Modeling and simulation of sedimentation processes in a lowland river

MATHEMATICS AND COMPUTERS 7- IN SIMULATION ELSEVIER

Mathematics and Computers in Simulation 39 (1995) 627-633

Modeling and simulation of sedimentation processes in a lowland river Christof Engelhardt *, Dieter Prochnow, H e i n z Bungartz Institute of Freshwater and Fish Ecology, Rudower Chaussee 5, 12484 Berlin, Germany

Abstract

Sediment load is an ecological criterion for the assessment of water quality as well as dissolved or particle-bound pollutants, governed by sediment transport. In a tidally unaffected section of the Elbe River upstream from Geesthacht a decreasing concentration of suspended particle load was observed during several measuring campaigns and for a few discharge situations. The reduction of suspended sediment seems to be caused by a seifpurification process due to particle settling. To verify this assumption the turbulent sediment transport in this Elbe River reach was simulated applying two- and three-dimensional mathematical models. The transport in rivers can be modeled by the turbulent momentum equations (Reynolds' equations) to determine the velocity field and by special convectiondiffusion equations to calculate concentrations of different particle fractions each characterized by a mean settling velocity. Aside from concentration measurements for this river section a few settling velocity spectra were measured simultaneously. To simulate the suspended sediment load in this case study, the program package S E D I F L O W was used. In a first step the three-dimensional version of S E D I F L O W was applied to check the concentration distribution for the transport problem. Because of the almost uniform concentration profiles in vertical direction in a second step, the same problem was solved by the vertically integrated model. It is shown that the observed sedimentation process in a real river reach could be confirmed with sufficient accuracy by model simulation.

1. Introduction Sediment load is an ecological criterion for the assessment of water quality as well as dissolved or particle-bound pollutants, governed by sediment transport. In a section of the Elbe River upstream from Geesthacht a decreasing concentration of suspended particle load was

* Corresponding author. 0378-4754/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0378-4754(95)00127-8

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C. Engelhardt et al. / Mathematics and Computers in Simulation 39 (1995) 627-633

observed during several measuring campaigns and for a few discharge situations [5]. The reduction of suspended sediment seems to be caused by a selfpurification process due to particle settling. To verify this assumption the turbulent sediment transport in this Elbe River reach was simulated applying two- and three-dimensional mathematical models [1-4,6].

2. Equations The transport in rivers can be modeled by the turbulent momentum equations (Reynolds' equations) to determine the velocity field and by special convection-diffusion equations to calculate concentrations of different particle fractions each characterized by a mean settling velocity. The governing equations of a depth-integrated (2D) model, see [2], are given as follows (sum convention on j = 1,2):

Momentum equations

or,

o [

av, ~

V'ox, ox,

~n

Dv

+gox, = -n v,

i=1,2,

- - - ~ 0 , ~Xj

1

H

v ~ = ~ f ° vi ax3,

H=h+,l

in which: Ui

component of the turbulent velocity in/-direction, i = 1, 2, 3,

V~(x) /-/(x) x=(x,, x2)

depth-averaged velocity in/-direction, i = 1, 2,

Dv DI~ g rl

water depth, horizontal cartesian coordinates, vertical eddy viscosity, horizontal eddy viscosity, gravity, vertical deviation of water surface above level h.

Boundary conditions Vi(x ) = V/B(x)

or

(aVi/ax1)n 1 d- (aVi/aXz)n 2 = O,

V/B

boundary value,

n = (n 1, n z )

outward directed normal vector.

C Engelhardt et aL ~Mathematics and Computers in Simulation 39 (1995) 627-633

Transport equation Oc

Vl-- + OX1

o(oHoc) o(oH c)

Oc

OX 1

0X 2

SZ ~X 1

-- ~X-~2

629

1

SZ ~X'~2 =

I-IfS

in which:

c(x) L sz

concentration of particulate/dissolved matter, sedimentation rate, turbulent Schmidt number.

Boundary conditions

c(x)=cB(x) ca

or

(ac/Oxl)n, + (Oc/Ox2)n2=O

boundary value.

A corresponding three-dimensional model is given in [3].

3. Computational domain and input data The area of interest is a tidally unaffected eight km section of the Elbe River (from km 578 to km 586) upstream the Geesthacht weir (Fig. 1). Neither flow nor concentration sinks or sources were observed. The lock was assumed to be closed. Aside from concentration measurements for this river section a few settling velocity spectra were measured simultaneously. Together with bathometric plans of the Elbe River these experiments permit a model simulation with following input values: constant inflow velocity

HAMBURG

:I

Z m

flow domsin

Fig. 1. Map of the lower Elbe River with simulated reach.

630

C. Engelhardt et al. ~Mathematics and Computers in Simulation 39 (1995) 627-633

Fig. 2a. Turbulent flow near surface ( 3 D - S E D I F L O W

simulation).

