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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 rD 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 i1,1 vz i1,1 (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 105 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.