a numerical wave tank for the generation and propagation

A numerical wave tank for the generation and propagation of bichromatic, nonlinear, long- crested surface waves

W. ~arsons' & R.E. add our^ l NSERC Visiting Fellow at National Research Council- Canada, Institute for Marine Dynamics, Canada 2 National Research Council - Canada, Institute for Marine Dynamics, Canada

Abstract

We are studying numerically the problem of generation and propagation of bichromatic gravity long-crested waves in a tank containing an incompressible inviscid homogeneous fluid initially at rest with a horizontal free surface of finite extent and of infinite and finite depths. A non-othogonal curvilinear coordinate system, which follows the free surface is constructed and the full nonlinear kinematic and dynamic free surface boundary conditions are utilized in the algorithm.

1 Introduction

A two-dimensional rectangular basin containing an incompressible inviscid homogeneous fluid initially at rest with a horizontal free surface of finite extent is considered to generate and propagate long-crested waves. On the left vertical boundary a wavemaker is positioned while at the right hand side is the radiation boundary. The same numerical beach is used as in Parsons and Baddour (2002), and a depth profile for the potential is assumed, giving us a waveform rekuration method, and thereby drastically reducing the computational cost of soIving Laplace's equation. A bichromatic deterministic wavemaker employing a Diriclet type boundary condition is applied; see Baddour and Parsons (2003). This method can easily be extended to the case of a wavemaker utilizing a Neumann type boundary condition.

Transactions on the Built Environment vol 71, © 2004 WIT Press, www.witpress.com, ISSN 1743-3509

408 Fluid Structure Interaction 11

Laplace's equation is solved using a non-orthogonal curvilinear coordinate system, which follows the free surface. This gives a more realistic "conti- nuity condition", since it involves the entire fluid domain. Also, the full nonlinear kinematic and dynamic free surface boundary conditions are em- ployed. Although, these ideas can be extended to finite depth tanks, we restrict our attention to the infinite depth case.

2 Problem formulation Consider a two-dimensional rectangular basin containing an incompressible inviscid homogeneous fluid of constant density p, in a Cartesian coordinate system ( X , z ) , with the origin at the still water level, the positive z-axis pointing upwards. The horizontal extent of the basin is 0 < X < L, so L > 0 is the length of the tank. The depth of the basin is infinite. See Figure 1 for the coordinate system configuration. This is an initial-value problem, since the surface is initially at rest for time, t < 0 and is disturbed at t = 0 giving rise to surface waves. See Debnath (1994).

If the @ ( X , 2 , t ) is the velocity potential and V ( X , t ) is the free surface elevation, the problem is defined by the following equations, where g is the acceleration due to gravity. The conservation of mass equation is Laplace's equation:

a2@ a2@ -+- = 0 , - m < z < q ( x , t ) , O < x < L , t > 0 (1) ax2 az2

For t > 0, the full nonlinear kinematic and dynamic free surface boundary conditions are given, respectively, by:

and

See Wehausen and Laitone (1960). We introduce the following function:

which represents the potential on the free surface. We also have


Fluid Structure Interaction I1 409

implying:

Also,

The initial conditions are given by:

where (l/p)ao ( X ) and ( l / p ) @ o ( ~ ) represent the given free surface im- pulse and qo(x) the initial displacement, where the "dot" represents a time derivative. Since we are considering a fluid in which the initial velocities are zero and the initial free surface is at rest at t = 0, we take

@o(x) = 0, Qo(x)=O, qo(x)=O for O < X < L . ( 9 ) Therefore we are left with two lateral boundary conditions for t 2 0 at X = 0 and X = L, which we call the LHS lateral boundary condition and the RHS lateral boundary condition, respectively.

