Wide Range Of Prandtl Numbers Research Articles

Turbulent heat flux modelling in forced convection flows using artificial neural networks

Fluid dynamics of liquid metals plays a central role in new generation liquid metal cooled nuclear reactors, for which numerical investigations require the use of an appropriate thermal turbulence model for low Prandtl number fluids. Given the limitations of traditional modelling approaches and the increasing availability of high-fidelity data for this class of fluids, we propose a machine learning strategy for the modelling of the turbulent heat flux. A comprehensive algebraic mathematical structure is derived and physical constraints are imposed to ensure attractive properties promoting stability and realizability. The closure coefficients of the model are predicted by an Artificial Neural Network (ANN) which is trained with the available Direct Numerical Simulation (DNS) data at different Prandtl numbers. The validation shows that the ANN provides an accurate representation of the heat flux in a wide range of Prandtl numbers (Pr = 0.01–0.71) and compares well with other existing thermal closures. Nevertheless, simulations with different auxiliary models to compute the inputs of the ANN revealed a certain sensitivity of the data-driven formulation on the models used in combination with it. In particular, the ANN model strongly relies on the Reynolds stress anisotropy. Such a sensitivity limits the robustness of the model to the inaccuracies of the underline momentum field and the momentum turbulence model applied in combination with it.

The bimodal instability of thermal convection in a tall vertical annulus

We pursue a three-dimensional linear stability analysis to investigate the convective stability in a cylindrical annulus imposed with a horizontal temperature gradient. The investigation focuses on the stability of both axisymmetric and asymmetric modes in a wide range of Prandtl number Pr and the radii ratio η between the inner to outer cylinders. Results show that, for the axisymmetric mode, the stability strongly depends on the Prandtl number. For 0≤Pr≤1.4, the instability sets in as the shear mode that the instability enhances as η increases while it is virtually independent of Pr. The critical Grashof number can be approximated by Grc(η)≈103×4η−0.68+3. For 1.5≤Pr≤12.5, the buoyant mode appears and competes with the shear mode to predominate the stability under various Pr and η, namely, the bimodal instability occurs. The transition between the two modes occurs at the specific radii ratio η1=0.21Pr0.727−0.31. For Pr≥12.6, the shear mode vanishes and the buoyant mode prevails. The asymmetric modes are less prevailing than the axisymmetric mode for small Prandtl numbers, except that the mode of an azimuthal wavenumber equal to 1 becomes dominant in two small regions of η. When the Prandtl number is large, the axisymmetric mode predominates over the asymmetric mode except when η is very small. Otherwise, the asymmetric mode of a larger azimuthal wavenumber is less prevailing.

Free convection boundary layer correlations for a vertical plate with nonuniform heating

An analytical study of a free convection boundary layer (BL) for a laminar and steady flow over a vertical flat plate (VFP) having nonuniform thermal conditions has been investigated. The plate is subjected to a variable wall heat flux of where is the distance along the VFP from the leading edge and is a reference heat flux. The boundary layer equations for nonuniform wall heat flux conditions have been formulated. These equations are then solved numerically for the range of exponents and Prandtl numbers () of −0.25 to 1 and 0.01 to 1000 respectively. The effects of two physical parameters and on the characteristic of thermal boundary layer (TBL) and hydrodynamic boundary layer (VBL) are described. Moreover, correlations of TBL and VBL thicknesses have been presented as a function of Pr and The TBL and VBL thickness are more influenced by Pr as compared to These correlations are reasonably accurate for uniform heat flux (n = 0) proposed by A. Bejan using a scale analysis and no other such correlations are available for nonuniform heat flux for a wide range of Prandtl numbers (Pr). It has been found that when both the BL thicknesses decrease with the Pr till a value of is reached. For the growth of VBL and TBL thickness over the VFP increases and decreases respectively. Consequently, the local skin friction coefficient decreases and the local Nusselt number increases with the increase in and at a fixed value of modified Rayleigh number.

