Chow 9-8 Water Surface Profile

WSE =1427.0'

1272.0'

1270.0'

So = .01

So = .0004So = .003171

downstreamWSE = 1266.0

500' 1,500' 1,000'

The rectangular channel above is 20 ft wide and consists of three reaches of different slopes. Thechannel has a roughness coefficient n = .015 and carries a discharge of 500 cfs. Determine:

a. normal and critical depth for each reachb. water surface profile - must be done using hec-ras 2.2c. Plot the entire water surface profile to some consistent scale

V.T. Chow, Open Channel Hydraulics, 1959problem 9-8

yn = 1.863'yc = 2.688'

yn = 5.436'yc = 2.688'

yn = 2.702'yc = 2.688'

yc and yn for each reach computed in file below and placed here

9-8 water surface profile 500 cfs.mcdlast save 10/7/2001 / 10:40 AM

C:\MyFiles\Mathcad application areas\Fluid Mechanics-hydraulics and open channel flow\CHOW\9-8 water surface profile 500

cfs.mcd

1 / 510/13/2001 / 8:17 AM

normal_depth1 1.863 ft= - normal depth on uppermost reach

normal_depth1 root Q1.49

ny b⋅( )⋅

y b⋅( )

2 y⋅( ) b+

2

3⋅ Sf

1

2⋅− y,

:=

solve for the normal depth using the root function

Q1.49

ny b⋅( )⋅

y b⋅( )

2 y⋅ b+

2

3⋅ Sf

1

2⋅= Mannings equation

Compute normal depth in the uppermost reach of the channel Sf .01:= , n .015:= , initial guess for y: y .5 ft⋅:=

Q 500ft

3

sec⋅:=b 20 ft⋅:= channel width

y 2 ft⋅:= water depth exiting from sluice gate

E0 4 ft⋅:= Required specific energy across sluice gate, assuming no losses

Even though the discharge is given as 500 cfs we will determine the discharge from the specific energy behind the sluice gate

elev1 500 ft⋅( ) 1265 ft= bottom elevation at the lower end of the uppermost reach

elevation at upper end of reach = 1270'

elev1 dist( ) Elev1 So1 dist⋅−:=

dist 0 ft⋅ 100 ft⋅, 500 ft⋅..:=Elev1 1270 ft⋅:=

∆ 1 ft=∆ So1 100⋅ ft⋅:=So1 .01ft

ft⋅:=

SECTION 1: So1 .01ft

ft⋅:=

Determine the elevation every 100' along the structure. Begin with most upstream section



cfs.mcd

2 / 510/13/2001 / 8:17 AM

Now compute the critical depth for the entire system. This is independent of slope and will be constant throughout system:

Q

yc b⋅

g yc⋅1= Froude number

yc1

b2

Q g⋅ b2

⋅( )2

3

g⋅:= yc 2.688 ft= critical depth, Fr =1.0

This depth combined with the normal depth, computed earlier, tells us that the flow issueing from the sluice gate is in the supercritical range. This means there will not be a critical section at the upstream end of the system because the flow depth issueing from the sluice gate is already less than critical and will drop even further as it moves down reach 1. The appropriate upstream boundary condition is then the know water surface elevation, 1272' and we will have a mixed water surface profile.

SECTION 2 So2 .0004:=

Elevo2 1265 ft⋅:= ∆ 004 So2 100⋅ ft⋅:= ∆ 004 0.04 ft=

dist 0 ft⋅ 100 ft⋅, 1500 ft⋅..:=

elevation at upperend of reach : elev1 500 ft⋅( ) 1265 ft=

elev2 dist( ) Elevo2 So2 dist⋅−:=

elev2 1500 ft⋅( ) 1264.4ft= - elevation at lower end of reach


ny b⋅( )⋅

y b⋅( )

2 y⋅( ) b+

2

3⋅ So2

1

2⋅− y,

:=

normal_depth1 5.436 ft= > yc therefore flow subcritical

Based on this we surmise that a hydraulic jump must occur somewhere on reach 1 or 2.



cfs.mcd

3 / 510/13/2001 / 8:17 AM

SECTION 3 So3 .003171:=

Elevation at upper end of reach Elevo3 1264.4 ft⋅:=

dist 0 ft⋅ 100 ft⋅, 1000 ft⋅..:= So3 100⋅ ft⋅ 0.317 ft=

elev3 dist( ) Elevo3 So3 dist⋅−:=

Elevation at lower end of reach elev3 1000 ft⋅( ) 1261.229 ft=

y 8 ft⋅:= Q 500ft

3

sec=


ny b⋅( )⋅

y b⋅( )

2 y⋅( ) b+

2

3⋅ So3

1

2⋅− y,

:=

normal_depth1 2.702 ft= note that this is quite close to the critical depth of yc 2.688 ft=

Thus, the water surface elevation assuming normal depth at the lower end of the reach will be 1261.23ft 2.702ft+ 1263.9ft=

normal_depth1 2.702 ft= > yc 2.688 ft= therefore normal depth flow regime will be subcritical

However, the problem points out that the depth at the lower end of the profile is 1266'. Thus we have an M-1 profile extending back from this known elevation.

Note: Flow never reaches uniform conditions on reach 2 for Q = 500 cfs. the reach is too short. Flow approaches critical depth on reach 3 but stays just above it as it transitions to an M-1 profile.



cfs.mcd

4 / 510/13/2001 / 8:17 AM

HEC - RAS water surface profile 500 cfs Problem 9-8 Chow

0 500 1000 1500 2000 2500 30001260

1262

1264

1266

1268

1270

1272

1274

Main Channel Distance (ft)

Ele

vatio

n (f

t)

Legend

WS 500 cfs

Crit 500 cfs

Groundhydraulic jumpM-2 profile

M-1 profileyn = 5.436' reach 2

yn = 2.702' reach 3 yn = 1.863' reach 1

S-2 profile

yc = 2.688'



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Chow 9-8 Water Surface Profile