Confluence of Density Currents Produced by Lock-Exchange Hassan Ismail University of South Carolina

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Theses and Dissertations

2017 Confluence of Density Currents Produced by Lock-Exchange Hassan Ismail University of South Carolina

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This Open Access Dissertation is brought to you by Scholar Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. Confluence of Density Currents Produced by Lock-Exchange

by Hassan Ismail

Bachelor of Science University of South Carolina 2011

Master of Science University of South Carolina 2015

Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Civil Engineering College of Engineering and Computing University of South Carolina 2017 Accepted by: Jasim Imran, Major Professor M. Hanif Chaudhry, Committee Member Enrica Viparelli, Committee Member Jamil Khan, Committee Member Cheryl L. Addy, Vice Provost and Dean of the Graduate School c Copyright by Hassan Ismail, 2017 All Rights Reserved.

ii Dedication

For Heather.

iii Acknowledgments

This work was possible thanks to funding support from the National Science Foun- dation. I would like to thank my dissertation committee for their time, constructive feed- back, and guidance in completing this work. Foremost is Jasim Imran, my advisor, mentor, major professor, and collaborator on research projects through my gradu- ate studies. His encouragement and flexibility allowed me to investigate a variety of scholarly projects through my time at the University, and what I have learned under his advisement goes far beyond the contents of the work presented here. Thank you to the members of my committee, M. Hanif Chaudhry, Enrica Viparelli, and Jamil Khan who have supported, questioned, and probed my work to ensure that I have earned this doctorate, and the work itself can be given the name "dissertation" in the truest meaning of the word. I would also like to thank the water resources research group, Department of Civil & Environmental Engineering, and the University of South Carolina. I can not think of any fellow graduate student or post-doctoral researcher in the water resources research group with whom I have not discussed, collaborated, or brain-stormed about my research. Specifically, I would like to thank Lindsey LaRocque for her guidance in the laboratory and in graduate school life. I have made a home at the University of South Carolina over the course of my studies, and I want to thank the University system which has treated me well for many years, and I hope I have and continue to represent the University with integrity and pride. Beyond academic faculty and researchers, I would like to thank the support staff

iv in the Department of Civil & Environmental Engineering. Specifically I want to ac- knowledge Anne Kawamoto, Patrick Blake, Russell Inglett, and Karen Ammarell. Anne has always been helpful with organization of the research group. Patrick has helped with information technology needs including helping me get started with us- ing Linux operating systems which became critical to this work. Russell provided laboratory support in all studies conducted in the laboratory. Finally, Karen has helped with any procedural needs to ensure I followed all departmental and univer- sity guidelines. I tell every student that I can, "If Karen is not your best friend yet in this department, make sure she becomes that soon." Most importantly, I want to acknowledge my family. My father, Mohamad, my late mother, Randa, and my two brothers, Ahmad and Bassem: thank you all for shaping me into the man I am today. And mom, I miss you and thank you for the time we did have together. I want to thank my wife and best friend, Heather. Nothing in this work would be possible without the support given by her; I could not ask for more. Finally I would like to thank Heather for giving to me my joy, my son Gabriel. Gabriel makes me happy. Thank you for that.

v Abstract

Density currents represent a broad classification of flows driven by the force of gravity acting on a fluid with variable density. With examples of density currents including turbidity currents, sand storms, salt wedges in tidal rivers, and oil spills, a great deal of attention has been previously given to understanding the underlying mechanisms of such flows and implications of those flows on fluid, species, and sediment transport. Although documented confluences occur naturally in terrestrial and submarine set- tings, little attention has been given to understanding the confluence of two density currents. This study furthers the state of knowledge on density current confluences by systematically studying the unsteady flow phenomena and providing a methodology for describing the flows based on the bulk properties in the pre- and post-confluence density currents. Numerical simulations were conducted with experimental validation in which the effect of the initial density difference, channel depth, and junction angle were studied. The simulations revealed that the junction played a critical role in the combined current’s bulk properties. In the junction zone, the density currents accelerated and became thicker. After the combined front continues downstream, an elevated plume of dense fluid remained in the junction zone for some time. It was concluded for the range of densities tested, that initial density difference has little effect on the bulk properties when nondimen- sionalized. The role of the junction angle was isolated, and it is concluded that higher junction angles result in higher peak velocity in the junction zone and a larger plume. This notwithstanding, the effect of the junction is short-lived and the bulk properties

vi of the combined current have little dependence on junction angle further downstream. The initial conditions (initial density, channel depth, and junction angle) are com- bined via a Reynolds number giving an indication of the downstream oriented inertia entering the junction zone from both upstream branches of the channel network. Trends in bulk properties as functions of this Reynolds number are presented, but at high values of Reynolds number, many bulk properties approach a constant value independent of Reynolds number.

vii Table of Contents

Dedication ...... iii

Acknowledgments ...... iv

Abstract ...... vi

List of Tables ...... x

List of Figures ...... xii

Chapter 1 Introduction ...... 1

1.1 Background ...... 1

1.2 Outline ...... 7

Chapter 2 Numerical Model Description ...... 9

2.1 Governing Equations ...... 9

2.2 Solution Algorithm ...... 11

Chapter 3 Physical Experiments & Model Validation ...... 13

3.1 Model Set-up ...... 13

3.2 Results ...... 14

3.3 Numerical Model Performance ...... 15

3.4 Summary ...... 16

viii Chapter 4 Confluence of Density Currents in a 45◦ Junction . 21

4.1 Initial & Boundary Conditions ...... 21

4.2 Numerical Experiments ...... 22

4.3 Results & Analysis ...... 23

4.4 Summary ...... 29

Chapter 5 The Effect of Junction Angle on Density Current Confluences ...... 53

5.1 Model Set-up ...... 53

5.2 Test Cases ...... 54

5.3 Results & Analysis ...... 54

5.4 Summary ...... 57

Chapter 6 Summary & Conclusions ...... 81

Bibliography ...... 86

ix List of Tables

Table 3.1 Experimental and numerical test cases and results of bulk properties. 17

Table 4.1 Case summary. D0/H0 = 1 indicates full-depth cases, and D0/H0 < 1 indicates partial-depth cases...... 31

Table 4.2 Constant upstream , peak, minimum, and constant downstream head thickness for each case. Thicknesses are normalized by channel depth and position is normalized by lock length mea- sured from the upstream wall. Empty entries indicate that no constant value was reached...... 32

Table 4.3 Plume thickness results for full-depth cases. The position is mea- sured from the upstream wall of the domain and is normalized by the lock length...... 33

Table 4.4 Results of the distance from the current nose to the location of maximum plume height, ∆X . The position is measured from the upstream wall of the domain and is normalized by the lock length. 34

Table 4.5 Nose height results for full-depth cases. The position is measured from the upstream wall of the domain and is normalized by the lock length...... 35

Table 4.6 Froude number summary in each phase of flow. Location val- ues are normalized by lock length and are measured from the upstream wall...... 36

Table 4.7 Summary of near-bed maximum horizontal velocity. Locations are normalized by the lock length and measured from the up- q 0 stream wall. Horizontal velocity is normalized by goHo...... 37

Table 5.1 Initial conditions for each junction angle...... 59

ˆ Table 5.2 Re0 for each case...... 59

x Table 5.3 Froude number results for each case. The second peak and con- stant Froude number correspond to the post-confluence reaccel- eration phase...... 60

Table 5.4 Head thickness results for each case...... 61

xi List of Figures

Figure 3.1 Plan and profile of the experimental flume for full- and partial- depth cases. Units are in meters...... 17

Figure 3.2 Comparison of front shape for the physical experiments and simulations at 5s and 14s after the lock-gates were released. . . . 18

Figure 3.3 Comparison of bulk properties (front position, front velocity, and front thickness) from the physical experiments and simulations. 18

Figure 3.4 Comparison of simulation output from the grid 1 and grid 2 for the full-depth case...... 19

Figure 3.5 Comparison of simulation output from the grid 1 and grid 2 for the partial-depth case...... 20

Figure 4.1 Computational grid and domain geometry...... 38

Figure 4.2 Typical propagation of the density currents for a full-depth case (case 5). Colors indicate distance from the bed...... 39

Figure 4.3 Typical propagation of the density currents for the partial- depth case (case 11). Colors indicate distance from the bed. . . . 40

Figure 4.4 Propagation of the density current for case 9 along the main- channel centerline visualized as isolines of nondimensional con- centration...... 41

Figure 4.5 Time series of the near-bed front shape for case 5. Each output is separated by 0.77 nondimensional time units. The dash line is the interpreted shear plane...... 42

Figure 4.6 Time series of near-bed horizontal velocity and vertical velocity direction for case 5. Light shading indicates vertical velocity is away from the bed...... 42

Figure 4.7 Nondimensional front position versus time along the main-channel centerline for the full-depth cases and partial-depth cases. . . . . 43

xii Figure 4.8 Head thickness versus front position for full-depth and partial- depth cases. The symbology follows figure 4.7...... 44

Figure 4.9 Plume thickness versus front position for full-depth and partial- depth cases. The symbology follows figure 4.7...... 45

Figure 4.10 Distance from the nose to the location of the plume, ∆X , versus front position for full-depth and partial-depth cases...... 46

Figure 4.11 Nose elevation versus front position along the main-channel cen- terline for the full-depth cases and partial-depth cases. The symbology follows figure 4.10...... 47

Figure 4.12 Froude number versus front position for the full-depth and partial- depth cases...... 48

Figure 4.13 Maximum near-bed horizontal velocity versus front position for full-depth and partial-depth cases. The symbology follows fig- ure 4.12...... 49

Figure 4.14 Pre-confluence trends in bulk properties with initial Reynolds number...... 50

Figure 4.15 Trends in ∆X with initial Reynolds number...... 51

Figure 4.16 Post-confluence constant Froude number versus initial Reynolds number...... 51

Figure 4.17 Trends in nose height with initial Reynolds number...... 52

Figure 5.1 Dimensions for the domain for each junction angle, θ. Linear dimensions are in meters...... 62

Figure 5.2 Typical propagation of the density currents for case 5 (H0 = 0.4 3 and ρ0 = 1025 kg/m ) for each junction angle. Colors indicate distance from the bed...... 65

Figure 5.3 Time series of the near-bed front shape for case 5 (H0 = 0.4 3 and ρ0 = 1025 kg/m ) for each junction angle. Each output is separated by 0.77 nondimensional time units...... 68

Figure 5.4 Nondimensional front position versus time for each junction angle. 71

xiii Figure 5.5 Nondimensional front velocity versus front position for each junction angle...... 74

Figure 5.6 Nondimensional front thickness versus front position for each junction angle...... 77

Figure 5.7 Nondimensional trends in head thickness, Hf and plume thick- ˆ ness with Re0. The front position is measured from the up- stream wall of the main-channel...... 78

Figure 5.8 Nondimensional trends in maximum near-bed horizontal veloc- ˆ ity, Uh, with Re0...... 78

ˆ Figure 5.9 Nondimensional trends in ∆X with Re0...... 79

ˆ Figure 5.10 Nondimensional trends in nose height with Re0...... 79

ˆ Figure 5.11 Nondimensional trends in F with Re0...... 80

xiv Chapter 1

Introduction

1.1 Background

Density currents are flows driven by the force of gravity acting on a fluid with vari- able density. They can occur in a number settings generated by both natural and man-made phenomena including ocean currents driven by salinity and temperature variation, turbidity currents, oil spills on sea surfaces, or salt wedges in tidal rivers to name a few [Simpson, 1997]. Although density currents have been studied in laboratory and numerical models [e.g., Keeulegan, 1949, Simpson and R. E. Britter, 1979, Parker et al., 1986, 1987, Garcia and Parker, 1993, Kneller et al., 1999, Peakall et al., 2000, Kneller and Buckee, 2000, Peakall et al., 2007, Huang et al., 2008, Islam and Imran, 2008, Sequeiros et al., 2010, Huang et al., 2012, Ezz et al., 2013, Janocko et al., 2013, Tokyay and Garcia, 2013] and limited field observation [e.g., Prior et al., 1987, Normark, 1989, Khripounoff et al., 2003, Paull et al., 2002, Xu et al., 2004, Vangriesheim et al., 2009, Cooper et al., 2013], attention is almost exclusively given to behavior of these currents in straight or sinuous channels. Although submarine channel confluences that convey episodic density currents are evident in the geologic record [e.g., Canals et al., 2000, Valle and Gamberi, 2011, Greene et al., 2002, Hesse, 1989, L’Heureux et al., 2009, Mitchell, 2004, Paquet et al., 2010, Straub et al., 2011], and deltaic channel networks passing salt wedges during tidal cycles have documented confluences as salt wedges move landward [e.g., Buschman et al., 2013, Warner et al., 2002, Sassi et al., 2011], to

