\(\renewcommand{\AA}{\text{Å}}\)

# fix lb/fluid command

## Syntax

    fix ID group-ID lb/fluid nevery viscosity density keyword values ...

- ID, group-ID are documented in fix command

- lb/fluid = style name of this fix command

- nevery = update the lattice-Boltzmann fluid every this many timesteps (should normally be 1)

- viscosity = the fluid viscosity (units of mass/(time*length)).

- density = the fluid density.

- zero or more keyword/value pairs may be appended

- keyword = dx or dm or noise or stencil or read_restart or write_restart or zwall_velocity or pressurebcx or bodyforce or D3Q19 or dumpxdmf or linearInit or dof or scaleGamma or a0 or npits or wp or sw

      dx values = dx_LB = the lattice spacing.
      dm values = dm_LB = the lattice-Boltzmann mass unit.
      noise values = Temperature seed
          Temperature = fluid temperature.
          seed = random number generator seed (positive integer)
      stencil values = 2 (trilinear stencil, the default), 3 (3-point immersed boundary stencil), or 4 (4-point Keys' interpolation stencil)
      read_restart values = restart file = name of the restart file to use to restart a fluid run.
      write_restart values = N = write a restart file every N MD timesteps.
      zwall_velocity values = velocity_bottom velocity_top = velocities along the y-direction of the bottom and top walls (located at z=zmin and z=zmax).
      pressurebcx values = pgradav = imposes a pressure jump at the (periodic) x-boundary of pgradav*Lx*1000.
      bodyforce values = bodyforcex bodyforcey bodyforcez = the x,y and z components of a constant body force added to the fluid.
      D3Q19 values = none (used to switch from the default D3Q15, 15 velocity lattice, to the D3Q19, 19 velocity lattice).
      dumpxdmf values = N file timeI
          N = output the force and torque every N timesteps
          file = output file name
          timeI = 1 (use simulation time to index xdmf file), 0 (use output frame number to index xdmf file)
      linearInit values = none = initialize density and velocity using linear interpolation (default is uniform density, no velocities)
      dof values = dof = specify the number of degrees of freedom for temperature calculation
      scaleGamma values = type gammaFactor
          type = atom type (1-N)
          gammaFactor = factor to scale the setGamma gamma value by, for the specified atom type.
      a0 values = a_0_real = the square of the speed of sound in the fluid.
      npits values = npits h_p l_p l_pp l_e
          npits = number of pit regions
          h_p = z-height of pit regions (floor to bottom of slit)
          l_p = x-length of pit regions
          l_pp = x-length of slit regions between consecutive pits
          l_e = x-length of slit regions at ends
      wp values = w_p = y-width of slit regions (defaults to full width if not present or if sw active)
      sw values = none  (turns on y-sidewalls (in xz plane) if npits option active)

## Examples

    fix 1 all lb/fluid 1 1.0 0.0009982071 dx 1.2 dm 0.001
    fix 1 all lb/fluid 1 1.0 0.0009982071 dx 1.2 dm 0.001 noise 300.0 2761
    fix 1 all lb/fluid 1 1.0 1.0 dx 4.0 dm 10.0 dumpxdmf 500 fflow 0 pressurebcx 0.01 npits 2 20 40 5 0 wp 30

## Description

Changed in version 24Mar2022.

Implement a lattice-Boltzmann fluid on a uniform mesh covering the LAMMPS simulation domain. Note that this fix was updated in 2022 and is not backward compatible with the previous version. If you need the previous version, please download an older version of LAMMPS. The MD particles described by group-ID apply a velocity dependent force to the fluid.

The lattice-Boltzmann algorithm solves for the fluid motion governed by the Navier Stokes equations,

\[\begin{split}\partial_t \rho + \partial_{\beta}\left(\rho u_{\beta}\right)= & 0 \\ \partial_t\left(\rho u_{\alpha}\right) + \partial_{\beta}\left(\rho u_{\alpha} u_{\beta}\right) = & \partial_{\beta}\sigma_{\alpha \beta} + F_{\alpha} + \partial_{\beta}\left(\eta_{\alpha \beta \gamma \nu}\partial_{\gamma} u_{\nu}\right)\end{split}\]

with,

\[\eta_{\alpha \beta \gamma \nu} = \eta\left[\delta_{\alpha \gamma}\delta_{\beta \nu} + \delta_{\alpha \nu}\delta_{\beta \gamma} - \frac{2}{3}\delta_{\alpha \beta}\delta_{\gamma \nu}\right] + \Lambda \delta_{\alpha \beta}\delta_{\gamma \nu}\]

where \(\rho\) is the fluid density, u is the local fluid velocity, \(\sigma\) is the stress tensor, F is a local external force, and \(\eta\) and \(\Lambda\) are the shear and bulk viscosities respectively. Here, we have implemented

\[\sigma_{\alpha \beta} = -P_{\alpha \beta} = -\rho a_0 \delta_{\alpha \beta}\]

with \(a_0\) set to \(\frac{1}{3} \frac{dx}{dt}^2\) by default. You should not normally need to change this default.

