Effect of azimuthal flow fluctuations on flow and flame dynamics of axisymmetric swirling flames Free

Thermoacoustic instabilities are a significant problem for premixed combustion systems.^1–7 These instabilities occur when unsteady heat release couples with one or more of the acoustic modes in the combustor, potentially causing high amplitude pressure and velocity oscillations.⁸ These oscillations then, in turn, lead to component fatigue and failure that eventually reduce combustor operability and increases overall operating costs. The focus of this paper is on combustors with swirl-stabilized flames. Swirling flows are subject to shear and centrifugal flow instability mechanisms, leading to a variety of unsteady flow features, such as the precessing vortex core and helical shear layer disturbances.^9–13 

The dominant mechanisms leading to heat release oscillations in low Mach number premixed flames are fuel/air ratio fluctuations and flow velocity fluctuations.^14–19 The latter mechanism is the focus of this study, whereby flow disturbances lead to flame surface wrinkling and surface area oscillations. The flow oscillations are comprised of both acoustic and vortical disturbances.²⁰ The direct excitation of the flame by these flow disturbances has previously been treated in detail by both experimental and modeling studies.^21–25 Acoustic waves are directly excited by the flame and reverberate in the combustor system. Vortical disturbances are generated by modulation of the separating shear layer, which organize themselves into concentrated regions of vorticity through the Kelvin-Helmholtz instability—the axisymmetric shear layer modes are initially composed only of azimuthal vorticity, while helical modes also initially contain axial vorticity. Several studies have also shown how these shear layer disturbances are subject to secondary instabilities, leading to axial vorticity and streamwise flow structures.^26,27

In addition, there exists an “indirect” mechanism for exciting axial vorticity that is unique to swirl flows. In this mechanism, acoustic waves propagating through swirlers excite axial vortical disturbances, leading to modulations in swirl number,^28–32 or “swirl fluctuations.” The effects of the different vortical disturbances are illustrated in Figure 1. The focus of this paper is to understand this indirect mechanism and its significance relative to shear generated vorticity.

The first study we are aware of that indirectly suggested the importance of this mechanism was performed by Straub and Richards,³² who noted the importance of swirler vane position on measured combustion instability limits. This mechanism was explicitly noted in computational studies by the simulations of Wang et al.^19,33,34 Hirsch et al.³⁵ similarly reported experiments showing the effect of swirler vane location on the flame transfer function (FTF). They also showed that the flame shape was unaffected by these swirl vane location changes, an important observation, as time averaged changes in flame shape (such as induced by changes in swirl) would also lead to changes in flame response and stability limits. They argued that axial acoustic flow fluctuations excited an azimuthal flow disturbance which is non-acoustic and therefore convected by the flow. They suggested that this azimuthal flow disturbance induces an axial velocity fluctuation, which then causes modulation of the heat release.

This basic idea was made further rigorous by Palies et al.,³⁶ using results developed by Cumpsty and Marble³⁷ for pulsating flow over an airfoil, showing how acoustic flow fluctuations lead to axial vorticity fluctuations. They developed explicit equations relating the strength of the excited axial vorticity, as well as reflected and transmitted acoustic wave as a function of swirler and incident acoustic wave parameters. They also reported experiments in both axial and radial swirlers³⁸ showing that the mode conversion process from the acoustic disturbances to the convective azimuthal flow disturbances is similar for both swirlers and so is their impact on the flame response characteristics. The effect of swirler geometry was also investigated by Bourgouin et al.,³⁹ drawing similar conclusions. Experiments by Durox et al.⁴⁰ used a radial swirler with variable vane angle. They showed that dynamic variation of the blade angle can be used to control the flame dynamics in the combustor through swirl fluctuations, similar to earlier computations by Stone and Menon.⁴¹ 

