Algebraic Heat Flux Model Research Articles

A modified algebraic turbulent heat flux model for non-equilibrium and rotating flow and its application in film cooling simulation of a rotating turbine blade

Thermal fields and cooling flows over rotating turbine blades are usually a non-equilibrium anisotropic flow, and most turbulence models in RANS simulations do not provide acceptable accuracy. A modified model is proposed and assessed here. This is based on an explicit algebraic heat flux model that is a low-Reynolds-number version of the higher order generalized gradient diffusion hypothesis turbulent heat flux model. The presented model, called “Non-equilibrium and Sensible to Rotation Algebraic Heat Flux Model” or “NSR-AHFM” includes additional production terms added to the second moment closure of turbulent heat flux transport equation that considers system rotation and non-equilibrium effects in velocity field, while maintaining the frame invariance during transformation to a rotating coordinate system. The model was initially tested through a-priori analysis, comparing the model to corresponding quantities obtained from Direct Numerical Simulations, allowing for the identification of suitable model parameters. Subsequently, the proposed modified model was implemented in a CFD code to a representative configuration (i.e., rotating turbine blade) and results were compared against available experimental data from literature. The present model was undergone a comparative analysis with other similar models in both stationary and rotating channel flow scenarios. Also, the present model was applied to a film cooling simulation of a rotating turbine blade in different rotational speeds. The findings indicate that the present model exhibits superior performance when compared to other models. Moreover, its behavior aligns consistently with the underlying physics governing the flow dynamics of the rotating channel.

Numerical heat transfer to LBE flow in a wire-wrapped 19-rod bundle using an OpenFOAM-based solver: fourParameterFoam

The LBE-cooled fast reactor (LFR) has received significant attention due to its diverse applications. However, numerical heat transfer simulations encounter challenges with lead-bismuth eutectic (LBE) due to its low Prandtl number and high molecular thermal diffusion, often resulting in an overestimation of heat transfer using the simple gradient diffusion hypothesis (SGDH). This study aims to improve the prediction accuracy of heat transfer to LBE in numerical models by developing a solver, named fourParameterFoam, based on the turbulent model SST k-ω and algebraic heat flux model (AHFM) kθ-ɛθ within the OpenFOAM framework. The performance of SGDH models and AHFMs is compared based on heat transfer experiments in vertical tube flow with LBE. The behavior of the turbulent Prandtl number (Prt) under various flow conditions is concluded based on the SST k-ω-kθ-ɛθ model. Additionally, the applicability of the SST k-ω-kθ-ɛθ model in complex flow is validated through heat transfer experiments in a wire-wrapped 19-rod bundle with LBE flow. Local temperature variations on the heated rod surface and sub-channels are analyzed to elucidate the thermal development process in the bundle. The impact of wire spacers on flow behavior is summarized using visualization techniques, and the suitability of corresponding empirical Nusselt number correlations in LBE flow is evaluated.

Numerical prediction of turbulent flow and heat transfer in buoyancy-affected liquid metal flows

The present paper investigates the capabilities of some selected Reynolds-Averaged Navier Stokes (RANS) based turbulence models in reproducing liquid metals thermal hydraulics Direct Numerical Simulation (DNS) data. For this purpose, forced and mixed convection conditions, both addressing buoyancy-aided and buoyancy-opposed flow, are considered. The paper mainly focuses on velocity and temperature fields estimation, providing a comparison between the RANS and DNS computations. The capabilities of the turbulence models are discussed with the aim to highlight which ones provide the best predictions.In particular, attention is paid to the approach adopted for the calculation of the turbulent heat flux contributions. Together with models assuming the commonly adopted Simple Gradient Diffusion Hypothesis (SGDH) approach and the Reynolds analogy, a model including the Algebraic Heat Flux Model (AHFM) approach is considered. While being a practical and robust approach to deal with turbulent heat fluxes, the SGDH approach shows intrinsic limitation in dealing with liquid metal thermal hydraulics, mainly because of their low-Prandtl number. The adoption of a more advanced AHFM method may instead relevantly improve the quality of the obtained predictions.The obtained results show that the selected model adopting the AHFM method provides definitively better predictions of the addressed phenomena with respect to the ones considering the SGDH approach. While some discrepancies are still observed for the velocity fields, the temperature fields are captured very well, suggesting a clear superiority of the AHFM model. The present paper thus provides further validation and supports the use of AHFM as a valuable tool to predict turbulent heat fluxes.

