Hydrodynamic instability leading to Monami

Hydrodynamic instability leading to Monami

by

Ravi Shanker Singh

B.Sc., Banaras Hindu University; Varansi, India, 2006

M.Sc. (Physics), Indian Institute of Technology-Bombay; Mumbai, India, 2008

M.S. (Applied Mathematics), Brown University; Providence, RI, 2013

A dissertation submitted in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

in Department of Physics at Brown University

PROVIDENCE, RHODE ISLAND

May 2016

c© Copyright 2016 by Ravi Shanker Singh

This dissertation by Ravi Shanker Singh is accepted in its present form

by Department of Physics as satisfying the

dissertation requirement for the degree of Doctor of Philosophy.

Date

Shreyas Mandre, Ph.D., Advisor

Recommended to the Graduate Council

Date

Jay Tang, Ph.D., Reader

Date

Thomas Powers, Ph.D., Reader

Approved by the Graduate Council

Date

Peter Weber, Dean of the Graduate School

iii

Vitae

Ravi Shanker Singh was born on 5th May 1986 in Gorakhpur district, Uttar

Pradesh, India. He graduated from Banaras Hindu University with Bachelor’s de-

gree in Physics major in year 2006. He graduated with master’s degree from Indian

Institute of Technology-Bombay in 2008. While pursuing his PhD at Brown Univer-

sity, he also earned master’s degree in Applied Mathematics in 2013.

iv

To my parents and family.

v

Acknowledgements

I would like to thank my advisor Professor Shreyas Mandre for the guidance and

support he has provided for the last six years. He gave me plenty of freedom to

work on problems on my own and was always available for providing me his valuable

feedback. His constant encouragement, especially in the initial few years of my

research, really helped me to remain motivated.

I would also like to thank Professor Jay Tang and Professor Thomas Powers for

agreeing to serve on my PhD committee and taking the time to read my thesis.

During last six years of my stay at Brown, many people have helped me with

various aspects of my project. In particular, I would like to thank Professor Mahesh

Bandi, Amala Mahadevan and L. Mahadevan, for their helpful discussion about my

research. I would like to thank all the group members for discussing both scientific

and non-scientific work during last 6 years. I would also like to thank CCV (Center

for Computation and Visualization) at Brown for providing me the computational

resources required for this project. I have used lot of other resources provided by

Brown as well, especially the Saferide service, which I used on a regular basis to

go home during night. I would like to convey my thanks to all the team members

of Saferide, the staff members of Library and all the staff members of Physics De-

partment. The secretary of Fluids group have helped me for many paperworks such

as refund of cost for attending various conferences. I would to thank Jeffrey and

Carolyn who worked as secretary of fluid group during last 5 year.

vi

I am very thankful to my parents, my aunt and other family members for their

constant encouragement, which have been valuable source of my motivation.

I am thankful to all my friends at Brown and also friends at other places for

making my life happy outside work. It was a wonderful experience knowing and

spending time with all of you and I hope we get to stay in touch after Brown as well.

I would like to convey my special thanks to some people from Fluids and Thermal

sciences group especially Kyohei Onoue and Xinjun Guo, with whom I had regular

interaction during Fluids seminar.

Finally, I would like to thank two very close friends Seshu and Rohini, who have

made significant difference to my life.

Thank you.

vii

Abstract of “ Hydrodynamic instability leading to Monami ” by Ravi Shanker Singh,Ph.D., Brown University, May 2016

The onset of monami - the synchronous waving of seagrass beds driven by a steady

flow is modelled as a linear instability of the flow. Unlike previous works, our model

considers the drag exerted by the grass in establishing the steady flow profile, and

in damping out perturbations to it. We find two distinct modes of instability, which

we label as mode 1 and mode 2. Mode 1 is closely related to Kelvin-Helmholtz

instability modified by vegetation drag, whereas mode 2 is unrelated to Kelvin-

Helmholtz instability and arises from an interaction between the flow in the vegetated

and unvegetated layers. The vegetation damping, according to our model, leads to

a finite threshold flow for both of these modes. Experimental observations for the

onset and frequency of waving compare well with the model predictions for instability

onset criteria and the imaginary part of the complex growth rate respectively, but

experiments lie in a parameter regime where the two modes can not be distinguished.

Full scale numerical simulation of the governing equation has been performed as

well. The result of this simulation agrees well with the prediction of linear stability

analysis. The full numerical simulation of governing equation shows that initially

the perturbation to the flow grows with rate predicted by linear stability analysis

and then reaches a saturation. At the saturation the flow can be described by a

background flow plus a steadily propagating wave whose mode shape, wavelength

and frequency are the same as the one predicted by the linear stability analysis.

Contents

Vitae iv

Acknowledgments vi

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Our Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Mathematical Model 10

3 Linear stability analysis for flow through submerged vegetation 173.1 Base velocity profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Governing equation for flow instability . . . . . . . . . . . . . . . . . 22

3.2.1 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.1 Unstable modes and critical parameters . . . . . . . . . . . . . 253.3.2 Mode 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.3 Mode 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.4 Comparison with the experimental data . . . . . . . . . . . . 36

4 Nonlinear Solution of flow through submerged grass 394.1 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2 Comparison with linear stability analysis . . . . . . . . . . . . . . . . 42

5 Linear stability analysis of flow around emergent vegetation 515.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2 Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2.1 Governing equation for flow instability . . . . . . . . . . . . . 545.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

viii

6 Discussion 606.1 Role of turbulence Model . . . . . . . . . . . . . . . . . . . . . . . . . 616.2 Impact of non-linear dynamics . . . . . . . . . . . . . . . . . . . . . . 636.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

A Matched asymptotic solution of base profile 66

ix

List of Tables

3.1 Comparison between KH instability and the two unstable modes re-sulting from solution of 3.6. . . . . . . . . . . . . . . . . . . . . . . . 35

4.1 Comparison between the estimates of frequency and wavelength be-tween prediction based on the solution of (3.6) and (2.6). The cal-culation is done for Ng = 2, hg/(2H) = 0.4. The critical Reynoldsnumber is R = 190 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1 Comparison between the estimate of frequency with that from exper-imental observation of White in [1] (Case X and Case I from Figure4.6 and Figure 4.15). The corresponding estimate of Ng is also shownin the table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

x

List of Figures

1.1 Importance of seagrass for marine life . . . . . . . . . . . . . . . . . . 31.2 Experimental observation of monami . . . . . . . . . . . . . . . . . . 6

3.1 Comparison of experimentally measured mean velocity profile withthe prediction of mathematical model . . . . . . . . . . . . . . . . . . 19

3.2 Numerical estimate of shear thickness for experimental data . . . . . 213.3 Collapse of velocity profile near the grass tip over boundary layer

thickness δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Growth rate (σr) of perturbation to the solution of (3.1) as predicted

by the solution of (3.6) . . . . . . . . . . . . . . . . . . . . . . . . . . 243.5 Critical Reynolds number, threshold Reynolds number for Mode 1

and Mode 2 and the corresponding marginally stable wave numberfor different submergence ratio . . . . . . . . . . . . . . . . . . . . . . 26

3.6 Re(σ) and the neutral curve (Re(σ)=0) as a function of k and R . . . 273.7 Growth rate for various values of δ/H, as a function of wavenumber k 283.8 Plot of the neutral Mode 1 shape |φ| in the limit of small δ/H for

hg/H = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.9 Plot of the neutral Mode 2 shape |φ| in the limit of small δ/H for

hg/H = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.10 Comparison of the experimentally measured dominant frequency fo

(in Hz) with the predictions fp = Im(σ) from the solution of equation(3.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.11 Comparison of experimental velocity profile of lab scale experimentswith the solution of equation (3.1) for Run 1 experimental observationof Ikeda [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.12 Comparison of experimental velocity profile of lab scale experimentswith the solution of (3.1) for Case A and Case B on experimentalobservation of Vivoni [3]. . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1 Staggered grid showing the grid points of u, v and p field. . . . . . . . 414.2 Graph showing growth rate of a random perturbation made to the

solution of (3.1), by explicitly solving (2.6) and its comparison withthe highest growth rate predicted by linear stability analysis. . . . . 43

4.3 Comparison of velocities and pressure predicted by Linear Stabilityanalysis and by explicitly solving equation (2.6). . . . . . . . . . . . 46

4.4 evolution of vorticity ω = ∇ × u as function of time for a randomperturbation over the steady flow arising from the solution of (3.1). . 47

xi

4.5 Plot of vorticity at the grass tip as function of time. . . . . . . . . . 484.6 Plot of vorticity at the grass tip as function of x. . . . . . . . . . . . 494.7 Comparison of velocities and pressure predicted by Linear Stability

analysis and deviation of velocity and pressure from its spatial averageby explicitly solving equation (2.6). . . . . . . . . . . . . . . . . . . 50

5.1 Schematic diagram of conveyance channel adjacent to an emergentvegetation plain, and the schematic velocity profile in both regions. . 52

5.2 Comparison of steady profile from the solution of equation (5.2) withthat from the experiments in White [1] (Case X and Case I . . . . . 56

5.3 Plot of the neutral Mode 2 shape |φ| in the limit of small δ/H forhg/H = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.4 Re(σ) and the neutral curve (Re(σ)=0) as a function of wavenumberand R for flow around emergent vegetation . . . . . . . . . . . . . . . 58

5.5 Critical Reynolds number and the corresponding marginally stablewave number for different submergence ratio as a function of vegeta-tion density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

xii

Chapter One

Introduction

2

1.1 Motivation

Seagrasses are submerged flowering plants found in shallow marine waters, mostly

in the littoral zones of freshwater and in large expanses of low lying coastal shore.

They occupy less than 0.05% of the ocean area [4] but contribute directly to about

15% of the total biomass production in the ocean [5]. Seagrasses plays crucial role in

protecting sea shores. The extensive root system in seagrasses, which extends both

vertically and horizontally, helps in stabilizing the sea bottom in a manner similar to

the way land grasses prevent soil erosion. In absence of seagrass numerous beaches,

businesses, and homes would be subject to greater damage from storms.

Seagrasses provide food, shelter, and essential nursery areas to commercial and

recreational fishery species and to countless invertebrates living in seagrass com-

munities. Some fish, such as seahorses and lizardfish can be found in seagrasses

throughout the year, while other fish spends their certain life stages in seagrass

beds. While some organisms, including the endangered Florida manatee and green

sea turtle, graze directly on seagrass leaves, others use seagrasses indirectly for nutri-

ents. Bottlenose dolphins are often found feeding on organisms that live in seagrass

areas. Detritus from the bacterial decomposition of dead seagrass plants provides

food for worms, sea cucumbers, crabs, and filter feeders such as anemones and ascid-

ians. Further decomposition releases nutrients (such as nitrogen and phosphorus),

which, when dissolved in water, are re-absorbed by seagrasses and phytoplankton.

The relative safety of seagrass meadows from strong water current provides an

ideal environment for the female fishes to lay and hatch their eggs [6]. The safety

of seagrass meadows are also useful for the juvenile fish and invertebrates to hide

themselves from predators. Seagrass leaves are also ideal for the attachment of larvae

3

Figure 1.1: Seahorse (left) uses seagrass for their habitat and Green sea turtle (right) grazing onseagrass leaves, pictures taken from amagintheseaworldpress.com and mikegil.com respectively

and eggs, including those of the sea squirt and mollusk. While seagrasses are ideal for

juvenile and small adult fish for escaping from the larger predators, many infaunal

organisms (animals living in soft sea bottom sediments) also live within seagrass

meadows. Species such as clams, worms, crabs and echinoderms, like starfishes, sea

cucumbers, and sea urchins use the buffering capabilities of seagrasses to provide a

refuge from strong currents. The dense network of roots established by seagrasses

also helps deter predators from digging through the substratum to find infaunal

prey organisms. Seagrass leaves provide a place of anchor for seaweeds and for filter-

feeding animals like bryozoans, sponges, and forams as well.

