A ship's hull of length moves through still water at (kinematic viscosity of water )

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

Fluid Dynamics, Ch Scaling, similarity and similarity solutions

Problem

A ship's hull of length L=10 mL=10\,\mathrm{m} moves through still water at U=10 m s−1U=10\,\mathrm{m\,s^{-1}} (kinematic viscosity of water ν=1.00×10−6 m2 s−1\nu=1.00\times10^{-6}\,\mathrm{m^2\,s^{-1}}), giving Re=UL/ν=1.0×108Re=UL/\nu=1.0\times10^{8}. A colleague argues: \"The viscous term in the non-dimensional Cauchy equation carries the factor Re−1=1.0×10−8Re^{-1}=1.0\times10^{-8} in front of it, so the true (viscous) velocity differs from the inviscid potential-flow velocity by only about one part in 10810^8 at every point in the flow.\" ((a)) State what the no-slip condition requires of the fluid velocity at the hull's wetted surface, and what the inviscid solution predicts there instead. ((b)) Use these two values to compute the actual fractional difference between the viscous and inviscid velocities at the wall, and compare it with the claimed 10−810^{-8}. ((c)) Estimate the thickness of the region next to the hull where the two solutions differ by order one, and explain in one sentence why the size of the coefficient in front of the highest derivative in an equation does not by itself measure the error everywhere.

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