Low Prandtl Number Fluids Research Articles

Journey to the center of stars: The realm of low Prandtl number fluid dynamics

Turbulence deep within the interior of stars bears many qualitative similarities with turbulence in the ocean and atmosphere on Earth. In both contexts, various forms of convective instabilities and shear instabilities govern the transport of heat, momentum (angular momentum) and chemical species. However, an important distinction between stellar fluids and geophysical fluids is the value of the Prandtl number, which is of order unity in oceanic and atmospheric flows, and asymptotically small in stars. As a result, many well-known scaling laws for turbulent transport in geophysical flows must be revisited at low Prandtl number, as reviewed in this paper.

Unsteady development of Marangoni convection in a sidewall moving open cavity

In this study, simultaneous effect of free, forced and Marangoni convection on fluid flow and heat transfer in a sidewall moving open cavity is investigated. The cavity is filled with low Prandtl number fluid (Pr = 0.054); such a situation can be find in the application of liquid metal and semiconductor melts. Free convection is induced due to the heating of fluid from the bottom of the cavity. Forced convection is originated due to the translation of the sidewalls either in upward or downward direction. Whereas, the Marangoni convection is generated in the upper portion of the fluid due to imposed temperature gradient along the top free surface. The transient phenomena of the combined effects are analyzed numerically in dimensionless forms utilizing an in-house CFD code. The analyses are conducted under different pertinent parameters such as Marangoni number, Biot number, Reynolds number and Richardson number. The study is conducted to explore the flow-field structures and heat transfer characteristics in the presence of three convection modes with their relative strengths.

Spiral defect chaos in Rayleigh-Bénard convection: Asymptotic and numerical studies of azimuthal flows induced by rotating spirals

Spiral defect chaos appears near the onset of Rayleigh-B\'enard convection for a low Prandtl number fluid. In this state, rotating spirals are continuously nucleated and eliminated, yielding a persistent dynamics. Here we derive an equation for the azimuthal flow induced by an effective body force originating from rotating spirals. This result is verified numerically using a two-dimensional generalized Swift-Hohenberg model and the three-dimensional Boussinesq equations. We identify a correlation between the appearance of spiral defect chaos and the balance between mean-flow advection and diffusive dynamics related to roll unwinding.

Modeling and simulation of a supercritical CO2-liquid sodium compact heat exchanger for sodium-cooled fast reactors

The study focuses on modeling and simulations of sodium-sCO2 intermediary compact heat exchangers for sodium-cooled fast reactors (SFR). A simplified 1-D analytical model was developed in companion with a 3-D CFD model. Using classic heat transfer correlations for Nusselt number, some simulation results using the 1-D model have achieved reasonable match with the CFD simulation results for longer channels (i.e., 40 cm and 80 cm). However, for short channel (10 cm) when axial conduction within the sodium fluid is significant, the 1-D model significantly over-predicted the heat transfer effectiveness. By incorporating the temperature-jump model, the 1-D model can extend its predictive capability for low-Prandtl number fluid/Peclet number flows. The results can help improve the understanding of heat transfer for sodium and low-Prandtl number fluids in general and improve designs of sodium-sCO2 compact heat exchangers. The results also confirmed that the sCO2 side dominates the overall heat transfer for Na-sCO2 heat exchangers. A preliminary attempt of optimizing the channel geometry shows mixing results – while heat transfer effectiveness was significantly increased for the wavy channel, much greater pressure drop was also predicted by the simulations.

A collaborative effort towards the accurate prediction of turbulent flow and heat transfer in low-Prandtl number fluids

This article reports the experimental and DNS database that has been generated, within the framework of the EU SESAME and MYRTE projects, for various low-Prandtl flow configurations in different flow regimes. This includes three experiments: confined and unconfined backward facing steps with low-Prandtl fluids, and a forced convection planar jet case with two different Prandtl fluids. In terms of numerical data, seven different flow configurations are considered: a wall-bounded mixed convection flow at low-Prandtl number with varying Richardson number (Ri) values; a wall-bounded mixed and forced convection flow in a bare rod bundle configuration; a forced convection confined backward facing step (BFS) with conjugate heat transfer; a forced convection impinging jet for three different Prandtl fluids corresponding to two different Reynolds numbers of the fully developed planar turbulent jet; a mixed-convection cold-hot–cold triple jet configuration corresponding to Ri = 0.25; an unconfined free shear layer for three different Prandtl fluids; and a forced convection infinite wire-wrapped fuel assembly. This wide range of reference data is used to evaluate, validate and/or further develop different turbulent heat flux modelling approaches, namely simple gradient diffusion hypothesis (SGDH) based on constant and variable turbulent Prandtl number; explicit and implicit algebraic heat flux models; and a second order turbulent heat flux model. Lastly, this article will highlight the current challenges and perspectives of the available turbulence models, in different codes, for the accurate prediction of flow and heat transfer in low-Prandtl fluids.

