An integrated numerical model for vegetated surface-saturated subsurface flow
interaction
K.S. Erduran
Department of Civil Engineering, Faculty of Engineering & Architecture,
University of Nigde, Campus, 51245, Nigde, Turkey.
Phone: ++903882252288
Fax: ++903882250112
[email protected]
Abstract: This study involves the development of an integrated numerical model to deal with
interactions between vegetated surface and saturated subsurface flows. The numerical model
is built up by integrating quasi three dimensional (Q3D) vegetated surface model with two
dimensional saturated groundwater flow model. The vegetated surface model is constructed
by coupling the explicit finite volume solution of the two dimensional shallow water
equations with the implicit finite difference solution of Navier stokes equations for vertical
velocity distribution. The subsurface model is based on the explicit finite volume solution of
two dimensional saturated groundwater flow equations. Ground and vegetated surface water
interaction is achieved by the introduction of source-sink terms into the continuity equations.
Two solutions are tightly coupled in a single code. The integrated model has been applied to
two test cases and the results are satisfactory.
Key Words: Vegetated surface flow, saturated groundwater flow, flow interactions, tight
coupling, finite volume method, finite difference method, flow resistance.
1
Abbreviations
h : water depth (m)
v x and v y : depth-averaged velocity components in x and y directions respectively (m/s)
q i : flow due to infiltration (m/s)
q sp : excess of water coming from the ground (m/s)
g : acceleration due to gravity (m/s²)
So x and So y : bed slope in x and y directions respectively
Sf x and Sf y : friction terms in x and y directions respectively


