Figure 2.6

The onset of turbulence as a function of velocity and water depth. The boundary is not precise, occurring at Reynolds numbers between Re = 500 and Re = 2000. Plot based on Reynolds (1992b) and redrawn, with permission, from Reynolds (1997a).

The key empirical quantification of turbulence is based upon the time-averaged velocity fluctuations in the horizontal and vertical directions (±u', ±w). Because the summation of positive and negative measurements must tend quickly to zero, the positive roots of their squares, [(±ur)2]1/2, [(±wr)2]1/2, are cumulated instead. The turbulent intensity, (u* )2, comes from the product of their root mean squares:

The square root (u*) has the dimensions of velocity and is known variously as the turbulent velocity, the friction velocity or the shear velocity. In natural systems, both (u*) and (u*)2 are extremely variable, in time and in space, depending upon the energy of the mechanical forcing and the speed at which it is dissipated through the eddy spectrum. By considering the effects of forcing in contrasted situations, relevant ranges of values for (u*) may be nominated.

Thus, the simplest model that may be proposed applies to a body of open water, of infinite depth and horizontal expanse and lacking any gradient in temperature. We subject it to a (wind) stress of constant velocity and direction. The momentum transferred across the surface must balance the force applied. So we may propose the following equalities:

where pa is the density of air (~1.2 kgm-3), U is the wind velocity (properly, measured 10 m above the water surface) and cd is a dimensionless coefficient of frictional drag on the surface (~1.3 x 10-3). The equation is imprecise for a number of reasons, one being the interference in transfer caused by surface waves. Nevertheless, the implied linear relationship, (u*) ~ U/800, is sufficiently robust in the wind speed range 5-20 m s-1 (u* x 10-3 to 2.5 x 10-2 ms-1) for it to stand as a good rule-of-thumb quantity.

A general derivation for water flowing down a channel in response to gravity relates (u*) to u, as has been developed, inter alia, by Smith (1975):

where H is the depth of the flow and rp is the roughness of the bed, as defined by the heights of projections from the bottom. The ratio between them is such that (u*) is generally 1/30 to 1/10 the value of u. A general relationship relating (u*) to channel form is:

where g is gravitational acceleration (in m s-2), sb is the gradient of the bed (in m m-1), Ax is the cross-sectional area of the channel (in m2) and p is its wetted perimeter (in m). In wide channels Ax/p approximates to the mean depth, H. Thus,

The turbulent velocity in a river 5 m deep and falling 0.2 m in every km, approximates to (u*) = 10-2 m s-1. Turbulent intensity increases in rivers with increasing relative roughness, with increasing depth and increasing gradient. Theoretical contours of (u*) are mapped in Fig. 2.7 in terms of gradient and water depth and links them to

Figure 2.7

Contour map of the approximate distribution of u* in aquatic environments (including rivers), in terms of water-column height, bed gradient and applied wind velocity. Original plot from Reynolds (1994b) and redrawn, with permission, from Reynolds (1997a).

Contour map of the approximate distribution of u* in aquatic environments (including rivers), in terms of water-column height, bed gradient and applied wind velocity. Original plot from Reynolds (1994b) and redrawn, with permission, from Reynolds (1997a).

those driven by surface wind stress: the dog-legs thus represent the 'switch points', where atmospheric forcing overtakes gravitational flow as the main source of turbulent energy in the water. Major aquatic habitats are noted on the map.

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