Numerical Investigation Of flow and Scour Around Avertical Circular Cylinder

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rsta.royalsocietypublishing.org Research Cite this article: Baykal C, Sumer BM, Fuhrman DR, Jacobsen NG, Fredsøe J. 2015 Numerical investigation of flow and scour around a vertical circular cylinder. Phil. Trans. R. Soc. A 373: 20140104. http://dx.doi.org/10.1098/rsta.2014.0104 One contribution of 12 to a Theme Issue ‘Advances in fluid mechanics for offshore engineering: a modelling perspective’. Subject Areas: fluid mechanics, ocean engineering, civil engineering Keywords: scour, morphology, sediment transport, turbulence modelling Author for correspondence: C. Baykal e-mail: [email protected] Present address: Middle East Technical University, Department of Civil Engineering, Ocean Engineering Research Centre, Dumlupinar Blvd. 06800, Cankaya, Ankara, Turkey. Electronic supplementary material is available at http://dx.doi.org/10.1098/rsta.2014.0104 or via http://rsta.royalsocietypublishing.org. Numerical investigation of flow and scour around a vertical circular cylinder C. Baykal 1,, B. M. Sumer 1 , D. R. Fuhrman 1 , N. G. Jacobsen 2 and J. Fredsøe 1 1 Technical University of Denmark, Department of Mechanical Engineering, Nils Koppels Alle 403, Kongens Lyngby 2800, Denmark 2 Deltares, Department of Coastal Structures and Waves, Boussinesqweg 1, 2629 HV Delft, The Netherlands Flow and scour around a vertical cylinder exposed to current are investigated by using a three-dimensional numerical model based on incompressible Reynolds- averaged Navier–Stokes equations. The model incorporates (i) k-ω turbulence closure, (ii) vortex- shedding processes, (iii) sediment transport (both bed and suspended load), as well as (iv) bed morphology. The influence of vortex shedding and suspended load on the scour are specifically investigated. For the selected geometry and flow conditions, it is found that the equilibrium scour depth is decreased by 50% when the suspended sediment transport is not accounted for. Alternatively, the effects of vortex shedding are found to be limited to the very early stage of the scour process. Flow features such as the horseshoe vortex, as well as lee-wake vortices, including their vertical frequency variation, are discussed. Large-scale counter-rotating streamwise phase-averaged vortices in the lee wake are likewise demonstrated via numerical flow visualization. These features are linked to scour around a vertical pile in a steady current. 1. Introduction Flow and scour around a vertical cylinder have been investigated extensively over the past four decades, with specific relevance to the flow and scour around bridge piers in river engineering, and around pile foundations in marine and coastal engineering. While much has been written on this subject, comparatively few studies have been presented involving detailed 2014 The Author(s) Published by the Royal Society. All rights reserved. on August 17, 2015 http://rsta.royalsocietypublishing.org/ Downloaded from

description

Flow and scour around a vertical cylinder exposed tocurrent are investigated by using a three-dimensionalnumerical model based on incompressible Reynolds averagedNavier–Stokes equations. The modelincorporates (i) k-ω turbulence closure, (ii) vortexsheddingprocesses, (iii) sediment transport (both bedand suspended load), as well as (iv) bed morphology.The influence of vortex shedding and suspendedload on the scour are specifically investigated. For theselected geometry and flow conditions, it is foundthat the equilibrium scour depth is decreased by50% when the suspended sediment transport is notaccounted for. Alternatively, the effects of vortexshedding are found to be limited to the very earlystage of the scour process. Flow features such asthe horseshoe vortex, as well as lee-wake vortices,including their vertical frequency variation, arediscussed. Large-scale counter-rotating streamwisephase-averaged vortices in the lee wake are likewisedemonstrated via numerical flow visualization. Thesefeatures are linked to scour around a vertical pile in asteady current.

Transcript of Numerical Investigation Of flow and Scour Around Avertical Circular Cylinder

Page 1: Numerical Investigation Of flow and Scour Around Avertical Circular Cylinder

rsta.royalsocietypublishing.org

ResearchCite this article: Baykal C, Sumer BM,Fuhrman DR, Jacobsen NG, Fredsøe J. 2015Numerical investigation of flow and scouraround a vertical circular cylinder. Phil. Trans.R. Soc. A 373: 20140104.http://dx.doi.org/10.1098/rsta.2014.0104

One contribution of 12 to a Theme Issue‘Advances in fluid mechanics for offshoreengineering: a modelling perspective’.

Subject Areas:fluid mechanics, ocean engineering,civil engineering

Keywords:scour, morphology, sediment transport,turbulence modelling

Author for correspondence:C. Baykale-mail: [email protected]

†Present address: Middle East TechnicalUniversity, Department of Civil Engineering,Ocean Engineering Research Centre,Dumlupinar Blvd. 06800, Cankaya, Ankara,Turkey.

Electronic supplementary material is availableat http://dx.doi.org/10.1098/rsta.2014.0104 orvia http://rsta.royalsocietypublishing.org.

Numerical investigation offlow and scour around avertical circular cylinderC. Baykal1,†, B. M. Sumer1, D. R. Fuhrman1,

N. G. Jacobsen2 and J. Fredsøe1

1Technical University of Denmark, Department of MechanicalEngineering, Nils Koppels Alle 403, Kongens Lyngby 2800, Denmark2Deltares, Department of Coastal Structures and Waves,Boussinesqweg 1, 2629 HV Delft, The Netherlands

Flow and scour around a vertical cylinder exposed tocurrent are investigated by using a three-dimensionalnumerical model based on incompressible Reynolds-averaged Navier–Stokes equations. The modelincorporates (i) k-ω turbulence closure, (ii) vortex-shedding processes, (iii) sediment transport (both bedand suspended load), as well as (iv) bed morphology.The influence of vortex shedding and suspendedload on the scour are specifically investigated. For theselected geometry and flow conditions, it is foundthat the equilibrium scour depth is decreased by50% when the suspended sediment transport is notaccounted for. Alternatively, the effects of vortexshedding are found to be limited to the very earlystage of the scour process. Flow features such asthe horseshoe vortex, as well as lee-wake vortices,including their vertical frequency variation, arediscussed. Large-scale counter-rotating streamwisephase-averaged vortices in the lee wake are likewisedemonstrated via numerical flow visualization. Thesefeatures are linked to scour around a vertical pile in asteady current.

1. IntroductionFlow and scour around a vertical cylinder have beeninvestigated extensively over the past four decades,with specific relevance to the flow and scour aroundbridge piers in river engineering, and around pilefoundations in marine and coastal engineering. Whilemuch has been written on this subject, comparativelyfew studies have been presented involving detailed

2014 The Author(s) Published by the Royal Society. All rights reserved.

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three-dimensional numerical modelling of such processes. Important contributions to thenumerical investigation of both flow and scour around vertical piles include [1–15].

