Turbulent Prandtl Number Model Research Articles

A Priori Tests of RANS Models for Turbulent Channel Flows of a Dense Gas

Dense gas effects, encountered in many engineering applications, lead to unconventional variations of the thermodynamic and transport properties in the supersonic flow regime, which in turn are responsible for considerable modifications of turbulent flow behavior with respect to perfect gases. The most striking differences for wall-bounded turbulence are the decoupling of dynamic and thermal effects for gases with high specific heats, the liquid-like behavior of the viscosity and thermal conductivity, which tend to decrease away from the wall, and the increase of density fluctuations in the near wall region. The present work represents a first attempt of quantifying the influence of such dense gas effects on modeling assumptions employed for the closure of the Reynolds-averaged Navier–Stokes equations, with focus on the eddy viscosity and turbulent Prandtl number models. For that purpose, we use recent direct numerical simulation results for supersonic turbulent channel flows of PP11 (a heavy fluorocarbon representative of dense gases) at various bulk Mach and Reynolds numbers to carry out a priori tests of the validity of some currently-used models for the turbulent stresses and heat flux. More specifically, we examine the behavior of the modeled eddy viscosity for some low-Reynolds variants of the $k-\varepsilon $ model and compare the results with those found for a perfect gas at similar conditions. We also investigate the behavior of the turbulent Prandtl number in dense gas flow and compare the results with the predictions of two well-established turbulent Prandtl number models.

Study of Variable Turbulent Prandtl Number Model for Heat Transfer to Supercritical Fluids in Vertical Tubes

In order to improve the prediction performance of the numerical simulations for heat transfer of supercritical pressure fluids, a variable turbulent Prandtl number (Prt) model for vertical upward flow at supercritical pressures was developed in this study. The effects of Prt on the numerical simulation were analyzed, especially for the heat transfer deterioration conditions. Based on the analyses, the turbulent Prandtl number was modeled as a function of the turbulent viscosity ratio and molecular Prandtl number. The model was evaluated using experimental heat transfer data of CO2, water and Freon. The wall temperatures, including the heat transfer deterioration cases, were more accurately predicted by this model than by traditional numerical calculations with a constant Prt. By analyzing the predicted results with and without the variable Prt model, it was found that the predicted velocity distribution and turbulent mixing characteristics with the variable Prt model are quite different from that predicted by a constant Prt. When heat transfer deterioration occurs, the radial velocity profile deviates from the log-law profile and the restrained turbulent mixing then leads to the deteriorated heat transfer.

Turbulent Convection Heat Transfer Analysis of Supercritical Pressure CO2 Flow in a Vertical Tube Based on the Field Synergy Principle

ABSTRACTTurbulent convection heat transfer of fluids at supercritical pressures can experience abnormal heat transfer phenomena as the fluid temperature approaches the pseudo-critical temperature. A better understanding of these abnormal heat transfer characteristics is essential to develop ways to assure the security and reliability of engineering designs. In this study, we analyze the turbulent convection heat transfer characteristics of supercritical pressure fluid flow in a vertical heated tube using the field synergy principle. A turbulent Prandtl number model that is a function of the flow conditions as well as the physical properties is used in a low Reynolds number turbulence model with validations against experimental data. The validity of the field synergy principle is extended here to analyze convection heat transfer of supercritical pressure fluids where the heat transfer rate is a function of the synergy between the velocity and temperature gradients. Numerical simulation results show that when heat transfer deterioration occurs, the synergy angle between the velocity and temperature gradient vectors increase. The field synergy number could be used as an indicator to quantitatively identify the degree of heat transfer deterioration. The results presented here provide a basis for further research on enhancing turbulent convection of supercritical pressure fluid.

Heat Transfer Characteristics and Prediction Model of Supercritical Carbon Dioxide (SC-CO2) in a Vertical Tube

Due to its distinct capability to improve the efficiency of shale gas production, supercritical carbon dioxide (SC-CO2) fracturing has attracted increased attention in recent years. Heat transfer occurs in the transportation and fracture processes. To better predict and understand the heat transfer of SC-CO2 near the critical region, numerical simulations focusing on a vertical flow pipe were performed. Various turbulence models and turbulent Prandtl numbers (Prt) were evaluated to capture the heat transfer deterioration (HTD). The simulations show that the turbulent Prandtl number model (TWL model) combined with the Shear Stress Transport (SST) k-ω turbulence model accurately predicts the HTD in the critical region. It was found that Prt has a strong effect on the heat transfer prediction. The HTD occurred under larger heat flux density conditions, and an acceleration process was observed. Gravity also affects the HTD through the linkage of buoyancy, and HTD did not occur under zero-gravity conditions.

