Effect of Combustion on Near-Critical Swirling Flow
Zvi Rusak, A. K. Kapila and Jung J. Choi
Rensselaer Polytechnic Institute
Troy, New York 12180
March 18, 2002
Abstract
This paper examines the manner in which heat release resulting from premixed
combustion alters the nature of near-critical, axisymmetric, swirling flow in a straight
circular pipe. Attention is confined to dilute premixtures so that exothermicity is weak
and a small-disturbance approach applicable. The weak exothermicity is found to have
a considerably larger effect on the flow. In the absence of combustion the columnar
solution loses stability via a transcritical bifurcation as the level of swirl rises beyond a
critical value. Exothermicity splits the bifurcation portrait into two branches separated
by a gap in the level of swirl; within this gap steady, near-columnar solutions cease to
exist. As a result the critical value of swirl for a combusting flow is smaller than that
for the cold flow. For a certain range of swirl below this critical value and for small
enough heat release, the solution branch is double-valued and yields two equilibria, one
corresponding to a near-columnar state and the other pointing to the appearance of
a large-amplitude structure. For larger heat release the double-valued branch loses its
fold, suggesting the gradual appearance of large-amplitude disturbances with increasing
levels of swirl. The mechanism that governs the behavior of the reactive flow with swirl,
and the relevance of the results to combustion states with vortex breakdown, are also
discussed.
1
1
Introduction
Combustion is susceptible to a variety of destabilizing influences (see, for example, Williams
1985, Kapila 1992, Gutmark et al 1992, McManus et al 1993, Paschereit et al 1998). These
can either be intrinsic to the mode of combustion or arise as a result of interaction between
the combustion process and its flow environment. A manifestation of intrinsic instability is
the spontaneous development of cellular structures in flames. Instabilities induced by the
coupling between combustion and the underlying flow are exhibited by reactive shear layers
and jets, while acoustic instabilities in propulsion devices and flame acceleration prior to
possible transition to detonation in closed vessels are examples where confining boundaries
and obstacles in the path of flames play crucial roles. In technological applications, instabilities can have an adverse effect (e.g., damage or flame extinction), a favorable effect (e.g.,
enhanced burning) or no consequence.
Instabilities caused by combustion interacting with the underlying flow, the boundaries,
or with processes occurring in other parts of the system can involve a broad range of physical
mechanisms such as acoustic oscillations, vortex-shedding, and deformation of the flame zone.
The situation is particularly complex when the underlying inert flow, such as a flow with
swirl, is itself susceptible to instabilities. It is important then to understand how combustion
affects, and is in turn affected by, the intrinsic instability of the underlying flow. This paper
addresses the behavior of near-critical swirl flow in the presence of combustion. Attention is
restricted to small heat release for which, it is found, the dominant aspect of the flame-flow
coupling, is the effect of the flame on the flow.
Swirl is of practical importance in combustion not only as a natural feature in actual
industrial configurations, but also as an agent that can enhance combustion efficiency by
inducing vortex breakdown; see, for example, Lefebvre 1998, McVey et al 1993, Snyder et al
1994, Stephens et al 1997, Wang and Yang 1997, and Paschereit et al 1998. The importance
stems from the fact that when the swirl component in the incoming flow is above a certain
critical level, a large and nearly stagnant separation region appears near the inlet of the
combustion chamber. In the premixed case, burned particles are trapped in this breakdown
zone where they create a region of higher temperature. Heat transfer from this hot region to
the surrounding swirling flow apparently helps to stabilize the flame and burn more of the
reactants, thereby improving combustion effectiveness (Sivasegaram and Whitelaw 1991).
In the nonpremixed case where liquid fuel is injected into a swirling flow of air, the hot
breakdown zone serves as a noninvasive flame holder where fuel particles are trapped for
longer residence times and mixing between air and fuel is improved.
A major problem in inducing vortex breakdown is the appearance of flow instabilities
inside the large separation zone and in its wake (Gupta et al 1984, 1998). The zone may
become unstable and disappear because of a variety of reasons such as changes in the distribution of incoming swirl, variations in the temperature field, or large perturbations in the
back pressure at the downstream end of the combustor. Then the hydrodynamic instabilities
may also induce flame instabilities and affect combustor performance. Gupta et al (1998)
have discussed the various modes of instabilities that can occur. A detailed study of the
dynamical behavior of combustion systems with swirl is thus in order, for better predictions
of the onset of instabilities, for improved understanding of the underlying mechanisms, and
2
ultimately, for devising suitable control strategies.
As a topic in fluid mechanics, vortex breakdown has an extensive literature, including
reviews by Hall (1972), Leibovich (1978, 1984), Escudier (1988), Sarpkaya (1995), Althaus
et al (1995), and Rusak (2000). Although several possible explanations have been advanced,
each clarifying some aspects of the problem, a consistent description of the phenomenon has
proved elusive. Until recently, the relationship between various theoretical and numerical
solutions had not been fully made clear, nor had precise criteria been given for the occurrence, stability, and dynamics of states of vortex breakdown. In a series of recent papers,
Rusak and co-authors have developed a new theoretical framework for predicting the axisymmetric vortex breakdown process (Wang and Rusak 1996a,b, 1997a,b, Rusak and Wang
1996, Rusak et al 1997, 1998a,b, and Rusak 1998, 2000) in a circular pipe of finite length.
The results, established through a rigorous nonlinear global analysis and complementary
asymptotic treatments, provide a fundamental and nearly complete mathematical description of the dynamics and stability of inert axisymmetric swirling flows. They show a good
correlation with numerical computations (Rusak et al 1998a,b) and experimental studies
(Malkiel et al 1996, Rusak and Lamb 1999, and Judd et al 2000). The new theory unifies
the major theoretical and numerical approaches that exist in the literature, and provides a
global understanding of the problem (Rusak 2000).
The analysis shows that in the inviscid limit the swirl rate ω of the incoming flow has two
critical values, ω0 and ω1 with ω0 < ω1 . Columnar flows with ω < ω0 are unconditionally
stable to any axisymmetric disturbance. For ω0 < ω < ω1 the flow may evolve into one of
two steady states depending on the size of the initial disturbance. When disturbances are
sufficiently small they decay in time and the flow returns to the columnar state, otherwise
they grow and evolve into a large stagnation zone similar to the breakdown states found in
very high Reynolds number flows (Sarpkaya 1995). When ω > ω1 , any initial disturbance
grows and evolves into a breakdown zone.
These special stability characteristics are related to the upstream propagation of both
small- and large-amplitude disturbances and their interaction with the flow conditions downstream of the vortex generator but ahead of the pipe inlet. The disturbances tend to propagate upstream with a speed that increases with ω . When ω < ω1 , small disturbances
are convected by the axial flow out of the pipe and the columnar flow is therefore stable.
