hydraulic resistance of vegetation

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University of Twente, Faculty of Engineering Technology, Water Engineering and ManagementP.O. Box 217, 7500 AE Enschede, The Netherlands

Final Project ReportPlanungsmanagement für Auen

(Project no. U2/430.9/4268)

HHyyddr r aauulliicc r r eessiissttaannccee oof f vveeggeettaattiioonn P P r r eed d i i c c t t i i oonnss oof f aav v eer r aag g ee f f l l oow w v v eel l ooc c i i t t i i eess bbaasseed d oonn aa

r r i i g g i i d d --c c y y l l i i nnd d eer r ss aannaal l oog g y y

FREDRIK HUTHOFF

DENIE AUGUSTIJN

(January 2006)



Hydraulic resistance of vegetation January 2006

ii



January 2006 Hydraulic resistance of vegetation

iii

Preface

This is the fourth and final report of the project “Planungsmanagementansatz für Auen” as

assigned by the Bundesanstalt für Gewässerkunde (German Institute of Hydrology).

(Project reference number U2/430.9/4268).

The aim of the presented research is to improve the understanding of vegetation roughness and todetermine an appropriate way for including vegetation resistance in hydraulic models used for

spatial planning or maintenance of river floodplains. Before the hydraulic effects of vegetation is

studied an overview is given of open channel flows and the associated resistance effects (chapter

2). Next, as an analogy to flow through vegetation, the flow resistance due to rigid cylinders is

investigated (chapter 3 and 4). Two different methods to describe the average flow field in such

situations are demonstrated: a depth-integrated vertical velocity profile (Klopstra et al. 1997) and a

new bulk description where the average flow field is estimated as based on turbulent energy

considerations and a simple force balance. The two methods are compared to one another and

also to independent results from flume experiments (chapter 5). The properties of the two methods

are discussed and preliminary recommendations are given on the appropriate treatment of

vegetation effects in hydraulic models used for river management. A follow-up study in 2006 will

focus on uncertainties involved with these issues.

The help and comments of several people were of great support in the realization of this work. We

would like to thank Volker Hüsing, Anne Wijbenga, Paul Termes, Detlef Lohse, Francisco

Fontanele, Suzanne Hulscher, Wim Uijttewaal and Martin Baptist for their critical remarks,

contributions of information or data, and pushes in the right direction.

F. Huthoff 1,2

D.C.M. Augustijn1

Enschede, January 26th, 2006.

1University of Twente

Faculty of Engineering Technology

Water Engineering and Management

P.O. Box 217

7500 AE Enschede

The Netherlands

2HKV consultants

PO-box 2120

8203 AC Lelystad

The Netherlands




iv




v

Summary

River flows can be described in all detail by the Navier-Stokes equation, however, for practical

purposes the complexity of the flow description needs to be reduced. The objective of this study is

to find an appropriate way of describing the vegetation roughness. For river management purposes

floodplain roughness is relevant because it has an effect on water levels. Too much resistance

causes high water levels, which increases flood risk. To be able to predict the effect on water levels

of future changes in floodplains an appropriate independent predictor is required. In this study we

compared existing and a newly developed descriptor for vegetation roughness on their

performance under different conditions. Based on this analyses the most appropriate description of

vegetation resistance is deduced for different conditions.

The Chézy or Manning equations are commonly used to describe steady uniform flows and

the associated flow resistance in open channels (e.g. rivers). Although these two equations are

fundamentally different, they behave similar when the relative flow depth (R/k) is large. A stronger

theoretical foundation is often attributed to the Chézy equation and the Manning equation is usually

considered to be entirely empirical. However, Gioia and Bombardelli (2002) showed that theManning/Strickler equation can be derived theoretically based on (i) a force balance between

gravitational pull and bottom shear stress and (ii) a relation between small scale velocities and the

mean velocity that includes relative roughness. Another requirement in the derivation is that flow is

hydraulically rough. In general river flows can be considered hydraulically rough, which justifies the

application of the Manning equation.

In chapter 2 it was pointed out that for flow over a flat (rough) surface the Manning

coefficient is a true measure of wall roughness, while the Chézy coefficient is inversely related to

the roughness via the White-Colebrook relation. The appearance of the hydraulic radius in this

formula complicates the applicability of Chézy for complicated river geometries, e.g. when the

calculation of a composite roughness for compound channels is desired. Therefore, it is easier to

construct a composite resistance parameter based on addition of Manning coefficients instead of

Chézy coefficients. The resultant composite Manning coefficient is no longer a measure of wallresistance only, but also depends on depth variations. Even if the material of the bounding wall is

the same everywhere, then the composite resistance parameter is depth-dependent. Therefore, if

field measurements indicate that hydraulic resistance is depth-dependent, this does not necessarily

imply that wall-roughness or friction is complicated. It could also be the result of the cross-section

geometry.

In chapters 3 and 4 two different descriptions for the hydraulic resistance of vegetation are

put forward. The principal assumption in both of these methods is that separate plants may be

treated as rigid cylinders (i.e. the rigid cylinder analogy ). Because the Manning and Chézy

coefficients are typically associated with flows affected by wall roughness, flow resistance caused

by rigid cylinders is described in terms of resulting average flow velocities. For the situation of

submerged vegetation, the flow field can be divided in flow over the vegetation, the surface layer,

and flow through the vegetation, the vegetation layer. Klopstra et al. (1997) derived an analytical

solution for the velocity profile for this case, which is the topic of chapter 3. Originally the turbulent

characteristics in this description were not treated consistently and the necessary turbulent length

scale α was described empirically. Therefore, the same data as used by Klopstra et al. (1997) was

analyzed further to arrive at the following expression for average velocity (equation numbering

refers to numbering in subsequent chapters):




vi

topveg

s

tot tot

h

k h

h

k

u

U Υ

−+Υ==Υ (3.66)

with the depth-integrated average velocities in the vegetation and surface layer:

+Υ

−Υ

−+−Υ+≈=Υ 1

14ln2

11

21 *

**

k

k k

s

veg veg

k hk u

U ll

(3.59)

( )

−

+−

−

+−+−+Υ==Υ ∗ 1ln

2 s

s s s

k

s

toptop

h

hk h

k h

hk h

b

hk h

u

U

κ (3.64)

where

−+=Υ

l

k hk 1* (3.56)

4/14/33.4 bh s α ≈ (3.65)

)(240.0

k hb

sh

−+=α (Table 3.3)

( ))/(1 D

D

mDC bbmDC

=== α α

l (3.9)

gibmDC

giu

D

s 22

== (3.10)

Where U tot is the depth-averaged flow velocity, h is total water depth, k is vegetation height,

κ the von Kármán constant (0.4), m is surface density of vegetation elements, D is the plant

diameter and CD is dimensionless drag coefficient. Advantages of the newly proposed relation for α

as compared to the previously existing ones are that (i) it is dimensionally correct and (ii) it

converges to the roughness height for relatively deep flow. Furthermore, it performs slightly better

than the expression by Klopstra et al. (1997), which is adopted as a standard for Dutch river

modeling (Van Velzen et al. 2003). Although the proposed adopted solution gives reasonable results, it still has some

drawbacks: (i) the closure relation for the length scale α is still poorly understood and (ii) the

expression for the average velocity is quite complicated. Therefore, in chapter 4 a different

approach was used that resulted in a simpler, yet physics-based description. This description is,

again, based on a two-layer approach and does not require a closure relation as difficult to interpret

as the ones found for the turbulent length scale α. For the surface layer an expression for the

average velocity in the surface layer was derived based on Kolmogorov scaling (related to turbulent

energy dissipation at different length scales). For the vegetation layer a simple force balance was




vii

used that takes into account cylinder form drag, shear stress due to flow over the cylinders and the

gravitation force that drives the flow. As a result an expression for the depth-averaged flow velocity

is found (along the two-layer scaling approach):

3/22/1

),min(

),min(

+−−+

==Υ

sk

sk k h

h

k h

h

k

u

U

s

tot tot (4.33)

where again the velocity is scaled against the velocity that would be present when the vegetation is

not overflown (i.e. emergent vegetation):

gibmDC

giu

D

s 22

== (3.10)

Which agrees well with the data of Klopstra et al. (1997) but is evidently much simpler than the

expression from chapter 3. Comparison with data for real vegetation shows that the predictions are

reasonable. Deviations are possibly due to the fact that the predictors are derived for rigid

cylinders, and that properties of real vegetation such as flexibility, foliage and inhomogeneous

spatial distributions were neglected.

In chapter 5 the description based on the depth-integrated and two-layer scaling approach

are compared on their performance for real vegetation characteristics, i.e. values for k , m, D and

C D as given by Van Velzen et al. (2003) for different types of vegetations are evaluated. It appears

that the two methods perform similarly for most vegetation types, although the difference at high

water levels may still be as large as 25%, where the two-layer method tends to give higher values

for the average velocity. Comparison to flow measurements, and empirical relations, of the average

flow velocity near real vegetation suggest that the new two-layer scaling method is best suited for

vegetation that is not extremely sparse or dense. For very sparse vegetation bed roughness effects

become active which were neglected in the current study. In case of very dense vegetation

(grassland) the two methods clearly deviate in their answers. A comparison to independent data of

flow over (natural) vegetation shows to be favorable for the depth-integrated method. More data isneeded to further substantiate this result.

Future studies will focus on translating the demonstrated rigid cylinder resistances to

descriptions of hydraulic resistance of natural vegetation. Also, the relative importance of

vegetation resistance will be studied in relation to other processes that cause energy losses in river

flows, such as bed pattern formations in the main channel (i.e. river dunes) or energy exchanges at

the floodplain–main channel interface.




viii

Contents

1 Introduction.................................................................................................................1

1.1 Aim and relevance ......................................................................................................... 1 1.2 Flow through vegetation................................................................................................. 1

2 Steady uniform river flows.........................................................................................3

2.1 Flow over rough surfaces............................................................................................... 3

2.1.1 Steady uniform flow formulas............................................................................. 3

2.1.2 Hydraulically rough or smooth? ......................................................................... 6

2.2 Composite channels....................................................................................................... 8

2.3 Conclusions ..................................................................................................................10

3 Hydraulic resistance of submerged cylindrical elements.....................................12

3.1 Analytical velocity profile of flow trough a vegetated layer.............................................12

3.1.1 Flow in the vegetation layer ..............................................................................13 3.1.2 Flow in the surface layer...................................................................................15

3.1.3 Calibration of analytical profile..........................................................................16

3.1.4 Vegetation solidity ............................................................................................18

3.2 Numerically determined velocity profile .........................................................................21

3.3 Turbulent length scales .................................................................................................23

3.3.1 Two definitions of turbulent length scales .........................................................23

3.3.2 Scaling properties of the flow field ....................................................................24

3.4 Average flow velocity based on vertical velocity profile .................................................29

3.4.1 Average velocity in the vegetation layer............................................................31

3.4.2 Average velocity in the surface layer ................................................................33

3.4.3

Overall average velocity ...................................................................................35 3.5 Resistance parameters .................................................................................................36

3.6 Conclusions ..................................................................................................................36

4 Hydraulic resistance based on a two-layer scaling approach..............................38

4.1 The average flow velocity in the surface layer...............................................................38

4.1.1 Shear stress in the surface layer ......................................................................39

4.1.2 Turbulent energy and Kolmogorov scaling........................................................39

4.1.3 Asymptotic behavior of hydraulic radii...............................................................41

4.2 A force balance in the vegetation layer .........................................................................43

4.3 Overall average flow velocity for submerged rigid cylinders ..........................................45

4.4 Real vegetation .............................................................................................................48

4.5 Conclusions ..................................................................................................................50

5 Evaluation of vegetation resistance descriptions .................................................51

5.1 Appropriate modelling in river flows...............................................................................51

5.2 Comparing vegetation roughness descriptions..............................................................52

5.3 Sparse or very dense vegetation...................................................................................57

5.4 Bed resistance ..............................................................................................................60

5.5 Conclusions ..................................................................................................................61




ix

6 Discussion & outlook ...............................................................................................62

6.1 Composite channel geometries.....................................................................................62

6.2 Variability in vegetation characteristics..........................................................................62




x



University of Twente Final Project Report

1

1 Introduction

1.1 Aim and relevance

The prospect of climate change in combination with several floods and threatening situations in the

past decade has changed the vision of river engineers and authorities. The general attitude in

many countries worldwide is now to look for alternative ways to enlarge the discharge capacity of

rivers, instead of increasing dike heights to meet protection standards (e.g. Favier 2003).

Floodplain lowering, dike shifting, side channels, and removal of hydraulic obstacles in floodplains

are some of the measures considered. A fortunate consequence of this trend in river management

is the opportunity for nature development and restoration of the ecological corridor function of

floodplains.

Ironically, nature rehabilitation in floodplains may pose potential flood threads because of

the changing hydraulic resistance of the vegetation in the floodplain: increased vegetation

abundance leads to a decrease in flow velocity and a rise of water levels. Simple calculations for

typical lowland river reveal that floodplain roughness, when varied within realistic boundaries, could

affect water levels in the order of several decimeters (e.g. Huthoff and Augustijn 2004). By Dutch

law, any increase in design water levels due to changes in the floodplain needs to be compensated

(Pluimakers andvan Rijswick 2003). It is therefore important to know what future effects nature

rehabilitation in the floodplains will have on water levels during flood conditions (e.g. Favier 2003).

For that purpose, it is essential to understand the interaction between vegetation and the flow field,

and eventually to understand how these processes change over time (vegetation succession,seasonal variability). The purpose of the current study is to develop a framework that enables

predictions of water level consequences in nature development areas.

1.2 Flow through vegetation

Above, James addresses the necessity to find methods other than the conventional roughness

formulations to describe flow in presence of vegetation. Furthermore, Kadlec (1990) points out that

flows in presence of vegetation are often in the transition region between laminar and turbulent and

that the usage of conventional roughness relations is therefore no longer justified. Examples to

more physically based descriptions of flow through vegetation are (i) cylindrical methods

(vegetation as an array of cylinders, e.g. Petryk and Bosmajian 1975, Klopstra et al. 1997, Nepf

1999 and Stone and Shen 2002) and (ii) flow through a porous medium (e.g. Hoffmann and Van

der Meer 2002). Also, several groups have performed detailed numerical simulations of flow

Wetlands remain one of the most complex and poorly understood ecosystems in

terms of water management. It is therefore necessary to develop a greater

understanding of the hydrodynamic processes involved in such ecosystems so that

effective management tools may be developed (Harris et al. 2003).

Conventional resistance equations (such as those of Manning, Chézy and

Darcy–Weisbach) are inappropriate for flow through emergent vegetation,

where resistance is exerted primarily by stem drag throughout the flow depth

rather than by shear stress at the bed (James et al. 2004).




2

through vegetation (e.g. Shimizu and Tsujimoto 1994, Neary 2003, Choi and Kang 2004).

Unfortunately, due to their complexity these are not suitable for implementation in commonly used

simulation packages for river flows. The other extreme of existing vegetation resistance methods

can be found in empirical relations (e.g. Copeland 200, Fischer 2001). These are indeed simple

expression that can easily be applied for fast rule-of-thumb calculations. The biggest problem in

this case is that they only guarantee good results in the situations where they were derived.

General utilization in river systems where vegetation characteristics are known, but where no flowvelocity measurements are available, is therefore not easily justified.

