when to use totally explicit schemes or dual time-stepping in transient flow?
 

Hi,guys.In the simulation of a transient phenomenon, we can choose explicit time marching schemes or implicit time stepping(dual time-stepping), but which schemes was better for the phenomenon? I saw some tips about this, for example,when the time scale are comparable to the spatial scales over the eigenvalue,i.e. when the CFL number dictated by the physics is of the order of unity. so what is the time scale? and the spatial scales is the length of computational domain or the grid length? and the eigenvalue of the problem is the u+c ??
thanks

I think that for transient simulation you should take care of the accuracy... implicit or explicit schemes is a secondary issue

While I agree with Prof Denaro, I would mention that understanding all of the timescales in your problem is really critical. These do not need to necessarily involve complex computations either. Just per-term dimensional analysis will give you a lot of guidance. Length scales are the smallest cell length and the domain length. Velocities are typical and maximum. Timescales (for transient simulations) are the timestep and the final simulation time. If you juggle those in as representative numbers in your transport equations, you will see timescale emerge. Generally, you can balance the transient term against each term in the transport equation (turn off the other effects and look at the time scale for each phenomena).

Once you have those timescales, you can see which effects group together. In the incompressible limit, you will see "flow" timescales that are relatively large compared to acoustic timescales. That leads us to chose pressure-implicit schemes like Fractional Step/Projection methods that implcitly treats the pressure/mass conservation and removes that acoustic timescale from the stability analysis but (as Prof Denaro notes) NOT from accuracy considerations.

Similarly, with reacting flows, the timescale of reactions can be very small relative to flow/acoustic timescales. That naturally leads to so-called stiff combustion solvers where computational cells are treated as independent reactors and time integrated with stiff ODE solvers to the next fluid timestep.

IMO, understanding the various timescales is critical for simulations that involve transport more complicated than single-phase cold non-reacting flow. This is the only way to both make a solver that captures all important phenomena while not wasting lots of CPU time resolving irrelevant details.

from my experience, using explicit time-marching schemes fulfilling the numerical stability constraints are also a requirement for a good physical resolution of the relevant scales of the flow (apart form formulation where the scales are filtered out and then modelled).

Thanks for your answer. If I have found the timescale of the transient problem, I can choose which time integration schemes is better for my problem, the method to determine the timescale is very helpful to me.But I am little confused by the first example, the "flow" timescales that are relatively large compared to acoustic timescales in incompressible flow, in this problem,we want to resolve the physical characteristics of this phenomenon, so use the acoustic timescale as the time step? I'm not really sure about this.

If you want to resolve acoustic waves, then you have no choice but to choose your timestep based on the dx/|u+c|. Physically, this implies that an acoustic wave cannot travel more than one computational cell in a timestep. And that implies a fully explicit scheme.

Using the incompressible model, your governing equations will be not well suitable...Often the acoustic problem is decoupled from the flow solutions and obeys to a different set of equations solved after the incompressible solution is obtained.
Alternatively, you can solve the full compressible problem if your Mach is not too low.

Does that mean，in a real flow, it involves many timescales,but the time march step size or the time integration scheme depends on our need? for example, in the finite rate reacting flow, the involves convection/diffusion and reacting of different time scale, but I only focus on the change of a specific species(if the reacting time scale of this species is neither the largest or the smallest),so we can ignore the other species which have smaller time scale by implicit way( if the can treat individually)?

So my opinion is ,in a transient flow simulation, we can only focus on the specific time scale we want to know, and ignore the other time scales?( fulfill the stability criterion)

Thanks.