Application of the Modified Log-Wake Law in Open-Channels

Transcription

1 Jornal of Applied Flid Mecanics, Vol., No. 2, pp. 7-2, 28. Available online at ISSN Application of te Modified Log-Wake Law in Open-Cannels Jnke Go and Pierre Y. Jlien 2 Department of Civil Engineering, University of Nebraska, Omaa, NE , USA 2 Department of Civil Engineering, Colorado State University, Fort Collins, CO 685, USA jgo2@nl.ed (Received December 4, 26; accepted Agst 7, 27) ABSTRACT Te modified log-wake law, wic was developed for trblent bondary layers and pipe flows, is extended to trblent flows in open-cannels. Trblent velocity profiles in open-cannels can be approximated wit tree components: () te law of te wall tat reslts from te constant bed sear stress; (2) te law of te wake tat reflects te effects of gravity, secondary crrents and bed rogness; and () te cbic correction near te maximm velocity. A procedre to determine te for model parameters from velocity measrements wile keeping κ =.4 is presented. Te modified log-wake law compares very well wit experimental data from Coleman, Lyn, Kironoto and Graf and Sarma et al. It also replicates te measred velocity profiles of te Mississippi River. In particlar, it can well fit te velocity dip penomenon in opencannels were te conventional log-wake law fails. Keywords: Open cannel flow, trblence, velocity dip penomenon, velocity distribtion, velocity profile. NOMENCLATURE a,b,c B Fr k s Re y y i y p S T fitting parameters of te parabolic law additive constant of te log law Frode nmber flow dept Nikradse rogness eigt Reynolds nmber distance from te bed discrete distance from te bed zero-velocity position over rog bed MatLab fitting parameter cannel slope Temperatre U i max W δ κ ν Π ξ vertical average velocity time-averaged velocity at a distance y time-averaged velocity at a distance y i maximm velocity at y = δ sear velocity cannel widt dip distance from te bed von Karman constant kinematic viscosity of water Coles wake strengt normalized distance,. INTRODUCTION Field measrements sowed tat most natral river flows are tree-dimensional de to te presence of large-scale free-srface secondary crrents (Nez, Tominaga and Nakagawa 99). Te measred maximm velocity sally appears below te free srface at a distance of.5 to.5 of te flow dept (Cow 959, p.24; Ceng and Gartner 2; Moramarco et al. 24), wic is called te velocity dip penomenon. Modeling te dip penomenon is significant for establising stage-discarge relationsips and for te analysis of resistance to flow and contaminant transport. It is also important to define te relationsip between srface and mean flow velocities (Lee and Jlien 26). Previos stdies focsed primarily on two-dimensional flows were secondary crrents can be neglected and te maximm velocity occrs at te free srface. For sc flows, velocity profiles can be approximated by te conventional log law or te log-wake law (Steffler et al. 985; Nez and Rodi 986; Kirkgoz 989; Cardoso et al. 989; Kironoto and Graf 994; Mste and Patel 997).

