Weakly nonlinear surface gravity waves in a depth-dependent background flow

2021 ◽  
Author(s):  
Zibo Zheng ◽  
Yan Li ◽  
Simen Ellingsen
Keyword(s):  
Shear Flow ◽  
Surface Waves ◽  
Second Order ◽  
Irregular Waves ◽  
Exceedance Probability ◽  
Wave Group ◽  
Background Flow ◽  
Geophysical Research ◽  
Weakly Nonlinear ◽  
Direct Integration Method

<p>An open ocean often has a wind driven shear-current near the surface that is able to significantly change the properties of surface waves. This work aims to investigate the effects of a vertically sheared background flow on weakly nonlinear waves with both a statistical study for irregular random waves and a deterministic study for a wave group.</p><p>We first extended the theory by Dalzell (1999) to allow for the effects of a horizontal background flow with arbitrary depth dependence. The extended theory is valid up to second order in wave steepness and is applicable for directional-spread waves of a broad bandwidth. The Direct Integration Method (Li & Ellingsen 2019) is used for the linear dispersion relation.</p><p>Using the theory, we examine the effects of an opposing and assisting shear, respectively, on the nonlinear properties of a short wave group on deep-water through comparisons to cases without a shear flow. A shear flow leads to wave crests (troughs) being either steepened or flattened, depending mainly on the direction of a shear relative to the propagation direction of the group and the strength of the depth-integrated velocity of a shear relative to the group velocity. We, furthermore, investigated skewness and kurtosis of a time record of the wave elevation for irregular waves in a background sheared flow, compared to a linear Gaussian random sea for surface waves only. We obtained the probability density function and exceedance probability for wave crests. Relevance for rogue wave formation is discussed.</p><p><strong>Key words</strong>: waves/free-surface flow, ocean surface waves, wave-current interaction</p><p> </p><p>References</p><p>Dalzell, J. F. "A note on finite depth second-order wave–wave interactions." Applied Ocean Research 21, no. 3 (1999): 105-111.</p><p>Li, Y., and Ellingsen, S. Å. "A framework for modeling linear surface waves on shear currents in slowly varying waters." Journal of Geophysical Research: Oceans 124, no. 4 (2019): 2527-2545.  </p>

scholarly journals An improved Lagrangian model for the time evolution of nonlinear surface waves

2019 ◽  
Vol 876 ◽  
pp. 527-552 ◽  
Author(s):  
Charles-Antoine Guérin ◽  
Nicolas Desmars ◽  
Stéphan T. Grilli ◽  
Guillaume Ducrozet ◽  
Yves Perignon ◽  
...  
Keyword(s):  
Surface Waves ◽  
Time Evolution ◽  
Forecast Accuracy ◽  
Second Order ◽  
Stokes Drift ◽  
Irregular Waves ◽  
Lagrangian Model ◽  
Simple Method ◽  
Lagrangian Models ◽  
Wave Forecast

Accurate real-time simulations and forecasting of phase-revolved ocean surface waves require nonlinear effects, both geometrical and kinematic, to be accurately represented. For this purpose, wave models based on a Lagrangian steepness expansion have proved particularly efficient, as compared to those based on Eulerian expansions, as they feature higher-order nonlinearities at a reduced numerical cost. However, while they can accurately model the instantaneous nonlinear wave shape, Lagrangian models developed to date cannot accurately predict the time evolution of even simple periodic waves. Here, we propose a novel and simple method to perform a Lagrangian expansion of surface waves to second order in wave steepness, based on the dynamical system relating particle locations and the Eulerian velocity field. We show that a simple redefinition of reference particles allows us to correct the time evolution of surface waves, through a modified nonlinear dispersion relationship. The resulting expressions of free surface particle locations can then be made numerically efficient by only retaining the most significant contributions to second-order terms, i.e. Stokes drift and mean vertical level. This results in a hybrid model, referred to as the ‘improved choppy wave model’ (ICWM) (with respect to Nouguier et al.’s J. Geophys. Res., vol. 114, 2009, p. C09012), whose performance is numerically assessed for long-crested waves, both periodic and irregular. To do so, ICWM results are compared to those of models based on a high-order spectral method and classical second-order Lagrangian expansions. For irregular waves, two generic types of narrow- and broad-banded wave spectra are considered, for which ICWM is shown to significantly improve wave forecast accuracy as compared to other Lagrangian models; hence, ICWM is well suited to providing accurate and efficient short-term ocean wave forecast (e.g. over a few peak periods). This aspect will be the object of future work.

scholarly journals Effects of an abrupt depth change on weakly nonlinear surface gravity waves: deterministic and stochastic analysis

2020 ◽  
Author(s):  
Yan Li ◽  
Samuel Draycott ◽  
Yaokun Zheng ◽  
Thomas A.A. Adcock ◽  
Zhiliang Lin ◽  
...  
Keyword(s):  
Gravity Waves ◽  
Second Order ◽  
Surface Gravity ◽  
Irregular Waves ◽  
Wave Steepness ◽  
Weakly Nonlinear ◽  
Wave Statistics ◽  
Surface Gravity Waves ◽  
Seabed Topography ◽  
Nonlinear Surface

