Lattice boltzmann collision operators enforcing isotropy and galilean invariance
地域：
RI
RI Johnston
摘要
A method comprising: simulating, in a lattice velocity set, movement of particles in a volume of fluid, with the movement causing collision among the particles; based on the simulated movement, determining relative particle velocity of a particle at a particular location within the volume, with the relative particle velocity being a difference between (i) an absolute velocity of the particle at the particular location within the volume and measured under zero flow of the volume, and (ii) a mean velocity of one or more of the particles at the particular location within the volume; and determining, based on the relative particle velocity, a non-equilibrium post-collide distribution function of a specified order that is representative of the collision.
說明書
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This application is a continuation of and claims priority to Ser. No. 14/829,933, filed on Aug. 19, 2015, which is a continuation of and claims priority to international patent application number PCT/US2014/048004 filed Jul. 24, 2014, published on Jan. 29, 2015, which claims priority to provisional application No. 61/858,051 filed on Jul. 24, 2013, the entire contents of each of which are incorporated herein by reference.
Collision process is one of the two fundamental dynamical processes in a many particle system—another one is advection process. Collision process is essential for individual particles to interact and form a collective behavior. During a collision process, mass, momentum and energy are exchanged among the particles obeying conservation laws. These conservation laws ensure that the overall mass and momentum (and sometimes energy) among the participating particles are unchanged before and after a collision.
權(quán)利要求
What is claimed is:1. A method comprising:simulating, in a lattice velocity set, movement of particles in a volume of fluid, with the movement causing collision among the particles; processing, by a data processing system, based on the simulated movement of particles in the volume of fluid, data indicative of a particle velocity at a lattice location to determine a relative particle velocity, with the relative particle velocity being a difference between an absolute particle velocity at the lattice location measured under zero flow in the volume of fluid, and a mean velocity of particles at the lattice location; and determining, by the data processing system, based on the relative particle velocity at the lattice location, a lattice Boltzmann post-collide distribution for a fluid system that is representative of a collision process at the lattice location. 2. The method of claim 1 , further comprising:providing, by one or more computer systems, the lattice velocity set that supports hydrodynamic moments up to a specified order of the particle velocity. 3. The method of claim 2 , wherein the supported order for the lattice velocity set is less than and different from the specified order of the lattice Boltzmann post-collide distribution; andwherein the specified order for the lattice Boltzmann post-collide distribution is determined by the order of the particle velocity. 4. The method of claim 1 , wherein the mean flow velocity is the mean velocity of the flow near to or at where the relative velocity is taken. 5. The method of claim 2 , wherein the lattice velocity set is a set of state vectors associated with the lattice Boltzmann method. 6. The method of claim 1 , wherein the lattice Boltzmann post-collide distribution retains non-equilibrium moments for predefined physical quantities up to a specified order, and eliminates non-equilibrium moments for undefined physical quantities beyond the specified order. 7. The method of claim 6 , wherein the specified order is an exponential value associated with a ratio of particle velocity to lattice sound speed. 8. The method of claim 2 , wherein the lattice velocity set comprises a set of momentum states in a space that is limited to a lattice. 9. The method of claim 1 , wherein the lattice Boltzmann post-collide distribution is a Galilean invariant filtered operator. 10. The method of claim 1 , further comprising:modeling, based on the lattice Boltzmann post-collide distribution, a collision process of particles in a volume of fluid. 11. The method of claim 1 , wherein the lattice Boltzmann post-collide distribution is produced by a lattice Boltzmann post-collide distribution function that is a collision operator Ci(1)(x,t) of a first order Galilean invariance in terms of Mach number for a lattice velocity set that provides first order support for hydrodynamic moments, with the collision operator Ci(1)(x,t) being a finite approximation of an exact form of the collision operator as a function of relative velocity; andwherein the collision operator is defined in accordance with: wherein x is the lattice location within a volume; wherein t is a particular point in time; wherein i is an index number of lattice velocities in the lattice velocity set; wherein T0 is a constant lattice temperature; wherein ci is a velocity vector of particles prior to collision; wherein u(x,t) is fluid velocity among the particles at lattice location x at time t; wherein I is a second rank unity tensor; wherein τ is collision relation time; wherein wi is a constant weighting factor; and wherein πneq is a non-equilibrium momentum flux. 12. The method of claim 1 , wherein the lattice Boltzmann post-collide distribution is produced by a lattice Boltzmann post-collide distribution function that is a collision operator Ci(x,t) for a lattice velocity set that provides an infinite order of support for hydrodynamic moments, and wherein the collision operator is defined in accordance with: wherein x is the lattice location within a volume; wherein t is a particular point in time; wherein i is an index number of lattice velocities in the set; wherein T0 is a constant lattice temperature; wherein I is a second rank unity tensor; wherein τ is collision relation time; wherein Ci(1)(x,t) is relative particle velocity; wherein ρ is fluid density; wherein fieq is an equilibrium distribution function; and wherein πneq is a non-equilibrium momentum flux. 13. The method of claim 1 , wherein the lattice Boltzmann post-collide distribution is produced by a lattice Boltzmann post collide distribution function that is a collision operator Ci(2)(x,t) of a second order Galilean invariance in terms of Mach number for a lattice velocity set that provides second order support for hydrodynamic moments, with the collision operator Ci(2)(x,t) being a finite approximation of an exact form of the collision operator as a function of relative velocity; andand wherein the collision operator is defined in accordance with: wherein x is the lattice location within a volume; wherein t is a particular point in time; wherein i is an index number of lattice velocities in the set; wherein T0 is a constant lattice temperature; wherein ci is a velocity vector of particles prior to collision; wherein u(x,t) is fluid velocity among the particles at lattice location x at time t; wherein I is a second rank unity tensor; wherein τ is collision relation time; wherein wi is a constant weighting factor; and wherein πneq is a non-equilibrium momentum flux. 14. The method of claim 6 , wherein a predefined physical quantity comprises at least one of mass of fluid in a particular volume, momentum of fluid in that particular volume or energy of fluid in that particular volume. 