Boussinesq Model Research Articles

Vortex shedding suppression in mixed convective flow past a square cylinder subjected to large-scale heating using a non-Boussinesq model

In the present work, numerical simulations are performed to investigate the vortex shedding suppression phenomenon for mixed convective flows past a square cylinder in the large-scale heating regime. A full compressible flow model with variable transport and thermo-physical properties is employed to capture large-scale heating effects. The Reynolds number, Prandtl number, and Mach number are kept constant at Re = 100, Pr = 0.71, and M = 0.1, while the cylinder inclination (ϕ), free-stream inclination (α), and over-heat ratio (ϵ) are varied in the range [0, 45°], [0, 90°], and [0, 2], respectively. The governing equations are solved numerically using the particle velocity upwind (PVU-M+) scheme. The buoyancy parameter which governs the vortex shedding suppression process in the non-Boussinesq model is identified as RiNB=ϵFr222+ϵ, where Fr is the Froude number. Using the Stuart-Landau model, the neutral curves separating the steady and unsteady flow regimes are generated in the ϵ–ϕ and ϵ–α parametric spaces. The neutral curves show qualitatively similar characteristics as observed for Boussinesq models. The relative contribution of various large-scale heating effects in suppression of vortex shedding is also highlighted. This reveals that buoyancy effects followed by variations in transport properties play a major role in suppression of vortex shedding. The findings are also applicable to a range of low Re (O(100)) as supported by data obtained at Re = 130 for ϕ = 40°. The mechanism of vortex shedding suppression has been analyzed and extended for the large-scale heating scenarios.

A spectral/[formula omitted] element depth-integrated model for nonlinear wave–body interaction

We present a depth-integrated Boussinesq model for the efficient simulation of nonlinear wave–body interaction. The model exploits a ‘unified’ Boussinesq framework, i.e. the fluid under the body is also treated with the depth-integrated approach. The unified Boussinesq approach was initially proposed by Jiang (2001) and recently analyzed by Lannes (2017). The choice of Boussinesq-type equations removes the vertical dimension of the problem, resulting in a wave–body model with adequate precision for weakly nonlinear and dispersive waves expressed in horizontal dimensions only. The framework involves the coupling of two different domains with different flow characteristics. Inside each domain, the continuous spectral/hp element method is used to solve the appropriate flow model since it allows to achieve high-order, possibly exponential, convergence for non-breaking waves. Flux-based conditions for the domain coupling are used, following the recipes provided by the discontinuous Galerkin framework. The main contribution of this work is the inclusion of floating surface-piercing bodies in the conventional depth-integrated Boussinesq framework and the use of a spectral/hp element method for high-order accurate numerical discretization in space. The model is verified using manufactured solutions and validated against published results for wave–body interaction. The model is shown to have excellent accuracy and is relevant for applications of waves interacting with wave energy devices.

Study of irregular wave run-up over fringing reefs based on a shock-capturing Boussinesq model

This paper presents the results of a parametric study of irregular wave run-up over fringing reefs using the shock-capturing Boussinesq wave model Funwave-TVD to better understand the role of fringing reefs in the mitigation of wave-driven flooding. Laboratory experiments were newly performed with a typical fringing reef profile and typical hydrodynamic conditions to validate the model. Experimental data shows irregular wave run-ups are dominated by the low-frequency motions and confirms the run-up resonant phenomenon over the back-reef slope, which has been revealed in previous numerical studies. It is demonstrated that irregular wave evolution and run-up over fringing reefs are reasonably reproduced by the present model with a proper grid size. However, the infragravity run-up height and highest 2% run-up height over the back-reef slope are under-predicted due to the underestimation of the infragravity wave height over the reef flat. The validated model was then utilized to model irregular wave transformations and run-ups under different conditions. Through a series of numerical experiments, the effects of key hydrodynamic and reef geometry parameters, including the reef flat width, water depth over the reef flat, fore-reef slope angle and back-reef slope angle, on the irregular wave run-up were investigated. Variations of spectral components of irregular wave run-ups were examined to better understand the physical process underlying the effect of each parameter.

