Premixed Flame Kinematics in an Axially Decaying,
Harmonically Oscillating Vorticity Field
Dong-hyuk Shin*, Santosh Shanbhogue*, Tim C. Lieuwen†
School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150
This paper describes experiments and analysis of the spatio-temporal dynamics of a
bluff-body stabilized flame responding to excitation from a harmonic velocity field. The
dependence of the flame response is shown to exhibit two fundamentally different
dependencies on underlying parameters, referred to here as “interference dominated” and
“dissipation dominated” behaviors. Which behavior dominates depends upon the phase
velocity and decay rate of the convecting disturbance and the flame angle. The magnitude of
the flame sheet wrinkling response initially increases with downstream distance, reaches a
maximum and then decays. This basic envelope of flame response can also exhibit short
length scale undulations, associated with wave interference effects. The bluff body nearfield
response is controlled by linear processes, with a magnitude that grows downstream at a
slope proportional to the local normal velocity of excitation. The peaking in response and
far-field behavior is influenced by both linear and nonlinear effects, and controlled by phase
interference, decay of the excitation, and kinematic restoration processes.
Nomenclature




 v
0



’
’peak
’K
’
f0
G
K
SL
u0
uc
uf
ut,0
u’
ua’
un’
uref’
v0
*
†
=
=
=


=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
Projection of mean tangential velocity in axial direction
Slope of mean flame front position
Excitation velocity decay rate
Axial velocity fluctuation ratio at the flame base, uref’/u0
Transverse velocity fluctuation ratio at the flame base, v’x=0 /u0
Convective wavelength of harmonically oscillating disturbance, u0 / f0
Mean flame angle
Instantaneous flame front position
Mean flame front position
Fluctuation of flame front
Maximum value of ’
Magnitude of ’ at x=xK
Magnitude of homogeneous part of ’
Excitation frequency
Scalar function representing flame position
Ratio of mean axial velocity to disturbance propagation velocity, u0/uc
Laminar Flame speed
Mean axial velocity
Disturbance propagation velocity
Axial velocity just upstream of flame
Mean velocity tangential to mean flame front
Fluctuation of axial velocity
Acoustic velocity magnitude of incident disturbance
Fluctuation of velocity normal to the mean flame front
Magnitude of fluctuating axial velocity at the flame base.
Mean transverse velocity
Graduate Research Assistant, Student Member AIAA
Associate Professor, Associate Fellow AIAA
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vf
v’
xpeak
xK
x
=
=
=
=
=
Transverse velocity just upstream of flame
Fluctuation of Transverse velocity
Axial location where ’ is maximum
Axial location where interferences are in-phase
Axial location where velocity fluctuation decays to e-3 of its initial value
I. Introduction
T
he objective of this research is to understand the flame dynamics of acoustically forced, two dimensional
flames. The dynamics of such acoustically forced flames involve complex interactions between gas expansion
effects and vorticity dynamics induced by both shear and combustion, as highlighted by, e.g., Poinsot and
Veynante1, Schadow et al.2, Rogers & Marble3, Cetegen4, and Coats5. This work is particularly motivated by the
problem of combustion instabilities3. In many such instances, it is known that vortical structures excited by
harmonic acoustic oscillations interact with the flame, causing its heat release to oscillate. This is illustrated in
Figure 1, which depicts instantaneous flame locations and vorticity fields of an acoustically excited flame.
a)
b)
c)
d)
Figure 1. Instantaneous images of the flame front location and the underlying vorticity field. a) without
excitation, b-d) with increasing amplitudes of excitation. (reproduced from Shanbhogue et al.6)
A number of prior studies have characterized the interaction of flames with harmonic waves arising due to both
acoustic7 and convecting, vortical disturbances8-11. The dynamics of the flame are controlled by flame kinematics,
i.e., the propagation of the flame normal to itself at the local burning velocity, and the flow field that the flame is
locally propagating into. This is mathematically described by the G equation1:
G
 u  G  S L | G |
t
(1)
In this equation, the flame position is described by the parametric equation G ( x , t )  0 . Also, u  u ( x , t ) and SL
denote the flow field just upstream of the flame and laminar burning velocity, respectively. In the unsteady case, the
flame is being continually wrinkled by the unsteady flow field, u . The action of flame propagation normal to itself,
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the term on the right side of Eq. (1), is to smooth these wrinkles out through “Huygens propagation” / “kinematic
restoration”. A wrinkle created at one point of the flame due to a velocity perturbation propagates downstream and
diminishes in size due to kinematic restoration. Indeed, the interaction between the excitation (acoustic/vortical
flow oscillations) and the damping (restoration property of the flame) can lead to a range of effects depending upon
