Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 7-Day Trial for You or Your Team.

Learn More →

Interface shape and superelevation in curved stratified flows

Interface shape and superelevation in curved stratified flows INTERFACE SHAPE AND SUPERELEVATION IN CURVED STRATIFIED FLOWS BY L. N. CHIKWENDU * shape of a free vortex, whereas equation (2) gives Introduction a linear profile. Also, the observed amount of trans­ verse superelevation was fOlmd by Shukry to be The existence of transverse pressure 0Tadien t is better approximated by a modified free vortex for­ . 1 t> an 111 lerent characteristic of ail rotational and irro- mula. tational curved flo"\vs. This pressure gradient is For a sufIiciently long bend, the transverse surface made manifest by an inward free surface inclina­ slope and the associated secondary circulations tion for the case of curved open channel flows in­ bring about a vertical and radial redistribution of v.olvin~ homogeneous fluids. From Euler's equa­ stream velo city by mechanism of momentum trans­ tIon, glven by: port. These changes in velocity pattern give rise to (oP/or) = pV2/ r 0) the graduaI shi ft of high velocity filament l'rom the inner wall to the outer wall, with the ultimate sup­ where oP/or is the radial pressure gradient and pression of free vortex and a development of forced V is the local velocity at a radius l', it is possible vortex pattern. In more recent work, Ippen and to carry out a one-dimensional analysis for the free Drinker (962) have shown that provided the mean surface superelevation by assuming that a cons­ radius to width ratio of a given channel bend is tant mean velocity V acis at the centre-line radius greater than unity, the magnitude of transverse rI' of the bend. Integration of equation 0) over superelevation in sub-critical homogeneous fluid the channcl width yields: flow is fairly approximated by the one-dimensional ho - hi = t:..h = (V 2/g ) (b/rJ (2) theOl'y of equation (2) irrespective of the l'orIn of latera,l distribution of axial velocity component. where ho and hi are the water levels at the outer The shape of the transverse free surface profile is, and inner walls respectively, and b is the width however, dépendent on the radial distribution of of the channel. axial velo city com ponen t. The experiments carried out by a number of As no work seems to have been reported on the early workers have, however, estàblished that at transverse interface shape and superelevation in a short distance l'rom the beginning of a bend both curved stratified flows, it is intended he rein to pre­ the velocity and pressure distributions in a sub­ sent the results of a series of laboratory tests carried critical homogeneous fluid flow are weil described out while the writer was on leave at the College of by a free vortex. Shukry (950), for instance, has Science & Technology, University of Manchester. compared the one-dimensional theOl'y given in In these tests, two forms of stratified flow were equation (2) with results obtained experimentally considered, (a) the « underflow » wherein a layer of and noted that the aciua1 transverse free surface fluid flowed beneath a fairly stationary layer of profile was nearly represented by the hyperbolic slightly less dense fluid; and (b) the « overflow-with­ underflow » in which the upper fluid and the lower Lecturer in Civil Engineering, University of Nigeria Nsukka (Nigeria). denser fluid layers both flowed in turbulent motion Article published by SHF and available at http://www.shf-lhb.org or http://dx.doi.org/10.1051/lhb/1966043 L. N. CHIKWENDU in the same general direetion. The density difTe­ gular channel section were 10 ins. and H 3/4 ins. respedively. The downstream end of the curved rences between the upper and the lower fluid layers covered in these tests are within the range found in unit had a 40-inch straight exit length which termi­ thermally stratified rivers. nated in a small outlet tank. The tank incorporated a tilting weil' for adjusting water level, and a ver­ tical recess for inserting a sluice gate used in controlling the pool of the upper layer during the Notation « underflow »tests. Schematic diagrams of the arrangements for flow control are shown in figs 1 The following symbols have been adopted for use and 2. Because of the transparency of the perspex in this paper: mate rial used for the bottom and walls of the entire channel, visual observations were made possible. b width of curved ehannel; The channel rested on supports which were levelled §i ~ : densimetric Froude number to keep the bed horizontal throughout the tests. = V /Y[.6.p/p) gh] ; The outlet end of a thermosiatic mixing valve H : total depth of the upper and lower layers in wasconnected to a rubber hose which delivered hot an underflow; water into the flume through a manifold-pipe distri­ h : depth of the flowing layer(s); butor. The hot water distributor spanned the width hi' ho : interfaee height at the inner and outer walls of the stilling basin and rested on the floating board respectively; in the basin j ust above the cold "mter level. Dis­ charge of hot and cold water were recorded by pre­ .6.h : total transverse superelevation (ho - hJ ; viously calibrated elbow meter and orifice plate res­ Pl' P : mean pressures experienced by the upper pectively. Precise and rapid measurements of tem­ and lower layer at a given radius respecti­ perature readings ,vere achieved by the use of a vely; commercial direct-reading multipoint thermistor P 1 : mean pressure in the lower laver due to its instrument which incorporated twelve sensing pro­ own depth at a given radius; v bes and had a scale graduated in degree Fahrenheit. ÀP 1 : pressure difTerence in the lower layer be­ Each miniature semi-conductor sensing element of tween the outer and the inner walls; the instrument was mounted at the last one-quarter Q : total diseharge of the flowing layer(s); inch length of stainless tubing, 1 ft. long and 0.0625 q,. : percent age ratio of the diseharge of the inch outside diameter. From the temperature read­ upper layer to that of the 100ver layer in an ings it was possible to determine the den si ties and overflow-with-underf1ow; locate interfaces. Velocitv measurements were not dl : Heynolds number = Voh/v; made, as the local velo~ities were generally too l' : radial distance to a point in eune; small to he recorded with the availahle instrument. The free surface levels were measured hy means of 1'0 : centreline radius of bend; micrometer point gauges. V : local axial velocity at a bend radius r; For the unclerflow tests, a sluice gate was inserted T • \ mean axial velocity of the Howing layer(s); o . in position to give an opening of 1 1/2 inch at the mean velocities of the upper and lower VI' V : end of the channel while the tilting weil' was raised layers respectively; so as to l'etain a pool of hot water in the entire Z : vertical distanee measured from the ehan­ channel (see Fig. 1). The desired amount of cold nel bed; watcr discharge was then slowly turned on until a p : mass density of lower fluid layer; quasi-steady state was attainecl ,vhereby the eolcl Àp : density difTerence between the lower and layer flowed underneath the layer of warm "mter, upper 'f1uid layers; through the sluice gate and over the weil' before clis­ charging to waste. The total depth of hot and cohl v : kinematic viscosity of Huid; water was kept eonstant at 6.5 ins. throughout the À : ratio of channel width to centreline radius tests. of bend = b/1'o' In the case of overflow-with-underflow