Application of the multi distribution function lattice Boltzmann approach to thermal flows

Numerical methods able to model high Rayleigh (Ra) and high Prandtl (Pr) number thermal convection are important to study large-scale geophysical phenomena occuring in very viscous fluids such as magma chamber dynamics (10⁴ < Pr < 10⁷ and 10⁷ < Ra < 10¹¹). The important variable to quantify the thermal state of a convective fluid is a generalized dimensionless heat transfer coefficient (the Nusselt number) whose measure indicates the relative efficiency of the thermal convection. In this paper we test the ability of Multi-distribution Function approach (MDF) Thermal Lattice Boltzmann method to study the well-established scaling result for the Nusselt number (Nu ∝ Ra ^1/3) in Rayleigh Bénard convection for 10⁴ ≤ Ra ≤ 10⁹ and 10¹ ≤ Pr ≤ 10⁴. We explore its main drawbacks in the range of Pr and Ra number under investigation: (1) high computational time N _c required for the algorithm to converge and (2) high spatial accuracy needed to resolve the thickness of thermal plumes and both thermal and velocity boundary layer. We try to decrease the computational demands of the method using a multiscale approach based on the implicit dependence of the Pr number on the relaxation time, the spatial and temporal resolution characteristic of the MDF thermal model.     

Experimental Study of Natural Convection from Electrically Heated Vertical Cylinders Immersed in Air

Abstract

An experimental study of laminar steady-state natural convection heat transfer from electrically heated vertical cylinders immersed in air has been undertaken. Three stainless steel (316 SS) test sections of different slenderness ratios were employed. Surface temperature profiles along the vertical cylinders were obtained using miniature thermocouples when the cylinders were heated with different power levels resulting in different total wall heat fluxes. After the mandatory corrections for the radiation heat fluxes were made, three correlation equations relating the local Nusselt number Nu_y with the local modified Rayleigh number Ra^* _y and the position-to-cylinder diameter y/d were developed. The correlation equations are valid for Ra^* _y ≤ 2 × 10¹².     

Temperature gradient spectra and temperature dissipation rate in a turbulent convective flow

Determining mixing coefficients in oceanographic flows relies on the form of temperature gradient spectra in turbulent water flows at large wavenumbers. Several recent investigations concluded that these spectra are best described by the functional form proposed by Kraichnan rather than by the Batchelor form, more commonly used in oceanography. In this study, we provide additional support for this conclusion using laboratory measurements of the temperature gradient spectra in a Rayleigh–Bénard convective cell, in order to avoid difficulties inherent in oceanographic field measurements. The range of Rayleigh numbers in experiments is between Ra = 3×10⁷ and Ra = 5×10⁹. In addition to a traditional method of traversing thermistors, a novel optical technique recently introduced for oceanic measurements was used to obtain the spectra; comparison between these two methods serves as a validation test for the new optical technique. The temperature measurements were also augmented by 2D particle image velocimetry (PIV) observations. The measured dependence of the Nusselt number on the Rayleigh number followed Nu ∝ Ra^0.29 at Pr = 6 and was consistent with the literature data. We observed the temperature dissipation rate to vary by an order of magnitude over a horizontal transect at Ra > 10⁹. The temperature dissipation spectra obtained by both methods were in agreement over the Ra interval considered. The location of the temperature dissipation peaks was also consistent with PIV measured energy dissipation rates. Our data suggest increasing importance of top/bottom boundaries for the momentum and the temperature dissipation with increasing Ra number. Applied to oceanic upper ocean convection, our results imply that most of the dissipation occurs close to the air–sea boundary. Thus, attempts to parameterise or measure air–sea turbulent convective fluxes have to reflect the dominant role of near boundary dissipation at large Ra.     

Scaling in convective evaporation and sidewall boundary layer

Turbulent convection at aspect ratios from 0.06 to 2 is investigated in the laboratory with evaporation experiments from vertical cylinders having different diameters and liquid levels. With alcohol, only diffusive evaporation takes place. With water, for small diameters, evaporation proceeds by diffusion whereas convective evaporation develops when the diameter is increased. This onset can be effectively interpreted in terms of a viscous sidewall boundary layer, whose thickness δ varies with respect to the available height h according to δ/h = 3.4 Ra^-0.28±0.01 versus Rayleigh number Ra. The Sherwood number Sh, analog of the Nusselt number, exhibits a power law variation Sh = 0.6 Ra^0.27±0.02 for Ra varying from 10⁴ to 3 ×10⁸. The scaling observed in this case of an open boundary is thus similar to the scaling measured in confined Rayleigh-Bénard convection.     

