INSTABILITY MECHANISM IN A SWIRL FLOW COMBUSTOR ...

Proceedings of ASME Turbo Expo 2015: Turbine Technical Conference and ExpositionGT2015

June 16-20, 2015, Montreal, Canada

GT2015-42985

INSTABILITY MECHANISM IN A SWIRL FLOW COMBUSTOR: PRECESSION OFVORTEX CORE AND INFLUENCE OF DENSITY GRADIENT

Kiran ManoharanIndian Institute of Science

Department of Aerospace EngineeringBangalore, India

Email: [email protected]

Samuel HansfordPenn State University

Department of Mechanical and Nuclear EngineeringUniversity Park, PA 16802


Jacqueline O’ ConnorPenn State University

Department of Mechanical and Nuclear EngineeringUniversity Park, PA 16802


Santosh HemchandraIndian Institute of Science

Department of Aerospace EngineeringBangalore, India


ABSTRACT

Combustion instability is a serious problem limiting the op-erating envelope of present day gas turbine systems using a leanpremixed combustion strategy. Gas turbine combustors employswirl as a means for achieving fuel-air mixing as well as flamestabilization. However swirl flows are complex flows comprisedof multiple shear layers as well as recirculation zones whichmakes them particularly susceptible to hydrodynamic instabil-ity. We perform a local stability analysis on a family of base flowmodel profiles characteristic of swirling flow that has undergonevortex breakdown as would be the case in a gas turbine combus-tor. A temporal analysis at azimuthal wavenumbers m = 0 andm = 1 reveals the presence of two unstable modes. A compan-ion spatio-temporal analysis shows that the region in base flowparameter space for constant density density flow, over whichm = 1 mode with the lower oscillation frequency is absolutelyunstable, is much larger that that for the corresponding m = 0mode. This suggests that the dominant self-excited unstable be-havior in a constant density flow is an asymmetric, m=1 mode.The presence of a density gradient within the inner shear layerof the flow profile causes the absolutely unstable region for them = 1 to shrink which suggests a possible explanation for the

suppression of the precessing vortex core in the presence of aflame.

NOMENCLATUREDimensional reference quantitiesUz,re f Maximum forward axial flow velocity at a given axial

locationRmax Radius at which maximum axial velocity occursρu Unburnt gas density

Non-dimensional symbolsUz Base flow velocity in the axial directionUθ Base flow velocity in the azimuthal directionS Local swirl numberr f Radial location of the flame r f , normalized by Rmaxγ Density ratio of burnt to unburnt gas ρb/ρuβ reverse flow ratio, i.e. normalized magnitude of max. reverse

flow velocityα Perturbation wave numberαr Real part of wavenumber,= Re(α)αi Spatial growth rate,= Im(α)

1 Copyright c© 2015 by ASME

ω Temporal eigenvalueωr Real part of temporal eigenvalue,= Re(ω)ωi Temporal growth rate, = Im(ω)ωi,max Maximum temporal growth rate for given base flow ve-

locity and density profilesΩ′r Vorticity fluctuation in radial directionΩ′

θVorticity fluctuation in azimuthal direction

Ω′z Vorticity fluctuation in axial directionρ Base flow density profileρ′ Density perturbationu′r Radial-component of velocity perturbationu′

θAzimuthal-component of velocity perturbation

u′z Axial-component of velocity perturbationp′ Hydrodynamic pressure perturbation

Modifiers() Fourier component() Base flow quantity()′ Perturbation around base flow value

1 IntroductionThe combustion instability in lean premixed gas turbine

combustors is caused by coupling between unsteady heat releasefluctuations and the combustor acoustic field. This coupling canpotentially result in amplification of acoustic pressure oscilla-tions in the combustor. This can cause undesirable levels of pol-lutant emissions, structural damage, loss of performance etc. [1].Unsteady heat release in lean premixed combustors is due towrinkling and distortion of the premixed flame sheet by unsteadyflow structures in the combustor flow field. These flow structuresarise from various hydrodynamic instability mechanisms associ-ated with the combustor flow field. Burners in modern gas tur-bine combustors use swirling flows as a means to achieve fuel-airmixing and flame stabilization [2]. Vortex breakdown in thesetypes of flows results in a flow field that is comprised of multipleshear layers and a central recirculation bubble, that are suscepti-ble to multiple instability mechanisms [3,4]. These flow instabil-ity mechanisms result in inherently unsteady self-excited flowstructures such as the Precessing Vortex Core (PVC) in theseflows [5–8]. Therefore, insight into hydrodynamic instabilitymechanisms causing flow unsteadiness in swirl flows is neces-sary, in order to develop physically realistic and quantitativelyaccurate reduced order models for the prediction of combustioninstability.

