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Numerical Models in Fluid-Structure Interaction

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Chapter 4

Cylinder in steady & oscillatory flow

E.A. Hansen1 & J. Dalheim2

1DHI, Hørsholm, Denmark 2DNV Consulting, Høvik, Norway 4.1 Introduction

The subject of flow around circular cylinders is fundamental in understanding the flow effects on more complex structures such as offshore structures. Even for such a simple problem, the solution is complicated. Numerous papers and articles may be found in the literature on the numerical simulation of flow around circular cylinders, from two-dimensional creeping flow to the simulation of three-dimensional flow at high Reynolds numbers.

It should be stressed at the outset that the flow field around the cylinder is rather complex and the success of numerical models has been limited, to say the least. While the mathematical formulation is well established, the numerical difficulties are numerous, partly because of enormous computing demand of the problem and partly because of shortcomings in the discretization techniques. Even in steady flow, such as wind around land-based structures, the flow effect is not very well understood, especially at high Reynolds numbers experienced by most structures in high winds. Unlike wind engineering, the forces in offshore engineering are not simply caused by constant flow, but a combination of oscillatory flow (waves) and constant flow (current).

In spite of this difference, important information may be obtained from wind-engineering studies. For example, the numerical models can be used for studying relative effects, such as, how much is the force increased when the cylinder is

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placed side by side with another structure, or how much is the force reduced when one cylinder is placed behind another. Most of the studies related to offshore problems have been initiated mainly because of the needs of the offshore industry. The results of these studies are needed directly or indirectly for determining the hydrodynamic design loads or fatigue damage of offshore structures.

This chapter briefly describes the fundamental theory behind the flow past cylinders and shows the basic equations involved in such a problem. A few of the numerical procedures used for solving the set of equations are shown. Some of the associated results are also presented. 4.2 Flow description and force

While the geometry of the problem is simple, the flow phenomena are highly complex. The complexity can be illustrated by uniform flow over a smooth cylinder, where the flow depends on a one-dimensional parameter only, namely the Reynolds number

ν

DURe c= , (4.1)

in which D is the cylinder diameter, Uc is the flow velocity, and ν is the kinematic viscosity of the fluid.

The flow comprises • Flow with no separation, such as creeping flow, flow with a fixed pair of

symmetric vortices, shedding of a laminar vortex street, and shedding of a turbulent wake.

• Fully laminar flow in the boundary layers, partly turbulent partly laminar flow in the boundary layer and fully developed turbulent flow in the boundary layers, see Schlicting (1979).

No numerical model of today is able to effectively simulate the flow and the flow induced forces from creeping flow for Re<5 to trans-critical flow for Re>4×106. 4.2.1 Morison equation The lack of numerical capabilities necessitates use of empirical formulas in conventional computing of forces on structures due to fluid flow. In steady flow the force in the flow direction consists of a drag force for which a drag coefficient is needed. The drag coefficient is derived as a function of Reynolds number from laboratory experiments. In offshore engineering the forces due to



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waves (and combined waves and current) are often described by the Morison equation. This equation is written as a sum of a drag force and an inertia force.

The unsteady wave force, F, is

dt

)t(dUCVolρ)t(U)t(UACρ½)t(F MD += , (4.2)

where CD and CM are the drag and inertia coefficients respectively, A is the cross-sectional area perpendicular to the flow. Vol is the volume of the structure, ρ is the density, and U is ambient velocity. The first term on the right-hand side represents drag and the second term represents inertia.

The understanding of the most fundamental case of separated steady flow past a single circular cylinder is far from complete and thus, numerous attempts have been made either to improve Morison’s equation or to devise new equations. So far no satisfactory results have been obtained. 4.2.2 Morison’s equation versus CFD Morison’s equation [Morison et al. (1950)] is easy to implement in a numerical algorithm, and determines the time-varying inline forces directly from the ambient fluid velocity and the fluid acceleration in combination with the drag and the inertia coefficients. The non-dimensional coefficients depend on the flow conditions.

The drag-force coefficient for a smooth cylinder in steady flows varies mainly with the Reynolds number. Additionally, it depends to some extent on the incoming turbulence level, the cylinder length/diameter ratio, and possible shear in the flow. The force coefficient is determined from experiments.

The alternative of using a simple drag force in steady flow or Morison’s equation in oscillatory flow is refined flow modeling/CFD calculations, which require access to high-performance computers and advanced numerical codes. Before the numerical modeling can be used with confidence, the results from the numerical models need to be verified. It has to be shown that the results are similar to those from experiments and that they are mesh and time independent.

Today, almost all analyses of non-potential forces on offshore structures use Morison’s equation. The refined numerical flow models are used as an alternative or supplement for studying details such as the determination of the force on a small cylinder placed in the vicinity of a large cylinder. The force coefficients for cylinders fitted from the numerical studies can hereafter be implemented as load generators for more complicated structures, such as jacket platforms.

