International journal of spray and combustion dynamics Volume 4 Number

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1 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 reprinted from International journal of spray and combustion dynamics Volume 4 Number 4 22 Multi-Science Publishing ISSN

2 International journal of spray and combustion dynamics Volume. 4 Number pages Transverse to longitudinal acoustic coupling processes in annular combustion chambers J. Blimbaum, M. Zanchetta 2, T. Akin 2, V. Acharya 3, J. O Connor 4, D. R. Noble 3 and T. Lieuwen,3 Department of Mechanical Engineering, Georgia Institute of Technology, Atlanta GA 338, USA 2 Institut Supérieur de l Aéronautique et de l Espace - École Nationale Supérieure de Mécanique et d Aérotechnique, Cedex, France 3 Department of Aerospace Engineering, Georgia Institute of Technology, Atlanta GA 338, USA 4 Engine Combustion Department, Sandia National Laboratories, Livermore, CA 94566, USA Submission May 7, 22; Revised Submission June 26, 22; Acceptance July 9, 22 ABSTRACT Combustion instability is a major issue facing lean, premixed combustion approaches in modern gas turbine applications. This paper specifically focuses on instabilities that excite transverse acoustic modes of the combustion chamber. Recent simulation and experimental studies have shown that much of the flame response during transverse instabilities is due to the longitudinal fluid motions induced by the fluctuating pressure field above a nozzle. In this study, we analyze the multi-dimensional acoustic field excited by transverse acoustic disturbances interacting with an annular side branch, emulating a fuel/air mixing nozzle. Key findings of this work show that the resultant velocity fields are critically dependent upon the structure of the transverse acoustic field and the nozzle impedance. Significantly, we also show that certain cases can be understood from relatively simple quasi one-dimensional considerations, but that other cases are intrinsically three-dimensional.. INTRODUCTION Emissions regulations, reliability, and fuel costs are significant factors that drive combustion technology. Combustion instabilities have arisen as one of the most critical problems facing development of robust, low emissions combustors for both power generation and aviation applications []. These instabilities can arise due to interactions between heat release and local flow perturbations. When these heat release oscillations are in-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

3 276 Transverse to longitudinal acoustic coupling processes in annular combustion chambers as velocity and fuel/air ratio, and much work has been done to understand these mechanisms in combustion systems [3 8]. This study focuses specifically on transverse oscillations in combustion chambers, which have been historically problematic in rockets [9 2] and jet engine afterburners [3 5]. In addition, they appear in gas turbines in both annular and can combustion chambers [6 22]. In order to motivate the approach taken in this study, it is useful to consider in more detail a typical arrangement of annular combustor systems and transverse instabilities in these systems. A simplified schematic of such a system is shown in Figure. It illustrates an annular ring around which regularly spaced nozzles are placed. Air from the compressor exits these nozzles and flows in the axial flow direction. These nozzles have their own acoustic characteristics, associated with distributed inertia/compressibility in the flow passage, and the inertia and resistivity of the swirlers [8]. For systems with sufficient wave transmission through the swirler, the upstream acoustic characteristics also 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, and standing waves in the radial direction. In many cases, nozzles could be situated at any point in the standing wave field, because nozzles are distributed around the combustor. For example, a nozzle situated near the velocity node and another near the pressure node experience significantly different disturbance fields, and possibly different flame excitation physics. Moreover, these disturbance wavelengths could be quite long relative to the nozzle dimension in the case of annular modes, or on the same order of the wavelength for radial modes. In many cases, the oscillations in the azimuthal direction are closely approximated by traveling waves [8]. In this case, the flame response would vary significantly in time as the wave spins around the combustion chamber. Transverse instability Nozzles Figure : Schematic of an annular combustor.

