TURBULENT SHEAR STRESS IN HETEROGENEOUS SEDIMENT-LADEN FLOWS

Transcription

1 TURBULENT SHEAR STRESS IN HETEROGENEOUS SEDIMENT-LADEN FLOWS By Hyoseop Woo, 1 Associate Member, ASCE, ad Pierre Y. Julie, 2 Member, ASCE INTRODUCTION Curret kowledge of the mechaics of alluvial chaels depeds very largely o calculatios of turbulet shear stresses; typical examples are the begiig of motio of sedimet particles ad sedimet trasport i alluvial chaels. If shear stress ca be well defied i clear-water flows, comparatively little is kow about shear stresses i sedimet-lade flows. Eistei ad Chie (1955) proposed a turbulet shear stress relatioship that does ot deped o the fall velocity of uiform sedimet particles. Opposite results were later proposed by Zagusti (1969) ad Willis (1972). This study briefly reviews existig relatioships for turbulet shear stress i homogeeous sedimet-lade flows, ad proposes a derivatio of turbulet shear stress i heterogeeous sedimet-lade flows. The aalysis of turbulet shear stress for steady-uiform flows carryig heterogeeous sedimet mixtures provides additioal isight o the role played by sedimet size gradatio ad fall velocity by size fractio. PREVIOUS DERIVATIONS Eistei ad Chie (1955) first derived the turbulet shear stress relatioship for sedimet-lade flows. Without providig further clarificatio they first assumed that T pfu'(l c)v p s u'c(v w) (1) i which the shear stress f depeds o three costats ad three variables. The costats are the fall velocity of uiform sedimet particles w, the mass desity of the fluid py, ad the mass desity of sedimet p s ; the variables are the istataeous velocity compoets u ad v, ad the volumetric sedimet cocetratio c. These variables fluctuate aroud their respective timeaverage values u, v, ad c by the amouts u', v', ad c'. The time-average values of the Reyolds stress compoets are obtaied from the Pradtl mixig theory as follows: -u'v' e m (2) dy 'Res. Fellow, Water Resour. Egrg. Div., Korea Ist, of Costr. Tech., Umyeodog 142, Socho-gu, Seoul, Korea Assoc. Prof., Dept. of Civ. Egrg., Egrg. Res. Ctr., Colorado State Uiv., Ft. Collis, CO Note. Discussio ope util April 1, To exted the closig date oe moth, a writte request must be filed with the ASCE Maager of Jourals. The mauscript for this paper was submitted for review ad possible publicatio o May 17, This paper is part of the Joural of Hydraulic Egieerig, Vol. 116, No. 11, November, ASCE, ISSN /90/ /S $.15 per page. Paper No

2 TABLE 1. Turbulet Shear Stress i Sedimet-Lade Flows Source (1) Eistei ad Chie (1955) Zagusti (1969) Willis (1972) Formula (2) T p f <z, du/dy(l + Ac) T pfe. m du/dy{\ + Ac) - p s mic T p f e m du/dy(l + Ac) + (p s - p f )ii[e tfc/dy + coc] d{\ - c) -u\\ - c)' f 0 (3) ax for streamwise uiform cocetratios dc -u'c' e, 0 (4) dx for streamwise uiform cocetratios where e /; e,, ad e s deote the diffusio coefficiets for fluid, mometum, ad sedimet, respectively. Usig Eqs. 3 ad 4, Eq. 1 ca be reduced to T -PJ-MV (1 + Ac) (5) where A ^ ^ (6) P/ Zagusti (1969) provided a slightly differet derivatio i order to explai the reductio of the vo Karma costat i sedimet-lade flows. His fial relatioship, give i Table 1, differs from Eq. 5 ad ivolves the fall velocity (0. Based o Taylor series expasio, Willis (1972) derived a third relatioship for turbulet shear stress i sedimet-lade flows. This relatioship give i Table 1 also depeds o co. Woo ad Julie (1989) reviewed these formulatios ad idicated that all three derivatios are questioable. For istace, Eistei ad Chie used Eq. 1 without justificatio ad erroeously eglected the correlatio betwee c ad v. Questioable steps i Zagusti's derivatio iclude the costat value of the sedimet cocetratio ad the use of the velocity fluctuatio v' istead of v i the vertical directio. Willis' equatio would reduce to Eistei ad Chie's relatioship if e, were substituted by e s. TURBULENT SHEAR STRESS FOR HETEROGENEOUS SEDIMENT-LADEN FLOWS The followig derivatio by size fractios (Woo 1985) clarifies the role played by the fall velocity of sedimet particles from a graded sedimet mixture. Cosider the steady ad uiform motio of a heterogeeous sedimet-lade flow i a ope chael. With referece to Fig. 1, the fluxes of water F f ad sedimet F s passig respectively through a horizotal plae of area f ad s parallel to the mai flow directio are F f PfV f (7) 1417

