Generalized Gradient Diffusion Hypothesis Research Articles

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

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

Extended Hypothesis for Reynolds Stress Tensor and Turbulent Heat Flux Modeling Within a Novel K‐ε Model for Prediction of Crystal Growth from the Melt

AbstractThe major technique of manufacturing silicon monocrystals is Czochralski (Cz) method. To obtain high‐quality crystals, it is necessary to control the melt flow that is turbulent. The most cost‐effective approach for modeling turbulence nowadays is RANS (Reynolds Averaged Navier‐Stokes) that allows for providing steady axisymmetric computations using reasonable grids. However, conventional eddy‐viscosity RANS models usually fail to predict spatial details of turbulence characteristics of the melt flow. To overcome the limitations of the eddy‐viscosity approach the Stress Tensor Reconstruction (STR) approach is developed that is aimed at the reproduction of anisotropy of the Reynolds stresses. The Generalized Gradient Diffusion Hypothesis (GGDH) in combination with the STR approach successfully takes into account the anisotropy of turbulent fluxes in a melt flow. Relying upon the STR/GGDH approach a novel RANS STR k‐ε model is developed. The model is validated using the ILES (Implicit Large Eddy Simulation) data of turbulent melt convection during Cz Si crystal growth, as well, as the experimental data of oxygen concentration in 100 and 200 mm diameter Cz crystals. The results are compared to the results obtained with the conventional low Reynolds number Chien k‐ε model extended by the buoyancy generation term.

Large Eddy Simulation of passive scalar transport in transitional boundary layer and development toward improving models

In this paper, the experiment of flat plate transition under freestream turbulence with heat transfer conducted by Blair (1983) is reproduced using subgrid dynamic scalar-flux models (Bader and Durbin, 2020; Moin et al., 1991). The advantage of adopting tensorial subgrid diffusivity is quantified. A set of Reynolds Averaged scalar-flux models is also evaluated using the LES data. Based upon the observations, an improved Higher Order Generalized Gradient Diffusion Hypothesis (HOGGDH) model is proposed with a modified, spatially varying model coefficient. The model coefficient is made a function of the turbulent Reynolds number RT, which enables it to detect the near wall region, thereby increasing the accuracy down to the wall. The mean temperature predictions and the scalar fluxes in the wall-normal as well as the streamwise directions by the improved version of the HOGGDH model are shown to be accurate compared to the HOGGDH model with a constant coefficient and other considered models.

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.

Improvements of turbulence model for analysis of hydrogen stratification erosion by turbulent buoyant jet

Light gas stratification erosion by steam turbulent buoyant jet is the typical phenomenon about containment thermal hydraulics and hydrogen safety. Overprediction of stratification erosion rate has been observed in computational fluid dynamics (CFD) analysis by using current turbulence model based on simple gradient diffusion hypothesis (SGDH). To solve this problem, the effects of buoyancy term, coefficient and turbulent Schmidt number are discussed and a modified k-ε model with buoyancy term based on generalized gradient diffusion hypothesis (GGDH) which contains vertical and horizontal density gradient is developed. In modified k-ε model, besides the value of coefficient, the buoyancy term based on SGDH is modified into GGDH. The strain tensor is modified into rotating tensor to suppress the overestimated local turbulent kinetic energy generated in flow stagnation zone. Comparison between analysis results and experimental data shows that improvement effect of modified model on prediction of light gas stratification erosion rate is significant.

Asymmetric reverse transition phenomenon in internal turbulent channel flows due to temperature gradients

