Proceedings of ASME Turbo Expo 2015: Turbine Technical Conference and Exposition
GT2015
June 16-20, 2015, Montreal, Canada
GT2015-42985
INSTABILITY MECHANISM IN A SWIRL FLOW COMBUSTOR: PRECESSION OF
VORTEX CORE AND INFLUENCE OF DENSITY GRADIENT
Kiran Manoharan
Indian Institute of Science
Department of Aerospace Engineering
Bangalore, India
Email: [email protected]
Samuel Hansford
Penn State University
Department of Mechanical and Nuclear Engineering
University Park, PA 16802
Email: [email protected]
Jacqueline O’ Connor
Penn State University
Department of Mechanical and Nuclear Engineering
University Park, PA 16802
Email: [email protected]
Santosh Hemchandra
Indian Institute of Science
Department of Aerospace Engineering
Bangalore, India
Email: [email protected]
ABSTRACT
suppression of the precessing vortex core in the presence of a
flame.
Combustion instability is a serious problem limiting the operating envelope of present day gas turbine systems using a lean
premixed combustion strategy. Gas turbine combustors employ
swirl as a means for achieving fuel-air mixing as well as flame
stabilization. However swirl flows are complex flows comprised
of multiple shear layers as well as recirculation zones which
makes them particularly susceptible to hydrodynamic instability. We perform a local stability analysis on a family of base flow
model profiles characteristic of swirling flow that has undergone
vortex breakdown as would be the case in a gas turbine combustor. A temporal analysis at azimuthal wavenumbers m = 0 and
m = 1 reveals the presence of two unstable modes. A companion spatio-temporal analysis shows that the region in base flow
parameter space for constant density density flow, over which
m = 1 mode with the lower oscillation frequency is absolutely
unstable, is much larger that that for the corresponding m = 0
mode. This suggests that the dominant self-excited unstable behavior in a constant density flow is an asymmetric, m=1 mode.
The presence of a density gradient within the inner shear layer
of the flow profile causes the absolutely unstable region for the
m = 1 to shrink which suggests a possible explanation for the
NOMENCLATURE
Dimensional reference quantities
Uz,re f Maximum forward axial flow velocity at a given axial
location
Rmax Radius at which maximum axial velocity occurs
ρ̄u Unburnt gas density
Non-dimensional symbols
Ūz Base flow velocity in the axial direction
Ūθ Base flow velocity in the azimuthal direction
S Local swirl number
r f Radial location of the flame r f , normalized by Rmax
γ Density ratio of burnt to unburnt gas ρ̄b /ρ̄u
β reverse flow ratio, i.e. normalized magnitude of max. reverse
flow velocity
α Perturbation wave number
αr Real part of wavenumber,= Re(α)
αi Spatial growth rate,= Im(α)
1
c 2015 by ASME
Copyright vortex breakdown, spiral vortex breakdown, helical vortex breakdown etc. [9]. These breakdown modes primarily determine the
mechanism of flame stabilization and fuel-air mixing in gas turbine combustors. However, swirl numbers in gas turbine combustors are nominally much higher than the critical swirl number
at which vortex breakdown occurs. Thus, the mechanism leading to vortex breakdown state has been considered as a primary
hydrodynamic instability mechanism and have been reported in
several prior studies [9–16]. Practical gas turbine combustors operate at high swirl numbers that are much higher than the critical
swirl number beyond which vortex breakdown occurs. Therefore, the unsteady PVC observed in these flows can be conceptualized as a secondary hydrodynamic instability associated with
the post vortex breakdown flow. Instabilities in flows such as
these can be classified as locally convective or locally absolute
on the basis of the nature of the flow response to an impulsive
perturbation at large subsequent times, i.e. t → ∞ [17, 18]. Impulsive perturbation at a point in a convectively unstable (CU)
flow, results in spatially growing disturbances that are convected
away from the point of disturbance. As such, the flow returns to
its former quiescent state at large times in the absence of continuous forcing. On the other hand, impulsive forcing of an absolutely unstable (AU) flow generates disturbances that grow both
temporally and spatially. Thus, an AU flow acts as a self-excited
oscillator with a well defined, characteristic frequency. CU and
AU flows interact with the combustor acoustic field through the
unsteady heat release that they generate [19].
Several prior studies have characterized AU/CU nature in
swirling jets on the basis of local swirl number and the ratio of reverse to forward flow using canonical baseflow models [20–25].
Post vortex breakdown, swirling flows have shear layers generated by axial and azimuthal velocity gradients and a reverse
flow close to centreline, which makes them a candidate for absolute instability. Further, these flows have instability mechanisms, apart from just shear layer instability, which can interact
with each other and determine the final unsteady nature of the
flow [3, 26].
Several theoretical and experimental studies have identified
multiple instability mechanisms occurring in purely rotating and
swirling jet flows [3, 26–29]. From these studies, the various instability mechanisms can be broadly classified into (1) Azimuthal
shear layer instability (2) Axial shear layer instability, (3) Centrifugal instability and (4) Kelvin’s instability. The azimuthal and
axial shear layer instability are equivalent to the classical KelvinHelmholtz type instability, due to the presence of shear layers in
the tangential and axial velocity profiles. These two types of
instability will be referred to as azimuthal KH and axial KH instability in the rest of the paper. The centrifugal and the Kelvin’s
instabilities are due to centrifugal and Coriolis forces acting on
rotating fluids respectively.
