Nonlinear Forcing Term Research Articles

Отсутствие математически обоснованных критериев в используемых крупнейшими мировыми центрами NOAA (USA) и ECMWF (UK) моделях прогноза ветрового волнения океана, основанных на численном моделировании кинетического уравнения Хассельмана, привело к ошибочной иерархии и аппроксимации членов нелинейного взаимодействия, ветровой накачки и затухания волн за счет обрушения волн. Существующие операционные модели ветровых волн работают в рамках парадигмы регулируемого «черного ящика», требуя каждый раз перенастройки.

The absence of mathematically justified criteria in the models of prediction of wind waves of the ocean, used by the world’s largest centers NOAA (USA) and ECMWF (UK), based on numerical modeling of the Hasselmann kinetic equation, led to erroneous hierarchy and erroneous nonlinear interaction approximation, wind forcing and waves dissipation terms due to wave-breaking. Existing models of wind waves operate in the paradigm of the adjustable «black box», each time requiring reconfiguration. On the basis of numerical simulation, we were able to construct a model, taking into account the wind forcing of the power type in combination with the «implicit» dissipation.

Wave-Packet Models for Jet Dynamics and Sound Radiation

Organized structures in turbulent jets can be modeled as wavepackets. These are characterized by spatial amplification and decay, both of which are related to stability mechanisms, and they are coherent over several jet diameters, thereby constituting a noncompact acoustic source that produces a distinctive directivity in the acoustic field. In this review, we use simplified model problems to discuss the salient features of turbulent-jet wavepackets and their modeling frameworks. Two classes of model are considered. The first, that we refer to as kinematic, is based on Lighthill's acoustic analogy, and allows an evaluation of the radiation properties of sound-source functions postulated following observation of jets. The second, referred to as dynamic, is based on the linearized, inhomogeneous Ginzburg–Landau equation, which we use as a surrogate for the linearized, inhomogeneous Navier–Stokes system. Both models are elaborated in the framework of resolvent analysis, which allows the dynamics to be viewed in terms of an input–ouput system, the input being either sound-source or nonlinear forcing term, and the output, correspondingly, either farfield acoustic pressure fluctuations or nearfield flow fluctuations. Emphasis is placed on the extension of resolvent analysis to stochastic systems, which allows for the treatment of wavepacket jitter, a feature known to be relevant for subsonic jet noise. Despite the simplicity of the models, they are found to qualitatively reproduce many of the features of turbulent jets observed in experiment and simulation. Sample scripts are provided and allow calculation of most of the presented results.

Reconsideration of spanwise rotating turbulent channel flows via resolvent analysis

We study the effect of spanwise rotation in turbulent channel flow at both low and high Reynolds numbers by employing the resolvent formulation proposed by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382). Under this formulation, the nonlinear terms in the Navier–Stokes equations are regarded as a forcing that acts upon the remaining linear dynamics to generate the turbulent velocity field in response. A gain-based decomposition of the forcing–response transfer function across spectral space yields models for highly amplified flow structures, or modes. Unlike linear stability analysis, this enables targeted analyses of the effects of rotation on high-gain modes that serve as useful low-order models for dynamically important coherent structures in wall-bounded turbulent flows. The present study examines a wide range of rotation rates. A posteriori comparisons at low Reynolds number ($\mathit{Re}_{\unicode[STIX]{x1D70F}}=180$) demonstrate that the resolvent formulation is able to quantitatively predict the effect of varying spanwise rotation rates on specific classes of flow structure (e.g. the near-wall cycle) as well as energy amplification across spectral space. For fixed inner-normalized rotation number, the effects of rotation at varying friction Reynolds numbers appear to be similar across spectral space, when scaled in outer units. We also consider the effects of rotation on modes with varying speed (i.e. modes that are localized in regions of varying mean shear), and provide suggestions for modelling the nonlinear forcing term.

Strongly damped wave equations with Stepanov type nonlinear forcing term

In this paper we investigate the existence and uniqueness of weighted pseudo almost automorphic mild solution for a class of strongly damped wave equations where the semilinear forcing term is a Stepanov weighted pseudo almost automorphic function.

