Density Inversion Parameter Research Articles

Experimental and numerical study of turbulent penetrative Rayleigh-Bénard flow of water near 4 °C in a box-shaped container

Understanding the reversal process of the large-scale circulation will facilitate system recognition of turbulent penetrative Rayleigh-Bénard convection. The present work describes experimental and numerical investigations on the Rayleigh-Bénard flow of water near 4 °C in a box-shaped container. The results indicate that the revised Nusselt number is slightly higher than average Nusselt number with adiabatic assumption. The Nusselt numbers from the simulations show good agreement with the experiment values. Large-scale circulation reversal becomes less likely with the increasing Rayleigh number and density inversion parameter. Reversal is not observed at large density inversion parameters and Rayleigh numbers. Large vertical velocity and temperature fluctuations at the center point are found during the reversal process. However, the Nusselt number is slightly affected by the occurrence of the reversal.

Instability in heated water-porous system

This paper investigates the stability in a heated fluid-porous system. Darcy’s law is used for the flow in the porous medium. Previous studies indicate that the classical Rayleigh–Bénard instability in this flow is independent of the mutual positions of the fluid layer and porous layer. This paper demonstrates that the instability depends on the mutual positions of the fluid layer and porous layer when the liquid density is nonlinearly dependent on temperature. When porous medium is located at the bottom of the system, onset of convection may occur in lower porous layer (pure-porous mode), upper water layer (pure-water mode) or across both layers (porous-water mode). However, onset of convection cannot take place only in the upper porous layer when porous medium is placed at the top of the system. The influence of depth ratio dˆ and the density inversion parameter θM on instability of bilayer system are investigated in detail and dependence of critical modes on parameters are summarized in phase diagrams. In addition, direct numerical simulation is used to analyse the instability mechanism and identify subcritical or supercritical instability for bilayer systems.

Effect of aspect ratio on three-dimensional Rayleigh-Bénard flow of a gas mixture with density maximum in rectangular containers

In order to understand the temperature distribution of Rayleigh-Bénard (R–B) flow of cyclohexane/oxygen mixture to avoid the potential hazard of deflagration, we presented a series of three-dimensional numerical simulations on the aspect ratio dependence of three-dimensional R–B flow of cyclohexane/oxygen mixture with the density maximum in rectangular containers. Results show that with increasing the aspect ratio (A), the sidewall's restriction weakens gradually, the critical Rayleigh (Ra) numbers for the steady flow onset and the oscillatory convection appearance decrease gradually. When A = 0.25 and 0.5, all the critical flow patterns are the single straight roll state. The types of the stable flow patterns are few and the evolutions of flow patterns are relatively simple. When A ≥ 1, the critical flow patterns change with the increasing density inversion parameter (Θm) and the evolution becomes more complicated. With the increasing Ra, the oscillation gets more apparent and the oscillatory frequency increases. When Θm = 0.4, the flow rolls swing back and forth along the sidewalls and the fundamental frequency is the highest at A = 0.25. With the increasing Ra and A, and the decreasing Θm, the heat transfer strengthens. Finally, a scaling relationship for the average Nusselt number in terms of Ra, Θm and A is proposed.

Rotating thermal flow of water near its density inversion point in a vertical annulus: flow regime transition and heat transfer enhancement

Three-dimensional simulations are performed to identify the flow regimes of the rotating thermal flow of cold water near its density inversion point within a vertical annulus and evaluate the commensurate heat transfer abilities. The development of flow regimes and the variation in heat transfer behaviors in the special case with the density inversion parameter Θm=0.5 are explored over a wide range of parameters, with the Rayleigh number 104≤Ra≤106 and the Reynolds number Re≤150. The convection flow is radially separated into two layers by the density-maximum isosurface, with the inner and outer layers dominated by a positive and negative radial density gradient, respectively. Three axisymmetric and four non-axisymmetric basic flow regimes with different generation mechanisms are defined to describe the double-layer convection flow. Eleven flow regimes are identified, each of which consists of one, two or three basic flow regimes. The spatio-temporal features of the flow regimes depend on Ra and Re. The mixed convection parameter is demonstrated to be capable of qualitatively measuring the relative strength of the flow between the inner and outer layers in certain ranges of Ra. The mechanisms of both the flow regime transitions and the mode transitions between steady and oscillatory states are explored in terms of the combined action of density inversion effect, centrifugal effect and buoyancy effect. The heat transfer behaviors depend on the spatio-temporal features of flow regimes. The mechanism of heat transfer enhancement and that for the non-monotonic variation of heat transfer ability are revealed. The limitation of the axisymmetric assumption is quantified.

