Question by Student 201312171 Professor, thank you for your awesome teaching. Sir, in this morning, I've learned that the Kelvin-Helmholtz instability occurs when two viscous flows, which have highly different speeds each, meet and when Reynolds number is near ${10}^{4}$;(you showed some examples with cylinders in flows with different Reynolds numbers.) Then, I'm curious about whether the Kelvin-Helmholtz instability can occur in transition region on the flat plane. In my thought, it can occur within a flow of which Reynolds number is near ${10}^{4}$ because boundary layer right before the transition is so thin that stiff rate of change of speed along the depth may cause a kind of mixing.
 10.19.16
Yes you are right, something similar to the Kelvin-Helmholtz instability occurs within the transition region of a boundary layer on a flat plate. But, the physical phenomenon is not the same because there is no clear shear layer separating two streams within a boundary layer. Rather, the boundary layer becomes turbulent often (but not always) due to the Tollmien-Schlichting instabilities. Those are rolling 2D vortices that form on the edge of the laminar boundary layer and that are amplified through sound waves reflecting off the nearby wall. I'll give you 2 points bonus boost for your question.
 Question by Student 201227103 Solving assignment #5, question #5, I have a question about relation between pressure and mass flow rate. As along as dimeter does not change $\frac{dP}{dx}$ is constant in Fully-Developed region however if pipe condition changes the mass flow rate will be changed compared to previous mass flow rate. Then bulk velocity and Reynolds Number will change. I'm wondering if I could assume $\frac{dP}{dx}$ constant nonetheless pipe condition changes. For example, rusted pipe to smooth pipe.
 11.15.16
Yes, you can assume $dP/dx$ to be the same when solving the rusted pipe and the smooth pipe. For this problem the pump can be assumed to yield constant pressure increase, and this pressure increase leads to a certain $dP/dx$ in the pipe which won't change as long as the pressure at the pipe exit and the pressure of the fluid entering the pump doesn't change. In this case, the pressure of the fluid entering the pump as well as the pressure at the exit of the pipe are atmospheric, hence why $dP/dx$ can be assumed not to vary. I'll give you 2 points bonus boost.
 Question by Student 201427115 Professor, I have a question about assignment #5, Question #6 (f). How we can know Re range if pressure change ten times higher at turbulent flow? from tables, friction factor is defined differently along range of Re at turbulent flow.
 11.16.16
You can proceed iteratively by trying out a correlation and checking if the correlation is valid once the answer is obtained. Or you can use the Moody diagram instead of using the correlations. I'll give you 1 point bonus boost.
 Question by Student 201227103 From Question part#3 (b), i cannot understand the meaning of 'grid-converged' and 'coarser grid'. I tried to found this meanings on google but i didn't get proper answer. Also I understood that the meaning of 0.1% variable is comparing values of ub from Prandtl versus Jones solution. Is it correct meaning?
 12.06.16
This is a good question. What you have to do is to make a table of $u_{\rm b}$ as a function of the number of nodes as follows:
 $N$ $u_{\rm b}$ 100 .. 200 .. 400 .. 800 .. 1600 .. 3200 .. etc.. ..
The solution is “grid-converged” on a grid level if $u_{\rm b}$ is within 0.1% of the one on the next-coarser grid level (a coarser grid is a grid with fewer nodes). For example, if the difference between $(u_b)_{N=800}$ and $(u_b)_{N=400}$ is less than 0.1%, then the solution is grid-converged at $N=800$. I'll give you 1.5 points bonus boost for your question.
 Question by Student 201427115 professor, I have a question about assignment5 #5 (b). I got answer of (a) using shear stress at wall. i'm wondering is it okay to use same shear stress at wall of (a) to (b). I think that shear stress is change with respect to eddies.
 12.18.16
If you use the same shear stress at the wall in (a) and (b), then you need to prove why it is the same (if it is!). You can start your proof by looking at the forces acting on a fluid element within the fully developed region. It's a good question, but a bit late and not so well typeset.. I'll give you 1 point bonus boost.
 12.19.16
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