Open Channel flow second year PDF

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in this chapter, we are concerned with the important topic of open channel flow...

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Applied Fluid Mechanics MACE 20002 Lecture Notes

2.

Open Channel flow



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1.

Introduction

2

2.

Governing Equations

3

3.

Flow over bumps

4

4.

Surface (Gravity) Waves

8

5.

Modes of Flow

11

6.

Energy Analysis

13

7.

Hydraulic jumps

18

8.

9.

7.1.

Introduction

18

7.2.

Jump Analysis

19

7.3.

Energy loss across a jump

21

Flow over Weirs and under Sluice gates

23

8.1.

Introduction

23

8.2.

Broad-crested weir

23

8.3.

Sharp-crested weir

26

8.4.

Sluice gates

28

Summary

31

2.1. Introduction In this chapter, we are concerned with the important topic of Open-channel flow. We shall see in the following chapter that this is closely related to compressible flow. In both cases the behaviour is intimately related to the Lecture Notes

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waves. In the case of open-channel flow the waves are gravity waves (see section 2.4) whilst in the case of compressible flow the waves are sound waves. We begin by considering the important features of the flow. Since we are concerned with the flow of water (which is incompressible), this implies that, for steady flow, the volume flow rate will be conserved. Thus, for constant width channels, changes in the channel height lead to changes in the flow velocity. However, hydrostatics implies that changes in depth also results in a 1 change in the pressure force ( ρgby 2 , b being the channel width, y the flow 2 depth and ρ and g being the fluid density and acceleration due to gravity, respectively). Hence changes in depth gives rise to a coupling between velocity changes and pressure changes. This coupling between the velocity and pressure results in surface (sometimes called gravity) waves. Thus the shape of the channel can affect the flow. In the next section we consider the equations of motion in more detail.

2.2. Governing Equations

control volume Pressure force 1!gby2 y0 0 2

pressur e

constan t on

2 0

!U

U0

surface Pressure force 1!gby2 2 y !U

"w

U

L 

Figure 2.1 The Governing equations of open-channel flow Figure 2.1 shows a general flow (with varying depth) along a channel of uniform width. Since the the surface is a free surface the pressure there is constant and equal to atmospheric pressure. The motion of the water is then Lecture Notes

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governed by the equations of continuity (conservation of mass), momentum and energy:

Continuity: AU = A0 U0 = constant

(2.1)

  1 Momentum: ρQ(U − U0 ) = ρgb y02 − y 2 − τw Lb 2

(2.2)

1 1 Energy: gy0 + U02 = gy + U 2 + ghf 2 2 

(2.3)

Equations (2.2) and (2.3) include a contribution due to viscous losses ( τw L and ghf respectively). They are included here for completeness. In practice, however, there are many situations (although not all) in which the viscous effects can be ignored. As an example of this, in the next section we consider the response of the surface of an open-channel flow to a bump on the base of the channel. As we shall see, the response is non-trivial and depends upon the speed of the flow.

2.3. Flow over bumps

z(x) U



d(x)

h(x) Figure 2.2 Flow over a bump in deep flow.

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d(x) z(x) U h(x) 

Figure 2.3 Flow over a bump in shallow flow.

Consider the flow over a bump in otherwise uniform flow, shown in figures 2.2 and 2.3. In figure 2.2 the surface is seen to drop as a result of a rise in the position of the base (a 'bump'). In figure 2.3 the reverse is seen to happen with the surface rising in response to the rise in the height of the base. The essential question, which we intend to answer here, is which of these two situation arises. As we shall see, the answer is not straightforward and depends upon both the flow speed and the depth of the flow. In the z − h). In equations below, h is the height of the surface and d is the depth (= order to solve this problem we must consider both the momentum and continuity equations. If we treat the flow as inviscid (a fair assumption provided the depth is not too shallow), then we saw in the previous section that the momentum equation along a streamline can be simplified (since the flow is also steady and incompressible) as Bernoulli's equation. It is helpful to choose a streamline in which the pressure is known everywhere, so we choose the surface streamline where the pressure is constant (and atmospheric). 1 Bernoulli’s equation along surface: pa + ρv 2 + ρgz = constant 2 

