Long-wave Model Research Articles

On Rayleigh expansion for nonlinear long water waves

We consider strongly nonlinear long waves on the surface of a homogeneous fluid layer. By modifying the formulation for the high-order spectral (HOS) method for waves in water of finite depth, we present a higher-order nonlinear system for the surface elevation and the velocity potential on the free surface to describe the two-dimensional evolution of large amplitude long waves. It is shown that the resulting system preserves the Hamiltonian structure of the Euler equations and can be transformed to the strongly nonlinear long-wave model for the depth-averaged velocity. Due to truncation of the linear dispersion relation for water waves, both the system for the surface velocity potential and that for the depth-averaged velocity are ill-posed when the order of approximation is odd and even, respectively. To avoid this ill-posedness, fully dispersive models are also proposed. Under the same order approximation, the long-wave model is found more effective for numeral studies of large amplitude long waves than the finite-depth model.

Experimental and numerical study of the propagation of focused wave groups in the nearshore zone

The propagation of focused wave groups in intermediate water depth and the shoaling zone is experimentally and numerically considered in this paper. The experiments are carried out in a two-dimensional wave flume and wave trains derived from Pierson-Moskowitz and JONSWAP spectrum are generated. The peak frequency does not change during the wave train propagation for Pierson-Moskowitz waves; however, a downshift of this peak is observed for JONSWAP waves. An energy partitioning is performed in order to track the spatial evolution of energy. Four energy regions are defined for each spectrum type. A nonlinear energy transfer between different spectral regions as the wave train propagates is demonstrated and quantified. Numerical simulations are conducted using a modified Boussinesq model for long waves in shallow waters of varying depth. Experimental results are in satisfactory agreement with numerical predictions, especially in the case of wave trains derived from JONSWAP spectrum.

Assessing the accuracy and efficiency of longwave radiative transfer models involving scattering effect with cloud optical property parameterizations

A radiative transfer model (RTM) that accurately and explicitly accounts for both absorbing and scattering effects requires a substantial amount of computational effort. In the longwave (LW) spectral regime, the atmosphere is optically opaque and absorbs a large portion of terrestrial thermal radiation. To alleviate the computational burden, clouds are assumed to be only absorptive in most general circulation models (GCMs) and their scattering effects are neglected. Using parameterizations of cloud bulk single-scattering properties derived from the latest cloud optical property models for satellite remote sensing, this study analyzes the numerical accuracy and efficiency of a variety of LW RTMs . The approaches considered in this study include the absorption approximation (AA), the absorption approximation with scattering parameterization (ASA), the two-stream approximation (2S), a hybrid two- and four-stream approximation (2/4S), and the discrete ordinate radiative transfer (DISORT) with multiple streams. These approximations are benchmarked against 128-stream DISORT calculation. After evaluating the full ranges of ice and water cloud optical and microphysical properties using these RTMs, we find that neglecting LW scattering effect causes simulation errors as large as ±15% by using the AA method. Among these RTMs, the 2/4S method provides an optimal balance between computational efficiency and accuracy, leading to the maximum cloud emissivity errors within ±5%, 25th to 75th percentile errors about ±1%, and almost zero net bias. Therefore, the 2/4S approximation is computationally accurate and yet affordable option for incorporating cloud longwave scattering effect into the radiation schemes used in weather and climate models.

The Tsunami Scenario Database of the Indonesia Tsunami Early Warning System (InaTEWS): Evolution of the Coverage and the Involved Modeling Approaches

This study reports on recent developments of the Indonesia Tsunami Early Warning System (InaTEWS), specifically the tsunami modeling components used in the system. It is a dual system: firstly, InaTEWS operates a high-resolution scenario database pre-computed with the finite element model TsunAWI; running in parallel, the system also contains a supra real-time modeling component based on the GPU-parallelized linear long-wave model easyWave, capable of dealing with events outside the database coverage. The evolution of the tsunami scenario database over time is covered in the first sections also touching on the involved capacity building efforts. Starting with a coverage of just the Sunda Arc region, the database now includes scenarios for 15 fault zones. The study is augmented by an investigation of warning products used for early warning; the estimated wave height (EWH) and the estimated time of arrival (ETA). These parameters are determined by easyWave and TsunAWI with model specific approaches. Since the numerical setup of the two models is very different, the extent of variations in warning products is investigated for a number of scenarios, where both pure database scenarios and applications to real events are considered. Finally, the performance of the system in past tsunami events is reviewed to point out major system updates.

