Abstract This paper reports an experimental investigation of premixed propane andmethane-air flames propagating freely in tubes 1.5 m long and with diameters rangingfrom 26 to 141 mm. The thermo-acoustic instability was eliminated by means of anovel acoustic absorber placed at the closed end of the tube. We first remark that theflame can adopt different shapes either quasi-axisymmetric and normal to the meandirection of propagation, or inclined with a larger propagation speed because of theincrease in flame surface area. The minima of the propagation speeds, correspondingto non-tilted flame propagation, are then analyzed using analytical models for theself-turbulent flame propagation. The concept of a cut-off wavelength appears to berelevant to explain the different behaviors observed on the rich side of methane-airand propane-air flames.
Since the pioneering work of Mallard and Le Chatelier [1], who determined theflammability limits and the propagating velocity of various combustible gaseousmixtures, there have been numerous studies concerning the prediction of turbulentflames speeds and safety limits. The result is generally written as
UT/UL = 1 + (u′/UL)α, (1)
J. Quinard (B) · G. Searby · B. DenetCNRS and Aix-Marseille Université, Marseille, Francee-mail: [email protected]
J. Graña-OteroThermodynamics Departement, ETSI Aeronauticos, Universidad Politécnica de Madrid,Madrid, Spain
232 Flow Turbulence Combust (2012) 89:231–247
where UT is the turbulent flame speed, UL the laminar flame speed, u′ the r.m.s.turbulent velocity of the incoming flow and α an exponent whose value variesbetween 0.5 and 2 depending on the modeling assumptions [2–4]. The low turbulencelimit would thus be UT ≈ UL. However it is well-known that laminar flames prop-agating freely in quiescent mixtures propagate at higher velocities due to curvatureeffects induced by hydrodynamic and possibly thermo-diffusive instabilities [5–7].There have been already attempts to incorporate flame instabilities into such a for-mula [8, 9] and the very existence of such an universal formula has been questionedrecently on the basis of both experiments and numerical modeling [10, 11]. Assuminga semi-cylindrical or hemispherical shape for the cells on the flame in a 2–D or3–D configuration, the ratio UT/UL would tend to π/2 or 2 respectively. Thesevalues are similar to the results of recent analytical and numerical investigations forflame propagation in narrow tubes that suggest that the maximum velocity ratio isapproximately 1.3 for two-dimensional flames, and approximately 1.7 in the three-dimensional case.
These investigations were limited to the case of tubes with an inner diametersmaller than 4 λc, where λc is the cut-off wavelength for flame front instability [8, 12–16]. For tubes of larger diameter, secondary instabilities occur, possibly leading to aself-similar behavior of the topology of the flame. Using a fractal dimension obtainedfor expanding spherical flames, a corresponding power-law behavior for the flamepropagation speed has been proposed [15]:
UT/UL = (�m/λc)D (2)
where �m is the largest characteristic length of the flame and D ≈ 1/3 is an exponentresulting from the fractal dimension of the flame [8, 17, 18]. This relation seemsto work well with weakly turbulent premixed flames [4, 12], but to the best ofour knowledge, apart from the ancient experimental observations of Coward andHartwell [19] performed in large horizontal tubes where buoyancy effects are impor-tant, such an increase of the flame speed with the tube radius has never been seenexperimentally for flames propagating freely in a quiescent gas. One probable reasonis the occurrence of violent thermo-acoustic instabilities that completely change theshape and propagation velocity [20, 21] of flames propagating in tubes.
In this paper, our objective is to observe the dependence of the propagation speedof cellular flames in a quiescent medium on the characteristic dimensions of theburner. In the next section we will briefly describe the very “simple” experiment.The results obtained with propane- and methane-air flames will then be presentedand discussed in the last section with some comments on the way to determine thecharacteristic lengths �m and λc.
2 Experimental Setup and Procedure
The propagation velocity of laminar cellular flames is measured in verticalPyrex tubes, 1.5 m long, with internal diameters ranging from 26 to 141 mm. Theequivalence ratio, φ, of the premixed gas is controlled via a PC-interface connectedto mass-flow regulators. The flame propagation is recorded using a video camera.The particularity of this experiment is related to the bottom part of the burner wherean acoustic damper is installed to prevent the onset of thermo-acoustic instabilities.
