Steady-Uniform Flow: Force BalanceHydraulic radiusWW sin DxabcdShear forceEnergy grade lineHydraulic grade line Shear force =________W cos Wetted perimeter = __Gravitational force = ________toP DxPgADx sinqDimensional analysisRelationship between shear and velocity? ___________________

Open Conduits:Dimensional AnalysisGeometric parameters_________________________________________________________Write the functional relationshipHydraulic radius (Rh)Channel length (l)Roughness (e)Uniform flow

Pressure Coefficient for Open Channel Flow?Pressure CoefficientHead loss coefficientFriction slope coefficient(Energy Loss Coefficient)Friction slopeThe friction slope is the slope of the EGL. The friction slope is the same as the bottom slope (So) for steady, uniform flow.

Dimensional AnalysisHead loss length of channel(like f in Darcy-Weisbach)

Open Channel Flow FormulasChezy formulaDimensions of n?Is n only a function of roughness?Manning formula (MKS units!)NO!T /L1/3

Manning FormulaThe Manning n is a function of the boundary roughness as well as other geometric parameters in some unknown way...___________________________________________________Hydraulic radius for wide channels

Channel curvature (bends)Cross section geometryP1 < P2Rh1 > Rh2hb

Why Use the Manning FormulaTraditionNatural channels are geometrically complex and the errors associated with using an equation that isnt dimensionally correct are small compared with our inability to characterize stream geometryMeasurement errors for Q and h are largeWe only ever deal with water in channels, so we dont need to know how other fluids would respond

Values of Manning nThe worst channel hasRoughness at many scales!

Lined Canals

n

Cement plaster

0.011

Untreated gunite

0.016

Wood, planed

0.012

Wood, unplaned

0.013

Concrete, trowled

0.012

Concrete, wood forms, unfinished

0.015

Rubble in cement

0.020

Asphalt, smooth

0.013

Asphalt, rough

0.016

Natural Channels

Gravel beds, straight

0.025

Gravel beds plus large boulders

0.040

Earth, straight, with some grass

0.026

Earth, winding, no vegetation

0.030

Earth , winding with vegetation

0.050

Example: Manning FormulaWhat is the flow capacity of a finished concrete channel that drops 1.2 m in 3 km? 123 m1.5 msolution

Depth as f(Q)Find the depth in the channel when the flow is 5 m3/sHydraulic radius is function of depthArea is a function of depthCant solve explicitlyUse trial and error or solver

Open Channel Energy RelationshipsTurbulent flow ( 1)z - measured from horizontal datumy - _____________Pipe flowEnergy Equation for Open Channel FlowFrom diagramdepth of flow

Specific EnergyThe sum of the depth of flow and the velocity head is the specific energy:If channel bottom is horizontal and no head lossy - _______ energy- _______ energyFor a change in bottom elevationpotentialkineticEGLHGL

Specific EnergyIn a channel with uniform discharge, Qwhere A=f(y)Consider rectangular channel (A = By) and Q = qBABy3 roots (one is negative)q is the discharge per unit width of channelHow many possible depths given a specific energy? _____2

Specific Energy: Sluice Gate12sluice gateEGLy1 and y2 are ___________ depths (same specific energy)Why not use momentum conservation to find y1?q = 5.5 m2/sy2 = 0.45 mV2 = 12.2 m/sE2 = 8 malternateGiven downstream depth and discharge, find upstream depth.vena contracta

Chart6

0.410.0460459184

0.56.6734693878

0.64.8871315193

0.73.8497292795

0.83.2115114796

0.92.8053917863

12.5433673469

1.12.3755102041

1.22.2717828798

1.32.2132351165

1.42.1874323199

1.52.1859410431

1.62.2028778699

1.72.2340371443

1.82.2763479466

1.92.3275255809

22.3858418367

2.12.4499699199

2.22.518877551

2.32.5917518614

2.42.66794572

2.52.7469387755

2.62.8283087791

2.72.9117101985

2.82.99685808

2.93.0835157369

33.1714852608

3.13.2606001402

3.23.3507194675

3.33.441723356

3.43.5335092861

3.53.6259891712

3.63.7190869866

3.73.8127368405

3.83.9068813952

3.94.0014705685

44.0964604592

4.14.1918124537

4.24.28749248

4.34.3834703811

4.44.4797193878

4.54.5762156715

4.64.6729379654

4.74.7698672407

4.84.86698643

4.94.9642801894

55.0617346939

5.15.1593374605

5.25.2570771948

5.35.3549436578

5.45.4529275496

5.55.5510204082

5.65.64921452

5.75.7475028423

5.85.8458789342

5.95.9443368959

66.0428713152

6.16.1414772198

6.26.240150035

6.36.3388855467

6.46.4376798669

6.56.5365294047

6.66.635430839

6.76.7343810948

6.86.8333773215

6.96.9324168735

77.0314972928

7.17.1306162933

7.27.2297717467

7.37.3289616691

7.47.4281842101

7.57.5274376417

7.67.6267203488

7.77.7260308205

7.87.8253676421

7.97.924729488

88.0241151148

8.18.1235233554

8.28.2229531134

8.38.3224033582

8.48.42187312

8.58.5213614858

8.68.6208675953

8.78.7203906374

8.88.8199298469

8.98.9194845013

99.0190539179

9.19.1186374514

9.29.2182344913

9.39.31784446

9.49.4174668102

9.59.5171010232

9.69.6167466075

9.79.7164030965

9.89.8160700473

00

y

E

y

Scenario Summary

Scenario Summary

Maximum dischargenormal depthcritical depth

Solver ModelSolver ModelSolver Model

Changing Cells:

y0.93818119560.5633490170.3552672635

Result Cells:

Fr0.13245723660.41344400940.9999996849

Rh0.28998611470.26850547280.1957624813

Qmanning0.70683645160.39999969910.1777019769

V0.92362045620.87742667950.7107688424

Notes: Current Values column represents values of changing cells at

time Scenario Summary Report was created. Changing cells for each

scenario are highlighted in gray.

