# NUMERICAL INVESTIGATION OF TURBULENT-DRIVEN SECONDARY FLOW

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# Abstract

Secondary flows of second type (also known as turbulent secondary flows) are one of the most important mechanisms responsible for sediment transport processes in fluvial streams. A new two-equation Reynolds-Averaged Navier-Stokes (RANS) based model is investigated in depth in this work, for modelling secondary flows of second type.

This thesis incorporates a new k − ω model with nonlinear fourth-order closure terms for modelling Reynolds stresses. The model has k − ω formulation which enables the model to solve for flow equations near the walls without the need for utilizing wall functions. Moreover, k − ω models are capable of applying roughness on boundaries by imposing rough boundary turbulent attributes through ω functions. The model was tested in two main categories on five case-studies to observe its ability in simulating turbulent secondary flow in various configurations. The tests carried out case scenarios identical to experimental case studies. First, the model was used in simulating turbulent secondary current in a simple case study conducted inside a rectangular duct with smooth boundaries. The model performed well in this part, when simulated data such as secondary velocity profiles, shear stresses, and secondary current vectors being compared with experimental data.

In the second test, the model was investigated in case scenarios to explore the model’s capacity for carrying out simulation of turbulent secondary currents over rough boundaries. While in case

studies with uniform rough boundaries the model functioned well (velocity profiles, and shear stress distribution, being compared to experimental data), in cases with non-uniform roughness distribution the model needed noticeable tweaking, tuning, and calibration for roughness modelling. However, for validation purposes, the model with calibrated function parameter were tested against other non-uniform distribution case scenarios, in which their results showed excellent agreement with experimental data. The numerical simulations were able to produce secondary velocity profiles very close to experimental studies which are difficult to capture. The proposed roughness height values in roughness modelling function for secondary flows over beds, with nonuniformly distributed roughness, are critically discussed and assessed.

During the tuning of the model, it was detected that the original approach for computing wall shear stresses, using law of the wall, provided dissimilar results compared to results which calculated shear stresses directly from velocity gradients at the wall. This disagreement was investigated in depth, and it was concluded that the calculation of wall shear stress, using law of the wall, was not accurate. Finally, providing accurate result along with computational efficiency (due to RANS-based formulation), and applicability to rough case scenarios, this model is advantageous in investigation of turbulent secondary current.

List of Figures                                                                                                                                                                                 ix

List of Tables                                                                                                                                                                                   xi

Acknowledgments                                                                                                                                                                      xii

Chapter 1

Introduction                                                                                                                                                                           1

1.1                                Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                   1

1.2      Sediment Transport                              . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                    2

1.3                                 Secondary Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                        6

1.3.1       Role of secondary flows                         . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                               6

1.3.2                         Secondary flows of second type . . . . . . . . . . . . . . . . . . . . . . . . .                               7

1.4       Thesis Aim and Objectives                            . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                 8

1.4.1                            Motivation of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                  8

1.4.2                              Scope of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                    9

1.4.3                             Structure of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                   9

Chapter 2

Literature Review                                                                                                                                                             11

2.1       Origin of secondary flow of second type                       . . . . . . . . . . . . . . . . . . . . . . . .                          12

2.2                  Effect of roughness and shallowness on secondary flows . . . . . . . . . . . . . . . .                       16

2.3                   Role of CFD simulations in studies of turbulent secondary flow . . . . . . . . . . .                      18

Chapter 3

Computational Details and Numerical Models                                                                                              23

3.1                               Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                    23

3.1.1                     The Navier-stokes and The Continuity Equations . . . . . . . . . . . . . . .                        24

3.1.2                   The Reynolds-Averaged Navier-stokes (RANS) Equations . . . . . . . . . .                      25

3.1.3                       Turbulence Modelling for RANS Equations . . . . . . . . . . . . . . . . . .                           25

3.1.3.1        Reynolds stress closure modelling                  . . . . . . . . . . . . . . . . . .                     26

3.1.3.2                        two-equation modelling . . . . . . . . . . . . . . . . . . . . . . . .                           28

3.2               New Two-Equations k − ω Turbulence Closure Model by Hellsten (2004) . . . . . .                   28

3.3      Roughness Implementation                           . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                               30

3.4                              Computational Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                               31

3.4.1 OpenFOAMr CFD Package . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                31

3.4.2        Pre-processing Techniques and Utilities                    . . . . . . . . . . . . . . . . . . . .                       32