flame stabilization and the relative values of the flow oscillations and flame speed.
Most recent theoretical analyses of flame dynamics in oscillating flows have focused upon predicting the
“global” unsteady heat release; i.e., the spatially integrated temporal response of the heat release to the
perturbations12. Characterization of the detailed spatio/temporal dynamics of the flame has received considerably
less attention. Baillot and co-workers did make such comparisons between experiments and theory (using solutions
of the G-equation13) for Bunsen flames, where they showed good agreement between predicted and measured flame
shapes. In addition, Shanbhogue et al.6 presented experimental measurements of the spatio/temporal gain and phase
response of bluff body stabilized flames to harmonic forcing. A typical set of their results are shown in Figure 2.
The convective wavelength of the flame front disturbances, 0=uo/fo, equals the distance a disturbance propagating
at the mean flow velocity travels in one acoustic period. The plot on the left shows the spectrum of the flame
response at several axial locations. This spectrum was determined from the time variation of the flame position,
(x,t) (see Figure 4), at each axial location. The envelope of the flame response at f = fo is also drawn. Figure 3 also
plots the gain/phase response at several other conditions at only the forcing frequency. From these plots, we can
observe several generic features of the flame response at the forcing frequency:
1) Very low amplitude of flame fluctuation near attachment point, with subsequent growth downstream, i.e.,.
|’| ~ x in the bluff body nearfield
2) A peak in amplitude of fluctuation, ’=’peak, at some axial location, x=xpeak
3) Decay in amplitude of flame response farther downstream, i.e., ’(x,f0) ~ 1/x
4) Shorter wavelength modulation in flame response manifested in “ripples” in gain curve, typically
downstream of peak location
5) Approximately linear phase-frequency dependence, implying a nearly constant axial convection speed of the
flame sheet disturbances
As will be shown in this paper, these features reflect the excitation of flame wrinkles by the velocity field, the
decay of these wrinkles by kinematic restoration, interference phenomenon between wrinkles on the flame excited
by different mechanisms, and the convection of these wrinkles by the velocity tangential to the flame front.
Briefly considering the flame response at frequencies other than the forcing frequency in Figure 2a, note that the
spectrum also exhibits a monotonic increase in broadband fluctuations with downstream distance. This reflects the
random flapping of the flame brush, which increases in magnitude with downstream axial distance, as is clearly seen
for the unforced case (see Figure 1a).
a)
b)
Figure 2. a) Dependence of flame front fluctuation spectrum, ’(x, f ) upon axial location (u0 = 4.5 m/s, f0 =
300 Hz). b) Phase dependence upon normalized axial location, where x0 indicates first axial location where
data was obtained (ua’/ u0 = 0.05, triangular bluff body, reproduced from Shanbhogue et al.6).
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a)
b)
Figure 3. Dependence of flame front fluctuation magnitude (left) and phase (right) at forcing frequency upon
axial location for 2D circular bluff-body (x0 indicates the first axial location where data was obtained. Data
were reproduced from Shanbhobue et al.14)
Some of these flame features were discussed and explained by Shanbogue et al.15, who showed that the flame
nearfield response is controlled by flame anchoring and its farfield response by kinematic restoration. However, the
velocity model considered in this study had a uniform axial magnitude (i.e., it did not decay downstream) - the
constant spatial magnitude velocity disturbance implies that the flame sheet continues to wrinkle and respond
indefinitely far downstream, rather than decaying as experimentally observed. The goal of the present work is to
further elucidate these flame-sheet dynamics by generalizing the prior model to include a velocity disturbance field
that decays downstream. It will be shown that this model can capture the observed spatio/temporal characteristics of
the flame sheet, both in the near and farfield.
II. Kinematic Model
A. Formulation
The investigated geometry is a 2-D bluff-body stabilized flame, as shown in Figure 4. The instantaneous location
of the flame surface is determined from the G-equation shown in Eq. (1). The principal assumptions made in this
analysis are the following: (i) the flame is a thin interface, dividing reactants and products, (ii) the flame base
remains fixed to the burner throughout the excitation cycle (i.e., no unsteady liftoff), and (iii) the flame speed is a
constant.
For small velocity fluctuation, the flame front position is single-valued (i.e., there is only one y axis flame front
location value at any axial direction), an approximation that breaks down at high amplitude fluctuations as in Figure
1d). In this case, G(x,y,t) is written as:
G  x, y, t     x, t   y
(2)
By definition, G=0 at the flame front, so that  represents the y location of the flame front, as depicted in Figure 4.
Substituting Eq. (2) into Eq. (1), and non-dimensionalizing leads to:
2
  


