Showing papers on "Boussinesq approximation (buoyancy) published in 2016"

Triple-Diffusive Natural Convection in a Square Porous Cavity

Abstract: The triple-diffusive flow, heat and mass transfer in a cavity filled with a porous medium and saturated with a mixture is theoretically studied in a cavity with differential temperature and concentrations at the side walls. The effect of buoyancy forces due to mass transfer of phases is also taken into account using the Boussinesq approximation. The governing equations are transformed into a non-dimensional form and numerically solved using the finite element method. Five groups of non-dimensional parameters including the Rayleigh number, the Lewis numbers for phases 1 and 2, and the buoyancy ratio parameters for phases 1 and 2 are obtained. The effect of each group of non-dimensional parameters on the heat and mass transfer in the cavity is discussed. The results show that for specific values of the Lewis number of one phase, the heat transfer of the mixture and the mass transfer of the other phase can be maximum. The presence of one phase could reduce or enhance the mass transfer of the second phase depending on the Lewis number of phases.

Vortex ring formation in starting forced plumes with negative and positive buoyancy

Abstract: The limiting process of vortex ring formation in starting forced plumes, with Richardson number in the range of −0.06 ≤ Ri ≤ 0.06, was studied numerically under the Boussinesq approximation. The examination of the dynamics of the starting flow evolution reveals that the plume-ambient density difference affects the vortex ring pinch-off mainly through three mechanisms, i.e., the baroclinic production of vorticity, the buoyancy acceleration (or deceleration) on the vortical structures, and its effect on the trailing shear layer instability. As Ri increases from negative to positive values, three regimes can be identified in terms of the vortex interaction patterns during the pinch-off process, i.e., the weak-interaction regime (−0.06 < Ri < − 0.02), the transition regime (−0.02 ≤ Ri < 0), and the strong-interaction regime (0 ≤ Ri < 0.06). By eliminating the influence of the baroclinic vorticity production, the circulation method proposed for the starting jets is revised to determine the buoyant formation number F in the starting forced plumes. Besides the formation number F, another dimensionless time scale (dubbed as the separation number S), which corresponds to the end of the pinch-off process, is identified by the time of vanishment of the vorticity flux feeding the leading vortex ring. The numerical results show that the variation trends of formation number and separation number against Ri change near the critical value of Ric ≈ − 0.02. In the weak-interaction regime, both formation number and separation number increase rapidly against Ri. While in the transition and strong-interaction regimes alike, the formation number increases at a much slower rate than in the weak-interaction regime, and the separation number declines dramatically as Ri increases. Finally, a qualitative explanation on the variation patterns of formation number and separation number is proposed based on the buoyancy effects on the dynamic properties of the leading vortex ring and the vortex interaction patterns.

