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Response time analysis for

suspended sediment transportPeter K. Stansby

a& M.A. Omar Awang

a

aHydrodynamics Research Group, Manchester School of

Engineering, The University, Manchester, M13 9PL

Available online: 13 Jan 2010

To cite this article: Peter K. Stansby & M.A. Omar Awang (1998): Response time analysis for

suspended sediment transport, Journal of Hydraulic Research, 36:3, 327-338

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Response time analysis for suspended sediment transportAnnalyse temporelle du transport de sediments ensuspensionP E T E R K. S T A N S B Y A N D M . A . O M A R A W A N G , Hydrodynamics Research Group, ManchesterSchool of Engineering, The University, M anchester M13 9PL

ABSTRACTThe way in which sediment characteristics and water depth determine the time taken for suspended sediment concentration profiles to approach a steady state has been investigated. The idealised situation of a bed of sediment ina steady current with initially clear water is considered. A nalytical solutions are inferred from previous work anda numerical scheme w ith a particular vertical mesh transformation has been found to be effective. Response time,ln however non-dimensionalised, is generally dependent on the Rouse parameter ,y, and the particle size todepth ratio, dlh. For y > 1 however, t,uJd depends only on y, where u, is the friction velocity. It is also shown thattrw/h, where w s is the fall velocity, is always less than about 2.4, at least for dlh a I0"6 . the smallest valueinvestigated. The numerical scheme is used to compare with some experimental measurements of non-equilibrium sediment transport in a steady current to confirm some of the physical modelling assumptions.RSUML'tude porte sur la maniere dont les caractristiques sdimentaires et la profondeur d'eau dterminent ladure ncessaire pour que des profils de concentration en sediment s'approchent d'un tat stable. Le cas idealtudi est celui d'un lit de sediments soum is a un courant perm anent d 'eau initialement non charge. Des solutions analytiques ont t tires d'tudes antrieures et un schema numrique avec un maillage volutif sur laverticale a t jug efficace. Le temps de rponse /,., sous forme adimensionnelle, dpend gnralement duparamtre de Rouse y et du rapport d/h (diamtre des particules sur tirant d'eau). Cependant pour y > 1, t,uJdne dpend que de y, si on dsigne par w. la vitesse de frottement. II est galement montr que trw/h (avec w svitesse de chute) est toujours infrieur a 2.4, au moins lorsque d/h a ICH'qui est la plus petite valeur tudie. Leschema numrique a t utilise pour des comparaisons avec des valeurs exprimentales de transport solide ensituation transitoire dans un courant d'eau permanent; cela a permis de eonfirmer certaines hypotheses de lamodification physique.Introduct ionThree-dimensional numerical model l ing of unsteady shal low-water f lows is now widely appl iedgiving the potent ial for accurate suspended sediment t ransport predict ion. The character is t ic lengthand t ime scales for sediment t ransport are however general ly very different f rom those character ising the f low, present ing a chal lenge to numerical model l ing. Interact ion of bed evolut ion with thef low is desirable and requires par t icular accuracy since erosion rate is proport ional to the sedimentconcentrat ion gradient at the bed and close to the bed concentrat ion general ly varies extremely rapidly. Important length scales for sediment t ransport are grain s ize and water depth and importantvelocity scales are bed friction velocity, fall velocity and flow velocity. Clearly the magnitudeswithin each category can be orders of magnitude different and i t i s important to know the minimumlength and t ime scales relevant to different condi t ions. For example, the t ime step in a numericalmodel must be small enough to resolve sediment t ransport as wel l as f low and turbulence t ransportprocesses in horizontal and vertical directions. A general situation is rather complex and as a starting point the relat ively s imple case of s teady uniform f low ( in a wide channel) with t ime-dependentsediment t ransport is considered. The vert ical eddy viscosi ty profi le is known to be parabol ic andRevision received July 24, 1997. Open for discussion till December 31,1998.

