3.5 Continuum models based for fluid flows
We summarize direct numerical simulations (DNS) and lattice Boltzmann
methods for solving transport equations.
3.5.1 Direct numerical simulation using finite element method
(FEM): A finite element based arbitrary Lagrangian-Eulerian (ALE)
technique can be used to directly solve equations such as (Eq. 3) handle
the movement of single or multiphase domains including particle motions
and fluid flow. An adaptive finite element mesh, generated by the
Delaunay-Voronoi method, enables a significantly higher number of mesh
points in the regions of interest (i.e., close to the particle and wall
surfaces compared to the regions farther away). This feature also keeps
the overall mesh-size computationally reasonable [42].
3.5.2 The lattice Boltzmann method (LBM): Vast number of
applications of the lattice Boltzmann method (LBM) in simulating heat
and mass transfer in fluids, particularly in complex geometries and with
multicomponents, have been demonstrated by previous researchers [43,
44]. This approach’s primary strategy is to incorporate the
microscopic physical interactions of the fluid particles in the
numerical simulation and reveal the mesoscale mechanism of
hydrodynamics. The LBM uses the density distribution functions
f(xi,vi,t) (similar to the Boltzmann or
Liouville equations) to represent a collection of particles with the
microscopic velocities vi and positions
xi at time t, and model the propagation and collision of
particle distribution taking the Boltzmann equations for flow and
temperature fields into consideration. The LBM solves the discretized
Boltzmann equation in velocity space through the propagation of the
particle distribution functions f(x,t) along with the discrete lattice
velocities ei and the collision operation of the local
distributions to be relaxed to the equilibrium distribution
fi0. The collision term is usually
simplified to the single-relaxation-time Bhatnagar-Gross-Krook (BGK)
collision operator, while the more generalized multi-relaxation-time
collision operator can also be adopted to gain numerical stability. The
evolution equation for a set of particle distribution function with a
single relaxation time is defined as:
fi(x-Δx,t+Δt)=fi(x,t)- (Δt/τ)
[fi(x,t)
-fi0(x,t)] + Fs,
(Eq. 13)
where Δt is the time step, Δx=Δt ei is the unit lattice
distance, and τ is the single relaxation time scale associated with the
rate of relaxation to the local equilibrium, and Fs is a
forcing source term introduced to account for the discrete external
force effect. The macroscopic variables such as density and velocity,
are then obtained by taking moments of the distribution function, i.e.,
ρ=Σi fieq and
ρv=Σi fieqei. As explained earlier, through averaging the mass and
momentum variables in the discrete Boltzmann equation, the continuity,
and Navier-Stokes equations may be recovered.
3.5.3 Fluctuating hydrodynamics method: As noted in (Eq. 4),
thermal fluctuations are included in the equations of hydrodynamics by
adding stochastic components to the stress tensor as white noise in
space and time as prescribed by the FHD method [12, 45]. Even though
the original equations of fluctuating hydrodynamics are written in terms
of stochastic partial differential equations, at a very fundamental
level, the inclusion of thermal fluctuations always requires the notion
of a mesoscopic cell in order to define the fluctuating quantities. The
fluctuating hydrodynamic equations discretized in terms of finite
element shape functions based on the Delaunay triangulation satisfy the
fluctuation-dissipation theorem. The numerical schemes for implementing
the thermal fluctuations in the FHD equations are delicate to implement,
and obtaining accurate numerical results is a challenging endeavor
[46].