment, the density of the position of the mass center is slowly adjusted upwards until the particles end the crossing state at the next moment of a certain time step; the density adjustment range is from the fluid density when the flow field outlet reverses the particle radius towards the inlet direction to the fluid density when the particle radius advances from the inlet to the outlet direction; along with the correction of the mass center position density of the particles, the distribution function of the fluid-solid connection lines around the particles is correspondingly corrected, and the resultant force and the torque are calculated through a momentum exchange method after correction;
5) Macro-scale calculation: and (3) obtaining updated fluid node mass and velocity distribution through a modified momentum exchange method and a macroscopic constraint equation after the step of time delta t evolution flow and boundary processing, wherein the step of time delta t+1 is carried out, the next step is carried out, the steps 2) to 4) are repeated, macroscopic density and velocity of the flow field are obtained, and a continuous simulation process is realized.
In step 2), the LBM evolution equation is a numerical simulation method derived according to the Boltzmann equation, and the Boltzmann differential integral equation is as follows:
in the formula (1), f and f 1 As a distribution function of any two particles before collision, f' and f 1 ' is the distribution function of the corresponding particles after collision, t is time, x is the space where two particles are located, v and v 1 Is the velocity vector of two particles, α is the acceleration vector, dθ is the solid angle, and σ (θ) is the probability of the particles scattering to a unit solid angle; the collision term at the right end of the equation in the formula (1) is complex and difficult to directly solve, and the approximation method is adopted to solve, so that the approximation requirement of the collision term meets the following conservation law of mass and momentum:
in the above formula (2) and formula (3), Ω (f) i ) Representing collision terms, e i Representing a discrete velocity vector; assuming that the fluid motion has an equilibrium state according to the Bhatnager-Gross-Krook (BGK) model, intermolecular collisions cause the distribution function to approach the Maxwell equilibrium state distribution when the distribution function f i Deviation from a balanced distribution functionNear this, the rate of change of the distribution function caused by particle collisions can be regarded as being equal to +.>The proportional, i.e., BGK equation is:
wherein τ is the relaxation time, and based on the BGK equation, an approximate lattice Boltzmann equation composed by adopting a single relaxation time model is expressed as:
assume thatSmall enough, performing Taylor expansion at the right end of equation (5) and ignoring second order and subsequent small terms to obtain a lattice Boltzmann equation which is discrete in time and space, namely, an LBM evolution equation is as follows:
in the step 2), the flow of the collision and distribution function is as follows:
and (3) collision:
and (3) flow steps:
in the above formula (7) and formula (8), f andthe distribution functions before and after collision are respectively represented, the trend of fluid towards equilibrium distribution in the motion process is considered based on a BGK equation, the equilibrium distribution function is the key of LBM evolution, the equilibrium distribution function of LBGK adopts Maxwell equilibrium distribution, and the distribution functions can be represented as follows based on a DnQm model:
in the formula (9) of the present invention,is the lattice sound velocity omega i Is a weight systemThe number, i.e. the fluid distribution function and density used for initialization in step 1).
In step 3), the open boundary (pipe passageway) is defined by a periodic boundary condition expressed in the form of:
wherein f i | in And f i | out The distribution functions in the directions of an inlet end and an outlet end i are respectively represented, p is the pressure difference at two ends of a pipeline, the upper pipeline solid wall and the lower pipeline solid wall adopt a half-way rebound condition, and the treatment form is as follows:
i and +.>In the opposite direction, the closed boundary of the particle surface adopts a Lallemann quadratic interpolation method, and the form is as follows:
wherein omega i The definition is as follows:
q represents the shortest distance from the intersection point between the curve boundary and the fluid-solid connection line to the adjacent fluid point, and the wall-solid speed between two lattice points is u ω ，x h 、x h′ 、x h″ Respectively representing the fluid point on the fluid-solid connection, the adjacent node in the i direction and the secondary adjacent node.
