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Journal articles on the topic 'Approximation de Boussinesq quadratique'

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Published: 7 December 2024

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1

Young, William R. "Dynamic Enthalpy, Conservative Temperature, and the Seawater Boussinesq Approximation." Journal of Physical Oceanography 40, no. 2 (February 1, 2010): 394–400. http://dx.doi.org/10.1175/2009jpo4294.1.

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Abstract A new seawater Boussinesq system is introduced, and it is shown that this approximation to the equations of motion of a compressible binary solution has an energy conservation law that is a consistent approximation to the Bernoulli equation of the full system. The seawater Boussinesq approximation simplifies the mass conservation equation to ∇ · u = 0, employs the nonlinear equation of state of seawater to obtain the buoyancy force, and uses the conservative temperature introduced by McDougall as a thermal variable. The conserved energy consists of the kinetic energy plus the Boussinesq dynamic enthalpy h‡, which is the integral of the buoyancy with respect to geopotential height Z at a fixed conservative temperature and salinity. In the Boussinesq approximation, the full specific enthalpy h is the sum of four terms: McDougall’s potential enthalpy, minus the geopotential g0Z, plus the Boussinesq dynamic enthalpy h‡, and plus the dynamic pressure. The seawater Boussinesq approximation removes the large and dynamically inert contributions to h, and it reveals the important conversions between kinetic energy and h‡.

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2

Wood, T. S., and P. J. Bushby. "Oscillatory convection and limitations of the Boussinesq approximation." Journal of Fluid Mechanics 803 (August 30, 2016): 502–15. http://dx.doi.org/10.1017/jfm.2016.511.

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We determine the asymptotic conditions under which the Boussinesq approximation is valid for oscillatory convection in a rapidly rotating fluid. In the astrophysically relevant parameter regime of small Prandtl number, we show that the Boussinesq prediction for the onset of convection is valid only under much more restrictive conditions than those that are usually assumed. In the case of an ideal gas, we recover the Boussinesq results only if the ratio of the domain height to a typical scale height is much smaller than the Prandtl number. This requires an extremely shallow domain in the astrophysical parameter regime. Other commonly used ‘sound-proof’ approximations generally perform no better than the Boussinesq approximation. The exception is a particular implementation of the pseudo-incompressible approximation, which predicts the correct instability threshold beyond the range of validity of the Boussinesq approximation.

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3

RAJAGOPAL, K. R., M. RUZICKA, and A. R. SRINIVASA. "ON THE OBERBECK-BOUSSINESQ APPROXIMATION." Mathematical Models and Methods in Applied Sciences 06, no. 08 (December 1996): 1157–67. http://dx.doi.org/10.1142/s0218202596000481.

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This paper deals with a derivation (using a perturbation technique) of an approximation, due to Oberbeck8,9 and Boussinesq,1 to describe the thermal response of linearly viscous fluids that are mechanically incompressible but thermally compressible. The present approach uses a nondimensionalization suggested by Chandrasekhar2 and utilizing the ratio of two characteristic velocities as a measure of smallness, systematically derives the Oberbeck-Boussinesq approximation as a third-order perturbation. In the present approach, the material is subjected to the constraint that the volume change is determined solely by the temperature change in the body and uses a novel approach in deriving the thermodynamical restrictions. Consequently, it is free from the additional assumptions usually added on in earlier works in order to obtain the correct equations.

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4

Téchené, Jean-Jacques. "Les aspects fondamentaux de l'admissibilité en approximation quadratique d'applications linéaires." Linear Algebra and its Applications 264 (October 1997): 389–419. http://dx.doi.org/10.1016/s0024-3795(96)00405-3.

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5

Barletta, Antonio, Michele Celli, and D. Andrew S. Rees. "On the Use and Misuse of the Oberbeck–Boussinesq Approximation." Physics 5, no. 1 (March 17, 2023): 298–309. http://dx.doi.org/10.3390/physics5010022.

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The Oberbeck–Boussinesq approximation is the most commonly employed theoretical scheme for the study of natural or mixed convection flows. However, the misunderstanding of this approximated framework is a possibility that may cause the emergence of paradoxes or, at least, incorrect conclusions. In this paper, the basic features of the Oberbeck–Boussinesq approximation are briefly recalled and three simple examples where this theoretical scheme may be misused are provided. Such misuses of the approximation lead to erroneous conclusions that, in the examples presented in this note, entail violations of the principle of mass conservation. A discussion about the Oberbeck–Boussinesq approximation as an asymptotic theory obtained by letting the product of the thermal expansion coefficient and the reference temperature difference tend to zero is also presented.