Peak response of frictional tuned mass dampers optimally designed to white noise base acceleration

Abstract This study investigates the characteristics of stochastic responses of a system having frictional tuned mass dampers (FTMDs) optimally designed to white noise base acceleration. First, with the aim of finding the optimum parameters, a statistical linearization technique is employed, in which a nonlinear force term is replaced with linearized one that is equivalent in a statistical sense. In order to improve accuracy in predicting root-mean-square (RMS) response of FTMD, this study adopted a statistical nonlinearization technique in which original nonlinear system is replaced with other already-solved nonlinear system. A correction factor defined as the ratio of RMS displacements between nonlinear and equivalent linear systems is also proposed together with an explicit formula for the peak response factor. Numerical simulations were conducted to examine the accuracy of the correction and peak factors proposed.

Dynamical behaviors of solutions to nonlinear wave equations with vanishing local damping and Wentzell boundary conditions

We are concerned with dynamical behaviors of solutions to nonlinear damped wave equations with nonlinear dampings and force terms, and subject to Wentzell boundary conditions which can be used to describe, for instance, the boundary behavior of a vibrating elastic body (resp. membrane) coated (resp. edged) with a thin layer (resp. coil) of high rigidity. Here the internal dampings are only assumed to be locally distributed and, especially, may disappear gradually over time. We find a new and effective method to overcome all the difficulties caused by the interplay of vanishing localized dampings, Wentzell boundary conditions, as well as nonlinear force terms. Ideal uniform decay rates of solution energies are obtained in terms of the exponents associated with the time-varying damping. Our result shows that the dynamical behavior of solutions is clearly stable without any bifurcation and chaos. To illustrate our theoretical results, we provide some numerical simulations.

Development of a projection-based SPH method for numerical wave flume with porous media of variable porosity

The paper presents a numerical wave flume with porous media of variable porosity. A fully-Lagrangian computational method is developed for simulation of fluid flow interactions with porous media based on a projection-based (or Incompressible) SPH (Smoothed Particle Hydrodynamics) method. The continuity and Navier-Stokes equations are reformulated with incorporation of volume fraction, based on two-phase mixture theory. The proposed method benefits from a set of enhanced schemes and is therefore referred to as Enhanced Incompressible SPH (ISPH). The effect of porous media is represented by considering linear and nonlinear resistance force terms as well as incorporation of a newly-derived source term of PPE (Poisson Pressure Equation). The results are characterized by smooth and continuous pressure fields, as well as regular particle distributions at the fluid-porous media interface in absence of any artificial smoothing schemes. Since the local volume fraction of fluid phase particles is consistently calculated based on the volume fraction of surrounding porous phase particles, the method is capable of accurately reproducing fluid flow interactions with spatially varying porous media. Validations are made through a set of benchmark tests including fluid flow in a U-tube with uniform and multi-layered porous media, rapid seepage flows in rockfill dam, solitary wave attenuation over a porous bed, solitary wave interaction with a submerged porous structure and wave propagation over a submerged triangular permeable bar.

Numerical solution of linear and nonlinear hyperbolic telegraph type equations with variable coefficients using shifted Jacobi collocation method

The telegraph equation is one of the important equations of mathematical physics. In this work, a spectral collocation scheme is proposed for the numerical solutions of one- and two-dimensional linear telegraph equations and telegraph equations with nonlinear forcing term. The homogeneous initial and boundary conditions are satisfied exactly by expanding the unknown variable using polynomial bases of functions which are built upon the Jacobi polynomials. The suggested scheme is successfully developed for the aforementioned problem with nonhomogeneous data. Extensive numerical experiments are presented to verify the efficiency of the proposed scheme.