Flow Turbulence Combust (2012) 89:231–247 233
Fig. 1 Schematic of the experimental arrangement. The dimensions D0, D1, l, h are explained inthe text
The damper consists of a small annular slit, of height h and length l that dissipatesacoustic energy by terminating the tube with a real (resistive) acoustic impedanceequal to the characteristic acoustic impedance of free air. The details of the damperare given in the Appendix. An expansion chamber is placed outside of the slit toavoid mixing of ambient air with the combustible mixture (Fig. 1).
The operating procedure is as follows: After each run, the air flow is opened andmaintained until the tube walls have cooled to ambient temperature. The flow ofcombustible, methane or propane, is then adjusted to the desired equivalence ratioand a lightweight plate is placed over the open end of the burner. The presence of theplate is sufficient to prevent mixing with ambient air, but does not prevent exhaustof the premixed gas. The flow is maintained for a time corresponding to at least tenfillings of the tube and then stopped by closing the valve at the bottom of the burner.A delay of at least one minute is allowed before gently withdrawing the upper plateand igniting the mixture with a lighter.
The video-movie is then digitized and post-processed using ImageJ 1.40 softwareto obtain the trajectory of the upstream tip of the flame.
3 Results
The first experimental result is that a given flame does not propagate systematicallyat a fixed speed, even if the measured velocity can be constant during all or partof a given experiment (see Fig. 2, left line). Another experiment made in thesame conditions with the same equivalence ratio, φ = 0.9, shows the flame initiallypropagating at a slightly smaller speed and then decelerating suddenly (at t ≈ 0.8 s)to reach a still smaller speed of propagation (see Fig. 2, right line). We find that thereis almost a continuum of flame speeds related to different flame shapes, see Fig. 3.
234 Flow Turbulence Combust (2012) 89:231–247
Fig. 2 Record of twopropane-air flame trajectoriesin the same tube, diameter90 mm, equivalence ratio 0.9.The dotted lines are parallel tothe experimental points of thetrajectory on the right. Theyhighlight the change in flamespeed
It has been found theoretically from the Sivashinsky equation [22] that multiplestationary solutions exist for the problem of a flame propagating in a (2–D) tube,including the two typical flame shapes similar to those we observe in our experiments:a flame inclined relative to its direction of propagation (Fig. 3, bottom), or a slightlyasymmetric flame (Fig. 3, top). These two different type of solutions have alsobeen found in direct 2–D numerical simulations or potential models [8, 12, 13, 23],with a velocity increase ranging from 1.3 UL to 1.58 UL. In our experiments wefind larger flame velocities, in the range from 1.5 UL to 3 UL see Fig. 4, but it isgenerally recognized [24] that the velocity increase relative to the laminar velocityis higher in 3–D than in 2–D. Another important result is that the inclined flamevelocity is larger than the slightly asymmetric one, a property which is not foundin the Sivashinsky equation [22], where these two solutions have the same velocity,but which is found in a potential model [23]. The increased velocity of inclinedflames has been also demonstrated in direct numerical simulations of flame dynamics[12]. Interesting discussions on the flame velocity in the Sivashinsky equation forwide tubes, compared to direct numerical simulations and results from the Frankelequation, can be found in [25] and [26]
It is generally believed that the fastest solution should dominate at long times,however this is not clear from our experiments since both slow to fast and fast to slowtransitions were observed, with either a quasi-axisymmetric flame evolving into aninclined flame, or an inlined flame evolving into a flame normal to the mean directionof propagation. The fact that both slow to fast, and fast to slow transitions can beobserved for flames propagating in the same mixture and in the same tube indicatesthat: a) the ignition protocol is not the (only) factor controlling the onset of slantedpropagation, and b) this transition is very sensitive to very small perturbations.