Several computational studies of this phenomenon have also been reported. Garcia-Villalba et al.³¹ investigated the effect of swirl fluctuations in non-reacting flows for a model combustor using LES (Large Eddy Simulations). They excited different velocity components at the inlet of the combustion chamber (post-swirler) and examined the differences in flow-field in the combustion chamber. They showed that the instantaneous structures were mainly influenced by the azimuthal velocity oscillations at the inlet. Work by Komarek and Polifke³⁰ considered both experiments and Unsteady Reynolds Averaged Navier-Stokes (URANS) simulations of a reacting swirling flow in a model combustor. They investigated the effect of swirl number fluctuations for different upstream axial positions of their axial swirl generator. They showed that the axial position of the swirler influenced the time delay between the acoustic disturbance at the swirler, and the heat release response, similar to the earlier observations of Straub and Richards.³² 

To summarize, it is clear that axial vortical disturbances stemming from the swirler, and not only the familiar vortical disturbances associated with vortex roll-up of the shear layers, have significant influences on the flame response. However, there is a fundamental difference between axial and azimuthal vorticity disturbances in terms of the flow oscillations they induce on the flame. In order to better understand this point, consider the disturbance pathways shown in Figure 2, which show how an acoustic disturbance leads to heat release oscillations. The familiar azimuthal vortical disturbances due to vortex roll-up of the shear layers, excited by acoustic disturbances, are shown in pathway (2a). Azimuthal vorticity fluctuations necessarily induce axial and radial flow fluctuations as shown in (2b). The presence of the swirler causes axial vorticity fluctuations (path 1a). Axial vorticity fluctuations that are not centered on the cylindrical axis will induce both azimuthal and radial flow fluctuations. However, an axisymmetric distribution of axial vorticity will in effect only induce azimuthal flow fluctuations (path 1b). This latter configuration is the focus of this paper. These azimuthal flow fluctuations indirectly excite axial and radial velocity fluctuations, due to bending/rotation of the axial vortex tube (1c). For example, the oscillatory azimuthal flow in the injector nozzle induces an oscillatory radial flow component at the rapid expansion point where the swirling nozzle flow enters the combustor—this vortex bending similarly induces axial and radial flow disturbances.

The differentiation between azimuthal flow disturbances, on one hand, and radial/axial flow disturbances, on the other hand, is significant because the flame itself is disturbed only by the velocity component normal to it (path 3a or 3c). This can be seen from the following expression, derived from the linearized G-equation for a flame whose position is a single valued function of the (r, θ) coordinate:⁴² 

Here, ξ′ is the fluctuation in the local flame position, $Ū→t$ is the mean velocity vector tangential to the mean flame surface, and $un′$ is the local normal component of the flow fluctuations. An important conclusion from this expression is that axisymmetric flames are not directly wrinkled by azimuthal flow fluctuations, since such flow disturbances have no component normal to the flame. Thus, it is the indirect azimuthal to radial/axial mechanism (1c) that controls the strength of the flow oscillations normal to the flame that, in turn, lead to heat release oscillations in axisymmetric flames, as indicated by pathways (1a-1b-1c-3a), which is indicated by the black dashed box in the figure under the heading of “I.” If the flame is non-axisymmetric, then azimuthal flow disturbances do directly induce wrinkles on the flame, as indicated by path (3c).

Our focus in this paper is on the former route to heat release oscillations that is indicated by the dashed black box in the figure—i.e., the generation of flow disturbances that are normal to the flame, which originate from axial vorticity, which is bent so as to induce radial and axial velocity disturbances. This coupling process is not easily amenable to analytical calculations and, as such, we report here a computational study of the role of these different flow fluctuations on the flame response in an axisymmetric framework. In addition, we present computations of a forced model combustor to understand the role played by axial vorticity fluctuations relative to azimuthal ones, in affecting the global flame response. In other words, we compare the relative strengths of the cumulative path I and path II indicated in the figure. This global flame response is quantified by means of the FTF^{28,30,43–46} and defined as

Here, $Q′ˆ(f)/Q0$ is the normalized heat release rate fluctuations (prime notation) in the frequency domain (superscript hat) and $uˆref′(f)/U0$ is the normalized reference velocity fluctuations chosen to be those at the exit plane of the swirler. A key part of this work is to understand how vorticity excited at the swirlers is reoriented in the inflow passage and at the rapid expansion into the combustor, and its significance relative to shear generated, azimuthal vorticity.