The algorithm involves tracking the time evolution of a set of partial distribution functions which evolve according to a velocity discretized version of the Boltzmann equation,

\[\left(\partial_t + e_{i\alpha}\partial_{\alpha}\right)f_i = -\frac{1}{\tau}\left(f_i - f_i^{eq}\right) + W_i\]

where the first term on the right hand side represents a single time relaxation towards the equilibrium distribution function, and \(\tau\) is a parameter physically related to the viscosity. On a technical note, we have implemented a 15 velocity model (D3Q15) as default; however, the user can switch to a 19 velocity model (D3Q19) through the use of the D3Q19 keyword. Physical variables are then defined in terms of moments of the distribution functions,

\[\begin{split}\rho = & \displaystyle\sum\limits_{i} f_i \\ \rho u_{\alpha} = & \displaystyle\sum\limits_{i} f_i e_{i\alpha}\end{split}\]

Full details of the lattice-Boltzmann algorithm used can be found in Denniston et al..

The fluid is coupled to the MD particles described by group-ID through a velocity dependent force. The contribution to the fluid force on a given lattice mesh site j due to MD particle \(\alpha\) is calculated as:

\[\mathbf{F}_{j \alpha} = \gamma \left(\mathbf{v}_n - \mathbf{u}_f \right) \zeta_{j\alpha}\]

where \(\mathbf{v}_n\) is the velocity of the MD particle, \(\mathbf{u}_f\) is the fluid velocity interpolated to the particle location, and \(\gamma\) is the force coupling constant. This force, as with most forces in LAMMPS, and hence the velocities, are calculated at the half-time step. \(\zeta\) is a weight assigned to the grid point, obtained by distributing the particle to the nearest lattice sites.

The force coupling constant, \(\gamma\), is calculated according to

\[\gamma = \frac{2m_um_v}{m_u+m_v}\left(\frac{1}{\Delta t}\right)\]

Here, \(m_v\) is the mass of the MD particle, \(m_u\) is a representative fluid mass at the particle location, and \(\Delta t\) is the time step. The fluid mass \(m_u\) that the MD particle interacts with is calculated internally. This coupling is chosen to constrain the particle and associated fluid velocity to match at the end of the time step. As with other constraints, such as shake, this constraint can remove degrees of freedom from the simulation which are accounted for internally in the algorithm.

Note

While this fix applies the force of the particles on the fluid, it does not apply the force of the fluid to the particles. There is only one option to include this hydrodynamic force on the particles, and that is through the use of the lb/viscous fix. This fix adds the hydrodynamic force to the total force acting on the particles, after which any of the built-in LAMMPS integrators can be used to integrate the particle motion. If the lb/viscous fix is NOT used to add the hydrodynamic force to the total force acting on the particles, this physically corresponds to a situation in which an infinitely massive particle is moving through the fluid (since collisions between the particle and the fluid do not act to change the particle’s velocity). In this case, setting scaleGamma to -1 for the corresponding particle type will explicitly take this limit (of infinite particle mass) in computing the force coupling for the fluid force.

Physical parameters describing the fluid are specified through viscosity and density. These parameters should all be given in terms of the mass, distance, and time units chosen for the main LAMMPS run, as they are scaled by the LB timestep, lattice spacing, and mass unit, inside the fix.

The dx keyword allows the user to specify a value for the LB grid spacing and the dm keyword allows the user to specify the LB mass unit. Inside the fix, parameters are scaled by the lattice-Boltzmann timestep, \(dt_{LB}\), grid spacing, \(dx_{LB}\), and mass unit, \(dm_{LB}\). \(dt_{LB}\) is set equal to \(\mathrm{nevery}\cdot dt_{MD}\), where \(dt_{MD}\) is the MD timestep. By default, \(dm_{LB}\) is set equal to 1.0, and \(dx_{LB}\) is chosen so that \(\frac{\tau}{dt} = \frac{3\eta dt}{\rho dx^2}\) is approximately equal to 1.