Transitions in overstable rotating magnetoconvection.

The classical Rayleigh-Bénard convection (RBC) system is known to exhibit either subcritical or supercritical transition to convection in the presence or absence of rotation and/or magnetic field. However, the simultaneous exhibition of subcritical and supercritical branches of convection in plane layer RBC depending on the initial conditions, to the best of our knowledge, has not been reported so far. Here, we report the phenomenon of simultaneous occurrence of subcritical and supercritical branches of convection in overstable RBC of electrically conducting low Prandtl number fluids (liquid metals) in the presence of an external uniform horizontal magnetic field and rotation about the vertical axis. Extensive three-dimensional (3D) direct numerical simulations (DNS) and low-dimensional modeling of the system, performed in the ranges 750≤Ta≤3000 and 0<Q≤1000 of the Taylor number (Ta, strength of the Coriolis force) and the Chandrasekhar number (Q, strength of the Lorenz force), respectively, establish the phenomenon convincingly. Detailed bifurcation analysis of a simple 3D model derived from the DNS data reveals that a supercritical Hopf bifurcation and a subcritical pitchfork bifurcation of the conduction state are responsible for this. The effect of Prandtl number on these transitions is also explored in detail.

Numerical Investigation of Turbulent Heat Transfer Properties at Low Prandtl Number

Sodium-cooled fast reactor (SFR) which is one of the most promising candidates to meet the Generation IV International Forum (GIF) declare has drawn a lot of attentions. Turbulent heat transfer in the liquid sodium which is one of those low Prandtl fluids is an extremely complex phenomenon. The limitations of the commonly used eddy diffusivity approach have become more evident for low-Prandtl fluids. The current study focuses on the assessment and optimization of the existing modeling closure for single-phase turbulence in liquid sodium based on the reference results provided by LES method. In this paper, a wall-resolved Large-Eddy Simulation was performed to simulate the flow and heat transfer properties in a turbulent channel at low Prandtl number. The simulation results are firstly compared with the DNS results obtained from literature. A good agreement demonstrated the capability of the employed numerical approach to predict the turbulent and heat transfer properties in a low Prandtl number fluid. Consequently, new reference results were obtained for typical Prandtl number and wall heat flux of SFR. A time-averaged process has been employed to evaluate temperature profile quantitatively as well as turbulent heat flux. Their dependency was also evaluated based on a systematic CFD simulation which covers the typical Reynolds numbers of SFR. Based on the obtained reference results, the coefficients employed in the algebraic turbulent heat flux model (AFM) are calibrated. The optimized coefficients provide a better prediction accuracy of heat transfer properties for typical flow conditions of SFR if comparing with the existing models found in the literature.

Influence of buoyancy in a mixed convection liquid metal flow for a horizontal channel configuration

This article presents the direct numerical simulation (DNS) of mixed convection turbulent heat transfer in a horizontal channel case for liquid lead. Cartesian mesh is used and the incompressible Navier-Stokes equations are discretized with highly accurate finite difference sixth-order compact schemes to perform the DNS. The influence of mixed convection in liquid metal with Prandtl number equal to 0.025 and Reynolds number equal to 4667 has been studied by varying the Richardson number (Ri = 0, 0.25, 0.50, 1.00). The obtained results are extensively analyzed and discussed in this article. In particular, large-scale circulation is observed under the influence of buoyancy. Compared to the forced convection case (Ri = 0), stronger velocity fluctuations are noticed that highlight the fact that turbulence is strongly enhanced with the increasing buoyancy. It also proves that the thermal plumes rising up from the hot wall of the channel activate the cross-stream eddies. Moreover, temperature fluctuations are found to be more homogeneously distributed with increasing buoyancy effects and mixing is more effective in the center of the channel. In addition, compared with forced convection, mixed convection has shown enlargement of the large-scale structures that only appear in the temperature field for low Prandtl number fluids. Extensive results of flow and temperature fields are analyzed and presented.

