Forces and dynamics - Physical Processes in Earth and Environmental Sciences - page 70

von Karman
vortices

Very low Reynolds number; no separation

Separation point

B

Cylinder axis
normal
to page

Cylinder axis
normal
to page

Separated wake

A

Stagnation
point

High Reynolds Number; separation

By Bernoulli:
Flow velocity at A << Flow velocity at B (evidence of
streakline “bunching”), therefore pressure at A >>
pressure at B

Flow pathlines visualize periodic von Karman vortices formed by
shear at the unstable margins to the separated fluid. They tend to
be shed alternately from one side to the other of the obstacle,
diffusing gradually downstream after intense turbulent mixing

Fig. 3.53 Lateral pressure gradients cause flow separation around obstacles at high particle Reynolds number: an important consequence of
the conservation of energy expressed in Bernoulli's equation.

speed up and then slow downstream. Bernoulli's equation
states that the pressure should decrease in the accelerated
flow section. This decrease of pressure produces a pressure
gradient and a lift force that may reach sufficient magni-
tude to exceed the downward acting weight force and so
cause upward movement. All flight and some forms of sed-
iment transport depend upon this Bernoulli effect for the
conservation of flow energy. When a convexity reaches a
certain critical height, the pressure gradients d p /d x

(a)

Aerofoil axis horizontal

(b)

Axis inclined 5 º

Separation point

0,

upstream, and d p /d x 0, downstream, have the greatest

effect on the lower-speed fluid near to the boundary. This
fluid retarded by the adverse pressure gradient may be
moved upstream at some critical point, a process known as
flow separation . Flow separation creates severe pressure
energy degradation and destroys the even pressure gradi-
ents necessary for lift (Fig. 3.54); a process known as stall
results. Flow separation also occurs when a depression
(negative step; Fig. 3.55) exists on a flow boundary; accen-
tuated erosion results due to energy degradation in the
separation and reattachment zones.
Another application of Bernoulli's equation occurs
when fluid flow occurs within another ambient fluid. In
such cases, with shear between the two fluids, the situation
becomes unstable if some undulation or irregularity
appears along the shear layer, for any acceleration of flow
on the part of one fluid will tend to cause a pressure drop
and an accentuation of the disturbance. Very soon a strik-
ing, more-or-less regular system of wavy vortices develops,
rotating about approximately stationary axes parallel to the
plane of shear. Such vortices are termed Kelvin-Helmholtz
instabilities that are important mixing mechanisms across a
vast variety of scales, from laboratory tube to the Gulf
Stream (Sections 4.9 and 6.4).

Fig. 3.54 In these symmetrical aerofoils, only a slight change
(5 here) in the angle of incidence can cause flow separation.

3.12.4 Real-world flows of increased complexity

For real-world flows of hydraulic, oceanographic, and
meteorological interest several additional terms are rele-
vant, including those for friction (viscous and turbulent),
buoyancy, radial, and rotational forces. We sample just a
few of the various possibilities here, to give the reader an
idea of the richness presented by Nature.
Frictionless oceanographic and meteorological flows: In
the open oceans and atmosphere, away from constraining
boundaries to flow, currents have traditionally been
viewed as uninfluenced by viscous or turbulent frictional
forces. This is because in such regions there was thought
to be very little in the way of spatial gradient to the
velocity flow field and therefore not much in the way of
viscous or turbulent forcing. Clearly this somewhat unre-
alistic scenario is inapplicable in regions of fast ocean sur-
face and bottom current systems, where dominant