Redistribution of Kinetic Energy in Turbulent Flows

Abstract

In statistically homogeneous turbulent flows, pressure forces provide the main mechanism to redistribute kinetic energy among fluid elements, without net contribution to the overall energy budget. This holds true in both two-dimensional (2D) and three-dimensional (3D) flows, which show fundamentally different physics. As we demonstrate here, pressure forces act on fluid elements very differently in these two cases. We find in numerical simulations that in 3D pressure forces strongly accelerate the fastest fluid elements, and that in 2D this effect is absent. In 3D turbulence, our findings put forward a mechanism for a possibly singular buildup of energy, and thus may shed new light on the smoothness problem of the solution of the Navier-Stokes equation in 3D.

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Plain Language Summary

Pressure plays an important role in the dynamics of incompressible turbulent flows. Even in the idealized case of spatially homogeneous flows, this role is not fully understood. For such flows, it is known that pressure, on average, does not contribute to the energy budget of fluid particles; it merely redistributes energy between fluid particles. We find that this energy redistribution strongly depends on the spatial dimension, and it unexpectedly leads to a possible runaway mechanism in three dimensions.

We use numerical simulations of turbulent flows in two and three dimensions to investigate energy redistribution. We find that, both in two and three spatial dimensions, pressure forces play a dominant role in the fluctuations of the rate of change of kinetic energy of individual fluid particles. Turbulence, however, displays fundamentally different physics in two and three spatial dimensions. In three dimensions, on average, pressure takes energy away from slower particles and gives it to faster particles, while in two dimensions there is no such effect. This mechanism of acceleration by pressure suggests a runaway process of velocity growth in three dimensions.

Additional calculations, necessary to reveal whether this mechanism may lead to an unbounded velocity growth, are expected to provide insights into possible singularities in the Navier-Stokes equations.