F x and F y : average drag forces in x and y directions respectively (N/m²).
u x , u y and u z :velocity components in x, y and z directions respectively (m/s)
 : density of water (kg/m³)
τ x and τ y : vertical shear stresses in x and y directions respectively (N/m²)
Fx and Fy : drag forces in x and y directions respectively (N/m²)
 x and  y : vertical eddy viscosities along x and y directions respectively (m²/s)
m : density of vegetation per m²
C d : drag coefficient (empirical constant)
d : diameter of a reed (m)
hr : effective reed height (m)
z : vertical space step(m)
S y : specific yield (constant)
H : groundwater head (m)
K x , K y and K z : hydraulic conductivities in x, y and z directions respectively (m/s)
E : thickness of fully saturated groundwater inside the aquifer (m)
2
Z : vertical elevation from the bottom (m)
t : time step (s)
f : groundwater fluxes normal to the cell interfaces (m²/s)
L : length of a cell side (m)
A : area of the cell (m²)
h up : updated water depth (m)
H up : updated groundwater head (m)
v xup and v up
y : updated depth averaged velocities in x and y directions respectively (m/s)
up
u xup , u up
y and u z : updated velocity components in x, y and z directions respectively (m/s)
h n 1 : next time step values of water depth (m)
H n 1 : next time step values of groundwater head (m)
v xn1 and v yn1 : next time step values of depth averaged velocities in x and y directions
respectively (m/s)
u xn1 , u yn1 and u zn1 : next time step values of velocity components in x, y and z directions
respectively (m/s)
3
1. INTRODUCTION
With the advent of fast digital computers and numerical methods, it has become possible to
solve numerically many engineering problems. In this study, an integrated numerical model,
which is constructed for simulation of vegetated surface - saturated subsurface flow
interactions, is introduced. Surface and subsurface flow interactions are common in nature
and these flow processes are the core of the hydrological cycle. The most frequent example
takes place between river and aquifer and it often occurs between overland flow and aquifers.
These flow processes get more complicated if the surface flow is obstructed by natural causes
or manmade structures. This study concentrates on numerical modeling of such flow
processes that occur in areas, where the surface flow is often covered and obstructed by
vegetation and there exists interaction between the surface and subsurface flow.
Many integrated groundwater and surface water models have been developed. These models
can be distinguished according to their spatial and time dimensions and the type of equations
and solutions methods used. These factors help us to understand the complexity of the model.
In nature, flow is truly three-dimensional. However, dimensional reduction as well as other
simplifications on the governing equations and their solutions are made. Hence, surface water
flow processes are generally described by 2D shallow water equations or 1D de Saint-Venant
equations. In the case of vegetated flow, such simplifications are limited as the flow velocity
generally shows considerable changes in all three spatial directions and also experimental
studies prove that the flow shows turbulent characteristic [1]. The main impact of vegetation
on flow is that it causes a drag resulting in momentum losses [2]. Vegetative characteristics
such as height, diameter, placement and stiffness of the vegetation play important role on the
flow components [3, 4, 5]. Flow is also controlled by the condition of the vegetation, i.e.
whether the vegetation is submerged or non-submerged. Palmer [6] classifies three flow
conditions low flows (vegetation is emergent and no bending occurs), intermediate flows
(vegetation is completely submerged and bent and flow resistance shows rapid changes with
changes in discharge) and high flows (vegetation is forced to a prone or nearly prone
position). As stated by Carollo et al. [7] the most design problems, corresponding to vegetated
flows fall within intermediate flows and low flow type.
Wu et al. [8] also show that
vegetation can cause not only a drag effect but also a blockage effect. All these parameters
either reduce or increase the drag effects induced by vegetation; hence, they result in a change
in flow components. In addition, vegetation plays significant role on dispersion [9]. These
4
parameters also indicate that the computation of flow through vegetation is not an easy task.
Many attempts, most of which are based on experiments, have been made to find an accurate
way to compute flow resistance induced by vegetation [1, 4, 7, 10 - 12]. The earliest approach
to compute uniform flow through rigid vegetation is to use Manning’s formula. Petryk and
Bosmajian III [13] introduce a rather simple approach for one-dimensional steady uniform
flow including drag forces caused by vegetation. A more mechanistic method for the
computation of flow through flexible vegetation is proposed by Kouwen [14]. Flow through
submerged and non-submerged vegetation is studied by Fischenich [2] and Wu et al. [4].
Helmiö [15] studies unsteady one- dimensional flow in a compound channel with vegetated
floodplains. The model is later applied to the Rhine River in order to assess the effects of
resistance caused by partially vegetated floodplains [16]. The effects of type, placement, and
density of vegetation on flow components are experimentally studied by Järvelä [10]. Stone
and Shen [17] introduce physically based formulas for computation of flow resistance valid
for submerged and emergent rigid vegetation. Among the previously developed numerical
approaches the following studies covers more general vegetated flow conditions. Kutija and
Hong [18] introduce one-dimensional model, suitable for computation of flow through
flexible, rigid, submerged and non-submerged vegetation. Darby [17] develops a 1D flow
model, which can be dealt with flexible and rigid vegetation. Vionnet et al. [20] determine the
flow resistance as well as eddy viscosity coefficients numerically by so called lateral
distribution method for flow through flexible plants. 2D depth averaged model with drag
effects is employed for the investigation of the effects of vegetation on flow [21]. Simoes and
Wang [22] introduce a quasi three-dimensional (Q3D) turbulence model. Their Q3D model is
suitable for simulation of flow through rigid vegetation. Shimizu and Tsujimoto [23] also
develop a turbulence model, applicable only for the rigid vegetation. Recently, Fischer-Antze
et al. [24] introduce a 3D k-  turbulence model, which is suitable for rigid submerged
vegetation. Zhang and Su [25] also use 3D LES model for the simulation of flow through
rigid, straight and smooth cylindrical vegetation. Erduran and Kutija [3] introduce a Q3D
simple turbulence model that is applicable to flexible, rigid, submerged and non-submerged
vegetation.
As for the subsurface water flow model, previously developed groundwater models also differ
according to the equations and the solutions methods used. Generally speaking, groundwater
flow is modelled by solving either Richards equation (saturated, unsaturated and variable-
5
saturated flow can be dealt with), or a rather simple equation, which is based on Darcy law
and only suitable for simulation of saturated flow processes [26, 27].
The problem gets more complex when a surface water model needs to be coupled with a
groundwater model. Two types of coupling can be achieved; external (loose) coupling and
internal (also known as tight or dynamic) coupling. In external coupling surface and
groundwater simulation is done simultaneously (one after another) whereas in internally
coupled models coupling is provided within the same time level. The latter requires a single
source written for both surface and ground water computations and this coupling is also rather
difficult to implement comparing with external coupling [28-30].
The main aim of this study is to construct a numerical model that internally couples surface
and subsurface flow solutions as well as that is capable of dealing with flow through flexible
or rigid and submerged or non-submerged vegetation. Hence, in this study, much attention is
paid to the coupling technique and the model features and limits rather then numerical
solution techniques as they are partially presented in details in the previous studies [3, 28, and
31].
In the following sections, the equations and their solutions used in the model are briefly
introduced. The developed model is applied to two test cases and the model ability for dealing
with vegetated surface-saturated subsurface flow has been demonstrated. Assumptions made
to develop the model and the model applicability are discussed. Finally, conclusions are
presented.
2. METHODOLOGY
The integrated model is constructed by internally coupling vegetated surface flow solution
with the solution of saturated groundwater flow. In order to avoid the repetition, the solutions
will not be given in detail but the corresponding references, where the solution in details can
be found, will be given.
6
2.1. Used Equations and Solution Methods
The vegetated surface flow computation is achieved by using two main modules within a
program. In the first module, the shallow water equations with drag forces are solved by the
finite volume method (FVM). The numerical fluxes are computed by Roe scheme [32] and
the upwinded technique [33, 34] is used to deal with the bottom slope. This provides better
flux balances in the existence of bottom slope. In this module, the following shallow water
equations with drag forces are solved.
h hv x  hv y 


 qi  q sp
t
x
y
hv x   hv x2  gh 2 2 hv x v y 


 gh So x  Sf x   F x
t
x
y

 hv y 
t

(1)

 hv x v y 
x


 hv y2  gh 2 2
y
  gh So
y
 Sf y  F y
(2)
(3)
where h is the water depth, v x and v y represent the depth-averaged velocity components in
the x and y directions respectively, qi is the infiltration from surface to ground, q sp is the
excess of water coming from the ground, g is the acceleration due to gravity, So x and Sf x are
the bed slope and friction terms respectively in the x direction and similarly So y and Sf y in


the y direction. F x and F y are the depth averaged drag forces in the x and y directions due to
vegetation.
The reader may refer to the following references [3, 31] for the solution to Equations (1)
through (3). In the second module, Navier Stokes equations are solved in the vertical direction
using the implicit finite difference method on grids, which lie vertically above the cell centres
of the finite volumes in the first module, see Figure 1. The Navier Stokes equations including
the drag forces can be given as
u x u y u z