Validation and application of numerical modeling for in-vessel melt retention in corium pools

In a hypothetical severe accident scenario in light water reactors (LWR), after the loss of primary coolant, the melted core moves to the lower plenum of the reactor pressure vessel and accumulate there. The melted core cautiously releases the decay heat which forms a pool of melted core, called corium, and turbulent natural convection starts. The decay heat transferred by the corium to the vessel may result in vessel failure due to the focusing effect in the absence of an effective cooling system. A numerical analysis is carried out using STAR-CCM+ commercial software to investigate the ability of the existing turbulence models and Algebraic Heat Flux Model (AHFM) to simulate the natural convective heat transfer phenomena in the corium pools. Efforts have been made to predict the BALI experimental results which are designed to simulate the heat transfer in corium pools, in the framework of severe accident studies. For BALI experiments various test campaigns are run varying the internal Rayleigh number, the viscosity of the simulant fluid, and test facility height. Various such experimental cases are verified and compared with numerical simulations. Temperature profiles along a vertical line, surface heat flux along the curved surface, average Nusselt number on the top, and curved surfaces are estimated. The effect of wall boundary conditions on heat transfer is analyzed. The numerical analysis is extended to Pressurized Water Reactor (PWR) and Boiling Water Reactor (BWR) geometries. The CFD analysis predicts the stratified zone and upper mixing zone very well, and the temperature profiles and average heat transfer values are in agreement with the experiments.

Heat transfer characteristics of supercritical water in channels: A systematic literature review of 20 years of research

• More experiments are required to validate the existing the turbulence model. • Turbulence model with Algebraic heat flux model (AHFM) may be promising. • Heat transfer peak occurs as the fluid approaches the pseudo-critical temperature. • To achieve the high accuracy, piecewise correlation may be an effective method. • Internally ribbed tube combined with inclined angles may be a research direction. Supercritical water (SCW) is one of the high-profile fluids between scholars, which has great potential in the thermal industry. Therefore, it may serve as a promising fluid utilized in nuclear and boiler systems. This paper presents a comprehensive review in terms of the published literature on supercritical water heat transfer in channels. Three types of heat transfer, involving normal, enhanced and deteriorated heat transfer, are classified in this article. Firstly, the thermo-physical properties of supercritical water are illustrated to explain the reasons for the different heat transfer behavior. Heat transfer enhancement and deterioration appear to have greater potential and draw some researchers’ interest. In short, transformation of geometry or operating conditions, such as, channel structure, mass flux and heat flux, would induce variations of thermo-physical properties. Consequently, heat transfer performance can be significantly affected. The heat transfer characterization and affecting factors were evaluated. Some unusual phenomena were explained. Then, the developed heat transfer coefficients (HTC) were compared with the experimental data available in the open literature. The results indicated that the reliable accuracy could be achieved through these correlations except the pseudo-critical region. Additionally, the mechanism and the onset of heat transfer deterioration were revealed and estimated. Finally, this review also identifies the current challenges regarding this research through analysis and comparison. These challenges deserve to be efficiently solved and optimized to enhance supercritical water heat transfer in the future.

Computation of upward pipe flows at supercritical pressures with blended turbulence model

A flow field at supercritical pressure may undergo a strong mixed convection at a certain combination of mass and heat flux, and there are more than one flow (or heat transfer) mode in such a flow field. Therefore, it is unlikely to be able to simulate such a complicated flow field with a single turbulence model. It is known that the flow (or heat transfer) modes are effectively distinguished through the degree of buoyancy parameter. In order to computationally reproduce the flow including various flow modes, two turbulence models that well reproduce the flow with weak and strong buoyancy were blended with buoyancy as a parameter. The algebraic heat flux model (AHFM) was also adopted to model the turbulence production by buoyancy. The temperature variance used in the AHFM was calculated by solving the transport equation for temperature variance. The coefficient of temperature variance used in the AHFM was treated as a function of dimensionless enthalpy. The turbulent Prandtl number was treated as a function of flow variables and physical properties. The proposed computational model was successfully validated against the DNS and experimental data for upward flow in vertical tubes, in which the wall temperatures were satisfactorily reproduced.

