# Study on SPH Viscosity Term Formulations
## 6 Conclusions  
For viscosity-dominated flow simulations, the viscous term in the N-S equations plays an important role. So an accurate treatment should be useful for fully exploring the great potentials of the mesh-free SPH method. One major difficulty arises from the fact that the viscosity term takes the form of a second-derivative and it is highly sensitive to the particle disorders. This work introduced a series of approaches to discretize the viscosity term and its main contributions lie in the following two aspects: firstly, by comparing with the formulations used to discretize the Laplacian of PPE, four benchmark calculation methods on the viscosity term were proposed; secondly, a comprehensive analysis on the different viscosity force models was carried out through vigorous benchmark tests, and the most accurate model was identified.  
Contrary to the common feeling that a higher-accuracy discretized form can achieve the most accurate result, this should be carefully interpreted based on the different model applications. According to the model test on a polynomial function, VICC-03 was shown to achieve the highest accuracy under both the uniform and irregular particle distributions. However, as further disclosed in the subsequent practical model applications, such as a lid-driven flow, VICC-04 performed the most desirably instead. In nature, VICC-04 is improved from the fundamental framework of VICC-01 and just corrects the first-order derivative so as to achieve the maximum potentials. For all of the numerical tests, it is concluded that VICC-03 consumes the most CPU time, while VICC-02 is the fastest model although it may easily induce the instability under a disordered particle configuration. Generally speaking, although these four viscosity force models can all be used in the viscous flow simulations, it is recommended that VICC-04 has the highest accuracy, as least for the present case studies.  
Although all the formulations discussed can be used for the practical free surface flow simulations in principle, it should be noted that the current study focuses a little more on the pure SPH numerical implementation technique, while the relevant underlying physics was not given sufficient attention. This should constitute our future research work.  
This research work is supported by the National Natural Science Foundation of China (Nos. 51739001, 51579056, 51639004, 51379051 and 51479087), Foundational Research Funds for the Central Universities (Nos. HEUCDZ1202 and HEUCF170104), Defence Pre Research Funds Program (No. 9140A14020712 CB01158) and Open Fund of the State Key Laboratory of Coastal and Offshore Engineering of Dalian University of Technology (LP1707). Author [PERSON] thanks the [PERSON] Visiting Chair Professorship Scheme of the Chinese Ministry of Education, hosted by HEU. Author [PERSON] thanks the Open Research Fund Program of the State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University (No. SKHLL701).  
Figure 14: Convergence and error analysis of four viscosity force models for the different particle resolutions of a lid-driven flow with an inside cylinder.  
**Author Contributions:** All authors contributed to the work. [PERSON] performed the numerical computations and data analysis; [PERSON] guided the research work and edited/proof-read the paper; [PERSON] drafted the manuscript and replied to the referees.  
**Conflicts of Interest:** The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.