A thin instrumented fin of length and large span is towed lengthwise through still…

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

Fluid Dynamics, Ch The boundary layer: Prandtl's equations and the Blasius solution

Problem

A thin instrumented fin of length

L = 1.0\,\text{m}

and large span is towed lengthwise through still water at constant speed

U = 0.20\,\text{m s}^{-1}

, so that in the fin's own frame a uniform stream of speed

U

passes over each face with a laminar boundary layer and no pressure gradient imposed by the outer flow. Water properties are

\rho = 998\,\text{kg m}^{-3}

\mu = 1.00\times10^{-3}\,\text{Pa s}

. ((a)) State the condition on

\partial p/\partial y

that the boundary-layer scaling gives for this flow, and use it together with the definition of the momentum thickness

\delta_2

to write the momentum integral equation for zero pressure gradient in the form

\tau_w = \rho U^2\,d\delta_2/dx

. ((b)) Using the Blasius result

\delta_2(x) = 0.664\sqrt{\nu x/U}

, integrate

\tau_w

from the leading edge to

x = L

to obtain the drag per unit width on one face of the fin, and show that the drag coefficient

C_D = D/(\tfrac12\rho U^2 L)

obeys

C_D = 1.328/\sqrt{Re_L}

. ((c)) Compute

Re_L

, check that the layer is laminar over the whole fin (transition is not observed below

Re_x \approx 3\times10^{5}

), and evaluate

C_D

and the drag per unit width on one face.

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