2.1. Experimental facility

This section describes the experimental facility in which flame density ratio and lip velocity can be independently varied. By varying the density ratio, the hydrodynamic global mode growth rate can be systematically varied. During a density ratio sweep, the flow evolves from a globally stable to a globally unstable system that exhibits intrinsic oscillations. In addition, by varying the lip velocity, the global mode frequency can be varied relative to the fixed forcing frequency. Great care is taken to operate this combustor well away from blowoff boundaries, where additional flame and flow dynamics can occur (Shanbhogue et al. Reference Shanbhogue, Husain and Lieuwen2009a ; Chaudhuri et al. Reference Chaudhuri, Kostka, Renfro and Cetegen2010a ,Reference Chaudhuri, Kostka, Tuttle, Renfro and Cetegen b ).

The experimental rig, shown in figures 2 and 3, consists of two premixed, methane-air combustors in series. The first combustor is used to vitiate the flow and raise its temperature. The second, bluff body stabilized combustor, consists of a rectangular section with a bluff body spanning the width of the combustor, creating a nominally 2-D flow. The aspect ratio of bluff body height to chamber width is 0.15. This combustor has quartz windows for optical access from all four sides. The bluff body used in the test section, shown in figure 3(b) has a 2-D ballistic shape. Additional fuel and air injection sources are located between the two burners, allowing substantial flexibility in density ratio and lip velocity. Further details of this facility and its design are provided in Emerson et al. (Reference Emerson, O’Connor, Juniper and Lieuwen2012).

Figure 2.

Schematic of the atmospheric pressure, vitiated, bluff body rig.

Figure 3.

Drawings of (a) the test section and (b) the bluff body.

A function generator is used to generate the forcing signal, which is then amplified and used to drive two loudspeakers which are mounted to tubes that enter the combustor one metre upstream of the test section. Because the first transverse mode of this rig has a frequency of 5000 Hz, multi-dimensional disturbances are evanescent and decay quickly, so that the acoustic excitation in the test section is effectively planar. The forcing frequency is held fixed at 515 Hz for all experiments, since a strong forced response is possible here, which is an important consideration in the $50~\text{m}~\text{s}^{-1}$ flow conditions.

Data acquisition parameters were optimized to effectively eliminate spectral leakage from the forcing frequency, a particularly important consideration when the global mode frequency and forcing frequency are close. This is accomplished by using a boxcar spectral windowing function, and setting the forcing frequency, $f_{f}$ , sampling frequency, $f_{sample}$ and number of samples, $N_{s}$ , such that (2.1) is satisfied (Lieuwen et al. Reference Lieuwen, Rajaram, Neumeier and Nair2002). In essence, this equation ensures that an integer number of cycles, $j$ , of the forced response are sampled.

The uncertainty in the relative forcing and sampling frequencies is about $10^{-6}~\text{Hz}$ . As such, when using a boxcar window, the bias errors in the spectral estimate due to leakage one frequency bin away from $f_{f}$ is of the order of $10^{-12}\,\%$ . Note that such a discrete frequency excitation problem is one instance where the boxcar window function is superior to other windows. These considerations are important when analysing the impact of narrowband forcing upon other frequencies, such as the power in the global mode, as it is otherwise difficult to differentiate leakage induced bias errors from physical changes in the spectral power of the phenomenon of interest.

The test matrix is designed to independently adjust the air flow rates and equivalence ratios in the two burners to achieve a velocity sweep at a fixed density ratio, for a number of different density ratios. The velocity sweep causes a sweep in global mode frequency. In these experiments, the test section inlet temperature (after vitiation and addition of secondary air and fuel) was varied from 770 to 1200 K; this enabled us to vary the density of the reactants, ${\it\rho}_{u}$ , as well as the density ratio across the flame, ${\it\rho}_{u}/{\it\rho}_{b}$ . The test section equivalence ratio was held in the range $0.70<{\it\phi}<0.75$ , and the adiabatic flame temperature remained in the range $1950~\text{K}<T_{b}<2030~\text{K}$ .

Bluff body lip velocities quoted in this paper are calculated using the formula $U_{lip}={\dot{m}}/({\it\rho}_{lip}A_{lip})$ , which is an average axial flow velocity. The density for the lip velocity calculation was determined from the temperature measured just before the test section and the gas composition of the secondary air and fuel adiabatically mixed with the equilibrium vitiated gas. An equilibrium gas solver was then used to calculate the adiabatic density ratio across the test section flame. The area, $A_{lip}$ , is the cross-sectional area of the flow at the plane of the bluff body trailing edge (accounting for the blockage of the bluff body).

The primary experimental diagnostics consisted of high speed, line of sight integrated (along the bluff body span) imaging of flame chemiluminescence, and high speed particle image velocimetry (PIV). High speed chemiluminescence imaging was performed with a high speed CMOS camera with $21.7~{\rm\mu}\text{m}$ pixels, operating at a 5000 Hz frame rate and 768 $\times$ 384 pixel resolution. The exposure time was $1/5000~\text{s}$ . The camera was outfitted with a 135 mm lens at $f/2.8$ , with a magnification of 1:17.1, such that a 286 mm $\times$ 80 mm region of the test section was imaged on 768 $\times$ 215 pixels. This region of the test section spanned from the bluff body trailing edge to 15 bluff body diameters downstream. The flame was imaged through a bandpass filter centred at 434 nm, with 90 % transmission between 425.5 and 442.5 nm. A total of 4000 images was acquired in each run. The chemiluminescence imaging was used to characterize the dynamical displacement of the flame, as well as the unsteady heat release (Gaydon & Wolfhard Reference Gaydon and Wolfhard1970) of the flame. The displacement of the flame was quantified by first performing edge detection on the chemiluminescence images. The threshold based edge detection scheme used Otsu’s method (Otsu Reference Otsu1975) to identify a threshold intensity contour level. At each axial position, the flame edge displacement, $L^{\prime }$ , is defined as the instantaneous transverse position of this contour level, relative to its mean transverse position (see figure 5 b).

Mie scattering images for high speed PIV were obtained as follows. The flow was seeded with $5~{\rm\mu}\text{m}$   $\text{Al}_{2}\text{O}_{3}$ particles, chosen due to their durability in reacting flows and their high refractive index (Melling Reference Melling1997). This particle size was chosen because it is large enough to slow down accumulation on and clouding of the test section windows, while retaining a sufficiently small Stokes number to accurately track the flow at the frequencies of interest. Following the work by Mei (Reference Mei1996), the Stokes number calculated for this flow and seed particle combination was 0.015. This Stokes number is calculated using a kinematic viscosity of $3\times 10^{-4}~\text{m}^{2}~\text{s}^{-1}$ , and oscillation frequency of 850 Hz (roughly the highest global mode frequency encountered). This Stokes number is comfortably below the cutoff Stokes number for solid particles in air (Mei Reference Mei1996), which is in the range of 0.02–0.04 and would correspond to a frequency of at least 1500 Hz.

