Seth B. Dworkin, Blair C. Connelly , Beth Anne V. Bennett ,
description
Transcript of Seth B. Dworkin, Blair C. Connelly , Beth Anne V. Bennett ,
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)