Seth B. Dworkin, Blair C. Connelly , Beth Anne V. Bennett ,

1. Journée des Doctorants du CMAP le mercredi 7 mars 2007 Ecole Polytechnique Palaiseau, France Application of a Modified Vorticity-Velocity Formulation to Steady and Unsteady Laminar Diffusion Flames Seth B. Dworkin, Blair C. Connelly, Beth Anne V. Bennett, Andrew M. Schaffer, Marshall B. Long, Mitchell D. Smooke Yale University, New Haven, CT, USA Maria P. Puccio, Brendan McAndrews, J. Houston Miller George Washington University, Washington, DC, USA

2. Outline • The vorticity-velocity formulation • Background, motivation and derivation • Mass conservation and the vorticity-velocity formulation • Derivation of a mass-conservative vorticity-velocity formulation • Numerical methods • Steady laminar methane/air diffusion flame • Comparison to experimental data • Periodically forced laminar methane/air diffusion flame • Comparison to experimental data • Conclusions • Future work

3. The Vorticity-Velocity Formulation:Derivation of the Vorticity Transport Equation • Elliptic set of PDEs • Used successfully for flame simulation since Ern et. al., (1995) • Governing equations are presented in cylindrical coordinates at steady state • Vorticity transport equation is derived by taking the curl of the momentum equations with negligible bulk viscosity eliminates the term • Any resulting terms having the form are replaced by Vorticity Transport Equation

4. The Vorticity-Velocity Formulation:Derivation of the Elliptic Velocity Equations two Poisson-like equations • Substituting into the axial and radial derivatives of the continuity equation Radial Velocity Equation: Axial Velocity Equation: • Continuity is not explicitly satisfied by this formulation • Some simulations employing these equations exhibit “mass loss or gain” Can mass loss or gain be avoided?

5. Derivation of the Modified Vorticity-Velocity Formulation • is substituted into the governing equations • Results in a stronger coupling between the field and the curl of the predicted v field Vorticity Equation Radial Velocity Equation Axial Velocity Equation

6. Modified Vorticity-Velocity Formulation • is substituted into the governing equations • Results in a stronger coupling between the field and the curl of the predicted v field Modified Vorticity Equation Radial Velocity Equation Modified Axial Velocity Equation

7. Laminar Diffusion Flame:Governing Equations • Modified vorticity-velocity equations are augmented by conservation equations for energy and species Energy Species

8. Numerical Methods • Centered differences are used to discretize diffusion terms at all interior mesh points on a two-dimensional mesh • First order upwind differences are used for convective terms • A second order one-sided difference is used to discretize the vorticity boundary condition at the inflow, far field and outflow • Pseudo-time terms are temporarily appended to one or more of the governing equations to aid convergence from a starting estimate • A damped, modified Newton’s method solves the nonlinear equations at each pseudo-time level and finally at steady state • A preconditioned (block Gauss-Seidel) Bi-CGSTAB method is used to solve the linear system within each Newton iteration

9. Application: Modified Vorticity-Velocity to a Steady Laminar Diffusion Flame Goal: • Compare experimental and computational data in order to validate the new modified vorticity-velocity formulation Problem definition: • Axisymmetric, laminar methane/air diffusion flame • Methane chemistry using a kinetic mechanism containing 16 species and 46 reversible reactions

10. Steady Laminar Diffusion Flame:Boundary Conditions Inlet Boundary Condition (2nd order) • Fuel Tube: Parabolic velocity profile with vavg = 35 cm/s, 35% CH4 (mole) in N2 • Oxidizer tube: Air with vavg = 35 cm/s Outlet Boundary Condition Far Field Boundary Condition (2nd order) Symmetry Boundary Condition (2nd order)

11. Steady Laminar Diffusion Flame:Comparison • Experimental data generated by Rayleigh and Raman scattering • Modified formulation better predicts overall flame structure • Predictions for temperature, O2, CO2 and CO concentrations agree well with experiment

12. Application: Modified Vorticity-Velocity to Periodically Forced Flame Goal: • Compare experimental and computational data in order to validate computational model of transient combustion Problem definition: • Axisymmetric, laminar methane/air diffusion flame • Previously observed lack of agreement in overall flame structure • Artificial viscosity/discretization error? • Lack of soot/radiation models • More accurate solution of the velocity field may help the comparison

13. Periodically Forced Flame:Problem Formulation • Employs the same governing equations except each PDE also contains one or more time-dependent terms, as needed • Methane chemistry using a kinetic mechanism containing 16 species and 46 reversible reactions • Second order, implicit temporal discretizations Boundary Conditions • Fuel tube inlet (transient boundary condition) • Parabolic velocity profile with vavg = 35 cm/s (averaged both spatially and temporally) and T = 298 K • Axial velocity is forced by a sinusoidal perturbation with amplitude of 30% or 50% at 20 Hz • 35% CH4 (mole) in N2 • Air flows in the oxidizer tube with vavg = 35 cm/s and T = 298 K • Boundary conditions are otherwise identical to the steady flame

14. Periodically Forced Flame:Results • Temperature fields • Forced at 20 Hz • Each cycle corresponds to 0.05 seconds of actual time 30% modulation 50% modulation

15. Periodically Forced Flame:Temperature Contours • 30% modulation • 10 ms intervals • Computational (top) and experimental (bottom) isotherms • Panels b, c, g and h between 3.5 cm and 5.0 not shown • Highest level of particulate interference in Rayleigh imaging • Lift-off heights remain constant • Flame height varies greatly • Lower temp in experiment

16. Periodically Forced Flame:CO Mole Fraction Contours • 30% modulation • 10 ms intervals • Computational (top) and experimental (bottom) isopleths for CO • 15% increase in YCO on the centerline

17. Periodically Forced Flame:CO2 Mole Fraction Contours • 30% modulation • 10 ms intervals • Computational (top) and experimental (bottom) isopleths for CO2 • CO is oxidized to form CO2 via CO+OH→CO2+Hdownstream of hydrocarbon oxidation

18. Conclusions and Future Work • Modified vorticity-velocity formulation conserves mass while maintaining the overall structure ofgoverning equations • Particularly useful when high are present (such as corners, walls, shear flows, etc.) • Original formulation has been used successfully for flames without such vorticity generators • Can be applied to a periodically forced methane/air diffusion flame • Good qualitative agreement with experiment Periodically Forced Flame; Future Objectives • Implementation of 31-species C2 chemical mechanism • Implementation of a 66 species ethylene mechanism coupled to a sectional soot model (total of 90 unknowns per grid point) • Parallel implementation • Implimentation of EGLIB, for multicomponent transport property evaluation

19. Acknowledgements Yale University, New Haven, CT, USA Prof. Mitchell D. Smooke Prof. Marshall B. Long Dr. Beth Anne V. Bennett Dr. Andrew M. Schaffer Blair C. Connelly George Washington University, Washington, DC, USA Prof. J. Houston Miller Maria P. Puccio Brendan McAndrews Funding US Department of Energy Office of Basic Energy Sciences (grant no. DE-FG02-88ER13966) National Science Foundation (grant no. CTS-0328296) Natural Sciences and Engineering Research Council of Canada National Defense Science and Engineering Graduate Fellowship (ASEE)