Date 2012 . Author Duz, Bulent, Rene H.M. Huijsmans e.a.
Address Delft University of Technology
j O l T f r
Ship Hydromechanics and Structures Laboratory \ ^ I I L
Mekelweg 2, 2628 CD Delft
Delft U n i v e r s i t y of T e c h n o l o g y
Application of an absorbing boundary condition
in a w a v e - s t r u c t u r e interaction problem
by
Duz, B., R.H.M. Huijsmans, M.J.A. Borsboom and
A.E.P. Veldman
Report No. 1852-P 2012
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APPLICATION OF AN ABSORBING BOUNDARY CONDITION IN A
WAVE-STRUCTURE INTERACTION PROBLEM
Bulent Duz*
Department of Ship Hydrodynamics Technical University of Delft
Mekelweg 2, 2628 CD Delft The Netherlands b.duz@tudelft.nl
Rene H.M. Huijsmans
Department of Ship Hydrodynamics Technical University of Delft
Mekelweg 2, 2628 CD Delft The Netherlands r.h.m.huijsmans@tudelft.nl
Mart J.A. Borsboom Peter R. Wellens
Deltares
P.O. Box 177, 2600 MH Delft The Netherlands
{mart.borsboom, peter.wellens}@deltares.nl
Arthur E.P. Veldman
Institute for Mathematics and Computer Science University of Groningen
PO. Box 407, 9700 AK Groningen The Netherlands
a.e.p.veldman@rug.nl
ABSTRACT
For the design of offshore structures, an accurate assessment of the ability of the structure to survive in extreme sea condidons is of prime importance. Next to scaled model tests on the struc-ture in waves, also CFD capabUities are at the disposal of the designer. However even with the fastest computers avaUable, it is sdll a challenge to use CFD in the design stage because of the large computational resources they require.
In this study we focus our attention on the implementation of an absorbing boundary condition (ABC) in a wave-structure interaction problem. Unlike the tradidonal approach where the boundaries are located far from the object to avoid reflection, we gradually locate them closer while at the same time obsen'ing the influence of the absorbing boundary condition on the solution. Numerical calculations are perfonned using the CFD simulation tool ComFLOW which is a volume-of-fluid (VOF) based Navier-Stokes solver. Comparisons with experimental results are also provided and the performance of the ABC is discussed.
'Address all correspondence to this author.
INTRODUCTION
Scientists and engineers regard the surface of the ocean as interesting due to one o f the broadest physical phenomena which has been intensively studied in various fields of science: waves. Although waves can occur i n all types amongst a variety depend-ing on the forces actdepend-ing on the water, one aspect exists i n aU the cases and attracts our attention: waves exert loads and stresses on numerous kinds of man-made structures i n the ocean. I f the other elements o f the environment are also hostile, major catas-trophic failures can appear threatening human safety and causing economic loss.
Although highly capable numerical features are at the dis-posal o f researchers, particular aspects o f numerically solv-ing wave-structure interaction problems in unbounded domains cause various bottlenecks. Typically the phenomena o f interest are local but embedded in a vast spatial domain. A t this point, the infinite domain, although sometimes it may not be tnily un-bounded, is truncated via artificial boundaries, thus introducing a finite computational domain and a residual infinite domain. One
of such bottlenecks is developing a robust and efficient boundary condition to be imposed on these artificial boundaries.
The Sommerfeld boundary condition [1] was the comer-stone of non-reflecting boundary conditions. Engquist and Ma-jda [2] presented a method to develop the first hierarchy of ab-sorbing boundary conditions. Higdon [3] generalized this the-ory and showed that Engquist and Majda boundary condition is a subset of the Higdon operators. Since high-order boundary operators include high-order derivatives both i n time and space, Collino and Joly [4] introduced the use of auxiliary variables to circumvent this difficulty. This idea has found widespread in-terest and has been used by Grote and Keller [5], GivoU and Neta [6], and Hagstrom and Warburton [7] among others. For a general review regarding high order local non-reflecting bound-ary conditions, see [8].
b l this paper, we study application of an absorbing boundary condition ( A B C ) in a wave-structure interaction problem where a fifth-order Stokes wave is traveling under an angle of incidence in a three-dimensional computational domain. Numerical com-putations are carried out using ComFLOW [9,10]. Here we focus our attention specifically on the performance of the A B C for the duration of the simulations. For this purpose, numerical results are compared to the experimental results. We end the paper with some concluding remarks.
