Tài liệu Open channel hydraulics for engineers. Chapter 4 non uniform flow ppt

This chapter will discuss the effect of change in any one of the above quantities, including specific energy, critical depth and slope, and flow types.. Significant in each one of the ab

Chapter NON-UNIFORM FLOW

_

4.1 Introduction

4.2 Gradually-varied steady flow

4.3 Types of water surface profiles

4.4 Drawing water surface profiles

_

Summary

Linking up with Chapter 2, dealing with uniform flow in open channels, it may be noted that any change in the flow phenomenon (i.e flow rate, velocity, flow depth, flow area, bed slope do not remain constant) causes the flow to be non-uniform This chapter will discuss the effect of change in any one of the above quantities, including specific energy, critical depth and slope, and flow types How to draw water surface profiles will also be introduced

Key words

Non-uniform; specific energy; critical; gradually-varied steady flow; water surface profiles _

4.1 INTRODUCTION

4.1.1 General

In the previous Chapter 2, the flow was uniform under all circumstances under consideration In many situations the flow in an open channel is not of uniform depth along the channel In this chapter the flow conditions studied relate to steady, but non-uniform, flow This type of flow is created by, among other things, the following major causes:

 Changes in the channel cross-section

 Changes in the channel slope

 Certain obstructions, such as dams or gates, in the stream’s path

 Changes in the discharge – such as in a river, where tributaries enter the main stream

A non-uniform flow is characterized by a varied depth and a varied mean flow velocity:

0 s

V 



0 s

 

If the bottom slope and the energy line slope are not equal, the flow depth will vary along the channel, either increasing or decreasing in the flow direction Physically, the difference between the component of weight and the shear forces in the direction of flow produces a change in the fluid momentum which requires a change in velocity and, from continuity considerations, a change in depth Whether the depth increases or decreases depends on various parameters of the flow, with many types of surface profile configurations possible Fig 4.1 illustrates some typical longitudinal free-surface profiles Upstream and downstream controls can induce various flow patterns In some cases, a hydraulic jump might take place A jump is a rapid-varied flow phenomenon; calculations were developed

in Chapter 3 However, it is also a control section and it affects the free surface profiles upstream and downstream

Fig 4.1 Examples of non-uniform flow

4.1.2 Accelerated and Retarded flow

An idealized section of a reach of a channel with accelerated and retarded flow conditions is shown in Fig 4.2a and Fig 4.2b, respectively As flow accelerates, with the rate of flow constant, the depth h must decrease form point 1 to point 2, and a water surface profile as shown in Fig 4.2a results Retarded flow will produce water surface profiles as shown in Fig 4.2b

Significant in each one of the above cases is the fact that now the water surface is a curved line and not longer parallel to the channel bottom and the energy line, as was the case for uniform flow The following points are made in connection with the above observations

upstream control

downstream control control

sluice

gate

rapid

varied

flow

weir

rapid varied flow

rapid varied flow

gradually varied flow

gradually varied flow upstream control

downstream control

hydraulic jump

supercritical

flow

critical

depth

overflow (critical depth)

control

hc

hc

rapid

varied

flow

rapid varied flow

rapid varied flow

gradually varied

varied flow

 The water surface, as will be shown later, can have a concave or a convex shape

 The energy line is not necessarily a straight line; however, it is assumed that the energy gradient is constant along the length of a reach and the energy line will be represented and considered to have a slope ie = HL/L

 As was done in the case of uniform flow, it is here also accepted that the depth of flow, h, is equal to the pressure head in the energy equation Obviously, this applies only when the slope of the channel bottom is small For very steep slopes, allowances for this discrepancy must be made

i

L

water surface

HL

1 1 p h





2 2 p h





2 V

2 V

2 g

energy-head line

hydraulic grade line

datum









Fig 4.2a Accelerated flow

i

L

water surface

HL

1 1 p h



2 p h





2 V

2 g

energy-head line

hydraulic grade line

datum









Fig 4.2b Retarded flow

4.1.3 Equation of non-uniform flow

Fig 4.3 Non-uniform flow

Consider a non-uniform flow in an open channel between section 1-1 and section 2-2, in which the water surface has a rising trend (i.e the energy-head gradient is less than the bed slope) as shown in Fig 4.3

Let V = velocity of water at section 1-1;

h = depth of water at section 1-1;

V+dV = velocity of water at section 2-2;

h+dh = depth of water at section 2-2;

ib = slope of channel bed;

ie = slope of the energy grade line;

dl = distance between section 1-1 and section 2-2;

b = average width of the channel,

Q = discharge through the channel,

zb = change of bottom elevation between section 1-1 and section 2-2, and

he = HL, change of energy grade line between section 1-1 and section 2-2

Since the depth of water at section 2-2 is larger than at section 1-1, the velocity of water at section 2-2 will be smaller than that at section 1-1

