# QUANTIFYING FLOW RESISTANCE IN NATURAL ENVIRONMENTS

• : Ms Word Format
• : 70 Pages
• : ₦3000
• : 1-5 Chapters
•

# Abstract

Quantifying flow resistance is important for understanding and predicting energy and momentum transfer processes in natural environments such as ice-melting in the polar ice sheet and sediment transport in mountain streams. Flow resistance represents the retarding forces experienced by flow and the associated energy loss (dissipated through heat) in these systems. Flow resistance is mainly due to skin friction and form drag. Among many factors, the geometry of flow conduit and channel plays the dominant role. Though important for predicting the ice-sheet dynamics and mountain stream bed evolution, a general methodology for accurate and rapid quantification of flow resistance in natural environments has not been established due to either the inaccessibility of harsh polar environments or the lack of accurate measurements of realistic mountain streams. Many of existing flow resistance predictors are empirical and indirect. To fill this knowledge gap, this thesis work tried to directly quantify the flow resistance in these two geophysical settings using a workflow combing structure-from-motion (SfM) photogrammetry and computational fluid dynamics (CFD) models.

Based on the high-resolution topographic data of a realistic subglacial conduit and three mountain streambeds, a series of CFD simulations were performed. The simulations based on a realistic subglacial conduit surface quantified the bulk flow resistance, in the form of Darcy-Weisbach friction factor, as around 2.41. Additional CFD simulations based on three simplified conduits revealed that cross-sectional shape and size variations of, and sinuosity in, this conduit dominate (95%) the bulk flow resistance, whereas the contribution from surface roughness due to bottom rocks and icy roof is relatively unimportant (5%). This result suggests that most glaciological models, which use surface roughness to quantify resistance and ignore the effects of cross-sectional variation and sinuosity, may significantly underestimate the flow resistance in realistic subglacial conduits.

To evaluate the implications of the CFD simulated flow resistance, an opensource one-dimensional subglacial conduit model, conduitFoam, was developed and

used to model the subglacial conduit dynamics and its dependence on different forcing conditions, such as entrance water head, ice-melting/creep-closure rate, discharge, water velocity, conduit size, and effective pressure. With the CFD simulated flow resistance, the results show that realistic subglacial conduits may have a smaller water velocity and effective pressure but larger conduit size compared to those currently predicted in glaciology models due to the choosing of a much smaller friction factor in the range 0.01-0.5. Real conduits also need longer time to reach a quasi-steady state because they have much larger flow resistance than the ones used in these models. This finding suggests that a re-evaluation of the effect of subglacial flow resistance may be necessary for ice-sheet or climate models.

Applying the same workflow of combining SfM and CFD to mountain streams, we quantified the microtopography of mountain streambeds and directly calculated the flow resistance. The roughness of the streambeds can mainly be represented by the standard deviation of the surface microtopography. A new resistance relationship between the standard deviation and the flow resistance was then established. This new resistance formula links the flow resistance directly to the surface features which are easily quantifiable with data acquired from SfM photogrammetry. With the workflow and the new formula, it is possible to rapidly quantify flow resistance in natural mountain streams which may be hard to access.

This thesis work further tested the applicability of traditional resistance relationship/formulas. CFD simulations with rough pipes reconstructed from different detrending/smoothing methods show that most traditional resistance relationships for rough pipes are still valid when relative surface roughness is less than 20%. Under this scenario, the hydraulic roughness required by traditional rough pipe theories can be estimated by 1.1-1.4 times of the surface roughness. For surfaces with relative surface roughness larger than 20%, direct simulations are necessary to determine the flow resistance.

List of Figures                                                                                                                   ix

List of Tables                                                                                                                    xi

List of Symbols                                                                                                                xii

Acknowledgments                                                                                                         xix

Chapter 1

Introduction                                                                                                              1

1.1                  Motivations and methodology . . . . . . . . . . . . . . . . . . . . .                    1

1.2                 Objective and scientific questions . . . . . . . . . . . . . . . . . . .                    7

1.3                        Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                          7

Chapter 2

Flow resistance in subglacial conduits                                                              10

2.1                       Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                       10

2.2                         Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                          12

2.2.1                  Structure-from-motion . . . . . . . . . . . . . . . . . . . . .                  13

2.2.2      Surface features decomposition and synthetic pipes       . . . . .       14

2.2.2.1              The Centerline of Surface S0 . . . . . . . . . . . . .              15

2.2.2.2              Generating surfaces S1 and S2 . . . . . . . . . . . .              16

2.3                    Computational model . . . . . . . . . . . . . . . . . . . . . . . . . .                    18

2.4                     Resistance theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .                      23

2.5                        Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                         25