In the step 4), the momentum exchange method satisfies Galileo invariance, and the form expression is:
in the formula (16), x s Is the physical boundary of the particle, x f Is equal to x b The fluid and solid points of the fluid-solid connection, respectively, and the total hydraulic force F and torque T applied to the particles are calculated by summing over all the fluid-solid connections, the calculation formula being as follows:
F＝∑F(x s ) (17)
T＝∑(x s -R)F(x s ) (18)
wherein R represents the spatial position of the centroid, and in order to calculate the hydraulic power to keep stable in the particle crossing process, the distribution function participating in calculation is corrected, and the correction standard is centroid density; the specific correction method is as follows:
the center of mass divides the particles into two parts, corrects the particles at the left part of the center of mass, and fluid points on the fluid-solid connecting line at the left side are provided with horizontal subscripts, one of which is provided with subscript i f The corresponding solid point subscript is i s At this time, the density of the centroid is ρ', the subscript of the centroid is x, and the horizontal offset from the fluid-solid line to the centroid is i f -x and i s -x, again because the two-dimensional two-parallel plate flow field inlet density is ρ 1 An outlet density ρ of 2 There is an entrance density difference Δρ=ρ 1 -ρ 2 The length L of the flow field has a density gradientAs a constant, subscript i f And i s The density after correction is +.>Andcorrection factor D f And D s Expressed as:
the modified momentum exchange method is expressed as:
the density after left correction is necessarily larger than rho ', the density after right correction is necessarily smaller than rho', and the translational and rotational actions of fluid on the particles are calculated stably when the particles are in a crossing state by combining formulas (17) and (18); the correction of the distribution function does not affect the data of the evolution of the fluid, and the macroscopic density and the velocity of the flow field are calculated in the same time step delta t according to the following macroscopic constraint equation:
ρ＝f i (21)
ρu＝Σe i f i (22)
the method of correcting the particles in the right part of the centroid is the same as the method of correcting the particles in the left part of the centroid.
The flow field processing method for simulating the periodical movement of particles based on the lattice Boltzmann model improves the existing simulation method that particles cannot directly flow out of one end of a flow field and flow in from the other end of the flow field in the simulated flow field. The existing method for simulating particles in a flow field has the defect that a pipeline with a complex shape (such as a snake shape, an annular shape, an asymmetric bent pipe and the like) cannot be conveniently simulated, and is characterized in that no matter in a momentum exchange or immersion boundary method, because pressure difference exists at two ends of the flow field, when one part of the surfaces of the particles are positioned at one end with high pressure, the other part of the surfaces are positioned at one end with low pressure, and correction is needed when the stress of the surfaces of the particles is calculated, so that the overall stress balance of the surfaces of the particles is realized, and the stress is equal to the stress when the particles are completely immersed in the flow field in the process of crossing out of an inlet. Therefore, the improvement scheme has the difficulty of finding a reasonable correction method to stabilize the calculated stress on the particle surface. Compared with the prior art, the invention has the following advantages:
1. compared with other fluid-solid coupling methods, the momentum exchange method adopted by the invention has the advantages of calculating the hydraulic power of the solid surface, being independent of the geometric shape of particles, having simple form and high calculation precision;
2. the invention can simulate the complex shape pipeline which can not be simulated by the traditional momentum exchange method and other fluid-solid coupling methods, increases the simulation scenes of different experimental modes such as particle sorting, aggregation and migration, and greatly reduces the consumption of the memory.
Drawings
FIG. 1 is a flow chart of a flow field processing method for simulating periodic movement of particles based on a lattice Boltzmann model;
FIG. 2 is a schematic diagram of a solid wall boundary process and velocity direction;
FIG. 3 is an overview of suspended particles in a circular pipe;
FIG. 4 is a graph of density where particles cross a boundary and are centroids;
figure 5 is a graph of the horizontal and vertical force analysis of the particles during the cross-out inlet stage.
Detailed Description
The present invention will now be further illustrated with reference to the drawings and examples, but is not limited thereto.