This paper is organized as follows. First, we present the model geometry being simulated along with the different models and parameters used for the simulation framework. This is followed by the discussion of the flow field dynamics in the annulus and combustion zone. The flow field dynamics discussion is then complimented by the discussion of vorticity evolution in the post swirler region of the combustor. Finally, the FTFs calculated from each simulation are compared for variations in input control parameters to illustrate the differences in global flame response due to azimuthal flow fluctuations.

The model combustor configuration used in this work is shown in Figure 3. It consists of an annular flow passage with a 45°, 8 vane swirler, connected to a larger combustor. The inner and outer annular dimensions are given by D_I and D, respectively. The dimensions of this geometry are detailed in Table I.

Two different types of calculations were performed in order to focus on different aspects of the problem. The first calculation approach is referred to as the “swirler-annulus calculation.” The computed domain consists of the annulus section upstream of the combustor dump. This geometry is used to capture the fully three-dimensional interactions of the upstream-incident acoustic disturbance and the swirler passages, including the acoustic-vortical coupling described by Palies et al.³⁶ The swirler annulus simulations are non-reacting and performed for an inlet axial velocity of 30 m/s and a Reynolds number Re = U₀D/ν, of 87 000. The swirler-annulus geometry is solved for a single blade passage, with rotational periodic boundary conditions. It is meshed into 1 × 10⁶ body-fitted hexagonal volume elements. The annular walls and the swirler vane surfaces have a zero velocity wall boundary condition and the exit of the swirler-annulus is a convective outflow boundary condition.

The second calculation approach is referred to as the “post-swirler calculation.” The calculation domain starts at the exit of the swirler passage and goes to the combustor exit. These are axisymmetric calculations. This geometry shown in Figure 4 is meshed into 0.5 × 10⁶ quad elements. Conditions for which the post-swirler simulations were performed are detailed in Table II. Here, φ denotes the equivalence ratio, T_u denotes the unburnt gas temperature, and the mean velocities (subscript 0) for the axial and swirl component are indicated. Note that in the swirler-annulus simulations, the swirl number and mean swirl velocity are controlled by the vane angle and vane geometry. However, for the post-swirler simulations, the swirl number and mean azimuthal inlet velocity are freely chosen.

These “post-swirler calculations” were reacting flow calculations performed in order to decompose the relative contributions of axial and azimuthal flow disturbances on the flame response (note that these flow components are coupled at the swirler exit plane in the “swirler-annulus calculations” and controlled by the swirler geometry). Thus, these results enable some more generic insights into the problem outside of the specific swirler geometry chosen. Separate calculations were performed for the following three different inlet forcing configurations, similar to Garcia-Villalba et al.:³¹ 

(a) axial flow forcing only

(b) azimuthal flow forcing only (c) both axial and azimuthal flow components forced Clearly, configuration (c) is the most representative for a real swirler, but the other two simulations allow insight into the isolated roles of disturbances in only azimuthal or axial flow forcing. Here, φ is the relative phase difference between the axial and azimuthal flow components at the inlet (from the exit of the swirler).

Apart from the inlet forcing and wall boundary conditions, the exit of the combustor is modeled with a pressure outlet boundary condition. The steady state solution for each case is used as the initial condition for the respective forced simulations. The axial flow forcing amplitude was kept fixed at 10% of the bulk axial flow velocity, ε_z = 0.1. The amplitude of forcing for the azimuthal flow was chosen based on the mean inlet azimuthal velocity and swirl number. The forcing frequency, f, is varied from 150 Hz to 1000 Hz. This corresponds to 0.27 < St < 1.85, where St = fD/U₀ is the Strouhal number.