2.5.1             Validations with smooth and rough pipes . . . . . . . . . . .             25

2.5.2                    Flow resistance . . . . . . . . . . . . . . . . . . . . . . . . .                    28

2.6                       Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                        30

2.6.1                Important surface features . . . . . . . . . . . . . . . . . . .                 30

2.6.1.1      Cross-sectional variations            . . . . . . . . . . . . . .            31

2.6.1.2     Sinuosity                 . . . . . . . . . . . . . . . . . . . . . . .                  33

2.6.1.3                Surface roughness . . . . . . . . . . . . . . . . . . .                36

2.6.2               Pressure and pressure gradient . . . . . . . . . . . . . . . . .               39

2.7                       Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                        42

Chapter 3

conduitFoam: an open-source subglacial conduit model                              43

3.1                       Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                       43

3.2      One-dimensional subglacial conduit model             . . . . . . . . . . . . . .            46

3.2.1     Governing equations                . . . . . . . . . . . . . . . . . . . . . .                 46

3.2.2                Boundary and initial conditions . . . . . . . . . . . . . . . .                49

3.3                     Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . .                     50

3.4                     Analytical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . .                     51

3.5                        Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                         55

3.5.1              Comparision with analytical solutions . . . . . . . . . . . . .              55

3.5.2         Impacts on hydrodynamics and ice-dynamics of polar ice-sheets 56

3.5.2.1              Effects of flow resistance . . . . . . . . . . . . . . .              57

3.5.2.2       Effects of variations in upstream input discharge      .     59

3.5.2.3             Effects of ice sheet thickness . . . . . . . . . . . . .             62

3.6      Discussion and Conclusions                  . . . . . . . . . . . . . . . . . . . . . .                 63

Chapter 4

Rapid assesment of flow resistance in mountain streams                             66

4.1                       Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                       66

4.2                        Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                         68

4.3                        Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                        70

4.3.1            SfM-based streambed microtopography . . . . . . . . . . . .             70

4.3.2     Surface roughness characterization             . . . . . . . . . . . . . .            73

4.3.3              Computational fluid dynamics model . . . . . . . . . . . . .              75

4.4                        Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                         80

4.4.1                 Grid independence study . . . . . . . . . . . . . . . . . . . .                 80

4.4.2      Roughness scales and complexity             . . . . . . . . . . . . . . .             81

4.4.3               Darcy-Weisbach friction factor . . . . . . . . . . . . . . . . .                83

4.5                       Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                        86

4.5.1       Empirical relationship between flow resistance and bed mi-

crotopography . . . . . . . . . . . . . . . . . . . . . . . . . .                    86

4.5.2             Secondary controls on flow resistance . . . . . . . . . . . . .              91

4.5.3                Distributed flow resistance . . . . . . . . . . . . . . . . . . .                 92

## Chapter 5 Insights into surface roughness and flow resistance in nature 93

5.1                       Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                       93

5.2               Roughness definition and quantification . . . . . . . . . . . . . . . .               96

5.2.1      Surface detrending and roughness definition          . . . . . . . . .         96

5.2.2                               Roughness quantification . . . . . . . . . . . . . . . . . . . . 100

5.2.2.1                 The structure function in cylindrical system . . . . 100

5.2.2.2                         Roughness quantification . . . . . . . . . . . . . . . 101

5.3     Numerical Modeling                                    . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.3.1     Governing equations                              . . . . . . . . . . . . . . . . . . . . . . 106

5.3.2                              Simulation cases and setups . . . . . . . . . . . . . . . . . . 108

5.3.3                                  Numerical schemes . . . . . . . . . . . . . . . . . . . . . . . 111

5.4                           Theories and models for flow resistance . . . . . . . . . . . . . . . . 111

5.5                                       Results and analyses . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.5.1                           Definition-based friction factor . . . . . . . . . . . . . . . . . 115

5.5.2      Pressure-gradient based friction factor                    . . . . . . . . . . . . 116

5.5.3                               Energy-based friction factor . . . . . . . . . . . . . . . . . . 118

5.5.4                             Influence of Reynolds number . . . . . . . . . . . . . . . . . 119

5.5.5                   Geometric roughness and hydraulic roughness . . . . . . . . 120

5.6                                            Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.6.1     On roughness definition                             . . . . . . . . . . . . . . . . . . . . 125

5.6.2                              On friction factor estimation . . . . . . . . . . . . . . . . . . 127

5.6.3                                   Future applications . . . . . . . . . . . . . . . . . . . . . . . 129

5.7                                            Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

## Chapter 6 Summary and future directions  131

6.1     Summary                                          . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.2     Future directions                                     . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Appendix A

Uncertainty of friction factor in subglacial conduits                                    136

Appendix B

Quantifying cross-sectional size                                                                        137

