Turbulent Thermal Diffusivity Research Articles

Verification and validation of electrostatic turbulence simulation for numerical Lie transform code

Abstract This paper reports the verification and validation of a new gyrokinetic (GK) code, numerical Lie transform (NLT), for electrostatic turbulence simulation in tokamak plasmas. Based on the standard cyclone base case (CBC), NLT has been verified against two typical GK codes, GK toroidal code (GTC) and GK numerical experiment of tokamak, by simulating the ion temperature gradient mode and trapped electron mode. The linear dispersion relation and mode structure agree well among all these codes. In nonlinear simulation, the time evolution of ion heat diffusivity obtained from NLT is consistent with the GTC results. Furthermore, utilizing experimental equilibria with obvious turbulence features from HL-2A shot #22388, the validation of NLT for real tokamak simulation has been carried out. The profiles of ion turbulent thermal diffusivity calculated by NLT match well with experimental results.

A new equilibrium temperature inflow profile for modelling the non-isothermal atmospheric boundary layer in CFD simulation

The atmospheric boundary layer (ABL) flow in urban morphologies, ranging from micro-scale to local-scale, will become non-isothermal due to the combined effects of solar radiation, wall heat transfer, the urban heat island, and anthropogenic heat, resulting in a mutually coupled urban thermal and wind environment. An appropriate temperature inflow profile is the key prerequisite for Computational Fluid Dynamics (CFD) simulations of the non-isothermal ABL flow and the urban thermal and wind environment, as it must coincide with actual thermal characteristics while being compatible with CFD. Based on the equilibrium ABL model, a new equilibrium temperature inflow profile compatible with the energy equation, the standard gradient diffusion hypothesis, and the turbulence model is derived, which is proportional to the turbulent kinetic energy inflow profile, with the ratio further determined by the wall heat flux, demonstrating that turbulent thermal diffusion is strongly related to turbulent characteristics and atmospheric stability. To address the scaled effect, its full expression is provided based on thermodynamic similarity criteria. Using a numerical model of empty domain based on the SST k-ω model, the horizontal homogeneity of the equilibrium thermal and wind field modeled by the new set of inflow profiles is verified. Taking the flow around a single square building model as an example, its applicability in simulating the urban thermal and wind environment around buildings has been numerically validated, where the comparison of numerical and test results reveals tolerable errors overall.

Experimental study of turbulent thermal diffusion of particles in an inhomogeneous forced convective turbulence

We investigate experimentally the phenomenon of turbulent thermal diffusion of micrometer-size solid particles in an inhomogeneous convective turbulence forced by one vertically oriented oscillating grid in an air flow. This effect causes the formation of large-scale inhomogeneities in particle spatial distributions in a temperature-stratified turbulence. We perform detailed comparisons of the experimental results with those obtained in our previous experiments with an inhomogeneous and anisotropic stably stratified turbulence produced by a one oscillating grid in the air flow. Since the buoyancy increases the turbulent kinetic energy for convective turbulence and decreases it for stably stratified turbulence, the measured turbulent velocities for convective turbulence are larger than those for stably stratified turbulence. This tendency is also seen in the measured vertical integral turbulent length scales. Measurements of temperature and particle number density spatial distributions show that particles are accumulated in the vicinity of the minimum of the mean temperature due to the phenomenon of turbulent thermal diffusion. This effect is observed in both convective and stably stratified turbulence, where we find the effective turbulent thermal diffusion coefficient for micrometer-size particles. The obtained experimental results are in agreement with theoretical predictions.

Interactions between two adjacent convection rolls in turbulent Rayleigh-Bénard convection

Neighboring convection rolls are experimentally found to be stable in either counter-rotating or co-rotating states, with stochastic switching between the two states. These states are quantitatively described by analytic solutions of a low-dimensional stochastic model in which turbulent thermal diffusion between the rolls provides the forcing that stabilizes both states, using the same turbulent thermal diffusivity parameter that characterizes heat transport. This is presented within a generalized model framework that can account for the geometry of the convection cells and other forces.

RANS thermal modelling of a natural convection boundary layer at low Prandtl number

The present work evaluates the performance of different RANS turbulence models in a natural convection boundary layer at three different Prandtl numbers: 0.71, 0.2 and 0.025. An assessment on the prediction of the boundary layer is presented in function of the wall fluxes, mean and turbulent flow fields. It is observed that the chosen turbulence model to close the momentum equation is relevant in order to capture the flow field at low Prandtl numbers. This is due to the strong interaction between the momentum and thermal fields in natural convection. From the thermal point of view, the analogy between the turbulent momentum and turbulent thermal diffusivities presents serious issues to correctly predict the flow field independently of the Prandtl number. Advanced models that employ algebraic or transport equations of the turbulent heat fluxes introduces the needed physics to correctly predict the natural convection boundary layer.

