J. Blimbaum, M. Zanchetta, T. Akin, V. Acharya, J. O ... · International journal of spray and...

reprinted from

International journal of spray and combustion dynamics

Volume 4 • Number 4 • 2012

Transverse to longitudinal acoustic coupling processes in annular combustion chambers

by

J. Blimbaum, M. Zanchetta, T. Akin, V. Acharya, J. O’Connor, D. R. Noble and T. Lieuwen

Multi-Science PublishingISSN 1756-8277

Transverse to longitudinal acousticcoupling processes in annular

combustion chambersJ. Blimbaum1, M. Zanchetta2, T. Akin2, V. Acharya3, J. O’Connor4, D. R. Noble3

and T. Lieuwen1,3

1Department of Mechanical Engineering, Georgia Institute of Technology, Atlanta GA 30318, USA2Institut Supérieur de l’Aéronautique et de l’Espace - Ecole Nationale Supérieure de Mécanique

et d’Aérotechnique, Cedex, France 3Department of Aerospace Engineering, Georgia Institute of Technology, Atlanta GA 30318, USA

4Engine Combustion Department, Sandia National Laboratories, Livermore, CA 94566, USA

Submission May 17, 2012; Revised Submission June 26, 2012; Acceptance July 09, 2012

ABSTRACTCombustion instability is a major issue facing lean, premixed combustion approaches in moderngas turbine applications. This paper specifically focuses on instabilities that excite transverseacoustic modes of the combustion chamber. Recent simulation and experimental studies haveshown that much of the flame response during transverse instabilities is due to the longitudinalfluid motions induced by the fluctuating pressure field above a nozzle. In this study, we analyzethe multi-dimensional acoustic field excited by transverse acoustic disturbances interacting withan annular side branch, emulating a fuel/air mixing nozzle. Key findings of this work show thatthe resultant velocity fields are critically dependent upon the structure of the transverse acousticfield and the nozzle impedance. Significantly, we also show that certain cases can be understoodfrom relatively simple quasi one-dimensional considerations, but that other cases are intrinsicallythree-dimensional.

1. INTRODUCTIONEmissions regulations, reliability, and fuel costs are significant factors that drivecombustion technology. Combustion instabilities have arisen as one of the most criticalproblems facing development of robust, low emissions combustors for both powergeneration and aviation applications [1]. These instabilities can arise due to interactionsbetween heat release and local flow perturbations. When these heat release oscillations arein-phase with the acoustic pressure oscillations, the flame adds energy to the acoustic field[2]. The instantaneous heat release rate of the flame is sensitive to several quantities, such

International journal of spray and combustion dynamics · Volume .4 · Number . 4 .2012 – pages 275 – 298 275

as velocity and fuel/air ratio, and much work has been done to understand thesemechanisms in combustion systems [3–8]. This study focuses specifically on transverseoscillations in combustion chambers, which have been historically problematic in rockets[9–12] and jet engine afterburners [13–15]. In addition, they appear in gas turbines in bothannular and can combustion chambers [16–22].

In order to motivate the approach taken in this study, it is useful to consider in moredetail a typical arrangement of annular combustor systems and transverse instabilities inthese systems. A simplified schematic of such a system is shown in Figure 1. It illustratesan annular ring around which regularly spaced nozzles are placed. Air from thecompressor exits these nozzles and flows in the axial flow direction. These nozzles havetheir own acoustic characteristics, associated with distributed inertia/compressibility inthe flow passage, and the inertia and resistivity of the swirlers [18]. For systems withsufficient wave transmission through the swirler, the upstream acoustic characteristicsalso come into play, such as the compressibility of the compressor discharge plenum.

Having discussed the geometry, next consider the transverse modes in these systems.These modes consist of standing and traveling waves in the azimuthal direction, andstanding waves in the radial direction. In many cases, nozzles could be situated at any pointin the standing wave field, because nozzles are distributed around the combustor. Forexample, a nozzle situated near the velocity node and another near the pressure nodeexperience significantly different disturbance fields, and possibly different flame excitationphysics. Moreover, these disturbance wavelengths could be quite long relative to the nozzledimension in the case of annular modes, or on the same order of the wavelength for radialmodes. In many cases, the oscillations in the azimuthal direction are closely approximatedby traveling waves [18]. In this case, the flame response would vary significantly in time asthe wave spins around the combustion chamber.

276 Transverse to longitudinal acoustic coupling processes in annular combustion chambers

Nozzles

Transverseinstability

Figure 1: Schematic of an annular combustor.

Recent work has shown that several paths exist through which a transverse mode mayexcite a flame [23–26]. As shown in Figure 2, transverse modes can directly excite theflame, they can excite hydrodynamic flow instabilities, and they can also lead to axialacoustic flow oscillations in the nozzle. Measurements and simulations have suggestedthat these axial oscillations are the dominant source of flame excitation during atransverse instability [23, 27, 28]. These axial acoustic oscillations are a wavediffraction effect, as the dominantly transverse mode leads to an oscillatory pressurefield across the nozzle. This oscillatory pressure field induces axial flow oscillations,referred to as “injector coupling” in the rocket literature [29, 30].

