### The Application of Smoothed Particle Hydrodynamics to Complex Problems in Fluid Dynamics

**
Stephen Vinay III, Carnegie Mellon University
**

Because SPH is a Lagrangian method, novel SPH algorithms will have a definite computational advantage over conventional methodologies: the elimination of a domain grid structure, which for complex flows, must be continuously rebuilt and realigned. In conventional methods, a significant amount of computation time is spent in creating and redistributing mesh points, and typical computational fluid dynamics calculations are inefficient and require mass data storage. The absence of a grid also means that 3-D calculations are as easy to implement as 1-D. Additionally, the extension of the SPH methodology to multi-phase systems could be inherently important to technologies such as particulate control and collection, combustion/chemical reaction processes, and the design of micro/nano-electromechanical systems.

To simulate any physical system, the best methodology could be an N-particle dynamics simulation. Even though an N-particle simulation is logically straightforward, it is extremely difficult to implement in engineering systems because of the enormous number of degrees of freedom; and strong, long-range interactions between particles make N-particle simulations virtually impossible. By reducing the number of degrees of freedom and averaging the N-particle dynamics, one often adopts continuum theory to tackle complex engineering problems. The continuum-mechanical approach is best-suited for the numerical simulation of single-phase systems. However, in handling hybrid systems composed of continua/dispersed particles or chemically-reacting flows, continuum theory may not be the optimal scheme for numerical studies. An alternative scenario for the description of physical systems is the SPH technique, which casts either the N-particle system or the continuum into a finite number “n” (n << N) of “pseudo-particle” aggregates. SPH not only reduces the degrees of freedom to a numerically-manageable number (from the N-particle picture), but it also eases the implementation of parallel algorithms (from the continuum picture) because the partial differential equation representation (field view) is converted into ordinary differential equations (particle view). Recently, there have been two frontier achievements in the application of SPH to fluid mechanics. The original work was initiated by Takeda, *et al*. [6], who converted the Navier-Stokes equations to the SPH framework and focused on relatively high-Reynolds number flows. Morris, *et al*. [7] extended Takeda’s analysis to low-Reynolds number situations and was the first to use an (anti)-symmetrized form for the viscous flow term, including artificial viscosity. In this paper, we will generalize and unify these two methodologies by focusing on and clarifying the following three unresolved issues found in the previous work. The first issue addresses boundary conditions. Both Takeda, *et al*. and Morris, *et al*. introduced fluid “ghost” particles outside of the actual fluid domain to simulate the fluid/solid boundary. The use of ghost particles may be reasonable and efficient in simple cases (e.g., rectangular coordinates) but is extremely difficult for more complex problems (curved coordinates, etc.). Therefore, there is a dire need to invent a new, physical, and numerically-elegant boundary condition methodology. This will be especially helpful when one expands the SPH fluid mechanics technique to include multi-phase flows.

- L.B. Lucy, Astron. J. 82, 1013 (1997).
- R. A. Gingold and J. J. Monaghan, Monthly Notices Royal Astron. Soc. 181, 375 (1977).
- W. Moran, W. G. Hoover, and S. Bestiale, J. Stat. Phys. 109, 709 (1993).
- W. G. Hoover and C. G. Hoover, Adv. Comp. Meth. Materials Modeling 180, 231 (1993).
- L. Libersky, A. G. Petschek, T. C. Carney, J. R. Hipp, and F. A. Allahidadi, J. Comp. Phys. 109, 67 (1993).
- H. Takeda, S. M. Miyama, and M. Sekiya, Progress Theoretical Phys. 92, 939 (1994).
- J. P. Morris, P. J. Fox, and Y. Zhu, J. Comp. Phys. 136, 214 (1997).

**Abstract Author(s): **Stephen Vinay III