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Turbulence
CEFRC Combustion Summer School
Prof. Dr.-Ing. Heinz Pitsch
2014
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 ↑
2
oxidizer fuel + = diffusion
Surface Area ↑
diffusion
Course Overview
3
• 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
Part II: Turbulent Combustion
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
6
• 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
Part II: Turbulent Combustion
Statistical Description of Turbulent Flows
Conventional Averaging/Reynolds Decomposition
• Averaging
Ensemble average
Time average
• For constant density flows:
Reynolds decomposition: mean and fluctuation, e.g. for the flow velocity ui
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N and Δt sufficiently large
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
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
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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
Course Overview
11
• 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
Part II: Turbulent Combustion
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:
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Scalar dissipation rate in statistically homogeneous turbulent flow
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)
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DNS of statistically homogeneous and isotropic turbulence: x1-component of the velocity
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
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Turbulent jet: magnitude of vorticity
„Temporally evolving shear layer“: Scalar dissipation rate χ (left), mixture fraction Z (rechts)
Quelle: www-ah.wbmt.tudelft.nl
Example: DNS of Homogeneous Shear Turbulence
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Close-up/detail Scalar dissipation rate in homogeneous shear turbulence 2048x2048x2048 collocation points
Example: DNS of a Shear Flow
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statistically homogeneous
Statistically homogeneous
inhomogeneous
Scalar dissipation rate
Course Overview
17
• 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
Part II: Turbulent Combustion
Mean-flow Equations
• Starting from the Navier-Stokes-equations for incompressible fluids
→ Four unknowns within four equations: u1, u2, u3, p
• Reynolds decomposition
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(continuity)
(momentum)
Averaged Continuity Equation
1. From continuity equation it follows and
→ Linearity of the continuity equation: no correlations of fluctuating quantities
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Averaged Momentum Equation
2. This does not apply for the momentum equation!
Convective term
Time-averaging yields
→ This term includes product of components of fluctuating velocities: this is due to the non-linearity of the convective term
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Contin.
Contin.
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
23
• 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
Part II: Turbulent Combustion
*Transport Equation for Reynolds Stress Tensor
Multiplication of the equation
with the fluctuating velocity and a corresponding equation for with leads after summation to
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The viscous terms on the right hand side of
can be transformed into
*Transport Equation for Reynolds Stress Tensor
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*Transport Equation for Reynolds Stress Tensor
Splitting of the pressure-terms in
with Kronecker delta
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*Transport Equation for Reynolds Stress Tensor
Averaging and rearranging leads to Six new equations, but far more new unknowns
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*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)
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*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
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*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“)
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Transport Equation for Reynolds Stress Tensor
Transport equation for Reynolds stress tensor Six new equations, but far more new unknowns
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Course Overview
33
• 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
Part II: Turbulent Combustion
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
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Transport Equation for Turbulent Kinetic Energy
• Continuity equation pressure-rate-of-strain correlation PSC = 0
• Dissipation
• Mean dissipation of turbulent kinetic energy
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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
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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
39
• 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
Part II: Turbulent Combustion
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
Turbulent-viscosity models
• Algebraic models: e.g. Prandtl´s mixing-length concept
• TKE models: e.g. Prandtl-Kolmogorov
• k-ε-Modell (Jones, Launder)
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Algebraic Model: Prandtl´s Mixing-length Concept
• Eddy viscosity
• Based on dimensional analysis
• All unknown proportionalities mixing-length
• Empirical methods for determining lm
• Assumption: lm = const.