tests, (see Fig. 2) the general procedure adopted in super­ posing the two streams consisted of first turning on Experimental equipment and method the required amount of cold ,vatel', allowing it to stabilise before turning on the hot water. A fair Hot water and eold water were used to simulate degree of initial mixing was eliminated as the hot the upper and lower layers respectively. water jets from the distributor fell on the floating The experimental channel was made up of a still­ wooden board within the stilling basin. The two ing basin which was Hared on the bottom and sides streams then flowecl along the channel after being to give smooth entry to the Hume. The basin was controlled in the straight approach length by a se­ fed through a 2-inch pipe which discharged cohl ries of parallel blades. The tilting weil' was used to water vertically dOWl1'Ward onto the bed at the cen­ achieve variations in total depth of flow. tre of the end wall. Surface waves within the basin In aH the tests, the temperature distribution and were suppressed by a floating ,vooden board plaeed the interface form were determined by vertical tem­ on the ,vatel' surfaee. perature traverses taken at the central seetion of the The stilling basin was joined onto a 1:35" eurved hend; except for a few tests in which further mea­ unit by a 40-inch straight approach length. The surernents were taken along the bend. The tempe­ centreline radius of curvature of the curved unit rature readings at a given section were recorded was 60 indles. The width and height of the rectan~ ",vith five probes spaced radially at 21/2 inches 692 LA HOUILLE BLANCHE/N° 6-1966 apart, and held by horizontal rigid perspex bar. The perspex bar ,vas in turn carried by a bent iron Stationary layer Couche stotionnaire rod which was fixed to a ll1icrometer screw gauge, x..-,! Sluice gate- Vanneplate thereby allowing simultaneous lowering or raising ", L '-,~~.;, :(P~-)-;zj.J ~rT~-.~:'":l~LPl~ of aIl five probes to any desired depth. Surface and ~ h .... ' P '-" \ ,~ 0---'----r--p?~~~\ bed tell1perature readings were taken al 0,25 ins. -'l: \ 1 i from the free surface and bed respectively. Inter­ x..- Tilting weir y .. Déversoir mobile mediate readings were taken at 0.5 ins. vertical intervals except near the interfacial zone where this Interface-Interface "~",,,v:L ' .. ". 1( C pl was reduced to 0.25 inch intervals. h,' , 6h The range of various variables covered in the lnn~~wLa-II-f--:--' lnne:wall ~ poute; wall tests is sumll1arised in Table 1. Paroi int. Paroi tnt. Paroi ext. Section YY Section XX Coupe " Coupe ""8 Interface shape 21 Schematic diagram ';i 1/ Schematic diagram of cunee! stratifiee! "under­ cUl'ved stratified "over­ flow". flow-with-underflow" . Undcrflow: Visual observation of the longitudi­ Schéma d'lin «solls-écoll­ Schéma d'lin écolllement nal and transverse interface profiles were made by lement» cOllrbe stratifié. mi:rie : « sllr-écolllement colouring the upper layer with a suitable dye. The avec SOliS - écoll/enzenl », longitudinal interface profile was fairly uniform cOllrbe et stratifié. over the bend but becall1e slightly depressed in the straight outlet length. Near the sluice end, where the cold water layer was eOll1pelled to How through Table 1 the l 112 inch opening, the longitudinal interface slope becall1e relatively steep showing resell1blance Range of tests to the "free overfall" profile in free surface How of a homogeneous fluid. A photograph of the longitu­ OVEHFLOW- dinal profile looking from the outer wall of the un­ VAHIABLE UNDEHFL<HV WITH- UNDEHFLOW derflow can be seen in Figure :3. Across the channel width, visual observations alone showed a deeper cold water region on the depth (ins) H = 6.5 Ih = 2.3 to 6.3 Overall .... outer ,:vall of the bend than on the inner wall, thus Total reflecting on the transverse interface superelevation. diseharge Q (in / see) 17.03 ta 52.9 3'2.6 ta 72.0 The experimental shapes of the interfaces at mid­ Diseharge ratio ...... - 40 0/0 beud for various values of densimetric Froude num­ gr and 80 0/0 ber (@ L! ) are shown in Figure 4. The interface posi­ Density differenee tions were obtained by plotting each temperature 8 8 0.49.10- ratio ....... 0.45.10- profile on an enlarged scale, conslrucling the tan­ C~ç/ç) 2 2 ta 0.469.10- ta 0.434.10- gent at the point of inflexion and hence determining (ins) - the interface height. Sluiee opening .... 1.5 As can be observed from Figure 4, there is a close Densimetrie similarity behveen aIl the transverse interface pro­ Fraude Number 0.38 ta 0.725 0.338 ta 2.42 (@L!) files obtained, except for slight differences which Reynolds Number .. (Ol) 1.1.10;' 3.84.10 arose from the varying wall efIects. Because a des­ 8 8 i and 4.65.10 ta 3.34.10 cription of the patterns of seconda l'y currents in curved stratified flows will be the subject of a forth­ coming paper, it will be rell1arked in passing here that aIthough these secondary currents are prima­ rily responsible for the interfacialmixing of the two layers, yet they etrected mixing only to a small extent and so permitted the interfaces to be located with SOlne degree of accuraey. Entrain ment of the upper layer was very small. Ovcrflow-with-Undcrllow : At a bend angle smaIl­ el' than reiS the interface height near the inner wall was found to increase downstreamwards with the opposite behaviour on the outer wall. The visual transverse superelevations in this region were also found to be large in comparison to that over the l'est of the channel bene!. Beyond reiS, however, the superelevalion appeared to be constant along the bend. These observations were coniirmed by actual measurell1ents of interface supereleva­ 3/135 test flume showing longitudinal interface profile in an "underllow". lions at bend angles of reiS, :3 reiS and 5 reiS respec­ Canal d'essai cOlldé à 135' mettant en évidence le profil tively, l'rom which the supcrelevation at 3 reiS and de l'interface longitudinale dans lin «solls-écoulement ». 5 re/8 ditrered only slightly whereas that at reiS was found to be ahout 50 % laI'ger than any of the 693 L. N. CHIKWENDU former. In the circumstanccs, it would appear that overflow-with-underflow with turbulent motion in the bend entry conditions influenced the interface the two layers, then AP 1 will be negative and there shape and superelevation at bend angles sm aller will be larger depth of the lower layer on the inner than 7t/8. This is to be expected since the flow wall than on the outer wall. 'Vhen VI -7 0, as could be thought to be then in its adjusting process. obtains in stratified underflow or in frce surface In Figure 5 are shown S(llUe of the experimental flow of a homogeneous fluid, then AP ] becomes transverse profiles of the interface for various positive and there will be larger depth of the lower values of densimetric Froude numbers, total depths layer on thc outer wall than on the inner wall. and discharge ratios. 