CFD Assisted Study of Multi-Chapels Greenhouse Vents Openings Effect on Inside Airflow Circulation and Microclimate Patterns

The aim of this work is to study and quantify the air mass flow exchanged between inside and outside of the greenhouse, in order to determine the ventilation openings layout and the design effect on greenhouse airflow and microclimate distribution. The study was conducted over a 945 m2 multi-chapels arched greenhouse with a polyethylene cover and has thirteen crop rows oriented from north to south; the greenhouse was equipped with side wall and roof vents openings. A simulation was performed using different arrangements and configurations of ventilation openings with the same wind direction. Numerical simulation has been adopted in three dimensions (CFD), using the Fluent computer code which relies on the resolution of the Navier-Stokes equations. These equations were solved in the presence of the turbulence model (k - e) and the Boussinesq model equation adopted to incorporate buoyancy forces. The effects of solar and atmospheric radiation were included by solving the radiative transfer equation (RTE), using Discrete Ordinate (DO) model. The effects of the roof openings, the presence of anti-insect screens and crops orientation were investigated and quantified. In a 3-span greenhouse with an anti-aphid in-sect screen in the vent openings, combining roof and sidewall vents gave a ventilation rate per unit opening area that was 1.4 times more than with only side vents. In the latter case, the difference of temperature between the inside and the outside of the greenhouse was greater than 3°C. Numerical simulations with an anti-insect screen having a porosity of 56% showed that the air exchange rate with combined ventilation was reduced by 48%. Finally, the paper focused on the effect of vent arrangement on the efficiency of the ventilation and the distribution of the microclimate inside the greenhouse. Results showed that computed ventilation rates varied from 53.43 to 70.95 kg/s, whereas temperature differences varied from 7.15°C to 10.14°C. This study also showed that other characteristics such as climate heterogeneity must be investigated in order to define the best ventilation configuration.

TSUNAMI PHASE SPEED REDUCTION DUE TO WATER COMPRESSIBILTY

Most existing tsunami propagation models consider the ocean to be an incompressible, homogenous medium. Recently, it has been shown that a number of physical features can slow the propagation speed of tsunami waves, including wave frequency dispersion, ocean bottom elasticity, water compressibility and thermal or salinity stratification. These physical effects are secondary to the leading order, shallow water or long wave behavior, but still play a quantifiable role in tsunami arrival time, especially at far distant locations. In this work, we have performed analytical and numerical investigations and have shown that consideration of those effects can actually improve the prediction of arrival time at distant stations, compared to incompressible forms of wave equations. We derive a modified Mild Slope Equation for Weakly Compressible fluid following the method proposed by Sammarco et al. (2013) and Abdolali et al. (2015) using linearized wave theory, and then describe comparable extensions to the Boussinesq model of Kirby et al. (2013). Both models account for water compressibility and compression of static water column to simulate tsunami waves. The mild slope model is formulated in plane Cartesian coordinates and is thus limited to medium propagation distances, while the Boussinesq model is formulated in spherical polar coordinates and is suitable for ocean scale simulations.

ESTIMATION OF BOUND AND RELEASED INFRAGRAVITY WAVES BASED ON WAVE OBSERVATION AND NUMERICAL SIMULATION IN SHALLOW WATER

It is mentioned that observed infragravity waves consist of bound waves propagating with short-wave groups, released waves due to reduction of short crest waves and free waves existing in a field. Though it is difficult to distinguish among them, a standard spectrum for infragravity waves is defined by using a relation to a wind wave spectrum. In this study, a comprehensive definition of standard spectrum is newly proposed to estimate infragravity wave heights with the relation between the ratio of wave height and the ursell number of observed wave property, represented by selected data of wave observation in shallow water. Moreover, the release process of bound waves at a harbor entrance is reproduced in numerical simulation using a Boussinesq model for short-wave transformation. These results are verified by comparison to infragravity waves observed at outside/inside of a harbor for a month.

EXTENDED BOUSSINESQ EQUATIONS FOR WAVES IN TWO POROUS LAYERS

In this study we continue the work of Vu et al. (2018) [Coastal Eng. 139, 85-97] to develop an extended Boussinesq model that predicts the propagation of water waves in two porous layers. The first and second layer can be a water layer or a porous layer. The inertial and drag resistances are considered in the developed model. After being successfully validated against the analytical solutions, the model is used to simulate waves propagating over a submerged triangular porous bar. The numerical results show good agreement with the physical experimental data of Hsiao et al. (1998) [Proc. Royal Society of London A 458, 1291-1322].

