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Predicting the Small Scales of Alpha Models for Turbulence

Evelyn Lunasin emanalo@cut.this.part.math.uci.edu

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.