UNIVERSITÀ DEGLI STUDI DI TRIESTE
XXIX CICLO DEL DOTTORATO DI RICERCA IN
Scienza della Terra e Meccanica del Fluidi
NUMERICAL AND EXPERIMENTAL
INVESTIGATION OF SUSPENDED
SEDIMENT TRANSPORT IN LAB-SCALE
TURBULENT OPEN CHANNEL FLOW
DOTTORANDO:
COORDINATORE:
SUPERVISORE DI TESI:
MAHMOUD JOURABIAN
PROF. PIERPAOLO OMARI
PROF. VINCENZO ARMENIO
ANNO ACCADEMICO 2016/2017
Abstract
Single-phase Euler-Euler based wall-resolving LES with the dynamic Smagorinsky model is used to investigate suspended sediment transport in a turbulent
open channel flow. Aspect ratio of the open channel is high. Bottom bed is
smooth. For the clear water flow, streamwise and vertical turbulence intensities in experiments of Muste et al. [2005] are much bigger than those in the
wall-resolving LES and DNS by Hoyas and Jimenez [2008]. Bulk velocity in the
sediment-laden flow is lesser than that in the clear water flow. Friction velocity
is same for both flows. While Muste et al. [2005] recorded that in the inner
region, the streamwise velocity of the sediment-laden flow was higher than that
of the clear water flow, wall-resolving LES shows no alteration. In the outer region, the depth-resolved streamwise velocity of the sediment-laden flow is lower
than that of the clear water flow. Introduction of suspended sand particles into
the turbulent open channel flow of the clear water results in the decrease of the
drag force while shear stress on the channel bed is constant. To get reduction
of the bulk velocity, the fast Eulerian method in the two-way coupling should
be employed.
Single-phase Euler-Euler based unresolved wall-function LES with the Smagorinsky model under the equilibrium stress assumption is implemented to understand interactions between turbulence and suspended sand particles in a turbulent open channel flow. Channel bed is rough. Aspect ratio of the open channel is low. To treat erosion from the bed, the reference concentration method
together with the Shields diagram is used. Results are compared against experiments of Cellino [1998]. Suspended particles engender reduction of the friction velocity, bulk velocity and roughness in contrast to the clear water flow.
Streamwise velocity is decreased in the outer region while it is expedited in the
super-saturated region near the channel bed. It is due to high inter-particle
collisions between sediment particles which are not bounded by viscosity. In
upper levels, remarkable weakening of the vertical turbulence intensity is seen.
When the buoyancy term is deactivated, the sediment concentration gets high
and unsatisfactory turbulence statistics are obtained. Wall shear stress on the
sidewalls of narrow open channels must be considered.
Effects of lateral and bottom macro-rough boundaries on the propagation of
a suspended sediment wave in a turbulent open channel flow are shown experimentally. Least decay of the normalized concentration is for the reference
i
ii
case in absence of trapping zone. Least sedimentation among lateral configurations is for case L4 with the highest roughness aspect ratio, cavity density and
medium flow discharge. Length of the inlet reach is lowest and it boosts mixing.
Highest deposition of the Polyurethane particles is for case L5 with the lowest
roughness aspect ratio and cavity density. Discharge is also low. Turbulence is
most attenuated for case L2 with the medium flow discharge due to adequate
lengths of the inlet and outlet reach. Deposition of the Polyurethane particles
and turbulence in the lateral macro-rough flows depend on the cavity aspect
ratio, flow discharge, roughness aspect ratio, cavity density and location of the
lateral cavities. In the bottom macro-rough flows, effects of spacing between
bottom macro-rough elements on the deposition and turbulence characteristics
are seen. Turbulence characteristics of upstream C1 and downstream C2 signals
are identical for reference, B1 and B1.5 cases. When spacing is 2, highest trapping and deposition take place. Most weakening of the turbulence is seen for
spacing 2. When spacing is augmented from 2 to 5, the turbulence is enhanced
and trapping becomes lesser. Bottom macro-rough elements change significantly
flow pattern and turbulence characteristics in a specific spacing. As spacing gets
larger, effects of bottom macro-roughness elements on each other reduces more.
Acknowledgements
I give glory to Almighty God for all
This thesis was fulfilled under the fund of the European Union via SEDITRANS
(REA grant agreement no. 607394), an initial training network (ITN) project
supported financially by the Marie Curie Actions of the EU's 7th Framework
Program.
The numerical part of this PhD thesis was done in PhD school of the Earth
Science and Fluid Mechanics (ESFM) at University of Trieste (UNITS) while
experimental part was accomplished at Hydraulic Constructions Laboratory of
the École Polytechnique Fédérale de Lausanne (EPFL).
The anther expresses thanks to his supervisor Prof. Vincenzo Armenio for
opportunity to work under his guidance and help in ESFM team.
The author is thankful of availability of the supercomputer at the University of
Malta to run the LES-COAST code.
Many thanks go to Prof. Anton Schleiss, Dr. Mario J. Franca, Dr. Carmelo
Juez and other technicians at Hydraulic Constructions Laboratory-EPFL for
supervision of experiments and providing infrastructure.
The author wishes to thank all scientific members in ESFM group, SEDITRANS
network and respectable reviewers of this PhD thesis.
Finally, my deep and sincere gratitude to my parents for their continuous and
unparalleled love, help and support in my life. I am grateful to my brothers,
Masoud and Mahyar, for always being there for me as a real friend. They selflessly encouraged me to seek my own destiny in life.
Thank you.
iii
Contents
Abstract
i
Acknowledgements
iii
List of Figures
vi
List of Tables
viii
1 Introduction
1
2 Suspended sediment transport (SST)
2.1 Definition of suspension mode . . . . . . . . . . . . . . . .
2.2 Settling velocity . . . . . . . . . . . . . . . . . . . . . . . .
2.3 SST in turbulent open channel flow . . . . . . . . . . . . .
2.4 Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 One-way coupling . . . . . . . . . . . . . . . . . .
2.4.2 Two-way coupling . . . . . . . . . . . . . . . . . .
2.4.3 Four-way coupling . . . . . . . . . . . . . . . . . .
2.5 Fast Eulerian method (FEM) . . . . . . . . . . . . . . . .
2.5.1 Scalar equation for sediment concentration in FEM
2.5.1.1 Rouse equation . . . . . . . . . . . . . . .
2.5.2 Continuity equation in FEM . . . . . . . . . . . .
2.5.3 Navier-Stokes (NS) equations in FEM . . . . . . .
2.6 Boundary condition for sediment concentration . . . . . .
2.6.1 Free surface . . . . . . . . . . . . . . . . . . . . . .
2.6.2 Channel bed . . . . . . . . . . . . . . . . . . . . .
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3 Large eddy simulation (LES)
3.1 Generalities on turbulence . . . . . . .
3.2 Filtering operation in LES . . . . . . .
3.3 Filtered governing equations . . . . . .
3.4 SGS models . . . . . . . . . . . . . . .
3.4.1 Smagorinsky model . . . . . . .
3.4.2 Dynamic eddy viscosity model
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3.5
3.4.3 Dynamic eddy diffusivity model . . . . . . . . . . . . . . .
Unresolved wall-function LES . . . . . . . . . . . . . . . . . . . .
24
24
4 LES-COAST code
4.1 Curvilinear coordinate . . . . . . . . . . . .
4.2 Discretization . . . . . . . . . . . . . . . . .
4.3 Fractional step method . . . . . . . . . . . .
4.4 Filtered equations in curvilinear coordinate
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5 Case study I: wall-resolving LES of SST
5.1 DNS-FEM for laminar sediment-laden flow
5.2 Experimental data in literature . . . . . . .
5.3 Simulation details . . . . . . . . . . . . . .
5.4 Validations and findings . . . . . . . . . . .
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6 Case study II: unresolved wall function LES of SST
44
6.1 Experimental data in literature . . . . . . . . . . . . . . . . . . . 44
6.2 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.3 Validations and findings . . . . . . . . . . . . . . . . . . . . . . . 47
7 Propagation of suspended sediment wave in macro-rough flow
7.1 Literature review on macro-rough flow . . . . . . . . . . . . . . .
7.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Configurations and procedure . . . . . . . . . . . . . . . . . . . .
7.4 Experimental findings . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Lateral macro-rough flow . . . . . . . . . . . . . . . . . .
7.4.2 Bottom macro-rough flow . . . . . . . . . . . . . . . . . .
53
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63
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8 Conclusions
72
References
75
Publications
83
List of Figures
2.1
2.2
3.1
Multifarious modes of the sediment transport taken from [Dey,
2014] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Shields diagram together with proposal of Vanoni [1975] . . . . .
4
16
Depiction of wide-range eddies in the flow taken from [Brown and
Roshko, 1974] . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
5.1
Comparison between DNS and analytic solution in terms of the
mean streamwise velocity . . . . . . . . . . . . . . . . . . . . . .
5.2 Comparison between the DNS and analytic solution in terms of
the mean volumetric concentration . . . . . . . . . . . . . . . . .
5.3 Grid refinement in the vertical direction in the wall-resolving LES
5.4 Dimensionless depth-resolved streamwise velocity for the clear
water flow against experimental [Muste et al., 2005] and theoretical [Nezu and Nakagawa, 1993] studies . . . . . . . . . . . . . .
5.5 Depth-resolved vertical velocity for the clear water flow against
[Muste et al., 2005] . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Depth-resolved Reynolds stress for the clear water flow against
Muste et al. [2005] . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Streamwise turbulence intensity for the clear water flow against
DNS data [Hoyas and Jimenez, 2008] and experiment of Muste
et al. [2005] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Vertical turbulence intensity for the clear water flow against DNS
[Hoyas and Jimenez, 2008] and experiment of Muste et al. [2005]
5.9 Imposed profile of the sediment concentration from experiment
NS3 of Muste et al. [2005] . . . . . . . . . . . . . . . . . . . . .
5.10 How suspdended sediment particles modify the streamwise velocity, taken with courtesy from [Yu et al., 2014] . . . . . . . . . . .
5.11 Effect of imposed profile of the sediment concentration in the
experiment NS3 on the dimensioneless depth-resolved streamwise
velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1
Comparison of the streamwise velocity against experiments of
Cellino [1998] . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
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42
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48
vii
6.2
6.3
6.4
6.5
6.6
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11
7.12
7.13
7.14
7.15
Comparison of the vertical velocity against experiments of Cellino
[1998] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of the Reynolds shear stress against experiments of
Cellino [1998] . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of the streamwise turbulence intensity against experiments of Cellino [1998] and Kironoto [1992] . . . . . . . . . .
Comparison of the vertical turbulence intensity against experiments of Cellino [1998] and Kironoto [1992] . . . . . . . . . . . .
Comparison of the sediment concentration against experiments
of Cellino [1998] . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dimensions of limestone bricks . . . . . . . . . . . . . . . . . . .
Details of setup: downstream (a) and upstream (b) sections of
the recirculating pumping system, manually regulated pump (c)
and magnetic flow meter (d) . . . . . . . . . . . . . . . . . . . . .
Details of setup: regulating mechanism in the downstream section
(a) and tranquilizing structure in upstream section (b) . . . . . .
Details of setup: transmitter b-line multi-amplifier (a), turbidimeter probe (b) and acquisition card NI-USB-6259 M series (c) . . .
Reference case (a) and lateral macro-roughness banks L1 (b), L2
(c), L3 (d), L4 (e) and L5 (f) - • and symbols representing
positions of turbidimeters T1 and T2 and water height measuring
device, respectively . . . . . . . . . . . . . . . . . . . . . . . . . .
Reference case (a) and bottom macro-roughness configurations
B1 (b), B1.5 (c), B2 (d), B3 (e) and B5 (f) - • and symbols representing positions of turbidimeters T1 and T2 and water
height measuring device, respectively . . . . . . . . . . . . . . . .
Bottom roughness: d-type (ϑ = 1.0) (a) and k-type (ϑ = 5.0) (b)
Schematic of the hydraulics system . . . . . . . . . . . . . . . . .
Lateral cavity flow with pertinent variables . . . . . . . . . . . .
Instantaneous normalized concentration for the reference (a) and
lateral macro-rough configurations L1 (b), L2 (c), L3 (d), L4 (e)
and L5 (f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Instantaneous auto-correlation for the reference (a) and lateral
macro-rough configurations L1 (b), L2 (c), L3 (d), L4 (e) and
L5 (f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cross-correlations of two sediment concentration signals for the
reference and lateral macro-rough configurations . . . . . . . . .
Instantaneous normalized sediment concentration for the reference (a) and bottom macro-roughness configurations B1 (b), B1.5
(c), B2 (d), B3 (e) and B5 (f) . . . . . . . . . . . . . . . . . . .
Instantaneous auto-correlation for the reference (a) and bottom
macro-roughness configurations B1 (b), B1.5 (c), B2 (d), B3 (e)
and B5 (f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cross-correlations of two concentration signals for bottom macrorough configurations and reference case . . . . . . . . . . . . . .
48
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62
64
65
66
68
69
70
71
List of Tables
5.1
Properties of the mixture flow in [Muste et al., 2005] . . . . . . .
35
6.1
6.2
Germane variables in experiments of [Cellino, 1998] . . . . . . . .
Key variables obtained from the unresolved wall-function LES . .
45
47
7.1
7.2
Dimensions of the flume and tanks in experiments . . . . . . . .
Number of bottom roughness elements and the length of the
rough area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Relevant flow variables of the SLF in the reference case . . . . .
germane variables and patterns for lateral macro-rough flows . .
56
7.3
7.4
viii
62
62
64
Chapter 1
Introduction
Sediment transport is a mechanism that relates hydrodynamic processes to morphological changes. Suspended sediment transport (SST) refers to particles that
move along a river completely supported by the flow. To keep sediment particles in the suspension mode, upward-directed forces related to the turbulence
in the flow have to become powerful enough to overcome the downward gravity
force acting on particles. During previous decades, many researchers working on
fluvial hydraulics elucidated partially interactions between suspended sediment
particles and turbulence in wall-bounded flows with flat and rough boundaries.
Since these interactions are not adequately quantified, common practice is to
treat suspended sediment particles and water flow as the mixed fluid.
In the numerical part of this study, the Navier-Stokes and scalar transport equations in the mixture flow model are solved iteratively using the LES-COAST
code which is developed for harbor and coastal areas. The curvilinear forms
of governing equations are integrated using a modified version of the finitedifference fractional-step method in a non-staggered grid. Periodic boundary
condition is set for the velocity, pressure and sediment concentration in the
streamwise and spanwise directions. Sediment-laden flows are Newtonian and
sediment particles are noncohesive. Four-way coupling is not taken into account. All complex phenomena such as preferential accumulation, turbophoresis
or clogging of sediment particles are neglected. Firstly, using the LES-COAST
code, this PhD thesis aims to unravel effects of suspended sediment particles
on hydrodynamics in a turbulent open channel flow with high aspect ratio and
bottom smooth wall. Data of clear water and sediment-laden flows from experiments of Muste et al. [2005] are selected for comparison. Single-phase EulerEuler based wall-resolving LES with the dynamic Smagorinsky model is used
to solve the filtered Navier-Stokes equations and scalar equation for the sediment concentration. Fast Eulearian method (FEM) in the two-way coupling
framework is implemented to reproduce influences of the sediment concentration on hydrodynamics of the flow. Non-cohesive sand particles with diameter
of d = 0.00023m are selected. To better explain interactions between turbulence
1
2
and suspended sand particles, the sediment-laden flow with highest concentration C = 0.00162 is chosen. It should be understood how suspended sand
particles could modify the bulk velocity and shear stress on the channel bed
compared to the clear water flow. Variations of the streamwise velocity in the
sediment-laden flow in contrast to the clear water flow have to be explored.
Secondly, influences of suspended noncohesive sand particles on the bulk velocity, friction velocity, Reynolds stress, roughness of the channel bed, streamwise
and vertical velocities in a turbulent open channel flow with rough bottom wall
and low aspect ratio are researched using the LES-COAST code. Single-phase
Euler-Euler based unresolved wall-function LES under the equilibrium stress assumption is employed. Smagorinsky model is used to calculate SGS stresses and
sediment fluxes in the mixture flow model. Results of clear water and sedimentladen flows are compared against experiments of Cellino [1998]. A constant
volumetric concentration C = 0.00077 of sand particles with the diameter of
d = 0.000135m is imposed to the fully developed clear water flow to obtain
statistics of the sediment-laden flow. The reference concentration method proposed by Smith and Mclean [1977] together with the Shields diagram is used
to treat the erosion of sediment particles from the channel bed. Buoyancy induced by suspended sediment particles may influence hydrodynamics of flow,
and hence the two-way coupling meaning introduction of modified gravity term
prescribed by Dallali and Armenio [2015] in the Navier-Stokes equations of the
mixture flow is taken. Wall shear stress on the sidewalls is included in the computation of imposed pressure gradient for clear water and sediment-laden flows.
The same roughness as expressed in experiments is applied for clear water and
sediment-laden flows. It has to be recognized whether suspended sand particles
cause attenuation or enhancement of streamwise and vertical turbulence intensities comparing to the clear water flow. Importance of the buoyancy effect in
the mixture flow model should be understood.
In natural rivers, morphological irregularities on banks or bed could change flow
patterns and SST. To do sediment management and to promote habitat suitability, hydraulics engineers artificially add cavities and lateral embayments to
lateral river banks. Sedimentation processes depend on the geometrical configuration of river banks. Laboratory works relating to hydrodynamics and
morphodynamics of lateral embayments in open channel flows are rare and
complicated to perform. Although some works in past were done on investigation of turbulence characteristics of turbulent open channel flows in presence
of bottom macro-rough elements, little effort is dedicated to study propagation
of a suspended sediment wave in these flows. Thirdly, effects of lateral and
bottom macro-rough boundaries on propagation of a suspended sediment wave
in a turbulent open channel flow are shown experimentally under the steady
state condition. Because of imposed volumetric flow rate Q = 0.015m3 /s and
flow depth h = 0.07m, Polyurethane particles with s = 1.16 and d = 0.00008m
are suspended in a wide open channel. Two turbidimeters are in upstream and
downstream parts to show temporal concentration. Sediment-laden flows are
Newtonian. Results are demonstrated in terms of normalized concentration of
suspended Polyurethane particles, corresponding auto-correlations and cross-
3
correlations. Based on Meile [2007], Maechler [2016] and Juez et al. [2017],
influences of lengths of inlet and outlet reach, aspect ratio of lateral cavity flow,
roughness aspect ratio, cavity density and flow discharge on flow patterns, turbulence and deposition of suspended sediment particles are investigated. For
bottom macro-rough flows, as Leonardi et al. [2004] and Leonardi et al. [2007]
reported, changing spacing between bottom macro-rough elements could alter
flow patterns and turbulence characteristics. Influence of spacing on propagation of a suspended sediment wave is examined here. Relationships between
spacing, turbulence characteristics and trapping of sediment particles in certain discharge and flow depth are stated. For example, it is shown that with
which spacing turbulence is most weakened and there is highest trapping and
difference between concentration signals in upstream and downstream parts.
Chapter 2
Suspended sediment
transport (SST)
2.1
Definition of suspension mode
Sediment transport is a process that links hydrodynamic processes and beach
morphology. Without transport of sediments, there would be no morphological
change and beach.
In principal, the bed materiel load, which is found on the bed of a stream, undergoes an exchange between the bed and fluid flow. It may move as bed-load
[van Rijn, 1984a] [Wu et al., 2000] or suspended load [van Rijn, 1984b]. They
are categorized by Mehta [2013].
Figure (2.1) shows that the bed-load transport (BLT) consists of sediment particles that slide, roll and saltate over the channel bed. The bed-load transport is
mainly governed by the water drag and collisions among bed material. Turbulence takes an auxiliary role in maintaining saltation . When upward-directed
Figure 2.1 – Multifarious modes of the sediment transport taken from [Dey,
2014]
4
2.2 SETTLING VELOCITY
5
forces, which are associated with the turbulence in the flow, are strong enough
to exceed significantly the downward gravity force, particles are kept in the suspension mode. Dey [2014] clarified that the suspension occurs when the ratio
between the settling velocity and shear velocity becomes lower than 0.6.
SST is a prevailing process in lower reaches of rivers. Particles are also fine.
