numerical simulation of scour

NUMERICAL SIMULATION OF SCOUR AROUND FIXED AND SAGGING

PIPELINES USING A TWO-PHASE MODEL

by

Zhihe Zhao

A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree

Doctor of Philosophy

ARIZONA STATE UNIVERSITY

December 2006

R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

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NUMERICAL SIMULATION OF SCOUR AROUND FIXED AND SAGGING

PIPELINES USING A TWO-PHASE MODEL

Zhihe Zhao

has been approved

September 2006

APPROVED:

, Chair

Supervisory Committee

ACCEPTED:

epartment

)ean, Division of Graduate Studies


ABSTRACT

A key aspect of design and the maintenance of underwater pipelines is the assessment of

local scour and its propagation. Scouring around objects placed on a sandy bottom is very

complex because it involves two-phase turbulent flows and a myriad of sediment

transport modes. This dissertation addresses two principal configurations of scour around

pipelines in two parts. First, clear-water scour around a long fixed pipeline placed just

above a non-cohesive sandy bed is numerically simulated. Second, live-bed scour around

a fixed pipeline and scour below a sagging pipeline are investigated. These two

simulations are conducted by using an Eulerian two-phase model that implements Euler-

Euler coupled governing equations for fluid and solid phases and a modified k - s

turbulence closure for the fluid phase, the modeling system being a part of software

FLUENT. Both flow-particle and particle-particle interactions are considered in the

model. During the simulations, the interface between sand and water is specified using a

threshold volume fraction of sand, and the evolution of the bedforms is studied in detail.

For clear-water scour around a fixed pipeline, the predictions of bedform evolution are

in agreement with previous laboratory measurements. Investigations into the mechanisms

of scour reveal that three sediment transport modes (bed-load, suspended-load and

laminated-load) are associated with the scour development. While some previously

proposed scour development formulae for cylindrical objects are in good agreement with

the simulations, scour predictions based on some operational mine-burial models show

disparities with present simulations.

m


For investigations of live-bed scour and scour under a sagging pipeline, the flow and

pipeline evolve in two steps: (1) the local live-bed scour around the pipeline developed

around a fixed pipeline; and (2) the pipeline is lowered to the scour hole in controlled

fashion until it reaches the bottom of the scour hole. Three sagging velocities are

simulated, and predicted scour profiles agree well with the laboratory data. General

characteristics of flow fields, including turbulence, suspension of particles and sediment

transport, are described paying attention to their dependence on pipeline sagging. Scour

profiles simulated are also in agreement with a LES-based numerical study reported

earlier.


ACKNOWLEDGEMENT

I would like to extend my thanks to my advisor Professor Harindra J.S. Fernando for

his guidance, patience and support on my research work. I specially thank him for the

many hours he spent going through my drafts and giving me resourceful thoughts to

improve my work. I also thank him for showing me what it takes to be successful in the

graduate studies and beyond.

I am thankful for Professor Don L. Boyer for his insightful suggestions on my research

work. In addition, it is also a privilege for me to have Professor Ronald Calhoun,

Professor Kangping Chen and Professor Mark Schmeeckle as committee members.

Finally, I appreciate the administrative supports from Ms. Gabrielle Stidham, Ms.

Jennifer McCulley and Mr. Richard Hampton.

This work was funded by the U.S. office of Naval Research through its Coastal

Geosciences and Mine Burial Programs.

v


TABLE OF CONTENTS

Page

LIST OF FIGURES................................................................................................................ ix

CHAPTER

1. SIMULATION OF SCOUR AROUND A FIXED PIPELINE.................................. 1

1.1. Motivation.............................................................................................................. 2

1.2. Overview of Scour Models...................................................................................2

1.3. Overview of Using Two-phase Models on Sediment Transport Calculations..4

1.4. Summary.................................................................................................................6

2. MATHEMATICAL DISCRETION OF THE TWO-PHASE MODEL................... 8

2.1. Governing Equations...........................................................................................8

2.2. Turbulence Closure for Fluid Phase................................................................ 10

2.3. Turbulence for Solid Phase............................................................................... 12

2.4. Transport Equation for Granular Temperature.................................................13

3. NUMERICAL SIMULATION AND VALIDATION: FIXED PIPELINES 14

3.1. Mao’s Experimental Set-up.............................................................................. 14

3.2. Numerical Configuration..................................................................................16

3.3. Simulation with Fluent.......................................................................................18

4. RESULTS AND DISCUSSION: SIMULATION WITH A FIXED PIPELINE....23

4.1. Clear-water Scour Simulation...........................................................................23

4.2. Scour Depth....................................................................................................... 27

4.3. Sediment Transport Modes............................................................................29

vi


CHAPTER Page

4.3.1. Bed-load, suspended-load and laminated-load..................................... 29

4.3.2. Sediment velocity.....................................................................................33

4.3.3. Calculation of bed-load and suspended-load............................................35

4.4. Comparison with NBURY and DRAMBUIE Models.................................... 38

4.4.1. Formulation of NBURY and DRAMBUIE models................................. 38

4.4.2. Comparison with NBURY and DRAMBUIE models..........................39

4.5. Conclusions: Scour Under fixed Pipelines.....................................................43

5. SIMULATION OF SCOUR BELOW A SAGGING PIPELINE.............................45

5.1. Background.........................................................................................................45

5.2. Overview of Sagging Pipeline Studies........................................................... 48

5.3. Summary............................................................................................................ 50

6. TWO-PHASE MODEL, SIMULATION AND VALIDATION: SAGGING

PIPELINES.................................................................................................................. 52

6.1. Fredsoe et al.’s Experimental Set-up................................................................. 53

6.2. Numerical Configuration.................................................................................... 55

7. RESULTS AND DISCUSSIONS: SAGGING PIPELINES.................................... 60

7.1. Live-bed scour around a fixed pipeline........................................................... 60

7.2. Scour around a sagging pipeline........................................................................62

7.3. Comparisons with Cheng and Li’s Simulation.................................................68

7.3.1. Vortex shedding in scouring process.......................................................68

7.3.2. Comparison with Cheng and Li’s simulation...................................... 69

vii


CHAPTER Page

7.4. Sediment Transport............................................................................................ 72

7.5. Conclusions: Scour under Sagging Pipelines.................................................81

REFERENCES..................................................................................................................... 83


LIST OF FIGURES

Figure Page

Fig 1. Numerical configuration for the simulation. X is in the streamwise direction, Y in

the cross-stream direction and 8s the thickness of the sand layer....................15

Fig 2. Mao’s physical experiment........................................................... 16

Fig 3. The Grid for the two-phase model calculations................................................... 17

Fig 4. The contours of volume fraction of the sediment at t = 0. Note the introduction of

an interfacial disturbance (arrow) at the beginning...........................................19

Fig 5. A schematic diagram showing the interface, bed-load, suspended-load and

laminated-load layers. Here ds is the diameter of the sediment particles, 77 the

depth of the water, 70 the level where the sediment volume fraction is at 0.5

21

Fig 6. An example of the grid that was used for the flow model. The bed profile is

specified as the contour with a s =0.5 obtained from the previous calculation

step conducted with the two-phase model..........................................................22

Fig 7. Bed profiles during the development of scouring...................................................25

Fig 8. Normalized turbulent intensity at a location 1cm above the bed; ws is the particle

settling velocity. The inset shows the location of turbulence measurements...26

Fig 9. The time evolution of scour depth in simulations and comparison with

equation (10)...................................................................................................... 28

Fig 10. Patterns of sediment motion from a flat bed, redrawn based on [8]................. 30

I X


Figure Page

Fig 11. Vectors of the sediment velocity u /U x at (a) t = 10 minutes (b) t = 100 minutes

(c) t = 200 minutes............................................................................................... 31

Fig 12. Normalized turbulence intensity profiles at various downstream locations (X

= 0.0, 0.1, 0.2, 0.3,0.4, 0.5, 0.6m) of the cylinder wake at t = 200 min.

a s = 0.5 is chosen as the bed profile (Fig.7). Note the scale for J u J / w s on

the upper left comer............................................................................................. 33

Fig 13. A. schematic diagram for the formulation of recirculation in the sediment zone.

The sediment movement is driven by interfacial shearing force and pressure

gradient................................................................................................................. 34

Fig 14. The bed load qb and suspended load qs . The arrow indicates the position of

the maximum height of the sand mound.............................................................37

Fig 15. Comparison of numerical calculations of the maximum scour depth with

those predicted by the NBURY model......................................................... 40

Fig 16. Comparison of present numerical results (for two sand-layer depth cases) with

DRAMBUIE model predictions. The inset shows DRAMBUIE model reaches

equilibrium after 1400 minutes........................................................................... 42

Fig 17. (i) The three-dimensional pipeline sagging process [14]. (ii) A Sketch for the

scour [38]. (iii) The variation in the position of pipeline at cross-section

B-B [14]. (iv) An underwater pipeline ready to be deployed in Port Kembla

harbor ...................................................................................................................46

x


Figure Page

Fig 18. Freds0e’s Experimental set-up that mimicked the sagging pipeline [14]..........55

Fig 19. The numerical configuration for the simulation. X is in the streamwise

direction and Y the cross-stream direction..................................................... 56

Fig 20. The contours of volume fraction a of the sediment at t = 0. Note the

introduction of an interfacial disturbance (arrow) at the beginning...............57

Fig 21. An example of the grid that was used for the flow model. The bed profile is

specified as the contour with a s - 0.5 obtained from the previous calculation

step conducted with the two-phase model........................................................ 58

Fig 22. An example of the grid that was used for the two-phase flow model................ 58

Fig 23. Bed profile after the development of scouring around a fixed pipeline for

60 minutes.........................................................................................................61

Fig 24. Comparison of the scour profiles between the present study and Fredsoe et al.

(1988)’s measurements before the sagging starts....................................61

Fig 25. Comparison of bed profiles between the present study and Fredsoe’s

measurements at Vp = l.Omm/min, 3.1mm/min and 12.4mm/min............... 63

Fig 26. Sagging process with Vp = 1.0 mm / m in ............................................................. 65

Fig 27. Maximum scour depth development at V = 1.0 mm/min, 3.lmm/min and

12.4m m /m in.................................................................................................... 67

Fig 28. Comparison of bed profiles between the present study and Cheng and Li’s

study when the pipe reaches the bottom of the scour sole atVp =1.0 mm/min,

3.lmm/min and 12.4mm/min............................................................................71

xi


Figure Page

Fig 29. Comparison of the scour profiles before sagging between the present study

and that of Cheng and Li, and the experimental results, before the sagging

starts............................................................... ...................................................... 72

Fig 30. Vectors of the sediment velocity u /U m during the sagging process with the

sagging speed 3.lmm/min. The pipeline’s center is located at (0, 0.54D), (0,

0.34D) and (0 ,0.085D) separately (see Figurel9)............................................ 73

Fig 31. Sediment transport rate through the gap underneath the pipe during the sagging

process.................................................................................................................. 74

Fig 32. Average Flow velocity (average velocity = volume/gap) in the gap between the

pipe and interface...............................................................................................75

Fig 33. Sediment transport (bed-load and suspended load) above the bed surface........ 77

Fig 34. Normalized turbulence intensity inside the gap between the lower side of the

pipe and the scour hole. The sagging speed is 3.lmm/min...........................80

Fig 35. Normalized turbulent intensity at a location midway between the cylinder and

the sand layer gap, as a function of downstream distance for

Vp = 3.1/wm/min.................................................................................................81

xii


1. Simulation of scour around a fixed pipeline

1.1. MOTIVATION

Continuous scouring around pipelines under the action of waves and currents has an

enormous influence on the structural stability of pipelines and their environs. An

understanding of scouring processes and our ability to predict scour around pipelines,

therefore, are important in the design of offshore pipelines [44]. Scouring around objects

placed on a sandy bottom is very complex because in most cases it involves two-phase

turbulent flows and various sediment transport modes. The interface between water and

sand bed is also intricate [29], wherein the flow alters the bedform, which, in turn, affects

the flow. Added to this complexity are the flow-particle and particle-particle interaction

mechanics. Therefore, the Navier-Stokes equations as well as pertinent turbulence closure

schemes need to be properly modified to account for the ensuing complex phenomena.