Experiments and computations of transversely excited flames have clearlydemonstrated the significance of this transverse to axial flow coupling process. Forinstance, Staffelbach et al. [27] presented LES results of an instability in an annularcombustor, clearly showing the presence of strong axial pulsations in nozzle flowaccompanying these transverse oscillations. This study suggested that it was theseaxial flow pulsations that dominated the flame response. In other words, thetransverse flow oscillations serve as the “clock” which controls the natural frequencyof the wave motions and the structure of the wave field, but it is the induced axialfluctuations which actually excite the heat release oscillations that, in turn, excite thetransverse modes.

Experimental observations of this same point were also made by O’Connor andLieuwen [23], who presented a number of non-reacting and reacting results oftransversely excited flames, using high speed planar velocimetry measurements andline of sight chemiluminescence. A set of results is presented in Figure 3 below,clearly showing the strong transverse motions away from the nozzle section, but alsothe significant axial flow oscillations in the nozzle region. These studies alsoemphasized the importance of the transverse mode structure relative to the nozzle incontrolling the flow field. For example, a nozzle located at a pressure nodeexperiences significant transverse flow oscillations, and a relatively weak unsteadypressure field. However, the phase of the pressure disturbances differs by 180 degreeson the left and right sides of the nozzle, implying that the excited axial flow

International journal of spray and combustion dynamics · Volume .4 · Number . 4 . 2012 277

11

Longitudinal (axial)acoustics

Flowinstabilities

Flame response (Q. ′)

23

6

z

x

p′

u′z

54

Transverse acoustic excitation (p′)

Figure 2: Pathways for velocity coupled combustion instabilities.

oscillations are highly non-symmetric. Different forcing conditions can lead tovarying flow response in the region of the nozzle. For example, the fluctuatingcomponent of the flow shown in Figure 3a is the result of forcing at a pressure nodeand velocity anti-node. The asymmetric velocity forcing results in a strong bias in the


2

1.5

1

0.5

x/D

0

2

1.5

1

0.5

x/D

0

Figure 3: Experimental results of PIV measurements of coherent velocityfluctuation for 400 Hz out-of-phase (left) and 400 Hz in-phase (right)forcing in non-reacting swirling flow at uo = 10 m/s, S = 0.85.

fluctuating velocity away from the nozzle, shown here by all the vectors pointing left.Closer to the nozzle, however, an asymmetric breathing in and out of the nozzlecavity in the axial direction can be seen. O’Connor and Lieuwen showed [20] thatthis led to the excitation of strong helical shear layer modes. Figure 3b shows thesymmetric forcing case, a result of a pressure anti-node and acoustic velocity node.Here, the velocity fluctuations away from the nozzle are very small, yet near thenozzle significant vortical velocity fluctuations are excited by bulk axial velocityfluctuations in and out of the nozzle. This led to the strong excitation ofaxisymmetric shear layer modes.

For these reasons, this study particularly focuses on the transverse to axial acousticcoupling processes, path 1, in Figure 2. This coupling process controls the “forcingfunction” for the hydrodynamic flow instabilities. While particularly motivated by thecombustor problem, this work also has more general relevance to duct acoustics in thepresence of side branches.

The goals of this study are accomplished by performing three-dimensional acousticsimulations for a non-flowing, inviscid, non-reacting environment. As such, thisanalysis is useful for studying the disturbance field away from the boundary layers,where shear induced instabilities may be dominant, and for low Mach number flows.Moreover, these results are useful for understanding the axial, “outer flow” forcingwhich excites the shear layers in an oscillatory manner.

The rest of the paper is organized in the following manner. First, we present themodel framework. Then, we present the basic characteristics of the pressure and three-dimensional acoustic velocity field in the vicinity of the nozzle. Results are shown forcases where the nozzle is located at a pressure and velocity node, as well as when it issubjected to a traveling wave. Results from either the two standing wave cases, or thesingle traveling wave case can then be combined to evaluate the disturbance field fornozzles located in any other location in a standing wave. Finally, we show that theupstream impedance of the nozzle has important influences on the axial acousticvelocity, and show how the impedance translation theorem can be used to provide auseful interpretation of the bulk axial disturbance field in several cases.

2. MODEL FRAMEWORKThis section details the finite element acoustic model. The physical domain was selectedto duplicate an existing experimental facility [31]. The physical domain is 114 cm inlength, 36 cm in height and 8 cm in depth. A nozzle section connects into the center ofthe box as illustrated in Figure 4, which has an outer radius, ro, of 16 mm and innerradius, ri, of 11 mm, and extends 51 mm from the bottom of the combustion chamber.COMSOL Multiphysics (version 4.2) was used to model, mesh, and analyze this system,shown in Figure 4. We present all results in dimensionless form, and so the results applyto other systems with the same ratios of dimensions.