High-resolution simulations of cylindrical gravity currents in a rotating system

Abstract: Cylindrical gravity currents, produced by a full-depth lock release, in a rotating system are investigated by means of three-dimensional high-resolution simulations of the incompressible variable-density Navier–Stokes equations with the Coriolis term and using the Boussinesq approximation for a small density difference. Here, the depth of the fluid is chosen to be the same as the radius of the cylindrical lock and the ambient fluid is non-stratified. Our attention is focused on the situation when the ratio of Coriolis to inertia forces is not large, namely , and the non-rotating case, namely , is also briefly considered. The simulations reproduce the major features observed in the laboratory and provide more detailed flow information. After the heavy fluid contained in a cylindrical lock is released in a rotating system, the influence of the Coriolis effects is not significant during the initial one-tenth of a revolution of the system. During the initial one-tenth of a revolution of the system, Kelvin–Helmholtz vortices form and the rotating cylindrical gravity currents maintain nearly perfect axisymmetry. Afterwards, three-dimensionality of the flow quickly develops and the outer rim of the spreading heavy fluid breaks away from the body of the current, which gives rise to the maximum dissipation rate in the system during the entire adjustment process. The detached outer rim of heavy fluid then continues to propagate outward until a maximum radius of propagation is attained. The body of the current exhibits a complex contraction–relaxation motion and new outwardly propagating pulses form regularly in a period slightly less than half-revolution of the system. Depending on the ratio of Coriolis to inertia forces, such a contraction–relaxation motion may be initiated after or before the attainment of a maximum radius of propagation. In the contraction–relaxation motion of the heavy fluid, energy is transformed between potential energy and kinetic energy, while it is mainly the kinetic energy that is consumed by the dissipation. As a new pulse initially propagates outward, the potential energy in the system increases at the expense of decreasing kinetic energy, until a local maximum of potential energy is reached. During the latter part of the new pulse propagation, the kinetic energy in the system increases at the expense of decreasing potential energy, until a local minimum of potential energy is reached and another new pulse takes form. With the use of three-dimensional high-resolution simulations, the lobe-and-cleft structure at the advancing front can be clearly observed. The number of lobes is maintained only for a limited period of time before merger between existing lobes occurs when a maximum radius of propagation is approached. The high-resolution simulations complement the existing shallow-water formulation, which accurately predicts many important features and provides insights for rotating cylindrical gravity currents with good physical assumptions and simple mathematical models.

Analysis of Mixed Convective Heat and Mass Transfer on Peristaltic Flow of Fene-P Fluid with Chemical Reaction

Abstract: This study presents the influence of heat and mass transfer on peristaltic transport of Finitely Extensible Nonlinear Elastic Peterlin (FENE-P) fluid in the presence of chemical reaction. It is assumed that all the fluid properties, except the density are constant. The Boussinesq approximation which relates density change to temperature and concentration changes is used in formulating buoyancy force terms in the momentum equation. Moreover, we neglect viscous dissipation and include diffusion-thermal (Dufour) and thermal-diffusion (Soret) effects in the present analysis. By the consideration of such important aspects the flow equations become highly nonlinear and coupled. In order to make the problem tractable we have adopted widely used assumptions of long wave length and low Reynolds number. An exact solution of the simplified coupled linear equations for the temperature and concentration has been obtained whereas numerical solution is obtained for dimensionless stream function and pressure gradient. The effects of different parameters on velocity field, temperature and concentration fields and trapping phenomenon are highlighted through various graphs. Numerical integration has been performed to analyze pressure rise per wavelength.

Convective heat transfer in a measurement cell for scanning electrochemical microscopy

Abstract: Electrochemical experiments, especially those performed with scanning electrochemical microscopy (SECM), are often carried out without taking special care to thermostat the solution; it is usually assumed that its temperature is homogeneous and equal to the ambient. The present study aims to test this assumption via numerical simulations of the heat transfer in a particular system – the typical measurement cell for SECM. It is assumed that the temperature of the solution is initially homogeneous but different from that of its surroundings; convective heat transfer in the solution and the surrounding air is taken into account within the framework of the Boussinesq approximation. The hereby presented theoretical treatment indicates that an initial temperature difference of the order of 1 K dissipates with a characteristic time scale of ∼1000 s; the thermal equilibration is accompanied by convective flows with a maximum velocity of ∼10−4 m s−1; furthermore, the temporal evolution of the temperature profile is influenced by the sign of the initial difference. These results suggest that, unless the temperature of the solution is rigorously controlled, convection may significantly compromise the interpretation of data from SECM and other electrochemical techniques, which is usually done on the basis of diffusion-only models.

