Basic Hydraulics: Channels Analysis and design – I

Basic Hydraulics: Channels Analysis and design – I

Terminology

Manning’s equation

• Manning’s equation originally developed for open channel flow (by an accountant, no less!)

• Usually written as

v = (1.486/n) R2/3Sf1/2

where v = velocity (ft/sec); n = Manning’s coefficient (also called Manning’s n); R = hydraulic radius (A/P in ft); P = wetted perimeter (ft); Sf = slope of energy gradient line (ft/ft) = hL / L

• Tables of n values available for various surfaces.

• Rearrange Manning's equation to solve for Sf = hL / L (head loss per unit length):

(hL / L) = [Vn / (1.486 R2/3)]2

Hydraulic radius

• The hydraulic radius (Rh) is the cross sectional area of the flow divided by the wetted perimeter. For a circular pipe flowing full, the hydraulic radius is one-fourth of the diameter. For a wide rectangular channel, the hydraulic radius is approximately equal to the depth.

A = cross–sectional area of the flowing fluid; P = wetted perimeter.

P

ARh

Slope-area method

• Provides a simple relationship for relating water surface elevation to discharge at a particular channel section.

• Often used for calculating tailwater at culvert outlets and storm drain outlets.

• TxDOT Hydraulic Design Manual suggests using this procedure for small stream crossings or situation for which no unusual flow characteristics are anticipated.

• If crossing is an important one, the Hydraulic Design Manual recommends using a “backwater method.” (More on that later.)

Slope-area data needs

• Channel cross section: Choose a typical cross section downstream from crossing

• Channel roughness• Channel slope

• Use average bed slope near site• Find from surveys or topographic maps

• Use Manning’s equation to calculate water surface elevation as a function of discharge

Hydraulic depth

• The Froude Number (Fr) represents the ratio of inertial force to gravitational force and is calculated by:

where dm is the hydraulic mean depth and is defined by dm = A/T where A is the cross-sectional area of the flow and T is the channel top width at the water surface.

m

rgd

vF

Backwater (Frontwater) Methods• Computing water surface profiles in cases more

complex than slope-area situations uses the energy equation to estimate the water surface elevation at different sections from a known location.

• The plot of the elevations is usually called a back-water curve for M1,M2, and S1 curves and a front-water curve for M3, S2, and S3 curves.

• The variable step method is illustrated by an example to familiarize the participant with the mechanics of the method, however for practical applications use of specialized software is recommended (HEC-RAS, WSPRO, SWMM, etc.)

Backwater (Frontwater) Methods• Backwater methods start with the specific

energy at two cross sections

€

E1 = y1 +v12

2g

€

E2 = y2 +v22

2g

Backwater (Frontwater) Methods• Then the bottom elevations are included as

is the head loss• Therefore the total head at both sections

are equal

€

E1 = y1 +v12

2g

€

E2 = y2 +v22

2g

€

E1 + z1 = E2 + z2 + hL

Backwater (Frontwater) Methods• Next incorporate the channel bottom slopes

and the energy grade line slope to replace the elevations in terms of these slopes

€

E1 + z1 = E2 + z2 + hL

€

E1 + (z1 − z2) = E2 + hL

€

E1 + S0Δx = E2 + S fΔx

€

S0 − S f =E2 − E1

Δx

Backwater (Frontwater) Methods• Now use some calculus and section geometry

to convert into discharge, area, and depth

€

S0 − S f = limΔx→0

E2 − E1

Δx=

dE

dx=

dE

dy

dy

dx

€

dE

dy=

d

dy[y +

v 2

2g] =

d

dy[y +

Q2

2gA2]

€

dE

dy=

d

dy[y +

Q2

2gA2] =1−

Q2

2gA3

dA

dy

Backwater (Frontwater) Methods• Now use some calculus and section geometry

to convert into discharge, area, and depth

€

S0 − S f =dE

dy

dy

dx

€

dE

dy=

d

dy[y +

Q2

2gA2] =1−

Q2

gA3

dA

dy

€

Q2

gA3

dA

dy= Fr2

Backwater (Frontwater) Methods• Finally insert the substitutions and the

result is an equation that relates depth taper to channel geometry and specific energy

€

S0 − S f = (1− Fr2)dy

dx

€

dy

dx=

S0 − S f

1− Fr2

Backwater (Frontwater) Methods• “Integrating” the GVF equation from a known

section forward in space or backward in space produces the front- or back-water curve (water surface profile)