The centrifugal instability can be explained as follows. The
net force on a fluid particle in a purely rotating flow is due to
ω Temporal eigenvalue
ωr Real part of temporal eigenvalue,= Re(ω)
ωi Temporal growth rate, = Im(ω)
ωi,max Maximum temporal growth rate for given base flow velocity and density profiles
Ω0r Vorticity fluctuation in radial direction
Ω0θ Vorticity fluctuation in azimuthal direction
Ω0z Vorticity fluctuation in axial direction
ρ̄ Base flow density profile
ρ0 Density perturbation
u0r Radial-component of velocity perturbation
u0θ Azimuthal-component of velocity perturbation
u0z Axial-component of velocity perturbation
p0 Hydrodynamic pressure perturbation
Modifiers
ˆ Fourier component
()
¯ Base flow quantity
()
()0 Perturbation around base flow value
1
Introduction
The combustion instability in lean premixed gas turbine
combustors is caused by coupling between unsteady heat release
fluctuations and the combustor acoustic field. This coupling can
potentially result in amplification of acoustic pressure oscillations in the combustor. This can cause undesirable levels of pollutant emissions, structural damage, loss of performance etc. [1].
Unsteady heat release in lean premixed combustors is due to
wrinkling and distortion of the premixed flame sheet by unsteady
flow structures in the combustor flow field. These flow structures
arise from various hydrodynamic instability mechanisms associated with the combustor flow field. Burners in modern gas turbine combustors use swirling flows as a means to achieve fuel-air
mixing and flame stabilization [2]. Vortex breakdown in these
types of flows results in a flow field that is comprised of multiple
shear layers and a central recirculation bubble, that are susceptible to multiple instability mechanisms [3,4]. These flow instability mechanisms result in inherently unsteady self-excited flow
structures such as the Precessing Vortex Core (PVC) in these
flows [5–8]. Therefore, insight into hydrodynamic instability
mechanisms causing flow unsteadiness in swirl flows is necessary, in order to develop physically realistic and quantitatively
accurate reduced order models for the prediction of combustion
instability.
The dynamics of swirling flows are to a large extent governed by its swirl number, i.e the ratio of the axial flux of tangential momentum and the axial flux of axial momentum. Hydrodynamic instability in swirling jet flows have been studied in
the context of vortex breakdown. Various kinds of vortex breakdown behaviour have been observed in swirling flows, eg. bubble
2
c 2015 by ASME
Copyright the resultant of the local centrifugal force and the force generated by the the local pressure gradient. Thus, when the fluid
particle is displaced radially away from the axis of rotation, the
imbalance between the centrifugal force and the force due to the
pressure gradient at the displaced location determines whether
the particle continues to move away or returns to the initial position [30]. Rayleigh has derived a stability criterion to identify centrifugal instability of rotating flows subjected to axisymmetric perturbations by taking into consideration the change
in square of circulation of the purely rotating laminar base state
(see ref. [30, 31]). Gallaire et al [32] have partially generalized
this criterion for both axi-symmetric and non axi-symmetric perturbations by showing that purely rotating flows in the large axial
wavenumber limit are centrifugally unstable, with the axisymmetric mode being the most centrifugally unstable.
Kelvin’s instability is due to the restoring action of the Coriolis force on a rotating fluid particles when displaced from their
radial location. This restoring action of the Coriolis force in a
pure rotating flow produces neutrally stable, inertial waves which
appear in a stability analysis as a continuous spectrum within
a well defined frequency range [33]. The interaction between
these neutral modes and axial KH modes have been studied by
Loiseleux et al [4] where Loiseleux et al has shown that a drastic
change in behaviour occurs when these two modes resonate (ie.
when both of the modes are having the same frequency). The instability mechanisms observed in a non-reacting flows have been
studied individually by several groups on various context, but
the interaction of these modes in a reacting flow (which is what
is seen in a typical swirl flow combustor) has not been reported
as per the authors knowledge.
In the present study we first perform an inviscid local temporal stability analysis in order to identify unstable modes for
the base flow profiles introduced by Oberleithner et al [34]. In
order to identify the dominant instability mechanism at various
axial (α) and azimuthal wavenumber (m) settings, we analyze the
source terms of the linearized vorticity transport equation generated by the unstable mode. These source terms can be identified as being related to each of the above instability mechanisms.
Thus, from the relative dominance of one set of terms over the
others allows us to determine which of the four mechanisms is
responsible for driving the instability. Next, we include a density
variation on the base flow to assess the effect that the presence of
a flame would have on instability characteristics. The density
variation introduces a fluctuating baroclinic torque which can
constructively or destructively interact with other mechanisms
and thereby, modify the instability characteristics of the flow. We
also perform a local spatio-temporal analysis to identify the absolute/convective [19] nature of the unstable modes. We identify
regions of absolute/convective instability in a parametric space
spanned by local swirl number (S), defined as the ratio of the
maximum azimuthal to maximum axial flow velocity and back
flow ratio (β), defined as the ratio of the magnitude of the cen-
terline axial velocity to the maximum axial velocity. We show
that a density jump within the inner shear layer of the annular
jet causes suppression of the absolute instability for the m = ±1
modes due to the effect the fluctuating baroclinic torque.