1 the author’s knowledge, the work of Ismail et al. [2016] represents the only controlled investigation into the confluence of density currents. Salt wedges, propagating due to tidal forcing, extend landward in coastal channels beneath ambient riverine discharge. In such coastal channel systems, bifurcating river channels act as confluences for salt wedge density currents during high tide conditions when the flow is in the landward direction. Limited field studies have been conducted with regard to tidal channels with salt wedge confluences [e.g., Buschman et al., 2013, Warner et al., 2002, Sassi et al., 2011]. Buschman et al. [2013] observed periodic variation in stratification in their field study of suspended sediment, salt, and water fluxes in a tidal channel junction. They found that the bed level gradient was relatively high in the junction compared with the straight-channel reaches due to periodically high sediment transport capacity. Warner et al. [2002] concluded that residual circulation in the junction zone of the Mare Island and Carquinez Straights in San Francisco Bay were altered by phasing of landward-migrating currents. Sassi et al. [2011] developed a numerical model of the Mahakam Delta channels in which they studied the effects of the tidal discharge. They found that the salt wedges in the system have a significant effect on the entire water column including the upper zone of riverine discharge. Submarine channel junctions which convey density currents have been identified through field observation [e.g., Canals et al., 2000, Valle and Gamberi, 2011, Greene et al., 2002, Hesse, 1989, L’Heureux et al., 2009, Mitchell, 2004, Paquet et al., 2010, Straub et al., 2011]. Gamboa et al. [2012] interpreted and analysed 3D seismic data to provide a detailed survey of submarine confluences and present a classification scheme similar to that used for river confluences. Hesse [1989] and Klaucke et al. [1998] identified extensive networks of converging drainage channels on the Labrador sea floor. Mitchell [2004] and Vachtman et al. [2013] presented analyses of erosion rates and submarine hydrology on the Atlantic continental slope. These researchers

2 applied fluvial hydrology principles to the submarine setting and recommended simple relationships for erosion rates as a function of channel gradient and contributing area. Straub et al. [2007] assessed submarine channel profiles and drainage areas on the Monterey, CA and Brunei Darussalam continental slopes and found that submarine scaling exponents were within the range of terrestrial observations despite differences in physical processes. Ismail et al. [2016] conducted a laboratory investigation on the confluence of con- tinuous release density currents over an erodible bed. They found through their phys- ical experiments, that models developed for confluences for subaerial rivers cannot adequately predict separation zone dimensions, streamline deviation along the junc- tion line, and maximum scour depth. They concluded that the upward convection of dense fluid in the junction had a major impact on the flow dynamics and sediment transport in the junction zone and downstream reach. The present work aims to further the understanding of density current confluences through investigation of the current front behavior in a confluence. In work presented here, converging density currents at a channel confluence are in- vestigated. Numerical experiments are conducted in a lock-exchange configuration to study the transient phenomena. The flow conditions in the confluence are described, and trends in the behavior at the junction based on initial and geometric conditions are assessed and presented with proposed relationships.

1.1.1 Background on lock-exchange flows

Since many natural density currents are episodic in nature, researchers often perform so called lock-exchange experiments to study their behavior. In a lock-exchange scenario a vertical barrier, the lock gate, separates denser and lighter fluids which are initially at rest. When the lock gate is removed, the denser fluid slumps below the lighter fluid; the lighter fluid, in turn, rises above the denser fluid. The exchange

3 produces a dense current propagating near the bed and a light current propagating above the dense current. Shin et al. [2004] provides a review of previous lock-exchange experimental work. Two cases of lock-exchange experiments are conducted – full- and partial-depth flow. In a full-depth condition, the dense fluid and light fluid have the same initial depth, Ho, typically equal to the total channel depth. In a partial-depth condition, the initial depth of the dense fluid, Do, is less than the entire channel depth, Ho (see Shin et al. [2004], their figure 1).

Phases of Flow

Huppert and Simpson [1980] described the spreading of density currents produced by lock exchange in three phases. The first phase is the slumping or constant velocity phase. Following this is an inertial phase where the front speed is affected by buoy- ancy and inertial forces. Finally, the current reaches the viscous phase during which viscous effects overwhelm inertial forces, and viscosity and buoyancy are dominant.

In general, the front velocity, uf , begins as time-independent in the constant velocity

−1/3 phase, followed by dependence as uf ∼ t in the inertial phase, then finally is an

−4/5 additional time dependence in the viscous phase as uf ∼ t . In addition to these three phases, an initial acceleration phase is described by some researchers [Martin and Moyce, 1952a,b, Hartel¨ et al., 1999, Cantero et al., 2007]. The acceleration phase is short lived and describes the flow between the initial, at-rest state of the dense fluid and the constant velocity phase. In the acceleration phase, the front velocity sharply increases to a peak value followed by a slight decrease to the constant value. Cantero et al. [2007] attribute the reduction at the end of the acceleration phase to interface friction causing roll-up of the current in the early stages of the release. Following the acceleration phase is the constant velocity phase. To describe front

4 velocity, it is convenient to define a Froude number as

u F = f (1.1) q 0 g hf

0 where uf is the front velocity, g = g(ρ − ρa)/ρ is the reduced gravity, hf is the front thickness, ρ is the current’s density and ρa is the ambient fluid density. Benjamin [1968] proposed an analytical solution for a two-dimensional cavity using a reference frame moving with the front. In his constant velocity analysis, Benjamin [1968] did not distinguish the current head versus the body and theorized that the

Froude number is only a function of the fractional current depth, hf /Ho, where hf is the current depth and Ho is the depth of the channel. Benjamin [1968] found that for energy-conserving currents, F = 0.5. For the case of maximum dissipation, Benjamin

[1968] found that F = 0.527. In the limit of infinite ambient water depth, hf → 0, the analytical theory of Benjamin [1968] results in F = 1.41. Huppert and Simpson [1980] proposed an empirical expression for Froude number as a function of the fractional current depth based on the body of the current. They found that F = 0.445 for the energy conserving case of hf /Ho = 0.5. For the limit of infinite ambient depth, their empirical expression yields F = 1.19. Shin et al. [2004] expanded the analysis of Benjamin [1968] by including the effects of the reverse ambient bore interaction with the dense current. Their model was based on the initial depth of dense fluid, Do, and they found that for full-depth cases (Do

= Ho), the theory was identical to that of Benjamin [1968]. The inertial phase of flow begins when the light current propagating in the up- stream direction reflects off the upstream boundary, and the resulting wave reaches the front [Rottman and Simpson, 1983]. The behavior of the density current in this stage has been studied by several researchers [Fay, 1969, Fannelop and Waldman, 1971, Hoult, 1972, Huppert and Simpson, 1980, Rottman and Simpson, 1983] result- ing in the following relations

5 ¯ 2 1/3 x¯f = ζ(hox¯ot¯ ) (1.2)

2 u¯ = ζ(h¯ x¯ )1/3t¯−1/3 (1.3) f 3 o o

where the overbar represents a nondimensional value (Ho is the length scale,

0 1/2 Uo = (g Ho) is the velocity scale, and Ho/Uo is the time scale), x¯f and u¯f are ¯ the dimensionless streamwise front position and front velocity, respectively, hox¯o rep- resents the initial dense fluid volume per unit width, and ζ is a shape factor and was prescribed by Hoult [1972] as 1.47 and by Huppert and Simpson [1980] as 1.6. As the current continues to propagate, the inertial phase causes the current to decelerate with time. Thus the ratio of inertial to viscous forces decreases, i.e., the

Reynolds number, Re = uf h/ν, decreases. Here, ν is the kinematic viscosity. Once the Reynolds number is sufficiently small, and the viscous forces become comparable to the buoyancy forces, the current enters the viscous phase. Similar to the solutions for the inertial phase, the viscous phase front position and velocity can be described by the initial volume of the release but with an additional dependence on Reynolds number and time. The equations will not be repeated here but can be found in Hoult [1972] and Huppert [1982].

Current Thickness

To avoid ambiguity in defining the current depth, a similar equivalent depth method is adopted here as was presented by Shin et al. [2004], Marino et al. [2005], Cantero et al. [2007]. First, a nondimensional density is defined as

ρ − ρ ρ¯ = a (1.4) ρo − ρa

where ρ is the measured density as a function of space and time, ρa is the ambient density and ρo is initial density in the lock. With this nondimensional density, an

6 equivalent depth can be computed by integration of the vertical density profile. In the present work, ρ¯ = 0.5 is used as the threshold for the dense-ambient interface to define the current depth as h = h(x, z, t)|ρ¯=0.5.

1.2 Outline

In work presented here, converging density currents at a channel confluence are in- vestigated. Numerical experiments are conducted in a lock-exchange configuration to study the unsteady phenomena. The flow conditions in the confluence are described, and trends in the behavior at the junction based on initial and geometric conditions are assessed and presented with proposed relationships. Chapter 2 introduces the governing equations and model. Solution algorithms utilized in this work are discussed. Chapter 3 presents the physical experiments conducted for model validation. Three tests were run of full- and partial-depth density current confluence in a 45◦ junction in the Hydraulics Laboratory at the University of South Carolina. Assessment of the front velocity and front thickness were made and compared to numerical model results. Additionally, refined grid numerical simulations were run to verify grid independence of trends in bulk properties of the flow. Chapter 4 provides insight into the unsteady phenomena of density current head confluences. Nine full-depth cases and three-partial depth cases are presented in which initial lock depth and density difference were varied for a 45◦ junction angle. A discussion of the observed phenomena and results are presented on the front velocity, current depth, nose height, and near-bed horizontal velocity before and after the confluence. Finally, the results are correlated to the initial conditions leading to predictive relationships for the effect of the junction. Chapter 5 continues the investigation into density current head confluences by systematically investigating the effect of the junction angle. Five junction angles are examined in a similar numerical procedure as is presented in Chapter 4 (for a total

7 of 45 cases). First, the effect of the junction angle on the bulk properties is isolated. Then, the initial conditions, including the junction angle, are combined into a single parameter representing the downstream inertia entering the junction zone for the development of predictive equations.

8 Chapter 2

Numerical Model Description

The three-dimensional, unsteady simulations were implemented in the open-source computational fluid dynamics toolbox, OpenFOAM [OpenFOAM, 2011a,b]. Open- FOAM is a compilation of modifiable C++ libraries which are used to create exe- cutable solvers and utilities. Solvers are compiled and run to solve specific continuum mechanics problems, and utilities are executables used for pre- and post-processing data for use with a solver (e.g., mesh generation, initial condition set-up, solution data visualization, etc.). The modifiable executables allow for the creation of customized solvers and utilities for specific problems not handled by pre-compiled solvers within the toolbox. The twoLiquidMixingFoam solver in OpenFOAM, which solves the flow equations for two miscible, incompressible fluids was used to solve the governing equations. Simulations were set up to consider water and salt water as the two fluids. Since the solver considers two completely miscible fluids, the resulting simulation models the behavior of density currents produced by a solute or temperature difference. A large eddy simulation (LES) approach was employed with a Smagorinsky-Lilly subgrid- scale (SGS) closure [Smagorinsky, 1963, Lilly, 1966], and dynamic time stepping was used to improve stability based on the Courant-Friedrichs-Lewy condition [Courant et al., 1928].