The dynamics of swirling flows are to a large extent gov-erned by its swirl number, i.e the ratio of the axial flux of tan-gential momentum and the axial flux of axial momentum. Hy-drodynamic instability in swirling jet flows have been studied inthe context of vortex breakdown. Various kinds of vortex break-down behaviour have been observed in swirling flows, eg. bubble

vortex breakdown, spiral vortex breakdown, helical vortex break-down etc. [9]. These breakdown modes primarily determine themechanism of flame stabilization and fuel-air mixing in gas tur-bine combustors. However, swirl numbers in gas turbine com-bustors are nominally much higher than the critical swirl numberat which vortex breakdown occurs. Thus, the mechanism lead-ing to vortex breakdown state has been considered as a primaryhydrodynamic instability mechanism and have been reported inseveral prior studies [9–16]. Practical gas turbine combustors op-erate at high swirl numbers that are much higher than the criticalswirl number beyond which vortex breakdown occurs. There-fore, the unsteady PVC observed in these flows can be concep-tualized as a secondary hydrodynamic instability associated withthe post vortex breakdown flow. Instabilities in flows such asthese can be classified as locally convective or locally absoluteon the basis of the nature of the flow response to an impulsiveperturbation at large subsequent times, i.e. t → ∞ [17, 18]. Im-pulsive perturbation at a point in a convectively unstable (CU)flow, results in spatially growing disturbances that are convectedaway from the point of disturbance. As such, the flow returns toits former quiescent state at large times in the absence of contin-uous forcing. On the other hand, impulsive forcing of an abso-lutely unstable (AU) flow generates disturbances that grow bothtemporally and spatially. Thus, an AU flow acts as a self-excitedoscillator with a well defined, characteristic frequency. CU andAU flows interact with the combustor acoustic field through theunsteady heat release that they generate [19].

Several prior studies have characterized AU/CU nature inswirling jets on the basis of local swirl number and the ratio of re-verse to forward flow using canonical baseflow models [20–25].Post vortex breakdown, swirling flows have shear layers gen-erated by axial and azimuthal velocity gradients and a reverseflow close to centreline, which makes them a candidate for ab-solute instability. Further, these flows have instability mecha-nisms, apart from just shear layer instability, which can interactwith each other and determine the final unsteady nature of theflow [3, 26].

Several theoretical and experimental studies have identifiedmultiple instability mechanisms occurring in purely rotating andswirling jet flows [3, 26–29]. From these studies, the various in-stability mechanisms can be broadly classified into (1) Azimuthalshear layer instability (2) Axial shear layer instability, (3) Cen-trifugal instability and (4) Kelvin’s instability. The azimuthal andaxial shear layer instability are equivalent to the classical Kelvin-Helmholtz type instability, due to the presence of shear layers inthe tangential and axial velocity profiles. These two types ofinstability will be referred to as azimuthal KH and axial KH in-stability in the rest of the paper. The centrifugal and the Kelvin’sinstabilities are due to centrifugal and Coriolis forces acting onrotating fluids respectively.

The centrifugal instability can be explained as follows. Thenet force on a fluid particle in a purely rotating flow is due to


the resultant of the local centrifugal force and the force gener-ated by the the local pressure gradient. Thus, when the fluidparticle is displaced radially away from the axis of rotation, theimbalance between the centrifugal force and the force due to thepressure gradient at the displaced location determines whetherthe particle continues to move away or returns to the initial po-sition [30]. Rayleigh has derived a stability criterion to iden-tify centrifugal instability of rotating flows subjected to axi-symmetric perturbations by taking into consideration the changein square of circulation of the purely rotating laminar base state(see ref. [30, 31]). Gallaire et al [32] have partially generalizedthis criterion for both axi-symmetric and non axi-symmetric per-turbations by showing that purely rotating flows in the large axialwavenumber limit are centrifugally unstable, with the axisym-metric mode being the most centrifugally unstable.

Kelvin’s instability is due to the restoring action of the Cori-olis force on a rotating fluid particles when displaced from theirradial location. This restoring action of the Coriolis force in apure rotating flow produces neutrally stable, inertial waves whichappear in a stability analysis as a continuous spectrum withina well defined frequency range [33]. The interaction betweenthese neutral modes and axial KH modes have been studied byLoiseleux et al [4] where Loiseleux et al has shown that a drasticchange in behaviour occurs when these two modes resonate (ie.when both of the modes are having the same frequency). The in-stability mechanisms observed in a non-reacting flows have beenstudied individually by several groups on various context, butthe interaction of these modes in a reacting flow (which is whatis seen in a typical swirl flow combustor) has not been reportedas per the authors knowledge.

In the present study we first perform an inviscid local tem-poral stability analysis in order to identify unstable modes forthe base flow profiles introduced by Oberleithner et al [34]. Inorder to identify the dominant instability mechanism at variousaxial (α) and azimuthal wavenumber (m) settings, we analyze thesource terms of the linearized vorticity transport equation gener-ated by the unstable mode. These source terms can be identi-fied as being related to each of the above instability mechanisms.Thus, from the relative dominance of one set of terms over theothers allows us to determine which of the four mechanisms isresponsible for driving the instability. Next, we include a densityvariation on the base flow to assess the effect that the presence ofa flame would have on instability characteristics. The densityvariation introduces a fluctuating baroclinic torque which canconstructively or destructively interact with other mechanismsand thereby, modify the instability characteristics of the flow. Wealso perform a local spatio-temporal analysis to identify the ab-solute/convective [19] nature of the unstable modes. We identifyregions of absolute/convective instability in a parametric spacespanned by local swirl number (S), defined as the ratio of themaximum azimuthal to maximum axial flow velocity and backflow ratio (β), defined as the ratio of the magnitude of the cen-

terline axial velocity to the maximum axial velocity. We showthat a density jump within the inner shear layer of the annularjet causes suppression of the absolute instability for the m =±1modes due to the effect the fluctuating baroclinic torque.

