Academic literature on the topic 'Quadratic Boussinesq approximation'

We construct approximate analytical solutions of the Boussinesq equation for horizontal unconfined aquifers in the buildup phase under constant recharge and zero-inflow conditions. We employ a variety of methods, which include wave solutions, self-similar solutions, and two classical linear approximations of the Boussinesq equation (linear and quadratic), to explore the behavior and performance of the solutions derived from each method against the Boussinesq equation and the exact (non-closed form) analytical solutions. We find that the wave approximation, which is of a conceptual nature, encapsulates quite faithfully the characteristics of the nonlinear Boussinesq equation solution and, overall, performs much better than the other methods, for which the relatively low performance can be attributed to the specific mathematical features of their construction. These endeavors might be useful for theoretical and modeling purposes related to this problem.

<abstract> <p>In current days, hybrid models have become more essential in a wide range of systems, including medical treatment, aerosol particle handling, laboratory instrument design, industry and naval academia, and more. The influence of linear, nonlinear, and quadratic Rosseland approximations on 3D flow behavior was explored in the presence of Fourier fluxes and Boussinesq quadratic thermal oscillations. Ternary hybrid nanoparticles of different shapes and densities were also included. Using the necessary transformation, the resulting partial differential system is transformed into a governing ordinary differential system, and the solution is then furnished with two mixed compositions (Case-Ⅰ and Case-Ⅱ). Combination one looked at aluminum oxide (Platelet), graphene (Cylindrical), and carbon nanotubes (Spherical), whereas mixture two looked at copper (Cylindrical), copper oxide (Spherical), and silver oxide (Platelet). Many changes in two mixture compositions, as well as linear, quadratic, and nonlinear thermal radiation situations of the flow, are discovered. Case-1 ternary combinations have a wider temperature distribution than Case-2 ternary mixtures. Carbon nanotubes (Spherical), graphene (Cylindrical), and aluminum oxide (Platelet) exhibit stronger conductivity than copper oxide (Spherical), copper (Cylindrical), and silver oxide (Platelet) in Case 1. (Platelet). In copper oxide (Spherical), copper (Cylindrical), and silver (Platelet) compositions, the friction factor coefficient is much higher. The combination of liquids is of great importance in various systems such as medical treatment, manufacturing, experimental instrument design, aerosol particle handling and naval academies, etc. Roseland's quadratic and linear approximation of three-dimensional flow characteristics with the existence of Boussinesq quadratic buoyancy and thermal variation. In addition, we combine tertiary solid nanoparticles with different shapes and densities. In many practical applications such as the plastics manufacturing and polymer industry, the temperature difference is remarkably large, causing the density of the working fluid to vary non-linearly with temperature. Therefore, the nonlinear Boussinesq (NBA) approximation cannot be ignored, since it greatly affects the flow and heat transport characteristics of the working fluid. Here, the flow of non-Newtonian elastomers is controlled by the tension of an elastic sheet subjected to NBA and the quadratic form of the Rosseland thermal radiation is studied.</p> </abstract>

On the basis of field experiments and modeling, the dependence of the dissipation of the energy of waves breaking by plunging and spilling on the frequency of wave spectra was investigated. It was shown that the modeling of wave breaking should take into account the compensation of the nonlinear growth of higher wave harmonics, which occurs in different ways for waves breaking with different types and for different methods of modeling a nonlinear source term. The study revealed that spilling breaking waves have a frequency selectivity of energy dissipation at frequencies of second and third harmonics for the Boussinesq and SWAN models for any method of modeling a nonlinear source term. Plunging breaking waves have a quadratic dependence of the dissipation coefficient on frequency in the Boussinesq model and SWAN model with the SPB approximation for a nonlinear source term. The SWAN model with default LTA approximation for plunging breaking waves also assumes frequency-selective energy dissipation. The discrepancy between the LTA default method and others can be explained by the overestimation of the contribution of the second nonlinear harmonic and by inaccurate approximation for the biphase. It is possible to improve the accuracy of LTA and SPB methods by tuning SWAN model coefficients.

