Global Stability and Forced Response Analysis of Swirling Flows in Aviation Combustors

Patki, Parth A.; Emerson, Benjamin L.; Lieuwen, Timothy C.

doi:10.1115/1.4066208

Abstract

Swirling jets are canonical flow fields used to stabilize flames in many gas-turbine combustion systems. These flows amplify background acoustic disturbances and perturb the flame, producing combustion noise and potentially combustion instabilities. Hydrodynamic flow response to acoustic excitations in reacting flows is an important modeling task for understanding combustion noise and combustion instabilities. This study utilizes a global hydrodynamic stability analysis in a biglobal and triglobal framework to model the vortical hydrodynamic modes of a reacting, swirling large eddy simulations (LES) mean-flow based on a commercial nozzle. After conducting an unforced, natural biglobal stability analysis to study the unstable eigenmodes of the flow, linearized equations of motion are used to study flow response of the flow to forcing. An empirical velocity transfer function is constructed by determining the flow response to a harmonically varying $150 - 1500$ Hz $u^{'}$ velocity disturbance at the inflow boundary. These disturbances are strongly amplified, with amplification factors ranging from about 20 to 50 and peaking at about 1050 Hz, or $St = 0.375$ ⁠. A comparison study with a Cartesian, three-dimensional (3D) reacting base-flow is also performed where the base-flow varies spatially in three-dimensions. The study utilizes a stable, high accuracy, but computationally effective centered finite difference scheme based on a three-dimensional structured mesh. Illustrative results present the qualitative modes and quantitative transfer functions for Biglobal and Triglobal stability analyses.

1 Introduction

Practical aviation combustor systems often use swirling flows for flame stabilization and static stability control. However, swirling flows are subject to several types of hydrodynamic instabilities, leading to large-scale organized vortical structures that can couple with combustor acoustics and lead to thermoacoustic instability. Thus, a key challenge to use swirling jets in practice is to model the vortical dynamics of the unsteady flow structures, especially in the presence of combustion acoustics.

Hydrodynamic stability analysis is a prominent tool to study the linear, leading order coherent dynamics of flows. In literature, many studies have focused on analysis of swirling flows [1–4]. Relevant literature related to this study includes work by Tammisola and Juniper [1] that studied the linear global modes in a swirl combustor. Douglas et al. [2] studied the nonlinear hydrodynamics of the fully developed swirling jets. Furthermore, Karmarkar et al. studied precessing vortex core oscillation dynamics of swirling flow motivated by the thermoacoustic instability in Ref. [3], while in a similar context Oberleithner et al. used stability analysis to track coherent structures of swirling flows with vortex breakdown in Ref. [4]. Several studies in literature also focus on linear, natural temporal modes obtained by local spatiotemporal analysis or weakly nonparallel flow analysis to observe sensitivity of different parameters. Oberleithner et al. further experimentally observed “self-excited” modes at critical $Sw = 0.88$ in Ref. [4], before testing the sensitivity of an eddy viscosity models to stability calculations in another study [5].

While there is an extensive literature on natural flow dynamics, the literature on external forcing of swirl flows is much more limited. An experimental forced response analysis of swirling flames was done by Manoharan et al. in Ref. [6]. A local and weakly, global forced response analysis with external acoustic forcing based on Wentzel–Kramers–Brillouin–Jeffreys assumption was carried out by Douglas et al. in Ref. [2]. In particular, little has been done on forced response analysis in a global framework. Thus, an overarching objective of this work is to develop understanding of swirling jet global mode response, as well as understand how modeling fidelity; i.e., weakly nonparallel versus Biglobal versus Triglobal, influences the results.

2 Methods

As described above, the objective of this study is to analyze the swirl flow response to harmonic forcing in a global framework. The objective of such a global framework is to analyze the stability of small amplitude disturbances to a “base” or undisturbed state. In reality, real flows of interest are inherently turbulent, unsteady, and possibly globally unstable, and so a common simplification is to analyze linear stability of the time-averaged flow [7]. This approach assumes that the stability of the time-averaged flow is equal to that of an equivalent flow with the same undisturbed base state. This so-called “mean-flow approximation” to analyze the stability and forced response of high Re unstable shear flows has been analyzed in detail and demonstrated to be accurate in several studies [1,4,8]. Fundamental work analyzing this assumption has shown that it is leading order accurate when the fundamental mode is much stronger in comparison to subsequent harmonic modes, which may be generated by the inclusion of nonlinear terms. The fundamental linear modes extracted from mean-flow stability have an amplitude much stronger than that of subsequent harmonic modes or their products. As such, words mean-flow and base-flow will be used interchangeably for the remainder of this study.

For this study, the time averages are extracted from large eddy simulations (LES) previously presented by Panchal and Menon in Ref. [9]. An instantaneous snapshot of the reacting, LES base-flow axial velocity is shown in Fig. 1. This snapshot illustrates the geometry along the axial direction and marks flow features present in the turbulent swirl induced flow such as corner recirculation zones, vortex break down bubble, and cooling quench jets. Tables 1 and 2 display the reacting LES base-flow air and fuel inflow properties, respectively. The 350 instantaneous snapshots of LES base-flow data with time-step of 0.1 ms are time-averaged to generate the “mean”-flow. The experimental progress and investigation of a combustor with the same geometric, fuel and air inflow specifications can be found in Ref. [10]. The flow-field generated by this effort is considered as an independent base-flow for future work.