3.4.2.1                           meshing process . . . . . . . . . . . . . . . . . . . . . . . . . . . .                              33

3.4.2.2                      boundary and initial conditions . . . . . . . . . . . . . . . . . . . .                         34

3.4.2.3                           system directory . . . . . . . . . . . . . . . . . . . . . . . . . . . .                              34

3.4.3                      Post-processing Techniques and Utilities . . . . . . . . . . . . . . . . . . . .                          36

3.4.3.1                              Paraview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                 36

3.4.3.2      sample                           . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                37

3.4.3.3                               Python . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                  38

Chapter 4

Simulation of Turbulent Secondary Flows Over Bed with Uniformly Distributed

Roughness                                                                                                                                                       39

4.1                                 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                      39

4.2             Rodríguez and García (2008) Uniform Rough Bed with Smooth Walls (Case FB1)                    40

4.2.1                           Description of Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                               40

4.2.2                                CFD Case Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                41

4.2.2.1                           mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                           41

4.2.2.2                       initial and boundary conditions . . . . . . . . . . . . . . . . . . . . .                      41

4.2.3                             Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                42

4.3                                   Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                        47

Chapter 5

Simulation of Turbulent Secondary Flows Over Bed with Alternate Rough

and Smooth Strips                                                                                                                                     49

5.1                                 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                      49

5.2                     Wang and Cheng (2006) Case S75 and Case S50 Setup . . . . . . . . . . . . . . . .                         50

5.2.1        Description of Case S75 Setup                        . . . . . . . . . . . . . . . . . . . . . . . . .                          50

5.2.2        Description of Case S50 Setup                         . . . . . . . . . . . . . . . . . . . . . . . . . .                       51

5.2.3                           Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . .                              52

5.3                               Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                   53

5.3.1                           mesh independence study . . . . . . . . . . . . . . . . . . . . . . . . . . . .                              53

5.3.2                         Wang and Cheng (2006) S75 case . . . . . . . . . . . . . . . . . . . . . . . .                            54

5.3.2.1                     traditional rough and smooth values . . . . . . . . . . . . . . . . .                       54

5.3.2.2                       transitional roughness values . . . . . . . . . . . . . . . . . . . . .                          55

5.3.2.3                       ksmooth = 0.00075m and krough = 0.005m . . . . . . . . . . . . . .                          56

5.3.3                         Wang and Cheng (2006) S50 case . . . . . . . . . . . . . . . . . . . . . . . .                            59

5.4                                   Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                        60

Chapter 6

Summary and Conclusion                                                                                                                                            65

6.1                                   Highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                       65

6.2                         Recommendations for Future Studies . . . . . . . . . . . . . . . . . . . . . . . . . .                              68

Appendix A

OpenFOAMr Case Setup Files                                                                                                                                    69

A.1 Overview                                   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                      69

A.2 Boundary and Initial Condition Setup . . . . . . . . . . . . . . . . . . . . . . . . .                                 69

A.2.1      U, velocity dictionary                           . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                              69

A.2.2      p, pressure dictionary                          . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                              70

A.2.3     omega, ω dictionary                          . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                               72

A.3 Libraries from system Directory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                  74

A.3.1     controlDict                              . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                  74

A.3.2                                fvOption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                    75

A.3.3     fvScheme                              . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                   76

A.3.4                                fvSolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                    77

Appendix B

Mesh Generation Scripts                                                                                                                                             79

B.1 m4 script . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                          79

B.2 The blockMesh Dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                    85

B.3 Mesh View From Paraview                              . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                               89

Chapter 1 |

# Introduction

## 1.1        Problem Statement

The need for appropriate numerical means for studying turbulent secondary flows is indisputable. This phenomena have significant role in fluvial geomorphology. The hydrodynamics of turbulent secondary flow and their interaction with geometry and roughness have been subject of abundant studies in the past (Stoesser et al., 2010). The expense of experimental studies and the limitations and restraints of current measurement tools, made the thorough investigation of such flows difficult. This is specifically more evident, when numerous case studies are needed to obtain insight into interaction of secondary flows with other element (e.g. geometry). Although, experimental studies, over the past decades have provided us with insightful knowledge about secondary flow mechanisms (Nezu, 2005). Developing a numerical framework to study this phenomena is necessary.