'Vhile there seems to be a Thus in curved stratified flovvs with slight density close similarity between the shapes of these inter­ dift'erences and in which both the upper and lower faces, it will be observed that some difference exists fluid laycrs are in turbulent motion, the interfacial when comparison is made with those found for the shape is such that the largest depth of the lower underflows. In particular, the direction of the inter­ layer will be f0lll1d on the inner wall and the largest face curvature has heen reversed in the ove rflow­ depth of the upper layer on the outer wall. with underflow, the deepest cold vvater layer being now on the inner wall. At small depths and large densimetric Froude Transverse superelevation number of the order of 2.5 or higher, the locus of the interfaee intersected the channel bed near the outer wall, with the cold water layer virtually Under/law: One characteristic of stratified flows exposed at the free surface on the inner wall. This which has emerged from the present studies is the behaviour of the interface in an overflow-with­ magnification of the interface transverse superele­ underflow tended to set the upper limit of the densi­ vation as compared to the corresponding free sur­ metric Froude number which could be investigated face superelevation in a homogeneous fluid flow. in smaH scale lahoratory experiments of this nature. From one-dimension al analysis for a homogeneous The rather peculiar interface shape of the over­ flow-with-underflow, in which the deepest cold water layer is found on the inner wall rather than Interface on the outer wall, is weIl in accord with a simple 3·5 Upper layer Interface Couche sup. theoretical argument which can be formulated as follows: (ins) F =0·380 Lower laye! Couche ln! H =6·5" Consider hvo co-current stratified layers flowing 3·0 F = 0890 within a curved channel of mean radius rc, and h = 4 3" 4,25, . (ins) assume that the lighter upper layer of density 3·0 CP - Ap) and the lower layer of density p move with z 1 (ins) mean velocities of VI and V 2 respectively. If the , F =0415 mean pressure experienced by the upper layer is Pl '''~ and that of the lower layer is P , then the mean 2 2·5 2·75 pressure P 1 of the lower layer due entirely to its z ' own depth will be given by: 4·5l'~ .. ('Ins), 2·5 F = ',360 , F6=O'450 Z (ins) <O· , ,·ôO· h =3 ·3" where Pl' P and P 1 aIl vary with l' only. 2 2 2·0 4'75~' i - From Euler's equation, and by carrying one­ Z ' , ·75 ­ dimensional analysis for the upper and lower (ins) , i.. F6= 0'510', 2·75 .. layers, the following are obtained: 1 H = 6·5" 4·25, , . 2·5 (4) and: (5) F = 1·95 So that: 2·0 .. 6 h =3· 3" (6) (ins) - For the order of density differences under conside­ '·5 .. 5'0~!",."", Z . F6=0.600,' H =6· 5" ration, (1 - Ap/p) = 1.0. Equation (6) is therefore (ins) .. re.duced to the form: 4·5 [~ner woll g Outer wall' '·0 .. (7) PalOi in! Paroi ex! Inner wall Outer wall Paroi inl Paroi ex! On integrating equation (7) across the channel 4/ Interface transverse pro­ 5/ Interface transverse pro­ yields: files in an "underllow". files in an "overllow­ (8) with-undertlow" . Profils en tra/Jers de l'in­ Profils en tra/Jers de where b is the width of the cbannel. terface dans un «sous­ l'interface dans un écou­ In equation (8) AP 1 is the positive pressure diffe­ écoulement ». lement mixte: «sur-écou­ lement a/Jec sous-écoule­ rence in the lower layer between the outer and the ment ». inner wall. lt is therefore clear from equation (8) that if V2 < Vl' as is generally the case in stratified 694 LA HOUILLE BLANCHE/N° 6-1966 o. Ollr---,-------,---,------,---~--~-_ 1·0 1·0 0·8 0·8 0·6 0·6 0.071-------i---+---+--+-----1---+---......I-:....-I 0·4 0·4 0·06 0·2 0·2 • Experiment Valeur expérimentale (6h/h) (6h/h) (6h/h) - Equation (9) équation ( 9 J 0·04 @:L'h 0·03 1nner wall Outer wall Paroi int. Paroi ext. • 4·65 0·02 t---,---- o 384 0·01 L,-,J------,--'--"--'--'---__'---...J C3 DA 0·6 DB 1·0 2·0 3·0 Ft> 0·01 1/ Correlation of interface 8/ Correlation of interface superelevation in an superelevation in an "overflow - with - und er­ "overflow - with - und er­ flow" for '1,. = 40 'Jo. flow" for 'l,. = 80 'Jo. o 0·1 0·2 0·3 0·4 0·5 0·6 0·7 Corrélation en t rel e s Corrélation en t rel e s Fil szzrélévations de l'interface surélévations de l'interface dans zzn écoulement mi,c/e dans un ecoulement mixte 6/ Correlation of interface superelevation in an "under­ (<< szzr - écoulement avec (<< szzr - écoulement avec flow". sous - écozzlement ») pour sous - écoulement») pour Corrélation entre différentes surélévations de l'interface 'Ir = ·'10 'Jo. 'Ir = 80 %. dans 1111 «sous-écozzlement». fluid, it has been shown that the total transverse metric Froude number were based on the mean superelevation is given by: velo city of the lower flowing layer and on the den­ sity difl'erences between the upper and lower layers (2) as measured at mid-bend section. There is a consid­ erable scatter in exprimental points which is whereas for a stratified underflow, similar analysis attributable to both the error involved in the evalua­ will give the interface transverse superelevation as: tion of the interface heights and to wal,l viscous !:lh = [V /(!:lp/p) g] (b/r ) o c ef1'ects. The increasing trend of the superelevation or: with densimetrie Froude number is however easily (!:lh/h) = À.F.:, 2 (9) discernibleand follows closely the curve given hy where: the one-dimensional theory. Bence, pl'ovided that an open-channel bend has a À. = (b/r,), F.:,2 =V /[!:lp/p) gh] moderate curvature (b/r < 1) and is sufficiently In view of the magnitude of density di/Terence long, the theoretical expression given in equation (9) ratios (!:lp/p) in the stratified media under conside­ will adequately predict the amount of transverse ration, it is apparent that the interface transverse interface superelevation in a stratified underflow. superelevation is of the order of about 10 lllUlti­ Attemps to measure the free surface superelevation plied by the free surface superelevation in a homo­ were unsuccessful as the transverse variation in geneous fluid flowing under similar conditions. total depth was extremely small. A number of theoretical expressions for trans­ Overflow-with-zwdertlow.· Any rigorous theOI'y for verse superelevation in stratified underflows was the interface superelevation in cUl'ved overflow-with derived by the writer by making several assump­ tions about the radial velocity distribution in the underflow will require a kno'\vledge of the distribu­ Euler's equation. A comparison of the results with tion of axial velocities in the radial direction for both layers. Because velocity traverses were not that given by the simple one-dimensional theory of made in the present experimental studies, it was equation (9) showed largest deviation of about 8 % considered that for practical convenience the results for the value of À. of one-sixth used in the present tests. Il would seem, therefore, that the magnitude could be correlated empirically hy the use of desi­ metric Froude number based on the mean velocity of the transverse superelevation in moderate'ly cu rv­ of the combined flow over the total depth, and on ed stratified underflow is not materiallv influenced the density ditrerence ratio between the two layers by the form of axial velocity distrilnltion in the at mid-bend section. This approach also made it radial direction. possible to check whether much error would be Figure 6 shows a plot of the experimental super­ introduced by maldng use of the one-dimensional elevations obtained at mid-section of the 135 bend. theoretical analysis of equation (9) for the interface In the same graph the theoretical ClIrVe as given by superelevation in overflow-with-underflow. equation (9) has been plotted for the value of À. of In Figures 7 and 8 are the experimental plots of one-sixth. The experimental values of the den si- 695 .... N. CHIKWENDU the dimensionless transverse superelevation in Summary of conclusions overflow-with-underfluw against the densimetric Froude number hased on the comhined flow. Each 1. In an underflow where the upper layer is relati­ of the two figures, which represents the discharge vely stationary, the interface slopes inwards with ratio of 40 % and 80 % respectively, contains the largest depth of the lower layer on the outer results ohtained from two difl'erent Beynolds num­ wall. bers, while keeping the dis charge ratio constant. 