PUBLS. INST. GEOPHYS. POL. ACA. SC. E-7 (4) 27 etermato of Sedmet Trasport Characterstc ameter for the Odra Rver Secto Rysard COUFAL ad Zygmut MEYER Scec Uversty of Techology Al. Pastów 5 7-3 Scec Polad e-mal: coufal@ps.pl; meyer@ps.pl Abstract The paper presets a method of calculato of the gra dameter whch should be used for the evaluato of sedmet stream accordg to Ackers-Whte formulae. Based upo seve curve for the sedmet take from the rver bottom the authors gve the formula how to calculate whch should be take to the Ackers-Whte method stead of the recommeded 35 whch s ot satsfactory.. Mathematcal descrpto of flow pheomeo The Chey relato for the rver flow the uform steady moto s defed by the relatoshp: υ = c R I. () For the rver breadth B 5 H t s assumed that R H = H s the depth ad I deotes free water surface slope the rver H 2 Q H B 2 H 2 B 2 () () { ; p } x (2) (2) { ; p } Fg.. Scheme of flow elemets assumed for aalyss.
42 I each cross-secto of the rver υ s the velocty of flowg water B s the breadth of the rver bed ad { ; p } s the set of values represetg seve curve.e. the dameter of certa fracto ad percetage. For further cosderato two cross-sectos the rver were take ad 2. I uform flow the slope of eergy le I * s equal to the free water surface slope the rver. To estmate the eergy losses geeral case the rver flow t s commoly accepted that f the dstace Δx betwee the two cross-sectos ad 2 s small ad 2 the term Δ( υ /2 g) Δx ca be eglected I * ca be take from Chey equato. So we have 2 2 Q I = s *. (2) 2 /3 BH I the above equato s deotes roughess coeffcet after Mag ad Q s the flow. The am of the paper s to cosder sedmet trasport the rver. The sedmet stream was estmated accordg to the Ackers-Whte formulae (Ackers ad Whte 973 Meyer 98). Varous formulae of dfferet authors were checked by Pluta ad Meyer (23) for Odra rver. The Ackers-Whte formulae are a good agreemet wth the other results; ths has a advatage because t cludes two flow factors such as mea velocty the gve cross-secto υ ad the shear velocty υ. So we have ad Q υ = (3) BH υ τ ρ * = b = ghi * (4) ad further calculato I * ca be take accordg to the prevous assumptos from Chey Eq. (2). To troduce to the aalyss the sedmet gras dameter Mag roughess coeffcet accordg to Strckler was appled: s 26. 6 = (5) Prevous research (Rosak 998 Kotas 2) dcates that the roughess coeffcet s s a fucto of the rato /H. So they proposed to modfy Strckler equato. Accordg to Kotas research verfed for lower Odra Rver we have s = M H /6 (6) where M vares practcally for lower Odra rver from.3 to.5. The represetatve dameter ca be evaluated accordg to Rosak (998) the followg form:
43 = p Π = (7) where the symbol Π deotes multplcato of all the terms from = tll = ad = ( p). (8) = Further research ams to estmate a approprate value of whch should be take to the sedmet stream calculato based o the seve curve data {p }. Ths ca be acheved by combg two curves: the oe represetg water flow ad the other represetg sedmet stream based upo samples take from the bottom sedmet. The curve for water flow based upo Eq. 2 s gve Fg. 2 ad the curve for sedmet stream wll be preseted the ext secto. I Fg. 2 H m ad I *m are the measured depth ad slope respectvely. H [m] H m I * m I * Fg. 2. Plot of the fucto H = H(I * ). 2. Model of sedmet trasport Assumg that flow testy sedmet trasport rate the cross-secto ad rver depth as well as the correspodg slope whch has bee measured the steady moto codtos are costat.e. Q= cost ω = cost H = cost I cost ad usg Ackers-Whte s method (Ackers ad Whte 973) the total sedmet trasport rate the rver ca be evaluated as follows: =
44 ω = Qgρ X (9) where: X s G = gr H υ υ m Fgr Ggr = C A () ( ) 2 gr logc = 2.86log log 3.53 gr 9.66 m.34 s = + ad s = ρ gr ρ w F gr υ υ α H 32log ( ) 32log * = g s H α υ () ad the values of υ ad υ were estmated earler (Eqs. 3 ad 4). The value of g s accelerato due to gravty ad s gra se dameter whch s the matter of cosderato ad s s the rato of desty of sedmet ρ s to the water desty ρ w. Furthermore.23 A = +.4 gr =.56log gr ( ) /3 g s gr = 2 ν ad v =.3 6 s the kematcs vscosty coeffcet of water [m 2 /s]. I depedece o the value of gr the followg cases wll be cosdered: for gr floated sedmet ( = ) for < gr 6 totally floated ad dragged sedmet for gr > 6 oly dragged sedmet ( = ; A =.7; m =.5; C =.25). I the orgal Ackers-Whte s method t s recommeded to put = 35 (from the seve curve of bottom sedmet). From the prevous research (Pluta ad Meyer 23)