Sediment concentration reduces as the vertical distance from the channel bed
increases. If the turbulence intensity is diminished and the flow expands, sediment particles return to the bed.
2.2
Settling velocity
Sediment particles become suspended in the flow when the gravity force, which
enforce them to fall towards bed, is surpassed by upwardly lifting forces generated by the turbulence. Particles denser than water such as sands inevitably
fall in a laminar flow since the gravity force is not opposed. As gravity force
becomes comparable to forces counteracting downward movement, a balance
may occur. Particles are characterized with their settling velocity.
The settling velocity ws is of prime importance in modeling of SST. When the
diameter of a suspended particle moving in a fluid is in the range 0.1mm < d <
1mm, equation (2.1) is used based on [van Rijn, 1984b] [Lin and Falconer, 1997]
[Dallali and Armenio, 2015]. There is a balance between the gravity force and
drag resistance,
"
#
0.5
0.01 (s − 1) gd3
10ν
1+
ws =
−1
(2.1)
d
ν2
ρs
(2.2)
ρ0
s is the specific gravity of suspended particles. g is the gravitational acceleration. ρs and ρ0 are densities of sediment particles and water, respectively. For
particles with d < 0.1mm, the settling velocity is estimated based on the Stokes
drag law [Cheng et al., 2015],
s=
ws =
1 (s − 1) gd2
18
ν
(2.3)
There are many factors affecting tunable settling velocity. For instance, Kawanisi
and Shiozaki [2008] experimentally found out that the turbulence intensity and
Stokes number St may influence mean settling velocity,
τp
St =
(2.4)
τλ
τp and τλ are Stokes the response time and Taylor time scale [Elghobashi, 1994],
respectively,
1
ws
15ν 2
λ
τp =
, τλ =
(2.5)
=
g
ǫ
urms
2.3 SST IN TURBULENT OPEN CHANNEL FLOW
6
For a fully developed turbulent open channel flow, most energetic eddies are
the Taylor scales. They are not largest (integral scales) or smallest ones (Kolmogorov scales). Dissipation takes place in the Kolmogorov scales. ǫ is the
energy dissipation. It is assessed from the energy spectrum by utilizing the
inertial subrange. For open channel flows, it is calculated from [Nezu and Nakagawa, 1993],
!
−3y
−0.5
y
ǫh
e h
= 9.8
(2.6)
u3∗
h
h is the water depth in open channel flows. The Taylor time scale is the characteristic timescale of the flow. λ is the Taylor length scale [Tennekes and Lumley,
1972] and urms is the root-mean-square (RMS) of the streamwise turbulent velocity fluctuation. Kawanisi and Shiozaki [2008] concluded that the settling
velocity was abated by 40 percent in a flow with weak turbulence. Settling velocity was enhanced substantially in a highly turbulent flow.
Additionally, the influence of the concentration on the settling velocity was
reported by Cheng [1997]. Lewis et al. [1949] used this formulation,
wm
m
= (1 − C)
ws
(2.7)
While ws is the settling velocity of a solitary particle in a fluid, wm is the settling
velocity of particles with the concentration C. The exponent m depends on the
effective density, particle Reynolds number and volumetric concentration of the
mixture.
The settling velocity depends on the inverse of the kinematic viscosity in the
Stokes law. Hence, doubling the kinematic viscosity would halve the settling
velocity. If fine sediments are in suspension, the viscosity of the sedimentladen flow may be higher than that of the clear water. Since the sediment
concentration is highest close to the river bed, falling sediment particles may
experience augmentation of the viscosity and their settling velocity may be
abated.
2.3
SST in turbulent open channel flow
There are many theoretical, numerical and experimental works in the literature that focused on understanding of interactions between suspended sediment
particles and turbulence in open channel flows. Simulation of the SST in the
presence of macro-rough boundaries is more complicated.
Wang and Qian [1989] experimentally compared turbulence characteristics of
a Newtonian flow with noncohesive particles to those of the clear water flow
in a recirculating tilting flume with hydraulic smooth surface. It was mentioned that the channel roughness could affect results in the vicinity of boundary even for the clear water flow. For both plastic particle-laden flows with
s = 1.052, d = 0.000266m and sand-laden flows with s = 2.64, d = 0.00015m,
the streamwise turbulence intensity was decreased with the enhancement of the
2.3 SST IN TURBULENT OPEN CHANNEL FLOW
7
concentration. Close to the bed region y/h < 0.2, streamwise turbulence intensities for plastic particle-laden flows and sand-laden flows were deviated mostly
from that for the clear water flow. In upper flow regions y/h > 0.2, the difference between the clear water flow and plastic particle-laden flows was very
low. Streamwise turbulence intensity for sand-laden flows was reduced obviously
with the increase of he tconcentration. Reduction of the turbulence intensity in
sand-laden flows was explained by the effect of the density gradient. Particle
delay played important role in damping of the turbulence near the bed in plastic particle-laden flows. Maximum of the streamwise turbulence intensity for
the clear water flow with h = 0.1m, u∗ = 0.0775m/s was 2.3 in y/h = 0.0535.
Autocorrelations revealed that sizes of the longitudinal eddy in sediment-laden
flows were bigger than those in the clear water flow. Using the discriminator
laser-Doppler velocimeter (DLDV), Muste and Patel [1997] measured the mean
velocity and turbulence characteristics in open channel flows. Suspended sand
particles were in the range of 0.21 − 0.25mm. Particle velocity lagged the water velocity and it was dependent on the concentration. A small decline in
streamwise particle velocity fluctuations was seen. Considerable lessening and
modification of the shape of vertical velocity fluctuations were reported.
Cellino [1998] and Graf and Cellino [2002] investigated experimentally the turbulence and concentration characteristics of suspension flows in open channels.
Flows were steady and uniform at the capacity condition. Sizes of sediment
particles were d = 0.000135, 0.00023 and the concentration was small. Clear
water formulation was used. Stratification effect induced by suspended particles affected the longitudinal mean velocity profile in the outer region of the
flow. Largest vertical velocities, which were seen close to the free surface, were
directed towards the bed. They were created by secondary currents. Suspended
particles engendered slight increase of the longitudinal turbulence intensity and
40 percent suppression of the vertical turbulence intensity. There was no effect
on the linear trend of the Reynolds stress.
Winterwerp [2001] employed one-dimensional (1D) k -ǫ numerical modeling to
study stratification effects generated by interactions between suspension of finegrained sediment and turbulence. Sediment-induced buoyancy effects resulted
in significant interactions between suspended sediment particles and turbulence.
Winterwerp [2001] also validated experiments of Coleman [1981]. Both decrease
in effective von Karman constant and alteration of the velocity defect law were
due to sediment-induced buoyancy effects. These results were also seen at low
concentration C = 0.001. Guo and Julien [2001] investigated theoretically effects of the sediment suspension on governing equations and turbulence characteristics. Boussinesq approximation in stratified flows was included. Effects of
the sediment concentration on the fluid density were neglected in continuity and
momentum equations except the gravity term. Viscosity was constant. Velocity profiles in sediment-laden flows were analogous to those in the clear water.
Modified log-wake law was valid for sediment-laden flows. Sediment suspension
augmented the mean flow energy loss and abated the vertical turbulence intensity. Both average concentration and density gradient (Richardson number)
diminished the von Karman constant in sediment-laden flows.
2.3 SST IN TURBULENT OPEN CHANNEL FLOW
8
Righetti and Romano [2004] used the phase Doppler anemometry (PDA) to detect the role of solid particles on the velocity profile in the near-wall turbulence
with 13000 < Re < 14500. Fluid and particle velocities were simultaneously
measured and compared to the clear water flow. Glass particles with s = 2.6,
diameters of 100−200×10−6 m and mean volumetric concentration of 10−3 were
used. For sediment-laden flows, the mean velocity in the logarithmic and outer
regions considerably was diminished. It was enlarged inside the inner region
y + < 5. RMS fluctuations of the vertical velocity were damped in the outer
region but were escalated within the inner region in contrast to the clear water
flow. Close to the wall, there was increase of the Reynolds stress. Far from the
wall there was a decrease.
Nezu and Azuma [2004] conducted simultaneous measurements of particles and
water in particle-laden open channel flows using the discriminator particletracking velocimetry (PTV). Friction velocity in the sediment-laden flow was
computed from the Reynolds stress. Normalized fluid velocity in the sedimentladen flow was described by the log law inside the inner wall region. Though,
the von Karman constant was lessened slightly with the increment of the concentration. Average particle velocity was slightly lesser than the fluid velocity
far from wall. On the other hand, the average particle velocity was bigger than
the fluid velocity. These differences caused fluid-particle interactions. Turbulence intensities were almost not changed by particles far from wall, irrespective
of the particle diameter and specific density. Conversely, turbulence intensities
of the fluid were escalated remarkably by particles inside the wall region. Turbulence intensities of particles got bigger than those of the fluid inside the wall
region. Vertical movements of particles were more increased due to ejections
and sweeps.
Muste et al. [2005] and Muste et al. [2009] employed simultaneous PIV-PTV
measurements to determine interactions between suspended particles and turbulent structures inside the flow. Bulk velocity in sediment-laden flows was
abridged with the sediment concentration irrespective of the particle density.
Suspensions of the natural sand and crushed Nylon particles extracted energy
from the flow. Also, the von Karman constant gradually was decreased with
the addition of particles. There was a strong particle flow turbulence within
the inner region. Mean streamwise velocity of the sand particles lagged those of
water flow inside the outer region. Mean streamwise velocities of particles were
bigger than those of water close to the channel bed because particle velocities
were not directly influenced by the no-slip condition. Vertical velocities of the
sand particles were higher than those of water. Traditional mixed-flow treatment of sediment-laden flows showed inconsistencies with regard to actual flow
of water and transport of suspended particles in sediment-laden flows.
Noguchi and Nezu [2009] examined the turbulence modulation in suspended
sediment-laden open-channel flows. They used the discriminator particle tracking velocimetry (D-PTV) for sediment particles and the discriminator particle
image velocimetry (D-PIV) for fluid tracers. It was concluded that inside region
y + < 15, the particle velocity was faster than carrier fluid whereas for y + > 15,
fluid became faster than particles. This relative velocity was aggrandized for
2.3 SST IN TURBULENT OPEN CHANNEL FLOW
9
bigger particles. Pittaluga [2011] developed analytic approach to account for
stratification effects on distributions of the vertical velocity and sediment concentration in fluvial channels. For a given shear stress, stratification effects
induced enhanced velocity gradient and augmented streamwise velocity. Stratification caused a decrease in the suspended sediment concentration relative to
the neutral condition. Increase of the sediment concentration caused enlargement of the depth-averaged velocity through stratification. It also resulted in
increase of the effective bed roughness, friction and abated flow velocity.
Zhu et al. [2013] applied the modified coherent dynamic eddy model of the
LES. Pick-up function was employed to simulate erosion and deposition of noncohesive sediment particles in turbulent channel flow with high concentration.
The rough wall model was used instead of high-cost LES with near-wall resolution. In the streamwise direction, sediment particles were picked up predominantly due to excess bottom stress, whereas in the spanwise direction,
the turbulent mixing engendered by streamwise vortexes contributed more to
the erosion. Kundu and Ghoshal [2014] developed mathematical modeling for
steady uniform turbulent open channel flow laden with sediments. They incorporated effects of stratification and secondary flows. Direction and strength
of secondary currents generated various mean vertical velocities and sediment
concentration patterns. Stratification caused a decrease in the sediment concentration relative to the neutral case. The concentration was boosted in the flow
with upward vertical velocity. It was diminished with downward vertical velocity. Yu et al. [2014] concluded that if sediment particles were in a turbulent flow
over a smooth bed, the von Karman constant was reduced. Sediment particles
engendered a velocity lag inside the outer region and velocity increment within
the inner region. Slope of the velocity distribution was increased.
Maechler [2016] experimentally highlighted interactions between discharge, capacity concentration and aspect ratio of lateral macro-rough cavities in turbulent open channel flows. Turbulent eddies were induced by interface with calm
zone and main flow. Bigger aspect ratio limited cavity sedimentation through
turbulent and oscillating processes. With highest discharge, sedimentation was
countered by turbulence and flow oscillations inside cavity flows.
Juez et al. [2017] carried out systematic laboratory tests in turbulent open channel flow with multifarious geometries of lateral macro-rough cavities. Idealized
artificial lateral macro-roughness elements perturbed flow. The sediments remained within the cavity flow, where turbulence levels were lower than in main
channel. Probability of settling was higher. As roughness was increased in lateral macro-rough flows, turbulence was enhanced and particles were more kept
in the suspension mode. Settling was hindered. Decay of the sediment concentration was quicker for lower discharges. Larger discharges boosted turbulence
levels and augmented sediment suspension.
Laboratory works relating to hydrodynamics and morphodynamics of lateral
embayments in open channel flows are rare and complicated to perform. Moreover, few works in the literature explored interactions between suspended fine
particles and turbulence in open channel flows with bottom macro-rough boundaries.
2.4 COUPLINGS
2.4
10
Couplings
Elghobashi [1994] classified turbulent sediment-laden flows (SLF) according to
interactions between sediment particles and turbulence. There are one-way
coupling (C < 10−6 ), two-way coupling (10−6 < C < 10−3 ) and four-way
coupling (C > 10−3 ). This classification depends on the amount of suspension
load.
2.4.1
One-way coupling
The interaction is called one-way coupling [Zedler and Street, 2001] [Zedler and
Street, 2006] when the volumetric concentration is lower than 10−6 . Dispersion
of particles is dependent on the state of the turbulence and no term is introduced in the Navier-Stokes equations. Just settling velocity is appeared in the
concentration equation.
2.4.2
Two-way coupling
As the volumetric concentration increases 10−6 < C < 10−3 , the two-way coupling [Ferry and Balachandar, 2001] happens and sediment particles may transfer momentum to the flow. It engenders alteration of turbulence characteristics.
It should be reminded that flows in already expressed criteria are called dilute
suspensions. Abating size of particles augments the surface area of the sediment phase and dissipation rate of the turbulence energy. On the other hand,
enlarging the particle response time with the particle Reynolds number in the
range Rep > 400 results in the vortex shedding and enhanced production of
the turbulence energy. More numerical discussions on the role of the size of
suspended particles in modulating turbulence could be seen in Cao et al. [2003].
Two-way coupling or interactions between base fluid and sediment particles
could be imposed by introducing a modified gravity term (2.25) in the NS equation (2.22).
2.4.3
Four-way coupling
In the four-way coupling framework [Breuer and Alletto, 2012], the volumetric
concentration goes beyond 0.001 and flow is a dense matter. In addition to the
two-way coupling, collisions between particles are also examined. Breuer and
Alletto [2012] carried out a LES-based comparison between various couplings
and found out that there was a satisfactory agreement between two-way and
four-way coupled simulations and experimental measurements. Because here
we just focus on dilute suspensions, the four-way coupling is neglected.
2.5 FAST EULERIAN METHOD (FEM)
2.5
11
Fast Eulerian method (FEM)
Numerical models for treating two-phase sediment transport are grouped as
Euler-Lagrange [Soldati and Marchioli, 2012] and Euler-Euler [Chou and Fringer,
2008] [Harris and Grilli, 2014].
The Euler-Lagrangian model can be used when the concentration of particles
is low. Motion of individual particles is tracked. The water is treated as a
continuum. Tracking a large number of particles is very expensive from a computational point of view and it may be impractical for fine sediment suspensions.
In the Euler-Euler-based models, which are more suitable for engineering applications, governing equations for both particulate and fluid phases are based on
the continuum approximation. The particles follow the fluid motion except for
their settling velocity which appears in the concentration equation.
Euler-Euler-based models are divided to two-phase models [Cheng and Hsu,
2014] [Chiodi et al., 2014] or single-phase models [Chou and Fringer, 2008] [Harris and Grilli, 2014] [Dallali and Armenio, 2015].
In the two-phase models, mass and momentum conservation equations for both
phases are computed. Phases are differentiated by a particle volume fraction.
As stated by Chiodi et al. [2014], forces, which are induced in the flow such as
viscous drag, inertia, contact force, solid friction and lubrication, or Archimedes
effect, could be investigated in the computation.
The single-phase or mixture model is adopted for fine particles in the suspension mode. In the mixture model, the vertical velocity is simply the sediment
particle settling velocity. One advection-diffusion equation is defined for the
sediment concentration in the mixture.
More specifically, Ferry and Balachandar [2001] developed the fast Eulerian
method (FEM) in the two-way coupling framework to simulate the sedimentladen flow as a two-phase flow. The particle response time should be low enough
compared to the characteristic time-scale of the turbulent flow. If particles are
very small, they may move with the fluid and are spatially well mixed. The
flow is treated as a single-phase medium with a modified density. Suppose
that the Eulerian fluid and particle velocities are represented, respectively, by
uf (x, t) and vp (x, t). Maxey [1987] and Druzhinin [1995] considered following
Lagrangian equation (2.8) for the particle motion,
1
dvp
(uf − vp ) + g
=
dt
τp
(2.8)
The particles are subjected to only viscous drag and gravity force g. τp is the
particle time-scale which can be obtained from Ferry and Balachandar [2001].
When expanding equation (2.8) in terms of uf (x, t) and vp (x, t), following
formulation could be gained,
Dψ
+ ψ.∇uf τp2
(2.9)
vp = uf − ψτp +
Dt
Higher orders of the approximation are disregarded. No influence of surrounding
particles on the velocity of a particle is assumed. The total derivative is defined
2.5 FAST EULERIAN METHOD (FEM)
12
only for fluid elements. ψ is the modified acceleration taken as,
ψ=
Duf
−g
Dt
(2.10)
Maxey [1987] stated that the first-order version of equation (2.9) could explain
the preferential accumulation of particles in regions of low vorticity and high
strain. When additional forces like added-mass, Basset and Saffman lift are
included in this expansion, following form could be acquired after some simplifications [Ferry and Balachandar, 2001],
Dψ
(2.11)
+ ψ.∇uf τp2
vp = uf + (1 − η) −ψτp +
Dt
3
(2.12)
2s + 1
η is the density ratio. With η=0.0, equation (2.11) reverts back to equation
(2.9). Finally, the particle velocity vp is approximated in terms of local fluid
velocity uf and its spatial and temporal derivatives.
η=
2.5.1
Scalar equation for sediment concentration in FEM
It is recalled from Winterwerp [2001] that if the settling velocity ws of fine sediment particles is much lower than vertical velocity fluctuations ws << vRM S , in
the range 0.1mm/s < ws < 2mm/s, the single-phase formulation is applicable.
The sediment concentration must be also low. Here, we focus on the Euler-Euler
single-phase or mixture model [Zhu et al., 2013] in which each fine particulate
phase follows the movement of the fluid flow except for the settling velocity.
Ferry and Balachandar [2001] demonstrated applicability of the FEM in the
two-way coupling framework meaning the volumetric loading is extremely below
unity. Accordingly, the conservation equation for the sediment concentration is,
∂C
+ ∇. (Cvp ) = 0.0
∂t
(2.13)
It has to be noticed that diffusive term relating to the volumetric concentration
is neglected in equation (2.13). After introducing expansion equation (2.11) into
equation (2.13), we get a simplified closed system of the concentration balance,
Duf
DC
= (1 − η) τp ∇. C
−g
(2.14)
Dt
Dt
It is interesting to acknowledge that the settling velocity term is appeared in
equation (2.14). Here, we implement some simplifications in equation (2.14)
while sustaining the diffusive part. Consequently, proposed advection-diffusion
equation (2.15) for the volumetric concentration C as the scalar field is [Dallali
and Armenio, 2015],
∂C
∂2C
∂
[(uj − ws δj2 ) C] = ǫ
+
∂t
∂xj
∂xj ∂xj
(2.15)
2.5 FAST EULERIAN METHOD (FEM)
13
ν
(2.16)
Sc
ǫ is the molecular diffusivity or molecular diffusion coefficient for sediment particles. Sc is the molecular Schmidt number which is equal to unity [Dallali and
Armenio, 2015]. And, δij is the Kronecker delta. Since the settling velocity
term is defined in the scalar transport equation (2.15), one-way coupling is regarded in this study. Moreover, because a highly turbulent open channel flow
is investigated in this study, the turbulent eddy diffusivity may be much higher
than the molecular one in equation (2.15).