2. In an overflow-with-underflow where the flows Sin ce there is no systematic scaUer in the graphs, in the two layers are in turbulent motion, the arising from the variations in Reynolds numher, it interface slopes outwards with the largest depth can he concluded that for a given channel hend, of the lower layer on the inner wall. given dis charge ratio and given density difIerence, 3. The magnitude of the interface superelevation is the magnitude of interface superelevation is not essentially dependent on the me an momentum of very sensitive to variations in Beynolds numher. ilow, the width to radius ratio of the channel, Generally, hovvever, the experimental points show and the difference in specific weights of the stra­ some scaUer which may again be due to the error tified media. For channel bends of moderate cur­ involved in evaluating the superelevations and on vature, the interface superelevation is not mate­ the use of densimetric Froude numhers hased on the rially influenced by the form of velocity distri­ mean velocity of the two layers wlüch paid no bution, the Beynolds number and the discharge regard to the momentum correction factor. Never­ ratio. theless, it has been possible to fit a mean Cluve 4. In a sufIiciently long bend with moderate cu rva­ through the points in- each graph. The agreement ture, one-diInensional analysis for interface between the results for the two discharge ratios is superelevation can adequately· predict the amount fairly dose, that they are weIl represented by the of superelevation in stratified underflows as weIl following elnpirical equation: as in stratified overflow-with-underflovv. (1::..11/11) = 0.16 §i~ (JO) " Acknowledgement Il is to be noted that equation Cl 0) is nearly the sa me as the one-dimensional theory of equation (9) The writer would like to express his sincere gra­ for the given value of À of one-sixth. titude to Professor J.R.D. Francis for his advice and the laboratory facilities provided. Thus, for a sufIiciently long bend with a mode­ rate curvature, the interface transverse supereleva­ tion in stratified overflow-with-underflow is not References materially affected by the discharge ratio and Rey­ nolds numher. The magnitude of the interface SHlJKHY (A.). - Flow around hends in an open thune. Trans. superelevation is primarily dependent on the mean ASCE, Vol. 115 (1%0), paper 2411, p. 751. momentum of flow, densitv difIerences and width to II'I'EN (A. T.) and DHINIŒI\ (P. A.). - Boundarv shear stresses radius ratio; and can be· evaluated by applying a in curved trapezoidal channels. Proc. :4SCE, Vol. 88, one-dimension al analysis to the overall flow. No. I-1Y5 (1!)(i2) , part 1, p. 143. Résumé Forme et surélévation de l'interface dans les écoulements stratifiés à trajectoire courbe par L. N. Chikwendu * Il s'agit, dans cette étude, de la forme et de la surélévation de l'interface dans les écoulements il stratification de densité, et suivant un parcours en courbe. Après un résumé succinct d'études antérieures portant sur les écoulements homogènes il trajectoires courbes, on décrit l'étude expérimentale des écoulements stratifiés en deux couches. Cette étude tient compte d'écoulements stratifiés de deux natures différentes, l'un cor'respondant il un courant d'eau froide s'écoulant sous une couche supérieure d'eau relati­ vement plus chaude et quasi-stationnaire et l'autre correspondant il un courant chaud, superposé sur un courant froid, tous les deux étant en régime turbulent, et s'écoulant sensiblement dans la mêlne direction dans un canal courbe. La gamme des différences de densité étudiées se situait il l'intérieur de celle rencontrée normalement dans les rivières présentant un écoulement il stratification thermique. On a observé les profils longitudinal et transversal de l'interface par visualisation, en teintant la couche liquide supé­ rieure il l'aide d'un colorant approprié . • Lecturer in Civil Engineering, University of Nigeria Nsukka (Nigeria). G9G LA HOUILLE BLANCHE/N° 6-1966 Dans le cas d'un courant froid s'écoulant sous une couche quasi stationnaire, on a pu constater que le profil longitu­ dinal de l'interfacc du courant inférieur était assez uniforme sur l'ensemble de la courbe du canal, mais qu'il se rabattait à l'extrémité aval de ce dernier, où la couche inférieure devait s'écouler sous une vanne située dans un pertuis. L'obser­ vation visuelle a révélé que ce même courant d'eau froide était plus profond du côté de la paroi à l'extérieur de la courbe, que du côté de la paroi intérieure. Dans le cas des deux courants superposés, ona pu constater que la forme transversale était contraire à la précédente, puisque la plus grande profondeur de la couche inférieure se situait, cette fois, du côté de la paroi intérieure. Des mesures expérimentales ont été effectuées; elles sont représentées sur les figures 4 et 5. Ces résultats ont amené à la conclusion que, pour les variables dont il a été tenu compte dans l'étude, la forme du profil transversal de l'interface n'est modifiée, ni par le nombre de Froude densimétrique, ni par le nombre de Heynolds, ni par le rapport entre la profondeur et la lar­ geur. L'auteur présente un argument théorique, vérifiant certaines de ces observations (voir les équations 3 à 8). Une analyse théorique unidirnensionnelle a montré qu'il est possible d'exprimer la surélévation transversale Al! de l'interface par une relation de la forme: (9) dans laquelle 11 correspond à la hauteur d'écoulement, )." au rapport entre la largeur du canal et le rayon moyen, et ~ ê> au nombre de Froude clensimétrique, ce dernier étant donné par la relation ~ ê> 2 = V /(Âp/ p. gh). La confrontation de l'équation (9), et d'un résultat analogue obtenu pour le cas d'un écoulement liquide homogène, indique que la valeur de la surélévation de l'interface dans un écoulement à stratification thermique, correspondrait à un multiple de l'ordre de 10: de la valeur de la surélévation ayant lieu dans un écoulement d'un liquide homogène correspondant. La figure 6 repré­ sente la surélévation mesurée dans un courant inférieur au droit de la section médiane de la courbe à 135". Cette dernière figure montre que les résultats s'accordent de près avec l'expresion théorique donnée dans l'équa­ tion (9). Des résultats expérimentaux analogues, correspondant à la surélévation de l'interface, dans le cas de deux cou­ rants superposés, sont représentés sous forme graphique dans les figures 7 et 8. La relation empirique déduite, et définissant la grandeur de la surélévation de l'interface dans le cas de deux courants superposés, est de la forme: (10) Puisqu'il a été tenu compte, dans ces essais, d'un rappOl',t égal à 1/16 entre la largeur et le rayon, on observe que les équa­ tions (9) et (H» sont identiques, et on aboutit à la conclusion que l'analyse théorique unidimensionnelle donnée dans l'équation (9) permet de déterminer d',avance les surélévations ayant lieu dans les écoulement stratifiés, quels que soient la valeur du rapport entre les débits des couches inférieure et supérieure, et le nombre de Reynolds, et quelle que soit la forme de la répartition des vitesses. 697 Barrage de Cap-de-Long. GILLES EHRMANN, PHOTOTHËQUE E.D.F. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png La Houille Blanche Taylor & Francis

