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Turbulence

Turbulence
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. Turbulent Mixing • Combustion requires mixing at the molecular level • Turbulence: convective transport ↑  molecular mixing ↑ Surface Area ↑ diffusion fuel oxidizer = + diffusion 2 Course Overview Part II: Turbulent Combustion • Characteristics of Turbulent Flows • Statistical Description of Turbulent Flows • Reynolds decomposition • Turbulence • Favre decomposition • Types of turbulence • Turbulent Premixed Combustion • Meanflow Equations • Turbulent NonPremixed • Reynolds Stress Equations Combustion • kEquation • Modelling Turbulent Combustion • Turbulence Models • Applications • Scales of Turbulent Flows/Energy Cascade • Kolmogorov Hypotheses • Scalar Transport Equations • Large Eddy Simulation 3 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 Characteristics of Turbulent Flows Characteristics of turbulent flows: • Random • 3D • Has Vorticity • Large Re 5 Course Overview Part II: Turbulent Combustion • Characteristics of Turbulent Flows • Statistical Description of Turbulent Flows • Reynolds decomposition • Turbulence • Favre decomposition • Types of turbulence • Turbulent Premixed Combustion • Meanflow Equations • Turbulent NonPremixed • Reynolds Stress Equations Combustion • kEquation • Modelling Turbulent Combustion • Turbulence Models • Applications • Scales of Turbulent Flows/Energy Cascade • Kolmogorov Hypotheses • Scalar Transport Equations • Large Eddy Simulation 6 Statistical Description of Turbulent Flows Conventional Averaging/Reynolds Decomposition • Averaging  Ensemble average N and Δt  Time average sufficiently large • For constant density flows:  Reynolds decomposition: mean and fluctuation, e.g. for the flow velocity u i 7 ReynoldsZerlegung • Mean of the fluctuation is zero (applies for all quantities) • Mean of squared fluctuation differs from zero: • These averages are named RMSvalues (root mean square) 8 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 Favre average ↔ conventional average • Favre average as a function of conventional mean and fluctuation • and for the fluctuating quantity → For nonconstant density: Favre average leads to much simpler expression 10 Course Overview Part II: Turbulent Combustion • Characteristics of Turbulent Flows • Statistical Description of Turbulent Flows • Reynolds decomposition • Turbulence • Favre decomposition • Types of turbulence • Turbulent Premixed Combustion • Meanflow Equations • Turbulent NonPremixed • Reynolds Stress Equations Combustion • kEquation • Modelling Turbulent Combustion • Turbulence Models • Applications • Scales of Turbulent Flows/Energy Cascade • Kolmogorov Hypotheses • Scalar Transport Equations • Large Eddy Simulation 11 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 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 DNS of statistically homogeneous and isotropic about turbulence turbulence: x component of the velocity 1  Turbulent motions on small scales are typically assumed to be isotropic (Kolmogorov hypotheses) 13 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: wwwah.wbmt.tudelft.nl „Temporally evolving shear layer“: Scalar dissipation rate χ (left), mixture fraction Z (rechts) Turbulent jet: magnitude of vorticity 14 Example: DNS of Homogeneous Shear Turbulence Scalar dissipation rate in homogeneous shear turbulence Closeup/detail 2048x2048x2048 collocation points 15 Example: DNS of a Shear Flow inhomogeneous Scalar dissipation rate statistically homogeneous Statistically homogeneous 16 Course Overview Part II: Turbulent Combustion • Characteristics of Turbulent Flows • Statistical Description of Turbulent Flows • Reynolds decomposition • Turbulence • Favre decomposition • Types of turbulence • Turbulent Premixed Combustion • Meanflow Equations • Turbulent NonPremixed • Reynolds Stress Equations Combustion • kEquation • Modelling Turbulent Combustion • Turbulence Models • Applications • Scales of Turbulent Flows/Energy Cascade • Kolmogorov Hypotheses • Scalar Transport Equations • Large Eddy Simulation 17 Meanflow Equations • Starting from the NavierStokesequations for incompressible fluids (continuity) (momentum) → Four unknowns within four equations: u , u , u , p 1 2 3 • Reynolds decomposition 18 Averaged Continuity Equation 1. From continuity equation it follows and → Linearity of the continuity equation: no correlations of fluctuating quantities 19 Averaged Momentum Equation 2. This does not apply for the momentum equation  Convective term Contin.  