0 UNTEY FLOW IN OEN CHNNEL 0. Introdution Unsteady flow in open hannels differs from that in losed onduits in that the eistene of a free surfae allows the flow ross-setion to freely hange, a fator whih has an important influene on the rate of transient hange propagation. Unsteady open hannel flow is enountered in flood flow in riers, in headrae anals supplying hydropower stations, in rier estuaries, and so on. 0.asi equations s in the ase of losed onduits the basi equations are deried from ontinuity and momentum onsiderations. In deriing these equations the following assumptions are made:. Hydrostati pressure preails at eery point in the hannel.. Veloity is uniformly distributed oer eah ross-setion. 3. The slope of the hannel bed is small and uniform. 4. The fritional resistane is the same as for steady flow. Fig 0. defines a ontrol olume and the dimensional parameters used to deelop the ontinuity equation. Control olume lateral inflow q + m m + m m 0 sin Fig 0. eferene diagram for the ontinuity equation The ontinuity equation balanes mass inflow and mass outflow with the rate of hange of the ontained mass within the ontrol olume: In time dt: Inflow - Outflow Change in mass of fluid in CV ρdt ρq dt ρ ρ + + dt dt +
iiding aross by ρ dt and negleting higher order terms: + + q (0.) Equation (0.) an also be written in the form or + q Q + q (0.) For a retangular hannel with zero lateral inflow, equation (0.) simplifies to Hene or y y y + + 0 (0.3) y y + 0 q y + 0 (0.4) Where q is the disharge per unit width of hannel. Fig 0., whih shows the fores ating on a fluid ontrol olume, defines the parameters required for the deelopment of the momentum equation. Control olume F W F + m F m t 0 0 sin Fig 0. eferene diagram for the momentum equation The momentum equation relates net fore to momentum hange: F d F F + + W sin θ τ o r ρ () dt where the pressure fore F ρgy ; the weight fore W ρgad; the wall shear stress τ o ρg h f ; r is the perimeter length. Equation () may therefore be epressed in terms of more basi flow parameters: diiding aross by ρg: ρg ( y) + ρg ρg ρ d o f dt
+ + y g o f (0.6) For a retangular setion this simplifies to or y + + o f g g + + g y + g( f o ) 0 (0.7) The terms of the momentum equation hae the dimension of aeleration or fore per unit mass. The first two terms on the left-hand side are the fluid aeleration terms, g y/ represents the pressure fore omponent, g f and g o represent the frition and graity fore omponents, respetiely. The forms of the ontinuity and momentum equations, represented in equations (0.3) and (0.7), respetiely, are known as the aint Venant equations; they relate the dependent ariables y and to the independent spae and time ariables and t, respetiely. 0.3olution by the harateristis method The same proedure as used in Chapter 6 for the solution of the orresponding pair of equations for unsteady flow in pipes, is applied here. Multiplying the ontinuity equation (0.3) by the fator λ and adding to the momentum equation (0.7): λ y y g g f -o 0 + + + λ + (0.8) ( λy + ) + This partial differential equation an be onerted to a total differential equation proided that or λy g/λ. Hene and dt λy + + g / λ λ ± g y dt ± gy ± (0.9) where is the graity waespeed; hene λ ±g/. Thus eqn (0.8) an be written in the equialent total differential form: subjet to and d dt g dy + + g( f o ) 0 (0.0) dt dt + ; (0.) d g dy + g( f o ) 0 (0.) dt dt 3
subjet to dt (0.3) Thus, the two partial differential equations (0.3) and (0.7) hae been onerted to their harateristi form i.e. linked pairs of ordinary differential equations. On integration of the latter oer the time interal t we get a pair of C + harateristi equations and a pair of C - harateristi equations: y t + g dy + g( f o ) dt 0 y t C + t ( + ) dt t y t g dy + g( f o ) dt 0 y t C - t ( ) dt t where and are the interpolated alues of at and, respetiely, as illustrated on the -t plane on Fig 0.3. The foregoing integrations may be approimated to a first order auray by assigning their known alues to, and f, giing the harateristi equations the following format: g + ( y y ) + g( o ) t 0 (0.4) C + + t () g + ( y y ) + g( o ) t 0 (0.6) C - where and are the alues of f at and, respetiely. t (0.7) t t Fig 0.3 The t plane The parameter alues at are found by linear interpolation in the interal and the parameter alues at are found by linear interpolation in the interal. eferring to the interal in Fig 0.4: 4
t + Fig 0.4 Linear interpolation eplaing by and ( - ) by, the following are the interpolated alues at : + θ( + ) + θ + θ( ) + θ (0.8) (0.9) y y θ y y + (0.0) where θ t/. Interpolated alues are similarly established at on the negatie harateristi side of : t olution of these equations gies the following interpolated alues at : + θ( ) θ + θ + + θ (0.) (0.) y y + θ y y (0.3) 5