Appendix C

Uncertainty of friction factor in mountain streams                                      140

Appendix D

Resistance formula simplification                                                                    142

Appendix E

Subglacial conduit surface detrending                                                            144

E.1                           Quantifying subglacial conduit centerline . . . . . . . . . . . . . . . 144

E.2                         Defining trend surface of subglacial conduit . . . . . . . . . . . . . . 146

Appendix F Matlab code for rapid quantification of mountain stream flow

reistance                                                                                              150

Bibliography  151

Chapter 1 |

# Introduction

## 1.1 Motivations and methodology

Flow resistance is important for understanding and predicting hydrodynamics in subglacial conduits and sediment transport in mountain streams because flow resistance controls the energy losses due to viscous dissipation and the retardation of fluid motion (Smith, 2014). Thus, accurate quantification of flow resistance in these natural systems is the key to understanding the energy and momentum transfer processes which play an important role for ice sheet dynamics and sediment motion. The two natural systems studied in this work are a mountain glacier in

Svalbard, Norway and three mountain streams with different geophysical settings.

Greenland hydrological system is usually divided into the accumulation zone

(regions higher than the Equilibrium Line Altitude (ELA), Figure 1.1(a)) and the ablation zone (regions below the ELA Line). The ablation zone can further be divided into four subsystems, including supraglacial system, englacial system, subglacial system, and proglacial system (see Figure 1.1 for the whole hydrological system and the appearance of the subsystems). Though the ice-melting water can be delivered to the ocean directly through the supraglacial streams, a large amount of water is delivered through the englacial channels and subglacial systems. In addition, the subglacial channels directly link the ice-sheet and the rock bottom. Thus, their pressurization and lubrication effects are important for regulating the ice-sheet sliding speed. When water flows in the subglacial conduit, the lubrication effect is directly controlled by the relative motion between the moving water and the boundary, i.e., the ice-surface and sediment bottom (Figure 1.1d). This effect is usually quantified as a function of flow resistance. In particular, the flow resistance regulates the ice-sliding and ice-dynamics by changing the basal motion which further changes the water pressure in the subglacial conduit (Bell, 2008; Covington et al., 2012; Flowers, 2015). Therefore, an accurate quantification of the flow resistance in subglacial conduits is important for predicting the ice-sliding and ice-dynamics in Greenland ice sheet.

On the other hand, flow resistance in mountain streams is important for flood routing and river morphology because it controls the bulk velocity and sediment transport (Ferguson, 2007; Katul et al., 2011; Rickenmann and Recking, 2011; Ferguson, 2013). This controlling effect is usually quantified with existing relationships such as the Chezy, Manning, and Darcy-Weisbach equations (Rickenmann

Figure 1.1. The Greenland hydrological system and its important components. a, the Greenland hydrological system, modified from (Chu, 2014). b, the appearance of a supraglacial stream on Greenland, ©Copyright 2011 Los Alamos National Security, LLC

All rights reserved. http://www.lanl.gov/discover/news-release-archive/2018/ January/0122-greenland-meltwater-drainpipe-formation.php. c, the real picture of input water to an englacial channel, credited to Roger Braithwaite from the University of Manchester. https://www.giss.nasa.gov/research/briefs/gornitz_09/. d, the appearance of subglacial conduit, credited to Robbie Shone. http://www.sidetracked. com/ice-caves/.

and Recking, 2011). The key problem of applying these empirical equations to mountain streams is how to determine the coefficients in this formulas, such as the Darcy-Weisbach friction factor, Manning’s roughness coefficient, or Chezy coefficient.

Common approaches for estimating flow resistance include both physics-based and data-based approaches (Ferguson, 2013). By assuming a logarithmic velocity distribution near the channel wall, Keulegan proposed the physics-based flow resistance equation, with the form of

s

### 8       U                        R

= = 5.75log( ) (1.1) f uτ             ks/a

where f is the Darcy-Weisbach friction factor, U and uτ are the bulk velocity and bed shear velocity, respectively. R is the hydraulic diameter, ks is the Nikuradse equivalent grain size, and a is a constant. The difficulty of applying equation 1.1 originates from the difficulty of determining the equivalent grain size ks which encapsulates the effects of both flow conditions and channel geometry into this single parameter. It is a common way to relate the equivalent grains size with characteristic grain sizes such as D50, D84, and D90 (Simões, 2010; Ferguson, 2013). However, both the relationship between ks and characteristic grain size and the approach of calculating characteristic grain size are subject to great uncertainty (Ferguson, 2007).

