Residual stress relation between averaged and filtered compressible flows

The statistical relation of residual stress between averaged and filtered compressible flow, known as Reynolds stress in the Reynolds-averaged Navier–Stokes equation (RANS) and subgrid-scale (SGS) stress in a large eddy simulation (LES), serves a significant role in high-Reynolds-numbers wall-bounded turbulence modelling. However, existing residual stress relations are not universally applicable due to additional assumptions or variables not directly derived from compressible turbulence modelling. To establish an effective and accurate residual stress relation, a theoretical study accompanied by numerical verification has been carried out. By introducing a novel pair of average and filter operators with commutative properties, the statistical relations of residual stress for compressible flows are derived. Then, a realisation and verification of the stress relation is carried out within the finite volume method framework to facilitate the application of the proposed stress relation in engineering turbulence modelling. The reliability of the residual stress relation is confirmed using the compressible channel turbulence at various Mach numbers and compressible boundary layer flow. The stress relation formula effectively establishes the decomposition between Reynolds stress and subgrid-scale stress of the compressible flows. The proposed residual stress relation and filter operators may contribute to the compressible turbulence modelling, including the development of the wall model, SGS model and RANS/LES hybrid strategy for high-Reynolds-number turbulence modelling.