2 Jnke Go and Pierre Y. Jlien / JAFM, Vol., No. 2, pp. 7-2, 28. By contrast, altog te velocity dip penomenon as been reported for a long time (Leigly 92), or nderstanding of it is poor and only a few stdies on it can be fond in te literatre (Sarma et al. 98, 2; Ci and Said 995; Ci and Tng 22; Moramarco et al. 24; Go and Jlien 2). Te velocity dip penomenon can ardly be modeled wit log-wake velocity profiles becase it imposes a velocity increase wit distance from te bondary. Recently, Go and Jlien (2) and Go et al. (25) proposed a modified log-wake law (MLWL) tat well represents experimental data in pipes and zero-pressregradient (ZPG) bondary layers. Since open-cannel flows associated wit te dip penomenon are similar to tose in pipes and bondary layers were a zero velocity gradient exists at te maximm velocity, te objective of tis paper is to extend te MLWL to open-cannels associated wit te velocity dip penomenon. 2. THE MODIFIED LOG-WAKE LAW (MLWL) According to Go and Jlien (2) and Go et al. (25), te modified log-wake law reads y 2Π 2 πξ ξ = ln + B + sin () κ ν κ 2 κ were = time-averaged velocity in te flow direction, = sear velocity, κ = von Karman constant, y = distance from te wall, ν = kinematic viscosity of te flid, B = additive constant tat relates to te wall rogness, Π = Coles wake strengt, and ξ = normalized distance relative to te dip position δ. Te terms in parenteses are te logaritmic law of te wall; te sine-sqare term is te law of te wake tat expresses te effects of te constant pressre-gradient in pipes or te convective inertia in ZPG bondary layers; and te cbic fnction forces te log law gradient to be zero at te maximm velocity.. TEST WITH FLUME DATA. DETERMINATION OF THE MODEL PARAMETERS For convenience, we replace te additive constant B in Eq. () wit te maximm velocity max, i.e. max 2Π 2 πξ ξ = lnξ + cos (2) κ κ 2 κ By convention, we assme κ =.4 in tis paper, ts leaving for parameters to be determined from a measred velocity profile. We need to know te parameters max,, δ and Π to plot a profile. Note tat δ is embedded in ξ. Since te MLWL redces to a parabolic law near te point of maximm velocity (Go and Jlien 2), we assme 2 = ay + by + c for data wit >.6, were a, b and c are fitting parameters. Eqation () gives te dip position at and te maximm velocity () b δ = (4) 2a max 2 b = c (5) 4a We ten normalize te distance y as. Applying Eq. (2) to data wit ξ <.2 gives te law of te wall, were = lnξ + B κ (6) B = max 2Π (7) κ We can get te sear velocity from te slope /κ and te wake strengt Π from te intercept B. Te data sets of Coleman (986), Lyn (986), Kironoto and Graf (994), and Sarma et al. (2) are sed to test te MLWL and te above procedres..2 DATA ANALYSIS Coleman's (986) data are widely sed in te literatre. Te tree clear water rns (Rns, 2 and 2) are sed in tis analysis. To illstrate te above procedres, take RUN for an example. Te measred velocity profile data and corresponding positions are: i = (.79,.77,.82,.849,.884,.927,.98,.26,.54,.5,.48,.9); y i = (6, 2, 8, 24,, 46, 69, 9, 22, 7, 52, 62 ); were i is in m/s and y i in mm. We can see tat te dip position is at abot y = 22 mm corresponding to te velocity =.54 m/s. To accrately locate te maximm velocity, we fit te last 6 data to te parabolic law, Eq. (), wic gives te dip position δ = 2 mm from Eq. (4) and te maximm velocity max =.56 m/s from Eq. (5). Te distance y i is ten normalized by δ. Te first 4 data satisfy te condition ξ ι = y i / δ <.2 and are sed to fit te log law, Eq. (6). Te slope /κ and intercept B are fond to be.24 and.25, respectively. Assming κ =.4, te sear velocity is ten =.42 m/s and te wake strengt from Eq. (7) is Π =.2, wic is larger tan.9 obtained by Coleman (986). Tis difference is de to te cbic correction term at te maximm velocity. Te experimental and calclated model parameters for te tree clear water rns are tablated in Table ; and te comparison of te MLWL wit te experimental data is plotted in Fig., were Fig. a is in rectanglar coordinates and Fig. b in semilog plot. 8

3 Jnke Go and Pierre Y. Jlien / JAFM, Vol., No. 2, pp. 7-2, 28. Similarly, we list te parameters of experiments of Lyn (986, 2) in Table, and Kironoto and Graf (994) as well as Sarma et al. (2) in Table 2. Te corresponding comparisons are sown in Figs. 2, and 4, respectively. All te for data sets confirm te fnctional strctre of te MLWL. However, from Tables and 2, we can see tat te wake strengt varies between and.48. Tis means a niversal wake strengt Π does not exist in open-cannel flows, nlike ZPG bondary layers and pipe flows. Table - Experimental and calclated parameters of Coleman and Lyn Data set Coleman (986) Lyn (986) RUN RUN2 RUN2 C C2 C C4 () (2) () (4) (5) (6) (7) (8) S ( - ) T ( C) (m) U (m/s) Re = 4U/ν ( 5 ) Fr = U/(g) / W/ δ, m, from (4) max, m/s, from (5) , m/s, from (6) П, from (7) δ/ Table 2- Experimental and calclated parameters of Kironoto and Graf, and Sarma et al. Data set Kironoto and Graf (994) Sarma, Prasad and Sarma (2) UGA UGA5 UGB UGB5 A B C D E () (2) () (4) (5) (6) (7) (8) (9) () S ( - ) T ( C) (m) &.85 U (m/s) to.92 Re = 4U/ν ( 5 ) to 8. Fr = U/(g) / W/ to 5.64 δ, m max, m/s , m/s П δ/ Data of Coleman (986) Te MLWL, Eq.(2) RUN RUN 2 Sift by 4 RUN 2 RUN Sift by 4 RUN 2 RUN Fig. - Comparison of te MLWL wit Coleman s data set 9