The goal of the present study is to simulate scour around long cylindrical objects using an

Eulerian two-phase model with the hope of understanding scour around and the burial of

antiship mines that typically have kindred shapes; these mines are usually placed in the

ocean bottom. This study is part of an integral team effort of the Mine Burial Prediction

(MBP) program sponsored by the U.S. Office of Naval Research, of which the long range

goal has been to develop mine scour burial models that incorporate dynamic coupled

environmental processes, seafloor material properties, and different mine types. Currently

available operational mine burial prediction models are known to perform poorly, in view

of which it deem necessary to investigate the efficacy of sour formulae used in these

models [41]. Over the past two decades, a large number of numerical models have been


developed for scour predictions, but the present effort is built upon more recently

developed two-phase flow theory.

1.2. OVERVIEW OF SCOUR MODELS

Mao [28] applied a modified potential flow theory and sediment continuity equation to

simulate scour below long cylinders placed on a sandy bottom subject to a mean current.

The model predictions were compared with complementary laboratory experiments, and

the potential flow model was able to simulate the upstream portion of the scour hole

satisfactorily. Li & Cheng [23] also developed a scour model based on the potential flow

theory. Instead of using an empirical sediment transport formula, they calculated the

equilibrium scour pit size by assuming that the bottom shear stress everywhere on the

seabed is equal to or less than the far field shear stress when the equilibrium state is

reached. A boundary adjustment technique based on Newton-Raphson method was

utilized for simulations. Their model also predicted the approximate upstream scour

depth reasonably well, but failed to give correct predictions for the downstream part of

the scour hole as a result of the limitations of potential flow theory. Later, the same

authors [24] solved flow equations by employing the Smagorinsky sub-grid scale (SGS)

closure. The predicted equilibrium scour hole agreed well with the experimental results.

This approach, however, solely relied on the above-mentioned assumption on the bed

shear stress, and it only produced equilibrium scour profiles, not the scour evolution.


Leeuwestein et al. [22] used a k - s turbulence model coupled with a sediment

transport equation to simulate scour around pipes. In this study, only the bed-load

transport was considered first, and in this case the main sediment transport occurred by

ripples. This prediction could not be corroborated by laboratory experiments. Suspended

sediments were then included in modeling, whence the ripples disappeared, indicating

that bed-load transport alone is not sufficient for representing complex sediment transport

processes around solid objects. Brerrs [3] utilized a finite element method to solve the

RANS equations with k - s closure, while simultaneously solving a sediment transport

model that included both the bed-load and suspended-load transports using a finite

difference method. The overall agreement between the predicted and measured scour

evolution of Mao [28] was good. However, the scour development almost stopped after

100 minutes into the simulation, although the measurements show continuous scouring

even after 300 minutes [28]. In addition, during the 6,000 bed profile updates conducted

in Brors' simulations, problems were often encountered with regard to numerical

instability of the bed update scheme as a result of the non-linear bed-load formula used.

Liang et al. [26] performed simulations of scouring around pipelines using a similar

approach. Two turbulence models, a standard k - s model and the Smagorinsky SGS

model, were applied. To avoid the appearance of unrealistically sharp irregular scour

profiles and numerical instabilities during their calculations, a special smoothing

technique, known as the sand-slide model, was employed in the simulation. Simulations

with the SGS model showed intense vortex shedding dining scour, but scour profiles

obtained with the k - s model were more realistic when compared with Mao's


measurements. Therefore, Liang et al. recommended the use of a standard k - s model

for scour predictions.

Dupuis & Chopard [10] proposed a Lattice Boltzman Method to simulate scour around

pipelines. In this method, fictitious fluid and sediment particles moved on a regular lattice

synchronously at discrete time steps, and time-dependent erosion processes involved

were simulated. However, only a portion of the equilibrium scour hole could be

quantitatively compared well with laboratory measurements. The model needed to be

“tuned” to match various other cases, thus limiting its utility as a robust predictive tool.

Ali & Karim [1] used CFD software FLUENT to predict the three-dimensional flow and

bed shear stress over a rigid bed. By employing experimental data and the one

dimensional sediment continuity equation, they derived the variation of maximum scour

depth with time as a function of the dimensionless bed-shear stress and streamwise

distance. Field measurements of scour around bridges were also applied to further verify

the latter result. Since the numerical simulation was only limited to a rigid bed, the

scouring was not simulated explicitly.

1.3. OVERVIEW OF USING TWO-PHASE MODELS ON SEDIMENT TRANSPORT

CALCULATIONS

In recent years, two-phase models, which consider the dynamics of particle and fluid

phases as well as interactions thereof, have been employed for sediment transport

calculations in the framework of Navier-Stokes equations. Such models predict sediment

transport from somewhat more fundamental (though modeled) dynamical equations,


thereby avoiding the use of purely empirical sediment transport formulae. Such formulae

are replete in literature and have been found to be case dependent, thus limiting their

general use to cover a broad range of flow configurations. The two phase formulations,

on the other hand, are developed based on more fundamental concepts, though naturally

some parameterizations are required for closure. As such, such models are expected to

have more general applicability to a range of problems.

With regard to two-phase models, Yeganeh et al. [48] used an Euler-Lagrange coupled

two-phase model to simulate bed-load transport under high bottom shear. Although the

experimental results have shown the existence of a three-layer type velocity profile, the

model produced only a two-layer velocity profile. The authors ascribed this discrepancy

to the neglect of inter-particle collisions in the model. Hsu et al. [18] employed a two-

phase model to simulate suspended sediment transport, and demonstrated the ability of

such models to predict the time-averaged concentration under a range of conditions.

Greimann [16] employed a two-phase model to compute the average velocity of bed-load

and suspended-load sediments in a laboratory flume under two-dimensional uniform flow

conditions. To calculate the coefficient of momentum loss a particle would experience

when it is in contact with the bed, a critical Shields number was specified based on the

particle shape and bed characteristics; the measured and calculated sediment velocities

showed a reasonably good agreement. In this study, the two-phase flow equations were

used to calculate the velocity and concentration profiles of the sediment phase only.

Greimann argued that there was no sufficient experimental data or analytical

understanding of particle-turbulence interactions to develop a reliable two-phase model


for the flow phase. Wanker et al. [43] calculated sedimentation and sediment transport

using an Euler-Euler coupled two-phase model. The numerical model predicted the

movement of a sand mound well, and they concluded that the bedforms are dependent

mainly on the momentum exchange and particle-particle interaction terms.

In the present study, an Eulerian two-phase model embedded in FLUENT software is

employed to simulate scour around pipelines. The aim is to evaluate the model efficacy

using available benchmark data and, if successful, to use the model to educe important

information on flow dynamics, especially those that could not be conveniently obtained

with available laboratory techniques. The flow-particle interaction and particle-particle

interactions are considered in the model formulation. Each of the two phases (solid and

fluid) is described using appropriately modified Navier-Stokes equations, and coupling

between the phases is achieved through pressure and an interphasial exchange term. For

the solid phase, Boltzman's kinetic theory for dense gases is modified to account for the

inelastic collisions between particles. In order to include the effects of granular friction

between particles for the cases of highly concentrated beds, the frictional viscosity

derived from plastic potential theory is used [32]. The simulation results are validated

using experimental data available in literature.

1.4. SUMMARY

It is well known that there is no general universally accepted formula to quantify

sediment transport over a range of conditions. The sediment transport rate is one of the

most important characteristic for the two-phase flow motion. Finding an expression for


the transport rate has fascinated so many scientists. Consequently, since the first bed-load

transport formula by Du Boys in eighteen century (especially in the past three decades),

numerous transport formulae have been produced by various authors. Among them, some

representatives are E.Meyer-Peter, R. Muller, R.Bagnold, H.A. Einstein and M.S.Yalin

[47].

Also, numerical results based on available sediment transport models tend to be

sensitive to the selection of the sediment formula. In the present simulations, however,

there is no need for the selection of an empirical formula, given that such transport is

handled using dynamical equations, underpinned to the extent possible by fundamental

flow and sediment interaction mechanics. The novel feature of our simulations is the use

of two-phase flow theory to compute scour below a pipeline placed transverse to the

flow. Although two-phase flow theory has been applied for sediment transport

calculations, it has not yet been applied for simulating scour. In the latter case, the

problem is more complex and due consideration should be given to the evolution of bed

profiles.


2. Mathematical description of the two-phase model

The mathematical model used in this simulation is an Eulerian two-phase model. It

assumes that the sediment-laden flow consists of solid s and fluid / phases, which are

separate, yet they form interpenetrating continua. The space occupied by each phase is

represented by the volume fraction a ( 0 < a <1). The laws for the conservation of mass

and momentum are satisfied by each phase individually. Coupling is achieved through

pressure and interphasial exchange coefficients. A symmetric drag model is employed to

describe the interaction between phases. In this Eulerian two-phase model, the equations

for the two phases are solved in an Eulerian frame. The model details are described in

[12], and for completeness, the essentials of the model development are outlined below.

Note that the model is intricate and thus an understanding of the fundamentals of the

parameterizations employed and implementation of various model components are

essential parts of the model usage.

2.1. GOVERNING EQUATIONS

The continuity equations for both the fluid / and solid s phases take the form:

^ - ( a tp t) + V *{a tp tvt) = 0, (1)ot

where t = s , f and a f + a s = 1; a f , a s = volume fraction for water and sediment and

p f , p = mass density of water and sediment, respectively.

The studies of the dynamics of a single particle in a fluid have identified the following

important forces: the static pressure gradient; the solid pressure gradient, which is the

normal force due to particle interactions; the drag force, caused by the velocity


differences between the two phases; and the viscous force and the body forces. Other

forces, such as the virtual mass force and Basset force, are assumed negligible [4]. The

momentum equations for the fluid and solid phases, respectively, are [11]:

Q =

— (af p f vf ) + V * ( a f p f vf vf ) = - a f VP + V * T f + a f p f g + K sf(vs - v f ), (2)

~ ( a sp svs) + V • (asp svsvs) = - a sVP - VPS + V • r s + a sp sg + K fi(vf - v s), (3)

in which vf , v s= the mean-flow velocity for flow and sediment; P = pressure shared by

t h e tw o p h a s e s ; r s = s t r e s s t e n s o r f o r t h e s o l i d p h a s e =

2 -

a sp s(yvs +Vv,s) + a s{Xs - —p sy7-vsI \ t f = stress tensor for the flu id phase =

a f p f (Vvf + V v J ) ; / = the identity te n so r;^ = bulk viscosity o f the sediment =

4 0—a ,p .d ,g 0.s(l + 0 (—~)U2 ’ Soss= radial distribution function, which is interpreted as 3 n

a 1the probability o f a particle touching another particle = [ l - ( — -—)3] ‘ ; a smia =

"̂s.max

maximum value of the sediment volume fraction = 0.63; © ,= granular temperature,

which is proportional to the kinetic energy of the fluctuating particle motion; eJS =

restitution coefficient; g = gravitational acceleration; ds = diameter of sediment; p f =

shear viscosity of water; p s = shear viscosity of sediment, p s = /jscol + ns kin + jus Jr, where

Pscoi= collisional viscosity = - a sPsdsg0ss{ 1 + e^X— )1/2, ^ = frictional viscosity =5 ’ n


“ d A » = U n e tic viscosity- 5 i M S i [l4. i (1 + e )(3e _ l)a g ]; I 1D = the2 i]I2D 6(3 - e „ ) 5

second invariant of the deviatoric stress tensor; Ps = solid pressure =

asPs®s + (1+ea)a2g0>11®, '■> $ = internal friction angle; K sf(= K fs) = interphasial

momentum exchange coefficient = CD(— U|v, -v ,}; CD= drag fimction^ & s Vrs

Re -- ,=(0.63 + 4.8(— -) 2) ; Res = relative Reynolds number between phase / and phase s =

^ r , s

Pf ds |v, - V / | .