The LHS boundary condition involves a wavemaker, which we assume is the superposition of two sinusoid waves, each of the general form

q(x , t ) = A cos(kz - wt), -m < X < 0, - oo < t < oo, (10)

where A = H/2 is the amplitude of the wave, tk is the wave number and W is the angular frequency of the wave. These progressive waves at X = 0 and t 2 0 are the sole sources of the disturbance that gives rise to water waves in the initially calm basin. Using standard water wave mechanics, the associated velocity h e a r potential corresponding to each wave is given by

@ ( X , t , t ) = %lir s in (k~-wt ) , X = 0, t 2 0, -m < t < q(O,t), W

(11) where k is the representative wave number, and the linear dispersion relation is

W = A. (12) Since we are solving the full nonlinear problem, we compare our solution

for V ( X , t ) with the superposition of the third-order Stokes expansion, where each has the form

1 q ( x , t ) = A 1 { c o s k ( ~ - et) + -A1kcos2k(x - ct) +

2


41 0 Fluid Structure Interaction 11

where A' = A{l + Q A ~ ~ ~ } , and the velocity C is given by

in which case we take the velocity potential for each wave to be

(15) See Wehausen and Laitone (1960). Clearly, equations (13) and (15) reduce to equations (10) and (11) if only the first order is taken (linear case).

If we assume a wall at X = L so the RHS lateral boundary condition

we will get reflections from waves going from the wavemaker at X = 0 toward the wall. This distracts from the objective of the present model and is to be avoided. To absorb these waves we place a "numerical beach" on the RHS. This is accomplished by including a damping term p in the dynamic free surface boundary condition equation. See Parsons and Baddour (2002).

3 The Laplacian and boundary conditions on the free surface coordinate system We define the free surface coordinate system (FSCS), (S, W), where S 2 0 is the arclength along the free surface z = V(X, t ) and W 5 0 is the vertical depth to any point in the fluid from the free surface. This is a non-orthogonal moving coordinate system that follows the free surface, and Laplace's equation becomes:

where

and ds = sec 8 dx .

We refer the reader to Baddour and Parsons (2003) for the details. The kinematic free surface boundary condition (KFSBC) given by equation (2), becomes

arl ax 2 a@ - + tan 8- - sec 8- = 0, at ax a2


Fluid Structure Interaction I1 41 1

and the dynamic free surface boundary condition (DFSBC), given by (3), becomes

dx 1 dx d@ - + - sec2 Q(-)" + +v = 0, dt 2 dx ax (21)

where all values are evaluated on the free surface, and since the free surface coordinate system tracks the free surface, these values are readily obtained. These derivatives allow for integration in time t. 4 Spatial approximations and waveform relaxation In this paper, we employ a curvilinear grid that follows the free surface. In light of equation (19), this grid is taken to be

where si = S;-l + sec sipl AI, and sec Qi = [Jm] , x=xi

where we take SO = 0, and w j = - ~ A w , where AX = L / M , so xi = iAx, and AW = h / N L , where h is the depth to which we solve Laplace's equation, taken to be greater than or equal to one wavelength of the wavemaker. We allow the possibility that AW f Ax. We must solve Laplace's equation, (17) over this grid.

The semi-discretized approximation of this potential @(S;, wj, t) is writ- t e n a s a i , j ( t ) i =0,1;-- , M ; j = 0 , l . - . , N L ; t > 0.

To implement the wavemaker, we assume two sinusoids of the form (10), for X 5 0, -CO < t < CO, and their associated (linear) velocity potentials, (15), where the celerity and dispersion relation of each is given by (14) and (12), respectively. This LHS boundary condition can be implemented as a Dirichlet or Neumann condition, and we refer to the resulting wavemakers as Dirichlet and Neumann wavemakers, respectively; see Baddour and Parsons (2003). In this paper we restrict our attention to the Dirichlet condition, and assure the reader that the extension to the Neumann wavemaker is straightforward. The Dirichlet wavemaker, being the superposition of two sinusoids, is given by:

j = 0 , l . - . , NL; t > 0, where Ail ki and ci, i = 1 , 2 are the corresponding properties of each sinusoid given and related by equations (15), (14) and (12), and vi, i = 1 ,2 , are the corresponding third-order Stokes expansion for the free surface elevation, each given by equation (13).

The horizontal bottom boundary condition, given by (4), will be automatically satisfied by the relaxation method, as we shall soon see. The



boundary condition at the free surface, z = ~ ( x , t),

{@i,o(t) : i = 1 , 2 - - - , M - I; t > 01, (24)

is given by the kinematic and dynamic boundary conditions, (20) and (21), respectively, and involve integration over time. This will be discussed in the next section.