4 International journal of spray and combustion dynamics Volume. 4 Number Transverse acoustic excitation (p ) Longitudinal (axial) acoustics 2 3 p 4 5 Flow instabilities 6 z x u z Flame response (Q. ) Figure 2: Pathways for velocity coupled combustion instabilities. Recent work has shown that several paths exist through which a transverse mode may excite a flame [23 26]. As shown in Figure 2, transverse modes can directly excite the flame, they can excite hydrodynamic flow instabilities, and they can also lead to axial acoustic flow oscillations in the nozzle. Measurements and simulations have suggested that these axial oscillations are the dominant source of flame excitation during a transverse instability [23, 27, 28]. These axial acoustic oscillations are a wave diffraction effect, as the dominantly transverse mode leads to an oscillatory pressure field across the nozzle. This oscillatory pressure field induces axial flow oscillations, referred to as injector coupling in the rocket literature [29, 3]. Experiments and computations of transversely excited flames have clearly demonstrated the significance of this transverse to axial flow coupling process. For instance, Staffelbach et al. [27] presented LES results of an instability in an annular combustor, clearly showing the presence of strong axial pulsations in nozzle flow accompanying these transverse oscillations. This study suggested that it was these axial flow pulsations that dominated the flame response. In other words, the transverse flow oscillations serve as the clock which controls the natural frequency of the wave motions and the structure of the wave field, but it is the induced axial fluctuations which actually excite the heat release oscillations that, in turn, excite the transverse modes. Experimental observations of this same point were also made by O Connor and Lieuwen [23], who presented a number of non-reacting and reacting results of transversely excited flames, using high speed planar velocimetry measurements and line 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 also the significant axial flow oscillations in the nozzle region. These studies also emphasized the importance of the transverse mode structure relative to the nozzle in controlling the flow field. For example, a nozzle located at a pressure node experiences significant transverse flow oscillations, and a relatively weak unsteady pressure field. However, the phase of the pressure disturbances differs by 8 degrees on the left and right sides of the nozzle, implying that the excited axial flow

5 278 Transverse to longitudinal acoustic coupling processes in annular combustion chambers 2.5 x/d x/d.5 Figure 3: Experimental results of PIV measurements of coherent velocity fluctuation for 4 Hz out-of-phase (left) and 4 Hz in-phase (right) forcing in non-reacting swirling flow at u o = m/s, S =.85. oscillations are highly non-symmetric. Different forcing conditions can lead to varying flow response in the region of the nozzle. For example, the fluctuating component of the flow shown in Figure 3a is the result of forcing at a pressure node and velocity anti-node. The asymmetric velocity forcing results in a strong bias in the

6 International journal of spray and combustion dynamics Volume. 4 Number 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 nozzle cavity in the axial direction can be seen. O Connor and Lieuwen showed [2] that this led to the excitation of strong helical shear layer modes. Figure 3b shows the symmetric 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 the nozzle significant vortical velocity fluctuations are excited by bulk axial velocity fluctuations in and out of the nozzle. This led to the strong excitation of axisymmetric shear layer modes. For these reasons, this study particularly focuses on the transverse to axial acoustic coupling processes, path, in Figure 2. This coupling process controls the forcing function for the hydrodynamic flow instabilities. While particularly motivated by the combustor problem, this work also has more general relevance to duct acoustics in the presence of side branches. The goals of this study are accomplished by performing three-dimensional acoustic simulations for a non-flowing, inviscid, non-reacting environment. As such, this analysis 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 forcing which excites the shear layers in an oscillatory manner. The rest of the paper is organized in the following manner. First, we present the model framework. Then, we present the basic characteristics of the pressure and threedimensional acoustic velocity field in the vicinity of the nozzle. Results are shown for cases where the nozzle is located at a pressure and velocity node, as well as when it is subjected to a traveling wave. Results from either the two standing wave cases, or the single traveling wave case can then be combined to evaluate the disturbance field for nozzles located in any other location in a standing wave. Finally, we show that the upstream impedance of the nozzle has important influences on the axial acoustic velocity, and show how the impedance translation theorem can be used to provide a useful interpretation of the bulk axial disturbance field in several cases. 2. MODEL FRAMEWORK This section details the finite element acoustic model. The physical domain was selected to duplicate an existing experimental facility [3]. The physical domain is 4 cm in length, 36 cm in height and 8 cm in depth. A nozzle section connects into the center of the box as illustrated in Figure 4, which has an outer radius, r o, of 6 mm and inner radius, r i, of mm, and extends 5 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 apply to 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 evanescent multi-dimensional disturbances occur at the nozzle-combustor interface, the nozzle length shown in Figure 5, h, was chosen in order to ensure that the disturbance field has reverted

