Numerical Simulation of Laminar Diffusion Flames

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Numerical Simulation of Laminar Diffusion Flames
Craig C. Douglas Alexandre Ern Mitchell D. Smooke
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• Goals
• Model high heat release combustion problems
• Predict the effects of combining chemical
  reactions with heat and mass transfer phenomena

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Improving Jet Engine Efficiency
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Improving Combustion and System
EfficienciesTurbulent Combustion
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Improving CombustionCompact Waste Incinerator
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Flame Treatment of Polymer Films
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Research Projects

• Combustion of Solid Rocket Propellants
• Microgravity Combustion of Structural Metals
• Combustion-Generated Air Pollution
• Air Quality Impacts of Alternative Fuels and
  Reformulated Gasoline
• Effects of Subsonic Aircraft Emissions on
  Chemistry in the Upper Troposphere

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Turbulent Premixed Flame
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Axisymmetric Laminar Diffusion Flames

• Flame type of several combustion devices
• Complications
• Large number of dependent unknowns
• Governing partial differential equations are
  nonlinear
• Different length scales due to regions of high
  activity

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Flame Sheet Model

• Adds only one field to the underlying flow fields
• Greatly reduces cost of computer simulations
• Coupling and nonlinearity features are the same
  in both models
• Temperature and major species profiles can be
  obtained from a conserved scalar equation

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Physical Configuration
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Formulations of the Navier-Stokes Equations

• Streamfunction-Vorticity
• eliminates pressure as a dependent variable
• reduces the number of equations by one
• continuity is explicitly satisfied locally
• accurate vorticity boundary conditions are hard
  to specify
• results in ill-conditioned Jacobians
• does not reasonably extend to 3 dimensions
• Primitive Variable
• Vorticity-Velocity

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Formulations of the Navier-Stokes Equations

• Streamfunction-Vorticity
• Primitive Variable
• easily extendible to three dimensions and
  unsteady problems
• allows for accurate boundary conditions
• staggered grid creates problems for multi-grid
  solvers and in complex geometries
• Vorticity-Velocity

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Formulations of the Navier-Stokes Equations

• Streamfunction-Vorticity
• Primitive Variable
• Vorticity-Velocity
• allows for more accurate vorticity boundary
  conditions than streamfunction-vorticity
• yields better conditioned Jacobians
• uses nonstaggered grids
• extends easily to three dimensions

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Notation
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Notation
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Notation
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Diffusion Flame Governing Equations
Radial Velocity
Axial Velocity
Vorticity
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Additional Equations
Equation of State
Shvab-Zeldovich Equation
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Boundary Conditions
Axis of Symmetry
Outer Zone
Inlet
Exit
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Solution
Grid
Dependent Variables
Nodes
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Finest Grid
Coarsest Grid
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• Spatial Operators finite
  differences
• Diffusion and Source Terms centered
  differences
• Convective Terms monotonicity
  preserving upwind scheme

(1)
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• Boundary conditions - first order back or forward
  differences
• Exceptions
• vorticity inlet boundary condition
• axial velocity boundary condition at r 0

(2)
(3)
(4)
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Discretization of the partial differential
equations and boundary conditions yield nonlinear
equations
The equations are solved using a damped Newton
method
(5)
Convergence tolerance
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• Algorithms
• Bi-CGSTAB/GS
• GMRES/GS

Parabolic Problem
(6)
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Multigrid Methods

• One Way Multigrid
• course grid problem is solved and the solution is
  interpolated onto the next finer grid as an
  initial guess
• time relaxation process only on coarsest level

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Multigrid Methods

• Damped Newton Multilevel Multigrid
• makes use of Jacobians computed at all levels
  coarser than the current level
• significantly faster than one way multigrid
• requires 62 Mb of storage as compared to 39 Mb
  for the one way multigrid

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Numerical Results
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Sources
Douglas, C. C., and A. Ern, Numerical Solution
of Flame Sheet Problems with and without
Multigrid Methods, in Sixth Copper Mountain
Conference on Multigrid Methods, N. D. Melson, T.
A. Manteuffel, and S. F. McCormick, eds., vol. CP
3224, Hampton, VA, 1993, NASA, pp.
143-157. Douglas, C. C., A. Ern, and M. D.
Smooke, Multigrid Solution of Flame Sheet
Problems on Serial and Parallel Computers,
1994. Douglas, C. C., A. Ern, and M. D. Smooke,
Numerical Simulation of Laminar Diffusion Flames,
1994. Ern, A., Vorticity-Velocity Modeling of
Chemically Reacting Flows, PhD Thesis, Yale
University, February 1994. Mechanical and
Engineering Department. Ern, A., C. C. Douglas,
and M. D. Smooke, Detailed Chemistry Modeling of
Laminar Diffusion Flames on Parallel Computers,
1994.
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Numerical Solution of Laminar Diffusion Flames
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