Turbulence. CEFRC Combustion Summer School. Prof. Dr.-Ing. Heinz Pitsch

Transcription

1 Turbulence CEFRC Combustion Summer School 2014 Prof. Dr.-Ing. Heinz Pitsch Copyright 2014 by Heinz Pitsch. This material is not to be sold, reproduced or distributed without prior written permission of the owner, Heinz Pitsch.

2 Turbulent Mixing Combustion requires mixing at the molecular level Turbulence: convective transport molecular mixing diffusion Surface Area fuel oxidizer + = diffusion 2

3 Course Overview Part II: Turbulent Combustion Turbulence Turbulent Premixed Combustion Turbulent Non-Premixed Combustion Modelling Turbulent Combustion Applications Characteristics of Turbulent Flows Statistical Description of Turbulent Flows Reynolds decomposition Favre decomposition Types of turbulence Mean-flow Equations Reynolds Stress Equations k-equation Turbulence Models Scales of Turbulent Flows/Energy Cascade Kolmogorov Hypotheses Scalar Transport Equations Large Eddy Simulation 3

4 Characteristics of Turbulent Flows Transition to turbulence From observations: laminar flow becomes turbulent Characteristic length d Flow velocity u Viscosity ν Dimensionless number: Reynolds number Re 4

5 Characteristics of Turbulent Flows Characteristics of turbulent flows: Random 3D Has Vorticity Large Re 5

7 Statistical Description of Turbulent Flows Conventional Averaging/Reynolds Decomposition Averaging Ensemble average Time average N and Δt sufficiently large For constant density flows: Reynolds decomposition: mean and fluctuation, e.g. for the flow velocity u i 7

8 Reynolds-Zerlegung Mean of the fluctuation is zero (applies for all quantities) Mean of squared fluctuation differs from zero: These averages are named RMS-values (root mean square) 8

9 Favre averaging (density weighted averaging) Combustion: change in density correlation of density and other quantities Reynolds decomposition (for ρ const.) Favre averaging By definition: mean of density weighted fluctuation 0 Density weighted mean velocity 9

10 Favre average conventional average Favre average as a function of conventional mean and fluctuation and for the fluctuating quantity For non-constant density: Favre average leads to much simpler expression 10

12 Types of Turbulence Statistically Homogeneous Turbulence All statistics of fluctuating quantities are invariant under translation of the coordinate system for averaged fluctuating quantities (more generally ) applies Constant gradients of the mean velocity are permitted: Scalar dissipation rate in statistically homogeneous turbulent flow 12

13 Statistically Isotropic Turbulence All statistics are invariant under translation, rotation and reflection of the coordinate system Mean velocities = 0 Isotropy requires homogeneity Relevance of this flow case: Simplifications allow theoretical conclusions about turbulence Turbulent motions on small scales are typically assumed to be isotropic (Kolmogorov hypotheses) DNS of statistically homogeneous and isotropic turbulence: x 1 -component of the velocity 13

14 Turbulent Shear Flow Relevant flow cases in technical systems Round jet Flow around airfoil Flows in combustion chamber Due to the complexity of these turbulent flows they cannot be described theoretically Quelle: www-ah.wbmt.tudelft.nl Temporally evolving shear layer : Scalar dissipation rate χ (left), mixture fraction Z (rechts) Turbulent jet: magnitude of vorticity 14

15 Example: DNS of Homogeneous Shear Turbulence Scalar dissipation rate in homogeneous shear turbulence 2048x2048x2048 collocation points Close-up/detail 15

16 Example: DNS of a Shear Flow inhomogeneous Scalar dissipation rate statistically homogeneous Statistically homogeneous 16

18 Mean-flow Equations Starting from the Navier-Stokes-equations for incompressible fluids (continuity) (momentum) Four unknowns within four equations: u 1, u 2, u 3, p Reynolds decomposition 18

19 Averaged Continuity Equation 1. From continuity equation it follows and Linearity of the continuity equation: no correlations of fluctuating quantities 19

20 Averaged Momentum Equation 2. This does not apply for the momentum equation! Convective term Contin. Time-averaging yields Contin. This term includes product of components of fluctuating velocities: this is due to the non-linearity of the convective term 20

21 Reynolds Stress Tensor Averaging of the other terms averaged momentum equation: The additional term, resulting from convective transport, is added to the viscous term on the right hand side (divergence of a second order tensor) is called Reynolds stress tensor 21

