Lattice Boltzmann collision operators enforcing isotropy and Galilean invariance
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.
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;
based on the simulated movement, processing, by a data processing system, data indicative of a lattice velocity at a lattice location to determine relative lattice velocity, with the relative lattice velocity being a difference between (i) an absolute lattice velocity at the particular lattice location, and (ii) a mean velocity at the lattice location; and
determining, based on the relative lattice velocity at the lattice location, a lattice Boltzmann post-collide distribution function 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, a lattice velocity set that supports hydrodynamic moments up to a specified order of the mean 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 function; and
wherein the specified order for the lattice Boltzmann post-collide distribution function is determined by the order of the mean velocity.
4. The method of claim 1 , wherein the mean velocity comprises a mean velocity of a particular type of particles at the lattice location.
5. The method of claim 2 , wherein the lattice velocity set is a set of state vectors associated with the lattice Boltzmann method.
6. There method of claim 1 , wherein the lattice Boltzmann post-collide distribution function (i) retains non-equilibrium moments for predefined physical quantities up to a specified order, and (ii) 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 fluid 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 relative lattice velocity comprises the mean velocity at the lattice location subtracted from the absolute lattice velocity at the lattice location.
10. The method of claim 1 , wherein the lattice Boltzmann post-collide distribution function is a Galilean invariant filtered operator.
11. The method of claim 1 , further comprising:
modeling, based on the lattice Boltzmann post-collide distribution function, a collision process of particles in a volume of fluid.
12. The method of claim 1 , wherein the lattice Boltzmann post-collide distribution function is a collision operator c i (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; and
wherein the collision operator is defined in accordance with:
C
i
(
1
)
(
x
,
t
)
=
(
1
-
1
τ
)
w
i
2
T
0
[
(
1
+
c
i
·
u
(
x
,
t
)
T
0
)
(
c
i
c
i
T
0
-
I
)
-
c
i
u
(
x
,
t
)
+
u
(
x
,
t
)
c
i
T
0
]
:
Π
neq
(
x
,
t
)
;
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 T 0 is a constant lattice temperature;
wherein c i is a velocity vector of particles prior to collision;
wherein u(x,t) is mean 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 w i is a constant weighting factor; and
wherein Π neq is a non-equilibrium momentum flux.
13. The method of claim 1 , wherein the lattice Boltzmann post-collide distribution function is a collision operator C i (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:
C
i
(
x
,
t
)
=
(
1
-
1
τ
)
f
i
eq
(
x
,
t
)
2
ρ
(
x
,
t
)
T
0
[
c
i
′
(
x
,
t
)
c
i
′
(
x
,
t
)
T
0
-
I
]
:
Π
neq
(
x
,
t
)
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 T 0 is a constant lattice temperature;
wherein I is a second rank unity tensor;
wherein τ is collision relation time;
wherein c′ i (x,t) is relative particle velocity;
wherein ρ is fluid density;
wherein ƒ i eq is an equilibrium distribution function; and
wherein Π neq is a non-equilibrium momentum flux.
14. The method of claim 1 , wherein the lattice Boltzmann post-collide distribution function is a collision operator C i (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; and wherein the collision operator is defined in accordance with:
C
i
(
2
)
(
x
,
t
)
=
(
1
-
1
τ
)
w
i
2
T
0
[
(
1
+
c
i
·
u
(
x
,
t
)
T
0
+
(
c
i
·
u
(
x
,
t
)
)
2
2
T
0
2
-
u
2
(
x
,
t
)
2
T
0
)
(
c
i
c
i
T
0
-
I
)
-
(
1
+
c
i
·
u
(
x
,
t
)
T
0
)
c
i
u
(
x
,
t
)
+
u
(
x
,
t
)
c
i
T
0
+
u
(
x
,
t
)
u
(
x
,
t
)
]
:
Π
neq
(
x
,
t
)
;
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 T 0 is a constant lattice temperature;
wherein c i is a velocity vector of particles prior to collision;
wherein u(x,t) is mean 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 w i is a constant weighting factor; and
wherein Π neq is a non-equilibrium momentum flux.
15. 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.
