Energy Transfer in a Linear Turbulence Model

Energy transfer is present in many natural and engineering systems which include different scales. It is important to study the energy cascade (which refers to the energy transfer among the different scales) of such systems. A well-known example is turbulent flow in which the kinetic energy of large vortices is transferred to smaller ones. Below a threshold vortex scale the energy is dissipated due to viscous friction. We introduce a mechanistic model of turbulence which consists of masses connected by springs arranged in a binary tree structure. To represent the various scales, the masses are gradually decreased in lower levels. The bottom level of the model contains dampers to provide dissipation. We define the energy spectrum of the model as the fraction of the total energy stored in each level. A simple method is provided to calculate this spectrum in the asymptotic limit, and the spectra of systems having different stiffness distributions are calculated. We find the stiffness distribution for which the energy spectrum has the same scaling exponent (−5/3) as the Kolmogorov spectrum of 3D homogeneous, isotropic turbulence.