4 Jnke Go and Pierre Y. Jlien / JAFM, Vol., No. 2, pp. 7-2, Data of Lyn (986) Te MLWL, Eq.(2) C C2 C C4 C C2 C C4.8.6 Sift 2.4 Sift Fig. 2- Comparison of te MLWL wit Lyn s data set.6.4 Data of Kironoto and Graf (994) Te MLWL, Eq.(2) UGA UGA5 UGB UGB5 Sift 5 UGA UGA5 UGB UGB5 Sift Fig. - Comparison of te MLWL wit Kironoto and Graf s data set 2 Data of Sarma, et al. (2) Te MLWL, Eq.(2) E RUN A B D C E.5 C D ξ = y/ δ RUN A B Fig. 4- Comparison of te MLWL wit Sarma s data set. 2

5 Jnke Go and Pierre Y. Jlien / JAFM, Vol., No. 2, pp. 7-2, APPLICATION TO FIELD MEASUREMENTS Altog a niversal vale of Π does not exist, te MLWL as a clear application in flow measrements. Given a few sampled velocities, it can provide te vertical average velocity wit dip penomenon. Figre 5 defines te teoretical bed in natral rivers. y y y Non-niform rogness Teoretical bed Nominal bed in wic [p, p2, p, p4] are initial vales of te optimization. Te for parameters are ten obtained from Eq. (9). Frtermore, we can get te maximm velocity max δ = ln + 2Π κ y () Integrating Eq. (8) between y and and divided by -y gives te vertical average velocity U ln κ y = 2 y y δ y Π δ y π sin π y + Π ( y ) ( δ y ) For >>y and δ >>y, te above can be simplified as () Π δ π = (2) U ln sin + Π κ y π δ 2 δ y y y Cannel wit bedforms Fig. 5- Sceme of teoretical bed in field measrements Referring to Fig. 5, we can rewrite Eq. () as y y y 2Π ln + κ y y δ κ = 2 π ( y y) sin 2( δ ) y (8) Application to a Mississippi River velocity profile measrement: Figre 6 sows a velocity profile measrement on a vertical tat is sitated at te deepest location in a cannel section of te Mississippi River (Ci and Said 995; Gordon 992). Fitting Eq. (8) to te measred data and applying te above MatLab fnctions, we obtain: =.2 m/s, y =.5 m, δ = 22.2 m, and Π =.2. Te maximm velocity from Eq. () is max =.2 m/s, and te average velocity is U =.97 m/s from Eq. () wile.96 m/s from Eq. (2). Te teoretical bed y relates to te eqivalent Nikradse rogness k s given k s = y, wic gives k s = m tat is definitely nreasonable and implies bed forms exist. Te dip position δ is located at abot a tird of water dept below te free srface. Te comparison in Fig. 6 sows excellent agreement between te MLWL and te real velocity distribtion measrements. 5 Mississippi River Data (Gordon 992) Fitted to te MLWL, Eq.(2) in wic y is based on an arbitrary vertical coordinate and y is te teoretical bed were te velocity is zero. Note tat te constant B in Eq. () or te maximm velocity max in Eq. (2) as been inclded in te vale y in Eq. (8). Given measrements (y i, i ), assming κ =.4, we ave for fitting parameters,, y, δ, and Π. Let p() = p() = δ, / κ, p(2) = y p(4) = 2Π / κ te for parameters can be fitted sing a nonlinear optimization program. For example, wit MatLab te parameters can be fond by te following fnctions: fn = inline('p().(log(zi./p(2)) - ((zi-p(2))./(p()-p(2))).^./) + p(4).sin(pi.(zi-p(2))./2./(p()- p(2))).^2','p','zi'); p = lsqcrvefit(fn,[p p2 p p4], zi, i); (9) y, m = m y =.5 m δ = 22.2 m =.2 m/s max =.2 m/s U =.97 m/s , m/s Fig. 6- A velocity profile in Mississippi River Note tat nlike flme experiments, tis field data set sows negative wake strengt, wic is close to zero and sold be considered negligible compared to te vales listed in Tables and 2. 2

