A Study of High Velocity Flow Around a Sharp Bend of a Rectangular Open Channel Citation Boothe, Raymond Hudson Ferris and Wagner, Warren O. (1937) A Study of High Velocity Flow Around a Sharp Bend of a Rectangular Open Channel. Master's thesis, California Institute of Technology. doi:10.7907/qg6j-8k23. https://resolver.caltech.edu/CaltechTHESIS:11112024-225842245 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Civil Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Civil Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 10 June 1937 Record Number: CaltechTHESIS:11112024-225842245 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11112024-225842245 DOI: 10.7907/qg6j-8k23 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16847 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 13 Nov 2024 18:40 Last Modified: 13 Nov 2024 18:55 Thesis Files PDF - Final Version See Usage Policy. 15MB Repository Staff Only: item control page

A Stu.oy of High Velocity Flow >.round a Sharp Bend of a Rectangular Open Channel .B y Ra,ymond Boothe and Warren Wa.gner In partial :fulfillment of the requirements for the degree of ?.last er of Science in Engineering at the California. Instit'Ute of Teehnology Report prepared 'by Raymond Boothe under the direction of June 10, 1937 I FORWARD ·unfortunately a thorough investigation of the problem of high velocity flow a.round sharp bends was n ot permitted du.a to lack of time. The time put in building the apparatus \11a.s mu.ch greater than that anticipated., with a result that only a pitifully short time wn.s available for making runs~ Actually only two runs were made. A thorough investigation would incl 'Ude runs on at lea.st three slopes (sa¥ 1.5%, 3•5% a..11d 10%) with at lea.st three ox· fOU1• different discharges each. Further, more angles should have been used, 37½ 0 , and 45°. S8tY 15°, 22½0 , 30°, 22½ 0 a.nd 45° a.re commonly used in canal design, and would be especially valuable in that regar.d. Investigation 1,hould also include velocity distribution in the cross-sections of the channel~ .A lso, it would have been interesting to plot con tours of the wat~ aurfa.ce as related to the channel bottom~ ill these investigations would. be neceesa.:ry as super- critical velocity flow is peculiar m1to itself. Thanks and appreciation is hereby extended to Dr. Ippen for his many helpful hints, suggestions, a.."ld influence, to Warren Wagner for his invaluable aid, to the u. s. Soil Goneervation group for their courtesy in uj,ending the use of their tools, to the Mechanical Engineering Division for the use of their tools, and to Le Van Griffis, without whose help this manuscript woul d n ot have been completed. Raymond 'Boothe ,Tune 10, 1937 2 Part Title Forward Table of Contents Statement of Pl",oblem Including a. diaouss1on of some brief theoretical considera.tiens, and some qualitative discussion on probable result .a . Discussion of Apparatus /o Including a. rather extended disc11ssion 0£ the &l)paratus and equipment used, some of the difficulties involved. in building the a.ppa.l'atus, and some of the limitation,s eneo,;mtered. 0 Discussion of Il'$sults 20 Including a very brief discussion of the results obtained, some plots of the cross-sections and pl''@file , and some of the inferences drawn. D :Si'tiliogra.phy Including a. 11st of some of the works oonsult ed with a brief commentary on each. 29 3 LIST OF li'IGUBES ~JD PHQrOS l!'igure Number Title 1 Diagram of Wave .Angle 2 Plan of a Typ ioal Channel 3 4 5 6 7 Profile of a Bu.ild~'P Resultant Pressure of a Buildup Velocity Distribution of Suboritical Flow Velocity Dhtribu.tion of Supercritical Flow Schematic DiagraJn of Circulation System Plan and. Profile of Fhlme stat iontl Calibrat ion Owte for 8 11 Venturi Met er Flume Entrance Streamline g 9 10 11 6 6 7 7 8 .3 // Detail of Joint 12 Development - l!'lume Entrance Section 13 Template of 30° Bend 14 Cross-}1eotions for B:un No. 1 15 16 17 Oro8s-Sections for Run No. 2 Profiles along Walls General View of Experimental Set-up 2S 18 Flume :Entrance 2-(p 