Elliptic Relaxation Model Research Articles

Formatting Elliptic Model From SST k-ω Closure

A single-equation turbulence model is devised from the shear-stress transport (SST) k-ω closure without an assumption of equal-diffusion coefficients; evolving diffusion/destruction terms are habitually avoided owing to their numerical stiffness and complexity. Rearranging retained terms yields a production term naturally with an elliptic identity in the R-equation and hence, the production term confronts an elliptic blending. A near-wall damping function is introduced with the elliptic-relaxation model in order to moderate the numerical burden on viscous length-scale coefficient. The overall approach represents a combination of wall-blocking (non-local) and wall-viscous (low-Reynolds number) effects encountered in anisotropic turbulence. A few well-documented test cases are utilised to validate the model competency; a good correlation is obtained with experimental and DNS (direct numerical simulation) data. Comparisons dictate that the present elliptic model remains competitive with widely used SST k-ω and Spalart-Allmaras models.

Improved Menter’s one-equation turbulence model

Compatible modifications to original Menter model (OMM) are incorporated using an elliptic relaxation approach to accurately accounting non-local characteristics accompanied by near-wall turbulence. Coefficients/functions of improved Menter model (IMM) are parameterized with the elliptic relaxation function to imitate the combined effects of near-wall turbulence and non-equilibrium. The characteristic length scale involved in the elliptic relaxation scheme is constructed in terms of viscous and turbulent length scales in conjunction with a non-singular strain-rate invariant. Non-local effects are consequently reinforced by the mean flow and turbulent variables. A non-linear eddy-viscosity damping function is invoked to relax the viscous length-scale coefficient pertaining to the elliptic relaxation model. Comparisons indicate that the IMM substantially improves the accuracy of flow simulations compared to the OMM and widely used Spalart-Allmaras (SA) model. A good consensus is obtained between IMM and DNS (direct numerical simulation)/measured data.

Elliptic Blending with One-Equation Model

A wall-distance-free modification to Wray-Agarwal (WA) one-equation turbulence model is convoked using an elliptic relaxation approach to accurately accounting for non-local characteristics of near-wall turbulence. Model coefficients/functions are parameterized with the elliptic relaxation function to preserve the combined effects of near-wall turbulence and nonequilibrium. The characteristic length scale associated with the elliptic relaxation equation is formulated in terms of viscous and turbulent length scales in conjunction with the invariant of strain-rate tensor. Consequently, non-local effects are explicitly influenced by the mean flow and turbulent variables. A near-wall damping function is introduced to relax the viscous length-scale coefficient adhering to the elliptic relaxation model. Comparisons indicate that the new model improves the accuracy of flow simulations compared to the widely used Spalart-Allmar as model and remains competitive with the SST model.A good correlation is obtained between the current model and DNS/experimental data.

Near-wall modelling in Eulerian–Eulerian simulations

The near-wall region in turbulent Eulerian–Eulerian (E–E) simulations has hitherto received little to no attention. A standard approach to modelling this region is through the employment of single-phase wall-functions in the fluid-phase and it is unclear whether such an approach is capable of capturing the turbulent fluid-particle interaction in the near-wall region. In order to both investigate and alleviate E–E models reliance on single-phase wall-functions we propose an E–E elliptic relaxation model to account for the near-wall non-homogeneity which arises in wall-bounded flows. The proposed model is derived within an E–E framework and enables the full resolution of the boundary layer and arbitrary wall sensitivity. The model is then compared against the conventional kf−εf turbulence model with standard single-phase wall-functions. Additionally, the modelling is compared against a low-Re number turbulence model. The elliptic relaxation model is implemented within the open-source CFD toolbox OpenFOAM, applied to a vertical downward-facing channel and validated against the benchmark experimental data of Kulick et al. [1]. Model results show marked improvements over the conventional turbulence model across mean flow and turbulence statistics predictions. The use of conventional single-phase wall functions were shown to negatively impede on the prediction of the velocity covariance coupling term and as a result the particle fluctuation energy. Moreover, this also lead to an underestimation of the near-wall volume fraction accumulation. Finally, the elliptic relaxation model, E-E model and accompanying validation cases are made open-source.

