Local energy losses at positive and negative steps in subcritical open channel flows

Transcription

1 Local energ losses at positive and negative steps in subcritical open channel flows Nura Denli Toka and A Burcu Altan-Sakara* Middle East Technical niversit, Department of Civil Engineering, 63, Ankara, Turke Abstract Local energ losses occur when there is a transition in open channel flow. Even though local losses in subcritical open channel flow due to changes in channel width have been studied, to date no studies have been reported for losses due to changes in bed elevations. Steps are commonl used in engineering applications to stabilise the flow in open channels. Hence, it is important to estimate local losses for design purposes. The aim of this stud was to formulate the local energ losses at positive and negative steps in subcritical open channel flows. Flow rates and water depths before and after the step were measured for varing step heights of abrupt and 4 o inclined steps. Empirical equations relating the local losses to the Froude number on the step and the relative step height are proposed for positive and negative steps. In addition, practical values of local loss coefficients are determined. Kewords: local (minor) loss, positive step, negative step, open-channel flow Notation a, a constants b, b constants C, C, constants E specific energ Fr Froude number g gravitational acceleration H total head local energ loss due to step K p local loss coefficient for positive steps defined in Eq. (4) K n local loss coefficient for negative steps defined in Eq. () K p local loss coefficient for positive steps defined in Eq. (6) K n local loss coefficient for negative steps defined in Eq. (7) average velocit water depth z bed elevation α energ correction coefficient f inclination of the step Dz step height Introduction The energ losses that result from local features such as weirs, gates, cross-sectional changes or changes in alignment are called local or minor losses. Local losses ma be computed as a fraction of the velocit head (Chow, 99). This fraction is usuall termed as the local loss coefficient and is usuall obtained experimentall. Although the local loss coefficients are well determined for pipe flow, there are onl a few cases where local loss coefficients have been studied in open channel flows. These are for the abrupt expansion or contraction in width of a rectangular channel. * To whom all correspondence should be addressed. (9)-3-477; fax: (9)-3-6; burcuas@metu.edu.tr Received April ; accepted in revised form 8 March. Steps are often used in control structures like stilling basins; in order to design these properl, the amount of head losses should be known accuratel. In the present stud, local energ losses at a step in a subcritical open channel flow in a rectangular channel were studied experimentall. The effects of the step shape and step height as well as the flow properties were investigated. Formica (9) conducted experiments on various designs for subcritical flow passing through sudden transitions. He presented tpical flow profiles and energ lines for the design of expansions and contractions. Since velocit cannot be measured easil, because of the turbulent condition of flow, near the section where the transition takes place, he simpl extended the energ lines. Formica also showed that, in general, the sudden contractions have higher head losses than the sudden expansions; because in a sudden contraction the flow is first contracted and then expanded. He stated that a process of conversion from potential to kinetic energ is followed immediatel b a process of re-conversion from kinetic to potential energ, and that, as a result, much less energ is recovered in a sudden contraction than in a sudden expansion. Formica concluded that this loss of energ could be greatl reduced b modifing the sharp-edged corners of the entrance of the reduced channel. Skogerboe et al. (97) studied the head loss occurring in open channel expansions. The presented a comparison of various methods studied b previous researchers, and showed that the coefficients used in local loss calculations are not constants. Morris and Wiggert (97) and Brater and King (976) determined the constant empirical local loss coefficients for channel transitions in width. Binnie (97) performed laborator experiments on the flow of water over a downward step in an open channel, focussing on hdraulic jumps formed because of the abrupt drop. Vittal and Chiranjevii (983) suggested a rational method of design for open channel transitions. In their stud, the existing design methods for the design of transitions between rectangular and trapezoidal channels for subcritical flows are carefull examined. The suggest a new and rational design method for such transitions. ISSN (Print) = Water SA Vol. 37 No. April ISSN (On-line) = Water SA Vol. 37 No. April 37