Physics-constrained machine learning for thermal turbulence modelling at low Prandtl numbers

Liquid metals play a central role in new generation liquid metal cooled nuclear reactors, for which numerical investigations require the use of appropriate thermal turbulence models for low Prandtl number fluids. Given the limitations of traditional modelling approaches and the increasing availability of high-fidelity data for this class of fluids, we propose a Machine Learning strategy for the modelling of the turbulent heat flux. A comprehensive algebraic mathematical structure is derived and physical constraints are imposed to ensure attractive properties promoting applicability, robustness and stability. The closure coefficients of the model are predicted by an Artificial Neural Network (ANN) which is trained with DNS data at different Prandtl numbers. The validity of the approach was verified through a priori and a posteriori validation for two and three-dimensional liquid metal flows. The model provides a complete vectorial representation of the turbulent heat flux and the predictions fit the DNS data in a wide range of Prandtl numbers (Pr=0.01−0.71). The comparison with other existing thermal models shows that the methodology is very promising.

Hydro-thermal characteristics of laminar flow through a square duct having transverse ribs and helical screw-tape inserts

The hydro-thermal characteristics of transverse ribs combined with helical screw tape inserts in square duct are numerically investigated and compared the numerical findings with very little available experimental results. Moreover, the experimental data is available only with circular duct. No other cross-sectioned duct has ever been investigated. The motivation and novelty of the present work is to fill this hiatus. Finite volume method-based solver, ANSYS Fluent has been used for simulation. The uniform heat flux boundary condition has been imposed and a wide range of Prandtl number with five different fluids has been used for this investigation. This is never ever possible in a single experimental programme. Only numerical simulation can do the job and this is the inherent beauty of the present investigation. The results of experimental and computational findings proved that the present configuration produces good mixing and enhanced boundary layer separation. The results of the effect of rib height, rib pitch and screw tape parameter on Nusselt number and friction factor are presented. The numerical investigation has shown that Servotherm oil is the best fluid. The Nusselt number is increasing with increase in rib height and Reynolds number and it is decreasing with increase in rib pitch and screw tape parameter value. Also, the friction factor is increasing with increase in rib height and Reynolds number but the increment in Nusselt number is larger than the increment in friction factor. Therefore, the present fin geometry is recommended for industrial use.

Boundary layers in turbulent vertical convection at high Prandtl number

Many environmental flows arise due to natural convection at a vertical surface, from flows in buildings to dissolving ice faces at marine-terminating glaciers. We use three-dimensional direct numerical simulations of a vertical channel with differentially heated walls to investigate such convective, turbulent boundary layers. Through the implementation of a multiple-resolution technique, we are able to perform simulations at a wide range of Prandtl numbers ${Pr}$ . This allows us to distinguish the parameter dependences of the horizontal heat flux and the boundary layer widths in terms of the Rayleigh number $\mbox {{Ra}}$ and Prandtl number ${Pr}$ . For the considered parameter range $1\leq {Pr} \leq 100$ , $10^{6} \leq \mbox {{Ra}} \leq 10^{9}$ , we find the flow to be consistent with a ‘buoyancy-controlled’ regime where the heat flux is independent of the wall separation. For given ${Pr}$ , the heat flux is found to scale linearly with the friction velocity $V_\ast$ . Finally, we discuss the implications of our results for the parameterisation of heat and salt fluxes at vertical ice–ocean interfaces.

High fidelity simulations in support to assess and improve RANS for modeling turbulent heat transfer in liquid metals: The case of forced convection

This paper attempts to give a brief overview of the work conducted through some recent EU funded projects under the Euratom research for innovative nuclear systems (THINS, SESAME and MYRTE) concerning the use of high fidelity (HiFi) simulations, namely Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES), to adapt Reynolds Averaged Navier-Stokes (RANS) models for the computation of turbulent heat transfer in liquid metal flows.Here the focus is on forced convection only, that prevails in normal reactor operation, through some selected cases performed at UCLouvain, e.g. the channel flow and the impinging jet. The considered RANS approaches are those based on the simple gradient diffusion hypothesis or SGDH, and those based on the algebraic heat flux formulation (AHFM). Among the AHFM models the two possible formulations are assessed, i.e. the explicit form (k-∊-kθ-∊θ of Manservisi and Menghini (2014) still based on the eddy diffusivity concept (gradient diffusion assumption), and the implicit form of the AHFM-NRG which is essentially a recalibration of the reference model of Kenjeres et al. (2005). The Results show the overall superiority of AHFM models, although SGDH-models using the Kay correlation for the turbulent Prandtl number and the dedicated thermal wall-function developed by Duponcheel et al. (2014) provide reasonable results at a much lower effort making them interesting for industrial applications. However further research is undoubtedly required to come closer to more universal models working for a wide range of Prandtl numbers and flow conditions.