Although the introduction of a detailed vegetation resistance description (based on physical

plant parameters) represents the field situation more realistically, it could also introduce additional

uncertainty into flow simulations. If the combined uncertainty of the relevant physical parameters

(due to natural variability and experimental error) is larger than the uncertainty of a bulk resistance

parameter (as in methods of, for example, Bettess 2003, Mastermann and Thorne 1992) then a

detailed resistance description could yield worse results. Kouwen (1992) also points out that these

detailed methods are usually not intended for channel design.

The aim of the proposed research is to improve the understanding of vegetation resistance

and to determine a way for including vegetation resistance in hydraulic models used for spatial

planning and maintenance of river floodplains. The emphasis in this work is on a simple yet realistic

model to describe vegetation resistance. A future objective is the application to nature rehabilitation

schemes. While still beyond the scope of the current research report, the influence of rehabilitation

schemes on flood potential should follow from the combination of vegetation succession

techniques, translations to vegetation resistance, incorporation of other dynamic resistance effects

(e.g. in main section, such as bed forms) and interactions between different flow sections (e.g. at

main channel-floodplain interface).

Towards the ultimate goal of achieving an appropriate method to describe vegetation

effects in hydraulic flow models, the principle issue that needs to be resolved is identification of

dominant processes in certain situations. In the current report, appropriate methods (for certain

ranges of applicability) are identified by comparing the predictions of different flow descriptions to

results of flume experiments. In order to get an overview of the relevant hydraulic processes in river

flows in the presence of vegetation, first a study is carried out on fundamentals of flow over roughsurfaces. Next, interaction processes between the flow field and rigid cylinders are investigated.

These will serve as an analogy for flow through and over a vegetated bed.




3

2 Steady uniform river flows

A close look at the details of the flow field in a river reveals that the flow velocity is slower near the

boundaries, and that throughout the flow many swirling motions and local accelerations exist. The

laws that govern these motions are well understood, but do not provide a practical tool for

assessment of phenomena at the temporal and spatial scales that are of interest to river engineers

(e.g. Alcrudo 2002). A simplification of the fundamental flow description (i.e. the Navier-Stokes

equation, see Pope 2000, Fox et al. 2002 or other textbooks on fluid dynamics) is therefore

necessary.

When overlooking a river, the overall flow field appears steady and uniform. This

observation may suggest that on a large scale simpler laws govern the behavior of the flow.

However, these laws are much less understood, and although flow seems uncomplicated, it

remains difficult to predict the overall flow behavior solely based on the characteristics of the

riverbed. In this chapter commonly used descriptors of average flow velocities in channel flows are

compared and evaluated for their preferred application. Towards the goal of describing vegetation

resistance in a suitable manner for river engineering purposes, particular emphasis is made on the

way that resistance to flow is incorporated in these flow relationships.

2.1 Flow over rough surfaces

Hydraulic resistance in a river system can be attributed to any process that somehow protrudes,

blocks or diverts the flow on its natural path through the system. Due to the large amount of

possible underlying processes, hydraulic resistance is generally classified into four components

(Rouse 1965): (i) surface/skin friction, (ii) form resistance (drag), (iii) wave resistance (free surface

distortions) and (iv) resistance associated with flow unsteadiness or local accelerations. While this

separation is merely artificial (the underlying processes interact continuously in the river system,

e.g.Yen 2002), it enables distinction between relevant processes in different situations. The most

basic conceivable situation in which hydraulic resistance can be studied is the case of steady

uniform flow over a flat (rough) surface. Before studying more complex manifestations of flow and

its response to roughness it is essential to understand how resistance is described and what

assumptions lead to the most basic relationships.

2.1.1 Steady uniform flow formulas

For open channels, three different formulas are commonly used to describe the relation between

the mean flow field and channel resistance in the steady uniform case:

1. Chézy: RiC U = (2.1)

2. Darcy-Weisbach: Ri f

g U

8= (2.2)

3. Manning: Rin

RU

6/1

= (2.3)




4

Since these equations are supposed to describe the same situations, but give different predictions

the question arises which of these descriptions reflects reality best? A disturbing observation is that

in different parts of the world different preferences for each of these equations exist. Here are some

commonly heard arguments used in favour for any of these:

1. Chézy description has a sound theoretical basis, while Manning is merely empirical and

Darcy-Weisbach has its origin in pipe flow.2. Manning’s resistance coefficient is a true measure of wall roughness, while Chezy’s and

Darcy-Weisbach relations are dependent on the water depth.

3. Only the Darcy-Weisbach’s friction factor is dimensionless and is therefore the most

general and widely applicable resistance parameter.

It is immediately apparent that in the Chézy equation and the Darcy-Weisbach equation the

average flow field (U ) is in the same manner dependent on the hydraulic radius R . Consequently,

any additional dependencies of C must also be existent in f . The fact that they are inversely

proportional is only a matter of definition but does not change their overall (equivalent) behaviour.

Darcy-Weisbach’s friction factor f is indeed dimensionless (while C is not), but the introduction of

the gravitational constant g can hardly be considered a useful scaling technique considering that

for all practical applications g is constant. Remains the advantage of f that it is the same

irrespective of the unit system chosen. Effectively, Darcy-Weisbach’s f and Chézy’s C can be

considered equivalent, and whether or not one of the equations finds its origin in pipe flow is

irrelevant.

Manning’s resistance parameter is fundamentally different from the Chézy parameter because

a different dependence between the mean flow field and the hydraulic radius is given. Any depth or

geometry dependence in (the inverse of) Manning’s n must therefore necessarily be different from

that in Chézy’s C . Not both of these expressions can be correct if the respective resistance

parameters are both true measures of wall roughness. To investigate this issue let us have a look

at the fundamental assumptions that lead to the Chézy equation:

1. Steady uniform flow2. Force balance between gravitational pull and bottom shear stress

3. Mixing length assumption (Prandtl, von Kármán): flow field produces a vertical logarithmic

velocity profile.

4. Calculate mean flow field by integrating velocity profile over depth (depth averaged

velocity).

The result is the following (see Jansen 1979 for derivation):

( ) Rik RU N /12log18= (2.4)

Where k N is the equivalent sand roughness height introduced by Nikuradse (1933). Consequently

Chézy’s equation is found with the condition that the resistance coefficient C behaves as (White-

Colebrook relation for hydraulically rough flow):

( ) N k RC /12log18= (2.5)

From this expression it becomes apparent that C is not a function of wall roughness only, but also

of the depth of flow (through the hydraulic radius R ). In that respect it would be more suitable to call

C a conveyance parameter (e.g. Chow 1959) since it reflects the resistance of flow as a measure




5

of wall roughness and cross-section geometry. In other words, the Chézy parameter is a function of

relative depth (R/k N ), expressed here as f C :

Rik R f RiC U C *)/(== (2.6)

If the Manning equation is equivalent to the relation above, then a similar relative depth

dependence should be present in the uniform flow formula. Again, the Manning equation reads:

n Rk R f Rin

RU M /)/( 6/1

6/1

∝⇒= (2.7)

Where the roughness descriptor f M now inhibits Manning’s coefficient n instead of Chézy’s C.

Assuming that Manning’s n is a measure of roughness height only, then, in order for f M to become

a function of relative depth only:

6/1)/()/( k Rk R f M ∝ (2.8)

Likewise, the relative depth dependence of the roughness descriptor in the Chézy formula behavesas:

)/12log()/( k Rk R f C ∝ (2.9)

In Figure 2.1 the ratio between these two (f M /f C ) is demonstrated as function of relative depth. As

can be seen, the ratio converges towards a constant value at high relative depth. Therefore, in this

range the two equations show equivalent behaviour.

Figure 2.1 Ratio between functions that describe the dependence on relative depth for the Manning and

the Chézy equations ( f M /f C ).

Above, an assumption was made about the relation between Manning’s n and the roughness

height k . Actually, the same functional form has been proposed by Strickler (1923) as:

25:

6/1

S k nStrickler = (2.10)

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 200 400 600 800 1000

Relative depth R/k

( f M a n n i g ) / ( f C h e z y )

r e l a t i v e d e p t h d e p e n d e

n c e




6

Where k S is the equivalent roughness height proposed by Strickler (1923). In the relation above,

Strickler actually proposed a constant of proportionality of 21.1 instead of 25. However, later works

consistently refer to the value 25 (e.g. van Rijn 1990). The Manning equation now becomes:

Rik

RU

s

6/1

25

= (2.11)

Or, when written in a dimensionally homogenous form through introduction of the gravitational

acceleration g (e.g. Yen 2002):

gRik

RU

s

6/1

8

= (2.12)

A derivation of Manning’s equation that predicts Strickler’s relation (apart from the proportionality

constant 8) is demonstrated by Gioia and Bombardelli (2002). Based on scaling arguments and the

concept of incomplete asymptotic similarity (Barenblatt 1996) they conclude the following:

• Flow over a surface that is steady, uniform and in the hydraulically rough regime is

described by an equation such as Manning’s.

• The Manning coefficient is a measure of absolute roughness and is related to the

roughness height as proposed by Strickler.

The Manning equation was first proposed as being entirely empirical (Manning 1889) but Gioia and

Bombardelli have shown that there is a justification for its form based on physical arguments. The

derivation that led to the Chézy equation (with the Chézy coefficient as the White-Colebrook

relation) has some similarities with the derivation that led Gioia and Bombardelli to the

Manning/Strickler relations. Both derivations are based on (i) a force balance between gravitational

pull and bottom shear stress and (ii) a relation between small-scale velocities to the mean velocity

that includes relative roughness. The essential difference between the two approaches lies in thedecisions made to arrive at point (ii). In order to arrive at the White-Colebrook equation, shear

stress is assumed constant throughout the flow field and a specific form for the mixing length is

assumed. The most crucial assumption in Gioia and Bombardelli’s derivation lies in the acceptance

of Kolmogorov’s hypothesis of constant energy dissipation rate, and the choice of velocities to

scale against characteristic length scales (roughness height and hydraulic radius).

2.1.2 Hydraulically rough or smooth?

Gioia and Bombardelli (2002) showed that the Manning/Strickler formula is found when the flow

equation is based on incomplete similarity in the relative roughness. They argue that this

assumption corresponds to the condition of hydraulically rough flow and that for hydraulically

smooth flow a different flow formula would be found derived (i.e. when assuming complete

asymptotic similarity in relative roughness). The resulting roughness descriptor in hydraulically

smooth flow yields the expression as first proposed by Blasius in 1912 (e.g. Hager 2003):

8/1Re~: f Blasius (2.13)

Where f is a roughness descriptor similar to those earlier presented in equations (2.8) and (2.9) for

the Manning and Chézy relations, respectively. Kolmogorov (1941a, 1941b) characterized the

hydraulically rough regime through introduction of a length scale η. At the scale of this Kolmogorov




7

length, molecular dissipation begins to dominate over inertial energy transfer from larger eddies to

smaller ones, i.e. viscous effects come into play. If however roughness elements (i.e. roughness

heights) are significantly larger than the Kolmogorov length, then viscous effects are negligible and

energy losses in the flow are determined by the irregularities at the wall. The flow is then

characterized as hydraulically rough. Conversely, if the roughness height is smaller or nearly equal

to the Kolmogorov length then viscous effects start to dominate, and flow is hydraulically smooth. In

that case the flow field no longer ‘feels’ the irregularities of the surface.Based on dimensional analysis, the Kolmogorov length η [m] is shown to be a function of

the kinematic viscosity ν [m2/s] and the energy dissipation rate ε [m/s

3]:

4/13

~:

ε

ν η length Kolmogorov (2.14)

Assuming that the energy dissipation rate ε at the Kolmogorov scale equals the energy input rate at

large scales (Kolmogorov 1941b), then the Kolmogorov length η can also be written as a function

of hydraulic radius R , viscosity and the average flow velocity U :

4/1

3

3

3

4/13

~

~

~

⇒

U

R

R

U

ν η

ε

ε

ν η

(2.15)

The Manning/Strickler formula is only valid in the hydraulically rough regime. This implies that the

(Strickler) roughness height should be larger than the Kolmogorov length:

( )326

67

6/1

4/1

3

3

88

gi Rk

Rgik

RU

U

Rk k

S

S

S S

ν

ν η

>⇒

=

>⇒>

(2.16)

Figure 2.2 shows for two different channel slopes (i ) the transition from hydraulically rough to

hydraulically smooth flow. It can be seen that for a channel slope of i =0.0001 (i.e. a 1 m drop over

10 km length) and a roughness height of about 50 µm, flow becomes hydraulically rough when the

depth exceeds 10 cm. This is no longer the case when the channel slope is smaller because flow

velocities become smaller at similar flow depths. When the channel slope is ten times smaller

(i =0.00001) then flow becomes hydraulically rough for a roughness height of 50 µm when the depth

is well beyond 2 m. Considering that sand grains have sizes that are typically 50-1000 µm and that

flow depths in lowland rivers are easily a few meters, most river flows are hydraulically rough. It istherefore justified to use the Manning equation for most river flows. Only when flow velocities

become very small (at low flow depths) Manning’s equation is no longer suitable because flow

becomes hydraulically smooth. In that case the roughness descriptor as proposed by Blasius (2.13)

should be used (e.g. Gioia and Bombardelli 2002), which yields for the average flow velocity:




8

Hydraulically smooth flow:( )ν

ν

457

8/1

Re

Re gi R

U UR

gRiU

∝⇒

=

∝ (2.17)

Figure 2.2 The transition from hydraulically smooth to hydraulically rough flow as a function of Strickler’s

roughness height (k S ) and the hydraulic radius (R) for two different channel slopes (i). Below

the dashed line flow is hydraulically smooth, above hydraulically rough. The viscosity of water

is taken as ν =10 -6

[m2 /s].

2.2 Composite channels

So far, only bottom resistance has been considered and resistance due to sidewalls of the channel

was neglected. Therefore, all arguments put forward strictly only apply to wide and shallow

channels, where it can be expected that sidewall effects have no significant contribution to the

mean flow field. However, in case of overbank flow, a significant increase in the complexity of the

flow occurs and a 1D approximation no longer automatically holds. A common method to describe

two-stage flows is by the divided channel method (e.g. Chow 1959, Jansen 1979, Yen 2002). This

method assumes that flow in each of the subsections is independent from other sections and that

the total discharge equals the sum of section-discharges. The approach is expected to be valid if

the different sections are wide, such that lateral transfer mechanisms at the interfaces can beneglected. In Chow (1959) an indicative value of 10 for the relative width (as compared to water

depth) is given for a wide channel .

When the depth of flow is of the same order of magnitude as the width of the channel then

the interaction between the main channel and the floodplain needs to be taken into account.

Several groups have performed studies of composite channels in order to quantify lateral exchange

processes at the main channel–floodplain interface (e.g. Knight and Shonio (1996), van Prooijen

(2004)). Here we will investigate whether the divided channel method may serve to describe flows

in composite channels. For this purpose, three divided channel methods are compared to data from

0

25

50

75

100

0 1 2 3 4

R [m]

k S

µ [ m ]

Hydraulically rough

Hydraulically smooth

i = 0.0001

i = 0.00001




9

flume experiments by Knight and Demetriou (1983), who measured average flow velocities in a

two-stage channel with adjustable side walls (see Yen 2002 for overview of resistance addition

methods).

The general equation for the composite Manning resistance parameter for a two-stage

channel is:

m

m

f

f

T

T

n

R

n

R

n

R 3/23/2

3/2

+= (2.18)

Where R T , R f and R m are the hydraulic radii of the total, the floodplain and the main channel cross-

sections respectively with n their corresponding Manning coefficients. In the flume experiments that

will be considered here the main channel and the floodplain were made of the same material. If the

Manning coefficient is indeed a measure of wall roughness then we may assume nf = nm = nwall .