Having described the two different sets of calculations, we next describe the computational approach which is common to both calculations. Similar to Komarek and Polifke,³⁰ an URANS approach is used, implemented with the C+ + toolbox OpenFOAM (Open-Field-Operations-And-Manipulations). This toolbox is an open-source collection of finite volume solvers and numerical methods tailored for Computational Fluid Dynamics simulations.⁴⁷ The non-reacting steady state flow fields are computed using the simpleFoam solver which uses the (Semi Implicit Pressure Linking of Equations) SIMPLE⁴⁸ pressure-coupling method in an incompressible framework. The forced unsteady simulations are performed using the pisoFoam solver which uses the (Pressure Implicit Splitting of Operators) PISO method for pressure-velocity coupling. For the flow forcing at the inlet, the “inlet velocity modulation” (IVM) technique is adopted, as outlined by Kaufmann et al.⁴⁹ A SSG (Speziale-Sarkar-Gatski) closure model⁵⁰ is used for turbulence closure, following the recommendations of Shamami and Birouk⁵¹ based on their assessment of different URANS models for swirling flows in can-combustors. The flame position is modeled using the G-equation^41,52 with Zimont and Lipatnikov’s turbulent flame speed closure.⁵³ The effect of the heat release/gas expansion induced by the flame on the flow field is applied using the ghost fluid method.^54,55 In this method, a small band of ghost cells are defined around the flame location which corresponds to the G = 0 iso-contour. The temperature increases from the unburned to burned value across these ghost cells in a smooth fashion using the interface preserving methods developed by Moureau et al.⁵⁶ and Nguyen et al.⁵⁷ The flame is anchored at the intersection of the combustor dump plane and the annulus, with the boundaries along the dump plane set to the Dirichlet boundary condition of G = 0 for this purpose. The remaining boundaries use a zero gradient Neumann boundary condition. The physics of these are handled using a user modified version of the XiFoam module in OpenFOAM. The original XiFoam module makes use of a progress variable approach with a turbulent Schmidt number-based diffusion of the progress variable. This was modified to accommodate the G-equation-based front tracking, without any diffusion, in a finite volume framework with re-initialization. The module was further modified to use the in-built 2D Cartesian solver and add the required terms and momentum equation for the azimuthal flow component. The spatial discretization is performed using a second order scheme. A second order backward Euler scheme was used with a time step that is 1/100th the time-period of acoustic forcing. The simulations are performed for 10 acoustic time periods. After the initial transient, the “steady-oscillating” state is reached and the time-signal from this set is used for further analysis.

This section describes the “swirler-annulus” simulations that were performed to understand the generation of azimuthal flow fluctuations at the swirler and its subsequent evolution in the annular mixing section. Typical streamlines from an unforced calculation are shown in Figure 5. The streamlines are colored by the azimuthal component of velocity. This figure clearly shows the change in flow direction across the swirler.

The downstream evolution of the mean axial and tangential flow components is shown in Figure 6. Note that the flow is uniform upstream of the swirler and accelerates as the flow traverses the swirler vanes. This is due to the volume constriction created by the swirler vanes within the annulus cross section. Since the swirler has 8 vanes, the immediate downstream region of the swirler is spatially periodic. However, this profile becomes uniform downstream, as indicated in the figure.

Next, consider the flow dynamics in the swirler annulus section when the axial flow at the inlet is forced. The axial forcing was performed with amplitude of 10% at frequencies of 250, 300, 350, and 400 Hz. The time-series of the axial and azimuthal flow at different locations downstream of the swirler is shown in Figure 7 for the 250 Hz forcing case. Note that there is negligible phase difference at the different locations for the axial component, as shown in Figure 7(a). This is a manifestation of incompressible simulations where the speed of the wave is infinite. However, notice the axial dependence of the azimuthal flow phase, as shown in Figure 7(b). The axial phase variation at the annulus center, r = 0.4D, is plotted in Figure 7(c). Note that the phase decreases almost linearly with axial coordinate, indicative of a nearly constant convection speed. This convection speed can be extracted from the slope of this plot, and equals about 40 m/s which is the same value as the local axial velocity at this radial position. This plots clearly indicates how the swirler converts the upstream axial disturbance into an azimuthal, convecting (vortical) flow disturbance, as previously described by Hirsch et al.,³⁵ Komarek and Polifke,³⁰ and Palies et al.^28,36 This is likely the reason why Straub and Richards³² noted the sensitivity of combustion instability limits to axial swirl vane location.

