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. SmookeYale University, New Haven, CT, USA

Maria P. Puccio, Brendan McAndrews, J. Houston MillerGeorge Washington University, Washington, DC, USA

Journée des Doctorants du CMAP le mercredi 7 mars 2007 Ecole PolytechniquePalaiseau, France

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

• 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 byv

zv

rvdiv

gr

v

zv

rv

rrzr

zr

rzr

v2

2

vv2

2

2

2

Vorticity Transport Equation

p

r

Kz

K

K where

The Vorticity-Velocity Formulation:Derivation of the Vorticity Transport Equation

Radial Velocity Equation:

Axial Velocity Equation:

v

zz

v

rrz

v

r

v rzz 12

2

2

2

• Substituting into the axial and radial derivatives of the continuity equation

v

rr

v

r

v

rzz

v

r

v rrrr22

2

2

2 1

The Vorticity-Velocity Formulation:Derivation of the Elliptic Velocity Equations

• 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?

v two Poisson-like equations

Vorticity Equation

zv

rvdivg

r

v

zv

rv

rrzr

zr

rzr

v22

vv

2

2

2

2

v

rr

v

r

v

rzz

v

r

v rrrr22

2

2

2 1

v

zz

v

rrz

v

r

v rzz 12

2

2

2

Axial Velocity Equation

Radial Velocity Equation

• is substituted into the governing equations• Results in a stronger coupling between the field and the curl

of the predicted v field

r

v

z

vω zr

Derivation of the Modified Vorticity-Velocity Formulation

Radial Velocity Equation

Modified Vorticity Equation

Modified Axial Velocity Equation

v

zz

v

rr

v

z

v

rz

v

r

v rzrzz 12

2

2

2

zv

rvdivg

r

v

z

v

r

v

z

ωv

r

ωvρ

r

v

z

v

r

μ

rzr

zr

zrrzr

zr

v22

vv

2

2

2

2

• is substituted into the governing equations• Results in a stronger coupling between the field and the curl

of the predicted v field

r

v

z

vω zr

Modified Vorticity-Velocity Formulation

v

rr

v

r

v

rzz

v

r

v rrrr22

2

2

2 1

• Modified vorticity-velocity equations are augmented by conservation equations for energy and species

22222

1

1

vdiv3

2222

1

r

v

z

v

z

v

r

v

r

vqwWh

z

TV

r

TVYc

z

T

zr

Tr

rrz

Tv

r

Tv

zrzrrR

N

nnnn

N

nznrnnnpzr

spec

spec

,,,

,,, nnznnrnnn

zn

r wWVYz

VYrrrz

Yv

r

Yv

1

specN

Nnn

nN YY

2

21

1

Species

Energy

Laminar Diffusion Flame:Governing Equations

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

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

Steady Laminar Diffusion Flame:Boundary Conditions

Outlet Boundary Condition

0

zz

V

z

V

z

Y

z

T zri

Far Field Boundary Condition (2nd order)

,0

r

V

r

Y

r

T zi

)( , 298 inletYYKT ii

Inlet Boundary Condition (2nd order)

Symmetry 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

0

rr

V

r

V

r

Y

r

T zri

continuity ,r

v

z

vω zr

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

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

Periodically Forced Flame:Problem Formulation

ftRrvz 2 and 3.or 5. wherecm/s )sin1(*1*70 22

• 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

Periodically Forced Flame:Results

– Temperature fields– Forced at 20 Hz – Each cycle corresponds

to 0.05 seconds of actual time

50% modulation

30% modulation

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

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

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+H

downstream of hydrocarbon oxidation

Conclusions and Future Work

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

• Modified vorticity-velocity formulation conserves mass while maintaining the overall structure of governing 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

AcknowledgementsYale University, New Haven, CT, USAProf. Mitchell D. SmookeProf. Marshall B. LongDr. Beth Anne V. BennettDr. Andrew M. SchafferBlair C. ConnellyGeorge Washington University, Washington, DC, USAProf. J. Houston MillerMaria P. PuccioBrendan McAndrewsFundingUS 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 CanadaNational Defense Science and Engineering Graduate Fellowship (ASEE)