3 777"/ > / y 777T7 '"/ 7 / ' 7~7 7 / / ~ FIG. 1. Two Dimesioal Diagram for Sedimet-Lade Flow ad (8) respectively. Here the subscript i deotes oe of the sedimet size fractios ad a), refers to the fall velocity of this size fractio. The turbulet shear stress T is thus obtaied from addig the turbulet mometum fluxes i the x-directio for both the fluid ad each fractio of sedimets: Pf v ~7I + ZJ Prf( u ~~ w <) "TT It is assumed that the iertia of sedimet particles is so small that the sedimet velocity i the x-directio equals the fluid velocity, u, whereas i the v-directio the sedimet velocity differs from the fluid velocity, v, by the settlig velocity a>, of the size fractio ;'. The followig idetities are defied: (9) 2 c, (10) (11) f 1 V sl, 1 ^ p m P/ + (P s ~ P/)c With these four relatioships (Eqs ), ad assumig that p s is costat for all size fractios (p s, p s ), Eq. 9 reduces to T -p,uv + p s ^ co,-c,.«(14) (12) (13) Istataeous values of the sedimet cocetratio ad the mass desity of the mixture are defied as follows: c c + c ' 2 c "< + 2 c 't (15a) 1418

4 p, p, + p; Pm P/ + (P* - P/)C (15&) (15c) p; (p s - P/)C' (15^) i which the overbar deotes the time-average value of the parameter ad the prime deotes fluctuatios of the same parameter. Substitutio of Eqs. 15a ad I5d ito Eq. 14 ad the applicatio of the Reyolds averagig process gives T ~(Pm + Pm)(«+ V ')(U + "') + Ps 2) M >( C i + c',){u + u') (16) (-1 The expressio for T is the time-average value of the shear stress. Uder equilibrium coditios, the time-average mass trasfer of water i the y- directio ca be writte as f 1 - X c ) v ( 17fl ) which ca be rewritte as 2 civ' 5(1 - c) (lib) For sedimets, the time-average mass trasfer i the v-directio is give by - 2) Ci(v - co,) 0 (18a) /i i.e. t) 2 w.c, ^ 0 (186) ;i This relatioship (Eq. 18b) demostrates that the average velocity v i the vertical directio is ot zero but proportioal to the fall velocity of sedimet particles ad the cocetratio of each size fractio. With the aid of Eqs. 3, 4, 17, ad 18, the shear stress relatioship (Eq. 16) ca be expaded (see Appedix I) ad simplified to T -p m ttv (19) This relatioship is valid without ay restrictio o the triple correlatio. Further simplificatio of Eq. 19 is possible whe eglectig the triple correlatio term p m u'v' ca be justified. This is possible from a order of magitude aalysis of the terms i Eq. 15b. Ideed, whe the mass desity fluctuatio p,' is egligible compared to p,, Eq. 19 further reduces to T - P/ 7t7 (1 + Ac) (20) which is idetical to the relatioship derived by Eistei ad Chie. This 1419