Examples:
as shown in fig. 1, a flow field processing method for simulating periodic movement of particles based on a lattice Boltzmann model comprises the following steps:
1) Initializing a flow field: according to a Navier-Stokes equation derived from a discretized grid gas model, a discrete grid model DnQm of n-dimensional discrete space and m-speed space is adopted, a simulated flow field simulating periodic movement of particles is established, a solving range of the flow field, boundary conditions and initial conditions are set, the boundary conditions are periodic pressure boundaries, and the initial conditions comprise a fluid distribution function, density, initial speed, node type, radius of the particles and rotational inertia of the particles;
2) And (3) flow field evolution: according to an LBM evolution equation, carrying out collision and flow of a distribution function on the fluid nodes in the whole simulation flow field established in the step 1), and completing simulation of most of non-boundary processing areas of the flow field in a time step delta t;
3) Boundary processing: processing the boundary conditions set in the step 1), wherein the boundary conditions comprise a closed boundary and a dynamic open boundary; as shown in fig. 2, the physical boundary of the fixed wall is positioned in the middle of the grid under the half-path rebound condition, and the treatment condition of the lower boundary of the pipeline is shown in the figure; in the same time step delta t, firstly, performing half-way rebound boundary processing on the upper and lower closed boundaries of the pipeline, solving to obtain upper and lower rows of distribution functions lost when the distribution functions are flowing due to the fact that no boundary points exist, and secondly, processing the curve boundaries of the particle surfaces by using a quadratic interpolation method; finally, a periodic pressure boundary condition is adopted at the left and right dynamic open boundaries, and the distribution functions of the inlet inflow and the outlet outflow are processed so as to maintain the stable evolution of the flow field; the solid wall boundary processing and the velocity direction are shown in fig. 2, taking a D2Q9 model as an example, the front and back velocity directions of the horizontal direction are respectively 1 and 3, the up and down velocity directions of the vertical direction are respectively 2 and 4, the left 4 velocity directions are respectively 5 from the vector sum of the 1 and 2 directions, the left 4 velocity directions are respectively 6, 7 and 8 directions after anticlockwise rotation by 90 degrees, namely, the distribution functions of 3, 6 and 7 directions of the fluid points at the inlet at the same horizontal position are sequentially determined, after collision, the distribution functions of 3, 6 and 7 directions of the fluid points at the outlet are converted and processed through a certain period of pressure boundary and are used as the distribution functions of the fluid points at the outlet, and likewise, the distribution functions of 1, 5 and 8 directions of the fluid points at the outlet are converted through the same method and are used as the distribution functions of the 1, 5 and 8 directions of the fluid points at the inlet after collision;
4) Momentum exchange method correction: the momentum exchange method acts on the fluid-solid connection line of the particle surface, the momentum flowing out from the fluid point in the unit time is expressed as the momentum increment when entering the surface particle point from the form, the momentum rebounded back to the fluid point from the surface particle point is taken as the momentum decrement, and the sum of the momentum increment and the momentum decrement is the force applied by the fluid-solid connection line. In order to stabilize the calculation stress of the round particles crossing the inlet, the round particle momentum exchange in the crossing state is corrected, and as shown in fig. 3, the distribution functions on the fluid-solid connecting lines at the two ends of the inlet and the outlet are respectively corrected in the crossing state; when the round particles enter a crossing state at the next moment, the density of the position of the mass center is slowly adjusted upwards until the particles end the crossing state at the next moment of a certain time step; the density adjustment range is from the fluid density when the flow field outlet reverses the particle radius towards the inlet direction to the fluid density when the particle radius advances from the inlet to the outlet direction; along with the correction of the mass center position density of the particles, the distribution function of the fluid-solid connection lines around the particles is correspondingly corrected, and the resultant force and the torque are calculated through a momentum exchange method after correction;
5) Macro-scale calculation: and (3) obtaining updated fluid node mass and velocity distribution through a modified momentum exchange method and a macroscopic constraint equation after the step of time delta t evolution flow and boundary processing, wherein the step of time delta t+1 is carried out, the next step is carried out, the steps 2) to 4) are repeated, macroscopic density and velocity of the flow field are obtained, and a continuous simulation process is realized.
In step 2), the LBM evolution equation is a numerical simulation method derived according to the Boltzmann equation, and the Boltzmann differential integral equation is as follows:
in the formula (1), f and f 1 As a distribution function of any two particles before collision, f' and f 1 ' is the distribution function of the corresponding particles after collision, t is timeBetween x is the space where two particles are located, v and v 1 Is the velocity vector of two particles, α is the acceleration vector, dθ is the solid angle, and σ (θ) is the probability of the particles scattering to a unit solid angle; the collision term at the right end of the equation in the formula (1) is complex and difficult to directly solve, and the approximation method is adopted to solve, so that the approximation requirement of the collision term meets the following conservation law of mass and momentum:
in the above formula (2) and formula (3), Ω (f) i ) Representing collision terms, e i Representing a discrete velocity vector; assuming that the fluid motion has an equilibrium state according to the Bhatnager-Gross-Krook (BGK) model, intermolecular collisions cause the distribution function to approach the Maxwell equilibrium state distribution when the distribution function f i Deviation from the equilibrium distribution function f i (eq) Not far away, the rate of change of the distribution function due to particle collisions can be regarded as being equal to f i -f i (eq) The proportional, i.e., BGK equation is:
wherein τ is the relaxation time, and based on the BGK equation, an approximate lattice Boltzmann equation composed by adopting a single relaxation time model is expressed as:
assume thatSmall enough, performing Taylor expansion at the right end of equation (5) and ignoring second and subsequent small terms to obtain time and spaceThe inter-discrete lattice Boltzmann equation, the LBM evolution equation, is:
in the step 2), the flow of the collision and distribution function is as follows:
and (3) collision:
and (3) flow steps:
in the above formula (7) and formula (8), f andthe distribution functions before and after collision are respectively represented, the trend of fluid towards equilibrium distribution in the motion process is considered based on a BGK equation, the equilibrium distribution function is the key of LBM evolution, the equilibrium distribution function of LBGK adopts Maxwell equilibrium distribution, and the distribution functions can be represented as follows based on a DnQm model:
in the formula (9) of the present invention,is the lattice sound velocity omega i The fluid distribution function and density are initialized for the weight coefficients, i.e. for use in step 1).