The rest of the paper is organized as follows. Section 2 outlines the mathematical formulation, baseflow model and numerical techniques used in the present study. Section 3 enumerates
the “Linearized vorticity transport” equations used in the present
study. Section 4 discusses the results obtained and section 5 concludes the paper with an outlook on future work.
2
Formulation
The hydrodynamic stability analysis in the present study is
performed on a nominally steady base flow that is representative of an axisymmetric post vortex breakdown swirling jet as
shown schematically in fig. 1. The analysis is performed in the
low Mach number M → 0 limit and the base flow is assumed
to be locally parallel, axisymmetric and inviscid. The base flow
density is allowed to vary spatially. We do this in order to capture the influence of a premixed flame on the stability characteristics. Thus, linearizing the Navier-Stokes equations about a
general and nominally steady base flow yields (eg. see ref. [30]).
∂u0r u0r 1 ∂u0θ ∂u0z
∂ρ0
+ ρ̄(r)
+ +
+
∂t
∂r
r
r ∂θ
∂z
+u0r
!
d ρ̄ Ūθ (r) ∂ρ0
∂ρ0
+
+ Ūz (r)
=0
dr
r ∂θ
∂z
∂u0r Ūθ (r) ∂u0r
∂u0 2Ūθ (r)u0θ
+
+ Ūz r −
ρ̄(r)
∂t
r ∂θ
∂z
r
−
ρ̄(r)
!
ρ0Ūθ2
∂p0
=−
(2)
r
∂r
∂u0θ
dŪθ (r) Ūθ (r) ∂u0θ
+ u0r
+
∂t
dr
r ∂θ
!
∂u0θ Ūθ (r)u0r
1 ∂p0
+Ūz (r)
+
=−
∂z
r
r ∂θ
∂u0z
∂u0
dŪz (r) Ūθ (r) ∂u0z
ρ̄(r)
+ u0r
+
+ Ūz (r) z
∂t
dr
r ∂θ
∂z
3
(1)
!
=−
(3)
∂p0
(4)
∂z
c 2015 by ASME
Copyright ∂u0r u0r 1 ∂u0θ ∂u0z
+ +
+
=0
∂r
r
r ∂θ
∂z
(5)
where the radial, azimuthal and the axial components of velocities in the cylindrical polar coordinate system are given by subscripts r, θ and z respectively. The primed quantities represent
small perturbations about their respective base flow values. The
axial and tangential derivatives of the base flow quantities as well
as the radial component of the base flow velocity (ie. Ūr ) have
been neglected in eqs. (1)-(5) (axi-symmetric, locally parallel
base flow assumption). Thus the base flow quantities Ūθ , Ūz , P̄
and ρ̄ are assumed to vary only in the radial direction. All lengths
in eqs. 1-5 are non-dimensionalized with the length Rmax given
by radial location of peak axial base flow velocity, Ūz,re f . The
latter is chosen as the velocity scale for non-dimensionalization.
Pressure and density have been non-dimensionalized using their
respective base flow values in the unburnt gas.
Next, the following boundary conditions are imposed at r →
∞,
q0 → 0
Figure 1.
these fluctuations can displace the axis of rotation of the central
vortex core, thereby, causing it to precess. The unsteady flow
field generated by PVC has been reported to have a helical mode
structure (i.e. |m| = 1) in several studies [5,6]. This suggests that
the presence of unstable |m| = 1 modes eventually results in the
formation of the PVC.
Next, the perturbed quantities in the eqs. 1-5 are expressed
in the normal mode form as follows,
(6)
q0 (r, θ, z,t) = q̂(r)ei(αz+mθ−ωt)
where q0 = {ρ0 , u0r , u0θ , u0z , p0 }. Kinematic compatibility conditions at the centerline (r = 0) proposed by Batchelor et al [35],
are imposed as follows,

u0r = u0θ = 0
if m =0
∂u0z
∂p0
∂ρ0
=
=
= 0
∂r
∂r
∂r

ρ0 = u0z = p0 = 0




∂u0θ
0
ur +
= 0 if |m| = 1
∂θ



∂u0r

= 0
∂r
)
ρ0 = u0r = u0θ = 0
u0z = p0 = 0
if |m| > 1
Schematic showing the variation of base flow velocity and den-
sity variation along the radial direction.
(10)
where q̂ = {ρ̂, ûr , ûθ , ûz , p̂}, α is the axial wavenumber and m is
the azimuthal wave number. Using eq. 10 in eqs. 1-9 yields an
eigenvalue problem where, in general, ω and α are the eigenvalues (parametrized by m) and q̂ is the eigenvector (see Appendix A). The solution to this eigenvalue problem is equivalent to solving the dispersion relation, D(α, m, ω) = 0 with the
baseflow specified. The azimuthal wavenumber, m, can in general take both positive and negative integer values. The positive
value corresponds to the co-rotating mode, i.e., fluctuations have
the same sense of rotation as the azimuthal base flow velocity and
the negative value corresponds to the counter rotating mode, i.e.
fluctuations have the opposite sense of rotation as the azimuthal
base flow velocity. While, the mathematical formulation in this
paper is valid for general m, we restrict our analysis in this paper to only co-rotating modes (m > 0) and axisymmetric modes
(m = 0). The quantity ω, is the complex temporal eigenvalue
whose real (ωr ) and imaginary (ωi ) parts represent the oscillation
frequency and the temporal growth rate of the flow perturbations.
