Spatial scale decomposition of shear layer turbulence and the sound sources associated with the missing scales in a large-eddy simulation

In turbulent flows, it is well known that there exists a range of spatial and temporal scales that increases monotonically with the Reynolds number. The major premise in the formulation of a large eddy simulation (LES) is that of this range of scales only the the energy containing eddies are dynamically important and that the small scales can be modeled in some appropriate fashion. When a LES is used for noise prediction, the question arises as to the importance of these small, or missing, scales in the noise generation process. Moreover, this question becomes increasingly important as the Reynolds number threshold in LES is pushed higher, implying a larger range of scales that must be accounted for in the subgrid scale model. To address the role of the small scales on noise generation, we perform an a priori analysis of a direct numerical simulation (DNS) database of a turbulent spatially developing shear flow between a cold stream at a Mach number of 1.2 and the stationary ambient. The kinematics of the small scales are discussed where it is found that the small scales have convection velocities similar to the large scales with decorrelation times roughly one-half of the full field statistics. The large scale fluctuations carry the entire effect of compressibility, leaving the small scales incompressible. The generation of sound by the interaction of the small scales and large scales is estimated in a proposed statistical noise model, with simplifications for a unidirectional transversely sheared mean flow. The model's source terms are seen to have characteristics distinct from the Lighthill source term, including different convection velocity and decorrelation time but with similar lengthscales.