When ω > ω1 , small disturbances tend to move upstream. Since the flow out of the vortex
generator at steady operation is fixed, the disturbances cannot move through it. They are
trapped, and they grow and stabilize as a large and steady stagnation zone. Subsequent to
their appearance, these large-amplitude zones tend to grow further and move upstream as
ω is increased. However, when ω is decreased to levels below ω0 , they are convected by the
axial flow out of the pipe and the flow returns to the columnar. The theory also presents
a consistent explanation of the physical mechanism leading to axisymmetric vortex breakdown, as also of the conditions for its occurrence. The results show that the phenomenon is a
necessary evolution from an initial columnar state to another, relatively stable, lower-energy
equilibrium describing a swirling flow around a large breakdown zone. This evolution is the
result of the interaction between waves propagating upstream and the incoming flow, which
leads to an absolute loss of stability of the base columnar state when the swirl ratio of the
incoming flow is near or above the critical level.
3
To the best of our knowledge, fundamental studies on combustion with swirling flows
are quite limited to experimental investigations and numerical simulations (Sivasegaram and
Whitelaw 1991, Wang and Yang 1997, Paschereit et al 1998 and Gupta et al 1998). Specifically, there does not appear to be a consistent mathematical treatment of the interaction
between combustion and vortex breakdown. The effect of heat release on the stability and
dynamics of swirling flows is also unclear. This paper takes a first step in this direction by
examining the way in which the onset of instability in a swirling flow is influenced by the
presence of combustion. Attention is confined to a dilute mixture so that the heat release is
weak. This allows for a small-disturbance but nonlinear analysis which finds that when the
incoming swirl level is near critical ( ω ≈ ω1 ), flow perturbations induced by the heat release
are in fact larger than the magnitude of the heat release itself. The transcritical bifurcation
portrait of the cold flow splits in the presence of combustion, and the upshot is that arrival
of instability is hastened.
The paper begins with the introduction of a mathematical model appropriate for premixed combustion in a circular pipe of finite length. The model is subjected to an asymptotic
analysis along the lines indicated above, the result of which is an algebraic equation for the
unfolding of the bifurcation singularity. The paper ends with a discussion of the physical
implications of this equation on combustion with swirl.
2
Mathematical model
Dimensional equations governing the steady, inviscid flow of a reactive premixed fluid in a
cylindrical, axisymmetric geometry are (Buckmaster and Ludford 1982),
p̄ = ρ̄RT̄ ,
ρ̄ū
+ (ρ̄w̄)x̄ = 0,
r̄
v̄ 2
ρ̄ ūūr̄ + w̄ūx̄ −
= −p̄r̄ ,
r̄
ūv̄
= 0,
ūv̄r̄ + w̄v̄x̄ +
r̄
ρ̄ (ūw̄r̄ + w̄w̄x̄ ) = −p̄x̄ ,
1
ρ̄ (ūYr̄ + w̄Yx̄ ) = D
(r̄Yr̄ )r̄ + Yx̄x̄ − W̄ ,
r̄
1 ρ̄Cp ūT̄r̄ + w̄T̄x̄ − (ūp̄r̄ + w̄p̄x̄ ) = λ
r̄T̄r̄ r̄ + T̄x̄x̄ + B W̄ ,
r̄
W̄ = Aρ̄Y exp(−Ē/RT̄ ).
(ρ̄ū)r̄ +
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Here p̄ is the pressure, ρ̄ the density, and T̄ the temperature of the fluid, while ū , v̄ ,
and w̄ are, respectively, the radial, circumferential, and axial components of the velocity.
Mass fraction of the reactant in the combustible gas is denoted by Y. The thermophysical
properties of the reacting fluid include the specific gas constant R, the diffusion coefficient
D, the specific heat at constant pressure Cp , the specific-heat ratio γ, thermal conductvity
4
λ , and the specific heat release B. These parameters are all assumed to be constants. A onestep first-order Arrhenius reaction is postulated, with A the reaction-rate pre-exponential
factor and Ē the activation energy.
It is assumed that the fluid enters a pipe of radius r̄0 and length x¯o at a nominally
uniform state of density ρ̄0 , temperature T̄0 and pressure p̄0 = ρ̄0 RT̄0 , and with a radiallyvarying axial velocity whose maximum is Ū0 . This state is taken to be the reference state for
purposes of nondimensionalization, while lengths are referred to the pipe radius r̄0 . Then
the equations take the dimensionless form
p = ρT,
(9)
ρu
(ρu)r +
(10)
+ (ρw)x = 0,
r v2
1
pr ,
(11)
=−
ρ uur + wux −
r
γM02
uv
uvr + wvx +
= 0,
(12)
r
1
px ,
(13)
ρ (uwr + wwx ) = −
γM02
1 1
(14)
ρ (uYr + wYx ) =
(rYr )r + Yxx − W,
Pe r
γ−1
L 1
ρ (uTr + wTx ) −
(15)
(upr + wpx ) =
(rTr )r + Txx + βW,
γ
Pe r
W = AρY exp(−θ/T ).
(16)
The dimensionless parameters appearing above are Mach number M0 = Ū0 /γ p̄0 /ρ̄0 , Peclet
number Pe = ρ̄0 Ū0 r̄0 /D, Lewis number L = λ/Cp D, heat-release parameter β = B/Cp T̄0 ,
scaled activation-energy θ = Ē/RT̄0 , and scaled frequency factor A = Ār̄0 /Ū0 .
The boundary conditions are as follows. Along the axis r = 0 the symmetry conditions
u(x, 0) = v(x, 0) = Tr (x, 0) = Yr (x, 0) = 0,
for 0 ≤ x ≤ xo
(17)
apply, where xo = x¯o /r¯0 . At the wall r = 1 the normal velocity vanishes, as do the fluxes
of heat and reactant, i.e.,
u(x, 1) = Tr (x, 1) = Yr (x, 1) = 0,
for 0 ≤ x ≤ xo .
(18)
At the inlet x = 0 we consider a steady state of flow created by a swirl generator ahead of
the pipe, for which are prescribed general profiles of incoming axial speed, circumferential
speed, azimuthal vorticity (η = ux − wr ) , temperature, and reactant mass fraction. Thus
w(0, r) = w0 (r), v(0, r) = ωv0 (r), η(0, r) = η0 (r),
T (0, r) = 1 + δT10 (r), Y (0, r) = δY10 (r), for 0 ≤ r ≤ 1,
(19)
where the maximum values of w0 (r), v0 (r) , and Y10 (r) are each taken to be unity. Here
ω is the swirl ratio of the incoming flow and δ the measure of the amount of incoming
5
reactant mass, and hence a measure of the intensity of combustion. Note that the radial
velocity component along the inlet is not prescribed. Instead, the state at the inlet is
given the freedom to develop a radial velocity, thus accommodating the upstream influence
of disturbances that may develop downstream of the inlet. We also take the incoming
azimuthal vorticity to be η0 = −w0r which implies a zero axial gradient of the radial velocity
at inlet, i.e., ux (0, r) = 0. This assumption is for simplicity and does not limit the scope
of the analysis, or the results, in any substantial way. Rusak (1998) describes how general
azimuthal vorticity profiles at the inlet can be included. At the outlet x = xo the radial
velocity is taken to be zero and the axial gradients are assumed to vanish, in accordance
with an expected columnar flow state, i.e.,
u(x0 , r) = wx (x0 , r) = vx (x0 , r) = 0,
Tx (x0 , r) = Yx (x0 , r) = 0,
for 0 ≤ r ≤ 1.
