Showing papers on "Boussinesq approximation (buoyancy) published in 2022"
TL;DR: In this article , the authors used Lie symmetry analysis and generalized Kudryashov method to obtain the invariant solutions of the Boussinesq-Burgers system.
Abstract: Water waves, a common natural phenomenon, have been influential in various fields, such as energy development, offshore engineering, mechanical engineering, and hydraulic engineering. To describe the shallow water waves near an ocean coast or in a lake, we use the (1 + 1)-dimensions Boussinesq–Burgers system. By means of Lie symmetry analysis, symmetry groups and infinitesimal generators are obtained for the (1 + 1)-dimension Boussinesq–Burgers system. For the sake of finding the invariant solutions of the Boussinesq–Burgers system, the optimal one-dimensional subalgebra system is computed. Furthermore, using similarity reduction and the generalized Kudryashov method, we attain the abundant wave solutions of the Boussinesq–Burgers system presented in this research paper. Additionally, the exact solutions, which illustrate the effectiveness of the proposed method, also reveal the physical interpretation of the nonlinear models. To demonstrate the significance of interaction phenomena, dynamical behaviors of some attained solutions are depicted geometrically and theoretically through suitable parameter values. Consequently, kink, singular, periodic, solitary wave solutions, and their elastic nature have been shown to validate these solutions with physical phenomena. With the aid of the obtained results, the researchers could gain an understanding of the different modes of shallow water waves nearby an ocean beach. The computational work ascertained that the imposed methods are sturdy, precise, modest, and widely applicable.
25 citations
TL;DR: In this article , the authors developed two new types of (1+1)-dimensional nonlocal Boussinesq equations to represent more situations in complex water waves by applying an ansatz method.
21 citations
TL;DR: In this paper , an exact travelling wave solution for the (2 + 1)-dimensional Boussinesq equation was derived by applying another analytical method, i.e., the (G′G′+G+A)-expansion approach.
13 citations
TL;DR: In this article , a simplified version of the manifold asymptotic analyses available in the literature is discussed and a simplified Boussinesq approximated governing equation for buoyant flows is presented.
9 citations
TL;DR: In this article , the authors examined the compound impact of electromagnetic induced force and internal heat source on a tangent hyperbolic fluid in quadratic Boussinesq approximation.
Abstract: The current investigation is to examine the compound impact of electromagnetic induced force and internal heat source on a tangent hyperbolic fluid in quadratic Boussinesq approximation. The current hyperbolic tangent liquid flow and heat transport formulation model adequately predicts and characterizes the shear-stricken event. The nonlinear dimensionless heat transfer flow equations are solved completely using weighted residual solution procedures coupled with Galerkin approximation integration approach. The results in the table and graphs revealed that the magnetic field strength has a substantial impact on the fluid flow and heat propagation, as well as the internal heat source. Therefore, the entropy generation is optimized through an enhanced thermodynamic equilibrium and adequate control of heat generating terms and energy loss.
7 citations
TL;DR: In this paper , a residual-based a posteriori error estimator for the 2D and 3D versions of the associated mixed finite element schemes is proposed for the Boussinesq and the Oberbeck-Boussineq models, along with suitable Helmholtz decomposition in nonstandard Banach spaces.
Abstract: Abstract In this paper we consider Banach spaces-based fully-mixed variational formulations recently proposed for the Boussinesq and the Oberbeck–Boussinesq models, and develop reliable and efficient residual-based a posteriori error estimators for the 2D and 3D versions of the associated mixed finite element schemes. For the reliability analysis, we employ the global inf-sup condition for each sub-model, namely Navier–Stokes and heat equations in the case of Boussinesq, along with suitable Helmholtz decomposition in nonstandard Banach spaces, the approximation properties of the Raviart–Thomas and Clément interpolants, further regularity on the continuous solutions, and small data assumptions. In turn, the efficiency estimates follow from inverse inequalities and the localization technique through bubble functions in adequately defined local Lp spaces. Finally, several numerical results including natural convection in 3D differentially heated enclosures, are reported with the aim of confirming the theoretical properties of the estimators and illustrating the performance of the associated adaptive algorithm.