Similarly , the real (αr ) and imaginary (αi ) parts of α represent
the axial wavenumber and the axial growth rate of flow perturbations. The base flow model used in this study is presented next.
(7)
(8)
(9)
Note that the kinematic compatibility conditions for |m| =
1 mode do not cause the radial and azimuthal components of
velocity fluctuations to go to zero at the centerline. Therefore,
2.1
Base flow model
Figure 1 schematically shows a typical time averaged flow
field, observed in a swirl flow combustor. The swirler induces an
4
c 2015 by ASME
Copyright 2
r
1
2
ρ̄
Ūz
Nf
(a)
N4
b3
b4
4.18
0.73
0.51
0.27
r 1
Table 1. Table showing the values of azimuthal velocity shear layer parameters (N3 and N4 ) and fitting parameters b3 and b4 used in the
present study
rf
0
−0.5
N3
0
Ūz
0.5
0
0
1
0.5
Ūθ
1
The parameters N1 and N2 in eq. 11 determine the thicknesses
of the ISL and the OSL respectively. We assume that the ISL
thickness and the OSL thickness (see Figure 1) are the same in
this paper, i.e. N1 = N2 = N. The parameter b j is determined for a
particular value of shear layer thickness by constraining the axial
velocity to be maximum at r = 1. The shear layer parameters N3
and N4 , along with the fitting parameters b3 and b4 , are given by
the values specified in table 1.
Flames in swirl flow combustors operating at various conditions have been observed to be stabilized within the ISL, the OSL
or both [8]. We examine the influence of ISL stabilized flames
on the hydrodynamic stability characteristics of the flow in this
paper. The premixed flame is modelled as a variation of density
along the radial direction as follows
Density profile:
(b)
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 numbers, the flow transitions into a vortex breakdown state which
creates a central recirculation zone (CRZ) (see fig. 1). The two
shear layers formed due to flow separating from the centerbody
edge and the burner lip, are referred to as the Inner Shear Layer
(ISL) and the Outer Shear Layer (OSL) respectively. In addition, there are two Azimuthal Shear Layers (ASL), arising from
the radial variation of U¯θ within and outside the vortex core as
shown schematically in fig. 1.
We use the following base flow velocity model for the axial
(Ūz (r)) and azimuthal (Ūθ (r)) velocity components,
Axial flow velocity profile ( [34, 36]):
Ūz (r) = 4BF1 [1 − BF1 ]
ρ̄(r) =
(12)
The parameter β, is the non-dimensional reverse flow velocity
magnitude at the center line. Azimuthal flow velocity profile is
given by [34]
Ūθ (r) = 4SF3 [1 − F4 ]
1−γ
2
tanh
r −r f
Nf
(15)
2.2
Numerical method
The physical space, r ∈ [0, rmax ], is mapped into the computational space [−1, 1] using the mapping function suggested by
Malik et al [37] as follows,
(13)
1+y
r
=
c
rc
1 − y + r2r
max
where, S is the local swirl number which is defined as the ratio
between the maximum azimuthal velocity to the maximum axial
velocity. The functions, F1 , F2 , F3 and F4 are defined as follows,
1
i , j = 1, 2, 3
Fj (r) = h
2b
r
1 + (e j − 1)N j
2
+
where, N f is the normalized half flame thickness, and r f is the
location of the inflection point in the density profile. Note that
N f = 0.2 is used for all the reacting flow cases presented in this
paper. 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. 19, yields an eigenvalue problem that cannot be solved in closed
form. Therefore, we use a numerical pseudospectral method
to solve the eigenvalue problem as discussed in the next subsection.
(11)
where quantity B determines the amount of back flow at the center line through the following relation,
B = 0.5[1 + (1 + β)1/2 ]
1+γ
(16)
The mapping parameter rc in eq. 16 distributes the grid points
between 0 to rmax in such a way that half of the grid points are
placed 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
(14)
5
c 2015 by ASME
Copyright paper. These values were found sufficient to achieve convergence
of eigenvectors in this study.
