Chapter 2 2D fully nonlinear numerical wave tanks M.H. Kim & W.C. Koo Ocean Engineering Program Department of Civil Engineering Texas A&M University College Station, TX, USA 2.1 Introduction It is of increasing interest to investigate nonlinear wave–wave and wave–body interactions in various ocean and coastal engineering projects. Recently, many floating vessels and production units have been installed for oil and gas exploration and production in deep water. In particular, many new and innovative floating units have been invented in a variety of challenging deepwater and ultra-deepwater projects. It is of vital importance to reliably predict the responses of such novel platforms in large waves. In this regard, fully nonlinear numerical wave tanks (NWTs) have been developed to study such highly nonlinear wave–body interaction problems.
Numerous researchers/scientists have studied various nonlinear wave–body interaction problems during the past two decades. For weakly nonlinear problems, perturbation theory has been used but it becomes extremely complicated as the order increases. On the other hand, a number of fully nonlinear NWTs have also been developed in parallel to deal with more nonlinear problems including breaking waves. However, most of the work has until recently been limited to the problems involving waves-only, fixed-body, and prescribed-body-motion cases [Grilli and Horrillo (1998), Boo and Kim (1997), Ferrant (1998), Celebi et al. (1998), Grilli et al. (2001), Hong and Kim
(2000), Dommermuth and Yue (1987), Kashiwagi (1996), Berkvens (1998), and Shirakura et al. (2000), etc.]. Kim et al. (1999), for example, presented a comprehensive review of the recent progress of NWTs. The fully nonlinear NWT simulations for freely floating bodies have to overcome a number of difficult numerical problems and are still considered to be a very challenging problem. Accordingly, fully nonlinear NWTs for freely floating bodies are very rare, especially in the case of 3D bodies. There also exist viscous NWTs [Tavassoli and Kim (2001), Yeung (2002)] and potential-viscous hybrid NWTs [Biausser et al. (2003)].
The fully nonlinear floating-body simulations against incident waves have been attempted by several researchers including Cointe et al. (1990), Sen (1993), Beck et al. (1994), Contento (1996), Kashiwagi et al. (1998), Tanizawa (1995), Tanizawa and Minami (1998), etc. They use different methodologies and different numerical schemes. In particular, the body boundary condition of a floating body is very complicated and the complete expression was not used until Tanizawa (1995). On the other hand, the accurate calculation of the time derivative of velocity potential, tφ , is very critical to obtain correct pressure and force on the body surface. The straightforward backward difference formulas are known to be numerically unstable in floating-body simulations, although the finite-difference formula is still useful for fixed-body and prescribed body-motion cases. For floating-body simulations, the use of acceleration potential is known to be the best in calculating tφ , as suggested by Tanizawa (1995). Regarding the acceleration potential method and the treatment of complicated body-boundary condition, there are four different methods: (1) iterative method by Sen (1993) and Cao et al. (1994), (2) mode-decomposition method by Vinje and Brevig (1981) and Cointe et al. (1990), (3) indirect method by Wu and Eatock Taylor (1996), and Kashiwagi et al. (1998), and (4) implicit boundary condition method by Tanizawa (1995).
In the present chapter, nonlinear wave–wave and wave–body interactions have been analyzed by a potential-based 2D fully nonlinear NWT. The NWT is developed based on the mode-decomposition method, mixed Eulerian–Lagrangian (MEL) material-node time-marching scheme [e.g. Longuet Higgins and Cokelet (1976)], and boundary element method (BEM). The time-marching scheme requires, at each time step, the following procedure: (i) solving the Laplace equation in the Eulerian frame, and (ii) updating the moving boundary points and values in a Lagrangian manner. To avoid non-physical saw-tooth instability on the free surface in time marching, smoothing and regriding schemes are also used.
To simulate the open-sea condition, a proper outgoing-wave condition at the radiation boundary needs to be imposed. The most physically plausible open
boundary condition is the Sommerfeld/Orlanski outgoing-wave condition
[Orlanski (1976)], which is given by ( ) 0C tt nφ φ∂ ∂
+ =∂ ∂
, where C is the celerity at
far field. The Orlanski radiation condition, for example, was used by Boo et al. (1994) for the simulation of nonlinear regular and irregular waves and by Isaacson and Cheung (1993) and Kim and Kim (1997) for wave–current–body interaction problems. There exist other open-boundary conditions, such as absorbing beaches by artificial damping on the free surface [Beck et al. (1994), Celebi et al. (1998), Hong and Kim (2000), Koo and Kim (2001), etc.] or matching with linear time-domain solutions at far field [Dommermuth and Yue (1987)]. When the matching technique (more useful for 3D problems) is used, the matching boundary has to be located far away from a body to minimize the difference between the inner nonlinear solution and outer linear solution, which consequently increases the size of the computational domain. On the other hand, it is well known that the properly designed artificial damping on the free surface is not necessarily far from the body and can damp out most wave energy if its length is greater than two wavelengths. Therefore, it is more effective than the matching technique and ideal to damp out relatively short waves. There exists another open-boundary scheme [Skourup and Schaffer (1997), Chatry et al. (1998)] by using controlled motion of the end wall, which is similar to piston-type wave absorbers. It is known to be effective for long waves. In this regard, Clement (1996) developed a very efficient open-boundary scheme called the hybrid method, which uses both the artificial damping on the free surface for short waves and the piston absorber for long waves. More detailed discussion including pros and cons of various open-boundary conditions is, for example, given in Kim (1995) and Clement (1996). In the present study, a special artificial damping scheme on the free surface is devised and implemented to minimize the reflection from both end wall and wave maker.
In this chapter, the theory and formulation for potential-based fully nonlinear NWTs are summarized. Various features and problems for numerical schemes are also explained. Using the developed fully nonlinear 2D NWT, nonlinear wave profiles and kinematics, wave generation by wavemakers, interactions with stationary and floating bodies are studied. The NWT results are compared with the corresponding linear and 2nd-order theoretical results. If possible, comparisons are also made with the published numerical and experimental results by other researchers.