A Multiple-Grid Lattice Boltzmann Method for Natural Convection under Low and High Prandtl Numbers

A multi-distribution lattice Boltzmann Bhatnagar–Gross–Krook (BGK) model with a multiple-grid lattice Boltzmann (MGLB) model is proposed to efficiently simulate natural convection over a wide range of Prandtl numbers. In this method, different grid sizes and time steps for heat transfer and fluid flow equations are chosen. The model is validated against natural convection in a square cavity, since extensive benchmark solutions are available for that problem. The proposed method can resolve the computational difficulty in simulating problems with very different time scales, in particular, when using extremely low or high Prandtl numbers. The technique can also enhance computational speed and stability while keeping the simplicity of the BGK method. Compared with the conventional lattice Boltzmann method, the simulation time can be reduced up to one-tenth of the time while maintaining the accuracy in an acceptable range. The proposed model can be extended to other lattice Boltzmann collision models and three-dimensional cases, making it a great candidate for large-scale simulations.

The Generalized Lévêque Equation and its application to flat tubes without and with passive inserts

Searching for an analogy concept between heat transfer and pressure drop is anything but new, nevertheless it is still the subject of researches as it enables the prediction of heat transfer from pressure drop data, which helps reducing experimental effort for heat transfer measurements. The Generalized Lévêque Equation (GLE) is a widely-used analogy concept, which allows the prediction of heat transfer from pressure drop data, if hydrodynamic boundary layer is developed or still developing. This analogy concept has been successfully applied to flow situations with practical relevance like chevron-type plate heat exchangers, packed beds and in tube bundles, where the boundary layers are forced to reform continuously. In this contribution, the GLE is applied to flat tube flow without and with passive inserts, investigating the impact of passive inserts, which force the boundary layers to reform continuously, on the application of the GLE.This paper presents results of an experimental investigation on pressure drop and heat transfer in flat tubes without and with passive inserts, covering a wide range of Prandtl number (8 < Pr < 60) and Reynolds number values (40 < Re < 3500). These parameter ranges and the passive inserts represent the typical conditions and design used for the liquid side of single-phase flat tube heat exchangers used in the automotive industry. 238 experimental data points for heat transfer coefficients are compared to heat transfer coefficients calculated from pressure drop data with the GLE.The results show that the GLE is not applicable to flat tubes without passive inserts, yet becomes applicable with passive inserts, which are forcing a re-formation of boundary layers in the whole range of investigated Reynolds and Prandtl numbers.

Numerical study on the effect of corrugation angle on thermal performance of cross corrugated plate heat exchangers

In the present work, the single-phase thermohydraulic performance of a cross-corrugated plate heat exchanger (PHE) with different corrugation angles 30°≤β≤80° is studied numerically. A three-dimensional numerical model using the finite volume method is developed and validated assuming laminar and turbulent flows in the Reynolds number range of 400–10000. The two-equation k-ω SST model is employed to analyse the turbulent flows in the PHE. It is found that for all the configurations of the PHEs, the pressure drop per unit length increases with the Reynolds number and the corrugation angle. The friction factor of the PHE with β = 80°–80° is higher by a maximum of 2.8 times than that of the flat PHE. The Nusselt number is found to be higher in the corrugated PHE as compared to that of the flat plate PHE. The Nusselt number is increased by a maximum of 3 times for the PHE with β = 80°–80° than that of the flat PHE. Further, single phase generalized correlations of Nusselt number with Reynolds number, geometrical parameters of the cross-corrugated plate heat exchanger and fluid properties are developed for high cross-corrugated angles and a wide range of Prandtl numbers within a maximum accuracy of 40%.