Equations 1-5 written in terms of q̂ are transformed into
computational space using the mapping function in eq.16. Next,
these equations are discretized in computational space using the
well known pseudospectral collocation method based on cardinal functions [38]. This yields a discrete form of the eigenvalue
problem given by eqs. 1-5, which can be written symbolically as
the following generalized eigenvalue problem,
∂Ω0θ U θ ∂Ω0θ
∂Ω0
+
+U z θ
∂t
r ∂θ
∂z
= ubr
|
imUbz
r
|
dU θ U θ
dρ
2iαU θ ubθ
1
dP
−
− iα pb
+
+ 2 iαb
ρ
dr
r
r
dr
dr
ρ
{z
} | {z } |
{z
}
7
(19)
2
= −ubr
|
10
d 2U θ 1 dU θ U θ
+
− 2 −
dr2
r dr
r
{z
}
dU θ U θ
+
+
dr
r
{z
}
11
dρ
1
dP
b
b
im
p
−
im
ρ
dr
dr
ρ2 r
|
{z
}
12
(20)
imubr dU z
dU θ
Uθ
=−
+ iαubr
+ iαubr
r
dr
dr
| {z } | {z } | {z r }
1
dU z
iαubθ
+ iαubz
| {zdr } |
Linearized vorticity transport equation
We use the linearized vorticity transport equation to gain insight into the relative influence of various instability mechanisms
on the characteristics of the unstable modes in this study. The
vorticity transport equation, linearized about a steady base flow,
can be written in the normal mode form as follows,
∂Ω0
∂Ω0z uθ ∂Ω0z
+
+ uz z
∂t
r ∂θ
∂z
9
3
∂Ω0
∂Ω0r U θ ∂Ω0r
+
+U z r
∂t
r ∂θ
∂z
8
(17)
The q̂d in equation 17 is the discrete equivalent of the vector
q̂ = {ρ̂, ûr , ûθ , ûz , p̂}, evaluated at the collocation points. The
matrices A and B are functions of axial wavenumber α and azimuthal wavenumber m. Boundary conditions are imposed by
replacing the rows in the matrix A and B with expressions corresponding to the r = 0 and r = 300 collocation points with eqs. 6
- 9. We solve this the discrete eigenvalue problem (eq. 17) using
the MATLAB eig function. Converged solutions for eigenvalues and eigenvectors, q̂d , are obtained by using 100 collocation
points for the constant density cases and 150 collocation points
for the variable density cases presented in this paper.
The absolute/convective nature of flow instability modes
is determined by the nature of response of the flow to an impulsive forcing. This is determined by finding (ωs , αs ) such
that dω/dα = 0, i.e. the saddle points of the dispersion relation [17,18]. If Im(ωs ) > 0 and Im(αs ) < 0, the flow is absolutely
unstable (AU). On the other hand if Im(ωs ) < 0 and Im(αs ) < 0
the flow is convectively unstable (CU). We adopt the algorithm
proposed by Deissler [39] to find (ωs , αs ) using eq. 17. Thus for
given values of N, γ and r f , the values of β and S where the flow
transitions from being AU to CU is given by ωs,i (β, S) = 0. This
equation is solved for different values of S by using newton iteration along with the saddle point search algorithm. The newton
iterations are continued until the residual is less than 10−6 for all
results presented in this study.
5
4
6
Aq̂d = ωBq̂d
imubθ dU z
d 2U z 1 dU z
−
−
+
2
dr
r dr
r dr
{z
} | {z }
3
(18)
6
The source terms on the RHS of eqs. 18-20 represent contributions to the net generation rate of vorticity fluctuations from
rearrangement of base flow vorticity by velocity disturbances,
unsteady vortex stretching due to base flow velocity gradients
and fluctuating baroclinic torque due to base flow density gradients. The contribution to the net generation rate of fluctuating vorticity from various fundamental instability mechanisms
can be deduced from eqs. 18-20 as follows. Consider first a
purely rotational flow (Ūz = 0) and axisymmetric perturbations
(ie. m = 0, α > 0). The only instability mechanism driving the
flow unsteadiness is due to the centrifugal instability in this case.
Hence the terms, 2 , 3 , 7 , 9 and 11 can be identified as
centrifugal instability contributions. Likewise if |m| > 0, α = 0,
only azimuthal KH instability mechanism drives flow unsteadiness through the source term 9 . Thus the centrifugal and azimuthal KH instability mechanisms are coupled through term 9
which appears as a non-zero source term when only one or the
other mechanism alone is present. This term is referred to as
the coupling term in the present study. The relative importance
of azimuthal KH and centrifugal instability due to this term can
be identified from the base flow characteristics and the perturbation wave vector orientations with respect to the azimuthal shear
layer. The baroclinic source terms in the above equations appear
c 2015 by ASME
Copyright LFmode
ωi 0
(a)
Figure 3.
ûz 0.5
1
ωr
2
0
0
3
1
r
2
(b)
(a)
(a) Typical eigenvalue spectrum and (b) unstable eigenmode
1
Axial KH
0.5
Azimuthal KH
Centrifugal
0
0.5
1
r
Mag. z-component
1
HFmode
−0.5
Mag. θ-component
0.5
1.5
0.4
Azimuthal KH
Axial KH
0.2
Centrifugal
0
0
0.5
1
1.5
r
(b)
Figure 4. Spatial budgets of fluctuating vorticity generation rate source
terms for (a) Ω0θ and (b) Ω0z , 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).
shapes 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 present
study, the contributions from the baroclinic torque to the vorticity field is through the terms 8 and 12 . The remaining source
terms in eqs. 18-20 can be identified as the contribution to the
vorticity field from the axial KH instability mechanism. Thus,
in the vorticity generation rate budget results presented in the
forthcoming section, the componentwise generation rate contributions, from any single instability mechanism, is the componentwise resultant of all unsteady source terms associated with the
mechanism. So in the rest of the paper, componentwise sum of
terms corresponding to centrifugal instability (ie. 2 , 3 , 7 , 9
and 11 ) is referred to as “centrifugal” in each component of the
unsteady vorticity generation rate. Likewise componentwise sum
of terms corresponding to azimuthal KH instability and axial KH
instability will be referred to as “azimuthal KH” and “axial KH”
respectively in each component of the unsteady vorticity generation rate. We compute source terms of the eqs. 18- 20 using
the eigenvectors corresponding to eigenvalues of interest in the
present paper. Thus, the dominant mechanism causing instability for that eigenmode can be identified from spatial source term
budgets.
the high frequency (HF) mode. The cluster of eigenvalues close
to the real axis is an artifact of the discrete representation of the
continuous spectrum of the dispersion relation, as well as, some
spurious eigenvalues arising from the numerical discretization.