The research presented in this paper is largely motivated by the previous work of Roulund et al.[3]. Their model was based on cohesionless sediment and simulated scour in the live-bed regime.However, unsteady flow structures such as lee-wake vortices (due to mesh resolution), as well assuspended sediment transport, were not incorporated in their simulations. Recently, Stahlmann[14] has investigated the effects of these processes on the scour case studied by Roulund et al. [3].Their study further improved the results of scour depths achieved by Roulund et al. [3].

This study focuses on incorporating the effects of lee-wake vortices, as well as suspendedsediment transport, within the three-dimensional numerical modelling of scour around a verticalpile. The fully coupled model provides a detailed overview of both phase-averaged (time-averaged over a certain number of periods of vortex shedding until no significant changesare observed in the phase-averaged quantities), as well as phase-resolved (instantaneous), flowstructures in the lee wake of a vertical pile placed on a plane bed. Kirkil et al. [16] investigatedcoherent structures present in the flow field around a circular cylinder located in a scourhole with large-eddy simulation and laboratory-flume visualizations. Recently, Petersen [17]experimentally investigated the edge scour around scour protections of offshore wind turbinefoundations, with special emphasis on the flow features in the lee-wake region. In their 2008study, Kirkil et al. [16] addressed the presence of the phase-averaged streamwise vorticity fieldsin the near-wake region of the pile for a scoured bed. Petersen [17] emphasized the prominenteffects of these vortices on the lee-wake scour around a scour protection of a vertical pile. Petersenshowed that the edge scour at the downstream side of the scour protection is mainly caused bythese vortices, two large-scale counter-rotating streamwise vortices in the lee wake, which havebeen visualized by sand. The effects of roughness (due to stone protection around the pile) onthe formation of these vortices could not be interpreted, however, as a comparable study has yetto be undertaken with a rigid plane bed. Regarding the lee-wake vortices, Jacobsen et al. [15]have studied transient effects from vortex shedding on scour patterns. They have shown thatthe process of vortex shedding is affected by the sheared flow itself and the scour depth aroundthe cylinder, and the shedding frequencies are found to vary along the height of the cylinder forvarious scour depths. In the present work, such a study will be directly carried out through three-dimensional numerical simulations of the flow around a cylinder placed on a rigid and planesmooth bed.

This paper is organized as follows. The hydrodynamic and turbulence models used aredescribed in §2, and the sediment transport and morphological models are described in §3.Simulations regarding the flow features around a vertical circular pile placed on a rigid bedunder steady flow conditions are described and discussed in §4. These will serve to validate themodel as well as to help visualize phase-averaged flow features for the given flow conditions.Simulations of scour around a circular pile, to investigate the effect of unsteady flow structuresand the suspended sediment transport, are subsequently presented and discussed in §5. Finally,conclusions are drawn in §6.

2. Hydrodynamic and turbulence models

(a) Governing equationsIn this section, a description of the computational model used throughout the present workis provided. The numerical model solves the incompressible Reynolds-averaged Navier–Stokesequations

∂ui

∂t+ uj

∂ui

∂xj= − 1

ρ

∂p∂xi

+ ∂

∂xj

[2νSij + τij

ρ

], (2.1)

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where the mean strain-rate tensor is

Sij = 12

(∂ui

∂xj+ ∂uj

∂xi

). (2.2)

These are combined with the local continuity equation

∂ui

∂xi= 0. (2.3)

Here, ui are the mean (phase-resolved) velocities, xi are the Cartesian coordinates, t is time, p isthe pressure, ν is the fluid kinematic viscosity, ρ is the fluid density and τij is the Reynolds stresstensor, which accounts for additional (normal and shear) stresses due to momentum transfer fromturbulent fluctuations. Throughout the present work, the Reynolds stress tensor will be definedaccording to the constitutive relation

τij

ρ= −u′

iu′j = 2νTSij − 2

3kδij, (2.4)

where δij is the Kronecker delta, νT is the eddy viscosity,

k = 12 u′

iu′i (2.5)

is the turbulent kinetic energy density, and the overbar denotes time averaging. To achieveclosure, the two-equation k-ω turbulence model of [18,19] is adopted. In this model, the eddyviscosity is defined by

νT = kω̃

, ω̃ = max

⎧⎨⎩ω, Clim

√2SijSij

β∗

⎫⎬⎭ , (2.6)

which incorporates a stress-limiting feature, with Clim = 78 . This model additionally uses transport

equations for the turbulent kinetic energy density k

∂k∂t

+ uj∂k∂xj

= τij

ρ

∂ui

∂xj− β∗kω + ∂

∂xj

[(ν + σ ∗ k

ω

)∂k∂xj

], (2.7)

as well as for the specific dissipation rate ω

∂ω

∂t+ uj

∂ω

∂xj= α

ω

k

τij

ρ

∂ui

∂xj− βω2 + σd

ω

∂k∂xj

∂ω

∂xj+ ∂

∂xj

[(ν + σ

)∂ω

∂xj

],

where

σd = H

{∂k∂xj

∂ω

∂xj

}σdo, (2.8)

and H{·} is the Heaviside step function, taking a value of zero when the argument is negativeand a value of unity otherwise. The standard model closure coefficients are used: α = 13

25 , β∗ = 9100 ,

σ = 12 , σ ∗ = 3

5 , and σdo = 18 , β = β0fβ and β0 = 0.0708, with

fβ = 1 + 85χω

1 + 100χω, χω ≡

∣∣∣∣ΩijΩjkSki

(β∗ω)3

∣∣∣∣ , Ωij = 12

(∂ui

∂xj− ∂uj

∂xi

). (2.9)

(b) Boundary conditionsThe hydrodynamic model described above is subject to the following boundary conditions.At friction wall boundaries, a no-slip condition is imposed whereby velocities are set to zero.Alternatively, Neumann conditions are applied to the three components of the velocity and scalarhydrodynamic quantities at the symmetry boundaries, which is in line with the approach in[3]. The ‘lid’ at the top boundary is not rigid and finite velocities are allowed perpendicular tothe upper boundary. The application of a ‘lid’ at the top boundary rather than a free surfacefacility is a commonly used and valid approach for the flow conditions without large free surface

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disturbances as in the case of wave modelling cases. Regarding the effect of the ‘lid’, this issuehas been discussed in [3] in conjunction with their numerical simulations of flow and scouraround a pile. With a free surface, and for sufficiently large Froude numbers, Fr = U/(gh)1/2,in which h is the height of the computational domain, U is the mean (undisturbed depth-averaged) flow velocity and g is the gravitational acceleration, the free surface will, in real life,exhibit some variation in the vicinity of the cylinder; there will be a run-up in front of thecylinder and a depression around the side edge and at the back of the cylinder. Roulund et al.[3, pp. 373–378] pointed out that this variation ceases to exist for Froude numbers smaller thanO(0.2). Now, in the present case, the Froude number, Fr = O(0.4), is not radically different fromO(0.2), and therefore the flow will be fairly well represented in the lid simulations althoughsome variations may be expected such as the absence of a flow in the radial direction, caused,otherwise, by the head difference between the surface elevation in front and at the side edge ofthe cylinder. Regarding the side boundaries, for all the quantities except the velocity, U, zero-gradient boundary conditions are applied, and for the velocity the slip type boundary condition,in which the normal component of the velocity is taken to be zero, is applied.

