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Clifford Arnold March 5, 1999 11:50

Fun on Friday: The Funnel Problem
I like to study very simple (looking) fluid systems and see how much can be learned without adding unnecessary complexity. Such cases can make for good benchmarks of software, solution algorithms, and as a general learning vehicle in fluid dynamics. These casses also make for a nice public demonstration.

Here is an example I have been studying that demonstrates nicely that 2D analysis is not always sufficient, even when the problem is geometrically 2Dand the BCs are trivial.

Check out:

for a summary of the problem plus a series of computational results. This may seem a bit light hearted, there is much to be learned in this case. Besides, it's Friday. Have a nice weekend.

Adrin Gharakhani March 5, 1999 16:43

Re: Fun on Friday: The Funnel Problem
The example you are presenting at your website is the "more complicated" :) version of the classic studies of divergent nozzles that one finds in Schlichting! The key here is convergent vs. divergent flow. The latter is stable, whereas the former is inherently unstable! It is also intuitively obvious that due to "sudden expansion" you are going to see recirculation of flow _back_ toward the nozzle inlet. How this recirculating flow interacts with the "incoming" jet depends on many factors - the geometric expansion ratio, the inlet jet momentum, density variation, etc..

Now, when a fluid system is inherently unstable (just like any other system), all it takes is to perturb the flow just slightly and the instability mechanisms feed on themselves and develop an unsteady 3-D motion - _irrespective_ of the geometry, whether 2D or 3D is unimportant.

In the real world, the perturbations are everywhere: the granularity of the walls (even if it is a "trivial" no-slip, no-flux BC), the yawning of the bored experimentalist because it takes forever to setup an "isolated" system :)), the vibrations in the building, etc. all initiate this instability.

In the numerical world, your code is basically mimicing the real world perturbations by round off errors and by the fact that your code is inherently _non-symmetric_. This is a crucial point. To be pedantic, one should not be too thrilled if the CFD code can match the experiments, because it shows that the code is poorly written or the algorithm itself is not conservative! Only after one has the control to introduce perturbations at will and gets a match with experiment should one claim "victory"! I claim this because there is proof (I forget by who) that the N-S solutions will remain symmetric forever for a symmetric geometry! If the symmetry is lost, it's because you have introduced (hopefully random) perturbations to the solution. Note that for fluid systems that are inherently stable (e.g., laminar channel flow) the perturbations will die down (and again we get "matching" with experiments)

Having said all this, if you run your test problem a number of times and in each case you introduce small perturbations you will (should) see different unsteady solutions. BUT, if you average out the solutions (from the different cases) or even if you just time-average the one case that you presented (assuming you let it run through enough cycles to have a statistically stationary state solution) you will see a fully symmetric soltuion that is compatible with the geometry at hand!

Adrin Gharakhani

Jay March 5, 1999 17:22

Re: Fun on Friday: The Funnel Problem
Very good explanation there, Adrin.

Moreover, the "funnel" problem as stated in Cliff's note generally invokes an image in the mind of flow from a large c/s thru to the smaller one, while the inverse really is presented on the web page. The web presentation treats a funnel not really as a "funnel," but rather as a diffuser. And, this is essentially the same point that Adrin made, diffuser flows are inherently more unstable than "funnel" or converging flows. The Friday fun was sure short-lived :)

John C. Chien March 5, 1999 18:46

Re: Fun on Friday: The Funnel Problem
It is a well recognized behavior in a separated diffuser flow; unsteady, asymmetric flow separation. Researchers at Stanford University have done a lot of work in the diffuser area. Most computations were carried out in 1-D, or axisymmetric flows condition. In early 70's, I have conducted a series of systematic CFD computation using body fitted coordinates system and low Reynolds number k-epslon model for both 2-D and axisymmetric diffusers with non-uniform inlet conditions. For 3-D simulation, it is a good idea to run the computation long enough to see whether the results are random or asymmetric ( steady). Normally, the diffuser half angle is around 8-10 degrees to avoid getting into this regime. There are diffuser map available which defines the boundary of flow separation.

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