The RHS boundary condition for the numerical beach is incorporated in the calculation of the free surface potential over the damping zone, and involves modification of equation (21) to,

see Parsons and Baddour (2002). Therefore, the potential should vanish everywhere at the wall, and as a convenience, we may employ equation (16) to get the RHS boundary condition; that is,

(27) where xi(t) is the potential at = T(X, t) (the free surface potential) and

Even with a uniform Ax, the curvilinear grid will have non-constant AS. Therefore, to apply finite difference formulas to discretize Laplace's equation (17), we will need appropriate finite difference operators. Clearly, applying these discrete operators to Laplace's equation (17), is best done using computer algebra, and the authors utilized Maple. We refer to the resulting equation, as the "semi-discretized Laplace equation". The details are to be given elsewhere.

Consider it's application over any S-coordinate curve for fixed W

in the interior of the domain. That is, we evaluate the semi-discretized Laplacian at (si, wj) , to calculate ai,j (t) i = 1 ,2 , - . - , M - 1; j = 1 , 2 , - - - , NL - 1 (fixed); t > 0. This will involve nine potentials, namely:

therefore satisfies equations (20) and (25), and since we are considering the

The sj-1-curve : @i-l,j-l(t) The S?-curve : Qi-1 (t) The sj+l-curve : j+l (t)

infinite depth basin, we take

See Figure 1. The idea behind the waveform relaxation (WR) method is to follow the lead of separation of variables methods and assume that the potential can be written as,

Gi (t) Qi,j (t) ai,j+1 (t)

ai+l j-1 (t) @i+l j (t) @i+l,,j+l (t) ,


Fluid Structure Interaction 11 41 3

where k* is the representative wave number, which for the monochromatic case we can take as the wave number of the wavemaker, see equation (15), and Baddour and Parsons (2003). In the bichromatic case, however, there are two wave numbers, kl and k2 to choose from. In this case we use an iterative approach and choose some "reasonable" k*, where k* E [min{kl, k2), max{kl, k2)] and use equation (28) as a "first guess" only. We then iterate to converge on the depth profile

4 i , j ( t ) = y j ( x i , t ) , i = 1 , 2 , . . . , M , j = 0 , 1 , - - - ,NL, t > 0 , (29)

where y i ,o (~ i , t) = 1, i = 1 , 2 , . . . , M , t > 0. Therefore the nine potentials that occur in the discretized Laplace equation become,

so clearly the only "unknowns" are the three potentials along the sj-coordinate curve, and we have successfully decoupled the system along dimensional lines, by "relaxing" the W dependence. Note that, in light of equations (28) and (29), the bottom boundary condition (4) is automatically satisfied. - ~incetheterms{~~(t),4~,~(t)),i = 1 , 2 , . - - , M , j = 0, l , . . . , NL, t 2

A 7

0 are known and become incorporated into the vector bfs, this semi-discretized Laplace equation gives rise to the iterated matrix equation

A (6) q+l (t) = (hs) (t) + L&), for q = 0,1, - - . , where the matrix, A is a square sparse tridiagonal matrix

of dimension (M- l) X (M- l ) , and 6j(t) = [@l,j ( t), G2,j(t), . . . , @cI-~,j(t)]Tj + 0

where j = 1 , 2 , . - . , NL - 1 ; t 2 0. The starting value, (b is) is ob- \ I

+ 4 tained from equation (28) and all subsequent (b in) , q = 1 , 2 , - - . are

\ /

. In general, two types of iteration schemes are possi- , ,

ble, Gauss- Jacobi in which "new values" are used only after a complete iteration has been completed and Gauss-Seidel in which "new values" are used as soon as they are available. Clearly, the Gauss-Jacobi method is fully parallel, but we employ the Gauss-Seidel method, since it generally converges faster. Note that the matrix A does not depend on the iteration parameter q, and therefore it must be inverted only once per time step.

The terms {Qo,j(t)), where j = (),l,. . - , NL, t 2 0 are known since they are given by the wavemaker condition (23) and are incorporated



-+ into the vector b,,(t). The inversion of the complete (M - l)(NL - 1) X (M - l)(NL - 1) system reduces to (NL - 1) inversions of the (M - 1) X (M - l) linear system given by (30). Furthermore, if AW is a constant, then the matrix A is identical for any S-coordinate curve, and the Gaussian elimination operations necessary to invert each system, using LU-decomposition, need be performed only once. This effectively reduces the dimension of the problem by one.