Fig. 1

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Instantaneous snapshot of LES base-flow [9]

Table 1

Air inflow

	Cooling	Swirler	Quench 1	Quench 2
${\dot{m}}_{air}$ (lbm/s)	0.103	0.207	0.258	0.826

Table 2

Fuel/spray inflow

	Primary injection	Secondary injection
${\dot{m}}_{fuel}$ (g/s)	2.4	6.89

A cylindrical coordinate transform is performed on a slice of the reacting three-dimensional (3D) base-flow velocity fields, i.e., $\bar{u} = \bar{u} e_{x} + \bar{v} e_{y} + \bar{w} e_{z}$ ⁠. The sliced data are used to construct an axisymmetric dataset by utilizing the symmetric properties along the axial centerline upstream of the quench jets. The axial, tangential, and radial velocity fields of the decomposed axisymmetric base-flow data to be used in this study, i.e., $\bar{u} = {\bar{u}}_{r} e_{r} + {\bar{u}}_{θ} e_{θ} + {\bar{u}}_{z} e_{z}$ ⁠, are shown in Figs. 2–4, respectively.

Fig. 2

Axisymmetric base-flow axial velocity field: u¯z (m/s)

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Axisymmetric base-flow axial velocity field: ${\bar{u}}_{z} (m / s)$

Fig. 3

Axisymmetric base-flow tangential velocity field: u¯θ (m/s)

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Axisymmetric base-flow tangential velocity field: ${\bar{u}}_{θ} (m / s)$

Fig. 4

Axisymmetric base-flow radial velocity field: u¯r (m/s)

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Axisymmetric base-flow radial velocity field: ${\bar{u}}_{r} (m / s)$

2.1 Linear Stability Analysis.

As discussed above, a linear mean-flow stability methodology, which works well for swirling flows, will be used in this study. As such, the flow-field can be decomposed into a mean base-flow and coherent fluctuations, i.e.,

g = \bar{g} + g^{'}

⁠. Linear stability analysis determines the growth/decay rates of disturbance imposed upon the linearized equations of motion expanded around the base-flow (⁠

\bar{u} = {\bar{u}}_{r} e_{r} + {\bar{u}}_{θ} e_{θ} + {\bar{u}}_{z} e_{z}

or

\bar{u} = \bar{u} e_{x} + \bar{v} e_{y} + \bar{w} e_{z}

⁠). Governing equations and display the linearized, incompressible continuity and momentum equations for this study. Since this is a reacting flow study, some discussion of the incompressible assumption is warranted. The key point here is that the focus of this analysis is not upon the overall velocity (which also includes acoustic modes), but upon the hydrodynamic, vortical modes. In subsonic flows, i.e.,

Ma ≪ 1

⁠, the feedback of the simultaneously present acoustic modes upon the vortical modes is weak except in very specific locations. Furthermore, the vortical mode amplitude is much larger than the acoustic mode amplitude as shown by numerical experiments of Luo and Sundham [11]. O'Connor and Lieuwen also conducted forced flame measurements to study relative amplitudes of acoustic and vortical modes in Ref. [12]

\nabla \cdot u^{'} = 0

(1)

\frac{\partial u^{'}}{\partial t} + \bar{u} \cdot \nabla u^{'} + u^{'} \cdot \nabla \bar{u} + \frac{\nabla p^{'}}{ρ} - \frac{1}{Re} \nabla^{2} u^{'} = 0

(2)

Upon linearization of the equations and retaining first-order terms, the unsteady disturbances are assumed to be harmonic with an amplitude and a phase. The most widely used technique to determine temporal frequencies at which the disturbance grows is to implement local stability analysis. In local stability analysis, the amplitude of the assumed disturbance varies only with a single dimension, while the phase captures periodic oscillations of the disturbance wave in remaining spatial dimensions and time. Local stability studies have been conducted for swirling flows in literature by Vasconcellos et al. [13] and Mistry et al. [14]. Local stability analyses are, strictly speaking, only valid for parallel flows that do not evolve in the streamwise direction. While the technique can be generalized for some level of weak axial variations, swirl flows' features evolve strongly in the axial direction and such an approximation is problematic. For example, the study by Mistry et al. [14] shows that accuracy of local stability is limited for nonparallel flows where flow is evolving spatially in multiple directions.

Biglobal stability analysis allows for modal features to vary in two spatial dimensions in an arbitrary manner. Note that the Wentzel–Kramers–Brillouin–Jeffreys, or weakly nonparallel approximation, requires weak variation in the axial dimension, a problematic assumption for swirling flows with vortex breakdown. For the application of swirling flow, the amplitude of the disturbance can be expressed as a function of radial and axial directions. The phase maintains periodicity in the tangential direction in terms of the azimuthal wave number m. As such, any disturbance

g^{'}

with a cylindrical Biglobal assumption takes the form of the following modal decomposition shown in the following equation:

g^{'} (r, θ, z, t) = \hat{g} (r, z) e^{- i (ω t - m θ)}

(3)

Finally, Triglobal stability analysis allows variations in all three spatial dimensions. The disturbance amplitude is three-dimensional, and the phase term only retains the temporal dependence. For this case, in a Cartesian coordinate system, the normal mode expands as shown in the following equation:

g^{'} (x, y, z, t) = \hat{g} (x, y, z) e^{- i ω t}

(4)

Computational expense of the three linear stability approaches varies accordingly; i.e., local stability is cheapest, and Triglobal is the most expensive. Use of triglobal techniques to efficiently track modes has gained traction with evolution of more powerful computers with greater processing power [15,16]. Paredes et al. compare the three techniques of linear stability in greater detail and further discuss the implications of computational expense for each method [17].

