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Modeling and experimental validation of unsteady impinging flames
Combustion and Flame 146 (2006) 674–686 www.elsevier.com/locate/combustflame
Modeling and experimental validation of unsteady impinging flames E.C. Fernandes ∗ , R.E. Leandro Center for Innovation, Technology and Policy Research, IN+ , Mechanical Engineering Department, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa Codex, Portugal Received 26 August 2005; received in revised form 25 May 2006; accepted 25 June 2006 Available online 8 August 2006
Abstract This study reports on a joint experimental and analytical study of premixed laminar flames impinging onto a plate at controlled temperature, with special emphasis on the study of periodically oscillating flames. Six types of flame structures were found, based on parametric variations of nozzle-to-plate distance (H ), jet velocity (U ), and equivalence ratio (φ). They were classified as conical, envelope, disc, cool central core, ring, and side-lifted flames. Of these, the disc, cool central core, and envelope flames were found to oscillate periodically, with frequency and sound pressure levels increasing with Re and decreasing with nozzle-to-plate distance. The unsteady behavior of these flames was modeled using the formulation derived by Durox et al. [D. Durox, T. Schuller, S. Candel, Proc. Combust. Inst. 29 (2002) 69–75] for the cool central core flames where the convergent burner acts as a Helmholtz resonator, driven by an external pressure fluctuation dependent on a velocity fluctuation at the burner mouth after a convective time delay τ . Based on this model, the present work shows that
τ=
2δω+(1+N )j ω2 −j ω02 + 2π K Re 2j tanh−1 2 2 2δω+(1−N )j ω −j ω0
ω
,
i.e., there is a relation between oscillation frequency (ω), burner acoustic characteristics (ω0 , δ), and time delay τ , not explicitly dependent on N, the flame-flow normalized interaction coefficient [D. Durox, T. Schuller, S. Candel, Proc. Combust. Inst. 29 (2002) 69–75], because ∂τ /∂N = 0. Based on flame motion and noise analysis, K was found to physically represent the integer number of perturbations on flame surface or number of coherent structures on impinging jet. Additionally, assuming that τ = β H U , where H is the nozzle-to-plate distance and U is the mean jet velocity, it is shown that βDisc = 1.8, βCCC = 1.03, and βEnv = 1.0. A physical analysis of the proportionality constant β showed that for the disc flames, τ corresponds to the ratio between H and the velocity of the coherent structures. In the case of envelope and cool central core flames, τ corresponds to the ratio between H and the mean jet velocity. The predicted frequency fits the experimental data, supporting the validity of the mathematical modeling, empirical formulation, and assumptions made. © 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Flame dynamics; Thermoacoustics; Instability; Impinging flame
* Corresponding author. Fax: +351 21849 6156.
E-mail address: [email protected] (E.C. Fernandes). 0010-2180/$ – see front matter © 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2006.06.008
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Nomenclature a A c Ci f f0 h H K LN LN m n
N P P0 Pr Q0
constant of proportionality instantaneous flame area (m2 ) speed of sound (346 m/s) traveling speed of a perturbation (m/s) frequency (Hz) frequency of resonance (Hz) perturbation path length (m) nozzle-to-plate distance (m) integer constant (number of vortices or wave crests) burner neck length (m) burner neck length with acoustic end correction (m) air mass at burner neck (kg) constant of proportionality between instantaneous flame area and velocity fluctuations (flame–flow interaction coefficient) flame–flow normalized interaction coefficient (= SnQ0 /m) pressure fluctuations inside burner (Pa) pressure fluctuations at burner mouth (Pa) Prandtl number constant of proportionality between instantaneous flame area and pressure fluctuations
1. Introduction Early studies on laminar impinging jet flames [1,2] identified two basic combustion modes, depending on the nozzle exit velocity and the fuel content of the burned mixture: rim-stabilized flames and platestabilized flames. Both works presented a stability diagram, as a function of the above-mentioned variables, for a fixed wall-to-plate distance (H /D = 1), in which extinction limits, transition between different combustion modes, and associated hysteresis were mapped. More recently, Zhang and Bray [3] introduced a classification of five different combustion modes for an impinging turbulent flame, designated as conical, envelope, cool central core, disc, and ring flames, and, like Foat et al. [4], added the nozzle-to-plate distance to the relevant set of variables. In Schuller et al. [5], the classification introduced by Zhang and Bray [3] was applied to laminar impinging flames. The transitions between different combustion modes were mapped at a constant equivalence ratio (Φ = 0.95), but for different flow velocities and impingement distances.
r R Ra Rr RU S s SPL t U V u u0
radius of burner neck (m) total resistance at burner neck (kg/s) acoustic thermoviscous resistance (kg/s) acoustic radiation resistance (kg/s) mean flow acoustic resistance (kg/s) burner mouth area (m2 ) stiffness of burner volume (kg/s2 ) sound pressure level (dB) time (s) average velocity at burner mouth (m/s) volume of burner (excluding neck) (m3 ) instantaneous velocity fluctuations at burner mouth (m/s) amplitude of velocity fluctuations at burner mouth (m/s)
Greek letters α, β, γ δ ε λ μ ρ τ φ ω ω0
constants of proportionality damping factor (= R/2m) displacement at burner mouth (m) wavelength (m) coefficient of dynamic viscosity (Pa s) specific weight (kg/m3 ) time delay (s) equivalence ratio angular frequency (rad/s) angular resonance frequency (rad/s)
By appropriate manipulation of the parameters Φ, H , and U , natural oscillating flames arise in this scenario. Durox et al. [6] have shown that a cool central core flame stabilized on a burner rim and impinging onto a plate is prone to self-sustained oscillations with oscillating characteristics depending on the burner acoustics, flame type, flow conditions, and burner-to-plate distance. Within the same classification of flames Foat et al. [4] report a disc flame that, without external acoustic actuation, produced strong acoustic emissions at low H /D values. Finally, Zhang and Bray [3] only mentioned the existence of an impinging turbulent flame with strong acoustic emissions with frequency dependence on flow rate. A systematic approach explaining the nature of self-induced combustion oscillations of a cool central core impinging flame was proposed by Durox et al. [6]. They modeled a convergent burner as a Helmholtz resonator with the external force applied at the burner mouth and the pressure fluctuations originating at the flame tip close to the wall. In a previous study Schuller et al. [5] confirmed that pressure fluctuations emitted by the flame correlate with the rate of change of chemiluminescence signal, i.e., P = k dI /dt, and
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showed that the chemiluminescence signal is proportional to the instantaneous flame area, I = A , resulting in P0 = Q0 ∂A∂t(t) . Using this model, together with the relation A (t) = nu (t − τ ), where the unsteady flame surface area is related to the unsteady velocity at the burner mouth after a convective time delay τ , Durox et al. [6] obtained the final cou ) . The pling model expressed as P0 = nQ0 ∂u (t−τ ∂t time delay τ represents the time taken by a velocity perturbation to travel from the burner mouth to the flame tip where sound is produced. Solving the Helmholtz resonator equation numerically with this coupling model, they [6] were successful in establishing a relation between τ and the flame resonant frequency. The time delay τ is a critical issue in this problem and Durox et al. [6] have proposed a nonlinear relation between τ and the impinging characteristics (nozzle-to-plate distance and jet velocity) based on phase analysis between the chemiluminescence and velocity fluctuation signals. In this context, this work aims at • identifying impinging flame types as a function of Re, φ, and H , to discriminate between steady and unsteady flames, and • developing and testing the above-mentioned model for the resonance feedback mechanism [6] for the three types of unsteady laminar flames, cool central core, envelope, and disc flames, with a linear relation between the time delay τ and the impinging characteristics (nozzle-to-plate distance and jet velocity) followed by a physical analysis of the proportionality constant value.
2. Experimental setup A premixed flame was stabilized above a convergent cylindrical nozzle, with an area contraction ratio of 6.53 and an exit diameter of 18 mm, as shown in Fig. 1. Due to the nature of the study, special attention was paid to the isolation of the downstream flow from any feed line pulsations or acoustic couplings, so the nozzle was attached to a decoupling unit that consisted of three discs, each 5 mm thick, made of packed glass spheres and equally spaced, through which the fuel/air mixture is injected. The effectiveness of the system was confirmed by measurements of pressure fluctuations inside the feeding line, upstream of the decoupling unit, and the outside noise field with acoustic resonant flames that showed the two signals to be uncorrelated. The impingement plate consisted of an aluminum container, filled with water at constant temperature and atmospheric pressure, kept boiling by the use of
Fig. 1. Diagram of the experimental setup showing the burner assembly, with main dimensions, and the impinging plate with controlled temperature. The microphone locations, PMic1 and PMic2 , are also identified.
two thermostatically controlled electric heating units of 2.5 kW each. The instantaneous pressure field was assessed through a condenser microphone (B&K-2230) with the tip installed at a radial distance r = 30 cm from the nozzle axis. A Fulcrum DT3808 data acquisition board acquired the microphone signal at 20 kHz, with 16-bits resolution and with a time series length of 20 × 32,768 data points. The frequency analysis was processed with an accuracy of ±1.22 Hz. Qualitative and quantitative flow visualization was performed using a high-speed CCD video camera (Kodak Motion Corder Analyzer SR-500), enabling flow pattern recording with a sampling rate of up to 10,000 images per second and with a maximum resolution of 512 × 512 pixels. Planar laser visualization was achieved by illuminating the flow with coherent laser light from a 2W argon-ion laser, expanded by means of a cylindrical lens. The fuel used was propane and the experiments were conducted at atmospheric pressure, with Re up to 6000 and 0.75 < Φ < 1.4. Both air and propane flow rates were controlled by calibrated rotameters with a reading error of less than 5%.
