hydraulic resistance of vegetation
TRANSCRIPT
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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)
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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
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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):
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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
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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.
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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
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6 Discussion & outlook ...............................................................................................62
6.1 Composite channel geometries.....................................................................................62
6.2 Variability in vegetation characteristics..........................................................................62
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University of Twente Final Project Report
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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).
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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.
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University of Twente Final Project Report
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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)
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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
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University of Twente Final Project Report
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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
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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
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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:
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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
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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
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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
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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.
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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
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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)
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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)
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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)
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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.
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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.
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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
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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)
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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
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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)):
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)(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.
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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]
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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
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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
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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.
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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
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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).
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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.
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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.
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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:
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+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.
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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 ]
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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.
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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)
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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.
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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.
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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)).
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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.
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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
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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
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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.
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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)
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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
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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:
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%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
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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]
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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
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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
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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
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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
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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
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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.
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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.
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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%.
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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.
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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
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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.
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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)
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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.
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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.
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