7 28 Transverse to longitudinal acoustic coupling processes in annular combustion chambers a b z x Figure 4: a b Detail C Schematic showing simulation domain and coordinate system. Point D z = R r i z x Surface E y x r o z = h Figure 5: Detail C from Figure 4 showing centerbody and annulus, as well as fillet and surface of integration. to a nearly one-dimensional field at the opposite end of the nozzle, z = h. This choice of h, coupled with the approach described later to specify the impedance at z = h, eliminates sensitivity of these results to h, and so removes it as an independent parameter influencing the acoustic field in the nozzle-combustor interface region. Forcing is employed by applying a spatially uniform pressure disturbance on the opposing faces of the chamber, shown as walls a-a and b-b in Figure 4. Three different velocity 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 fields in the system, where the combustor centerline is nominally a pressure anti-node and node, respectively. The velocity field exhibits a node and antinode, respectively. In the third 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 forcing scenarios lead to the magnitude profiles shown in Figure 6. A spatially uniform impedance boundary condition, Z, is applied at the lower end of the nozzle section, z = h. As noted above, the acoustic field reverts to a onedimensional field at this end of the nozzle section, because the forcing frequency remains well below the transverse mode cutoff frequency for all frequencies considered in this study. Once the acoustic field is radially uniform in the nozzle, the relationship between the axial velocity and pressure in the nozzle is uniquely related to the nozzle impedance through the impedance translation theorem [32]:

8 International journal of spray and combustion dynamics Volume. 4 Number Normalized magnitude In phase pressure/ out of phase velocity Out of phase pressure/ in phase velocity Traveling wave pressure/ traveling wave velocity Figure 6:.4.2 x/λ.2.4 Acoustic pressure and velocity fields for three different forcing cases, solved in the absence of the side branch. Zo ik( z+ h) Zo e ( ) e Z c c tr () z ρ + + ik ( z + h ) ρ = () ρc Z o ik ( z + h ) Z + e o e ik z c + c + ( + h) ( ) ρ ρ We chose Z values such that Z tr (z = ) approximates pressure release, anechoic, and rigid boundary conditions by using Z tr /ρc values of.,, and, respectively. For this reason, a frequency dependant Z value is applied in order to maintain a fixed Z tr. The impedance translation relates the impedance at any axial location in a plane wave field 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 required at the inside and outside corners of the annulus to insure that the simulated results are grid-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 radius should be on the order of the boundary layer thickness (that is, in turn, a function of the Reynolds number and swirl number). The velocity field in the vicinity of the corner is then a function of R. To illustrate, Figure 7 shows the dependence of the axial velocity at location D indicated in Figure 5 upon fillet radius. For reference, a R line is also indicated in the figure, representing the theoretical result for a twodimensional 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 uz = π r r π ro + R 2 2 ( ) o i ri R u ( r, θ) rdrdθ z (2)

9 282 Transverse to longitudinal acoustic coupling processes in annular combustion chambers u z ρc u /p max D/ λ =.4 D/ λ =.9 D/ λ =.5 /R /2 u z 2 R/D Figure 7: Dependence of axial velocity at point D in Figure 5 (left) and spatially averaged over surface E in Figure 5 (right) upon fillet radius at three frequencies, for in-phase forcing with Z tr /ρc =. The vertical dashed line indicates the fillet radius used for the results presented in this paper. This result shows that while the local velocity exhibits a significant dependence on R near the fillet, the spatially averaged axial velocity results are almost independent of fillet radius. It is important to point out the steps taken to define the normalizing area used in equation (2). In the fillet region, the volume flow rate remains virtually constant as z varies from z = to R. However, the averaging surface area does not remain constant. For this reason, the averaged axial velocity does vary with z, and similarly varies at z = with fillet radius. This effect is nothing more than a manifestation of the dependence of nozzle outlet area at z = on R. In order to account for this geometry effect, we rescale the average axial velocity by the area ratio at z = and z = R, to arrive at the formula shown in equation (2). All results shown in this paper use a radius of.2d, which is indicated by the vertical dashed line in Figure 7. The mesh is comprised of 374,75 free tetrahedral mesh elements and 529,592 degrees of freedom, and employs minimum interior boundaries. The fillet region requires particular care in meshing. The mesh density in this region is quite high and then smoothly transitioned to the required density needed to simulate the rest of the system. This configuration allows the mesh elements to grow freely from the small radius to the large combustor box with maximum efficiency and accuracy. This final mesh was settled on after a grid independence study using four meshes of increasing resolution showed variations of less than.%. 3. RESULTS AND DISCUSSION This section presents typical results. Results were obtained for a range of frequencies between 2 3 Hz, corresponding to non-dimensional D λ values ranging from.2 to.3. These frequencies were simulated for the three forcing configurations and using the three upstream nozzle impedance values described in the previous section.