3. Results and discussions 3.1. Flame stabilization modes—stability diagrams Through direct visualization of impinging laminar flames, six types of flames were identified and
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Fig. 2. Typical flame modes observed for different conditions of H /V and Φ: (a) conical flame; (b, c) envelope flame; (d, e) cool central core flame; (f, g) disc flame; (h) asymmetric side-lifted flame; (i) asymmetric disc flame; (j) ring flame.
grouped into nonlifted and lifted flames, as shown in Fig. 2. The designation is the same as that of Zhang and Bray [3] with added asymmetric flame modes. The flames are a conical free jet flame; an envelope flame, in which the upper zone of the flame is deformed due to impingement of the reaction zone on the plate; a cool central core flame, where a central region of unburned mixture impinges directly onto the plate; a symmetrical or asymmetrical disc flame in which a circular flame is stabilized at the impingement plate; a ring flame in which there is cool central stagnation region surrounded by an annular flame; and a side-lifted flame, similar in shape to a disc flame but tilted and anchored on the burner rim.
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To access the stabilization modes of impinging flames (Fig. 2), the mapping of the different combustion modes and extinction limits is presented in Fig. 3. The maps in Fig. 3a were obtained in the following way: for each H /D and constant φ the procedure was to rim-stabilize the flame and then to map the various combustion modes as the flow rate changed, starting with the lowest possible Re. The stability diagrams of Fig. 3b were obtained by taking full advantage of hysteretic phenomena by first stabilizing a cool central core flame (rich condition with 2000 Re 4000) and then changing Re and φ as indicated by arrows in Fig. 3b. Following this procedure, the extended combustion regimes are explored until specific blowoff limits or regimes previously encountered in Fig. 3a for the same H /D are reached. Similar procedures and results were used by other authors, such as those published for impinging turbulent flames [1,3,4] and for laminar impinging flames at a specific equivalence ratio [5,6]. The influence of initial conditions and the way they change were shown to be key factors in obtaining different types of flames. According to the results of the stability diagram in Fig. 3, a conical flame exists when the flame height, conditioned by Φ and Re, is less than the nozzleto-plate distance. In this case, combustion products impinge upon the plate. Otherwise, the upper zone of the flame is deformed by impingement onto the plate, giving rise to an envelope or cool central core flame. The latter flame, originally described as a premixed open flame [2], is typically favored by high Re and high equivalence ratios, while the envelope flame depends on the stretch applied close to the stagnation point, quantified by the H /U ratio as described by Foat et al. [4]. When the dynamic equilibrium at the burner rim between laminar flame speed and local velocity is lost for low Re and lean conditions, a side-lifted flame asymmetric in shape is obtained (Fig. 3c4), while for higher Re numbers the flame either blows off (H /D 4) or stabilizes at the plate (H /D < 4) in a disc (Fig. 3c5) or ring (Fig. 3c6) mode configuration, depending on flame mode before blowoff. The side-lifted flame has a small portion of its outer disc edges attached to the burner rim, with an angular tilt depending on the Re and the transition between disc and ring mode flames depending on H /U . The flashback boundary had no noticeable variation over the distances swept, regardless of the flame stabilization mode. 3.2. Analysis of impinging resonant flames The naturally induced self-sustained oscillations do not generate any new type of flame structure; in-
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Fig. 3. Mapping of the different combustion modes, sound-producing regions (gray areas), and extinction limits as a function of Re and Φ for different nozzle-to-plate separation distances H /D. (a) Maps obtained by increasing flow velocity (Re) at a constant equivalence ratio (Φ). (b) Maps obtained by stabilizing an initial cool central core flame, and changing Re and Φ, where white arrows indicate start of exploration (hysteretic regions are followed by blow off regions or combustion regimes of (a)). (c) Diagram of combustion modes: (c1) conical flame, (c2) envelope flame, (c3) cool central core flame, (c4) side-lifted flame, (c5) disc flame, and (c6) ring flame.
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679
(a)
(a)
(b)
(b)
Fig. 4. Evolution of the resonant frequency and respective sound pressure level, SPL, as a function of nozzle-to-plate distance H /D, for (a) disc flame (Re = 2765, Φ = 0.85); (b) envelope flame (Re = 3230, Φ = 1.15) and cool central core flame (Re = 3093, Φ = 1.2).
Fig. 5. Evolution of the resonant frequency and respective sound pressure level, SPL, as a function of jet velocity (U ), with a fixed nozzle-to-plate distance H /D = 2 for (a) disc flame (Φ = 0.85); (b) envelope flame (Φ = 1.15) and cool central core flame (Φ = 1.2).
stead, they symmetrically or asymmetrically modulate the shape of the steady state flame, as shown by typical images in Fig. 2. Experiments conducted with unsteady flames show, in general, that as H /D is reduced the dominant oscillation frequency rises, in all type of flames, to a limit value of about 500 Hz, while the sound pressure level (SPL) emitted by these flames is in the range of 70–90 dB (Fig. 4). In addition, resonant frequencies and SPL increase monotonically with flow velocity for all types of flames (e.g., Fig. 5), but no significant influence was established by changing the equivalence ratio (see Fig. 6). The observed jumps in frequency and SPL evolutions are found associated with vortex pairing in disc flames and changes in the number of visual waves in the case of envelope and cool central core flames. The limit frequency of 500 Hz is close to the natural resonant frequency of the burner, modeled as a Helmholtz resonator with a resonant frequency of 484 Hz given c by f0 = 2π
S , where c is the sound velocity, S
LN V
the burner mouth area, LN the burner neck length with acoustic end correction [7], and V the burner volume (excluding neck).
The trend of the dominant peak frequency of the sound emitted by the unsteady flames, as a function of nozzle-to-plate distance, is similar to the results of Wagner [8], Lau et al. [9], Ho and Nosseir [10], and Tam and Ahuja [11] for subsonic isothermal impinging jets (Mach numbers = 0.65–0.95), to the subsonic and sonic impinging jet results of Neuwerth [12] and Ho and Nosseir [10], and to the cool central core oscillating flames of Durox et al. [6]. This substantiates the belief that in both reactive and isothermal impinging jets, oscillations are sustained by the same basic mechanism, i.e., there is a feedback mechanism related to nozzle-to-plate distance and jet velocity. 3.3. Burner acoustic characteristics The burner acoustics is modeled, due to its geometry, as a Helmholtz resonator with the equation (Kinsler et al. [13]) ∂ε ∂ 2ε + sε = −SP0 , +R (1) ∂t ∂t 2 where ε is the axial displacement of air mass m at the burner mouth, P0 is the external pressure exerted at the burner exit over its area S, and R and s, respec-
m
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(a)
Fig. 7. Comparison between analytical and experimental burner acoustic response for different flow velocities under isothermal conditions.
