$\text {The pressure drop in $10 \mathrm {~m}$ of a turbulent flow in a pipe is $200 \mathrm {~Pa}$. The time-average velocity,}$ $\text {in the pipe at three radial locations, is measured to be:}$
$$\begin{array} { c c c c }
\bar { u } ( \mathrm {~m} / \mathrm { s } ) & 7.24 & 7.07 & 6.72 \
r ( \mathrm {~mm} ) & 23 & 25 & 27
\end{array}$$
$\text {The eddy viscosity at $r = 25 \mathrm {~mm}$ is nearest (neglect the kinematic viscosity):}$
A) $0.00163 \mathrm {~m} ^ { 2 } / \mathrm { s }$
B) $0.00181 \mathrm {~m} ^ { 2 } / \mathrm { s }$
C) $0.00192 \mathrm {~m} ^ { 2 } / \mathrm { s }$
D) $0.00227 \mathrm {~m} ^ { 2 } / \mathrm { s }$

Question

$\text {The pressure drop in $10 \mathrm {~m}$ of a turbulent flow in a pipe is $200 \mathrm {~Pa}$. The time-average velocity,}$ $\text {in the pipe at three radial locations, is measured to be:}$
$$\begin{array} { c c c c } 
\bar { u } ( \mathrm {~m} / \mathrm { s } ) & 7.24 & 7.07 & 6.72 \
r ( \mathrm {~mm} ) & 23 & 25 & 27
\end{array}$$
$\text {The eddy viscosity at $r = 25 \mathrm {~mm}$ is nearest (neglect the kinematic viscosity):}$
A) $0.00163 \mathrm {~m} ^ { 2 } / \mathrm { s }$
B) $0.00181 \mathrm {~m} ^ { 2 } / \mathrm { s }$
C) $0.00192 \mathrm {~m} ^ { 2 } / \mathrm { s }$
D) $0.00227 \mathrm {~m} ^ { 2 } / \mathrm { s }$

Quizplus · Accepted Answer

$0.00192 \mathrm {~m} ^ { 2 } / \mathrm { s }$
$\text {The relation between the eddy viscosity and the pressure gradient is given by Eq.}$$\text { 7.6.9, which provides}$
$$- \frac { r } { 2 } \frac { \Delta \bar { p } } { L } = \eta \frac { d \bar { u } } { d r } . \quad - \frac { 0.025 } { 2 } \times \frac { 200 } { 10 } = \eta \frac { 6.72 - 7.24 } { 0.027 - 0.023 } . \quad \therefore \eta = \underline { 0.00192 \mathrm {~m} ^ { 2 } / \mathrm { s } }$$
$\text {We used central differences to approximate $d \bar { u } / d r$ along with $d \bar { p } / d x = - \Delta \bar { p } / L$.}$