代写CVEN90051 CIVIL HYDRAULICS Module 1: CHANNEL HYDRAULICS AND HYDRAULIC STRUCTURES Topic 2代做回归

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CVEN90051 CIVIL HYDRAULICS

Module 1: CHANNEL HYDRAULICS AND HYDRAULIC STRUCTURES

LEARNING GUIDE Topic 2

Topic 2: Flow transitions

Gradually varied flow, backwater curves, standard step method

Learning objectives

• Explain and quantify the effect of channel structures and transitions on water depths;

• Recognise when a hydraulic jump will occur, determine the depth after the jump and the energy loss that occurs;

• Estimate height and length of the jump;

• Consolidate understanding of water surface profiles in gradually varied flow (GVF);

• Implement the standard step and/or direct step methods to determine water depth in GVF.

Introduction

In Topic #1, we discussed uniform flows, which occur when the water depth and cross section do not change with distance. In natural channels, however, cross sections and bottom slopes are not constant. This also happens in constructed channels, when there is the need to suit the existing topographical conditions for economic reasons (slope change) or deploy structures to control the discharge. Changes in the channel geometry or obstructions produce non- uniform flows, which develop while transitioning from one uniform flow condition to another. These flow conditions are called gradually varying flows if the rate of variation of depth with respect to distance is small, and rapidly varying flows if the rate of variation is large. In other words, the flow depth changes gradually over a long distance in gradually varying flows and in a short distance in rapidly varying flows.

QUESTION: Can you name a case of rapidly varying flow?

The complexity of natural or man-made channels is shown in Figure 1 below, where several structures/obstructions affecting the flow are reported. The aim of Module #1 is to provide enough knowledge to compute water levels and design hydraulic structures. The former are fundamental to make sure water remains within the river/channel and it does not overflow the banks. The latter are key to control discharge and, hence, water levels.

FOOD-FOR-THOUGHT: What happens to the water when it encounters a barrier (e.g. a dam)? Are there any analogies in transport engineering? (Think of what happens when an accident occur on the freeway).

Questions to guide your reading

• What determines water levels in a channel?

• How would you know if water level in a channel is being influenced by a structure?

• What is the hydraulic jump and when does it occur?

• What are control points? How do you locate them?

• How do you implement the standard step and direct step methods, and under what circumstances?

Reading guide

The reading material is the primary source to understand the subject. The reading material for Topic #2 is available through “Reading Online” in LMS. The material is extrapolated from the textbook: Hamill, L., 2011. Understanding hydraulics. Macmillan International Higher Education. Read part on non-uniform flow pp. 266-296.

Figure 1: Types of flow

The reading starts with Section 8.10, which introduces the concept of flow transition (i.e. from slow to fast & from fast to slow). The acceleration of the flow is straightforward and means the flow switch from sub to super critical. Less trivial is the transition from super to sub critical as this require a dissipation of energy. This latter transition is discussed in detail in Section 8.10.2, which introduces a very important type of rapidly varying flow: the hydraulic jump. In general terms, the hydraulic jump induces a dissipation of energy that slows down the flow, allowing the transition from super to sub critical conditions. The momentum equation is the fundamental background knowledge that is used to describe the hydraulic jump. Section 8.10.2 provides key equations without any derivation. An example of hydraulic jump is presented in Figure 2 below.

NOTE: You have probably studied the hydraulic jump already in Fluid Mechanics. Here we recall the concept to refresh your memory and provide some key equations that will help with some exercises.

Figure 2: Example of hydraulic jump.

The key concept of the hydraulic jump is provided in Figure 8.25 of the reading material (also replicated in Figure 3 below). The change of depth occurs through an intense turbulent dissipation.

OBSERVATION: If we are in a super-critical regime, a reduction of energy would increase the depth. This is exactly what we want to switch from super to sub-critical, right? But once we reach the critical depth, we need to inject energy into the system to force the flow to become sub-critical. Is it possible????

Pay attention to this extract from the reading:

“ … the depth initially increases gradually from D1 towards DC [switching from super-critical to critical], then before it attains this depth the hydraulic jump occurs at energy level E2 with the flow switching to a depth D2 on the upper, subcritical part of the depth–specific energy curve (Fig. 8.25b). As a result of the loss of energy, DE, in the jump, the sequent or conjugate depth of flow after the jump, D2, is less than the alternate depth vertically above E1 that would otherwise have occurred. This energy loss means that the hydraulic jump cannot be analysed simply with either the depth–specific energy curve or the energy equation.”