Note that such an assumption is not possible if Chézy coefficients were to be combined because

they are measures of the relative roughness. Expression (2.18) may now be simplified to:

+=3/23/2

3/2

m f

T

wall T R R

R

nn (2.19)

The expression above shows that the total (composite) Manning coefficient of a two-stage channel

is no longer a measure of wall-roughness only: the composite Manning coefficient is a function of

three hydraulic radii, which themselves are dependent on the flow depth. Consequently, the

composite Manning coefficient is depth dependent. If this is the case for a two-stage channel, then

this would also be the case for a three-stage channel, or, a channel with any number of stages

greater than 1. In general, when the depth of flow is not the same throughout the cross-section of a

channel, then the hydraulic resistance coefficient is depth-dependent. This is even the case when

the surface characteristics of the bounding wall are the same everywhere.

In (2.19) the value of nwall was determined in experiments where only 1-stage flow was

considered (i.e. for a prismatic channel with constant depth throughout). Now, depending on theused definition of the hydraulic radius of each of the cross-section compartments, the total

(composite) Manning coefficients can be determined. Since the hydraulic radius is equal to the

cross-section area divided by its wetted perimeter, the value is directly related to chosen definitions

of the wetted perimeters:

Divided channel methods (see Figure 2.3 for parameter dimensions):

1. The wide channel approximation: in flow formula the wetted perimeter is equal to the

bottom width of the channel (Wf + Wm, side walls are neglected)

2. True wetted perimeters: all surfaces in contact with water are included in wetted

perimeter (walls and channel bottom, Hf + Wf + Hm + Wm).

3. Virtual interface in wetted perimeter: the virtual interface between the main channel and the

floodplain is included in the wetted perimeter (Hf + Wf + Hv + Hm + Wm).

Figure 2.3 Possible compartments of perimeter accounted for in calculation of composite resistance. H m is

the bankfull depth and H f =H v is the depth in the floodplain.

Hm

Hf

Wm

Wf Hv




10

Figure 2.4 shows the results when composite Manning resistance coefficients are predicted

according to the three divided channel methods. Between the three methods, the composite

resistance is smallest (least rough) in the wide channel approximation and largest in the method

that includes virtual interfaces. Larger wetted perimeters (greater contact surface) result in more

resistance to flow. Overall it can be concluded that in the considered experiments of Knight and

Demetriou (1983) method 2 works best: resistance due to sidewalls is included but extra energy

losses at the main channel-floodplain interface are neglected. However, this outcome should betaken with some caution. A closer look at the results tells us that none of the methods predict

composite resistance values adequately over the entire range of depths. At low depths method 3

seems to work best (virtual interface, extra resistance), while at large depths method 1 performs

better (wide channel, no side wall or lateral transfer effects). Apparently, in narrow channels (as

was the case for the used data) the practice of composing one resistance value that comprises

geometry and overall resistance is a tricky task. In that case it seems more appropriate to leave the

field of 1D modeling and revert to more detailed methods that also describe the transfer processes

at the interface (e.g. methods proposed by Knight and Shonio (1996) or van Prooijen (2004)).

Nevertheless, the outcomes give most confidence to method two, which is therefore recommended

for further use.

Figure 2.4 Composite resistance values from flume experiment (black dots) by Knight and Demetriou(1983) and predicted composite resistance values as based on divided channel methods. Two

data sets are shown (each in separate graph) corresponding to different floodplain widths.

2.3 Conclusions

Common uniform steady resistance relations are only valid in the turbulent hydraulically rough

regime. This condition should always be checked for when applying corresponding flow models.

Furthermore, the Manning equation and the Chézy equation for steady uniform flow, show

practically the same behavior when the channel geometry is simple (i.e. for constant bed level in

cross-section). However, because the Manning coefficient for f low over a flat surface is a true

measure of wall-roughness, composite resistance techniques are preferably based on addition of

Manning coefficients instead of Chézy coefficients. The latter complicates addition of roughnesses

because it is dependent on the relative roughness (and therefore is depth-dependent).

The hydraulic resistance coefficient of a composite channel, or any channel with depth

variations across the lateral direction, is depth-dependent. This is even the case when the surface

characteristics of the bounding wall are the same everywhere. Therefore, photographic methods

that suggest certain constant Manning coefficients for specific river appearances (e.g. Chow 1957,

Fisher 2001) are inadequate. When composing a composite roughness coefficient that reflects the

geometry and wall-roughness from different sections in the river cross-section a divided channel




11

method may be used when the channel is sufficiently wide. The most suitable method is addition of

discharges per river section where only the true wetted perimeters are included in corresponding

hydraulic radii.

In narrow channels, where the depth of flow is of same order of magnitude as the width of

the channel, side-wall roughness plays a significant role in the overall flow field. Consequently, 1D

descriptions are no longer adequate and lateral transfer mechanisms should be included in the flow

description.




12

3 Hydraulic resistance of submerged cylindricalelements

The hydraulic resistance of vegetation can play a major role in the hydrodynamics of rivers with

extensive natural floodplains. Contrary to commonly used wall roughness methods, vegetationpenetrates the flow field and thereby causes drag and, subsequently, additional energy losses.

Here, these processes are studied in an idealized form by treating vegetation as cylindrical

elements with homogeneous geometrical dimensions. That way, complications associated with the

natural variability of plants and bending or streamlining effects are circumvented. At a later stage

(in a future investigation) the effect of these simplifications will be investigated.

The method demonstrated in this chapter is to a large extent based on the findings of

Klopstra et al. (1997) who derived an analytical solution of the flow velocity profile through a field of

homogeneously distributed rigid cylinders. Their theoretical work was complemented by flume

studies with natural and artificial vegetation, performed at the WL|Delft Hydraulics facility De Voorst

in 1996 (e.g. Meijer 1998, see Figure 3.1). The same data set and the analytical solution is re-

examined in the current chapter. Even though the experimental results showed very good

agreement with the analytically derived velocity profile, it is shown that some assumptionsoversimplify the problem. Here, numerical solutions of the more general system equations are

compared to the analytical solution. The numerical analysis gives new insights on the properties of

a free parameter in the analytical solution: the turbulent length scale α.

Figure 3.1. In 1997 flume experiments were conducted by HKV Consultants at the WL Delft Hydraulics

facility ‘De Voorst’ (Meijer 1998). Several configurations of different water levels, cylinder

heights and cylinder surface densities were used.

3.1 Analytical velocity profile of flow trough a vegetated layer

Figure 3.2 shows a schematic view of the geometric parameters that are involved in flow near

submerged cylinders. The cylinders (or rods, stems) are considered nonflexible (stiff), have

constant diameter D and are homogeneously distributed over the bed surface. When determining

the velocity profile over the entire depth, the flow above and between the cylinders will be treated




13

separately. First in section 3.1.1 the flow between the cylinders will be considered and in section

3.1.2 the velocity profile in the surface layer will be matched at the interface (at z = k ). Although

rigid cylinders are considered instead of real vegetation, the flow layer between the rigid cylinders

will in subsequent sections be referred to as the vegetation layer .

Figure 3.2. A schematic view of the geometrical parameters involved when describing flow in presence of

rigid cylinders.

3.1.1 Flow in the vegetation layer

The following force balance governs the flow between the cylinders:

g F

D F

dz

d −=

τ (3.1)

Where the momentum loss in the fluid due to the turbulent shear stress (τ ) is balanced by the

difference of the gravitational driving force F g and the drag force F D due to vegetation. The

Boussinesq hypothesis is a common method to describe energy dissipation in turbulent flows by

introduction of an eddy viscosity ν ε . The shear stress is in the same manner related to ν ε as the

kinematic viscosity in laminar flows (e.g. Pope 2000):

z

u

∂

∂= ε ν τ (3.2)

Following Tsujimoto and Kitamura (1990), it is assumed that the product of the velocity u and a

characteristic turbulent length scale α represents the eddy viscosity:

u ρα ν ε = (3.3)

Which results in an expression for τ :

dz duu ρα τ = (3.4)

The drag force and gravitation force are taken as:

2

2

1umDC F D D ρ = (3.5)




14

gi F g ρ = (3.6)

In expressions (3.4) and (3.5) u is the longitudinal velocity (as function of the height from the

bottom z ), ρ is the water density and α is a turbulent length scale that represents the size of the

dominating eddies. The gravitation force (3.6) is proportional to the bed slope i and, of course, the

gravitational constant g . In the expression for the drag force (3.5) m represents a surface density of

vegetation elements that have a diameter D and C D is the dimensionless drag coefficient. The

surface density m is related to the separation between elements s as:

m s

1= (3.7)

Inserting the expressions (3.4), (3.5) and (3.6) into the force balance (3.1) results in:

02

2

2

2

2

22

=+−∂

∂

ll

suu

z

u (3.8)

Where the introduced length scale l and the characteristic velocity us are defined as:

( ))/(1 D

D

mDC bbmDC

=== α α

l (3.9)

gibmDC

giu

D

s 22

== (3.10)

Where b carries the dimension [m] (length scale), and can be interpreted as a ‘drag length’. This

length scale reflects the flow resistance due to vegetation form drag. The characteristic velocity us

corresponds to the flow velocity that is approached by u when no bottom or surface layer effectsare present. In that case the shear stress term vanishes from the force balance (3.1), which results

in a constant vertical velocity profile of magnitude us.

The solution to (3.8) has the form:

2/

2

/

1 s

z z ueC eC u ++= − ll (3.11)

Choosing appropriate boundary conditions now enables quantification of the unknown constants C 1

and C 2 with which the velocity profile through the vegetation layer becomes fixed (Klopstra et al.

1997). A natural choice for a bed level boundary condition is the no-slip condition (i.e. u(0)=0). At

the top of the vegetation layer a shear stress is imposed of:

ik h g )( −= ρ τ (3.12)

Which corresponds to the shear stress that balances the gravitational force that is acting on the

water in the surface layer. With the no-slip boundary condition, and the shear stress condition

(3.12), expression (3.11) becomes:

ll

l

l

l

//

)/cosh(

)/sinh(1

z k

s ek

z e

k huu −

+

−+= (3.13)




15

Which results in a velocity at the top of the vegetation layer (when z=k) of:

lll

l

// )/tanh(1 k k

sk ek ek h

uu −

+

−+= (3.14)

Alternatively, when imposing complete slip as bed level boundary condition (i.e. when u(0)=us) then

expression (3.13) simplifies to:

)/cosh(

)/sinh(1

l

l

l k

z k huu s

−+= (3.15)

The velocity at the top of the vegetation layer now becomes:

)/tanh(1 ll

k k h

uu sk

−+= (3.16)

Now that the velocity profile in the vegetation layer is specified we can continue to the situation in

the surface layer, and match the two profiles at the interface.

3.1.2 Flow in the surface layer

Above the cylinders the flow velocity is no longer determined by the force balance (3.1) but is

assumed to follow the universal logarithmic velocity profile of flow over a rough surface (e.g.

Keulegan 1938):

)(ln)( h z k

h

hk z uu z u

s

sk ≤≤

+−+= ∗

κ

(3.17)

Where κ is the von Kármán constant which is often taken to be 0.4. The friction velocity is denoted

by u* and the distance between the top of the cylinders and a virtual bed is characterized by the

artificial roughness height hs. The definition for the friction velocity is:

)( shk h giu +−=∗ (3.18)

An inconsistency between the derived velocity profile in the vegetation layer and in the free flowing

region above is the chosen turbulence model. While the flow in between the cylinders is based on

the shear stress function (3.4), the derivation that leads to the logarithmic velocity profile adopted in

(3.17) is based on Prandtl’s mixing length concept (e.g. Nikuradse 1932 and citations therein). The

logarithmic velocity profile corresponds to a turbulence model that follows an alternative expression

for the shear stress:

2

2

∂

∂=

z

u ρλ τ (3.19)




16

Here λ is the characteristic length scale of the turbulent eddies. In boundary layer theory the

parameter λ is often taken to be proportional to the distance from the bottom (i.e. λ = κ z, with κ von

Kármán’s constant).

Although physically inconsistent we continue by matching the analytical solution of the

velocity profile in the vegetation layer (3.15) with the logarithmic profile of the surface layer (3.17).

This requires for hs that:

Υ

−++

−

Υ=

b

k h

k h

bh

k

k s 22

3

22

22)(2

11)( α

κ

κ

α (3.20)

Where k Υ is the dimensionless velocity at the interface between vegetation and surface layer (at

z=k ), defined as in (3.21):

)/tanh(1 ll

k k h

u

u

s

k k

−+==Υ (3.21)

The logarithmic velocity profile (3.17) together with the condition for the artificial roughness height

(3.20) now specifies the flow in the surface layer. Moreover, the velocity profile gives a continuous

match with the profile for the vegetation layer as derived in section 3.1.1.

3.1.3 Calibration of analytical profile

With flume experiments Klopstra et al. (1997) and Meijer (1998) calibrated the analytical solutions

and found that the complete-slip boundary condition gave better results than the no-slip condition

(see also Figure 3.3). This may seem a surprising result because numerous experimental works

have shown that the no-slip boundary condition is a universal property of wall-bounded flows (e.g.

Lauga 2005). However, an essential property of solution (3.11) is that the turbulent length scale α

remains constant over depth. This may be a reasonable assumption for most parts of the

vegetation layer (in the cylindrical elements analogy), but near the bottom this is probably no longervalid. Boundary layer theory proposes near-wall turbulent length scales that are proportional to the

distance from the wall. Consequently, the near-wall behaviour of the solution (3.11) is based on a

turbulent length scale that is too large because it is not bounded by the distance to the wall.

Apparently, this can be compensated when choosing a boundary condition that also does not

account for the presence of the wall: the complete-slip.




17

Figure 3.3 The analytical velocity profile (black line, shown with 2 different bed level boundary conditions)

that fits the data for one of the experiments (RUN2) from Klopstra et al. (1997). The legend also

shows the calibrated value for the turbulent length scales α. Note that the complete-slip

boundary condition gives better agreement with the data than the no-slip boundary condition.

The value for the drag coefficient C D was determined in a separate experiment where the

cylinders were not completely submerged.

In addition to the preferred complete-slip condition, it was found that α correlated with the depth of

flow and the height of the vegetation layer. The following empirical expressions (e.g. Baptist 2005)

were found to describe the relation between α , the depth h and vegetation height k reasonably well

(see also Figure 3.4).

hk 015.0=α (3.22)

7.00227.0 k =α (3.23)

Unfortunately, for the expressions above no justification is known based on physical principles. A

disturbing observation is also that equation (3.23) is not dimensionally correct. Nevertheless,

equation (3.23) is the relation adopted by the Dutch Institute for Inland Water Management and

Waste Water Treatment (van Velzen 2003). The reason for this choice lies in the more realistic

behavior of relation (3.23) at very low flow depths (e.g. Baptist 2005). In section 3.3 we will have a

new look at the properties of the unknown turbulent length scale α.




18

Figure 3.4 Relations (3.22) and (3.23) compared with experimentally derived values for the turbulent

length scale α. Data included are taken from Klopstra et al (1997), Tsujimoto et al. (1993) and

Shimizu and Tsujimoto (1994).

3.1.4 Vegetation solidity

A more detailed description of the vertical velocity profile (3.15) can be derived when the solidity of

the vegetation is also taken into account, i.e. a factor is introduced that corrects for the surface

area and volume occupied by the vegetation (e.g. Kaiser 1984, Stone and Shen 2002, Hoffmann

and Van der Meer 2004). The solution of the velocity profile in the vegetation layer becomes (3.24)

and the length scale l and the characteristic velocity us are modified to (3.25) and (3.26)

respectively.