6 citations
TL;DR: In this paper , the authors compare various statistical properties of the Boussinesq, AA and FC simulations in 2-D simulations and show that in the infinite Prandtl number case, solving the fully compressible (FC) equations of convection with a realistic equation of state (EoS) is however not much more difficult or numerically challenging than solving the approximate cases.
Abstract: SUMMARY The numerical simulations of convection inside the mantle of the Earth or of terrestrial planets have been based on approximate equations of fluid dynamics. A common approximation is the neglect of the inertia term which is certainly reasonable as the Reynolds number of silicate mantles, or their inverse Prandtl number, are infinitesimally small. However various other simplifications are made which we discuss in this paper. The crudest approximation that can be done is the Boussinesq approximation (BA) where the various parameters are constant and the variations of density are only included in the buoyancy term and assumed to be proportional to temperature with a constant thermal expansivity. The variations of density with pressure and the related physical consequences (mostly the presence of an adiabatic temperature gradient and of dissipation) are usually accounted for by using an anelastic approximation (AA) initially developed for astrophysical and atmospheric situations. The BA and AA cases provide simplified but self-consistent systems of differential equations. Intermediate approximations are also common in the geophysical literature although they are invariably associated with theoretical inconsistencies (non-conservation of total energy, non-conservation of statistically steady state heat flow with depth, momentum and entropy equations implying inconsistent dissipations). We show that, in the infinite Prandtl number case, solving the fully compressible (FC) equations of convection with a realistic equation of state (EoS) is however not much more difficult or numerically challenging than solving the approximate cases. We compare various statistical properties of the Boussinesq, AA and FC simulations in 2-D simulations. We point to an inconsistency of the AA approximation when the two heat capacities are assumed constant. We suggest that at high Rayleigh number, the profile of dissipation in a convective mantle can be directly related to the surface heat flux. Our results are mostly discussed in the framework of mantle convection but the EoS we used is flexible enough to be applied for convection in icy planets or in the inner core.
5 citations
TL;DR: In this paper , the formation of undular bores in Riemann problems of the good generalized Kaup-Boussinesq equation is investigated by Whitham modulation theory.
5 citations
TL;DR: In this article , a coupled mathematical model based on the nonlinear Boussinesq equation (BE) for shallow water waves is developed to investigate the influence of the periodic non-linear long waves inside the irregular domain.
4 citations
TL;DR: In this article , the authors investigated the interactions of waves governed by a Boussinesq system with a partially immersed body allowed to move freely in the vertical direction, and showed that the whole system of equations can be reduced to a transmission problem with transmission conditions given in terms of the vertical displacement of the object and of the average horizontal discharge beneath it; these two quantities are in turn determined by two nonlinear ODEs with forcing terms coming from the exterior wave-field.
4 citations
TL;DR: In this article, a linear stability analysis is conducted for horizontal natural convection under a Gay-Lussac (GL) type approximation in a relatively shallow enclosure cavity, and the GL type approximation is developed based on extending density variations to the advection term as well as gravity term through the momentum equation.
TL;DR: In this paper , the Boussinesq-intermediate long-wave model was proposed for algebraic Rossby solitary wave models in stratified fluids, and several common conservation laws were proposed for exploring the properties of the model.
TL;DR: In this paper , the authors studied the stability of the two-dimensional Navier-Stokes Boussinesq system around the Couette flow with small viscosity ν and small thermal diffusion μ.
TL;DR: In this paper , the authors considered a Whitham-Boussinesq-type system for modeling surface water waves of an inviscid incompressible fluid layer and proved that the system is locally wellposed on the time scale of order O 1/ϵ$$ \mathcal{O}\left(1/\sqrt{\epsilon}\right) $$ , where ϵ>0$$ \epsilon >0 $$ is the shallowness parameter measuring the ratio of amplitude of the wave to mean depth of fluid.
