I wonder if there is any relation between laminar,turbulent and stable,unstable.

A laminar flow that is stable will continue to be laminar.

If a lmainar flow is unstable under certain conditions, and these conditions are reached, then there can be actually two possibilities:

-1) the laminar flow will directly transit to turbulence. IT can be the whole flow that becomes turbulent, or just a region of the flow is laminar and the other region is turbulent. All the modes (or scale lengths if you prefer) are unstable and grow exponentially (so the turbulence is at all scale lengths, short and small) and eventually saturate.

-2) The laminar flow is unstable only for a few number of modes (scale lengths), and this can be observed as wave patterns for example or a stable flow that is actually not in steady state (d/dt is not zero). The best example are the famous Taylor vortices in the Couette flow (flow between rotating cylinder). At this stage the flow is not defined as turbulent. Under certain conditions this flow itself can be again unstable and transit to turbulence, but not especially.

You can differentiate between locally and globally v stable (2 cases), and linearly and non-linearly stable. linearly means that an infinitesimal pertubation will make the flow unstable. non-linear means that you need a perturbation strong enough to perturb the flow. So all in all you have 2X2=4cases of instability, etc..

If you wish to know more just have a look at

Landau and Lifshitz, Fluid Dynamics, Pergamon PRess.

or any other book on hydrodynamic stability of flows ,etc..

Or if you have any specific question, just post it here.

Patrick

Thank you very much. If I want to model the 'local' and 'linear' flow, what kind of model (laminar or turbulent as a whole) should I take? By the way, in such situation, is FVM or SM preferred? Thanks in advance.

Wood

Not quite sure I understand your question:

what do you mean by: to model the local and linear flow? Do you mean that you solve for the flow only locally and with no finite amplitude perturbation?

Then this is not sufficient to evaluate if your flow is laminar or turbulent.

What is exactly your flow problem?

According to your first reply, I think in the whole field, some of the flow is in laminar mode and the other is in turbulent mode. Is it possible? Then how to simulate it?

Yes, this can be possible. The cases that I know are when the flow is linearly stable, i.e. if there is no perturbation the flow is laminar and stable. However, the same flow is non-linearly unstable: i.e. if there is a perturbation (strong enough) the flow can transit to turbulence. The turbulence can be confined locally and it can depands on many parameters (e.g. if the Reynolds number - the velocity - is not the same over the entire domain, etc..). A good example is the one in the water table experiment, where (as the name indicates it) water flows over a table and forms thin layer. Because the table is not horizontal the water flows over it (say at a constant speed). Now if there are some objects or irregularities on the table (over which the water flows), then these obstacles can induces turbulence. In this case the turbulence starts where the obstacle is located and continues streamwise from there, forming a V shape (a little bit like what is seen behind a motor boat). In this specific case there are streamwise vortices to carry energy from the laminar flow (flowing on the table) to the turbulent region. In some other cases there are turbulent 'spots' in the flow.

So I am still not quite sure what is the exact problem you are talking about, but in short you can certainly have laminar regions and turbulent regions in the same flow.

In my case, a jet issued from a nozzle hits the target in the downstream. Due to the confined geometry, two big vortexes are generated in front of the target and another two small ones behind the target. A disturbance will cause the jet oscillation begin. When the velocity at the nozzle exit, which is proportional to the flowrate, is high, it is sure that the flow is turbulent. Now,what I am not sure is at very low flowrate. Especially I am interested in the start point when the oscillation begins.

Well it seems that there are here two components.

First the jet. Even at a low flow rate you can have a Kelvin Helmoltz (KH) instability which will make the jet wavy (small amplitude oscillation) with usually only one wave length dominant. And this will happen even without the the target downstream. At higher speed the KH instability is stronger and can drive the jet to a turbulent regime. If the jet is supersonic then in addition to the KH instability there are also unstable pressure modes (since at supersonic speeds the compressibility becomes important). The KH modes affect mainly the envelop of the jet while the p (pressure) modes affect the body of the jet.

Second: the target. This sounds like when you have a cylinder in a stream (or a cylinder moving in a medium). At a low speed you can have a wavy pattern developping behind the targed downstream (it is itself a bit similar to a jet, with the difference that the speed there is smaller than the speed of the medium - while in the jet the speed is higher than the speed of the medium). So here again it is a KH instability (shearing instability - the shear between the fast moving flow and the stationary one). At higher speed you can have turbulence. In the case of the cylinder the situation involves also the detachment of the boundary layer and the formation of a Von Karman vortex street downstream (behind the cylinder).

So all this can involve many different processes working together.

And you asked me how to simulate that. Again it depends on the symmetry of the problem, the target, the flow, the boundary and initial conditions, and the value of the physical parameter, and also what you are looking for exactly, etc...

I can try to help if you have a specific question, but I am not quite sure what you mean by "how to simulate" (in a previous posting). Are you starting from scratch?

You might want to consider posting a new message with a new title to explicitly reformulate the question and the problem clearly, there are many people out there who can help and will be interested.

Thank you very much!