J O U R N A L O K H Y D R A U L I C ' R H S L A R C H , V O L . 3 6 . 19 98 . N O . 3 327

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the velocity profile logarithmic [10] (although the latter will be seen not to be significant). In thesteady state the vertical concentration distribution has the well known Rouse profile which isderived an alytically. When the initial conc entration is zero everywh ere (apart from the bed w here itis constant) analytical solutions for temporal and spatial concentration variations may also beinferred from previous work [7], A response time scale is defined as the time for the erosion rate tobecome 99% of the (constant) deposition rate. Although this definition is somewhat arbitrary acriterion close to the steady state is desirable since, if the response time scale is small in relation topredominant flow time scales (e.g. the tidal period), quasi-steady concentration profiles may beassumed locally.It is desirable to know how well numerical schemes reproduce the analytical results. It is expectedthat vertical mesh transformation is needed to resolve the steep concentration gradients near thebed. Although the problem specification is simple it will be seen to present a severe test for numerical modelling. Finally the preferred numerical scheme will be compared with experimental measurements of non-equilibrium sediment transport in steady flow conditions, made in the controlledlaboratory environment [8].Problem Definition and Analytical SolutionThe equation for the transport of sediment concentration c in uni-directional flow of velocity u inthe x direction is given by

Dc dc dc dc d I dc\ , , , = + u = w. + (!)Dt dt dx sdz dz\ dz)where t is time, z isdistance above the bed, w, is fall velocity and eddy diffusivity is given by theparabolic profile

e = PKM,Z(1 -z/h) (2)

where K is Karman's constant (0.435), u, is bed friction velocity (Jxn/p), T is bed shear stress, pis water density, /; is water depth and 1/(5 is the turbulent Schmidt number (with fj = 1, eddy diffusivity equals eddy viscosity). Horizontal diffusion is ignored since it is very small in relation to vertical diffusion.Once a steady state is reached, the Rouse equation is valid: = l--h-^ \ (3)ca \n-a z I

where ca is the concen tration a small height a above the bed and y is the Rouse parameter equal toW ( / (P K . ) . The definition of c and a has been the subject of much research since the originalwork of Einstein [3] who proposed a value of a equal to two grain diameters. Engelund andFredsoe[4] later proposed a value of one grain diamter. Clearly the physics of the flow/sedimentinteraction ishighly complex close to the bed even when the bed is flat and ismore complex whenbed forms develop. For the latter reference concentrations at half the bed wave height have been

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suggested [11]. Reference concentrations at some fraction of the water depth have also been suggested for cases with and without bed forms [11]. A recent review is given in Fredsoe [5]. Zyser-man and Fredsoe [12] argue that it is more reasonable from a physical point of view to definereference concentrations a few grain diameters from the bed since particles even at such low levelsare kept in suspension by turbulence rather than by grain interactions between themselves. It is thusnecessary for a numerical method for unsteady sediment transport to resolve concentrations accurately very close to the bed, as close as one grain d iameter for the Engelund and Fredsoe formulation. This has been chosen because it is a well tried formulation and is the most severe test of anumerical scheme.The boundary condition at the bed is given by c = ca at z = a.At the surface, z = h, we require zero flux,

dc + W.Cdzand, since E = 0 from Eq. 2, c - 0The initial conditions are c = 0 when t = 0 for a < z s /? and c = ca at z = aSince velocity and bed concentration are independent of x, the concentration everywhere will alsobe independent of x and Eq. 1 simplifies to

dc dc d I dc\ , , = w + z (4)dt dz dz\ dz)An equation of similar form

,,dcilex = wdc

sdz +d iHdt i dcE ^ dz. (5)

has been solved analytically [7]. This is a steady problem where U is the constant flow velocity; thebed is fixed for x < 0 and sediment is entrained for x a 0 (without change in bed profile). Putting

dt dx

gives Eq.4 above with the same initial and boundary conditions. The equation was solved using themethod of separation of variables which produced a hypergeometric equation. Hypergeometricfunctions are then used to obtain solutions. This involves the summation of some infinite series.Hjelmfelt and Lenan non-dimensionalised time ast = - ^ l