PfVrs = terminal velocity correlation for solid phase

=0.5(A - 0.06 Res + J(0.06 Ref )2 + 0.12Re, (2B - A) + A2), where A = a 4/ 4, B = 0.8a}'28 for

a , < 0.85 and A = a } 14 ,B = a l 65 for a f > 0.85 .

2.2. TURBULENCE CLOSURE FOR FLUID PHASE

Predictions for turbulent quantities for the fluid phase are obtained using a standard k - s

model (Launder and Spalding [21]), supplemented by additional terms that take into

account the interfacial turbulent momentum transfer. The Reynolds stress tensor for the

2fluid phase / is: r / + p f P t j V - U f ) I + p f +VC/J), w h e re !/3 /

klthe phase-weighted velocity, /ut f = turbulent viscosity = — , C^ =0.09. The

ef


predictions for the turbulent kinetic energy kf and its rate of dissipation s f are obtained

from the following transport equations:

— (af p f kf ) + V» ( a f p f Uf kf ) = V » ( a f - y - V k f ) + a f Gk J - a f p f ef +a f p f U kf , (4)at <rk

j t (af pf sf ) + 'V»(af pfU/ £f ) = V»(a/ ^ - W £ f ) + af ^-(CuGkJ- C2epf sf ) + a / p/ IlSf- (5)s f

Here FI* and Y[£f represent the influence of the solid phase s on the fluid phase / .

n t/ = (ksf - 2 kf +vsf-vdr),a f P f

^ £/ = Cse ~k~^k/ ’Kf

where vdr = the drift velocity = - D tsf{— -— V a s ----- — V a , ) ; v , = the relative’ <*#<*/

velocity between fluid phase and solid phase; Dtsf = the binary turbulent diffusion

coefficient (see definition in Sec. 2.3); <rsf = 0.75; ksf = the covariance of the velocities

of the fluid phase / and the solid phase s (see definition in Sec. 2.3); K fs = the interphase

momentum exchange coefficient (see definition in Sec. 2.1); Gk f = the production of

turbulent kinetic energy in the flow; Cle =1.44; C2s =1.92; C3e =1 .2 ; crk =1.0 and

<r£ = 1.3.


12

2.3. TURBULENCE FOR SOLID PHASE

To predict turbulence in the solid phase [33, 34], Tchen’s theory [17] on the dispersion of

discrete particles in homogeneous and steady turbulent flows are used. Dispersion

coefficients, correlation functions, and turbulent kinetic energy of the solid phase are

represented in terms of the characteristics of continuous turbulent motions of the fluid

phase based on two time scales. The first time scale is relevant to the inertial effects

acting on the particle, which is represented by rFsf = a sp f K~l{-^J- + Cr ), where Cv =Pf

0.5 = the added mass coefficient. The second one is the characteristic time of correlated

turbulent motions or eddy-particle interaction time, which is written as

-i 2 3 k fr =rt / [l + C ^ 2] 2, where £ = Vr / J —kf , rt f = —Cfl— = a characteristic time of

V 3 2 s j

energetic turbulent eddies, Vr = the averaged value of the local relative velocity between

a particle and the surrounding fluid, and Cp = 1. 8 - 1.35 cos2 6 . Here 6 is the angle

between the mean particle velocity and the mean relative velocity. The ratio between

these two characteristic times is written as

(6)T F ,s f

The turbulent kinetic energy for the solid phase ks is as follows

b2 + rj.ks =k f ( - £■), (7)

1 + 7,/

and the eddy viscosity for the solid phase is specified as


13

(8)

where b = (\ + CY) (ps / pf + C„)“' and ksf is the covariance of the velocities of the fluid

phase.

2.4. TRANSPORT EQUATION FOR GRANULAR TEMPERATURE

The granular temperature ©s for the solid phase describes the kinetic energy of random

motions of sediment particles. The transport equation derived from the kinetic theory

takes the form [12]:

where (~PSI + r s) : Vvs= the generation of energy by the solid stress tensor, k@ = the

energy dissipation rate within the solid phase due to collisions between particles;

(f>fs = -3 K fs<ds , the transfer of the kinetic energy of random fluctuations in particle

velocity from the solid phase s to the fluid phase / .

diffusion coefficient (41 - 3 3 7 j ) ? i a s g 0^ ]

diffusive flux of granular energy, and rj = —(l + ess);

Y& f" )g°'" psa]®2s , the collisional dissipation of energy, which represents the


3. Numerical simulation of scour: Fixed Pipelines

Here, simulations are conducted with a spatially fixed pipeline to study scour around it.

The flow configurations in the experiments of Mao [28] are used, given that the data of

these experiments have been utilized many times for benchmarking numerical codes [3,

10, 23-24,26].

3.1. MAO’S EXPERIMENTAL SET-UP

A photograph of Mao's experiment is shown in Figure 2a and a schematic of the

experiment is shown in Figure 2b. A pipe with a diameter D = 0.1m was initially placed

just above a sand layer of thickness ^ - O . l r n (diameter of the sand particle ds = 0.36

mm); the sand layer depth was said to vary in 0.1 ~ 0.15 m, but careful examination

shows that it is close to 0.1m (also see Section 4.2). A turbulent channel flow with

Shields parameter 6 = 0.048 was introduced at time t = 0. Here the Shields parameter 0

Xis defined as 0 = ------------------ , where x is the bed shear stress. The pipe was held fixed

g { P s - P f )ds

by the end supports and the scour below the pipe was studied. The channel was 2m wide,

23m long with a height 0.5m. The water depth was H„ = 0.35 m (Fig. 2b). The time

variation of scour profiles was measured dining Mao's experiments.


15

Sym m etry^

Velocity inlet Y W a t e r Pressure outlet

X

Sediment 5 *

Wall

Figure 1. Numerical configuration for the simulation. X is in the streamwise direction, Y in the cross-stream direction and Ss the thickness of the sand layer.


16

(i) The Flume in Mao’s experiment.

...; : ■ ?------------------- —— ---------—— . .

£ > ;H<*> Pipe

.................—---------—-----— 8m — ■—------- —-----------«• — 2m — *-

(ii) Sketch o f Mao’s experimental setup.

Figure 2. Mao’s physical experiment.

3.2. NUMERICAL CONFIGURATION

In the numerical computations, the two-phase model described in Section 2 was set up to

match the experimental configuration. A logarithmic velocity profile with U„ =0.31 m/s

was applied at the flow inlet. The profile for k and £ are given by referring to Brers [3].

Pressure outlet boundary conditions, which require specification of gauge pressure at the


17

outlet boundary, was applied at the flow exit. The water surface is defined as the

symmetry boundaries, wherein zero normal velocity and zero normal gradients of all

variables are satisfied. Wall boundary conditions were applied to the bottom of the sand

layer.

In the simulations, a two-dimensional grid system with 9803 nodes and 9575 cells was

generated with the grid generator GAMBIT of the FLUENT package. The grid consisted

of two zones, the water and the sediment. A 105x60 non-uniform grid was mapped in the

water zone with dimensions 2mx0.4m, and a 105x31 grid was mapped in the sediment

zone with dimensions of 2mx0.1m (Fig. 3). To match Mao’s experiment, the main

simulations were performed with a sand-layer thickness of Ss = ID, which constitutes

the main results to be described in Section 4. Nevertheless, to document the possible

influence of Ss , simulations were also performed with 5s - 1.5 D, the upper limit for 5S

in Mao [28]. The results of the latter are also presented, as appropriate.

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ I \

, Water Zone

I. Sediment I Zone

Figure 3. The Grid for the two-phase model calculations.

sediment zone


18

The inlet and exit boundaries, respectively, were placed 5D and 15D (D being the

diameter of the cylinder) from the center of the cylinder. At the beginning of the

simulations, a sinusoidal profile perturbation with amplitude 0.1D was introduced as a

small disturbance to the initial bed profile (Fig. 4).

3.3. SIMULATION WITH FLUENT

FLUENT uses the segregate solver to solve equations (1) - (9) sequentially. Firstly, the

fluid properties are updated based on the current solution. In order to update the velocity

field, each of the momentum equations is solved using current values of pressure and

mass fluxes at the faces. A Poisson-type equation for the pressure correction is derived

from the continuity and linearized momentum equations, which is then solved to obtain

necessary corrections to the pressure and velocity fields in such a way that the continuity

equation is satisfied. Finally, the equations for turbulent quantities and granular

temperature are solved using previously updated values of other variables.


19

Figure 4. The contours of volume fraction of the sediment at t = 0. Note the introduction of an interfacial disturbance (arrow) at the beginning.

As pointed out by Greimann [16], a bane of two-phase models is the inadequacy of the

parameterization of particle-turbulence interaction terms; these terms are too strong and

produce an unrealistically strong local reduction in the flow. This, in our case, caused

particles to settle rapidly, leading to an unrealistic pile-up of particles near the pipeline.

Another issue is the time delay of flow adjustment following the scour. When the bed

profile varies, the flow needs time to adjust to the bed profile variation. In the present

simulations, this flow adjustment takes place on a time scale on the order of time that a

fluid parcel takes to travel over the computation domain, which has a mismatch with the

time scale where particle-turbulence interactions are taking place in the model. This

disparity of time scales can cause significant errors in scour calculations, aggravated by

the fact that the response of flow to scour development is only approximately represented

in two-phase model dynamics.


20

To avert the above problem, we opted to calculate the single-phase velocity field using

the Navier-Stokes equations and k - s closure (Launder & Spalding [21]) without taking

into account the effects of particles. The fully developed velocity field for the fluid phase

so calculated (as a single-phase flow) was then used as the input field to conduct

supervening two-phase model calculations (rather than obtaining the velocity field of the

fluid phase using the two-phase model itself). Thereafter, the steady single-phase velocity

field of the flow was calculated again with an updated bedform. The procedure was

repeated until the equilibrium state of scour was reached. The interface between water

and sand in the physical experiments was taken as that corresponding to the sediment

volume fraction a s « 0.5 of the numerical experiments, as shown in Fig.5 (see Section

4.1. for justification). Figure 6 shows a typical example of a grid used in the flow

calculations at an intermediate time, where 9567 nodes and 9300 cells are included; the

bed profile shown corresponds to the contour level of a s « 0.5 obtained from the

previous calculation step conducted using the two-phase model. To obtain the fully-

developed flow field, an adaptive grid that responds to the bedform evolution was used.