All frequencies analyzed in this study are below the cutoff frequency of the side branch,so that only one-dimensional waves propagate in this region. However, since evanescentmulti-dimensional disturbances occur at the nozzle-combustor interface, the nozzle lengthshown in Figure 5, h, was chosen in order to ensure that the disturbance field has reverted


to a nearly one-dimensional field at the opposite end of the nozzle, z = – h. This choice ofh, coupled with the approach described later to specify the impedance at z = – h, eliminatessensitivity of these results to h, and so removes it as an independent parameter influencingthe acoustic field in the nozzle-combustor interface region.

Forcing is employed by applying a spatially uniform pressure disturbance on theopposing faces of the chamber, shown as walls a-a and b-b in Figure 4. Three differentvelocity forcing fields were used, and are referred to as “in-phase”, “out-of-phase”, and“traveling wave” scenarios. The first two disturbance fields lead to standing wave fieldsin the system, where the combustor centerline is nominally a pressure anti-node andnode, respectively. The velocity field exhibits a node and antinode, respectively. In thethird forcing scenario, an anechoic boundary is applied to the right side of the domain.In the absence of the nozzle, the acoustic field is one-dimensional and the three forcingscenarios lead to the magnitude profiles shown in Figure 6.

A spatially uniform impedance boundary condition, Z0, is applied at the lower end ofthe nozzle section, z = – h. As noted above, the acoustic field reverts to a one-dimensional field at this end of the nozzle section, because the forcing frequencyremains well below the transverse mode cutoff frequency for all frequencies consideredin this study. Once the acoustic field is radially uniform in the nozzle, the relationshipbetween the axial velocity and pressure in the nozzle is uniquely related to the nozzleimpedance through the impedance translation theorem [32]:


Detail C

a

a b

b

z

x

Figure 4: Schematic showing simulation domain and coordinate system.

Point D

Surface Ey

z = 0z

x

z = − h

R ri

ro

x

Figure 5: Detail C from Figure 4 showing centerbody and annulus, as well as filletand surface of integration.

(1)

We chose Z0 values such that Ztr (z = 0) approximates pressure release, anechoic, andrigid boundary conditions by using Ztr /ρc values of 0.01, 1, and 100, respectively. Forthis reason, a frequency dependant Z0 value is applied in order to maintain a fixed Ztr.The impedance translation relates the impedance at any axial location in a plane wavefield to its value at some prescribed location.

We next discuss the treatment of the nozzle-combustor interface geometry.Because singularities occur at sharp corners for inviscid flows, special care is requiredat the inside and outside corners of the annulus to insure that the simulated results aregrid-independent. Singularities are avoided by adding a fillet radius, R, to the corners,as shown in Figure 5. In order to model the actual, viscous flow, this fillet radiusshould be on the order of the boundary layer thickness (that is, in turn, a function ofthe Reynolds number and swirl number). The velocity field in the vicinity of thecorner is then a function of R. To illustrate, Figure 7 shows the dependence of theaxial velocity at location D indicated in Figure 5 upon fillet radius. For reference, a

line is also indicated in the figure, representing the theoretical result for a two-dimensional corner [32].

Figure 7 also shows the average axial velocity at the nozzle-combustor junction,calculated on the top half annulus of the nozzle, shown in Figure 5, as

(2)′ =−( ) ′ ( ) ⋅

−

+

∫∫ur r

u r rdrdzo i

z

r R

r R

i

o22 2

0πθ θ

π

,

1 R

Z

c

Z

ce

Z

ce

ztr

o ik z h o

( )( )

ρρ ρ

= −− −

+ − +− +( )1 1 iik z h

o ik z h o ik zZ

ce

Z

ce

+( )

− +( )+

+ − +ρ ρ

1 1( ) ++( )

h


1.2

1

0.8

0.6

0.4

0.2

00

In phase pressure/out of phase velocity

Out of phase pressure/in phase velocity

Traveling wave pressure/traveling wave velocity

Nor

mal

ized

mag

nitu

de

0.2 0.4−0.4 −0.2

x/λ

Figure 6: Acoustic pressure and velocity fields for three different forcing cases,solved in the absence of the side branch.

This result shows that while the local velocity exhibits a significant dependence on Rnear the fillet, the spatially averaged axial velocity results are almost independent offillet radius. It is important to point out the steps taken to define the normalizing areaused in equation (2). In the fillet region, the volume flow rate remains virtually constantas z varies from z = 0 to − R. However, the averaging surface area does not remainconstant. For this reason, the averaged axial velocity does vary with z, and similarlyvaries at z = 0 with fillet radius. This effect is nothing more than a manifestation of thedependence of nozzle outlet area at z = 0 on R. In order to account for this geometryeffect, we rescale the average axial velocity by the area ratio at z = 0 and z = − R, toarrive at the formula shown in equation (2). All results shown in this paper use a radiusof 0.02D, which is indicated by the vertical dashed line in Figure 7.