The particle-laden flow was illuminated with a double-pulsed Nd:YLF laser, with wavelength of 527 nm, pulse duration of 100 ns and $4~\text{mJ}~\text{pulse}^{-1}$ pulse energy. Each of the two lasers in the double-pulsed system is operated at 5000 Hz. The laser beams were expanded with a $-50~\text{mm}$ cylindrical lens into an expanding sheet, and the sheet was shaped with a 750 mm cylindrical lens to have a nominally 4 mm thickness. An expanding sheet was chosen instead of a collimated sheet because of the large test area. The time delay, $\text{d}t$ , between the two illuminating laser pulses of a given pair was $12~{\rm\mu}\text{s}$ . The short $\text{d}t$ is responsible for much of the velocity uncertainty, as discussed below. Mie scattering image pairs were captured by a high speed CMOS camera with $20~{\rm\mu}\text{m}$ pixels. The camera was outfitted with a 55 mm lens, with aperture at $f/5.6$ . The camera was operated with a pixel resolution of 640 $\times$ 448, with image pairs acquired at a frame rate of 5 kHz. The optical configuration produced a magnification of 1:12.5, such that the measurement domain covered a 640 pixel $\times$ 320 pixel region of the CMOS chip, which corresponded to a region of the combustor that was roughly 160 mm long in the axial direction and 80 mm wide in the transverse direction. The measurement plane was located at the centrespan of the bluff body. The diffraction limited spot size for this configuration was $7~{\rm\mu}\text{m}$ . Slight defocusing, together with the natural aberration and imperfections of the imaging optics, distributed the scattered light from the particles over a nominal 3 pixel diameter. A bandpass optical filter was placed in front of the camera in order to reject the majority of flame and other background luminosity. The filter passband is centred at 527 nm and has a transmission of greater than 93 % from 517 to 537 nm.

Velocity vectors were computed from the Mie scattering image pairs using a digital cross-correlation algorithm (Willert Reference Willert1991) with a multi-pass approach (Soria Reference Soria1996) and 2-D Gaussian subpixel interpolation (Willert Reference Willert1991). The multi-pass analysis consisted of a single pass with 64 $\times$ 64 pixel interrogation windows and 50 % overlap between interrogation windows, followed by 2 passes with 32 $\times$ 32 pixel interrogation windows and 50 % overlap between the interrogation windows. The spacing between vectors was 4 mm.

Post-processing algorithms were used to discard and replace a small number of spurious vectors, and to validate the PIV data. First, any velocity vector with an axial component outside of a $20\,\pm \,90~\text{m}~\text{s}^{-1}$ range or a transverse component outside of a $\pm 45~\text{m}~\text{s}^{-1}$ range was discarded. These ranges were based on the expected range of mean axial and transverse velocities, and additionally, they conservatively accommodate coherent structures with unsteady velocity on the order of the nominal $50~\text{m}~\text{s}^{-1}$ lip velocity. Next, any velocity vector whose velocimetry calculation had a peak correlation coefficient less than 0.2 was discarded and a local validation method discarded neighbouring vectors whose ratio of difference to average velocity magnitude was greater than 20 % of the r.m.s. of the nearest neighbours. This threshold is consistent with values recommended in the literature (Nogueria, Lecuona & Rodriguez Reference Nogueria, Lecuona and Rodriguez1997). In all, less than 1 % of the total velocity vectors were discarded and their values filled with the instantaneous average of neighbouring vectors.

The PIV measurement was calibrated by imaging a black anodized aluminum calibration plate with a laser etched pattern, which was located coincidently with the laser light sheet. The laser etched pattern consisted of a regularly spaced grid of crosses, composed of 0.3 mm thick lines, with spacing of 6 mm and with positional uncertainty of 0.004 %. The spatial coordinates of the Mie scattering images and resultant velocity fields were mapped to this grid. The worst-case calibration error was estimated to be one pixel over the large 160 mm calibration plate, which would result in a small 0.2 % bias error. The phase jitter between the pair of laser illumination pulses was measured to be on the order of nanoseconds, which is significantly less than the relatively long 100 ns duration at half-maximum of each illumination. Therefore, the duration of the laser illumination dominates the temporal measurement uncertainty, which is less than 1 % of the time delay between the pair of illumination pulses. Peak locking during subpixel interpolation (Fincham & Spedding Reference Fincham and Spedding1997), combined with the relatively short $\text{d}t$ , was estimated to provide a velocity uncertainty of 4 %. In/out of plane motion dominates the velocity uncertainty over much of the measurement domain, and increases with distance from the measurement centre (located at $x/D=4$ ). For example, at axial positions 20 mm from the measurement centre (i.e. at $x/D=3$ and $x/D=5$ ), this uncertainty is 5 %. As the measurement position moves to the extreme axial extents of the measurement, i.e. to $x/D=0$ and $x/D=8$ , this uncertainty increases linearly to 20 %. However, the uncertainty due to in and out of plane motion applies to turbulence and secondary instabilities in this flow, but does not directly impact the nominally 2-D hydrodynamic motion and time-averaged velocity field of interest. Finally, following the seeding particle Stokes number discussion above, the seeding particles are expected to respond at the forcing frequency with 92 % of the flow response amplitude and with less than a $20^{\circ }$ phase delay.

The PIV data was supplemented with two component laser Doppler velocimetry (LDV) (Durst, Melling & Whitelaw Reference Durst, Melling and Whitelaw1976) and a single wire constant temperature hot wire anemometer, to characterize the flow upstream of the test section. The LDV measurement was obtained at the bluff body trailing edge by scanning the flow transversely in steps of 0.1 mm. LDV was selected for measurement in this region because it is well suited to fine spatial resolution, and has less trouble than PIV with laser light reflecting from the bluff body. The LDV measurement captured the streamwise, $u$ , and transverse, $v$ , velocity components at the inflow. Seeding material consisted of $5~{\rm\mu}\text{m}$ aluminum oxide, whose flow tracking fidelity characteristics were discussed previously. For each measurement volume, data acquisition was limited to either 30 s or 5000 s counts, whichever came first. The acquisition channel used for axial velocity measurements used a laser wavelength of 514.5 nm, a fringe spacing of $7.9196~{\rm\mu}\text{m}$ , a beam waist of $70.19~{\rm\mu}\text{m}$ and a Bragg cell frequency of 40 MHz. The resulting low and high velocity limits were $-24~\text{m}~\text{s}^{-1}$ and $119~\text{m}~\text{s}^{-1}$ , respectively. The acquisition channel used to measure the transverse velocity used a laser wavelength of 488 nm, a fringe spacing of $7.5117~{\rm\mu}\text{m}$ , a beam waist of $66.57~{\rm\mu}\text{m}$ and a Bragg cell frequency of 40 MHz. For this channel, the low and high velocity limits were $-68$ and $68~\text{m}~\text{s}^{-1}$ , respectively. Both channels had a focal length of 300 mm, a beam separation of 19.5 mm, a beam diameter of 2.8 mm and used a 20 MHz high pass filter. The measurement volume for LDV was $150~{\rm\mu}\text{m}$ long, aligned in the axial direction, and $80~{\rm\mu}\text{m}$ in diameter.

The hot wire measurement was obtained in the optically inaccessible region upstream of the test section. This measurement was used to estimate the inflow conditions, but was limited to non-vitiated conditions with the test section removed. The hot wire measurement scanned the channel in steps of 5 mm, with 10 000 samples recorded at each position at an acquisition rate of 8186 Hz.