10 International journal of spray and combustion dynamics Volume. 4 Number Figure 8 illustrates the coordinate system and various cuts used to represent the threedimensional disturbance field. Figure 9 presents representative instantaneous pressure contours and velocity vector fields 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, generates zero transverse velocity at the nozzle outlet, but large pressure fluctuations are present x-z x-y y y z x x x y Figure 8: Schematics showing coordinate system used to define various cuts at which data is plotted z/d.4 z/d x/d x/d z/d x/d.8 Figure 9: Instantaneous disturbance fields at D/λ.4 (4 Hz) with an anechoic boundary condition at the nozzle for in-phase (left), out-of-phase (right), and traveling wave (bottom) scenarios. Colors represent instantaneous pressure, while arrows denote the instantaneous total velocity field.

11 284 Transverse to longitudinal acoustic coupling processes in annular combustion chambers 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, the nozzle response for this in-phase forcing case can be understood from quasi onedimensional concepts. In contrast, the out-of-phase case exhibits large transverse velocity fluctuations in the center of the combustor. Because of the centerline pressure node, the pressure fluctuations have a 8 degree phase difference on the two sides of the annulus. Similarly, the axial velocity fluctuations are phased 8 degrees apart on the left and right sides of the annulus. This nozzle response is intrinsically three-dimensional. The traveling wave case shows an intermediate behavior. For the illustrated case, the wavelength is long relative to the nozzle, so the disturbance field is nearly uniform across the nozzle. However, a slight phase difference in axial velocity fluctuations exists on the two sides of the annulus, evident in Figure 9. In addition, there is a slight amount of 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 an incident wave by a side branch. For example, a quasi one-dimensional analysis, using continuity of volume flow rate and pressure at the interface, leads to the following predicted result for the reflection coefficient, R, from the side branch [33, 34]: R = 2( S / S )( Z / c) + b b ρ (3) where S b and Z b 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 not exactly correct inside an annular combustor with a spinning mode. With multiple nozzles, there will always be reflected waves between two nozzles, so the traveling wave will 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 the annulus, the traveling wave approximation is a good one. The azimuthal distributions of axial velocity magnitude and phase at the nozzle interface are shown in Figure, along with the accompanying pressure distributions along the bottom surface of the combustor shown in Figure. From this view, we can clearly see how the pressure anti-node at the center of the combustor excites symmetrically distributed axial velocity in the nozzle region. The phase reversal in axial velocity 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 axial response on the right side of the nozzle. The axial distribution of pressure and axial velocity along the centerline of the annulus are plotted in Figure 2 for the in-phase and out-of-phase cases. Notice that, for the out-of-phase forcing case, the pressure and velocity disturbance fields decay to zero in the nozzle region. Thus, although the axial velocity fluctuations at the nozzle outlet are non-negligible, they are of opposite phases on the opposite sides of the annulus and cancel each other, leading to a progressive decay in disturbance amplitude and fluctuation energy in the nozzle. For this reason, the nozzle impedance has no influence