Figure 2.1: Illustrative sketch of a numerical wave tank for the calculation of
freely floating barge motion with spring-type mooring. Local radius of round corner = 0.064 m, spring constant (k) = 197.58 N/m, damping coefficient (c) = 19.8 N/m/s.
2.2 Mathematical formulation 2.2.1 Boundary value problem An ideal fluid is assumed so that the fluid velocity can be described by the gradient of the velocity potential φ. A Cartesian coordinate system is chosen such that the z=0 corresponds to the calm water level and z is positive upwards.
Then the governing equation of the velocity potential is given by
02 =∇ φ (2.1)
The boundary conditions consist of 1) Fully nonlinear dynamic free-surface condition,
ρφηφ aP
gt
−∇−−=∂∂ 2
21 satisfied on the exact free surface, (2.2)
where aP is the pressure on the free surface, and we assume that it is set to be zero from now on. 2) Fully nonlinear kinematic free surface condition
3) Body-boundary condition (fluid velocity equals the body velocity in normal direction)
BV nnφ∂
= ⋅∂
(2.4)
at the instantaneous body position. 4) No normal-flux condition
0=∂∂
nφ (2.5)
on the rigid and impervious bottom and the vertical end-wall of NWT. 5) Input boundary condition: At the inflow boundary, a theoretical particle- velocity profile can be fed along the vertical input boundary or the actual wave-maker motion can be described. The exact velocity profile of a truly nonlinear wave under the given condition is not known a priori. Therefore, the best we can do is to input the best theoretical wave profile along the input boundary. Since the fully nonlinear free-surface condition is applied in the computational domain, the input wave immediately takes the feature of fully nonlinear waves. However, the mismatch between input velocity profiles and real water-particle velocity may cause unnecessary spurious waves (e.g. 2nd-order free waves) inside the domain [Koo and Kim (2001) and Ryu et al. (2003)]. On the other hand, we can also numerically simulate the nonlinear waves by actually moving piston or flap- type wave makers, as in the physical wave tanks. In this case, the effects of spurious free waves are more pronounced due to larger mismatch.
When linear or 2nd-order Stokes regular waves are prescribed, the following equations are used.
24
cosh ( ) cos( ) (Linear)cosh
3 cosh 2 ( )(2 ) cos 2( ) (Stokes' 2nd)8 sinh
x x
x
gAk k z hn n kx tn x kh
k z hn A k kx tkh
φ φ ωω
ω ω
∂ ∂ += = −
∂ ∂+
+ −
(2.6)
where A, ω , k, and h are wave amplitude, frequency, wave number, and water depth, respectively.
At each time step, the velocity potential (φ ) and the corresponding free-surface profiles are obtained by solving the discretized form of the following integral equation with the given boundary values.
where G is the Green function satisfying Laplace equation and α is a solid angle (α =0.5 when singularities are on the boundary). For two-dimensional problems, the simple source G is given by
1ln),,,( RzxzxG ii = , (2.8)
where 1R is the distance between source ( ,i ix z ) and field points (x, z) [Brebbia and Dominguez (1992)]. When the bottom is flat, an image source can be added to eqn. 2.8 to automatically satisfy the no-flux bottom condition. 2.2.2 Time marching for fully nonlinear free surface conditions To update the fully nonlinear kinematic and dynamic free-surface conditions at each time stop, Runge–Kutta 4th-order time-integration scheme and the MEL (mixed Eulerian–Lagrangian) approach are adopted. In the present NWT, both semi-Lagrangian and full Lagrangian (material-node approach, for which the free-surface nodes move with water-particle motion (v=∇φ)) are used. In the case of fully Lagrangian approach, the free-surface nodes need to be updated and rearranged every time step. The regriding scheme prevents the free-surface nodes from crossing or piling up on the free surface, and thus makes the integration scheme more stable. When the free-surface nodes move with velocity vr ,
∇⋅+∂∂
= vtt
r
δδ , and the fully nonlinear free-surface conditions can be modified
as follows in the Lagrangian frame
212
g vt
δφ η φ φδ
= − − ∇ + ∇ ⋅r (2.9)
( )vt z
δη φ φ ηδ
∂= − ∇ − ⋅∇
∂r (2.10)
Particularly, when v φ= ∇r
(material-node approach), eqns. (2.9) and (2.10) can be simplified as
Compared to the semi-Lagrangian approach [Contento (1996), Hong and Kim (2000), etc.], in which the free-surface nodes are allowed to move only in
the vertical direction, i.e. nodal velocity (0, )vt
δηδ
=r , the free-surface conditions
of material-node approach become much simpler and represent the fluid-particle motions more realistically. On the other hand, when the semi-Lagrangian approach is used, the regriding step is not necessary but numerical errors related to the evaluation of η∇ may be introduced. At any rate, the bottom line is that both methods should produce the same results. When simulating plunging breakers, the semi-Lagrangian approach is difficult to be applied because of double potential values in the vertical direction. 2.2.3 Ramp function When the simulation is started, a ramp function at the input boundary is applied. The ramp function prevents the impulse-like behavior of the wavemaker, and consequently reduces the corresponding unnecessary transient waves. As a result, the simulation can be more stable and reach the steady state earlier. In the present numerical examples, the ramp function is applied in 2T or 4T (wave period) depending on input wave height. The ramp function is given by
≤−
>= 2Tfor t , 2/)
2cos(1
2Tfor t, 1)(
Tttr π
. (2.13)
2.2.4 Numerical beach (artificial damping zone) Toward the end of the computational domain, an artificial damping zone was applied on the free surface so that the wave energy is gradually dissipated in the direction of wave propagation. The profile and magnitude of the artificial damping have to be designed to minimize possible wave reflection at the entrance of the damping zone, while maximizing wave-energy dissipation. After comprehensive tests, the length of the damping zone (ld) was determined to be at least 2 wavelengths. In general, the longer ld is needed for more nonlinear waves. In this study, both nφ and η -type damping terms were added to the fully nonlinear dynamic and kinematic free-surface conditions to maximize its effectiveness.