0
x
y
z
(4)
u x 1 τ x
u
u
u
h

 Fx  u x x  u y x  u z x  g
 gSo x  0
t
ρ z
x
y
z
x
(5)
u y
t

u y
u y
u y
1 τ y
h
 Fy  u x
 uy
 uz
g
 gSo y  0
ρ z
x
y
z
y
7
(6)
where u x , u y and u z are the velocity components in the x, y and z directions respectively, 
is the density of water, τ x and τ y are the vertical shear stresses in the x and y directions
respectively, Fx and Fy are the additional drag forces per unit area due to vegetation in the x
and y directions respectively. The vertical shear stresses are represented in terms of vertical
viscosity and the vertical gradient of horizontal velocities as shown in Equation (7).
τ
u
 ε   ,  = x, y
ρ
z
(7)
where  x and  y are the vertical eddy viscosities along the x and y directions respectively.
For computation of the vertical viscosity values for vegetated flow, Kutija and Hong [18]
approach is opted here.
As seen in Equations (4) to (6), the momentum equation in the vertical direction is omitted.
Hence, a solution is quasi three-dimensional (Q3D). Drag forces, Fx and Fy , in the x and y
directions due to vegetation are zero above the vegetation and inside the vegetative
watercourse they can be computed as
Fx  m
hr C d u x u x2  u y2 d
2Δz
Fy  m
hr C d u y u x2  u y2 d
2Δz
(8)
where m is the density of vegetation, C d is a drag coefficient, d is diameter of a reed, hr is
effective height of a reed, see Figure 2.
In the solution of equations (4) to (6), different discretisation techniques are used in order to
increase the stability as well as ease the computation. The implicit finite difference
approximations are employed to the following terms; acceleration, drag forces, and the shear
stresses. Remaining terms are treated explicitly to decrease the computational effort. The
advective terms are discretisied by so called upwind technique. The horizontal gradients of
water depth are approximated using forward difference approximations. Resulting
approximated equations produce a system of linear algebraic equations. The number of
equations is equal to the number of grid points in the vertical direction. The unknown
8
horizontal velocities are computed using a double-sweep algorithm [35] since the matrix of
the system is tri-diagonal and they are computed at all the dicretisation points but the vertical
shear stresses are computed halfway between each two grid points.
The depth averaged drag forces given in Equations (2) and (3) can be computed by
kk

F 
 F 
k 0
i,j,k 
kk  1
,  = x, y
(9)
Equation (9) shows that the depth averaged drag forces are first computed in every vertical
grid point, k in the second module and they are later passed to the first unit. In order to add a
solution for flow through flexible vegetation, cantilever beam theory [36] is used.
The 2D groundwater flow equation including the infiltration term for homogeneous fluid with
constant density can be given as:
Sy
H