During the simulations, the grid has been generated manually only once. Thereafter, the

grid was regenerated with the updated scour profile. This was accomplished by running

the journal file (a sequential list of geometry, mesh, zone, and tools commands executed

during the first grid generation) using the grid generator GAMBIT, thus minimizing the

total computation time.


21

Suspended-load Layer 3ds ~ H

Pipeline

a „ « 0 .5

0 ~ 3*/,, Bed-load Layer

Laminated Load Layer (developing slowly)

Figure 5. A schematic diagram showing the interface, bed-load, suspended-load and laminated-load layers. Here ds is the diameter of the sediment particles, H the depth of the water, Y0 the level where the sediment volume fraction is at 0.5.


22

Figure 6. An example of the grid that was used for the flow model. The bed profile is specified as the contour with a s = 0.5 obtained from the previous calculation step conducted with the two-phase model.

In this way, more realistic flow velocities could be maintained in the domain, thus

alleviating rapid velocity profile changes characteristic of pure two-phase calculations.

During the simulations, the time step size was chosen based on the number of iterations

per time step [12], which was 30 - 40 to guarantee satisfactory results. For the single

phase flow model, the time step was on the order of 10“' s whereas the time step for the

two-phase model was on the order of 10“3 s. The flow model for a given scour state was

run for a period of about three to four times the time it takes the flow to travel the

computation domain to ensure that the flow is fully developed. The corresponding

duration for running the two-phase model was chosen so that the maximum change of

scour depth along the sand-water interface is less than 0.03D for the first 10 mins and less

than 0.01D thereafter.


4. Results and discussion: Simulations with a fixed pipeline

4.1. CLEAR-WATER SCOUR SIMULATION

Figure 7 shows the results of bed profiles in the two-phase flow simulation described in

Section 3. As mentioned, the volume fraction of sediment contour a s « 0.5 was chosen

as the bed profiles corresponding to the laboratory experiments in Mao [28]. This

selection was made in accordance with the experimental observations of Wang & Chien

[42], which indicated the ‘laminar behavior’ of two-phase flows for a s > 0.5. Note that

our selection is consistent with the scour models in the literature [3, 26, 31], which have

considered only the bed-load and suspended-load transports above the bed surface. Our

study, however, revealed that the sediments could still be in motion in the region below

the bed-load layer although, in usual scour modeling literature, this region is assumed

immobile. Detailed measurements and analysis, however, have shown that the layer

immediately below the bed-load layer can be in motion, leading to a “laminated load” [8,

42]. This aspect is further addressed in Section 4.3.

The bed profile so determined computationally (Figure 7) is compared with Mao's

experimental data in the right column; the agreement is very satisfactory. Initially, the

flow is subjected to blockage due to the existence of the transversal pipeline, and the flow

beneath and above the pipe tends to accelerate. Hence, the sediment particles underneath

the pipeline have a tendency to be ejected fast. The ejected sediments are supported by

strong turbulent fluctuations, but further downstream, with the decay of turbulence (see

Figure 8), the particles are deposited to form a mound. As the scour depth continues to

increase slowly at later times, the mound slowly moves away from the pipeline as a result

of further downstream sand transport from the sand mound, above which the local flow


24

speed is larger (also see [41]). Finally, an equilibrium situation is achieved in such a way

that particles flown into and carried out from the scour pit are in balance.


020.19

01 02 09X(m)

02

0.16

i><

-o*-0.1 02 0 A

02

0.19

01

1

-0102 06

X(m)

0.19

+ t=200 mln

-oi•0.1

X(m)

02

0.19

1

-006■Ha 01 02 M 06

X (a)

Figure 7. Bed profiles during the development of scour.


26

1.5— *— M O mln — $— KlOmln — KtOOmln — t«2Q0min - h— t=300m1n

0.5

*w0.8 0.8 1.2

X(m)

Figure 8. Normalized turbulent intensity at a location 1cm above the bed; ws is the particle settling velocity. The inset shows the location of turbulence measurements.

The agreement between the predicted and measured scour shown on the right hand side

of Figure 7 is highly encouraging, given the complexity of the model and the nascent

nature of this work in simulating scour without invoking a purely empirical sediment

transport formula. The deviations in scour hole depth between the observations and

predicted scour occurred at earlier times (t = 10 min), which could be attributed, at least

in part, to the transient forcing of initially imposed sinusoidal disturbance.

The evolution of normalized turbulent intensity at a distance 1cm above the interface is

shown in Figure 8. Because of the accelerating flow above and below the cylinder, initial

turbulence levels therein are large, but with the development of scour, the flow velocity

under the cylinder decreases, so do the turbulent velocity fluctuations. Note that at large

times the fluid turbulence intensity in the proximity of the scour pit approaches a


27

value » ws , where ws is the particle settling velocity, at a magnitude commensurate

with that is necessary to keep particles in suspension. This observation is consistent with

the arguments of Stommel [37] and Boothroyd [5] that particles can be in a continuous

state of suspension when the background turbulence velocities exceed the settling

velocity [also see Noh & Fernando [30] and Biihler & Papantoniou [6]].

4.2. SCOUR DEPTH

Figure 9 shows a plot of scour depth (i.e. maximum depths of the scour hole as measured

from the initial level) as a function of time. Note the good agreement between the

simulated scour depth and that measured by Mao [28]. Also shown in the figure is the

simulated depth with a sand layer of Ss = 1.5D = 0.15 m, which shows a faster scouring

rate during period between 10 min and 140 min, indicating that the bottom layer depth is

a factor that determines the initial scour rate at least in the range of 8S = 0.1m ~ 0.15 m.

The solid line shown is the commonly used scour prediction formula in literature [44]

S = S , [ l - « p ( ~ ) ] , (10)

where Se denotes the equilibrium depth and T is the time at which the scour depth

reaches 63% of its equilibrium value. The scouring rate calculated using a deeper sand

layer agrees well with (10) over the entire time period. Therefore, we infer that in Mao’s

experiments the sand-layer thickness has been close to 1.0D. Also note that equation (10)

has been derived using measurements made with fairly thick sand layers, which may

explain why the thicker sand layer showed a better agreement with it.


28

Also note that there is a slight oscillation of the scour depth about the equilibrium value

at large times. This has also been observed in the simulation of scour with Lattice-

Boltzman method by Dupuis & Chopard [10]. Sumer and Fredsoe [38] compiled data

from four previous investigations and suggested the average equilibrium scour depth Se

under a fixed pipeline subjected to current as

— = 0.6 ± 0 .1 . ( 11)D v J

The numerical simulation result of the present study gives S J D » 0.6 (Figure 9), which

agrees well with experimental observations.

E. 0.05 0 o 0 O 0( £ 0 0 O'

+ simulation with the thickness of a sand layer = 1.5D Empirical formula (10)• Mao's measurementO simulation with the thickness of a sand layer = 1D

100 150Time(min)

200 250 300

Figure 9. The time evolution of scour depth in simulations and comparison with equation (10).


29

4.3. SEDIMENT TRANSPORT MODES

4.3.1. BED-LOAD, SUSPENDED-LOAD AND LAMINATED-LOAD

In practice, sediment transport is subdivided into several modes, and the common

mechanisms often found in literature are the bed-load and suspended-load transports. No

precise definitions that help demarcate these modes clearly have been proposed thus far,

although they represent two different mechanisms of sediment transport in the flow. The

bed-load is the part of the sediment load that is traveling immediately above the bed,

supported by intergranular collisions rather than fluid turbulence. According to Einstein

[11], the bed-load layer is confined to a few grain diameters, within ( 2 - 5 ) ^ [45]. On

the other hand, the suspended sediment load is supported by fluid turbulence [13]. These

definitions, nonetheless, are rather qualitative and inadequate to describe complex

dynamics of sediment transport near movable beds. For example, when the shear stress is

high, not only the particles at the interface but also those immediately beneath it start

moving due to the penetration of momentum into the sediment layer by gradient transport

and intergranular collisions [8]. Unlike the bed and suspended loads, these sediments

move in “laminar-like” layers, producing a laminated sediment load [8,42].

Figure 10 illustrates the nature of sediment motion including the bed-load, suspended-

load and laminated load at a river bed, based on Chien & Wan [8]. On the same provisos,

three sediment transport modes could be identified in the present simulations, as

exemplified in Figure 11. For a better appreciation of these layers, the normal horizontal

turbulence intensity profiles for t = 200 min are plotted in Figure 12, along different

locations downstream of the cylinder. It is clear that the turbulence dies off quickly below


30

the interface ( a s «0.5), indicating that the drifting flow below the interface is of laminar

nature. According to Wang & Chien [42], when the volume fraction of sediment is

greater than the threshold value of sand volume fraction, the particles are so closely

packed that turbulence in the fluid is almost suppressed and the two-phase flow behaves

as a laminar one, leading to the layers of laminated transport [8].

/* Suspended load_ 0 > x «

; . e x} Bed load

} Laminated load

| Still bed

Figure 10. Patterns of sediment motion from a flat bed, redrawn based on [8].


Figure 11. Vectors of the sediment velocity u /U x at (a) t = 10 (b) t = 100 minutes


32

9.57«-01 9.094-01 8.614-01 8.134-01 7.654-01 7.174-01 6.704-01 6.224-01 5.744-01 5.260-014.780-01 4,30o-01 3.83O-01 3.35O-01 2.87O-01 2.39O-01 1.910-01 1.430-01 9.57O-024.780-02 0.004400

t = 200 min

(c)

a, = 0.5

Figure 11. Vectors of the sediment velocity u /U m at (c) t = 200 minute.

Note that most of the numerical scour models in literature have considered only the

bed-load and suspended-load transports. For example, in Brors' simulations [3] discussed

in Section 1, the sediment particles below the bed-load layer were assumed to be

stationary and the results showed that bed development stopped completely after 200

mins, an observation that is at odds with laboratory results [28]. The lack of laminated

load may partly explain why the scour development stopped earlier in Brors' simulations

vis-a-vis the experiments. From Figure 11 it is clear that the laminated load, at least that

in the layer immediately below the interface, plays a certain role in scour development.

This observation suggests that frequently used sediment continuity equation [3, 26, 31]

ought to incorporate the laminar load, in addition to bed and suspended loads, in scour

calculations.


33

/■», =1.5 I-------- H

04Pro SicProfileProfile Profile

X=0.5m00

> 0.18

<108

a , =0.5-008

-0.10 02 04 OS OS

m

Figure 12. Normalized turbulence intensity profiles at various downstream locations (X = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6m) of the cylinder wake at t = 200 min.a s = 0.5 is chosen as the bed profile (Fig.7). Note the scale for ^JuJ/ws on theupper left comer.

4.3.2. SEDIMENT VELOCITY

A noteworthy feature of the particle motion in Figure 11 is the presence of a

recirculation zone beneath the scour pit and the adjacent sediment mound. Analysis of

computational output shows that this is a result the shear stresses near the water-sediment

interface (which is transmitted downward through intergranular collisions and gradient

transport, parameterized in terms of collisional viscosity, for example) and pressure

gradients induced by flow surrounding the sand mound. As shown in Figure 13, under the

pipeline, the accelerating flow exacerbates interfacial stresses, which, aided by the


34

negative pressure gradient induced by the converging flow toward the sand mound is

expected to cause enhanced laminated transport beneath the interface. The decelerating

flow downstream of the sand mound neither exerts a high interfacial stress nor does it

produce a favorable pressure gradient for laminated flow. As such, the laminated load

that can be supported downstream of the sand mound is small. To maintain the

continuity, the flow under the sand mound forms a recirculating flow as shown in the

Figure 11, but it is spatially confined to the region where the driving forces are

substantial.

s h e a r s t r e s s d r i v e n ( a i d i n g p r e s s u r e g r a d i e n t ) s h e a r s t r e s s d r i v e n

( o p p o s i n g p r e s s u r e g r a d i e n t )

' S t r e a m l i n eP i p e l i n e

Figure 13. A schematic diagram for the formulation of recirculation in the sediment zone. The sediment movement is driven by interfacial shearing force and pressure gradient.

