Gradient Diffusion Hypothesis Research Articles

Lateral edge effects on heat/mass transfer on a finite width surface within a turbulent boundary layer

Numerical simulations of the local heat/mass transfer on a finite width surface within a turbulent boundary layer are presented. Different approaches to the RANS modelling of the turbulent heat/mass flux are compared to Large Eddy Simulations (LES). Mass transfer experiments conducted with the naphthalene sublimation technique are used as validation. The isotropic eddy viscosity model, Simple Gradient Diffusion Hypothesis (SGDH), is shown to underestimate the span-wise effects. Its anisotropic extension, Generalized Gradient Diffusion Hypothesis (GGDH), improves the prediction, but still does not account for near-wall contribution in strongly dissimilar velocity and temperature/concentration fields, even in combination with a wall-sensitive second-moment closure model such as the Elliptic Blending Reynolds Stress Model (EB-RSM). A more complete turbulent heat flux model based on the elliptic blending approach, the Elliptic Blending GGDH (EB-GGDH) presents very good agreement with LES and with the experiments, confirming the need for more advanced turbulent heat flux modelling in applications with strong three-dimensional effects.

Application of compressible Reynolds-averaged governing equations to turbulent mixed convection in supercritical fluids in heated vertical tubes

Accurate estimation of thermal-hydraulic characteristics of supercritical flows has long been an attractive but elusive subject to many researchers in spite of tremendous effort devoted to the development of suitable turbulence models. One of the key reasons for the difficulty is a lack of measured turbulence data, which might have been used to formulate adequate turbulence models suitable for highly buoyant fluids. Turbulence models are typically based on the log-law, while the velocity profile in buoyant fluids substantially deviates from the log-law because of significant density variation in a turbulent boundary layer. In this paper, axisymmetric compressible Reynolds-Averaged governing equations were employed together with the property-dependent turbulent Prandtl number to reproduce experimental data representing heat transfer deterioration and consequential sudden temperature increase. The additional turbulence terms associated with turbulent mass flux appeared in the governing equations were modeled using the simple gradient diffusion hypothesis (SGDH). The proposed model successfully reproduced the experimental data. The various turbulence properties are presented and discussed.

Physical Interpretation of Machine Learning Models Applied to Film Cooling Flows

Current turbulent heat flux models fail to predict accurate temperature distributions in film cooling flows. The present paper focuses on a machine learning (ML) approach to this problem, in which the gradient diffusion hypothesis (GDH) is used in conjunction with a data-driven prediction for the turbulent diffusivity field αt. An overview of the model is presented, followed by validation against two film cooling datasets. Despite insufficiencies, the model shows some improvement in the near-injection region. The present work also attempts to interpret the complex ML decision process, by analyzing the model features and determining their importance. These results show that the model is heavily reliant of distance to the wall d and eddy viscosity νt, while other features display localized prominence.

A Statistical Dynamic Model for the Turbulent Transport of Kinetic Energy in Shear Layers

Dynamic statistical equations for the turbulent fluxes of scalars and momentum in incompressible flows are derived after time wise integration of the equation for the oscillating transported property, decomposing the turbulent fluxes in terms representing distinct features of the main and fluctuating flow that influence the respective turbulent transport. These expressions provide a means for discussing the gradient diffusion hypothesis for the turbulent transport, for reconsidering the mixed length model in entirely continuous terms, and for seeking possible alternatives or corrections. Applying this methodology to the turbulent transport flux of kinetic energy, two dominating terms are found: one identified with a kinetic energy gradient model for shear layers; the other related to the main velocity gradient. Accordingly, a composed, Statistical Dynamic model is proposed for the turbulent transport of kinetic energy in shear layers, adding a velocity derivative term to Daly and Harlow’s generalized gradient model. This velocity derivative term is calibrated in a nearly homogeneous turbulent shear flow, and the resulting Statistical Dynamic model is proved superior to Daly and Harlow’s and other gradient models in channel and boundary layer flows.

