Part 2: Introduction to Open-Channel Flow SPRING 2005

Transcription

1 Part : Introduction to Open-Cannel Flow SPRING 005. Te Froude number. Total ead and specific energy 3. Hydraulic jump. Te Froude Number Te main caracteristics of flows in open cannels are tat: tere is a free surface, were te pressure is atmosperic; te flow is driven by gravity. Te most important examples are down-slope flows and waves. If te flow as caracteristic lengt scale L and velocity scale ten te ratio of inertial forces (mass acceleration to gravitational forces (mass g is of order acceleration ( / L g g gl Te square root of tis quantity is called te Froude number Fr. Since te caracteristic lengt scale involved in tese flows is typically (but not always te water dept, te Froude number is usually written Fr ( g A flow wit Fr < is called subcritical or tranquil. A flow wit Fr > is called supercritical or rapid. Since Fr determines te ratio of inertial to gravitational forces, one migt expect very different beaviour according as Fr < or Fr > and tis is indeed te case. A second interpretation comes from noting tat te speed of long waves in a cannel of dept is given by c g. Ten te Froude number can be regarded as te ratio flow velocity Fr ( wave speed If Fr > ten te flow is moving faster tan any disturbance can propagate upstream. Hence te upstream flow cannot be canged by downstream conditions. If, on te oter and, Fr < ten circumstances downstream (e.g. weirs can control te upstream flow. Te interpretation as a ratio of velocities (equation ( is analogous to te situation of sound waves in a compressible fluid, were te corresponding ratio is te Mac number: flow velocity Ma sound speed We don t usually pus te analogy in undergraduate civil-engineering courses, but tere is considerable similarity between te equations governing flow in open cannels and tose in ig-speed compressible flow. Introduction to Open-Cannel Flow David Apsley

2 . Total Head and Specific Energy If te cannel cross-section and slope are fairly uniform and te flow is slowly-varying (so tat any vertical acceleration << g ten te pressure is approximately ydrostatic: p g( z z surface and at any orizontal position (or stage te piezometric pressure is constant; i.e. p + gz or, in terms of ead: p + z g gz surface z surface datum z surface z bed piezometric ead constant along ere In oter words, since te pressure is ydrostatic, any cange in eigt is exactly offset by a cange in pressure ead. Hence: (i Te piezometric ead at any stage is independent of te particular streamline but depends only on te eigt of te free surface. (ii Te ydraulic grade line coincides wit te free surface. At any point in te flow te total ead (energy per unit weigt is p H + z + g g Hence H zsurface + (3 g Te eigt of te surface, z surface, is te sum of te local eigt of te bed, z bed, and te vertical dept of water,. Hence, H zbed + + (4 g Te quantity + (i.e. te ead relative to te local bed eigt is called te specific g energy. In tis first-year course we will analyse te ydraulic jump in a orizontal cannel (so tat z bed can be taken as zero and te specific energy is te same as te total ead. Hydraulics will look at uniform flow in sloping cannels and Hydraulics 3 at more general flow wit varying bed eigts. If tere are no energy losses ten H is constant. In practice tere are: (i bed friction losses, giving rise to te gradually-varying flow equation (Hydraulics 3; (ii large energy losses at ydraulic jumps. Introduction to Open-Cannel Flow David Apsley

3 3. Hydraulic Jump A ydraulic jump is an abrupt cange from a sallow ig-speed flow to a deep low-speed flow of lower energy. It occurs wen eiter te bed slope and canges to water dept are insufficient to compensate for te ig frictional losses associated wit rapid flow, or wen a eigt differential is imposed by upstream and downstream conditions (sluice gates, weirs,... Rapid flow may be created by, for example, a steep spillway or sluice gate. Te formation of a ydraulic jump at te base of a spillway may be desirable to remove surplus energy and reduce downstream erosion. Across a ydraulic jump: mass is conserved; te momentum principle is satisfied; mecanical energy is lost (mostly as eat. u u Assume (for simplicity: orizontal cannel bed; rectangular cross-section of uniform widt w; velocity uniform over cross-section; negligible bed friction over te lengt of te jump. Continuity Flow rate per unit widt is constant: u u (5 Momentum Consider a control volume encompassing te jump. Since streamlines are parallel, te pressure at stations and is ydrostatic and te average pressure is given by p av g. Bed friction may be neglected if te jump region is sort. Hence, te momentum principle (rate of cange of momentum force gives: ( uw u ( uw u pav, ( w pav, ( w or, dividing by te widt w, u u p p av, g av, g u u g( (6 Introduction to Open-Cannel Flow 3 David Apsley

4 Velocities Eliminate u using continuity ( u u / : u ( g( u ( g( ( + u g ( + (7 An exactly similar expression for u may be derived by using continuity or, more simply, by noting tat equations (5 and (6 are uncanged by intercanging and : u g ( + (8 Energy Cange Te total ead (energy per unit weigt cange in te transition is u u H ( + ( + g g sing te expressions above for u and u : H ( u u + g ( + ( + 4 wic, after some simple but tedious algebra (exercise, gives 3 ( H (9 4 It follows, tat: Since mecanical energy cannot increase in te transition one must ave H < 0 or < i.e. te jump must be from sallow to deep flow; Te energy loss rises sarply wit difference in depts. Froude Numbers Finally, we sow tat te upstream flow must be supercritical ( rapid and te downstream flow subcritical ( tranquil and tat te ratio of a downstream quantity ( or u to te corresponding upstream quantity is dependent only on te Froude number. Dividing equations (7 and (8 by g or g, respectively, u Fr ( + (0 g Introduction to Open-Cannel Flow 4 David Apsley

5 u Fr ( + ( g Since / >, it follows tat Fr > and Fr < ; i.e. te upstream flow is supercritical and te downstream flow subcritical. and are called sequent depts or conjugate depts. (0 can be rearranged as a quadratic for teir ratio, te jump, r : r + r Fr 0 wit solution (te negative root is impossible: r ( + + 8Fr ( By intercanging subscripts and we can also use: ( + 8Fr + r Hydraulic Jump Summary A ydraulic jump is a rapid transition from supercritical (Fr > to subcritical (Fr < flow wit consequent loss of mecanical energy. Te jump (r / is given by: ( + r + 8Fr or Fr r ( + r Te loss of ead is given by 3 ( r H H 4r and te power loss by gq( H H Introduction to Open-Cannel Flow 5 David Apsley

6 Example. Flow in a 50 mm wide cannel jumps from a dept of 50 mm to 50 mm. Wat are te upstream and downstream velocities, te volumetric flow rate and te rate of energy loss at te jump? Example (Exam, June 004 re-worded sligtly. Water is flowing along a narrow cannel of widt w 00 mm and undergoes a ydraulic jump. Te dept and speed of te flow before te jump are and u respectively, wilst te dept and speed after te jump are and u respectively. (a (b If te dept before and after te jump are 0 mm and 90 mm, wat is te net orizontal force due to te ydrostatic pressure acting on te control volume? Give an expression for te rate of cange of momentum for te fluid passing troug te jump in terms of u. Hence find te speeds u and u, and tus te volumetric flow rate. Introduction to Open-Cannel Flow 6 David Apsley