2.2 Biglobal Stability.

For this study, Biglobal and Triglobal stability frameworks will be employed. Biglobal stability analysis applies for axisymmetric base-flow, where the flow develops only with radial and axial coordinates of the domain. A Biglobal formulation also balances the computational cost, while maintaining a global framework. Thus, introducing the Biglobal normal mode shown in Eq. into Eqs. and results in the Biglobal linearized Navier–Stokes equations (LNSE). The frequency domain Biglobal LNSE consisting of radial, tangential, axial momentum and continuity equations are displayed as follows:

\begin{matrix} ({\bar{u}}_{r} \frac{\partial}{\partial r} + i m \frac{{\bar{u}}_{θ}}{r} + {\bar{u}}_{z} \frac{\partial}{\partial z} + \frac{\partial {\bar{u}}_{r}}{\partial r} \\ - \frac{1}{Re} (\frac{1}{r} \frac{\partial}{\partial r} + \frac{\partial^{2}}{\partial r^{2}} - \frac{1}{r^{2}} (m^{2} + 1) + \frac{\partial^{2}}{\partial z^{2}})) {\hat{u}}_{r} \\ + (\frac{1}{r} \frac{\partial {\bar{u}}_{r}}{\partial θ} + \frac{2}{Re} (\frac{i m}{r^{2}}) - \frac{2 {\bar{u}}_{θ}}{r}) {\hat{u}}_{θ} \\ + (\frac{\partial {\bar{u}}_{r}}{\partial z}) {\hat{u}}_{z} + (\frac{\partial}{\partial r}) \hat{p} = i ω {\hat{u}}_{r} \end{matrix}

(5)

\begin{matrix} (\frac{{\bar{u}}_{θ}}{r} + \frac{\partial {\bar{u}}_{θ}}{\partial r} - \frac{2}{Re} (\frac{i m}{r^{2}})) {\hat{u}}_{r} \\ + ({\bar{u}}_{r} \frac{\partial}{\partial r} + i m \frac{{\bar{u}}_{θ}}{r} + {\bar{u}}_{z} \frac{\partial}{\partial z} + \frac{1}{r} \frac{\partial {\bar{u}}_{θ}}{\partial θ} + \frac{{\bar{u}}_{r}}{r} \\ - \frac{1}{Re} (\frac{1}{r} \frac{\partial}{\partial r} + \frac{\partial^{2}}{\partial r^{2}} - \frac{1}{r^{2}} (m^{2} + 1) + \frac{\partial^{2}}{\partial z^{2}})) {\hat{u}}_{θ} \\ + (\frac{\partial {\bar{u}}_{θ}}{\partial z}) {\hat{u}}_{z} + (\frac{i m}{r}) \hat{p} = i ω {\hat{u}}_{θ} \end{matrix}

(6)

\begin{matrix} (\frac{\partial {\bar{u}}_{z}}{\partial r}) {\hat{u}}_{r} + (\frac{1}{r} \frac{\partial {\bar{u}}_{z}}{\partial θ}) {\hat{u}}_{θ} \\ + ({\bar{u}}_{r} \frac{\partial}{\partial r} + i m \frac{{\bar{u}}_{θ}}{r} + {\bar{u}}_{z} \frac{\partial}{\partial z} + \frac{\partial {\bar{u}}_{z}}{\partial z} \\ - \frac{1}{Re} (\frac{1}{r} \frac{\partial}{\partial r} + \frac{\partial^{2}}{\partial r^{2}} - \frac{m^{2}}{r^{2}} + \frac{\partial^{2}}{\partial z^{2}})) {\hat{u}}_{z} \\ + (\frac{\partial}{\partial z}) \hat{p} = i ω {\hat{u}}_{z} \end{matrix}

(7)

(\frac{\partial}{\partial r} + \frac{1}{r}) {\hat{u}}_{r} + (\frac{i m}{r}) {\hat{u}}_{θ} + (\frac{\partial}{\partial z}) {\hat{u}}_{z} = 0

(8)

The four coupled Biglobal LNSE can be segregated by convection, diffusion, and absorption type terms. Upon resolving the different terms, Biglobal LNSE can be solved using the coefficient-form eigenvalue solver of comsol. The computational domain is meshed using Taylor–Hood finite elements in comsol to obtain numerical solutions for the Biglobal LNSE.

2.3 Triglobal Stability.

Acknowledging its computational expense, Triglobal stability analysis is also considered in this study. Inserting Eq. into Eqs. and results in Triglobal LNSE for a Cartesian coordinate system. The frequency domain Triglobal LNSE consisting of x-, y-, z-momentum and continuity equations are displayed as follows:

\begin{matrix} (\bar{u} \frac{\partial}{\partial x} + \bar{v} \frac{\partial}{\partial y} + \bar{w} \frac{\partial}{\partial z} + \frac{\partial \bar{u}}{\partial x} - ν (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}})) \hat{u} \\ + (\frac{\partial \bar{u}}{\partial y}) \hat{v} + (\frac{\partial \bar{u}}{\partial z}) \hat{w} + (\frac{1}{ρ} \frac{\partial}{\partial x}) \hat{p} = i ω \hat{u} \end{matrix}

(9)

\begin{matrix} (\frac{\partial \bar{v}}{\partial x}) \hat{u} + (\bar{u} \frac{\partial}{\partial x} + \bar{v} \frac{\partial}{\partial y} + \bar{w} \frac{\partial}{\partial z} + \frac{\partial \bar{v}}{\partial y} - ν (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}})) \hat{v} \\ + (\frac{\partial \bar{v}}{\partial z}) \hat{w} + (\frac{1}{ρ} \frac{\partial}{\partial y}) \hat{p} = i ω \hat{v} \end{matrix}

(10)

\begin{matrix} (\frac{\partial \bar{w}}{\partial x}) \hat{u} + (\frac{\partial \bar{w}}{\partial y}) \hat{v} + (\bar{u} \frac{\partial}{\partial x} + \bar{v} \frac{\partial}{\partial y} + \bar{w} \frac{\partial}{\partial z} + \frac{\partial \bar{w}}{\partial z} \\ - ν (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}})) \hat{w} + (\frac{1}{ρ} \frac{\partial}{\partial z}) \hat{p} = i ω \hat{w} \end{matrix}

(11)

(\frac{\partial}{\partial x}) \hat{u} + (\frac{\partial}{\partial y}) \hat{v} + (\frac{\partial}{\partial z}) \hat{w} = 0

(12)

Triglobal LNSE appear more symmetric in comparison to the Biglobal LNSE. In order to solve the Triglobal LNSE, a centered finite difference approach can be employed. If the flow domain is arranged on a cubical structured grid, a finite difference approximation to the Triglobal LNSE can be efficiently achieved by utilizing properties of symmetry. In a Cartesian coordinate system, the interior nodes can be discretized about a centered-difference seven-point stencil for all $n^{3}$ interior nodes as shown in Fig. 5. The interior nodes can be discretized using an equal spacing in all directions, i.e., $h = Δ x = Δ y = Δ z$ ⁠.