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.

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.

An overview of the AHFM-NRG formulations for the accurate prediction of turbulent flow and heat transfer in low-Prandtl number flows

Turbulent heat transfer is a complex phenomenon, which is ubiquitous in engineering applications and has challenged turbulence modellers for several decades. In an attempt to simplify the problem it is often assumed that turbulent heat transfer can be inferred from the knowledge of the turbulent momentum transport, in what is known as the Reynolds analogy. This approach presents well-known drawbacks that limit its applicability to low-Prandtl fluids such as liquid metals. In an effort to overcome such limitations, an implicit Algebraic Heat Flux Model named AHFM-NRG has been recently proposed by the Nuclear Research and Consultancy Group (NRG). In the framework of the EU THINS project, this model was initially tested for a limited number of academic test cases in all three flow regimes (i.e.: natural, mixed and forced convection) and showed encouraging results. Further assessment and development of this or any other turbulent heat flux model with low-Prandtl fluids was hampered by the lack of accurate and reliable reference data. Thanks to the extensive reference database generated within the subsequent EU SESAME and MYRTE projects, the AHFM-NRG formulation has been further tested and developed. This article reports the development of the AHFM-NRG approach and its assessment against some representative test cases in all three flow regimes. It is shown that the AHFM-NRG formulations result in significant improvement with respect to the classical Reynolds analogy in all considered flow configurations.

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.

Prandtl number effect on thermal behavior in volumetrically heated pool in the high Rayleigh number region

The in-vessel retention by external reactor vessel cooling (IVR-ERVC) is a severe accident management strategy to hold or mitigate the molten radioactive material in the reactor pressure vessel (RPV). Highly turbulent natural convection is one of the most important phenomenon determining the thermal behavior of the molten pool. In the safety evaluation of IVR-ERVC, thermal behavior of the corium pool has been determined using experimental correlations. Property differences between the experimental simulant and the corium have been considered negligible. The effect of property differences represented by Prandtl number (Pr) on the thermal behavior of oxide pool is numerically investigated. The algebraic heat flux model (AFM) is used to simulate the turbulent natural convection and is validated with the BALI experiment. The Pr effect on the flow characteristics and the thermal behavior is investigated in the high Rayleigh number region. The present results show that the upward heat transfer tends to increase at the same Ra′ in the case of the molten pool which is lower Pr fluid than the simulant material.

Towards the accurate prediction of the turbulent flow and heat transfer in low-Prandtl fluids

In the numerical simulation of turbulent heat transfer, a reliable representation of the turbulent flow field is a prerequisite in order to obtain an accurate prediction of the thermal field. In the framework of RANS modelling approach, classical two-equation eddy-viscosity models present intrinsic limitations for applications characterised by complex flow fields where significant turbulence anisotropy can be expected. In this respect, differential Reynolds stress models represent the natural choice for dealing with such highly anisotropic flows. On the other hand, the modelling of the turbulent heat flux term almost universally relies on the so-called Reynolds analogy, due to its simplicity and robustness. Nevertheless, this approach presents several well-known limitations, especially when dealing with low-Prandtl fluids. In this context, algebraic heat flux models are regarded as a promising approach to overcome the shortcomings of the Reynolds analogy. In particular, an implicit algebraic heat flux closure, called as AHFM-NRG, has been proposed for applications involving low-Prandtl fluids. In its original formulation, the algebraic turbulent heat flux closure was coupled with a low-Reynolds linear k-ε model for the closure of the Reynolds stress tensor. In the present work, the applicability of the AHFM-NRG model is extended to an advanced low-Reynolds differential Reynolds stress model. This new version of the model is validated against different planar channel flows at unity and low-Prandtl number; successively, the model is applied to two relatively complex flows, i.e. a planar impinging jet and a bare rod bundle configuration. It is shown that the coupling of the AHFM-NRG with an advanced closure for the Reynolds stresses gives encouraging results for these cases, where an accurate prediction of the anisotropy in the flow field results in a noticeable improvement in the prediction of the turbulent heat transfer.

