numerical simulation of wave-induced scour and...
TRANSCRIPT
Coastal Engineering 94 (2014) 10–22
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Coastal Engineering
j ourna l homepage: www.e lsev ie r .com/ locate /coasta leng
Numerical simulation of wave-induced scour and backfilling processesbeneath submarine pipelines
David R. Fuhrman a,⁎, Cüneyt Baykal a, B. Mutlu Sumer a, Niels G. Jacobsen b, Jørgen Fredsøe a
a Technical University of Denmark, Department of Mechanical Engineering, DK-2800 Kgs. Lyngby, Denmarkb Deltares, Department of Coastal Structures and Waves, Rotterdamseweg 185, 2629 HD Delft, The Netherlands
⁎ Corresponding author.E-mail address: [email protected] (D.R. Fuhrman).
http://dx.doi.org/10.1016/j.coastaleng.2014.08.0090378-3839/© 2014 Elsevier B.V. All rights reserved.
a b s t r a c t
a r t i c l e i n f oArticle history:Received 1 April 2014Received in revised form 17 August 2014Accepted 18 August 2014Available online 18 September 2014
Keywords:ScourBackfillingPipelinesSediment transportMorphologyWavesTurbulence modeling
A fully-coupled hydrodynamic/morphodynamic numerical model is presented and utilized for the simulation ofwave-induced scour and backfilling processes beneath submarine pipelines. The model is based on solutions toReynolds-averaged Navier–Stokes equations, coupled with k − ω turbulence closure, with additional bed andsuspended load descriptions forming the basis for sea bed morphology. The morphological evolution is updatedcontinuously, rather than being based e.g. on period- or other time-averaging techniques. Simulations involvingwave-induced scour over the rangeof Keulegan–Carpenter number 5.6≤KC≤ 30demonstrate reasonablematchwith previous experiments, both in terms of the equilibrium scour depth as well as the scour time scale. Wave-induced backfilling processes are additionally studied by subjecting initial conditions taken from scour simula-tionswith larger KC to newwave climates characterized by lower KC values. The simulations considered demon-strate the ability of themodel to predict backfilling toward expected equilibrium scour depths based on the newwave climate, in line with experimental expectations. The simulated backfilling process is characterized by twostages: (1) An initial re-distribution phase involving re-organization of sediments in the immediate vicinity of thepipeline, potentially followed by (2) a more lengthy backfilling evolution toward equilibrium scour depth. Thesimulated backfilling time scales are of the same order of magnitude as in experiments, though the multi-stageprocess complicates a more systematic characterization. The simulated sequences of scour and backfillingachieved within the present work are estimated to represent temporal durations of up to approximately 12 hat full practical scales.
© 2014 Elsevier B.V. All rights reserved.
1. Introduction
Submarine pipelines are commonly used to transport water, wastewater, oil, and other hydrocarbons across marine environments. An im-portant aspect in their design andmaintenance is the local scour whichdevelops due to the action of waves and/or currents. This scour createsfree-spanning regions, which in turn can increase stress and structuralfatigue. To improve the understanding of these processes, significant re-search efforts have been devoted to studying the scour processes be-neath pipelines over the past few decades. Such efforts have primarilyfocused on laboratory experimentation (e.g. Sumer et al., 1988; Sumerand Fredsøe, 1990; Çevik and Yüksel, 1999; Kızılöz et al., 2013; Chenget al., 2014), the development of stochastic engineering approachesfor scour prediction (e.g. Myrhaug et al., 2009), as well as the develop-ment of sophisticated numerical modeling tools for predicting thescour evolution beneath pipelines induced by currents (e.g. Brørs,1999; Zanganeh et al., 2012) or waves (e.g. Liang and Cheng, 2005b;Kazeminezhad et al., 2012). For general treatises on scour the interested
reader is referred to e.g. Hoffmans and Verheij (1997), Whitehouse(1998), and Sumer and Fredsøe (2002).
Most studies investigating wave-induced scour processes beneathpipelines have focused on the use of fixed wave climates, typicallystarting from a zero-scour initial bed profile (or small initial scour, inthe case of numerical models). In engineering practice, however, it islikewise of interest to understand the scour profile development in-duced by changes in wave climate, as local weather and wave condi-tions will inevitably vary over time e.g. during the transition fromstorm to calmer conditions. The bed profile response beneath pipelines,and associated backfilling time scales, under such changes in wave cli-mate have been investigated experimentally by Fredsøe et al. (1992).While the number of testswas limited, they found thatwhen an initiallylarge scour holewas subject to less violentwave conditions, a new equi-libriumwould develop whichwas predominantly governed by the newwave climate alone. The recent experiments of Sumer et al. (2013) haveconfirmed similar findings for backfilling aroundmonopiles. The abilityof numerical models to simulate such backfilling processes induced bychanges in wave climate has yet to be established, however. This isthemotivation of the present work, which will demonstrate simulationof both wave-induced scour, as well as backfilling, processes beneath
11D.R. Fuhrman et al. / Coastal Engineering 94 (2014) 10–22
submarine pipelines using an advanced fully-coupled hydrodynamicand morphodynamic CFD model.
The present paper is organized as follows: The hydrodynamic andturbulence models utilized are described in Section 2, which drive thesediment transport and morphological models described in Section 3.Simulations involving scour beneath pipelines are described anddiscussed in Section 4. These will serve to both validate the model, aswell as to establish equilibrium scour configurations for a given set ofwave conditions. Selected simulations involving backfilling processes,induced by changes in wave climate, are subsequently presented anddiscussed in Section 5. Discussion on the practical relevance of the sim-ulations at full scales is provided in Section 6, with conclusions finallydrawn in Section 7.
2. Hydrodynamic and turbulence models
2.1. Governing equations
In this section a description of the computational model usedthroughout the present work is provided. The numerical model solvesthe incompressible Reynolds-averaged Navier–Stokes equations
∂ui
∂t þ uj∂ui
∂xj¼ − 1
ρ∂p∂xi
þ ∂∂xj
2νSij þτijρ
� �ð1Þ
where the mean–strain-rate tensor is
Sij ¼12
∂ui
∂xjþ ∂uj
∂xi
!: ð2Þ
These are combined with the local continuity equation
∂ui
∂xi¼ 0: ð3Þ
Here ui are themean (phase-resolved) velocities, xi are the Cartesiancoordinates, t is time, p is the pressure, ν is the fluid kinematic viscosity,ρ is the fluid density, and τij is the Reynolds stress tensor, which ac-counts for additional (normal and shear) stresses due to momentumtransfer from turbulent fluctuations.
Throughout the present work the Reynolds stress tensor will be de-fined according to the constitutive relation
τijρ
¼ −u0iu
0j ¼ 2νTSij−
23kδij; ð4Þ
where δij is the Kronecker delta, νT is the eddy viscosity,
k ¼ 12u0iu
0i ð5Þ
is the turbulent kinetic energy density, and the overbar denotes timeaveraging.
