Fluid Mechanics Questions & Answers  
Please ask your questions related to Fluid Mechanics in this thread, I will answer them as soon as possible. To insert mathematics use LATEX. For instance, let's say we wish to insert math within a sentence such as $C_f$ is equal to $\tau_w /\frac{1}{2} \rho q_\infty^2 $. This can be done by typing \$C_f\$ is equal to \$\tau_w /\frac{1}{2} \rho q_\infty^2\$. Or, if you wish to display an equation by itself out of a sentence such as: $$ \frac{d}{dt}\int_V \rho dV + \int_S \rho (\vec{v} \cdot \vec{n}) dS =0 $$The latter can be accomplished by typing \$\$\frac{d}{dt}\int_V \rho dV + \int_S \rho (\vec{v} \cdot \vec{n}) dS =0\$\$. You can learn more about LATEX on tug.org. If the mathematics don't show up as they should in the text above, use the Chrome browser instead of Internet Explorer. Ask your question by scrolling down and clicking on the link “Ask Question” within the page footer. 













In this problem, it is stated that the center velocity is measured using a pitotstatic tube. So, in this case what is measured is not $q_\infty$ but $u$ in the middle of the pipe. Recall from the class notes that we derived an expression for $u$ as a function of $u_{\rm b}$ for fullydeveloped laminar flow in a pipe:$$ \frac{u}{u_{\rm b}}=2\left(1\frac{r^2}{R^2} \right) $$ If the flow is laminar, then use the latter to find $u_{\rm b}$ from the center velocity (velocity at ${r=0}$). For fullydeveloped turbulent flow, recall that the velocity profile in the pipe is almost uniform and varies only near the walls. Therefore, if the flow is turbulent, you can assume that the velocity in the center of the pipe is equal to the bulk velocity. Also, keep in mind that the flow in the pipe is air, not H$_2$O. Therefore, you cannot use directly the expression $u_{r=0}=\sqrt{2g\Delta {H}}$ to find the center velocity. You need to draw the Pitot tube, analyze the problem with Pascal's law, and then find out a new expression for the center velocity as a function of the height difference. Hint: the height difference in mm H$_2$O is proportional to the pressure difference between the pitot and static tubes, which itself is related to the difference between the stagnation and static pressure of the air.





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