Turbulent Prandtl Number Research Articles

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.

Scaling of Turbulent Viscosity and Resistivity: Extracting a Scale-dependent Turbulent Magnetic Prandtl Number

Abstract Turbulent viscosity ν t and resistivity η t are perhaps the simplest models for turbulent transport of angular momentum and magnetic fields, respectively. The associated turbulent magnetic Prandtl number Pr t ≡ ν t /η t has been well recognized to determine the final magnetic configuration of accretion disks. Here, we present an approach to determining these “effective transport” coefficients acting at different length scales using coarse-graining and recent results on decoupled kinetic and magnetic energy cascades. By analyzing the kinetic and magnetic energy cascades from a suite of high-resolution simulations, we show that our definitions of ν t , η t , and Pr t have power-law scalings in the “decoupled range.” We observe that Pr t ≈ 1–2 at the smallest inertial-inductive scales, increasing to ≈5 at the largest scales. However, based on physical considerations, our analysis suggests that Pr t has to become scale independent and of order unity in the decoupled range at sufficiently high Reynolds numbers (or grid resolution) and that the power-law scaling exponents of velocity and magnetic spectra become equal. In addition to implications for astrophysical systems, the scale-dependent turbulent transport coefficients offer a guide for large-eddy simulation modeling.

Data-driven algebraic models of the turbulent Prandtl number for buoyancy-affected flow near a vertical surface

The behaviour of the turbulent Prandtl number (Prt) for buoyancy-affected flows near a vertical surface is investigated as an extension study of Gibson & Leslie, Int. Comm. Heat Mass Transfer, Vol. 11, pp. 73–84 (1984). By analysing the location of mean velocity maxima in a differentially heated vertical planar channel, we identify an infinity anomaly for the eddy viscosity νt and the turbulent Prandtl number Prt, as both terms are divided by the mean velocity gradient according to the standard definition, in vertical buoyant flow. To predict the quantities of interest, e.g. the Nusselt number, a machine learning framework via symbolic regression is used with various cost functions, e.g. the mean velocity gradient, with the aid of the latest direct numerical simulation (DNS) dataset for vertical natural and mixed convection. The study has yielded two key outcomes: (i) the new machine learnt algebraic models, as the reciprocal of Prt, successfully handle the infinity issue for both vertical natural and mixed convection; and (ii) the proposed models with embedded coordinate frame invariance can be conveniently implemented in the Reynolds-averaged scalar equation and are proven to be robust and accurate in the current parameter space, where the Rayleigh number spans from 105 to 109 for vertical natural convection and the bulk Richardson number Rib is in the range of 0 and 0.1 for vertical mixed convection.

Numerical investigation for the heat transfer mechanisms between subchannels of bar rod bundles cooled by liquid sodium

The flow field in the fuel assembly of Sodium cooled Fast Reactor (SFR) is highly non-uniformed which results in quite complex heat transfer phenomena between subchannels of rod bundles.From the design and safety purpose of the reactor core, mechanisms behind these heat transfer phenomena must be analysed thoroughly under all working conditions. However, due to the decrease of the fast reactor activity and the huge experimental costs with liquid sodium, few researches have been conducted to deepen the understanding of the heat transfer mechanisms after the 1980’s in spite of the large uncertainty and limited parameter ranges of existing empirical correlations. In this paper, a Computational Fluid Dynamic (CFD) model of two subchannels with adjacent fuel rods is used to investigate the heat transfer mechanisms under a wide range of geometry and flow conditions. The turbulent heat transfer of the liquid metal is resolved through the SST k-ω turbulent model with modified turbulent Prandtl number, and the CFD model is verified with experimental data. Data analysis of the simulation results shows clearly dependence of the heat transfer mechanisms on the geometry parameters and the flow parameters. By fitting of the numerical results with the least square method, new correlations for the turbulent mixing and the rod conduction are developed.

Direct numerical simulation of thermal channel flow for medium–high Prandtl numbers up to [formula omitted

A new set of DNS of a thermal channel flow have been conducted for friction Reynolds and Prandtl numbers up to 2000 and 10, respectively, reaching the Prandtl number of water, 7, for new Reynolds numbers, never simulated before. The Mixed Boundary Condition has been used as the thermal boundary condition. A new scaling of the thickness of the conductive sublayer is presented for medium–high Prandtl number values. The maximum of the intensity of the thermal field does not increase with the Reynolds number for the highest Prandtl numbers. This entails a good scaling near the wall of the viscous diffusion and the dissipation terms in the budgets of the temperature variance. The Nusselt number shows a power function behaviour with respect to the Prandtl number in a certain range of the friction Péclet number. Finally, the turbulent Prandtl number presents an increase near the wall for highest Prandtl numbers due to the reduction of the thermal eddy diffusivity. The statistics of all simulations can be downloaded from the web page of our group: http://personales.upv.es/serhocal/.

