CHAPTER 46 SEDIMENT TRANSPORT IN RANDOM WAVES AT CONSTANT WATER DEPTH

Transcription

1 CHAPTER 46 SEDIMENT TRANSPORT IN RANDOM WAVES AT NSTANT WATER DEPTH Hsiang Wang and S. S. Liang* University f Delaware, Newark, Delaware, U.S.A. by Abstract Sediment transprt in randm waves at cnstant water depth is analyzed by dividing the flw field int tw regins the internal regin and the bundary layer regin. The suspensin and transprt f sediment in these tw regins are treated separately. The ttal transprt is then btained as the ttal f these regins thrugh matching bundary. In the bed layer, the lad cncentratin is assumed t be prprtinal t the specific weight f the sediment and t the prbability that the fluctuating lifting frce exceeds the weight f the sediment. The bed lad is then transprted by the secndary flw (which is unidirectinal) in the bundary. In the internal flw regin, the sediment suspensin is treated as a diffusin prblem with the intensity f diffusin t be prprtinal t the amplitude f fluid particle mtin. The transprt velcity is assumed t be the same as the mass transprt velcity f the wave. The predminant mde f transprt is fund t be suspended lad. The ttal transprt in a wind-generated wave field can be expressed as a pwer law f wind speed. Fr the case tested, the pwer shuld be f the rder f INTRODUCTION The analysis presented in this paper applies nly t regins well beynd the surf zne where the flw field, thugh irregular in appearance, can be described in reasnable detail withut ging int micr-structures. In ther wrds, the water surface variatins, the internal flw characteristics, and the pressure field can still be expressed in manageable mathematical expressins in grss terms. T facilitate calculatins f sediment mtin, the flw field is divided int tw znes. The upper zne, where viscsity plays a negligible rle except *Nw at Stne and Webster, Bstn, Massachusetts. 795

2 796 ASTAL ENGINEERING n water mass transprt, is termed zne f suspended lad. In this zne, sediment is in a state f suspensin duj2 t the cmbined effects f turbulence and rbital mtin f fluid mass. The lwer zne, which is a thin turbulent bundary layer near the sea bed created by wave mtin verhead, is termed bed lad layer fllwing Einstein's terminlgy. In this zne, sediment particles rll, slide, and smetimes jump. As stated by Einstein (1972), the measureable thickness f this turbulent bundary layer is usually rather small; 5-10 m.m. are very cmmn values. Hwever, since it is directly adjacent t the sea bed, it culd be very effective in sediment transprt. The suspended lad can be cnsidered t ccupy the full water depth. Therefre, althugh the sediment cncentratin may be relatively small, the transprt rate may still be appreciable. Frm these bservatin, sediment transprt in bth znes need prper attentin. Therefre, the analysis naturally breaks dwn int investigating the fllwing fur parts systematically: 1) Bed Lad Cncentratin, 2) Bed Lad Transprt, 3) Suspended Lad Cncentratin, and '4) Suspended Lad Transprt. The prblems are first frmulated fr mncrmatic wave train and the results are then generalized fr randm waves. The ttal sediment transprt rate is btained thrugh matching bundary cnditins f the tw znes. Because f the cmplicated mathematical frmulatins, cmputer prgrams are develped t facilitate numerical cmputatins. 2. BED LOAD Cncentratin Let the bed-lad cncentratin be expressed as C 0 = N g W where N s is the number f particles per unit vlume inside the bed layer and W = A]D3y s is the dry weight f sediment particle. Since in a unit vlume N s is inversely prprtinal t the size f the sediment particle and is prprtinal t the prbability, p, f a sediment particle t be set in mtin, we have C " A P Y s»> where AQ is an empirically determined cnstant. The prbability, p, is equal t the prbability that the instantaneus lift exceeds the weight f the particle (Einstein, 1950), i.e., p = p (L + L'> W) where L is the mean lift frce and L' is the fluctuating cmpnent f the lift frce.

