Shape of water in a rotating container along A vertical axis

[Momo 6aye3]

Hi!
I am currently working on a project that includes rotating a water-filled container. The container is NOT spinning about its vertical axis, but about the vertical axis of the rotating disc.
I am aware that the surface shape of water in a rotating bucket takes the shape of a parabola when it is rotated about its axis, but what is the effect when rotated on a vertical axis that isn't its own?
My intuition tells me that it will be a part of a parabola whose vertex lies on the y-axis of the disc, but I am not sure.
Also, will the shape of the container affect the shape of the surface of the water? (The cylinder is placed horizontally on the disc).

Thank you!

 

[berkeman]

Hi!
I am currently working on a project that includes rotating a water-filled container. The container is NOT spinning about its vertical axis, but about the vertical axis of the rotating disc.
I am aware that the surface shape of water in a rotating bucket takes the shape of a parabola when it is rotated about its axis, but what is the effect when rotated on a vertical axis that isn't its own?
My intuition tells me that it will be a part of a parabola whose vertex lies on the y-axis of the disc, but I am not sure.
Also, will the shape of the container affect the shape of the surface of the water? (The cylinder is placed horizontally on the disc).

Thank you!

Welcome to the PF.

Could you Upload a figure that shows the situation? I think I understand the question, but the wording is hard for me to parse so far. Thanks.

 

[Momo 6aye3]

Welcome to the PF.

Could you Upload a figure that shows the situation? I think I understand the question, but the wording is hard for me to parse so far. Thanks.

Yo,
Imagine the following cylinder filled with water placed where the box is.

 

[Nidum]

Not difficult to solve in principle . This particular problem though has added difficulty because of the geometry of the container . Depending on the rotational speed the contained water could take up at least two different basic volume shapes and both of these would have difficult to define variable dimensions .

Do you really want to get straight into what would be a messy problem to deal with or would you like to start by finding the solution to a simplified version of the problem ?

 

[Dr.D]

Not difficult to solve in principle . This particular problem though has added difficulty because of the geometry of the container . Depending on the rotational speed the contained water could take up at least two different basic volume shapes and both of these would have difficult to define variable dimensions .

Why do you think the shape of the container has anything to do with the surface shape? I'm not seeing that. Would you explain, please?

 

[Nidum]

The water surface will tend to slope with the outboard end rising and the inboard end dropping .

Two conditions are possible . These are where the water in the outboard region has :

(a) not reached the top inner surface of the tube . The water has a free top surface
(b) reached the top inner surface of the tube . The water then has a partly constrained and a partly free top surface .

The top free surface will have curvature¹ . This curvature and the curvature of the tube interact . The geometric shape of the water volume and the way it changes shape under action of a pressure gradient varying with table spinning speed would in general be difficult things to determine analytically .

The problem can probably be solved analytically for two specific conditions where the spinning speed is :

(c) low and the water top surface is free and not much off level .
(d) very high and the water volume has become a near cylindrical slug at the outboard end . (The remaining free surface is then on the inboard end of the slug) .

Real problems of this nature can usually be solved more easily using direct numerical methods or CFD .

Note 1 : How complex this curvature is considered to be depends on the level of detail to which this problem is being looked into .

On a related topic and just for interest :

The way that leakage oil flows in the fast spinning complex shaped components of gas turbine rotors is a fascinating subject . The oil does not always do what you would expect it to .

Anyone curious have a think about this situation :

A spinning component has an empty space in it's construction which is essentially a C section void of revolution .

If trace oil gets into this void how does it distribute over the surface of the void ?

 

[A.T.]

The water surface will tend to slope with the outboard end rising and the inboard end dropping .

The water surface will be an equipotential surface of the combined gravitational and centrifugal potential. So the shape of the surface doesn't depend on the container shape, but it's elevation for a given spin rate can depend on it.

 

[Dr.D]

It seemed to me to be implicit in the original problem statement that the volume of water is less than the maximum interior volume of the container. If that is not true, that is, if the container is absolutely filled with water, then the shape of the water is the shape of the container under any and all conditions.

