Page 370 - Civil Engineering Formulas

HYDRAULICS AND WATERWORKS FORMULAS 299

where F Froude number (dimensionless)
V velocity of fluid, ft/s (m/s)
L linear dimension (characteristic, such as depth or diameter), ft (m)
2
2
g acceleration due to gravity, 32.2 ft/s (9.81 m/s )
For hydraulic structures, such as spillways and weirs, where there is a rapidly
changing water-surface profile, the two predominant forces are inertia and gravity.
Therefore, the Froude numbers of the model and prototype are equated:
F m F p V m Vp (12.10)
L m g L p g

where subscript m applies to the model and p to the prototype.
The Reynolds number is
VL
R (12.11)

2
2
R is dimensionless, and is the kinematic viscosity of fluid, ft /s (m /s). The
Reynolds numbers of model and prototype are equated when the viscous and
inertial forces are predominant. Viscous forces are usually predominant when
flow occurs in a closed system, such as pipe flow where there is no free surface.
The following relations are obtained by equating Reynolds numbers of the
model and prototype:
V m L m V p L p v r
V r (12.12)
v m v p L r
The variable factors that fix the design of a true model when the Reynolds
number governs are the length ratio and the viscosity ratio.
The Weber number is
2
V L
W (12.13)

4
2
4
2
where density of fluid, lb s /ft (kg s /m ) (specific weight divided by g);
2
and surface tension of fluid, lb/ft (kPa).
The Weber numbers of model and prototype are equated in certain types of
wave studies.
For the flow of water in open channels and rivers where the friction slope is
relatively flat, model designs are often based on the Manning equation. The
relations between the model and prototype are determined as follows:
2/3 1/2
V m (1.486/n m )R m S m
(12.14)
2/3 1/2
V p (1.486/n p )R p S p
1/3
where n Manning roughness coefficient (T/L , T representing time)
R hydraulic radius (L)
S loss of head due to friction per unit length of conduit (dimensionless)
slope of energy gradient