Four equation k-omega based turbulence model with algebraic flux for supercritical flows

Hydrocarbon and nuclear fuels will contribute substantially to the growing global energy portfolio for some time to come. At the same time atmospheric carbon levels and economic concerns are increasingly shaping the power generation environment. Supercritical fluid technologies are being investigated to address both concerns. For example, direct fired supercritical CO2 power cycles offer key advantages for carbon capture and thermal efficiency. Additionally, the supercritical water reactor Generation IV nuclear reactor allows higher power densities and more efficient operation than current pressurized or boiling water reactors. As these technologies are developed, a need exists for accurate, yet cost-effective modeling and simulation solutions to assist with design and safety calculations. Supercritical fluid properties are strongly dependent on temperature which results in a number of unique flow features. Among these features are the complex buoyancy and buoyancy-turbulence interaction behaviors exhibited by supercritical flows that are not found in liquid or gas flows. Therefore, standard turbulence models with current model coefficient values are not likely to be applicable to supercritical fluid flows.The subject of this paper is efficient modeling and simulation of supercritical flows to support nascent technologies growing to maturation and operational deployment. We present a framework for developing Reynolds-Averaged Navier-Stokes turbulence models specifically equipped for the challenges of supercritical flows. A novel formulation of the algebraic heat flux model of the buoyancy production of turbulence term is used with a traditional shear stress transport model. To produce a new turbulence model for supercritical channel flows, the empirical coefficients of the resulting four equation model were calculated from data previously published by other authors regarding upward supercritical flow through heated pipes. The presented SST kt-ωt approach was validated against heated tube experiments to show predictive capabilities in moderate flow conditions, where buoyancy effects are important but not dominating of inertial effects.

The influence of low Prandtl numbers on the turbulent mixed convection in an horizontal channel flow: DNS and assessment of RANS turbulence models

In this paper, the turbulent mixed convection in a channel flow with differentially heated walls is considered for a fixed Richardson number (Rib=0.5) and three different Prandtl numbers (Pr=1,0.1and0.01). Numerical simulations are conducted assuming constant fluid properties and the effect of the buoyancy is taken into account by means of the Boussinesq approximation. Direct Numerical Simulations (DNS) are performed first and the effect of the buoyancy on the first and second order statistics of the fluid and thermal fields is highlighted. Furthermore, it is found that in mixed convection the Prandtl number has a much larger effect on the results than in the case of forced convection. The obtained DNS results are then used as a validation database for two different RANS turbulent heat flux models, i.e. the classical Reynolds analogy and a recently proposed three-equation algebraic heat flux models called AHFM-NRG+. It is observed that, as expected, the Reynolds analogy fails to predict the thermal field even for unitary Prandtl numbers fluids. On the other hand, it is shown that the AHFM-NRG+ is in a reasonable agreement with the reference DNS over the entire range of Prandtl numbers considered in the study.

Application of a modified algebraic heat-flux model and second-moment-closure to high blowing-ratio film-cooling and corrugated heat-exchanger simulations

The present paper outlines the application of the recently proposed heat-flux model (Mazaheri et al., 2017) to high blowing-ratio film-cooling and corrugated heat-exchanger simulations. Here, the focus is mainly on the accuracy of the predicted thermal fields, while to find out the sources of inaccuracy detailed analysis of the adopted second-moment-closure hydrodynamic model is provided. To do so, fundamental benchmarks which contain the dominant phenomena in the main cases are thoroughly analyzed to identify the anomalies. Then, the main cases including leading-edge film-cooling, antivortex film-cooling and corrugated heat-exchanger are investigated. The numerical predictions indicate that the erroneous evaluation of the surface temperature near the leading-edge stagnation-line, caused by constant Prandtl assumption, is markedly improved through a more accurate prediction of heat-fluxes within the impinging region. In this regards, for a high blowing ratio case, i.e. BR=2, the error in prediction of average adiabatic cooling effectiveness (η‾) is reduced more than 30% and 75% near the first and second row of cooling holes, respectively. In the antivortex film-cooling, the lateral diffusion of the coolant jet, which is essentially depending on heat-transfer between kidney vortices and surface, is reasonably predicted which results in more than 90% error reduction in prediction of η‾ in vicinity of the holes with respect to constant Parndtl assumption case. Also, more than 50% improvement can be observed in predicting the average surface temperature in the corrugated heat-exchanger case, which contains numerous recirculation zones, with respect to constant Prandtl simulations. The results demonstrate that the anisotropic hydrodynamic model may offer potentially improved predictions, however, in order to achieve high level of accuracy, adopting a proper heat-flux model is essential for the high blowing-ratio film-cooling cases which contain highly non-equilibrium flow characteristics.