12 International journal of spray and combustion dynamics Volume. 4 Number x x x Figure : Instantaneous pressure along the x-y surface at D/λ.28 (3 Hz) for in-phase (top), out-of-phase (middle), and traveling wave (bottom) cases for anechoic nozzle impedance. on the disturbance field characteristics in the out-of-phase case, as discussed further in the next section. As alluded to in the above discussion, important insights into the character of the axial velocity at the nozzle exit can be gained from the pressure field. As such, we next discuss the characteristics of the pressure field in the combustor-nozzle interface in more detail. Figure 3 presents plots of the magnitude of the pressure along the cut lines for a Ztr ρ c = case. For reference, the solid line denotes the value of the disturbance field along the x-x cut line that would exist in the absence of the nozzle. Note that for the anechoic and rigid nozzle impedance conditions, the nozzle causes only a slight distortion of the disturbance field.

13 286 Transverse to longitudinal acoustic coupling processes in annular combustion chambers Figure : Spatial distribution of axial velocity magnitude at the nozzle-combustor junction at D/λ.28 (3 Hz) for in-phase (top), out-of-phase (middle), and traveling wave (bottom) scenarios for anechoic nozzle impedance. Color represents instantaneous axial velocity. In contrast, the nozzle significantly distorts the pressure field from the -D result when Ztr ρ c =. for the in-phase and traveling wave cases, as shown in Figure 4. Little distortion occurs for the out-of-phase case where the nominal pressure field is zero. When the combustor acoustic field exhibits a significantly non-zero pressure near the nozzle note how the nozzle pulls the pressure amplitude toward zero. Having discussed the pressure, we next consider the velocity field at the nozzle exit for the Ztr ρ c = case. Figure 5 plots the in-phase result, along with the -D result on the x-x cut line. Similar to the nominal, one-dimensional result, the transverse velocity is low everywhere except near the annulus corners. Here, the sharp area change leads to a strong transverse velocity field. Note that the magnitudes of these velocity values near the corners are functions of the fillet radius, R. The axial velocity field is nearly uniform at the nozzle exit, reflecting a nearly plug flow disturbance field, except for overshoots near the corners, which are again functions of the fillet radius.

14 International journal of spray and combustion dynamics Volume. 4 Number u z Normalized magnitude.5 In-phase Out-of-phase p u z p Figure 2: z/h Axial distribution of pressure and axial velocity along the annulus centerline, normalized by their value at the nozzle outlet (D/λ.4 or 4 Hz, Z tr /ρc = ). Figure 3: Pressure distributions at D/λ.4 (4 Hz) with an anechoic nozzle for the in-phase, out-of-phase, and rightward traveling wave scenarios. Cut lines x-x and y-y are shown in Figure 8.

15 288 Transverse to longitudinal acoustic coupling processes in annular combustion chambers Figure 4: Pressure distribution at D/λ.4 (4 Hz) a pressure release condition at the nozzle for the in-phase, out-of-phase, and rightward traveling wave scenarios. Cut lines x-x and y-y are shown in Figure 8. Axial velocity magnitudes for the out-of-phase and traveling wave forcing cases are shown in Figure 6. Note that the largest axial velocities are observed near the outer edge of the annulus for the out-of-phase forcing case. As discussed previously, the values on the left and right sides are 8 degrees out-of-phase. The traveling wave case results are asymmetric on the x-axis, also discussed previously. Along the y-y cut, the traveling wave displays the same shape and similar magnitude as those two sides of the nozzle experience the same disturbance field. As mentioned in the introduction, experiments have shown that completely different shear layer modes are excited during in-phase and out-of-phase forcing [2]. The resultant fluctuations in flame position are indicated in Figure 7, showing the symmetric and asymmetric wrinkling of the flame as a result. The staggered flame wrinkling in the asymmetric forcing case, as seen in Figure 7 on the left, results from

16 International journal of spray and combustion dynamics Volume. 4 Number Figure 5: In-phase transverse (top) and axial (bottom) velocities at D/λ.4 (4 Hz) for an anechoic nozzle. 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 at this condition. Conversely, the symmetric flame wrinkling, shown in Figure 7 on the right, stems from the rollup of axisymmetric vortex rings. This rollup is a result of the symmetric bulk forcing of the in-phase acoustic mode shape. While computing this behavior requires a viscous flow calculation that captures the excitation and convection of vortical disturbances, the basic fundamentals leading to

17 29 Transverse to longitudinal acoustic coupling processes in annular combustion chambers Figure 6: Axial velocity at D/λ.4 (4 Hz) with an anechoic nozzle for the outof-phase (top) and traveling wave (bottom) scenarios. 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 convectively unstable shear layers. The vortical field features are a simple manifestation of the fact that 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 layer modes in the two cases.