In step 3), the open boundary (pipe passageway) is defined by a periodic boundary condition expressed in the form of:
wherein f i | in And f i | out The distribution functions in the directions of an inlet end and an outlet end i are respectively represented, p is the pressure difference at two ends of a pipeline, the upper pipeline solid wall and the lower pipeline solid wall adopt a half-way rebound condition, and the treatment form is as follows:
i and +.>In the opposite direction, the closed boundary of the particle surface adopts a Lallemann quadratic interpolation method, and the form is as follows:
wherein omega i The definition is as follows:
q represents the shortest distance from the intersection point between the curve boundary and the fluid-solid connection line to the adjacent fluid point, and the wall-solid speed between two lattice points is u ω ，x h 、x h′ 、x h″ Respectively representing the fluid point on the fluid-solid connection, the adjacent node in the i direction and the secondary adjacent node.
In the step 4), the momentum exchange method satisfies Galileo invariance, and the form expression is:
in the formula (16), x s Is the physical boundary of the particle, x f Is equal to x b The fluid and solid points of the fluid-solid connection, respectively, and the total hydraulic force F and torque T applied to the particles are calculated by summing over all the fluid-solid connections, the calculation formula being as follows:
F＝∑F(x s ) (17)
T＝∑(x s -R)F(x s ) (18)
where R represents the spatial location of the centroid.
The existing pressure driving model enables the density and the pressure difference at two sides of the particle to be inconsistent, the equation (10) is used for calculating boundary stress, and when the particle straddles the inlet, the density (distribution function) at two sides of the particle is huge because one side of the particle is positioned at the inlet and the other side of the particle is positioned at the outlet, so that the hydraulic power calculation is inaccurate. The particles always move from one end close to the inlet to the outlet due to the driving of the pressure difference, the density of fluid points on the same column where the mass center of the particles is located during the movement is acted by the density difference, and the density tends to decrease towards the outlet, so that the particles start to end from the crossing state, the density change of a column where the centroid is located has huge difference, thus the fluctuation of calculated hydraulic force is necessarily brought, and as shown in fig. 4, the change curve diagram of the centroid density of the spherical particles in a crossing state is shown, wherein 0 represents a horizontal subscript of the initial position of the flow field, n represents a horizontal subscript of the end of the flow field, and r represents the radius of the spherical particles. The highest part of the black line segment represents the density of the inlet end, the lowest part represents the density of the outlet end, and the red line segment represents the mass center position density to be adjusted along with the crossing process, so that the mass center of particles before and after crossing is exposed to the density of the flow field can not jump, but rather, the particles are smoothly transited to ensure stable stress; in order to calculate the hydraulic power to keep stable in the particle crossing process, correcting the distribution function participating in calculation, wherein the corrected standard is centroid density; the specific correction method is as follows:
the center of mass divides the particles into two parts, corrects the particles at the left part of the center of mass, and fluid points on the fluid-solid connecting line at the left side are provided with horizontal subscripts, one of which is provided with subscript i f The corresponding solid point subscript is i s At this time, the density of the centroid is ρ', the subscript of the centroid is x, and the horizontal offset from the fluid-solid line to the centroid is i f -x and i s -x, again because the two-dimensional two-parallel plate flow field inlet density is ρ 1 An outlet density ρ of 2 There is an entrance density difference Δρ=ρ 1 -ρ 2 The length L of the flow field has a density gradientAs a constant, subscript i f And i s The density after correction is +.>Andcorrection factor D f And D s Expressed as:
the modified momentum exchange method is expressed as:
the density after left correction is necessarily larger than rho ', the density after right correction is necessarily smaller than rho', and the translational and rotational actions of fluid on the particles are calculated stably when the particles are in a crossing state by combining formulas (17) and (18); the correction of the distribution function does not affect the data of the evolution of the fluid, and the macroscopic density and the velocity of the flow field are calculated in the same time step delta t according to the following macroscopic constraint equation:
ρ＝∑f i (21)
ρu＝∑e i f i (22)
the method of correcting the particles in the right part of the centroid is the same as the method of correcting the particles in the left part of the centroid.