But for the partially filled container, the original question seemed to me to ask the shape of the free surface, not the shape of those parts that are bounded by the container walls.

wo conditions are possible . These are where the water in the outboard region has :

(a) not reached the top inner surface of the tube . The water has a free top surface
(b) reached the top inner surface of the tube . The water then has a partly constrained and a partly free top surface .

The top free surface will have curvature

Are these really two separate conditions, or simply slight variations on the same condition? I think that they are the same.

The final statement about the curvature (and the associated foot note) seem to relate to meniscus effects. Was this your intent @Nidum ? I would not think the meniscus would be considered in must such problems, but it does depend to a great extent on the container dimensions compared to the meniscus radius.

 

[Nidum]

Axis of spin is to the left of cylinder . Cylinder initially half filled with water . Water is red . Air is blue .

Cylinder dimensions were made large enough for surface tension effects to be safely ignored . This allowed a fairly basic CFD model to be used .

If I have an opportunity I'll do a more refined CFD model sometime with a smaller diameter cylinder .

 

[Dr.D]

@Nidum The figures you show in #9 are certainly interesting, but they are rather difficult to interpret since they attempt to show in 2D what is a complicated 3D surface. I'd like to raise two questions:

1) Imagine a cylindrical coordinate system aligned with the spins axis with the r-theta plane horizontal. Does your CFD software permit you to show a 2D plot in an r-z plane for various values of theta to sweep across the free surface?

2) Earlier, in #6p, you said,

The top free surface will have curvature1 . This curvature and the curvature of the tube interact

Can you provide a physical principle, law, or some such to support this statement? On what basis do you think that the curvature of the free surface interacts with the (submerged) surface of the container (or is it an interaction with the non-wetted portion of the container surface?)? I cannot begin to justify this in my own mind, and I'd like to be enlightened.

 

[A.T.]

Axis of spin is to the left of cylinder . Cylinder initially half filled with water . Water is red . Air is blue .

Can you also show the boundary surface between them (iso-surface of the alpha.phase parameter at 0.5)?

 

[Nidum]

I need to conclude a different CFD/Fluid mechanics exercise being worked on with another PF member before getting too involved in this one .

Give me a couple of days and then I'll attempt to deal with both your questions .

 

[A.T.]

I need to conclude a different CFD/Fluid mechanics exercise being worked on with another PF member before getting too involved in this one .

Give me a couple of days and then I'll attempt to deal with both your questions .

Okay, basically my question is:

Is the simulated water surface eqipotential in the rotating frame?

The effective potential in the rotating frame is the sum of the gravitational and centrifugal potentials:
https://en.wikipedia.org/wiki/Gravitational_potential
http://scienceworld.wolfram.com/physics/CentrifugalPotential.html

 

[Momo 6aye3]

@Nidum Hi there. I was interested in how the surface curvature of the water would effect the rollover velocity of the model. No analytics. I noticed when conducting experiments that surface curvature does indeed "The water surface will tend to slope with the outboard end rising and the inboard end dropping". From intuition, I know that the center of mass would shift outwards increasing the model's propensity to rollover. I can verify this experimentally as collected the results and they were much different compared to what the theory predicted. I just wanted an explanation on why the surface curvature took that shape.

I have researched about why water takes on a parabolic shape when rotated about its central vertical axis, I am assuming the same forces will apply but only the shape will different correct?

 

[Dr.D]

Give me a couple of days and then I'll attempt to deal with both your questions .

Do we have a follow up yet?

 

[A.T.]

I have researched about why water takes on a parabolic shape when rotated about its central vertical axis, I am assuming the same forces will apply but only the shape will different correct?

If the vehicle goes on an circular path at constant speed for a while, the surface shape will tend towards an Paraboloid of revolution. But this equilibrium analysis ignores sloshing around after abrupt transition between going straight and turning. That is the actual danger to vehicle stability.