• grid resolution for 3-D spreading of wave is better to be at least:

• the recommended time-step is:

(If you are using implicit unsteady solver, try other CFL conditions by changing time-step value.)

You need to fulfill two criteria:

1) the domain length must contain the largest wavelenght characteristic of your problem;

2) the grid size h must be small enough to represent the smallest wavelenght of your problem. This means that the Nyquist frequency pi/h must be of order of the inverse value.

Dear Prof. Filippo Maria Denaro

Thanks for suggestions. As usual, you recommendation are helpful.

May I seek a little more? If possible, can you recommend literature for grid resolution requirement. I am wondering, how small grid size is small enough to resolve the flow features. For example: If L is wavelength; How many grids/nodes shall I place to capture this wave.

The Nyquist theorem states that the smallest wavelength can be represented on a grid of size h using 3 grid nodes, that is over a computational lenght 2h. This is because the 3 nodes can represent the smallest oscillation of a sine wave. Hence, if you know the value of L you can set 2h=L to compute h.

There is a 3rd issue to consider:

It is not only problem of mesh size and domain size. Some wave lengths (Frequencies) are attenuated by your numerical scheme. And this damping is related to some factors like the nature of your numerical descretization scheme and mesh size. As you change the mesh you kill different frequencies. Having a frequency preservative scheme is challenging, one way is using multi-grid schemes

The Nyquist criterion is simply understood by noting that a wave has three unknowns: The amplitude, the wavelength and a phase shift with respect to datum. So to find these three unknowns, you need to pass the curve through three points, at zero, h and 2h. Hence the maximum respoved wavenumber is \pi/ h. This is strictly for spatial discretization with uniform spacing.

People loosely interchange wavenumber with circular frequency. This is not proper. While Nyquist critierion applies to spatial discretiztion, for circular frequency there is no such restrictions, as one can change time step at will, from one step to another.

To capture small length scale in space, one way is to take non-uniform spacing, when you know where the gradient in maximum. For space-time dependent problems, one can alternately use mesh refinement that can change with time step via some algorithm. This can make such computations very expensive.

Hope this helps to keep the matter in perspective! Happy computing!

Thanks Prof. Sengupta and Dr. Zamani for informative responses.

Hope it helped Kuldeep! Sorry for the typos in my last response though.

• grid resolution for 3-D spreading of wave is better to be at least:

• the recommended time-step is:

(If you are using implicit unsteady solver, try other CFL conditions by changing time-step value.)

Reza Khazaee, why should one take so many points per wave (ppw)? These seems excessive, and must be for a very low accuracy method, which filters the signal so much, that virtually one is forced to take such excessive ppw! I am also presuming that your wave is in the xy-plane. Correct me please, if I am wrong.

I would add some details about the issue of the smoothing of the content in the resolved range of wavenumbers.

Using FV methods based on the integral form, this is not a numerical error at all but it is the correct answer to the fact we solve for a local averaged variable. The numerical error is shown by the difference to the ideal smoothing. On the other hand, using SM there is not smoothing at all up to Nyquist, at least if de-aliasing or filtering techniques (that are very commonly used) are superimposed. Somehow in the between, one can considet the FD method, the smoothing tending to disappear going from low to very higher accuracy order. The expressions of the transfer functions for a certain scheme can easily show these behaviors.

Finally, the excessive numbers of nodes that Reza Khazaee addressed has not a theoretical reason, as correctly addressed by Tapan K. Sengupta, but I know it is somehow a common practice in solving shock wave problems. We could argue what is the meaning of doing that in case of a mathematical discontinuity (Euler flow) or in case of a viscous shock layer (NS) but this is a different topic.

Dear Tapan Sengupta, The main reasons of these suggestions are the CFL condition and Capturing Scheme in RANS equation solvers. For better capturing of the 3D free surface waves, the wave should be refined well in every direction. if the sharpness of the captured wave is out of interest you can neglect the above values. if the transverse flux of every cell is not important and the wave is 2-Dimensional you can neglect the y-direction refinement.

x ---> Propagation Direction

y ---> Transverse Direction

z ---> Normal to the undisturbed water surface

Filippo Maria Denaro is absolutely right in classifying the requirements of different classes of methods. For all practical purposes, Fourier spectral methods (SM) is not used, despite its superior resolution properties, due to the requirement of uniform grid and periodicity. Compact schemes are best bet, also for the reason of better dispersion properties. However, before use, one must characterize the numerical properties and not use multi-time level methods which bring in spurious numerical methods. All results obtained by Adams-Bashforth method are thus very suspect!

Filippo is also right in pointing out that for shock-capturing one requires large number of points. But in most such calculations, where is the wave? To view the waves, one would have to perform DNS with dispersion relation preserving methods, as described for a very limited number of high accuracy compact schemes. Also, the major issue of shock-capturing is related to avoiding Gibbs' phenomenon. To take care of such dispersive waves from the discontinuities, one takes a very large number of points.

The question above is saying any kind of wave type. It's better to answer this question focused on it's subject and concern.

The above suggestions are outcome of grid independence study of my thesis and has theoretical base. The grid resolution is the most important part of capturing waves. If you are disregarding the proper mesh for this problem, you are making a mistake.

https://ocw.mit.edu/courses/mechanical-engineering/2-20-marine-hydrodynamics-13-021-spring-2005/lecture-notes/lecture1.pdf

check page 2

Dear Prof. Tapan K. Sengupta and Prof. Filippo Maria Denaro thanks for your suggestions and very relevant information. I have gone thru a document which discussed about grid requirement based on critical wavelength (which differentiate Capillary waves and Gravity waves). The recommendation in the documents were something like Reza Khazaee is suggesting. Although, there were some loop-holes in the document which I was referring. I was expecting a lower node requirement as suggested by Prof. Filippo Maria Denaro and Prof. Tapan K. Sengupta. Thanks to both of them to add scientific explanation. I am planning grids as per the suggestions. Thanks to Dr. Kaveh Zamani for check-points. Reza Khazaee, if you can suggest me some literature based on the criterion which you suggested, it might be helpful to optimise my simulation plan.

Thanks to all for the contribution and support.

Interesting