5 Time integration Equation (30) must be fully discretized to complete the numerical model. For T a positive finite real number, we solve our initial-value problem over

T the time interval [O, T]. For a natural number NT, we let At = K , which gives the sequence of time steps {t,}, where, t, = nAt, n = 0,1, - - - , NT. The approximation of aij(tn), xi(tn) and the free surface elevation q(xi, tn) is denoted by atj, X: and v:, respectively, for i = 0 , 1 , - . - ,M, j = 0 , 1 , - - - , N L , n = O , l , - - - ,NT.Aforth-order Adams predictor-corrector method is used to generate {v;}, {X:}, see Baddour and Parsons (2003) for the details.

6 Test problems 6.1 Laplace solver - monochromatic case

To test and illustrate the Laplace solver alone, we consider the following monochromatic test problem defined by a potential

on the domain V, where

and k = 1. See Figure 2 for the curvilinear coordinate system appropriate to this example. In this present monochromatic case, the waveform relaxation method (30) converges in only one iteration for k* = k = 1. However, to demonstrate this convergence in the more eneral bichromatic case, we & choose k* = 10k. Figure 3 gives a plot of (dz)t=17(zl against X showing that we get good convergence in only q = 3 iterations. The percentage error between the numerical and exact value of this derivative at the place shown in the graph was calculated to be 3.5%. Since this derivative is used in our numerical wave generation and propagation model, this adds confidence to our modeling. In fact, this is the only use our model makes of the Laplace solver 6.2 Depth profile for monochromatic and bichromatic waves

To demonstrate the validity of the assumption implicit in equation (28) for the monochromatic case, and to study the properties of the function


Fluid Structure Interaction 11 41 5

in equation (29) in the bichromatic case, we performed a number of tests with the complete numerical tank model, where for all tests, L = l, h = X*, where X* = 2 ~ / k * , and we implemented a damping zone equal to 2X*. Both monochromatic and bichromatic wavemakers were used and we generated a depth profile yj (xi, t,), where

and xi = L/2 and t, = 30 periods, where the period, T* is calculated from W* = m. In Figure 4 we give a plot for the case of a monochromatic wavemaker, showing that the waveform relaxation method converges to (28), but not on the first iteration.

We next show the results for a test involving a bichromatic wavemaker in Figure 5. Note that the "waveform relaxation" profile becomes "less steep", with increasing iterations, exhibiting good agreement with Xi(t)e0.6sW3, at least closer to the free surface. Of course, waveform relaxation does not assume a'priori that the depth profile is a single "exponential", and the method is therefore not restricted to this function form alone, but allowed to converge on the appropriate profile for the particular problem being solved. In light of this, it should not be surprising if yj(xi, t,) cannot be closely

approximated by Xi(t)ekewj, for any k* over the whole depth [O, -h]. Also, the profile is not required to be the same everywhere in the

tank and at all times throughout the simulation. Since the tank contains a wavemaker at one end and a damping zone at the other and the problem involves a transient phase and a steady-state phase, this flexibility allows the method to adapt the depth profile to a wide varieties of general situ- ations. 6.3 Generation and propagation of steep bichromatic waves We performed a number of tests with the complete numerical tank model, designed to show that our model can generate and propagate steep bichromatic waves with reasonable accuracy. We set L = 1, h = X*, where X* = 2 ~ / k * , and we implemented a damping zone equal to 2X*, and gave results after t, = 30 periods, where the period, T * is calculated from W* = m, and we implemented spatial Fourier filtering to eliminate spatial aliasing, see Baddour and Parsons (2003).

In Figure 6, we show the free surface elevation, 7, generated by the numerical model, including, for comparison, the Stokes third order solution, which is the superposition of two waves of the form given by equation (13). Near the LHS (wavemaker) the agreement is good, but appears to degrades as we go toward the RHS (damping zone) of the tank. This is due to a simple phase difference, resulting from a difference in celerity for each wave relative to the Stokes solution, and reduces as the spatial discretization is refined, see Parsons and Baddour (2002) and Baddour and Parsons (2003). Note that the steeper kl wave exhibits the sharper raised crests and broader



raised troughs that are well known for nonlinear waves, see Wehausen and Laitone (1960). Figure 7 gives a plot of the spatial Fourier transform of the free surface elevation for this case and shows that both waves are well represented.