Natural convection heat transfer coefficients for a horizontal cylinder with vertically attached circular fins

Experiments were performed to investigate the natural convection heat transfer characteristics of a vertically finned horizontal isothermal cylinder. The apparatus was designed so that the fin spacing, fin length, and the cylinder Rayleigh number could be varied within limits of practical interest. The Rayleigh number (based on the cylinder diameter) ranged from 10⁵ to 5 × 10⁷. It was found that the interaction of the fin boundary layer with the cylinder brought about a reduction of the cylinder Nusselt number relative to that for the classical case of the long isolated horizontal cylinder. The reduction was strong at Rayleigh numbers below the critical value and at spacing ratios less than 0.125. However, at Rayleigh numbers beyond the critical, the Nusselt number was quite insensitive to the fin spacing and length, with the typical data spread being in the 5–8% range.     

Laminar Natural Convection in a Square Cavity with a Partition on the Heated Vertical Wall

Abstract

The laminar natural convection in an air-filled square cavity with a partition on the heated vertical wall was experimentally investigated. Temperature measurements and flow visualizations were performed for cases with heated and cooled vertical walls (corresponding to a global Grashof number Gr _H of approximately 1.4 × 10⁸) and non dimensional top wall temperatures θ _T of 0.57 (insulated) to 2.3. Experiments were performed with an aluminum partition with non dimensional height H _P /H of 0.0625 and 0.125 attached to the heated vertical wall at y/H = 0.65 and 0.95. The blockage effect and/or the thermal effect of the partition resulted in changes to the temperature and flow fields but were mainly limited to the vicinity of the partition. For the cases with the heated top wall, the change in the height of the partition at y/H = 0.95 resulted in changes to the ambient temperature outside the boundary layer due to the reduction of the size of the recirculating flow in the corner region. The changes in the partition height and the top wall temperature affected the blockage effect of the partition, resulting in the local Nusselt number near the corner region to be affected. The local Nusselt number over most of the heated vertical wall of the partitioned cavity (y/H < 0.7) was correlated to the local Rayleigh number in the form Nu = C · Ra ⁿ .     

Natural convection in a square cavity containing a nanofluid and an adiabatic square block at the center

The problem of free convection fluid flow and heat transfer of Cu–water nanofluid inside a square cavity having adiabatic square bodies at its center has been investigated numerically. The governing equations have been discretized using the finite volume method. The SIMPLER algorithm was employed to couple velocity and pressure fields. Using the developed code, a parametric study was conducted and the effects of pertinent parameters such as Rayleigh number, size of the adiabatic square body, and volume fraction of the Cu nanoparticles on the fluid flow and thermal fields and heat transfer inside the cavity were investigated. The obtained results show that for all Rayleigh numbers with the exception of Ra = 10⁴ the average Nusselt number increases with increase in the volume fraction of the nanoparticles. At Ra = 10⁴ the average Nusselt number is a decreasing function of the nanoparticles volume fraction. Moreover at low Rayleigh numbers (10³ and 10⁴) the rate of heat transfer decreases when the size of the adiabatic square body increases while at high Rayleigh numbers (10⁵ and 10⁶) it increases.     

Natural-Convection Thermosyphon Heat Transfer from a Stack of Horizontal Plates

Steady laminar natural convection of water about vertically stacked, two-sided, horizontal heated plates was studied experimentally over a range of plate gap to plate half-width ratio, H/a, from 0.078 to 0.94. Measurements were made of power input per plate, plate temperatures, and inlet and outlet bulk fluid temperatures. From these data were determined average Nusselt and Rayleigh numbers for each plate, which were correlated with a power law Nu = C Raⁿ for each gap height. The Nusselt numbers increased with gap height up to H/a 0.47 (from 8 to 48), and then became independent of H/a. The exponent n increased from n 0.2 at the lowest gap height tested (H/a 0.078), to a high value of n approximately equal to 0.4 at H/a = 0.24 and then decreased back to n = 0.2 for H/a greater than or equal to 0.47.     