19 View of First lla.xima 2to 20 Constant Head Tank 27 21 27 23 View of Bend View of Point Gauge Closeup of Pist'Ul"banoee ?rod:uced. 24 View of Wave Pattern 28 22 :Z c, 27 :28 A. SI' ATEMENT OF PR OBI.EM The problem under investigation concerns t he c onditions incident to water flowing at supercritical velocities encountering a sharp horizontal bend in an open channel of rectangular croH-ae.c t ion. It is first necessary ·~o define the critical velocity, and then place a limitat ion on the bend. Critical velocity, with which we are dealing , is defined by the condition where ditcharge takes place with a mini.mum amount of energy, for a finite depth. This condition is beet indicated by a wave of translation on the surf.ace of water at a. finite depth, the wave being relatively small. This crit ical velocity nJ83 be found by diff erentiating the specific energy of a. unit width of fluid with respect to its depth i, E The specific energy E is gi'van by eq. (l). (1) v is the velocity of the water , ru1d g is the 6-ravitational constant . Differentiation gives the result of the required oritica.l velocity, in eq •. (2). (2) V = Jgd This &ame relationship can also be derived on a b a 1:ds of momentum, but will not be done here for laok of space. Now we come to the limitation 011 the bend, assuming the water flowing a supercritical velocities. If there were a sheet of water flowing at a velocity above the critical, a.nd a. c ondition of free discharge (or disturbance) were set up, it is clear, tr.at t he resulting wave front would not travel in a direction opposite to the direction of flow, (assuming, of course, a relatively small wave of translation, and a stable condition of flow) but. would assume some angle a. with the direction of flow, a.s demonstrated. in the diagram below: S/,Yl__, (gel V Fig. l Di~ram of Wave Angle It is clear from a.n examination of the diagram, that the angle a. (between the direction of the flow of the sheet and wave front) is defined by and. where v is the velocity of the sheet. The :problem at hand is a. eonsid.e ra.tion of what happeit1l when water flowing at s1.1percritical velocity down a. rectangular channel enccranters a. bend a.t an angle greater than that defined by dn -I {id:. T as indicated below. 13 A C Plan of a Typical Channel Fig. 2 If~ were leas than a.. it is clear that the first infinitesimal parti.on of water flowing along .Ai would erea.te a wave front when it hit 13 which would travel back a.cross the channel in a condition of discharge such that there would be a coneid.erable decrease in the piling -up along the outside edge th.an would normally be expected. The object of making t8 > this lessening of the pile up. a. 18 to eliminate Thus we are able. more or leas, to get the .net effect of a. sharp bend., as th$ first infinitesimal out side portion hitting Bis not able to set up a wave front. It seems probable that the first outside element piling up, will. due to to a change in the wo11,entnm in ·thf) 1-<),tt.ei· resulting from the differential press'\>J.'e acting on it. It also seems evident that the water will aontirr~.e to pile up to a certain maximum and then decrease. Thh maximwn would be some distance downstr.;,run from th~ bend.. The preceding diaoussion is illus tr at ed f.rnmew.b :tt roughly 'by the diat;rama below: 13 C. 1 d t d,, J .,.,,. • 11g P:r essui·e of a lmildup 3 .Assuming in Fig. 2 that C is the point of maxim:-u.m height• and drawing a. profile along the outside wa,11, we would probably obtain a figure similar to Fig. 3. Fig. 4 illustrates the idea of a change in level producing a. change in direct ion. P and PO 1·ttpr(:s 0-nt resultrui.t pressures per unit nil.th at 0 42 . d. 2 and A respectively, being the products of p 2 and p 2 den.sity. wllere p is the The existence of a g:rea.t er p:i.