(b) Fig. 6. Evolution of the resonant frequency and respective sound pressure level, SPL, as a function of the equivalence ratio Φ, with fixed Re = 2800 and H /D = 2 for (a) disc flames and (b) envelope flames.
tively, are the total resistance and volume stiffness, given according to R = Ra + Rporous + Rr + RU = (a + 1)Ra + Rr + RU , γ −1 2m μω 0.5 1+ √ Ra = , r 2ρ Pr ρ
(1b)
2
(ωS) , Rr = c 4π RU = ρU S, ρC 2 S 2 . s= V
(1a)
(1c) (1d) (1e)
In this model there are two additional acoustic resistance terms: Rporous and RU . The term RU quantifies the interactions between mean flow and acoustics in the burner neck [7]. The term Rporous is due to a porous plate in the burner’s base (see Fig. 1) modeled as a collection of capillary tubes with an overall acoustic resistance similar to resistance Ra [14]. Hence, Rporous = aRa , where the constant a is obtained by fitting the burner response (P ) to an external excitation (P0 ). The analytical burner response is 2 SP0 is the |P | = ρc VS|ε| , where ε = j ω(R+j (ωm−s/ω))
solution of Eq. (1). The burner volume stiffness s, together with the air mass m at the burner neck, defines a more compact expression for the previously mentioned Helmholtz resonant frequency now given by ω0 = 2πf0 = (s/m)0.5 . The experimental data are obtained by exciting the burner with a loudspeaker placed near the burner mouth (at a distance of 300 mm), i.e., generating P0 . The ratio between pressure fluctuations measured inside the burner (P ) to those near the mouth (P0 ) is plotted in Fig. 7 for a range of frequencies from 50 to 1500 Hz. Two sets of results are plotted, for 2.5 and 5 m/s, corresponding to typical working conditions. The burner’s response to an external excitation shows one distinct peak at 477.8 Hz, lower than the natural undamped Helmholtz resonator oscillation frequency (f0 = 484 Hz) due to the acoustic damping effect, which forces the pressure peak amplitude to occur to the left of f0 , the shift being larger for larger damping values [13]. On the other hand, the damping effect due to mean flow (e.g., Durox et al. [6]) does not significantly influence the transfer function (the resonance bandwidths are the same). In the same figure, the analytical acoustic transfer function is also plotted, for which the constant a was found equal to 9.46. Hence, the complete burner (cavity and porous discs) behaves as a Helmholtz resonator and for the peak frequency of 477.8 Hz, the damping coefficient, δ = R/2m, is equal to 296 s−1 , which defines a characteristic of this system. 3.4. Flame–burner acoustic coupling In what follows P0 , in Eq. (1), is now related to the flame behavior. Consider the scheme shown in Fig. 8a, in which a typical vortex or bulge trajectory is represented along a curved pathline imposed by the adverse pressure gradient exerted by the wall, traveling with an instantaneous velocity Ci from its
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τ function of a characteristic velocity and a characteristic distance as proposed by [6,15,16]. The coupling between the unsteady flame and the burner acoustic response can be modeled following the approach suggested by Durox et al. [6], who showed, for an axisymmetric cool central core type flame, that the external force in Eq. (1) can be related to the velocity at the burner mouth as P0 =
(a)
) nQ0 ∂u (t−τ . This coupling model will be tested ∂t with the other types of axisymmetric flames in this work. Substituting this equation in Eq. (1) and applying the time derivative yields ∂u ∂ 2 u ∂ 2 u + su = −SnQ0 2 , m 2 +R (2) ∂t ∂t ∂t t−τ
originally presented by [6] and valid for the cases where the same dominant frequency peak is present outside and inside the burner during resonance (see Fig. 8b), excluding the vortex pairing regimes in unsteady disc flames. A general solution of Eq. (2) is proposed as u˜ = u0 ej ωt , where the real part of this is the particular solution for Eq. (2) (e.g., [13]). Replacing u by u˜ and τ by τ˜ , the respective complex forms, in Eq. (2) and solving it in order to obtain τ˜ yields 2j tanh−1 τ˜ =
2δω+(1+N )j ω2 −j ω02 2δω+(1−N )j ω2 −j ω02
+ 2π K
. ω The real part of this equation is therefore the solution for the original Eq. (2) as
τ=
2δω+(1+N )j ω2 −j ω02 Re 2j tanh−1 + 2π K 2 2
K ∈ ℵ, (b)
2δω+(1−N )j ω −j ω0
ω
, (3)
Fig. 8. (a) Schematic diagram of the feedback mechanism for an impinging jet, where Ci is the flow perturbation velocity, U the mean exit flow velocity, H the nozzle-to-plate distance, and Ca the feedback acoustic wave velocity. The dashed line represents the path traveled by a given flow perturbation from the nozzle exit to the acoustic wave birth location. (b) Power spectral density (PSD) of pressure fluctuations measured inside (PMic1 ) vs outside (PMic2 ) for envelope and disc flames.
where N = SnQ0 /m. Coefficient N represents the flame–flow normalized interaction coefficient [6] and, in spite of its appearing in the first term of Eq. (3), further analysis dτ˜ = −i ; i.e., Eq. (3) does not depend showed that dN Nω explicitly on N . Therefore, in Eq. (3), the term
2δω + (1 + N )j ω2 − j ω02 Re 2j tanh−1 2δω + (1 − N )j ω2 − j ω02
origin (the shear layer nozzle exit) to a specific region (represented by an asterisk). According to Schuller et al. [5], sound is produced in this region at the upper tips of the flame front, those nearer the stagnation surface, when each of the upward propagating bulges is finally consumed. The generated feedback pressure wave then propagates upstream, reaching the nozzle exit zone and triggering a new perturbation. The total time taken by this process corresponds to a time delay
holds the burner acoustic characteristics and 2π K gives the periodic solution. For simplicity, N was set equal to 1 (one) in the numerical solution of Eq. (3). In Fig. 9 are plotted Eq. (3) and the burner-related term for three different damping factors δ: 0.1R/2m, R/2m (present case), and 10R/2m. The coefficient R is given by Eq. (1a). Coupling between flame and burner acoustics occurs around ω0 with a bandwidth governed by the damping factor δ (Fig. 9a). For high
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(a)
(b) Fig. 9. Analysis of the “f –τ ” mathematical model. (a) Behavior of the term containing the burner acoustic characteristics, as a function of the damping factor δ; (b) influence of the burner damping factor δ on “f –τ ” evolution.
values of δ the evolution of f –τ is smooth from a low- to a high-frequency band (Fig. 9b). For very small values of δ, the burner only amplifies a narrow bandwidth around f0 , and consequently the evolution of f –τ has an abrupt transition around f0 , which vanishes as K increases. The contribution of the burner acoustics term, shown in Fig. 9a, ranges between two asymptotes, 0 and −π , automatically fulfilling the phase criterion of Schuller et al. [15], which based on an acoustic energy balance, showed that ωτ belongs to [π, 2π ] modulo 2π as a necessary condition for the onset of oscillations. In the same study, Schuller et al. [15] have shown that the energy fed by the flame to the system (burner), proportional to the normalized interaction coefficient N , should be greater than the energy dissipated inside the system, as a necessary condition for obtaining a self-sustained oscillation. The role of N is therefore to dictate the occurrence or not of selfsustained oscillations, at the frequencies predicted by Eq. (3). To plot Eq. (3) against experimental data, τ is taken as a function of a characteristic velocity and a characteristic distance as mentioned before. In this
context, different models were proposed for the τ function. Among the first, we have the experiments of Neuwerth [12] with a high-speed subsonic impinging jet with τ = H /Ci , where H is the distance from the injector to the wall and Ci the coherent structures velocity. Recently, Schuller et al. [15] have shown that τ = L/U in a study conducted with mutual flame interactions driving self-sustained oscillations. In [15] L is a characteristic distance where two flame front elements pinch off suddenly and U is the nozzle flow velocity. Also, Durox et al. [16] suggested the same type of model in experiments to study self-sustained oscillations driven by vortex shedding from a burner lip. For this particular configuration L is the distance where the vortex, initially surrounded by the flame, extinguishes it and U is replaced by Ci , the coherent structure velocity. Generally, it was shown that the choice of a characteristic distance and velocity is critical for modeling success, and Durox et al. [6] proposed a nonlinear model for τ in experiments similar to that shown here. This model was successfully applied for small flame-to-wall distances, or equivalently for ωH /U 10, as estimated from data presented in [6], and, as pointed out by the authors [6], probably not applicable to larger distances, where the predicted frequencies were not consistent with experimental results. In the present work, since ωH /U > 10, and recalling that τ is the time taken by a perturbation with velocity U to travel a path L, we propose τ as a linear function of U and H , using the initial model proposed by Neuwerth [12] when studying impinging air jets or equivalently identifying in the model of a resonant, nonimpinging, reactive flame of Schuller et al. [15] or Durox et al. [16], L with H . The τ function is then given by H (4) , U where β must be defined for each flame type. Substituting Eq. (4) into Eq. (3) yields 2 2 −1 2δω+(1+N)j ω −j ω0 + 2π K H 1 Re 2j tanh 2δω+(1−N)j ω2 −j ω02 = . ω U β (5) Equation (5) establishes a relation between the burner acoustic characteristics, resonant frequency, jet velocity, and nozzle-to-plate distance. It was applied systematically to the three types of unsteady laminar flames, namely the envelope flame, cool central core flame, and disc flame, sweeping a set of self-sustained resonant conditions for impingement distance H , average exit flow velocity U , and frequency f . For each resonant situation, the parameter K is obtained from the high-speed film sequences by counting the wavelengths (number of waves) or number of τ =β
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(a)
(b) Fig. 10. Tracking of a vortex trajectory in resonant disc flame images by laser plane visualization (H /D = 1.2, Φ = 0.85, Re = 2880). (a) Sequence of a resonant disc flame oscillating with a frequency of f = 162 Hz, with K = 3. The images were acquired at a rate of 1 kHz. (a1) to (a6) are images at phases 0, π , 2π , 8π/3, 10π/3, 4π , where a characteristic vortex (marked with crosses) is followed from near the nozzle exit (a1) until its total consumption at the flame front (a6); (b) dependence of the vortex pathline on nozzle-to-plate distance (H /D).
