younis 2006 ocean-engineering

8/3/2019 Younis 2006 Ocean-Engineering

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Prediction of hydrodynamic loading on a mini TLPwith free surface effects

Bassam A. Younisa,*, Vlado P. Przuljb

a

Department of Civil and Environmental Engineering, University of California, Davis, CA 95616, USAbRicardo Software, Shoreham-by-sea, West Sussex BN43 5FG, UK

Received 20 October 2004; accepted 6 April 2005

Available online 25 July 2005

Abstract

This paper describes the extension of a fluid-flow simulations method to capture the free surface

evolution around a full-scale Tension Leg Platform (TLP). The focus is on the prediction of the

resulting hydrodynamic loading on the various elements of the TLP in turbulent flow conditions and,

in particular, on quantifying the effects of the free surface distortion on this loading. The basic

method uses finite-volume techniques to discretize the differential equations governing conservation

of mass and momentum in three dimensions. The time-averaged forms of the equations are used, and

the effects of turbulence are accounted for by using a two-equation, eddy-viscosity closure. The

method is extended here via the incorporation of surface-tracking algorithm on a moving grid to

predict the free-surface shape. The algorithm was checked against experimental measurements from

two benchmark flows: the flow over a submerged semi-circular cylinder and the flow around a

floating parabolic hull. Predictions of forces on a model TLP were then obtained both with and

without allowing for the deformation of the free surface. The results suggest that the free surface

effects on the hydrodynamic loads are small for the values of Froude number typically encountered

in offshore engineering practice.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Computational fluid dynamics; TLP; Free surface effects; Turbulence

Ocean Engineering 33 (2006) 181204

www.elsevier.com/locate/oceaneng

0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.oceaneng.2005.04.007

* Corresponding author. Tel.: C1 530 754 6417; fax:C1 530 752 7872.

E-mail address: [email protected] (B.A. Younis).
http://www.elsevier.com/locate/oceanenghttp://www.elsevier.com/locate/oceaneng


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1. Introduction

Tension Leg Platforms (TLP) are structures that are frequently deployed for deepwater

oil and gas operations in benign sea areas. They consist of a floating structure formed ofa working platform and floatation and storage members which are anchored to the sea bed

by pretensioned lines. An example of one, and the focus of the present study, is shown in

Fig. 10. An important parameter that influences the design of such structures is the

maximum offset which arises from the combined action of wind, currents and waves.

While it is difficult to ascertain the exact contribution of each of these effects, it is

generally agreed that, for deepwater operations in benign areas, no less than 60% of the

total offset is due to the action of currents alone and hence the need for accurate prediction

of loading due to a steady uniform current. Even for the case of a steady uniform current,

the flow past the TLP is quite complex and difficult to predict to the degree of accuracy

required in engineering design. The complexity arises from the large-scale and oftenunsteady separation from the various members of the TLP and from the non-linear wake

interactions that are often made more complicated by the presence of vortex shedding. The

sources of uncertainty in the numerical predictions are many. These include factors that

determine numerical uncertainty (e.g. grid resolution and choice of discretization scheme)

and others that determine physical realism such as the choice of mathematical model to

account for the effects of turbulence. To these must be added here the precise role played

by the free surface in determining the details of the pressure field in its vicinity and,

consequently, the hydrodynamic forces exerted on the various components of the TLP.

Younis et al. (2001) used a three-dimensional NavierStokes method to estimate the forces

on the same full-scale TLP considered in this study. As is often the practice in engineering

computations, the free surface was treated as a fixed plane of symmetry with slip boundary

conditions (the so-called rigid-lid approximation). This allowed the calculations to be

carried out in steady-state mode. That study was aimed at assessment of the numerical and

modeling factors that influence the quality of the predictions. The assumption of a fixed

free surface was recognized as being a major limitation whose precise role was worthy of

further study.