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round conduit

parameterequationunits

Q0.4m^3/s

radius0.5m

y0.9381811956m0.81280143

So0.001

n0.015

theta2.6390534835radians

A0.765288866m^2

T0.4816523221m

Fr0.1324572366

P2.6390534835m

Rh0.2899861147m

Qmanning0.7068364516m^3/s

V0.9236204562m/s

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specific energy

q5.5Ey

g9.810.04604591840.410.0460459184

6.67346938780.56.6734693878

4.88713151930.64.8871315193

3.84972927950.73.8497292795

3.21151147960.83.2115114796

2.80539178630.92.8053917863

2.543367346912.5433673469

2.37551020411.12.3755102041

2.27178287981.22.2717828798

2.21323511651.32.2132351165

2.18743231991.42.1874323199

2.18594104311.52.1859410431

2.20287786991.62.2028778699

2.23403714431.72.2340371443

2.27634794661.82.2763479466

2.32752558091.92.3275255809

2.385841836722.3858418367

2.44996991992.12.4499699199

2.5188775512.22.518877551

2.59175186142.32.5917518614

2.667945722.42.66794572

2.74693877552.52.7469387755

2.82830877912.62.8283087791

2.91171019852.72.9117101985

2.996858082.82.99685808

3.08351573692.93.0835157369

3.171485260833.1714852608

3.26060014023.13.2606001402

3.35071946753.23.3507194675

3.4417233563.33.441723356

3.53350928613.43.5335092861

3.62598917123.53.6259891712

3.71908698663.63.7190869866

3.81273684053.73.8127368405

3.90688139523.83.9068813952

4.00147056853.94.0014705685

4.096460459244.0964604592

4.19181245374.14.1918124537

4.287492484.24.28749248

4.38347038114.34.3834703811

4.47971938784.44.4797193878

4.57621567154.54.5762156715

4.67293796544.64.6729379654

4.76986724074.74.7698672407

4.866986434.84.86698643

4.96428018944.94.9642801894

5.061734693955.0617346939

5.15933746055.15.1593374605

5.25707719485.25.2570771948

5.35494365785.35.3549436578

5.45292754965.45.4529275496

5.55102040825.55.5510204082

5.649214525.65.64921452

5.74750284235.75.7475028423

5.84587893425.85.8458789342

5.94433689595.95.9443368959

6.042871315266.0428713152

6.14147721986.16.1414772198

6.2401500356.26.240150035

6.33888554676.36.3388855467

6.43767986696.46.4376798669

6.53652940476.56.5365294047

6.6354308396.66.635430839

6.73438109486.76.7343810948

6.83337732156.86.8333773215

6.93241687356.96.9324168735

7.031497292877.0314972928

7.13061629337.17.1306162933

7.22977174677.27.2297717467

7.32896166917.37.3289616691

7.42818421017.47.4281842101

7.52743764177.57.5274376417

7.62672034887.67.6267203488

7.72603082057.77.7260308205

7.82536764217.87.8253676421

7.9247294887.97.924729488

8.024115114888.0241151148

8.12352335548.18.1235233554

8.22295311348.28.2229531134

8.32240335828.38.3224033582

8.421873128.48.42187312

8.52136148588.58.5213614858

8.62086759538.68.6208675953

8.72039063748.78.7203906374

8.81992984698.88.8199298469

8.91948450138.98.9194845013

9.019053917999.0190539179

9.11863745149.19.1186374514

9.21823449139.29.2182344913

9.317844469.39.31784446

9.41746681029.49.4174668102

9.51710102329.59.5171010232

9.61674660759.69.6167466075

9.71640309659.79.7164030965

9.81607004739.89.8160700473

00

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specific energy

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y

E

y

Specific Energy: Raise the Sluice Gate12sluice gateEGLas sluice gate is raised y1 approaches y2 and E is minimized: Maximum discharge for given energy.

Chart7

0.410.0460459184

0.56.6734693878

0.64.8871315193

0.73.8497292795

0.83.2115114796

0.92.8053917863

12.5433673469

1.12.3755102041

1.22.2717828798

1.32.2132351165

1.42.1874323199

1.52.1859410431

1.62.2028778699

1.72.2340371443

1.82.2763479466

1.92.3275255809

22.3858418367

2.12.4499699199

2.22.518877551

2.32.5917518614

2.42.66794572

2.52.7469387755

2.62.8283087791

2.72.9117101985

2.82.99685808

2.93.0835157369

33.1714852608

3.13.2606001402

3.23.3507194675

3.33.441723356

3.43.5335092861

3.53.6259891712

3.63.7190869866

3.73.8127368405

3.83.9068813952

3.94.0014705685

44.0964604592

4.14.1918124537

4.24.28749248

4.34.3834703811

4.44.4797193878

4.54.5762156715

4.64.6729379654

4.74.7698672407

4.84.86698643

4.94.9642801894

55.0617346939

5.15.1593374605

5.25.2570771948

5.35.3549436578

5.45.4529275496

5.55.5510204082

5.65.64921452

5.75.7475028423

5.85.8458789342

5.95.9443368959

66.0428713152

6.16.1414772198

6.26.240150035

6.36.3388855467

6.46.4376798669

6.56.5365294047

6.66.635430839

6.76.7343810948

6.86.8333773215

6.96.9324168735

77.0314972928

7.17.1306162933

7.27.2297717467

7.37.3289616691

7.47.4281842101

7.57.5274376417

7.67.6267203488

7.77.7260308205

7.87.8253676421

7.97.924729488

88.0241151148

8.18.1235233554

8.28.2229531134

8.38.3224033582

8.48.42187312

8.58.5213614858

8.68.6208675953

8.78.7203906374

8.88.8199298469

8.98.9194845013

99.0190539179

9.19.1186374514

9.29.2182344913

9.39.31784446

9.49.4174668102

9.59.5171010232

9.69.6167466075

9.79.7164030965

9.89.8160700473

00

y

E

y

Scenario Summary

Scenario Summary

Maximum dischargenormal depthcritical depth

Solver ModelSolver ModelSolver Model

Changing Cells:

y0.93818119560.5633490170.3552672635

Result Cells:

Fr0.13245723660.41344400940.9999996849

Rh0.28998611470.26850547280.1957624813

Qmanning0.70683645160.39999969910.1777019769

V0.92362045620.87742667950.7107688424

Notes: Current Values column represents values of changing cells at

time Scenario Summary Report was created. Changing cells for each

scenario are highlighted in gray.

