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# ES 240 project: Sloshing of a liquid in a cylindrical tank

Although this is a class in solid mechanics, I chose a project which focuses on fluid dynamics. I did this because I think this project will be very educational with respect to the Finite Element Analysis and mechanics in general.

Project motivation: When I was working in an office this past summer, one day I walked around with a cup of coffee in my hand. I noticed that the amplitude of vibration of the liquid in the cup, due to my shaking hand, was visibly depedent on how fast I was walking. Unfortunately the resonant frequency seemed to be about the same as my walking pace, and to avoid spilling the coffee I had to either be very careful (keep my hand very still) or walk much faster or slower. I started wondering if there was another way of decreasing the amplitude of vibration of the liquid.

Project description: I will investigate the sloshing of a liquid in a cylindrical tank. I will focus on the modes and harmonic frequencies of vibration due to harmonic forces acting on the tank. I will investigate responses to the following forces: horizontal, circular horizontal, vertical and perhaps a combination of these. I will also look into possibility of reducing the amplitude of vibration using simple alterations of the tank’s geometry – e.g. using baffles of various shapes.

FEM contribution: The equations involved in solving this problem are non-linear PDEs and therefore it is extremely hard to obtain an analytical solution. When the cylindrical symmetry of the system is interrupted by adding an additional geometric feature (e.g. a baffle) the problem’s complecity increases significantly – FEM becomes the only reasonable, and certainly the easiest and the quickest, method to solve the problem.

Rsearch done on the problem so far: This problem is in no way novel – it has been researched thoroughly, as it has multiple importan applications in designing maritime structures and transportation tanks, e.g. for the Liquified natural gas. I chose this problem because I believe that the purpose of this assignement is not to create something novel but rather to gain insight into a physical phenomenon.

One of the large number of journal articles published on this subject is:

[1] Liu, Dongming, and Pengzhi Lin. "A numerical study of three-dimensional liquid sloshing in tanks". Journal of Computational Physics. Vol. 227, 8 (April 2008), pp 3921-3939.

Note: When writing the report I was working under some time constraints, and unfortunately I didn't have time to fully proof-read it. As a result, there might be multiple spelling, grammar and other mistakes. I will upload a revised version as soon as possible. The basic content, however, is unlikely to change.

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## Comments

## Why Man Created The Travel Mug...

Pawel,

I nearly spill my coffee all the time too.

This is an interesting topic, and one that is fairly important to many areas of mechanical engineering. Espescially important to vehicle dynamics, violent sloshing can radiaclly change a vehicle's performance envelope in dynamic circumstances.

I found an interesting article on Google Scholar that talks about seismically-excited sloshing. It might be a good reference for dealing with multiple simultaneous forcing functions; it's linked below.

-Tom

http://doi.wiley.com/10.1002/(SICI)1096-9845(199607)25:7%3C653::AID-EQE513%3E3.0.CO;2-H

## "Contents are Hot"

Hot coffee is a serious problem with much work being done on it in terms of legality

http://64.233.169.132/search?q=cache:GbYEBcPI-x0J:www.okbar.org/public/judges/mcdonaldsoutline.pdf+warning+contents+hot+coffee&hl=en&ct=clnk&cd=4&gl=us

and design

http://www.gizmodo.com.au/2007/06/because_contents_are_hot_warni.html

Actually in all seriousness I have found this topic (sloshing of a fliud) interesting since I did a problem on a partially filled milk tanker in introductory fluid mechanics. One thing that you might want to think about though is how ABAQUS will handle this problem. I looked through the documentation and initially found that solving this problem might not be possible to do in ABAQUS. If you look in the ABAQUS 6.8 documentation under Abaqus Analysis User's Manual in Analysis Techniques and then Section 11.5.1, one can see that it is very easy to model a fluid-filled cavity in Abaqus. However, it mentions in this section that the technique used to do so cannot be extended to a partially-filled container; it even directly says that this cannot be applied to sloshing. However, upon further investigation, I found that in Section 13.1.1 that an Eulerian approach can effectively handle liquid sloshing. So, this will be very important to your project. I have also found that this section was not included in the 6.6 documentation, so this feature may not be available in that version. You should definitely look into this and see if you need to find access to Version 6.8.

http://www.hlrs.de/organization/aw/services/cae/app/abaqus/Documentation/docs/v6.8/books/usb/default.htm

Also, I have found an article that may be of interest to you. It gives an overview of different numerical techniques (11 of them) that are used to model sloshing of liquid cargo in ship tanks. Here is the link

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V41-3XRG6T9-5&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=33b663f98abf83a045e45543293e0e1f

-Matt

## Comment on Project

Dear Pawel,

Your project is a neat one, and one about which I've given a bit of thought. I think it will be more about the fluid mechanics than about fluid solid interactions, but it would be neat to simulate this in the low-modulus limit of the cup, where some of the inertial energy in the fluid goes into elastic deformation of the container.

An alternative approach would be to change the container geometry, to see if you can mitigate the formation of the waves. It appears as though the wavelength in a coffee cup is proportional to its diameter; perhaps by making something like a soda bottle, with a narrow top, the sloshing will stop. Alternatively, you could try to divert the waves, by reflecting them off of an angled surface.

Good luck!

John