River engineering

f10 September 1993

,Pi

TUDelft

250

River Engineering

Lecture notes f10 Prof.dr. M.de Vries

250

CONTENTS

page

l. Introduetion 3

2. One-dimensional models (principle)

2.1 General 5

2.2 Basic equations 5

2.3 Analytical models

2.3.1 General 12

2.3.2 Practical applications 19

2.3.3 Aggradation due to overloading 27

2.4 Numerical models

2.4.1 General 31

2.4.2 Boundary conditions 32

2.4.3 Internal boundaries 34

2.4.4 Confluences and bifurcations 36

2.4.5 Numerical schemes 37

2.4.6 Practical aspects 38

3. One-dimensional models (extension)

3.1 General 43

3.2 Influence of suspended-Ioad changes

3.2.1 Introduetion 43

3.2.2 Basic equations 44

3.2.3 Solving 2DV-equations 48

3.2.4 Asymptotic approach 50

3.2.5 First-order approach 57

3.3 Roughness and resistance of river beds

3.3.1 General 59

3.3.2 Alluvial roughness and sediment transport 60

3.3.3 Notes on statistics 65

page

4. .Solving river problems

4.1 Introduetion 73 4.2 Tools 4.2.1 Introduetion 76 4.2.2 Numerical models 77 4.2.3 Scale models 82 4.2.4 Hybride models 85 4.3 Flood mitigation 4.3.1 Introduetion 87 4.3.2 Detention reservoirs 88

4.3.3 Flood mitigation middle-river 91

4.3.4 Design of flood way 96

4.3.5 Applications 102 4.4 Bank proteetion 4.4.1 Introduetion 106 4.4.2 Types of structures 107 4.4.3 Morphological aspects 109 4.4.4 Alignment etc. 112 Main symbols 117 Literature 119 Annex I Annex 11, Dimensional analysis

On the design of scale models

127 131

1 INTRODUCTION

The aim of river engineering is to 'improve' rivers. This implies human interference in river systems. Before the actual interference takes place it is necessary to predict whether the improvement anticipated will indeed be reached. It is then also possible to investigate whether side effects may take place leading to a disimprovement.

The forecasting of the effects of interference is carried out by using modeIs. This is a three-step approach:

(i) The river is schematised into a model-river (ii) The problem is solved for the model-river (iii) This solution is interpreted for the real river

These three steps are linked. The schematisation has to be reliable otherwise interpretation of the model-solution is not possible.

In all cases the model has to be calibrated. This means that field measurements have to be used to adjust coefficients (such as Chézy-roughness). In the ideal case other field measurements have to be used for the verification of the model. This is necessary to judge the accuracy of the model.

At this stage it is not yet necessary to specify whether a numerical model or a scale model is used. Both types are used in river engineering. Also combina-tions are used. For a general reference to the application of models for river problems De Vries (1993b) can be mentioned.

In these lecture notes the main attention is on the prediction of morphological changes in rivers due to natural causes or human interference. These changes are of a time-depending nature. At present (1993) for these predictions mainly numerical models are used. These models are available at various degrees of sophistication. Besides numerical models also some analytical models are described. They provide insight into the nature of morphological processes and are therefore indispensable tools (Section 2.3).

For the sake of completeness some information on scale models is provided.

These lecture notes are based on the assumption that the reader is familiar with the basic knowledge of fIuvial processes especially with respect to the

2 ONE-DIMENSIONAL MODELS (pRINCIPLE)

2.1 Genera!

In the one-dimensional approach the morphological parameters (flow velocity, sediment transport, depth and bed level) are averaged over the cross-section. The discussion in this Chapter on one-dimensional models will mainly focus on models with two additional underlying assumptions.

Uniform bed material is supposed to be present. At any rate the absence .of grain sorting is assumed. Hence the grain size is not a function of

time and space.

Fixed banks are postulated. Or, in other words the erodibility of the banks is much smaller than that of the bed due to natura! causes or human interference (bank protection).

After a discussion of the basic equations (Section 2.2) some simplifications can be made for subcritical flow with moderate Froude numbers.

Then in Section 2.3 analytical models are treated. This can only be done after additional schematisation of the basic equations. Moreover it appears that analytical solutions can only be obtained for specific boundary conditions. Nevertheless some practical applications can be given.

In Section 2.4 numerical models are considered. In this Section also some practical problems and their solution are mentioned.

2.2 Basic equations

a

The dependent variables are (Fig. 2.1) flow velocity u(x,t)

sediment transport stx.t)

water depth a(X,t)

bed level z(x,t)

In some cases the water level h =z

+

a

will be considered. The width of the h

Considering a constant width leads to the possibility to take the basic equations for the unit of width. The basic equations are:

au

au

aa

az

- + u- + g- +

g-at

ax

àx êx

=

_gulul

_Cla (2.1)

az

as

= 0

at

+

ax

(2.2)

aa

aa

au

- + u- + a-

=

0

at

Bx àx (2.3)

s

=

.f{u,

parameters) (2.4) Remarks

(i) Equation (2.3) is the continuity equation for water if sediment is absent. The equation can here be used ifs/q

< <

1, which is usually the case. (ii) At this stage no specific transport formula is used. Equation (2.4)

expresses the essential assumption that the loeal transport is governed by the loeal hydraulic parameters of which only u is supposed to vary in time and space.

(iii) The alluvial roughness appears twice in the basic equations. The momentum equation (Eq. (2.1» contains the Chézy value in the

hydraulic friction term. Moreover the roughness is one of the parameters in the transport formula, Eq. (2.4). For this first analysis it is assumed

C(x,t)

=

C

=

constant.

The Eqs (2.2) and (2.4) can easily be combined into

az

+ d

.f{u) au =

0

at

d

u

ax

(2.5)

Hence Eqs (2.1), (2.3) and (2.5) form a system of three partial differential equations in the three dependent variables u, a and z.

The three celerities cd~ dx/dt of this hyperbolic system follow from the cubic equation (de Vries, 1959, 1965)

- c3 + 2 U c2 + (g a - u2 + g

r» -

u g

f..

=

0 (2.6) in which j, = dJtu)/du

It is convenient to define three dimensionless parameters cP = c/u =relative celerity

Fr

=

uN ga

=

Froude number

t/t

=

hl

a

=

dimensionless transport parameter Inserting this in Bq. (2.6) yields

cP3 - 2 cP2 + (1 - Fr-2 -

t/t

Fr-2) cP + t/tFr-2

=

0 (2.7)

The relevanee of the introduetion of the parameter

t/t

can be seen when the transport formula is approximated by a power law s

=

fïu)

=

m u", in which locallym and n are supposed to be constant.

This gives

m n un-1 S

t/t

= --- =

n-a q (2.8)

Hence the parameter

t/t

is proportional to the usually small ratio between sediment transport and water transport. Note that Otn) = 5.

It is worthwhile to analyse Bq. (2.7). The following cases can be considered .

• Fixed bed

The fixed bed case follows fromt;

=

0 (no transport) or

t/t =

O. Inserting this in Bq. (2.7) leads to three roots (Fig. 2.2).

1 10 , I I 1.1

'\

" ,~-1+Fr \ -T-t- --

-_

.