2.1 Governing Equations

The unsteady, incompressible Navier-Stokes equations modeled by the solver are

9 ∂ρ ∂(ρu ) + i = 0 (2.1) ∂t ∂xi

" ! # ∂(ρui) ∂(ρujuj) ∂ ∂ui ∂uj 2 ∂uj ∂p ∂(ρτij) + = µ + − µ δij − − + ρgj (2.2) ∂t ∂xj ∂xj ∂xj ∂xi 3 ∂xi ∂xj ∂xj

where ρ is the bulk density, t is time, uj is velocity in the j direction, xj is a

Cartesian direction, µ is the dynamic viscosity, δij is the Kronecker delta, p is the pressure, τij is the Reynolds stress, and gj is the gravity vector. The Reynolds stress term in LES modeling is handled using spatial filtering and is taken to be

¯ τij = −2νtSij (2.3)

¯ where νt is the turbulent eddy viscosity, and Sij is the rate of strain tensor. By including equation 2.3 into the momentum equation (equation 2.2), the LES momentum equation takes the form

∂ui ∂uj ∂p ∂ρ ∂σ ρ + ρui = − + σ + ρ + ρgj (2.4) ∂t ∂xj ∂xj ∂xj ∂xj

! ∂ui ∂uj 2 ∂ui σ = (ν − νt) + − ν δij (2.5) ∂xj ∂xi 3 ∂xj To close the momentum equation, the Smagorinsky-Lilly SGS model is used where

2 νt = (C∆) |S| (2.6)

where C is a constant and ∆ is the filter length computed as the cube root of the cell volume. The bulk density for each cell is given by

10 X ρ = ckρk (2.7)

where ck and ρk are the concentration and density of species k, respectively. For one fluid and a species within that fluid, i.e., water and dissolved salt, equation 2.7 becomes

ρ = csρs + (1 − cs)ρw (2.8)

where cs and ρs are the concentration and density of the salt water, respectively, and ρw is the density of the ambient water.

The concentration, cs, of the species is conserved and solved for using the species continuity equation as

! ∂cs ∂ ∂ νt ∂cs + (uj − vsδijcs) = (2.9) ∂t ∂xj ∂xj Sct ∂xj

where vs is the fall velocity and equal to zero for a dissolved species or heat and

Sct is the turbulent Schmidt number.

2.2 Solution Algorithm

For efficient solution of the governing equations, the PIMPLE (PISO-SIMPLE) al- gorithm is utilized [OpenFOAM, 2011a,b]. The PIMPLE algorithm is a merger of the Pressure Implicit with Splitting of Operators (PISO) algorithm [Issa, 1986] and Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm [Ferziger and Perić, 2001]. The PISO algorithm splits operators into an implicit predictor step and mul- tiple explicit corrector steps. In OpenFOAM, velocity is first predicted using the known pressure from the previous time step, then the pressure is corrected using the predicted velocity, followed then by correcting the velocity using the new pres-

11 sure. SIMPLE uses relaxation factors to iteratively adjust the corrected variables for smooth convergence. The PIMPLE algorithm takes advantage of the PISO method with the addition of relaxation factors for smooth convergence from the SIMPLE method. This allows for stable solution of the coupled pressure-velocity equations to a prescribed threshold at Courant numbers greater than one. The generalized geometric-algebraic multi-grid (GAMG) and Gauss-Seidel smooth solvers were utilized to solve discretized equations. The advantage of GAMG is that it first generates a solution for a geometrically coarsened grid which is then used as an initial solution for the specified finer grid resulting in a faster solution of matrices. Discretized terms in partial differential equations are solved using the standard second-order accurate Gaussian finite volume integration method. The first order accurate Euler time discretisation scheme and second order accurate Gauss linear divergence scheme were utilized. Cell face values are interpolated between cell center values using a central difference scheme [OpenFOAM, 2011a].

12 Chapter 3

Physical Experiments & Model Validation

Before applying the numerical model introduced in Chapter 2 to the test cases, phys- ical experiments were conducted for model validation. Three experiments were con- ducted in a horizontal, 45◦: two involving full-depth lock-exchange cases and one partial-depth case. The results from the physical experiments, including front posi- tion, front velocity, front thickness, and front shape are compared with the results from the numerical model.

3.1 Model Set-up

The physical model geometry can be seen in figure 3.1. The bed of the horizontal flume was made of wood, and the walls were made of wood except for the longest wall along the main- and downstream-channels which was acrylic glass for flow vi- sualization. The flume consisted of two upstream reaches each with 1.83 m length, 0.305 m width, and 0.61 m depth. The upstream channels meet at the upstream junction point and continue to the downstream reach. The downstream reach then discharges into a large tank with an overflow pipe to ensure constant ambient water depth during testing. Each upstream reach was equipped with a thin gate at the midpoint of the upstream reaches (0.915 m from the extreme upstream walls) and lifting mechanism. The main-channel centerline and wall of the flume were marked with a scale to track the current’s progression and size. Each test involved closing and sealing the two gates and filling the flume down- stream of the gates with ambient water (density = 996-996.5 kg/m3) to a depth of

13 0.437 m. For the full-depth tests, each lock was filled with a premixed saltwater solution with a density of 1030 kg/m3 to the same depth as the ambient water. For the partial-depth test, the lock was filled with saltwater for half the depth with am- bient water above. Mixing of the lock fluid in the partial-depth case was avoided by first filling the ambient water to the prescribed depth followed by slowly injecting the dene fluid near the bed of the flume within the locks. Samples of the dense fluid were obtained for the partial-depth case before release to verify the density had not changed. Each test was initiated by simultaneously lifting the two lock gates. To ensure the gates lifted simultaneously, they were tethered together using steel cable, and a single counter-weight was released to lift the gates. The density currents then formed in a relatively short time after the release before reaching the junction zone. Then, they combined in the junction zone and were allowed to continue downstream. The currents were tracked visually by dying the saltwater before testing and tracking movement with two high-definition video cameras positioned along the main channel perpendicular to the main- and downstream-channels’ acrylic glass wall. The cap- tured video for each case was split into individual frames (at a rate of 60 frames/s), and the current position as a function of time along with the height of the current were digitized for analysis.

3.2 Results

A summary of the test cases and results of bulk properties of the physical experiments can be seen in table 3.1. The full-depth case was repeated to ensure repeatability of the tests, and it can be seen from the results that there was only 3.7% difference between the two test runs in terms of maximum front velocity and 3.4% difference in maximum head thickness.

14 3.3 Numerical Model Performance

The numerical model was applied to the test cases seen in table 3.1. Figure 3.2 shows a qualitative comparison between the front shape from the physical experiment and numerical model at 5s and 14s from the initiation of the current. At 5s, the current had not yet reached the junction zone, and at 14s the front is in the downstream reach. The general trend in the front shape between the experimental and numerical results does not greatly vary in these two snapshots. Figure 3.3 presents the time series of front position, front velocity, and head thickness along the main-channel centerline for the physical experiments and numerical model. Although some variability in the data is noted, the bulk properties and trends in the data agree well. Agreement between the physical experiments and numerical model in terms of the bulk properties are good with differences in maximum front velocity and maximum head thickness of 8.1% and 3.3% for the full-depth cases and 4.8% and 13.9% for the partial-depth cases. The numerical model was applied to the physical experiments using a proposed grid resolution (grid 1) that is later to be applied to the numerical test cases presented in the remainder of this work. To ensure the grid resolution was sufficient to capture the bulk property characteristics of the flow phenomena, the validation tests were reassessed with a refined grid (grid 2). Grid 2 consists of quadruple the number of grid points as grid 1 by refinement of the horizontal resolution by a factor of two in each direction. From figure 3.2 it can be seen that grid 2, the refined grid, captures greater detail in the interface between the dense and ambient fluid, but bulk properties do not differ. Figures 3.4 and 3.5 present a comparison of the simulations with very close agreement between the two grids. This indicates that the grid 1 sufficiently captures the flow phenomena is terms of bulk properties which is of interest in this study.

15 3.4 Summary

This chapter presented the methodology, results, and analysis of physical experiments intended to validate the proposed numerical model and solution grid. Experiments were conducted in a horizontal, asymmetric junction flume with gates equipped in each upstream reach. Three physical experiments were conducted: two were full- depth lock-exchange tests which were repeated to verify the consistency of the exper- imental results, and the third was a partial-depth case. Video cameras were used to capture the front position, front velocity, and head thickness as a function of time. Dye was used for flow visualization and to iden- tify the interface between the density current and ambient fluid. The repeatability test showed good agreement indicating the experimental apparatus returns consistent results across all tests. The numerical model was applied to identical cases as the physical experiments with a grid resolution equal to that presented in the later chapters of this work. The shape of the current captured during the physical experiments matched well with the numerical model results, and bulk properties were captured with the highest difference being the maximum head thickness for the partial-depth case with 13.9% difference. The rest of the bulk properties investigated were well below 10% difference in the experimental and numerical results. The numerical simulations were repeated for both the full- and partial-depth cases with a refined grid (grid 2). The results indicate that the simulation output had little variation between grid 1 and grid 2, thus the coarser grid, grid 1, is used for the remainder of this work.

16 Table 3.1: Experimental and numerical test cases and results of bulk properties.

3 3 Test ρa (kg/m ) ρ0 (kg/m ) H0 (m) Depth Ratio Max Uf (m/s) Max Hf (m) 1 996 1030 0.437 1 0.222 0.317 2 996.5 1030 0.437 1 0.231 0.306 3 996 1030 0.219 0.5 0.188 0.208 Num - Full depth 996 1030 0.437 1 0.204 0.327 Num - Partial depth 996 1030 0.219 0.5 0.179 0.237

0.91 0.91 0.40 2.33 OVERFLOW PIPE

GLASS WALL 0.31 MAIN-CHANNEL LOCK LOCK DOWNSTREAM GATES JUNCTION PINT DISCHARGE TANK UPSTREAM 45° JUNCTION POINT

SIDE-CHANNEL LOCK

LOCK GATE 0.91 LOCK GATE

0.61 AMBIENT 0.44 DENSE AMBIENT AMBIENT DENSE

FULL-DEPTH PARTIAL-DEPTH

Figure 3.1: Plan and profile of the experimental flume for full- and partial-depth cases. Units are in meters.

17 EXP Grid 1 Grid 2 0.5 0.4 5 s 0.3 0.2 0.1 0

Distance Bed from (m) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Downstream Distance (m)

0.5 14 s 0.4

0.3

0.2

0.1

0 Distance Bed from (m) 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 Downstream Distance (m)

Figure 3.2: Comparison of front shape for the physical experiments and simulations at 5s and 14s after the lock-gates were released.

Full-Depth Partial-Depth 3 3

2.5 a 2.5 d

2 2 (m)

(m) 1.5 1.5

f f

x x 1 1

0.5 0.5

0 0 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 Time (s) Time (s)

Full-Depth Partial-Depth 0.25 0.2

0.2 b e 0.15

0.15 (m)

(m) 0.1

f f U U 0.1 0.05 0.05

0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Time (s) Time (s)

Full-Depth Partial-Depth 0.4 0.3

0.3 c f

0.2 (m)

(m) 0.2

f f H H 0.1 0.1

0 0 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 Time (s) Time (s)

Exp: Full-depth Num: Full-depth Exp: Partial-depth Num: Partial-depth

Figure 3.3: Comparison of bulk properties (front position, front velocity, and front thickness) from the physical experiments and simulations.