2 Swamee and Basak (99) presented an equation for the design of rectangular open channel transitions based on minimising the energ loss. Since such transitions are usuall involved in channels for power generation and irrigation purposes, the emphasised that, b minimising the energ loss throughout the sstem, the sstem efficienc (as well as the life of the structure) ma be increased. Based on optimal control theor, the presented a methodolog for the optimal design of rectangular subcritical expansion transitions. Swamee and Basak (99) suggested a new method for the design of trapezoidal expansion transitions. Similar to their work on the design of rectangular transitions, the based the method on minimising the energ loss incurred in the sstem. The pointed out that in such transitions as the flow decelerates a positive pressure gradient occurs causing separation and hence considerable head loss. The suggested that if the sstem is designed for minimum head loss, considerable savings could be achieved. The evolved a methodolog, based on the optimal control theor, for the design of an expansion transition between a rectangular flume and a trapezoidal channel for subcritical flow. Analsing a large number of designed optimal transitions, the obtained empirical equations for the bed width and side slope profiles. Bhallamudi and Chaudr (99) presented a stud on the computation of flows in open channel transitions. The solved -dimensional, depth-averaged, unstead flow equations in a transformed coordinate sstem using the MacCormack scheme to analse flows in expansions and contractions. The found that the computed results were in satisfactor agreement with the experimental data for cases where the hdrostatic pressure distribution is valid. Swamee and Basak (993) discussed a methodolog, based on optimal control theor, for designing expansion transitions for graduall varied subcritical flow, called the comprehensive open channel expansion transition design. As in their previous works introduced in this paper, the pointed out that subcritical flow through a transition can result in significant head loss due to separation of flow and subsequent edd formation. The also emphasised that the reduction of head loss is desirable, for it results in more power generation in the case of power channels and increased service area for irrigation canals, as well as being desirable from the viewpoint of increasing the life of the transition structure. All of the studies mentioned above refer to the flow over transitions; when required the used approximate values for local loss coefficients or obtained them experimentall. Some of these studies report on numerical modelling of flow over transitions. There are also studies in the literature aimed at finding empirical formulas for the local loss coefficients in pipe flow; but no such studies could be found for flows in open channels. If empirical formulas for the calculation of the local loss coefficients in open channel flow can be obtained, subsequent analtical and numerical works on the subject would be more accurate. The present stud aims to find such equations for positive and negative steps in subcritical open channel flows. The present stud addresses the flow over positive and negative steps in a subcritical open channel. Two tpes of step shape, an abrupt step and a 4 inclined step, were used in the experiments. For the positive steps, the abrupt step heights used were: cm, 7 cm, cm, and 3 cm; and for the inclined steps: 7 cm, cm, and 3 cm. Similarl, for the negative steps, the abrupt step heights used were: 7 cm, cm, and 3 cm; and for the inclined steps: 7 cm, cm, and 3 cm. The experimental data has been analsed to obtain relations to represent the local losses as a function of the Froude number on the step and the relative step height defined for positive and negative steps. Flow over a step Sudden transitions will induce rapidl varied flow. Assuming that the frictional losses are negligible, the energ equation between Sections () and () (Figs. and ) is: H H H i is the total head at i-th section, i=, is the local energ loss due to the step. Eq. () ma be written in open form as: z () z g g z,, and represent the bed elevation, the water depth, and the average velocit, respectivel g is the acceleration due to gravit a is the energ correction coefficient. Formica (9) showed that the values of a are ver close to unit for sudden contractions (between.