Boussinesq Hypothesis Research Articles

Numerical Modeling of Solar Ponds

A SGSP is a basin of water where solar energy is trapped due to an artificially imposed salinity gradient. In a SGSP three zones can be identified: the surface and bottom zones that are both convective and an intermediate zone in between which is intended to be non-convective. This zone acts as a transparent insulation and allows the storage of solar energy at the bottom where it is available for use.A numerical model where the SGSP dynamics is described in terms of velocity u, pressure p, temperature and salt concentration σ is presented. It is based on the Navier-Stokes equations for an incompressible fluid coupled to one advection-diffusion equation for θ and one advection-diffusion equation for σ. The fluid density ρ is taken to depend on θ and σ and the Boussinesq hypothesis is adopted: the fluid density appearing in the LHS of the Navier-Stokes equation is supposed constant and equal to some reference value whereas it is assumed to be variable in the RHS.The space discretization of the governing equations is based on the respective weak formulations and the discretization employs finite elements with a pressure correction method used to decouple velocity and pressure. Integration in time is accomplished by a BDF (Backward Differentiation Formula) method with the above PDEs treated sequentially within each time step. A computer code was developed employing the finite element class library deal.II.Comparisons with available experimental results are made to validate this numerical model.

Analysis and Optimization of the Fan-Pad Evaporative Cooling System for Greenhouse Based on CFD

A CFD model was presented to simulate the distribution of air velocity and temperature in a greenhouse adopting the fan-pad cooling system in summer. The Boussinesq hypothesis was applied for the simulation of gravitation; the k-ε turbulent model and discrete ordinates model were selected to predict the distribution of air velocity and temperature inside greenhouse using the commercial software Fluent. The differences between simulated and measured air temperature varied from 0.9 to 4°C and the differences of air velocity were less than 0.15 m/s, which proved that the CFD method can estimate the distribution of air velocity and temperature in the greenhouse rationally and effectively. The validated CFD model was then used to evaluate the cooling effect and design the installment of fan and pad in terms of the crop size. The results implied that Case 3 and Case 5 should be chosen when the height of crop canopy varies from 2 m to 3 m. When it varies from 1 m to 2 m, all the cases can be effective except Case 1. When the canopy height is below 1 m, all the cases can be selected. This paper suggested that the CFD model can be used as an optimal tool for fan-pad evaporative cooling system in the greenhouse.

Investigation effect of nanoparticle mean diameter on mixed convection Al2O3-water nanofluid flow in an annulus by two phase mixture model

In this paper, laminar mixed convection of nanofluid (Al2O3–water) in horizontal concentric annulus with constant heat flux boundary condition has been studied. Two thermal boundary conditions were investigated, one in which a uniform heat flux at the inner wall and an adiabatic at the other wall, and the other inner and outer walls were heated in a same heat flux. Two phase mixture model employed to investigate effect of mean diameter of nanoparticle on the hydrodynamics and thermal characteristic. The fluid flow properties are assumed constant except for the density in the body force, which varies linearly with the temperature (Boussinesq's hypothesis), thus the fluid flow characteristics are affected by the buoyancy force. Three dimensional elliptical governing equations have been discretized using the finite volume approach (FVM) using SIMPELC algorithm to investigate fluid flow throughout of an annulus duct. Numerical simulations have been carried out for the nanoparticle volume fraction (ϕ=0.02) and various mean diameters of nanoparticles (dp) between 13 and 72nm and different values of the Grashof and Reynolds numbers. The calculated results demonstrate that Nusselt number decreases with increasing nanoparticle mean diameter while it does not influence significantly the hydrodynamic parameters. Also this results show that nanoparticle distribution at the annuluses cross section is non-uniformity.