Numerical simulation and experiments on turbulent air flow around the semi-circular profile at zero angle of attack and moderate Reynolds number

Two-dimensional flow around a semi-circular profile at the zero angle of attack and at Re = 50000 on the self-oscillatory period is extensively studied by the URANS method involving the standard semi-empirical SST turbulence models, the SST turbulence model with the correction for streamline curvature modified within the Rodi-Leschziner-Isaev and Smirnov-Menter approaches, as well as involving Hanjalic's four-parameter eddy viscosity elliptic relaxation model and its analog - eddy viscosity elliptic blending model proposed in the present work. This has been done with the use of different-structure grids (multiblock with structured overlapping and unstructured composite). Different numerical approximation methods realized in six codes (VP2/3, SigmaFlow, Fluent, CFX, OpenFOAM, and StarCCM+) are used. An underestimation (up to 30%) of time-averaged integral aerodynamic loads is revealed by means of the standard near-wall SST model. This is explained by the high vortex viscosity production in the profile wake. Wind tunnel tests show that the location of cutoff washers on the semi-circular profile provides a quasi-two-dimensional flow around it and allows applying measurement data to verify two-dimensional turbulent flow. The best agreement of experimental results and numerical predictions when comparing the Strouhal number and time-averaged surface pressure coefficient distributions is achieved using both the modified SST model with the correction for streamline curvature and the modified eddy viscosity elliptic blending model. When the SST model with the correction for streamline curvature, modified within the Rodi-Leschziner-Isaev, Smirnov-Menter and Durbin approaches, is used, all the above codes yield close predictions of a vertical aerodynamic load on the oscillation period.

Elliptic relaxation model for stably stratified turbulence

The article presents a new elliptic relaxation procedure for Second Moment Closure (SMC) turbulence modeling of wall-bounded, stably stratified flows. Specifically, it directly introduces buoyancy effects in the elliptic relaxation equation. The new formulation results in an additional length scale that produces a coupled set of Helmholtz equations. The study then extends the new technique to Reynolds scalar fluxes. Furthermore, it examines the budgets of the different terms of their transport equations near the wall and systematically derives the relevant boundary conditions. Comparison of model predictions with Direct Numerical Simulation (DNS) data for stably stratified plane channel flows available in the literature rounds off this paper.

Analysis of Turbulent Natural Convection by an Elliptic Relaxation Model in Tall Vertical Cavities with Linear Temperatures on Sidewalls

Turbulent natural convection of air is studied, by the elliptic-relaxation model v^2-f, in a tall vertical cavity whose hot and cold walls are maintained at linear temperatures of slopes γ_1 and γ_2, respectively. The average temperatures of the active walls are located at mid-height of the cavity. Four situations are analyzed, corresponding to γ_1=γ_2=γ (case I), γ_1=-γ_2=γ (case II), γ_1=0 and γ_2=γ (case III), γ_1=γ and γ_2=0 (case IV). These boundary conditions may be more representative or used to control heat transfer for certain systems. The effects of the slope (-1≤γ≤1), the aspect ratio of the cavity (10≤A≤80) and the average Rayleigh number (5×〖10〗^4≤〖Ra〗_m≤〖10〗^6 ) on the streamlines, isotherms, contours of the turbulent kinetic energy, heatlines, local and average Nusselt numbers are investigated. It is shown that the local and average heat transfers of cases III and IV can be deducted from those of cases I and II. The obtained dynamic and thermal fields as well as local and average heat transfers of the studied cases are quite different of those of the classical case corresponding to γ=0. A simplified procedure for calculating the average Nusselt number is also developed for each case.

On eddy viscosity transport models with elliptic relaxation

ABSTRACTAn eddy viscosity transport (EVT) model is formulated based on the transformation of the standard k–ϵ model without the assumption of equal diffusion coefficients. These terms that are usually neglected, due to their complexities and numerical difficulties, are regrouped so that an elliptic production term naturally emerges. This term forms the basis of an elliptic relaxation model that avoids the need for wall-damping functions. All of the EVT model's constants follow from the transformation, maintaining the closest relationship to its parent k–ϵ model. To assess the ability of the elliptic relaxation equation to account for the non-local wall effects, another version of the EVT model that neglects the diffusion terms and retains only the Laplace operator term, i.e., , is proposed and compared to Menter one-equation model, which utilises wall-damping functions. The performance of this version of the EVT model is also compared to the fully transformed EVT model in order to assess the importance of the neglected diffusion terms. All three models are tested in detail over a wide class of internal and external turbulent flows.

