**The Diffusion and Drag on Flows through Vegetated Floodplains**

# Abstract

Vegetation in floodplains are a significant contributor to diffusion and drag on flood flows. The interference of plant stems modelled by cylinders can be used to describe the impact of the presence of vegetation on flood flows.

The research presented in this paper explores the use of different density spreads of vegetation by changing the diameter and the quantity of the vegetation stems. By measuring the spread of a tracer through different vegetation spreads the concentration gradient can be used in order to determine differences in the diffusion and bulk drag coefficient.

Despite limitations in this approach due to the experimental method and procedures adopted, trends can still be drawn from the data collected. Lower population densities exhibited more ambiguous results since the influence of bed shear was more significant. None the less, it could still be seen that the diffusion increased as the population density increased.

Higher population densities showed significantly more patterns, where diffusion increased and bulk drag coefficient decreased with increasing stem diameter and population. In addition, diffusion was greater at higher velocities indicating that the effects of turbulence and stems still caused significant mixing.

Comparing both population densities against the bulk drag coefficient, the bulk drag coefficient is shown to decrease with increasing population density. The influences of sheltering on downstream cylinders are more significant at greater densities.

When considering flooding the research suggests that greater densities of vegetation are likely to increase the diffusion of fluid and therefore increase the time taken for the fluid to move across the floodplain as well as reducing the mean flow depth. The bulk drag coefficient however is likely to vary significantly dependant on the channel and floodplain properties and should be considered independent for each case.

Table of Contents

Abstract

1. Introduction

2. Literature Review

2.1. Plant Geometry and Dimensions

Cylinder Analogy

Leafless Vegetation

Foliage

Roughness

2.2. Flexibility

Rigid vegetation

Flexible vegetation

2.3. Spatial distribution

Individual Plant Elements

Plant arrays

2.4. Degree of submergence

2.5. Density

2.6. Flow conditions

3. Methodology

3.1. Background

3.2. Experimental Apparatus

Flume

Vegetation zone

Apparatus

Tracer & Spectrophotometer

3.3. Experiment Basis

3.4. Experimental Procedure

Variables

3.5. Anticipated results

4. Results

4.1. Raw data: Concentrations

4.2. Diffusion Coefficient

4.3. The Bulk Drag Coefficient

5. Limitations

Cylindrical Analogy

Flexibility

Flow Conditions

Spacing

Channel Properties

6. Discussion

7. Conclusion

8. Bibliography

** **

# 1. Introduction

Flooding is a highly topical issue at present, due to the increased uncertainty in the climate due to global warming (Worley, 2016). Given the weaknesses of meteorological models it is extremely difficult to accurately foresees extreme weather events beyond a couple of weeks. Coupled with lengthy and costly recovery processes, it is essential to extend research to improve our understanding and enable us to mitigate against the catastrophic consequences of extreme weather (Ralph, 2016), including disruption and damage to infrastructure (capital losses), as well as social losses (Chatterton, et al., 2010).

A floodplain is typically an area of flat low lying land alongside a river or stream, where the size is generally dictated by the topography of the surrounding area and magnitude of previous flood events. A floodplain can be separated into two areas: the floodway and the flood fringe. The floodway is the main channel of the river or stream and the flood fridge is the area that extends from the outer banks of the channel to the bluff lines, which are the outer limits of a rivers floodplain where the ground level starts to rise (National Geographic, 2016) . Flow in the main channel of a river or stream is usually dependant on topography and climate of an area, since these influence factors such as precipitation rates and volumes, channel networks for runoff and storage of water. During flood events, the rate of flow and volume as well as the rate of sediment transport in rivers and streams is much greater than normal. This often results in the maximum storage of the channel being exceeded and once the floodway is breached, it overflows onto the floodplain, where sediments and debris are deposited (French, 1985).

The accumulation of deposits increases the fertility soils on floodplains, which for decades has been realised for its agricultural importance. This has led to floodplains becoming a source of thriving vegetation, and hence emergent vegetation became a common feature of wetlands and floodplains (Birkhead, et al., 2003).

As a result, the opportunities in agriculture and transport presented by waterways led to the growth of communities and urbanisation of the surrounding land. Over time, investments made in these flood catchment areas have increased the need for mitigation against flooding. Hence an increase in research into our rivers, floodplains and climate has improved our understanding and therefore risk management strategies against such events (Cassidy, et al., 1998).

The urbanisation of flood fringes has resulted in a decline in the density of vegetation and changes in the properties of the floodplains bed (ground conditions), and hence the rate of infiltration and resistance against flooding has decreased. Therefore, solutions are currently being proposed suggesting that floodplains should be restored to their original state, as it would seem that their natural properties are best at mitigating the consequences of flooding (National Wildlife Federation, n.d)

One of the most important features and assets of natural floodplains is vegetation, since it decreases flow rate, thus increasing the residence time on the floodplain. Its presence acts beneficially, increasing flow resistance through the dissipation of energy caused by the interference of stems and foliage (Fathi-Maghadam & Kouwen, 1997; Fathi-Maghadam & Kouwen, 1997; Fathi-Maghadam & Kouwen, 1997), as well as changing backwater profiles and modifying sediment transport deposition (Järvelä, 2004).

As a result, there has been an increase in interest and research into the properties of vegetation in rivers and floodplains. The complex nature of floodplains and vegetation (which will be discussed in more detail at a later stage) has meant that many researchers have been forced to make considerable assumptions and simplifications to allow them to replicate different scenarios (Birkhead, et al., 2003).

Resistance and roughness coefficients of the channel bed and vegetation have been studied on numerous occasions in an attempt to provide models, formulas and even constant values in order to describe the impact that vegetation on the floodplain has on the flow.

This research has resulted in the derivation and use of parameters such as the drag coefficient (

CD) and friction factor (

f) in order to help quantify resistance. They are commonly used in the computation of resistance including that of the well-known drag force (Equation 2‑1) where

CDis taken as a dimensionless estimated or fitted parameter (Cassidy, et al., 1998). The majority of models and formulas for

CDonly stand for a given range of Reynolds numbers and specified shapes. Since each floodplain is unique, in terms of topography and hence vegetation, assumptions that can be carried forward are limited (Birkhead, et al., 2003).

This research attempts to build on a previous study conducted by Nepf (1999), ‘Drag, turbulence, and diffusion in flow through emergent vegetation’, where the drag through emergent vegetation was described by diffusion. Using this concept, this research will focus of establishing a correlation, and therefore determining if it is possible to measure the drag coefficient, C_{D}, for a population density of vegetation by relating it to total diffusivity (Nepf, 1999).

To extend the research carried out by Nepf (1999), this investigation will go beyond the use of a single diameter cylinder as representation for vegetation in order to determine if cylinders of mixed diameter, hence another variation on population density, will have an influence the drag exerted. Due to the variation of vegetation on a floodplain, this research should be more representative of natural vegetation and allow further research into the resistance of mixed vegetation, thus helping us to improve our understanding of floodplain and the nature of drag coefficient.

# 2. Literature Review

The drag coefficient is a dimensionless parameter dependent on the shape of an object and its flow regime (NASA, n.d). It quantifies and characterises complex factors that influence drag, including the amount of resistance and force exerted on an object in a fluid environment (Dukler, et al., n.d). For simple shapes, research has enabled a range of values for

CDassociated with particular flow regimes to be determined (NASA, n.d). The resistance is commonly expressed in terms of the drag coefficient using the standard drag force formula from fluid mechanics:

FD= 12CDρU2Ap

Equation 2.1. Drag force formula. (Chadwick, et al., 2004)

where

CDis the drag coefficient,

ρis the water density, U is the average velocity and

Apis the momentum absorbing area.

The drag that occurs in a fluid is due to a combination of skin friction and form drag. Skin friction describes the transition in flow regime due to properties associated with the roughness of the surface of an object (Clancy, 1986) (Nepf, 1999). Conventional approaches typically use reference sources, publications or experimental research when selecting a roughness coefficient such as manning’s number

(n), the Chezy coefficient

(C)and the friction factor

fin order to describe the objects surface (Hu & Hui, 2010).

Form drag occurs due to the shape and space occupied by a particular object and will vary depending on the momentum absorbing area of the object (NASA, n.d). Complexities arise when quantifying the momentum absorbing area due to variations in its definition (this will be discussed later in the chapter).

The drag is influenced by a range of factors, some of which are dependent on fluid properties, whilst others are dependent on the properties of the obstructing object. In attempts to define

CD, experimental methods are commonly used in which certain fluid and object properties are known.