Importance of direction of vibration on the onset of Soret-driven convection under gravity or weightlessness

This paper considers the influence of the direction of vibration on the stability threshold of two-dimensional Soret-driven convection. The configuration is an infinite layer filled with a binary mixture, which can be heated from below or from above. The limiting case of high-frequency and small-amplitude vibration is considered for which the time-averaged formulation has been adopted. The linear stability analysis of the quasi-mechanical equilibrium shows that the problem depends on five non-dimensional parameters. These include the thermal Rayleigh number ( Ra_T), the vibrational parameter (R), the Prandtl number ( Pr), the Lewis number (Le), the separation ratio (S) and the orientation of vibration with respect to the horizontal heated plate (). For different sets of parameters, the bifurcation diagrams are plotted Ra_c = f (S) and k_c = g(S), which are the critical thermal Rayleigh and the critical wave numbers, respectively. Our results indicate that, relative to the classical case of static gravity, vibration may affect all regions in Ra_c-S stability diagram. In the case of mono-cellular convection, by using a regular perturbation method, a closed-form relation for the critical Rayleigh number is found. Several physical situations in the presence or in the absence of gravity (micro-gravity) are discussed.     

Turbulent convective heat transfer in an inclined tube filled with sodium

Turbulent free convection of liquid sodium in a straight thermally insulated tube with a length equal to 20 diameters and with end heat exchangers ensuring a fixed temperature drop is investigated experimentally. The experiments are performed for a fixed Rayleigh number Ra = 2.4 × 10⁶ and various angles of inclination of the tube relative to the vertical. A strong dependence of the power transferred along the tube on the angle of inclination is revealed: the Nusselt number in the angular range under investigation changes by an order of magnitude with a maximum at the angle of 65° with the vertical. The characteristics of large-scale circulation and turbulent temperature pulsations show that convective heat transfer is mainly determined by the velocity of large-scale circulation of sodium. Turbulent pulsations are maximal for small angles of inclination (α = 20°–30°) and reduce the heat flux along the channel, although in the limit of small angles (vertical tube), there is no large-scale circulation, and the convective heat flux, which is an order of magnitude larger than the molecular heat flux, is ensured only by small scale (turbulent) flow.     

Dependence of the Compton to Rayleigh intensity ratio on the scatterer atomic number in the range of 4(Be) to 31(Ga)

Compton to Rayleigh scattering intensity ratios (I_C/I_R) have been measured using X-rays with energy 17.44 keV for single-component materials with atomic number Z from 4 (Be) to 31 (Ga) and binary compounds of stoichiometric composition. The measurements have been performed using two optical schemes: an energy-dispersive X-ray fluorescence scheme with a molybdenum secondary target and wavelength-dispersive X-ray fluorescence one. The processing of the spectra was carried out by fitting with Pearson VII functions. For single-component and binary standards, the experimental dependence of the scattering intensity ratio on the atomic number was found to be the same. This confirms the additivity of the contribution of different atoms to the scattering. The dependence has a complex shape but is well described by the theoretical relationship for I_C/I_R with correction on the difference between Compton and Rayleigh radiation absorption coefficients. Two ranges of atomic number values are defined, in which the effective atomic number Z_eff can be determined by the calibration method using this dependence: for Z from 4 to 7 with low error of ΔZ_eff =±0.15 and for Z_eff from 10 to 18 with low error of ΔZ_eff =±0.69. A change in the shape of the Compton peak and an overestimated value of the of the Compton and Rayleigh peak intensity ratio when passing from a single-component scatterer (Al or Si) to their oxides Al₂O₃ or SiO₂, respectively, have been revealed.     

Rotating turbulent Rayleigh–Bénard convection subject to harmonically forced flow reversals