Nusselt number and friction factor in thermally stratified turbulent channel flow under Non-Oberbeck–Boussinesq conditions

In stably stratified turbulence, computations under Oberbeck–Boussinesq (OB) hypothesis of temperature-independent fluid properties may lead to inaccurate representation of the flow field and to wrong estimates of momentum/heat transfer coefficients. This is clearly assessed here comparing direct numerical simulations of stratified turbulence under OB conditions to simulations under NOB (Non-Oberbeck–Boussinesq) conditions of temperature-dependent fluid viscosity and thermal expansion coefficient. Compared to the OB case, NOB conditions may induce local flow relaminarization with significant variations (up to 30%) of heat and momentum transfer coefficients. Together with DNS results, we propose a phenomenological model (based on turbulent bursts) for heat transfer prediction in stratified turbulence under OB and NOB conditions. Implications of NOB assumptions on mixing efficiency (i.e. flux Richardson number Rif) and turbulent Prandtl number (Prt) are also discussed. These results are of specific importance in RANS modelling, where the condition Prt=1 is usually assumed (Reynolds analogy). Although this assumption is valid in some situations (i.e. boundary layer, pipe flow) there is uncertainty about its validity for stably-stratified turbulence. We demonstrate that this assumption is inaccurate when NOB effects become significant.

Numerical Simulation of Compressible Thermo-Buoyant Flow in a Partially Opened Enclosure with Localized Heater from Below

We present two numerical models for the study of unsteady two-dimensional flow of air subjected to a heat source in a cell divided into two areas of different size by a vertical lintel. The heat source placed on the floor of the enclosure produces hot gases for a finite time. First numerical model based on solving the equations of conservation of mass, momentum and energy, uses an implicit scheme in time and a finite difference hybrid space. The injection of hot gas in the two areas led to the abandonment of the Boussinesq hypothesis. This model also takes into account variations in the physical properties of the fluid. The second model is a finite element approach which gave more detailed results, thanks to the high density of the mesh node and the incorporation of the boundary conditions in the general equations of the problem. Dynamically, the results are very significant in terms of the flow direction, circulation areas and creation of vortices. The results for two simple cases of a cell partitioned allowed us to know the characteristics of the flow adjacent to the heat source and through the two compartments of the enclosure.

Turbulent Flowfield Analysis in a Bluff-Body Burner Using PIV

The structure of inert turbulent flows, stabilized in a Bluff-Body burner, is studied considering different volumetric flows for Nitrogen jet and annular air in coflow configuration. Flowfield analysis on Bluff-Body burner is essential to improve the knowledge about this burner, which plays an important role in industrial applications. Thus, vector velocity field is performed, employing Particle Image Velocimetry technique. Also, an uncertainty analysis is performed considering parameters involved in this technique yielding 6% to velocity measurements. The acquired information produces the results based in flowfield structure, which are presented in terms of statistical momentum and Reynolds stress, in which Boussinesq Hypothesis is considered to incompressible flows. However, this hypothesis fails in certain conditions. In this way, is possible to comprehend and provide experimental data from the turbulent effects on the flowfield and also contribute to predict the combustion flows, in order to enable the validation and develop numerical models.

Influence of thermal performance on design parameters of a He/LiPb dual coolant DEMO concept blanket design

Spanish Breeding Blanket Technology Programme TECNO_FUS is exploring the technological capabilities of a Dual-Coolant He/Pb15.7Li breeding blanket for DEMO and studying new breeding blanket design specifications. The progress of the channel conceptual design is being conducted in parallel with the extension of MHD computational capabilities of CFD tools and the underlying physics of MHD models. A qualification of MHD effects under present blanket design specifications and some approaches to their modelling were proposed by the authors in [1]. The analysis was accomplished with the 2D transient algorithm from Sommeria and Moreau [2] and implemented in the OpenFOAM CFD toolbox [3]. The thermal coupling was implemented by means of the Boussinesq hypothesis. Previous analyses showed the need of improvement of FCI thickness and thermal properties in order to obtain a desirable liquid metal temperature gain of 300°C. In the present study, an assessment through sensitivity and parametric analyses of the required FCI thickness is performed.Numerical simulations have been carried out considering a Robin-type thermal boundary condition which assumes 1D steady state thermal balance across the solid FCI and Eurofer layers. Such boundary condition has been validated with a fluid–solid coupled domain analysis.Results for the studied flow conditions and channel dimensions show that, in order to obtain a liquid metal temperature gain of about 300°C, the required FCI material should have a very small effective heat transfer coefficient ((k/δ)≤1W/m2K) and fluid velocities should be about 0.2m/s or less. Moreover, special attention has to be placed on the temperature difference across the FCI layer. However, for a maximised liquid metal thermal gain, higher velocities would be preferable, what would also imply a reduced temperature difference across the FCI layer.