Vortex ring formation in starting forced plumes with negative and positive buoyancy

Abstract: The limiting process of vortex ring formation in starting forced plumes, with Richardson number in the range of −0.06 ≤ Ri ≤ 0.06, was studied numerically under the Boussinesq approximation. The examination of the dynamics of the starting flow evolution reveals that the plume-ambient density difference affects the vortex ring pinch-off mainly through three mechanisms, i.e., the baroclinic production of vorticity, the buoyancy acceleration (or deceleration) on the vortical structures, and its effect on the trailing shear layer instability. As Ri increases from negative to positive values, three regimes can be identified in terms of the vortex interaction patterns during the pinch-off process, i.e., the weak-interaction regime (−0.06 < Ri < − 0.02), the transition regime (−0.02 ≤ Ri < 0), and the strong-interaction regime (0 ≤ Ri < 0.06). By eliminating the influence of the baroclinic vorticity production, the circulation method proposed for the starting jets is revised to determine the buoyant formation number F in the starting forced plumes. Besides the formation number F, another dimensionless time scale (dubbed as the separation number S), which corresponds to the end of the pinch-off process, is identified by the time of vanishment of the vorticity flux feeding the leading vortex ring. The numerical results show that the variation trends of formation number and separation number against Ri change near the critical value of Ric ≈ − 0.02. In the weak-interaction regime, both formation number and separation number increase rapidly against Ri. While in the transition and strong-interaction regimes alike, the formation number increases at a much slower rate than in the weak-interaction regime, and the separation number declines dramatically as Ri increases. Finally, a qualitative explanation on the variation patterns of formation number and separation number is proposed based on the buoyancy effects on the dynamic properties of the leading vortex ring and the vortex interaction patterns.

Natural convection in a horizontal cylinder with axial rotation.

Abstract: We study the problem of thermal convection in a laterally heated horizontal cylinder rotating about its axis. A cylinder of aspect ratio Γ=H/2R=2 containing a small Prandtl number fluid (σ=0.01) representative of molten metals and molten semiconductors at high temperature is considered. We focus on a slow rotation regime (Ω<8), where the effects of rotation and buoyancy forces are comparable. The Navier-Stokes and energy equations with the Boussinesq approximation are solved numerically to calculate the basic states, analyze their linear stability, and compute several secondary flows originated from the instabilities. Due to the confined cylindrical geometry-the presence of lateral walls and lids-all the flows are completely three dimensional, even the basic steady states. Results characterizing the basic states as the rotation rate increases are presented. As it occurred in the nonrotating case for higher values of the Prandtl number, two curves of steady states with the same symmetric character coexist for moderate values of the Rayleigh number. In the range of Ω considered, rotation has a stabilizing effect only for very small values. As the value of the rotation rate approaches Ω=3.5 and Ω=4.5, the scenario of bifurcations becomes more complex due to the existence in both cases of very close bifurcations of codimension 2, which in the latter case involve both curves of symmetric solutions.

Conjugate Heat Transfer in a Closed Volume with the Local Heat Sources and Non-Uniform Heat Dissipation on the Boundaries of Heat Conducting Walls

Abstract: Is solved the problem of heat transfer in the closed volume, limited by heat-conducting walls, with the local source of heat emission and the heterogeneous conditions of heat sink on the outer boundaries of solution area. The problem of convective heat transfer is solved with using a system of differential Navier-Stokes equations in the Boussinesq approximation. The simulation of turbulent flow conditions of heated air is carried out within the framework to k -e model. On the basis the analysis of the obtained temperature field and the contour lines of stream functions is made conclusion about the essential transiency of the process in question. The obtained values of temperatures and speeds in different sections of region illustrate turbulence of the process. Are investigated laws governing the formation of temperature fields in closed areas with a local heat emission source under the conditions of intensive local heat sink into environment and accumulation of heat in the enclosing constructions.

When is natural convection completely passive

Abstract: Momentum and energy equations for vertical flow, including viscous dissipation and pressure work, are derived and shown to require that the cross-section mean density is taken as the reference density for calculation of buoyancy forces under the Boussinesq approximation. Solutions are obtained for flow between parallel plane walls, with and without the pressure work as an explicit term in the energy equation. Both walls are at the same temperature, so there is no thermal forcing, but solutions are obtained for all admissible values of dynamic pressure gradient. The passive convection condition, whereby the flow is driven entirely by buoyancy forces resulting from heat generated by the flow's own viscous dissipation, is found on one branch of the dual solutions. However, while theoretically possible, passive convection is not physically realisable with any real fluid.