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6

Gomes, Diogo A., and Claudia Valls. "Approximation of ill-posed boussinesq equations." Dynamical Systems 19, no. 4 (December 2004): 345–57. http://dx.doi.org/10.1080/1468936042000269587.

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7

Baron-Pertuz, Cristian-Fabian, Ana-Magnolia Marin-Ramirez, and Ruben-Dario Ortiz-Ortiz. "An approximation to the Boussinesq equations." International Journal of Mathematical Analysis 8 (2014): 2433–37. http://dx.doi.org/10.12988/ijma.2014.48274.

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8

PRUŠA, VÍT, and K. R. RAJAGOPAL. "ON MODELS FOR VISCOELASTIC MATERIALS THAT ARE MECHANICALLY INCOMPRESSIBLE AND THERMALLY COMPRESSIBLE OR EXPANSIBLE AND THEIR OBERBECK–BOUSSINESQ TYPE APPROXIMATIONS." Mathematical Models and Methods in Applied Sciences 23, no. 10 (July 12, 2013): 1761–94. http://dx.doi.org/10.1142/s0218202513500516.

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Viscoelastic fluid like materials that are mechanically incompressible but are compressible or expansible with respect to thermal stimuli are of interest in various applications ranging from geophysics and polymer processing to glass manufacturing. Here we develop a thermodynamical framework for the modeling of such materials. First we illustrate the basic ideas in the simpler case of a viscous fluid, and after that we use the notion of natural configuration and the concept of the maximization of the entropy production, and we develop a model for a Maxwell type viscoelastic fluid that is mechanically incompressible and thermally expansible or compressible. An important approximation in fluid mechanics that is frequently used in modeling buoyancy driven flows is the Oberbeck–Boussinesq approximation. Originally, the approximation was used for studying the flows of viscous fluids in thin layers subject to a small temperature gradient. However, the approximation has been used almost without any justification even for flows of non-Newtonian fluids induced by strong temperature gradients in thick layers. Having a full system of the governing equations for a Maxwell type viscoelastic mechanically incompressible and thermally expansible or compressible fluid, we investigate the validity of the Oberbeck–Boussinesq type approximation for flows of this type of fluids. It turns out that the Oberbeck–Boussinesq type approximation is in general not a good approximation, in particular if one considers "high Rayleigh number" flows. This indicates that the Oberbeck–Boussinesq type approximation should not be used routinely for all buoyancy driven flows, and its validity should be thoroughly examined before it is used as a mathematical model.

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9

Siddiqui, Perwez. "Density Modelling in Mixed Convection Flow in a Vertical Parallel Plate Channel." International Journal of Heat and Technology 39, no. 4 (August 31, 2021): 1294–304. http://dx.doi.org/10.18280/ijht.390428.

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In this paper, a novel way of modelling the density in buoyancy term of mixed convection flow problem is presented using equation of state and Boussinesq approximation without first-order approximation of density with respect to temperature. The presented density model is used to investigate the laminar mixed convection flow in a vertical parallel plate channel under symmetric constant wall heat flux. The results obtained are compared with the results obtained using first-order approximation of density with Boussinesq approximation, and also compared with the results obtained using variable thermophysical properties with negligible viscous dissipation. Investigation is performed on the basis of flow and thermal fields for Re=150 and 300, Ri=0.1 to 25. It is found that the presented density model produces relatively better results, which is able to describe the case of developing flow under constant heat flux condition that is not evident if Boussinesq approximation with first-order approximation of density is used. An appearance of recirculatory cells when reverse flow takes place is also witnessed in vertical channel flow with constant heat flux boundary condition which was not reported earlier.

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10

Grandi, Diego, and Arianna Passerini. "On the Oberbeck–Boussinesq approximation for gases." International Journal of Non-Linear Mechanics 134 (September 2021): 103738. http://dx.doi.org/10.1016/j.ijnonlinmec.2021.103738.

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11

Danchin, Raphaël, and Lingbing He. "The Oberbeck–Boussinesq approximation in critical spaces." Asymptotic Analysis 84, no. 1-2 (2013): 61–102. http://dx.doi.org/10.3233/asy-131170.

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12

Churilov, S. M. "Holmboe instability beyond the Boussinesq approximation revisited." Physics of Fluids 31, no. 9 (September 2019): 094104. http://dx.doi.org/10.1063/1.5115826.

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13

Demuren, Ayodeji, and Holger Grotjans. "Buoyancy-Driven Flows—Beyond the Boussinesq Approximation." Numerical Heat Transfer, Part B: Fundamentals 56, no. 1 (June 9, 2009): 1–22. http://dx.doi.org/10.1080/10407790902970080.