The ‘ultimate goal’ for using CFD in the offshore industry is to be able to predict the total/local hydrodynamic forces on a complicated offshore structure



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when the structure is exposed to a given sea state. At present, we are far from this goal. Unfortunately, there is no known step between the use of the traditional force coefficients and the use of CFD modeling. This is why most offshore structures are still being designed based on the use of simple equations like the Morison equation. However, CFD analysis is slowly moving towards the level of today’s structural design, which can determine structural stresses for complicated structures exposed to a given external force. Today, the level of potential flow analysis has reached a point similar to the level of structural design, so that it is possible to determine the first-order/linear potential forces on rather complicated structures. See, for example, Chapter 6. 4.3 Navier–Stokes equations

The governing equations to be solved by the CFD program are the Navier–Stokes equations for incompressible flow. These equations can be written in several different ways, one of which is given below:

,

0

2 upuutu

u

∇ν+ρ

−∇=∇+∂∂

=∇ (4.3)

The upper equation in eqn (4.3) is the continuity equation, and the lower

equation describes the momentum equation, with (the vector) u being the velocity, p the pressure, ν the kinematic viscosity, and ρ is the fluid density.

All numerical studies solve the Navier–Stokes equations and the continuity equation. It should be noted that for the numerical solutions of the steady flow past a cylinder in a compressible fluid, the compressible Navier–Stokes equation is solved. Such simulations are often carried out by wind engineers/researchers at high Mach numbers. At low Mach numbers the flow becomes almost identical to the corresponding incompressible flow.

For an incompressible flow, the pressure term in the Navier–Stokes equation can be eliminated. Moreover, by introducing vorticity, the vorticity transport equation can be derived. In two dimensions, the number of equations can be reduced from three to two by introducing the stream function. The vorticity transport equation can be solved either directly or by moving discrete vortex particles. This alternative formulation is briefly treated in the following.



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4.3.1 Stream-function vorticity formulation It may be more convenient to describe the Navier–Stokes equations in two dimensions in terms of the stream function, ψ, and the vorticity function, ω.

,yu

xv

xu

yu

y

x

∂∂

−∂∂

=ω

∂ψ∂

−=

∂ψ∂

=

(4.4)

The pressure can be eliminated from the Navier–Stokes equations and the

following equation is obtained

∂

ω∂+

∂ω∂

=∂ω∂

+∂ω∂

+∂ω∂

=ω

2

2

2

2

yx yxv

yu

xu

tdtd

, (4.5)

This is known as the vorticity transport equation in two dimensions. The Poisson equation may be derived from eqn 4.4.

2

2

2

2

yψ

xψω

∂∂

+∂∂

= , (4.6)

4.3.2 The vortex method The vortex method is based on the theorem that in an inviscid incompressible fluid, vorticity is a kinematic property of fluid particles. It can neither be created nor destroyed. It can only undergo convection and deformation. In a viscous fluid, however, vorticity is generated along boundaries and is subjected to diffusion as well as convection and deformation. For two-dimensional flows Chorin (1973) introduced a split-time approach to solve the Navier–Stokes equations formulated by the vorticity transport equation.

A viscous part of the split-time step evaluated the change of vorticity given by



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ωνyω

xων

tω 2

2

2

2

2

∇=

∂∂

+∂∂

=∂∂

, (4.7)

and the convective part

0=∂ω∂

+∂ω∂

+∂ω∂

=ω

yu

xu

tdtd

yx , (4.8)

The convected part can be solved by moving vortex particles (discrete particles conserve their circulation). In Chorin’s (1973) split approach, the diffusive part was satisfied by a random-walk simulation. 4.3.3 CFD boundary conditions Different types of boundary conditions have to be applied:

a) Wall and cylinder surface b) Far field c) Inflow d) Outflow e) Periodic

They are applied to the two-dimensional/three-dimensional velocity

components, as well as to the pressure. In the primitive variable, where the velocity and the pressure are unknown: a) At a wall or at the cylinder surface, the u, v, and w velocity components

are set equal to zero. The pressure gradient in the normal direction is zero.

b) At the far field, the velocity components and the pressure are set equal to the free-stream values.

c) At the inflow, the u, v, w and the pressure gradient are specified. d) At the outflow boundary, the velocity gradients (in the normal direction)

are set to zero, while the pressure is set equal to zero. e) At two periodic boundaries the velocities and the pressure gradient are

identical at the two boundaries.

Using the stream function, ψ, and the vorticity, ω, as unknown: a) At a wall or cylinder surface ψ is constant. b) At the far field, ψ is set equal to the free-stream values. c) At the inflow, the stream function is set equal to the free-stream values.



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d) At the outflow boundary, the velocity and vorticity gradients (in the normal direction) are set to zero.

e) At two periodic boundaries, the stream vorticity functions are identical. In the vortex model there is a relation between the surface pressure and the circulation carried by the vortices generated on the surface. Stansby & Smith (1991) showed that the pressure change along an element could be found from

)/( tp ii ∂∆Γ∂ρ=∆ .

By defining ttii ∆∂∆Γ∂=Γδ )( , iΓδ presents the circulation carried by the

vortices crossing the surface element in the time increment t∆ . The pressure distribution relative to a reference pressure (taken at θ=0) can be written as

∑=

Γ∆

+=i

jji t

pp1

0 δρ , (4.9)

Because the pressure is the same at θ=0 and θ=2π one can see that the sum of the circulation carried by the vortices crossing the cylinder surface is zero.

