SOLUTION OF THREE-CONSTRAINT ENTROPY-BASED VELOCITY DISTRIBUTION

Transcription

1 SOLUTION OF THREECONSTRAINT ENTROPYBASED VELOCITY DISTRIBUTION By D. E. Barbe,' J. F. Cruise, 2 and V. P. Singh, 3 Members, ASCE ABSTRACT: A twdimensinal velcity prfile based upn the principle f maximum entrpy (POME) fr wide pen channel flws is presented. The derivatin is based n the cnservatin f mass and mmentum. The resulting prfile invlves three parameters that are determined frm bservatins f mean velcity and the velcity at the water surface. The velcity prfile is verified using field data in a river with a live bed. A cmparisn with three existing methds shws that the prfile presented is the mst accurate f the three, especially near the bed. INTRODUCTION There are presently tw basic methds and a recently prpsed methd t btain a timeaveraged hrizntal velcity prfile: the lgarithmic distributin law, the pwer law, and the tw cnstraint entrpy methds by Chiu (1987). The entrpy methd uses the principle f maximum entrpy (POME) t maximize the infrmatin cntent f the data. The entrpy methd prduces fur integral equatins in fur unknwns. These equatins are derived frm the physical cnstraints n the system and are nt slvable by exact analytical means except fr the tw cnstraint cases (Chiu 1987). Here, the entrpy methd with three cnstraints is used, and an apprximate slutin t the resulting integral equatins is determined. The resulting equatin is then cmpared t the ther methds stated previusly using actual field data. DERIVATION OF VELOCITY DISTRIBUTION USING POME The cncept f entrpy can be applied in mdeling the vertical prfile f the hrizntal velcity in pen channel flw. Fur cnstraints can be develped fr use in the entrpy methd, namely, cnstraints fr prbability, cntinuity, mmentum, and energy. Chiu (1987, 1989) used this methd t derive a velcity distributin fr the hrizntal velcity in a wide, pen channel with unifrm flw. Chiu (1987) btained an exact slutin fr the entrpy methd when nly tw f the cnstraints n the system were used. The cnstraints he used were the prbability and the cntinuity cnstraints. Here the cnstraint btained frm mmentum cnsideratin prpsed by Chiu (1989) is added and the apprximate slutin t the resulting integral equatins is btained. Frm bundary shear cnsideratins, the classical methd f describing 'Asst. Prf., Dept. f Civ. Engrg., Univ. f New Orleans, New Orleans, LA Assc. Prf., Dept. f Civ. Engrg., Luisiana State Univ., Batn Ruge, LA Prf., Dept. f Civ. Engrg., Luisiana State Univ., Batn Ruge, LA Nte. Discussin pen until March 1,1992. T extend the clsing date ne mnth, a written request must be filed with the ASCE Manager f Jurnals. The manuscript fr this paper was submitted fr review and pssible publicatin n August 9, This paper is part f the Jurnal f Hydraulic Engineering, Vl. 117, N. 10, Octber, ASCE, ISSN $ $.15 per page. Paper N

