UNIFORM FLOW

CRITICAL FLOW

GRADUALLY VARIED FLOW

Derivation of uniform flow equation

Dimensional analysis

Computation of normal depth

UNIFORM FLOW1. Uniform flow is the flow condition obtained from a

balance between gravity and friction forces.

2. Uniform flow is often used as a design condition to determine channel dimensions

3. It requires the use of an empirical resistance coefficient

4. This coefficient has been subject of research since the 19th century

5. Design parameters include: channel slope, channel shape, soil conditions, topography and availability of land, risk and frequency analysis

6. The uniform flow depth is called NORMAL DEPTH

MOMENTUM AND ENERGY ANALYSISFOR UNIFORM FLOW IN OPEN CHANNELS

MOMENTUM ANALYSIS

FORCE BALANCE

Gravity Force = Friction force

P = wetted perimeter

τ0 = mean boundary shear stress

A = cross sectional area

γ = specific weight of water

Dividing by PΔL gives:

LPLA 0sin

0sinR 0 0 /S R

• Assuming small channel slope:

S0= sinθ, where S0=channel bed

slope, gives

W=γA ΔL

θ

ΔL

τ0 P ΔL

Wsenθ

Fp1

Fp2

A y

6

2 2

2 2 1 12 2 1 1

2 2f

y V y Vz z h

g g

Dr. Walter F. Silva

Energy

Potential Energy

+

Pressure Energy

+

Kinetic Energy

1

EGL

HGL

2

2

V

g

z

2

2

V

g

z

y

fh

2

y

z

y

2

2

V

g

ENERGY ANALYSIS

𝑆0

𝑆𝑓

𝑆𝑤

ENERGY ANALYSISFor uniform flow the bed slope is parallel to the water slope and to the energy slope as shown next

Energy Slope

Recalling that for uniform flow the velocity is constant in the channel reach, dV2 /dx = 0, therefore:

2

( )2

f

dH dy dz d VS

dx dx dx dx g

0

dx

dz

dx

dy

dx

dHS f

0f

dH dzS S

dx dx

lf

hS

L

Since the water depth is constant in uniform flow, dy/dx = 0, then:

The energy slope is equal to:

where hl is the energy loss

ENERGY ANALYSISExpressing the slope in terms of the shear stress results in

This equation relates the shear stress with the channel slope, however, it does not includes the flow characteristics.

There most be a relation between the shear stress and the flow velocity.

This relationship is obtained from dimensional analysis

RL

hS l

00

W=γA ΔL

θ

ΔL

τ0 P ΔL

Wsenθ

Fp1

Fp2

A y

DIMENSIONAL ANALYSISFOR UNIFORM FLOW

DIMENSIONAL ANALYSIS• Consider that the shear stress is a function of

Fluid density

Fluid viscosity

Acceleration of gravity

Flow velocity

Channel geometry (represented by the hydraulic radius)

Channel roughness (represented by a parameter k)

• Other variables could also be included such as channel meandering and unsteady effects.

• The functional relation for the previous parameters is given by:

kRVgf ,,,,,0

DIMENSIONAL ANALYSIS

• Applying Buckingham Π Theorem to this relation we get the following results

• Non-dimensional parameters namely: the Reynolds number (Re), the Froude number (Fr) and a relative roughness parameter.

• Not all of them are included in commonly used equations.

• Reynolds number and a roughness parameter are used for pipe flow.

• Others use the roughness parameter as the most important factor, assuming fully rough flow, where the viscous effects are not important

• There are procedures to include meandering and vegetation effects in the estimation of the roughness parameter

R

k

Frf

V,

1,

Re

1222

0

UNIFORM FLOW FORMULAS

UNIFORM FLOW FORMULAS• There is an expression for the energy losses extensively

used for pipe flow called the Darcy-Weisbach friction formula, given by:

where L is the pipe length, D is the pipe diameter and f is the Darcy-Weisbach friction factor.

f is a function of the Reynolds number and the relative roughness.

• According to this equation the slope of the energy line is

gD

LfVh f

2

2

gD

fV

L

h f

2

2

D

UNIFORM FLOW FORMULAS• For a direct use of this equation to an open channel, it is

necessary to change the geometric characteristic.

• For a channel the convention is to replace the diameter by the hydraulic radius.

• For a pipe the hydraulic radius is:

• Therefore, for a channel:

44

2 D

D

D

P

AR

)4(2

2

0

Rg

fV

RL

h f

0

2 8

f

V

D

UNIFORM FLOW FORMULAS• Also

• This equation is called the Chezy’s formula in honor to a French Engineer called Antoine Chezy who first proposed this formula in 1775.

