3. GRADUALLY-VARIED FLOW (GVF) AUTUMN EGL (energy grade line) weir change of slope

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1 3. GRADUALLY-ARIED FLOW (GF) AUTUMN Normal Flow vs Gradually-aried Flow /g EGL (eergy grade lie) ictio slope Geometric slope S I ormal flow te dowslope compoet of weigt balaces bed frictio. As a result, te water dept ad velocity are costat ad te total-ead lie (or eergy grade lie) is parallel to bot te water surface ad te cael bed; i.e. te frictio slope is te same as te geometric slope S. GF RF GF RF GF RF GF RF GF UF sluice gate ydraulic jump weir cage of slope As a result of disturbaces due to ydraulic structures (weir, sluice, etc.) or cages to cael widt, slope or rougess te dowslope compoet of weigt may ot locally balace bed frictio. As a result, te frictio slope ad bed slope S will be differet ad te water dept ad velocity will cage alog te cael. Te gradually-varied-flow equatio gives a expressio for d/ ad allows oe to predict te variatio of water dept alog te cael. Hydraulics 3 Gradually-aried Flow - 1 Dr David Apsley

2 3. Derivatio of te Gradually-aried-Flow Equatio Provided te pressure distributio is ydrostatic te, at ay streamwise locatio x: p z zs ( x) ρg Te total ead is te H zs zb (1) g g were z s is te level of te free surface ad z b is te level of te bed. z s g z b cos Altoug ot crucial, we make te small-slope assumptio ad make o distictio betwee te vertical dept (wic forms part of te total ead) ad tat perpedicular to te bed, cos θ (wic is used to get te flow rate). Te total ead may be writte H zb E () were E is te specific eergy, or ead relative to te bed: E (3) g I frictioless flow, H = costat; i.e. te eergy grade lie is orizotal. I reality, H decreases over large distaces due to bed frictio; te eergy grade lie slopes dowward. Differetiate (): dh dzb (4) Defie: dh (5) dz b S (6) is te dowward slope of te eergy grade lie, or frictio slope; (more about ow tis is calculated later). S is te actual geometric slope. Te S (7) Tus, te specific eergy oly cages if tere is a differece betwee te geometric ad frictio slopes, i.e. betwee te rates at wic gravity drives te flow ad frictio retards it. Oterwise we would ave ormal flow, i wic te dept ad specific eergy are costat. Hydraulics 3 Gradually-aried Flow - Dr David Apsley

3 Equatios (5) ad (7) are two forms of te gradually-varied-flow equatio. However, te tird, ad most commo, form rewrites / i terms of te rate of cage of dept, d/. Q E were (8) g A Q E ga Differetiatig wit respect to streamwise distace x (usig te cai rule for te last term): d Q da 3 ga If b s is te widt of te cael at te surface: bs d A b d s ad A bs Hece, d d Q b A E s (1 ) 3 ga d (1 ) g d (1 ) Combiig tis wit (7) gives, fially, d Gradually-aried-Flow Equatio d S 1 (9) 3.3 Fidig te ictio Slope Sice te flow (ad ece te velocity profile ad bed stress) vary oly gradually wit distace, frictio is primarily determied by te local bulk velocity. Te local frictio slope ca te be evaluated o te quasi-uiform-flow assumptio tat tere is te same rate of eergy loss as i ormal flow of te same dept; e.g. usig Maig s equatio: 1 / 3 1/ R Ivertig for te frictio slope: S f Q (1) R R A 4 / 3 4 / 3 Bot A ad R (wic deped o te cael sape) sould be writte i terms of dept. I geeral, te deeper te flow te te smaller te velocity ad frictio losses. Qualitatively, greater dept lower velocity smaller ; smaller dept iger velocity greater. Hydraulics 3 Gradually-aried Flow - 3 Dr David Apsley