Although the origin of laminated load can be explained as above, there are some issues

related to the magnitude of sediment velocity below the bed surface. Figure 11 displays

vector plots of dimensionless sediment velocities. The sediment velocity above the


35

surface ranges from 0.7U0 to l.3U0, where U0 is the background flow velocity, which is

reasonable given the small response time (d* / l8 vf ~ 0.007s, where vf = KT6 m2 Is is

the kinematic viscosity of water) of particles that allow them to approximately follow the

fluid phase. The sediment velocity of recirculation zone inside the sand layer (« 0.2Uo),

however, is larger than that one would expect based on intuition, although there are no

available measurements within the sand layer to corroborate our suspicion. Perhaps this

overprediction of sand velocity reflects the difficulty of modeling highly concentrated

sediment flow as well as simplifying assumptions made in our simulations. For example,

frictional viscosity parameterizations employed could have yielded too low of a value,

allowing excessive momentum diffusion below the interface. Also, following previous

works [3, 26, 31], it was assumed in the calculations that the scour profile does not vary

during the flow adjustment (Section 3), which perhaps may not be tenable in reality albeit

this assumption works well in scour profile calculations. With sediment velocity higher

than the normal, sediment particles in the recirculation zone (driven by velocity input

from the single-phase flow simulation) may expedite the scour profile changes, thus

compensating for the plausible reduction of scour velocity resulting from this assumption.

These explanations, however, are speculative at best given that no relevant observational

results exist on the laminated load. Future studies should be directed at such studies.

4.3.3. CALCULATION OF BED-LOAD AND SUSPENDED-LOAD

Sediment loads in various layers were calculated as follows using the simulation results.

First, the effective sediment-water interface, which separates the laminated-transport


36

layer from the turbulent layer, was chosen as a s « 0.5. As shown in Figure 5, the

sediment transport above this interface within a layer of thickness As = 3ds was

considered as bed load [45], and the suspended transport occurred aloft this layer. The

bed-load flux was calculated as qb = a SASUS and the suspended-load flux as

q =T UsasdY, where Y0 is the vertical coordinate corresponding to the surface* JYq+ScI.

a s » 0.5, a s the volume fraction of the sediment (see definitions in Sec. 2.1), Us the

horizontal velocity of the particles and As the bed-load layer thickness.

Figure 14 shows the bed load qb and the suspended load qs during the scour evolution

at different times and distances. At the beginning, the bed-load transport is somewhat

non-uniform, but it became more spatially uniform at later stages of scour evolution. The

peak of the suspended load is roughly consistent with the peak of the sediment mound

formed by deposited sediment particles (see Figure 7), and this peak moves downstream

as the mound moves away from the pipeline. In general, the suspended load was found to

make a profound contribution to the development of scour around the pipe, and this result

agrees well with field observations of Johns et al. [20]. Given the uncertainty of

laminated flow velocity, the laminated loads are not shown in Figure 14, but it is worth

noting that the integral of laminated load over the entire layer depth is near zero in the

region where the driving force is substantial.


Bed-

load

and

Susp

ende

d-lo

ad

(rrf/s

) Be

d-loa

d an

d Su

spen

ded-

load

(rr

f/s)

37

t = 10minx 10

Bedload Suspended load

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8

t = 30minx 10

Bedload Suspended load■a 0.8

•o 0 .6

to 0.4

to 0.2

0.2 0.4 0.6 0.8X(m) X(m)

t = 100minx 10


0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8X(m)

t = 200minx 10


-o 0 .8

T3 0 .6 -

w 0.4

a 0.2

0.2 0.6 0.80.4X(m)

t = 300minx 10'

Bed load Suspended load

-o 0 .8

■o 0 .6

0.2 0.4 0.6 0.8X(m)

Figure 14. The bed load qb and suspended load qs . The arrow indicates the position of the maximum height of the sand mound.


38

4.4. COMPARISON WITH NBURY AND DRAMBUIE MODELS

4.4.1. FORMULATION OF NBURY AND DRAMBUIE MODELS

At present, several models are in use for practical predictions of mine burial in the coastal

zone, which includes Defense Research Agency Mine Burial Environmental model or

DRAMBUIE (developed by H.R.Walingford, U.K., 1994), NBURY (German Navy,

Stender [36], 1980) and Wave-Induced Spread Sheet Prediction or WISSP (U.S. Navy,

1960's). Because of the operational convenience, these models use (sometimes overly)

simple scour parameterizations mostly derived using field and laboratory observations

[15, 41], Of these, DRAMBUIE and NBURY use scour formulae based on (limited) data

collected in the presence of currents, and it is instructive to compare their scour

predictions with the present numerical calculations. NBURY implements the Carstens &

Martin equation [15] derived using U-tube tests. The sediment transport here is

characterized by the sediment Froude numberF = Um[(s- \)g d s~\ 2, Um being the

(orbital) velocity above the boundary layer and a Froude number threshold for the mine

burial is defined as Fx =5.0A(ds ID )v u . For the present study, F = 4.126

andF, = 3.3719.For the caseF > Fx, the NBURY model calculates the maximum (scour)

depth Ym by solving the following equation,

0.01F d.D

0.5

D0.786 m 4 4.45 Ymmtan2 </> D

1tan^ D

3

+ 7.07 Y.D

(12)

where ds is the grain size, D the diameter of the cylinder, (f) the angle of repose of the

sand and t the (tidal) current duration. In the NBURY model, ifF > Fx, the scour is


39

assumed to occur by the suspended-load transport, which is consistent with the results

shown in Fig. 13. An alternative expression is used when F < Fx.

4.4.2. COMPARISON WITH NBURY AND DRAMBUIE MODELS

Note that the NBURY formula does not incorporate a time period of current oscillations,

and thus can be construed as applicable to steady currents of velocity Um, although the

model has been often used in the context of flow under waves with a maximum orbital

velocity of Um. Although the main mechanism of mine burial in NBURY is assumed to

be sand-ripple migration, it is instructive to investigate whether the main formula of this

model is valid under conditions for which it was originally derived (i.e. scour). Figure 15

shows a comparison of the present numerical model results for scour with those predicted

by NBURY. The latter shows a higher scour level compared to numerical results.

On the other hand, DRAMBUIE uses a current-induced scour formula based on

observations around pilings, where

5 = S . [ l - e x p ( - ( ^ - r ) ] . (13)

A 0 bD 2The time scale here is TD = —= = = = , where A = 0.095, B = -2.02, g = 9.81 m /s 2, s

= sediment density relative to water (2.65 for silicious sediment) and 0 = Shields

parameter as defined in Sec. 3, which is related to the ambient flow away from the object.

This empirical equation is a variant of (10), and its equilibrium scour depth is given by


40

SL =

0U - 0.15Ucr

oomax 0.5 U„

i f 0<U<0.75Ucr,

i f 0.75Z7cr <U <\.25Ucr,

i f \25U cr<U,

(14)

where 5'00max = 1.15D is the maximum depth of scour at large free stream velocities (U )

and Ucr is the value of U for the initiation of grain movement without the object.

10'

■*— Present Model -e— NBURY Model

■1• § 10'

-210' 2000 4000 6000 8000 10000 12000 14000 16000 180000Time(s)

Figure 15. Comparison of numerical calculations of the maximum scour depth with those predicted by the NBURY model.

According to Whitehouse [44], it is reasonable to assume that(£/cr /U )2 - 6cr! 0 ,

where 9cr is the critical Shields parameter for the initial grain movement given by

Soulsby [35],


41

0 = °-3Q + 0.055[1 - exp(-0.02D,)], (15)1 + 1.2D, v '

where D, = J[(s - l)g / v2 ]1/3 and v = 10-6 m2 / s is the kinematic viscosity of water.

Figure 16 shows present numerical results vis-a-vis the DRAMBUIE model

predictions. It appears that the initial scour in DRAMBUIE is taking place at a lower

pace. DRAMBUIE also shows an overprediction of the equilibrium depth and a large

relaxation time. Similar disparities have also been noted in the recent work of Testik et al.

[41] who compared DRAMBUIE predictions with laboratory field observational results

of cylinder burial in wave shoaling zone. In summary, it appears that the scour formulae

used in practical mine burial models NBURY and DRAMBUIE needed to be revisited in

future studies.


42

+ P re s e n t M odel w ith th e th ic k n e s s o f a s a n d la y e r = 1.5D0 P re s e n t M odel w ith th e th ic k n e s s of a s a n d la y e r = 1D DRAMBUIE M odel

0 .0 8

0 .0 7

0 .0 6

N,

E'W '.cQ .<1)"Ou.3OOCO

0 .0 5 0 0 0 0 0

DRAMBUIE Model0 .0 4

0 .0 3

0.02

500

0.01

2 5 0 3 0 0100 150Time(min)

200

Figure 16. Comparison of present numerical results (for two sand-layer depth cases) with DRAMBUIE model predictions. The inset shows DRAMBUIE model reaches equilibrium after 1400 minutes.


4.5. CONCLUSIONS: SCOUR UNDER FIXED PIPELINES

A two-phase flow simulatioh of scour around a long (2-D) circular cylinder held fixed on

a sandy bed and subjected to a transverse channel flow was reported in this paper. The

simulation was conducted using the CFD software FLUENT, which allows

implementation of momentum equations for both solid and fluid phases individually, with

Euler-Euler coupling between them and a modified k - s turbulence closure scheme for

the fluid phase. Both flow-particle and particle-particle interaction mechanics were

considered and their effects are parameterized in the modeling system. The model was

applied to simulate an experiment conducted by Mao [28], which is often used as a

benchmark for scour simulations. The computation time for a single simulation was

260hrs on a 2.4GHz PC.

Several special features were needed to be adopted for successful simulations. The

method used for the direct coupling of two phases caused a strong reduction of fluid

phase intensity, which, in turn, caused unrealistic particle settling. In addition, the time

scales for flow adjustment and particle-turbulence interactions were different, which

caused unrealistic bedform predictions. Therefore, the flow simulations were conducted

independently, and the results were used to drive the two-phase system at each

computation step. The ensuing calculations were found to predict the bedform evolution

well. To our knowledge, this is the first time where scour calculations have been

performed using a two-phase model. The satisfactory prediction of the bedform evolution

clearly shows the efficacy of the modeling paradigm used in Euler-Euler coupled two-


44

phase models for simulating scour around a fixed pipeline under a current. In this

approach, no sediment transport formula needs to be invoked. Thus, it preludes the

necessity of selecting an empirical sediment transport formula for scour modeling out of

many available; no standard formula is available in this regard.

Detailed study of sediment motions in the bed shows three main types of sediment

loads related to bedform development. Suspended-load and bed-load are above the bed

surface and laminated-load is below the bed surface. The suspended load was found to

dominate sediment transport above the surface, but the effects of laminated load on scour

development need to be further studied in future, considering that no concrete inferences

could be made on the laminated layer due to nominally unrealistic sediment velocities

appeared in the recirculation zone. To our knowledge, no detailed study, either numerical

or experimental, on the motion of sediment particles inside the sediment bed during scour

evolution, here we make a first attempt to study the motion of these sediment particles

while successfully simulating the scour under a fixed pipeline.