Laminarization of a turbulent flow due to wall heating has been known for more than 50 years, to the point that it is sometimes used as means of reducing friction. However this phenomenon has been mainly studied for cylindrical pipes and with imposed heat flux but not for channel flows and with imposed temperature boundary conditions, especially with asymmetric ones (that is to say in presence of a transverse thermal gradient). Based on the recent success of some Reynolds-averaged Navier-Stokes (RANS) models to correctly describe the influence of a strong transverse temperature gradient on turbulent Poiseuille flows, when compared to similar direct numerical simulations (DNS) or large eddy simulations (LES) results, these approaches are used here to investigate reverse transition. Since the choice of the turbulence model has a non-negligible influence on the results, however, it is necessary to use different models to get an indication of the uncertainty associated with them. The proposed methodology is based on the use of RANS closures that do not involve any wall functions due to the strong gradient in the wall layer that has to be modeled. Thus, two first-moment closures and a second-moment closure are considered: the k−ω−SST and the k−ε−v2¯/k, and the EB-RSM. The latter two rely on an elliptic blending. The turbulent heat flux is modeled with a simple gradient diffusion hypothesis (SGDH) and a generalized gradient diffusion hypothesis (GGDH) for the first-moment and second-moment closures respectively. In summary, more than 800 calculations are performed for the above three models in order to analyze the reverse transition, and to open room for debate on the possibility for such approaches to correctly reproduce the experimentally observed behavior.

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.

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.

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.

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.

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.

URANS computations with buoyancy corrected turbulence models for turbulent thermal plume

The aim of the present paper is to evaluate the capabilities of the unsteady Reynolds-averaged Navier–Stokes (URANS) method to simulate a turbulent plume. Turbulent plume is mainly driven by thermal buoyancy and is characterized by unsteadiness, energy nonequilibrium, counter-gradient diffusion and lack of universal scaling. All these phenomena are believed to be associated with distinct large coherent eddy structures, which are hardly captured by steady RANS equations. In URANS approach, the large scale structures are fully resolved, whereas the stochastic small scales are modeled. Therefore large coherent eddy structure can be captured well by the URANS computations. The present simulation is carried out using an open source code called buoyantPimpleFOAM which is based on OpenFOAM platform. For modeling the effect of turbulence, the buoyancy-corrected standard k–ε model and RNG k–ε model are employed. Since buoyancy plays a significant role in production and destruction of turbulence in a plume, additional turbulence production term due to buoyancy effects is introduced on the basis of simple gradient diffusion hypothesis (SGDH) and generalized gradient diffusion hypothesis (GGDH). A comparison of computational results with the experimental data reported in the literature has been presented in detail. We are able to capture the well-known Gaussian nature of the plume and empirical plume spread coefficients.

Computational Models for Turbulent Thermal Plumes: Recent Advances and Challenges

Thermal plumes have been the subject of research due to their technological and environmental importance in many physical processes, such as spread of smoke and toxic gases from fires, and release of gases from volcanic eruptions and industrial stacks. In this paper, history and current trends in the computation of turbulent buoyant plume have been reviewed. First an introduction to the turbulent buoyant plume is presented, which includes its importance, the flow physics, and heat transfer characteristics. Subsequently, a brief review of the experimental works reported in the literature is presented, followed by a review of the computational methods. Several subgrid-scale models used in large eddy simulation performed in the literature to simulate buoyant plume are discussed. Further, the boundary conditions, computational schemes, and computational grid sizes are discussed. The efficacy of various Reynolds-averaged Navier–Stokes (RANS)-based modeling techniques to simulate turbulent buoyant plumes has been reviewed. It has been concluded that the dynamic subgrid-scale models perform fairly well in predicting the statistics of turbulent buoyant plume. However, no subgrid-scale model has been able to capture the inverse energy cascade, called backscatter, which is an important phenomenon in the evolution of a plume. In case of RANS modeling of plumes, models based on the generalized gradient diffusion hypothesis capture the flow accurately as compared with the models based on the simple gradient diffusion hypothesis.

Development of an explicit algebraic turbulence model for buoyant flows – Part 1: DNS analysis

An analysis of DNS databases of vertical plane channel flow for forced, mixed and natural convection is proposed. This analysis aims to assess the main features needed to develop an algebraic model for buoyant flows. First, the weak equilibrium assumption, at the root of algebraic models, is investigated. This hypothesis is shown to fail near the velocity maximum and close to the walls but remains valid otherwise, whatever the convection regime. The models for the redistribution term and the pressure scrambling term are then analyzed on the same configurations. A linear form of the Speziale et al. (1991) model is retained for the redistribution term. No model for the pressure scrambling term is fully satisfactory; nevertheless some models are recommended. The buoyant contributions to the pressure term are investigated. Finally, the generalized gradient diffusion hypothesis, which could be used to model the turbulent heat fluxes in order to avoid the coupling with the Reynolds stresses, is shown to be inaccurate.