The present paper reports on the outcome of work carried out to remove the limitation

in the method ofYounis et al. (2001) in order to capture the evolution of the free surface

and thus determine its influence on the lift and drag forces acting upon the TLP. Themotivation is to assess whether the computation of the hydrodynamic loading is

significantly influenced by free surface effects at values of Froude number that are

representative of those found in offshore engineering practice. It would be reasonable to

expect, and important to confirm that, for sufficiently low values of Froude number, the

curvature of the free surface would not be so severe as to invalidate the rigid-lid

approximation. This is the case, for example, when a pair of vortices generated below the

free surface interact with it-as can be seen from the experimental study ofOhring and Lugt

(1991) and from the numerical simulations by Yu and Tryggvason (1990). There is, of

course, no reason to assume that the same would apply for the more complex case of the

TLP which, to date, does not appear to have been studied in much detail. In this paper, wedescribe the extension of the Younis et al. (2001) method to handle a moving free surface

and show the extent to which this impacts the computed results.

B.A. Younis, V.P. Przulj / Ocean Engineering 33 (2006) 181204182


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2. Computational method

2.1. Governing equations

The method ofYounis et al. (2001) is based on the solution of the differential equations

that govern the conservation of fluid mass and momentum in three dimensions. For the

case of an unsteady incompressible flow these equations are:

Continuity:

vUi

vxiZ 0 (1)

Momentum:

vUi

vtCUj

vUi

vxjZ

v

vxjn

vUi

vxjKuiuj

K

1

r

vp

vxi(2)

In the above Ui is the mean-velocity vector, p is the static pressure and r and n are,

respectively, the fluids density and kinematic viscosity. Conventional Cartesian-tensor

notation is used wherein repeated indices imply summation.

As is usual in the simulation of practically relevant flows at high Reynolds numbers, the

continuity and momentum equations have been averaged over a time interval Dt:

UiZ 1Dt

tCDtt

Uidt (3)

where Ui signifies instantaneous value. Dtis taken here to be the computational time step.

The implication of this averaging process is that fluid motions having time scale greater

than Dt are captured directly while motions having time scale smaller than the

computational time step would be filtered out and their effect will then need to be

accounted for via a turbulence closure. This averaging is quite distinct from the Reynolds

averaging (which is appropriate only for steady flows, i.e. in the limit ofDt/N). It is also

distinct from ensemble averaging which requires the period of oscillations to be known a

priori. The sole requirement for the validity of the present averaging practice is that thecomputational time step should be significantly greater than the time scale associated with

the turbulent fluctuations. This is immediately satisfied in the present flow where the

mean-flow Reynolds number is of O(106). Nevertheless, and irrespective of the precise

interpretation placed on the averaging process, the final outcome is the same; namely, the

appearance in the resulting Reynolds-averaged NavierStokes equations of unknown

Reynolds stress correlations Kuiuj which will first need to be determined before solutionof the governing equations becomes possible.

2.2. Turbulence modeling

To determine the unknown Reynolds stresses, we adopt the same turbulence model as

Younis et al. (2001); namely a two-equation model based on the Boussinesq assumption of

B.A. Younis, V.P. Przulj / Ocean Engineering 33 (2006) 181204 183


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linear stressstrain relation:

Kuiuj Z ntvUi

vxj

CvUj

vxi

K2

3dijk (4)

The eddy viscosity nt is defined in terms of two turbulence parameters: the turbulence

kinetic energy (k) and its rate of dissipation by viscous action (3):

ntZCmk2

3(5)

k and 3 are determined from the solution of their own differential transport equations.

These equations have the form:

vk

vtCUj

vk

vxj

Z

v

vxj

nt

sk

vk

vxj

CPkK3 (6)v3

vtCUj

v3

vxjZ

v

vxj

nt

s3

v3

vxj

CC31

3

kPkKC32

32

kKR

where Pk is the production of the turbulent kinetic energy:

PkZKuiujvUi

vxj(7)

The term R in the above is absent from the standard k3 model (Launder and Spalding,

1974). It is included in a widely used variant of the k3 model; the RNG (Renormalization

Group) formulation where it is defined as:

RZCmh

31Kh=ho1Cbh3

32

k; hZ S

k

3; SZ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2

vUi

vxjC

vUj

vxi

2s(8)

The complete turbulence models contain a number of coefficients whose value depend on

the variant used:

Model Cm sk s3 C31 C32 b ho

Standard 0.09 1.0 1.3 1.45 1.90

RNG 0.0845 0.72 0.72 1.42 1.68 0.012 4.38

Both turbulence models are of the high turbulence Reynolds number variety and are

thus only applicably in the fully turbulent regions of the flow, away from the walls. The

near-wall region is treated by assuming that the flow there can be described by the

universal, logarithmic, law of the wall. The alternative is to use a low Reynolds-number

version of the model and carry out the computations through the viscous sub-layer directly

to the wall. However, this would not be a viable proposition for the computation of flows

around full-scale structures at high Reynolds numbers.