Correlating laminar convection in slots with developing flow

The problem of combined entry convection between the walls of a duct and a hydrodynamically developing flow is revisited for the geometry of a two-dimensional slot. Accurate calculations of the local and average Nusselt numbers are reported for an extended range of inverse Graetz number (ZH≥10−6), and a wide range of Prandtl number (0.1 ≤ Pr ≤ 500), while considering both constant temperature and constant heat flux wall conditions. Accuracy in this work is achieved by applying a transformation of variables to remove the singularity along the perimeter of the duct inlet before integrating the governing equations. Results of this investigation are correlated for all inverse Graetz numbers and Prandtl numbers Pr ≥ 0.5. Correlations for the local and average Nusselt numbers, and both wall conditions, are accurate to ±2.1% when compared to the benchmark results of this work.

Laminar Convection in Circular Tubes With Developing Flow

Abstract The combined entry problem for the simultaneous development of heat and momentum transfer in a circular tube has been resolved over an extended range of inverse Graetz number ZH≥10−6 and for a wide range of Prandtl numbers 0.1≤Pr≤500. For the historical range of ZH≥5×10−4 and 0.7≤Pr≤50, earlier studies are within 5% of the current benchmark calculations, but for the new extended range of conditions, the best authoritative sources were in error by as much as 33%. Four new correlations are proposed for the local and average Nusselt numbers, and for the constant temperature and constant heat flux wall condition, which are accurate to 2.2% for all values of inverse Graetz number and Pr≥0.5. In contrast, legacy correlations typically had a 10–20% error range when compared to the results of this work, with many exhibiting larger errors and only few achieving errors as low as 5–10%.

Minimization of entropy generation due to MHD double diffusive mixed convection in a lid driven trapezoidal cavity with various aspect ratios

Entropy generation minimization has significant importance in fluid flow, heat and mass transfer in an enclosure to get the maximum efficiency of a system and to reduce the loss of energy. In the present study, the analysis of mixed convection fluid flow, heat and mass transfer with heat line and mass line concept and entropy generation due to the effects of fluid flow, heat flow, mass flow and magnetic field in a trapezoidal enclosure with linearly heated and diffusive left wall, uniformly heated and diffusive lower wall, cold and nondiffusive right wall, adiabatic and zero diffusion gradient top wall have been reported. Parametric studies for the wide range of Prandtl number (Pr = 0.7 for air cooling system and Pr = 1000 for the engines filled with olive or engine oils), Rayleigh number (Ra = 103–105), aspect ratio (A = 0.5–1.5) and inclination angle of the cavity (ϕ = 45°–90°) have been performed, which help to construct the perfect shape of cavity in many engineering and physical applications so that the entropy is minimum to get the maximum efficiency of any system. The finite-difference approximation has been used to find out the numerical solutions. Biconjugate Gradient Stabilized (BiCGStab) method is used to solve the discretized nonhomogeneous system of linear equations.

MODELING HEAT EXCHANGE DEPENDING ON THE PRANDTL NUMBER FOR VARIOUS GEOMETRIC AND REGIME PARAMETERS

Objectives. The aim is to study the dependency of the distribution of integral heat transfer during turbulent convective heat transfer in a pipe with a sequence of periodic protrusions of semicircular geometry on the Prandtl number using the calculation method based on a numerical solution of the system of Reynolds equations closed using the Menter’s shear stress transport model and the energy equation on different-sized intersecting structured grids.Method. A calculation was carried out on the basis of a theoretical method based on the solution of the Reynolds equations by factored finite-volume method closed with the help of the Menter shear stress transport model, as well as the energy equation on different-scaled intersecting structured grids (fast composite mesh method (FCOM)).Results. The calculations performed in the work showed that with an increase in the Prandtl number at small Reynolds numbers, there is an initial noticeable increase in the relative heat transfer. With additional increase in the Prandtl number, the relative heat transfer changes less: for small steps, it increases; for median steps it is almost stabilised, while for large steps it declines insignificantly. At large Reynolds numbers, the relative heat transfer decreases with an increase in the Prandtl number followed by its further stabilisation.Conclusion. The study analyses the calculated dependencies of the relative heat transfer on the Pr Prandtl number for various values of the relative h/D height of the turbulator, the relative t/D pitch between the turbulators and for various values of the Re Reynolds number. Qualitative and quantitative changes in calculated parameters are described all other things being equal. The analytical substantiation of the obtained calculation laws is that the height of the turbuliser is less for small Reynolds numbers, while for large Reynolds numbers, it is less than the height of the wall layer. Consequently, only the core of the flow is turbulised, which results in an increase in hydroresistance and a decrease in heat transfer. In the work on the basis of limited calculation material, a tangible decrease in the level of heat transfer intensification for small Prandtl numbers is theoretically confirmed. The obtained results of intensified heat transfer in the region of low Prandtl numbers substantiate the promising development of research in this direction. The theoretical data obtained in the work have determined the laws of relative heat transfer across a wide range of Prandtl numbers, including in those areas where experimental material does not currently exist.