ITB formation in gyrokinetic flux-driven ITG/TEM turbulence

The formation mechanism of internal transport barriers (ITBs) in flux-driven turbulence is studied by means of the full-f gyrokinetic code GKNET. In the adiabatic electron case with a weak magnetic shear configuration, toroidal momentum injection can change the radial mean electric field through radial force balance, leading to a kind of driven ITB formation in which the ion thermal diffusivity by ion temperature gradient (ITG) turbulence decreases to the neoclassical transport level. Only cocurrent toroidal rotation in the outer core region can benefit the ITB formation, and this mechanism is identified to originate from a positive feedback loop between the radial shear and resultant momentum flux. On the other hand, in the kinetic electron case with a reversed magnetic shear configuration, robust co-intrinsic rotation is driven near the surface in ITG turbulence and sustains the shear through the radial force balance, leading to the spontaneous reduction of ion turbulent thermal diffusivity, while this is not observed in the adiabatic electron case. In the presence of electron heating, counter-intrinsic rotation by trapped electron mode turbulence is selectively driven in the negative magnetic shear region, which provides steeper shear formation and a resultant larger reduction of ion turbulent thermal diffusivity. This indicates that the co-existence of different modes can trigger the ‘discontinuity’ of mode structure, intrinsic rotation, and resultant mean near , leading to spontaneous ITB formation.

Turbulent Prandtl number from isotropically forced turbulence

Turbulent motions enhance the diffusion of large-scale flows and temperature gradients. Such diffusion is often parameterized by coefficients of turbulent viscosity ( $\nu _{t}$ ) and turbulent thermal diffusivity ( $\chi _{t}$ ) that are analogous to their microscopic counterparts. We compute the turbulent diffusion coefficients by imposing sinusoidal large-scale velocity and temperature gradients on a turbulent flow and measuring the response of the system. We also confirm our results using experiments where the imposed gradients are allowed to decay. To achieve this, we use weakly compressible three-dimensional hydrodynamic simulations of isotropically forced homogeneous turbulence. We find that the turbulent viscosity and thermal diffusion, as well as their ratio the turbulent Prandtl number, $\textit {Pr}_{t} = \nu _{t}/\chi _{t}$ , approach asymptotic values at sufficiently high Reynolds and Péclet numbers. We also do not find a significant dependence of $\textit {Pr}_{t}$ on the microscopic Prandtl number $\textit {Pr} = \nu /\chi$ . These findings are in stark contrast to results from the $k{-}\epsilon$ model, which suggests that $\textit {Pr}_{t}$ increases monotonically with decreasing $\textit {Pr}$ . The current results are relevant for the ongoing debate on, for example, the nature of the turbulent flows in the very-low- $\textit {Pr}$ regimes of stellar convection zones.

Physics-informed machine learning based RANS turbulence modeling convection heat transfer of supercritical pressure fluid

Solving flow problems based on the Reynolds-averaged Navier-Stokes equations is a dominant method in terms of efficiency and accuracy for design and analysis of engineering applications, while the performance of RANS models in predicting the turbulent heat transfer of supercritical pressure fluid could be severely poor. To a great extent, the models are plagued by imperfect closures on higher order turbulence quantities with strongly varying thermal physical properties. This paper reports on an alternative approach to model the turbulent heat transfer at supercritical pressure, including the direct turbulent production models and the indirect transport closures using deep neural networks (DNN). This paper presents a method of modifying turbulence models of supercritical pressure fluid from high-fidelity simulation (DNS) data. An iterative DNS-DNN-RANS framework is proposed to establish explicit closures for turbulent momentum diffusion and turbulent thermal diffusion of turbulent heat transfer at supercritical pressure. Priori physics knowledge, feature engineering strategy and establish existing constitutive theory are involved to embed the special characteristics of supercritical pressure fluid into the turbulent closures to establish a proper regression system for machine learning (ML) algorithm. By embedding the explicit ML models, the low Reynolds number k-ε model was modified and was trained under a cross-case strategy with abundant DNS data. The modified model was successfully validated against DNS and experimental data for upward pipe flows, in which wall temperature were satisfactorily reproduced. The posterior simulations showed that the modified ML-KTVT model not only has a good performance in predicting the convection heat transfer of supercritical pressure fluid but also performs a favorable reproducibility of turbulence development in heat transfer deterioration cases.