•enure tends to force neighboring eleme:its to change direction with.out actually- coming in contact with the wall. }3eyond. this qualitative discussion~ the probl@ro of what goes on qu:,,.ntitatively is 8t>J:>arently qui.te oomplex. is anot her' difficult problem.. vrnere the trJP..Ximum. heig.'lt will occur Were the increases in water level very small in relation to the depth, M1a pro·blem would be very much simplified, but such is not the ca.ee. The height of water at C uan easily be two or three times the depth at A.. Various factors which might enter into consideration a.re (1) The :possibility that the vertical acceleration of the water have a marked effect 7 (2). The effect of turbulence. to changes in tempera.t-ure. (3). The effect of clkwges in viscosity due (4). Consideration of the negative front set up at Din Fig. 2. (5). The changes in longitudinal velocities which would probably occv.r. These changes are indicated when one substitutes different depths in the Ohezy :formula {V = o T m,a, c being a constant, m being the hydraulic depth or radi'us, s being the slope) and the S:peoif'ic Energy equation. (V = J2g ( E-d)) notati on a.s previously ueed) (6). The effect of friction .. (7) The increase in energ which oecurs because of drop in the channel. Reflection leads one to beli.e ve that numbers 2 and 3 could be eliminat ed. It is also probable tr.at number 5 could be eliminated., beca-u.se, as Dr·. Ippe.n indicated, the velocity tends to increase with increase of depth, referring to the Chezy formula, while the velocity, considering the ,Specific .Ehere;:r eq_ua.tion , tends to de,crease, the two effects probably canceling each other. It a.ho seems lik ely that n'Ulllbers 6 and 7 could be eliminated. The increa B& in energy d:ue to drop in the ch..9.llnel will partly (if not wholly) overcome the effects of friction and turbulence. This is dependent, of course, on the ~lo:pe, and whether the velocity is increasing a.s it goes down ·the channel. It ia seen, however, that even if these simplifying asaumptions a.re made, we still have a very complex p:ro'blem if we are to get the theor$tioal 1·elations that exist. One very helpful cha.ra.oterist ic of S'lq)ercrit.ical :flow is the velocity distribution. Fig. Water in su'bcritical flow has a distribution as indicated in 5. while that of S\.<--percritica.1 flow is shown 1n l!'ig. 6. Swbcri/J~/ Sw/e'/"cr1 ft ·cal - Ye,,llu::d V ve: !,nd'I n?rl;-c4/ V //elocrly -0;:S/d ,6,,,,/}a,,, 01',J/ri ,i,w//OYI I / :Hg. 5 Fig. 6 Hence, it is clear that the velocity may be treated as const aut throughout the en.tire c ores-section with.out appreciable error. simpltfying assumption. This is a very gre~t ,o On the preceding page& will be found a schema.tic diagram of the No attempt wa.s made to picture the exact circulation of water system. relationships, only the rel.at ive ones. Discussion will be taken up in order of the letters on the sheet. ·?he water was draw.n from a stora.ge re$ervoir (A.) of about 1400 cubic feet ca.pa.city D'J a. centrifugal )pump (:B) to a constant head tank (0) which was 5' high, and a.b01lt 61 in diameter. it works. The schematic diajraro indicates h ow When the central tank is full, it overflows into the ou.ter basin, which, as is seen, has a. pipe connected which leads back to the well. Al. condition of overflow was ma.1nta.ined during the runs. The water :from th£t constant head ta..,"'lk was taken through a. 10 1 pipe then through a 8" the fhune. J½• Venturi meter by whieh the inflow was Ill$a,Sured into The measuring device was a mercury manometer (D). calibrated in feet of mercury. It was 'll}iere were sevg:ra.l connections at 1;he throat and full cross-seotions, thus i neuring an oocvrate reading. The ma..11ometer was equipped with valve$ whidh allowed the line to be bled be:f'Ql'e ea.ch reading, further inst1ring accuracy. On the following sheet is e. calibration curve f or this pa.r t icv.lar ~1ent-uri mater. During the seccnd run, for some mllcnown reason, t.he d:i.scr...arge changed from a reading corresponding to 1. 265 cd. s. to l.235 c.f.s. This b a change of -2. >+%1 but, cons i d erin g the accm·a.c y with be which t he cross-secfions could/lread (after t he bend) the variation ns so small it could be, and was, neglected.. A picture: of the 'box containing t h e manometer is se-en below the constant ...helil,GLta.nk in Fig . 