The collective behaviour of ensembles of condensing liquid drops on heterogeneous inclined substrates

Employing a long-wave mesoscopic hydrodynamic model for the film height evolution we study ensembles of pinned and sliding drops of a volatile liquid that continuously condense onto a chemically heterogeneous inclined substrate. Our analysis combines, on the one hand, path continuation techniques to determine bifurcation diagrams for the depinning of single drops of nonvolatile liquid on single hydrophilic spots on a partially wettable substrate and, on the other hand, time simulations of growth and depinning of individual condensing drops as well as of the long-time behaviour of large ensembles of such drops. Pinned drops grow on the hydrophilic spots, depin and slide along the substrate while merging with other pinned drops and smaller drops that slide more slowly, and possibly undergo a pearling instability. As a result, the collective behaviour converges to a stationary state where condensation and outflow balance. The main features of the emerging drop size distribution can then be related to single-drop bifurcation diagrams.

Combined Numerical Model of Tsunami

A numerical model describing the dynamics of the surface gravity waves and acoustic waves induced in the ocean by small dynamic deformations of the bottom is developed. The model is based on the linear potential theory. The model represents a combination of two dynamically coupled blocks: deep-water and shallow. The deep-water block solves a three-dimensional problem of potential wave theory in the sigma-spherical coordinates; the shallow block solves a two-dimensional problem of shallow water theory in the spherical coordinates. The results of testing the numerical model using the analytical solution of the problem for the flat horizontal bottom are presented. A comparative analysis of the simulation results of tsunamis on November 15, 2006 and January 13, 2007 on the Central Kuril Islands with the use of the newly developed and conventional long-wave models is performed.

Robust low-dimensional modelling of falling liquid films subject to variable wall heating

Accurate low-dimensional models for the dynamics of falling liquid films subject to localized or time-varying heating are essential for applications that involve patterning or control. However, existing modelling methodologies either fail to respect fundamental thermodynamic properties or else do not accurately capture the effects of advection and diffusion on the temperature profile. We argue that the best-performing long-wave models are those that give the surface temperature implicitly as the solution of an evolution equation in which the wall temperature alone (and none of its derivatives) appears as a source term. We show that, for both flat and non-uniform films, such a model can be rationally derived by expanding the temperature field about its free-surface values. We test this model in linear and nonlinear regimes, and show that its predictions are in remarkable quantitative agreement with full Navier–Stokes calculations regarding the surface temperature, the internal temperature field and the surface displacement that would result from temperature-induced Marangoni stresses.

Long-wave evolution model of thermocapillary convection in an evaporating thin film of pseudoplastic fluids

Purpose By solving a long-wave evolution model numerically for power-law fluids, the authors aim to investigate the hydrodynamic and thermal characteristics of thermocapillary flow in an evaporating thin liquid film of pseudoplastic fluid. Design/methodology/approach The flow reversal attributed to the thermocapillary action is manifestly discernible through the streamline plots. Findings The thermocapillary strength is closely related to the viscosity of the fluid, besides its surface tension. The thermocapillary flow prevails in both Newtonian and pseudoplastic fluids at a large Marangoni number and the thermocapillary effect is more significant in the former. The overestimate in the Newtonian fluid is larger than that in the pseudoplastic fluid, owing to the shear-thinning characteristics of the latter. Originality/value This study provides insights into the essential attributes of the underlying flow characteristics in affecting the thermal behavior of thermocapillary convection in an evaporating thin liquid film of the shear-thinning fluids.

Relative periodic solutions in conducting liquid films flowing down vertical fibres

The dynamics of a conducting liquid film flowing down a cylindrical fibre, subjected to a radial electric field, is investigated using a long-wave model (Ding et al., J. Fluid Mech., vol. 752, 2014, p. 66). In this study, to account for the complicated interactions between droplets, we consider two large droplets in a periodic computational domain and find two distinct types of travelling wave solutions, which consist of either two identical droplets (type I) or two slightly different droplets (type II). Both are ‘relative’ equilibria, i.e. steady in a frame moving at their phase speed, and are stable in smaller domains when the electric field is weak. We also study relative periodic orbits, i.e. temporally recurrent dynamic solutions of the system. In the presence of the electric field, we show how these invariant solutions are linked to the dynamics, where the system can evolve into one of the steady travelling wave states, into an oscillatory state, or into a ‘singular structure’ (Wray et al., J. Fluid Mech., vol. 735, 2013, pp. 427–456; Ding et al., J. Fluid Mech., vol. 752, 2014, p. 66). We find that the oscillation between two similarly sized large droplets in the oscillatory state is well represented by relative periodic orbits. Varying the electric field strength, we demonstrate that relative periodic solutions arise as the dynamically important solution once the type-I or type-II travelling wave solutions lose stability. Oscillation can be either enhanced or impeded as the electric field’s strength increases. When the electric field is strong, no relative periodic solutions are found and a spike-like singular structure is observed. For the case where the electric field is not present, the oscillation is instead caused by the interaction between a large droplet and a nearby much smaller droplet. We show that this oscillation phenomenon originates from the instability of the type-I travelling wave solution in larger domains, and that the oscillatory state can again be represented by an exact relative periodic orbit. The relative periodic orbit solution is also compared with experimental study for this case. The present study demonstrates that the relative periodic solutions are better at capturing the wave speed and oscillatory dynamics than the travelling wave solutions in the unsteady flow regime.