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14

Passerini, Arianna, and Gudrun Thäter. "Boussinesq-type approximation for second-grade fluids." International Journal of Non-Linear Mechanics 40, no. 6 (July 2005): 821–31. http://dx.doi.org/10.1016/j.ijnonlinmec.2004.07.019.

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15

Nadolin, K. A. "Boussinesq approximation in the Rayleigh-Benard problem." Fluid Dynamics 30, no. 5 (September 1995): 645–51. http://dx.doi.org/10.1007/bf02079380.

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16

Lopez, Jose M., Francisco Marques, and Marc Avila. "The Boussinesq approximation in rapidly rotating flows." Journal of Fluid Mechanics 737 (November 15, 2013): 56–77. http://dx.doi.org/10.1017/jfm.2013.558.

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AbstractIn commonly used formulations of the Boussinesq approximation centrifugal buoyancy effects related to differential rotation, as well as strong vortices in the flow, are neglected. However, these may play an important role in rapidly rotating flows, such as in astrophysical and geophysical applications, and also in turbulent convection. Here we provide a straightforward approach resulting in a Boussinesq-type approximation that consistently accounts for centrifugal effects. Its application to the accretion-disc problem is discussed. We numerically compare the new approach to the typical one in fluid flows confined between two differentially heated and rotating cylinders. The results justify the need of using the proposed approximation in rapidly rotating flows.

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17

Lamb, K. G. "Energetics of internal solitary waves in a background sheared current." Nonlinear Processes in Geophysics 17, no. 5 (October 8, 2010): 553–68. http://dx.doi.org/10.5194/npg-17-553-2010.

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Abstract. The energetics of internal waves in the presence of a background sheared current is explored via numerical simulations for four different situations based on oceanographic conditions: the nonlinear interaction of two internal solitary waves; an internal solitary wave shoaling through a turning point; internal solitary wave reflection from a sloping boundary and a deep-water internal seiche trapped in a deep basin. In the simulations with variable water depth using the Boussinesq approximation the combination of a background sheared current, bathymetry and a rigid lid results in a change in the total energy of the system due to the work done by a pressure change that is established across the domain. A final simulation of the deep-water internal seiche in which the Boussinesq approximation is not invoked and a diffuse air-water interface is added to the system results in the energy remaining constant because the generation of surface waves prevents the establishment of a net pressure increase across the domain. The difference in the perturbation energy in the Boussinesq and non-Boussinesq simulations is accounted for by the surface waves.

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18

Castro, Ángel, Diego Córdoba, and Daniel Lear. "On the asymptotic stability of stratified solutions for the 2D Boussinesq equations with a velocity damping term." Mathematical Models and Methods in Applied Sciences 29, no. 07 (June 24, 2019): 1227–77. http://dx.doi.org/10.1142/s0218202519500210.

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We consider the 2D Boussinesq equations with a velocity damping term in a strip domain, with impermeable walls. In this physical scenario, where the Boussinesq approximation is accurate when density or temperature variations are small, our main result is the asymptotic stability for a specific type of perturbations of a stratified solution.

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19

Selmi, Ridha. "Global Well-Posedness and Convergence Results for the 3D-Regularized Boussinesq System." Canadian Journal of Mathematics 64, no. 6 (December 1, 2012): 1415–35. http://dx.doi.org/10.4153/cjm-2012-013-5.

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Abstract Analytical study of the regularization of the Boussinesq systemis performed in frequency space using Fourier theory. Existence and uniqueness of weak solutions with minimum regularity requirement are proved. Convergence results of the unique weak solution of the regularized Boussinesq system to a weak Leray-Hopf solution of the Boussinesq system are established as the regularizing parameter α vanishes. The proofs are done in the frequency space and use energy methods, the Arselà-Ascoli compactness theorem and a Friedrichs-like approximation scheme.

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20

Greatbatch, Richard J., Youyu Lu, and Yi Cai. "Relaxing the Boussinesq Approximation in Ocean Circulation Models." Journal of Atmospheric and Oceanic Technology 18, no. 11 (November 2001): 1911–23. http://dx.doi.org/10.1175/1520-0426(2001)018<1911:rtbaio>2.0.co;2.

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21

Zeytounian, Radyadour Kh. "Joseph Boussinesq and his approximation: a contemporary view." Comptes Rendus Mécanique 331, no. 8 (August 2003): 575–86. http://dx.doi.org/10.1016/s1631-0721(03)00120-7.

