coupling between vorticity transport eq. and stream function eq. (urgent help)
I am solving impinging jet by vorticity- stream function method. I have doubt how to solve the coupling between vorticity transport equation and stream function equation. I do it in the following way:
1. I solve stream function until its residual becomes less than 10E-6
2. I solve vorticity transport equation once.
3. I repeat step 1 and 2 until vorticity residual for consecutive values becomes less than 10E-5.
It converges for cavity and channel flow problem, But it doesn't converge for impinging jet case. Is my algorithm correct?
Thanks in advance.
This is a classic method for classic problems. Patrick Roache published several papers and two books (his first two I think) on (among other techniques and problems) various ways to solve the stream function - vorticity problem. Early to mid-1970's.
One method required substituting the vorticity equation into the stream function equation (using the defintiions) to eliminate the vorticity altogether, then solving the resulting bi-harmonic equation for the stream function.
My memory is that the application of boundary conditions was critical, both the mathematical statement AND the details of the differencing at the boundaries. And (dimly) that the methods were all dependent on the Reynolds number - method might work for lower but not higher values (or the other way round!).
Good luck with your problem!
Thank you for your response.
I’ve read in other papers that “Patrick Roache” has crucial publications on vorticity-stream function method. Unfortunately I don’t have access to them. Would you please send me if you have access to his book (in pdf file) which is entitled ” Roache PJ. Computational Fluid Dynamics (revised edn). Hermosa: Albuquerque, NM, 1982.” Or any other his publications.
I have been trying to solve Impinging jet problem with this method for two months, but I could not obtain correct results. In other word, I am working on this paper “A. Sivasamy, V.Selladurai and P. Rajesh Kanna, Numerical simulation of two-dimensional laminar slot-jet impingement ﬂows conﬁned by a parallel wall Int. J. Numer. Meth. Fluids 2007; 55:965–983” which I can send to your e-mail.
I really appreciate it if you help me to solve my problem. I am eagerly looking forward to hearing from you. This is my e-mail: firstname.lastname@example.org
I really can't help more than I have. Roache's book is a self-published paperback. To copy it, you'd need to destroy it - separating the pages to scan one at a time. So I doubt that a pdf is available.
When I was interested in stream function - vorticity, I was paid to do other things. So I read the papers but never tried to code it up. My previous message was what I remember (from the 1970's). The details of boundary condition application (especially for vorticity) were debated for several years.
I can only recommend that you take the time in the library to look at the old CFD publications to sort it out.
PS: I just checked amazon.com. Roache's oldest books are no longer for sale there. But there is a more recent book by him - Fundamentals of ... . That apparently is 1976 and is hardback.
Used Roache books are listed by http://www.alibris.com/
Sorry I can't be of more help.
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