(20)
Similar boundary conditions on the flow have been considered in the cold-flow analysis of
Wang and Rusak (1997a) and in the various numerical simulations of non-reacting swirl flows
in a pipe. These conditions may also reflect the physical situation as reported in the coldflow experiments of Malkiel et al (1996) and Bruecker and Althaus (1995). These conditions
also formulate a basic problem which focuses on the interaction between the lean combustion
state and the near-critical swirling flow.
Equations (9)-(16) with boundary conditions (17)-(20) constitute a well-defined mathematical problem for the fields of velocity, temperature, pressure, density, and mass fraction.
It is nonlinear, and one expects that for a certain set of inlet conditions there may exist multiple steady-state solutions. The present study concentrates on dilute mixtures, 0 ≤ δ 1,
and near-critical swirl ratios, |ω − ω1 |/ω1 1. As is typical of low-speed combustion, the
Mach number M0 is presumed small as well.
3
Small-disturbance approach
It is expected that energy released by combustion of the O(δ) amount of reactant entering
the pipe creates correspondingly small perturbations in temperature, pressure and density,
as also in the velocity components. Yet, previous studies have shown that near criticality,
velocity perturbations can be larger than those in the thermodynamic variables. Accordingly,
we postulate the following asymptotic expansions:
Y
T
p
ρ
w
v
u
K
=
=
=
=
=
=
=
≡
δY1 + · · · ,
1 + δT1 + δ2 T2 + · · · ,
1 + γM02 (p0 + !1 p1 + !2 p2 + · · · ),
1 + δρ1 + δ2 ρ2 + · · · ,
w0 (r) + !1 w1 + !2 w2 + · · · ,
ωv0 (r) + !1 v1 + !2 v2 + · · · ,
! 1 u1 + ! 2 u2 + · · · ,
rv = K0 (r) + !1 K1 + !2 K2 + · · · ,
6
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
where K denotes the circulation. We assume the ordering
0 < δ2 δ 1 and 0 ≤ M02 δ !1 1,
(29)
specifically as the incoming flow swirl ratio ω approaches its critical level ω1 . All perturbation functions depend on x and r .
3.1
Leading-order pressure
In view of the above expansions, the radial and axial momentum equations, (11) and (13),
yield at leading order,
p 0r =
ω 2 v02 (r)
, p0x = 0,
r
(30)
whence the leading-order pressure is determined to be
r
2
v02 (r )/r dr .
p0 = ω
0
3.2
The basic flame structure
Similarly, the equation of state (9) and the species, energy and rate equations, (14) - (16),
reduce at leading order to
ρ + T1 = 0,
1
1 1
w0 Y1x =
(rY1r )r + Y1xx − W,
Pe r
L 1
w0 T1x =
(rT1r )r + T1xx + βW,
Pe r
W = AY1 exp(−θ/T ).
(31)
(32)
(33)
Here, T = 1 + δT1 is used. Typically, P e is large and effects of diffusion may be neglected
everywhere except in the vicinity of the flame front. Boundary conditions for Y1 and T1
come from (17)-(19), and are
Y1 (0, r) = Y10 (r), Y1x (xo , r) = 0, for
Y1r (x, 0) = 0, Y1r (x, 1) = 0, for
T1 (0, r) = T10 (r), T1x (xo , r) = 0, for
T1r (x, 0) = 0, T1r (x, 1) = 0. for
0 ≤ r ≤ 1,
0 ≤ x ≤ xo ,
0 ≤ r ≤ 1,
0 ≤ x ≤ xo .
(34)
(35)
Equations (32) and (33), together with boundary conditions (34) and (35), constitute the
standard, premixed flame-structure problem, and show that the reactant mass fraction perturbation Y1 and the temperature perturbation T1 can be computed for any given inlet
axial flow w0 (r) , inlet temperature profile T10 (r) , and reactant mass-fraction profile Y10 (r) .
7
Moreover, to orders considered, the solution of the fields of Y1 , T1 , and ρ1 = −T1 is independent of the swirl level ω of the incoming flow and is decoupled from the solution of
the flow-speed perturbations. In the present paper we consider the parameters L, P e, and
β to be in the domain for which the solution for the free standing flame is stable (see, for
example, Buckmaster and Ludford 1982, chapter 11).
For the case where the inlet profiles are uniform, i.e., w0 (r) = 1, T10 (r) = 0 , and
Y10 (r) = 1 , the solution is also independent of r . Then, the problem (31)-(33) with conditions (34) and (35) reduces to a system of two ordinary differential equations which can
be solved numerically. A representative solution of the flame structure is shown in Figure
1 for the parameters β = 2 , P e = θ = 10 , L = 1 , A = 50, 000 , and inlet reactant mass
fraction δ = 0.075 . For this case, the flame is diffused along the pipe with an adiabatic
flame temperature Tb = 1.15.
3.3
Flame-induced flow perturbations at near-critical swirl
The above solution for the temperature will now be used to determine the effect of heat
addition on the flow.
3.3.1
The stream functions
Turning first to the continuity equation (10) and using the expansions (24), (25) and (27),
we find at leading order !1 ,
1
(ru1 )r + w1x = 0.
r
The velocity perturbations u1 and w1 may now be written in terms of a perturbation stream
function ψ1 as
ψ1x
u1 = − √ , w1 = ψ1y .
2y
(36)
y = r2 /2
(37)
where
is a modified radial coordinate, varying in the interval 0 ≤ y ≤ 1/2 . We note that ψ1
is known only to within an arbitrary constant. However, the first of the inlet boundary
conditions (19) implies that ψ1y (0, y) = 0, so that ψ1 is a constant at the inlet. Furthermore,
the first of the symmetry conditions (17) on the pipe axis implies that ψ1x (x, 0) = 0, so that
ψ1 is a constant on the axis as well; in fact the same constant as that at inlet. We take this
constant to be zero, so that
ψ1 (0, y) = ψ1 (x, 0) = 0.
At the next order the continuity equation (10) reduces to
u2
+ w2x + δw0 ρ1x = 0,
!2 u2r +
r
8
(38)
and proceeding as above, we can define a second stream function ψ2 via the relations
ψ2x
δ
u2 = − √ , w2 + w0 ρ1 = ψ2y .
