The overflow spillway of a dam is to be tested as a geometrically similar model at a…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity · Viscous flow and dynamical similarity

Fluids and Continuum Physics, Ch Dimensional analysis, dynamical similarity and the dimensionless groups

Problem

The overflow spillway of a dam is to be tested as a geometrically similar model at a scale of

1{:}36

, in the same water as the prototype (

\rho = 1000

kg m

^{-3}

\mu = 1.0\times10^{-3}

Pa s). At the design condition the water leaves the prototype crest with a mean speed

U_p = 7.2

m s

^{-1}

and the discharge is

Q_p = 900

^{3}

^{-1}

. The flow has a free surface open to the atmosphere. Take

g = 9.8

m s

^{-2}

, and use the definitions

Fr = U/\sqrt{gL}

and

Re = \rho U L/\mu

with

L

the corresponding length of the geometry. ((a)) Which dimensionless group must be matched here, and which two forces does it compare? ((b)) Find the model crest speed that gives dynamical similarity. ((c)) Find the discharge the laboratory pump must supply to the model. ((d)) Evaluate the ratio

Re_m/Re_p

and say what it means for the trustworthiness of the test. ((e)) The prototype flow over the crest is

3.0

m deep. Taking that depth as

L

, and trusting a model only while its

Re

stays above

10^{4}

, find the largest scale ratio

L_p/L_m

at which a Froude-matched model in the same water could still be used.

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