To achieve closure, the two-equation k − ω turbulence model ofWilcox (2006, 2008) is adopted. In this model the eddy viscosity is de-fined by
νT ¼ keω ; eω ¼ max ω;Clim
ffiffiffiffiffiffiffiffiffiffiffiffi2SijSijβ�
s8<:9=;; ð6Þ
which incorporates a stress limiting feature, with Clim=7/8. This modeladditionally utilizes transport equations for the turbulent kinetic energydensity k
∂k∂t þ uj
∂k∂xj
¼ τijρ∂ui
∂xj−β�kω þ ∂
∂xjν þ σ � k
ω
� � ∂k∂xj
" #; ð7Þ
as well as for the specific dissipation rate ω
∂ω∂t þ uj
∂ω∂xj
¼ αωkτijρ∂ui
∂xj−βω2 þ σd
ω∂k∂xj
∂ω∂xj
þ ∂∂xj
ν þ σkω
� � ∂ω∂xj
" #;
ð8Þ
where
σd ¼ H ∂k∂xj
∂ω∂xj
( )σdo; ð9Þ
and H{⋅} is the Heaviside step function, taking a value of zero whenthe argument is negative, and a value of unity otherwise. The stan-dard model closure coefficients are used: α = 13/25, β = β0fβ,β0 = 0.0708, β∗ = 9/100, σ = 1/2, σ∗ = 3/5, and σdo = 1/8. Notethat for two-dimensional problems, as considered throughout thepresent work, fβ = 1 and hence β = β0; For the generalization tothree spatial dimensions see Wilcox (2006). It can finally be notedthat the basic hydrodynamic model used herein represents asingle-phase variant of the two-phase (air–water) model utilizedby Jacobsen et al. (2012).
2.2. Boundary conditions
Thehydrodynamicmodel described above is subject to the followingboundary conditions: At friction wall boundaries a no-slip condition isimposed whereby velocities are set to zero. Alternatively, the topboundary is treated as a frictionless slip wall i.e. with vertical velocitiesset to zero, and horizontal velocities and scalar hydrodynamic quanti-ties having zero gradient. It is therefore emphasized that the top bound-ary does not represent the free surface of real waves, the flow beneathwhich will be approximated by an oscillatory flow as described below.
At the bottom seabed boundary, where sediment transport pro-cesses will be modeled, a hydraulically rough-wall will be assumed.Hence, the friction velocityUf is determined from the tangential velocityat the nearest cell center based on an assumed logarithmic velocitydistribution
uU f
¼ 1κln30ycks
; ð10Þ
where yc = Δy/2 is the normal distance from the wall to the cell center,with Δy being the cell thickness, κ = 0.4 is the von Karman constant,and ks = 2.5d is Nikuradse's equivalent sand roughness. The frictionvelocity is then utilized within standard wall functions for k and ω inthe cells nearest to the wall. In dimensional, or equivalent dimen-sionless, forms these read:
k ¼ U2fffiffiffiffiffiffiβ�p or
kU2
f
¼ 1ffiffiffiffiffiffiβ�p ð11Þ
ω ¼ U fffiffiffiffiβ
p �κΔyor
ωνU2
f
¼ 1ffiffiffiffiffiffiβ�p
κΔyþð12Þ
where Δy+ = ΔyUf/ν is the thickness of the near-wall cell in wallcoordinates. It can be noted that the former yields k/Uf
2 ≈ 3.33,which is in good agreement with flume measurements of k presentedby Fuhrman et al. (2010), based on all three fluctuating velocitycomponents.
Pipeline boundaries will be modeled as smooth walls, formallyemployed within a generalized wall function approach. Accordingly,the friction velocity at pipeline walls is determined from the tangential
12 D.R. Fuhrman et al. / Coastal Engineering 94 (2014) 10–22
velocity at the nearest cell center based on the profile of Cebeci andChang (1978):
uU f
¼ 2Z yþc
0
dyþ
1þ 1þ 4κ2 yþ þ Δyþcc� �2
C� �1=2 ; ð13Þ
C ¼ 1−exp − yþ þ Δyþcc� �
=25� �h i2
; ð14Þ
Δyþcc ¼ 0:9ffiffiffiffiffiffikþs
q−kþs exp − kþs
6
� �� �; ð15Þ
who generalized the van Driest (1956) profile to incorporate potentialroughness effects, with yc
+ = ycUf/ν.Similarly, the generalized wall functions for k and ω are defined ac-
cording to
kU2
f
¼ min AΔyþ2;
1ffiffiffiffiffiffiβ�p( )
; ð16Þ
ωνU2
f
¼ maxB
Δyþ2 ;1ffiffiffiffiffiffi
β�pκΔyþ
( ): ð17Þ
The first arguments in these functions ensure that these variablesfollow their proper scaling k ∼ y2 andω ∼ 1/y2 for near wall cells within
the viscous sub-layer (see e.g. Wilcox, 2006). The values A ¼ 1=
δþ2ffiffiffiffiffiffiβ�p� �
¼ 0:02466 and B ¼ δþ=ffiffiffiffiffiffiβ�p
κ� �
¼ 96:885 are utilized,
which ensure a continuous transition to the (fully-turbulent) second ar-guments at Δy+ = δ+, where δ+ = 11.626 is taken as the viscous sub-layer thickness (in dimensionless wall coordinates). It is emphasizedthat, while the full description of these boundary conditions is providedhere for completeness, throughout the present work the pipelines aremodeled as smooth walls, by keeping both the pipeline roughness ksand near-wall cell sizes small, such that it is the first arguments in theabove functions that typically dominate. Note that reasonably similaruse of k−ω turbulence closures within flow around and scour beneathpipelines can be found e.g. in Liang and Cheng (2005a,b).
To drive the desired oscillatory wave flow conditions within themodel, at the left hand boundary the following boundary conditions,taken from Liang and Cheng (2005a), are imposed:
u ¼ Umsin2πTw
t� �
; v ¼ 0 ð18Þ
k ¼ km sin2πTw
t� �� �2
; km ¼ 0:0005U2m ð19Þ
ω ¼ ωm sin2πTw
t� � ; ωm ¼ km
100νð20Þ
where Um is the maximum free stream velocity, and Tw is the waveperiod. At the opposite right hand boundary zero-gradient boundaryconditions are imposed. Note that preliminary validation of the basichydrodynamic model described above for a case involving oscillatoryflow around a near-wall cylinder is provided in Appendix A.
3. Sediment transport and morphological models
This section describes the sediment transport and morphologicalmodels utilized herein. As the implementation of these models, in-cluding a detailed account of numerical aspects, has been describedin detail by Jacobsen (2011), as well as in the recent publication ofJacobsen et al. (2014), only essential details are provided in whatfollows.
3.1. Sediment transport model
The model makes use of separate bed and suspended sediment loaddescriptions. The rate of bed load sediment transport qB is calculatedbased on the method described in detail by Engelund and Fredsøe(1982), as well as Roulund et al. (2005), who generalized the well-known transport formulation of Engelund and Fredsøe (1976) to ac-count for three-dimensional effects as well as bed-slope modifications.
The suspended sediment model is, alternatively, based on aturbulent-diffusion equation (see e.g. Fredsøe and Deigaard, 1992,p. 238) of the form:
∂c∂t þ uj−wsδ j3
� � ∂c∂xj
¼ ∂∂xj
ν þ βsνTð Þ ∂c∂xj
" #; ð21Þ
where c is the suspended sediment concentration, ws is the settlingvelocity, and βs = 1 is utilized throughout i.e. the sediment diffusivityis taken as equal to the eddy viscosity. The settling velocity is solvedfor a given grain diameter d using the drag-coefficient approachdescribed in e.g. Fredsøe and Deigaard (1992). Eq. (21) is solved on asub-set of the main computational mesh i.e. where near-bed cellsbelow a so-called reference level b are removed, as described in detailby Jacobsen et al. (2014), see e.g. their Fig. 2.