ǫ=
2.5.1.1
Rouse equation
The vertical distribution of the eddy viscosity in uniform unidirectional open
channel flows can be determined by [Cellino, 1998],
νt = κu∗
y
(h − y)
h
(2.17)
h, κ and u∗ are the water depth, von Karman constant and shear velocity, respectively. It could be also expressed using the log-law as [Nezu and Nakagawa,
1993],
hu′ v ′ i
νt = −
(2.18)
∂ hU i
∂y
hu′ v ′ i is the depth-averaged Reynolds stress and hU i is the depth-averaged
streamwise velocity. Coleman [1981] comprehended that the von Karman constant (κ = 0.4) remains unchanged for both clear water case (CWC) and
sediment-laden flow (SLF). It is recalled here that the eddy diffusivity and eddy
viscosity are assumed to be identical. When substituting equation (2.17) into
equation (2.26), the well-known Rouse equation [Cellino, 1998] being valid in
the range a < y < h, can be derived,
α
C
h−y a
=
(2.19)
Cref
y h−a
ws
(2.20)
α=
κu∗
α is the Rouse number. It is understood that Cref which is computed on a
layer of sediment with the height of a = 0.05 × h. It quantifies the erosion rate
from the channel bed. For y > a, sediment particles are transported in the
suspension mode whereas for y < a, sediments are part of the bed-load. When
a sediment-laden flow manifests this distribution of the sediment concentration,
suspended matter is in the full capacity. It means any further insertion of sediment particles into the flow field results in the deposition of sediment particles
without any augmentation of the suspended load.
Pay attention that the steady-state condition, one dimensionality of the sediment concentration and single settling velocity in deriving Rouse equation are
assumed.
2.5 FAST EULERIAN METHOD (FEM)
2.5.2
14
Continuity equation in FEM
Some assumptions are taken regarding the derivation of the two-way coupled
equations in the FEM. There is no volume transfer between phases. Ferry
and Balachandar [2001] wrote a balance for total mass of a system within an
arbitrary control volume in the limit of small volumetric loading (C << 1).
Since the fluid is incompressible and there are insignificant density variations,
velocity field is solenoidal or divergence-free. Mass conservation or continuity
equation is,
∂u ∂v
∂w
∂uj
=0
(2.21)
+
+
=0⇔
∂x ∂y
∂z
∂xj
Indexes j=1, 2 and 3 represent the streamwise, wall-normal and spanwise directions, respectively. x1 , x2 and x3 also refer to x, y and z, respectively. Moreover,
velocities u1 , u2 and u3 represent u, v and w, respectively.
2.5.3
Navier-Stokes (NS) equations in FEM
The Navier-Stokes equations [Dey, 2014] for an incompressible viscous Newtonian fluid [Pope, 2000] in Cartesian coordinate are illustrated by [Fakhari,
2015],
∂ui
1 ∂p
ν ∂ 2 ui
1
∂ (ui uj )
=−
+
+ Π − gm δi2
+
∂t
∂xj
χρ0 ∂xi
χ ∂xj ∂xj
χ
(2.22)
The Boussinesq approximation [Armenio and Sarkar, 2002] is valid here because
the density variation is small with respect to the reference density ρ0 which is
the density of water. Π is the imposed mass flow rate or pressure gradient
divided by ρ0 for open channel flows. ν is the kinematic viscosity of fluid and p
is the hydrodynamic pressure. The coefficient χ is measured by,
χ = 1 + (s − 1) C
(2.23)
Note that for the clear water flow, we have χ = 1.0. Although the volumetric
loading (C) is small, the mass loading (ρC) is not. Ferry and Balachandar
[2001] communicated that main effect of the particulate phase on the momentum
equation is to introduce the mixture density,
ρm = χρ0
(2.24)
It can be interpreted that suspended particles are in the equilibrium. Though,
the particle velocity in the FEM is different from the fluid velocity. Though,
their acceleration is same. Similarly, Winterwerp [2001] and Dallali and Armenio [2015] stated that to account for effects of the sediment concentration on
hydrodynamics of the flow or two-way coupling, a modified gravity term gm
must be introduced in the NS equations,
gm = (χ − 1) g
(2.25)
2.6 BOUNDARY CONDITION FOR SEDIMENT CONCENTRATION
2.6
15
Boundary condition for sediment
concentration
Since the turbulent field is homogeneous in the streamwise (x) as well as spanwise directions (z), the periodic boundary condition is imposed for the velocity,
pressure and sediment concentration fields. One of important steps in modeling
SST is the definition of the boundary condition for the sediment concentration
at the free surface and bottom wall.
2.6.1
Free surface
Equation (2.15) represents the distribution of the suspended sediment concentration by considering an advection-diffusion equation in a general form. This
can be simplified if there is a steady-state condition with no change of the sediment concentration in x (streamwise) and z (spanwise) directions, [Cellino, 1998]
[Graf and Cellino, 2002],
∂C
ws C + ǫ
= 0.0
(2.26)
∂y
In the mixture model, the streamwise velocity of the fluid is equivalent to that of
sediment particles. Though the vertical velocity of sediment particles is equal to
the settling velocity of a single particle in a pure still clear water. Furthermore,
the molecular diffusion coefficient ǫt for the sediment concentration goes to zero
at the free surface.
Cellino [1998] and Graf and Cellino [2002] indicated that equation (2.26) is
valid when the volumetric concentration is small C < 0.05. Pay attention that
Bai et al. [2013] considered the sum of the molecular and turbulent diffusivities
ǫtot = ǫ+ǫt instead of ǫ for the boundary condition of the sediment concentration
at the free surface.
2.6.2
Channel bed
From a physical perspective, sediments are entrained into the flow when the
shear stress gets higher than its critical value. Thus, the bed boundary condition for the sediment concentration should outline this entrainment arising from
excess shear stress. Additionally, because of the settling velocity of sediment
particles and stratification effects, there is a hyper concentration layer close to
the channel bed so that scalar equation (2.15) fails to accurately account for
the sediment concentration. Therefore, in the literature multitudinous empirical reference concentration Cref formulations and pick-up rate functions are
proposed to parameterize this region. Zedler and Street [2006] stated that the
reference concentration formula were developed for flows in the equilibrium condition and hence they may not be appropriate for unsteady flows. Besides, they
are sensitive to the reference height. The alternative procedure is the pick-up
rate function which was developed to model the unsteady sediment transport.
2.6 BOUNDARY CONDITION FOR SEDIMENT CONCENTRATION
16
However, Villaret and Davies [1995] established that these reference concentration formula had been used mainly to assess the instantaneous volumetric
concentration in unsteady problems. Preference of the pick-up rate function
over the reference concentration formula is a subject of discussion. Hence, here
we adopt the reference concentration formulation derived by Smith and Mclean
[1977],
γ 0 S0
(2.27)
Cref = C0
1 + γ 0 S0
τw − τcr
S0 =
(2.28)
τcr
It has to be emphasized that corresponding values of γ0 and C0 are 2.4 × 10−3
and 0.65, respectively. C0 is the maximum of the concentration of sediment
particles. Cref depends on the temporal bed shear stress and critical bed shear
stress. The erosion is absent when the bed shear stress is lower than critical one.
Practically speaking, Cref , which is assumed at 0.05×h, represents the potential
of the turbulent flow to keep sediment particles in the suspension mode.
Figure 2.2 – Shields diagram together with proposal of Vanoni [1975]
Figure (2.2) depicts the Shields diagram. It shows relationships between dimensionless particle diameter d∗ and critical bed shear stress τcr or critical Shields
variable θcr ,
τcr
θcr =
(2.29)
(ρs − ρ0 ) gd
u∗ d
(2.30)
ν
Dallali and Armenio [2015] declared that the Shields diagram underestimates
the critical bed shear stress. To overcome this in the suspension mode, Vanoni
d∗ =
2.6 BOUNDARY CONDITION FOR SEDIMENT CONCENTRATION
17
[1975] introduced an upper limit shown by inclined lines in figure (2.2),
Ss =
0.5
ρs
d
0.1
− 1 gd
ν
ρ0
(2.31)
Keep in mind that the temporal shear stress on the channel bed is obtained
form corresponding components in streamwise (x) and spanwise (z) directions,
p
τw = τx2 + τz2
(2.32)
Chapter 3
Large eddy simulation
(LES)
3.1
Generalities on turbulence
Based on relative importance of viscous and inertia forces, fluid flows can be
classified into laminar, turbulent and transitional. Laminar or streamline flow
takes place when the velocity and flow depth are low. It is seen as a layered
flow in which layers of fluid slide over each other. Concerning hydraulics of open
channel flows, the laminar flow does not happen from a practical point of view.
The transitional regime happens when the Reynolds number is in this range,
2300 < Re =
4Rh U
< 4500
ν
(3.1)
As the velocity of the flow enhances more, the laminar flow gets highly unstable
and instantaneous eddies form in the flow. This three-dimensional (3D) irregular and fluctuating behavior of fluid particles in free shear and wall-bounded
flows is called turbulence [Pope, 2000]. The large-scale movements are significantly affected by the boundary condition and they are mainly responsible for
transport and mixing. In contrast to the laminar flow, mixing and transport
of fluid parcels in turbulent flows are highly effective. On contrary, small-scale
eddies, which have a universal behavior and are independent of the geometry,
dissipate the energy which is received from large-scale eddies by viscosity.
The turbulent flows are grouped into three classes, smooth turbulent flows, fully
rough turbulent flows and transitional turbulent flows. Shifting from smooth
turbulent flow to fully rough turbulent flow is quite different from the moment
when the laminar flow gets turbulent. The transition between the smooth turbulent flow and fully turbulent flow depends on the Reynolds number Re and
roughness height k. As the size of roughness elements gets bigger and approaches the flow depth or hydraulic radius, flow is called macro-rough flow.
Water surface may be disturbed for lateral macro-rough flows. In the smooth
18
3.2 FILTERING OPERATION IN LES
19
turbulent open channel flow, roughness elements are adequately tiny that are
located inside the viscous boundary layer. In the fully rough turbulent flow,
roughness elements are big enough to be inside the turbulent boundary layer
(TBL).
If the Reynolds number Re is in low-to-moderate range, governing equations
could be solved by the direct numerical simulation (DNS) [Kim et al., 1987] using fine meshing. This method is completely free of any modeling assumptions
and all lengthscales and timescales of the flow down to the Kolmogorov scale
should be computed and computational cost elevates by Re3 correspondingly.
DNS provides an improved knowledge of the physics of turbulent flows such as
the role of the coherent structures.
Reynolds Averaged Navier-Stokes equations (RANS) are frequently used in applied engineering, although the accuracy of the simulation depends on the closure model. It may require tuning of model constants.
Large-eddy simulations (LES) constitute an intermediate approach between
DNS and RANS. In the LES, large-scales, energy-carrying, structures are resolved while more isotropic, small scales, are modeled using a subgrid-scales
(SGS) model. The scale separation in the LES is accomplished using the filtering
operation. Depending on the treatment of the viscous sub-layer in wall-bounded
flows, there are two variations of the LES, wall-resolving LES and unresolved
wall-function LES.
In the wall-resolving LES, the resolution is comparable to the DNS while in the
unresolved wall-function LES, a coarse grid is used. Nodes next to the wall are
modeled with a wall function which mimics the presence of a solid wall.
3.2
Filtering operation in LES
To solve governing equations of turbulent open channel flows (TOCF), the large
eddy simulation (LES) becomes an alternative. Large scale eddies, figure (3.1),
are directly resolved and smaller scales, figure (3.1) are parameterized using subgrid scale (SGS) models. In the LES, the scale separation is accomplished by
Figure 3.1 – Depiction of wide-range eddies in the flow taken from [Brown and
Roshko, 1974]
using a low-pass filtering [Leonard, 1974] operation in which the filtered velocity
field could be applied on a coarse grid. Taking for instance Z as a turbulent
3.3 FILTERED GOVERNING EQUATIONS
variable, the grid filtering operation is defined as,
Z
Z (x, t) = G(r, x)Z (x − r, t) dr
20
(3.2)
x in equation (3.2) manifests general spatial positioning (x, y, z). Keep in mind
that the integration is extended to the entire flow domain and selected grid
filter kernel G(r, x), which specifies the size and structure of small eddies, has
to satisfy the normalization condition [Pope, 2000]. In the LES, each variable
Z (x, t) is divided into two parts, filtered part Z (x, t) and residual (SGS) part
z ′ (x, t) like,
(3.3)
Z (x, t) = Z (x, t) + z ′ (x, t)
Based on the Reynolds decomposition [Guo and Julien, 2001], each variable is
composed of the time-averaged Z (x, t) and fluctuating parts z (x, t),
Z (x, t) = Z (x, t) + z (x, t)
(3.4)
While time-averaging of the fluctuation part z is zero, filtering the residual
(SGS) part z ′ is not necessarily zero.
Multifarious types of the filter function are proposed and the top-hat filter
[Ghosal and Moin, 1995] is used,
1
∆
(3.5)
G(r) = ∆ , if | r |< 2
0,
otherwise
∆ is the smoothing operator or filter width. It plays a key role in modeling
of unresolved stresses. Depending on the SGS model implemented, various
smoothing operators in the LES were proposed.
3.3
Filtered governing equations
Considering the two-phase coupled equations in the FEM, when all turbulent
variables velocity, pressure and concentration are specified in terms of their corresponding filtered and SGS components, following filtered governing equations
could be gained,
∂uj
=0
(3.6)
∂xj
∂ui
1 ∂p
ν ∂ 2 ui
1
∂ (ui uj )
∂τij
=−
+
+ Π − gm δi2 −
+
∂t
∂xj
χρ0 ∂xi
χ ∂xj ∂xj
χ
∂xj
∂2C
∂ωj
∂
∂C
−
+
(uj − ws δj2 ) C = ǫ
∂t
∂xj
∂xj ∂xj
∂xj
(3.7)
(3.8)
The influences of smaller scales are represented by SGS stresses τij and SGS
turbulent fluxes for the concentration ωj ,
τij = ui uj − ui uj
(3.9)
3.4 SGS MODELS
21
ωj = uj C − uj C
(3.10)
Utilizing the gradient diffusion hypothesis [Bai et al., 2013], the SGS sediment
flux is estimated by,
νt
∂C
, ǫt =
(3.11)
ωj = −ǫt
∂xj
Sct
ǫt is the SGS concentration diffusivity. Sct is the turbulent Schmidt number.
Theses appeared terms should be modeled using SGS models for momentum
and concentration equations. As stated by Piomelli and Balaras [2002], SGS
terms just contribute a small fraction of total turbulent stresses.
3.4
SGS models
In the literature, there are many SGS models for treating SGS stresses and turbulent fluxes for the scalar field. As Germano et al. [1991] elucidated, majority
of models to treat SGS stresses is based on the eddy viscosity. Smagorinsky
model can be applied to account for both SGS stress and turbulent fluxes for
the concentration. A single, universal constant is assumed in the Smagorinsky
model to compute the eddy viscosity. Though, it is impractical for modeling
various phenomena in flows. This model cannot take into account energy transfer from small scales to large scales or backscatter. And, SGS stresses may not
disappear at wall. Modifications to the Smagorinsky model were made close to
walls in channel flows in order to vanish SGS stresses at walls. Also, the eddy
viscosity and state of the flow must be determined locally. For example, while
a dynamic eddy viscosity model developed by Armenio and Piomelli [2000] can
be implemented to compute SGS stresses, a dynamic eddy diffusivity model
proposed by Armenio and Sarkar [2002] can be used to compute SGS turbulent
fluxes for the concentration.
3.4.1
Smagorinsky model
In the isotropic Smagorinsky model [Smagorinsky, 1963], it is assumed that
small scales of the motion are in the equilibrium and there is a balance between
production and dissipation of the turbulent energy. The eddy viscosity νt is
calculated from,
2
(3.12)
ν t = Cs ∆ | S |
∆ is the grid filter width. The Deardroff equivalent length scale [Roman et al.,
2010] is commonly used for the unequal-sided cells,
∆ = (∆x∆y∆z)
0.33
(3.13)
Cs is Smagorinsky constant in range 0.065 < Cs < 0.2. The large-scale strainrate tensor | S | is taken from
| S |= 2S ij S ij
0.5
(3.14)
3.4 SGS MODELS
22
The appearing components are defined by,
1 ∂ui
∂uj
S ij =
+
2 ∂xj
∂xi
(3.15)
Eventually, SGS stresses, last term in equation (3.7), which represent influences
of small-scale motions, could be computed from,
τij = −2νt S ij
(3.16)
The Smagorinsky model can be applied to calculate the SGS sediment flux ω j
like the study of the wave-induced sediment transport done by Gilbert et al.
[2007]. In this study, the turbulent Schmidt number Sct is set to unity νt = ǫt
and Smagorinsky constant Cs for both momentum and concentration equations
is same. Nevertheless, Le Ribault et al. [2001] and Horiuti [1992] speculated
that the constant for the concentration equation must be bigger than that for
the momentum equation.
The Smagorinsky constant is a highly flow-dependent coefficient [Bai et al.,
2013]. Physically speaking, the eddy viscosity must be zero close to walls.
In the Smagorinsky model, it is not because the strain-rate tensor is big as
a disadvantage. Smagorinsky model also cannot justify the energy transfer
from small scales to large scales or backscatter. Accordingly, researchers have
introduced various strategies like the dynamic Smagorinsky model (DSM) to
force SGS stresses to vanish at walls of channel flows.
3.4.2
Dynamic eddy viscosity model
In dynamic SGS models, model coefficients are estimated dynamically in each
cell. It depends on the energy content of smallest resolved scales rather than
inputting a fixed constant. All inefficiencies of the fixed-coefficient model or
Smagorinsky model could be removed because the coefficient disappears in laminar flows as well as close to walls. To accomplish this, like equation (3.2), a
test filtering operation is done by,
Z
b x)Z (x − r, t) dr
Zb (x, t) = G(r,
(3.17)
Germano et al. [1991] stated that new filter kernel is selected as,
b x) = G(r,
b x)G(r, x)
G(r,
(3.18)
b x), new versions of the filtered continuity
Once we apply the filtering kernel G(r,
and Navier-Stokes equations could be realized,
bj
∂u
=0
∂xj
bj
bi u
u
∂
b
bi
∂ ui
p
1 ∂b
ν ∂2u
1
∂Tij
=−
+
+ Π − gm δi2 −
+
∂t
∂xj
χρ0 ∂xi
χ ∂xj ∂xj
χ
∂xj
(3.19)
(3.20)
3.4 SGS MODELS
23
Note that the test filter is supposed to be twice of the grid filter,
b = 2∆ = 2 (∆x∆y∆z)0.33
∆
(3.21)
bi u
bj
Tij = ud
i uj − u
(3.22)
bi u
bj
Γij = ud
i uj − u
(3.23)
Tij = Γij + τbij
(3.24)
The SGS stresses Tij are explained by,
The resolved turbulent stresses Γij are specified by,
The stresses Γij [Germano et al., 1991] could be measured explicitly. They are
indicative of contribution to SGS stresses by scales whose length is considered
b The identity between these stresses is,
intermediate between ∆ and ∆.
This identity equation (3.24) could be utilized to obtain the Smagorinsky coefficient depending on the type of turbulent flows. To parameterize both Tij and
SGS stresses τij in the Smagorinsky model, assume Mij and mij designating
anisotropic parts of Tij and τij , respectively,
δij
2
τkk ≃ mij = −2Cds ∆ | S | S ij
(3.25)
τij −
3
δij
b2 | S
b|S
b
Tij −
Tkk ≃ Mij = −2∆
(3.26)
ij
3
!
0.5
b
b
1
∂
∂
u
u
i
j
b S
b
b =
b |= 2S
+
,S
(3.27)
|S
mn mn
ij
2 ∂xj
∂xi
b is the filter width related to G(r,
b x).