Effect of Heat Dissipation on Thermocapillary Convection of Low Prandtl Number Fluid in the Annular Pool Heated from Inner Cylinder

This paper presents a series of numerical simulations on the effect of heat dissipation on the thermocapillary convection of a low Prandtl number (Pr = 0.011) fluid in an annular pool heated from its inner cylinder. The radius ratio and the aspect ratio of the annular pool set as η = 0.5 and e = 0.1, respectively. The results show that with the increase of Biot number, the isotherms of the two-dimensional basic flow are seriously compressed near the hot inner cylinder and the thermocapillary flow pattern will be impaired. With the increase of Marangoni number, the thermocapillary convection first evolves from the basic flow to a three-dimensional stationary flow with longitudinal stationary stripes on the free surface, and then to a three-dimensional oscillatory flow featured with the combination of stationary stripes and azimuthal waves, which is caused by the mutual effect of the radial convection, azimuthal local flow and heat dissipation. After the flow destabilization, the amplitude of temperature fluctuation increases and the wave number decreases with the increase of Marangoni number when the free surface is adiabatic. Furthermore, the temperature fluctuation pattern is changed due to the surface heat dissipation. In addition, with the increase of Biot number, the wave number almost keeps constant when Marangoni number is small, but it increases significantly at a large Marangoni number.

Thermal fluctuations in low-Prandtl number fluid flows over a backward facing step

The backward facing step geometry (BFS) is a representative geometry for sudden expansions in pipe, duct and channel flows. While this type of geometry by itself is not a part of engineering components, the flow separation and the accompanying flow features present in a BFS are of great importance when designing manifolds, heat exchangers or fuel bundles. In the frame of EU Horizon 2020 project SESAME, an extensive effort has been put forward to gain more insights into the flow and thermal features in a BFS geometry for low-Prandtl number fluids. The main motivation behind this effort is two-fold: to generate a reference database by means of experiments and high fidelity simulations, and accordingly utilize the reference database to validate and/or improve the turbulence models in engineering applications. In this paper, we present a broad description of the experimental facility and its expected capabilities, as well as the results of numerical efforts. The experimental results will be obtained in the DITEFA 2 facility of KIT with a GaInSn eutectic alloy. The expansion ratio of the BFS in the experiment is set to2 and the geometry has one heated wall. Unlike the vast majority of the BFS experiments found in the literature, the present BFS experiment has an outflow in a shape of a square, that is, the width and height of the outflow are approximately the same. Second, a direct numerical simulation (DNS) is performed with a passive scalar and for an expansion ratio of 2.25. Similar to the experiment, the shape of the outflow is a square and the average flow is three-dimensional. Conjugate heat transfer DNS is performed for the heated solid walls, while the unheated walls were neglected. Finally, this reference DNS data is used to validate a LES and an advanced RANS modelling approach.

Thermo-solute natural convection with heat and mass lines in a uniformly heated and soluted rectangular enclosure for low Prandtl number fluids

Thermo-solute natural convection with heat and mass lines in a uniformly heated and soluted rectangular enclosure for low Prandtl number fluids is studied. The left and right walls are maintained at constant temperature and solute while the bottom and top walls are adiabatic and non-diffusive. The finite difference method, together with the successive over-relaxation (SOR) technique, is used to solve the flow governing equations after converted into the vorticity-stream function form. Heat and mass lines visualization techniques are used for the better visualization of energy and solute distribution in the enclosure. The results obtained in this study are compared with experimental and numerical those from literature and found to be in good agreement. The influence of Rayleigh number Ra=103,104,105, Prandtl number Pr=0.015,0.025,0.71, Lewis number Le=1,2,5, and Buoyancy ratio N=−1,0,1 on streamlines, isotherms, isosolutes, heatlines, masslines, total heat and mass transfer are examined. The parameters mentioned above considerably influence the total heat and solute transfer rates; moreover, the heat and mass lines play significant roles in understanding the distribution of energy and solute transfers in the enclosure.

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.

Comparison of the quasi-steady-state heat transport in phase-change and classical Rayleigh-Bénard convection for a wide range of Stefan number and Rayleigh number

We report the first comparative study of the phase-change Rayleigh–Bénard (RB) convection system and the classical RB convection system to systematically characterize the effect of the oscillating solid-liquid interface on the RB convection. Here, the role of Stefan number Ste (defined as the ratio between the sensible heat to the latent heat) and the Rayleigh number based on the averaged liquid height Raf is systematically explored with direct numerical simulations for low Prandtl number fluid (Pr = 0.0216) in a phase-change RB convection system during the stationary state. The control parameters Raf (3.96 × 104 ≤ Raf ≤ 9.26 × 107) and Ste (1.1 × 10−2 ≤ Ste ≤ 1.1 × 102) are varied over a wide range to understand its influence on the heat transport and flow features. Here, we report the comparison of large-scale motions and temperature fields, frequency power spectra for vertical velocity, and a scaling law for the time-averaged Nusselt number at the hot plate Nuh¯ vs Raf for both the RB systems. The intensity of solid-liquid interface oscillations and the standard deviation of Nuh increase with the increase in Ste and Raf. There are two distinct RB flow configurations at low Raf independent of Ste. At low and moderate Raf, the ratio of the Nusselt number for phase-change RB convection to the Nusselt number for classical RB convection Nuh¯/NuhRB¯ is always greater than one. However, at higher Raf, the RB convection is turbulent, and Nuh¯/NuhRB¯ can be less than or greater than one depending on the value of Ste. The results may turn out to be of immense consequence for understanding and altering the transport characteristics in the phase-change RB convection systems.