!2
2y
(39)
By the same argument as above, ψ2 can be taken to be zero on the axis, i.e.,
ψ2 (x, 0) = 0.
(40)
At the inlet, however, the first of the boundary conditions (19) allows the second of the
equations (39) to be written as
ψ2y (0, y) =
δ
ρ1 (0, y)w0 (y),
!2
(41)
which must be integrated subject to (40). Note that ρ1 (0, y) = −T1 (0, y) = −T10 (y) . Also,
note that in the example of the flame structure studied above, T10 (y) = 0 for 0 ≤ y ≤ 1/2
and therefore, ρ1 (0, y) = 0 and ψ2 (0, y) = 0.
3.3.2
The circulations
We now turn to the circumferential momentum equation (12) and rewrite it in terms of the
circulation K and the new radial coordinate y as
2yuKy + wKx = 0.
(42)
At leading order !1 it yields, in view of (25)-(27),
2yu1 K0y + w0 K1x = 0,
and on applying the first of equations (36), can be re-expressed as
−ψ1x K0y + w0 K1x = 0.
An integration, subject to the first of the boundary conditions (38), leads to
K1 =
K0y
ψ1 .
w0
(43)
At order !2 , (42) has the form
!2 2yu2 K0y + w0 K2x + 1 ( 2yu1 K1y + w1 K1x ) = 0.
!2
Upon introducing the stream functions ψ1 and ψ2 via (36) and (39), and the circulation
K1 via (43), the above equation can be cast into
2
ψ1
!21 K0y
−ψ2x K0y + w0 K2x −
= 0.
!2 w0 y 2 x
Since K2 and ψ1 vanish at x = 0 (see the second of conditions (19) and the second of
conditions(38)), the above equation integrates to yield
K0y
!21 K0y
ψ12
K2 =
[ψ2 − ψ2 (0, y)] +
.
(44)
w0
!2 w0 y 2w0
9
3.3.3
The eigenproblem for critical swirl
The radial and axial momentum equations, (11) and (13), have already been used at order
unity to determine the leading-order pressure p0 . We now write them each at the next two
orders. Thus, (11) yields
2K0
2K0
!1 w0 u1x − 3 K1 + !2 w0 u2x − 3 K2
r
r
2
2
K
K
+!21 u1 u1r + w1 u1x − 31 − δ 30 ρ1 = −!1 p1r − !2 p2r .
r
r
Upon substituting for K1 from (43) and for K2 from (44), the above equation may be
rewritten as
2K0 K0y
2K0 K0y
!1 w0 u1x − 3
ψ1 + !2 w0 u2x − 3
[ψ2 − ψ2 (0, y)]
r w0
r w0
2
K
1
K
K
0
0
0y
y
+!21 u1 u1r + w1 u1x − 3
+
ψ12
r
w0
w0 w0 y
−δ
K02
ρ1 = −!1 p1r − !2 p2r .
r3
It is convenient to replace K0 in favor of K0 = ω K̃0 and let Ω = ω 2 . Then the above
equation may be expressed yet again as
2K̃0 K̃0y
2K̃0 K̃0y
!1 w0 u1x − Ω 3
ψ1 + !2 w0 u2x − Ω 3
[ψ2 − ψ2 (0, y)]
r w0
r w0


2
 

K̃0y
1
K̃0 K̃0y  2 
!21 u1 u1r + w1 u1x − Ω 3
+
ψ
 1
r  w0
w0
w0
y
K̃02
−δΩ 3 ρ1
r
= −!1 p1r − !2 p2r .
(45)
We now turn to the axial momentum equation (13) which reads
!1 (w0r u1 + w0 w1x ) + !2 (w0r u2 + w0 w2x ) + !21 (u1 w1r + w1 w1x ) = −!1 p1x − !2 p2x ,
and, upon a slight rearrangement, as
δ
!1 (w0r u1 + w0 w1x ) + !2 w0r u2 + w0 w2 + w0 ρ1
!2
x
2
2
+!1 (u1 w1r + w1 w1x ) − δw0 ρ1x = −!1 p1x − !2 p2x .
(46)
Elimination of pressure from (45) and (46) by cross-differentiation and subtraction leads to
an equation for the azimuthal vorticity. At order !1 , this equation reads
2K̃0 K̃0y
ψ1 − (w0r u1 + w0 w1x )r = 0.
w0 u1x − Ω 3
r w0
x
10
The use of (36), followed by some rearrangement, allows it to be rewritten as
ψ1xx
K̃0 K̃0y w0yy
ψ1yy +
+ Ω 2 2 −
ψ1 = 0.
2y
2y w0
w0
x
An integration with respect to x is immediate. Upon using the third boundary condition in
(19), which gives ψ1xx (0, y) = 0, and the second condition in (38), which gives ψ1 (0, y) = 0,
we obtain the partial differential equation for the solution of ψ1
ψ1xx
K̃0 K̃0y w0yy
ψ1yy +
(47)
+ Ω 2 2 −
ψ1 = 0,
2y
2y w0
w0
with boundary conditions
ψ1 (x, 0) = 0,
ψ1 (0, y) = 0,
ψ1 (x, 1/2) = 0, for 0 ≤ x ≤ x0 ,
ψ1x (xo , y) = 0, for 0 ≤ y ≤ 1/2.
(48)
(49)
This boundary-value problem possesses nontrivial solutions only for specific values of Ω, the
eigenvalues. Wang and Rusak (1997a) defined the first eigenvalue Ω1 = ω12 as the “critical
level of swirl for a straight pipe of finite length.” The corresponding eigenfunction can be
written as
πx
ψ1 = ψ1c (x, y) = Φ(y) sin
(50)
2xo
where Φ is determined by
Φyy +
K̃0 K̃0y w0yy π 2 /4x20
Ω1 2 2 −
−
2y w0
w0
2y
Φ(0) = 0,
Φ = 0,
Φ(1/2) = 0.
(51)
The critical swirl ω1 is a transcritical bifurcation point of the equilibrium solutions of the
reactionless equations, and has been discussed extensively by Wang and Rusak (1996a,b,
1997a) in their study of the vortex breakdown phenomenon. Note that when the pipe length
xo increases to infinity the critical swirl Ω1 and the related eigenfunction approache those
found by Benjamin (1962).
3.3.4
Near-critical behavior and the solvability condition
We now return to equations (45) and (46) to determine !1 , the amplitude of the leadingorder flame-induced flow perturbation for swirl ratios ω near the critical value ω1 . Let
Ω = Ω1 + ∆Ω where |∆Ω|/Ω1 1 . Then the radial momentum equation (45), upto the
11
order kept, rearranges into
2K̃0 K̃0y
2K̃0 K̃0y
ψ1c + !2 w0 u2x − Ω1 3
[ψ2 − ψ2 (0, y)]
!1 w0 u1cx − Ω1 3
r w0
r w0



2



K̃0y
Ω1
K̃0 K̃0y
+!21 u1c u1cr + w1c u1cx − 3
ψ2 
+
 1c
r  w0
w0
w0
y
−δΩ1
K̃02
2K̃0 K̃0y
ρ1 − !1 ∆Ω 3
ψ1c = −!1 p1r − !2 p2r .