Capillary–gravity waves on a dielectric fluid of finite depth under normal electric field

Abstract In this work we consider two-dimensional capillary–gravity waves propagating under the influence of a vertical electric field on a dielectric of finite depth bounded above by a perfectly conducting and hydrodynamically passive fluid. Both linear and weakly nonlinear theories are developed, and long-wave model equations are derived based on the analyticity of the Dirichlet–Neumann operator. Fully nonlinear computations are carried out by using a time-dependent conformal mapping method. Solitary waves are found, and their stability characteristics subject to longitudinal perturbations are studied numerically. The shedding of stable solitary waves is achieved by moving a Gaussian pressure on the free surface with the speed close to a phase speed minimum and removing the pressure after a period of time. The novel result shows that a depression bright solitary wave and an elevation generalized solitary wave co-exist in the solitary-wave excitation.

Numerical dispersion in non-hydrostatic modeling of long-wave propagation

Numerical discretization with a finite-difference scheme is known to introduce frequency dispersion in depth-integrated long-wave models commonly used in tsunami research and hazard mapping. While prior studies on numerical dispersion focused on the linear shallow-water equations, we include the non-hydrostatic pressure and vertical velocity through a Keller box scheme and investigate the properties of the resulting system in relation to a hydrostatic model. Fourier analysis of the discretized governing equations gives rise to a dispersion relation in terms of the water-depth parameter as well as the time step, grid size, and wave direction. The dispersion relation is illustrated by its lead-order approximation derived from Taylor series expansions as well as numerical experiments involving standing and progressive waves with uniform water depth. Interaction between the spatial discretization and non-hydrostatic terms leads to significant reduction of numerical dispersion outside the shallow-water range. Numerical dispersion also decreases for wave propagation oblique to the computational grid due to effective increase in spatial resolution. The time step, which counteracts numerical dispersion from spatial discretization, only has secondary effects within the applicable range of Courant numbers. Since the governing equations of the non-hydrostatic model derived from the Keller box scheme tend to underestimate dispersion in shoaling water, the numerical effects are complementary in producing a solution closer to Airy wave theory. A case study of the 2011 Tohoku tsunami illustrates the grid sensitivity and convergence properties in real-world applications. A properly selected grid size can achieve an accurate description of wave propagation over a wide range of water-depth parameters across the ocean.

Long wave dynamics for a liquid CO2 lake in the deep ocean

A long-wave model for the evolution of long waves at the interface of a deep and a shallow fluid is put forward. The model allows for a uniform stream in one of the layers, and the existence of interfacial capillarity. The model can be used to study the dynamics of the interface between liquid CO2 and seawater in the deep ocean, including the evolution of the hydrate layer.If restricted to unidirectional waves the model has the form of a Benjamin-type equation found by Benjamin [2]. Steady periodic solutions of the Benjamin equation are found using a numerical bifurcation code based on a pseudo-spectral projection. The bifurcation patterns are complex, with some branches featuring turning points and secondary bifurcations.

Pseudo bond graph tunnel greenhouse model with accurate longwave/shortwave radiations model

ABSTRACT This research work is focused on the construction of an accurate longwave/shortwave radiation model on a tunnel greenhouse pseudo-bond graph model, widely used in Tunisia. This model includes sun position, useful incoming solar radiation model, sky longwave radiation model, inside longwave and shortwave radiation model. The key idea is to use bond graphs allowing a lumped modelling approach which is suitable for control applications. Furthermore, an evaluation of some longwave radiative model components was made, noting that these components are particularly sensitive regarding to the thermal behaviour of the model. Experimental tunnel greenhouse data are used as validation elements for the present model with globally good results. A comparative study was also performed between the present model and a previous bond graph model containing a simplistic radiative model. Practical simulation results show a clear improvement compared with the previous model.