Selection of Suitable Turbulence Models for Numerical Modelling of Hydrocyclones

The capability of Computational Fluid Dynamics (CFD) alternates the interest of researcher from the empirical models into the numerical approaches for studying hydrocyclones. This paper presents a comprehensive survey on the influences of turbulence model options in the 3D simulation of the hydrocyclone flow pattern. The required grid resolution was selected through a grid independency study. Four categories of turbulence models involving models based on the Boussinesq hypothesis, the Reynolds Stress Model (RSM), the Large Eddy Simulation (LES) model, and the Detached Eddy Simulation (DES) model were investigated for prediction of velocity components within the hydrocyclone. The methodology was validated by experimental data. The results confirm that both RSM and LES models are efficient turbulent model choices for the simulation of swirling flow of hydrocyclones.

Comparison of Standard k-ε Model and RSM on Three Dimensional Turbulent Flow in the SEN of Slab Continuous Caster Controlled by Slide Gate

The fluid flow in the SEN and its outflow characteristic at nozzle port, after installation of slide gate system to control molten steel flow rate, have been investigated by mathematical simulation using standard k-ε model, RSM and experimental measurement by Ultrasonic Doppler Velocimeter (UDV) respectively, and the adaptability of turbulence models has been discussed. The research results indicate that, due to the throttle action of slide gate, a secondary flow and a separated flow are generated under the slide gate, and a swirl flow appears at nozzle port, with the swirl direction from the clogging side via nozzle bottom to the opening side of the slide gate. The swirl direction of outflow at nozzle port calculated by standard k-ε model is contrary to the measurement result by UDV and, the calculated result obtained by RSM is close to the UDV data. Based on kinetic theory of molecular, theoretical deficiencies of the Boussinesq hypothesis has been discussed.

난류 경계층에 놓인 공동 내부유동에 관한 수치해석적 연구

본 논문은 공동 주위 난류유동특성을 LES 기법으로 수치해석을 수행하여 알아보았다. 본 연구에 적용된 레이놀즈수는 공동 깊이만큼의 높이 h 에서의 유속을 기준으로 1.0×105 이며 3 차원 공동에서의 유동특성을 알아보았다. 적절한 비압축성 Filtered Navier-Stokes 방정식을 적용하기 위해, 계산격자를 공동 표면 근처에는 조밀하게 멀어질수록 성기게 생성하였으며, 이는 계산시간을 단축시키며 빠른 수렴을 도와준다. 또한, Boussinesq 가설을 subgrid-scale 난류모델에 적용하였고, Subgrid-scale 난류점성을 얻기 위해 smagorinsky-Lilly SGS 모델을 적용하였으며, 그 때의 CFL 수는 1.0 이다. 또한, 본 논문은 서로 다른 4 가지 형상의 공동의 및 입구조건의 변화에 따른 유동 특성도 함께 연구되었다.

Predictions of Turbulent Flow for the Impeller of a NASA Low-Speed Centrifugal Compressor

The turbulent flow inside a low-speed centrifugal compressor at design condition is investigated using large-eddy simulation (LES) comprising of up to 26×106 computational volume cells. Unlike in the past, the current study’s special emphasis is placed on the turbulence field evolution inside the impeller. LES predictions suggest that the Boussinesq hypothesis does not seem to be valid, especially near the exit of the impeller where the blade unloading takes place. Reynolds stress variations show a tendency toward an “axisymmetric expansion” type of turbulence after the impeller exit for which the subgrid-scale stress contribution shows a monotonic decrease. Probability density function analysis for the leakage flow show that instantaneous velocities in the wake region are less intermittent as compared with those in the jet. Time spectra analysis display also another feature that the energy cascade proceeds at a higher rate and lasts longer in the wake region than in the tip jet region.