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and defined variation of clca with t' as a function of the Rouse parameter y and alh. It is interestingto note that the analytical solution can be more computationally demanding than the numericalsolutions given below for y > 1 due to the need for the accurate evaluation of the summation of theinfinite series. Hjelmfelt and Lenan only presented results for alh = 0.05. Results are obtained hereto an accuracy of 10 significant figures and typical variations of mass flux at the bed (z. = a) in non-dimensional formm' = (wsca + E ) / ( P K C Q )

with t' for alh = 10~3 and different y are shown in Fig. 1. It can be seen that there is a singularity att = 0 and that rri -* 0 as t > oo (as a steady state is approac hed).

0.30 -rJ d/h - 0.001

0.20 - | \

0.10

0.000.00 2.00 4.00 6.00 8.00

Fig. 1. Variation of the non-dimensional sediment flux at the bed m' with non-dimensional time /' for d/h = 0.001and Y = 0.001, 0.1, 0.5, based on analytical solutions.Numerical SolutionsEq.4 may be solved numerically in a time-stepping computation with implicit discretisation toprovide stability. It will be shown below that the particular transformation

do _ Idz E

has desirable properties but causes problems when = 0 at z = 0,/?. A value of E or eddy viscosity v,,of zero is of course physically unrealistic and v f is more correctly given by v (. = v + v, where v ismo lecular viscosity (k inematic) and v, is the Boussinesq turbulen t viscosity. Usually v, v and v isignored. To avoid E = 0 at the bed and surface the z = 0 origin is taken at a distance a above the stationary bed and the value of E at the surface is taken as the small value at z = 0. The latter is purelya numerical device and the magnitude of E at the surface should not affect results provided it is3 3 0 JO UR N AL DU RECHERCHES HYDRAULIQUES, VOL. 36, 1998, NO. 3

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small enough. It should be mentioned that the physics of turbulence close to a free surface is notwel l understood and i t i s even doubtful whether a s imple eddy viscosi ty approach is appropriate.The e profi le becomesE = pKw.(z + a ) ( ^ | - 2 ) ( 6 )

which is symmetical about mid depth. The zero flux condition is imposed at the surface (c 0 ) .In the steady state the erosion rateE = - E dz

should equal the deposition rate D - w sca and the accuracy with which E is obtained numerical ly isa useful f i rs t check for a t ime-stepping numerical scheme. Example resul ts for a range of y aregiven in Table I for different mesh transformations with ca = 1 (for co nv en ien ce) an d fall veloc ityw s given by the formula for drag coefficient CD = 24(1 + 0.15 Re0M 1)IRe w her e Re = w/l/v [9] . Thetransformations appl ied are discussed below. The resul ts are independent of t ime step, At , i.e. theyhave ceased to change as t ime step is reduced within the level of accuracy presented (5 s ignif icantfigures). K is the number of ver t ical mesh points . Resul ts for a uniform mesh (no t ransformation)are very poor for all K values; a central (finite) difference scheme was used (equivalent to a finitevolume scheme for a uniform mesh) . Meshes with exponent ial s t retching were also tested (commonin boun dary layer com pulat io ns) but resul ts were s ti ll very inaccurate and are not presen ted.Table I. Steady state values of erosion rate E obtained numerically.

( = 0.0569 m/s. K = 0.435, p = 1, h = 1.0 m , g = 9.81 m/s2, p /p = 2.65)E = -e dc/di at z = 0

d(m ).0001.0002.0005.001.002

.001.0002.0005.001.002.0001.0002.0005.001.002

y

0.32201.0023.1696.26611.450.32201.0023 1696.26611.450.32201.0023.1696.26611.45

D= W s C a

7.9696 x 10"32.4795 x 10'27.8447 x 10"20.155100.283477.9696 x 10"32.4795 x 10"27.8447 xlO "20.155100.283477.9696 x 10"32.4795 x 10"27.8447 x 10"20.155100.28347

new meshtransformation7.9630 x 10"32.4643 x 10'27.5495 x 10"20.140980.234677.9794 x 10"32.4789 x 10'27.8282 xlO "20.154120.279107.9696 x 10"32.4794 x 10'27.8404 x 10"20.154840.28223

uniformmeshf 6 7 9 8 x 1 0 "47.2647 x 10"43.0628 xlO"37.3556 xlO"31.6258 x 10"27.8082 x lO *3.3259 xlO"31.3659xl0"23.2073 x 10"26.9034 xlO' 21.4450 xlO"36.0083 x 10"02.4010 x 10"25.5108 xlO"20.11558

