Direct and Large-Eddy Simulation of Compressible Rectangular Jet Flow Benjamin Rembold Dissertation ETH No. 15081 Diss. ETH No. 15081 DIRECT AND LARGE-EDDY SIMULATION OF COMPRESSIBLE RECTANGULAR JET FLOW A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY (cid:127) ZURICH for the degree of Doctor of Technical Sciences presented by Benjamin Rembold Dipl.-Ing. (University of Karlsruhe, TH) born on September 5, 1973 citizen of Germany accepted on the recommendation of Prof. Dr. L. Kleiser, examiner Prof. Dr. N. A. Adams, co-examiner Prof. Dr. N. D. Sandham, co-examiner 2003 The picture on the title page shows a density isosurfaceof the transitional jet. Colour coding denotes the jet downstream velocity component. Acknowledgments I would like to thank Prof. L. Kleiser for the supervision of my research at the Institute of Fluid Dynamics (IFD) and for always supporting me and my ideas throughout this work. In particular, I am grateful to Prof. N. A. Adams (Institute of Fluid Mechanics at the Technical University of Dresden) for his true help and patience during this work and (cid:12)nally for acting as co-examiner for my thesis. He spent an enormous amount of time in discussing open prob- lems, proofreading reports and teaching me the knowledge about com- pressible (cid:13)ow simulations. Furthermore, I would like to thank Prof. N. D. Sandham (Depart- ment of Aeronautics and Astronautics, School of Engineering Sciences, University of Southampton) for acting as second co-examiner and for his numerous helpful comments about my work. During my stay at the Center for Turbulence Research for the Sum- mer School 2002 I had the great opportunity to work together with Prof. S. Lele (Department of Aeronautics and Astronautics and Depart- ment of Mechanical Engineering, Stanford University), Prof. J. B. Fre- und (Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign) and Dr. M. Wang (Center for Turbulence Research, Stanford University). I learned a lot about aeroacoustics dur- ing that time thanks to them since they had always time to discuss open questions. I would also like to heartily thank all my friends and colleagues at IFD who not only contributed to my work through numerous instructive discussions, but also made the time at IFD unforgettable through an endless list of extracurricular activities. This work was supported by the Swiss National Science Foundation. Calculations were performed at the Swiss Center for Scienti(cid:12)c Comput- ing (CSCS). Zu(cid:127)rich, May 2003 Benjamin Rembold Contents Nomenclature VII Abstract XI Kurzfassung (in German) XI 1 Introduction 1 1.1 Non-axisymmetric jet (cid:13)ow . . . . . . . . . . . . . . . . . . 1 1.2 LES modelling using approximate deconvolution . . . . . 4 1.3 Jet-(cid:13)ow acoustics . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Objectives and outline of the present work . . . . . . . . . 7 2 Simulation method 11 2.1 Physical model . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Discretization . . . . . . . . . . . . . . . . . . . . . 12 2.2.2 Boundary conditions . . . . . . . . . . . . . . . . . 13 2.2.3 Validation . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 LES modelling . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.1 Approximate Deconvolution Model . . . . . . . . . 20 2.3.2 Filter de(cid:12)nition . . . . . . . . . . . . . . . . . . . . 23 3 DNS of a transitional rectangular jet 25 3.1 Flow description . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 In(cid:13)ow treatment . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Out(cid:13)ow sponge solution . . . . . . . . . . . . . . . . . . . 28 3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4.1 Instantaneous data . . . . . . . . . . . . . . . . . . 29 3.4.2 Statistically averaged data . . . . . . . . . . . . . . 30 4 LES results 39 4.1 The transitional rectangular jet . . . . . . . . . . . . . . . 39 4.1.1 Flow parameters . . . . . . . . . . . . . . . . . . . 39 4.1.2 In(cid:13)ow treatment . . . . . . . . . . . . . . . . . . . 39 4.1.3 Comparison of DNS and LES results . . . . . . . . 41 4.2 Turbulent rectangular jet . . . . . . . . . . . . . . . . . . 47 VI Contents 4.2.1 In(cid:13)ow generation . . . . . . . . . . . . . . . . . . . 47 4.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . 52 5 Acoustic analysis 59 5.1 Far-(cid:12)eld sound computation . . . . . . . . . . . . . . . . . 59 5.2 DNS results . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.2.1 Database description . . . . . . . . . . . . . . . . . 60 5.2.2 Acoustic analysis . . . . . . . . . . . . . . . . . . . 61 5.3 LES results . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.3.1 LES database . . . . . . . . . . . . . . . . . . . . . 66 5.3.2 Acoustic analysis . . . . . . . . . . . . . . . . . . . 67 5.3.3 Analysis of spurious noise predicted by LES . . . . 71 6 Summary and conclusion 79 A Description of the linear stability eigensolver 83 A.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . 83 A.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 B Duct (cid:13)ow simulations 87 B.1 Flow setup . . . . . . . . . . . . . . . . . . . . . . . . . . 87 B.2 Simulation method . . . . . . . . . . . . . . . . . . . . . . 88 B.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Bibliography 99 Curriculum vitae 107 Nomenclature Roman symbols A constant in Taylor-Green problem A amplitude of disturbance 0 A , A , B matrices in the eigenproblem s t s B length of computational domain in i coordinate direction i c1 ambient speed of sound E total energy E(n ;n ) modal energy (cid:20);i (cid:20);i f frequency f, f vector of (cid:13)uxes, components i f , f momentum forcing terms (cid:26)u1 (cid:26)u1;2 f energy forcing term E G, G (cid:12)lter transfer function 2 h velocity half-widths 1 ;i 2 i imaginary unit k , l , l mapping parameter i 2 3 L , L (L ;L ;L ) jet nozzle (duct) dimensions 2 3 1 2 3 Mk (cid:12)lter moments i M Mach number M convective Mach number c n , n parameters of the in(cid:13)ow velocity pro(cid:12)le 2 3 n coe(cid:14)cient denoting multiples of spatial (cid:20);i base frequency N deconvolution order Pr Prandtl number q heat (cid:13)ux components i Q deconvolution operator N R, R direction vector magnitude, components i Re Reynolds number Sd Sutherland’s constant t time T Lighthill source tensor i;j T temperature tke turbulent kinetic energy