Bayesian source term estimation of atmospheric releases in urban areas using LES approach

The estimation of source information from limited measurements of a sensor network is a challenging inverse problem, which can be viewed as an assimilation process of the observed concentration data and the predicted concentration data. When dealing with releases in built-up areas, the predicted data are generally obtained by the Reynolds-averaged Navier-Stokes (RANS) equations, which yields building-resolving results; however, RANS-based models are outperformed by large-eddy simulation (LES) in the predictions of both airflow and dispersion. Therefore, it is important to explore the possibility of improving the estimation of the source parameters by using the LES approach. In this paper, a novel source term estimation method is proposed based on LES approach using Bayesian inference. The source-receptor relationship is obtained by solving the adjoint equations constructed using the time-averaged flow field simulated by the LES approach based on the gradient diffusion hypothesis. A wind tunnel experiment with a constant point source downwind of a single building model is used to evaluate the performance of the proposed method, which is compared with that of the existing method using a RANS model. The results show that the proposed method reduces the errors of source location and releasing strength by 77% and 28%, respectively.

A Machine Learning Approach for Determining the Turbulent Diffusivity in Film Cooling Flows

In film cooling flows, it is important to know the temperature distribution resulting from the interaction between a hot main flow and a cooler jet. However, current Reynolds-averaged Navier–Stokes (RANS) models yield poor temperature predictions. A novel approach for RANS modeling of the turbulent heat flux is proposed, in which the simple gradient diffusion hypothesis (GDH) is assumed and a machine learning (ML) algorithm is used to infer an improved turbulent diffusivity field. This approach is implemented using three distinct data sets: two are used to train the model and the third is used for validation. The results show that the proposed method produces significant improvement compared to the common RANS closure, especially in the prediction of film cooling effectiveness.

On the Values for the Turbulent Schmidt Number in Environmental Flows

Computational Fluid Dynamics (CFD) has consolidated as a tool to provide understanding and quantitative information regarding many complex environmental flows. The accuracy and reliability of CFD modelling results oftentimes come under scrutiny because of issues in the implementation of and input data for those simulations. Regarding the input data, if an approach based on the Reynolds-Averaged Navier-Stokes (RANS) equations is applied, the turbulent scalar fluxes are generally estimated by assuming the standard gradient diffusion hypothesis (SGDH), which requires the definition of the turbulent Schmidt number, Sct (the ratio of momentum diffusivity to mass diffusivity in the turbulent flow). However, no universally-accepted values of this parameter have been established or, more importantly, methodologies for its computation have been provided. This paper firstly presents a review of previous studies about Sct in environmental flows, involving both water and air systems. Secondly, three case studies are presented where the key role of a correct parameterization of the turbulent Schmidt number is pointed out. These include: (1) transverse mixing in a shallow water flow; (2) tracer transport in a contact tank; and (3) sediment transport in suspension. An overall picture on the use of the Schmidt number in CFD emerges from the paper.

Simultaneous kHz-rate temperature and velocity field measurements in the flow emanating from angled and trenched film cooling holes

To design more efficient film cooling geometries for gas turbines, non-intrusive measurements of the flow temperature, velocity and derived quantities like the turbulent heat flux are needed in well-defined, generic flow configurations. With this aim we have applied thermographic particle image velocimetry (thermographic PIV) to investigate the flow emanating from angled and trenched cooling holes in a closed-loop optically-accessible wind tunnel facility. BAM:Eu2+ thermographic phosphor particles were seeded into the flow as a tracer. A pulsed high-speed UV laser was used to excite the particles and the luminescence was detected using two high-speed cameras to determine the temperature field by a two-colour ratiometric approach. The velocity field was measured using ordinary high-speed PIV. The simultaneously measured fields were sampled at a rate of 6kHz in a vertical plane through the centreline of the symmetrical single-row cooling holes. The flowrate and temperature of the cooling air and heated main flow were chosen to achieve density and momentum flux ratios of 1.6 and 8 respectively. For these conditions the average and RMS temperature fields show that for ordinary angled holes the jet is detached from the surface. In contrast, the trenched geometry leads to a cooling film attached to the surface. However, time-resolved image sequences show instances where hot air breaks through the cooling film and almost reaches the surface. Similar image sequences for the angled holes show that the detached coolant jet becomes unstable downstream and pockets of cold air are ejected into the main flow. This intermittency may in part explain the observation that the measured turbulent heat flux is oriented towards the cold core, but deviates from the direction of the mean temperature gradient, thereby contradicting the simple gradient diffusion hypothesis commonly used in RANS simulations.