Seagrasses help in trapping fine sediments and particles that are suspended in

the water column, which increases water clarity. When a sea floor area lacks sea-

grass communities, the sediments are more frequently stirred by wind and waves,

decreasing water clarity, affecting marine animal behavior and generally decreasing

the recreational quality of coastal areas. Seagrasses are also known to sequester CO2

[7], mix and recycle the nutrients necessary for life of its inhabitants. Seagrasses also

work to filter nutrients that come from land-based industrial discharge and stormwa-

ter runoff before these nutrients are washed out to sea and to other sensitive habitats

such as coral reefs.

4

The primary requirement for active marine eco-systems are presence of sunlight

and a mechanism to mix and transport the nutrients. Since sunlight can penetrate

only near the surface of water, the sea-shores are ideal place for rich marine ecology.

One of the other important factor in proliferating the marine ecology near the ocean

shores is the presence of regular wave and tides which constantly stir the region.

This mechanism which is not present in most lakes. Despite shallow average depth

of lakes (e.g. average depth of Lake Erie is 18.6 m and that of lake Chad is 4 m)

with plenty of sunlight, lakes are not able to support a rich ecology as compared to

the coastal shore due to absence of tidal mixing. The coastal oceans are about 10

times as productive as the lakes [8]. Along with marshes and mangroves, seagrasses

meadows rank the highest in terms of biomass production.

The ability of seagrass meadows to engineer the habitat for effective ecosystems is

directly related to its ability to influence hydrodynamic processes. This requires

a balance between two competing requirements such as flow should be sufficiently

slow so that species don’t get flushed along with the water, but not stagnant and

thus allowing the nutrients and other material to be transported by the flow. It

is widely believed that many systems, including seagrass, rely on the flow for the

transportation and mixing of nutrients, pollens, sperms etc. A simple estimate of

mixing-strength in absence of any flow instability can be estimated to help us un-

derstand the importance of existence of flow instability. A typical seagrass patch

extend from 102 m to 103 m with a typical flow speed of 0.1 m/s to 1 m/s, indicating

that any tracer particle carried by the flow would take about τ = 102−104 s to cross

the patch. In the absence of any flow instability mean flow is horizontal and vertical

transportation and mixing of material can happen only through turbulent diffusivity

κ = 0.1Ud, where U ≈ 1 cm/s is the mean flow in canopy, and d ≈ 1 cm is char-

acteristic length scale of plant such as its diameter or leaf width [9]. This estimate

indicates that transportation of material above the grass bed can only penetrate

5

about√κτ = 3 − 30 mm, compared to the canopy height of 10-100 cm. However,

in the presence of flow instability even a modest 10% conversion of horizontal flow

to vertical velocity results in penetration length scale of about 10-1000 cm. Indeed,

It is widely believed that the phenomenon of large amplitude coherent oscillation of

marine grass, known as Monami is a result of flow instability, much like coherent

waves commonly observed on terrestrial grass field, known as Honami in strong wind

[10, 11, 12, 13]. While the two cases seems superficially similar, there are major dif-

ference such as atmospheric flow is essentially unbounded. Another major difference

between the two is the considerable difference of stiffness of canopies.

1.2 Previous Work

While considerable research is done to understand the phenomenon associated with

atmospheric flows through terrestrial canopies, research for the case of flow through

aquatic vegetation is not prevalent. Evidence of the effect of aquatic plants on unidi-

rectional flow emerges from various study of research groups interested in conveyance

of water through vegetated canals [14], the cycling of particulate and dissolved mat-

ter etc. One such systematic study on the topic of blue mussels larvae settlement

attributed the excess presence of blue-muscel larvae on the tip of grass to the presence

of Monami [15]. The most notable research work related to Monami are laboratories

study of open channel flow through flexible and rigid canopies by Nepf and Ghisal-

berti [2, 16], which shows existence of coherent eddies propagating on canopy top.

The current explanation of Monami is proposed by Ghisalberti and Nepf [16, 17],

6

Figure 1.2: Sketches made by Grizzle, et al. [15] from direct observations of eelgrass blades in themouth of the Jordan river, ME, USA, in response to a tidal flow through half a tidal cycle. Thehorizontal velocity profile is also plotted with each of the sketches. The grass spontaneously startsto wave in frame E, when the flow speed exceeds a threshold of 10-15 cm/s.

which is inspired by the work of Raupach [18, 12] for the case of terrestrial canopies.

The work of Raupach [18, 12] is based on Kelvin-Helmholtz (KH) instability of free

shear flow. According to this mechanism, the essential role of vegetation is to exert

drag on the flow slowing it down, and giving rise to velocity differential between the

canopy and the region above it. This so called shear layer profile is expected to be

hydrodynamically unstable to KH instability and break down into coherent vortices.

The flexibility of grass blades merely aids in visualizing these vortices by deforming

as they are advected past by the flow. According to this shear layer model, the

velocity profile of flow through vegetation can be approximated to a characteristic

shear like profile - “ one where shear does not arise from the boundary condition” e.g.

U(y) = ∆U [1 + tanh(y/δ)], in an approximately unbounded domain −∞ < y <∞.

Using the shear layer thickness δ determined by fitting to the measured mean velocity

profile (e.g. see 3.1 ), the frequency predicted by KH instability agrees well with the

7

laboratories observations.

However, several aspects of existing theory remain unexplained. First, existing

theory assumes the existence of shear layer near the grass top due to presence of

drag discontinuity in the vertical direction caused by the vegetation; however, it

does not take into account the effect of vegetation drag while considering instability

analysis of shear flow, i.e., the shear layer is treated as a free shear as far as instability

analysis is concerned. In other words, the assumption of instability of perturbation

to shear layer through a mechanism of Kelvin-Helmholtz relies on absence of any

interaction between the flow perturbation and vegetation drag, making shear layer

an inconsistent theory. Second, classical free shear flow is known to be unstable for

all the Reynolds number. On the contrary, a threshold flow speed is observed in the

field for the grass to spontaneously wave [15]; below this threshold speed the grass

bends steadily in response to steady current. This threshold speed corresponds to a

critical Reynolds number ≈ O(1000), based on turbulent viscosity.

Another closely related problem is the flow around emergent patches, which have

many similarities with the flow through submerged vegetation. The flow around

emergent patches develops a strong shear in the velocity profile in the horizontal

direction near the patch boundary similar to the one developed in the flow through

submerged vegetation near the grass top in the vertical direction. The primary

reason for the existence of a region of strong shear near the patch boundary in

case of flow around emergent vegetation and near the grass top in case of flow

through submerged vegetation is the discontinuous drag experienced by the flow

due to presence of obstruction in the horizontal and vertical direction, respectively.

However, the two cases have some differences as well. One difference between these

two cases is the presence of isotropic drag in case of flow around emergent patches

as compared to the anisotropic drag experienced by the flow through submerged

8

vegetation. The presence of isotropic drag in case of flow through emergent patch

can be understood by observing that both the dominant component of flow is in

the horizontal direction, hence, across the vegetation. In contrast, in the case of

flow through submerged vegetation only one of the dominant components of the

flow is across the vegetation, the other component is along the vegetation and it

experiences negligible drag. An analysis of shallow flow around emergent patches

has been carried out by White & Nepf [19]. However, in their analysis they also

assumed steady profile to be hyperbolic tangent in the horizontal direction and the

flow domain to be infinite.

These drawbacks of existing theory suggest that both the flow through submerged

vegetation and the flow around emerged patches require further investigation for a

better understanding of the phenomenon.

1.3 Our Approach

In this study, we have developed a two dimensional mathematical model for the

linear stability analysis of flow through submerged vegetation. We used continuum

approximation for the drag experienced by flow due to presence of grass. A linear

stability analysis of this model shows that a competition between destabilizing effect

of shear and stabilizing effect of drag dissipation leads to a critical flow condition

characterized by Reynolds number, above which flow becomes unstable, leading to

Monami. Our linear stability analysis predicts existence of two different modes of

flow instability which we termed as Mode 1 and Mode 2. Flow instability associated

with Mode 1 is found to be localized on the length scale of shear layer formed near

the canopy top, whereas Mode-2 is represented by the flow instability on the scale of

9

full water column. We found that Mode 1 shares characteristics of Kelvin-Helmholtz

instability such as instability on the scale of shear layer thickness but is found to

have significant differences as well. The prediction of critical Reynolds number and

waving frequency associated with Monami is found to compare well the experimental

observations. All the experimental data we have found so far corresponds to grass

density at which we are unable to distinguish if the flow instability are due to Mode

1 or Mode 2.

We have also performed full numerical simulation of the mathematical model pre-

sented in equation (2.6), and compared the prediction of the linear stability analysis

such as growth rate, wavelength and the resultant frequency of dominant mode with

the solution of equation (2.6) in section 4.2.

In chapter 5, we have also shown results from the linear stability analysis pf

flow around emergent vegetation. Our calculation indicate that due to presence of

isotropic drag, the flow instability associated with Mode 1 experiences significant

dissipation, and is not observed in the limit of asymptotically large grass density.

Chapter Two

Mathematical Model

11

Given the range of length scales in the canopy (leaf thickness < 1mm, leaf width ∼

1cm, canopy height ∼ 1m, vegetation patch 102 − 103m), a fully resolved direct nu-

merical solution is challenging and unnecessary. We instead exploit this separation

of length scale between the canopy scale and its microstructure, and resort to mod-

eling the aquatic vegetation as a homogenized multiphase medium which occupies

no volume but exert drag on the fluid flow. Because the leaves are wide but thin

, the volume fraction of vegetation is less than 10% [20] in the densest of seagrass

canopies, and dominant interaction of vegetation with the surrounding water is by

exerting a drag on the flow. Hence to leading order, we ignore the space occupied

by the grass in the fluid mass and momentum balance. In our model, vegetation is

represented as a continuum field h(x, t).

The governing equation of flow in and above the grass can be represented by the

fluid continuity equation incorporating the mass conservation and fluid momentum

equation. The equation for incompressible fluid can be expresses as

∇ · u = 0,

ρ

[∂u

∂t+ u · ∇u

]= −∇p+ µm∇2u,

(2.1)

where u is velocity field, µm is molecular viscosity of fluid , ρ is fluid density and

p is the pressure field. For a typical flow in vegetation, the Reynolds number for

flow on the scale of leaf is O(1000), hence a typical flow in presence of vegetation

can be treated as a turbulent flow. In order to obtain equations for mean flow,

we use analysis similar to Pedras & de Lemos [21], we first average the flow over a

time interval τ , which is longer than the vegetation scale velocity fluctuation time,

then perform spatial averaging over an area A, where A is much larger than cross

sectional area of grass. Denoting overbar to represent the time averaged properties

and the angel brackets to denote the spatially averaged properties, for any quantity

12

of interest, represented here by φ such as velocity, pressure etc.

φ(x, y, z, t) =1

τ

∫ t+τ

t

φ dt, 〈φ〉 =1

A

∫A

φ dx dz,

φ = φ+ φ′, φ = 〈φ〉 + φ

′′,

where φ denotes temporal average, 〈φ〉 denotes temporally and spatially averaged

quantity, φ′

denotes temporal fluctuation and φ′′

denotes spatial fluctuation. Here

x and z represents co-ordinates in the horizontal plane. Upon performing temporal

averaging of Navier-Stokes and fluid continuity equation, we arrive at the following

equations for fluid continuity and momentum balance.