The shear stress at the cylinder surface is simply taken as oµωτ = .

Wall

OutFlow

Inflow

Figure 4.1: Example of a numerical grid and boundary conditions for calculation

of steady flow around a free cylinder.



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4.3.4 Turbulence modeling Direct numerical simulation (DNS) resolves all important scales directly. It does not involve any ad hoc assumptions usually associated with turbulence modeling. The relation of the largest to the smallest scales determines the number of degrees of freedom required by DNS, which depends on the Reynolds number. The number of degrees of freedom can be prohibitively large for high Reynolds number-flow, so DNS has (until now) been able to solve few (if any) practical offshore-engineering problems.

Turbulence models, such as mixing length, or two equation models, like the kε or kω models, have been used intensively in numerical simulation of turbulent flows. Turbulent models based on the Reynolds-averaged Navier–Stokes (RANS) equations have several shortcomings. They are designed to model all scales in the flow. Because the large scales are highly dependent on the particular geometry it is therefore difficult/impossible to develop a universal turbulence model that accounts for all scales. The models are, in principle, designed for flows that are steady and the parameters in the models have been calibrated from those steady flows.

Large-eddy simulation, LES, is a compromise between DNS and RANS and is basically a three-dimensional model. LES computes large scales directly and models the small scales. The key idea is that the small scales are more universal than the large scales. Because they are universal it is easier to construct a “turbulence model” for them. The LES requires reduced spatial resolution compared with DNS, and this also leads to less restrictive numerical stability requirements.

It should be mentioned that the flow can be modeled by the Reynolds stress closures, which are more advanced than the RANS models, see Wilcox (1994). 4.3.5 Which numerical model to choose If an offshore engineer has identified a critical flow situation and he wants to quantify this situation using CFD, he has to make a number of choices:

1) Is the flow problem one that can be modeled in two dimensions or is it necessary to go to complete 3D simulations?

2) How to model the turbulence? a) Directly (DNS) b) Solve the large scales directly and model the small scales (LES) c) Model the turbulence by Reynolds-averaged Navier–Stokes equation

(RANS) for example by two equation models kε or kω models, or use of algebraic mixing lengths

d) Reynolds-stress model



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3) Which variables to be solved: a) A velocity–pressure formulation (primitive variable) b) A vorticity–velocity formulation c) In the case of 2D flow: a stream–function vorticity formulation

4) Time-stepping procedure

a) Fractional step procedure b) Spectral models c) Implicit models d) Possible other choices

5) Numerical discretization

a) By finite volumes b) By finite elements c) By finite differences d) By grid-free models e) Possible other choices

6) Introduce possible boundary-layer approximations

a) Slip-boundary conditions b) Possible other choices

After having selected among the above-listed possibilities, the engineer can set up the numerical mesh (if not a mesh-free model), choose the time step, set up the boundary conditions and the wave-current conditions he has to simulate.

If commercial codes are used some of the choices are limited or have already been made. It should be noted that some Journal papers present findings obtained by commercial codes. For example Rocchi & Zasso (2002) have studied vortex shedding from a circular cylinder in a smooth and wired-configuration, using the commercial CFD code (FLUENT) with a 3D large-eddy simulation (LES) approach.

It is not possible to produce a recipe for choices that will generally give the best result. Neither is it possible to present/discuss all combinations that have been presented in the literature. So in the following part of the chapter only a selection of the above choices is presented. 4.3.6 Example of numerical solution algorithm The fluid equations constitute a coupled system of the primitive variables u and p. Coupled systems are not straightforward to solve with ordinary numerical solvers, and indeed the Navier–Stokes equations are impossible to solve fully explicitly in time because the continuity equation lacks the time dimension for



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incompressible flows. Additionally, the coupled system is prone to spurious pressure-oscillation modes, which occur, e.g., when equal-order interpolation is used for the pressure and the velocity. To avoid this pressure instability the so-called Babuska–Brezzi condition must be satisfied. In order to overcome these stability challenges an uncoupled (or segregated) approach can be employed. Two main variants of this method have matured simultaneously in finite elements over the past decades. One method has come to be known as the fractional-step method, and the other is sometimes referred to as the velocity-correction method. The essential difference between the two is that for the former the segregation of pressure and velocity is effected after the Galerkin FEM discretization of the differential equations, while for the latter, the segregation is made at the differential equation stage. It can be shown that when the velocity-correction method is used, the Babuska–Brezzi condition is circumvented and no spurious pressure modes will appear. An example of the velocity-correction method is shown below. 4.3.6.1 The velocity-correction method The main feature of the velocity-correction method is that one time step is calculated in three computational steps:

1. An intermediate velocity is computed at time tn+1 by subtracting the pressure term of the Navier–Stokes equations and solving the remaining convection diffusion equations by a temporal integration:

dtxs

xu

uuun

n

t

t j

ij

j

ij

ni

ni ∫

+

∂

∂−

∂∂

−=+

1

1~ , (4.10)