Mixing at high Schmidt number in a complex porous structure

Highly porous structures (porosity ≥75%), as metal foams or designed foam-like structures, are often used in industry to augment heat and mass transfer. In the present study, we focus on the mixing in highly porous structures in continuous millireactor characterized with a relatively low Reynolds number (ReD=240 – based on the ligament diameter and bulk velocity) and a high Schmidt number fluid (Sc=2400). We apply a combined numerical and experimental approach. The numerical simulations employ a Large Eddy Simulation (LES) method with dynamic Lagrangian approach for subgrid-scale turbulent stress and turbulent mass flux. The results of the numerical simulation are compared with our own combined Particle Imaging Velocimetry (PIV) and Laser Induced Fluorescence (LIF) measurements. The application of combined PIV/LIF makes it possible to simultaneously measure velocity components, turbulent stresses, concentration and concentration fluxes. The mechanism of the mixing is analyzed in detail with specific focus on validity of a simple gradient diffusion hypothesis (SGDH) in modeling of the turbulent mass transfer in complex porous structure.

Numerical Investigation of Compound Angle Effusion Cooling Using Differential Reynolds Stress Model and Zonal Detached Eddy Simulation Approaches

Effusion cooling is one of the most effective and widespread techniques to prevent combustor liner from being damaged. However, most recent developments in combustion techniques, resulting from increasingly stricter air pollution regulations, have highlighted the necessity of reducing the amount of air available for effusion cooling while keeping an adequate level of protection. Adoption of compound angles in effusion cooling is increasingly recognized by jet engine manufacturers as a powerful solution to meet new combustor requirements. Therefore, understanding the flow behavior and developing methods able to provide accurate estimates of wall temperatures is of a major importance. This study assesses the capability of a high-level Reynolds-averaged Navier–Stokes (RANS) method, differential Reynolds stress model (DRSM), in conjunction with a generalized gradient diffusion hypothesis (GGDH), and of a hybrid RANS–large eddy simulations (LES) method, zonal detached eddy simulation (ZDES), to predict overall film effectiveness. Both approaches are compared with the experimental data from Zhang et al. (2009, “Cooling Effectiveness of Effusion Walls With Deflection Hole Angles Measured by Infrared Imaging,” Appl. Therm. Eng., 29(5), pp. 966–972) and with a classical well-known RANS model (k–ω shear-stress transport (SST) model). Despite the fact that some discrepancies are found, both approaches have proved suitable and reliable for predicting wall temperatures (relative errors of about 5%). Moreover, a new method to deal with ZDES length scales for unstructured grids is proposed. ZDES applicability and its general advantages and drawbacks are also discussed. Finally, an in-depth analysis of the film structure is carried out on the basis of the ZDES simulations. The principal structures are identified (an asymmetric main vortex (AMV) and a counter rotating vortex pair, CRVP), and the film formation mechanisms are presented. Significant spanwise-homogeneous distributions of surface overall film cooling effectiveness are observed.