The linear stability threshold of the Rayleigh–Bénard configuration is analysed with compressible effects taken into account. It is assumed that the fluid under investigation obeys a Newtonian rheology and Fourier’s law of thermal transport with constant, uniform (dynamic) viscosity and thermal conductivity in a uniform gravity field. Top and bottom boundaries are maintained at different constant temperatures and we consider here mechanical boundary conditions of zero tangential stress and impermeable walls. Under these conditions, and with the Boussinesq approximation, Rayleigh (Phil. Mag., vol. 32 (192), 1916, pp. 529–546) first obtained analytically the critical value $27\unicode[STIX]{x03C0}^{4}/4$ for a dimensionless parameter, now known as the Rayleigh number, at the onset of convection. This paper describes the changes of the critical Rayleigh number due to the compressibility of the fluid, measured by the dimensionless dissipation parameter ${\mathcal{D}}$ and due to a finite temperature difference between the hot and cold boundaries, measured by a dimensionless temperature gradient $a$. Different equations of state are examined: ideal gas equation, Murnaghan’s model (often used to describe the interiors of solid but convective planets) and a generic equation of state with adjustable parameters, which can represent any possible equation of state. In the perspective to assess approximations often made in convective models, we also consider two variations of this stability analysis. In a so-called quasi-Boussinesq model, we consider that density perturbations are solely due to temperature perturbations. In a so-called quasi-anelastic liquid approximation model, we consider that entropy perturbations are solely due to temperature perturbations. In addition to the numerical Chebyshev-based stability analysis, an analytical approximation is obtained when temperature fluctuations are written as a combination of only two modes, one being the original symmetrical (between top and bottom) mode introduced by Rayleigh, the other one being antisymmetrical. The analytical solution allows us to show that the antisymmetrical part of the critical eigenmode increases linearly with the parameters $a$ and ${\mathcal{D}}$, while the superadiabatic critical Rayleigh number departs quadratically in $a$ and ${\mathcal{D}}$ from $27\unicode[STIX]{x03C0}^{4}/4$. For any arbitrary equation of state, the coefficients of the quadratic departure are determined analytically from the coefficients of the expansion of density up to degree three in terms of pressure and temperature.

This study examines the effect of non-linear radiation and viscous dissipation on the convective Newtonian fluid flow with temperature-dependent viscosity and the quadratic Boussinesq approximation around a cylinder with uniform heat flux at the wall. The coupled partial differential equations of the problem are non-dimensionalized with appropriate variables and reduced via stream functions. Regular perturbation technique is employed to transform the nonlinear coupled partial differential equations into a system of nonlinear coupled ordinary differential equations solved using the Trapezoidal method. A surge in the radiative parameter was found to heighten the fluid’s velocity and temperature, while an increase in the dissipative effect enhances the skin friction and heat transfer distributions. The limiting cases of the model considered and the results obtained in this study are consistent with those in the literature.

On a weakly nonlinear basis, we revisit the pattern formation problem in the Boussinesq convection, for which nonlinear terms of the quadratic order are known to vanish from amplitude equations. It is thus necessary to proceed to the quintic-order approximation in order for the amplitude equations to be generic. By deriving the quintic amplitude equations from the governing PDEs, we examined the bifurcation of steady solutions under rigid–free, rigid–rigid and free–free boundary conditions. Right above the criticality, all the axial solutions are obtained including up- and down-hexagons under the asymmetric boundary conditions and hexagons and regular triangles under the symmetric conditions. Hexagons and regular triangles are unstable whereas rolls are stable as has already been predicted by the cubic-order amplitude equations. Irrespective of the boundary conditions, quintic-order terms stabilize hexagons except near the criticality; rolls and hexagons thus coexist stably in an open region. This suggests that amplitude equations of higher order are possible to predict re-entrant hexagons .

