Numerical Simulation of Laminar Diffusion Flames by lap14150

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									Numerical Simulation of Laminar Diffusion Flames




               Craig C. Douglas
                Alexandre Ern
              Mitchell D. Smooke
Goals:
• Model high heat release combustion problems
• Predict the effects of combining chemical
  reactions with heat and mass transfer
  phenomena
Improving Jet Engine Efficiency
Improving Combustion and System Efficiencies:
            Turbulent Combustion
 Improving Combustion:
Compact Waste Incinerator
Flame Treatment of Polymer Films
            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
Turbulent Premixed Flame
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
            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
Physical Configuration




RI  0.2 cm      L f  3 cm
RO  2.5 cm      L  30 cm
Rm ax  7.5 cm   inlet velocity 35 cm/s
  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
 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
 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
               Notation


v  velocity vector  vr , v z 

                vr v z
  vorticity     
                z r
  mass density of the mixture
  shear viscosity of the mixture
                     Notation

T  temperature

Vk  diffusion velocity of k species
                                th


    
   Vkr , Vk z   
g  gravitational acceleration
                  Notation


           
     ,  
       z  r 


div (v)  cylindrical divergence of the velocity
          1            
               rvr   vz
          r r          z
      Diffusion Flame Governing Equations
Radial Velocity
              2vr  2vr  1 vr vr   v   
                    2          2       
             r  2
                    z   z r r r   r        
Axial Velocity
                2 vz  2 vz  1 vr   v   
                       2                
               r   2
                       z    r r z z         
 Vorticity
 2   2                     vr           v2
                     vr     vz               g 
 r  2
         z 2
               r  r        r       z   r           2
                                                           
                       2 div(v)     vr    vz   
                                                  r        z 
                Additional Equations
Equation of State                 pW
                               
                                  RT

        density            R  universal gas constant
       p  pressure          T  absolute temperature
      W  mean molecular weight of the mixture

Shvab-Zeldovich Equation
          1       S     S     S        S
                rD    D   vr     v z
          r r     r  z  z     r        z


      S  conserved scalar   D  diffusion constant
                 Boundary Conditions
Axis of Symmetry (r  0)
                                        vz                 S
                           vr  0           0     0         0
                                        r                  r
Outer Zone (r  Rmax )
                           vr           vz          vr
                               0            0             S 0
                           r            r           z
Inlet ( z  0)
                                               vr vz
                  vr  0     vz  v (r )
                                    0
                                                          S  S 0 (r )
                                               z r
                                    z



Exit ( z  L)
                                        vz               S
                           vr  0           0       0        0
                                        z        z        z
                     Solution

Grid:
0  r    r
        1     i 1    ri  ri 1    rnr  Rmax    
0  z    z
        1        j 1    z j  z j 1    znz    L

Dependent Variables:            Si , j

Nodes:       (ri , z j )
Finest Grid   Coarsest Grid
  • Spatial Operators                                       finite differences
  • Diffusion and Source Terms                                     centered
                                                                  differences
  • Convective Terms                              monotonicity preserving
                                                     upwind scheme

(1) vr
        S
        r
                       
            max vr i  1 , j ,0
                           2
                                       
                                    Si , j  Si 1, j
                                       ri  ri 1
                                                                
                                                       max  vr i  1 , j ,0
                                                                        2
                                                                                  
                                                                                 Si 1, j  Si , j
                                                                                    ri 1  ri
Boundary conditions - first order back or forward differences
 Exceptions:
* vorticity inlet boundary condition
                                   vr i,2 vz i1,1  vz i1,1
   (2)
                1
                  i,1  i,2           
                2                  z2  z1       ri 1  ri 1

* axial velocity boundary condition at r = 0

         vz 2, j  vz 1, j  r2  r1 2   2vz 
                                              2            
                                                          Ο r2  r1 
                                                                      2
                                                                          
   (3)                                              
                                    2        r 1, j
                    2vz    2 v z    vz  
   (4)                    2           z 
                                                
                   r  2
                           z      r z       
Discretization of the partial differential equations and
boundary conditions yield nonlinear equations:
                         F (U )  0

The equations are solved using a damped Newton method:

        (5)       J (U n )U n  n F (U n )
                  0  λ n  1,   n  0,1,

Convergence tolerance:
                          U n  105
Algorithms:
     •Bi-CGSTAB/GS
     •GMRES/GS


Parabolic Problem:

              n 1   U n 1  U n
    (6)   F (U )  D        n 1
                                  0
                        t
                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
               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
Numerical Results
                                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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