H

H
 (K x E
)  (K y E
)  qi
t
x
x
y
y
(10)
where S y is the specific yield, H is the groundwater head, K x and K y are the hydraulic
conductivity in x and y directions respectively, and E is the thickness of fully saturated
groundwater inside the aquifer.
Equation (10) without the term, qi is solved by the finite volume method. In the solution of the
equations by the FVM, the key element is to compute the fluxes through cell interfaces. The
Roe scheme is chosen for the surface water flux calculation whereas in the groundwater
solution, the fluxes are calculated by using Darcy’s Law. It is noted that the same finite
volume cells employed in the solution of 2D shallow water equations are used in the
groundwater flow computation, Figure 1.
9
2.2. Coupling Vegetated Surface-Saturated Subsurface Solutions
The vegetated surface water solution is coupled with the subsurface solution by considering
three cases. Although the computation always starts with the solution of the groundwater
equations, these cases determine the computational steps and are descried below;
Case A: The surface is wet and the water depth is prescribed, but the groundwater head is
below the ground level elevation for that cell, Figure 3a. In this case, there will be a flow from
surface to ground due to infiltration, computed by Darcy’s Law in the z direction:
q i = K z h  Z  H /Z
(11)
where q i is flow due to infiltration, K z is a hydraulic conductivity in z direction.
The splitting technique is applied to each continuity equation, Equations (1) and (10). Each
application then produces an ODE, which is solved by the first order Euler method. The
solutions update the groundwater head and the shallow water depth. Note that there should
not be the terms qi and q sp at the same time in the continuity equation of the shallow water
equations. In other words, while there is an infiltration, there should not be the flow from
ground to surface. As seen in Figure 4, the computation starts with the splitting technique.
After solving the resulting ODEs, the updated values for H and h over a time step, t are
obtained. In Figure 4, the superscript ‘up’ denotes for the updated values. While H up is later
used in the solution of the groundwater equation, h up is used in the solution of the shallow
water equations. As shown in Figure 4, the second splitting technique is employed to the
friction terms, and the bottom slope is treated by the upwinding technique. For this case, the
groundwater computation is completed with the explicit finite volume solution of the
groundwater equation and the solution gives the final values for the groundwater head and the
lateral groundwater unit discharges over a time step t. The solution to the shallow water
equations gives the final values for the water depth over a time step, t but not for the depthaveraged velocities as the solution does not cover the drag effects yet. That is why, in Figure
4, the depth-averaged velocities are shown as v xup and v up
y . The vegetated surface computation
continues with the solution of the Navier Stokes equations where the previously computed
next time step values of water depth is used. The solution to the Navier Stokes equations
10
up
produces the updated values of u xup , u up
in the vertical direction. Using these
y and u z
velocities, the depth-averaged drag forces are computed and they are sent to the first module,
where the third splitting technique is used to include the drag effects and the final values of
the depth-averaged velocities are computed. In a brief, the first and the second modules in
Q3D solutions are interconnected through the water depth, and the averaged drag forces. For
this case, the computation ends with the correction of the velocities u x , u y and u z , that gives
the final values, i.e. u xn1 , u yn1 and u zn1 . The correction is made using the depth-averaged
velocities obtained from the first module. This correction has to be done to avoid the errors
caused by differences between the solutions of the shallow water equations (written in
conservative form) and the Navier stokes equations (in primitive form).
In the most of subsurface and surface water interactions described by Case A, the
groundwater is either not saturated or partially saturated. In such cases, obviously Equation
(11) cannot be valid. In this study, coupling of surface and subsurface solutions is the main
concern and in the further development of the model, this problem will be overcome by
solving Richard equation or at least Green and Ampt type infiltration equation is going to be
adopted [37, 38].
Case B: The groundwater head is above the ground level and is equal to h+Z, Figure 3b. The
surface is wet and there is no infiltration. In this case, an interaction is assured by
compatibility of the groundwater head and the water level. Again, the groundwater equation is
first solved and the change in storage, q sp , is known. Hence, the updated groundwater head
values are obtained over a time step, t. After the application of the finite volume method to
the groundwater equation, the change in storage is computed as
q sp 
1 mm
f j L j
A j 1
(12)
where mm is the number of sides of a finite volume cell, f is groundwater fluxes normal to the
cell interfaces, L is the length of a cell side, and A is the area of the cell.
As the groundwater head is above the ground surface, any change in the groundwater head
will affect the surface water depth. Therefore, the updated values of the water depth should be
calculated in a similar way that described in Case A, i.e. application of the splitting technique
11
first to Equation (1) and solving the resulting ODE with the Euler method. The updated water
depth values, h up are then used in the solution of the shallow water equations, as shown in
Figure 5. The computation continues as explained in Case A. However, the groundwater head
values should be recomputed as the groundwater head is above the ground and the final
shallow water depth values h n 1 are different than h up . This is done by summing h n 1 and Z
as illustrated in Figure 5.
Case C: There is no water on the surface and the cells are effectively dry, Figure 3c. To avoid
the zero-division problem, water depth is the prescribed value 0.00001m. There is no
integration between ground and surface as no infiltration occurs and groundwater head is
below the ground level. However, both the groundwater equation and the surface water
equations are still solved in order to compute the lateral flow movements. The cells around
the dry cell on the surface could be wet (there could be water on the neighbouring cells) and
therefore the surface water computation has to be carried out. In this case, the number of use
of splitting technique reduces to two; one for the treatment of the friction terms and one for
the drag forces. The computational steps for this case are illustrated in Figure 6.
3. RESULTS
The integrated model is applied to the vegetated surface and the saturated sub-surface flow
conditions. Two examples are chosen to test the model performance. In these examples, the
vegetated surface flow condition is set up according to the experiment conducted by
Tsujimoto and Kitamura [1] in order to compare the model results with those of the
experiment. However, as the original experiment does not include groundwater test data, in
addition to the experimental flow condition, subsurface flow condition for each test is also
artificially introduced in order to demonstrate the model ability to deal with flow interactions
between the surface and subsurface flows. The first test is set up to produce the coupling type,
Case A whereas the second one represents the occurrence of Case B.
Test 1: A computational domain, 12m long and 0.4m wide, is divided into 96 finite volume
cells, each of which has a size of 1x0.05m. In other words, there are 12 cells in the x direction
and 8 cells in the y direction. Initially, the groundwater head is assumed to have a constant
12
value of 7m everywhere except that it is 7.5m in the middle of the domain as shown in
Figures 7a and 7b. Although this hump in the middle of the domain may not physically occur,
such an abrupt initial groundwater flow condition is provided so that the model ability for the