The shape of the damping magnitude is also important to minimize the entrance reflection. In this regard, the damping is designed to grow gradually to the target constant value:
≤
>
−−=
l
ll
lx
di
i
for x 0
for x ]2
cos1[0πµµ , (2.16)
where l is the length of the computational domain (no damping zone) and ld is the length of the damping zone.
Through linear stability analysis, the optimal values of 01µ and 02µ (optimized damping coefficients) were obtained to be 5.101 =µ and 0102 µµ k= . The latter condition minimizes the dispersion error at the discontinuity of the free-surface condition (entrance of damping zone). The performance and efficiency of the artificial damping coefficients were numerically tested and confirmed. Note that the damping coefficient, 01µ , may increase in cases of shallow water depth or higher wave amplitude. 2.2.5 Damping for reflected waves in incident wave zone When long-time simulations are required with surface-piercing bodies, the re-reflection from the wave maker has to be controlled and eliminated to best represent the open-sea condition. When simulating the physical wave tank, however, the same phenomenon (re-reflection) occurs and no control is needed. In the present study, a special damping scheme in front of the wave maker is employed to prevent the re-reflection from the wave maker. This damping scheme has to be designed to damp out only the reflected waves from the body, while preserving the original incident waves. In this regard, the damping is applied to the difference between the total and incident waves:
∂, *η are the solutions (incident waves) in the absence of
bodies.
The reference values of *nφ∂
∂ and *η can be computed with the same
computational condition without the body. When the incident waves are moderately nonlinear and computational efficiency is of concern, proper analytic solutions (e.g. 2nd-order Stokes wave in the case of nonlinear simulation) can also be used. In the present numerical examples, 2nd-order Stokes waves are used
for *nφ∂
∂ and *η , the efficacy of which is also tested. For many practical
applications, even the linear analytic solution can be used for *nφ∂
∂ and *η , as
pointed out by Tanizawa and Naito (1997). They showed that the scheme works even for higher wave steepness. The shape of the artificial damping in front of the wave maker is the same as that of the damping zone in front of the end wall. Only the direction of damping growth is reversed since reflected waves from the body have to be eliminated. 2.2.6 Smoothing scheme It is well known that the so-called saw-tooth instability may occur on the free surface during the simulation of highly nonlinear waves. It is caused either by variable mesh size/high-order aliasing or inherent singular behavior at the intersection of the wave maker and free surface. To avoid the non-physical saw-tooth instability problem, a smoothing scheme can be introduced. In the present case, a Chebyshev 5-point smoothing scheme was used along the free surface during time marching. The smoothing scheme was applied typically at every 5 time step. In our preliminary study [Koo (2003)], it is confirmed that the smoothing scheme little affects the higher-order components up to third order. Longuet-Higgins and Cokelet (1976) first introduced the evenly spaced Chebyshev 5-pt smoothing scheme. Here, the scheme is modified and extended to variable-node-space cases [See Koo (2003)].
2.2.7 Wave force on the body The accurate calculation of the time derivative of the velocity potential, tφ , is crucial in obtaining the correct pressure and force on the body surface at each time step. There are several ways to obtain tφ . Backward difference is the simplest way using the potential values of previous time steps. However, this method is unstable in most floating-body cases [Tanizawa (1995)]. For a stationary structure and prescribed-body-motion simulations, more accurate finite-difference formulas can be used as a post-processor. In the case of freely floating bodies, the finite-difference scheme causes problems because tφ on the body surface has to be obtained simultaneously with the body motion. In other words, to calculate floating-body motions correctly, the equations of fluid particle and body motion should be solved simultaneously. The acceleration potential method is known to be the most accurate and consistent way to solve
tφ of the floating-body motion. The wave force on the body surface can be calculated by integrating
Bernoulli’s pressure over the instantaneous wetted surface. Including gravitational force, external restoring force by spring, and artificial damper, the total force in the i-th direction can be calculated as follows:
where k=spring constant, c=damping coefficient, x=sway, x& =horizontal body velocity z=vertical coordinates on a body surface, W is the weight of body (=mg), and ijδ denotes the Kronecker delta function. 2.2.8 Acceleration potential method The acceleration potential method is used for calculating the time derivative of the velocity potential in the present study. The acceleration potential means that the acceleration of fluid particles can be determined from the gradient of acceleration potential Φ ,
The acceleration is the total time derivative of the fluid velocity and is given by
V Va V Vt t
δδ
∂= = + ⋅∇
∂
r rr rr . (2.22)
Combining eqn. (2.20) and (2.21) yields
21( )2
at tφ φφ φ φ∂∇ ∂ = + ∇ ⋅∇ ∇ = ∇ + ∇ ∂ ∂
r . (2.23)
Therefore, the acceleration potential is defined as
212t
φ φ∂Φ = + ∇
∂. (2.24)
Note that the acceleration potential does not satisfy the Laplace equation
( 2 0∇ Φ ≠ ) because of the nonlinear term ( 212
φ∇ ) [Tanizawa (1995)]. In view
of the numerical application, the nonlinear term can be explicitly calculated from the solution of the velocity-field equation. So, we do not need to solve Φ directly; instead we solve tφ ,
212t t
φφ φ∂= = Φ − ∇
∂. (2.25)
Then, tφ can be solved in the entire fluid domain since tφ satisfies the Laplace equation. 2.2.9 Body boundary conditions for acceleration field The kinematic body-boundary conditions for velocity and acceleration potentials are
V n nnφ φ∂
= ⋅ = ∇ ⋅∂
r r r (2.26)
a n nn
∂Φ= ⋅ = ∇Φ ⋅
∂r r r , (2.27)
where x zn n i n k= +r is the unit normal vector of the body surface.