It should be emphasized that ∆
When replacing equations (3.25) and (3.26) in the identity equation, we get a
dynamic constant Cds over parallel planes (shown by <>) to the wall [Germano
et al., 1991],
Cds = −
Γkl S kl
1
D
D
E
E
2
2
2b
b S
b S
b|S
b|S
∆ |S
|S
mn mn − ∆
pq pq
(3.28)
An error [Foroozani, 2015] appears in this substitution which can be minimized
using the least-squares approach [Lilly, 1992]. Finally, the coefficient is calculated dynamically [Foroozani, 2015],
Cds = −
1 hΓij Mij i
2 hMij Mij i
(3.29)
3.5 UNRESOLVED WALL-FUNCTION LES
3.4.3
24
Dynamic eddy diffusivity model
Le Ribault et al. [2001] applied the LES with the dynamic eddy diffusivity
model for the scalar transport in a plane jet. In the sediment-laden flow, the
SGS concentration flux is retained like [Armenio and Sarkar, 2002],
2
ωj = −Cρ ∆ | S |
∂C
∂xj
(3.30)
Analogous to the dynamic eddy viscosity model, constant Cρ could be ascertained utilizing grid and test filters in the concentration equation,
Cρ = −
1 hΛi Υi i
2 hΥk Υk i
(3.31)
According to Armenio and Sarkar [2002], corresponding resolved SGS sediment
fluxes Λi and anisotropic part Υi are characterized by,
d −C
bu
bi
Λi = Cu
i
d
[
∂C
2
b | ∂C − ∆
b2 | S
|S|
Υi = ∆
∂xi
∂xi
(3.32)
(3.33)
To prevent unphysical back-scattering [Armenio and Sarkar, 2002], dynamic
coefficients are equal to zero when they get negative.
3.5
Unresolved wall-function LES
There are two distinct ways to treat wall-bounded flows using the LES called
near-wall resolution (LES-NWR) and near-wall modeling (LES-NWM). Considering smooth walls, if the grid and filter are selected so that nearly 80 percent
of the energy everywhere including in the viscous sub-layer are resolved, the
consequence is a LES with the near wall resolution. This method necessitates
a very fine grid close to walls especially in very high Reynolds number flows in
aeronautical and meteorological applications. Hence, it is highly expensive from
a computational cost point of view.
Chapman [1979] theorized that resolving the inner layer (il) in a plane channel
flow needs this meshing,
(Nx Ny Nz )il ∝ Re1.8
(3.34)
In the outer layer (ol), in which the the direct influence of the viscosity on the
mean velocity is insignificant, the resolution is less demanding,
(Nx Ny Nz )ol ∝ Re0.4
(3.35)
Piomelli [2008] visually clarified that even at moderate Re = 104 over 50 percent
of computer resources has be used to resolve just 10 percent of the flow.
The alternative is to use the LES-NWM in which the inner layer is bypassed
3.5 UNRESOLVED WALL-FUNCTION LES
25
and its effects are modeled in a global sense. The grid size in the LES-NWM
may be reckoned by outer-flow eddies and the cost of calculation becomes negligibly dependent on Re. The model should be able to determine the wall shear
stress to the outer layer. The grid chosen close to walls is too coarse to solve
small-scale motions contributing to the momentum transfer. Time-step is much
bigger than the time-scale of these motions so the filtering operation becomes almost identical to the Reynolds averaging as established by Piomelli and Balaras
[2002] and Piomelli [2008].
There are many wall-layer models proposed for channel flows but here the equilibrium stress model is adopted. Deardorff [1970] and Schumann [1975] firstly
bypassed the inner layer by assuming an equilibrium layer in which stress is constant. This approximate boundary condition may be applied in environmental
and geophysical flows where the geometry is so simple and Re can be high. It
has to be kept in mind that corrections [Piomelli, 2008] to this model may be
accomplished when considering stratification effects or roughness.
Assuming a constant stress layer near the bottom smooth wall implies that horizontal velocities at the first grid off the bottom wall and in the outer layer (ol)
satisfies a logarithmic profile [Piomelli, 2008],
Ψ+
1 =
1
Ψ1
= log y1+ + B
u∗
κ
(3.36)
y1 u∗
(3.37)
ν
Note that variables with superscript + are shown in wall units. Ψ1 is the
modulus of the velocity at the first node off the bottom wall. It is obtained
from,
p
(3.38)
Ψ1 (x, y1 , z, t) = u2 (x, y1 , z, t) + w2 (x, y1 , z, t)
y1+ =
Assuming the clear water flow, the von Karman constant κ is equal to 0.41 and
constant is B = 5.1.
Piomelli [2008] commented that if the vertical position of the first node in wall
units (y1+ ) is lower than 11.0, the logarithmic law switches to the linear law
+
Ψ+
1 = y1 . From this law-of-the-wall, the friction velocity is computed based on
+
y1 and accordingly, the wall shear stress is obtained from the friction velocity
by,
τw = ρ0 u2∗
(3.39)
Deardorff [1970] defined second-order derivatives of filtered streamwise (u) and
spanwise (w) velocities in a turbulent channel flow at infinite Re,
1
∂2u
∂2u
=− 2 + 2
2
∂y
κy1
∂z
(3.40)
∂2w
∂2w
=
∂y 2
∂x2
(3.41)
Equation (3.40) forces the average velocity over the horizontal plane at the first
node off the bottom wall to satisfy the logarithmic law. Normalization of all
3.5 UNRESOLVED WALL-FUNCTION LES
26
variables in equations (3.40) and (3.41) can be done by the friction velocity and
viscosity.
Schumann [1975] calculated the mean shear stress from the momentum balance equation. Wall shear stress is iteratively balanced by the imposed pressure
gradient in channel flows. Schumann [1975] proposed following boundary conditions to relate the plane-averaged wall shear stress hτw ixz to Ψ1 at the first
node in channel flows,
hτw ixz
u (x, y1 , z, t)
(3.42)
τw,x (x, z, t) =
Ψ1
hτw ixz
τw,z (x, z, t) =
w (x, y1 , z, t)
(3.43)
Ψ1
It is obvious that u (x, y1 , z, t) and w (x, y1 , z, t) are temporal filtered velocity
components at the first node off the bottom wall. The equilibrium stress model
is applicable where the turbulence is in the equilibrium. Geometry of the flow
under investigation is simple.
Piomelli and Balaras [2002] set cost of computations using the equilibrium stress
model as Re0.5 . It is known that the Smagorinsky model may not be appropriate in reproducing correctly the eddy viscosity near the bottom smooth wall.
Although velocity gradient terms are derived from the no-slip condition, they
are wrongfully non-zero. In this regard, the strain rate tensor must be carefully analyzed to apply a proper SGS eddy viscosity at smooth walls in the
Smagorinsky model. If the first computational node off the bottom smooth wall
is positioned in the logarithmic region, corresponding prevailing terms in the
strain rate tensor are prescribed by [Fakhari, 2015],
S 12 =
u∗ u(x, y, z, t)
κy1 u(x, y1 , z, t)
(3.44)
u∗ w(x, y, z, t)
(3.45)
κy1 u(x, y1 , z, t)
If the first computational node is located in the viscous layer, dominant terms
are,
u2 u(x, y, z, t)
S 12 = ∗
(3.46)
ν u(x, y1 , z, t)
S 32 =
u2∗ w(x, y, z, t)
(3.47)
ν u(x, y1 , z, t)
For flows over a rough surface with a constant roughness height ks , the velocity
is expressed by [Fakhari, 2015],
1
y1
+
Ψ1 = ln
, y1 > ks
(3.48)
κ
ks
S 32 =
As Shamloo and Pirzadeh [2015] discussed, if the ratio h/ks is less than 10, the
boundary layer theory collapses. When there are isolated roughness elements,
flow patterns depend strongly on the pitch-to-height ratio [Cui et al., 2003].
Chapter 4
LES-COAST code
In this study, the LES-COAST code, which is available at the PhD school of
Earth Science and Fluid Mechanics (ESFM) of University of Trieste (UNITS)
is used to do the LES of the SST in a turbulent open channel flows. Mixture
flow model is used.
The curvilinear forms of governing equations are integrated using a modified
version of the finite-difference fractional-step method in a non-stggered grid.
It was developed by Zang et al. [1994]. The spatial derivatives are performed
using central second-order methods, except for advective terms treated by the
third-order QUICK scheme. The integration over time is done utilizing the
Adams-Bashforth scheme for convective and off-diagonal diffusive terms. For
diagonal diffusive terms, Crank-Nicolson scheme is adopted. To solve the Poisson equation, a multigrid technique is used. Periodic boundary conditions are
defined in the streamwise (x) and spanwise (z) directions. No-slip conditions are
applied at stationary walls. Totally speaking, written algorithm has a secondorder accuracy both in time and space.
The LES-COAST code found many applications in the wind-driven sea circulation [Petronio et al., 2013], water mixing and renewals in Barcelona harbor
area [Galea et al., 2014], mobile bed [Kyrousi et al., 2016] using level-set method
(LSM) and immersed boundary method (IBM), oil spill dispersion [Zanier et al.,
2017] in coastal areas, cubical Rayleigh-Benard convection [Foroozani et al.,
2017] and so forth.
4.1
Curvilinear coordinate
To simulate complicated flow geometries, curvilinear coordinates are preferred
due to better managing of boundaries. The transformation from Cartesian coordinates xi to curvilinear ones ξi is established. Under strong-conservation law
form [Zang et al., 1994] two-phase coupled equations in the FEM are trans-
27
4.2 DISCRETIZATION
28
formed in curvilinear coordinates [Zang et al., 1994],
∂Um
=0
∂ξm
(4.1)
∂Fim
∂J −1 ui
= J −1 Bi
+
∂t
∂ξm
(4.2)
The flux Fim takes this form [Zang et al., 1994],
Fim = Um ui +
ν
∂ui
J −1 ∂ξm
p − Gmn
χρ0 ∂xi
χ
∂ξn
(4.3)
Bi manifests all other terms [Foroozani, 2015] like the modified gravity term gm
appearing in the Navier-Stokes equations (2.22). J −1 is the reciprocal of the
Jacobian. It is equivalent to the volume of each cell. Um represents the volume
flux [Zang et al., 1994] and Gmn is the mesh skewness tensor,
∂ξm
uj
∂xj
∂xi
= det
∂ξj
Um = J −1
(4.4)
J −1
(4.5)
Gmn = J −1
∂ξm ∂ξn
∂xj ∂xj
(4.6)
When writing the advection-diffusion equation (2.15) for the volumetric concentration as the scalar field in curvilinear coordinates, we could get,
∂J −1 C
∂
∂Um C
∂C
=
+
ǫGmn
+ Qj
(4.7)
∂t
∂ξm
∂ξm
∂ξn
Q2 represents the vertical convective flux. It is related to the settling velocity
of sediments in curvilinear coordinates,
−1
∂J ξm
∂
(ws δj2 ) C
(4.8)
Qj =
∂ξm
∂xj
4.2
Discretization
A non-staggered grid [Zang et al., 1994] is taken in this study. It means pressure and velocities are defined at the center of each cell. Volume fluxes are
defined at central points of their corresponding faces of control volumes. Spatial discretizaion is done using the second-order centered finite difference method
(FDM). A semi-implicit time advancement is accomplished utilizing the accurate
explicit Adams-Bashforth scheme for convective and off-diagonal viscous terms.
4.3 FRACTIONAL STEP METHOD
29
Implicit Crank-Nicolson method is used for diagonal viscous terms. Accordingly,
discretizations of the continuity and Navier-Stokes equations are done,
∂Um
= 0.0
∂ξm
(4.9)
3
un+1
− uni
i
= (Xin + YE (uni ) + Bin ) −
∆t
2
1
(4.10)
Xin−1 + YE uin−1 + Bin−1 +
2
1
Ri pn+1 +
YI un+1
+ uni
i
2
Superscript n depicts advancement in time. The time step ∆t could be fixed or
adaptive but it must satisfy minimum Courant-Friedrichs-Lewy (CFL) condition. Xi shows convective terms. Ri is the gradient operator for pressure gradient terms. YI characterizes implicitly treated diagonal viscous terms whereas
YE are explicitly treated off-diagonal viscous terms. They all are,
J −1
∂
(Um ui )
∂ξm
∂ξm
∂
J −1
Ri = −
∂ξm
∂xi
∂
∂
νGmn
,m = n
DI = −
∂ξm
∂ξn
∂
mn ∂
DE = −
νG
, m 6= n
∂ξm
∂ξn
Xi = −
(4.11)
(4.12)
(4.13)
(4.14)
Discretization of the concentration equation as the scalar field can be performed
similarly.
4.3
Fractional step method
The numerical code LES-COAST solves governing equations using the fractional
step method. In each iteration, two steps, predictor and corrector (projection)
solutions, are done. For example, for Cartesian velocities, instead of directly
computing un+1
, an intermediate velocity u∗i is introduced in the predictor step
i
by,
∆t
∆t 3
∗
n
I − −1 (ui − ui ) = −1
(Xin + YE (uni ) + Bin ) −
2J
J
2
(4.15)
1
1
n−1
n−1
n−1
n+1
n+1
n
+ Ri p
+
+ Bi
Xi
+ YE ui
YI ui + ui
2
2
4.4 FILTERED EQUATIONS IN CURVILINEAR COORDINATE
30
I is the identity matrix. In the second step, corrector, the velocity is updated
and pressure is taken to enforce the continuity condition,
un+1
− u∗i =
i
∆t
Ri φn+1
−1
J
(4.16)
As assumed by Zang et al. [1994], the intermediate velocity u∗i is not restricted
by the continuity condition. The projection variable φ is related to the pressure
with,
Ri (φ)
∆t
−1
(4.17)
YI
Ri (p) = J −
2
J −1
The intermediate volume flux U∗m is defined like Zang et al. [1994],
∂ξm
∗
−1
u∗j
Um = J
∂xj
Contravariant fluxes at successive time instant are determined by,
n+1
n+1
∗
mn δφ
Um = Um − ∆t G
δξn
(4.18)
(4.19)
It must be emphasized that u∗j is defined at cell centers while contravariant
fluxes U∗m and Un+1
are specified on cell faces. To calculate U∗m on cell face,
m
∗
uj must be interpolated on corresponding cell face using the third-order accurate
upwind quadratic interpolation. It is analogous to the QUICK scheme. Finally,
the elliptic pressure Poisson equation for the projection variable φn+1 is solved
iteratively using the unstructured multigrid method [Perng and Street, 1989].
Smoothing relaxation algorithm called successive line over relaxation (SLOR)
is used,
∂φn+1
1 ∂U∗m
∂
(4.20)
Gmn
=
∂ξm
∂ξn
∆t ∂ξm
Regarding stability of the LES-COAST code, the Courant number could be fixed
or changeable for the fixed time step ∆t. The local CFL number is established
using,
| u 1 | | u2 | | u3 |
∆t
CF L =
+
+
(4.21)
= (| U1 | + | U2 | + | U3 |) −1
∆x
∆y
∆z
J
∆x, ∆y and ∆z are the grid spacing in Cartesian coordinates (x,y,z). ui are
Cartesian velocities while Ui are contravariant fluxes in curvilinear coordinates.
4.4
Filtered equations in curvilinear coordinate
Filtering aforementioned governing equations in curvilinear coordinates yields,
∂Um
=0
∂ξm
(4.22)
4.4 FILTERED EQUATIONS IN CURVILINEAR COORDINATE
31
!
1 ∂J −1 ξm
∂ ∂σmi
p −
+
χρ0 ∂xi
∂ξm ∂xi
∂
ν mn ∂ui
G
+ J −1 Bi
−
∂ξm χ
∂ξn
(4.23)
∂J −1 ui
∂
∂Um ui
=−
+
∂t
∂ξm
∂ξm
As communicated by Fakhari [2015], SGS stresses σmi in curvilinear coordinates is defined like corresponding SGS stresses τij in Cartesian coordinates in
equation (3.9),
σmi = Um ui − Um ui = J −1
∂ξm
∂ξm
uj ui − J −1
uj ui
∂xj
∂xj
(4.24)
SGS stress models demonstrated in Cartesian coordinates are also applicable in
curvilinear coordinates. When doing the filtering of the concentration equation
(4.7) in curvilinear coordinates, we obtain,
∂
∂Um C
∂J −1 C
=
+
∂t
∂ξm
∂ξm
∂
∂ξm
∂J −1 ξm
(ws δj2 ) C
∂xj
ǫGmn
!
−
∂C
∂ξn
+
∂ξm ∂ωj
∂xj ∂ξm
(4.25)
Chapter 5
Case study I: wall-resolving
LES of SST
5.1
DNS-FEM for laminar sediment-laden flow
The accuracy of the code in solving two-phase coupled equations in the FEM is
assessed using the direct numerical simulation (DNS) of a laminar close channel
flow laden with sediment particles. The numerical results are compared against
the analytic solution. The lengths of the close channel flow are,
Lx = 2π(m), Ly = 2(m), Lz =
2π
(m)
3
(5.1)
There is no effect of the sidewall or secondary flow on the SST. The number
of cells is 32 in all directions. The flow is driven with a constant pressure
gradient divided by the water density Π = 1.0 in the streamwise direction. The
gravity term is deactivated and the density ratio between sediment particles
and water in this laminar case is s = 2.65. In x and z directions, the periodic
boundary condition for both momentum and scalar transport equations are
specified. Smooth rigid walls are assumed on top and bottom boundaries. The
Courant number, molecular Schmidt number, kinematic viscosity and time step
are,
m2
CF L = 0.5, Sc = 0.7, ν = 0.1
, ∆t = 1.96 × 10−2 s
(5.2)
s
The fluid is incompressible and flow in the close channel is fully developed in the
streamwise direction. Since the flow is constrained by parallel top and bottom
plates, the vertical velocity is zero v = 0.0. Under these circumstances, the
Navier-Stokes equation (2.22) is simplified,
ν ∂2u
1
Π=
χ
χ ∂y 2
32
(5.3)
5.1 DNS-FEM FOR LAMINAR SEDIMENT-LADEN FLOW
33
It is apparent that Π is inherently negative. One significant finding here is that
vertical profiles of the streamwise velocity u in both clear water and sedimentladen flows are equivalent regardless of the imposed concentration profile. Using
integration twice, we could easily get the vertical profile of the streamwise velocity,
1
u (y) =
y 2 − 2hy Π
(5.4)
2ν
Under the steady state condition, the vertical profile of the concentration is
obtained from the advection-diffusion equation (2.15),
− ws
ν ∂2C
∂C
=
∂y
Sc ∂y 2
(5.5)
When supposing C = 1.0 at the bottom wall and C = 0.0 at the top wall,
following vertical profile of the volumetric concentration is achieved,
−ws Sc
y
1−e ν
C (y) = 1 +
−ws Sc
2h
−1
e ν
(5.6)
Figure (5.1) illustrates vertical profiles of the mean streamwise velocity for the
laminar close channel flow using the DNS and analytic solution. Averaging is
done over horizontal planes and in time.
Figure 5.1 – Comparison between DNS and analytic solution in terms of the
mean streamwise velocity
When the concentration profile manifested by equation (5.6) is introduced into
the clear fluid, no change in the mean streamwise velocity is seen as expected.
5.2 EXPERIMENTAL DATA IN LITERATURE
34
Figure (5.2) demonstrates that there is an exact similarity between the analytic
solution and DNS when evaluating the depth-averaged volumetric concentration.
Figure 5.2 – Comparison between the DNS and analytic solution in terms of the
mean volumetric concentration
5.2
Experimental data in literature
Muste et al. [2005] and Muste et al. [2009] researched experimentally the dilute
dispersion of suspended sediment particles in a highly turbulent open channel
flow by means of the particle image velocimetry (PIV) and particle-tracking
velocimetry (PTV) techniques. The aim was to understand how suspended sediment affected flow structures and turbulence. The results were illustrated in
terms of streamwise and vertical velocities, Reynolds stresses, turbulence intensities and volumetric concentration. The volumetric concentration of sediment
particles was generally small and hence collisions between particles were negligible. All sediment particles were kept suspended and there was no deposition
on the channel bed due to the imposed mass flow rate. The experimental flume
was 6m long and 0.15m wide with water depth h = 0.021m. All measurements
were conducted along the centerline of the flume, in a section positioned 5.3m
from the entrance. The bed was made of smooth stainless steel and sidewalls
were of the glass. From an experimental viewpoint, a flow-conditioning honeycomb structure was used at the entrance of experimental flume to facilitate
the development of the flow turbulence. The sediment-laden flow was created
by adding sand particles to the clear water flow. The slope of the channel was
set at 0.0113 in all experiments. Flow uniformity was also ensured. Due to the
large aspect ratio AR = 7.5, no effective secondary flows were reported. Faint
5.2 EXPERIMENTAL DATA IN LITERATURE
35
streaks were seen for the sediment-laden flow called NS3 where the bulk flow
velocity was diminished 20 percent. The water surface waviness was 2 − 3mm.
