Direct and large-eddy simulation of inert and reacting compressible turbulent shear layers

Transcription

1 Technische Universität München Fachgebiet Strömungsmechanik Direct and large-eddy simulation of inert and reacting compressible turbulent shear layers Inga Mahle Vollständiger Abdruck der von der Fakultät für Maschinenwesen der Technischen Universität München zur Erlangung des akademischen Grades eines Doktor-Ingenieurs genehmigten Dissertation. Vorsitzender: Univ.-Prof. Dr.-Ing. habil. N. A. Adams Prüfer der Dissertation:. Univ.-Prof. Dr.-Ing., Dr.-Ing. habil. R. Friedrich i.r. 2. Univ.-Prof. W. H. Polifke Ph.D. (CCNY) Die Dissertation wurde am bei der Technischen Universität München eingereicht und durch die Fakultät für Maschinenwesen am.7.27 angenommen.

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3 Abstract In the first part of the thesis, Direct Numerical Simulations (DNS) of temporally evolving, turbulent, compressible shear layers are discussed. Simulations at three different convective Mach numbers (.5,.7 and.) were performed for both, inert and infinitely fast reacting gases. All simulations were continued beyond the onset of a self-similar state in order to guarantee statistics of general value. Self-similarity manifested itself by a collapse of suitably normalized profiles of flow variables and a constant momentum thickness growth rate. During this state, the Reynolds number based on the vorticity thickness of the shear layers was between and 4 and therefore in a fully turbulent regime. The relevance of the achieved results and parameter ranges for practical applications can be seen from the fact that shear (or mixing) layers develop when injecting fuel into the combustion chamber of an engine. Here, a good mixing of fuel and oxidizer is of great interest for an efficient combustion process. The focus of the DNS data analyses was on the effects of compressibility and heat release due to combustion on turbulence and scalar mixing. Both phenomena, compressibility and heat release, were studied separately as well as in combination. Increasing compressibility, i.e. increasing convective Mach number, resulted in a stabilization of the mixing layers: Instantaneous fields of flow quantities became smoother, there were less turbulent fluctuations and the growth rate of the mixing layers reduced. The latter effect was related to a reduction in the production rate of the streamwise Reynolds stress and a reduction in the pressure-strain correlations caused by changes in the fluctuating pressure field. When heat release was present, the effects of compressibility were similar as for the inert mixing layers, but they were less distinct, e.g. the reduction of the growth rate with increasing Mach number was comparatively smaller. At first sight, heat release alone had similar consequences as compressibility: A stabilization of the shear layers, flow fields with lower levels of fluctuations and smaller spreading rates. However, when studied in more detail, it could be seen that the consequences of heat release, were mainly mean density effects, i.e. a result of the reduction of the mean density by the high temperatures in the vicinity of the flame sheets. This was not the case for the compressibility effects. Therefore, it is important to distinguish between compressibility and heat release effects, even though they share the property to be both detrimental for the turbulent mixing process. In the second part of the thesis, Large Eddy Simulations (LES) of shear layers at a convective Mach number of.5 were performed. By a coarsening of the grid, large reductions of computational time were achieved. A deconvolution approach in the form of a single explicit filtering step was validated successfully for inert and reacting mixing layers by comparison with DNS data. For the LES with chemical reactions, two differently detailed chemistry models were used for the filtered chemical source term: one model taking into account the same infinitely fast, irreversible, global reaction as in the DNS and one flamelet model. The particular formulation of the flamelet equations allowed not only to take into account multistep Arrhenius chemistry, but also detailed diffusion mechanisms. The evaluation of the results obtained with two different descriptions of these mechanisms - one with Soret and Dufour effects as well as multicomponent diffusion and

4 one without - showed differences for both, laminar flamelets and turbulent mixing layers, in quantities related to the flame dynamics and in the extinction behaviour.

5 Acknowledgements First of all, I would like to thank my supervisor Prof. Dr.-Ing. habil. Rainer Friedrich for the opportunity to perform this work at the Fachgebiet Strömungsmechanik. He gave me constant guidance and support and was always ready to answer questions or to discuss various aspects. I am also thankful to Prof. Dr. Joseph Mathew for giving me the opportunity to spend four months in his lab at the Indian Institute of Science in Bangalore. I appreciate his gracious hospitality and the fruitful discussions that we had. The final outcome of our joint work has been very rewarding. I would also like to thank Prof. W. Polifke Ph.D. (CCNY) for taking over the role of the second examiner and Prof. Dr.-Ing. habil. N. Adams for leading the board of examiners. My thanks go also to the High Performance Computing Group of the Leibniz Rechenzentrum (LRZ) for providing help and support at nearly all times of the day, all days of the week. The computations of this work were performed on the Hitachi-SR8 and the Altix 47 of the LRZ. Financial contributions came from the Federal Ministry of Education and Research (BMBF) under grant number 3FRAAC and from Bayerischer Forschungsverbund für Turbulente Verbrennung (FORTVER). Without my colleagues and the mutual support and encouragement within our group, this work would not have been imaginable. In this context, I would like to mention especially Dr.-Ing. Holger Foysi who gave me a lot of help and advice concerning the numerical codes and evaluation programs. I would also like to thank Prof. Alexandre Ern from CERMICS, ENPC (France) for providing the code EGlib. Last but not least, I am deeply grateful to my family, in particular to my parents and grandparents, for supporting me throughout the process of this work and throughout all my life. Garching, March 27 Inga Mahle