Computation of Turbulent Natural Convection in a Rectangular Cavity with the Lattice Boltzmann Method

A numerical study of a turbulent natural convection in a rectangular cavity with the lattice Boltzmann method (LBM) is presented. The primary emphasis of the present study is placed on investigation of accuracy and numerical stability of the LBM for the turbulent natural-convection flow. A HYBRID method in which the thermal equation is solved by the conventional Reynolds-averaged Navier-Stokes equation (RANS) method while the conservation of mass and momentum equations are resolved by the LBM is employed in the present study. The elliptic-relaxation model is employed for the turbulence model and the turbulent heat fluxes are treated by the algebraic flux model. All the governing equations are discretized on a cell-centered, nonuniform grid using the finite-volume method. The convection terms are treated by a second-order central-difference scheme with the deferred correction method to ensure accuracy and stability of solutions. The present LBM is applied to the prediction of a turbulent natural convection in a rectangular cavity and the computed results are compared with the experimental data commonly used for the validation of turbulence models and those by the conventional finite-volume method. It is shown that the LBM with the present HYBRID thermal model predicts mean velocity components and turbulent quantities which are as good as those by the conventional finite-volume method. It is also found that the accuracy and stability of the solution is significantly affected by the treatment of the convection term, especially near the wall.

유한체적법을 기초한 레티스 볼쯔만 방법을 사용하여 직사각형 공동에서의 난류 자연대류 해석

A numerical study of a turbulent natural convection in an enclosure with the lattice Boltzmann method (LBM) is presented. The primary emphasis of the present study is placed on investigation of accuracy and numerical stability of the LBM for the turbulent natural convection flow. A HYBRID method in which the thermal equation is solved by the conventional Reynolds averaged Navier-Stokes equation method while the conservation of mass and momentum equations are resolved by the LBM is employed in the present study. The elliptic-relaxation model is employed for the turbulence model and the turbulent heat fluxes are treated by the algebraic flux model. All the governing equations are discretized on a cell-centered, non-uniform grid using the finite-volume method. The convection terms are treated by a second-order central-difference scheme with the deferred correction way to ensure accuracy and stability of solutions. The present LBM is applied to the prediction of a turbulent natural convection in a rectangular cavity and the computed results are compared with the experimental data commonly used for the validation of turbulence models and those by the conventional finite-volume method. It is shown that the LBM with the present HYBRID thermal model predicts the mean velocity components and turbulent quantities which are as good as those by the conventional finite-volume method. It is also found that the accuracy and stability of the solution is significantly affected by the treatment of the convection term, especially near the wall.

A hybrid numerical scheme for a new formulation of delayed detached-eddy simulation (DDES) based on elliptic relaxation

A new formulation of Delayed Detached-Eddy Simulation (DDES) based on the φ − f elliptic relaxation model has been derived and calibrated. A suitable correction function Ψ and a model constant CDDES have been formulated and validated using decaying isotropic turbulence (DIT). The model has been applied to a 2D wall-mounted hump at high-Reynolds number where it has shown promising improvements over the common SST-DDES model. The improved RANS modelling of the near-wall physics results in an improved prediction of the levels of turbulent Reynolds stresses which leads to a better approximation of the recirculation zone and flow physics. A hybrid numerical scheme has also been developed for use with DDES; this blends an upwind based scheme and a centred scheme based explicitly on the blending function of the DDES.

Reynolds-averaged modeling of polymer drag reduction in turbulent flows

A novel eddy viscosity model for predicting friction drag reduction induced by polymers in turbulent wall-bounded flows is presented. The approach is based on the elliptic relaxation model modified to account for the modified Reynolds-stress equilibrium established by the presence of elastic polymer chains in the fluid. The increased wall damping of the turbulent fluctuations is obtained by modifying the pressure–strain redistribution term. Polymer solutions are represented using the Finite Extensibility Non-linear elastic FENE-P dumbbell model; only one transport equation for the elongation of the polymer chains is considered. The model reproduces the level of drag reduction observed over a wide range of rheological parameters. In addition, both the mean velocity and the turbulent fluctuations are predicted with good accuracy. The approach is computationally attractive because of its limited increase in computational cost in comparison with its Newtonian counterpart.

