Flow, deformation, and transport - Physical Processes in Earth and Environmental Sciences - page 113

corresponding to a narrow zone in the free stream for a
Newtonian fluid and a broad zone for a non-Newtonian
fluid, there is no internal deformation. The principle is
wonderfully illustrated by a serendipity exposure of an
ancient intruded magma body, Coward's dyke (Fig. 4.17;
see Section 5.1 on magma intrusions generally), whose
internal shear deformation has been “fixed” in time due to

the flow deformation of included gas bubbles (geologists
call these vesicles ). Notice how the form of the bubbles
changes from highly sheared at the margins to virtually
spherical and unsheared in the middle of the former flow.
The form of the profile, with a marked rigid central plug,
indicates that the magma was behaving as a
non-Newtonian fluid at the instant of cessation of flow.

4.5 Turbulent flow

Leonardo made many sketches of the swirling patterns of
turbulent flows (Fig. 4.18a). Our own casual observations
show fluid to be moving around in what initially seem con-
fusing and possibly random patterns. However, careful
analysis (Fig. 4.18b-f), including Leonardo's time-average
observations, shows that the motions seem to have a well-
defined coherence or structure; flow visualizations show
clearly that eddies have a 3D nature with vorticity and
“strands of turbulence.” Turbulence is a strongly rota-
tional phenomenon, characterized by fluctuating vorticity.
Such 3D turbulent fluctuations cause local velocity gradi-
ents to be set up in the flow which work against the mean
velocity gradient to remove energy from the flow. This
turbulent energy is ultimately dissipated by the action of
viscosity on the turbulent fluctuations.

2 At the same time, relatively slow “lumps” of strongly
rotating fluid move out from the inner to the outer flow.
These are termed burst motions.
In the xy plane:
3 Close to the bed there exist flow-parallel lanes of relatively
slow and fast fluid that alternate across the flow. The low-
speed lanes are termed streaks .
4 The streaks become increasingly less defined and
“tangled-up” as we ascend the flow.
A 3D reconstruction of the eddies of turbulent shear
flows shows the presence of large-scale coherent vortex
structures within the boundary layer (Fig. 4.19). These are
rather far removed from Prandtl's “lumps” of fluid and
more similar to Reynolds' “sinuous” description. They
have the shape of hairpin vortices whose “legs” are formed
from the low-speed streaks (Fig. 4.20). The streak pattern
is quasi-cyclic, with new streaks forming and reforming
constantly across the flow as the hairpin vortices rise up,
advect through the boundary layer and are destroyed in
the outer flow.

4.5.1 More about turbulent eddies

So, we now have evidence from flow visualization that the
“sinuous motion” of Reynolds (Section 4.2) is dominated
by 3D eddy motions with vorticity (Figs 4.18 and 4.19).
Prandtl noted these eddies and from his surface flow
visualization experiments regarded them as moving fluid
“lumps” ( Flussigkeitsballen ) that transferred momentum
throughout the boundary layer. We use this concept to
derive the basic flow law for turbulent flows in Cookies 10
and 11. We would like to know a little more about the
nature of fluid eddies. For example, the following
questions arise:
1 Are turbulent eddies coherent entities?
2 How do eddies relate to momentum transfer and tur-
bulent stresses in turbulent flows?
Looking at the various flow visualizations in Fig. 4.18b-f
(see also Sections 4.2 and 4.3), we observe that:
In the xz plane:
1 Relatively fast fluid is displaced downward from the outer
to the inner flow region in curved 3D vortices. These are
termed sweep motions.

4.5.2 The distribution of velocity in turbulent flows

We previously derived the distribution of velocity in
laminar flow. We now need to make a similar attempt for
turbulent flows, where inevitably the situation is more
complicated and we must take advantage of both experi-
mental observations and intuition. For example, at a flow
boundary, eddies will be vanishingly small since here the
velocity components, u and v , must be zero because of the
no-slip boundary condition (Section 4.3) and for two-
dimensional (2D) flow in the xz -plane no net vertical
velocity is possible. In this region we would expect viscos-
ity to still dominate flow resistance. Experiments confirm
this, establishing a linear increase of velocity with height
and negligible turbulent shear stresses just above the
boundary. This very thin zone of flow closest to the bed
(Figs 4.21 and 4.22) is known as the viscous sublayer .