CEFRC Combustion Summer School Lecture Notes/Pitsch/Lecture7...• Combustion requires mixing at the molecular level ... • Types of turbulence • Mean-flow Equations • Reynolds

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



Combustion


• Applications








• k-Equation







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

7

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

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

11

• Turbulence



Combustion


• Applications








• k-Equation







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:

12

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)

13

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

14

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

15

Close-up/detail Scalar dissipation rate in homogeneous shear turbulence 2048x2048x2048 collocation points

Example: DNS of a Shear Flow

16

statistically homogeneous

Statistically homogeneous

inhomogeneous

Scalar dissipation rate

Course Overview

17

• Turbulence



Combustion


• Applications








• k-Equation







Mean-flow Equations

• Starting from the Navier-Stokes-equations for incompressible fluids

→ Four unknowns within four equations: u1, u2, u3, p


18

(continuity)

(momentum)

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

Time-averaging yields

→ This term includes product of components of fluctuating velocities: this is due to the non-linearity of the convective term

20

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



Combustion


• Applications








• k-Equation







*Transport Equation for Reynolds Stress Tensor

24


Multiplication of the equation

with the fluctuating velocity and a corresponding equation for with leads after summation to

25

The viscous terms on the right hand side of

can be transformed into


26


Splitting of the pressure-terms in

with Kronecker delta

27


Averaging and rearranging leads to Six new equations, but far more new unknowns

28


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


• „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


• „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

33

• Turbulence



Combustion


• Applications








• k-Equation







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


• Continuity equation pressure-rate-of-strain correlation PSC = 0

• Dissipation

• Mean dissipation of turbulent kinetic energy

35


• 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

example: pipe-flow example: free jet


• Transport equation

• BUT: Closure problem is not solved

Triple correlations

Derivation of equations for such correlations even higher correlations…

38

Course Overview

39

• Turbulence



Combustion


• Applications








• k-Equation







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

46

• Turbulence



Combustion


• Applications








• k-Equation







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

47

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

48

correlation function

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

50

energy density

wave number

Energy Spectrum (logarithmic) Energy Cascade

Energy

Transfer of Energy

Dissipation of Energy

Course Overview

51

• Turbulence



Combustion


• Applications








• k-Equation







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

53

Course Overview

54

• Turbulence



Combustion


• Applications








• k-Equation







Scalar Transport Equations

• Transport equation for mixture fraction Z

• Favre averaging

55

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

56


• Variance equation

• First step: equation for

57


• 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

58

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

59

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)

60


• Temperature equation is similar to the equation for the mass fraction of component α

61


• 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


• Derivation of a transport equation for

• Favre decomposition and averaging of leads to

63

not closed

molecular transport

turbulent transport

averaged source term


• 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

65

• Turbulence



Combustion


• Applications








• k-Equation







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

66


• Example: Turbulent Jet with Re = 15000

• This is for one integral length scale only!

67 Pope, „Turbulent Flows“


Large-Eddy Simulation (LES)

• Spatial filtering as opposed to RANS-ensemble averaging

• Sub-filter modeling as opposed to DNS

68


69


• Spatial filtering rather than ensemble average

70

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


• For example:

• Box filter in 1D:

• Sharp spectral filter:

71


72 Pope, „Turbulent Flows“


• Filtered momentum equation:

• Define residual stress tensor:

73


Sub-filter Modeling

• Eddy viscosity model for

• Filtered strain rate tensor

74


• Smagorinsky model for

(in analogy to mixing length model)

• Sub-filter eddy viscosity

• Sub-filter velocity fluctuation

with filtered rate of strain

75


• Smagorinsky length scale

• Similar equations can be derived for scalar transport

76

System of equations closed!

Summary

77

• Turbulence



Combustion


• Applications








• k-Equation







Documents

CEFRC Combustion Summer School Lecture Notes/Pitsch/Lecture7...• Combustion requires mixing at the molecular level ... • Types of turbulence • Mean-flow Equations • Reynolds