To drive the flow conditions within the model, prior to the actual simulation, for the samecalculation domain without the cylinder, the flow is first driven by a horizontal pressuregradient specified as a constant based on the desired undisturbed friction velocity, Uf = [−(h/ρ) ×(∂p/∂x)]0.5. As the steady-state flow condition is reached, the U, k and ω profiles are taken to beused as the inlet boundary conditions in the actual simulation for the calculation domain with thecylinder. At the inlet, for the remaining quantities, Neumann boundary conditions are applied.At the opposite right-hand boundary (outlet), zero-gradient boundary conditions are imposed forall quantities except the pressure term, p, which has been taken as zero at the outlet boundary.

At the bottom as well as the cylinder, a generalized wall function approach developed byFuhrman et al. [20] is used. Under this approach, the friction velocity Uf is determined from thetangential velocity at the nearest cell centre based on the profile of Cebeci & Chang [21],

uUf

= 2∫ y+

c

0

dy+

1 + [1 + 4κ2(y+ + �y+cc)2C]1/2

, (2.10)

C =[

1 − exp

(− (y+ + �y+

cc)25

)]2

(2.11)

and �y+cc = 0.9

[√k+

s − k+s exp

(− k+

s6

)]; (2.12)

Cebeci & Chang [21] generalized the van Driest [22] profile to incorporate potential roughnesseffects. Here, yc = �y/2 is the normal distance from the wall to the cell centre, expressed in termsof wall coordinates as y+

c = ycUf/ν, �y is the near-wall cell thickness, κ = 0.4 is the von Karmanconstant and ks is Nikuradse’s equivalent sand grain roughness. At the bed ks = 2.5d is used,whereas at the cylinder ks is kept sufficiently small (ks = 1 × 10−5 m) such that it is effectivelymodelled as hydraulically smooth. Here, d is the grain size. The friction velocity is then usedwithin the generalized wall functions for k and ω in the cells nearest to the wall, according to

k

U2f

= min{

A�y+2,1√β∗

}(2.13)

andων

U2f

= max{

B�y+2 ,

1√β∗κ�y+

}. (2.14)

The first arguments in these functions ensure that these variables follow their proper scalingk ∼ y2 and ω ∼ 1/y2 for near-wall cells within the viscous sub-layer (e.g. [18]). The valuesA = 1/(δ+2√β∗) = 0.02466 and B = δ+/(

√β∗κ) = 96.885 are used, which ensure a continuous

transition to the (fully turbulent) second arguments at �y+ = δ+, where δ+ = 11.626 is taken asthe viscous sub-layer thickness (in dimensionless wall coordinates).

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zxy

bed: wall

side surface:symmetryboundary

outlet

top surface:symmetry boundary

side surface:symmetryboundary

inlet

pile surface:wall

Figure 1. Boundaries of the computational domain on an example of mesh used for scour calculation.

Table 1. Characteristics of mesh in the numerical simulations.

calculation type rigid bed or live-bed scour rigid bed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

height of calculation domain, h 2D 3D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

length of calculation domain, l 20D 20D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

width of calculation domain,w 15D 15D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

total number of cells 104 832 237 248. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

number of cells across the water depth 16 22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

number of cells around the pile perimeter 96 128. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(c) Computational meshThe calculation domain is discretized into finite volumes of quadrilateral blocks in varying shapesand dimensions. Figure 1 shows the boundaries of the domain on an example of computationalmesh used for scour calculations. During the live-bed calculations, the mesh is continuouslyupdated to adjust the changes of bed topography. For further details, see §3b.

The calculation domain has the following dimensions: length, l = 20D, width, w = 15D, andheight, h = 2D (or 3D), in which D is the pile diameter. The total number of cells is O(105). Table 1summarizes the mesh characteristics for different domain heights and calculation types. The pileis located at the centre of the domain (x = 0 and y = 0). As an indication of computational expense,a fully coupled hydrodynamic and morphological calculation lasting 1 min of physical time forthe three-dimensional scour around a vertical pile problem requires approximately 10 days ofCPU time, when simulated in parallel on eight modern processors. To give an indication of theCPU times, the authors would like to note that, for the same computational mesh and using thesame computational power, only the hydrodynamic calculations for 1 min of physical time lastapproximately 1.5 days, and 3 days if the suspended sediment calculations are also included.In all the calculations, the time step is kept as variable to ensure that certain Courant numbersselected separately for the numerical stability in each of hydrodynamic, suspended sediment andmorphological calculations are not exceeded. Throughout the study, it has been seen that themorphological calculations have governed the speed of the simulations. The authors also would

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like to note that considerable effort has been put into optimizing the computational mesh for thebest grid and time convergence. For the scour simulations, finer computational meshes, as usedin rigid bed simulation with h = 3D, could not be used as CPU times increased significantly dueto numerical stability considerations.

In this study, basically, two kinds of calculations are carried out: the rigid bed calculations toinvestigate the flow features only; and scour calculations. In rigid bed calculations, the bottomof the calculation domain is kept rigid by turning off the morphological model, whereas in scourcalculations the bed is continuously updated. The rigid bed simulations are carried out for twodifferent domain heights (h = 2D and 3D), whereas in the scour simulations only the calculationmesh with h = 2D is used.

3. Sediment transport and morphological modelsThis section describes the sediment transport and morphological models used herein. As theimplementation of these models, including a detailed account of numerical aspects, has beendescribed in detail by Jacobsen [23], as well as in the recent publication of Jacobsen et al. [24], onlyessential details are provided in what follows.

(a) Sediment transport modelThe model makes use of separate bed and suspended sediment load descriptions. The rateof bed load sediment transport qB is calculated based on the method described in detail byEngelund & Fredsoe [25], as well as Roulund et al. [3], who generalized the well-known transportformulation of Engelund & Fredsoe [26] to account for three-dimensional effects as well asbed-slope modifications.

The suspended sediment model is, alternatively, based on a turbulent-diffusion equation(e.g. [27, p. 238]) of the form

∂c∂t

+ (uj − wsδj3)∂c∂xj

= ∂

∂xj

[(ν + βsνT)

∂c∂xj

], (3.1)

where c is the suspended sediment concentration, ws is the settling velocity, and βs = 1 is usedthroughout, i.e. the sediment diffusivity is taken as equal to the eddy viscosity. The settlingvelocity is solved for a given grain diameter d using the drag-coefficient approach described in,for example, [27]. Equation (3.1) is solved on a sub-set of the main computational mesh, i.e. wherenear-bed cells below a so-called reference level b are removed, as described in detail by Jacobsenet al. [24, fig. 2].