.

-

r--i\4>-1.Fr

I-/'7

V \

\'(

4>1,2

=

1

±

Fr-I}

(2.9) 4>3

=

0 ~ 0 ~ 10 UI ·1 > 10 F

:s

w .2 a:: 10 Or in dimensional form ol 10 0 0.2 0.6 1 1.4 --- Fr C1,2

=

U

+

s: }

(2.10) C₃

=

0

Fig. 2.2 Celerities for fixed bed

Thus the familiar celerities of small disturbances of the water level for a fixed bed are found again .

• Mobile bed, moderate Froude numbers

Equation (2.7) has for physically relevant values of if; and

Fr

always three real roots. In Fig. 2.3 the three roots are plotted graphically. Only small values of if; are used as usually if; = ns/q

< <

1. Apparently for moderate Froude numbers (say

Fr

< 0.6 to 0.8) the celerities of the water levels are not influenced by the mobility of the bed.

Thus like in the case of if; = 0 it holds

,I..

=

1 +

Fr-

I

0/1,2 - (2.11)

Equation (2.7) can be written as

0 10 ~

a:

w

_·1 ...J W 10

o

W

>

~ -2 10 W

a:

·3 10

I

<t>

I

t

-4 10 ~ ~

---

_-,

-"~

~ ~

V

~

SYMBOl~EGARDS SIGN

d_

Fr<1 FT'>

-

WATERlEVEL+ + _~\

;(

--

WATERlEVEL- + /~ ..

_

--

--_

.

_{BEDlEVEL +}

-"

I

_" "_" " _'... '1'-10-2 '«1'=10.2 _{---_...,}", "'... _---- I , ""

..

I ,

..

..

I \ _... I \ I \ _....... I \ I \ I ,

"

I , I;'" 1 ,I , " ," 1_" •

-,

...' 1 • ~, '«1'-10-3 ...'" I • '1'-10-3 ,'" 1 •

_"

" I , _

..".

--

I • '...

---

I

•

... I \

....

I \

..

..

I \

....

I \ ... I • \ I • \ I • \ I _" \ , " , " , " " , 1 • _~, '«1'-10-4 ...' I • '1'-10-4 ,,'" 1 , 1 • ... -_

...

.,., 1 • ... 1 •

....

---

I • I , ... I \

....

I \

..

..

I \ ... I \ I \ I , /' ,, " " ...' _" '«1'-10-5 ,'" '«1'-10-5 " '... "

.

..

-.--- ---

... ·5 10 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

..

Fr

FROUDE NUMBER

Hence for the last term of the left-hand side of Bq. (2.7) holds with Bq. (2.12)

(2.13)

Or, with Bq. (2.11)

(2.14)

Thus in this case

= _....:....1/;_

1 - Fr2

(2.15)

For small Froude numbers (Fr-

<

< 1)it follows cP3

=

1/;. As I/;

< <

1it yields cP3

< <

1 or C3

< <

u. Apparently cP3

< < 1 cP1

,

21

as can be seen

from Fig. 2.3.

Note that according to this analysis for subcritical flow (Fr

<

1) the value of

cP3 ispositive .

• For crisical flow (Fr = 1) the celerity cP1 is not influenced by the mobility of the bed (see Fig. 2.3). Hence for Fr = 1 it follows

cPI

= 2. Moreover Bq. (2.7) becomes

(2.16)

Using now Eqs (2.12) and (2.16) leads to

Figure 2.3 shows that

1 <P21 .,

1 <P31

for Fr = 1. Hence with

<PI

=

2 this gives here

(2.18)

• For supercritical flow (Fr> 1) and not too close tocritica1 flow Fig. 2.3 shows that two roots are again not influenced by the mobility of the bed. The two celerities of the water surface are now both positive. However, as again

(2.19)

the third root, that cao be identified with the celerity of a small disturbaoce at the bed is here negative. This is in accordaoce with the behaviour of

antidunes propagating opposite to the flow direction.

The aoalysis is particularly of importance for moderate Froude numbers

(Fr

<

0.6 to 0.8) because it follows that in this case the water movement and the sediment movement cao be decoupled. Moreover for this case yields c

3 < < 1 CI,21

or <P3

< < 1 <PI,21

(Fig. 2.3). This mak~s a further schematisation possible (Fig. 2.4).

GENERAL TIDALCOMP. MORPH. COMP.

dt-O t t t

T

t

"~

t

'~

Cz

t

I ! Cz c3 c3 : C3 II

--

X --- X

----

X

Figure 2.4 shows thex-t plane for the general case of the propagation of disturbances in an open channel with a mobile bed. In this case

(Fr

<

0.6 to 0.8) the values of Cl and C2are not influenced by the mobility of

the bed. Hence if only the water movement has to be computed (for tidal waves or flood waves in rivers) it can be stated

I

Cl

,21

> >

c3or C3 = O. In other

words during the time interval important for the computations of the water movement (say days) it can be assumed that the bed is fixed.

However, if time depending changes of a river bed are studied it can be concluded from

I

Cl

.

21

>

>

C3 that Cl•

2

-+

±

00. Thus the flow can then be

considered as quasi-steady. This means that for rivers (not for tidal rivers) the terms au/at and aa/at can then be neglected with respect to the other terms in their respective equations.

The water equations become then

au aa az u2

u-

+ g- + = - g-ax ax ax (!la (2.20) u aa + a au

=

aq

=

0 ax ax ax (2.21) Remark

Note that integration of öq/öx = 0 leads toq = q(t). Thus the discharge may still vary in time. At any time t, however, the discharge is constant for every x and the Eqs (2.20) and (2.21) are the differential equations for theflow profile present at time t.

2.3 Analytical models

2.3.1 General

It is attractive to study analytical solutions based on the schematised one-dimensional morphological equations. After the schematisation carried out in

Section 2.2 these equations read:

uau

+

gaa

+

gaz= _ g~

ax

Bx àx

Cla

(2.22)

uaa

+

aau

=

0

ax

ax

(2.23)

az

+ ds

au =

0

at

du Bx (2.24)

The disadvantage of analytical solutions is that usually they can only be obtained after linearisation of the equations. This makes that the analytica1 solutions can only be rough estimates of the real solutions. The linearisation regards three terms.

• Thefriction term in Eq. (2.22) is non-linear in the flow velocity u. As

u'la = u3/q and q = constant, it means that this term varies with the third power of u.

• The convective term u àulê» in Eq. (2.22) is also a non-linear term. However, the influence of this non-linearity is small. This can be seen from a comparison of the first two terms of Eq. (2.22) and using Eq. (2.23):

au

aa

u-+g-=

ax

ax

u2 àa + g

aa

= { _

Fr2 + 1 } g

aa

a àx

ax

àx (2.25)

In many cases

Fr

< <

1 which restricts the influence of the term

u au/ax.

• The transport derivative (ds/du) in Eq. (2.24) is a highly non-linear term.

With the approximation s = m u' it followsds/du - ",-1, with

O(n) = 5.

they give insight into the physics of morphological processes.