Le stockage d'énergie est essentiel à la transition énergétique car il permet de découpler la production de l'énergie de sa consommation. Le stockage de chaleur thermocline en eau, utilisé dans les réseaux de chaleur à moyenne ou basse température, repose sur la stratification thermique dans une cuve. Les performances de ce type de stockage sont fortement liées à la bonne stratification du fluide qui peut être perturbée par l'injection et le soutirage du liquide, des aspects peu explorés dans la littérature.L'objectif de cette thèse est de modéliser un tel stockage de manière fiable pour analyser la distribution du fluide. En effet, le but est de mieux appréhender les phénomènes physiques gouvernant la thermocline pendant les cycles de fonctionnement et d'accroître ses performances énergétiques par un design ou un pilotage amélioré. Pour ce faire, des études numériques utilisant la CFD (Computational Fluid Dynamics) ont été réalisées et comparées à des données expérimentales disponibles dans la littérature et obtenues via une nouvelle section d'essais.Dans un premier temps, un modèle CFD a été développé basé sur un cas expérimental existant de la littérature. Dans un stockage thermocline en eau, il y a bien souvent coexistence entre une région laminaire dans la cuve et turbulente à proximité des distributeurs. Cette coexistence est un enjeu majeur de la modélisation car la plupart des modèles de turbulence ne sont pas capables de représenter fiablement la transition d'un écoulement turbulent vers laminaire. Pour ces travaux, une méthode statistique RANS (Reynolds Average Numerical Simulation) est adoptée et le modèle k-omega-SST est sélectionnée car il permet de représenter les écoulements en proche paroi. Concernant la flottabilité, il existe deux méthodes pour considérer ses effets : utiliser une masse volumique variable dans l'ensemble des équations, ou constante sauf dans le terme de flottabilité . Cette dernière est connue sous le nom de l'approximation de Boussinesq mais n'est valable que sur une faible gamme de ΔT. La précision de l'approximation de Boussinesq a été remise en question et une approche au second ordre de ce modèle est employée. Celle-ci permet d'obtenir le même terme de flottabilité qu'un modèle à masse volumique variable mais avec un temps de calcul réduit de moitié. La comparaison avec des données expérimentales a permis de souligner l'impact de l'état initial en température (stockage stratifié ou homogène). Une étude exploratoire de l'impact d'une injection progressive selon une rampe en débit a montré son impact sur la réduction de l'épaisseur de la thermocline au moment de sa création.Dans une démarche de validation du modèle et de vérification des observations numériques, un nouveau dispositif expérimental a été conçu. Celui-ci mesure la température grâce à 300 thermocouples disposés dans la cuve et permet un contrôle précis des conditions opératoires. Des études en phase statiques pour évaluer les pertes thermiques ont été réalisées. Des études dynamiques ont permis de faire varier les paramètres opératoires pertinents : la vitesse de propagation axiale, l'écart de température, le dispositif de soutirage ou encore l'injection progressive. Pour ce système, les résultats montrent qu'il est possible d'obtenir une stratification à forte vitesse (&gt; 2 mm/s) tant que le ΔT est suffisamment élevé.Enfin, l'écoulement dans la section d'essais a été étudié numériquement avec un modèle CFD actualisé. Les champs de variables ont montré que les résultats numériques et expérimentaux sont cohérents, en particulier lors de la formation de la thermocline. Toutefois, un excès de diffusion lors de la propagation du gradient thermique à faible débit est notable. Pour tous les essais réalisés les écarts expérimentaux et numériques ont été quantifiés: à l'exception des conditions critiques, l'écart sur l'épaisseur de thermocline est de ±50% et se situe entre 0 et -10% pour le taux de restitution<br>Energy storage is essential to the energy transition as it allows decoupling energy production from its consumption. Water-based thermocline heat storage, used in medium or low-temperature heating networks, relies on thermal stratification in a tank. The performance of this type of storage is strongly linked to the proper stratification of the fluid, which can be disrupted by the injection and extraction of the liquid, aspects that are scarcely explored in the literature.The objective of this thesis is to reliably model such storage to analyze the fluid distribution. The aim is to better understand the physical phenomena governing the thermocline during operating cycles and to enhance its energy performance through improved design or control. To achieve this, numerical studies using CFD (Computational Fluid Dynamics) were conducted and compared with experimental data available in the literature and obtained via a new experimental setup.Initially, a CFD model was developed based on an existing experimental case from the literature. In water thermocline storage, there is often coexistence between a laminar region in the tank and a turbulent region near the distributors. This coexistence is a major challenge in modeling because most turbulence models cannot reliably represent the transition from turbulent to laminar flow. For this work, a RANS (Reynolds Average Numerical Simulation) statistical method is adopted, and the k-omega-SST model is selected as it can represent near-wall flows. Regarding buoyancy, there are two methods to consider its effects: using a variable density in all equations or a constant density except in the buoyancy term. The latter is known as the Boussinesq approximation but is only valid over a narrow range of ΔT. The accuracy of the Boussinesq approximation has been questioned, and a second-order approach of this model is employed. This allows obtaining the same buoyancy term as a variable density model but with a calculation time reduced by half. Comparison with experimental data highlighted the impact of the initial temperature state (stratified or homogeneous storage). An exploratory study of the impact of progressive injection according to a flow ramp showed its effect on reducing the thermocline thickness at the time of its creation.As part of the model validation and verification of numerical observations, a new experimental setup was designed. It measures the temperature using 300 thermocouples placed in the tank and allows precise control of operating conditions. Static phase studies to evaluate thermal losses were conducted. Dynamic studies allowed varying relevant operating parameters: axial propagation speed, temperature difference, extraction device, and progressive injection. For this system, the results show that it is possible to obtain stratification at high speed (&gt; 2 mm/s) as long as the ΔT is sufficiently high.Finally, the flow in the test section was numerically studied with an updated CFD model. The variable fields showed that the numerical and experimental results are consistent, especially during the formation of the thermocline. However, excessive diffusion during the propagation of the thermal gradient at low flow is notable. For all the tests carried out, the experimental and numerical discrepancies were quantified: except for critical conditions, the discrepancy in thermocline thickness is ±50% and ranges from 0 to -10% for the restitution rate

Abstract The static and dynamic stability of Rayleigh-Bénard convection in a rectangular flow domain is computationally investigated. Sinusoidal vertical oscillations are applied to the system to provide dynamic flow stabilization. Stability maps are produced for a range of flow and heating conditions, and are compared to experimental measurements and linear stability analysis predictions from existing literature. Density variation is introduced through: 1) the Boussinesq approximation, 2) a linearly varying temperature dependent equation of state (EOS) and 3) the perfect gas EOS. Significant effects of choice of EOS on dynamic stability are observed. These weakly compressible flows are solved efficiently using an implicit numerical method that has been developed to solve the momentum, continuity, enthalpy and state equations simultaneously in fully coupled fashion. This block coupled system of equations is linearized with Newton’s method, and quadratic convergence is achieved. The details of these numerics are presented.