∂ui∂xi

= 0,

ρ

[∂ui∂t

+ uj∂ui∂xj

]= − ∂p

∂xi+ µm

∂2ui∂x2

j

− ρ∂u

′iu

′j

∂xj,

(2.2)

and further spatial averaging gives,

∂〈ui〉∂xi

= 0,

ρ

[∂〈ui〉∂t

+ 〈uj〉∂〈ui〉∂xj

]= −∂〈p〉

∂xi+ µm

∂2〈ui〉∂x2

j

+∂〈τij〉∂xj

−Di.(2.3)

Where τij = −ρ〈u′iu

′j〉 − ρ〈ui

′′uj

′′〉 is macroscopic shear stress tensor, which consists

of Reynolds stresses (−ρ〈u′iu

′j〉) due to temporal fluctuation and dispersive stresses

(−ρ〈ui′′uj

′′〉) due to spatial fluctuation. The drag force Di(x, y) = 〈∂p′′

∂xi〉− 〈µm∂2ui

′′

∂x2j〉

is resistance due to vegetation, the sum of form and viscous drag over the averaging

scale, which is zero above the vegetation, i.e. Di = 0 for y > h(x, t).

Both molecular stress (µm ∂ui/∂xj) and dispersive stress(−ρ〈ui′′uj

′′〉) are negli-

gible compared to Reynolds stress (−ρ〈u′iu

′j〉). The former is usual assumption in

13

analysis of a turbulent flow, such as flow in vegetation and the latter can be under-

stood by observing that the lateral dimensions of canopy elements are much smaller

than the canopy height so that the correlation between spatial deviation of both

components of velocity on horizontal plane is small [18, 12]. With this assumption

the equation can be further simplified into,

∂〈ui〉∂xi

= 0,

ρ

[∂〈ui〉∂t


]= −∂〈p〉

∂xi− ρ

∂〈u′iu

′j〉

∂xj−Di.

(2.4)

In this work, we parametrized the Reynolds stress by an eddy viscosity, i.e.,

−ρ〈u′

iu′

j〉 = µ

(∂ui∂xj

+∂uj∂xi

),

where µ is eddy viscosity. In a real grass canopy, the measured value of eddy viscosity

is found to vary with depth in the water column [22]. However, in this work, in-order

to develop essential mathematical model we have used a constant representative value

of eddy viscosity throughout the water column. With this simplification, equation

(2.4) further simplifies into,

∂〈ui〉∂xi

= 0,

ρ

[∂〈ui〉∂t


]= −∂〈p〉

∂xi+ µ

∂2〈ui〉∂x2

j

−Di.(2.5)

Which is similar to Navier-Stokes equation with molecular viscosity replaced by a

constant eddy viscosity, with an extra drag term due to presence of vegetation. From

now onwards we will drop angle bracket and over bar for notational simplicity.

∇ · u = 0,

ρ

[∂u

∂t+ u · ∇u

]= −∇p+ µ∇2u−D.

(2.6)

14

When the Reynolds number of the flow on the microstructure and molecular vis-

cosity is small, the drag force D can be modeled according to Darcy’s law D ∝ u.

While there is no generic way to model the drag force when Reynolds number on

microstructure is large ( O(100-1000) ), general principals of high-Reynolds number

flows guide us towards a phenomenological model for drag. Firstly, in this range of

Reynolds number, the flow separates from grass blade, forming a wake. Therefore, it

is reasonable to expect the drag to scale with dynamic pressure ρu2r, where ur is the

relative velocity between the fluid and the grass blade. Moreover, we assume that

the drag is anisotropic, since the flow across the grass blade is expected to experience

more drag than the flow along the glass blade. Mathematically, the drag law uses

projection of relative velocity along the blade and normal to the blade as.

uN = ur.n, uT = ur.t, D = −ρd(x, y)(CNuN |uN |n + CTuT |uT |t

), (2.7)

where CN and CT are drag coefficient in the normal and tangential directions respec-

tively, Ng is the grass number density, d(x, y) is average grass diameter at location

specified by co-ordinates (x, y), t is average unit vector tangent to the grass element

at (x, y), and n is average unit vector normal to the grass blade at (x, y). In a real

grass canopy CT � CN . The relative magnitude of CN and CT can also be under-

stood by observing that in high Reynolds number flow, the normal drag is dominated

by the form drag ∝ ρu2N , due to pressure difference between the front and back of

grass blade caused by flow separation across the grass, hence, DN ≈ −ρd|uN |uN n,

where DN = −ρCNd|uN |uN n is the normal component of drag force and CN ≈ 1.

On the other hand, the drag in tangential direction is caused by viscous drag, which

can be approximated by DT ≈ −(µmuT/δbl)d, where µm is molecular viscosity of

fluid, δbl is thickness of boundary layer formed near the surface of grass blade due

to no slip boundary condition at the glass blade surface. We can easily estimate

15

δbl ≈√µmd/ρ. Using the estimate of δbl, the tangential drag can be estimated to

be DT ≈ − 1√Rblρu2

T t, where Rbl is Reynolds number on the scale of grass thick-

ness, which is O(100− 1000). Hence, CT ≈ CN/√Rbl. In order to develop essential

mathematical model we take CT = 0 in this analysis. So, the drag force can be

approximated as

D = −ρd(x, y)CNuN |uN |n, (2.8)

for the flow through submerged grass bed.

In the case of flow around emergent vegetation both dominant components of the

flow are in the horizontal direction and hence across the vegetation, which suggests

that vegetation drag in case of flow around emergent vegetation can be modeled by

an isotropic drag, i.e,

D = −ρd(x, y)Cu|u|, (2.9)

for the flow around emergent vegetation. Where C is the drag coefficient for flow

around emergent vegetation.

Boundary condition for (2.6) include no-shear ( uy = 0 ) and no normal velocity

(v = 0) on the both bottom surface and top surface. In real flow the true boundary

condition at the bottom surface is no slip velocity and no normal velocity; however,

the boundary layer formed due to no slip condition is so thin (≈ 0.5cm) that it’s

hardly visible in experiments as can be seen from the experimental data of [22]

in Figure 3.1. The presence of thin boundary layer near the grass bed can also

be understood by observing that in the limit of dense vegetation, the shear stress

exerted by the bottom surface is expected to be negligible compared to the vegetation

drag [22], hence the bottom boundary condition can be approximated by no shear

(uy(0) = 0) and no normal flow (v(0) = 0) condition. On the top surface, a rigid lid

condition ( no shear and no normal velocity ) or free surface is suitable. In order to

16

model free interface we have used no shear at top as well. Since changes in water

surface elevation due to presence of monami is small compared to the depth of water

channel, we can approximate the top surface by no normal flow (v(2H) = 0) as well.

The usual response of submerged grass to steady current is to remain bend

steadily in some fixed configuration. However, above certain flow velocity the grass

beds shows coherent large amplitude oscillation as well, such response of time de-

pendent large amplitude oscillation exhibited by the vegetation under the influence

of otherwise steady current is thought to be understood as a result of hydrodynamic

instability. Lab scale experiments also show that the flow structures resulting into

monami are present even when the flexible grass is replaced by rigid dowels [16, 22],

so, to develop essential understanding of flow instability leading to monami , we

assume grass to be rigid and oriented vertically.

In this work, we have used linear stability analysis in order to understand flow

instability leading to monami .

Chapter Three

Linear stability analysis for flow

through submerged vegetation

18

3.1 Base velocity profile

In order to understand nature of the flow instability associated with monami, we first

calculate the fully developed steady solution u = U(y)x of (2.6) driven by constant

pressure gradient dP/dx, and use it to non-dimensionalize the mathematical model.

The momentum balance (2.6) for steady U(y) simplifies to

−dPdx

+ µU ′′(y)− S(y)ρCNdNgU |U | = 0,

S(y) = 1, 0 < y < hg,

S(y) = 0, hg < y < 2H.

(3.1)

Eq. (3.1) is solved subject to no shear at the boundaries, i.e., U ′(0) = U ′(2H) = 0.

A comparison of the steady flow profile from the solution of (3.1) with experimental

measurement is shown in Fig 3.1. The profile U(y) has three distinct regions. Within

the vegetation, it is approximately uniform with U(y) ≈ Ug =√−dP/dxρCNdNg

, which arise

from the balance of the drag with the pressure gradient. Outside the vegetation, the

velocity has a simple parabolic profile similar to a Poissueille profile, arising from the

balance between pressure and viscous forces together with the free shear condition at

the top surface. At the grass top, continuity of shear stresses results in a boundary

layer of thickness δ in the velocity profile. Denoting Ubl to be the velocity scale in the

boundary layer, and U0 = (−dP/dx) H2/µ is the velocity scale in the unvegetated

region, the balance between viscous forces and vegetation drag with in grass implies

(µUbl/δ2 ∼ ρCNdNgU

2bl), and the continuity of shear stress across the grass top

implies (Ubl/δ ∼ U0/H). Solving for δ and Ubl yields δ/H ∼ Ubl/U0 ∼ (RNg)−1/3,

where Ng = (CNdHNg) is the vegetation frontal area per bed area, and R = ρU0H/µ

is the Reynolds number of the flow. A numerical estimate of δ (estimated as U/Uy

at y = hg) is compared with this prediction in figure 3.2 (right). Since the existence

19

0 3 6 90

5

10

15

20

25

30

35

40

45

y

x

v

u

U(y) (cm/s)

y(cm)

y = hg

y = 2H

δ

ModelNepf

Figure 3.1: Schematic setup and comparison of our steady flow profile with that from the exper-iments in ref. [22] (Case I from Table 1) and its approximation with U0 = 7.28 cm/s and δ = 5.02cm in our model. The grass extends up to y = hg in the water column of depth 2H. The steadyvelocity profile can be decomposed into a parabolic profile in the unvegetated region, a uniformprofile deep within the vegetation, and a boundary layer of thickness δ near the grass top.

20

of this boundary layer is not influenced by the boundary of the flow domain, we

identify it as similar to the shear layer invoked by the Ghisalberti & Nepf [16, 22],

i.e. “ one where shear does not arise from boundary conditions”.

We can also understand the scales for Ubl and boundary layer thickness δ by

integrating equation (3.1) from (hg − ε) to 2H.

∫ 2H

hg−ε

(−dPdx

+ µU ′′(y)− S(y)ρCNdNgU |U |)dy = 0 (3.2)

−dPdx

(2H − hg + ε) + µ(Uy(2H)− Uy(hg − ε))−∫ hg

hg−ερCNdNgU |U |dy = 0

The above equation can be further simplified by observing that Uy(2H) = 0 due to

zero shear boundary condition and Uy(hg − ε) ≈ 0 due to approximately constant

velocity below the boundary layer, we also observe that∫ hghg−ε U |U |dy ∼ O(U2

blδ).

Further, since ε� H we can ignore ε in comparison to 2H − hg.

−dPdx

(2H − hg) ∼ O(ρCNdNgU2blδ) (3.3)

In the above equation the left term −dPdx

(2H−hg) is the total shear stress applied at

the grass tip by flow above the canopy, which is balanced by the drag ρCNdNgU2blδ

within the boundary layer. By observing that Ubl/δ ∼ U0/H due to continuity of

shear stress at the grass top and U0 = (−dP/dx)H2/µ, equation (3.3) can be solved

for δ and Ubl

δ = β(RNg)−1/3 Ubl =

δ

HU0 (3.4)

where β depends only on submergence ratio. For another presentation of the bound-

ary layer analysis see Appendix A. The numerical estimate of δ (estimated as U/Uy

at y = hg) for the experimental data of [22] is shown in Figure 3.2 (left). we have

also shown the comparison of numerical estimate of δ with the prediction of (3.4) on

21

0 0.2 0.4

0

10

20

30

40

δ

y = hg

y = 2H

Uy(y) (1/s)

y(cm)

Model10

0

102

104

106

10−2

10−1

100

RNg

δ/H

˜(RNg)−1/3

δ/H

Figure 3.2: Graph showing numerical estimate of δ for the experimental data of [22] (Case Ifrom Table 1) from the numerical solution of (3.1)). The dependence of boundary layer thickness(estimated as |U/Uy| at y = hg from numerical solution of 3.1 on the vegetation density parameter

RNg is shown on the right.

