Fluid problem - periodic forcing over a finite region

[avenged*7]

Hello,
I am working on a solo project outside my domain of expertise (Physics PhD student). I am trying to analyze/replicate the wave phenomena shown in the following video:



To summarize what I am doing:

• I need to analyze a simple (cylindrical) pool, say 17.5" wide, 4" deep
• Figure out how to periodically force over a finite region
• Find the expression for surface height η
• Find the expression for wavelength λ
• Determine the decay time of the wave (not sure the expression for half-life τ=ln(2)/λ works here)
• Scale up to larger dimensions and repeat

Fluid flow problems have so many approximations and domains of validity that I am not quite sure where to start. After exhaustive searches, I have concluded that Airy Wave Theory is likely my best starting point. I looked through slosh dynamics books and found some expressions for η in cylindrical and conical (what I need to analyze once I have figured out the cylindrical case) containers. I will need to use the appropriate initial and boundary conditions to get the constants correct, but that is another story. The most important parts of my analysis include what happens when I scale up the dimensions, how does that affect the decay time, and what is the best way to produce this phenomenon.

Since my primary interest is for shallow and wide pools, I am assuming that scaling up the calculations should be rather trivial as long as λ >> h. However, I do not have an expression for λ. I have only observed the "nodal circles" to be near (but not at) the boundaries. I can't be sure until I have an expression. I believe that I found the appropriate dispersion relation. If these waves are resonant capillary-gravity waves then I should have something like:

ω² = |k|(ρ_water + ρ_air)^-1[(ρ_water-ρ_air)g + σk²],

where ω is the frequency, k is the wavenumber, ρ are the densities, g is gravitational acceleration, and σ is the surface tension. This comes from Airy wave theory. For a cylindrical geometry, I have something like

η(r,θ,t) = (1/g) Σ_m∑_n α_mncos(mθ)J_m(λ_mnr)cosh(λ_mnh)[ω_mncos(ω_mnt)],

where α_mn is TBD, J_m is a Bessel function of 1st kind (order m), h is the water depth, m=(0,...,inf), and n=(1,...,inf). Furthermore, since we appear to be generating a fundamental mode, I think I can take m = 0.

Hopefully, this much is correct. The next and most formidable part is how to add the forcing over a finite region of the surface (like over a circle of radius R). The problem seems simple at first, but turns out to be quite difficult. I spoke with a mathematician who specializes in fluid flow and he said "that's hard". He recommended incorporating it into the pressure, but I am having a hard time figuring out how to make that happen. The only idea I have is to partition the pool into regions (one with gravity, and one with gravity plus the forcing) and use derivatives to stitch them together, as I have done in certain problems in Quantum Mechanics.

The slosh dynamics books and papers focused on moving the container itself, not moving the fluid with the container fixed. I assume that moving to the coordinate system of the container would be similar, but I'm not sure you can simply discard the inertial terms.

I guess I am asking if I am on the right track and if there are any simple ways to incorporate the forcing. I tried doing stuff computationally and quickly realized it was over my head. Then, I tried to use ELMER to do some work for me, but it quickly got too technical. Any help is welcome.

 

[DEvens]

You can probably ignore surface tension. Imagine putting some dish soap in the pool. It shouldn't change the wave noticeably. Also, air is about 1000 times lower density than water. You can probably ignore it also. So you get $w^2 = K g$. Just eye-balling the setup this formula isn't vastly far off.

The pool walls are clearly distorting during the wave. Whether or not this is important is difficult to say. It would certainly simplify the problem to consider rigid walls. Moving walls is often a very difficult thing in fluids because it involves a feedback. The walls don't look particularly linear here.

I'm thinking you can't ignore viscosity. Viscosity will be how the force on your finite region is transmitted to the water near it. It's probably also a major source of damping the wave.

Probably to solve this in any reasonable degree of accuracy you would need a 2-D fluid dynamics system. You could treat the system as a radius and depth, assuming rotational symmetry. If that turns out to be easy you could add back the moving walls. You could model them as an elastic barrier.

 

[avenged*7]

You can probably ignore surface tension. Imagine putting some dish soap in the pool. It shouldn't change the wave noticeably. Also, air is about 1000 times lower density than water. You can probably ignore it also. So you get $w^2 = K g$. Just eye-balling the setup this formula isn't vastly far off.

The pool walls are clearly distorting during the wave. Whether or not this is important is difficult to say. It would certainly simplify the problem to consider rigid walls. Moving walls is often a very difficult thing in fluids because it involves a feedback. The walls don't look particularly linear here.