Variable Turbulent Prandtl Number Model for Shock/Boundary-Layer Interaction

Interaction of shock waves with turbulent boundary layers can enhance the surface heat flux dramatically. Reynolds-averaged Navier–Stokes simulations based on a constant turbulent Prandtl number often give grossly erroneous heat transfer predictions in shock/boundary-layer interaction flows. This is due to the fact that the underlying Morkovin’s hypothesis breaks down in the presence of shock waves; thus, the turbulent Prandtl number cannot be assumed to be a constant. In this paper, a new variable turbulent Prandtl number model based on linearized Rankine–Hugoniot conditions applied to shock–turbulence interaction is developed. The turbulent Prandtl number is a function of the shock strength, and a shock function is proposed to identify the location and strength of shock waves. The shock function also simulates the postshock relaxation of the turbulent heat flux, which is akin to that observed in canonical shock–turbulence interaction. The model is combined with the well-validated shock-unsteadiness model and is applied to the complex shock topology observed in oblique shock/turbulent boundary-layer interactions. Comparison with experimental data shows significant improvement in the surface heat transfer rate in the interaction region, both for attached and separated shock/boundary-layer interaction cases. The shock function is also used to propose a robust form of the existing shock-unsteadiness model that simplifies the numerical implementation enormously.

Development of an analytical model for pure vapor downflow condensation in a vertical tube

In-tube condensation of pure vapor are widely adopted by passive safety systems of advanced nuclear reactors to remove the decay heat. Accurate calculation of the in-tube condensation heat transfer coefficient is important for evaluating the heat removal ability. A better understanding of the mechanism of pure vapor condensation heat transfer and its modelling are essential for the design and assessment of those passive safety systems. An analytical model for pure vapor condensation in a vertical tube is developed based on mass, momentum and energy conservation equations. The total axial pressure drop is calculated through the momentum equation of the vapor core assuming a uniform radial pressure distribution. An optimized transition area between the laminar and turbulent film flow is defined to avoid discontinuity of heat transfer coefficients. Based on the sensitivity analyses, Kay’s turbulent eddy diffusivity model and his turbulent Prandtl number model were recommended for the newly proposed model. The predicted results were compared with results of Shah (2016a,b), Chen et al. (1987) and Cavallini et al. (2002) correlations as well as Kuhn’s and KAIST experimental data. It was found that the proposed model predicts the heat transfer coefficient well with a mean absolute deviation of 11.99% for Kuhn’s data and 8.08% for KAIST’s data.

Numerical study of supercritical and transcritical injection using different turbulent Prandlt numbers: A second law analysis

The purpose of this investigation is to study the supercritical and transcritical injection of cryogenic fluids by means of a Computational Fluid Dynamics analysis using a RANS approach and different turbulent Prandtl number models. Besides, with the aid of the differential equation for entropy, the local entropy generation is pursued. Using this method, the areas where the irreversibilities occur are detected, and the mechanisms concerning entropy generation identified. The novelty of the numerical procedure presented herein is the use of two different turbulent Prandtl number (PrT) models that affect strongly the result produced: a constant PrT and a molecular dependent PrT. The main results show that the constant PrT predicts better the flow structure when compared to experimental data, being PrT=0.5 the optimum one. This result suggest that supercritical and transcritical fluid injection are more dissipative processes than normal pressure injection, where PrT≈1.