Experimental study of turbulent thermal diffusion of particles in inhomogeneous and anisotropic turbulence

We experimentally studied the turbulent thermal diffusion of small particles in inhomogeneous and anisotropic stably stratified turbulence produced by one oscillating grid in the air flow. The velocity fields have been measured using Particle Image Velocimetry (PIV). We have determined various turbulence characteristics: the mean and turbulent velocities, two-point correlation functions of the velocity field, and an integral scale of turbulence from the measured velocity fields. The temperature field has been measured with a temperature probe equipped with 12 E thermocouples. Spatial distributions of micrometer-sized particles have been determined by a PIV system using the effect of the Mie light scattering by particles in the flow. The experiments have demonstrated that particles are accumulated at the minimum of mean fluid temperature due to the phenomenon of turbulent thermal diffusion. Using measured spatial distributions of particles and temperature fields, we have determined the effective turbulent thermal diffusion coefficient of particles in inhomogeneous temperature-stratified turbulence. This experimental study has clearly detected the phenomenon of turbulent thermal diffusion in inhomogeneous turbulence.

Experimental study of turbulent transport of nanoparticles in convective turbulence

We perform an experimental study of transport of nanoparticles in convective turbulence with the Rayleigh number ∼108 in the air flow. We measure a temperature field in many locations by a temperature probe equipped with 11 E-thermocouples. Nanoparticles of the size ∼70 nm in diameter are produced by an advanced electrospray aerosol generator. To determine the number density of nanoparticles, we use a condensation particle counter. The joint action of turbulent effects and the large-scale circulations in convective turbulence, which are important in the core flow, and molecular effects, which are essential near the boundaries of the chamber, results in an effective accumulation of nanoparticles at the cold wall of the chamber. The turbulent effects are characterized by turbulent diffusion and turbulent thermal diffusion of nanoparticles, while the molecular effects are described by the Brownian diffusion and thermophoresis, as well as the adhesion of nanoparticles at the cold wall of the chamber. In different experiments in convective turbulence in a chamber with the temperature difference ΔT between the bottom and top walls varying between ΔT=29 and ΔT=61 K, we find that the mean number density of nanoparticles decreases exponentially in time. For instance, the characteristic decay time of the mean number density of nanoparticles varies from 12.8 min for ΔT=61 K to 24 min for ΔT=29 K. For better understanding of experimental results, we perform one-dimensional mean-field numerical simulations of the evolution of the mean number density of nanoparticles for conditions pertinent to the laboratory experiments. The obtained numerical results are in agreement with the experimental results.

Inversion learning of turbulent thermal diffusion for film cooling

Film cooling is a typical three-dimensional fluid phenomenon, where the coolant with lower temperature is ejected from discrete holes to protect metal walls from being burnt by the hot mainstream. It is a great challenge for Reynolds-averaged Navier–Stokes (RANS) methods to accurately predict the coolant coverage on the wall because the turbulent thermal diffusion tends to be under-predicted due to inherent assumptions behind RANS models. In this paper, a framework of integrated field inversion and machine learning is built to enhance RANS prediction of turbulent thermal diffusion. A neural network (NN) is trained in this framework to predict the spatially varying turbulent Prandtl number (Prt) and to improve the prediction of RANS models. The temperature distribution obtained from the large eddy simulation is used as the learning target, and the discrete adjoint method is used as the inverse model that helps calculate derivatives of mean square error of the temperature distribution to NN parameters. The training process of NN shows good convergence properties. The results show that the obtained NN effectively increases the insufficient turbulent thermal diffusion by predicting much lower Prt than the commonly used value of 0.9. The NN-enhanced RANS provides significant improvements on predicting experimental temperature distributions compared with general RANS models not only on the training data but also on the unseen testing data. In addition, the obtained NN can be implemented into general-purpose software with minimal effort and no numerical stability problem.

Development and Assessment of an Isotropic Four-Equation Model for Heat Transfer of Low Prandtl Number Fluids

In the simple gradient diffusion hypothesis, the turbulent Prandtl number (Prt) with a constant of 0.85 is difficult to accurately predict for liquid metals having low Prandtl numbers (Pr), while a four-equation model can improve this solution by introducing the turbulence time-scale into the calculation of turbulent thermal diffusivity. However, the four-equation model’s transport form and numerical stability are so complex that suitable commercial code is lacking. Therefore, an isotropic four-equation model with simple Dirichlet wall boundary conditions is built in the present work. Based on the open-source computational fluid dynamics program OpenFOAM, the fully developed velocity, temperature, Reynolds stress, and heat flux of low Pr fluids (Pr = 0.01–0.05) in the parallel plane are obtained by numerical simulation. The results show that the time-average statistics predicted using the present four-equation model are in good agreement with the direct numerical simulation data. Then, the isotropic four-equation model is used to analyze the flow and heat of liquid metal (Pr = 0.01) in a quadrilateral infinite rod bundle. The numerical results are compared with the various and available experimental relationships. The Nusselt numbers calculated using the isotropic four-equation model are betweenness the available correlations, while the turbulent Prandtl number model using a constant of 0.85 over predicts heat transfer. More detailed local heat transfer phenomena and distribution of low Pr fluids are obtained using the present isotropic four-equation model.

