Boussinesq Model Research Articles

Exact solutions of a class of generalized nanofluidic models

Abstract Nanofluid, a significant branch of fluid mechanics, plays a pivotal role in thermal management, optics, biomedical engineering, energy harvesting, and other fields. The nanoparticles present in the fluid render the continuum mechanics ineffective, necessitating the adoption of fractional calculus to elucidate the effects of nanoparticles on the motion properties of the nanofluid. This article applies the modified extended tanh-function technique to solve two classical Schrödinger equations, the fractional Phi-4 model and the conformable fractional Boussinesq model, for nanofluids. Multiple exact solutions are obtained, and the corresponding graphical representations are provided to elucidate the basic properties of the nanofluid. This article provides new research perspectives for the development of nanofluids.

A novel energy-bounded Boussinesq model and a well balanced and stable numerical discretisation

Many Boussinesq models suffer from nonlinear instabilities, especially in the context of rapid variations in the bed topography. In this work, a Boussinesq system is put forward which is derived in such a way as to be both linearly and nonlinearly energy-stable.The proposed system is designed to be robust for coastal simulations with sharply varying bathymetric features while maintaining the dispersive accuracy at any constant depth. For constant bathymetries, the system has the same linear dispersion relation as Peregrine's system ([22]). Furthermore, the system transitions smoothly to the shallow-water system as the depth goes to zero.In the one-dimensional case, we design a stable finite-volume scheme and demonstrate its robustness, accuracy and stability under grid refinement in a suite of test problems including Dingemans's wave experiment.Finally, we generalise the system to the two-dimensional case.

Local stability of differential rotation in magnetized radiation zones and the solar tachocline

ABSTRACT We study local magnetohydrodynamical instabilities of differential rotation in magnetized, stably stratified regions of stars and planets using a Cartesian Boussinesq model. We consider arbitrary latitudes and general shears (with gravity direction misaligned from this by an angle $\phi$), to model radial ($\phi =0$), latitudinal ($\phi =\pm 90^\circ$), and mixed differential rotations, and study both non-diffusive [including magnetorotational instability (MRI) and Solberg–Høiland instability] and diffusive instabilities [including Goldreich–Schubert–Fricke (GSF) and MRI with diffusion]. These instabilities could drive turbulent transport and mixing in radiative regions, including the solar tachocline and the cores of red giant stars, but their dynamics are incompletely understood. We revisit linear axisymmetric instabilities with and without diffusion and analyse their properties in the presence of magnetic fields, including deriving stability criteria and computing growth rates, wave vectors, and energetics, both analytically and numerically. We present a more comprehensive analysis of axisymmetric local instabilities than prior work, exploring arbitrary differential rotations and diffusive processes. The presence of a magnetic field leads to stability criteria depending upon angular velocity rather than angular momentum gradients. We find MRI operates for much weaker differential rotations than the hydrodynamic GSF instability, and that it typically prefers much larger length-scales, while the GSF instability is impeded by realistic strength magnetic fields. We anticipate MRI to be more important for turbulent transport in the solar tachocline than the GSF instability when $\phi \gt 0$ in the Northern (and vice versa in the Southern) hemisphere, though the latter could operate just below the convection zone when MRI is absent for $\phi \lt 0$.

Self-induced transparency of long water waves over bathymetry: the dispersive shock mechanism

The shoreline hazard posed by ocean long waves such as tsunamis and meteotsunamis critically depends on the fraction of energy transmitted across the shallow near-shore shelf. In linear setting, bathymetric inhomogeneities of length comparable to the incident wavelength act as a protective high-pass filter, reflecting long waves and allowing only shorter waves to pass through. Here, we show that, for weakly nonlinear waves, the transmitted energy flux fraction can significantly depend on the amplitude of the incoming wave. The basis of this mechanism is the formation of dispersive shock waves (DSWs), a salient feature of nonlinear evolution of long water waves, often observed in tidal bores and tsunami/meteotsunami evolution. Within the framework of the Boussinesq equations, we show that the DSWs efficiently transfer wave energy into the high wavenumber band, where reflection is negligible. This is phenomenologically similar to self-induced transparency in nonlinear optics: small amplitude long waves are reflected by the bathymetric inhomogeneity, while larger amplitude waves that develop DSWs blueshift into the transparency regime and pass through. We investigate this mechanism in a simplified setting that retains only the key processes of DSW disintegration and reflection, while the effects such as bottom dissipation and breaking are ignored. The results suggests that the phenomenon is a robust, order-one effect. In contrast, the increased transmission due to the growth of bound harmonics associated with the steepening of the wave is weak. The results of the simplified modelling are validated by simulations with the FUNWAVE-TVD Boussinesq model.

