CFD modeling of combustionPart 2

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Rixin Yu

2017-09-01

A general guide for CFD of reacting flow

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Know the major physics of governing your problem:Important physics includes:

Low/high speed flow, non-negligible acoustic interaction? Combustion modes: premixed / non-premixed/ auto-ignition? Laminar/ turbulent flow

Find the characteristic scales (in time and space) of your physical problem. Turbulence: estimate the largest and smallest flow scale.Combustion:

Flame dominance: flame thickness/speed, inner-reaction-zone thicknessAuto-ignition dominance : Ignition delay timeKinetic dominance: time-scale of various elementary reactions

Check the overlapping in scales from different physicsDecouple scales differing by order of magnitude(stiffness remover)Otherwise, either resolve those scales or use a good model.

Chose an modelling framework0D, 0.5D, 1D, 2D, Laminar, turbulent, (RANS, LES, DNS)

Estimate how many grid cells you can afford based your accessible computing power

Chose appropriated numerical schemes and solution method, boundary condtionsTVD scheme for problem contain discontinty, FD method for smooth problem

Scales of various nature phenomena

3While most natural phenomena affecting human survival are either at large scales (firestorm, glacier movement) or small scales (lightning, mites) occurring at large or small scale velocities, respectively, technical combustion devices operate at the humanscale of the order of 1 m and at velocities comparable to the laminar burning velocity which is of the order of 1 m/s. [Peters]

Scales in the system of turbulent reacting flow

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What we mean by saying “reacting flow”

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Chemical Reaction(Gas phase) It involves a mixture of multi-components species

Different thermodynamic proprieties Heat capacity, Molecular weight,…

It is governed by a large (detail) chemical kinetic mechanismMulti-elementary reactions, Nonlinear reaction rates…

……Transport-coupling of flow and reaction

Multi-component species different mass diffusivities, heat conductivity

Reaction releases heatdilation, density and viscosity variation…

…….Flow:

Laminar flow of various typeFlow instability, transition to turbulence, ….

Turbulent flowA wide range of cascading scales, ….

High speed compressible flow shock wave and rarefaction wave …..

…..

Let first look an “isolated”+“stationary” 0-D reacting systemNeglect transport, or in other words, neglect derivative in space

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It is a non-linear dynamical system (Examples:pendulum system, three-body problem, lots of math):

A set of ODE equations solved for ( ), , ( ), k=1,,,N) , starting at = 0.

The solution to the ODEs is a trajectory in high dimensional phase space, spanned by N+2 unknown coordinates.

a) Simple algebraic constraints given by conservation of elements and total mass conservation can help to reduce the number of unknowns.

A note from theoretical chemistry: chemical reactions do not have to be dominated by a equilibriumthermodynamic behavior! (It may have some type of limit-cycle or chaotic orbit) .

YouTube showing Belousov-zhabotinsky reaction!

https://www.youtube.com/watch?v=0Bt6RPP2ANI#t=00m29s

The famous “butterfly” trajectory shown in 3-dimentional phase space

The Belousov-zhabotinsky reaction!

Theoretic and numerical aspects for 0-D reacting system

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A set of ODE equations solved for ( ), , ( ), k=1,,,N) , starting at = 0.

1) For most gas-phase combustion, there often exists fast and slow elementary reactions, the time scales may differ by several order of magnitude. It is a mathematical multiple-time-scales “stiff” system, an expensive adaptive-time-step ODE solver must be used to perform numerical time-integration.

1) Such calculation is usually performed by “Popular” software package: such as Chemkin(not free), Cantera (free) and Flamemaster … . Note, evolutions of all thermodynamic and transport coefficients ( , , Δℎ , ,.. , ) are usually based on based NASA polynomials, the chemical kinetic mechanism including all elementary reactions and the reacting constants can be downloaded together with a published journal article.

2) For common gas-phase reaction, there often exist certain “intrinsic lower-dimentionalmanifolds” (ILDM) in the phase space, towards which a trajectory will be quickly attracted. When the trajectory come close to the vicinity of such “manifold” region, the solution along trajectory then stay parallel and move slowly within such “manifold”.