Instantaneous Heat Flux Simulation of a Motored Reciprocating Engine: Unsteady Thermal Boundary Layer With Variable Turbulent Thermal Conductivity

Due to the inherently unsteady environment of reciprocating engines, unsteady thermal boundary layer modeling may improve the reliability of simulations of internal combustion engine heat transfer. Simulation of the unsteady thermal boundary layer was achieved in the present work based on an effective variable thermal conductivity from different turbulent Prandtl number and turbulent viscosity models. Experiments were also performed on a motored, single-cylinder spark-ignition engine. The unsteady energy equation approach furnishes a significant improvement in the simulation of the heat flux data relative to results from a representative instantaneous heat transfer correlation. The heat flux simulated using the unsteady model with one particular turbulent Prandtl number model agreed with measured heat flux in the wide open and fully closed throttle cases, with an error in peak values of about 6% and 35%, respectively.

A CFD four parameter heat transfer turbulence model for engineering applications in heavy liquid metals

In ordinary fluids, such as water or air, similarity between thermal and dynamical fields holds and it is commonly accepted that implementing a Computational Fluid Dynamics (CFD) code for a two-equation turbulence model with the hypothesis of a constant turbulent Prandtl number in the range 0.85–0.9 is sufficient to obtain reliable results both for velocity and temperature fields. In heavy liquid metals such as sodium, lead and Lead–Bismuth Eutectic (LBE) with low Prandtl number (Pr≈0.025) the time scales of temperature and velocity fields are rather different, because heat transfer is due mainly to molecular diffusion. In these fluids a standard constant turbulent Prandtl number model fails to reproduce correlations build from experimental data and predicts a too high heat transfer. Heavy liquid metals are promising coolant fluids for achieving the necessary requirements of fast nuclear reactors and many European projects have been started with the purpose of developing CFD codes able to correctly predict turbulent heat transfer for these fluids. The present work addresses an effort to improve the prediction of turbulent heat transfer for liquid metal flows in plane and cylindrical geometries assessing a κ–∊–κθ–∊θ four parameter turbulence model. In particular the simulations aim to reproduce fully developed thermal and velocity profiles by using a standard finite element implementation of the Navier–Stokes equations coupled with the energy and momentum turbulence models. A modified κ–∊ system with low-Reynolds model functions is used for the turbulent velocity field while a κθ–∊θ system is employed to compute the turbulent temperature field. The results of the simulations are compared with Direct Numerical Simulations (DNS) data and with heat transfer experimental correlations in order to validate the four parameter turbulence model. Different uniform heat flux boundary conditions with zero and constant temperature fluctuations at the wall are presented.

Algebraic Turbulent Heat Flux Model for Prediction of Thermal Stratification in Piping Systems

The effect of stratification on the flow in bounded geometries is studied through computational fluid dynamics and two different modelings of the turbulent heat flux: constant turbulent Prandtl number and Algebraic Heat Flux Model (AHFM). The main feature of the work is evaluation of the effect of buoyancy on the thermal quantities, velocity field, and related pressure drop. For evaluation of the turbulent heat flux and temperature field, AHFM has been demonstrated to be superior to the simple eddy diffusivity approach. However, serious concerns remain for the prediction of the velocity field in both isothermal and nonisothermal conditions, since greater uncertainties for the obtained pressure drop and related Fanning friction factor can be introduced. Incremental pressure drop is also investigated in conditions deviating from fully developed flows, in order to study stratification effects qualitatively using an engineering method.

New development of the turbulent Prandtl number models for the computation of film cooling effectiveness

This paper presents the new development on the turbulent Prandtl number distribution models which are used to improve the computation accuracy of film cooling effectiveness distribution in the frame of the isotropic turbulence models. In the new developed turbulent Prandtl number distribution models, the turbulent Prandtl number varies with some characteristic quantity of the film cooling to make the models applicable for more film cooling cases effectively. A new VTG-Pr t model is developed for the computation of film cooling cases in which the wall is mainly influenced by the mixing layer region between the jet core and the mainstream. The Pr t value in this model varies with dimensionless temperature gradient which is suggested to be the characteristic quantity of the mixing layer region. The VTG-Pr t model has been used in the computations of long cylindrical hole film cooling, short cylindrical hole film cooling, compound angle cylindrical hole film cooling and expansion shaped hole film cooling in the present paper. And it is validated to be an effective and easy way to obtain accurate film cooling effectiveness distributions for this kind of film cooling. But for the kind of film cooling, in which the wall is mainly influenced by the jet core region and the influence of heat diffusion in the mixing layer region on the cooling effectiveness is very small, the VTG-Pr t model is disabled. Another turbulent Prandtl number distribution model, named VT-Pr t model, is developed for this kind of film cooling in the present paper. The Pr t value in this model varies with dimensionless temperature which is suggested to be the characteristic quantity of the jet core region. Good agreement between the computed and measured results is obtained through the use of VT-Pr t model in the computation of converging slot hole film cooling.