18 xf(t) 3

2.5 a

2

1.5

, Grid 2 Grid , f x 1

0.5

0 0 0.5 1 1.5 2 2.5 3

xf, Grid 1

Uf(t) 0.25 b 0.2

0.15 , , 2 Grid

f 0.1 U

0.05

0 0 0.05 0.1 0.15 0.2 0.25

Uf, Grid 1

Hf(t) 0.5 c 0.4

0.3 , Grid 2 Grid ,

f 0.2 H

0.1

0 0 0.1 0.2 0.3 0.4 0.5

Hf, Grid 1

Figure 3.4: Comparison of simulation output from the grid 1 and grid 2 for the full-depth case.

19 xf(t) 3.5 3 a 2.5 2

, Grid 2 Grid , 1.5

f x 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5

xf, Grid 1

Uf(t) 0.2 b 0.15

0.1

, , 2 Grid

f U 0.05

0 0 0.05 0.1 0.15 0.2

Uf, Grid 1

Hf(t) 0.25

0.2 c

0.15 , Grid 2 Grid ,

f 0.1 H

0.05

0 0 0.05 0.1 0.15 0.2 0.25

Hf, Grid 1

Figure 3.5: Comparison of simulation output from the grid 1 and grid 2 for the partial-depth case.

20 Chapter 4

Confluence of Density Currents in a 45◦

Junction

This chapter presents numerical tests and results of the confluence of density currents in a horizontal channel with 45◦ junction angle. Nine cases of full-depth lock-exchange and three cases of partial-depth lock-exchange confluences were tested and assessed. First, the test conditions are presented followed by a discussion of the general trends in the flow phenomena, then the bulk properties of the flow are analyzed as a function of the initial conditions. Finally, the initial conditions are combined into an initial Reynolds number and predictive equations are presented.

4.1 Initial & Boundary Conditions

The computational domain consisted of an asymmetrical junction of two horizontal channels and a downstream channel (figure 4.1). The lock length, Lo, was 1 m, the distance from the upstream edge of each channel to the upstream junction point was 2 m, the channel width, b, was 0.5 m, and the junction angle was 45◦. At the start of each simulation, each lock was filled with dense water, and the rest of the domain with ambient water. For each case, the only changes in the domain were the height of the channel (0.2, 0.4, and 0.5 m) and the number of vertical grids so that the vertical resolution remained consistent across all cases (the same grid resolution established in Chapter 3). The grid consisted of 312 000, 624 000, and 780 000 cells respectively for the

21 cases of 0.2 m, 0.4 m, and 0.5 m channel height. The grid size was 0.005 m, 0.03 m, and 0.015 m in the vertical, down-channel, and cross-channel directions, respectively (relative to the main-channel). All boundaries were considered walls. Thus no mass flux through any boundary was allowed. The bottom and side walls were rough boundaries and treated with a no-slip condition for velocity. The initial condition for each case included a stationary volume of dense water in each upstream channel’s lock. The release of the dense fluid was instantaneous, and output was collected for analysis every one second of flow time. The initial time step was set to 0.1 seconds which was dynamically adjusted during the simulations to limit the Courant number, Co ≤ 1 for improved numerical stability.

4.2 Numerical Experiments

The numerical model was applied to 12 density current cases. The first nine cases are full-depth releases in which initial density difference and channel depth were varied. The next three cases are partial-depth cases in which only initial density difference was varied. Table 4.1 lists a summary of the conditions for each case. The channel

0 1/2 depth, H0, is used as the length scale, U0 = (g H0) is used as the velocity scale, and t0 = H0/U0 is used as the time scale throughout this work. The initial density difference and channel depth are combined to assess the combined impact of these q 0 3 2 variables through the initial Reynolds number,Re0, defined as Re0 = (g0H0 )/ν

0 where g0 is the reduced gravity and ν is the kinematic viscosity of water.

22 4.3 Results & Analysis

4.3.1 Typical Behavior

This section presents the typical behavior of the junction cases. The general trends in the full- and partial-depth cases are assessed in terms of propagation and confluence, and nose shape.

Current Propagation

Figure 4.2 illustrates the typical propagation of the density currents as they are released, combine in the junction zone, and continue downstream visualized by the dense-ambient interface (ρ¯ = 0.5). The color contours indicate elevation from the bed. The density currents form in a short distance in the upstream reaches where the dense-ambient interface is relatively smooth. As the currents combine in the junction zone, the interface rises behind the front. After the head has cleared the junction zone, the interface remains high in the junction zone near the downstream junction point, and the interface is no longer smooth except just behind the combined current’s nose. The elevated plume in the junction falls thereafter as the bulk of the fluid continues downstream with a distinct and detached head. Figure 4.3 shows a typical partial-depth case. As compared with the full-depth case, the partial-depth case behaves similarly but propagates more slowly in the downstream reach and with a more distinct head throughout. The typical behavior of the current along the main-channel centerline can be seen in figure 4.4. The plot shows isolines of constant nondimensional density. Note that at 10s, the front has already passed the junction zone and interaction with the side-channel current can be seen by the enlarged current head and chaotic body.

23 Near-bed Front Shape % Velocity

Figure 4.5 shows the near-bed location of the density current in time. In this plot, each output interval is 0.77 nondimensional time units. The near-bed plane is defined as the upper edge of the first cells in the bed-normal direction. In each upstream reach, the front shape is relatively smooth. In the junction zone, the side-channel current extends into the junction first near the upstream junction point and begins to rotate as the main-channel current enters the junction. Note the retreat of the side-channel current as the fronts reach the downstream junction point. Further downstream, the front has a more distinct lobe-and-cleft shape as is typical of density currents. The absence of the lobe-and-cleft shape in the upstream reaches is due to the length of the upstream channels. The lobe-and-cleft pattern becomes more pronounced as a density current travels longer distances as reported by Cantero et al. [2007]. There is a clear shear plane highlighted in the figure. This shows the separation of the contributing currents from each upstream reach as they pass through the junction zone. Further downstream, the currents mix and form a combined current. There was no appreciable difference in the trends observed in the full-depth versus partial-depth cases. Figure 4.6 shows the near bed horizontal velocity vectors with shading indicating the direction of the vertical velocity component; the light shade means the vertical velocity is away from the bed. Superimposed in this figure is the near-bed front position. It is clear that as the currents combine in the junction zone there is a distinctive three-dimensional pattern in velocity with fluid along the shear plane moving towards the bed. Further downstream the pattern of vertical velocity becomes less obvious. In the downstream reach, where the lobe-and-cleft pattern in the front shape is observed, vertical velocity can be seen to move towards the bed in the lobes and away from the bed in the clefts. Additionally, the horizontal velocity vectors converge at the clefts and diverge in the lobes. Both observations agree with the

24 earlier work of Cantero et al. [2007].

Bulk Property Behavior

The bulk properties of all test cases are presented in figures 4.7 - 4.13. In these plot, the junction zone extends from xf /L0 = 2 to xf /L0 = 2.71. As observed in figure 4.7, the position versus time collapses over initial density difference and only differs for different values of H0. The same collapse is observed for the partial-depth cases. As the currents reach the junction zone, the current thicknesses increases (figures 4.8 and 4.9) and continues to increase until the head exits the junction zone. From the plot of ∆X (figure 4.10), the location of the maximum thickness is near the head when it enters the junction (∆X approaches 0 near xf /L0 = 2). This is followed by increasing ∆X for the full-depth cases indicating that the elevated plume observed in figure 4.2 remains in the vicinity of the junction as the head continues downstream for some time. For the partial-depth cases, ∆X remains small indicating the head of the current is always the thickest part of the current and no elevated plume is observed (see figure 4.3). For most full- and partial-depth cases the nose height, defined as the vertical dis- tance from the bed to the downstream-most location of the current front, experiences a slight peak in the junction zone. This observation follows that of Ismail et al. [2016] that the main-channel current lifts above the side-channel current in the initial mixing in the junction zone. The front velocity behavior, normalized as a Froude number,F , shows distinctive behavior atypical of straight-channel lock-exchange flows. Figure 4.12 shows F versus nondimensional front position along the main-channel centerline for the full- and partial-depth cases. Initially, the currents experience an acceleration phase to a first peak value followed by deceleration to a constant velocity phase. This behavior is

25 expected since the currents have not yet reached the junction zone so they behave as typical lock-exchange flows. In the junction zone however, the currents in all cases experience a distinctive reacceleration to a second, global peak velocity. This peak velocity always occurs near the downstream junction point (near xf /L0 = 2.71) and is followed by a post-confluence constant velocity phase. This reacceleration is attributed to energy input from the side-channel current in the junction zone. Far downstream, the currents enter the inertial phase where deceleration occurs gradually as a function of time. The maximum near-bed horizontal velocity is assessed and nondimensionalized

q 0 similarly to the front velocity (nondimensionalized by goHo). The maximum near- bed horizontal velocity plot (figure 4.13) show similar trends in the early stages of the release in which the currents undergo a sharp acceleration of horizontal velocity followed by a constant value until reaching the junction zone. In the junction, a small peak in the near-bed horizontal velocity occurs near the upstream junction point (xf /L0 = 2). A third peak is then observed near the downstream junction point (xf /L0 = 2.71). The second peak is attributed to increased flux of dense fluid into the junction zone causing the currents to accelerate, and the third peak is associated with the confinement of the combined mass of dense fluid to the width of the downstream channel. In general, unlike the trends in Froude number, the global maximum near-bed horizontal velocity occurs after the initial acceleration phase, but the junction causes the currents to locally accelerate in all cases. Far downstream in the inertial phase of the front velocity, all full-depth cases show convergence of near-bed horizontal velocity to a value of approximately 0.27 regardless of initial conditions. The partial-depth cases also appear to converge to a value of 0.2 far downstream.

26 4.3.2 Effect of Initial Conditions

The bulk properties are compared quantitatively in terms of front position, head thickness, plume thickness in the junction, nose elevation, front velocity, and max- imum near-bed horizontal velocity along the main-channel centerline. In addition, the distance from the nose to the location of the maximum plume thickness, ∆X , is assessed. Nondimensionalized results appear in figures 4.7 - 4.13. Extracted bulk properties from these figures are presented in tables 4.2 - 4.7. These bulk properties include constant and peak values of head thickness, plume thickness, ∆X , nose eleva- tion, Froude number, and maximum near-bed horizontal velocity. Additionally, the front position at which these parameters occur is listed in the tables. In general, almost no trends appear in the bulk properties as a function of the initial density, ρ0. Most plots (see figures 4.7 - 4.13) show collapse over ρ0. The peak Froude number (see table 4.6) does decrease with increasing initial density for the cases of H0 = 0.5 m, but this trend weakens for H0 = 0.4 m and is nonexistent for

H0 = 0.2 m. This trend also occurs for the partial-depth cases. Trends appear in the bulk properties as a function of the initial channel depth,

H0. It can be seen in the tabulated results that the pre-confluence Froude number is a function of H0. As expected, before the junction effects the flow properties, higher

H0 shifts the first peak Froude number downstream, and the constant velocity phase Froude number is higher.

After the confluence, the front thickness, plume thickness, and ∆X are functions of H0. Both the front thickness and plume thickness peak values do not change significantly with H0, but the peaks occur further downstream with increasing H0.

Additionally, at lower H0, the minimum head thickness after the peak is lower with smaller H0. With higher initial H0, the ∆X results suggest that the thickest part of the current, the plume thickness, remains further behind the nose. This is intuitive since higher H0 should result in a larger head in length and thickness.

27 4.3.3 Effect of Initial Reynolds Number

Data extracted from the results are used to assess trends based on the combined effects of the initial density difference and channel depth using the initial Reynolds q 0 3 2 number defined as Re0 = (g0H0 )/ν , where ρ0 is the initial density of the dense

fluid in the lock, ρa is the ambient fluid density, and ν is the kinematic viscosity of water. Tables 4.2 - 4.7 list the extracted data.

