bifurcation of numerical solution
hi there, what is bifurcation and why it happens? i came across this term in curved duct flow studies, where it is used to explain the formation of two and more vortex solutions when the Dean number (similarity parameter) exceeds certain critical value. any information would be greatly appreaciated. thanks. regards, yfyap

Re: bifurcation of numerical solution
A bifurcating solution is one which does not exist over a range of parameter values(in this case the Dean number) but does exist when the parameter reaches a certain value. The point of bifurcation can usually be discerned by looking at the stability of the linearized problem. An example in structural dynamics is buckling..another in aeroelasticity is flutter.

Re: bifurcation of numerical solution
Peter Attar, isn't it that there should be a unique solution when the Dean number is smaller than the critical value, and beyond that there are more than one solution? thanks. regards, yfyap

Re: bifurcation of numerical solution
In the way I'm familiar with when the critical parameter value is reached there are a family of solutions of which some are stable and some aren't..numericallly you won't be able to see the unstable solution(s) though. I am not familiar with the case you are speaking of though. Perhaps you can point out the article which you are referring to.

Re: bifurcation of numerical solution
hi peter,
do you mean that when the critical parameter value is reached, there will be two or more distinct solutions for a given problem? if this is the case then, how do we decide which one is the solution for the physical problem that is modeled? or all the solutions are valid ? What will happen if beyond the critical parameter value? Actually, why bifurcation happens? i have two papers on bifurcation study of laminar flow in curved square ducts: Winters, K. (1987). "A bifurcation study of laminar flow in curved tube of rectangular cross section." J. of Fluid Mechanics, Vol. 180, pp. 343369. Daskopoulos, P. and Lenhoff, .M. (1989). "Flow in curved ducts: bifurcation structure for stationary ducts." J. of Fluid Mechanics, Vol. 203, pp. 125148. thank you. regards, yfyap 
Re: bifurcation of numerical solution
You should take a look at a text which deals with nonlinear ordinary ODE's..most will go over bifurcation theory in detail. Like I said there are a family of solutions but usually only one is stable..ie only one will persist with time. I'm only speaking from my personal experience in dealing with Limit Cycle Oscillations in aeroelasticity where at certain flow speeds a Hopf bifurcation will appear. I will take a look at those articles today and see what they are speaking of.

Re: bifurcation of numerical solution
The bifurcation of solution is a characteristic of nonlinear dynamic system. When some parameters, say, Re or Dean number, exceeds a critical value, there may be multiple solutions of the system (Euler, NS or other equations). There are may be more than one stable states in supercritical regime. Typical examples of flow bifurcation are cylindrical and spherical Couette flows. There much literatures on "flow bifurcation". A discretzied equation also forms a dynamic system. If it is good approximation to Euler or NS, it will produces the same bifurcation. However, it may produce erroneous bifurcations of itself, or can not reproduce the bifurcation of original equation.

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