86 水力學91-101) 6/12/2025

 86 水力學91-101) pp. 63

 Figure 3.1.3 Streamlines. (a) Flow in a conduit; (b) Flow from a slot; (c) Open-channel flow (uniform); (d) Open-channel flow (nonuniform). When streamlines are not straight there is a directional change in velocity. If they are not parallel, there is a change in speed along the streamlines. Under such circumstances the flow is nonuniform flow and 

 qV qx 不 =0 nonuniformflow (3:1: 16)

So this flow pattern has streamlines that are curved in space (converging or diverging) as shown in Figure 3.1.3(d) for nonuniform open-channel flow. 

     The variation invelocity with respect to time at a given point in a flow field is also used to classify flow. Steady flow occurs when thevelocity at a point in the flow field does not vary in magnitude or direction with respect to time:             qV qt = 0 steady flow            (3:1:17)

 Unsteady flow occurs when the velocity does vary in magnitude or direction at a point in the flow field with respect to time.          

3.1.7 Laminar and Turbulent Flow

 Turbulent flow is caused by eddies of varying size within the flow that create a mixing action. The fluid particles follow irregular and erratic paths and no two particles have similar motion. Turbulent flow then is irregular with no definite flow patterns. Flow in rivers is a good example of turbulent flow. The index used to relate to turbulence is the Reynolds number         Re = VD/u =  VDr      (3:1:18)

where D is a characteristic length such as the diameter of a pipe. For pipe flow, the flow is generally turbulent for Re > 2000. 

Laminar flow does not have the eddies that cause the intense mixing and therefore the flow is very smooth. The fluid particles move in definite paths and the fluid appears to move by the sliding of laminations of infinitesimal thickness relative to the adjacent layers. The viscous shear of the fluid particles produces the resistance to flow.  The resistance to flow varies with the first power of the velocity. For pipe flow, the flow is generally laminar for Re < 2000. 

        Flow can be one-, two-, or three-dimensional flow, for which one, two, or three coordinate direction, respectively, are required  to describe the velocity and properityflow field.

3.1.8  Discharge

Hydraulic Processes: Open-Channel Flow Sf The difference in water surface elevations is referred to as the fall. The k is a factor to account for a contraction and expansion of a reach. For a contracting reach Vu < Vd so k ¼ 1.0 and for an expanding reach (Vu > Vd)sok ¼ 0.5. The first approximation would compute the discharge using Q¼K ffiffiffiffiffi p withthe friction slope computed ignoring the velocity heads. 

Using the first approxmation of Q,     the upstream and downstream velocity head sare computed for the next approximation of the friction slope, which is used to compute the second approximation of the discharge. The procedure continues computing the new friction slope using the last discharge approximation to compute the new discharge. This process continues until the discharges approximations do not change significantly. It should be    emphasized that in the energy equation the Ff(loss due to friction) is a measureofthe internal energy dissipated in the entire mass of water in the control volume, whereas F0 f in the 5.2 Specific Energy, Momentum, and Specific Force 131 momentum equation measures the losses due to external forces exerted on the water by the wetted perimeter of the control volume. Ignoring the small difference between the energy coefficient a and the momentum coefficient b in gradually varied flow, the internal energy losses are practically identical with the losses due to external forces (Chow, 1959). For uniform flow, Fg ¼ F0 f . Application of the energy and momentum principles in open-channel flow can be confusing at f irst. It is important to understand the basic differences, even though the two principles may produce identical or very similar results. Keep in mind that energy is a scalar quantity and momentum is a vector quantity and that energy considers internal losses in the energy equation and momentum considers external resistance in the momentum equation. The energy principle is simpler and clearer than the momentum principle; however, the momentum principle has certain advantages in application to problems involving high internal-energy changes, such as the hydraulic jump (Chow, 1959), which is discussed in section 5.5 on rapidly varied flow. 5.2.3 Specific Force Forashorthorizontalreach(controlvolume)withq ¼ 0andthegravityforceFg ¼ W sinq ¼ 0,the external force of friction F0 f , can be neglected so F0 f ¼ 0 and Fg ¼ 0. Also assuming b1 ¼ b2, the momentum equation (5.2.22) reduces to gA1y1 gA2y2 ¼ rV1QþrV2Q ð5:2:23Þ Substituting V1 ¼ Q=A1 and V2 ¼ Q=A2, dividing through by g and substituting 1=g ¼ r=g, and then rearranging yields Q2 gA1 þA1y1 ¼ Q2 gA2 þA2y2 The specific force F (Figure 5.2.6) is defined as 2 F ¼Q gA þAy ð5:2:24Þ ð5:2:25Þ Figure 5.2.6 Specific force curves