Fluid properties that cause variation in the drag coefficient include flow velocity, fluid density and fluid viscosity (Clancy, 1986) (Chadwick, et al., 2004). Reynolds number

(Re)is a fundamental term in fluid mechanics, which provides a dimensionless way of describing the flow regime by using a ratio of inertia forces and viscous forces, where inertia forces are due to the fluids momentum, whereas viscous forces tend to occur from the resistance of the fluid (Spurk & Aksel, 2008).

Re= ρulμ= ulν

Equation 2.2. Reynolds number (Chadwick, et al., 2004).

Reynolds number can be quantified using the equation above, where

ρis the density of the fluid (kg/m^{3}),

uis the velocity (m/s),

lis the length (m),

μis the absolute viscosity (kg/m s) and

νis the kinematic viscosity (m^{2}/s) (Chadwick, et al., 2004).

Since the flow regime has dependence on velocity, bed slope, fluid density and fluid viscosity these factors will define

Reand hence the drag coefficient

(CD)will be a function of Reynolds number (Spurk & Aksel, 2008). This in turn means that

CDcannot be taken as a constant, unless the range of values of each of the parameters that influence flow regime are constant or extremely small. Studies have shown that the drag coefficient is dependent on the type of flow and therefore its relation with

Reis largely dependant on whether the flow is laminar, turbulent or transitional (Spurk & Aksel, 2008). The obstruction caused by vegetation along with the flow properties of floods, tend to mean that turbulent flow is of most interest in this situation.

In this case, it has been shown that

CDgenerally decreases as a result of an increasing Reynolds number, due to the layer of fluid that remains close to the body of the object, causing a narrower wake at a higher

Re(Clancy, 1975-1991) (Clancy, 1986) & (Hu, 2010).

Vegetation in the channel causes an obstruction in the flow path and according to Lee (2004), the main properties of vegetation that influence the resistance of a flow, include geometry and dimensions of the plants, flexibility, density, spatial distribution and the degree of submergence, which in turn impact the mean velocity. Large variations and complexities in the properties of vegetation and floodplains result in difficulties when attempting to quantify the exact influence on resistance this generates (Fathi-Maghadam & Kouwen, 1997).

## 2.1. Plant Geometry and Dimensions

Drag caused by vegetation is largely dependent on the momentum absorbing area (also referred to as the reference area) of the plant, and as a result the manner in which it is defined will directly impact the quantification of drag. A major challenge when exploring the drag forces exerted within a flow, is that the momentum absorbing area is particularly difficult to define. Järvelä (2013) states that mechanical plant properties that influence size of the momentum absorbing area and as a result the energy loss through viscous and pressure drag include topology, age, foliage, volumetric and areal porosities, density and patchiness. However, the range in methods and definitions when calculating the momentum absorbing area cause significant variability in its prediction, which in turn cause variation in the computation of the energy dissipated and thus that of the drag coefficient.

Difficulties in computing the momentum absorbing area occur because of the variation in distribution and geometry of naturally occurring vegetation (Hasimoto, et al., 2009). Vogel (1994) argued that no published figure for the drag coefficient was of any real significance or applicability unless the reference area was indicated. The use of previous field studies to determine the drag coefficient is therefore limiting due to the varying nature of floodplains. The immense variation in characteristics means that many assumptions and properties, including that of the momentum absorbing area of vegetation, tend to be case specific and will not be suitable in the application of studies of other floodplains. Hence, this often leads to computed values of

CDbeing considered independently for each scenario.

*Cylinder Analogy*

In order to simplify the complex geometry associated with most species of vegetation, flow resistance formulas and/or models have been developed, treating vegetation as simply static rigid cylinders. These facilitate the computation of the momentum absorbing area since the properties of cylinders are well known and defined, and this allows for a more precise quantification of the geometry (Hasimoto, et al., 2009). The repercussion in using this approach is that cylinders offer limited applicability in exploring vegetation, as they fail to provide an accurate representation of vegetal features.

Cylinder analogy is frequently used when initially establishing the potential effect of vegetation on fluid dynamics. In order to investigate the influence on drag, (Thompson, et al., 2003) used well-defined shapes (cylinders and rectangles) to establish a value of

CD.For isolated elements, the average value for the drag coefficient was found to be 0.8 for cylinders and 1.55 for rectangles. As a result of the differences in drag between the two shapes, it was concluded that

CDhas a strong dependence on shape and hence the geometry of stems will have a major influence on the drag for that type of vegetation.

In later research, Järvelä (2013) used cylinders as a represention of stems of riparian trees and reeds and concluded that in addition to being influenced by the shape and size of the cylinder, the drag coefficient was also dependant on the roughness, Stem Reynolds number, array and density.

*Leafless Vegetation*

The dominant cause of drag from vegetation is due to the influence of the stems, this is further impacted by the stems morphology (Kadlec, 1990).

Hashimoto (2009) modelled the stem resistance

(τD)due to drag (shown in Equation 2‑3.) where the main contributors to resistance were the drag coefficient

(CD)for a single stem place in an array of stems, the number of stems per unit plan area of the bed

Nand the vegetation layer velocity

Vcwhichuses the concept of flow continuity and is equal to the discharge divided by gross cross sectional flow area. The following terms describe basic vegetation properties including stem diameter,

d,and the height of the stem below the water surface,

l.

τD=0.5ρCDNdlVc2

Equation 2.3. Stem resistance (Hashimoto, 2009).

*Foliage*

Foliage in vegetation is a significant contributor to the momentum absorbing area, so to accurately consider the impact of vegetation on drag, foliage should ultimately be included.

Birkhead (2003) conducted experiments comparing the resistance of harvested reed stems fully foliated to those with no foliage. To give an indication of the foliage area for each case they traced the plant outlines on to squared paper. This was used along with the drag force equation (Equation 2‑1) and Reynolds number (Equation 2‑2) to compute the drag coefficient and from this it was found that the area of foliage considerably increased

CD.

The same pattern was established for isolated vegetation elements, where the drag increased by approximately 40% in the presence of foliage in comparison to when it was leafless (Fathi-Maghadam & Kouwen, 2000). Järvelä (2013) suggested that foliage increased the momentum absorbing area and used the Leaf area index (LAI) to describe the additional surface area providing resistance. The foliage also increased the density of the vegetation and therefore the friction factor was shown to vary with plant stand density, where large LAI values resulted in more fricition.

Fathi-Maghadam (1997) carried out a flume based study to determine the relationship for velocity and drag using naturally occuring vegetation. They derived a drag equation which relating the drag force, shear stresses and area occupied by vegetation as:

τ0=FDa=12CD(A∀)ρV2

Equation 2.4. Relationship between drag and vegetation. M Fathi-Maghadam (1997).

where

τ0is the shear stress,

FDis the drag force absorbed by vegetation,

ais the area of the horizontal bed slope occupied by vegetation,

Ais the momentum absorbing area and

∀=aynwhere

yn=h(vertical height of vegetation) and

Vis the velocity.

Using this model, it was found that for rigid elements, the drag increased proportionally to the square of the velocity. Since they had no direct way to calculate the drag coefficient, they coupled it with the deflection of the momentum absorbing area as a dimensionless parameter

CD(Aa). This enabled the calculation of the parameter, using experimental methods where the velocity and drag were known. The results indicated that the drag coefficient reduced when the velocity increased as this caused the foliage to deflect, reducing its area.

*Roughness*

Commonly, the effect of roughness of vegetation elements on drag has been described using modified roughness coefficients such as a modified Manning’s number (Guardo & Tomasello, 1995). Nepf (1999) mentions that research by Kadlec (1990) shows that although the adaptation of Manning’s number is convenient as it provides a quick simplification in quantifying the resistance due to the roughness of vegetation, it reveals little detail about the influence on flow structure or regions of emergent vegetation.

## 2.2. Flexibility

The stiffness of vegetation will influence the magnitude of drag experienced each plant element. Stiffness determines the amount each element deflects therefore governing the its profile and contact area to the incoming flow (Hasimoto, et al., 2009).

*Rigid vegetation*

Rigid naturally occurring vegetation includes mostly branches and trees and tends to be denser that flexible vegetation. In experimental research cylinder analogy models normally use rigid cylinder elements to avoid deflection, therefore keeping the momentum absorbing area constant and as a result the drag force is proportional to the velocity squared (Hasimoto, et al., 2009).