-.4), but slightl higher than unit for sudden expansions (up to.). In the present stud, the value of energ correction coefficient, a, is determined experimentall as a =. for positive steps (Orsel, ). Throughout the stud, the energ correction coefficient is taken as unit. Solving Eq. () for the local loss, : z z On the other hand, in the literature the local loss ma be z g z Figure Subcritical flow over a positive step (dashed line shows the inclined step) z z Figure Subcritical flow over a negative step (dashed line shows the inclined step) () (3) 38 ISSN (Print) = Water SA Vol. 37 No. April ISSN (On-line) = Water SA Vol. 37 No. April

3 represented as a fraction of velocit head, /g (Chow, 99). In the present stud, the following local loss equations were used: h for positive steps (4a) m Kp g The above equation can be non-dimensionalised b dividing each term with the flow depth as: Fr K for positive steps p Kp g Fr g is the Froude number at Section. K n for negative steps g Similarl, Eq. (a) can be non-dimensionalised as: Fr K for negative steps n K n g Fr is the Froude number at Section. g Another form of local loss is defined as (Chow, 99): (4b) (a) (b) h for positive steps (6a) m K p g Dividing each term b the flow depth, the non-dimensionalised form of Eq. (6a) is obtained as: K p g K n g for positive steps for negative steps (6b) (7a) Similarl, K for negative steps (7b) n g In the above equations subscripts p and n refers to positive and negative steps in coefficients K and K, respectivel. Experimental stud The experiments were carried out in a horizontal rectangular channel having cm width, cm depth, and. m length (Fig. 3). The channel is made of concrete and fibreglass. The section where the observations were done has fibreglass sidewalls and a concrete bottom. The steps used are made of fibreglass. The step heights used were 3 cm, cm, 7 cm, and cm for abrupt steps, and 3 cm, cm, and 7 cm for 4 o inclined steps. The water depths were measured b a movable point gauge, which has an accurac of ±. cm. Water was supplied from a constant head tank having a maximum capacit of l/s through an 8 cm diameter pipe. The maximum capacit of the channel was 3. l/s. Discharge entering the sstem was controlled b a valve in the deliver pipe. The depth of flow was adjusted b a sluice gate at the end of the channel. After the sluice gate, water is collected in a small basin, which directs the water to a return channel having a 3 o triangular weir at the end. This triangular weir is used to measure the discharge with an accurac of ±.%. The limiting values of experimental data for positive and negative steps are given in Tables and, respectivel. In the experiments conducted the range of the Froude number on the step was <Fr<.. Table Limiting values of experimental data for positive steps Tpe Abrupt Abrupt Abrupt Abrupt Inclined Inclined Inclined Dz (cm) # of data Q min (l/s) Q max (l/s) min (cm) max (cm) min (cm) max (cm) min (mm) max (mm) Table Limiting values of experimental data for negative steps Tpe Abrupt Abrupt Abrupt Inclined Inclined Inclined Dz (cm) # of data Q min (l/s) Q max (l/s) min (cm) max (cm) min (cm) max (cm) min (mm) max (mm) ISSN (Print) = Water SA Vol. 37 No. April ISSN (On-line) = Water SA Vol. 37 No. April 39

4 SIDE VIEW. m Depth gauge Fibreglass section Gate step.7 m Discharge gauge Discharge regulation valve PLAN VIEW Figure 3 Schematic view of the experimental setup 4. m. m. m. m 3. m.4 m Analsis of data Positive steps The dimensional analsis applied to variables f,, z,g, gives the following dimensionless parameters: z F Fr,, (8) Dz is the positive step height f is the inclination of the step Fr is the downstream Froude number (Froude number on the step). Positive abrupt steps For the local energ loss, relationships which have the following forms have been analsed: b a z C Fr (9) a, b, and C are the constants to be determined from the data. These constants are determined b using the best fit curves as: a=., b=., C =.7. Local loss coefficients for positive abrupt steps In order to calculate the numerical values of the local loss coefficients Eqs. (4b) and (6b) are used for positive steps. K p defined in Eq. (4b) is calculated from the graph of / versus Fr / (Fig. ). The slope of the best fit line is nothing / (Fr ) (z/ ).7 Figure 4 / versus (Fr ) (Dz/ ).7 for positive abrupt steps / =.7Fr / Data Eq. () + SD - SD Therefore, the local loss for a positive abrupt step ma be formulated as:. 7 z. Fr () The plot of Eq. () is shown in Fig. 4 together with the standard deviation lines and data points. The best fit line has a correlation coefficient of R=.8 and a standard deviation of.4. / Fr / Figure / versus (Fr ) / for positive abrupt steps 4 ISSN (Print) = Water SA Vol. 37 No. April ISSN (On-line) = Water SA Vol. 37 No. April