3
r
r w0
(52)
Once again, we cross-differentiate and eliminate the pressure between equations (46) and
(52). Following some algebraic manipulations we arrive at the result
ψ1cxx
K̃0 K̃0y w0yy
!1 ψ1cyy +
+ Ω1 2 2 −
ψ1c
2y
2y w0
w0
x
K̃0 K̃0y w0yy
ψ2xx
ψ2
+ !2 ψ2yy +
+ Ω1 2 2 −
2y
2y w0
w0
 x   2

2
w0yy
K̃0y
Ω1 K̃0 K̃0y
Ω1
K̃0 K̃0y  ψ1c
2 1
+ !1
−
+ 2
+

w0
2y 2 w02
w0
2y w0  w0
w0
w0
2 x
y
y
K̃0 K̃0
Ω1 K̃02 ρ1
+ δ
− 2w0y ρ1 − w0 ρ1y + !1 ∆Ω 2 2y ψ1cx = 0.
(53)
2
4y w0
2y w0
x
As we have already seen, the leading term in the above equation vanishes. The remainder
integrates with respect to x and yields
ψ2xx
K̃0 K̃0y w0yy
!2 ψ2yy − ψ2yy (0, y) +
+ Ω1 2 2 −
{ψ2 − ψ2 (0, y)}
2y
2y w0
w0

 2
 
2
w0yy
Ω1 K̃0 K̃0y
K̃0y
Ω1
K̃0 K̃0y  ψ1c
2 1
+ !1
−
+ 2
+

w0
2y 2 w02
w0
2y w0  w0
w0
w0
2
y
y
Ω1 K̃02 (ρ1 − ρ1 (0, y))
+ δ
− 2w0y (ρ1 − ρ1 (0, y)) − w0 (ρ1y − ρ1y (0, y))
4y 2 w0
+ !1 ∆Ω
K̃0 K̃0y
ψ1c = 0,
2y 2 w02
(54)
where we have used the boundary conditions
ψ2 (x, 0) = 0, ψ2 (x, 1/2) = 0, 0 ≤ x ≤ xo ,
ψ2xx (0, y) = 0, ψ2x (xo , y) = 0, 0 ≤ y ≤ 1/2,
12
(55)
(56)
while ψ2 (0, y) is given by (41). Equation (54) implies that
!2 = O(!21 ) = O(!1 ∆Ω) = O(δ).
(57)
This important result confirms that the amplitude !1 of the flow perturbation near critical
swirl is much greater than the amplitude δ of the thermal perturbation due to reaction, as
anticipated in (21)-(28).
It is convenient to introduce ψ̃2 via the expression
ψ̃2 = ψ2 − ψ2 (0, y),
(58)
whence (54) can be rewritten as
ψ̃2xx
K̃0 K̃0y w0yy
+ Ω1 2 2 −
!2 ψ̃2yy +
ψ̃2
2y
2y w0
w0


2 K̃0 K̃0y
1 w0yy  ψ1c
Ω1
= −!21  3/2
−
3/2
w0 w0 y
2
yw0
yw0
y
Ω1 K̃02 (ρ1 − ρ1 (0, y))
− δ
− 2w0y (ρ1 − ρ1 (0, y)) − w0 (ρ1y − ρ1y (0, y))
4y 2 w0
− !1 ∆Ω
K̃0 K̃0y
ψ1c .
2y 2 w02
Here we have also taken the opportunity to rearrange the O(!21 ) term above. The above
equation is a nonhomogeneous partial differential equation for ψ̃2 , subject to homogeneous
boundary conditions for ψ̃2 similar to those in (48) - (49). Since the corresponding homogeneous problem has the nontrivial solution ψ1c , solvability requires that the forcing term
above be orthogonal to ψ1c . Upon multiplication of the above equation by ψ1c , followed
by an integration over the domain (0 ≤ x ≤ xo , 0 ≤ y ≤ 1/2) and use of the homogeneous
conditions for ψ̃2 , we obtain the solvability condition
!21 M1 − !1 ∆Ω M2 + δ M3 = 0.
Here,
M1 = −
0
xo

1/2
Ω1
0
1
K̃0 K̃0y
3/2
xo
yw0
M2 =
0
xo
M3 =
0
0
1/2
−
3/2
yw0
0
1/2
y
1
w0
w0yy
w0
(59)

3
 ψ1c dydx = 4xo N1 ,
2
3π
y
K̃0 K̃0y 2
1
x 0 N2 ,
ψ
dydx
=
1c
2y 2 w02
2
2w0y (ρ1 − ρ1 (0, y)) + w0 (ρ1y − ρ1y (0, y))
K̃02
−Ω1 2 (ρ1 − ρ1 (0, y)) ψ1c dydx
4y w0
13
(60)
(61)
(62)
where
N1 = −
0

1/2
Ω1
1
3/2
K̃0 K̃0y
−
3/2
yw0
yw0
N2 =
0
1/2
y
1
w0
w0yy
w0
K̃0 K̃0y 2
Φ (y)dy.
2y 2 w02

3
 Φ (y) dy,
2
y
(63)
(64)
Equation (59) is the principal result of the analysis; it is a quadratic equation for flowdisturbance amplitude !1 . For relevant base flows such as the Rankine vortex, the Burgers’
vortex, or the “Q-vortex,” it can be shown that N1 and N2 are positive. Now according
to equation (31), ρ1 = −T1 , where T1 , the solution of the flame-structure equations. This
allows one to rewrite M3 as
xo 1/2
[2w0y (T1 − T1 (0, y)) + w0 (T1y − T1y (0, y))
M3 = −
0
0
2
K̃
(65)
−Ω1 2 0 (T1 − T1 (0, y)) ψ1c dydx.
4y w0
We observe that T1 increases with x along the pipe axis ( T1 ≥ T1 (0, y) , see the example
in Figure 1). Also, for axial jet flows coming into the pipe, w0 ≥ 0 and w0y ≤ 0 for
0 ≤ y ≤ 1/2 . It is expected, therefore, that in most relevant cases M3 is also positive.
For (59) to have real solutions for !1 the inequality
M1 (δM3 )
|∆Ω| ≥ 2
(66)
M2
√
must hold. This means that there exists an O( δ) interval of swirl around ω1 where
no near-columnar, steady, axisymmetric state can exist and the flow must develop large
disturbances and a vortex breakdown. Outside this interval the solutions are
∆ΩM2 ± (∆Ω)2 M22 − 4M1 (δM3 )
!1 (δ, ∆Ω) =
.