In all boundaries except the bottom boundary, a zero-flux condition for c is used. At the bottomboundary so-called reference concentration boundary conditions are imposed. More specifically,the method of Engelund & Fredsoe [26] is used, with the concentration at the reference level set to

cb(θ ) = c0

(1 + 1/λb)3 , (3.2)

where c0 = 0.6, and the linear concentration λb is

λ2b = κ2α2

10.013sθ

(θ − θc − π

6μdpEF

), (3.3)

where

pEF =[

1 +(

πμd

6(θ − θc)

)4]−1/4

(3.4)

is the probability of moving grains, and

θ = τ0

ρg(s − 1)d= U2

f(s − 1)gd

(3.5)

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is the Shields parameter, in which τ0 = ρUf2 is the wall shear stress, s = ρs/ρ is the specific gravity

of the sediment grains, and g is the gravitational acceleration. Throughout the present work, thecoefficient of dynamic friction is set to μd = 0.51. The critical Shields parameter θc is computedfrom a base value θc0 = 0.05, accounting for bed-slope effects as in [3]. Following [20], throughoutthe present work, the reference level is taken as b = α1d = 3.5d.

To prevent un-physical ‘overloading’ conditions (i.e. where reference cb is forced to be smallerthan the concentration immediately above) from occurring in the model, the solution suggestedby Justesen et al. [28] is finally invoked. That is, if the concentration close to the bed exceeds thereference concentration cb(θ ), the value in practice is taken from the cell adjacent to the boundary.

(b) Morphological modelThe morphology of the bed elevation hb is based on the sediment continuity (Exner) equation

∂hb

∂t= 1

1 − n

[−∂qBi

∂xi+ D + E

], i = 1, 2, (3.6)

where n = 0.4 is the bed porosity, and D and E are the deposition and erosion stemming fromthe suspended sediment model, respectively. Further details of the evaluation of deposition anderosion terms in the above equation are given in [24], including corrections for the non-slopingbeds. It is stressed that the simulated bed morphology within the present work is continuous,always being based on the instantaneous sediment transport fields, i.e. the model does not makeuse of morphological rates averaged over a certain duration or any other time scale. Accordingly,morphological and hydrodynamic times are equivalent. The temporal integration of (3.6) is hereinbased on the explicit Euler method. For more specific details regarding the numerical evaluationof the three terms within the right-hand side of (3.6), the interested reader is referred to [24]. Itis finally noted that, to prevent excessive erosion induced by the imposed uniform flow at theboundary, the sea bed is fixed at the left (inlet) boundary and relaxed to full morphology over adistance spanning a few pile diameters.

Experience has shown that, if left unchecked, the morphological model can lead to local bedslopes in excess of the angle of repose. To combat such un-physical steepening, the physicallybased sand slide model described in detail in [3] is implemented. In the present work, thissand slide model is activated at positions where the local bed angle exceeds the angle of reposeφs = 32◦ and is de-activated once the local bed angle is reduced to 30.0◦. Additionally, some localfiltering of the bed was necessary in the scour runs for stability reasons. For this purpose, afiltering algorithm based on the least-squares method is applied to the morphological rates inthe near vicinity of the pile to smooth out small-sized (i.e. high wavenumber) ripples that canoccur inside the scour hole adjacent to the pile, where the numerical stability of the mesh motioncomputations becomes critical due to increased rates of change in bed elevations. The strength offiltering inside and around the scour hole is kept variable with the use of a hyperbolic functionbased on the radial distance from the origin of the domain, so that the far-field bed ripples areresolved but the small-sized ripples inside the scour hole are filtered.

The equations constituting the fully coupled model outlined above are solved numericallyusing the open-source CFD toolbox OpenFOAM, v. 1.6-ext, making use of a finite volume spatialdiscretization with a collocated variable arrangement, in conjunction with a standard PIMPLEalgorithm. Again, for further details see [24]. For the simulation in which the vortex shedding isturned off, more diffusive schemes are used for the divergence terms in (2.1).

4. Rigid bed simulationsIn this section, the flow features around a vertical cylindrical pile mounted on a horizontal planesoil bed and subjected to a steady current are investigated through numerical simulations for thetwo meshes listed in table 1. For the simulations in this section, the bed is kept rigid by turningoff the morphological model. The test conditions of the simulations (for both rigid bed and scour

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Table 2. Test conditions in the numerical simulations.

calculation type rigid bed or live-bed scour rigid bed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

domain height, h (cm) 8 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

boundary layer thickness, δ (cm) 8 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

mean flow velocity, U (cm s−1) 41.3 42.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

pile diameter, D (cm) 4 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Reδ,U = Uδ/ν 3.3 × 104 5.1 × 104. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ReD,U = UD/ν 1.7 × 104 1.7 × 104. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

grain size, d50 (mm) 0.17 0.17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

friction velocity, Uf (cm s−1) 1.9 1.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

shields parameter, θ 0.13 0.13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

bed Nikuradse eq. sand roughness, ks,bed = 2.5d50 (cm) 0.0425 0.0425. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

dimensionless bed roughness, k+s,bed = ks,bedUf/ν 8.1 8.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

cylinder Nikuradse eq. sand roughness, ks,cyl (cm) 0.001 0.001. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

simulations) with respect to domain height are given in table 2. As shown previously in table 1, therigid bed simulations are carried out for two different domain heights (h = 2D and 3D), whereasin the scour simulations only the computational mesh with h = 2D is used.

(a) Horseshoe vortexThe main flow features around a vertical cylindrical pile exposed to a steady current on a planebed are the horseshoe vortex formed in front of the pile, the lee-side vortices (usually in the formof vortex shedding), contracted streamlines around the sides of the pile and the down-flow infront of the pile due to deceleration of the flow. The horseshoe vortex in front of the pile is anend result of the adverse pressure gradient on the bed upstream of the pile, which is actuallyinduced by the presence of the pile in a steady current. This adverse pressure gradient enforcesa three-dimensional boundary layer separation just in front of the pile and results in a vorticalstructure surrounding the upstream perimeter of the pile and fading away downstream. The sizeof the horseshoe vortex in front of a circular pile with a smooth bed depends mainly on [29] theratio of the bed-boundary-layer thickness to the pile diameter, δ/D; the pile Reynolds number,ReD = UD/ν (or alternatively the bed-boundary-layer-thickness Reynolds number, Reδ = Uδ/ν),where U is the free-stream velocity; and the ratio of bed roughness to pile diameter, ks/D, in thecase of rough beds.