The mesh is comprised of 374,705 free tetrahedral mesh elements and 529,592 degreesof freedom, and employs minimum interior boundaries. The fillet region requiresparticular care in meshing. The mesh density in this region is quite high and then smoothlytransitioned to the required density needed to simulate the rest of the system. Thisconfiguration allows the mesh elements to grow freely from the small radius to the largecombustor box with maximum efficiency and accuracy. This final mesh was settled onafter a grid independence study using four meshes of increasing resolution showedvariations of less than 0.1%.

3. RESULTS AND DISCUSSIONThis section presents typical results. Results were obtained for a range of frequenciesbetween 200–3000 Hz, corresponding to non-dimensional values ranging from0.02 to 0.30. These frequencies were simulated for the three forcing configurations andusing the three upstream nozzle impedance values described in the previous section.

D λ


10−2

R/D

u−′z

u ′z

10−110–1

100

1/R1/2

100

D/ = 0.04λ

D/ = 0.15λD/ = 0.09λ

cu

/pm

axρ

Figure 7: Dependence of axial velocity at point D in Figure 5 (left) and spatiallyaveraged over surface E in Figure 5 (right) upon fillet radius at threefrequencies, for in-phase forcing with Ztr /ρc = 1. The vertical dashed lineindicates the fillet radius used for the results presented in this paper.

Figure 8 illustrates the coordinate system and various cuts used to represent the three-dimensional disturbance field.

Figure 9 presents representative instantaneous pressure contours and velocity vectorfields for the different forcing configurations, using an anechoic nozzle impedance.Results are shown on the x-z cut plane. The in-phase case, shown on the left, generateszero transverse velocity at the nozzle outlet, but large pressure fluctuations are present


y

y

y xz

x-yx-z

xx

Figure 8: Schematics showing coordinate system used to define various cuts atwhich data is plotted.

−0.8 0 0.8

−0.64

0

0.64

1.28

−0.8 0 0.8

−0.64

0

0.64

1.28

z/D

z/D

z/D

x/D x/D

x/D

1

0.8

0.4

0

−0.4

−0.8

−1

1

0.8

0.4

0

−0.4

−0.8

−1

−0.8 0.80

−0.64

0

0.64

1.281

0.8

0.4

0

−0.4

−0.8

−1

Figure 9: Instantaneous disturbance fields at D/λ ≈ 0.04 (400 Hz) with an anechoicboundary condition at the nozzle for in-phase (left), out-of-phase (right),and traveling wave (bottom) scenarios. Colors represent instantaneouspressure, while arrows denote the instantaneous total velocity field.

that are symmetric across the centerline. These pressure fluctuations lead to symmetric,axial velocity disturbances on both sides of the annulus. As we will discuss later, thenozzle response for this in-phase forcing case can be understood from quasi one-dimensional concepts. In contrast, the out-of-phase case exhibits large transverse velocityfluctuations in the center of the combustor. Because of the centerline pressure node, thepressure fluctuations have a 180 degree phase difference on the two sides of the annulus.Similarly, the axial velocity fluctuations are phased 180 degrees apart on the left and rightsides of the annulus. This nozzle response is intrinsically three-dimensional.

The traveling wave case shows an intermediate behavior. For the illustrated case, thewavelength is long relative to the nozzle, so the disturbance field is nearly uniformacross the nozzle. However, a slight phase difference in axial velocity fluctuations existson the two sides of the annulus, evident in Figure 9. In addition, there is a slight amountof asymmetry in wave magnitudes on the two sides of the nozzle due to wave reflection.This traveling wave scenario simulates the classical problem of wave reflection of anincident wave by a side branch. For example, a quasi one-dimensional analysis, usingcontinuity of volume flow rate and pressure at the interface, leads to the followingpredicted result for the reflection coefficient, R, from the side branch [33, 34]:

(3)

where Sb and Zb describe the cross sectional area and impedance of the side branch,respectively. This analysis also explains why a traveling wave scenario such as this is notexactly correct inside an annular combustor with a spinning mode. With multiplenozzles, there will always be reflected waves between two nozzles, so the traveling wavewill actually lead to a field with a slight standing wave structure at each nozzle.However, as long as the nozzle cross-sectional area is small relative to that of theannulus, the traveling wave approximation is a good one.

The azimuthal distributions of axial velocity magnitude and phase at the nozzleinterface are shown in Figure 11, along with the accompanying pressure distributionsalong the bottom surface of the combustor shown in Figure 10. From this view, we canclearly see how the pressure anti-node at the center of the combustor excitessymmetrically distributed axial velocity in the nozzle region. The phase reversal in axialvelocity for the out-of-phase scenario can also be inferred from the node in the center.The traveling wave velocity distribution is asymmetric, exhibiting a much larger axialresponse on the right side of the nozzle.

