Turbulent flow of a fluid of density and kinematic viscosity is driven along the…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Instability, transition and turbulence · Instability, transition and turbulence

Fluid Dynamics, Ch Transition to turbulence

Problem

Turbulent flow of a fluid of density

\rho

and kinematic viscosity

\nu

is driven along the

x

-direction between two parallel plates a distance

2h

apart, by a constant mean pressure gradient

d\bar p/dx

. The flow is steady and fully developed, so every mean quantity depends only on

y

, the coordinate measured from the mid-plane (

-h\le y\le h

), the mean flow is purely in the

x

-direction, and by the symmetry of the geometry both

d\bar u/dy

and

\overline{u'v'}

vanish at

y=0

. ((a)) Starting from the mean streamwise momentum equation

0 = -\frac{1}{\rho}\frac{d\bar p}{dx} + \frac{d}{dy}\left(\nu\frac{d\bar u}{dy} - \overline{u'v'}\right),

integrate once and use the stated symmetry at

y=0

to show that the total (kinematic) shear stress

\tau(y)/\rho \equiv \nu\, d\bar u/dy - \overline{u'v'}

is the linear function

\tau(y) = (d\bar p/dx)\,y

, and hence that its magnitude is

|\tau(y)| = \tau_w\,|y|/h

, where

\tau_w = |\tau(\pm h)|

is the magnitude of the stress at the walls

y=\pm h

. ((b)) Water (

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

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

) flows between plates with half-gap

h = 1.5\,\mathrm{cm}

, and the wall shear stress is measured to be

\tau_w = 2.9\,\mathrm{Pa}

. Find the friction velocity

u_\tau = \sqrt{\tau_w/\rho}

, the distance from the mid-plane at which the total stress has fallen to half its wall value, and the estimated thickness

5\nu/u_\tau

of the wall region within which viscous stress dominates. Comment on what comparing these two lengths shows about where the Reynolds stress carries the momentum flux.

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