vortices visible on the perturbation path (see, for example, Figs. 2 and 10). The frequency f corresponds to the dominant peak on the sound spectrum signal, which matches the flame front oscillation frequency taken from the high-speed films. This matching was already checked by Schuller et al. [5] and Durox et al. [6] for a flame impinging on a plate, by Schuller et al. [15] for mutual flame interactions, and by Durox et al. [16] for flame vortex interactions on a bluff body configuration. Analytical results are plotted against experimental data in Fig. 11. The results show a clear superimposition for constant beta factors of βDisc = 1.80, βCCC = 1.03, and βEnv = 1.00, obtained by data best fit. The most noticeable difference is that βDisc is very different from the other two. In the former, we have an isothermal jet ending in a disc flame, while in the latter the flame extends towards the nozzle. This approach confirms that the mathematical model and the approximation given by Eq. (5) are both valid. 3.5. Analysis of τ for each type of resonant flame In what follows, the physical meaning of β in Eq. (4) is discussed. Consider the time delay given
Fig. 11. Resonant frequency as a function of time delay H /U . Comparison between predicted frequency and experimental data with the proposed relation τ = βH /U .
by Eq. (4) as follows: 1 1 dl. dl + τ= Ci c pathline-1 pathline-2
This total time is divided into pathline-1 C1 dl i
1 pathline-2 c dl, which are the travel times of branches of the feedback cycle. The first term
(6)
and two cor-
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Fig. 12. The influence of Re and nozzle-to-plate distance H /D on the integrated path length of the downstream propagating instability. (a) Disc flame (Re = 2880, Φ = 0.85); (b) disc flame (H /D = 2, Φ = 0.85); (c) envelope (Re = 3230, Φ = 1.15) and cool central core (Re = 3093, Φ = 1.2) flames; (d) envelope (H /D = 2, Φ = 1.15) and cool central core (H /D = 2, Φ = 1.2) flames.
responds to the time taken by each flow perturbation to travel from the nozzle exit, where it is generated, through the intrinsically unstable thin shear layer [10] to the sound-producing region, while the second term corresponds to the time taken by an upstream propagating feedback acoustic wave to travel from this region to the nozzle exit, where it further excites the shear layer of the jet and closes the feedback loop. This last term is insignificant compared to the first one, because pathline-1 is longer than pathline-2 and the flow conditions are subsonic, i.e., c Ci . Returning to Eq. (6) and despite the expected changes in Ci , in modulus and direction along the curved path line, it is proposed that Ci = γ U . The remaining integral that
gives the curved pathline length is scaled to H , as pathline-1 dl = αH . Inserting these two simplifications into Eq. (6), yields β = γα , which is the ratio of two weight factors, one related to the length and the other to the perturbation travel speed. Hence, an improved understanding of the feedback mechanism is achieved by isolating the contribution of α and γ in the global parameter β. For each resonant flame, the pathline was obtained by analyzing a sequence of high-speed digital images and recording the spatial coordinates of a given perturbation as it moves from the nozzle to the acoustic
wave birth locus, as shown in Fig. 10a. The pathline obtained by this process is plotted in Fig. 10b for three example cases of a disc flame and shows that the ratio between vertical and horizontal path lengths depends strongly on the imposed distance H . In spite of this behavior the total length h is a linear function of the nozzle-to-plate distance H , with a slope αDisc = 1.02, as shown in Fig. 12a. A similar evolution is found for all cool central core and envelope flames, e.g., Fig. 12c, but with different slopes: αEnv = 0.94, αCCC = 1.12. The influence of Re on the above results was found to be comparatively small, as shown for example in Fig. 12b. Now the value of γ is obtained straightforwardly from the definition of β. As a result, we have γDisc = 0.57, γEnv = 0.94, and γCCC = 1.09. The α factor is on the order of unity, which means that the perturbation’s trajectory length, from the burner mouth to the flame tip where it produces sound, is on the order of the nozzle-to-plate distance H , independent of the flame type or deformation imposed by the adverse pressure gradient. The velocity γ weight is on the order of unity for the cool central core and envelope type of flames, which means that a perturbation (those waves modulating the flame front; see Fig. 2) travels with a speed close to the velocity U . For the disc flame, this factor is 0.57, corresponding
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685
Table 1 Diagram of the three oscillating impinging flames with the main variables of the process: Physical identification of the relation between time delay (τ ) and nozzle-to-plate distance (H ) and jet velocity (U )
α = h/H γ = Ci /U β = α/γ τ = βH /U
Disc flame
Env flame
CCC flame
1.02 0.57 1.80 ∼ H /0.56U
0.94 0.94 1.00 ∼ H /U
1.12 1.09 1.03 ∼ H /U
to the travel speed of coherent structures in free jets [9,10,17]. The results show the validity of the approach suggested by Eq. (4) with a specific constant of proportionality depending on the type of flame to be studied.
4. Conclusions The present experimental work studies the behavior of laminar flames impinging onto a controlled temperature plate, with special emphasis on oscillating flames. The experimental techniques used were microphones, photography, high-speed digital cinematography, and planar laser visualization. Six types of flame structures were found, namely conical, envelope, disc, cool central core, ring, and side-lifted flames, based on parametric variations of nozzle-to-plate distance, Re, and equivalence ratio. Of these, three types of flames—disc, cool central core, and envelope—were found to oscillate, with frequency and SPL clearly increasing with Re and the inverse of nozzle-to-plate distance. To summarize, and according to the scheme shown in Table 1, the disc unstable flame consists of an isothermal mixing jet impinging onto a flat flame. The jet has coherent shear layer structures that travel downstream, interacting with the flame surface. The unstable cool central core flame has a tulip shape, while the unstable envelope flame has a closed tulip shape, both attached to the burner, with velocity perturbations traveling from the burner mouth up to the flame tip, deforming the flame surface. For the analysis of these unstable flames, the mathematical model derived by [6] to study unsteady cool central core flames was adopted here, due to flame and burner setup similarities. The burner was therefore modeled as a Helmholtz resonator with the external force being the sound produced by the impinging flames. Analytical solutions derived in this work showed, for all types of flames, how the predicted oscillation
frequency is related to the generic time delay taken by these perturbations to travel from the burner mouth to the flame tip, the burner acoustic characteristics, and the integer number of perturbations (vortices or waves), but with no explicit dependency on the flame– flow normalized interaction factor N . What does depend on N is the actual onset of self-sustained resonant conditions, where for a given working regime with oscillation frequencies given through Eq. (3), the energy fed by the flame, proportional to N , must be greater than the energy dissipated inside the system, as shown by Schuller et al. [15]. The most relevant burner acoustic characteristic in this process is the damping factor (energy dissipation). A larger damping factor increases the frequency bandwidth that the burner can amplify and hence it reinforces the burner’s role in the frequency results. On the other hand, the experimental analysis conducted with oscillating flames reveals that the time taken by perturbations to travel from the burner mouth to the flame tip τ is linearly related to the ratio H /U , i.e., the ratio between nozzle-to-plate distance to jet velocity, regardless of flame type. This contrasts with [6], which obtained a nonlinear dependence for the time delay for the case of CCC flames in the range of ωH /U < 10. The proportionality constant, β, is represented in Table 1 and has almost the same value for the cool central core and envelope flames and a different value for the disc flame. The main implication of these results, based on physical analysis of β, is that the travel time for envelope and cool central core flames can be obtained directly from the H /U ratio, while for the disc flame this time is the ratio between distance H and the traveling speed of the coherent structures. Incorporating this information into the mathematical model, the analytical solution fits experimental data if the integer number K (from 2π K of Eq. (2)) is taken as the number of vortices and/or flame surface waves observed under resonant regimes.
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References [1] C.K. Law, S. Ishizuka, M. Mizomoto, Proc. Combust. Inst. 18 (1981) 1791–1798. [2] S. Ishizuka, K. Miyasaka, C.K. Law, Combust. Flame 45 (1982) 293–308. [3] Y. Zhang, K.N.C. Bray, Combust. Flame 116 (1999) 671–674. [4] T. Foat, K.P. Yap, Y. Zhang, Combust. Flame 125 (2001) 839–851. [5] T. Schuller, D. Durox, S. Candel, Combust. Flame 128 (2002) 88–110. [6] D. Durox, T. Schuller, S. Candel, Proc. Combust. Inst. 29 (2002) 69–75. [7] A. Hirschberg, in: Advances in Aeroacoustics, Lecture Series, March 12–16, Von Karman Institute for Fluid Dynamics, 2001. [8] F.R. Wagner, NASA TT F-13942, 1971.