I'm thinking you can't ignore viscosity. Viscosity will be how the force on your finite region is transmitted to the water near it. It's probably also a major source of damping the wave.

Probably to solve this in any reasonable degree of accuracy you would need a 2-D fluid dynamics system. You could treat the system as a radius and depth, assuming rotational symmetry. If that turns out to be easy you could add back the moving walls. You could model them as an elastic barrier.

I dropped the viscosity since it is on the order of 10^-4 Pa*s and the atmospheric pressure is 101 kPa and most resources have considered only inviscid fluids. I figured surface tension to contribute a bit more to the dampening since it is considered to be pretty strong in water. I am definitely only considering rigid, regular cylinder at this point. I may explore dynamic boundaries as an alternative to centrally-driven waves. Thanks for your input

 

[Andy Resnick]

Hello,
I am working on a solo project outside my domain of expertise (Physics PhD student). I am trying to analyze/replicate the wave phenomena shown in the following video:
<snip>

An analysis of the pool geometry as shown is the video is going to be highly complicated- the elasticity of the pool walls are an essential component of the fluid motion and forcing. The free surface makes the problem even messier. One simplification is to make it a 2-D problem (radial and height coordinates), but this type of problem is typically attacked numerically.

 

[avenged*7]

An analysis of the pool geometry as shown is the video is going to be highly complicated- the elasticity of the pool walls are an essential component of the fluid motion and forcing. The free surface makes the problem even messier. One simplification is to make it a 2-D problem (radial and height coordinates), but this type of problem is typically attacked numerically.

Thanks for your reply. The type of geometry is not exactly the same as shown in the video. In the case I am interested, it is a perfect cylinder with rigid walls.

 

[Andy Resnick]

Thanks for your reply. The type of geometry is not exactly the same as shown in the video. In the case I am interested, it is a perfect cylinder with rigid walls.

Ah- that simplifies the problem immensely. It's called 'Faraday resonance' and has been studied for a while:

http://www.uvm.edu/~pdodds/files/papers/others/1984/miles1984a.pdf

 

[avenged*7]

Ah- that simplifies the problem immensely. It's called 'Faraday resonance' and has been studied for a while:

http://www.uvm.edu/~pdodds/files/papers/others/1984/miles1984a.pdf

Thanks again! I looked into Faraday waves, but not Faraday resonance. The effect generated in the video above very much mirrors the one generated in the pool with rigid sides, which leads me to believe that much approximation can be done without removing the desired effect. I need to study the paper in detail. I'm wondering why there is no Bessel functions in there for the surface heights, though it may not appear since they are working in phase space.

 

[boneh3ad]

The problem I see with using the Faraday problem as a model here is that the forcing mechanism is entirely different. Faraday waves are forced by a vibration of the container, whereas the waves from the video are forced by an oscillation at the center of the container. It seems to me that the better simplified model would be to take a rigid, cylindrical container, and apply a forcing function at a point at the center of the cylinder that takes the form of an oscillation of the surface height. You wouldn't get the feedback from the walls, but you would get the essential forcing mechanism more correct than using Faraday resonance. If you wanted to get more complicated, it's probably more realistic to use a periodic body force at that point, though it would be more complicated to solve.

 

[avenged*7]

The problem I see with using the Faraday problem as a model here is that the forcing mechanism is entirely different. Faraday waves are forced by a vibration of the container, whereas the waves from the video are forced by an oscillation at the center of the container. It seems to me that the better simplified model would be to take a rigid, cylindrical container, and apply a forcing function at a point at the center of the cylinder that takes the form of an oscillation of the surface height. You wouldn't get the feedback from the walls, but you would get the essential forcing mechanism more correct than using Faraday resonance. If you wanted to get more complicated, it's probably more realistic to use a periodic body force at that point, though it would be more complicated to solve.

Thanks for your input. I understand that Faraday waves are generally produced by vibrating the container. Since there is no inertial forces involved, I am wondering if using the container frame would be equivalent to having the fluid move (though that would probably mean ALL of the fluid would move, instead of just the surface).

I have tried to figure out how to include a periodic force such as the one in the video. The problem is that I have no idea how to impose that force over a finite area, aside from splitting the pool into regions (one forced, one under atmospheric pressure) and splicing the regions via continuity of the derivatives. If you have any ideas on how to do that, I am wide open to suggestion. I am hoping that a single-mode Faraday wave is sufficient for our purposes since extensive work has already been done.

The other problem I have is determining the decay time of the wave. I have some ideas, but I have a lot of work to do in that regard.