Differentiating wrt x: 

∂v ∂z ∂v ∂z ∂pa =0=v +g + ρv + ρg ∂x ∂x ∂x ∂x ∂x

(2.4) (2.5)

Equation (2.5) is a differential equation relating the variation in the flow velocity to the height of the surface. A second equation relating the flow velocity to the depth of the flow can now be found using the continuity equation:

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(2.6)

Continuity: ρvd = constant v ∂d ∂v ∂v ∂d =− =0⇒ +d d ∂x ∂x ∂x ∂x ∂d ∂z ∂h − But d = z − h ⇒ = ∂x ∂x ∂x  Differentiating wrt x: v

(2.7) (2.8)

Equation (2.7) is a differential equation relating the flow velocity to the depth of the flow, whilst equation (2.8) relates the variation in depth to the variation in surface height (unknown) and base (bump) height (known). Substituting for

∂v  ∂x from equation (2.5) and for the depth ( d) from equation (2.8) into equation (2.7) we have:  ⇥ v2 ∂z ∂h ∂z (2.9) =0 − − +g ∂x d ∂x ∂x

⇥  ∂z v2 ∂h v2 Rearranging: =− g− d ∂x d ∂x



∂z ⇒ = ∂x



−v2 (gd − v2 )

⇥

∂h ∂x

(2.10)

(2.11)

Equation (2.11) relates the surface height to the height of the base (bump). In particular whether the surface rises in response to rises in the base (as in figure 2.3) or dips in response to rises in the base height (as in figure 2.2) depends upon the sign of the right hand side of equation (2.11). 2 This in turn depends upon the relative values of  v and  gd. Hence for deep, 2 slow flow,  v < gd so the denominator is positive and the flow dips as it goes 2 over the bump. For shallow, fast flow, in contrast, v > gd and the denominator is negative and the level rises as the flow goes over the bump. 2

When v = gd, then the level experiences a discontinuity and a sudden rise in height is observed. This is called a hydraulic jump (see section 2.7). We can explain this rather odd behaviour in a number of different waves. We begin by considering momentum.

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When flow passes over the bump, its momentum changes. There are two possibilities. If the flow is to move upwards in response to the bump, then it has to overcome gravity and thus the momentum drops. Since the mass flow rate is constant, this can only be achieved if its depth increases which in turn results in a rise in the pressure-area force (the pressure integrated from the surface to the base). This rise in the pressure-area force creates the horizontal force needed to reduce the x-component of the momentum. If the flow is initially shallow and fast moving, then it starts with high momentum and low pressure-area force. In this situation it has sufficient momentum to lose and create a higher pressure-area force. If, in contrast, the flow is initially deep and slow-moving then it has very little momentum and a large amount of pressure force. In this situation it would need to lose more momentum than it has and so, instead, the level drops resulting in a shallower flow which in turn means greater speed, and hence momentum, but lower pressure force. Next we consider this in terms of energy. Before the bump, the flow has both potential and kinetic energy, with the relative quantities depending upon how high it is and how fast and deep it is. In order to move over the bump there will be an exchange between kinetic and potential energy. Notice, however, that it cannot simply displace over the bump as this would result in equal kinetic energy but an increase in potential energy, which would violate the first law of thermodynamics (conservation of energy). It can, instead, rise up by more than the size of the bump; resulting in a rise in potential energy but a drop in kinetic energy (since the velocity drops for deeper flow); or the surface can drop; resulting in a reduction in potential energy, but a rise in kinetic energy. For initial flow which is deep and slow-moving, it does not have sufficient kinetic energy to rise up over the bump, so the surface level drops with the corresponding rise in kinetic energy and drop in potential energy. When, in contrast, the initial flow is shallow and fast-moving, it has plenty of kinetic energy, but insufficient potential energy. As as result the surface level rises with the associated rise in potential energy but drop in kinetic energy. Whilst both momentum and energy considerations explain why the level rises in one situation, but falls in another, the reason for 2 the sudden rise when  v = gd is more subtle. Mathematically, it occurs at the point in which any change in level can occur without a bump being present at the bottom of the channel. To understand this physically, it is helpful to consider the surface waves (sometimes called gravity waves).  Surface waves move at a speed of  gd in undisturbed flow (see section 2.4). With flow they are convected by the flow and so move at a speed of  v + gd downstream and gd − v upstream. Hence surface waves can only Lecture Notes