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Page &P

round conduit

parameterequationunits

Q0.4m^3/s

radius0.5m

y0.9381811956m0.81280143

So0.001

n0.015

theta2.6390534835radians

A0.765288866m^2

T0.4816523221m

Fr0.1324572366

P2.6390534835m

Rh0.2899861147m

Qmanning0.7068364516m^3/s

V0.9236204562m/s

&A

Page &P

specific energy

q5.5Ey

g9.810.04604591840.410.0460459184

6.67346938780.56.6734693878

4.88713151930.64.8871315193

3.84972927950.73.8497292795

3.21151147960.83.2115114796

2.80539178630.92.8053917863

2.543367346912.5433673469

2.37551020411.12.3755102041

2.27178287981.22.2717828798

2.21323511651.32.2132351165

2.18743231991.42.1874323199

2.18594104311.52.1859410431

2.20287786991.62.2028778699

2.23403714431.72.2340371443

2.27634794661.82.2763479466

2.32752558091.92.3275255809

2.385841836722.3858418367

2.44996991992.12.4499699199

2.5188775512.22.518877551

2.59175186142.32.5917518614

2.667945722.42.66794572

2.74693877552.52.7469387755

2.82830877912.62.8283087791

2.91171019852.72.9117101985

2.996858082.82.99685808

3.08351573692.93.0835157369

3.171485260833.1714852608

3.26060014023.13.2606001402

3.35071946753.23.3507194675

3.4417233563.33.441723356

3.53350928613.43.5335092861

3.62598917123.53.6259891712

3.71908698663.63.7190869866

3.81273684053.73.8127368405

3.90688139523.83.9068813952

4.00147056853.94.0014705685

4.096460459244.0964604592

4.19181245374.14.1918124537

4.287492484.24.28749248

4.38347038114.34.3834703811

4.47971938784.44.4797193878

4.57621567154.54.5762156715

4.67293796544.64.6729379654

4.76986724074.74.7698672407

4.866986434.84.86698643

4.96428018944.94.9642801894

5.061734693955.0617346939

5.15933746055.15.1593374605

5.25707719485.25.2570771948

5.35494365785.35.3549436578

5.45292754965.45.4529275496

5.55102040825.55.5510204082

5.649214525.65.64921452

5.74750284235.75.7475028423

5.84587893425.85.8458789342

5.94433689595.95.9443368959

6.042871315266.0428713152

6.14147721986.16.1414772198

6.2401500356.26.240150035

6.33888554676.36.3388855467

6.43767986696.46.4376798669

6.53652940476.56.5365294047

6.6354308396.66.635430839

6.73438109486.76.7343810948

6.83337732156.86.8333773215

6.93241687356.96.9324168735

7.031497292877.0314972928

7.13061629337.17.1306162933

7.22977174677.27.2297717467

7.32896166917.37.3289616691

7.42818421017.47.4281842101

7.52743764177.57.5274376417

7.62672034887.67.6267203488

7.72603082057.77.7260308205

7.82536764217.87.8253676421

7.9247294887.97.924729488

8.024115114888.0241151148

8.12352335548.18.1235233554

8.22295311348.28.2229531134

8.32240335828.38.3224033582

8.421873128.48.42187312

8.52136148588.58.5213614858

8.62086759538.68.6208675953

8.72039063748.78.7203906374

8.81992984698.88.8199298469

8.91948450138.98.9194845013

9.019053917999.0190539179

9.11863745149.19.1186374514

9.21823449139.29.2182344913

9.317844469.39.31784446

9.41746681029.49.4174668102

9.51710102329.59.5171010232

9.61674660759.69.6167466075

9.71640309659.79.7164030965

9.81607004739.89.8160700473

00

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specific energy

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y

E

y

Step Up with Subcritical FlowNO! Calculate depth along step.Short, smooth step with rise h in channelhGiven upstream depth and discharge find y2Is alternate depth possible? __________________________Energy conserved

Chart7

0.410.0460459184

0.56.6734693878

0.64.8871315193

0.73.8497292795

0.83.2115114796

0.92.8053917863

12.5433673469

1.12.3755102041

1.22.2717828798

1.32.2132351165

1.42.1874323199

1.52.1859410431

1.62.2028778699

1.72.2340371443

1.82.2763479466

1.92.3275255809

22.3858418367

2.12.4499699199

2.22.518877551

2.32.5917518614

2.42.66794572

2.52.7469387755

2.62.8283087791

2.72.9117101985

2.82.99685808

2.93.0835157369

33.1714852608

3.13.2606001402

3.23.3507194675

3.33.441723356

3.43.5335092861

3.53.6259891712

3.63.7190869866

3.73.8127368405

3.83.9068813952

3.94.0014705685

44.0964604592

4.14.1918124537

4.24.28749248

4.34.3834703811

4.44.4797193878

4.54.5762156715

4.64.6729379654

4.74.7698672407

4.84.86698643

4.94.9642801894

55.0617346939

5.15.1593374605

5.25.2570771948

5.35.3549436578

5.45.4529275496

5.55.5510204082

5.65.64921452

5.75.7475028423

5.85.8458789342

5.95.9443368959

66.0428713152

6.16.1414772198

6.26.240150035

6.36.3388855467

6.46.4376798669

6.56.5365294047

6.66.635430839

6.76.7343810948

6.86.8333773215

6.96.9324168735

77.0314972928

7.17.1306162933

7.27.2297717467

7.37.3289616691

7.47.4281842101

7.57.5274376417

7.67.6267203488

7.77.7260308205

7.87.8253676421

7.97.924729488

88.0241151148

8.18.1235233554

8.28.2229531134

8.38.3224033582

8.48.42187312

8.58.5213614858

8.68.6208675953

8.78.7203906374

8.88.8199298469

8.98.9194845013

99.0190539179

9.19.1186374514

9.29.2182344913

9.39.31784446

9.49.4174668102

9.59.5171010232

9.69.6167466075

9.79.7164030965

9.89.8160700473

00

y

E

y

Scenario Summary

Scenario Summary

Maximum dischargenormal depthcritical depth

Solver ModelSolver ModelSolver Model

Changing Cells:

y0.93818119560.5633490170.3552672635

Result Cells:

Fr0.13245723660.41344400940.9999996849

Rh0.28998611470.26850547280.1957624813

Qmanning0.70683645160.39999969910.1777019769

V0.92362045620.87742667950.7107688424

Notes: Current Values column represents values of changing cells at

time Scenario Summary Report was created. Changing cells for each

scenario are highlighted in gray.