The rest of the paper is organized as follows. Section 2 out-lines the mathematical formulation, baseflow model and numer-ical techniques used in the present study. Section 3 enumeratesthe “Linearized vorticity transport” equations used in the presentstudy. Section 4 discusses the results obtained and section 5 con-cludes the paper with an outlook on future work.

2 FormulationThe hydrodynamic stability analysis in the present study is

performed on a nominally steady base flow that is representa-tive of an axisymmetric post vortex breakdown swirling jet asshown schematically in fig. 1. The analysis is performed in thelow Mach number M → 0 limit and the base flow is assumedto be locally parallel, axisymmetric and inviscid. The base flowdensity is allowed to vary spatially. We do this in order to cap-ture the influence of a premixed flame on the stability charac-teristics. Thus, linearizing the Navier-Stokes equations about ageneral and nominally steady base flow yields (eg. see ref. [30]).

∂ρ′

∂t+ ρ(r)

(∂u′r∂r

+u′rr+

1r

∂u′θ

∂θ+

∂u′z∂z

)

+u′rdρ

dr+

Uθ(r)r

∂ρ′

∂θ+Uz(r)

∂ρ′

∂z= 0 (1)

ρ(r)

(∂u′r∂t

+Uθ(r)

r∂u′r∂θ

+Uz∂u′r∂z−

2Uθ(r)u′θr

)

−ρ′U2

θ

r= −∂p′

∂r(2)

ρ(r)

(∂u′

θ

∂t+u′r

dUθ(r)dr

+Uθ(r)

r∂u′

θ

∂θ

+Uz(r)∂u′

θ

∂z+

Uθ(r)u′rr

)= −1

r∂p′

∂θ(3)

ρ(r)

(∂u′z∂t

+u′rdUz(r)

dr+

Uθ(r)r

∂u′z∂θ

+Uz(r)∂u′z∂z

)=−∂p′

∂z(4)


∂u′r∂r

+u′rr+

1r

∂u′θ

∂θ+

∂u′z∂z

= 0 (5)

where the radial, azimuthal and the axial components of veloci-ties in the cylindrical polar coordinate system are given by sub-scripts r,θ and z respectively. The primed quantities representsmall perturbations about their respective base flow values. Theaxial and tangential derivatives of the base flow quantities as wellas the radial component of the base flow velocity (ie. Ur) havebeen neglected in eqs. (1)-(5) (axi-symmetric, locally parallelbase flow assumption). Thus the base flow quantities Uθ, Uz, Pand ρ are assumed to vary only in the radial direction. All lengthsin eqs. 1-5 are non-dimensionalized with the length Rmax givenby radial location of peak axial base flow velocity, Uz,re f . Thelatter is chosen as the velocity scale for non-dimensionalization.Pressure and density have been non-dimensionalized using theirrespective base flow values in the unburnt gas.

Next, the following boundary conditions are imposed at r→∞,

q′→ 0 (6)

where q′ = ρ′,u′r,u′θ,u′z, p′. Kinematic compatibility condi-tions at the centerline (r = 0) proposed by Batchelor et al [35],are imposed as follows,

u′r = u′θ = 0

∂u′z∂r

=∂p′

∂r=

∂ρ′

∂r= 0

if m =0 (7)

ρ′ = u′z = p′ = 0

u′r +∂u′

θ

∂θ= 0

∂u′r∂r

= 0

if |m|= 1 (8)

ρ′ = u′r = u′θ = 0

u′z = p′ = 0

if |m|> 1 (9)

Note that the kinematic compatibility conditions for |m| =1 mode do not cause the radial and azimuthal components ofvelocity fluctuations to go to zero at the centerline. Therefore,

Figure 1. Schematic showing the variation of base flow velocity and den-sity variation along the radial direction.

these fluctuations can displace the axis of rotation of the centralvortex core, thereby, causing it to precess. The unsteady flowfield generated by PVC has been reported to have a helical modestructure (i.e. |m|= 1) in several studies [5,6]. This suggests thatthe presence of unstable |m| = 1 modes eventually results in theformation of the PVC.

Next, the perturbed quantities in the eqs. 1-5 are expressedin the normal mode form as follows,

q′(r,θ,z, t) = q(r)ei(αz+mθ−ωt) (10)

where q = ρ, ur, uθ, uz, p, α is the axial wavenumber and m isthe azimuthal wave number. Using eq. 10 in eqs. 1-9 yields aneigenvalue problem where, in general, ω and α are the eigen-values (parametrized by m) and q is the eigenvector (see Ap-pendix A). The solution to this eigenvalue problem is equiva-lent to solving the dispersion relation, D(α,m,ω) = 0 with thebaseflow specified. The azimuthal wavenumber, m, can in gen-eral take both positive and negative integer values. The positivevalue corresponds to the co-rotating mode, i.e., fluctuations havethe same sense of rotation as the azimuthal base flow velocity andthe negative value corresponds to the counter rotating mode, i.e.fluctuations have the opposite sense of rotation as the azimuthalbase flow velocity. While, the mathematical formulation in thispaper is valid for general m, we restrict our analysis in this pa-per to only co-rotating modes (m > 0) and axisymmetric modes(m = 0). The quantity ω, is the complex temporal eigenvaluewhose real (ωr) and imaginary (ωi) parts represent the oscillationfrequency and the temporal growth rate of the flow perturbations.Similarly , the real (αr) and imaginary (αi) parts of α representthe axial wavenumber and the axial growth rate of flow perturba-tions. The base flow model used in this study is presented next.