3 NSTANT DEPTH TRANSPORT 797 On the further assumptin that in a turbulent bundary layer L' is a randm prcess fllwing a nrmal distributin f zer mean and standard deviatin prprtinal t the mean lift (a = ri L, where ri is a cnstant f prprtinality), Einstein arrived at the fllwing expressin fr the prbability distributin functin = 1 * r 1 n 2 z e " 2 dz (2) where B A = 2A 3 ; D is equivalent diameter f the sediment particle; C is the lift cefficient f sediment particle; A is a cnstant and = s p g -2 P u is knwn as the flw intensity functin in which u is the amplitude f the scillatry flw velcity in the directin f wave prpagatin. Therefre, the prblem f determining the bed-lad cncentratin, aside frm a number f cnstants f prprtinality, becmes a prblem f determining the hrizntal velcity in the bundary layer. Analgus t the laminar case (Schlichting, 1960), Kalkanis (1964) prpsed the fllwing expressin fr the hrizntal velcity cmpnent in a fluctuating furbulent bundary layer. u = U Q {1 + f ± 2-2f csf 2 } 1/2 cs(wt + 6) (3) f, sinf. -13^ 9/^ -112 in which f = 0.5 e ^^- y, f = 0.5 (By) ' ~ 0.3$y, 8 = tan 1 abd f csf 2, and 3 = /in. Therefre, the bed-lad cncentratin in a mncrmatic waye train / 2y can be determined thrugh Eqs. Q-), (21, and QL. Fr, randm waves, the.-velcity is expressed by u = Xu [cs (k.x - (D.t + e.) - f,. cs (k.x v - u.t + f. + e.)l (4). i I l i' li i l 2i I where U. = / S(u)di and S((u) the spectral density functin f wave energy.

4 798 ASTAL ENGINEERING Transprt Within the bundary layer f a wave field, the first rder effect is t relieve the sediment particles f all r part f their weight s as t bring them int a state f incipeint equilibrium. At this state, if there is n ther incidental current, the net transprt will result nly frm higher rder effect. Nw treat the mass transprt velcity as the secnd rder effect such that, t the secnd rder, the velcity field inside the bundary can be expressed as; (u + U)e ex + (v + V e ) e y where U and V are the secnd rder velcity cmpnents and are f the rder f eu and ev, respectively. The e is a perturbatin parameter,f the rder f ak fr deep water wave and hk fr shallw water wave. Substituting the abve expressin int the bundary layer equatin in the hrizntal directin yields, 3(u v + U e'+ ). 8(u + U e' ) 2 +, _ 3(u + U e^j )(v + V e/_ ) = -, 1 a &g_ +,,J) 2, (u +, U ), (5), c. e 3t 3x 3y T the secnd rder, the pressure inside the bundary layer is equal t that at the uter edge f the bundary layer, the mean mmentum equatin inside the bundary layer reduces t a balance between shear stress and Reynlds stress: 2 u 3(uv) a e (6) "^7 Frm Eq. (3) r (4) the vertical velcity cmpnent inside the bundary layer can be btained thrugh cntinuity relatin. The shear stress uv can then be slved thrugh the fact that u and v are rthgnal functins. The secnd rder hrizntal velcity U is then slved frm Eq. (6) when the fllwing bundary cnditins are specified: (1) At the uter edge f the bundary layer, the mean shear stress is negligible. e = 0 at y =» 3y

5 NSTANT DEPTH TRANSPORT 799 (2) At the bttm, the velcity is zer "e " at y - 0 The slutin btained is in the Eulerian reference frame. The true mass transprt velcity,which is a Lagrangian prperty, is btained thrugh transfrmatin f crdinate (Lnguet-Higgins, 1953) by U = U e + (f udt) ll + <Hl7 (?) The slutin is (Liang and Wang, 1973) 0.5 k u 2 -I; 1.5 E x 01 U = I 5- {-e 1 sin0.36,y[e y + j±-] i 2v(E e t ) E P ± -F v 1 * 2 -^ E i -e i y cs0.3g.y[0.36.y + ± ] l I 2 2 E ± ^ , -2E y x + 5-=-i (e 1-1) + X ' 2 g i E ^ E l. r. k U? -v _2E^ 1 x + I i i {1 - e cs0.3g,y e i 2 t 1 -E.y x e y[e jc s0.38.y sin0.3b.y]} i x i i k. u -E y 2 -E y 2 1 Pl + 1, {0.5(E e 1 ) - 0.5(0.3B.e *) i 2u ± CE B ± ) i " E ix y e cso.sb^t-e + (.3g ± )^] -E.y x - 2(0.5)E 0.3B ± e sino.36^} (8), _ 133 sinh( k -; h > Where E 1 i a^.d

