AFC-100Drive Amplifier
3.4 Experimental Results
3.4.5 The Forced Rayleigh Index
In chapter 1, the formulation of Rayleigh’s Criterion as derived by Culick (1976, 1987, 1997) was presented in equation (1-1). It is repeated here for convenience.
(3-12) ( 1)
( , ) ( , )
t
E dV t p x t Q x t dt p
.Once again, p x t( , ) is the varying portion of the pressure field and Q x t( , ) is the varying portion of the heat release. The energy added to the system over one cycle is then given byE. For
100.0 1000.0
1.0E-1 1.0E+0 1.0E+1 1.0E+2 1.0E+3
20.0
Frequency (Hz)
Magnitude
Notch at 202 Hz in dataset 86 Notch at 314 Hz in dataset 116
Frequency (Hz)
Phase (deg)
100.0 1000.0
-2520 -2160 -1800 -1440 -1080 -720 -360
20.0
Dataset 116
Dataset 86 Bifurcation at 238 Hz
positive E the closed-loop system is self-driving (although not necessarily unstable), and for negative E the closed-loop system is self-damping.
In 2000, Pun, Palm and Culick proposed the forced Rayleigh index for acoustically forced systems:
(3-13)
1
0
( ) ( ) ( )
f
rms
p q
R d
p q
.Here, variable substitutions have been made so that the fluctuating pressure and heat release are ( )
p and q( ) respectively. The new argument, , spans from 0 to 1, constituting a full cycle of the acoustic forcing. The pressure value (p)rms, appearing in the denominator, is the root-mean- square value of the fluctuating pressure component.
The selection of this relation for the forced Rayleigh index is arguably a poor choice as it eliminates any dependence on average system pressure. On closer examination it also becomes apparent that the relation produces an index that is not independent of the forcing amplitude in the linear regime as is desired. Consequently, a reevaluation of the forced Rayleigh index is warranted.
A relation, in the spirit of equation (3-13), is desired that is properly normalized such that it produces a dimensionless index that is independent of the forcing amplitude. The exploration begins by assuming forms for the acoustic forcing and the response of the heat release:
(3-14) ( )
( , ) 2 p rms cos( )
p t p t
p
,
(3-15) q( , )t 2( )p rms qQ( ) cos
t ( )
p
.
Here, as before, p and q are the average pressure in the test section and the average intensity or heat release rate in the examined flame region, respectively. And, again, the quantity (p)rms is
the root-mean-square value of the fluctuating portion of the pressure. The group ( ) 2 p rms
p
is
the amplitude multiplier for the acoustic forcing. Experimentally, efforts are made to hold this value constant. Nonetheless, for the following analysis it can be allowed to vary with
for the sake of generality.Similarly, Q( ) is the amplitude multiplier in the heat release equation while ( ) is the phase offset between the fluctuating pressure and heat release. Indeed, the quantities Q( ) and
( ) are just the magnitude and phase components of the combustion response function which have already been produced from the collected data.
Multiplying (3-14) and (3-15) and then integrating over a single cycle of the forcing gives:
(3-16) f ( )2rms Q( )2t cos( ) cos
( )
t
R p q t t dt
p
.Using 2
and substituting t
gives:(3-17) 2 1
0
( )
2 rms Q( ) cos(2 ) cos 2 ( )
f
R p q d
p
.It quickly becomes apparent (by inspection) that the proper normalizing factor is:
(3-18) 2
N ( )
rms
p
p q
. So that the appropriate forced Rayleigh index becomes:
(3-19)
1 2
0
( ) ( ) ( )
f
rms
R p p q d
p q
.Applying the normalizing factor to equation (3-17) produces:
(3-20) 1
0
2 Q( ) cos(2 ) cos 2 ( ) Rf
d and solving the integral produces the beautifully simple relation:(3-21) Rf Q( ) cos
( )
.As mentioned earlier, the two parameters involved have already been computed in determining the combustion response function. As a result, the generation of the forced Rayleigh index from the experimental data collected becomes a triviality.