Figures 8 and 9 shows plots of the free surface elevation and the tem- poral Fourier transform of the free surface elevation, respectively, at a fixed location in the tank. Clearly, both waves are evident in these plots. Also, we indicated in both spectrum plots the "aliasing limits", K, = r /Ax and Kt = r /At , which are the highest wave number and frequency, respectively that can be resolved by our grid.

7 Conclusions and future work We have demonstrated that the iterated Laplace matrix equation that results from the application of the waveform relaxation method is able to converge on the appropriate depth profile for both the monochromatic and bichromatic cases. This profile is general and dynamic and allows the method to adapt to a wide varieties of problems. Coupled with the bichromatic wavemaker, this dynamic Laplace solver permits us to generate and propagate transient steep bichromatic waves, which are then completely ab- sorbed by the same numerical beach given in Parsons and Baddour (2002). Our next step will be to construct a "multichromaticl~ wavemaker, involving a finite set of wave numbers. Clearly, Fourier methods will then facilitate the extension of these ideas to "sea-states".

Finally, we stated in the introduction of this paper, that these ideas can be extended to the finite depth case. In Parsons and Baddour (2002) we used an appropriate "hyperbolic profile" to facilitate this extension on a Cartesian coordinate system for the linear problem. Using methods from Riemannian geometry, an extension of the free surface coordinate system used in this paper is being developed to construct a curvilinear coordinate system for a domain with a finite depth and an irregular shaped bottom. A paper is to follow.

References

[l] Debnath, L. (1994) Nonlinear water waves, Academic Press, New York.

[2] Baddour, R.E. and Parsons, W. (2003) A comparison of Dirichlet and Neumann wavemakers for the numerical generation and propagation of transient long-crested surface waves, Proceed- ings of 22nd International Conference on Offshore Mechanics and Artic Engineering.

[3] Parsons, W. and Baddour, R. E. (2002), The generation and propagation of transient long-crested surface waves using a wave-



form relaxation method, Proceedings Advances in Fluid Mechanics 2002, Wessex Institute of Technology Press.

[4] Wehausen, J.V. and Laitone, E.V. (1960) Encyclopedia of Physics, Vol. IX, Fluid Dynamics 111, Springer-Verlag, Berlin.

Acknowledgements To College of the North Atlantic, Newfoundland, Canada, for giving

Wade Parsons a one year sabbatical. To Natural Sciences and Engineering Research Council of Canada and

the National Research Council's Institute for Marine Dynamics, Newfound- land, Canada, for awarding Wade Parsons a Visiting Fellowship.

Figure 1. Free surface coordinate system

X (m1

Figure 2. Curvilinear coordinate system for Laplace test problem



z(m)

Figure 4. Depth profile for monochromatic wave at x=0.5L

2

- E l - - m_ * 0 5 -

0

-0 5

1

-1 5

2 (m)

Figure 5. Depth profile for bi-chromatic case at x=0.2L

X (m)

Figure 3. The z-derivative of the free surface potential for Laplace test problem

-

-

-

-

2 W 2 z at w=O

j 5 - /

1 error calculated here

0 10 20 30 40 50 60 70



Figure 6. Free surface elevation for a steep bi-chromatic

k,=2.Ok.A ,=0.5A k2=0.5k,A 2=0.5A steepness of k , wave=0.0700 steepness d k wave=0.0175

aner 30 periods

l 5 0 100 150 200 250 300

wavenumber (m -' t

Figure 7. FFT of the free surface elevation of a steep bi-chromatic wave

4 waves, A X= 1'125. A r= 1'115, A t=T '140 s'=007, q=lO

0.1227429, w.=111198

igure 8. Free surface elevation at x=0.2L


420 Fluid Structure Interaction 11

.thsahamnsl,ms. K ,

&30psriodrdr=0.2L

($0 200 250 300 350

,roq"on.y(r .' )

Figure 9. FFT of the free surface elevation at x=0.2L


a numerical wave tank for the generation and propagation

Documents