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7 Contents Introduction 2 DNS of inert compressible turbulent shear layers 4 2. Introduction and literature survey The DNS code for inert gas mixtures Navier-Stokes equations for a gas mixture The numerical method Test cases Results and analysis The structure of the compressible shear layers Inert shear layer at M c = Inert shear layer at M c = Inert shear layer at M c = The self-similar state Check of resolution, domain sizes and filtering The effect of compressibility Turbulence characteristics Mean flow variables Reynolds stresses, turbulent kinetic energy and anisotropies Reynolds stress transport equations Analysis of the reduced growth rate Pressure-strain terms TKE transport equation Thermodynamic fluctuations Correlations of thermodynamic fluctuations Behaviour of the pressure-strain correlations Turbulent and gradient Mach numbers Spectra

8 II CONTENTS Scalar mixing Mean profile and variance Scalar pdfs Mixing efficiency Scalar variance transport equation Scalar fluxes Transport equations of scalar fluxes Behaviour of the pressure-scrambling terms Spectra Entrainment Measurement of volumes Visual thickness Measurement of densities Particle statistics Fractal nature of the mixing layer interface Shocklets Summary and conclusions DNS of infinitely fast reacting compressible turbulent shear layers Introduction and literature survey The DNS code with an infinitely fast chemical reaction Infinitely fast chemistry Transport equations for infinitely fast reacting flows The numerical method Test cases Results and analysis The structure of the infinitely fast reacting shear layers Infinitely fast reacting shear layer at M c = Infinitely fast reacting shear layers at M c =.7 and M c = The self-similar state Check of resolution and domain sizes Effects of compressibility and heat release Mean heat release term

9 CONTENTS III Turbulence characteristics Mean flow variables Reynolds stresses, turbulent kinetic energy and anisotropies Reynolds stress transport equations Analysis of the reduced growth rate Pressure-strain terms TKE transport equation Thermodynamic fluctuations Correlations of thermodynamic fluctuations Behaviour of the pressure-strain correlations Turbulent and gradient Mach numbers Spectra Scalar mixing Mean profile and variance Scalar pdfs Mixing efficiency Scalar variance transport equation Scalar fluxes Transport equations of scalar fluxes Spectra Entrainment Measurement of volumes Visual thickness Measurement of densities Particle statistics Fractal nature of the mixing layer interface Shocklets Summary and conclusions

10 IV CONTENTS 4 LES of inert and infinitely fast reacting mixing layers Introduction and literature survey Description of the LES method Implicit Modeling Approach Applied filters Filtered equations Modeling of the filtered heat release term The filtered density function The conditionally filtered scalar dissipation rate The filtered scalar dissipation rate Test cases Results and analysis Inert mixing layers Instantaneous fields Profiles of averaged flow variables Spectra Effect of filtering on dissipation rates Refinement of the grid Infinitely fast reacting mixing layers Instantaneous fields Profiles of averaged flow variables The filtered heat release term Spectra Effect of filtering on dissipation rates Refinement of the grid Summary and conclusions

11 CONTENTS V 5 LES of shear layers with chemical kinetic and detailed diffusion effects Introduction and literature survey LES approach LES equations and models Filtered heat release term and filtered species mass fractions The flamelet database Detailed reaction scheme and Arrhenius chemistry Computation of detailed diffusion fluxes and of heat flux by EGlib The mixture fraction and its diffusivity Steady flamelet solutions Test cases Results and analysis Flamelets with detailed and simplified diffusion Evaluation of the LES results Summary and conclusions Conclusions and outlook 86 A Appendix: The characteristic form of the Navier-Stokes equations 9 A. The one-dimensional equations A.2 The three-dimensional equations in Cartesian coordinates A.3 The source terms in the transport equations A.4 Specification of the transport equations for an ideal gas mixture A.5 Non-reflecting boundary conditions

12 List of Tables 2. Geometrical parameters of the simulations inert-.5, inert-.7, inert-.. The computational domain has the dimensions L, L 2 and L 3 with N, N 2 and N 3 grid points, respectively. The reference vorticity thickness δ ω, is chosen such that it results in Re ω, = Dimensionless times and Reynolds numbers at the beginning (index: B) and end (index: E) of the self-similar state Integral length scales Values used in the analysis linking momentum thickness growth rate with pressurestrain rate Π for the inert test cases Actual and approximated momentum thickness growth rates and relative errors for the inert test cases Particle parameters: N P particles are initialized at τ ω,p B. They are situated initially between x 3 = x 3,P and x 3,P 2 and between x 3 = x 3,P 3 and x 3,P Statistics of displacements and elapsed times for growth of vorticity and mixture fraction along particle pathlines Fractal dimensions D of isosurfaces determined from the slopes of the curves in Figs to 2.76 and corresponding ones for ω =.2 ω max and Y = Geometrical parameters of the simulations inf-.5, inf-.7, inf-.. The computational domain has the dimensions L, L 2 and L 3 with N, N 2 and N 3 grid points, respectively. The reference vorticity thickness δ ω, is chosen such that it results in Re ω, = Dimensionless times and Reynolds numbers at the beginning (index: B) and end (index: E) of the self-similar state Integral length scales Values used in the analysis linking momentum thickness growth rate with pressurestrain rate Π for the infinitely fast reacting test cases Actual and approximated momentum thickness growth rates and relative errors for the infinitely fast reacting test cases Averaged values (respective inert and reacting cases taken into account) used in the analysis for each convective Mach number Actual and approximated momentum thickness growth rates and relative errors.