Figure 3-16 shows the forced Rayleigh index for the test case presented in Figure 3-4. In this run, the burner equivalence ratio was 0.85 and the nozzle exit velocity ratio was 4.0. As can be seen, the strong combustion response at low frequencies in combination with the ever-increasing phase lag generated by the transport-delay-like behavior results in strong, oscillatory behavior of the Rayleigh Index between 40 Hz and 200 Hz. The data is shown again in Figure 3-17 where the amplitude axes are chosen to be logarithmic to enhance the detail at small amplitudes. In this
Figure 3-16: Forced Rayleigh index, Rf, for test case with equivalence ratio of 0.85 and a nozzle exit velocity ratio of 4.0. Strong driving bands (i.e. large, positive Rayleigh Index) lie from 40 to 60 Hz, 78 to 97 Hz, and 118 to 140 Hz.
f = 54.1 Hz Rf = 1150
f = 88.0 Hz Rf = 1355
f = 124.5 Hz Rf = 351
100.0 1000.0
-2000.0 -1000.0 0.0 1000.0 2000.0
20.0
Forced Rayleigh Index (Rf)
Frequency (Hz)
100.0 1000.0 1.0E-1
1.0E+0 1.0E+1 1.0E+2 1.0E+3
20.0 100.0 1000.0
1.0E-1 1.0E+0 1.0E+1 1.0E+2 1.0E+3
20.0
Forced Rayleigh Index (Rf)
Frequency (Hz)
Positive Branches (Driving)
Forced Rayleigh Index (Rf)
Frequency (Hz)
Negative Branches (Damping)
Figure 3-17: Forced Rayleigh index, Rf, for test case with equivalence ratio of 0.85 and a nozzle exit velocity ratio of 4.0. Results are shown with logarithmic amplitude axes and with positive and negative branches presented on separate graphs.
Figure 3-18: Forced Rayleigh index, Rf, for a nozzle exit velocity ratio of 4.0 and equivalence ratios from 0.75 to 0.90.
100.0 -2000.0
-1000.0 0.0 1000.0 2000.0
300.0 20.0
φ = 0.90 φ = 0.85 φ = 0.80 φ = 0.75
Increasing Equivalence
Ratio
Forced Rayleigh Index (Rf)
Frequency (Hz)
Figure 3-19: Forced Rayleigh index, Rf, for an equivalence ratio of 0.85 and for nozzle exit velocity ratios, VR, varying from 3.5 to 6.5.
presentation, the positive and negative branches of the forced Rayleigh index have been separated onto adjacent plots.
It is now possible to generate plots in the same vein as Figures 3-7 and 3-8, this time revealing the variation in forced Rayleigh index versus burner equivalence ratio and nozzle exit velocity ratio. These graphs are shown in Figures 3-18 and 3-19. Note that the horizontal axes of the graphs have been truncated at 300 Hz and stretched accordingly to improve visibility of the features present at lower frequencies.
As expected for the previously presented data, both Figures 3-18 and 3-19 show the Rayleigh Index curves shifting to the right for increasing equivalence ratio, as well as increasing
100.0 -2000.0
-1000.0 0.0 1000.0 2000.0
300.0 20.0
Increasing Velocity Ratio
Forced Rayleigh Index (Rf)
VR = 3.5 VR = 4.5 VR = 5.5 VR = 6.5
Frequency (Hz)
nozzle exit velocity ratio. In general, they also display increasing peak magnitudes with increased equivalence ratio and exit velocity ratio.
To examine these features more closely, Figures 3-20 through 3-23 are presented. The datasets for the entire tested operating regime of the burner were processed to discern the magnitude and location of the extrema of the forced Rayleigh index for each test condition. Figure 3-20 shows level-set contours indicating the peak positive (driving) value of Rf observed as a function the burner equivalence ratio, φ, and the nozzle exit velocity ratio, VR. Cross-hairs on the plot mark the actual locations of the experimental data points. The companion plot in Figure 3-21 indicates the frequency (in Hertz) at which these Rf peaks are observed. As an example, for the operating condition at φ = 0.90 and VR = 5.0, Figure 3-20 shows that the peak forced Rayleigh index observed is approximately 1700. Figure 3-21 then indicates that this peak Rf value is observed at a drive frequency of approximately 110 Hertz.