0.4Numerial omputation proedure The foregoing finite differene formulation of the harateristi form of the unsteady flow equations an be used where there are no abrupt hanges in the water surfae profile and where onditions are sub-ritial. The omputational proedure adopted is similar to that outlined in Chapter 6 for the solution of the orresponding set of pipe flow equations. The hannel length is diided into N reahes, eah of length. The orresponding alue of the time step t is set by the so-alled Courant ondition: t + (0.4) This ensures that the harateristi ures plotted on the -t plane (Fig 0.3) remain within a single -t grid. t time zero the alues of y and are known at eah hannel node point. Their alues at internal nodes, at one time interal t later, are found by solution of equations (0.4) and (0.6) and are as follows: y + y + y + g t( ) () g y ( y ) ( ) g t (0.6) o The updated alues of y (y ) and ( ) at the upstream end of the hannel are goerned by the negatie harateristi equations (0.6) and (0.7) and the preailing upstream boundary ondition equation, whih is typially in the form of a defined ariation of either y or Q with time. olution of equation (0.7) and the boundary ondition equation yields the required alues for and y. The new alues for and y at the downstream end of the hannel are found in the same manner as their orresponding alues at the upstream end, the defining equations being the positie harateristi equation (0.8) and the preailing downstream boundary ondition equation. The foregoing analysis relates to onditions of tranquil flow only, that is, where the Froude number F r is less than unity. s the flow depth approahes the ritial alue (F r ), the numerial omputation beomes unstable. t ritial depth, and hene the negatie harateristi on the -t plane beomes ertial, that is, points and are oinident. Worked eample The foregoing analysis has been applied to the omputation of the ariation of flow depth and flow rate in the following eample. The outlet sluie gate in a 4m diameter ulert is losed at a rate that linearly redues an initial steady disharge rate of 4 m 3 s - to zero oer a period of 60 seonds. The ulert length is 500m, bottom slope is 0.0005 and the Manning n-alue is 0.05. The depth of water at the upstream end of the hannel remains onstant at its steady flow alue. The output of the omputation is plotted in Fig, whih shows the ariations of the following parameters with time: upstream depth (onstant at initial steady flow alue of.57m) downstream depth (osillating alue) inflow to hannel (osillating alue) outflow from hannel (redues to zero in 60s) 6
4 4 3 upstream inflow Flow depth (m) depth at downstream end downstream outflow 0 Flow rate (m 3 s - ) depth at upstream end - 0-4 0 4 6 8 0 Time (min) Fig Worked eample: plotted output data implifiation of the t Venant equations The aint Venant equations an be made more amenable to solution by omitting seleted terms from the momentum equation (0.). The latter may be written in the form y g o f + + + g (0.7) Henderson has pointed out that the aeleration terms (3rd. and 4th. on the right-hand side of (0.7)) are usually two orders of magnitude less than the graity ( o ) and frition ( f ) terms and one or two orders of magnitude less than the remaining term y/. This suggests that the solution of the simplified equation obtained by dropping the aeleration terms may proide a good approimation to the solution based on the full equations. The resulting simplified momentum equation beomes dy (0.8) On ombination with the ontinuity equation (0.3) the resulting unsteady open hannel flow equation for a retangular hannel has the form o f y y + f o (0.9) further simplifiation of the momentum equation is obtained by omission of the dy/ term (this term represents the unbalaned pressure fore omponent), reduing the momentum equation to its steady uniform flow form: f o (0.30) Using the Manning epression of frition slope equation (0.30) beomes and hene we an write o nq 0.67 h (0.3) 7
Q f() and dq dq d d F() d Combining this form of simplified momentum equation with the ontinuity equation (0.), where q 0, the resulting open hannel unsteady flow equation beomes: F + 0 (0.3) This equation is known as the kinemati wae equation beause the dynami terms of the momentum equation hae been omitted in its deelopment. The solution of equation (0.3) is learly of the form: ϕ t (0.33) F where the form of the funtion ϕ is determined by the boundary ondition for 0. 0.6apidly aried unsteady flow apidly aried unsteady flow gies rise to a surge or wae front, whih moes as a step-hange in water depth along the hannel. positie surge is defined as one whih leaes an inreased water depth in its wake as the wae front passes, while a negatie surge is one whih leaes a shallower depth in its wake as the wae front passes. In the following simplified analysis of surge front moement the effet of fritional resistane is negleted. 