2.1 Base flow modelFigure 1 schematically shows a typical time averaged flow

field, observed in a swirl flow combustor. The swirler induces an


(a)

−0.5 0 0.5 10

1

2

Uz

r

Uz

Nf

ρ

rf

(b)

0 0.5 10

1

2

Uθ

r

Figure 2. Typical base flow profile: (a) axial velocity and density profiles,and, (b) azimuthal velocity profile (β = 0.1, N = 4, γ = 0.3, r f = 1.0,N f = 0.2, S = 1.0, N3 = 4.18 and N4 = 0.73).

azimuthal component to the velocity vector. At high swirl num-bers, the flow transitions into a vortex breakdown state whichcreates a central recirculation zone (CRZ) (see fig. 1). The twoshear layers formed due to flow separating from the centerbodyedge and the burner lip, are referred to as the Inner Shear Layer(ISL) and the Outer Shear Layer (OSL) respectively. In addi-tion, there are two Azimuthal Shear Layers (ASL), arising fromthe radial variation of Uθ within and outside the vortex core asshown schematically in fig. 1.

We use the following base flow velocity model for the axial(Uz(r)) and azimuthal (Uθ(r)) velocity components,Axial flow velocity profile ( [34, 36]):

Uz(r) = 4BF1[1−BF1] (11)

where quantity B determines the amount of back flow at the cen-ter line through the following relation,

B = 0.5[1+(1+β)1/2] (12)

The parameter β, is the non-dimensional reverse flow velocitymagnitude at the center line. Azimuthal flow velocity profile isgiven by [34]

Uθ(r) = 4SF3[1−F4] (13)

where, S is the local swirl number which is defined as the ratiobetween the maximum azimuthal velocity to the maximum axialvelocity. The functions, F1, F2, F3 and F4 are defined as follows,

Fj(r) =1[

1+(er2b j −1)N j] , j = 1,2,3 (14)

N3 N4 b3 b4

4.18 0.73 0.51 0.27

Table 1. Table showing the values of azimuthal velocity shear layer pa-rameters (N3 and N4) and fitting parameters b3 and b4 used in thepresent study

The parameters N1 and N2 in eq. 11 determine the thicknessesof the ISL and the OSL respectively. We assume that the ISLthickness and the OSL thickness (see Figure 1) are the same inthis paper, i.e. N1 =N2 =N. The parameter b j is determined for aparticular value of shear layer thickness by constraining the axialvelocity to be maximum at r = 1. The shear layer parameters N3and N4, along with the fitting parameters b3 and b4, are given bythe values specified in table 1.

Flames in swirl flow combustors operating at various condi-tions have been observed to be stabilized within the ISL, the OSLor both [8]. We examine the influence of ISL stabilized flameson the hydrodynamic stability characteristics of the flow in thispaper. The premixed flame is modelled as a variation of densityalong the radial direction as followsDensity profile:

ρ(r) =(1+ γ

2

)+(1− γ

2

)tanh

( r− r f

N f

)(15)

where, N f is the normalized half flame thickness, and r f is thelocation of the inflection point in the density profile. Note thatN f = 0.2 is used for all the reacting flow cases presented in thispaper. The density ratio across the flame is given by γ = ρb/ρu(see fig. 2). Using these base flow profiles (eqs. 11-15) in eq. 1-9, yields an eigenvalue problem that cannot be solved in closedform. Therefore, we use a numerical pseudospectral methodto solve the eigenvalue problem as discussed in the next sub-section.

2.2 Numerical methodThe physical space, r ∈ [0,rmax], is mapped into the compu-

tational space [−1,1] using the mapping function suggested byMalik et al [37] as follows,

rrc

=1+ y

1− y+ 2rcrmax

(16)

The mapping parameter rc in eq. 16 distributes the grid pointsbetween 0 to rmax in such a way that half of the grid points areplaced in between 0 to rc and the other half between rc to rmax.We have set rc = 2 and rmax = 300 for all results presented in this


paper. These values were found sufficient to achieve convergenceof eigenvectors in this study.

Equations 1-5 written in terms of q are transformed intocomputational space using the mapping function in eq.16. Next,these equations are discretized in computational space using thewell known pseudospectral collocation method based on cardi-nal functions [38]. This yields a discrete form of the eigenvalueproblem given by eqs. 1-5, which can be written symbolically asthe following generalized eigenvalue problem,

Aqd = ωBqd (17)

The qd in equation 17 is the discrete equivalent of the vectorq = ρ, ur, uθ, uz, p, evaluated at the collocation points. Thematrices A and B are functions of axial wavenumber α and az-imuthal wavenumber m. Boundary conditions are imposed byreplacing the rows in the matrix A and B with expressions corre-sponding to the r = 0 and r = 300 collocation points with eqs. 6- 9. We solve this the discrete eigenvalue problem (eq. 17) usingthe MATLAB eig function. Converged solutions for eigenval-ues and eigenvectors, qd , are obtained by using 100 collocationpoints for the constant density cases and 150 collocation pointsfor the variable density cases presented in this paper.