20. After leaving the Venturi meter, the -water went through the S" pipe into the bottom of the stilling basin~ (E). The trtilli.ng basin was neceseary 1n order that turbulent conditions would not be :present at the flume entram1:e. The di schaxge into the basin was aoaomplbhed by a long pipe which 1~ the 1/ length of the stilling basin, and was perforated throughout. At (F) is indicated a gate valve, which facilitated emptying the stilling basin. Considerable difficulty was had in constructing this basin, and consumed rnost of the time spent on the project. location and the materials used. The chief troo.ble was its It would have been easier to have ma.de the ba.ain walls out of concrete, but this was barred for va:rtw:a reasons. Essentially it was a. sheet metal a.1d1tion or topping on concrete. 'Water- proofing the juncture W,' -!l accomplished thro'Ug'h the use of tar p i-a.per, mastic, and pla..cit er. 41.'here were no leaks of meas'lll'able quantities in it, however, when it was finished. After run number 1, it wa.s noticed th~t the water surface dropped• probably due to leakage through the gate valve F. 2. 6 cm. in four minutes. The a.mo'Ullt of drop was Since the surfac.e of the basin was 2.82' x 14.20 1 or 400 s q . ft.• the leakage corresponded to a flow of about 0.142 second :feet• which value was su1$ract&d from the discharge obtained from the manometer re.a.din~. In the second l"'Un, there was 110 leaka.gs whatever. The stilling basin was very satisfactorr. As a check on opera tions in general, stage readings were t ~ in one corner of the st illing basin. 1.l.1he variation obtained. in the first run wag 1.2 millimeters, and l.8 millimeters in the second. This amount of variation would have a negligible effect on the quantity of water carried by the channel. 't he water enter ad the flume by a speoially designed stream-line entrance (G). 'fhis precaution we.s necessary in order to avoid turbi;ilent flow i n the chazmel, so that decent ree:ults could be obtained. Fig. 18. The basis of the entrance design wae a two dimensional st;raaml:tne as plotted frorG a l)oint. som:ce. page. This entrance is shmm in The actual cu:rve ia $hovm. on the following It was necessary to dN.1w 1111 asymptote to the c-urve a.s 1:1hown, because the streamline would not ha~e become parallel to the walls of the straight ,; z 0 0 0: . . "'f/1 w 14 section of' the channel (assuming t;he channel walls parallel to the x axis) mtil an infinite distance out. In the making of the entrance, the sheet metal was worked until it was parallel to the x a.xit. It was felt necessary to extend. the end of' the sheet metal beyond the ~heoretical end opposite ,. point O on the x axis. Thia portion of the curve was put in by eye. Since the curve merely represents the cross-section as obta ined from a plan or side view, it was necessary to make a. development of the two dimensiona l stream. line ta fit three dimensions. A development was ma.de coI"res:p ond ing to the arr is as laid out on a plane. This development is shown in l! igure 12. From t his development was made a tetq)late which in 1 turn wa s 1., sed to cut out the sheet metal shapes. These sheet metal plat es were curved according to the two dimensional stream line, and soldered tog et,h er to form the entrance. }~langes were added of course. It vms noted with satisfacti.on that the entrance worked perfectly. The flume was a sheet metal rectangular channel l' x 11" in cross-section, supported at frequent intervals (about every 4 1 Joints between sections as sh ow in Figure 11. 5') throughout its length. Gaskets were made of tar paper, which were covered with axle grease. Bolts were used in joining the sections. The f'l'Ul'lle was set approximately 1n place, and then set accurately with the use O O O Fi g. 11 Detail of Joint end of the bend.:. sect ion. 0 of a precise level. The elevations were determined 'by taking 3½ %of t he length of the ch~nel, relative to the beginning and The error in t his procedure would amo,mt to only 3.5% x versine (ain- 1 .035) x length or 0.000577• in the entire length which. wa.3 closer than the flume could be set anyway. A.t ea ch leveling section or Developernent - r/ume y y y / / / / / / V / V / / / V / / / / V / V V ~ Entrance V V / V ...........V -- 5ect1on i..