Applying Bernoulli’s equation at section 1-1 and section 2-2:



V dV V



V.dV

g

2 (dV) 2g (small of second order) (4-4)

or ib dh V.dV ie

dl g.dl

1

1

2

2

ie

ib

dl

flow

he water surface

2

V

2g

h

h+dh

zb

b e

We know that the quantity of water flowing per unit width is constant, therefore

or d(Vh) 0

Differentiating the above equation (treating both V and h as variables),

V.dh h.dV

0

Substituting the above value of dV

dl in Eq (4-6), yields 2

2

b e

2

i i dh

1

gh





(4-14)

Notes: The above relation gives the slope of the water surface with respect to the bottom

of the channel Or in other words, it gives the variation of water depth with respect to the distance along the bottom of the channel The value of dh/dl (i.e zero, positive or negative) gives the following important information:

 If dh/dl is equal to zero, it indicates that the slope of the water surface is equal to the bottom slope Or in other words, the water surface is parallel to the channel bed

 If dh/dl is positive, it indicates that the water surface rises in the direction of flow

The profile of water, so obtained, is called backwater curve

 If dh/dl is negative, it indicates that the water surface falls in the direction of flow

The profile of water, so obtained, is called downward curve

Example 4.1: A rectangular channel, 20 m wide and having a bed slope of 0.006, is discharging water with a velocity of 1.5 m/s The flow is regulated in such a way that the slope of the water energy gradient is 0.0008 Find the rate at which the depth of water will

be changing at a point where the water is flowing 2 m deep

Solution:

Given: width of the channel: b = 20 m

velocity of water: V = 1.5 m/s

slope of energy line: ie = 0.0008

depth of water: h = 2 m

Let dh

dl be the rate of change of water depth Using equation in (4-14):

b e 2

i i dh

1

gh





4.2.1 Backwater calculation concept

Gradually varied flow is a steady, non-uniform flow in which the depth variation in the direction of motion is gradual enough to consider the transverse pressure distribution as being hydrostatic This allows the flow to be treated as one-dimensional with no transverse pressure gradients other than those due to gravity

For subcritical flows the flow situation is controlled by the downstream flow conditions A downstream hydraulic structure (e.g bridge piers, gate) will increase the upstream depth and create a “backwater” effect This concept has been introduced shortly in section 4.1.3 The term “backwater calculation” refers more generally to the calculation of the longitudinal free-surface profile for both subcritical and supercritical flows The backwater calculation is developed assuming:

 a non-uniform flow

 a steady flow

 that the flow is gradually varied, and

 that, at a given section, the flow resistance is the same as for a uniform flow with

the same depth and discharge, regardless of trends of the depth

4.2.2 Equation of gradually-varied flow

In addition to the basic gradually-varied flow assumption, we further assume that the flow occurs in a prismatic channel, or one that is approximately so, and that the slope

of the energy grade line can be evaluated from uniform flow formulas with uniform flow resistance coefficients, using the local depth as though the flow were locally uniform Referring to Fig 4.4., the total energy head at any cross-section is

2 V

2g



in which z = channel bed elevation; h = water depth,  = kinetic-energy correction coefficient as introduced in Chapter 2, and V = mean flow velocity

Fig 4.4 Definition sketch for gradually-varied flow

If this expression for H is differentiated with respect to x, the coordinate in the flow direction, the following equation is obtained:

dx

dE i i dx

dH

b

e  



2 V

2g



in which ie is defined as the slope of the energy grade line; ib is the bed slope (= - dz/dx); and E is the specific-energy head (i.e the energy head with respect to the bottom) Solving for dE/dx gives the first form of the equation of gradually varied flow:

e

b i i dx

dE



It appears from this equation that the specific-energy head can either increase or decrease

in the downstream direction, depending on the relative magnitudes of the bed slope and the slope of the energy grade line Yen (1973) showed that, in the general case, ie is not the same as the friction slope if (= 0/R, this equation will be introduced again in Chapter 7)

or the energy dissipation gradient Netherless, we have no better way of evaluating this slope than applying uniform-flow formulas such as those of Manning or Chezy It is incorrect, however, to mix the friction slope, which clearly comes from a momentum analysis, with terms involving , the kinetic-energy correction (Martin and Wiggert, 1975) Note: The bed slope ie and the friction slope if are defined as:

o

i = sin tan and i



respectively, where H is the mean total energy-head, z is the bed elevation,  is the channel slope and o is the bottom shear stress

bed

h

2

V

2 g

z

slope of energy grade line, ie

datum

dH

H

bed slope ib



dx





The second form of the equation of gradually-varied flow can be derived if it is recognized that dE dE dh

dx  dh dx and that, applying equation (4-11), dE 2

1 Fr

dh   , provided that the Froude number is properly defined Then, equation (4-17) becomes:

b e 2

i i dh

dx 1 Fr





The definition of the Froude number in equation (4-18) depends on the channel geometry For a compound channel, it should be the compound-channel Froude-number, while for a regular, prismatic channel, in which d/dh is negligible, it assumes the conventional energy definition given by Q2