2. A pressure Poisson equation is calculated implicitly on the basis of the

predicted velocities in order to find the new pressure contributions that will make the velocity field divergence free:

j

nj

jj

n

xu

txxp

∂

∂

∆=

∂∂∂ ++ 112 ~ρ

, (4.11)

3. The final divergence-free velocities are calculated by updating the

predicted velocities with the contribution from the new pressure field:



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i

nn

ini x

ptuu∂

∂∆−=

+++

111 ~

ρ, (4.12)

The pressure Poisson equation (PPE) can be obtained from taking the divergence of eqn (4.12) and introducing the continuity equation. Since the predicted velocities are projected onto a convergence-free space, these methods are sometimes referred to as projection methods. If the known pressure pn is included in the prediction step, eqn (4.10), a higher-order scheme in time is achieved, however, this may also result in spurious pressure modes unless the order of interpolation is lower for the pressure than for the velocities (satisfying the Babuska–Brezzi condition).

4.3.6.2 The finite element formulation Given that the computational domain can be divided into a finite number of smaller subdomains (elements), all scalar quantities can be approximated by test functions and nodal values on every element. In mathematical terms this can be expressed as

∑=

=⋅=≈NPOINTS

iii txtxtx

1)()(),(),( NUuNuu , (4.13)

and similarly,

NP≈),( txp , (4.14)

The matrix N contains the interpolation polynomials (test functions) for all elements, and P and U represent the nodal values of the pressure and fluid velocity, respectively. 4.3.6.3 Galerkin FEM A Galerkin weighted residual formulation can be employed to obtain an integral form of the Navier–Stokes equations. From writing eqn (4.3) in vector form, weighting with the test function, and integrating over the computational domain, the following weighted integral form of the Navier–Stokes equations is obtained:

,1)( 2∫∫∫∫ΩΩΩΩ

Ω∇ν+Ω∇ρ

−=Ω⋅∇+Ω∂∂ dpddd

tTTTT uNNuuNuN

(4.15)



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The last term on the right-hand side can be transferred using the Green–Gauss theorem and applying the boundary conditions of zero viscous stress tensor (i.e. zero gradients on normal velocity):

,

2

∫

∫∫∫

Ω

ΓΩΩ

Ω∇ν∇−=

=Γ∇+Ω∇ν∇−=Ω∇ν

d

ddd

T

nTTT

uN

uNuNuN (4.16)

The weak form of the Navier–Stokes equations is obtained by introducing eqn (4.16) into eqn (4.15):

,1)( ∫∫∫∫ΩΩΩΩ

Ω∇ν∇−Ω∇ρ

−=Ω⋅∇+Ω∂∂ dpddd

tTTTT uNNuuNuN

(4.17) The weak form contains only the first-order derivatives, thus the velocity and pressure can be continuously modeled with equal order on the interpolation functions. By applying the spatial discretization of velocity, eqn (4.13), and pressure, eqn (4.14), an algebraic equation system of the Navier–Stokes equation can be obtained from eqn (4.17)

LUDPAUUM νρ

−−=+∂∂ 1

t, (4.18)

with the incompressibility constraint 0=UDT , (4.19) The matrices in eqn (4.18) are defined in the following way: ∫

Ω

Ω= dT NNM , (4.20)

( )∫Ω

Ω∇⋅= dT NNUNA , (4.21)



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∫Ω

Ω∇= dT NND , (4.22)

∫Ω

Ω∇∇= dT NNL , (4.23)

where M is referred to as the mass matrix, A the convection (advection) matrix, D the discrete divergence matrix, and L the weak form of the discrete Laplacian, ∇2, which is symmetrical and positive definite. NU is the spatial discretization of the convective fluid velocity. From inspecting the definition of the advection matrix, eqn (4.21), it is obvious that this non-linear expression with a resulting triple product of test functions is costly to solve on a computer. Simplifications of this term are therefore desirable, especially when considering that the convection velocity NU is time varying, even in problems where the fluid mesh is fixed. One simple and effective way to simplify the advection matrix is to use the centroid convection method. By this method the convective velocity ui is assumed constant over each element, which leads to the following simplification of the convection matrix:

( ) ,∑∫ ∑ ∫ =Ω∇≈Ω∇⋅=Ω Ω elementsall

eeelementsall

Te

T

e

dd DuNNuNNUNA

(4.24) where eu is the mean convective velocity of element e, and Ωe denotes the element domain. By using the expression (4.24) the already-calculated matrix D can be utilized in the advection calculation and this constitutes a significant computational speedup. An alternative way to evade the problems of integrating the non-linear advection term is to apply the group finite-element formulation where the product of the scalar quantities instead of the quantities themselves is interpolated when calculating the advection term. However, this method requires the Navier–Stokes equations to be written in conservative form, which is inconvenient in cases when the continuity equation does not apply to the convective velocity, as would be the case when the fluid mesh is moving arbitrarily. 4.3.6.4 Solution of the algebraic equations By employing the velocity-correction method on the spatially discretized Navier–Stokes equations the following solution scheme for the algebraic equations can be obtained:



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Step 1 (prediction):