Effects of Buoyancy Flux on Upper‐Ocean Turbulent Mixing Generated by Non‐Breaking Surface Waves Observed in the South China Sea

AbstractThe surface waves are the most energetic motions, which have great contribution to the turbulent mixing in the upper ocean. In this study, a novel turbulent mixing scheme is proposed in terms of the non‐breaking wave velocity shear module with the buoyancy flux. In the scheme, the mixing coefficients can be calculated empirically from the significant wave height, the wave number, the wave frequency, the buoyancy frequency, and the turbulence Prandtl number. The buoyancy fluxes, as well as the non‐breaking wave velocity shear production, are important for the upper‐ocean turbulent mixing, and make the calculated turbulence dissipation rate closer to in situ observations, which are provided by the “Responses of Marine Hazards to Climate Change in the Western Pacific (ROSE)” Project. The effects of the buoyancy flux on the turbulent mixing are examined based on three numerical experiments using the MASNUM ocean model. In the experiments, the topography, the lateral boundaries, initialization conditions, and the surface forcing fluxes are taken from the GEBCO, HYCOM/NCODA, and ERA5 data, respectively. Comparing to the AVHRR remote sensing sea surface temperature (SST) products and in situ observations, the simulated results with the non‐breaking wave‐generated turbulent mixing gain significant improvement in the SST, upper‐ocean temperature structure, and the mixed layer compared with the classic Mellor‐Yamada scheme. The results show that the buoyancy flux is able to suppress the enhanced non‐breaking wave‐generated turbulent mixing, so that the improved model simulates the observations better than that without the buoyancy effects.

난류 유동에서 다변수 인자에 대한 기계 학습 기법 비교 연구

It is important to predict spatially varying parameters to model turbulent flows. In this study, the spatially varying parameters are modeled via machine learning techniques using experiment-based turbulent bubble flow data and DNS-based turbulent Prandtl number data. The prediction and generalization errors of machine learning models are evaluated, and the different techniques are compared. Among the artificial neural network (ANN) techniques, the regular ANN using the full-batch training and the stochastic gradient descent (SGD) ANN based on mini-batch training are compared with the random forest (RF) method. The prediction and generalization errors show different characteristics according to the data resolution. For the coarsest bubbly flow data set, SGD ANN shows stable training and prediction, which leads to the smallest prediction and generalization errors. For the data sets with a finer resolution, the generalization error of SGD ANN is smallest, whereas the average prediction errors of regular ANN and RF method are smaller compared to SGD ANN. When evaluated using the trained models, all machine learning techniques show similar spatial distribution to the original data.

Large-eddy simulation of turbulent channel flow at transcritical states

We present well-resolved large-eddy simulations (LES) of a channel flow solving the fully compressible Navier–Stokes equations in conservative form. An adaptive look-up table method is used for thermodynamic and transport properties. A physically consistent subgrid-scale turbulence model is incorporated, that is based on the Adaptive Local Deconvolution Method (ALDM) for implicit LES. The wall temperatures are set to enclose the pseudo-boiling temperature at a supercritical pressure, leading to strong property variations within the channel geometry. The hot wall at the top and the cold wall at the bottom produce asymmetric mean velocity and temperature profiles which result in different momentum and thermal boundary layer thicknesses. Different turbulent Prandtl number formulations and their components are discussed in context of strong property variations.