Large Eddy Simulations on a natural convection boundary layer at Pr = 0.71 and 0.025

The development of Gen IV nuclear reactors entails research oh heavy liquid metals, which are considered as appropriate coolant agents due to their high thermal conductivity and favorable nuclear safety characteristics. These fluids are characterized by a very low Prandtl number (between 0.006 and 0.025). During the design of such reactors, the scenarios of maintenance or loss-of-flow accidents consider a pure natural convection loop inside the reactor. The CFD modeling of the momentum and heat transfer must contemplate natural convection in low Prandtl number fluids, however in literature these flow configurations are not usual. This is why in the present article, Large Eddy Simulations over a natural convection boundary layer at Pr=0.025 are presented and described. Turbulence is triggered by adding a perturbation in the boundary layer. The adopted numerical approach is validated through a Large Eddy Simulation at Pr=0.71, where heat transfer, friction coefficient and mean and turbulent flow fields are compared with measurements in literature. Once turbulence is reached in the Pr=0.025 boundary layer, flow characteristics as Nusselt number, skin friction coefficient and turbulent profiles are calculated and compared with the Pr=0.71 simulation to assess the effect of Prandtl number on heat and momentum transfer.

Status and perspectives of turbulent heat transfer modelling in low-Prandtl number fluids

Thermal-hydraulics is recognized as a key safety challenge in the development of liquid metal cooled reactors. At nominal operating conditions, the Prandtl number of liquid metals which are used as primary coolants, such as lead and sodium, is very low: typically of the order of 0.025–0.001. Obtaining an accurate prediction of the turbulent heat transfer at such a low Prandtl number is not an easy task for the standard turbulence models and has challenged the modellers over several decades. In the framework of the EU SESAME project, an effort has been put forward to assess and/or further develop/calibrate different turbulent heat flux closures. In this regard, the present article reports an assessment of four different turbulent heat flux closures for applications involving low-Prandtl fluids. These closures include: (i) the Reynolds analogy based on a constant turbulent Prandtl number (ii) a four-equation explicit algebraic heat flux model (AHFM) (ii) a three-equation implicit AHFM called AHFM-NRG and (iv) a non-linear second-order heat flux model called Turbulence Model for Buoyant Flows (TMBF). The performance of these turbulence models has been assessed in three different test cases against high-fidelity numerical reference data been generated within the SESAME project. The three test cases are: a natural Rayleigh-Bénard convection flow, a mixed convection planar channel flow and a forced convection impinging jet flow. The shortcomings of the classical Reynolds analogy approach for low-Prandtl fluids in all flow regimes are highlighted; hence, more advanced and well-calibrated closures are recommended.

Azimuthally inhomogeneous thermal boundary conditions in turbulent forced convection pipe flow for low to medium Prandtl numbers

The effect of azimuthally inhomogeneous surface heat flux on thermal statistics of turbulent forced convection pipe flow is studied via direct numerical and large eddy simulations. Two Prandtl numbers of Pr=0.71 and Pr=0.025 are considered together with bulk Reynolds numbers of Reb=5300,11700,19000,37700. A zero temperature gradient condition is imposed on one half of the pipe’s surface, while on the other either a constant or a sinusoidal heat flux distribution is imposed. The azimuthally averaged Nusselt number does not change with the different boundary conditions and is close to the one calculated with an appropriate correlation valid for uniform heat flux. Contrarily, the local Nusselt number depends on the wall thermal condition. The turbulent Prandtl number is anisotropic and strongly dependent on the radial location, especially for low Prandtl number fluids at low Reynolds numbers. For higher Reynolds numbers, the turbulent Prandtl number attains an isotropic state with a value of approximately 1 in the core region of the flow.