Linearisation can be carried out for cases in which variation from an originally equilibrium situation (subscript 0) is studied. The original bed slope is ioand the x-axis is now taken along the original bed slope, still positive downstream. The basic equations then become

(2.26)

u(h - z)

=

q (2.27)

s =j{u) (2.28)

(2.29)

For the linearisation it is assumed (see Ribberink and Van der Sande, 1984)

u

=

u_o + UI with (2.30)

h

=

ao + 11 with (2.31)

a = a

o +

t

with (2.32)

This gives

(2.33)

Moreover small bed changes are assumed, thus

Thus Eq. (2.26) becomes:

o

0 . (u + U~3

(u + u~ - {u + u' } + g- {a +,,}

=

g '0 - g. 0C2q

o êx 0 Bx 0

(2.35)

As oujox

=

0 and oajox

=

0 as weIl as io = this gives

u Bu' 0"

o ox + g OX =::

-3u~ u'

C2q g (2.36)

From Eq. (2.27) follows:

(2.37)

or

u' =:: (2.38)

Combination of Eqs (2.36) and (2.38) gives

u; {O'f/

oz}

O'f/ _ 3u;

{U

o } (2.39)

- - - + g - - - g - - - ('f/ - z) ao OX OX êx C2q ao or { 2 } O'f/ 2 OZ 3i0 g 1 - Fro - + g • Fro -

=

g • - ('f/ - z) ox OX ao (2.40)

Thus with ao = 1- Fr~ andAo = 3i)ao

O'f/ OZ

a - + (1 - a \ -

=

A (" - Z)

o Bx ol Bx 0

Now also the sediment equations, Eqs (2.28) and (2.29) can be linearised

az

+ [

dJtU)]

•

au'

=

0

at

du 0 Bx (2.42) or with Eq. (2.38)

.

{

_BxBz _

a.,.,

_êx } = 0 (2.43) or with (2.44) àz + c

az _

c

a.,.,

=

0

at

0

ax

0

ax

(2.45)

Thus the combination of Eqs (2.41) and (2.45) gives the general linearised morphological equations in the dependent variables .,.,and

z.

The following two special cases can be distinguished.

• Theparabolic modelis valid if backwater effects can be neglected. This means in practice if only large values ofx (and t) are considered. Thus the assumption

au/ax

= 0and

aa/ax

= 0are made. This can be reached by putting CXo

=

0 in Bq. (2.41) without stating, however, that Fr~

=

1. Thus

az

-

=

A (.,.,- z)

ax

0 (2.46) or

az }

ax

(2.47)

Elimination of

a.,.,/ax

from Eqs (2.45) and (2.47) gives

az

+ c

az _

c

{A

-1

a

2

z

+

az }

=

0 (2.48)

Thus

OZ _ K 02Z

=

0

at

0 OX2

(2.49)

• The simple-wave equation is found if for smallx (and t) the resistance term is neglected. Equation (2.41) becomes then

a 0'1] + (1 - a \ OZ

=

0 o Bx ol êx

(2.50)

Combination with Eq. (2.45) gives

• The hyperbolic model expressed in Eqs (2.41) and (2.45) can also be written as a second order partial differential equation in

z.

Combining Eqs (2.41) and (2.45) gives:

az

+ C

az _

c { __1_-_a_o

_az

+ _Ao ('1]

!lt 0

ax

0

ax

u

«

«: (2.52) or

oz

Co

az

c A + - - - ~ ('1] - z)

=

0

at

=, ax

=,

(2.53) Thus (2.54)

Or, again using Bq. (2.45) [ -1 Co àz +

az ]

at

ax

CoAo

a

z

+ -- -

=

0 (2.55) 010

ax

Hence

az

010

a

2

z

at

Ao

axat

(2.56) Or (2.57)

if again K; = cfA, is used.

Ithas been shown by Vreugdenhil (1982) that indeed if a small lengthL has to be considered that the hyperbolic model expressed in Bq. (2.57) reduces to the

simple-wave model of Bq. (2.51) and reduction to theparabolic model of Bq. (2.49) if large values ofL are of importance. In dimensionless form the validity of the models can be judged by means of the Péclet number P = CJ)OI~o.

type equation validity

az

Co

az

= 0

simple wave model

-

_at

+ --- small x and t

CXO OX

az

a

2

z

= 0

parabolic model -

_at

-K-_oOX2 large x and t

CO

az

a [az

CD

az

1

0

hyperbolic model

---

«,«, at

-

ax

at

+ CXo

ax

= general

In Table 2.1 a summary of the three models discussed is given. Therefore Eq. (2.57) has been rewritten in a slightly different way. For small Péclet numbers the simple-wave model follows whereas for large Péclet numbers the parabolic model applies.

2.3.2 Practical applications

In Chapter 1 it is stated that analytica1 models can be used to gain insight into morphologica1 phenomena. In the first place an application of the parabolic model of Eq. (2.49) can be given. Itregards a morphological time-scale

(De Vries, 1975).

• Theparabolic model can be derived straight forward from the basic equations in the following way. The derivation is based on the assumption that the flow is not only quasi-steady but also quasi-uniform. Hence the momentum

equation reduces to

=

_g~

=

gaZ

Cl-q

êx

(2.58)

Differentiation with respect tox leads to

(2.59)

This can be combined with the combined sediment equations

az

+

df(u) au

= 0 (2.60)

at

du Bx into

az

a

2

z

= 0 (2.61)

at

-K-

ax

2

with

(2.62)

See also Jansen et al. (1979).

Only differentation with respect to x has been carried out. Hence the time t is treated as a parameter. Therefore the merit of the parabolic model is that it can also be used for time-dependent discharges with

az _

K(t)

a

2

z

=

0

at ax2

(2.63)

Consider now a river discharging into a (hypothetical) lake (Fig. 2.5).

1:.11

--=---

---At t = 0 the lake level is supposed

to drop over a distance.:lh. The

neglect of draw-down effects

implies that for t

>

0 the flow is

LAKE assumed to be uniform. The x-axis

is taken along the original bed level positive upstream. That does not change Eq. (2.61) but it

simplifies the initial condition into

Fig. 2.5 Morphological time-scale

Z(x,o)

=

0 (2.64)

The boundary conditions read

lim z(x,t) = 0

x-oo (2.65)

and

in which H(t) is the Heaviside function (unit-step function). The solution reads (see for details De Vries, 1975):

z(x,t)

=

-Ah • erfc [ x ]

2{KJ

(2.67) in which IlO erfc y

= ~

i

exp

(-!"l

d,

(2.68)

For

a

varying discharge the solution is z(x,t) = -Ah • erfc

(2.69)

The parabolic model is valid for A =xi/a> 2 to 3 (Vreugdenhil and De Vries, 1973).

Consider now in Fig. 2.5 a standard length L; from the mouth upstream. How long does it take before the river bed is lowered by 50 % at the station x

=

Lm' i.e. before z(Lm, T

J

=

-1/2J~.h?