The differences in wave propagation speed of the axial and azimuthal components also imply that their relative phase difference evolves axially. In fact, these components of the flow fluctuations are not in phase at the exit of the swirler. This phase difference, defined as $ϕ=∠uˆx′−∠uˆθ′$⁠, at the swirler exit is a function of the forcing frequency, as shown in Figure 8. The relative phase exhibits a monotonically increasing trend (roughly linear) with an increase in forcing frequency. This relative phasing is incorporated in the parameter ϕ shown in Eq. (5) and is a very important parameter for the FTF, which quantifies the gain/phase response to an incident axial flow disturbance. Section II B considers the evolution of the flow field in the region downstream of the swirler.

Having demonstrated how the swirler converts axial flow disturbances into azimuthal flow disturbances, this section further considers the evolution of these disturbances in the annular flow passage and combustor. The generation of radial and axial flow fluctuations due to azimuthal flow fluctuations at the inlet is best understood by considering simulations that use the forcing conditions described in Eq. (4) that enable us to decouple the relative significance of axial and azimuthal flow disturbances. The initial condition used for these simulations is the steady-state solution shown in Figure 9.

Consider the forcing configuration (b), consisting of forcing of only the azimuthal flow velocity (i.e., zero axial velocity fluctuations at the inflow/swirler exit plane). Figure 10 shows how these purely azimuthal disturbances evolve axially, leading to generation of axial and radial flow fluctuations in the annulus. The plots show the spatial evolution of the disturbances at different instances in an acoustic cycle. Note that since only the azimuthal flow is forced, the axial and radial fluctuations grow from an initially zero value at the inlet, z/H = 0. The axial component has a maximum amplitude of about 25% of the inlet azimuthal forcing amplitude while the radial component reaches a maximum of about 50%. Note that the inlet azimuthal forcing amplitude is 10% of the mean axial flow. Consequently, the axial component has a maximum of 2.5% of the mean axial flow and the radial has a maximum of 5%. These correspond to the transfer of velocity fluctuations from their purely azimuthal component origin, to having axial and radial component fluctuations, due to vortex tube re-orientation. The plots also clearly indicate the convective nature of the disturbance. Based on the Strouhal number for this example (St = 1.2) and the disturbance wavelength, the disturbance convection speed is 1.4U₀. This velocity corresponds to the center-line mean axial velocity in the annulus.

These fluctuations further evolve downstream of the combustor dump due to changes in the cross-sectional area. Since the flame is located in this region, it is important to consider the evolution of these disturbances along the flame. Figure 11 shows the spatial evolution of the normalized axial velocity fluctuations along the mean flame position shown in Figure 9 for case (1) in Table II, using the azimuthal only forcing boundary condition in Eq. (4). The plots are shown separately for the upper and lower branches of the flame (mean flame shown in Figure 9). There are two important features to notice from these plots. First, they show the presence of radial and axial velocity disturbances, although neither of them is forced at the inflow point. These velocity disturbances are a superposition of the vortical nature of the source disturbance (azimuthal flow fluctuations) and the vortical fluctuations generated at the shear layer as the flow enters the combustor dump from the annulus. The maximum fluctuations on both flame branches reach about 60% of the azimuthal forcing amplitude at the inlet (6% of the mean axial velocity).

Similar qualitative features can be seen for the radial flow fluctuations shown in Figure 12. The radial flow fluctuations reach maximum amplitude of 125% of the inlet azimuthal forcing amplitude (or 12.5% fluctuation amplitude), indicating that they are the primary contributor to the unsteady flame response and dominate relative to the axial fluctuations. Note that these maximum amplitudes and the wavelength of the disturbances are a function of forcing Strouhal number. An important takeaway from this example is that, despite the presence of only azimuthal flow fluctuations at the inlet, the axial and radial flow fluctuations affecting the flame are not negligible and can even exceed the forcing amplitude at the inlet.