Quantitative results of scour (burial) depth variation with time and the maximum scour

depth agree well with the results of previous research, which have been complied by

Whitehouse [44] and Sumer & Fredsoe [39]. The results also show the dependence of

scouring rate on the sediment layer depth for the depth range investigated, but this

dependence is expected to disappear at larger sediment layer depths. The scour predicted

by two commonly used (operational) mine burial models did not agree well with the

present results, pointing to the need of further research on scour predictions around solid

objects.


5. Simulation of scour below a sagging pipeline

5.1. BACKGROUD

The ability to transport petroleum products across waterways has always been an

important factor in the successM development of an oil or natural gas field, both onshore

and offshore, and historically pipelines (also known as hydropipelines) have been the

most common means of such transport. For example, in the U.S., products from offshore

oil and gas fields located in the shallower waters of the Gulf of Mexico (GOM) are

transported by some 45,000 km of pipeline laid on the seafloor of the GOM. Most of

these pipelines support shelf and near-shelf facilities and a small percentage supports

deepwater operations. The technology and methods involved in handling petroleum

pipelines have evolved to the level of routine and commonplace, but critical scientific

issues still linger with regard to the response of pipelines to sustained wave and current

loadings. For example, survey of pipelines in service has shown significant regional and

local scour occurs in the vicinity of pipelines. The scour leaves pipelines unsupported

over some sections, which has a considerable effect on the stability of the pipeline and

nearby beaches. As shown in Figure 17 (i), when the pipeline lies on the seabed, the

scour starts locally and then spreads along the pipeline. When the scour hole is

sufficiently long, the pipeline sags into the scour hole due to structural deflection. The

scour may cease when the sagging pipeline reaches the bottom of the scour hole, and

finally it may be covered partly by the sediment. This sequence of processes is called the

“self-burial”. Pipeline sagging disturbs the flow field, which, in turn, may exacerbate the

erosion in the scour hole. As shown in Figure 17 (ii), the scour in the free-span areas is

two-dimensional, whereas in the neighborhood of span shoulders, it is three-dimensional.


46

Figure 17 (iii) illustrates an end view of scour and sagging process, through a cross

section in the middle of the pipeline. A typical marine pipeline is shown in Figure 17 (iv)

before the deployment.

- a)

b)

d)

rr:=-l3D scour 2D scour ' 3D scour longitudinal section

(ii)

Figure 17. (i) The three-dimensional pipeline sagging process [14]. (ii) A Sketch for the scour [38].


47

t>i

CD

(iii)

(iv)

Figure 17. (iii) The variation in the position of the pipeline at cross-section B-B [14]. (iv) An underwater pipeline ready to be deployed in Port Kembla harbor (from Australian National Library).


48

5.2. OVERVIEW OF SAGGING PIPELINE STUDIES

Local scour and sagging is an extremely complex phenomenon due to interplay among

various components: the flow, bed material and pipeline. The goal of this part of our

study is to simulate scour around long cylindrical objects using the Eulerian two-phase

model described in Chapter 2 with the hope of understanding how sagging can change the

scour around a pipeline. This work is a natural extension of the work described in

Chapters 3 and 4 and also in [49], wherein the efficacy of an Eulerian two-phase model

associated with the CFD package FLUENT was demonstrated, at least partly, by

evaluating its predictions against experimental data taken around a fixed horizontal

pipeline initially placed on a sandy bed in a turbulent flow. A methodology was

developed to simulate scour, paying special attention to particle-turbulence interaction

mechanics, which is critical to realize effective scour around objects resting on a sandy

bed. The general predictions of the model were also used to infer aspects of mine burial

in the ocean bottom. In addition to obtaining new information, the work presented in this

Chapter on a related but different problem of pipeline sagging is helpful to further

evaluate the efficacy of two-phase Eulerian model described in Chapter 2 as well as the

scour prediction methodology used therein.

During the past three decades, considerable research effort has been devoted to

developing scour models, both experimentally and numerically. Bemetti et al. [2] created

an integrated empirical model considering both waves and a steady current. The model

was applied to scour around a pipe and a pipe sagging into its own scour hole. The model

was tested against laboratory experiments and a good agreement was noted. In this


49

model, however, the scour was mainly realized by implementing empirical relationships,

and the necessity of more refined models was emphasized.

Only a handful of studies exist on scour around sagging pipelines. Fredsoe et al. [14]

investigated the effects of sagging on the depth of the underlying scour-hole depth using

a two-dimensional laboratory model. In a series of laboratory flume experiments, the

sagging process was simulated in two steps: (i) the scour was allowed to develop around

a fixed pipeline, and (ii) the model pipe was lowered artificially into the scour hole at

different speeds to mimic pipe sagging. The scour profile around a fixed pipe (step 1) was

employed as the initial seabed profile for the pipeline sagging in step 2. The scour profile

before sagging was recorded and final scorn profiles when the pipe reached the bottom of

the scour hole (with different sagging speeds) were also measured.

Cheng et al. [7] simulated pipeline sagging by combining a flow model and a sediment

transport model. The former solved the Navier-Stokes equation using Smagorinsky

Subgrid Scale (SGS) closure and the sediment transport model consisted of equations for

mass conservation of sediments above the bed surface. To smooth out numerical

irregularities of scour profiles, a sliding procedure was incorporated. A well-defined

vortex shedding was found to exist in the flow field during sagging. The predicted final

scour profiles generally agree with Fredsoe et al.’s (1988) experimental results. The scour

depth was found to be sensitive to the sagging velocity; for example, the maximum scour

depth increased ~ 20% as the sagging velocity decreased from 12.4 mm/min to 1.0

mm/min.


50

As mentioned before, no general universally accepted formula exists to quantify

sediment transport over a range of conditions, and numerical results based on available

sediment transport models tend to be sensitive to the selection of the sediment formula. In

the present simulations, as in our previous fixed pipeline’s simulation, no such empirical

sediment transport formula was needed because the sediment transport is realized using

dynamical equations governing solid and liquid phases. If the model can be validated for

different flow configurations without ‘adjusting’ parameterizations, then it can be

employed for general use involving scour in two-phase turbulent flows.

5.3. SUMMARY

The scour occurs as a result of complex interactions between flow, turbulence, objects

and sediments, the full treatment of which is currently untenable. To simplify the

problem, previous studies [3, 26, 31] have executed the flow and sediment transport

models separately, thus decoupling two main contributors. Although the results have

often shown satisfactory agreements with experimental measurements, at least for certain

dependent parameters such as scour-hole depth, over the past three decades, the studies

have been limited to scour under the action of currents in laboratory test rigs. If robust

scour models that have wide applicability are to be built, it will be important to couple

sediment and fluid phases in a realistic way while accounting for the object that leads to

local scour. Recently developed two-phase models that consider flow and sediment

particle interaction mechanics are some alternatives in this regard, and such models are

beginning to be applied sediment transport and sedimentation [18, 43, 48] calculations.


Such a two phase model (Chapter 2) has been applied for the first time for scour

calculations under fixed pipelines in Chapter 3 and 4, and in this work the same two-

phase model is used for simulating scour below a sagging pipeline. In the latter case, the

problem is more complex and due consideration should be given to the interaction of the

moving pipe and the evolving bed profiles.


6. Two-phase model, simulation and validation: Sagging pipelines

The mathematical model used in this simulation is the Eulerian two-phase model

described in Chapter 2. It assumes that the sediment-laden flow consists of solid s and

fluid / phases, which are separate, yet they form interpenetrating continua. The space

occupied by each phase is represented by the volume fraction a (0 < a <l) . The laws for

the conservation of mass and momentum are satisfied by each phase individually. For

example, the continuity equations for both the fluid / and solid s phases take the form:

^ ( « (A ) + v , k M ) = ° . (16)at

where t = s , f and a f + a s - 1; a f , a s = volume fraction for water and sediment and

p f , p s = mass density of water and sediment, respectively. The important forces acting

on single particle are the static pressure gradient, the solid pressure gradient (a normal

force due to particle interactions), the drag force, viscous and body forces. Assuming

negligible virtual mass and Basset force [4], the momentum equations for the fluid and

solid phases, respectively, are [12]:

3 ~— (a f p f vf ) + V * ( a f p f vf vf ) = -ccf VP + V * T f + a f p f g + K sf(vs - v / ), (17)

^ K A v , ) + = -a ,V P -V F J + V * r , + asp sg + K fi(vf - v s), (18)

in which vf , v s = the mean-flow velocity for flow and sediment; P = pressure shared by

the two phases; t s = stress tensor for the solid phase; t f = stress tensor for the fluid

phase; and K sf (= K fi) = interphase momentum exchange coefficient. The coupling is

achieved through pressure and interphase exchange coefficients. A symmetric drag model


53

is employed to describe the interaction between phases. Turbulent quantities for the fluid

phase are obtained using a standard k - s model (Launder and Spalding [21]),

supplemented by additional terms that take into account the interfacial turbulent

momentum transfer. To predict turbulence in the solid phase [33, 34], Tchen’s theory

[17] on the dispersion of discrete particles in homogeneous, steady turbulent flows is

used. Dispersion coefficients, correlation functions, and turbulent kinetic energy of the

solid phase are represented in terms of the characteristics of continuous turbulent motions

of the fluid phase based on two time scales. The model closure is realized through the

modified kinetic theory for dense gas and plastic potential theory. In this Eulerian two-

phase model, the equations for the two phases are solved in an Eulerian frame. The model

details are described in Chapter 2 and hence are not repeated here.

6.1. FREDS0E ET AL.’S EXPERIMENTAL SET-UP

The numerical set-up followed the laboratory experiments of Fredsoe et al. [14], which

were designed to simulate sagging of the middle section of a pipeline in real situations

(Figure 17 (i)). In their experiments (Figure 18), the pipe was held fixed by the end

supports and the scour below the pipe was studied, starting from an initially flat bed.

After scour develops around the pipeline, the sagging of the pipeline was simulated by

lowering the model pipe vertically downward at a constant (controlled) speed. This was

achieved by a long screw bolt attached to the frame which was holding the pipe. By

turning the handler, the pipe could be moved up and down smoothly.


54

The sand used in the experiments had median diameter dSQ = 0.36mm . The diameter of

pipe D was 0.1m. There is a gap e between the pipe and the original sea bed, the gap ratio

e lD = 0.1 and the Shields parameter 6 = 0.098; here the Shields parameter 6 is defined

in the usual way as q = -------1------- , where t is the bed shear stress. Since the Shieldsg(ps - p f ¥ s

parameter is greater than the critical Shields parameter, the scour is in live-bed scour

regime (i.e. sediment transport takes place over the entire sand bed).


55

Figure 18. Freds0e’s Experimental set-up that mimicked the sagging pipeline [14].

6.2. NUMERICAL CONFIGURATION

As mentioned, the two-phase model (Section 2) was set up to match the experimental

configurations. As shown in Figure 19, a logarithmic velocity profile with Um =0.5 m/s

was applied at the flow inlet, with a Shields parameter of 0.098. The model constants and

parameterizations used were identical to those used in Chapter 2 and in [49]. The

pressure outlet boundary condition, which requires specification of gauge pressure at the

outlet boundary, was applied at the flow exit. The water surface is defined as the


56

symmetry boundary, wherein zero normal velocity and zero normal gradients of all

variables are satisfied. Wall boundary conditions were used at the sandy bottom.

Symmetry

Velocity infet w |— Water Piearaie outlet

e /D -0 ! A 40

Sediment 1.5D

10 D ^ 20 D

Figure 19. The numerical configuration for the simulation. X is in the streamwise direction and Y the cross-stream direction.

