F. Ilinca

Positivity Preservation and Adaptive Solution for the k- Model of Turbulence

A simple change of dependent variables that guarantees positivity of turbulence variables in numerical simulation codes is presented. The approach consists of solving for the natural logarithm of the turbulence variables, which are known to be strictly positive. The approach is valid for any numerical scheme, be it finite difference, a finite volume, or a finite element method. The work focuses on the advantages of the proposed change of dependent variables within the framework of an adaptive finite element method. The turbulence equations in logarithmic variables are presented for the standard κ-e model. Error estimation and mesh adaptation procedures are described. The formulation is validated on a shear layer case for which an analytical solution is available. This provides a framework for rigorous comparison of the proposed approach with the standard solution technique, which makes use of k and e as dependent variables. The approach is then applied to solve turbulent flow over a NACA0012 airfoil for which experimental measurements are available. The proposed procedure results in a robust adaptive algorithm. Improved predictions of turbulence variables are obtained using the proposed formulation

An adaptive finite element method for a two-equation turbulence model in wall-bounded flows

SUMMARY This paper presents an adaptive finite element method for solving incompressible turbulent flows using a k‐ model of turbulence. Solutions are obtained in primitive variables using a highly accurate quadratic finite element on unstructured grids. A projection error estimator is presented that takes into account the relative importance of the errors in velocity, pressure and turbulence variables. The efficiency and convergence rate of the methodology are evaluated by solving problems with known analytical solutions. The method is then applied to turbulent flow over a backward-facing step and predictions are compared with experimental measurements.

Adaptive Remeshing for the k-Epsilon Model of Turbulence

An adaptive ® nite element method for solving incompressible turbulentows using the k± ≤model of turbulence is presented. Solutions are obtained in primitive variables using a highly accurate quadratic ® nite element on unstructured grids. Turbulence is modeled using the k± ≤ model of turbulence. Two error estimators are presented that take into account in a rigorous way the relative importance of the errors in velocity, pressure, turbulence variables, and eddy viscosity. The ef® ciency and convergence rate of the methodology are evaluated by solving problemswith known analytical solutions. Themethod isthen applied to turbulent free shearowsand predictions are compared to measurements. Nomenclature Cl , C1, C2 = k±≤ model constants e = error f = body force h = element size K = element in the mesh k = turbulent kinetic energy n = outward unit vector P(u) = production of turbulence p = pressure u = velocity vector

On stabilized finite element formulations for incompressible advective–diffusive transport and fluid flow problems

Abstract A new approach is presented to obtain stabilized finite element formulations such as streamline-upwind/Petrov–Galerkin (SUPG) and Galerkin-least-squares (GLS). The procedure consists in modifying the equations to be solved and then obtaining the variational equations by the standard Galerkin method. The new formulation generates additional terms involving boundary integrals to standard stabilization techniques. These terms compensate for the lack of consistency of the traditional SUPG and GLS methods for which stabilization terms are added only on the element interiors, while jumps of the residual across element faces are neglected. A physical interpretation is provided of how the modified equations are obtained. It is shown how stabilized formulations such as streamline-upwind (SU) and SUPG are recovered as special cases. Stabilization terms defined on the element interiors are always accompanied by additional boundary integrals. The presence of the boundary integrals is shown to improve the numerical prediction for various viscous and nearly inviscid flows.

Adaptive finite element method for thermal flow problems

This paper presents an adaptive finite element method based on remeshing to solve incompressible viscous flow problems including heat transfer effects by forced or free convection. Conjugate heat transfer problems are also considered. Solutions are obtained in primitive variables by an Uzawa algorithm using a highly accurate finite element approximation on unstructured grids. Two error estimators are presented and compared on problems with known analytical solutions. The methodology is then applied to a problem of practical interest and predictions are compared with experimental measurements and show very good agreement.