In a similar fashion, Figures 3-22 presents the level-set contours for the peak negative (damping) values of the forced Rayleigh index over the operating range. The companion plot indicating the drive frequency at these minima is shown in Figure 3-23. Trending behavior in the two cases is similar and is, in general, consistent with the discussion above; the magnitude of the extrema tends to grow with increasing equivalence ratio, and increasing nozzle exit velocity ratio. Furthermore, the frequencies at which these extrema occur tend to increase, likewise, with these parameters.
As an aside, it is interesting to plot the Rayleigh Index extrema presented above on a scatterplot showing their position as a function of the frequency at which the peak value occurs versus the magnitude of the peak. This data is presented in Figure 3-24.
For low nozzle exit velocity ratios (corresponding to VR = 2.0 and VR = 2.5) the extrema appear to lie on their own independent branches. These can be seen as the dotted lines in the figure.
(Note that these are linear fits and only appear curved due to the logarithmic frequency
0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.0
2.0 3.0 4.0 5.0 6.0 7.0
Equivalence Ratio
Nozzle Exit Velocity Ratio
40
60 80
100
120
Contour level-sets indicating frequency (Hz) at peak positive (driving) Forced Rayleigh Index.
0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
1.0 2.0 3.0 4.0 5.0 6.0 7.0
Equivalence Ratio
Nozzle Exit Velocity Ratio
600 800
1000 1200
1400 1600
1600
1800 1800
1800 1800
Contour level-sets indicating peak positive (driving) Forced Rayleigh Index, Rf.
Figure 3-20: Interpolated contour level-set showing the peak positive (driving) forced Rayleigh index observed over the burner operating envelope. Cross-hairs indicate the locations of experimental data points.
Figure 3-21: Interpolated contour level-set showing the frequency (in Hertz) of the peak positive (driving) forced Rayleigh index observed over the burner operating envelope. Cross- hairs indicate the locations of the experimental data points.
0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.0
2.0 3.0 4.0 5.0 6.0 7.0
Equivalence Ratio
Nozzle Exit Velocity Ratio
40
60
80
100
Contour level-sets indicating frequency (Hz) at peak negative (damping) Forced Rayleigh Index.
0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
1.0 2.0 3.0 4.0 5.0 6.0 7.0
Equivalence Ratio
Nozzle Exit Velocity Ratio
-1800
-1700 -1600
-1600
-1400 -1200
-1000 -800
-600
Contour level-sets indicating peak negative (damping) Forced Rayleigh Index, Rf.
Figure 3-22: Interpolated contour level-set showing the peak negative (damping) forced Rayleigh index observed over the burner operating envelope. Cross-hairs indicate the locations of experimental data points.
Figure 3-23: Interpolated contour level-set showing the frequency (in Hertz) of the peak negative (damping) forced Rayleigh index observed over the burner operating envelope.
Cross-hairs indicate the locations of the experimental data points.
Figure 3-24: Scatter plot of the extrema of the forced Rayleigh index for all experimental test conditions. These are plotted as extrema location (frequency) versus extrema magnitude.
Both positive and negative branches are shown.
axis.) However, for all higher velocity ratios, the data points appear to collapse into collective branches. A single “continuous” negative branch appears near the bottom of the plot. The term
“negative” is used to identify this as the branch formed by the peak negative (damping) values of the forced Rayleigh index. At the top of the plot, two distinct positive branches are present. There is a very noticeable gap between the end of the first positive branch and the beginning of the second positive branch. This gap in frequency spans 54.1 to 71.4 Hertz. The Rf magnitude of the last data point in the first positive branch is approximately 1430 while the magnitude of the first data point in the second branch is approximately 990.
The reason for this discontinuity is unclear and requires further investigation. However, the resulting implication is this: for the given burner configuration, there is a band of frequencies
100.0 -2000.0
-1000.0 0.0 1000.0
20.0 200.0
Forced Rayleigh Index (Rf)
Frequency (Hz)
VR = 2.0
VR = 2.5 VR = 2.5
Positive Branch II
Negative Branch
lying between 54 and 71 Hertz for which the burner has no maximum sensitivity, regardless of operating condition (i.e. equivalence ratio and nozzle exit velocity ratio.)