ch9 non-uniform flow in open channels

Chapter 9Non-uniform Flow in Open Channels

Overview

9.1 Critical , Sub-critical and super critical flow

9.2 Specific energy

9.3 Gradually Varied Flow

9.4 Hydraulic jump

NON-UNIFORM FLOW IN OPEN CHANNELS

• Definition: By non-uniform flow, we mean that the velocity varies at each section of the channel.

• Velocities vary • Non-uniform flow can be caused by

i) Differences in depth of channel and ii) Differences in width of channel. iii) Differences in the nature of bed iv) Differences in slope of channel and v) Obstruction in the direction of flow.

Non-uniform Flow In Open ChannelsContd.

• In the non-uniform flow, the Energy Line is not parallel to the bed of the channel.

• The study of non-uniform flow is primarily concerned with the analysis of Surface profiles and Energy Gradients.

Hydraulic characteristics of the non-uniform flow in open channels: The bed slope i, water surface gradient Jp and the hydraulic slope J are not equal, i.e. show as fig.

9.1 Critical , Sub-critical and super critical flow

Sub-critical Flow: If the small surface wave can propagate upstream as well as downstream, it will lead to a backwater zone of great distances before the obstacles. This flow is called a sub-critical flow, in which the velocity of flow is less than that of the wave propagation, namely, V<C.

C C

C- v C+ v

Sub-critical Flow

Supercritical Flow

The small surface wave will be formed when the flow in the open channel is disturbed. If the wave can only propagate downstream, and can't propagate against the flow, the backwater zone is formed only around the obstacles. This kind of flow in open channels is called supercritical flow, in which the velocity of flow is greater than that of the wave propagation, namely, v>c.

C + v

v > C

Supercritical Flow

Critical Flow

If the velocity of the small surface wave propagating upstream is zero, which is just the critical situation to distinct the supercritical flow and the sub-critical flow, this kind of flow is called the critical flow, namely, V=C .

2 C

v= C

Critical Flow

9.2 Specific Energy

• Total mechanical energy of the liquid in a channel in terms of heads

z is the elevation heady is the gage pressure headV2/2g is the dynamic head

• Taking the datum z=0 as the bottom of the channel, the specific energy Es is

Specific Energy

In a channel with constant discharge, Q2211 VAVAQ ==

2

2

2gAQyEs +=g

VyEs 2

2

+= where A=f(y)

Consider rectangular channel (A = By) and Q = qB

2

2

2gyqyEs +=

A

B

y

q is the discharge per unit width of channel

How many possible depths given a specific energy? _____2

Specific Energy Curve

• For a channel with constant width b,

• Plot of Es vs. y for constant V and b

ybVVAc ==Q

22

2

2 ygbQyEs +=

Specific Energy Curve

For a given flow, if

Es < Emin No solution is possible

Es = Emin Flow is critical, y = yc, V = Vc Fr = 1

E s > Emin , V < Vc :

Flow is subcritical, y > yc , Fr < 1 , disturbances can propagate upstream as well as downstream

Es > Emin , V > Vc :

Flow is supercritical, y < yc , Fr > 1, disturbances can only propagate downstream

Froude Number

• This is a Dimensionless Ratio Characterizing Open Channel Flow.

• Froude Number, gh

VF r =

forcegravity forceinertial

Kinetic energyPotential energy

Froude Number and Wave Speed

• Critical depth yc occurs at Fr = 1

• At low flow velocities (Fr < 1)• Disturbance travels upstream• y > yc

• At high flow velocities (Fr > 1)• Disturbance travels downstream• y < yc

2

22

cc gA

Qg

Vy ==

Multiple Choices

When the flow in open channel is supercritical flow:

A. Fr>1B. h>hcC. v<vcD.

Judgement

The specific energy must increase with the increase of water depth.

Your answer: True false

9.3 Gradually Varied Flow

• In GVF, y and V vary slowly, and the free surface is stable

• In contrast to uniform flow, Sf ≠S0. Now, flow depth reflects the dynamic balance between gravity, shear force, and inertial effects

• To derive how how the depth varies with x, consider the total head

Gradually Varied Flow

• Take the derivative of H

• Slope dH/dx of the energy line is equal to negative of the friction slope

• Bed slope has been defined

• Inserting both S0 and Sf gives

Gradually Varied Flow

• Introducing continuity equation, which can be written as

• Differentiating with respect to x gives

• Substitute dV/dx back into equation from previous slide, and using definition of the Froude number gives a relationship for the rate of change of depth

Gradually Varied Flow

• This result is important. It permits classification of liquid surface profiles as a function of Fr, S0, Sf, and initial conditions.

• Bed slope S0 is classified as• Steep : yn < yc• Critical : yn = yc• Mild : yn > yc• Horizontal : S0 = 0• Adverse : S0 < 0

• Initial depth is given a number• 1 : y > yn• 2 : yn < y < yc• 3 : y < yc

Gradually Varied Flow

• 12 distinct configurations for surface profiles in GVF.

Gradually Varied Flow

• Typical OC system involves several sections of different slopes, with transitions

• Overall surface profile is made up of individual profiles described on previous slides

9.4 Hydraulic Jump

• Flow is called rapidly varied flow (RVF) if the flow depth has a large change over a short distance• Sluice gates• Weirs• Waterfalls• Abrupt changes in cross

section• Often characterized by

significant 3D and transient effects• Backflows• Separations

Hydraulic Jump

• Used for energy dissipation• Occurs when flow transitions from supercritical to

subcritical• base of spillway• Steep slope to mild slope

• We would like to know depth of water downstream from jump as well as the location of the jump

• Which equation, Energy or Momentum?

Fig. Flow under a sluice gate accelerates from subcritical to critical to supercritical and then jumps back to subcritical flow

subcriticalflow

supercritical

A hydraulic jump in a flume is illustrated below.

Hydraulic jump in the laboratory.

Itaipu dam spillway withhydraulic jump.

Hydraulic jump variations: (a) jump caused by a change in channel slope,(b) submerged jump.

CONJUGATE DEPTH RELATION FOR THE HYDRAULIC JUMP

The illustrated channel below carries constant water discharge Q in a wide, rectangular channel of constant width B. The flow makes the transition from supercritical to subcritical through a hydraulic jump.

Momentum balance in the illustrated control volume is considered (width B out of the page).

hydraulic jump

Fr > 1Fr < 1

CONJUGATE DEPTH RELATION FOR THE HYDRAULIC JUMP

• Continuity equation

• momentum equation

• Substituting and simplifying

Quadratic equation for y2/y1

CONJUGATE DEPTH RELATION FOR THE HYDRAULIC JUMP

• Solving the quadratic equation and keeping only the positive root leads to the depth ratio

• Energy equation for this section can be written as

• Head loss associated with hydraulic jump

Hydraulic Jump:Energy Loss and Length

No general theoretical solution

Experiments show

26 yL = 14.5 13Fr< <

äLength of jump

äEnergy Loss

2

2

2gy

qyE +=

LhEE += 21

significant energy loss (to turbulence) in jump

( )

21

312

4 yy

yyhL

−=algebra

for

Hydraulic Jump

• Often, hydraulic jumps are avoided because they dissipate valuable energy

• However, in some cases, the energy must be dissipated so that it doesn’t cause damage

• A measure of performance of a hydraulic jump is its fraction of energy dissipation, or energy dissipation ratio

Hydraulic Jump

• Experimental studies indicate that hydraulic jumps can be classified into 5 categories, depending upon the upstream Fr