&A

Page &P

round conduit

parameterequationunits

Q0.4m^3/s

radius0.5m

y0.9381811956m0.81280143

So0.001

n0.015

theta2.6390534835radians

A0.765288866m^2

T0.4816523221m

Fr0.1324572366

P2.6390534835m

Rh0.2899861147m

Qmanning0.7068364516m^3/s

V0.9236204562m/s

&A

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specific energy

q5.5Ey

g9.810.04604591840.410.0460459184

6.67346938780.56.6734693878

4.88713151930.64.8871315193

3.84972927950.73.8497292795

3.21151147960.83.2115114796

2.80539178630.92.8053917863

2.543367346912.5433673469

2.37551020411.12.3755102041

2.27178287981.22.2717828798

2.21323511651.32.2132351165

2.18743231991.42.1874323199

2.18594104311.52.1859410431

2.20287786991.62.2028778699

2.23403714431.72.2340371443

2.27634794661.82.2763479466

2.32752558091.92.3275255809

2.385841836722.3858418367

2.44996991992.12.4499699199

2.5188775512.22.518877551

2.59175186142.32.5917518614

2.667945722.42.66794572

2.74693877552.52.7469387755

2.82830877912.62.8283087791

2.91171019852.72.9117101985

2.996858082.82.99685808

3.08351573692.93.0835157369

3.171485260833.1714852608

3.26060014023.13.2606001402

3.35071946753.23.3507194675

3.4417233563.33.441723356

3.53350928613.43.5335092861

3.62598917123.53.6259891712

3.71908698663.63.7190869866

3.81273684053.73.8127368405

3.90688139523.83.9068813952

4.00147056853.94.0014705685

4.096460459244.0964604592

4.19181245374.14.1918124537

4.287492484.24.28749248

4.38347038114.34.3834703811

4.47971938784.44.4797193878

4.57621567154.54.5762156715

4.67293796544.64.6729379654

4.76986724074.74.7698672407

4.866986434.84.86698643

4.96428018944.94.9642801894

5.061734693955.0617346939

5.15933746055.15.1593374605

5.25707719485.25.2570771948

5.35494365785.35.3549436578

5.45292754965.45.4529275496

5.55102040825.55.5510204082

5.649214525.65.64921452

5.74750284235.75.7475028423

5.84587893425.85.8458789342

5.94433689595.95.9443368959

6.042871315266.0428713152

6.14147721986.16.1414772198

6.2401500356.26.240150035

6.33888554676.36.3388855467

6.43767986696.46.4376798669

6.53652940476.56.5365294047

6.6354308396.66.635430839

6.73438109486.76.7343810948

6.83337732156.86.8333773215

6.93241687356.96.9324168735

7.031497292877.0314972928

7.13061629337.17.1306162933

7.22977174677.27.2297717467

7.32896166917.37.3289616691

7.42818421017.47.4281842101

7.52743764177.57.5274376417

7.62672034887.67.6267203488

7.72603082057.77.7260308205

7.82536764217.87.8253676421

7.9247294887.97.924729488

8.024115114888.0241151148

8.12352335548.18.1235233554

8.22295311348.28.2229531134

8.32240335828.38.3224033582

8.421873128.48.42187312

8.52136148588.58.5213614858

8.62086759538.68.6208675953

8.72039063748.78.7203906374

8.81992984698.88.8199298469

8.91948450138.98.9194845013

9.019053917999.0190539179

9.11863745149.19.1186374514

9.21823449139.29.2182344913

9.317844469.39.31784446

9.41746681029.49.4174668102

9.51710102329.59.5171010232

9.61674660759.69.6167466075

9.71640309659.79.7164030965

9.81607004739.89.8160700473

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Chart7

0.410.0460459184

0.56.6734693878

0.64.8871315193

0.73.8497292795

0.83.2115114796

0.92.8053917863

12.5433673469

1.12.3755102041

1.22.2717828798

1.32.2132351165

1.42.1874323199

1.52.1859410431

1.62.2028778699

1.72.2340371443

1.82.2763479466

1.92.3275255809

22.3858418367

2.12.4499699199

2.22.518877551

2.32.5917518614

2.42.66794572

2.52.7469387755

2.62.8283087791

2.72.9117101985

2.82.99685808

2.93.0835157369

33.1714852608

3.13.2606001402

3.23.3507194675

3.33.441723356

3.43.5335092861

3.53.6259891712

3.63.7190869866

3.73.8127368405

3.83.9068813952

3.94.0014705685

44.0964604592

4.14.1918124537

4.24.28749248

4.34.3834703811

4.44.4797193878

4.54.5762156715

4.64.6729379654

4.74.7698672407

4.84.86698643

4.94.9642801894

55.0617346939

5.15.1593374605

5.25.2570771948

5.35.3549436578

5.45.4529275496

5.55.5510204082

5.65.64921452

5.75.7475028423

5.85.8458789342

5.95.9443368959

66.0428713152

6.16.1414772198

6.26.240150035

6.36.3388855467

6.46.4376798669

6.56.5365294047

6.66.635430839

6.76.7343810948

6.86.8333773215

6.96.9324168735

77.0314972928

7.17.1306162933

7.27.2297717467

7.37.3289616691

7.47.4281842101

7.57.5274376417

7.67.6267203488

7.77.7260308205

7.87.8253676421

7.97.924729488

88.0241151148

8.18.1235233554

8.28.2229531134

8.38.3224033582

8.48.42187312

8.58.5213614858

8.68.6208675953

8.78.7203906374

8.88.8199298469

8.98.9194845013

99.0190539179

9.19.1186374514

9.29.2182344913

9.39.31784446

9.49.4174668102

9.59.5171010232

9.69.6167466075

9.79.7164030965

9.89.8160700473

00

y

E

y

Scenario Summary

Scenario Summary

Maximum dischargenormal depthcritical depth

Solver ModelSolver ModelSolver Model

Changing Cells:

y0.93818119560.5633490170.3552672635

Result Cells:

Fr0.13245723660.41344400940.9999996849

Rh0.28998611470.26850547280.1957624813

Qmanning0.70683645160.39999969910.1777019769

V0.92362045620.87742667950.7107688424

Notes: Current Values column represents values of changing cells at

time Scenario Summary Report was created. Changing cells for each

scenario are highlighted in gray.