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22

Anufriev, Alexander P., and Pavel Hejda. "Earth's core convection: Boussinesq approximation or incompressible approach?" Geophysical & Astrophysical Fluid Dynamics 104, no. 1 (February 2010): 65–83. http://dx.doi.org/10.1080/03091920903517863.

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23

Bowker, Jordan A., David W. Hughes, and Evy Kersalé. "Incorporating velocity shear into the magneto-Boussinesq approximation." Geophysical & Astrophysical Fluid Dynamics 108, no. 5 (June 30, 2014): 553–67. http://dx.doi.org/10.1080/03091929.2014.917296.

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24

Durran, Dale R., and Akio Arakawa. "Generalizing the Boussinesq approximation to stratified compressible flow." Comptes Rendus Mécanique 335, no. 9-10 (September 2007): 655–64. http://dx.doi.org/10.1016/j.crme.2007.08.010.

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25

Kagei, Yoshiyuki, and Michael Růžička. "The Oberbeck–Boussinesq approximation as a constitutive limit." Continuum Mechanics and Thermodynamics 28, no. 5 (December 12, 2015): 1411–19. http://dx.doi.org/10.1007/s00161-015-0483-9.

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26

Duchêne, Vincent. "Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation." ESAIM: Mathematical Modelling and Numerical Analysis 46, no. 1 (October 3, 2011): 145–85. http://dx.doi.org/10.1051/m2an/2011037.

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27

El-Gendi, Mahmoud M., and Abdelraheem M. Aly. "Numerical simulation of natural convection using unsteady compressible Navier-stokes equations." International Journal of Numerical Methods for Heat & Fluid Flow 27, no. 11 (November 6, 2017): 2508–27. http://dx.doi.org/10.1108/hff-10-2016-0376.

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Purpose Boussinesq approximation is widely used in solving natural convection problems, but it has severe practical limitations. Using Boussinesq approximation, the temperature difference should be less than 28.6 K. The purpose of this study is to get rid of Boussinesq approximation and simulates the natural convection problems using an unsteady compressible Navier-Stokes solver. The gravity force is included in the source term. Three temperature differences are used namely 20 K, 700 K and 2000 K. Design/methodology/approach The calculations are carried out on the square and sinusoidal cavities. The results of low temperature difference have good agreement with the experimental and previous calculated data. It is found that, the high temperature difference has a significant effect on the density. Findings Due to mass conservation, the density variation affects the velocity distribution and its symmetry. On the other hand, the density variation has a negligible effect on the temperature distribution. Originality/value The present calculation method has no limitations but its convergence is slow. The current study can be used in fluid flow simulations for nuclear power applications in natural convection flows subjected to large temperature differences.

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28

Zhang, Yao, Andrew Brian Kennedy, Joannes Westerink, Nishant Panda, and Client Dawson. "NEW BOUSSINESQ SYSTEM FOR NONLINEAR WATER WAVES." Coastal Engineering Proceedings 1, no. 33 (October 12, 2012): 4. http://dx.doi.org/10.9753/icce.v33.waves.4.

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In this paper, a new Boussinesq water wave theory is derived which can simulate highly dispersive nonlinear waves, their depth-varying velocities, and wave-induced currents, from very deep, but still finite, depths through the surf zone to the shoreline.. Boussinesq scaling is employed. We removed the irrotationality assumption by using polynomial basis functions for velocity profile which are inserted into basic equations of motion. Keep terms up to the desired approximation level and solve the coupled weighted residual system together with vertically integrated mass equation. The computational cost is similar to normal Boussinesq theories although there are more unknown variables to be solved than that in normal Boussinesq models. Because we can reduce the number of the coupled equations by multiplying some coefficients and subtracting from each other which means the matix to be solved is in similar size as normal Boussinesq models. The models show rapid convergence to exact solutions for linear dispersion, shoaling, and orbital velocities; however, properties may be simultaneously and substantially improved for a given order of approximation using asymptotic rearrangements. This improvement is accomplished using the large numbers of degrees of freedom inherent in the definitions of the polynomial basis functions either to match additional terms in a Taylor series, or to minimize errors over a range. Future work will be focused on rotational performance in 2D model by including viscosity,breaking and turbulence.

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29

Dhar, A. K., and K. P. Das. "Stability analysis from fourth order evolution equation for small but finite amplitude interfacial waves in the presence of a basic current shear." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 35, no. 3 (January 1994): 348–65. http://dx.doi.org/10.1017/s0334270000009346.