IfTm is expressed by the number of years (Nm) then the following expression can be obtained (2.70) where Iyear Iyear Y

=

f

K(t)dt

= ~

;i

f

S(t)dt o 0 (2.71)

STATION

RIVER (approx. D i 3ali Nm

distance mm *1ü4 km centuries

to sea)

Rhine Zaltbommel

(Netherlands) (100 km) 2 1.2 100 20

Magdalena Puerto Berrïo

(Colombia) (730km) 0.33 5 30 2 Dunaremete (1826 km) 2 3.5 40 10 Nagymaros Danube (1581 km) 0.35 0.8 180 1.5 (Hungary) Dunaujvaros (1581 km) 0.35 0.8 180 1.5 Baja (1480 km) 0.26 0.7 210 0.6 Tana Bura (Kenya) (230 km) 0.32 3.5 50 2 Apure

(Venezuela) San Fernando 0.35 0.7 200 4.4

Mekong Pa Mong (Thailand) (1600 km) 0.32 1.1 270 1.3 Serang (Indonesia) Godong 0.25 2.5 50 2.0 Rufiji Stiegler's (Tanzania) Gorge 0.4 3.2 20 4.0

In Table 2.2 information is given on the morphological time scale for some rivers. In this table Lm

=

200 km is taken. For the Mekong River this is even too small. Note the large differences found for the morphological time scale for various rivers.

• An application for the simple-wave model is found in the deformation of a trench dredged across a river (Fig. 2.6).

:;;:;:.' .

h

a

~.._I ••

LI"IUM

Fig. 2.6 Dredged trench across a river

In this case the friction can be neglected because of the small value of

x

involved.

The momentum equation reduces now to

êa + oz = oh = 0

êx àx ox (2.72)

Hence h(x,t) = constant (rigid-lid approximation).

Here bed load has to be postulated in order to allow the use of s = mu', Itis attractive to rewrite Eq. (2.51) with ao = 1 using a(x,t) as the only

dependent variabie left. With Eq. (2.72) this gives

oa + c(a) oa

=

0

in which

e(a)

=

ds

da

(2.74)

If

m

and

n

are constant then

(2.75)

Hence in this (exceptional) case linearisation is not required.

Now the modification of the downstream slope can be studied.

j ; : ~u i

_

___

__

J

_Q

_i

~

_

~

~

_

:

_

I.

A0SITST---- ~-; ,

I

p

I

_I

i

; ! !

Figure 2.7 Downstream slope

Taking the original bed level as x-axis the downstream slope at t = 0 is

given by

[

L -

x]

a(x,o)

=

ao + p °Lo (2.76)

a(x,o) = ao x > L_o (2.77)

A certain depth

a

with

a;

<

a

<

a

,

+

p is for t

>

0present at alocation

which is a distancee(a) . t more downstream than at t = O.

The question can be raised at what time t a prescribed depth will be present

This simply leads to

(2.78)

Remarks

(i) The change of the downstream slope can easily be computed for a trench because it regards an expansion wave. With time the slope becomes flatter. For the upstream slope the opposite is the case: it is a shock wave. In theory the slope gets steeper here until the angle of repose of the sediment is reached (Fig. 2.8).

~ EXPANSION-WAVE SHOCK-WAVE

~ SHOCK-WAVE EXPANSION-WAVE

Fig. 2.8 Deformation of a trench and a hump (bed load)

(ii) If instead of a trench there is a hump present, then the deformation is in the opposite way. Now the upstream slope is an expansion wave and the

downstream slope behaves as a shock wave (Fig. 2.8). Note the similarity with a propagating dune.

• An application for the hyperbolic model is not easily obtained. On the one hand this model is only valid for a constant discharge. On the other hand the hyperbolic equations (Eqs (2.41) and (2.45» can only be solved for specific boundary conditions.

For small Froude numbers

Fr

-+0 hence a;-+ 1 the Eqs (2.41) and (2.45)

become

071 - A (71 - z)

=

0

ox

0

and

az + c az _ c 01'/

=

0

at 0 ax 0 ox

(2.80)

This hyperbolic model is applied to the following case (Fig. 2.9).

Fig. 2.9 Hyperbolic model

A river with a constant discharge q is flowing into a lake. Att = 0 the lake level is dropped over a certain distance Sh. Solutions for 1'/(x,t) and z(x,t) are sought.

For Ilh

< <

ho Eqs (2.79) and (2.80) apply with the initial conditions:

z(x,O) = 0 and 11(x,O) :: 0 (2.81)

The boundary conditions are: lim z(x,t) = 0 x-oo and lim 1'/(x,t)

=

0 x-oo (2.82) 1'/(O,t)

= -

ll.h • H(t) (2.83)

in which H(t) is the Heaviside function (unit-step function). Itis possible to arrive at an analytical solution for the relative variation of the depth at the

river mouth HO,t) (De Vries, 1980).

in which

T

=

2Ao

cJ

=

dimensionless time

I, = modified Bessel function of the fust kind and the vlh order.

,

t

Fig. 2.10 Dimensionless depth reduction ~ = HO,T)

This solution has been used to investigate how errors in the 'submodels' i.e.

the predietors for s and C propagate into an error in the morphological prediction (here

n.

Here only the main line of thinking is given. For details see De Vries (1982).

(i) The lowering of the lake level (Fig. 2.9) induces a temporary reduction of the available depth in the mouth expressed in ~. Accepting a certain reduction (with ~

<

1) means that a certain (dimensionless) time (T) is present during which navigation is restricted.

(ii) Equation (2.84) expresses the result of a deterministic model. By

considering that T = 2Ao

cJ

errors in Ao and Copropagate into an error in

t. The errors inAo and Cocome from the errors in s and C as it can be shown that

(2.85)

2.3.3. Aggradation due to overloading

due to overloading. The problem sterns from the Indian subcontinent. Earthqua

-kes in the Himalayas lead locally to subsidence causing the supply of (some-times huge amounts of) sediment to a river. This causes aggradation of the river bed and consequently of a raising water level during floods. Inundations may occur.

By Sony et al (1980) a study was reported measuring the aggradation in a

laboratory flume with steady uniform flow S and Q adding ilS for t ~ 0 at

x = O. The authors used the parabolic model as a theoretical model to explain the tentendency of their measurements. However, the parabolic model is not valid for small

x

and tand the solution of the authors could only be adjusted to the measurements by changing the parameter K. But then the model loses the capability of making forecasts.

The general (linearised)hyperbolic model suitable for this problem is described by Eqs (2.41) and (2.45).

The boundary conditions for a supply .6.s atx = 0 for t

>

0 are

z(x,O)

=

0 and 7](x,O)

=

0 for x

>

0 and t ~ 0 (2.86)

and

lim z(x,t) = 0

x~oo x-oolim 7](x,t)

=

0 (2.87)

Moreover the continuity of sediment requires

C)O

ss • t

=

f

z(x,t) dx

o

(2.88)

Taking the time derivative of this equation and substitutingaz/at from Eq. (2.57) using (2.45) and (2.88) yields the upstream condition:

Ribberink and Van der Sande (1984, 1985) present solutions using the following dimensionless parameters

location

x

-

=

• x (2.90)

time (2.91)

bed level (2.92)

The overloading atx = 0 leads to the propagation of a shock-front described by

i = 112

i.