Before the confluence, trends in Re0 appear to be similar to those observed in terms of H0. At high Re0, the initial peak Froude number occurs further downstream for both full- and partial-depth cases, and the constant velocity phase Froude number also increases with higher Re0. Additionally, the first peak head thickness linearly decreases with increasing Re0 for the full-depth cases (figure 4.14). In the partial- depth cases, the first peak Froude number was independent of Re0.

During and after the confluence, many results show some dependence on Re0 but only at certain values for the full-depth cases. Specifically, the minimum ∆X in the junction zone after the first peak shows two modes, one for low Re0 which is constant around 0.2, and another for high values of Re0 around 0.48 (figure 4.15). The second peak value of ∆X has similar bimodal behavior for low and high Re0. However, at higher Re0, the location of minimum ∆X and second peak ∆X both shift downstream slightly with higher Re0. The post-confluence constant Froude number appears to increase with increasing Re0 at lower Reynolds numbers (figure 4.16). This trend diminishes at higher values of Re0 where an approximately constant value of F = 0.56 is observed. The behavior of the nose height along the main-channel centerline for the full- depth cases has the most dependence on Re0. With increasing Re0, the maximum nose height, second constant nose height, and second peak nose height all linearly decrease with increasing Re0. Additionally, the front location at the maximum nose height moves upstream with increasing Re0 (see figure 4.17). Unlike the full-depth

28 cases, the partial-depth cases showed no appreciable dependence on Re0 during and after the confluence.

4.4 Summary

Results and analysis of simulations of density current confluences in a 45◦ junction in a lock-exchange arrangement have been presented. The focus of this study is to describe the behavior of the head of the currents as they form upstream, combine in the junction zone, and progress downstream as a single current. In addition to describing the behavior, implications of changing initial conditions (excess density and channel depth) are included in the analysis. Trends in initial Reynolds number dependence were identified and presented. After the release of the density currents, they formed in a relatively short down- stream distance before reaching the junction zone. The currents combined in the junction zone while they were in the constant velocity phase. They reaccelerated to a global peak value near the downstream junction point with a distinctive rise in thickness in the junction. As the combined current continued downstream, the full- depth cases left an elevated plume of dense fluid in the junction which later subsided. The partial-depth cases showed different behavior in the sense that the thickest part of the current remained near the head at all times, and no elevated plume was ob- served. The reacceleration and post-confluence constant velocity phases are unique to the confluence phenomena versus straight-channel lock-exchange flows indicating the combined current re-initiates after the initial mixing in the junction. A faster and larger current is formed by the combining currents which propagates downstream. When nondimensionalized, all bulk properties of the pre- and post-confluence current collapsed over initial density, ρ0. Some dependence on the initial lock-depth,

H0 was observed. As expected, the front velocity, nondimensionalized as a Froude number, reached a higher constant-velocity phase value for higher values of H0. After

29 the confluence, the front thickness, plume thickness and ∆X showed dependence on

H0. The two initial conditions for each case were combined as an initial Reynolds number, Re0, to assess trends. Before the confluence, higher values of Re0 caused a downstream shift of the peak Froude number for all cases. Also, the pre-confluence peak head thickness linearly decreases with increasing Re0 for the full-depth cases.

After the confluence, most bulk properties showed some dependence on Re0 but with diminishing dependence at high values of Re0. The post-confluence constant Froude number increases with Re0 at lower Reynolds numbers, but the trend diminishes at high values of Re0 where approximately constant values of F = 0.56 is observed.

Increasing Re0 also resulted in higher nose height in the junction, constant nose height after the junction, and post-confluence peak nose height. In general, the input of energy from the side-channel density current results in reacceleration of the current and increase in the current thickness. The combined effect of the initial density difference and channel depth effects the pre-confluence currents as expected, and the post-confluence current in terms of the downstream constant front velocity which increases with increasing Re0. The strongest role of the combined initial conditions is in terms of nose height in the junction zone where the main-channel current rises less at higher Re0. This suggests the effect from the side-channel current in terms of competition in the junction zone becomes less at higher values of Re0. In the next chapter, the study will be expanded to include variable junction angle. The role of the junction angle is isolated, and the combined effect of the main- and side-channel inertia into the junction on the combined current’s bulk properties are assessed.

30 Table 4.1: Case summary. D0/H0 = 1 indicates full-depth cases, and D0/H0 < 1 indicates partial-depth cases.

3 Cases ρ0 (kg/m ) H0 (m) D0/H0 Re0 1 1010 0.2 1.0 27 880 2 1010 0.4 1.0 78 840 3 1010 0.5 1.0 110 190 4 1025 0.2 1.0 43 750 5 1025 0.4 1.0 123 750 6 1025 0.5 1.0 172 940 7 1040 0.2 1.0 54 940 8 1040 0.4 1.0 155 400 9 1040 0.5 1.0 217 170 10 1010 0.4 0.5 78 840 11 1025 0.4 0.5 123 750 12 1040 0.4 0.5 155 400

31 Table 4.2: Constant upstream , peak, minimum, and constant downstream head thickness for each case. Thicknesses are normalized by channel depth and position is normalized by lock length measured from the upstream wall. Empty entries indicate that no constant value was reached.

Pre-Confluence Post-Confluence Case ρ0 H0 D0/H0 Re0 First Peak Position of Upstream Second Peak Position of Minimum after Downstream First Peak Second Peak Second Peak 1 1010 0.2 1.0 27900 0.5831 1.279 0.5618 0.7357 3.086 0.5445 - 2 1010 0.4 1.0 78800 0.5373 1.299 0.508 0.7756 3.511 0.5617 0.5762 32 3 1010 0.5 1.0 110200 0.5297 1.335 0.5073 0.7597 3.579 0.5608 0.5803 4 1025 0.2 1.0 43800 0.578 1.332 0.557 0.7279 3.071 0.5392 - 5 1025 0.4 1.0 123700 0.5502 1.315 0.5327 0.7591 3.345 0.5612 0.578 6 1025 0.5 1.0 172900 0.54 1.353 0.5092 0.764 3.641 0.562 0.5816 7 1040 0.2 1.0 54900 0.5855 1.275 0.559 0.714 2.833 0.5405 - 8 1040 0.4 1.0 155400 0.5098 1.407 0.5078 0.7571 3.277 0.5605 0.5782 9 1040 0.5 1.0 217200 0.5072 1.457 0.5236 0.7572 3.564 0.5601 0.5793 10 1010 0.4 0.5 78840 0.3469 1.345 0.3739 0.5314 2.856 0.4585 0.4264 11 1025 0.4 0.5 123750 0.3475 1.263 0.3732 0.5308 2.846 0.4523 0.4505 12 1040 0.4 0.5 155400 0.3478 1.341 0.3688 0.526 2.953 0.4488 0.4206 Table 4.3: Plume thickness results for full-depth cases. The position is measured from the upstream wall of the domain and is normalized by the lock length.

Case ρ0 H0 D0/H0 Re0 Maximum Position of Maximum Downstream Constant 1 1010 0.2 1.0 27900 0.749 3.383 - 2 1010 0.4 1.0 78800 0.7952 3.733 0.5811 3 1010 0.5 1.0 110200 0.8052 4.199 0.5862 4 1025 0.2 1.0 43800 0.7541 3.425 - 5 1025 0.4 1.0 123700 0.7891 3.695 0.578 6 1025 0.5 1.0 172900 0.7985 4.229 0.5872 7 1040 0.2 1.0 54900 0.7483 3.447 - 8 1040 0.4 1.0 155400 0.7859 3.719 0.58 9 1040 0.5 1.0 217200 0.7889 4.064 0.5904 10 1010 0.4 0.5 78840 0.5314 2.856 0.4546 11 1025 0.4 0.5 123750 0.5308 2.846 0.4552 12 1040 0.4 0.5 155400 0.526 2.953 0.4502

33 Table 4.4: Results of the distance from the current nose to the location of maximum plume height, ∆X . The position is measured from the upstream wall of the domain and is normalized by the lock length.

Case ρ0 H0 D0/H0 Re0 Peak Position of Peak Minimum Position of Second Peak Position of Constant Minimum Second Peak Downstream 1 1010 0.2 1.0 27900 1.84 1.88 0.21 2.18 1.58 3.98 - 2 1010 0.4 1.0 78800 1.97 1.97 0.45 2.42 1.93 4.29 - 3 1010 0.5 1.0 110200 1.97 1.97 0.51 2.48 1.83 4.32 0.73 34 4 1025 0.2 1.0 43800 1.92 1.92 0.21 2.28 1.59 4.13 - 5 1025 0.4 1.0 123700 2.02 2.02 0.44 2.37 1.87 4.22 - 6 1025 0.5 1.0 172900 1.95 1.95 0.51 2.56 1.86 4.43 0.72 7 1040 0.2 1.0 54900 1.81 1.87 0.20 2.17 1.59 4.04 - 8 1040 0.4 1.0 155400 1.75 1.85 0.46 2.53 2.08 4.38 - 9 1040 0.5 1.0 217200 1.96 1.96 0.50 2.47 1.82 4.31 0.69 10 1010 0.4 0.5 78840 1.72 1.72 0.22 1.82 0.39 3.07 0.31 11 1025 0.4 0.5 123750 1.71 1.71 0.23 1.86 0.38 3.02 0.32 12 1040 0.4 0.5 155400 1.72 1.72 0.24 1.91 0.39 2.95 0.32 Table 4.5: Nose height results for full-depth cases. The position is measured from the upstream wall of the domain and is normalized by the lock length.

Case ρ0 H0 D0/H0 Re0 Constant Peak Front Post- Second Peak Front Constant Upstream Position at confluence Position at Downstream Peak Constant Second Peak 1 1010 0.2 1.0 27900 0.104 - - 0.082 - - 0.082 2 1010 0.4 1.0 78800 0.072 0.104 2.667 0.072 0.093 4.175 -

35 3 1010 0.5 1.0 110200 0.066 0.100 2.614 0.065 0.091 4.199 - 4 1025 0.2 1.0 43800 0.104 - - 0.082 0.093 3.190 0.082 5 1025 0.4 1.0 123700 0.072 0.093 2.560 0.072 0.087 4.219 - 6 1025 0.5 1.0 172900 0.066 0.089 2.560 0.066 0.089 4.229 - 7 1040 0.2 1.0 54900 0.104 - - 0.082 0.098 3.297 0.082 8 1040 0.4 1.0 155400 0.081 0.093 2.527 0.072 0.090 4.383 - 9 1040 0.5 1.0 217200 0.066 0.074 2.468 0.066 0.083 4.311 - 10 1010 0.4 0.5 78840 0.072 0.083 2.610 0.062 0.072 3.568 0.070 11 1025 0.4 0.5 123750 0.070 0.083 2.649 0.062 0.070 3.645 0.069 12 1040 0.4 0.5 155400 0.072 0.083 2.718 0.062 0.072 3.942 0.066 Table 4.6: Froude number summary in each phase of flow. Location values are normalized by lock length and are measured from the upstream wall.

pre-confluence post-confluence Case ρ0 H0 D0/H0 Re0 Acc. Phase Position of Constant- Reaccel. Location of Post- Position Peak First Peak Phase Phase Peak Reaccel. confluence Inertial Peak Constant Phase Begins 1 1010 0.2 1.0 27900 0.603 1.127 0.537 0.653 2.755 0.535 3.911 2 1010 0.4 1.0 78800 0.632 1.183 0.565 0.742 2.813 0.563 - 36 3 1010 0.5 1.0 110200 0.629 1.202 0.576 0.764 2.783 0.566 5.098 4 1025 0.2 1.0 43800 0.570 1.215 0.540 0.651 2.813 - 3.664 5 1025 0.4 1.0 123700 0.607 1.316 0.569 0.716 2.781 0.568 - 6 1025 0.5 1.0 172900 0.613 1.353 0.576 0.732 2.813 0.569 5.042 7 1040 0.2 1.0 54900 0.552 1.275 0.541 0.656 2.833 0.545 3.747 8 1040 0.4 1.0 155400 0.589 1.407 0.570 0.705 2.801 0.564 - 9 1040 0.5 1.0 217200 0.601 1.457 0.582 0.693 2.769 0.572 5.081 10 1010 0.4 0.5 78840 0.524 1.154 0.478 0.625 2.733 0.528 4.057 11 1025 0.4 0.5 123750 0.496 1.263 0.480 0.637 2.846 0.495 0.496 12 1040 0.4 0.5 155400 0.489 1.341 0.486 0.607 2.953 0.501 4.326 Table 4.7: Summary of near-bed maximum horizontal velocity. Locations are normalized by the lock length and measured from q 0 the upstream wall. Horizontal velocity is normalized by goHo.