13 LIST OF TABLES VII 3.8 Particle parameters: N P particles are initialized at τ ω,p B. They are situated initially between x 3 = x 3,P and x 3,P 2 and between x 3 = x 3,P 3 and x 3,P Statistics of displacements and elapsed times for growth of vorticity and mixture fraction along particle pathlines Fractal dimensions D of isosurfaces LES simulations Reaction scheme for H 2 /O 2 combustion with pre-exponential factors A, temperaturedependence coefficients β and activation energies E [3]

14 List of Figures 2. The configuration of temporally evolving shear layers Case inert-.5: Instantaneous mass fraction field of O 2, x -x 3 -plane in the middle of the computational domain at τ ω = 83, isolines Y O2 =. and.9 are shown Case inert-.5: Instantaneous mass fraction field of O 2, x -x 3 -plane in the middle of the computational domain at τ ω = 286, isolines Y O2 =. and.9 are shown Case inert-.5: Instantaneous mass fraction field of O 2, x -x 3 -plane in the middle of the computational domain at τ ω = 49, isolines Y O2 =. and.9 are shown Case inert-.5: Instantaneous mass fraction field of O 2, x -x 3 -plane in the middle of the computational domain at τ ω = 533, isolines Y O2 =. and.9 are shown Case inert-.5: Instantaneous mass fraction field of O 2, x -x 2 -plane in the middle of the computational domain at τ ω = Case inert-.5: Instantaneous mass fraction field of O 2, x -x 2 -plane in the middle of the computational domain at τ ω = Case inert-.5: Instantaneous mass fraction field of O 2, x -x 2 -plane in the middle of the computational domain at τ ω = Case inert-.5: Instantaneous mass fraction field of O 2, x -x 2 -plane in the middle of the computational domain at τ ω = Case inert-.5: Instantaneous mass fraction field of O 2, x 2 -x 3 -plane through a braid (left) and a roller (right) at τ ω = 83, isolines Y O2 =. and.9 are shown Case inert-.5: Instantaneous mass fraction field of O 2, x 2 -x 3 -plane through a braid (left) and a roller (right) at τ ω = 286, isolines Y O2 =. and.9 are shown Case inert-.5: Instantaneous mass fraction field of O 2, x 2 -x 3 -plane through a braid (left) and a roller (right) at τ ω = 49, isolines Y O2 =. and.9 are shown Case inert-.5: Instantaneous mass fraction field of O 2, x 2 -x 3 -plane through a braid (left) and a roller (right) at τ ω = 533, isolines Y O2 =. and.9 are shown Case inert-.7: Instantaneous mass fraction field of O 2, x -x 3 -plane in the middle of the computational domain at τ ω = 46, isolines Y O2 =. and.9 are shown. 3

15 LIST OF FIGURES IX 2.5 Case inert-.7: Instantaneous mass fraction field of O 2, x -x 3 -plane in the middle of the computational domain at τ ω = 48, isolines Y O2 =. and.9 are shown Case inert-.7: Instantaneous mass fraction field of O 2, x -x 3 -plane in the middle of the computational domain at τ ω = 697, isolines Y O2 =. and.9 are shown Case inert-.7: Instantaneous mass fraction field of O 2, x -x 3 -plane in the middle of the computational domain at τ ω = 98, isolines Y O2 =. and.9 are shown Case inert-.7: Instantaneous mass fraction field of O 2, x -x 2 -plane in the middle of the computational domain at τ ω = Case inert-.7: Instantaneous mass fraction field of O 2, x -x 2 -plane in the middle of the computational domain at τ ω = Case inert-.7: Instantaneous mass fraction field of O 2, x -x 2 -plane in the middle of the computational domain at τ ω = Case inert-.7: Instantaneous mass fraction field of O 2, x -x 2 -plane in the middle of the computational domain at τ ω = Case inert-.7: Instantaneous mass fraction field of O 2, x 2 -x 3 -plane through a braid (left) and a roller (right) at τ ω = 46, isolines Y O2 =. and.9 are shown Case inert-.7: Instantaneous mass fraction field of O 2, x 2 -x 3 -plane through a braid (left) and a roller (right) at τ ω = 48, isolines Y O2 =. and.9 are shown Case inert-.7: Instantaneous mass fraction field of O 2, x 2 -x 3 -plane through a braid (left) and a roller (right) at τ ω = 697, isolines Y O2 =. and.9 are shown Case inert-.7: Instantaneous mass fraction field of O 2, x 2 -x 3 -plane through a braid (left) and a roller (right) at τ ω = 98, isolines Y O2 =. and.9 are shown Case inert-.: Instantaneous mass fraction field of O 2, x -x 3 -plane in the middle of the computational domain at τ ω = 62, isolines Y O2 =. and.9 are shown Case inert-.: Instantaneous mass fraction field of O 2, x -x 3 -plane in the middle of the computational domain at τ ω = 38, isolines Y O2 =. and.9 are shown Case inert-.: Instantaneous mass fraction field of O 2, x -x 3 -plane in the middle of the computational domain at τ ω = 735, isolines Y O2 =. and.9 are shown Case inert-.: Instantaneous mass fraction field of O 2, x -x 3 -plane in the middle of the computational domain at τ ω = 98, isolines Y O2 =. and.9 are shown Case inert-.: Instantaneous mass fraction field of O 2, x -x 2 -plane in the middle of the computational domain at τ ω = Case inert-.: Instantaneous mass fraction field of O 2, x -x 2 -plane in the middle of the computational domain at τ ω = Case inert-.: Instantaneous mass fraction field of O 2, x -x 2 -plane in the middle of the computational domain at τ ω =

