Dear Steve,

I appreciate your help.
So I just followed your recommendation. I lowered the residuals as shown in the figure below.

I made a simple linear function to account for pressure ramping during the start-up of the simulation.
Compressible fluid as you recommended.
Quasi-Newton converges within a few iterations, where both transient structural and transient Fluent converges from the first coupling-iteration.

By postprocessing the results, I took the time-step n 44 which corresponds to the flow-time 4.4e-5sec. Pressure inlet should be = 880Pa

I monitor the pressure at the inlet and outlet and have this :
Where in the inlet the average static pressure is = 94 Pa and the pressure distribution is like
Plane section view of the fluid domain, flow-time = 4.4e-5sec, P inlet = 880 Pa.
This is the convergence problem that I get. I don't have this kind of weird behavior when I increase the solid density, say by 2 orders of magnitude.
Similar behavior observed when I use incompressible fluid.
As I mentioned earlier, the problem is not convergence on itself, but the results I get after convergence. I could run successfully this case when I applied Explicit relaxation within the pressure-based coupled solver of momentum and pressure. However, the factors I used must be less or equal to 10^-4 which in my opinion is not convincing. What do you think?