The axial distribution of pressure and axial velocity along the centerline of theannulus are plotted in Figure 12 for the in-phase and out-of-phase cases. Notice that, forthe out-of-phase forcing case, the pressure and velocity disturbance fields decay to zeroin the nozzle region. Thus, although the axial velocity fluctuations at the nozzle outletare non-negligible, they are of opposite phases on the opposite sides of the annulus andcancel each other, leading to a progressive decay in disturbance amplitude andfluctuation energy in the nozzle. For this reason, the nozzle impedance has no influence

R = −+

1

2 1( / )( / )S S Z cb b ρ


on the disturbance field characteristics in the out-of-phase case, as discussed further inthe next section.

As alluded to in the above discussion, important insights into the character of theaxial velocity at the nozzle exit can be gained from the pressure field. As such, we nextdiscuss the characteristics of the pressure field in the combustor-nozzle interface inmore detail. Figure 13 presents plots of the magnitude of the pressure along the cut linesfor a case. For reference, the solid line denotes the value of the disturbancefield along the x-x cut line that would exist in the absence of the nozzle. Note that forthe anechoic and rigid nozzle impedance conditions, the nozzle causes only a slightdistortion of the disturbance field.

Z ctr ρ = 1


1

0.8

0

0.4

−0.4

−0.8

−1

10.8

0

0.4

−0.4

x

x

x

1.3

1.2

1.1

1

0.9

0.8

0.7

−0.8

−1

Figure 10: Instantaneous pressure along the x-y surface at D/λ ≈ 0.28 (3000 Hz) for in-phase (top), out-of-phase (middle), andtraveling wave (bottom) cases for anechoic nozzle impedance.

In contrast, the nozzle significantly distorts the pressure field from the 1-D resultwhen for the in-phase and traveling wave cases, as shown in Figure 14.Little distortion occurs for the out-of-phase case where the nominal pressure field iszero. When the combustor acoustic field exhibits a significantly non-zero pressure nearthe nozzle note how the nozzle “pulls” the pressure amplitude toward zero.

Having discussed the pressure, we next consider the velocity field at the nozzle exitfor the case. Figure 15 plots the in-phase result, along with the 1-D result onthe x-x cut line. Similar to the nominal, one-dimensional result, the transverse velocityis low everywhere except near the annulus corners. Here, the sharp area change leads toa strong transverse velocity field. Note that the magnitudes of these velocity values nearthe corners are functions of the fillet radius, R. The axial velocity field is nearly uniformat the nozzle exit, reflecting a nearly plug flow disturbance field, except for overshootsnear the corners, which are again functions of the fillet radius.

Z ctr ρ = 1

Z ctr ρ = 0 01.


0.9

0.7

0.5

0.3

0.1

0

0.1

0.08

0.04

2

2.0

1.5

0.5

0

0

1

−0.04

−0.08

−0.1

0

Figure 11: Spatial distribution of axial velocity magnitude at the nozzle-combustorjunction at D/λ ≈ 0.28 (3000 Hz) for in-phase (top), out-of-phase(middle), and traveling wave (bottom) scenarios for anechoic nozzleimpedance. Color represents instantaneous axial velocity.


−0.8

Out-of-phase

In-phase

z/h

p′

p′

u′z

u′z

−10

0.5

Nor

mal

ized

mag

nitu

de

1.5

1

−0.6 −0.4 −0.2 0

Figure 12: Axial distribution of pressure and axial velocity along the annuluscenterline, normalized by their value at the nozzle outlet (D/λ ≈ 0.04 or400 Hz, Ztr /ρc = 1).

Figure 13: Pressure distributions at D/λ ≈ 0.04 (400 Hz) with an anechoic nozzle forthe in-phase, out-of-phase, and rightward traveling wave scenarios. Cutlines x-x and y-y are shown in Figure 8.

Axial velocity magnitudes for the out-of-phase and traveling wave forcing casesare shown in Figure 16. Note that the largest axial velocities are observed near theouter edge of the annulus for the out-of-phase forcing case. As discussed previously,the values on the left and right sides are 180 degrees out-of-phase. The travelingwave case results are asymmetric on the x-axis, also discussed previously. Along they-y cut, the traveling wave displays the same shape and similar magnitude as thosetwo sides of the nozzle experience the same disturbance field.