2.3. Solution methodology and boundary conditions

The solution methodology, which is described in detail in Younis and Przulj (1993),

utilizes a finite-volume method for the solution of the generalised flow equations in



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arbitrary domains. The method is based on a term-by-term integration of the governing

equations over micro control volumes sub-dividing the solution domain. In the original

formulation, temporal discretization was by using the first-order accurate Euler scheme.

This proved to be inadequate for the accurate tracking the deformable free surface and wasreplaced here by an implicit second-order accurate scheme. In this scheme, the time

derivative at the new time level tiC1 was evaluated by fitting a quadratic function through

the values of the dependent variable at three time levels (Ferziger and Peric, 2002), viz.

tiC1C

Dt2

tiC1KDt2

vF

vtdtZ

3FiC1K4FiCFiK1

2(9)

where F stands for any of the dependent variables in the above equations. The

implementation of this scheme was checked by the prediction of vortex shedding fromsquare and circular cylinders at high Reynolds numbers and was found to be both robust

and accurate. The diffusive fluxes were approximated using second-order accurate central

differencing. The convective fluxes were approximated using the third-order accurate

SMART scheme (Gaskell and Lau, 1988) which was found by Teigen et al. (1999) to

produce accurate results with the least number of grid nodes. The resulting set of linear

algebraic equations was solved using a modified version of Stones (1968) lowerupper

decomposition algorithm. The overall solution strategy is iterative and is based on the

SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm of Patankar

and Spalding (1972) which ensures that the calculated flow field satisfies, simultaneously,

both the Continuity and momentum equations. The method was modified as detailed inFerziger and Peric (2002) to prevent pressurevelocity decoupling when used in

conjunction with colocated variable storage. A multi-block solution methodology was

adopted wherein each block was meshed separately with no requirement for the separate

grids to match at the block interfaces. Both block-structured and unstructured grids could

be used, with arbitrary polyhedral control volumes. Example grid for a full-scale TLP can

be seen in Fig. 10. The complete mesh was constructed within 47 separate blocks in order

to avoid the unnecessary placement of nodes within the TLP itself. The resulting node

distribution is highly non-uniform, with the greatest concentration being in the vicinity of

the TLP where sharp gradients exist.

Prediction of the free-surface movement was achieved by implementation of an

interface-tracking algorithm based on that described in Muzaferija and Peric (1997). This

entailed re-casting the governing equations above in a form applicable to a domain

bounded by a moving surface. Thus, for surface Smoving at velocity Vs, the continuity and

momentum equations become:

d

dt

V

d VolC

S

UiKVsnid SZ 0;

ddt

V

Uid VolCS

UjUiKVsnid SZS

Tijnid S

(10)



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where ni is the outward-pointing unit vector normal to the surface S and Tij is the stress

tensor (Eq. (2)).

In order to prevent the generation of a spurious velocity due to grid movement, an

additional equation was introduced to enforce space conservation at each time step(Demirdzic and Peric, 1990):

d

d t

V

d VolK

S

Vsnid SZ 0 (11)

Two boundary conditions were imposed at the free surface: the kinematic condition

which ensures that there is no convective mass transfer through the free boundary:

UiKVsniZ 0 (12)

and the dynamic condition which requires the forces acting on either side of the freesurface to be in equilibrium. By neglecting the normal components of the total stress

across the free surface, this requires setting the static pressure at the free surface to equal

the atmospheric pressure. This value was then used to calculate the fluid velocity at the

free surface from a reduced momentum balance (Ferziger and Peric, 2002). This is an

iterative procedure designed to ensure that both kinematic and dynamic conditions are

simultaneously satisfied. The kinematic condition is subsequently used in conjunction

with the space conservation law to displace the boundary cell faces to a new position. This

sequence forms part of an overall iteration cycle which includes the SIMPLE algorithm.