45 t comes that fxed value of 35 does ot allow to calculate properly the sedmet trasport for varous rver cross-sectos (Coufal 995 Coufal ad Meyer 997). The referece value of whch should be take to the evaluato of sedmet stream should be the matter of further research to specfy =. The depedece of the depth H o the value I * for costat ω for varous gras se dameters s gve Fg. 3 (Meyer ad Skorupska 24 25). O ths fgure the cure H(I * ) from Fg. 2 s addtoally plotted. So oe ca see that f the measured depth s equal to H m ad the correspodg slope s equal to I *m must be equal to. H [m] =5 mm =55 mm 2 =6 mm H 3 = 7 mm m I * m Fg. 3. Optmal dameters satsfyg the Chey equato. I * 3. Evaluato of The aalyses show that the mal sedmet dameter whch wll satsfy the Chey codto ad wll close the equato of Ackers-Whte s sedmet trasport model by terms of quatty ca be descrbed for each rver cross-secto based o the determed value of depth ad slope. The mal sedmet dameter was evaluated from the full seve curve ad ca be descrbed by the statstcal parameters (mea value stadard devato ad skewess of the curve) as follows: k = c + δ c2 + εc3 = f( δ ε ) (2) where the seve curve dstrbuto factors were chose as follows: stadard devato 3 δ = ( ) skewees ε = 3 ( ) p ad s gve by Eq. 8. = 2 p =
46 The results of calculatos are preseted both Table ad some examples Fgs. 4 ad 5. I Table the values are: Q = 85 m 3 /s ad I m =.283. Table The results of calculatos for the set of 554 566 km km B [m] H [m] / 552 3.5 2.25.5379.6335.777 55 7.2 2.2.5447.69.2668 548 5.85 2.7.767.8468.85 546 6.75 2.5.5874.528.899 544 9. 2.2.5327.494.926 542 9.25 2.3.589.67484.457 6 4 2 8 6 4 2 f( δε) 5 5 2 f( δε) 578 67 km 554 566 km 8 6 4 2 8 6 4 2 5 5 2 Fg. 4. Statstcal mato results for sets of 578 67 km ad 554 566 km. f( δε) 542.4 67 km f( δε)= 3 Fg. 5. Statstcal mato results for all the sets.
47 Combg all the data for all cross-sectos over the whole aalyed dstace we have: C =.622; C 2 = 497.33; C 3 = 537.6 ad k =.3. The resultg formulae takes the form.3 =.622 497.33 δ + 537.6 ε. (3) 4. Coclusos. The expermetal research of sedmet trasport lower Odra Rver s preseted. The expermets cocer estmato of represetatve gra dameter sedmet trasport calculato. 2. The basc assumpto of the method was the steady flow ad costat sedmet stream alog tested rver dstace. The seve curve for sedmet samples take from the bottom of the rver was the backgroud for verfcato. 3. The statstcal aalyss of the seve curves leads to the cocluso that t s possble to relate the mal gra-se dameter for sedmet stream calculato upo seve curve dstrbuto factors. The mathematcal formulae for ths relatoshp s also gve. It gves good agreemet wth expermetal data for the whole aalyed dstace. Refereces Ackers P. ad W.R. Whte 973 Sedmet trasport. New Approach ad Aalyss. Joural of the Hydraulcs vso ASCE Vol. 99. Coufal R. 995 Zmay położea da w ujścowym odcku rek wywołae ruchem rumowska Wyd. Ucelae Poltechk Scecńskej Scec. Coufal R. ad Z. Meyer 997 Badae ależośc pomędy atężeem prepływu a średcą marodają dla odcka Odry Środkowej Istytut Budowctwa Wodego Polskej Akadem Nauk w Gdańsku XVII Ogólopolska Skoła Hydraulk Gdańsk-Sobesewo. Kotas W. 2 Wpływ abudowy regulacyjej koryta a may położea da rek Ph Thess Poltechka Scecńska. Meyer Z. 98 Hydraulka stosowaa c. I. Podstawy ruchu rumowska ora oblcae stablośc da rek Wyd. Poltechk Scecńskej Scec. Meyer Z. ad W. Skorupska 24 Aalyss of substtutve sedmet dameter chages uder crcumstaces of the Lower Odra XXIV Iteratoal School of Hydraulcs Jastręba Góra September 3-7 Polad. Meyer Z. ad W. Skorupska 25 Aalyss of represetatve dameter descrbg sedmet trasport rate the Mddle Odra Rver secto XXV Iteratoal School of Hydraulcs ebryo September 2-6 Polad.
48 Pluta M. ad Z. Meyer 23 Ocea prydatośc worów emprycych dla określea atężea rumowska w Odre XXIII Ogólopolska Skoła Hydraulk Współcese Problemy Hydraulk Wód Śródlądowych Tleń wreseń 2-6. Rosak A. 998 Hydraulce waruk rodału strumea rumowska w rowdleu recym w warukach prepływu welkch wód a prykłade węła wodego Wduchowa a Odre Ph Thess Poltechka Scecńska.