(67)
2M1
In order to represent the solutions on a bifurcation diagram we select w(xo , 0), the axial
velocity along the centerline at the pipe outlet, as the physical quantity that characterizes
the solutions. To order !1 it is given by
∆ΩM2 ± (∆Ω)2 M22 − 4M1 (δM3 )
w(xo , 0) = w0 (0) + !1 Φy (0) = w(xo , 0) +
Φy (0), (68)
2M1
where we have used equations (25), (36), and (50)). Figure 2 displays w(xo , 0) as a function
of ∆Ω . It can be seen that for the branch of solutions with ∆Ω < 0 ( ω < ω1 ), the solution
with the (+) sign in (67) describes a flow state with a small-amplitude disturbance and a
14
slight flow deceleration near the centerline. The solution with the (−) sign in (67) corresponds to a flow state with a large-amplitude disturbance and a significant flow deceleration
near the centerline.
When (66) is an equality, i.e., for
M1 (δM3 )
2
ω1δa ≡ Ω1δa = Ω1 − 2
M2
M1 (δM3 )
2
ω1δb
≡ Ω1δb = Ω1 + 2
,
(69)
M2
there exist special equilibrium states which are saddle-fold bifurcation (limit) points of solutions of the axisymmetric reactive flow problem. We reiterate that no near-columnar
solutions exist when Ω1δa < Ω < Ω1δb , see again Figure 2. The quantity ω1δa is an asymptotic estimate of “the critical swirl for a premixed combusting flow in a straight pipe of finite
length.” It is lower than the corresponding value for the inert flow, and decreases with an
increase in the intensity of combustion.
4
Examples
The results found above are demonstrated for the case where the inlet profiles for w , T ,
and Y are uniform and where the flame structure is described by the numerical solution of
the r -independent problem of (32) and (33) with boundary conditions (34) and (35). We
concentrate on the case where β = 2 , P e = θ = 10 , L = 1 , and A = 50, 000 . Similar
results are found for other values of these parameters. We also use the Burgers vortex model
for the profiles of the incoming axial velocity and scaled circulation:
w0 (y) = 1, K̃0 (y) = 1 − exp(−2by).
(70)
√
Here b is a constant related to the vortex core radius, rc = 1.12/ b; we adopt b = 4 as a
representative example. A pipe of length xo = 2 is considered. Using a standard ordinary
differential equation solver from Maple for (51) (with the numerical condition Φy (0) = 1 ),
the eigenfunction Φ(y) is computed. Then the constants N1 and N2 are calculated from
(63) and (64). For b = 4 , we find that ω1 = 0.8976 , N1 = 0.00683 and N2 = 0.02831 .
The computation of δM3 uses the numerical solution of the flame structure. For example,
for δ = 0.025 the adiabatic flame temperature is Tb = 1.05 and δM3 = 0.00253 . Note that
the bifurcation diagram of Figure 2 corresponds to this case. When the incoming reactant
mass fraction is increased to δ = 0.075 , the adiabatic temperature rises to Tb = 1.15. (The
flame structure for this case is shown in Figure 1). Then, δM3 increases to 0.00801 .
Using these values, the bifurcation diagram of w(xo , 0) versus ω for various incoming
reactant mass fractions was computed and is shown in Figure 3. Each solution branch
corresponds to a fixed value of δ . Only the branches for 0 < ω < ω1 are considered. It can
be seen that when δ = 0.0 (no combustion) the critical swirl is a transcritical bifurcation
point. When δ = 0.025 , the solutions exhibit a fold behavior where in a certain range
of swirl two steady states appear for the same ω. To illustrate the nature of such states,
15
streamlines are presented in Figures 4a and 4b which correspond to points (a) and (b) in
Figure 3, respectively. The two states are found at ω = 0.721 and have the same flame
structure. In each frame in these figures the upper streamline represents the pipe wall, the
lower streamline the pipe centerline, and the flow runs from left to right. There are 11 equispaced levels of ψ = constant with ∆ψ = 0.05 . Also shown are vertical lines representing
10 equi-spaced levels of the reactant mass fraction Y = constant decreasing from δ to
zero. State (a), which corresponds to the solution with the (+) sign in (67), is a nearly
columnar flow (figure 4a) and is characterized by a relatively small flow deceleration along
the centerline. State (b), which corresponds to the solution with the (-) sign in (67), shows
a larger flow divergence with a stronger flow deceleration along the centerline. Actually, in
this state the axial speed on the centerline at the outlet is near zero.
Also shown in Figure 3 are branches of solutions for larger values of δ. It can be seen that
as δ is increased above a limiting value δlimit , the fold behavior in the solutions disappears
and only one solution can be found for each ω . As ω increases, large-amplitude perturbations may gradually appear in the flow as the swirl ratio ω is increased, and stagnation may
be found at lower levels of swirl. Figure 4c illustrates the flow nature at point (c) along such
a branch, where δ = 0.075 , Tb = 1.15 , and ω = 0.556 . Note that the flame structure in
this case moves upstream with respect to that in Figures 4a and 4b since more mass fraction
of the reactant is supplied at the inlet. The figure shows a significant flow deceleration along
the pipe centerline with almost zero axial speed at the outlet.
Equation (68) can be used to estimate the limit value of δ at which the solution branches
change their nature from a fold to a no-fold behavior. It is found that δlimit ∼ w02 (0)M1 /M3 .
In the case where δ = 0.05 we find M1 = 0.00579 and M3 = 0.1046 which shows that at
the given flow and combustion parameters δlimit ∼ 0.055 .
5
Conclusions and discussion
Near-critical swirling flow in a straight circular pipe of finite length has been examined for a
premixed, exothermically reacting fluid. Attention is confined to a dilute mixture, for which
the thermal energy imparted to the fluid (as measured by the amount of reactant) is small
and the problem is amenable to an asymptotic treatment. Exothermicity of order δ produces same-order perturbations in temperature and√density, but near the critical swirl level,
disturbances to the flow are much larger, of order δ. In the absence of exothermicity the
basic, columnar flow undergoes a transcritical bifurcation as the swirl ratio ω is increased
through the critical value ω1 . Exothermicity acts as an imperfection and splits the bifurcation portrait into two branches, folding at swirl levels ω1δa and ω1δb , and thereby creating a
gap, ω1δa < ω < ω1δb , within which no steady, near-columnar, axisymmetric states can exist.
The new critical swirl, ω1δa , decreases with increase in exothermicity. For sufficiently small
values of δ there exists an interval of ω bounded above by ω1δa for which there are two
equilibria, one describing a near-columnar state and the other a large-amplitude disturbance.
However, when exothermicity is increased above a limit value, δlimit , this behavior changes
uniformly into a branch of steady states with no fold. These states describe a gradually
increasing steady disturbance with flow deceleration near the pipe centerline as the swirl is
16
increased.