-- ~ ~ i.-- ~ .--- station , shims were p la.oed under the SU:pports to make the flume level •. :Both sides were l eveled. The accuracy of these elevations was n ot more th&n 0 , 005 1 off , and. the l evel of ea.oh side to within 0.002 ' of the other~ Good pictures of the flume are sho'11ll in Fi{~$• 17, 22 , 23, and 24. A di,agram showing the plan and profile of the £1ume is shown in Jig. S on p age .!J ., t (B) is shom the bend. sho·wn in Fi g. 13. A drawing of the temph>.te of this angl e is It will be noticed that it is $0 m:td.e (ref 01-r in.g to the fol d lines on the Vertical ,Rrall) that the f loor of the bend will be levE;l I..,, when in place. Thia is neoess ,try to avoid a1..ewing the chrumel near the bend , ;;;.s a little re!lection wil l snow. lf the channel with the bend were l a.id out level, and then the plane on which it wer e lai d tilted to a 3}% slope , ?ither t h e .1.patream section of the channel, or the downstream s ection, or both s ect ions would not ha:ve a horizontal cer :osa-section, which is untemable i1 t his e:;tperiment . Skewi ng should have to be resorted to in order to f;et t hem level, which is undesirable. I t was thot1ght beat to make t :he floor of the 1lend level in or der that the cban,tel e m.J .d have '!l. horizona.tal cross-section throughout its length , even i:f it did add. some complicati ons to the conditions giving rise to the standing wav es. as small as possible. For this reas on , the ~ e fl:)or wa"' ?'.Mlf.ie A 30° angle was used. Phe water was d i s charged a.bout 9.9 • downstream from the bend ont o a concrete s1.1rface. The water was kept in ita p l a ce by a s mall dam shovm. at (I). The co'Ul'se which th i s water took in g,ing to its u ltimate des tination o:f the gate valve at (J ) was extr emely interesti. g in its elf , becau$e of the WSJ/ it shift .ea. about for no app~ent reason. t he sma.11 scope of the eXperime.nt. However , such consid~ration. 11as O\l.tside From the valve at (J). the water returned. to t h e well, tlm.s compl eting t:i1e cir(luit of water .. Outside 01' t.he one leak menti on ed in con.necrcion with the stilling basin , there wer e no leaks of sufficient m':il.gnitude to ev~m consider • and hence they 17 - - -, \ I I I I I I I I I I \ I \ \ I I \ I \ \ I \ \ I \ I \ I I I \ I \ I I I \ I \ \ I \ \ I \ I I I \ I I I \ - \ \ --"- J--_______ - J_ I/ . ,- -I I I I \ I \ I I \ I I I \ I \ I \ I I I I I I I ~ I ~ I I \ I ~ I I I ~ I \ I I I I I I I 5ymmetr1ca/ nmpla,te of 30 ° Be/?d 18 were neglect ed. The whole outfit hold water very well considering the d.ifficultiar. in ita construction.. In order to make i:ooasw·em{mts of the cross- section tmd. :profHes o! the wate~ surface~ stations were est!'1.blished different f.rom those used to level the flume. The !Gheroatic diagram of the oe stations is shewn in Fig. 7. It was t hought neoe.ssary to establish stations "'bcve the bend, in order to indieaiie the condition of the \-;ater entering the bend in rege.rd to cross-sec·tion , and also whether it he.d rea ched a stable veloci y i.s indicated by the area. o:f oross- seetion. lfhe. sta.ti 0ns at and below the bend. were put a.t such intervals as thought necess1.::l'y to giv e a. fairly complete $tud.y of the problem. Belew about 4, o:i: 5' from the <;h .nn~l , bend., moi1t of the phcnomm.~ of impor tance had occurrecl, n.nd. a o stations ·were placed more !lpa.ringl y from there on. he layout of' these stations was referred to the intersections of the 1 •~ cent er lines o:f the upst:rearu ~.nd domistream aections of the channel. u.