The kinematic boundary condition for the acceleration on the body surface can be described as follows [Tanizawa (1995)],
where 0a , and ω& are the translational, and angular accelerations, 0υ , and ω , translational and angular velocities of a body, respectively.
The origin of the body-fixed coordinate is set to the center of gravity of a body in the present study, whereas those motions are observed with respect to the space-fixed frame at the center of gravity of a body. r is the position vector from the center of gravity to the body surface. nk is the normal curvature of the body
surface along the path of fluid (*
1nk
ρ= − , where *ρ is the radius of the local
center). The first term of eqn. (2.28) can be rearranged in the following form,
5 5x x z za n a n a n+ + , where subscript 5 means roll acceleration and roll normal vector. The other terms can be defined as a q term.
20 0( ) ( ) 2 ( )nq n r k r n rω ω φ υ ω ω φ υ ω= ⋅ × × − ∇ − − × + ⋅ × ∇ − − ×
r r . (2.29) From eqns. (2.25), (2.28), and (2.29), the body-boundary condition for the acceleration field can be expressed as,
2 2
20
1 12 2
1 ( )2
t
n n n n
n a r qn
φ φ φ
ω φ
∂ ∂ ∂Φ ∂ = Φ − ∇ = − ∇ ∂ ∂ ∂ ∂ ∂ = ⋅ + × + − ∇ ∂
r&
(2.30)
0( )tBn a r q
nφ ω∂
= ⋅ + × +∂
r& (2.31)
where 212Bq q
nφ∂ = − ∇ ∂
.
The second term of Bq can then be written as [Tanizawa (1995)]
Therefore, the body-boundary condition for acceleration potential field can be completely described from eqns. (2.31) and (2.33). The Bq represents the contribution of the velocity field to the acceleration field and all terms are
determined explicitly after the velocity field is calculated, i.e.φ and nφ∂
∂are
determined. The time derivative of the velocity potential on a body surface can then be
obtained after solving the discretized form of the following integral equation.
( )tj ijti ij tj
GG ds
n nφ
αφ φΩ
∂ ∂= −
∂ ∂∫∫ . (2.34)
The tφ can be obtained by solving the implicit loop of acceleration and velocity field equations simultaneously. To calculate the implicit loop, there have been several methods developed so far; iterative method [Sen (1993), Cao et al. (1994)], mode-decomposition method [Vinje and Brevig (1981), Cointe et al. (1990)], indirect method [Wu and Eatock Taylor (1996), Kashiwagi et al. (1998)], and implicit boundary-condition method [Tanizawa (1995)]. In the present study, the mode-decomposition method has been used. The indirect method is also employed to confirm the direct method of mode decomposition. 2.2.10 Mode decomposition method The idea of the mode-decomposition method was first introduced by Vinje and Brevig (1981), though their expression of the acceleration field was not
complete. They developed a 2D NWT by using a complex potential. In their 2D NWT, the acceleration field was decomposed into four modes corresponding to three unit accelerations for sway-heave-roll and the acceleration due to the velocity field. Each mode can be obtained by solving the respective boundary integral equation. Using these four modes and the body-motion equation, body acceleration can be determined.
The tφ for 2D case is given by 3
41
t i ii
aφ ϕ ϕ=
= +∑ , (2.35)
where ia is the i-th mode component of generalized body acceleration (1=sway, 2=heave, 3=roll).
The body-surface boundary condition for each mode is
, 1 3, 4
ii
B
n iq in
ϕ = ∂= =∂
. (2.36)
The free-surface boundary condition is also decomposed as
2
0, 1 31 , 4 2
i
i
g iϕ
η φ
= =
− − ∇ =
. (2.37)
The input boundary condition is
0, 1 3
, 4 i
t
i
n in
ϕφ
= ∂ = ∂ ∂ = ∂
. (2.38)
The other rigid boundary condition is
0, 1 4i inϕ∂
= =∂
. (2.39)
After solving the boundary integral equation with the given conditions, iϕ (on
the body surface), i
nϕ∂
∂ (on the free surface), and iϕ (on the rigid boundary) are
obtained. Then, the remaining unknown values in the eqn. (2.35) are ia only.
In order to calculate the generalized acceleration ( ia ) for each mode (sway, heave, and roll), the wave force equation, which integrates Bernoulli’s pressure over the instantaneous wetted surface, should be combined with the Newton’s 2nd law. Including the gravitational force and external spring (k) and damper (c), the total force in the i-th direction can be calculated as follows:
21 1 2 2 3 3 4
1( )2
P a a a gzρ ϕ ϕ ϕ ϕ φ= − + + + + + ∇ (2.40)
Bx xS
F P n ds kx cx= ⋅ − −∫ &
Bz zS
F P n ds W= ⋅ −∫
5BS
M P n ds= ⋅∫ (2.41)
1 xma F=
2 zma F=
3ma M= (2.42)
From eqns. (2.41) and (2.42), we can determine the generalized acceleration ( ia ) of each mode, and then the time derivative of velocity potential is finally determined. After obtaining tφ , the nonlinear wave forces on a body can be easily obtained by eqn. (2.43) at each time step. The formulation of the nonlinear wave force for each mode with external spring and damper in this study is shown in eqn. (2.19).
