Proceedings of the 27th International Conference on Offshore Mechanics and Arctic Engineering OMAE2008 June 15-20, 2008, Estoril, Portugal
OMAE2008-57652 SPATIAL VS. TEMPORAL EVOLUTION OF NONLINEAR WAVE GROUPS – EXPERIMENTS AND MODELING BASED ON THE DYSTHE EQUATION Lev Shemer and Boris Dorfman School of Mechanical Engineering, Faculty of Engineering, Tel-Aviv University Tel-Aviv 69978 Israel, E-Mail:
[email protected]
ABSTRACT The evolution along the tank of unidirectional nonlinear wave groups with narrow spectrum is studied both experimentally and numerically. Measurements of the instantaneous surface elevation within the tank are carried out using digital processing of video-recorded sequences of images of the contact line movement at the tank side wall. The accuracy of the video-derived results is verified by measurements performed by conventional resistance-type wave gauges. An experimental procedure is developed that enables processing of large volumes of video images and capturing the spatial structure of the instantaneous wave field in the whole tank. The experimentally obtained data are compared quantitatively with the solutions of the modified nonlinear Schrödinger (MNLS, or Dysthe) equation written in either temporal or spatial form. Results on the evolution along the tank of wave frequency spectra and on the temporal evolution of the wave number spectra are presented. It is demonstrated that accounting for the 2nd order bound (locked) waves is essential for getting a qualitative and quantitative agreement between the measured and the computed spectra.
INTRODUCTION Rapid advancement in both theoretical and experimental studies of water waves that occurred in recent decades were prompted by the discovery by Benjamin and Feir [1] of the sideband instability of weakly nonlinear Stokes waves. Important theoretical model for studying the long time behavior of the nonlinear water waves was developed by Zakharov [2]. The Zakharov integral equation describes near-resonant interactions between waves and was originally derived for infinitely deep water. Zakharov also showed that under the assumption that the wave spectrum is narrow, the nonlinear Schrödinger (NLS) equation can be deduced from the Zakharov equation. The NLS equation, accurate to the 3rd order in the wave steepness ε, therefore describes resonant interactions pertaining to a weakly nonlinear wave train with a narrow band of frequencies and
wave lengths, and governs the slow modulation of the wave group envelope. Later the NLS equation for water gravity waves in water of finite depth was derived by using the multiple scales method [3] and by applying the averaged Lagrangian formulation [4]. An agreement between the experimentally found growth rates of the unstable sidebands with the theoretical predictions based on the NLS equations was obtained [5]. Shemer et al. [6] performed quantitative comparison of the numerical simulations based on the NLS equation with experiments in a laboratory wave flume. They demonstrated that while the NLS equation is adequate for qualitative description of the global properties of the group envelope evolution, such as focusing observed for water waves in sufficiently deep water, it is incapable of capturing more subtle features, for example the emerging front-tail asymmetry observed in experiments. Due to nonlinear interactions, considerable widening of the initially narrow spectrum can occur; therefore more advanced models are required for an accurate description of nonlinear wave group evolution. The modified nonlinear Schrödinger (MNLS) equation derived by Dysthe [7] is a higher (4th) order extension of the NLS equation, where the higher order terms account for finite spectrum width [8]. Further modification of the NLS equation appropriate for wider wave spectra was presented by Trulsen and Dysthe [9] and Trulsen et al. [10]. It was shown by Kit and Shemer [11] that this modification can be easily derived by expanding the dispersion term in the Zakharov equation into the Taylor series. The effect of each one of the 4th order terms in the Dysthe equation was studied in [12]. They demonstrated that for steep waves all these terms contribute significantly to the accuracy of the solution. The theoretical models mentioned above were derived to describe the evolution of the wave field in time. Complete information on the surface elevation distribution along the tank at a prescribed instant constitutes the required for the solution of the problem initial condition. In laboratory experiments, however, waves are generated by a wavemaker usually placed at the end of the experimental facility. The experimental data are commonly accumulated using sensors
The MNLS coupled system of equations, which describes the evolution of the complex envelope a(x, t) and of the potential of the induced mean current φ (x ,z, t) was in fact derived by Dysthe for the surface velocity potential amplitude, aψ. It was demonstrated by Hogan [16], see also [11], that while at the 3rd order the governing equation for both amplitudes, of the surface elevation, aη, and of the free surface velocity potential, aΨ, are identical, and thus there is no difference in the NLS equation for either of those amplitudes, at the 4th order the governing equations differ somewhat. For quantitative comparison of the model predictions with the experiment that directly provides data on the surface elevation variation, the equation describing the variation of aη is applied in sequel, with the index “η” omitted. In fixed coordinates, the governing system of equations has the following form:
placed at fixed locations within the tank. Hence, to perform quantitative comparison of model predictions with results gained in those experiments, the governing equations have to be modified to a spatial form, to describe the evolution of the temporally varying wave field along the experimental facility. Such a modification of the Dysthe model was carried out by Lo and Mei [13] who obtained a version of the equation that describes the spatial evolution of the group envelope. Numerical computations based on the Dysthe model for wave groups propagating in a long wave tank indeed provided good agreement with experiments and exhibit the front-tail asymmetry [12]. The spatial version of the Dysthe equation was derived in [11] from the spatial form of the Zakharov equation. [14, 15] that is free of any restrictions on the spectrum width. In the present work, the evolution of unidirectional nonlinear wave groups along the tank is studied using digital processing of video-recorded sequences of images of the contact line movement at the tank side wall. The technique allows accurate measurements of both the spatial variation of the instantaneous surface elevation along the whole tank, and of the temporal variation of the surface elevation at any prescribed location within the tank. The comparison of the experimentally obtained data thus can be carried out with the solutions of the model equations presented in their temporal or the spatial forms. Narrow-spectra nonlinear deep-water wave groups excited by a wavemaker are studied. The Dysthe MNLS equation describes evolution of the complex nonlinear wave group envelope and constitutes an appropriate theoretical model for studying such wave groups. The advantages and disadvantages of the spatial and temporal forms of the model equation are discussed. The Dysthe equation in both temporal and spatial forms is solved numerically, and the results of both versions are compared quantitatively with the experimental data.