Extension of equilibrium turbulent heat flux models to high-speed shear flows

An algebraic heat flux truncation model was derived for high-speed gaseous shear flows. The model was developed for high-temperature gases with caloric imperfections. Fluctuating dilatation moments were modelled via conservation of mass truncations. The present model provided significant improvements, up to 20%, in the temperature predictions over the gradient diffusion model for a Mach number ranging from 0.02 to 11.8. Analyses also showed that the near-wall dependence of the algebraic model agreed with expected scaling, where the constant Prandtl number model did not. This led to a simple modification of the turbulent Prandtl number model. Compressibility led to an explicit pressure gradient dependency with the present model. Analyses of a governing parameter indicated that these terms are negligibly small for low speeds. However, they may be important for high-speed flow.

Effect of turbulent Prandtl number on the computation of film-cooling effectiveness

This paper presents a method to improve the accuracy of computed film cooling effectiveness spanwise distribution. The effect of turbulent Prandtl number in the flow field outside the near-wall region on the computation is studied. Realizable k – ε model with a one-equation model in near-wall region is employed. The results show that the variation of turbulent Prandtl number has great influence on the computation. Reducing turbulent Prandtl number increases film cooling effectiveness of the whole spanwise region remarkably under large blowing ratios. Under small blowing ratios, the reduction of turbulent Prandtl number decreases the cooling effectiveness of the center region, and increases the effectiveness of the lateral region off the centerline. Compared with the single value turbulent Prandtl number computation the agreement between computation and measured results is improved notably with varied one in different spanwise regions. A new laterally varying turbulent Prandtl number (LV- Pr t) model dependent on lateral location and blowing ratio has been suggested. Computation accuracy is improved greatly by LV- Pr t model. Compared with the TLVA- Pr model of Lakehal [D. Lakehal, Near-wall modeling of turbulent convective heat transport in film cooling of turbine blades with the aid of direct numerical simulation data, ASME J. Turbomach. 124 (2002) 485–498] which provides the best results in the calculated cases LV- Pr t model is an effective way to improve computation accuracy in the frame of the traditional isotropic turbulence models. More work on the information of the turbulent Prandtl number has to be done for the further development of the LV- Pr t model.

The Turbulent Prandtl Number for Temperature Analysis in Rod Bundle Subchannels

The turbulent Prandtl number (PrT) model for the temperature field analysis in rod bundle subchannels is proposed based on the experimental data. The proposed turbulent Prandtl number is a function of the position and has the characteristic of an anisotropy which may not be found in simple geometries such as circular tubes and flat plates. Also, the proposed turbulent Prandtl number is modeled by considering the effects of the pitch to diameter (P/D) and the molecular Prandtl number (Pr). The turbulent temperature field is analyzed by a detailed analysis code. The Nusselt numbers (Nu) obtained by the new model are compared with those obtained by the well-known experimental correlations. The new turbulent Prandtl number model can predict the experimental data more accurately than the previous turbulent Prandtl number model.

The Turbulent Prandtl Number for Temperature Analysis in Rod Bundle Subchannels

The turbulent Prandtl number (PrT) model for the temperature field analysis in rod bundle subchannels is proposed based on the experimental data. The proposed turbulent Prandtl number is a function of the position and has the characteristic of an anisotropy which may not be found in simple geometries such as circular tubes and flat plates. Also, the proposed turbulent Prandtl number is modeled by considering the effects of the pitch to diameter (P/D) and the molecular Prandtl number (Pr). The turbulent temperature field is analyzed by a detailed analysis code. The Nusselt numbers (Nu) obtained by the new model are compared with those obtained by the well-known experimental correlations. The new turbulent Prandtl number model can predict the experimental data more accurately than the previous turbulent Prandtl number model.