21( )2BS
F gz ndstφρ φ∂
= − − ∇ − ⋅∂∫
r , (2.43)
where BS is the wetted body surface. Then, using the Runge–Kutta–Nystrom 4th-order method [see Koo (2003)],
the body velocities and displacements are determined. The body displacements, such as sway (horizontal), heave (vertical), and roll (rotational) motions, will be used for updating body geometry for the next time step. 2.2.11 Indirect method The Haskind–Newmann relation can be utilized to calculate the wave-exciting force from the radiation damping in the linear theory. Wu and Eatock Taylor
(1996) first introduced the same idea called the ‘indirect method’. In this method, the contribution of 4ϕ can be obtained from iϕ (i=1,2,3) indirectly, instead of
directly solving for 4ϕ . If Green’s 2nd identity is applied to 4ϕ and iϕ (i=1,2,3)
2 2 44 4 4
ii i iS
d dSn nϕ ϕϕ ϕ ϕ ϕ ϕ ϕ
Ω
∂ ∂ ∇ − ∇ Ω = − ∂ ∂ ∫ ∫ . (2.44)
Since 2iϕ∇ =0 and 2
4ϕ∇ =0, Eq. 2.44 becomes
44 0
B F A R
iiS S S S
dSn nϕ ϕϕ ϕ
+ + +
∂ ∂ − = ∂ ∂ ∫ , (2.45)
where BS , FS , AS , and RS are body surface, free surface, input boundary, and other rigid boundaries, respectively.
Equation. (2.45) can be divided into each mode and the result can be written as
When we calculate the body force by eqns. (2.35) and (2.41), the 4ϕ integral can be replaced by eqn. (2.47). Therefore, the computational time is reduced because the direct evaluation of 4ϕ can be avoided, i.e. comparing to the direct mode decomposition method, only 3 acceleration terms ( ia ) need to be determined. When we use the indirect method, however, the local quantities on the body, such as body-surface pressure, cannot be obtained.
Basically, the mode-decomposition method and the indirect method are mathematically identical. Only the 4ϕ integral term can be replaced by other terms in the indirect method. In the present study, both direct and indirect methods are implemented and the two independent results are cross-checked for verification. 2.2.12 Frozen coefficient method When the Runge–Kutta 4th-order scheme is used for time marching, it requires 3 internal sub-step calculations including the respective boundary conditions for free surface and body boundaries. For the complete Runge–Kutta 4th-order scheme, the body geometry should be updated and the new updated influence matrix has to be inverted even for every internal sub-step calculation. So, the computational burden increases four-fold compared with other simpler one-step schemes, such as the leap-frog method, using the same t∆ .
During the internal sub-step calculation, the change of boundary shape and influence matrix is expected to be small. From this assumption, we can skip updating the influence matrix, i.e. the same influence matrix can be used within one time step. It is called frozen-coefficient method. The frozen-coefficient scheme is known to be a good approximation, which reduces CPU time significantly. The scheme is valid for fixed-body and prescribed-body motion simulations.
For the freely floating body simulation, however, the frozen-coefficient scheme does not work very well especially for large waves and body motions. As mentioned earlier, for the floating-body simulation, the equation of fluid particle and the equation of body motion should be solved simultaneously, and the calculation has to be consistent even at every sub-step. For instance, wave- particle motion affects pressure on the instantaneous wetted body surface and the resulting body force will make body geometry, velocity, and acceleration change. Therefore, even if the difference in the body geometry is small, it may appreciably affect the input for the next time step resulting in a difference in body motions. The comparison of frozen-coefficient and fully updated methods will be shown in the coming section.
2.3 Numerical results and discussions To illustrate the performance of the developed fully nonlinear NWT, four examples are selected (1) nonlinear propagating waves and their kinematics, (2) nonlinear waves propagating over a submerged stationary cylinder, (3) nonlinear wave generation by wedge-type wave makers, and (4) nonlinear wave interactions with floating bodies.
0 2 4 6 8 10
x(m)
0
0.4
0.8
1.2
1.6
Har
mon
ic A
mpl
itude
(cm
)
Figure 2.2: Comparison of spatial variation of Fourier amplitudes from the
present NWT (rectangle) with Goda’s 3rd-order theoretical (solid line) and experimental measurement (small circle).
2.3.1 Nonlinear propagating waves and their kinematics Figure 2.2 shows the spatial variation of each harmonic amplitude (1st-, 2nd-, and 3rd-order) when linear wave of period T=1.697 s and height H=2.5 cm is fed along the input boundary. Computational parameters are water depth h=0.25 m, wavelength L=2.5 m, free surface (computational domain)=9 m, damping zone (artificial beach zone)=6 m, x∆ on the free surface=0.1 m(first 3 m) and 0.05 m(rest 6m) and t∆ is T/64. The corresponding wave steepness H/L=0.01 and depth to wavelength ratio h/L=0.1(intermediate depth).
Figure 2.3: Case of H=5cm. Stokes 2nd-order input (big circle). (Other legend and parameters are the same as Figure 2.2).
Figure 2.3 shows similar plots when wave heights are doubled (H=5.0 cm).
As wave height increases from 2.5 cm to 5.0 cm, the 2nd- and 3rd- harmonic amplitudes grow rapidly and the fluctuation of the 1st- harmonic amplitude increases as well. For both wave heights, the present nonlinear NWT results are in good agreement with Goda’s 3rd-order theoretical and experimental results [Goda (1998)]. In this case, the 2nd-order magnitude can reach 45% of the first-order magnitude, i.e. the 2nd-order free wave becomes very important despite the fact that the wave itself is not very steep (H/L=0.02).
The spatial variation of wave profiles along the propagation distance is mainly caused by the generation of 2nd-order free waves due to the mismatch between actual wave velocity and prescribed wave-maker motions. If it is true, the phenomenon should disappear by matching the wave-maker motion as close as possible to the actual velocity profile of the generated nonlinear waves. To confirm this fact more clearly, the 2nd-order Stokes wave velocity profile resembling the kinematics of the actual nonlinear wave more closely is fed along the input boundary. The results are represented by the white circle, as shown in Figure 2.3 and the spatial variation is greatly reduced as expected. In this regard,
care needs to be taken in interpreting experimental/simulation data when undesirable 2nd-order free waves are present.