2.2. Linear stability calculation approach

This section introduces the background and methodology behind the stability analysis, which closely follows Juniper, Tammisola & Lundell (Reference Juniper, Tammisola and Lundell2011). These analyses use measured, time-averaged velocity profiles as inputs into the stability analysis in order to obtain insight into the role of linear processes in disturbance growth. A number of studies have similarly used the time-averaged velocity profiles from turbulent flows as inputs to laminar flow stability analyses to make predictions about flow stability behaviour such as linear disturbance amplification rates, nonlinear saturation and disturbance frequencies (Barkley Reference Barkley2006; Meliga, Pujals & Serre Reference Meliga, Pujals and Serre2012; Mettot, Sipp & Bezard Reference Mettot, Sipp and Bezard2014). This ‘quasi-laminar’ approach is taken in this study as well, but the authors caution that a number of questions remain around the bounds of applicability of such analyses.

First, the measured base flow is inserted into the linearized continuity and 2-D Navier–Stokes equations, which results in a system of PDE with the linear velocity and pressure perturbations as state variables. This system of PDE is reduced to ODE by projecting the state variables onto a basis of Fourier modes in time and axial position. Next, the system of ODE is reduced to a coupled system of algebraic equations, expressed as a generalized eigenvalue problem by projecting the state variables onto Chebyshev polynomials in the transverse dimension. Boundary conditions at the wall and the flow centreline are included in this algebraic system. At the confining wall, the boundary conditions enforce the no-slip and impermeability conditions. The boundary conditions at the flow centreline establish either a varicose or sinuous mode. At this location, the varicose mode has zero transverse velocity and zero cross-stream gradients of pressure and axial velocity. In contrast, the sinuous mode has centreline boundary conditions of zero axial velocity and pressure, and zero cross-stream gradient of transverse velocity. This generalized eigenvalue problem is defined for a given base flow and complex wavenumber, $k=k_{r}+\text{i}k_{i}$ . Its eigenvalue is the complex frequency, ${\it\omega}={\it\omega}_{r}+\text{i}{\it\omega}_{i}$ , and its eigenvector gives the radial dependence of the state variables, $u$ , $v$ and $p$ . The spatial stability analysis is solved by first performing a temporal stability analysis, which computes the dependence of the complex frequency on the real valued wavenumber, ${\it\omega}(k_{r})$ . The temporal analysis is used to identify the most amplified solution branch, after removing high frequency non-physical solutions. The temporal stability analysis defines a path through the complex ${\it\omega}$ plane, with a corresponding path along the real $k$ axis. Integration of the solution along this frequency-domain path would represent an inverse Fourier transform. This inverse Fourier transform would provide the time domain solution for the flow, if it were excited with the broadband dynamics defined by the values of $k$ and ${\it\omega}$ that lie along the integration path.

In order to construct a meaningful spatial stability analysis, this integration contour is deformed to lie along the real ${\it\omega}$ axis, following the procedure outlined by Huerre & Monkewitz (Reference Huerre and Monkewitz1985). The resulting path along the real ${\it\omega}$ axis has a corresponding path through the complex $k$ plane. To conclude the local spatial stability analysis, the complex wavenumber that corresponds to the real valued forcing frequency is extracted from this path. The negative of the imaginary part of this wavenumber represents the linear spatial growth rate of harmonic disturbances travelling in the positive $x$ (downstream) direction. These local analyses are used to estimate the axial dependence of the local spatial growth rate, $k(x,f)$ .

The process outlined above is performed as a local stability analysis, and is repeated at each axial position. In real flows, the time-averaged flow evolves axially. Therefore, following Chrighton (Reference Chrighton1973), the Wentzel–Kramer–Brillioun–Jeffreys (WKBJ) approximation is used to assemble the local stability analyses into a growth envelope, ${\it\zeta}(x,f)$ , which is a function of axial position and the frequency of interest using the following expression.

This approximation is valid for weakly non-parallel flows.

3.1. Inlet conditions

This section provides the properties of the velocity field at the inlet of the test section. A hot wire anemometer, under non-vitiated conditions, was traversed across the channel height, $H$ , to measure the mean velocity profile and turbulence properties upstream of the bluff body. The hot wire measurements do not resolve the thin boundary layers of the channel walls. Figure 4(a) shows the mean axial velocity and r.m.s. velocity profiles across the channel. The results show that the mean velocity in the channel is relatively uniform, and is roughly $3/4$ of the bluff body lip velocity (due to acceleration through the blockage of the bluff body). The turbulence intensity ranges from roughly 1 % in the channel centre to 3 % near the walls. Figure 4(b) shows the turbulent energy spectrum at the channel centreline just upstream of the bluff body.

Figure 4.

Hot wire measurements in unvitiated, unforced conditions with $u_{lip}=50~\text{m}~\text{s}^{-1}$ . Results show (a) mean axial velocity and unsteady velocity profiles across the channel just upstream of the bluff body, with dashed line indicating $\bar{u}=0.75u_{lip}$ and (b) turbulent energy spectrum measured at the centreline of the channel just upstream of the bluff body.

LDV measurements in the channel between the bluff body and the confining wall, at the plane of the bluff body trailing edge, were obtained to measure the thin shear layer separating from the trailing edge of the bluff body. Near the surface of the bluff body, the shear layer is 1 mm thick with 15–20 % turbulence intensity over the range of conditions reported in the paper. Outside of this boundary layer, the velocity profile (in the channel) is uniform with 8–10 % turbulence intensity. At this axial position, the contribution of the global mode to the turbulent kinetic energy is less than 1 %. These shear layer measurements, along with further characterization of the time-averaged flow, are reported in more detail in previous work (Emerson et al. Reference Emerson, O’Connor, Juniper and Lieuwen2012).

3.2. Density ratio influences

This section reviews the dynamics of the unforced flame response, summarizing results from Emerson et al. (Reference Emerson, O’Connor, Juniper and Lieuwen2012). The coordinate system and nomenclature used for flame edge displacement are presented in figure 5(b). For brevity, $L^{\prime }$ will be used to denote the upper flame branch displacement; when both the upper and lower flame branches are considered together, the subscripts lower and upper are added.

Figure 5.

Instantaneous edges of unforced flames at (a) ${\it\rho}_{u}/{\it\rho}_{b}=2.5$ , (b) ${\it\rho}_{u}/{\it\rho}_{b}=1.9$ , extracted from chemiluminescence images.

Figure 6.

(a) Flame edge displacement spectra for two density ratios, (b) dependence of the global mode amplitude on density ratio, shown for the flame edge displacement and the centreline transverse velocity. All results are at $x/D=3$ .