Timeaveraging yields Contin. → This term includes product of components of fluctuating velocities: this is due to the nonlinearity of the convective term 20 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 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 Course Overview Part II: Turbulent Combustion • Characteristics of Turbulent Flows • Statistical Description of Turbulent Flows • Reynolds decomposition • Turbulence • Favre decomposition • Types of turbulence • Turbulent Premixed Combustion • Meanflow Equations • Turbulent NonPremixed • Reynolds Stress Equations Combustion • kEquation • Modelling Turbulent Combustion • Turbulence Models • Applications • Scales of Turbulent Flows/Energy Cascade • Kolmogorov Hypotheses • Scalar Transport Equations • Large Eddy Simulation 23 Transport Equation for Reynolds Stress Tensor 24 Transport Equation for Reynolds Stress Tensor Multiplication of the equation with the fluctuating velocity and a corresponding equation for with leads after summation to 25 Transport Equation for Reynolds Stress Tensor The viscous terms on the right hand side of can be transformed into 26 Transport Equation for Reynolds Stress Tensor Splitting of the pressureterms in with Kronecker delta 27 Transport Equation for Reynolds Stress Tensor Averaging and rearranging leads to   Six new equations, but far more new unknowns 28 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 Reynoldsstress tensor and the gradient of timeaveraged velocity) 29 Transport Equation for Reynolds Stress Tensor • „DS“: (Pseudo)dissipation of Reynolds stresses • „PSC“: pressurerateofstrain correlation. It contributes to the redistribution of Reynolds stresses in a similar way the diffusion term does 30 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 Transport Equation for Reynolds Stress Tensor Transport equation for Reynolds stress tensor  Six new equations, but far more new unknowns 32 Course Overview Part II: Turbulent Combustion • Characteristics of Turbulent Flows • Statistical Description of Turbulent Flows • Reynolds decomposition • Turbulence • Favre decomposition • Types of turbulence • Turbulent Premixed Combustion • Meanflow Equations • Turbulent NonPremixed • Reynolds Stress Equations Combustion • kEquation • Modelling Turbulent Combustion • Turbulence Models • Applications • Scales of Turbulent Flows/Energy Cascade • Kolmogorov Hypotheses • Scalar Transport Equations • Large Eddy Simulation 33 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 Transport Equation for Turbulent Kinetic Energy • Continuity equation pressurerateofstrain correlation PSC = 0 • Dissipation • Mean dissipation of turbulent kinetic energy 35 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 Transport Equation for Turbulent Kinetic Energy example: pipeflow example: free jet 37 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 Course Overview Part II: Turbulent Combustion • Characteristics of Turbulent Flows • Statistical Description of Turbulent Flows • Reynolds decomposition • Turbulence • Favre decomposition • Types of turbulence • Turbulent Premixed Combustion • Meanflow Equations • Turbulent NonPremixed • Reynolds Stress Equations Combustion • kEquation • Modelling Turbulent Combustion • Turbulence Models • Applications • Scales of Turbulent Flows/Energy Cascade • Kolmogorov Hypotheses • Scalar Transport Equations • Large Eddy Simulation 39 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 Turbulentviscosity models • Algebraic models: e.g. Prandtl´s mixinglength concept • TKE models: e.g. PrandtlKolmogorov • kεModell (Jones, Launder) 41 Algebraic Model: Prandtl´s Mixinglength Concept • Eddy viscosity • Based on dimensional analysis • All unknown proportionalities  mixinglength • Empirical methods for determining l m • Assumption: l = const. m 42 TKE model: PrandtlKolmogorov • Eddy viscosity  Model constant C (often: C = 0,09) μ μ  l : characteristic length scale  determined empirically pk • Equation for TKE 43 Twoequationmodel: kεmodel • Eddy viscosity • Solving one equation each for  TKE  dissipation  the model parameters need to be determined empirically 44 Twoequationmodel: kεmodel 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 Course Overview Part II: Turbulent Combustion • Characteristics of Turbulent Flows • Statistical Description of Turbulent Flows • Reynolds decomposition • Turbulence • Favre decomposition • Types of turbulence • Turbulent Premixed Combustion • Meanflow Equations • Turbulent NonPremixed • Reynolds Stress Equations Combustion • kEquation • Modelling Turbulent Combustion • Turbulence Models • Applications • Scales of Turbulent Flows/Energy Cascade • Kolmogorov Hypotheses • Scalar Transport Equations • Large Eddy Simulation 46 Scales of Turbulent Flows/Energy Cascade TwoPoint Correlation • Characteristic feature of turbulent flows: eddies exist at different length scales x + r x Turbulent round jet: Reynolds number Re ≈ 2300 • Determination of the distribution of eddy size at a single point  Measurement of velocity fluctuation and  Twopoint correlation 47 Correlation Function • Homogeneous isotropic turbulence: , • Twopoint correlation normalized by its variance • Degree of correlation of stochastic signals correlation function 48 Integral Turbulent Scales • Largest scales: physical scale of the problem  Integral length scale l (largest eddies) t  Integral velocity scale  Integral time scale 49 Energy Spectrum Energy Spectrum (logarithmic) Energy Cascade energy density Energy Transfer of Energy Dissipation of Energy wave number 50 Course Overview Part II: Turbulent Combustion • Characteristics of Turbulent Flows • Statistical Description of Turbulent Flows • Reynolds decomposition • Turbulence • Favre decomposition • Types of turbulence • Turbulent Premixed Combustion • Meanflow Equations • Turbulent NonPremixed • Reynolds Stress Equations Combustion • kEquation • Modelling Turbulent Combustion • Turbulence Models • Applications • Scales of Turbulent Flows/Energy Cascade • Kolmogorov