David T. Walker

Shear-free turbulence near a flat free surface

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 96 3 . 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.

TURBULENT STRUCTURE IN A CHANNEL FLOW WITH POLYMER INJECTION AT THE WALL

Two-component laser velocimeter measurements in a fully developed turbulent water channel flow with polymer injection were used to examine the effect of polymer injection on the Reynolds stresses and the production terms in the Reynolds stress transport equations. These measurements show that while the root-mean-square fluctuation level of the streamwise velocity was increased, the r.m.s. of the wall-normal velocity and the Reynolds shear stress were reduced.

Radar Backscatter and Surface Roughness Measurements for Stationary Breaking Waves

In this study the surface features and the radar backscatter associated with breaking waves generated by a uniform flow past a stationary submerged hydrofoil were examined. The level of energy dissipation due to breaking was varied by changing the foil angle of attack. Time series of surface elevation profiles were obtained for the breaking crest region and the following waves. Radar backscatter (X-band) was also measured for an incidence angle of 45° with the radar looking both upwave and downwave for HH and VV polarizations. These measurements were compared to model predictions of radar backscatter using the surface elevation data as inputs to the model. The breaking crest region exhibited the largest surface disturbances, as measured by the temporal variance of the surface elevation. The maximum in the variance was associated with large low-frequency disturbances in the ‘toe’ region. Downstream-moving waves appear just ahead of the crest and, due primarily to interaction with the spatially varying current set up by the stationary wave, decrease in amplitude by an order of magnitude as they propagate downstream. These surface disturbances remain at a low level thereafter. A maximum radar cross-section per unit area of about 0.5 was observed near the breaking crest, for both HH and VV polarization in the upwave look direction. The maximum value for the upwave look direction was about twice as large as for the downwave look direction. Downstream of the breaking crest, the radar cross-section decreased rapidly and then leveled off, and an increasing difference between the VV and HH backscatter was observed as the overall backscatter level decreased. Near the second crest, there was a small increase in the height variance and in the radar cross-section. The surface-elevation measurements were used as inputs for a Bragg-scattering model and the expected radar backscatter was calculated. The variations in the observed radar cross-section downstream of the breaking crest are satisfactorily explained by the Bragg model when surface-tilt effects are taken into account. However, the backscatter from the breaking crest itself is not accurately predicted since, in this region, the small-scale surface roughness exceed the limits of validity for the Bragg model.

Turbulent structure in free-surface jet flows

Results of an experimental study of the interaction of a turbulent jet with a free surface when the jet issues parallel to the free surface are presented. Three different jets, with different exit velocities and jet-exit diameters, all located two jet-exit diameters below the free surface were studied. At this depth the jet flow, in each case, is fully turbulent before significant interaction with the free surface occurs. The effects of the Froude number ( Fr ) and the Reynolds number ( Re ) were investigated by varying the jet-exit velocity and jet-exit diameter. Froude-number effects were identified by increasing the Froude number from Fr = 1 to 8 at Re = 12700. Reynolds-number effects were identified by increasing the Reynolds number from Re = 12700 to 102000 at Fr = 1. Qualitative features of the subsurface flow and free-surface disturbances were examined using flow visualization. Measurements of all six Reynolds stresses and the three mean velocity components were obtained in two planes 16 and 32 jet diameters downstream using a three-component laser velocimeter. For all the jets, the interaction of vorticity tangential to the surface with its ‘image’ above the surface contributes to an outward flow near the free surface. This interaction is also shown to be directly related to the observed decrease in the surface-normal velocity fluctuations and the corresponding increase in the tangential velocity fluctuations near the free surface. At high Froude number, the larger surface disturbances diminish the interaction of the tangential vorticity with its image, resulting in a smaller outward flow and less energy transfer from the surface-normal to tangential velocity fluctuations near the surface. Energy is transferred instead to free-surface disturbances (waves) with the result that the turbulence kinetic energy is 20% lower and the Reynolds stresses are reduced. At high Reynolds number, the rate of evolution of the interaction of the jet with the free surface was reduced as shown by comparison of the rate of change with distance downstream of the local Reynolds and Froude numbers. In addition, the decay of tangential vorticity near the surface is slower than for low Reynolds number so that vortex filaments have time to undergo multiple reconnections to the free surface before they eventually decay.