Analytical study of heat transfer for liquid metals in a turbulent tube flow

Research on turbulent heat transfer of liquid metals has been lasting for decades since 1940s. Although it had developed various heat transfer relations of liquid metals, most of them are based on the structure of semi-empirical correlation proposed by Lyon in 1949 and fitted according to their respective experimental data. Due to the differences between the experimental data, there are inevitably obvious differences between the empirical correlations, especially in the range of high Peclet numbers. Therefore, reliable relation for turbulent heat transfer of liquid metals is still missing. Focused on this problem, a theoretical analysis of heat transfer for liquid metals in a fully developed turbulent tube flow has been performed following the method of Lyon, but use the velocity and vortex flow diffusivity profile obtained by theoretical analysis. Based on a reasonably determined turbulent Prandtl number, a theoretical heat transfer relation for liquid metals is obtained and validated with experimental data of various liquid metals. The most difference between the theoretical relation and existing relations is the “thermal conductive term” of theoretical relation has a relationship with the Peclet number, while for most of the existing relations this term is constant. Further, it reveals the turbulent Prandtl number has significant impacts on turbulent heat transfer of liquid metals while the molecular Prandtl number has little impact. Therefore, the turbulent heat transfer of Heavy Liquid Metals (HLM) is different from other liquid metals because their turbulent Prandtl number is found to be larger than other liquid metals by existing research.

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.

Application of higher-order heat flux model for predicting turbulent methane-air combustion

The present study addresses a new effort to improve the prediction of turbulent heat transfer and NO emission in non-premixed methane-air combustion. In this regard, a symmetric combustion chamber in a stoichiometric condition is numerically simulated using the Reynolds averaged Navier-Stokes equations. The Realizable k-? model and discreate ordinate are applied for modelling turbulence and radiation, respectively. Also, the eddy dissipation model is adopted for predicting the turbulent chemical reaction rate. Zeldovich mechanism is applied for estimating the NO emission. Higher-order generalized gradient diffusion hypothesis is employed for predicting the turbulent heat flux in turbulent reactive flows. Results show that the higher-order generalized gradient diffusion hypothesis model is capable of predicting temperature distribution in good agreement with the available experimental data. Comparison of the results obtained by the simple eddy diffusivity and HOGGDH models shows that applying the higher-order generalized gradient diffusion hypothesis significantly improves the over-prediction of NO emission. Finally, the average turbulent Prandtl number for the non-premixed methane-air combustion has been calculated.

Shock-induced heating and transition to turbulence in a hypersonic boundary layer

Abstract

Hot-wire experimental investigation on turbulent Prandtl number in a rotating non-isothermal turbulent boundary layer

This experiment used a parallel array of hot wire probes to simultaneously measure the temperature and velocity fields in the non-isothermal turbulent boundary layer of a rotating straight channel. The Reynolds numbers are 15,000 and 25,000, respectively. The rotation numbers are 0, 0.07, 0.14, 0.21 and 0.28, respectively. The purpose of this study is to calculate the turbulent Prandtl number in a rotating non-isothermal turbulent boundary layer. Due to the difficulty in measuring local turbulent Prandtl numbers, this study focuses on the average turbulent Prandtl numbers in the logarithmic region instead. Under static conditions, this value is taken as 0.9 normally. This research finds that rotation conditions can affect the turbulent Prandtl number by affecting the properties of velocity and temperature boundary layers. The change range of the turbulent Prandtl number is roughly 0.6–1.1. The influence of the leading side is greater than that of the trailing side, especially at high rotation numbers. This can provide validation and guidance for numerical simulation. Other information within the turbulent boundary layer is also discussed. It is hoped that this study would enhance our understanding of the mechanism of turbulent flow in the turbulent layer at rotating conditions.

Numerical Study of the Influence of Turbulent Diffusion Coefficients and Turbulent Prandtl Number on the Reactive Flow Simulation in a Combustor

The influence of the k–e–RNG model parameters is studied in terms of simulation of processes in the combustor with the gas oxygen and methane inside including the conjugate heat transfer with the wall. A sufficient influence of the empirical diffusion coefficients of turbulence model on the reactive flow simulation is observed. Recommendations for the choice of the turbulent diffusion coefficients and the turbulent Prandtl number are provided.

Large-Aspect-Ratio Structures in Simulated Ocean Surface Boundary Layer Turbulence under a Hurricane

AbstractA large-eddy simulation (LES) initialized and forced using observations is used to conduct a process study of ocean surface boundary layer (OSBL) turbulence in a 2-km box of ocean nominally under Hurricane Irene (2011) in 35 m of water on the New Jersey shelf. The LES captures the observed deepening, cooling, and persistent stratification of the OSBL as the storm approaches and passes. As the storm approaches, surface-intensified Ekman-layer rolls, with horizontal wavelengths of about 200 m and horizontal-to-vertical aspect and velocity magnitude ratios of about 20, dominate the kinetic energy and increase the turbulent Prandtl number from about 1 to 1.5 due partially to their restratifying vertical buoyancy flux. However, as the storm passes, these rolls are washed away in a few hours due to the rapid rotation of the wind. In the bulk OSBL, the gradient Richardson number of the mean profiles remains just above (just below) 1/4 as the storm approaches (passes). At the base of the OSBL, large-aspect-ratio Kelvin–Helmholtz billows, with Prandtl number below 1, intermittently dominate the kinetic energy. Overall, large-aspect-ratio covariance modifies the net vertical fluxes of buoyancy and momentum by about 10%, but these fluxes and the analogous diffusivity and viscosity still approximately collapse to time-independent dimensionless profiles, despite rapid changes in the forcing and the large structures. That is, the evolutions of the mean temperature and momentum profiles, which are driven by the net vertical flux convergences, mainly reflect the evolution of the wind and the initial ocean temperature profile.