In Fig. 2.11 an indication is given for the validity of the various analy-tical models. Itappears that only for a small interval for 1 no analytical solution of this problem could be obtained.

The authors used the linearised hyperbolic model also for the case in which the over-loading is substantial. In that case adapted expressions for c and K were used.

Fig. 2.11 Validity diagram (after

Ribberink&Van der Sande, 1985) At the shock locally the simple-wave

;

continuity equation yields

I---,..._~::-"""""""'-T I I I I I I ~ -u

Fig. 2.12 Finite shock

equation is used. Combination with the

c

az

=

as

ax

ax

(2.93)

In discrete form (Fig. 2.12) this becomes 6.S

C = C = - (2.94)

s 6.z

Using locally the equation for the flow profile without friction a

aa

+

az

=

0

ax

ax

(2.95)

it follows that over the shock

-a . 6a + 6Z

=

0 (2.96)

Eliminating ~ from Eqs (2.94) and (2.96) and assuming that upstream of the shock front S = s;

+

Ils = SI (the subscript 1 used for t - 00) it follows that a = 1/2(ao

+

al) and the adapted celerity results

1 _(2.97)

Similarly also an adapted diffusion coefficient KI can be found. Expressed in terms of Ils the results are:

(2.98)

and

(2.99)

In Fig. 2.13 the results for the hyperbolic model for large disturbances are compared with the experimental data of Soni et al (1980). The dimensionless relative bed level change is plotted against ()=

sb/i.

Considering the usually large scatter in morphological measurements, this can be called a fair agreement between the model and the experiments.

1.8

t-1

experIments ~ SonIetal (1980) io o 1 <; < 10

t

1.2 .6.10<; <100 1.0 100 0.' 8

Fig. 2.13 Hyperbolic model for large disturbance (after Ribberink & Van der Sande, 1985)

2.4 Numerical models

2.4.1 General

For practical river problems the basic equations for the one-dimensional morphological process have to be solved numerically.

Here only the case will be treated for which the equations for water and sediment can be decoupled i.e. for quasi-steady flow.

For the unit of width the equations are

(2.100)

and

(2.101)

This implies that the following restrictions apply

(i) Only for Fr

<

0.6 to 0.8.

(ii) Only if

au/at

and

aa/at

can be neglected in Eq. (2.1) and Eq. (2.2) . respectively. Hence morphological computations for tidal rivers are excluded. The same holds for rapid discharge-changes in non-tidal rivers.

Moreover it excludes the case in which au/at

< <

0 but aa/at has to be considered. This holds forinstanee for a river with a reservoir where due

to Q(t) there is storage govemed by aa/at.

(iii) An essential restrietion is that the local sediment transport is govemed

by the loeal hydraulic conditions. Hence a sediment transport formula can

be used, expressed in a general way by s =f(u).

(iv) The alluvial roughness has to be known or supplied by a sub-model: a

roughness predietor.

The Eqs (2.100) and (2.101) are written in the dependent variables u(x,t) and

z(x,t). The other dependent variables can be found as follow

•

•

•

a(x,t) s(x,t) h(x,t) from from from q

=

u.a s = .f(u) h=z+a

To solve the basic equations means that boundary conditions have to be

available (Sub-sections 2.4.2 ... 2.4.4). A numerical scheme is required for the

equation. This aspect is treated in Sub-section 2.4.5.

In Sub-section 2.4.6 some practical aspects of the use of numerical models for

morphological problems are given attention.

2.4.2 Boundary conditions

First of all we consider a standard case of a river reach 0

<

x

<

L for which

Eqs (2.100) and (2.101) have to be solved. The following conditioris have to be

known.

(i) Initial eondition: z(x,O)

(ii) Downstream eondition: h(L,t) (subcritical flow)

(iii) Upstream condttion: q(O,t) and s(O,t)

The following information can be given ad (i) Initial eondition

Even for a meandering river with constant width (B) it is not so clear how z(x,y,O) with -JI2B

<

Y

<

+

JI2Bhas to be schematised into z(x,O). Itseems logical to take the average bed level across the river.

ad (ii) Downstream condition

For subcritical flow the downstream water level has to be known to determine the flow profile by solving Eq. (2.100) for known values of z and q. In practice usually h(L,t) follows from the discharge rating-curve h =h(Q). Obviously when the bed level atx= Lchanges then the Q-h curve known at t = 0is not valid anymore. In practice the downstream

.boundary is placed so far downstream that bed level changes do not take place during the time of interest.

ad (iii) Upstreamconditions

Two boundary conditions are necessary

• Q(o,t) for the solution of Eq. (2.100)

• S(o,t) for the solution of Eq. (2.101).

The upstream boundary conditions have to be selectedby the user of the model. The model is not more than a tool in the hands of the user.

Some general guidelines can be given.

• The selection of Q(t) can be from historical records. The selection can also be based on records of the rainfall R(t). A hydrological model is then required for the translation of R(t) into Q(t).

• Which historical record is taken cannot be answered in a simple way. It

depends on the purpose of the model study. Wet years will in generallead to quicker morphological changes than dry years. One may also conclude that various records Q(t) have to be used to investigate the sensitivity to the morphological changes.

• Mathematically s(o,t) has to be known. As here via s =./{u) a unique rela-tionship is assumed betweens and u, it is also sufficient to know u(o,t). Only as an exception it is possible to selects(o,t) without difficulty (see remark below).

• The fact has to be faced thats(o,t) cannot be selected properly. There is, however, a simple way to overcome errors in the morphological predictions due to erroneous selections of s(o,t).

...

<::::

«:

~gron

...:.:.:::

:

:::

:::

:

::

::

:

:

:

::

Irl~U~

:

::

:::::::::::::::

Fig. 2.14 The

x-t

diagram

-x

Any error in s(o,t) will propagate into the

x-t diagram (Fig. 2.14) above the characteristic through the origin (0,0).

Therefore at x =Xo errors can be expected for t

>

to' The location ofx =

°

has then to be selected in such a way that for interesting points in the interval

°

<

x

<

L forecasts can be made over a sufficiently long time interval.

• There is one case in which the (upstream) boundary condition for the sediment creates no difficulty. Consider a dam built in a river. In case no sediment passes the dam the condition downstream of the dam is then u = u.;

(Shields). The depth is there a = qlu.; and the bed level follows from

z = h-a.

2.4.3 Internal boundaries

For practical problems the boundary conditions are not sufficient. Consider again the interval

°

<

x

<

L. In this interval one of the relevant parameters may be discontinuous. Now two branches have to be considered. Standard Pro-blems are

(i) Withdrawal of a discharge AQ(t) at a point.

(ii) Withdrawal of part of the sedimentAS(t) at a point. (iii) A change of the width of the river at a point.

For the three cases indicated above the following additional information can be given.

ad (i) Withdrawal of water

In case the withdrawal takes place viapumping than the only internal boundary condition that has to be introduced is Ql = Q2

+

AQ.