16 X LIST OF FIGURES 2.33 Case inert-.: Instantaneous mass fraction field of O 2, x -x 2 -plane in the middle of the computational domain at τ ω = Case inert-.: Instantaneous mass fraction field of O 2, x 2 -x 3 -plane through a braid (left) and a roller (right) at τ ω = 62, isolines Y O2 =. and.9 are shown Case inert-.: Instantaneous mass fraction field of O 2, x 2 -x 3 -plane through a braid (left) and a roller (right) at τ ω = 38, isolines Y O2 =. and.9 are shown Case inert-.: Instantaneous mass fraction field of O 2, x 2 -x 3 -plane through a braid (left) and a roller (right) at τ ω = 735, isolines Y O2 =. and.9 are shown Case inert-.: Instantaneous mass fraction field of O 2, x 2 -x 3 -plane through a braid (left) and a roller (right) at τ ω = 98, isolines Y O2 =. and.9 are shown Temporal development of the momentum thickness, normalized by initial momentum thickness δ θ,, : inert-.5, : inert-.7, : inert-., dashed lines show linear regressions for the self-similar state Dependence of shear layer growth rate on M c : solid line: Langley curve, +: Debisschop & Bonnet [36], : Samimy & Elliot [55], : Chambres, Barre & Bonnet [25], : Papamoschou & Roshko [26], : Clemens & Mungal [27], : Hall, Dimotakis & Rosemann [7], : Pantano & Sarkar [23], : Present DNS Case inert-.5: Spatially averaged profiles of the Reynolds shear stress R 3 at different times, +: τ ω = 83, : τ ω = 23, : τ ω = 64, : τ ω = 24, : τ ω = 245, : τ ω = 286, : τ ω = 327, : τ ω = Case inert-.7: Spatially averaged profiles of the Reynolds shear stress R 3 at different times, +: τ ω = 39, : τ ω = 474, : τ ω = 557, : τ ω = 64, : τ ω = 725, : τ ω = 89, : τ ω = 866, : τ ω = Case inert-.: Spatially averaged profiles of the Reynolds shear stress R 3 at different times, +: τ ω = 735, : τ ω = 87, : τ ω = 878, : τ ω = 949, : τ ω = 23, : τ ω = 98, : τ ω = 74, : τ ω = Streamwise velocity, solid line: inert-.5, dashed line: DNS Pantano & Sarkar M c =.3 [23], [52], +: Experiments Bell & Mehta [7], : Experiments Spencer & Jones [72] Streamwise rms velocity, solid line: inert-.5, dashed line: DNS Pantano & Sarkar M c =.3 [23], dotted line: DNS Rogers & Moser, [52], +: Experiments Bell & Mehta [7], : Experiments Spencer & Jones [72] Spanwise rms velocity, curves as in Fig Transverse rms velocity, curves as in Fig Velocity computed from Reynolds shear stress, curves as in Fig Turbulent kinetic energy budget, +: production, : transport, : dissipation, normalized by u 3 δ θ, solid lines: inert-.5, dashed lines: DNS Pantano & Sarkar M c =.3 [23], dotted line: DNS Rogers & Moser, [52]

17 LIST OF FIGURES XI 2.49 Streamwise rms velocity, solid line: inert-.7, dashed line: DNS Pantano & Sarkar M c =.7 [23], +: Experiments Elliott & Samimy [49] Transverse rms velocity, curves as in Fig Velocity computed from Reynolds shear stress, curves as in Fig Turbulent kinetic energy budget, +: production, : transport, : dissipation, normalized by u 3 /δ θ, solid lines: inert-.7, dashed lines: DNS Pantano & Sarkar M c =.7 [23] Dimensionless momentum thickness growth rate of the inert-.7 case, computed with Eq. (2.2) Turbulent kinetic energy budget, +: production, : transport, : dissipation, normalized by u 3 /δ θ, solid lines: inert-., dashed lines: DNS Pantano & Sarkar M c =. [23] Two-point correlation R with f = u, in the middle of the computational domain, averaged over the self-similar state, : inert-.5, : inert-.7, : inert Two-point correlation R 2 with f = u, in the middle of the computational domain, averaged over the self-similar state, symbols as in Fig Two-point correlation R with f = u 3, in the middle of the computational domain, averaged over the self-similar state, symbols as in Fig Two-point correlation R 2 with f = u 3, in the middle of the computational domain, averaged over the self-similar state, symbols as in Fig Two-point correlation R with f = Y O2, in the middle of the computational domain, averaged over the self-similar state, symbols as in Fig Two-point correlation R 2 with f = Y O2, in the middle of the computational domain, averaged over the self-similar state, symbols as in Fig Test of compact filter, solid: filter dissipation, dashed: ɛ Test of compact filter, solid: filter scalar dissipation, dashed: ɛ Y Averaged temperature, normalized by T =.5 (T + T 2 ), : inert-.5, : inert-.7, : inert Averaged density, normalized by ρ, symbols as in Fig Averaged pressure, normalized by ρ u 2, symbols as in Fig Favre averaged streamwise velocity, normalized by u, symbols as in Fig Reynolds stress ρ R, normalized by ρ u 2, curves as in Fig Reynolds stress ρ R 22, normalized by ρ u 2, curves as in Fig Reynolds stress ρ R 33, normalized by ρ u 2, curves as in Fig Reynolds stress ρ R 3, normalized by ρ u 2, curves as in Fig