At the top and pipeline boundaries a zero-flux condition for c isutilized. At the bottom boundary so-called reference concentrationboundary conditions are imposed. More specifically, the method ofEngelund and Fredsøe (1976) is utilized, with the concentration atthe reference level set to
cb θð Þ ¼ c01þ 1=λbð Þ3 ; ð22Þ
where c0 = 0.6, and the linear concentration λb is
λ2b ¼ κ2α2
1
0:013sθθ−θc−
π6μdpEF
� �ð23Þ
where
pEF ¼ 1þ πμd
6 θ−θcð Þ� �4� �−1
4
ð24Þ
is the probability of moving grains, and
θ ¼ τbρg s−1ð Þd ¼ U2
f
s−1ð Þgd ð25Þ
is the Shields parameter. Throughout the present work the coeffi-cient of dynamic friction is set to μd=0.7. The critical Shields param-eter θc is computed from a base value θc0 = 0.05, accounting for bed-slope effects as in Roulund et al. (2005).
Throughout the present work the reference level is taken as b =α1d = 3.5d, which is somewhat larger than the more traditionallyused b≈ 2d (e.g. Fredsøe andDeigaard, 1992). Preliminary testingwith-in the present study has found that such larger values are necessary, atleast in the present context, to promote enough suspended sediment toinduce the scour process. The presently used value for b turns out to bevery close to that used within the previous modeling of pipeline scour
Table 1Summary of cases considered for wave-induced scour beneath submerged pipelines. Allcases use pipeline diameter D = 0.03 m and grain diameter d = 0.19 mm.
KC T (s) Um (m/s) θmax S=D ¼ 0:1ffiffiffiffiffiffiKC
p(S/D)meas Reference
5.6 1.10 0.153 0.13 0.24 0.29–0.30 Fredsøe et al. (1992)11 1.22 0.240 0.19 0.33 0.47 Sumer and Fredsøe
(1990)15 2.50 0.177 0.10 0.39 0.50 Sumer and Fredsøe
(1990)19.6 3.00 0.196 0.092 0.44 0.44 Fredsøe et al. (1992)21.1 2.64 0.239 0.12 0.46 0.41 Fredsøe et al. (1992)25.3 3.51 0.216 0.094 0.50 0.38 Fredsøe et al. (1992)30 3.50 0.257 0.11 0.55 – –
13D.R. Fuhrman et al. / Coastal Engineering 94 (2014) 10–22
by Liang and Cheng (2005a), who utilized a reference level correspond-ing to b = 3.6d, though it is unclear if their value was based on similarexperience. The reference levels mentioned above also correspondquite closely to the top mean vertical position reached by bed load par-ticles, 3.0d − 3.7d, found by Sumer and Deigaard (1981); see alsoAbbott and Francis (1977). Hence, the reference levels mentionedabove are within a range that is consistent with the near-bed physics,regarding the conceptual separation of bed and suspended load.
To prevent un-physical “overloading” conditions (i.e. where ref-erence cb is forced to be smaller than the concentration immediatelyabove) from occurring in the model, the solution suggested byJustesen et al. (1986) is finally invoked. That is, if the concentrationclose to the bed exceeds the reference concentration cb(θ), thevalue in practice is taken from the cell adjacent to the boundary.
3.2. Morphological model
The morphology of the bed elevation h is based on the sedimentcontinuity (Exner) equation
∂h∂t ¼
11−n
−∂qBi∂xi
þ Dþ E� �
; i ¼ 1;2 ð26Þ
where n = 0.4 is the bed porosity and
D ¼ ws−u3ð Þcb; E ¼ ν þ βsνTð Þ ∂c∂x3
x3¼b
ð27Þ
are, respectively, the deposition and erosion stemming from thesuspended sediment model. It is stressed that the simulated bedmorphology within the present work is continuous, always beingbased on the instantaneous sediment transport fields i.e. the modeldoes not make use of morphological rates averaged over the waveperiod or any other time scale. Accordingly, morphological and hy-drodynamic times are equivalent. The interpolation of the computedbed level change in the face centers on the bed to the vertices for theboundary condition for the mesh motion (also on the bed) needsspecial attention. As described in Jacobsen et al. (2014), the inverselinear interpolation scheme is used, as this is the one, which uniquelypreserves mass. The temporal integration of Eq. (26) is herein basedon the Explicit Euler method. For more specific details regarding thenumerical evaluation of the three terms within the right-hand-sideof Eq. (26) the interested reader is referred to Jacobsen et al.(2014). It is finally noted that, to prevent excessive erosion inducedby the imposed uniform flow at the boundary Eq. (18), the sea bedis fixed at the left/right boundaries, and relaxed to full morphologyover a distance spanning a few pipeline diameters.
Experience has shown that, if left un-checked, the morphologicalmodel can lead to local bed slopes in excess of the angle of repose. Tocombat such un-physical steepening, the physically-based sand slidemodel described in detail by Roulund et al. (2005) is implemented. Inthe present work, this sand slide model is activated at positions wherethe local bed angle exceeds the angle of repose ϕs = 32∘, and is de-activated once the local bed angle is reduced to 31.9∘. Apart from thisfeature, no other ad hoc smoothing of the bed and/or sediment trans-port fields is utilized.
The equations comprising the fully-coupled model outlinedabove are solved numerically using the open-source CFD toolboxOpenFOAM®, version 1.6-ext, making use of a finite volume spatialdiscretization with a collocated variable arrangement, in conjunc-tion with a standard PISO algorithm. Again, for further details seeJacobsen et al. (2014).
4. Simulation of wave-induced scour
We will begin the present study by considering the simulation ofwave-induced scour processes beneath submarine pipelines. Thesesimulations will serve as model validation, as well as provide a seriesof benchmark equilibrium conditions useful for the subsequentstudy of backfilling processes. For this purpose a series of sevenscour cases have been selected, based primarily on previously con-sidered experimental conditions of Sumer and Fredsøe (1990) andFredsøe et al. (1992). The selected conditions are summarized inTable 1, and have been conveniently chosen to utilize a fixed pipelinediameter D = 30 mm, as well as a fixed (fine) sediment grain sized= 0.19 mm. Individual cases will then be characterized by their re-spective Keulegan–Carpenter number
KC ¼ UmTw
D¼ 2π
aD
ð28Þ
which is well-known to be the predominant physical parametergoverning the equilibrium scour depth S/D and profile (e.g. Sumerand Fredsøe, 2002). In Eq. (28) a = UmTw/(2π) is the amplitude ofthe orbital motion of particles in the free stream. Fixing D and d en-ables the same initial mesh to be utilized, while use of fine sedimentenables relatively fast bed morphology, which in turn helps maintainreasonable (though still rather large) computational expense.
Each simulation uses a computational mesh spanning− 20D≤ x≤ 20D i.e. a total horizontal span of 40D, and with a domain heightof 10D. This has been selected to more than double the expectedhorizontal extent of the lee-wake according to Jensen et al. (1989)
LD¼ 0:3KC ð29Þ
on either side of the pipeline, even for the largest KC considered (takingKC=30 indeed yields L=9D). The pipeline is placed such that its bot-tom corresponds to the origin (x, y) = (0, 0). For numerical reasons asmall (sinusoidal) scour holewith initial depth S0/D=0.15 is placed di-rectly beneath the pipeline. The computational mesh is graded, suchthat near the pipeline the smallest cells have thickness equal to0.003D, whereas at the seabed cells are set to have a thickness of 0.5d.A section of the initial computational mesh in the near-vicinity of thepipeline is depicted in Fig. 1. The total computational domain uses8732 cells, which has been found to yield a reasonable balance betweencomputational expense and physical detail. Utilizing this mesh, as anindication of the computational times required, a fully-coupled mor-phological simulation spanning 1min of physical time requires approx-imately 1 day of CPU time on a single modern processor.