[9] J.C. Lau, M.J. Fisher, H.V. Fuchs, J. Sound Vibration 22 (1972) 379–406. [10] C.M. Ho, N.S. Nosseir, J. Fluid Mech. 105 (1981) 119– 142. [11] C.K.W. Tam, K.K. Ahuja, J. Fluid Mech. 214 (1990) 67–87. [12] G. Neuwerth, NASA TT F-15719, 1974. [13] L.E. Kinsler, A.R. Frey, A.B. Coppens, J.V. Sanders, Fundamentals of Acoustics, fourth ed., Wiley, New York, 2000. [14] A. Craggs, J.G. Hildebrandt, J. Sound Vibration 92 (1984) 321–331. [15] T. Schuller, D. Durox, S. Candel, Combust. Flame 135 (2003) 525–537. [16] D. Durox, T. Schuller, S. Candel, Proc. Combust. Inst. 30 (2005) 1717–1724. [17] S.C. Crow, F.H. Champagne, J. Fluid Mech. 48 (1971) 547–591.
Modeling and experimental validation of unsteady impinging flames E.C. Fernandes ∗ , R.E. Leandro Center for Innovation, Technology and Policy Research, IN+ , Mechanical Engineering Department, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa Codex, Portugal Received 26 August 2005; received in revised form 25 May 2006; accepted 25 June 2006 Available online 8 August 2006
Abstract This study reports on a joint experimental and analytical study of premixed laminar flames impinging onto a plate at controlled temperature, with special emphasis on the study of periodically oscillating flames. Six types of flame structures were found, based on parametric variations of nozzle-to-plate distance (H ), jet velocity (U ), and equivalence ratio (φ). They were classified as conical, envelope, disc, cool central core, ring, and side-lifted flames. Of these, the disc, cool central core, and envelope flames were found to oscillate periodically, with frequency and sound pressure levels increasing with Re and decreasing with nozzle-to-plate distance. The unsteady behavior of these flames was modeled using the formulation derived by Durox et al. [D. Durox, T. Schuller, S. Candel, Proc. Combust. Inst. 29 (2002) 69–75] for the cool central core flames where the convergent burner acts as a Helmholtz resonator, driven by an external pressure fluctuation dependent on a velocity fluctuation at the burner mouth after a convective time delay τ . Based on this model, the present work shows that
τ=
2δω+(1+N )j ω2 −j ω02 + 2π K Re 2j tanh−1 2 2 2δω+(1−N )j ω −j ω0
ω
,
i.e., there is a relation between oscillation frequency (ω), burner acoustic characteristics (ω0 , δ), and time delay τ , not explicitly dependent on N, the flame-flow normalized interaction coefficient [D. Durox, T. Schuller, S. Candel, Proc. Combust. Inst. 29 (2002) 69–75], because ∂τ /∂N = 0. Based on flame motion and noise analysis, K was found to physically represent the integer number of perturbations on flame surface or number of coherent structures on impinging jet. Additionally, assuming that τ = β H U , where H is the nozzle-to-plate distance and U is the mean jet velocity, it is shown that βDisc = 1.8, βCCC = 1.03, and βEnv = 1.0. A physical analysis of the proportionality constant β showed that for the disc flames, τ corresponds to the ratio between H and the velocity of the coherent structures. In the case of envelope and cool central core flames, τ corresponds to the ratio between H and the mean jet velocity. The predicted frequency fits the experimental data, supporting the validity of the mathematical modeling, empirical formulation, and assumptions made. © 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Flame dynamics; Thermoacoustics; Instability; Impinging flame
* Corresponding author. Fax: +351 21849 6156.
E-mail address: [email protected] (E.C. Fernandes). 0010-2180/$ – see front matter © 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2006.06.008
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Nomenclature a A c Ci f f0 h H K LN LN m n
N P P0 Pr Q0
constant of proportionality instantaneous flame area (m2 ) speed of sound (346 m/s) traveling speed of a perturbation (m/s) frequency (Hz) frequency of resonance (Hz) perturbation path length (m) nozzle-to-plate distance (m) integer constant (number of vortices or wave crests) burner neck length (m) burner neck length with acoustic end correction (m) air mass at burner neck (kg) constant of proportionality between instantaneous flame area and velocity fluctuations (flame–flow interaction coefficient) flame–flow normalized interaction coefficient (= SnQ0 /m) pressure fluctuations inside burner (Pa) pressure fluctuations at burner mouth (Pa) Prandtl number constant of proportionality between instantaneous flame area and pressure fluctuations
1. Introduction Early studies on laminar impinging jet flames [1,2] identified two basic combustion modes, depending on the nozzle exit velocity and the fuel content of the burned mixture: rim-stabilized flames and platestabilized flames. Both works presented a stability diagram, as a function of the above-mentioned variables, for a fixed wall-to-plate distance (H /D = 1), in which extinction limits, transition between different combustion modes, and associated hysteresis were mapped. More recently, Zhang and Bray [3] introduced a classification of five different combustion modes for an impinging turbulent flame, designated as conical, envelope, cool central core, disc, and ring flames, and, like Foat et al. [4], added the nozzle-to-plate distance to the relevant set of variables. In Schuller et al. [5], the classification introduced by Zhang and Bray [3] was applied to laminar impinging flames. The transitions between different combustion modes were mapped at a constant equivalence ratio (Φ = 0.95), but for different flow velocities and impingement distances.
r R Ra Rr RU S s SPL t U V u u0
radius of burner neck (m) total resistance at burner neck (kg/s) acoustic thermoviscous resistance (kg/s) acoustic radiation resistance (kg/s) mean flow acoustic resistance (kg/s) burner mouth area (m2 ) stiffness of burner volume (kg/s2 ) sound pressure level (dB) time (s) average velocity at burner mouth (m/s) volume of burner (excluding neck) (m3 ) instantaneous velocity fluctuations at burner mouth (m/s) amplitude of velocity fluctuations at burner mouth (m/s)
Greek letters α, β, γ δ ε λ μ ρ τ φ ω ω0
constants of proportionality damping factor (= R/2m) displacement at burner mouth (m) wavelength (m) coefficient of dynamic viscosity (Pa s) specific weight (kg/m3 ) time delay (s) equivalence ratio angular frequency (rad/s) angular resonance frequency (rad/s)
By appropriate manipulation of the parameters Φ, H , and U , natural oscillating flames arise in this scenario. Durox et al. [6] have shown that a cool central core flame stabilized on a burner rim and impinging onto a plate is prone to self-sustained oscillations with oscillating characteristics depending on the burner acoustics, flame type, flow conditions, and burner-to-plate distance. Within the same classification of flames Foat et al. [4] report a disc flame that, without external acoustic actuation, produced strong acoustic emissions at low H /D values. Finally, Zhang and Bray [3] only mentioned the existence of an impinging turbulent flame with strong acoustic emissions with frequency dependence on flow rate. A systematic approach explaining the nature of self-induced combustion oscillations of a cool central core impinging flame was proposed by Durox et al. [6]. They modeled a convergent burner as a Helmholtz resonator with the external force applied at the burner mouth and the pressure fluctuations originating at the flame tip close to the wall. In a previous study Schuller et al. [5] confirmed that pressure fluctuations emitted by the flame correlate with the rate of change of chemiluminescence signal, i.e., P = k dI /dt, and
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showed that the chemiluminescence signal is proportional to the instantaneous flame area, I = A , resulting in P0 = Q0 ∂A∂t(t) . Using this model, together with the relation A (t) = nu (t − τ ), where the unsteady flame surface area is related to the unsteady velocity at the burner mouth after a convective time delay τ , Durox et al. [6] obtained the final cou ) . The pling model expressed as P0 = nQ0 ∂u (t−τ ∂t time delay τ represents the time taken by a velocity perturbation to travel from the burner mouth to the flame tip where sound is produced. Solving the Helmholtz resonator equation numerically with this coupling model, they [6] were successful in establishing a relation between τ and the flame resonant frequency. The time delay τ is a critical issue in this problem and Durox et al. [6] have proposed a nonlinear relation between τ and the impinging characteristics (nozzle-to-plate distance and jet velocity) based on phase analysis between the chemiluminescence and velocity fluctuation signals. In this context, this work aims at • identifying impinging flame types as a function of Re, φ, and H , to discriminate between steady and unsteady flames, and • developing and testing the above-mentioned model for the resonance feedback mechanism [6] for the three types of unsteady laminar flames, cool central core, envelope, and disc flames, with a linear relation between the time delay τ and the impinging characteristics (nozzle-to-plate distance and jet velocity) followed by a physical analysis of the proportionality constant value.