Flame edge spectra, $\hat{L}(x,f)$ , are presented in figure 6(a) for high and low density ratios. Note that the low density ratio flame has strong, narrowband energy centred around $St_{D}=0.24$ , as opposed to the much more spectrally distributed energy in the high density ratio flame. This is indicative of the presence of a global instability for the low density ratio cases (this peak is not associated with an acoustic mode of the chamber, as evidenced by its frequency scaling linearly with flow velocity). In order to quantify the limit cycle amplitude of the global mode, the integrated power under the spectral peak between Strouhal numbers of 0.20 and 0.28 is computed as a function of density ratio and converted into a r.m.s. amplitude using Parseval’s theorem, shown in (3.1) for a time series $s(t)$ with duration $T$ and power spectrum $|{\hat{s}}(f)|^{2}$ . Figure 6(b) plots the flame displacement and transverse velocity amplitude as functions of the density ratio, illustrating its strong influence on the amplitude of the global mode. Figure 7 presents spectra of the flame edge displacement decomposed into sinuous and varicose components, demonstrating that the global mode at $St_{D}=0.24$ is dominated by sinuous disturbances.

Figure 7.

Decomposition of flame edge displacement into sinuous and varicose disturbances for ${\it\rho}_{u}/{\it\rho}_{b}=1.9$ , $x/D=3$ .

4.1. Basic flow response features

Figure 9 shows typical flow power spectra at two forcing frequencies. The three columns show the spectra of the transverse velocity (figure 9 a,d) and its decomposition into varicose (figure 9 b,e) and sinuous (figure 9 c,f) disturbances. The first row shows results for a forcing frequency right at the global mode frequency, and the second row for a forcing frequency 30 % higher than the global mode frequency. The spectra show clear narrowband energy at the forcing frequency ( $f/f_{f}=1$ ), at harmonics of the forcing frequency and at the global mode frequency (at $St_{D}=0.24$ ). For example, figure 9(f) shows a case where the sinuous mode contains significant energy at both the forcing and global mode frequency. Figure 9(b) shows a case where the varicose mode contains significant energy at the forcing frequency and its first harmonic. When the flow is near the global stability boundary (i.e. at low ${\it\rho}_{u}/{\it\rho}_{b}$ ), such multi-modal behaviour is typical of the forced response dynamics.

The spectra in figure 9 reveal general trends about each of the spectral features. First, just as in the unforced case (e.g. see figure 7), the global mode at $St_{D}=0.24$ is almost completely composed of sinuous motions; very little energy at this frequency is present in the varicose mode. Second, the forced response (at $f/f_{f}=1$ ) is composed of both sinuous and varicose motions. Third, while the first harmonic of the forced response (at $f/f_{f}=2$ ) is present in both varicose and sinuous modes, its amplitude is similar to the fundamental’s amplitude only in the varicose mode when the system is forced at $f_{n}/f_{f}=1$ (referred to as resonant forcing in the rest of this paper). The rest of this paper focuses on how these spectral features evolve as functions of axial position, $x/D$ , and proximity of the forcing frequency to the global mode frequency, $f_{n}/f_{f}$ .

Figure 9.

(a,d) Shows typical forced flow power spectra at $x/D=3$ , ${\it\rho}_{u}/{\it\rho}_{b}=1.9$ . (b,e) Shows same power spectra but for varicose mode only, and (c,f) shows same power spectra but for sinuous mode only. (a–c) Shows results for $f_{n}/f_{f}=1.0$ , (d–f) shows results for $f_{n}/f_{f}=1.3$ .

Figure 10.

Forced flame and flow spectra for ${\it\rho}_{u}/{\it\rho}_{b}=1.9$ , $x/D=3$ , showing a sweep (a–o) of the global mode frequency from $f_{n}/f_{f}=0.7$ (a–c), 9 (d–f), 1.2 (g–i) to 1.3 (m–o). (a,d,g,j,m) Instantaneous line of sight chemiluminescence with phase-averaged iso-vorticity contours overlaid. (b,e,h,k,n) Transverse velocity spectra at $x/D=3.0$ . The ● symbol hovers over the global mode at $f_{n}$ . The solid lines in the (c,f,i,l,o) are envelopes of the forced response energy, decomposed into sinuous and varicose modes. These envelopes are laid over shaded regions, which are linear growth envelopes (shaded over a range of uncertainty) from local stability analysis.

Figure 10 summarizes the effect of $f_{n}/f_{f}$ on the flow spectra, the flame structure and the forced flow topology. The rows in figure 10 from (a–o) sweep the ratio of the global mode to forcing frequency, from $f_{n}/f_{f}=0.7$ to $f_{n}/f_{f}=1.3$ . The first column shows instantaneous flame images at the different forcing frequencies. The flame serves as a convenient form of flow visualization near the bluff body but there are important precautions for interpreting its behaviour farther downstream. Specifically, its behaviour reflects not only the local flow dynamics but also the dynamics from all upstream positions (Boyer & Quinard Reference Boyer and Quinard1990; Lieuwen Reference Lieuwen2012); i.e. it exhibits ‘memory’, which renders its interpretation farther downstream more complex. This memory effect has significant influence on its heat release dynamics, as discussed later in § 5.3. Iso-vorticity contours are also overlaid on the flame images, providing an image of the ensemble-averaged flow topology. The iso-vorticity contours are obtained by phase-averaging 400 instantaneous velocity measurements at the same phase as the flame image. The contour levels were hand-selected to illustrate the spatial distribution of coherent structures. The figure clearly shows the strong varicose flow and flame disturbance fields at global mode frequencies much lower than the forcing frequency, and the strong sinuous motions when $f_{n}\approx f_{f}$ .

Figure 10(b,e,h,k,n) shows flow spectra at a fixed axial position for the same frequency sweep test conditions. As $f_{n}/f_{f}$ increases, the broader hump associated with the global mode at $St_{D}=0.24$ and the much narrower peak at $f_{f}$ clearly move across each other. Note the strong resonant amplification of the narrowband forced response (the ordinate scales are kept consistent for all values of $f_{n}/f_{f}$ ) when $f_{n}/f_{f}\approx 1$ , and the appearance of the harmonic of the forcing frequency at $f/f_{f}=2$ .

The solid lines plotted in figure 10(c,f,i,l and o) show the axial development of the energy in the sinuous and varicose disturbance fields at $f/f_{f}=1$ using (4.1). The shaded areas on these plots represent linear stability calculations, which are discussed in the next section. The figure shows that the acoustic forcing initially deposits energy in the varicose mode, as would be expected for this axially forced configuration, evidenced by this mode’s dominance near the bluff body for all cases. The energy in the sinuous mode also rises with downstream distance, but starting from a smaller value. This growth is expected, since both the sinuous and varicose modes are spatially amplified, as shown later in § 4.2. The relative growth in energy of these two disturbance modes is, however, strongly dependent upon $f_{n}/f_{f}$ . In particular, when $f_{n}\approx f_{f}$ , the sinuous mode grows very rapidly and quickly dominates the overall disturbance energy. Thus, the input acoustic disturbance energy quickly moves into sinuous disturbances under resonant conditions. At non-resonant conditions, the forced response energy remains in the varicose mode for a larger axial duration, although the sinuous mode may eventually dominate at some downstream axial location, quantified later.

An important question in regards to the growth of the sinuous and varicose disturbances is the relative roles of linear and nonlinear effects in their axial development, as well as the degree of their nonlinear interaction. For example, can the eventual dominance of the sinuous mode in the resonant case be explained purely by linear instability arguments due to the rapid linear spatial growth rates of slight asymmetries in the excitation field, or is its growth associated with the nonlinear transfer of energy from the varicose mode which is directly excited? The next two sections further examine the role of linear and nonlinear processes in the phenomena shown above. The analysis will show that the two disturbance modes initially grow independently of each other, and that linear processes are responsible for the much faster growth of the sinuous mode relative to the varicose mode. However, the sinuous mode begins to extract energy from the varicose mode farther downstream.