The results in Fig. 4 present the maxima (U+T ) and minima (U−
T ) of the self-turbulent flame speed in tubes of different diameters. They were obtained from 2to 4 runs per configuration, retaining only events where the flame was propagatinguniformly during typically 30% of the tube length, and then adjusting two curves to
Flow Turbulence Combust (2012) 89:231–247 235
Fig. 3 Different shapes ofpropane-air flames during freepropagation in tubes. Left:φ = 1.2, Ø tube = 40 mm;right: φ = 0.9, Ø tube =140 mm (not the same scale)
interpolate between the extrema of the measured values. In some cases the flamewas always inclined, whereas in some other cases, it was only weakly tilted. More ex-periments are thus needed, and probably with a more sophisticated and reproducibleigniter, to reproduce the whole range of possible flame velocities. Nevertheless therange of flame speed spanned by this set of experiments is wide enough to distinguishclearly the two limits, the scatter of the experimental measurements on the curve fitsbeing less than 10%.
The reference speed UL is taken from measurements of Bosshaart and De Goey[27] which were found to be very close to our own measurements on some planarflames in tubes. The general trends seem to be the same for both propane- andmethane-air flames: the increase in flame speed near to the extinction limits is onlysmall: for slow flames the stabilizing effect of gravity is important and forces theflame to be almost flat. For faster flames, in the midrange of equivalence ratios,both the maximum and the minimum flame speeds increase with tube diameter withno indication of any saturation effect in the largest tube investigated here (141 mmdiameter).
236 Flow Turbulence Combust (2012) 89:231–247
Fig. 4 Minimum (U−T : dotted lines with open symbols) and maximum (U+
T : solid symbols) methane-air (left) and propane-air (right) self-turbulent flame speed in tubes of different diameter. UL is thelaminar flame speed. The numerical values in the legend give the tube diameter in mm
4 Analysis
4.1 Reduced results
Retaining only the minima of the measured flame speeds, U−T , corresponding to non-
tilted flame propagation, the results are presented in Fig. 5 as the normalized velocityratio U−
T /UL. For methane-air flames, there is no dramatic change of the flame speedwhen the tube diameter is increased from 26 to 74 mm. The only significant increasein normalized flame speed occurs for methane flames in the 141 mm tube. It willbe seen later that the largest tube is the only one for which the cut-off wavelengthof methane flames is very much smaller than the tube diameter. However it is also
Fig. 5 Normalized velocity ratio of methane-air (left) and propane-air (right) flame speed in tubesof different diameter. The numerical values in the legend give the tube diameter in mm
Flow Turbulence Combust (2012) 89:231–247 237
difficult to exclude a bias due to the fact that it is very difficult to obtain a non-tiltedfast methane flame in the smaller tubes, which have a larger Froude number for thesame flame speed. Nevertheless all these curves present a maximum on the lean side,the maximum velocity being 2.2 UL in the 141 mm tube near φ ≈ 0.8. This value isalso close to the normalized burning velocity that can be extrapolated to u′/UL = 0from the experimental data in reference [4].
The normalized velocity ratio drops to a value even less than unity for the richestflames, φ ≥ 1.4, showing that heat loss effects are significantly strong for these veryslow flames (see also Fig. 4).
For propane-air flames, the increase in flame speed with the tube diameter is muchmore regular. There is a maximum on the lean side near to φ ≈ 0.8, but there is alsoa second maximum on the rich side and the maximum velocity ratio in the largesttube reaches the value ≈ 2.5 over a large range of equivalence ratios from φ = 0.8 toφ = 1.4. The curves are truncated at φ = 1.5 because the value of the laminar flamespeed, UL is not given in ref. [27] for equivalence ratios beyond 1.5. However it isclear from Fig. 4 that the limiting value of the velocity ratio will be close to unity.
The reason for the difference in behaviour of rich methane and propane flamesis clearly seen in Fig. 6. Rich methane-air flame have large relatively smooth cells,whereas the rich propane-air flame have numerous secondary cells superimposed onthe larger cells, leading to a greater increase in flame surface area. A usual propertyof the hydrodynamic instability is that small cells are convected towards the cusps ofthe larger cells. They finally merge with the cusp, thus increasing the amplitude ofthe larger cell. However for large tubes (compared to λc) the front is very sensitiveto any form of noise in the system (here residual turbulence), and new small cellsare continuously created on the front. A WKB argument [28] suggests that the orderof magnitude of these small cells is the most amplified wavelength (approximately2λc), so that a smaller λc leads to the creation of smaller new cells. Furthermore,if simulations of the Sivashinsky equation are to be believed [22], for wide tubes(versus λc), very complicated stationary solutions exist, which could be close to therich propane-air flame shown in Fig. 6.