Direct Numerical Simulation of a Low Prandtl Number Rayleigh–Bénard Convection in a Square Box

Direct numerical simulations for low Prandtl number fluid (Pr = 0.0216) are used to study the steady-state Rayleigh–Bénard convection (RB) in a two-dimensional unit aspect ratio box. The steady-state RB convection is characterized by analyzing the time-averaged temperature-field, and flow field for a wide range of Rayleigh number (2.1 × 105 ⩽ Ra ⩽ 2.1 × 108). It is seen that the time-averaged and space-averaged Nusselt number (Nuh¯) at the hot-wall monotonically increases with the increase in Rayleigh number (Ra) and the results show a power law scaling Nuh¯∝Ra0.2593. The current Nusselt number results are compared with the results available in the literature. The complex flow is analyzed by studying the frequency power spectra of the steady-state signal of the vertical velocity at the midpoint of the box for different Ra and probability density function of dimensionless temperature at various locations along the midline of the box.

Numerical simulation of a turbulent Lead Bismuth Eutectic flow inside a 19 pin nuclear reactor bundle with a four logarithmic parameter turbulence model

Computational Fluid Dynamic allows scientists and engineers to investigate fluid flow in complex geometries and evaluate heat transfer between a solid body and a fluid. In the present paper we study the heat transfer of a Lead Bismuth Eutectic (LBE) turbulent flow in a bare 19 pin nuclear reactor bundle. When dealing with low Prandtl number fluids, like LBE for which Pr = 0.025, proper turbulence models are needed to improve the prediction of heat transfer. We use here a four logarithmic parameter turbulence model in order to calculate Reynolds stresses and turbulent heat flux. In particular, an equation for temperature fluctuations and one for their dissipation are solved. These variables are used to model thermal characteristic time scales. The results are reported for different values of the Peclet number and a fixed value of the pitch to diameter ratio. The obtained values of the Nusselt number are compared with experimental correlations, that can be found in literature, and with the ones obtained using a Simple Eddy Diffusivity model, where the eddy thermal diffusivity is calculated as proportional to eddy viscosity through a modeled turbulent Prandtl number.

Bifurcations from steady to quasi-periodic flows in a laterally heated cavity filled with low Prandtl number fluids

This study deals with the transition toward quasi-periodicity of buoyant convection generated by a horizontal temperature gradient in a three-dimensional parallelepipedic cavity with dimensions $4\times 2\times 1$ (length $\times$ width $\times$ height). Numerical continuation techniques, coupled with an Arnoldi method, are used to locate the steady and Hopf bifurcation points as well as the different steady and periodic flow branches emerging from them for Prandtl numbers ranging from 0 to 0.025 (liquid metals). Our results highlight the existence of two steady states along with many periodic cycles, all with different symmetries. The bifurcation scenarios consist of complex paths between these different solutions, giving a succession of stable flow states as the Grashof number is increased, from steady to periodic and quasi-periodic. The change of these scenarios with the Prandtl number, in connection with the crossing of bifurcation points, was carefully analysed.

Heat transfer regimes in fully developed plane-channel flows

This paper investigates heat transfer regimes in fully developed plane channel flows without considering the buoyancy effect. Analyzing the governing thermal energy equation, aided by direct numerical simulation (DNS) data, six heat transfer regimes are identified including (i) laminar flow and laminar heat transfer (LamF-LamH), (ii) transitional flow and laminar-like heat transfer (TraF-LamH), (iii) transitional flow and transitional heat transfer (TraF-TraH), (iv) turbulent flow but laminar-like heat transfer (TurF-LamH), (v) turbulent flow and transitional heat transfer (TurF-TraH), and (vi) turbulent flow and turbulent heat transfer (TurF-TurH). One key result is the clarification of a TurF-LamH regime which exists only for low Prandtl number fluid (Pr≪1). A critical non-dimensional number is determined as ReτPr1/2≲50 where Reτ is the Reynolds number defined by the channel half-height δ and the frictional velocity uτ. In the TurF-LamH regime, the simplified thermal energy equation yields a prediction of Nusselt number as Nu≈6.0. The clarification of the TurF-LamH regime provides valuable insight into the understanding of the Nusselt number data for liquid metals, which have very low Prandtl number. Another major finding is that, in the TurF-TurH regime, the Kader-Yaglom-style equation is shown to be better than the traditional power law correlations at predicting Nusselt number, for either low or high Prandtl numbers. No separate correlations are needed for the prediction of Nusselt number at low Prandtl numbers and high Prandtl numbers. Another advantage of the Kader-Yaglom-style equation is that the equation can be theoretically connected to the log-law for the mean velocity and the mean temperature. More importantly, in the TurF-TurH regime the prediction of Kader-Yaglom-style equation can be reliably extended to ultra high Reynolds numbers, for either low or high Prandtl numbers.