As seen in Table 1, the selected cases cover the range 5.6≤KC≤ 30 inapproximately ΔKC = 5 increments. The series of simulations is con-ducted as follows. For each case considered, the initially still domain isintroduced to the specific wave climate via the left-hand boundary, asdefined by Um and Tw, for a total warm-up duration of 10Tw. Duringthis warm-up period hydrodynamic and sediment transport fields are
0 1 2 30
0.2
0.4
0.6
t*
S/D
(a) KC=5.6
0 1 2 3 4 5 60
0.2
0.4
0.6
t*
S/D
(c) KC=15
0 1 2 3 4 50
0.2
0.4
0.6
t*
S/D
(e) KC=21.1
0 1 2 3 4 50
0.2
0.4
0.6
t*
S/D
(g) KC=30
Fig. 2. Computed time series the scour depth (full lines), taken at full wave-period Tw time interthe horizontal dashed line.
Fig. 1. Initial mesh in the vicinity of the pipeline.
14 D.R. Fuhrman et al. / Coastal Engineering 94 (2014) 10–22
allowed to develop, with bed morphology switched off. The end of thiswarm-up periodwill be denoted as t=0, after which time themorpho-logical model is activated, and the bed is allowed to evolve freely. Indi-vidual simulations have been monitored, and will be characterized bytheir scour depths S taken directly below the pipeline center. Simula-tions have been stopped after an apparent equilibrium scour is reached,as evidenced by a flattening of the temporal scour evolution, as well asvisual inspection of the profile evolution as a whole.
The computed time series of dimensionless scour depth S/D for eachof the seven scour cases is depicted as a function of dimensionless time
t� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig s−1ð Þd3
qD2 t ð30Þ
in Fig. 2. Note that the given parameters, taking gravitational accelera-tion g = 9.81 m/s2 and relative sediment density s = 2.65, yield a
0 1 2 3 40
0.2
0.4
0.6
t*
S/D
(b) KC=11
0 1 2 3 4 50
0.2
0.4
0.6
t*
S/D
(d) KC=19.6
0 1 2 3 4 5 60
0.2
0.4
0.6
t*
S/D
(f) KC=25.3
vals. On each sub-plot the equilibrium scour value according to Eq. (31) is also indicated by
15D.R. Fuhrman et al. / Coastal Engineering 94 (2014) 10–22
time scaling t/t∗ = 85.4 s. The plotted scour depths are monitored foreach case in increments of the wave period Tw. As can be seen, foreach of the cases, the scour process begins once the bed morphologyis activated. In all cases considered, the bed gradually evolves from itsinitial profile to one with larger scour depth, before reaching an equilib-rium value. Individual simulations were monitored, and stopped aftersuch equilibrium is maintained for a reasonable amount of time (ap-proximately equal to at least one t∗ unit). The computed equilibria arebest described as dynamic, rather than static, as the bed does not remaincompletely still. This is likely due, in part, to our use of instantaneousmorphological updating, and is evidenced e.g. by the secondary fluctua-tions in S/D even after significant long-term trends in the scour depthhave vanished. The fluctuations are due to quasi-cyclic formation anderosion of small scale bed features that emerge due to local differencesin sediment transport patterns which drive the morphology.
On each sub-plot within Fig. 2 a horizontal dashed line correspond-ing to the equilibrium scour depth predicted by the regression equationof Sumer and Fredsøe (1990)
SD¼ 0:1
ffiffiffiffiffiffiKC
pð31Þ
is also depicted. This equation is based on the extensive laboratory datasets of Sumer and Fredsøe (1990), as well as Lucassen (1984), after re-casting the latter in dimensionless terms. Collectively these data setstotal over 50 cases, spanning the range 2≤ KC≤ 1050, with pipeline di-ameters 1 cm≤D≤ 18 cm. As can be seen, for nearly all of the KC values
−4 −2 0 2 4−1
0
1
2
x/D
y/D
(a) KC=5.6
−4 −2 0 2 4−1
0
1
2
x/D
y/D
(c) KC=15
−4 −2 0 2 4−1
0
1
2
x/D
y/D
(e) KC=21.1
−4 −2 0 2 4−1
0
1
2
x/D
y/D
(g) KC=30
Fig. 3. Computed equilibrium scour profiles (full lines) for various KC. In (g) the additi
considered, the equilibrium scour obtained in the simulations is in rea-sonable agreementwith the target value predicted by Eq. (31). The onlydramatic deviation from the target value occurs in the case with KC =11 (Fig. 2b), where a much larger scour than predicted by Eq. (31) isfound. This particular case will be discussed in more detail in whatfollows.
The scour profiles predicted by the model at their respective endstates are likewise shown in Fig. 3. From these plots, as expectedbased on Fig. 2, there is a clear tendency for increased scour as KC is in-creased. As can be seen, the scour holes in the direct vicinity of the pipe-lines remain largely symmetric, even after considerable time. In somecases, asymmetry does develop further away from the pipeline, typical-ly via the formation of asymmetric “shoulders” on either side of themain scour hole. The degree of asymmetry is reasonably similar tothat in the physical experiments, based on visual inspection of the pro-filesmeasured by e.g. Sumer and Fredsøe (1990). Note also that the left/right asymmetry does not appear in any way systematic i.e. dependingon the case, either the left or right shoulder may develop into the largerof the two, which tends to be dynamically balanced by two smallershoulders on the opposite side.
The importance of including suspended sediment in the simula-tion of the scour process is also emphasized. Indeed, re-simulationof selected wave-induced scour cases considered herein, but withthe suspended sediment model switched off, actually results in de-position directly beneath the pipeline, rather than scour. The reasonfor this behavior in such bed load-only simulations is due toconverging-diverging effects (studied previously within oscillatory
−4 −2 0 2 4−1
0
1
2
x/D
y/D
(b) KC=11
−4 −2 0 2 4−1
0
1
2
x/D
y/D
(d) KC=19.6
−4 −2 0 2 4−1
0
1
2
x/D
y/D
(f) KC=25.3
onal dashed line represents the initial profile used within each of the simulations.
16 D.R. Fuhrman et al. / Coastal Engineering 94 (2014) 10–22
flows e.g. by Sumer et al., 1993; Fuhrman et al., 2009a,b, 2013) in theflow beneath the pipeline. These promote period-averaged bed shearstresses, as well as near-bed flow, directed in the convergent direction,hence driving bed load toward the pipeline center. This is illustratedfurther e.g. in Fig. 4, which depicts an example (using KC = 11)period-averaged velocity field (often called streaming) computedwith the bed fixed at its initial position. The inward directed near-bedflow described above is especially evident in Fig. 4b, which depicts azoomed in region directly beneath the pipeline. Such features are notdiscussed by Liang and Cheng (2005a), who report period-averaged ve-locities always pointing away from the pipeline in the near-bed region.Notably, however, inward-directed streaming very close to the bed(though not discussed), seems to be present upon close examinationof many of the results presented in An et al. (2011) (e.g. theirFigs. 11e,f; 12e; and 13e), who considered steady streaming due to os-cillatory flow around circular cylinders near a plane boundary. Strongeroutward-directed flow develops further away from the bed (approxi-mately above the reference concentration level, depicted as the dashedline in Fig. 4b), which is largely responsible for transporting suspendedsediments away from the pipeline. So, inclusion of suspended sedimentis necessary to promote a realistic scour process, at least in the condi-tions considered: Bed load alonewill form a bump just below the center
−1 −0.5 0 0.5 1−0.5
0
0.5
1
1.5
2
x/D
y/D
(a)
−0.2 −0.1 0 0.1 0.2−0.2
−0.1
0
0.1
x/D
y/D
(b)
Fig. 4. Example computed period-averaged velocity field with the bed fixed at its initialposition, with KC = 11. Subplot (b) depicts a zoomed in portion of subplot (a) beneaththe pipeline. The additional dashed line in (b) corresponds to the reference concentrationlevel b = 3.5d.
of the pipe due to inward streaming just above the bed; Suspended sed-iment, which is transported further from the bed, prohibits formation ofthis bump due to the change in the streaming direction further from thebed.