Iterations were performed at each new time level till a fully converged surface profile was

obtained and the total mass and momentum conserved over the entire solution domain.The normal gradients of k across the free surface were set equal to zero. For 3, there is

some uncertainty about the best choice of boundary conditions at the free surface. A

number of different alternatives were therefore used with the objective of quantifying the

sensitivity of the computations to the choice of boundary condition. Amongst the

alternatives examined was the specification of a zero normal gradient. Another alternative

was based on a simplified form of its transport equation there (Cokljat and Younis, 1995).

By neglecting convection and retaining diffusion only in the direction normal to the free

surface (which would be consistent with a fully developed flow in a broad open channel),

the 3-transport equation simplifies to:

v

vz

nt

s3

v3

vz

CC31

3

kPkKC32

32

k(13)

where z is the coordinate direction normal to the free surface. Furthermore, if kis taken as

constant across the thin layer adjacent to the free surface (which would be consistent with

the zero normal gradient condition) and 3 assumed to vary with zKn then Eq. (13) reduces

to the following simple expression for the values of 3 at the free surface:

3fZ 0:197n1=2 k

3=2f

Dnf(14)

where the subscript fdenotes value at the free surface and Dn is the normal distance from

the cell centre to the free surface. The value of the index n cannot be determined with great



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confidence from the few detailed measurements of the turbulence field below a free

surface reported in the literature though a value of unity seems to produce an acceptable

match. An assessment of the sensitivity of the results to the value ofn will be presented in

Section 3.The remaining boundary conditions for the TLP case were specified as follows. The

inlet to the computation domain was located a distance of 14 column diameters (D)

upstream of the TLP. There, the incident current was assumed to be uniform in the vertical

direction with velocity Uo. The turbulence kinetic energy was deduced by prescribing a

uniform local relative intensity level (hu 0/Uo) of 0.05 and then by assuming thatturbulence at inlet is isotropic which implies kZ3/2u 02. The dissipation rate was specifiedby inverting the eddy-viscosity relation (Eq. (5)) and by specifying the ratio of turbulent to

molecular viscosity to be 10. The assumption of finite turbulence intensity in the incident

current is necessary to prevent the turbulence kinetic energy from becoming negative at

small distances downstream of the inlet where the dissipation rate is finite but theturbulence production rate is still zero due to the absence of shear. At exit from the

computational domain, which was located at distance of 32D from the TLP, the flow was

assumed to be fully re-established in the streamwise direction and thus all the gradients in

that direction were set equal to zero.

The planes that define the width of the computational domain were located at distance

16.3D from each side of the TLPs centerline. The boundary conditions applied there are

similar to those for the exit plane. Values of the wall friction at the sea bed (which was

located at distance 6.5D from the base of the pontoons) and on the TLP itself were used to

provide the wall boundary conditions for the momentum equations. Both surfaces were

assumed to be smooth and the wall friction was thus evaluated from the standard log-law:

U

utZ

1

kln E

utn

n

(15)

where ut is the friction velocity and n is the normal distance from the wall. k (the von

Karman constant) and E were assigned the usual values of 0.41 and 9.0, respectively.

3. Results and discussion

We first checked the complete computational model together with the free-surface

tracking algorithm by obtaining predictions for two benchmark flows: the flow in an open

channel with a semi-circular cylinder placed at the bottom and the flow around a parabolic

hull. The geometry of the first test flow is defined in Fig. 1. Different flow regimes may

develop downstream of the cylinder depending on the value of Froude number upstream of

it. The interest here is in the critical case with sub-critical flow FrhU0=ffiffiffiffiffiffiffi

gHp

!1upstream of the cylinder and super-critical flow FrdhUd=

ffiffiffiffiffiffiffigH

pO1 downstream of it.