Fig. 5

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Seven-point stencil of an interior node for a centered finite difference scheme

The centered finite difference scheme with truncation error

O (Δ x^{2} + Δ y^{2} + Δ z^{2})

is employed in all three directions to compute the spatial derivatives. Dropping the hats from the variables for simplicity, the discretized forms of Eqs. – with centered finite difference spatial derivatives read as follows:

\begin{matrix} (- \frac{h}{2} \bar{u} - ν) u_{i - 1, j, k} + (- \frac{h}{2} \bar{v} - ν) u_{i, j - 1, k} + (- \frac{h}{2} \bar{w} - ν) u_{i, j, k - 1} \\ + (h^{2} \frac{\partial \bar{u}}{\partial x} + 6 ν) u_{i, j, k} + (\frac{h}{2} \bar{u} - ν) u_{i + 1, j, k} + (\frac{h}{2} \bar{v} - ν) u_{i, j + 1, k} \\ + (\frac{h}{2} \bar{w} - ν) u_{i, j, k + 1} + (h^{2} \frac{\partial \bar{u}}{\partial y}) v_{i, j, k} + (h^{2} \frac{\partial \bar{u}}{\partial z}) w_{i, j, k} \\ + (- \frac{h}{2 ρ}) p_{i - 1, j, k} + (\frac{h}{2 ρ}) p_{i + 1, j, k} = h^{2} i ω u_{i, j, k} \end{matrix}

(13)

\begin{matrix} (h^{2} \frac{\partial \bar{v}}{\partial x}) u_{i, j, k} + (- \frac{h}{2} \bar{u} - ν) v_{i - 1, j, k} + (- \frac{h}{2} \bar{v} - ν) v_{i, j - 1, k} \\ + (- \frac{h}{2} \bar{w} - ν) v_{i, j, k - 1} + (h^{2} \frac{\partial \bar{v}}{\partial y} + 6 ν) v_{i, j, k} + (\frac{h}{2} \bar{u} - ν) v_{i + 1, j, k} \\ + (\frac{h}{2} \bar{v} - ν) v_{i, j + 1, k} + (\frac{h}{2} \bar{w} - ν) v_{i, j, k + 1} + (h^{2} \frac{\partial \bar{v}}{\partial z}) w_{i, j, k} \\ + (- \frac{h}{2 ρ}) p_{i, j - 1, k} + (\frac{h}{2 ρ}) p_{i, j + 1, k} = h^{2} i ω v_{i, j, k} \end{matrix}

(14)

\begin{matrix} (h^{2} \frac{\partial \bar{w}}{\partial x}) u_{i, j, k} + (h^{2} \frac{\partial \bar{w}}{\partial y}) v_{i, j, k} + (- \frac{h}{2} \bar{u} - ν) w_{i - 1, j, k} \\ + (- \frac{h}{2} \bar{v} - ν) w_{i, j - 1, k} + (- \frac{h}{2} \bar{w} - ν) w_{i, j, k - 1} + (h^{2} \frac{\partial \bar{w}}{\partial z} + 6 ν) w_{i, j, k} \\ + (\frac{h}{2} \bar{u} - ν) w_{i + 1, j, k} + (\frac{h}{2} \bar{v} - ν) w_{i, j + 1, k} + (\frac{h}{2} \bar{w} - ν) w_{i, j, k + 1} \\ + (- \frac{h}{2 ρ}) p_{i, j, k - 1} + (\frac{h}{2 ρ}) p_{i, j, k + 1} = h^{2} i ω w_{i, j, k} \end{matrix}

(15)

\begin{matrix} \begin{matrix} - u_{i - 1, j, k} + u_{i + 1, j, k} - v_{i, j - 1, k} + v_{i, j + 1, k} - w_{i, j, k - 1} + w_{i, j, k + 1} = 0 \end{matrix} \end{matrix}

(16)

The discretized disturbance equations resulting from Triglobal LNSE at each interior node consist of a system of linear, coupled equations. The system of equations at a particular node can be arranged to formulate a generalized-eigenvalue problem (GEVP) to obtain eigensolutions at each interior node. The eigenvalue

ω

represents the natural frequency of the GEVP consisting coefficients of a stiffness matrix

A

and a mass matrix

B

as shown in the following equation:

A v = ω B v

(17)

For any arbitrary interior node (i, j, k), matrices $A = [A_{1} | A_{2} | A_{3} | A_{4}]$ and $B$ can be assembled as shown in Eqs. (19)–(23). Upon glancing at matrices $A$ and $B$ ⁠, it is clear that $B$ is a singular matrix, i.e., it is not invertible. As such, the GEVP cannot be explicitly solved as $B^{- 1} A v = ω v$ to obtain the eigenvalue solutions.

