Chapterg13gOpen-ChannelgFlow
Solutions Manual for g g
Fluid Mechanics: Fundamentals and Applications
g g g g
Third Edition g
Yunus A. Çengel & John M. Cimbala
g g g g g g g
McGraw-Hill, 2013 g
Chapter 13 OPEN g g
-CHANNEL FLOW g
PROPRIETARY AND CONFIDENTIAL g g
This Manual is the proprietary property of The McGraw-Hill Companies, Inc. (“McGraw-
g g g g g g g g g g g
Hill”) and protected by copyright and other state and federal laws. By opening and using t
g g g g g g g g g g g g g g g
his Manual the user agrees to the following restrictions, and if the recipient does not agree t
g g g g g g g g g g g g g g g g
o these restrictions, the Manual should be promptly returned unopened to McGraw-
g g g g g g g g g g g
Hill: This Manual is being provided only to authorized professors and instructors for
g g g g g g g g g g g g
guse in preparing for the classes using the affiliated textbook. No other use or distribu
g g g g g g g g g g g g g g
tion of this Manual is permitted. This Manual may not be sold and may not be distrib
g g g g g g g g g g g g g g g g
uted to or used by any student or other third party. No part of this Manual may be r
g g g g g g g g g g g g g g g g g g
eproduced, displayed or distributed in any form or by any means, electronic or other
g g g g g g g g g g g g g
wise, without the prior written permission of McGraw-Hill.
g g g g g g g
13-1
PROPRIETARYgMATERIAL.g©g2014gbygMcGraw-
HillgEducation.g Thisgisgproprietarygmaterialgsolelygforgauthorizedginstructorguse.gNotgauthorizedgforgsalegorgdistributionginganygmanner.g
Thisgdocumentgmaygnotgbegcopied,gscanned,gduplicated,gforwarded,gdistributed,gorgpostedgongagwebsite,gingwholegorgpart.
, Chapterg13gOpen-ChannelgFlow
Classification,gFroudegNumber,gandgWavegSpeed
13-1C
Solution Wegaregtogdefinegnormalgdepthgandghowgitgisgestablished.
Analysis Ingopengchannelsgofgconstantgslopegandgconstantgcross-
section,gthegfluidgacceleratesguntilgthegheadglossgdueg togfrictionalgeffectsgequalsgthegelevationgdrop.gThegfluidgatgthisgpointgrea
chesgitsgterminalgvelocity,gandguniformgflowgisgestablished.gThegflowgremainsguniformgasglonggasgthegslope,gcross-
section,gandgthegsurfacegroughnessgofgthegchannelgremaingunchanged.gThegflowgdepthginguniformgflowgisgcalledgthegnormalg
depthgyn,gwhichgisgangimportantgcharacteristicgparametergforgopen-channelgflows.
Discussion Thegnormalgdepthgisgagfairlygstronggfunctiongofgsurfacegroughness.
13-2C
Solution Wegaregtogdiscussghowgpressuregchangesgalonggthegfreegsurfacegingopen-channelgflow.
Analysis
Thegfreegsurfacegcoincidesgwithgtheghydraulicggradeglineg(HGL),gandgthegpressuregisgconstantgalonggthegf
reegsurface.
Discussion Atgagfreegsurfacegofgagliquid,gthegpressuregmustgbegequalgtogthegpressuregofgtheggasgabovegit.
13-3C
Solution Wegaregtogdeterminegifgthegslopegofgthegfreegsurfacegisgequalgtogthegslopegofgthegchannelgbottom.
Analysis
Nog ing general.g Theg slopeg ofg theg freeg surfaceg isg notg necessarilyg equalg tog theg slopeg ofg theg bottomg surface
g evengduringgsteadygfullygdevelopedgflow.
Discussion However,gtheregaregsituationsgcalledguniformgflowgingwhichgthegconditionsgheregaregmet.
13-4C
Solution
Wegaregtogdiscussgsomegreasonsgforgnonuniformgflowgingopengchannels,gandgthegdifferencegbetweengrapidlygv
ariedgflowgandggraduallygvariedgflow.