We will now return our attention to the scour evolution for the casewith KC = 11, with the scour profile depicted in Fig. 3b. The simulatedtwo-stage scour process for this case (see again Fig. 2b) differsmarkedlyfrom the other KC values considered herein, andmerits further explana-tion. It is believed that this qualitative difference is due to a “resonance”phenomenon triggeredwithin themodel, as described below. The initialscour process for this case, in fact, reasonably resembles that observedin the other cases. This can be seen e.g. from the profile near the timewhen the computed scour depth initially levels (t∗ = 0.5), presentedin Fig. 5a. At this time the profile still bears some resemblance tothose developed under the considered neighboring KC values (e.g.Fig. 3a and c). The scour depth achieved during this initial stage is like-wise in quite good agreement with the target (Eq. (31)), see e.g. Fig. 2b.In contrast to the other cases, however, the computed profile has devel-oped features with trough-to-trough wavelength λ, as identified direct-ly on Fig. 5a, that are very near the natural length of vortex ripples (e.g.Brøker, 1985)
λ≈1:2a orλD≈1:2
2πKC≈2:1 ð32Þ
that can be expected within this regime. Consequently, the profile issusceptible to an instabilitymechanism, closely resembling that respon-sible for the well-known formation of vortex ripples. The developmentof such a feature, with crest beneath the pipeline center and maintain-ing similar wavelength λ, is indeed readily apparent in the profiledepicted somewhat later at t∗ =0.9 (Fig. 5b). At this time, the scour di-rectly beneath the pipeline has remained nearly the same as in Fig. 5a.The mechanisms discussed above, however, have now lead to deeptrough regions on both sides of the pipeline. As time elapses further,the exposed crest is eroded, corresponding to the second scouringstage apparent in Fig. 2b, and eventually leading to the profile presentedpreviously in Fig. 3b. Remnants of the evolution history are still appar-ent at this late stage, e.g. as the secondary crest directly beneath thepipeline. The resonance phenomenon described above is seemingly par-ticular to the KC ≈ 10 regime; Indeed, additional testing (not shownhere for brevity)with similarKC≈ 10 values, butwith lower Shields pa-rameter, has demonstrated similar tendencies. To avoid giving a mis-leading presentation of the simulated process for KC = 11, relative tothe other cases which do not exhibit the two-stage process described
−4 −2 0 2 4−1
0
1
2
x/D
y/D
(a) KC=11, t*=0.5
λ
−4 −2 0 2 4−1
0
1
2
x/D
y/D
(b) KC=11, t*=0.9
λ
Fig. 5. Computed scour profiles at intermediate stages (a) t∗ = 0.5 and (b) t∗ = 0.9 fromthe simulation with KC= 11.
0 1 2 3 4 5 60
0.2
0.4
0.6
S/D
t*shift
(a)t′*max
0 1 2 3 4 5 60
0.5
1
T*
(Smax
−S)/S
max
t′*
(b)
Fig. 7. Example calculation of the scour time scale T∗ for the case with KC= 15.
17D.R. Fuhrman et al. / Coastal Engineering 94 (2014) 10–22
above, in what follows the characteristic features (scour depth and timescale) for this particular case will be based only on the initial scour timeseries in Fig. 2b (i.e. to t∗ = 1). In all other cases results from the entiresimulation will be considered.
The equilibrium scour depths from the model simulations, heretaken as the average of the final 10Tw from the time series in Fig. 2 (ex-cept for KC = 11, as described immediately above), are summarizedversus their respective KC values in Fig. 6 (circles). Also shown on thisfigure are the measured values (Sumer and Fredsøe, 1990; Fredsøeet al., 1992) corresponding to the selected cases simulated in the pres-ent work (see Table 1). Eq. (31), again considered as a target value, islikewise depicted on this figure by the full line. As can be seen, thescour depths predicted by the model compare reasonably with the tar-get expression (Eq. (31)), and seem to exhibit approximately the samevisual scatter as do the experiments.
The simulated time scale of scour, representing the time required forsignificant scour to develop, is also of interest. This value will be esti-mated via integration of the scour curves. This is consideredmost robustin the present context, as the full time series of the initial scour processwill beutilized. In order tomakedirect comparisonwith the experimen-tally-based time scales (starting from zero scour), it is necessary to com-pensate in some way for the presence of the initial scour hole in thesimulations. To account for this, the computed scour curves have beenextrapolated backwards in time to a zero value, based on the estimatedslope of the initial scour curve. The time used for the integration thencorresponds to a shifted time t′ ∗ = t∗ + tshift
∗ . After adjusting the scourtime series as described above, the time scale is calculated according to
T� ¼Z t0�max
0
Smax−SSmax
dt0� ð33Þ
An example demonstrating the extrapolation, aswell as the resulting in-tegrated quantity and resulting time scale T∗, are respectively providedin Fig. 7a and b. As seen in this example, the additional time addedfrom the extrapolation is not particularly significant. Because in somecases the initial scour process gives rise to a local maximum which isslightly larger than the equilibrium scour value at longer times, for con-sistency from case to case, we have elected to use the maximum scourdepth Smax within Eq. (33), and to perform the integral to the time tmax′∗where the local maximum occurs, thus characterizing the initial scourprocess. As can be surmised from Fig. 7, none of these decisions affectthe estimated time scale very significantly.
100 101 10210−1
100
KC
S/D
Present − scourMeasured0.1(KC)0.5
Fig. 6. Comparison of computed and empirical scour depths versus KC. Note that the re-sults for KC= 11 are based on the computed scour curve in Fig. 2b to t∗ = 1.
The resulting time scale values are summarized in Fig. 8 versus theirrespective (far field) Shields parameter θ taken from Table 1. Alsoshown is a line depicting the empirical relation, based on the study ofFredsøe et al. (1992)
T� ¼ 150
θ−5=3: ð34Þ
While there is considerable scatter in the resulting values, and therange of simulated Shields parameter is limited, based on this compar-ison it can be surmised that the simulated scour time scales are of theproper order of magnitude; Some are larger and some are smallerthan expected based on Eq. (34).
The series of seven wave-induced scour simulations consideredwithin this section demonstrates the ability of the fully-coupled CFDmodel to simulate the scour processes beneath submarine pipelinesover a fairly wide range of Keulegan–Carpenter number. Based on theprevious comparisons, the morphological evolutions of the scour pro-cess seem to be reasonably captured, both in terms of the predicted
10−2 10−1 10010−1
100
101
θ
T*
Present − scour1/50(θ)−5/3
Fig. 8. Comparison of computed and empirical scour time scales. Note that the results forKC = 11 (θ = 0.19) are based on the computed scour curve in Fig. 2b to t∗ = 1.