<p>This work focuses on two different aspects of the effect of an abrupt depth transition on weakly nonlinear surface gravity waves: deterministic and stochastic. It is known that the kurtosis of waves can reach a maximum near the top of such abrupt depth transitions. The analysis is based on three different approaches: (1) a novel theoretical framework that allows for narrow-banded surface waves experiencing a step-type seabed, correct to the second order in wave steepness; (2) experimental observations; and (3) a numerical model based on a fully nonlinear potential flow solver. To reveal the fundamental physics, the evolution of a wave envelope that experiences an abrupt depth transition is examined in detail; (a) we show the release of free waves at second order in wave steepness both for the super-harmonic and sub-harmonic or ‘mean’ terms; (b) a local wave height peak that occurs near the top of a depth transition – whose exact position depends on several nondimensional parameters – is revealed; (c) furthermore, we examine which parameters affect this peak. The novel physics has implications for wave statistics for long-crested irregular waves experiencing an abrupt depth transition. We show the connection of the second-order physics at work in the deterministic and stochastic cases: the peak of wave kurtosis and skewness occurs in the neighborhood of the deterministic wave peak in (b) and for the same parameters set composed of a seabed topography, water depths, primary wave frequency and steepness, and bandwidth.</p>

Balanced ellipsoidal vortex equilibria in a background shear flow at finite Rossby number

2021 ◽  
Vol 926 ◽  
Author(s):  
William J. McKiver
Keyword(s):  
Shear Flow ◽  
Strain Rate ◽  
Second Order ◽  
Rossby Number ◽  
Strain Rates ◽  
Background Flow ◽  
Flow Parameters ◽  
Critical Strain Rate ◽  
Balanced Model ◽  
Background Shear

We consider a uniform ellipsoid of potential vorticity (PV), where we exploit analytical solutions derived for a balanced model at the second order in the Rossby number, the next order to quasi-geostrophic (QG) theory, the so-called QG+1 model. We consider this vortex in the presence of an external background shear flow, acting as a proxy for the effect of external vortices. For the QG model the system depends on four parameters, the height-to-width aspect ratio of the vortex, $h/r$ , as well as three parameters characterising the background flow, the strain rate, $\gamma$ , the ratio of the background rotation rate to the strain, $\beta$ , and the angle from which the flow is applied, $\theta$ . However, the QG+1 model also depends on the PV, as well as the Prandtl ratio, $f/N$ ( $f$ and $N$ are the Coriolis and buoyancy frequencies, respectively). For QG and QG+1 we determine equilibria for different values of the background flow parameters for increasing values of the imposed strain rate up to the critical strain rate, $\gamma _c$ , beyond which equilibria do not exist. We also compute the linear stability of this vortex to second-order modes, determining the marginal strain $\gamma _m$ at which ellipsoidal instability erupts. The results show that for QG+1 the most resilient cyclonic ellipsoids are slightly prolate, while anticyclonic ellipsoids tend to be more oblate. The highest values of $\gamma _m$ occur as $\beta \to 1$ . For large values of $f/N$ , changes in the marginal strain rates occur, stabilising anticyclonic ellipsoids while destabilising cyclonic ellipsoids.

Towards a Technique for Nonlinear Modal Analysis

2012 ◽  
Author(s):  
Simon A. Neild ◽  
Andrea Cammarano ◽  
David J. Wagg
Keyword(s):  
Normal Form ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Transformation Process ◽  
Second Order ◽  
Identity Transformation ◽  
Modal Testing ◽  
System Response ◽  
Weakly Nonlinear ◽  
Nonlinear Modal Analysis

In this paper we discuss a theoretical technique for decomposing multi-degree-of-freedom weakly nonlinear systems into a simpler form — an approach which has parallels with the well know method for linear modal analysis. The key outcome is that the system resonances, both linear and nonlinear are revealed by the transformation process. For each resonance, parameters can be obtained which characterise the backbone curves, and higher harmonic components of the response. The underlying mathematical technique is based on a near identity normal form transformation. This is an established technique for analysing weakly nonlinear vibrating systems, but in this approach we use a variation of the method for systems of equations written in second-order form. This is a much more natural approach for structural dynamics where the governing equations of motion are written in this form as standard practice. In fact the first step in the method is to carry out a linear modal transformation using linear modes as would typically done for a linear system. The near identity transform is then applied as a second step in the process and one which identifies the nonlinear resonances in the system being considered. For an example system with cubic nonlinearities, we show how the resulting transformed equations can be used to obtain a time independent representation of the system response. We will discuss how the analysis can be carried out with applied forcing, and how the approximations about response frequencies, made during the near-identity transformation, affect the accuracy of the technique. In fact we show that the second-order normal form approach can actually improve the predictions of sub- and super-harmonic responses. Finally we comment on how this theoretical technique could be used as part of a modal testing approach in future work.

