You'll have to get it a bit indirectly. For instance if you start with fluid flow through a pipe, the Reynolds Number S = 4 v A p / P u , where v is the mean fluid velocity, A is the cross sectional area of the duct, p is the fluid density, u is the dynamic viscocity of the fluid and P is the perimeter of duct that comes into contact with the fluid.
Your surface roughness would presumably be in the surface of your pipe, so it would enter your S equation through the P variable. Thus, the rougher the surface, the longer the effective perimeter P and also the higher the effective cross sectional area A.
For example, consider a relatively small duct with a smooth surface and then compare that same duct with a rough surface. The area scales with the square of the linear cross section of the duct. Thus the ratio A/p of the rough duct will be larger than a smiliar smooth duct. The larger A/p will increase S, and lead to a higher Reynold's number, effectively increasing the likelihood that the system will enter the transitional or turbulent regime out of the laminar regime.
In other words, the rougher the surface of the duct, the more likely that the fluid moving through it will be turbulent for a given flow. The smoother the surface the more likely that the fluid moving through it will be laminar for a given flow. However, for a very large duct, small differences in surface roughness may have negligible effect on the Reynold's number and the fluid may remain in the laminar flow in large pipes even with very rough surfaces, presuming the mean velocity of the fluid is low enough.
You'll have to get it a bit indirectly. For instance if you start with fluid flow through a pipe, the Reynolds Number S = 4 v A p / P u , where v is the mean fluid velocity, A is the cross sectional area of the duct, p is the fluid density, u is the dynamic viscocity of the fluid and P is the perimeter of duct that comes into contact with the fluid.
Your surface roughness would presumably be in the surface of your pipe, so it would enter your S equation through the P variable. Thus, the rougher the surface, the longer the effective perimeter P and also the higher the effective cross sectional area A.
For example, consider a relatively small duct with a smooth surface and then compare that same duct with a rough surface. The area scales with the square of the linear cross section of the duct. Thus the ratio A/p of the rough duct will be larger than a smiliar smooth duct. The larger A/p will increase S, and lead to a higher Reynold's number, effectively increasing the likelihood that the system will enter the transitional or turbulent regime out of the laminar regime.
In other words, the rougher the surface of the duct, the more likely that the fluid moving through it will be turbulent for a given flow. The smoother the surface the more likely that the fluid moving through it will be laminar for a given flow. However, for a very large duct, small differences in surface roughness may have negligible effect on the Reynold's number and the fluid may remain in the laminar flow in large pipes even with very rough surfaces, presuming the mean velocity of the fluid is low enough.
Respected Michael Wofsey Sir,
Thank you for giving such a very good Answer......
I have been studied performance losses due to surface roughness in Axial turbine. please check the attachment file. i think that theorotical relation between surface roughness and Raynold's number was determined.
As mentioned by Michael Wofsey, there is no direct correlation between surface roughness and Reynold number. However, you can find the indirect correlation between these two through Moody chart that is used to determine friction factor for turbulent region or to use Colebrook formula. In turbulent region, the friction factor is independent of the relative roughness. The chart shows that relative roughness would increase the friction factor and so does the losses due to friction. While as the Reynold number increased, the friction factor would decrease and so does the losses due to friction. It should also be noted that at rough turbulent region, friction factor is independent of the Reynold number. You may check this out in Fundamentals of Fluid Mechanics by Munson Chapter 8 page No. 476 - 478.
Respected Michael Wofsey Sir, here you mention that in the equation of S one of the variable is capital P which is perimeter of circular tube or duct.
but i have dought that the perimeter of smooth tube is (π×Diameter of tube) and for inside corrugated pipe same formula one can use.........
now my question is how the ration of A/P varies for corrugated pipe???
For a perfectly circular pipe, A/P = pi r^2 / 2 pi r = r/2. Now look at the corrugated pipe:
For corrugation, there are two possbilities, that the corrugation ribs run around the circumference of the pipe, and that the corrugation runs the length of the pipe.