--pstrea.r. stations were mea.S1ll'ed in :feet -·:tnd marked minus. were rnu1·kcd plus. mentioned. T'ne Those doi'1l'.lstream ~he elevations were made ·r ela.t.ive to t ne intersection just Ati e ach st.at i on ·~as established a grid iron.. Thes e :places were, m."lde at every inch across the cl:ia.nn~l . and. were me.rked to facilitate making the cross- zection ~~am.1r ements . It is iutereuting t o note that theae marks were ::if no part icular go0d i n the second run because they couldn ' t be s een. A ruler a.c:ross the to:p ,1<?,d to be '.!:'~sorted to. 'l1he actual measurements of tb.e c1·oss-section were made by means of 0. point ga.1.1{!:e v:hich is shown i n :i:"igu.r e 22. It wa.s simply a pointer which moved up and down relative to a vernier on the c1·osa piece. wan attH.ched to he pointer. whole ga"U{:e h.::l.d t o be 1.iovecl.. plotting theme A s.cnle in centir.:1_eters In. o:r.-der to move the poi.nt e:c: horizontally, the ~C1H} ree.dings o·bt ained were changed to feet before Although the pointer coo.ld. be rea d to the nearest 1/100 of a centimeter by virtu e of the vernier , a ct1.:1.al measurements were made only to the nearest 1/10 of a centimeter , or one millimeter. :Below the bend. , this a ccura cy wa.s hard.ly justified, be cause of t h e difficulty in telling j ust exe.etly where the su.rfa.oe of the water was, a.s there were mild fluct'li.a.ti ons , usuall,y less than½ cent i meter. However, an honest att en::rpt was made to get accurate readings. l)ue to the fa.ct t hat the fl anges of the channel on which t he crossp iece of the gauge rested were not always level, although t he 'bottom ,vas , or even a constant heigp.t above th e bottom, it was necessary to ta..'lce a reading on ea.ch point of the bottom for any giv en cros s-secti on., and the read i ng thus made would have t h e reading to the surface subtracted from it in order to get the depth. (.Aot11a.lly the reverse . as regards subtra cti on was nec es sa,1·y , because ·t he reading increased as the pointer wa s rai s ed). I:n reading tbe point gauge . t he attempt was made to get the distance p erpendiC'lUa"I:' to the bottom ot the channel. Considerat ion· of the versine (sin-1 . 035) shows t hat the •rror is not worth troubling over, being about . 0006136 times the length to the point. ln making measuremen ts, it was found to be &~1Jed ient to make mea.$'t.1l'ementa at oth er than t he planned point s on llhe cros s-section, as other changes in the wa.te:r s'l1.r:face than were anticipated oocu:rred,. It was thought that a. va.lu$ o:f n of 0. 007 could be used ag the coefficl.ent oi roughness in calculations of the channel in using the Kut ·t er eq_Wlt i on of flowo i.he maximum flow which wa:s a.vaih.ble to the flume was about 0. 5 second. feet, although, at t he base of the tower , about 2.5 seccmd feet could have been obtained. This difference is explained ma.inly by fr iction in the :p ipes and outlet in the st illing basin. 20 The followin g dia,cussion is concerned with t he :pl>..,ysic:a l appea:r-anc e of the dei'orme.tion of surfaoe , as there is no :t"..athematical t heory at the hand of the writer by which t he reeu.lte can be chec:k et\ 4lthough roi attempt was made to find a mathema.t iea.1 rela.t i onsh.ip , no succ~s1i'l wa s had.. The reason for t h e rather long channel be.for e the sh~p bend was t o give the water sufficient time to reach a c ondition of cons,t ant velocity before striking the bend . An eJ1;amina.tion of t h e cross-sections of rt1ll nun:iber 2 giv e the following d.at.a: -17.00 -9.00 -1. 00 -0. 20 +o.30 +1.20 45.00 +9o00 .. 355 .. 210 . 192 . 192 . 21+4 .206 .201 .196 Av. Velocity ft/ $-eO. 3.48 5.ss 6.43 6.43 5. 05 6.oo 6.15 6.30 Station .Area in Ftw -11 It is apparent that , by t h e time the bend was reached, the water had attained l"- more or les s consta nt velocity , which was what was s ought. It is n ext to impossible . i n the lig.,'ltt of present knowledge. ta cor:rela t. a this r ath.er sudd en velocity change (after t he bend ) with turbulence or v~rtiaa.l Col!iPonent •. The ma,jor portion m...qy be ascribed. it seems, to t he mo,ementu.in change. It i s noticed in the cross-section at st ati on 0.00 (]'igure 15) 011 t h e outside, a r a ising of the wat er surface . which had tommenoed a.bou.t 0.12 1 abav'e the bend p:roper . It is rather hard to explain this , because there oug..rit not to be any b'd.ld. i.."tp before t he bend. 