Several numerical studies have investigated, modified, and developed Computational Fluid Dynamics (CFD) models in studying turbulent secondary flows(Ansari et al., 2011; Choi et al., 2007, 2008; Demuren and Rodi, 1984; Naot and Rodi, 1982; Nezu and Azuma, 2004; Stoesser et al., 2010). This paragraph examines present CFD models, considering roughness and geometry effects which require running great deal of case studies. Direct Numerical Simulation (DNS) studies of this phenomena is almost impossible, given the fact that they are computationally expensive, and the current resources would not allow running these simulations. Large Eddy Simulation (LES) of turbulent secondary flows are accurate, and they provide us with perceptive results. Even though considering running numerous case studies, LES studies are not computationally affordable. RANS-based Reynolds Stress Models (RSM) demand lower computational resources and they are much affordable, compared to LES. Yet, they need a lot of tuning which make them problematic. The alternative to all of previous models are RANS two-equation models.

Standard two equation models suffer simulating secondary currents due to their simplified linear modelling of Reynolds stresses (Nezu, 2005). For the aforementioned reasons, the necessity to develop a new numerical framework which is computationally fast, straightforward to implement roughness, and numerically accurate and reliable is obvious.

## 1.2       Sediment Transport

Sediment transport is one of the most important processes in engineering studies. It has been studied in various scales and diverse fields. From geological and civil engineering studies investigating the transportation of gigantic rocks by glaciers (García, 2008) to built environmental studies exploring the transport of micron size particles in indoor air (Salimifard et al., 2015). It occurs in response to different surface dynamics processes such as glacier movement, forced by gravity on hill slopes, flows of wind, streams, and oceans. Figure 1.1 and 1.2 illustrate two examples of earth surface changes caused by sediment transport, under different mechanisms. They show Mississippi Delta in Louisiana and Barchan dunes in Peru, respectively. Depending on the mechanism, sediments of various sizes can be moved by these processes. Glaciers can move sediments up to several meters large in diameter, while small-scale stream can only transport sediments as small as sand grains. Scientist have scrutinized these processes in diverse areas such as civil engineering, environmental engineering, earth sciences and oceanography to understand their mechanism and their effects. Having vital impacts on the environment, researchers have studied sediment transport in fluvial geomorphology for decades.

In fluvial rivers and streams, sediment transport has been a research subject for over decades due to its significant influence on the environment. Just to name a few, it is responsible for reservoir sedimentations (Buckley, 2003), changing river paths, providing sediment budget for river deltas, transporting and depositing fine chemical particles and consequently changing ecology balance in environmental habitats. Various mechanisms in fluvial geomorphology are responsible in sediment transport processes. Therefore, understanding these mechanisms plays a crucial role in many engineering practices including river restoration projects, reservoir sediment management, delta management projects, habitat ecological studies and even social and economic studies of cities situated near rivers (MacArthur et al., 2008).

Initiation of motion is the beginning part of sediment transport processes. This is the part in which sediment particles get detached from the bottom or side surfaces. It is important to uncover the physical procedure occur in order for a particle to start its motion. Moreover, the dynamics of the particles at the start point and their potential routes after being dislodged from the ground are vital

Figure 1.1: Mississippi Delta in Louisiana; (Photo Credit: ©NASA image created by Jesse Allen , Retrieved from http://www.landcover.org/)

for understanding sediment transport dynamics. Depending on the various influential factors such as type of fluid, particle characteristics, and flow field researchers have characterized the initiation of motion. Researchers have correlated initiation of motion for particles at rest in fluids to the shear stress (τbed) applied by the fluid on their surface. This shear stress should exceed a critical shear stress (τc) for initiation of motion of particles in order for them to be dislodged from the bed and move with the flow (García, 2008).

Figure 1.2: Barchan dunes developed by wind action predominately from one direction in crescent shapes, Paracas National Park, Peru; (Photo Credit: ©Steinmetz, George, Anastasia Photo, Retrieved from http://www.anastasia-photo.com/george-steinmetz-desert-air )

Depending on the forces exerted by the flow on the suspended particle, the particle size and the flow hydrodynamics, the particle might be deposited after a few number of saltation steps or continue moving along the flow to farther locations (García, 2008). Numerous elements of flow hydrodynamics and bed roughness characterization take part in calculating the critical shear stress and bed shear stress to determine the initiation of motion for particles. This study will look into secondary flows of second type (also called turbulent secondary flow) as one of the flow hydrodynamic mechanisms which has a crucial influence on dislodging particles from the bed and transferring the suspended

particles.

