Spectral modeling of turbulent flows and the role of helicity

We present a version of a dynamical spectral model for large eddy simulation based on the eddy damped quasinormal Markovian approximation [S. A. Orszag, in, edited by R. Balian, Proceedings of Les Houches Summer School, 1973 (Gordon and Breach, New York, 1977), p. 237; J. P. Chollet and M. Lesievr, J. Atmos. Sci. 38, 2747 (1981)]. Three distinct modifications are implemented and tested. On the one hand, whereas in current approaches, a Kolmogorov-like energy spectrum is usually assumed in order to evaluate the non-local transfer, in our method the energy spectrum of the subgrid scales adapts itself dynamically to the large-scale resolved spectrum; this first modification allows in particular for a better treatment of transient phases and instabilities, as shown on one specific example. Moreover, the model takes into account the phase relationships of the small scales, embodied, for example, in strongly localized structures such as vortex filaments. To that effect, phase information is implemented in the treatment of the so-called eddy noise in the closure model. Finally, we also consider the role that helical small scales may play in the evaluation of the transfer of energy and helicity, the two invariants of the primitive equations in the inviscid case; this leads as well to intrinsic variations in the development of helicity spectra. Therefore, our model allows for simulations of flows for a variety of circumstances and a priori at any given Reynolds number. Comparisons with direct numerical simulations of the three-dimensional Navier-Stokes equation are performed on fluids driven by an Arnold-Beltrami-Childress (ABC) flow which is a prototype of fully helical flows (velocity and vorticity fields are parallel). Good agreements are obtained for physical and spectral behavior of the large scales.