3D topology optimization design of air natural convection heat transfer fins

This study focuses on the fin-tube heat exchanger and utilizes topology optimization methods to design a completely new fin structure. In this optimization process, complete Navier-Stokes (N-S) equations were used to describe the steady-state incompressible flow, and the Boussinesq model was employed to simulate natural convection. The flow equations were coupled with the heat convection–diffusion equation to achieve topology optimization for natural convection heat transfer. Topology optimization was conducted using density-based optimization methods, and interpolation was performed on the permeability and conductivity of the distributed materials. Given the initial fin structure, an interpolation progressive approach was adopted to obtain a new “ airfoil-shaped” optimized structure through density-based topology optimization method for natural convection. The new structure enhances the convective heat transfer by perforating the fins. The perforations are mainly concentrated in the central region of the heat exchanger and the upper half of the fins. The new structure, compared to the prototype structure, not only has a reduced volume but also exhibits a decrease in convective thermal resistance within a larger range of heat flux densities, as revealed by CFD simulations. Moreover, as the heat flux density increases, the rate of reduction in convective thermal resistance shows an upward trend for the new structure compared to the prototype structure.

Convergence rates and central limit theorem for 3-D stochastic fractional Boussinesq equations with transport noise

This work investigates the 3-D stochastic modified fractional Boussinesq equations driven by transport noise on the torus. The existence of weak solutions is initially established through Galerkin approximation and compactness techniques. Subsequently, we demonstrate that the weak solutions of stochastic modified fractional Boussinesq equations converge to the unique solution of a deterministic Boussinesq model with fractional dissipation, subject to an appropriate noise scaling, along with providing the quantitative estimates with explicit convergent rates. Finally, the central limit theorem with an explicit convergence rate is established based on the aforementioned scaling limit. It is noteworthy that the scaling limit result of stochastic modified fractional Boussinesq equations to deterministic Boussinesq models with additional fractional dissipation implies that the transport noise regularizes the modified fractional Boussinesq equations, thus leading to “approximate weak uniqueness”.

Warning signs for boundary noise and their application to an ocean Boussinesq model

In this paper, we construct and discuss early-warning signs of the approach of a parameter to a deterministic bifurcation on a stochastic partial differential equation (SPDE) model with Gaussian white-noise on the boundary. We specifically focus on the qualitative behaviour of the time-asymptotic autocovariance and autocorrelation of the solutions of the linearized system. We also discuss the reliability of the tools from an analytic perspective and through various examples. Among those, the application of the early-warning signs for an ocean Boussinesq model is explored through numerical simulations. The analytic results obtained expand on previous work and present valuable early-warning signs for various applications.

Influence of thermal buoyancy on the wake dynamics of a heated square cylinder

Direct numerical simulation of the three-dimensional (3-D) wake transition of a heated square cylinder subjected to horizontal cross-flow is performed in the presence of buoyancy. In order to capture the effects of large-scale heating, a non-Oberbeck–Boussinesq model is utilized, which includes the governing equations for compressible gas flow. All computations are performed at low free stream Mach number $M=0.1$ using air (free stream Prandtl number, $Pr=0.71$ ) as the working fluid. The 3-D instability modes A and B, which correspond to free stream Reynolds numbers of 180 and 250, are observed with longer and shorter spanwise wavelengths, respectively, and the onset of three-dimensionality is triggered at a Reynolds number of 173. In the presence of buoyancy, baroclinic vorticity production in the near-wake plays an important role for streamwise vorticity generation. The chaotic wake of the Mode-A instability bifurcates into periodic and quasiperiodic wakes at various heating levels, expressed by the overheat ratio, $\varepsilon =(T_w-T_\infty )/T_\infty$ , where $T_w$ and $T_\infty$ are the temperature of the cylinder surface and the ambient air, respectively. At low heating ( $\varepsilon =0.2$ ), the 3-D Mode-A instability is suppressed leading to a two-dimensional wake flow. Further increase in heating, again brings back the three-dimensionality in the wake through Mode-E instability. The variation of thermophysical properties and the effective Reynolds number with increase in heating level around the cylinder is examined. It is shown that the effect of thermophysical properties competes with the baroclinic streamwise vorticity generation at higher levels of heating ( $\varepsilon \geqslant 0.4$ ) to control the 3-D modes and wake dynamics.