*Flexible vegetation*

Fathi-Magahdam (1997) suggested that the momentum absorbing area, drag and velocity directly influenced each other since they were able to show that a reduction in foliage area caused by deflection, reduced the drag and increased the mean velocity.

Following this, Kouwen (2000) used naturally occurring tree samples to demonstrate that the additional influence of roughness alongside flexibility, where it was noted that the mean velocity and flow depth were the main factors causing bending and increasing the submerged momentum absorbing area, thus increasing the drag.

In order to account for bending due to the flexibility of vegetation, the mean velocity and flow depth should be considered as these will directly impact the submerged momentum absorbing area and this will cause variations in resistance (Fathi-Maghadam & Kouwen, 2000) (Järvelä, 2004) and (Järvelä, 2002).

Determining the effect of flexibility on drag remains difficult since streamlining alters the frontal and wetted areas of the element thus modifying the momentum absorbing area (Hasimoto, et al., 2009).

## 2.3. Spatial Distribution

Focus in research to date, tends to concentrate on the impact of flow due to a singular cylinder or plant, and therefore the drag is generally determined for individual elements. Arrays of cylinders and/or plants have been researched but these generally focus on the isolated effect of the group on an individual downstream element rather the implications of an array of cylinders on the flow.

For cases where an array of vegetation has been accounted for, the focus has been on defining the effects of either sparse or high density conditions.

For densities between these extremes, it is normally assumed that sparse and high density conditions are the boundaries and that resistances will lie within this range.

*Individual Plant Elements*

When vegetation is the main cause of resistance in a flow, the drag coefficient of an individual cylindrical element is often used as a fitted parameter or chosen and adapted using reference sources (Hasimoto, et al., 2009).

Lindner (1982) compiled a method to compute

CDfor an individual plant element in an array using cylindrical analogy, resulting in the derivation of an empirical formula:

CD=(1+1.9dayCD∞)(0.02025axd)0.46CD∞+(2ayay-d-2)

Equation 2.5. Drag coefficient for an array of vegetation (Lindner, 1982).

where

CD∞is the drag coefficient of a single cylinder in an ideal two-dimensional flow,

dis the diameter of an element and

axand

ayare the longitudinal and lateral distances respectively. The volume of vegetation per square meter was kept constant the

CD∞was found to range between 1.0-1.2 for a typical range of Reynolds numbers.

Based on this approach sub-sequential research determined a series of drag coefficients with the studies indicating that

CDwould remain constant. (Helmiö, 2004) and DVWK (1991) suggested that a

CD=1.5would be a sufficiently representative value for a practical range of flows.

Morris, H (1995) suggested that the resistance caused by dense non-submerged vegetation was dominated by drag forces exerted on individual plant elements within an array. The total resistance was said to be due to an accumulation of roughness elements that exhibit primary profile drag characteristics. Thus in order to determine the drag of a dense population of vegetation, the drag coefficients for individual elements were summed.

(Järvelä, 2002) showed that Equation 2‑5 underestimated the value of the drag coefficient. Moreover, even after modifying

dand

axby using averages,

CD,the drag coefficient was still underestimated. It has been proposed that this is due to the complex three-dimensional nature of plants structures since there was nothing implemented to consider the random factors such as the positioning, orientation, geometry of vegetation. In contrast to Morris (1995), Järvelä (2004) concluded that it was not feasible to compute the total drag based on the addition of drag coefficients for each individual branch.

The density of stems in a given area were found to cause

CDto increase rapidly, whilst an increasing Reynolds number had the adverse effect, although this was at a much lower rate (Hasimoto, et al., 2009).

The spacing pattern also had a significant influence on drag. When several configurations were trialled, their influence on

CDled to the implementation of a parameter to represent the variation in drag due to different staggering patterns. The effect of drag on a stem placed in a triangular or square staggering patterns is evident from the difference in the pattern staggering parameter

ξ,where

ξwas equated to 1.0 and 0.8 respectively.

From this the drag coefficient was quantified using a common fluid and vegetation properties as shown by Equation 2‑6 for a Froude number

(Fr)greater than 1, where

Rdis the stem Reynolds number and

λis concentration of stems per area (Hasimoto, et al., 2009).

CD={1.8ξRd-3501+0.45ln1+100λ{0.8+0.2Fr-0.15Fr2}

Equation 2.6. Drag coefficient derivation considering stem Reynolds number, Froude number and stem staggering pattern. (Hasimoto, et al., 2009)

*Plant Arrays*

Nepf (1999) extended the research and modelling of the drag induced by individual cylinders by defining the effect of a stem population density on the drag coefficient. Physical modelling was used to establish a relationship turbulence intensity and a drag, resulting in the implications of vegetation on drag, turbulence intensity and turbulent diffusion to be derived. Predictions of the drag coefficient were initiated by considering the interaction of a pair of cylinders. It was suggested that the drag coefficient of the trailing cylinder would be based on the upstream velocity and thus the effect would increase as both lateral and longitudinal spacing between cylinders’ decreases due to a reduction in ‘sheltering’ effects.

The effects of stem density and diameter on drag were found to have a strong influence on the resistance coefficient, in turn impacting the drag coefficient. Birkhead (2003) concluded that the resistance increased significantly when stems were spaced closer together, as well as adding that depth contributed to this variation. This contribution stems from bed shear, causing greater resistance at shallower depths and hence becoming minimal and relatively constant as depth increases. The flow depth at which bed shear resistance became relatively constant increased as spacing between stems increased. Therefore, when exploring the effects of drag from predominantly vegetation, it is more reliable to use denser spacing patterns as this weakens the influence of bed shear (Birkhead, et al., 2003).

By exploring the staggering pattern of vegetation using a square grid and varying the lateral and longitudinal dimensions, Ishikawa (2000) found that the distribution of vegetation greatly influenced the vegetation drag coefficient.

Further research conducted by Järvelä (2013) summerised the drag coefficient due to an array of stems as the bulk drag coefficient. It was found that a constant bulk drag coefficient could be achieved in a cylinder array when the distance to the leading edge was greater than

20ax, where

axstands for the longitudinal spacing between elements, and below this distance there would be considerable variation due to inconsitencies in the approach velocity

(uc).

The staggering pattern influenced the density per area of the vegetation and thus the effects of plant drag alone were increasing difficult to define at lower densities due to the contribution of bed shear to the total drag (Righetti, 2008).

## 2.4. Degree of Submergence

The degree of submergence significantly effects the hydrodynamics. It impacts not only on sediment transport and deposition, but also the deposit of nutrients and pollutant transport. All this in turn impacts on the evolution of the floodplain.

Research by Fathi-Maghadam (1997) indicated that for rigid, non-submerged vegetation, the density of vegetation would be the governing parameter regardless of the species, foliage shape and distribution.

Non-submerged vegetation consumes significant amounts of energy and thus is often found to be a region of high roughness, thus resulting in a high manning’s number, therefore increasing the resisting capacity of the floodplain (Barnes, 1967).

For submerged vegetation the nature of the flows, become more complex, as there are transition zones need to be considered. In this instance there are usually two zones, in the lower zone, submerged vegetation will reduce the velocity of the flow, whilst the upper zone will have a greater velocity since there are no obstructions therefore less energy losses. When vegetation is submerged, there will be significantly more bending in vegetation due to forces being exerted from the volume of fluid (Järvelä & Jochen, 2013).

** **

## 2.5. Density

Studies concerning the impact of mixed density vegetation on drag is generally relatively limited. Experimental research using real vegetation tends to consider the effects of momentum absorbing area on drag and the influence of increasing in the plants surface area in comparison to the effects of a stripped stem or cylinder.

Focus when using the simplified cylinder analogy has been on using spacing and spacing patterns as variables, where the cylinder diameters remain constant. Given that there are already considerable concerns with the reliability of using this cylinder analogy due to the simplification of the geometric properties of the plant, the use of a singular diameter only adds to this discrepancy.

Lindner (1982) touched on density by concluding that the resistance for an array of cylinders of identical volume was greater when the array was composed of cylinders of a smaller diameter since it led to a greater quantity of obstacles.

For stems of a constant diameter, the general trends show that at low stem density concentrations,

CDis small and varies significantly as the influence of bed shear on drag is greater. The drag coefficient increases more or less logarithmically with stem concentration and therefore at large densities

CDtends towards a constant value (Hasimoto, et al., 2009).