Numerical predictions (based on potential flow analysis) and experimental results for this

flow have been reported by Forbes (1988).The inlet plane was located at distance 10D upstream of the cylinder. Uniform

distributions of velocity and of turbulence intensity (Z0.05) were assumed there and



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the dissipation rate 3 deduced as described earlier. At the free-surface, the normal

gradients of all the dependent variables were set equal to zero. The outlet plane was at

distance 3D downstream of the cylinder. There, the streamwise gradients of all dependent

variables were set equal to zero and a self-adjusted pressure boundary condition was

employed wherein the outlet pressure was obtained by linear extrapolation of values

computed at two adjacent interior nodes. In order to enforce the critical flow solution, the

computations were started with a pre-specified pressure at the outlet, whose value

corresponds to the super-critical conditions. This value is calculated from the approximate

relation (Naghdi and Vongsarnpigoon, 1986):

poutZ rgHg; gZh

Hz

Fr2

41C 1C

8

Fr2

1=2 : (16)

Once the downstream level hzgH was reached, (which, in the present calculations,

occurred afterz1.4 s), the values of pressure at the outlet plane were switched to the self-

adjusted values. At tZ0, the velocity was set everywhere to its value at the inlet plane and

the free-surface was taken to be horizontal.

In order to compare with the data ofForbes (1988), predictions were obtained for three

sets of the upstream conditions. Those are defined in Table 1.

The values of R/H are quite close to those reported in the experiments. The upstreamvalues of Froude number Frwere not reported. The measured values ofh/Hwere therefore

used to determine the approximate values of Fr. This was done by using Bernoullis

equation, written for the upstream and downstream sections, to obtain:

FrZh

H

2

1C hH

!1=2(17)

From knowledge of R/H and Fr, the height H and the inlet velocities could then be

calculated. The radius was assigned the same value as in the experiments, viz. RZ0.03 m.

Two grids were used for this test flow. One had a total of 2096 active nodes and anotherwith 7744 active nodes. Both grids were generated in three separate blocks. The finer grid

had a total of 80 nodes in direct contact with the cylinders wall. Inspection of the computed

Fig. 1. Geometry for the free surface flow over a semi-circular cylinder.



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results showed that the nodes closest to the wall were at distances (in wall coordinates) in

the range 6!Y*!23. The time-step size was DtZ0.001 s.

The initial distribution of the finer grid is shown in Fig. 2. The final (steady state)

distribution of the same grid, which was arrived at after a period T of 21.4 s, is shown in

the same figure. Although the final grid is very skewed near the free surface downstream

of the cylinder, no convergence problems were encountered during the solution cycle and

the resulting free-surface profile showed no signs of discontinuity. Tests were carried out

to check the sensitivity of the solutions on the grid size and on the choice of

discretization scheme. Sample results are presented in Fig. 3. Plotted there are the free-

surface profiles as obtained on the two grids and using both first- and third-order accurate

differencing schemes. While the effects of grid refinement are small, there are no visible

differences when the two differencing schemes are used with the fine grid. The present

results predict separation to occur at about 1508a value which is in good accord with

observations.

All the remaining runs for this test case (Table 1) were performed using the finer gridtogether with the third-order accurate scheme. The predicted piezometric pressure

distribution for the case of FrZ0.336 is shown in Fig. 4. The predicted free-surface

profiles are presented in Fig. 5. The figure also shows the results for three sets of upstream

values of Fr. As expected, the downstream water level increases with the upstream level

(i.e. with Fr). For the case ofR/HZ0.435, the computational and experimental conditions

are very similar and the predicted and measured profiles can be compared. It is clear that

the computations are in close accord with the data, especially in the region of rapid free-

surface variation. The computed downstream free-surface levels corresponding to the

three inlet conditions are compared in Fig. 6 with the measurements ofForbes (1988). The

agreement is again fairly good with maximum relative difference being of the order of 5%.This is quite an acceptable result especially bearing in mind the uncertainty associated

with the absence of experimentally reported values of Fr. The potential-flow results of

Forbes are plotted in the same figure; their proximity to both the measurements and the

turbulence-model predictions suggests that turbulent mixing plays only a secondary role in

determining the behavior of this flow.

The second test flow considered is the Wigley parabolic hull for which extensive

experimental data exist (Anon, 1983). The shape of this hull is made of parabolic curves in

(x, y) and (y, z) coordinate planes described by Eq. (18):

yZB

21K

2x

L

2 1K

z

D

2 ; (18)

Table 1

Upstream flow conditions for the flow over a semi-circular cylinder

R/H Fr H (m) U0 (m/s) ReZU02R/n

0.300 0.540 0.100 0.535 32,100

0.400 0.373 0.075 0.320 19,200

0.435 0.336 0.069 0.276 16,560



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The length (in the x-direction) is LZ1 m, the width (in the z-direction) is BZ0.1Land

the depth is DZ0.0625L. Comparisons are presented for the case with Froude number

(based on L) of 0.267. This corresponds to a uniform inlet velocity ofU0Z0.8363 m/s and

a Reynolds number (also based on L) of 836,300.

