A Mathematical Analysis of Scale Similarity

Scale similarity is found in many natural phenomena in the universe, from fluid dynamics to astrophysics. In large eddy simulations of turbulent flows, some sub-grid scale (SGS) models are based on scale similarity. The earliest scale similarity SGS model was developed by Bardina et al., which produced SGS stresses with good correlation to the true stresses. In the present study, we perform a mathematical analysis of scale similarity. The analysis has revealed that the ratio of the resolved stress to the SGS stress is $γ^2$ , where $γ$ is the ratio of the second filter width to the first filter width, under the assumption of small filter width. The implications of this analysis are discussed in the context of large eddy simulation.