Figure 3b shows the spatial variation of |ûz | eigenvector magnitude (normalized with the maximum of all eigenvectors) corresponding to the LF and HF unstable modes. Notice that the
amplitude of these modes is large at locations within the two axial shear layers, suggesting that the instability of these modes
is driven primarily by the KH instability caused by shear in the
axial flow profiles. This fact can be understood from spatial budgets of fluctuating vorticity generation rate source terms. Figure 4 shows the spatial variation of source terms of Ω0θ and Ω0z
components corresponding to a disturbance identical to the LF
eigenmode in fig. 4a. The source terms of both the components
have been decomposed into contributions from axial KH instability, azimuthal KH instability and centrifugal instability as discussed in section 3. All data shown in fig. 4 are normalized by
the maximum magnitude of axial KH instability source term associated with the Ω0θ component. Figures 4a-b clearly shows that
the axial KH instability source term in Ω0θ dominates over the
source terms from all other instability mechanisms - implying
that this is the dominant driving mechanism for the combination
of parameters corresponding to the result in fig. 3.
We next present results concerning the absolute/convective
nature of these unstable modes for the constant density case, as
determined from the spatio-temporal analysis. The HF mode was
found to be convectively unstable over the entire range of S, β
and m considered in this study. Figure 5 shows the transition
boundaries between flow profiles showing AU and CU behavior
of the LF mode for m=0 and m=1, corresponding to values of
N = 3 and 4 in each case (γ = 1). The region to the left of each
of the boundaries corresponds to CU flow profiles.
Consider first the m = 0 mode. Figure 5 shows a large
change in the location of the AU/CU transition boundary at high
values of S with decreasing N. This is because, for m=0, the
4
Results
Figure 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 is
seen for other values of α and m as well. The two physically relevant unstable egenvalues are shown using ‘+’ and ‘x’ markers in
fig. 3. The two unstable eigenvalues are due to the stretching and
rearrangement of vortex tubes within the inner and outer shear
layers induced by various instability mechanisms (see section 3).
The eigenvalue corresponding to the marker ‘+’ is denoted as the
low frequency (LF) mode as it always has a lower frequency than
the other mode for all cases studied in this paper. Therefore, the
eigenvalue corresponding to ‘x’ maker in fig. 3a is denoted as
7
c 2015 by ASME
Copyright S 0.5 m = 1
m = 0, N=3
CU
P
AU
AU
0.2
0.4
0.6
(a)
β
(a)
1 Axial KH
Mag. θ-component
Mag. θ-component
Figure 5. Boundary between absolutely unstable and convectively unstable behavior for various values of m and axial shear layer parameters
N (γ = 1). The arrows marked AU and CU respectively point into regions
of absolutely and convectively unstable flow.
Centrifugal
0.5
0
0.5
r
1
1.5
1 Centrifugal
0.5
1
Azimuthal
KH
0.5 Centrifugal
0
0.5
1
r
1.5
0.4
Axial KH
Azimuthal KH
0.2
0
0.5
Centrifugal
1
r
1.5
(b)
Consider next the m=1 case shown in fig. 5. The change in
N has a minimal effect on the location of the AU/CU boundary.
Further, the region of convective instability is very small when
compared to that of m=0 and occurs at much smaller values of
both β and S. Figure 7 shows the contribution of each instability mechanism to Ω0θ and Ω0z 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 7a
shows that the axial KH instability mechanism dominates over all
other instability mechanisms and hence axial KH instability determines the instability characteristics of m=1 mode. This plot is
typical and has been found to have the same behaviour for other
values of base flow parameters. Next, fig. 5 also shows that this
mode can be expected to dominate the overall unsteady dynamics of the flow since it is AU over the entire region over which
the m=0 low frequency mode is convectively unstable. This is
a clear indication of the fact that the predominant self-excited
behavior in a nominally axi-symmetric swirl flow would be expected to be a helical, m=1 instability. This fact is in qualitative
agreement with recent experimental observations [40]. Next, we
consider the influence of a density variation generated by a premixed flame on these absolute/convective instability transition
boundaries.
Figure 8 shows the AU/CU transition boundaries for the density jump located within the ISL, i.e. r f = 1.0, 0.5, γ = 0.3 for
the m=0 low frequency mode (N=4). The boundary for the γ = 1
case is also shown for reference. The large change in the location of the boundary is due to the additional influence of fluctuating baroclinic torque on the net generation rate of fluctuating vorticity in the flow (see eq.19). The strong stabilization of
the absolute instability for the r f = 1.0 case can be understood
as follows. Figure 9a which shows the spatial variation of the
magnitude of the source terms due to rearrangement and stretching and the fluctuating baroclinic torque for the Ω0θ component
due to the m=0 LF mode. This plot is typical and corresponds
to (β, S) = (0.35, 0.50) (point Q in fig.8). The corresponding
phase difference between these two source terms is also shown
Axial KH
r
Axial KH
Figure 7. Spatial budgets of fluctuating vorticity generation rate source
terms, showing contributions from various instability mechanisms at the
saddle point (a) Ω0θ (b) Ω0z (m = 1, β = 0.1, S = 0.5, N = 4, γ = 1,
N3 = 4.18 and N4 = 0.73).