Analysis Thegpresencegofgangobstructiongingagchannelgsuchgasgaggategorgagchangegingslopegorgcross-
sectiongcausesgthegflowgdepthgtogvary,gandgthusgthegflowgtogbecomegvariedgorgnonuniform.gThegvariedgflowgisgcalledgrapidlyg
variedgflowg(RVF)gifgthegflowgdepthgchangesgmarkedlygovergagrelativelygshortgdistancegingthegflowgdirectiong(suchgasgthegflo
wgofgwatergpastgagpartiallygopenggategorgshortlygbeforegagfalls),gandggraduallygvariedgflowg(GVF)gifgthegflowgdepthgchangesggr
aduallygovergaglonggdistancegalonggthegchannel.
Discussion ThegequationsgofgGVFgaregsimplifiedgbecausegofgthegslowgchangesgingthegflowgdirection.
13-2
PROPRIETARYgMATERIAL.g©g2014gbygMcGraw-
HillgEducation.g Thisgisgproprietarygmaterialgsolelygforgauthorizedginstructorguse.gNotgauthorizedgforgsalegorgdistributionginganygmanner.g
Thisgdocumentgmaygnotgbegcopied,gscanned,gduplicated,gforwarded,gdistributed,gorgpostedgongagwebsite,gingwholegorgpart.
, Chapterg13gOpen-ChannelgFlow
13-5C
Solutiong g g Wegaregtogdiscussgthegdrivinggforcegingopen-channelgflowgandghowgflowgrategisgdetermined.
Analysis
Flowgingagchannelgisgdrivengnaturallygbyggravity.gWatergflowgingagriver,gforgexample,gisgdrivengbygthegelevat
iongdifferencegbetweengthegsourcegandgthegsink.gThegflowgrategingangopengchannelgisgestablishedgbygthegdynamicgbalancegbet
weenggravitygandgfriction.gInertiagofgthegflowinggfluidgalsogbecomesgimportantgingunsteadygflow.
Discussiong g g Ingpipegflow,gongthegotherghand,gtheregmaygbegangadditionalgdrivinggforcegofgpressuregduegtogpumps.
13-6C
Solution Wegaregtogdiscussgthegdifferencegbetweenguniformgandgnonuniformgflow.
Analysis
Thegflowgingagchannelgisgsaidgtogbeguniformgifgthegflowgdepthg(andgthusgthegaveragegvelocity)gremainsgcons
tant.gOtherwise,gthegflowgisgsaidgtogbegnonuniformgorgvaried,gindicatinggthatgthegflowgdepthgvariesgwithgdistancegingthegflowg
direction.gUniformgflowgconditionsgaregcommonlygencounteredgingpracticeginglonggstraightgsectionsgofgchannelsgwithgconstan
tgslopegandgconstantgcross-section.
Discussion Inguniformgopen-channelgflow,gthegheadglossgduegtogfrictionalgeffectsgequalsgthegelevationgdrop.
13-7C
Solution Wegaregtogexplainghowgtogdeterminegifgagflowgisgtranquil,gcritical,gorgrapid.
Analysis Knowinggthegaveragegflowgvelocitygandgflowgdepth,gthegFroudegnumbergisgdeterminedgfromg Frg=gVg/ gy
.gThengthegflowgisgclassifiedgas
Frg<g1
Subcriticalgorgtranquilgflow
Frg=g1
g Criticalgflow
Frg>g1 Supercriticalgorgrapidgflow
Discussion ThegFroudegnumbergisgthegmostgimportantgparametergingopen-channelgflow.
13-8C
Solution
Weg areg tog discussg whetherg theg flowg upstreamg ofg ag hydraulicg jumpg mustg beg supercritical,g andg whetherg the
g flowgdownstreamgofgaghydraulicgjumpgmustgbegsubcritical.
Analysis
Upstreamg ofg ag hydraulicg jump,g theg upstreamg flowg mustg beg supercritical.g Downstreamg ofg ag hydraulic
g jump,gthegdownstreamgflowgmustgbegsubcritical.