[ ]dtn

n

t

t

nnnn ∫+

+−=++

1

11 ~ LUAUUMUM ν , (4.25)

Step 2 (pressure Poisson):

1111 ~ ++++

∆= nnnn

tUDPL ρ

, (4.26)

Step 3 (correction):

111111 ~ ++++++ ∆−= nnnnnn t PDUMUM

ρ, (4.27)

Using the above equations will ensure that a stable solution to the flow field can be obtained. A variety of numerical solutions to the eqn (4.25) to eqn (4.27) have been presented over recent decades, and the efficiency of the solver will be strongly dependent on the selected solution. The accuracy of the solution, however, will be more dominated by the spatial discretization than by the numerical solution algorithm. 4.3.7 Surface roughness, near wall treatment In the case of rough cylinders, the flow becomes not only a function of Re number, but also a function of the roughness parameter, Dks / , in which ks is the Nikuradse equivalent sand roughness. The change in drag force can be quite dramatic. Achenbach & Heinecke (1981) have, for example, found from physical experiments the 5.0≈DC for a smooth cylinder and 0.1≈DC for a relative roughness 31075.0/ −×=Dk s both at Re = 106. It should be noted that the results have not been corrected for flume blocking.

Almost all offshore structures are designed for roughness (marine growth). Therefore, surprisingly little effort has been put into improving the description of roughness compared with the effort concerning smooth cylinders as reflected in the large number of articles on the subject.

4.3.7.1 Wall function Wall functions in which the viscous sublayer is not resolved but the first grid point is located outside this layer are used. Basically,



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the quantities at this grid point are related to the friction velocity based on the assumption of a logarithmic velocity distribution and local equilibrium of turbulence (the production of turbulent energy equals the dissipation of turbulent energy). 4.3.7.2 The kω model The kω model has an advantage compared with the kε model. By choosing the dissipation rate ω as a function of the Reynolds number based on the Nikuradse roughness and the friction velocity, it is possible to apply no-slip conditions for the velocities and get a correct logarithmic velocity, see Wilcox (1994).

It should be mentioned that all wall models, also the wall value for ω in the kω model, basically are developed for attached flows, and their application in separated flow regions is somewhat questionable.

Figure 4.2: Drag coefficient of the single cylinder in cross flow at various

surface roughness parameters, ks/d: x, Smooth: ∆, 75×10-5: Ο, 300×10-5; , 900×10-5; ∇, 3000×10-5. Reprinted from Achenbach & Heinecke ‘On vortex shedding from smooth and rough cylinders in the range of Reynolds numbers 6×103 to 5×106’, Journal of Fluid Mechanics, 109, pp. 239–251, 1981. With permission of Cambridge University Press.



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4.3.7.3 LES Ideally, the no-slip conditions should be used at walls in the LES calculations, but this demands very high resolution near the wall at high Reynolds numbers. At high Reynolds numbers, a near-wall model similar to the wall functions used in RANS calculations can be used. Various such models have been proposed. 4.3.8 Incoming turbulence The incoming turbulence level can, like the roughness, have a large effect on the drag (and lift) forces, as illustrated by Figure 4.3 presented by Zan & Matsuda (2002), which shows measured drag coefficients as a function of the Reynolds number for three different incoming turbulence levels. This effect has (as far as the author is aware) until now not been studied numerically.

Figure 4.3: Variation in drag coefficient with Reynolds numbers. For three

different incoming turbulence levels. ◊ Smooth flow, ∆ 5% turbulence, x 13% turbulence. Reprinted from Zan & Matsuda ‘Steady and unsteady loading on a roughened circular cylinder at Reynolds numbers up to 900,000’, Journal of Wind Engineering and Industrial Aerodynamics, 90, pp. 567–581, 2002. With permission of Elsevier.



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4.4 Numerically obtained results

The remaining part of the chapter presents selected results for steady and oscillatory flow in two and three dimensions. 4.4.1 2D steady flow around a single cylinder at low Reynolds numbers Numerous studies have been made for low Reynolds number cases (Re<1000) by solving the Navier–Stokes equations for two-dimensional flow directly. For a cylinder exposed to a constant flow for the range of Reynolds number 40<Re<170 the vortex street is laminar. The shedding is essentially two-dimensional, see Schlicting (1979) and the flow around a smooth cylinder can today be simulated very accurately with two-dimensional numerical models:

Stansby & Slaouti (1993) used the discrete-vortex model to compute the two-dimensional flow around a circular cylinder for Reynolds numbers from 60 to 180. They were able to reproduce the Reynolds number dependence as obtained experimentally by Williamson (1989).

Figure 4.4 presents the calculated drag and lift forces for Re=60 and Re=180. Posdziech & Grundmann (2001) studied the development of three-

dimensional flow structures around a circular cylinder at Reynolds numbers 190<Re<330. Based on two-dimensional calculation, a stability analysis was performed to obtain wavelengths that are unstable against spanwise perturbations and the critical Reynolds number for the onset of three-dimensionality.



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Figure 4.4: Force variation with time upper Re=60 and lower Re=180.

