A submarine of overall length m cruises fully submerged at m s in sea water of…

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

A submarine of overall length

L_p = 12

m cruises fully submerged at

U_p = 4.0

m s

^{-1}

in sea water of density

\rho = 1025

kg m

^{-3}

and dynamic viscosity

\mu = 1.2\times10^{-3}

Pa s. Treat the flow as steady and incompressible, and take every model mentioned below to be geometrically similar to the boat. Reynolds numbers here are built on the overall length. ((a)) The drag

F

is taken to depend only on

L

U

\rho

and

\mu

. Using the Buckingham Pi counting rule, how many independent dimensionless groups can be formed from these five quantities? ((b)) Estimate the size of the inertial term of the Navier-Stokes equation as

\rho U^{2}/L

and of the viscous term as

\mu U/L^{2}

for the full-size boat, and give the ratio of the first to the second. ((c)) A

1/8

scale model is towed in the same sea water. What speed does it need for its Reynolds number to equal the boat's?

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