An adaptive finite element method for turbulent heat transfer

This paper presents an adaptive finite element method for solving incompressible turbulent flows, including heat transfer effects, using a k e model of turbulence. Solutions are obtained in primitive variable using a highly accurate quadratic finite element on unstructured grids. Turbulence is modeled using the k e model. A projection error estimator is presented that takes into account the relative importance of the errors in velocity, temperature, pressure and turbulence variables, including the eddy viscosity. The efficiency and convergence rate of the methodology are evaluated by solving problems with known analytical solutions. The method is then applied to heated jets, heat transfer in a channel and finally, to heat transfer over a backward facing step. In all cases predictions are compared to experiments. Nomenclature c r i' 1' 2' e E f h k K n P P Pe Pr u V, W, S, T P E T k 6 model constants error roughness parameter body force element size turbulent kinetic energy Karman constant outward unit vector pressure production term Peclet number Prandtl number velocity vector friction velocity test function temperature density turbulent dissipation stress tensor V Vn A. a 8 0 av h T strain rate tensor wall shear stress gradient operator divergence viscosity thermal conductivity boundary element size for new mesh Subscripts initial value average finite element solution turbulent Copyright © 1996 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. INTRODUCTION Many aerodynamic and industrial flows involve the solution of the energy equation coupled to momentum and continuity to determine either their thermodynamic state or their heat transfer characteristics. Adaptive finite element methods provide a powerful approach for tackling such complex computational fluid dynamics problems. They can provide accurate solutions at a reasonable cost by automatically clustering elements around flow features of interest such as 1 American Institute of Aeronautics and Astronautics shear and boundary layers and reattachment points. The adaptive process is also cost effective in the sense that the best numerical solution is obtained at the least computational cost. Moreover, such approaches provide flexibility in modeling and algorithm development. The ability of the methodology to produce uniformly accurate solutions makes it possible to obtain 'numerically exact solutions' (grid independent) to the equations of motion, so that mathematical models of the physical phenomenon of interest can be evaluated with some confidence. Initial breakthroughs in adaptive computation were achieved in aerodynamics because of the pressing need for accurate computations of shock waves '. However, little work has been done for incompressible flows and even less for turbulent flow problems. Proof of concept computations for laminar incompressible flow were reported in Ref. [2,3]. A paper discussing adaptivity and the k e is that of Ref. [4] where structured grid are adapted by both moving nodes and imbedding a finer grid in the coarse one. This approach has led to solution improvements. However, the authors performed only one pass of adaptation. The degree of solution improvement is thus limited by the structured nature of the mesh and the limited number of refinements that are easily be implemented (only one step of refinement was implemented). Reference [5] presents applications of an adaptive finite element method to turbulent compressible flows with shocks. Turbulence is modeled by a low Reynolds number &-e model. The adaptive remeshing procedure significantly improved the accuracy of the predictions. However, adaptation was driven by an error estimator that is only sensitive to velocity gradients. Large variations in the turbulence kinetic energy, its dissipation and the eddy viscosity were ignored. The methodology proposed here is free of such ethic limitations. The use of unstructured grids provides for very highly localized grid resolution at a reasonable cost. The remeshing procedure also makes it possible to achieve any preset level of accuracy. The method can thus be viewed as a technique for generating 'numerically exact (grid independent) solutions to the differential equations modeling turbulent flow and heat transfer. In Ref. [6-9] the methodology proposed by the authors was quantitatively validated by solving laminar flows with known analytical solution and by computing cases for which experimental measurements were available. The methodology was further extended to convective heat transfer flows with variable fluid properties 10 and to zero-equation and two-equation models of turbulence for free shear flows l . The authors recently demonstrated the applicability of the proposed methodology for the k e and k u> models of turbulence applied to internal isothermal flows. The current work extends the methodology to turbulent flows with heat transfer effects. The methodology is based on adaptive remeshing coupled to a finite element solver for steady-state incompressible turbulent flows for which turbulence is represented by the k-€ model. The proposed error estimation technique and adaptive methodology are applicable to a wider class of problems than is treated here. The approach is valid for aerodynamic flows, such as those treated by Vemaganti, and to internal flows such as those found in combustors. The paper is organized as follows: First we describe the modeling of the problem. The equations of motion and the finite element solver are reviewed. The turbulence model is discussed and details of the nonlinear equation solver and wall boundary conditions are presented. The methodology section describes the error estimator and the adaptive remeshing strategy. The proposed methodology is then validated by solving problems with known analytical solutions to clearly quantify the accuracy improvements due to adaptivity. The method is then applied to turbulent flow in a heated channel and over a heated backward facing step for which experimental data are available. The paper closes with conclusions. MODELING OF THE PROBLEM Reynolds-averaged Navier-Stokes equations The flow regime of interest is modeled by the Reynoldsaveraged Navier-Stokes and energy equations: American Institute of Aeronautics and Astronautics