0.6. Upstream positie surge n upstream positie surge may be reated in hannel flow, for eample, by the rapid losure of a gate, resulting in a step redution in flow rate. The effet of this on the upstream side of the gate is the deelopment of a wae front, whih traels upstream, as illustrated on Fig 0.6. gate y y Fig 0.6 Upstream positie surge eferring to Fig 0.6, the surge front is seen to leae in its wake an inreased depth y, hene the desription positie. y superimposing a downstream eloity on the flow system, the flow regime is onerted to an equialent steady state, in whih the wae front is now stationary. pplying the ontinuity and momentum priniples to the ontrol olume between setions and, under the transformed steady state onditions: Hene Continuity ( + ) ( + ) (0.34) 8
and Negleting the flow frition fore: ( ) Q Q Momentum ρgy ρgy ρ ( )( ) + (0.35) oling equations (0.34) and (0.35) for and : g y y ( ) (0.36) ( )( ) g y y (0.37) For a retangular hannel: gy ( y + y ) y (0.38) If is assumed equal to zero in equation (0.38) the resulting relation between y and y is that for a hydrauli jump. Thus, the hydrauli jump an be onsidered to be a stationary surge. It should be noted that the ontinuity and momentum equations are not suffiient on their own to define the flow regime sine there are three unknowns,, y and (or Q ). One of these must therefore be known to enable omputation of the remaining two parameters. 0.6. ownstream positie surge downstream positie surge is aused, for eample, by the sudden opening of a gate, whih results in an instantaneous inrease in disharge and flow depth downstream of the gate, as illustrated on Fig 0.7. gate y y Fig 0.7 ownstream positie surge pplying the same analytial approah as used for the analysis of the upstream positie surge, the flow regime is transformed to an equialent steady state by superimposing a bakward eloity of magnitude on the system. Thus, referring to Fig 0.7, the ontinuity and momentum priniples an be applied to the ontrol olume defined by setions and : Continuity ( - ) ( - ) (0.39) 9
Hene Negleting the flow frition fore: Q Q (0.40) Momentum ρgy ρgy ρ ( )( ) (0.4) oling equations (0.39) and (0.4) for and : g y y ( ) + (0.4) ( )( ) g y y + (0.43) For a retangular hannel: gy ( y + y ) y ( + ) g yy y y yy + (0.44) + (0.45) 0.6.3 Upstream negatie surge negatie surge is seen by the obserer as a wae front moement whih leaes a lowered water surfae leel in its wake. n upstream negatie surge may be aused, for eample, upstream of a rapidly opened gate, as illustrated on Fig 0.8. The waefront flattens as it traels along the hannel, due to the top of the wae haing a greater eloity than the bottom. It is neessary, therefore, to alulate two waespeeds, one for the wae rest and the seond for the wae trough. y y gate Fig 0.8 Upstream negatie surge Consider a small rapid disturbane, giing rise to a small negatie surge, moing upstream as illustrated on Fig 0.9. pplying the ontinuity and momentum priniples as before: Continuity (+)y (y-y)(-+) (0.46) 0
my y y-my -m Fig 0.9 Negatie surge propagation Momentum ρg y ( y y) ρy( )( ) + (0.47) from (0.46) y - y /(+); from eqn(0.47) y - (+)/g. Hene and also gy (0.48) y g gy whih, as y approahes zero, an be written dy y d g Integrating for a wae of finite height gy + onstant (0.49) For the upstream negatie surge, illustrated on Fig 0.8, we hae the known boundary ondition, when y y; using these alues in equation (0.49), the integration onstant is found to be gy +. Hene, from eqn (0.49) : gy gy + (0) and from equation (0.48) gy Hene from (0) 3 gy gy () 0.6.4 ownstream negatie surge downstream negatie surge is propagated downstream of a rapidly losed gate, for eample, as illustrated on Fig 0.0.
gate y y Fig 0.0 ownstream negatie surge Using the same analytial proedure as applied in the ase of the upstream negatie surge, the wae front eloity an be shown, in the ase of a downstream negatie surge, to be: and the eloity is Hene the alues of and are found to be gy + () gy gy + (3) gy gy + (4) 3 gy gy + (5) elated reading bbott, M (966) n introdution to the method of harateristis, Elseier, New York. Chaudhry, M. H. (987) pplied hydrauli transients, (nd. Ed). Van Nostrand einhold, New York. Chow, Ven Te (959). Open hannel hydraulis, MGraw-Hill ook, New York. ooge, J C I (986) Theory of flood routing, in ier Flow Modelling and Foreasting (ed Kraijenhoff and J.. Moll), eidel ublishing Co., ordreht. Featherstone,. E. and Nalluri, C. (98). Ciil Engineering Hydraulis, Collins, London. Henderson, F. M. (966) Open hannel flow, Mamillan, New York. Wylie, E.. and treeter, V. L. (978) Fluid Transients, MGraw Hill, New York.