)/cosh(

)/sinh(

)1(1

'

'

'

'

l

l

l k

z k huu sc

−

−+=

σ (3.24)

)1(' σ α −= bl (3.25)

)1(2' σ −= gibu s (3.26)

Where σ gives the relative ground coverage of the vegetation:

2

4

1 Dmπ σ = (3.27)

First of all, the effect of neglecting solidity is investigated for the case of emergent vegetation and is

largely due to the difference in su . An estimate for the largest error is thus:

σ

σ

−

−−=

−=

1

11'

'

s

s s

u

uuerror (3.28)

If σ is small this can be approximated by (first order Taylor expansion):

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

α [m] predicted

α

[ m ]

0.0227k^0.7

0.015sqrt(hk)

R 2= 0.87

R 2= 0.89




19

2

8

1

2

1 Dmerror π σ =≈ (3.29)

Therefore, using this expression a rough estimate can be made for the error in the flow velocity

when solidity is neglected. The expression can also be written in terms of the separation between

elements s and the diameter D:

2

2

8 s

Derror

π ≈ (3.30)

Consequently, errors in average flow velocity due to neglect of solidity is smaller than 5% (or 1%)

when:

3.501.0

8.105.0

>⇒<

>⇒<

D

serror

D

serror

(3.31)

This means that if the spacing between elements is only twice as large as the diameter of the

stems themselves, then still the estimated flow velocity in the vegetation layer has an error of ~5%

if solidity is neglected. The velocity in the surface layer is determined by the resistance experienced

in the vegetation layer, and thus directly related to the velocity us. Therefore, neglect of solidity has

also only a minor impact on the flow velocity in the surface layer.

Figure 3.5. The velocity error function when neglecting vegetation solidity as function of s/D (the ratio of

spacing between elements to the stem diameter).

Next, we will have a look at the influence on water levels when solidity is neglected. Without taking

solidity into account the discharge Q over a width of flow B is written as:

BhuQ s= (3.32)

Where h is the depth of flow through emergent vegetation and us the corresponding average flow

velocity determined by the force balance between vegetative drag and the gravitational force.

When solidity is taken into account us changes to us‘ (3.26), and the flow width B changes to an

effective flow width B’:

σ −=′ 1 B B (3.33)




20

In (3.33) it is assumed that vegetation elements are distributed homogeneously and that correction

on the effective flow width is equal to the correction on the effective flow length (per unit length). A

square of width B gives a surface area A:

2 B A = (3.34)

The correction due to vegetation ground coverage (3.27) gives an effective bed surface area A’ andthe corresponding effective flow width B’ :

( )2)1( B A A ′=−=′ σ (3.35)

The discharge Q in terms of parameters that are corrected for the vegetation solidity now becomes:

h BuQ s ′′′= (3.36)

Equating expressions (3.32) and (3.36), and by using the corrections on the flow velocity (3.26) and

the flow width (3.33) yields the corrected flow depth h’ :

σ −=′

1

hh (3.37)

The relative error in flow depth by neglecting solidity is thus (for emergent vegetation):

2

2

4 s

D

h

hh π σ −=−=

′

′− (3.38)

The error made by not taking solidity into account is approximately twice as large for depth

estimation (3.38) as compared to the relative error in the estimation of the flow velocity (3.29). As

could be expected, (3.38) shows that neglect of solidity gives an underestimation of the flow depth.

However, as was the case for the flow velocity correction, for typical vegetation surface densitiesthe water level difference due to solidity is small (see also Figure 3.5). James et al. (2004) and

Baptist (2005) also already pointed out that solidity could usually be neglected, especially in view of

uncertainties associated with vegetation dimensions.

Next, we will consider the effect of solidity when the vegetation is overflown (i.e.

submerged vegetation) and question whether solidity may be neglected in most of such cases. For

flow over submerged vegetation two effects influence the total water level when solidity is taken

into account:

1. The volume occupied by the vegetation reduces the available volume for flow in the

vegetation layer.

2. Reduced water volume in the vegetation layer results in a smaller driving force per unit

volume (gravitational force) in the vegetation layer.

In comparison to neglect of solidity, the first effect causes a water level rise (for a certaindischarge) because of the increased water volume in the surface layer. However, this effect is not

as large as in the case of emergent vegetation, because a discharge increase in the surface layer

also increases the average flow velocity and thereby dampens the water level rise. For emergent

vegetation (in the rigid cylinder analogy) a water level rise does not increase the average flow

velocity because the drag due the cylinders is the same at all depths. The effect described at point

2 decreases the velocity in the vegetation layer. Consequently, the shear stress at the top of the

vegetation elements becomes larger, which reduces the friction velocity experienced in the surface




21

layer. As a result the depth of the surface layer increases. Taking both effects together, the

estimated water level increases when solidity is taken into account.

Assume that for a certain total discharge Q a part Qtop flows in the surface layer and a part

Qveg in the vegetation layer when solidity is neglected. For the same total discharge the partial

discharges between the layers are Q’ top and Q’ veg when solidity is taken into account. Therefore

topveg topveg QQQQQ ′+′=+= (3.39)

When taking h’ as total flow depth with inclusion of solidity effects and h as the depth when solidity

is neglected, then the total discharge equality (3.39) yields:

)()( k h Buk Buk h Bu Bk uQ topveg topveg −′′+′′=−+= (3.40)

Where the average flow velocities in the two layers are utop and uveg (‘primed’ velocities correspond

to inclusion of solidity effects), and B is the flow width. Inserting expression (3.33) for the effective

flow width and rearranging terms yields:

)1()()( σ −′−+−=−′′ veg veg toptop uuk k huk hu (3.41)

Further specification of this relation requires accurate estimation of average flow velocities in both

layers in terms of plant parameters and total flow depth. The average velocities in the vegetation

and the surface layer will depend on the depth of the surface layer. It is therefore already clear that

relation (3.41) is implicit in h’ and h and will most likely not have a straightforward solution.

3.2 Numerically determined velocity profile

In this section we will reconsider the shape of the vertical velocity profile without making

assumptions that previously led to an analytical solution of the force balance. In particular, the

assumed shear stress function will be chosen to agree with the assumed logarithmic velocity profilein the surface layer. That way, a consistent treatment of turbulence characteristics over the entire

water depth is followed. Furthermore, turbulent length scales will be bounded by the distance to the

bed and the no-slip condition will be imposed.

The expression for the shear stress that corresponds to the logarithmic velocity profile in the

surface layer, is given as (see also expression (3.19)).

2

2

∂

∂=

z

u ρλ τ (3.42)

This shear stress expression will also be used for evaluation of the force balance in the vegetation

layer, i.e. expression (3.1). In the vegetation layer it seems natural that the size of eddies is

restricted by the available space between vegetation elements, and, therefore, that λ remains

constant throughout the vegetation layer. One could argue that this is no longer the case for natural

vegetation because the presence of leaves may vary along the height of the plant. However, here

we are still considering rigid cylinders that do not have different geometrical properties at different

depths. Assuming λ to be a constant results in the force balance (e.g. substitution of (3.42) into

(3.1)):




22

)(024

12

2

22

2

k z gi

ub z

u

z

u<=+−

∂

∂

∂

∂

λ λ (3.43)

However, near the bottom λ is no longer restricted by the separation between vegetation elements

but by the distance to the bed z . In boundary layer theory the parameter λ is often taken to be

proportional to the distance from the bottom (i.e. λ = κ z, with κ von Kármán’s constant ~ 0.4). For

the region where κ z is smaller that the separation between vegetation elements s, the force

balance becomes:

)(01

24

122

2

222

2

s z z

u

z z

giu

z b z

u

z

u<=

∂

∂++−

∂

∂

∂

∂κ

κ κ (3.44)

The equations (3.43) and (3.44) can no longer be solved analytically, but have to be determined

numerically. Figure 3.6 illustrates the difference between the numerically determined profile and the

analytical description. While the physically sound no-slip boundary condition performs badly in the

analytical case (see Figure 3.3), the numerical profile follows the data points very well. The

numerical profile follows the data points slightly better in the ‘knee’ just below the top of the

vegetation layer (k ). This trend is also visible in nearly all other experiments (not shown here). Forthe purpose of calculating the overall conveyance (discharge capacity) over submerged vegetation,

the analytical solution with the complete-slip boundary condition may still be used because the

error in over-predicting flow velocities near the bottom is only very small (see Figure 3.6).

In the velocity profiles the parameters α and λ are used for calibration. The parameter α is

calibrated on the total discharge, while λ is chosen such that the profile in the vegetation layer

connects smoothly to the logarithmic velocity profile from the (calibrated) analytical case. This way

both solutions give the same shear stress at the top of the vegetation layer, which allows

comparison between length scales α and λ (see section 3.3.1). A drawback of this approach is that

the logarithmic profile in the surface layer is assumed to be correct, and that the numerical fit to the

log profile results in a (small) deviation from the overall measured discharge.

In the following section we will have a closer look at the characteristics of both turbulent

length scales, and whether either one of them can be understood in terms of measurable geometric

parameters.

Figure 3.6. The analytical velocity profile (thin line) and the numerical velocity profile (thick line) that fit the

data for one of the experiments from Klopstra et al. (1997). For both cases the same

logarithmic profile in the surface layer is assumed. The legend also shows the determined

values for both turbulent length scales α and λ.




23

3.3 Turbulent length scales

As mentioned before, in 1997 HKV Consultants and WL| Delft Hydraulics carried out 48

experiments with submerged rigid cylinders under varying hydraulic conditions (Klopstra et al.

1997, Meijer 1998). The determined values for α where chosen such that the overall profile (within

and above vegetation) fit the data as good as possible, while preserving the overall measured

discharge value (with bottom boundary condition a constant velocity us). Agreement with measuredvelocity profile was very good. However, expressions to predict the value of α based on

geometrical quantities, (3.22) and (3.23), both are not satisfying. One expression is dimensionally

incorrect and the other shows undesired behavior in case of very low flow depths (e.g. Baptist

2005).

Here, a new attempt is made to describe the turbulent length scale α, that solves

drawbacks of the available descriptions. For this purpose we do not only consider the properties of

α, but also of the newly introduced turbulent scale λ. That way, if either can be understood in terms

of geometrical quantities, the other can also be derived. Figure 3.7 shows values for turbulent

length scales α and λ (data from experiments by Tsujimoto et al. (1993) and Shimizu and Tsujimoto

(1994) are also included) and shows that there is a clear relationship between the two turbulent

scales.

Figure 3.7. Turbulent length scales α and λ for all experiments by Klopstra et al (1997). The shown

experiments that correspond to separations s of 0.93 and 1.85 cm are from Tsujimoto et al.

(1993) and Shimizu and Tsujimoto (1994).

3.3.1 Two definitions of turbulent length scales

In order to evaluate the relationship between α and λ we will compare the definitions of the shear

stress τ at the top of the vegetation layer . Using both definitions of turbulent length scales gives the

following shear stress equalities (at the top of the vegetation layer, i.e. where z=k):

( )

k

k

k z

k

k z

u

z u

z

uu

z

u

∂∂=⇒

∂

∂=

∂

∂=

=

= /2

2

2

λ

α

ρα τ

ρλ τ

α

λ

(3.45)

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.00 0.02 0.04 0.06 0.08

λ [m]

α

[ m ]

s = 5.45[cm]

s = 11.7[cm]

s = 0.93[cm]

s = 1.85[cm]




24

The equality between α and λ (3.45) applied to the logarithmic profile (3.17) reveals the following

relation between α and λ:

b

k h

k 2

−

Υ= λ

α (3.46)

Relation (3.46) is implicit in α because the dimensionless velocity k Υ (see (3.21)) also inhibits the

turbulent scale α. However, if k Υ is approximated as:

( )( ) 4/1

2/1

)/tanh(1b

k hk hk

k hk

α

−=

−≈

−+=Υ

ll

l (3.47)

Then α can be related to λ in the following manner:

3/14

4

≈

b

λ α (3.48)

If a scaling relation can be found for either α or λ, then the other can be readily determined through

the approximated relation (3.48). This relation is a good approximation if the flow velocity at the top

of the vegetation layer is considerably larger than the velocity in the vegetation layer, i.e. when h-k

is larger than l . If k /l is larger than 1.1 and (h-k )/ l is larger than 10 then the error made in the

approximation (3.47) is less than 5%. Figure 3.8 shows that, overall, the approximation leading to

(3.48) is indeed a reasonable one.

Figure 3.8. The approximate relation between turbulent scales α and λ.

3.3.2 Scaling properties of the flow field

Earlier attempts to scale the turbulent length scale α to geometric properties of the system were not

very successful. In analogy to a method often used for flow through porous media (e.g. Bentz and

Martys 1995), we will look whether the geometric boundaries in the flow field are suitable for

constructing a spacing hydraulic radius (R S) that scales with the turbulent length scales α and λ. In

the study of Tollner et al. (1982), who investigated sediment transport processes in flow through a

0.00

0.01

0.02

0.03

0.04

0.05

0.00 0.01 0.02 0.03 0.04 0.05

( λ4 /4b )

1/3 [m]

α

[ m ]

s = 5.45[cm]

s = 11.7[cm]

s = 0.93[cm]

s = 1.85[cm]

Perfect agreement




26

Figure 3.9 Turbulent length scales λ plotted vs. different types of spacing hydraulic radii R s. Trend linesfitted to the data are also shown. The data points in the lower left corner of each graph

(corresponding to low values of turbulent length scales) are from the experiments by Tsujimoto

et al. (1993) and Shimizu (1994).

Table 3.1 shows that both turbulent length scales α and λ scale equally well on specific hydraulic

radii when considering all data from Klopstra, Shimizu and Tsujimoto. Considering only Klopstra’s

data results in the following best-fit functions (Table 3.2):

Best fit to λ Best fit to α

Rs KR R2 Rs KR R

2

ks/(2k+b) 3.38 0.74 sh/(2b+(h-k)) 0.39 0.69

bh/(2b+(h-k)) 0.12 0.72 kh/(2k+h) 0.82 0.62sh/(2(h-k)+b) 1.54 0.71 ks/(2b+k) 0.04 0.62

kb/(2b+k) 0.25 0.50 k(h-k)/(2(h-k)+h) 0.09 0.60

Table 3.2 Regression characteristics of hydraulic radii functions fitted to Klopstra’s data.

Note that the four best performing hydraulic radii functions are still the same as in Table 3.1. The

functions that are marked grey in Table 3.2 will be used for further analysis, because these perform

significantly better than the remaining possibilities. One scaling expression of α can compete with




27

the quality of the scaling expressions found for λ. The hydraulic radius function that scales best

with α is written as:

)(211k hb

hs K

−+= α α (3.50)

In expression (3.50) the cross section area (i.e. the numerator in the fraction) shows to be thecross-section area in between cylinders, extended all the way to the water surface. The wetted

perimeter (i.e. the denominator) is more difficult to interpret: the sum of twice the drag length and

once the depth of the surface layer. Note that similar wetted perimeters show up in the hydraulic

radii that scale best with λ. Taking the three hydraulic radii functions that scale best with λ, and

inserting these into expression (3.48), yields three more expressions of α in terms of measurable

geometric quantities:

3/4

3/14

3/4

3/13

3/4

3/12

)(2)4(

1

)(2)4(

1

2)4(

1

4

3

2

+−=

−+=

+=

bk h

hs

b K

k hb

hb

b K

bk

ks

b K

α

α

α

α

α

α

(3.51)

Figure 3.10 shows the four new functions for predicting α together with the experimentally

determined values.