Because natural floodplains contain a range of species of flora, from grasses to bushes to trees, it is understandable that difficulties will arise when defining the momentum absorbing area. As simplifications have already been made by modelling vegetation as cylinders, and the reference area is easier to define, varying the diameters of the cylinders should be considered in order to provide a slightly more accurate representation of vegetation occurring in nature.

## 2.6. Flow Conditions

Increase in drag caused by vegetation can lead to an increase in flow depth and resistance, thus slowing the flow and increasing the residence time on the flood plain.

Conventional methods to indicate resistance include the use of Manning’s number, the Chézy coefficient and Darcy-Weisbach equation which been modified numerous times in order to specifically account for the roughness of vegetation. Due to variation in velocity caused by vegetation, the velocity is often substituted for the mean velocity. The speed of flow is it still greatly dependant on flow depth, which in turn leads to requiring depth-dependant resistant coefficients to be also be included when considering drag (Birkhead, et al., 2003).

When accounting for the roughness of vegetation, the flow is an important factor to consider, since the flow through vegetation it is often transitional between laminar and turbulent, meaning that it is not possible to use manning’s number since this accounts for turbulent flow only (Kadlec, 1990).

When considering a vegetated channel, Kadlec (1990) stated that the discharge per unit width

(q)(shown in Equation 2‑7) was influenced by flow depth

(y), bed slope

(S)and empirical factors dependant on vegetation and flow conditions.

q=aSbyc

Equation 2.7. Flow per unit width. (Kadlec, 1990)

Vegetation was described by

cwhich accounted for the effects caused by the density and bed topography, whilst the flow regime was defined by

bwhich was equal to 1.0 when the stem Reynolds number exhibited laminar flow and 0.5 for turbulent flow (Kadlec, 1990).

For naturally occurring vegetation (i.e. including foliage), high values of Reynolds number caused variation in the momentum absorbing area and thus it was shown to influence the drag coefficient (Birkhead, et al., 2003).

Fathi-Maghadam (1997) found that most channel flows involving vegetation were subcritical. The experimental research conducted, assumed a range of ‘practical cases’ for which the flow through dense non-submerged vegetation was in the fully turbulent zone and therefore ignored the Reynolds number.

Ishikawa (2000) proved that isolated stems in subcritical open channel flow were also independent of the stem Reynolds number and furthered it by estimating the drag coefficient to be approximately 0.9. For subcritical flow conditions

CDremained relatively constant for a given density of stems and it was only as the density became significantly large that

CDand Froude’s number

(Fr)started to decline.

The opposite was shown for supercritical flow, where

CDwas shown to decrease as

Frincreased (Ishikawa, et al., 2000), although Hashimoto (2009) stated that in the presence of vegetation supercritical flows were uncommon and that they only really occurred when vegetated sloped had high inclination. Data produced by Thompson (2003) seemed to comply by showing that for an isolated cylindrical stem with a stem Reynolds number between 14,152 to 26,054,

Frvaried between 1.285 to 1.45, whilst

CDwas found to vary between 0.63 and 1.06.

The derivation of the drag coefficient from known values of drag force showed that for a majority of cases,

CDhad a tendency to decrease exponentially with the square of the flow velocity (Hasimoto, et al., 2009). Other impacts such as streamlining altered the friction and thus the drag and Järvelä (2013) found that as the velocity increased there was less friction.

The inclination of the slope had minimal effect on resistance and manning’s number was only slightly smaller for increased inclinations, this is potentially due to an increase in velocity and therefore drag (Birkhead, et al., 2003).

In natural floodplains, the density of naturally occurring vegetation is relatively small, however this is dependent on a range of factors such as topography and climate (Hasimoto, et al., 2009). Therefore, when using manning’s number, calculated from previous flood events the characteristics of the floodplain and event itself should be defined to enable it to be applied to other scenarios Fathi- Maghadam (2000).

# 3. Methodology

## 3.1. Background

To further examine the implications of vegetation on flows, experimental research will be based on the previous findings of Nepf (1999) in ‘Drag, Turbulence and Diffusion in flow through emergent vegetation’.

The model relates the effect of stem population density on drag coefficient by relating it to diffusion for an applicable range of flow conditions and spatially varied vegetation. By considering the influence on drag and the interaction between pairs of cylinders, Nepf (1999) concluded that the upstream cylinder suppresses the drag coefficient of the trailing cylinder. It was also identified that as the longitudinal and lateral spacing between cylinders decreased the drag on the trailing cylinder would also decrease. This indicates that the trailing cylinder experiences a lower impact velocity due to reduction in wake. Thus the drag coefficient will be dependent on the upstream velocity and lateral and longitudinal spacing between cylinders.

To reach these conclusions, a dimensionless vegetation population density

(ad)was defined as a fractional volume of the flow field occupied by plants:

ad=nd

Equation 3.1. Vegetation population density. Nepf (1999)

Where

ais the vegetation density per m,

nis the number of cylinders per unit and

dis the cylinder diameter.

Since the drag exerted on a group of cylinders varies within a flow, the bulk drag coefficient

(C̅D)was used to define the drag coefficient of a specified population density of vegetation.

Nepf (1999) assumed that energy losses due to stem drag from the mean flow resulted in the production of turbulent kinetic energy within stem wakes. Thus it was possible to scale and equate the dissipation rate in terms of stem geometry and turbulent kinetic energy to the production of turbulent kinetic energy within stem wakes (shown in Equation 3‑2).

kU= α1[C̅Dad]13

Equation 3.2. Relationship between dissipation rate from stem geometry and turbulent kinetic energy to production of turbulent kinetic energy. Nepf (1999).

where

kis the turbulent kinetic energy per unit mass,

Uis the mean velocity and

α1is a

O(1)-scale coefficient. From this, it was concluded that turbulence intensity increased with the bulk drag coefficient and with population density.

Furthermore, the net diffusion within a cylindrical array was shown to be the result of contributions from both mechanical and turbulent diffusion within a zone.

The turbulence intensity was related to the bulk drag coefficient through turbulent diffusion

(Dt)as shown in Equation 3‑3.

Dt=α2kd

Equation 3.3. Turbulent Diffusion. Nepf (1999)

Nepf (1999) established the coefficients using experimental methods of flow and dispersion through a cylindrical array. It was established that

α1=0.9for

400<Red<900where

Reddenotes the stem Reynolds number. The value of

α2was obtained as

0.9by fitting experimental results where mechanical diffusion was observed to be negligible. The coefficient

βwas taken as a

O(1)-Scale factor therefore tending towards

1, except in the instance of laminar wake in which case it was noticed to marginally underestimate

β.

Mechanical diffusion

(Dm)was defined by considering the dispersion of fluid particles due to the variability in flow paths instigated by vegetation (Nepf, 1999). The diffusion of particles generally illustrated a Fickian diffusion process, a way of describing how concentration under steady state reduces from high initial concentrations to expanses of low concentration relative to the concentration gradient with time or space (Crank, 1975).

By expressing the diffusion of particles in respect to space, the variance of particle distribution over time was accounted for and enabled the derivation of mechanical diffusion as shown in Equation 3‑4.

Dm=β22adUd

Equation 3.4. Mechanical Diffusion. Nepf (1999)

Since mechanical and turbulent diffusivity are independent of one another their contribution to the total diffusivity is additive. Therefore, the total horizontal diffusion was defined as shown in Equation 3‑5.

DUd=α[CD̅ad]13+β22ad

Equation 3.5. Total horizontal diffusion. Nepf (1999).

where

Dis the net diffusivity and

α=α1α2.Value for

awere tailored to laboratory and field experiments for the following ranges of stem Reynolds numbers;

400<Red<2000and

300<Red<600for laboratory and field scenarios respectively, resulting in

αbeing equal to 0.8 for these ranges.

Nepf (1999) based observations on

CD, on the interaction of pairs of cylinders, thus an individual local drag coefficient for each cylinder was found based on the proximity of its nearest neighbouring upstream cylinder. The total drag

(FT)_{,} was estimated as the sum of individual cylinder drags and used to estimate the drag coefficient using Equation 2‑1 and results implied that C_{D} declined with an increasing population density of vegetation.

## 3.2. Experimental Apparatus

*Flume*

The experiments were conducted in a hydraulic flume measuring

7.5min length,

0.45min height and

0.3min width. A rectangular adjustable sluice gate is located

6.85mfrom the inflow source and the bed slope remained at a

0%inclination throughout.

