Mathematics deals with pure numbers: 1, 2, 3, fractions and more exotic numbers like π. Since π depends on lengths, we might think its value depends on our units. But it is the ratio of the circumference of a circle to its diameter and, as long as both are measured in the same units — … Continue reading Dynamic Similarity and the Reynolds Number
Tag: Fluid Dynamics
Simple Models of Atmospheric Vortices
Atmospheric circulation systems have a wide variety of structures and there is no single mechanistic model that describes all their characteristics. However, we can construct simple kinematic models that capture some primary aspects of the flow. For simplicity, we will concentrate on idealized extra-tropical depressions. We will not consider hurricanes and tropical storms in any … Continue reading Simple Models of Atmospheric Vortices
Zhukovsky’s Airfoil
A simple transformation with remarkable properties was used by Nikolai Zhukovsky around 1910 to study the flow around aircraft wings. It is defined by $latex \displaystyle \omega = \textstyle{\frac{1}{2}}\displaystyle{\left(z +\frac{1}{z}\right)} &fg=000000$ and is usually called the Joukowsky Map. We begin with a discussion of the theory of fluid flow in two dimensions. Readers familiar with … Continue reading Zhukovsky’s Airfoil
Using Maths to Reduce Aircraft Noise
If you have ever tried to sleep under a flight-path near an airport, you will know how serious the problem of aircraft noise can be. Aircraft noise is amongst the loudest sounds produced by human activities. The noise is over a broad range of frequencies, extending well beyond the range of hearing. The problem of … Continue reading Using Maths to Reduce Aircraft Noise
Stokes’s 200th Birthday Anniversary
Next Tuesday, the 30th of August, is the 200th anniversary of the birth of George Gabriel Stokes. This extended blog post is to mark that occasion. See also an article in The Irish Times. Whether we are designing aircraft, modelling blood flow, studying propulsion, lubrication or the dynamics of swimming, constructing wind turbines or forecasting … Continue reading Stokes’s 200th Birthday Anniversary
Spin-off Effects of the Turning Earth
On the rotating Earth, a moving object deviates from a straight line, being deflected to the right in the northern hemisphere and to the left in the southern hemisphere. The deflecting force is named after a nineteenth century French engineer, Gaspard-Gustave de Coriolis [TM164 or search for “thatsmaths” at irishtimes.com]. Coriolis was interested in the dynamics of machines, … Continue reading Spin-off Effects of the Turning Earth
A Zero-Order Front
Sharp gradients known as fronts form in the atmosphere when variations in the wind field bring warm and cold air into close proximity. Much of our interesting weather is associated with the fronts that form in extratropical depressions. Below, we describe a simple mechanistic model of frontogenesis, the process by which fronts are formed. Life-cycle … Continue reading A Zero-Order Front
Tides: a Tug-of-War between Earth, Moon and Sun
All who set a sail, cast a hook or take a dip have a keen interest in the water level, and the regular ebb and flow of the tides. At most places the tidal variations are semi-diurnal, with high and low water twice each day [see TM144, or search for “thatsmaths” at irishtimes.com]. Equilibrium Tides In the … Continue reading Tides: a Tug-of-War between Earth, Moon and Sun
Stan Ulam, a mathematician who figured how to initiate fusion
Stanislaw Ulam, born in Poland in 1909, was a key member of the remarkable Lvov School of Mathematics, which flourished in that city between the two world wars. Ulam studied mathematics at the Lvov Polytechnic Institute, getting his PhD in 1933. His original research was in abstract mathematics, but he later became interested in a … Continue reading Stan Ulam, a mathematician who figured how to initiate fusion
Waves Packed in Envelopes
In this article we take a look at group velocity and at the extraction of the envelope of a wave packet using the ideas of the Hilbert transform. Interference of two waves A single sinusoidal wave is infinite in extent and periodic in space and time. When waves interact, the dynamics are more interesting. The … Continue reading Waves Packed in Envelopes
The Heart of Mathematics
At five litres per minute the average human heart pumps nearly 200 megalitres of blood through the body in a lifetime. Heart disease causes 40 percent of deaths in the EU and costs hundreds of billions of Euros every year. Mathematics can help to improve our knowledge of heart disease and our understanding of cardiac … Continue reading The Heart of Mathematics
Energy Cascades in Van Gogh’s Starry Night
"Big whirls have little whirls that feed on their velocity, And little whirls have lesser whirls, and so on to viscosity." We are all familiar with the measurement of speed, the distance travelled in a given time. Allowing for the direction as well as the magnitude of movement, we get velocity, a vector quantity. In … Continue reading Energy Cascades in Van Gogh’s Starry Night
Euler and the Fountains of Sanssouci
When Frederick the Great was crowned King of Prussia in 1740 he immediately revived the Berlin Academy of Sciences and invited scholars from throughout Europe to Berlin. The most luminous of these was Leonhard Euler, who arrived at the academy in 1741. Euler was an outstanding genius, brilliant in both mathematics and physics. Yet, a … Continue reading Euler and the Fountains of Sanssouci
Inertial Oscillations and Phugoid Flight
The English aviation pioneer Frederick Lanchester (1868--1946) introduced many important contributions to aerodynamics. He analysed the motion of an aircraft under various consitions of lift and drag. He introduced the term ``phugoid'' to describe aircraft motion in which the aircraft alternately climbs and descends, varying about straight and level flight. This is one of the … Continue reading Inertial Oscillations and Phugoid Flight
The Water is Rising Fast
Seventy percent of the Earth is covered by water and three quarters of the world's great cities are on the coast. Ever-rising sea levels pose a real threat to more than a billion people living beside the sea. As the climate warms, this is becoming a greater threat every year [TM113 or search for “thatsmaths” … Continue reading The Water is Rising Fast