Fredsoe etal [6] mesh8.1465 xlO"32.4614 xlO"27.7733 x l O ' 2-

-8.0793 x 10 -32.4777 xlO"27 . 8 3 1 4 x l 0 '2--8.0580 x 10"32.4793 x 10'27.8387 xlO"2--

K

101

501

1001

Note: where a numerical value is not given, overflow occurred, caused by dividing by a very small number whensetting up the transformed coordinates; p s is sediment density, K is number of vertical mesh points.

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Fredsoe et al. [6| used a transformation where the concentration of mesh points matches the steady-state sediment concen tration. In the coordinate system used here the transformed coordinate becomes

o =1

+ dh + d (7)

A finite-volume discretisation gives better accuracy than finite-difference and the values of Eshown in Table I are now quite close to their correct values. The local Peclet number (R\Ar/e whereAz is mesh spacing) was always well below 2 which is considered a main criterion for computationwithout spurious o scillations.However a new transformation is introduced which simplifies the form of Fq.4. thereby reducing thepotential for numerical errors, and makes the Peclet number constant and very much smaller than themaximum values in the above transformation. It also causes the maximum Courant number (\\\Al/Az) tobe reduced by an order of m agnitude or more w hich is desirable from consideration of numerical dissipation and dispersion which both increase as Courant num ber increases. The transformation is given by

dodz (8)

for 0 < a < amu where aEq.4 now becom esdc _ w sdc 1 d'cUt E dz fJa 2 (9)

Substituting Eq.6 into Eq.8 gives

a = fjKU In Z + aa+ h (10 )

Eq.9 is solved by finite (central) differencing (on a uniform o mesh) and, overall, results for E inTable 1 are improved over the Fredsoe et al. transformation. For high accuracy a large number ofmesh points is still required for the largery values, although lory > 3 suspended sediment transportis usually ignored.The response time requires an accurate prediction of transient behaviour and thus gives a moresevere test for the numerical schemes.Response Time A nalysisThe definition of response time /,. was given in the Introduction and

/,- = t,.(u*, h, d, w s, p\ K) ( 1 1 )

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where |5 and K are constants and a = d. By dimensional analysist,.u* t,.u, t,.n\ t,w\ i w\ d\ , n , - or or - or - = h j , T) ( | 2 )(/ h d h \pKM. hi

In [7] response time would be non-dimensionalised asBKM

which is equivalent to (u J,)lh in Eq.12. Plots of (l,n,,)lh against y(= w/fiKu*) for different dlh are shownin Fig.2. The range 10 6 1 the variation is almost independent of dlh for I0" 6s(/ / / ;s 10~3. On the other hand it would be expected that the suspension of lightersediment is affected by water depth and fall velocity and Fig.4 shows plots of trwjh against y for different dlh. It can be seen that trwjh < 2.4 always. In other words the reponse time is always less than abouttwice the time taken for a grain to fall from the surface to the bed in still water. Fig. 4 also shows thattrwjh -* 0 as 7 - 0 (to be expected as vr, - 0). The dependence of tr on y and dlh is clearly complex.Results from the numerical model are included in the figures and agreement with the analytical resultscan be seen to be very close. In particular comparisons for a wide range of y are shown for dlh = 10"3 inFig. 4. For the smaller dlh with y > 1 accurate analytical solutions are difficult to obtain and only resultsfrom converged numerical solutions are shown in some regions. Numerical convergence required a verysmall time step in some cases, much smaller than that required to give a converged steady state solution.The agreement with the analytical results indicates that the very small but non-zero E value at the surfacein the numerical method has an insignificant influence on the results.