Volcano tsunamis and their effects on moored vessel safety: the 2022 Tonga event

Abstract. The explosion of the Hunga Tonga–Hunga Ha'apai volcano on 15 January 2022 (Tonga 2022) was the origin of a volcano-meteorological tsunami (VMT) recorded worldwide. At a distance exceeding 10 000 km from the volcano and 15 h after its eruption, the moorings of a ship in the port of La Pampilla, Callao (Peru), failed, releasing over 11 000 barrels of crude oil. This study delves into the profound implications of the Tonga 2022 event, investigating whether it could have led to the breaking of the mooring system. We conducted a comprehensive analysis of this significant event, examining the frequency content of the time series recorded at tide gauges, DART (Deep-ocean Assessment and Reporting of Tsunamis) buoys, and barometers in the southern Pacific Ocean. Our findings revealed that the maximum energy of the spectra corresponds to the 120 min wave period off the coast of Peru, with the arrival time of these waves coinciding with the time of the accident in the port. We used a Boussinesq model to simulate the propagation of the volcano-meteorological tsunami from the source to the port in Peru to study the impact of those waves on the mooring system. We used the synthetic tsunami recorded in the port as input for the model that simulates mooring line loads based on the ship's degrees of freedom. The results suggest that the 120 min wave triggered by the VMT could significantly increase mooring stresses due to the resulting hydrodynamic effects, exceeding the minimum breaking load (MBL). We conclude that the propagation of the long wave period generated by the VMT caused overstresses in moored lines that triggered accidents in port environments. This event showed the need to prepare tsunami early warning systems and port authorities for detecting and managing VMTs induced by atmospheric acoustic waves. The work provides new insights into the far-reaching impacts of the Tonga 2022 tsunami.

General boundary conditions for a Boussinesq model with varying bathymetry

AbstractThis paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq–Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ordinary differential equation. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order, respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data are not known.

Boussinesq model for two-fluid system with surface- and interfacial tension

A Boussinesq model associated with surface and internal waves with surface tension and interfacial tension in a two-layer fluid is presented in the application range of weakly dispersive nonlinear. The nonlinear form of model definitions in terms of surface and interfacial tensions along with velocity potentials are provided. Based on the perturbation technique, the Boussinesq equations for both layers are derived. The present Boussinesq result is validated against existing published results and it has a good level of agreement. Further, the present model results of crests of surface and interface displacements are compared with existing published numerical OpenFOAM simulations. The dispersion relations for fast- and slow modes are derived in the quadratic form of wave frequencies in the presence of surface and interfacial tensions. The second-order surface and internal wave displacements along with velocity potentials for upper- and lower-layer fluids are presented. The effects of surface and interfacial tensions on dispersion characteristics, harmonic transfer functions, and shoaling gradients on different physical parameters associated with the present model are studied by analyzing several numerical results. In the end, a real physical problem is demonstrated in the application of surface tension with solvents in the gas-liquid interface.

Geometric Approaches to Lagrangian Averaging

Lagrangian averaging theories, most notably the generalized Lagrangian mean (GLM) theory of Andrews and McIntyre, have been primarily developed in Euclidean space and Cartesian coordinates. We reinterpret these theories using a geometric, coordinate-free formulation. This gives central roles to the flow map, its decomposition into mean and perturbation maps, and the momentum 1-form dual to the velocity vector. In this interpretation, the Lagrangian mean of any tensorial quantity is obtained by averaging its pull-back to the mean configuration. Crucially, the mean velocity is not a Lagrangian mean in this sense. It can be defined in a variety of ways, leading to alternative Lagrangian mean formulations that include GLM and Soward and Roberts's volume-preserving version. These formulations share key features that the geometric approach uncovers. We derive governing equations both for the mean flow and for wave activities constraining the dynamics of the perturbations. The presentation focuses on the Boussinesq model for inviscid rotating stratified flows and reviews the necessary tools of differential geometry.

Solvability analysis for the Boussinesq model of heat transfer under the nonlinear Robin boundary condition for the temperature.