Numerical study of supercritical octane flows with multicomponent effects by pyrolysis

This study focuses on supercritical octane flow simulation with pyrolysis reactions in a heated circular pipe to determine the multicomponent effects of changes in thermophysical properties. The reaction rate of n-octane for pyrolysis was estimated by a zeroth-dimensional simulation of hydrocarbon pyrolysis. The resulting mole fractions of the decomposed components were compared with the experimental data. The mole fraction results were then used to develop three thermophysical property models for the mixture of unreacted n-octane and decomposed hydrocarbons, i.e., 1-, 4-, and 12-component models. In the supercritical octane flow simulations, the multicomponent thermophysical properties of each component model were determined using the polynomial equations in the Reference Fluid Thermodynamic and Transport Properties Database. The conventional k−ω SST model and k−ωSST+Mτ model served as the Reynolds-averaged Navier–Stokes turbulence models. In cases with a low wall temperature, the outlet temperatures in all component models were consistent with the experimental data, in which pyrolysis did not occur. In cases with a high wall temperature above 973 K, the density change and production of kinetic energy affected the turbulent thermal diffusivity in the heated pipe. In cases where the multicomponent effects of thermophysical properties and turbulence were considered, the numerical method reproduced a reasonable outlet temperature and conversion rate, whereas the experimental results were underestimated in all the other cases. These numerical results indicate that the accurate prediction of density changes and the production of turbulent kinetic energy are the key issues in reproducing thermal fluid flows of supercritical hydrocarbons with pyrolysis reactions in uniform wall temperature cases.

A POD-Galerkin reduced order model of a turbulent convective buoyant flow of sodium over a backward-facing step

A Finite-Volume based POD-Galerkin reduced order modeling strategy for steady-state Reynolds averaged Navier–Stokes (RANS) simulation is extended for low-Prandtl number flow. The reduced order model is based on a full order model for which the effects of buoyancy on the flow and heat transfer are characterized by varying the Richardson number. The Reynolds stresses are computed with a linear eddy viscosity model. A single gradient diffusion hypothesis, together with a local correlation for the evaluation of the turbulent Prandtl number, is used to model the turbulent heat fluxes. The contribution of the eddy viscosity and turbulent thermal diffusivity fields are considered in the reduced order model with an interpolation based data-driven method. The reduced order model is tested for buoyancy-aided turbulent liquid sodium flow over a vertical backward-facing step with a uniform heat flux applied on the wall downstream of the step. The wall heat flux is incorporated with a Neumann boundary condition in both the full order model and the reduced order model. The velocity and temperature profiles predicted with the reduced order model for the same and new Richardson numbers inside the range of parameter values are in good agreement with the RANS simulations. Also, the local Stanton number and skin friction distribution at the heated wall are qualitatively well captured. Finally, the reduced order simulations, performed on a single core, are about 105 times faster than the RANS simulations that are performed on eight cores.

Discussion on improved method of turbulence model for supercritical water flow and heat transfer

The turbulence model fails in supercritical fluid-flow and heat transfer simulation, owing to the drastic change of thermal properties. The inappropriate buoyancy effect model and the improper turbulent Prandtl number model are several of these factors lead to the original low-Reynolds number turbulence model unable to predict the wall temperature for vertically heated tubes under the deteriorate heat transfer conditions. This paper proposed a simplified improved method to modify the turbulence model, using the generalized gradient diffusion hypothesis approximation model for the production term of the turbulent kinetic energy due to the buoyancy effect, using a turbulence Prandtl number model for the turbulent thermal diffusivity instead of the constant number. A better agreement was accomplished by the improved turbulence model compared with the experimental data. The main reason for the over-predicted wall temperature by the original turbulence model is the misuse of the buoyancy effect model. In the improved model, the production term of the turbulent kinetic energy is much higher than the results calculated by the original turbulence model, especially in the boundary-layer. A more accurate model for the production term of the turbulent kinetic energy is the main direction of further modification for the low Reynolds number turbulence model.