The data-based approaches include the non-dimensional hydraulic geometry approach and the variable-power-equation (VPE). The hydraulic geometry correlates the non-dimensional bulk flow velocity with the non-dimensional discharge using a power law, which has a general form of

U= kq∗m                                                                         (1.2)

√                                  q

where U= U/          gSD84 and q= q/        gSD843 . k and m are data fitted coefficients.

The application of hydraulic geometry requires calculating the characteristic grain size D84 and fitted coefficients of k and m. The variable-power-equation (VPE) method is another data-based method trying to combine the Manning-Strickler equation for deep water and the roughness layer method for shallow water (Ferguson,

2007). The form of VPE is

s

8                              2         2                   5/3 1/2

= a1a2(d/D84)[a1 + a2(d/D84)        ]                                (1.3)

f

where a1 and a2 denote data fitted constants. d and D84 are water depth and

characteristic grain size (Ferguson, 2007). When the coefficients a1 and a2 are determined from data, the key of applying equation 1.3 is determining the characteristic grain size.

According to equations 1.1-1.3, it is clear that the characteristic grain size is a key parameter for applying these existing formulas. Due to the complexity and diversity of realistic mountain streambeds (see for example Figure 1.2), the definition and calculation of characteristic grain size usually contain great uncertainty. In addition, a large amount of data is required to determine the constant coefficients in these equations. Due to these reasons, it is difficult to estimate flow resistance in complex and realistic mountain streams. Though the data-based approaches are widely used in river research, this thesis work mainly focuses on the physics-based approach because this model is derived from the boundary layer theories and can be easily linked into computational fluid dynamics (CFD) models.

Figure 1.2. The real appearance (a) and point cloud (b) model of the Garner Run, Susquehanna Shale Hills CZO, Pennsylvania. Data is provided by Roman A. DiBiase.

With the technological advancement in structure-from-motion (SfM) photogrammetry, high-resolution topographic data of realistic subglacial conduits and mountain streams become available (Micheletti et al., 2015; Mankoff et al., 2017). With these high-resolution topographic data, computational fluid dynamics can be used to directly simulate and quantify the flow resistance in subglacial conduit and mountain stream environments, two important geophysical environments which are important for the planet Earth. This thesis work focuses on proposing a general approach to parameterize the rough surfaces and flow resistance in subglacial conduits and mountain streambeds.

## 1.2 Objective and scientific questions

Following the introduction in Section 1.1, the objective of this thesis is to answer following questions:

1. What is the bulk flow resistance in a realistic subglacial conduit?
2. What are the important sources contribute to the bulk flow resistance in subglacial conduit?
3. How does the flow resistance in subglacial conduit impact the hydrodynamics within conduits and ice-sheet dynamics?
4. How can we rapidly estimate the flow resistance in mountain streams?

## 1.3 Outline

In order to answer the questions in Section 1.2, this thesis is organized as follows:

Chapter 2 aims to quantify the bulk flow resistance and identify the important contributions of flow resistance. The bulk flow resistance is obtained by conducting CFD simulations based on a 10-meter mm-resolution topographic surface of a real subglacial conduit. The contributions of different sources such as sinuosity, cross-sectional variation, and surface roughness are determined by conducting CFD simulations based on three simplified conduit surfaces.

Chapter 3 aims to study the implications of the simulated flow resistance of a real subglacial conduit by using a new open-source one-dimensional subglacial conduit model. This model is based on the mass conservation of ice and water, momentum conservation of water, modeled ice-melting rate due to basal friction, and an ice-closure rate model. A lake-conduit or moulin-conduit model can be used to impose a pressure boundary condition at the conduit entrance.

Chapter 4 aims to propose a workflow for rapid quantification of flow resistance in mountain streams. The roughness scales on rough streambeds are firstly quantified with the structure function, a relationship between these quantified roughness scales and the CFD simulated flow resistance is then established. This new resistance relationship directly connects the surface roughness scales to the flow resistance. Since high-resolution topographic data can be quickly obtained with newly developed photogrammetry technology, this new resistance relationship thus provides a rapid and accurate way to estimate the flow resistance in realistic mountain streams.

Chapter 5 aims to evaluate the effects of surface detrending on the resistance relationship. Surface detrending is always necessary when the mean surface is not flat. This is true for a realistic subglacial conduit whose cross-section shape and size significantly vary along the streamwise direction. As roughness is defined as the difference between the original surface and the mean trend, the surface detrending directly affects the quantification of surface roughness and associated flow resistance. This chapter evaluates the applicability of structure function in characterizing the surface roughness under different detrending scenarios. The traditional resistance formula for rough pipes is then evaluated based on the quantified surface roughness and CFD simulated friction factor.

Chapter 6 summarizes the findings in each chapter and provides a discussion on future application of the workflow of combining structure-from-motion and computational fluid dynamics models.