> Note

> Care must be taken when choosing both a value for \(dx_{LB}\), and a simulation domain size. This fix uses the same subdivision of the simulation domain among processors as the main LAMMPS program. In order to uniformly cover the simulation domain with lattice sites, the lengths of the individual LAMMPS subdomains must all be evenly divisible by \(dx_{LB}\). If the simulation domain size is cubic, with equal lengths in all dimensions, and the default value for \(dx_{LB}\) is used, this will automatically be satisfied.

If the noise keyword is used, followed by a positive temperature value, and a positive integer random number seed, the thermal LB algorithm of Adhikari et al. is used.

If the keyword stencil is used, the value sets the number of interpolation points used in each direction. For this, the user has the choice between a trilinear stencil (stencil 2), which provides a support of 8 lattice sites, or the 3-point immersed boundary method stencil (stencil 3), which provides a support of 27 lattice sites, or the 4-point Keys’ interpolation stencil (stencil 4), which provides a support of 64 lattice sites. The trilinear stencil is the default as it is better suited for simulation of objects close to walls or other objects, due to its smaller support. The 3-point stencil provides smoother motion of the lattice and is suitable for particles not likely to be to close to walls or other objects.

If the keyword write_restart is used, followed by a positive integer, N, a binary restart file is printed every N LB timesteps. This restart file only contains information about the fluid. Therefore, a LAMMPS restart file should also be written in order to print out full details of the simulation.

Note

When a large number of lattice grid points are used, the restart files may become quite large.

In order to restart the fluid portion of the simulation, the keyword read_restart is specified, followed by the name of the binary lb_fluid restart file to be used.

If the zwall_velocity keyword is used y-velocities are assigned to the lower and upper walls. This keyword requires the presence of walls in the z-direction. This is set by assigning fixed boundary conditions in the z-direction. If fixed boundary conditions are present in the z-direction, and this keyword is not used, the walls are assumed to be stationary.

If the pressurebcx keyword is used, a pressure jump (implemented by a step jump in density) is imposed at the (periodic) x-boundary. The value set specifies what would be the resulting equilibrium average pressure gradient in the x-direction if the system had a constant cross-section (i.e. resistance to flow). It is converted to a pressure jump by multiplication by the system size in the x-direction. As this value should normally be quite small, it is also assumed to be scaled by 1000.

If the bodyforce keyword is used, a constant body force is added to the fluid, defined by it’s x, y and z components.

If the keyword D3Q19 is used, the 19 velocity (D3Q19) lattice is used by the lattice-Boltzmann algorithm. By default, the 15 velocity (D3Q15) lattice is used.

If the dumpxdmf keyword is used, followed by a positive integer, N, and a file name, the fluid densities and velocities at each lattice site are output to an xdmf file every N timesteps. This is a binary file format that can be read by visualization packages such as Paraview . The xdmf file format contains a time index for each frame dump and the value timeI = 1 uses simulation time while 0 uses the output frame number to index xdmf file. The later can be useful if the dump vtk command is used to output the particle positions at the same timesteps and you want to visualize both the fluid and particle data together in Paraview .

The scaleGamma keyword allows the user to scale the \(\gamma\) value by a factor, gammaFactor, for a given atom type. Setting scaleGamma to -1 for the corresponding particle type will explicitly take the limit of infinite particle mass in computing the force coupling for the fluid force (see note above).

If the a0 keyword is used, the value specified is used for the square of the speed of sound in the fluid. If this keyword is not present, the speed of sound squared is set equal to \(\frac{1}{3}\left(\frac{dx_{LB}}{dt_{LB}}\right)^2\). Setting \(a0 > (\frac{dx_{LB}}{dt_{LB}})^2\) is not allowed, as this may lead to instabilities. As the speed of sound should usually be much larger than any fluid velocity of interest, its value does not normally have a significant impact on the results. As such, it is usually best to use the default for this option.