MATHEMATICAL MODELING
I f we consider water as a homogeneous, incompressible, vis-cous fluid, we can describe fluid motion i n a three-dimensional domain Q (see Fig. 1) by the continuity equation and the Navier-Stokes equations in a conservative f o r m as.
(a)
u • n £?r = 0, (1)
j^^da+j u u ^ • ndT =
r
- j{pn~ pVu • ii)dr+ ƒ FdQ.. (2)
In Eqns. (1) and (2), Q. denotes a volume with boundary F and normal vector n, u = (;/, v , w ) ^ is the flow velocity, p is the fluid density, p is the pressure, p is the dynamic viscosity, V is the gradient operator and F = {Fx,Fy,F^Y represents external body forces acting on the fluid such as gravity. For discretization of Eqns. (1) and (2), see, e.g., [11,12].
To solve Eqns. (1) and (2) i n Q, we impose three types of boundary conditions: the Dirichlet boundary, the generating and
r . v
f /
(b)
w r ,
FIGURE 1. A C O M P U T A T I O N A L D O M A I N W I T H A STRUC-T U R E L O C A STRUC-T E D A STRUC-T STRUC-T H E CENSTRUC-TER. L^v, F f , F^ and Fiv A R E A R STRUC-T L F I C I A L B O U N D A R I E S O N W H I C H A N A B C IS TO B E A P P L I E D .
absorbing boundary and a free surface conditions. Below we describe these boundary conditions.
On the north and east boundaries Ff^ and FE:
On the north and east boundaries FN and F ^ the absorbing boundary condition ( A B C ) is prescribed. The boundary con-dition on F/i/ and FE should aUow the waves to move out of the computational domain without generating spurious reflec-tion. Here, we will derive this boundary condition only f o r FE since it is rather straightforward to extend the idea to F^. Now, consider the foUowing boundary operator:
coso;-^ -l-c"
dt dx (3)
Equation (3) is perfectly absorbing i f the boundary parameter
a is equal to the angle of incidence 6 of a wave (see Fig. 1(a))
characterized by the velocity or wave potential and traveling out of the domain with the phase speed c"'". To make Eqn. (3) independent of the phase speed c"'", we introduce the following rational expression,
and tangential stresses the following conditions are implemented for the velocity components and the pressure.
(8)
~ao + ai{k'""hf \+bi{k°"'li)
which approximates the dispersion relation.
2 ' (4)
tanhik"'"!!)
(5)
In Eqn. (4), a proper choice of coefficients ao. "1 and bi leads to a close approximation for a wide range of k"'")! values. Thus, reflection from the boundary w i l l be minimized over that specific range. Although this range can be specified by adjusting the co-efficients, their values are bounded by stability constraints. For an extensive study regarding the stability analysis and choosing a set of coefficients to ensure stability, see [13].
Next, exploiting the exponential behavior of i n the z direction, we can replace the wave number k°"' with
(6)
By means of Eqns. (4) and (6), the wave number is resolved using the velocity potential 5>°"'.
Since Eqns. (1) and (2) are specified as the governing equa-tions, the time and spatial derivatives of the velocity potential i n Eqn. (3) must be interpreted i n terms of the velocity components and pressure. Recalling the linearized Bernoulli equation and potential theory, we have d^/dt = —pt, — gZp and d^/dx =
lib-Finally, combining Eqns. (3), (4) and (6), we obtain
cos a 1+bih Pb
'gh [ao + aih'-^]iib+gZp=Q. (7)
In Eqn. (7) the subscript b indicates that the quantity is defined at the boundary and the subscript p indicates that the quantity is evaluated at the elevation of the pressure point. For the discrete f o r m of the ABC, see [14].
