Summer Scholarship Programme
Project Summary
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All the World's a Cube: Computational Fluid Dynamics on a Sphere without Coordinate Singularities

Student

Ian Mulvany, University of Edinburgh

Supervisor

David Henty, EPCC

The equations of Computational Fluid Dynamics (CFD) are relatively straightforward to solve on a parallel computer for the case of steady-state, incompressible flow. For example, one of last year's SSP students wrote an MPI CFD solver from scratch with no prior experience of either CFD or parallel computing. Despite this, however, the equations can still describe a wide range of interesting physical phenomena.

Although the equations themselves are simple to discretise on a regular grid, problems arise when a more complicated coordinate system is required. The prime example is when a 2D problem is solved on the surface of a sphere, for example to study global ocean properties (where the finite depth of the oceans is ignored) or perhaps the formation of Jupiter's red spot. The standard approach is to use a latitude / longitude grid, but this leads to coordinate singularities at the poles. This can be avoided by having two mutually rotated grids, one with its coordinate poles at the physical North and South pole and the other with its poles at the physical equator. However, this leads to problems of marrying the two grids where they overlap and interpolating physical quantities between them.

A radically new, but beautifully simple, solution was proposed by an Italian group within the last couple of years. The sphere is mapped to a cube which leads to surprisingly small distortion even at the corners of the cube, and no singularities at all in the grid. The regularity of the cube makes parallelisation remarkably simple and perfectly suited to SIMD machines (the original target architecture) and, by analogy, HPF on any architecture.

The project is therefore to write an HPF code that solves for the steady state flow of an incompressible fluid on the surface of a sphere. There will be extensive opportunity to perform visualisations of the resulting velocity fields, perhaps using DIVE as and easy way of doing this in real time.

The final report for this project is available here.
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