Figure       1.3:        Channel      stabilization;       (Photo      Credit:        ©Intuition     &     Logic,      n.d.          Web.

,        Retrieved                   from http://intuitionandlogic.com/Third%20Pages/FuvialGeomorphology/

FluvialGeomorphology.html ); Left: Galaxy/Meteor/Brierhall Channel, Restoration City of

Maryland Heights, Missouri; Right: Kirkwood Park – Walker Lake Channel, City of Kirkwood, Missouri

## 1.3       Secondary Flows

### 1.3.1       Role of secondary flows

Secondary flows have significant impact on fluvial streams including altering their path and changing their boundary configurations Einstein and Li (1958); García (2008); Gessner (1973); Gulliver and Halverson (1987); Nezu (2005); Nezu and Nakagawa (1984); Parker (2008). Given the fact that the mechanism behind the generation and maintenance of turbulent secondary flow is not yet fully understood, this phenomenon in different configurations should be studied in order to obtain better understanding of its generation and maintenance mechanism. The insight into turbulent secondary current mechanism is necessary in many engineering practices projects and scientific studies including river restoration (Rodríguez and García, 2008), reservoir sediment management, and establishing erosion processes’ criteria and theories.

Secondary flows in fluvial streams have been long recognized as a substantial mechanism for dislodging and picking up sediment particles from erodible boundaries, transferring them in helical routes along streamwise direction and depositing them elsewhere downstream. They have been recognized to alter the path of sediment particle motion and consequently change the river and land surface evolution process. Furthermore, many researchers have discovered noticeable coevolution between sediment transport and secondary flows. This is especially more obvious where erodible boundaries exist (Colombini, 1993; Ikeda, 1981).

### 1.3.2        Secondary flows of second type

Secondary flows also known as turbulent secondary flows in open channels exist in various forms. They are perpendicular to streamwise direction. In other word, secondary flows are secondary currents (velocity components) in the cross sectional plane at each location. There are two different kinds of secondary flows: the first kind due to curvature in a river bend and the second kind which is believed to be caused by turbulence anisotropy. This study deals with the second kind in straight open channels. For the purpose of simplification, in this study, secondary flow refers to the secondary flow of second type.

Figure 1.4: Schematic view of secondary currents of second type.

## 1.4         Thesis Aim and Objectives

### 1.4.1        Motivation of This Thesis

Considering the significant impacts turbulent secondary flow has on fluvial geomorphology, the need for appropriate tools to investigate the mechanisms behind its generation and maintenance is substantial. Researchers have explored these mechanisms throughout the past century, utilizing experimental and numerical methods. Numerical studies are of great importance in shedding light on secondary flow mechanisms and complementing the experimental studies by explaining mechanisms in which experimental apparatuses are restricted and limited. Developing numerical approaches to be capable of carrying out more realistic case scenarios while maintaining and improving the computational efficiencies of the models is necessary.

### 1.4.2        Scope of This Thesis

This study aims to map the details of secondary flows and their interaction with the boundaries, in straight open channels, in a numerical framework. The purpose is to develop, modify and implement a numerical framework which can simulate secondary currents with high accuracy in a computationally affordable time-line. This numerical framework will be used to accurately study the secondary flows in open channels with different configurations in order to investigate the mechanisms behind the generation and maintenance of secondary flows in straight open channels.

### 1.4.3         Structure of This Thesis

This thesis is divided into seven chapters. The first chapter is a brief introduction to the problem statement, importance of secondary flows, and aim and objectives of this thesis. Providing critical literature review, chapter 2 will focus on the advancements and progresses in the study of turbulent secondary flows. Then, to define the research methodology of this research, chapter 3 outlines the computational details and describes the utilized numerical model. Chapter 4 investigates the performance of the numerical model in simulating turbulent-induced secondary flows in open channels with smooth boundary. In this chapter, the efficiency of the model without consideration of roughness is analyzed. Subsequently, the performance of the model in conjunction with roughness function, in presence of boundaries with uniformly distributed roughness is investigated in chapter 5. Then, chapter 6 examine the combination of the model and roughness function in simulating cellular secondary currents in open channels over beds with alternate rough and smooth boundaries. The modifications required to deploying the function for these configurations is explained. Finally, chapter 7 provides the concluding remarks of the work presented in this thesis, and recommendations for the future research studies.