Assessment of Buoyancy-Corrected Turbulence Models for Thermal Plumes

A computational investigation of thermal plume is important because such flows are encountered in various industrial applications. The investigation is also important because thermal plume can be considered as a test case for modeling of fire which can help designers and safety engineers to develop preventive measures and fire safety systems. The present study primarily focuses on the investigation of thermal buoyant plume in the self-similar region. In the present study, an assessment of three buoyancy-corrected turbulence models, namely k-ω, the standard k-ε model and the RNG k-ε model, has been conducted for a thermal buoyant plume. Modifications to the turbulence models have been made to account for the effect of buoyancy on the production and dissipation of turbulent kinetic energy and these modifications are based on the simple gradient-diffusion hypothesis and generalized gradient-diffusion hypothesis. The model based on the simple gradient diffusion hypothesis is shown to under-predict contribution to the generation of turbulence kinetic energy due to buoyancy. A comparison with the experimental measurements reported in the literature shows that the generalized gradient-diffusion hypothesis along with both turbulence models correctly predicts the mean flow field, temperature field and spread rates. The results of the present simulations using the RNG k-ε model with the generalized gradient-diffusion hypothesis are shown to be in good agreement with the corresponding experimental results reported in the literature for thermal plumes.

Turbulence modeling of natural convection in enclosures: A review

In this paper a review of recent developments of turbulence models for natural convection in enclosures is presented. The emphasis is placed on the effect of the treatments of Reynolds stress and turbulent heat flux on the stability and accuracy of the solution for natural convection in enclosures. The turbulence models considered in the preset study are the two-layer k − ɛ model, the shear stress transport (SST) model, the elliptic-relaxation (V2-f) model and the elliptic-blending second-moment closure (EBM). Three different treatments of the turbulent heat flux are the generalized gradient diffusion hypothesis (GGDH), the algebraic flux model (AFM) and the differential flux model (DFM). The mathematical formulation of the above turbulence models and their solution method are presented. Evaluation of turbulence models are performed for turbulent natural convection in a 1:5 rectangular cavity ( Ra= 4.3×1010) and in a square cavity with conducting top and bottom walls ( Ra= 9 1.58×109) and the Rayleigh-Benard convection ( Ra = 2 × 106 ∼ Ra = 109). The relative performances of turbulence models are examined and their successes and shortcomings are addressed.

Forced turbulent heat convection in a square duct with non-uniform wall temperature

In the present work, the numerical simulation to calculate the problem of the turbulent convection with non-uniform wall temperature in a square cross-section duct was adopted. To solve this problem some assumptions for the flow, such as: the condition of fully developed turbulence and incompressible flow have been assumed. The methodology of the dimensionless energy equation was used to calculate the fluid temperature field in the square cross-section in function of the non-uniform wall temperatures prescribed. Numerical simulations were done using two different turbulent models to resolve the momentum equations and two more models to resolve the energy equation. The models of turbulence k–ε Nonlinear Eddy Viscosity Model (NLEVM) and the Reynolds Stress Model (RSM) were used to determine the turbulent intensities as well as the profiles of axial and secondary mean velocities. The turbulence model RSM was simulated using a commercial software. The thermal field was determined from other two models: Simple Eddy Diffusivity (SED), based in the hypothesis of the constant turbulent Prandtl number; and Generalized Gradient Diffusion Hypothesis (GGDH). In this last model, as the turbulent heat transfer depends on the shear tensions, the anisotropy is considered. These two last equation models of the energy equation of the fluid have been implemented in FORTRAN, a code of programming. The performances of the models were evaluated by validating them based in the experimental and numerical results published in the literature. Two important parameters of great interest in engineering are presented: the friction factor and the Nusselt number. The results of this investigation allow the evaluation of the behavior of the turbulent flow and convective heat fluxes for different square cross-sectional sections throughout the direction of the main flow, which is mainly influenced by the temperature distribution in the wall.