18 XII LIST OF FIGURES 2.7 Turbulent kinetic energy ρ k, normalized by ρ u 2, curves as in Fig Reynolds shear stress anisotropy, b 3, curves as in Fig Streamwise Reynolds stress anisotropy, b, curves as in Fig Spanwise Reynolds stress anisotropy, b 22, curves as in Fig Transverse Reynolds stress anisotropy, b 33, curves as in Fig Budget of R, normalized by ρ u 3 /δ ω, symbols as in Fig. 2.63, solid: production, dashed: dissipation rate Budget of R, normalized by ρ u 3 /δ ω, symbols as in Fig. 2.63, solid: pressurestrain rate, dashed: turbulent transport Budget of R 22, normalized by ρ u 3 /δ ω, symbols as in Fig. 2.63, solid: production, dashed: dissipation rate Budget of R 22, normalized by ρ u 3 /δ ω, symbols as in Fig. 2.63, solid: pressurestrain rate, dashed: turbulent transport Budget of R 33, normalized by ρ u 3 /δ ω, symbols as in Fig. 2.63, solid: production, dashed: dissipation rate Budget of R 33, normalized by ρ u 3 /δ ω, symbols as in Fig. 2.63, solid: pressurestrain rate, dashed: turbulent transport Budget of R 3, normalized by ρ u 3 /δ ω, symbols as in Fig. 2.63, solid: production, dashed: dissipation rate Budget of R 3, normalized by ρ u 3 /δ ω, symbols as in Fig. 2.63, solid: pressurestrain rate, dashed: turbulent transport Production, integrated in transverse direction, normalized by ρ u 3 : +: P, : P 22, : P 33, : P Pressure-strain rate, integrated in transverse direction, normalized by ρ u 3 : +: Π, : Π 22, : Π 33, : Π Dissipation rate, integrated in transverse direction, normalized by ρ u 3 : +: ɛ, : ɛ 22, : ɛ 33, : ɛ Ratios of integrated budget terms: +: Π / P, : Π 3 / P 3, : P3 / Π, : Π 3 / Π Reynolds stress ρ R, normalized by ρ u 2, curves as in Fig Rms value of p, normalized by ρ u 2, curves as in Fig Rms value of u / x, normalized by δ ω, / u, curves as in Fig Correlation coefficient R (p, u / x ) between pressure and density fluctuations, curves as in Fig

19 LIST OF FIGURES XIII 2.92 Suppression of integrated p rms (+), integrated ( u / x ) rms ( ) and integrated pressure-strain rate Π ( ), normalized by the respective incompressible value at M c = Budget of ρ k, normalized by ρ u 3 /δ ω, symbols as in Fig. 2.63, solid: production, dashed: dissipation rate Budget of ρ k, normalized by ρ u 3 /δ ω, symbols as in Fig. 2.63, solid: pressure dilatation, dashed: turbulent transport Ratio of the integrated pressure dilatation and the TKE production versus M c Ratio of the integrated dissipation rate and the TKE production versus M c Decomposition of TKE dissipation rate, normalized by ρ u 3 /δ ω, symbols as in Fig. 2.63, solid: ɛ, dashed: ɛ 2, dotted: ɛ Decomposition of ɛ, normalized by ρ u 3 /δ ω, symbols as in Fig. 2.63, solid: ɛ s, dashed: ɛ d, dotted: ɛ I Ratio of the integrated compressible and solenoidal dissipation rates versus M c Ratio of the integrated dilatational and total dissipation rates versus M c Rms value of the density fluctuations, normalized by ρ, symbols as in Fig Rms value of the pressure fluctuations, normalized by p, symbols as in Fig Rms value of the temperature fluctuations, normalized by T, symbols as in Fig Rms value of the molecular weight fluctuations, normalized by W, symbols as in Fig Integrated rms values, +: p rms / p, : ρ rms / ρ, : T rms / T, : W rms / W Acoustic (solid line) and entropic part (dashed line) of the density fluctuations, normalized by ρ, symbols as in Fig Acoustic (solid line) and entropic part (dashed line) of the temperature fluctuations, normalized by T, symbols as in Fig Correlation coefficient R (ρ, p) between pressure and density fluctuations, symbols as in Fig Correlation coefficient R (ρ, W ) between pressure and molecular weight fluctuations, symbols as in Fig Case inert-.5: Parts of the pressure-strain correlation Π computed with the Green function, normalized by ρ u 3 /δ ω, +: f = f (A ), : f = f (A 2 ), : f = f (A 3 ), : f = f (A 4 ), : f = f (B ), : f = f (B 2 ), : f = f (B 3 ), : f = f (C ), : f = f (C 2 ), : f = f (C 3 ), : f = f (C 4 ), : f = f (C 5 ) Case inert-.5: Pressure-strain correlation Π, normalized by ρ u 3 /δ ω, solid: computed with the help of the Green function with f = f ( 4 i= A i + 3 i= B i + 5 i= C i), dashed: computed exactly