2 the twdimensinal velcity prfile in wide channels (vn Karman 1935) is by relating it t the depth. In wide, pen channel flw with depth D, the velcity mntnusly increases frm zer at the bed t a maximum value at the surface when the waterair interface shear is neglected. Let u be the velcity at a distance y abve the channel bed. Then, the prbability f the velcity being less than r equal t u is yid and the cumulative distributin functin is p{u) = ^ :...(i) and the prbability density functin is n^(m *> Chiu (1987, 1989) used the POME and the cnstraints n the system based n prbability and the three cnservatin principles, namely cnservatin f mass, mmentum, and energy, t btain the prbability density functin f u as (u) = exp(a + L 2 u + L 3 u 2 + L 4 u 3 ) (3) and the velcity prfile as I exp(a + L 2 u + L 3 u 2 + L 4 u 3 ) du = ~ + C (4) where A = L 1 1; L u L 2, L 3, and L 4 = Lagrange multipliers; and C = the cnstant f integratin t be evaluated by the bundary cnditin, u = 0 at y; = 0. Chiu (1987) then let L 3 = L 4 = 0 and slved this equatin using nly the first tw cnstraints t btain u = ~ mjl + [exp(l 2 u D ) 1] gj (5) where u D and L 2 = the parameters. u D and L 2 are related t u m by u m = u D exp(l 2 u D ) [exp(l 2 u D ) l]" 1 (6) L 2 Eq. (5) is the entrpybased, twcnstraint, velcity prfile equatin, fr flw in a wide channel, develped by Chiu (1987). APPROXIMATION OF ENTROPY DISTRIBUTION Chiu (1989) shwed that the value fr L 4 was small fr the data that he used in his analysis. If in general L 4 = 0, the entrpybased, threecnstraint, velcity distributin based n mmentum is btained [(3) with L 4 = 0]. This is evaluated using Lagrange multipliers t maximize the entrpy subject t the three cnstraints representing prbability, cntinuity, and mmentum. These equatins are nt slvable by exact analytic means. An expnential functin can be apprximated by a Maclaurin series. An apprximate slutin fr the threecnstraint entrpy methd is btained by expansin f the term invlving the third parameter, L 3. Using the first 1390

3 tw terms f the Maclaurin series expansin, and slving the cnstraints by integratin by parts gives three equatins in the three unknwn Lagrange multipliers, A, L 2, and L 3. Slving these equatins simultaneusly leads t the velcity prfile equatin fr the twterm Maclaurin series expansin f the threecnstraint entrpy methd based n mmentum as exp(l)j exp(l 2 w) + L 3 exp(l 2 H)( u 2 j + exp(^4) _L ih L, where L 2 is btained frm the fllwing equatin exp(l 2 u D ) u D _1_ u 7A_ 14 exp(l 2 w D ) K x u 2 D 2^ 24 exp(l 2 M Z) ) u 2 D 2u U m U D 2u L L, 2u D K l 2u m u D J_ U f2 U D T, L,2 J*2 K x 2u, Q K, u 3 3w D + U Uu 2 D T 2 6w D 24M D 6_ L\ and K x = MI(pD) = ^uf n, (p = the mmentum cefficient). The value f L 3 is btained by substitutin f L 2 int the equatin L, = exp(l 2 u D ) \u D 1 \ exp(l 2 u D )( U, U 2 D f 2 + JJ ~ K L, 3K 2 D 6»p 6 Q LI 2u, 6_ Zi The cefficient A is then btained by substitutin f L 2 and L 3 int the equatin frm the slutin f the prbability cnstraint: L 2 exp( A) = [exp(l 2 w fl ) 1] + L exp(l 2 «D ) u 2 D 2Ur LJJ + ^ ''2, 2_ L\ (7) (8) (9) (10) COMPARISON OF VELOCITY PROFILE METHODS The basic methds that are cmpared are: (1) Lgarithmic distributin law f Prandtlvn Karman; (2) pwer law; (3) twcnstraint entrpy 1391