Abstract: In this paper, we consider a Whitham–Boussinesq‐type system modeling surface water waves of an inviscid incompressible fluid layer. The system describes the evolution with time of surface waves of a liquid layer in the two‐dimensional physical space. Using fixed point argument, we prove that the system is locally well‐posed on the time scale of order O1/ϵ$$ \mathcal{O}\left(1/\sqrt{\epsilon}\right) $$ , where ϵ>0$$ \epsilon >0 $$ is the shallowness parameter measuring the ratio of amplitude of the wave to mean depth of fluid. We also show that the solution to the Whitham–Boussinesq system approximates the solution of a Boussinesq system on the time scale of order O1/ϵ$$ \mathcal{O}\left(1/\sqrt{\epsilon}\right) $$ .
TL;DR: In this paper , the global existence and stability of perturbation of 2D Boussinesq equations with partial dissipation near hydrostatic equilibrium on a flat strip Ω≔T×[0,1] are obtained with the help of a time-weight energy estimate.
Abstract: The global existence and stability of perturbation of 2D Boussinesq equations with partial dissipation near hydrostatic equilibrium on a flat strip Ω≔T×[0,1] are obtained with the help of a time-weight energy estimate. Meanwhile, we also establish the explicit decay rates of the solution of 2D Boussinesq equations.
TL;DR: In this article, the global existence and stability of perturbation of 2D Boussinesq equations with partial dissipation near hydrostatic equilibrium on a flat strip Ω ≔ T × [ 0, 1 ] are obtained with the help of a time-weight energy estimate.
Abstract: The global existence and stability of perturbation of 2D Boussinesq equations with partial dissipation near hydrostatic equilibrium on a flat strip Ω ≔ T × [ 0 , 1 ] are obtained with the help of a time-weight energy estimate. Meanwhile, we also establish the explicit decay rates of the solution of 2D Boussinesq equations.
TL;DR: In this paper , the authors considered the higher-order/extended Boussinesq equations over a flat bottom topography in the well-known long wave regime and provided an existence and uniqueness of solution on a relevant time scale of order 1/√ε and showed that the solution's behavior is close to the solution of the water waves equations with a better precision corresponding to initial data.
Abstract: This study deals with higher-order asymptotic equations for the water-waves problem. We considered the higher-order/extended Boussinesq equations over a flat bottom topography in the well-known long wave regime. Providing an existence and uniqueness of solution on a relevant time scale of order 1/√ε and showing that the solution’s behavior is close to the solution of the water waves equations with a better precision corresponding to initial data, the asymptotic model is well-posed in the sense of Hadamard. Then we compared several water waves solitary solutions with respect to the numerical solution of our model. At last, we solve explicitly this model and validate the results numerically.
TL;DR: In this paper , the problem of gravity surface waves for the ideal fluid model in (2+1)-dimensional case was studied, and a systematic procedure for deriving the Boussinesq equations for a prescribed relationship between the orders of four expansion parameters, the amplitude parameter $\alpha$, the long-wavelength parameter $\beta$, the transverse wavelength parameter $\gamma, and the bottom variation parameter $\delta$.
Abstract: We study the problem of gravity surface waves for the ideal fluid model in (2+1)-dimensional case. We apply a systematic procedure for deriving the Boussinesq equations for a prescribed relationship between the orders of four expansion parameters, the amplitude parameter $\alpha$, the long-wavelength parameter $\beta$, the transverse wavelength parameter $\gamma$, and the bottom variation parameter $\delta$. We also take into account surface tension effects when relevant. For all considered cases, the (2+1)-dimensional Boussinesq equations can not be reduced to a single nonlinear wave equation for surface elevation function. On the other hand, they can be reduced to a single, highly nonlinear partial differential equation for an auxiliary function $f(x,y,t)$ which determines the velocity potential but is not directly observed quantity. The solution $f$ of this equation, if known, determines the surface elevation function. We also show that limiting the obtained the Boussinesq equations to (1+1)-dimensions one recovers well-known cases of the KdV, extended KdV, fifth-order KdV, and Gardner equations.
TL;DR: In this article , the authors used three different techniques to obtain abundant analytical optical soliton solutions to the (3+1)-Boussinesq equation (BE) for the first time.
TL;DR: In this paper , a finite element model for depth integrated form of Boussinesq equations is presented, where the equations are solved on an unstructured triangular mesh using standard Galerkin method with mixed interpolation scheme.
