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Finite-Element Simulation of Buoyancy-Driven Turbulent Flows

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Finite-Element Simulation of Buoyancy-Driven Turbulent Flows Finite-element simulation of buoyancy-driven turbulent flows Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨aten der Georg-August-Universit¨atzu G¨ottingen vorgelegt von Tobias Knopp aus L¨ubeck G¨ottingen2003 D7 Referent: Prof. Dr. G. Lube Korreferent: Prof. Dr. R. Schaback Tag der m¨undlichen Pr¨ufung:4. Juni 2003 2 Contents Preface 7 Epitome . 8 Acknowledgements . 9 I. Turbulence modelling for buoyancy driven flows 11 1. The laminar model 13 1.1. Laminar thermally coupled flow problems . 13 1.2. Boundary conditions for thermally coupled flows . 15 1.3. A model for non-isothermal flow problems . 18 1.4. Modelling turbulent boundary layers using a fully overlapping DDM . 19 2. Fundamentals, modelling and simulation of turbulent flows 23 2.1. Aspects of randomness and statistical description of turbulent flows . 23 2.2. The scales of turbulent flows . 26 2.3. Criteria for appraising approaches in CFD . 27 3. The k/ turbulence model 29 3.1. The Reynolds averaged Navier-Stokes equations . 29 3.2. Turbulent-viscosity and gradient-diffusion hypotheses . 30 3.3. Production and dissipation of turbulent kinetic energy in RANS models . 33 3.4. A two-equation model : The k/ model . 34 4. Large-eddy simulation 39 4.1. Filtering . 39 4.2. Differential filtering . 41 4.3. The space averaged non-isothermal Navier-Stokes equations . 42 4.4. Modelling the residual stress tensor and the residual fluxes . 44 4.5. System of equations for non-isothermal LES . 49 5. Near-wall treatment in turbulence modelling 51 5.1. Fundamentals of turbulent boundary-layer theory . 51 5.2. Boundary-layer equations and singular perturbation methods . 53 5.3. Algebraic turbulence models for non-isothermal boundary layers . 59 5.4. Algebraic turbulence models for natural convection boundary layers . 63 5.5. On the near-wall behaviour of the k/ model . 65 5.6. On LES in the near-wall region . 66 3 6. A computational k/ model using wall functions 67 6.1. A two-domain approach . 67 6.2. The wall function concept as a fully overlapping DDM . 68 6.3. The wall function concept using boundary-layer theory . 70 7. A computational LES model 75 7.1. Wall stress models . 77 7.2. Hybrid RANS/LES approaches . 80 8. Some analytical results for LES with near wall modelling 81 8.1. Some simplifications of the coupled problem . 82 8.2. A separate study of global and local subproblem . 84 8.3. The coupled steady state problem . 94 8.4. Some closing remarks . 106 II. Numerical solution scheme and numerical tests 111 9. Semidiscretisation in time, decoupling and linearisation 113 9.1. Semidiscretisation in time using the discontinuous Galerkin method . 113 9.2. Semidiscretisation, decoupling, and linearisation for the k/ model . 114 9.3. Semidiscretisation, decoupling, and linearisation for the LES model . 117 9.4. Variational formulation of the arising model problems . 120 10.Discretisation in space using stabilised FEM 121 10.1. Finite-element discretisation for ADR-problems . 121 10.2. Stabilisation techniques for ADR-problems . 121 10.3. Discontinuity capturing for ADR-problems . 123 10.4. Finite-element discretisation for Oseen problems . 124 10.5. SUPG- und PSPG-stabilisation for Oseen problems . 124 11.Non-overlapping domain decomposition methods 127 11.1. The Robin-Robin algorithm for advection-diffusion-reaction problems . 128 11.2. Choice of the interface function in the R-R-algorithm for ADR problems . 129 11.3. The Robin-Robin algorithm for Oseen type problems . 130 12.Turbulent channel flow 133 12.1. Fundamentals of isothermal channel flow . 133 12.2. Isothermal channel flow computations using the k/ model . 136 12.3. Quasi a priori testing of the SGS model . 138 13.Turbulent natural convection in an air filled square cavity 151 13.1. Introduction . 151 13.2. Description of the flow configuration . 152 13.3. Testing the wall iteration concept . 154 13.4. A posteriori testing for the k/ model without DDM . 157 13.5. A posteriori testing for the k/ model with DDM . 160 13.6. Appraisal of the k/ model predictions . 166 13.7. First results for the posteriori testing using LES . 167 14.Summary and future prospects 175 III. Appendix 177 A. Mathematical tools for residual stress modelling in LES 179 A.1. Fourier transformation, convolution and distributions . 179 A.2. Closure approximations for LES . 183 B. Some mathematical tools for the Navier-Stokes equations 187 B.1. Functional analytic fundamentals . 187 B.2. Analytical results for some turbulence models . 191 C. Turbulent boundary-layer theory 193 C.1. Natural convection turbulent boundary layers . 193 C.2. Forced convection boundary-layer equations in non-dimensional form . 195 C.3. The universal log law by Prandtl and van Karman . 196 C.4. A non-isothermal wall law for forced convection problems by Neitzke . 