The surface layer for free-surface turbulent flows

Abstract

Direct numerical simulation (DNS) is used to examine low Froude number free-surface turbulence (FST) over a two-dimensional mean shear flow. The Navier–Stokes equations are solved using a finite-difference scheme with a grid resolution of 1283. Twenty separate simulations are conducted to calculate the statistics of the flow. Based on the velocity deficit and the vertical extent of the shear of the mean flow, the Reynolds number is 1000 and the Froude number is 0.7. We identify conceptually and numerically the surface layer, which is a thin region adjacent to the free surface characterized by fast variations of the horizontal vorticity components. This surface layer is caused by the dynamic zero-stress boundary conditions at the free surface and lies inside a thicker blockage (or ‘source’) layer, which is due to the kinematic boundary condition at the free surface. The importance of the outer blockage layer is manifested mainly in the redistribution of the turbulence intensity, i.e. in the increase of the horizontal velocity fluctuations at the expense of the vertical velocity fluctuation. A prominent feature of FST is vortex connections to the free surface which occur inside the surface layer. It is found that as hairpin-shaped vortex structures approach the free surface, their ‘head’ part is dissipated quickly in the surface layer, while the two ‘legs’ connect almost perpendicularly to the free surface. Analysis of the evolution of surface-normal vorticity based on vortex surface-inclination angle shows that both dissipation and stretching decrease dramatically after connection. As a result, vortex structures connected to the free surface are persistent and decay slowly relative to non-connected vorticities. The effects of surface and blockage layers on the turbulence statistics of length scales, Reynolds-stress balance, and enstrophy dynamics are examined, which elucidate clearly the different turbulence mechanisms operating in the respective near-surface scales. Finally we investigate the effect of non-zero Froude number on the turbulence statistics. We show that the most significant effect of the presence of the free surface is a considerable reduction of the pressure–strain correlation at this surface, compared to that at a free-slip at plate. This reduction is finite even for very low values of the Froude number.

Free-surface turbulence and air-water gas exchange

“…The blockage effect has been described theoretically with the source theory of Hunt and Graham [61], which treats the turbulent field as a superposition of homogeneous turbulence and an irrotational velocity field driven by a source distribution at the interface that causes the vertical velocity fluctuations to vanish there. Numerical simulations confirm this behavior (e.g., Shen et al [120]; Tsai [132]), as do experimental observations like those of Brumley and Jirka [13] and the present work. For example, figure 6-23 shows the slight amplification of the horizontal velocity fluctuations at the expense of the vertical, the latter beginning to fall off at a depth near fl = 2£HT as predicted.…”

“…This layer gives rise to a "pancake" eddy structure (Handler et al [47]) near the interface where the vertical extent of the eddies is reduced and the lateral extent changes little. Similar results for the Taylor microscales were found in the direct numerical simulation work of Shen et al [120].…”

“…Therefore, the free-surface vortical motions at a clean interface are predominated by normally connected vortices (e.g., figure 7-18c). The stress-free dynamic boundary condition results in a viscous sublayer of thickness ()(flRe~1/2), where Reo is based on the turbulent scales, U o and fl (Brumley and Jirka [14]; Hunt [60]; Shen et al [120]). Over this viscous sublayer thickness, there is typically a small change (reduction) in the horizontal velocity of an approaching eddy of ()(uoRe~1/2) (Hunt [60]).…”

Turbulence in the swash and surf zones: a review

“…Some hairpin-shaped vortex structures develop inside the surface layer and evolve losing their heads, connecting to the free surface and orienting the two legs almost perpendicularly to the free surface. For those vortices, the dissipation rate is extremely low after connection to the free surface, inducing a strong persistence (Shen et al, 1999). The existence of these vertical vortices, having random number and location, was documented by Chang and Liu (1998), through PIV measurements under breaking waves.…”

Large-eddy simulations of ducts with a free surface

“…Close to the sidewall and within 20 viscous lengths of the free surface, all three components of turbulent motions were reported to decrease to a minimum at the free surface. This reduction was explained by Shen et al (1999) through two mechanisms: the annihilation of w due to the constraint on the vertical motion at the free surface, and the vanishing of ∂ u /∂z caused by the shear-free boundary condition. Our calculations agree only in part with the experimental findings.…”