Discussion
Otherwise,gthegsecondglawgofgthermodynamicsgwouldgbegviolated.gNotegthatgaghydraulicgjumpgisganalogousg t
ogagnormalgshockgwaveg–gingthatgcase,gthegflowgupstreamgmustgbegsupersonicgandgthegflowgdownstreamgmustgbegsubsonic.
13-3
PROPRIETARYgMATERIAL.g©g2014gbygMcGraw-
HillgEducation.g Thisgisgproprietarygmaterialgsolelygforgauthorizedginstructorguse.gNotgauthorizedgforgsalegorgdistributionginganygmanner.g
Thisgdocumentgmaygnotgbegcopied,gscanned,gduplicated,gforwarded,gdistributed,gorgpostedgongagwebsite,gingwholegorgpart.
, Chapterg13gOpen-ChannelgFlow
13-9C
Solution Wegaregtogdefinegcriticalglength,gandgdiscussghowgitgisgdetermined.
Analysis ThegflowgdepthgycgcorrespondinggtogagFroudegnumbergofgFrg=g1gisgthegcriticalgdepth,gandgitgisgdetermined
from Vg =g g gyc or ycg =gVg2g /gg .
Discussion Criticalgdepthgisgagusefulgparameter,gevengifgthegdepthgdoesgnotgactuallygequalgycganywheregingthegflow.
13-10C
Solution WegaregtogdefinegandgdiscussgthegusefulnessgofgthegFroudegnumber.
Analysis Froudeg number,g definedg as Frg =gVg /g g gy ,g isg ag dimensionlessg parameterg thatg governsg theg characterg of
flowgingopengchannels.gHere,gggisgtheggravitationalgacceleration,gVgisgthegmeangfluidgvelocitygatgagcross-
section,gandgLcgisgagcharacteristicglengthg(Lcg=gflowgdepthgygforgwidegrectangulargchannels).g Frgrepresentsgthegratiogofginertia
gforcesgtogviscousgforcesgingopen-
channelgflow.gThegFroudegnumbergisgalsogthegratiogofgthegflowgspeedgtogwavegspeed,gFrg=gVg/co.
Discussiong g g ThegFroudegnumbergisgthegmostgimportantgparametergingopen-channelgflow.
13-11
Solution
Agsinglegwavegisginitiatedgingagseagbygagstronggjoltgduringgangearthquake.gThegspeedgofgthegresultinggwavegisg t
ogbegdetermined.
Assumptions Thegdepthgofgwatergisgconstant,
Analysis Surfacegwavegspeedgisgdeterminedgthegwave-speedgrelationgtogbe
c0g = ghg= (9.81m/s2g)g(2000gm) =g140gm/s
Discussion
Noteg thatg waveg speedg dependsg ong theg waterg depth,g andg theg waveg speedg increasesg asg theg waterg depthg
increases.gAlso,gthegwavesgeventuallygdiegoutgbecausegofgthegviscousgeffects.
13-12
Solution
Theg flowg ofg waterg ing ag wideg channelg isg considered.g Theg speedg ofg ag smallg disturbanceg ing flowg forg twog
differentgflowgdepthsgisgtogbegdeterminedgforgbothgwatergandgoil.
Assumptions
Theg distanceg acrossg theg waveg isg shortg andg thusg frictiong atg theg bottomg surfaceg andg airg dragg atg theg topg areg
negligible,
Analysis Surfacegwavegspeedgcangbegdeterminedgdirectlygfromgthegrelationg c0g = gh .
(a) c0g = ghg= (9.81m/sg2g)g(0.25gm) =g 1.57g m/s
(b) c0g ghg= (9.81m/sg2g)g(0.8gm) =g 2.80g m/s
=
13-4
PROPRIETARYgMATERIAL.g©g2014gbygMcGraw-
HillgEducation.g Thisgisgproprietarygmaterialgsolelygforgauthorizedginstructorguse.gNotgauthorizedgforgsalegorgdistributionginganygmanner.g
Thisgdocumentgmaygnotgbegcopied,gscanned,gduplicated,gforwarded,gdistributed,gorgpostedgongagwebsite,gingwholegorgpart.