22 Closure Problem in Statistical Turbulence Theory This leads to the closure problem in turbulence theory! The Reynolds Stress Tensor needs to be expressed as a function of mean flow quantities A first idea: derivation of a transport equation for 22

24 *Transport Equation for Reynolds Stress Tensor 24

25 *Transport Equation for Reynolds Stress Tensor Multiplication of the equation with the fluctuating velocity with and a corresponding equation for leads after summation to 25

26 *Transport Equation for Reynolds Stress Tensor The viscous terms on the right hand side of can be transformed into 26

27 *Transport Equation for Reynolds Stress Tensor Splitting of the pressure-terms in with Kronecker delta 27

28 *Transport Equation for Reynolds Stress Tensor Averaging and rearranging leads to Six new equations, but far more new unknowns 28

29 *Transport Equation for Reynolds Stress Tensor The meaning and name of the single terms are listed below: L : Local change C : Convective transport P : Production of Reynolds stresses (negative product of Reynolds-stress tensor and the gradient of time-averaged velocity) 29

30 *Transport Equation for Reynolds Stress Tensor DS : (Pseudo-)dissipation of Reynolds stresses PSC : pressure-rate-of-strain correlation. It contributes to the redistribution of Reynolds stresses in a similar way the diffusion term does 30

31 *Transport Equation for Reynolds Stress Tensor DF : diffusion of the Reynolds stresses. It includes all terms under the divergence operator In this balance production and dissipation are the most important terms The mean velocity gradients are responsible for the production of turbulence ( P ) 31

32 Transport Equation for Reynolds Stress Tensor Transport equation for Reynolds stress tensor Six new equations, but far more new unknowns 32

34 Transport Equation for Turbulent Kinetic Energy Derivation of an equation for the turbulent kinetic energy (TKE) TKE is defined as Contraction j = k ( k: index, not TKE) in Reynolds equation yields 34

35 Transport Equation for Turbulent Kinetic Energy Continuity equation pressure-rate-of-strain correlation PSC = 0 Dissipation Mean dissipation of turbulent kinetic energy 35

36 Transport Equation for Turbulent Kinetic Energy The transport equation for turbulent kinetic energy can be interpreted just as the transport equation for the Reynolds stress tensor Local change and convection of turbulent kinetic energy (lhs) Production, dissipation and diffusion (rhs) PSC 0 36

37 Transport Equation for Turbulent Kinetic Energy example: pipe-flow example: free jet 37

38 Transport Equation for Turbulent Kinetic Energy Transport equation BUT: Closure problem is not solved Triple correlations Derivation of equations for such correlations even higher correlations 38

40 Turbulence Models Turbulent Viscosity The derived averaged equations are not closed turbulent stress tensor has to be modeled Analogy to Newton approach for molecular shear stress gradient transport model: is eddy viscosity/turbulent viscosity (important: molecular viscosity!) 40

41 Turbulent-viscosity models Algebraic models: e.g. Prandtl s mixing-length concept TKE models: e.g. Prandtl-Kolmogorov k-ε-modell (Jones, Launder) 41

42 Algebraic Model: Prandtl s Mixing-length Concept Eddy viscosity Based on dimensional analysis All unknown proportionalities mixing-length Empirical methods for determining l m Assumption: l m = const. 42

43 TKE model: Prandtl-Kolmogorov Eddy viscosity Model constant C μ (often: C μ = 0,09) l pk : characteristic length scale determined empirically Equation for TKE 43

44 Two-equation-model: k-ε-model Eddy viscosity Solving one equation each for TKE dissipation the model parameters need to be determined empirically 44

45 Two-equation-model: k-ε-model Assumptions: Turbulent transport term Influence of correlation between velocity- and pressure fluctuations is not considered Molecular transport is assumed to be much smaller than turbulent transport and is therefore neglected Production 45

47 Scales of Turbulent Flows/Energy Cascade Two-Point Correlation Characteristic feature of turbulent flows: eddies exist at different length scales x x + r Turbulent round jet: Reynolds number Re 2300 Determination of the distribution of eddy size at a single point Measurement of velocity fluctuation and Two-point correlation 47

48 Correlation Function Homogeneous isotropic turbulence:, Two-point correlation normalized by its variance Degree of correlation of stochastic signals correlation function 48

49 Integral Turbulent Scales Largest scales: physical scale of the problem Integral length scale l t (largest eddies) Integral velocity scale Integral time scale 49

50 Energy Spectrum Energy Spectrum (logarithmic) Energy Cascade energy density Energy Transfer of Energy wave number Dissipation of Energy 50