Effect of Nonthermal Electrons and Ions on Longitudinal Dust Acoustic Solitary Waves in a Strongly Coupled Dusty Plasma

The propagation properties of longitudinal dust acoustic waves (LDAWs) have been investigated in a strongly coupled dusty plasma containing negatively charged dust fluid, and nonthermal electrons and ions using the generalized hydrodynamic model in “kinetic regime.” The nonlinear dynamics of the system are shown to be governed by a modified KdV equation in the presence of viscoelastic forces. The modified KdV equation so obtained is found to possess a nonlinear force term and a linear damping term. The effects of various physical parameters on the phase velocity and coefficients of modified KdV equation have been analyzed. Furthermore, the effect of nonthermality of electrons and ions is realized in the numerical solution of the equation so obtained to study the time evolution of LDAWs under the influence of dust correlation forces.

Integral Manifolds of a Class of Delayed Partial Differential Equations with Nondense Domain

ABSTRACTWe prove the existence of some integral manifolds for the nonautonomous delayed partial differential equationwhere (A,D(A)) satisfies the Hille–Yosida condition, is a family of operators in satisfying some measurability and boundedness conditions and the nonlinear forcing term f satisfies ∥f(t,ϕ)−f(t,ψ)∥≤φ(t)∥ϕ−ψ∥𝒞, where φ belongs to an admissible space and ϕ, ψ∈𝒞: = C([−r,0],X). To achieve our target, we resort to the properties of admissible spaces, the extrapolation theory, the characterization of exponential dichotomy and exponential trichotomy, Lyapunov–Perron method, and other technical structures. Finally, an example is given to illustrate our results.

Theoretical consideration of free convective heat transfer from a round isothermal plate slightly inclined from the vertical

A semi-analytical solution of simplified Navier-Stokes and Fourier-Kirchhoff equations describing free convective heat transfer from a round isothermal surface slightly inclined from the vertical is presented. The solution is based on the assumption, typical for natural convection, that the velocity component normal to the surface is negligibly small in comparison to the tangential one. Next we neglect the nonlinear inertia force term, but more real mass continuity is taken into account in control volume approach. This assumption do not permit to use stream function. The results for a vertical round plate in the form of the boundary layer thickness and mean Nusselt number are obtained in explicit form. They are in good agreement with literature solutions for vertical rectangular or square plates. The correction function to the Nusselt number for inclined plate is obtained in the analytical integral form that is calculated numerically and compared with the experimental values. Analysis of the results for the correction function is proportional to tanα, hence heat transfer intensifies for a positive inclination. The solution analysis also suggests extreme heat transfer at some angle for a given range of Rayleigh numbers.

Traction curves for the decohesion of covalent crystals

We study, by first principles, the energy versus separation curves for the cleavage of a family of covalent crystals with the diamond and zincblende structure. We find that there is universality in the curves for different materials which is chemistry independent but specific to the geometry of the particular cleavage plane. Since these curves do not strictly follow the universal binding energy relationship (UBER), we present a derivation of an extension to this relationship that includes non-linear force terms. This extended form of UBER allows for a flexible and practical mathematical description of decohesion curves that can be applied to the quantification of cohesive zone models.

Entropy production by active particles: Coupling of odd and even functions of velocity.

Nonequilibrium stochastic dynamics of several active Brownian systems are modeled in terms of nonlinear velocity dependent force. In general, this force may consist of both even and odd functions of velocity. We derive the expression for total entropy production in such systems using the Fokker-Planck equation. The result is consistent with the expression for stochastic entropy production in the reservoir that we obtain from probabilities of time-forward and time-reversed trajectories, leading to fluctuation theorems. Numerical simulation is used to find probability distribution of entropy production, which shows good agreement with the detailed fluctuation theorem.

Improved algorithm for solving nonlinear parabolized stability equations**Project supported by the National Natural Science Foundation of China (Grant Nos. 11332007 and 11402167).

Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations (NPSE) approach has been widely used to study the stability and transition mechanisms. However, it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation (DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers.