Computations were performed on three numerical grids that were generated within six

blocks. Only the nodes of the blocks immediately below the free surface were moved at

each time step to track the evolution of the free surface. Grid details are given in Table 2.

The values ofY* shown there denote the range of the distances, in wall coordinates, from

the hull to the center of the control volumes in contact with it.All the computations which were carried on the three different grids, with both first- and

second-order accurate schemes and with time-step size in the range 0.01!Dt!0.002

Fig. 2. Free surface flow over a semi-circular obstacle at FrZ0.336 and R/HZ0.435: initial grid (top) and

(middle) and the final shape of the grid (bottom).



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yielded steady-state solutions around the Wigley hull. This can be seen from Fig. 7 which

presents the time histories of the total (piezometric pressureCviscous) drag coefficient CD

and the viscous drag coefficient CDv. The plots show that the flow becomes essentiallysteady after approximately 60 s from the start of the calculations. Beyond this time, the

normalized residuals of all the dependent variables fell below 4.0!10K5 after only one

iteration.

The results obtained with the three grids in conjunction with the third-order

accurate scheme and the RNG kK3 model are presented in Fig. 8 where the

Fig. 3. Semi-circular obstacle. Effects of the grid refinement and convective schemes on the computed free-

surface shape.

Fig. 4. Pressure distribution for the flow over a semi-circular obstacle (FrZ0.336, R/HZ0.435).



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computed free surface profiles along the hull are compared with experimental data.

The predicted profiles with the fine or medium grids agree very well with the

measured one. The coarse-grid results show some variation, especially in the regionahead of the hull. A further check grid independence is provided with reference to the

Fig. 5. Semi-circular obstacle. The free-surface shape as computed at different upstream Froude numbers and

upstream water levels.

Fig. 6. Flow over a semi-circular obstacle. Comparison of computed free-surface downstream levels with

experimental and numerical data of Forbes (1988).



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hulls viscous drag coefficient. In ITTC-57-Formel (1957), the following expression

for this quantity is proposed:

CDvZ0:075

log

Re

K2

2

: (19)

Table 2

Grid details for the Wigley hull

Grid size Active CVs CVs over hull Y*

37!12!22 9700 20 48227

72!24!42 72500 40 1125

108!38!68 279000 60 824

Fig. 7. Wigley hull. Time histories of the total and viscous drag coefficients.



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Table 3 compares the present results with the above equation. The differences

between the medium and fine grid results are clearly very small; both are also in

close agreement with the ITTC correlations.

Next, we assess the sensitivity of the predictions to the choice of turbulence model andfree-surface boundary conditions for the dissipation rate equation. Of particular interest is

the effect of the above parameters on the predicted free-surface profile. Sample results are

presented in Fig. 9 (top). Plotted there are the results for the relative free surface elevation

z/L(zZ0 defines the initial free surface level) as obtained with the standard and the RNG

versions of the kK3 model. It is clear that the free surface profile is not particularly

affected by the choice of turbulence model. This turns out to be the case also for the choice

of boundary condition for 3, as can be seen from Fig. 9 (bottom). The designation FS-BC0

there refers to the boundary condition proposed by The et al. (1994) and Hagiwara (1989)

wherein the value of the dissipation rate at the near-surface is determined from:

3fZC3=4m k

3=2f

kDnf; (20)

where CmZ0.09 and kZ0.41. FS-BC1 is the boundary condition given by Eq. (14) with

nZ1. Note that the boundary condition given by Eq. (20) is obtained simply by setting

Table 3

Predicted and measured viscous drag coefficient CDv

Grid Computed CDv!103 (A) ITTC-57 CDv!10

3 (B) (AKB)/B (%)

9700 3.189 4.875 K34.672500 4.363 4.875 K10.5

279000 4.435 4.875 K9.0

Fig. 8. Effect of the grid refinement on the free surface profile along the hull.