To obtain a solution to the closed GEVP, the shift inverse power method is used. With this algorithm, a user-defined shift

μ

can be used to transform the GEVP as shown in the following equation:

{(A - μ B)}^{- 1} B v = \frac{1}{ω - μ} v

(18)

The shift inverse power method is designed such that it converges to the eigenvalue

ω

that is closest to the shift

μ

⁠, or “guess.” For the particular application of triglobal stability, the search region is restricted where if

ω \in C

as

ω = ω_{r} + i ω_{i}

⁠, then

ω_{i} > 0

represent the unstable modes with real frequency

ω_{r}

⁠. Thus, if a particular frequency

ω_{r}

is provided as a guess value in the shift inverse power method, the algorithm will converge to the closest eigenvalue, which can be filtered by the criterion

ω_{i} > 0

for unstable modes

\begin{matrix} A_{1} = [\begin{matrix} - \frac{h}{2} \bar{u} - ν & - \frac{h}{2} \bar{v} - ν & - \frac{h}{2} \bar{w} - ν & h^{2} \frac{\partial \bar{u}}{\partial x} + 6 ν & \frac{h}{2} \bar{u} - ν & \frac{h}{2} \bar{v} - ν & \frac{h}{2} \bar{w} - ν \\ 0 & 0 & 0 & h^{2} \frac{\partial \bar{v}}{\partial x} & 0 & 0 & 0 \\ 0 & 0 & 0 & h^{2} \frac{\partial \bar{w}}{\partial x} & 0 & 0 & 0 \\ - 1 & 0 & 0 & 0 & 1 & 0 & 0 \end{matrix}] \end{matrix}

(19)

\begin{matrix} A_{2} = [\begin{matrix} 0 & 0 & 0 & h^{2} \frac{\partial \bar{u}}{\partial y} & 0 & 0 & 0 \\ - \frac{h}{2} \bar{u} - ν & - \frac{h}{2} \bar{v} - ν & - \frac{h}{2} \bar{w} - ν & h^{2} \frac{\partial \bar{v}}{\partial y} + 6 ν & \frac{h}{2} \bar{u} - ν & \frac{h}{2} \bar{v} - ν & \frac{h}{2} \bar{w} - ν \\ 0 & 0 & 0 & h^{2} \frac{\partial \bar{w}}{\partial y} & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 & 0 & 1 & 0 \end{matrix}] \end{matrix}

(20)

\begin{matrix} A_{3} = [\begin{matrix} 0 & 0 & 0 & h^{2} \frac{\partial \bar{u}}{\partial z} & 0 & 0 & 0 \\ 0 & 0 & 0 & h^{2} \frac{\partial \bar{v}}{\partial z} & 0 & 0 & 0 \\ - \frac{h}{2} \bar{u} - ν & - \frac{h}{2} \bar{v} - ν & - \frac{h}{2} \bar{w} - ν & h^{2} \frac{\partial \bar{w}}{\partial z} + 6 ν & \frac{h}{2} \bar{u} - ν & \frac{h}{2} \bar{v} - ν & \frac{h}{2} \bar{w} - ν \\ 0 & 0 & - 1 & 0 & 0 & 0 & 1 \end{matrix}] \end{matrix}

(21)

\begin{matrix} A_{4} = [\begin{matrix} - \frac{h}{2 ρ} & 0 & 0 & 0 & \frac{h}{2 ρ} & 0 & 0 \\ 0 & - \frac{h}{2 ρ} & 0 & 0 & 0 & \frac{h}{2 ρ} & 0 \\ 0 & 0 & - \frac{h}{2 ρ} & 0 & 0 & 0 & \frac{h}{2 ρ} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] \end{matrix}

(22)

\begin{matrix} B = [\begin{matrix} 0 & 0 & 0 & i & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & i & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & i & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] \end{matrix}

(23)

2.4 Forcing Techniques.

A key goal of this study is to analyze the forced response of the swirling flow to imposed forcing. In literature, multiple methods for flow forcing exist. One of the methods used in numerical applications is to implement an inhomogenous Dirichlet velocity (⁠ $u^{'}$ ⁠) at the inlet. The fluctuating velocity profile at the inflow boundary simulates forcing as done by Polifke et al. in Ref. [18]. Another technique that is commonly used in computational studies is adding a forcing source term in the governing equations. This technique is usually appropriate when volumetric forcing is preferred over boundary forcing. In this study, forcing will be implemented by defining a harmonically varying the velocity disturbance normal to the inflow boundary of the flow domain. This method is commonly used in computational studies across literature as the inhomogenous $u^{'}$ boundary condition simulates “velocity disturbances” caused by noise. Polifke et al. studied this method in Ref. [18] for construction of an acoustic transfer function of a thermoacoustic system.

2.5 Boundary Conditions.

The Biglobal and Triglobal stability frameworks developed above result in four coupled disturbance equations. To complete the system, the disturbance equations need to be accompanied with appropriate boundary conditions. For the Biglobal case, the azimuthal wave number, m, can vary in integer multiples, i.e., $m = 0, \pm 1, \pm 2$ ⁠. The $m = 0$ wave number represents the axisymmetric modes, while $m < 0$ and $m > 0$ represent the co-rotating and counter-rotating modes, respectively. For the axisymmetric case of $m = 0$ considered in this study, the boundary conditions for the velocity and pressure disturbances are listed in Table 3. At the walls, all three velocity disturbances vanish, while at the centerline, only the radial and tangential velocity disturbances vanish. At the outlet, no-flux boundary conditions are used for three velocity and pressure disturbances following the strategy implemented by Vanierschot et al. in Ref. [19] for domain truncation with nonzero perturbations at the truncation region. Furthermore, an inhomogenous Dirichlet boundary condition $A_{f} = 0.2$ m/s is implemented for the radial velocity disturbance to simulate boundary forcing for forced response.

Table 3

Disturbance boundary conditions for $m = 0$ Biglobal stability

Disturbance	Axis	Walls	Inlet	Outlet
Radial velocity	${\hat{u}}_{r} = 0$	${\hat{u}}_{r} = 0$	${\hat{u}}_{r} = A_{f}$	$\partial_{n} {\hat{u}}_{r} = 0$
Tangential velocity	${\hat{u}}_{θ} = 0$	${\hat{u}}_{θ} = 0$	${\hat{u}}_{θ} = 0$	$\partial_{n} {\hat{u}}_{θ} = 0$
Axial velocity	$\partial_{n} {\hat{u}}_{z} = 0$	${\hat{u}}_{z} = 0$	${\hat{u}}_{z} = 0$	$\partial_{n} {\hat{u}}_{z} = 0$
Pressure	$\partial_{n} \hat{p} = 0$	$\partial_{n} \hat{p} = 0$	$\partial_{n} \hat{p} = 0$	$\partial_{n} \hat{p} = 0$

Figure 6 shows the geometrical representation of the Biglobal axisymmetric flow domain on which the boundary conditions are imposed. The centerline axis, walls, inlet, and outlet are marked. The inhomogenous radial velocity boundary shown in Table 3 is conveniently placed normal to the inlet plane to simulate external forcing.