However, if the withdrawal takes place via an intake upstream of a weir than not only thedischarge but also the water level is discontinuous at

x

=Xo' The characteristics of the weir have to be introduced at the

internal boundary to determine the water level at the downstream end of the upstream branch.

t

t

branch 1 branch 2

o

_{Xo_x}

L

Fig. 2:15 Intemal boundary

In Fig. 2.15 atx =Xo a discontinuity is present. Itis then necessary to place atx

=

Xo an intemal boundary to handle

the discontinuity properly. In this way the interval 0

< x <

L is divided in

two branches linked by condition(s) at

the intemal boundary.

ad (ii) Withdrawal of sediment

In this case ('sediment mining') the sediment transport is discontinuous atXo. Again two branches are used. At the intemal boundary the condition has to be given that

(2.102)

Note:

If the sediment mining takes place in areach rather than in a point then it is not necessary to use an intemal boundary. In this reach the sediment withdrawal is represented by a lowering of the bed with a velocity W(x,t). For this reach this

W is then introduced as a souree term in the sediment continuity equation: B

az

+

as

= -

W • B

at

ax

(2.103)

Outside the reach W = 0 has to be taken. For this problem three branches are required.

ad (iii) Change of widtb

Again two branches are necessary with B

=

BI andB

=

B2 respectively.

The intemal boundary condition is now

Remark

The discontinuities discussed here create discontinuities in other parameters notably in the bed level. For instance the sediment withdrawal at a point will gives a downward stepàz in the bed level with

(2.105)

At every time t the value of àz is directly related to the local conditions.

2.4.4 Confluences and bifurcations

Confluences and bifurcations in the river considered require a special treatment. Now at least three branches have to be used.

Confluences are relatively simple to introduce. At the conjluence there is continuity for water and sediment. As the values of Q and S for the two

upstream branches are known, the values of Q and S for the downstream branch follow from these two equations.

A bifurcation is more difficult to handle.

In Fig. 2.16 a sketch of a bifurcation is given. Now Qo and S, are known and there

arefour unknowns namely QI, Q2, SI and

S2. Hence besides the two equations

2 2

L

Qj

=

0 and

L

s,

=

0 (2.106)

Fig. 2.16 Bifurcation

(withi = 0,1,2) there are two more equations required.

(i) At the bifurcation there is only one water level.

Hence the distribution of Qo into Ql and Qz is such that given the conveyance of the two downstream rivers and their stage-discharge relations both rivers lead to the same water level at the bifurcation.

the bifurcation (BulIe, 1926). Hence the bifurcation determines SI/S2' In practice it is not easy to establish this ratio.

2.4.5 Numerical schemes

A number of schemes are in use to solve the basic morphological equations numerically. Here only the cases for which Eqs (2.100) and (2.101) apply will be considered. Hence for the decoupled equations valid for subcritical flow with

Fr

<

0.6 to 0.8.

A complete numerical solution of Eqs (2.100) and (2.101) is obtained by altemating steps

Step I: Solve Eq. (2.100)for given values z Step 11:Solve Eq. (2.101)for given values u

In this Sub-section only some attention is given to step 11. For the first step i.e. the determination of the flow profile any suitable methode can be used.

For a detailed overview of varlous possible schemes for Eq. (2.101) reference

can be made to Vreugdenhil (1982). Explicit schemes can be usedbecause the

celerity C is rather small so large time steps can be made. A class of difference

schemes can be indicated based on the following difference equations.

n+1

n n

n-I

Zk - Zk Sk+1 - Sk

+ +

M 2t.x

- _1_ [

{aZ+1

+

aZ}

{Z:+I

- Z:} -

{aZ

+

aZ

_

I}

{Z:

- Z:_I} ]

=

0 (2.107)

4M

in which z: = z(ktu, nAt).

The coefficient a can be chosen in different ways for (J =

coü

Sx

(i) a = 1 gives the original scheme by Lax (1954)

(ii) a = (J gives a form of an upstream difference (Godunov) scheme

(iii) a =

al

gives a form of the Lax-Wendroff (1960) scheme

intermediate type (Vreugdenhil & De Vries, 1967).

These schemes except (iii) give a first-order approximation of Bq. (2.101). These and other (second-order) schemes are analysed by Vreugdenhil (1982) on a number of aspects: (i)lineair wave propagation, (ii) stability,

(iii)conservation, (iv)shock waves and (v)numerical diffusion.

The analysis is among other things based on the comparison of analytical solutions and numerical solutions of the simple-wave equation (in a dimensionless form)

àz

a {n}

= 0

-+-z

at

ax

(2.108)

This equation describese.g. the deformation of a dredged trench in the case of bedload transport c.f. Bq. (2.75).

Remarks

(i) The solution of a numerical scheme has to be made with care and the characteristics of the scheme should be analysed properly. This is not always done. Itwas shown by Croat (1986) that the HEC-6

morphological model publishedby Thomas (1979) is unstable.

(ii) No general rules can be given as to the actual selection of ÀX and !l.t in a specific case. Obviously the accuracy of the computational results will increase the smaller!l.t and ÀX are selected. Moreover !l.t has to be small enough to reproduce adequately the boundary condition Q(o,t).

(iii) The scheme applied for Bq. (2.101) has to beconservative also in the treatment at the (internal) boundaries.

2.4.6 Practical aspects

The results of numerical computations can hardly be judged in detail in their correctness. There are, however, two aspects that can be helpful in assessing numerical computations. These are:

(i) The ultimate solution (t -+ 00).

These aspects will be treated seperately here .

• New equilibrium situation (t - 00).

The final equilibrium solution with respect to the induced changes can give a guidance to the judgement of the computation for t

>

0 if t

=

0 is the time at which the changes start. It has to be recalled that three 'standard' cases can be distinghuised

(i) withdrawal of water (ii) withdrawal of sediment (iii) long constriction

For a constant discharge Q both depth and slope will change eventually. For a varying discharge, expressed in the probability density p{Q} only the change of the slope (from ioto it) is of practical importance.

Table 2.3summarises the results for the three standard cases. A practical example of predicting bed-level changes due to the withdrawal of water from a river can be found for the Tana River (Kenya) in Jansen et al.

(1979, pp. 433 .. .440) .

• Magnitude of discontinuities

In various cases discontinuities are introduced in Q, S, and/or B. This leads to discontinuities in the river characteristics, notably in the bed level (z). The magnitude of the discontinuities can be estimated by means of simple conside-rations on conservation of mass.

As an example the withdrawal of water (dQ from Qo)atx = 0 for t

>

0 can be taken. For the discharge holds

Q=uoBoa (2.109)

For the sediment holds (m and n supposed to be constant)

problem for constant varying discharge discharge withdrawal of _{i} Q} 3/n

--

=

_f

Qn/3Po{Q} dQ water _io

o-eo

i1 ₀ -

₌

io

..

AQ or _-a}

₌

1- flQ f Qn/3PI{Q} dQ Po{Q} - PI{Q} ao Q 0 withdrawal of _i}

: [1-

~r

I} [ fl

vr

n - -

₌

I-V

,with: sediment _{io io} a} [ flSr,n

..

ASor AV V

=

fS(Q) p{Q} dQ -

=

1- -ao S 0 long constriction i}

:[!:F

B}-n/3.inf3f Qn/3 p{Q} dQ

..