u
  v  SL     2
t
x
 x 
(3)
where u, v, SL, , x and t are normalized by u0, u0, u0, 0/ 0 and f respectively. The constants, u0, f0, 0=u0/f0,
and  are mean axial velocity, excitation frequency, mean flow wavelength and the mean flame aspect ratio (in the
absence of excitation), respectively. The superscript “tilde” represents non-dimensional quantities.
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Figure 4. Schematic of bluff-body stabilized flame with its corresponding coordinates. red solid :
instantaneous flame front, red dash : mean flame position,  = aspect ratio of mean flame position.
The velocity field is written as:
u  x, y, t   u0  x, y     u   x, y, t 
v  x, y, t   v0  x, y     v  x, y, t 
(4)
where u0 , u  , v0 , v  , and y are variables normalized by u0, uref’, u0, uref’, and 0/ , respectively and  = uref’/u0.
We consider both the linear and nonlinear response character of the flame. As such,  is written as :
  x, t    0  x        x , t   O   2 
(5)
Note that ’ becomes    O( ) when normalized by uref’/( f0 ). By expanding the solution in powers of 16, the
zeroth and first order equations for the flame fluctuation are
2
u0
  
0
 S L  2   0    v0  0
x
 x 


  
  u0 
t





    

 u  0   v    0

2
 x

    x
2   0  
 x  

SL 0
x
(6)
(7)
Insight into the solution of the first order solution can be obtained from the methods of characteristics17. Note
that it is a sub-class of the more general equation:
    x, t 

t
    x, t 
x
 f  x, t 
(8)
where  is assumed to be a constant here (but not in general). Define the function H(x,t) as:

H  x, t   


x

   
f  ,
d 

 

  t  x
Then, the solution of Eq. (8), subject to boundary condition  (0, t )  0 is:
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(9)
   x, t   
 t  x 
1
H  0,
H  x, t 
 
 
 

1
(10)
Particular part
Homogeneous part
This solution shows that the flame response at each spatial position and time is a superposition of two waves that
propagate along the flame front. The first wave, denoted as the “Homogeneous part” is excited at the flame
anchoring point and propagates along the flame sheet at an axial velocity of . The second wave propagates at the
excitation phase velocity. Note that the particular solution is a convolution of the velocity at all points upstream of x
while the homogeneous solution is only influenced by the velocity field near the attachment point. The spatial
variation in amplitude and phase of these two solutions leads to interesting interference phenomenon, as will be
described later.
B. Numerical Method
For the fully nonlinear case, Eq. (3) is solved numerically. Spatial derivatives are discretized using a Weighted
Essentially Non-Oscillatory (WENO)18 scheme designed specifically for Hamilton-Jacobi equations. This scheme is
uniformly fifth order accurate in regions wherein the spatial gradients are smooth and third order accurate in
discontinuous regions. Derivatives at the boundary nodes are calculated using fifth order accurate upwinddifferencing schemes so that only the nodes inside the computational domain are utilized. A Total Variation
Diminishing (TVD) Runge-Kutta scheme19, up to third order accurate, is used for time integration. The spatial and
temporal grid size are 0/1000 and 1/(1000f0), respectively. Sensitivity studies performed with a grid ten times finer
demonstrated that the difference between the two grid density results was less than 0.1%. The flame front
perturbation at the forcing frequency is determined from the Fourier transform of the computed solution at f=f0.
C. Velocity Model
Several velocity fields were utilized in this study, ranging from very simple to more complex, experimentally
measured fields. A model velocity profile was used for most of these calculations that captures many of the key
features this paper is interested in exploring, i.e., the response of a flame to an axially convecting, decaying velocity
disturbance:

