# Review on the Recent Numerical Studies of Liquid Atomization
## 7 Smoothed Particle Hydrodynamics  
As a non-grid method, Smoothed Particle Hydrodynamics (SPH) has some advantages in tracking multiphase flow over grid-based method, mainly because the diffusive effects at the interface can be avoided. The SPH method was originally developed in the context of astrophysics [82; 83]. The numerical discretization elements are particles fulfilled into the whole domain and the motivation for using Lagrangian framework was to avoid the discretization of large, empty interstellar spaces. In many two-phase flows, the less important gas phase is allowed to be omitted without particles representing it, thus saving massive computational effort. This is not the case for the atomization process since the gas flow plays an essential role. Particles representing either liquid or gas phase are tracked with identical equations and are discriminated through properties such as density. Density ratios of liquid to gas less than 1000 can be handled without any stability issues. No explicit interface tracking or capturing is needed, and interface diffusion is inherently avoided, so SPH is suitable for simulating multiphase flow with severe interface deformations like liquid atomization. The main idea behind the SPH is to evaluate the physical property of a particle or its derivative by interpolating over neighbor particles within a certain cut-off radius, and the basic interpolation formalism for a particle with index \(i\) is:  
\[\left\langle\Phi\right\rangle_{i}=\sum_{j}V_{j}\Phi_{j}W\big{(}\mathbf{x}_{i}-\bm {x}_{j},h\big{)} \tag{42}\]  
where \(\Phi\) is a physical quantity, \(V_{j}\) is the volume of a neighbor particle \(j\), \(W\) is a weighting function, depending on the distance between particle \(i\) and \(j\), and \(h\) is the smoothing length, which defines the cut-off radius. The density formulation is proposed by [PERSON] and [PERSON] [84] as:  
\[\left\langle\rho\right\rangle_{i}=m_{i}\sum_{j}W_{ij} \tag{43}\]  
The contributions by the pressure gradient [85] and viscous force [86] to the conservation of momentum can be described by the following Equations (44) and (45):  
\[\left\langle\overset{\rightharpoonup}{F}\right\rangle_{i,\
abla p}=-\sum_{j} \frac{m_{j}\big{(}p_{i}+p_{j}\big{)}}{\rho_{i}\rho_{j}}\
abla W_{ij} \tag{44}\]  
\[\left\langle\overset{\rightharpoonup}{F}\right\rangle_{i,\mu}=\sum_{j}2m_{j}( d+2)(\frac{v_{i}+v_{j}}{\rho_{i}+\rho_{j}}\frac{\overset{\rightharpoonup}{v_{j}} \overset{\rightharpoonup}{r_{ij}}}{\tau_{ij}^{2}+\eta^{2}})\
abla W_{ij} \tag{45}\]  
where \(d\) denotes the dimensionality and \(\eta\) is a small constant to avoid divisions by zero. The SPH method also has some disadvantages. For instance, neither the consistency of the method nor the convergence have been proven mathematically. The interpolation accuracy is strongly affected by the particle arrangement. This feature resembles the influence of grid quality in grid-based methods. The introduction of the background pressure described previously, in order to regularize the particle arrangement, can improve interpolation accuracy only to a certain extent. However, due to the extremely high numerical efficiency, it is often possible to carry out simulations with a much finer resolution than with conventional methods. This high spatial resolution is indispensable if primary atomization is to be simulated. [PERSON] et al. [87] used an incompressible SPH method in their simulation of the liquid flow scattering from a rotary atomizer. The influence of grooves at the edges of the atomizers on the formation of ligaments and droplets is investigated. It is found that small droplets are likely to be generated when the number of grooves is large and the depth of grooves is deep, and the grooves work more effectively in bell-cup atomizers than in disk-type atomizers. [PERSON] et al. [88] performed SPH simulation of an air-assisted pre-filming atomization in a dimension of 6 mm \(\times\) 4 mm \(\times\) 6 mm.