The characteristics of turbulent flow in a cylindrical Rayleigh–Bénard convection cell which can be modified considerably in case rotation is included in the dynamics. By incorporating the additional effects of an Euler force, i.e., effects induced by non-constant rotation rates, a remarkably strong intensification of the heat transfer efficiency can be achieved. We consider turbulent convection at Rayleigh number Ra = 10⁹ and Prandtl number σ = 6.4 under a harmonically varying rotation, allowing complete reversals of the direction of the externally imposed rotation in the course of time. The dimensionless amplitude of the oscillation is taken as 1/Ro^* = 1 while various modulation frequencies 0.1 ≤ Ro_ω ≤ 1 are applied. Both slow and fast flow-structuring and heat transfer intensification are induced due to the forced flow reversals. Depending on the magnitude of the Euler force, increases in the Nusselt number of up to 400% were observed, compared to the case of constant or no rotation. It is shown that a large thermal flow structure accumulates all along the centreline of the cylinder, which is responsible for the strongly increased heat transfer. This dynamic thermal flow structure develops quite gradually, requiring many periods of modulated flow reversals. In the course of time, the Nusselt number increases in an oscillatory fashion up to a point of global instability, after which a very rapid and striking collapse of the thermal columnar structure is seen. Following such a collapse is another, quite similar episode of gradual accumulation of the next thermal column. We perform direct numerical simulation of the incompressible Navier–Stokes equations to study this system. Both the flow structures and the corresponding heat transfer characteristics are discussed at a range of modulation frequencies. We give an overview of typical time scales of the system response.     

高瑞利数下水平自然热对流的幂律关系

采用数值模拟方法，研究了高度和宽度比为1∶10的狭长矩形腔内的水平自然热对流. 根据对瑞利数(Rayleigh数)Ra在10⁴ <Ra<10¹¹内情形的计算结果，将流动分为三个不同的区间：线性区、连续过渡区、1/5次幂律区. 虽然流量和努塞尔数(Nusselt数)Nu随瑞利数的变化都包括了三个参数演化区间，但从一个区间到另外一个参数区间的转变时并不是同步的，其中努塞尔数的转变总是超前流量的转变. 对比前人的研究发现，流量1/3次幂律的结果是由于瑞利数不够高所致. 此外，模拟结果也表明Siggers等的理论分析过高估计了热通量强度，实际的温度边界层内努塞尔数和瑞利数为1/5次幂律关系.     

Numerical simulation of turbulent natural convection of oxide heat generating melt in a core catcher at NPP with VVER-1000

The problem statement and simulation results are presented concerning turbulent natural convection in a vertical cylindrical molten pool with internal heat generation and other parameters (inner Rayleigh number Ra _i ∼ 10¹⁶–10¹⁷) corresponding to oxide core melt in a core catcher for NPP with VVER-1000. Commercial code FLUENT 6.3 was used for CFD calculations. The results on heat transfer are approximated by power law correlations for mean Nusselt numbers vs. Rayleigh number and pool height, describing the heat transfer at upper, lateral, and total boundaries of the cylinder. The influence of volumetric heat generation and material properties is studied. Spatial distribution of wall heat transfer is analyzed for different pool heights possible in the real core catcher. Along with serial calculations with isothermal boundary conditions, the cases with heat radiation conditions are considered. The results may be used for estimations of heat transfer and melt overheating in a VVER core catcher and for coefficient identification of simplified models of integrated system severe-accident codes.     

Thermal Lattice Boltzmann Flux Solver for Natural Convection of Nanofluid in a Square Enclosure

In the present study, mathematical modeling was performed to simulate natural convection of a nanofluid in a square enclosure using the thermal lattice Boltzmann flux solver (TLBFS). Firstly, natural convection in a square enclosure, filled with pure fluid (air and water), was investigated to validate the accuracy and performance of the method. Then, influences of the Rayleigh number, of nanoparticle volume fraction on streamlines, isotherms and average Nusselt number were studied. The numerical results illustrated that heat transfer was enhanced with the augmentation of Rayleigh number and nanoparticle volume fraction. There was a linear relationship between the average Nusselt number and solid volume fraction. and there was an exponential relationship between the average Nusselt number and Ra. In view of the Cartesian grid used by the immersed boundary method and lattice model, the immersed boundary method was chosen to treat the no-slip boundary condition of the flow field, and the Dirichlet boundary condition of the temperature field, to facilitate natural convection around a bluff body in a square enclosure. The presented numerical algorithm and code implementation were validated by means of numerical examples of natural convection between a concentric circular cylinder and a square enclosure at different aspect ratios. Numerical simulations were conducted for natural convection around a cylinder and square in an enclosure. The results illustrated that nanoparticles enhance heat transfer in higher Rayleigh number, and the heat transfer of the inner cylinder is stronger than that of the square at the same perimeter.     