At the free surface
Tps'-A t the free surface F^^ resulting f r o m the continuity of normal
^ da,,
-p + 2 p ^ = -po + aK, (9)
where »„ and a, correspond to the normal and tangential compo-nent of the velocity, respectively, po is the atmospheric pressure,
a is the surface tension and K is Are total curvature o f the free
sur-face. I f we describe the position of the free surface hy s{x,t)= 0, the displacement of the free surface can be computed via,
Dt
dt
+ {u-V)s = 0. (10)For the reconstruction and advection of the free surface, see, e.g., [15,16].
On the bottom and structure Tg and
FST'-On the bottom and structure TB and TST we specify a slip no-penetration condition which is simply the Dirichlet condition, i.e.
u = 0.
On the west and south boundaries Tw and Vs:
On the west and south boundaries Tw and the generating and absorbing boundary condition (GABC) is applied. The G A B C should play the role of an open boundary condition permitting waves to move into and out of the computational domain. When the total wave signal at the inflow boundary is decomposed as (J) = <j)(" _|_<j)Oi''^ tijgn it is straightforward to substitute this rela-tion in Eqn. (3). Here, represents the incoming wave which is known beforehand and given in Tab. 1(a) whereas rep-resents the outgoing wave which is diffracted f r o m the structure and propagating in all directions. For the discrete form of the G A B C , see [17].
By means of all the boundary conditions explained above we complete the statement of the problem.
NUMERICAL SIMULATION
We apply the mathematical model, expressed i n the pre-vious section, to a problem in three dimensions where a fifth-order Stokes wave is travehng under an angle of incidence i n a computational domain with a structure located at the center (see Fig. 1 f o r an illustration of the problem). The structure is a semi-submersible consisting of two columns and a pon-toon under water, see Fig. 5(a) for the dimensions of the semi-submersible at f u l l scale. Now, we define a parameter d which
F I G U R E 2. P A R A M E T E R d CONTROLS T H E D I M E N S I O N S OF T H E C O M P U T A T I O N A L D O M A I N O N T H E x - y P L A N E .
controls the dimensions of the computational domain on the x —y plane as shown in Fig. 2. Basically, we choose two values for d:
d =\L= 130;?? and d = 0.5L = 65m where L denotes the wave
length. The first choice results i n a domain with the length and width of Rx = Ry = 265m whereas in the second case the length and width are = Ry = 173»!. Table 1 (b) gives configurations of the test cases. As we locate the boundaries closer to the structure, we w i l l investigate how the GABC influences the results. Note that in both cases, the dimension of the computational domain in the z direction is R^ = 210/h and the water depth is h = 180/n.
To capture the details of the flow near the stmcture, a finely spaced grid i n all directions is used as sketched i n Fig. 3(a). How-ever, far away f r o m the stmcture, the grid becomes progressively coarser.
The characteristics o f the fifth-order Stokes wave are given in Tab. 1(a). A t f = 0 this wave is generated and initialized every-where in the computational domain as depicted in Fig. 3(b). A t every time step starting from the initial condition, the solution variables are updated according to the fifth-order Stokes wave theoiy [18] and prescribed at the inflow boundaries.
S - 1 0 0 v - 1 5 0 - J
(a) A VIEW OF COMPUTATIONAL GRID.
(a) CHARACTERISTICS OF THE WAVE.
WAVE
5th-order Stokes w a v e
WAVE period helgth length s t e e p n e s s water depth WAVE
T ( s ) H (m) L ( m ) H/L h (m) W9 9 7,5 130 0,058 180
(b) CONFIGURATIONS OF THE COMPUTATIONAL DOMAIN IN CASE 1 AND CASE 2.