B/gA3

The ratio dh/dx in Eq (4-18) represents the slope or the tangent to the water surface at any point along the channel This relationship therefore indicates whether at any point along the channel the water surface is rising (backwater condition) or dropping (drawdown condition) Immediately the following deductions can be made:

 When dh 0

dx  , the slope of the water surface is dropping in the downstream direction and the depth decreases downstream

 When dh 0

dx  , the slope of water surface is parallel to the channel bottom and uniform flow exists This can be readily seen from Eq (4-18) since, for this condition, ib = ie must equal zero

 When dh 0

dx  , the slope of water surface rises in the downstream direction and the depth h increases downstream

 When dh

dx   , which requires that 1 – Fr2

= 0 or Fr = 1, the slope of the water surface must theoretically be vertical This flow occurs when the flow changes from subcritical to supercritical, or vice versa, as indicated by the value of the Froude number The formulas derived do not actually apply any longer due to the assumptions made A vertical water surface also does not occur in reality; however,

a very noticeable change in the water surface takes place This is especially so when the flow changes from below hc to above hc In such instance a phenomenon known as the hydraulic jump occurs

4.3.1 Classification of flow profiles

From the foregoing, it is evident that the relationship expressed in Eq (4-18) provides a considerable amount of information as to the shape of the water surface profile

in an open channel Investigation of this formula yields the following results:

1 The relationship between the slope of the channel bottom and the slope of the energy grade line determines whether the numerator of the equation is positive or negative

2 The denominator of the equation is positive if Fr < 1.0 and vice versa In other words, if the flow is subcritical (Fr smaller than 1) the denominator is positive, and

if the flow is supercritical (Fr greater than 1) the denominator is negative

The conditions at which flow in an open channel can take place and the possible

relationships between the observed depth ho, the normal depth at which flow is uniform hn, and the critical depth hc are illustrated in Fig 4.5 It is evident from this figure that there

are three zones of channel depths at which flow can be observed:

Zone 1, with ho greater than hn and hc (i.e ho > hn > hc)

Zone 2, with ho between hn and hc (i.e hn > ho > hc)

Zone 3, with ho less than hn and hc (i.e hn > hc > ho)

Fig.4.5 Three zones of channel depths

The relative bottom slope defines whether uniform flow is subcritical or supercritical Determine the associated Froude-number Fre

e

Fr

where R is the hydraulic radius of the open channel flow Subcript e denotes the equilibrium flow The bottom/wall shear stress is defined as:

2

o cf Ve gR ie f

gR c  c (the friction slope if = the bed slope ib )

e

e f

R i Fr

h c





We have: Ae = Be.he  e e e

e

A P R h

  , where Pe is the equilibrium wetted perimeter

ho hn

hc

ho > hn > hc

ho

hn

hc

hn > ho > hc

ho

hn

hc

hn > hc > ho

 2 e b

e

e f

B i Fr

P c





In case of turbulent flow:   1

For two-dimensional flow: e e

1

h  P 

So, a proper approximation for Fre is: 2 b

e f

i Fr c



 If ib < cf, we have a mild slope (M – type)

The “uniform flow” is subcritical: Fre2 < 1, he > hc

 If ib > cf, we have a steep slope (S – type)

The “uniform flow” is supercritical: Fre2 > 1, he < hc

 If ib = cf, we have a critical slope (C – type) Fre2 = 1, he = hc

Note: It can easily be derived that 2 13

g

C



  , where C is Chezy coefficient and n

is Manning’s

Two conditional channel bottom conditions or slopes exist These do not really constitute open channel flow, but gravity flow can take place along them They are as follows:

 If ib < 0, we have an adverse slope (A – type)

 If ib = 0, we have a horizontal slope (H – type)

It should be noticed that hn = he

Note: The actual flow depends on the boundary condition, i.e “mild”, “steep”, etc does not tell us anything about the actual flow

4.3.2 Sketching flow profiles

In theory, for each of the five slope descriptions above there are three zones in which flow can be observed It follows then that a total of 15 theoretical water surface profiles are possible, presented in Table 4.1 These profiles, together with illustrations of practical applications, are shown in Fig 4.6

While this figure is for the most part self-explanatory, the following observations and explanations are presented for further clarification

 Mild slope (ib < cf) The M1 curve is generally very long and asymptotic to the horizontal and the line representing ho The M2- and M3-curves end in a sudden drop through the line representing hc and a hydraulic jump, respectively

 Critical slope (ib = cf) Since hc = hn in this case, there is no zone 2, and only two water surface profiles exist, C1 and C3 The C2-curve coincides with the water surface that corresponds to uniform flow at critical depth