Fig. 6

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Flow domain for axisymmetric Biglobal stability

The Triglobal framework of this study utilizes a Cartesian coordinate system to be consistent with the original Cartesian base-flow with flow variation in all three directions. As such, the original Cartesian base-flow is interpolated on to a coarser and structured cubical mesh to balance with the greater computational expense of triglobal stability. The interpolated base-flow used in this study involves a $20 \times 20 \times 20$ 3D grid with constant grid spacing of $h = 3.28$ mm in each direction. Figure 7 displays the axial velocity of the base-flow on interpolated grid, but in the 3D Cartesian coordinate system.

Fig. 7

Interpolated Cartesian base-flow axial velocity field: u¯z (m/s)

View large Download slide

Interpolated Cartesian base-flow axial velocity field: ${\bar{u}}_{z} (m / s)$

As discussed above, the finite difference solution discretizes the $18 \times 18 \times 18$ interior grid points leading to matrices $A$ and $B$ with a dimensions of $23, 328 \times 23, 328$ ⁠. However, the advantage of utilizing a finite difference method for the expensive Triglobal calculation becomes evident by considering the sparsity pattern of $A$ and $B$ shown in Fig. 8. Exploiting the sparse characteristics of $A$ and $B$ significantly aids the computational cost while obtaining the solution to the GEVP.

Fig. 8

View large Download slide

Sparsity patterns of matrices $A$ and $B$

As for the Biglobal case, the Triglobal framework needs to be completed with appropriate boundary conditions. In the Triglobal study of this paper, only the interior nodes away from the boundaries are discretized. However, an item for future work in the Triglobal study is to model the in-flow and out-flow boundary conditions with more careful consideration. As for the Biglobal study, a harmonically varying inflow boundary condition will be implemented to observe Triglobal forced response. Furthermore, a “high-viscosity” sponge layer can be implemented as discussed by Pant and Bhattacharya in Ref. [20] for a nonreflecting, open boundary to model the outlet. With the discussion of formulations and methodology for Biglobal and Triglobal stability analyses complete, the Illustrative Results section focuses on results of applying the two frameworks to the base-flow.

3 Illustrative Results

In this section, the results obtained by applying Biglobal and Triglobal frameworks to the axisymmetric and Cartesian base-flows are discussed. First, Biglobal stability applied to the reacting, axisymmetric base-flow is used to study the unforced, natural modes of the flow. Then, the same Biglobal framework is used to understand the forced global modes. Finally, Triglobal stability is used to study the natural modes of the same reacting, three-dimensional base-flow in a Cartesian coordinate system. As discussed in methods, the operating conditions (preheat temperature, equivalence ratio, etc.) and meshing details used to the reacting base-flow can be found in Ref. [9].

3.1 Biglobal Natural Hydrodynamics.

Utilizing the Biglobal framework developed above, a traditional linear hydrodynamics stability analysis is conducted on the axisymmetric base-flow without the boundary forcing condition. This effort is used to identify unforced, natural hydrodynamic modes, i.e., modes representing the instability that is naturally generated within the domain in absence of inlet boundary forcing.

The unforced case study produces a spectrum of natural eigenmodes shown in Fig. 9. Each vortical mode detected by the code contains a numerical eigenvalue $ω \in C$ as $ω = ω_{r} + i ω_{i}$ ⁠. The eigenvalue spectrum as plotted in Fig. 9 can be comprehended by correlating the growth rates (⁠ $ω_{i}$ ⁠) to the natural frequencies (⁠ $ω_{r}$ ⁠) of the modes. The unstable modes, represented by $ω_{i} > 0$ ⁠, are marked in Fig. 9. The spectrum of modes includes a low-frequency unstable mode associated with global instability in swirl-stabilized combustors as discussed by Terhaar et al. in Ref. [21]. This low-frequency mode is considered to be most energetic by Terhaar et al. The Biglobal natural hydrodynamics study also detects three additional unstable modes at higher natural frequencies.

Fig. 9

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Eigenspectrum (growth rates versus natural frequencies) of unforced Biglobal modes

The fact that the flow is linearly unstable implies that at “steady-state” it is executing nonlinear, limit cycle oscillations. Since the linear, forced response of this time varying, nonlinear flow is being analyzed, this also requires some discussion as it initially appears to be contradictory in terms of assumptions. First, note that these linear instabilities are accounted for in the LES and the nonlinear distortion of the mean-flow is thereby captured; i.e., the mean-flow differs from the base-flow due to both turbulence effects and the nonlinear oscillations of the global mode(s). This is the essence of the mean-flow approximation discussion above where an explanation of its theoretical justification is provided. Also worth discussion is that these growth rates computed from the LES are still nonzero. In contrast, in some flows, such as laminar, nonreacting wakes, the global mode grows to an amplitude where the instability growth rate calculated using the amplitude-dependent mean-flow (which deviates from the base-flow due to nonlinearities) becomes zero. Formal analysis of the instability growth rate calculated around the linearized, time-averaged flow-field has been performed for several canonical flows, and depending upon the relative amplitudes of higher-order nonlinear interactions is negligible in some cases [1] and nonzero in other applications such as cavity flows [22]. As such, these nonzero natural Biglobal mode growth rates calculated in this analysis were expected.

