Flux Reconstruction for Future High Accuracy Aerospace Design Tools

In computational fluid dynamics, higher order-of-accuracy methods have been shown to be at least as, if not significantly more accurate than standard second order accuracy schemes while using significantly less computing power. Therefore, these methods could present opportunities to better model highly complex fluid flows, such as in prime movers or power generation equipment for example, through utilising the fast computer speeds available currently.

However, individual processor speeds have not increased substantially in recent years and faster computers are as a result of manufacturers simply increasing the number of processors. This causes a major issue for evaluating fluid flows since the analysis must be split over many processors and a large proportion of computing time is devoted solely to transferring data between different processors rather than on calculations. Higher order methods are particularly susceptible to this and their performance is much poorer when multiple cores are used. For very large jobs, it is a huge issue with maybe only 5% of a computer's processing power actually being used for calculations, so there are severe limitations in using higher order equations in practical fluid dynamic design using both current computing and fluid analysis technology.

Flux Reconstruction (FR) seeks to maintain the benefits of using higher order methods but enable them to make use of computer hardware that is available today. The fluid domain is divided into elements and high order equations are used to solve for the parameter of interest and its flux (rate of movement) within each element. However, because each element is solved individually with no relation to neighbouring elements, the solution is discontinuous through the domain. It is corrected using the calculated flux values at each element interface which is used to 'reconstruct' a continuous solution throughout the domain.

Modern fluid analysis packages used in industry are mainly based on the second order equations mentioned previously, which require significantly more computing power to achieve the same level of accuracy as higher order equations. FR provides a means to conduct fluid analysis using these higher order equations while still using current computing technology without the issues associated with transferring data between many processors. It is therefore a novel solution if it can be successfully applied to practical fluid analysis and though some effort has gone into current understanding of the FR method, there are some limitations surrounding its complexity, robustness and applicability to commercial software packages.

The aim of this PhD is to build on the current understanding of FR, conduct further research and attempt to both resolve some of these issues and improve the efficiency of the method. It can then be determined whether there is potential in developing it into a useful computational fluid dynamics tool for industrial applications. This project is fundamental research, so the direction it takes will be led by results, but likely areas of exploration include developing the methods, and applying them to real problems such as the Navier-Stokes Equations. Typical issues associated with computational fluid dynamics would also need to be addressed throughout the course of the project such as stability and accuracy of the FR method.

The project is of direct relevance to aerospace companies where highly accurate knowledge of dynamic, highly complex fluid flows is required. Current state-of-the-art technology is used to model these flow fields but it is extremely expensive in both computational and monetary terms. If it proved to be viable, FR could significantly reduce the cost of running simulations on for example gas turbine engines. It would also make it easier to implement turbulence modelling techniques that could lead to revolutionary new designs that cannot currently be used since the results may not be accurate.