[katiuskas]

Dear all,

I am experiencing some issues with the wave generation module. I am running some cases which I created from the waveFlume tutorial (so same solver settings, just changes in domain size and wave period and height). However, both in the tutorial and in my cases I find this error: the surface elevation is not centred at zero (see attached image, the amplitude should be 0.55). I tried using first and second-order wave theory, but the problem persists.

In my case setup, I am using a wave height equal to 1.5% of the wavelength whereas depth is 15% of the wavelength (shallow water). As for the tutorial case, the water depth is 0.4 and the wave height is 0.1. I've seen from section 3.1 of the original work by Jacobsen that linear waves in shallow water do not satisfy the non-linear wave problem ultimately resulting in energy transfer to higher harmonics. This, in my opinion, should not explain the offset in surface elevation I presented earlier.

Does anyone have a hint on what's going on here?

Thank you!

[ZJS67373]

Hi Pere,

I am facing the same problem now. In my 2D waveFlume case, d=1m，H=0.2m, T=2s, delta x = λ/100 and delta y = H/20. When I use the stokesFirst ,the wave surface rises as you do, and it doesn't match the theoretical solution. But when I use stokesSecond, the results are very close to the theoretical ones. After reading some papers, I found that this phenomenon is relatively common, but how can we reduce the error caused by this field to the first order wave.

So, do you have a deeper understanding of the problem now? Any suggestions will be of great help to me and I am looking forward to your reply.

Thank you!

[katiuskas]

Dear Zhang,

Indeed, I think I found an explanation which I hope satisfies your question.

When we apply stokesFirst waves onto the wave tank's free surface, we impose an initial condition on velocity, pressure and surface elevation.
But in reality, Navier-Stokes equations predict a value that generally is non-linear and different from first-order theory. Thus the initial, first-order condition will evolve into the true solution.

Nonetheless, I found that this solution gets closer to first-order for increased water depth and shorter wave steepness.
You can find some engineering criteria on what values to use, which I tried to summarise in the attached figure representing the linear theory validity region as a function of those two parameters.
In my original picture, the increased peaks and lower valleys are indeed a common characteristic of second-order harmonics.
This suggests that waves propagate along my wave tank as second-order theory predicts, and there is nothing I can do to make the results match first-order apart from changing the tank geometry and wave height.

Of course, none of this makes sense if your simulation suffers from numerical dissipation and dispersion, so make sure your spatial and temporal discretizations are fine before playing around with the wave tank geometry.

Some resources that helped me out were:

• Hedges, T. S., & URSELL. (1995). Regions of validity of analytical wave theories. Proceedings of the Institution of Civil Engineers-Water Maritime and Energy, 112(2), 111-114.
  
• Larsen, B. E., Fuhrman, D. R., & Roenby, J. (2019). Performance of interFoam on the simulation of progressive waves. Coastal Engineering Journal, 61(3), 380-400.

Hope this helped, good luck with your project!

Best regards,
Pere

[ZJS67373]

Dear Pere，

First of all, thank you for replying to my message in your busy schedule. Your answer is really helpful to me.

Earlier, I also read Hedges' paper（1995）and ended up choosing second-order waves in my simulations. But I really don't understand why the wave front is nonlinear when I choose stokesFirst. Your explanation has provided me with new ideas. I will carefully read the literature you provided and think about this problem.

Finally, thank you again for your help and wish you a smooth project.

Best wishes，
Zhang

[katiuskas]

Dear Zhang,

The Stokes first-order theory is based upon the assumption of infinite depth and very small perturbations at the free surface. Think of water steepness and finite depth as sources of non-linearity. If your case where λ = 6.24m, you have a wave steepness H/λ = 0.03 and relative water depth d/λ = 0.16. These values are enough to introduce higher-order effects, see the previously attached figure.

I would recommend increasing the water depth and comparing again to Stokes' first order.

Best,
Pere