The absolute/convective nature of flow instability modesis determined by the nature of response of the flow to an im-pulsive forcing. This is determined by finding (ωs,αs) suchthat dω/dα = 0, i.e. the saddle points of the dispersion rela-tion [17,18]. If Im(ωs)> 0 and Im(αs)< 0, the flow is absolutelyunstable (AU). On the other hand if Im(ωs)< 0 and Im(αs)< 0the flow is convectively unstable (CU). We adopt the algorithmproposed by Deissler [39] to find (ωs,αs) using eq. 17. Thus forgiven values of N, γ and r f , the values of β and S where the flowtransitions from being AU to CU is given by ωs,i(β,S) = 0. Thisequation is solved for different values of S by using newton iter-ation along with the saddle point search algorithm. The newtoniterations are continued until the residual is less than 10−6 for allresults presented in this study.

3 Linearized vorticity transport equationWe use the linearized vorticity transport equation to gain in-

sight into the relative influence of various instability mechanismson the characteristics of the unstable modes in this study. Thevorticity transport equation, linearized about a steady base flow,can be written in the normal mode form as follows,

(∂Ω′r∂t

+Uθ

r∂Ω′r∂θ

+U z∂Ω′r∂z

)=− imur

rdU z

dr︸︷︷︸1

+ iαurdUθ

dr︸︷︷︸2

+ iαurUθ

r︸︷︷︸3

(18)

(∂Ω′

θ

∂t+

Uθ

r∂Ω′

θ

∂θ+U z

∂Ω′θ

∂z

)= ur

(d2U z

dr2 −1r

dU z

dr

)︸︷︷︸

4

− imuθ

rdU z

dr︸︷︷︸5

+

imUz

r

(dUθ

dr−Uθ

r

)︸︷︷︸

6

+2iαUθuθ

r︸︷︷︸7

+1ρ

2

(iαρ

dPdr− iα p

dρ

dr

)︸︷︷︸

8

(19)

(∂Ω′z∂t

+uθ

r∂Ω′z∂θ

+uz∂Ω′z∂z

)=−ur

(d2Uθ

dr2 +1r

dUθ

dr−Uθ

r2

)︸︷︷︸

9

−

iαuθ

dU z

dr︸︷︷︸10

+ iαuz

(dUθ

dr+

Uθ

r

)︸︷︷︸

11

+

1ρ

2r

(imp

dρ

dr− imρ

dPdr

)︸︷︷︸

12

(20)

The source terms on the RHS of eqs. 18-20 represent con-tributions to the net generation rate of vorticity fluctuations fromrearrangement of base flow vorticity by velocity disturbances,unsteady vortex stretching due to base flow velocity gradientsand fluctuating baroclinic torque due to base flow density gra-dients. The contribution to the net generation rate of fluctuat-ing vorticity from various fundamental instability mechanismscan be deduced from eqs. 18-20 as follows. Consider first apurely rotational flow (Uz = 0) and axisymmetric perturbations(ie. m = 0,α > 0). The only instability mechanism driving theflow unsteadiness is due to the centrifugal instability in this case.Hence the terms, 2 , 3 , 7 , 9 and 11 can be identified ascentrifugal instability contributions. Likewise if |m| > 0,α = 0,only azimuthal KH instability mechanism drives flow unsteadi-ness through the source term 9 . Thus the centrifugal and az-imuthal KH instability mechanisms are coupled through term 9which appears as a non-zero source term when only one or theother mechanism alone is present. This term is referred to asthe coupling term in the present study. The relative importanceof azimuthal KH and centrifugal instability due to this term canbe identified from the base flow characteristics and the perturba-tion wave vector orientations with respect to the azimuthal shearlayer. The baroclinic source terms in the above equations appear


(a)

1 2 3−0.5

0

0.5

ωr

ωi

(b)

0 1 20

0.5

1

uz

r

LFmodeHFmode

Figure 3. (a) Typical eigenvalue spectrum and (b) unstable eigenmodeshapes corresponding to the the LF mode (‘+’ marker) and the HF mode(‘x’ marker). (α = 2, m = 1, β = 0.1, N = 4, S = 1.0, γ = 1, N3 =4.18 and N4 = 0.73).

when there are density gradients across a flame in the presentstudy, the contributions from the baroclinic torque to the vortic-ity field is through the terms 8 and 12 . The remaining sourceterms in eqs. 18-20 can be identified as the contribution to thevorticity field from the axial KH instability mechanism. Thus,in the vorticity generation rate budget results presented in theforthcoming section, the componentwise generation rate contri-butions, from any single instability mechanism, is the componen-twise resultant of all unsteady source terms associated with themechanism. So in the rest of the paper, componentwise sum ofterms corresponding to centrifugal instability (ie. 2 , 3 , 7 , 9and 11 ) is referred to as “centrifugal” in each component of theunsteady vorticity generation rate. Likewise componentwise sumof terms corresponding to azimuthal KH instability and axial KHinstability will be referred to as “azimuthal KH” and “axial KH”respectively in each component of the unsteady vorticity gener-ation rate. We compute source terms of the eqs. 18- 20 usingthe eigenvectors corresponding to eigenvalues of interest in thepresent paper. Thus, the dominant mechanism causing instabil-ity for that eigenmode can be identified from spatial source termbudgets.

