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May 4, 2013, 03:04 
correct formula of Cf

#1 
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Mehdi
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hi friends;
I face a mindboggling problem in calculation of Cf (skin friction coef.) over a flat plate which is imposed by a nonzero pressure gradient. this pressure gradient implement by changing domain width that cause each section has an specific velocity in its freestream. now my question is how should I calculate Cf. should I used velocity (U) in each section or velocity at leading edge? Cf(i)=wall shear stress/(0.5Uinf^2) or Cf(i)=wall shear stress/(0.5Uinf(i)^2) Uinf: freestream velocity at leading edge Uinf(i): freestream velocity at each section thanks in advance;
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Best Regards; Mehdi Email: mb.pejvak[at]Gmail[dot]com 

May 4, 2013, 05:15 

#2 
Senior Member

Cf is computed relatively U at infinity (i.e. free stream velocity Uinf). It is quite logical since, due nonslip condition, velocity at any point of surface of the body/plate is zero and Uinf(i) is associated with ith point of the surface only in the bl approximation, while Cf can be used in more general situation.


May 4, 2013, 10:26 

#3  
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Mehdi
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Quote:
Thank you for your reply. I know what you mean, but I did not mean Uinf(i) is velocity in boundary layer. the attached pic. shows a flat plate with variable domain width. by considering continuity equation, velocity on freestream should be changed. the pic shows freestream velocity in each section. my question is Cf should be calculated by using velocity at leading edge or Cf in each section should make use of Uinf at each section?
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Best Regards; Mehdi Email: mb.pejvak[at]Gmail[dot]com 

May 5, 2013, 05:26 

#4 
Senior Member

Dear Mehdi,
Sorry, I did not understand what you meant. Everything depends on your purpoce. For instance, momentum integral equation uses Cf which is defined relatively local velocity Uinf(i). This is convenient to connect Cf with shape factors etc. I think it is your case. Truffaldino 

May 5, 2013, 11:56 

#5 
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Mehdi
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Thank you truffaldino. I think I found the correct answer. because the formula is
Cf=Wall shear stress/ dynamic pressure, and dynamic pressure is obtains from Bernouli Equation (ru*u(x)*du(x)/dx=dp/dx), so after integration from this equation, Pressure dynamic= 0.5*U(x)**2.0 so for calculation of Cf , U should be the velocity of each section (Uinf(i)). I think it is reasonable, because it is obtain from the NS equation over a stream line.
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May 6, 2013, 20:04 

#6 
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Mehdi
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yesterday I argued with one of my friends about this problem. he mentioned a point which I had not considered it. he said that if in calculation of Cf variable Uinf (Uinf(i)) is used, effect of pressure gradient will also enter in the calculations and it is not reasonable. this point caused me to think more about this problem, but unfortunately I have not reach any clear and reasonable result. because as I mentioned before, using Uinf(i) was achieved from Bernoulli equation, and in another side, this point can pose a problem in results.
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Best Regards; Mehdi Email: mb.pejvak[at]Gmail[dot]com 

May 7, 2013, 03:10 

#7 
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Dear Mehdi,
Cf that is defined through Uinf(i) is routinely used in the boundary layer theory see e.g. pg 15 of http://www.engineering.uiowa.edu/~me...lec32_2012.pdf In integral momentum equations there is no problem with gradients. Truffaldino P.S. Could you tell why do you need Cf, what is the purpoce of your work? 

May 7, 2013, 10:35 

#8 
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Mehdi
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I need it ti evaluate results of my code in modeling ERCOFTAC T3 cases as benchmarks in modeling transition regime and specifically for T3C that is flat plate faces a nonzero pressure gradient.
today I asked it from a person how wrote a paper related to this subject and he confirmed your point, but I have not faced it. in the file you sent me, where Cf was defined is U= cons in X direction, because it is wrote d(U^2*theta)/dx=U^2*d(theta)/dx and in the case U=U(x) Cf is not defiend. isn't it?
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Best Regards; Mehdi Email: mb.pejvak[at]Gmail[dot]com 

May 8, 2013, 03:55 

#9 
Senior Member

Perhaps I gave you not the best reference, see, for instance, this one (which considers exact solution to a problem that is similar to yours)
http://www.cambridge.org/us/engineer...sformation.pdf (especially Eqs. 1251, 1252). Here U(x)=const*x^m I have just googled FalknerSkan solutions and Friction Coefficient. P.S. Sometimes, in engeneeric applications, people uses Cf defined through Uinf, rather than through Uinf(i). For instance Xfoil uses such a definition. They write in their manual "This differs from the standard boundary layer theory definition which uses the local Ue rather than Qinf for the normalization. Using the constant freestream reference makes Cf(x) have the same shape as the physical shear stress tau(x)." Last edited by truffaldino; May 8, 2013 at 04:17. 

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