To separately investigate linear and nonlinear effects, figure 11 plots phase portraits (as scatter plots) of the instantaneous relationship between the sinuous and varicose disturbances at fixed axial locations. A positive value for the varicose mode corresponds to flow moving away from the centreline. Non-zero values of the sinuous mode indicate that the flow is washing across the centreline, either towards the upper channel wall (positive values) or towards the lower wall (negative values). Figure 11(a,b) shows phase portraits at an axial position of $x/D=1$ for cases without and with resonant amplification, respectively. The case without resonant amplification, shown in figure 11(a), exhibits a flattened shape due to the low amplitude of sinuous motion. In general, the analysis of all cases at $x/D\leqslant 1$ shows no correlation between the sinuous and varicose motions. At this condition, the sinuous and varicose disturbances are uncorrelated with each other, which therefore suggests that their spatial evolutions simply reflect independent linear amplification processes. Figure 11(c,d) shows phase portraits for the same conditions, but farther downstream at $x/D=3$ . In particular, figure 11(d) presents a phase portrait from $x/D=3$ for a case with clear resonant amplification, and shows a phase-coupled relationship between the sinuous and varicose disturbances. Interestingly, this trajectory is not circular or ovular, which would be the case for superposition of two synchronized harmonic signals. Rather, the varicose mode completes two cycles for every sinuous cycle, a manifestation of the prominent harmonic ( $f=2f_{f}$ ) in the spectrum for the $f_{n}/f_{f}=1$ case. The behaviour of the harmonic is discussed further in the next section and appendix B.

Figure 11.

Phase portraits of the varicose mode energy plotted as a function of the sinuous mode energy at, ${\it\rho}_{u}/{\it\rho}_{b}=2.4$ , for a wake at (a) $f_{n}/f_{f}=1.3$ , $x/D=0.5$ , with weak resonant amplification, (b) $f_{n}/f_{f}=1.0$ , $x/D=0.5$ , with strong resonant amplification, (c) $f_{n}/f_{f}=1.3$ , $x/D=3$ , with weak resonant amplification, (d) $f_{n}/f_{f}=1.0$ , $x/D=3$ , with strong resonant amplification.

4.2. Growth of sinuous and varicose flow disturbances: linear effects

The discussion of figure 11 suggests that the forced sinuous and varicose modes grow independently, linearly, at axial positions close to the bluff body. Therefore, this section explores a spatial stability analysis of these modes. This section will show that the sinuous mode is highly spatially amplified, relative to the varicose mode, during resonant conditions. To begin, figure 10 plots the predicted forced hydrodynamic kinetic energy response based upon the linear stability analysis and WKBJ approximation outlined in § 2.2. The predicted amplitudes, ${\it\zeta}(x,f)$ are denoted ${\it\zeta}_{f,v}(x)$ for the forced varicose response and ${\it\zeta}_{f,s}(x)$ for the forced sinuous response. These envelopes are matched to $e_{v}(f_{f})$ and $e_{s}(f_{f})$ at $x/D=0.3$ , extend to $x/D=3$ and are overlaid as shaded areas on the measured modal energies in the figure 10(c,f,i,l,o). These stability predictions include an uncertainty analysis, which accounts for the effects of one PIV interrogation window of uncertainty in the shear layer density gradient offset. Therefore, these growth envelopes are shown as a broad function of axial position. In almost all cases, the linear stability analysis clearly underpredicts the spatial growth of the sinuous and varicose modes. At resonant conditions (e.g.  $f_{n}/f_{f}=0.9$ and $f_{n}/f_{f}=1.0$ ), the measured spatial growth of the sinuous mode far exceeds the predicted linear spatial growth of the sinuous mode at non-resonant conditions (e.g. at $f_{n}/f_{f}=0.7$ and $f_{n}/f_{f}=1.3$ ). Likewise, the linear analysis predicts that the sinuous mode will grow to an amplitude at $x/D=3$ that is nearly an order of magnitude higher at resonant conditions than at non-resonant conditions. However, despite the prediction of higher spatial growth rates at resonance, the linear analysis systematically underpredicts the forced response for all cases, for both the sinuous and varicose modes. Uncertainty in the base flow does not account for this underprediction, and the authors suspect that it is an inherent bias error of the local quasi-laminar method.

In order to more closely investigate the linear spatial amplification characteristics, the local spatial growth rates are directly compared to each other, again calculated using measured velocity profiles, this time plotted without using a logarithmic scale and the smoothing effects of the WKBJ integration. Figure 12 presents the axial dependence of the local spatial growth rates, calculated for several values of $f_{n}/f_{f}$ . The figure shows several noteworthy features. The first feature is that the sinuous spatial growth rate tends to exhibit a ‘bump’ near $x/D=1.5$ . The second feature is the sensitive, non-monotonic variation of the sinuous spatial growth rate as the value of $f_{n}/f_{f}$ changes. The third feature is the low-valued spatial growth rate of the varicose mode, and its relative insensitivity to the value of $f_{n}/f_{f}$ (over the studied range of this parameter). The details of these features are explored next.

Figure 12.

Local spatial growth rates computed for the measured base flow, showing growth rates of oscillations at $f_{f}$ for the (a) sinuous and (b) varicose modes at ${\it\rho}_{u}/{\it\rho}_{b}=1.9$ for five $f_{n}/f_{f}$ values.

The bumps in the curves of $-k_{i,s}(x)D$ in figure 12(a) are resonant effects associated with the nearby weakly damped absolute instability mode. The absolute instability mode occurs at a saddle point in the complex ${\it\omega}(k)$ surface. As the real ${\it\omega}$ axis (i.e. the ${\it\omega}_{i}=0$ contour) passes near this saddle point, the associated value of $k_{i}$ is minimized, and as such the spatial growth rate, $-k_{i}$ , is maximized. Figure 13 shows an example of this, by illustrating how the absolute instability mode’s saddle point draws the spatial stability analysis (the ${\it\omega}_{i}=0$ contour labeled as $k^{+}$ ) to lower values of $k_{i}$ . The frequency of the absolute instability mode (which is nominally $f_{n}$ ), and the topological location of its saddle point, are strong functions of $x/D$ and $f_{n}/f_{f}$ . Therefore, the resonant effect of the absolute instability mode is highly dependent on $x/D$ and $f_{n}/f_{f}$ and occurs when the forcing frequency, located on the real ${\it\omega}$ axis, is near where this axis crosses the topological ridge associated with the absolute instability mode’s saddle point. For the sinuous mode, this occurs close to $f_{n}/f_{f}=1$ at $x/D=1.5$ . For the varicose mode, this would occur at a frequency far from the studied range, such that the varicose mode is neither highly amplified nor highly sensitive to $f_{n}/f_{f}$ over the range of forcing frequencies studied.

Figure 13.