As mentioned in the introduction, experiments have shown that completely differentshear layer modes are excited during in-phase and out-of-phase forcing [20]. Theresultant fluctuations in flame position are indicated in Figure 17, showing thesymmetric and asymmetric wrinkling of the flame as a result. The staggered flamewrinkling in the asymmetric forcing case, as seen in Figure 17 on the left, results from


Figure 14: Pressure distribution at D/λ ≈ 0.04 (400 Hz) a pressure release conditionat the nozzle for the in-phase, out-of-phase, and rightward traveling wavescenarios. Cut lines x-x and y-y are shown in Figure 8.

a helical disturbance in the shear layers, excited by the asymmetric acoustic forcing atthis condition. Conversely, the symmetric flame wrinkling, shown in Figure 17 on theright, stems from the rollup of axisymmetric vortex rings. This rollup is a result of thesymmetric bulk forcing of the in-phase acoustic mode shape.

While computing this behavior requires a viscous flow calculation that captures theexcitation and convection of vortical disturbances, the basic fundamentals leading to


Figure 15: In-phase transverse (top) and axial (bottom) velocities at D/λ ≈ 0.04 (400Hz) for an anechoic nozzle. Cut lines x-x and y-y are shown in Figure 8.

this behavior can be understood from these inviscid, purely acoustic calculations.Namely, the acoustic field acts as the forcing function that excites the convectivelyunstable shear layers. The vortical field features are a simple manifestation of the factthat the axial velocity in the left and right sides of the nozzle are in phase in one case,and out-phase in the other, leading to excitation of completely different shear layermodes in the two cases.


Figure 16: Axial velocity at D/λ ≈ 0.04 (400 Hz) with an anechoic nozzle for the out-of-phase (top) and traveling wave (bottom) scenarios. Cut lines x-x and y-y are shown in Figure 8.

4. FURTHER ANALYSIS OF THE AXIAL VELOCITYThis section expands the analysis of the axial velocity which, as discussed in theintroduction, has been proposed as a particularly significant feature influencing howflames are excited during transverse instabilities. This section will further emphasize therole of nozzle impedance on these characteristics. As discussed previously, the acousticfield in the nozzle quickly reverts to a one-dimensional field because the frequency isbelow the duct cut-off frequency. Once one-dimensional, the axial velocity and pressurein the nozzle are directly related by the translated nozzle impedance, Ztr, given byequation (1). Thus, it is useful to define the impedance ratio, RZ, through which tocompare simulated results to quasi one-dimensional results, as

(4)

where the spatially averaged pressure field is given by

(5)

While this expression and equation (2) describes the pressure and velocitydisturbance evaluated over one half of the nozzle, slightly different forms were used fordifferent cases. It was shown in Figure 6 that the in-phase case leads to symmetricresults on the two halves of the nozzle, and the out-of-phase case leads to anti-symmetric results. As such, integrating the pressure or velocity over the entire annulusarea for the out-of-phase case leads to zero, because of cancellation of results on the two

′ =+( ) − −( )( ) ′( ) ⋅

−

+

∫∫pr R r R

p r ro i r R

r R

i

o22 2

0πθ

π

, ddrdθ

R

pu

ZZz

tr

=

′′


Figure 17: Experimental flame luminescence images of 400 Hz out-of-phase (left)and 400 Hz in-phase (right) forcing of a flame in a swirling flow at uo = 10 m/s, S = 0.5, and an equivalence ratio of 0.9. Arrows point to thewrinkles resulting from a) helical and b) ring vortices.

halves. As such, we use half the nozzle area, equation (5), for the standing wave cases.The traveling wave case is integrated over the entire annulus face.

Figure 18 and Figure 19 plot the calculated dependence of RZ upon the dimensionlessfrequency for both the in-phase and traveling wave cases, respectively. Note how themagnitude of RZ is quite close to unity. In both of these cases, the unsteady pressure fieldhas nearly uniform phase across the entire face of the annulus, so it is expected thatmulti-dimensional effects in the nozzle are quite small. The growing, but slight,deviation of RZ magnitude from unity with increasing frequency is a manifestation of the


Figure 18: Impedance ratio magnitude (top) and phase (bottom) for various nozzleboundary conditions at the nozzle for the in-phase forcing case.

increasing phase and magnitude variation in unsteady pressure in the traveling and in-phase cases, respectively.

Although not shown, RZ deviates substantially from unity for the out-of-phaseforcing case. Recall that for this case, there exists a 180 degree phase change in axialvelocity and pressure on opposite sides of the annulus centreline. If the two halves of thenozzle were physically separated by a rigid barrier, then RZ could be a useful quantity.In actuality, the phase cancellation causes a vanishing of the acoustic field in the nozzle,


Figure 19: Impedance ratio magnitude (top) and phase (bottom) for various nozzleboundary conditions at the nozzle for the traveling wave case.

as shown in Figure 12. Because the disturbance field is essentially zero at z = − h, thisrenders the z = 0 results effectively independent of the nozzle impedance. As such, RZis not a useful quantity for characterizing the axial velocity response. These points canbe seen from Figure 20, which plots the ratio of the spatially averaged pressure and axialvelocity over one half of the nozzle face as a function of frequency. Note that all threenozzle impedance values give the same pressure-velocity relationship at the nozzle exit.