&A

Page &P

round conduit

parameterequationunits

Q0.4m^3/s

radius0.5m

y0.9381811956m0.81280143

So0.001

n0.015

theta2.6390534835radians

A0.765288866m^2

T0.4816523221m

Fr0.1324572366

P2.6390534835m

Rh0.2899861147m

Qmanning0.7068364516m^3/s

V0.9236204562m/s

&A

Page &P

specific energy

q5.5Ey

g9.810.04604591840.410.0460459184

6.67346938780.56.6734693878

4.88713151930.64.8871315193

3.84972927950.73.8497292795

3.21151147960.83.2115114796

2.80539178630.92.8053917863

2.543367346912.5433673469

2.37551020411.12.3755102041

2.27178287981.22.2717828798

2.21323511651.32.2132351165

2.18743231991.42.1874323199

2.18594104311.52.1859410431

2.20287786991.62.2028778699

2.23403714431.72.2340371443

2.27634794661.82.2763479466

2.32752558091.92.3275255809

2.385841836722.3858418367

2.44996991992.12.4499699199

2.5188775512.22.518877551

2.59175186142.32.5917518614

2.667945722.42.66794572

2.74693877552.52.7469387755

2.82830877912.62.8283087791

2.91171019852.72.9117101985

2.996858082.82.99685808

3.08351573692.93.0835157369

3.171485260833.1714852608

3.26060014023.13.2606001402

3.35071946753.23.3507194675

3.4417233563.33.441723356

3.53350928613.43.5335092861

3.62598917123.53.6259891712

3.71908698663.63.7190869866

3.81273684053.73.8127368405

3.90688139523.83.9068813952

4.00147056853.94.0014705685

4.096460459244.0964604592

4.19181245374.14.1918124537

4.287492484.24.28749248

4.38347038114.34.3834703811

4.47971938784.44.4797193878

4.57621567154.54.5762156715

4.67293796544.64.6729379654

4.76986724074.74.7698672407

4.866986434.84.86698643

4.96428018944.94.9642801894

5.061734693955.0617346939

5.15933746055.15.1593374605

5.25707719485.25.2570771948

5.35494365785.35.3549436578

5.45292754965.45.4529275496

5.55102040825.55.5510204082

5.649214525.65.64921452

5.74750284235.75.7475028423

5.84587893425.85.8458789342

5.94433689595.95.9443368959

6.042871315266.0428713152

6.14147721986.16.1414772198

6.2401500356.26.240150035

6.33888554676.36.3388855467

6.43767986696.46.4376798669

6.53652940476.56.5365294047

6.6354308396.66.635430839

6.73438109486.76.7343810948

6.83337732156.86.8333773215

6.93241687356.96.9324168735

7.031497292877.0314972928

7.13061629337.17.1306162933

7.22977174677.27.2297717467

7.32896166917.37.3289616691

7.42818421017.47.4281842101

7.52743764177.57.5274376417

7.62672034887.67.6267203488

7.72603082057.77.7260308205

7.82536764217.87.8253676421

7.9247294887.97.924729488

8.024115114888.0241151148

8.12352335548.18.1235233554

8.22295311348.28.2229531134

8.32240335828.38.3224033582

8.421873128.48.42187312

8.52136148588.58.5213614858

8.62086759538.68.6208675953

8.72039063748.78.7203906374

8.81992984698.88.8199298469

8.91948450138.98.9194845013

9.019053917999.0190539179

9.11863745149.19.1186374514

9.21823449139.29.2182344913

9.317844469.39.31784446

9.41746681029.49.4174668102

9.51710102329.59.5171010232

9.61674660759.69.6167466075

9.71640309659.79.7164030965

9.81607004739.89.8160700473

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y

Max Step UpShort, smooth step with maximum rise h in channelhWhat happens if the step is increased further?___________Choked flow

Chart7

0.410.0460459184

0.56.6734693878

0.64.8871315193

0.73.8497292795

0.83.2115114796

0.92.8053917863

12.5433673469

1.12.3755102041

1.22.2717828798

1.32.2132351165

1.42.1874323199

1.52.1859410431

1.62.2028778699

1.72.2340371443

1.82.2763479466

1.92.3275255809

22.3858418367

2.12.4499699199

2.22.518877551

2.32.5917518614

2.42.66794572

2.52.7469387755

2.62.8283087791

2.72.9117101985

2.82.99685808

2.93.0835157369

33.1714852608

3.13.2606001402

3.23.3507194675

3.33.441723356

3.43.5335092861

3.53.6259891712

3.63.7190869866

3.73.8127368405

3.83.9068813952

3.94.0014705685

44.0964604592

4.14.1918124537

4.24.28749248

4.34.3834703811

4.44.4797193878

4.54.5762156715

4.64.6729379654

4.74.7698672407

4.84.86698643

4.94.9642801894

55.0617346939

5.15.1593374605

5.25.2570771948

5.35.3549436578

5.45.4529275496

5.55.5510204082

5.65.64921452

5.75.7475028423

5.85.8458789342

5.95.9443368959

66.0428713152

6.16.1414772198

6.26.240150035

6.36.3388855467

6.46.4376798669

6.56.5365294047

6.66.635430839

6.76.7343810948

6.86.8333773215

6.96.9324168735

77.0314972928

7.17.1306162933

7.27.2297717467

7.37.3289616691

7.47.4281842101

7.57.5274376417

7.67.6267203488

7.77.7260308205

7.87.8253676421

7.97.924729488

88.0241151148

8.18.1235233554

8.28.2229531134

8.38.3224033582

8.48.42187312

8.58.5213614858

8.68.6208675953

8.78.7203906374

8.88.8199298469

8.98.9194845013

99.0190539179

9.19.1186374514

9.29.2182344913

9.39.31784446

9.49.4174668102

9.59.5171010232

9.69.6167466075

9.79.7164030965

9.89.8160700473

00

y

E

y

Scenario Summary

Scenario Summary

Maximum dischargenormal depthcritical depth

Solver ModelSolver ModelSolver Model

Changing Cells:

y0.93818119560.5633490170.3552672635

Result Cells:

Fr0.13245723660.41344400940.9999996849

Rh0.28998611470.26850547280.1957624813

Qmanning0.70683645160.39999969910.1777019769

V0.92362045620.87742667950.7107688424

Notes: Current Values column represents values of changing cells at

time Scenario Summary Report was created. Changing cells for each

scenario are highlighted in gray.