u f  1    e  x cos 2  Kx  t 

vf  0
(11)
Note that while the full vector velocity field is specified here, the linear flame response is controlled by the scalar
velocity field component normal to the flame, un’. The above velocity model is equivalent to the normal fluctuation:
un  sin     e x cos(2 ( Kx  t ))
(12)
III. Results and Discussion
This section describes the processing controlling the key flame response features identified in the Introduction.
The nearfield behavior is described in section A and the farfield behavior in section B.
A. Near Field Behavior
1. Controlling Processes
The magnitude of the nearfield flame response is inherently linear. This is due to the   =0 boundary condition,
which specifies that the flame fluctuation amplitude is very small in the vicinity of the bluff body, so that finite
( / x)2   2 term in Eq. (3) are negligible. Thus, a great deal of
insight into this nearfield behavior can be gained from the analytically tractable linearized solutions of Eq. (7).
amplitude effects, such as associated with the
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We first derive an expression for the slope of   vs. x dependence in the flame nearfield. Defining the local
mean flame angle () with respect to the axial direction (see Figure 4), note that the zeroth order flame front
equation is given by:
u0 sin   v0 cos  SL
(13)
In the same way, the first order perturbation Eq. (7) can be written as:
 
 

 cos  u0 cos   v0 sin  

 u  sin   v cos   0
t
x
cos 
 un
ut ,0
(14)
 
 

 cos  ut ,0

un  0
t
x
cos 
where un and ut ,0 denote a fluctuation of velocity normal to the flame and a mean velocity in the tangential
direction of the flame, respectively. Very near the bluff body, the temporal fluctuation of flame response,   / t is
negligible due to the flame anchoring condition at the bluff body. Thus, the slope of   vs. x response is:

 / 
x

1 un
cos 2  ut ,0
(15)
Note that division by  occurs because  and x are normalized by 0/ and 0, respectively. The cos2 term in Eq.
(15) reflects the choice of a coordinate which is rotated with respect to that of the flame; i.e.,   cos /  and
x / cos  describe the normal and the tangential coordinate, respectively, of a rotated coordinate system that is
locally aligned with the mean flame front. Thus:

   cos  / 
  x / cos  

un
ut ,0
(16)
This equation shows that the fluctuation of the flame sheet in the direction normal to the mean flame front is equal to
the ratio of the fluctuation perturbation velocity normal to the flame front and mean velocity tangential to the flame
front. Note also that this equation makes no assumption about the nature of the velocity field.
This equation describes the rate of growth of the flame front wrinkling in the very nearfield of the flame sheet.
However, axial variations in perturbation velocity magnitude and non-negligible values of the   / t term cause a
departure from this behavior farther downstream. Flame behaviors in the nearfield of the bluff body that are valid
over a larger axial range can be determined by specifying a velocity field and solving the linearized G-equation.
2. Numerical and Experimental results
This section compares the results developed in Sec. 1 to data and fully nonlinear computed solutions using a
model velocity profile. The basic velocity model shown below in Eq. (17) was used. The objective of this
calculation was not to use what is necessarily a physically accurate or realizable velocity field, but only to compare
the above described nearfield flame behavior with explicit calculations for various values of the perturbation and
mean field parameters (details in Table 1).
u f  1    cos  2  Kx  t  
v f  v0   v  cos  2  Kx  t  
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(17)
where, v represents the magnitude of fluctuating transverse velocity. The resulting solutions for the magnitude of
  , normalized by | un | / ut ,0 cos2  , are shown in Figure 5. Note that all of the results converge to the theoretical
solution from Eq. (15) in the nearfield, indicated by the line y=x. The solutions diverge downstream due to the nonnegligible temporal fluctuation,   / t and nonlinear effects.
Figure 5. Dependence of flame response normalized by | un | / ut ,0 cos2  upon axial location. (K=1.2, details
are listed in Table 1)
Similar results were observed experimentally. Figure 6a plots the magnitude of the flame sheet fluctuations,
normalized by ua’/u0. The flame angle( for each experiment is less than 15°, making the cos2 term close to unity,
and therefore not included in the normalization. Again, note the collapse of these data to a common curve in the
bluff body nearfield, and the divergence of these curves in the farfield, again demonstrating that the nearfield of the
flame response is essentially linear.
Figure 6. Measured amplitude response normalized by acoustic velocity amplitude upon normalized axial
distance, 0=u0/f0 : (o) f0 = 150 Hz, ua’/u0 = 0.028/2.27, () f0 = 150 Hz, ua’/u0 = 0.01/2.27, () f0 = 180 Hz,
ua’/u0 = 0.015/2.27, () f0 = 150 Hz, ua’/u0 = 0.021/3.37 (cylindrical bluff body, reproduced from Shanbhogue
et al.6)
Efforts were also made to quantitatively compare the measured and predicted slope, see also Shanbhogue et al.6
PIV velocity field measurements were used to estimate the velocity field near the flame. Obtaining good velocity
field measurements requires excitation of the flame with sufficient magnitude perturbations. This causes the flame
sheet to move around, requiring that the conditioned velocity perturbation just upstream of the flame must be used
for the un’ specification – i.e., the velocity field cannot be determined at a fixed location, but at a location that moves
with the mean flame front.
Nonetheless, reasonable quantitative comparisons are possible in the flame nearfield, see Figure 7. The value of
RHS of Eq. (15) was determined from the local measured flame angle and velocity field quantities. The results are
shown in Figure 7b. Note that this ratio appears to asymptote to a value of | un ( x, f0 ) | / (ut ,0 ( x) cos 2  )  0.15.
This is in good agreement to the slope measured from Mie scattering images of the flame shape, illustrated in Figure
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 |  ( x  0, f 0 ) |
 0.14. Note that the derivative was not divided by shown in Eq. (15) because these
x
calculations used dimensional variables.
7a,
a)
b)
Figure 7. a) Dependence of flame front fluctuation spectrum, |  ( x, f0 ) | / 0 upon axial location. b)
Dependence of normal velocity fluctuation amplitude, | un ( x, f0 ) | / (ut ,0 cos 2  ) upon axial direction. (u0 = 4.5
m/s, f0 = 300 Hz, reproduced from Shanbhogue et al.6)
B. Farfield Response
We next turn to the flame response farther downstream, in order to consider the peaking, decay, and modulation
of the flame front amplitude. As suggested by the data and computational results above, this farfield region is
inherently nonlinear and, thus, less amenable to analytical insight. However, some insight into various features of
the flame response can be gained by first considering the linear response. This analysis highlights some of the
controlling physics which nonlinearity acts upon – specifically, it provides insight into the roles of interference of
the waves propagating along the flame (see Eq. (11)), and the role of decay of the velocity field downstream.
1. Linear Response
The first order flame response for the velocity model in Eq. (11) is

 