3.2 Biglobal Forced Modes.

Upon studying the unforced, natural modes of the axisymmetric base-flow, the same flow-field is utilized as base state for forced Biglobal stability analysis. As discussed above, the imposed boundary forcing at the domain inlet is utilized to simulate noise amplification. The velocity component normal to the inlet plan is harmonically forced with amplitude $A_{f} = 0.2$ m/s (scalable with linear analysis) and forcing frequency sweeps $f = 150 - 1500$ Hz. The mode-shape solutions for axial velocity ${\hat{u}}_{z}$ disturbances are shown in Fig. 10 for various forcing frequencies. The mode-shapes indicate that forcing causes the bulk of disturbance activity in the swirling flow shear layers.

Fig. 10

View large Download slide

Mode-shapes of axial velocity with harmonic inlet forcing at various frequencies

Having identifying that inlet introduced disturbances causes fluctuations predominantly in the shear layer, the flow domain is next probed to evaluate a point-to-point transfer function. First, these empirical transfer functions are constructed by placing probes at a particular axial location downstream of the inlet and marching radially through the shear layer. At this particular axial location, probes are implemented at nine radial locations to capture axial velocity disturbances across the shear layer. The axial velocity disturbance normalized by the amplitude of the inlet velocity forcing (considering the linear nature of the study) is formulated into an empirical transfer function,

H_{e}

⁠, as shown in the following equation:

H_{e} (i 2 π f) = \frac{| {\hat{u}}_{z} |}{A_{f}}

(24)

$H_{e}$ is evaluated at the nine different radial probes spanning the shear layer for multiple forcing frequencies. Figure 11 shows the one-dimensional (1D) empirical transfer function at an axial location of $z / d = 1.32$ for frequencies spanning $f = 150 - 1500$ Hz. The band of forcing frequencies ranging between 600 and 1400 Hz amplifies the axial velocity disturbance at most radial locations in the shear layer. Particularly at $r = 14$ mm, the 1050 Hz forcing frequency causes the most amplification and generates a response of an axial velocity disturbance that has an amplitude close to 60 times that of the inflow disturbance.

Fig. 11

View large Download slide

One-dimensional empirical transfer function $H_{e}$ at axial location $z / d = 1.32$

Axial location of $z / d = 1.32$ was sufficiently downstream to capture measurements of axial velocity disturbances through the radially varying probes across the shear layer. In the next stage of this study, the 1D empirical transfer function is extended along the axial direction to further quantify this transfer function spatially. Additional probes are placed in the flow domain to measure response at varying axial and radial locations downstream the inlet. The probe mesh refinement allows the placement of 16 radial probes at 24 different axial locations to measure the velocity disturbance at each frequency sweep. The radial, axial, and tangential velocity disturbances are evaluated for all 384 probes at various frequencies. A comparison of the flow domain probes used for the 1D empirical transfer versus the two-dimensional (2D) empirical transfer function is displayed in Fig. 12.

Fig. 12

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Comparison of flow response probes for 1D (left) and 2D (right) empirical transfer functions

In order to properly scale the large datasets obtained by the 2D empirical transfer function, the frequency can be nondimensionalized using a Strouhal number as shown in Eq. with average swirler exit velocity

u_{0} = 70

m/s and swirler diameter

d = 0.025

m.

St = \frac{f d}{u_{0}}

(25)

Then, the 2D empirical function can be visualized over a range of radial and axial locations and Strouhal numbers as displayed in Figs. 13 and 14. Each subplot in Figs. 13 and 14 shows a contour plot of $H_{e}$ as a function of radial coordinate and Strouhal number at a different axial location downstream the inlet. By looking at the colorbars for each subplot, the axial velocity disturbance is more strongly amplified by the inlet forcing at downstream locations. A key result from this figure indicates that the 1050 Hz inlet-forced disturbance is dominant in amplification of axial velocity disturbances at all axial locations as previewed by the 1D empirical transfer function. Furthermore, excitations around this corresponding $St = 0.375$ mode lead to axial velocity disturbances with amplification factors varying between 20 and 50 times at axial locations $0.24 d < z < 1.5 d$ and over 60 times at downstream axial locations $z > 1.5 d$ ⁠.

Fig. 13

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Two-dimensional empirical transfer function $H_{e}$ at axial locations $0.24 < z / d < 1.04$

Fig. 14

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Two-dimensional empirical transfer function $H_{e}$ at axial locations $1.2 < z / d < 2$

The forced Biglobal framework and the corresponding empirical transfer functions work well as a model where the inlet forcing frequency can be supplied to observe the forcing amplitude scaled, linear flow response. However, a significant limitation of the current model is the restriction to an axisymmetric base-flow. For practical combustor flow fields, the base-flow is never truly axisymmetric, especially downstream of the quench jets. The Biglobal analysis carried out in this study works well to study stability in the shear layers upstream of the quench jets, but the application of the triglobal framework developed above will be discussed in Sec. 3.3 which can be used to relax the axisymmetric flow assumption used in the Biglobal study.

3.3 Triglobal Natural Hydrodynamics.

Upon completion of natural and forced Biglobal studies, an effort was made to extract the dimensional natural Triglobal modes using the framework developed above. Using the shift inverse power method, the finite difference Triglobal solution converged to multiple dimensional low-frequency unstable modes around a supplied guess value. In the low-frequency search region, three unstable modes with the highest growth rate are listed in Table 4 in ascending order.

Table 4

Unstable low-frequency triglobal modes

	$ω_{r} + i ω_{i}$
Mode 1	$0.3234 + 0.0275 i$
Mode 2	$0.2989 + 0.0276 i$
Mode 3	$0.3198 + 0.0277 i$

In order to visualize a few preliminary results of the newly developed tool, contours of axial velocity disturbances for mode 1, mode 2, and mode 3 are displayed in Figs. 15–17, respectively. The contours display the behavior of solution away from the boundaries for the interior nodes of the domain discretized in this study. In fact, for the three unstable modes with the highest growth rate, no significant interaction near the boundary was observed, i.e., mode development was significantly away from the boundary. Triglobal modes 1 and 3 appear to be eigenmode pairs at approximately the same oscillatory frequency. The propagation of modes 1 and 3 is similar along the axial direction, and both modes nearly complement each other by obtaining symmetry about the base-flow centerline. Mode 2 displayed in Fig. 16 appears to be an independent mode, but has a similar growth rate to modes 1 and 3. From the mode-shapes displayed in all figures, the axial velocity disturbance is noticed to have strong presence in the shear layers, which motivates a separate forced Triglobal study to re-inforce the conclusions of forced biglobal modes. As discussed in the methods, future work of triglobal study involves a more detailed analysis by enforcing more comprehensive in-flow and out-flow boundary conditions, while observing the sensitivity of boundary effects on the Triglobal modes.