The increase in flame speed in wide tubes is thus related to the cellular instabilityas already proposed by several authors [8, 12–16]. Akkerman and Bychkov [15] usedmeasurements of the Markstein lengths taken from the literature [29] to calculate the
Fig. 6 Rich methane-air (left:φ = 1.3) and propane-airflame (right: φ = 1.4) in a tube140 mm inner diameter
238 Flow Turbulence Combust (2012) 89:231–247
cut-off wavelengths and to determine flame propagation speeds resulting from theeffect of the Darrieus–Landau instability. This set of Markstein numbers for propaneand methane-air flames was chosen because it was obtained at the stability limitof planar premixed flames propagating downwards, in accordance with theoreticalsimplified models used in their numerical approach. However further work [30]demonstrated that this way of determining the Markstein number is valid only whenthe Lewis number is very close to one. Moreover this work also showed that goodagreement between different experimental measurements of Markstein numberscan be obtained only by correct extrapolation of the flow velocity to the reactionzone. This work also demonstrated the need to apply a correction factor when thegas velocity is extrapolated from the burnt gas side, as for instance, when usingmeasurements on spherically expanding flames.
The value of the Markstein number for different mixture is still a subject ofcontroversy and it is thus interesting to compare relation (2) using different waysto calculate the characteristic lengths �m and λc.
4.2 Characteristic lengths of unstable flames
It is reasonable to suppose that the characteristic lengths �m and λc are related tothe minimum and maximum unstable wavelengths of the reactive mixture, as givenby the linear theory. These wavelengths are obtained from the roots of the dispersionrelation, which in turn depends on the Markstein number, Ma, of the mixture. In thefollowing we will compare three different ways to obtain Markstein number. Thesethree estimation of Ma will be denoted by Ma I, Ma II and Ma III respectively.
According to analytical calculations of the stability of premixed planar flamesincluding expansion effects, gravity and preferential diffusion [2, 31], the rate ofgrowth, σ , of small perturbations with a wavenumber k = 2π/� is given by
σ = kUL�, (3)
with
� = EE − 1
{[E2 + E − 1
E+ Ma dLk(Ma dLk − 2E) − E2 − 1
E2
g
kU2L
]1/2
− Ma dLk − 1
},
(4)
where E is the expansion ratio ρu/ρb , Ma is the Markstein number, dL the laminarflame thickness and g the acceleration of gravity. The subscripts u and b referrespectively to unburnt and burnt gases.
For downward propagating flames above the threshold of cellular instability,
UL >√
8Ma dLg/(E − 1),
there is a band of unstable wavelengths limited by two neutral wavenumbers:
k±n = E − 1
4EMa dL
(1 ±
√1 − 8Ma dLg
(E − 1)U2L
). (5)
Flow Turbulence Combust (2012) 89:231–247 239
At the threshold of stability for a planar flame, k+n = k−
n = 2π/�∗M where �∗
Mis the most unstable wavelength, and the Markstein number for these mixtures canbe determined experimentally in two ways:
– either by using the critical flame speed U∗L at the stability limit as was done in
[29]:
Ma∗I = (E∗ − 1)U∗2L
8gd∗L
, (6)
– or by using the critical wavelength measured at the stability limit
Ma∗II = (E∗ − 1)
8π E∗�∗
M
d∗L
(7)
In the above expressions, the superscript ∗ denotes values at the threshold of stability.The second relation has never been used because the published results are
relatively scarce [32, 33], but it can be expected to be more appropriate to the presentproblem since it is directly related to the cell size.
