Forces and dynamics - Physical Processes in Earth and Environmental Sciences - page 71

(c)

(a)

Dials indicate
> pressure clockwise

Separation point

Separated wake

Leeside flow separation

(b)

Fig. 3.55 (a) Many situations occur in nature where downstream slope gradients lead to a decrease of mean flow velocity. Desert and subaqueous
dunes and ripples are obvious examples. By Bernoulli's equation we can predict that such decelerations should cause a rise in the fluid pressure
or flow work. Given the right conditions this downstream increase in pressure can be sufficient to reverse the flow locally and cause flow
separation: see sketches below; (b) Time-mean upstream recirculating flow in the leeside; (c) Pressure guages measure downstream increases in
pressure over the steeper leeside slope. This negative pressure gradient causes upstream flow close to the boundary and hence flow separation.

turbulent mixing occurs. Nevertheless, the geostrophic
approximation has enabled major progress in understand-
ing the large-scale oceanic circulation substantially
affected by the Coriolis force:

incompressible, nonrotating, straight-line flow, such as in a
straight river channel or along a local wind, as

F F (pressure) F (gravity) F (Viscous)

mass acceleration

F F (pressure) F (gravity) F (Coriolis)

Turbulent friction flows: Reynolds' approach : As we have
seen in the previous chapter, Reynolds' neatly decon-
structed turbulent flow velocities into mean and fluctuat-
ing components. The latter are responsible for a very large
increase in the resisting forces to fluid motions on account
of the immense accelerations produced in the flow. Thus
through most of the flow thickness these fluctuating tur-
bulent forces dominate over viscous frictional forces.
However, as we shall see subsequently, there still remain
strong residual viscous resisting forces close to any flow
boundary and so we keep the viscous contribution in the
equation of motion for turbulent flows, written here for a
simple case of straight channel flow:

mass acceleration

In situations of steady flow with no acceleration, where
there are no density changes and where gravity is balanced
in the hydrostatic condition, this expression becomes an
equality between the pressure and Coriolis forces:

F (pressure) F (Coriolis)

Viscous friction flows: Navier-Stokes approach : The incor-
poration of frictional resistance via viscous forces into the
Euler-Bernoulli versions of the equations of motion was a
major triumph in science, attributed jointly to Navier and
Stokes. Refer back to Section 3.10 for an account of the
derivation of the viscous stress and the net viscous force
resulting in a flow boundary layer. The simplest form of
the Navier-Stokes equation may then be written for an

F F (pressure) F (gravity)

F (Viscous) F (turbulent)

mass acceleration

3.13 Solid stress

We have seen (Section 3.3) that vectorial force, F , is
defined in classical Newtonian physics as an action which
tends to alter the state of rest or uniform straight line
velocity, u , of any object of a certain mass, m . Also, forces

can be defined in terms of changes in momentum, p m u

in time or space, F d( m u )/d t . In relation to the solid

deformation accompanying plate tectonics it is more likely
that spatial changes in velocity are responsible, produced