Interface shape and superelevation in curved stratified flows

Chikwendu, L. N.
La Houille Blanche , Volume 52 (6): 8 – Oct 1, 1966
Download PDF
8 pages

Loading next page...
 
/lp/taylor-francis/interface-shape-and-superelevation-in-curved-stratified-flows-0KYhMVVFfS

References

References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.

Publisher
Taylor & Francis
Copyright
© Société Hydrotechnique de France, 1966
ISSN
0018-6368
eISSN
1958-5551
DOI
10.1051/lhb/1966043
Publisher site
See Article on Publisher Site

Abstract

INTERFACE SHAPE AND SUPERELEVATION IN CURVED STRATIFIED FLOWS BY L. N. CHIKWENDU * shape of a free vortex, whereas equation (2) gives Introduction a linear profile. Also, the observed amount of trans­ verse superelevation was fOlmd by Shukry to be The existence of transverse pressure 0Tadien t is better approximated by a modified free vortex for­ . 1 t> an 111 lerent characteristic of ail rotational and irro- mula. tational curved flo"\vs. This pressure gradient is For a sufIiciently long bend, the transverse surface made manifest by an inward free surface inclina­ slope and the associated secondary circulations tion for the case of curved open channel flows in­ bring about a vertical and radial redistribution of v.olvin~ homogeneous fluids. From Euler's equa­ stream velo city by mechanism of momentum trans­ tIon, glven by: port. These changes in velocity pattern give rise to (oP/or) = pV2/ r 0) the graduaI shi ft of high velocity filament l'rom the inner wall to the outer wall, with the ultimate sup­ where oP/or is the radial pressure gradient and pression of free vortex and a development of forced V is the local velocity at a radius l', it is possible vortex pattern. In more recent work, Ippen and to carry out a one-dimensional analysis for the free Drinker (962) have shown that provided the mean surface superelevation by assuming that a cons­ radius to width ratio of a given channel bend is tant mean velocity V acis at the centre-line radius greater than unity, the magnitude of transverse rI' of the bend. Integration of equation 0) over superelevation in sub-critical homogeneous fluid the channcl width yields: flow is fairly approximated by the one-dimensional ho - hi = t:..h = (V 2/g ) (b/rJ (2) theOl'y of equation (2) irrespective of the l'orIn of latera,l distribution of axial velocity component. where ho and hi are the water levels at the outer The shape of the transverse free surface profile is, and inner walls respectively, and b is the width however, dépendent on the radial distribution of of the channel. axial velo city com ponen t. The experiments carried out by a number of As no work seems to have been reported on the early workers have, however, estàblished that at transverse interface shape and superelevation in a short distance l'rom the beginning of a bend both curved stratified flows, it is intended he rein to pre­ the velocity and pressure distributions in a sub­ sent the results of a series of laboratory tests carried critical homogeneous fluid flow are weil described out while the writer was on leave at the College of by a free vortex. Shukry (950), for instance, has Science & Technology, University of Manchester. compared the one-dimensional theOl'y given in In these tests, two forms of stratified flow were equation (2) with results obtained experimentally considered, (a) the « underflow » wherein a layer of and noted that the aciua1 transverse free surface fluid flowed beneath a fairly stationary layer of profile was nearly represented by the hyperbolic slightly less dense fluid; and (b) the « overflow-with­ underflow » in which the upper fluid and the lower Lecturer in Civil Engineering, University of Nigeria Nsukka (Nigeria). denser fluid layers both flowed in turbulent motion Article published by SHF and available at http://www.shf-lhb.org or http://dx.doi.org/10.1051/lhb/1966043 L. N. CHIKWENDU in the same general direetion. The density difTe­ gular channel section were 10 ins. and H 3/4 ins. respedively. The downstream end of the curved rences between the upper and the lower fluid layers covered in these tests are within the range found in unit had a 40-inch straight exit length which termi­ thermally stratified rivers. nated in a small outlet tank. The tank incorporated a tilting weil' for adjusting water level, and a ver­ tical recess for inserting a sluice gate used in controlling the pool of the upper layer during the Notation « underflow »tests. Schematic diagrams of the arrangements for flow control are shown in figs 1 The following symbols have been adopted for use and 2. Because of the transparency of the perspex in this paper: mate rial used for the bottom and walls of the entire channel, visual observations were made possible. b width of curved ehannel; The channel rested on supports which were levelled §i ~ : densimetric Froude number to keep the bed horizontal throughout the tests. = V /Y[.6.p/p) gh] ; The outlet end of a thermosiatic mixing valve H : total depth of the upper and lower layers in wasconnected to a rubber hose which delivered hot an underflow; water into the flume through a manifold-pipe distri­ h : depth of the flowing layer(s); butor. The hot water distributor spanned the width hi' ho : interfaee height at the inner and outer walls of the stilling basin and rested on the floating board respectively; in the basin j ust above the cold "mter level. Dis­ charge of hot and cold water were recorded by pre­ .6.h : total transverse superelevation (ho - hJ ; viously calibrated elbow meter and orifice plate res­ Pl' P : mean pressures experienced by the upper pectively. Precise and rapid measurements of tem­ and lower layer at a given radius respecti­ perature readings ,vere achieved by the use of a vely; commercial direct-reading multipoint thermistor P 1 : mean pressure in the lower laver due to its instrument which incorporated twelve sensing pro­ own depth at a given radius; v bes and had a scale graduated in degree Fahrenheit. ÀP 1 : pressure difTerence in the lower layer be­ Each miniature semi-conductor sensing element of tween the outer and the inner walls; the instrument was mounted at the last one-quarter Q : total diseharge of the flowing layer(s); inch length of stainless tubing, 1 ft. long and 0.0625 q,. : percent age ratio of the diseharge of the inch outside diameter. From the temperature read­ upper layer to that of the 100ver layer in an ings it was possible to determine the den si ties and overflow-with-underf1ow; locate interfaces. Velocitv measurements were not dl : Heynolds number = Voh/v; made, as the local velo~ities were generally too l' : radial distance to a point in eune; small to he recorded with the availahle instrument. The free surface levels were measured hy means of 1'0 : centreline radius of bend; micrometer point gauges. V : local axial velocity at a bend radius r; For the unclerflow tests, a sluice gate was inserted T • \ mean axial velocity of the Howing layer(s); o . in position to give an opening of 1 1/2 inch at the mean velocities of the upper and lower VI' V : end of the channel while the tilting weil' was raised layers respectively; so as to l'etain a pool of hot water in the entire Z : vertical distanee measured from the ehan­ channel (see Fig. 1). The desired amount of cold nel bed; watcr discharge was then slowly turned on until a p : mass density of lower fluid layer; quasi-steady state was attainecl ,vhereby the eolcl Àp : density difTerence between the lower and layer flowed underneath the layer of warm "mter, upper 'f1uid layers; through the sluice gate and over the weil' before clis­ charging to waste. The