3. Gradually-Varied Flow

Transcription

1 5/6/18 3. Gradually-aried Flow Normal Flow vs Gradually-aried Flow Normal Flow /g EGL (energy grade line) iction slope Geometric slope S Normal flow: Downslope component of weigt balances bed friction Uniform dept ( dept ) and velocity Bed slope or geometric slope (S ) is te same as te slope of te total ead line or friction slope ( ) Preferred dept, to wic flow tends given sufficient fetc 1

2 5/6/18 Gradually-aried Flow GF RF GF RF GF RF GF RF GF UF sluice gate ydraulic jump weir cange of slope Gradually-varied flow (GF): Component of weigt does not balance bed friction Geometric slope (S ) is different from friction slope ( ) Dept canges wit distance Te gradually-varied-flow equation gives te cange of dept wit distance Gradually-aried-Flow Equation 1 Assumptions: Small slopes Quasi-1d Hydrostatic pressure Depends on: Difference between geometric and friction slopes (S ) Sub- or supercritical flow () Derivation of te Gradually- aried Flow Equation

3 5/6/18 Derivation of te GF Equation (1) Total ead: H zs zb g g H z b E z s z b cos g dh dzb Define: GF equation (specific-energy form): dh dz b S d x friction slope geometric slope S S f Derivation of te GF Equation () Specific energy: E g Q E ga Q da 3 ga Q bs (1 ) 3 ga (1 ) g d S (1 S f ) GF equation (dept form): Q A da b s d A b s 1 S d x bs A Finding te iction Slope 3

4 5/6/18 Finding te iction Slope, 1 Quasi-uniform-flow assumption: rate of energy loss is te same as uniform flow of te same dept. 1 R n n S 4/3 R / 3 1/ n Q 4/3 R A function of dept f, greater dept lower velocity smaller smaller dept iger velocity greater Profile Classification Slope Classification Critical dept c : dept at wic = 1. Normal dept : dept of uniform flow. e.g. wide cannel: c ( q / g ) 1/ 3 ( nq / S ) 3/ 5 (For a given discarge) a slope is: steep, if te dept is less tan te critical dept (i.e. te flow is supercritical) mild, if te dept is greater tan te critical dept (i.e. te flow is subcritical) 4

5 5/6/18 Canging Dept 1 S > if and only if is greater tan dept 1 > if and only if is greater tan critical dept dept decreasing if and only if lies between and critical depts. Water-Profile Classification caracters (e.g. S1, M3 etc.): S, C, M, H, A (Steep, Critical, Mild, Horizontal, Adverse) 1,, 3 (were lies wit respect to c and ) Type Symbol Definition Sketces Examples S1 > c > c S1 Hydraulic jump upstream wit obstruction or reservoir controlling water level downstream. STEEP ( flow supercritical) S c > > n S S3 Cange to steeper slope. S3 c > > Cange to less steep slope. CRITICAL C1 > c = c= n C1 (undesirable; undular unsteady C3 flow) C3 c = > M1 > > c n M1 Obstruction or reservoir controlling water level downstream. MILD ( flow M > > c c M Approac to free overfall. subcritical) M3 M3 > c > Hydraulic jump downstream; cange from steep to mild slope or downstream of sluice. HORIZONTAL H > c H Approac to free overfall. (limiting mild c slope; ) H3 c > H3 Hydraulic jump downstream; cange from steep to orizontal or downstream of sluice. A > c A ADERSE (upslope) c A3 A3 c > 5

6 5/6/18 Qualitative Examples of Open- Cannel-Flow Beaviour Control Points Definition: locations at wic tere is a known relationsip between dept and flow rate (stage-discarge relationsip) Examples: Critical flow points: weir, venturi, free overfall,... Sluices Entry/exit from reservoir Hydraulic jump A control point often yields a boundary condition from wic to start a GF calculation General Principles Supercritical controlled by upstream conditions. Subcritical controlled by downstream conditions. Given a long-enoug fetc te flow will try to revert to flow. A ydraulic jump occurs between regions of supercritical and subcritical gradually-varied flow at te point were te jump condition for te sequent depts is correct. Were te slope is mild (i.e. te flow is subcritical), and any downstream control is far away, a ydraulic jump can be assumed to jump directly to te dept. 6

