Generalized Energy Budget Equations for Large-Eddy Simulations of Scalar Turbulence

The energy transfer between different scales of a passive scalar advected by homogeneous isotropic turbulence is studied by an exact generalized transport equation for the second moment of the scalar increment. This equation can be interpreted as a scale-by-scale energy budget equation, as it relates at a certain scale r terms representing the production, turbulent transport, diffusive transport and dissipation of scalar energy. These effects are analyzed by means of direct numerical simulation where each term is directly accessible. To this end, a variation of the Taylor microscale based Reynolds number between 88 and 754 is performed. Understanding the energy transport between scales is crucial for Large-Eddy Simulation (LES). For an analysis of the energy transfer in LES, a transport equation for the second moment of the filtered scalar increment is introduced. In this equation new terms appear due to the interaction between resolved and unresolved scales, which are analyzed in the context of an a priori and an a posteriori test. It is further shown that LES using an eddy viscosity approach is able to fulfill the correct inter-scale energy transport for the present configuration.