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AbstractA fourth-order nonlinear evolution equation is derived for a wave propagating at the interface of two superposed fluids of infinite depths in the presence of a basic current shear. On the basis of this equation a stability analysis is made for a uniform wave train. Discussions are given for both an air-water interface and a Boussinesq approximation. Significant deviations are noticed from the results obtained from the third-order evolution equation, which is the nonlinear Schrödinger equation. In the Boussinesq approximation, it has been possible to compare the present results with the exact numerical analysis of Pullin and Grimshaw [12], and they are found to agree rather favourably.

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30

Melinand, Benjamin. "Long wave approximation for water waves under a Coriolis forcing and the Ostrovsky equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 6 (July 19, 2018): 1201–37. http://dx.doi.org/10.1017/s0308210518000136.

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This paper is devoted to the study of the long wave approximation for water waves under the influence of the gravity and a Coriolis forcing. We start by deriving a generalization of the Boussinesq equations in one (spatial) dimension and we rigorously justify them as an asymptotic model of water wave equations. These new Boussinesq equations are not the classical Boussinesq equations: a new term due to the vorticity and the Coriolis forcing appears that cannot be neglected. We study the Boussinesq regime and derive and fully justify different asymptotic models when the bottom is flat: a linear equation linked to the Klein–Gordon equation admitting the so-called Poincaré waves; the Ostrovsky equation, which is a generalization of the Korteweg–de Vries (KdV) equation in the presence of a Coriolis forcing, when the rotation is weak; and the KdV equation when the rotation is very weak. Therefore, this work provides the first mathematical justification of the Ostrovsky equation. Finally, we derive a generalization of the Green–Naghdi equations in one spatial dimension for small topography variations and we show that this model is consistent with the water wave equations.

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31

Bannon, Peter R., Jeffrey M. Chagnon, and Richard P. James. "Mass Conservation and the Anelastic Approximation." Monthly Weather Review 134, no. 10 (October 1, 2006): 2989–3005. http://dx.doi.org/10.1175/mwr3228.1.

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Abstract Numerical anelastic models solve a diagnostic elliptic equation for the pressure field using derivative boundary conditions. The pressure is therefore determined to within a function proportional to the base-state density field with arbitrary amplitude. This ambiguity is removed by requiring that the total mass be conserved in the model. This approach enables one to determine the correct temperature field that is required for the microphysical calculations. This correct, mass-conserving anelastic model predicts a temperature field that is an accurate approximation to that of a compressible atmosphere that has undergone a hydrostatic adjustment in response to a horizontally homogeneous heating or moistening. The procedure is demonstrated analytically and numerically for a one-dimensional, idealized heat source and moisture sink associated with moist convection. Two-dimensional anelastic simulations compare the effect of the new formulation on the evolution of the flow fields in a simulation of the ascent of a warm bubble in a conditionally unstable model atmosphere. In the Boussinesq case, the temperature field is determined uniquely from the heat equation despite the fact that the pressure field can only be determined to within an arbitrary constant. Boussinesq air parcels conserve their volume, not their mass.

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32

Ahmad, Samaila K., Basant k. Jha, Ayyub M. Hussaini, and Muhammad M. Altine. "MHD Free-Convective Couette flow in a Vertical Porous Microchannel Using Non-Linear Boussinesq Approximation." Journal of Applied Science, Information and Computing 3, no. 2 (December 31, 2022): 25–34. http://dx.doi.org/10.59568/jasic-2022-3-2-05.

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An analytical solution for free convection flow of an electrically conducting fluid in a vertical micro-porous-channel, in the existence of transversely applied magnetic-field and nonlinear Boussinesq approximation is carried out in this article. The governing equations representing stated objective are obtained and solved analytically using method of undetermiend coefficients and direct integration. Pictorial and tabularlar representions of solutions obtained are carried out, so as to ascertain the role of various governing parameters entering flow formation. During the course of numerical simulation of results, it is found that the volumetric flow rate increases with increase in Couette flow parameter, asymmetric heating parameter and suction/injection parameter but decreases with increase in nonlinear Boussinesq approximation parameter.

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33

Artemov, Mikhail A., and Evgenii S. Baranovskii. "Solvability of the Boussinesq Approximation for Water Polymer Solutions." Mathematics 7, no. 7 (July 10, 2019): 611. http://dx.doi.org/10.3390/math7070611.

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We consider nonlinear Boussinesq-type equations that model the heat transfer and steady viscous flows of weakly concentrated water solutions of polymers in a bounded three-dimensional domain with a heat source. On the boundary of the flow domain, the impermeability condition and a slip condition are provided. For the temperature field, we use a Robin boundary condition corresponding to the classical Newton law of cooling. By using the Galerkin method with special total sequences in suitable function spaces, we prove the existence of a weak solution to this boundary-value problem, assuming that the heat source intensity is bounded. Moreover, some estimates are established for weak solutions.