Experimental conditions and pertinent flow variables in the single-phase mixture are shown in table (5.1).
Muste et al. [2005] called experiments with the natural sand as NS1, NS2 and
NS3 corresponding to three separate volumetric concentrations, C = 0.00046,
C = 0.00092 and C = 0.00162, respectively. As the volumetric concentration
was increased, the deviation of the streamwise velocity in the sediment-laden
flow from that in the clear water flow got bigger.
Subsequently, the experiment NS3 is used in present thesis to better explain
interactions between turbulence and suspended sand particles.
Table 5.1 – Properties of the mixture flow in [Muste et al., 2005]
Specific gravity s
Diameter d(m)
m
Settling velocity ws ( )
s
Volum. concentration C
Reynolds number Re
Froude number F r
m
Shear velocity u∗ ( )
ms
Bulk velocity Ũ ( )
s
von Karman κ
Temperature T(0 C)
Clear water
-
Sediment-laden flow (NS3)
2.65
0.00023
0.024
17670
1.89
0.042
0.00162
17340
1.75
0.043
0.839
0.792
0.402
23
0.367
23
The sediment particles were the natural sand with d = 0.23mm. The settling
velocity ws was estimated from the empirical formulation proposed by Dietrich
[1982]. It took into account effects of size, density, shape, and roundness.
The bulk Reynolds number Re and Froude number F r are computed from,
Re =
4Rh Ũ
Ũ
,Fr = √
νm
gh
(5.7)
Rh and νm are the hydraulic radius and kinematic viscosity of the mixture,
respectively.
As categorized by Muste et al. [2005], the shear velocity u∗ for open channel
flows could be computed from two methods, (1) momentum balance equation
and (2) computed Reynolds stresses. From an experimental point of view, uncertainties are larger in the first method, hence the extrapolation of the Reynolds
stresses was adopted.
High ratio between the shear velocity and settling velocity indicated that in this
highly turbulent flow all sand particles are suspended. The insertion of sediment
particles did not affect the bottom shear stress. Shear velocity were identical for
5.3 SIMULATION DETAILS
36
both clear water and sediment-laden flows. The kinematic viscosity was altered
less than 0.9 percent when sediments were introduced into the clear water flow.
So, νm ≈ ν. The bulk flow velocity Ũ for the sediment-laden flow was lesser
than that for the clear water and hence, the particle suspension extracted the
energy from the flow and the turbulence was attenuated.
The von Karman constant κ was diminished with the addition of sediment particles meaning inside the inner region, there was a strong particle flow turbulence.
Close to the bottom wall, the base fluid was bound by the viscous shear and
particles were not. As a result, close to the channel bed, the streamwise velocity
for the mixture was higher than that for the clear water.
Finally, no clear trend between the reduction of the von Karman constant and
density variations demonstrated. The linear trend of the Reynolds stress was
also preserved for the sediment-laden flow.
5.3
Simulation details
The two-phase coupled equations in the FEM framework [Ferry and Balachandar, 2001] are considered. Wall-resolving LES in the LES-COAST code is implemented to solve equations. It elucidates interactions between suspended sand
particles and turbulence in experiments of Muste et al. [2005].
The effects of the viscous dissipation and gravity are totally insignificant. Preferential accumulation, turbophoresis or clogging of cohesive sediment particles
are disregarded.
The dimensions of the computational domain are 2 × π × 0.021m, 0.021m and
π×0.021m in the streamwise x, vertical y and spanwise z directions, respectively.
The number of cells in x,y and z directions are 136, 128 and 192, respectively.
According to mesh requirements in the wall-resolving LES, grids in x and z
directions are homogeneous with ∆x+ = 40.0 and ∆z + = 14.0. The grid in the
vertical direction is stretched so that the first centroid is located at y1+ = 0.5
and nine cells are embedded below y + = 10.0. The grid requirement in the
vertical direction is depicted in figure (5.3).
The periodic boundary condition for all key variables is imposed in x and z
directions. A free-slip boundary condition is imposed at the top surface. Noslip boundary condition is defined at the channel bed. The Courant number is
0.5 and corresponding time step is ∆t = 3.917 × 10−6 s. Both molecular and
turbulent Schmidt numbers, Sc and Sct , are unity in the numerical simulation.
The dynamic eddy viscosity and diffusivity models are used to treat the SGS
stresses and sediment fluxes.
Firstly, statistics of the turbulent flow of the clear water are gathered by imposing a constant mass flow rate. After that, a Rouse-fitted profile of the sediment
concentration relating to the experiment NS3 of Muste et al. [2005] is introduced
to the clear water flow.
More importantly, to ensure a fully-developed state for clear water and sedimentladen flows, instantaneous streamwise velocities at four different points in the
vertical direction are checked.
5.4 VALIDATIONS AND FINDINGS
37
Figure 5.3 – Grid refinement in the vertical direction in the wall-resolving LES
5.4
Validations and findings
Figure (5.4) shows comparisons of the dimensionless streamwise velocity of the
clear water flow between the present wall-resolving LES, theoretical study of
Nezu and Nakagawa [1993] and experiment of Muste et al. [2005].
Figure 5.4 – Dimensionless depth-resolved streamwise velocity for the clear water flow against experimental [Muste et al., 2005] and theoretical [Nezu and
Nakagawa, 1993] studies
Nezu and Nakagawa [1993] presented the log-law velocity distribution using
5.4 VALIDATIONS AND FINDINGS
38
κ = 0.41 and B = 5.5. It must be highlighted that U , V and W are the
averaged velocities in time. Symbol hi indicates averaging over horizontal planes.
The streamwise velocity is made dimensionless using the shear velocity which
is 0.042m/s in the wall-resolving LES. Shear velocity is computed at the first
centroid off the bottom wall. It is located in the viscous sub-layer region where
+
there is hU i = y + . As would be seen, a satisfactory agreement is achieved
regarding the dimensionless depth-resolved streamwise velocity for the clear
water flow. Figure (5.5) manifests discrepancies between results of the wallresolving LES and Muste et al. [2005] in terms of the depth-resolved vertical
velocity for the clear water flow. A positive value means that the vertical velocity
is directed upwardly to the free surface. Generally speaking, vertical velocities
are too small compared to streamwise velocities. Largest downward depthresolved vertical velocities are observed near the free surface complying with
outcomes of Cellino [1998] and Graf and Cellino [2002]. It is found out in
experiments that the depth-averaged vertical velocity was constant. It is much
higher than that in the current wall-resolving LES. Reynolds shear stresses
normalized with the shear velocity u∗ for the clear water flow are compared in
figure (5.6). Second-order statistics calculated here are the sum of corresponding
resolved and SGS parts. The turbulent velocity fluctuations in streamwise and
vertical directions are computed by,
u′ (x, y, z, t) = u(x, y, z, t) − U (x, y, z)
(5.8)
v ′ (x, y, z, t) = v(x, y, z, t) − V (x, y, z)
(5.9)
Figure 5.5 – Depth-resolved vertical velocity for the clear water flow against
[Muste et al., 2005]
The linear variation of the Reynolds shear stress after some distance off the bottom wall demonstrates that flow is fully developed. The streamwise turbulence
5.4 VALIDATIONS AND FINDINGS
39
intensities for the clear water flow are illustrated in figure (5.7). The RMS is
computed from,
p
p
(5.10)
urms = hu′ u′ i, vrms = hv ′ v ′ i
The streamwise turbulence intensity is the RMS of the streamwise turbulent
velocity fluctuation normalized by the shear velocity.
Figure 5.6 – Depth-resolved Reynolds stress for the clear water flow against
Muste et al. [2005]
Figure 5.7 – Streamwise turbulence intensity for the clear water flow against
DNS data [Hoyas and Jimenez, 2008] and experiment of Muste et al. [2005]
5.4 VALIDATIONS AND FINDINGS
40
The comparison is fulfilled against experimental data [Muste et al., 2005] and
DNS database of a close turbulent channel flow at turbulent Reynolds number
Reτ = 944. This DNS was provided by Hoyas and Jimenez [2008]. The turbulent
Reynolds number in the present wall-resolving LES of the clear water flow is
Reτ = 882. As could be comprehended, the streamwise turbulence intensity in
experiment [Muste et al., 2005] is much bigger than those in the wall-resolving
LES and DNS. Even the DNS with a higher Reτ gives a lower level of fluctuations
with respect to experiments. It is worthy to mention that the underestimation of
the streamwise turbulence intensity was also recorded by Chauchat and Guillou
[2008]. They validated experimental data using a 2D two-phase formulation
with k − ǫ closure model. k is the turbulent kinetic energy. More specifically,
the maximum of the streamwise turbulence intensity in the DNS and wallresolving LES is equal to 2.82 and 3.0, respectively. In experiments, it is 3.3.
Furthermore, the apex of the streamwise turbulence intensity in experiments
is well below y + = 10. However, the maximum of the streamwise turbulence
intensity was at y + = 14 [Chauchat and Guillou, 2008], y + = 15 [Hoyas and
Jimenez, 2008] or y + = 17 [Nezu and Rodi, 1986]. In figure (5.8), the vertical
turbulence intensity for the clear water flow is compared against existing DNS
and experiments.
Figure 5.8 – Vertical turbulence intensity for the clear water flow against DNS
[Hoyas and Jimenez, 2008] and experiment of Muste et al. [2005]
Again, DNS and wall-resolving LES highly underestimate the vertical turbulence
intensity obtained in experiment of Muste et al. [2005] except near the free
surface. Close to the free surface, vertical turbulence intensities from the wallresolving LES and experiment are same. On contrary, Chauchat and Guillou
[2008] reported the overestimation of the vertical turbulence intensity with the
friction velocity u∗ = 0.047m/s using a 2D k − ǫ numerical model.
5.4 VALIDATIONS AND FINDINGS
41
Like the numerical study of Chauchat and Guillou [2008], the profile of the
sediment concentration in the experiment NS3, which is demonstrated in figure
(5.9), is imposed to the fully developed clear water flow.
The maximum of the volumetric sediment concentration is above the channel
bed. The experimental profile of the sediment concentration was matched by a
Rouse-fitted profile,
Rn =
1
ws
= 0.19, β =
= 0.22
βκu∗
Sct
(5.11)
Rn is the fitted Rouse number. β is the inverse of the turbulent Schmidt number.
Kundu and Ghoshal [2014] theoretically categorized relationships between the
direction of vertical velocities induced by secondary currents and the shape of
the concentration profile.
Figure 5.9 – Imposed profile of the sediment concentration from experiment NS3
of Muste et al. [2005]
In type (I), the maximum of the sediment concentration is on the channel bed
and the vertical velocity is downward. In type (II), the maximum is at some
distance off the bottom wall. It is seen in figure (5.9). Vertical velocity is upward. In contrast to explications of Kundu and Ghoshal [2014], Muste et al.
[2005] recorded negative vertical velocities and absence of any secondary currents while the concentration profile for experiment NS3 was acquired. Yu et al.
[2014] presented figure (5.10) for clarifying better results of experiment NS3.
When sediment particles are introduced to the flow, the streamwise velocity of
the mixture is decelerated inside the outer region. It is due to the velocity lag
between particles and water. On the other hand, the streamwise velocity of the
mixture is expedited within the inner region because sediment particles without viscosity move faster than water. The gradient of the streamwise velocity
5.4 VALIDATIONS AND FINDINGS
42
distribution gets steeper inside the buffer layer near the bed. Accordingly, the
von Karman constant declines.
Figure 5.10 – How suspdended sediment particles modify the streamwise velocity, taken with courtesy from [Yu et al., 2014]
Figure (5.11) shows influences of suspended natural sand particles on the dimensionless depth-resolved streamwise velocity in a highly turbulent open channel
flow. Once sediment particles are in the flow, the bulk flow velocity Ũ is reduced
from 0.839m/s to 0.799m/s in the wall-resolving LES. Error is less than one
percent compared to Ũ = 0.792m/s in the experiment NS3.
Figure 5.11 – Effect of imposed profile of the sediment concentration in the
experiment NS3 on the dimensioneless depth-resolved streamwise velocity
Besides, for clear water and sediment-laden flows, the friction velocity is u∗ =
5.4 VALIDATIONS AND FINDINGS
43
0.042m/s and u∗ = 0.041m/s, respectively. Contrary to Yu et al. [2014] and
Muste et al. [2005], in the wall-resolving LES, no acceleration of the depthresolved streamwise velocity inside the inner region is seen when sediment particles are introduced. It substantiates that the friction velocities for clear water
and sediment-laden flows are same. Also inside the outer region, the depthresolved streamwise velocity is abated for the sediment-laden flow. Suspending
sand particles reduces the drag force. Similar to conclusions of Muste et al.
[2005], imposing the buoyancy term gm does not have any effect on the depthresolved streamwise velocity. There is no influence of suspended sediment particles on the Reynolds stresses.
Chapter 6
Case study II: unresolved
wall function LES of SST
6.1
Experimental data in literature
Cellino [1998] and Graf and Cellino [2002] performed experimental studies on
the SST in a recirculating tilting open channel flow with bottom rough wall.
The length and width of the channel were 16.8m and 0.6m, respectively. To
measure data, a section was chosen 13m from the entrance of the channel where
the boundary layer established. Measurements were performed by a sonar instrument (Acoustic Particle Flux Profiler, APFP) which was located at the
center-line of this cross section. The imposed flow discharge, which was continuously checked using a magnetic flowmeter, provided h = 0.12m. Graf and
Cellino [2002] emphasized that the aspect ratio (AR) of the channel was big
enough to consider the flow bidimensional. Channel bed was rough but infiltration of sediments on that was totally prevented. The mixture flow was steady
and uniform. It was in the capacity condition meaning a sediment layer with
a constant thickness of 2mm covered the bottom rough wall. Remaining sediments were in the suspension mode. Bed slope was equal to the channel slope.
The sediment particles were of non-cohesive natural sand with,
d = 0.135mm, s = 2.65, ws = 0.012
m
s
(6.1)
Table (6.1) presents pertinent variables for the clear water and sediment-laden
flows in experiments of Cellino [1998].
The APFP instrument provided the instantaneous vertical profiles of longitudinal (streamwise) u(y, t) and vertical v(y, t) velocities. In the experiment of
Cellino [1998], the bulk velocity was computed from,
Ũ =
1
h
Z
h
U (y)dy
0
44
(6.2)
6.1 EXPERIMENTAL DATA IN LITERATURE
45
Note that U (y) is the average streamwise velocity in time. Shear velocity u∗ can
be approximated using the energy-gradient method or Reynolds-stress method.
The extrapolation of the Reynolds stresses towards the channel bed gives the
shear velocity.
Table 6.1 – Germane variables in experiments of [Cellino, 1998]
CW S015
SLF (Q40S003)
0.05
0.049
0.12
5.0
0.15
0.726
0.12
5.0
0.03
0.68
248900
0.67
0.045
233000
0.63
0.028
0.031
0.0012
54.0
-
0.014
0.00005
1.4
0.000135
-
2650
-
0.00012
-
0.00077
1000.0
1001.28
0.000001
0.000001002
0.4
0.4
3
m
)
s
Flow depth h(m)
Aspect ratio AR
Bed slope Sb (%)
m
Bulk velocity Ũ ( )
s
Reynolds number Re
Froude number F r
m
Friction velocity u∗ ( )
s
Friction factor f
Equivalent roughness ks (m)
Dimensionless roughness ks+
Particle diameter d50 (m)
kg
Sediment density ρs ( 3 )
m
m
Settling velocity ws ( )
s
Volum. concentration C
kg
Density ρm ( 3 )
m
m2
)
Kinematic viscosity νm (
s
von Karman κ
Discharge Q(
In agreement with Coleman [1981], the von Karman constant was κ = 0.4 for
clear water and sediment-laden flows. The friction factor f is estimated by,
2
u∗
(6.3)
f =8
Ũ
The equivalent bed roughness ks is given by the phenomenological ColebrookWhite equation,
r
1
ks
bf
√
(6.4)
+
= −2.log
f
a f Rh
Re f
The constants af and bf are equal to 11.5 and 3.0, respectively. In order to
specify the type of turbulent flows, the roughness height in the dimensionless
form is calculated,
ks u∗
(6.5)
ks+ =
νm
6.2 SIMULATION DETAILS
46
ks+ is called the roughness Reynolds number in experiments of Cellino [1998]. If
it is lower than 5.0, the bed is considered hydraulically smooth and the constant
of the integration in the log-law equation (3.36) is B = 5.5. If it is between 5.0
and 70.0, the bed is hydraulically in the transitional condition. For ks+ > 70.0,
the bed is hydraulically rough with B = 8.5. As Cellino [1998] showed, the
classical log-law in transitional and fully rough turbulent open channel flows is,
y
1
U
= ln
+B
(6.6)
u∗
κ
ks
Appropriate length-scale in hydraulically smooth open channel flow is the viscous length-scale [Pope, 2000]. In transitional and fully rough ones, it is equivalent to the bed roughness ks . In experiments, for the clear water flow, ks+ is 54.0
and hence, the channel bed is hydraulically transitional. For the sediment-laden
flow (Q40S003), it is 1.4 so the bed is hydraulically smooth. While the mixture
density is taken from equations (2.23) and (2.24), the kinematic viscosity νm is
modified using [Cellino, 1998],
νm = ν ×
ρ0 (1 + 2.5C)
ρ0 + (ρs − ρ0 ) C
(6.7)
Kinematic viscosity is changed around 0.2 % so physical conclusions are not related to the viscosity change. Cellino [1998] stated that clear water formulations
still can give a reasonable and theoretical-empirical description of the suspension flow if the volumetric concentration is small C < 0.01, size of particles is
so tiny, sediments are not cohesive and Newtonian rheological law is valid. If
sediment particles are too smaller than the micro-scale of the turbulence λ, they
may follow smallest eddies in the flow.
6.2
Simulation details
In the LES-COAST code, the single-phase Euler-Euler based unresolved wall
function LES with the equilibrium stress model is selected to analyze interactions between suspended sediment particles and turbulence. Results are compared against experiments of Cellino [1998]. In the numerical simulation, channel dimensions in x, y and z directions are 2 × π × 0.12m, 0.12m and π × 0.12m,
respectively. Number of cells is 32 × 32 × 32. Both the molecular and turbulent
Schmidt numbers, Sc and Sct are 1.0. For the SLF and CWC, the constant
in Smagorinsky model is Cs = 0.065. The Courant number in this study is
0.2. Time step (∆t) changes over time around 2 × 10−3 s to get a fixed Courant
number. The flow is driven by imposing a constant pressure gradient divided
by the water density and it is equal to 0.021 and 0.0081 for the CWC and
SLF, respectively. It is found that the wall shear stress on the sidewalls should
be included in the calculation of the imposed pressure gradient. According to
Yang et al. [2012], the shear stress on each sidewall is equal to 0.63 × τw . τw is
the bottom shear stress obtained from the energy-gradient method. The same
6.3 VALIDATIONS AND FINDINGS
47
roughness as expressed in experiments is applied differently for the CWC and
SLF. In addition, no-flux condition for the sediment concentration is defined at
the free surface. Periodic boundary condition is defined in the streamwise and
spanwise directions. The initial state of the open channel flow is obtained by
interpolating from a highly turbulent closed channel flow. After reaching the
steady state, statistics of the CWC are gathered. Then, a constant profile of
the sediment volumetric concentration C = 0.000773 is imposed to get statistics of the SLF. The reference concentration formulation proposed by Smith and
Mclean [1977] together with the Shields diagram is used to treat the erosion of
sediment particles from the channel bed. It is supposed that the buoyancy induced by suspended sediment particles could affect hydrodynamics of the flow
and hence the two-way coupling is applied. Modified gravity term derived by
Winterwerp [2001] in the Navier-Stokes equations of the SLF is introduced.