C h a r a c t e r i s t i c s of t h e t e s t c a s e s Distance d (m) Domain size Rx X Ry X Rz Number of cells Boundary condition on Fw and Fs Boundary condition on Fn and Fe Case 1 I L 265ni«265mx210m 1.5 million GABC ABC Case 2 0,5L 173mxl73mx210m 0.9 million GABC ABC
T A B L E 1. INPUTS FOR T H E N U M E R I C A L S I M U L A T I O N S .
t =
O.Os
x - a x I s
(b) INITIAL CONDITION FOR THE NUMERICAL COMPUTA-TION.
F I G U R E 3. B A S I C SETUP FOR T H E S I M U L A T I O N I N C A S E 2. T H E S A M E PROCEDURE IS F O L L O W E D FOR C A S E 1 AS W E L L .
EXPERIMENTAL MODELING
The tests were carried out at Maritime Research Institute in the Netherlands ( M A R I N ) in 2008. Several experimental param-eters were specified, such as the design o f the semi-submersible and the generated sea states, to provide a comprehensive analy-sis regarding the physics behind the wave-structure interaction. In this section, we present some details of the test considered i n this smdy. The 1:50 scale of the semi-submersible was modeled and its position was fixed i n the High Speed Basin with dimen-sions 200m long by 4m wide by 4m deep (see Figs. 4(a) and 4(b) for the photographs of the semi model i n the work shop and in the experiment, respectively, and also, see Fig. 5(a) for the general plan o f the semi-submersible at f u l l scale). The basin is equipped with a flap-type wave generator at one end and a beach at the other end. The center o f the test set-up was located 100m away f r o m the wave flap. For simplicity, i n the remainder o f this paper all the details w i l l be presented in prototype values unless stated otherwise.
(a) THE 1:50 MODEL OF THE SEMI-SUBMERSIBLE I N THE WORK SHOR
(b) WAVE RUN-UP AND IMPACT ON THE FORWARD COLUMN AND DECK IN REGULAR WAVES.
The waves were produced with a flap-type wave generator at one end of the basin. The surface elevations were measured using resistance type wave probes at various locations on and around the model. Additionally, piezo-type pressure transducers were attached to the struchire to monitor wave impact pressures for the duration of the tests, see Fig. 5(b) for the positions of the measurement instruments which w i l l be utilized to compare the numerical results to the experimental results. Also, video recordings were made during the model tests using various types of cameras. The measurements ceased as soon as reflected waves f r o m the beach started to arrive at the semi model i n the basin.
+
-(a) GENERAL PLAN OF THE SEMI-SUBMERSIBLE AT FULL SCALE.
I Pressure i Sensors ; Pressure i Sensor 4 Pressure Sensors Pressure ; Sensor 2 : Pressure i Sensor 1 : • • • • i • Pressure Sensor 6 : Wave i Sensor 2 Wave Sensor4 t i i
; Wave i ; Wave i Wave i Wave i
\ Sensor 1 ; : Sensor 3 ! i Sensor 5 i i Sensor 6 i (b) POSITIONS OF THE WAVE PROBES AND PRESSURE TRANSDUC-ERS. F I G U R E 4. PHOTOGRAPHS OF T H E S E M I - S U B M E R S I B L E M O D E L I N T H E W O R K SHOP A N D I N T H E E X P E R I M E N T . F I G U R E 5. DETAILS OF T H E S E M I S U B M E R S I B L E A N D M E A -S U R E M E N T I N -S T R U M E N T -S . 5 Copyright © 2012 by A S M E
t O 22,86s C a s e 2 C a s e 2 C a s e 1 C a s e 2 t = 2 S , E 1 s C a s e l C a s e 2 C a s e 1 C a s e l C a s e 1 C a s e 2 F I G U R E 6. SNAPSHOTS OF T H E N U M E R I C A L S I M U L A T I O N S I N C A S E 1 A N D C A S E 2 AT VARIOUS INSTANCES ESI T I M E .
Several sea states were generated including regular and ir-regular long crested head waves. Here, we w i l l present the results for the wave given i n Tab. 1 (a).
RESULTS AND DISCUSSION
In this section, we w i l l compare two simulations given in Tab. 1(b) to each other and to the experiment. For this purpose, we w i l l utiUze measurements o f the wave elevation and wave impact pressures at various positions during the experiment and numerical simulations where the entire structure was restrained.