4 ResultsFigure 3a shows a typical eigenvalue spectrum, determined

from temporal analysis (α = 2, m = 1, β = 0.1, S = 1.0, N = 4,γ = 1, N3 = 4.18 and N4 = 0.73). This result is typical and isseen for other values of α and m as well. The two physically rel-evant unstable egenvalues are shown using ‘+’ and ‘x’ markers infig. 3. The two unstable eigenvalues are due to the stretching andrearrangement of vortex tubes within the inner and outer shearlayers induced by various instability mechanisms (see section 3).The eigenvalue corresponding to the marker ‘+’ is denoted as thelow frequency (LF) mode as it always has a lower frequency thanthe other mode for all cases studied in this paper. Therefore, theeigenvalue corresponding to ‘x’ maker in fig. 3a is denoted as

(a)

0.5 1 1.50

0.5

1

r

Mag.θ-component

Axial KH

Centrifugal

Azimuthal KH

(b)

0 0.5 1 1.50

0.2

0.4

r

Mag.z-component

Azimuthal KHAxial KH

Centrifugal

Figure 4. Spatial budgets of fluctuating vorticity generation rate sourceterms for (a) Ω′

θand (b) Ω′z, showing contributions from various instability

mechanisms for the LF mode (α = 2, m = 1, β = 0.1, N = 4, S = 1.0,γ = 1, N3 = 4.18 and N4 = 0.73).

the high frequency (HF) mode. The cluster of eigenvalues closeto the real axis is an artifact of the discrete representation of thecontinuous spectrum of the dispersion relation, as well as, somespurious eigenvalues arising from the numerical discretization.Figure 3b shows the spatial variation of |uz| eigenvector mag-nitude (normalized with the maximum of all eigenvectors) cor-responding to the LF and HF unstable modes. Notice that theamplitude of these modes is large at locations within the two ax-ial shear layers, suggesting that the instability of these modesis driven primarily by the KH instability caused by shear in theaxial flow profiles. This fact can be understood from spatial bud-gets of fluctuating vorticity generation rate source terms. Fig-ure 4 shows the spatial variation of source terms of Ω′

θand Ω′z

components corresponding to a disturbance identical to the LFeigenmode in fig. 4a. The source terms of both the componentshave been decomposed into contributions from axial KH insta-bility, azimuthal KH instability and centrifugal instability as dis-cussed in section 3. All data shown in fig. 4 are normalized bythe maximum magnitude of axial KH instability source term as-sociated with the Ω′

θcomponent. Figures 4a-b clearly shows that

the axial KH instability source term in Ω′θ

dominates over thesource terms from all other instability mechanisms - implyingthat this is the dominant driving mechanism for the combinationof parameters corresponding to the result in fig. 3.

We next present results concerning the absolute/convectivenature of these unstable modes for the constant density case, asdetermined from the spatio-temporal analysis. The HF mode wasfound to be convectively unstable over the entire range of S, β

and m considered in this study. Figure 5 shows the transitionboundaries between flow profiles showing AU and CU behaviorof the LF mode for m=0 and m=1, corresponding to values ofN = 3 and 4 in each case (γ = 1). The region to the left of eachof the boundaries corresponds to CU flow profiles.

Consider first the m = 0 mode. Figure 5 shows a largechange in the location of the AU/CU transition boundary at highvalues of S with decreasing N. This is because, for m=0, the


0 0.2 0.4 0.60

0.5

1

β

S

AU

m = 0, N=3

AU

CU

m = 1

m = 0, N=4

P

CU

S

Figure 5. Boundary between absolutely unstable and convectively un-stable behavior for various values of m and axial shear layer parametersN (γ = 1). The arrows marked AU and CU respectively point into regionsof absolutely and convectively unstable flow.

(a)

0.5 1 1.50

0.5

1

r

Mag.θ-component

Centrifugal

Axial KH

(b)

0.5 1 1.50

0.5

1

r

Mag.θ-component

Centrifugal Axial KH

Figure 6. Spatial budgets of Ω′θ

source term, showing contributions fromvarious instability mechanisms at the saddle point (a) S= 0.3 (b) S= 0.9in fig. 5 (m = 0, β = 0.4, N = 3, γ = 1, N3 = 4.18 and N4 = 0.73).

centrifugal instability mechanism becomes increasingly impor-tant at larger swirl numbers. This fact can be seen from fig. 6which shows the spatial variation of fluctuating vorticity genera-tion rate source terms for disturbance identical to the eigenmodescorresponding for (ωs,αs) at (β,S) = (0.4,0.3) and (0.4,0.9),i.e., points ‘P’ and ‘S’ in fig. 5. The contribution from cen-trifugal instability terms to the generation rate of the Ω′

θcom-

ponent is comparable to the corresponding contribution from theaxial KH mechanism for S = 0.9 case (see fig. 6b), as opposedto the S = 0.3 case (see fig. 6a) where the Axial KH contribu-tion is dominant. Thus, decreasing in the value of N, causesthe shear layers in the axial base flow velocity profile to weakenand hence, weakens the quantitative influence of the axial KHinstability mechanism at all values of S. Thus, the centrifugalinstability mechanism becomes dominant at smaller values of Sresulting in a larger change in the AU/CU boundary location forS > 0.2 when compared to S < 0.2 (see fig. 6). Note that con-tributions from the azimuthal KH mechanism do not exist for anaxi-symmetric (m = 0) type disturbances. The same behaviouris seen for the Ω′z component as well and is not shown in theinterest of brevity.