0 1 2 3 4 5 6 7 8 90
0.2
0.4
0.6
t*
S/D
(a) KCi=30 to KCf=5.6
0 1 2 3 4 5 60
0.2
0.4
0.6
t*
S/D
(b) KCi=25.3 to KCf=5.6
Fig. 9. Computed scour depth time series for backfilling cases with KCf = 5.6. The dashedlines indicate the equilibrium scour predicted by Eq. (31) with KC = KCf.
−4 −2 0 2 4−1
0
1
2
x/D
y/D
−4 −2 0 2 4−1
0
1
2
x/D
y/D
(a) KCi=30 to KC
f=5.6
(b) KCi=25.3 to KC
f=5.6
Fig. 10. Computed backfilling scour profiles (full lines) with KCf = 5.6. The additionaldashed lines indicate the profile used as the initial condition.
18 D.R. Fuhrman et al. / Coastal Engineering 94 (2014) 10–22
scour depths as well as the rate of the scour evolution, as evidenced bythe computed time scales.
5. Simulation of wave-induced backfilling
An important, but seemingly less-studied, process is the pipelinescour profile response to a change in wave climate. For example, ascour profile developed under more violent (storm) wave conditionscan be expected to backfill when calmer conditions emerge. Fredsøeet al. (1992) demonstrated experimentally that in such transitional sit-uations, the equilibrium scour depth is ultimately determined by thenewKC value. The ability ofmodern CFD tools to simulate the backfillingprocess under a change of wave climate has yet to be demonstrated. It isnoted, however, that backfilling in the form of self-burial of pipelines intrenches has been studied numerically by Liang et al. (2005).
To investigate the ability of the present fully-coupled model to sim-ulate the backfilling process caused by a change inwave climate, wewillnow consider the evolution of scour beds developed under an initiallylarge KCi value, after being subject to a wave climate corresponding tosmaller final KCf values. Attention will be restricted to a sub-set of thewave climates studied within the previous section, as model equilibri-um scour values have already been systematically established forthese conditions.More specifically, wewill consider the simulatedmor-phological evolution of scoured beds developed from cases having thelargest two KC values considered (i.e. KCi = 30 and 25.3) subject tonew wave climates corresponding to two smaller KC values: KCf = 5.6and 15. Note that the KC = 11 climate is not considered for backfilling,to avoid the resonance phenomenon discussed previously.
For the present purposes, model initial conditions are taken directlyfrom the previously described scour simulations at a time after the re-spective equilibrium scour depths have been achieved. These initial pro-files will be illustrated directly in forthcoming figures. Following asimilar methodology as before, the initially scoured bed is first intro-duced to the new wave climate for a warm-up period correspondingto 10Tw, withmorphology switched off. The end of this warm-up periodwill again be denoted t = 0, after which time the bed morphology isagain allowed to evolve freely. Individual simulations have been moni-tored, and are stopped when an apparent new equilibrium is reached,similar to before.
Results from the two backfilling simulations with the lowest KCf =5.6 will first be considered. The resulting time series of the simulatedscour depth are presented in Fig. 9. From these cases it is seen that,when subject to the lower KC value, the model indeed tends to backfillto a reduced scour depth. More specifically, given sufficient time, themodel tends to backfill to a scour depth similar to the previouslyestablished equilibrium value (compare with Fig. 2a). The equilibriumvalue is again in good agreement with the target predicted by Eq. (31)with KC = KCf, depicted by the horizontal dashed line, in accordancewith the findings of Fredsøe et al. (1992). In the present simulations,the backfilling seems to follow two distinct phases: The first (relativelyrapid) backfilling stage corresponds to an initial redistribution of sedi-ments in the immediate vicinity of the pipeline, whereas the secondpost-redistribution stage demonstrates a much slower progression to-ward the new equilibrium value.While the evolution of both time seriesis initially smooth, secondary fluctuations in the bed elevation are morepronounced in Fig. 9a than b. Animation of these two cases has revealedthat the bed is in fact similarly dynamic; For the simulation shown inFig. 8b, however, it turns out that the bed directly beneath the pipelinecenterline happens to be relatively stable in this particular case, thoughlarger fluctuations are more evident a small distance to the side.
The final backfilled bed profiles for the two cases with KCf = 5.6 arepresented in Fig. 10 as the full lines. For direct comparison, the initialcondition profiles taken from the scour simulations using larger KCivalues are likewise presented by the dashed lines, making the amountof backfilling readily apparent. From these profiles it is seen that, afterbeing subject to the lower KCf, the model consistently evolves to a
backfilled profile closely resembling the shape of the equilibrium profilefrom the previously considered scour simulation (compare withFig. 3a). This is true not only in terms of the scour depth, as alreadydiscussed (see Fig. 9), but also for other profile features e.g. the build-up and location of multiple shoulders on either side of the pipeline.Hence, these simulations support the contention of Fredsøe et al.(1992) that the features of a scour hole under an altered wave climatewill be largely dominated by the new KCf value. Similar experimentalfindings have been reported for backfilling of scour holes around pilesby Sumer et al. (2013).
As in the previously discussed scour simulations, the importanceof including suspended sediment in the simulation of the backfillingprocess is emphasized. As a demonstration of this, Fig. 11 depicts abed profile from a re-simulation of the backfilling case from KCi =30 to KCf = 5.6, but now with suspended sediment switched off.Here it is seen that the backfilled scour profile differs qualitativelyin shape from the full model simulation (Fig. 10a), instead fairly rapidlyevolving to a “w” shape,with a pronounced secondary peak beneath the
−4 −2 0 2 4−1
0
1
2
x/D
y/D
KCi=30 to KC
f=5.6 (no suspended sediment)
Fig. 11. Computed backfilling scour profile (full lines, taken at t∗=1) going from KCi=30to KCf = 5.6, but with suspended sediment switched off. The additional dashed line indi-cates the profile used as the initial condition. This figure can be compared to Fig. 10a,which is based on the full model i.e. including suspended sediment.
19D.R. Fuhrman et al. / Coastal Engineering 94 (2014) 10–22
pipeline center. The reason for this development is, again, due toconverging–diverging effects in the flow beneath the pipeline. Thesecreate period-averaged bed shear stresses directed toward the pipelinecenter which are in turn responsible for the large deposition corre-sponding to the secondary peak. Clearly, suspended sediment plays amajor role in continuously re-distributing sediment around the pipelineduring the backfilling process, at least in the conditions considered.
The time series of the scour depth for the other two backfilling caseswith KCf = 15 are similarly shown in Fig. 12. As the differences in KCiand KCf are not as significant as in the previous cases, the level ofbackfilling is expectedly not nearly as pronounced. Nevertheless, thetime series in Fig. 12 again demonstrate a backfilling evolution to anequilibrium scour depth in line with the new KCf. Likely owing to thegreater similarity in the initial and final profiles, the evolution is notcharacterized by distinct redistribution and longer-term backfillingphases, in contrast to the time series shown previously in Fig. 9. Rather,the simulated backfilling in these cases seems to be dominated only byan initial redistribution phase, with the new equilibrium scour depthseemingly reached during this process. This difference may not be alto-gether surprising, again, due to the smaller differences in KCi and KCf. Itcan be noted that the redistribution time (t∗=2− 3) actually takes lon-ger in these simulations, than observed e.g. in Fig. 9. This is seeminglyconsistent with expectations based on the lower Shields parameter inthe present situation (θ = 0.10, versus θ = 0.13 for the cases withKCf = 5.6; see again Table 1).