Contour plot of complex ${\it\omega}_{i}(k)$ surface for the sinuous mode from local stability analysis of measured base flow for ${\it\rho}_{u}/{\it\rho}_{b}=1.9$ , $f_{n}/f_{f}=1.2$ and $x/D=1.3$ . At this axial position, the local flow is convectively unstable. Thick lines labeled $k^{+}$ and $k^{-}$ are contours of ${\it\omega}_{i}=0$ . The ‘ $\boldsymbol{\times }$ ’ marks the saddle point. Vertical thick line is a branch cut.

Figure 14 shows the dependence of the calculated linear spatial growth rate of the sinuous mode on the frequency ratio and density ratio at $x/D=1$ . The figure shows that the sinuous linear spatial growth rate peaks near $f_{n}/f_{f}=1$ , as expected. The fact that this peak occurs at a value of $f_{n}/f_{f}$ slightly greater than one is due to the fact that the local stability analysis predicts an absolute frequency slightly greater than the measured value of $f_{n}$ at this axial position. The figure also shows the influence of the density ratio, which does not have a clear influence over the range of density ratios studied here. Note that the influence of the density ratio on this forced spatial problem is substantially different than for the spatio-temporal problem, where the absolute growth rate of the sinuous mode is highly sensitive to the density ratio, such as shown in figure 1(b) (Emerson et al. Reference Emerson, O’Connor, Juniper and Lieuwen2012; Yu & Monkewitz Reference Yu and Monkewitz1990). The reason for this is that the density ratio rescales the values of the iso- ${\it\omega}_{i}$ lines in figure 13, but has relatively little influence on their shapes.

Figure 14.

Local spatial growth rates computed for the measured base flow versus $f_{n}/f_{f}$ , showing values for the (a) sinuous and (b) varicose modes at $x/D=1$ and $f=f_{f}$ .

Figure 15.

(a,b) Distribution of energy at the forcing frequency between the sinuous and varicose modes, for (a) ${\it\rho}_{u}/{\it\rho}_{b}=2.4$ and (b) ${\it\rho}_{u}/{\it\rho}_{b}=1.9$ . Vertical axis represents fraction of the forced kinetic energy that is sinuous. (c,d) Varicose and sinuous forced response energies for (c) ${\it\rho}_{u}/{\it\rho}_{b}=2.4$ and (d) ${\it\rho}_{u}/{\it\rho}_{b}=1.9$ . Following a curve counter-clockwise (see arrows) from the origin corresponds to increasing $x/D$ . In (c,d), axial positions of $x/D=1$ are marked by circles, positions of $x/D=3$ are marked by squares, and positions of $x/D=5$ are marked by diamonds.

4.3. Interactions between sinuous and varicose flow disturbances: nonlinear effects

In order to better visualize the axial dependence of how energy is partitioned between sinuous and varicose modes, figure 15(a,b) plots the axial dependence of the fraction of the forced response energy that is in the sinuous mode, $e_{s}(f_{f})/[e_{s}(f_{f})+e_{v}(f_{f})]$ . Nominally, values greater than 0.5 indicate that the forced response is predominantly sinuous, and values less than 0.5 indicate that a varicose response dominates. The figure reinforces the observation from figure 10 that the varicose mode dominates the total forced disturbance energy near the bluff body. It also shows that the fraction of energy in the sinuous mode generally grows with downstream distance. Additionally, it demonstrates that the forced sinuous mode dominates in near-resonant cases, and that the varicose mode dominates in off-resonant cases. Comparing figure 15(a,b) shows a relative insensitivity of this varicose to sinuous disturbance mode phenomenon to density ratio, as expected based on the spatial growth rates shown in figure 14. Figure 15(c,d) provides an alternate view of these results by plotting the sinuous mode energy as a function of the varicose mode energy using axial position as a parameter. The curves begin at the origin and loop counter-clockwise as the axial position proceeds downstream. The figure shows that at resonance, the sinuous mode amplitude is augmented and the varicose mode amplitude is reduced. For the resonant cases in particular, the curves clearly show that the varicose and sinuous modes both grow initially, but that the varicose mode saturates and begins to decay while the sinuous mode continues to grow.

The growth of the sinuous mode relative to the varicose mode that occurs during resonant amplification manifests itself as vortex staggering. During longitudinal forcing, acoustic waves excite the two shear layers simultaneously, which roll up into larger structures that are symmetrically placed on the upper and lower half of the flow. These structures are initially symmetric, but stagger with downstream distance. This effect is evident in the iso-vorticity contours shown previously in the first column of figure 10. The degree and rate of staggering is a strong function of the forcing frequency. Although the density ratio is held constant, the vortical structures clearly evolve axially in very different manners depending on $f_{n}/f_{f}$ . In off-resonant cases, the vorticity iso-contours remain symmetrically configured to $x/D=5$ and beyond. For cases with strong resonant amplification, the vortices quickly stagger into a sinuous configuration.

The spatial dependence of the staggering phenomenon can be quantified by defining a reference length scale where vortex staggering occurs. Of course, staggering does not occur at a discrete location, but instead the degree of flow response symmetry varies smoothly with axial position, as shown in figure 15. An independent approach to quantify the degree of staggering can be obtained by considering the phase difference between the unsteady vorticity for two spatial positions straddling the flow centreline, chosen here as $y/D=\pm 0.3$ . The axial variation of the vorticity phase at the forcing frequency is presented in figure 16(a). The nearly linear phase roll-off of each curve indicates a nearly constant phase speed of the vortical disturbances. Notice, however, that the phase difference, ${\rm\Delta}{\it\varphi}$ , is approximately $180^{\circ }$ near the bluff body, and decreases to almost zero farther downstream. The phase difference between these two lines is shown in figure 16(b) at two frequency ratios. Based on this vorticity phase difference, a characteristic stagger length, $L_{stagger}$ , is defined as the axial location where ${\rm\Delta}{\it\varphi}=90^{\circ }$ (illustrated in figure 16 a).

Figure 16.

(a) Axial dependence of the phase of the vorticity from the upper and lower side of the flow centreline. The phase is extracted from the Fourier coefficient at the forcing frequency. Flow conditions are ${\it\rho}_{u}/{\it\rho}_{b}=1.9$ , and $f_{n}/f_{f}=1.1$ . (b) Vorticity phase difference, across the flow centreline, for two values of $f_{n}/f_{f}$ at ${\it\rho}_{u}/{\it\rho}_{b}=1.9$ .

Figure 17 plots the dependence of the stagger length, $L_{stagger}$ , upon $f_{n}/f_{f}$ for a number of density ratios. Note that $L_{stagger}$ is dramatically reduced when $f_{n}\approx f_{f}$ . Similar to the growth rate discussion, ${\it\rho}_{u}/{\it\rho}_{b}$ does not have a discernible influence on $L_{stagger}$ . Staggering is observed at density ratios as high as 2.5, where the flow is globally stable (see figure 6), and has a weakly damped global mode.

Figure 17.

Stagger length as a function of $f_{n}/f_{f}$ for all density ratios.