28

Figure 3.10 Predicted values for the turbulent length scale α compared with measured data.

In the limiting case that the water depth is much larger than the vegetation height (i.e. h>>k) and

that k, s and b are of the same order of magnitude (k≈s≈b) all four expressions reduce to

k ∝α (3.52)

The turbulent length scale α in situations of large flow depths, with small but densely packed

roughness elements, therefore represents the roughness height (k). This property is also present in

the empirical relation (3.23). Because of this property, expression (3.23) (although dimensionally

incorrect) is often favored above the earlier mentioned dimensionally correct expression for α (e.g.

Baptist 2005, see equation (3.22)), which is proportional to the square root of the flow depth. Table

3.3 gives an overview of the four newly proposed formulas and also the two relations (3.23) and(3.22) with their respective properties.

Table 3.3 shows that the newly proposed expressions for α give equally good predictions

as the already existing ones α5 and α6 (see also (3.22) and (3.23)). However, the newly proposed

formulas comprise the properties of being dimensionally correct with the advantage that at large

flow depths the turbulent length scale α reduces to a roughness height. Therefore, all four of them

provide an improvement to the already existing expressions. Even though it is easier to scale

geometrical boundaries to turbulent length scale λ as compared to α (see Table 3.2), the final

scaling relations of α based on λ perform worse than the relation that scaled directly to α. This is




29

due to the extra uncertainty that is introduced by relation (3.48), which is only an approximated

relation between α and λ. Because function α1 is also dimensionally correct and gives a realistic

value when the depth of the surface layer is small, expression α1 is recommended as best closure

relation for the turbulent length scale.

Prediction of α R s R2 Dim.

correct?

Limit h>>k

s R)40.0(1 =α

)(2 k hb

sh

−+

0.91 OK OK

3/1

3/4

2 )86.2(b

R s=α bk

ks

+2

0.88 OK OK

3/1

3/4

3 )036.0(b

R s=α )(2 k hb

bh

−+

0.87 OK OK

3/1

3/4

4 )97.0(b

R s=α bk h

sh+− )(2

0.84 OK OK

7.0

5 )0227.0( k =α 0.87 OK

hk )015.0(6 =α 0.89 OK

Table 3.3 Characteristics of functions to predict α. The expression in the first column performs best in

comparison to the available data sets and is therefore recommended for further use.

3.4 Average flow velocity based on vertical velocity profile

While details about the velocity profile of the flow field through vegetation are important for

sediment transport studies, for discharge capacity studies (i.e. conveyance studies) the average

flow velocity is of main interest. Having established a relation that describes the vertical velocity

profile through submerged vegetation, the average flow velocity can be determined by integrating

that expression over depth. When treating the velocity profile in and above the vegetation layer

separately we get for the overall average flow velocity:

topveg tot U h

k hU

h

k U

−+= (3.53)

Where veg U and lopU are the average velocities in and above the vegetation layer respectively.These will be determined separately. In the 1996 experiments by HKV and WL|Delft Hydraulics

(Klopstra et al. 1997, Meijer 1998) flow velocities were measured at depths 10 cm apart. Linear

interpolation between the velocities at these depths and integration over depth gives the average

velocity in the vegetation layer, the surface layer and the total flow depth. The analytical velocity

profile as derived in section 3.1 cannot be integrated analytically without making further

simplifications. Steps towards analytical integration are covered in subsequent sections. For a first

comparison with measured average velocities, the average velocities based on the analytical

profile are determined through numerical integration (Figure 3.12).




30

Figure 3.11 Velocity profile shown together with calculated average velocities in the surface and the

vegetation layer (magnitude of average velocities are depicted by arrows). Average velocities

were calculated from trapezoidal interpolation between the measured points of the velocity

profile. The fitted analytical velocity profile with complete-slip boundary condition is also shown.

Figure 3.12 Average velocities in the vegetation and in the surface layer (top layer) as determined from (i)

the calibrated analytical profile (horizontal axis) and (ii) the measured velocity profile. The

graph on the right shows the overall average velocities as constructed from the 2 layers

through equation (3.66).

Figure 3.12 shows the average velocities in the vegetation and surface layer determined from both

the measured velocities at different depths and the (numerical) integration of the fitted analytical

profile (see for example Figure 3.11). From the figures it becomes clear that the average velocities

in the surface layer are described quite well by the chosen log-profile. However, in the vegetation

layer the analytical profile systematically underestimates velocities when the influence of the

surface layer is small and overestimates velocities when the influence of the surface layer is large.

The graph on the right in Figure 3.12 shows that the overall average velocity (over entire depth of

flow) based on the analytical profile is near perfect. This is no surprise since the analytical profile

was chosen such that the measured discharge is represented (by calibration of α). It also shows

that, although the prediction of the average flow velocity in the vegetation layer is at t imes quite

poor, its small contribution to the overall average velocity results in negligible error as long as the

surface layer prediction is accurate.




31

3.4.1 Average velocity in the vegetation layer

In Figure 3.12 the average velocity in the vegetation layer as predicted by the analytical velocity

profile was determined through numerical integration over depth (trapezoidal interpolation). This

method gives the most accurate value of the average velocity described by the profile but has to be

done numerically. Unfortunately, there is no exact solution to the integral of (3.15), so we will

search for an approximated analytical solution to the integral of (3.15). Following Klopstra et al.

(1997), a simplification is made by assuming that )/exp( l z − is negligible as compared to

)/exp( l z ). The velocity profile in the vegetation layer now becomes (instead of expression

(3.15)):

l

l

/)(1

k z

sveg ek h

uu −

−+≅ (3.54)

It is now possible to integrate expression (3.54) analytically (see also Klopstra et al. 1997):

∗

∗

Υ

Υ

∗

∗

∗

+Υ

−Υ+Υ=⇒

+

−+== ∫

k

k u

U

uu

uu

k

uu

k

dz u

k

U

s

veg

k

sveg

sveg

sveg

k

veg veg

0

1

1ln

2

12

ln21

0

'

'

'

0

l

ll

(3.55)

where the integration boundaries are given by:

−+=Υ

−+=Υ

Υ

−

l

l

l

k h

ek h

k

k

1

1

*

/*

0

* (3.56)

Figure 3.13 shows how this result compares with the numerical integration procedure, and how well

measured velocities in the vegetation layer are represented.




32

Figure 3.13 Average velocities in the vegetation layer based on the exact solution compared to the

integrated simplified velocity profile as in expression (3.55) (left graph). The graph on the right

show how analytical integration of the simplified velocity profile compares with measured

average velocities.

A fortunate result seen in Figure 3.13 is that the approximated velocity profile and the subsequent

average velocity in the vegetation layer is closer to the measured average velocity as compared to

the previously shown numerically integrated value. The approximation (3.54) compensates for

overestimation of the average velocity when the surface layer has a large influence on the

vegetation layer (i.e. in situations where the dimensionless velocity in the vegetation layer is

significantly larger than unity). This compensation appears to be especially effective when the

surface density of roughness elements (m) is small (i.e. here in the case where m = 64 m-2

).

In the previous section it was pointed out that the error in the average velocity of the

vegetation layer has only little effect on the average velocity over the entire depth. We will therefore

proceed to find a more simplified expression of the average velocity that still performs reasonably.

When assuming complete slip for the velocity profile near the bed, then the dimensionless velocity

at z=0 is expected to approach 1. However, if this value is inserted into (3.55) then the logarithmic

term becomes undetermined. In order to further simplify expression (3.55) it is therefore necessary

to make a first order approximation of*

0Υ (Taylor expansion):

ll

ll

//*

02

111 k k e

k he

k h −−

−+≈

−+=Υ (3.57)

The logarithmic term from expression (3.55) evaluated at z=0 may then be approximated as (first

order approximation):

lll

l

l l

l

l

k k he

k h

ek h

ek h

k

k

k

−

−=

−≈

+−

−

≈+Υ

−Υ −

−

−

4ln

4ln

22

2ln1

1ln /

/

/

*

0

*

0 (3.58)

Inserting (3.58) into (3.55) yields:




33

+Υ

−Υ

−+−Υ+≈=Υ

1

14ln

2

11

21

*

**

k

k k

s

veg veg

k hk u

U ll (3.59)

Figure 3.14 (right graph) shows that approximation of the integrated average velocity in the

vegetation layer (i.e. expression (3.59)), compensates for the systematic over-prediction at high

velocities as present in Figure 3.13. In effect, the approximations that were made to arrive atexpression (3.59) from the integration of (3.15) only increased the quality of the predicted value.

Figure 3.14 Average velocities in the vegetation layer based on the approximated relation (3.59).

3.4.2 Average velocity in the surface layerThe average velocity in the surface layer can be determined without any further modifications of the

velocity profile. The integral of expression (3.17) becomes:

∫

+−+

−= ∗

h

k s

sk top dz

h

hk z uu

k hU ln

1

κ (3.60)

Where hs and u* are given by the definitions in (3.20) and (3.18). The result of (3.60) is:

−

+−

−

+−+= ∗ 1ln

s

s sk top

h

hk h

k h

hk huuU

κ

(3.61)

Figure 3.12 (left graph) shows that this exact solution compares very well with the average velocity

as based on velocity measurements at different depths.




34

Figure 3.15 The behaviour of the turbulent length scale λ in comparison with the artificial roughness height

hs as the depth of the surface layer (h-k) increases. When (h-k)/ hs becomes large then the

ratio between λ and hs approaches von Kármán’s constant κ (~0.41). Also shown are the

values that correspond with the used data sets.

For relatively deep flow over low vegetation, the turbulent length scale λ in the vegetation layer

reflects the artificial roughness height hs. This is demonstrated in Figure 3.15, where the ratio

between hs and λ is shown to converge towards von Kármán’s constant. Therefore, If the artificial

roughness height hs is small compared to the depth of the water column above the vegetation (h-k ),

then the relation between hs and λ may be approximated as:

λ κ

1≈ sh (3.62)

Consequently, by expression (3.48) hs is related to α as:

4/14/3 bh s α ∝ (3.63)

Figure 3.16 shows that the corresponding constant of proportionality is approximately 4.3.

Figure 3.16 The relation between the artificial roughness height hs and the turbulent length scale α.

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.0 10.0 20.0 30.0 40.0 50.0

(h-k)/h s

λ / h

s

data points

function

κ

y = 4.32x

R 2 = 0.97

0.00

0.10

0.20

0.30

0.40

0.50

0.00 0.02 0.04 0.06 0.08 0.10

α3/4

b1/4

[m]

h s

[ m ]




35

In dimensionless form (3.61) becomes:

( )

−

+−

−

+−+−+Υ==Υ ∗ 1ln

2 s

s s s

k

s

toptop

h

hk h

k h

hk h

b

hk h

u

U

κ (3.64)

Where

4/14/33.4 bh s α ≈ (3.65)

Figure 3.17 Average velocities in the surface layer based on the approximated relation (3.64).

3.4.3 Overall average velocity

Making (3.53) dimensionless by dividing it by the characteristic velocity us yields for the average

velocity over the entire flow depth:

topveg

s

tot tot

h

k h

h

k

u

U Υ

−+Υ==Υ (3.66)

Inserting (3.59) and (3.64) into (3.66) gives a complete description of the average velocity for flow

through submerged vegetation. The only unknown is the turbulent length scale α. In section 3.3.2 it

was argued that the best available closure relation for α is expression α1, as stated in Table 3.3:

)(240.0

k hb

hs

−+=α (3.67)

Figure 3.18 shows that using this closure relation for α gives very good agreement with measured

average velocities.




36

Figure 3.18 Average velocities in the surface layer, the vegetation layer and the total flow depth (from left to

right) as predicted with the new expression for α (3.67) and from the measured velocities at

different depths (vertical axis).

3.5 Resistance parameters

In many commonly used software packages for river flows vegetation effects are required in the

form of resistance parameters. In the previous sections descriptors for the average (dimensionless)

flow velocity were presented instead. The earlier found average flow description (3.66) translated to

the most commonly used resistance parameters Chézy, Manning and Darcy-Weisbach, become:

Chézy: tot

D DhC

s

g

C Υ=

22 (3.68)

Darcy-Weisbach: tot

D DhC

s

f

Υ=2

2

11 (3.69)

Manning: tot

D DhC

s

h

g

nΥ=

2

6/1

21 (3.70)

Since the resistance parameters above are usually associated with wall roughness, it may be

misleading to write vegetation resistance in terms of (3.68), (3.69) or (3.70). It was shown that

vegetation resistance is not purely a wall resistance effect. Moreover, in section 5.4 it is

demonstrated that bed roughness can often be neglected. Nevertheless, when required for (1D)

flow simulation packages the relations above may be used to include vegetation resistance effects.

3.6 Conclusions

The analytical velocity profile as proposed by Klopstra et al. (1997) gives a good description of the

flow over submerged cylinders even though an inconsistent treatment of turbulent characteristics of

the flow is used. A numerically determined velocity profile of flow through vegetation was

presented, with realistic boundary conditions and an acceptable turbulence model, which agreed

very well the analytical profile. Comparison with the analytical solution of Klopstra et al. (1997)




37

revealed that an unrealistic bottom boundary condition (complete slip) compensates a turbulent

length scale that is assumed constant over the entire depth of flow.

The turbulent length scale α that fixes the shape of the velocity profile is determined by

choosing a function that resembles the spacing hydraulic radius between the cylinders. Through

fitting with many conceivable spacing hydraulic radii a new empirical closure for this length scale

was found. Besides the (slightly) improved predictive power of the newly found expression, other

advantages as compared to the previously existing ones are that (i) it is dimensionally correct and(ii) it converges to the roughness height for relatively deep flow.

Furthermore, the analytical velocity profile is integrated over depth to give the average flow

velocity. A few assumptions were used to simplify the final expression. In combination with the

newly found relation for α, the method predicts average flow velocities accurately.




38

4 Hydraulic resistance based on a two-layerscaling approach

In the previous section a method was shown by which the average flow velocity through (and over)

vegetation was determined through vertical integration of the velocity profile. It turned out thatsome simplifications of the flow characteristics could lead to an analytical description of the velocity

profile, which did not differ much from the outcomes of the physically more correct numerical

prediction. However, the analytical solution still had some drawbacks. First of all, an unknown

turbulent length scale needed to be determined which was already difficult in the physically more

consistent approach. This remained so even when making simplifying assumptions. Eventually,

several possibilities for the closure relations of this length scale are put forward, but none of these

is fully understood in physical terms. Another drawback of the analytical velocity profile is that,

although vertical integration can be done analytically, the average velocity function is still very

complicated. In short, the method demonstrated in chapter 3 gives reasonable results but (i)

inhibits a poorly understood closure relation and (ii) has a complicated functional form. Therefore,

the outcome of the previous chapter provides the motivation to search for a simple, yet physically

sound, description that does not require a closure relation as difficult to interpret as the ones foundfor the turbulent length scale α.

In this chapter we will investigate the possibility to describe flow through submerged stiff

cylinders by means of a bulk flow description. Instead of considering the detailed depth-

dependency of the flow field (i.e. the velocity profile), we will merely consider depth-averaged

velocities and their relation to geometrical boundaries. First, in section 4.1, turbulent energy

considerations are used to find a description of the average flow velocity in the surface layer. Next,

the average flow velocity in the vegetation layer is estimated in section 4.2 by means of a simple

force balance. Combining the average velocities in the two layers yields the overall average

velocity (i.e. the average velocity over entire depth, see section 4.3).

