Calculate the max speed at which a submarine rises/dives

I'm trying to figure out how I would calculate the rate at which a submarine (or any vaguely cylindrical object) ascends or descends in seawater given a certain buoyancy or change in buoyancy. For example, if my submarine is 2000 feet below sea level and weighs 393 metric tons (with empty ballast tanks) and displaces 13,583 cubic feet of water, what's the maximum rate it will rise to the surface (assuming it can reach that rate before breaking the surface) and it's acceleration/deceleration to that rate? I'd like to know if there's a reliable and accurate way to solve a problem like this that takes into account factors like the change in sea water pressure and density with depth, as well as the increase in water resistance against the submarine as it accelerates upward. If not, would a CFD program like Ansys CFX or Autodesk CFD be capable of simulating a situation like this accurately?

Welcome to PF.

You ask about submarines, but it sounds like you are wanting to do this for unpowered objects in water, not powered vehicles like submarines? Submarines can do 30 knots under power while submerged, so you would need to add that to whatever you can get from just buoyancy alone.

https://en.wikipedia.org/wiki/Underwater_speed_record

berkeman said:

Welcome to PF.

You ask about submarines, but it sounds like you are wanting to do this for unpowered objects in water, not powered vehicles like submarines? Submarines can do 30 knots under power while submerged, so you would need to add that to whatever you can get from just buoyancy alone.

https://en.wikipedia.org/wiki/Underwater_speed_record

My goal is to do this for a powered submarine, albeit a very small and slow one like the NR-1; it's supposed to have a maximum speed of about 6 knots submerged. For context, I'm drafting an idea for a submarine simulator game with a heavy focus on realism, but this obviously requires doing a lot of complex math that I'm unfamiliar with.

Massimo said:

TL;DR Summary: How would I calculate the rate at which a submarine ascends or descends in seawater given a certain buoyancy or change in buoyancy?

I'd like to know if there's a reliable and accurate way to solve a problem like this that takes into account factors like the change in sea water pressure and density with depth, as well as the increase in water resistance against the submarine as it accelerates upward.

First, get a years experience in fluid dynamics. You will need to know the drag coefficient of the submarine as it moves through the water. That will depend on orientation.

Seawater pressure increase significantly with depth, but pressure is not critical in determining speed. Water density changes submarine buoyancy, more with salinity and temperature variation, than with pressure.

The greatest speed would be under power, in an emergency breach.

Massimo said:

My goal is to do this for a powered submarine, albeit a very small and slow one like the NR-1; it's supposed to have a maximum speed of about 6 knots submerged. For context, I'm drafting an idea for a submarine simulator game with a heavy focus on realism, but this obviously requires doing a lot of complex math that I'm unfamiliar with.

Really Nice Idea, but perhaps conceptual ideas may be the best here. There is a lot of variables and unknowns that would come into modelling a physical system like the one your describing. The best area of physics would probably be a mix of Fluid Dynamics and Classical Dynamics (Specifically Lagrangian and Hamiltonian approaches). One major problem with trying to define this system is that it will be very hard to calculate quantitative results from people online , i also think for a game a qualitative approach can be just as detailed and a satisfactory answer.

While the calculations are possible other things are needed to be known from a physics perspective until you can solve this type of problem. Your system would have a fair amount of degrees of freedom and the constraints would make for some very complicated equations of motion. Secondly not only would the fact that the constraints on this system change in time, but they will not change in a way that is predictable in time either.

This is what we call Chaotic Dynamics where in this case the path of the Submarine would only be described by 1 curve in 1 instance with 1 set of initial conditions and changing the initial conditions of your problem(e.g. the subs position, the water density at different points, the waters direction and velocity, the dynamics of the submarines geometry with respect to the water, etc etc etc would drastically change the mathematics that describe the path of the submarine, from a physics perspective.

In other words its unrealistic for us to try and calculate this type of problem for you online. Good Luck with the game.

It would be best to generalise the problem for calculations e.g make the velocity of the sub constant , assume a constant density of water, assume the waters velocity doesn't affect the submarines velocity, assume the mass of the fuel/the sub is const etc..

I would recommend modelling this as a block sliding on a surface with a coefficient friction of that of water to find you a speed my friend. complicated physics = bad because physicist like me = lazy

If the submarine is moving forward and pointed up, then the solution is straightforward. The submarine has a coefficient of drag, which can be estimated by searching coefficient drag submarine. The total drag is calculated from the frontal area, the speed, and the drag coefficient.

You have a thrust, which is the sum of any driving power plus the net buoyancy. If the net thrust is greater than the drag, it is accelerating. If less, it is decelerating. The speed at which the net thrust equals the drag is the steady state speed. The rate of climb is the vertical component of the forward speed. If you just want the steady state speed, that is easily calculated. If you are accelerating or decelerating to a steady state speed from a different initial speed, then the calculation is easily done using numerical techniques. In either case, the calculation will be easier, faster, and more accurate than CFD.

If the submarine is dead in the water, in a horizontal orientation, and rising vertically due to buoyance forces, the same calculation is used. The only change is the drag coefficient, which is the drag coefficient for a cylinder in sideways flow. The drag coefficient of a cylinder moving sideways is a function of the Reynolds number as shown in the figure below, from Fluid-Dynamic Drag by Hoerner.

Hint: Do some steady state speed Reynolds number calculations before you try to build a drag coefficient lookup table in your program. You may find that you can get good results by using a fixed value.