We consider the new boundary value problem for the generalized Boussinesq model of heat transfer under the inhomogeneous Dirichlet boundary condition for the velocity and under mixed boundary conditions for the temperature. It is assumed that the viscosity, thermal conductivity and buoyancy force in the model equations, as well as the heat exchange boundary coefficient, depend on the temperature. The mathematical apparatus for studying the inhomogeneous boundary value problem under study based on the variational method is being developed. Using this apparatus, we prove the main theorem on the global existence of a weak solution of the mentioned boundary value problem and establish sufficient conditions for the problem data ensuring the local uniqueness of the weak solution that has the additional property of smoothness with respect to temperature. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Numerical simulation of local and upwind temperature stratification on flow and dispersion in a bi-dimensional street canyon.

The thermal effect mainly includes boundary temperature stratification and the local thermal effect. The combined effect of these factors on flow and dispersion in a bi-dimensional canyon was investigated by the RANS and LES methods to evaluate their performance. The results, including the flow field, turbulent kinetic energy, temperature, heat flux, pollutant concentration and fluxes, were compared with the data from wind tunnel experiments. The comparison results showed that the RANS method severely overestimated the impact of windward heating on the flow in the canyon because of the lack of simulated flow separation ability and the limitation of the Boussinesq model, leading to an incorrect flow field and an incorrect temperature and concentration. In contrast, LES performed better mainly because of its ability to simulate flow separation. LES regenerated the right vortexes, flow field and low wind velocity. LES slightly overestimates the overall temperature in the canyon because heat exchange is eliminated in LES but difficult to avoid in the experiment. The difference in the air exchange rate at the roof level between the LES and wind tunnel data was no more than 5%, and the pollutant concentration distribution of the LES was almost the same as that of the experiments. This work emphasizes that the RANS method has limited ability to simulate flow and dispersion when the thermal effect is considered even at a reduced-scale, while LES can simulate the combined effects of incoming flow temperature stratification and local thermal effects. It is therefore suggested that if computing resources are limited and the temperature difference is not large, a steady-state calculation RANS can be used. Otherwise, LES must be performed.

Deep neural network-based prediction of tsunami wave attenuation by mangrove forests

The goal of this research is to develop a model employing deep neural networks (DNNs) to predict the effectiveness of mangrove forests in attenuating the impact of tsunami waves. The dataset for the DNN model is obtained by simulating tsunami wave attenuation using the Boussinesq model with a staggered grid approximation. The Boussinesq model for wave attenuation is validated using laboratory experiments exhibiting a mean absolute error (MAE) ranging from 0.003 to 0.01. We employ over 40,000 data points generated from the Boussinesq numerical simulations to train the DNN. Efforts are made to optimize hyperparameters and determine the neural network architecture to attain optimal performance during the training process. The prediction results of the DNN model exhibit a coefficient of determination (R2) of 0.99560, an MAE of 0.00118, a root mean squared error (RMSE) of 0.00151, and a mean absolute percentage error (MAPE) of 3 %. When comparing the DNN model with three alternative machine learning models— support vector regression (SVR), multiple linear regression (MLR), and extreme gradient boosting (XGBoost)— the performance of DNN is superior to that of SVR and MLR, but it is similar to XGBoost.•High-accuracy DNN models require hyperparameter optimization and neural network architecture selection.•The error of DNN models in predicting the attenuation of tsunami waves by mangrove forests is less than 3 %.•DNN can serve as an alternate predictive model to empirical formulas or classical numerical models.

Optimal error estimates of a second-order fully decoupled finite element method for the nonstationary generalized Boussinesq model

In this paper, we develop and analyze a fully decoupled finite element method for the non-stationary generalized Boussinesq equations, where the viscosity and thermal conductivity depend on the temperature. Based on some subtle implicit-explicit treatments for the nonlinear coupling terms, we develop a second-order in time, fully decoupled, linear and unconditionally energy stable scheme for solving this system. The unconditional stability of the fully discrete scheme with finite element approximation is proved. The optimal L2-error estimates are analyzed for temperature-dependent thermal conductivity system. Numerical experiments are presented to illustrate the convergence, accuracy and applicability of the proposed numerical scheme.