=

constant

-io ₀ Bo-Bl a}

: [!J'

thus: i}

:

[!J'

-

-ao _io

Table 2.3 Changes of slope and depth for n = constant (after De Vries, 1986)

Upstream ofx =0 it holds Qo = u, .B, . a;and downstream UI = BI . al'

As there can be no sedimentation at a point, for reasons of continuity it has to be stated S; = SI' This leads via Eq. (2.110) to u; = UI' Eq. (2.109) gives

then for BI = Bo. Qo Qo -

AQ

=

ao _al or al

=

1

-

AQ ao Qo (2.111) (2.112)

By definition the water level (h) is given by

h

=

a + z

If the water is withdrawn via pumping then h is continuous (hl = hJ.

Hence

or

Combining Eqs (2.112) and (2.115) leads to

(2.113)

(2.114)

(2.115)

3. ONE-DIMENSIONAL MODELS (EXTENSION)

3.1 Genera!

Inthe previous Chapter the basis of one-dimensional modelling is treated. However, a number of restrictions are made there.

In a random order these restrictions are:

(i) The sediment transport is supposed to be dependent on the local hydraulic conditions, expressed as s =ft...u).

(ii) The sediment is supposed to be (nearly) uniform. This implies that sorting effects are excluded.

(iii) The erodibility of the river banks is assumed to be negligible like in the case of protected banks.

(iv) The alluvial roughness (e.g. expressed in the C-value) is supposed to be known.

The following initial remarks can be made.

ad (i) This point gets more attention in Section 3.2.

ad (ii) Research is going on to include grain-sorting in the numerical modeIs. Ribberink (1987) carried out detailed studies on this matter. Olesen (1987) used the research of Ribberink to include this in his two-dimensional (horizontal) model.

ad (iii) The erodibility of the river banks is especially of importance to two-dimensional (horizontal) models.

ad (iv) Roughness and resistance are discussed in Section 3.3.

3.2 Influence of suspended-Ioad changes

3.2.1 Introduction

The presence of suspended load not necessarily means that s =ft...u) cannot be

used anymore. The presence of suspended load means that in space (and time) it takes for the transport a certain distance (or time) to adapt itself to the

hydraulic conditions. If the adaptation length is much smaller than the space-step ÄX of the one-dimensional model, then the adaptation takes place 'sub-grid'

that means that still s

=

/(u) can be used. A similar situation is present in time. If theadaptation time is much smaller than the time-step t::.tof the one-dimen-sional model, then the adaptation can be assumed to be instantaneous. In other

words s =fïu) can be used.

3.2.2 Basic equations

Por steady uniform flow the sediment concentration (cp)is described by the

differential equation

Wcp+ e dcp

=

0

s s dz (3.1)

in which Wsis the fall velocity of the particles. Usually the sediment diffusion

coefficient (e.) is taken equal to the turbulent diffusion coefficient for

momentum (e"J with

em(z)

=

K U. Z

{1 -

zla} (3.2)

Combining Eqs (3.1) and (3.2) gives after integration

cp(z) = [a _ z • ~]

W/

KU •

CPI a -

z,

Z

(3.3)

This equation, known as theRouse equation, contains an integration coefficient:

at Z

=

ZI the concentration CP(ZI) = CPI has to be known.

Remarks

(i) Equation (3.1) holds for dilute suspensions i.e. cp

< <

1. If this is not

(l - cf»W cf> + E dcf>

=

0

s s dz (3.4)

.Consequently the value of z, in Eq. (3.3) should not be taken too close to the bed as there cf>

< <

1 is not valid.

(ii) Equation (3.2) is an approximation which according to Coleman does not hold too good in the upper half of the vertical. Moreover there seems to be an influence of Z = W)KU.on Es. In Fig. 3.1 the results of

the measurements (Coleman, 1970) are depicted. The scatter in this figure is due to the fact that dé/dz was determined from flume measure-ments and used in Eq. (3.1) to determine Es.

1.0 0.6 0.4 0.2 _i _0.1

u.a

0.06

t

0.040.02 0.01 0.006 0.004

•

_*

+ * ~

•

+ 11

..

... ~ A

.4

<I

•

<I ~

.

<I

...

_-8 • -.! I ...• ,_

.

-

o •

.

~ _~ ~~~

--

-

..._ ~ _~ 0 o·b~~ (..~ 0 _, ::I

.

:~

_e-

.

'~~.,r\ ~

'K

••

_°0 Oe __ Von Karman --PowerLaw --- Logarlthmlc wju • wju. 0.3470 0.641. 0.4140 0.672• 0.43211 0.702-0.4390 0.705+ 0.475Ó 0.818* 0.513<I 0.840... 0.542 ... 0.864'" 0.570+ 0.908x 0.002 0.001

o

ru

U U M U U V U M 1~

-

z/a

Fig. 3.1 Sediment diffusion coefficient Es (after Coleman, 1970)

In his experiments Coleman flushed the sediment through the flume; hence no alluvial bed was present.

By matching the Coleman measurements with a parabola in the lower half of the depth and a constant in the upper half theparabolie-constant function of Eswas

This reads

Es(Z)

=

Emu

=

{O.I3 + O.2(W/u.)} u ,a

Es(Z) =

4{z/a} {I - va}

Emax

for zla ~ ~

for zla < ..!. 2

(3.5)

For steady uniform flow

a4>/ax

= O. However, for the steady and non-uniform situation

a4>/ax ~

O. For this case the mass balance for the sediment has to be reconsidered. Consider the control volume Ax dl. The momentary velocity components are Ui and the momentary concentration is 4>.

The mass balance gives

a4>

a

a

- + - {UI 4>} + - {U3 4> - W 4>} = 0

at

ax

az

s

(3.6)

Defining as usual in turbulent water-movement Ui

=

Ui

+

U'i and 4>

=

4>

+

4>'

makes averaging over a suitable time-interval () possible. It is assumed that () is large enough to get rid of the turbulent fluctuations.

The averaging leads to

It can now be assumed

- ui

4>'

= E1l

4>

x

+ En

4>

z

- U;

4>'

= E31

4>

x

+ E33

4>

z

(3.8)

in which the subscripts x and z denote differentiations. The continuity equation for the fluid reads

(3.9)

oe/> oe/> oe/> oe/> 0 0

-+UI-+U3--W---{EIIe/>

_ot

+EI3e/>}--{E31e/> +E33e/>}

=

0

OX OZ s OZ OX x Z OZ x Z

(3.10)

In order to make Eq. (3.10) manageable it is common to assume that the axes of the coordinate system are taken along the main axes of the tensor E;j(see e.g.

Graf, 1971,p. 164). This is a (weak) argument to neglect EI3and E31. The result

is

(3.11)

in which EI

=

Eli and E3

=

E33is taken for simplicity.

A further simplification is possible by assuming nearly uniform flow (u3 =: 0).

Moreover it can be assumed that EI and E3 are of the same order of magnitude.