52 Kolmogorov Hypotheses First Kolmogorov Hypothesis At sufficiently high Reynolds numbers, small-scale eddies have a universal form. They are determined by two parameters Dissipation Kinematic viscosity Dimensional analysis Length η Time τ η Velocity u η 52

53 Second Kolmogorov Hypothesis At sufficiently high Reynolds numbers, the statistics of the motions of scale r in the range η << r << l t have a universal form that is uniquely determined by Dissipation But independent of kinematic viscosity Inertial subrange Integral length scale Ratio η/l t 53

55 Scalar Transport Equations Transport equation for mixture fraction Z Favre averaging not closed molecular transport turbulent transport 55

56 Transport Equation for Mixture Fraction Neglecting molecular transport (assumption: Re ) Gradient transport model for turbulent transport D t : Turbulent diffusivity Sc t : Turbulent Schmidt number Transport equation for mean mixture fraction 56

57 Transport Equation for Mixture Fraction Variance equation First step: equation for 57

58 Transport Equation for Mixture Fraction By neglecting the derivatives of ρ and D and their mean values, then multiplying this equation by, applying continuity equation, averaging and neglecting the molecular transport results in not closed Favre averaged scalar dissipation 58

59 Modeling of Scalar Dissipation Scalar dissipation rate has to be modeled Integral time τ Z (dimensional analysis) with Typically proportional to τ and This leads to 59

60 Transport Equation for Reactive Scalars Assumptions: Specific heat c p,α = c p = const. Pressure p = const., heat transfer by radiation is neglected Lewis number Le α = Le = Sc/Pr = 1 Temperature equation Source term due to chemical reactions (heat release) 60

61 Transport Equation for Reactive Scalars Temperature equation is similar to the equation for the mass fraction of component α 61

62 Transport Equation for Reactive Scalars The term reactive scalar includes Mass fractions Y α of all components α = 1, N Temperature T Balance equations for D i : mass diffusivity, thermal diffusivity S i : mass/temperature source term 62

63 Transport Equation for Reactive Scalars Derivation of a transport equation for Favre decomposition and averaging of leads to not closed molecular transport turbulent transport averaged source term 63

64 Transport Equation for Reactive Scalars Neglecting the molecular transport (assumption: Re ) Gradient transport model for the turbulent transport term Averaged transport equation Not closed chapter Modelling of Turbulent Combustion 64

66 Large-Eddy Simulation Direct Numerical Simulation (DNS) Solve NS-equations No models For turbulent flows Computational domain has to be at least of order of integral length scale l Mesh spacing has to resolve smallest scales η Minimum number of cells per direction n x = l/η = Re 3/4 t Minimum number of cells total n t = n 3 x = Re 9/4 t 66

67 Large-Eddy Simulation Example: Turbulent Jet with Re = This is for one integral length scale only! 67 Pope, Turbulent Flows

68 Large-Eddy Simulation Large-Eddy Simulation (LES) Spatial filtering as opposed to RANS-ensemble averaging Sub-filter modeling as opposed to DNS 68

69 Large-Eddy Simulation 69

70 Large-Eddy Simulation Spatial filtering rather than ensemble average Representation taken from Pope (2000) 70 Computational Grid Scales smaller than filter scale absent from the filtered quantities Filtered signal can be discretized using a mesh substantially smaller than the DNS mesh

71 Large-Eddy Simulation For example: Box filter in 1D: Sharp spectral filter: 71

72 Large-Eddy Simulation 72 Pope, Turbulent Flows

73 Large-Eddy Simulation Filtered momentum equation: Define residual stress tensor: 73

74 Large-Eddy Simulation Sub-filter Modeling Eddy viscosity model for Filtered strain rate tensor 74

75 Large-Eddy Simulation Smagorinsky model for (in analogy to mixing length model) Sub-filter eddy viscosity Sub-filter velocity fluctuation with filtered rate of strain 75

76 Large-Eddy Simulation Smagorinsky length scale Similar equations can be derived for scalar transport System of equations closed! 76

77 Summary Part II: Turbulent Combustion Turbulence Turbulent Premixed Combustion Turbulent Non-Premixed Combustion Modelling Turbulent Combustion Applications Characteristics of Turbulent Flows Statistical Description of Turbulent Flows Reynolds decomposition Favre decomposition Types of turbulence Mean-flow Equations Reynolds Stress Equations k-equation Turbulence Models Scales of Turbulent Flows/Energy Cascade Kolmogorov Hypotheses Scalar Transport Equations Large Eddy Simulation 77