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2

travel upstream in the flow in regions in which v < gd. Suppose, therefore, 2 we have a flow in which  v > gd (called a super-critical flow) which is slowed as a result of an obstruction or other reason. Now suppose that the slowing is sufficient for there to be a region downstream in which the flow is sub2 critical (v < gd). In this downstream region, surface waves are free to travel upstream, whereas in the upstream region they are not. The question then arises: what happens to these upstream-travelling waves when they get to the upstream region where they are prevented from travelling upstream? At 2 the intersection, the flow speed passes through the point at which v = gd. At this point the upstream-travelling waves must stop. In a real flow, there will be waves caused by slight flow disturbances, noise etc. All such waves, must 2 stop at the point where v = gd giving rise to the jump. In the next section we discuss these surface (gravity) waves in more detail.

2.4. Surface (Gravity) Waves In the previous section, we saw that the simple problem of flow over a bump resulted in two completely different types of behaviour depending upon the starting conditions. Furthermore  a singularity occurred at the boundary between the two flows (at a speed  gd)  suggesting that there is an important gd. It turns out, that, as mentioned phenomenon associated with the speed   above (but not proved), speed  gd is the speed of surface waves in undisturbed flow. Since they are related to gravitational effects, these waves are termed gravity waves by some. In this section we consider the implications of these waves. We begin by presenting a simplified proof of there propagation speed. In particular, we assume that the waves are steady. For a more rigorous proof see Aero-acoustics course. Figure 2.4 shows a surface wave propagating across a stationary fluid at speed  c. The fluid below the wave is disturbed, so we assume that it moves at speed  V . In order to turn this into a steady problem, it is convenient to use a relative frame moving at speed c. In this frame, the surface wave remains in a fixed position, the flow moves at speed  c and the fluid below the wave moves at a speed c − V . This is shown in figure 2.5. In both figure 2.4 and figure 2.5, the width of the channel is taken to be b .

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c y

still water

!y

!V

 Figure 2.4 A wave disturbance on still fluid in the stationary frame.

!y y

c c!!V

 Figure 2.5 A wave disturbance on still fluid in the relative frame.

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Since we now have a steady problem, we can analyse the flow, as we did before for the flow over a bump, using the continuity and momentum equations. Beginning with continuity: (2.12)

⇥cyb = ⇥(c − V )(y + y)b





y ⇒ V = c y + y

⇥

(2.13)

Equation (2.13) gives us an equation relating the motion of the fluid below the disturbance to the height of the disturbance and the propagation speed of the disturbance. Another equation relating these quantities can be obtained by considering the momentum equation. Neglecting viscous effects, the only force results from the pressure-area force so we have: ⇥  1 ⇥gb y 2 − (y + y )2 = ⇥cby ((c − V ) − c) 2  ⇥ y ⇒ cV = g 1 + y 2y



(2.14) (2.15)

Equation (2.15) gives us a second equation for V  . Eliminating V  using equation (2.13) then leads to:





y c = gy 1 + y 2

⇥

y 1+ 2y

⇥

(2.16)

Equation (2.16) tells us the waves speed for a given depth. We note the following observations:

•

The flow speed depends upon the size of the oscillation, with large waves (large y ) having higher speeds than small waves.