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34

Mikaelian, Karnig O. "Boussinesq approximation for Rayleigh-Taylor and Richtmyer-Meshkov instabilities." Physics of Fluids 26, no. 5 (May 2014): 054103. http://dx.doi.org/10.1063/1.4874881.

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35

Lannes, D., and Jean-Claude Saut. "Weakly transverse Boussinesq systems and the Kadomtsev–Petviashvili approximation." Nonlinearity 19, no. 12 (November 3, 2006): 2853–75. http://dx.doi.org/10.1088/0951-7715/19/12/007.

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36

Paolucci, S., and D. R. Chenoweth. "Departures from the Boussinesq approximation in laminar Bénard convection." Physics of Fluids 30, no. 5 (1987): 1561. http://dx.doi.org/10.1063/1.866218.

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37

Katsaounis, Theodoros, Dimitrios Mitsotakis, and Georges Sadaka. "Boussinesq-Peregrine water wave models and their numerical approximation." Journal of Computational Physics 417 (September 2020): 109579. http://dx.doi.org/10.1016/j.jcp.2020.109579.

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38

Tarman, I. Hakan. "A Karhunen-Loève based Galerkin approximation to Boussinesq equation." Computer Methods in Applied Mechanics and Engineering 137, no. 3-4 (November 1996): 275–84. http://dx.doi.org/10.1016/s0045-7825(96)01048-1.

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39

Eden, Carsten. "Revisiting the Energetics of the Ocean in Boussinesq Approximation." Journal of Physical Oceanography 45, no. 3 (March 2015): 630–37. http://dx.doi.org/10.1175/jpo-d-14-0072.1.

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AbstractFollowing a suggestion by Tailleux, a consistent formulation of internal energy, the first law of thermodynamics, and the thermodynamic potentials for an ocean in Boussinesq approximation with a nonlinear equation of state is given. A modification of the pressure work in the first law is the only necessary modification from which all thermodynamic potentials and thermodynamic relations follow in a consistent way. This treatment of thermodynamics allows for a closed and explicit formulation of conservation equations for dynamic and potential reservoirs of both enthalpy and internal energy, which differentiate approximately reversible from irreversible effects on internal energy, and allows for a formulation of a closed energy cycle on which energetically consistent ocean models can be based on.

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40

ROTUNNO, R., J. B. KLEMP, G. H. BRYAN, and D. J. MURAKI. "Models of non-Boussinesq lock-exchange flow." Journal of Fluid Mechanics 675 (April 8, 2011): 1–26. http://dx.doi.org/10.1017/jfm.2010.648.

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Nearly all analytical models of lock-exchange flow are based on the shallow-water approximation. Since the latter approximation fails at the leading edges of the mutually intruding fluids of lock-exchange flow, solutions to the shallow-water equations can be obtained only through the specification of front conditions. In the present paper, analytic solutions to the shallow-water equations for non-Boussinesq lock-exchange flow are given for front conditions deriving from free-boundary arguments. Analytic solutions are also derived for other proposed front conditions – conditions which appear to the shallow-water system as forced boundary conditions. Both solutions to the shallow-water equations are compared with the numerical solutions of the Navier–Stokes equations and a mixture of successes and failures is recorded. The apparent success of some aspects of the forced solutions of the shallow-water equations, together with the fact that in a real fluid the density interface is a free boundary, shows the need for an improved theory of lock-exchange flow taking into account non-hydrostatic effects for density interfaces intersecting rigid boundaries.

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41

Zhang, Lifang, Jianmin Zhang, Yong Peng, Jiangyang Pan, and Zhongxian Peng. "Numerical Simulation of Flow and Temperature Fields in a Deep Stratified Reservoir Using Water-Separating Curtain." International Journal of Environmental Research and Public Health 16, no. 24 (December 16, 2019): 5143. http://dx.doi.org/10.3390/ijerph16245143.