6 800 ASTAL ENGINEERING 3. SUSPENDED LOAD Cncentratin Generally, sediment suspensin in a fluid media is treated as a diffusindispersin prcess. This line f apprach is fllwed here. The diffusin equatin takes the fllwing frm: I7 + V (CY ) = V (e VC) (9) dt ~s m where C is sediment cncentratin (Mass/vlume); V is the particle velcity vectr and e is the diffusin cefficient. Tr Suspended sediment in a wave field, the cncentratin and velcity can bth be divided int three cmpnents: C = C(x,y) + C (x.y.t) + C (x,y,t) V = V + V + V' _s ~w Here. C and V are the mean values. C and V are the wave induced cmpnents, ~ w _w and C", V are the turbulent fluctuatins. Substituting C and V int the diffusin equatin and taking the time average yields V- (C V + C V + c*v') = V- (e VC) - W ~ W - m (10) Integrating and assuming C = 0, VC = 0 at free surface the fllwing equatin is arrived at: CV = e m VC - cft" w_w - (Pv* (11) The right hand side.f the abve equatin represents the effects f mlecular diffusin, wave agitatin and turbulent diffusin. In the usual cntext f the diffusin prcess (Hinze, 1962) it is further assumed that C V - -e VC ; CV = e' VC w,w w ' where e and e' are, respectively, the diffusin cefficients f wave mtin and turbulence. Since mlecular diffusin is usually negligible, the task f slving Eq. (11) rests n the estimatin f e and e". Fllwing Prandtl's mixing thery, these cefficients are assumed t be prprtinal t certain

7 NSTANT DEPTH TRANSPORT 801 characteristic velcities and mixing lengths. Fr sediment suspensin in a wave field, it is reasnable t assume that the characteristic velcities fr bth e^, and e"- are prprtinal t the amplitude f the vertical velcity cmpnent f the wave field. Furthermre, if mixing lengths are treated as cnstant in the flw field, a cmmnly adpted assumptin, the diffusin equatin becmes dct CV = v H m dy (12) where a is a cnstant f prprtinality and Vm is the amplitude f the vertical velcity cmpnent. Fr small amplitude waves it is expressed Vm =,, sinh ky sinh ky where a is the wave amplitude; w is the angular frequency; k is the wave number, h is the water depth and y is the vertical crdinate measured frm eea bttm, as shwn belw: Figure 1. Definitin Sketch" Fr regular waves, the slutin f Equatin (12) is: p ky tanh tf-) - t y- ] R as) r tanh ( ) in which y is a reference level where C = C and R = ^2 where G is r r cr k a t the mean particle settling velcity. Fr randm wave systems, the suspended sediment distributin functin can be shwn as equal t hi C r r Tanh 2.R. C i Vr X (14) r 1 Tanh -± -

8 802 ASTAL ENGINEERING where R. = i and 1 a." - I S(u))dt. Fr shallw water, bth k^a JO. i X ' > 0 Equatins (13) and (14) reduce t pwer law. Transprt In a wave field, it has been shwn by Lnguet-Higgins (1953) that the field equatin fr mass transprt, crrect t the secnd rder, takes the fllwing frm: v *< u i + v i> j' *s dt + v (15) J ^f dt^ where 1 is the mass-transprt velcity stream functin, u and v are the first rder velcity cmpnents, and \fr is first rder stream functin. In the present case f randm wave field, the first rder velcity cmpnents and stream functins are, respectively: csh k y U = T U..,, cs(k.x -u.t +!) v L. i sinh k.y x x i x x a.. sinh k.y,; ^, ^ \. _ v i i j/_ cs (k.x -»t + E.) V s~ ~ L k. sinh k.h i i x i x x where E is the randm phase angle. Substituting the abve equatins int Equatin (15), we have 2 V 4 T = V 4 (I J^- sinh 2 k.y) ("> i ^i Let i <(0) = 0 0-7) then the requirement f zer net mass transprt in the whle field yields Kh) = 0 0-8) at the uter edge f the interir regin. Furthermre, at the bttm f the interir regin (y = 0), the hrizntal velcity must be matched with the uter edge velcity f the bundary layer; i.e., 3x fl U ly=0 "

