Background
The LBM, an innovative numerical method based on kinetic theory is a reasonable candidate for simulating turbulence apart from other phenomena, such as flow-induced noise, and sound propagation. It is well known that the LBM is often used as a direct numerical simulation tool without any assumptions for the relationship between the turbulence stress tensor and the mean strain tensor. Thus, the smallest captured scale in the LBM is the lattice unit, and the largest scale depends on the characteristic length scale in the simulation. These scales are often determined by the available computer memory. Consequently, the LBM is able to resolve relatively low Reynolds number flows (Haiqing & Yan 2015). Unlike the traditional Computational fluid dynamics (CFD) methods, which solve the conservation equations of macroscopic properties (i.e., mass, momentum, and energy) numerically, in LBM models the fluid is modelled as fictive particles, and such particles perform consecutive propagation and collision processes over a discrete lattice mesh. Due to its particulate nature and local dynamics, LBM has several advantages over other conventional CFD methods, especially in dealing with complex boundaries.
Numerical studies have shown that LBM can result in the numerical instability for simulating high Reynolds number flows if unresolved small-scale effects on large-scale dynamics are not considered. A better option is to combine the LBM and large eddy simulation (LES) model in order to solve the problem at high Reynolds numbers. A sub grid model is often used as LES model in the numerical simulation for traditional Navier–Stokes equation (Nieuwstadt & Keller 1973), (Dennis & Chang 1970), (Ghia, Ghia & Shin 1982). Anyway, this novel approach has some limitations. High-Mach number flows in aerodynamics are still difficult for LBM, and a consistent thermo-hydrodynamic scheme is absent. However, as with Navier–Stokes based CFD, LBM methods have been successfully coupled to thermal-specific solutions to enable heat transfer (solids-based conduction, convection and radiation) simulation capability. Simulating multiphase/multicomponent flows has always been a challenge to conventional CFD because of the moving and deformable interfaces. More fundamentally, the interfaces between different phases (liquid and vapor) or components (e.g., oil and water) originate from the specific interactions among fluid molecules.
Therefore, it is difficult to implement such microscopic interactions into the macroscopic Navier–Stokes equation. However, in LBM, the particulate kinetics provides a relatively easy and consistent way to incorporate the underlying microscopic interactions by modifying the collision operator. Several LBM multiphase/multicomponent models have been developed. Here phase separations are generated automatically from the particle dynamics and no special treatment is needed to manipulate the interfaces as in traditional CFD methods. Successful applications of multiphase/multicomponent LBM models can be found in various complex fluid systems, including interface instability, bubble/droplet dynamics, wetting on solid surfaces, interfacial slip, and droplet electrohydrodynamic deformations.
Large eddy simulation (LES) is a mathematical model for turbulence used in computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents, (Smagorinsky, 1963) and first explored by Deardorff, later in 1970 for engineering applications (Deardorff, 1970). LES is currently applied in a wide variety of engineering applications, including combustion (Pitsch, 2006), acoustics (Wagner, Hüttl & Sagaut, 2007), and simulations of the atmospheric boundary layer (Sullivan, McWilliams & Moeng, 1994).
The principal idea behind LES is to reduce the computational cost by ignoring the smallest length scales, which are the most computationally expensive to resolve, via low-pass filtering of the Navier–Stokes equations. Such a low-pass filtering, which can be viewed as a time- and spatial-averaging, effectively removes small-scale information from the numerical solution. This information is not irrelevant, however, and its effect on the flow field must be modeled, a task which is an active area of research for problems in which small-scales can play an important role, such as near-wall flows (Piomelli & Elias, 2002), Spalart, 2009), reacting flows (Pitsch, 2006), and multiphase flows (Fox, 2012).
The simulation of turbulent flows by numerically solving the Navier–Stokes (N-S) equations requires resolving a very wide range of time and length scales, all of which affect the flow field. Such a resolution can be achieved with direct numerical simulation (DNS), but DNS is computationally expensive, and its cost prohibits simulation of practical engineering systems with high Reynolds numbers and complex geometry or flow configurations, such as turbulent jets. The treatment of the turbulence will be shown next.