4 methd [(5)]; and (4) threecnstraint entrpy methd based n mmentum [(7)]. In each cmparisn, it will be assumed that nly the average and mean velcities are knwn. All the prfiles are cmputed fr the variables and cmpared t the data presented in the wrk f Davren (1985). This is actual field data fr a river with a live bed. Davren measured velcities f the flw dwnstream frm a hydrpwer plant. This prvided steady unifrm flws ver several hurs fr his measurements. A cmparisn t Davren's run 1 will be shwn fr all fur methds. Als, the twcnstraint entrpy methd [(5)] and the threecnstraint entrpy methd based n mmentum [(7)] are cmpared t Davren's run 6 and run 10. The Prandtlvn Karman universal lgarithmic velcity distributin can be stated as fllws (Daugherty and Franzini 1977): u = u D K In (11) where S = the slpe f the energy grade line; and K = the vn Karman cnstant, having a value f abut 0.40 fr clear water and a value as lw as 0.2 fr sediment laden water (Daugherty and Franzini 1977). This is a distributin with ne parameter that is determined by the maximum r the mean velcity nly. In practice, the value f K and S are nt knwn. Therefre t cmpare the different methds, the bed slpe is used as an estimate f S fr unifrm flw and the range f K frm Daugherty and Franzini (1977) is shwn in Fig. 1. The pwerlaw velcity distributin fr flw in an pen channel can be stated as fllws (Sarma et al. 1983): u u n (i«) * K0 2 _~~ *" ^^"" S"'"^ uum R 2 =0.935 k=0.4 D=3.14 m Um=2.38 ms NFO.429 NR=7.45X10 S * ^ 1.5 UCms5 bserved data * * (12) FIG. 1. Prandtlvn Karman Velcity Distributin Pltted Against Observed Prfile (Run 1) 1392

5 where n = a parameter determined by the frierinal resistance at the bed that usually is in the range f 67 (Karim and Kennedy 1987). In practice, n is nt knwn and its value is ften estimated frm surces as stated previusly. This is als a distributin with ne parameter that is determined by the maximum velcity nly. The velcity distributin btained frm this equatin is pltted against the bserved prfile in Fig. 2. The fit fr the lg and pwer distributins culd be imprved if the values f K and n are knwn. In cases where bserved prfiles are available, ptimum values f K and n culd be btained by leastsquares methds. Hwever, in actual practice these values are nt knwn, therefre generally accepted estimates f their value are used fr cmparisn. T use the threecnstraint entrpy methd based n mmentum a value f p = 1 is used fr a first estimate f K t in (8). The first iteratin f the velcity prfile is then btained using (7), (8), (9), and (10). The velcity prfile is then used t btained a secnd estimate f (3 and therefre K x. An iterative prcess yields the velcity prfile f the threecnstraint entrpy methd based n mmentum. The velcity prfile btained frm the threecnstraint entrpy methd [(7)] and the twcnstraint entrpy methd [(5)] are pltted against the bserved prfile in Figs. 3,4, and 5. REMARKS AND CONCLUSIONS The fit f the Prandtlvn Karman lgarithmic velcity prfile was nly gd clse t the surface f the flw. As the value f y is decreased, the departure frm the bserved data f the equatin becmes apparent (R 2 = 0.935). The lgarithmic velcity prfile was ttally unacceptable near the bed f the channel. Therefre it seems reasnable t cnclude that this velcity prfile wuld nt be apprpriate fr any evaluatin f nearbed prcesses such as scur. Hwever, a mdified versin such as by Christensen (1972) des nt have this shrtcming. Still, a value fr K is needed fr its * n6 0 urn n=7 R 2 =0.975 D=3.14 m Um=2.38 ms N F=0.429 NR =7.45X10 0 ^^' V T.*^^^', 2.0 UCms) bserved data v7 2.5 J 11 FIG. 2. Pwer Law Velcity Distributin Pltted Against Observed Prfile (Run 1) 1393

6 l.'o 0.5 * EQN. (5) pnjrfl i EQN.C 73 D=3.14 m Um=2.38 ms NF=0.429 NR=7.45X10 R U(ms) bserved data OK R =0.990 FIG. 3. Entrpy Velcity Distributins Pltted Against Observed Prfile (Run 1) M S l.4 " * EQN.C5) EQN.C7) 1.2 l.o D=l.09 m Um=2.68 NF=0.820 ms NR=2.91X10 R 2 =0 bservbd data 1.998, ji R E = UCms) FIG. 4. Entrpy Velcity Distributins Pltted Against Observed Prfile (Run 6) use. Of curse, the fit culd have been imprved by btaining K by least squares. The velcity prfile fr the pwer law was gd fr a value f n = 6 (R 2 = 0.975). Again, this prfile fit the bserved data best near the channel surface and departed frm this fit as the value f y is decreased. Even thugh the fit f the pwerlaw velcity distributin was better than that f the lgarithmic velcity distributin near the channel bed, mre accuracy fr use in the determinatin f nearbed prcesses is still desirable. The twparameter entrpy velcity prfile [(5] had a superir fit t the bserved data than the tw previusly mentined velcity prfiles (R 2 = 1394