CFD modeling and analysis of the behavior of 30° and 45° inclined dense jets – new numerical insights

A three-dimensional numerical model of inclined turbulent jets with negatively buoyant discharge into stationary ambient water is presented in this paper to study certain jet parameters with turbulence schemes that have not been employed before in this context such as standard Boussinesq gradient diffusion hypothesis and general gradient diffusion hypothesis to account for the buoyancy-induced turbulence generation. Two jet discharge angles have been chosen for this study: 30° and 45° with the horizontal. These two angles are chosen in this study due to lower terminal rise heights for 30° and 45°, a fact which is critically important for discharges of effluent into shallow waters compared to higher angles than these values. The spatio-temporal jet evolutions for these cases have been modeled using OpenFOAM open-source CFD code, which is based on Finite-Volume Method. Results presented in this paper deal with the geometrical and flow properties of the inclined dense jets. The densimetric Froude number of the effluent at the nozzle ranges between 10 and 34. Two Reynolds-Averaged Navier–Stokes turbulence models are applied to evaluate the accuracy of the numerical predictions: the realizable k–ε (a two-equation model) and the Launder Reece Rodi (a Reynolds Stress Model – RSM).

Analysis of Turbulent Scalar Flux Models for a Discrete Hole Film Cooling Flow

Algebraic closures for the turbulent scalar fluxes were evaluated for a discrete hole film cooling geometry using the results from a high-fidelity large eddy simulation (LES). Several models for the turbulent scalar fluxes exist, including the widely used gradient diffusion hypothesis (GDH), the generalized GDH (GGDH), and the higher-order GDH (HOGGDH). By analyzing the results from the LES, it was possible to isolate the error due to these turbulent mixing models. Distributions of the turbulent diffusivity, turbulent viscosity, and turbulent Prandtl number were extracted from the LES results. It was shown that the turbulent Prandtl number varies significantly spatially, undermining the applicability of the Reynolds analogy for this flow. The LES velocity field and Reynolds stresses were fed into a Reynolds-averaged Navier–Stokes (RANS) solver to calculate the fluid temperature distribution. This analysis revealed in which regions of the flow various modeling assumptions were invalid and what effect those assumptions had on the predicted temperature distribution.

Analysis of Turbulence Modelling Approaches to Simulate Single-phase Buoyancy Driven Counter-current Flow in a Tilted Tube

In the present paper, both Large Eddy Simulation (LES) and unsteady-RANS (uRANS) CFD studies of single-phase buoyancy-driven counter-current flow in a pipe are presented for an Atwood number of 1.15×10−2 and an inclination angle of 15 degrees from the vertical. The basic flow phenomena involved are fundamentally the same as those encountered in various industrial applications including passive coolant flow in the loops of a nuclear reactor. Earlier work on this type of flow was focused on its physical aspects using both experimental measurements and Direct Numerical Simulation (DNS). The present work investigates the performance of several commonly used LES eddy viscosity subgrid-scale models. The results show that LES is able to reproduce accurately the experimental results and that the dynamic Smagorinsky subgrid-scale model gives the best predictions. The results of those calculations were then used to obtain more information on the physics of such flow. As regards uRANS simulations, a range of classic two-equation linear eddy viscosity models, based on low-y + formulation at the wall and the single gradient diffusion hypothesis for the turbulent density fluxes were compared against experimental data available in the literature. An elliptic blending Reynolds stresses model (EBRSM) with generalised gradient diffusion hypothesis (GGDH) for the scalar fluxes was then implemented and compared with experiments. The latter showed very good agreement for the first order moment with the experimental data whereas all the two-equation eddy viscosity based models showed rather high levels of discrepancy. The main cause of discrepancy was found to be the underprediction of the axial turbulent buoyancy production effect, which has a more detrimental impact on the eddy viscosity models than the second moment closure one.

Large eddy simulation of passive scalar transport in a high Schmidt number turbulent incompressible wake with experimental validation

Large eddy simulation of the passive scalar transport in a high Schmidt number turbulent confined wake flow has been performed. The results are evaluated by comparison to particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) data, including point-wise data as well as spatial correlations. In the LES simulations, the gradient diffusion hypothesis is used to close the transport equation for the passive scalar. Different discretization schemes are investigated in order to determine the best choice for ensuring boundedness of the passive scalar and to accurately predict the mixing rate. The simulation results compare well to experimental data, demonstrating that the transport mechanisms in this high Schmidt number turbulent flow are well predicted by the LES method. Two-point spatial correlations of passive scalar with velocity predicted by the simulation show good agreement with the experimental results, indicating that the turbulent coherent structures of the flow are reproduced by the simulation.

