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 sudden jumps in the surface level.
When fluid at high speed surges into a zone of slower flow, an abrupt change of depth, called a hydraulic jump, may occur. Hydraulic jumps are found in some river estuaries, below dams and weirs and even in the kitchen sink (see next post).
The Severn Bore
When the tide floods into a funnel-shaped river estuary, it can form a wall of water called a tidal bore that travels rapidly upstream against the current. The most notable example in this neighbourhood is the River Severn, where bores are observed several times each year under suitable conditions of Spring tide.
Surfers on the Severn bore (photograph by Mark Humpage, with thanks)
Typically, the flow is smooth or laminar upstream from the bore. Downstream it may be either wavelike (an undular bore) or chaotic (a turbulent bore).
Tidal bores propagating briskly upstream are popular with intrepid surfers, who ride them for miles up the river. But there are also stationary hydraulic jumps on a river, often found in the spillway of a dam or below a weir. These have recirculating water that traps flotsam and they provide entertainment for kayakers prepared to risk the hazard of being caught in the back-flow.
Characteristics of a hydraulic jump (image Wikimedia).
Conservation Principles
A relatively simple mathematical analysis yields information on the height of a hydraulic jump and the speed of propagation of a travelling bore. An early study was carried out by Jean-Baptiste Bélanger (1790-1874), a French engineer, who derived an expression for the difference in height by means of conservation principles. Gustave Eiffel was one of his students, and Bélanger’s name is inscribed on the Eiffel Tower.
Let us denote the water depths on each side of the hydraulic jump by H1 and H2. It might seem reasonable to apply the principle of conservation of energy to derive an expression for the depth ratio H1 / H2. The problem is that there is energy dissipation at the hydraulic jump, so the conservation principle does not hold. This dissipation is exploited by engineers who design spillways to generate bores that remove destructive energy.
In fact, Bélanger originally used energy conservation, obtaining the wrong answer. He later corrected his analysis, applying the fundamental physical principles of mass and momentum conservation across the jump to obtain the ratio of water depths upstream and downstream:
( H1 / H2 ) = ½ [ √( 1 + 8 Fr2) – 1 ]
Where Fr = V / √(gH) is the Froude Number. This is what we now call Bélanger’s Equation.
The Froude number Fr is a ratio of speeds
Fr = (Characteristic flow speed) / (Gravitational wave speed)
where inflow values are used for the flow speed V and the depth H, which determines the wave speed. Although Bélanger used this ratio in his work, he did not give it a name and it is now called after William Froude (1810-1879), an English engineer, hydrodynamicist and naval architect who formulated laws for the resistance of water on ships.
A hydraulic jump occurs if the Froude number exceeds 1. The condition Fr > 1 is called supercritical flow. Above the bore, the water is shallow so the wave speed c = √(gH) is small, making the Froude number large. Downstream, Bélanger’s equation implies that H2 > H1 so there is deeper water. Moreover, the flow speed is less, so we have Fr < 1, a condition called subcritical flow. Thus there is a transition from supercritical to subcritical flow across the bore.
Hydraulic jumps continue to provide opportunities for engineers, sport for surfers and challenges for physicists and applied mathematicians. Who would have thought that bores could be so interesting?
Addendum
I was unaware of observations of a bore on any Irish river. When the above post appeared, I received a mail from Frank Larkin of Limerick, with the following information:
“I once saw a tidal bore in Limerick City about 50 years ago. I was on the north shore river bank opposite Limerick Docks one afternoon at a low spring tide when I heard a strange rustling sound. I looked downriver to see a wall of water across the entire river about 1 ft high coming upstream at a quick walking pace. The foot high wall of water rushing upstream seemed to overlay the static low water underneath. The noise I heard was the wall of water rushing over the mud flats. The bore rushed past and seemed to dissipate near Sarsfield Bridge.”
Limerick city is at the point where the river water meets the sea water of the estuary.
* * *
Peter Lynch’s book about walking around the coastal counties of Ireland is now available as an ebook (at a very low price!). For more information and photographs go to http://www.ramblingroundireland.com/
ThatsMaths 