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


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


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


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



• 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



• 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



• 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.



• 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.

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Basic Hydraulics: Channels Analysis and design – I