A HIGH-ORDER SPECTRAL ELEMENT UNIFIED BOUSSINESQ MODEL FOR FLOATING POINT ABSORBERS

Nonlinear wave-body problems are important in renewable energy, especially in case of wave energy converters operating in the near-shore region. In this paper we simulate nonlinear interaction between waves and truncated bodies using an efficient spectral/hp element depth-integrated unified Boussinesq model. The unified Boussinesq model treats also the fluid below the body in a depth-integrated approach. We illustrate the versatility of the model by predicting the reflection and transmission of solitary waves passing truncated bodies. We also use the model to simulate the motion of a latched heaving box. In both cases the unified Boussinesq model show acceptable agreement with CFD results – if applied within the underlying assumptions of dispersion and nonlinearity – but with a significant reduction in computational effort.

On the conditions for existence of unidirectional motions of binary mixtures in the Oberbeck–Boussinesq model

On the conditions for existence of unidirectional motions of binary mixtures in the Oberbeck–Boussinesq model

Shock-Capturing Boussinesq Modelling of Broken Wave Characteristics Near a Vertical Seawall

Broken wave characteristics in front of a vertical seawall were modeled and studied using a shock-capturing Boussinesq wave model FUNWAVE-TVD. Validation with the experimental data confirmed the capability of FUNWAVE-TVD in predicting the wave characteristics via the shock-capturing method. Compared to the results obtained from the Boussinesq model coupled with an empirical breaking model, the advantage of the present shock-capturing model for the broken waves near a vertical seawall was clearly revealed. A preliminary investigation of the effects of the key parameters, such as the incident wave height, water level at the seawall, and seabed slope, on the wave kinematics (i.e., the root mean square of the surface fluctuations and depth-averaged horizontal velocity) near the seawall was then conducted through a series of numerical experiments. The numerical results indicate the incident wave height and the water depth at the seawall are the important parameters in determining the magnitude of the wave kinematics, while the effect of the seabed slope seems to be insignificant. The role of the breaking point locations is also highlighted in this study, in which case further breaking can reduce the wave kinematics significantly for the coastal structures predominately subjected to broken waves.

High-resolution coastal hazard assessment along the French Riviera from co-seismic tsunamis generated in the Ligurian fault system

The French Riviera is a densely populated and touristic coast. It is also one of the most seismically active areas of the Western Mediterranean. This is evidenced by the Mw 6.7–6.9, 1887 earthquake and tsunami, that was triggered nearshore, rupturing the easternmost 40 km of the 80-km-long Ligurian fault system, which runs parallel to and offshore of the Riviera. Here, coastal hazard from co-seismic tsunamis is assessed along the French and part of the Italian Riviera by simulating three Ligurian earthquake scenarios: (1) the 1887 event offshore Genoa, Italy; (2) a similar event transposed to the westernmost 40-km segment of the fault, offshore Nice, France; and (3) the rupture of the entire 80-km fault, which constitutes an extreme case scenario for the region. Simulations of tsunami propagation and coastal impact are performed by one-way coupling with the Boussinesq model FUNWAVE-TVD, in a series of nested grids, using new high-resolution bathymetric and topographic data. Results obtained in 10-m coastal grids provide the highest resolution predictions to date for this section of the French Riviera of co-seismic tsunami coastal hazard, in terms of inundation, runup, and current velocity. In general, the most impacted areas are bays (near Cap d’Antibes and Cap Ferrat), due to wave buildup and shoaling within semi-enclosed shallow areas, enhanced by possible resonances. In contrast to earlier work, which was based on coarser resolution grids, the area of Nice harbor is found to be rather well sheltered. It should be noted that uniform fault slip was used in the ruptures and runup estimates could locally be enhanced in case of more complex ruptures, such as segmented and heterogeneous ruptures.

