Author: kvan637 (page 1 of 8)

Chapter 2: Dimensional Analysis, Buckingham Pi Theorem, and the physics of flight


Chapter 2 of A Guided Tour of Mathematical Methods for the Physical Sciences deals with dimensional analysis; a great way to test the accuracy of equations that describe physical laws. It also introduces the Buckingham Pi Theorem, allowing us — under the right conditions — to find the relationship between physical parameters based on such a dimensional analysis. At the end of this post is a link to a jupyter notebook to explore the topic yourself.

The relation between weight and flying speed

One of the examples examines the relationship between physical parameters involved in flight. The book by Henk Tennekes contains a collection of the cruising speed $v$ and weight $W$ of heaps of flying objects and animals. Let us have a look at the data first:

In [ ]:
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np

url="https://ndownloader.figshare.com/files/9764671"
df = pd.read_csv(url,skiprows=5, index_col=False)
speed = df['cruising speed (m/s)'].values
weight = df['weight (newtons)'].values

As the weights $W$ and speeds $v$ of these fliers ranging from small insects to large aircraft span orders of magnitude, it makes sense to plot on log-log scale:

In [8]:
plt.plot(np.log10(speed), np.log10(weight),'k.')
plt.xlabel('10log(v)')
plt.ylabel('10log(W)')
plt.show()

These data appear to follow a line (on this log-log scale), with the largest outliers involving “exotic” fliers such as a large dynosaur and a human-powered aircraft. Nevertheless, a regression on log-log scale of the data, results in:

In [6]:
from scipy.stats import linregress
slope, intercept, r_value, p_value, std_err = linregress(np.log10(speed), np.log10(weight))

# set up an array to plot the regression line from the fitted parameters:
xfid= np.linspace(0,2.5)

plt.plot(np.log10(speed), np.log10(weight),'k.',label='fliers')
plt.plot(xfid, xfid*slope+intercept,label='regression with slope {:1.2f}'.format(slope))
plt.xlabel('10log(speed)')
plt.ylabel('10log(weight)')
plt.legend(loc=0)
plt.show()

The slope of the best-fitting line is approximately 5.58. Is this a coindidence? No! If you want to learn why, we encourage you to work your way through section 2.3-5 of our book. A dimensional analysis with the Buckingham Pi Theorem shows

$$W \propto v^{6}.$$

On a logarithmic scale, we find that $$\log(W) \propto \log(v^{6}) = 6 \log (v),$$ so that our experimentally determined slope of 5.58 is not half bad! Of course, the relationship of weight and speed of flight involve other variables. if you want to learn how the density of the object and the air, gravity, and the shape and size of the wings come in to play, please read Chapter 2 of our book. And if you want to have a play with these data yourself, here is the jupyter notebook.

A 3D view of compressional wave speed under New Zealand

The movie below is a 3D visualization of estimates of the p-wave speed under New Zealand, from supplementary material of this paper. The three orthogonal planes cross approximately 50 km under Auckland.

 

 

These graphics are made with the python package mayavi.

Jami got engaged!

Jami Johnson sent us word of her engagement to her Kiwi  boyfriend Sam! Here is a picture of the happy couple on a recent trip to Norway. We wish Sam and Jami all the best (and for them to move back to New Zealand, of course).

Apple Seismology

Today, our paper on Apple Seismology appeared in Physics Today. It was a follow up on Sam and Zoe’s experiments, highlighting the similarities between seismic waves and normal modes in the Earth, and their acoustic equivalents in a Braeburn apple.

In the panel on the left, seismic (surface) waves traverse Earth, while in the right panel laser-generated and detected surface waves circle the apple.

Congratulations, Dr. Jami Johnson!

It’s the end of an era… Jami Johnson successfully completed her PhD thesis in photoacoustic imaging.  She is now off to The University Pierre and Marie Curie, in Paris (France) to pursue postdoctoral research in medical imaging. We wish Jami all the best, and we will follow her in what will undoubtedly be a very exciting career. The PALs want to thank Jami for all her efforts and the wonderful memories she leaves behind.

 

Postgraduate opportunities in our lab

Below are short descriptions of research projects available for postgraduate students in our group at the University of Auckland. If you are interested in any of these, please contact us for technical questions. For questions around enrollment at the University of Auckland, please go here. Scholarships may be available for the best applications through the University of Auckland, or the Dodd Walls Centre.

ResonanT Ultrasound Spectroscopy

Resonant ultrasonic spectroscopy fills an important gap between our ultrasonic and seismic research. Together with the PORO lab, we have rock samples to complement laser ultrasound and a host of other petrophysical data with new RUS results.

Laser ultrasonic rock physics under high pressure and/or temperature

The PAL and PORO lab join forces by combining our respective strengths in laser ultrasound and rock physics to improve data quality and quantity. In this project, we are building the expertise to do laser ultrasound in a pressure vessel with optical windows. Source/receiver locations are varied under computer control with an arduino-controlled servo rotational stage.

unraveling the mysteries of the Auckland volcanic field

Part of an active volcanic field, questions surround the nature of Auckland’s past, present and future. Using a suite of seismic techniques that range from ambient seismic noise tomography, to receiver functions and body wave tomography, we aim to build a representative model that helps us explain the geodynamics of the Auckland Volcanic Field.

quality control of timber and fruit products with laser ultrasound

Laser ultrasound can be applied to products of interest to a wide community. Current methods of testing the quality of fruit and timber, for example, can often be described by one or more of the following terms: sparse, contacting, expensive, and often destructive. In this project, we aim to explore the opportunities for laser ultrasound in estimating the quality of  fruit and timber in a non-contacting and non-destructive matter.

Our first glimpses into the Auckland Volcanic Field

Today, the New Zealand Journal of Geology and Geophysics published Josiah‘s first results using ambient seismic noise to peek deep into the lithosphere under the Auckland Volcanic Field (AVF). This noise, created by the oceans surrounding the AVF, suggests the lithosphere under Auckland is made up of oceanic and continental components. Further studies aimed at increasing the resolution of our results may help explain why this active volcanic field is where it is.

Laser ultrasonics on ice cores

Many years ago, Andrei Kurbatov asked us if we could do laser ultrasound on ice cores, during half time of an international football match we were watching in Golden, Colorado. Dylan Mikesell started these measurements in a home-made refridgerator in the Physical Acoustics Labd as a research project toward his undergraduate degree at Colorado School of Mines. We have come a very long way since, and today — with the additional help of Thomas Otheim and Hans Peter Marshall, our first paper on laser ultrasound on ice cores from Greenland and the Antarctic appeared in the journal Geosciences.

A new paper on Marchenko-equation medical imaging

Our collaboration with TU Delft has led to a new publication in the Journal of the Acoustical Society of America. Led by Joost van der Neut and Jami Johnson, this article adds the Marchenko equation to our tool set to image ultrasonic and photoacoustic data for medical applications. Perfect timing, Jami, for your PhD thesis defense tomorrow!

Lab photos

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