3) Very expensive calculations of stiff-ODE solver for every CFD-cells. Ideal: Tabulation

The In-situ adaptive-tabulation (ISAT), by S.B. Pope.

Let we look an ½-D reacting system composed of multiple “stationary” zones but sharing the same pressure (i.e. drop “isolate”)

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Now enable transport, (≥ 1D) laminar reacting system

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Premixed flame Diffusion flame

Different color, because of light emitted by different excited radicals

Turbulent D) reaction flow

10OH-PLIF image of turbulent premixed flame

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A YouTube video showing the difference between (1) igniting a premixed H2/O2 balloon (Premixed combustion )(2) igniting a pure-H2 balloon (Non-premixed combustion)

https://www.youtube.com/watch?v=qOTgeeTB_kA#t=03m52s

Non-premixed (diffusion) combustion

12Photograph of a non-premixed n-decaneflame stabilized in the counterflow

Fuel and oxidizer does not mix prior to chemical reaction

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Non-premixed combustion in DIESEL engine

High-speed video of DIESEL combustion.

(Injected atomized fuel burn in compressed hot air)

Non-premixed laminar jet flames

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Fuel and oxidizer does not mix prior to chemical reactionHow much percentage of mass coming from the fuel stream?

A simple model of non-premixed combustionmixture fraction equation Z

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Non-premixed laminar flames (example of using mixture fraction)

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A certain point in the domain has a

mixture faction ( , ) = 0.1 10% mass percentage contributed by fuel (CH4),

90% mass percentage contributed by oxidizer(O2)

1 ⋅ + 2 ⋅ ⇒ 1 ⋅ + 2 ⋅1Δ ∶ 2Δ ∶ 1Δ ∶ 2Δ1W 16 ∶ 2W (64): 1W (44) ∶ 2W (36)

(10-10*1)%= 0% mass of CH4(90-10*4)%=50% mass of O2(0+10* )%=27.5% mass of CO2

(0+10* )%=22.5% mass of H2O

Fuel and O2 can not co-existDo not need solve species equations.

mole:mass:

1 : 4 : :

Temperature is determined similar to computing adiabatic temperature

Initial unburned: 10% CH4 ( ) + 90% O2 ( )

First law of thermodynamic⇒ ?(50% , 27.5% , 22.5% )

Non-premixed combustionAn simplified transport equation for mixture-fraction

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+ = ( )No reaction-rate type of source term!! Easy to solve

N-1 species mass equations

+ Energy equations

reduce

Reaction quantities ( , ) is expressed as functions of . ( , ) = ( , )( , ) = ,

Laminar premixed flame

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Premixed combustion in Spark-ignition (SI) enginePropagating flames!

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https://www.youtube.com/watch?v=xflY5uS-nnw#t=04m50shttps://www.youtube.com/watch?v=xflY5uS-nnw#t=05m23s

A true transparent engine, high-speed video showing deflagration wave after spark

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Laminar flame speed (deflagration wave)the self-propagation speed relative (normal) to the side of fresh reactants

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is the laminar flame (deflagration) speed .

is a characteristic diffusivity , is time scale of chemical reaction

Analytic solution of is given by Zeldovich, Frank-Kamenetskiand von Karman (ZFK) analysis.

~

ℎ and D varies with , therefore S changes!

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What is the physics for the self-propagation of deflagration front?

Preheating: heat and radical diffused from hot product side

Structure of laminar premixed flameVarious forms of thickness characterization

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Structure of a lean methane/air flame (equivalence ratio Φ=0.6), including definition of different layers: inert preheating layer δT, reaction layer δrconsisting of an inner layer with thickness δ and oxidation layer with thickness ϵ; dotted line indicates the heat release profile.

Overall laminar flame thickness = / :~ ∶Thickness of inner reaction zone :~Non-dimentional Zel’vodich number ≡ Also notes, the thickness for different species layer are also different!