0 1 2 3 4 50
0.2
0.4
0.6
t*
S/D
0 1 2 3 4 50
0.2
0.4
0.6
t*
S/D
(a) KCi=30 to KC
f=15
(b) KCi=25.3 to KC
f=15
Fig. 12. Computed scour depth time series for backfilling cases with KCf =15. The dashedlines indicate the equilibrium scour predicted by Eq. (31) with KC= KCf.
The corresponding initial and final computed bed elevation profilesare depicted in Fig. 13 for the two cases with KCf = 15. The computedbackfilled profiles again reasonably resemble the equilibrium scour pro-file obtained for this KC in Fig. 3c. This is especially the case for thebackfilling profile depicted in Fig. 13b. Notably, in this case, even thoughthe change in the actual scour depth is not very pronounced, the overallprofile has changed much more dramatically.
The equilibrium backfilling depths from the four cases consideredare summarized versus their respective KC= KCf values in Fig. 14, indi-cated by the asterisks. Also repeated for comparison, are the computedequilibrium scour data from the present study taken from Fig. 6 (cir-cles). As a whole, the values clearly support that Eq. (31), indicated bythe full line, is generally valid for both the equilibrium scour, as wellas backfilling depths after a change in wave climate beneath submarinepipelines.
The backfilling time scale is also of interest. Similar to the previouslyestimated scour time scale, this has been estimated from the backfillingtime series via the integral
T� ¼Z t�min
0
S−Smin
S0−Smindt�: ð35Þ
The resulting (dimensionless as well as dimensional) values are tab-ulated in Table 2. The dimensionless time scales are also plotted versustheir respective backfilling Shields parameter θ, corresponding to theKCf conditions, in Fig. 15. While the present comparison is limited toonly a few cases, the results suggest that the backfilling time scales,for the present conditions, are larger than the corresponding expectedscour time scale, represented by the full line corresponding toEq. (34). The dimensionless backfilling time scales are generally T∗ =O(1). This is the same order of magnitude as found experimentally byFredsøe et al. (1992), who under reasonably similar backfilling condi-tions with the present grain size d = 0.19 mm, report the range T∗ =1.9 − −5.5. The results for both of the two KCf values considered alsosuggest that, for a given KCi, the backfilling time scale expectedly be-comes smaller as the differences in KCi and KCf are reduced. This is qual-itatively also in accordance with the experimental findings of Fredsøeet al. (1992).
While there are certainly similarities in the computed time scaleswith the previous experimental observations of Fredsøe et al. (1992),there are also notable discrepancies with the experimentally observedbackfilling process. These should also be discussed, as they make amore systematic parameterization of the backfilling time scale difficult,
−4 −2 0 2 4−1
0
1
2
x/D
y/D
−4 −2 0 2 4−1
0
1
2
x/D
y/D
(a) KCi=30 to KC
f=15
(b) KCi=25.3 to KC
f=15
Fig. 13. Computed backfilling scour profiles with KCf=15. The additional dashed lines in-dicate the profile used as the initial condition.
10−2 10−1 10010−1
100
101
θ
T*
Present − backfilling1/50(θ)−5/3
Fig. 15. Summary of computed backfilling time scales. The full line represents the regres-sion curve (Eq. (34)) for the scour time scale. It is emphasized that this curve serves only asa reference, rather than a target, for the backfilling time scale.
100 101 10210−1
100
KC
S/D
Present − scourPresent − backfilling0.1(KC)0.5
Fig. 14. Summary of computed equilibrium scour depth for scour (circles) and backfilling(asterisks) cases.
20 D.R. Fuhrman et al. / Coastal Engineering 94 (2014) 10–22
at least based on the present results. Most notably, the simulatedbackfilling process seems heavily influenced by the initial redistributionof sediments in the vicinity of the pipeline. This consistently leads tobackfilling time series that are initially downward-concave, as seen inFigs. 9 and 12. In contrast, the experimentally observed backfillingtime series, e.g. Fig. 5 of Fredsøe et al. (1992), are initially upward-concave, tending to their equilibrium value more smoothly than pre-dicted by any of the present simulations. The suspected influence ofthe initial profile conditions on the backfilling time is further supportedby the reduction in T∗ by a factor of 2 to 3, induced by only changingfrom KCi = 30 to 25.6. This is much more dramatic than would be ex-pected based on the experimental observations of Fredsøe et al. (1992).
Due to these qualitative discrepancies with experimental obser-vations, as well as the limited number of backfilling cases considered,an attempt at formulating a more precise KC dependence on thebackfilling time scale will not be made here, though this remainsan important and open question. Nevertheless, the present workdemonstrates the ability of fully-coupled hydrodynamic andmorphodynamic CFD models to predict both wave-induced scour,as well as backfilling (due to a change in wave climate) beneath sub-marine pipelines to similar profile equilibria. This is an important,and by no means trivial step, given the well-known difficulties inmaking accurate sediment transport and morphological predictionsin the coastal environment. Establishing a methodology for system-atically determining the backfilling time scale is necessarily left asfuture work.
6. Remarks on practical application
It is finally of interest to consider the practical interpretation of thevarious simulations considered herein. The present work has focusedexclusively on simulating scour and backfilling processes beneath sub-marine pipelines at physical model scales i.e. specifically with pipelinediameter D = 0.03 m and sediment size d = 0.19 mm. The grain size
Table 2Summary of backfilling time scales.
KCi KCf T (s) T∗
30 5.6 340 4.025.3 5.6 105 1.230 15 231 2.725.3 15 112 1.3
and other sediment characteristics can conveniently be taken to alreadyrepresent full scale. Differences in the scour area (proportional to D2)and scour rates (principally governed by θ)must be accounted for, how-ever, when scaling morphological times. Based on the time scalings inEqs. (30) and (34), model and full scale morphological timesmay be re-lated according to:
tfulltmodel
¼ Dfull
Dmodel
� �2 θfullθmodel
� �−5=3
: ð36Þ
Defining a geometric scale factor χ, and making use of standardFroude scaling, wemay then relate various characteristic length and ve-locity scales according to
χ ¼ Dfull
Dmodel¼ afull
amodel¼ U2
m;full
U2m;model
: ð37Þ
For hydraulically rough conditions, the wave friction factor can betaken as (Fredsøe and Deigaard, 1992)
f w ¼ 2U2
fm
U2m
¼ 0:04akN
� �−0:25ð38Þ
where Ufm is themaximum (far field) friction velocity. From this Shieldsparameter can be taken to scale according to θ ∼ Ufm
2 ∼ fwUm2 ∼ a−0.25Um
2 ,whence
θfullθmodel
¼ afullamodel
� �−0:25 Um;full
Um;model
!2
¼ χ0:75: ð39Þ
Invoking Eqs. (37) and (39) within Eq. (36) finally leads to themor-phological time scaling
tfulltmodel
¼ χ0:75: ð40Þ
which has maintained strict Froude similitude for the governing waveparameters and pipeline dimensions.