The flow in the flume is re-circulating with a rate limited to the range produced by the hydraulic flume with maximum and minimum discharges confined to

0.0015m3/sto

0.03m3/s.respectively. Using the maximum and minimum flow depths of

0.45mand

0.26m, the maximum and minimum cross sectional areas are

0.135m2and

0.078m2can be quantified and by using continuity it is evident that the range of velocities for a horizontal bed slope are confined to

0.00577m/sto

0.2̇m/s. Hence using Equation 2‑2 Reynold number can be identified and will thus also be limited to the range

42750-1665000.

*Vegetation Zone*

The leading edge of the vegetation board was positioned

4.5mdownstream from the flow inlet occupying the entire width of the flume

(0.3m)over a length of

0.55mwith a thickness of

25mm.

Vegetation stems were simulated using cylindrical wooden dowels

450mmin height and

6mmand/or

8mmin diameter which were arranged randomly on the grid. In plan the vegetation grid displays a triangular spacing pattern (see Figure 3.1) with a capacity of

173stems (see Figure 3.2).

Figure 3.1. Triangular staggering pattern on the vegetation grid. Units (mm).

The spacing is denoted by x in the longitudinal direction whilst the width in the lateral direction is denoted by y.

*Apparatus*

A vertical tracer injector with

5outlets spaced vertically in increments of

50mmfrom the base of the flume was used in order to achieve a planar injection of the tracer in the z-axis. The lowest outlet was placed on the base of the flume, with the overall position being

4.2mfrom the inlet with a capacity of

25mlof diluted methylene blue tracer.

The sample collector was placed centrally 5.15m from the inlet with the locations of syringes at 30mm, 90mm, 150mm, 210mm and 270mm across the width of the flume (y-axis) at a height of 0.13m from the base in order to be at the mid flow depth.

Figure 3.2. Elevation of the experimental apparatus. Dimension labels: cm.

Figure 3.3. Plan of the experimental apparatus. Dimension Labels: mm.

*Tracer & Spectrophotometer*

Loefflers Methylene Blue is the tracer used throughout this series of experiments since it is unreactive when combined with water and therefore suitable for use in the flume. In solution it appears a dark blue making it clearly visible when injecting and diffusing around elements in the flume. Furthermore, its light absorption is estimated at approximately 650nm when in an aqueous solution and this means that it can be identified in the spectrophotometer (Cenensand & Schoonheydt, 1988).

The spectrophotometer uses the intensity of light emitted and measure the amount reflected back to the receive in order to measure the amount of light absorption by a substance at a given range of wavelengths (Chemistry Libre Texts, 2015). The light intensity absorbed by each sample can be compared in order to determine a relative absorbance. In order to quantify absorbance in terms of concentration, the Beer-Lambert law is used, which states that there is a linear relationship between the two, such that concentration

(C)can be defined as:

C=A∈l

Equation 3.6. The linear relationship between Concentration and Absorbance. (Chemistry Libre Texts, 2015).

where

Ais the absorbance,

∈is the absorption coefficient

Lmol-1cm-1and

lis the path length

(cm). In this instance the dimensions are generally in centimetres since this is the standard size of a cuvette, used to hold the samples in the spectrophotometer.

** **

## 3.3. Experiment Basis

By only considering the interaction of cylindrical pairs, rather than a group, in order to determine the drag coefficient (Nepf, 1999), the representation of vegetation between the experiment and reality is somewhat inaccurate.

Although the effect of drag on each cylinder may not change dramatically when considering the interaction of multiple surrounding cylinders, the accumulation of differences when determining the total drag may have a substantial effect on the outcome when estimating

CD_{.}

Since a model relating the total diffusivity and drag already exists, this research aims to determine if it is possible to exploit it in order to obtain the drag coefficient for a given population density of vegetation. By relating the change in concentration over a sample of vegetation to a diffusion coefficient, this research intends to determine if it is possible to measure the drag coefficient for a population density of vegetation by relating it to the total diffusivity.

The bulk drag coefficient can be re-arrange and written as shown by Equation 3.7.

CD̅= (1α(DUd-β22ad))3ad

Equation 3.7. Rearrangement of Nepfs model for the bulk drag coefficient.

The equation above (Equation 3.7) indicates that in order to determine the bulk drag coefficient, other parameters need to be defined beforehand.

Experiments were conducted using several variables in order to establish the diffusion coefficient and hence if it is possible to use this model to quantify the drag caused by a population of vegetation and the influence of using a combination of densities and spacing patterns.

## 3.4. Experimental Procedure

To ensure a suitable basis, the flume was filled with fresh water before commencing the experiments. In addition, before each test was run, a control sample was taken as a reference concentration in order to compare the samples against, since the water in the flume was not dumped and re-filled after each experiment.

The first series of tests were run at

0%population density providing another control measure and allowing the calculation of the diffusion coefficient for an environment of uniform flow without the presence of obstructions.

For each series the methylene blue dye was diluted with water to a

1:2ratio and this concentration was kept consistent throughout all experiments to ensure consistency in absorption readings. The injector had a capacity of

25ml which allowed for it to flow continuously, reaching the sample collection location well before the volume of tracer had diminished.

*Variables*

In order to determine the characteristics of drag, different vegetal and flow properties have been changed in order to examine the difference this causes.

*Velocity/ Reynolds number*

Since the flow was required to be greater than

0.25min order for the tracer injector to be completely submerged, it was decided to keep a constant depth of

0.26mthroughout and hence the cross sectional area also remains the same, measuring

0.078m2.

Two different flow rates were trailed for each experimental layout and using the concept of continuity the velocity can be derived by dividing the flow rate by the flow depth (see Equation 3.7 in order to determine the velocity by continuity).

Provided that the flow depth domains constant across the vegetation zone, this will also yield the mean velocity

(U)required for the quantification of the bulk drag coefficient in Equation 3.8.

Q=VA

Equation 3.8. Continuity equation

Flow rates of

0.0035m3/sand

0.0045/swere used, equivalent to velocities of

0.0449m/sand

0.0577m/srespectively.

According to RH. Kadlec (2008) the stem Reynolds number

Resfor the respective flows can be defined using

Res=dρuμ

Equation 3.9. Stem Reynolds number.

where

dis the diameter of the cylinders/stems

(m), ρis the density of fluid

(kgm3),

uis the superficial flow

msand

μis the water viscosity

(kgm. s).

For the following experiments a range for which the stem Reynolds number will lie between has been quantified. The lowest stem Reynolds number will occur at the lowest diameter of cylinder vegetation where

d=6mmwhen the velocity is at its lowest and the largest will occur at

d=8mmwhen velocity is highest, when vegetation diameters are mixed the mean diameter will fall within this range of stem Reynolds numbers for both velocities.

Taking the density of water as

1000 kg/m3and the viscosity as

0.001002 kg/m. sand assuming a superficial flow of

0.0449m/sand a

6mmdiameter, the lower bound stem Reynolds number for these experiments will be

264. Whilst for the upper bound stem Reynolds number the same fluid properties will be used, with the exception of the superficial flow which will be taken as

0.0577m/sand the diameter of the cylinders will increase to

8mmgiving a stem Reynolds number of

461.

Since the stem Reynolds number is greater than 200 (boundary condition for laminar flow) for both the lower and upper flow in the vegetation zone, in particular around the stems in all experiments is most probably going to be in the transition region (between laminar and turbulent) or completely turbulent (Kadlec & Wallace, 2008) (Nepf, 1999).

*Population Density*

The second variable throughout the experiments was the density of the vegetation. Experiments were conducted at 0%, 25% and 50% population density since vegetation on flood plain is often relatively sparse and doesn’t often consist of excessively high coverage For each population density, the percentage of both the 6mm and 8mm diameter cylinders was varied, along with their positioning on the staggering grid where each layout was tested at both velocities.

Since the board contains 173 vegetation positions, values were rounded in give whole values of cylinders. The following number of cylinders will be used for each density:

Table 3‑1. Quantity of cylinders to be used for each density population.

Density | Density x Positions on board | Number of cylinders |

10% | 17.3 | 17 |

15% | 25.95 | 26 |

25% | 43.25 | 43 |

35% | 60.55 | 61 |

50% | 86.5 | 87 |

The first purpose of these experiments was to consider whether a relationship can be identified between the diffusivity model on a set of cylinders of the same diameter but varying population density from 25% to 50% and their layout. At 25% density the layouts tested can be seen in Figure 3.3 and 50% density can be seen in Figure 3.4, both were tested with vegetation of uniform diameters in order to analyse and compare the effects of an increase in diameter of cylinders on the diffusion and drag.