As described in the context of Figure 10, the reorientation of axial vorticity into radial/azimuthal vorticity is critical for the swirl fluctuation mechanism in terms of exciting oscillations in heat release. As such, this section focuses on the vorticity dynamics in the inlet section and combustor inlet and the processes leading to axial vorticity reorientation. Consider the vorticity transport equation for a constant density, incompressible flow,

Here, the vorticity in cylindrical coordinates is expressed as

Expanding out, the evolution equations for the three components of the vorticity are given by

Consider the vorticity source terms (describing stretching or reorientation/bending) for a flow with only axial vorticity, an approximation that is a useful descriptor of the core flow in the inlet, outside of the two annular boundary layers. In this case, the source terms on the right sides of these equations that are multiplied by ω_r or ω_θ are zero and the source terms revert to $ωz∂uz/∂z$⁠, ω_z(∂u_θ/∂z), $ωz∂ur/∂z$ for the axial, azimuthal, and radial vorticity, respectively. It can be seen by inspection that the first term is a vortex stretching term, while the latter two are due to vortex bending. Thus, azimuthal and radial vorticities are generated by axial gradients in azimuthal and radial velocities, respectively. Outside of the boundary layers, these terms are nonzero but quite weak. For this reason, there is little alteration of the axial vorticity in the inlet core flow. This point can be seen from Figure 13, which plots the downstream evolution of vortex lines in the post swirler region for the inlet forcing corresponding to case (c) with the mean and disturbance flow field information obtained from the swirler exit plane in the “swirler-annulus” calculations. The orange and blue lines originate in the core. Note how these lines remain vertical all the way to the combustor inlet, showing that essentially no axial vorticity reorientation occurs in the core flow.

Note how the vortex lines are rotated into azimuthal and radial vorticities at the combustor inlet. This occurs because the rapid flow expansion leads to a sudden increase in the values of $∂uθ/∂z$ and $∂ur/∂z$⁠. This vortex tube reorientation leads to the generation of azimuthal and radial vorticity fluctuations which in-turn induces axial and radial flow velocity fluctuations. The magnitudes of these two gradients are shown in Figure 14. Note how for the colorbar shown that these two gradients do not show up in the inlet section but appear prominently in the combustor section. Also, note that the $∂ur/∂z$ is an order of magnitude larger than $∂uθ/∂z$⁠, as might be expected since the radial velocity is essentially starting from zero.

Returning to Figure 13, consider next the evolution of vorticity in the inner and outer boundary layers of the inlet annulus. In this region, azimuthal vorticity exists, i.e., ω_θ≠0, and there are strong velocity gradients and so significant reorientation of the vorticity vectors occurs in the inlet itself. This can be seen from the red, green, and black vortex lines. Moreover, this azimuthal vorticity in the boundary layers induces axial and radial flow disturbances in the core flow, as shown earlier in Figure 10. These axial flow fluctuations are also shown in Figure 15 for this post-swirler calculation, demonstrating the same point—namely, the generation of core flow disturbances by boundary layer vorticity.

To summarize—strong conversion of axial vorticity originating from the swirlers into radial and axial vorticities occurs in the inlet section boundary layers and combustor inlet. For this reason, even for an axisymmetric flame that is insensitive to purely axial vortical disturbances, there are efficient mechanisms for conversion of this axial vorticity into velocity disturbances that wrinkle the flame. In Sec. III, we quantify the relative significance of this vortical disturbance mechanism relative to azimuthal vorticity originating from the separating boundary layers in terms of exciting unsteady heat release.

In this section, we shall discuss the impact of azimuthal flow fluctuations on the global flame response using the FTF defined earlier in Eq. (2). The instantaneous heat release rate is calculated as

Here, ρ_u is the unburnt gas density, h_R is the heat of reaction, s_T is the turbulent flame front speed, and A is the flame surface area. The flame front speed is obtained from a closure model⁵³ as mentioned earlier and the area is obtained from the instantaneous G = 0 contour. The reference velocity fluctuation is chosen at the inlet. Using these quantities, the FTF is obtained as defined in Eq. (2).