9 NSTANT DEPTH TRANSPORT 803 where U ra can be deduced frm Equatin (18). With the bundary cnditins s defined, slutin f <\> can be btained frm Equatin (16). The mass transprt 1 velcity can then be btained thrugh differentiatin. The slutin is 'ivi V - Z i 4 sinh^h 3 sinh2 k h 2 2 csh 2k i^ + 2kh <h- 2 J) i n V A. i i + I i 4v(E 2 ± ) Ei 0.3 "i 2 " 4 E. E "+ 0.09$,. l (^- 2 b:> + 1 (20) 4. RESULTS Befre examining the case f sediment transprt in randm waves, it is instructinal t cmpare ur results with sme f the existing infrmatin which, unfrtunately, is limited t mncrmatic wave trains. Behattacharya (1971) studied the sediment suspensin in shaling waves experimentally. Thse data btained near hrizntal bed were cmpared with Equatin (13) and the value f a was fund t be 5.15 m. (p-/p = 2.83)(Fig. 2). The mass transprt velcity fr a regular wave train in a cnstant water depth is illustrated in Fig. (3) with a rughness f D = t The theretical results f Lnguet-Higgins (1953) and Hwang (1970), and the experimental results f Russell(1957), are als shwn in the same figure t ffer sme cmparisns. Fr randm waves, the case illustrated here assumes the fllwing input cnditin: water depth (h) = 7.5 m., density rati (Y /Y) = 2.83, viscsity (v) = _ c s x 10 m /sec, mean sand grain size (D) = 0.65 m.m. The randm wave is generated in accrdance with the deepwater wave spectrum f Piersn and Mskwitz (1964). Fr the bed lad, Table 1 shws the cncentratin fr varius wind speed: Figure 4 illustrates the mass transprt velcity distributin inside the bundary layer. Fr the suspended lad, Figure 5 plts the cncentratin fr varius wind speeds; the mass transprt velcities are shwn in Figure 6. The ttal transprt f sediment (the summatin f bed lad and suspended lad) is shwn in Figure 7.

10 804 ASTAL ENGINEERING Bed-Lad Cncentratin TABLE 1 h = 7.5 m, ps/p = 2.83, V = x Iff 6 m2/sec, D ,3, 8(d)) = ( g /a ) EXP [-0.74 (g/vu) ] v (knts) (PPM) Althugh the wave spectrum chsen here may nt be cmpatible with the finite depth case, the results illustrated abve revealed a number f interesting facts: 1) The bed lad is f minr imprtance in sediment transprt if the wave is the nly factr under cnsideratin. 2) The suspended lad cncentratin increases rapidly with increasing wind velcity. Hwever, the gradient f the suspended lad in the water clumn decreases as wind speed increases. 3) Fr small wind velcity (r small waves) the shape f the mass transprt velcity distributin in the vertical directin is similar t that btained in the labratry; the velcity pssesses frward cmpnents at the surface and bttm layers with return flw in the middle. As the wind speed increases,the frward velcity cmpnent at the surface layer diminishes and eventually becmes return flw, thus, the wave mass transprt advances with the wave celerity at the lwer half f the water clumn and returns frm the upper half. 4) The directin f the sediment transprt is with the directin f wave prpagatin; the reate f transprt can be apprximated by a pwer law f wind speed; i.e., Q s = A(V) n where A is a cnstant. Fr the case studied, the value f n is apprximately equal t 4. Acknwledgment This wrk was supprted by the Office f Naval Research, Gegraphy Prgrams, under cntract N. N00014-A0407.