x


x

x i 2 K    t
i  2   t  
 

  
i
i
  
  
e
e

e    
2  K   1    i
2  K   1    i
(18)
where the parameter,    2 /(1   2 ) physically represents the projection of the tangential mean velocity in the
axial direction. Note that the solution consists of two parts, as also shown in Eq. (10). The first term is the particular
solution that is excited by the unsteady velocity field at all positions upstream of the position, x. The second term is
the homogeneous solution, which is determined by the zero response of the flame at the anchoring point to the nonzero velocity disturbance at that point. Mirroring the underlying perturbation velocity field, the particular solution
decays downstream and propagates with the same phase velocity as the excitation‡. In contrast, the homogenous
solution has a constant axial magnitude and propagates at a phase velocity equal to the projection of the mean flow
on the mean flame front. Note that these two disturbances propagate at the same phase velocity when K=1. This
equation shows that the flame response in the axial coordinate x /  is controlled by two parameters, K and
whose influence is discussed next.
Figure 8 illustrates the influence of the decay parameter, . First, notice that all five solutions converge to a
common solution in the very nearfield, as described in the previous section. Next, note that all the results exhibit
flame responses that grow, reach a local maximum, and then oscillates with constant local maximum value (for the
‡ Note that the particular solution is actually a superposition of infinitesimal, non-decaying waves (within the
constant flame speed assumption, decay of flame wrinkling occurs at O(2) and higher – i.e., it cannot be captured
with a linear analysis), whose superposition sums to a decaying field due to interference effects.
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no decay case, =0) or with decaying local maximum (for >0) until settling at a constant magnitude given by the
homogenous solution. The highest  case shows the lowest response due to the lower spatially integrated
disturbance field. For the slowest decaying cases, such as  = 0 and 0.5, the flame response has local minimum that
are even lower than this =1 case, due to the negative interference between the waves propagating along the front.
As the decay coefficient increases, this interference phenomenon plays a monotonically lesser role in the flame
response, until the flame response is primarily controlled by the homogeneous solution (referred to as “dissipation
dominated” behavior in this paper), as shown in Figure 8b.
a)
b)
Figure 8. a) Dependence of |   | upon axial coordinate for different values of velocity decay rate parameter,
. B) Dependence of maximum value of |   | upon velocity decay rate parameter,  (K= 1.28).
The above results illustrate the important role of interference phenomenon between the waves propagating along
the flame sheet. Such interference occurs when K≠1, due to differing propagation speeds of the two waves along
the flame sheet. As such, the value of K has important influences upon the solution. Figure 9 plots typical
solutions of Eq. (18) for various values of K. As Ka increases from zero, the overall response increases, reaches a
maximum at K = 1, and then decreases for the higher Ka. This can be seen in Figure 9b, which plots the
 , upon K. These plots clearly show
dependence of the magnitude of the local maximum in flame response,  peak
the role of K and interference processes in controlling the spatial character of the flame response. Note also that
when K=1, the flame response is not oscillatory, showing the role of interference phenomenon in inducing an
oscillatory gain in flame response.
The linear flame response is symmetric about |K-1|. This shows that the magnitude of the flame’s spatial
response is controlled by the magnitude of the difference in phase velocity, and not their individual values. This
symmetry does not extend into the non-linear regime.
a)
b)
Figure 9. a) Dependence of |   | upon axial coordinate for different values of disturbance convection velocity
parameter, K b) Dependence of maximum value of |   | upon K ( = 1.87).
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From these results we can see that the magnitude of the flame’s spatial response is controlled by interference and
dissipation. In general, both of these processes control the flame response, but for certain parameter combinations
one or the other is controlling. We will next exploit this point to derive insight into the scaling for the axial location
in the maximum of flame response, x peak . As will be shown next, the flame response at approximate axial locations
of 1/2(K – 1) and 3/is close to the maximum for the interference and dissipation dominant regimes,
respectively. Note that in the dissipation dominated case, there is no strong peak as seen in Figure 7a and Figure 8a.
Rather, x peak describes how fast the flame response approaches the maximum. Which regime is more dominant in
controlling x peak is determined by which length is shorter. This implies the following criterion for dissipation vs.
interference dominance:
Dissipation dominant :  
6 | K 1|
(19)
Interference dominant :  
6 | K  1|
(20)
In the interference dominated regime, the maximum and minimum flame magnitude response occurs when the
relative phase between the two flame front waves is 0º or 180º, respectively (the phase between these two waves is
180º at x =0, due to the   =0 boundary condition). In contrast, the dissipation dominant regime response exhibits a
far less distinct local maximum, because the particular solution magnitude already has negligible values at the
spatial point where it is in phase with the homogeneous solution. In fact, a key difference between the interference
dominant shape and the decaying dominant shape is the relative magnitudes of the two waves at the spatial location
where they are in phase. Analytic approximations of each regime are listed in appendix.
The two waves are first in phase at the location, xK , indicated in Eq. (21). Figure 10a plots the linearized flame
response, the magnitude of the sum of the two waves, and the relative phase between the two waves. It also
indicates the actual maximum in flame response and the estimated response based upon Eq. (22). Note that
 |  |  parti
 | at the axial locations where the respective phase of the two terms is zero. This result shows
|   |  | homo
that the estimated and actual peak locations are quite close for the chosen parameter values. They differ due to the
non-zero  value which causes the actual peak to occur earlier, because of the decaying amplitude of the particular
solution.
xK /  
 |x  xK  |  parti
 | x  xK 
|  K |  |  homo
1
(21)
| 2  K  1 |
1
4 2  K  1    
2
2
x
   K 


1

e




(22)
The dissipation dominant regime is next considered. In this parameter regime, the axial location of xpeak is
controlled by how fast the particular solution decays, because the particular and homogeneous solutions are out of
phase in the very nearfield. This decay length scale is 1/. If the criterion is set to the point where the excitation
amplitude is lower that e-3 (about 5%) of its initial amplitude, this leads to the estimate for an axial location in Eq.
(23) and its corresponding estimated response in Eq. (24). A typical solution is shown in Figure 10b. Note also that
defining a “local maximum” for this case is often not a good representation of the results, as the curves really do not
have a peak (see Figure 10), but are better described as asymptotically (but with some small level of fluctuation)
approaching their maximum, given by |   |max = 1/. Similarly, x is also better interpreted as the length scale
describing the location where |   | asymptotes to its maximum value.
x /  
3