Natural convection in a differentially heated enclosure filled with low Prandtl number fluids with modified lattice Boltzmann method

The convective motion in a three- and two- dimensional square cavity driven by a temperature gradient is analyzed. The cavity is filled with a low-Prandtl-number (Pr) fluid, typical for liquid metals. The vertical walls have constant but different temperatures, while the horizontal boundaries are adiabatic. Low Prandtl number exhibits strong non-linearity, where the advection term dominates the diffusion term in the momentum equations. Therefore, a Multi-Relaxation-Time, modified Lattice Boltzmann (MRT-LBM) approach is used to overcome the stability of the numerical solutions. Also, by using the mentioned method a wide range of Prandtl numbers, Pr, (0.01–0.5) and a wide range of Rayleigh numbers, Ra, (104 to 108) are investigated. It was found that the flow field exhibits periodic oscillations at the critical Rayleigh numbers, which are dependent on the Prandtl number. The periodic flow becomes aperiodic as the Prandtl number decreases and/or the Rayleigh number decreases. Also, the average Nusselt number are presented and correlated for the investigated range of the controlling parameters.

Interface control in vertical Bridgman growth by shaped and flat rotating baffles

We present numerical results of a study on the influence of a submerged rotating baffle on the control of the crystal-melt interface shape in vertical Bridgman crystal growth. 3D simulations were performed for four materials: CdTe, InP, Ga2O3 and Al2O3, having a wide range of Prandtl numbers, 0.015 < Pr < 33. Bowed baffle was considered for the first time. The rotation rate was varied from stagnant up to ω = 50 rpm. The results indicate that slightly convex crystal-melt interface can be obtained for all studied materials. The higher the Pr number of melts, the stronger is the influence of the baffle rotation.

New combined heat transfer enhancement techniques used in laminar flow through non-circular ducts

The problem of heat transfer enhancement using different combinations of transverse ribs, transverse corrugations and twisted-tape insert with oblique teeth, twisted tape with centre clearance and helical screw-tape insert in square and rectangular channel was investigated. The study also involved the thermohydraulic performance of inserts in non-circular channels in the laminar regime. Non-circular channels having aspect ratios of 1, 0.5 and 0.33 were considered. The techniques used for heat transfer augmentation in the present study were and the problem was (1) transverse ribs with twisted tape with oblique teeth, (2) integral transverse corrugation and centre-cleared twisted-tape insert, (3) transverse rib and helical screw-tape insert, (4) axial ribs and centre-cleared twisted-tape insert and (5) integral transverse rib and centre-cleared twisted-tape insert. These combined fin geometry are supposedly important from possible heat transfer enhancement point of view. In the channels, the combined influence of insert geometry such as rib pitch, rib height, twist ratio of twisted tape, twisted tape tooth horizontal length and twisted tape tooth angle was studied. The experiment was carried out using Servotherm oil as the working fluid, which had a wide range of Prandtl number (430–530). Uniform heat flux boundary condition was used. The methods used in the experiment have been discussed in the text of the presentation. The important results of the experimental investigation showed that the thermohydraulic performance using a combination of inserts was better than that of bare ducts and that of the individual enhancement technique acting alone. Both the increase in pressure drop and heat transfer augmentation occurred due to the addition of inserts, but heat transfer enhancement dominated over the increase in pressure drop. Nusselt number and friction factor correlations have also been developed and are presented in this paper. The experimental program considered different fin arrangements with various parameters and therefore, it should be very helpful in designing heat exchangers of different shapes and sizes used for industrial purpose. It is concluded from the present investigation that 31–52 per cent increase in heat duty at constant pumping power and 25–36 per cent reduction in pumping power at constant heat duty are achievable. This is the novelty of the present work since no such study and the observations have been made earlier. These types of combined fin configurations have not been tried before. The findings are likely to have good impact in the industry.