−5 −4 −3 −2 −1 0

−1.4

−1

−0.6

−0.2

(y − hg)/δ

(U−

Uhg)/(U

0δ/H

)

δ/H =1.5δ/H =0.24δ/H =0.049δ/H =0.011δ/H =0.0049

Figure 3.3: Collapse of velocity profile near the grass tip over boundary layer thickness δ

the right panel in figure 3.2. The prediction of (3.4) with the solution of (3.1) are

also shown in Figure 3.3.

The dependence of δ on Ng gives us a way to systematically investigate the effect

of the shear layer thickness on the instability mechanism. Figure 3.2 also shows that

the asymptotic regime of a thin boundary layer is expected to hold for RNg & 100.

In this notation, Ug/U0 = (RNg)−1/2.

22

3.2 Governing equation for flow instability

In order to understand the nature of perturbation to the flow arising from the solution

of 3.1, we substitute u = (U + u, v), p = P + p in (2.6) and expand to linear order

to investigate the evolution of small perturbations (u, v), which obey

ρ(ut + Uux + vUy) = −px + µ∇2u− 2SρCNdNgUu,

ρ(vt + Uvx) = −py + µ∇2v, ∇ · u = 0,

(3.5)

where the tilde are dropped for simplicity. We non-dimensionalized these equa-

tion using half channel height H, velocity U0, and time H/U0, leading to three

non-dimensional parameters, namely R, Ng, and the vegetation submergence ra-

tio hg/H. We also use δ/H in lieu of Ng to parametrize the vegetation den-

sity in order to elucidate the instability mechanism. Using a stream function ψ

with u = ψy, v = −ψx to satisfy mass balance, we seek a solution of the form

(u, v, ψ) = (u(y), v(y), φ(y)) eikx+σt. After substituting (u, v) in equation (3.5) and

eliminating pressure term, we obtain a modified Orr-Sommerfeld equation [23, 24, 25]

R−1(D2 − k2

)2φ =

[(σ + ikU)

(D2 − k2

)− ikUyy

]φ+ NgD (2SUDφ) , (3.6)

where D = d/dy. The equation (3.6) is solved with the boundary conditions φ =

D2φ = 0 at y = 0 and y = 2, which are consistent with the zero shear and no normal

flow at both boundary. The growth rate σ for a given wave number k arises as an

eigenvalue that allows a non-trivial solution φ of equation (3.6).

23

3.2.1 Numerical Method

In order to solve equation (3.6), we used finite difference method with N equally

spaced grid points (yi = i∆y,∆y = 2N

), and used second order approximation for

various terms in equation (3.6) at these grid points. We used following approximation

for evaluating various derivatives appearing in equation (3.6)

D4φ∣∣i

=φi+2 − 4φi+1 + 6φi − 4φi−1 + φi−2

∆y4− ∆y2

6

∂6φ

∂y6

∣∣∣∣i

,

D2φ∣∣i

=φi+1 − 2φi + φi−1

∆y2− ∆y2

12

∂4φ

∂y4

∣∣∣∣i

,

Dφ|i =φi+1 − φi−1

2∆y− ∆y2

6

∂3φ

∂y3

∣∣∣∣i

,

(3.7)

where the terms O(∆y2) are truncation error to the approximation of the derivatives

at the grid points. The boundary conditions at the top and bottom surface are

approximated by φ0 = 0, φN = 0 to satisfy φ = 0 at the top and bottom surface.

The zero shear condition at the top and bottom surface (D2φ = 0) are expressed

by φ−1 = −φ1 and φN+1 = −φN−1. The discontinuity of S(y) across the grass tip

give rise to the dirac-delta term 2NgUgDφgδ(y − hg) in equation (3.6), where Ug is

velocity at the grass tip and φg is value of φ at the grass tip. We approximate the

term with dirac-delta as 2NgUgDφg1

∆y. The base velocity profile U(y) at these grid

points are evaluated by solving equation (3.1). Using these approximation we can

rearrange terms in equation (3.6) to yield

Aφ = σBφ,

A = R−1(D2 − k2

)2φ− ikU

(D2 − k2

)φ+ ikUyyφ− NgD (2SUDφ) ,

B = σ(D2 − k2

)φ,

(3.8)

24

0 1 2 3 4 5−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

k

σr

Figure 3.4: Growth rate (σr) of perturbation to the solution of (3.1) as predicted by the solutionof (3.6). The Reynolds number, grass number density and submergence ratios are R = 500, Ng = 2and hg/(2H) = 0.4 respectively.

where the operator A and B are approximated at the grid points using finite dif-

ference method. This results in a finite dimensional generalized eigen-value prob-

lem, whose solution provide growth rate σ and eigen mode φ for a perturbation of

wavenumber k to the base velocity profile U(y). A plot of <(σ) as a function of

wavenumber k for a base profile arising from R = 500, Ng = 2 and hg/(2H) = 0.4 is

shown in figure 3.4

25

3.3 Results

3.3.1 Unstable modes and critical parameters

In order to understand the critical criteria for flow instability leading to waving,

we solve equation (3.6) numerically for σ and φ, which predicts a threshold in R,

above which the flow is unstable (<(σ) > 0) for at least one k. The dependence of

this threshold R, and the corresponding marginally stable wavenumber k, on δ/H

and hg/H is shown in Fig. 3.5. We observe that the threshold R increases with the

vegetation density which indicates a competition between the destabilizing shear in

the flow and the stabilizing effect of damping due to vegetation drag.

To better understand the instability mechanism, we consider the dependence of

the fastest growing wavenumber on δ. Our calculation indicates that the fastest

growing wavenumber first increases proportional to H/δ, but at a critical δ discon-

tinuously jumps and remains O(1) (see Fig. 3.5). To understand this behavior in

greater detail, we have also plotted heat maps of <(σ) as a function of R and k, for

various hg/H and Ng in Fig. 3.6. On top of heat map of <(σ), we have also plotted

the contour of neutral curve <(σ) = 0. The threshold R is set by the smallest R on

this neutral curve. We observe that as Ng increases, the unstable region splits into

two; we refer to the region with the higher k as “Mode 1”, and the one with the lower

k as “Mode 2”. For hg/H . 0.9, the unstable region for Mode 1 recedes to higher

R, and for hg/H & 0.9, the region shrinks to zero size. In either case, due to such

behavior the most unstable mode exhibits discontinuous transition from Mode 1 to

Mode 2. So far, all the experimental data we have found corresponds to vegetation

density at which the unstable region in the R−k has not split into two, so with these

data we are unable to determine if flow instability in the lab scale experiments [22]

26

101

102

103

101

102

103

104

105

106

H/δ

R

hg/H = 0.4, Mode 1hg/H = 0.4, Mode 2hg/H = 0.4, Criticalhg/H = 0.8, Mode 1hg/H = 0.8, Mode 2hg/H = 0.8, Critical(H/δ)2

(H/δ)3/2

No WavingWaving

101

102

103

100

101

H/δ

k

hg/H = 0.4, Mode 1hg/H = 0.4, Mode 2hg/H = 0.4, Critical(H/δ)hg/H = 0.8, Mode 1hg/H = 0.8, Mode 2hg/H = 0.8, Critical

Figure 3.5: Critical Reynolds number, threshold Reynolds number for Mode 1 and Mode 2 (left)and the corresponding marginally stable wave number (right) for different submergence ratio asa function of vegetation density parametrized by the boundary layer thickness. Parameters fromexperiments reported by [16] to exhibit or suppress synchronous waving are also included in theleft panel. In order to estimate the R for these experiments, a representative value of µ = 0.1 Pa swas assumed.

27

R

k

Unstable

Stable

hg/(2H) = 0.3

Ng = 200

5000 10000 15000

1

4

7

R

k

Mode 1

Mode 2

Ng = 500

1 2 3

x 104

1

4

7

R

k

Mode 1

Mode 2

Ng = 800

1 3 5

x 104

1

4

7

R

k

Unstable

Stable

hg/(2H) = 0.4

Ng = 500

0.3 1 2

x 104

1

3

5 Mode 1

Mode 2

Ng = 850

1 2

x 104

1

3

5Mode 1

Mode 2

Ng = 925

2 3 4

x 104

1

5

9

R

k

Unstable

Stable

hg/(2H) = 0.5

Ng = 900

1 2

x 104

1

3

5

Mode 1

Mode 2

Ng = 1400

1 2 3

x 104

1

3

5

Mode 1

Mode 2

Ng = 1700

1 2 3

x 104

1

3

5

R

k Unstable

Stable

hg/(2H) = 0.6

Ng = 4000

1 3 5

x 104

1

3

5

Mode 1

Mode 2

Ng = 5000

1 4 8

x 104

1

3

5

Mode 2

Ng = 6000

1 4 8

x 104

1

3

5−0.03

−0.02

−0.00

0.01

0.03

Figure 3.6: Re(σ) and the neutral curve (Re(σ)=0) as a function of wavenumber and R forparameters shown in the corresponding panel. As Ng increases, the unstable region splits into two

labeled as “Mode 1” and “Mode 2”. For Ng below (above) a critical value, Mode 1 (Mode 2) setsthe threshold R.

28

1 2 3 4 5 6 7 8 9 10−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

k

Re(σ)

δ/H = 0.0052δ/H = 0.0059δ/H = 0.0063δ/H = 0.0070

Figure 3.7: Growth rate for various values of δ/H, as a function of wavenumber k. The wavenum-ber with highest growth for Mode 2 (first peak) is always O(1), whereas for Mode 1 the wavenumberwith highest growth increases as ∼ (H/δ). The submergence ratio and grass number density forabove graph are 0.3 and Ng = 300 respectively.

are arising due to Mode 1 or Mode 2. We numerically observe that the Mode 1 and

Mode 2 exhibit different asymptotic behavior for large Ng, which distinguishes them

from each other and facilitate their comparison with the KH instability mechanism.

3.3.2 Mode 1

We numerically observe that the threshold Reynolds number for Mode 1 instability

scales as R ∼ (H/δ)2 (or R ∝ N2g ). We also observe that Mode 1 is asymptotically

localized to the boundary layer near the grass top, and exhibits highest growth

for a perturbation of k ∼ H/δ (see Fig. 3.5). We can understand the behavior of

critical Reynolds number by performing asymptotic and dominant balance analysis

29

in the limit R � 1 and Ng � 1. In order to understand the behavior of critical

R, we estimate the size of various term of equation (3.6) in the boundary layer.

Observing that in the limit Ng � 1 and R � 1, Mode 1 is localized over a length

of δ near the grass top (see figure 3.8), we estimate D ∼ H/δ, we also observe that

σ ∼ O(1) and U = Ubl ∼ δ/H (see appendix A) in the boundary layer; so, the

magnitude of the advection term (σ + ikU) (D2 − k2)φ is (H/δ)2 (or R2/3N2/3g ), the

magnitude of the viscous term (1/R) (D2 − k2)2φ is (1/R)(δ/H)−4 (or (R1/3N

4/3g )),

and the magnitude of the drag term NgD (2SUDφ) is Ng(H/δ) or (R1/3N4/3g ). The

advection term, viscous term and vegetation drag terms balance when R ∼ (H/δ)2

(or R ∼ N2g ).