Numerical investigation on Rayleigh-Bénard convection of cyclohexane-oxygen mixture near its maximum density in a rectangular container

The oxidation of cyclohexane is an important route in the chemical industries for producing cyclohexanol, cyclohexanone, plasticizers, surfactants and so on. However, it will be in danger of explosion once the concentration of cyclohexane in the mixture exceeds the explosion limit. Therefore, it is necessary to research the temperature distribution and flow characteristics in Rayleigh-Bénard (RB) convection of cyclohexane/oxygen mixture to avoid the potential hazard of deflagration. In this paper, a set of three-dimensional simulations were performed to understand RB flow characteristics of cyclohexane/oxygen mixture near the maximum density in a rectangular container with an aspect ratio of 2. Rayleigh (Ra) number ranges from 2 × 103 to 6 × 105. Results show that the density inversion has a notable influence on RB convection of cyclohexane/oxygen mixture. The critical Rayleigh numbers for the onset of stable flow and for the oscillatory flow show an upward tendency with increasing density inversion parameter (Θm). When Θm is small, the flow occurs in the whole cavity. When Θm is large, the fluid layer near the top wall is static in the region. Multiple flow states and their evolutions are obtained at various Θm. The heat transfer ability is enhanced with increasing Ra, while it weakens with increasing Θm.

Oscillatory natural convection of Al2O3-water nanofluid near its density maximum in a narrow horizontal annulus

In order to reveal the oscillatory characteristics of the natural convection of nanofluid, this paper presents a comprehensive numerical work on the oscillatory natural convection of Al2O3-water nanofluid near its density maximum in a narrow horizontal annulus. The influences of nanoparticle volume fraction, Rayleigh number and density inversion parameter on the flow pattern evolution and heat transfer ability are analyzed. Results show that the density inversion of the working fluid affects the velocity and temperature distributions of the natural convection of Al2O3-water nanofluid significantly. Adding nanoparticles is beneficial to stabilizing the flow of nanofluid and keeping the symmetry of the flow structure. The convective heat transfer is enhanced with the increase of the volume fraction of nanoparticles. Furthermore, the evolution of the flow regime with the Rayleigh number at different volume fractions of nanoparticles is summarized.

Flow patterns of Rayleigh‐Bénard convection in cold water at maximum density in vertical cylindrical containers under conducting sidewall conditions

To understand the effects of aspect ratio and density inversion parameters on the critical conditions at the onset of convection in vertical cylindrical containers with perfectly-conducting side walls, a series of linear stability analyses (LSA) and three-dimensional numerical simulations were performed. The working fluid was cold water at maximum density and Prandtl number at ∼11.57. Results indicated that, with increased aspect ratio (A, ratio of radius to height), the critical Rayleigh number (Racri) was reduced and the number of convective rolls was more at a higher A than that at a lower one. The effects of A and density inversion (Θm=(Tm-Tc)/(Th-Tc)) were discussed in detail. The Θm had a dramatic influence on the Racri as well as flow structure at the primary threshold. With increased Θm, Racri sharply increased at higher values. Moreover, flow motion was only occurred near the bottom wall and a motionless fluid layer was found near the top wall, which were quite different from the fact that flow motion would occur in the whole container in a Boussinesq fluid. The depth of the convective fluid layer was thinner at a higher Θm. Notably, more convective rolls were found in the Z-direction at high Θm due to fluid viscosity. Racri obtained by LSA were satisfyingly consistent with those by three-dimensional numerical simulations.