Possible causes of electrical resistivity distribution inhomogeneity in Czochralski grown single crystal silicon

Electrical resistivity distribution maps have been constructed for single crystal silicon wafers cut out of different parts of Czochralski grown ingots. The general inhomogeneity of the wafers has proven to be relatively high, the resistivity scatter reaching 1–3 %. Two electrical resistivity distribution inhomogeneity types have been revealed: azimuthal and radial. Experiments have been carried out for crystal growth from transparent simulating fluids with hydrodynamic and thermophysical parameters close to those for Czochralski growth of silicon single crystals. We show that a possible cause of azimuthal electrical resistivity distribution inhomogeneity is the swirl-like structure of the melt under the crystallization front (CF), while a possible cause of radial electrical resistivity distribution inhomogeneity is the CF curvature. In a specific range of the Grashof, Marangoni and Reynolds numbers which depend on the ratio of melt height and growing crystal radius, a system of well-developed radially oriented swirls may emerge under the rotating CF. In the absence of such swirls the melt is displaced from under the crystallization front in a homogeneous manner to form thermal and concentration boundary layers which are homogeneous in azimuthal direction but have clear radial inhomogeneity. Once swirls emerge the melt is displaced from the center to the periphery, and simultaneous fluid motion in azimuthal direction occurs. The overall melt motion becomes helical as a result. The number of swirls (two to ten) agrees with the number of azimuthally directed electrical resistivity distribution inhomogeneities observed in the experiments. Comparison of numerical simulation results in a wide range of Prandtl numbers with the experimental data suggests that the phenomena observed in transparent fluids are universal and can be used for theoretical interpretation of imperfections in silicon single crystals.

Effect of the Prandtl number on the instabilities of the thermocapillary flow in an annular pool

This paper is devoted to the instability mechanisms of the thermocapillary flow in an annular pool. The stability limit of the axisymmetric basic state was studied over a wide range of Prandtl numbers (0.001 ≤ Pr ≤ 6.7) using linear stability analysis based on the spectral element method. The results demonstrate five types of instabilities, and the corresponding instability mechanisms were revealed by disturbance energy analysis. In particular, in the narrow range 1.4 ≤ Pr ≤ 1.53, with the increase in the Marangoni number, three transitions between the axisymmetric steady flow and the three-dimensional oscillatory flow were found, owing to the coupling and interaction of the hydrodynamic and hydrothermal instability mechanisms.

Heat and mass transfer phenomena due to a rotating non-isothermal wavy disk

Augmentation in the heat transfer rate through wall roughness is a passive technique which is extensively being used in the modern technologies. For rotating disk geometries its implementation is very rare as the literature does not include any significant works on this topic except a very few. The isothermal rotating wavy disk is available in literature (and has been cited in the introduction part) in which the author reported 15% increase in the rate of heat transfer compared to the flat disk case. Such an increased heat transfer rate can further be increased by assuming certain other flow assumptions. An attempt in this regard has been made in order to acquire the increased heat transfer coefficient when the rotating wavy disk temperature is distributed non-uniformly. The disk temperature is assumed to depend on radial distance in the form of power-law function. In the situation of radially increasing disk temperature following a quatratic variations along the radius up to 110% intensification in the overall heat exchange is observed from the hot wavy disk (with two waves and surface roughness ratio equal to 0.1) to the surrounding quiescent air in comparison to an isothermal flat disk. Apart from this, rate of heat transfer from the wavy disk has also been estimated for a very wide range of Prandtl number (10-2 to 104) in this study. In the current analysis some threshold values of the power-law exponent of the disk temperature distribution have also been captured for which the heat transfer is seized. The invariant threshold value (n∗=-2) of the flat disk is observed to change for a non-flat disk case. However, this variation is found to be small for a variety of fluids. Due to the analogy between convective heat transfer and mass transfer phenomena the obtained findings of this study are two fold as they also characterize the mass transfer rate from a wavy rotating disk with a uniform distribution of mass at the disk surface.