Thank Heaven for Turbulence
The chaotic flow of water cascading down a mountainside is known as turbulence. It is complex, irregular and unpredictable, but we should count our blessings that it exists. Without turbulence, we would gasp for breath, struggling to absorb oxygen or be asphyxiated by the noxious fumes belching from motorcars, since pollutants would not be dispersed … Continue reading Thank Heaven for Turbulence
Modelling Rogue Waves
There are many eyewitness accounts by mariners of gigantic waves – almost vertical walls of water towering over ocean-going ships – that appear from nowhere and do great damage, sometimes destroying large vessels completely. Oceanographers, who have had no way of explaining these 'rogue waves', have in the past been dismissive of these reports [TM090, or search for … Continue reading Modelling Rogue Waves
Richardson’s Fantastic Forecast Factory
Modern weather forecasts are made by calculating solutions of the mathematical equations that express the fundamental physical principles governing the atmosphere [TM083, or search for “thatsmaths” at irishtimes.com] The solutions are generated by complex simulation models with millions of lines of code, implemented on powerful computer equipment. The meteorologist uses the computer predictions to produce … Continue reading Richardson’s Fantastic Forecast Factory
The Flight of a Golf Ball
Golf balls fly further today, thanks to new materials and mathematical design. They are a triumph of chemical engineering and aerodynamics. They are also big business, and close to a billion balls are sold every year. [TM081: search for “thatsmaths” at Irish Times ]. The golfer controls the direction and spin of the ball by … Continue reading The Flight of a Golf Ball
Life’s a Drag Crisis
The character of fluid flow depends on a dimensionless quantity, the Reynolds number. Named for Belfast-born scientist Osborne Reynolds, it determines whether the flow is laminar (smooth) or turbulent (rough). Normally the drag force increases with speed. The Reynolds number is defined as Re = VL/ν where V is the flow speed, L the length … Continue reading Life’s a Drag Crisis
The Hodograph
The Hodograph is a vector diagram showing how velocity changes with position or time. It was made popular by William Rowan Hamilton who, in 1847, gave an account of it in the Proceedings of the Royal Irish Academy. Hodographs are valuable in fluid dynamics, astronomy and meteorology. The idea of a hodograph is very simple. … Continue reading The Hodograph
Plateau’s Problem and Double Bubbles
Bubbles floating in the air strive to achieve a spherical form. Large bubbles may oscillate widely about this ideal whereas small bubbles quickly achieve their equilibrium shape. The sphere is optimal: it encloses maximum volume for any surface of a given area. This was stated by Archimedes, but he did not have the mathematical techniques … Continue reading Plateau’s Problem and Double Bubbles
Falling Bodies [1]: Sky-diving
Aristotle was clear: heavy bodies fall faster than light ones. He arrived at this conclusion by pure reasoning, without experiment. Today we insist on a physical demonstration before such a conclusion is accepted. Galileo tested Aristotle's theory: he dropped bodies of different weights simultaneously from the Leaning Tower of Pisa and found that, to a … Continue reading Falling Bodies [1]: Sky-diving
El Niño likely this Winter
This week’s That’s Maths column in The Irish Times (TM056 or search for “thatsmaths” at irishtimes.com) is about El Niño and the ENSO phenomenon. In 1997-98, abnormally high ocean temperatures off South America caused a collapse of the anchovy fisheries. Anchovies are a vital link in the food-chain and shortages can bring great hardship. Weather … Continue reading El Niño likely this Winter
How Big was the Bomb?
By a brilliant application of dimensional analysis, G.I.Taylor estimated the explosive energy of the first atomic blast, the Trinity Test (see this week’s That’s Maths column in The Irish Times, TM053, or search for “thatsmaths” at irishtimes.com). Physicists, engineers and applied mathematicians have an arsenal of problem-solving techniques. Computers crunch out numerical solutions in short … Continue reading How Big was the Bomb?
White Holes in the Kitchen Sink
A tidal bore is a wall of water about a metre high travelling rapidly upstream as the tide floods in. It occurs where the tidal range is large and the estuary is funnel-shaped (see previous post on this blog). The nearest river to Ireland where bores can be regularly seen is the Severn, where favourable … Continue reading White Holes in the Kitchen Sink
Interesting Bores
This week’s That’s Maths column in the Irish Times ( TM036 ) is about bores. But don't be put off: they are very interesting. According to the old adage, water finds its own level. But this is true only in static situations. In more dynamic circumstances where the water is moving rapidly, there can be … Continue reading Interesting Bores
A Mathematical Dynasty
The idea that genius runs in families is supported by many examples in the arts and sciences. One striking case is the family of Johann Sebastian Bach, the most brilliant star in a constellation of talented musicians and composers. In a similar vein, several generations of the Bernoulli family excelled in science and medicine. More … Continue reading A Mathematical Dynasty
Paddling Uphill
Recently, I kayaked with two friends on the River Shannon, which flows southward through the centre of Ireland. Starting at Dowra, Co. Cavan, we found it easy paddling until we reached Lough Allen, when the going became very tough. It was an uphill struggle. Could we really be going uphill while heading downstream? That seems … Continue reading Paddling Uphill
The Pitch Drop Experiment
Later this year a big black blob of sticky pitch will plummet from a funnel and plop into a beaker. The story is recounted in this week's That's Maths ( TM017 ) column in the Irish Times. In one of the longest-running physics experiments, the slow-flowing pitch, under a bell-jar in the University of Queensland … Continue reading The Pitch Drop Experiment
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