-0.04
-0.02
0
0.02
0.04
0.06
0 0.5 1
t/T
elev
atio
n (m
) x=12m H=0.0812m
-0.04-0.02
00.020.040.06
0 0.5 1elev
atio
n (m
) x=9m H=0.0781m
-0.04
-0.020
0.020.04
0.06
0 0.5 1elev
atio
n (m
) x=4m H=0.0668m
-0.04
-0.020
0.02
0.04
0.06
0 0.5 1elev
atio
n (m
) x=15m H=0.0902m
-0.04
-0.02
0
0.02
0.04
0.06
0 0.5 1elev
atio
n (m
) x=18m H=0.0931m
-0.04
-0.020
0.020.04
0.06
0 0.5 1
t/T
elev
atio
n (m
) x=22m H=0.0849m
Figure 2.4: Examples of wave profiles with secondary crests at various locations
(water depth=0.32 m, T=3.07 s, dt=T/64, dx=0.1 m).
Figure 2.4 shows the change of wave profiles and corresponding heights at various locations and the appearance of the secondary crest at the trough. The nonlinear phenomenon observed by Goda’s experiment can be accurately reproduced here by using the present fully nonlinear NWT with the identical condition. The overall pattern is very similar to the experimental measurement. The secondary crest appears behind the main crest, propagates with slower speed than the main crest, and is overtaken by the next crest as the wave propagates along the wave tank.
Using the fully nonlinear NWT, the wave kinematics of Figure 2.2 under the wave crest are calculated and they are compared in Figure 2.5 with linear and 2nd-order Stokes wave results. For the NWT computation, the 2nd-order Stokes waves are fed along the input boundary. The actual maximum horizontal velocity turned out to be greater than that of 2nd-order Stokes waves. The difference is more pronounced above the mean water level. Below the mean water level, the
maximum (crest) horizontal velocity is greater than the minimum (trough) value and it causes some mean transport flow in the direction of wave propagation. Detailed results for this section are seen in Koo and Kim (2001).
0.12 0.16 0.2 0.24
Velocity(m/sec)
-8
-6
-4
-2
0
2
dept
h(cm
)
Figure 2.5: Comparison of maximum horizontal velocity-wave kinematics of Figure 2.2 (white rectangle, +=minimum horizontal velocity) by the present NWT with Stokes 2nd-order (dashed line) and linear theory (solid line).
2.3.2 Nonlinear waves propagating over a submerged stationary cylinder Next, we consider nonlinear wave propagation over a submerged stationary cylinder, which was experimentally studied by Chaplin (1984) and also theoretically studied by many researchers.
Mean Vertical ForcePresent ResultTheoreticalExperimental
Figure 2.6: Mean vertical forces on a fixed submerged cylinder against KC
values with log scale. Forces are normalized by 2 3Rρω (Water depth=0.85 m, free surface (no damping zone)=4 m, length of damping zone=3 m, cylinder diameter (D=2R)=0.102 m, center of cylinder=(2,-D) m, wave frequency= 2π (rad/s), dt=T/64, discretized free surface element size=0.025 m).
Fourier analysis was applied to the portion of the steady-state wave-force time series to obtain respective harmonic components. Then, the mean, 1st-, 2nd-, and 3rd- harmonic wave forces were compared with other theoretical (2nd-order theory; Ogilvie (1963)) and numerical (high-order spectral method; Liu et al. (1992)) results as well as experimental results of Chaplin (1984). Figure 2.6 shows that the calculated mean vertical forces are in good agreement with both theoretical and experimental results for different KC numbers
( 02 kyAKC e
Dπ
= , 0y =-D). The corresponding wave steepness ranges from 0.008
to 0.024. It appears that the mean vertical force is linearly proportional to the KC number on a log scale.
The 1st- harmonic horizontal and vertical forces obtained from NWT simulations are shown in Figure 2.7. They agree well with the potential-based
nonlinear calculation using the high-order spectral method by Liu et al. (1992). However, there exists a big difference between the potential-flow-based nonlinear simulations and Chaplin’s experimental results, especially for larger KC values. The discrepancy is mainly due to the viscous effects, i.e. the presence of clockwise circulation around the body, as observed by Chaplin (1984). To verify Chaplin’s observation, another independently developed viscous-flow-based NWT [Tavassoli and Kim (2001)] was run for the same case. The viscous NWT results actually reproduce the clockwise circulation and show the same trend (decrease of 1st- harmonic force with KC number) compared to Chaplin’s experiment.
Figure 2.7: 1st- harmonic forces normalized by 2 3R KCρω . Horizontal and
vertical forces are very close to each other (other conditions are the same as Figure 2.6).
Figure 2.8 shows the comparisons of 2nd- and 3rd- harmonic horizontal and vertical forces among the present NWT simulation, high-order spectral method, and Chaplin’s experimental results. All the results are in good agreement with
each other, which means that viscous effects are not important in the case of high-order forces. The reason is that the high-order forces are mainly caused by the free-surface distortion, which can be reasonably described by the potential theory. Detailed results for this section are seen in Koo and Kim (2003) and Koo et al. (2004).
-2 -1.5 -1 -0.5 0 0.5 Ln(Kc)
-8
-6
-4
-2
0
Ln
FhρR
3 W2
2nd harmonic forces
3rd harmonic forces
Figure 2.8: 2nd- and 3rd-harmonic forces on a log scale. Present NWT (2nd and 3rd
horizontal-black circle), high-order spectral method (2nd and 3rd=solid line), and experimental measurement (small white rectangle), (other conditions are the same as Figure 2.6).
2.3.3 Nonlinear wave generation by wedge-type wave makers Next, we consider a nonlinear radiation (outgoing-wave generation) problem by forced prescribed oscillations of a body. In the ensuing examples, nonlinear waves are generated by a wedge-type wave maker, for which experiments were also conducted by Kashiwagi (1996). This case was selected by the ISOPE NWT sub-committee for a comparative study [Tanizawa and Clement (1999)]. The comparison of surface elevation between the present NWT simulation and
Kashiwagi’s experiment is shown in Figures 2.9 and 2.10. There is a big deviation of wave elevation and phase between the linear calculation and the experimental measurement. The discrepancy is much more noticeable in higher and steeper waves. On the other hand, the results of fully nonlinear NWT are in good agreement with the measured values both in elevation and phase. From these results, it is clearly demonstrated that fully nonlinear NWT simulations can very accurately reproduce the nonlinear waves generated by forced body motions. For the present example, both material-node (full Lagrangian) approach and semi-Lagrangian schemes were used to confirm that they produce almost identical results, which should be so unless programming errors are involved.
