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 mb.pejvak May 4, 2013 03:04

correct formula of Cf

hi friends;
I face a mind-boggling problem in calculation of Cf (skin friction coef.) over a flat plate which is imposed by a non-zero 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

 truffaldino May 4, 2013 05:15

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

 mb.pejvak May 4, 2013 10:26

1 Attachment(s)
Quote:
 Originally Posted by truffaldino (Post 425133) Cf is computed relatively U at infinity (i.e. free stream velocity Uinf). It is quite logical since, due non-slip condition, velocity at any point of surface of the body/plate is zero and Uinf(i) is associated with i-th point of the surface only in the bl approximation, while Cf can be used in more general situation.
Dear truffaldino;

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?

 truffaldino May 5, 2013 05:26

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

 mb.pejvak May 5, 2013 11:56

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.

 mb.pejvak May 6, 2013 20:04

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.

 truffaldino May 7, 2013 03:10

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?

 mb.pejvak May 7, 2013 10:35

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 non-zero 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?

 truffaldino May 8, 2013 03:55

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. 12-51, 12-52). Here U(x)=const*x^m

I have just googled Falkner-Skan solutions and Friction Coefficient.

P.S. Sometimes, in engeneeric applications, people uses Cf defined through Uinf, rather than through Uinf(i). For instance X-foil 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)."

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