In this paper we investigate the large-eddy simulation (LES) of the interaction between
a turbulent shear flow and a free surface at low Froude numbers. The benchmark
flow field is first solved by using direct numerical simulations (DNS) of the Navier–Stokes
equations at fine (1282 × 192 grid) resolution, while the LES is performed
at coarse resolution. Analysis of the ensemble of 25 DNS datasets shows that the
amount of energy transferred from the grid scales to the subgrid scales (SGS) reduces
significantly as the free surface is approached. This is a result of energy backscatter
associated with the fluid vertical motions. Conditional averaging reveals that the
energy backscatter occurs at the splat regions of coherent hairpin vortex structures
as they connect to the free surface. The free-surface region is highly anisotropic
at all length scales while the energy backscatter is carried out by the horizontal
components of the SGS stress only. The physical insights obtained here are essential
to the efficacious SGS modelling of LES for free-surface turbulence. In the LES,
the SGS contribution to the Dirichlet pressure free-surface boundary condition is
modelled with a dynamic form of the Yoshizawa (1986) expression, while the SGS
flux that appears in the kinematic boundary condition is modelled by a dynamic
scale-similarity model. For the SGS stress, we first examine the existing dynamic
Smagorinsky model (DSM), which is found to capture the free-surface turbulence
structure only roughly. Based on the special physics of free-surface turbulence, we
propose two new SGS models: a dynamic free-surface function model (DFFM) and
a dynamic anisotropic selective model (DASM). The DFFM correctly represents the
reduction of the Smagorinsky coefficient near the surface and is found to capture the
surface layer more accurately. The DASM takes into account both the anisotropy
nature of free-surface turbulence and the dependence of energy backscatter on specific
coherent vorticity mechanisms, and is found to produce substantially better surface
signature statistics. Finally, we show that the combination of the new DFFM and
DASM with a dynamic scale-similarity model further improves the results.
Scale-similarity model for Lagrangian velocity correlations in isotropic and stationary turbulence