20 40 60 80t*sqrt(g/a)
-2
-1
0
1
2
A/Y
ExperimentCalculationLinear x/a=9.629
20 40 60 80t*sqrt(g/a)
-2
-1
0
1
2
A/Y
Fully Nonlinear x/a=9.629
Figure 2.9: Comparison of free-surface elevation at x/a=9.629 between present numerical and Kashiwagi’s experimental results. Oscillation period (T*sqrt(g/a))=4.895 and amplitude (Y/a)=0.0988 (dt=T/64, 22 nodes per wavelength), a=wave-maker width on SWL =0.3776 m, initial draft d=0.45 m, and wedge angle (half)=40 degrees.
Figure 2.10: Comparison of free-surface elevation at x/a=9.629. Oscillation
period is T*sqrt(g/a)=3.671 and amplitude Y/a=0.0525 (dt=T/64, 20 nodes per wavelength) (other conditions are the same as Figure 2.9).
In the next figures, the mean and 2nd- harmonic body forces and added mass of the wedge-type wave maker are plotted as a function of oscillation frequency. The added mass was calculated by analyzing the steady-state portion of the force-time histories. The present NWT results are compared with the database collected by Tanizawa and Clement (1999) that also includes Yamashita’s (1977) experimental results. The FVM is the only viscous-flow-based NWT results. Present results show good correlation with the experimental and other numerical results. The detailed results for this section are seen in Koo (2003).
2.3.4 Nonlinear wave interactions with floating bodies Finally, the present NWT is used to calculate barge-type floating-body motions and drift forces in waves. The simulated results are compared with the experimental results of Nojiri and Murayama (1975), the theoretical drift-force result of Maruo (1960), and the independent nonlinear NWT results of Tanizawa and Minami (1998). The present fully nonlinear results are also compared with the linear results. In the present NWT study, the convergence with respect to t∆ and x∆ is verified.
-0.08
-0.04
0
0.04
0.08
0 20 40 60 80time (sec)
Rol
l ang
le(r
ad)
FreezingFull updated
Figure 2.14: Comparison of roll for different time-marching scheme:
wavelength=3.93 m, period (T)=1.586 s, and incident wave height (H)=0.01 m, solid line= Runge–Kutta with frozen-coefficient scheme (dt=T/128), dotted line= fully updated Runge–Kutta method (dt=T/40).
The comparison of fully updated and frozen-coefficient Runge–Kutta
methods in the case of floating-body simulations is shown in Figure 2.14. The frozen-coefficient scheme does not reach steady state and grows with time even for much smaller time steps. The fully updated RK scheme, on the other hand, gradually reaches the steady state. From this comparison, it can be shown that the fully updated RK scheme is more robust for fully nonlinear floating-body simulations. It is also confirmed that both direct and indirect mode-decomposition methods produce identical results, which should be so unless programming errors are involved.
Figure 2.15: Sway comparisons; Incident wave height (H)=7 cm,
xsi=w^2*B/(2g), barge width (B)=0.5 m. Initial draft (d)=0.25 m, and KG (keel to center of gravity)=0.135 m. Other conditions are shown in Figure 2.1.
00.10.20.30.40.50.60.70.8
0.5 0.7 0.9 1.1 1.3 1.5xsi
X/A
Linear (freq)
H=7cm
H=1cm
H=3cm
H=5cm
Figure 2.16: Sway comparisons of fully nonlinear simulations with different
incident wave heights. Other conditions are the same as Figure 2.15.
Sway-motion comparisons with experimental and other numerical results are shown in Figures 2.15 and 2.16. The present NWT results are for the 1st harmonics and they are in good agreement with the experimental results of Nojiri
and Murayama (1975) and the fully nonlinear numerical results of Tanizawa and Minami (1998). As expected, the difference between linear (freq=frequency- domain results) and fully nonlinear results is large near the resonance area (around xsi=0.6). The experimental results were obtained about 30 years ago, so their accuracy may be questioned. If very accurate, the difference between the experimental and fully nonlinear simulation results can be attributed to viscous effects, particularly in roll motions. In Figure 2.16, the fully nonlinear NWT results converge monotonically to the linear results as the incident wave height decreases, which shows that the present fully nonlinear simulation scheme is robust even for freely floating bodies. In the present case study, the natural frequencies of the sway, heave, and roll are all close to each other. As input-wave height increases, the nonlinear effects near the resonance region, particularly caused by large roll motion (See Figure 2.18), make the resulting sway motion deviate from the linear results.
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2xsi
Z/A
Linear (freq)presentexperimentTanizawa
Figure 2.17: Heave comparisons. Other conditions are the same as Figure 2.15.
Figure 2.17 shows the comparison of heave motions. The experimental and numerical results are in good agreement over the entire frequency range except near the roll-sway resonance area. When xsi (ξ )=0.6 (close to heave resonance), the fully nonlinear results are calculated to be smaller than the linear results; while at xsi (ξ )=0.65, the nonlinear, linear and experimental results are very close to each other. As the incident wave height decreases, the nonlinear heave result monotonically converges toward the linear result.
The roll-motion comparison is similar to the sway and heave cases. The present nonlinear and experimental results again agree well except near the resonance area. When xsi (ξ )=0.6 or 0.65, a big difference in roll angle between
the linear and nonlinear results is observed, and the experimental result at xsi (ξ )=0.65 supports the nonlinear results. At the roll resonance, the potential-based NWT over-predicts the measured value, which emphasizes the importance of the viscous effect in resonant roll motions. When the incident wave height decreases, the nonlinear results again converge to the linear results, which was also seen in the sway and heave cases.