5 derivatio shows that turbulet shear stress i sedimet-lade flows is similar to that of the water flow except that the fluid mass desity p f is replaced with the desity of the water sedimet mixture p,. Eq. 19 is the chief result of this aalysis, which demostrates that the sedimet size distributio ad the fall velocity of sedimet particles do ot affect turbulet shear stress of heterogeeous sedimet mixtures uder steadyuiform flow coditios. ACKNOWLEDGMENTS The writers are grateful to Prof. E. V. Richardso for his support of the first author's graduate studies at Colorado State Uiversity. We are also thakful to Mr. Y. La for his careful review of the mauscript. APPENDIX I. DERIVATION Applyig the Reyolds averagig method to Eq. 16 yields -T p m UV + p m u'v' + V p' m U f + U p' m v' + p' m u'v' -p.«^wia-p.x u i e i»' ( 21 ) ;i 11 The last term of Eq. 21 ca be dropped due to Eq. 4 because dc/dx 0 uder steady-uiform flow coditios. Note that Eq. 4, which is critical to this derivatio, is also substatiated by the fidigs of Fischer et al. (1979), who stated that "the total mass trasport i the streamwise directio is proportioal to the cocetratio gradiet i the streamwise directio." With Eq. 18b, the remaiig summatio reduces to p s iiv. Now expadig p' m from Eq. 15c? gives -T p m uv + p m u'v' + v u'(p s - p f )c' + u v'(p s - p f )c' - p uv (22) The third term o the right-had side vaishes due to Eq. 4 ad the followig term reduces to uv{\ c)(p s p f ) accordig to Eq. lib. Expadig p m from Eq. 15c yields the fial form of Eq. 22 T PfUV + p s UVC p f UVC + p m u'v' + p s UV p s iivc pfuv + PfUvc p s uv (23) which simply reduces to T -p m u'v' (24) APPENDIX II. REFERENCES Eistei, H. A., ad Chie, N. (1955). "Effects of heavy sedimet cocetratio ear the beds o velocity ad sedimet distributio." MRD Series No. 8, U.S. Army Egrg. Div., Missouri River, Corps of Egrs., Omaha, Nebr., Aug. Fischer, H. B., et al. (1979). Mixig i ilad ad coastal waters. Academic Press, New York, N.Y. Willis, J. C. (1972). "Mathematical models for suspesio from a mometum diffusio viewpoit." Sedimetatio (Eistei), H. W. She, ed., Fort Collis, Colo. Woo, H. (1985). "Sedimet trasport i hypercocetrated flows," thesis preseted 1420

6 to Colorado State Uiversity, at Fort Collis, Colo., i partial fulfillmet of the requiremets for the degree of Doctor of Philosophy. Woo, H. S., ad Julie, P. Y. (1989). "Mathematical models for turbulet shear stress i sedimet-lade flows." Proc. It. Symp. o Sedimet Trasport Modelig, Aug., Zagusti, A. (1969). "Mechaics of turbulet flow i sedimet-lade streams." Proc. 13th Cogress of It. Associatio for Hydr. Res., Vol. 2. APPENDIX III. The followig NOTATION symbols are used i this paper: A c s f Ff F s u V X y / e m e, P/ Pm Ps T 0) desity quotiet; A (p s p/)/p/; volumetric sedimet cocetratio; area of elemetary plae parallel to mai flow directio; portio of occupied by sedimet; portio of occupied by water; mass flux of fluid; mass flux of sedimet; umber of sedimet size fractios; logitudial fluid velocity; vertical fluid velocity; logitudial distace i dowstream directio; vertical distace from chael bed; diffusio coefficiet for fluid mass (water); diffusio coefficiet for fluid mometum; diffusio coefficiet for sedimet; desity of fluid (water); desity of sedimet-water mixture; desity of sedimet; shear stress; ad settlig velocity of sedimet i flow. Subscripts ad Superscripts ' fluctuatig value of parameter; time-average value of parameter; ad i size fractio of sedimet http