5 but the corresponding loss coefficient which is calculated to be.7. Similarl, K p is determined from the graph of / drawn vs. ( )/g as shown in Fig. 6. The slope of the best fit line is.3 which is the local loss coefficient defined b Eq. (6b) for positive steps. Positive inclined steps In the present stud, the local loss for positive inclined steps is studied for onl one value of f=4 o, to compare with that of abrupt steps. Therefore, the relationship does not include the inclination angle and an analsis similar to that for positive abrupt steps was performed. The equation obtained is as follows:. 4. z. 4Fr () The plot of Eq. () and its standard deviation lines are shown in Fig. 7 together with data points. The correlation coefficient of the Eq. () is R=.78 and the standard deviation is.. Local loss coefficients for positive inclined steps Local energ losses at 4 inclined steps can also be represented with local loss coefficients, K p and K p, which are defined in Eqs. (4b) and (6b), respectivel. Figure 8 shows the variation of / with respect to Fr /. The corresponding slope of the best fit line is K p which is equal to.. Similarl, Figure 9 shows the change of / with ( )/g and the corresponding slope of the best fit line is the local loss coefficients, K p, which takes a value of.. Discussion of results for positive steps / / / =.3( - )/(g ) ( - )/(g ) Figure 6 / versus ( - )/(g ) for positive abrupt steps (Fr ). (z/ ).4 Data Eq. () + SD - SD Figure 7 / versus (Fr ). (Dz/ ).4 for positive inclined steps Formica (9) obtained the values of K for sudden and gradual contraction in width for rectangular channels. Based on Formica s results, Chow (99) suggests to use an approximate median value of K =. and K =.6 for abrupt and gradual contractions, respectivel. In the present stud, the values of the local loss coefficient expressed with Eq. (4) are:.7 and. for positive abrupt and inclined steps, respectivel. Morris and Wiggert (97) suggested using a value of.4 for sharp transitions and. for wedge transitions in the case of contracting sections. Similarl, Brater and King (976) obtained the local loss coefficient expressed b Eq. (6), K =. for abrupt contractions and suggested using a value of. for designing contracting channels. These values are compatible with the local loss coefficients obtained from the present experimental studies of.3 and. for positive abrupt and inclined steps, respectivel. nfortunatel, there is no published stud which gives the local loss coefficients for contractions due to positive steps, and which could have been used for comparison purposes. It is believed that this stud fills that gap and gives the average values of local loss coefficients, K and K. The present stud also provides a relation, which computes the local loss as a function of downstream Froude number, and relative step height, Dz/. Hence, it is suggested that Eqs. () and () are used for positive abrupt and 4 inclined steps, respectivel. Negative steps The dimensional analsis applied to variables gives the following dimensionless parameters: f,, z,g, / /.... / =.Fr /... Fr /..... Figure 8 / versus (Fr ) / for positive inclined steps / =.( - )/(g )... ( - )/(g ) Figure 9 / versus ( - )/(g ) for positive inclined steps ISSN (Print) = Water SA Vol. 37 No. April ISSN (On-line) = Water SA Vol. 37 No. April 4

6 z F Fr,, () Dz is the negative step height, f is the inclination of the step and Fr is the upstream Froude number (Froude number on the step). Negative abrupt steps / Data Eq. (4) + SD - SD For the local energ loss, relationships which have the following forms have been analsed: b' h m a' z C (3) ' Fr a, b, and C, are the constants to be determined from the data. These constants are determined b using the best fit curves as: a =, b =., C =.34. Therefore the local loss for an abrupt step ma be formulated as:. z. 34Fr (4) The plot of Eq. (4) is shown in Fig., together with data points and the standard deviation lines. The correlation coefficient of the Eq. (4) is R=.98 and the