0
1
2
3
4
5
6
0 0.5 1 1.5 2xsi
roll
(nor
mal
ized
)
Linear (freq)presentexperimentTanizawa
Figure 2.18: Roll-angle (radian) comparisons. Roll angle is normalized by
(ω^2/g)A.. Other conditions are the same as Figure 2.15.
0
0.4
0.8
1.2
1.6
0 0.5 1 1.5 2xsi
Drif
t For
ce (n
orm
aliz
ed)
TheoreticPresentExperimentTanizawa
Figure 2.19: Drift force comparisons; solid line represents Maruo’s theoretical
results. Incident wave height =0.07 m. Drift force is normalized by (1/2)*rho*g*A^2. All the numerical solutions were obtained from the pressure integral on the body. (Other conditions are the same as Figure 2.15)
Figure 2.19 shows mean sway drift forces. The present fully nonlinear simulation results are compared with the 2nd-order theoretical result, Tanizawa and Minami’s nonlinear simulations, and experimental results. The fully nonlinear simulation (H=0.07 m) near the resonance area (xsi (ξ )=0.6) gives 50% larger values compared with the 2nd-order theoretical values. When 0.7 ξ≤ ≤ 1.2, the present nonlinear simulation has a good agreement with both theoretical and experimental results. When the input frequency is higher (ξ ≥ 1.5), the present nonlinear results are larger than the theoretical (2nd-order) and experimental values. A similar trend was also found in Tanizawa and Minami’s fully nonlinear results. The deviation between the fully nonlinear and the 2nd-order results in the high-frequency region may be due to the effect of higher wave steepness (the same incident wave height (H=0.07 m) was used over the entire frequency region). It is found that present fully nonlinear results near the resonance area (ξ =0.6) also converge to 2nd-order theoretical results monotonically, as the incident-wave height decreases. This fact supports the view that the present NWT simulation is a robust and accurate numerical tool to investigate wave-drift forces. In general, wave-drift forces are much more sensitive to wave steepness than body motions. The magnitudes of the mean drift forces are, in general, very small and are known to be difficult to measure accurately in the experiment.
-1.5
-1
-0.5
0
0.5
1
1.5
80 82 84 86 88 90t (sec)
Fx/(r
ho*g
*A*d
)
Figure 2.20: Time series of sway force on the barge (ξ =0.6, T=1.295 s, H=0.07
m). Forces are normalized by rho*g*d*A. Other conditions are the same as Figure 2.15.
To investigate the resonance region in more detail, sway forces are plotted in Figure 2.20. It is seen that the contribution of higher harmonics in the horizontal force is significant. Due to the strong nonlinearity of the horizontal force, the sway motions exhibit rather complicated nonlinear and parametric behaviors restrained in certain range of motion amplitude. Detailed results for this section are seen in Koo (2003) and Koo and Kim (2004).
2.4 Concluding remarks A fully nonlinear 2D NWT was developed based on the potential theory, mixed Eulerian–Lagrangian (MEL) time-marching scheme, and boundary-element method (BEM). A robust and stable 4th-order Runge–Kutta time-integration scheme was used with regriding (every time step) and smoothing (every 5 steps). When the semi-Lagrangian approach is used, the regriding step is not necessary with the penalty of more complicated terms on the free-surface condition. A special nφ η− -type numerical beach applied both in kinematic and dynamic free-surface conditions was devised and implemented to minimize wave reflection from end-wall and wave maker. In front of the wave maker, only the difference between the incident and reflected waves was damped out so that the artificial damping may not influence the incident wave field, the performance of which was shown to be satisfactory. At the input boundary, either feeding theoretical particle velocities along the fixed boundary or actual prescribed wave-maker motions with moving boundary can be used. The mismatch between the wave-maker-motion-induced velocities and the actual nonlinear water- particle velocities causes spatial variation of the wave profile on the free surface. The high-order free waves are major contributors to the spatial variability. For NWT simulations for floating bodies, the evaluation of the time derivative of the velocity potential is critical for numerical accuracy and stability. The acceleration-potential formulation and direct mode-decomposition method were used for calculating the time derivative of the velocity potential. The indirect mode-decomposition method was also independently developed for verification. Both direct and indirect methods, which are mathematically equivalent, produced the same numerical results. For the freely floating body simulations, the frozen-influence-coefficient method does not work well, and thus is not recommended.
To illustrate the performance of the developed fully nonlinear NWT, four examples are selected (1) nonlinear propagating waves and their kinematics, (2) nonlinear waves propagating over a submerged stationary cylinder, (3) nonlinear wave generation by wedge-type wave makers, and (4) nonlinear wave interactions with floating bodies. It is shown that the fully nonlinear results converge to the corresponding linear results as incident wave heights decrease. In particular, the simulated motion, wave-force, and drift-force results for freely
floating bodies, which is known to be the most difficult problem, were in reasonable agreement with the experimental and other independent nonlinear simulation results. The examples show that the developed NWT can reproduce various nonlinear-wave features and wave–body interactions very accurately. However, in certain cases, such as the 1st-harmonic force on a submerged cylinder and resonant roll motions, the viscous effects should be taken into consideration. Under these kinds of circumstances, empirical tuning is necessary. Otherwise, viscous or viscous-potential-hybrid NWTs are recommended.
One important nonlinear wave problem not considered here is the evolution in a long tank. If the tank length is greatly increased, the matrix size and computational burden in the BEM method increase drastically. For the long-tank simulation for evolution study, multi-block methods are recommended, in which the long tank is divided into many blocks and the boundary-value problem is solved in each block. The continuity of the potential and normal velocities is applied at the overlapped boundaries to form a banded-block matrix instead of a full matrix [Wang et al. (1993), De Haas and Zandbergen (1996)]. References [1] Beck, R.F., Cao, Y., Scorpio, S.M. & Schultz, W.W., Nonlinear Ship
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