Question about the characteristic length calculation
Hi
When we need to calculate the Reynolds number, we need to know the characteristic length firstly. May i ask how to calculate the characteristic length? 
Characteristic length is the length of the obstacle in the flow field. The conventions vary from field to field so characteristic length can be: diameter of a cylinder around which fluid flows, length of a vehicle, length of the wheelbase of a vehicle, height of an obstacle etc. All in all, it's not really calculated, it's just a value you take as a factor because others years ago agreed on it. :)
In fact, most of the time it's listed on what is the Re based such as "Reynolds number in relation to the height of the barrier", or a similar quotation with "length" or "diameter". 
Thanks. I have fluid flow through a cube. The length l_x is 100. The width is l_y=10. The height l_z=0.01. The fluid flow at y direction. May I ask what's the characteristic length for calculating the Re?
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If I've understood your problem correctly, you are interested in flow through a rectangular channel/pipe. In this case (and this only holds true for turbulent flow) you need to calculate the "equivalent hydraulic diameter" for which I've written the formula and drew a rough sketch in this picture:
http://i.imgur.com/ImM4g.png I assume your units are meters (since they ought to be for calculating the Reynolds number), so just put in 0.02 in the . This is the basis for Re calculation in nonround hydraulic calculation of pipelines and again, it's only valid for turbulent flows. Convetion is (as far as I'm aware) that internal flows with Re<2300 are considered laminar, period between Re 2300 and 4000 can be laminar but can easily transition into turbulent flow, and for Re>4000 you can almost be sure when you say it's fully turbulent. 
Thank you very much for your answer. It's very helpful! The unit is actually micrometer. One question: we need to know the characteristic length firstly, then we can calculate the Re. Then we can know if it's turbulent flow or not. If I don't know whether it's turbulent or not beforehand, shall I still use this formula to detect the regime of the flow?
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You can use this formula, but if the calculated Re is below 2300 then the validity of the result is "questionable". :)

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Just some facts:
Re is a nondimensional number > you just need to use consistent units. If the length is based on micrometers than you just have to add 1e6 factor or use consistent units for the other quantities. Re number and its effects exist independently from the state of the flow. For incompressible isothermal flows, Re number is the only parameter affecting the flow (actually you can define many of them based on different length scales). If for certain ducts with a certain roughness the transition is established around some Re number... well than that's it. You compute that Re number and, according to its value you know if you are turbulent or not. It's completely non sense to discard the Re number if it comes out to be below some treshold value. At most, if it's in the transitional regime you can have some doubts about the flow being laminar or not, but the Re number is definetely remaining with the same value. Hydraulic diameter is just a reference length. There is no way for it to become invalid in defining a Re number just because it comes out too low. 
My purpose is to know whether my flow is laminar or turbulent. Do you mean i can completely use hydraulic diameter as the characteristic length to calculate Re to know my regime?
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Long story short: YES.
More correct way to express my way of thinking: "if you have some rule (based on experimental evidence or whatever) which tells you that, for a certain flow, when Re (based on a certain length, which can be the hydraulic diameter, it usually is but depends on the rule) is below some number then the flow is laminar" that rule is necessarily valid for all the values which can possibly come out from the Re number defined above. Does this sentence makes sense to you: "if Re is above X the flow is turbulent, otherwise i don't know"? And this one:" if Re is above X the flow is turbulent; if Re is below Y the flow is laminar; if Re is between Y and X the flow is transitional"? If the last one makes sense to you (i hope so), do you expect the "Re" appearing in it being defined always in the same way all the times it appears in the sentence, or i should have written something like: " if Re1 is above X the flow is turbulent; if Re2 is below Y the flow is laminar; if Re3 is between Y and X the flow is transitional"? I imagine my point is clear now. Going to your problem, that rule of thumb exists for duct flows and is based on the hydraulic diameter. Do i expect it to be accurate enough independently from the real shape of the duct section and its roughness? Hell no, it's a rule of thumb based on pure experimental evidence, not a theory. 
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Thank you. May I ask you another question? If I have a very complex geometry and I have no experience about the relation between Re and the flow regime, then how to decide whether laminar model, or turbulent mode, should be used for this case? Thank you very much! 
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