total depth of hot and cohl v : kinematic viscosity of Huid; water was kept eonstant at 6.5 ins. throughout the À : ratio of channel width to centreline radius tests. of bend = b/1'o' In the case of overflow-with-underflow tests, (see Fig. 2) the general procedure adopted in super­ posing the two streams consisted of first turning on Experimental equipment and method the required amount of cold ,vatel', allowing it to stabilise before turning on the hot water. A fair Hot water and eold water were used to simulate degree of initial mixing was eliminated as the hot the upper and lower layers respectively. water jets from the distributor fell on the floating The experimental channel was made up of a still­ wooden board within the stilling basin. The two ing basin which was Hared on the bottom and sides streams then flowecl along the channel after being to give smooth entry to the Hume. The basin was controlled in the straight approach length by a se­ fed through a 2-inch pipe which discharged cohl ries of parallel blades. The tilting weil' was used to water vertically dOWl1'Ward onto the bed at the cen­ achieve variations in total depth of flow. tre of the end wall. Surface waves within the basin In aH the tests, the temperature distribution and were suppressed by a floating ,vooden board plaeed the interface form were determined by vertical tem­ on the ,vatel' surfaee. perature traverses taken at the central seetion of the The stilling basin was joined onto a 1:35" eurved hend; except for a few tests in which further mea­ unit by a 40-inch straight approach length. The surernents were taken along the bend. The tempe­ centreline radius of curvature of the curved unit rature readings at a given section were recorded was 60 indles. The width and height of the rectan~ ",vith five probes spaced radially at 21/2 inches 692 LA HOUILLE BLANCHE/N° 6-1966 apart, and held by horizontal rigid perspex bar. The perspex bar ,vas in turn carried by a bent iron Stationary layer Couche stotionnaire rod which was fixed to a ll1icrometer screw gauge, x..-,! Sluice gate- Vanneplate thereby allowing simultaneous lowering or raising ", L '-,~~.;, :(P~-)-;zj.J ~rT~-.~:'":l~LPl~ of aIl five probes to any desired depth. Surface and ~ h .... ' P '-" \ ,~ 0---'----r--p?~~~\ bed tell1perature readings were taken al 0,25 ins. -'l: \ 1 i from the free surface and bed respectively. Inter­ x..- Tilting weir y .. Déversoir mobile mediate readings were taken at 0.5 ins. vertical intervals except near the interfacial zone where this Interface-Interface "~",,,v:L ' .. ". 1( C pl was reduced to 0.25 inch intervals. h,' , 6h The range of various variables covered in the lnn~~wLa-II-f--:--' lnne:wall ~ poute; wall tests is sumll1arised in Table 1. Paroi int. Paroi tnt. Paroi ext. Section YY Section XX Coupe " Coupe ""8 Interface shape 21 Schematic diagram ';i 1/ Schematic diagram of cunee! stratifiee! "under­ cUl'ved stratified "over­ flow". flow-with-underflow" . Undcrflow: Visual observation of the longitudi­ Schéma d'lin «solls-écoll­ Schéma d'lin écolllement nal and transverse interface profiles were made by lement» cOllrbe stratifié. mi:rie : « sllr-écolllement colouring the upper layer with a suitable dye. The avec SOliS - écoll/enzenl », longitudinal interface profile was fairly uniform cOllrbe et stratifié. over the bend but becall1e slightly depressed in the straight outlet length. Near the sluice end, where the cold water layer was eOll1pelled to How through Table 1 the l 112 inch opening, the longitudinal interface slope becall1e relatively steep showing resell1blance Range of tests to the "free overfall" profile in free surface How of a homogeneous fluid. A photograph of the longitu­ OVEHFLOW- dinal profile looking from the outer wall of the un­ VAHIABLE UNDEHFL<HV WITH- UNDEHFLOW derflow can be seen in Figure :3. Across the channel width, visual observations alone showed a deeper cold water region on the depth (ins) H = 6.5 Ih = 2.3 to 6.3 Overall .... outer ,:vall of the bend than on the inner wall, thus Total reflecting on the transverse interface superelevation. diseharge Q (in / see) 17.03 ta 52.9 3'2.6 ta 72.0 The experimental shapes of the interfaces at mid­ Diseharge ratio ...... - 40 0/0 beud for various values of densimetric Froude num­ gr and 80 0/0 ber (@ L! ) are shown in Figure 4. The interface posi­ Density differenee tions were obtained by plotting each temperature 8 8 0.49.10- ratio ....... 0.45.10- profile on an enlarged scale, conslrucling the tan­ C~ç/ç) 2 2 ta 0.469.10- ta 0.434.10- gent at the point of inflexion and hence determining (ins) - the interface height. Sluiee opening .... 1.5 As can be observed from Figure 4, there is a close Densimetrie similarity behveen aIl the transverse interface pro­ Fraude Number 0.38 ta 0.725 0.338 ta 2.42 (@L!) files obtained, except for slight differences which Reynolds Number .. (Ol) 1.1.10;' 3.84.10 arose from the varying wall efIects. Because a des­ 8 8 i and 4.65.10 ta 3.34.10 cription of the patterns of seconda l'y currents in curved stratified flows will be the subject of a forth­ coming paper, it will be rell1arked in passing here that aIthough these secondary currents are prima­ rily responsible for the interfacialmixing of the two layers, yet they etrected mixing only to a small extent and so permitted the interfaces to be located with SOlne degree of accuraey. Entrain ment of the upper layer was very small. Ovcrflow-with-Undcrllow : At a bend angle smaIl­ el' than reiS the interface height near the inner wall was found to increase downstreamwards with the opposite behaviour on the outer wall. The visual transverse superelevations in this region were also found to be large in comparison to that over the l'est of the channel bene!. Beyond reiS, however, the superelevalion appeared to be constant along the bend. These observations were coniirmed by actual measurell1ents of interface supereleva­ 3/135 test flume showing longitudinal interface profile in an "underllow". lions at bend angles of reiS, :3 reiS and 5 reiS respec­ Canal d'essai cOlldé à 135' mettant en évidence le profil tively, l'rom which the supcrelevation at 3 reiS and de l'interface longitudinale dans lin «solls-écoulement ». 5 re/8 ditrered only slightly whereas that at reiS was found to be ahout 50 % laI'ger than any of the 693 L. N. CHIKWENDU former. In the circumstanccs, it would appear that overflow-with-underflow with turbulent motion in the bend entry conditions influenced the interface the two layers, then AP 1 will be negative and there shape and superelevation at bend angles sm aller will be larger depth of the lower layer on the inner than 7t/8. This is to be expected since the flow wall than on the outer wall. 'Vhen VI -7 0, as could be thought to be then in its adjusting process. obtains in stratified underflow or in frce surface In Figure 5 are shown S(llUe of the experimental flow of a homogeneous fluid, then AP ] becomes transverse profiles of the interface for various positive and there will be larger depth of the lower values of densimetric Froude numbers, total depths layer on thc outer wall than on the inner wall. and discharge ratios. 