A two-dimensional grid system with 10378 nodes and 10145 cells was generated with

the grid generator GAMBIT. The grid consisted of two zones, the water and the

sediment. The inlet and exit boundaries, respectively, were placed 10D and 20D (D being

the diameter of the cylinder) from the center of the cylinder. A small initial sinusoidal-

shape scour hole with an amplitude 0.1D underneath the pipeline was introduced at the

beginning of the simulation (Figure 20).


57

Figure 20. The contours of volume fraction a of the sediment at t = 0. Note the introduction of an interfacial disturbance (arrow) at the beginning.

Following the methodology used for the clear-water scour around a fixed pipeline

described in Chapter 3 and 4, we calculated the steady single-phase velocity field using

the Navier-Stokes equations and k - s closure (Launder, Spalding [21]) without taking

into account the effects of particles. The fully developed velocity field for the fluid phase

so calculated (as a single-phase flow) was then used as the input field to conduct the two-

phase model calculations. Thereafter, the steady single-phase velocity field of the flow

was calculated again with an updated bedform. The interface between water and sand in

the physical experiments was taken as that corresponding to the sediment volume fraction

a s ~ 0.5 of the numerical experiments [42]. Figure 21 shows a typical example of a grid

used in the flow calculations at an intermediate time, where 11848 nodes and 12146 cells

are included. The bed profile shown corresponds to the contour level of a s * 0.5

obtained from the previous calculation step of the two-phase model. To obtain the fully


58

developed flow field, an adaptive grid that responds to the bedform evolution was used.

Figure 22 shows an example o f the grid for the two-phase flow simulations.

Figure 21. An example of the grid that was used for the flow model. The bed profile is specified as the contour with a s =0.5 obtained from the previous calculation step conducted with the two-phase model.

Figure 22. An example of the grid that was used for the two-phase flow model.

During the simulations, the grid has been generated manually only once. After the first

generation, the grid was regenerated with the updated bed profile data and updated pipe

position. This was accomplished by running the journal file (a sequential list of


geometry, mesh, zone, and tools commands executed during the first grid generation)

using the grid generator GAMBIT, thus minimizing the total computation time. During

the sagging process, the mesh was updated, as appropriate, when the pipeline sagging

exceeded 0.005D ~ 0.015D, where D is the diameter of cylinder. The remeshing in the

second step was achieved by adjusting appropriately the parameters in the journal file

from the first step and then rerunning the journal file.


7. Results and discussions: sagging pipelines

7.1. LIVE-BED SCOUR AROUND A FIXED PIPELINE

In the first step, the scour is developed around a fixed pipeline. Initially, the flow is

subjected to partial blockage due to the existence of the transversal pipeline, and the flow

beneath and above the pipe tends to spatially accelerate, thus causing sediment particles

underneath the pipeline to be ejected preferentially. The ejected sediments are initially

supported by strong turbulent fluctuations, but further downstream, the turbulent intensity

decreases and particles are deposited to form a mound. As the scour depth continues to

increase slowly at later times, the mound slowly moves away from the pipeline as a result

of downstream sand transport from the sand mound. When the pipeline was sagging into

the scour hole at a constant speed, which is the second step of simulations, the turbulence

intensity and scour rate around the pipe decline rapidly, given the flow inside the scour

hole is significantly weakened because the pipe is partly protected against the flow.

Finally, when the pipe reaches the bottom of the scour hole, the sagging is stopped.

According to Fredsoe et al.’s experiments, the scoured bed at the end of the first step is

highly reproducible. Therefore, different sagging speeds are assumed to start from the

same scorned bed. In present study, the first step was taken as complete when the scour

hole is developed for about 60 minutes, as shown in Figure 23, where the interface

between water and sediment is taken as a s ~ 0.5 . The present calculation (Figure 24)

shows a very good agreement with Fredsoe et al.’s measurements except that there are

some deviations in the downstream.


Figure 23. Bed profile after the development of scouring around a fixed pipelinefor 60 minutes.

Hmwfcil SfcraihrtiDri

o j 04- m

Figure 24. Comparison of the scour profiles between the present study and that measured by Fredsoe et al. (1988) before the sagging starts.


7.2. SCOUR AROUND A SAGGING PIPELINE

In the second step, the pipe was allowed to descend into the scour hole with three

controlled sagging speeds (vp= l.Omm/min, 3.1mm/min and 12.4mm/min), and the

computationally determined bed profiles (with a s = 0.5 as the interface; Figure 25) were

compared with Fredsoe's experimental data, which is shown in the right column of

Figure 25. The agreement is satisfactory, except that disparities appear in the downstream

in some cases. Although in Fredsoe et al.’s experiments, the sediment particles were

found to accumulate near the pipeline to some extent. In model results the water flow

tends to wash out the sediment particles in the vicinity of the pipeline.


o Fredsae et al's M easurement (t = 93 minutes)

Present Simulation (t = 9 3 5 minutes)

Sagging spaed V = 3.1 mmAnin

F redsae et al's M easurement (t = 31 minutes)

P resen t Simulation (t = 29.5 minutes)

Sagging sp e ed V = 12.4 mmfmin

o F re d sa e e t a l's M easurem ent (t = 9 minutes)

P re se n t Simulation (t = 6.1 minutes)0.15

J . 0.05 -

-0.05

-0.1 0.2 0.3 0.5x(m)

Figure 25. Comparison of bed profiles between the present study and Fredsoe’s measurements at Vp = l.Omm/min, 3.1mm/min and 12.4mm/min.


Figure 26 shows the sagging process of the pipeline at a speed vp= l.Omm/min. The

first image corresponds to the case where the motion of the pipeline has just begun and

the last one shows the state where the pipeline is about to touch the sandy bed. During the

sagging process, the scour hole deepens and the sediment particles are continuously

transported downstream. No experimental data is available, however, to compare with

these profile evolution calculations.


65

t = 0 min

t = 23.0 min

t = 65.0 min

t = 80.5 min

t = 93.5 min

Figure 26. Sagging with Vp = 1.0 mm / m in .


66

Figure 27 illustrates the computed scour depth variation during sagging at different

speeds. As the pipe is sagging into the scour hole, the maximum scour depth of the bed

increases, given that the flow velocity below the pipe, and thus the erosion under the

pipeline, increases as the pipe sags into the scour hole. At a glance, for the relatively fast

sagging case of vp =12.4 mm/min, the increase in the maximum scour depth is limited.

For the intermediate velocity of v/J=3.1 mm/min, the scour depth almost stops increasing

after the pipe sags into about half of the scour hole. For the lowest speed case vp=1.0

mm/min, the scour depth continues to show an increase even when the pipe is close to the

bottom of scour hole. Sagging with vp =1.0 mm/min increases in scour hole depth by

15.56% relative to the maximum scour depth after scour developed in the first step, but

the enhancement of the maximum scour depth with the sagging speed of

vp =12.4mm/min is less than 1%. This can be explained by considering the response of

the flow beneath the pipe to disturbances induced by sagging. When the sagging speed is

as large as 12.4mm/min, the flow distortions inside the scour hole is large, the resultant

flow perturbations inside the scour hole are rapid and the sediment particles do not have

sufficient time to respond to the change of flow and turbulence.


y(m

)

67

0.04

0.0 2 -

0- 0.02 •

-0.04

-0.06 O -0.08 '

- 0.1

- 0.12

-0.14

0,1 0*

Maximum scour depth in present study♦ Maximum scour depth In Fredsee’s Measurement

20 40 60 80 100T(min)

Sagging speed Vp = 3.1 mm/min

0.04 — Maximum scour depth in present study • Maximum scour depth in Fredsee’s Measurement0.02

-0.02

-0.08

- 0.1

-0.12

-0.14

-0.16,

T(min)

Sagging speed Vp * 12.4 mm/min

0.04

0.02

0-- 0.02

-0.04

i-0 .0 6o-0.08

-0.1 - -0.12 -0 .14-

-0.16L

- Maximum scour depth in present study Maximum scour depth in Fredsee's Measurement

4T(min)

Figure 27. Maximum scour depth development at Vp = 1.0 mm/min,3.1 mm/min and 12.4mm/min.


68

7.3. COMPARISONS WITH CHENG AND L I ’S SIMULATION

7.3.1. VORTEX SHEDDING IN SCOURING PROCESS

The role of vortex shedding on scour surrounding objects is still subjected to debate,

and there is evidence for and against the notion that vortex shedding behind the pipeline

plays a significant role in scour. Sumer et al. [40] conducted a series of experiments on

the scour around pipelines. They first placed the pipe 2D (D is diameter of the pipe)

above the bed. The scour developed downstream showed great difference when the

distance between the bed and the pipe was varied. Based on these observations, Sumer et

al. concluded that vortex shedding plays an important role in shaping the downstream

scour profiles. In the numerical context, Li & Cheng [23][25] presented a local boundary

adjustment technique to calculate the equilibrium scour hole. They first employed

potential flow theory to calculate the flow, but only the predictions for the upper stream

of the scour hole compared well with experimental measurements. When the potential

flow model was replaced by a large eddy simulation (LES) model, however, the scour

hole compared favorably with experimental data. Since the LES model permits well-

defined vortex structures to develop, they inferred that vortex shedding behind the

pipeline can be a key mechanism that determines equilibrium scour profiles.

On the other hand, k - e models have difficulty of predicting vortex shedding in bluff

body wakes (Brers, 1999; Liang et al., 2005), given that they have a tendency to ‘smooth

out’ the fluctuations produced by vortex shedding (due to “averaged” nature of the

approach when compared to the ability of LES to predict instantaneously resolved large-


69

scales structures). Therefore k - s models are expected to underestimate the interaction

between shed vortices from the pipe and the bed. Liang et al’s study [26] based on k - s

closure, on the other hand, showed that vortex shedding does not affect scour profile

predictions at later stages of scour development. A similar conclusion was made by Lii et

al. [27], who studied scour around pipelines by employing renormalized-group (RNG)

and standard k - s models. Both models did not exhibit vortex shedding signatures, but

their predictions still compared well with experimental measurements. It was therefore

concluded that vortex shedding is not a decisive factor in scour that occurs

downstream of pipelines.

7.3.2. COMPARISON WITH CHENG AND LI’S SIMULATION

As mentioned, Cheng & Li [7] simulated scour around a sagging pipeline with a LES

turbulence model. Well-defined vortex shedding patterns were found to form

continuously as the pipe started to sag into the scour hole, but the vortex shedding was

suppressed as half of the pipeline is under the original bed level. Although Cheng & Li’s

LES study delivered more details on the flow field (regular vortex shedding, for example)

than the present model, a comparison shown in Figure 28 illustrates that their final bed

profiles compare well with those of the present study. There is a very good agreement for

the slow and intermediate sagging velocities of l.Omm/min and 3.1 mm/min. In contrast,

deviations were noted in the upstream profile of Cheng & Li’s simulation when the

sagging velocity is increased to 12.4mm/min. The scour profiles used in the two studies

before the sagging is initiated are different, as shown in Figure 29, which may explain the


70

disparities observed at later times. In this regard, the present simulations clearly

outperform the LES simulation of Cheng and Li [7].

Cheng and Li’s simulation started from a state where the scour hole reached about 0.6

times the pipeline diameter (Fredsoe et al. claimed that the sagging initiated when the

scour hole depth is about 0.6 times the pipeline diameter. However, according to their

quantitative measurements taken during the experiments, it appears that they actually

started from a slightly deeper scour hole). The present study was initiated from a scour

profile which has already shown a satisfactory agreement with Fredsoe et al.’s

measurements during the first step of the experiment (Figure 24).

