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].