C a s e 1 C a s e 2
F I G U R E 7. C O N T I N U E D F R O M FIG. 6.
The duration o f the numerical simulations is 40 seconds, which is approximately 4 wave periods. However, the experi-ment continued for 1850 seconds. Therefore, at first, we had to specify a time frame of 40 seconds f r o m the experiment during which the wave field is f u l l y developed but the effect o f the side walls and the beach is not dominant. After analyzing time traces of the measured signals, r = 1105* is selected as the starting point and during the subsequent 40 seconds the comparisons are made between the numerical simulations and the experiment.
Two simulations are carried out to investigate the perfor-mance o f the absorbing boundary condition which is applied on the boundaries surrounding the structure. Table 1(b) gives details
1
—E x periment —Case 1 —Case 2 j —m^
i
1105 1110 1115 1120 1125 1130 Time (5) (b) WAVE SENSOR 2 1135 1140 1145 16 12f a
I "
I
"I 1 —Experiment —C a s e l —C a s e 2 | i -4 1105 1110 1115 1120 1125 1130 1135 1140 1145 Time (s) (b) WAVE SENSOR 5I
0•-Experiment Case 1 —Case 2 i L .
-—IF
1-IA
— /' V ^ 1' \Lu^
J 1 1 Time (8) 1135 1140 1145 (c) WAVE SENSOR 3 F I G U R E 8. W A V E E L E V A T I O N AT VARIOUS L O C A T I O N S . I 4 J 2 oI
0 -2 -41 —Experiment —Case 1 —Case 2 |
1 1 „ \ ^ 1105 1110 1115 1120 1125 1130 1135 1140 1145 Time (a) (c) WAVE SENSOR 6 F I G U R E 9. C O N T I N U E D F R O M F I G . 8.
of these simulations. The difference between Case 1 and Case 2 is the dimensions o f the computational domain on the x—y plane. As we locate the boundaries closer to the strucmre i n Case 2, the number o f cells used in the simulation reduces substantially, which as a result saves memory consumption as well as compu-tational effort. In this section, we w i l l demonstrate whether this advantage comes at a price.
Figures 6 and 7 show snapshots of the simulations at numer-ous instances for Case 1 and Case 2. Throughout both o f the simulations, we detect no apparent differences at the free
sur-faces between two computations, which was the objective o f our approach.
Figures 8 and 9 show the wave elevation and Figs. 10 and 11 show the wave impact pressure at various locations, see Fig. 5(b) for the positions o f the wave probes and pressure transducers. Observing Figs. 8 to 11, we first notice that the results f r o m the numerical simulations agree well with each other although some-times they do not with the experiment. As a first requirement, we expect the solution i n Case 2 to be reasonably close to that in Case 1. Essentially, the procedure of reducing the size of the
500 400 g- 300 * I 200 £ loo; 0 —Experiment — Case 1 —Case 2 —Experiment — Case 1 —Case 2 •i ^ i 05 1110 1115 1120 1125 1130 1135 1140 1145 Time (s)
(a) PRESSURE SENSOR 1
—Experiment — Case 1
\
A
i
j
L
i 05 1110 1115 1120 1125 1130 1135 1140 1145 Time (s)(a) PRESSURE SENSOR 4
300 250 200 I 150 it 100 50 —Experiment —Case 1 — Case 2
1
—Experiment —Case 1 — Case 21
r
fc—.L
r
\
IN
...r
1105 1110 1115 1120 1125 1130 1135 1140 1145 Time (s) (b) PRESSURE SENSOR 2 100 80 i 60 I 40 a 20 0 — E x —C a }eriment se 1 se 2 —C a }eriment se 1 se 2Ik
i
h
liv
\
A
i S 0 5 1110 1115 1120 1125 1130 1135 1140 1145 Time (s) (b) PRESSURE SENSOR 5 80 60 I AO e i 20I
0 -20 1^ —Experiment — Case 1 — Case 21
—Experiment — Case 1 — Case 21
i
1 .