197 D. Nomenclature 199 Bibliography 206 Curriculum vitae 219 5 6 Preface Turbulent flows driven or significantly affected by buoyancy occur in a variety of problems including building ventilation, cooling of electrical equipment, and environmental science. The fundamental mathematical model are the non-isothermal Navier-Stokes equations, gov- erning the time-evolution of velocity u~, pressurep ~, and temperature T~. The phenomenon of turbulence reveals that their solutions can become very complex if a critical parameter, e.g., the Reynolds number or the Rayleigh number, becomes large. A proper numerical resolution of the random motion of all scales of u~,p ~, and T~ (called Direct Numerical Simu- lation) is feasible only for a very limited number of flows. Thus the major task in turbulence modelling is to reduce the complexity of the Navier-Stokes equations in a manner which is appropriate to the needs of science and engineering. The goal is to develop models that are computationally simpler than the Navier-Stokes equations but "whose predictions are close to those of the Navier-Stokes equations". In this thesis we pursue two strategies: The first approach is a statistical approach which is based on a statistical averaging procedure for the Navier-Stokes equations. The objective is to obtain a set of equations for the statistical mean values for u~,p ~, and T~, which requires an empirical modelling of the terms involving statistical fluctuations. The second approach is called large-eddy simulation (LES). The idea of LES is to apply a spatial averaging filter to the Navier-Stokes equations in order to extract the large-scale structures of u~,p ~, and T~, and to attenuate their small-scale struc- tures. Then only the random motion of the large scales is resolved and the effects of the small scales on the large scales are modelled. This thesis is involved into a longlasting cooperation with the Institute for Thermodynam- ics and Building Energy Systems at Dresden University of Technology. A major result of this cooperation is our research code ParallelNS, see e.g. [Mue99] and [KLGR02]. ParallelNS is intended for the numerical solution of indoor-air flow problems, see e.g. [Gri01]. The building blocks of this code are the k/ model (which is a statistical turbulence model), an improved wall-function concept for the treatment of the near-wall region, and a stabilised finite-element method together with an iterative substructuring method as a domain decomposition method for the numerical solution process. The first objective of this thesis is a critical review of the theoretical background of these building blocks. Both the turbulence model and the numerical solution scheme used in Par- allelNS are described in a manner which is more convenient to mathematicians than the presentations in engineering textbooks. Secondly, the aim is to investigate the accuracy of our research code. The near-wall treatment in ParallelNS conceived by [Nei99] had not yet been assessed by reference with experimental data from other research groups. We will investigate a natural convection flow in an air filled cavity. For this test case Karayiannis et al. (see [TK00a] and [AK02]) provided widely accepted experimental data. Moreover the accuracy of the domain decomposition method for this three-dimensional test case has to be investigated, since the numerical tests in [Mue99] are restricted to two-dimensional problems. 7 The k/ model is the most widespread turbulence model, but it suffers from several well-known deficiencies. Thus an additional objective of this thesis is to recommend al- ternative turbulence models which are amenable for use in ParallelNS. A successful improvement of the standard k/ model is the so called k--v2 model, which was de- vised by Durbin, see [Dur91]. However, this model requires resolving the near-wall re- gion, which is infeasible for three-dimensional problems of practical relevance. Therefore we study LES, which has the additional advantage of being much closer to the Navier- Stokes equations than statistical turbulence models. Modern advances in computer power have allowed LES to become more and more interesting for engineering applications, see, e.g., the current projects in Prof. Dr. Lars Davidson's research group at the Depart- ment of Thermo and Fluid Dynamics at Chalmers University of Technology G¨oteborg (http://www.tfd.chalmers.se/ lada/projects/proind.html) and the homepage of the Flow Physics and Computation Division∼ at the Department of Mechanical Engineering at Stan- ford University (http://www-fpc.stanford.edu/). The objective of this thesis is not to devise new LES models but to review current models in order to employ them in ParallelNS. LES models are often referred to as residual stress models. Three residual stress models have been studied in this thesis, viz., the well-known Smagorinsky model, the Iliescu-Layton model (see [IL98]), and the Galdi-Layton model (see [GL00]), including a modification de- vised by Eidson, cf.
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