# Study on SPH Viscosity Term Formulations
## 1 Introduction  
Free surface flow simulations have always generated substantial theoretical and practical interests. [PERSON] [1] used the spectral spatial discretization method for the laminar film flow along a periodic wall. [PERSON] et al. [2] used the Finite Element Method (FEM) and complex variable method for modeling the film over corrugated surfaces. [PERSON] et al. [3] developed a potential-velocity formulation of the Navier-Stokes (N-S) equations by using the least square FEM approach. More recently, [PERSON] et al. [4] proposed a generalized complex-valued first integral of the N-S equations for the unsteady Couette flows in a corrugated channel system. Compared with these, the Smoothed Particle Hydrodynamics (SPH) method is equipped with the unique advantage of tracking the free surfaces with much larger deformation even leading to violent breaking. Since the concept of the mesh-free SPH method was introduced to model hydrodynamic flows [5], SPH has shown its promising performance in a wide range of free surface flows. The benchmark model applications have been documented in the study of dam-break flow [6], wave-structure interaction [7], inflow and outflow [8], porous flow [9] and multi-phase flow [10]. Besides, the shallow water equations (SWEs) were also solved by SPH for the flows in a large area where the vertical flow variations can be reasonably ignored [11; 12]. More complex SPH model applications involving the movable bed, sediment and pollutant transport were reported by [13] and [PERSON] [14; 15]. Besides, some recent weakly compressible SPH (WCSPH) works include a robust procedure to filter thehigh-frequency oscillations in the pressure field arising from the use of the artificial sound speed, in the field of both strong dynamics [16] and slow dynamics [17]. These were based on the improvement of certain diffusive approaches (e.g., [18]) in which the numerical noise was just alleviated through the additional post-processing of the pressure data. The above-mentioned free surface flows are dominated by the gravity and pressure influence. To improve the modeling accuracy of such flows, quite a few studies have been done to enhance the SPH pressure formulation [19] and the pressure solution in either the WCSPH [20] or incompressible SPH (ISPH) [21]. On the other hand, there is another kind of the flow, which is also of great engineering interest, but dominated by the influence of the flow viscosity, such as a lid-driven cavity flow. To realistically simulate these, the treatment of the viscosity force term in the N-S equations becomes important. As compared with the established studies on the pressure formulation, the research on the viscosity term was paid much less attention in the SPH field, probably due to its minor role in the medium-to-high Reynolds number flows commonly found in the free surface applications. Besides, one technical difficulty arises from the fact that the viscosity term involves a second-order derivative calculation, while the pressure term is only of the first-order, and therefore the numerical treatment becomes more complex. The earliest viscosity term treatment in SPH adopted the artificial viscosity approach proposed by [PERSON] [5] to avoid the particle instability and thus to maintain the computational stability. The effect of relevant parameters in the artificial formulation was extensively explored by [PERSON] et al. [22] in their SPH simulation of six different regular waves. Following the establishment of the projection SPH [23] and ISPH [24], the real viscosity term form based on [PERSON] et al. [25] has been more widely used. Compared with the artificial viscosity, the real viscosity has a clearer physical implication and it also performs much better in the practical SPH applications. To improve viscosity performance in reproducing the broken free surfaces, [PERSON] et al. [26] made one step-forward correction by comprehensively addressing the linear and angular momentum conservations.