Assumption Of Constant Eddy Viscosity Research Articles

Interference of wind-driven and pressure gradient-driven flows in shallow homogeneous water bodies

This study explores the characteristics of wind-driven steady-state flows in water bodies of constant density focusing on situations in which the surface and bottom Ekman layer interfere. Under the assumption of constant eddy viscosity in conjunction with a zero-flow bottom boundary condition, such flows can be linearly decomposed into wind-driven and pressure gradient-driven flow components, each affiliated with a frictional boundary layer. The resultant interference patterns, including the creation of undercurrents, are discussed using a one-dimensional water-column model. The second part of this paper employs a three-dimensional hydrodynamic model to study interferences for an idealized large and shallow oceanic bay at low latitudes under the action of a uniform wind stress. Lee effects trigger a surface pressure field that tends to slope against the wind direction. The associated pressure gradient force creates an undercurrent in deeper portions of the bay, while unidirectional flows prevail in shallower water. It is demonstrated that such undercurrents can operate as an effective upwelling mechanism, moving sub-surface water into a bay over large distances (~100 km). Based on estimates of eddy viscosities, it is also shown that Ekman layer dynamics play a central role in the dynamics of most mid-latitude lakes. On the continental shelf of the modern ocean, inferences between the surface and bottom Ekman layers leading to undercurrents do currently only exist in shallow shelf seas at low latitudes, such as the Arafura Sea.

Impact of estuarine convergence on residual circulation in tidally energetic estuaries and inlets

Estuarine convergence (landward reduction of width and/or depth) is known to have the potential to significantly enhance estuarine circulation, a result theoretically derived under the assumption of constant eddy viscosity. Recent studies of longitudinally uniform energetic tidal channels indicate that tidal straining, a process driven by tidally varying eddy viscosity, is a major driver of estuarine circulation. The combined effect of estuarine convergence and tidal straining is investigated, for the first time, in this paper. The present idealized numerical study shows that estuarine convergence is reducing or even reversing tidal straining circulation in such a way that estuarine circulation can be weakened. This is a counterintuitive hydrodynamic effect of estuarine convergence, which may reduce (rather than increase) up‐estuary particulate matter transport in estuaries and tidal inlets.

Feasibility Study of Constant Eddy-Viscosity Assumption in Gradient-Based Design Optimization

Feasibility Study of Constant Eddy-Viscosity Assumption in Gradient-Based Design Optimization

On the vertical structure of tidal currents in a homogeneous sea

The vertical variation of tidal currents caused by friction at the sea-bed is investigated in both qualitative and quantitative ways with the Coriolis force being taken into account. The simple model with the assumption of constant eddy viscosity is employed to study the effects of friction on the vertical structure of tidal currents. The model explains these effects in vertical change of the maximum velocity, the ellipticity (the ratio of the minor to the major axis) of current ellipse, the time and the direction of the maximum velocity, and the ratio of diurnal current to semidiurnal current. To reach a quantitative agreement with the data observed by Bowden & Fairbairn in the Irish Sea, the mixing length theory is applied to numerical calculation of the vertical distribution of tidal currents with a finite difference scheme. A fair agreement with the observations is obtained in the calculation. The results show the following features: (1) the vertical profile of amplitude of the main component for the current is almost logarithmic throughout the depth; (2) the corresponding profde for the shearing stress is almost linear with depth; (3) the maximum stress lags the maximum velocity with the delay increasing with elevation; (4) the eddy viscosity coefficient has its maximum around the mid-depth; and (5) the vertically averaged viscosity lags the velocity magnitude. The dependence of the quadratic resistance coefficient and the coefficient k defined by Bowden's formulav̄=κhū(h— depth, v̄ and ū— depth-averaged eddy viscosity and velocity) on the roughness length is given for a steady flow in a non-rotating system. The numerical solution shows that these relations can be approximately applied to the tidal currents in shallow waters.

On currents, internal and inertial waves in a stratified ocean due to variable winds. Part 1

The response of a continuously stratified viscous ocean to a variable wind air pressure field has been studied during recent years by means of analytical methods. This requires the assumption of constant eddy viscosity and exponential stratification. In shallow water constant eddy viscosity leads to irrealistically thick boundary layers compared to the water depth and, therefore, eliminates internal waves. In this first part of a two-part study a theoretical model is developed for arbitrary density stratification and eddy viscosity. The results of the integration are dealt with in part 2.