However, it is known that changes in the gas expansion ratio affect the value ofthe Markstein number. Since the stability limits of planar flames were measuredusing diluted flames with low flame speeds and small expansion ratios, and ourmeasurements were performed for non-diluted flames over a large range of flamespeeds and expansion ratios, the differences in expansion ratio must be taken intoaccount in the evaluation of the characteristic lengths of the flame. Assuming a hard-sphere model for the gas mixture and a one-step irreversible Arrhenius reaction,Clavin and Garcia [34] obtained the following analytical expression for the Marksteinnumber:
Ma = 2E√E + 1
+ β(Le − 1)
[2√
E + 1− E
E − 1ln
(√E + 1
2
)](8)
where Le is the Lewis number of the limiting reactant and β is the Zeld’ovichnumber. This expression can be used to correct the Markstein number for flameswith the same equivalence ratio (and Lewis number), but different expansion ratios(dilutions). Knowing the Markstein number at the threshold of stability, the reducedLewis number β(Le − 1) can be determined from Eq. 8. The same equation can thenbe used to calculate the Markstein number for the mixture of interest using the ap-propriate expansion ratio E given by GASEQ [35]. The Markstein numbers obtainedfrom experimental measurements of the instability threshold, using relations (6) and(7), corrected using Eq. 8, will be called Ma I and Ma II respectively.
Finally, some values of Markstein numbers obtained by direct numerical simula-tions of stretched methane and propane flames with detailed chemical kinetics areavailable in the literature [30]. These values from numerical simulation will be calledMa III.
Tables 1 and 2 resume the parameters we have used to calculate λc = 2π/k+ and�m = 2π/k− in order to test Eq. 2. The flame thickness is taken equal to Dth/UL
with Dth = 0.2 cm2s−1.In these tables, U∗
L, �∗M and the expansion ratio at the threshold of instability, E∗,
are taken from [32, 33]. They are used to determine Ma∗ from Eqs. 6 and 7, and theeffective value of β(Le-1) in Eq. 8.
240 Flow Turbulence Combust (2012) 89:231–247
Table 1 Parameters used to calculate characteristic lengths of methane-air flames (units: cm, s)
Labels: ‘*’ = results from [33]; ‘I’ = Eq. 6; ‘II’ = Eq. 7; ‘III’ = results from [30]
It can be seen that the Markstein numbers calculated using Eqs. 6 and 7 are veryclose so, for the sake of clarity, only Ma II and Ma III (from ref [30]) will be usedin the following to calculate the characteristic velocities of the flames. The values ofλc = 2π/k+
n and �m = 2π/k−n are then calculated using Eq. 5. The resulting values
are plotted in Fig. 7.
Table 2 Parameters used to calculate characteristic lengths of propane-air flames (units: cm, s)
Labels: ‘*’ = results from [33]; ‘I’ = Eq. 6; ‘II’ = Eq. 7; ‘III’ = results from [30]
Flow Turbulence Combust (2012) 89:231–247 241
Fig. 7 Longest, �m, and shortest, λc, unstable wavelengths calculated for methane and propaneflames using Eq. 5 for the three different methods of evaluating the Markstein length
The longest unstable wavelength, �m, reaches a maximum value for equivalenceratios close to 1.1 and decreases towards the extinction limits. Its value is higherfor propane flames than for methane flames with the same equivalence ratio. Thisis a consequence, firstly of the higher laminar flame speed of propane-air flames,and secondly of preferential diffusion that is less stabilizing for the rich propane-airflames. In general, the long wavelength cut-off �m, is larger than the diameter of thetube, except for slow flames in large tubes.
For methane flames, the short wavelength cut-off, λc, has a minimum valuearound stoichiometry with an order of magnitude of 0.5 to 1 cm. The range ofunstable wavelengths is only a factor ≈ 2 for slow flames in small tubes, and ≈ 30 forstoichiometric flames in the widest tube. Model Ma III predicts cut-off wavelengthsthat are about 50% larger than Ma II, with a corresponding decrease in the range ofunstable wavelengths.
For propane flames, models Ma I and Ma II predict a short wavelength cut-off thatis quite large (≈ 1.5 cm) for very lean flames, decreasing rapidly to a very small orzero value for rich flames. This is consistent with the observation that rich propaneflames have many small cells on the flame front. It results from the decrease of theLewis number as the oxygen becomes the limiting reactant so that all flames withφ > 1.1 were intrinsically unstable even at the lowest flame speed attainable in theexperiments of [29, 32]. For this reason the further increase of the cut-off wavelengthis not evaluated from these measurements. Model Ma III predicts a slightly greatervalue for λc that remains positive for all equivalence ratios, with a minimum valueof ≈ 0.4 cm at an equivalence ratio of 1.4. According to Ma III the range of unstablewavelengths varies from ≈ 1.5 for lean propane flames in the smallest tube, to ≈ 35for rich propane flames in the largest tube.
