Recent advancement in the usage and deployment of large supercomputing resources require the need for algorithmic improvements to make use of the increased parallelism architecture. The Alternating Anderson-Richardson (AAR) method has been recently shown to exhibit good performance when solving problems in distributed parallel computers. This research will extend and investigate the performance of the AAR method to solve CFD problems using a modern compressible flow solver. This work will compare its performance and scalability against commonly used linear solvers, such as the Richardson method and the Generalised Minimal RESidual (GMRES), for solving large, sparse linear systems of equations arising from CFD applications. Results using a range of turbomachinery test cases demonstrate that the current AAR implementation offers significant performance improvement over the Richardson method. The speedup of AAR with respect to GMRES is less significant due to the load imbalance across partitions.