simulation of 2D ground water motion can be clearly demonstrated. The ground elevation is
8m in the centre of upstream cells at x = 0.5m and reduces with a constant slope of 0.001 to
7.989m in the centre of downstream cells at x = 11.5m. Hence, the groundwater head is below
the ground surface everywhere. Hydraulic conductivity values in all directions are chosen to
be 1x10 5 m/s, which is within a range of 103 md-1 for gravels - 10-5md-1 for compact clays
[39]. The specific yield values are set to unity but the models allow to use any number
instead, generally this value changes, i.e. 0.01 for clay to 0.46 for sand [40]. The surface water
depth is 0.095m everywhere and initial velocities are set to zero. The surface of the channel is
covered with vegetation. The diameter of each plant is 0.15cm. The density of vegetation is
2500 per m². The stiffness value of 2Nm² is applied to produce rigid vegetation as used in the
experiment. Manning coefficient is chosen to be 0.025. Drag coefficient is set to be 1.1. The
number of vertical grid points is 21. The boundaries for the groundwater are assumed to be
closed. For the surface water, the water depth of 0.095m is applied as the upstream boundary
condition and the downstream boundary is assumed open. The initial conditions described
above and expected flow motions are schematized and given in Figure 7. The model is run
60000s with a time step of 0.02s.
As shown in Figure 7 with arrows, the expected flow motions will be as follows; the surface
water will flow from upstream to downstream of the channel due to the bottom slope whereas
the lateral groundwater flow occurs in both horizontal directions (x and y) around the points
where the groundwater head is 7.5m. The volume of groundwater is also expected to rise
gradually due to infiltration from surface to ground.
Figures 8 and 9 show the groundwater head changes caused by the lateral and vertical flows.
The most significant changes occur in the horizontal directions around the points where the
initial groundwater head is 7.5m. As expected, the volume of groundwater increases vertically
due to the infiltration. The lateral groundwater movements get slower as the time proceeds
and the groundwater head shows very small changes along the channel at 60000s.
After simulation of 100s, the surface flow becomes almost steady as the surface water
velocities and the water depth show very small differences (the maximum difference in water
13
depth values is less than 0.0001m) along the channel. Figure 10a illustrates the vertical
velocity profiles obtained from the experiment and the model results at 20000s. The
experimental result is shown as A11, which is named after Tsujimoto and Kitamura [1]. The
velocities in the vertical grid points on the remaining finite volume cells show a very similar
trend, as they are almost the same. This profile given in Figure 10a is a typical velocity profile
observed or computed for flow through submerged vegetation [1]. The profile has three
distinctive regions; near bed, vegetated region, and region above the vegetation. The profile in
the first region is formed mainly by the bed friction. The shape of the profile in the second
region is dominantly affected by the drag forces caused by the vegetation. In the third region,
water flows freely, no obstruction, and the velocity increases rapidly towards to the free
surface. The most significant change in the profile occurs around the top of the vegetation, in
other words, at the boundary between the second and third regions. This is obviously due to
transition from slower flow caused by vegetation to the faster flow above. For this test, no
deflection is observed as the stiffness value of 2Nm² is quite large and the velocities so the
loads acting on the vegetation are small. It may be worth mentioning again that this stiffness
value is chosen in order to reproduce the experimental condition, where the vegetation
(cylinders made of bamboo) is rigid. The trend of drag force profile is given in Figure 10b.
The drag effects increase upwards since the velocities increase. The typical shear stress
profile is illustrated in Figure 10c. The shear stress distribution is mainly controlled by the
gradient of the velocities in the vertical direction and so the shear stress values are larger at
the transition region, between second and third regions, whereas the velocity gradients below
the top of the vegetation are small so does the shear stress values. However, on the bottom,
the shear stress (more precisely, the bottom friction) computation does not depends on the
velocity gradient but it is computed by Manning friction formula [3]. On the top of the
vegetation, the velocity gradient decreases upwards and approaches zero, so does the shear
stress values. Another reason for decreasing shear stress values in this region is the use of
mixing length theory to compute the shear stress values [3].
Test 2: In this test, the computational domain is the same as Test 1 and also the data are
almost the same. However, the following changes are made. The domain is divided into 12x4
finite volume cells, each of which has a size of 1x0.1m. Initial water depth of 0.0895m is
applied everywhere and the bottom slope of 0.007 in the x direction is applied, producing
experimental case A71 introduced by Tsujimoto and Kitamura [1]. The upstream boundary
condition is a water depth of 0.0895m. Initially, the groundwater head is assumed to have a
14
constant value of 7.5m everywhere except that it coincides with the ground level between 5m
and 7m along the channel as shown in Figures 11a and 11b. The ground elevation is 8m in the
centre of upstream cells at x = 0.5m and reduces to 7.923m in the centre of downstream cells
at x = 11.5m. Hence, this test produces a coupling type, called Case B. Hydraulic conductivity
values in all directions are chosen to be 2 x10 4 m/s. The simulation is completed at 10000s
with a time step of 0.02s.
The groundwater head changes over a 10000s are shown in Figure 12. The groundwater
movements occur in both horizontal and vertical directions. The groundwater head rises and
reaches the surface water level everywhere at 10000s. The velocity profile obtained from the
model is compared with those of the experiment and shown in Figure 13a. The computed
horizontal velocities along the channel have almost a constant value of 0.2711m/s. The model
results again agree with the experimental ones. The drag force profile for this test is also given
in Figure 13b.
In order to demonstrate other features of the model, Test 2 is modified that both the stiffness
value and the diameter of the vegetation are reduced. They are taken to be 0.00001 Nm² and
0.001m respectively. These modifications are made to provide flexible vegetation in the
channel. The model is run for 80s and the velocities, deflections, drag forces as well as the
shear stresses at 10s, 20s, 30s, 50s, and 80s are demonstrated in Figure 14. The velocities in
the channel show almost no changes after 50s as illustrated in Figure 14a. The reduced
stiffness and the diameter result in deflection of the vegetation. The maximum lateral
deflection is around 0.35cm and the reduction in the effective vegetation height is about
0.1cm. Although the deflection and the reduction in the vegetation height are small, the
horizontal velocities increase from 0.2711m/s to 0.311m/s. In other words, 2 % reduction in
the vegetation height causes around 14.7 % increase in the horizontal velocities. As shown in
Figure 14, when the flow velocity increases so does the drag force, shear stress and the
deflection.
15
4. DISCUSSION
The results presented show that the model can deal with the vegetated surface saturated
subsurface flow interactions. It also provides user-friendly environment (i.e. it has buttons,
menu bar etc.) since it is written in Delphi, which is an object-oriented language.
In the developments of the model, apart from the principles assumptions made to drive the
equations, additional assumptions are made. In this section, these assumptions and the model
limitations are discussed.
Turbulence closure is achieved in a rather simple way. Eddy viscosity values are computed by