4.1 The average flow velocity in the surface layer

Gioia and Bombardelli (2002) have shown that the well-known Manning equation can be derived

based on Kolmogorov scaling for the case of rough channel flow. Here we will follow the same line

of reasoning to get a bulk description for flow over submerged rigid cylinders (i.e. for the surface

layer). The three assumptions made in Gioia and Bombardelli’s derivation were:

1. Spaces between roughness elements are occupied by eddies that are of the same size as

the roughness elements themselves.

2. The roughness elements are small compared to the hydraulic radius

3. Turbulent eddies in the vicinity of the roughness elements are governed by Kolmogorov

scaling.

Assumption (1) needs to be modified in the case of cylindrical roughness elements: while Gioia and

Bombardelli assumed that the roughness elements could be represented by a roughness height

that reflected the size of the elements, here we will use a spacing hydraulic radius that reflects the

spatial properties of the field of cylinders. In the case of flow over submerged vegetation

assumption (2) is often not met. However, if we consider the spacing hydraulic radius instead of the

actual size of the roughness elements as a measure of hydraulic resistance, then assumption (2)

may often remain valid.




39

Gioia and Bombardelli point out that assumption (3) is still valid in inhomogeneous

turbulence. It is therefore expected to still hold just above cylindrical roughness elements (or

natural vegetation). In between the roughness elements (or vegetation) turbulent mixing is also

affected by the presence of vortices in the wake of the flow behind the protruding stems (e.g. Akilli

and Rockwell 2002). Therefore assumption (3) is no longer valid in the vegetation layer and a

different approach will be used (section 4.2).

The main principles behind the reasoning of Gioia and Bomdardelli are (i) a simple forcebalance and (ii) the concept of a constant energy dissipation rate when large flow patterns break up

into smaller patterns (eddies). In section 4.1.1 and 4.1.2 these will be treated separately.

4.1.1 Shear stress in the surface layer

In stationary flow, the shear stress at the top of the roughness layer τ k balances the gravitational

pull on the water volume in the surface layer. This gives an expression for the shear stress (R top is

the hydraulic radius of the surface layer):

gi Rtopk ρ τ = (4.1)

Furthermore, the Reynolds averaged Navier-Stokes equation tells us that the (Reynolds)

shear stress at any location in a (2D) flow field is (e.g. Pope 2000):

t nvv ρ τ = (4.2)

Here nv and t v are velocity fluctuations perpendicular and parallel to the direction of the average

flow field, respectively. The overbar denotes time-averaging of these fluctuations. Gioia and

Bombardelli (2002) argue that for the shear stress near the wall, t v is associated with the largest

eddies in the surface layer and that the perpendicular component is dominated by eddies that fit in

between the roughness elements. The average velocity in the surface layer is topU and the

characteristic velocity at the top of the roughness layer is denoted by ur . In analogy with Gioia and

Bombardelli (2002) the shear stress at the top of the roughness layer then scales as:

topr k U u ρ τ ~ (4.3)

Together with the force balance (4.1) this yields:

topr top U u gi R ρ ρ ~ (4.4)

In order to find a relation for topU an independent expression for ur needs to be found. For that

purpose, energy considerations of the turbulent flow field will be used (Kolmogorov scaling).

4.1.2 Turbulent energy and Kolmogorov scaling

In the Kolmogorov view on turbulent flow turbulent energy is created through external forcing at the

largest scales of the system (energy containing range) and flows to smaller and smaller scales until

eventually viscosity damps the smallest flow patterns (viscous dissipation). The rate of production

of turbulent kinetic energy per unit mass is denoted by ε , and is independent from viscosity at large

scales. Consequently, for the energy input at the largest scale (the energy containing scale) a

scaling expression for ε can be obtained that is constructed with representative geometrical

parameters (hydraulic radius) and the representative flow velocity (see also (2.15)).




40

top

top

R

U 3

~ε (4.5)

Furthermore, if we assume that the turbulent intensity at the top of the vegetation layer is a result of

the energy cascade to smaller scales and that eddies of a specific size R s dominate the flow field at

the top of the roughness elements, then the dissipation rate at the top of the vegetation layershould scale as:

s

r

R

u3

~ε (4.6)

Where now ε is related to the characteristic velocity ur and where R s is the spacing hydraulic radius

that bounds the extent of the largest local eddies. Regardless of the exact function that describes

R s, the characteristic velocity in the roughness layer can be related to the average velocity in the

surface layer as (using (4.5) and (4.6)):

top

top

sr U

R Ru

3/1

~

(4.7)

Expression (4.7) can now be inserted into relation (4.4). This yields a scaling expression for the

average velocity in the surface layer:

gi R R

RU top

s

top

top

6/1

~

(4.8)

Or, when making this expression dimensionless by scaling it against the characteristic velocity us

(equation (3.10)):

b

R

R

R

u

U top

s

top

s

toptop

2~

6/1

=Υ (4.9)

Relation (4.8) (or (4.9)) reduces to the Manning/Strickler equation if the total relative flow depth is

large (R top large) and the spacing hydraulic radius is small (R s becomes an equivalent roughness

height). Using expression (4.7) and (4.9) a scaling equation for the characteristic velocity at the top

of the vegetation layer can be established:

2/1

3/16/1

)2(~

b

R R

u

u top s

s

r r =Υ (4.10)

Next, the two scaling expressions (4.9) and (4.10) will be analyzed for their behavior in certain

limiting cases (asymptotic analysis). The expected limits will impose properties of the hydraulic radii

as they appear in these two relations.




41

4.1.3 Asymptotic behavior of hydraulic radii

When the total depth of flow approaches the height of the roughness elements (i.e. when h

approaches k) then (4.9) should yield 1, because the dimensionless velocity is scaled against the

velocity in the vegetation layer in absence of extra surface drag.

1)2(

~ 2/16/1

3/2

→Υ → Υ → topk h

s

toptop

b R

R (4.11)

In that case the characteristic velocity in the surface layer is equal to the characteristic velocity at

the top of the vegetation layer. Therefore, the characteristic velocity at the top of the vegetation

layer should also approach 1 when the flow depth h is equal to the vegetation height k :

1)2(

~2/1

3/16/1

→Υ → Υ →r

k htop sr

b

R R (4.12)

We will assume that (4.11) and (4.12) show similar behavior when the depth of the surface layer

vanishes (i.e. asymptotic similarity, Barenblatt 1996). Introducing an unknown length scale k’ against which we scale the depth of the surface layer (h-k ), then in case of incomplete similarity in

(h-k+k’ ), (4.11) and (4.12) admit the following power law asymptotics (e.g. Gioia and Bombardelli

2002):

γ γ

+−⇒

+−=Υ

→ '

'~

)2('

'lim

2/16/1

3/2

k

k k h

b R

R

k

k k h

s

toptop

k h (4.13)

η η

+−⇒

+−=Υ

→ '

'~

)2('

'lim

2/1

3/16/1

k

k k h

b

R R

k

k k h top sr

k h (4.14)

Note that in the power law the unknown length k’ is added to the depth of the surface layer. This is

necessary because in the case of flow over vegetation the velocity in the surface layer will never

become 0. In effect, k’ can be interpreted as the extent to which the profile of the surface layer

extends into the vegetation layer. Now both equations approach 1 when h approaches k , but the

incomplete similarity condition allows different rates of convergence. Demanding that r top Υ=Υ

when the surface layer vanishes (i.e. in the limit that h approaches k ), yields:

If

γ η +

+−=⇒→

'

'2

k

k k hb Rk h top (4.15)

If

γ η 24

'

'2

−

+−=⇒→

k

k k hb Rk h s (4.16)

Next, we will determine values for the two unknowns in the power law, η and γ, by investigating the

expected behavior in the case that the water depth becomes much larger than the vegetation

height k . Experimental observations (e.g. Manning (1898), Strickler (1923)) have shown that when

the depth of flow is much larger than the roughness height, then the magnitude of the average flow

is well described my Manning’s formula. We will impose limiting values on R s and Rtop in order to




42

get the conventional Manning equation when h is much larger than k . In that case the hydraulic

radius in the surface layer approaches the total flow depth while the spacing hydraulic radius is

represented by a roughness height that is independent from the depth h:

If h>>k:

?)(~

~

sor k R

h R

s

top (4.17)

A priori it is not quite clear which parameter, or combination of parameters, R s should scale with

when h becomes much larger than k , but as long as R s is no longer dependent on the depth h then

the power law for R s in (4.16) should vanish:

0241~24 =−⇒− γ η γ η h (4.18)

The hydraulic radius in the surface layer scales with h, therefore the power law in (4.15) should

behave as:

1~ =+⇒+ γ η γ η hh (4.19)

Solving for η and γ by combining expression (4.18) and (4.19) yields η =1/3 and γ =2/3. As a result,

the power law expression for R s and R top (4.15) and (4.16) become:

)2(22

)1(2

2

DC

sb R

k

k k hb R

D

s

top

==

′

′+−=

(4.20)

It was mentioned before that R s reflects the extent of the largest eddies at the top of the vegetation

layer. This means that when the separation between roughness elements (s) is equal to the

dynamic diameter of the cylinders (C D*D) then the largest eddies are bounded by s. However,when the diameters of the elements are significantly smaller than the spacing between elements

then (4.20) shows that R s exceeds s. In effect, when the spacing between elements is relatively

large as compared to the size of the elements themselves, then the eddies are not limited to the

spaces in between the roughness elements.

When these expression are inserted into the scaling expressions for the average velocity in

the surface and at the top of the vegetation layer, then we arrive at:

3/1

3/2

'

'

~

'

'~

+−

Υ

+−Υ

k

k k h

k

k k h

r

top

(4.21)

Next, the unknown scaling length k’ needs to be specified. The parameter k’ reflects the extent to

which the profile of the surface layer extends into the vegetation layer. Therefore, k’ may never

exceed the height of the vegetation elements (k ). For the available data several combinations of

available length scales were attempted (D, s, b and combinations of these) and it turned out that

when for the surface layer the scaling length is taken equal to the separation between vegetation

elements (i.e. k’ = s) the proposed scaling law gives very good agreement with laboratory data (see




43

Figure 4.1). Apparently, the spacing between the roughness elements is a suitable measure for the

extent to which the flow in the surface layer influences the flow field in the vegetation layer.

Furthermore, by imposing a limit of height k for the scaling length k’ we propose the following

scaling expression for the average velocity in the surface layer:

3/2

),min(

),min(

~

+−

Υ sk

sk k htop (4.22)

Figure 4.1 Scaled velocity function using a (h-k+s)/s power law.

4.2 A force balance in the vegetation layer

Nezu and Onitsuka (2002) measured turbulent structures in partly vegetated open channel flows

and found that the energy dissipation rate does not remain constant at all scales. They argue that

energy losses in the flow field through vegetation are strongly influenced by drag processes while

turbulent mixing dominates the energy losses in wall-bounded flows. Therefore, in order to estimate

the average flow field in the vegetation layer an approach is followed in which a simple force

balance of the flow in the vegetation layer is considered.




44

Figure 4.2. The forces involved when describing the average flow through a (submerged) vegetation layer.

A faster flowing region above the vegetation layer produces a shear stress, which adds an

additional term to the force balance (bed resistance is neglected).

In the vegetation layer four forces are acting on the fluid: (i) gravitation, (ii) the drag force due to the

vegetation elements, (iii) bed resistance (iv) and a shear stress at the top of the vegetation layer

(see Figure 4.2). The latter is due to the higher flow velocities in the surface layer. We will neglect

bed resistance for now, assuming that resistance caused by the cylinders dominates. Estimation of

the gravitation and drag force in the vegetation layer is straightforward (see also (3.5) and (3.6)):

gi F g ρ = (4.23)

b

U U gmDC F

veg

veg D D22

12

2 ρ

ρ == (4.24)

In the expression of the drag force we assume that the average velocity in the vegetation layer may

be used as the dependent variable. The shear stress at the top of the vegetation layer (at z=k )should balance the gravitational pull of the surface layer and is therefore given by:

ik h g k )( −= ρ τ (4.25)

Following the result of the previous section that the flow in the surface layer affects the flow below

over a vertical height s (or k if k<s), then the corresponding extra shear force on the layer

becomes:

s

ik h g

y F

)( −=

∆

∆=

ρ τ τ (4.26)

This shear force only acts over a depth of s, while the drag force and the gravitational force act

over the entire vegetation layer with depth k . When distributing the extra shear force uniformly over

the depth over the vegetation layer we get:

k

ik h g F

k

s )( −= ρ

τ (4.27)




45

Now a bulk force balance for the entire depth of the vegetation layer can be established where the

drag force is balanced against the shear force and the gravitational pull:

gik

ik h g

b

U veg ρ

ρ ρ +

−=

)(

2

2

(4.28)

Solving for the (average) velocity in the vegetation layer yields:

k

hu

k

bghiU sveg ==

2 (4.29)

Or, in dimensionless form:

k

hveg =Υ (4.30)

Figure 4.3 shows how this scaling law compares with the measured average velocities, and the

result is quite extraordinary: it appears even better than the predicted values of the integratedvelocity profile shown earlier in section 3.4 (Figure 3.14, right graph).

Figure 4.3 The average flow velocity in the vegetation layer from experiments by Klopstra et al (1997)

compared to predictions with expression (4.30).

4.3 Overall average flow velocity for submerged rigid cylinders

Our analysis revealed that turbulent energy considerations (Kolmogorov scaling, energy

dissipation) lead to a relation that describes the average flow field in the surface layer quite well.

This method proved useful in estimating overall flow conditions in the surface layer, but cannot be

used for the vegetation layer because an additional energy loss mechanism is active: form drag

due to protruding cylinders. However, a simple force balance that takes form drag into account

describes the overall flow field in the vegetation layer well. This demonstrates that energy

dissipation in flow over rough surfaces is dominated by internal mixing (energy cascade), while for




46

a flow layer that is penetrated by roughness elements the form drag accounts for most energy

losses. Summarizing, the scaling expressions for the surface (top) and the vegetation layer

become:

2/1

=Υ

k

hveg (4.31)

3/2

),min(

),min(

+−=Υ

sk

sk k htop (4.32)

Combining the scaling law expressions for the surface and vegetation layer into (3.66) yields:

3/22/1

),min(

),min(

+−−+

=Υ

sk

sk k h

h

k h

h

k tot (4.33)

Which in case that the water depth is much larger than the cylinder height reduces to Manning’sequation for rough channel flow. Figure 4.4 shows that the scaling function (4.33) gives very good

agreement with the measured average velocities and may even compete with the predictive power

of the average flow description given earlier in section 3.4.3. Naturally, the equations given in

section 3.5 may also be used to determine resistance parameters based on the average velocities

predicted by the two-layer scaling approach.

Figure 4.4 The new average velocity function for the entire flow depth (4.33) based on the two-layerscaling approach in comparison to measured average flow velocities.