4 3.4 Profile Classificatio For ay give discarge (but idepedet of slope) tere is a particular critical dept c, at 1/ 3 wic = 1. For example, i a wide or rectagular cael, c ( q / g). For ay give discarge ad slope tere is a particular ormal dept, associated wit 3/ 5 uiform flow. For example, i a wide cael, ( q / S ). Te ormal dept is tat to wic te flow would ted give a log eoug udisturbed fetc. A dowward slope is classed as steep if te ormal dept is less ta te critical dept (i.e. te ormal flow is supercritical) ad mild if te ormal dept is greater ta te critical dept (i.e. te ormal flow is subcritical). 1 Note tat, i priciple at least, a particular slope could be described as eiter steep or mild depedig o te flow rate. I geeral, give te actual dept, critical dept c ad ormal dept, simple ispectio of sigs of umerator ad deomiator o te RHS of te GF equatio d S 1 will tell us weter dept is icreasig or decreasig wit distace. I te special case = S we ave d/ = ; i.e. ormal flow. Oterwise: S > if ad oly if is greater ta ormal dept; (slower, ece less frictio) 1 > if ad oly if is greater ta critical dept Hece, d (dept decreasig) if ad oly if lies betwee ormal ad critical depts. Cosiderig te wole rage of possibilities allows a two-caracter classificatio of slopes (S1, M3 etc) were: te first caracter is S, C, M, H, A for Steep, Critical, Mild, Horizotal, Adverse; te secod caracter is 1,, 3 depedig o were lies wit respect to c ad. Typical profiles ad examples of were tey occur are give i te table overleaf. A backwater curve is a streamwise icrease of dept because of a dowstream obstructio or because te slope is isufficiet to maitai tat rate of flow. A reductio i dept is called a drawdow curve. Profiles wit secod caracter 1 or 3 are backwater curves (umerator ad deomiator of te GF equatio ave te same sig); profiles wit secod caracter are drawdow curves (umerator ad deomiator of te GF equatio ave opposite sigs). Note tat: te ormal dept (were S = ) is approaced asymptotically (d/ ); te critical dept (were = 1) is approaced at rigt agles (d/ ). above ormal dept a backwater asymptotes a orizotal surface (M1, S1) 1 A alterative statemet is tat a dowward slope is steep if it exceeds te critical slope (tat slope at wic te ormal flow is critical). Hydraulics 3 Gradually-aried Flow - 4 Dr David Apsley

5 Type Symbol Defiitio Sketces Examples STEEP (ormal flow is supercritical) S1 S S3 c c c c S 1 S S 3 Hydraulic jump upstream wit obstructio or reservoir cotrollig water level dowstream. Cage to steeper slope. Cage to less steep slope. CRITICAL (udesirable; udular usteady flow) C1 C3 c c = c C 1 C 3 MILD (ormal flow is subcritical) M1 M M3 c c c M 3 M 1 M Obstructio or reservoir cotrollig water level dowstream. Approac to free overfall. c Hydraulic jump dowstream; cage from steep to mild slope or dowstream of sluice gate. HORIZONTAL (limitig mild slope; ) H H3 c c c H H 3 Approac to free overfall. Hydraulic jump dowstream; cage from steep to orizotal or dowstream of sluice gate. ADERSE (upslope) A A3 c c A c A3 Hydraulics 3 Gradually-aried Flow - 5 Dr David Apsley

6 3.5 Qualitative Examples of Ope-Cael-Flow Beaviour A cotrol poit is a locatio were tere is a kow relatiosip betwee water dept ad discarge (aka stage-discarge relatio ). Examples iclude critical-flow poits (weirs, veturi flumes, sudde cages i slope, free overfall), sluice gates, etry or discarge to a reservoir. A ydraulic jump ca also be classed as a cotrol poit. Cotrol poits ofte provide a locatio were oe ca start a GF calculatio; i.e. a boudary coditio. Some geeral rules: (i) Supercritical cotrolled by upstream coditios. Subcritical cotrolled by dowstream coditios. (ii) (iii) (iv) Give a log-eoug udisturbed fetc te flow will revert to ormal flow. A ydraulic jump occurs betwee regios of supercritical ad subcritical graduallyvaried flow at te poit were te jump coditio for te sequet depts is correct. Were te slope is mild (i.e. te ormal flow is subcritical), ad ay dowstream cotrol is a log way away, a ydraulic jump ca be assumed to jump directly to te ormal dept. Flow over a weir (mild slope) ormal M1 1 c WEIR M3 ydraulic jump ormal Flow uder a sluice gate (a) mild slope ormal M1 1 M3 ydraulic jump ormal Flow uder a sluice gate (b) steep slope ormal S1 1 S3 ormal Hydraulics 3 Gradually-aried Flow - 6 Dr David Apsley

7 Flow from a reservoir (a) mild slope ormal RESEROIR Flow from a reservoir (b) steep slope RESEROIR c S ormal Flow ito a reservoir (mild slope) ormal M1 RESEROIR ee overfall (mild slope) ormal M c critical Hydraulics 3 Gradually-aried Flow - 7 Dr David Apsley