ω ∂ 2a i ∂a ω0 ∂a 1 ω0 ∂ 3a 2 + + i 02 2 + ωk 2 a a − ∂t 2k0 ∂x 2 16 k03 ∂x 3 8k0 ∂x ∂a * 3 ∂φ 2 ∂a a + + ω0 k0 a + ik0 a =0 4 ∂x 2 ∂x ∂x z =0
ω0 k0
∂ 2φ ∂ 2φ + =0 ∂x 2 ∂z 2
∂φ ω0 ∂ a = ∂z 2 ∂x
∂φ = 0 ( z → −∞ ) ∂z
2
( z = 0)
(5)
(6)
The first four terms in (3) constitute the cubic Schrödinger equation for deep water in the fixed frame of reference. Dysthe’s equations that can be derived from the Zakharov integral equation that is of the 3rd order in steepness by adding the narrow-band assumption with spectral width O( ε ) [8]. The sign of the term a 2∂a * / ∂x in (3) is positive, while in the velocity potential version used in [7] and [13] it is negative. The opposite signs of this term constitute the only difference between the two versions of the 4th order envelope evolution equation. The problem of wave field evolution in a tank admits two different formulations. In the so-called temporal formulation, the spatial distribution of the complex envelope a(x) is presumed to be known at a prescribed instant t0, and its variation in time is obtained by numerical solution of the model equation. Alternatively, the variation of the complex envelope in time, a(t), can be specified at a prescribed location x = x0, and the variation of a(t) along the tank is then studied in the spatial formulation using the appropriately modified model equations. It should be
(1)
where g is the acceleration due to gravity. Evolution of the wave group in a wave flume can be represented by variation in time and space of either the surface elevation η (x, t), or of the velocity potential φ at the free surface, ψ (x, t) = φ (x, z=η, t). For a narrow-banded wave group it is convenient to express the variation of η and ψ at the leading order in terms of their complex envelope amplitudes:
ψ(x, t) = Re[- i 2ω aψ( x, t) exp i(k0x - ω0t)] 0
(4)
and at the bottom
Consider a narrow-banded unidirectional deep-water wave group with the dominant frequency ω0 and wave number k0 that are related by the deep-water dispersion relation for gravity waves:
η(x, t) = Re[aη( x, t) exp i(k0x - ω0t)]
( − h < z < 0) .
These equations are subject to the boundary conditions at the free surface
THEORETICAL BACKGROUND
ω02 = k0g,
(3)
2
(2a) (2b)
2
Copyright ©2008 by ASME
elevation at the leading order. With A(ξ, τ) known, application of (2a) represents the so-called free waves only. The bound, or locked, waves can also be determined from A(ξ, τ) using:
stressed that the spatial formulation is routinely applied in the experiment-related studies [12, 13], since the wave gauges provide information on the temporal variation of the surface elevation at fixed locations. The experimental approach of the present study makes it possible to measure the variation with time of the instantaneous complex group envelope along the tank, as well as the variation of the surface elevation with time at any location within the tank. Both temporal and spatial formulations of the Dysthe equation are therefore employed. Consider first the temporal model. In analogy to [13], in a coordinate system moving at the group velocity cg = ω0/2k0, the following dimensionless scaled variables are introduced:
τ = ε ω0t, ξ = εk0(x-cgt), A = a/a0, Φ = ω 2
B(A) = ½ ε A2
The surface elevation accurate to the 2nd order for both temporal and spatial formulation is thus obtained as
(
η / a0 = Re A e (
φ, Z = ε k0z (7)
In these variables, the equations for A and Φ are:
∂ 2Φ ∂ 2Φ + =0 ∂ξ 2 ∂ Z 2
( Z < 0)
(9)
with Φ satisfying the following boundary conditions: 2
∂Φ 1 ∂ A = , Z = 0, ∂Z 2 ∂ξ
∂Φ = 0, ∂Z
Z → −∞
(10)
The set of equations (7) - (10) and the appropriate initial conditions constitute the temporal version of the Dysthe model. The corresponding spatial version can be obtained either from (3) as in [13], or from the spatial version of the Zakharov equation [11]. The scaled dimensionless space and time variables in (7) are replaced for the spatial version by
ξ = ε2k0x, τ = εω0(x/cg-t )
(11)
The governing equations then assume the following form ∂A ∂ 2 A 2 +i 2 + i A A ∂ξ ∂τ ⎛ ∂A∗ ∂Φ 2 ∂A +ε ⎜8 A + 2 A2 + 4iA τ τ ∂ ∂ ∂τ ⎝
4 ∂Φ ∂Z
∂ 2Φ ∂ 2Φ + =0 ∂τ 2 ∂Z 2
Z =0
=
∂A ∂τ
2
,
Z =0
⎞ ⎟=0 ⎠
Z