two approaches; in the vegetated watercourse, a formula introduced by Tsujimoto and
Kitamura [1] is used whereas above the vegetation mixing length theory is applied. The idea
to use these two approaches together is first introduced by Kutija and Hong [18] and later it is
used by [3]. Although the method is simple, it gives quite satisfactory results. Rodi [41] states
that the most universal turbulence model does not mean that it is also the most suitable one
for a particular problem. He suggests to use an easy and the most economical one among the
available solution algorithms if the satisfactory results can be obtained. In this study, in order
to solve the governing equations, to couple the surface and surface flow solutions and also to
capture the different flow conditions that previously stated, it was inevitable to make some
simplifications. Hence, simple algorithms were used for the solution of the turbulence closure
problem. Further simplifications were also made that the model is Q3D, which is provided by
assuming that the pressure distribution is hydrostatic. This assumption let us to omit the
vertical momentum equation of the Navier Stokes equations and that ease the overall
solutions. However, this assumption may not be acceptable particularly for flow around the
tip of the flexible vegetation under submerged condition. Although it is not particularly a
problem here as the results well agree with those of the experiment, generally speaking,
around this point, the flow shows highly turbulent characteristics and that can require a fully
3D solution in order to deal with non-hydrostatic pressure distribution.
Bending of vegetation is based on an assumption that the linear elastic theory is valid. In other
words, when the load acting on the vegetation disappears the vegetation goes back to the
original shape. This assumption is not always true. Depending on the vegetation type, after
bending the vegetation may not turn back to its original shape. That means the vegetation may
16
show non-elastic behaviour. The model, therefore, requires further alterations in order to
cover more general cases.
It is also word noting that groundwater flow solution is valid only for the saturated subsurface
flow condition. Furthermore, the solution is applicable to those flow conditions that Darcy
Law is valid. It is known that the moisture above the water table due to capillary forces is
neglected then the model will overestimate the surface-groundwater fluxes. The amount of
moisture above the water table is dependent on the soil type, the finer the soil (clays, muds)
the greater the capillary forces and hence the more moisture [42]. Therefore, the applications
of the model to such conditions require additional developments that addressed in section 2.2.
As in all numerical models, the number of grid points is important that the accuracy increases
with an increase in the number of grid points. It is worth reminding here that the number of
grid points in the vertical direction is particularly important. The computation of the load is
based on an assumption that the load between the points k+½ and k-½ is uniform and the
velocity at the point k is used to compute the load. If there is large interval between k+½ and
k-½ , and the velocities at points k -1, k, and k+1 show significant changes, the load
distribution between these points would no longer be uniform or close to uniform and cannot
be represented using the velocity in the middle (k point). Thus, the deflection of the vegetation
would be estimated incorrectly.
The simulation time for Test 1 is compared with the actual simulation time and plotted in
Figure 15a. The relationship between the simulation time and the actual simulation time is
linear for a particular Test. In these examples, the simulation times are shorter than the actual
simulation times, Figure 15c. However, more tests with increasing number of grid points are
necessary to reach a conclusion whether or not the above statement is always true. Figure 15b
shows the relationship between the number of finite volume cells and the actual simulation
time. The values for actual simulation time are corresponding to 10000s simulation for both
Test 1 and Test 2. It has been seen that increasing number of finite volume cells (as the
number of vertical grid points are the same for both tests) dramatically increases the actual
simulation time. It is word reminding here that the number of finite volume cells for Test 1 is
twice the number of finite volume cells for Test 2.
17
The drawbacks of the model and its applicability limits have been addressed. It is because in
selecting a numerical method, it is important to understand the assumptions and
simplifications under which the model is developed. Although as for the other numerical
tools, the model still has limitations, there are many areas that the current model can be
applied and it can produce quite accurate results. For example, the model is suitable for
simulation of surface flow with submerged or nonsubmerged, flexible or rigid vegetation. It
allows only 2D saturated groundwater flow simulation or only 2D surface water flow
simulation including shock wave resulting from dam break problems. It has been
demonstrated that surface and saturated groundwater surface flow interactions can be
modeled.
5. CONCLUSION
From this study, the following conclusions can be drawn. An integrated model with the
solution steps for simulation of flow interactions between vegetated surface and saturated
subsurface flows is introduced. The surface flow part can deal with flow through flexible,
rigid, submerged and non-submerged vegetation. The surface water solution is internally
coupled with the sub-surface solution. In the solution of the governing equations, both the
explicit finite volume and implicit finite difference methods are employed. The coupling
technique and the computational steps are explained. The model is tested and it has been
shown that the results are satisfactory. The model limitations are explained and some of the
model features are demonstrated.
18
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21
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22
List of Figures
Fig1 Grids used in the solution of equations
Fig2 Effective height of vegetation used in the computation of drag forces
Fig3 The vegetated surface-saturated subsurface flow interaction processes;
a) Case A, b) Case B, c) Case C
Fig4 Computational steps for Case A
Fig5 Computational steps for Case B
Fig6 Computational steps for Case C
Fig7 Computational domain with initial flow conditions and expected flow motions (arrows);
a) a view of the whole domain, b) a view of any section across the channel between x = 4m
and x = 8m c) a view of a section between x = 3m and x = 9m along the channel and between
y = 0.150m and y = 0.250m across the channel
Fig8 Groundwater head profiles and contours at; a) t = 10s and b) t = 50s
Fig9 Groundwater head profiles and contours at; a) t = 10000s and b) t = 60000s
Fig10 Vegetated surface flow results for Test 1; a) velocity profile, b) drag force profile, c)
Shear Stress profile
Fig11 Initial flow conditions and expected flow motions (arrows); a) a view of any crosssection between x = 5m and x = 7m, b) a view of any section between x = 4m and x = 8m
along the channel
Fig12 Groundwater head profiles along the channel
Fig13 Vegetated surface flow results for Test 2; a) velocity profile, b) drag force profile
Fig14 Vegetated surface flow results for modified Test 2;
a) velocity profile, b) deflection profile, c) drag force profile, d) shear Stress profile
Fig15 Comparisons of a) simulation and actual simulation times for Test 1, b) number of
finite volume cells and actual simulation time, c) simulation and actual simulation times for
Tests 1 and 2
23
Vertical Grid for Solution
of Navier Stokes
Equations
k= kk
Surface water Grid
k=1
i,j
i,j
Groundwater Grid
Fig1 Grids used in the solution of equations
24
Height of vegetation before bending
Flow
hr : Effective height of vegetation
(Height of vegetation after bending)
Fig2 Effective height of vegetation used in the computation of drag forces
25


