Comparsion of different Channels
 

Hey guys,

i have a stupid question but i m kinda confused abt this problem. I have 3 different rectangular channels with different hight and different wide. So for hydrodynamic investigation, is it better to set a few different Reynoldnumbers or to set a constant volumeflow and compare it with the other designs ?

Thank you very much!

abt an answer i would be very happy

Dear Benjamin,

despite the appearance, your problem is really far from obvious. I worked for some time on this topic and i ended up with these conclusions:

1) The first obvious conclusion is that the problem only has two non-dimensional numbers, the Re and the aspect ratio of the section, AR.

2) What is much less obvious is: how are you going to fix the Re number when studying the dependence from AR? That is, from a dimensional point of view, what are you going to change in order to keep Re fixed while the section is changing? There are up to 6 different (non-)dimensional quantities which you can (ideally) manipulate. These are:

- mean wall shear stress over the section perimeter
- driving pressure gradient
- mass flow rate
- mean velocity
- mean wall shear stress * perimeter of the section
- friciton Reynolds number based on the hydraulic diameter

it really depends from how you define your Reynolds number or, in Buckingham theorem parlance, how you choose your control variable
for the velocity (velocity, pressure gradient, etc.)

3) Whatever you want to study in non-dimensional terms for the rectangular channel as a function of AR, it is going to depend from how you define the Re number. Some non-dimensional quantities are less dependent than others from this. Also, the dependence dies out for high AR (i.e., >6).

4) In my case, i was interested in studying the friction velocity in the symmetry plane. I found that i could also define the relative non-dimensional number in 3 different ways, each with its own dependence from the above choices.

Please don't take these as facts as these were just my conclusions from a limited study for a Ph.D. exam. Still, you might want to take them as a possible point of view (and maybe rule it out as possibly wrong). I would pass you my final report but, for some reason i can't remember, it is not in english. So it would not be helpful (firm theoretical facts appear in formula just like pure speculative hypotheses used to buil up a model, and a non native reader could not distinguish them).

Despite the above conclusions (which are, mostly, theoretical speculations) i think you should change the Re number by let varying the quantity you can actually control (it has no point in considering quantities you could not control as "control parameters").

just to spend time ... would be useful to build a modified Re number taking into account for both parameters (h eight, L spanwise extension)?

Reh = U*h/ni

ReL = V*L/ni

maybe the value of the AR implies a diferent choice of characteristic lenght and velocity?

To tell the truth, at first insight I was thinking that the only relevant parameter is the Re number, no matter of the AR value... but in Re you should consider the correct reference quantities...

:confused:

Well, according to a purely geometric reasoning, symmetry would imply that for AR > 2 the zones near the short sides of the section somehow behave independently from those in the middle. At some point (AR>6) these are far enough to not affect the central zone anymore.

Moreover, the side zones are affected by the secondary motions, which probably have their own Re number (based on what velocity scale?).

Thus, to follow your reasoning, i think that the answer is probably yes. Secondary motions have their own length (the shortest side?) and velocity (which one?) scales. The entity of the secondary motions (a secondary Re number?), their distance from the channel centre (AR) and the global major Re number then probably determine the overall interaction.

However, the main issue (which caused me a lot of trouble) is that, beside speculations, Buckingham theorem still says that only one Re exists here (and i had to stick to this, because the exam was on the Buckingham theorem itself!!). Thus, the question is: once you select a velocity scale for your Re, in order to perform an experiment (in which you keep Re fixed while changing AR), what's really happening to the flow physics (besides the obvious effects for large AR)?

I found that none of the quantities mentioned above leads to a Re number which, once fixed in the experiment, leads to a really predictable dependence from the AR. Barely because when one of the above quantities is fixed to enter the Re definition, all the others are likely to change (some of them also a lot). Moreover, the same quantities also affect the nondimensionalization of the quantity you actually want to study as a function of the AR.

Just to make a practical example. In studying the friction velocity in the channel center, you can non-dimensionalize it by the mean velocity, which can also be used to define the Re number. You do the experiment and, for 1<AR<4 what you get is really something unpredictable (this was the original question i was asked during the exam and i was unable to give an answer; more than a year later i came back showing that i couldn't have given one because of this).

Then you do the same experiment by using the mean friction velocity of the section as velocity scale (and the hydraulic diameter of the section as length in the Re definition) and what you get is completely different and much better behaved. Still, the behavior is not monotone and the slope changes sign across AR=4.

This was, in the end, my best shot. Also, for AR->Inf, the plane channel definitions are fully recovered (in my case this also meant that the same quantity could be used to estimate the required number of cells in LES simulations).

Still, notice the contradiction of fixing the mean stress over the section while studying its local value in the channel center. Not really insightful.