As shown in fig. 5, in the case of the horizontal and vertical stress of the spherical particles in the crossing state, by adopting the method, the size of the simulated flow field is set to 300 x 100, the particles are released in 10 grid positions which are downwards offset in the horizontal direction center and the vertical direction center of the flow field, and after three crossing, the stress situation of the spherical particles in the relatively stable fourth crossing state is obtained, as can be seen from fig. 5, the method of less correction in the stability of calculating the stress of the corrected momentum exchange is significantly improved.
Claims (4)
1. A flow field processing method for simulating periodic movement of particles based on a lattice Boltzmann model is characterized by comprising the following steps:
1) Initializing a flow field: according to a Navier-Stokes equation derived from a discretized grid gas model, a discrete grid model DnQm of n-dimensional discrete space and m-speed space is adopted, a simulated flow field simulating periodic movement of particles is established, a solving range of the flow field, boundary conditions and initial conditions are set, the boundary conditions are periodic pressure boundaries, and the initial conditions comprise a fluid distribution function, density, initial speed, node type, radius of the particles and rotational inertia of the particles;
2) And (3) flow field evolution: according to an LBM evolution equation, carrying out collision and flow of a distribution function on the fluid nodes in the whole simulation flow field established in the step 1), and completing simulation of most of non-boundary processing areas of the flow field in a time step delta t;
3) Boundary processing: processing the boundary conditions set in the step 1), wherein the boundary conditions comprise a closed boundary and a dynamic open boundary; in the same time step delta t, firstly, performing half-way rebound boundary processing on the upper and lower closed boundaries of the pipeline, and solving to obtain a distribution function of which the upper and lower rows are lost when the distribution function flows due to the fact that no distribution function exists at boundary points; secondly, processing the curve boundary of the particle surface by using a quadratic interpolation method; finally, a periodic pressure boundary condition is adopted at the left and right dynamic open boundaries, and the distribution functions of the inlet inflow and the outlet outflow are processed so as to maintain the stable evolution of the flow field;
4) Momentum exchange method correction: in order to stabilize the calculation stress of the round particles crossing the inlet, correcting the momentum exchange of the round particles in the crossing state, and respectively correcting the distribution functions on the fluid-solid connecting lines at the two ends of the inlet and the outlet in the crossing state; when the round particles enter a crossing state at the next moment, the density of the position of the mass center is slowly adjusted upwards until the particles end the crossing state at the next moment of a certain time step; the density adjustment range is from the fluid density when the flow field outlet reverses the particle radius towards the inlet direction to the fluid density when the particle radius advances from the inlet to the outlet direction; along with the correction of the mass center position density of the particles, the distribution function of the fluid-solid connection lines around the particles is correspondingly corrected, and the resultant force and the torque are calculated through a momentum exchange method after correction; the momentum exchange method satisfies Galileo invariance and is expressed as follows:
in the formula (16), x s Is the physical boundary of the particle, x f And x b The fluid and solid points of the fluid-solid connection, respectively, and the total hydraulic force F and torque T applied to the particles are calculated by summing over all the fluid-solid connections, the calculation formula being as follows:
F＝∑F(x s ) (17)
T＝∑(x s -R)F(x s ) (18)
wherein R represents the spatial position of the centroid, and in order to calculate the hydraulic power to keep stable in the particle crossing process, the distribution function participating in calculation is corrected, and the correction standard is centroid density; the specific correction method is as follows:
the centroid divides the particle into two pairsThe particles at the left part of the mass center are corrected, the fluid points on the left fluid-solid connecting line are provided with horizontal subscripts, and one of the fluid points is provided with the subscript as i f The corresponding solid point subscript is i s At this time, the density of the centroid is ρ', the subscript of the centroid is x, and the horizontal offset from the fluid-solid line to the centroid is i f -x and i s -x, again because the two-dimensional two-parallel plate flow field inlet density is ρ 1 An outlet density ρ of 2 There is an entrance density difference Δρ' =ρ 1 -ρ 2 The length L of the flow field has a density gradientAs a constant, subscript i f And i s The density after correction is +.>Andcorrection factor D f And D s Expressed as:
the modified momentum exchange method is expressed as:
the density after left correction is necessarily larger than rho ', the density after right correction is necessarily smaller than rho', and the translational and rotational actions of fluid on the particles are calculated stably when the particles are in a crossing state by combining formulas (17) and (18); the correction of the distribution function does not affect the data of the evolution of the fluid, and the macroscopic density and the velocity of the flow field are calculated in the same time step delta t according to the following macroscopic constraint equation:
ρ＝∑f i (21)
ρu＝∑e i f i (22)
the correction method of the particles at the right part of the mass center is the same as that of the particles at the left part of the mass center;
5) Macro-scale calculation: and (3) obtaining updated fluid node mass and velocity distribution through a modified momentum exchange method and a macroscopic constraint equation after the step of time delta t evolution flow and boundary processing, wherein the step of time delta t+1 is carried out, the next step is carried out, the steps 2) to 4) are repeated, macroscopic density and velocity of the flow field are obtained, and a continuous simulation process is realized.