Interaction of magnetic islands with turbulent electron temperature fluctuations in DIII-D and in GENE nonlinear gyrokinetic simulations

We present localized measurements of interactions between neoclassical tearing modes (NTMs) and turbulent electron temperature fluctuations () in the expected kθρs = 0–0.5 wave-number range of the ion temperature gradient (ITG) instability measured via correlation electron cyclotron emission (kθ and ρs are the poloidal wave-number and ion-sound Larmor radius, respectively). Comparison to GENE gyrokinetic simulations, shows qualitative agreement with being reduced (increased) inside (outside) of the islands due to modified local gradients, and being higher (lower) in the region outside the island where lower (higher) flow shear is expected due to radially asymmetric island shape. These results are consistent with previously reported modifications of long and intermediate wavelength turbulent density fluctuations and cross-field electron thermal diffusivity reduction at the O-point of magnetic islands. Interestingly, a 30% increase of is detected at the NTM rational surface when the island width is a few times ρs, which is replicated by GENE through zonal flow damping due to the destroyed magnetic field topology of the NTM rational surface.

On the use of LES-based turbulent thermal-stress models for rod bundle simulations

An alternative methodology is proposed here to overcome the excessive cost of large eddy simulations (LES) of full-length heated rod bundle calculations, while improving the inaccurate results typically obtained with Reynolds-averaged Navier–Stokes equations (RANS). While the cost of the full-length LES is generally too high, LES of a small section of a single rod is usually affordable. The idea presented here consists of using the information granted by the small LES calculation to determine the appropriate turbulence viscosity or turbulent thermal diffusivity that can be used to solve only for the temperature field in a pseudo-RANS approach. The study has been performed with single-rod simulations with a P/D of 1.12 and 1.24, considering rod lengths that are representative of reactor applications, for the cases of uniform heat flux and a more realistic cosine-like axial heat distribution. The spectral element code Nek5000 has been used for all LES, RANS, and pseudo-RANS simulations. The recently proposed Nek5000 steady-state solver has been used for solving the temperature field in the pseudo-RANS approach and has proved significantly faster than transient schemes. Prediction of thermal quantities is compared with classical linear and nonlinear RANS models. LES for the full-length rods has also been performed and is used as a reference. Results of the proposed method show significant improvements with respect to those obtained with RANS.

Simplified Prediction Method of Vertical Temperature Distribution for Impinging Jet Ventilation System

The impinging jet ventilation (IJV) system is an air-distributing strategy to generate a thermal stratification in a room by supplying the air vertically toward the floor, which provides higher ventilation effectiveness than mixing ventilation. To date, however, no simplified prediction model of indoor thermal environment for an impinging jet ventilated room has been established. Therefore, this study aims to propose a simplified calculation model based on “Block Model”. This paper first presents a parametric study using Computational Fluid Dynamics (CFD), where total supply airflow rate of a test room is changed as a parameter. The number of terminal is also changed. Secondly, the paper presents the simplified calculation model of vertical temperature distribution based on block model that solves heat flow by advection and diffusion. Here, in this prediction model, the most important parameter is the turbulent thermal diffusivity between two room-space blocks in vertical direction. The room-space blocks are classified into two types, i.e., lower and upper part of a room. Based on parametric study using CFD, turbulent thermal diffusivity is arranged for lower and upper block respectively, and it is finally shown that the vertical temperature distribution from block model well agrees with CFD result.

Renormalization group analysis of heat transfer in the presence of endothermic and exothermic chemical reactions.

In the present paper, renormalization group methods are used to develop a macroscopic turbulence model for thermal diffusivity in turbulent fluid flow under conditions of endothermic and exothermic chemical reactions in flow. The temperature field is divided into slow (large-scale) and fast (small-scale) modes. With the help of the renormalization procedure, energy equations for the large-scale modes and relations for effective turbulent thermal diffusivity were obtained. It was shown how the type of the chemical reaction affects turbulent thermal diffusivity. In addition, the conditions were identified where effective thermal diffusivity undergoes a sharp growth.

Large-Scale Magnetic Field Fragmentation in Flux-Tubes Near the Base of the Solar Convection Zone

Magnetic quenching of turbulent thermal diffusivity leads to instability of the large-scale field with the production of spatially isolated regions of enhanced field. This conclusion follows from a linear stability analysis in the framework of mean-field magnetohydrodynamics that allows for thermal diffusivity dependence on the magnetic field. The characteristic growth time of the instability is short compared to the 11-year period of solar activity. The characteristic scale of the increased field regions measures in tens of mega-meters. The instability can produce magnetic inhomogeneities whose buoyant rise to the solar surface forms the solar active regions. The magnetic energy of the field concentrations coincides in order of magnitude with the energy of the active regions.