Laminar convective heat transfer combining buoyancy and thermal radiation of Carreau fluid within a concentric and eccentric annulus

The present study concerns a steady-state laminar combined thermal radiation and buoyancy-driven conjugate magnetohydrodynamic (MHD) natural convective flow and heat transfer inside a fully porous medium eccentric circular annulus bounded by two infinite horizontal cylinders occupied by a Carreau fluid permeated by a uniform transverse magnetic field (B0 ) and inclined by an angle (α). The convective and thermal radiation are approximated by the Boussinesq and Roseland models, respectively. The numerical integration was performed using the implicit finite volume method. The convective terms in the momentum and energy equations were Discretized using the second-order upwind method. On the other hand, the diffusive terms were Discretized using the central difference scheme. The pressure-velocity coupling was solved using the SIMPLE approach. The findings suggest a clear trend of enhancement in the heat transfer rate when the aspect ratio and annulus eccentricity increase. This pattern is more pronounced in the negative vertical eccentricities. It is worth noting that the eccentric annulus, which has an annulus aspect ratio of 1.6 and a vertical eccentricity of −0.8, offers the best performance regarding the heat transfer mechanism inside the annulus gap. The heat transfer mechanism turns out to be strongly affected by thermal radiation, Hartmann number, and magnetic field inclination angle. The increased thermal radiation parameter and the decreased flow behavior index improve the heat transfer mechanism between the Carreau fluid and the inner cylinder in the annular gap. Furthermore, the average Nusselt number increases considerably as the induced magnetic field inclines.

Multigoal-oriented a posteriori error control for heated material processing using a generalized Boussinesq model

In this work, we develop a posteriori error control for a generalized Boussinesq model in which thermal conductivity and viscosity are temperature-dependent. Therein, the stationary Navier–Stokes equations are coupled with a stationary heat equation. The coupled problem is modeled and solved in a monolithic fashion. The focus is on multigoal-oriented error estimation with the dual-weighted residual method in which an adjoint problem is utilized to obtain sensitivity measures with respect to several goal functionals. The error localization is achieved with the help of a partition-of-unity in a weak formulation, which is specifically convenient for coupled problems as we have at hand. The error indicators are used to employ adaptive algorithms, which are substantiated with several numerical tests such as one benchmark and two further experiments that are motivated from laser material processing. Therein, error reductions and effectivity indices are consulted to establish the robustness and efficiency of our framework.

The dynamical perspective of soliton solutions, bifurcation, chaotic and sensitivity analysis to the (3+1)-dimensional Boussinesq model

In this study, we examine multiple perspectives on soliton solutions to the (3+1)-dimensional Boussinesq model by applying the unified Riccati equation expansion (UREE) approach. The Boussinesq model examines wave propagation in shallow water, which is derived from the fluid dynamics of a dynamical system. The UREE approach allows us to derive a range of distinct solutions, such as single, periodic, dark, and rational wave solutions. Furthermore, we present the bifurcation, chaotic, and sensitivity analysis of the proposed model. We use planar dynamical system theory to analyze the structure and characteristics of the system’s phase portraits. The current study depends on a dynamic structure that has novel and unexplored results for this model. In addition, we display the behaviors of associated physical models in 3-dimensional, density, and 2-dimensional graphical structures. Our findings demonstrate that the UREE technique is a valuable mathematical tool in engineering and applied mathematics for studying wave propagation in nonlinear evolution equations.

Exploring the excess of cloud condensation nuclei and rain suppression using a minimal three-dimensional Boussinesq model with bulk cloud microphysics

Over the years, there have been discussions about the possibility of air pollution affecting the process of rain formation. In this study, we have developed a simplified model that represents the atmospheric dynamics and cloud microphysics to explore the impact of pollution on rain formation. We used an existing three-dimensional minimal model consisting of five equations, for which we added a simple bulk parametrization that represents the role of cloud condensation nuclei (CCN) in cloud formation processes. We conducted numerical tests using two CCN profiles, with either one or two accumulation layers and modified their abundance to explore the effects of different CCN concentrations and distributions. We conducted four numerical tests corresponding to the two aforementioned profiles with polluted and low-polluted scenarios. The numerical simulations suggested that a layer with high CCN concentration close to the surface tends to suppress precipitation, while the same concentration distributed over two layers tends to enhance the efficiency of rain formation. The simulations also showed that CCN particles far from the surface produced higher cloud tops and longer events, consistent with previous research. Although the model includes a stable representation of precipitating turbulent convection with bulk cloud microphysics, we expect its simplicity and conservation properties to allow for deeper theoretical analyses that can help us better understand the physical processes involved in the studied phenomenon. We hope this model will serve as a tool to explore different aerosol-related scenarios within the context of minimal models.