(a)

0.5 1 1.50

0.5

1

r

Mag.θ-component

Centrifugal

Azimuthal KH

Axial KH

(b)

0.5 1 1.50

0.2

0.4

r

Mag.z-component

Azimuthal KH

Axial KH

Centrifugal

Figure 7. Spatial budgets of fluctuating vorticity generation rate sourceterms, showing contributions from various instability mechanisms at thesaddle point (a) Ω′

θ(b) Ω′z (m = 1, β = 0.1, S = 0.5, N = 4, γ = 1,

N3 = 4.18 and N4 = 0.73).

Consider next the m=1 case shown in fig. 5. The change inN has a minimal effect on the location of the AU/CU boundary.Further, the region of convective instability is very small whencompared to that of m=0 and occurs at much smaller values ofboth β and S. Figure 7 shows the contribution of each instabil-ity mechanism to Ω′

θand Ω′z components at (β,S)=(0.1,0.5) in

fig. 5 (m = 1, N = 4, γ = 1, N3 = 4.18 and N4 = 0.73). Figure 7ashows that the axial KH instability mechanism dominates over allother instability mechanisms and hence axial KH instability de-termines the instability characteristics of m=1 mode. This plot istypical and has been found to have the same behaviour for othervalues of base flow parameters. Next, fig. 5 also shows that thismode can be expected to dominate the overall unsteady dynam-ics of the flow since it is AU over the entire region over whichthe m=0 low frequency mode is convectively unstable. This isa clear indication of the fact that the predominant self-excitedbehavior in a nominally axi-symmetric swirl flow would be ex-pected to be a helical, m=1 instability. This fact is in qualitativeagreement with recent experimental observations [40]. Next, weconsider the influence of a density variation generated by a pre-mixed flame on these absolute/convective instability transitionboundaries.

Figure 8 shows the AU/CU transition boundaries for the den-sity jump located within the ISL, i.e. r f = 1.0,0.5, γ = 0.3 forthe m=0 low frequency mode (N=4). The boundary for the γ = 1case is also shown for reference. The large change in the loca-tion of the boundary is due to the additional influence of fluc-tuating baroclinic torque on the net generation rate of fluctuat-ing vorticity in the flow (see eq.19). The strong stabilization ofthe absolute instability for the r f = 1.0 case can be understoodas follows. Figure 9a which shows the spatial variation of themagnitude of the source terms due to rearrangement and stretch-ing and the fluctuating baroclinic torque for the Ω′

θcomponent

due to the m=0 LF mode. This plot is typical and correspondsto (β,S) = (0.35,0.50) (point Q in fig.8). The correspondingphase difference between these two source terms is also shown


0 0.1 0.2 0.3 0.4 0.50

0.5

1

β

S

AU

CU

γ=0.3

rf = 0.5

γ=0.3

rf = 1.0

γ=1Q

Figure 8. Boundary between absolutely unstable and convectively un-stable behavior for various locations of the base flow density gradient(N = 4, γ = 0.3 and m = 0). The boundary for γ = 1 (thick blackline) is also shown for reference.

on the vertical axis on the right. Clearly the baroclinic torqueterms peaks close to where the resultant of all other source termspeak while being out of phase with the latter. Thus the baroclinictorque counteracts the production of vorticity by rearrangementand stretching causing the mode to become temporally stable(and hence CU) for the (ωs,αs) values corresponding to pointQ. The opposite result can be seen from fig. 9b which is a similarsource term plot for r f = 0.5. For r f = 0.5 case, the fluctuat-ing baroclinic torque peaks at a location (ie. r ≈ 0.5) where thecontribution from resultant of other source terms are negligible.Thus, in this case, the fluctuating baroclinic torque increases thenet fluctuating vorticity generation rate causing the flow to be-come temporally unstable for the (ωs,αs) values at Q and hence,absolutely unstable. Thus, it is clear that the density gradientcan have a significant stabilizing or destabilizing impact on theabsolute/convective nature of the instability corresponding to them=0 low frequency mode, depending on the location of the den-sity gradient.

Similar results can be seen for the m=1 mode from fig. 10which shows the location of the AU-CU transition boundary forvarious values of γ and r f (N=4). Also shown is the γ = 1 casefor comparison. The presence of the baroclinic torque term, re-sults in the m=1 low frequency mode becoming CU over a largerregion of the β−S plane when compared to the constant densitycase. The reason for this can again be seen in fig. 11(a)-(b) whichplots the spatial variation of the magnitude of the vorticity gener-ation rate source terms, of Ω′

θand Ω′z. These plots correspond to

(β,S) = (0.3,0.5) (point R in fig. 10) for γ = 0.3,r f = 1.0. Notethat the baroclinic contribution is in phase with the net contribu-tion from all other source terms resulting in the flow becomingAU (see fig. 11(a)). However, the opposite is true for the cor-responding result shown in fig. 12(a)-(b) for γ = 0.3, r f = 0.75.Further, the result in fig. 10 shows that the region of absolute in-stability is greatly diminished due to the influence of baroclinictorque generated by the density variation. This suggests that the

(a)

0

0.5

1

r

Mag.θ-component

0.6 0.8 1−100

0

100

Phas

e D

iff.

(deg

rees

)

Baroclinic

Rearrangement+Stretching

(b)

0.2 0.4 0.6 0.8 1 1.20

0.5

1

r

Mag.θ-component

−300

−200

−100

0

Phas

e D

iff.