Hydrodynamic Models and Confinement Effects by Horizontal Boundaries

Confinement effects by rigid boundaries in the dynamics of ideal fluids are considered from the perspective of long-wave models and their parent Euler systems, with the focus on the consequences of establishing contacts of material surfaces with the confining boundaries. When contact happens, we show that the model evolution can lead to the dependent variables developing singularities in finite time. The conditions and the nature of these singularities are illustrated in several cases, progressing from a single layer homogeneous fluid with a constant pressure free surface and flat bottom, to the case of a two-fluid system contained between two horizontal rigid plates, and finally, through numerical simulations, to the full Euler stratified system. These demonstrate the qualitative and quantitative predictions of the models within a set of examples chosen to illustrate the theoretical results.

Inclined film boiling: Film stability and heat transfer

An inertially modified long-wave model is used to analyze the stability of inclined saturated film boiling. By performing quasi-parallel linear stability analysis on this model and combining the result with a heat-transfer model, an explicit expression for the heat-transfer coefficient is obtained. The results appear to fit all relevant data for film-boiling heat-transfer coefficient within an error of 15%.

Rule of long-wave radiation in enclosed building space

To study the trend of long-wave radiation in urban spaces surrounded by dense buildings, a calculation model for long-wave radiation was established. A simplified calculation of view factor was built in the model. The model of long-wave radiation formed by direct and reflected components and Kirchhoff's current law was used to calculate the reflected radiation. The accuracy of model was verified using the measured experimental data obtained from three locations in Xi'an, Sanya, and Harbin. The model was used to calculate the long-wave radiation on the external surface of a building in both enclosed and open spaces. Using the long-wave radiation model, the rule of long-wave radiation in an enclosed space was obtained by analyzing different forms of building enclosure.

Nonlinear waves in marginally stable flows of Antarctic Bottom Water

In this paper, we study internal waves and stratified flows in the Romanche Fracture Zone of the Mid Atlantic Ridge. Theoretical analysis involves mathematical formulation of stratified flows which uses LADCP- and CTD-data obtained from field observation in the cruises of RV Akademik Sergey Vavilov. We discuss the 2.5-layer model of the fluid with exponential stratification of both layers. The long-wave approximate model describing travelling waves is constructed by means of scaling procedure with a small Boussinesq parameter. It is demonstrated that solitary wave regimes can be affected by the Kelvin — Helmholtz instability arising due to interfacial velocity shear in upstream flow.

Two-dimensional pulse dynamics and the formation of bound states on electrified falling films

The flow of an electrified liquid film down an inclined plane wall is investigated with the focus on coherent structures in the form of travelling waves on the film surface, in particular, single-hump solitary pulses and their interactions. The flow structures are analysed first using a long-wave model, which is valid in the presence of weak inertia, and second using the Stokes equations. For obtuse angles, gravity is destabilising and solitary pulses exist even in the absence of an electric field. For acute angles, spatially non-uniform solutions exist only beyond a critical value of the electric field strength; moreover, solitary-pulse solutions are present only at sufficiently high supercritical electric-field strengths. The electric field increases the amplitude of the pulses, can generate recirculation zones in the humps and alters the far-field decay of the pulse tails from exponential to algebraic with a significant impact on pulse interactions. A weak-interaction theory predicts an infinite sequence of bound-state solutions for non-electrified flow, and a finite set for electrified flow. The existence of single-hump pulse solutions and two-pulse bound states is confirmed for the Stokes equations via boundary-element computations. In addition, the electric field is shown to trigger a switch from absolute to convective instability, thereby regularising the dynamics, and this is confirmed by time-dependent simulations of the long-wave model.

Internal waves in marginally stable abyssal stratified flows

Abstract. The problem on internal waves in a weakly stratified two-layer fluid is studied semi-analytically. We discuss the 2.5-layer fluid flows with exponential stratification of both layers. The long-wave model describing travelling waves is constructed by means of a scaling procedure with a small Boussinesq parameter. It is demonstrated that solitary-wave regimes can be affected by the Kelvin–Helmholtz instability arising due to interfacial velocity shear in upstream flow.

Thermocapillary instability of self-rewetting films on vertical fibers

The self-rewetting fluid has a quadratic temperature dependence of surface tension with a well-defined minimum, which is distinct from that of normal fluids whose surface tension decreases linearly with temperature. The instability of a self-rewetting film on a vertical hot or cold fiber is investigated using the long-wave asymptotic model, which discovers that the role of the thermocapillarity varies in three different regimes. In the normal regime, the behavior of this fluid is similar to that of a normal fluid while in another anomalous regime, it shows a reverse feature of instability. It is this reverse behavior that guarantees the potentials of this fluid in heat transfer applications. Transient numerical simulations are also performed to be compared with the linear theory and to explore the saturate states as well as the breakup for this special fluid. Apart from supporting the conclusions obtained in the linear analysis, noticeably, the numerical results predict a flatter saturate shape and a weaker breakup in the anomalous regime.