Universal properties of penetrative turbulent Rayleigh-Bénard convection

Penetrative turbulence, which occurs in a convectively unstable fluid layer and penetrates into an adjacent, originally stably stratified layer, is numerically and theoretically analyzed. We chose the most relevant example, namely thermally driven flow of water with a temperature around $T_m\approx 4^\circ\rm{C}$, where it has its density maximum. We pick the Rayleigh-B\'enard geometry with the bottom plate temperature $T_b > 4^\circ\rm{C}$ and the top plate temperature $T_t \le 4^\circ\rm{C}$. Next to the overall thermal driving strength set by the temperature difference $\Delta = T_b - T_t$ (the Rayleigh number $Ra$ in dimensionless form), the crucial new control parameter as compared to standard Rayleigh-B\'enard convection is the density inversion parameter $\theta_m \equiv (T_m - T_t ) / \Delta$. The crucial response parameters are the relative mean mid-height temperature $\theta_c$ and the overall heat transfer (i.e., the Nusselt number $Nu$). We theoretically derive the universal (i.e., $Ra$-independent) dependence $\theta_c (\theta_m) =(1+\theta_m^2)/2$, which holds for $\theta_m$ below a $Ra$-dependent critical value, beyond which $\theta_c (\theta_m)$ sharply decreases and drops down to $\theta_c=1/2$ at $\theta_m=\theta_{m,c}$. Our direct numerical simulations with $Ra$ up to $10^{10}$ are consistent with these results. The critical density inversion parameter $\theta_{m,c}$ can be precisely predicted by a linear stability analysis. The heat flux $Nu(\theta_m)$ monotonically decreases with increasing $\theta_m$ and we can theoretically derive a universal relation for the relative heat flux $Nu(\theta_m)/Nu(0)$. Finally, we numerically identify and discuss rare transitions between different turbulent flow states for large $\theta_m$.

Linear stability analysis of Rayleigh–Bénard convection for cold water near its density maximum in a cylindrical container

In this paper, the onset of Rayleigh–Bénard convection of cold water near its density maximum in a vertical cylindrical container and the stability of the steady axisymmetric flow were studied by using linear stability analysis. The results show that the critical Rayleigh number for the onset of convection increases with increasing the density inversion parameter Tm (Tm=(Фmax-Фc)/(Фh-Фc), where Фmax is the temperature at the maximum density, and Фh and Фc are the temperature at the bottom wall and top wall, respectively), and some new flow patterns not found in the Rayleigh–Bénard convection of common fluids are presented.Within a certain range of density inversion parameters, the onset of convection is steady and axisymmetric. The steady axisymmetric basic flow is found to depend on the initial condition, and two distinctly different basic flow are obtained at identical parameters, due to the top-bottom symmetry breaking. For the upward basic flow, four different flow patterns are observed after the axisymmetry-breaking instability. In addition, a hysteresis phenomenon of the flow pattern transition is found when Tm=0.57. For the downward basic flow, six flow patterns are observed after the axisymmetry-breaking instability. In particular, it is found that there still exists stable axisymmetric flow beyond the second bifurcation in certain ranges of Rayleigh numbers. The energy analysis shows that both the primary instability and the axisymmetry-breaking instability are caused by the thermal-buoyancy mechanism.

Rayleigh-Bénard convection of a gas-vapor mixture with abnormal dependence of thermal expansion coefficient on temperature