Figure 20: Averaged nozzle impedance magnitude (top) and phase (bottom) for out-of-phase forcing. Nozzle impedance values of Ztr /ρc = 0.01, 1, and 100are plotted.

These results show that the nozzle response exhibits a strong sensitivity to its upstreamimpedance in certain cases and is totally independent of it in others. For example,consider azimuthal standing modes in an annular combustor. Because of the spatiallyvarying pressure/velocity field, different nozzles will be located in different parts of thestanding wave and exhibit different response characteristics and nozzle impedancesensitivities. For the first radial mode in an annular combustor, nozzles located along theradial centerline, such as shown in Figure 1, will always be located at a velocity anti-node/pressure node and, as such, exhibit no sensitivity to nozzle impedance.

5. CONCLUDING REMARKSThis paper has described an analysis of the coupling between transverse acoustic motionsand the induced axial motions in a small area side channel, simulating the complex acousticfield generated by a fuel/air mixing nozzle inside an annular combustion chamber. Theseresults illustrate a critical dependence of the near-field acoustics on “macro” features of theacoustic field, such as the general waveform of the disturbance in the absence of the nozzle,or the location of the nozzle with respect to global velocity or pressure nodes. In addition,it was shown that nozzle impedance has a very significant effect on this transverse to axialcoupling for in-phase and traveling wave acoustic excitation. The bulk features of thepressure field at the nozzle exit can be understood from 1-D acoustic considerations forseveral of the cases, due to the small cross sectional area of the side branch relative to thatof the main chamber. An important exception to this occurs when the nozzle is nominallylocated in a pressure anti-node and the nozzle impedance attempts to force that samelocation into a pressure node, as was seen for an in-phase wave with a pressure releasenozzle. Similarly, we show that the spatially averaged pressure to axial velocity relationshipis quite close to the one-dimensional, translated impedance value at the end of the sidebranch. The notable exception to this result occurs in the out-of-phase forcing case, whoseaxial velocity characteristics are independent of the nozzle impedance.

ACKNOWLEDGMENTSThis work has been partially supported by the US Department of Energy under contractDEFG26-07NT43069, contract monitor Mark Freeman.

REFERENCES[1] Lieuwen, T.C. and V. Yang, Combustion Instabilities in Gas Turbine Engines,

Operational Experience, Fundamental Mechanisms, and Modeling. Progress inAstronautics and Aeronautics, ed. T.C. Lieuwen and V. Yang 2005.

[2] Rayleigh, B.J.W.S., The Theory of Sound. Vol. 2. 1896: Macmillan.

[3] Kedia, K., S. Nagaraja, and R. Sujith, Impact of Linear Coupling onThermoacoustic Instabilities. Combustion Science and Technology, 2008. 180(9):p. 1588–1612.

[4] Ducruix, S., T. Schuller, D. Durox, and S. Candel, Combustion Dynamics andInstabilities: Elementary Coupling and Driving Mechanisms. Journal ofPropulsion and Power, 2003. 19(5): p. 722–734.


[5] Venkataraman, K., L. Preston, D. Simons, B. Lee, J. Lee, and D. Santavicca,Mechanism of Combustion Instability in a Lean Premixed Dump Combustor.Journal of Propulsion and Power, 1999. 15(6): p. 909–918.

[6] Palies, P., D. Durox, T. Schuller, and S. Candel, The Combined Dynamics ofSwirler and Turbulent Premixed Swirling Flames. Combustion and Flame, 2010.157(9): p. 1698–1717.

[7] Hirsch, C., D. Fanaca, P. Reddy, W. Polifke, and T. Sattelmayer, Influence of theSwirler Design on the Flame Transfer Function of Premixed Flames. Volume 2Turbo Expo 2005, 2005: p. 151–160.

[8] Paschereit, C.O., E. Gutmark, and W. Weisenstein, Excitation of ThermoacousticInstabilities by Interaction of Acoustics and Unstable Swirling Flow. AIAAjournal, 2000. 38(6): p. 1025–1034.

[9] Culick, F. and V. Yang, Prediction of the Stability of Unsteady Motions in Solid-Propellant Rocket Motors. 1992.

[10] Harrje, D.T. and F.H. Reardon, Liquid Propellant Rocket Combustion Instability1972: Scientific and Technical Information Office, National Aeronautics andSpace Administration.

[11] Price, E.W., Solid Rocket Combustion Instibility - An American HistoricalAccount, in Nonsteady Burning and Combustion Stability of Solid Propellants1992. p. 1–16.

[12] Chehroudi, B., D. Talley, J.I. Rodriguez, and I.A. Leyva, Effects of a Variable-Phase Transverse Acoustic Field on a Coaxial Injector at Subcritical and Near-Critical Conditions, in 47th Aerospace Sciences Meeting 2008: Orlando, FL.

