CIVIL ENGINEERING FORMULAS

HYDRAULICS AND WATERWORKS FORMULAS 299

whereFFroude number (dimensionless)
Vvelocity of fluid, ft/s (m/s)
Llinear dimension (characteristic, such as depth or diameter), ft (m)
gacceleration due to gravity, 32.2 ft/s^2 (9.81 m/s^2 )

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:

(12.10)

where subscript mapplies to the model and pto the prototype.
TheReynolds numberis

(12.11)

Ris dimensionless, and is the kinematic viscosity of fluid, ft^2 /s (m^2 /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:

(12.12)

The variable factors that fix the design of a true model when the Reynolds
number governs are the length ratio and the viscosity ratio.
TheWeber numberis

(12.13)

wheredensity of fluid, lbs^2 /ft^4 (kgs^2 /m^4 ) (specific weight divided by g);
andsurface tension of fluid, lb/ft^2 (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:

(12.14)

wherenManning roughness coefficient (T/L1/3,Trepresenting time)
Rhydraulic radius (L)
Sloss of head due to friction per unit length of conduit (dimensionless)
slope of energy gradient

Vm

Vp



(1.486/nm)R2/3mS1/2m

(1.486/np)R2/3pS1/2p

W

V^2 L



VmLm

vm



VpLp

vp

Vr

vr

Lr

R

VL

FmFp

Vm

Lmg



Vp

Lpg