Generation of turbulence

Despite the rather self-evident relationship that plankton mostly goes where the water takes it, the large-scale motion of water bodies tells us frustratingly little about what the conditions of life are like at the spatial scales appropriate to individual species of phytoplankton (generally <2 mm), or about the trajectories followed by the individual phytoplankter whose survival depends on its passing a reasonable fraction of its life in the insolated upper reaches of the water column. Although it has long been appreciated that the energy of the major circulations is dissipated through cascades of smaller and smaller gyratory structures, now called the Kolmogorov eddy

North Eq^itorial C^ Equatorial Counter C.

i^onsooo

Equatorial Counter C"

Equatorial

Equatorial

Weddell C.

JS^ Wind Drift.

Figure 2.3

The currents at the surface of the world's oceans in the northern winter. Redrawn, with permission, from Harvey (1976).

(b) Wind direction =217"

(b) Wind direction =217"

(c) Wind direction ■ 3T

1 Model reconstructions of the near-surface currents generated in Esthwaite Water, UK, by steady winds of 5 m s-1 and various orientations. The longest axis of the lake is approximately 2.2 km. Redrawn, with permission, from Falconer et al. (1991).

spectrum after one of its most famous investigators (Kolmogorov, 1941), the impact of the lower end of the series on the behaviour of plankton had remained philosophically and mathematically obscure. My cumbersome attempts to overcome this deficiency in an earlier text (Reynolds, 1984a) only emphasise this frustration. Looking back from the present standpoint, they serve as a point of reference for just how far the appreciation and quantification of turbulence, together with their impacts on particle entrain-ment, have moved on during the subsequent couple of decades.

'Turbulence' testifies to the failure of the molecular structure of a fluid to accommodate introduced mechanical energy. The mysteries of turbulence have gradually given way to a developing turbulence theory (Levich, 1962; Ten-nekes and Lumley, 1972) but it is the remarkable progress in instrumentation and direct sensing of turbulence (see especially Imberger, 1998), together with the rapid assimilation of its quantification into physical oceanography and limnology (see, for example, Denman and Gargett, 1983; Spigel and Imberger, 1987; Imberger and Ivey, 1991; Mann and Lazier, 1991; Imboden and Wuest, 1995; Wuest and Lorke, 2003) that have given most to the characterisation of the environment of the phytoplankton.

A helpful starting point is to envisage a completely static column of water (Fig. 2.5a). If its upper surface is subjected to a mild horizontal force, t , then water molecules at the air-water interface are dragged across the surface in the direction of the force. Their movement is transmitted to the layer below, which also begins to move, albeit at a lesser velocity. Further downward propagation soon leads to a configuration envisaged in Fig. 2.5b, each layer of molecules sliding smoothly over the one below, in what is described as laminar flow. The structure conforms to a vertical gradient of horizontal velocities, u, the steepness of which is defined by the differential notation, du/dz (literally the increment or decrement of horizontal velocity for a small increment in the vertical direction, z). While the condition of laminar flow persists, the ratio between the applied force (per unit area) and the velocity gradient corresponds to the absolute viscosity of the water, n. That is, n = t (du/dz)-1 (2.1)

Adopting the appropriate SI units for force (N = newtons, being the product, mass x acceleration, may be expressed as kgms-2) per unit area (m2), for velocity (m s-1) and for vertical distance (m), the absolute viscosity is solved in poises (P = kgm-1 s-1). The values plotted in Fig. 2.1b approximate to 10-3 kg m-1 s-1. Caution over units is urged because it is common in hydrodynamics to work with the kinematic viscosity of a fluid, v, which is equivalent to the absolute viscosity with the density (pw) divided out:

Figure 2.5

The generation of turbulence by shear forces. In

(a), the water beneath the horizontal surface is unstressed and at rest. In (b), a mild force, t, is applied which that drags water molecules at the surface in the direction of the force;

(b) their movement serves to drag those immediately below, and so on, giving rise to an ordered structure of laminar flow. In (c), the transmitted energy of the intensified force can no longer be dissipated through the velocity gradient which breaks down chaotically into turbulence. Redrawn, with permission, from Reynolds (1997a).

Figure 2.5

The generation of turbulence by shear forces. In

(a), the water beneath the horizontal surface is unstressed and at rest. In (b), a mild force, t, is applied which that drags water molecules at the surface in the direction of the force;

(b) their movement serves to drag those immediately below, and so on, giving rise to an ordered structure of laminar flow. In (c), the transmitted energy of the intensified force can no longer be dissipated through the velocity gradient which breaks down chaotically into turbulence. Redrawn, with permission, from Reynolds (1997a).

The transition between ordered and turbulent flow patterns has long been supposed to depend upon the ratio between the driving and viscous forces; this ratio is expressed by the dimension-less Reynolds number, Re:

Given the density of water of ~103 kgm-3 (Fig. 2.1a), its kinematic viscosity approximates to 10-6 m2 s-1.

If the applied force is now increased sufficiently, it begins to shear molecules from the upper surface of the water column. Thus, the smooth, ordered velocity gradient fails to accommodate the applied energy; the structure breaks up into a complex series of swirling, recoiling eddies (Fig. 2.5c). A new, turbulent motion is superimposed upon the original direction of flow. The layer now assumes a net mean velocity in the same direction (U ms-1). Now, at any given point within the turbulent flow, there would be detected a series of velocity fluctuations, accelerating to (U + u!) and decelerating to (U - u') ms-1. Simultaneously, the displacements in the vertical (z) direction introduce a velocity component which fluctuates between (0 + W) and (0 -W) ms-1. This pattern is maintained for so long as the appropriate level of forcing persists. The driving energy is, as it were, extracted into the largest eddies, is progressively dissipated through smaller and smaller eddies of the Kolmogorov spectrum and is finally discharged as heat, as the smallest eddies are overwhelmed by viscosity.

where la is the length dimension available to the dissipation of the energy, usually the depth of the flow. Turbulence will develop wherever there is a sufficient depth of flow with sufficient horizontal velocity. Solving Eq. (2.3) for a notional small stream travelling at 0.1 ms-1 in a channel 0.1 m deep, Re ~ 104; for a 10-mm layer in a well-established thermocline in a small lake subjected to a horizontal drift of 10mms-1, Re ~ 102. The former is manifestly turbulent but the latter maintains its laminations. There is no unique point at which turbulence develops or subsides; rather there is a transitional range which, for water, is equivalent to Reynolds numbers between 500 and 2000. The depth-velocity dependence of turbulence is sketched in Fig. 2.6.

The point in the spectrum where the eddies are overwhelmed by molecular forces and collapse into viscosity is more difficult to predict without information on their velocities. As suggested above, it is now possible to measure the velocity fluctuations directly with the aid of sophisticated accoustic sensors but the quantities still need to be interpreted within a theoretical context. More significantly, the theoretical framework can be used estimate the intensity of the turbulence from properties of the flow which are measurable with relative ease.

Was this article helpful?

0 0

Post a comment