In order to gain further insight into the mechanism of Mode 1, we rescale equation

(3.6) near the grass-top using the scaling of various variables in the boundary layer,

η = y/(δ/H), U(y) = (δ/H)U(η) and k = (H/δ)k. With these scalings equation

(3.6) simplifies to

(D

2 − k2)2

φ = (R/N2g )1/3

[(σ + ikU

) (D

2 − k2)− ikUηη

]φ+D

(2SUDφ

),

(3.9)

in a region of thickness O(δ) near y = hg, where D = d/dη. Since (R/N2g ) is the only

remaining parameter in equation (3.9), we expect the mode shape and solution to

converge in the limit R� 1, Ng � 1 with R/N2g fixed. Our numerical findings also

confirm this expectation; we can see that the critical R scales as (H/δ)2 as shown

in Fig. 3.5 and the mode shapes are self-similar with length scale δ, as shown in Fig

3.8.

There are many similarities between the KH instability and the Mode 1 (see

Table 3.1). In case of Mode 1 the fastest growing wavenumber at the critical R

30

scales as k ∝ (H/δ), similar to KH instability. In both cases the extent of the

unstable mode is also localized to the boundary layer region. Due to the porous

nature of the vegetation only a weak flow of magnitude Ubl = U0δ/H penetrates

a thin boundary layer region δ, and therefore the shear gradient Uyy ∼ U0/δH is

largest in this region. The existence of strong shear gradient Uyy in the boundary

layer plays a central role in destabilizing the flow and localizing the flow instability

to the boundary layer. However, there are some key differences between Mode 1 and

KH instability as well. Our detailed description of Mode 1, given by equation (3.9)

highlights important differences between Mode 1 and the formulations of KH. KH is

usually described using the inviscid Rayleigh’s equation,

(σ + ikU)(D2 − k2

)φ = ikUyyφ, (3.10)

which is not parametrized by the Reynolds number. If we describing the instability

using the Orr-Sommerfeld equation, we do get Reynolds number as a parameter, but

the shear flow with the tanh-profile are unstable at all values of Reynolds number

[23]. Therefore, based on the inviscid formulations of KH instability or with the

tanh-shear profile the origin of the threshold flow conditions observed in the lab

scale experiments and the field observation is unclear.

In our model, the scale of boundary layer is set by the (turbulent eddy) viscosity,

and therefore for Mode 1 as well. However, unlike KH model, the boundary layer

in the present model is established only in the vegetated region. In the unvegetated

region the velocity profile does not saturate on the scale of δ (see figure 3.1), rather, in

the unvegetated region the velocity profile has length scale O(H−hg). The threshold

flow condition arises from a competition between the destabilizing role of fluid inertia,

which is very similar to the one played in KH mechanism and the stabilizing role of

dissipation due to vegetation drag. We also observe that the vegetation drag may

31

not be neglected within this boundary layer, and therefore plays a central role in the

Mode 1 instability mechanism.

3.3.3 Mode 2

Our numerical observation for Mode 2 shows that threshold Reynolds number for

Mode 2 scales as R ∝ (δ/H)−3/2, which is shown in Figure 3.5. The behavior of this

critcal reynolds number can be understood by doing asymptotic analysis of equation

(3.6) in the limit R � 1 and Ng � 1, we also observe that Mode 2 shows highest

growth to a perturbation of k ∼ O(1) and the eigen value σ is O(1) as well. In the

limit R � 1 and Ng � 1, the non-dimensional flow in the grass bed is Ug/U0 ∼

(RNg)−1/2 � 1, and therefore ikU � σ, hence, ikU may be neglected in comparison

to σ. Furthermore, since U(y) is approximately constant in the vegetation, Uyy

decays to zero within the grass outside the boundary layer. Using D ∼ 1 and U ∼ 1,

outside the grass; the magnitude of advection term (σ + ikU) (D2 − k2)φ is O(1)

and the magnitude of turbulent viscous stress term (1/R) (D2 − k2)2φ is O(1/R).

In the limit R� 1, the turbulent viscous term is negligible compared to the inertial

term. Thus, (3.6) simplifies to

σ(D2 − k2

)φ = −2(Ng/R)1/2D2φ, for y < hg (3.11a)

(σ + ikU)(D2 − k2

)φ = ikUyyφ, for y > hg. (3.11b)

The above equation (3.11) has only one parameter R/Ng. So, for fixed R/Ng, we

expect that the mode shape converges in the aforementioned limit, which we can

confirm from our numerical findings shown in Fig. 3.9. The convergence of mode

shape in this limit indicates that we have identified the correct asymptotic limit to

investigate Mode 2. We therefore expect that the threshold is set by parameterR/Ng,

32

−1 −0.5 0 0.5 10

0.01

0.02

0.03

y/H

|φ|

δ/H =0.005

δ/H =0.0022

δ/H =0.0013

−200 −100 0 100 2000

0.2

0.4

0.6

0.8

1

(y − hg)/δ

|φ|/|φ| m

ax

δ/H =0.005δ/H =0.0022δ/H =0.0013

Figure 3.8: Plot of the neutral Mode 1 shape |φ| in the limit of small δ/H for hg/H = 0.2 in thetop panel. The approach of mode shapes to each other for these small values of δ/H indicates thatthe dense vegetation asymptote is reached. Mode 1 shapes appear self-similar in shape as δ → 0.The bottom graphs shows rescaled |φ| for Mode 1 as a function of (y − hg)/δ approach a universalshape, indicating that an asymptotic limit has been reached.

33

which can be further justified by the numerically observed asymptotic behavior R ∝

Ng (or R ∼ (δ/H)−3/2; see Fig. 3.5 for comparison with numerical results). The

structure of this mode in the aforementioned limit is such that φ is continuous at

y = hg, but Dφ undergoes a rapid transition on the scale of boundary layer thickness

δ ( see figure 3.9). Furthermore, the eigenvalues and the mode shapes have very weak

dependence on δ. Since u(x, y, t) = Dφe(ikx+σt), we conclude that the effect of rapid

transition of the Dφ on the scale of boundary layer near the grass tip have secondary

role of regularizing the discontinuity in tangential velocity arising at y = hg in this

instability mechanism. The enhanced shear in the boundary layer plays no role for

this mode of instability.

The dependence of Mode 2 on the perturbation of wavenumber k (see Figure 3.6)

and the structure of mode (see Figure 3.9) suggest that Mode 2 has characteristics

distinct from KH. We can also understand the difference between Mode 2 and the KH

instability by observing that outside the grass, the unstable mode shape is governed

by the inviscid Rayleigh’s equation (3.10). According to Rayleigh’s criteria [26], an

inflection point in U(y) is a necessary condition for instability arising from (3.10).

However, with the current model Uyy(y) = −1 above the grass and therefore does not

change sign for y > hg. Instead, the dynamics are coupled with the flow in the grass

bed described by (3.11a) in y < hg. The absence of Uyy in (3.11a) indicates that Uyy

is approximated to be zero in y < hg, and therefore the positive values of Uyy that

occurs in the boundary layer do not affect this mode of instability to the leading

order. Therefore, we conclude that Mode 2 is distinct from the KH instability as

well, and owes its existence to the vegetation drag.

34

0 0.5 1 1.5 20

0.002

0.004

0.006

0.008

0.01

y/H

|φ|

δ/H =0.0012

δ/H =0.00018

0 0.5 1 1.5 20

0.005

0.01

0.015

0.02

0.025

y

|φy|

δ/H =0.0012δ/H =0.00018

−20 0 200

20

40

60

(y − hg)/δ

|φy|/|φ

y(h

g)|

Figure 3.9: Plot of the neutral Mode 2 shape |φ| in the limit of small δ/H for hg/H = 0.2 inthe top panel. The approach of mode shapes to each other for these small values of δ/H indicatesthat the dense vegetation asymptote is reached. The bottom graphs shows that φy undergoes rapidtransition on the scale of boundary layer thickness δ.

35

KH Mode 1 Mode 2

Base velocity profile U(y) = U0 tanh(y/δ) Equation (3.1)

Domain −∞ < y <∞ −1 < y < 1

Inflection point exists at y = 0 U ′′(y) discontinuous at y = hg

Shear layer thickness δ δ ∼ H(RNg

)−1/3

Linearized dynamics Equation (3.10) Equation (3.6)

Dense grass limit no grass included Equation (3.9) Equation (3.11)

Critical parameters none R ∝ N2g R ∝ Ng

Most unstable k as δ → 0 ∝ H/δ ∝ H/δ O(1)

Mode localized? yes, near y = 0 yes, near y = hg no, spans water column

Table 3.1: Comparison between KH instability and the two unstable modes resulting from solutionof 3.6.

36

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

fp

fo

Ghisalberti

Vivoni

Ikeda

Figure 3.10: Comparison of the experimentally measured dominant frequency fo (in Hz) with thepredictions fp = Im(σ) from the solution of equation (3.6). The experimental data in the inset isobtained from [16] (Ghisalberti) and [3] (Vivoni). In order to estimate the R for these experiments,a representative value of µ = 0.1 Pa s was assumed.

3.3.4 Comparison with the experimental data

We have compared the prediction of critical Reynolds number with the data observed

in the lab scale experiments. It is found to compare well [16] (see Figure 3.5). The

frequency (Im(σ)/2π) of the fastest growing mode also agrees well with observed

frequency of monami which is estimated from maxima in the velocity spectra, and

frequency of vortex passage in lab scale experiments [16], for cases where the vege-

tation was sufficiently dense to be modeled by a continuum drag field as shown in

Fig. 3.10. The comparison of solution of (3.1) with the experimentally measured

velocity profile in the lab scale experiment of Vivoni [3] are shown in 3.12. The ve-

locity profile for experiment of Ghisalberti [16] are not available for comparison. The

calculated velocity profiles for these experiments are estimated using the constraint

of given flow rate, grass number density and the submergence ratio. We have used a

constant representative value of eddy viscosity µ = 0.1 Pa s. We observe that there

37

10 20 30 40 50 60 70 800

5

10

15

U (y)cm/s

y(cm

)

Solution of 3.1

Ikeda

Figure 3.11: Comparison of experimental velocity profile of lab scale experiments with the solutionof equation (3.1) for Run 1 experimental observation of Ikeda [2]. The graph with solid linerepresents experimental data and the dashed line represent the estimated velocity profile from thesolution of (3.1).

are large variation between the experimentally measured frequency of Ikeda [2] and

the frequency predicted by the solution of 3.6, the reason for this disagreement might

be applicability of the current model to the experimental set up of Ikeda [2], which

can be seen from the comparison between the experimentally measured velocity and

the velocities estimated from the solution of (3.1) (see Figure 3.11). We see that for

the case of Ikeda [2], the boundary condition at the bottom of the grass can not be

approximated by zero shear, hence we do not expect the prediction of current model

to agree well with the experimental data of Ikeda [2].

Although the experimentally observed monami wavelengths are not available for

comparison, the predicted wavelength for lab scale experiments [16] are found to be

O(H).

38

2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

U (y)cm/s

y/h

g

2H/hg = 1.25Solution of 3.12H/hg = 1.50Solution of 3.12H/hg = 1.75Solution of 3.1

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

U (y)cm/s

y/h

g

Q=6.31

Solution of 3.1

Q=10.72

Solution of 3.1

Q=5.14

Solution of 3.1

Figure 3.12: Comparison of experimental velocity profile of lab scale experiments with the solutionof (3.1) for Case A (on top) and Case B on (bottom ) experimental observation of Vivoni [3]. Thegraph with solid line represents experimental data and the dashed line with same color representthe estimated velocity profile from the solution of (3.1).

