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Predicting the Small Scales of Alpha Models for Turbulence

Evelyn Lunasin
Susan Kurien

The use of numerical models for the Navier-Stokes equations of fluid dynamics is an established practice in the study of turbulent flows. The calculation of flow from a model, instead of the underlying Navier-Stokes equations, allows for the use of less computing resources for a given flow. The so-called `alpha-models' for turbulence typically use a smoothed velocity field to transport a rough velocity field. The smoothing is performed over a length scale alpha. The expectation is that the smoothed field recovers the large-scale (greater than alpha) statistical properties, such as the energy spectrum, of the `true' turbulent flow which is not smooth. However, for scales smaller than alpha the statistics are not easily deduced analytically since there are two participating `velocities' which have different characteristic timescales and presumably different dynamics. In this work we use numerical simulations to arrive at an empirical hypothesis as to how one can predict the scaling of the energy spectrum of an alpha model. We have thus deduced that the ambiguity in choice of dynamical timescales in alpha-models, due to the presence of two velocities, disappears when one simply considers the relevant conserved quantity and uses its timescale to govern the evolution of all other quantities of interest, including the energy.