6.3
Validations and findings
Figure (6.1) demonstrates influences of suspended sediment particles on the
streamwise velocity. Table (6.2) presents the bulk velocity, friction velocity
and position of the first centroid off the bottom wall. They are obtained from
the unresolved wall-function LES. The friction velocity is obtained from the
extrapolation of the Reynolds stress inside the upper layer.
Table 6.2 – Key variables obtained from the unresolved wall-function LES
m
Bulk velocity Ũ ( )
s m
Friction velocity u∗ ( )
s
Position of first centroid y1+
CW S015
SLF (Q40S003)
0.73
0.69
0.049
0.03
91.87
56.25
When comparing the bulk velocity from the unresolved wall function LES to
that in experiments, it is found that the error is around 2 percent.
Compared to the CWC, it is seen that the suspension of sand particles engenders
the reduction of the friction velocity and roughness. From a theoretical point of
view, inside the inner region y < 0.2 × h the universal law-of-the-wall (log-law)
and inside the outer region y > 0.2 × h law-of-the-wake (Cole’s law) could be
applied to get the streamwise velocity of the flow over smooth and rough beds.
As could be observed from experimental data in figure (6.1), the maximum of the
streamwise velocity was not at the free surface. Yang et al. [2004] theoretically
proved that secondary currents induced the non-zero mean vertical velocity
and the deviation of the Reynolds shear stress from the conventional linear
distribution. Furthermore, the maximum streamwise velocity occurs below the
free surface. Yan et al. [2011] mentioned that for narrow open channels (AR ≤
5.0), the flow becomes actually 3D. Toorman [2003] hypothesized that deviations
6.3 VALIDATIONS AND FINDINGS
48
from the logarithmic law close to the free surface in experiment of Cellino [1998]
were probably created by secondary currents and friction with the sidewalls.
Figure 6.1 – Comparison of the streamwise velocity against experiments of
Cellino [1998]
Figure 6.2 – Comparison of the vertical velocity against experiments of Cellino
[1998]
More importantly, as is seen in the experiment of Cellino [1998], suspending
sand particles in the flow results in the decrease of the bulk velocity like Muste
et al. [2005]. More specifically, in accordance with conclusions of Muste et al.
6.3 VALIDATIONS AND FINDINGS
49
[2005], the streamwise velocity is reduced inside the outer layer. Flow is accelerated in the super-saturated region close to the bottom wall because there are
high inter-particle collisions between sediment particles which are not bounded
by viscosity.
Figure (6.2) demonstrates effects of suspended sediments on the vertical velocity.
A positive value means the velocity is directed upward to the free surface. Generally speaking, vertical velocities are too small in comparison with streamwise
velocities. In experiments of Cellino [1998], largest downward vertical velocities
were recorded near the free surface. Secondary flows created non-zero vertical
velocities based on Yang et al. [2004]. Unresolved wall-function LES shows that
vertical velocities for the sediment-laden flow in lower levels are significantly
higher than vertical velocities for the clear water flow. Close to the free surface,
vertical velocities for clear water and sediment-laden flows are equal.
Figure 6.3 – Comparison of the Reynolds shear stress against experiments of
Cellino [1998]
The Reynolds stresses obtained from experiments and unresolved wall-function
LES for the CWC and SLF are shown in figure (6.3).
Pay attention that in the viscous sub-layer region, the Reynolds shear stress is
negligible compared to the viscous stress. For y > 0.05×h, the total shear stress
may be well approximated by the Reynolds shear stress. The linear variation
of the Reynolds shear stress after some distance from the bottom wall indicates
that there is uniform flow. Analogous to Cellino [1998], the Reynolds stress
deviates from the linear trend inside the inner region y < 0.2 × h. Erosion and
deposition take place in this region and the sediment concentration is high.
The streamwise turbulence intensity from the unresolved wall-function LES is
compared against experimental data of Cellino [1998] and Kironoto [1992] in
figure (6.4). Kironoto [1992] experimentally reported that for the uniform open
6.3 VALIDATIONS AND FINDINGS
50
channel flow of the clear water over a rough bed, streamwise and vertical turbulence intensities are estimated by,
urms
= 2.04 × e
u∗
urms
= 1.14 × e
n
u∗
−0.97×
y
h
(6.8)
y
h
(6.9)
−0.76×
Kironoto [1992] showed distributions of turbulence intensities by the universal
exponential law. 2D flow variables in experiments of Kironoto [1992] and Cellino
[1998] are comparable. Cellino [1998] illustrated that inside the inner region
y < 0.2 × h, the presence of sand particles induced the intensification of the
streamwise turbulence intensity. Within the outer region y > 0.2 × h, there is
no detectable difference between the CWC and SLF. This is what Muste et al.
[2005] also reported.
Figure 6.4 – Comparison of the streamwise turbulence intensity against experiments of Cellino [1998] and Kironoto [1992]
Inside the outer region, data of Cellino [1998] totally overestimates the streamwise turbulence intensity obtained from experimental correlation of Kironoto
[1992]. Regarding the unresolved wall function LES, it is seen that the streamwise turbulence intensity for the sediment-laden flow is bigger than that for the
clear water case. Close to the free surface they are identical to data of Kironoto
[1992] and Cellino [1998]. This similarity is due to noticeable decrease of the
sediment concentration near the free surface.
The vertical turbulence intensity from the unresolved wall-function LES is compared against experimental data of Cellino [1998] and Kironoto [1992] in figure
(6.5).
6.3 VALIDATIONS AND FINDINGS
51
Figure 6.5 – Comparison of the vertical turbulence intensity against experiments
of Cellino [1998] and Kironoto [1992]
As Cellino [1998] found out, suspending sand particles in the clear water flow
caused the attenuation of the turbulence more remarkably close to the bed. It
is seen from the unresolved wall-function LES that near the bed, the presence of
suspended sand particles results in the slight enhancement of the vertical turbulence intensity. In upper levels, significant weakening of the vertical turbulence
intensity is seen. Adjacent to the free surface, vertical turbulence intensities
for the CWC and SLF are identical since the sediment concentration is low. In
contrast to Cellino [1998], a better convergence in the middle of the channel
is achieved using the unresolved wall-function LES against results of Kironoto
[1992]. In figure (6.6), the sediment concentration from the unresolved wallfunction LES is compared against experimental data of Cellino [1998]. The
experimental variation of the sediment concentration was matched by a Rousefitted profile like,
Rn =
1
ws
= 2.15, β =
= 0.498, Cref = 0.00929
βκu∗
Sct
(6.10)
When the buoyancy term is deactivated, the sediment concentration is high
and unsatisfactory higher-order turbulence statistics are gathered with the unresolved wall-function LES. Analogous to the study of Toorman [2003], experimental results are underestimated close to the free surface. In experiments,
the streamwise velocity was deviated from the logarithmic profile due to the
existence of secondary currents. Near the free surface, large velocity gradients
produced more turbulence and kept more sediments in the suspension mode
compared to the theory of Toorman [2003] for infinitely wide channels. From
an experimental viewpoint, the accuracy of very low values of the concentration
near the free surface is highly questionable. Understanding of highly saturated
6.3 VALIDATIONS AND FINDINGS
52
layer of the sediment close to the bed is done only qualitatively [Toorman, 2003].
Figure 6.6 – Comparison of the sediment concentration against experiments of
Cellino [1998]
Chapter 7
Propagation of suspended
sediment wave in
macro-rough flow
7.1
Literature review on macro-rough flow
Channelized rivers from a geometrical point of view are simple configurations
because they have a straight path. In natural morphology, however, a high
diversity across the channel bed or bank may be seen. In river restoration applications, hydraulics engineers artificially alter the natural path and geometry
of rivers by creating lateral cavities. Lateral macro-roughness cavities are generated to improve the habitat diversity. The dead zone areas are refugees for
fishes and are targets for spawning. They have implications for the sediment
management in rivers and propagation of contaminants. Large-scale irregularities may change the SST in rivers. Macro-rough boundaries could influence the
propagation of a suspended sediment wave in a turbulent open channel flow.
Bathurst [1985] defined if ratio between flow depth (h) and characteristic size of
the roughness element (k) was less than 4, flow was a macro-rough one. According to Meile [2007] and Meile et al. [2011], when the size of lateral roughness
elements was comparable to the flow depth or hydraulic radius of the channel,
there was a macro-rough flow. Lyapin [1994] found out experimentally that
canals with increased bottom roughness were those in which the ratio between
h and k was lower than 5. These macro-roughness elements could be mounted
laterally or on the bottom surface. The macro-roughness elements may cover
entire channel section or only a part of it so arrangement is important.
Considering lateral macro-rough flows, Meile [2007] stated that longitudinal
waves were created in the streamwise direction (x) while the transversal waves
were oscillated in the crosswise direction (z). By increasing the flow depth,
the effect of the bank roughness got more prominent. The cavity flows, which
53
7.1 LITERATURE REVIEW ON MACRO-ROUGH FLOW
54
are generated by lateral macro-roughness elements, are frequently examined in
terms of their trapping efficiency and corresponding aspect ratio.
When the cavity aspect ratio ϑcf is between 0.15 and 0.6, a primary vortex and
a secondary vortex with a much lower velocity are detected. For ϑcf = 0.8,
a single circular vortex is seen inside the cavity flow. Meile [2007] and Meile
et al. [2011] recorded the existence of unstable recirculation zones, wake regions,
coherent structures, mixing layers and oscillations of the flow in the transverse
direction (z). As prescribed in the literature, for the bottom roughness, the
more the velocity or water depth augments, the more the influence of roughness
elements abates. Ashrafian and Andersson [2006] concluded that mounting of
roughness elements on the bottom wall resulted in the reduction of the size of
large-scale flow structures within the roughness sub-layer using flow visualization techniques. Moreover, extra eddies were generated that led to the consumption of the mechanical energy and an elevated resistance to the flow. Ashrafian
et al. [2004] characterized this roughness sub-layer as a non-equilibrium shear
layer in which sturdy inhomogeneities, particularly close to roughness elements
were observed. Tachie et al. [2000] hypothesized that the presence of the wall
roughness might be a source of secondary flows. It was resulted in the damping
of vertical fluctuations of the velocity under the free surface. The decrease of the
vertical velocity of the flow was another consequence. Correspondingly, the flow
below the free surface could be retarded and the maximum streamwise velocity
was seen some distance below the free surface. It is obvious that the influence
of the roughness of the bottom wall could be comprehended well further than
roughness sub-layer. As a result, the wall-similarity hypothesis [Bisceglia et al.,
2001], fails to describe flows over bottom rough walls. It is known that the
distribution and size of macro-rough elements existing on the bed have a more
prevailing role than the viscous length scale on near-wall turbulent structures.
To understand the hydraulic resistance and turbulent characteristics of bottom
macro-rough turbulent flows in a simplified way, the homogeneous arrangement
of bottom elements is considered in this study. Homogeneity [Romdhane et al.,
2017] implies that crosswise strips have identical length as the channel width.
Both d-type roughness [Djenidi et al., 1994] [Bisceglia et al., 2001] in which
spacing is equal to the height of the roughness element and k-type roughness
[Perry et al., 1969], in which spacing is bigger than the height of the roughness
element, are investigated experimentally here.
The ratio ϑ of the distance w between bottom roughness elements and height
of bottom roughness elements k is defined by,
ϑ=
w
k
(7.1)
The variable γ represents the ratio between the roughness height k and flow
depth h,
k
(7.2)
γ=
h
Leonardi et al. [2007] emphasized that with the d-type roughness, the frictional
drag was much larger than the pressure drag. Lyapin [1994] did experimental
7.1 LITERATURE REVIEW ON MACRO-ROUGH FLOW
55
visualization of fluid flows over rough surfaces. It was confirmed that the appreciable space between bottom roughness elements was occupied by stagnant
vortex zones. More specifically, for a d-type rough wall, because elements were
closely spaced, stable vortices were formed within cavities, and no eddy shedding into the flow was above bottom elements. It was discovered by Romdhane
et al. [2017] that the wall skin friction coefficient cf in the d-type roughness type
was identical to that in the smooth case. The flow visualizations for the square
bar roughness [Djenidi et al., 1999] showed that near-wall structures called longitudinal streaks, which were observed the over smooth wall, were also seen over
roughness elements. Jimenez [2004] specified that walls with grooves bigger than
3 or 4 times of the roughness height behaved like a k-type roughness. Extra
form drag was generated and the viscous cycle was entirely destroyed. Meile
[2007] reported that no streaky structures in the fully rough flow were present
in case of the k-type roughness. Leonardi et al. [2004] performed the DNS of
a fully turbulent channel flow with square bars with the height of 0.2 × h on
the bottom wall. Turbulent Reynolds number Reτ ranged from 180 (smooth
case) to 460 (ϑ = 7.0). When the ratio ϑ was so small below 3, turbulence
structures and intensities were like those in the smooth channel case. When
the flow was driven by the imposed pressure gradient or constant mass flow
rate, Leonardi et al. [2003] and Leonardi et al. [2004] observed numerically that
the mean streamwise velocity in the d-type and k-type roughness cases was decreased compared to the smooth case. The maximum decrease was reported for
the bottom rough case with ϑ = 7.0 which showed the maximum form drag.
Using the LES, Cui et al. [2003] ascertained that for the k-type roughness, the
separation and reattachment took place between two consecutive bottom roughness elements and eddies were ejected into the outer flow.
Maechler [2016] highlighted experimentally relationships between the discharge,
suspended sediment concentration and aspect ratio of cavities in a channel
with lateral macro-roughness elements. The experimental channel represented
a channelized river where artificial banks were adapted. Clear water flow was
noticeably affected by the lateral embayments. It had been shown that the creation of the dead zones in the lateral macro-rough flows induced the deposition
of suspended sediment particles. Deposition areas of sediment particles matched
with the flow pattern analysis. Configuration with high discharge showed lowest
decay of the sediment concentration due to enhanced flow turbulence and strong
oscillations. Finally, preliminary tests on configurations showed that the flow
was perturbed up to 3m from entrance and hence the deposition of sediments
was affected. Moreover, the regulating downstream gate modified values shown
by the turbidimeter. So, the probe 2 was shifted 1m to the upstream part.
Juez et al. [2017] analyzed influences of lateral macro-roughness elements on the
transport of fine sediments in open channel flows. Banks were equipped with
large-scale rectangular roughness elements. Influences of the flow shallowness
and geometric ratios of cavities were discussed. Sheltered areas within cavities
were identified by streamlines. They were areas where particles settled. The
longer sediment particles remained inside the cavity, where the turbulence was
weaker than in the main channel, the higher probability of the settling. For
7.2 EXPERIMENTAL SETUP
56
highest discharge, small-scale fluctuations were homogenized, the vertical mixing promoted and sediment particles were hindered from the settling.
As could be seen, works relating to hydrodynamics and morphodynamics of
lateral embayments in open channel flows are rare and complicated to perform. Sedimentation processes depend on the geometrical configuration of river
banks. Also, effects of bottom macro-rough boundaries on the propagation of a
suspended sediment wave in turbulent open channel flows should be elucidated.
Turbulence and the decay of the suspended sediment concentration are related
to spacing between bottom macro-rough elements.
7.2
Experimental setup
The hydraulic system consists of tanks in the upstream and downstream parts
and rectangular flume. Dimensions are shown in table (7.1). Lateral macroroughness configurations are built using the limestone bricks. Their dimensions
are demonstrated in figure (7.1). Square wooden bars with the width of 1.75cm,
height of 1.75cm and length of 60cm are bottom macro-roughness elements.
Figure (7.2) shows recirculating water pipes, two manually regulated pumps
and two magnetic flow meter devices.
Table 7.1 – Dimensions of the flume and tanks in experiments
Upstream tank
Downstream tank
Flume
Length(x)
2m
3.5m
7.5m
Height(y)
1m
1m
0.47m
Width(z)
1m
1m
1m
Figure 7.1 – Dimensions of limestone bricks
Figure (7.2b) depicts locations of outlets in the upstream tank. It engenders
the enhancement of the mixing by inducing an upward flow.
7.2 EXPERIMENTAL SETUP
57
(a)
(b)
(c)
(d)
Figure 7.2 – Details of setup: downstream (a) and upstream (b) sections of the
recirculating pumping system, manually regulated pump (c) and magnetic flow
meter (d)
(a)
(b)
Figure 7.3 – Details of setup: regulating mechanism in the downstream section
(a) and tranquilizing structure in upstream section (b)
7.2 EXPERIMENTAL SETUP
58
The maximum allowable mass flow rate in this hydraulic system is 18 liter per
second to eradicate the cavitation. There is a pump inside the downstream tank
making effectively the mixing of water and sediment.
To set the water height in the flume, a regulating mechanism consisting of the
four rotating blades, which restricts the flow in the downstream part, is manufactured. Also, to direct appropriately the flow into the channel, a tranquilizing
structure is mounted at the entrance. Both mechanism and structure are illustrated in figure (7.3).
The experimental flume in absence of any macro-roughness elements is exhibited in figure (7.3).
A slope of 0.1% is used for the experimental flume. Side and bottom walls are
made of glass and Bakelite, respectively. There is a porous plate in figure (7.3a)
in the downstream tank for collecting dirt from the hydraulic system.
To obtain the instantaneous sediment concentration, two turbidimeters COSMOSR 25 are embedded vertically in the flow. The probe of turbidimeters (T1 and T2 )
is linked to a transmitter b-line multi amplifier by wires (Zullig AG, Switzerland).
To connect the probe to a computer, there is an acquisition card NI-USB-6259
M series (NI Instruments, USA) between transmitter and computer. All of these
components are shown in figure (7.4).
Turbidimeter demonstrated in figure (7.4b) enables the measurement of a wide
range of suspended solids from 0.001 to 400 gram per liter. The sensor in this
device utilizes infrared beams transmitted and received to and from medium.
Surface for the measurement of the concentration is planar and the sensor surface is flat and polished. This device could be used in open channel flows.
(a)
(b)
(c)
Figure 7.4 – Details of setup: transmitter b-line multi-amplifier (a), turbidimeter
probe (b) and acquisition card NI-USB-6259 M series (c)
The diameter of the turbidimeter is equivalent to 0.04m. Minimum distance of
the bottom surface of the turbidimeter off the bottom wall must be 0.01m in
order to eradicate erroneous reflections from the bed and high temporal concentration.
The card in figure (7.4c) is connected to a computer with the LABVIEW software installed on to visualize the temporal sediment concentration.
7.3 CONFIGURATIONS AND PROCEDURE
7.3
59
Configurations and procedure
Figure (7.5) demonstrates top-view sketches of the reference case (without any
macro-roughness) and lateral macro-roughness banks under investigation in this
study.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 7.5 – Reference case (a) and lateral macro-roughness banks L1 (b), L2
(c), L3 (d), L4 (e) and L5 (f) - • and symbols representing positions of
turbidimeters T1 and T2 and water height measuring device, respectively
Bottom macro-roughness configurations together with the reference case are
shown in figure (7.6).
7.3 CONFIGURATIONS AND PROCEDURE
60
(a)
(b)
(c)
(d)
(e)
(f)
Figure 7.6 – Reference case (a) and bottom macro-roughness configurations B1
(b), B1.5 (c), B2 (d), B3 (e) and B5 (f) - • and symbols representing positions
of turbidimeters T1 and T2 and water height measuring device, respectively
The direction of the flow is in the streamwise direction from left to right. To
build the reference case, two rows of the limestone bricks are located longitudinally on each side of the experimental flume. The width of the channel is 0.6m.
The limestone bricks are aligned longitudinally in lateral roughness banks L1
and L5. They are positioned spanwisely in lateral roughness banks L2, L3 and
L4. T1 is the upstream turbidimeter and T2 is the downstream one. Based on
previous study [Maechler, 2016] of the SST in lateral macro-rough flows, it is
supposed that up to 3m from the entrance, the flow could be perturbed and
the deposition of sediment particles could be modified remarkably. Likewise,
7.3 CONFIGURATIONS AND PROCEDURE
61
the sediment concentration shown by the downstream turbidity could be biased up to 1m from the downstream regulating gate. For the reference case,
the downstream probe T2 is located 2.85m upstream of the exit gate and the
upstream probe T1 is positioned 2.87m downstream of the entrance. It circumvents totally perturbations from tranquilizing entrance structure and regulating
exit gate. For the reference case, the deposition certainly takes place since
there are many dead and trapping zones between bricks and walls and under
regulating gates as well. Regarding lateral macro-rough cases, this necessary
circumstance is not satisfied in some configurations in downstream or upstream
parts. Maechler [2016] recorded that with the flow rate of 15 liter per second,
free surface oscillations inside lateral cavities were detected. Also, there was a
relationship between the amplitude of oscillating waves and sedimentation pattern and aspect ratio of lateral cavities. Here, to capture these oscillations of
the free surface for lateral macro-rough flows, a camera is located inside the
second cavity flow from the left side.