10 2 j

10 i

i 1

10 'ic :

~>10 " 2 ,

10 - 3]

10 ~'i

10 "5 i0.001 0.01 0.1 17

Fig. 2. Variation of non-dimensional response time t,u /h with y for d/h = 0.05, 10~3, 10~\ I0"3, 10"6.The solid lines are from analytical solutions and the symbols from the numerical scheme.

T 1 ' t I ' M 1 f

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d/h=1CT*

Variation of non-dimensional response time t,uJd with y for d/h = 0.05, 10~-\ I0~*, 10~5, I(H'. Thesolid lines are from analytical solutions and the symbols from the numerical scheme.a) Log-log scale.b) Log-linear scale to show convergence for y > I more clearly.

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1 =

1 0 ' i-_10 "2- |

o

10 " 4 i

-10 - 5 3

^ ^ S ^ > ^ ^ \ ^ ^ - ^ ^ ^ 5 \ \ \ ^ \ d / h = 0.0 5

^^^^^^^ \ \ \ \ 1 0 " J

\ \ V " \

\ W 5 \

i o - e

*1 1 1 1 1 1 1 1 1 1 1 1 1 1 [ 1 1 1 1 1 1 1 1

0 .0 01 0 .01 0 .17 1Fig. 4. Variation of non-dimensional response time t,.wjh with y for d/h = 0.05, 10~3, KH, I0~5, 10 -6. Thesolid lines are from analytical solutions and the symbols from the numerical scheme.From Fig. 3b, the converged values for t,uJd for y > 1 (excluding results for d/h = 0.05) may beapproximated as

i / ' < " 0.235y - 1.85v + 3.93 (13)

Experimental ComparisonsExperimental comparisons of response time are not possible but experimental measurements havebeen made of non-equilibrium sediment transport in steady uniform flow [8]. Since this is similar tothe situation considered here it is a useful test for the physical assumptions inherent in Eq. 1 and itsnumerical solution. Sand was introduced into the flow in a channel at a constant rate as a line sourceat the surface. Large roughness elements (of height 7% of the water depth) were used on the bedand bed boundary conditions were cited for each test. Here the two most appropriate cases aremodelled, the details of which are specified in Table 2. The velocity was measured and may be fitted acceptably by a power law of the form

M (14)where w is the mean velocity. The unusually large exponent is presumably due to to the very largeroughness height.JOURNAL OF HYDRAULIC' RESEARCH, VOL. 36. 1998, NO. 3 335

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Table 2. Experimental specification (Jobson and Sayre [8]).Case

12

h(m)0.4070.405

(m/s)0.321.00(m/s).0308.1033

pjp2.652.42

d(mm).390.123w,(m/s).063O il

Commentno erosion at bederosion rate = deposition rate at bed

To simulate the experimental conditions a rectangular mesh was set up in the vertical plane. The advec-tion terms in Eq. 1 must now be included and are determined here by a Lagrangian scheme which isaccurate and straightforward to use [2]. A mesh was typically set up with K = 501 and a horizontal spacing of 0.05h although mesh-independent results were also produced with coarser mesh spacings.Concentrat ion profi les at several posi t ions downstream are shown in Fig. 5 for Case I . (x is the horizontal dis tance downstream of the point of introduct ion of sediment . ) The sediment is coarse(y = 4.7) and it is argued that there is no erosion at the bed, only deposition, presumably because thecoarse sediment fal ls between the large roughness elements and does not escape. In the computations sediment was introduced as a l ine source positioned just below the surface to match the firstm easu red co nce ntra tion profile as well as pos sible . Th at the position is below the surface is presumably because the sand is dropped into the water in the experiments with a cer tain veloci ty making an effective origin be low the surface . Fig. 6 sho ws results for Ca se 2 wh ere it wa s argued thatthe fine sediment (y = 0.245) is eroded or re-entrained by the turbulence in the flow as soon as it isdeposi ted. I t can be seen that the computat ional resul ts are in reasonable agreement with experiment . Similar agreement was obtained previously using a k- t turbulence model [ 1 ].