which has units of ft3 or m3. The minimum value of the specific force with respect to the depth is determined using 2 d Q dF dy ¼ which results in gA dy þ dF dy ¼ dAy ð Þ dy ¼0 TQ2 gA2 þA¼0 Refer to Chow (1959) or Chaudhry (1993) for the proof and further explanation of d Ay Equation (5.2.27) reduces to ð5:2:26Þ ð5:2:27Þ ð Þ=dy ¼A. V2=gþA=T ¼ 0where the hydraulic depth D ¼ A=T,so 2 V g ¼D or V2 gD ¼1 ð5:2:28Þ which we have already shown is the criterion for critical flow (equation (5.2.9) or (5.2.10)). Therefore, at critical flow the specific force is a minimum for a given discharge. Summarizing, critical flow is characterized by the following conditions: . Specific energy is minimum for a given discharge. . Specific force is minimum for a given discharge. . Velocity head is equal to half the hydraulic depth. . Froude number is equal to unity. Two additional conditions that are not proven here are (Chow, 1959): . The discharge is maximum for a given specific energy. . Thevelocityofflowinachannelofsmallslopewithuniformvelocitydistributionisequaltothe celerity of small gravity waves in shallow water caused by local disturbances. Whenflowisatornearthecriticalstate, minorchanges inspecificenergynear critical flow cause major changes in depth (see Figures 5.2.1 or 5.2.2), causing the flow to be unstable. 

Figure 5.2.7 illustrates examples of locations of critical flow

5.3.1.  Gradually Varied Flow Equations 

Several types of open-channel flow problems can be solved in hydraulic engineering practice using the concepts of nonuniformflow. The first  to be discussed are gradually varied flow problems in which the change in the water surface profile is small enough that it is possible to integrate the relevant differential equation fromone section to an adjacent section for the change in depth or change inwater surface elevation.  Consider the energy equation (5.1.6) previouslyderivedfornonuniformflow(Figure5.1.1)usingthecontrolvolumeapproach(with a1¼a2¼1): y1þz1þV2 1 2g¼y2þz2þV2 2 2gþhL ð5:1:6Þ BecausehL¼SfL¼SfDxlettingDy¼y2 y1andDz¼z1 z2¼S0Dx, thenequation(5.1.6) canbeexpressedas S0DxþV2 1 2g¼DyþV2 2 2gþSfDx ð5:3:1Þ Rearrangingyields Dy¼S0Dx SfDx V2 2 2g V2 1 2g ð5:3:2Þ andthendividingthroughbyDxresultsin Dy Dx¼S0 Sf V2 2 2g V2 1 2g 1 Dx ð5:3:3Þ TakingthelimitasDx!0,weget lim Dx!0 Dy Dx ¼dy dx ð5:3:4Þ and lim Dx!0 V2 2 2g V2 1 2g 1 Dx ¼ d dx V2 2g ð5:3:5Þ Substitutingtheseintoequation(5.3.3)andrearrangingyields dy dxþ d dx V2 2g ¼S0 Sf ð5:3:6Þ Thesecondterm d dx V2 2g canbeexpressedas d V2 2g dy 2 6 6 4 3 7 7 5 dy dx , sothatequation(5.3.6)canbe simplifiedto dy dx 1þ d V2 2g dy 2 6 6 4 3 7 7 5¼S0 Sf ð5:3:7Þ 134 Chapter 5 HydraulicProcesses:Open-ChannelFlow or dy dx¼ S0 Sf 1þ d V2 2g dy 2 6 6 4 3 7 7 5 ð5:3:8Þ Equations(5.3.7)and(5.3.8)aretwoexpressionsofthedifferentialequationforgraduallyvaried flow.Equation(5.3.8)canalsobeexpressedintermsof theFroudenumber.Firstobservethat d dy V2 2g ¼ d dy Q2 2gA2 ¼ Q2 gA3 dA dy ð5:3:9Þ Bydefinition, the incremental increase incross-sectional areaofflowdA, due toan incre mentalincreaseinthedepthdy,isdA¼Tdy,whereTisthetopwidthofflow(seeFigure5.3.1). AlsoA=T¼D,whichisthehydraulicdepth.Equation(5.3.9)cannowbeexpressedas d dy V2 2g ¼ Q2 gA3 Tdy dy ¼ Q2 gA2 T A ¼ Q2 gA2D ð5:3:10aÞ ¼ F2 r ð5:3:10bÞ whereFr¼ V ffiffiffiffiffiffi gD p ¼ Q A ffiffiffiffiffiffi gD p .Substitutingequation(5.3.10b)into(5.3.8)andsimplifying,wefind that thegraduallyvariedflowequationintermsof theFroudenumber is dy dx¼S0 Sf 1 F2 r ð5:3:11Þ EXAMPLE5.3.1 Consideraverticalsluicegateinawiderectangularchannel(R¼A=P¼By=Bþ2Y ð Þ ybecause B 2y).Theflowdownstreamofasluicegateisbasicallyajet thatpossessesavenacontracta (seeFigure5.3.2).Thedistancefromthesluicegatetothevenacontractaasaruleisapproximatedas thesameasthesluicegateopening(Chow,1959).Thecoefficientsofcontractionforverticalsluice gatesareapproximately0.6,rangingfrom0.598to0.611(Henderson,1966).Theobjectiveofthis problemistodeterminethedistancefromthevenacontractatoapointbdownstreamwherethe depthofflowisknowntobe0.5mdeep.Thedepthofflowatthevenacontractais0.457mforaflow rateof4.646m3/spermeterofwidth.Thechannelbedslopeis0.0003andManning’sroughness factorisn¼0.020. Figure