Figure 3.4. 25% Vegetation Density Layouts (Left: Layout 1 & Right: Layout 2).

Figure 3.5. 50% Vegetation Density Layouts (Left: Layout 3 & Right: Layout 4).

After the two densities and diameters had been tested, the total density in terms of positions occupied remained the same (25% of 50%). The composition however was varied and rather than having a singular diameter configuration, they were mixed arrangements with both 6mm and 8mm diameter cylinders where the sum of the positions occupied totalled either 25% or 50% of the total density of the grid.

For a total density of 25% vegetation; mixed diameter experiments were conducted for 10% total occupation by one diameter of cylinders and the remaining 15% occupied by the other as can be seen in Figure 3.5. The locations indicated by the blue marker were trialled first with 6mm diameter cylinders, whilst the remaining orange markers were occupied with 8mm diameter cylinder and then the inverse was tested for both layouts 5 and 6.

Figure 3.6 25% Total density represented by positions indicated by orange and blue circles. Left: Layout 5 (10% Blue & 15% Orange) & Right: Layout 6 (10% Blue & 15% Orange).

Three different combinations of mixed density were trialled for the 50% total density experiments. The division of the total density went as follows; 15% occupation by 6mm diameter cylinders with 35% by the 8mm diameter cylinders (as seen in Figure 3.6) followed by the inverse to this (15% occupied by 8mm diameter cylinders and 35% by 6mm)

and the third category was 25% occupation of each diameter.

In order to best replicate field conditions, the configurations for each of the layouts above defined randomly from a random number generating grid representation of the vegetation board.

Figure 3.7. 50% Total density represented by positions indicated by orange and blue circles. Left: Layout 7 (15% Orange & 35% Blue) Centre: Layout 8 (15% Orange & 35% Blue) Right: Layout 9 (25% Orange 25% Blue).

## 3.5. Anticipated Results

Based on previous literature, it is known that drag is proportional to the square of the velocity (Hasimoto, et al., 2009) (NASA, 2017). Therefore, when the velocity is increased it would be expected that the drag exerted on the cylindrical stems would increase.

However, this variable is the least predictable since it is easily influenced by factors such as the shape and size of the obstruction, inclination and the mass and viscosity of the fluid (NASA, 2017).

The impact of cylinders as a representation for vegetation is likely to influence the drag in various ways as they will cause obstructions within the flow thus increasing the drag. The bulk drag coefficient is likely to increase with population density, since this increased the frequency of interaction between the tracer and cylinders. Furthermore, as the diameter of the cylinders are increased it is predicted that this will increase the drag as it increases the momentum absorbing area.

# 4. Results

## 4.1. Raw Data: Concentrations

In order to determine the drag coefficient, the diffusion coefficient was required and hence the concentrations were measured in each experiment in order these coefficients to be quantified.

In the graphs presented below, solid lines represent a velocity of 0.0577m/s whilst the dashed lines denote a velocity of 0.0449m/s and colours indicate the vegetation layout and density which can be seen in section 3.4.

Figure 4.1. A graph displaying the concentration distribution at 0% vegetation density (please note that the concentration scale differs from that used in the rest of the figures to follow).

At 0% vegetation density the peak concentration occurs at the lower of the two velocities which is as expected since theoretically there should be less turbulence, therefore less mixing of the tracer at lower velocities. The Gaussian curve shows little variance indicating that there was little distribution of concentration along the width of the flume (in the Y-direction).

In comparison, the concentration distribution at 25% vegetation density using 6mm diameter cylinders shows a peak concentration of roughly half of that at 0% vegetation. The decrease in peak concentration and increase in variance indicates that the interference of stems caused a greater distribution of the tracer.

Figure 4.2 shows that for all case but one, the faster velocities has lower peak concentrations. It is probable that the value at 50% density with a velocity of 0.449m/s for layout 4 is an anomaly since the concentration is significantly higher than the others and at that density it would be expected that the peak concentration be lower than the values at 25% vegetation density, but further experiments may be required to justify this.

Figure 4.2. Concentration Distribution of 6mm Diameter Cylinders at 25% and 50% Vegetation Density.

Figure 4.3. Concentration Distribution of 8mm Diameter Cylinders at 25% and 50% Vegetation Density.

Alike the results shown in Figure 4.2 when the diameter of the cylinders are augmented and kept constant using only 8mm diameter cylinders the same velocity trend is apparent. However, in contrast the greater the area occupied by cylinders, the greater the peak concentration. It is possible that this effect could be due to the effects of sheltering and the alteration of wake flow conditions caused by the upstream cylinders, with a small chance of it being due to the configuration of the vegetation, although the latter is less likely given the results of Figure 4.2.

Figure 4.4. Concentration Distribution at 25% Total Vegetation Density.

For a 25% mixed cylindrical density the results demonstrate the expected trend. When the density comprised of a greater percentage of 8mm cylinders, the peak concentration was generally lower and the variance was greater, which suggests that there was a greater dispersal of tracer. Similarly, in comparison to the concentration distribution using constant diameters of cylinders throughout the same velocity trend seemed to occur, where the lower velocities produced a higher peak concentration.

At 50% total density the same trends mentioned above are exhibited. For the 25% even split between the two diameter cylinders, it is probable that the result at a velocity of 0.0577m/s using layout 9 (denoted in Figure 4.4 by a solid blue line) is an anomaly since it does not follow the general trend as shown by the data thus far, not only is it greater than that of its identical layout lower velocity result, but the peak concentration is significantly greater than that present in all other data for this density. Furthermore, the result for the same layout at a velocity of 0.0499m/s also exhibits some unexpected behaviours since theoretically it would have been expected to lie between the 15% 6mm and 35% 8mm density and the 35% 6mm and 15% 8mm density layout, whereas instead it has the lowest peak concentration.

In terms of density, results are relatively varied. For a velocity of 0.0499m/s, the peak concentrations are in relative proximity to one another when the proportions of stems of varied diameter are not equal. When the vegetation density is dominated by 8mm diameter cylinders, as expected the peak concentration of tracer is lower.

When the velocity increases, the difference in peak concentration increases considerably, with much greater variety between results. In general, it would be expected that the same trends would occur, which would indicate that the Layout 8 at densities of 35% by 6mm and 15% by 8mm may also be an anomaly.

Figure 4.5. Concentration Distribution at 50% Total Vegetation Density.

## 4.2. Diffusion Coefficient

Figure 4.6. A diagram representing the idealised diffusion process of tracer from injection to sample collection.

The diffusion of the tracer can be quantified using the concentration data collected at the sample collection point. The intensity of concentration will be high at the point of injection, with theoretically 100% concentration since the tracer has yet to mix. As the tracer moves along the flume with the fluid, it will go from a region of high concentration and spread due to effects such as friction and turbulence. The outcome is a concentration gradient along the length of the flume, and in theory when the tracer reaches the collection point its concentration should be at its lowest.

Since the tracer is injected in the centre of the flume, an ideal dispersion should produce a Gaussian distribution. However, in reality obstructions, flows and responses in the field are far from ideal and the likelihood of achieving this in practise is minimal. Similarly, in laboratory experiments, there are many additional factors that may result in an uneven distribution, of which these will be discussed in more detail in the following section.

The diffusion coefficient can be described by the change in concentration relative to the distance moved by the substance and therefore shows the spread of a plume of tracer along the width of the flume. In order to quantify diffusion, the following method of integrating moments is used in order to give a practical range of values.

The amount of tracer can be quantified by Equation 4.1, where the resulting value of the integral (M) remains constant in each individual experiment.

M=∫-∞∞c dy

Equation 4.1. Amount of tracer – M Diffusion Integral (Thayer School of Engineering, 2017)

The second term shown in Equation 4.2 describes the effects of distance with respect to concentration, where

ydenotes the location of the sample collection points across the width of the flume.

M1=∫-∞∞yc dy

Equation 4.2. Concentration relative to distance – M_{1} Diffusion Integral (Thayer School of Engineering, 2017)

The mean

(x̅)refers to the position of the plume and can be taken as the product of Equation 4.1 and Equation 4.2 (shown in Equation 4.3).

x̅=M∫-∞∞yc dy

Equation 4.3. Position of the mean of the plume of tracer –

x̅Diffusion Integral. (Thayer School of Engineering, 2017)

Equation 4.4 illustrates the spread of the plume across the width of the flume (in the y-direction).