Radar backscatter from stationary breaking waves

Radar backscatter measurements from stationary breaking waves were used to examine how the surface roughness generated by wave breaking affects radar backscatter at moderate incidence angles. Stationary breaking waves were generated by submerging a stationary hydrofoil in a uniform flow. X band radar backscatter measurements were made at numerous streamwise positions along the stationary breaking waves from an incidence angle of 45° for horizontal transmit and receive polarization (HH) and vertical transmit and receive polarization (VV) looking both upwave and downwave. The radar returns increased substantially, and the HH-to-VV polarization ratio approached unity near the breaking crests. This radar signature is consistent with those observed in the field. Detailed optical measurements of the breaking surfaces revealed that the observed radar returns near the breaking crests were the result of increased incoherent backscatter from the small-scale surface roughness generated by the breaking waves, although surface tilt effects also modified the radar return. Scattering models based upon the small perturbation solution performed well in the wake of the stationary breaking crest, but they significantly underestimated the HH-to-VV polarization ratio near the breaking crest. More advanced scattering solutions such as the integral equation method produced more accurate results in regions containing the largest surface roughness. These findings suggest that incoherent backscatter from surface disturbances produced by deep water breaking waves may be the source of the high radar returns and small polarization ratios observed from the ocean at moderate incidence angles.

Reynolds-averaged equations for free-surface flows with application to high-Froude-number jet spreading

The goals of this study were to develop a set of Reynolds-averaged governing equations for turbulent free-surface flow, and to use the resulting equations to determine the origin of the surface current in high-Froude-number jet flows. To develop the Reynolds-averaged equations, free-surface turbulent flow is treated as a two-fluid flow separated by an interface. The general Navier-Stokes equations written for variable property flow embody the field equations applicable to each fluid, as well as the boundary conditions for the interface and, therefore, can be applied across the entire fluid domain, including the interface. With this as a starting point, a formulation of the Reynolds-averaged governing equations for turbulent free-surface flows can be developed rigorously. The resulting Reynolds-averaged equations are written in terms of density-weighted averages, their derivatives, and the probability density function for the free-surface position. These equations are similar to the conventional Reynolds-averaged equations, but include additional terms which represent the average effect of the forces acting instantaneously on the free surface, forces normally associated with the boundary conditions

Reynolds Stress Modeling for Drag Reducing Viscoelastic Flows

A Reynolds-stress transport equation model for turbulent drag-reducing viscoelastic flows, such as that which occurs for dilute polymer solutions, is presented. The approach relies on an extended set of Reynolds-Averaged Navier-Stokes equations which incorporate additional polymer stresses. The polymer stresses are specified in terms of the mean polymer conformation tensor using the FENE-P dumbbell model. The mean conformation tensor equation is solved in a coupled manner along with the Navier-Stokes equations. The presence of the polymer stresses in the equations of motion results in additional explicit polymer terms in the Reynolds-stress transport equations, as well as implicit polymer effects in the pressure-strain redistribution term. Models for both the explicit and implicit effects have been developed and implemented in a code suitable for boundary layer, rectangular channel and pipe-flow geometries. Calibration and validation is has been carried out using results from recent direct numerical simulation of viscoelastic turbulent flow.Copyright

A Simple Model for Marine Radar Images of the Ocean Surface

This letter describes a simple model for the backscattered signal received from the ocean surface by a horizontal-polarization marine radar operating at low grazing angles, including the effects of the logarithmic amplifiers conventionally used in these radars. Shadowing is accounted for via geometric optics. The range dependence of the mean radar signal predicted by this model compares well with that observed in two marine radar data sets collected under different environmental conditions. The model also indicates that the radar signal modulation (the deviation from the mean normalized by the mean) is proportional to the radial component of the surface slope in unshadowed regions. This suggests the possibility of direct quantitative wave observations using marine radar modulation data, without the need for calibration.

Reynolds stress modeling for rough wall turbulence: An invited Paper

A full Reynolds stress tensor model for rough wall turbulent flow is developed and presented in this paper. The basis for the model development is a two-phase averaging approach wherein the rough solid boundary is treated as the second phase. This approach leads to a set of averaged equations similar to the canonical Reynolds averaged equations, but which contain additional terms reflecting the inclusion of the roughness. The full stress model is developed from a variant of the Speziale, Sakar annd Gatski model. The primary modifications are the addition of a model for the production of turbulent stresses and kinetic energy by the roughness elements and a modification to the pressure strain modeling near the wall. The model is then reduceed to a k model. In the k model, the eddy viscosity model contain an additional term representing the roughness production of turbulence. The model is calibrated by reference to established equivalent sand grain roughness scaling.