The O’KEYPS Equation and 60 Years Beyond

Some 60 years ago, six researchers obtained a semi-empirical equation that describes how the stability correction function for the mean velocity profile ( $$\phi _\mathrm{m}$$ ) in the atmospheric surface layer varies with the stability parameter—the famous O’KEYPS equation. Their derivations are essentially based on interpolation of the turbulent eddy viscosity between neutral and convective conditions. Comparing the O’KEYPS equation with new theoretical developments—such as phenomenological and cospectral budget models—suggests that Heisenberg’s eddy viscosity provides a unifying framework for interpreting the behaviour of $$\phi _\mathrm{m}$$ . The empirical coefficient in the O’KEYPS equation, which is on the order of 10 based on data fitting to observations, is found to be primarily linked to the increase of the size of turbulent eddies as instability increases. The ratio of the sizes of turbulent eddies under convective and neutral conditions is on the order of $$1{/}\kappa $$ , where $$\kappa $$ is the von Karman constant, and is modulated by the turbulent Prandtl number.

Interpreting neural network models of residual scalar flux

Abstract

A Simple Description of the Turbulent Transport in a Stratified Shear Flow as Applied to the Description of Thermohydrodynamics of Inland Water Bodies

A way to parameterize the turbulent Prandtl number is proposed based on the model of turbulent transport in a stratified fluid, which allows for the two-sided transformation of the kinetic and potential energies of turbulent fluctuations. Numerical experiments aimed at studying the influence of the proposed parameterization on characteristic features of thermohydrodynamic processes in inland water bodies have been carried out.

Numerical simulation on temperature in wood crib fires

Temperature from burning wood cribs will be simulated in this paper by subgrid scale model in fire dynamics simulator. A baseline gas phase uncertainty is determined for simulating wood crib fire spread scenarios. This uncertainty is based on a sensitivity analysis of key input parameters and their subsequent effect on key output variables that are important for fire spread. Effects of different grid systems, computing domains and moisture contents on the predictions were studied first and then used to study the gaseous phase sensitivity. The gaseous phase input variables considered are: Smagorinsky constant, Prandtl number, and Schmidt number. The results show that the predictions for temperature have good agreement with experiment with the values of 0.25, 0.7, 0.4, and 5 for Smagorinsky constant, turbulent Schmidt number, and turbulent Prandtl number, respectively.

Simulation of the GOx/GCH4 Multi-Element Combustor Including the Effects of Radiation and Algebraic Variable Turbulent Prandtl Approaches

Multi-element thrusters operating with gaseous oxygen (GOX) and methane (GCH4) have been numerically studied and the results were compared to test data from the Technical University of Munich (TUM). A 3D Reynolds Averaged Navier–Stokes Equations (RANS) approach using a 60° sector as a simulation domain was used for the studies. The primary goals were to examine the effect of the turbulent Prandtl number approximations including local algebraic approaches and to study the influence of radiative heat transfer (RHT). Additionally, the dependence of the results on turbulence modeling was studied. Finally, an adiabatic flamelet approach was compared to an Eddy-Dissipation approach by applying an enhanced global reaction scheme. The normalized and absolute pressures, the integral and segment averaged heat flux were taken as an experimental reference. The results of the different modeling approaches were discussed, and the best performing models were chosen. It was found that compared to other discussed approaches, the BaseLine Explicit Algebraic Reynolds Stress Model (BSL EARSM) provided more physical behavior in terms of mixing, and the adiabatic flamelet was more relevant for combustion. The effect of thermal radiation on the wall heat flux (WHF) was high and was strongly affected by spectral models and wall thermal emissivity. The obtained results showed good agreement with the experimental data, having a small underestimation for pressures of around 2.9% and a good representation of the integral wall heat flux.