(deg

rees

)

Baroclinic


Figure 9. Spatial budgets of fluctuating vorticity generation rate sourceterms for the Ω′

θcomponent due to the m = 0 impulse response mode at

point Q (see fig. 8), from rearrangement, stretching and baroclinic mech-anisms (a) r f = 1.0 (b) r f = 0.5 (N = 4).

0 0.1 0.2 0.3 0.4 0.50

0.5

1

β

S

γ=0.3

rf = 1.0

γ=1.0

CU

AU

Rγ=0.1

rf = 1.0

γ=0.3

rf = 0.75

Figure 10. Boundary between absolutely unstable and convectively un-stable behavior for different locations of the base flow density gradientand density change parameter γ (N = 4 and m = 1). The boundary forγ = 1 (thick black line) is also shown for reference.

PVC, which is associated with a m = 1 type instability, shouldbe suppressed in the presence of a density gradient in the ISLwhich is in qualitative agreement with prior experimental obser-vations [41].

5 ConclusionThis paper analyzes local convective/absolute instability

transition for velocity and density profiles characteristic of pre-


(a)

0.6 0.8 10

0.5

1

r

Mag.θ-component

0

100

200

300

Phas

e D

iff.

(deg

rees

)

Baroclinic


(b)

0.6 0.8 10

0.5

1

r

Mag.z-component

−200

0

200

400

Phas

e D

iff.

(deg

rees

)

Baroclinic


Figure 11. Spatial budgets of fluctuating vorticity generation rate sourceterms at point R in fig. 10 due to rearrangement, stretching and baroclinictorque for r f = 1.0 and γ = 0.3 (a) Ω′

θsource term (b) Ω′z source term

(N = 4).

(a)

0.4 0.6 0.8 10

0.5

1

r

Mag.θ-component

0

100

200

Ph

ase

Dif

f. (

deg

rees

)

Baroclinic


(b)

0.4 0.6 0.8 10

0.1

0.2

r

Mag.z-component

−100

0

100

200

300

Ph

ase

Dif

f. (

deg

rees

)

Baroclinic


Figure 12. Spatial budgets of fluctuating vorticity generation rate sourceterms at point R in fig. 10 due to rearrangement, stretching and baroclinictorque for r f = 0.75 and γ = 0.3 (a) Ω′

θsource term (b) Ω′z source term

(N = 4).

mixed swirl stabilized combustors. We do this via a spatio-temporal hydrodynamic stability analysis about a locally paral-lel base flow with spatially varying density (as would be gener-ated by a premixed flame), in the inviscid and low Mach numberlimit. Flows in swirl stabilized combustors are inherently com-plex as they are comprised of multiple shear layers and recircula-tion zones. In addition to the presence of shear layer instabilities,swirl flows can also be destabilized by other instability mecha-nisms such as the centrifugal instability and Kelvin’s instability.These different mechanisms are in general coupled and manifestthemselves with varying importance at different flow conditions.

The initial temporal analysis of the base flow profiles con-sidered, revealed the presence of two unstable modes. Of these,only the mode with the lower oscillation frequency showed tran-sition between absolute and convective behavior. Interestingly,this study revealed that the first asymmetric mode, correspondingto an azimuthal wavenumber, m = 1, showed a very large regionof absolute instability behavior when compared to the symmet-ric m = 0 mode. This suggests that the eventual, self-excited,unsteady behavior of the flow would be dominated by an m = 1instability - a fact borne out by several observations of helicalPrecessing Vortex Core (PVC) phenomena in constant densityswirl flows.

The inclusion of a flame induced density gradient in the in-ner shear layer (ISL) resulted in a dramatic stabilization of theabsolute instability for the m=1 mode due to the influence ofbaroclinic torque. Thus, the present analysis suggests a possi-ble reason to explain the suppression of the PVC in the presenceof a flame - a fact that has been borne out by recent experiments.However, the actual location of the AU-CU transition boundaryin the parameter space considered in this study is a strong func-tion of the density gradient magnitude as well as its dispositionrelative to the shear layers in the flow. The same fact is seen tobe true for the symmetric, m = 0 mode as well.

Appendix A: Linearized inviscid Navier-Stokes equa-tions: Normal mode form

Using eq. 10 in eqs. 1-9 yields the following system of equa-tions,

−iωρ+ ρ

(imuθ

r+ iαuz

)+

imUθρ

r+ ur

dρ

dr

iαUz(r)ρ = 0 (21)

ρ

(− iωur +

imUθur

r+ iαUzur−

2Uθuθ

r

)

−U2

θρ

r= −d p

dr(22)


ρ

(− iωuθ + ur

dUθ

dr+

imUθuθ

r

+iαUzuθ +Uθur

r

)= − imp

r(23)

ρ

(− iωuz + ur

dUz

dr+

imUθuz

r+ iαUzuz

)=−iα p (24)

dur

dr+

ur

r+

imuθ

r+ iαuz = 0 (25)

Next, the boundary conditions in the normal mode form atr→ ∞ and at the centerline (r = 0) are obtained using eq. 10,

q→ 0 (26)

ur = uθ = 0duz

dr=

d pdr

=dρ

dr= 0

if m =0 (27)

ρ = uz = p = 0ur + imuθ = 0

dur

dr= 0

if |m|= 1 (28)

ρ = ur = uθ = 0uz = p = 0

if |m|> 1 (29)

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