We should remark that the values Ma III are obtained from numerical measure-ments of the speed of stretched planar flames in a divergent flow, whereas the valuesMa II were obtained from the stability limits in a uniform flow. There is numericaland experimental evidence that the Markstein numbers of curved and stretchedflames can be different [36–39].
242 Flow Turbulence Combust (2012) 89:231–247
Fig. 8 Normalised self-turbulent flame speed of methane- (left) and propane- (right) air flames. Thenumerical values in the legend give the tube diameter in mm. Symbols: experimental values. Thinlines: propagation speeds calculated using �m and λc from Ma II [32, 33]. Thick lines: �m and λcfrom Ma III [30]
The resulting normalised turbulent flame speeds are then calculated using Eq. 2with D = 1/3. When the diameter of the tube is smaller than the calculated value of�m then we have used the former as the upper limit for �m. The results are shown inFig. 8.
The agreement between experimental results and calculated values is not perfect,but the general trends are relatively well reproduced. All curves decrease towardsunity near the extinction limits, reflecting the reduced range of unstable wavelengths.
The self-similar flame velocity calculated using Ma II systematically overestimatesthe experimental values, particularly for methane flames. However, it successfullypredicts a velocity maximum on the lean side of methane-air flames. For propaneflames, the curves are truncated beyond φ ≈ 1 because all richer propane-air flameswere systematically unstable and thus it is not possible to obtain a Markstein numberor a cut-off wavelength from the threshold of stability.
The values of Ma III [30] are higher than Ma II, leading to a larger cut-offwavelength. The velocity maximum on the lean side is less pronounced, but thecalculated values of self-similar flame speed are closer to experimental results. Infact, since the largest length scale is generally the diameter of the burner, thesecurves reproduce the inverse tendency of the cut-off wavelengths of both fuels, seeFig. 9. Such measurements of intrinsically unstable flames speed in tubes of differentdiameters could thus be used to determine the characteristic cut-off length-scale ofcombustible mixtures in a simpler way than by measuring the onset of instability onplanar flames or the Markstein number on expanding spherical flames. However, thecalculated increase of the flame speed with the tube diameter is still larger than thatobserved experimentally, particularly for methane-air flames.
We have used results of the linear theory of flame stability to calculate the largestpossible flame scale and we generally obtain values much larger than the tubediameter. The largest cells should thus have the dimension of the tube diameter.This prediction is not confirmed by visual observation of the flames, particularly forthe case of rich propane-air flames (see Fig. 6) where the maximum cell size seems
Flow Turbulence Combust (2012) 89:231–247 243
Fig. 9 Cut-off wavelengths forpropane- and methane-airflames
to be approximately 1/3 of the tube diameter. This observation implies that relation(2) overestimates the self turbulent flame speed.
A better caracterisation of the flame geometry is probably needed to determinecharacteristic length scales. Nevertheless, there is sufficient agreements betweentheory and experiment to support the influence of cut-off wavelengths on the speedof flame propagation in wide tubes.
5 Conclusion
The propagation velocities of self-turbulent premixed flames propagating in quies-cent mixtures were measured in tubes having diameters ranging from 26 to 141 mmin order to test the assumption of a self-similar behaviour in wide tubes.
An unexpected and striking result is that the free flame can propagate withdifferent velocities in a given configuration, depending on the angle of tilt of thefront with respect to the burner axis. Propagation velocities as high as 3.5 UL weremeasured. This could be a relevant result for determining the limits of flash-back.
Supposing that the minimum value of the measured flame velocity is close to thefundamental self-turbulent flame speed, the experimental results were comparedto a simple model equation for the propagation speed of cellular flames using twodifferent sets of Markstein numbers. There is a reasonable qualitative agreement,and the calculated cut-off wavelengths explain the differences observed betweenrich methane- and propane-air flames. However, the predicted values of propagationvelocity are generally significantly larger than the measured values. This differenceis probably related to an overestimation of the largest characteristic lengths of theflame as evidenced by the topology of rich propane-air flames (Fig. 6) and, we mayalso question the validity of determining cut-off wavelengths from a linear model ofstability of planar flames. Moreover, the fractal exponent D ≈ 1/3 was determinedfrom experiments on large freely expanding spherical laminar flames. There is no
244 Flow Turbulence Combust (2012) 89:231–247
solid argument to justify that flames in tubes will have exactly the same fractalexponent than freely expanding spherical flames, so the agreement is surprisinglygood.