c/Cref

Fig. 5. Com parison of measured concentration profiles [8 | with num erical results at different downstreamlocations: Case 1 (no bed erosion, deposition only); experimen t. comp utation.a) x/h = 2.0 c) x/h = 3.5b) x/h = 2.75 d).v//; = 5.0

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0 .00.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0 0 5 1.0 1.5 2.0 2.5 3.0c/cref

Fig. 6. Com parison of measured concentration profiles [8] with numerical results at different downstreamlocations: Case 2 (bed erosion rate = deposition rate); experiment, com putation.a) x/h = 5.0b) x/h = 7.0c) x/h = 9.0d)x/h= 15.0e) x/h = 67.0Conclus ionsSome progress has been made in determining how the t ime taken for suspended sediment t ransportto approach a s teady state is dependent on sediment character is t ics and water depth. The ideal isedproblem of a steady current with initially clear water is a severe test of a numerical scheme, withsingular behaviour at t = 0, but excel lent agreement with analyt ical resul ts is obtained provided a

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part icular ver t ical mesh t ransformation is used. For y > I the numer ica l scheme is actual ly morecomputat ional ly eff icient than the analyt ical method while for y < 0.1 the numer ica l schemerequires very small t ime steps for conve r gence and the analyt ical method is quite efficient. Knowledge of these response t ime scales is impor tant for general numerical models of sediment t ransportas well as being of fundamental interest .A c k n o w l e d g e m e n t sWe acknow l edge the cont r ibut ion made by a referee in br inging the paper by Hjelmfelt and Lenan[7 ] to our at tent ion. We also grateful ly acknowledge the scholarship from the Malaysian governmen t suppor t i ng M. A . O mar A w ang .Ref erences

1. CELIK,I. and RODI,W. 1988 Modelling suspended sediment transport in non-equilibrium situations Am.Soc. Civ. Engrg. Jour. Hyd. Engrg. 114(10), 1157-1191.2. CHENG, R.T., CASULLI, V. and MILFORD, S.N. 1984 Eulerian-Langrangian solution of' the convection-disdersion equation in natural coordinates Water Resources Research, 20(7), 944-952 .3. EINSTEIN, H.A. 1950 Th e bed load function for sediment transport in open-channel flow Tech. Bull. 1026US Dept. of Agriculture, Washington DC.4. ENGELUND, F. and FREDSOE, J. 1976 A sediment transport model for straight alluvial channels NordicHydrology 7, 293-306.5. FREDSOE, J. 1993 Modelling of non-cohesive sediment transport processes in the marine environmentCoastal Engrg., 21, 71-103.6. FREDSOE, J., ANDERSON, O.H. and SILBERG, S. 1985 Distribution of suspended sediment in large wavesAm. Soc. Civ. Engrg. Jour. Wat. Port, Coastal and Ocean Eng rg. 11 1(6), 1041-1059.7. HJELMFELT, A.T. and LENAN, C.W. 1970 Non-equilibrium transport of suspended sediment Proc. Am.Soc. C iv. Engrg. HY7, 1567-1586.8. JOBSON, H.E. andSAYRE, W.W. 1970 Vertical transfer in open channel flow Am. Soc. Civ. Engrg. J. Hyd.Engrg. 96(3), 703-72 4.9. RAUDKIVl, K. 1976 Loose boundary hydraulics, 2nd ed., Pergamon. Oxford.10. RODI, W. 1984 Turbulence models and their application in hydraulics, IAHR monog ragh, Delft, The Netherlands.1 1. VAN RUN, L.C. 1984 Sediment transport . Part II: suspended load transport Am . Soc. Civ. Engrg. Jour.Hyd. Engrg. 110(11), 1613-1641.12. ZYSERMAN, J.A. and FREDSOE, J. 1994 Data analysis of bed concen tration of suspended sediment Am. Soc.Civ. Engrg. Jour. Hyd. Engrg. 120(9), 1021-1042.

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