7 5/6/18 Qualitative Examples: Weir (Mild Slope) M1 1 c WEIR M3 ydraulic jump Qualitative Examples: Sluice Mild slope M1 1 M3 ydraulic jump Steep slope S1 1 S3 Qualitative Examples: Flow om Reservoir Mild slope RESEROIR Steep slope RESEROIR c S 7

8 5/6/18 Qualitative Examples M1 Flow into reservoir (mild slope) RESEROIR ee overfall (mild slope) M c critical Numerical Solution of te GF Equation Te GF Equation Tree forms: Total ead: Specific energy: Dept: dh d x S S 1 f 8

9 5/6/18 Solving Te GF Equation 1 Impossible to solve analytically (in most circumstances) Find depts 1,, 3, at discrete points x 1, x, x 3, approximated by Δ Δx were Δ i1 i Δx x i1 x i Starting Point and Direction 1 Start at a control point. Proceed: Forward in x if supercritical (upstream control); flow Backward in x if subcritical (downstream control). flow Types of Metod 1 1. Standard-step metods Solve for dept i at fixed distance intervals Δx x x x x. Direct-step metods Solve for distance x i at fixed eigt intervals Δ 1 S x x 1 x x 3 9

10 5/6/18 Standard-Step Metod: Total Head x x x x dh H zb g H i H i Δx S ( S 1 f, i f, i1 Adjust dept i+1 (iteratively) at eac step until LHS = RHS. ) Direct-Step Metod: Specific Energy x x 1 x x 3 S d x E g 1 S ΔE Δx ( S ) av ΔE E i E 1 i Direct-Step Metod: Dept x x 1 x x3 1 1 S Δx av Δ Write n S f R 4/3 g as functions of 1

11 5/6/18 Example A long, wide cannel as a slope of 1:747 wit a Manning s n of.15 m 1/3 s. It carries a discarge of.5 m 3 s 1 per metre widt, and tere is a free overfall at te downstream end. An undersot sluice is placed a certain distance upstream of te free overfall wic determines te nature of te flow between sluice and overfall. Te dept just downstream of te sluice is.5 m. (a) Determine te critical dept and dept. (b) Sketc, wit explanation, te two possible gradually-varied flows between sluice and overfall. (c) Calculate te particular distance between sluice and overfall wic determines te boundary between tese two flows. Use one step in te gradually-varied-flow equation. Direct-Step Metod: Reprise x x 1 x x S Δx av Δ 1 /3 R S n g 1/ n S f R 4/3 Q A A b s Example A sluice gate discarges water at 9 m 3 s 1 into a 6 m wide rectangular cannel laid on a slope of.4 wit n =.15 m 1/3 s. Te dept at te vena contracta is.15 m. (a) Find te and critical depts. (b) Compute te position of te ydraulic jump, assuming dept downstream. Use one step in te GF equation. 11

12 5/6/18 Example An undersot sluice is used to control te flow of water in a long wide cannel of slope.3 and Manning s rougness coefficient.1 m 1/3 s. Te flow rate in te cannel is m 3 s 1 per metre widt. (a) Calculate te dept and critical dept in te cannel and sow tat te cannel is ydrodynamically steep at tis flow rate. (b) Te dept of flow just downstream of te sluice is.4 m. Assuming no ead losses at te sluice calculate te dept just upstream of te sluice. (c) Sketc te dept profile along te cannel, indicating clearly any flow transitions brougt about by te sluice and indicating were water dept is increasing or decreasing. (d) Use steps in te gradually-varied flow equation to determine ow far upstream of te sluice a ydraulic jump will occur. 1