16. The method of claim 1 , wherein the non-equilibrium post-collide distribution function is a collision operator C i (x,t) pertaining to energy flux, and
wherein the collision operator is defined in accordance with:
C
i
(
x
,
t
)
=
(
1
-
1
τ
e
)
f
i
eq
(
x
,
t
)
6
ρ
T
0
3
[
c
i
′
(
x
,
t
)
c
i
′
(
x
,
t
)
c
i
′
(
x
,
t
)
-
3
c
i
′
(
x
,
t
)
T
0
I
]
⋮
W
neq
(
x
,
t
)
;
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 T 0 is a constant lattice temperature;
wherein I is a second rank unity tensor;
wherein τ is collision relation time;
wherein c′ i (x,t) is relative particle velocity;
wherein ƒ i eq is an equilibrium distribution function; and
wherein W neq is a non-equilibrium energy flux; and
wherein ρ represents density.
17. 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 lattice velocity at a lattice location to determine relative lattice velocity, with the relative particle lattice velocity being a difference between (i) an absolute lattice velocity at the lattice location and (ii) a mean velocity at the lattice location; and
determining, based on the relative lattice velocity at the lattice location, a lattice Boltzmann post-collide distribution function for a fluid system that is representative of a collision process at the lattice location.
18. The one or more machine-readable hardware storage devices of claim 17 , wherein the operations further comprise:
providing a lattice velocity set that supports hydrodynamic moments up to a specified order of the mean velocity.
19. The one or more machine-readable hardware storage devices of claim 18 , 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 function; and
wherein the specified order for the lattice Boltzmann post-collide distribution function is determined by the order of the mean velocity.
20. The one or more machine-readable hardware storage devices of claim 17 , wherein the mean velocity comprises a mean velocity of a particular type of particles at the lattice location.
21. The one or more machine-readable hardware storage devices of claim 18 , wherein the lattice velocity set is a set of state vectors associated with the lattice Boltzmann method.
22. The one or more machine-readable hardware storage devices of claim 17 , wherein the lattice Boltzmann post-collide distribution function (i) retains non-equilibrium moments for predefined physical quantities up to a specified order, and (ii) eliminates non-equilibrium moments for undefined physical quantities beyond the specified order.
23. The one or more machine-readable hardware storage devices of claim 22 , wherein the specified order is an exponential value associated with a ratio of fluid velocity to lattice sound speed.
24. The one or more machine-readable hardware storage devices of claim 18 , wherein the lattice velocity set comprises a set of momentum states in a space that is limited to a lattice.
25. The one or more machine-readable hardware storage devices of claim 17 , wherein the relative lattice velocity comprises the mean velocity at the lattice location subtracted from the absolute lattice velocity at the lattice location.
26. The one or more machine-readable hardware storage devices of claim 17 , wherein the lattice Boltzmann post-collide distribution function is a Galilean invariant filtered operator.
27. The one or more machine-readable hardware storage devices of claim 17 , wherein the operations further comprise:
modeling, based on the lattice Boltzmann post-collide distribution function, a collision process of particles in a volume of fluid.
28. The one or more machine-readable hardware storage devices of claim 22 , 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.
29. 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 lattice velocity at a lattice location to determine relative lattice velocity, with the relative lattice velocity being a difference between (i) an absolute lattice velocity at the lattice location and (ii) a mean velocity at the lattice location; and
determining, based on the relative lattice velocity at the lattice location, a lattice Boltzmann post-collide distribution function for a fluid system that is representative of a collision process at the lattice location.
30. The method of claim 1 , wherein processing comprises processing, by the data processing system, data indicative of a lattice velocity for a portion of particles represented at the lattice location to determine relative lattice velocity for the portion of particles, with the relative lattice velocity being a difference between (i) an absolute lattice velocity for the portion of particles represented at the lattice location, and (ii) the mean velocity at the lattice location; and
wherein the lattice Boltzmann post-collide distribution function comprises a lattice Boltzmann post-collide particle distribution function for the fluid system that is representative of a particle collision process at the lattice location.
31. The method of claim 1 , further comprising:
simulating, with a lattice velocity set, movement of particles in a volume of fluid, with the movement causing collision among the particles.
32. The method of claim 1 , 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.
33. The method of claim 10 , 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.