[Werne]

Hello,

I'm currently trying to do a global stability analysis of a flow around a cylinder to find the critical Reynoldsnumber for the flow behind the cylinder to get unstable.

I'm following the approach that has been given in "Computation of eigenvalue sensitivity to base flow modifications in a discrete framework: Application to open-loop control" by Denis Sipp et al.

The general procedure is:
1) Getting an (U)RANS solution of the flow
2) Build the Flux Jacobian by disturbing the flow at individual cells and evaluating the resulting changes
3) Compute the Eigenvalues of the Jacobian with ARPACK

The real part of the dominant Eigenvalue should be negative for stable flow and positive for unstable flow. However, no matter what Reynoldsnumber I choose, all Eigenvalues have a positive real part. I even spent a weekend and calculated the full Eigenvalue spektrum for one Jacobian, and still there were no negative real parts for my flux Jacobian.

I analysed several Reynoldsnumbers from 40 to 120 (literature says the critical Reynoldsnumber should be around 47), and all showed the same behaviour. The flowfield itself does become unsteady, of course. For higher Reynoldsnumbers a Karman vortex street can be observed.

I'm wondering if maybe somebody else has observed this kind of behaviour?
I'm using the Python implementation of ARPACK (scipy.sparse.linalg) without any special options. The only thing I tried was using the shift-invert parameter with a negative real part, however, this case didn't converge, even after a whole week of computation.

So, if somebody has any idea what my mistake might be, I'd be glad for any help.

[FMDenaro]

What I do not understand is why using the RANS solution as basis field...I suppose you are searching for a critical Re number that makes transitional the flow from its laminar state, is it correct?

[Werne]

It's not the transition from laminar to turbulent flow that I'm interested in, its the onset of large-scale unsteadiness such as the development of a vortex street.

I'm using RANS solution as a baseline for the cases where the flow is steady, for unsteady cases I'm using URANS.

[FMDenaro]

Sorry bu I do not understand ...let me try to explain better: I assume you have: 1) laminar steady flow for Re below a certain value 2) laminar vortex shedding 3) transition to turbulence 4) fully developed turbulent flow.

Now you can use RANS or URANS only for regime 4). But RANS does not mean you have a steady flow as happens in 1) but on,y that you are searching the statistically averaged solution. And even using URANS you are still looking for a statistically averaged solution. URANS makes sense if you have an external time-depending forcing.

[Werne]

Yes, I'm basically looking for the border between 1) and 2). But I don't know why I shouldn't be able to use RANS in this case? Of course, I neglect the influence of turbulence, since the flow is laminar, but apart from that, it shouldn't be a problem.

Also, there are multiple examples of this approach in literature. For example the one from Denis Sipp et al, mentioned above, or the paper from J. D. Crouch: "Predicting the onset of flow unsteadiness based on global stability".

They also use (U)RANS for their computations, and they get reasonable results from it.

As I said, the (U)RANS simulations are only used to derive the flux Jacobian of a given flow. The stability analysis is then carried out on the basis of this Jacobian.

Or maybe we're misunderstanding each other?

[FMDenaro]

Originally Posted by Werne

Yes, I'm basically looking for the border between 1) and 2). But I don't know why I shouldn't be able to use RANS in this case? Of course, I neglect the influence of turbulence, since the flow is laminar, but apart from that, it shouldn't be a problem.

Also, there are multiple examples of this approach in literature. For example the one from Denis Sipp et al, mentioned above, or the paper from J. D. Crouch: "Predicting the onset of flow unsteadiness based on global stability".

They also use (U)RANS for their computations, and they get reasonable results from it.

As I said, the (U)RANS simulations are only used to derive the flux Jacobian of a given flow. The stability analysis is then carried out on the basis of this Jacobian.

Or maybe we're misunderstanding each other?

RANS (and URANS) cannot be used to solve a laminar regime, is simply theoretically wrong.... you are solving a statistacally averaged N-S system not the original one and you need to supply a closure model to have a mathematically well posed problem. When you do that, you superimpose something to the solution (the turbulence model) that does not exist in the laminar solution! The RANS approach is not able to automatically check for a laminar flow region and set the model to vanish. This happens using LES.

[FMDenaro]

Originally Posted by Werne

Yes, I'm basically looking for the border between 1) and 2). But I don't know why I shouldn't be able to use RANS in this case? Of course, I neglect the influence of turbulence, since the flow is laminar, but apart from that, it shouldn't be a problem.

Also, there are multiple examples of this approach in literature. For example the one from Denis Sipp et al, mentioned above, or the paper from J. D. Crouch: "Predicting the onset of flow unsteadiness based on global stability".

They also use (U)RANS for their computations, and they get reasonable results from it.

As I said, the (U)RANS simulations are only used to derive the flux Jacobian of a given flow. The stability analysis is then carried out on the basis of this Jacobian.

Or maybe we're misunderstanding each other?

From what I understand from the paper (https://pdfs.semanticscholar.org/8d9...c094c8310e.pdf), the statistically steady field q_bar is the RANS field which is obtained for a real turbulent flow at a moderately high Reynolds number. This is not the laminar field you would to analyse ...
In Sec.4.1 it is clearly stated that:
"at low Reynolds numbers the flow is laminar, so the state vector reduces to q". Therefore, the steady solution used in the paper is not a RANS but a laminar solution of the NS equations.

[Werne]

Ah, now I see your point. Yes, you are right, I'm not solving the RANS equations, I'm solving the laminar NS equations. What I meant when I said I'm neglecting turbulence terms is that I set a switch in our CFD code to use the laminar equations.
I was using RANS and URANS to distinguish between steady and unsteady simulations.

[Kader]

Hi Werne
I would know if you have solved your problem, I have similar problem.