Now that a simple expression is available that describes the average flow velocity in presence of

submerged rigid cylinders, we can more easily investigate which processes dominate at specific

conditions. In the scaling function (4.33) the first term gives the contribution of the flow in the

vegetation layer to the overall average velocity and the second term the contribution of the flow

field in the surface layer. Therefore, the relative contribution of the vegetation layer to the overall

average velocity can be written as:




47

%1001

2/1

∗Υ

=Ρ

tot veg

h

k (4.34)

Likewise, the relative contribution of the velocity in the surface layer to the overall average flow

velocity is:

%1001

),min(

),min(3/2

∗Υ

+−−=Ρ

tot top

sk

sk k h

h

k h (4.35)

In Figure 4.5 Ptop (surface layer, (4.35)) and Pveg (vegetation layer, (4.34)) are plotted as functions

of relative cylinder heights (k /h) for three configurations of cylinder surface density. When the

separation between cylinders equals the cylinder height (i.e. when s=k ) then the average velocity in

the surface and vegetation layers contribute nearly equally to the overall average velocity (over

total depth) when the depth of flow is about twice the cylinder height (k /h = 0.5). To be precise, in

this specific case 46% of the overall average velocity is determined by the flow in the vegetation

layer, and 54% is determined by the flow in the surface layer. When the cylinders are packed

closer together then the flow in the surface layer becomes more dominant in the overall average

velocity. If the separation between cylinders is only one tenth of the cylinder height (s=0.1*k ), and

surface layer and vegetation layer are equally large (k /h = 0.5), then the flow in the surface layer

determines the total depth-averaged velocity for 79%. In other words, in such cases the velocity in

the surface layer is nearly 4 times more important for an accurate determination of the total

average flow velocity. In flows over vegetation where the surface layer is significantly larger (i.e.

deeper) than the vegetation layer the dominance of the flow field in the surface layer becomes

even greater.

Figure 4.5 The contribution of the average velocity in the vegetation layer and the surface layer to the total

depth-averaged flow velocity ( Pveg (4.34) and Ptop (4.35)) as function of relative cylinder height

(k/h). Three surface density configurations of the cylinders are considered (s=k, 0.1*k and

0.01*k).

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1k /h

C o n t r i b u t i o n t o d e p t h - a v e r a g e d v e l o c i t y

( P )

Surface layer

Vegetation layer

s = k s = 0.1*k

s = 0.01*k




48

4.4 Real vegetation

So far, the experiments that where considered for validation of the average flow description were

actually carried out with rigid cylinders and lack at least two properties that are often found in real

vegetation elements: flexibility and the presence of foliage (leaves). In this section a comparison

will be made to results form experiments with real vegetation. Are the found expressions suitable to

describe these situations as well?The product of the mean flow velocity times the flow depth (or hydraulic radius) is a

commonly used flow parameter to describe the variation of Manning’s n when related to vegetation

resistance (e.g. Fisher 2001, Copeland 2000). Rearranging expression (3.70) to this form yields:

Manning:( )

ihU

hn

tot

3/5

= (4.36)

In Figure 4.6 the predicted Manning coefficient is plotted against the predicted flow parameter

1/U tot R using the scaling expression (4.33) (for the data from Klopstra et al.1997). The figure shows

that the proposed scaling law predicts results that agree with relations of the type n=n0 + K/U tot R .

Figure 4.6 Manning’s resistance coefficient as function of the flow parameter 1/UR.

Fisher (2001) gives an overview of several empirical relations of the form n=n0 + K/U tot R and claims

that n-U tot R relations give sound results in the majority of cases. Among these is the classification

of the US Soil Conservation Service (1954) who proposed 5 different classes of vegetation

resistance with corresponding relations between Manning’s n and the flow parameter U tot R . The

main criterion for membership of any of these classes is the vegetation height. Green and Garton

(1983) provide equations for each of these 5 vegetation resistance classes (Figure 4.6 and Table

4.1). We will compare these equations to predictions of Manning coefficients based on the two-

layer scaling approach, where it is assumed that the vegetation is homogeneously distributed and

non-flexible.

y = 0.08x + 0.06

R 2 = 0.83

y = 0.04x + 0.05

R 2 = 0.75

y = 0.01x + 0.05

R 2 = 0.67

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2 2.5 3 3.5

1/UtotR

M a n n i n g ' s n

k = 1.5 [m]

k = 0.9 [m]

k = 0.45 [m]




49

Vegetation

resistance class

Height k

[m]

Equation for Manning’s n Limits of U tot R

[m2/s]

( ) RU n tot 62.144.0 −= 1542.0< RU tot

Class A > 0.76

( ) RU n

tot

0223.0046.0 += 1542.0> RU tot

( ) RU n tot 3356.3403.0 −= 0535.0< RU tot

( ) RU n

tot

0096.0046.0 += 1792.00535.0 << RU tot

Class B 0.28 – 0.61

( ) RU n

tot

0115.00354.0 += 1792.0> RU tot

( ) RU n

tot

0046.0034.0 += 0833.0< RU tot

Class C 0.15 – 0.25

( ) RU n

tot

0051.0028.0 += 0833.0> RU tot

( ) RU n

tot

002.0038.0 += 100.0< RU tot

Class D 0.05 – 0.15

( ) RU ntot

0028.003.0 += 100.0> RU tot

( ) RU n

tot

0007.0029.0 += 123.0< RU tot

Class E < 0.05

( ) RU n

tot

0015.00225.0 += 123.0> RU tot

Table 4.1 Vegetation resistance classes (US Soil Conservation Service, 1954) and their respective

equations and limits (Green and Garton, 1983).

The empirical relations proposed by Green and Garton (1983) give larger Manning coefficients forvegetation classes with large average heights. This is exactly what one would expect because

taller vegetation causes more resistance due to the larger surface area that blocks the flow.

According to the classification of the US Soil Conservation Service (1954) the experiments from

Klopstra et al. (1997) should correspond with vegetation resistance class A, or when the roughness

height is k=0.45 [m], to class B. Figure 4.7 shows the predicted Manning values when using the

new scaling expression (4.33). It can be seen that predicted Manning coefficients in flow situations

where k is large (k = 0.9 [m] and k= 1.5 [m]) are consistently larger than the Manning-relation

attributed to vegetation resistance of class A. These differences are considerably smaller when




50

comparing predictions corresponding to k = 0.45 [m] to the Manning relation of class B. In the latter

case agreement is actually quite good.

The finding that hydraulic resistance of vegetation is better predicted by the proposed

scaling expression (4.33) when vegetation heights are smaller is possibly due to the neglect of

bending and streamlining effects. Tall vegetation experiences a greater drag force and may,

depending on its stiffness, bend over towards the direction of flow. Consequently, in the new more

streamlined position resistance to flow decreases and a smaller Manning coefficient is expected.Stephan and Gutknecht (2002) already pointed out that the average deflected plant height is a

suitable parameter for the vegetation’s resistance to flow. Whether the deflection effect is indeed

more severe for taller vegetation will be investigated in future studies.

Figure 4.7 Predicted Manning coefficients based on the new scaling expression (4.33) compared to the

empirical relations proposed by Green and Garton (1983) (i.e. vegetation resistance class A

and B). Hydraulic resistance of cylinders with the smallest roughness height (k = 0.45 [m])

corresponds best to the associated empirical relation of resistance of natural vegetation.

4.5 Conclusions

Based on scaling considerations of the forces involved, depth-averaged flow velocities within the

roughness layer and in the free flowing layer above the roughness elements are estimated.

Consequently, conditions in the two separate flow layers yield a new description of the overall

average flow field, which is entirely determined by measurable geometrical boundaries. The new

description shows to give good agreement with laboratory flume experiments of rigid cylinders.

Whether the expression also reflects the resistance of real vegetation adequately needs to be

investigated in greater detail.

First comparisons to empirical relations that are based on real vegetation resistance

indicate that the scaling expression performs better when (flexible) vegetation is not extremely tall

(k<60 cm). It is hypothesized that the larger discrepancy for taller vegetation is due to stronger

streamlining and bending effects. These effects are not present in the proposed scaling expression

and will be the topic of subsequent research.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2 2.5 3 3.5

UR

M

a n n i n g ' s n

k = 1.5 [m]

k = 0.9 [m]

k = 0.45 [m]

class A

class B




51

5 Evaluation of vegetation resistancedescriptions

In chapter 3 and 4 two methods were shown to determine the average flow velocity when a field of

homogeneously distributed rigid cylinders obstructs the flow. In this chapter the methods arecompared and situations are given when each method is more suitable for practical application.

Some concepts that describe a framework for appropriate modelling are also put forward. However,

these concepts cannot be deployed at this stage because uncertainties associated with the two

descriptions are currently not available. In a future uncertainty analysis these will be determined.

Here the evaluation of the two methods is restricted to an analysis that identifies conditions when

the two predict practically the same outcome, or give significantly different results.

5.1 Appropriate modelling in river flows

The concept of appropriate modelling was first introduced by Vreugdenhil (2002) as a strategy that

guarantees process description of a model with minimal effort, or, a way to model a process in the

most (technically) efficient way. In the case of changing water levels due to changing vegetation

characteristics in floodplains, it should be added that not only the technical complexity should be

appropriate, but also that the desired accuracy of water level predictions should be met. If,

furthermore, the development costs are weighed against the quality of the model, then three levels

of model appropriateness can be discerned (see also Figure 5.1):

1. Technically appropriate: modelling errors are smaller than the effect of uncertainties in

parameters, data and required output (Vreugdenhil 2002). Every model should fulfil this demand in

order to capture the process to be studied.

2. Practically appropriate: outcomes of the model should meet the desired accuracy for practical

application. Currently, the desired accuracy for water level predictions in river systems is of the

order of a few cm.3. Cost/benefit appropriate: the costs, or the amount of effort, to be put into further development

of the model should be in balance with potential gain of quality.

Technically appropriate Practically appropriate Cost/benefit appropriate

Figure 5.1 Three levels of ‘appropriateness’ in modelling practice.

In particular in the field of fluid mechanics the concept of appropriate modelling is

important. It has generally been accepted that the Navier-Stokes equation describes all flows, and

should, in combination with the appropriate boundary conditions, be able to predict all imaginable

flow situations. In other words, the laws that govern all flows are already known, then why look

further for simpler expressions? The problem is twofold: first off all, boundaries may show natural




52

variability (irregularities, unevenness) or even interact dynamically with the flow field (e.g. flexible

vegetation or bed pattern formation through sediment transport), secondly, the Navier-Stokes

equation is non-linear. Because of this non-linearity, small variations in the boundaries can have

large influences on the predicted outcomes. Moreover, in order to calculate the behaviour of the

average flow field it is necessary to consider small-scale motions because these determine the

energy losses. In Direct Numerical Simulation techniques (DNS) these small-scale processes are

modelled all the way to the Kolmogorov scale where molecular processes become active(molecular viscosity) and damp out the smallest motions. Although DNS most closely follows the

laws of fluid flows, DNS shows to be computationally cumbersome and even with today’s most

powerful computers remains practically unusable for predicting real river flows (e.g. Hirsch 2000).

For practical purposes, it is therefore an absolute necessity to develop flow equations that average

out small scale motions: detail of flow descriptions must be reduced.

When Manning published his well-known formula in 1889 many competing flow formulas

existed, some of which were simple and some rather complex. One of the arguments that Manning

put forward in favour of his proposed formula was its simple form. This immediately triggered fellow

researchers to point out that the truth-value should be the leading factor when deciding which

method to use, not its elegance or simple form. In the discussion following Manning’s paper

Vernon-Hartcourt commented: “[Manning] objects to the complexity of [the Ganguillet and Kutter]

formula; but if greater accuracy is obtained, complexity should not be complained of” (Manning

1889). Exactly here the concept of appropriate modelling comes in. Increased accuracy is always

desired, but if it requires much effort with little or no effect to serve a specific purpose, then a

complexity complaint is justified. In view of the many uncertain influences present in river flows, the

Manning equation proved to be useful in many situations and the more complex descriptions of, for

example, Ganguillet and Kutter did not add to significant improvement of flow predictions.

Eventually, Manning was proved right by the acceptance of his proposed formula by river

engineers worldwide in favour of descriptions that were still in use at the beginning of the 20th

century, and are now all practically extinct.

However, for river flows in presence of vegetation the conventional Manning equation with

a constant roughness factor oversimplifies the problem (e.g. Green 2005) and a new description is

called for. In the previous chapters two descriptions were demonstrated that describe the flowthrough and over a field of rigid cylinders. Can these be used to model vegetation effects in rivers,

and if so, which of the two requires least effort for satisfying results? Here we will compare the flow

predictions of the two methods, and determine in which conditions the two predict practically the

same values. In such situations it would be logical to choose the simplest method for further use. In

situations where the two methods give substantially different outcomes, a comparison with further

experimental data is needed: regardless of model complexity, which method reflects reality best

and is in that case the appropriate model?

5.2 Comparing vegetation roughness descriptions

In the previous chapter two descriptions were presented for the average flow velocity in presenceof submerged vegetation (or rigid cylinders ). One method was based on a depth-integrated

velocity profile, which required a closure parameter in the form of a turbulent length scale. Through

trial and error a suitable closure description was identified. Next, energy and force balance

considerations of the bulk (average) flow led to an alternative description of the average flow field.

In this section we will compare the two methods for realistic ranges of input parameters and identify

the regimes where they perform equivalently and the regimes where a choice between the two is

necessary. As a reference we use the Dutch handbook for vegetation roughness in floodplains (van

Velzen et al. 2003), where different types of vegetation found along rivers in the Netherlands are




53

characterized by their geometrical properties. Table 5.1 gives an overview of the used reference

data selected from van Velzen et al. 2003 (English translations taken from Favier 2003).

Nature type k m D CD

[m] [m-2

] [m] [-]

1. Pioneer vegetation 0.15 50 0.003 1.8

2. Production grassland 0.06 15000 0.003 1.8

3. Natural grassland 1 0.1 4000 0.003 1.8

4. Natural grassland 2 0.2 5000 0.003 1.8

5. Thistle bushes 0.3 3000 0.003 1.8

6. Bushes (with biodiversity) 0.56 46 0.005 1.8

7. Reed bushes 2 40 0.004 1.8

8. Bramble bushes 0.5 112 0.005 1.8

9. Spiraea bushes 0.95 26 0.005 1.8

10. Dune reed bushes 0.35 90 0.004 1.8

11. Reed 2.5 80 0.005 1.8

12. Reed grass 1 200 0.002 1.8

13. Reed-mace 1.5 20 0.0175 1.814. Pipe grass 0.5 300 0.004 1.8

15. Sedges 0.3 200 0.006 1.8

16. Young brushwood 3.5 19.6 0.011 1.5

17. Orchard (low) 2 0.16 0.1 1.5

18. Orchard (high) 3 0.16 0.2 1.5

Table 5.1 Reference parameters for appropriate modeling tests (selected from van Velzen et al. 2003).

Shown parameters are the vegetation (roughness) height (k), the surface density (m), the stem

diameter (D) and the drag coefficient (C D ).

In Figure 5.2 and Figure 5.3 the integrated velocity profile and the bulk flow method are

compared for the different reference values (Table 5.1) and for the earlier used flume experiments.

The relative difference is calculated as

int

int

U

U U

U

U bulk −=∆

(5.1)

Where U bulk is the average flow velocity that corresponds with the scaling approach from

Chapter 4 (expression (4.33)) and U int corresponds with the depth-integrated velocity profile from

Chapter 3. For the vegetation layer the two expressions that are compared are:

2/1

,

=

k

h

u

U

s

veg bulk (5.2)

+Υ

−Υ

−+−Υ+=

1

14ln

2

11

21

*

**int,

k

k k

s

veg

k hk u

U ll (5.3)

For the surface layer the corresponding expression are




54

3/2

,

),min(

),min(

+−=

sk

sk k h

u

U

s

topbulk (5.4)

( )

−

+−

−

+−+−+Υ= ∗ 1ln

2

int,

s

s s s

k

s

top

h

hk h

k h

hk h

b

hk h

u

U

κ

(5.5)

For the total flow depth the average velocity in the surface layer and vegetation layer are, again,

combined as:

topveg tot U h

k hU

h

k U

−+= (5.6)

In Figure 5.2 the relative differences between the two vegetation-resistance descriptions

are set out against changing water depths (to a maximum water depth of 4 [m].) and in Figure 5.3

other parameter variations are considered. The figures show that the predicted values that

correspond with the flume experiments (large dots) are nearly the same between the two methods,

which is what would be expected because both methods were validated against precisely these

experiments. The points that correspond to the reference parameters (Table 5.1) give an indication

in which ranges significant differences in predictions can be expected.

