Energy cascade in turbulent flows: quantifying effects of Reynolds number and local and nonlocal interactions
Seminar Room 1, Newton Institute
The classical Kolmogorov theory of three-dimensional turbulence is based on the concept of the energy transfer from larger to progressively smaller scales of motion. The theory postulates that bulk of the energy transfer in the inertial range of turbulence occurs between scales of similar size, a process known as the local energy cascade. The locality allows to postulate that after multiple cascade steps the small scale dynamics become universal, i.e., independent of particulars of large scales that are determined by geometry, boundary conditions, and forces causing a flow. Yet despite its central role in the Kolmogorov theory the locality assumption cannot be easily verified, neither analytically nor experimentally. This is because the energy transfer is a result of interactions among different scales of motion originating from the nonlinear term in the Navier-Stokes equation that couples all scales. Relevant questions have been productively addressed for the first time using databases generated in large scale numerical simulations. We revisit and extend previous work and use such databases to compute detailed energy exchanges between scales of motion obtained by decomposing numerical velocity fields using banded filters, and investigate how the detailed transfers contribute to the global quantities such as the classical energy transfer, the energy flux, and the subgrid-scale transfer. We address two questions in detail. First, for the purposes of quantitative analyzes, various definitions of scales of motion can be used. This non-uniqueness leads to the possibility, raised in the literature on the subject, that properties of the energy transfer deduced from such analyzes can be qualitatively affected by the employed scale definitions. We address this question by computing detailed energy exchanges between different scales of motion defined by decomposing velocity fields using three specific filters: sharp spectral, Gaussian, and tangent hyperbolic. Second, we quantify the locality of the energy transfer and address a persistent controversy concerning the role of nonlocal interactions in the energy transfer process, i.e., the role of much larger scales than those transferring energy. The analysis of detailed interactions reveals that the individual nonlocal contributions are always large but significant cancellations lead to the global quantities asymptotically dominated by the local interactions. The detailed locality functions are computed and their behavior compared with the asymptotic scaling laws valid for infinite Reynolds numbers turbulence. Apart from an intellectual challenge of clarifying these issues, obtained results have bearing on practical questions of turbulence modeling that will also be addressed in the talk.