Solutions Manual for g g
Fluid Mechanics: Fundamentals and Applications
g g g g
Third Edition g
Yunus A. Çengel & John M. Cimbala
g g g g g g g
McGraw-Hill, 2013 g
Chapter 13 OPEN g g
-CHANNEL FLOW g
PROPRIETARY AND CONFIDENTIAL g g
This Manual is the proprietary property of The McGraw-Hill Companies, Inc. (“McGraw-
g g g g g g g g g g g
Hill”) and protected by copyright and other state and federal laws. By opening and using t
g g g g g g g g g g g g g g g
his Manual the user agrees to the following restrictions, and if the recipient does not agree t
g g g g g g g g g g g g g g g g
o these restrictions, the Manual should be promptly returned unopened to McGraw-
g g g g g g g g g g g
Hill: This Manual is being provided only to authorized professors and instructors for
g g g g g g g g g g g g
guse in preparing for the classes using the affiliated textbook. No other use or distribu
g g g g g g g g g g g g g g
tion of this Manual is permitted. This Manual may not be sold and may not be distrib
g g g g g g g g g g g g g g g g
uted to or used by any student or other third party. No part of this Manual may be r
g g g g g g g g g g g g g g g g g g
eproduced, displayed or distributed in any form or by any means, electronic or other
g g g g g g g g g g g g g
wise, without the prior written permission of McGraw-Hill.
g g g g g g g
13-1
PROPRIETARYgMATERIAL.g©g2014gbygMcGraw-
HillgEducation.g Thisgisgproprietarygmaterialgsolelygforgauthorizedginstructorguse.gNotgauthorizedgforgsalegorgdistributionginganygmanner.g
Thisgdocumentgmaygnotgbegcopied,gscanned,gduplicated,gforwarded,gdistributed,gorgpostedgongagwebsite,gingwholegorgpart.
, Chapterg13gOpen-ChannelgFlow
Classification,gFroudegNumber,gandgWavegSpeed
13-1C
Solution Wegaregtogdefinegnormalgdepthgandghowgitgisgestablished.
Analysis Ingopengchannelsgofgconstantgslopegandgconstantgcross-
section,gthegfluidgacceleratesguntilgthegheadglossgdueg togfrictionalgeffectsgequalsgthegelevationgdrop.gThegfluidgatgthisgpointgrea
chesgitsgterminalgvelocity,gandguniformgflowgisgestablished.gThegflowgremainsguniformgasglonggasgthegslope,gcross-
section,gandgthegsurfacegroughnessgofgthegchannelgremaingunchanged.gThegflowgdepthginguniformgflowgisgcalledgthegnormalg
depthgyn,gwhichgisgangimportantgcharacteristicgparametergforgopen-channelgflows.
Discussion Thegnormalgdepthgisgagfairlygstronggfunctiongofgsurfacegroughness.
13-2C
Solution Wegaregtogdiscussghowgpressuregchangesgalonggthegfreegsurfacegingopen-channelgflow.
Analysis
Thegfreegsurfacegcoincidesgwithgtheghydraulicggradeglineg(HGL),gandgthegpressuregisgconstantgalonggthegf
reegsurface.
Discussion Atgagfreegsurfacegofgagliquid,gthegpressuregmustgbegequalgtogthegpressuregofgtheggasgabovegit.
13-3C
Solution Wegaregtogdeterminegifgthegslopegofgthegfreegsurfacegisgequalgtogthegslopegofgthegchannelgbottom.
Analysis
Nog ing general.g Theg slopeg ofg theg freeg surfaceg isg notg necessarilyg equalg tog theg slopeg ofg theg bottomg surface
g evengduringgsteadygfullygdevelopedgflow.
Discussion However,gtheregaregsituationsgcalledguniformgflowgingwhichgthegconditionsgheregaregmet.
13-4C
Solution
Wegaregtogdiscussgsomegreasonsgforgnonuniformgflowgingopengchannels,gandgthegdifferencegbetweengrapidlygv
ariedgflowgandggraduallygvariedgflow.