Reprinted from Stansby & Slaouti ‘Simulation of vortex shedding including blockage by the random-vortex and other methods’, International Journal for Numerical Methods in Fluids, 17, pp. 1003–1013, 1993. With permission of John Wiley & Sons, Inc.

4.4.2 2D Steady flow around two cylinders at low Reynolds numbers Stansby (1981) investigated two cylinders in side-by-side arrangements using an inviscid discrete-vortex method, and Stansby et al. (1987) used the random-vortex method to investigate flow around two cylinders in tandem and staggered arrangements. This method was developed further by Slaouti & Stansby (1992) to study the flow around two cylinders in side-by-side and tandem arrangements at a Reynolds number of 200.

The numerically calculated mean drag and lift for the two cylinders placed side by side (at Re=200) was presented versus the transverse spacing ratio, T/D, where T is the center-to-center distance and D is the diameter, see Figure 4.5. In the same figure, experiments at different Reynolds numbers (Re=8×103 Hori (1959), Re=1.2×105 Okajima (1986), Quadflieg (1977) Re=1.5×105 and Re = 1.6×105 by Zdravkovich & Pridden (1977)) were presented. Numerical results by Chang & Song (1990) are also presented in the figure.



153

Figure 4.5: Side-by-side cylinders mean drag and lift coefficients, versus

transverse spacing ratio, T/D. Reprinted from Slaouti & Stansby ‘Flow around two circular cylinders by the random-vortex method’, Journal of Fluids and Structures, 6, pp. 641–670, 1992. With permission of Elsevier.



154

Figure 4.6: Tandem mean drag coefficient, versus longitudinal spacing ratio, L/D. Reprinted from Slaouti & Stansby ‘Flow around two circular cylinders by the random-vortex method’, Journal of Fluids and Structures, 6, pp. 641–670, 1992. With permission of Elsevier.



155

The numerically calculated drag coefficients for the two cylinders placed in tandem (at Re=200) were presented versus the longitudinal spacing ratio, L/D, where L is the center-to-center distance, see Figure 4.6. In the same figure experiments at high subcritical Reynolds numbers (Re=3.4×103 Tanida (1973), Re 2.1×104 by Imaichi (1973) and Re = 3.1×104 by Zravkovich & Pridden (1977)) were presented.

Slaouti & Stansby (1992) concluded that the two-dimensional numerical simulations at Re=200 have revealed many of the features characterizing the flow around cylinders in the side-by-side and tandem arrangements observed experimentally at high subcritical Reynolds numbers. Seen from a practical point of view their findings may have the important implication that laminar-flow computations for other cylinder configurations can produce flow mechanisms and corresponding force variations of comparable complexity. Such simulations can thus be used for evaluating (without making experiments) the relative force reduction/increase for cylinders located close to each other. 4.4.3 3D steady flow around a cylinder in shear/non-shear flow using the LES model

Figure 4.7: Flow configuration and co-ordinate system. Reprinted from Xu &

Dalton ‘Computation of force on a cylinder in a shear flow’, Journal of Fluids and Structures, 6, pp. 941–954, 2001. With permission of Elsevier.



156

Xu & Dalton (2001) have presented results with steady flow at Reynolds number Re=1000, 10,000, and 20,000 using the LES turbulence model. They defined the

non-dimensional shear parameter zu

UD

o ∂∂

=β .

They investigated two cases a) a sinusoidal velocity variation with maximum shear parameter equal 0.0393 and b) flow with no shear. By using the sinusoidal variation of the incoming velocity they were able to apply periodic boundary conditions along the cylinder axis, see Figure 4.7. At the low Reynolds number Re=1000, they used a mesh 129×129×16, at the high Reynolds numbers Re=10,000 and Re=20,000 they used 129×193×16 points. They used a combination of a spectral and finite-difference approximation. The finite-difference approximation was used in the direction of the approach flow, and the spectral description (16 grid points) was used in the axial direction.

Figure 4.8 presents the axial vorticity distribution for Re = 20,000 and β = 0.0393. The figure shows that the mixing in the wake for the shear-flow case is more prominent than in the uniform-flow case. They found that the length-averaged drag force has a slight axial dependence and that the only noticeable difference in the length-averaged lift coefficients is a phase difference between the shear and the non-shear flow results.

Figure 4.8: Axial vorticity distribution at Re = 20,000. Top: uniform flow;

bottom: shear flow. Reprinted from Xu & Dalton ‘Computation of force on a cylinder in a shear flow’, Journal of Fluids and Structures, 6, pp. 941–954, 2001. With permission of Elsevier.



157

4.4.4 2D steady flow around a single cylinder at post-critical flow Smith & Stansby (1989) used a two-dimensional vortex-in-cell method with an algebraic boundary-layer turbulence model to simulate postcritical flow. They compared the result by measurements made by Achenbach (1968) at Reynolds numbers of 3.6×106. They introduced two assumptions: 1) They divided the flow into two areas: inside the inner region εD/2 adjacent to the cylinder they used the algebraic boundary-layer model; outside the inner region the vorticity transport was assumed to be dominated by convection. And 2) they introduced an exponential decay to the circulation of each vortex located in the outer region, in order to compensate three-dimensional effects. By ‘calibrating’ ε, they were able to obtain a good prediction of the mean pressure around the cylinder, see Figure 4.9. It should be noted that Smith & Stansby (1989) write that the assumptions: should clearly not be used to predict flows that differ appreciably from the calibration conditions.