Hypotheses • Scalar Transport Equations • Large Eddy Simulation 51 Kolmogorov Hypotheses First Kolmogorov Hypothesis • At sufficiently high Reynolds numbers, smallscale 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 Second Kolmogorov Hypothesis • At sufficiently high Reynolds numbers, the statistics of the motions of scale r in the range η r l have a universal form that is uniquely determined by t  Dissipation  But independent of kinematic viscosity → Inertial subrange  Integral length scale  Ratio η/l t 53 Course Overview Part II: Turbulent Combustion • Characteristics of Turbulent Flows • Statistical Description of Turbulent Flows • Reynolds decomposition • Turbulence • Favre decomposition • Types of turbulence • Turbulent Premixed Combustion • Meanflow Equations • Turbulent NonPremixed • Reynolds Stress Equations Combustion • kEquation • Modelling Turbulent Combustion • Turbulence Models • Applications • Scales of Turbulent Flows/Energy Cascade • Kolmogorov Hypotheses • Scalar Transport Equations • Large Eddy Simulation 54 Scalar Transport Equations • Transport equation for mixture fraction Z • Favre averaging  not closed molecular turbulent transport transport 55 Transport Equation for Mixture Fraction • Neglecting molecular transport (assumption: Re↑) • Gradient transport model for turbulent transport  D : Turbulent diffusivity t  Sc : Turbulent Schmidt number t → Transport equation for mean mixture fraction 56 Transport Equation for Mixture Fraction • Variance equation • First step: equation for 57 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 Modeling of Scalar Dissipation Scalar dissipation rate has to be modeled • Integral time τ (dimensional analysis) Z with • Typically proportional to τ and • This leads to  59 Transport Equation for Reactive Scalars • Assumptions:  Specific heat c = c = const. p,α p  Pressure p = const., heat transfer by radiation is neglected  Lewis number Le = Le = Sc/Pr = 1 α • Temperature equation • Source term due to chemical reactions (heat release) 60 Transport Equation for Reactive Scalars • Temperature equation is similar to the equation for the mass fraction of component α 61 Transport Equation for Reactive Scalars • The term „reactive scalar“ includes  Mass fractions Y of all components α = 1, … N α  Temperature T • Balance equations for  D : mass diffusivity, thermal diffusivity i  S : mass/temperature source term i 62 Transport Equation for Reactive Scalars • Derivation of a transport equation for • Favre decomposition and averaging of leads to not closed molecular turbulent averaged transport transport source term 63 Transport Equation for Reactive Scalars • Neglecting the molecular transport (assumption: Re↑) • Gradient transport model for the turbulent transport term Not closed  chapter → Averaged transport equation „Modelling of Turbulent Combustion“ 64 Course Overview Part II: Turbulent Combustion • Characteristics of Turbulent Flows • Statistical Description of Turbulent Flows • Reynolds decomposition • Turbulence • Favre decomposition • Types of turbulence • Turbulent Premixed Combustion • Meanflow Equations • Turbulent NonPremixed • Reynolds Stress Equations Combustion • kEquation • Modelling Turbulent Combustion • Turbulence Models • Applications • Scales of Turbulent Flows/Energy Cascade • Kolmogorov Hypotheses • Scalar Transport Equations • Large Eddy Simulation 65 LargeEddy Simulation Direct Numerical Simulation (DNS) • Solve NSequations • 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 3/4 • Minimum number of cells per direction n = l/η = Re x t 3 9/4 • Minimum number of cells total n = n = Re t x t 66 LargeEddy Simulation • Example: Turbulent Jet with Re = 15000 • This is for one integral length scale only Pope, „Turbulent Flows“ 67 LargeEddy Simulation LargeEddy Simulation (LES) • Spatial filtering as opposed to RANSensemble averaging • Subfilter modeling as opposed to DNS 68 LargeEddy Simulation 69 LargeEddy Simulation • Spatial filtering rather than ensemble average Representation taken from Pope (2000) 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 70 LargeEddy Simulation • For example: • Box filter in 1D: • Sharp spectral filter: 71 LargeEddy Simulation Pope, „Turbulent Flows“ 72 LargeEddy Simulation • Filtered momentum equation: • Define residual stress tensor: 73 LargeEddy Simulation Subfilter Modeling • Eddy viscosity model for • Filtered strain rate tensor 74 LargeEddy Simulation • Smagorinsky model for (in analogy to mixing length model) • Subfilter eddy viscosity • Subfilter velocity fluctuation with filtered rate of strain 75 LargeEddy Simulation • Smagorinsky length scale • Similar equations can be derived for scalar transport  System of equations closed 76 Summary Part II: Turbulent Combustion • Characteristics of Turbulent Flows • Statistical Description of Turbulent Flows • Reynolds decomposition • Turbulence • Favre decomposition • Types of turbulence • Turbulent Premixed Combustion • Meanflow Equations • Turbulent NonPremixed • Reynolds Stress Equations Combustion • kEquation • Modelling Turbulent Combustion • Turbulence Models • Applications • Scales of Turbulent Flows/Energy Cascade • Kolmogorov Hypotheses • Scalar Transport Equations • Large Eddy Simulation 77
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