6 Jnke Go and Pierre Y. Jlien / JAFM, Vol., No. 2, pp. 7-2, CONCLUSIONS Te following conclsions can be drawn from te above analysis: ) Te modified log-wake law (MLWL) can be applied to describe trblent velocity profiles in open-cannel flows. 2) Besides te von Karman constant κ =.4, te MLWL incldes for additional model parameters: (i) te dip position δ from te bed; (ii) te bed sear velocity ; (iii) te wake strengt Π; and (iv) eiter te integration constant B, te maximm velocity max, or te teoretical bed elevation y for rog cannels. ) Te MLWL compares very well wit flme data from Coleman (986), Lyn (986), Kironoto and Graf (994) and Sarma et al. (2). In particlar, it can well fit te velocity dip penomenon near te free srface. 4) A procedre of applying te MLWL to field measrements is proposed. Te application to a Mississippi River velocity profile measrement sows a good agreement between te MLWL and te field data. In general, te empirical procedre to determine te for parameters of te modified log-wake law (MLWL) reslts in excellent profiles compared wit laboratory and field measrements. Except for te von Karman constant set at κ =.4, te model does not yield niversal vales of te for oter model parameters for generalized predictive prposes. ACKNOWLEDGEMENTS Te ators are gratefl to Dr. D. A. Lyn at Prde University, Dr. M. Mste at te University of Iowa, and Dr. W. Graf at te Swiss Federal Institte of Tecnology at Lasanne for providing teir laboratory experimental data for tis stdy. Tis stdy was spported by te University of Nebraska Academic Researc Fnd nder Grant No for te first ator. REFERENCES Cardoso, A. H., Graf, W. H. and Gst, G. (989). Uniform flow in smoot open-cannel. J. Hydr. Res., IAHR, 27(5), Ceng, R. T., and Gartner, J. W. (2). Complete velocity distribtion in river cross-sections measred by acostic instrments. Proceedings of te IEEE/OESS 7t Working Conference on Crrent Measrement Tecnology, San Diego, CA, Marc -5, 2, p Ci, C. L. and Said, C. A. A. (995). Maximm and mean velocities and entropy in open-cannel flow. J. Hydr. Engrg., ASCE, 2(), Ci, C. L. and Tng, N. C. (22). Maximm velocity and reglarities in open-cannel flow. J. Hydr. Engrg., ASCE, 28(4), Cow, V. T. (959). Open-Cannel Hydralics. McGraw- Hill, 68p. Coleman, N. L. (986). Effects of sspended sediment on te open-cannel velocity distribtion. Water Resorces Researc, AGU, 22(), Gordon, L. (992). Mississippi river discarge. RD Instrments, San Diego, CA. Go, J. and Jlien, P. Y. (2). Trblent velocity profiles in sediment-laden flows. J. Hydr. Res., IAHR, 9(), -2. Go, J. and Jlien, P. Y. (2). Modified log-wake law for trblent flows in smoot pipes. J. Hydr. Res., IAHR, 4(5), Go, J., Jlien, P. Y. and Meroney, R, N. (25). Modified Log-Wake Law in Zero-Pressre-Gradient Trblent Bondary Layers. J. of Hydr. Researc, IAHR, 4(4), Kirkgoz, S. (989). Trblent velocity profiles for smoot and rog open-cannel flow. J. Hydr. Engrg., ASCE, 5(), Kironoto, B. A. and Graf, W. H. (994). Trblence caracteristics in rog niform open-cannel flow. Proc. Instn Civ. Engrs Wat. Marit. & Energy, 6(2), -44. Lee, J.S. and P.Y. Jlien (26). Electromagnetic Wave Srface Velocimetry. J. Hydr. Engrg., ASCE, 2(2), Leigly, J. B. (92). Toward a teory of te morpologic significance of trblence in te flow of water in streams. University of California Pblications in Geograpy, 6(), -22, Cambridge University Press. Lyn, D. A. (986). Trblence and Trblent Transport in Sediment-Laden Open-Cannel Flows. W. M. Keck Laboratory of Hydralics and Water Resorces, California Institte of Tecnology, Pasadena, CA. Lyn, D. A. (2). Regression Residals and mean profiles in niform open-cannel flows. J. Hydr. Engrg., ASCE, 26(), Moramarco, T., Saltalippi and Sing, V. P. (24). Estimation of mean velocity in natral cannels based on Ci's velocity distribtion eqation. J. Hydrologic Engrg., ASCE, 9(), Mste, M. and Patel, V. C. (997). Velocity profiles for particles and liqid in open-cannel flow wit sspended sediment. J. Hydr. Engrg., ASCE, 2(9),

7 Jnke Go and Pierre Y. Jlien / JAFM, Vol., No. 2, pp. 7-2, 28. Nez, I., Tominaga, A. and Nakagawa, H. (99). Field measrements of secondary crrents in straigt rivers. J. Hydr. Engrg., ASCE, 9(5), Nez, I. and Rodi, W. (986). Open-cannel flow measrements wit a Laser Doppler Anemometer. J. Hydr. Engrg., ASCE, 2(5), Sarma, K. V. N., Laksminarayana, P., and Rao, N. S. L. (98). Velocity distribtion in smoot rectanglar open-cannels., J. Hydr. Engrg., ASCE, 9(2), Sarma, K. V. N., Prasad, B. V. R. and Sarma, A. K. (2). Detailed stdy of binary law for open cannels. J. Hydr. Engrg., ASCE, 26(), Steffler, P. M. Rajaratnam, N. and Peterson, A. W. (985). LDA measrement in open cannel. J. Hydr. Eng., ASCE, (), 9-. 2