The top row of graphs in Figure 5.2 shows the situation for the vegetation layer, the surface

layer and the total depth respectively as a function of the total depth. The row below shows the

same situations for changing relative depths (h/k). When we look at the vegetation layer only (2

graphs in the first column) we can conclude that the two descriptions gradually diverge with

increasing water depth. In the surface layer some combinations of input parameters already give a

large difference between the two descriptions at low (relative or absolute) depths. Apparently, the

surface layer descriptions are very sensitive to certain ranges of input parameters. This observation

requires that a clear distinction is made between situations where there is a preference for either

one of the descriptions. The relative difference between the two descriptions exceeds –0.5, whichis also directly transmitted to the relative difference over the total flow depth (graphs in the right

column). Overall it can be concluded that in the vegetation layer the bulk description of the average

velocity tends to give larger values than the depth-integrated description while in the surface layer

the situation is reversed. The velocity differences considered over the total depth (graphs in the

right column) indicate that in particular the flow descriptions in the surface layer affect the overall

outcome.




55

Figure 5.2 Evaluation of difference between methods to predict the average flow velocity for varying

absolute (top row) and relative (bottom row) depths. From left to right the situation is shown

separately for the vegetation layer, the surface layer and the total water column. The large data

points correspond to the parameter combinations that were present in the flume experiments.

The reference data points (small dots) represent typical geometries of vegetation types found in

the Netherlands (taken from van Velzen et al. 2003).

In order to understand the characteristics of situations where the two descriptions give predictions

that are significantly different, in Figure 5.3 velocity predictions are set out against the geometrical

parameters k, s and CDD (roughness height, spacing between elements and dynamic stem width,

respectively). Again, the situation in the surface layer is almost identically reflected in the

predictions of the flow over the entire depth. The figures in the bottom row, where predictions are

compared against the ratio between the separation s and the equivalent flow diameter of the

roughness elements CDD, show the most pronounced general trend. If the ratio s/CDD is smaller

than about 5, the bulk description gives significantly larger velocity predictions in the vegetation

layer and smaller predictions in the surface layer. These situations correspond to tightly packed

roughness elements, where the separation between elements (s) is only a few times their width

(CDD). Also, when the roughness elements are very loosely packed and the separation is more

than about 25 times the stem width, then predictions in the vegetation layer become increasingly

different between the two methods. The top row of Figure 5.3, where the aspect ratio between

vegetation height and separation between vegetation elements is considered (k/s), does not show

a clear general trend between the performances of the two velocity prediction methods.




56

Figure 5.3 The same appropriate model tests as in Figure 5.2 but here shown as functions of vegetation

height (k), width (D) and spacing between roughness elements (s).

In Table 5.2 nature types are highlighted that correspond to situations with extreme packing of the

roughness elements (i.e where s/CDD<5 or s/CDD>25). For the remaining vegetation types, the two

methods give similar predictions and, therefore, the most appropriate choice would be the simplest

expression: the bulk flow method based on the two-layer scaling approach. Figure 5.4 shows the

behaviour of the two models as a function of total flow depth when the nature types with extreme

packing are left out. At very large depths of flow (near 4 [m]) the two methods may still give

average velocity predictions with differences as large as 25%. As pointed out earlier this difference

finds its origin mainly in different predictions of flow in the surface layer. In the vegetation layer the

two models give better agreement; even at large total depths differences are smaller than 10%.




57

Figure 5.4 Same as Figure 5.2 but without the nature types with extreme (very dense or sparse packing).

The two methods to predict the overall average flow velocity give values within 25% of each

other.

5.3 Sparse or very dense vegetation

Nature types that are associated with substantial differences in predicted flow velocities include

very dense types as grasslands and thistle bushes and the very sparse pioneer vegetation (see

Table 5.2). These nature types are abundant in the floodplains of Dutch and German rivers and

therefore necessitate special treatment. Comparison between the two methods alone cannot clarify

which method is most suitable for these nature types. Therefore, a comparison with further

experimental data is made in order to determine which of the two descriptions reflects reality more

closely. In this case the predicted average velocities (for the entire flow depth, for each of the two

methods) is compared to the actual measured average velocities (U ref ) as:

ref

ref

U

U U

U

U −=

∆ (5.7)

Figure 5.5 shows such a comparison; data sets are used that reflect situations with extremely

sparse and extremely dense packing. The figure shows that the predictions of the bulk flow

description easily mispredicts by as much as 50% for both sparse and dense packing (graphs in

bottom row). On the other hand, the depth-integrated description gives relatively good predictions

for the dense vegetation measurements, especially as the total water depth increases. The

velocities over sparse vegetation are also poorly predicted by the depth-integrated method.




58

Figure 5.5 Validation test of the predicted overall average velocity (total depth) of the two studied

methods. Predictions are compared with data from Murota et al. (1984), Kouwen et al. (1969),

Järvelä (2003) and Ikeda and Kanazawa (1996) (compilation taken from Baptist 2005).

Triangles correspond to sparse vegetation (Ikeda and Kanazawa (1996), Murota et al. (1984))

and stars to the remaining dense vegetation (Kouwen et al. (1969), Järvelä (2003)).

The observation that neither of the average flow descriptions describes the situation for very sparse

vegetation well, may be understood when considering the relative importance of an additional

source of energy loss: bed roughness. The column on the outer right of Table 5.2 shows for which

nature types bed irregularities may play an important role in the average velocity field (by means ofa transitional roughness height kbed). When bed irregularities exceed the size indicated by kbed, then

bed roughness is no longer negligible (see section 5.4). This is very likely to be the case for

pioneer vegetation and low orchards, where sand grains or gravel may be expected to significantly

exceed the size of 10 µm or 0.22 mm, respectively. According to the British standard of M.I.T.

typical grain sizes for fine to medium sand are in the range of 60 µm – 0.6 mm (e.g. Jansen 1979).

Moreover, the bed level itself or short vegetation in between tall plants may show elevation

irregularities that are even much larger. Therefore, because of the large separation between




59

vegetation elements, for both pioneer vegetation and low orchards the bed roughness is expected

to significantly influence the velocity profile in the vegetation layer.

In section 5.4 it was pointed out that the local roughness height should be much larger than

kbed for the flow to be entirely determined by bed roughness. Table 5.2 shows that it is not realistic

to find irregularities of such sizes in between the vegetation. We therefore conclude that the

average flow field in presence of sparse vegetation cannot be described by a single roughness

height but is determined by both vegetative drag and bed roughness. This is also consistent withthe observed velocities in Figure 5.5. Here average flow velocities are overestimated because the

neglect of bed roughness results in underestimation of the total resistance.

Nature type k s (m-1/2

) CDD s/CDD kbed

[m] [m] [m] [-] 10-3

[m]

1. Pioneer vegetation 0.15 0.14 0.0054 26.2 0.01

2. Production grassland 0.06 0.008 0.0054 1.5 7800

3. Natural grassland 1 0.1 0.016 0.0054 2.9 940

4. Natural grassland 2 0.2 0.014 0.0054 2.6 33000

5. Thistle bushes 0.3 0.018 0.0054 3.4 38000

6. Bushes (with biodiversity) 0.56 0.15 0.009 16.4 6.3 7. Reed bushes 2 0.16 0.0072 22.0 360

8. Bramble bushes 0.5 0.094 0.009 10.5 66

9. Spiraea bushes 0.95 0.20 0.009 21.8 9.3

10. Dune reed bushes 0.35 0.11 0.0072 14.6 3.2

11. Reed 2.5 0.11 0.009 12.4 16000

12. Reed grass 1 0.071 0.0036 19.6 360

13. Reed-mace 1.5 0.224 0.0315 7.1 1300

14. Pipe grass 0.5 0.058 0.0072 8.0 610

15. Sedges 0.3 0.071 0.0108 6.5 80

16. Young brushwood 3.5 0.23 0.0165 13.7 4600

17. Orchard (low) 2 2.5 0.15 16.7 0.22

18. Orchard (high) 3 2.5 0.3 8.3 9.0

Table 5.2 Nature types from the vegetation roughness handbook by van Velzen et al. (2003) with

corresponding geometric dimensions. Vegetation with extreme volume packing (either very

dense or very sparse) is highlighted. The column on the outer right shows the roughness height

that indicates when bed roughness starts to play a role. Considering that sand or gravel grains

in between vegetation could be as large as 1 mm, for pioneer vegetation and orchards bed

roughness should be taken into account.

The fact that neither of the two velocity descriptions performs well over the entire range of possible

vegetation characteristics may be due to several reasons. First of all, flow in the vegetation layer

may be slowed down to a level that the flow field becomes laminar. In that case energy transfer in

the vegetation layer is of a different nature than in the turbulent case, thereby affecting the shape ofthe velocity profile. Other effects that are neglected in the proposed descriptions are (i) additional

drag due to foliage (leaves and side branching), and resistance decrease due to streamlining and

bending of roughness elements. It is beyond the scope of this report to further reflect on these

processes but it is expected that they are present in the data sets used for validation in Figure 5.5.




60

5.4 Bed resistance

In the derivation of the average velocity in the vegetation layer in section 4.2 the effect of bed

resistance (sand roughness) was neglected because of the assumption that energy losses due to

drag effects are considerably larger. Here we will reconsider the validity of this statement. First of

all, in the absence of drag effects (absence of vegetation) we adopt the Manning/Strickler equation

(see also section 2.1.1) to describe the average flow velocity over a rough surface (with roughnessheight k s):

gRik

RU

s

6/1

8

= (5.8)

The expression above describes flow where energy losses are generated primarily through bottom

boundary irregularities. On the other hand, flow through emergent vegetation with no influence of

the bottom-boundary yields the characteristic velocity us (see expression(3.10)). Therefore, if for

specific conditions the value of us is considerably smaller than the expected velocity in absence of

vegetation (expression (5.8), bottom boundary dominated flow) then roughness effects of the bed

can be neglected. By replacing the hydraulic radius with the vegetation height k (i.e. roughnesselements penetrate the entire depth of flow), this condition can be expressed as

gkik

k bgi gki

k

k u

s s

s

6/16/1

828

<<⇒

<< (5.9)

With expression (5.9) it is possible to define a bed roughness height (k bed ) that indicates when bed

roughness starts to play a role in the overall resistance. When we choose as a criterion that us is

about 5.7 times smaller than the average velocity in absence of vegetation (i.e. replace the

proportionality constant by unity), we get:

Transition to bed roughness influence: k k

k bbed

6/1

≈ (5.10)

or

Transition to bed roughness influence:( )

6

343

s

DC k k

b

k k D

bed =

≈ (5.11)

Which means that bed roughness should only be taken into account when the separation between

roughness elements (s) is very large, and vegetation height (k ) or stem diameter (D) are relatively

small. Only then the critical value for k bed may come close to realistic values of sand grain

dimensions.

When us is equal to or exceeds the average velocity as given by (5.8), then drag due to

protruding vegetation is no longer significant and the average flow velocity is determined by bottom

roughness only.

gkik

k bgi gki

k

k u

s s

s

6/16/1

828

≈⇒

≈ (5.12)




61

In that case, complicated vegetation resistance models become obsolete and standard

roughness height methods (e.g. Nikuradse or Strickler roughness heights) should be used.

Quantitatively this is the case when the local roughness height is more than 33000 times larger

than the value that marks the transition to bed roughness influence (as defined in (5.9)):

Bed roughness only: bed s k k 33000> (5.13)

In section 5.2 an overview is given of typical vegetation dimensions. It may be concluded that only

in case of very sparse vegetation (pioneer vegetation and orchards) the bed roughness starts to

play a role (see Table 5.2), but never to an extent that vegetative drag may be neglected.

5.5 Conclusions

Two earlier derived methods to describe the average flow field in presence of homogeneously

distributed cylinders are compared. Vegetation characteristics that correspond to commonly found

vegetation types in lowland floodplains were used as input parameters. It is shown that for a wide

range of known vegetation characteristics the newly proposed bulk flow method predicts flowvelocities that are similar to the predictions of the depth-integrated method. In those cases it is

recommended to use the simplest flow equation, the one based on scaling considerations.

For very dense nature types, such as grasslands or dense bushes, the bulk flow method is

no longer suitable because it systematically overestimates the flow resistance. In such situations

the depth-integrated method based on Klopstra’s approach gives better results. In case of very

sparse vegetation (e.g. pioneer vegetation or orchards) bed roughness can no longer be neglected

and neither of the investigated description gives satisfying results.




62

6 Discussion & outlook

6.1 Composite channel geometries

What are the consequences for compound river channels when floodplains are covered withvegetation? This is the key question in the research on hydraulic resistance of vegetation. The

current report only reflected on the hydraulic resistance of vegetation in cases of completely

homogeneous surface coverage. However, what role does vegetation resistance play in more

complex channel geometries, for example for a compound channel with a main channel and a

vegetated floodplain. When, for example, the main channel is very wide and free of vegetation then

a large resistance in a small floodplain may not be very significant. If, however, the floodplain is

very large and wide and the main channel not very deep then a little increase in floodplain

resistance could have very large effects. In order to truly understand the relevance of vegetation

resistance it is essential to understand the influences of the overall geometry of the river channel.

The flume experiments that were investigated did not include regions free of cylindrical

elements that could be interpreted as main channels. Future studies need to focus on the relative

importance of vegetation resistance compound channel geometries. These will be compared withtheoretical predictions made my resistance addition techniques, for example by using a divided

channel method. In section 2.2 it was already pointed out that such a divided channel method may

be used as long as the channel is relatively wide as compared to its depth. Fortunately, this is

usually the case for lowland rivers. For narrow channels possibly more complex methods have to

be used that also included energy losses due to transverse flow near the floodplain-main channel

interface. Knight and Shonio (1996) and van Prooijen (2004) have suggested methods to describe

such processes.

An earlier theoretical analysis investigated effects of increased roughness values in

floodplains of compound channels (Huthoff and Augustijn 2004). This study provided combinations

of geometrical parameters (of the rivers cross-section) that may be used to describe the channel’s

sensitivity to resistance changes. With the newly found velocity description of flow through

vegetation a more realistic vegetation resistance analysis can be carried out.

6.2 Variability in vegetation characteristics

Vegetation shows a large diversity in a natural environment. In the current study only situations

were considered where vegetation characteristics of individual plants (cylinders) were identical. In

future studies the effect of mixed vegetation types and random variability in plant parameters will

be investigated. Furthermore, the importance of vegetation characteristics such as flexibility and

the presence of leaves will be reflected upon.

Because of the many factors that are involved in natural environments it is important to

describe idealized vegetation resistance in as few possible parameters, but still to an extent that

resistance is representative of the real situation. Only then is it feasible to make general statements

about the relative importance of vegetation resistance in view of the many additional external

factors. When a method is found to realistically describe the hydraulic response of a natural

vegetated environment, vegetation succession models can be coupled to a hydraulic resistance

prediction model. Then finally long term effects of nature development plans can be evaluated.




63

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