pre-confluence post-confluence Case ρ0 H0 D0/H0 Re0 First Peak Position of Constant Second Peak Position of Third Peak Position of First Peak Second Peak Third Peak 1 1010 0.2 1.0 27900 0.3761 1.127 0.3237 0.3557 1.952 0.3554 3.161 2 1010 0.4 1.0 78800 0.3479 1.183 0.2816 0.2952 1.966 0.3128 3.4

37 3 1010 0.5 1.0 110200 0.3389 1.202 0.2695 0.2824 2.218 0.3105 3.327 4 1025 0.2 1.0 43800 0.4005 1.09 0.3237 0.3518 1.921 0.3572 3.071 5 1025 0.4 1.0 123700 0.3601 1.128 0.2845 0.2963 2.016 0.3136 3.345 6 1025 0.5 1.0 172900 0.3387 1.142 0.2717 0.2982 2.56 0.3074 3.245 7 1040 0.2 1.0 54900 0.3835 1.124 0.3244 0.3593 2.018 0.3538 3.297 8 1040 0.4 1.0 155400 0.3515 1.179 0.29 - - 0.3135 3.277 9 1040 0.5 1.0 217200 0.3415 1.196 0.2725 0.2847 2.209 0.3075 3.315 10 1010 0.4 0.5 78840 0.2758 1.154 0.2341 0.2521 2.008 0.2592 3.173 11 1025 0.4 0.5 123750 0.3211 1.11 0.2588 0.2765 2.01 0.2868 3.02 12 1040 0.4 0.5 155400 0.2801 1.151 0.2358 0.246 2.096 0.2592 2.953 MAIN-CHANNEL CENTERLINE

L0 = 1m b = 0.5m

DOWNSTREAM JUNCTION PINT

LOCK UPSTREAM GATES JUNCTION POINT 45° SIDE-CHANNEL CENTERLINE

= 1m L 0

b = 0.5m

Figure 4.1: Computational grid and domain geometry.

38 Figure 4.2: Typical propagation of the density currents for a full-depth case (case 5). Colors indicate distance from the bed.

39 Figure 4.3: Typical propagation of the density currents for the partial-depth case (case 11). Colors indicate distance from the bed.

40 Base Grid 1s

2s

3s

4s

5s

10s

Figure 4.4: Propagation of the density current for case 9 along the main-channel centerline visualized as isolines of nondimensional concentration.

41 Figure 4.5: Time series of the near-bed front shape for case 5. Each output is sep- arated by 0.77 nondimensional time units. The dash line is the interpreted shear plane.

0 s 4 s 8 s

12 s 16 s 20 s

24 s 28 s 32 s

Figure 4.6: Time series of near-bed horizontal velocity and vertical velocity direction for case 5. Light shading indicates vertical velocity is away from the bed.

42 6

5

Case 1

o 4 /L f Case 2 Case 3 3 Case 4 Case 5 Case 6

Front Position, x Position, Front 2 Case 7 Case 8 Case 9 1

0 0 10 20 30 40 50 60 t*

6

5

o /L

f 4

3 Case 10 Case 11 2

Front Position, x Position, Front Case 12

1

0 0 5 10 15 20 25 30 35 40 t*

Figure 4.7: Nondimensional front position versus time along the main-channel cen- terline for the full-depth cases and partial-depth cases.

43 1

0.9

0.8 o

/H 0.7 f

0.6

0.5

0.4

0.3 Front Thickness, H 0.2

0.1

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x/L f o

1

0.8

o

/H f

0.6

0.4 Front Thickness, H Thickness, Front 0.2

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

Figure 4.8: Head thickness versus front position for full-depth and partial-depth cases. The symbology follows figure 4.7.

44 1

0.9

0.8

0.7

0.6 o

/H 0.5 max Y 0.4

0.3

0.2

0.1

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x/L f o

1

0.8

o 0.6

/H

x

a m

Y 0.4

0.2

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

Figure 4.9: Plume thickness versus front position for full-depth and partial-depth cases. The symbology follows figure 4.7.

45 4 Case 1 Case 2 3.5 Case 3 Case 4 Case 5 3 Case 6 Case 7 Case 8

2.5 Case 9 o

2 deltaX/L 1.5

1

0.5

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

4 Case 10 3.5 Case 11 Case 12 3

2.5 o

2

deltaX/L 1.5

1

0.5

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

Figure 4.10: Distance from the nose to the location of the plume, ∆X , versus front position for full-depth and partial-depth cases.

46 0.25

o 0.2 /H f

0.15

0.1 Nose Height, y

0.05

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x/L f o

0.3

0.25 o

/H 0.2 f

0.15

0.1 Nose Height, y Height, Nose

0.05

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

Figure 4.11: Nose elevation versus front position along the main-channel centerline for the full-depth cases and partial-depth cases. The symbology follows figure 4.10.

47 1

0.9

0.8

0.7

0.6

0.5 Case 1 0.4 Case 2 Froude Number Froude Case 3 0.3 Case 4 Case 5 Case 6 0.2 Case 7 Case 8 0.1 Case 9

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

1 Case 10 Case 11 0.8 Case 12

0.6

0.4 Froude Number Froude

0.2

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

Figure 4.12: Froude number versus front position for the full-depth and partial-depth cases.

48 0.5

0.4

0.3

0.2

0.1 Maximum Near−bed Horizontal Velocity 0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x/L f o

0.6

0.5

0.4

0.3

0.2

0.1 Maximum Near-bed Horizontal Velocity Horizontal Near-bed Maximum 0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

Figure 4.13: Maximum near-bed horizontal velocity versus front position for full- depth and partial-depth cases. The symbology follows figure 4.12.

49 0.7 0.6 a

f 0.5

0.4 Peak Hf = -4E-07Re0 + 0.59 0.3

Firs Peak H Peak Firs 0.2 0.1 0 0 50000 100000 150000 200000 250000

Re0

1.6 f b 1.4

1.2 xf|peak Hf = 8E-07Re0 + 1.25 1

0.8

0.6

0.4

0.2

Front Position at Firs Peak H Peak Firs at Position Front 0 0 50000 100000 150000 200000 250000

Re0

0.7 0.6 c 0.5 Fr|constant = 2E-07Re0 + 0.5351

0.4

phaseFr - 0.3 0.2

0.1 Constant 0 0 50000 100000 150000 200000 250000

Re0

Figure 4.14: Pre-confluence trends in bulk properties with initial Reynolds number.

50

0.6 2.5

X X

0.5 Δ 2 0.4 1.5 0.3 1 0.2 0.1 0.5 Minimum Δ Minimum 0 0

0 50000 100000 150000 200000 250000 Second Peak 0 50000 100000 150000 200000 250000

Re0 Re0

Figure 4.15: Trends in ∆X with initial Reynolds number.

0.7 0.6 0.5 0.4 0.3 0.2

0.1

confluence Constant Fr Constant confluence - 0

Post 0 50000 100000 150000 200000 250000

Re0

Figure 4.16: Post-confluence constant Froude number versus initial Reynolds number.

51 0.14 3

0.12 2.5 0.1 2 y = -1E-06x + 2.7534 0.08 1.5 0.06 y = -2E-07x + 0.1206 1

0.04 Nose Height Nose 0.02 0.5 Peak Nose Height Nose Peak a b 0 Max. at Position Front 0 0 50000 100000 150000 200000 250000 0 50000 100000 150000 200000 250000

Re0 Re0

0.09 0.12 0.08 0.1 0.07 0.06 0.08 0.05 y = -1E-07x + 0.0838 y = -6E-08x + 0.0982 0.06 0.04

0.03 0.04

Nose Height Nose confluence Constant confluence

- 0.02 0.02

0.01 c d Post

0 Height Nose Peak Second 0 0 50000 100000 150000 200000 250000 0 50000 100000 150000 200000 250000

Re0 Re0

Figure 4.17: Trends in nose height with initial Reynolds number.

52 Chapter 5

The Effect of Junction Angle on Density

Current Confluences

This work builds from the work presented in Chapter 4 by investigating the effect of the junction angle on the confluence of two density currents produced by full-depth lock-exchange. The same solver and similar domain is considered with the addition of varying the junction angle to include 30◦, 45◦, 60◦, 75◦, and 90◦ junctions. Trends in the bulk properties are assessed and relationships are established between the bulk properties and all initial conditions in terms of an initial Reynolds number combining the initial density difference, channel depth, and junction angle .

5.1 Model Set-up

The twoLiquidMixingFoam solver in OpenFOAM was used for this work. The reader is directed to Chapter 2 for details on the solver, governing equations, and solution algorithm and Chapter 3 for details on model validation. The initial and boundary conditions are similar to those presented in section 4.1. For each junction angle, the upstream junction point was kept stationary, and the downstream junction point location was adjusted to achieve the desired junction angle (figure 5.1).

53 5.2 Test Cases

A total of 45 tests were run in which each of the five junction angles were tested with variable lock height (H0 = 0.2, 0.4, and 0.5 m) and initial density behind the

3 locks (ρ0 = 1010, 1025, and 1040 kg/m ). A summary of initial conditions for each junction angle can be found in table 5.1. ˆ A parameter combining all initial conditions is introduced in table 5.2 as Re0 = q 0 3 2 Re0[1 + cos(θ)] where Re0 = (g0H0 )/ν is the Reynolds number based on the initial

0 lock-depth, H0, θ is the junction angle, g is the reduced gravity, and ν is the kinematic ˆ viscosity of water. Re0 gives a measure of the relative inertial forces in the main- channel downstream direction entering the junction based on the combination of all initial conditions.

5.3 Results & Analysis

The typical behavior of the variable-angle junction simulations was identical to those presented in section 4.3.1. It was observed that the junction angle plays a role in the bulk properties, but the general behavior of the confluence does not change. Specifically, the currents experience a reacceleration phase in the junction zone, the currents become thicker, and the main-channel nose height rises in the junction.

Figure 5.2 shows the typical current behavior for case 5 for each junction angle (H0

3 = 0.4 and ρ0 = 1025 kg/m ). In general the trends are similar, but with higher junction angle there is a slight rise in the plume thickness in the junction zone. ˆ Trends in the bulk properties and specific effects of the junction angle and Re0 are presented in the following sections. Figure 5.3 shows the near-bed front position in time for case 5 for each junction

3 angle (H0 = 0.4 and ρ0 = 1025 kg/m ). Each line is separated by 0.77 nondimensional time units. For all junction angles, there is a distinct shear plane separating the two

54 density currents in the junction zone before combining in the downstream reach. As the junction angle increases, the main-channel current is squeezed further towards the opposite side of the junction zone, and the side-channel current becomes more pronounced in the junction with little rotation about the junction line while entering the junction zone. Further downstream, there is little difference in shape of the front on the near-bed plane. Nondimensionalized simulation results can be seen in figures 5.4 - 5.6 for the front position versus time, Froude number versus front position, and head thickness versus time for each junction angle. To quantitatively assess the role of the junction angle, ˆ θ, and Re0, extracted data from the plots are shown in tables 5.3 and 5.4.