The impact of the Boussinesq approximation on plasma turbulence in the scrape-off layer

Abstract: Plasma turbulence in the tokamak scrape-off layer (SOL) region, where magnetic field lines are open and intersect the reactor inner walls, determines the heat loads on the limiter or divertor targets. This is one of the most crucial issues on the way towards a fusion reactor. Since SOL plasma is colder compared to the tokamak core, it is reasonable to use a fluid approximation to describe its dynamics. In particular the drift-reduced Braginskii equations are chosen to study the SOL plasma turbulence. To further simplify the drift-reduced Braginskii equations, the Boussinesq approximation is also applied in a number of numerical codes. This approximation consists in considering the plasma density constant in the evaluation of the divergence of the polarisation current, which simplifies substantially the solution of the Poisson equation necessary to evaluate the electric potential. In this study a new formulation of the drift-reduced Braginskii equations is presented together with a new numerical implementation that allow us to relax the Boussinesq approximation in the plasma turbulent code GBS. We show the energy conservation properties of the new system of equations. Also we compare the results of three-dimensional turbulent simulations with and without the Boussinesq approximation.

Effects of axial magnetic field and thermal convection on a counterrotating von karman flow

Abstract: The effects of thermal convection and of a constant axial magnetic field on a von Karman flow driven by the exact counter-rotation of two lids are investigated in a vertical cylinder of aspect ratio Gamma(= height/radius) = 2 at a fixed Reynolds number Re(= Omega R-2/v) = 300. Direct numerical simulations are performed when varying separately the Rayleigh and Hartmann numbers in the range [0, 1800] and [0, 20], respectively, in the limit of the Boussinesq approximation and of a small magnetic Reynolds numbers, Re-m << 1. Without a magnetic field, the base flow symmetries of the von Karman flow are broken by thermal convection that becomes dominant in the range of Ra [500, 1000]. Three-dimensional solutions are characterized by the occurrence of a steady, m = 1, azimuthal mode exhibiting a cat's eye vortex in the circumferential plane. When increasing the Rayleigh number in the range [500, 1000], the vortex pulsates in an oscillatory manner, due to variations of the flow intensity. Otherwise, increasing the axial magnetic field intensity stabilizes the flow, and the oscillatory motion can be inhibited. Numerical solutions show that the critical Rayleigh number for transition increases linearly with the Hartmann number. Finally, results show that when varying the Rayleigh number, the structure of the electric potential can be strongly modified by thermal convection. Such an observation suggests new induction mechanisms in the case of small nonzero values of the magnetic Reynolds number.

Vertical momentum transfer by weakly nonlinear inertia-gravity internalwaves

Abstract: Free inertia-gravity internal waves in a plane-parallel flow are considered in the Boussinesq approximation with account of the horizontal turbulent viscosity and diffusion. The dispersion relation and the wave decay rate are derived in the linear approximation. The effect of critical layers, in which the wave frequency with the Doppler shift is equal to the inertial frequency, on the dispersion curves is considered. It is shown that the dispersion curves are cut off in the low-frequency domain due to the critical layers mentioned above. The verticalwave momentum fluxes are nonzero and can be greater than the corresponding turbulent fluxes. It is shown that the Stokes drift velocity component transverse to the direction of propagation of the wave is nonzero with account of the turbulent viscosity and diffusion.

Passive Heat Transfer in Solid-State Lighting: Relaxing the Boussinesq Approximation Analytically in the Vertical Channel

Abstract: Analytical relations between the heat flux, temperature rise, thermal boundary layer thickness, and characteristic velocity have been derived for the two-dimensional vertical channel, without use of the Boussinesq approximation. Results have been put into the context of well-established scaling behavior in the literature. In addition, useful implications of the analytical results have been described, including a criterion to determine the suitability of a heat-sink configuration to a particular application.