Long relaxation times and tilt sensitivity in Rayleigh Bénard turbulence

We study the influence of a small tilt angle ( rd) on the Nusselt number in a 1/2 aspect ratio Rayleigh-Bénard cell, at high Rayleigh number (5 x 10¹¹ < Ra < 4 x 10¹²). The small decrease observed is interpreted as revealing a two rolls structure of the flow. Transitions between different global flows are also observed, on very long times, comparable to the diffusion time on the whole cell. The consequence is that the Nusselt number observed in most high Ra experiments should significantly depend on initial conditions.Received: 19 May 2004, Published online: 12 August 2004PACS: 92.60.Ek Convection, turbulence, and diffusion - 47.27.Te Convection and heat transfer - 44.25. + f Natural convection     

Laminar natural convection between vertical isothermal heated plates with different temperatures

Numerical investigation of laminar free convection heat transfer in the vertical parallel plate channel with asymmetric heating is presented. Both inlet and exit effects are included into the analysis. A numerical solution is obtained for a Prandtl number of 0.71 and for modified Rayleigh number [`(Ra)]\overline {Ra} = 10⁻¹ ÷ 10⁵, and varying heating ratio T_R = 0 ÷ 1 and aspect ratio A = 10. Fully elliptic Navier-Stokes and energy equations are solved using the finite volume techniques with staggered grid arrangements. The obtained results show a strong influence of the temperature ratio on local and average heat transfer coefficient on the hot and cold plates. With reduction of T_R the heat transfer parameter on the hot wall grows, and on the cold one, on the contrary, it decreases. As a result, the total heat exchange from two plates depends poorly on the parameter T_R.     

Numerical study of the heat transfer and electro-thermo-convective flow patterns in dielectric liquid layer subjected to unipolar injection

In this article we study the electro-thermal convection in a dielectric liquid layer placed between two electrodes and subjected to the simultaneous action of an electric field and a thermal gradient. The full set of equations describing the electro-thermo-convective phenomena is directly solved using a finite volume method. We first heat the liquid from below at time t = 0, wait for the thermal steady state and then inject the electric charges by applying the electric potential. The development of the electro-convective motion is analysed in detail in two cases: 1) strong injection from the lower electrode, 2) strong injection from the upper one. We also study the heat transfer enhancement due to electro-convection. The evolution in time of the Nusselt number Nu for different combinations of the two usual non-dimensional parameters associated to the electro-thermo-convection phenomena (Rayleigh number Ra and the electrical parameter T) is also given and analysed.     

Computational approaches to aspect-ratio-dependent upper bounds and heat flux in porous medium convection

Direct numerical simulation (DNS) has shown that Rayleigh–Bénard convection in a fluid-saturated porous medium self-organizes into narrowly spaced plumes at (ostensibly) asymptotically high values of the Rayleigh number Ra. In this Letter a combination of DNS and upper bound theory is used to investigate the dependence of the Nusselt number Nu on the domain aspect ratio L at large Ra  . A novel algorithm is introduced to solve the optimization problems arising from the upper bound analysis, allowing for the best available bounds to be extended up to Ra≈2.65×10⁴$\mathit{Ra}\approx 2.65\times {10}^4$. The dependence of the bounds on L(Ra)$L(\mathit{Ra})$ is explored and a “minimal flow unit” is identified.     

Experimental and Numerical Study of Free Convection on an Isothermal Downward Cone

Abstract

The laminar free-convection heat transfer from an isothermal downward cone in air is investigated experimentally and numerically. The experimental investigation is carried out by Mach-Zehnder interferometery technique and the numerical simulation was done by Fluent. The cone tip angle has been kept constant to 45° and it was suspended from its base throughout the experiment. This article focuses on the effect of Rayleigh number variation on the local and average free-convection heat transfer coefficient over the conical surface. The local and average Nusselt numbers were determined for the Rayleigh number range of 4.9×10⁵ to 1.1×10⁶. Also, the experiment and the numerical simulation were carried out on a vertical isothermal cylinder of circular cross section in order to compare results with other researchers for the verification of our experimental and numerical results. The significant influence of the upper end surface of the cone, both in the experimental and numerical studies, indicated a recirculation region above the upper end surface which affects the local convection heat transfer at the slant trailing edge and causes it to increase. Also, a correlation for the calculation of the local Nusselt number over the cone is proposed.