Numerical solver for laminar premixed flame

• Chemistry (t) + transport (x) = P.D.E. – Determine the steady-state flame-speed is an eigenvalue problem.

• Chemistry software package usually offers premixed-flame solver. – The “PRIMIX” package in “Chemkin”

• A guess-and-trial algorithm – Adaptively refined the spatial mesh to capture large gradient

• Require users to adjust some parameters– Need a reasonable initial guesses – Sometime it can be quite difficult for complicate kinetic mechanism.

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A modelling framework for (deflagration) premixed flamethe “flame-let” assmption

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If( >> ) :Assume the profiles of combustion quantitates ( , , ) along the normal direction

of any local flame front elements do not deviate far from its laminar solution in a pseudo-planar configuration(which is an easier, subset problem); the main quantity remains unknown and need to be modeled is the local self-propagation speed ( ) of flame front interface which separates the hot product from cold fresh reactants.

cold, fresh fuel

hot, products

How to model premixed flame ?(a) the level-set equation

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( >> ) :One such a model can be implemented using the

“leve-set” equation for a distance function ( , )representing the distance from local point to the nearest interface. + ⋅ = ⋅

| | = 1The flow( ) and combustion quantitates ( , , ) can

be determined from G. cold, fresh fuel

hot, products

How to model premixed flame ?(b) The reaction-progress variable equation (ANSYS Fluent)

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Another approach is so called the reaction progress variable (c) approach, c=0 denotes fresh gas, c=1 denotes products and 0<c<1 describe the middle flame zone. An transport equation for c is used to describe the evolution of c-field, which is basically similar to a global representive specie (or T) equitation, model should be introduced for the reaction source term to yield a correct .

cold, fresh fuel

hot, products

= 0

= 1= 0.5 = 0.9= 0.1 = − ( )( ) − ( ) = 1 − ( )+ →

+ = + General in Arrieuhusform

+ ⋅ ≈ ⋅Goal of the model

Premixed combustion at low and high speedFuel and oxidizer are mixed prior to combustion.

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Low speed, incompressible flow(deflagration wave)

High speed compressible flow. Shockwave+heat relase Detonation wave

also responsible for engine knock!

A illustrative video, showing the difference between deflagration and detonation in a spark igntion (SI)engine

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-Deflagration (Normal combusiton)-Detonation (Abnormal)

https://www.youtube.com/watch?v=4ZysyokEU60

Theory: detonation and deflagrationRayleigh line, Hugoniot curve in the p-v digaram

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Slides taken from Matalon’s princeton lecture (2011).

(with heat release)(no heat release)

Deflagration to Detonation Transition (DDT)

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What is the structure of in a 1D detonation wave?Zel’dovich, von Neuman, Doring (ZND) structure

31Numerical requirement for resolve an detonation ?

What is the physics for the self-propagation of detonation front?Shockwave compress heating!

Premixed combustion at low and high speed

• Fuel and oxidizer are fully mixed prior to combustion – The flame fronts propagates into the fresh reactants mixture in a self-sustained

manner• Low speed : Deflagration:

– First ignition. » external heat source (Spark)

– Self-sustained propagation » Preheating: heat from the hot product side diffuses to the unburned side.

• High speed: Detonation– First starting of detonation:

» Auto-ignition of high-reactivity reactant mixture pocket. » Compression by the pressure spike due to travelling pressure waves

wall reflected at wall ; or local hot spots heating. – Self-sustained propagation

» The leading (non-reacting) shock compresses and heat up the reactants.

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Combustion instability in premixed combustionWhen self-propagating premixed flame goes above “1D”

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Intrinsic flame instability

– Mechanisms of premixed flame instability• 1) Landau-Darrieus instability(hydrodynamic

instablity)– Heat relased caused dillation

gas expansion and density difference

• 2) Diffussive-thermal instabilty– Heat diffuses differently with the reactant mass

• 3) Rayleigh-Taylor instability– Accelerate light matter into heavy matter

(velocity difference + density difference)

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Landau-Darrieus instabilitySketch explanation of mechanism

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A01A>A01u<SL

u0=SL

A02A<A02u>SL Flame

SL

Landau-Darrieus instability in planar flameSimulations

Shape characteristics:

Cusps, troughs,Cells (3D)

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Numerical simulation of fractal flame front structure in wide channel developed due to Landau-Darrieus instability

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Diffusion-Thermal instabilitySketch explanation of the mechanism

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Hot productscold fresh reactant

A

B

Le<1Le=1

Where is your tip?