8 3.6 Numerical Solutio of te GF Equatio Aalytical solutios of te GF equatio are very rare ad it is usual to solve it umerically. Te process yields a series of discrete pairs of distace x i ad dept i. Itermediate poits ca be determied, if required, by iterpolatio. All metods employ a discrete approximatio to oe of te followig forms of GF equatio: dh S d S 1 (total ead H zs ) (11) g (specific eergy E ) (1) g (dept ) (13) I ay of tese te frictio slope ca be obtaied by ivertig Maig s equatio: S f Q 4 / 3 4 / 3 R R A (14) ad te oude umber is (15) g were A/ b is te mea dept (= actual dept for a rectagular or wide cael). s Pysically, itegratio sould start at a cotrol poit ad proceed: forward i x if te flow is supercritical (upstream cotrol) flow backward i x if te flow is subcritical (dowstream cotrol). flow Tere are two mai classes of metod: Stadard-step metods: solve for dept at specified distace itervals. 3 1 x x x x 4 Direct-step metods: solve for distace x at specified dept itervals Δ. (Oe advatage is tat tey ca calculate profiles startig from a critical poit, were 1 = ad stadard-step metods would fail). x x 1 x x 3 Hydraulics 3 Gradually-aried Flow - 8 Dr David Apsley

9 3.6.1 Total-Head Form of te GF Equatio dh Tis is solved as a stadard-step metod (fid dept at specified distace itervals ). Te equatio is discretised as H, i1 H S f i S i f, i1 ( ) (16) solvig sequetially for 1,, 3, startig wit te dept at te cotrol poit. Sice bot H ad are fuctios of, te metod operates by adjustig i+1 iteratively at eac step so tat te LHS ad RHS of (16) are equal. Tis is a good metod, but sice it requires iterative solutio at eac step it is better suited to a computer program ta ad or spreadseet calculatio Specific-Eergy Form of te GF Equatio S (were E ) g Sice icremets i E are determied by successive values of, tis is solved as a direct-step metod (fid displacemet x at specified dept itervals Δ). First ivert to make E te idepedet variable: 1 S Te equatio is te discretised ( ) ad rearraged for distace icremets as: ΔE ΔE, were Δ E E i1 Ei (17) ( S ) av Tere are various ways of estimatig te average slope differece. Te example to follow uses te average of values at depts i ad i+1. Hydraulics 3 Gradually-aried Flow - 9 Dr David Apsley

10 3.6.3 Dept Form of te GF Equatio d S 1 Here we sall solve tis by a direct-step metod (fid displacemet x at specified dept itervals Δ). First, ivert to make te idepedet variable: 1 d S Te fuctio o te RHS is first writte as a fuctio of. Te equatio is te discretised ( ) ad rearraged for distace icremets as: Δ d Δ (18) d av As before, te bracketed term o te RHS ca be take as te average of values at te start ad ed of a iterval or (my preferece) by evaluatio at te iterval mid-dept: 1 ( 1). mid i i Commet. Differet autors adopt differet ways of solvig te GF equatio umerically, particularly i coosig weter to use te specific-eergy or dept form, ad ow to form te average derivative (e.g., average of values at te eds of te iterval or simply te sigle value at te midpoit). All sould give te same aswer we te step size Δ becomes very small, but may differ for te larger step sizes typical of ad calculatios. Te specificeergy form seems to be sligtly more commo i te literature, but my ow tests suggest tat te dept form, wit derivative evaluated just oce at te mid-poit of te iterval, gives sligtly better results for large step sizes. Note tat surface profiles become igly curved ear critical poits ad more steps, wit a smaller Δ, sould be used tere. Hydraulics 3 Gradually-aried Flow - 1 Dr David Apsley

11 Example. (Exam 7 modified) A log rectagular cael of widt 4 m as a slope of 1:5 ad a Maig s of.15 m 1/3 s. Te total discarge is 8 m 3 s 1. Te cael arrows to a widt of 1 m as a veturi flume over a sort legt. (a) (b) Determie te ormal dept for te 4 m wide cael. Sow tat critical coditios occur at te arrow 1 m wide sectio. (c) Determie te dept just upstream of te veturi were te widt is 4 m. (d) Determie te distace upstream to were te dept is 5% greater ta te ormal dept usig two steps i te gradually-varied flow equatio give below; (you may use eiter form). Data I stadard otatio, d S 1 or S S f Solutio. (a) For te ormal dept, 1 / 3 1/ Q A were R S, R b b 1 /, A b b 5 / 3 b S Q / 3 ( 1 / b) Rearragig as a iterative formula for to fid te ormal dept at te cael slope S : 3/ 5 Q / 5 ( 1 / b) b S Here, wit legts i metres: / (1.5) Iteratio (from, e.g., = 1.57) gives ormal dept:.9 m Aswer:.9 m. (b) To determie weter critical coditios occur, compare te total ead i te approac flow wit tat assumig critical coditios at te troat. Te total ead, assumig ormal flow ad measurig eigts from te bed of te cael is Q H a Ea.137 m g gb Hydraulics 3 Gradually-aried Flow - 11 Dr David Apsley