Chapter Four

Nonlinear Solution of flow through

submerged grass

40

The analysis of linear stability analysis is helpful in understanding mechanism leading

to monami , hence is a good approximation for predicting threshold flow speed beyond

which flow leads to monami . However, in order to predict mixing of flow near the

grass tip, we need to know the detailed flow structure, when flow become turbulent

and can not be described linear dynamics. In this chapter, we have developed a

numerical method to simulate flow for large time and compared its prediction with

that of linear stability analysis. We have found that, when the Reynolds number of

flow is close to the threshold value, the flow can be described by a background flow

plus a wave, whose frequency and wavelength is similar to the the dominant mode

(mode with the highest growth rate ) predicted by linear stability analysis.

4.1 Numerical Method

We solved equation (2.6) with a finite difference method on a staggered grid using

a MAC (marker and cell) scheme, subject to periodic boundary condition in x.

On this staggered grid, pressure variables are defined at the center of the cell, the

x-component of velocity u is defined on the center of vertical faces and the vertical

velocity v is defined on the center of horizontal faces, as shown in the Figure 4.1. In

this method, each time step is divided into two substeps. In the first step, we ignore

pressure and solve convection and diffusion equation,

un+1 − un

∆t+ (un · ∇)un − 1

R∇2un − Ng(u

2)nx, (4.1)

where ∆t is the size of time discretization, n denotes time label and R is the Reynolds

number. In the second step, we add pressure gradient operator and impose the

41

u2,5/2 u3,5/2

v5/2,2

v5/2,3

p3/2,3/2

Figure 4.1: Staggered grid showing the grid points of u, v and p field. The horizontal velocityu is defined at the center of vertical faces of the cell, vertical velocity v is defined at the center ofhorizontal faces and the pressure is defined at the center of the cell.

incompressibility of flow.

un+1 − ˜un+1

∆t+∇P n+1 = 0

∇ · un+1 = 0

(4.2)

Using the staggered grid the two components of equation (4.1) can be written as

un+1i,j+1/2 = uni,j+1/2 −∆t

(uux + vuy −

1

R∇2u

)ni,j+1/2

−∆tNg(u2)ni,j+1/2

vn+1i+1/2,j = vni+1/2,j −∆t

(uvx + vvy −

1

R∇2v

)ni,j+1/2

(4.3)

where(uux + vuy − 1

R∇2u

)ni,j+1/2

and(uvx + vvy − 1

R∇2v

)ni,j+1/2

needs to be eval-

uated at the grid location for the x-component of velocity at (i, j + 1/2) and for

y-component of the velocity at (i + 1/2, j), respectively. The equations (4.2) are

discretized as

un+1i,j+1/2 = un+1

i,j+1/2 −1

R

∆t

∆x

(pn+1i+1/2,j+1/2 − p

n+1i−1/2,j+1/2

)vn+1i+1/2,j = vn+1

i+1/2,j −1

R

∆t

∆y

(pn+1i+1/2,j+1/2 − p

n+1i+1/2,j−1/2

) (4.4)

42

where ∆x and ∆y are the uniform grid spacing in the x and y directions, respectively.

The continuity equation on this staggered can be written as

un+1i+1,j+1/2 − u

n+1i,j+1/2

∆x+vn+1i+1/2,j+1 − v

n+1i+1/2,j

∆y= 0 (4.5)

Substituting the velocity components from equation (4.4) into equation (4.5), we get

discretized poisson equation for pressure.

∆t

[pn+1i+3/2,j+1/2 − 2pn+1

i+1/2,j+1/2 + pn+1i−1/2,j+1/2

∆x2

]+ ∆t

[pn+1i+1/2,j+3/2 − 2pn+1

i+1/2,j+1/2 + pn+1i+1/2,j−1/2

∆y2

]=(

un+1i+1,j+1/2 − u

n+1i,j+1/2

)∆x

+

(vn+1i+1/2,j+1 − v

n+1i+1/2,j

)∆y

(4.6)

The zero normal velocity on the top and bottom surface provides the boundary

condition for pressure field, which are used to solve for pressure using (4.6). Sub-

stituting the pressure from the solution of equation (4.6) into equation (4.4) we get

the velocity field at the next time step.

4.2 Comparison with linear stability analysis

We have explicitly solved equation (2.6) with the initial condition u(x, y, 0) = U0(y)+

u0(y) and v(x, y, 0) = 0, where U0(y) is velocity profile arising from the solution of

equation (3.1) and u(y) = εf(y) is a random perturbation to U0(y), where ε� 1 and

f(y) is an arbitrary function. We expect this perturbation to grow exponentially

as ||u(t)|| = ||u(t0)||eσr(t−t0) with the maximum growth rate (σr) as predicted by

the solution of equation (3.6), where ||.|| represents L2 norm of perturbation. A

comparison of growth rate computed from the solution of equation (2.6) with the

43

0 200 400 600 800 1000 1200

−10

−5

0

5

10

15

T = U0

HT

log( ||u||U0

)

log( ||v||U0

)

σrT

Figure 4.2: Graph showing growth rate of a random perturbation made to the solution of (3.1), byexplicitly solving (2.6) and its comparison with the highest growth rate predicted by linear stabilityanalysis. The σr is the real part of growth rate predicted by linear stability analysis. The Reynoldsnumber, grass number density and submergence ratios are R = 250, Ng = 2 and hg/(2H) = 0.4respectively.

maximum growth rate σr predicted from the solution of equation (3.6) is shown in

Figure 4.2. We can see that initially when the perturbation to the U0(y) is small,

the perturbation grow with the rate predicted by equation (3.6). However, once the

magnitude of perturbation becomes comparable to the U0(y), non-linear effect also

becomes important and the growth rate can not be described by the solution of linear

stability analysis.

In the regime when ||u−U0|| � ||U0||, we also expect that the profiles of pertur-

bation velocities u(x, y, t) = u(x, y, t)−U0(y) and v(x, y, t) = v(x, y, t) can be easily

described by the dominant eigen mode φ(k, y) corresponding to k with the highest

44

growth rate

u(x, y, t) = <(φy(k, y)eikx+σt+iθ

),

v(x, y, t) = <(ikφ(k, y)eikx+σt+iθ

).

(4.7)

Where θ is a constant phase between u(0, y, t) and <(φy(k, y)eσt). A comparison

between initial evolution of perturbation variables (u, v, p) with a random pertur-

bation to the base flow U0(y) arising from the solution of equation (2.6) with the

prediction based on the dominant eigen-mode φ(k, y) is shown in Figure 4.3; which

shows expected agreement between prediction based on the dominant eigen mode

analysis and the prediction based on the full numerical simulation of equation (2.6).

By carrying out numerical simulation of equation (2.6) for large time, we observe

that initially the amplitude of perturbation variable grow with the rate predicted

by linear stability analysis and then saturates to constant value ( See Figure 4.2 ).

After the perturbation saturates to an approximately constant value, the flow can

be described as a steadily propagating wave (see figure 4.4 ). The frequency and

wavelength of the wave arising from the flow instability can be easily calculated by

observing the vortex passage frequency near the grass tip and the vortex pattern

over the flow domain (see figure 4.5 and figure 4.6 ). If the Reynolds number is close

to the critical value, we expect the frequency and wavelength of the wave arising due

to flow instability to be comparable to the one predicted by linear stability analysis.

A comparison between prediction of wavelength and frequency based on the solution

of equation (3.6) and equation (2.6) is shown in table 4.1. In figure 4.7, we have

also shown comparison between u(x, y, t) − 〈u(y, t)〉 and <(φy(k, y)ei(kx+σit+θ)g(t)

)and between v(x, y, t)−〈v(y, t)〉 and <

(φ(k, y)ei(kx+σit+θ)g(t)

), for Reynolds number

close to its critical value. Where u(x, y, t) and v(x, y, t) are solution of equation

(2.6), σi is the imaginary part of the eigenvalue predicted by the solution of equation

45

Reynolds Number Linear Stability Analysis Solution of (2.6)Hk ω(H/U0) Hk ω(H/U0)

R = 250 1.35 0.535 1.256 0.505R = 300 1.50 0.567 1.256 0.468R = 400 1.50 0.541 1.256 0.386R = 500 1.50 0.523 1.256 0.325R = 600 1.50 0.509 1.570 0.363R = 700 1.65 0.538 1.570 0.311

Table 4.1: Comparison between the estimates of frequency and wavelength between predictionbased on the solution of (3.6) and (2.6). The calculation is done for Ng = 2, hg/(2H) = 0.4. Thecritical Reynolds number is R = 190 .

(3.6) corresponding to the maximum growth rate (<(σ)), φ(k, y) and φy(k, y) are the

dominant eigen mode and its derivative corresponding to k with the highest growth

rate, g(t) is amplitude of u and v and 〈〉 denotes horizontally averaged variables, i.e.,

〈u(y, t)〉 =1

L

∫ L

0

u(x, y, t)dx.

Similar comparison for pressure is also shown in figure 4.7. From the agreement

between the explicit numerical calculation and the prediction based on the solution

of Orr-Sommerfeld equation (3.6), we can infer that for Reynolds number R close

to its critical value, the flow in the saturation region ( for T (U0/H) > 600 ) can

be described as steadily propagating wave whose wavelength, frequency and mode

shape is close to that predicted by linear stability analysis.

46

0 0.5 1 1.5 2−6

−4

−2

0

2

4

6x 10

−4

y/H

u/U

0

Linear Stability Analysis

CFD

0 0.5 1 1.5 2−6

−4

−2

0

2x 10

−4

y/H

u/U

0


CFD

0 0.5 1 1.5 2−1

−0.5

0

0.5

1

1.5

2x 10

−4

y/H

v/U

0


CFD

0 0.5 1 1.5 20

1

2

3

4

5

6x 10

−4

y/H

v/U

0


CFD

−1 −0.5 0 0.5 10.006

0.008

0.01

0.012

0.014

y/H

p/P0


CFD

−1 −0.5 0 0.5 1−0.015

−0.014

−0.013

−0.012

−0.011

−0.01

−0.009

y/H

p/P0


CFD

Figure 4.3: Comparison of velocities and pressure predicted by Linear Stability analysis and byexplicitly solving equation (2.6). The comparison are made at T = 360 and two different locations,where velocities are π/2 out of phase with each other. The parameters used are R = 250, Ng = 2,hg/(2H) = 0.4 .

47

T=0.00

x/H

y/H

0 5 10 15

0

0.5

1

1.5

2

T=377.93

x/H0 5 10 15

0

0.5

1

1.5

2

T=566.89

x/H

y/H

0 5 10 15

0

0.5

1

1.5

2

T=755.86

x/H0 5 10 15

0

0.5

1

1.5

2

T=944.82

y/H

x/H0 5 10 15

0

0.5

1

1.5

2

T=1133.79

x/H

0 5 10 15

0

0.5

1

1.5

2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 4.4: evolution of vorticity ω = ∇× u as function of time for a random perturbation overthe steady flow arising from the solution of (3.1). The perturbation grow with the rate predictedby linear stability analysis for T (U0/H) < 400 and then saturates to an approximately constantvalue. For T (U0/H) > 400 flow can be described as a steadily propagating wave, whose wavelengthand frequency matches with the one predicted by linear stability analysis. The simulation is donefor R = 250, Ng = 2 and hg/(2H) = 0.4 .