0.5
0
1
Mag. z-component
S
m = 0, N=4
0 CU
0
Mag. θ-component
1
1.5
(b)
Figure 6. Spatial budgets of Ω0θ source term, showing contributions from
various instability mechanisms at the saddle point (a) S = 0.3 (b) S = 0.9
in fig. 5 (m = 0, β = 0.4, N = 3, γ = 1, N3 = 4.18 and N4 = 0.73).
centrifugal instability mechanism becomes increasingly important at larger swirl numbers. This fact can be seen from fig. 6
which shows the spatial variation of fluctuating vorticity generation rate source terms for disturbance identical to the eigenmodes
corresponding 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 centrifugal instability terms to the generation rate of the Ω0θ component is comparable to the corresponding contribution from the
axial KH mechanism for S = 0.9 case (see fig. 6b), as opposed
to the S = 0.3 case (see fig. 6a) where the Axial KH contribution is dominant. Thus, decreasing in the value of N, causes
the shear layers in the axial base flow velocity profile to weaken
and hence, weakens the quantitative influence of the axial KH
instability mechanism at all values of S. Thus, the centrifugal
instability mechanism becomes dominant at smaller values of S
resulting in a larger change in the AU/CU boundary location for
S > 0.2 when compared to S < 0.2 (see fig. 6). Note that contributions from the azimuthal KH mechanism do not exist for an
axi-symmetric (m = 0) type disturbances. The same behaviour
is seen for the Ω0z component as well and is not shown in the
interest of brevity.
8
c 2015 by ASME
Copyright Q
γ=1
CU
0.1
0.2
0.3
0.4
100
Rearrangement+
Stretching
0.5
0
Baroclinic
0
0.6
0.8
r
−100
1
(a)
Mag. θ-component
β
Figure 8. Boundary between absolutely unstable and convectively unstable behavior for various locations of the base flow density gradient
(N = 4, γ = 0.3 and m = 0). The boundary for γ = 1 (thick black
line) is also shown for reference.
on the vertical axis on the right. Clearly the baroclinic torque
terms peaks close to where the resultant of all other source terms
peak while being out of phase with the latter. Thus the baroclinic
torque counteracts the production of vorticity by rearrangement
and stretching causing the mode to become temporally stable
(and hence CU) for the (ωs , αs ) values corresponding to point
Q. The opposite result can be seen from fig. 9b which is a similar
source term plot for r f = 0.5. For r f = 0.5 case, the fluctuating baroclinic torque peaks at a location (ie. r ≈ 0.5) where the
contribution from resultant of other source terms are negligible.
Thus, in this case, the fluctuating baroclinic torque increases the
net fluctuating vorticity generation rate causing the flow to become temporally unstable for the (ωs , αs ) values at Q and hence,
absolutely unstable. Thus, it is clear that the density gradient
can have a significant stabilizing or destabilizing impact on the
absolute/convective nature of the instability corresponding to the
m=0 low frequency mode, depending on the location of the density gradient.
Similar results can be seen for the m=1 mode from fig. 10
which shows the location of the AU-CU transition boundary for
various values of γ and r f (N=4). Also shown is the γ = 1 case
for comparison. The presence of the baroclinic torque term, results in the m=1 low frequency mode becoming CU over a larger
region of the β − S plane when compared to the constant density
case. The reason for this can again be seen in fig. 11(a)-(b) which
plots the spatial variation of the magnitude of the vorticity generation rate source terms, of Ω0θ and Ω0z . These plots correspond to
(β, S) = (0.3, 0.5) (point R in fig. 10) for γ = 0.3, r f = 1.0. Note
that the baroclinic contribution is in phase with the net contribution from all other source terms resulting in the flow becoming
AU (see fig. 11(a)). However, the opposite is true for the corresponding 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 instability is greatly diminished due to the influence of baroclinic
torque generated by the density variation. This suggests that the
1
0
Rearrangement+
Stretching
0.5
−100
Baroclinic
−200
0
0.2
0.4
0.6
0.8
1
−300
1.2
Phase Diff. (degrees)
0
0
AU
0.5
1
Phase Diff. (degrees)
γ=0.3
rf = 1.0
γ=0.3
rf = 0.5
S 0.5
Mag. θ-component
1
r
(b)
Figure 9. Spatial budgets of fluctuating vorticity generation rate source
terms for the Ω0θ component due to the m = 0 impulse response mode at
point Q (see fig. 8), from rearrangement, stretching and baroclinic mechanisms (a) r f = 1.0 (b) r f = 0.5 (N = 4).
