In Fig. 9.16 the destabilizing effect of the forcing on the flow field is clearly visible. For the investigated cases with high forcing amplitudes (|uc|/U0 = 1.04; 1.11; 1.23), the onset of absolute instability is shifted upstream and the absolute growth rates considerably increase.
Consequently, the global stability criterion yields considerably higher global growth rates and frequencies. The predicted frequencies rise from 73.8 Hz at |uc|/U0 = 1.04, to 87.6 Hz at
|uc|/U0 = 1.111, and to 91.67 Hz at |uc|/U0 = 1.23. This compares reasonably well with the measured frequencies of 68.1, Hz, 77.1 Hz, and 89.1 Hz, respectively. Since the effect of the forcing on the angular velocity, which is very important for the frequency but less important for the growth rate, was not measured, the achieved frequency match is rather remarkable and provides further credibility to the stability calculations based on the substitute profiles for the tangential velocities.
The results of the stability analysis clearly demonstrate that the excitation of the PVC in the attached flame case is caused by changes of the mean flow field induced by axial acoustic forcing. The spatial shape of the PVC, shown in Fig. 9.11, indicates that the PVC is considerably shifted downstream compared to the isothermal case and the detached flame.
This is qualitatively in line with the wavemaker location predicted by the stability analysis of xs= 1.62Dh for the highest forcing amplitude compared to xs = 0.34Dh in the detached flame case.
9.5 Analogy of the Acoustically Forced PVC to a Parametric
9.5 Analogy of the Acoustically Forced PVC to a Parametric Oscillator 169 forcing cycle, the flow field undergoes very strong changes. It seems plausible that during this forcing cycle, both the eigenfrequency of the helical mode (PVC) and its growth rate vary. If one considers a very low acoustic forcing frequency, where the forcing frequency is an order of magnitude lower than the frequency of the PVC, the PVC can fully develop during the different phases of the forcing. It will adapt to the new phase-averaged flow field with changes of its shape, frequency, and amplitude. In this case, a linear stability analysis of the phase-averaged flow fields (with respect to the forcing)
v(x,Ψ) =V(x) +vc,a(x,Ψ) (9.9) could accurately predict the modulation of the PVC.
Similar studies have been carried out successfully for slowly varying base flows (e.g., Thiria et al., 2007). In the present case, however, the forcing frequency and the frequency of the PVC are of the same order. Thus, for an exact study, the framework of a Floquet analysis would be necessary. Since the main interest in this chapter is placed on the qualitative effect of the oscillating flow on the PVC, the implementation of the Floquet analysis into the local stability analysis is beyond the scope of this work. Instead, it is assumed that a strictly qualitative idea of the modulation of the PVC can be obtained from the calculation of the stability characteristics of the flow field at different phases of the forced oscillation. This assumption is based on the idea that the mechanisms remain the same, even though the PVC cannot be expected to fully develop into its theoretical limit-cycle at every phase of the forcing cycle.
The results of the stability analysis on the phase-averaged flow fields are presented in Fig. 9.17 for an axial location of x/Dh = 0.4, which is close to the predicted wavemaker location. It is evident that the absolute growth rate ω0,i, from which also the global growth rate ωg,i is deduced, strongly oscillates along the forcing cycle. Since the tangential velocity was not measured for the stability analyses presented in Fig. 9.17, only the forced oscillations of the axial velocity and the density field are considered. Thus, no valid information about the oscillation of the absolute frequency ω0,r can be obtained. In contrast to the growth rates, the absolute frequency strongly depends on the tangential velocities (see Chapter 6).
Nevertheless, since the tangential velocity component oscillates at the forcing frequency, the same is expected from the absolute frequencyω0,r.
The validity of the presented phase-averaged stability calculations can be assessed by con-sidering the amplitude modulation of the PVC from the experiments. To obtain the am-plitude of the helical oscillation at different forcing phases, first the phase-averaged fluctu-ations with respect to the forcing phase (Ψ) and the PVC phase (θ) are calculated. Sub-sequently, the difference of these doubly phase-resolved fluctuations (vc,ha(x, θ,Ψ)) and the coherent fluctuations, phase-averaged with respect only to the forcing (vc,a(x,Ψ)) are calcu-lated. Thereby, the virtual oscillation of the PVC at a fixed forcing phase Ψ is obtained as vc,ha(x, θ,Ψ)−vc,a(x,Ψ). Next, the modulation of the amplitude of the helical oscillation is
fa= 110 Hz
0 2 4 6
−0.2 0 0.2 0.4 0.6
Forcing phase Ψ ω0,i(Ψ)
fa= 158 Hz
0 2 4 6
Forcing phase Ψ
0 0.5 1
|uc |/U0
(a) Isothermal fa= 110 Hz
0 2 4 6
−0.2 0 0.2 0.4 0.6
Forcing phase Ψ ω0,i(Ψ)
fa= 158 Hz
0 2 4 6
Forcing phase Ψ
0 0.5 1
|uc |/U0
(b) Detached flame
Figure 9.17: Results of the phase-resolved stability analysis at x/Dh = 0.4 of the isothermal flow and the detached flame forced atfa= 110 Hz (left) and fa= 158 Hz (right).
obtained as the magnitude of the Fourier transform of this difference.
ˆ
vc,h(x,Ψ) = F
vc,ha(x, θ,Ψ)−vc,a(x,Ψ) (9.10) Finally, ˆvc,h(x,Ψ) is integrated over the region of interest (−2.5 < y/Dh < 2.5 and 0 <
x/Dh <4) and yields the integral amplitude of the PVC, phase-averaged with respect to the forcing.
