The development of lean-burn gas turbine combustors, which will lead to a reduction of emissions from aviation, relies upon resolving significant problems associated with this new technology, namely the emergence of combustion-induced pressure oscillations that can damage the engine, and the danger of flame extinction, which may not allow the full benefits of lean-burn to be realised. Apart from their very significant practical importance, these phenomena are very challenging scientifically and a complete theoretical treatment is not available at present.In this proposal, simulation tools for advanced cooling methods, for thermoacoustics, and for flame extinction will be developed, enabled by recent theoretical, computational, and experimental developments. These tools will be based on low-order models and more complex Computational Fluid Dynamics methods, both validated against focused experiments in single-sector, multi-sector, and full annular geometries. The results will assist engine manufacturers assess ideas and designs at low TRL levels, which will decrease the development time and the high cost associated with testing at high TRL.