48

200 400 600 800 1000 12000.9

0.95

1

1.05

1.1

1.15

TU0/H

ωH/U0

1100 1110 1120 1130 1140 11500.93

0.94

0.95

0.96

0.97

0.98

0.99

TU0/H

ωH/U0

Figure 4.5: Plot of vorticity at the grass tip as function of time, from this plot we can estimatefrequency of vortex passage at any time, the calculated frequency matches with the imaginary partof eigen value arising from the solution of equation (3.6) corresponding to k with the highest growthrate. The parameter used in this simulation is R = 250, Ng = 2, hg/(2H) = 0.4 .

49

0 5 10 15 200.93

0.94

0.95

0.96

0.97

0.98

0.99

x/H

ωH/U0

Figure 4.6: Plot of vorticity at the grass tip as function of x. The vorticity is calculated in theregime when the flow is saturated T (U0/H) = 800, from this plot we can estimate wavelength ofvortex passage at the grass tip. The calculated wavelength matches with wavenumber k for whichsolution of equation (3.6) predicts highest growth rate. The parameter used in this simulation isR = 250, Ng = 2, hg/(2H) = 0.4

50

0 0.5 1 1.5 2−0.015

−0.01

−0.005

0

0.005

0.01

0.015

y/H

(u−

<u>)/U0


CFD

0 0.5 1 1.5 2−5

0

5

10x 10

−3

y/H

(u−

<u>)/U0


CFD

0 0.5 1 1.5 2−4

−3

−2

−1

0

1x 10

−3

y/H

(v−

<v>)/U0


CFD

0 0.5 1 1.5 2−0.01

−0.008

−0.006

−0.004

−0.002

0

y/H

(v−

<v>)/U0


CFD

−1 −0.5 0 0.5 10.9

1

1.1

1.2

1.3

y/H

p/P0


CFD

−1 −0.5 0 0.5 1

0.4

0.5

0.6

0.7

0.8

y/H

p/P0


CFD

Figure 4.7: Comparison of velocities and pressure predicted by Linear Stability analysis anddeviation of velocity and pressure from its spatial average by explicitly solving equation (2.6). Thecomparison are made at T = 800 and two different locations (x/H = 10 and x/H = 35 ), wherevelocities are π/2 out of phase with each other. The parameters used are R = 250, Ng = 2,hg/(2H) = 0.4 .

Chapter Five

Linear stability analysis of flow

around emergent vegetation

52

y = 2H

Vegetated Plain

Main PlainU(y)

yx

Figure 5.1: Schematic diagram of conveyance channel adjacent to an emergent vegetation plain,and the schematic velocity profile in both regions. The green dot represents vegetation which comesout of the plane of the paper. The flow is directed in the x direction.

5.1 Introduction

There are many situation in nature where there exists an interface between dense

vegetation and a high conveyance channel, such as tidal creek with fringing man-

groves, salt marshes and tidal channels etc. The difference of hydrodynamic drag

between the vegetated plain and the main plain is characterized by strong shear

in the velocity profile near the boundary of vegetated plain and the main channel,

similar to the flow through submerged vegetation. Lab scale experiments simulating

flow around emergent vegetation shows existence of coherent periodic vortices at

the vegetation interface [19]. These coherent structures causes large momentum flux

across the vegetation due to strong lateral motion which consists of sweeps toward

vegetation and ejections away from the vegetation. The coherent structures also in-

fluence the exchange of mass and momentum between the two regions. The material

exchange have been found to be of great importance for the hydrologic and ecolog-

ical processes in the system. For example, the exchange of fresh water between a

tidal creek and fringing mangrove is known to be extremely important in preventing

hypersaline conditions which are harmful to the mangroves.

53

5.2 Linear stability analysis

The formation of coherent eddies near the vegetation interface under the influence

of an otherwise steady current can be understood as result of some hydrodynamic

instability. In order to understand the flow instability in the presence of flow around

emergent vegetation, we perform a linear stability analysis of the system. In this

analysis we use a continuum approximation of vegetation and represent the effect

of vegetation as a continuum drag field. Since both component of flow (u, v) in the

vegetated region are across the vegetation, we approximate the effect of drag due to

presence of vegetation by an isotropic drag field in the flow equation (2.6), which

can be represented mathematically as,

D = −C|u|u. (5.1)

In order to perform linear stability analysis we first calculate fully developed steady

solution u = U(y)x of equation (2.6) and equation (5.1) driven by a constant pressure

gradient dP/dx. Since the steady profile is in x direction, the momentum balance

equation (2.6) simplifies to the same from as equation (3.1), i.e.,

−dPdx

+ µU ′′(y)− S(y)ρCdNgU |U | = 0,

S(y) = 1, 0 < y < hg,

S(y) = 0, hg < y < 2H,

(5.2)

with the understanding that hg represents the extent of vegetation in the y direction

rather than vegetation height (see Figure 5.1). Since we are solving the same equation

as equation (3.1), the base profile U(y) from the solution of equation (5.2) has the

same structure as the one arising from the solution of equation (3.1). The base profile

U(y) is approximately uniform within vegetated region with U(y) ≈ Ug =√−dP/dxρCdNg

,

54

and has parabolic velocity profile in the unvegetated region. The discontinuity of

drag across the vegetation boundary along with the continuity of shear results in a

boundary layer δ near the vegetation boundary. By performing analysis similar to

the one done in section 3.1 we can derive the scaling of boundary layer δ ∼ Ubl/U0 ∼

(RNg)−1/3, where Ubl is the scale of velocity in the boundary layer, µ is turbulent

eddy viscosity, U0 = (−dP/dx)H2/µ is velocity scale in unvegetated region and

R = ρU0H/µ is the Reynolds number of the flow.

5.2.1 Governing equation for flow instability

We substitute u = (U + u, v), p = (P + p) in (2.6) and expand to linear order to

investigate the evolution of small perturbations (u, v), which obey

ρ(ut + Uux + vUy) = −px + µ∇2u− 2SρCdNgUu,

ρ(vt + Uvx) = −py + µ∇2v − SρCdNgUv, ∇ · u = 0,

(5.3)

we have drop tilde for convenience. Comparing (5.3) with the (3.5), we observe that

the use of isotropic drag leads to an additional term in the v momentum equation

which lead to addition of an extra term in the equation 3.6, implying that to leading

order the flow perturbation (u, v) experiences additional drag for flow in vertical

direction (v component) as well due to CdNgUv term in the v component of the

momentum equation. With stream function ψ to satisfy mass balance and seeking a

solution of the form (u, v, ψ) = ( ˆu(y), ˆv(y), φ(y))eikx+σt, we arrive at a modified form

of Orr-Sommerfeld equation [23, 24, 25].

R−1(D2 − k2

)2φ =

[(σ + ikU)

(D2 − k2

)− ikUyy

]φ+ NgD (2SUDφ)− NgUk

2φ,

(5.4)

55

5.2.2 Results

Comparing equation (5.4) with (3.6), we notice that due to isotropic drag equation

(5.4) have an additional dissipative term Dy = NgUk2φ. A heat map of <(σ) as a

function of R and k for different hg/H and Ng is shown in figure 5.5. The smallest

R on the neutral curve (<(σ) = 0) sets the threshold for flow instability. We observe

that as Ng increases, the unstable region splits into two. Similar to anisotropic

drag case we refer to the region with higher k as “Mode 1” and one with lower

k as “Mode 2”; however the instability associated with large k gets damped due

to this additional drag dissipation and hence at large Ng, Mode 1 doesn’t set the

threshold for flow instability. The damping of Mode 1 at higher grass density can

also be understood by observing that the additional drag Dy ∝ k2; hence, the flow

instability associated with Mode 1 ( k ∝ H/δ ) gets significantly damped compared

to the instability associated with Mode 2 (k ∼ O(1)). We numerically observe that

threshold Reynolds number for large Ng to be R ∝ Ng or R ∝ (δ/H)−3/2 ( see figure

5.5 ), similar to the case with anisotropic drag ( see figure 3.5 Mode 2 asymptote ).

An analysis similar to one in section 4.3 can be done here as well to simplify the 5.4

σ(D2 − k2

)φ = −2(Ng/R)1/2D2φ+ (Ng/R)1/2k2φ, for y < hg (5.5a)

(σ + ikU)(D2 − k2

)φ = ikUyyφ, for y > hg. (5.5b)

The only remaining parameter in (5.5) is R/Ng. The parameter R/Ng sets the

threshold justifying the numerically observed asymptotic behavior R ∝ Ng or R ∝

(δ/H)−3/2. For fixed R/Ng, the mode shape converges in the aforementioned limit,

in agreement with the numerical result shown in figure 5.3. The structure of mode

in the aforementioned limit is such that φ is continuous at y = hg, but φy undergoes

a rapid transition on the scale of boundary layer thickness δ ( see figure 5.3 ).

56

0 5 10 15 20 25 30 350

10

20

30

40

50

60

70

80

U(y)(cms−1)

y(cm)

White, Case X

Solution of 5.2

White, Case I

Solution of 5.2

Figure 5.2: Comparison of steady profile from the solution of equation (5.2) with that from theexperiments in White [1] (Case X and Case I from figure 4.6 and figure 4.15) .

Experimental details Observed frequency Calculated frequency Ng

Case X 0.112 0.068 3.5Case I 0.066 0.043 0.2

Table 5.1: Comparison between the estimate of frequency with that from experimental observationof White in [1] (Case X and Case I from Figure 4.6 and Figure 4.15). The corresponding estimateof Ng is also shown in the table.

A comparison of steady profile and the frequency of resultant flow instability with

that of experimental observation of White in [1] is shown in Figure 5.2 and in table

5.1. The steady profile is calculated using the given flow rate constrain. Since these

experimental data correspond to vegetation density for which both the mode have

not split into two in the R− k space ( see Ng in 5.1 and column 1 in Figure 5.4), we

are unable to determine if the flow instability observed in the lab scale experiments

are the result of Mode 1 instability or Mode 2 instability.

57

0 0.5 1 1.5 20

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

y/H

|φ|

δ/H =0.01

δ/H =0.0069

δ/H =0.0025

δ/H =0.0013

0 0.5 1 1.5 20

0.01

0.02

0.03

0.04

0.05

y/H

|φy|

δ/H =0.01δ/H =0.0069δ/H =0.0025δ/H =0.0013

−20 0 200

5

10

15

(y − hg)/δ

|φy|/|φ

y(h

g)|

Figure 5.3: Plot of the neutral Mode 2 shape |φ| in the limit of small δ/H for hg/H = 0.2 inthe top panel. The approach of mode shapes to each other for these small values of δ/H indicatesthat the dense vegetation asymptote is reached. The bottom graphs shows that φy undergoes rapidtransition on the scale of boundary layer thickness δ.

58

R

k

Unstable

Stable

hg/(2H) = 0.2

Ng = 70

0.5 1 1.5 2

x 104

1

4

7

Mode 2

Ng = 100

1 2 3

x 104

1

5

9

Mode 2

Ng = 200

1 2

x 104

1

5

9

R

k

Unstable

Stable

hg/(2H) = 0.3

Ng = 100

0.3 1 2

x 104

1

4

7

Mode 2

Ng = 150

0.3 1 2

x 104

1

3

5

Mode 2

Ng = 200

1 2

x 104

1

3

5

R

k

Unstable

Stable

hg/(2H) = 0.4

Ng = 100

0.3 1 2

x 104

1

3

5

Mode 2

Ng = 150

0.5 1.5 2.5

x 104

1

4

Mode 2

Ng = 200

0.1 1 2

x 104

1

3

5

R

k

Unstable

Stable

hg/(2H) = 0.5

Ng = 100

2000 4000 6000 8000

1

3

5

Mode 2

Ng = 200

5000 10000 15000

1

3

5

Mode 2

Ng = 400

5000 10000 15000

1

3

5

−0.06

−0.03

0.00

0.03

0.06

Figure 5.4: Re(σ) and the neutral curve (Re(σ)=0) as a function of wavenumber and R forparameters shown in the corresponding panel. As Ng increases, the unstable region with higher kvalue, i.e. the region corresponding to “Mode 1” become stable due to increase dissipation.