Frictional Currents in the Mixed Layer at the Sea Surface

Abstract The wind-driven flow in the mixed layer between the free surface and the top stable sheet of a thermocline is usually turbulent. The frictional component of such a flow may be separated from a depth-independent general current. The turbulent frictional current is governed by a small number of external parameters and should be subject to quite simple laws. Explicit approximate solutions may be obtained with the assumption of constant eddy viscosity and a lower boundary condition of zero stress at the bottom of the mixed layer: these show a variety of different “Ekman spirals,” depending only on the parameter fh/u* where f is the Coriolis parameter, h the mixed layer depth, and u* the shear velocity. Experimental data collected within the mixed layer on Lake Huron are reported and compared with the simple theory. Encouraging agreement between theory and observation is found and some quantitative relationships are tentatively established.

Turbulent Rayleigh shear flow

Townsend has derived a relation between mean vorticity and Reynolds stress valid in the wall layer of a turbulent flow. The vorticity Ω appears as a function of the local stress τ and its gradient. Such a relation is better suited for use in the vorticity equation than in the momentum equation. The Rayleigh problem, whose vorticity equation is simply ∂Ω/∂t = ∂2τ/∂y2, is introduced as a setting for Townsend's theory. Certain wall speed programmes are shown to generate Rayleigh layers that are exactly self-similar in the fully turbulent part of the flow. Those layers correspond to Clauser's equilibrium boundary layers. A formal analogy between the two families is found; the analogy becomes quantitatively exact in the limit of infinite Reynolds number. The Rayleigh problem is posed in similarity form. A composite non-linear, ordinary differential equation for the stress profile is deduced from a two-layer model incorporating Townsend's relation for the wall layer and Clauser's constant eddy-viscosity assumption for the outer layer. The profile depends on the wall-speed programme selected and on two empirical constants: the combination λ = √(k)/k of Clauser's k and Kármán's k, and Townsend's constant B. Closed-form solutions for arbitrary λ and B are obtained in two important cases: constant wall stress, analogous to constant pressure above a boundary layer, and zero wall stress, corresponding to continuous separation. The velocity profile in the wall region of a continuously separating Rayleigh layer is found to depend sensitively on B.

ESTUARINE CURRENTS AND TIDAL STREAMS

Fresh water spreading out from the mouth of a river as it enters a salt sea may preserve its identity for a considerable distance on the surface if wind-generated waves, longshore currents and tidal streams are capable of producing only weak mixing. Fig. 1 shows the three dimensional shape of a fresh-water tongue overlying more dense salt water, derived by Takano (1954) on the assumption of constant eddy viscosity and constant density in the fresh water layer, below which the density increases according to an assumed law, making an asymptotic approach to the density of salt water. Takano's model is thus a water jet entraining salt from around and below it.
 Salt or brackish water may penetrate along the deep channels of an estuary in the shape of a wedge complementary to the fresh water tongue, the salt wedge retreating seawards as heavy rainfall increases the river discharge, and advancing in dry weather intervals. Tidal streams cause a regular oscillation of both fresh and braok water in flood and ebb directions but the seasonal movements of the sloping boundary between fresh and salt water may still be important in low-lying delta regions. Strong tidal streams lead to intense mixing, when neither a fresh water tongue nor a salt wedge can be distinguished, but the isohalines (salinity contours) preserve the general wedge pattern - see Figs. 3 to 6. In the upper reaches of an estuary it is possible to study the effect of the tidal motion by treating it as a simple harmonic perturbation of the uni-directional river flow. Even in the middle portion of the estuary where there is reversal of the horizontal motion, one may seek a "quasi steady" solution for the net effect (seaward movement of fresh water) while allowing for the increased turbulence due to the tidal action. At the seaward end of the estuary there is little deviation from the astronomical tidal rhythm, so the problem reduces to simple harmonic oscillations of salt water. Higher harmonics may be introduced as an extension of the simple solution. For a first approximation it is sufficient to consider flow in the longitudinal vertical plane, to assume that the pressure distribution is hydrostatic as in long wave theory, and even to neglect inertia terms when investigating net effects.