1
γ=0.3
rf = 0.75
γ=0.3
rf = 1.0
S 0.5
R
γ=1.0
γ=0.1
rf = 1.0
AU
0
0
CU
0.1
0.2
0.3
0.4
0.5
β
Figure 10. Boundary between absolutely unstable and convectively unstable behavior for different locations of the base flow density gradient
and density change parameter γ (N = 4 and m = 1). The boundary for
γ = 1 (thick black line) is also shown for reference.
PVC, which is associated with a m = 1 type instability, should
be suppressed in the presence of a density gradient in the ISL
which is in qualitative agreement with prior experimental observations [41].
5
Conclusion
This paper analyzes local convective/absolute instability
transition for velocity and density profiles characteristic of pre9
c 2015 by ASME
Copyright 300
Rearrangement+
Stretching
200
0.5
100
Baroclinic
0
0.6
0.8
0
1
mixed swirl stabilized combustors. We do this via a spatiotemporal hydrodynamic stability analysis about a locally parallel base flow with spatially varying density (as would be generated by a premixed flame), in the inviscid and low Mach number
limit. Flows in swirl stabilized combustors are inherently complex as they are comprised of multiple shear layers and recirculation zones. In addition to the presence of shear layer instabilities,
swirl flows can also be destabilized by other instability mechanisms such as the centrifugal instability and Kelvin’s instability.
These different mechanisms are in general coupled and manifest
themselves with varying importance at different flow conditions.
The initial temporal analysis of the base flow profiles considered, revealed the presence of two unstable modes. Of these,
only the mode with the lower oscillation frequency showed transition between absolute and convective behavior. Interestingly,
this study revealed that the first asymmetric mode, corresponding
to an azimuthal wavenumber, m = 1, showed a very large region
of absolute instability behavior when compared to the symmetric m = 0 mode. This suggests that the eventual, self-excited,
unsteady behavior of the flow would be dominated by an m = 1
instability - a fact borne out by several observations of helical
Precessing Vortex Core (PVC) phenomena in constant density
swirl flows.
The inclusion of a flame induced density gradient in the inner shear layer (ISL) resulted in a dramatic stabilization of the
absolute instability for the m=1 mode due to the influence of
baroclinic torque. Thus, the present analysis suggests a possible reason to explain the suppression of the PVC in the presence
of a flame - a fact that has been borne out by recent experiments.
However, the actual location of the AU-CU transition boundary
in the parameter space considered in this study is a strong function of the density gradient magnitude as well as its disposition
relative to the shear layers in the flow. The same fact is seen to
be true for the symmetric, m = 0 mode as well.
Phase Diff. (degrees)
Mag. θ-component
1
r
1
400
Rearrangement+
Stretching
200
0.5
Baroclinic
0
0
0.6
0.8
r
−200
1
Phase Diff.(degrees)
Mag. z-component
(a)
(b)
1
Rearrangement+
Stretching
200
0.5
100
Baroclinic
0
0.4
0.6
0.8
1
0
Phase Diff. (degrees)
Mag. θ-component
Figure 11. Spatial budgets of fluctuating vorticity generation rate source
terms at point R in fig. 10 due to rearrangement, stretching and baroclinic
torque for r f = 1.0 and γ = 0.3 (a) Ω0θ source term (b) Ω0z source term
(N = 4).
r
0.2
Rearrangement+
Stretching
300
200
Baroclinic
0.1
100
0
0
0.4
0.6
0.8
1
−100
Appendix A: Linearized inviscid Navier-Stokes equations: Normal mode form
Using eq. 10 in eqs. 1-9 yields the following system of equations,
Phase Diff. (degrees)
Mag. z-component
(a)
!
imŪθ ρ̂
d ρ̄
imûθ
+ iαûz +
+ ûr
−iωρ̂ + ρ̄
r
r
dr
iαŪz (r)ρ̂ = 0
(21)
r
(b)
imŪθ ûr
2Ūθ ûθ
ρ̄ − iωûr +
+ iαŪz ûr −
r
r
Figure 12. Spatial budgets of fluctuating vorticity generation rate source
terms at point R in fig. 10 due to rearrangement, stretching and baroclinic
torque for r f
(N = 4).
= 0.75 and γ = 0.3 (a) Ω0θ source term (b) Ω0z source term
−
10
!
Ūθ2 ρ̂
d p̂
=−
r
dr
(22)
c 2015 by ASME
Copyright dŪθ imŪθ ûθ
+
dr
r
!
Ūθ ûr
im p̂
+iαŪz ûθ +
=−
r
r
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ρ̄ − iωûθ + ûr
dŪz imŪθ ûz
ρ̄ − iωûz + ûr
+
+ iαŪz ûz
dr
r
(23)
!
= −iα p̂
d ûr ûr imûθ
+ +
+ iαûz = 0
dr
r
r
(24)
(25)
Next, the boundary conditions in the normal mode form at
r → ∞ and at the centerline (r = 0) are obtained using eq. 10,
q̂ → 0
(26)

ûr = ûθ = 0
if m =0
d ûz
d p̂ d ρ̂
=
=
= 0
dr
dr
dr
(27)

ρ̂ = ûz = p̂ = 0


ûr + imûθ = 0 if |m| = 1


d ûr
= 0
dr
(28)
)
ρ̂ = ûr = ûθ = 0
ûz = p̂ = 0
if |m| > 1
(29)
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Copyright