7 * EQN.C5) mm n EQN.C7) bserved data D=0.70 m Um=2.24 ms NF=0.855 NR=1.56X10 R = = U(ms) t; ; " * 2 R 0.992,, FIG. 5. Entrpy Velcity Distributins Pltted Against Observed Prfile (Run 10) fr run 1). The fit f this velcity prfile was nt nly gd at the surface f the channel but als near the channel bed. The velcity prfile btained by the threecnstraint entrpy methd based n mmentum [(7)] had the best fit t the bserved data f any f the methds cmpared (R 2 = fr run 1). The fit f this prfile was particularly gd near the bed f the channel. It is f interest t cmpare the tw and threecnstraint entry methds in rder t determine the relative value f the third cnstraint. In the twcnstraint methd, nly the prbability and masscnservatin cnstraints are bserved. In the threecnstraint methd, the hydrdynamics f the flw are intrduced in the frm f the mmentum cnservatin principle. Then, the questin arises as the relative benefit f this imprvement in the case f unifrm flw. Of curse, this questin cannt be cnclusively answered based upn the limited data samples analyzed in this study. Hwever, taken as a whle, it appears that the threecnstraint methd des nt ffer a significant imprvement t the fit f the verall prfile in any f the cases analyzed. As expected, the maximum imprvement is greatest near the bttm f the prfile, r near the channel bed. This may prve t be significant if nearbed prcesses, such as scur r sediment transprt in the frm f bed lad, are t be analyzed. APPENDIX I. REFERENCES Chiu, C. (1987). "Entrpy and prbability cncepts in hydraulics.". Hydr. Engrg,, ASCE, 113(5), Chiu, C. (1989). "Velcity distributin in pen channel flws.". Hydr. Engrg., ASCE, 115(5), Christensen, B. A. (1972). "Incipient mtin n chesinless channel banks." Prc. f Symp. t hnr H. A. Einstein, University f Califrnia, Daugherty, R. L., and Franzini, J. B. (1977). Fluid mechanics with engineering applicatins. McGrawHill Bk C., New Yrk, N.Y., Davren, A. (1985). "Lcal scur arund a cylindrical bridge pier." Publicatin N. 1395

8 3 f the Hydrlgy Centre, Nat. Water and Sil Cnservatin Authrity. Ministry f Wrks and Dev., Christchurch, New Zealand. Karim, M, F., and Kennedy, J. F. (1987). "Velcity and sediment cncentratin prfiles in river flws.". Hydr. Engrg., ASCE, 113(2), Sarma, K. V. N., Lakshminarayana, P., and Ra, N. S. L. (1983). "Velcity distributin in smth rectangular pen channels.". Hydr. Engrg., ASCE, 109(2), vn Karman, T. (1935). "Sme aspects f the turbulence prblem." Mech. Engrg., 57(7), APPENDIX II. NOTATION The fllwing symbls are used in this paper: A, L 1; L 2, L 3, and L 4 = Lagrange multipliers; C = cnstant f integratin; D = depth f flw in the channel; g = acceleratin due t gravity; K = vn Karman universal cnstant, which has value f 0.40 fr clear water and value as lw as 0.2 in flws with heavy sediment lads; K t = M(pD) = pt4; M = the mmentum flux per unit width; n = parameter determined by frictinal resistance at bed («is usually in range f 67); S = slpe f energy grade line; u = hrizntal velcity at distance y frm channel bed; u D = maximum velcity f flw; u m = mean velcity (depth averaged); (J = mmentum cefficient; and p = mass density f water. 1396