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
• Turbulent Premixed Combustion
• Types of turbulence
• Turbulent Non-Premixed
• Mean-flow Equations
Combustion
• Modelling Turbulent Combustion
• Applications
• Reynolds Stress Equations
• k-Equation
• Turbulence Models
• 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
• Turbulent Premixed Combustion
• Types of turbulence
• Turbulent Non-Premixed
• Mean-flow Equations
Combustion
• Modelling Turbulent Combustion
• Applications
• Reynolds Stress Equations
• k-Equation
• Turbulence Models
• 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
 Time average
N and Δt
sufficiently large
• For constant density flows:
 Reynolds decomposition: mean and fluctuation, e.g. for the flow velocity ui
7
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
9
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
Part II: Turbulent Combustion
• Characteristics of Turbulent Flows
• Statistical Description of Turbulent Flows
• Reynolds decomposition
• Turbulence
• Favre decomposition
• Turbulent Premixed Combustion
• Types of turbulence
• Turbulent Non-Premixed
• Mean-flow Equations
Combustion
• Modelling Turbulent Combustion
• Applications
• Reynolds Stress Equations
• k-Equation
• Turbulence Models
• 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: x1-component of the velocity
 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: www-ah.wbmt.tudelft.nl
„Temporally evolving shear layer“: Scalar dissipation rate χ (left), mixture fraction Z (rechts)
14
Turbulent jet: magnitude of
vorticity
Example: DNS of Homogeneous Shear Turbulence
Scalar dissipation rate in homogeneous shear turbulence
2048x2048x2048 collocation points
15
Close-up/detail
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
• Turbulent Premixed Combustion
• Types of turbulence
• Turbulent Non-Premixed
• Mean-flow Equations
Combustion
• Modelling Turbulent Combustion
• Applications
• Reynolds Stress Equations
• k-Equation
• Turbulence Models
• Scales of Turbulent Flows/Energy Cascade
• Kolmogorov Hypotheses
• Scalar Transport Equations
• Large Eddy Simulation
17
Mean-flow Equations
• Starting from the Navier-Stokes-equations for incompressible fluids
(continuity)
(momentum)
→ Four unknowns within four equations: u1, u2, u3, p
• 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.

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
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
• Turbulent Premixed Combustion
• Types of turbulence
• Turbulent Non-Premixed
• Mean-flow Equations
Combustion
• Modelling Turbulent Combustion
• Applications
• Reynolds Stress Equations
• k-Equation
• Turbulence Models
• 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
with
25
and a corresponding equation for
leads after summation to
*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 pressure-terms 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 Reynolds-stress tensor and
the gradient of time-averaged velocity)
29
*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
*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
• Turbulent Premixed Combustion
• Types of turbulence
• Turbulent Non-Premixed
• Mean-flow Equations
Combustion
• Modelling Turbulent Combustion
• Applications
• Reynolds Stress Equations
• k-Equation
• Turbulence Models
• 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 pressure-rate-of-strain 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: pipe-flow
37
example: free jet
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
• Turbulent Premixed Combustion
• Types of turbulence
• Turbulent Non-Premixed
• Mean-flow Equations
Combustion
• Modelling Turbulent Combustion
• Applications
• Reynolds Stress Equations
• k-Equation
• Turbulence Models
• 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
Turbulent-viscosity models
• Algebraic models: e.g. Prandtl´s mixing-length concept
• TKE models: e.g. Prandtl-Kolmogorov
• k-ε-Modell (Jones, Launder)
41
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.
42
TKE model: Prandtl-Kolmogorov
• Eddy viscosity
 Model constant Cμ (often: Cμ = 0,09)
 lpk: characteristic length scale  determined empirically
• Equation for TKE
43
Two-equation-model: k-ε-model
• Eddy viscosity
• Solving one equation each for
 TKE
 dissipation
 the model parameters need to be determined empirically
44
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
Course Overview
Part II: Turbulent Combustion
• Characteristics of Turbulent Flows
• Statistical Description of Turbulent Flows
• Reynolds decomposition
• Turbulence
• Favre decomposition
• Turbulent Premixed Combustion
• Types of turbulence
• Turbulent Non-Premixed
• Mean-flow Equations
Combustion
• Modelling Turbulent Combustion
• Applications
• Reynolds Stress Equations
• k-Equation
• Turbulence Models
• Scales of Turbulent Flows/Energy Cascade
• Kolmogorov Hypotheses
• Scalar Transport Equations
• Large Eddy Simulation
46
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
 Two-point correlation
47
and
Correlation Function
• Homogeneous isotropic turbulence:
,
• Two-point 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 lt (largest eddies)
 Integral velocity scale
 Integral time scale
49
Energy Spectrum
Energy Spectrum (logarithmic)
Energy Cascade
energy density
Energy
Transfer
of Energy
wave number
50
Dissipation
of Energy
Course Overview
Part II: Turbulent Combustion
• Characteristics of Turbulent Flows
• Statistical Description of Turbulent Flows
• Reynolds decomposition
• Turbulence
• Favre decomposition
• Turbulent Premixed Combustion
• Types of turbulence
• Turbulent Non-Premixed
• Mean-flow Equations
Combustion
• Modelling Turbulent Combustion
• Applications
• Reynolds Stress Equations
• k-Equation
• Turbulence Models
• 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, small-scale eddies have a universal
form. They are determined by two parameters
 Dissipation
 Kinematic viscosity
• Dimensional analysis
 Length η
 Time τη
 Velocity uη
52