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In this work, the flow and temperature fields of a thermally stratified reservoir under different settings of a water-separating curtain are simulated by using the standard k-ε turbulence model. In the simulation, two different equations of state including Boussinesq approximation and the density-temperature function have been used and compared. This study shows that Boussinesq approximation is more time-saving, and the density-temperature function has higher computational accuracy. Thus, the standard k-ε turbulence model with two equations of state is applied to study the effect of adding a water-separating curtain in the stratified reservoir on the Discharged Water Temperature (DWT). It is found that adding the Water-Separating Curtain (WSC) can effectively increase the discharged water temperature. Moreover, the different arrangements of WSC have obvious effects on the discharged water temperature. For example, the increased temperature by adding a WSC with full sealing is 1 °C higher than that by using the WSC with a bottom opening height of 2 m. However, the maximum pressure difference acting on the WSC for the former WSC is 100 Pa higher than that for the latter WSC. In addition, this study shows that the different equations of state have little effect on the simulation results. Considering the calculation efficiency, equations of state using the Boussinesq approximation can be recommended to save the calculation time.

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Mahanty, Debarati, Reeba Babu, and B. Mahanthesh. "Theoretical and analytical analysis of convective heat transport of radiated micropolar fluid over a vertical plate under nonlinear Boussinesq approximation." Multidiscipline Modeling in Materials and Structures 16, no. 5 (May 26, 2020): 915–36. http://dx.doi.org/10.1108/mmms-05-2019-0099.

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PurposeIn heat transfer problems, if the temperature difference is not sufficiently so small then the linear Boussinesq approximation is not adequate to describe thermal analysis. Also, nonlinear density variation with respect to temperature/concentration has a significant impact on heat and fluid flow characteristics. Because of this reason, the impact of nonlinear density variation in the buoyancy force term cannot be neglected. Therefore in this paper, the unsteady flow and heat transfer of radiating magneto-micropolar fluid by considering nonlinear Boussinesq approximation is investigated analytically.Design/methodology/approachThe flow is fully developed and time-dependent. Heat and mass flux boundary conditions are also accounted in the analysis. The governing equations of transport phenomena are treated analytically using regular perturbation method. To analyze the tendency of the obtained solutions, a parametric study is performed.FindingsIt is established that the velocity field is directly proportional to the nonlinear convection parameter and the same trend is observed with the increase of the value of Grashof number. The micro-rotational velocity profile decreases with increase in the nonlinear convection parameter. Further, the temperature profile increases due to the presence of radiative heat aspect.Originality/valueThe effectiveness of nonlinear Boussinesq approximation in the flow of micropolar fluid past a vertical plate in the presence of thermal radiation and magnetic dipole is investigated for the first time.

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Gravanis, Elias, Evangelos Akylas, and Ernestos Nikolas Sarris. "Approximate Solutions for Horizontal Unconfined Aquifers in the Buildup Phase." Water 16, no. 7 (April 2, 2024): 1031. http://dx.doi.org/10.3390/w16071031.

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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.

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Manaa, Saad A., and Nergiz M. Mosa. "Adomian Decomposition and Successive Approximation Methods for Solving Kaup-Boussinesq System." Science Journal of University of Zakho 7, no. 3 (September 30, 2019): 101–7. http://dx.doi.org/10.25271/sjuoz.2019.7.3.582.

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The Kaup-Boussinesq system has been solved numerically by using two methods, Successive approximation method (SAM) and Adomian decomposition method (ADM). Comparison between the two methods has been made and both can solve this kind of problems, also both methods are accurate and has faster convergence. The comparison showed that the Adomian decomposition method much more accurate than Successive approximation method.

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Korre, L., N. Brummell, and P. Garaud. "Boussinesq convection in a gaseous spherical shell." EAS Publications Series 82 (2019): 373–82. http://dx.doi.org/10.1051/eas/1982033.

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In this paper, we investigate the dynamics of convection in a spherical shell under the Boussinesq approximation but considering the compressibility which arises from a non zero adiabatic temperature gradient, a relevant quantity for gaseous objects such as stellar or planetary interiors. We find that depth-dependent superiadiabaticity, combined with the use of mixed boundary conditions (fixed flux/fixed temperature), gives rise to unexpected dynamics that were not previously reported.

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Xie, Jing, Chen Miao, Zhi Li Gao, Jian Bing Shi, and Tai Wang. "Numerical Simulation of Air Curtain Used in the Cold Store through Different Models." Advanced Materials Research 749 (August 2013): 554–60. http://dx.doi.org/10.4028/www.scientific.net/amr.749.554.