€

dy

dx=

S0 − S f

1− Fr2

€

y(x) =S0 − S f

1− Fr2∫ dx

Depth

Position Geometry

GravityFriction

Backwater (Frontwater) Methods• Recall the original expression of the Froude

Number

• Notice the area and topwidth are incorporated, some algebra and an alternate expression is

€

Fr =V

gA

T

€

Fr2 =V 2

gA

T

=V 2A2T

gA3=

Q2T

gA3

Backwater (Frontwater) Methods• Now we have the relationships for computing

a water surface profile from some known condition

• Such computation involves:• Select a location where depth is known (or assumed)

• Determine the slope designation, and profile type (M1, S2, etc.)

• Use the designation to decide if integration is downstream or upstream.

Backwater (Frontwater) Methods• Once these steps are completed, the simplest

method is the constant depth change, variable distance method.

• The energy equation is rearranged to solve for the spatial step as

€

E1 + S0Δx = E2 + S fΔx

€

Δx =E2 − E1

S0 − S f

Backwater (Frontwater) Methods• This form of the equation suggests the

following algorithm

1.Start at the known section, Q must be specified.

2.Calculate specific energy for the starting section (section 1)

3.Calculate friction slope at the section (Manning’s equation solved for slope is typically used)

€

E1 = y1 +Q2

2gA12

€

S f 1=

Q2n2

1.4921

A12

1

R14 / 3

Backwater (Frontwater) Methods• This form of the equation suggests the

following algorithm

4.Change the depth slightly, use that value as the depth at section 2

5.Calculate specific energy at section 2

€

y2 = y1 ± Δy

E2 = y2 +Q2

2gA22

Backwater (Frontwater) Methods• This form of the equation suggests the

following algorithm

6.Calculate friction slope at section 2

7.Compute average friction slope for the reach

€

S f 2=

Q2n2

1.4921

A22

1

R24 / 3

€

S f =S f1 + S f 2

2

Backwater (Frontwater) Methods• This form of the equation suggests the

following algorithm

8.Solve for the distance to section 2

9.Move to next section and repeat.

€

Δx =E2 − E1

S0 − S f

Backwater Example• The figure below is a backwater curve in a

rectangular channel with discharge over a dam (somewhere to the right of the figure)

Backwater Example• The channel is 5 meters wide, bottom slope

is 0.001, Manning’s n is 0.02 and channel discharge is 55.4 cubic meters per second

• Our goal is to compute the water surface profile and locate the distance upstream where the water flow depth is nearly at normal depth for the channel

Backwater Example• The channel is 5 meters wide, bottom slope

is 0.001, Manning’s n is 0.02 and channel discharge is 55.4 cubic meters per second

• Select a location where depth is known (or assumed)

• Use the pool on the right y= 8 meters• We will call this location x=0

Backwater Example• The channel is 5 meters wide, bottom slope

is 0.001, Manning’s n is 0.02 and channel discharge is 55.4 cubic meters per second

• Determine the slope designation, and profile type (M1, S2, etc.)

• Compute normal depth in the channel from Manning’s equation, yn = 5 meters

• Compute critical depth in the channel by setting Froude number to unity and solving for depth, yc = 2.3 meters

Backwater Example• The channel is 5 meters wide, bottom slope

is 0.001, Manning’s n is 0.02 and channel discharge is 55.4 cubic meters per second

• Determine the slope designation, and profile type

• yn = 5 meters, yc = 2.3 meters, y0=8.

• Using the slope designation the channel is Mild slope.

• Using the type designation the curve will be a Type 1.

• Thus this is an M1 curve.

Backwater Example• The channel is 5 meters wide, bottom slope

is 0.001, Manning’s n is 0.02 and channel discharge is 55.4 cubic meters per second

• Use the designation to decide if integration is downstream or upstream.• Curve is M1• Downstream control• Integrate upstream (-x direction)

Backwater Example• Use Excel to build a spreadsheet to

facilitate the computations. A portion of such a sheet is shown

Backwater Methods• The variable step method was illustrated by

an example to familiarize the participant with the mechanics of the method.

• The example geometry is intentionally simple, and other simplifications are imbedded into the example.

Backwater Methods• For practical applications use of

specialized software is recommended (HEC-RAS, WSPRO, SWMM, etc.)

• These program are computationally similar to the method presented herein

• These programs have user interfaces to facilitate the data entry

• These programs allow spatial locations to be fixed and depths to be computed, which is far more practical for engineering application.