convergence in transient flow
Hi friends. There are a lot about convergence in this forum but i still have some doubt. In one thread Mr. Alex mentioned that it is necessary for solution to converge in every n time step except the first few time steps and the time step size should be chosen sufficiently small to reach convergence within 510 iterations.
My question is, these 510 iterations per time step for convergence should be obeyed from the start or it will start from high and must reach 510 somewhere in middle of process and onward (of course without first few steps)? Well, Generally it starts from high and decreases. Thanks in advance. 
This is what I think:
The amount of coefficient loops necessary to reach convergence within a timestep is an indicator for the time step size. If it takes a large amount of inner iterations, the time step size is probably too large. Of course there may be exceptions when many equations are involved, for example for reactive multiphase flows where generally more iterations are required. If on the other hand the the amount of inner iterations to reach convergence within the timestep is very small, the time step size is probably too small. This is generally not a problem but will lead to unnecessary high computing times. 510 coefficient loops as a rule of thumb provide a balance between quick time advancement and accuracy for many applications. The exception for the first few time step arises from the initial conditions. If they do not "fit" the solution smoothly (for example if you initialize with zero velocity and pressure although there is a nonzero velocity art the inlet) convergence cannot necessarily be reached with a fixed time step size and number of iterations. If you can provide perfectly smooth initial conditions (for example from a steady.state solution) convergence should still be reached even for the first few iterations. If in doubt, adaptive time step methods homing in on 510 coefficient loops will deal with selecting a suitable time step size. 
Thanks Mr. Alex. what i saw in my cases whenever i run transient case with fixed time step it starts convergence with large no. of iterations but as the time goes on the Max no. of iterations/time steps for convergence decreases. e.g one of my case (which is still in process) my time step is 1e5, convergence criteria for residuals is 1e7. so at start it was taking about 40 iterations/time step for convergence but as time goes on (i checked after about 1500 time steps) it was taking 25 iterations/time step to converge and later it decreased to 21 iterations (after 5000 time steps) but at the current time, Max iterations/time step again increased to 26. The drop in Residuals in every time step is about 1e3 to 1e7.
So does it means my time step is still high if my accuracy criteria is 2025 coefficients loops? And why there is increase in max iterations for convergence from 21 to 26 in my case or it is normal? Thanks again Edit: Can we guess suitable time step by using Adaptive time step method for just a few time steps? If Yes then what Min time step size should be used in Adaptive time step method? 
Hi friends. Mr. Oj.bulmer wrote in one thread that "The residuals should be flat towards the end of the time step. But the flatness should not be prolonged which indicates the timestep can be further increased to accelerate the solution."
Can someone explain what is meant by flatness here? Generally Residuals in transient problems are sinusoidal (saw tooth shape) which repeats in some fixed no. of iterations but i am not getting what flatness refers here. Thanks 
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Here is a figure that should explain it better. After the first 100 iterations, the underrelaxation factors are changed leading to faster convergence per time step. Here you can clearly see the residuals flat out towards the end of the time step, indicating that the solution does not change within the last iterations per time step. Hence the 20 iterations per time step performed here are a waste of time. From this point on, the iterations per time step are decreased after every time step. The level of convergence is not altered until some "optimal" value for the number of iterations is reached. Further decreasing the number of iterations per time step leads to a lower level of convergence. Decreasing the number of iterations per time step has a similar effect on the residuals as increasing the time step size. 
Thanks Alex. It's really informative. here the drop in residuals at the end is about of the of 10e4, there is sharp end at every time step and iterations for convergence will be less than 20 so it's best time step setting...

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