Advective Removal of Localized Convective Structures in a Porous Medium

Thermal convection in a planar horizontal layer of a porous medium, which is saturated with a viscous incompressible liquid pumped along the layer and has solid impermeable boundaries with a set heat flux at them, is considered. In certain physical systems, the first instability in Rayleigh–Benard convection between thermally insulated horizontal plates is long-wave. The equations characterizing large-scale thermal convection in a horizontal layer in a homogeneous liquid and in a porous medium are similar: they differ in just one term, which vanishes under certain conditions (e.g., for two-dimensional flows or an infinite Prandtl number). In the system in question with a vertical heat flux that is nonuniform along the layer, localized convective structures may emerge in the region where the heat flux exceeds the critical value corresponding to uniform heating from below and to the onset of convection within the layer. When the rate of the longitudinal pumping of a liquid through the layer changes, the system may enter either a state with stable localized convective structures and a monotonic (or oscillatory) instability or a state in which the localized convective flow is removed completely from the region of its excitation. Calculations are carried out based on the amplitude equations in the long-wave approximation using the Darcy–Boussinesq model and the approximation of small deviations of the heat flux across the boundaries from the critical values under uniform heating. The results of the numerical modeling of the process of the removal of a localized flow from the region of its excitation at higher rates of the longitudinal pumping of the liquid through the layer are detailed. Stability maps for monotonic and oscillatory instabilities of the base state of the system are presented.

Restrictions of the Boussinesq model using the example of laminar natural gas convection between vertical isothermal walls

Restrictions of the Boussinesq model using the example of laminar natural gas convection between vertical isothermal walls

Numerical investigation on effects of fringing reefs on low-frequency oscillations within a harbor

The low-frequency (LF) oscillations inside an elongated rectangular harbor near the offshore fringing reef excited by bi-chromatic short wave groups are simulated using a fully nonlinear Boussinesq model. Based on a LF wave analysis procedure, effects of reef-face slope and reef ridge on the maximum LF wave amplitude, the free and the bound LF waves and their relative components inside the harbor are investigated systematically under the conditions of the 2nd to the 5th modes and wave breaking occurring over the reef. Results show that the variation trends of the maximum LF wave amplitude and the free LF wave amplitude with respect to the reef-face slope are almost identical with each other for all the four modes studied in this paper, no matter whether the reef ridge exists or not. For the 4th and 5th modes, the reef ridge has obvious influences on the LF waves inside the harbor. Under these two modes, the existence of the reef ridge always decreases the bound LF wave amplitude and its ratio to the free LF wave amplitude inside the harbor, while its influences on the maximum LF wave amplitude and the free LF wave amplitude both depend on the reef-face slope.

On the multi-symplectic structure of Boussinesq-type systems. I: Derivation and mathematical properties

The Boussinesq equations are known since the end of the XIXst century. However, the proliferation of various Boussinesq-type systems started only in the second half of the XXst century. Today they come under various flavors depending on the goals of the modeler. At the beginning of the XXIst century an effort to classify such systems, at least for even bottoms, was undertaken and developed according to both different physical regimes and mathematical properties, with special emphasis, in this last sense, on the existence of symmetry groups and their connection to conserved quantities. Of particular interest are those systems admitting a symplectic structure, with the subsequent preservation of the total energy represented by the Hamiltonian. In the present paper a family of Boussinesq-type systems with multi-symplectic structure is introduced. Some properties of the new systems are analyzed: their relation with already known Boussinesq models, the identification of those systems with additional Hamiltonian structure as well as other mathematical features like well-posedness and existence of different types of solitary-wave solutions. The consistency of multi-symplectic systems with the full Euler equations is also discussed.

Validation of a numerical model with a benchmark experiment for melting governed by natural convection in latent thermal energy storage

We report a combined experimental and numerical investigation of a melting process representative of latent thermal energy storage systems. The purpose of the work is to assess the accuracy of numerical models of melting governed by natural convection with a benchmark experiment. The experiment consists of a rectangular box filled with a model liquid (n-octadecane) and heated symmetrically from both sides such as to allow access for shadowgraph imaging and particle image velocimetry to measure the phase state and velocities, respectively. Our numerical method for computing fluid flow, temperature, and phase state involves two different approaches: the first is a detailed model using variable thermophysical properties and the volume of fluid method to allow volume expansion in an additional air phase that we solve in two dimensions. The second is a simplified model using constant thermophysical properties and the Boussinesq approximation that we solve either in two or in three dimensions. In the first part of the work, we systematically compare the simplified (Boussinesq) with the detailed (volume of fluid) model. We find that for the given set of parameters (Ra = 2·108, A = 4, Ste = 0.092, Pr = 52), the difference between the detailed and the simplified model in predicting global quantities such as the liquid phase fraction and the total heat flow rate is smaller than 4%, whereas velocities differ up to 20%. In the second part of the work, we compare the simulations of the simplified Boussinesq model in three dimensions with the benchmark experiment. We find that the simulation predicts the liquid phase fraction and temperatures with deviations below 4%, but significantly overestimates the velocity magnitudes. Our experimental and numerical tools provide a rational framework in which the accuracy of latent thermal energy storage simulations can be systematically and comprehensively assessed.