Figure (7.7a) shows the d-type bottom roughness in which ϑ = 1.0. Figure
(7.7b) manifests the k-type bottom roughness in which ϑ > 1.0.
(a)
(b)
Figure 7.7 – Bottom roughness: d-type (ϑ = 1.0) (a) and k-type (ϑ = 5.0) (b)
The distance between each turbidimeter, T1 or T2 , and the last bottom roughness element in both sides is equal to 0.15m. The number of bottom roughness
elements and the length of the bottom rough area are shown in table (7.2). The
water height measuring device shown by the square symbol is over the middle
bar. Figure (7.8) manifests moment when both pumps are switched off and
there is no water flowing in the open channel. Total volume of water inside the
hydraulics system is equivalent to 3045 liter. In this study, because the concentration of suspended particles is generally lower than 3 gram per liter, the
fluid is Newtonian. The imposed volumetric flow rate in all configurations is 15
liter per second under the steady state condition. It was mentioned by Maechler
[2016] that the mean velocity took place at 40% of the flow depth from the bed.
Accordingly, vertical positions of both turbidimeters, T1 and T2 , are set 0.028m
off the bed. In this experimental work, we select the frequency acquisition of
100Hz consequently 100 time steps are set in one second. The concentration of
7.3 CONFIGURATIONS AND PROCEDURE
62
the clear water in all configurations is less than 0.07 gram per liter at the beginning of the recording. For all configurations, 1.6kg of the Polyurethane particles
is added to the clear water in the downstream tank exactly 5 minutes after the
initiation of the recording on the LABVIEW. The total time of all experiments is
2 hours. The instantaneous concentration reduces more as the process proceeds
more due to depositing sediments in the recirculating hydraulics system.
Table 7.2 – Number of bottom roughness elements and the length of the rough
area
Number of elements
Length of rough area
B1
44
1.54m
1.5
35
1.53m
B2
29
1.5m
B3
22
1.48m
B5
15
1.5m
Figure 7.8 – Schematic of the hydraulics system
Table 7.3 – Relevant flow variables of the SLF in the reference case
Flow depth h(m)
Aspect ratio AR
Hydraulic radius Rh (m)
0.07
8.5
0.0313
Channel width (m)
0.6
Volum. concentration C
m3
)
Discharge Q(
s
m
Bulk velocity Ũ ( )
s
Reynolds number Re
0.00045
Froude number F r
Friction factor f
0.4268
0.0192
0.015
8.5
43773
Density ratio s
Particle diameter d(m)
m
Settling velocity ws ( )
s
m
Friction velocity u∗ ( )
s
Particle Reynolds number Rep
1.16
0.00008
0.0005
Equivalent roughness ks (m)
kg
Mixture density ρm ( 3 )
m2
m
Mixture viscosity νm (
)
s
Temperature T (. C)
Total time t(s)
0.000089
0.0175
1.39
1000.072
0.000001
20
7200
7.4 EXPERIMENTAL FINDINGS
63
Table (7.3) displays pertinent flow variables for the sediment laden flow in the
reference case. Each pump has a discharge of 7.5 liter per second. It is assumed that in the current flume with a aspect ratio of 8.5, there is no effect of
the sidewalls on the velocity and sediment concentration. The settling velocity
is obtained from equation (2.1) based on the particle diameter. The particle
Reynolds number is less than 5.0, so the bed is hydraulically smooth according
to experiments of Cellino [1998]. The equivalent roughness height is obtained
from the well-known Colebrook and White equation. Based on the volumetric concentration, there is insignificant change in the kinematic viscosity and
density. The friction velocity is approximated utilizing the energy-gradient procedure.
7.4
Experimental findings
In this study, the dependence of the deposition of Polyurethane particles on perturbations of the flow are confirmed by the video-recording. Some sedimentation
patterns could be found in Maechler [2016]. The temporal auto-correlation b(θ)
and cross-correlation q(θ) of sediment concentration signals C1 and C2 are calculated as,
C ′ (t)C ′ (t + θ)
(7.3)
b(θ) =
C ′ 2 (t)
C ′ (t)C2′ (t + θ)
q
q(θ) = q 1
2
′
C1 (t) C2′ 2 (t)
(7.4)
θ is the time interval in second. The cross correlation provides a measure of how
closely two signals of the sediment concentration measured with turbidimeters
T1 and T2 , are related. It is constructed as a scaled inner product of two
functions integrated over a temporal domain of interest. In all reference and
macro-rough flow cases, the imposed volumetric flow rate is identical and analysis is based on the steady flow condition. The flow is fully rough in the presence
of bottom macro-rough elements.
7.4.1
Lateral macro-rough flow
Figure (7.9) shows a lateral cavity flow with pertinent variables. According to
Juez et al. [2017], the aspect ratio ϑcf of the cavity flow, roughness aspect ratio
RR and cavity density CD are,
ϑcf =
Wcf
Wcf
lcf
, RR =
, CD =
lcf
Lcf
lcf + Lcf
(7.5)
From a flow pattern analysis point of view [Weitbrecht et al., 2008] [Akutina,
2015], when ϑcf is between 0.15 and 0.5, one large oval eddy is seen in cavity
flows recirculating in the clockwise direction, while other one rotating in the
counter-clockwise direction. Cavities with a large aspect ratio 0.8 demonstrate
7.4 EXPERIMENTAL FINDINGS
64
a single circular eddy [Weitbrecht et al., 2008] [Akutina, 2015] rotating in the
clockwise direction. It occupies nearly 80 percent of the length of the cavity
flow and there is smallest length for eddies to develop.
Figure 7.9 – Lateral cavity flow with pertinent variables
Because of the clockwise circulation of sediment particles, they mostly trap in a
circular pattern in the middle of cavity flows. These were experimentally established in the literature using the particle tracking velocimetry (PTV) [Akutina,
2015] and particle image velocimetry (PIV) [Weitbrecht et al., 2008]. According
to Meile [2007], for configurations L1 and L5 there is a low discharge. Ratio between the water depth 0.07m and the width of the narrow section of the channel
0.59m is lower than 0.12. For configurations L2, L3 and L4 there is a medium
discharge. Ratio is between 0.12 and 0.3. With low discharge, the water surface
is almost flat but tiny ripples may be developed at the free surface. On the other
hand, with medium discharge, more waviness of the free surface is present and
the cross-section area of the flow may be still homogeneous. Plus, longitudinal
and transversal waves are present in the flume. Generally speaking, for lateral
macro-rough cases, it is hard to depict the role of waves in filing and emptying
of cavity flows.
Table (7.4) summarizes all germane variables and patterns for lateral macrorough flows and reference case.
Table 7.4 – germane variables and patterns for lateral macro-rough flows
Configuration
Reference
L1
L2
L3
L4
L5
ϑcf
–
0.2
0.5
0.5
0.5
0.8
RR
–
0.39
0.5
0.62
1.25
0.39
CD
–
0.66
0.5
0.55
0.71
0.33
Vortexes
None
One big oval-One small
One big oval-One small
One big oval-One small
One big oval-One small
Single circular eddy
Discharge
Low
Low
Medium
Medium
Medium
Low
Figure (7.10) presents the temporal evolution of the normalized sediment con-
7.4 EXPERIMENTAL FINDINGS
65
centration for the reference case and lateral macro-rough configurations. C0 is
the maximum of the sediment concentration for each configuration.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 7.10 – Instantaneous normalized concentration for the reference (a) and
lateral macro-rough configurations L1 (b), L2 (c), L3 (d), L4 (e) and L5 (f)
7.4 EXPERIMENTAL FINDINGS
66
Figure (7.11) illustrates the temporal evolution of the auto-correlations b(θ) for
the reference case and lateral macro-rough configurations.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 7.11 – Instantaneous auto-correlation for the reference (a) and lateral
macro-rough configurations L1 (b), L2 (c), L3 (d), L4 (e) and L5 (f)
7.4 EXPERIMENTAL FINDINGS
67
The decay of the time-dependent sediment concentration would not be identical from case to case. Regarding the upstream concentration signal C1 , the
corresponding decrease of the normalized sediment concentration over the two
hour experiment for reference, L1, L2, L3, L4 and L5 configurations is approximately 0.2, 0.4, 0.45, 0.35, 0.3 and 0.5, respectively. Least decay of the
normalized sediment concentration in the two hour experiment is seen for the
reference case in absence of cavity flows. Sediment trapping and turbulence for
lateral macro-rough flows depends on the lengths of inlet and outlet reach, flow
discharge, aspect ratio ϑcf of the cavity flow, roughness aspect ratio RR and
cavity density CD.
Juez et al. [2017] mentioned that lower sedimentation was for configurations
with higher roughness RR and cavity density CD. Here, it is also seen that the
least sedimentation or decrease of normalized sediment concentration among
lateral configurations is for case L4 with highest RR = 1.25, CD = 0.71 and
medium flow discharge.
Figure (7.10e) manifests that for case L4 there is almost no difference between
instantaneous evolutions of normalized concentration signals C1 and C2 . For
case L4, the cavity aspect ratio is 0.5 and eddies, which are induced inside
the cavity flow, are stretched into the main flow. Counter-rotating behavior of
vortexes within the cavity flow generates mixing and removal of Polyurethane
particles into the main flow. Length of the inlet reach for case L4 is 1.78m which
is lowest amid other cases. It boosts the mixing of Polyurethane particles and
turbulence in open channel flows. In contrast to other cases, the integral scale
of concentration signals for case L4 is lower. It is understood from autocorrelations in figure (7.11e).
On the other hand, highest deposition of the Polyurethane particles is detected
for case L5 with lowest RR = 0.39 and CD = 0.33. For case L5, the discharge
is low and the Polyurethane particles accumulate in a circular pattern in the
middle of the cavity flow. Comparing to the reference, L1, L3 and L4 cases, the
turbulence is reduced for case L5. It could be observed from autocorrelations in
figure (7.11f). Deposition of the Polyurethane particles for case L2 is also high.
Figure (7.11c) demonstrates that the turbulence is most attenuated for case
L2 with medium discharge in comparison with the reference, L1, L3, L4 and
L5 cases. It is due to adequate lengths of the inlet and outlet reach. Consequently, the widening of the channel in case L2 is highly effective to deposit the
Polyurethane particles in open channel flow.
The cross-correlations of two sediment concentration signals for the reference
and lateral macro-rough configurations are depicted in figure (7.12). Crosscorrelation demonstrates similarity between two turbulent signals of the sediment concentration. As would be detected, the cross-correlation of two turbulent
signals of the sediment concentration for case L2 is high. It means that inserting configuration L2 induces similar signals since the widening of the channel in
this configuration is highly effective in reducing the turbulence. Furthermore,
cross-correlations for lateral cases L1, L3 and L4 are same but lower than that
for the reference case. Cross-correlations for lateral cases L2, L5 are bigger than
that for the reference case.
7.4 EXPERIMENTAL FINDINGS
68
Figure 7.12 – Cross-correlations of two sediment concentration signals for the
reference and lateral macro-rough configurations
7.4.2
Bottom macro-rough flow
Influences of spacing 1 ≤ ϑ ≤ 5 between bottom macro-rough elements on
the instantaneous normalized sediment concentration and auto-correlation and
cross-correlation of sediment concentration signals are discussed.
The instantaneous evolution of the normalized sediment concentration for the
reference and bottom macro-rough configurations is presented in figure (7.13).
C0 is the maximum of the concentration of suspended Polyurethane particles
detected in each configuration.
Figure (7.14) illustrates temporal auto-correlations of sediment concentration
signals for the reference case and bottom macro-rough flows.
Regarding the downstream concentration signal C2 , corresponding reduction in
the normalized sediment concentration over the two hour experiment for reference, B1, B1.5, B2, B3 and B5 configurations is approximately 0.2, 0.3, 0.3,
0.55, 0.3 and 0.3, respectively. For bottom macro-rough flows, the Polyurethane
particles are trapped between bottom elements. As can be understood, bottom
elements cause highest deposition and trapping of suspended Polyurethane particles in a specific spacing ϑ = 2 (B2). Decrease in the normalized concentration signal C2 is identical for all other bottom macro-rough flows. It is seen
that difference between concentration signals C1 and C2 , for cases ϑ = 1.0 (B1)
and ϑ = 1.5 (B1.5) is lesser than that for the reference case. It can be said
from auto-correlations that bottom elements lessen slightly the integral scale of
turbulent signals of the concentration in cases B1 and B1.5 compared to the
reference case. So, building cases B1 and B1.5 enhances a little mixing and
turbulence in contrast to the reference case. According to autocorrelations, turbulence characteristics of C1 and C2 signals are similar for the reference, B1 and
B1.5 cases. For spacing ϑ = 2 (B2), the turbulence is most attenuated since
7.4 EXPERIMENTAL FINDINGS
69
the integral scale of the downstream concentration signal C2 in figure (7.14d) is
biggest among other cases.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 7.13 – Instantaneous normalized sediment concentration for the reference
(a) and bottom macro-roughness configurations B1 (b), B1.5 (c), B2 (d), B3
(e) and B5 (f)
7.4 EXPERIMENTAL FINDINGS
70
It engenders the highest difference between concentration signals C1 and C2 as
process advances more. The trapping becomes less when the spacing increases
from ϑ = 2.0 to ϑ = 5.0.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 7.14 – Instantaneous auto-correlation for the reference (a) and bottom
macro-roughness configurations B1 (b), B1.5 (c), B2 (d), B3 (e) and B5 (f)
7.4 EXPERIMENTAL FINDINGS
71
This is in agreement with the increase of the turbulence or reduction of the
integral scale of the downstream concentration signal C2 . It is seen from figures
(7.14e) and (7.14f). As spacing is augmented from ϑ = 2 (B2) to ϑ = 5 (B5),
auto-correlations computed from C1 signal does not change anymore. Turbulence characteristics of C1 and C2 signals are similar for ratio ϑ = 5 (B5).
It could be stated that bottom macro-rough elements change significantly flow
patterns and turbulence characteristics in a specific spacing ϑ. After that, as
spacing gets larger, influences of bottom macro-roughness elements on each
other in terms of flow patterns between elements diminish more. They start to
be isolated elements.
In this study, the maximum effect of bottom macro-rough elements on concentration signals is observed for spacing ϑ = 2 (B2) instead of spacing ϑ = 7
which reported earlier by Leonardi et al. [2003]. In examinations of Leonardi
et al. [2003] and Leonardi et al. [2004], from a turbulence analysis viewpoint,
there was no difference between ratios ϑ = 0.3, 0.6, 1, 2.07. This discrepancy
stems from fact that in their study, the turbulent Reynolds number Reτ for
the reference case was equal to 180 while in the present study, it is equivalent
to 540. Therefore, the more turbulent Reynolds number increases in bottom
macro-rough flows, the more sensitivity of turbulence structures to ϑ enhances.
Cross-correlations of two concentration signals for the reference and bottom
macro-rough configurations are shown in figure (7.15). Cross-correlations for
cases B1 and B1.5 are same but lower than that for the reference case. The
maximum of the cross-correlation is seen for B2. Thus, highest attenuation
of the turbulence induces most similar turbulent concentration signals. As we
increase spacing from ϑ = 2, the cross-correlation reduces.
Figure 7.15 – Cross-correlations of two concentration signals for bottom macrorough configurations and reference case
Chapter 8
Conclusions
SST in uniform turbulent open channel flows includes many complexities that
complicate understanding of experimental data. Most of formulations developed
for analyzing SST are based on the mixture perspective.
In this study, first of all, the single-phase Euler-Euler based wall-resolving LES
with the dynamic Smagorinsky model in the LES-COAST code was implemented to understand interactions between suspended sand particles and turbulence in a turbulent open channel flow with high aspect ratio and bottom
smooth wall. Fast-Eulerian method in the two-way coupling framework was
used to explain effects of the sediment concentration on hydrodynamics of the
mixture. Streamwise and vertical turbulence intensities for the clear water flow
in experiment of Muste et al. [2005] were much bigger than those in the present
wall-resolving LES and DNS by Hoyas and Jimenez [2008]. The bulk flow velocity in the sediment-laden flow was lesser than that in the clear water flow which
illustrated that the suspension of sand particles resulted in the extraction of the
energy from the flow. Moreover, the shear stress on the channel bed was same
for both clear water and sediment-laden flows. Muste et al. [2005] from experimental point of view reported that the depth-averaged streamwise velocity of
the mixture in inner region was higher than that of the clear water flow. There
was strong particle flow turbulence and particle velocities were not affected by
the no-slip condition. On other hand, in this study in the inner region it was
seen that the depth-resolved streamwise velocity was not affected when sand
particles were introduced into the clear water flow. In the outer region, the
depth-resolved streamwise velocity was reduced.
Second of all, the single-phase Euler-Euler based unresolved wall-function LES
with the Smagorinsky model under the equilibrium stress assumption in the
LES-COAST code was employed to investigate the SST in a turbulent open
channel flow with a low aspect ratio and a bottom rough wall. The reference
concentration method proposed by Smith and Mclean [1977] together with the
Shields diagram was used to treat the erosion of sediment particles from the
channel bed. Wall shear stress on sidewalls must be considered in the com-
72
73
putation of the mixture flow in narrow open channels. When the modified
gravity term, which was proposed by Dallali and Armenio [2015] in the NavierStokes equations of the mixture flow, was deactivated, the sediment concentration became large and unsatisfactory higher-order turbulence statistics for the
sediment-laden flow were recorded. Hence, the buoyancy induced by suspended
sand particles influenced remarkably hydrodynamics of the flow. Suspending
sediment particles caused the reduction of the friction velocity, bulk velocity
and roughness in contrast to the clear water flow. Streamwise velocity was decreased in the outer layer while it was enlarged in the super-saturated region
near the channel bed. There were high inter-particle collisions between sediment particles which were not bounded by viscosity. Vertical velocities for the
sediment-laden flow in lower levels were significantly higher than vertical velocities for the clear water flow. Close to free surface, vertical velocities for clear
water and sediment-laden flows were equal. Streamwise turbulence intensity for
the sediment-laden flow was bigger than that for the clear water flow except
close to free surface where they were identical to experimental data of Cellino
[1998]. Compared to the clear water flow, suspended sand particles generated
slight enhancement of the vertical turbulence intensity near the channel bed
while in upper levels, significant weakening of the vertical turbulence intensity
occurred. Adjacent to the free surface, vertical turbulence intensities for clear
water and sediment-laden flows were same since the sediment concentration was
low.
It was concluded that systematic measurements and computations of the SST
in turbulent open channel flows should be performed simultaneously. Experimental data could be used for the validation and development of the mixture
flow model.
Third of all, impacts of lateral and bottom macro-rough boundaries on the propagation of a suspended sediment wave in a turbulent open channel flow were
investigated experimentally under the steady state condition. In the reference
case, there was no trapping and deposition zones, so there was the least decay of
the normalized concentration of suspended Polyurethane particles. Least sedimentation amid lateral configurations was for case L4 with the highest roughness
aspect ratio, cavity density and medium flow discharge. Cavity aspect ratio was
0.5 and counter-rotating vortexes inside the cavity flow removed Polyurethane
particles into the main flow. Lowest length of the inlet reach was for case L4
which elevated mixing and turbulence. Autocorrelation showed that the integral scale of concentration signals for case L4 was lower than those for other
cases. Highest deposition of the Polyurethane particles was for L5 case with the
lowest roughness aspect ratio, cavity density and flow discharge. Polyurethane
particles accumulated in a circular pattern in the middle of the cavity flow.