The first harmonic of the forcing frequency appears in both the varicose and sinuous modes and was noted earlier in the context of figures 10 and 11. The behaviour of the harmonics is discussed further in appendix B, which argues that resonant amplification involves a nonlinear interaction between the forced varicose mode and the forced sinuous mode, and that the first harmonic of the varicose mode is integral to this nonlinear interaction. This nonlinear relationship is likely due to the fact that the varicose mode sheds vortical structures from the bluff body once per cycle of the mode, while in contrast the sinuous mode sheds vortical structures from the bluff body at a rate that is twice the mode’s frequency (i.e. it sheds structures from both sides of the bluff body, alternately, in one cycle). Therefore, the varicose mode must operate at twice the frequency of the sinuous mode for these shedding events to be temporally aligned. This concept has been explored previously by (Leontini, Jacono & Thompson Reference Leontini, Jacono and Thompson2013), who showed that the sinuous mode is highly receptive to longitudinal excitation at twice its global mode frequency. Interestingly, this work suggests that forcing at $2f_{n}$ could directly excite the sinuous mode, without depending on a coupling to the varicose mode.

To summarize this section, resonant amplification appears to occur due to the following series of processes. First, the symmetric forcing excites the varicose mode in the separating shear layers of the bluff body. Simultaneously, small unavoidable asymmetries in the flow and the geometry allow the forcing to excite the sinuous mode. Initially, these modes are independently linearly amplified, as suggested by figure 17(a). When $f_{n}/f_{f}\approx 1$ , the sinuous mode is preferentially linearly amplified relative to the varicose mode, near $x/D=1.5$ . As the sinuous mode grows, it begins to nonlinearly interact with the varicose mode, extracting energy from it via the varicose mode’s first harmonic. Through this nonlinear interaction, the growth of the sinuous mode is boosted beyond its linear growth, and it becomes the dominant fluid dynamic mode.

Figure 18.

Axial dependence of spectra of the forced sinuous and varicose responses, for (a,c) flow and (b,d) flame. Conditions are ${\it\rho}_{u}/{\it\rho}_{b}=1.9$ . (a,b) $f_{n}/f_{f}=1.1$ and (c,d) $f_{n}/f_{f}=1.3$ .

5.1. Forced flame response structure

This section focuses on the flame response at the forcing frequency, $f=f_{f}$ . Understanding the ultimate relationship between the ‘input’ acoustic forcing and the ‘output’ heat release fluctuations is the key remaining question in the combustion instability problem. To begin, figure 18 plots flame response spectra, decomposed into sinuous and varicose disturbances, as functions of axial position. The figure shows these spectra for the flame dynamics at two values of $f_{n}/f_{f}$ , and compares them to spectra of the flow dynamics at the same conditions. This comparison illustrates the fact that the local flame and flow dynamics, although related, are not the same. The flame wrinkling at any given spatial location is a convolution integral of the time-delayed disturbance field at all upstream locations. For example, (5.1) shows a solution of the linearized G-equation for a single valued anchored flame front reproduced from equation (11.43) in Lieuwen (Reference Lieuwen2012), that relates the local flame position and velocity field, clearly illustrating this non-local relationship:

In (5.1), $u_{t,0}^{u}$ represents the local time-averaged unburned velocity tangent to the flame front, $v_{F,n}^{\prime }$ is the forced local velocity oscillation normal to the flame front and the convolution integral is performed across the transverse coordinate, $y_{s}$ . Thus, the local flame behaviour reflects not only the local flow dynamics, but also the dynamics from all upstream positions along the flame (Boyer & Quinard Reference Boyer and Quinard1990). For example, figure 18(a,c) reveals that the varicose mode of the flow fluctuations is significantly weaker than the sinuous mode at $x/D=6$ . However, the flame spectra (b,d) show that a strong varicose component remains at $x/D=6$ , which the flame ‘remembers’ because the strong varicose flow structure farther upstream excite symmetric flame wrinkles which subsequently convect downstream. Therefore, the strongly sinuous structure of the forcing flow field is not completely mirrored in the flame response – rather, the flame exhibits significant varicose oscillations as well, even at spatial locations where the varicose disturbance field is negligible and completely dominated by sinuous disturbances. However, sinuous and varicose features on the flame exhibit completely different heat release response characteristics, a topic that is emphasized in the rest of this section. Despite this fact, resonant amplification can still significantly alter the forced response structure of the flame.

5.2. Local flame heat release response

This section considers the forced response magnitude of the flame position displacement and the axial distribution of heat release rate fluctuations, and their sensitivities to frequency ratio and density ratio. As would be expected, resonant forcing manifests itself in large fluctuations in flame edge displacement. For example, figure 19 plots the dependence of the forced response magnitude of flame wrinkling amplitude upon forcing frequency, showing that it peaks near $f_{n}=f_{f}$ .

Figure 19.

Flame edge displacement amplitude at $x/D=3$ as a function of $f_{n}/f_{f}$ .

The local heat release rate of the flame is proportional to the flame area. The flame position fluctuations directly cause unsteady flame area perturbations, $S$ , according to the relationship: $\text{d}S\approx \sqrt{1+(\text{d}L/\text{d}x)^{2}}\,\text{d}x$ . Therefore, for a given flame branch, the magnitude of flame flapping is roughly proportional to the local heat release rate fluctuations. However, the heat release rate fluctuations at each axial location are controlled by the relative phasing of the upper and lower flame branches. As such, the degree of symmetry of the two branches has an important impact on the relationship between local flame flapping and the associated heat release fluctuations. In particular, varicose flow/flame oscillations cause the heat release oscillations of the two flame branches to be in phase, while sinuous oscillations cause phase-cancellation of flame surface area between the two flame branches. To visualize this, figure 20 illustrates differential lengths of flames perturbed by varicose and sinuous structures. The figure demonstrates how the sinuous structure elongates one flame branch while shortening the other. Thus, in terms of heat release oscillations, resonant amplification leads to two competing phenomena: large flame displacement and surface area fluctuations of the individual flame branches, but a sinuous flame structure. This is illustrated with the following sum of the heat release rate of the two flame branches:

The heat release rate of one flame branch may be related to the heat release rate of the other by a phase offset, ${\it\theta}$ , such that $q_{-}(x,f)=q_{+}(x,f)\text{cos}({\it\theta}(x,f))$ . Equation (5.2) then reduces to the following expression:

This expression illustrates the competing effects of flame edge displacement and staggering. Flame edge displacement drives the unsteady heat release rate of a given branch, and staggering affects the phase offset between the two branches. Note that a phase offset of ${\it\theta}=180^{\circ }$ drives this expression to zero, even if the local heat release of the individual flame branches is large.

Figure 20.

Notional sketch of a differential length of flame, perturbed by (a) varicose structure and (b) sinuous structure. Flame area change, relative to the mean, is visualized by comparing dashed lines to solid lines.

The phase difference between the upper and lower flame branches at the forcing frequency is denoted by ${\rm\Delta}{\it\varphi}=\angle \hat{L}_{upper}(f_{f})-\angle \hat{L}_{lower}(f_{f})$ . The flame is sinuous when ${\rm\Delta}{\it\varphi}=0^{\circ }$ , since the upper and lower branches are moving ‘up’ in synchronization. It is important to notice that this sinuous configuration, ${\rm\Delta}{\it\varphi}=0^{\circ }$ , occurs when the heat release of the two flame branches is out of phase, ${\it\theta}=180^{\circ }$ .