15. The method of claim 1 , wherein the lattice Boltzmann post-collide distribution is produced by a lattice Boltzmann post collide distribution function that is a collision operator Ci(x,t) pertaining to energy flux, andwherein the collision operator is defined in accordance with: wherein x is the lattice location within a volume; wherein t is a particular point in time; wherein i is an index number of lattice velocities in a lattice velocity set; wherein T0 is a constant lattice temperature; wherein I is a second rank unity tensor; wherein τ is collision relation time; wherein Ci′(x,t) is relative particle velocity; wherein fieq is an equilibrium distribution function; and wherein Wneq is a non-equilibrium energy flux; and wherein ρ represents density. 16. One or more machine-readable hardware storage devices storing instructions that are executable by one or more processing devices to perform operations comprising:simulating, in a lattice velocity set, movement of particles in a volume of fluid, with the movement causing collision among the particles; based on the simulated movement, processing data indicative of a particle velocity at a lattice location to determine relative particle velocity, with the relative particle velocity being a difference between an absolute particle velocity at the lattice location measured under zero flow in the volume of fluid, and a mean velocity of particles at the lattice location; and determining, based on the relative particle velocity at the lattice location, a lattice Boltzmann post-collide distribution for a fluid system that is representative of a collision process at the lattice location. 17. The one or more machine-readable hardware storage devices of claim 16 , wherein the operations further comprise:providing a lattice velocity set that supports hydrodynamic moments up to a specified order of the particle velocity. 18. The one or more machine-readable hardware storage devices of claim 17 , wherein the supported order for the lattice velocity set is less than and different from the specified order of the lattice Boltzmann post-collide distribution; andwherein the specified order for the lattice Boltzmann post-collide distribution is determined by the order of the particle velocity. 19. The one or more machine-readable hardware storage devices of claim 16 , wherein the mean local flow velocity is the mean velocity of the flow near to or at where the relative velocity is taken. 20. The one or more machine-readable hardware storage devices of claim 17 , wherein the lattice velocity set is a set of state vectors associated with the lattice Boltzmann method. 21. The one or more machine-readable hardware storage devices of claim 16 , wherein the lattice Boltzmann post-collide distribution retains non-equilibrium moments for predefined physical quantities up to a specified order, and eliminates non-equilibrium moments for undefined physical quantities beyond the specified order. 22. The one or more machine-readable hardware storage devices of claim 21 , wherein the specified order is an exponential value associated with a ratio of particle velocity to lattice sound speed. 23. The one or more machine-readable hardware storage devices of claim 16 , wherein the lattice velocity set comprises a set of momentum states in a space that is limited to a lattice. 24. The one or more machine-readable hardware storage devices of claim 16 , wherein the lattice Boltzmann post-collide distribution is produced by a lattice Boltzmann post-collide distribution function that is a Galilean invariant filtered operator. 25. The one or more machine-readable hardware storage devices of claim 16 , wherein the operations further comprise:modeling, based on the lattice Boltzmann post-collide distribution, a collision process of particles in a volume of fluid. 26. The one or more machine-readable hardware storage devices of claim 21 , wherein a predefined physical quantity comprises at least one of mass of fluid in a particular volume, momentum of fluid in that particular volume or energy of fluid in that particular volume. 27. A system comprising:one or more processing devices; and one or more machine-readable hardware storage devices storing instructions that are executable by the one or more processing devices to perform operations comprising: simulating, in a lattice velocity set, movement of particles in a volume of fluid, with the movement causing collision among the particles; based on the simulated movement, processing data indicative of a particle velocity at a lattice location to determine relative particle velocity, with the relative particle velocity being a difference between an absolute particle velocity at the lattice location measured under zero flow in the volume of fluid, and a mean velocity of particles at the lattice location; and determining, based on the relative particle velocity at the lattice location, a lattice Boltzmann post-collide distribution for a fluid system that is representative of a collision process at the lattice location. 28. The system of claim 27 , wherein processing further comprises processing, by the data processing system, data indicative of a particle velocity for a portion of particles represented at the lattice location to determine the relative particle velocity for the portion of particles. 29. The system of claim 27 wherein the mean flow velocity is the mean velocity of the flow near to or at where the relative velocity is taken. 30. The system of claim 27 , wherein at least one of the one or more physical conditions comprises flow rate in the fluid system, temperature of the fluid system, or pressure in the fluid system. 31. The method of claim 9 , where the Galilean invariant filtered operator is supported by a lattice velocity set with infinite order, and Galilean invariant up to an order in accordance to a lattice set of a specified order.
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