A Non-Linear Dynamical System: Flow Past a Sluice Gate
Binder, Ben J., Australasian Journal of Engineering Education
There are many examples throughout the world where infrastructure such as dams, weirs and sluice gates play an important role in controlling the flow of water in rivers and irrigation channels. More specifically, in Australia, the Great River Murray that passes through New South Wales, Victoria and South Australia has numerous weirs, locks and sluice gates that restrict the downstream flow of water toward the sea.
Traditionally sluice gates are wooden or metal plates that slide in grooves in the sides of a channel. They are commonly used to control water levels and flow rates in rivers and canals (see figure 1(a)). In general, flow past a sluice gate is three-dimensional and difficult to treat mathematically. However, when the disturbance (eg. sluice gate) in the channel or river is wide, the flow problem can be considered two-dimensional in the neighbourhood of the disturbance centre-plane. A sketch of a two-dimensional approximation is shown in figure 1(b) for flow past a curved plate or sluice gate.
The interface or unknown boundary between the air and the water in the channel is referred to as the free-surface. In most cases disturbances such as sluice gates and weirs generate water waves on the free-surface. For a given shape of sluice gate BC, we will determine the water height or free-surface elevation in the regions [A,B] and [C,D], as shown in figure 1(b).
We assume that the flow far downstream approaches a uniform stream with constant velocity U and constant depth H. We can then define the Froude number:
F = U/[(gH).sup.1/2] (1)
Here g is the acceleration due to gravity. The Froude number is the ratio of the flow velocity to the wave velocity and characterises the flow. If F > 1 with the same waveless depth H far upstream and downstream, the flow is called supercritical. If F < 1 with uniform flow far downstream and a train of waves far upstream, the flow is called subcritical. The terms supercritical and subcritical are analogous to the terms supersonic and subsonic in gas dynamics.
We will use the Korteweg-de Vries equation to model the steady two-dimensional irrotational flow of an inviscid and incompressible fluid past a sluice gate in an open channel. This results in studying a two-dimensional non-linear dynamical system. By examining phase plane diagrams for F > 1 and F < 1 with straight and curved sluice gates, we will determine solutions for the free-surface elevation in the corresponding physical plane. The sluice gate problem and exercises that follow in this paper serve as excellent examples that can be used by lecturers in courses on differential equations, fluid mechanics and dynamical systems. The author has utilised some of the material in this paper for his second-year course on differential equations at the University of Adelaide.
Consider the steady two-dimensional iaotational flow of an inviscid and incompressible fluid past a sluice gate in an open channel (see figure 1(b)). The fluid is bounded from below by the bottom of the channel, and from above by the two free surfaces AB, CD and the gate BC. A system of Cartesian coordinates ([x.sup.*],[y.sup.*]) is defined with the [x.sup.*]-axis along the bottom of the channel and the [y.sup.*]-axis passing Australasian journal of Engineering Education, Vol 15 No 2 through the point B of the sluice gate. The horizontal distance from the [y.sup.*]-axis to the point C is denoted by [x.sup.*.sub.c]. The flow is assumed to enter and leave the plate tangentially. The equation of the streamline ABCD is [y.sup.*]([x.sup.*]) = H + [[eta].sup.*]([x.sup.*]), where [[eta].sup.*]([x.sup.*]) is the elevation of the free-surface on top of the level H. The function [[eta].sup.*]([x.sup.*]) is uniquely defined by requiring [[eta].sup.*]([x.sup.*]) [right arrow] 0 as [x.sup.*] [right arrow] [infinity]. The dimensionless quantities (x, [eta], [x. …