Shear-free turbulence near a flat free surface

Abstract

In this study the evolution of initially homogeneous and isotropic turbulence in the presence of a free surface was investigated. The Navier–Stokes equations were solved via direct pseudo-spectral simulation with a resolution of 963. The Reynolds number based on the volume-averaged turbulence kinetic energy and dissipation rate was 147. Periodic boundary conditions were used in two dimensions, and the top and bottom sides of the domain were flat and shear-free. A random, divergence-free velocity field with a prescribed spectrum was used as the initial condition. An ensemble of sixteen separate simulations was used to calculate statistics.Near the surface, the Reynolds stresses are anisotropic and the anisotropy extends a distance from the surface roughly equal to the turbulent lengthscale. The tangential vorticity fluctuations also vanish near the surface, owing to the no-shear condition, causing a corresponding decrease in the fluctuating enstrophy. The thickness of the region in which the surface affects the vorticity distribution is roughly one-tenth the turbulent lengthscale. The stress anisotropy near the surface appears to be maintained by reduced dissipation for the tangential velocity fluctuations, reduced pressure–strain transfer from the tangential to surface-normal velocity fluctuations, and rapid decay of the surface-normal velocity fluctuations due to dissipation. The turbulence kinetic energy rises in the near-surface region owing to a decrease in dissipation at the surface. This decrease in dissipation results from the local reduction in enstrophy owing to the vanishing of the tangential vorticity fluctuations at the surface. At the free surface, the mean pressure rises. This is also due to the reduction in enstrophy.While the tangential vorticity must vanish at the free surface, the flow is fully three-dimensional up to the surface and the production of surface-normal vorticity by vortex stretching attains a maximum at the free surface. The contribution to the total enstrophy by the surface-normal vorticity fluctuations remains relatively constant over depth. The production of the surface-normal enstrophy component due to vortex stretching is roughly balanced by turbulent transport of enstrophy away from the surface. Near the surface, there are elevated levels of production of tangential vorticity by both vortex-stretching and fluctuating shear strains.

Large eddy simulation of stably stratified open channel flow

“…It has been shown that the flow at the free surface is highly anisotropic. 18 Also, as mentioned in the Introduction, Walker et al 8 found that the vertical gradient of the horizontal vorticity vanished only in a very thin layer near the free surface, requiring a fine grid to resolve the mean profile in that region. Shen and Yue 18 show that the energy backscatter ͑transfer from subgrid scales to larger scales͒ is maximal at the free surface.…”

“…I, previous studies 8,9 have shown that turbulence at a free surface without the presence of stratification cannot be well represented by twodimensional dynamics. It is generally thought that stratification tends to make turbulence more two-dimensional, so it might be anticipated that, with sufficient stratification, the dynamics at the free surface might be approximated by a two-dimensional model.…”

“…It was originally conjectured that the dynamics of free-surface turbulence would be quasi-twodimensional. However, Walker et al 8 asserted that turbulence is three-dimensional up to the surface, and even at the surface does not conform to two-dimensional dynamics. In support of this, they demonstrated that vortex stretching is maximal at the free surface, and the tangential vorticity vanishes only in a very thin layer.…”

Large-eddy simulations of ducts with a free surface

“…A difference between the results of the present work and those of Hsu et al (2000) is that the recovery of u u occurs across almost all the surface, except at the lower-wall bisector where ∂ u /∂y is equal to zero and ∂ u /∂z values are almost constant in the region 0.8 < z/D < 1. Previous studies of Perot & Moin (1995) and Walker et al (1996) revealed significant differences between shear-free boundary layers near solid walls and free surfaces, such as the higher tangential Reynolds stress and lower dissipation near a free surface. Walker et al (1996) attribute the apparent increase of turbulence kinetic energy near a rigidlid boundary to the reduced dissipation of the tangential velocity fluctuations.…”

“…velocity-fluctuation profiles exhibit trends similar to open channel flows. Indeed, u rms dominates, and near the free surface, turbulence moves towards a quasi-twodimensional state (see Walker et al 1996). From a mathematical point of view, this behaviour at the centre of the duct is obvious since the Reynolds stresses depend on the shear components, most of which are zero at the centre of the duct since symmetry and fully developed conditions are assumed.…”