Analysis Thegpresencegofgangobstructiongingagchannelgsuchgasgaggategorgagchangegingslopegorgcross-
sectiongcausesgthegflowgdepthgtogvary,gandgthusgthegflowgtogbecomegvariedgorgnonuniform.gThegvariedgflowgisgcalledgrapidlyg
variedgflowg(RVF)gifgthegflowgdepthgchangesgmarkedlygovergagrelativelygshortgdistancegingthegflowgdirectiong(suchgasgthegflo
wgofgwatergpastgagpartiallygopenggategorgshortlygbeforegagfalls),gandggraduallygvariedgflowg(GVF)gifgthegflowgdepthgchangesggr
aduallygovergaglonggdistancegalonggthegchannel.
Discussion ThegequationsgofgGVFgaregsimplifiedgbecausegofgthegslowgchangesgingthegflowgdirection.
13-2
PROPRIETARYgMATERIAL.g©g2014gbygMcGraw-
HillgEducation.g Thisgisgproprietarygmaterialgsolelygforgauthorizedginstructorguse.gNotgauthorizedgforgsalegorgdistributionginganygmanner.g
Thisgdocumentgmaygnotgbegcopied,gscanned,gduplicated,gforwarded,gdistributed,gorgpostedgongagwebsite,gingwholegorgpart.
, Chapterg13gOpen-ChannelgFlow
13-5C
Solutiong g g Wegaregtogdiscussgthegdrivinggforcegingopen-channelgflowgandghowgflowgrategisgdetermined.
Analysis
Flowgingagchannelgisgdrivengnaturallygbyggravity.gWatergflowgingagriver,gforgexample,gisgdrivengbygthegelevat
iongdifferencegbetweengthegsourcegandgthegsink.gThegflowgrategingangopengchannelgisgestablishedgbygthegdynamicgbalancegbet
weenggravitygandgfriction.gInertiagofgthegflowinggfluidgalsogbecomesgimportantgingunsteadygflow.
Discussiong g g Ingpipegflow,gongthegotherghand,gtheregmaygbegangadditionalgdrivinggforcegofgpressuregduegtogpumps.
13-6C
Solution Wegaregtogdiscussgthegdifferencegbetweenguniformgandgnonuniformgflow.
Analysis
Thegflowgingagchannelgisgsaidgtogbeguniformgifgthegflowgdepthg(andgthusgthegaveragegvelocity)gremainsgcons
tant.gOtherwise,gthegflowgisgsaidgtogbegnonuniformgorgvaried,gindicatinggthatgthegflowgdepthgvariesgwithgdistancegingthegflowg
direction.gUniformgflowgconditionsgaregcommonlygencounteredgingpracticeginglonggstraightgsectionsgofgchannelsgwithgconstan
tgslopegandgconstantgcross-section.
Discussion Inguniformgopen-channelgflow,gthegheadglossgduegtogfrictionalgeffectsgequalsgthegelevationgdrop.
13-7C
Solution Wegaregtogexplainghowgtogdeterminegifgagflowgisgtranquil,gcritical,gorgrapid.
Analysis Knowinggthegaveragegflowgvelocitygandgflowgdepth,gthegFroudegnumbergisgdeterminedgfromg Frg=gVg/ gy
.gThengthegflowgisgclassifiedgas
Frg<g1
Subcriticalgorgtranquilgflow
Frg=g1
g Criticalgflow
Frg>g1 Supercriticalgorgrapidgflow
Discussion ThegFroudegnumbergisgthegmostgimportantgparametergingopen-channelgflow.
13-8C
Solution
Weg areg tog discussg whetherg theg flowg upstreamg ofg ag hydraulicg jumpg mustg beg supercritical,g andg whetherg the
g flowgdownstreamgofgaghydraulicgjumpgmustgbegsubcritical.
Analysis
Upstreamg ofg ag hydraulicg jump,g theg upstreamg flowg mustg beg supercritical.g Downstreamg ofg ag hydraulic
g jump,gthegdownstreamgflowgmustgbegsubcritical.