Turbulence was most weakened for case L2 with the medium discharge due to
adequate lengths of the inlet and outlet reach. Deposition of the Polyurethane
particles was also high for case L2. From crosscorrelations it was found out that
building case L2 engendered similar concentration signals since the turbulence
was effectively abated. Consequently, the sediment trapping and turbulence in
lateral macro-rough flows were related to the lengths of inlet and outlet reach,
74
flow discharge, aspect ratio of the cavity flow, roughness aspect ratio and cavity density. In the bottom macro-rough flows, the Polyurethane particles were
trapped between bottom elements. According to autocorrelations, turbulence
characteristics of C1 and C2 signals were identical for reference, B1 and B1.5
cases. When the spacing between bottom elements was ϑ = 2 (B2), highest
trapping of suspended Polyurethane particles occurred. Most weakening of turbulence and highest difference between concentration signals C1 and C2 was
seen for spacing ϑ = 2 (B2). As spacing was enhanced from ϑ = 2 to ϑ = 5, the
integral scale of the downstream concentration signal C2 was reduced, turbulence was enhanced and trapping became lesser. Crosscorrelations for cases B1
and B1.5 were identical but lesser than that for the reference case. Crosscorrelations established that most similar turbulent concentration signals took place
for case B2 since the turbulence was highly attenuated. Bottom macro-rough
elements altered significantly flow pattern and turbulence characteristics in a
specific spacing ϑ. After that, as spacing got larger, the influence of bottom
macro-roughness elements on each other abated more and they were isolated
elements. The more turbulent Reynolds number was enhanced in flumes with
bottom macro-rough elements, the more sensitivity of turbulence structures to ϑ
was increased. It has prime importance for hydraulics engineers who artificially
modify natural paths of rivers to do river restoration, sediment management
and to control propagation of contaminants.
References
Y. Akutina. Experimental investigation of flow structures in a shallow embayment using 3D-PTV. PhD thesis, McGill University, 2015.
V. Armenio and U. Piomelli. Lagrangian mixed subgrid-scale model in generalized coordinates. Flow, Turbul. Combust., 65:51–81, 2000.
V. Armenio and S. Sarkar. An investigation of stably stratified turbulent channel
flow using large-eddy simulation. J. Fluid Mech., 459:1–42, 2002.
A. Ashrafian and H.I. Andersson. The structure of turbulence in a rodroughened channel. Int. J. Heat Fluid Flow, 27:65–79, 2006.
A. Ashrafian, H.I. Andersson, and M. Manhart. Dns of turbulent flow in a rod
roughened channel. Int. J. Heat Fluid Flow, 25:373–383, 2004.
J. Bai, H. Fang, and T. Stoesser. Transport and deposition of fine sediment in
open channels with different aspect ratios. Earth Surf. Process. Landforms,
38:591–600, 2013.
J.C. Bathurst. Flow resistance estimation in mountain rivers. J. Hydraul. Engrg., 111:625–643, 1985.
S. Bisceglia, R.J. Smalley, R.A. Antonia, and L. Djenidi. Rough-wall turbulent boundary layers at relatively high reynolds number. In B. Dally, editor,
Proceedings of the 14th Australasian Fluid Mechanics Conference. Adelaide
University, 2001.
M. Breuer and M. Alletto. Efficient simulation of particle-laden turbulent flows
with high mass loadings using LES. Int. J. Heat Fluid Flow, 35:2–12, 2012.
G.L. Brown and A. Roshko. On density effects and large structure in turbulent
mixing layers. J. Fluid Mech., 64:775–816, 1974.
Z. Cao, S. Egashira, and P.A. Carling. Role of suspended-sediment particle size
in modifying velocity profiles in open channel flows. Water Resour. Res., 39:
1–15, 2003.
M. Cellino. Experimental study of suspension flow in open channels. PhD thesis,
École Polytechnique Fédérale de Lausanne (EPFL), 1998.
75
76
D.R. Chapman. Computational aerodynamics development and outlook. AIAA
J., 17 :1293–1313, 1979.
J. Chauchat and S. Guillou. On turbulence closures for twophase sediment-laden
flow models. J. Geophys. Res., 113 :C11017, 2008.
N.S. Cheng. Effect of concentration on settling velocity of sediment particles.
J. Hydraul. Engrg., 123 :728–731, 1997.
Z. Cheng and T.J. Hsu. A turbulence-resolving eulerian two-phase model for
sediment transport. Coast. Eng. Proc., 1 :74, 2014.
Z. Cheng, X. Yu, T.J. Hsu, and S. Balachandar. A numerical investigation of
fine sediment resuspension in the wave boundary layeruncertainties in particle
inertia and hindered settling. Comput. Geosci., 83 :176–192, 2015.
F. Chiodi, P. Claudin, and B. Andreotti. A two-phase flow model of sediment
transport : transition from bedload to suspended load. J. Fluid Mech., 755 :
561–581, 2014.
Y.J. Chou and O.B. Fringer. Modeling dilute sediment suspension using largeeddy simulation with a dynamic mixed model. Phys. Fluids, 20 :115103, 2008.
N.L. Coleman. Velocity profiles with suspended sediment. J. Hydraul. Res., 19 :
211–229, 1981.
J. Cui, V.C. Patel, and C.L. Lin. Large-eddy simulation of turbulent flow in a
channel with rib roughness. Int. J. Heat Fluid Flow, 24 :372–388, 2003.
M. Dallali and V. Armenio. Large eddy simulation of two-way coupling sediment
transport. Adv. Water Resour., 81 :33–44, 2015.
J.W. Deardorff. A numerical study of three-dimensional turbulent channel flow
at large reynolds numbers. J. Fluid Mech., 41 :453–480, 1970.
S. Dey. Fluvial Hydrodynamics : Hydrodynamic and Sediment Transport Phenomena. Springer Berlin Heidelberg, 2014.
W.E. Dietrich. Settling velocity of natural particles. Water Resour. Res., 18 :
1615–1626, 1982.
L. Djenidi, F. Anselmet, and R.A. Antonia. Lda measurements in a turbulent
boundary layer over a d-type rough wall. Exp. Fluids, 16 :323329, 1994.
L. Djenidi, R. Elavarasan, and R.A. Antonia. The turbulent boundary layer
over transverse square cavities. J. Fluid Mech., 395 :271–294, 1999.
O.A. Druzhinin. On the two-way interaction in two-dimensional particle-laden
flows : the accumulation of particles and flow modification. J. Fluid Mech.,
297 :49–76, 1995.
77
S. Elghobashi. On predicting particle-laden turbulent flows. Appl. Sci. Res.,
52 :309–329, 1994.
A. Fakhari. Wall-Layer Modelling of massive separation in large eddy simulation
of coastal flows. PhD thesis, Universit degli studi di Trieste, 2015.
J. Ferry and E. Balachandar. A fast eulerian method for disperse two-phase
flow. Int. J. Multiphase Flow, 27 :1199–1226, 2001.
N. Foroozani. Numerical Study of Turbulent Rayleigh-Bnard Convection with
Cubic Confinement. PhD thesis, Universit degli studi di Trieste, 2015.
N. Foroozani, J.J. Niemela, V. Armenio, and K.R. Sreenivasan. Reorientations
of the large-scale flow in turbulent convection in a cube. Phys. Rev. E, 95 :
033107, 2017.
A. Galea, M. Grifoll, F. Roman, M. Mestres, V. Armenio, A. Sanchez-Arcilla,
and L. Zammit Mangion. Numerical simulation of water mixing and renewals
in the barcelona harbour area : the winter season. Environ. Fluid Mech., 14 :
1405–1425, 2014.
M. Germano, U. Piomelli, P. Moin, and W.H. Cabot. A dynamic subgrid-scale
eddy viscosity model. Phys. Fluids A, 3 :1760–1765, 1991.
S. Ghosal and P. Moin. The basic equations for the large eddy simulation of
turbulent flows in complex geometry. J. Comput. Phys., 118 :24–37, 1995.
R.W. Gilbert, E.A. Zedler, S.T. Grilli, and R.L. Street. Progress on nonlinearwaveforced sediment transport simulation. IEEE J. Oceanic Engrg., 32 :
236–248, 2007.
W.H. Graf and M. Cellino. Suspension flows in open channels : experimental
study. J. Hydraul. Res., 40 :435–447, 2002.
J. Guo and P.Y. Julien. Turbulent velocity profiles in sediment-laden flows. J.
Hydraul. Res., 39 :11–23, 2001.
J.C. Harris and S.T. Grilli. Large eddy simulation of sediment transport over
rippled beds. Nonlin. Processes Geophys. Discuss., 1 :755–801, 2014.
K. Horiuti. Assessment of two-equation models of turbulent passive-scalar diffusion in channel flow. J. Fluid Mech., 238 :405–433, 1992.
S. Hoyas and J. Jimenez. Reynolds number effects on the reynolds-stress budgets
in turbulent channels. Phys. Fluid, 20 :1–8, 2008.
J. Jimenez. Turbulent flows over rough walls. Annu. Rev. Fluid Mech., 36 :
173196, 2004.
C. Juez, I. Buhlmann, G. Maechler, A.J. Schleiss, and M.J. Franca. Transport
of suspended sediments under the influence of bank macro–roughness. Earth
Surface Processes Landforms, 2017. doi : 10.1002/esp.4243.
78
K. Kawanisi and R. Shiozaki. Turbulent effects on the settling velocity of suspended sediment. J. Hydraul. Engrg., 134 :261–266, 2008.
J. Kim, P. Moin, and R. Moser. Turbulence statistics in fully developed channel
flow at low reynolds number. J. Fluid Mech., 177 :133–166, 1987.
B. Kironoto. Turbulence characteristics of non-uniform flow in rough openchannel. PhD thesis, École Polytechnique Fédérale de Lausanne (EPFL),
1992.
S. Kundu and K. Ghoshal. Effects of secondary current and stratification on
suspension concentration in an open channel flow. Environ. Fluid Mech., 14 :
1357–1380, 2014.
F. Kyrousi, A. Leonardi, F. Zanello, and V. Armenio. Efficient coupling between
a turbulent flow and mobile bed by means of the level-set method and immersed boundaries. In The 26th International Ocean and Polar Engineering
Conference. International Society of Offshore and Polar Engineers, June-July
2016.
C. Le Ribault, S. Sarkar, and S.A. Stanley. Large eddy simulation of evolution
of a passive scalar in plane jet. AIAA J., 39 :1505–1516, 2001.
A. Leonard. Energy cascade in large eddy simulation of turbulent fluid flow.
Adv. Geophys., 18A :237–248, 1974.
S. Leonardi, P. Orlandi, R.J. Smalley, L. Djenidi, and R.A. Antonia. Direct
numerical simulations of turbulent channel flow with transverse square bars
on one wall. J. Fluid Mech., 491 :229–238, 2003.
S. Leonardi, P. Orlandi, L. Djenidi, and R.A. Antonia. Structure of turbulent
channel flow with square bars on one wall. Int. J. Heat Fluid Flow, 25 :
384–392, 2004.
S. Leonardi, P. Orlandi, and R.A. Antonia. Properties of d- and k-type roughness in a turbulent channel flow. Phys. Fluid., 19 :125101, 2007.
W.K. Lewis, E.R. Gilliland, and W.C. Bauer. Characteristics of fluidized particles. Ind. Engrg. Chem., 41 :1104–1114, 1949.
D.K. Lilly. Energy cascade in large eddy simulation of turbulent fluid flow.
Phys. Fluids, A4 :633–635, 1992.
B. Lin and R.A. Falconer. Numerical modeling of threedimensional suspended
sediment for estuarine and coastal waters. J. Hydraul. Res., 34 :435–456,
1997.
V.Y. Lyapin. Hydraulic analysis of channels with increased artificial bottom
roughness. Hydrotech. Constr., 28 :290–293, 1994.
79
G. Maechler. Influence of bank macro-roughness on the transport of suspended sediments. Master’s thesis, École Polytechnique Fédérale de Lausanne
(EPFL), 2016.
M.R. Maxey. The gravitational settling of aerosol particles in homogeneous
turbulence and random flow fields. J. Fluid Mech., 174 :441–465, 1987.
A.J. Mehta. An Introduction to Hydraulics of Fine Sediment Transport. World
Scientific Publishing Company, 2013.
T. Meile. Influence of macro-roughness of walls on steady and unsteady flow in
a channel. PhD thesis, École Polytechnique Fédérale de Lausanne (EPFL),
2007.
T. Meile, J.L. Boillat, and A.J. Schleiss. Flow resistance caused by large-scale
bank roughness in a channel. J. Hydraul. Engrg., 137 :1588–1597, 2011.
M. Muste and V.C. Patel. Velocity profiles for particles and liquid in openchannel flow with suspended sediment. J. Hydraul. Engrg., 123 :742–751,
1997.
M. Muste, K. Yu, I. Fujita, and R. Ettema. Two-phase versus mixed-flow perspective on suspended sediment transport in turbulent channel flows. Water
Resour. Res., 41 :1–22, 2005.
M. Muste, K. Yu, I. Fujita, and R. Ettema. Two-phase flow insights into openchannel flows with suspended particles of different densities. Environ. Fluid
Mech., 9 :161–186, 2009.
I. Nezu and R. Azuma. Turbulence characteristics and interaction between
particles and fluid in particle-laden open channel flows. J. Hydraul. Engrg.,
130 :988–1001, 2004.
I. Nezu and H. Nakagawa. Turbulence in Open-Channel Flows. IAHR Monograph Series. A.A. Balkema, Rotterdam, 1993.
I. Nezu and W. Rodi. Open-channel flow measurements with a laser doppler
anemometer. J. Hydraul. Engrg., 112 :335–355, 1986.
K. Noguchi and I. Nezu. Particle–turbulence interaction and local particle
concentration in sediment-laden open-channel flows. J. Hydro-Environ. Res.,
3 :54–68, 2009.
C.Y. Perng and R.L. Street. 3-d unsteady flow simulation : alternative strategies
for a volume-average calculation. Int. J. Numer. Methods Fluids, 9 :341–362,
1989.
A.E. Perry, W.H. Schofield, and P.N. Joubert. Rough wall turbulent boundary
layers. J. Fluid Mech., 37 :383–413, 1969.
80
A. Petronio, F. Roman, C. Nasello, and V. Armenio. Large eddy simulation
model for wind-driven sea circulation in coastal areas. Nonlin. Processes
Geophys., 20 :1095–1112, 2013.
U. Piomelli. Wall-layer models for large-eddy simulations. Prog. Aerosp. Sci.,
44 :437–446, 2008.
U. Piomelli and E. Balaras. Wall-layer models for large-eddy simulations. Annu.
Rev. Fluid Mech., 34 :349–374, 2002.
M.B. Pittaluga. Stratification effects on flow and bed topography in straight
and curved erodible streams. J. Geophy. Res., 116 :F03026, 2011.
S.B. Pope. Turbulent flows. Cambridge University Press, 2000.
M. Righetti and G.P. Romano. Particle-fluid interactions in a plane nearwall
turbulent flows. J. Fluid Mech., 505 :93–121, 2004.
F. Roman, G. Stipcich, V. Armenio, R. Inghilesi, and S. Corsini. Large eddy
simulation of mixing in coastal areas. Int. J. Heat Fluid Flow, 31 :327–341,
2010.
H. Romdhane, A. Soualmia, L. Cassan, and L. Masbernat. Evolution of flow
velocities in a rectangular channel with homogeneous bed roughness. Int. J.
Engrg. Res., 6 :120–125, 2017.
U. Schumann. Subgrid-scale model for finite difference simulation of turbulent
flows in plane channels and annuli. J. Comput. Phys., 18 :376–404, 1975.
H. Shamloo and B. Pirzadeh. Analysis of roughness density and flow submergence effects on turbulence flow characteristics in open channels using a large
eddy simulation. Appl. Math. Model., 39 :1074–1086, 2015.
J. Smagorinsky. General circulation experiments with the primitive equations.
Mon. Weather Rev., 91 :99–165, 1963.
J. Smith and S. Mclean. Spatially averaged flow over a wavy surface. J. Geophys.
Res., 82 :1735–1746, 1977.
A. Soldati and C. Marchioli. Sediment transport in steady turbulent boundary
layers : Potentials, limitations, and perspectives for lagrangian tracking in
DNS and LES. Adv. Water Resour., 48 :18–30, 2012.
M.F. Tachie, D.J. Bergstrom, and R. Balachandar. Rough wall turbulent boundary layers in shallow open channel flow. J. Fluid. Engrg., 122 :533–541,
2000.
H. Tennekes and J.L. Lumley. A First Course in Turbulence. MIT Press,
Cambridge, Mass., 1972.
81
E.A. Toorman. Validation of macroscopic modeling of particle-laden turbulent
flows. In E. Dick, J. Vierendeels, F. Cantrijn, G. Campion, and C. Lacor,
editors, Proceedings 6rd Belgian National Congress on Theoretical and Applied
Mechanics. Ghent University, May 2003.
L.C. van Rijn. Sediment transport, part I : bed load transport. J. Hydraul.
Engrg., 110 :1431–1455, 1984a.
L.C. van Rijn. Sediment transport, part II : suspended load transport. J.
Hydraul. Engrg., 11 :1613–1641, 1984b.
V.A. Vanoni. Sedimentation Engineering. ASCE Manuals and Reports on Engineering Practices, 1975.
C. Villaret and A.G. Davies. Modelling sediment-turbulent flow interactions.
Appl. Mech. Rev., ASME, 48 :601–609, 1995.
X. Wang and N. Qian. Turbulence characteristics of sediment-laden flow. J.
Hydraul. Engrg., 6 :781–800, 1989.
V. Weitbrecht, S.A. Socolofsky, and G.H. Jirka. Experiments on mass exchange
between groin fields and the main stream in rivers. J. Hydraul. Engrg., 134 :
173–183, 2008.
J.C. Winterwerp. Stratification effects by cohesive and noncohesive sediment.
J. Geophys. Res., 106 :22559–22574, 2001.
W. Wu, W. Rodi, and T. Wenka. Role of suspended-sediment particle size in
modifying velocity profiles in open channel flows. J. Hydraul. Engrg., 126 :
4–15, 2000.
J. Yan, H. Tang, Y. Xia, K. Li, and Z. Tian. Experimental study on influence
of boundary on location of maximum velocity in open channel flows. Water
Sci. Engrg., 4 :185–191, 2011.
S. Yang, S. Tan, and X. Wang. Mechanism of secondary currents in open channel
flows. J. Geophys. Res., 117 :1–13, 2012.
S.Q. Yang, S.K. Tan, and S.Y. Lim. Velocity distribution and dip phenomenon
in smooth uniform open channel flows. J. Hydraul. Engrg., 130 :1179–1186,
2004.
K. Yu, B. Yoon, and D. Kim. Re-evaluation of change of mean velocity profiles
in turbulent open-channel flows due to introducing sediment particles. KSCE
J. Civil Engrg., 18 :2261–2267, 2014.
Y. Zang, R.L. Street, and J.R. Koseff. A non-staggered grid, fractional step method for time-dependent incompressible navierstokes equations in curvilinear
coordinates. J. Comput. Phys., 114 :18–33, 1994.
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G. Zanier, A. Petronio, and V. Armenio. The effect of coriolis force on oil slick
transport and spreading at sea. J. Hydraul. Res., 55 :409–422, 2017.
E.A. Zedler and R.L. Street. Large-eddy simulation of sediment transport :
currents over ripples. J. Hydraul. Engrg., 127 :444–452, 2001.
E.A. Zedler and R.L. Street. Sediment transport over ripples in oscillatory flow.
J. Hydraul. Engrg., 132 :180–193, 2006.
H. Zhu, L.L. Wang, and H.W. Tang. Large-eddy simulation of suspended sediment transport in turbulent channel flow. J. Hydrodyn. Ser. B, 25 :48–55,
2013.
Publications
[1] M. Jourabian, V. Armenio. Large eddy simulation (LES) of suspended
sediment transport (SST) at a laboratory scale, The 26th International
Ocean and Polar Engineering Conference, Rhodes, Greece, 26 June-2 July 2016.
[2] M. Jourabian, V. Armenio. Wall-layer model for large eddy simulation (LES) of suspended sediment transport (SST) in a lab-scale
turbulent open channel flow, 13th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics (HEFAT2017), Portoroz, Slovenia,
17-19 July 2017.
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