Week 12 GVF Analysis

TOPIC 1: PIPE TOPIC 2: PIPE TOPIC 3: OPEN TOPIC 4:
FLOW FLOW ANALYSIS CHANNEL FLOW HYDRAULIC
MACHINERY

LAMINAR FLOW IN UNIFORM FLOW PUMP
FLOW PIPELINES

TURBULENT FLOW IN PIPE NON-UNIFORM TURBINES
FLOW NETWORKS FLOW (RAPIDLY
VARIED FLOW)
POINT LOSSES WATER
RETICULATION NON-UNIFORM
FLOW
DESIGN
(GRADUALLY
VARIED FLOW)

ECW 435 Hydraulics
Week 12

Topic 3.3: Non-Uniform Flow
(Gradually Varied Flow-GVF)

TOPIC 3:
OPEN CHANNEL FLOW

SUB-TOPIC 3.3:
GRADUALLY VARIED FLOW

SUB-TOPIC 3.3:
GRADUALLY VARIED FLOW

LEARNING OUTCOMES

To evaluate the effects of channel slope, friction slope, and
Froude number on flow depths.
To identify and describe the 12 types of water surface
profiles.
To solve GVF problems using direct step method and
standard step method.

CLASSIFICATION OF NON-UNIFORM OCF

RVF GVF

Rapidly Varied flow Gradually Varied Flow

d/dx ~ 1 d/dx <<1

Observable Usually not observable

Localised acceleration Not localised
Momentum equation is
Momentum equation is
used not used

e.g. flow over obstacles, Backwater curve
constriction, hydraulic
RVF: jump

INTRODUCTION TO GVF

Natural channel is characterised by GVF; uniform flow is rare (why?).
> Water depth and velocity change gradually

Define GVF > Flow is non-uniform
> Water surfaces changes smoothly and continuously
> Friction loss along the channel is not negligible

GVF RVF

Effect of a broad-crested weir on channel flow will give
RVF & GVF condition.

Illustration of gradually water surfaces changes due to effect of broad-crested weir

GVF ANALYSIS

GVF analysis is performed by applying uniform flow

concept on successive channel reaches where the flow

condition within the reach may be approximated as

uniform.

GVF RVF

Uniform flow Analytical solution possible if assuming:
concept ~ Straight, rectangular channel with constant roughness
~ Small bed slope (i.e. d ~ y)
~ Steady flow, where streamlines are approximately parallel
so that pressure distribution is hydrostatic

GVF EQUATION AND ANALYSIS

E1 +z1= E2 +z2+HL H1 = H2 + HL

dE = E2 − E1 = (z1 − z2 ) − HL dH = H2 − H1 = −HL

dE =S 0 −Sf Standard step dH = −Sf
dx method dx
•
Fixed distance difference x
• Solve incremental depth y

E = y + v 2 = y + (Q / By )2 dE = dE dy
2g 2g dx dy dx

dE = 1 − Q2B = 1− Fr2 dy = dy dE = S0 − Sf Direct step
dy gA3 dx dE dx 1− Fr 2 method

• Fixed depth difference y
• Solve incremental distance x

Same as:

D l = (s − i ) 1− Fr2

*Repetitious & tedious analysis
(USE SIMULATION MODEL)

https://www.youtube.com/watch?v=pFUSdfbbtVY

BED SLOPE AND FRICTION SLOPE

D = s−i Or y = s0 − s f
l 1− Fr2 x 1− Fr2

Change of water depth D over distance l
is dependent on relative difference between

bed slope S (S0) and friction slope i (Sf)

Effects of bed slope and friction slope are
important in GVF.
Their effect in RVF is negligible since RVF occurs
over a short distance.

FOR UNIFORM FLOW,

bed slope = friction slope, i.e. s = i (S0 = Sf),
From Manning eq.:

Q = 1 A5 / 3 S01/ 2  S0 =  nQ P2/3 2 = Sf z
n P2/3  A5 / 3 
STEADY & UNIFORM FLOW
FOR NON-UNIFORM FLOW,

bed slope  friction slope,i.e. s  i (S0  Sf),

Slope of EGL, Sf = Energy
loss, hL

Change of water depth D over distance l
is dependent on relative difference between

bed slope S (S0) and friction slope i (Sf)

Bed slope, S0 D = s − i y = s0 − s f
l 1− Fr2 x 1− Fr2
z = Or

STEADY & NON-UNIFORM FLOW (∂D due to bed level
/slope/lateral

changes)

CHANGES OF WATER SURFACE PROFILE

TYPES OF WATER SURFACE PROFILES:

Type 1 • D > Dn
• D > Dc

Type 2 • D between Dn & Dc

Type 3 • D < Dn
• D < Dc

Dn = Normal Depth
Dc = Critical Depth
D = Measuring Depth

Liquid Surface Profiles in Open Channels, y(x)

It is important to be able to predict the flow depth for a specified flow rate and
specified channel geometry.

A plot of flow depth versus downstream distance is the surface profile y(x) of the
flow.

The general characteristics of surface profiles for gradually varied flow depend
on the bottom slope and flow depth relative to the critical and normal depths.

Designation of the letters S, C, M, H, Designation of the numbers 1, 2, and 3 for liquid
and A for liquid surface profiles for surface profiles based on the value of the flow depth
different types of slopes. relative to the normal and critical depths.

12 TYPES OF WATER SURFACE PROFILES

1− (Dn D)c 1− (DC D)3

Mid Slope D = s 1− (Dn D)c
Steep Slope l 1− (Dc D)3

Type • D > Dn
1 • D > Dc

Critical Slope *Cannot have type 2 flow Type •D
2
between

Dn & Dc

D l = (s − i ) 1− Fr2 S−i 1− (DC D)c Type • D < Dn
3 • D < Dc

Horizontal *Cannot have
Slope Type 1 flow

Adverse
Slope

always (–)

TYPES OF WATER SURFACE PROFILES

1− (Dn D)c 1− (DC D)3

Mid Slope D = s 1− (Dn D)c
Steep Slope l 1− (Dc D)3

Type • D > Dn
1 • D > Dc

Type •D
2
between

Dn & Dc

Type • D < Dn
3 • D < Dc

*Change of water depth D over distance l
is dependent on relative difference between
bed slope S (@S0) and friction slope i (@Sf)

https://www.youtube.com/watch?v=pFUSdfbbtVY





EXAMPLE OF GVF IN HYDRAULIC JUMP

GVF may occur before or after a jump to match TW.

y1 y2 = TW (for stable
hydraulic jump)
(a) GVF before jump
Consider 2 mild slope cases with
same sluice gate opening & TW

y1 y2 < TW TW

(b) GVF after jump Baffle blocks

EXAMPLE OF GVF IN HYDRAULIC JUMP

GVF may occur before or after change of slope.

= D2

((ab))Jump on mild slope

((ba))Jump on steep slope

JUNE 2015
DEC 2015
JULY 2017

Solution: JULY 2017

Given: Actual water depth at measuring location, D = 1.4 m. Type • D > Dn
For GVF profiles, you need to do comparison D with Dc & Dn: 1 • D > Dc

Hint: flow condition_Manning Equation), Q = 1 A5 / 3 S01/ 2 Type • D between
Dn (uniform n P2/3 2 Dn & Dc

Dc (for rectangular channel), Dc = 3 q2 =3 Q2 Type • D < Dn
g gB 2 3 • D < Dc

Answer: Dn > Dc > D (4.22 m > 1.73 m > 1.4 m) → M3-Type curves