2. Experimental setup A premixed flame was stabilized above a convergent cylindrical nozzle, with an area contraction ratio of 6.53 and an exit diameter of 18 mm, as shown in Fig. 1. Due to the nature of the study, special attention was paid to the isolation of the downstream flow from any feed line pulsations or acoustic couplings, so the nozzle was attached to a decoupling unit that consisted of three discs, each 5 mm thick, made of packed glass spheres and equally spaced, through which the fuel/air mixture is injected. The effectiveness of the system was confirmed by measurements of pressure fluctuations inside the feeding line, upstream of the decoupling unit, and the outside noise field with acoustic resonant flames that showed the two signals to be uncorrelated. The impingement plate consisted of an aluminum container, filled with water at constant temperature and atmospheric pressure, kept boiling by the use of
Fig. 1. Diagram of the experimental setup showing the burner assembly, with main dimensions, and the impinging plate with controlled temperature. The microphone locations, PMic1 and PMic2 , are also identified.
two thermostatically controlled electric heating units of 2.5 kW each. The instantaneous pressure field was assessed through a condenser microphone (B&K-2230) with the tip installed at a radial distance r = 30 cm from the nozzle axis. A Fulcrum DT3808 data acquisition board acquired the microphone signal at 20 kHz, with 16-bits resolution and with a time series length of 20 × 32,768 data points. The frequency analysis was processed with an accuracy of ±1.22 Hz. Qualitative and quantitative flow visualization was performed using a high-speed CCD video camera (Kodak Motion Corder Analyzer SR-500), enabling flow pattern recording with a sampling rate of up to 10,000 images per second and with a maximum resolution of 512 × 512 pixels. Planar laser visualization was achieved by illuminating the flow with coherent laser light from a 2W argon-ion laser, expanded by means of a cylindrical lens. The fuel used was propane and the experiments were conducted at atmospheric pressure, with Re up to 6000 and 0.75 < Φ < 1.4. Both air and propane flow rates were controlled by calibrated rotameters with a reading error of less than 5%.
3. Results and discussions 3.1. Flame stabilization modes—stability diagrams Through direct visualization of impinging laminar flames, six types of flames were identified and
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Fig. 2. Typical flame modes observed for different conditions of H /V and Φ: (a) conical flame; (b, c) envelope flame; (d, e) cool central core flame; (f, g) disc flame; (h) asymmetric side-lifted flame; (i) asymmetric disc flame; (j) ring flame.
grouped into nonlifted and lifted flames, as shown in Fig. 2. The designation is the same as that of Zhang and Bray [3] with added asymmetric flame modes. The flames are a conical free jet flame; an envelope flame, in which the upper zone of the flame is deformed due to impingement of the reaction zone on the plate; a cool central core flame, where a central region of unburned mixture impinges directly onto the plate; a symmetrical or asymmetrical disc flame in which a circular flame is stabilized at the impingement plate; a ring flame in which there is cool central stagnation region surrounded by an annular flame; and a side-lifted flame, similar in shape to a disc flame but tilted and anchored on the burner rim.
677
To access the stabilization modes of impinging flames (Fig. 2), the mapping of the different combustion modes and extinction limits is presented in Fig. 3. The maps in Fig. 3a were obtained in the following way: for each H /D and constant φ the procedure was to rim-stabilize the flame and then to map the various combustion modes as the flow rate changed, starting with the lowest possible Re. The stability diagrams of Fig. 3b were obtained by taking full advantage of hysteretic phenomena by first stabilizing a cool central core flame (rich condition with 2000 Re 4000) and then changing Re and φ as indicated by arrows in Fig. 3b. Following this procedure, the extended combustion regimes are explored until specific blowoff limits or regimes previously encountered in Fig. 3a for the same H /D are reached. Similar procedures and results were used by other authors, such as those published for impinging turbulent flames [1,3,4] and for laminar impinging flames at a specific equivalence ratio [5,6]. The influence of initial conditions and the way they change were shown to be key factors in obtaining different types of flames. According to the results of the stability diagram in Fig. 3, a conical flame exists when the flame height, conditioned by Φ and Re, is less than the nozzleto-plate distance. In this case, combustion products impinge upon the plate. Otherwise, the upper zone of the flame is deformed by impingement onto the plate, giving rise to an envelope or cool central core flame. The latter flame, originally described as a premixed open flame [2], is typically favored by high Re and high equivalence ratios, while the envelope flame depends on the stretch applied close to the stagnation point, quantified by the H /U ratio as described by Foat et al. [4]. When the dynamic equilibrium at the burner rim between laminar flame speed and local velocity is lost for low Re and lean conditions, a side-lifted flame asymmetric in shape is obtained (Fig. 3c4), while for higher Re numbers the flame either blows off (H /D 4) or stabilizes at the plate (H /D < 4) in a disc (Fig. 3c5) or ring (Fig. 3c6) mode configuration, depending on flame mode before blowoff. The side-lifted flame has a small portion of its outer disc edges attached to the burner rim, with an angular tilt depending on the Re and the transition between disc and ring mode flames depending on H /U . The flashback boundary had no noticeable variation over the distances swept, regardless of the flame stabilization mode. 3.2. Analysis of impinging resonant flames The naturally induced self-sustained oscillations do not generate any new type of flame structure; in-
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Fig. 3. Mapping of the different combustion modes, sound-producing regions (gray areas), and extinction limits as a function of Re and Φ for different nozzle-to-plate separation distances H /D. (a) Maps obtained by increasing flow velocity (Re) at a constant equivalence ratio (Φ). (b) Maps obtained by stabilizing an initial cool central core flame, and changing Re and Φ, where white arrows indicate start of exploration (hysteretic regions are followed by blow off regions or combustion regimes of (a)). (c) Diagram of combustion modes: (c1) conical flame, (c2) envelope flame, (c3) cool central core flame, (c4) side-lifted flame, (c5) disc flame, and (c6) ring flame.
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679
(a)
(a)
(b)
(b)
Fig. 4. Evolution of the resonant frequency and respective sound pressure level, SPL, as a function of nozzle-to-plate distance H /D, for (a) disc flame (Re = 2765, Φ = 0.85); (b) envelope flame (Re = 3230, Φ = 1.15) and cool central core flame (Re = 3093, Φ = 1.2).
Fig. 5. Evolution of the resonant frequency and respective sound pressure level, SPL, as a function of jet velocity (U ), with a fixed nozzle-to-plate distance H /D = 2 for (a) disc flame (Φ = 0.85); (b) envelope flame (Φ = 1.15) and cool central core flame (Φ = 1.2).
stead, they symmetrically or asymmetrically modulate the shape of the steady state flame, as shown by typical images in Fig. 2. Experiments conducted with unsteady flames show, in general, that as H /D is reduced the dominant oscillation frequency rises, in all type of flames, to a limit value of about 500 Hz, while the sound pressure level (SPL) emitted by these flames is in the range of 70–90 dB (Fig. 4). In addition, resonant frequencies and SPL increase monotonically with flow velocity for all types of flames (e.g., Fig. 5), but no significant influence was established by changing the equivalence ratio (see Fig. 6). The observed jumps in frequency and SPL evolutions are found associated with vortex pairing in disc flames and changes in the number of visual waves in the case of envelope and cool central core flames. The limit frequency of 500 Hz is close to the natural resonant frequency of the burner, modeled as a Helmholtz resonator with a resonant frequency of 484 Hz given c by f0 = 2π
S , where c is the sound velocity, S
LN V
the burner mouth area, LN the burner neck length with acoustic end correction [7], and V the burner volume (excluding neck).
The trend of the dominant peak frequency of the sound emitted by the unsteady flames, as a function of nozzle-to-plate distance, is similar to the results of Wagner [8], Lau et al. [9], Ho and Nosseir [10], and Tam and Ahuja [11] for subsonic isothermal impinging jets (Mach numbers = 0.65–0.95), to the subsonic and sonic impinging jet results of Neuwerth [12] and Ho and Nosseir [10], and to the cool central core oscillating flames of Durox et al. [6]. This substantiates the belief that in both reactive and isothermal impinging jets, oscillations are sustained by the same basic mechanism, i.e., there is a feedback mechanism related to nozzle-to-plate distance and jet velocity. 3.3. Burner acoustic characteristics The burner acoustics is modeled, due to its geometry, as a Helmholtz resonator with the equation (Kinsler et al. [13]) ∂ε ∂ 2ε + sε = −SP0 , +R (1) ∂t ∂t 2 where ε is the axial displacement of air mass m at the burner mouth, P0 is the external pressure exerted at the burner exit over its area S, and R and s, respec-
m
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(a)
Fig. 7. Comparison between analytical and experimental burner acoustic response for different flow velocities under isothermal conditions.