20 XIV LIST OF FIGURES 2.2Case inert-.7: Parts of the pressure-strain correlation Π computed with the Green function, normalized by ρ u 3 /δ ω, symbols as in Fig Case inert-.7: Pressure-strain correlation Π, normalized by ρ u 3 /δ ω, solid: computed with the help of the Green function with f = f ( 4 i= A i + 3 i= B i + 5 i= C i), lines as in Fig Case inert-.: Parts of the pressure-strain correlation Π computed with the Green function, normalized by ρ u 3 /δ ω, symbols as in Fig Case inert-.: Pressure-strain correlation Π, normalized by ρ u 3 /δ ω, solid: computed with the help of the Green function with f = f ( 4 i= A i + 3 i= B i + 5 i= C i), lines as in Fig Case inert-.5: Parts of the pressure-strain correlation Π computed with the Green function and with constant density ρ, normalized by ρ u 3 /δ ω, symbols as in Fig Case inert-.7: Parts of the pressure-strain correlation Π computed with the Green function and with constant density ρ, normalized by ρ u 3 /δ ω, symbols as in Fig Case inert-.: Parts of the pressure-strain correlation Π computed with the Green function and with constant density ρ, normalized by ρ u 3 /δ ω, symbols as in Fig Turbulent Mach number M t, symbols as in Fig Gradient Mach number M g, symbols as in Fig Turbulent Mach number M t plotted as a function of the gradient Mach number M g. Symbols as in Fig. 2.63, solid line: M t =.286M g, dashed line: M t =.59M g One-dimensional, streamwise spectrum of u / u at the beginning of the selfsimilar state, solid: inert-.5, dashed: inert-.7, dotted: inert One-dimensional, streamwise spectrum of TKE k/ u 2 at the beginning of the self-similar state, solid: inert-.5, dashed: inert-.7, dotted: inert-., the straight line has 5/3 slope One-dimensional, streamwise dissipation spectrum (spectrum of u / u multiplied with (k δ ω, ) 2 ) at the beginning of the self-similar state, solid: inert-.5, dashed: inert-.7, dotted: inert Favre averaged oxygen mass fraction, : inert-.5, : inert-.7, : inert Scalar variance, symbols as in Fig Case inert-.5: pdfs of oxygen mass fraction in planes with various Y, +:., :.2, :.3, :.4, :.5, :.6, :.7, :.8, : Case inert-.7: pdfs of oxygen mass fraction in planes with various Y, +:., :.2, :.3, :.4, :.5, :.6, :.7, :.8, :

21 LIST OF FIGURES XV 2.29Case inert-.: pdfs of oxygen mass fraction in planes with various Y, +:., :.2, :.3, :.4, :.5, :.6, :.7, :.8, : Pdfs of oxygen mass fraction in the plane with Y =.3 (solid) and Y =.5 (dashed), symbols as in Fig Mixing efficiency, : ɛ =.2, : ɛ = Major terms in the scalar variance transport equation, normalized by ρ u/δ ω, solid: turbulent production, dashed: turbulent transport, dotted: dissipation rate, symbols as in Fig Parts of the scalar dissipation rate, solid: ρy α V αi Y α x i, dashed: ρy α V αi Yα f x i, normalized by ρ u/δ ω, symbols as in Fig Scalar flux ρu Y α of oxygen, normalized by ρ u, symbols as in Fig Scalar flux ρu 3 Y α of oxygen, normalized by ρ u, symbols as in Fig Part of the streamwise scalar flux production, normalized by ρ u 2 /δ ω, symbols as in Fig Part of the streamwise scalar flux production, normalized by ρ u 2 /δ ω, symbols as in Fig Part of the transverse scalar flux production, normalized by ρ u 2 /δ ω, symbols as in Fig Major part of the diffusion of the transverse scalar flux, normalized by ρ u 2 /δ ω, symbols as in Fig Part of the dissipation rate of the streamwise scalar flux, normalized by ρ u 2 /δ ω, symbols as in Fig Part of the dissipation rate of the transverse scalar flux, normalized by ρ u 2 /δ ω, symbols as in Fig Part of the dissipation rate of the streamwise scalar flux, normalized by ρ u 2 /δ ω, symbols as in Fig Part of the dissipation rate of the transverse scalar flux, normalized by ρ u 2 /δ ω, symbols as in Fig Turbulent transport of the streamwise scalar flux, normalized by ρ u 2 /δ ω, symbols as in Fig Turbulent transport of the transverse scalar flux, normalized by ρ u 2 /δ ω, symbols as in Fig Pressure-scrambling term in streamwise direction, Π Y, normalized by ρ u 2 /δ ω, symbols as in Fig Pressure-scrambling term in transverse direction, Π Y 3, normalized by ρ u 2 /δ ω, symbols as in Fig