In order to reveal basic characteristics of Rayleigh-Bénard convection (RBC) of a gas mixture, which has abnormal dependence of thermal expansion coefficient on temperature, we presented a direct numerical simulation on RBC of a gas mixture of oxygen and cyclohexane near its maximum density. The influences of the density inversion parameter and the Rayleigh (Ra) number on the critical conditions of the flow onset and destabilization were analyzed. The flow evolution with Ra was exhibited. The results show the critical value of Rayleigh number of the flow onset is approximately proportional to the fourth power of the thickness of the upper stable sublayer. Regardless of density inversion parameter, the flow pattern at the flow appearance always has a multicellular structure. With increasing Ra, the multicellular flow structure would be transformed into various steady flow patterns, including parallel flow and pendant pattern. When the density inversion parameter is small, the unsteady flow cells exhibit a back and forth motion along the border. However, at a large density inversion parameter, the cells swing around the centers of the flow rolls. We also found that the local Nusselt (Nu) number on the bottom depends on the flow pattern. The average Nu increases gradually with increasing Ra under a small density inversion parameter. However, an abrupt rise of the average Nu appeared under a large density inversion parameter.

Direct numerical simulations of Rayleigh-Bénard convection of a gas-liquid medium near its density maximum

Oxidation of hydrocarbons is a commonly used chemical reaction due to the prominent economic value of its products in plasticizers, resins and other intermediate manufacturing processes. However, there is a significant explosion danger of the gas-liquid medium in this chemical reaction. Thus, it is necessary to make sense of the flow and thermal fields in the gas-liquid medium to manage the convection, avoid overheating and prevent explosion in the industry. A comprehensive numerical work of Rayleigh-Bénard convection of a gas-liquid medium near its density maximum was presented in this paper. The influences of the density inversion parameter and Rayleigh number on the flow stability, flow pattern evolution, and heat transfer ability of Rayleigh-Bénard convection were analyzed. The results show that the critical Rayleigh numbers for onsets of a steady convection and an unsteady convection rise with the increase of the density inversion parameter. The flow pattern evolution strongly relies on the density inversion parameter. The strength and heat transfer ability of Rayleigh-Bénard convection are enhanced as the Rayleigh number rises, while they are suppressed along with the density inversion parameter. A scaling relationship for the average Nusselt number on the bottom wall in terms of the Rayleigh member and density inversion parameter is successfully proposed. In addition, the predicted results based on the scaling correlation are consistent with those from simulations.

Natural convection of water-based nanofluids near the density maximum in an annulus

This paper presents a series of numerical simulations on natural convection of a water-based nanofluid near its density maximum in an annulus. The volume fraction of the nanoparticles ranged from 0 to 0.08, the nanoparticle diameter from 10 nm to 100 nm, the radius ratio of the annulus from 0.1 to 0.8, and the density inversion parameter from 0 to 1. The results indicate that Brownian motion should be considered in natural convection of water-based nanofluid with the density inversion phenomenon. Both the solid volume fraction and the radius ratio affect greatly the flow and thermal fields, while the nanoparticle diameter has a little impact. Adding nanoparticles can weaken or even overcome the negative effect of the density inversion phenomenon on the heat transfer performance, possibly making it higher than that of the common fluid. Additionally, the heat transfer performance of the nanofluid with metal or metal oxide nanoparticles is better than that of the nanofluid with non-metallic compound nanoparticles.

Flow patterns and heat transfer correlations in partially active cavity filled with water at its density maximum

The work presents a numerical study of natural convection of water at its density maximum in a partially active square cavity, nine relatives locations of the partially heated/cooled zone are investigated as well as the effect of different thermal boundary conditions represented as the inversion parameter Two behaviors of average Nusselt number has been detected, at Top–Middle, Top–Bottom, and Middle–Bottom configurations give rise to a maximum Conversely, at Middle–Top, Bottom–Top, and Bottom–Middle configurations that take precedence. A heat transfer correlation based on a geometric parameter describing the locations of the active zone and the density inversion parameter is proposed.