ADAPTIVE COMPUTATIONS OF TURBULENT FORCED CONVECTION

This article presents an adaptive finite element method for solving incompressible turbulent flows with heat transfer. Solutions are obtained in primitive variables using a highly accurate quadratic finite element method on unstructured grids. Turbulence modeling is achieved using the k-e model. A projection error estimator is presented that incorporates errors from various sources: velocity, temperature, pressure, and turbulence variables, including the eddy viscosity. The efficiency and reliability of the methodology are studied by solving a problem with a known analytical solution. The method is then applied to heat transfer over a backward facing step and to a heated jet. In all cases, predictions are compared to experiments.

Adaptive Finite Element Solution of Compressible Turbulent Flows

´An adaptive e nite element method for solving compressible turbulent e ows up to transonic regime is presented. Pressure-based methods previously developed for laminar compressible e ows and turbulent incompressible e ows are combined to solve compressible turbulent e ows. Turbulence is incorporated via the k‐≤ model. The algorithm uses the logarithms of k and≤ as computational variables to preserve positivity. Solutionsare obtained in primitive variables using quadratic e nite elements on unstructured grids. The error is estimated by a local projection method. The solution algorithm and error estimation are validated on problems with known analytical solutions. The method is then applied to compressible subsonic and transonic e ows and predictions are compared with experimental measurements.

Finite element solution of three‐dimensional turbulent flows applied to mold‐filling problems

This paper presents a finite element solution algorithm for three-dimensional isothermal turbulent flows for mold-filling applications. The problems of interest present unusual challenges for both the physical modelling and the solution algorithm. High-Reynolds number transient turbulent flows with free surfaces have to be computed on complex three-dimensional geometries. In this work, a segregated algorithm is used to solve the Navier–Stokes, turbulence and front-tracking equations. The streamline–upwind/Petrov–Galerkin method is used to obtain stable solutions to convection-dominated problems. Turbulence is modelled using either a one-equation turbulence model or the κ–e two-equation model with wall functions. Turbulence equations are solved for the natural logarithm of the turbulence variables. The change of dependent variables allows for a robust solution algorithm and good predictions even on coarse meshes. This is very important in the case of large three-dimensional applications for which highly refined meshes result in untreatable large numbers of elements. The position of the flow front in the mold cavity is computed using a level set approach. Finally, equations are integrated in time using an implicit Euler scheme. The methodology presents the robustness and cost effectiveness needed to tackle complex industrial applications. Copyright

Adaptive remeshing for convective heat transfer with variable fluid properties

This article presents an adaptive finite element method based on remeshing to solve incompressible viscous flow problems for which fluid properties present a strong temperature dependence. Solutions are obtained in primitive variables using a highly accurate finite element approximation on unstructured grids. Two general purpose error estimators are presented, which take into account the temperature dependence of fluid properties. The methodology is applied to a problem of practical interest: the thermal convection of corn syrup in an enclosure with localized heating. Predictions are in good agreement with experimental measurements. The method leads to improved accuracy and reliability of finite element predictions. Nomenclature cp = specific heat

A universal formulation of two-equation models for adaptive computation of turbulent flows

Abstract This paper describes the application of a recently developed universal adaptive finite element algorithm to the simulation of several turbulent flows. The objective of the present work is to show how the controlled accuracy of adaptive methods provides the means to perform careful quantitative comparisons of two-equation models. The formulation uses the logarithmic form of turbulence variables, which naturally leads to a simple algorithm applicable to all two-equation turbulence models. The new methodology is free of ad-hoc stability enhancement measures such as clipping and limiters which may often differ from one model to the other. Such techniques limit the predictive capability of a turbulence model and cloud the issues of a comparison study. The present procedure results in one adaptive solver applicable to all two-equation models. The approach is demonstrated by comparing three popular turbulence models on a few non-trivial compressible and incompressible flows. We have chosen the following models: the standard k−ϵ model, the k−τ model of Speziale and the k−ω model of Wilcox. Results show that accurate solutions can be obtained for all models, and that systematic comparison of turbulence models can be made.