2. The flow field processing method for simulating periodic movement of particles based on a lattice Boltzmann model according to claim 1, wherein in the step 2), the LBM evolution equation is a numerical simulation method derived from the Boltzmann equation, and the Boltzmann differential integral equation is:
in the formula (1), f and f 1 As a distribution function of any two particles before collision, f' and f 1 ' is the distribution function of the corresponding particles after collision, t is time, x is the space where two particles are located, v and v 1 Is the velocity vector of two particles, α is the acceleration vector, dθ is the solid angle, and σ (θ) is the probability of the particles scattering to a unit solid angle; the collision term at the right end of the equation in the formula (1) is complex and difficult to directly solve, and the approximation method is adopted to solve, so that the approximation requirement of the collision term meets the following conservation law of mass and momentum:
in the above formula (2) and formula (3), Ω (f) i ) Representing collision terms, e i Representing a discrete velocity vector; assuming that the fluid motion has an equilibrium state according to the Bhatnager-Gross-Krook model, intermolecular collisions promote the distribution function to approach the Maxwell equilibrium state distribution when the distribution function f i Deviation from the equilibrium distribution function f i (eq) Not far away, the rate of change of the distribution function due to particle collisions can be regarded as being equal to f i -f i (eq) The proportional, i.e., BGK equation is:
wherein τ is the relaxation time, and based on the BGK equation, an approximate lattice Boltzmann equation composed by adopting a single relaxation time model is expressed as:
assume thatSmall enough, performing Taylor expansion at the right end of equation (5) and ignoring second order and subsequent small terms to obtain a lattice Boltzmann equation which is discrete in time and space, namely, an LBM evolution equation is as follows:
3. the flow field processing method for simulating periodic movement of particles based on lattice Boltzmann model according to claim 1, wherein in step 2), the flow of collision and distribution functions, the equation of collision and flow process is:
and (3) collision:
and (3) flow steps:
in the above formula (7) and formula (8), f andthe distribution functions before and after collision are respectively represented, the trend of fluid towards equilibrium distribution in the motion process is considered based on a BGK equation, the equilibrium distribution function is the key of LBM evolution, the equilibrium distribution function of LBGK adopts Maxwell equilibrium distribution, and the distribution functions can be represented as follows based on a DnQm model:
in the formula (9) of the present invention,is the lattice sound velocity omega i The fluid distribution function and density are initialized for the weight coefficients, i.e. for use in step 1).
4. The flow field processing method for simulating periodic movement of particles based on lattice Boltzmann model according to claim 1, wherein in step 3), the open boundary adopts a periodic boundary condition expressed in the form of:
wherein f i | in And f i | out The distribution functions in the directions of an inlet end and an outlet end i are respectively represented, p is the pressure difference at two ends of a pipeline, the upper pipeline solid wall and the lower pipeline solid wall adopt a half-way rebound condition, and the treatment form is as follows:
i and +.>In the opposite direction, the closed boundary of the particle surface adopts a Lallemann quadratic interpolation method, and the form is as follows:
wherein omega i The definition is as follows:
q represents the shortest distance from the intersection point between the curve boundary and the fluid-solid connection line to the adjacent fluid point, and the wall-solid speed between two lattice points is u ω ，x h 、x h′ 、x h″ Respectively representing the fluid point on the fluid-solid connection, the adjacent node in the i direction and the secondary adjacent node.
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