'Vhile there seems to be a Thus in curved stratified flovvs with slight density close similarity between the shapes of these inter­ dift'erences and in which both the upper and lower faces, it will be observed that some difference exists fluid laycrs are in turbulent motion, the interfacial when comparison is made with those found for the shape is such that the largest depth of the lower underflows. In particular, the direction of the inter­ layer will be f0lll1d on the inner wall and the largest face curvature has heen reversed in the ove rflow­ depth of the upper layer on the outer wall. with underflow, the deepest cold vvater layer being now on the inner wall. At small depths and large densimetric Froude Transverse superelevation number of the order of 2.5 or higher, the locus of the interfaee intersected the channel bed near the outer wall, with the cold water layer virtually Under/law: One characteristic of stratified flows exposed at the free surface on the inner wall. This which has emerged from the present studies is the behaviour of the interface in an overflow-with­ magnification of the interface transverse superele­ underflow tended to set the upper limit of the densi­ vation as compared to the corresponding free sur­ metric Froude number which could be investigated face superelevation in a homogeneous fluid flow. in smaH scale lahoratory experiments of this nature. From one-dimension al analysis for a homogeneous The rather peculiar interface shape of the over­ flow-with-underflow, in which the deepest cold water layer is found on the inner wall rather than Interface on the outer wall, is weIl in accord with a simple 3·5 Upper layer Interface Couche sup. theoretical argument which can be formulated as follows: (ins) F =0·380 Lower laye! Couche ln! H =6·5" Consider hvo co-current stratified layers flowing 3·0 F = 0890 within a curved channel of mean radius rc, and h = 4 3" 4,25, . (ins) assume that the lighter upper layer of density 3·0 CP - Ap) and the lower layer of density p move with z 1 (ins) mean velocities of VI and V 2 respectively. If the , F =0415 mean pressure experienced by the upper layer is Pl '''~ and that of the lower layer is P , then the mean 2 2·5 2·75 pressure P 1 of the lower layer due entirely to its z ' own depth will be given by: 4·5l'~ .. ('Ins), 2·5 F = ',360 , F6=O'450 Z (ins) <O· , ,·ôO· h =3 ·3" where Pl' P and P 1 aIl vary with l' only. 2 2 2·0 4'75~' i - From Euler's equation, and by carrying one­ Z ' , ·75 ­ dimensional analysis for the upper and lower (ins) , i.. F6= 0'510', 2·75 .. layers, the following are obtained: 1 H = 6·5" 4·25, , . 2·5 (4) and: (5) F = 1·95 So that: 2·0 .. 6 h =3· 3" (6) (ins) - For the order of density differences under conside­ '·5 .. 5'0~!",."", Z . F6=0.600,' H =6· 5" ration, (1 - Ap/p) = 1.0. Equation (6) is therefore (ins) .. re.duced to the form: 4·5 [~ner woll g Outer wall' '·0 .. (7) PalOi in! Paroi ex! Inner wall Outer wall Paroi inl Paroi ex! On integrating equation (7) across the channel 4/ Interface transverse pro­ 5/ Interface transverse pro­ yields: files in an "underllow". files in an "overllow­ (8) with-undertlow" . Profils en tra/Jers de l'in­ Profils en tra/Jers de where b is the width of the cbannel. terface dans un «sous­ l'interface dans un écou­ In equation (8) AP 1 is the positive pressure diffe­ écoulement ». lement mixte: «sur-écou­ lement a/Jec sous-écoule­ rence in the lower layer between the outer and the ment ». inner wall. lt is therefore clear from equation (8) that if V2 < Vl' as is generally the case in stratified 694 LA HOUILLE BLANCHE/N° 6-1966 o. Ollr---,-------,---,------,---~--~-_ 1·0 1·0 0·8 0·8 0·6 0·6 0.071-------i---+---+--+-----1---+---......I-:....-I 0·4 0·4 0·06 0·2 0·2 • Experiment Valeur expérimentale (6h/h) (6h/h) (6h/h) - Equation (9) équation ( 9 J 0·04 @:L'h 0·03 1nner wall Outer wall Paroi int. Paroi ext. • 4·65 0·02 t---,---- o 384 0·01 L,-,J------,--'--"--'--'---__'---...J C3 DA 0·6 DB 1·0 2·0 3·0 Ft> 0·01 1/ Correlation of interface 8/ Correlation of interface superelevation in an superelevation in an "overflow - with - und er­ "overflow - with - und er­ flow" for '1,. = 40 'Jo. flow" for 'l,. = 80 'Jo. o 0·1 0·2 0·3 0·4 0·5 0·6 0·7 Corrélation en t rel e s Corrélation en t rel e s Fil szzrélévations de l'interface surélévations de l'interface dans zzn écoulement mi,c/e dans un ecoulement mixte 6/ Correlation of interface superelevation in an "under­ (<< szzr - écoulement avec (<< szzr - écoulement avec flow". sous - écozzlement ») pour sous - écoulement») pour Corrélation entre différentes surélévations de l'interface 'Ir = ·'10 'Jo. 'Ir = 80 %. dans 1111 «sous-écozzlement». fluid, it has been shown that the total transverse metric Froude number were based on the mean superelevation is given by: velo city of the lower flowing layer and on the den­ sity difl'erences between the upper and lower layers (2) as measured at mid-bend section. There is a consid­ erable scatter in exprimental points which is whereas for a stratified underflow, similar analysis attributable to both the error involved in the evalua­ will give the interface transverse superelevation as: tion of the interface heights and to wal,l viscous !:lh = [V /(!:lp/p) g] (b/r ) o c ef1'ects. The increasing trend of the superelevation or: with densimetrie Froude number is however easily (!:lh/h) = À.F.:, 2 (9) discernibleand follows closely the curve given hy where: the one-dimensional theory. Bence, pl'ovided that an open-channel bend has a À. = (b/r,), F.:,2 =V /[!:lp/p) gh] moderate curvature (b/r < 1) and is sufficiently In view of the magnitude of density di/Terence long, the theoretical expression given in equation (9) ratios (!:lp/p) in the stratified media under conside­ will adequately predict the amount of transverse ration, it is apparent that the interface transverse interface superelevation in a stratified underflow. superelevation is of the order of about 10 lllUlti­ Attemps to measure the free surface superelevation plied by the free surface superelevation in a homo­ were unsuccessful as the transverse variation in geneous fluid flowing under similar conditions. total depth was extremely small. A number of theoretical expressions for trans­ Overflow-with-zwdertlow.· Any rigorous theOI'y for verse superelevation in stratified underflows was the interface superelevation in cUl'ved overflow-with derived by the writer by making several assump­ tions about the radial velocity distribution in the underflow will require a kno'\vledge of the distribu­ Euler's equation. A comparison of the results with tion of axial velocities in the radial direction for both layers. Because velocity traverses were not that given by the simple one-dimensional theory of made in the present experimental studies, it was equation (9) showed largest deviation of about 8 % considered that for practical convenience the results for the value of À. of one-sixth used in the present tests. Il would seem, therefore, that the magnitude could be correlated empirically hy the use of desi­ metric Froude number based on the mean velocity of the transverse superelevation in moderate'ly cu rv­ of the combined flow over the total depth, and on ed stratified underflow is not materiallv influenced the density ditrerence ratio between the two layers by the form of axial velocity distrilnltion in the at mid-bend section. This approach also made it radial direction. possible to check whether much error would be Figure 6 shows a plot of the experimental super­ introduced by maldng use of the one-dimensional elevations obtained at mid-section of the 135 bend. theoretical analysis of equation (9) for the interface In the same graph the theoretical ClIrVe as given by superelevation in overflow-with-underflow. equation (9) has been plotted for the value of À. of In Figures 7 and 8 are the experimental plots of one-sixth. The experimental values of the den si- 695 .... N. CHIKWENDU the dimensionless transverse superelevation in Summary of conclusions overflow-with-underfluw against the densimetric Froude number hased on the comhined flow. Each 1. In an underflow where the upper layer is relati­ of the two figures, which represents the discharge vely stationary, the interface slopes inwards with ratio of 40 % and 80 % respectively, contains the largest depth of the lower layer on the outer results ohtained from two difl'erent Beynolds num­ wall. bers, while keeping the dis charge ratio constant. 