Reconstruction of the electric field of the Helmholtz equation in three dimensions

In this paper, we rigorously investigate the truncation method for the Cauchy problem of Helmholtz equations which is widely used to model propagation phenomena in physical applications. The truncation method is a well-known approach to the regularization of several types of ill-posed problems, including the model postulated by Regińska and Regiński (2006). Under certain specific assumptions, we examine the ill-posedness of the non-homogeneous problem by exploring the representation of solutions based on Fourier mode. Then the so-called regularized solution is established with respect to a frequency bounded by an appropriate regularization parameter. Furthermore, we provide a short analysis of the nonlinear forcing term. The main results show the stability as well as the strong convergence confirmed by the error estimates in L2-norm of such regularized solutions. Besides, the regularization parameters are formulated properly. Finally, some illustrative examples are provided to corroborate our qualitative analysis.

The Evolution of Nonlinear Wave Statistics through a Variable Medium

AbstractIn coastal areas and on beaches, nonlinear effects in ocean waves are dominated by so-called triad interactions. These effects can result in large energy transfers across the wave spectrum and result in non-Gaussian wave statistics, which is important for coastal wave propagation and wave-induced transport processes. To model these effects in a stochastic wave model based on the radiative transfer equation (RTE) requires a transport equation for three-wave correlators (the bispectrum) that is compatible with quasi-homogeneous theory. Based on methods developed in optics and quantum mechanics, the authors present a general approach to derive a transport equation for higher-order correlators. The principal result of this work is a coupled set of equations consisting of the radiative transfer equation with a nonlinear forcing term and a new, generalized transport equation for bispectrum. This study discusses the implications and characteristics of the resulting equations and shows that the model contains various shallow- and deep-water asymptotes for nonlinear wave propagation as special cases.

Global observations and modeling of nonmigrating diurnal tides generated by tide‐planetary wave interactions

AbstractAdvective processes that couple planetary waves with tides have long been proposed as sources of nonmigrating diurnal tides. This paper reports observations of short‐term variability in global observations of nonmigrating tides predicted to arise from the interaction of the migrating diurnal tide (DW1) with a quasi‐stationary planetary wave number one (PW1). PW1 and tidal definitions are extracted from satellite temperatures and high‐altitude meteorological analyses. During winter months, the evolution of westward traveling diurnal tides with zonal wave number 2 (DW2) generally tracks that of strong‐amplitude stratospheric PW1. DW1 and PW1 spectra are used to compute nonlinear tidal forcing terms arising from advection. We then examine the response of a primitive equation model to the observation‐based nonlinear forcing. The model experiments indicate that meridional advection of PW1 zonal momentum by DW1 is a significant source of lower thermospheric DW2. Modeled DW2 amplitudes are very consistent with observed DW2 amplitudes when stratospheric PW1 penetrates to equatorial latitudes. The model experiments also indicate that the interaction can imprint short‐term variability associated with wintertime PW1 upon DW2 in the summer hemisphere and the lower thermosphere.

Propagation and extinction of fronts for semilinear parabolic equations on Riemannian manifolds

We study the Cauchy problem for the semilinear heat equation on Riemannian manifolds. Propagation and extinction of solutions are addressed, supposing that the nonlinear forcing term is either of KPP type or of bistable type. In particular, we highlight the influence both of sectional curvatures and of Ricci curvature on such phenomena.

Regularity of the attractor for strongly damped wave equations with nonlinearity

The long time behavior of the solutions for the strongly damped wave equation is considered with nonlinear damping, a nonlinear forcing term, and with a periodic boundary condition. We prove that the global attractor which captures all trajectories in is a compact set in . MSC:35B40, 34A35, 37L30.

Long-time behavior for a plate equation with nonlocal weak damping

This paper is devoted to the long-time behavior of solutions for a class of plate equations with nonlocal weak damping $$ u_{tt} + \Delta^2 u + g(u) + M \Big (\int_{\Omega}|\nabla u|^2 dx \Big )u_t =f\quad \mbox{in} \quad \Omega\times\mathbb{R}^{+}, $$ where $\Omega$ is a bounded domain of $\mathbb{R}^N$. Under suitable conditions on the nonlinear forcing term $g(u)$ and Kirchhoff damping coefficient $M (\int|\nabla u|^2 ),$ the existence of a global attractor with finite Hausdorff and fractal dimensions is proved.