The implementation of Computational Fluid Dynamics (CFD) in Mechanical Engineering applications is, almost without exception, based on the Eulerian description of fluid flow. Eulerian based modelling techniques have been heavily studied since the 1950's and are clearly understood. Commercial codes, especially, seem to have been developed using only finite volume or finite element approaches. The result is that Eulerian based models are used even for applications for which they are not suited.
One major assumption of the Eulerian approach is that the fluid is homogeneous. Any fluid that is not homogeneous, for example one with multiple phases, requires additional complex internal mathematical modelling to represent these phases. Eulerian models are grid based. The grid must be constructed before the analysis can commence, implying that the location of any free surfaces or fluid interfaces must be known in advance. In many cases, this partially defeats the purpose for computational modelling of flows involving these features. The Eulerian approach of monitoring the fluid properties in fixed volumes is also ill suited to dealing with particulate flows, as the particles simply pass through these volumes with the flow.
The analytical study of fluid mechanics acknowledges two descriptions of fluid flow, Eulerian and Lagrangian. One description can be more suitable for certain applications than the other. Fox and McDonald (1994) in their introductory fluid mechanics text book, state,
"Clearly the type of analysis depends on the problem. Where it is easy to keep track of identifiable elements of mass, we utilise a method of description that follows the particle."
This method is known as the Lagrangian description. Smoothed Particle Hydrodynamics (SPH) is a grid free Lagrangian based method in which the fluid flow is represented by fluid pseudo-particles. These individual particles interact with one other, moving with the flow and carrying with them all of the computational information about the fluid. Fluid properties are then interpolated between the particles. The technique is relatively new, being first introduced by Lucy and Monaghan (1977), in the context of Astrophysical modelling.
Being a particle based method, SPH is very well suited to modelling particles, such as bubbles entrained in flows. The use of Lagrangian based methods effectively eliminates the homogeneous fluid assumption of the Eulerian methods. The result is that in the areas where Eulerian methods struggle, Lagrangian methods can effectively model multiple phases naturally, even with greatly differing densities. These qualities make it very well suited to multiple phase Mechanical Engineering applications such as: fluid jets, tank sloshing, metal casting and extrusion, multiple phase pipe flow, fluid impacts and cavitation. A limitation to the grid based approach to modelling complex three dimensional geometry is that the meshes tend to become awkward to manage. The Lagrangian approach of SPH extends to complex three dimensional geometry extremely well.
The future for Lagrangian based methods does not simply lie in a choice between the two different fluid descriptions. A further possibility of these methods is a coupling with Eulerian methods in such a way that the best aspects of both approaches can be incorporated into a single model. For example, fluid interfaces could first be identified before employing an Eulerian based solution.
As a result of its origins, to date most applications of SPH have been in astrophysical domains, and most of the research that has been carried out has been from a purely mathematical perspective. One of the first real attempts at applying SPH techniques to Mechanical Engineering applications was by Steinmetz et al. (1992). A series of tests of the capabilities of SPH were formulated, one of which was a one dimensional test problem for compressible hydrodynamics codes, stating,
"The double shock tube results show that, contrary to the commonly maintained opinion, SPH can accurately handle problems which involve strong shock waves. Although the spatial resolution is inferior to modern higher-order Gudonov type schemes e.g., the Piecewise Parabolic Method (PPM), the state values on both sides of the discontinuities are represented well even with a relatively small number of particles, while the discontinuities are smeared out over a few smoothing lengths."
This conclusion was also reached in a Departmental undergraduate thesis, Robinson (1994). Effective one and two dimensional models of shock tube diaphragm rupture were used. The SPH approach allowed the dynamic solid material behaviour to be modelled along with the resulting unsteady compressible flow. This thesis demonstrated that SPH techniques can be effectively used to model compressible flow, he stated,
"One and two dimensional unsteady compressible flow shock tube problems, for which exact solutions exist were simulated using SPH. The SPH simulations gave reasonably accurate results for these problems, thus verifying the model's accuracy and validity."
SPH can also be extended to nearly incompressible flows by the use of an appropriate equation of state. This equation is constructed such that the compressibility effects are below the required level. The following treatment is then straight forward, with free surface problems begin modelled easily. Flows with phases of differing density can be modelled effectively (Monaghan 1992). Problems that are being studied include high pressure die Casting (Cleary 1998) and liquid jet impingement (Reichl 1998). Incompressible flow problems have been successfully verified with standard benchmarks such as couette flow problems, by Cleary et al. (1993), who stated,
"The extension to incompressible free surface flows with boundaries is a recent development. The results so far have been impressive, but there remain aspects that are not fully understood."
A PhD thesis in the Department, Delabbio (1999), used a commercial finite element based code (FIDAP) to model the flow from a liquid jet impinging on a cut hole. As part of the investigation, the pressure distribution in the hole was modelled. Delabbio identified two major errors in the CFD modelling of the flow, both of which were based on the continuum code's inability to model multiple phases. By looking at the problems encountered with this modelling we can demonstrate the need for a Lagrangian based approach to this type of project. He stated,
"The first reason for the discrepancies between the experimental results and the CFD modelling is that the CFD modelling was unable to effectively model air entrainment within the water jet..."
SPH, being a particle based method, has a much greater potential for accurately modelling the entrainment of particles by eliminating the assumption of an homogeneous fluid. Also implied in the method is a more intuitive approach to modelling the physics of bubble formation and behaviour in multiple phase flow fields. He stated further,
"The second reason for the discrepancies between the CFD modelling and the experimental results is that in the CFD modelling the cavity was always completely full of liquid."
The location of any fluid interfaces must be known in advance with finite element and finite volume based modelling. In this case, the location of the free surface was a function of the jet velocity, but this could not be modelled by these techniques. The result was that the boundary conditions were incorrect. With its natural modelling of free surfaces, this problem could be eliminated by using SPH, without the need for any other models or assumptions.
Similar problems to those described by Delabbio (1999) were also encountered in the undergraduate thesis Goozee (1998). A commercial, incompressible finite volume code (Phoenics) was used for the modelling of a cavitating liquid jet. This Eulerian type of code is, in many respects, unable to deal with these types of two phase flows. SPH methods, however, appear to have the potential to explicitly model flow separation and cavitation in the liquid and may provide a more accurate representation of the flow field.
Despite the promise of SPH, it is far from being the ideal approach. Until recently problems with the SPH technique have prevented its application to practical problems. In particular, SPH codes were known to be troublesome in the implementation of solid boundary conditions. Hernquist (1993) outlined these and some other cautions regarding the use of SPH, in particular the use of variable smoothing lengths and adaptivity in time. These errors were, however, relatively easy to estimate and in all but extreme cases only led to small violations of the fluid conservation laws. Hernquist (1993) stated that,
"while SPH remains a useful tool for many problems of astrophysical interest, a rigorous formulation of it, which is adaptive but still satisfies conservation properties, is clearly wanting."
Randles (1996) discusses the great number of improvements that have been made to the SPH technique as it became more widely used over the period 1991 to 1995, that is in the period since the discussion by Hernquist (1993). Most notably, suitable generalised methods were developed for dealing with solid boundaries. Randles (1996) conclusion was that over that period the technique matured, leading to a method that he summarised with the statement,
"The power of the method lies in its conceptual simplicity which gives rise to such desirable features as robustness, ease of adding new physics, a natural treatment of void regions, and the ease of simulating three dimensional problems."
The project will concentrate on solving some of the problems associated with the implementation of multiple phase physics in the SPH technique. Solving these problems will hopefully enable the method to be more widely used in Mechanical Engineering applications for which it appears to be most suited.