5.3.1 Effect of Junction Angle

In general, the front position plots (figure 5.4) shows collapse of the curves over initial density and only remains dependent on initial channel depth and junction angle. The most notable difference in terms of junction angle dependence is the acceleration-deceleration which occurs around t∗ ' 5 to t∗ ' 12 depending on the case. The steepening of the front position curves becomes more pronounced at higher junction angles. The implications on front velocity can be seen in figure 5.5. The pre-confluence F does not vary based on junction angle, as expected. The reacceleration peak F , however, results in a higher peak velocity with higher junction angle (see table 5.3). After the post-confluence peak F , higher junction angles also resulted in a lower minimum front velocity before transitioning to the post-confluence constant velocity phase. Although the peak Froude number is dramatically larger at higher junction angles, the effect is short-lived, as the downstream constant F shows no dependence on junction angle. The behavior of the front thickness along the main-channel centerline shows trends with junction angle (see figure 5.6). Again, he results collapse over ρ0 leaving only

55 dependence on H0 and junction angle. With larger junction angles, the peak front thickness increases. However, this trend diminishes at higher junction angles (see table 5.4). Likewise, the plume thickness shows the same behavior. Following the peak head thickness, larger junction angles also showed lower minimum head thickness corresponding to the dip observed in the Froude number after the reacceleration peak. As was seen with the Froude number plot (figure 5.5), the downstream values of head thickness show little dependence on junction angle.

5.3.2 Effect of Reˆ 0

To assess the combined effects of the initial density difference, ρ0, channel depth, H0, q ˆ ˆ 0 3 2 and junction angle, θ, Re0 is defined as Re0 = Re0[1+cos(θ)] where Re0 = (g0H0 )/ν to give a measure of the relative inertial forces in the main-channel downstream direction. ˆ Before the density currents reach the junction zone, trends in Re0 are similar to q 0 3 2 those observed for Re0 = (g0H0 )/ν in Chapter 4. During and after the confluence, the head and plume thicknesses, ∆X , nose height, Froude number, and near-bed ˆ horizontal velocity all trend with Re0. ˆ With increasing Re0, the front and plume maximum thicknesses, and the peak values of near-bed horizontal velocity all decrease (figures 5.7 and 5.8). Also with ˆ increasing Re0, the downstream constant value of ∆X increases for the cases that reached a constant value (figure 5.9). ˆ At lower values of Re0, the nose height and Froude number showed trending be- havior (figures 5.10 and 5.11). The peak nose hight along the main-channel centerline ˆ in the junction zone decreased with Re0 and so did the constant downstream nose height. The peak value of F after reacceleration has clear dependence on θ, but this ˆ trend was less evident as a function of Re0. However, the downstream location at ˆ which the peak F occurs was further upstream at low values of Re0. Additionally,

56 the post-confluence constant F values were lower and the location of the beginning ˆ of the inertial phase was further upstream at low values of Re0. At higher values of ˆ ˆ Re0, these variables all became independent of Re0.

5.4 Summary

This chapter included the results and analysis of simulations of density current conflu- ences at different junction angles. Along with varying the initial density of the dense fluid and channel depth, five junction angles were tested including 30◦, 45◦, 60◦, 75◦, and 90◦ for a total of 45 simulations. Generally, the behavior of the currents in the confluence is similar to that observed in Chapter 4. When nondimensionalized, bulk properties collapsed over initial density and only remained dependent on the channel depth and junction angle. As seen in the previous chapter, the currents undergo a sharp reacceleration in the junction zone in conjunction with a rise in front and plume thickness. This is then followed by a return to typical lock-exchange behavior downstream where the current thickness becomes constant and later decelerates in the inertial flow phase. It was observed that the junction angle plays a critical role in the behavior of the density current in the vicinity of the junction by dramatically increasing the front velocity, nondimensionalized as a Froude number, at higher junction angles. In conjunction with this was a rise in maximum current thickness with higher junction angles. Of particular note is that although the peak front velocity and current thick- ness are larger at larger junction angles, the downstream conditions were independent of the junction angle. The effect of the junction therefore was short-lived after the front continued downstream regardless of junction angle. ˆ A parameter was developed to combine all the initial conditions as Re0 defined q ˆ 0 3 2 as Re0 = Re0[1 + cos(θ)] where Re0 = (g0H0 )/ν . This parameter combines the initial density difference, channel depth, and junction angle giving an indication of

57 the inertial forces entering the junction in the main-channel downstream direction. ˆ Re0 effectively stretched the bulk property data to establish trends in the head and plume thicknesses, near-bed horizontal velocity, distance from the nose to the location of maximum plume thickness (∆X ), nose height, and Froude number. ˆ At larger values of Re0, the peak front and plume thickness both decrease lin- early. Additionally, the position at which those peaks occur shift downstream with ˆ increasing Re0. The peak values of maximum near-bed horizontal velocity inversely ˆ ˆ trend with Re0, and the constant downstream nose height directly correlates to Re0 linearly. The peak nose height in the junction and constant downstream nose height both ˆ become constant at large values of Re0. Finally, although the peak Froude number rose with junction angle, the post-confluence constant velocity Froude number and ˆ inertial phase starting position both become constant at large Re0. These findings suggest that there are significant implications on the flow phenom- ena in the junction zone based primarily on the channel depth and junction angle which is evident in the size and velocity of the current in the junction, but the effects do not carry far downstream via the combined current. This is further supported by the collapse of the Froude number results in the post-confluence current to constant ˆ values at large Re0.

58 Table 5.1: Initial conditions for each junction angle.

Case ρ0 H0 Re0 1 1010 0.2 27 900 2 1010 0.4 78 800 3 1010 0.5 110 200 4 1025 0.2 43 800 5 1025 0.4 123 700 6 1025 0.5 172 900 7 1040 0.2 54 900 8 1040 0.4 155 400 9 1040 0.5 217 200

ˆ Table 5.2: Re0 for each case. ˆ ˆ Case Re0 Case Re0 30-1 52 000 75-1 35 100 30-2 147 100 75-2 99 200 30-3 205 600 75-3 138 700 30-4 81 600 75-4 55 100 30-5 230 900 75-5 155 800 30-6 322 700 75-6 217 700 30-7 102 500 75-7 69 200 30-8 290 000 75-8 195 600 30-9 405 200 75-9 273 400 45-1 47 600 90-1 27 900 45-2 134 600 90-2 78 800 45-3 188 100 90-3 110 200 45-4 74 700 90-4 43 800 45-5 211 200 90-5 123 700 45-6 295 200 90-6 172 900 45-7 93 800 90-7 54 900 45-8 265 300 90-8 155 400 45-9 370 700 90-9 217 200 60-1 41 800 60-2 118 300 60-3 165 300 60-4 65 600 60-5 185 600 60-6 259 400 60-7 82 400 60-8 233 100 60-9 325 800

59 Table 5.3: Froude number results for each case. The second peak and constant Froude number correspond to the post-confluence reacceleration phase.

Pre-Confluence Post-Confluence Case First Peak First Constant Second Peak Second Constant 30-1 0.603 0.536 0.611 0.539 30-2 0.632 0.562 0.659 0.572 30-3 0.629 0.574 0.691 0.577 30-4 0.701 0.684 0.798 0.700 30-5 0.607 0.567 0.658 0.575 30-6 0.612 0.575 0.672 0.578 30-7 0.552 0.541 0.604 0.542 30-8 0.589 0.568 0.648 0.577 30-9 0.601 0.577 0.670 0.580 45-1 0.603 0.537 0.653 0.535 45-2 0.632 0.565 0.742 0.563 45-3 0.629 0.576 0.764 0.566 45-4 0.570 0.540 0.651 - 45-5 0.607 0.569 0.716 0.568 45-6 0.613 0.576 0.732 0.569 45-7 0.552 0.541 0.656 0.545 45-8 0.589 0.570 0.705 0.564 45-9 0.601 0.582 0.693 0.572 60-1 0.603 0.535 0.845 0.535 60-2 0.632 0.569 0.954 0.567 60-3 0.629 0.575 0.948 0.575 60-4 0.570 0.539 0.824 0.539 60-5 0.607 0.567 0.862 0.569 60-6 0.612 0.574 0.850 0.576 60-7 0.552 0.541 0.790 0.542 60-8 0.589 0.569 0.765 0.574 60-9 0.601 0.576 0.837 0.577 75-1 0.554 0.489 1.196 0.492 75-2 0.632 0.565 1.408 0.573 75-3 0.629 0.573 1.446 0.575 75-4 0.570 0.536 1.197 - 75-5 0.607 0.567 0.904 0.572 75-6 0.612 0.574 1.103 0.581 75-7 0.552 0.541 1.158 - 75-8 0.589 0.569 1.098 0.575 75-9 0.600 0.582 1.028 0.581 90-1 0.603 0.538 1.855 0.527 90-2 0.632 0.564 2.018 0.560 90-3 0.629 0.569 1.978 0.571 90-4 0.570 0.541 1.701 - 90-5 0.607 0.563 1.547 0.561 90-6 0.581 0.571 1.455 0.572 90-7 0.552 0.544 1.562 0.525 90-8 0.588 0.567 1.277 0.562 90-9 0.601 0.569 1.280 0.576

60 Table 5.4: Head thickness results for each case.

Pre-Confluence Post-Confluence Case First Peak Constant U/S Second Peak Constant D/S 30-1 0.583 0.552 0.695 0.557 30-2 0.537 0.534 0.747 0.571 30-3 0.530 0.529 0.736 0.575 30-4 0.578 0.557 0.698 0.561 30-5 0.550 0.533 0.747 0.575 30-6 0.540 0.526 0.737 0.580 30-7 0.586 0.550 0.699 0.554 30-8 0.510 0.508 0.739 0.586 30-9 0.507 0.517 0.738 0.581 45-1 0.583 0.562 0.736 - 45-2 0.537 0.508 0.776 0.576 45-3 0.530 0.507 0.760 0.580 45-4 0.578 0.557 0.728 - 45-5 0.550 0.533 0.759 0.578 45-6 0.540 0.509 0.764 0.582 45-7 0.586 0.559 0.714 - 45-8 0.510 0.508 0.757 0.578 45-9 0.507 0.524 0.757 0.579 60-1 0.583 0.562 0.765 - 60-2 0.537 0.537 0.784 0.577 60-3 0.530 0.507 0.771 0.563 60-4 0.578 0.557 0.762 - 60-5 0.550 0.540 0.774 0.577 60-6 0.540 0.506 0.767 0.564 60-7 0.586 0.559 0.760 - 60-8 0.510 0.541 0.768 0.575 60-9 0.507 0.517 0.770 0.570 75-1 0.576 0.587 0.847 - 75-2 0.537 0.534 0.802 0.568 75-3 0.530 0.530 0.797 0.555 75-4 0.578 0.560 0.798 0.530 75-5 0.550 0.540 0.801 0.567 75-6 0.540 0.506 0.792 0.550 75-7 0.586 0.559 0.789 0.544 75-8 0.510 0.509 0.793 0.567 75-9 0.507 0.506 0.796 0.550 90-1 0.583 0.557 0.836 - 90-2 0.537 0.509 0.815 0.549 90-3 0.530 0.508 0.808 0.541 90-4 0.578 0.548 0.815 - 90-5 0.550 0.537 0.814 0.548 90-6 0.540 0.508 0.808 0.544 90-7 0.586 0.559 0.831 - 90-8 0.486 0.507 0.768 0.558 90-9 0.479 0.517 0.807 0.540

61 1.00 1.00 A B

θ

1.00

1.00

Junction Angle, ϴ A B (deg) 30 1.00 4.00 45 0.71 4.29 60 0.58 4.42 75 0.52 4.48 90 0.50 4.50

Figure 5.1: Dimensions for the domain for each junction angle, θ. Linear dimensions are in meters.