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nZ4 in Eq. (14). FS-BC2 refers to the case where 3 is not fixed at the free-surface nodes

but is calculated there by applying the zero-gradient condition. The results for all three

types of boundary conditions are indistinguishable which again suggests that the

turbulence effects play only a minor role in determining the shape of the free surface.

Attention is now turned to consideration of the flow around the full-scale TLP. A

schematic representation of this structure is shown in Fig. 10 which also serves to definethe coordinate system. This is the same TLP that was the subject of earlier studies (Teigen

et al., 1999; Younis et al., 2001) in which the rigid-lid approximation was employed.

Fig. 9. Effects of the turbulence model (top) and free surface boundary conditions for the dissipation (bottom) on

the free surface profile along the hull.



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In Fig. 10, S is the columns center-to-center separation. B is the pontoon height and H is

the column height. The actual dimensions used in the present simulations are given in

Table 4.The TLP members are identified by the numbers shown in Fig. 10. The pontoons are

numbered (1, 2, 3, 4), the circular bases are numbered (5, 6, 7, 8) and the and columns are

numbered (9, 10, 11, 12).

The computational grid conformed to the boundaries of the TLP (see Fig. 10) and

consisted of 415,444 active nodes arranged in 47 separate blocks. Only the nodes that lie

within the blocks above the pontoons (i.e. the top layer of blocks below the free surface-13

blocks in total) were moved during the solution process. The nodes that were common to

both the inlet and the free surface planes were moved by the same extent as the

neighboring free surface nodes. It proved impractical to conduct a systematic grid-

independence test in this flow as the computations were performed in unsteady mode andrequired considerable time to adjust to the effects of the initial conditions. Teigen et al.

(1999) in their study of the same TLP found that when using the SMART scheme,

Fig. 10. TLP geometry and grid.

Table 4

Dimensions of TLP

Member Symbol Size (m)

Column diameter D 8.75

Column height H 22.25

Column separation distance S 28.5

Pontoon height B 6.25Pontoon width W 6.25

Draught T 28.5



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grid-independent solutions for the same TLP were obtained for a grid of 330,000 nodes.

The present results were also obtained with the SMART scheme and hence it would be

reasonable to expect that the solutions with 415,444 nodes are not far from being sensibly

free of grid effects. Moreover, the computations for both the fixed and the deformable freesurface cases were obtained with the same numerical conditions and thus it would be

reasonable to attribute similarities and differences in their results to the treatment of the

free surface.

Predictions were obtained for two values of inlet velocity, viz. U0Z1 and 2 m/s. The

corresponding Froude and Reynolds numbers, defined as:

FrZU0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigHCBp ; ReZ

rU0D

m(21)

were (0.06, 3.7!106), and (0.12, 1.46!107), respectively1.

An impression of the distortion to the free-surface wrought by the surface-piercing

columns and the submerged pontoons can be gained from Fig. 11. The results shown are

for the higher of the two values of Froude number where the extent of the perturbation

above the mean water level is more prominent. The distortion to the free surface is most

clearly manifested by a significant run-up along the stagnation line of the front columns

followed by rapid fall in the wake regions to levels below the mean water level. The

pattern is repeated for the downstream column, albeit to a lesser extent. Overall, these

perturbations are quite small relative to the overall dimensions of the TLP. Thus, for

example, the largest changes relative to the initial level (zZ

0) were approximately 1.0 m

Fig. 11. View of the free-surface shape around the TLP model for FrZ0.12.

1 Sea water: (rZ1026 kg/m3, mZ1.2312!10K3 kg/m s).



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for FrZ0.12 (falling to 0.15 m for the FrZ0.06 case). Nevertheless, the distorted free

surface can assume quite a complicated shape, as can be seen from the figure.

Fig. 12 shows the mean velocity and the piezometric pressure distributions at a

horizontal plane parallel to the free surface and immediately below it (zz0). The pressureis normalized by the mean-flow kinetic energy at inlet. Immediately discernible are small

regions of reversed flow in the wakes of the TLP columns. Also evident is the uneven fluid

acceleration on either side of the columns which leads to a shift in the location of the

separation bubble away from the TLP core. This, in turn, leads to a marked skewness of the

flow towards the downstream columns. The predicted contours of static pressure suggest

that interference effects are quite substantial with the pressure levels on the stagnation line

of the downstream column attain only about 20% of the levels found for the upstream

cylinders.