With the sagging velocity of V =12.4 mm/min, it takes the pipeline about 6 minutes to

reach the bottom of the scour hole. The scour profiles change so little during this fast

descent that the differences of initial scour profiles are still partly retained in the final

scour profile. For the two smaller velocities (F =3.1 mm/min, l.Omm/min) the bed

profile changes over a longer time, and thus the two simulations show a better agreement.

Considering the agreement between the LES study of Cheng and Li [7] and ours

performed with k - s model, which does not produce well- defined vortex shedding, we

conclude, as was by Lii et al. [27], that the effects of vortex shedding are not sufficiently

significant to alter downstream bed profiles in a sagging pipeline simulation.


71

Sagging s ps«d V_ * 1.0 mm/min

F rtd ss t r t afs Measurement

0.15 Present Simulation

Cheng and Us Simulation

•0.1•0.2 ■0.1 0.1 0.2 0.4

Sagging s peed * 3.1 mm/min0.2

Fredsse et afs Measurement


Cheng and Us Simulation0.1

A 0.05

-0.05

■0.1•0.1 0.50.1 0.2 0.4

Sagging s peed = 12.4 mm/min0.2

F red s se e ta fs Measurement


Cheng and U s Simulation

•0.05 -

•0.1 -0.1•0.2 0.1 0.3 0.5

Figure 28. Comparison of bed profiles between the present study and Cheng and Li’s study when the pipe reaches the bottom of the scour sole at Vp = 1.0 mm/min,3.1 mm/min and 12.4mm/min.


72

Scaur profile before s a going in Cheng and Lfs simulation0.2

Scour profile before sagging in present simulation□.15Fredsee et afs Measurement

0.1

K 0.05

* * x x t- i t-0.05

-0.1-0.2 0.1 0.2 0.5

Figure 29. Comparison of the scour profiles before the initiation of sagging for the present study and that of Cheng and Li. The experimental measurement of Fredsoe et al. is also shown.

7.4. SEDIMENT TRANSPORT

Figure 30 shows the spatial distribution of sediment velocity at different times. The

sediment velocity above the bed surface has been significantly suppressed as the pipe

sags into the scour hole. As for the sediment particles below the bed surface, the

recirculation is formed which is spatially confined to the region where the driving force is

substantial. Note that a pressure gradient is developed due to the curved flow path near

the cylinder and particle mound downstream. The velocity of the sediment particles is

about 50% of that above the mound, which is larger than that one would expect

intuitively. However, there is no experimental data to evaluate the prediction on the

sediment velocity below the bed interface. A detailed discussion on the possible reasons


73

which may have led to the recirculation and hence the large calculated velocity prediction

below the bed interface was given in Chapter 4.3.2.

9.77*41

9.13*41

9,47*41

742*41

7.18*41

841*41

8.88*41

5.21*41

4.88*41

3.91*41

338*41

3.81*41

1.98*41

1.30*41

83 1 * 4 2

i o.oo*+oo

832*41

7.98*01

7.36*01

| 832*01

8.28*01

i 8.88*01 8.11*41

| 4.84*01

338*01

331*01

2.84*01

237*01

1.70*01

1.14*01

8.88*03

; o.oo*+oo

8/48*41

7.92*01

738*41

6.79*41

632*01

936*41

8.08*01

432*01

3.98*41

339*41

233*41

2.26*41

1.70*41

1.13*01

8.88*02

030*400

m m m

Figure 30. Vectors of the sediment velocity u /U x during the sagging process with the sagging speed 3.1mm/min. The pipeline’s center is located at (0 ,0.54D), (0, 0.34D) and (0, 0.085D), where the vertical coordinate is measured from the original bed level at the beginning (see Figure 20).


74

Figure 31 shows the sediment transport rate S through the gap between the bed and the

pipe, where S m!a is the largest sediment transport rate during the entire sagging process.

During the early period of sagging, there appears to be some fluctuations in the sediment

transport rate, but as the pipe sagged into the scour hole, the sediment transport rate starts

to decrease continuously, so does the average flow velocity in the gap (Figure 32).

□ .8

0.3CD55 0.4

S a g g in g s p e e d V = 1 .Q m rrfm in0.2

20 40 30T im e(m in)

Figure 31. Sediment transport rate through the gap underneath the pipe during the sagging process.


75

0.30

0.34

^ 0.2B

0.20

Avaragc Flaw Velocity in lh a gap b e to aan th api pa and intirfaca w h an Sagging S p aad V_ * lOm nVm in

□.22 20

Figure 32. Average Flow velocity (average velocity = volume/gap) in the gap between the pipe and interface.

Figure 32 illustrates the sediment transport above the bed surface downstream (x) of the

pipeline when the center of the pipeline is located at (0, 0.54D), (0, 0.34D) and (0,

0.085D) (The center of the pipeline is located at (0, 0.6D) before sagging). Three sagging

speeds (V =1.0mm/min,3.1mm/mmand\2Amm/mm) are investigated and suspended

and bed loads are calculated as was discussed in Chapter 4.3. As expected for live-bed

scour, the suspended-load is the dominating sediment transport mode above the bed

surface, the bed-load being much smaller than the suspended-load. For the slowest

sagging speed (V = 1.0mm/nun), the peak of the suspended-load moved further


downstream as the sagging proceeded. However, for the two faster sagging speeds

( V = 3.1 mm/ min and Vp = YlAmm / m in), there is no obvious change in the location of

the maximum suspended-load for the time period investigated. This can be explained by

the slow time scale during the slow descent (Vp = 1.0mm/min, for example), which

allows suspended particles to respond and adjust to the varying flow. The non

equilibrium flow situation prevalent during the faster descent of pipelines does not allow

such response. For all the three sagging cases, the suspended-load is reduced in the

vicinity of the pipe (X = -0.1m ~ 0.2m) due to sagging. The relatively faster sagging

(V = 12.4mm/min, for example) has more effect on the suspended-load in comparison

to the two slower ones (V = 1.0mm / min and V = 3.1mm / min ).


77

Vp = \ .O m m /m m ; Pipeline center is located at: a-(0,0.54D), b-(0,0.34D) and c-(0,0.085D)

0.2-0.2 0.4 0.6 1.2 1.4X(m)

x 10'

0> -0.2 0.2 0.4 0.6 0.8 1.2 1.4X(m)■o

<D x 10 CO 3i--------

210>--0.2 0 0.2 0.4 0.6 0.6 1 1.2 1.4

(a)

(b)

(c)

X(m)

Vp = 3.1 mm / m in ; Pipeline center is located at: a-(0, 0.54D), b-(0, 0.34D) and c-(0, 0.085D)

•0.2 0.2 □.0 0.B 1.4

x 1 0 '

0.2 0.4 0.Q 12T3

CO 3

-0.2 0.2 0.4 0.0 1.4


Vp = \2Am m l m in; Pipeline center is located at: a-(0,0.54D), b-(0,0.34D) and c-(0,0.08SD)

78

0.2-0.2 0.4 0.6 1.2 1.4X(m)

Suspended-load Bed-load

<0 -0.2 0.2 0.6 0.60.4

•0.2 0.2 0.4 0.6 0.8

(a)

(b)

(c)

Figure 33. Sediment transport (bed-load and suspended load) above the bed surface.

Figure 34 shows the normalized turbulence intensity inside the gap between the pipe

and the scour hole. As the pipeline is lowered into the scour hole, the flow and turbulence

become increasingly weaker and thus reducing the suspended load, as shown in Figure

33. This distribution of turbulence has several connotations. As pointed out by Davila &

Hunt [9], heavy particle-laden flows are governed by the dimensionless parameters

Fp = Tp/ \ r / was+lU a~x~\la and VT = . Fp (the rescaled Stokes Number) is defined as

the ratio between the relaxation time of the particle tp(= d] /18v/ ) and the time for the

particle to move around a vortex with circulation T and the characteristic velocity U


79

— x w2when VT is of order unity or small. For the typical case of a = 1, Fp becomes —̂ J ,

where T = UL is a characteristic circulation, L the length scale and ws the settling

velocity. For the cases shown in Figure 31, U = u (horizontal averaged velocity between

the pipe and the bed) and L = 8 (gap width), Fp becomes 5.9808e-004, 7.695le-004 and

0.0012, respectively. Therefore, the particles suspended in the flow under the pipe are not

sensitive for the flow disturbance and approximately follow the main current flow until

they settle downstream due to the reduction of turbulent intensities.

The turbulence can effectively sustain particles in suspension (thus creating a

suspended load) only when VT < 1. Figure 34 shows the normalized turbulence intensity

in two-phase flow cases (under the pipe), which corroborates the conclusion that VT <1

is satisfied when the particles are in suspension. Also shown in Figure 35 are the

normalized turbulent intensities as a function of x at a height equal to half of the gap

width above the interface.


The center of the pipeline is at (0, 0.54D)

The center of the pipeline is at (0, 0.34D)

The center of the pipeline is at (0, 0.085D)- 0.01

- 0.02

-0.03

-0.04

£>-0.05

-0.06

-0.07

-0.08

-0.090.4 0.6 0.8 2.2 2.4

lw sFigure 34. Normalized turbulence intensity inside the gap between the lower side of the pipe and the scour hole. The sagging speed is 3.1 mm/min.


81

1.6

1.4-

1.2

0.8

0.6

0.4 The center of the pipeline at (0 ,0.54D)

The center of the pipeline at (0, 0.34D)

The center of the pipeline at (0, 0.085D)0.2

1.20.80.2 0.4- 0.2

Figure 35. Normalized turbulent intensity at a location midway between the cylinder and the sand layer gap, as a function of downstream distance for Vp = 3.1mm/m in.

7.5. CONCLUSIONS: SCOUR UNDER SAGGING PIPELINES

Scour below a sagging pipeline was simulated using an Eulerian two-phase model,

coupled with a modified k - s turbulence closure scheme for the fluid phase. The

sim ulation was conducted using the CFD software FLUENT, which allows

implementation of momentum equations for both solid and fluid phases individually, with

Euler-Euler coupling between them. Both flow-particle and particle-particle interaction

mechanics were considered and their effects were parameterized in the modeling system.

The particular interest here was an experiment conducted by Fredsoe et al. [14], wherein


82

the scour was allowed to develop below a fixed 2-D pipeline in the first stage followed by

lowering of the pipe at a controlled speed to mimic sagging. The simulations agree well

with Fredsoe et al.’s measurements.

One of the advantage of using the two-phase flow theory is that no sediment transport

formula needs to be invoked, thus avoiding necessity of selecting an empirical sediment

transport formula for scour modeling. Many such formulae are available, but no standard

or high fidelity formula has emerged yet. The two-phase theory was successfully used for

scour calculations below a fixed pipeline in our previous work [49], and a present work

extends this work to the case of a moving pipe. The success of the two-phase model

demonstrated here adds further credence to the versatility of the model.

By comparing the present results with those of Cheng & Li [7], which was conducted

using LES (and produces well defined vortex shedding behind the pipe), it was concluded

that vortex shedding is not a key factor in determining scour profiles below a sagging

pipeline.

Sagging can enhance erosion under the pipeline, especially at smaller sagging

velocities. For the cases investigated, the turbulence intensity under the pipeline is

sufficiently high so that sediments can be maintained in suspension, thus facilitating

suspended-load sediment transport. Sediment transport calculations also clearly showed

the dominance of suspended sediment transport.


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numerical simulation of scour

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