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Nomenclature

A localised area of ﬂame front

A(s), F (s) acoustic transfer functions

A(n) logarithmic growth rate of perturbation



empirical constant = 16, in relationship between λ and 

constant in Eq. (3.25)

area of iso-ϕ surface

a perturbation of ﬂame radius,

amplitude of sin component

turb

turbulent strain rate 1/τ

t,sg

subgrid tangential strain rate

B numerical constant for a mixture; see Eq. (3.13)

b amplitude of cos component

anisotropy tensor

, C

∗

, C

∗

Reynolds stress closure constants

, C

S

, C

1

, C

2

dissipation equation constants

, C

∗

time-scale ratio constants

eddy viscosity model parameter = 0.09

c reaction progress variable

c mean reaction progress variable

c acoustic speed





Favre variance of c

speciﬁc heat capacity at constant pressure for the mixture

p,i

speciﬁc heat capacity at constant pressure for species i

D molecular diffusivity

D fractal dimension

turbulent diffusivity, subgrid scale turbulent diffusivity

molecular diffusivity for species i

Da Damk

ohler number (≡  S

/(u

rms

))



Damk

ohler number at ﬁlter scale (≡  S

/(u



))

local Damk

ohler number

407

408 Nomenclature

∗

density-weighted local Damk

ohler number (ρ

/ρ)

d jet exit diameter; distance to pressure measurement

d data vector, [p

ref

s + z

]

dd/δ



activation energy for reaction r

k

symmetric part of strain tensor ∂u



/∂x

, e

eigenvalues of e

k

with e

> e

F ratio of ﬂame speeds, S

f lag factor for n

; frequency in Hz

body force per unit volume for species i

G(s) overall open-loop transfer function

g variance parameter (







c(1 −



c))

sign of the high-frequency gain

H(s) ﬂame transfer function

h speciﬁc enthalpy of the mixture

speciﬁc enthalpy of species i

√

−1

i,k

molecular diffusion ﬂux vector

K strain rate; mean stretch rate; ﬂame stretch parameter

(= 0.157 Ka

); turbulent Karlovitz stretch factor,

rms

/λ)(δ



)

K(s) controller transfer function

0.2

turbulent Karlovitz stretch factor (K) for probability of

ﬂame propagation of 0.2

0.8

turbulent Karlovitz stretch factor (K) for probability of

ﬂame propagation of 0.8



laminar Karlovitz stretch factor, α

∗



laminar Karlovitz stretch factor, α

∗



, for ﬂame curvature

laminar Karlovitz stretch factor, α

∗



, for aerodynamic

strain

laminar critical Karlovitz stretch factor at the onset of

instability

c,sg

stretch rate due to subgrid ﬂame curvature

Ka Karlovitz number [(δ/η

)

= τ

/τ

]

local Karlovitz number

k wave number ω/

c; turbulent kinetic energy; control

parameter



k Favre-averaged turbulent kinetic energy

, k

, z

, p

controller parameters

k controller vector [k

]

L length of Rijke tube

characteristic length scale of the acoustic perturbations

burned-gas Markstein length

characteristic length scale of the turbulent ﬂame

Lewis number for species i (ˆα/D

)

eff

effective neck length of Helmholtz resonator

Nomenclature 409

M,M Mach number

Ma general, undifferentiated, Markstein number

ﬂame-speed Markstein number

Markstein number for ﬂame curvature

Markstein number for aerodynamic strain rate



−∂



c/∂x



/|∇



mass ﬂow rate of air

mass ﬂow rate of fuel

N size of the nonlinear eigenvalue problem

number of particles

instantaneous SDR ( ˆα∂c/∂x

∂c/∂x

)

n wave number and pressure exponent

unit normal in direction k for an iso-scalar

surface pointing towards smaller scalar values

smallest unstable wave number

largest unstable wave number

∗

relative degree of transfer function

ﬁeld of interaction index

ref,1

, n

ref,1

, n

ref,1

) reference unit vector

BC,1

, n

BC,1

, n

BC,1

) outward unit vector normal to the boundary

wave number at Pe

s,cl

largest unstable wave number at Pe

P RANS or LES PDF



P Favre (density-weigted) PDF

Reynolds stress production term

Pe Peclet number

critical Peclet number

Pe at t = 0

Pr Prandtl number (μ c

λ)

p pressure



ﬂuctuating pressure

probability of ﬂame propagation (f = 0.2 and 0.8)

ref

measured sensor pressure

Q heat of reaction

heating value of the fuel

rad

radiative energy exchange per unit volume

q heat release rate per unit valume

tot

volume averaged heat release rate used for scaling purpose



ﬂuctuating volumetric heat release rate

R universal gas constant

turbulent Reynolds number (u

rms

/ν

)

r ﬂame or hot-spot radius, difference in heat capacities (c

− c

)