Aspect ratio dependence of Rayleigh-Bénard convection of cold water near its maximum density in box-shaped containers

The aim of this research is to understand the effect of the aspect ratio on the heat transfer ability and hydrodynamics characteristics of Rayleigh-Bénard convection of cold water near its maximum density in box-shaped containers. The Rayleigh number is fixed at 109, density inversion parameters are 0.3, 0.5 and 0.7, and the aspect ratio ranges from 1/60 to 1. Results indicate that the average Nusselt number presents a weak dependence on the aspect ratio at the large aspect ratio (A > 0.3). However, it reaches the maximum and then drops when the aspect ratio decreases from A = 0.3. Large scale circulations are observed for containers at the large aspect ratio, and the confinement of sidewalls weakens the large-scale circulation and eventually destructs it. At the large aspect ratio, the velocity fluctuation near the sidewalls is stronger than that in the center zone, because plumes primarily move along the sidewalls of the container. At a small aspect ratio, more plumes appear in the center of the container, where the fluctuation is stronger than that near sidewalls. The effect of cold plumes on the flow is reduced as the density inversion parameter increases. Therefore, the flow is mainly driven by hot plumes, and the velocity magnitude and fluctuation decrease significantly.

Penetrative turbulent Rayleigh–Bénard convection in two and three dimensions

Penetrative turbulent Rayleigh–Bénard convection which depends on the density maximum of water near $4^{\circ }\text{C}$ is studied using two-dimensional and three-dimensional direct numerical simulations. The working fluid is water near $4\,^{\circ }\text{C}$ with Prandtl number $Pr=11.57$. The considered Rayleigh numbers $Ra$ range from $10^{7}$ to $10^{10}$. The density inversion parameter $\unicode[STIX]{x1D703}_{m}$ varies from 0 to 0.9. It is found that the ratio of the top and bottom thermal boundary-layer thicknesses ($F_{\unicode[STIX]{x1D706}}=\unicode[STIX]{x1D706}_{t}^{\unicode[STIX]{x1D703}}/\unicode[STIX]{x1D706}_{b}^{\unicode[STIX]{x1D703}}$) increases with increasing $\unicode[STIX]{x1D703}_{m}$, and the relationship between $F_{\unicode[STIX]{x1D706}}$ and $\unicode[STIX]{x1D703}_{m}$ seems to be independent of $Ra$. The centre temperature $\unicode[STIX]{x1D703}_{c}$ is enhanced compared to that of Oberbeck–Boussinesq cases, as $\unicode[STIX]{x1D703}_{c}$ is related to $F_{\unicode[STIX]{x1D706}}$ with $1/\unicode[STIX]{x1D703}_{c}=1/F_{\unicode[STIX]{x1D706}}+1$, $\unicode[STIX]{x1D703}_{c}$ is also found to have a universal relationship with $\unicode[STIX]{x1D703}_{m}$ which is independent of $Ra$. Both the Nusselt number $Nu$ and the Reynolds number $Re$ decrease with increasing $\unicode[STIX]{x1D703}_{m}$, the normalized Nusselt number $Nu(\unicode[STIX]{x1D703}_{m})/Nu(0)$ and Reynolds number $Re(\unicode[STIX]{x1D703}_{m})/Re(0)$ also have universal relationships with $\unicode[STIX]{x1D703}_{m}$ which seem to be independent of both $Ra$ and the aspect ratio $\unicode[STIX]{x1D6E4}$. The scaling exponents of $Nu\sim Ra^{\unicode[STIX]{x1D6FC}}$ and $Re\sim Ra^{\unicode[STIX]{x1D6FD}}$ are found to be insensitive to $\unicode[STIX]{x1D703}_{m}$ despite of the remarkable change of the flow organizations.

Bifurcations in penetrative Rayleigh-Bénard convection in a cylindrical container

The bifurcations of penetrative Rayleigh-Benard convection in cylindrical containers are studied by the linear stability analysis (LSA) combined with the direct numerical simulation (DNS) method. The working fluid is cold water near 4°C, where the Prandtl number Pr is 11.57, and the aspect ratio (radius/height) of the cylinder ranges from 0.66 to 2. It is found that the critical Rayleigh number increases with the increase in the density inversion parameter θm. The relationship between the normalized critical Rayleigh number (Rac(θm)/Rac(0)) and θm is formulated, which is in good agreement with the stability results within a large range of θm. The aspect ratio has a minor effect on Rac(θm)/Rac(0). The bifurcation processes based on the axisymmetric solutions are also investigated. The results show that the onset of axisymmetric convection occurs through a trans-critical bifurcation due to the top-bottom symmetry breaking of the present system. Moreover, two kinds of qualitatively different steady axisymmetric solutions are identified.