Appendix: The Acoustic Damper
The purpose of the acoustic damper is to eliminate thermo-acoustic instabilities byabsorbing acoustic perturbations arriving at the base of the tube. This is done byintroducing a viscous loss (real acoustic impedance) at the base of the tube equalto the acoustic impedance of the gas in the tube. When this condition is fulfilled,propagating acoustic perturbations are dissipated at the base and not reflected. Theprinciple is a transposition of that used in electronic transmission lines (e.g. 50 and 75ohm cables) where reflection of the signal is eliminated by terminating the line witha real resistance whose value is equal to the impedance of the cable.
The characteristic impedance of the gas in the tube, Z = p′/u′, is equal to iρc,where p′, u′ are respectively the acoustic pressure and displacement velocity, ρ isthe density of the gas, and c is the speed of sound. For the mixtures used here,this impedance is very close to that of free air and has a value approximately equalto 410 Pa.s/m. The pressure p′ and the velocity u′ are in phase quadrature, so theimpedance of free air is imaginary and there is negligible energy dissipation.
A thin annular slit of height h and length l is introduced at the base of the tube,see Fig. 1. An acoustic pressure perturbation p′ introduces flow through the slit withan unsteady velocity us. The amplitude and phase of this flow is determined by theviscous resistance to flow in the slit (in-phase or real component) and by the inertiaof the gas in the slit (phase quadrature, or imaginary component)
The acoustically induced flow velocity in the annular slit is higher than theacoustic displacement velocity at the base of the tube. If the tube diameter is smallcompared to the acoustic wavelength, mass conservation imposes that the meanacoustic velocity u′ at the base of the tube (diameter Ø = D0) and the mean flowvelocity us in the annular slit (Ø = D1) are related by
us(t)π D1h = u′(t)π D2
0
4(9)
Assuming a Poiseuille flow in the slit, the mean velocity us is just 2/3 the maximumvelocity, us = 2/3umax, and the viscous contribution to the instantaneous pressuredrop across the slit, p′(t) = p cos(ωt), is then given by [40]
p cos(ωt) = 12μlh2 us cos(ωt), (10)
where μ is the shear viscosity and p′/ l is the pressure gradient across the slit. Sincethe flow is unsteady, there is also a contribution arising from the inertia of the fluid inthe slit. The mass of fluid in the slit is m = ρπ D1hl and its instantaneous accelerationis −ωus sin(ωt). Equating the total force on the gas in the annular slit to the unsteadyforce required to overcome the viscous and inertial resistance of the flow, and in theapproximation that the inertial contribution is small, we obtain
p cos(ωt) = 12μlh2 us cos(ωt) − ρlωus sin(ωt), (11)
Flow Turbulence Combust (2012) 89:231–247 245
where u′ can be substituted for us from Eq. 9 to give the acoustic impedance of theslit:
Z = p′
u′ = D20l
hD1
(3μ
h2 − iωρ
4
)(12)
Equating the real part of this impedance, Re[Z ], to the imaginary part of theimpedance of air, ρc, leads to the relation:
h =(
3μD20l
ρcD1
)1/3
. (13)
For the acoustic damper to be effective, the imaginary part of the impedance of theslit (inertial contribution) must be negligible compared to the real part, i.e.
|Im[Z ]||Re[Z ]| � 1, (14)
which implies
h �(
12μ
ρω
)1/2
, (15)
and from Eq. 13 this also imposes
l � 8cD1
D20
√3μ
ρω3 . (16)
Relation (16) imposes that the length of slit be relatively short, l ≈ 1 mm, andaccording to Eq. 13, the height of slit has to be tuned to a value that is a functionof both the slit length l and the tube diameter, D0. Typically h ≈ 0.2 mm. Despitethe short dimensions of the slit, the Poiseuille approximation appears to be sufficientand this device has proved to be very efficient in suppressing the thermo-acousticinstability otherwise encountered with premixed flames propagating in tubes.
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