r dy dx¼ S0 Sf 1þ d V2 2g dy 2 6 6 4 3 7 7 5 ð5:3:8Þ Equations(5.3.7)and(5.3.8)aretwoexpressionsofthedifferentialequationforgraduallyvaried flow.Equation(5.3.8)canalsobeexpressedintermsof theFroudenumber.Firstobservethat d dy V2 2g ¼ d dy Q2 2gA2 ¼ Q2 gA3 dA dy ð5:3:9Þ Bydefinition, the incremental increase incross-sectional areaofflowdA, due toan incre mentalincreaseinthedepthdy,isdA¼Tdy,whereTisthetopwidthofflow(seeFigure5.3.1). AlsoA=T¼D,whichisthehydraulicdepth.Equation(5.3.9)cannowbeexpressedas d dy V2 2g ¼ Q2 gA3 Tdy dy ¼ Q2 gA2 T A ¼ Q2 gA2D ð5:3:10aÞ ¼ F2 r ð5:3:10bÞ whereFr¼ V ffiffiffiffiffiffi gD p ¼ Q A ffiffiffiffiffiffi gD p .Substitutingequation(5.3.10b)into(5.3.8)andsimplifying,wefind that thegraduallyvariedflowequationintermsof theFroudenumber is dy dx¼S0 Sf 1 F2 r ð5:3:11Þ EXAMPLE5.3.1 Consideraverticalsluicegateinawiderectangularchannel(R¼A=P¼By=Bþ2Y ð Þ ybecause B 2y).Theflowdownstreamofasluicegateisbasicallyajet thatpossessesavenacontracta (seeFigure5.3.2).Thedistancefromthesluicegatetothevenacontractaasaruleisapproximatedas thesameasthesluicegateopening(Chow,1959).Thecoefficientsofcontractionforverticalsluice gatesareapproximately0.6,rangingfrom0.598to0.611(Henderson,1966).Theobjectiveofthis problemistodeterminethedistancefromthevenacontractatoapointbdownstreamwherethe depthofflowisknowntobe0.5mdeep.Thedepthofflowatthevenacontractais0.457mforaflow rateof4.646m3/spermeterofwidth.Thechannelbedslopeis0.0003andManning’sroughness factorisn¼0.020.