Finite element simulation of buoyancy-driven turbulent flows [Elektronische Ressource] / vorgelegt von Tobias Knopp

Sujets

• Physik

Informations

• Publié par: georg-august-universitat_gottingen
• Publié le: 01 janvier 2003
• Nombre de lectures: 34
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Finite-element simulation
of
buoyancy-driven turbulent ows
Dissertation
zur Erlangung des Doktorgrades
der Mathematisch-Naturwissenschaftlichen Fakult aten
der Georg-August-Universit at zu G ottingen
vorgelegt von
Tobias Knopp
aus
Lub eck
G ottingen 2003D7
Referent: Prof. Dr. G. Lube
Korreferent: Prof. Dr. R. Schaback
Tag der mundlic hen Prufung: 4. Juni 2003
2Contents
Preface 7
Epitome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
I. Turbulence modelling for buoyancy driven ows 11
1. The laminar model 13
1.1. Laminar thermally coupled ow problems . . . . . . . . . . . . . . . . . . . 13
1.2. Boundary conditions for thermally coupled ows . . . . . . . . . . . . . . . 15
1.3. A model for non-isothermal ow problems . . . . . . . . . . . . . . . . . . . 18
1.4. Modelling turbulent boundary layers using a fully overlapping DDM . . . . 19
2. Fundamentals, modelling and simulation of turbulent ows 23
2.1. Aspects of randomness and statistical description of turbulent ows . . . . 23
2.2. The scales of turbulent ows . . . . . . . . . . . . . . . . . . . . . . . . . . 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-di usion 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. Di erential ltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3. The space averaged non-isothermal Navier-Stokes equations . . . . . . . . . 42
4.4. Modelling the residual stress tensor and the residual uxes . . . . . . . . . . 44
4.5. System of equations for 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. 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
36. 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 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 simpli cations 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. in time using the discontinuous Galerkin method . . . . 113
9.2. Semidiscretisation, decoupling, and linearisation for the k/ model . . . . . 114
9.3. and 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 . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . 124
11.Non-overlapping domain decomposition methods 127
11.1. The Robin-Robin algorithm for advection-di usion-reaction problems . . . . 128
11.2. Choice of the interface function in the R-R-algorithm for ADR problems . . 129
11.3. The algorithm for Oseen type problems . . . . . . . . . . . . . 130
12.Turbulent channel ow 133
12.1. Fundamentals of isothermal channel ow . . . . . . . . . . . . . . . . . . . . 133
12.2. Isothermal channel ow computations using the k/ model . . . . . . . . . . 136
12.3. Quasi a priori testing of the SGS model . . . . . . . . . . . . . . . . . . . . 138
13.Turbulent natural convection in an air lled square cavity 151
13.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
13.2. Description of the ow con guration . . . . . . . . . . . . . . . . . . . . . . 152
13.3. Testing the wall iteration concept . . . . . . . . . . . . . . . . . . . . . . . . 154
13.4. A posteriori testing for the k/ model without DDM . . . . . . . . . . . . . 15713.5. 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
56Preface
Turbulent ows driven or signi cantly a ected 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 velocityu~, pressure p~, 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 ofu,p~, andT (called Direct Numerical Simu-
lation) is feasible only for a very limited number of ows. 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 rst
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 foru~, p~, and T , which requires an empirical modelling of the terms involving
statistical uctuations. The second approach is called large-eddy simulation (LES). The
idea of LES is to apply a spatial averaging lter to the Navier-Stokes equations in order to
~~extract the large-scale structures ofu, p~, and T , and to attenuate their small-scale struc-
tures. Then only the random motion of the large scales is resolved and the e ects 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 ow problems, see e.g.
[Gri01]. The building blocks of this code are thek/ model (which is a statistical turbulence
model), an improved wall-function concept for the treatment of the near-wall region, and
a stabilised nite-element method together with an iterative substructuring method as a
domain decomposition method for the numerical solution process.
The rst objective of this thesis is a critical review of the theoretical background of these
building blocks. Both the turbulence model and the numerical solution sche