In the present work, combined scour and backfilling simulationsspanning dimensionless times t⋆≈ 15 have been achieved (e.g. combin-ing the scour with KC=30 from Fig. 2g with the subsequent backfilling
21D.R. Fuhrman et al. / Coastal Engineering 94 (2014) 10–22
with KC = 5.6 from Fig. 9a), corresponding to a temporal model-scaleduration of tmodel ≈ 1280 s. Now, full scale high-capacity submarinepipelines commonly have diameters of O(1 m). Taking the geometricscale factor χ = Dfull/Dmodel = 100 for simplicity within Eq. (40), thenimplies that the simulated durations can be taken to represent physicaldurations of up to say tfull≈ 12 h=O(1 day) at full practical scales. Sim-ilar values may also be obtained directly from Eq. (36) based on morefreely assumed ratios for D and θ.
Hence, provided that model scales are utilized and that simulationsare restricted to two-dimensions, the fully-coupled CFDmodel present-ed herein seems capable of simulating combined scour and backfillingevolutions over temporal durations representative of individual storms.(Consider e.g. that sea states are commonly taken to represent 6 h inter-vals in coastal engineering practice.) Such temporal durations can beachieved in simulations lasting O(1 week) on a single modern proces-sor. The ability to simulate such processes is of major practical impor-tance e.g. within the fatigue life assessment of industrial submarinepipelines. It is again emphasized that the present work has notemployed any morphological acceleration techniques, though thesecould potentially increase computational efficiency further.
Similar scour or backfilling simulations at full scale have not beenattemptedhere, but can be expected to require increased computationaldemands. This is due both to the larger flow velocities, which can re-quire smaller time steps to maintain a given Courant number, as wellas to the significant increase in the ratio of large- (scaling approximatelywith D) and small-scales (scaling approximately with d) requiring res-olution. Nevertheless, this paper represents a step in the advanced useof CFD codes for simulating coupled flow and morphology (both scourand backfilling) beneath submarine pipelines at relevant practicalscales, as envisioned recently e.g. by Sumer (2014). Obviously, the scal-ing suggested within this section does not account for e.g. three-dimensional effects on the scour/backfilling processes or scaling issuesassociatedwith larger Reynolds numbers at full versus laboratory scales.
−4 −2 0 2 4−1
0
1
2
3
x/D
y/D
Fig. A.16. Physical set up for the flow around a near wall cylinder with gap-to-diameterratio e/D = 1.
7. Summary and conclusions
A fully-coupled hydrodynamic and morphodynamic CFD model ispresented for simulating wave-induced scour and backfilling processesbeneath submarine pipelines. The hydrodynamic model is based onReynolds-averaged Navier–Stokes (RANS) equations, coupled withtwo-equation k − ω turbulence closure. The sediment transportmodel consists of separate bed and suspended load descriptions, the lat-ter based on a turbulent diffusion equation coupled with a referenceconcentration function near the sea bed boundary. Bed morphology isbased on the sediment continuity (Exner) equation. The present simula-tions have utilized continuous morphological updating in time, ratherthan e.g. averaging morphological rates over selected time scales. Inthis fashion, the simulations illustrate the ability to simulate fully-coupled hydrodynamic and morphological evolutions based on contin-uous feedback, similar to what occurs in reality.
The fully-coupled model has been tested for wave-induced scourevolution beneath submarine pipelines for cases having Keulegan–Carpenter numbers ranging from KC = 5.6 to 30. The model is dem-onstrated to generally lead to increased scour with increasing KC,with equilibrium scour depths that are consistently in line with pre-viously established relations based on experimental observations.The lone exception is the case considered with KC = 11, which fol-lows a markedly different two-stage scour process. This qualitativedifference has been explained as due to a resonance phenomenonwhich excites a vortex ripple-like instability beneath the pipeline.The results for the various KC generally maintain reasonable symme-try about the pipeline, both in the near and far field regions. Analysisof the scour time scales, while scattered, are also reasonably in linewith experimental expectations for the (albeit limited) range ofShields parameter considered.
The model has subsequently been tested for simulating backfillingprocesses beneath submarinepipelines induced by a change inwave cli-mate. This has been done by subjecting initial conditions from selectedscour simulations having larger KC values, to previously consideredwave climates having lower KC values. The presented simulations illus-trate the ability of the present fully-coupledmodel to predict backfillingevolution to equilibrium depths closely corresponding to the new KCvalue. Thedeveloped profiles also typically resemble those from the cor-responding scour simulations having the same KC. Hence, themodel re-sults support the notion of Fredsøe et al. (1992) that the scour depth (aswell as overall profile features), will be governed by the new wave cli-mate in such backfilling situations.
Investigation of the simulated backfilling time scales reveals thatthey are roughly the correct order ofmagnitude as suggested by the lim-ited experimental data in the literature. However, qualitative differ-ences in the simulated/experimental processes are noted, with thesimulated backfilling evolution (and therefore the time scale) demon-strating likely dependence on the specific initial conditions used.Hence, a more systematic quantification regarding the KC dependenceon the backfilling time scale is left as future work.
While the present work has focused exclusively on simulating thescour and backfilling processes beneath submarine pipelines at modelscales, the simulated sequences of scour and backfilling achieved hereinhave been estimated to represent temporal durations of up to approxi-mately 12 h at full practical scales.
Acknowledgments
The first three authors acknowledge support from the EuropeanUnion project ASTARTE—Assessment, STrategy And Risk Reductionfor Tsunamis in Europe, Grant no. 603839 (FP7-ENV-2013.6.4-3).The second author also acknowledges the support of a postdoctoralgrant from The Scientific and Technical Research Council of Turkey(TUBITAK, Grant No. 2219). The second and third author additionallyacknowledge Innovative Multi-purpose Offshore Platforms: Plan-ning, Design and Operation (MERMAID), 20122016, Grant Agree-ment No. 288710 of European Commission, 7th FrameworkProgramme for Research.
Appendix A. Oscillatory flow around a cylinder near a wall
In this Appendix A, results from a simulation involving oscillatoryflow around a cylinder near a wall will be presented, which bears obvi-ous similarity tomany of the scour and backfilling conditions developedwithin the paper. The physical case considered is depicted in Fig. A.16,where the gap-to-diameter ratio is e/D=1. The oscillatory flow consid-ered corresponds to that with KC = 11 within the paper. The modelsetup, including domain size and mesh resolutions, is as described inSection 4. As a check of convergence, it can be noted that further refine-ments in the mesh yield the same results as shown below.
0 180 360 540 720−1
0
1u/U
1m
(a)
0 180 360 540 720−2
−1
0
1
2
ωt (o)
c L
(b)
Fig. A.17. (a) Reference free stream velocity and (b) comparison of computed (full line)and measured (circles, digitized from Sumer et al., 1991) lift coefficient cL for the flowaround a cylinder near a wall with gap-to-diameter ratio e/D=1 for flow conditions hav-ing KC = 11.
22 D.R. Fuhrman et al. / Coastal Engineering 94 (2014) 10–22
As validation, results from the experimental work of Sumer et al.(1991) will be utilized, who considered a case having almost identicalconditions (e/D= 1 and KC = 10). As preliminary (qualitative) valida-tion, visual inspection of the flow patterns and series of vortex sheddingthat occurs around the pipeline have first been confirmed as essentiallythe same as described by Sumer et al. (1991), see their Fig. 12a. This isfurther evidenced quantitatively through the comparison of the com-puted and measured lift coefficients cL provided in Fig. A.17. As seenthere, the model predicts reasonably both the amplitude and phase ofthe lift. The model results are generally comparable to those achievedpreviously by Liang andCheng (2005a)whomade a similar comparison,with the present results having a slightly larger lift amplitude.
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