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Air curtains are commonly used to cut off air flow between the cold store and hot environment, and reduce heat and mass transfer in order to maintain the low temperature in the cold store. CFD can be used to predict operation rule of air curtain used in the cold store intuitively and modelling is the most important part of numerical simulation of air curtain used in the cold store. In this study, different numerical models including standardmodel (with and without boussinesq approximation) and RSM (with and without boussinesq approximation) were used to simulate the temperature field and air flow field in the cold store and operation rule of air curtain used in the cold store after air curtain opened for 60s. Meanwhile, the actual operation of air curtain used in the cold store was tested and the simulation values were compared with the experimental values. The results showed that the velocity of the central mainstream decayed slowly, but the velocity of both sides of air curtain decayed fast. The optimal model used to predict the temperature field and air flow field in the cold store and operation rule of air curtain used in the cold store was standardmodel with boussinesq approximation, the relative error was within 20%. The optimal model can be used to predict the temperature field and air flow field in the cold store during different times in the future research.

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Wei, Ge, James T. Kirby, Stephan T. Grilli, and Ravishankar Subramanya. "A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves." Journal of Fluid Mechanics 294 (July 10, 1995): 71–92. http://dx.doi.org/10.1017/s0022112095002813.

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Fully nonlinear extensions of Boussinesq equations are derived to simulate surface wave propagation in coastal regions. By using the velocity at a certain depth as a dependent variable (Nwogu 1993), the resulting equations have significantly improved linear dispersion properties in intermediate water depths when compared to standard Boussinesq approximations. Since no assumption of small nonlinearity is made, the equations can be applied to simulate strong wave interactions prior to wave breaking. A high-order numerical model based on the equations is developed and applied to the study of two canonical problems: solitary wave shoaling on slopes and undular bore propagation over a horizontal bed. Results of the Boussinesq model with and without strong nonlinearity are compared in detail to those of a boundary element solution of the fully nonlinear potential flow problem developed by Grilli et al. (1989). The fully nonlinear variant of the Boussinesq model is found to predict wave heights, phase speeds and particle kinematics more accurately than the standard approximation.

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Wang, Yinxia. "On the Cauchy problem for one dimension generalized Boussinesq equation." International Journal of Mathematics 26, no. 03 (March 2015): 1550023. http://dx.doi.org/10.1142/s0129167x15500238.

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In this paper, we study the Cauchy problem for one dimension generalized damped Boussinesq equation. First, global existence and decay estimate of solutions to this problem are established. Second, according to the detail analysis for solution operator the generalized damped Boussinesq equation, the nonlinear approximation to global solutions is established. Finally, we prove that the global solution u to our problem is asymptotic to the superposition of nonlinear diffusion waves expressed in terms of the self-similar solution of the viscous Burgers equation as time tends to infinity.

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Dewar, W. K., J. Schoonover, T. J. McDougall, and W. R. Young. "Semicompressible Ocean Dynamics." Journal of Physical Oceanography 45, no. 1 (January 2015): 149–56. http://dx.doi.org/10.1175/jpo-d-13-0268.1.

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AbstractThe equations of motion are reexamined with the objective of improving upon the Boussinesq approximation. The authors derive new equations that conserve energy, filter out sound waves, are more accurate than the Boussinesq set, and are computationally competitive with them. The new equations are partly enabled by exploiting a reversible exchange between internal and gravitational potential fluid energy. To improve upon these equations appears to require the inclusion of acoustics, at which point one should use full Navier–Stokes. This study recommends the new sets for testing in general circulation modeling.

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Romps, David M. "Weak Pressure Gradient Approximation and Its Analytical Solutions." Journal of the Atmospheric Sciences 69, no. 9 (September 1, 2012): 2835–45. http://dx.doi.org/10.1175/jas-d-11-0336.1.

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Abstract A weak pressure gradient (WPG) approximation is introduced for parameterizing supradomain-scale (SDS) dynamics, and this method is compared to the relaxed form of the weak temperature gradient (WTG) approximation in the context of 3D, linearized, damped, Boussinesq equations. It is found that neither method is able to capture the two different time scales present in the full 3D equations. Nevertheless, WPG is argued to have several advantages over WTG. First, WPG correctly predicts the magnitude of the steady-state buoyancy anomalies generated by an applied heating, but WTG underestimates these buoyancy anomalies. It is conjectured that this underestimation may short-circuit the natural feedbacks between convective mass fluxes and local temperature anomalies. Second, WPG correctly predicts the adiabatic lifting of air below an initial buoyancy perturbation; WTG is unable to capture this nonlocal effect. It is hypothesized that this may be relevant to moist convection, where adiabatic lifting can reduce convective inhibition. Third, WPG agrees with the full 3D equations on the counterintuitive fact that an isolated heating applied to a column of Boussinesq fluid leads to a steady ascent with zero column-integrated buoyancy. This falsifies the premise of the relaxed form of WTG, which assumes that vertical velocity is proportional to buoyancy.

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