[13] Rogers, D.E. and F.E. Marble, A Mechanism for High-Frequency Oscillation inRamjet Combustors and Afterburners. Jet Propulsion, 1956. 26(6): p. 456–464.

[14] Kaskan, W.E. and A.E. Noreen, High-Frequency Oscillations of a Flame Held bya Bluff Body. ASME Transactions, 1955. 77(6): p. 855–891.

[15] Elias, I., Acoustical Resonances Produced by Combustion of a Fuel-Air Mixturein a Rectangular Duct. Journal of the Acoustical Society of America, 1959. 31(3):p. 296–304.

[16] Smith, K., L. Angello, and F. Kurzynske, Design and Testing of an Ultra-LowNO/Sub x/Gas Turbine Combustor, 1986, Solar Turbines Inc., San Diego, CA.

[17] Krebs, W., S. Bethke, J. Lepers, P. Flohr, and B. Prade, Thermoacoustic DesignTools and Passive Control: Siemens Power Generation Approaches, inCombustion Instabilities in Gas Turbine Engines, T.C. Lieuwen and V. Yang,Editors. 2005, AIAA: Washington D.C. p. 89–112.

[18] Dowling, A.P. and S.R. Stow, Acoustic Analysis of Gas Turbine Combustors.Journal of Propulsion and Power, 2003. 19(5): p. 751–764.

[19] Sewell, J. and P. Sobieski, Monitoring of Combustion Instabilities: Calpine’sExperience, in Combustion Instabilities in Gas Turbine Engines, T.C. Lieuwenand V. Yang, Editors. 2005, AIAA: Washington D.C. p. 147–162.


[20] O’Connor, J. and T. Lieuwen, Further Characterization of the Disturbance Fieldin a Transversely Excited Swirl-Stabilized Flame. Journal of Engineering for GasTurbines and Power - Transactions of the ASME, 2012. 134(1).

[21] Cohen, J., G. Hagen, A. Banaszuk, S. Becz, and P. Mehta, Attenuation OfCombustor Pressure Oscillations Using Symmetry Breaking, in 49th AIAAAerospace Sciences Meeting including the New Horizons Forum and AerospaceExposition 2011: Orlando, Florida.

[22] Hauser, M., M. Lorenz, and T. Sattelmayer. Influence of Transversal AcousticExcitation of the Burner Approach Flow on the Flame Structure. in ASME TurboExpo. 2010. Glasgow, Scotland.

[23] O’Connor, J. and T. Lieuwen, Disturbance Field Characteristics of a TransverselyExcited Burner. Combustion Science and Technology, 2011. 183(5): p. 427–443.

[24] Stow, S.R. and A.P. Dowling. Low-Order Modelling of Thermoacoustic LimitCycles. 2004.

[25] Acharya, V., Shreekrishna, D.H. Shin, and T. Lieuwen, Swirl Effects onHarmonically Excited, Premixed Flame Kinematics. Combustion and Flame,2012. 159(3): p. 1139–1150.

[26] Worth, N.A. and J.R. Dawson, Cinematographic OH-PLIF Measurements of TwoInteracting Turbulent Premixed Flames with and without Acoustic Forcing.Combustion and Flame, 2011.

[27] Staffelbach, G., L.Y.M. Gicquel, G. Boudier, and T. Poinsot, Large EddySimulation of Self Excited Azimuthal Modes in Annular Combustors. Proceedingsof the Combustion Institute, 2009. 32: p. 2909–2916.

[28] Wolf, P., G. Staffelbach, A. Roux, L. Gicquel, T. Poinsot, and V. Moureau,Massively Parallel LES of Azimuthal Thermo-Acoustic Instabilities in AnnularGas Turbines. Comptes Rendus Mecanique, 2009. 337(6-7): p. 385–394.

[29] Hutt, J.J. and M. Rocker, High-Frequency Injection-Coupled CombustionInstability, in Liquid Rocket Engine Combustion Instability, V. Yang and W.E.Anderson, Editors. 1995, American Institute of Aeronautics and Astronautics.p. 345–355.

[30] Davis, D., B. Chehroudi, D. Talley, R. Engineering, and C.A. Consulting IncEdwards Afb, The Effects of Pressure and an Acoustic Field on a CryogenicCoaxial Jet, in 42nd Aerospace Sciences Meeting and Exhibit 2004: Reno, NV.

[31] O’Connor, J., J. Mannino, C.Vanatta, and T. Lieuwen, Mechanisms for FlameResponse in a Transversely Forced Flame, in 7th US National Technical Meetingof the Combustion Institute 2011: Atlanta, GA.

[32] Pierce, A.D., Acoustics: An Introduction to its Physical Principles andApplications 1989: Acoustical Society of America.

[33] Lighthill, J., Waves in Fluids 2001: Cambridge Univ Pr.

[34] Rienstra, S.W. and A. Hirschberg, An Introduction to Acoustics. EindhovenUniversity of Technology, 2003.


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