M2=∫-∞∞y2c dy

Equation 4.4. Spread of the plume relative to concentration – M_{2} Diffusion Integral (Thayer School of Engineering, 2017)

The mean can be related to the integrals in Equation 4.1 and Equation 4.4 in order to determine the variance

(σ2)which describes the length of the plume width as a normalized distribution as shown by Equation 4.5.

σ2= M2M- x̅2

Equation 4.5. Variance of tracer with respect to distance, width and mean. (Thayer School of Engineering, 2017)

Finally, the diffusion coefficient

Dcan be expressed as being approximately proportional to the variance with respect to the time expelled as shown by Equation 4.6.

D≈σ2Δt

Equation 4.6. The Diffusion coefficient in terms of variance and time. (Thayer School of Engineering, 2017)

For each experiment, the concentration distribution was used in order to calculate a value for the diffusion coefficient.

Figure 4.7. The Diffusion coefficients at multiple 25% total vegetation density combinations.

When analysing the diffusion coefficient at 25% total density, the results are relatively grouped. This is in contrast to the concentration data, which indicated the general trend of increasing velocities having lower peak concentrations and slightly greater variance, therefore suggesting a greater spread of tracer along the width of the flume. The diffusion coefficients also seem somewhat scattered making it is difficult to draw any conclusions on trends for this group of data, although they suggest that the peak was lower at faster velocities and that the tracer did not spread laterally as much as expected. The majority of values for the diffusion coefficient at 25% vegetation density tend to range between 0.8-1.2, with only a few going beyond this range. Considering the values for only a velocity of 0.0577m/s, it can be seen that there is a slight increase in the diffusion coefficient as the stem size is increased.

Figure 4.8. The Diffusion coefficients at multiple 50% total density combinations.

The 50% density demonstrates a similar pattern of increasing diffusion as the area occupied by cylinders’ increases. And as in Figure 4.7, this is more evident for the faster of the two velocities. However, the indication that the same trend may follow is moderately clearer at a velocity 0.0499m/s at this density.

Overall, the diffusion coefficients are more varied, with the majority ranging between 0.7-1.4 on average. In the majority of cases presented the diffusion is greater at a higher velocity. This would broadly be as expected since there is greater surface area causing interference within the flow.

## 4.3. The Bulk Drag Coefficient

The bulk drag coefficient has been quantified using the model derived by Nepf (1999) as stated in section 3.3. The vegetation population density is quantified using Equation 3.1, whilst the bulk drag coefficient using the rearrange total horizontal diffusion equation shown by Equation 3.7. At 25% vegetation density, the bulk drag coefficient ranged between 0.25-0.75, with only a couple of values beyond this range. It is likely these are anomalies given their magnitude in comparison to the stated range.

Figure 4.9 .The Bulk Drag Coefficients at 25% Total Vegetation Density.

At 50% vegetation density, the bulk drag coefficient had a much narrower range varying between 0-0.35 and in comparison, to 25% density the values were significantly smaller.

Figure 4.10. The Bulk Drag Coefficients at 50% Total Vegetation Density.

The bulk drag coefficients for 25% and 50% vegetation density (shown in Figure 4.9 and Figure 4.10) are too erratic to give sufficient indication as to whether there is a trend between the velocity and the bulk drag coefficient. In order to improve the understanding, further research into the relationship between velocity and drag should be conducted, using a greater range of velocities in order to determine if it is possible to establish a relationship between the two.

The bulk drag coefficients for both densities have been collated and plotted together against population density (

ad)shown in Figure 4.11Figure 4.11, in order to provide a point of comparison of all the data. The orange markers indicate 25% vegetation density and blue markers 50% (shown in Figure 4.11). The relationship between the bulk drag coefficient and population density exhibit a similar general trend as that suggested by Nepf, whereby the denser the population density of vegetation the greater the bulk drag coefficient.

Figure 4.11. Bulk Drag Coefficients relative to the vegetation population density.

# 5. Limitations

*Cylindrical Analogy*

From the literature mentioned in Section 2, a clear problem with the modelling is that there are significant limitations to using the concept of cylindrical analogy when modelling vegetation. The vast range of vegetation that grows on floodplains and the numerous properties exhibited by each specimen has led to major simplifications resulting in the idealization of shapes (Järvelä & Jochen, 2013).

The differences in geometric characteristics including differences in roughness, foliage and non-constant stem morphology are considerable features that will limit the reliability of these results when comparing them to natural vegetation (Hasimoto, et al., 2009). An indication of how much simplifications can affect the outcome can be seen in a study conducted by Thompson (2003) where it was found that the drag coefficient of a rectangle was 0.75 greater than that of a cylinder, hence demonstrating how perhaps assuming a constant diameter could differ from the drag of natural stems.

As a result of using cylinders, the applicability of this research is more ambiguous and it is likely to only be indicative of the response of vegetation in the field.

*Flexibility*

Alongside the inadequate modelling of shape comes the consideration of flexibility. The cylinders used in the majority of research tend to be rigid cylinders which again are only representative of a very small proportion of vegetation of equivalent flexibility. For a large proportion of vegetation of small diameters, but also other species, it is likely that stems will differ in flexibility. This will differ further and influence the drag dependant on the degree of submergence. The loss of momentum when using rigid cylinders is likely to be smaller than that of natural vegetation since the momentum absorbing area is smaller as there is almost no deflection (Grant & Nickling, 1998).

In the field, particularly during flood events, the damage and bending experienced by a plant is significantly greater than that modelled by the conditions in the experiment and thus when applying research to field situations, consideration should be taken to account for changes in loss, deflection or movement of part or the entirety of the plant as these will influence the momentum absorbing area and therefore the drag (Järvelä, 2004).

*Flow Conditions*

The specifications to which flumes are manufactured provide limitations and boundaries for experiments since they govern the channel dimensions, flow depth and flow rate. In addition, the temperature and fluid used will affect the dispersion of tracer as well as any potential reactions, therefore, it is usually assumed that the diffusion coefficient is independent of pressure and temperature, which is not necessarily the case (Comsol, 2017).

In addition, the method in which the flume is calibrated will also influence the results when comparing them to other research conducted. For the purpose of this research, since all tests have taken place in the same flume and using the same equipment any errors should be relative to each in terms of calibration. There is also likely to be a certain error percentage associated with the equipment, since the injector and collector were made for the purpose of this research, the percentage error is unknown – positioning of the equipment

The small leak in the base of the flume is an instrumental error (Civil Engineering Dictionary, 2014)

Finally, human errors such as measuring the flow depth and flow rates are likely to have some errors associated with. Measurements for height were taken to the nearest millimetre and hence this was used to calculate the velocity and modify the depth of the sluice gate and therefore any errors associated with this measurement were carried forward without factoring.

*Spacing*

Regular patterned stem spacing’s and the use of grid even with random positioning is far from ideal. Spacing and plant geometry influence velocity and streamlining and hence the wake flow conditions. This is far more crucial for the downstream cylinders since it will alter what the experience based on the layout and geometry of the upstream cylinders (Järvelä & Jochen, 2013).

Plant characteristics are complex to accurately model since even the same species will differ between sites with different growth patterns and genetic differences, meaning that even modelling for a particular type of plant can have limited applicability elsewhere (Järvelä & Jochen, 2013).

*Channel Properties*

One of the primary considerations not accounted for in the flume experiment is the roughness of the channel bed and the additional drag this induces (Birkhead, A. et al, 2003).

Bed shear is a contributing factor to drag in any situation, but more so as spacing between stems decreases as the effects of bed shear become more significant (Hasimoto, et al., 2009).

The smooth properties of the flume can differ significantly from a practical situation where the boundaries including the channel walls and bed can be a major contributor to the overall drag. The rougher the surface, the greater the contribution to drag and hence for locations with high presence of granular materials, the contribution to drag will be more dominant than for fine materials (Järvelä, 2004).

# 6. Discussion

Following the study conducted by Nepf (1999) using singular diameter cylinders, this research provides the basis for suggesting that there is likely to be a correlation between the increase in surface area occupied shown by the use of mixed and singular sized cylinders and diffusion and the bulk drag coefficient.

At lower densities, the results presented in section 4 are more ambiguous and it is likely that the influence of bed shear on the diffusion and drag was considerably more significant on the outcome. However, none the less it is still possible to see a general increase in diffusion as the density of the stems increases hence showing that wider stems cause more disruption to the flow regime and generate more diffusion.