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
53
Course Overview
Part II: Turbulent Combustion
• Characteristics of Turbulent Flows
• Statistical Description of Turbulent Flows
• Reynolds decomposition
• Turbulence
• Favre decomposition
• Turbulent Premixed Combustion
• Types of turbulence
• Turbulent Non-Premixed
• Mean-flow Equations
Combustion
• Modelling Turbulent Combustion
• Applications
• Reynolds Stress Equations
• k-Equation
• Turbulence Models
• 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
transport
55
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
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 τZ (dimensional analysis)
with
• Typically proportional to τ
and
• This leads to 
59
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
60
due to chemical reactions (heat release)
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 Yα of all components α = 1, … N
 Temperature T
• Balance equations for
 Di: mass diffusivity, thermal diffusivity
 Si: mass/temperature source term
62
Transport Equation for Reactive Scalars
• Derivation of a transport equation for
• Favre decomposition
and averaging of
leads to
not closed
molecular
transport
63
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
64
Not closed  chapter
„Modelling of Turbulent
Combustion“
Course Overview
Part II: Turbulent Combustion
• Characteristics of Turbulent Flows
• Statistical Description of Turbulent Flows
• Reynolds decomposition
• Turbulence
• Favre decomposition
• Turbulent Premixed Combustion
• Types of turbulence
• Turbulent Non-Premixed
• Mean-flow Equations
Combustion
• Modelling Turbulent Combustion
• Applications
• Reynolds Stress Equations
• k-Equation
• Turbulence Models
• Scales of Turbulent Flows/Energy Cascade
• Kolmogorov Hypotheses
• Scalar Transport Equations
• Large Eddy Simulation
65
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 = Ret9/4
66
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
68
Large-Eddy Simulation
69
Large-Eddy 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
Large-Eddy Simulation
• For example:
• Box filter in 1D:
• Sharp spectral filter:
71
Large-Eddy Simulation
Pope, „Turbulent Flows“
72
Large-Eddy Simulation
• Filtered momentum equation:
• Define residual stress tensor:
73
Large-Eddy Simulation
Sub-filter Modeling
• Eddy viscosity model for
• Filtered strain rate tensor
74
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
Large-Eddy Simulation
• Smagorinsky length scale
• Similar equations can be derived for scalar transport

76
System of equations closed!
Summary
Part II: Turbulent Combustion
• Characteristics of Turbulent Flows
• Statistical Description of Turbulent Flows
• Reynolds decomposition
• Turbulence
• Favre decomposition
• Turbulent Premixed Combustion
• Types of turbulence
• Turbulent Non-Premixed
• Mean-flow Equations
Combustion
• Modelling Turbulent Combustion
• Applications
• Reynolds Stress Equations
• k-Equation
• Turbulence Models
• Scales of Turbulent Flows/Energy Cascade
• Kolmogorov Hypotheses
• Scalar Transport Equations
• Large Eddy Simulation
77