Chapter 4Bernoulli’s equationFor st

Chapter 4
Bernoulli’s equation
For steady inviscid flow under external forces which have a potential Ω such that F = −∇Ω Euler’s equation reduces to
and for an incompressible fluid
We may regard p + pΩ as a more general dynamic pressure; but for the particular case of gravitation potential, Ω = gzand F = −∇Ω=−(0,0,g)=−gk. We note that
using the fact that u·∇is a scalar differential operator. Hence,
and it follows that
1 2u2 + p/ρ +Ωis constant along each streamline (as u·∇is proportional to the rate-of-change in the direction u of streamlines). Thus for steady, incompressible, inviscid flow
1 2u2 + p/ρ +Ωis a constant on a streamline, although the constant will generally be different on each different streamline.
4.1 Application of Bernoulli’s equation
(i) Draining a reservoir through a small hole If the draining opening is of much smaller cross-section than the reservoir (Fig. 4.1), the water surface in the tank will fall very slowly and the flow may be regarded as approximately steady. We may take the outflow speed uA as ap- proximately uniform across the jet and the pressure pA uniform across the jet and equal to the atmospheric pressure p0 outside the jet (for, if this were not so, there would be a difference in pressure across the surface of the jet, and this would accelerate the jet surface radially, which is not observed, although the jet is accelerated downwards by its weight). Hence, on the streamline AB,
Figure 4.1: Draining of a reservoir.
This is known as Toricelli’s theorem. Note that the outflow speed is that of free fall from B under gravity; this clearly neglects any viscous dissipation of energy.
(ii) Bluff body in a stream; Pitot tube Suppose that a stream has uniform speed U0 and pressure p0 far from any obstacle, and that it then flows round a bluff body (Fig. 4.2). The flow must be slowed down in front of the body and there must be one dividing streamline separating fluid which follows past one side of the body or the other. This dividing streamline must end on the body at a stagnation point at which the velocity is zero and the pressure
Figure 4.2: Flow round a bluff body in this case a cylinder.
This provides the basis for the Pitot tube in which a pressure measurement is used to obtain the free stream velocity U0. The pressure p = p0 + 1 2ρU2 0 is the total or Pitot pressure (also known as the total head) of the free stream, and differs from the static pressure p0 by the dynamic pressure 1 2ρU2 0. The
Figure 4.3: Principal of a Pitot tube.
Pitot tube consists of a tube directed into the stream with a small central hole connected to a manometer for measuring pressure difference p−p0 (Fig. 4.3). At equilibrium there is no flow through the tube, and hence the left hand pressure on the manometer is the total pressure p0+ 1 2ρU2 0. The static pressure p0 can be obtained from a static tube which is normal to the flow. The Pitot-static tube combines a Pitot tube and a static tube in a single head (Fig. (4.4). The difference between Pitot pressure (p0 + 1 2ρU2 0) and static pres- sure (p0) is the dynamic pressure 1 2ρU2 0, and the manometer reading therefore provides a measure of the free stream velocity U0. The Pitot-static tube can also be flown in an aeroplane and used to determine the speed of the aeroplane through the air.
(iii) Venturi tube This is a device for measuring fluid velocity and discharge (Fig. 4.5). Suppose that there is a restriction of cross-sections in a pipe of cross-section S, with
Figure 4.4: A Pitot-static tube.
Figure 4.5: A Venturi tube.
velocities v, V and pressures p, P in the two sections, respectively, the pipe being horizontal. Then
The discharge
and substitution gives