Near-field pollutant dispersion in an actual urban area: Analysis of the mass transport mechanism by high-resolution Large Eddy Simulations

Large-Eddy Simulation of near-field pollutant dispersion from stacks on the roof of a low-rise building in downtown Montreal is performed. Two wind directions are considered, with different wind-flow patterns and plume behavior. The computed mean concentration field is analyzed by means of the convective and turbulent (including subgrid-scale) mass fluxes. This decomposition provides insight into the dispersion process and allows an evaluation of common turbulent transport models used with the Reynolds-Averaged Navier–Stokes approach, such as the standard gradient-diffusion hypothesis. Despite the specific character of the flow and dispersion patterns due to the complex geometry of the urban area under study, some similarities are found with the generic case of dispersion around an isolated simple building. Moreover, the analysis of dispersion in downtown Montreal is facilitated by the physical insight gained by the study of the generic case. In this sense, the present study supports the use of generic, simplified cases to investigate and understand environmental processes as they occur in real and more complex situations. Reciprocally, the results of this applied study show the influence on the dispersion process of the rooftop structures and of the orientation of the emitting building with respect to the incoming wind flow, providing directions for further research on generic cases.

Partially-averaged Navier–Stokes method for turbulent thermal plume

In this paper, the partially-averaged Navier–Stokes (PANS) simulation is performed for a turbulent thermal plume. The aim of the paper is to assess the PANS method for modeling buoyancy-driven flows at a reasonable computational cost. PANS is a turbulence closure model which is developed to be used as a bridging model ranging from the direct numerical simulation to the Reynolds-averaged Navier–Stokes simulation by varying the level of resolution. The PANS computations are performed for various values of the filter-width to evaluate the sensitivity of the filter-widths to the computed flow statistics. The present simulations have been carried out employing a source code buoyantPimpleFOAM based on the OpenFOAM platform. In order to capture the effect of buoyancy on turbulence, the generalized gradient diffusion hypothesis is employed to model the production of turbulence due to buoyancy. A detailed comparison of the time-averaged and turbulent statistics obtained from the PANS simulations with the experimental data and LES results reported in the literature has been presented. The present results have also been compared with the results of the unsteady Reynolds-averaged Navier–Stokes solutions. The PANS model is shown to enhance the computing capability significantly in predicting buoyancy-driven flows compared with those of URANS model. Finally, various important unsteady flow structures of turbulent thermal plume have been visualized from the instantaneous flow statistics obtained using the PANS simulations.

Direct numerical simulation of low-Prandtl number turbulent convection above a wavy wall

Turbulent forced convection is investigated by Direct Numerical Simulation in a channel with one sinusoidal wavy wall and one flat wall. Fluid flow and heat transfer are periodically fully developed, the simulated Reynolds number of the bulk velocity and the hydraulic diameter is Re=18, 880 while three Prandtl numbers are considered, i.e. Pr=0.025, Pr=0.2, and Pr=0.71. The fluid flow is characterized by separation, reattachment and a shear layer downstream the wave peak, these are conditions relevant for turbulent heat transfer and passive scalar transport applications.In the range of Péclet numbers investigated, the most important heat transfer mechanism is by mean flow advection. Accordingly, the peak heat transfer region is in the upslope part of the domain. The separation bubble instead acts as a barrier to convection and the heat transfer rate is minimum close to separation. An a priori analysis is performed in order to assess the accuracy of turbulent heat transfer models based on the Generalized Gradient Diffusion Hypothesis.