Water Depth, h
Water Depth, h
Vegetation
Vegetation
Ground
Infiltration, qi
Ground

Excess water, qsp
Groundwater Head, H
Z
Groundwater Head, H
Datum
Datum
(a)
(b)
Vegetation
Ground

Groundwater Head, H
Datum
(c)
Fig3 The vegetated surface-saturated subsurface flow interaction processes;
a) Case A, b) Case B, c) Case C
26
H n , hn
qi First Splitting Technique,
Solution to ODEs
H up , h up
Solution to Shallow water
Equations by FVM + Upwinding Technique +
Solution to Groundwater
Second splitting technique, solution to ODE
Equations by FVM
h n 1
H n1 , q xn1 , q yn1
h
n 1
up
x
,v ,v
up
Solution to Navier Stokes
up
y
up
u x , u up
y ,uz
Equations by FDM
F x,F y
Third Splitting Technique,
Solution to ODE
v xn1 , v yn1
h
n 1
,v
n 1
x
,v
n 1
y
Correction for Velocities
Fig4 Computational steps for Case A
27
ux
n 1
, u yn1 , u zn1
Hn
Solution to Groundwater
Equations by FVM
q sp
H up , q xn1 , q yn1 , qsp
h up
First splitting technique
Solution to ODE
Solution to Shallow water
Equations by FVM + Upwinding Technique +
Second splitting technique, solution to ODE
h n1 , v xup , v up
y
h n 1
up
Solution to Navier Stokes
up
u x , u up
y ,uz
Equations by FDM
F x,F y
Third Splitting Technique,
Solution to ODE
v xn1 , v yn1
h n1 , v xn1 , v yn1
Correction for Velocities
H n 1  h n 1  Z
Fig5 Computational steps for Case B
28
ux
n 1
, u yn1 , u zn1
Hn
Solution to Groundwater
Equations by FVM
H n1 , q xn1 , q yn1
Solution to Shallow water
Equations by FVM + Upwinding Technique +
First splitting technique, solution to ODE
h
n 1
up
x
,v ,v
h n 1
up
y
up
Solution to Navier Stokes
up
u x , u up
y ,uz
Equations by FDM
F x,F y
Second Splitting Technique,
Solution to ODE
v xn1 , v yn1
h
n 1
,v
n 1
x
,v
n 1
y
Correction for Velocities
Fig6 Computational steps for Case C
29
ux
n 1
, u yn1 , u zn1
0.4m