First consider the effects of the different forcing configurations (Eqs. (3)–(5)) on the FTF. For convenience of notation, we shall denote the FTFs using the axial only forcing boundary conditions (BC) in Eq. (3) as FTF_axial, using the azimuthal only forcing BC in Eq. (4) as FTF_swirl and using BCs in Eq. (5) as FTF_both. Figure 16 shows the variation in amplitude and phase of the FTF for the different inlet forcing configurations, for case (1) from Table II. As discussed earlier, the presence of azimuthal-only forcing at the inlet can generate comparable axial and radial flow disturbances depending on the frequency of forcing. This is reflected in the non-negligible value of FTF_swirl. FTF_axial is larger than FTF_swirl for the most part, although there are Strouhal number values where FTF_swirl dominates. This can be attributed to interference phenomena in the flow-field coupling effects and thus the flame response at those frequencies. As an example, Figure 17 plots the evolution of the flame front at different instances in a forcing cycle for the azimuthal only forcing case with the left plot corresponding to a peak in FTF_swirl and the right plot to a minima in FTF_swirl. Finally, Figure 16 plots the combined effect of both forcing components of the inlet showing that FTF_both takes values that oscillate about the FTF_axial curve.

Next, consider the effect of the relative phasing between the axial and azimuthal components at the inlet on the FTF. It was mentioned earlier, in the context of Figure 8, that this relative phase was a function of forcing frequency. However, in that geometry, this relative phasing is a function of the swirler vane design as well. In the post-swirler simulations, since the swirler is not included, we have the freedom to vary the relative phase so as to indirectly account for the effect of differences in swirler vane design, as well as location of the swirler upstream of the combustor. For this comparison, we use the inlet BCs from Eq. (5). The simulations are performed for φ varying between 0° and 270°, in steps of 90°. The amplitude and phase of the FTF are shown in Figure 18. Notice that as the relative phase is changed, the qualitative nature of the FTFs remains the same but the interference locations are shifted in Strouhal number space. This indicates that the relative phasing has a strong control over the frequencies at which the FTF exhibits minima or maxima. This is also reflected in the phase of the FTF, wherein the qualitative nature remains the same but the locations of phase jump are changed. A second comparison is shown in Figure 19 for a different set of control parameters corresponding to case (2) in Table II.

In this work, we used numerical simulations to understand how axial vorticity evolved in flow mixing passages and combustor entrance, and their role on the global FTF. Specifically, we considered axisymmetric mean flame configurations, where the azimuthal flow fluctuations do not have a direct influence on the flame response, but rather an indirect one, through coupling with other flow components.

Two different types of simulations were performed. The swirler-annulus simulations were used to understand the generation of azimuthal flow fluctuations from the axial flow fluctuations at the upstream inlet. Specifically, it was shown that these azimuthal flow disturbances propagated at the local axial mean flow velocity. The phasing between the azimuthal flow disturbances and the axial flow disturbances at the exit plane of the swirler region was shown to be dependent on the forcing frequency, as expected. This phasing was shown to control the FTF through interference.

A second set of simulations, referred to as “post-swirler” simulations, were used to study the vorticity dynamics by considering the effect of axial vorticity fluctuations and how these translate to fluctuations in radial and azimuthal vorticity fluctuations, which then induce axial and radial flow fluctuations. The key flow process responsible for axial vortex tube reorientation occurs in approach flow boundary layers and at the combustor entrance, due to the strong radial flow gradients that occur at the rapid expansion. These simulations were also used to independently vary the relative amplitudes of post swirler axial and azimuthal velocity fluctuations. This allowed for understanding the coupling between different flow components and how azimuthal flow fluctuations could change axial and radial flow fluctuations, thus causing heat release rate fluctuations. Specifically, the azimuthal flow fluctuations can generate radial and axial flow fluctuations of the same order, whose values are quite sensitive to the Strouhal number. These results indicate that while azimuthal, shear generated vorticity generally has the largest impact on the flame response, the swirler-induced axial vorticity can also exert non-negligible and comparable influences on the overall FTF.

This work has been partially supported by the US Department of Energy under Contract No. DE-NT0005054, contract monitor Mark Freeman, as well as the National Science Foundation through Contract No. CBET-1235779, contract monitor Professor Ruey-Hung Chen. The numerical simulations were performed on the Georgia Tech PACE cluster and the clusters (Kraken, Nautilus at NICS, Stampede at TACC, Comet at SDSC, and Blacklight at PSC) offered through the NSF XSEDE program, under Charge Nos. TG-CTS130016 and TG-DMS130001.