'hydraulic j ump. It seems likely that U is a case of As was noted. in the description of apparatus , it was necessary to ha:ve the bottom of the bend. level, in order to have the ch a.n.".lel make l>- bend without being skewed.. It s e ems probabl e that the sudden ch ange of 3½% slope to level vm.s sufficiently lar ge to cause this jump , and especially at the edge , where ther e are tw·o :ret arding surfaces of friction, 'With res,alt ing lower velocities near t o t hem. :2.( It wa s als o noticed that there wa.s a quite rapid dying out of the disturbances set up, downst:-eeam from the bend. for this. The vl.liiter has no e7.jpla.nation In the e:icperiments of Dr. !ppen on curved channels, these \'HlVea were of rel at ively large size even qu.ite a weyra downstream.. Altho the pattern of wave fronts is clearly dhcernible in the :pictures, their magnitude is not very large, as can be seen by consulting the cross-sect ion dra:wings. _4,nother interesting thing is a small ba.ekwater wa:.re which v.tas at station It was a'bou.t one half the width, and e.x;tended. fr.om the inside edge. +1. 200 It was perpendicular to the direction of :flow. No absolute explanation can be g iven t but it again seems to be a ma.tiar of hydraulie j'Ul'l'lp. It will be· not iced that at station 0.90, that the sur:fa.c e of the water on the inside edgelis vecy low. It may be tha.t friction on the bottom l1a.s sufticiently gree.t to slow the v elocity down to th e critic al velocity at which it would j't.:l.nq)$ The critical velocity at which jump occurs is not the same as that for "i: he velocity of a wave. (4) lt is giv en by equation (4) , Ve: Vgq where de is the critical depth, vc is the cri't ica.l velocity, and ci is the discharge • .Another interesting comparison is to plot the profiles along the wel];~ as is done in Figure 16. mind.: A remark will be ma.de which it is well to bear in fast ead of being as sho·mi , the out side edge (LEFl' WALL) is ?.ctua.11.y o. 6• long er than the inside wall. These profiles are p l ott~d in reference ·t; o the stations as laid down along the centerline of the channeL For convenience in plotting, the profil es vrere l aid out a ccording to the stations. At any :rate . 0. 6• o:f a foot is hardly noticeable on the hor izontal scale selected. ~ e ,,ertical exaggeration is 12.5 times the horizontal . Leading t..l]? to the be nd , is noticed the characteristic "down drop" curve which is encountered in open channels on a down slope. It seems app~..rent 22. i n both :runa 1 t:hat the increase i n friction due to increased. velocity resnltin g :from the drop in the chrumel . was matched by the :rate of i n crease of specific energ;r ~ result ing in a stable condition • .l'he theoretical wave angle of :run number l shoul d be I3 1 '1'111e 'W[!)}le angle for r un number 2 should be 5/M - t S (;, 7 (S_ ( / f .3 -f3 o .r 22" . So 1 Referring to F i gur e 24 , it i s s een that t he wav e angle i s equal or less than 30° .. There 1s a:n obvio,.1.s i n con!'! i st ency here , as the I.>hot ograph was· n1a.d.a of nm number l.. Consider ing the large l osse~ connect ed with t he stilling basin in r un number 1, it is probably best to l•efra.in from ma.'\-cing any definite conclus ions from ru11 num'ber 1. Unf ortunately . no angl e rneam:rre:uents '#ere taken on run number 2. lt is rather i nter e s t i ng, howev er, .,o note tl1e JAery olos,e corr es:pondence between t he profile on the hrn ru s on both the m:i.t slde und inside l'l'all s , Another interesting point is the r a t her r apid decrease in t he 111'3.t:.:,~itu.d.e of the sta;1di.ng wave a s it progresses down the chann el. It is rather u.nf ortuna;te that the channel was :not longer on the downstream end . so that the waYe pattern could. be f u.rth<".-r s t ·udied. 'l1he pho t ograp-hs ·t ell their own story , i,and will not be further discussed her e .