&A

Page &P

round conduit

parameterequationunits

Q0.4m^3/s

radius0.5m

y0.9381811956m0.81280143

So0.001

n0.015

theta2.6390534835radians

A0.765288866m^2

T0.4816523221m

Fr0.1324572366

P2.6390534835m

Rh0.2899861147m

Qmanning0.7068364516m^3/s

V0.9236204562m/s

&A

Page &P

specific energy

q5.5Ey

g9.810.04604591840.410.0460459184

6.67346938780.56.6734693878

4.88713151930.64.8871315193

3.84972927950.73.8497292795

3.21151147960.83.2115114796

2.80539178630.92.8053917863

2.543367346912.5433673469

2.37551020411.12.3755102041

2.27178287981.22.2717828798

2.21323511651.32.2132351165

2.18743231991.42.1874323199

2.18594104311.52.1859410431

2.20287786991.62.2028778699

2.23403714431.72.2340371443

2.27634794661.82.2763479466

2.32752558091.92.3275255809

2.385841836722.3858418367

2.44996991992.12.4499699199

2.5188775512.22.518877551

2.59175186142.32.5917518614

2.667945722.42.66794572

2.74693877552.52.7469387755

2.82830877912.62.8283087791

2.91171019852.72.9117101985

2.996858082.82.99685808

3.08351573692.93.0835157369

3.171485260833.1714852608

3.26060014023.13.2606001402

3.35071946753.23.3507194675

3.4417233563.33.441723356

3.53350928613.43.5335092861

3.62598917123.53.6259891712

3.71908698663.63.7190869866

3.81273684053.73.8127368405

3.90688139523.83.9068813952

4.00147056853.94.0014705685

4.096460459244.0964604592

4.19181245374.14.1918124537

4.287492484.24.28749248

4.38347038114.34.3834703811

4.47971938784.44.4797193878

4.57621567154.54.5762156715

4.67293796544.64.6729379654

4.76986724074.74.7698672407

4.866986434.84.86698643

4.96428018944.94.9642801894

5.061734693955.0617346939

5.15933746055.15.1593374605

5.25707719485.25.2570771948

5.35494365785.35.3549436578

5.45292754965.45.4529275496

5.55102040825.55.5510204082

5.649214525.65.64921452

5.74750284235.75.7475028423

5.84587893425.85.8458789342

5.94433689595.95.9443368959

6.042871315266.0428713152

6.14147721986.16.1414772198

6.2401500356.26.240150035

6.33888554676.36.3388855467

6.43767986696.46.4376798669

6.53652940476.56.5365294047

6.6354308396.66.635430839

6.73438109486.76.7343810948

6.83337732156.86.8333773215

6.93241687356.96.9324168735

7.031497292877.0314972928

7.13061629337.17.1306162933

7.22977174677.27.2297717467

7.32896166917.37.3289616691

7.42818421017.47.4281842101

7.52743764177.57.5274376417

7.62672034887.67.6267203488

7.72603082057.77.7260308205

7.82536764217.87.8253676421

7.9247294887.97.924729488

8.024115114888.0241151148

8.12352335548.18.1235233554

8.22295311348.28.2229531134

8.32240335828.38.3224033582

8.421873128.48.42187312

8.52136148588.58.5213614858

8.62086759538.68.6208675953

8.72039063748.78.7203906374

8.81992984698.88.8199298469

8.91948450138.98.9194845013

9.019053917999.0190539179

9.11863745149.19.1186374514

9.21823449139.29.2182344913

9.317844469.39.31784446

9.41746681029.49.4174668102

9.51710102329.59.5171010232

9.61674660759.69.6167466075

9.71640309659.79.7164030965

9.81607004739.89.8160700473

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Chart7

0.410.0460459184

0.56.6734693878

0.64.8871315193

0.73.8497292795

0.83.2115114796

0.92.8053917863

12.5433673469

1.12.3755102041

1.22.2717828798

1.32.2132351165

1.42.1874323199

1.52.1859410431

1.62.2028778699

1.72.2340371443

1.82.2763479466

1.92.3275255809

22.3858418367

2.12.4499699199

2.22.518877551

2.32.5917518614

2.42.66794572

2.52.7469387755

2.62.8283087791

2.72.9117101985

2.82.99685808

2.93.0835157369

33.1714852608

3.13.2606001402

3.23.3507194675

3.33.441723356

3.43.5335092861

3.53.6259891712

3.63.7190869866

3.73.8127368405

3.83.9068813952

3.94.0014705685

44.0964604592

4.14.1918124537

4.24.28749248

4.34.3834703811

4.44.4797193878

4.54.5762156715

4.64.6729379654

4.74.7698672407

4.84.86698643

4.94.9642801894

55.0617346939

5.15.1593374605

5.25.2570771948

5.35.3549436578

5.45.4529275496

5.55.5510204082

5.65.64921452

5.75.7475028423

5.85.8458789342

5.95.9443368959

66.0428713152

6.16.1414772198

6.26.240150035

6.36.3388855467

6.46.4376798669

6.56.5365294047

6.66.635430839

6.76.7343810948

6.86.8333773215

6.96.9324168735

77.0314972928

7.17.1306162933

7.27.2297717467

7.37.3289616691

7.47.4281842101

7.57.5274376417

7.67.6267203488

7.77.7260308205

7.87.8253676421

7.97.924729488

88.0241151148

8.18.1235233554

8.28.2229531134

8.38.3224033582

8.48.42187312

8.58.5213614858

8.68.6208675953

8.78.7203906374

8.88.8199298469

8.98.9194845013

99.0190539179

9.19.1186374514

9.29.2182344913

9.39.31784446

9.49.4174668102

9.59.5171010232

9.69.6167466075

9.79.7164030965

9.89.8160700473

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Scenario Summary

Scenario Summary

Maximum dischargenormal depthcritical depth

Solver ModelSolver ModelSolver Model

Changing Cells:

y0.93818119560.5633490170.3552672635

Result Cells:

Fr0.13245723660.41344400940.9999996849

Rh0.28998611470.26850547280.1957624813

Qmanning0.70683645160.39999969910.1777019769

V0.92362045620.87742667950.7107688424

Notes: Current Values column represents values of changing cells at

time Scenario Summary Report was created. Changing cells for each

scenario are highlighted in gray.

