SPH General Remarks

SPH smoothed particle hydrodynamics is a powerful particle method for the solution of complex fluid dynamical problems. It was developed in 1977 by Bob Gingold and myself and independently by Leon Lucy. SPH has a number of attractive features. The first of these is that spatial gradients can be calculated without a grid. The second is that it is easy to arrange for the resolution to change automatically. This is especially useful in astrophysical problems where a gaseous region can fragment to produce dense clumps surrounded by less dense material. It is also useful problems involving volcanic outbursts where the hot gas rises up and expands while the dust it carries eventually sinks to the ground. In this case SPH automatically adjusts the resolution of the hot gas fluid and the dense dust fluid. Another advantage of SPH is that it is often very easy to incorporate complex physics without great difficulty.

A useful review is:

J. J. Monaghan Smoothed particle hydrodynamics 1990. Ann. Rev. Astron and Astrophysics. (1992).

Some recent papers focussing on algorithms

J. P. Morris and J. J. Monaghan A switch to reduce SPH viscosity
J. Computat. Phys. 136, 41, (1997)

J. J. Monaghan. SPH and Riemann solvers
J. Computat. Phys. 136, 298, (1997)

J E. Chow and J. J. Monaghan Ultra relativistic SPH
J. Computat. Phys. 134, 296, (1997)

J. J. Monaghan Implicit drag and SPH dusty gas dynamics
J. Computat. Phys. 138, 801, (1997)

P. Cleary and J. J. Monaghan Conduction modelling with SPH
J. Computat. Phys. 148, 227, (1999)

J. Gray, J. J. Monaghan, and R. Swift. Elastic SPH dynamics.

Solid body impact with Water

The impact of a moving solid body with water is considered to be a difficult problem. Even for bodies of simple symmetry impacting a body of water assumed to be infinite in all directions (thus neglecting boundaries), and neglecting gravity, there are no analytic solutions except for the initial stages ( A. A.Korobkin Ann.Rev.Fluid Mech. vol 20, 159 (1988) ). For numerical solutions for the entry of a wedge see Greenhow (Ocean Res. vol 9, 214 (1987)) but these do not continue beyond the first splash.

Using SPH we can easily numerically simulate the impact problem for arbitrary shaped bodies, including gravity and a finite volume of fluid. This work is being used in combination with laboratory experiments to model block avalanches. In these simulations the rigid body is replaced by boundary particles equi-spaced around the body. These boundary particles exert forces on the fluid SPH particles and, in turn, experience forces from the fluid SPH particles. The total force and torque then determines how the rigid body moves. The agreement with experiment is very satisfactory. There are many fascinating problems in this area which can be tackled using SPH. For example the calculation of the pressure on a ship's hull when it slams into the water or a wave strikes an oil rig. More exotic problems are associated with asteroid or comet impact on the ocean.

J. J. Monaghan and A. Kos A solitary wave on a Cretan beach
J. Waterways, Ports, Coastal and Ocean Eng. 125, 145, (1999).

J. Monaghan and A. Kos Scott Russell's wave generator
Phys. of Fluids. 12, 622, (2000)

J.J.Monaghan, A. Kos and Nader Issa. ' Fluid motion generated by Impact '.

The initial stages of an impact where the box has run up the ramp, moved through the air, then impacted the fluid in the tank. The colour coding is according to speed. Low speed is blue, high speed is yellow to green.

Elastic Fracture

Real materials break more easily than theory based on atomic bonding would suggest. The reason for the discrepancy is that real materials have flaws. Small flaws can grow into larger flaws under stress. The first applications of SPH to these problems were carried out by W. Benz and A. Esphaug with outstanding success. Their calculations use a theory due to Grady and Kipp which specifies the damage of a piece of material. The theory provides an equation to calculate the damage. This equation assumes that the calculation follows the elements of material. This is just what SPH does. In the first applications of SPH to elastic problems it was noticed that the particles would clump. This is called the tensile instability. This has now been solved.

James Gray and I are studying fracture in materials using SPH. Our aim is to calculate the waves produced by the Bronze Age collapse of the northern section of the Santorini (Thera) caldera. James is exploring the way a magma chamber beneath the caldera might collapse. There are a wide variety of problems in this area including: 1. Following the fracture of rock as they tumble down a mountain side. 2. Studying how mountains and other features of the earth's surface are formed by stress moving rock on a large scale.

D. E. Grady and M. E. Kipp Dynamic Rock Fragmentation
Fracture Mechanics of Rock. Academic Press. (1987).

W. Benz and E. Asphaug Impact simulations with fracture I Method and Tests
Icarus 107, 98, (1994).

J. Monaghan SPH without a tensile instability
J. Computat. Phys 12, 622, (2000)

J. P. Gray, J. J. Monaghan, and R. P. Swift ' Elastic dynamics with SPH '

Leonardo da Vinci's famous drawing of turbulence generated by flow into a pool.

Turbulence

Turbulence has been classed as one of the most difficult problems of the last 100 years. The standard approach for the last 50 years has been through estimating averages often based on spectral or Fourier expansions. Over the last 2 years Darryl Holm and his colleagues at Los Alamos have worked out a new approach called the alpha model. One of the features of this model is that uses two velocities - one the normal velocity and one which is smoothed. However, the key feature of the model is that the equations of motion conserve circulation and have a quadratic invariant in the absence of dissipation.

There is an SPH version of this which uses a simple Lagrangian which mimics the features of Holm's alpha model. In the continuum limit it becomes Holm's alpha model. Particle methods like SPH do not preserve the circulation but they preserve a quantity which is related to the circulation. This approximate invariant is conserved with smoothing as well as it is conserved when there is no smoothing. This SPH model is currently being used for a number of applications in Astrophysics and Fluid dynamics. At present it is aimed at mechanical turbulence but even with this restriction it offers a practical approach to including turbulence in cosmological simulations.

J. J. Monaghan ' SPH and the alpha turbulence model '

Multi phase flow

In nature and industry many flows are multi phase. A good example is a dust storm or a river carrying sediment. To handle these problems we use SPH generalised to several fluids with the solid particles treated as a fluid with a stress tensor which is still a subject for research.

Our main research interest is in geological problems. For example

1. Flow of granular material into a lake
2. The eruption of a volcano

Projects

We have a number of opportunities for students interested in applying SPH techniques to a wide range of problems. These include

1. Fast iteration methods for SPH algorithms.
2. Studies of turbulence in astrophysical problems.
3. Water-solid body impact allowing for fracture and breakup.
4. Studies of turbulence in multi-phase flow.
5. Debris flows into alpine lakes.
6. Volcanic outbursts and related multi-phase problems.

If you are interested in projects like these send me an email.