The local chemiluminescence amplitude, $q^{\prime }(x,f)$ , is quantified as the Fourier transformed unsteady luminosity from the chemiluminescence images, and is non-dimensionalized by the time-averaged chemiluminescence, $Q_{0}(x)$ . Relating heat release oscillations to chemiluminescence fluctuations is known to be a good approximation in cases where the fuel/air ratio of the approach flow is constant (Gaydon & Wolfhard Reference Gaydon and Wolfhard1970; Lee & Santavicca Reference Lee and Santavicca2003). The dependence of the local heat release fluctuations, $q^{\prime }(x,f)$ , on $f_{n}/f_{f}$ is plotted in figure 21, at a fixed axial location. This figure shows that the local heat release oscillations decrease when the forcing frequency is close to the global mode frequency, an effect that is more pronounced at lower density ratios.

Figure 21.

Local heat release amplitude at $x/D=3$ versus $f_{n}/f_{f}$ .

This local minimum in heat release reflects the fact that the phase cancellation effect dominates the increased flame displacement effect when $f_{n}\approx f_{f}$ . To better visualize these competing effects, figure 22(a,b) overlay the frequency dependence of the flame displacement amplitude, the degree of stagger between the two flame branches ( ${\rm\Delta}{\it\varphi}$ ) and the local heat release amplitude. Notice that the flame displacement amplitude is maximized when $f_{n}\approx f_{f}$ . However, when $f_{n}\approx f_{f}$ , the phase difference between the two flame branches is close to a sinuous configuration (the phase is about $45^{\circ }$ ; the reason that it is not $0^{\circ }$ is due to the flame ‘memory effect’ discussed above), and the local heat release amplitude is minimized. Figure 22(c) shows the axial dependence of the relationship of local heat release amplitude to frequency ratio. As the figure suggests, the minimum in local heat release amplitude when $f_{n}\approx f_{f}$ persists at all axial positions downstream of the initial staggering. Thus, the local heat release response of low density ratio flames under resonant forcing conditions is less than that of flames forced off-resonance.

Figure 22.

Frequency dependence of flame edge displacement amplitude, local heat release amplitude and flame stagger at $x/D=3$ , (a) ${\it\rho}_{u}/{\it\rho}_{b}=1.9$ and (b) ${\it\rho}_{u}/{\it\rho}_{b}=2.5$ . (c) Dependence of local heat release amplitude on frequency at three axial positions for ${\it\rho}_{u}/{\it\rho}_{b}=1.9$ .

Figure 23.

Spatially integrated, forced heat release rate amplitude versus flame length for (a) ${\it\rho}_{u}/{\it\rho}_{b}=2.5$ and (b) ${\it\rho}_{u}/{\it\rho}_{b}=1.9$ .

This phase cancellation effect has important and very non-intuitive implications for the motivating phenomenon of combustion instability. It shows that, for longitudinal acoustic modes, alignment of the hydrodynamic global mode frequency with the acoustic excitation frequency inhibits the flame’s forced heat release response due to the flame’s resulting sinuous response structure. Thus, for low density ratio flames, a situation where the frequency of the hydrodynamic global mode overlaps a longitudinal acoustic mode frequency is not such a dangerous situation as previously thought (Schadow & Gutmark Reference Schadow and Gutmark1992).

5.3. Global heat release response

The response of the spatially integrated heat release, referred to here as the global heat release, is of particular interest from a combustion instability standpoint. The global heat release is defined as

where $L_{f}$ is the axial flame length. The global heat release is a strong function of $L_{f}$ , due to axial phase variation in the local heat release. Figure 23 plots the global heat release magnitude at the forcing frequency as a function of axial integration length. In order to understand these behaviours, note that the flame wrinkle wavelength, ${\it\lambda}_{f}$ , is related to the flame wrinkle propagation speed and frequency according to ${\it\lambda}_{f}=u_{f}/f$ , where $u_{f}$ is a reference flame wrinkle propagation speed. Equation (5.4) is rewritten to consider only the component of the local heat release at the forcing frequency:

The normalizing reference length scale, ${\it\lambda}_{f}$ is determined experimentally from the spatial evolution of the Fourier coefficients of the flame edge displacement according to (5.6), and represents an axially-averaged quantity from the domain $2<x/D<14$ .

Equation (5.5) shows that the global heat release is an oscillatory function of the flame length. In fact, if the local heat release magnitude was a constant function of axial position, i.e.  $|q(x,f_{f})|=|q(f_{f})|$ , then the global heat release would have a perfect pattern of nodes and antinodes spaced at integer multiples of ${\it\lambda}_{f}$ , $|\hat{Q}(L_{f},f_{f})|=|\text{sin}({\rm\pi}L_{f}/{\it\lambda}_{f}(f))|$ . Such a pattern is easily recognized in the data shown in figure 23, which shows the global heat release magnitude at the forcing frequency as a function of flame length (integration length).

The nodes and antinodes in this pattern arise due to destructive/constructive phase interference between the highs and lows on the instantaneous local heat release wave, $q^{\prime }(x)$ . Figure 23 shows the globally integrated forced heat release rate amplitude as a function of flame length for two density ratios and two values of $f_{n}/f_{f}$ . The figure shows that during resonant forcing ( $f_{n}/f_{f}=1.0$ ) there is a slight reduction in global heat release response for most flame lengths compared to $f_{n}/f_{f}=0.8$ . The effect is not as pronounced as it is for the local heat release response (see figure 22 c) since the varicose portion of the flame structure at upstream axial positions is included in the spatially-averaged heat release. Interestingly, the global heat release response of the flame shows sensitivity to the density ratio. As shown in the previous sections, the flow symmetry (namely, the transition from varicose to sinuous structure) is insensitive to the density ratio over the range of density ratios studied. However, the overall amplitude of the sinuous mode is slightly higher at lower density ratios. This is evidenced at resonant conditions by comparing figure 15(c,d). The convolution integral, (5.1), that describes the local flame response integrates this into a more noticeable effect, which can be seen by comparing the $L^{\prime }/D$ values between figure 22(a,b). This effect is further integrated by (5.5) to non-negligible proportions in the global heat release response (see figure 23 a,b). Thus, the role of the density ratio is to slightly increase the amplitude of the sinuous mode, to noticeably increase the magnitude of local flame flapping during resonance and to significantly increase the global heat release response of the flame. A brief investigation of higher density ratios, where the sinuous global mode is heavily damped, suggests that no level of forcing amplitude or alignment of $f_{n}$ and $f_{f}$ results in resonant amplification. Not surprisingly, $f_{n}/f_{f}$ has little effect on the global heat release in this case.

In summary, the global heat release is a strong function of the integration length (the flame length). At some flame lengths, global heat release response amplitude is very small while at others it is significant (as much as 10 % of the time-averaged heat release rate). For flame lengths associated with significant global heat release, the global heat release response magnitude is slightly minimized when the forcing frequency equals the natural asymmetric vortex shedding frequency, and when the density ratio is lowered only enough for the forced shedding to stagger to an asymmetric pattern. An important observation is that alignment of the forcing and hydrodynamic global mode frequencies does not cause a maximum in global heat release response and therefore does not promote combustion instability. This conclusion is valid only for longitudinal acoustic modes and is likely to differ for transverse acoustic modes.