Although the data at this density indicates a range in which the bulk drag coefficient is likely to range, there is no strong correlation between the increase in stem diameter and bulk drag coefficient, nor the velocity. Since at sparse densities the influence of shear from the channel bed become more important and contributes significantly to the drag, it is likely that the bulk drag coefficients do not display a clear trend at this density for this reason.

At greater densities, slightly more patterns emerged from the data, where increasing stem diameter resulted in a greater diffusion coefficient and lower bulk drag coefficient.

Since at increased density the effects of shear from the channel bed become less important in contrast to that caused by vegetation, the decrease in bulk drag coefficient would indicate that the upstream cylinders cause an alteration in the flow conditions reducing the drag experienced by the downstream cylinders.

The diffusion coefficient was generally greater at higher velocities implying that there was more turbulence and mixing and hence the tracer spread across a greater portion of the flume.

Additionally, the increase in stem diameter provoked an increase in diffusion, once again demonstrating that the interference of larger stems will increase the amount of diffusion within a system. The point of separation of the tracer on the upstream cylinders will define the flow properties as it continues through the population density of cylinders. It is likely that the upstream stems will shelter those downstream generating regions of lower pressure and diverting the flow path of the tracer, hence explaining why the bulk drag coefficient decreases, whereas the drag coefficient on each cylindrical stem is likely to vary considerably depending on its positioning on the vegetation grid.

The results when compared at each individual density were relatively varied and only suggest a trend. But when collating the data, as shown in REF _Ref479973665 h * MERGEFORMAT Figure 4.11Figure 4.11, it is evident that there is a clear trend between the bulk drag coefficient and the vegetation population density. As the surface area occupied by stems on the grid increases, the bulk drag coefficient decreases due to the increase in sheltering.

In terms of flooding, greater densities of vegetation are likely to increase the dispersal of the water, thus spreading it across the floodplain and reducing the flow depth. This increases the time it takes for the flow to move through the array of vegetation. The increase in residence time will be beneficial since it will increase the time permitted for mitigation techniques and strategies to be implanted.

The bulk drag coefficient is likely differ in a field situation since the floodway has a much greater span and varying roughness. Therefore, in addition to the drag caused by vegetation the influence of bed shear will be more significant. Since the bulk drag coefficient decreases with increasing population density, it suggests that a sparser array should be used. However, in reality, the most beneficial solution would be to have a lower density along the flood fringe in close proximity to the body of water with increasing density as the distance from the body of water increases. By implementing this, when the channel is breached, the stems closest to the waterway will cause some disturbance and considerable drag which should help in reducing the speed and increase the time in which it takes the fluid to disperse. As the density increases, this effect will increase along with the sheltering effect. Therefore the bulk drag coefficient will decrease, but since the velocity will have been reduced the fluid is likely to disperse more reducing the volume and hopefully the resulting impact.

Although this research gives some indication as to what the reaction of natural vegetation may be in the field, there are many factors which limit its reliability for real scenarios and cases.

One of the most crucial considerations that limits the usefulness of the data recorded is that it only provides information about the diffusion and the effects of drag in one-dimension. Whereas, evidently a field scenario will always need to be considered as a three-dimensional problem. Hence further research will be needed to give a comprehensive overview of the likely implications of vegetation on flow regime in the x and z axes.

Another contributing factor that is likely to have an impact on the accuracy of the results, is reflectance of the tracer within the flume. Because the channel bed is not adequately modelled in comparison to a natural waterway in these experiments, the role of reflectance is likely to be greater in the experimental process due to the nature of the surfaces and flume. This will affect the distribution of the tracer and should be considered alongside the layouts, since some are likely to cause more reflectance than others (for example if there are dense regions of vegetation in the centre of the grid the tracer is likely to diffuse out towards the wall of the flume and hence there is likely to be greater reflectance because of the direction of the flow path of the tracer). Reflectance will cause an increase in the concentration closer to the centre of the flume and hence the tracer will appear to have spread less. The reflectance, may help to explain why some of the results do not follow the general trends and some of the variation between results since the layout were changed, therefore altering the amount of reflectance.

A final consideration is that the fluid in the flume was not dumped at the end of each experiment. Base measurements were taken at a location upstream of the injector at the start of each experiment as a point of reference. However, there is a small chance that the initial additional increase in concentration as the experiments continued may have caused some mild changes in the resulting concentration as it may have impacted the flow properties and fluid density as additional tracer was mixed into the water.

Despite the evidence in this research, alongside that of Nepfs (1999), suggesting that an increase in the population density of vegetation should decrease the bulk drag coefficient , the limitations to the experimental procedure means that for further use and applicability the data presented is only suggestive and further research should be conducted considering some of the limitations mentioned above.

# 7. Conclusion

Despite the limitations associated with the experimental procedure, it is still possible to reach a conclusion about some trends that can be modelled relating diffusion, bulk drag coefficient, velocity and population density.

At lower densities the diffusion and bulk drag coefficient due to vegetation is less evident due to the additional contribution of bed shear. Whereas at higher densities, the results show more of a trend, as it is likely that at 50% total density the effects of bed shear start becoming negligible.

For both densities, as the diameter of the cylinder’s increase, the diffusion coefficient also increases, indicating more dispersion of the tracer particles due to a greater area of stems. This in turn leads to more interference and therefore more disruption within the flow. At a greater density, the faster velocity also proved to diffuse more as turbulence is likely to have been greater, therefore increasing the amount of mixing and spread of tracer.

The bulk drag coefficient also decreases as the population density increases. As the percentage of cylinders of 8mm diameter is increased, the bulk drag coefficient was generally seen to decrease. This, indicates that the greater frontal surface area displayed by a vegetation density the smaller the bulk drag coefficient tends to be. As such the flow path of the tracer is more likely to be forced outwards, or away from the stems, thus sheltering the downstream cylinders and reducing the overall drag experienced by each cylindrical element within the array.

Since this research has been carried out with the assumption of uniform stem length and diameter, as well as constant rigidity, there is undoubtedly the need for consideration of these factors, or alternatively the implementation of the same study on real vegetation prior to using this knowledge in the field (Hasimoto, et al., 2009).

For future investigation and analysis of vegetation on drag and diffusion some of the following suggestions could be implemented in order to improve the validity and reliability of the results. The use of more modern and accurate equipment in order to inject the tracer, as well as collecting the samples would improve the results. It would also ensure that the devices are then repositioned to the nearest millimetre as using more precise measuring equipment will aid the accuracy of the results.

The use of real vegetation rather than cylinder models would also increase our understanding of the impact of vegetation as a whole, rather than just considering the influence of stems, and therefore increase the applicability of this field of research. Modern equipment such as a terrestrial laser scanner can be used in order to quantify the momentum absorbing area with a greater accuracy, therefore aiding researchers to define vegetation geometry (Antonarakis, et al., 2009). This advance in this technology alongside 3D printing and electromagnetic methods can allow for accurate representations of vegetation to be replicated and used for experimental purposes, therefore allowing a plant to be represented and analysed as a whole rather than individual elements (Sinoquet, et al., 1997).

Finally, an in depth study where the range of velocities analysed is increased would improve the understanding of diffusion and drag behaviours at various vegetation densities. This would enable a more detailed proposal of the potential relationship between the flow properties and vegetation drag and diffusion and further identify if there are any trends that have been missed by the smaller range used in this research.

Overall, the research conducted in this study provides a good basis for suggesting that there are some trends, such as decreasing bulk drag coefficient and increasing diffusion, when the population density of stems increases. However, it should be noted that due to the use of simplifications and assumptions, this may only be indicative of the response caused by real vegetation in the field. Therefore, when carried forward to considering real scenarios, further elements should be considered and additional research conducted as the channel and vegetation properties will vary significantly between locations thus influencing the response of the diffusion and drag.

List of Abbreviations/symbols

C_{D} Drag Coefficient

f Friction Factor

F_{r} Froude Number

Re Reynolds Number

ρ

Density (kg/m^{3})

u Velocity (m/s)

l Length (m)

μ

Absolute viscosity

ν

Kinematic viscosity

List of Figures

List of Tables

List of Equations

List of Definitions

**Backwater Profile** The longitudinal profile of the surface of a fluid in open channel non-uniform flow conditions, where the water surface is not parallel to the depth of water but instead increasing due to the interposition of an obstruction.https://definedterm.com/backwater_curve

**Residence Time** The average length of time for which fluid remains on the floodplain.

**Wake**

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