• •

The minimum wave speed occurs when y → 0 For small waves this minimum speed is c0 =

√ gy

As discussed previously, √ the surface waves, which for low amplitude waves travel at a speed of  gy , controls the flow. Furthermore, the speed V relative to the surface wave speed (√gd ) is an important quantity called the

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Froude number. When the Froude number is less than 1, the flow is termed sub-critical, and the surface level drops as it passes over a bump on the base. When the Froude number is greater than 1 the flow is termed supercritical, and the level rises in response to bumps on the base. Hydraulic jumps occur at a Froude number of 1 and separates a supercritical region of flow from a sub-critical region. This behaviour has many parallels with compressible flow (see next chapter) in which the flow is controlled by the Mach number. In the next section which look at the implications of this for disturbances in a channel with flow.

2.5. Modes of Flow



a) Fr 1. Now the waves cannot propagate in any-direction outside a wave cone, as shown. The angle of the cone is a simple function of the Froude number. To see this, consider figure 2.8.

a) Fr = 1



b) Fr > 1

Figure 2.7 Critical and super-critical (Froude number (Fr) ≥  1) pipe modes

c ! Usin(!) ! U

 Figure 2.8 Cone angle for super-critical flow. The cone side does not move so the component of its velocity normal to the cone side must be zero. The actual speed is given by  c − U sin φ. For c c this to equal zero we require sinφ =U . Hence the cone angle is 2 sin−1 U .

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This mode behaviour is summarised in the table below. Open channel flow

Compressible flow

Surface waves

Sound waves

Fr < 1 (subcritical)

M < 1 (subsonic)

Fr = 1 (critical)

M = 1 (sonic)

Fr >1 (supercritical)

M > 1 (supersonic)

Wave cone

Mach cone

Undisturbed region outside cone for Fr > 1

Zone of silence outside cone for M>1

2.6. Energy Analysis In discussing the flow over bumps, we found that the behaviour could be explained in terms of the distribution between potential and kinetic energy. In this section we look at this distribution, and the overall energy in more detail. In our analysis, we will consider the overall energy per unit weight (e

e=y+ 

U2 q2 =y+ 2gy 2 2g

E =ρg ):

(2.17)

Equation (2.17) describes the energy in the flow in terms of the depth and the volume flow rate. It is useful here to consider flows with the same volume flow rate but different depths. Figure 2.9 describes the energy in the flow for fixed volume flow rate q and with varying depth, y. At low  y, the flow velocity is large and so the flow has large energy, almost entirely kinetic. As the depth increases, the potential energy increases but the drop in kinetic energy is more dramatic resulting in an overall energy drop. This behaviour continues until the depth reaches a critical depth, y  ∗ , at which point the energy is minimum.

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y

8

e

8

y

y e e

0

y2

y1

emin 

8

y*

Figure 2.9 A plot of the total energy in a flow as a function of height, for fixed flow rate, q As the depth increases further, the rise in potential energy exceeds the drop in kinetic energy, resulting in an overall rise in the energy of the flow. It is worth noting, that at a given energy level there are two possible depths,  y1 and  y2. These depths are called conjugate depths. As we shall see later, one of these depths corresponds to subcritical flow and the other to supercritical flow. In order to understand the significance of the critical depth,  y∗, and minimum, energy,  emin, we differentiate equation (2.17) to find at what depth this occurs: q2 ∂e = 0 = 1 − ∗3 ∂y gy∗

(2.18)

q2∗ U 2y 2 = ∗ ∗ g g

(2.19)

⇒

y∗3 =

U∗2 g 3y∗ = e∗ = 2



⇒ y∗ = 

emin

(2.20) (2.21)

It is clear, then, that the minimum occurs when the Froude number is unity and the flow is critical.

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2

U 2g

y2  Figure 2.10 An unrestricted sluice gate In practice, unrestricted flow will always take the minimum depth (i.e.  y1) and thus will be supercritical (see figure 2.10). However, if the flow is restricted, the flow will slow down and the depth will increase to give subcritical flow (as in figure 2.11). Let us now consider how this graph relates to the flow obstruction.

past an

Figure 2.12 shows the flow of water past an obstacle compared with flow o...