Figure 2 compares the amplification of the bed shear stress (ατ ) along the x-axis calculatedfrom the model (for h = 2D) with that measured in the smooth bed experiments by Roulund et al.[3] and Hjorth [30] (see fig. 6.18.b on p. 133 of [30] and the Test-1 data in fig. 16 on p. 376 of[3]). Here, the amplification of the bed shear stress is defined as ατ = τ0/τ∞, in which τ0 is the bedshear stress and τ∞ is the undisturbed bed shear stress. The negative bed shear stress correspondsto the location of the horseshoe vortex in front of the pile. Figure 2 shows that the numericaland experimental results agree well outside the horseshoe vortex for x/D < −1.2. However, for−1.2 < x/D < −0.5, the bed shear stress is apparently under-predicted by the numerical modelwith respect to the data given by Hjorth [30], the maximum difference between the model andthe experiment being around 30%. The latter is consistent with the findings given in [3]. No clearexplanation has been given in [3] or found for the observed discrepancy between the numericalmodel and the experiments.

Figure 3 compares the amplification of the bed shear stress defined as the ratio of magnitudeof the bed shear stress vector to the value of the undisturbed bed shear stress (|ατ | = |τ0|/τ∞)

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.........................................................

–4–2.50 –2.25 –2.00 –1.75 –1.50

x/D

–1.25 –1.00 –0.75 –0.50

–3

–2

–1

t 0/t

•0

1

2

data ReD

present model 2 1.7 × 104

1.5 × 104

1.7 × 105

2

1

Hjorth [30]

Roulund et al. [3]

Hjorth [30]

d/D

Figure 2. Bed shear stress amplification along the symmetry line upstream of the pile. Comparison with experimental datagiven by Roulund et al. [3] and Hjorth [30].

1.0(a) (b)

0.8

0.6

0.4

0.2

0–1.0 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0

x/D

–1.0 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0

x/D

y/D

flow

modelflow

experiment

10

10

99

7

71

11

1

1

3

3

3

55

3

Figure 3. Magnitude of bed shear stress amplification. (a) Model. (b) Experiment [30].

obtained from the present model (for h = 2D) and that from [30] smooth-bed experiments (see fig.6.18.b on p. 133 of [30]). The present calculations and the measurements are seen to be in goodagreement in terms of the magnitude of the bed shear stresses around the cylinder. However,the radial location of the maximum bed shear stress is found to be rather different from themeasurements, which might be a possible explanation for the difference between the model andHjorth’s results [30] in figure 2.

(b) Lee-wake flowThe lee-wake flow around the cylinder depends mainly on the pile Reynolds number, ReD. ForReD > 40 (e.g. [31]), the flow in the lee wake becomes unsteady and the vortex-shedding regimebegins to occur with vortices shed alternately at either side of the pile at a certain frequency. Inthis study, ReD is equal to 1.7 × 104, for which the wake flow is given as completely turbulentby Sumer & Fredsoe [31]. These vortices are caused mainly by the rotation in the boundary layerover the surface of the pile. The shear layers emanating from the side edges of the pile roll up toform these vortices in the lee wake of the pile. The emanating position of these shear layers onthe cylinder, the so-called position of boundary layer separation, changes over the height of thecylinder when the velocity profile is not uniform over the depth of the domain.

To investigate the flow features in the lee wake, first, the position of the boundary layerseparation over the height of the cylinder is drawn for the rigid bed numerical model simulations

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−1.5 −1.0 −0.5 0 0.5 1.0 1.50

0.2

0.4

0.6

0.8

1.0

1.2

x/D

z/h

h = 2Dh = 3D

flow

cylinder

Figure 4. Position of boundary layer separation around the cylinder over the height of the computational domain. (Onlineversion in colour.)

in which the bed roughness is in transitional range (close to hydraulically smooth conditions)and the cylinder roughness is taken as hydraulically smooth (table 2). Figure 4 shows the sideview of the position of boundary layer separation around the cylinder for both computationalmeshes (h = 2D and 3D), where the velocity on the cylinder surface becomes zero or negative withrespect to the streamwise direction. It can be seen in figure 4 that the boundary layer separationclose to the bottom starts earlier as the bottom velocities are smaller in magnitude than theupper section of the domain, and the bed roughness is almost smooth and does not result in adelay in boundary layer separation close to the bottom as in the case of flow around a verticalhydraulically smooth cylinder placed on a rough bottom. As in the latter case, the rough bottomincreases the exchange of momentum between the outer flow region and the wall, and, thus, inthe streamwise momentum in the boundary layer resulting in a delay in the separation [32]. Asthe velocities close to the bottom are smaller in magnitude, the magnitude of the vorticity supplyin the near lee-wake region becomes less than the upper parts of the domain. As an importantimplication of this, the vortex-shedding mechanism slows down close to the bottom, causing adelay in the shedding regime at the bottom, and, thus, the shed vortical structures along theheight of the domain are broken apart into two cells in the vertical direction.

Figures 5 and 6 give sequences of power spectra of the lift force applied on the cylinder inthe y-direction obtained for different values of z/h, for the computational mesh with h = 2D and3D, respectively. Here, φL and σL

2 are the power spectrum and the variance of the lift force,respectively, and St is the Strouhal number (St = fU/D), where f is the vortex-shedding frequency,U is the mean flow velocity and D is the pile diameter. Both simulations show a clear shift fromlower to higher St as z/h increases, indicating more frequent shedding of vortices further from thebed. The shift in Strouhal number clearly shows that the vortex shedding occurs in two separatecells, rather than one cell of domain height. This shift is observed to be around z/h = O(0.2) forboth domains (h = 2D and 3D). Regarding the effect of domain height on the scour, Sumer &Fredsoe [33, fig. 3.26] show that the boundary layer thickness has a clear effect on the scour depthup to a boundary layer thickness to pile diameter ratio equal to 4. Also, it can be seen from figure 5that the spectral peaks for h = 2D are rather irregular (composed of multiple peaks rather than justone peak at one frequency) compared with the ones given for h = 3D. However, the highlight ofthis section might be given as the behaviour of vortex-shedding changes close to the bottom dueto the bottom shear effect. One final note with regard to the vortex-shedding frequency is that thevalues of the Strouhal number are apparently different from the familiar value 0.2. This is linkedwith the fact that the mean flow velocity U is used when normalizing the frequency, rather thanthe ‘local’ approach velocity.

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100

f L U

/sL2

D100

100

100

50

50

50

50

0 0.1 0.2 0.3 0.4 0.5

0 0.1 0.2 0.3 0.4 0.5 z/h = 1.0

z/h = 0.26

z/h = 0.14

z/h = 0

0 0.1 0.2 0.3 0.4 0.5

0 0.1 0.2 0.3

St = fD/U

0.4 0.5

Figure 5. Power spectra of lift at various heights of the computational domain with h= 2D. The quantity σL2 is the variance

of the lift fluctuations.

200

100

0

0

0.1 0.2

0.1 0.2

0.3 0.4 0.5

0.3 0.4 0.5

0 0.1 0.2 0.3 0.4 0.5

0

100

200

200

100

200

100

0.1 0.2 0.3 0.4 0.5

St = fD/U

z/h = 1.0

z/h = 0.32

z/h = 0.15

z/h = 0

f L U

/sL2

D

Figure 6. Power spectra of lift at various heights of the computational domain with h= 3D. The quantity σL2 is the variance

of the lift fluctuations.