In order to provide some more insight into the physical meaning of the critical swirl
ω1δa it is useful to consider the expected stability characteristics of the solutions found in
the present analysis. We recall that in the absence of exothermicity ( δ = 0 ) the critical
swirl is also a point of exchange of stability (Wang and Rusak 1996a,b). Specifically, the
branch of columnar states has a decaying mode of disturbance when 0 ≤ ω < ω1 since the
convection of azimuthal vorticity perturbations dominates the stretching effects resulting
from the swirl. At the critical level ω = ω1 the stretching and convective effects are in a
critical balance and render a neutrally stable columnar state. When ω > ω1 the stretching
effects dominate, interact with the inlet conditions, and result in an unstable mode that is
related to the critical eigenfunction. We expect that these stability characteristics carry over
to the exothermic case. Thus we anticipate that for 0 < δ < δlimit , the limit swirl levels
ω1δa and ω1δb found above are points of exchange of stability (see Figure 5 for a schematic
representation of the expected behavior), i.e., the branch of near-columnar states is linearly
stable to small axisymmetric disturbances when ω < ω1δa , whereas the branch of largedisturbance solutions is unstable. At ω = ω1δa a neutral mode of disturbance exists. Also,
as ω approaches ω1δa the near-columnar states lose their stability margin. For ω > ω1δa
near-columnar states can not exist and the flow develops an axisymmetric vortex breakdown
process similar to that described in Wang and Rusak 1997a and Rusak et al 1998a. It is
also expected that when δ ≥ δlimit the solutions along the branches with no fold are linearly
stable.
Within the general context of flows with heat addition, the decrease in critical swirl
resulting from exothermicity may appear counter-intuitive at first sight. From classical
flame theory with no swirl one expects that increase of temperature along the pipe resulting
from the reaction causes a decrease in density and a corresponding increase in axial speed
along the pipe. Such an effect may strengthen the convection of perturbations out of the
domain and thereby raise the level of swirl needed for a critical balance. However, the
situation changes significantly when swirl is employed and the swirl level is high and near
critical. Then, the present analysis shows that flow perturbations induced by exothermicity
are much larger than changes in the thermodynamic properties.
The decrease of the critical swirl may be explained by the following arguments. The
vorticity transport equation for a compressible, inviscid, reacting flow (Thompson 1988, p.
73) is
* t̄ + V̄* · ∇ω̄
* = ω̄
* · ∇V̄* − ω̄
* (∇ · V̄* ) + ∇T̄ × ∇s̄
ω̄
(71)
* = ∇ × V̄* the vorticity, while the specific entropy
where V̄* is the velocity vector and ω̄
*,
is denoted by s̄ . This equation describes the balance between unsteady changes of ω̄
* , stretching and tilting of ω̄
* , vorticity changes resulting from expansion
convection of ω̄
*
*
∇ · V̄ = −(1/ρ̄)(ρ̄ + V̄ · ∇ρ̄) , and vorticity production resulting from nonalignment of
t̄
gradients of the thermodynamic properties, temperature and specific entropy (a baroclinic
effect). Here we denote by ζ̄, η̄ , and ξ¯ the radial, azimuthal and axial components of the
vorticity, respectively. For an axisymmetric and steady flow in a cylindrical geometry, the
17
azimuthal component of the above equation is
v̄
¯ x̄ + η̄ ū + η̄ (ūρ̄r̄ + w̄ρ̄x̄ ) + T̄x̄ s̄r̄ − T̄r̄ s̄x̄ .
w̄η̄x̄ + ūη̄r̄ + ζ̄ = ζ̄ v̄r̄ + ξv̄
r̄
r̄ ρ̄
(72)
Upon using the relations for an axisymmetric flow, ζ̄ = −v̄x̄ and ξ¯ = (r̄v̄)r̄ /r̄, and with
η̄ = r̄χ̄, the above equation takes the form
w̄χ̄x̄ + ūχ̄r̄ =
2v̄ v¯x̄ χ̄
1
+
+
w̄
ρ̄
)
+
s̄
−
T̄
s̄
.
(ūρ̄
T̄
r̄
x̄
x̄
r̄
r̄
x̄
r̄2
ρ̄
r̄
(73)
Using Gibbs equation T̄ ds̄ = Cp dT̄ − dP̄ /ρ̄ − µdY (where µ, the chemical potential, has
the form ∂G/∂Y for a single reaction, where G(p, T, Y ) is the Gibbs free energy and the
partial derivative is taken at fixed p and T ) and the radial and axial momentum equations
(3) and (5), we find that
1
T̄r̄
v̄ 2
s̄r̄ = Cp +
ūūr̄ + w̄ūx̄ −
− µYr̄ ,
r̄
T̄
T̄
1
T̄x̄
s̄x̄ = Cp + (ūw̄r̄ + w̄w̄x̄ ) − µYx̄ .
(74)
T̄
T̄
Then, (73) becomes
w̄χ̄x̄ + ūχ̄r̄ =
2v̄ v¯x̄ v̄ 2 T̄x̄ χ̄
T̄x̄
T̄r̄
+
(ūū
(ūw̄r̄ + w̄w̄x̄ )
(ūρ̄
−
+
w̄
ρ̄
)
+
+
w̄ū
)
−
r̄
x̄
r̄
x̄
r̄2
r̄2 T̄
ρ̄
T̄ r̄
T̄ r̄
µ
− T̄x̄ Yr̄ − T̄r̄ Yx̄ .
r̄
(75)
This equation shows that convection of the property χ̄ = η̄/r̄ is balanced by several terms
on the right hand side of the equation. The first term represents a stretching effect resulting
from the swirl axial gradient. The second term represents a baroclinic effect resulting from
interaction between swirl and axial temperature gradient. Note that both terms are functions
of the square of the swirl level, ω 2 . The third term represents vorticity changes resulting from
interaction between the azimuthal vorticity and expansion (density changes). The forth and
fifth terms also represent baroclinic effects which result from the interaction between radial
and axial gradients of the velocity components and radial and axial temperature gradients.
The last term represents vorticity changes resulting from the reaction.