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TKE model: Prandtl-Kolmogorov
• Eddy viscosity
Model constant Cμ (often: Cμ = 0,09)
lpk: characteristic length scale determined empirically
• Equation for TKE
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Two-equation-model: k-ε-model
• Eddy viscosity
• Solving one equation each for
TKE
dissipation
the model parameters need to be determined empirically
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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
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Course Overview
46
• 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
Part II: Turbulent Combustion
Scales of Turbulent Flows/Energy Cascade
Two-Point Correlation
• Characteristic feature of turbulent flows: eddies exist at different length scales
• Determination of the distribution of eddy size at a single point
Measurement of velocity fluctuation and
Two-point correlation
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x x + r
Turbulent round jet: Reynolds number Re ≈ 2300
Correlation Function
• Homogeneous isotropic turbulence: ,
• Two-point correlation normalized by its variance
• Degree of correlation of stochastic signals
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correlation function
Integral Turbulent Scales
• Largest scales: physical scale of the problem
Integral length scale lt (largest eddies)
Integral velocity scale
Integral time scale
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Energy Spectrum
50
energy density
wave number
Energy Spectrum (logarithmic) Energy Cascade
Energy
Transfer of Energy
Dissipation of Energy
Course Overview
51
• 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
Part II: Turbulent Combustion
Kolmogorov Hypotheses
52
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 τη
Velocity uη
Second Kolmogorov Hypothesis
• At sufficiently high Reynolds numbers, the statistics of the motions of scale r in the range η << r << lt have a universal form that is uniquely determined by
Dissipation
But independent of kinematic viscosity
→ Inertial subrange
Integral length scale
Ratio η/lt
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Course Overview
54
• 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
Part II: Turbulent Combustion
Scalar Transport Equations
• Transport equation for mixture fraction Z
• Favre averaging
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not closed
molecular transport
turbulent transport
Transport Equation for Mixture Fraction
• Neglecting molecular transport (assumption: Re↑)
• Gradient transport model for turbulent transport
Dt: Turbulent diffusivity
Sct: Turbulent Schmidt number
→ Transport equation for mean mixture fraction
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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
• Favre averaged scalar dissipation
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not closed
Modeling of Scalar Dissipation
Scalar dissipation rate has to be modeled
• Integral time τZ (dimensional analysis)
• Typically proportional to τ
• This leads to
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with
and
Transport Equation for Reactive Scalars
• Assumptions:
Specific heat cp,α = cp = const.
Pressure p = const., heat transfer by radiation is neglected
Lewis number Leα = Le = Sc/Pr = 1
• Temperature equation
• Source term due to chemical reactions (heat release)
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Transport Equation for Reactive Scalars
• Temperature equation is similar to the equation for the mass fraction of component α
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Transport Equation for Reactive Scalars
• The term „reactive scalar“ includes
Mass fractions Yα of all components α = 1, … N
Temperature T
• Balance equations for
Di: mass diffusivity, thermal diffusivity
Si: mass/temperature source term
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Transport Equation for Reactive Scalars
• Derivation of a transport equation for
• Favre decomposition and averaging of leads to
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not closed
molecular transport
turbulent transport
averaged source term
Transport Equation for Reactive Scalars
• Neglecting the molecular transport (assumption: Re↑)
• Gradient transport model for the turbulent transport term
→ Averaged transport equation
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Not closed chapter „Modelling of Turbulent Combustion“
Course Overview
65
• 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
Part II: Turbulent Combustion
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 nx = l/η = Ret3/4
• Minimum number of cells total nt = nx3 = Ret
9/4
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Large-Eddy Simulation
• Example: Turbulent Jet with Re = 15000
• This is for one integral length scale only!
67 Pope, „Turbulent Flows“
Large-Eddy Simulation
Large-Eddy Simulation (LES)
• Spatial filtering as opposed to RANS-ensemble averaging
• Sub-filter modeling as opposed to DNS
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Large-Eddy Simulation
• Spatial filtering rather than ensemble average
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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
Large-Eddy Simulation
Sub-filter Modeling
• Eddy viscosity model for
• Filtered strain rate tensor
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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
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Large-Eddy Simulation
• Smagorinsky length scale
• Similar equations can be derived for scalar transport
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System of equations closed!
Summary
77
• 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
Part II: Turbulent Combustion