The two-cell structures of the shed vortices are much more visible in figure 7, in whichthe magnitude of the vorticity in the vertical direction (ωz = ∂uy/∂x − ∂ux/∂y, in which ωz is thevorticity in the vertical direction, ux is the velocity component in the x-direction and uy is thevelocity component in the y-direction) around the cylinder at a selected instant in time is given(for h = 3D).

The upper and lower bounds of the colour scale of figure 7 are limited to small values ofvorticity to make the opposite fields easily distinguishable. As seen from figure 7, the verticalstructure of the vorticity in the z-direction is broken simply into two parts close to the bottom,and the shed vortical structure looks like either a tornado with a tail that follows the main bodyfrom behind or a broken tornado where the top and bottom parts rotate in opposite directionswith respect to each other.

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vorticity z8

4

0

–4

–8

Figure 7. Magnitude of the vorticity in the vertical direction (z) around the cylinder at a selected instant in time for one-halfof the computational domain. The units of the magnitude of the vorticity are given in 1 s−1. (Online version in colour.)

The flow around the cylinder over a rigid bed can be further visualized in an accompanyingvideo (see electronic supplementary material). The video is composed of four views, eachpresenting the motion of a different quantity to visualize the coherent structures in the flow. Inthe upper left and right quadrants of the video, the Q-criterion [34] is used to educe the maincoherent structures such as the horseshoe vortex and the lee-wake vortices. Here, an overall lookat the vortical structures around the cylinder is given. One can see the horseshoe vortex formedin front of the pile and the lee-wake vortices that are dispersed in the flow as they progress. In theupper right view, the threshold for Q is set to higher values so that a detailed picture of lee-wakevortices just behind the cylinder is given. Here, it is easily seen that the vortex shedding occursin two cells, where the bottom cell extends approximately up to z/D = 1.0 from the bottom. In thelower left view of the video, iso-surfaces of vorticity in the z-direction are visualized. In this view,it can be seen that the bottom part of the vortical structure moves more slowly than the upper part,like the swing of a skirt, and that the shedding occurs more slowly close to the bottom. The lowerright view shows iso-surfaces of vorticity in the streamwise direction. Here, again, the two-cellstructure of vortex shedding is prominent and it gives hints about the formation of large-scalecounter-rotating streamwise phase-averaged vortices (LSCSVs), which are discussed further indetail below.

Figure 8 depicts a sequence of bed shear stress (τ0) vectors and contour lines of magnitudeof bed shear stress amplification (|ατ | = |τ0|/τ∞) at the bottom (for h = 2D), obtained from theunsteady solution of the present model, over one-half period of vortex shedding. Here, theunsteady behaviour of the wake is evident. Incidentally, the figure further indicates the presenceof the horseshoe vortex.

The present model compares well with the pressure measurements of Dargahi [35] as givenin [3]; these particular results are not repeated here for brevity. Figure 9 shows the phase-averaged pressure field around the cylinder along the symmetry plane (for h = 3D). The quantitydepicted is the ratio of the pressure coefficient, Cp = p/( 1

2 ρU2), at any point in the domain, to thepressure coefficient computed at the stagnation point of the top boundary, Cp,stag = pstag/( 1

2 ρU2)(at x = −0.5D, y = 0 and z = h), in which p is the pressure (p = p̄ − p0, where p̄ is the time-averagedpressure and p0 is the hydrostatic pressure) and U is the mean flow velocity. The colour scalein the figure is selected to make the pressure gradients clearly visible. One can see from thisfigure that there exists a pressure gradient in the vertical direction over the symmetry plane. Thispressure gradient drives a mean flow in the upward direction. The same would hold true if thephase-resolved pressure values are used. This is simply due to the far-field velocity profile whichcreates a higher pressure field close to the bottom. Similarly, there exists a pressure gradient at the

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1.5

3

71

31

1

1

13

3

9

117

3

5

5

9

97

113

1

1

5

5

3

5

5

1.0

(a) (b)

(c)

0.5

0

–0.5

–1.0

–1.5–1.0 –0.5 0 0.5 1.0

x/D

y/D

1.5

1.0

0.5

0

–0.5

–1.0

–1.5

y/D

1.5

1.0

0.5

0

–0.5

–1.0

–1.51.5 2.0 2.5 –1.0 –0.5 0 0.5 1.0

x/D1.5 2.0 2.5

–1.0 –0.5 0 0.5 1.0x/D

1.5 2.0 2.5

11109876543210

11109876543210

11109876543210

Figure 8. Bottom wall shear stress (τ0) vectors and contour lines of the magnitude of bed shear stress amplification (|ατ |)during one-half period of vortex shedding at the bottom. (a) Phase= t; (b) phase= t + T/4; and (c) phase= t + T/2with T being the vortex-shedding period. (Online version in colour.)

Cp/C

p,stag

0.4 ~ 1.0

0

YZ

X

–0.4

–0.8

–1.2 ~ –2.0

Figure 9. Pressure coefficient ratio (Cp/Cp,stag) field in three dimensions. (Online version in colour.)

top boundary driving again a mean flow towards the side boundaries at the relatively far fieldwith respect to the cylinder.

Figure 10 depicts the pressure field around the cylinder (for h = 3D) at the horizontal crosssection corresponding to z = 1D. The quantity depicted is the same as that in figure 9. One can

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0 1 2 3 4 5 6 7 8 9

1

2

3

4

−0.6

−0.4

−0.2

−0.1

−0.05

−0.05

0

x/D

y/D

−0.6

Figure 10. Pressure coefficient ratio (Cp/Cp,stag) contour lines at z = 1D above the bottom.

0

0.5

1.0

1.5

2.0

2.5

3.0

z/D

x = 0.75D

−6 −4 −2 0 2 4 6y/D

x = 8.5D

−6 −4 −2 0 2 4 60

0.5

1.0

1.5

2.0

2.5

3.0

y/D

−6 −4 −2 0 2 4 6y/D

−6 −4 −2 0 2 4 6y/D

z/D

x = 5.5D

x = 2.5D

Figure 11. Phase-averaged velocity streamlines over transverse planes at (a) x = 0.75D, (b) x = 2.5D, (c) x = 5.5D and(d) x = 8.5D.

again see a pressure gradient which drives a mean flow towards the centre of the domain inthe transverse direction. The mean pressure gradients in the relatively far-field of the cylinderpresented in figures 9 and 10 creates LSCSVs downstream of the cylinder.

Figure 11 shows phase-averaged velocity streamlines over transverse planes at x = 0.75D,x = 2.5D, x = 5.5D and x = 8.5D (for h = 3D). As seen, LSCSVs become much more evident as ourfocus moves further away from the cylinder. At x = 0.75D, one can see a region in the middlewith upward directed velocity streamlines. This is the region with a strong vertical pressuregradient just behind the cylinder (figure 9). As the shed vortices disperse and the velocity field

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mean vorticity in x-dir.