Use of the vorticity transport equation (75) and the asymptotic expansions (23)-(28)
provides an alternate route to deriving (47) and (54). It shows that (47) and (54) are
essentially the linearized and second-order reductions of the steady vorticity transport equation (75). Moreover, when (23)-(28) are used, it can be shown that at orders !1 and
!2 = O(!21 ) = O(!1 ∆Ω) = O(δ) considered in the present analysis, the forth, fifth, and
sixth terms on the right hand side of (75) are of order !1 δ or δ 2 , which are much smaller
than the other terms in (75) and may be neglected. The third term is of order δ , fixed by
the base flow properties and the flame structure, and is independent of the swirl level. Then,
to the orders considered, only the first two terms on the right hand side of (75) depend on
the swirl level. The present analysis shows that a flow perturbation with a positive radial
18
speed ( ū > 0 ) and a negative axial gradient of swirl ( v̄x̄ = −[(r̄v̄)r̄ /r̄]ū/w̄ < 0 ) is the most
dominant disturbance. Such a disturbance produces from the first term on the right hand
side of (75) a negative gradient of χ̄ , reduces the azimuthal vorticity η̄ along the pipe axis,
and thereby acts to decelerate the flow around the pipe centerline and causes divergence of
streamlines. As the swirl increases, the flow deceleration also increases and reaches a critical
balance at a certain level of swirl. The second term provides an additional production of the
property χ̄ which also depends on swirl. The flame-structure problem (31)-(35) shows that
temperature rises, and therefore density ρ̄ decreases along the pipe, i.e., ρ̄x̄ < 0 . These density changes increase with the level of exothermicity, δ, and increase the negative gradients
of χ̄ (or η̄ ) and flow deceleration around the pipe centerline. Therefore, a lower level of the
swirl ratio ω is needed to create a critical balance. This means that the critical swirl ω1δa
decreases with increase of δ. Also, for a fixed level of swirl below the critical, the increase
of δ causes decrease of ρ̄ and a larger perturbation to the columnar flow with a stronger
deceleration near the pipe centerline. Similarly, for perturbations with the same size, an
increase of δ causes decrease of ρ and less swirl is needed to generate such a perturbation.
The computed examples in Section 4 (Figure 3) demonstrate this interesting and unusual
behavior of combustion states with swirl. Note that within the present small-disturbance
theory the classical effect of flow acceleration along the pipe resulting from the temperature
increase may appear only in the second order O(!2 ) ∼ δ velocity perturbations.
The theory of Wang and Rusak (1997a) and the numerical computations of Rusak et
al (1998a) showed the appearance of axisymmetric breakdown states in cold (δ = 0) , incompressible, and inviscid vortex flows in a straight circular pipe. These states develop as
a branch of linearly stable solutions that is connected to the branch of the unstable largedisturbance states found in the present analysis for δ = 0 . This branch starts from another
limit swirl level ω0 (where ω0 < ω1 ) which is the threshold level of swirl for the appearance
of axisymmetric breakdown states (see Figure 6). It is expected that similar breakdown
states appear in the reactive flow, specifically when the combustion is lean and δ is sufficiently small, 0 ≤ δ < δlimit . In such cases, it is expected that the reactive flow problem
defined by (9)-(16) with boundary conditions (17)-(20) has three steady-state solutions for
every ω in the range of swirl ω0 ≤ ω ≤ ω1δa (see Figure 6). One state is a near-columnar
stable state described by the positive root in (67), the second is an unstable large-disturbance
state described by the negative root in (67), and the third is a stable breakdown state which
contains a finite-size near-stagnation zone around the pipe centerline and may be computed
numerically. However, when the amount of the incoming reactant is above δlimit and the
fold behavior disappears, it is expected that the branch of stable solutions described by the
positive root in (67) is directly connected to the branch of breakdown states (see Figure 6).
Then, the reactive flow problem (9)-(16) with boundary conditions (17)-(20) has only one
solution for every ω . This discussion clarifies the possible steady reactive flow states that
may appear at different amounts of the incoming reactant and at various swirl levels below
ω1 .
The above arguments suggest the existence of hysteresis loops and limit-cycle oscillations
between various combustions states as either the amount of the incoming reactant δ or the
incoming swirl level ω are slightly changed in time from a base design point (see arrows
in Figure 6 which indicate for such situations). For example, when 0 ≤ δ < δlimit such
19
oscillations can occur as ω may oscillate around the range between ω0 and ω1δa . Then, the
flow may oscillate between near-columnar states and breakdown states. Another possible
situation is when ω is fixed and δ oscillates between two levels. Then, the flow may oscillate between perturbed states according to (67) and breakdown states or between different
breakdown states that correspond to the different levels of δ at the given ω . Such predicted
global instabilities of combustion states should be computed by numerical schemes of the
unsteady reactive flow problem and may shed new light on the dynamics of combustion with
swirl.
Acknowledgement
This research was carried out with the support of the National Science Foundation under
Grant CTS-9904327.
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1.2
δ T1
1
T
0.8
4Y , T
0.6
0.4
Y
0.2
δY
1
0
−0.2
0
0.2
0.4
0.6
0.8
1
x
1.2
1.4
1.6
1.8
2
Figure 1: The flame structure for β = 2 , P e = θ = 10 , L = 1 , A = 50, 000 , and
δ = 0.075 .
23
2
(−)
1.8
Ω1 δ b
1.6
1.4
w(xo,0)
1.2
(+)
1
(+)
0.8
0.6
Ω1 δ a
0.4
0.2
(−)
0
−1
−0.8
−0.6
−0.4
−0.2
0
∆Ω
0.2
0.4
0.6
0.8
1
Figure 2: The bifurcation diagram of steady-state solutions of near-critical reactive flows
with swirl in a pipe for a Burgers vortex (70) with b = 4 and δ = 0, 0.025 .
24
2
δ=0.075
1.8
1.6
δ=0.05
1.4
δ=0
1.2
w(xo,0)
δ=0.025
1
δ=0.025
0.8
0.6
δ=0.05
(a)
δ=0.075
0.4
0.2
(c)
0
0.4
0.5
0.6
(b)
0.7
0.8
0.9
ω
1
1.1
1.2
1.3
1.4
Figure 3: The bifurcation diagram of steady-state solutions of near-critical reactive flows
with swirl in a pipe for a Burgers vortex (70) with b = 4 and various values of δ .
25
a) Positive Root (domega<0)
r
1
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
b) Negative Root (domega<0)
1.4
1.6
1.8
2
x
0
0.2
0.4
0.6
0.8
1
1.2
c) Positive Root (domega<0)
1.4
1.6
1.8
2
x
0
0.2
0.4
0.6
1.4
1.6
1.8
2
x
r
1
0.5
0
r
1
0.5
0
0.8
1
1.2
Figure 4: (a) Streamline contours of a nearly columnar state (positive root of (67) for a
Burgers vortex (70) with b = 4 and ω = 0.721 and δ = 0.025 . (b) Streamline contours of
a large-disturbance state (negative root of (67)) for a Burgers vortex (70) with b = 4 and
ω = 0.721 and δ = 0.025 . (c) Streamline contours of a perturbed state (positive root of
(67)) for a Burgers vortex (70) with b = 4 and ω = 0.556 and δ = 0.075 .
26
1.2
δ=0
Stable State
Unstable State
1
w(xo,0)
0.8
0.6
δ
0.4
0.2
0
0.4
0.5
0.6
0.7
ω
0.8
0.9
1
Figure 5: Expected stability characteristics of the reactive flow equilibrium states.
27
Figure 6: Expected steady-state solutions of the reactive flow problem, their stability characteristics, and possible limit-cycle oscillations between states.
28