standard deviation is.. Local loss coefficients for negative abrupt steps Local loss coefficients are calculated for negative steps as was done for the positive step. The numerical values of the local loss coefficients are calculated b using Eqs. (b) and (7b) for negative steps. K n defined in Eq. (b) is calculated from the graph of / vs. Fr / (Fig. ). The slope of best fit line is the corresponding loss coefficient which is calculated to be.4. Similarl, K n is determined from the graph of / vs. ( )/g as shown in Fig.. The slope of the best fit line is.89 which is the local loss coefficient defined b Eq. (7b) for negative steps. Negative inclined steps In this stud, onl an inclination of 4 o was investigated for the negative step. Examination of the data revealed the following equation:. z. 37Fr () The plot of Eq. () together with the experimental data is shown in Fig. 3. The correlation coefficient of Eq.() is R=.99 and the standard deviation is.. Local loss coefficients for negative inclined steps Similar to the positive inclined steps, local losses at 4 inclined steps can also be represented with local loss coefficients, K n and K n, which are defined in Eqs. (b) and (7b), respectivel. Figure (4) shows the variation of / with respect to Fr /. The slope of the best fit line is K n, which is equal to.49. Similarl, Fig. () shows the change of / with ( )/g and the corresponding slope of the best fit line is the local loss coefficient, K n, which takes a value of.97. / / / (Fr ) (z/ ). Figure / versus (Fr ) (Dz/ ). for negative abrupt steps / =.4Fr /..4 Fr.6.8. / Figure / versus (Fr ) / for negative abrupt steps / =.89( - )/(g ) ( - )/(g ) Figure / versus ( - )/(g ) for negative abrupt steps.... (Fr ) (z/ ). Data Eq. () Figure 3 / versus (Fr ) (Dz/ ). for negative inclined steps + SD - SD 4 ISSN (Print) = Water SA Vol. 37 No. April ISSN (On-line) = Water SA Vol. 37 No. April

7 Discussion of results for negative steps Comparing the results obtained for negative abrupt and negative inclined steps (f=4 ), it is seen, contrar to expectation, that the local loss in negative abrupt steps is less than the local loss in negative inclined steps, b approximatel %. The formation of the separation zone in abrupt steps might cause the flow to occur smoothl leading to less energ loss. Formica (9) obtained the average values of the local loss coefficient for 8 different configurations of expanding channel widths. He defined the loss coefficient as a function of the square of the upstream and downstream velocit difference. For the abrupt expansion the average value of.8 is obtained. The minimum value of.7 is suggested for one of the configurations. It should be noted that the 4 expansion gives the highest loss coefficient of.87. No stud could be found in the literature for negative steps either. Comparison of the present data was made with the studies done for rectangular channels having expansion in width. Brater and King (976) reported the local loss coefficient, K =. for abruptl expanding channels and suggested use of the. value for design purposes in expanding channels. Morris and Wiggert (97) suggested using.7 for sharp transitions and. for wedge transitions in the case of expanding sections. In the present stud, as explained above, the average values for the local loss coefficients for abrupt negative steps and 4 inclined negative steps were found to be.89 and.97, respectivel. This stud thus fills the gap in the literature for negative steps. It is seen that local loss occurring due to negative steps depend on the Froude number and the relative step height. Hence, it is suggested that Eqs. (4) and () be used for negative abrupt and 4 inclined steps, respectivel. Relative energ loss for positive and negative steps In the literature, it is commonl assumed that the magnitude of local energ loss will be small. To determine how small these losses are, relative local energ loss, /E was calculated for both positive and negative steps for the range of Froude numbers used in this stud. E is the specific energ on the step for both cases, defined as the sum of depth and velocit head. For both of the step configurations, the Froude number and the specific energ on the step were used. Fig. 6 shows the relative energ loss ( /E )% vs. the Froude number (Fr ) on the step for positive abrupt and inclined steps. It can be seen from Fig. 6 that both losses are compatible, but that the abrupt loss is slightl higher than the inclined loss value. Similarl, Fig. 7 shows the relative energ loss ( /E )% versus the Froude number (Fr ) on the step for negative abrupt and inclined steps. This time, it is seen that, though the losses on the abrupt and inclined steps, are comparable, the abrupt step gives a loss that is slightl lower than that for the inclined step. It can be concluded that the relative loss increases with increasing Froude number. Skogerboe et al. (97) showed that the local loss coefficients are not constants. Also, in the present stud it was found that the negative step ields more loss when compared to the positive step. This is also validated b the work of Brater and King (976). The local loss coefficient, K, was suggested to be. and, for abrupt contractions and expansions, respectivel, even though Formica (9) showed that in general sudden contractions have higher head losses than sudden expansions. / / ( /E )% ( /E ) % / =.49 Fr / Fr / / =.97( - )/(g ) ( - )/(g ) Fr Figure 4 / versus (Fr ) / for negative inclined steps Figure / versus ( - )/(g ) for negative inclined steps Fr Abrupt Inclined Figure 6 Relative energ loss for positive abrupt and inclined steps Abrupt Inclined Figure 7 Relative energ loss for negative abrupt and inclined steps ISSN (Print) = Water SA Vol. 37 No. April ISSN (On-line) = Water SA Vol. 37 No. April 43

8 Conclusions Based on the experiments performed and the obtained results for positive and negative steps in a subcritical open channel flow, for a Froude number range of <Fr<., the following conclusions can be drawn: The local energ loss caused b positive and negative steps in a subcritical open channel flow is a function of the Froude number on the step and relative step height Local energ losses at abrupt positive steps ma be computed b using Eq. ():. 7 z. Fr Local energ losses at 4 inclined positive steps ma be computed b using Eq. ():. 4. z. 4Fr For all practical purposes, K p can be taken as equal to.7 for positive abrupt steps and. for 4 inclined positive steps K p can be taken as equal to.3 and. for positive abrupt steps and 4 inclined positive steps, respectivel Abrupt positive steps generall introduce slightl higher local losses than the inclined positive steps; and the loss increases with increasing downstream Froude number Local energ losses at abrupt negative steps ma be computed b using Eq. (4):. z. 34Fr Local energ losses at 4 inclined negative steps ma be computed b using Eq. ():. z. 37Fr For all practical purposes, K n can be taken as equal to.4 for negative abrupt steps and.49 for 4 inclined positive steps K p can be taken as equal to.89 and.97 for negative abrupt steps and 4 inclined negative steps, respectivel Abrupt negative steps introduce slightl lower local losses than the inclined positive steps and the loss increases with increasing downstream Froude number The relative energ loss occurring in negative steps is higher than that occurring in positive steps Acknowledgement The authors would like to thank to İlker Örsel and Deniz Özdemir for performing experiments at Hdromechanics Laborator of Civil Engineering Department, Middle East Technical niversit, Ankara. References BHALLAMDI SM and CHADRY MH (99) Computation of flows in open-channel transitions. J. Hdraul. Res BINNIE AM (97) laborator experiments on the flow of water over a downward step in an open channel. Int. J. Mech. Sci BRATER EF and KING HW (976) Handbook of Hdraulics. McGraw-Hill, New York. CHOW VT (99) Open-Channel Hdraulics. McGraw-Hill, New York. FORMICA G (9) Esperienze preliminar sulle perdite di carico nei canali, dovute a cambiamenti di sezione (preliminar test on head losses in channels due to cross-sectional changes) L Energia Elettrica 3 (7) MORRIS HM and WIGGERT JM (97) Applied Hdraulics in Engineering. John Wile & Sons, New York. ÖRSEL SI () Local Losses at a Step in a Sub-critical Open Channel Flow. M.Sc. Thesis, Department of Civil Engineering, Middle East Technical niversit, Turke. SKOGERBOE GV, ASTIN LH and BENNETT RS (97) Energ loss analsis for open channel expansions. J. Hdraul. Div. ASCE 97 (HY) SWAMEE PK and BASAK BC (99) Design of rectangular openchannel expansion transitions. J. Irrig. Drain. Eng. 7 (6) SWAMEE PK and BASAK BC (99) Design of trapezoidal expansion transitions. J. Irrig. Drain. Eng. ASCE 8 () SWAMEE PK and BASAK BC (993) Comprehensive open-channel expansion transition design. J. Irrig. Drain. Eng. ASCE 9 () -6. VITTAL N and CHIRANJEVII VV (983) Open channel transitions: Rational method of design. J. Hdraul. Engin. ASCE 9 () ISSN (Print) = Water SA Vol. 37 No. April ISSN (On-line) = Water SA Vol. 37 No. April