22 XVI LIST OF FIGURES 2.48Case inert-.5: Parts of the pressure-scrambling term Π Y 3 computed with the Green function, normalized by ρ u/δ ω, +: f = f (A ), : f = f (A 2 ), : f = f (A 3 ), : f = f (A 4 ), : f = f (B ), : f = f (B 2 ), : f = f (B 3 ), : f = f (C ), : f = f (C 2 ), : f = f (C 3 ), : f = f (C 4 ), : f = f (C 5 ) Case inert-.5: Pressure-scrambling term Π Y 3, normalized by ρ u/δ ω, solid: computed with the help of the Green function with f = f ( 4 i= A i + 3 i= B i + 5 i= C i), dashed: computed exactly Case inert-.7: Parts of the pressure-scrambling term Π Y 3 computed with the Green function, normalized by ρ u/δ ω, symbols as in Fig Case inert-.7: Pressure-scrambling term Π Y 3, normalized by ρ u/δ ω, solid: computed with the help of the Green function with f = f ( 4 i= A i + 3 i= B i + 5 i= C i), lines as in Fig Case inert-.: Parts of the pressure-scrambling term Π Y 3 computed with the Green function, normalized by ρ u/δ ω, symbols as in Fig Case inert-.: Pressure-scrambling term Π Y 3, normalized by ρ u/δ ω, solid: computed with the help of the Green function with f = f ( 4 i= A i + 3 i= B i + 5 i= C i), lines as in Fig Case inert-.5: Parts of the pressure-scrambling term Π Y 3 computed with the Green function and with constant density ρ, normalized by ρ u/δ ω, symbols as in Fig Case inert-.7: Parts of the pressure-scrambling term Π Y 3 computed with the Green function and with constant density ρ, normalized by ρ u/δ ω, symbols as in Fig Case inert-.: Parts of the pressure-scrambling term Π Y 3 computed with the Green function and with constant density ρ, normalized by ρ u/δ ω, symbols as in Fig One-dimensional, streamwise spectrum of the oxygen mass fraction Y, solid: inert-.5, dashed: inert-.7, dotted: inert-., the straight line has 5/3 slope One-dimensional dissipation spectrum of the oxygen mass fraction Y (spectrum of Y multiplied with (k δ ω, ) 2 ), solid: inert-.5, dashed: inert-.7, dotted: inert Case inert-.5: Instantaneous vorticity field, normalized by ω max, x -x 3 -plane in the middle of the computational domain at τ ω = 286, isoline at. is shown Case inert-.5: Instantaneous mass fraction field of O 2, x -x 3 -plane in the middle of the computational domain at τ ω = 286, isolines Y O2 =.5 and.95 are shown Based on vorticity thresholds: Mixing layer volume V ml (solid) and engulfed volume V en (dashed) vs. the normalized time passed since the beginning of the self-similar state at t B. Volumes are normalized with the mixing layer volume at the beginning of the self-similar state, V ml,b, : inert-.5, : inert-.7, : inert-. 62

23 LIST OF FIGURES XVII 2.62Based on mass fraction thresholds: Mixing layer volume V ml (solid) and engulfed volume V en (dashed) vs. the normalized time passed since the beginning of the self-similar state at t B. Volumes are normalized with the mixing layer volume at the beginning of the self-similar state, V ml,b. Symbols as in Fig Based on vorticity thresholds: Thickness computed from mixing layer volume δ vol (solid) and visual thickness δ vis (dashed). Thicknesses are normalized by the visual thickness at the beginning of the self-similar state, δ vis,b. Symbols as in Fig Based on mass fraction thresholds: Thickness computed from mixing layer volume δ vol (solid) and visual thickness δ vis (dashed). Thicknesses are normalized by the visual thickness at the beginning of the self-similar state, δ vis,b. Symbols as in Fig Based on vorticity thresholds: Mixing layer density ρ ml /ρ (solid), density of the engulfed volume, ρ en /ρ (dashed), and density of the mixed volume, ρ mix /ρ (dotted), vs. the normalized time passed since the beginning of the self-similar state at t B. Symbols as in Fig Based on mass fraction thresholds: Mixing layer density ρ ml /ρ (solid), density of the engulfed volume, ρ en /ρ (dashed), and density of the mixed volume, ρ mix /ρ (dotted), vs. the normalized time passed since the beginning of the selfsimilar state at t B. Symbols as in Fig Particle tracks in the upper half of the computational domain during the selfsimilar state of inert Pdfs of the local Mach number magnitude at the time when the particles are crossing the upper vorticity threshold. Symbols as in Fig Case inert-.5: Instantaneous isosurface of vorticity ω =.2 ω max at τ ω = Case inert-.7: Instantaneous isosurface of vorticity ω =.2 ω max at τ ω = Case inert-.: Instantaneous isosurface of vorticity ω =.2 ω max at τ ω = Case inert-.5: Instantaneous isosurface of oxygen mass fraction Y =.9 at τ ω = Case inert-.7: Instantaneous isosurface of oxygen mass fraction Y =.9 at τ ω = Case inert-.: Instantaneous isosurface of oxygen mass fraction Y =.9 at τ ω = Number of squares N covering the interface ω =. ω max vs. δ ω, /r, averaged over the self-similar state, solid: inert-.5, dashed: inert-.7, dotted: inert Number of squares N covering the interface Y =.95 vs. δ ω, /r, averaged over the self-similar state, solid: inert-.5, dashed: inert-.7, dotted: inert Temporal development of the maximum pressure gradient, normalized by p av /δ ω,, : inert-.5, : inert-.7, : inert