If the length scale of the problem area is longer than its depth scale then

(3.12)

The result of all these simplifications is

o</>+ U(I) a</> -

.i_

{W

e/> + E(3) a</>}

=

0

at

ax

Bz s àz

(3.13)

Remarks

(i) For steady uniform flow Eq. (3.13) reduces to (with E3 = Es)

.i_

az

{W

</>+ E ae/>}

=

0

s s àz

(3.14)

Integration in the z-direction gives

Ws</>+ E d</>

=

constant

Equation (3.15) expresses the sediment flux in the vertical direction. As this flux equals zero at the water surface, Eq. (3.1) follows from Eq. (3.15). (ii) Whether in Eq. (3.13) the terms with

acJ>/at

and/or

acJ>/ax

can be neglected

depends on the problem involved.

3.2.3 Solving 2 DV-equation

The solution of Eq. (3.13) will be continued here for the case

acJ>/at

= O. Hence the following equation has to be solved (with E = Es)'

acJ>

a {

acJ>}

u - - -

w

cJ>

+ E(Z) - = 0

ax

az

s

az

(3.16)

To simplify the understanding, the solution is sought for a case of uniform flow (Fig. 3.2).

For any upstream boundary condition

cJ>(o,z)

for increasing

x the concentration

cJ>(x,

z

)

will gradually approach the equili-brium concentration Fig. 3.2 Adaptation of

cJ>(x,

z)

_cJ>

_e

₍

_z

₎

of the flow in

the rectangie.

The solving of Eq. (3.16) requires the following boundary conditions:

(i) At

x

= 0 the value of

cJ>(o

,

z)

has to be known.

(ii) At z = a (water surface) the sediment flux has to be equal zero. Hence

(iii) Atx - 00 no condition is required. The value of cJ>(00.z) is a result of the com-putation.

(iv) At the bed (z

=

Zb) the flux is non-zero. Sediment is settled or picked-up. lim [WscJ> + e(z) OcJ>]

=

0

~""

oz z =

Z

b

(3.18)

The bed-boundary condition requires special attention. The reasoning made by Kerssens (1974) is as follows:

• For bed-lood transpon it caobe assumed that thetranspon reacts instantaneously on a change in the flow conditions.

• For suspended load it can similarly be assumed that the concentration near the

bed reacts instantaneously on a change in the flow conditions. This implies the assumption

(3.19)

in which Zbois close to the bed not at the bed. Equation (3.3) is then determined

by

z,

= Zbo and cJ>1= cJ>"and as the transport s is known for cJ>

=

cJ>"via a

transport formule and

a

s - lim

J

u(x,z) • cJ>(x,z) dl

x"''''' Zbo

(3.20)

the value of cJ>"is known from Eqs (3.3), (3.5) and (3.20).

An alternative to the assumption made in Eq. (3.19) is the one made by O'Connor (1971) that the concentration gradient reacts instantaneously on the change of the hydraulic conditions. Or

ocJ>

I

oZ Z=Z

bo

Remarks

(i) In order to save computer time it is customary to use the following transformation in the vertical direction.

(3.22)

az e(z) àz'

A constant step àz' gives then automatically that in the z-direction the grid points are concentrated near the bed where the largest concentration gradients are found.

(ii) A complete one-dimensional morphological computation using the 2-DV convection-diffusion equation requiresfour alternating steps.

Step I: Compute the flow field for given bed level.

Step 11: Compute the concentrations c/>(x,z)

Step 111: Step IV:

a Compute the transport from

J

uc/>dz Compute the new bed level

(iii) The solution of Eq. (3.16) requires per time step not much computer time. However, as in practice many time steps have to be taken it is worthwhile to look for approximative solutions. This topic is treated in Sub-section 3.2.4.

3.2.4 Asymptotic approach

Consider Eq. (3.13) rewritten as Eq. (3.23).

ac/>+

_!

{uc/>}-

i.

{wc/> + eac/>} = 0

at ax az s az

(3.23)

The equation is made dimensionless by using corresponding scales to the various parameters:

Inserting this in Bq. (3.23) gives

a ae/> aU { ,ae/>} ae/> E a { 1ae/>}

WsT

aT

+ LWs u

ar

=

ar

+

Wp

ar

E

ar

(3.24)

An asymptotic approach can be used to solve this equation. This is possible if the concentration profiles in the vertical do not differ too much from the equilibrium profiles (Galappatti, 1983; Galappatti and Vreugdenhil, 1985; Wang, 1984; Wang and Ribberink, 1986).

Galappatti and Vreugdenhil (1985) state that the scale E of E is of the order

1,4 K u. a. Hence 1 K u. a

=

4 Wa s U 0.005 -Ws (3.25)

which can weU be of the order one.

Therefore the right-hand terms of Bq. (3-24) are supposed to be of 0(1). They

are responsible for the vertical readjustment of the concentration profiles.

If the dimensionless parameters a~T and Ua/L Ws are small then asymptotic

solutions for Bq. (3.23) are possible provided the concentration profiles do not differ too much from the equilibrium profile.

In an introduetion to the method Ribberink (1986) considers the case of Bq.

(3.21) without the convective term. Itis assumed that a small parameter

5 = a/WsT

< <

1 is present and that the following asymptotic expansion can be

found for e/>(r,T)

e/>

=

e/>0 + 5 e/>1 + 52 e/>2 + ..• + fi e/>j + ... (3.26)

Inserting this in the equation

(the value of Eand hence of E' = E/Wsa assumed for simplicity sake to be

constant) gives

(3.28)

As

öi</>

I

o

< <

lil </>

,

0

-

I

the following equations can be found

O-order a </>0 +E--

I

a2</>0

=

0 (3.29)

ar

af

1st-order a</>

I

+E--

I a2</>1

₌

a </>0 (3.30)

ar

af

ar

flh-order a</>j +E-

I

a2</>j

₌

a</>j

_

1

(3.31)

a:;:-ar

af

Equation (3.29) is the equation for the equilibrium profile (steady state). Thus

</>o(r,r)

=

</>e

(r,r)

and also </»(/>0

=

</»(/>e·

a

a

2

By using the operator D[]

= -

+ EI - the

ï"

order equation can be

ar

af

written as

(3.32)

or with the inverse operator D -I

(3.33)

The important assumption made by Galappatti (1983) is that only the zero-order

1

1

1

q;(T)

=! ePG",T)dr=!

{ePo+OeP

l

+024>2

+ ...

}dr= !

ePodr=q;:(T)

(3.34)

A shape function I/;

ir.

T) is introduced with

(3.35)

Using Eq. (3.33) gives

eP

(r

T) =D

-

1

[a4>o] =D-

1

[i.{1/; •

q;}] =D-

1

[q;. al/;o

+1/;

aq;]

(3.36)

1 '

aT

aT

0

aT

0Br

As an approximation

(3.37)

In which

1/;1

is the first-order shape function.

In genera!

(3.38)

The complete asymptotic solution reads

+ ... (3.39)

Equation (3.39) shows that

4>(r,

T) can be described by an ordinary differential

equation for the depth-averaged concentration (if».

The equation contains the shape-functions I/;j. Now the boundary conditions have

to be considered. At the bed (here it can be taken

r

= 0) it can be assumed

either