(b) Fig. 6. Evolution of the resonant frequency and respective sound pressure level, SPL, as a function of the equivalence ratio Φ, with fixed Re = 2800 and H /D = 2 for (a) disc flames and (b) envelope flames.
tively, are the total resistance and volume stiffness, given according to R = Ra + Rporous + Rr + RU = (a + 1)Ra + Rr + RU , γ −1 2m μω 0.5 1+ √ Ra = , r 2ρ Pr ρ
(1b)
2
(ωS) , Rr = c 4π RU = ρU S, ρC 2 S 2 . s= V
(1a)
(1c) (1d) (1e)
In this model there are two additional acoustic resistance terms: Rporous and RU . The term RU quantifies the interactions between mean flow and acoustics in the burner neck [7]. The term Rporous is due to a porous plate in the burner’s base (see Fig. 1) modeled as a collection of capillary tubes with an overall acoustic resistance similar to resistance Ra [14]. Hence, Rporous = aRa , where the constant a is obtained by fitting the burner response (P ) to an external excitation (P0 ). The analytical burner response is 2 SP0 is the |P | = ρc VS|ε| , where ε = j ω(R+j (ωm−s/ω))
solution of Eq. (1). The burner volume stiffness s, together with the air mass m at the burner neck, defines a more compact expression for the previously mentioned Helmholtz resonant frequency now given by ω0 = 2πf0 = (s/m)0.5 . The experimental data are obtained by exciting the burner with a loudspeaker placed near the burner mouth (at a distance of 300 mm), i.e., generating P0 . The ratio between pressure fluctuations measured inside the burner (P ) to those near the mouth (P0 ) is plotted in Fig. 7 for a range of frequencies from 50 to 1500 Hz. Two sets of results are plotted, for 2.5 and 5 m/s, corresponding to typical working conditions. The burner’s response to an external excitation shows one distinct peak at 477.8 Hz, lower than the natural undamped Helmholtz resonator oscillation frequency (f0 = 484 Hz) due to the acoustic damping effect, which forces the pressure peak amplitude to occur to the left of f0 , the shift being larger for larger damping values [13]. On the other hand, the damping effect due to mean flow (e.g., Durox et al. [6]) does not significantly influence the transfer function (the resonance bandwidths are the same). In the same figure, the analytical acoustic transfer function is also plotted, for which the constant a was found equal to 9.46. Hence, the complete burner (cavity and porous discs) behaves as a Helmholtz resonator and for the peak frequency of 477.8 Hz, the damping coefficient, δ = R/2m, is equal to 296 s−1 , which defines a characteristic of this system. 3.4. Flame–burner acoustic coupling In what follows P0 , in Eq. (1), is now related to the flame behavior. Consider the scheme shown in Fig. 8a, in which a typical vortex or bulge trajectory is represented along a curved pathline imposed by the adverse pressure gradient exerted by the wall, traveling with an instantaneous velocity Ci from its
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τ function of a characteristic velocity and a characteristic distance as proposed by [6,15,16]. The coupling between the unsteady flame and the burner acoustic response can be modeled following the approach suggested by Durox et al. [6], who showed, for an axisymmetric cool central core type flame, that the external force in Eq. (1) can be related to the velocity at the burner mouth as P0 =
(a)
) nQ0 ∂u (t−τ . This coupling model will be tested ∂t with the other types of axisymmetric flames in this work. Substituting this equation in Eq. (1) and applying the time derivative yields ∂u ∂ 2 u ∂ 2 u + su = −SnQ0 2 , m 2 +R (2) ∂t ∂t ∂t t−τ
originally presented by [6] and valid for the cases where the same dominant frequency peak is present outside and inside the burner during resonance (see Fig. 8b), excluding the vortex pairing regimes in unsteady disc flames. A general solution of Eq. (2) is proposed as u˜ = u0 ej ωt , where the real part of this is the particular solution for Eq. (2) (e.g., [13]). Replacing u by u˜ and τ by τ˜ , the respective complex forms, in Eq. (2) and solving it in order to obtain τ˜ yields 2j tanh−1 τ˜ =
2δω+(1+N )j ω2 −j ω02 2δω+(1−N )j ω2 −j ω02
+ 2π K
. ω The real part of this equation is therefore the solution for the original Eq. (2) as
τ=
2δω+(1+N )j ω2 −j ω02 Re 2j tanh−1 + 2π K 2 2
K ∈ ℵ, (b)
2δω+(1−N )j ω −j ω0
ω
, (3)
Fig. 8. (a) Schematic diagram of the feedback mechanism for an impinging jet, where Ci is the flow perturbation velocity, U the mean exit flow velocity, H the nozzle-to-plate distance, and Ca the feedback acoustic wave velocity. The dashed line represents the path traveled by a given flow perturbation from the nozzle exit to the acoustic wave birth location. (b) Power spectral density (PSD) of pressure fluctuations measured inside (PMic1 ) vs outside (PMic2 ) for envelope and disc flames.
where N = SnQ0 /m. Coefficient N represents the flame–flow normalized interaction coefficient [6] and, in spite of its appearing in the first term of Eq. (3), further analysis dτ˜ = −i ; i.e., Eq. (3) does not depend showed that dN Nω explicitly on N . Therefore, in Eq. (3), the term
2δω + (1 + N )j ω2 − j ω02 Re 2j tanh−1 2δω + (1 − N )j ω2 − j ω02
origin (the shear layer nozzle exit) to a specific region (represented by an asterisk). According to Schuller et al. [5], sound is produced in this region at the upper tips of the flame front, those nearer the stagnation surface, when each of the upward propagating bulges is finally consumed. The generated feedback pressure wave then propagates upstream, reaching the nozzle exit zone and triggering a new perturbation. The total time taken by this process corresponds to a time delay
holds the burner acoustic characteristics and 2π K gives the periodic solution. For simplicity, N was set equal to 1 (one) in the numerical solution of Eq. (3). In Fig. 9 are plotted Eq. (3) and the burner-related term for three different damping factors δ: 0.1R/2m, R/2m (present case), and 10R/2m. The coefficient R is given by Eq. (1a). Coupling between flame and burner acoustics occurs around ω0 with a bandwidth governed by the damping factor δ (Fig. 9a). For high
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(a)
(b) Fig. 9. Analysis of the “f –τ ” mathematical model. (a) Behavior of the term containing the burner acoustic characteristics, as a function of the damping factor δ; (b) influence of the burner damping factor δ on “f –τ ” evolution.
values of δ the evolution of f –τ is smooth from a low- to a high-frequency band (Fig. 9b). For very small values of δ, the burner only amplifies a narrow bandwidth around f0 , and consequently the evolution of f –τ has an abrupt transition around f0 , which vanishes as K increases. The contribution of the burner acoustics term, shown in Fig. 9a, ranges between two asymptotes, 0 and −π , automatically fulfilling the phase criterion of Schuller et al. [15], which based on an acoustic energy balance, showed that ωτ belongs to [π, 2π ] modulo 2π as a necessary condition for the onset of oscillations. In the same study, Schuller et al. [15] have shown that the energy fed by the flame to the system (burner), proportional to the normalized interaction coefficient N , should be greater than the energy dissipated inside the system, as a necessary condition for obtaining a self-sustained oscillation. The role of N is therefore to dictate the occurrence or not of selfsustained oscillations, at the frequencies predicted by Eq. (3). To plot Eq. (3) against experimental data, τ is taken as a function of a characteristic velocity and a characteristic distance as mentioned before. In this
context, different models were proposed for the τ function. Among the first, we have the experiments of Neuwerth [12] with a high-speed subsonic impinging jet with τ = H /Ci , where H is the distance from the injector to the wall and Ci the coherent structures velocity. Recently, Schuller et al. [15] have shown that τ = L/U in a study conducted with mutual flame interactions driving self-sustained oscillations. In [15] L is a characteristic distance where two flame front elements pinch off suddenly and U is the nozzle flow velocity. Also, Durox et al. [16] suggested the same type of model in experiments to study self-sustained oscillations driven by vortex shedding from a burner lip. For this particular configuration L is the distance where the vortex, initially surrounded by the flame, extinguishes it and U is replaced by Ci , the coherent structure velocity. Generally, it was shown that the choice of a characteristic distance and velocity is critical for modeling success, and Durox et al. [6] proposed a nonlinear model for τ in experiments similar to that shown here. This model was successfully applied for small flame-to-wall distances, or equivalently for ωH /U 10, as estimated from data presented in [6], and, as pointed out by the authors [6], probably not applicable to larger distances, where the predicted frequencies were not consistent with experimental results. In the present work, since ωH /U > 10, and recalling that τ is the time taken by a perturbation with velocity U to travel a path L, we propose τ as a linear function of U and H , using the initial model proposed by Neuwerth [12] when studying impinging air jets or equivalently identifying in the model of a resonant, nonimpinging, reactive flame of Schuller et al. [15] or Durox et al. [16], L with H . The τ function is then given by H (4) , U where β must be defined for each flame type. Substituting Eq. (4) into Eq. (3) yields 2 2 −1 2δω+(1+N)j ω −j ω0 + 2π K H 1 Re 2j tanh 2δω+(1−N)j ω2 −j ω02 = . ω U β (5) Equation (5) establishes a relation between the burner acoustic characteristics, resonant frequency, jet velocity, and nozzle-to-plate distance. It was applied systematically to the three types of unsteady laminar flames, namely the envelope flame, cool central core flame, and disc flame, sweeping a set of self-sustained resonant conditions for impingement distance H , average exit flow velocity U , and frequency f . For each resonant situation, the parameter K is obtained from the high-speed film sequences by counting the wavelengths (number of waves) or number of τ =β
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(a)
(b) Fig. 10. Tracking of a vortex trajectory in resonant disc flame images by laser plane visualization (H /D = 1.2, Φ = 0.85, Re = 2880). (a) Sequence of a resonant disc flame oscillating with a frequency of f = 162 Hz, with K = 3. The images were acquired at a rate of 1 kHz. (a1) to (a6) are images at phases 0, π , 2π , 8π/3, 10π/3, 4π , where a characteristic vortex (marked with crosses) is followed from near the nozzle exit (a1) until its total consumption at the flame front (a6); (b) dependence of the vortex pathline on nozzle-to-plate distance (H /D).