Analysis on the Performance of Deckwetness of Sandglass-type and Cylindrical Floating Bodies

2016 ◽  
Vol 60 (03) ◽  
pp. 145-155
Author(s):  
Ya-zhen Du ◽  
Wen-hua Wang ◽  
Lin-lin Wang ◽  
Yu-xin Yao ◽  
Hao Gao ◽  
...  
Keyword(s):  
Transfer Functions ◽  
Linear Response Theory ◽  
Second Order ◽  
Irregular Waves ◽  
Wave Loads ◽  
Floating Bodies ◽  
Drift Motion ◽  
First Order ◽  
Series Of Experiments ◽  
Deck Wetness

In this paper, the influence of the second-order slowly varying loads on the estimation of deck wetness is studied. A series of experiments related to classic cylindrical and new sandglass-type Floating Production, Storage, and Offloading Unit (FPSO) models are conducted. Due to the distinctive configuration design, the sand glass type FPSO model exhibits more excellent deck wetness performance than the cylindrical one in irregular waves. Based on wave potential theory, the first-order wave loads and the full quadratic transfer functions of second-order slowly varying loads are obtained by the frequency-domain numerical boundary element method. On this basis, the traditional spectral analysis only accounting for the first-order wave loads and time-domain numerical simulation considering both the first-order wave loads and nonlinear second-order slowly varying wave loads are employed to predict the numbers of occurrence of deck wetness per hour of the two floating models, respectively. By comparing the results of the two methods with experimental data, the shortcomings of traditional method based on linear response theory emerge and it is of great significance to consider the second-order slowly drift motion response in the analysis of deck wetness of the new sandglass-type FPSO.

Extreme Response of Very Large Floating Structure Considering Second-Order Hydroelastic Effects in Multidirectional Irregular Waves

2010 ◽  
Vol 132 (4) ◽  
Author(s):  
Xujun Chen ◽  
Torgeir Moan ◽  
Shixiao Fu
Keyword(s):  
Second Order ◽  
Irregular Waves ◽  
Bending Moments ◽  
Floating Structure ◽  
First Order ◽  
Fluid Forces ◽  
Principal Coordinates ◽  
Vertical Displacements ◽  
Exciting Forces ◽  
Nonlinear Fluid

Hydroelasticity theory, considering the second-order fluid forces induced by the coupling of first-order wave potentials, is introduced briefly in this paper. Based on the numerical results of second-order principal coordinates induced by the difference-frequency and sum-frequency fluid forces in multidirectional irregular waves, the bending moments, as well as the vertical displacements of a floating plate used as a numerical example are obtained in an efficient manner. As the phase angle components of the multidirectional waves are random variables, the principal coordinates, the vertical displacements, and the bending moments are all random variables. Extreme values of bending moments are predicted on the basis of the theory of stationary stochastic processes. The predicted linear and nonlinear results of bending moments show that the influences of nonlinear fluid forces are different not only for the different wave phase angles, but also for the different incident wave angles. In the example very large floating structure (VLFS) considered in this paper, the influence of nonlinear fluid force on the predicted extreme bending moment may be as large as 22% of the linear wave exciting forces. For an elastic body with large rigidity, the influence of nonlinear fluid force on the responses may be larger than the first-order exciting forces and should be considered in the hydroelastic analysis.

Subharmonic resonance of short internal standing waves by progressive surface waves

1996 ◽  
Vol 321 ◽  
pp. 217-233 ◽  
Author(s):  
D. F. Hill ◽  
M. A. Foda
Keyword(s):  
Surface Wave ◽  
Internal Wave ◽  
Surface Waves ◽  
Internal Waves ◽  
Growth Rates ◽  
Evolution Equations ◽  
Standing Waves ◽  
Second Order ◽  
Inviscid Limit ◽  
Initial Growth

Experimental evidence and a theoretical formulation describing the interaction between a progressive surface wave and a nearly standing subharmonic internal wave in a two-layer system are presented. Laboratory investigations into the dynamics of an interface between water and a fluidized sediment bed reveal that progressive surface waves can excite short standing waves at this interface. The corresponding theoretical analysis is second order and specifically considers the case where the internal wave, composed of two oppositely travelling harmonics, is much shorter than the surface wave. Furthermore, the analysis is limited to the case where the internal waves are small, so that only the initial growth is described. Approximate solution to the nonlinear boundary value problem is facilitated through a perturbation expansion in surface wave steepness. When certain resonance conditions are imposed, quadratic interactions between any two of the harmonics are in phase with the third, yielding a resonant triad. At the second order, evolution equations are derived for the internal wave amplitudes. Solution of these equations in the inviscid limit reveals that, at this order, the growth rates for the internal waves are purely imaginary. The introduction of viscosity into the analysis has the effect of modifying the evolution equations so that the growth rates are complex. As a result, the amplitudes of the internal waves are found to grow exponentially in time. Physically, the viscosity has the effect of adjusting the phase of the pressure so that there is net work done on the internal waves. The growth rates are, in addition, shown to be functions of the density ratio of the two fluids, the fluid layer depths, and the surface wave conditions.

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