A methodology to quantify the nonlinearity of the Reynolds stress tensor

Turbulent models provide closure equations that relate the Reynolds stress with kinematic tensors. In this study, we present a methodology to quantify the dependence of the Reynolds stress tensor on mean kinematic tensor basis. The methodology is based upon tensor decomposition theorems which allow to extract from the anisotropic Reynolds stress tensor the part that is linear or nonlinear in the strain rate tensor D, and the parts that are in-phase (sharing the same eigenvectors) and out-of-phase with the strain rate. The study was conducted using direct numerical simulation (DNS) data for turbulent plane channel (from Reτ=180 to Reτ=1000) and square duct flows (Reτ=160). The results have shown that the tensorial form of the linear Boussinesq hypothesis is not a good assumption even in the region where production and dissipation are in equilibrium. It is then shown that the set of tensor basis composed by D, D2 and the persistence-of-straining tensor D·(W-ΩD)-(W-ΩD)·D, where W is the vorticity tensor and ΩD is the rate of rotation of the eigenvectors of D, is able to totally reproduce the anisotropic Reynolds stress. With the proposed methodology, the scalar coefficients of nonlinear algebraic turbulent models can be determined, and the adequacy of the tensorial dependence of the Reynolds stress can be quantified with the aid of scaled correlation coefficients.

Velocity profile modeling in rectangular and compound open-channel cross sections

A 3-D hydraulic model was developed for computing velocity profiles, surface velocity coefficients, and discharge under steady, uniform flow conditions for rectangular and compound open-channel cross sections. Reynolds-averaged Navier–Stokes equations, Reynolds stress equations, and kinetic energy and dissipation equations were applied in the model using the finite-volume method with the k–e turbulence model. Many previously unpublished approaches to solving the numerical details of this type of hydraulic model are presented herein. Four different sets of Reynolds stress equations (one using the Boussinesq hypothesis and three algebraic stress models of varying complexity) were tested. Only one of the four stress models was successful in predicting the depression of the maximum stream-wise velocity below the water surface. The model was verified using data collected at the Utah Water Research Laboratory. A companion paper (Marjang and Merkley in Irrig Sci, 2009, in press) describes the application of this model to the calculation of surface velocity coefficients for the float method to estimate discharge in rectangular and compound irrigation canals.

Turbulence in vegetated flows: Volume-average analysis and modelling aspects

Two approaches for the modelling of turbulence in vegetated flows have been developed in the past. The “microscopic” approach which is straightforward but limited to simple cases and the “macroscopic” approach which is based on Volume Average Theory (VAT). In this study, aspects of Volume-Average (VA) analysis and modelling are investigated for turbulent vegetated flow using computed three-dimensional results from the solution of the Reynolds-Averaged Navier-Stokes (RANS) equations around a representative vegetal element. In particular (a) the VA transport equations for k and e, based on VAT, are properly derived, (b) the Boussinesq hypothesis for the VA quantities, employed in 〈k〉-〈e〉 turbulence models is tested, and (c) the values of the coefficients used in such turbulence models are assessed in comparison with those used in the classical turbulence models.

Collocated finite volume schemes for the simulation of natural convective flows on unstructured meshes

AbstractWe describe here a collocated finite volume scheme that was recently developed for the numerical simulation of the incompressible Navier–Stokes equations on unstructured meshes, in two or three space dimensions. We recall its convergence in the case of the linear Stokes equations, and we prove a convergence theorem for the case of the Navier–Stokes equations under the Boussinesq hypothesis. We then present several numerical studies. A comparison between a cluster‐type stabilization technique and the more classical Brezzi–Pitkäranta method is performed, the numerical convergence properties are presented on both analytical solutions and benchmark problems, and the scheme is finally applied to the study of the natural convection between two eccentric cylinders. Copyright © 2007 John Wiley & Sons, Ltd.

About Boussinesq's turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity

Boussinesq's hypothesis is at the heart of eddy viscosity models, which are used in many different fields to model turbulent flows. In its present time formulation, this hypothesis corresponds to an alignment between the Reynolds stress and mean strain tensors. We begin with historical remarks on Boussinesq's results and recall that he introduced a local averaging twenty years before Reynolds, but using an approach that prevented him from discovering Reynolds' stress tensor. We then introduce an indicator that characterizes the validity of this hypothesis. For experimental and numerical databases, when the tensors are known, this can be used to directly estimate the validity of this hypothesis. We show, using several different databases, that this hypothesis is almost never verified. We address, in conclusion, the analogy with kinetic theory, and the reason why this analogy cannot be applied, in general, for turbulent flows. To cite this article: F.G. Schmitt, C. R. Mecanique 335 (2007).

