(1). It is really an interesting feature of fluid dynamics, that is, you don't always get symmetric solutions from symmetric configuration. (2). The book by Schlichting, The Boundary Layer Theory has a couple of chapters on the origin of turbulence. It is an excellent book for studying fluid dynamics and I think the book is also very useful for who are involved in turbulent flow computations.

I would caution you in saying

"the equations know the flow should be a turbulent one"

1. The continuous differential equations "know this" or probably better to say govern the transition to turbulence. This is of course simulated by DNS.

2. A series of discrete, linearized appoximations to the nonlinear differential equations, discretized on the space and time scales of mean flow (the flow solution which spurred this discussion) also may have imbedded "transitionary behaviour".

3. Does the "transitionary behaviour" in 2 have anything to do with that in 1?

Answer: It depends upon how closely the linearization, spatial and time discretizations can represent the behaviour in 1. In DNS this is well approximated. In a typical mean flow discretization of order (100 x 100 x 100) spacial nodes ie. spacial resolution of dx, dy, dz = 0.01 L with L the characteristic domain length scale and time resolution of 0.01 T, perturbations of the order of 0.1 L and 0.1 T can be resolved. The "transitionary behaviour" of this approximation to the continuous solution is likely a very poor approximation to the transition to turbulence! The same issue has been around for decades as to the validity of a linearized stablility analysis for the prediction of transition to turbulence. In this case, using continuous solutions which can resolve all of the wavelengths (much better than a poor discrete approximation) but none of the non-linear mode interactions, the prdictions are order of magnitude accuracy and not of much use.

4. Another issue which was mentioned was forcing a steady solution by removing the transient term. The result is a s stated usually a solvable system but often has multiple solutions above a given forcing parameter.

Regards.....................................Duane

I started off this discussion because in nature we can see that flow becomes turbulent and we have assigned a criterion from experimental data in terms of Re number. That if Re becomes greater than some number (again depending upon differnt flows) flow is turbulent. Now numerically we accept that N-S eqns. describe a fluid flow turbulent or laminar. To solve these pde's we put them into a linearised, discretised form. As you suggested the solution may have imbedded "transitionary behaviour". I also put this question is the "transitionary behaviour" due to a physical transition to turbulence (due to high Re) or due to the approximation of continuous pde's into discrete ode's. and then comes the numerical errors.

With respect to your point 4. Does the multiple solutions one gets has resemblance to behaviour of Poincare attractors. Where solution move from one point to another in a specified region. Can we suggest that multiple solutions indicate that the it is turbulent flow? And we better solve taking turbulence into account, either using a turbulence model or making spatial , time resolution very fine (DNS) which might not be practical.

I also would like to refer the situation which applies the condition in the Schlichting book

For a symmetric configuration of contra-rotating discs for example, laminar computation could give symmetric recirculation(converge and satisfied the conservation of mass and momentum) even at very low disc speed, each rotating in opposite direction. As Stewartson K(1953) suggest that it is instrinsic stable.

There is a research on this area, and the laminar computation does not agree with the measurement, the turbulent computation seem to show the physic of the flow

You can refer the work below for further discussion

Kilic, M. 1993 Flow between contra-rotating discs PhD Thesis University of Bath England

AAJ