18 International journal of spray and combustion dynamics Volume. 4 Number Figure 7: Experimental flame luminescence images of 4 Hz out-of-phase (left) and 4 Hz in-phase (right) forcing of a flame in a swirling flow at u o = m/s, S =.5, and an equivalence ratio of.9. Arrows point to the wrinkles resulting from a) helical and b) ring vortices. 4. FURTHER ANALYSIS OF THE AXIAL VELOCITY This section expands the analysis of the axial velocity which, as discussed in the introduction, has been proposed as a particularly significant feature influencing how flames are excited during transverse instabilities. This section will further emphasize the role of nozzle impedance on these characteristics. As discussed previously, the acoustic field in the nozzle quickly reverts to a one-dimensional field because the frequency is below the duct cut-off frequency. Once one-dimensional, the axial velocity and pressure in the nozzle are directly related by the translated nozzle impedance, Z tr, given by equation (). Thus, it is useful to define the impedance ratio, R Z, through which to compare simulated results to quasi one-dimensional results, as R Z = p uz Z tr (4) where the spatially averaged pressure field is given by 2 p = 2 2 π (( r + R) ( r R) ) π ro + R o i ri R p ( r, θ) rdrdθ (5) While this expression and equation (2) describes the pressure and velocity disturbance evaluated over one half of the nozzle, slightly different forms were used for different cases. It was shown in Figure 6 that the in-phase case leads to symmetric results on the two halves of the nozzle, and the out-of-phase case leads to antisymmetric results. As such, integrating the pressure or velocity over the entire annulus area for the out-of-phase case leads to zero, because of cancellation of results on the two

19 292 Transverse to longitudinal acoustic coupling processes in annular combustion chambers 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 8 and Figure 9 plot the calculated dependence of R Z upon the dimensionless frequency for both the in-phase and traveling wave cases, respectively. Note how the magnitude of R Z is quite close to unity. In both of these cases, the unsteady pressure field has nearly uniform phase across the entire face of the annulus, so it is expected that multi-dimensional effects in the nozzle are quite small. The growing, but slight, deviation of R Z magnitude from unity with increasing frequency is a manifestation of the Figure 8: Impedance ratio magnitude (top) and phase (bottom) for various nozzle boundary conditions at the nozzle for the in-phase forcing case.

20 International journal of spray and combustion dynamics Volume. 4 Number Figure 9: Impedance ratio magnitude (top) and phase (bottom) for various nozzle boundary conditions at the nozzle for the traveling wave case. increasing phase and magnitude variation in unsteady pressure in the traveling and inphase cases, respectively. Although not shown, R Z deviates substantially from unity for the out-of-phase forcing case. Recall that for this case, there exists a 8 degree phase change in axial velocity and pressure on opposite sides of the annulus centreline. If the two halves of the nozzle were physically separated by a rigid barrier, then R Z could be a useful quantity. In actuality, the phase cancellation causes a vanishing of the acoustic field in the nozzle,

21 294 Transverse to longitudinal acoustic coupling processes in annular combustion chambers as shown in Figure 2. Because the disturbance field is essentially zero at z = h, this renders the z = results effectively independent of the nozzle impedance. As such, R Z is not a useful quantity for characterizing the axial velocity response. These points can be seen from Figure 2, which plots the ratio of the spatially averaged pressure and axial velocity over one half of the nozzle face as a function of frequency. Note that all three nozzle impedance values give the same pressure-velocity relationship at the nozzle exit. Figure 2: Averaged nozzle impedance magnitude (top) and phase (bottom) for outof-phase forcing. Nozzle impedance values of Z tr /ρc =.,, and are plotted.