Surface Water
0.095m
Flow
8m

0.095m
7m
Groundwater
12m
Bottom
Datum
7.989m
z
(0m)
y
(a)
 Surface Water
 Surface Water
0.095m
0.095m
0.0459m
0.0459m

7m
x

7.5m
7m
0.150m 0.1m 0.150m
1m
(b)
7.5m
4m
1m
(c)
Fig7 Computational domain with initial flow conditions and expected flow motions (arrows);
a) a view of the whole domain, b) a view of any section across the channel between x = 4m
and x = 8m c) a view of a section between x = 3m and x = 9m along the channel and between
y = 0.150m and y = 0.250m across the channel
30
(a)
(b)
Fig8 Groundwater head profiles and contours at; a) t = 10s and b) t = 50s
31
(a)
(b)
Fig9 Groundwater head profiles and contours at; a) t = 10000s and b) t = 60000s
32
0.050
9
0.045
8
0.040
Vegetation Height (m)
Water Depth (cm)
10
7
6
5
4
3
2
Model
1
A11
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
0
0
10
20
30
0.0
0.5
1.0
1.5
2.0
2
Drag Force N per m
Velocity (m/s)
(a)
(b)
10
9
8
Water Depth (cm)
7
6
5
4
3
2
1
0
0
0.5
1
1.5
Shear Stress N per m2
(c)
Fig10 Vegetated surface flow results for Test 1; a) velocity profile, b) drug force profile, c)
shear Stress profile
33
Surface Water also Groundwater Head

Surface Water

0.0895m
0.0895m
0.0459m
0.0459m


7.5m
1m
0.4m
(a)
2m
1m
(b)
Fig11 Initial flow conditions and expected flow motions (arrows); a) a view of any crosssection between x = 5m and x = 7m, b) a view of any section between x = 4m and x = 8m
along the channel
34
8.1
Groundwater Head (m)
8.0
7.9
7.8
7.7
7.6
7.5
0
2
4
6
8
10
x (m)
2000s
4000s
6000s
8000s
10000s
Ground Level
Fig12 Groundwater head profiles along the channel
35
12
10
9
Water Depth (cm)
8
7
6
5
4
3
2
Model
1
A71
0
0
20
40
60
Velocity (m/s)
(a)
0.050
0.045
Vegetation Height (m)
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
0
5
10
15
2
Drag Force N per m
(b)
Fig13 Vegetated surface flow results for Test 2; a) velocity profile, b) drug force profile
36
5.0
9
4.5
8
4.0
Vegetation Height (cm)
Water Depth (cm)
10
7
6
5
4
3
2
3.5
3.0
2.5
2.0
1.5
1.0
1
0.5
0
0.0
0
20
40
60
0.00
0.10
Velocity (m/s)
10s
20s
30s
0.20
0.40
Deflection (cm)
50s
80s
10s
20s
(a)
30s
50s
80s
(b)
0.050
10
0.045
9
0.040
8
0.035
7
Water Depth (cm)
Vegetation Height (m)
0.30
0.030
0.025
0.020
0.015
6
5
4
3
0.010
2
0.005
1
0.000
0
0
5
10
15
0
5
2
20s
30s
50s
15
2
Shear Stress N per m
Drug Force N per m
10s
10
10s
80s
(c)
20s
30s
50s
80s
(d)
Fig14 Vegetated surface flow results for modified Test 2; a) velocity profile, b) deflection
profile, c) drug force profile, d) shear Stress profile
37
6000
30000
5000
Actual Simulation Time (s)
Actual Simulation Time (s)
35000
25000
20000
15000
10000
4000
3000
2000
5000
1000
0
0
0
20000
40000
60000
0
Simulation Time (s)
50
100
Number of Finite Volume Cells
(a)
(b)
Actual Simulation Time (s)
6000
5000
4000
3000
2000
1000
0
0
5000
10000
Simulation Time (s)
Test 1
Test 2
(c)
Fig15 Comparisons of a) simulation and actual simulation times for Test 1, b) number of
finite volume cells and actual simulation time, c) simulation and actual simulation times for
Tests 1 and 2
38