Compute Buoyancy Force on Irregular 3D Model in Real Time

[frs]

Hi everyone,
Its my first post and am not sure if my trivial question really deserves to be on this forum. But it is troubling me since few days and hence would really appreciate if someone help me out.

I am computing the buoyancy force on an irregular shaped object (tessellated 3D model of boat with triangular facets). Specifics of the computing scenario are as follows:
1. The model is floating and I am exactly (to numeric precision) computing the wetted region of the model. The wetted region is nothing but the set of triangles which is a subset of the triangles representing the model.
2. The water level is changing in time and in space (an ocean surface). It follows a law such as
z = A₁cos(B₁x + B₂y + B₃t)
Where, A, B, C are constants representing the amplitude, direction, and frequency respectively,
x,y is coordinate of the point on the ocean nominal plane,
z is water height, and
t is any given time

3. For each wet triangle, I compute the height of the water (at its centroid at that time) and compute the volume of water column above the wet triangle. Then I sum them to get the displaced volume of water and use it to compute the buoyancy force.

Now my question pertains to the step 3. Is it right to assume that the water height just above the wet triangles in the bottom of the model (boat) is same as the wave height had the boat not been there (i.e by using the equation given in the step 2)? Or I should rather ask, what could be the errors due to this assumption?
Although I am computing things this way and results looks okay visually, I think that it is not correct from physics point of view.
I think it could potentially be solved accurately in Navier-Stokes formalism, but is there some faster (computationally) way which doesn't require me to solve the movement of fluid due to the boat motion explicitly.
I am sure physics people must have solved this problem in numerous ways. If someone knows of a good reference or some formula (faster computation), please let me know. I am emphasizing fast computing as this computation needs to run in real time.
Thanks
-frs

 

[Curl]

You are making an error assuming that, although I don't know if the error is significant. If you go back and think about what causes the water waves, you'll see that it is a little trickier than just looking at surface effects. In essence, there is motion in the fluid which varies with x,y,z,t, and what you will need to find out is what is "hitting" the triangular area element at the bottom of the boat. It isn't as simple as a hydrostatic problem since the motion of water down there imparts impulses to the boat, which isn't the same as if the boat was not there (and water was there instead).

But like I said, the error is probably not significant. In fact, if the water waves are small compared to the area of the boat (as seen from above), I don't see why you are not just approximating the buoyant force as constant; of course that depends on the constants B1 B2 and A1. In essence the waves will average out to a constant with small deviation, its a statistical effect.

 

[frs]

You are making an error assuming that, although I don't know if the error is significant. If you go back and think about what causes the water waves, you'll see that it is a little trickier than just looking at surface effects. In essence, there is motion in the fluid which varies with x,y,z,t, and what you will need to find out is what is "hitting" the triangular area element at the bottom of the boat. It isn't as simple as a hydrostatic problem since the motion of water down there imparts impulses to the boat, which isn't the same as if the boat was not there (and water was there instead).

But like I said, the error is probably not significant. In fact, if the water waves are small compared to the area of the boat (as seen from above), I don't see why you are not just approximating the buoyant force as constant; of course that depends on the constants B1 B2 and A1. In essence the waves will average out to a constant with small deviation, its a statistical effect.

Thanks for the reply!
About not using average value: I need to simulate the boat motion in real time (visually, about 20Hz or faster) in all the six degrees of freedom (three linear and three angular). If I use the statistical measure, the boat is not balanced when I move it by applying forces.
And, what I am exactly looking for is the estimation of additional forces than the hydrostatic one as you pointed out (the impact force).
In other words, some kind of surrogate model, which not only include the additional fluid effects but also preserves the restoring effect of buoyancy in all six degrees of freedom.
-frs

 

[Curl]

I'm not sure what you mean by "the boat is not balanced when I move it by applying forces." Since the mass of the boat is large, the impulses from random water motion will have little effect. The way of looking at balance is to find the resultant force of the buoyancy force, which is acting through the centroid of the boat volume underneath water. It will act essentially straight up (with tiny deviations). That's how you do balance on floating objects. If you apply a force to the boat which generates a moment about the boat's CG (most likely the case) then the boat will "tip" until its CG is far enough away from the displaced water's centroid to generate an equal counter-moment.

I don't know if this is what you're asking, you probably know this already.

 

[pmacias]

I understand your concerns about the accuracy of your approach and the potential for errors. It is important to consider the assumptions and limitations of any method used in scientific research.

In this case, your approach of calculating the buoyancy force by summing the volumes of water displaced by each wet triangle is a reasonable approximation. However, it is not entirely accurate as it assumes that the water height above the wet triangles is the same as the wave height without the boat present. This can introduce some errors, particularly for irregularly shaped objects.

To improve the accuracy of your calculations, you could consider using a more sophisticated approach such as the Navier-Stokes equations, which take into account the movement of fluid due to the boat's motion. This would likely involve a more computationally intensive process, but it may provide more accurate results.

As for references or formulas, I suggest looking into literature on computational fluid dynamics (CFD) and specifically, the study of buoyancy forces on irregularly shaped objects. There may be specific methods or equations that have been developed for this type of problem.

In terms of faster computation, you could also consider using parallel computing methods or optimizing your code for efficiency. However, it is important to balance speed with accuracy, so it may be worth investing in a more accurate approach if it is feasible.

I hope this helps and I wish you all the best in your research. Good luck!