• Several researchers tried to develop rational procedures for estimating the value of Chezy’s constant, C.

)4(2

2

0

Rg

fV

RL

h f

0

2

1 SR

VC

L

h f

RSCV 0

𝐶1 =𝑓

8𝑔

UNIFORM FLOW FORMULAS• Gauckler (1867) showed that

• Latter, Robert Manning, an Irish engineer, proposed in 1889 the following formula

• Nowadays the Manning’s equation form is:

• and should be called Gauckler-Manning formula. Kn is 1.0 for the metric units system and 1.49 for English units system.

61

RC

21

32

1 SRCV 2 1

3 2nKV R S

n

UNIFORM FLOW FORMULAS

That Manning’s equation “has endured for more than a century as a uniform flow formula would seems to indicate that Manning’s labors were not in vain, although the formula that bears his name probably would be surprising to him” (Sturm)

21g

VyZH

2

2

TOTAL ENERGY HEAD

g

VyE

2

2

SPECIFIC ENERGY

g

VyE

2

2

2

2

2gA

QyE

y1 and y2 are called alternate depths

y1= supercritical flow, Fr>1

y2= subcritical flow, Fr<1

yc= critical depth, Fr =1 (minimum specific energy)22

y1

y2

y1

y1

y2y2

Flow Regimes Froude Number

Subcritical <1

Critical =1

Supercritical >1

23

V = Average flow velocity

g = Acceleration of gravity

Dh = Hydraulic depth

Critical Depth is the flow depth corresponding to the minimum specific energy. Corresponds to Fr =1

gD

VFr 1

)/( TAgV

gDAQ

24

D=y A=by

cc Ey3

2

2

23

gb

Qyc

25

y

b

0

0.2

0.4

0.6

0.8

1

1.2

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3W

ate

r D

ep

th (

m)

Specific Energy (m)

SPECIFIC ENERGY CURVE FOR CIRCULAR

CONDUIT

Y Theta Area KE Spec. E.

0.32 1.73 0.30 2.59 2.91

0.36 1.85 0.36 1.91 2.27

0.41 1.98 0.43 1.52 1.92

0.45 2.09 0.50 1.27 1.72

0.50 2.21 0.57 1.13 1.62

0.54 2.32 0.64 1.03 1.57

0.59 2.43 0.72 0.98 1.57

0.63 2.53 0.79 0.95 1.58

0.68 2.64 0.87 0.94 1.62

0.72 2.74 0.95 0.95 1.67

0.77 2.84 1.03 0.96 1.72

0.81 2.94 1.11 0.98 1.79

0.86 3.04 1.19 1.00 1.85

0.90 3.14 1.27 1.03 1.93

0.95 3.24 1.35 1.06 2.00

Pipe Diameter = 1.8

Discharge = 2

Units System = SI

Number of div = 40

Total Area = 2.54

Fundamental equations

Computation methods

Definition: Steady, Non-Uniform flow in which the depth

variation in the direction of motion is gradual enough that the pressure distribution can be considered hydrostatic

Additional assumptions

Small channel slope

Energy losses could be estimated by using

Manning’s equation

FUNDAMENTAL EQUATION

Total Head

We are interested in the variation of y with x

dx

dy

𝑧1 + 𝑦1 +𝑉12

2𝑔= 𝑧2 + 𝑦2 +

𝑉22

2𝑔+ ℎ𝑓

EGL

HGL

fh

2

𝑦1

𝑦2

1

𝑧1𝑧2

𝑉12

2𝑔𝑉22

2𝑔

𝑆0

𝑆𝑓

𝑆𝑤

Recalling that𝑑𝐻

𝑑𝑥= −𝑆𝑓 (slope of the energy equation)

And𝑑𝑧

𝑑𝑥= −𝑆0 (slope of the channel bottom)

The derivative of the total head becomes:

F = Froude number

𝑑𝑦

𝑑𝑥=𝑆0 − 𝑆𝑓

1 − 𝐹2

• The normal depth together with the critical depth are used to classify the channel slope as mild, steep or critical. This classification is associated with the gradually flow profiles notation as M, S or C.

• Mild slope (M) yn > yc , in this case the uniform flow is subcritical

• Steep slope (S) yn < yc , in this case the uniform flow is supercritical

• Critical slope (C) yn = yc , , in this case the uniform flow is critical

Normal and critical depths divides the space above the channel bottom into three regions

There are 13 different types of surface profiles:

3 mild, 3 steep, 2 critical, 2 horizontal, 2 adverse

Zone 3

Zone 2

Zone 1NDL or CDL

CDL or NDL