Fig. 15

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Mode-shape of axial velocity disturbance corresponding to Mode 1 of Table 4

Fig. 16

View large Download slide

Mode-shape of axial velocity disturbance corresponding to Mode 2 of Table 4

Fig. 17

View large Download slide

Mode-shape of axial velocity disturbance corresponding to Mode 3 of Table 4

In a very preliminary comparison of Biglobal and Triglobal natural modes, it can be seen that Triglobal study similarly predicts modes around the low-natural frequency detected by the biglobal study. The natural frequencies of three Triglobal modes are in the vicinity of the low-frequency mode predicted in the Biglobal study; however, in this preliminary comparison, the differences between both natural hydrodynamic studies could stem from a variety of reasons such as contrasting geometry/boundary conditions or interpolation in stability and base-flow meshes, etc. The limitation of the Triglobal is that it requires a guess frequency for convergence of the shift-and-invert algorithm, and thus the entire spectrum of natural modes cannot be efficiently obtained even for the $50 \times 50 \times 50$ coarse grid point base-flow mesh. As such, a much closer and detailed comparison between Biglobal and Triglobal modes still must be done once the Triglobal solutions are resolved on a more refined mesh with better boundary condition treatment. A more thorough comparison between the two frameworks remains as an item of future work once the Triglobal solver is further developed and validated.

4 Conclusion

In summary, this study develops a Biglobal and Triglobal analysis framework to understand flow stability and forced flow response of swirling flows commonly used in aviation gas-turbine combustors. Biglobal stability analysis is used to compute unforced, natural eigenspectrum of vortical hydrodynamic modes of an axisymmetric base-flow dataset. A forced Biglobal study was used to conduct a linear forced response analysis to visualize qualitative modes and evaluate a quantitative point-to-point 1D and 2D empirical transfer function. The empirical transfer functions serve as an excellent tool to detect flow response to harmonic inlet boundary forcing, simulating combustion noise present in practical combustors. Furthermore, the empirical transfer recognizes forcing frequencies for which the inflow disturbance is amplified. Finally, preliminary results of an extension to a natural Triglobal stability analysis are shown by utilizing a three-dimensional base-flow data. Possible future work of this study includes a forced Triglobal study with enforcement of upstream and downstream boundary conditions. In summary, Biglobal and Triglobal stability analyses emerge as effective tools to study hydrodynamic modal characteristics of flows spatially varying in multiple dimensions, which has been a key limiting point of local stability analysis in swirling flow literature.

Acknowledgment

This research was funded by the U.S. Federal Aviation Administration Office of Environment and Energy through ASCENT, the FAA Center of Excellence for Alternative Jet Fuels and the Environment, Project 55 through FAA Award No. 13-C-AJFE-GIT-058 under the supervision of Brandon Steele. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the FAA.

Funding Data

Federal Aviation Administration, Center of Excellence for Alternative Jet Fuels and Environment (ASCENT), Project 055 Noise Generation and Propagation from Advanced Combustors (Award No. 13-C-AJFE-GIT-058; Funder ID: 10.13039/100006282).

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Nomenclature

$A_{f}$ =: amplitude of inlet forcing (m $s^{- 1}$ ⁠)
c =: speed of sound (m $s^{- 1}$ ⁠)
d =: swirler diameter (m)
f =: forcing frequency (⁠ $s^{- 1}$ ⁠)
$H_{e}$ =: $| {\hat{u}}_{z} | / A_{f}$ , empirical transfer function
i =: x-component of finite difference node
j =: y-component of finite difference node
k =: z-component of finite difference node
m =: azimuthal wave number (⁠ $m^{- 1}$ ⁠)
$n$ =: normal vector
$u_{0}$ =: average swirler exit velocity (m $s^{- 1}$ ⁠)

$ν$ =: kinematic viscosity (m² $s^{- 1}$ ⁠)
$ω$ =: natural temporal frequency (⁠ $s^{- 1}$ ⁠)

Ma =: $u / c$ , Mach number
Re =: $u_{0} d / ν$ , Reynolds number
St =: $f d / u_{0}$ , Strouhal's number

GEVP =: generalized-eigenvalue problem
LES =: large eddy simulations
LNSE =: linearized Navier–Stokes equations

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Global Stability and Forced Response Analysis of Swirling Flows in Aviation Combustors

Abstract

1 Introduction

2 Methods

2.1 Linear Stability Analysis.

2.2 Biglobal Stability.

2.3 Triglobal Stability.

2.4 Forcing Techniques.

2.5 Boundary Conditions.

3 Illustrative Results

3.1 Biglobal Natural Hydrodynamics.

3.2 Biglobal Forced Modes.

3.3 Triglobal Natural Hydrodynamics.

4 Conclusion

Acknowledgment

Funding Data

Data Availability Statement

Nomenclature

References

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Global Stability and Forced Response Analysis of Swirling Flows in Aviation Combustors

Abstract

1 Introduction

2 Methods

2.1 Linear Stability Analysis.

2.2 Biglobal Stability.

2.3 Triglobal Stability.

2.4 Forcing Techniques.

2.5 Boundary Conditions.

3 Illustrative Results

3.1 Biglobal Natural Hydrodynamics.

3.2 Biglobal Forced Modes.

3.3 Triglobal Natural Hydrodynamics.

4 Conclusion

Acknowledgment

Funding Data

Data Availability Statement

Nomenclature

References

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