22 International journal of spray and combustion dynamics Volume. 4 Number These results show that the nozzle response exhibits a strong sensitivity to its upstream impedance in certain cases and is totally independent of it in others. For example, consider azimuthal standing modes in an annular combustor. Because of the spatially varying pressure/velocity field, different nozzles will be located in different parts of the standing wave and exhibit different response characteristics and nozzle impedance sensitivities. For the first radial mode in an annular combustor, nozzles located along the radial centerline, such as shown in Figure, will always be located at a velocity antinode/pressure node and, as such, exhibit no sensitivity to nozzle impedance. 5. CONCLUDING REMARKS This paper has described an analysis of the coupling between transverse acoustic motions and the induced axial motions in a small area side channel, simulating the complex acoustic field generated by a fuel/air mixing nozzle inside an annular combustion chamber. These results illustrate a critical dependence of the near-field acoustics on macro features of the acoustic 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 axial coupling for in-phase and traveling wave acoustic excitation. The bulk features of the pressure field at the nozzle exit can be understood from -D acoustic considerations for several of the cases, due to the small cross sectional area of the side branch relative to that of the main chamber. An important exception to this occurs when the nozzle is nominally located in a pressure anti-node and the nozzle impedance attempts to force that same location into a pressure node, as was seen for an in-phase wave with a pressure release nozzle. Similarly, we show that the spatially averaged pressure to axial velocity relationship is quite close to the one-dimensional, translated impedance value at the end of the side branch. The notable exception to this result occurs in the out-of-phase forcing case, whose axial velocity characteristics are independent of the nozzle impedance. ACKNOWLEDGMENTS This work has been partially supported by the US Department of Energy under contract DEFG26-7NT4369, contract monitor Mark Freeman. REFERENCES [] Lieuwen, T.C. and V. Yang, Combustion Instabilities in Gas Turbine Engines, Operational Experience, Fundamental Mechanisms, and Modeling. Progress in Astronautics and Aeronautics, ed. T.C. Lieuwen and V. Yang 25. [2] Rayleigh, B.J.W.S., The Theory of Sound. Vol : Macmillan. [3] Kedia, K., S. Nagaraja, and R. Sujith, Impact of Linear Coupling on Thermoacoustic Instabilities. Combustion Science and Technology, 28. 8(9): p [4] Ducruix, S., T. Schuller, D. Durox, and S. Candel, Combustion Dynamics and Instabilities: Elementary Coupling and Driving Mechanisms. Journal of Propulsion and Power, 23. 9(5): p

23 296 Transverse to longitudinal acoustic coupling processes in annular combustion chambers [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, (6): p [6] Palies, P., D. Durox, T. Schuller, and S. Candel, The Combined Dynamics of Swirler and Turbulent Premixed Swirling Flames. Combustion and Flame, 2. 57(9): p [7] Hirsch, C., D. Fanaca, P. Reddy, W. Polifke, and T. Sattelmayer, Influence of the Swirler Design on the Flame Transfer Function of Premixed Flames. Volume 2 Turbo Expo 25, 25: p [8] Paschereit, C.O., E. Gutmark, and W. Weisenstein, Excitation of Thermoacoustic Instabilities by Interaction of Acoustics and Unstable Swirling Flow. AIAA journal, 2. 38(6): p [9] Culick, F. and V. Yang, Prediction of the Stability of Unsteady Motions in Solid- Propellant Rocket Motors [] Harrje, D.T. and F.H. Reardon, Liquid Propellant Rocket Combustion Instability 972: Scientific and Technical Information Office, National Aeronautics and Space Administration. [] Price, E.W., Solid Rocket Combustion Instibility - An American Historical Account, in Nonsteady Burning and Combustion Stability of Solid Propellants 992. p. 6. [2] 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 28: Orlando, FL. [3] Rogers, D.E. and F.E. Marble, A Mechanism for High-Frequency Oscillation in Ramjet Combustors and Afterburners. Jet Propulsion, (6): p [4] Kaskan, W.E. and A.E. Noreen, High-Frequency Oscillations of a Flame Held by a Bluff Body. ASME Transactions, (6): p [5] Elias, I., Acoustical Resonances Produced by Combustion of a Fuel-Air Mixture in a Rectangular Duct. Journal of the Acoustical Society of America, (3): p [6] Smith, K., L. Angello, and F. Kurzynske, Design and Testing of an Ultra-Low NO/Sub x/gas Turbine Combustor, 986, Solar Turbines Inc., San Diego, CA. [7] Krebs, W., S. Bethke, J. Lepers, P. Flohr, and B. Prade, Thermoacoustic Design Tools and Passive Control: Siemens Power Generation Approaches, in Combustion Instabilities in Gas Turbine Engines, T.C. Lieuwen and V. Yang, Editors. 25, AIAA: Washington D.C. p [8] Dowling, A.P. and S.R. Stow, Acoustic Analysis of Gas Turbine Combustors. Journal of Propulsion and Power, 23. 9(5): p [9] Sewell, J. and P. Sobieski, Monitoring of Combustion Instabilities: Calpine s Experience, in Combustion Instabilities in Gas Turbine Engines, T.C. Lieuwen and V. Yang, Editors. 25, AIAA: Washington D.C. p