24 XVIII LIST OF FIGURES 2.78Case inert-.: Pressure gradient normalized by p av /δ ω, on a line parallel to the x -axis through x 2 =.3L 2 and x 3 =.32L 3 at τ ω = 23. Every 4th grid point is shown Case inert-.: Pressure normalized by p av on a line parallel to the x -axis through x 2 =.3L 2 and x 3 =.32L 3 at τ ω = 23. Every 4th grid point is shown Case inert-.: Density in x -direction normalized by ρ on a line parallel to the x -axis through x 2 =.3L 2 and x 3 =.32L 3 at τ ω = 23. Every 4th grid point is shown Case inert-.: Temperature normalized by T on a line parallel to the x -axis through x 2 =.3L 2 and x 3 =.32L 3 at τ ω = 23. Every 4th grid point is shown Case inert-.: Dilatation normalized by u/δ ω, on a line parallel to the x -axis through x 2 =.3L 2 and x 3 =.32L 3 at τ ω = 23. Every 4th grid point is shown Case inert-.: Vorticity normalized by u/δ ω, on a line parallel to the x -axis through x 2 =.3L 2 and x 3 =.32L 3 at τ ω = 23. Every 4th grid point is shown Case inert-.: Pressure gradient normalized by p av /δ ω, on a line through x =.2L 2, x 2 =.38 and x 3 =.3L 3 (in x -x 2 -plane, inclined 45 to the x -x 3 - plane) at τ ω = 295. Every 4th grid point is shown Case inert-.: Pressure normalized by p av on a line through x =.2L 2, x 2 =.38 and x 3 =.3L 3 (in x -x 2 -plane, inclined 45 to the x -x 3 -plane) at τ ω = 295. Every 4th grid point is shown Case inert-.: Density in x -direction normalized by ρ on a line through x =.2L 2, x 2 =.38 and x 3 =.3L 3 (in x -x 2 -plane, inclined 45 to the x -x 3 - plane) at τ ω = 295. Every 4th grid point is shown Case inert-.: Temperature normalized by T on a line through x =.2L 2, x 2 =.38 and x 3 =.3L 3 (in x -x 2 -plane, inclined 45 to the x -x 3 -plane) at τ ω = 295. Every 4th grid point is shown on Case inert-.: Dilatation normalized by u/δ ω, on a line through x =.2L 2, x 2 =.38 and x 3 =.3L 3 (in x -x 2 -plane, inclined 45 to the x -x 3 -plane) at τ ω = 295. Every 4th grid point is shown Case inert-.: Vorticity normalized by u/δ ω, on a line through x =.2L 2, x 2 =.38 and x 3 =.3L 3 (in x -x 2 -plane, inclined 45 to the x -x 3 -plane) at τ ω = 295. Every 4th grid point is shown Case inert-.: Instantaneous dilatation field and pressure isolines, x -x 2 -plane through x 3 =.49L 3 at τ ω = 295. Dilatation is normalized by δ ω, / u Case inert-.: Instantaneous magnitude of vorticity and velocity vectors, x -x 2 - plane through x 3 =.49L 3 at τ ω = 295. Vorticity is normalized by δ ω, / u.. 74

25 LIST OF FIGURES XIX 2.92Case inert-.: Instantaneous dilatation field and pressure isolines, x -x 3 -plane through x 2 =.59L 2 at τ ω = 295. Scale as in Fig Case inert-.: Instantaneous magnitude of vorticity and velocity vectors, x -x 3 - plane through x 2 =.59L 2 at τ ω = 295. Scale as in Fig Case inert-.: Instantaneous dilatation field and pressure isolines, x 2 -x 3 -plane through x =.3L at τ ω = 295. Scale as in Fig Case inert-.: Instantaneous magnitude of vorticity and velocity vectors, x 2 -x 3 - plane through x =.3L at τ ω = 295. Scale as in Fig Case inert-.: Instantaneous dilatation field, x -x 3 -plane through x 2 =.59L 2 at τ ω = 295. Dilatation is normalized by δ ω, / u Case inert-.: Instantaneous pressure gradient field, x -x 3 -plane through x 2 =.59L 2 at τ ω = 295. Pressure gradient is normalized by δ ω, / p av Burke-Schumann relations, : Y O, : Y F, : Y P Frozen chemistry, : Y O, : Y F Case inf-.5: Instantaneous mixture fraction field, x -x 3 -plane in the middle of the computational domain at τ ω = 573, isolines z =., z s =.3 and z =.9 are shown Case inf-.5: Instantaneous temperature field, x -x 3 -plane in the middle of the computational domain at τ ω = 573, isolines z =., z s =.3 and z =.9 are shown Case inf-.7: Instantaneous mixture fraction field, x -x 3 -plane in the middle of the computational domain at τ ω = 76, isolines z =., z s =.3 and z =.9 are shown Case inf-.7: Instantaneous temperature field, x -x 3 -plane in the middle of the computational domain at τ ω = 76, isolines z =., z s =.3 and z =.9 are shown Case inf-.: Instantaneous mixture fraction field, x -x 3 -plane in the middle of the computational domain at τ ω = 83, isolines z =., z s =.3 and z =.9 are shown Case inf-.: Instantaneous temperature field, x -x 3 -plane in the middle of the computational domain at τ ω = 83, isolines z =., z s =.3 and z =.9 are shown Temporal development of the momentum thickness, normalized by the initial momentum thickness δ θ,, : inf-.5, : inf-.7, : inf-., dashed lines show linear regressions for the self-similar state Temporal development of the product mass thickness, normalized by the initial product mass thickness δ θ,, symbols as in Fig. 3.9, dashed lines show linear regressions for the self-similar state