Forces and dynamics - Physical Processes in Earth and Environmental Sciences - page 56

Meander bends, R. Wabash,
Grayville, Illinois, USA

A

r = Radius of curvature

v

r

Center of curvature

f

B

u

Angular speed, v = df / dt

100 m Linear speed, u , at any point along AB = rv

Fig. 3.31 The speed of flow in channel bends.

3.7.1 Radial acceleration in flow bends

hence an inward angular acceleration is set up. This
inward-acting acceleration acts centripetally toward the
virtual center of radius of the bend. To demonstrate this,
refer to the definition diagrams (Fig. 3.32). Water moves
uniformly and steadily at speed u around the centerline at
90

Consider the flow bend shown in Fig. 3.31. Assume it to
have a constant discharge and an unchanging morphology
and identical cross-sectional area throughout, the latter a
rather unlikely scenario in Nature, but a necessary restric-
tion for our present purposes. From continuity for
unchanging (steady) discharge, the magnitude of the
velocity at any given depth is constant. Let us focus on
surface velocity. Although there is no change in the length
of the velocity vector as water flows around the bend, that
is, the magnitude is unchanged, the velocity is in fact
changing - in direction. This kind of spatial acceleration
is termed a radial acceleration and it occurs in every
curved flow.

to lines OA and OB drawn from position points A and
B . In going from A to B over time t the water changes

direction and thus velocity by an amount u u B u A

with an inward acceleration, a t . A little algebra

gives the instantaneous acceleration inward along r as
equal to

u /

u 2 / r . This result is one that every motorist
knows instinctively: the centripetal acceleration increases
more than linearly with velocity, but decreases with
increasing radius of bend curvature. For the case of the
River Wabash channel illustrated in Fig. 3.31, the
upstream bend has a very large radius of curvature,
c .2,350 m, compared with the downstream bend, c .575 m.
For a typical surface flood velocity at channel centerline of
u

3.7.2 Radial force

c .1.5 m s 1 , the inward accelerations are 9.6 · 10 4
and 4.5 · 10 3 ms 2 respectively.

The curved flow of water is the result of a net force being
set up. A similar phenomena that we are acquainted with is
during motorized travel when we negotiate a sharp bend
in the road slightly too fast, the car heaves outward on its
suspension as the tires (hopefully) grip the road surface
and set up a frictional force that opposes the acceleration.
The existence of this radial force follows directly from
Newton's Second Law, since, although the speed of
motion, u , is steady, the direction of the motion is con-
stantly changing, inward all the time, around the bend and

3.7.3 The radial force: Hydrostatic force imbalance
gives spiral 3D flow

Although the computed inward accelerations illustrated
from the River Wabash bends are small, they create a flow
pattern of great interest. The mean centripetal acceleration
must be caused by a centripetal force. From Newton's