vortices visible on the perturbation path (see, for example, Figs. 2 and 10). The frequency f corresponds to the dominant peak on the sound spectrum signal, which matches the flame front oscillation frequency taken from the high-speed films. This matching was already checked by Schuller et al. [5] and Durox et al. [6] for a flame impinging on a plate, by Schuller et al. [15] for mutual flame interactions, and by Durox et al. [16] for flame vortex interactions on a bluff body configuration. Analytical results are plotted against experimental data in Fig. 11. The results show a clear superimposition for constant beta factors of βDisc = 1.80, βCCC = 1.03, and βEnv = 1.00, obtained by data best fit. The most noticeable difference is that βDisc is very different from the other two. In the former, we have an isothermal jet ending in a disc flame, while in the latter the flame extends towards the nozzle. This approach confirms that the mathematical model and the approximation given by Eq. (5) are both valid. 3.5. Analysis of τ for each type of resonant flame In what follows, the physical meaning of β in Eq. (4) is discussed. Consider the time delay given
Fig. 11. Resonant frequency as a function of time delay H /U . Comparison between predicted frequency and experimental data with the proposed relation τ = βH /U .
by Eq. (4) as follows: 1 1 dl. dl + τ= Ci c pathline-1 pathline-2
This total time is divided into pathline-1 C1 dl i
1 pathline-2 c dl, which are the travel times of branches of the feedback cycle. The first term
(6)
and two cor-
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Fig. 12. The influence of Re and nozzle-to-plate distance H /D on the integrated path length of the downstream propagating instability. (a) Disc flame (Re = 2880, Φ = 0.85); (b) disc flame (H /D = 2, Φ = 0.85); (c) envelope (Re = 3230, Φ = 1.15) and cool central core (Re = 3093, Φ = 1.2) flames; (d) envelope (H /D = 2, Φ = 1.15) and cool central core (H /D = 2, Φ = 1.2) flames.
responds to the time taken by each flow perturbation to travel from the nozzle exit, where it is generated, through the intrinsically unstable thin shear layer [10] to the sound-producing region, while the second term corresponds to the time taken by an upstream propagating feedback acoustic wave to travel from this region to the nozzle exit, where it further excites the shear layer of the jet and closes the feedback loop. This last term is insignificant compared to the first one, because pathline-1 is longer than pathline-2 and the flow conditions are subsonic, i.e., c Ci . Returning to Eq. (6) and despite the expected changes in Ci , in modulus and direction along the curved path line, it is proposed that Ci = γ U . The remaining integral that
gives the curved pathline length is scaled to H , as pathline-1 dl = αH . Inserting these two simplifications into Eq. (6), yields β = γα , which is the ratio of two weight factors, one related to the length and the other to the perturbation travel speed. Hence, an improved understanding of the feedback mechanism is achieved by isolating the contribution of α and γ in the global parameter β. For each resonant flame, the pathline was obtained by analyzing a sequence of high-speed digital images and recording the spatial coordinates of a given perturbation as it moves from the nozzle to the acoustic
wave birth locus, as shown in Fig. 10a. The pathline obtained by this process is plotted in Fig. 10b for three example cases of a disc flame and shows that the ratio between vertical and horizontal path lengths depends strongly on the imposed distance H . In spite of this behavior the total length h is a linear function of the nozzle-to-plate distance H , with a slope αDisc = 1.02, as shown in Fig. 12a. A similar evolution is found for all cool central core and envelope flames, e.g., Fig. 12c, but with different slopes: αEnv = 0.94, αCCC = 1.12. The influence of Re on the above results was found to be comparatively small, as shown for example in Fig. 12b. Now the value of γ is obtained straightforwardly from the definition of β. As a result, we have γDisc = 0.57, γEnv = 0.94, and γCCC = 1.09. The α factor is on the order of unity, which means that the perturbation’s trajectory length, from the burner mouth to the flame tip where it produces sound, is on the order of the nozzle-to-plate distance H , independent of the flame type or deformation imposed by the adverse pressure gradient. The velocity γ weight is on the order of unity for the cool central core and envelope type of flames, which means that a perturbation (those waves modulating the flame front; see Fig. 2) travels with a speed close to the velocity U . For the disc flame, this factor is 0.57, corresponding
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Table 1 Diagram of the three oscillating impinging flames with the main variables of the process: Physical identification of the relation between time delay (τ ) and nozzle-to-plate distance (H ) and jet velocity (U )
α = h/H γ = Ci /U β = α/γ τ = βH /U
Disc flame
Env flame
CCC flame
1.02 0.57 1.80 ∼ H /0.56U
0.94 0.94 1.00 ∼ H /U
1.12 1.09 1.03 ∼ H /U
to the travel speed of coherent structures in free jets [9,10,17]. The results show the validity of the approach suggested by Eq. (4) with a specific constant of proportionality depending on the type of flame to be studied.
4. Conclusions The present experimental work studies the behavior of laminar flames impinging onto a controlled temperature plate, with special emphasis on oscillating flames. The experimental techniques used were microphones, photography, high-speed digital cinematography, and planar laser visualization. Six types of flame structures were found, namely conical, envelope, disc, cool central core, ring, and side-lifted flames, based on parametric variations of nozzle-to-plate distance, Re, and equivalence ratio. Of these, three types of flames—disc, cool central core, and envelope—were found to oscillate, with frequency and SPL clearly increasing with Re and the inverse of nozzle-to-plate distance. To summarize, and according to the scheme shown in Table 1, the disc unstable flame consists of an isothermal mixing jet impinging onto a flat flame. The jet has coherent shear layer structures that travel downstream, interacting with the flame surface. The unstable cool central core flame has a tulip shape, while the unstable envelope flame has a closed tulip shape, both attached to the burner, with velocity perturbations traveling from the burner mouth up to the flame tip, deforming the flame surface. For the analysis of these unstable flames, the mathematical model derived by [6] to study unsteady cool central core flames was adopted here, due to flame and burner setup similarities. The burner was therefore modeled as a Helmholtz resonator with the external force being the sound produced by the impinging flames. Analytical solutions derived in this work showed, for all types of flames, how the predicted oscillation
frequency is related to the generic time delay taken by these perturbations to travel from the burner mouth to the flame tip, the burner acoustic characteristics, and the integer number of perturbations (vortices or waves), but with no explicit dependency on the flame– flow normalized interaction factor N . What does depend on N is the actual onset of self-sustained resonant conditions, where for a given working regime with oscillation frequencies given through Eq. (3), the energy fed by the flame, proportional to N , must be greater than the energy dissipated inside the system, as shown by Schuller et al. [15]. The most relevant burner acoustic characteristic in this process is the damping factor (energy dissipation). A larger damping factor increases the frequency bandwidth that the burner can amplify and hence it reinforces the burner’s role in the frequency results. On the other hand, the experimental analysis conducted with oscillating flames reveals that the time taken by perturbations to travel from the burner mouth to the flame tip τ is linearly related to the ratio H /U , i.e., the ratio between nozzle-to-plate distance to jet velocity, regardless of flame type. This contrasts with [6], which obtained a nonlinear dependence for the time delay for the case of CCC flames in the range of ωH /U < 10. The proportionality constant, β, is represented in Table 1 and has almost the same value for the cool central core and envelope flames and a different value for the disc flame. The main implication of these results, based on physical analysis of β, is that the travel time for envelope and cool central core flames can be obtained directly from the H /U ratio, while for the disc flame this time is the ratio between distance H and the traveling speed of the coherent structures. Incorporating this information into the mathematical model, the analytical solution fits experimental data if the integer number K (from 2π K of Eq. (2)) is taken as the number of vortices and/or flame surface waves observed under resonant regimes.
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