Page 87 - Fundamentals of Geomorphology

70 INTRODUCING LANDFORMS AND LANDSCAPES

Box 3.4

REYNOLDS AND FROUDE NUMBERS

Reynolds number is a dimensionless number that where v is the flow velocity, g is the acceleration of
includes the effects of the flow characteristics, veloc- gravity, d is the depth of flow, and gd is the veloc-
ity, and depth, and the fluid density and viscosity. It ity of the gravity waves. When F < 1 (but more than
may be calculated by multiplying the mean flow veloc- zero) the wave velocity is greater than the mean flow
ity, v, and hydraulic radius, R, and dividing by the velocity and the flow is known as subcritical or tran-
kinematic viscosity, ν (nu), which represents the ratio quil or streaming. Under these conditions, ripples
between molecular viscosity, µ (mu), and the fluid propagated by a pebble dropped into a stream create
density, σ (rho): an egg-shaped wave that moves out in all directions
from the point of impact. When F = 1 flow is crit-
ρνR ical, and when F > 1itis supercritical or rapid or
R e = shooting. These different types of flow occur because
µ
changes in discharge can be accompanied by changes
in depth and velocity of flow. In other words, a given
For stream channels at moderate temperatures, the discharge can be transmitted along a stream channel
maximum Reynolds number at which laminar flow is either as a deep, slow-moving, subcritical flow or else
sustained is about 500. Above values of about 2,000, as a shallow, rapid, supercritical flow. In natural chan-
flow is turbulent, and between 500 and 2,000 laminar nels, mean Froude numbers are not usually higher
and turbulent flow are both present. than 0.5 and supercritical flows are only temporary,
The Froude number is defined by the square root of since the large energy losses that occur with this type
the ratio of the inertia force to the gravity force, or the of flow promote bulk erosion and channel enlarge-
ratiooftheflowvelocitytothevelocityofasmallgravity ment.This erosion results in a lowering of flow velocity
wave (a wave propagated by, say, a tossed pebble) in still and a consequential reduction in the Froude num-
water. The Froude number is usually computed as:
ber of the flow through negative feedback. For a fixed
velocity, streaming flow may occur in deeper sections
v
of the channel and shooting flow in shallower
gd sections.
F =

named after English scientist and engineer Osborne kind of turbulent flow. It occurs at a waterfall, when
Reynolds, may be used to predict the type of flow water plunges in free fall over very steep, often verti-
(laminar or turbulent) in a stream (Box 3.4). cal or overhanging rocks. The water falls as a coherent
In natural channels, irregularities on the channel bed mass or as individual water strands or, if the falls are very
induce variations in the depth of flow, so propagating high and the discharge low, as a mist resulting from the
ripples or waves that exert a weight or gravity force. The water dissolving into droplets.
Froude number, F, of the flow, named after the English Flow velocity controls the switch between subcrit-
engineer and naval architect William Froude, can be used ical and supercritical flow. A hydraulic jump is a
to distinguish different states of flow – subcritical flow sudden change from supercritical to subcritical flow.
and critical flow (Box 3.4). Plunging flow is a third It produces a stationary wave and an increase in water