The computed flow field are next presented in two vertical planes: one passing through

the centers of the flotation pontoons (yZ0) and another passing through the middle of thecolumns (9 and 10) and pontoon 1 (y/SZ0.5). The plots are presented in Fig. 13 for the

mean velocity, Fig. 14 for the static pressure and Fig. 15 for the turbulence kinetic energy.

The velocity plots indicate the formation of a region of reversed flow in between the two

columns, centered close to the pontoon with a smaller counterpart near the junction with

the free surface. Flow reversal is obtained in the wakes of both pontoons. The resulting

flow is slightly skewed towards the free surface. The contours of the turbulent kinetic

energy are seen to rise towards the free surface.

Fig. 16 shows the predicted variation with time of the in-line force coefficients on the

circular columns [9] and [10] (numbers in square brackets refer to the designation in

Fig. 10). The in-line force coefficient is defined as:

CDZFx

12

rU2oA(22)

where A is the projected area of the column, i.e. D!H. Also plotted there (as solid lines)

are the values of these coefficients obtained when the moving free surface is replaced by a

rigid lid, i.e. by employing boundary conditions that are appropriate to a fixed plane of

symmetry. The peak-to-peak variations in the magnitude of the in-line forces are, as

expected, more pronounced for the smaller Froude number flow. However, the

dependence of the long-time averaged value of CD on Fr is very weak viz. 1.150 forthe smaller value of Fr compared to 1.12 for the greater value. The same also applies to

column [10] where the average CD values are 0.68 and 0.63, respectively. These figure also

demonstrate the extent of the shielding on the downstream column [10] due to

modification of the flow field brought about by the upstream column. A measure of the

coupling between the in-line force fields is provided by the correlation coefficient g:

gZC0D9C

0D10

C0D9C0D10

(23)

For the case of FrZ0.06, g is obtained as 0.59. This relatively high value for thecorrelation coefficient is consistent with the streamwise re-alignment of the flow

downstream of column [9].



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20/24

applications (Younis et al., 2001). The results obtained with both treatments of the free

surface are within about 5% of each other which confirms the relatively minor infleunce of

the free-surface curvature in this flow. As expected, the DNV result is somewhat greater asthe calculation of the global drag coefficient does not take account of the shielding effects

mentioned earlier.

Fig. 13. Predicted contours of mean velocity through the mid-plane (yZ0, top) and at pontoon half-width (yZ

0.5S).



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4. Conclusions

The paper described the extension of a well-validated three-dimensional NavierStokessolver to capture the evolution of a free surface with a moving grid. The solver was previously

used in a detailed study of the hydrodynamic loading on a submerged full-scale TLP.

Fig. 14. Predicted contours of piezometric pressure through the mid-plane (yZ0, top) and at pontoon half-width

(yZ0.5S).



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Predictions reported here demonstrate the validity of the free-surface tracking algorithm via

comparisons with experimental data obtained in the turbulent flow over a semi-circularcylinder and around a parabolic hull. The present results show that the predicted

hydrodynamic loading on a full-scale mini TLP is not too dependent on curvature of

Fig. 15. Predicted contours of turbulence kinetic energy through the mid-plane (yZ0, top) and at pontoon half-

width (yZ0.5S).



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the free surface, at least not at the low values of Froude number encountered in practice. Apart

from some small changes in the loading on the surface-piercing columns, loading on the

pontoons, which represents a significant proportion of the total loading, was hardly affected by

movement of the free surface. This finding is of immense relevance to the practical

computations for design purposes for it indicates that the use of the computationally more

efficient and robust no-slip boundary condition is quite appropriate in these applications.

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Table 5

Predicted CD for model TLP

Technique Re CD

Rigid lid approximation 6!107 1.39

Deformable free surface 6!107 1.32

DNV design code 8.84!106 1.61

Fig. 16. Predicted time history of in-line loading on columns 9 and 10. FrZ0.06, FrZ0.12. Horizontal lines show

results for rigid-lid approximation.



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