62 a) 30° 4s 12s

20s 28s

b) 45° 4s 12s

20s 28s

63 c) 60° 4s 12s

20s 28s

d) 75° 4s 12s

20s 28s

64 e) 90° 4s 12s12s

20s 28s

Figure 5.2: Typical propagation of the density currents for case 5 (H0 = 0.4 and ρ0 = 1025 kg/m3) for each junction angle. Colors indicate distance from the bed.

65 a

30°

b

45°

66 c

60°

d

75°

67 e

90°

Figure 5.3: Time series of the near-bed front shape for case 5 (H0 = 0.4 and ρ0 = 1025 kg/m3) for each junction angle. Each output is separated by 0.77 nondimensional time units.

68 30° 6 (a) Case 1 Case 2 Case 3 5 Case 4 Case 5 Case 6 Case 7

o 4 Case 8 /L f Case 9

3

Front Position, x Position, Front 2

1

0 0 10 20 30 40 50 60 t*

45° 6 (b)

5

o 4

/L f

3

Front Position, x Position, Front 2

1

0 0 10 20 30 40 50 60 t*

69 60° 6 (c)

5

o 4

/L f

3

Front Position, x Position, Front 2

1

0 0 10 20 30 40 50 60 t*

75° 6 (d)

5

o 4

/L f

3

Front Position, x Position, Front 2

1

0 0 10 20 30 40 50 60 t*

70 90° 6 (e)

5

o 4

/L f

3

Front Position, x Position, Front 2

1

0 0 10 20 30 40 50 60 t*

Figure 5.4: Nondimensional front position versus time for each junction angle.

71 30° 1 (a) 0.9

0.8

0.7

0.6

0.5 Case 1 0.4 Case 2 Froude Number Froude Case 3 0.3 Case 4 Case 5 0.2 Case 6 Case 7 Case 8 0.1 Case 9

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

45° 1 (b) 0.9

0.8

0.7

0.6

0.5

0.4 Froude Number Froude

0.3

0.2

0.1

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

72 60° 1 (c) 0.9

0.8

0.7

0.6

0.5

0.4 Froude Number Froude

0.3

0.2

0.1

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

75° 1.5 (d)

1 Froude Number Froude 0.5

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

73 90° 2 (e)

1.8

1.6

1.4

1.2

1

Froude Number Froude 0.8

0.6

0.4

0.2

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

Figure 5.5: Nondimensional front velocity versus front position for each junction angle.

74 30° 1 (a) 0.9

0.8

o 0.7

/H f 0.6

0.5 Case 1 0.4 Case 2 Case 3

Front Thickness, H Thickness, Front 0.3 Case 4 Case 5 0.2 Case 6 Case 7 Case 8 0.1 Case 9

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

45° 1 (b) 0.9

0.8

o 0.7

/H f 0.6

0.5

0.4

Front Thickness, H Thickness, Front 0.3

0.2

0.1

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

75 60° 1 (c) 0.9

0.8

o 0.7

/H f 0.6

0.5

0.4

Front Thickness, H Thickness, Front 0.3

0.2

0.1

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

75° 1 (d) 0.9

0.8

o 0.7

/H f 0.6

0.5

0.4

Front Thickness, H Thickness, Front 0.3

0.2

0.1

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

76 90° 1 (e) 0.9

0.8

o 0.7

/H f 0.6

0.5

0.4

Front Thickness, H Thickness, Front 0.3

0.2

0.1

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Front Position, x /L f o

Figure 5.6: Nondimensional front thickness versus front position for each junction angle.

77

1 5 f

0.8 4 f 0.6 3

0.4 y = -8E-08x + 0.7841 2 Peak H Peak y = 3E-06x + 2.7993 0.2 a 1 b 0 H Peak of Position 0 0 100000 200000 300000 400000 500000 0 100000 200000 300000 400000 500000

푅푒෢0 푅푒෢0 1 5

0.8 4

0.6 3

0.4 y = -3E-08x + 0.791 2 y = 3E-06x + 2.9913

0.2 1 Position of Peak Peak of Position c Thickness Plume d 0 0 Peak Plume Thickness Plume Peak 0 100000 200000 300000 400000 500000 0 100000 200000 300000 400000 500000

푅푒෢0 푅푒෢0

Figure 5.7: Nondimensional trends in head thickness, Hf and plume thickness with ˆ Re0. The front position is measured from the upstream wall of the main-channel.

0.5 0.5

0.4 0.4

h 0.3 0.3

0.2 0.2 y = -1E-07x + 0.3845 Peak U y = -2E-07x + 0.4035 0.1 a ThirdPeak Uh 0.1 b 0 0 0 100000 200000 300000 400000 500000 0 100000 200000 300000 400000 500000

푅푒෢0 푅푒෢0

Figure 5.8: Nondimensional trends in maximum near-bed horizontal velocity, Uh, ˆ with Re0.

78 1

x 0.9 Δ 0.8 0.7 0.6 0.5 0.4 0.3 y = 2E-06x + 0.2363 0.2

Constant Downstream Downstream Constant 0.1 0 0 100000 200000 300000 400000 500000

푅푒෢0 ˆ Figure 5.9: Nondimensional trends in ∆X with Re0.

0.16 0.12 0.14 0.1 0.12 0.1 0.08 0.08 0.06 0.06 Height 0.04 0.04 0.02 Peak Peak Nose Height 0.02 a b 0 0 0 100000 200000 300000 400000 500000 Nose Downstream Constant 0 100000 200000 300000 400000 500000

푅푒෢0 푅푒෢0 ˆ Figure 5.10: Nondimensional trends in nose height with Re0.

79 3.5 3 2.5 2 1.5 1

Location of Peak F Peak of Location 0.5 a 0 0 100000 200000 300000 400000 500000

푅푒෢0 0.8 7 0.7 6 0.6 5 0.5 4 0.4 3 0.3

0.2 2 Viscous Viscous Phase 1 0.1 b PositionStarting of c 0 0 Constant Downstream F Downstream Constant 0 100000 200000 300000 400000 500000 0 100000 200000 300000 400000 500000

푅푒෢0 푅푒෢0 ˆ Figure 5.11: Nondimensional trends in F with Re0.

80 Chapter 6

Summary & Conclusions

Unsteady density current confluences are studied in this work, where two identical density currents initiated by lock-exchange combine in a horizontal, asymmetrical junction before continuing downstream as a combined current. The study focuses on simulations which are validated experimentally. Findings are assessed in terms of the initial conditions including the initial density difference, depth of the channel, and junction angle. The unsteady phenomena was studied through numerical simulation using the continuum mechanics toolbox, OpenFOAM. The solver, twoLiquidMixingFoam, solves the unsteady, incompressible Navier-Stokes equations. Turbulence modeling was han- dled with a large-eddy-simulation approach and Smagorinsk-Lilly subgrid scale clo- sure. In addition to the Navier-Stokes equations, the solver evaluated the species continuity equation to conserve mass of a dissolved species or heat in a separate con- servation equation. The species concentration determines the bulk density of each cell in the computational grid which drives density current flows. The governing equations were solved using the PIMPLE algorithm which merges the PISO and SIMPLE algorithms. PIMPLE takes advantage of the implicit predictor- explicit corrector steps of PISO while incorporating relaxation factors from the SIM- PLE algorithm for smooth solution convergence. To solve discretized equations, the GAMG and Gauss-Seidel smooth solvers were used. Time discretization was handled using the first-order accurate Euler scheme. To validate the numerical model, limited physical experiments were conducted.

81 The 45◦ junction flume was equipped with a gate along each upstream reach of the channel network with salt water (density = 1035 kg/m3) in each lock to a prescribed depth and ambient water downstream of the lock gates. Three experiments were run: two experiments involved a full-depth lock-exchange test with equal initial conditions to verify repeatability of the experiments, and the third test was a partial-depth release with the dense fluid equal to half the depth of ambient fluid. Numerical simulation of the conditions of the physical experiments matchethe results well, thus it was concluded that the numerical model is capable of capturing the flow phenomena in terms of the bulk properties. To ensure the proposed solution grid accurately captured the flow phenomena, the numerical simulations were retested with a refined grid. When plotted together, little difference was observed in the bulk property results between the base grid and refined grid. Once the numerical model and proposed grid were validated, the numerical model was applied to the case of 45◦ junction angles with variable initial density and channel depth. Nine cases of full-depth density currents and three cases of partial-depth density currents were tested. It was observed that when the two currents reach the junction zone, the front velocity and thickness of the current increase in all cases. Once the combined current continues downstream, an elevated plume of dense fluid remains in the junction zone for the full-depth cases, whereas the thickest part of the current is always close to the front position for the partial-depth cases. When the bulk property results are nondimensionalized, no dependence on initial density was observed. The channel depth does play a role in the bulk properties; specifically, the front position is further downstream with increasing channel depth. Additionally, the thickest part of the current remained further behind the nose of the current with higher channel depth. The initial conditions are combined in terms of an initial Reynolds number. It was found that increased initial Reynolds number resulted in higher nose height in the

82 junction. The post-confluence constant front velocity approaches a constant value of 0.56 at large initial Reynolds numbers, but decreases at lower values of Re. The study of density current confluences is extended to cases of variable junction angle. Along with varying initial density and channel depth, five junction angles are tested for a total of 45 simulations. The trends in the bulk properties at each junction angle are similar to those observed for the 45◦ case with little dependence on initial density. The most apparent effect of increasing the junction angle is a higher peak front velocity during reacceleration, and larger front thickness in the junction zone. A parameter is proposed to combine the initial density, channel depth, and junc- tion angle which gives a representation of the inertia entering the junction in the ˆ main-channel downstream direction. It is defined as Re0 = Re0[1 + cos(θ)]. The ˆ peak current and plume thicknesses both are indirectly correlated to Re0, and trend lines are fit to the results. The peak horizontal velocity associated with the influx of energy from the side channel and the peak horizontal velocity associated with con- finement of the combined mass of dense fluid to the width of the downstream channel ˆ ˆ both decrease with increasing Re0. At high values of Re0, the peak nose height and constant downstream nose height both are constant, and downstream front velocity ˆ also becomes constant. At lower values of Re0, the nose heights increase, and the constant downstream froude number decreases. Combining the observations from this work results in several general properties of the role of the confluence in the pre- and post-confluence density currents which can be summarized as follows.

• In all cases, the junction causes reacceleration of the front as well as an increase in current thickness; both the peak front velocity and peak current thickness increase at higher junction angles.

• The peak front velocity always occurs when the front is near the downstream

83 junction point.

• For the range of densities tested, the effect of the junction is independent of the initial density difference when bulk properties are nondimensionalized by the initial conditions.

• The only major distinction between the full-depth and partial-depth cases is that the latter does not leave an elevated plume in the junction zone, thus the front is the thickest part of the current at all times.

• The peak current thickness and peak near-bed horizontal velocity both de- ˆ ˆ crease at larger values of Re0. This should be expected since Re0 increases with decreasing junction angle where the competition in the junction zone is less or- thogonal, thus horizontal velocity components from the main- and side-channels are superimposed.

• The effect of the initial conditions on the confluence behavior is only local at high Reynolds numbers which is demonstrated by the peak Froude number being dependent on channel depth and junction angle in the junction, but the constant post-confluence downstream front velocity remains unchanged with Reynolds number. Furthermore, the near-bed horizontal velocity converges to a single value in the downstream reach regardless of initial conditions.

The work presented here gives a deeper understanding of the role of the junction and initial conditions on the confluence of unsteady density current fronts. With the conclusions and understanding presented in terms the general behavior of the currents and bulk properties, further studies can now be conducted to assess specific problems related to the flow phenomena. These can include detailed studies of mixing in the junction, confluences of steady density current flows, cases with sloping and erodible channel beds, and the effects of the confluence occurring in different flow phases

84 other than the constant velocity phase. With this and further studies, the findings can be applied to specific observations of density currents in natural systems such as salt-wedge flows into coastal channel networks and turbidity current confluences in submarine channel systems.

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