L L:~

.....~:~ii

~!!!i!i~i~ii!i~ii!ii~ii!¸¸!i!i~ i!~iii~~!!i~¸!II~I~:i?~!i!i~!i~iii~iili~ii!iii~ !i~iii~i!ii !i~i~ii!i~ i i i i~!ii~¸¸¸¸iiiiiiiill¸

Fig. 2b. Three-dimensional flow in detail ( 3 D - S E D I F L O W

simulation).

m3/s,

Ui = 0,4 m / s (discharge Q = 286 inflow cross-section area A = 715 m2); eddy viscosities D n = 1 m2/s, D r = 0,0015 m2/s; inflow suspended sediment concentration c = 35 mg/l; turbulent Schmidt number s z = 0.5.

Fig. 3. Depth-averaged distribution of suspended sediment concentration ( 2 D - S E D I F L O W

simulation).

C. Engelhardt et al. / Mathematics and Computers in Simulation 39 (1995) 627-633

631

4. Simulation and results To simulate the suspended sediment load in this Elbe River reach, the program package S E D I F L O W was used [3,4]. In a first step the three-dimensional version of S E D I F L O W was

J

O. 0t5

Y

I I /

0. 020

0. 025

I IIllJ I llJJJJJJJNilJJJlJ/JllJJ llJ

0. 030

0. 035

[g/l]

Fig. 4. Three-dimensional distribution of suspended sediment concentration (3D-SEDIFLOW simulation); horizontal cross-section at surface (a), in 2m depth (b) and in 4m depth (c).

C. Engelhardt et al. /Mathematics and Computers in Simulation 39 (1995) 627-633

632

applied to check the concentration distribution for the transport problem given above (for turbulent flow field see Fig. 2a,b). Because of the almost uniform concentration profiles in vertical direction (see Fig.'s 4a,b,c) in a second step, the same problem was solved by the vertically integrated model (see Fig. 3). In SEDIFLOW the sedimentation rate fs is modeled by sedimentation rate,

fs=S(C--Ceq)

mean settling velocity of particulate matter, C,

Ceq =

O,

UCR U, <~UCR , VT >

equilibrium concentration (written in case of non-erodible bed),

v3R = O , 1 5 g v ( Q v / p ~ - 1)critical bed shear velocity in which: v - kinematic viscosity;

pp, Qw - density of particles and water, respectively. The measured settling velocity spectra yield a characteristic value s = 10 - 4 m / s of the settling velocity [5]. Using this value and the observed [5] typical particle diameter of 25/~m a mean particle density Qv = 1320 k g / m 3 results by applying Stokes' law. From here a critical shear velocity of UCR = 0.078 m / s is received (gravitational acceleration g = 9.81 m/s2; kinematic viscosity v = 10 - 6 m2//s; water density Ow = 1000 kg/m3). The bottom shear velocities obtained from steady state solution of the three-dimensional problem are almost everywhere below the critical value. Thus here the suspended sediment transport was dominated by deposition. In two-dimensional simulation the flow solution was forced to yield bed shear velocities similar to those of three-dimensional simulation. Fig. 5 depicts the surface suspended sediment concentration at several river cross sections computed by the 3D- and 2D-SEDIFLOW model in comparison with the experimental results of Puls and Kiihl [5]. It is shown that the observed sedimentation process in a real river reach could be confirmed with sufficient accuracy by model simulation.

413

8 8

577 578 579 580

581 582 583 584 585 586 km Elbe River Fig. 5. Comparision of 3D-SEDIFLOW and 2D-SEDIFLOW simulation with suspended sediment concentration data [5] observed in the Elbe River

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Acknowledgements T h e w o r k r e p o r t e d h e r e i n was s u p p o r t e d by the G K S S F o r s c h u n g s z e n t r u m G e e s t h a c h t GmbH.

References [1] I. Celik and W. Rodi, Mathematical modelling of suspended sediment transport in open channels, 21st Congress, Int. Association for Hydraulic Research, Melbourne, Australia, August 1985. [2] W. Chu, I.-Y. Liou and K.D. Flenniken, Numerical modeling of tide and current in Central Puget Sound: comparison of three-dimensional and depth-averaged model, Water Resour. Res. 24 (1989) 721-734. [3] D. Prochnow, H. Bungartz and H.-J. Friedrich, Ein modifizierter Chorin'scher Algorithmus zur Berechnung von Gew~isserstr/~mungen, Acta Hydrophys. 34 (1990) 97-129. [4] D. Prochnow, H. Bungartz and Ch. Engelhardt, Modeling and simulation of contaminant transport and sedimentation processes in fluvial systems, in: R. Vichnevetsky and J.J.H. Miiller, eds., Proceedings of the 13th IMACS World Congress on Computation and Applied Mathematics, Vol. 4 (Criterion Press, Dublin 1991) 1970-1971. [5] W. Puls and H. Kiihl, Die Gesehwindigkeit von Elbeschwebstoff bei Lauenburg und Bunthaus, GKSS 89/E/54 (1989). [6] L.C. van Rijn, Sediment transport, part II: suspended load transport, J. Hydraulic Eng. 110 (1984) 1613-1641.