TL;DR: In this article , the nonlocal symmetry of the integrable (3+1)-dimensional Boussinesq equation is obtained with the truncated Painlevé method, which can be localized to the Lie point symmetry for the prolonged system by introducing auxiliary dependent variables.
Abstract: The nonlocal symmetry of the new (3+1)-dimensional Boussinesq equation is obtained with the truncated Painlevé method. The nonlocal symmetry can be localized to the Lie point symmetry for the prolonged system by introducing auxiliary dependent variables. The finite symmetry transformation related to the nonlocal symmetry of the integrable (3+1)-dimensional Boussinesq equation is studied. Meanwhile, the new (3+1)-dimensional Boussinesq equation is proved by the consistent tanh expansion method and many interaction solutions among solitons and other types of nonlinear excitations such as cnoidal periodic waves and resonant soliton solution are given.
TL;DR: In this paper , the authors detect unpredicted conducts to the soliton solutions of the (2+1)-Boussinesq equation by using three distinct algorithms: (G′G)expansion method, the extended direct algebraic method (EDAM), and the extended simple equation method (ESEM).
TL;DR: In this paper , a nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method, which is converted into two local equations which contain the local Bousinsineq equation.
Abstract: A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method. To study various exact solutions of the nonlocal Boussinesq equation, it is converted into two local equations which contain the local Boussinesq equation. From the N-soliton solutions of the local Boussinesq equation, the N-soliton solutions of the nonlocal Boussinesq equation are obtained, among which the (N = 2,3,4)-soliton solutions are analyzed with graphs. Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from the known solutions of the local Boussinesq equation. Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.
TL;DR: In this paper , the existence and uniqueness of a global solution of the Boussinesq-MHD equations with partial viscosity and damping were proved for β≥4.
TL;DR: In this article , the state of the art for computational fluid dynamics (CFD) relevant to the Navier-Stokes equations with the Boussinesq approximation is presented.
TL;DR: In this article , a numerical implementation of a Hamiltonian Boussinesq wave-body interaction for irrotational flow is described, with a restriction of one horizontal coordinate and a cross section of the body.
Abstract: This paper describes a numerical implementation of a Hamiltonian Boussinesq wave-body interaction for irrotational flow as formulated in van Groesen and Andonowati (2017), with a restriction of one horizontal coordinate and a cross section of the body. Part of the HAWASSI (Hamiltonian Wave-Ship-Structure Interaction) software we developed allows for numerical discretisation of the surface waves using spectral methods. Non-smooth effects from the body-fluid interaction are included in the design of a virtual wave in the body area, which is determined by the boundary conditions on the body hull. Except for a comparison with standard cases in the literature, the performance of the code is shown by comparison with measurements of an experiment on the slow-drift motion of a rectangular barge moored above a sloping beach and interacting with irregular waves, in the barge beam direction, including the infra-gravity waves from the runup on the shore.
TL;DR: In this article, the amplitude equation for the wave amplitude is obtained by a regular asymptotic procedure, which incorporates a complicated nonlinearity and Korteweg-de Vries dispersion.
Abstract: Large amplitude solitary internal waves of permanent form propagating in a stratified shallow fluid between the free surface and a horizontal bottom are described by the amplitude equation obtained by a regular asymptotic procedure, which incorporates a complicated nonlinearity and Korteweg-de Vries (KdV) dispersion. It is discussed how the structure of stratification and shear affects wave properties. The particular case of a constant buoyancy frequency and a quadratic polynomial for the ambient shear for the flow under free surface is considered in detail analytically. It is shown that for such profiles, the equation for the wave amplitude reduces to the mixed-modified KdV equation and finite amplitude waves obey it up to the breaking level. Rogue waves could appear in this case, and the condition for their generation is identified. More complicated shear profiles lead to higher-order nonlinearities, which produce the multiscaled pyramidal wave patterns, asymmetric bores, and various instabilities. Such wave structures are studied numerically. An analytical bore-like solution having both exponential and algebraic asymptotes is presented.
TL;DR: In this paper , a linear stability analysis is conducted for horizontal convection under a Gay-Lussac (GL) type approximation in a relatively shallow enclosure cavity, and the GL type approximation is developed based on extending density variations to the advection term as well as gravity term through the momentum equation.