Discussion
Otherwise,gthegsecondglawgofgthermodynamicsgwouldgbegviolated.gNotegthatgaghydraulicgjumpgisganalogousg t
ogagnormalgshockgwaveg–gingthatgcase,gthegflowgupstreamgmustgbegsupersonicgandgthegflowgdownstreamgmustgbegsubsonic.
13-3
PROPRIETARYgMATERIAL.g©g2014gbygMcGraw-
HillgEducation.g Thisgisgproprietarygmaterialgsolelygforgauthorizedginstructorguse.gNotgauthorizedgforgsalegorgdistributionginganygmanner.g
Thisgdocumentgmaygnotgbegcopied,gscanned,gduplicated,gforwarded,gdistributed,gorgpostedgongagwebsite,gingwholegorgpart.
, Chapterg13gOpen-ChannelgFlow
13-9C
Solution Wegaregtogdefinegcriticalglength,gandgdiscussghowgitgisgdetermined.
Analysis ThegflowgdepthgycgcorrespondinggtogagFroudegnumbergofgFrg=g1gisgthegcriticalgdepth,gandgitgisgdetermined
from Vg =g g gyc or ycg =gVg2g /gg .
Discussion Criticalgdepthgisgagusefulgparameter,gevengifgthegdepthgdoesgnotgactuallygequalgycganywheregingthegflow.
13-10C
Solution WegaregtogdefinegandgdiscussgthegusefulnessgofgthegFroudegnumber.
Analysis Froudeg number,g definedg as Frg =gVg /g g gy ,g isg ag dimensionlessg parameterg thatg governsg theg characterg of
flowgingopengchannels.gHere,gggisgtheggravitationalgacceleration,gVgisgthegmeangfluidgvelocitygatgagcross-
section,gandgLcgisgagcharacteristicglengthg(Lcg=gflowgdepthgygforgwidegrectangulargchannels).g Frgrepresentsgthegratiogofginertia
gforcesgtogviscousgforcesgingopen-
channelgflow.gThegFroudegnumbergisgalsogthegratiogofgthegflowgspeedgtogwavegspeed,gFrg=gVg/co.
Discussiong g g ThegFroudegnumbergisgthegmostgimportantgparametergingopen-channelgflow.
13-11
Solution
Agsinglegwavegisginitiatedgingagseagbygagstronggjoltgduringgangearthquake.gThegspeedgofgthegresultinggwavegisg t
ogbegdetermined.
Assumptions Thegdepthgofgwatergisgconstant,
Analysis Surfacegwavegspeedgisgdeterminedgthegwave-speedgrelationgtogbe
c0g = ghg= (9.81m/s2g)g(2000gm) =g140gm/s
Discussion
Noteg thatg waveg speedg dependsg ong theg waterg depth,g andg theg waveg speedg increasesg asg theg waterg depthg
increases.gAlso,gthegwavesgeventuallygdiegoutgbecausegofgthegviscousgeffects.
13-12
Solution
Theg flowg ofg waterg ing ag wideg channelg isg considered.g Theg speedg ofg ag smallg disturbanceg ing flowg forg twog
differentgflowgdepthsgisgtogbegdeterminedgforgbothgwatergandgoil.
Assumptions
Theg distanceg acrossg theg waveg isg shortg andg thusg frictiong atg theg bottomg surfaceg andg airg dragg atg theg topg areg
negligible,
Analysis Surfacegwavegspeedgcangbegdeterminedgdirectlygfromgthegrelationg c0g = gh .
(a) c0g = ghg= (9.81m/sg2g)g(0.25gm) =g 1.57g m/s
(b) c0g ghg= (9.81m/sg2g)g(0.8gm) =g 2.80g m/s
=
13-4
PROPRIETARYgMATERIAL.g©g2014gbygMcGraw-
HillgEducation.g Thisgisgproprietarygmaterialgsolelygforgauthorizedginstructorguse.gNotgauthorizedgforgsalegorgdistributionginganygmanner.g
Thisgdocumentgmaygnotgbegcopied,gscanned,gduplicated,gforwarded,gdistributed,gorgpostedgongagwebsite,gingwholegorgpart.