Improvements in the prediction of heat transfer to supercritical pressure fluids by the use of algebraic heat flux models

The paper discusses the latest improvements obtained in performing heat transfer calculations by RANS turbulence models when dealing with fluids at supercritical pressure.An algebraic heat flux model (AHFM) is adopted as an advanced tool for calculating the turbulent Prandtl number distribution to be used in the energy equation. Though maintaining a simple gradient approach, the proposed model manages to obtain interesting results when dealing with temperatures spanning from low to supercritical values.This is due to the introduction of a correlation for defining one of the relevant AHFM parameters. As stated in previous works, in fact, single fixed constant values could not be sometimes sufficient for dealing with very different operating conditions such as the ones occurring with supercritical fluids. In order to make the relation suitable for different fluids, a dependence on a non-dimensional quantity which proved to be relevant by parallel work is assumed.Some sensitivity analyses are also performed, showing some interesting capabilities in reproducing a sort of threshold behaviour when working in transition regions.Buoyancy induced phenomena are much better captured than in past attempts, though incomplete accuracy is observed for some boundary conditions.

Simulation of Rotating Channel Flow With Heat Transfer: Evaluation of Closure Models

A direct numerical simulation (DNS) of spanwise-rotating turbulent channel flow was conducted for four rotation numbers: Rob=0, 0.2, 0.5, and 0.9 at a Reynolds number of 8000 based on laminar centerline mean velocity and Prandtl number 0.71. The results obtained from these DNS simulations were utilized to evaluate several turbulence closure models for momentum and heat transfer transport in rotating turbulent channel flow. Four nonlinear eddy viscosity turbulence models were tested and among these, explicit algebraic Reynolds stress models (EARSM) obtained the Reynolds stress distributions in best agreement with DNS data for rotational flows. The modeled pressure–strain functions of EARSM were shown to have strong influence on the Reynolds stress distributions near the wall. Turbulent heat flux distributions obtained from two explicit algebraic heat flux models (EAHFM) consistently displayed increasing disagreement with DNS data with increasing rotation rate.

Prediction of heat transfer to supercritical fluids by the use of Algebraic Heat Flux Models

The paper discusses capabilities and limitations of Algebraic Heat Flux Models in predicting heat transfer to supercritical fluids. The model was implemented in a commercial code and used as a basis for obtaining an advanced definition of the turbulent Prandtl number and an improved estimate of the buoyancy production of turbulence kinetic energy. A comparison between the obtained results and experimental data available in literature is performed highlighting promising features, in particular when dealing with trans-pseudo-critical conditions. Experimental conditions using different fluids where analysed showing improvements with respect to two-equation turbulence models; a reference DNS calculation is considered as well for comparison. Calculated wall temperature values are in general well reproduced by the methodology and sensitivity analyses show that improvements may be obtained in future works by selecting case-specific AHFM parameters in association with different turbulence models.

Implicit Algebraic Heat Flux Model Linked up with Elliptic Blending Reynolds Stress Transport Equation Model

Implicit Algebraic Heat Flux Model Linked up with Elliptic Blending Reynolds Stress Transport Equation Model

Turbulent heat transport and its anisotropy in an impinging jet

The turbulent heat transport is anisotropic in many cases as reported by several researchers. RANS- based turbulence models use the turbulent viscosity when expressing the turbulent heat flux in the energy balance (analogy of the Reynolds stresses in the momentum balance). The turbulent (eddy) viscosity calculation comes from the Boussinesq analogy mainly and it represents just a scalar value, hence a possible anisotropy in the turbulent flow field cannot be simply transferred to the temperature field. The computational cost of a LES-based approach can be too prohibitive in complex cases, therefore simpler explicit algebraic heat flux models describing the turbulent heat flux in the time-averaged energy equation could be used to get more accurate CFD results. This paper compares several turbulence models for the case of a turbulent impinging jet and deals with a methodology of implementing a user-defined function describing the anisotropic turbulent heat flux in a CFD code.