Influence of density inversion and sinusoidal heating on dual diffusive convection in a water saturated square porous box

The motive of the present numerical study is to analyze the density maximum effect of double diffusive buoyancy-driven convection in a square porous box. The horizontal walls are treated as adiabatic and the vertical walls are maintained with different temperatures. The right wall is heated sinusoidally while the left wall kept with constant temperature. The concentration on the left and right walls of the box are Cl and Ch, respectively, with Ch> Cl. The present study uses finite volume method to solve the governing equations. The effects of buoyancy ratio, density inversion parameter, Darcy number and porosity are studied by using the Brinkmann-Forchheimer extended Darcy model. It is observed that the bicellular structure is formed due to density inversion. When Tm =1, the average mass and thermal energy transfer rates are high for every values of buoyancy ratio. When Tm =0.4, the average mass and thermal energy transfer rates are low for all values of N due to density maximum of water.

Turbulent Rayleigh-Bénard convection in a cubical container filled with cold water near its maximum density

In order to understand the effects of Rayleigh number and the density inversion parameter on the flow structures, the turbulent characteristics and the heat transfer ability, a series of large eddy simulations on turbulent Rayleigh-Bénard convection in a cubical container with cold water near its maximum density were carried out. Results indicate that the large scale circulation (LSC) moves along the diagonal plane at small and medium density inversion parameters. However, at a large density inversion parameter, the LSC disappears. With the increase of the density inversion parameter, there are an increase of temperature in the bulk region and a decrease of penetration depth. Furthermore, with the increase of Rayleigh number, the deformation of isothermal surfaces becomes intense until the plumes are observed. At the same time, the scale of vortices reduces and the fluctuation of Nusselt number amplifies with the increase of Rayleigh number. Based on the numerical results, the dependence of Nusselt number on Rayleigh number and the density inversion parameter is achieved within a satisfactory uncertainty.

Free Convection of Water near its Density Maximum in a Heat Generating Porous Cavity with Sinusoidal Heating

The aim of the present numerical investigation is to understand the unsteady free convective flow and heat transfer of cold water near its density maximum temperature in a two-dimensional square porous cavity with internal heat generation. The present study uses the Brinkmann-Forchheimer extended Darcy model for porous medium to study the effects of density inversion parameter, heat generation parameter, Darcy number and porosity. The finite volume method is used to solve the governing equations. The nonlinear behavior of local heat transfer is observed due to sinusoidal heating. The heat transfer enhances for heat absorption case than that of heat generation case in the presence of density maximum.

Large eddy simulation on Rayleigh–Bénard convection of cold water in the neighborhood of the maximum density

In order to understand the effect of the Rayleigh number, the density inversion phenomenon and the aspect ratio on the flow patterns and the heat transfer characteristics of Rayleigh–Bénard convection of cold water in the neighborhood of the maximum density, a series of large eddy simulations are conducted by using the finite volume method. The Rayleigh number ranges between 106 and 109, the density inversion parameter and the aspect ratio are varied from 0 to 0.9 and from 0.4 to 2.5, respectively. The results indicate that the reversal of the large scale circulation (LSC) occurs with the increase of the Rayleigh number. When there exists a density inversion phenomenon, the key driver for the LSC is hot plumes. When the density inversion parameter is large enough, a stagnant region is found near the top of the container as the hot plumes cannot move to the top wall. The flow pattern structures depend mainly on the aspect ratio. When the aspect ratio is small, the rolls are vertically stacked and the flow keeps on switching among different flow states. For a moderate aspect ratio, different long-lived roll states coexist at a fixed aspect ratio. For a larger aspect ratio, the flow state is everlasting. The number of rolls increases with the increase of the aspect ratio. Furthermore, the aspect ratio has only slight influence on the time averaged Nusselt number for all density inversion parameters.