On Applicability of Second Moment Closure Based on Durbin's Elliptic Relaxation Model to Free-surface Turbulence

The elliptic relaxation model of Durbin [J. Fluid Mech., 249 (1993) 465], which has had remarkable success in predicting wide variety of flows, is applied to prediction of free-surface turbulence for the first time. A set of appropriate free-surface boundary conditions for the elliptic relaxation equations is derived by taking account of the balance of dominant terms in the Reynolds-stress transport equations in the vicinity of a free surface. Test calculations for a fully-developed turbulent open-channel flow is conducted. The flow consists of two kinds of impermeable boundaries, a solid wall and a free surface, and hence is quite suitable to the present benchmark flow. It is demonstrated that the proposed boundary conditions do enable elliptic relaxation models to reproduce the correct surface-induced stress anisotropy and that the model properly describes the differences in feature between wall-bounded flows and free-surface flows by its main quality in reproducing the non-local blocking effect, kind of an elementary process of boundary-turbulence interaction. It is also found that the turbulent-transport term plays a central role in characterizing free-surface turbulence, which is therefore quite suitable for critical evaluation of the model performance.

Evaluation of Turbulence Models for Thermal Striping in a Triple Jet

A computational study for an evaluation of the current turbulence models for the prediction of a thermal striping in a triple jet is performed. The tested turbulence models are the two-layer model, the shear stress transport model, and the elliptic relaxation model. These three turbulence models are applied to the prediction of a thermal striping in a triple jet in which detailed experimental data are available. The predicted time-averaged and root-mean-square values of the temperature are compared with the experimental data, and the capability of predicting the oscillatory behavior of the ensemble-averaged temperature is investigated. From these works, it is shown that only the elliptic relaxation model is capable of predicting the oscillatory behavior of the ensemble-averaged temperature. It is also shown that the elliptic relaxation model predicts best the time-averaged temperature and the root mean square of the temperature fluctuation. However, this model predicts a slower mixing at the far downstream of the jet.

The Role of Turbulence Models for Predicting a Thermal Stratification

A numerical study of the evaluation of turbulence models for predicting the thermal stratification phenomenon is presented. The tested models are the elliptic blending turbulence model (EBM), the two-layer model, the shear stress transport model (SST), and the elliptic relaxation model (V2-f). These four turbulence models are applied to the prediction of a thermal stratification in an upper plenum of a liquid metal reactor experimented at the Japan Nuclear Cooperation (JNC). The EBM and V2-f models predict properly the steep gradient of the temperature at the interface of the cold and hot regions that is observed in the experimental data, and the EBM and V2-f models have the capability of predicting the temporal oscillation of the temperature. The two-layer and SST models predict the diffusive temperature gradient at the interface of a thermal stratification and fail to predict a temporal oscillation of the temperature. In general, the EBM predicts best the thermal stratification phenomenon in the upper plenum of the liquid metal reactor.

Inhomogeneity and anisotropy effects on the redistribution term in Reynolds-averaged Navier–Stokes modelling

A channel flow DNS database at Reτ = 590 is used to assess the validity of modelling the redistribution term in the Reynolds stress transport equations by elliptic relaxation. The model assumptions are found to be globally consistent with the data. However, the correlation function between the fluctuating velocity and the Laplacian of the pressure gradient, which enters the integral equation of the redistribution term, is shown to be anisotropic. It is elongated in the streamwise direction and strongly asymmetric in the direction normal to the wall, in contrast to the isotropic, exponential model representation used in the original elliptic relaxation model. This discrepancy is the main cause of the slight amplification of the energy redistribution in the log layer as predicted by the elliptic relaxation equation. New formulations of the model are proposed in order to correct this spurious behaviour, by accounting for the rapid variations of the length scale and the asymmetrical shape of the correlation function. These formulations do not rely on the use of so-called ‘wall echo’ correction terms to damp the redistribution. The belief that the damping is due to the wall echo effect is called into question through the present DNS analysis.

Near-wall turbulence closure modeling without “damping functions”

An elliptic relaxation model is proposed for the strongly inhomogeneous region near the wall in wall-bounded turbulent shear flow. This model enables the correct kinematic boundary condition to be imposed on the normal component of turbulent intensity. Hence, wall blocking is represented. Means for enforcing the correct boundary conditions on the other components of intensity and on the k — ɛ equations are discussed. The present model agrees quite well with direct numerical simulation (DNS) data. The virtue of the present approach is that arbitrary “damping functions” are not required.