Figure 4.9: The mean pressure coefficient (Cp) around a circular cylinder. An

exponential decay is applied to the circulation of vortices outside the annular region. The numerical results are compared with experiment. —— Numerical results; ------ Experiment. Achenbach (1968). Reprinted from Smith & Stansby ‘Postcritical flow around a circular cylinder by the vortex method’, Journal of Fluids and Structures, 3, pp. 275–291, 1989. With permission of Elsevier.



158

4.5 Oscillatory flow

4.5.1 2D planar flow For a cylinder with a planar oscillatory ambient (far-field flow), the Keulegan–Carpenter (KC) number is an important parameter defined by

D

TUKC w= , (4.28)

in which T is the period and Uw is the orbital velocity. The associated Reynolds number is

ν

DURe w= , (4.29)

The Reynolds number is often replaced by the frequency parameter ß given by

T

DKCRE

νβ

2

== , (4.30)

A theoretical analysis of attached oscillatory flow around a circular cylinder,

valid at small KC numbers, was first presented by Stokes (1851) and then by Wang (1968) who obtained a solution of the Navier–Stokes equations, valid for KC<< 1 and β>> 1, Wang’s theoretical force coefficients for drag and inertia are

( ) ( ) ( )[ ]5.115.03

25.023 −−− −+= πβπβπβπKC

CD , (4.31)

( ) ( ) 5.15.042 −− ++= πβπβMC , (4.32)

At very small values of KC, no separation occurs. The first separation appears when KC=1.1 at Re=1000. (The KC number changes as a function of Re.) This occurs in the form of the so-called Honji instabilities where the purely two-dimensional flow over the cylinder surface breaks into a three-dimensional flow pattern, where equally spaced, regular streaks are formed over the cylinder surface.

The theoretical force coefficients by Wang (1968) serve as an excellent two-dimensional numerical ‘basic’ benchmark at low KC. Several numerical solutions of the two-dimensional Navier–Stokes equations at low KC numbers and relatively low ß parameters have been reported: Stansby & Smith (1991)



159

used the random-vortex method together with the vortex-in-cell method. Justesen (1991) made a finite-difference solution of the vorticity/stream function version of the Navier–Stokes equations. Sun & Dalton (1996) solved the vorticity/stream function version of the Navier–Stokes equations using a large-eddy simulation. Zang & Zang (1997) solved the Navier–Stokes equations in a primitive variable formulation. 4.5.2 2D orbital flow The number of studies presented in the literature is significantly higher for planar flow than for orbital flow, even if a large part of offshore elements, for example horizontal elements placed close to the water surface, is exposed to orbital (or slightly elliptical) flow. One of the few works presented in the literature was carried out by Stansby & Smith (1991) who used the random-vortex method together with the vortex-in-cell method to study the viscous force and torque at circular cylinders in orbital flow. The flow was limited to KC numbers below 2. Only one Stoke parameter value, β=483, was studied. The force on the cylinder was taken as a sum of 1) potential forces, 2) a shear force viscous, and 3) the viscous component of the pressure force

pspot ffFFrrrr

++= , (4.33)

where the potential force is given as

aDFpotrr

ρπ 225.0= , (4.34)

For small KC numbers, it can be shown that sfr

and pfr

have equal magni- tudes.

50

51

22

225050 .

.ps

KcDU.f

DU.f

βπ

=ρ

=ρ

, (4.35)

Figure 4.10 illustrates the calculation of the non-dimensional (force normalized with 25.0 DUρ ) viscous force and shear force with time in cycles t/T for KC=0.1, 0.5, 1, 1.5 and 2 with circular onset flow.

The flow structures for KC<=1.5 are attached to the cylinders, small separation bubbles occur at KC=1.5. As presented in Figure 4.11 the bubbles are associated with the oscillations in the forces shown in Figures 4.10 and 4.11.



160

Figure 4.10: Variation of non-dimensional, viscous force magnitude F and shear force magnitude Fs with time in cycles, t/T, for K = 0.1, 0.5, 1, 1.5 and 2 with uniform, circular, onset flow. Reprinted from Stansby & Smith ‘Viscous forces on a circular cylinder in orbital flow at low Keulegan–Carpenter numbers’, Journal of Fluid Mechanics, 229, pp. 159–171, 1991. With permission of Cambridge University Press.



161

Figure 4.11: Streamline contours close to the cylinder surface in uniform,

circular, onset flow with K = 1.5 at values of t/T shown by the number in the circle. The arrow indicates the onset flow direction. Reprinted from Stansby & Smith ‘Viscous forces on a circular cylinder in orbital flow at low Keulegan–Carpenter numbers’, Journal of Fluid Mechanics, 229, pp. 159-171, 1991. With permission of Cambridge University Press.