2. In an overflow-with-underflow where the flows Sin ce there is no systematic scaUer in the graphs, in the two layers are in turbulent motion, the arising from the variations in Reynolds numher, it interface slopes outwards with the largest depth can he concluded that for a given channel hend, of the lower layer on the inner wall. given dis charge ratio and given density difIerence, 3. The magnitude of the interface superelevation is the magnitude of interface superelevation is not essentially dependent on the me an momentum of very sensitive to variations in Beynolds numher. ilow, the width to radius ratio of the channel, Generally, hovvever, the experimental points show and the difference in specific weights of the stra­ some scaUer which may again be due to the error tified media. For channel bends of moderate cur­ involved in evaluating the superelevations and on vature, the interface superelevation is not mate­ the use of densimetric Froude numhers hased on the rially influenced by the form of velocity distri­ mean velocity of the two layers wlüch paid no bution, the Beynolds number and the discharge regard to the momentum correction factor. Never­ ratio. theless, it has been possible to fit a mean Cluve 4. In a sufIiciently long bend with moderate cu rva­ through the points in- each graph. The agreement ture, one-diInensional analysis for interface between the results for the two discharge ratios is superelevation can adequately· predict the amount fairly dose, that they are weIl represented by the of superelevation in stratified underflows as weIl following elnpirical equation: as in stratified overflow-with-underflovv. (1::..11/11) = 0.16 §i~ (JO) " Acknowledgement Il is to be noted that equation Cl 0) is nearly the sa me as the one-dimensional theory of equation (9) The writer would like to express his sincere gra­ for the given value of À of one-sixth. titude to Professor J.R.D. Francis for his advice and the laboratory facilities provided. Thus, for a sufIiciently long bend with a mode­ rate curvature, the interface transverse supereleva­ tion in stratified overflow-with-underflow is not References materially affected by the discharge ratio and Rey­ nolds numher. The magnitude of the interface SHlJKHY (A.). - Flow around hends in an open thune. Trans. superelevation is primarily dependent on the mean ASCE, Vol. 115 (1%0), paper 2411, p. 751. momentum of flow, densitv difIerences and width to II'I'EN (A. T.) and DHINIŒI\ (P. A.). - Boundarv shear stresses radius ratio; and can be· evaluated by applying a in curved trapezoidal channels. Proc. :4SCE, Vol. 88, one-dimension al analysis to the overall flow. No. I-1Y5 (1!)(i2) , part 1, p. 143. Résumé Forme et surélévation de l'interface dans les écoulements stratifiés à trajectoire courbe par L. N. Chikwendu * Il s'agit, dans cette étude, de la forme et de la surélévation de l'interface dans les écoulements il stratification de densité, et suivant un parcours en courbe. Après un résumé succinct d'études antérieures portant sur les écoulements homogènes il trajectoires courbes, on décrit l'étude expérimentale des écoulements stratifiés en deux couches. Cette étude tient compte d'écoulements stratifiés de deux natures différentes, l'un cor'respondant il un courant d'eau froide s'écoulant sous une couche supérieure d'eau relati­ vement plus chaude et quasi-stationnaire et l'autre correspondant il un courant chaud, superposé sur un courant froid, tous les deux étant en régime turbulent, et s'écoulant sensiblement dans la mêlne direction dans un canal courbe. La gamme des différences de densité étudiées se situait il l'intérieur de celle rencontrée normalement dans les rivières présentant un écoulement il stratification thermique. On a observé les profils longitudinal et transversal de l'interface par visualisation, en teintant la couche liquide supé­ rieure il l'aide d'un colorant approprié . • Lecturer in Civil Engineering, University of Nigeria Nsukka (Nigeria). G9G LA HOUILLE BLANCHE/N° 6-1966 Dans le cas d'un courant froid s'écoulant sous une couche quasi stationnaire, on a pu constater que le profil longitu­ dinal de l'interfacc du courant inférieur était assez uniforme sur l'ensemble de la courbe du canal, mais qu'il se rabattait à l'extrémité aval de ce dernier, où la couche inférieure devait s'écouler sous une vanne située dans un pertuis. L'obser­ vation visuelle a révélé que ce même courant d'eau froide était plus profond du côté de la paroi à l'extérieur de la courbe, que du côté de la paroi intérieure. Dans le cas des deux courants superposés, ona pu constater que la forme transversale était contraire à la précédente, puisque la plus grande profondeur de la couche inférieure se situait, cette fois, du côté de la paroi intérieure. Des mesures expérimentales ont été effectuées; elles sont représentées sur les figures 4 et 5. Ces résultats ont amené à la conclusion que, pour les variables dont il a été tenu compte dans l'étude, la forme du profil transversal de l'interface n'est modifiée, ni par le nombre de Froude densimétrique, ni par le nombre de Heynolds, ni par le rapport entre la profondeur et la lar­ geur. L'auteur présente un argument théorique, vérifiant certaines de ces observations (voir les équations 3 à 8). Une analyse théorique unidirnensionnelle a montré qu'il est possible d'exprimer la surélévation transversale Al! de l'interface par une relation de la forme: (9) dans laquelle 11 correspond à la hauteur d'écoulement, )." au rapport entre la largeur du canal et le rayon moyen, et ~ ê> au nombre de Froude clensimétrique, ce dernier étant donné par la relation ~ ê> 2 = V /(Âp/ p. gh). La confrontation de l'équation (9), et d'un résultat analogue obtenu pour le cas d'un écoulement liquide homogène, indique que la valeur de la surélévation de l'interface dans un écoulement à stratification thermique, correspondrait à un multiple de l'ordre de 10: de la valeur de la surélévation ayant lieu dans un écoulement d'un liquide homogène correspondant. La figure 6 repré­ sente la surélévation mesurée dans un courant inférieur au droit de la section médiane de la courbe à 135". Cette dernière figure montre que les résultats s'accordent de près avec l'expresion théorique donnée dans l'équa­ tion (9). Des résultats expérimentaux analogues, correspondant à la surélévation de l'interface, dans le cas de deux cou­ rants superposés, sont représentés sous forme graphique dans les figures 7 et 8. La relation empirique déduite, et définissant la grandeur de la surélévation de l'interface dans le cas de deux courants superposés, est de la forme: (10) Puisqu'il a été tenu compte, dans ces essais, d'un rappOl',t égal à 1/16 entre la largeur et le rayon, on observe que les équa­ tions (9) et (H» sont identiques, et on aboutit à la conclusion que l'analyse théorique unidimensionnelle donnée dans l'équation (9) permet de déterminer d',avance les surélévations ayant lieu dans les écoulement stratifiés, quels que soient la valeur du rapport entre les débits des couches inférieure et supérieure, et le nombre de Reynolds, et quelle que soit la forme de la répartition des vitesses. 697 Barrage de Cap-de-Long. GILLES EHRMANN, PHOTOTHËQUE E.D.F.

Journal

La Houille Blanche – Taylor & Francis

Published: Oct 1, 1966

There are no references for this article.