Rayleigh-Taylor instability

Top heavier

Wikipedia

”fingers” ”mushroom”

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Various types of Combustion instabilityDiffusive-thermal, Darrius-Landau, Rayleigh-Taylor

40Spiral waves over propagating H2/Air flames (Jomaas, Law & Bechtold ,2007)

Law, 2000

Rayleigh-Taylor

Darrius-Landau

Diffusive-thermal

Turbulent combustionTurbulence

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A range of flow scales, at “each scale” has a “charactering velocity” and “time”

… . ≡ /…. = / = … . = /

Length [ ]velocity [ ]time[ ]

~ /

= /= /= /

Cascading

Turbulence eddies interact with the premixed flame

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Preheat material reaction zone

Direct interactions happens if two phenomena have overlapped scales

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Turbulent premixed combustionNon-dimensional numbers for characterizing the interaction between turbulence

and combustion

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Assume: = = == == =( = )= =

….….

Premixed combustion regime diagram

Recap of turbulence modellingRANS, LES, DNS

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RANS models all

LES model smaller eddy

DNS models nothing

Average a turbulent flame front

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Your eye indeed does the averaging

High speed single-shot

Average using math

Equations to model turbulent reacting flowwe need introduce Favre-average

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For example: Take average

Average breaks, enters derivative

+ =. .For two variables, this isalready solved issuefor incompressible eq.

Use a general average-operator directly on all the governing equations RANS : Reynolds (time) average operator LES : Spatial-filtering operatorDNS : null operator

Problems

Introduce Favre averaging ( )

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⋅ ⋅ = + + + = ⋅ ⋅ + ⋅ + ⋅ ′ +….+ ′⋅ = ( + ′)( + ) = + ′ + ′ + ′ = ⋅ + 0 + 0 + ′Previously: = . apply Reynold average ( )

Now : = ++ ( ) = − + Too many non-zero terms !

Still want to keep as simple as the two-variables-average?

Solution: Faver average ( ):

A new decompsoition ( ′′):

+ ( ) = −1 + 1

Equations for turbulent reacting flowSolve for Favre-averaged unknowns quantiles, put all complexity into a turbulent flux term

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+ = ⋯( )

Exact

Averaged formed

Turbulence flux term to be modeled

= [1, ℎ , , ]⋅ ⋅ = [ + + ]= + 0 + 0 + ′′

Equations for turbulent reacting flowOne more issue: the average reaction rate term

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+ = ⋯+( ) +

This term is highly nonlinear, difficult to model.Some methods: (1) Assumed we know the PDF , then do the averaging

(2) For turbulent premixed flame, it is often the grid is too coarse to resolve the flame thickness, this term is then usually model together with the turbulent flux term to yield a correct turbulent flame speed. …

Other combustion modes

• Partially premixed combustion – Some regions are premixed combustion, other

regions are non-premixed combustion.

• Quenching and Auto-ignition– Toward “clean”, low-emission combustion

• Combustion at very lean ( <1) condition, low temperature

– Development of partially premixed compression ignition (PPCI) engine.

• Diesel + SI + HCCI

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diffusionflame

premixedflame

Other reacting-flow topics• Combustion acoustic instability

– The Rayleigh criterion , which measures the correlation between pressure and heat release (Resonance).

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supersonic combustion (denotation?)

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The scramjets 6 ≤ ≤ 15 , ramjet 3 < < 6, normal jet engine0 < < 3Combustion of fuel release in a supersonic flow https://www.youtube.com/watch?v=COCDKWeU2Fw#t=04m52s