Question

Explain the role of the Komogorov scale in terms of the cascade of energy AND in terms of the size of particles in a “smoothie”. Provide the equation. 10% Comment on how the Kolmogorov scale changes when the energy dissipation rate increases by 10, 100, and 1000 folds. Explain the implications on the energy needs for lowering the Kolmogorov scale.10%

Answer 1

For the sustainability of turbulent flow, a continuous source of energy supply is required as turbulence dissipates rapidly as the kinetic energy is converted into internal energy by viscous shear stress. Turbulence causes the formation of eddies of many different length scales. Most of the kinetic energy of the turbulent motion is contained in the large-scale structures. The energy "cascades" from these large-scale structures to smaller scale structures by an inertial and essentially inviscid mechanism. This process continues, creating smaller and smaller structures which produces a hierarchy of eddies. Eventually this process creates structures that are small enough that molecular diffusion becomes important and viscous dissipation of energy finally takes place. The scale at which this happens is the Kolmogorov length scale.

Turbulent flow is composed by eddies of different sizes. The sizes define a characteristic length scale for the eddies, which are also characterized by flow velocity scales and time scales (turnover time) dependent on the length scale. The large eddies are unstable and eventually break up originating smaller eddies, and the kinetic energy of the initial large eddy is divided into the smaller eddies that stemmed from it. These smaller eddies undergo the same process, giving rise to even smaller eddies which inherit the energy of their predecessor eddy, and so on. In this way, the energy is passed down from the large scales of the motion to smaller scales until reaching a sufficiently small length scale such that the viscosity of the fluid can effectively dissipate the kinetic energy into internal energy.