Numerical simulations of fire-induced doorway flows in a small scale enclosure

Field simulation results of a fire, centered in a small scale enclosure (1.04m3) with an open doorway, performed with the CFD computer code ISIS, are presented. Three heat release rates of 10.6, 15.5 and 21.7kW, provided with a propane gas burner, are numerically studied and compared to experimental results. Turbulence and soot modeling are first validated by simulating the gas burner fire in an open atmosphere for the three fire powers. Due to the complex role of buoyancy in production of turbulence inside a pool fire plume, anisotropic modeling, through the generalized gradient diffusion hypothesis, is considered in the standard k–ε model. The comparisons between experimental measurements and numerical simulation results, for the enclosure fire, concern temperature and velocity profiles at the doorway and temperature profiles inside the enclosure. Velocity measurements at the open doorway are performed using a stereoscopic particle image velocimetry (SPIV) technique, allowing a full comparison with computational fluid dynamics (CFD) results. For the three heat release rates, the simulation results agree well with experimental measurements. General flow patterns, provided by CFD simulations, are reported for the highest fire power and supply useful information for understanding enclosure fires.

Reverse flow region associated to a heat transfer in a turbulent wall jet

A numerical study is performed in order to investigate the effects of the inlet flow structure on the flow and heat transfer characteristics in the reattachment zone downstream a backward facing step. Indeed, two inlet flow configurations are considered; a turbulent wall jet (WJ) which presents a particular structure with two sources of turbulence production (one is due to the shear flow related to the inner layer with small scale eddies and another one corresponds to the free shear jet flow with entrainment of large turbulence scales) and a standard channel flow (CF). The numerical approach is based on two-scale energy-flux model coupled with a full Reynolds stress closure valid at low turbulence Reynolds numbers. To calculate the turbulent heat-flux vector, we considered two models: a Single Gradient Diffusion Hypothesis (SGDH) and a Generalized Gradient Diffusion Hypothesis under a fixed heat flux boundary condition. Numerical predictions for the CF configuration are compared with experimental data of Vogel and Eaton (1985). So the present study would contribute emphasizing the role of the large eddies in the outer flow on the convective heat transfer in the recirculation zone.

The computation of buoyant flows in differentially heated inclined cavities

This paper discusses the application of RANS models to the computation of two-dimensional natural convection flows in different types of differentially heated cavities. They include a square cavity with differentially heated vertical walls and tall rectangular cavities, with differentially heated long walls, which are either vertical, or inclined. For the modelling of the turbulent stresses, eddy-viscosity-based two-equation models, as well as second-moment closures, have been employed. The eddy-diffusivity approach has been used for modelling the turbulent heat fluxes in the former, while second-moment-closure computations have employed both the generalised gradient diffusion hypothesis and a more elaborate algebraic model, which involves solving additional transport equations for the scalar variance and its dissipation rate. For the modelling of the near-wall turbulence, the eddy-viscosity computations involved the use of either a low-Reynolds-number model with fine near-wall grids, or a high-Reynolds-number version with wall functions. The standard log-law-based wall function (SWF) and also the more advanced analytical wall function (AWF), which does not rely on a prescribed near-wall velocity profile, have been tested. Only high-Reynolds number versions of the second-moment closures have been employed, with the two types of wall function mentioned above.For the square cavity, where the flow is turbulent in only the boundary layer regions, use of a low-Reynolds-number k–ε model resulted in the prediction of entirely laminar flow, while of the two high-Re versions, the one with the analytical wall function produced the more satisfactory predictions. Introduction of second-moment closures improved the predictions of the turbulence field.For the tall cavities, the vertical and the inclined cavity at 60° to the horizontal, with the upper side heated, produce similar mean fields, and most of the models tested perform reasonably well over much of the domain. The k–ε model with all the near-wall treatments tested produces reasonable predictions of the local Nusselt number variation with levels which are slightly lower than the experimental data. The stress transport schemes tend to over-predict the mixing near the end walls, although this can be improved by adopting a more complex model for the turbulent heat fluxes. This more elaborate thermal model has little effect on the predicted local Nusselt number, which is already in close agreement with the measurements. There are subtle differences in the turbulence fields between the 60° and 90° cases, some of which are captured by the stress transport models. When the cavity is further inclined to an angle of 5°, still under stable heating arrangements, the turbulence levels are quite low and, as a result, most of the models predict fairly similar mean flow profiles, in reasonable agreement with available LES data.