24 International journal of spray and combustion dynamics Volume. 4 Number [2] O Connor, J. and T. Lieuwen, Further Characterization of the Disturbance Field in a Transversely Excited Swirl-Stabilized Flame. Journal of Engineering for Gas Turbines and Power - Transactions of the ASME, (). [2] Cohen, J., G. Hagen, A. Banaszuk, S. Becz, and P. Mehta, Attenuation Of Combustor Pressure Oscillations Using Symmetry Breaking, in 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition 2: Orlando, Florida. [22] Hauser, M., M. Lorenz, and T. Sattelmayer. Influence of Transversal Acoustic Excitation of the Burner Approach Flow on the Flame Structure. in ASME Turbo Expo. 2. Glasgow, Scotland. [23] O Connor, J. and T. Lieuwen, Disturbance Field Characteristics of a Transversely Excited Burner. Combustion Science and Technology, 2. 83(5): p [24] Stow, S.R. and A.P. Dowling. Low-Order Modelling of Thermoacoustic Limit Cycles. 24. [25] Acharya, V., Shreekrishna, D.H. Shin, and T. Lieuwen, Swirl Effects on Harmonically Excited, Premixed Flame Kinematics. Combustion and Flame, (3): p [26] Worth, N.A. and J.R. Dawson, Cinematographic OH-PLIF Measurements of Two Interacting Turbulent Premixed Flames with and without Acoustic Forcing. Combustion and Flame, 2. [27] Staffelbach, G., L.Y.M. Gicquel, G. Boudier, and T. Poinsot, Large Eddy Simulation of Self Excited Azimuthal Modes in Annular Combustors. Proceedings of the Combustion Institute, : p [28] Wolf, P., G. Staffelbach, A. Roux, L. Gicquel, T. Poinsot, and V. Moureau, Massively Parallel LES of Azimuthal Thermo-Acoustic Instabilities in Annular Gas Turbines. Comptes Rendus Mecanique, (6-7): p [29] Hutt, J.J. and M. Rocker, High-Frequency Injection-Coupled Combustion Instability, in Liquid Rocket Engine Combustion Instability, V. Yang and W.E. Anderson, Editors. 995, American Institute of Aeronautics and Astronautics. p [3] Davis, D., B. Chehroudi, D. Talley, R. Engineering, and C.A. Consulting Inc Edwards Afb, The Effects of Pressure and an Acoustic Field on a Cryogenic Coaxial Jet, in 42nd Aerospace Sciences Meeting and Exhibit 24: Reno, NV. [3] O Connor, J., J. Mannino, C.Vanatta, and T. Lieuwen, Mechanisms for Flame Response in a Transversely Forced Flame, in 7th US National Technical Meeting of the Combustion Institute 2: Atlanta, GA. [32] Pierce, A.D., Acoustics: An Introduction to its Physical Principles and Applications 989: Acoustical Society of America. [33] Lighthill, J., Waves in Fluids 2: Cambridge Univ Pr. [34] Rienstra, S.W. and A. Hirschberg, An Introduction to Acoustics. Eindhoven University of Technology, 23.

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