4.5.3 Combined waves and current Zhou & Graham (2000) used a 2D discrete vortex model to calculate the planar oscillatory flow plus a small inline steady current at KC numbers from 0.2 to 26 and with Stokes parameter = 200. They found that the presence of a small current in an oscillatory flow can reduce the drag coefficient significantly and that the current tends to bring the whole vortex wake downstream and tries to form the stable Karman asymmetrical form in the downstream wake. They investigated the numbers of vortices shed per cycle and the conditions for symmetric and asymmetric vortex shedding. Figure 4.12 presents an example of their calculations (KC=2 and Re=400) where a pair of symmetric vortices is shed per cycle. The vortex pattern in the wake is presented together with analogous experiments visualized by Couder & Basdevant (1986). 4.6 Summary

The flow around circular cylinders has been studied and presented in numerous papers. Some results from the literature have been briefly presented in the present chapter. However, none of these studies addresses and answers the question: what are the forces on circular cylinders, maybe covered by marine growth, and exposed to offshore waves and current conditions? The reason is that even today such analyses are too computer demanding.

However, numerical models can be used for studying relative effects, such as: how much is the force increased when the cylinder is placed side by side with another structure, or how much is the force reduced when a cylinder is placed behind another structure.



162

As the computer power increases, flow at higher and higher Reynolds numbers can be simulated directly. The dramatic changes in the forces due to surface roughness (marine growth) have until now only been discussed briefly even though it may change the forces by a factor of more than 3, as shown in Figure 4.2. The authors of this chapter recommend that some research of the flow around rough cylinders should be carried out.

Figure 4.12: (a) Symmetric vortex shedding, a pair of symmetric vortices per

cycle of combined flow with KC=2 and Re=400 (B=1). (b,c) comparison of vortex pattern in the wake: (b) presents computational results and (c) experimental visualization by Couder & Basdevant (1986). Reprinted from Zhou & Graham ‘A numerical study of cylinders in waves and current’, Journal of Fluids and Structures, 14, pp. 403–428, 2000. With permission of Elsevier.



163

References

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[3] Chang, K. & Song, C., Interactive vortex shedding from pair of circular cylinders in a transverse arrangement. International Journal for Numerical Methods in Fluids, 11, pp. 317–329, 1990.

[4] Chorin, A. J., Numerical study of slightly viscous flow. Journal of Fluid Mechanics, 57, pp. 785–796, 1973.

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[6] Hori, E., Experiments on flow around a pair of parallel circular cylinders. Proceedings 9th Japan National Congress for Applied Mechanics, Tokyo, pp. 231–234, 1959.

[7] Imaichi, K. Preprint for JSME, 734–5, pp. 104–106 (in Japanese), 1973. [8] Morison, J.R., O'Brien, M.P., Johnson, J.W. & Schaaf, S.A., The force

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[9] Okajima, A., Sugitani, K. & Mizota, T., Flow around a pair of circular cylinders arranged side by side a high Reynolds numbers. Transactions of the JSME, 52, pp. 2844–2850 (in Japanese), 1986.

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[11] Quadflieg V.H., Induced load on pair of cylinders in incompressible flow at high Reynolds number. Forschung Ingeniurwesen, 43, pp. 9–18 (in German), 1977.

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[13] Schlicting, G., Boundary Layer Theory. 7th Edn., McGraw-Hill Book Company, 1979.

[14] Slaouti, A. & Stansby, P.K., Flow around two circular cylinders by the random-vortex method. Journal of Fluids and Structures, 6, pp. 641–670, 1992.



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[15] Smith, P.A. & Stansby, P.K., Postcritical flow around a circular cylinder by the vortex method. Journal of Fluids and Structures, 3, pp. 275–291, 1989.

[16] Stansby, P.K., A numerical study of vortex shedding from one and two circular cylinders. Aeronautical Quarterly, 32, pp. 48–71, 1981.

[17] Stansby, P.K., Smith, P.A. & Penoyre, R., Flow around multiple cylinders by the vortex method. In proceedings International Conference on Flow Induced Vibration, Bowness-on-Windermere, England, pp. 41–50. Cranfield, U.K.: BHRA, 1987.

[18] Stansby, P.K. & Smith, P.A., Viscous forces on a circular cylinder in orbital flow at low Keulegan–Carpenter numbers. Journal of Fluid Mechanics, 229, pp. 159–171, 1991.

[19] Stansby, P.K. & Slaouti, A., Simulation of vortex shedding including blockage by the random-vortex and other methods. International Journal for Numerical Methods in Fluids, 17, pp. 1003–1013, 1993.

[20] Stokes, G.G., On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc., 9, pp. 8–106, 1851.

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[26] Zan, S.J. & Matsuda, K, Steady and unsteady loading on a roughened circular cylinder at Reynolds numbers up to 900,000. Journal of Wind Engineering and Industrial Aerodynamics, 90, pp. 567–581, 2002.

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[29] Zang, H.L. & Zang, X., Flow structure analysis around an oscillating circular cylinder at low KC number: A numerical study. Computers & Fluids, 26(1), pp. 83–106, 1997.

[30] Zhou. C.Y. & Graham, M.R., A Numerical study of cylinders in waves and current. Journal of Fluids and Structures, 14, pp. 403–428, 2000.


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