2

1

ZYX

Z

(a) (b)

YX

0

–1

–2

Figure 12. Visualization of the streamwise phase-averaged vortices. (a) Phase-averaged velocity streamlines where thex-component is multiplied by 0.02. (b) Phase-averaged iso-surfaces of vorticity in the x-direction. (Online version in colour.)

downstream becomes more uniform, secondary circulation cells disappear, while the two largecirculation cells remain.

Figure 12 depicts the LSCSVs, visualizing the phase-averaged velocity streamlines infigure 12a, and the iso-surfaces of the phase-averaged vorticity in the streamwise direction (x) infigure 12b (for h = 3D). To visualize the LSCSVs in the form of streamlines in figure 12a, the phase-averaged velocities are used. As the flow is mainly unidirectional and the velocity componentin the streamwise (x) direction is much stronger than the other two components (in the y andz directions) in most parts of the domain, here, the phase-averaged velocity component in thex-direction only is reduced by multiplying by a factor of 0.02, which is found by trial and error,and the other two components are kept as they are. By doing so, the phase-averaged flow isslowed down in the streamwise direction only, so that the LSCSVs can be presented in the formof spirals in the three-dimensional domain. In addition to the LSCSVs shown in figure 12, anotherpair of vortices close to the bottom and adjacent to the cylinder rotating in the opposite directionmerits attention. These vortices are partly due to the horseshoe vortex and the change in thevortex-shedding frequency at the bottom, resulting in a delay of almost one shedding period ofthe upper cell.

It should be noted that, as mentioned previously, recently Petersen [17] studied edge scour atthe scour protection around the foundation of a wind turbine by physical model experiments.He found that the edge scour at the downstream side of the scour protection is mainly causedby the LSCSVs in the lee wake of scour protection and the pile system. These vortices were alsoclearly visualized by his video taken downstream where the LSCSVs were visualized by the sand.The bed morphology obtained after the experiments in [17] consisted of two elongated scourholes with a longitudinal ridge between them, and the bed form (associated with the edge scour)appears to be the ‘signature’ of the LSCSVs.

5. Simulation of the scour processScour around a vertical pile structure has been previously investigated in detail in [3], wheresteady-state flow calculations around a vertical pile over an initially plane bed were incorporatedwith a morphology model based on bed load sediment transport and sand slide processes only. Inthis study, the additional effects of unsteady flow structures and suspended sediment transportare investigated for the test conditions (for h = 2D) indicated in table 2. For this purpose, threedifferent scenarios with the same flow conditions and the same computational mesh are studied.In scenario 1, both the unsteady flow features (vortex shedding) and the suspended sedimenttransport are turned on. In scenario 2, only the vortex shedding is turned on and the suspendedsediment transport is turned off. In scenario 3, both the vortex shedding and suspended sediment

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Table 3. Scenarios in scour simulations.

vortex shedding suspended load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

scenario 1 on on. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

scenario 2 on off. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

scenario 3 off off. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

transport are turned off. The three scenarios are summarized in table 3. Note again that vortexshedding can be conveniently and effectively switched off by using more diffusive schemes (e.g.Gauss upwind) for the divergence terms in (2.1). The computational mesh used in the scoursimulations is with the domain height, h = 2D (table 1). The flow conditions in the simulationsare given in table 2 for h = 2D.

The undisturbed Shields parameter in the test conditions is given as θ = 0.13, which is largerthan the critical value for the initiation of motion, θc = 0.05, implying that the scour is in thelive-bed regime. The calculated maximum Shields parameter around the cylinder is aroundθ ≈ 1.3; therefore, suspended sediment transport can be expected to be active in scour around thecylinder. In the simulations, bed ripples are resolved. No extra treatments are applied to triggerthe formation of the bed ripples, i.e. they arise naturally within the computational domain as theripples form as a result of the physical instability of the sediment bed when exposed to a steadyboundary layer flow provided that the grain Reynolds number, dUf/ν, is smaller than O(20) [36].The sediment transport description in the present model is based on uniform-size sand, i.e. anyeffects of sediment gradation are not accounted for.

Figure 13 shows a sequence of pictures illustrating the time evolution of the scour holeobtained in scenario 1. In the figure, one can see (i) the semi-circular shape (in plan view) ofthe upstream part of the scour hole with a slope equal to the angle of repose, (ii) the formationof a ‘bar’ downstream of the pile (the deposited sand), and its downstream migration with thecontribution of LSCSVs, (iii) the formation of a gentler slope of the downstream side of thescour hole, (iv) the formation and migration of bed ripples at the downstream side of the scourhole, and, finally, (v) the bed material eroded and piled up along the edges of the scour holedownstream by means of the counter-rotating streamwise phase-averaged vortices close to thebottom shown in figure 12a.

The equilibrium scour depth obtained in scenario 1 is S/D = 0.91, where S is the scour depthin front of the pile. This value agrees well with the existing data, for the present water-depth-to-pile-diameter ratio h/D = 2.0 (see fig. 3.26, p. 180, in [33]) in the simulation and the ratio betweenthe approach velocity and the critical velocity for initiation of sediment motion is U/Ucr ≈ 1.6 (seefig. 3.25, p. 179, in [33]).

Figure 14 shows the time development of the scour depth at the upstream side (figure 14a)and at the downstream side (figure 14b) of the pile for the three scenarios. Figure 15, on the otherhand, shows the scour-hole profiles along the upstream–downstream symmetry line after 2 minof simulation time, again for all the three scenarios. In figure 14a, one can see that the upstreamequilibrium scour depth decreases by almost 50% when the suspended sediment transport is notincorporated in the computations, regardless of vortex shedding being enabled or disabled. Infigure 14b, the equilibrium scour depth ratio at the downstream side of the pile reaches to S/D =0.52 for scenario 1. For scenarios 2 and 3, however, the scour depth ratio reaches to S/D = −0.23and S/D = −0.16, respectively, the negative values indicating accretion rather than scour. This isbecause the sediment could not be picked up and carried further away by the flow, but insteadpiled up adjacent to the pile downstream. The initiation of scour at the downstream side of thepile for scenario 2 within the first 10 s might be attributed to the larger bed shear stresses underthe shed vortices.

In figure 15, the slope of the scour hole at the upstream side of the pile is equal to the internalangle of friction of the sediment grains, and the downstream slope is relatively less steep, which

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time = 00:05S/D = 0.49

1

5.0

2.5

–2.5 –2.50

2.55.0

7

0.60.40.20–0.2–0.4–0.6–0.8–1.0–1.2

0.60.40.20–0.2–0.4–0.6–0.8–1.0–1.2–5.0 –5.0

–7.5

0

0–1

z/D

y/D

x/D

flow time = 00:15S/D = 0.70

1

5.0

2.5

–2.5 –2.50

2.55.0

7

–5.0 –5.0–7.5

0

0–1

z/D

y/D

x/D

flow

time = 00:30S/D = 0.81

1

5.0

2.5

–2.5 –2.50

2.55.0

7

0.60.40.20–0.2–0.4–0.6–0.8–1.0–1.2

0.60.40.20–0.2–0.4–0.6

–0.8

–1.0–1.2–5.0 –5.0

–7.5

0

0–1

z/D

y/D

x/D

flow time = 01:30S/D = 0.91

1

5.0

2.5

–2.5 –2.50

2.55.0

7

–5.0 –5.0–7.5

0

0–1

z/D

y/D

x/D

flow

Figure 13. Scour development. Model results. Time=min:s. Maximum scour depth indicated in the panels. (Online version incolour.)