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(23)
 |
|  |  |  homo
a)
Figure 10.
1
4
2
 K  1   
2
2
(24)
b)
Dependence of |   | upon axial coordinate for (a) interference dominant case (K=1.28,  =
0.75, estimation of peak is calculated by Eq. (21) and (22)) and b) dissipation dominant case (K=0.89,  =
0.75, estimation of max is calculated by Eq. (23) and (24))
This interference and dissipation dominant behavior can also be used to understand the “ripples” in flame
response discussed in the Introduction section. These ripples are due to interference effects and, thus, can be
expected to be more prominent as K departs from unity. This is clearly seen in our computations, such as Figure
9a and Figure 11a. We are currently evaluating these K values for experimental data as well to assess this
prediction about when ripples will and will not appear experimentally.
2. Nonlinear response
The farfield flame response is intrinsically nonlinear and, thus, the above linear treatment provides some
guidelines on the role of interference and dissipation, but does not capture the critical nonlinear effect – kinematic
restoration. Kinematic restoration is flame propagation normal to itself which smoothes out wrinkles7 and acts as a
source of flame wrinkling dissipation. Note that the dissipation referred to in the linear discussion was in the
excitation velocity – in the linear, constant flame speed regime, flame wrinkles propagate with constant amplitude.
The rate of flame sheet dissipation increases with amplitude and is inversely proportional to flame front wrinkling
length scale. This nonlinear effect exerts two key influences upon the farfield – it causes the location of peak flame
response to be amplitude dependent and the flame disturbance magnitude to decay to zero (as opposed to the
constant values shown in Figure 10).
We begin first by illustrating results showing this latter effect. Figure 11a) and b) depict typical fully nonlinear
computations showing the flame response at different K and  values. The black line is associated with
dissipation dominant parameter values, whereas blue and red cases with interference dominated. In general, the
farfield non-linear flame response is smaller than the linear solution. Moreover, the reduction in interference
generated oscillations in flame response, as well as the monotonic reduction in magnitude of flame response with
downstream distance is clearly evident. This dissipation in flame wrinkling is amplitude dependent, as can be seen
by comparing the scaled linear solution with the =0.1 and 0.3 cases.
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a)
Figure 11.
b)
a) Dependence of |   | upon axial distance for at different excitation amplitude, solid : linear
solution, dash : =0.1, dot : =0.3 (=0.93). b) Dependence of |   | upon axial distance at different excitation
amplitudes, solid : linear solution, dash : =0.1, dot : =0.3 (=1.17).
Figure 12 plots experimental flame response measurements, illustrating similar points – i.e., the faster decay of
the flame wrinkle with increasing amplitude. Note that these results are dimensional, hence the growing peak
response and nearfield slope with amplitude of excitation. In contrast, the results in Figure 11 are dimensionless and
scaled by the magnitude of excitation.
Figure 12. Measured dependence of flame front fluctuation magnitude upon normalized axial distance,
0=u0/f0. (o) ua’= 0.028, (+) ua’ = 0.016, () ua’ = 0.010. (u0 = 2.27 m/s, f0 = 150 Hz, cylindrical bluff body,
reproduced from Shanbhogue et al.6)
The amplitude dependence of x peak is illustrated in Figure 13. For the interference dominant parameter sets,
depicted on the left, the peak location of the linear solution is a good indicator for the non-linear solution, as shown
in Figure 13a. This is also shown in Figure 14a) and c), which compares the location and magnitude of the peak
flame response for different perturbation amplitudes over a range of different parametric conditions (detailed in
Table 2). Increasing deviation between the two solutions occurs with increasing amplitude of excitation, as expected.
 in the nonlinear case is higher and lower than the linear solution when K>1 and K<1,
The value of  peak
respectively.
In the dissipation dominant regime, the peak in the linear solution is not a good indicator for x peak for the
nonlinear solution. This is because the linear solution does not necessarily have a strong peak, while the nonlinear
solution has a very well defined peak due to kinematic restoration. However, x in Eq. (23) is a good estimate for
the location of x peak . Note also that as excitation amplitude increases, nonlinear x peak shifts upstream because
nonlinearity starts to smooth out wrinkles earlier. Furthermore, as shown in Figure 14d, the linear value of |  |
 .
provides a good estimation of nonlinear  peak
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a)
Figure 13.
b)
Dependence of |   | upon axial distance illustrating variation in location of peak response upon
excitation amplitude for (a) interference dominant regime (K=2, =0.47,) and (b) dissipation dominant
regime ( K=0.95, =2.8).
a)
b)
c)
d)
Figure 14. Comparison of location and magnitude of peak flame response in interference ((a) and (c)) and
dissipation ((b) and (d)) dominated regimes. Conditions are listed in Table 2
IV. Conclusion
This paper has described the features and parameters that control the spatio-temporal dynamics of flame front
fluctuations of a harmonically excited flame. It has been shown that the magnitude of the flame response grows,
reaches peak, and then, decays with axial distance. There may also be short wavelength fluctuations superposed
upon this behavior.
In the nearfield, the flame response grows linearly with downstream distance, at least for well anchored flames
that do not fluctuate in position at the attachment point. The slope of this growth region is proportional to the ratio
of the fluctuating normal and mean tangential velocity at the flame.
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Farther downstream, the key flame features are controlled by interference phenomenon and the dissipation in
magnitude of the velocity excitation. Interference between waves propagating along the flame sheet is controlled by
the parameter |K-1|, which represent the difference between mean flow and excitation phase velocities. Dissipation
is controlled by the rate of decay of the velocity fluctuations, . Depending upon the relative values of these two
parameters, the farfield response can be “Interference dominated” or “Dissipation Dominated”. In the interference
dominant regime, the flame response is oscillatory and possesses local maxima and minima. In the dissipation
dominant regime, the flame response monotonically grows or exhibits no recognizable peak.
Nonlinearity plays an important role downstream due to kinematic restoration effects, which are responsible for
dissipation of flame front wrinkles. This effect causes the relative decay rate of flame wrinkles to increase with
disturbance amplitude.
The computational and analytical results presented above were obtained with relatively simple model velocity
fields. Although not shown here, additional results mirroring those presented above were obtained with a more
complex velocity field model, obtained from fits to data described in Shanbhogue et al.6 and given by:
u f  1    (1   x)e  x cos  2  Kx  t  
v f  v0   
2
c1
x e  x cos  2 ( Kx  t )   
(25)
The above results on the nearfield and farfield behavior, role of wave interference and dissipation, and peak location
estimates were reproduced with similar results.
Appendix
Table 1. Lists of Numerical simulation conditions depicted in Figure 5 for the velocity Model, Eq. (17).
Case No.
u0
v0
v


