680 MODELS IN HYDRAULIC ENGINEERING
similarity. The main exceptions are distorted models of rivers, river train-
ing and coastal engineering projects (Section 16.1.4).
The main causes of scale effects are model roughness and model
approach conditions associated with turbulent boundary layer develop-
ment, surface tension effects and associated aeration and vortex-formation
problems and cavitation phenomena. Some of these scale effects can be
overcome, or at least minimized, by using model scales giving sufficiently
high model Reynolds numbers (which are reduced against the prototype
when using the Froude scaling law by M3/2l for the same model viscosity as
in prototype) and Weber numbers (reduced on the model for the same
liquid as in prototype by M^2 l).
For example, for correct extrapolation of the shape of a nappe the
head on a sharpcrested notch should be at least 60 mm; for a head below
20 mm the overflow parabola of the free jet (see Section 4.7) is deformed
almost in to a straight line. Hence it is advisable to choose a spillway
model scale so that the head on the spillway crest in the model is at least
60 mm (for the maximum discharge). Similar conditions apply for the
shape of the outflow under a gate where the minimum gate opening should
also be about 60 mm. The diameter of model bottom outlets should prefer-
ably be larger than 50 mm to avoid scale effects in the entry loss coeffi-
cients. The Reynolds number for smooth models should be such that it
corresponds to the fully turbulent hydraulically rough prototype value to
obtain the correct friction losses. Thus, for a pipe with a relative roughness
of k/D0.001, the friction factor is independent of the Reynolds
number for ReVD/v 106. To achieve the same value of for a very
smooth model pipe we need a Reynolds number of about 70 000 (see
Moody diagram). Thus if the prototype Reynolds number is 10^7 , from
M3/2l 107 /(7 104 ), the model scale should be Ml27.5 – a condition
which may be difficult to fulfil. On models of outlet works with spillway
and bottom outlet(s) we may thus have to accept some minor scale effects
in the reproduction of the bottom outlet(s) performance. If flow in open
channels is involved, the Reynolds number on the model (ReVR/v)
should be greater than
Table 16.1 Scale factors
Parameter Scale factor Parameter Scale factor
Velocity MM1/2l Area MAM^2 l
Volume MVM^3 l Mass Mm(M )M^3 lM^3 l
Time MtM1/2l Discharge MQM5/2l
Force MP(M )M^3 lM^3 l Specific discharge Mq M3/2l
Pressure MP(M )MlMl Energy ME(M )M^4 lM^4 l
(intensity) Momentum MM(M )M7/2l M7/2l