J l l
i
y
05 1110 1115 1120 1125 1130 1135 1140 1145 Time (s) (c) PRESSURE SENSOR 3 F I G U R E 10. W A V E I M P A C T PRESSURE AT VARIOUS L O C A -T I O N S . i S 0 5 1110 1115 1120 1125 1130 1135 1140 1145 Time (s) (c) PRESSURE SENSOR 6 F I G U R E 11. C O N T I N U E D F R O M F I G . 10.domain i n Case 2 aimed to reduce the computational effort while replicating the solution in Case 1 regardless of how accurate it is. Close examination of the results demonstrate that both the wave elevation and wave impact pressure at numerous positions behave very similarly in Case 1 and Case 2.
The second step is to validate the numerical calculations. Overall, the numerical results are in reasonable agreement with the experimental results. However, we notice relatively large dif-ferences between the simulations and the experiment for a short
time frame following the starting instant, i.e., t = 1105i. As men-tioned earlier, the initial condition i n the numerical simulations is the fifth-order Stokes wave prescribed throughout the com-putational domains, see Fig. 6 for an illustration o f the initial condition i n Case 1 and Case 2. Since the Stokes wave theory does not include the effects of diffraction, the initial wave kine-matics of the numerical simulations do not have one-to-one cor-respondence with those of the experiment. As a result, until the diffracted waves from the structure start affecting the flow behav-ior, high deviations occur in terms of both surface elevation and
impact pressure. In addition, close scrutiny of the results shown in Fig. 10 reveals that the high peaks of the wave impact pres-sure are underpredicted by the numerical method. To capture the measurement values more precisely, several modifications, such as local grid refinement and a high-order volume of fluid (VOF) method, are under development.
Particular attention should be paid to designating a proper computational domain size. The interaction between the incom-ing wave and the diffracted wave from the structure is a signifi-cant factor for determining the location of the boundaries. I f this interaction causes highly nonlinear behavior at the boundaries, the secondorder vertical derivative i n Eqn. (7) may produce i n -stabilities. I n this regard, characteristics of the incoming wave and the reflection parameter of the structiire play an important role and should be taken into account when specifying the size of the computational domain.
Although the absorbing boundary condition performs fairly well, a grid convergence study is needed to provide a compre-hensive analysis f o r error assessment. Essentially, performing a complex numerical simulation on grids that are too coarse can give rise to anomalously large or smaU en'ors. As we already observe the effects of insufficient resolution on the results at par-ticular locations, we prefer grid refinement rather than grid coars-ening to display the resolution effects on the comparison. How-ever, i f we take ƒ as a refinement factor and include time in the refinement as well, we w i l l have a work increase of a factor o f / ' * between any two levels. Considering the duration and total point counts of the simulations presented in the current work, a grid refinement procedure causes substantial computational cost. To alleviate this cost, a paraflel version of the numerical algorithm is under development [9]. We hope to report about the effects of the above modifications i n a future pubhcation.
CONCLUSIONS
We have presented numerical and experimental results for a wave-structure interaction problem where one half of a typical semi-submersible was restrained in a regular head sea. Wave elevation and wave impact pressures were measured at various locations for comparisons.
In the numerical computations, an absorbing boundary con-dition is incorporated into the CFD simulation tool ComFLOW. The performance of the A B C is assessed by means of the mea-surements. Throughout the major part of the simulations, nu-merical results are i n reasonable agreement with the experiment. Calculation of the wave elevation and impact pressure exerted by the waves on the structure is not significantiy affected by the ap-phcation of the absorbing boundaiy condition when the bound-aries are located closely to the structure. Consequently, computa-tional costs are reduced considerably without compromising the numerical solution.
ACKNOWLEDGMENT
The research is supported by the Dutch Technology Founda-tion STW, applied science division of N W O and the technology programma of the Ministiy of Economic Affairs i n The Nether-lands (contracts GW1.6433 and 10475). The authors kindly thank the Maritime Research Institute i n the Netherlands ( M A R I N ) for the experimental results.
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