Scalar Variance Transport in the Turbulence Modeling of Propulsive Jets

A variable turbulent Prandtl number model is applied to the prediction of laboratory jets with signie cant temperature variations. The model involves solving supplemental scalar equations for the temperature variance anditsdissipationratealongwiththe k‐equations.Themodelisalsoappliedtojetswithvariablecompositionwith the aim of building a consolidated approach to model e uctuations of both temperature and species mass fraction. Comparisons are presented for subsonic jets for which scalar e uctuation measurements are available. The same coefe cientsareusedfortemperature/speciesvarianceanddissipationrateequations.Excellentagreementwithdata isobtained forround jets with respect to scalare uctuation intensitiesand prediction of turbulent Prandtl/Schmidt numbers.

Test of an Eulerian–Lagrangian simulation of wall heat transfer in a gas-solid pipe flow

An Eulerian–Lagrangian model is proposed to numerically predict the heat transfer between a vertical pipe wall and a turbulent gas-solid suspension. The whole problem lies in the combined solution of mass, momentum, and energy equations written for each phase. Closure equations devoted to turbulence and heat transfer simulation are also needed for the continuous phase. This problem is treated using a k–ε model and a turbulent Prandtl number model for the gaseous phase. The coupling terms standing for the fluid–particle interactions are taken into account. The simulation of the dispersed phase dynamics is addressed using a Lagrangian model, taking the particle–wall and particle–particle interactions into account. Additionally, the temperature of each particle is tracked using a model for the convective heat exchange between the two phases. The accuracy of the thermal and dynamic solutions has been tested for particles of diameter 200 and 500 μm, and for a wide range of loading ratios (0–20). Corresponding velocity and turbulent kinetic energy profiles are presented for the dynamic modelling validation, while Nusselt numbers are calculated for the study of the thermal problem.

Numerical Prediction of Transitional Features of Turbulent Forced Gas Flows in Circular Tubes With Strong Heating

In order to treat strongly heated, forced gas flows at low Reynolds numbers in vertical circular tubes, the k-ε turbulence model of Abe, Kondoh, and Nagano (1994), developed for forced turbulent flow between parallel plates with the constant property idealization, has been successfully applied. For thermal energy transport, the turbulent Prandtl number model of Kays and Crawford (1993) was adopted. The capability to handle these flows was assessed via calculations at the conditions of experiments by Shehata (1984), ranging from essentially turbulent to laminarizing due to the heating. Predictions forecast the development of turbulent transport quantities, Reynolds stress, and turbulent heat flux, as well as turbulent viscosity and turbulent kinetic energy. Overall agreement between the calculations and the measured velocity and temperature distributions is good, establishing confidence in the values of the forecast turbulence quantities—and the model which produced them. Most importantly, the model yields predictions which compare well with the measured wall heat transfer parameters and the pressure drop.

Turbulent Heat Transport in a Perturbed Channel Flow

The recovering boundary layer downstream of a separation bubble is known to have a highly perturbed turbulence structure which creates difficulty for turbulence models. The present experiment addressed the effect of this perturbed structure on turbulent heat transport. The turbulent diffusion of heat downstream of a heated wire was measured in a perturbed channel flow and compared to that in a simple, fully developed channel flow. The turbulent diffusivity of heat was found to be more than 20 times larger in the perturbed flow. The turbulent Prandtl number increased to 1.7, showing that the turbulent eddy viscosity was affected even more strongly than the eddy thermal diffusivity. This result corroborates previous work showing that boundary layer disturbances generally have a stronger effect on the eddy viscosity, rendering prescribed turbulent Prandtl number models ineffective in perturbed flows.

An extended Kays and Crawford turbulent Prandtl number model

For theoretical predictions of turbulent heat transfer in boundary layers and in duct flows, the knowledge of the turbulent Prandtl number is crucial. This is especially true for fluids with low molecular Prandtl numbers (liquid metals). The present formulation, which is an extended Kays and Crawford ( Convective Heat and Mass Transfer, 3rd edn. McGraw-Hill, New York, 1993) turbulent Prandtl number model, can be used for accurately predicting the heat transfer for liquid metal flows. The Nusselt numbers calculated with the modified model for Pr t are found to be in good agreement with experimental data for fully-developed pipe flows as well as for thermally developing pipe flows for various wall boundary conditions. The present model reduces to the one given in the above reference for liquids and gases with higher molecular Prandtl numbers.