Uniform Channel Flow – Basic Concepts

Hydromechanics VVR090

ppt by Magnus Larson; revised by Rolf L Feb 2014

SYNOPSIS

1. Definition of Uniform Flow2. Momentum Equation for Uniform Flow 3. Resistance equations 4. Flow Resistance Coefficients 5. Selecting a Manning’s roughness6. Examples/Problems

1. Definition of Uniform Flow

Uniform flow occurs when:

1. The depth, flow area, and velocity at every cross section is constant

2. The energy grade line, water surface, and channel bottom are all parallel:

f w oS S S

Sf = slope of energy grade line

Sw = slope of water surface

So = slope of channel bed

Definition Sketch for Uniform Flow

Depth for uniform flow is denoted ”Normal depth ” (y0 or yn)

If normal depth y0 < yc (supercritical flow) then slope is ”steep”If normal depth y0 > yc (subcritical flow) then slope is ”mild”

Profiles

Mild slope

Steep slope

Conditions that allow uniform flow to develop are rarely satisfied in practice.

However, it is a concept of great significance in understanding and solving most problems in open-channel hydraulics.

Uniform flow occurs in long, straight, prismatic channel where a terminal velocity can be achieved =>

ENERGY balance between head loss due to turbulent flow and reduction in potential energy

FORCE Balance between gravity and boundary shear forces

2. Momentum Equation for Uniform Flow

Gravity force (driving motion):

sin sinmF W AL

Boundary shear force (resisting motion):

R oF LP

Shear stress proportional to bottom velocity squared:2

o ku

VVR170. 5 Feb 2013. 8 (43)

Momentum Equation for Uniform Flow cont’d

Steady state conditions: gravity force = shear forces

2

1/ 2

sin

m RF F

AL ku LP

u RSk

ARP

3. Resistance equations. a) the Chezy Equation

The Chezy equation is given by:

1/ 2

u C RS

Ck

C has the dimensions L1/2/TAntoine Chezy

b) The Manning Equation

The Manning equation is given by:

2/31u R Sn

n has the dimensions T/L1/3

Compare with the Chezy equation:

1/ 6RCn

Robert Manning

General Equation for Uniform Flow

Most semi-empirical equations for the average velocity of a uniform flow may be written:

x yu CR S

Manning equation is the most commonly employed equation in open channel flow (x=2/3, y=1/2).

It will be used for calculations in the present course.

4. Flow Resistance Coefficients I

Difficult to estimate an appropriate value on the resistance coefficient in the Manning or Chezy equations.

Should depend on:

• Reynolds number

• boundary roughness

• shape of channel cross section

Compare with the Darcy-Weisbach formula for pipe friction:

2

4 2LL uh fR g

Flow Resistance Coefficients II

Slope of the energy line:

Compare with Manning and Chezy equation:

2

4 2Lh f uS

L R g

1/ 6

8

8

fn Rg

gCf

Types of Turbulent Flow

Two main types of turbulent flow:

• hydraulically smooth turbulent flow:

Roughness elements covered by viscous sublayer (resistance depends on Reynolds number Re)

• hydraulically rough flow:

Roughness elements penetrates through the viscous sublayer (resistance coefficient depends on roughness height ks)

Transitional region in between these flows (dependence on Re and ks)

Example of Roughness Heights (ks)

Definition of Reynolds Number

Definitions of Reynolds number:

**

*

4Re

Re s

oo

u R

u k

u gRS

*

*

*

0 Re 4 smooth

4 Re 100 transition

100 Re rough

Criteria for Turbulent Flow Types

Pipe Flow Friction Factors

Hydraulically smooth flow:

0.25

0.316 Re 100,000Re

Re1 2.0log Re 100,0002.51

f

ff

Hydraulically rough flow:

1 122.0logs

Rkf

Colebrook’s formula applicable for the transition region:

1 2.52.0log12 Re

skRf f

Plots of f versus ks/4R and Re (analogous to a Moody diagram).

Re number

Fric

tion

Fact

or

Relative Roughness

Selecting a Suitable Roughness

5. Selecting a Manning’s roughness

Difficult to apply f from pipe flow.

Manning’s n is often determined based on empirical knowledge, including the main factors governing the flow resistance:

• surface roughness

• vegetation

• channel irregularity

• obstruction

• channel alignment

• sedimentation and scouring

• stage and discharge

Soil Conservation Service (SCS) Method for n

Determine a basic n for a uniform, straight, and regular channel, then modify this value by adding correction factors.

Each factor is considered and evaluated independently.

Channel Characteristics Basic n

In earth 0.020

Cut in rock 0.025

In fine gravel 0.024

In coarse gravel 0.028

Procedure:

1. Select basic n

2. Modify for vegetation

3. Modify for channel irregularity

4. Modify for obstruction

5. Modify for channel alignment

6. Estimate n from step 1 to 5

A total n is obtained as the sum of the different contributions.

Influence of Vegetation

Influence of Cross-Section Size and Shape, and Irregulariy

Influence of Obstruction and Channel Alignment

Example of Manning’s n from Chow (1959)

(illustrative pictures in the following)

0.012

0.014

0.016

Manning’s Roughness n

0.018

0.018

0.020

Manning’s Roughness n

0.020

0.022

0.024

0.024

0.026

0.028

Manning’s Roughness n0.040

0.040

0.045

0.029

0.030

0.035

Manning’s Roughness n

0.110

0.125

0.150

0.050

0.060

0.080

Example 5.1Given a trapezoidal channel with a bottom width of 3 m, side slopes of 1.5:1, a longitudinal slope of 0.0016, and a resistance coefficient of n = 0.013, determine the normal discharge if the normal depth of flow is 2.6 m.

Example 5.2

Given a trapezoidal channel with a bottom width of 3 m, side slopes of 1.5:1, a longitudinal slope of 0.0016, and a resistance coefficient of n = 0.13, find the normal depth of flow for a discharge of 7.1 m3/s.