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Computational Fluid Dynamics!
Modeling Complex Flows!
Grétar Tryggvason!Spring 2011!
http://www.nd.edu/~gtryggva/CFD-Course/!Computational Fluid Dynamics!
Combustion Modeling!
Computational Fluid Dynamics!
Gas combustion!!Gasoline engines, !!gas burners!
!Spray combustion!
!Jet engines, !!diesel engines!
!Combustion of solids!
!coal, !!wood, !!polymers!
Examples!
Computational Fluid Dynamics!
Diffusion flames!Most burners, candle!!Flame stays at the boundary between the fuel and the oxidizer!
Premixed flames!Some burners!Hazards!IC engines!!The flame separates unburned and burned mixture of fuel and oxidizer!
Computational Fluid Dynamics!
Diffusion Flames!
Computational Fluid Dynamics!
Diffusion flames!
Fuel, CH4 (methane), for example!
Oxidizer, O2!
The thickness of the flame depends on the ratio of the reaction rates to the diffusion times (Dahmköler number)!
CH4! O2!
Interim species!
CH4! O2!
Interim species!Fast Reaction!
Slow Reaction!
Computational Fluid Dynamics!
In turbulent flows the flame sheet usually folds in complex ways!
P = 4, inj =
0.3
P = 4, inj =
0.1
P = 4, inj =
0.5
Computational Fluid Dynamics!
In general, the combustion is a very complex process involving O(100) species and reaction rates. These are reasonably well known for a number of reaction, but still an open research field in general.!!For combustion of natural gas, GRIMech, for example!
Computational Fluid Dynamics!
For the full problem it is necessary to track the mass fraction of every species along with the momentum, mass, and energy conservation equations!
!!t
"mi + # $u"mi = # $ J + R
!!tci + " #uci = R
R = TnAE!Ek /RT "cii
Arrenius reaction rates!
Only a handful of computations of the full problem have been done so far!
Computational Fluid Dynamics!
For realistic situations, the problem must be simplified!
Diffusion Flames!• Use one-step (overall) reaction rates ! (if Da -> ∞, Burke Shuman limits)!• Use a reduced set of chemical reactions!
Can work very well for laminar flames. For turbulent flows the reaction rates have to be modified to account for stretching and folding of flame sheets.!
Computational Fluid Dynamics!
This is a conserved variable that is simply advected with the flow!
Given f, we can find each species fraction by!
For a simple one step reaction it can be shown that it is sufficient to follow one variable, called the mixture fraction!
!i = !i( f )
!!t("f ) + # $ ("fu) = # $D#f
f =mf
mf + mo
Computational Fluid Dynamics!
The Φ function can be constructed either assuming infinitely fast reactions (flame sheets) or equilibrium. The library is constructed once only. If the system is non-adiabatic, Φ is a function of the enthalpy also!
• The chemical system must be a diffusion flame and consists of a fuel and an oxidizer!• The Lewis number must be unity (all diffusion coefficients equal)!• Only one fuel type (can be a mixture)!• Only one type of oxidizer (can be a mixture)!• Incompressible turbulent flow!
Computational Fluid Dynamics!
For turbulent flows we solve for both f and the fluctuations of f and use those to determine the species fraction!
Determines how f is distributed and therefore how the species are distributed!
In the actual code a 2D look up table is first constructed, given the shape of the pdf!
f '( )2
smaller!
pdf!
f!
! = ! f , f '( )2" # $ %
& '
Computational Fluid Dynamics!
pdf!
f!
f!
t!
Constructing the pdf from measurements!
Computational Fluid Dynamics!
Premixed Flames!
Computational Fluid Dynamics!
Premixed flames!
!!t
"G + u # $"G =Uf $"GMotion due to fluid flow!
Motion due to burning!
For turbulent flow the flame speed is different from laminar flow due to wrinkling!
The flame speed is found experimentally or by detailed computations !
Uf
Flame is marked by G=0!
G<0!
G<0!
Computational Fluid Dynamics!
Other combustion models!
Droplets! Solid particles!
Evaporation!
Burning (usually)!
Gasification!
Burning!
Computational Fluid Dynamics!
As for multiphase flows, many issues are still unresolved in modeling of combustion and these models should be applied with care!
Computational Fluid Dynamics!
Multiscale—!Isolated “defects”!
Computational Fluid Dynamics!
Continuum theory, supplemented by constitutive assumptions, equations of state and transport coefficients describes the average motion of molecules much smaller than any length scale that we are usually considering when using continuum theory. !!Usually this works very well!!But, occasionally it does not. This often happens when the flow contains small regions where physics that can be ignored in most of the domain becomes important!
Computational Fluid Dynamics!
Current approaches usually consist of combining solutions to the Navier-Stokes or the Euler equations with much more expensive approaches for a small region of the flow. These are most often:!!Moleculear dynamics for singular regions such as a moving contact line!!Boltzmann equation solver for non-equilibrium regions in gas flow (including DSMC)!!Phase field equations provide a “mesoscale” description of some effects not usually included in continuum theories!
Computational Fluid Dynamics!
Example: !Compressible flow with regions far from thermodynamic equilibrium. The compressible Euler equations are solved for most of the domain but the Boltzmann equation is solved where needed. The solution to the Boltzmann is much more expensive so it is important to keep the regions where it is used as small as possible.!! Such situations can be found in shocks and boundary layers in hypersonic flows or in micro-electro-mechanical devices, for example!
Computational Fluid Dynamics!
Example: !Continuum theory fails to describe the motion of a contact line (where the interface between a liquid and a gas meets a solid boundary). Molecular Dynamics has been used to capture this dynamics, in simulations where the Navier-Stokes equations are used for most of the flow. Phase fields models have also been used to describe the motion of the contact line, but not in a hybrid simulation.!
Computational Fluid Dynamics!
Example: !Continuum theory fails to describe the rupture of thin films in multiphase flows. Molecular Dynamics as has been used to capture this dynamics, in simulations where the Navier-Stokes equations are used for most of the flow. As for the contact line, phase field models should be able to capture the motion.!
Computational Fluid Dynamics!
http://en.wikipedia.org/wiki/Orders_of_magnitude_(length)!
Computational Fluid Dynamics!
In computations we usually prefer a single physics, small range of scales and modest variation in physical parameters. Nature is the opposite: it is usually multi physics, with broad range of scales and extreme range of physical parameters. Developing computational strategies for problems occurring in nature is one of the great current scientific challenges and while it is already an active field, much more progress is likely to be made in the years to come!