1
1
0.1
0.1
0.01
15
2
1
0.1
0.1
0.01
30
3
1
0.2
0.05
0.02
30
4
1
0.5
0.1
0.05
45
5
1
1
0.1
0.1
60
Table 2. Tabulation of conditions used for parameter sweep results depicted in Figure 14 for the velocity
model described in Eq.(11).
Interference dominant regime
Dissipation dominant regime
K
K


0 - 0.5
0.47
0.9 - 1.1
2.80
1.5 - 2
0.47
0.9 - 1.1
3.73
0 - 0.5
0.93
1.5 - 2
0.93
(1) Approximation of Interference dominant Regime (Taylor Series expansion of Eq. (18) about (=0)
x 
 2 i Kax t 
2 i   t  
i


 
 e   
e

2  K  1 

2 i  K  1

x

e
x 
 2 i Ka x t 
2 i   t  
  e     e   

 
   2 



O
 
 
4 2  K  1
K  1
  K  1  
x 

2 i  Ka t 
 

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(26)
(2) Approximation of Dissipation dominant Regime (Taylor Series expansion of Eq. (18) about (=0)
 
e
  a
x


1
e
  x 
i  2   t  
   
x
x
  a


x   a
2 i    e   e   1   x  
  K  1  2 
i  2  t    K  1



   

e

 O

    




(27)
Acknowledgments
This research was supported by the US-DOE & NSF under contracts DE-FG26-07NT43069 and CBET0651045; contract monitors Rondle Harp and Dr. Phil Westmoreland, respectively.
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