The npits keyword (followed by integer arguments: npits, h_p, l_p, l_pp, l_e) sets the fluid domain to the pits geometry. These arguments should only be used if you actually want something more complex than a rectangular/cubic geometry. The npits value sets the number of pits regions (arranged along x). The remaining arguments are sizes measured in multiples of dx_lb: h_p is the z-height of the pit regions, l_p is the x-length of the pit regions, l_pp is the length of the region between consecutive pits (referred to as a “slit” region), and l_e is the x-length of the slit regions at each end of the channel. The pit geometry must fill the system in the x-direction but can be longer, in which case it is truncated (which enables asymmetric entrance/exit end sections). The additional wp keyword allows the width (in y-direction) of the pit to be specified (the default is full width) and the sw keyword indicates that there should be sidewalls in the y-direction (default is periodic in y-direction). These parameters are illustrated below:

    Sideview (in xz plane) of pit geometry:
    ______________________________________________________________________
      slit                          slit                          slit     ^
                                                                           |
    <---le---><---------lp-------><---lpp---><-------lp--------><---le---> hs = (Nbz-1) - hp
                                                                           |
    __________                    __________                    __________ v
              |                  |          |                  |           ^       z
              |                  |          |                  |           |       |
              |       pit        |          |       pit        |           hp      +-x
              |                  |          |                  |           |
              |__________________|          |__________________|           v

    Endview (in yz plane) of pit geometry (no sw so wp is active):
    _____________________
                          ^
                          |
                          hs
                          |
    _____________________ v
        |          |      ^
        |          |      |          z
        |<---wp--->|      hp         |
        |          |      |          +-y
        |__________|      v

For further details, as well as descriptions and results of several test runs, see Denniston et al.. Please include a citation to this paper if the lb_fluid fix is used in work contributing to published research.

## Restart, fix_modify, output, run start/stop, minimize info

Due to the large size of the fluid data, this fix writes it’s own binary restart files, if requested, independent of the main LAMMPS binary restart files; no information about lb_fluid is written to the main LAMMPS binary restart files.

None of the fix_modify options are relevant to this fix.

The fix computes a global scalar which can be accessed by various output commands. The scalar is the current temperature of the group of particles described by group-ID along with the fluid constrained to move with them. The temperature is computed via the kinetic energy of the group and fluid constrained to move with them and the total number of degrees of freedom (calculated internally). If the particles are not integrated independently (such as via fix NVE) but have additional constraints imposed on them (such as via integration using fix rigid) the degrees of freedom removed from these additional constraints will not be properly accounted for. In this case, the user can specify the total degrees of freedom independently using the dof keyword.

The fix also computes a global array of values which can be accessed by various output commands. There are 5 entries in the array. The first entry is the temperature of the fluid, the second entry is the total mass of the fluid plus particles, the third through fifth entries give the x, y, and z total momentum of the fluid plus particles.

No parameter of this fix can be used with the start/stop keywords of the run command. This fix is not invoked during energy minimization.

## Restrictions

This fix is part of the LATBOLTZ package. It is only enabled if LAMMPS was built with that package. See the Build package page for more info.

This fix can only be used with an orthogonal simulation domain.

The boundary conditions for the fluid are specified independently to the particles. However, these should normally be specified consistently via the main LAMMPS boundary command (p p p, p p f, and p f f are the only consistent possibilities). Shrink-wrapped boundary conditions are not permitted with this fix.

This fix must be used before any of fix lb/viscous and fix lb/momentum as the fluid needs to be initialized before any of these routines try to access its properties. In addition, in order for the hydrodynamic forces to be added to the particles, this fix must be used in conjunction with the lb/viscous fix.

This fix needs to be used in conjunction with a standard LAMMPS integrator such as fix NVE or fix rigid.

## Related commands

fix lb/viscous, fix lb/momentum

## Default

dx is chosen such that \(\frac{\tau}{dt_{LB}} = \frac{3\eta dt_{LB}}{\rho dx_{LB}^2}\) is approximately equal to 1. dm is set equal to 1.0. a0 is set equal to \(\frac{1}{3}\left(\frac{dx_{LB}}{dt_{LB}}\right)^2\). The trilinear stencil is used as the default interpolation method. The D3Q15 lattice is used for the lattice-Boltzmann algorithm.

(Denniston et al.) Denniston, C., Afrasiabian, N., Cole-Andre, M.G., Mackay, F. E., Ollila, S.T.T., and Whitehead, T., LAMMPS lb/fluid fix version 2: Improved Hydrodynamic Forces Implemented into LAMMPS through a lattice-Boltzmann fluid, Computer Physics Communications 275 (2022) 108318 .

(Mackay and Denniston) Mackay, F. E., and Denniston, C., Coupling MD particles to a lattice-Boltzmann fluid through the use of conservative forces, J. Comput. Phys. 237 (2013) 289-298.

(Adhikari et al.) Adhikari, R., Stratford, K., Cates, M. E., and Wagner, A. J., Fluctuating lattice Boltzmann, Europhys. Lett. 71 (2005) 473-479.