11 NSTANT DEPTH TRANSPORT 805 References 1. Bhattacharya, P. K. (1971), Sediment Suspensin in Shaling Waves, Ph.D. Thesis, University f Iwa. 2. Einstein, H. A. (1950), The Bed-Lad Functin fr Sediment Transprtatin in Open Channel Flws, U. S. Dept. Agric, Sil Cnserv. Serv., Tech. Bull. N Einstein, H. A. (1972), Sediment Transprt by Wave Actin, Prc. 13th Cnf. n Castal Engineering. 4. Hinze, J. 0. (1959), Turbulence, McGraw-Hill, New Yrk. 5. Huang, N. W. (1970), Mass Transprt Induced by Wave Mtin, J. Mar. Res., Vl Kalkanis, G. (1964), Transprtatin f Bed Material Due t Wave Actin, Tech. Mem. N. 2, U. S. Army Castal Engineering Res. Center. 7. Liang, S. S., and Wang, H. (1973), Sediment Transprt in Randm Waves, TR26, Cllege f Marine Studies, University f Delaware. 8. Lnguet-Higgins, M. S. (1953), Mass Transprt in Water Waves, Phil. Trans. Ryal Sc. f Lndn, (A) Piersn, W. J., and Mskwitz, L. (1964), A Prpsal Spectral Frm fr Fully Develped Wind Seas Based n the Similarity Thery f S. A. Kitaigrdskii, J. Gephys. Res., Vl. 69, N Russell, R. C. H., and Osri, J. D. C. (1957), An Experimental Investigatin f Drift Prfile in a Clsed Channel, Prc. 6th Cnf. Castal Engineering. 11. Schlichting, H. (1960), Bundary Layer Thery, McGraw-Hill, New Yrk.

12 806 ASTAL ENGINEERING CM 2 3 a. a. lu?*r Z S 6 a 3 lu ec in II u M-l II *^ T c X SQ^ ^y^ ^* H 4-1 O i-l T3 T-l 0) O * H X S ; -«U <D M O SQO e 4-1 tfl T) i-l cd P + -* (3 0) S H. O >>! = d b d «U T" X SQ»/ B 0) a T* a,. (0 O r q * X T q a) T" O C > P. e 0) 3 60

13 NSTANT DEPTH TRANSPORT 807 \ \ \ ^-N /~\ u C 4) H )H a 9 el cfl i~. LTl 01 PI CTi H u-i TH M ON» O rh 1-1 0) *- rh 4) 4-1 c c t n c ih r < 3 a) 60 n 1*5 <u 60!>% ^ S ih /-N r- 4. u w «cu r-~ c : w (^,Q 1 CTi 0 ) c rh <D PM 4-> CD a) c S ih 3 H 3 a ) M Jt u 60 <D H c ed CD (X p O \ 3 n X O \ K ft M ib" "" i-h CU > \ «? 4J \ 1 O U O 1 1 c a, 1 t <0 a \ \ \\\ 9\\\ in 6 4 H* Nfd > 8 \ / c C1 O a) e c 0 H r) # / a) / d i ' CJ CU ^ H

14 808 ASTAL ENGINEERING 20. y.5 20 y 5 V=20 :. V=25 10 ' ) 10 : j, u «v sec 0 m/ S ec V = Figure 4 Mass transprt velcity distributin inside the bundary layer under a randm wave train.

15 NSTANT DEPTH TRANSPORT 809 V=25 C(ppm) C(ppm) V = 35 2 C(ppm) 10 3 C(ppm) 30 Figure 5 Suspended Sediment Cncentratin vs. Wind Velcity

16 810 ASTAL ENGINEERING y m V = 30 ^U ««/see >, i "i i i i i i i i Figure 6 Mass transprt veracities under randm waves

17 NSTANT DEPTH TRANSPORT h «7.5m p /p = 2.83 v x 10 _6 m 2 /sec D «0.65 m.m ' I I J!_ Figure 7 Rate f Sediment Transprt vs. Wind Speed