59

100

101

102

103

102

103

104

105

106

H/δ

R

hg/H = 0.4hg/H = 0.6hg/H = 0.8

(H/δ)3/2

100

101

102

103

0.6

0.8

1

1.2

1.4

1.6

1.8

2

H/δ

k

hg/H = 0.4hg/H = 0.6hg/H = 0.8

Figure 5.5: Critical Reynolds number ( top ) and the corresponding marginally stable wavenumber ( bottom ) for different submergence ratio as a function of vegetation density parametrizedby the boundary layer thickness.

Chapter Six

Discussion

61

6.1 Role of turbulence Model

In our mathematical model we have used constant eddy viscosity, with this simple

assumption about eddy viscosity, we are able to capture essential features of the flow

instability. However, in a real flow the eddy viscosity varies with position, so this

simple assumption about constant eddy viscosity only provides qualitatively accurate

description of the flow. In this section, we present an account of the advantages and

shortcomings of assuming a constant eddy viscosity to model the turbulence.

One of the consequence of using constant eddy viscosity is that the boundary

layer thickness δ scales as H(RNg)−1/3; however, lab scale experimental observations

carried by Nepf [27] show that the boundary layer thickness scales instead as N−1g [27].

Since the scaling of boundary layer arises due to balance of viscous stress and drag

force in the boundary layer, the precise nature of scaling is dependent on the precise

nature of eddy viscosity used in the model. While the precise scaling of boundary

layer thickness is governed by the details of the turbulence model, the existence of

this boundary layer for dense vegetation is independent of the turbulence model. In

this analysis we have captured one possible realization of this feature using constant

eddy viscosity. Experiments have also shown that a model based on constant mixing

length l better approximates the turbulent characteristics of the flow with l ∼ δ

[1, 27]; i.e., assuming that the boundary layer itself establishes eddies to transport

momentum. The eddy viscosity corresponding to this model can be approximated

as µ ∼ ρUδ, and the dominant balance between turbulent momentum transport

and vegetation drag in the boundary layer is µU/δ2 ∼ ρCNdNgU2. Substituting µ

and solving for δ yields δ/H ∼ N−1g , which is in agreement with the experimental

observations.

62

The scaling of experimentally observed boundary layer thickness δ ∝ N−1g can

also be understood with in the framework of our model. The mixing length model

implies that near the grass tip, eddy viscosity scales as µ ∼ ρUblδ, which in turn

corresponds to an effective R ∼ U0H/Ublδ in the boundary layer. Furthermore,

continuity of shear stress at the grass top implies Ubl/U0 ∼ δ/H, and therefore

R ∼ (H/δ)2. Substituting this relation in the scaling predicted by our model, i.e.

δ/H ∼ (RNg)−1/3 and solving for δ yields δ/H ∼ N−1

g , which is consistent with the

experimental observation. This simple scaling analysis shows that the boundary layer

thickness depends on the turbulence model, and indicates that turbulence models

based on mixing lengths will yield more realistic scalings for boundary layer thickness.

At the same time, the qualitative features of the instability are represented by our

analysis.

We observe that the Mode 1 instability is driven by the intense shear on the

scale of the boundary layer. The driving mechanism for this instability is found to

be similar to the KH, and relies only on the presence of this shear as presented in

the Uηη term in equation (3.9). Since the existence of boundary layer near the grass

tip is independent of the turbulence model used, we expect Mode 1 instability to be

exhibited irrespective of the exact turbulence modeling used in analysis. We further

expect that the fastest growing wavenumber to be proportional to 1/δ, and the mode

to be localized to the boundary layer, irrespective of the turbulence model, because

these results have a basis in dimensional analysis. However, the exact scaling of

critical parameters such as threshold R for flow instability may depend on the precise

turbulence model used.

For the Mode 2 instability we have found that to leading order turbulent momen-

tum transport is irrelevant. In the asymptotic limit of dense grass, equation (3.11)

shows that the instability is driven by the interaction of the unvegetated flow with

63

the vegetation drag. The influence of the turbulence model is limited in only regu-

larizing the sharp transition in tangential velocity across the grass top. Therefore we

expect Mode 2 and its features to be preserved even if a different turbulence model

is used.

From the analysis of Mode 1 and Mode 2 in section 3.3, we observe that, in

order to distinguish Mode 2 from Mode 1, one needs to perform experiment with

asymptotically high grass density. For example, for a typical flow with R = 1000,

grass diameter d = 5 mm, H = 50 cm, one would require a really high grass density

Ng ≈ 4.0 × 105/m2. While performing experiment with such a large grass density

might not feasible, an elaborate analysis with more realistic model for eddy viscosity

and the drag coefficient with the framework of our model will be helpful in determin-

ing if these two modes can be distinguished by performing experiments at natural

grass density, i.e, at Ng ≈ 103/m2.

6.2 Impact of non-linear dynamics

While linear stability analysis can provide great insight into the mechanism of flow

instability by predicting threshold Reynolds number and the dominant wavelength

corresponding to the resulting flow instability etc, its analysis usually is not sufficient

for predicting mixing as a result of flow instability. In order to predict the mixing due

to resulting hydrodynamic instability, we need to know the flow structure when the

perturbation to the flow has grown such that the approximation of linear stability

analysis does not hold. So, one needs to perform full numerical simulation of the

flow with vegetation. Such calculation are quite expensive in terms of numerical

simulation. In chapter-4, we have performed such calculations for various Reynolds

64

number at given grass density. Our result indicates that when the Reynolds number

is close to the corresponding critical values, the flow can be described as a steadily

propagating wave, whose wavelength, frequency and mode shape is similar to the one

predicted by linear stability analysis. This indicates that for Reynolds number close

to the corresponding critical value, the result of linear stability analysis is very useful

in predicting detailed flow structure and hence predicting mixing near the grass tip

due to flow .

6.3 Future Work

In this work, we have developed a mathematical model to understand hydrody-

namic instability leading to monami . In order to understand the essential feature

of flow instability, we have made number of approximations such as use of constant

eddy viscosity, use of constant drag coefficient etc. While these approximations are

good starting in understanding the essential features of flow, a detailed analysis

with more realistic approximation of eddy viscosity and drag coefficient is expected

to have much better agreement with the experimental data. In this analysis, we

have also assumed that the grass is rigid and the dominant interaction between

flow and vegetation is through vegetation drag and vegetation responds passively

to the flow structure. While this a good approximation for the cases where time

scale associated with the resulting flow structure and the time scale associated with

natural frequency of grass are well separated, this assumption breaks down when

the natural frequency of vegetation and frequency associated with the resulting flow

structure are similar. Indeed, previous investigation have found that when the natu-

ral frequency of vegetation and the frequency associated with the flow structure are

similar, the flexibility of vegetation modifies the details of flow instability in the form

65

of frequency-locking of the hydro-elastic oscillations with the hydrodynamic insta-

bility with different strength, depending on canopy being terrestrial [13] or aquatic

[28]. However, in the analysis of Gosselin & De Langre[13, 28] , they have made an

ad-hoc assumption of piece wise linear base velocity profile and adhoc assumption

about mode shape [13, 28], a more detailed analysis taking into account the impact

of vegetation flexibility without making any ad-hoc assumption about base velocity

profile or mode shape can lead to better agreement with the experimental data.

Appendix A

Matched asymptotic solution of

base profile

67

Here we present an asymptotic solution of equation (3.1) for Ng � 1, which should be

helpful in further understanding of boundary layer near the grass top. For simplicity

of calculation in this particular calculation we assume the grass to extend from y = 0

to y = H, i. e. a submergence ratio is 0.5. Using U0 = (−dP/dx)H2/µ as velocity

scale, H as length scale, we rescale velocity as U = U0U and y = Hy. Equation (3.1)

is then transformed to

Uyy + 1− ρU0H

µCNdNgU

2= 0, for y < 0

Uyy + 1−RNgU2

= 0, for y > 0.

(A.1)

For notational convenience, we will drop the overbar and define a drag parameter as

Dg = RNg for −1 < y ≤ 0, Dg = 0 for 0 < y ≤ 1, grass top is at y = 0 and the

bottom surface is at y = −1. Equation (3.1) now reads

Uyy + 1−DgU2 = 0. (A.2)

We use the zero shear boundary condition on both the boundaries (y = −1, 1) as

mentioned in the manuscript.

Asymptotic Solution of 3.1 below the grass

We observe that U =√

(1/Dg) is a stationary point of above equation, where both

Uyy = 0 and Uy = 0. By multiplying with Uy and integrating once, we get

1

2

(dU

dy

)2

+ U − 1

3DgU

3 + C = 0 (A.3)

Where C is a constant of integration, which can be determined by observing that as U

approaches 1/√Dg, Uy approaches zero. Applying this condition gives C = − 2

3√Dg

.

68

Using the value of C, (A.3) becomes

dU

dy=

√2

3DgU3 +

4

3√Dg

− 2U (A.4)

We further use the substitution U =1√Dg

u in the above equation, which simplifies

it to

du

(u− 1)√u+ 2

= D1/4g

√2

3dy (A.5)

Above equation can be integrated to obtain the solution in the vegetated region as

u = 3 coth

(C1 − yD1/4

g√2

)2

− 2

U =1√Dg

(3 coth2

(C1 − yD1/4

g√2

)− 2

) for −1 ≤ y ≤ 0, (A.6)

where C1 is constant of integration, which can be found by matching the shear stress

applied by the flow above the grass at the tip of the grass. Deep inside grass the

solution saturates to U(y) = 1√Dg

independent of the value of C1 due to the dominant

balance between drag and pressure gradient.

Solution above the grass

Above the grass, the dimensionless form of equation (3.1) translates to

Uyy + 1 = 0 (A.7)

which can be integrated together with the zero shear boundary condition at top to

69

provided

U(y) = y − y2/2 + C2 for 0 < y < 1 (A.8)

Where C2 is constant of integration.

Matching the solution obtained by (A.6) and (A.8)

The value of C1 and C2 can be determined by matching the solution (A.6) and (A.8)

at the grass tip. Matching the shear stress at the grass top results in

3√

2

D1/4g

coth

(C1√

2

)csch2

(C1√

2

)= 1; (A.9)

Above equation can be solved numerically, but here we will solve it asymptotically

for Dg � 1. We expect C1 � 1 in this limit, and the terms in (A.9) may be expanded

in this limit to get

3√

2

D1/4g

(√2

C1

)(2

C21

)= 1

C1 =(12)1/3

D1/12g

(A.10)

The boundary layer length scale may be derived from examining the scale of y for

which the argument of (A.6) changes asymptotic order. This examination implies

C1 ∼ O(yD1/4g ) or y = O(D

−1/3g ). The scale for velocity in the boundary layer is the

70

value of (A.6) at y = 0.

Utop =1√Dg

(3 coth2

(C1√

2

)− 2

)(A.11)

≈ 1√Dg

(6

C21

− 2

)(A.12)

≈ 1√Dg

6D1/6g

122/3(A.13)

≈(

3

2

)1/3

D−1/3g . (A.14)

We have employed one of the many possible methods to understand the structure

of the solution of (3.1); other methods can also be used to reach the same conclusions.

For a general treatment of the topic, we refer the book by Hinch [29].

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Hydrodynamic instability leading to Monami