Assignment 1(a): Roll added-inertia of a submerged body¶

In this assignment you will extend the functions from the panel method notebook to predict the flow and forces induced by roll motion.

Sketch a body subject to roll motion $\Omega_z=U_6$ around the point $(x_m,y_m)$ . Determine the equation for the normal body velocity $U_n$ at any point $(x,y)$ on the body surface. Using this, define the RHS of the body boundary condition $\frac{\partial\phi_6}{\partial n} = b_6$ . Define the equation for the roll added-inertia $m_{66}$ in terms of these variables.
Implement a function roll which takes in the coordinate arrays x,y defining a shape and the roll center xm,ym and returns the source strength vector q6 and added-inertia m66. Make sure you explain the linear algebra equations you have implemented for q6,m66. Validate the roll function by comparing to a known $m_{66}$ for a fully submerged shape (cite your reference) and discuss the convergence behaviour. Plot the induced flow and discuss it's physical characteristics.

Due 23:59, 24 Feb 2022¶

Write-up your solution for both Q1 & 2 and submit as a PDF or HTML. Note: If you make a PDF, you need to include images of important functions and the results plots. Exporting a notebook as HTML will contain everything automatically.
Submit the working python code needed to reproduce those plots as a PY or IPYNB. Note: Explain your method/results in the write-up, not in the code. Any time you are tempted to write "see code for ____", put the important information in the write-up instead.

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# Import functions
import requests
url = 'https://raw.githubusercontent.com/weymouth/MarineHydro/master/src/PanelMethod.py'
exec(requests.get(url).content)

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# Complete function
# def roll(x,y,xm=0,ym=0,G=source,args=()):
#     return q6,m66

# Create validation geometry, validate m66, and plot flow