A posteriori error analysis of an augmented fully-mixed formulation for the stationary Boussinesq model

In this paper we undertake an a posteriori error analysis along with its adaptive computation of a new augmented fully-mixed finite element method that we have recently proposed to numerically simulate heat driven flows in the Boussinesq approximation setting. Our approach incorporates as additional unknowns a modified pseudostress tensor field and an auxiliary vector field in the fluid and heat equations, respectively, which possibilitates the elimination of the pressure. This unknown, however, can be easily recovered by a postprocessing formula. In turn, redundant Galerkin terms are included into the weak formulation to ensure well-posedness. In this way, the resulting variational formulation is a four-field augmented scheme, whose Galerkin discretization allows a Raviart–Thomas approximation for the auxiliary unknowns and a Lagrange approximation for the velocity and the temperature. In the present work, we propose a reliable and efficient, fully-local and computable, residual-based a posteriori error estimator in two and three dimensions for the aforementioned method. Standard arguments based on duality techniques, stable Helmholtz decompositions, and well-known results from previous works, are the main underlying tools used in our methodology. Several numerical experiments illustrate the properties of the estimator and further validate the expected behavior of the associated adaptive algorithm.

Limitations of the Boussinesq Model on the Example of Laminary Natural Convection of Gas between Vertical Isothermal Walls

Limitations of the Boussinesq Model on the Example of Laminary Natural Convection of Gas between Vertical Isothermal Walls

A full discretisation of the time-dependent Boussinesq (buoyancy) model with nonlinear viscosity

In this article, we study the time dependent Boussinesq (buoyancy) model with nonlinear viscosity depending on the temperature. We propose and analyze first and second order numerical schemes based on finite element methods. An optimal a priori error estimate is then derived for each numerical scheme. Numerical experiments are presented that confirm the theoretical accuracy of the discretization.

Performance of various shock-capturing-type reconstruction schemes in the Boussinesq wave model, FUNWAVE-TVD

In this study, we assessed the performance of five shock-capturing-type reconstruction schemes in modeling shallow water waves implemented in the public-domain Boussinesq model, FUNWAVE-TVD. The five shock-capturing schemes include two fourth-order Monotonic Upwind Scheme for Conservation Laws - Total Variation Diminishing (MUSCL-TVD) schemes with the Minimod and van-Leer limiters, respectively, a second-order MUSCL-TVD scheme with the van-Leer limiter, a fifth-order Weighted Essentially Non-Oscillatory (WENO) scheme, and a third-order Multi-dimensional Limiting Process (MLP) scheme. Numerical simulations on solitary waves and laboratory measurements showed that for one-dimensional simulations all five shock-capturing schemes maintain non-oscillatory properties, but exhibit different numerical dissipations. Numerical dissipation can be affected by both the order of shock-capturing schemes and choice of limiters. The van-Leer limiter is more dissipative than Minimod when used in fourth-order schemes, which is different from what would be inferred from the linear analysis for the second order. For two-dimensional problems, the shock-capturing schemes can generate either fewer or more high-frequency oscillations, depending on the applied Courant–Friedrichs–Lewy (CFL) number. The most stable results occur in the second-order MUSCL-TVD scheme with the van-Leer limiter, which has large numerical dissipation, and the third-order MLP scheme, which has the ability to control the oscillations in multi-dimensional simulations. The wavelet analysis showed that high-frequency oscillations usually occur at later time and thus affect long-time simulation. Decreasing the CFL number is an effective way of reducing high-frequency oscillations in two-dimensional applications. Models with different shock-capturing schemes can predict similar strength and patterns of wave-induced currents in the breaking wave application. However, the analysis based on the vorticity or enstrophy (integral of the square of the vorticity) calculations show that the schemes with higher numerical dissipation have more significant damping effect on the vorticity field, which may affect the model accuracy in predicting wave-current interaction in a long time simulation.