&A

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round conduit

parameterequationunits

Q0.4m^3/s

radius0.5m

y0.9381811956m0.81280143

So0.001

n0.015

theta2.6390534835radians

A0.765288866m^2

T0.4816523221m

Fr0.1324572366

P2.6390534835m

Rh0.2899861147m

Qmanning0.7068364516m^3/s

V0.9236204562m/s

&A

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specific energy

q5.5Ey

g9.810.04604591840.410.0460459184

6.67346938780.56.6734693878

4.88713151930.64.8871315193

3.84972927950.73.8497292795

3.21151147960.83.2115114796

2.80539178630.92.8053917863

2.543367346912.5433673469

2.37551020411.12.3755102041

2.27178287981.22.2717828798

2.21323511651.32.2132351165

2.18743231991.42.1874323199

2.18594104311.52.1859410431

2.20287786991.62.2028778699

2.23403714431.72.2340371443

2.27634794661.82.2763479466

2.32752558091.92.3275255809

2.385841836722.3858418367

2.44996991992.12.4499699199

2.5188775512.22.518877551

2.59175186142.32.5917518614

2.667945722.42.66794572

2.74693877552.52.7469387755

2.82830877912.62.8283087791

2.91171019852.72.9117101985

2.996858082.82.99685808

3.08351573692.93.0835157369

3.171485260833.1714852608

3.26060014023.13.2606001402

3.35071946753.23.3507194675

3.4417233563.33.441723356

3.53350928613.43.5335092861

3.62598917123.53.6259891712

3.71908698663.63.7190869866

3.81273684053.73.8127368405

3.90688139523.83.9068813952

4.00147056853.94.0014705685

4.096460459244.0964604592

4.19181245374.14.1918124537

4.287492484.24.28749248

4.38347038114.34.3834703811

4.47971938784.44.4797193878

4.57621567154.54.5762156715

4.67293796544.64.6729379654

4.76986724074.74.7698672407

4.866986434.84.86698643

4.96428018944.94.9642801894

5.061734693955.0617346939

5.15933746055.15.1593374605

5.25707719485.25.2570771948

5.35494365785.35.3549436578

5.45292754965.45.4529275496

5.55102040825.55.5510204082

5.649214525.65.64921452

5.74750284235.75.7475028423

5.84587893425.85.8458789342

5.94433689595.95.9443368959

6.042871315266.0428713152

6.14147721986.16.1414772198

6.2401500356.26.240150035

6.33888554676.36.3388855467

6.43767986696.46.4376798669

6.53652940476.56.5365294047

6.6354308396.66.635430839

6.73438109486.76.7343810948

6.83337732156.86.8333773215

6.93241687356.96.9324168735

7.031497292877.0314972928

7.13061629337.17.1306162933

7.22977174677.27.2297717467

7.32896166917.37.3289616691

7.42818421017.47.4281842101

7.52743764177.57.5274376417

7.62672034887.67.6267203488

7.72603082057.77.7260308205

7.82536764217.87.8253676421

7.9247294887.97.924729488

8.024115114888.0241151148

8.12352335548.18.1235233554

8.22295311348.28.2229531134

8.32240335828.38.3224033582

8.421873128.48.42187312

8.52136148588.58.5213614858

8.62086759538.68.6208675953

8.72039063748.78.7203906374

8.81992984698.88.8199298469

8.91948450138.98.9194845013

9.019053917999.0190539179

9.11863745149.19.1186374514

9.21823449139.29.2182344913

9.317844469.39.31784446

9.41746681029.49.4174668102

9.51710102329.59.5171010232

9.61674660759.69.6167466075

9.71640309659.79.7164030965

9.81607004739.89.8160700473

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Step Up with Supercritical flowShort, smooth step with rise h in channelhGiven upstream depth and discharge find y2What happened to the water depth?______________________________Increased! Expansion! Energy Loss

Chart7

0.410.0460459184

0.56.6734693878

0.64.8871315193

0.73.8497292795

0.83.2115114796

0.92.8053917863

12.5433673469

1.12.3755102041

1.22.2717828798

1.32.2132351165

1.42.1874323199

1.52.1859410431

1.62.2028778699

1.72.2340371443

1.82.2763479466

1.92.3275255809

22.3858418367

2.12.4499699199

2.22.518877551

2.32.5917518614

2.42.66794572

2.52.7469387755

2.62.8283087791

2.72.9117101985

2.82.99685808

2.93.0835157369

33.1714852608

3.13.2606001402

3.23.3507194675

3.33.441723356

3.43.5335092861

3.53.6259891712

3.63.7190869866

3.73.8127368405

3.83.9068813952

3.94.0014705685

44.0964604592

4.14.1918124537

4.24.28749248

4.34.3834703811

4.44.4797193878

4.54.5762156715

4.64.6729379654

4.74.7698672407

4.84.86698643

4.94.9642801894

55.0617346939

5.15.1593374605

5.25.2570771948

5.35.3549436578

5.45.4529275496

5.55.5510204082

5.65.64921452

5.75.7475028423

5.85.8458789342

5.95.9443368959

66.0428713152

6.16.1414772198

6.26.240150035

6.36.3388855467

6.46.4376798669

6.56.5365294047

6.66.635430839

6.76.7343810948

6.86.8333773215

6.96.9324168735

77.0314972928

7.17.1306162933

7.27.2297717467

7.37.3289616691

7.47.4281842101

7.57.5274376417

7.67.6267203488

7.77.7260308205

7.87.8253676421

7.97.924729488

88.0241151148

8.18.1235233554

8.28.2229531134

8.38.3224033582

8.48.42187312

8.58.5213614858

8.68.6208675953

8.78.7203906374

8.88.8199298469

8.98.9194845013

99.0190539179

9.19.1186374514

9.29.2182344913

9.39.31784446

9.49.4174668102

9.59.5171010232

9.69.6167466075

9.79.7164030965

9.89.8160700473

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y

Scenario Summary

Scenario Summary

Maximum dischargenormal depthcritical depth

Solver ModelSolver ModelSolver Model

Changing Cells:

y0.93818119560.5633490170.3552672635

Result Cells:

Fr0.13245723660.41344400940.9999996849

Rh0.28998611470.26850547280.1957624813

Qmanning0.70683645160.39999969910.1777019769

V0.92362045620.87742667950.7107688424

Notes: Current Values column represents values of changing cells at

time Scenario Summary Report was created. Changing cells for each

scenario are highlighted in gray.