12 At te troat te discarge per uit widt is Q 1 8 m s b q m mi Te critical dept ad critical specific eergy at te troat are 1/ 3 q m c g m 3 E c c.84 m Sice te bed of te flume is flat (z b = ), te critical ead H c = E c. Sice te approac-flow ead H a is less ta te critical ead H c (te miimum ead required to pass tis flow rate troug te veturi), te flow must back up ad icrease i dept just upstream to supply tis miimum ead. It will te udergo a subcritical to supercritical trasitio troug te troat. Te total ead trougout te veturi is H = H c =.84 m. (c) I te viciity of te veturi te total ead is H =.84 m. Just upstream (were widt b = 4 m), we seek te subcritical solutio of Q H zs g gb Rearrage for te deeper solutio: Q H gb Here, wit legts i metres: Iterate (from, e.g., =.84) to get te dept just upstream of te veturi:.778 m Aswer:.78 m (d) Do a GF calculatio (subcritical, so pysically it sould start at te fixed dowstream cotrol ad work upstream, altoug matematically it ca be doe te oter way) from te pre-veturi dept ( =.778 m) to were =.194 m (i.e. 1.5 ). Usig two steps te dept icremet per step is Δ.9 m =.194 = =.778 Step Step 1 x x 1 x Bot dept ad specific-eergy metods are sow o te followig pages. Hydraulics 3 Gradually-aried Flow - 1 Dr David Apsley

13 METHOD 1: usig te dept form of te GF equatio d S 1 1 d S Δ d mid Δ d mid ( mid meas mid-poit of te iterval: alf way betwee i ad i+1 ; sometimes writte i+½.) For coveiece, work out umerical expressios for ad i terms of : Q / b g g 5 / 3 b S Maig s equatio (see earlier) gave Q. Assumig tat te rate of loss / 3 ( 1 / b) of eergy ( ) at a geeral dept is te same as te cael slope tat would give ormal flow at tat dept, rearragemet for te slope gives Q b (1 / b) 1/ 3 4 / (1.5) 1/ 3 Hece, / 3 d S S f (1.5) / 3 Wit Δ ad Δ =.9 m d mid workig may te be set out i tabular form. (All depts assumed to be i metres.) i i x i mid (/d) mid Tis gives a distace of about 8.7 km upstream. 4 / 3 Hydraulics 3 Gradually-aried Flow - 13 Dr David Apsley

14 METHOD : Usig te specific-eergy form of te GF equatio S 1 S ΔE ( S ( S 1 ΔE ) ) av av ( av is take as te average of values calculated at start ad ed of eac iterval i.) Here: E Q / b g g.39 ad te same expressio as before may be used for, so tat: 4 / (1.5) S S f / 3 Wit ΔE ( S ) av workig may te be set out i tabular form. (All depts assumed to be i metres.) i i x i E i S ΔE (S ) av Tis gives a distace of about 9. km upstream. Smaller steps Δ will give more accurate results (ad closer agreemet betwee te two metods). Hydraulics 3 Gradually-aried Flow - 14 Dr David Apsley

15 Example. (Exam 8 reworded) A log, wide cael as a slope of 1:747 wit a Maig s of.15 m 1/3 s. It carries a discarge of.5 m 3 s 1 per metre widt, ad tere is a free overfall at te dowstream ed. A udersot sluice is placed a certai distace upstream of te free overfall wic determies te ature of te flow betwee sluice ad overfall. Te dept just dowstream of te sluice is.5 m. (a) (b) (c) Determie te critical dept ad ormal dept. Sketc, wit explaatio, te two possible gradually-varied flows betwee sluice ad overfall. Calculate te particular distace betwee sluice ad overfall wic determies te boudary betwee tese two flows. Use oe step i te gradually-varied-flow equatio. Example. A sluice gate discarges water at 9 m 3 s 1 ito a 6 m wide rectagular cael laid o a slope of.4 wit =.15 m 1/3 s. Te dept at te vea cotracta is.15 m. (a) (b) Fid te ormal ad critical depts. Compute te positio of te ydraulic jump, assumig ormal dept dowstream. Use oe step i te GF equatio. Example. (Exam 14) A udersot sluice is used to cotrol te flow of water i a log wide cael of slope.3 ad Maig s rougess coefficiet.1 m 1/3 s. Te flow rate i te cael is m 3 s 1 per metre widt. (a) (b) (c) (d) Calculate te ormal dept ad critical dept i te cael ad sow tat te cael is ydrodyamically steep at tis flow rate. Te dept of flow just dowstream of te sluice is.4 m. Assumig o ead losses at te sluice calculate te dept just upstream of te sluice. Sketc te dept profile alog te cael, idicatig clearly ay flow trasitios brougt about by te sluice ad idicatig were water dept is icreasig or decreasig. Use steps i te gradually-varied flow equatio to determie ow far upstream of te sluice a ydraulic jump will occur. Hydraulics 3 Gradually-aried Flow - 15 Dr David Apsley