APVC(Ψ) = 1 DhV˙
Z 2πr
q
0.5 (ˆuc,h(x,Ψ)2+ ˆv(x,Ψ)2)dA (9.11)
This integral PVC amplitude oscillates during the forcing cycle at a specific phase with respect to the forcing signal. In Fig. 9.18, the phase-lag of the integral PVC amplitude to the forcing signal is shown for various forcing amplitudes. Additionally, the phase-lag of the calculated phase-dependent linear absolute growth rate ω0,i to the forcing signal is provided.
For forcing at 110 Hz, the average difference of the phase of the PVC amplitude oscillation to the phase of the growth rate oscillation is 101◦. In the 158 Hz case, the mean phase difference is 115◦.
9.5 Analogy of the Acoustically Forced PVC to a Parametric Oscillator 171
0 0.2 0.4 0.6 0.8 1 1.2
−4
−3
−2
−1 0 1 2 3
101◦
115◦
|uc|/U0 6(ω0,i),6(APVC)
Figure 9.18: Phase of the absolute growth rate predicted by the stability analysis in the wavemaker region (ω0,i, filled symbols) and phase of the amplitude fluctuation of the PVC (APVC, hollow symbols). Forcing at 110 Hz (black circles) and 158 Hz (red squares). Dashed lines are the average phases and arrows indicate the average phase differences betweenAPVC and ω0,i.
In the linear framework, the amplitudeAof an oscillation is determined by the linear global growth rate as
ωi = ∂A
∂t. (9.12)
That is, the amplitudeA is increased as long as the growth rateωi is positive. Ifωi oscillates around zero at a fixed frequency, it follows from Eqn. 9.12 that the phase-lag between A and ωi is exactly 90◦. In the present investigation of the amplitude of the PVC APVC, it is assumed that the phase of the global growth rate ωg,i is qualitatively similar to the phase of the absolute growth rate ω0,i in the vicinity of the wavemaker. Thus, the phase of ω0,i can be used as a substitute to the phase of ωg,i in order to avoid the cumbersome and fault-prone task of identifying the wavemaker at each forcing phase (see Section 2.4.2). The reasonable proximity of the observed phase-difference between the PVC amplitudeAPVCand the predicted absolute growth rateω0,ito the theoretical value of 90◦indicates that the results of the linear stability analysis capture the effect of the oscillating flow on the excitation of the PVC at least qualitatively. The linear growth rate (and eigenfrequency) seem to oscillate at the forcing frequency.
9.5.2 Dynamics of the Parametric Van der Pol Oscillator
The consideration of the phase-averaged linear stability analysis suggests that the parametric oscillator with an oscillating linear damping and eigenfrequency is a viable model to describe
fh∗ fh∗+fa∗
fh∗−fa∗ fa∗−fh∗ 2fa∗−fh∗
0 0.2 0.4 0.6 0.8 1 0
0.5 1 1.5 2
ζˆ
f∗
0 0.2 0.4 0.6 0.8 1
PSD(q)
fh∗ fh∗+fa∗
fh∗−fa∗ fa∗−fh∗ 2fa∗−fh∗
0 0.2 0.4 0.6 0.8 1 ζˆ
−8
−6
−4
−2 0
log(PSD(q))
(a) Parametric excitation atfa∗= 0.8
fh∗ fh∗+fa∗
fh∗−fa∗ fa∗−fh∗ 2fa∗−fh∗ 3fa∗−fh∗
0 0.2 0.4 0.6 0.8 1 0
0.5 1 1.5 2
ζˆ
f∗
0 0.2 0.4 0.6 0.8 1
PSD(q)
fh∗ fh∗+fa∗
fh∗−fa∗ fa∗−fh∗ 2fa∗−fh∗ 3fa∗−fh∗
0 0.2 0.4 0.6 0.8 1 ζˆ
−8
−6
−4
−2 0
log(PSD(q))
(b) Parametric excitation atfa∗= 1.2
Figure 9.19: Spectra of the parametrically forced VdP oscillator.
the PVC dynamics at axially forced conditions. The dynamics of the parametrically forced oscillator are obtained by the numerical integration of Eqn. 9.6 and are presented in Fig. 9.19 for a nonlinearity of µ= 0.4. Note that the value of µ = 0.4 is exemplarily chosen and the results presented in the following remain similar for reasonable variations ofµ. The oscillator is parametrically excited with amplitudes from ˆζ = 0 to ˆζ = 1 at one frequency lower than the eigenfrequency (fa∗ = 0.8) and one frequency higher than the eigenfrequency (fa∗= 1.2).
At increasing amplitudes of the parametric excitation ˆζ, the peak amplitude is considerably decreased and almost completely vanishes for high parametric forcing amplitudes. Further-more, the peak frequency is slightly reduced and peaks at the difference and sum of the self-excited and the modulation frequency occur. No peak at the modulation frequency itself can be observed, what is in strong contrast to a (non-parametrically) forced oscillator, where a peak at the forcing frequency is always present, which may be accompanied by the natural peak and several interaction frequencies. The observed spectral behavior of the parametric VdP oscillator resembles very well the antisymmetric spectra observed in Figs. 9.6b and 9.9b.
It may be concluded that the PVC at axially forced conditions shows a similar dynamical behavior as a parametric nonlinear oscillator. This analogy to a parametric oscillator allows for the explanation of the pronounced frequency sensitivity of the damping of the PVC due