0 20 40 60 80 100 120time (s)

1.60

1.40

1.20

1.00

0.80

0.60

0.40

0.20

0

S/D

0 20 40 60 80 100 120time (s)

1.20

1.00

0.80

0.60

0.40

0.20

0

–0.20

–0.40

(a) (b)

Figure 14. Scour development. Solid line, scenario 1; dashed line, scenario 2; dotted line, scenario 3. (a) Scour depth at theupstream edge of the pile. (b) Scour depth at the downstream edge of the pile.

agrees well with the observations [33]. The formation of the ‘bar’ accumulated adjacent to the pilein scenarios 2 and 3 has been explained previously. The ‘bar’ formed at the downstream side inscenario 1, which is further away than scenarios 2 and 3, is in agreement with the observations ofscour around a vertical pile in steady current.

The present simulation results for scenario 3 differ from the results in [3], where the simulationconditions resemble scenario 3. Two major differences may be noted, namely (i) the equilibriumscour is, in the present simulation, reached much faster than in [3] and (ii) the equilibrium scourdepth is a factor of 2 smaller than in [3] although the Shields parameter and the boundary-layer-thickness-to-diameter ratio of the present simulation are much the same as in [3]. No clearexplanation has been found for this discrepancy. Nevertheless, this is not a problem as the main

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1.5

1.0

0.5

0

–0.5

–1.0

–1.5–8 –6 –4 2 0 2 4 6 8

x/D

S/D

Figure 15. Equilibrium scour profile along the length of the longitudinal symmetry axis. Solid line, scenario 1; dashed line,scenario 2; dotted line, scenario 3.

objective of the present scour simulations is to investigate the influence of the suspended load aswell as the vortex shedding on the scour process, and the present simulations serve the purpose.

At this juncture, it is also interesting to elaborate on the influence of the suspended load onscour. Laboratory observations in the previous research indicated that scour around a pile insteady currents is not influenced very significantly by the presence of suspended load (e.g. fig.3.25 in [33]). The latter source shows that the scour depth increases by a factor of 1.3 (for a nearlyuniform size sediment) when the Shields parameter is increased to a value corresponding tothe suspension regime including the sheet flow transport. By contrast, the present simulationsindicate that the increase in the scour depth is a factor of 2 when the suspended load is switchedon. This aspect of the problem appears to be worth exploring in future research; see also thediscussion in [37].

6. ConclusionA three-dimensional hydrodynamic model, incorporated with a k-ω turbulence model, sedimenttransport (bed and suspended load) description and a morphological model is used to investigatethe flow and scour around a vertical circular pile exposed to a steady current. The model,tested and validated against the experimental data in the literature, has been used to studyflow processes to visualize the phase-averaged flow structures and to study the influence ofthe unsteady flow features and incorporation of the suspended sediment transport on the scouraround a vertical pile.

The numerical model results show that the vortex shedding around a vertical pile placed on aplane bed occurs namely in a two-cell fashion, one cell very close to the bottom z < 1.0D and theother one above the first cell, of which the shedding frequency is almost doubled with respect tothe bottom. The variation of the shedding frequency over the height of the pile is found to be dueto the early boundary layer separation on the surface of the pile close to the bottom because ofbottom roughness, a smooth wall and the decrease in the magnitude of the vorticity feeding thelee wake as a result of smaller velocities at the bottom.

The phase-averaged flow features were first studied by visualizing the phase-averagedpressure fields in the lee-side of the pile. Here, a vertical pressure gradient continuous along thesymmetry plane normal to the transverse direction at y = 0D, strongest at −0.5 < x/D < −1.5 anddriving an upward-directed mean flow, is observed. Additionally, there exist horizontal pressuregradients close to the bottom and top boundaries that drive horizontal mean flows towards thecentre close to the bottom and towards the sides of the domain close to the top boundary. Theformation of the LSCSVs in the far-wake region is attributed to these mean pressure gradients.

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Rather small-scale streamwise phase-averaged vortices adjacent to the pile close to the bottom atthe near-wake region could be attributed to the inherited vorticities from the horseshoe vortexin front of the pile and the increase in the vortex-shedding period (almost doubling the periodin the upper cell) close to the bottom. The presence of the phase-averaged streamwise vortices isvisualized through streamlines and the iso-surface of the vorticity magnitude in the streamwisedirection.

The influence of unsteady flow structures (vortex shedding) and suspended sedimenttransport on the scour around a vertical pile has been investigated by considering three scenarios,in which these individual features are systematically switched on and off. When using thefull model, i.e. with both vortex shedding and suspended load enabled, an equilibrium scourdepth upstream of the pile S/D = 0.91 has been obtained, which agrees well with existing data[33]. Alternatively, the upstream equilibrium scour depth is decreased by almost 50% whenthe suspended sediment transport is not incorporated in the simulations, regardless of whethervortex shedding is enabled or disabled.

With the full model, it can be seen that the bed material is eroded and piled up along theedges of the scour hole downstream by means of the small-scale counter-rotating streamwisephase-averaged vortices close to the bed, while the LSCSVs contribute to the formation of a ‘bar’downstream of the pile (the deposited sand) and its downstream migration. The slopes of thescour hole at both the upstream and downstream sides of the pile in the simulations are found tobe in agreement with observations in [33].

Alternatively, turning off the suspended sediment transport decreased the overall transportand resulted in accumulation at the downstream perimeter of the pile for the first 120 s ofsimulation. The initiation of scour downstream of the pile within the first 10 s, where the vortexshedding is active but the suspended transport is not, is attributed to the larger bed shear stressesunder the shed vortices. In the cases considered, including or excluding vortex shedding has hadonly a minimal impact within the duration of the simulations.

Funding statement. C.B., B.M.S. and D.R.F. acknowledge support from FP7-ENV-2013.604-3 European Unionproject ASTARTE (Assessment, STrategy And Risk Reduction for Tsunamis in Europe), grant no. 603839. C.B.also acknowledges the support of a post-doctoral grant from The Scientific and Technical Research Councilof Turkey (TUBITAK, grant no. 2219). C.B. and B.M.S. additionally acknowledge Innovative Multi-purposeOffshore Platforms: Planning, Design and Operation (MERMAID), 2012–2016, grant agreement no. 288710 ofthe European Commission 7th Framework Programme for Research.

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