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round conduit

parameterequationunits

Q0.4m^3/s

radius0.5m

y0.9381811956m0.81280143

So0.001

n0.015

theta2.6390534835radians

A0.765288866m^2

T0.4816523221m

Fr0.1324572366

P2.6390534835m

Rh0.2899861147m

Qmanning0.7068364516m^3/s

V0.9236204562m/s

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specific energy

q5.5Ey

g9.810.04604591840.410.0460459184

6.67346938780.56.6734693878

4.88713151930.64.8871315193

3.84972927950.73.8497292795

3.21151147960.83.2115114796

2.80539178630.92.8053917863

2.543367346912.5433673469

2.37551020411.12.3755102041

2.27178287981.22.2717828798

2.21323511651.32.2132351165

2.18743231991.42.1874323199

2.18594104311.52.1859410431

2.20287786991.62.2028778699

2.23403714431.72.2340371443

2.27634794661.82.2763479466

2.32752558091.92.3275255809

2.385841836722.3858418367

2.44996991992.12.4499699199

2.5188775512.22.518877551

2.59175186142.32.5917518614

2.667945722.42.66794572

2.74693877552.52.7469387755

2.82830877912.62.8283087791

2.91171019852.72.9117101985

2.996858082.82.99685808

3.08351573692.93.0835157369

3.171485260833.1714852608

3.26060014023.13.2606001402

3.35071946753.23.3507194675

3.4417233563.33.441723356

3.53350928613.43.5335092861

3.62598917123.53.6259891712

3.71908698663.63.7190869866

3.81273684053.73.8127368405

3.90688139523.83.9068813952

4.00147056853.94.0014705685

4.096460459244.0964604592

4.19181245374.14.1918124537

4.287492484.24.28749248

4.38347038114.34.3834703811

4.47971938784.44.4797193878

4.57621567154.54.5762156715

4.67293796544.64.6729379654

4.76986724074.74.7698672407

4.866986434.84.86698643

4.96428018944.94.9642801894

5.061734693955.0617346939

5.15933746055.15.1593374605

5.25707719485.25.2570771948

5.35494365785.35.3549436578

5.45292754965.45.4529275496

5.55102040825.55.5510204082

5.649214525.65.64921452

5.74750284235.75.7475028423

5.84587893425.85.8458789342

5.94433689595.95.9443368959

6.042871315266.0428713152

6.14147721986.16.1414772198

6.2401500356.26.240150035

6.33888554676.36.3388855467

6.43767986696.46.4376798669

6.53652940476.56.5365294047

6.6354308396.66.635430839

6.73438109486.76.7343810948

6.83337732156.86.8333773215

6.93241687356.96.9324168735

7.031497292877.0314972928

7.13061629337.17.1306162933

7.22977174677.27.2297717467

7.32896166917.37.3289616691

7.42818421017.47.4281842101

7.52743764177.57.5274376417

7.62672034887.67.6267203488

7.72603082057.77.7260308205

7.82536764217.87.8253676421

7.9247294887.97.924729488

8.024115114888.0241151148

8.12352335548.18.1235233554

8.22295311348.28.2229531134

8.32240335828.38.3224033582

8.421873128.48.42187312

8.52136148588.58.5213614858

8.62086759538.68.6208675953

8.72039063748.78.7203906374

8.81992984698.88.8199298469

8.91948450138.98.9194845013

9.019053917999.0190539179

9.11863745149.19.1186374514

9.21823449139.29.2182344913

9.317844469.39.31784446

9.41746681029.49.4174668102

9.51710102329.59.5171010232

9.61674660759.69.6167466075

9.71640309659.79.7164030965

9.81607004739.89.8160700473

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Hydraulic Jumpy1y2EnergyMassPer unit widthUnknown lossescs1cs2

Hydraulic Jumpy1y2Momentumor

SummaryOpen channel flow equations can be obtained in a similar fashion to the Darcy-Weisbach equation (based on dimensional analysis)The dimensionally incorrect Manning equation is the standard in English speaking countriesThe free surface (an additional unknown) makes the physics more interesting!

Turbulent Flow Losses in Open ConduitsMaximum shear stressNo shear stress

Example

Grand Coulee Damhttp://users.owt.com/chubbard/gcdam/html/gallery.html

Columbia Basin ProjectThe Columbia Basin Project is a major water resource development in central Washington State with Grand Coulee Dam as the project's primary feature. Water stored behind Grand Coulee Dam is lifted by giant pumps into the Banks Lake Feeder Canal and then into Banks Lake. The water stored in Banks Lake is used to irrigate 0.5 million acres of land stretching 125 miles from Grand Coulee Dam.

PumpsAt the time of original construction the pumping plant contained six 65,000 horsepower pumps. In 1973 work began on extending the plant. The pump bay was doubled in length to the south and six 67,500 horsepower pump/generators were added (the last in 1983) providing 12 pumps in all. Each pump lifts water from Lake Roosevelt up through a 12 foot diameter discharge pipe to the feeder canal above. For most of their length the discharge pipes are buried in the rocky cliff to the west but at the top of the hill they emerge and can be seen as 12 silver pipes leading to the headworks of the feeder canal. The original pumps can supply water to the feeder canal at a rate of 1,600 cubic feet of water a second while the newer units can supply 2,000 cubic feet of water a second. They also have the advantage of being reversible. During times of peak power need the new pumps can be reversed thus turning them into generators. Water flows back down through the outlet pipes, through the generators and into Lake Roosevelt. When operating in this mode each pump can produce 50 megawatts of electrical power.

Grand Coulee Feeder CanalThe Grand Coulee Feeder Canal is a concrete lined canal which runs from the outlet of the pumping plant discharge tubes to the north end of Banks Lake. The original canal was completed in 1951 but has since been widened to accommodate the extra water available from the six new pump/generators added to the pumping plant. The canal is 1.8 miles in length, 25 feet deep and 80 feet wide at the base. It has the capacity to carry 16,000 cubic feet of water per second.

Columbia Basin Irrigation Project

Unsteady Hydraulics!The base width of the feeder canal was increased from 50 to 80 feet; however, the operating capacity remained at 16,000 cubic feet per second. Water depth was reduced from 25 to about 20 feet to safely accommodate wave action when the water flow is reversed as the pump-generators are changed from pumping to generating and vice-versa.

Gates

Gates

Banks Lake