Developing mixed convection of a nanofluid in a horizontal tube with uniform heat flux

PurposeThis paper seeks to show the effect of using nanofluid on mixed convection heat transfer in a horizontal tube.Design/methodology/approachThree‐dimensional elliptic governing equation has been solved using finite volume approach. Grid independence test has been performed to find the suitable grids. Obtained numerical results have been validated with the available experimental and numerical results in the literature. Parametric study has been done to see the effects of Reynolds number, Grashof number and volume fraction of the nanoparticles on the hydrodynamic and thermal parameters in a horizontal tube.FindingsThe nanoparticles volume fraction does not have a direct effect on the secondary flow and the skin friction coefficient. However, its effect on the entire fluid temperature causes the strength of the secondary flow to reduce. For a given Grashof number, increasing the particles' concentration augments convective heat transfer coefficient. It does not have a significant effect on the skin friction coefficient at the low Grashof number. However, skin friction coefficient is slightly affected at the higher Grashof numbers.Research limitations/implicationsThe Grashof number is limited for which the Boussinesq hypothesis for the variation of density with the temperature would be valid.Practical implicationsThis paper promotes designing heat exchangers, solar collectors, cooling electronic devices.Originality/valueNanofluid mixed convection in a horizontal tube has been studied and the effects of nanoparticles concentration on the hydrodynamic and thermal parameters have been shown and discussed.

Linear stability of flow in an internally heated rectangular duct

The linear stability of flow in a vertical rectangular duct subject to homogeneous internal heating, constant-temperature no-slip walls and a driving pressure gradient is investigated numerically in the framework of the Boussinesq hypothesis. A Galerkin method based upon modified Chebyshev polynomials is used to discretize the linearized Navier-Stokes equations in their most compact form, i.e. eliminating the pressure and the streamwise velocity and making use of the intrinsic symmetry properties. A classification of the basic flow in Grashof and Reynolds space in terms of inflectional properties is proposed. It is found that the flow loses stability at all aspect ratios for a combination of finite thermal buoyancy and pressure forces with opposed signs. In the square duct, the unstable region lies inside the range where the basic velocity profile exhibits additional inflection lines. Unstable eigenfunctions are obtained for all basic symmetry modes, consisting of rapidly propagating large-scale structures located in the vicinity of the inflection lines near the centre of the duct.

Boussinesq modeling of breaking waves: Description of turbulence

Improved turbulent closures for use in fully nonlinear Boussinesq‐type models are described here. The approach extends previous works in order to give a more flexible and accurate description of the turbulence due to a breaking wave. Turbulent stresses are handled by means of the Boussinesq hypothesis, and the eddy viscosity is assumed to vary over the water depth according to different laws. The model is described in detail, and its performances are evaluated both against available analytical solutions and against experimental data of regular waves breaking over a slope. The influence of the vertical structure of turbulence under a breaking wave is analyzed by means of four different vertical profiles of eddy viscosity; the differences in terms of hydrodynamic features are also discussed. Among the four selected profiles, two of them (the uniform one and that with uniform eddy viscosity over the top half of the water column which linearly decreases to zero over the lower half) give better overall performances when compared with experimental data concerning velocity profiles.

Turbulent magnetohydrodynamic elasticity: Boussinesq-like approximations for steady shear

We re-examine the Boussinesq hypothesis of an effective turbulent viscosity within the context of simple closure considerations for models of strong magnetohydrodynamic turbulence. Reynolds-stress and turbulent Maxwell-stress closure models will necessarily introduce a suite of transport coefficients, all of which are to some degree model-dependent. One of the most important coefficients is the relaxation time for the turbulent Maxwell stress, which until recently has been relatively ignored. We discuss this relaxation within the context of magnetohydrodynamic turbulence in steady high fluid Reynolds-number, high magnetic-Reynolds-number shearing flows. The relaxation time for the turbulent Maxwell stress is not constrained by the shear time scale, in contrast with Reynolds-stress closure models for purely hydrodynamical turbulence. As a result, the anisotropy of the turbulent stress tensor for magnetohydrodynamic turbulence, even for the case of zero mean-field considered here, cannot be neglected. This shear-generated anisotropy can be interpreted as being due to an effective turbulent elasticity, in analogy to the Boussinesq turbulent viscosity. We claim that this turbulent elasticity should be important for any astrophysical problem in which the turbulent stress in quasi-steady shear has been treated phenomenologically with an effective viscosity.