Figure 3: Initial depth D1, sequent depth D2 and energy loss DE: (left panel) at the hydraulic jump, and (right panel) on the depth–specific energy diagram [after Webber (1971)]

The hydraulic jump is extremely turbulent. It is characterized by the development of large-scale turbulence, surface waves and spray, energy dissipation and air entrainment. The large-scale turbulence region is usually called the “ roller”. The flow within a hydraulic jump is extremely complicated and it is not required usually to consider its details. To evaluate the basic flow properties and energy losses in such a region, the momentum principle is used.

ADDITIONAL REFERENCE: For more rigorous mathematical details on the hydraulic jump, see Chaudhry, M.H., 2007. Open- channel flow. Springer Science & Business Media (available through the library).

Applications of the momentum equation allow the derivation of quadratic equations that links the depth before and after the hydraulic jump (see equations 8.36 and 8.37).

where F is the Froude number. These equations form. the base for solving exercises involving the hydraulic jump.

ADD-ON: An interesting addition to the reading material is the table reported in Figure 3 below. This provides some details on the different types of jump.

What about the length of the hydraulic jump? There are several empirical expressions based on laboratory experiments. Table 8.5 in the reading material an easy approach. What if you are in between those conditions? Just interpolate!

FOOD-FOR-THOUGHT: How can we force the jump to occur? See an example in figure 8.26 in the reading material. Can you think of a simpler solution?

Section 8.11 introduces the gradually varied flows. The key concept to remember is that an obstruction induces a gradual variation of the flow with a consequent changes of water profile, which is no longer parallel to the bed slope. There are methods to compute the water surface profile. It is also important to get a qualitative idea of the water profile. A general discussion is presented in Section 8.11.1, after which you should be able to produce an educated guess of water profiles. A summary of the different type of profiles is reported in Figure 4 below. To draw profiles (or compute them), you will need to start from control points. These are discussed in Section 8.11.2.

FOOD-FOR-THOUGHT: So, go back to an earlier question: what happens when the water flow impact against a dam? What profile do you think it will form?

There are 2 methods to compute the water profiles with very little difference between them Sections 8.11.3-4 introduce the standard step method. Section 8.11.5 introduces the direct step method. These sections are better comprehended through exercise. Practice before your workshop and ask question to clarify any doubts.

QUESTIONS: What are the differences between these two methods? Which is the best one?

Section 8.12 introduce the concept of surge. This is a perturbation that propagate over the surface. Go through it during your reading, but it is a bit marginal in this module. (But, of course, do NOT neglect this section!).

Figure 3: Classification of the hydraulic jump. (Note that y1 andy2 are equivalent to D1 and D2 in previous figures).

Figure 4: Classification of water profiles. (Note that y is the water depth D).

Practice problems

1. A colleague claims that the flow resistance (shear stress) of channel surfaces can be neglected in the analysis of rapidly varied flow but should be considered in the analysis of gradually varied flow. Do you agree with this claim?

2. A hydraulic jump occurs in a rectangular channel 5 m wide. The depths immediately upstream and downstream of the jump are observed to be, respectively, 0.25 m and 1.0 m.

a) Calculate the flow rate in the channel.

b) Calculate the fractional energy loss across the hydraulic jump.

c) Can you estimate the channel slope?

3. A rectangular channel of width 2 m carries a flow of 1.2 m3/s. The Manning’s n is 0.02. As shown in the following diagram, the water flows down a slope S1 = 0.002, then the slope changes to S2 =0.0015, then the water eventually flows over a sharp crested overflow. Assume the second section is long enough for the flow to come to normal depth.

In week 1 (practice Exercise #5), we calculated normal and critical depth for each slope. We also determined the flow condition on each slope. Now, draw the water surface profile, labelling each GVF profile type.

4. For the channel in the previous question, calculate how far upstream of the overflow a depth of 0.45 m occurs (only use one step for simplicity).

5. Consider gradually varied flow of water in a wide rectangular irrigation channel with a discharge per unit width of 5 m2/s, a slope of 0.01, and a Manning’s n coefficient of 0.02. The flow is initially at uniform depth. At a given location, x = 0, the flow enters a 200 m length of channel where lack of maintenance has resulted in an increased Manning’s n of 0.03. Following this stretch of channel, the roughness returns to the initial (maintained) value.

(a) Calculate the normal and critical depths of the flow for the two distinct segments.

(b) Describe, qualitatively, the behaviour of the water surface along the channel.