First the length corrugation, say that corrugation doubles the average perimeter from (2 pi r) to (4 pi r) but the average area remains pi r^2 ... that would make A/P = r/4, and it could potentially decreases the Reynolds number if other things are equal. That would work for relatively large corrugations, and it's possible that corrugations the length of the pipe could maintain a laminar flow longer than a smooth pipe.
But that's an unusual configuration, the more common one is when the corrugations run normal to the pipe and flow direction, this is conventional corrugated pipe. In this case, Reynold's number is going to be more highly dependent on mean fluid velocity, because the "pockets" will offer very low fluid flow for the fluid that is essentially trapped in there, and depending on the size of the corrugation, you may get something closer to a smooth pipe. (For instance researchers were surprised to find that a pickup truck with a tailgate removed and with the tailgate up had similar drag coefficients, the air became trapped in the closed bed, which allows the faster air to smoother over the top of it.) In this case though, for the S equation, it would probably be a decent first approximation to take the average perimeter and average area (i.e. using the mean diameter from the center of each corrugation) and work from there.
The interplay of the area and perimeter has an optimization point for each geometry. Some testing will be necessary to determine the parameters. If I were to test the model I would probably try to measure the flow rate (related to the mean fluid velocity) as a function of S for different systems.
I think that this is a terrific area of research because corrugated pipe is somewhat new and it's an incredible and promising material. For instance I was able to recently purchase 75 psi, 25-feet of 4-inch corrugated pipe for less than US$20, and the whole length came collapsed into a small length before unfurling it. We currently use these corrugted pipes mainly for drains, but it seems to me that there is a lot more potential applications for these in Developing Nations where the need for water and sanitary infrastructure is extreme, and corr-pipe (i.e. in HDPE) offers the possibility to add this infrastructure at very low cost.
Here, I just want to revise a bit about my previous comment on "In turbulent region, the friction factor is independent of the relative roughness", it should be in laminar region not in turbulent region.
I have attached the file for surface roughness relation with Reynolds Number
If your flow is laminar you do not need to bother about it, in turbulent case you have to calculate the friction of surface
it depends on your geometry. in microscale (ie microfludics) surface roughness can affect the onset of vortex shedding for flow around objects
Let me interject on this roughness discussion. Point is, for a strighforward and empirical relation between Re and r/D see Nikuradse's work tabled down by Moody, sometime during WWII. The Moody chart can be used on any PC (as it was digitized some 30 years ago) by way of interpolation, interation, and the ways how to do it are covered in any Fluid Mechanics 101 course Worldwide. For gases and liquids there are approximate relations available, look up Darcy-Weisbach equation and the rest of the same chapter in any engineering book. Most formulae require iterations, but some do not. However, Mr Wofsey brought in an important factor, that is the surface microgeometry. What we perceive as roughnes may, in fact, be from the fluid flow viewpoint a deterministic roughnes .... and that is a whole new ball game. Given one is very selective some roughness types for some pipe as well as external flows create eddies close to the wall, in fact destroying the viscous sublayer, to the exted that the surface sheer drops substantially. Axial ribs in a pipe are a good exapmple of this, they act as skate blades. Over 20 years ago same trick was used during America's Cup race by Americans by way of putting ribs on a hull of a racing yacht. For some wings balls from a ball bearing attached to the upper surface give amazing results. Finally, shark skin suits used in swimming competitions or strips of a material that looks as sand paper on winngs of several passanger jets testify best to the importance of the surface phenomenon and its intricate characteristics. For non-Newtonian fluids surface roughness problem is open. Sheer could be reduced, but the mechanism seems different, in not totaly awkward.
It`s everywhere. Just google up Nikuradse Moody. Better yet get any Fluid Mechanics 101 textbook. Why a medical doctor needs to know these things? Jerzy
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