initial hot-spot radius

S stretched ﬂame speed; cross-section area; neck area of

Helmholtz resonator

unstable ﬂame speed

410 Nomenclature

ﬂame speed at zero stretch

laminar ﬂame speed

dispalcement speed

turbulent ﬂame speed

T,SG

propagation speed of resolved interface

unstrained planar laminar ﬂame speed

κ,L

curvatured-induced laminar ﬂame s peed

k

mean ﬂuid dynamic strain tensor

Schmidt number for species i

s entropy per mass unit; Laplace transform variable

T absolute temperature

burning velocity activation temperature

burnt side temperaure

pilot ﬂame temperature

unburnt side (reactant) temperature

t time

time for sound wave to propagate

acoustic residence time (= r

t dimensionless time ( tu



/ν)

dimensionless time for acoustic wave

blowout velocity

bulk

bulk velocity of the burner (scaling purpose)

ref

a reference velocity

auto-ignition front propagation velocity

unstable burning velocity

velocity component in direction x



, u

unstretched laminar burning velocity

stretched laminar burning velocity

rms

turbulence rms velocity

turbulent burning velocity

turbulent burning velocity at ﬂame surface deﬁned by c



rms velocity at ﬁlter scale 

V net volume generated at reaction front; cavity volume of

Helmholtz resonator

v speciﬁc volume

actuator signal used to drive fuel valve

Kolmogorov velocity

molecular weight of species i

actuator transfer function

∗

closed-loop transfer function resulting from stabilising

controller

k

mean vorticity tensor

mole fraction of species i

x(x

, x

) position vector

mass fraction of species i

Z mixture fraction; reduced boundary impedance

Nomenclature 411

stoichiometric mixture fraction

···

surface average; ﬁltered surface average; gradient weighted

average

··· Favre average

Greek Symbols

α relaxation coefﬁcient; efﬁciency factor; constant in Eqs. (3.25)

and (3.26); efﬁciency function

ˆα thermal diffusivity (

λ/ρc

); diffusivity of c

ˆα

eddy mass diffusivity

∗

contribution of ﬂame curvature to ﬂame stretch rate

∗

total stretch rate at Pe

∗

contribution of aerodynamic strain to ﬂame stretch rate

∗

total ﬂame stretch rate = α

∗

+ α

∗

α, β, γ weights in the BML PDF for c

β constant in Eqs. (3.25) and (3.26)

β Zeldovich number



stretch efﬁciency function; diagonal matrix containing

convergence coefﬁcients

γ ratio of speciﬁc heats (c

)

∗

sample space variable for scalar gradient magnitude

 ﬁlter scale

p increase in pressure above ambient

δ Zeldovich thickness ( ˆα/S

)

effective 1D premixed ﬂame thickness



laminar ﬂame thickness, given by ν/u



k

Kronecker delta

unstrained planar laminar ﬂame thermal thickness

 small parameter of the order of p



Favre-averaged scalar dissipation rate of

progress variable (

ρ ˆα∂c



/∂x

∂c



/∂x

/ρ)



Favre-averaged scalar dissipation rate of

mixture fraction (

ρ ˆα∂Z



/∂x

∂Z



/∂x

/ρ)



Favre-averaged cross dissipation rate (ρ ˆα∂c



/∂x

∂Z



/∂x

/ρ)

ε acoustic residence/excitation time (t

/τ

)

ε dissipation rate of



ζ sample space (stochastic) variable for c

Kolmogorov length scale

inner cut-off scale

 turbulence integral length scale

 normalised wavelength



largest normalised unstable wavelength



smallest normalised unstable wavelength

λ Taylor length scale of turbulence, excess air factor = 1/φ

412 Nomenclature

λ thermal conductivity

μ molecular dynamic viscosity

dynamic eddy viscosity

ν kinematic viscosity; stoichiometric coefﬁcient

ˆν subgrid-scale viscosity

kinematic eddy viscosity

 ﬂame wrinkling factor

ξ stochastic variable for Z,

c/u

lower detonation limit of ξ

upper detonation limit of ξ

ρ ﬂuid mixture density

reactant density

 FSD



subgrid FSD

 generalised FSD

σ unburned-, burned-gas density

turbulent Schmidt number for quantity i

τ heat release parameter [(T

− T

)/T

]; time delay; wave

period

chemical time scale

excitation time

ignition delay time

Kolmogorov time scale

scalar integral time scale (





/

)

turbulence time scale

ﬁeld of time delays

shear stress tensor

Kolmogorov time scale

φ equivalence ratio

subgrid variance for ϕ

ϕ Reynolds mean of ϕ (resolved in LES)

ϕ Favre mean of ϕ (resolved in LES)

instantaneous cross dissipation rate (D∇Y

·∇Y

)

ψ sample-space variable for stretch rate K

 instability parameter; computational domain

 heat release rate per unit valume

ω growth rate parameter; angular frequency

˙ω

mass reaction rate of species i



molar reaction rate of species i

∂ boundary of the computational domain

∂

subset of the boundary where ∇ ˆp · n

= 0

∂

subset of the boundary where ˆp = 0

∂

subset of the boundary where the reduced impedance

Z is imposed