[american institute of aeronautics and astronautics 46th aiaa aerospace sciences meeting and exhibit...

American Institute of Aeronautics and Astronautics

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AIAA 2008-1052 Compressible Swirling Flows with Lean Premixed Chemical Reaction

By Z. Rusak1

Dept. of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY

And J.J. Choi2

School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA

The theoretical foundation for the global analysis of vortex flows is

extended to the case of compressible, swirling flows with lean premixed reaction in a finite-length, straight, circular pipe. A novel nonlinear partial differential equation for the solution of the reactive flow stream function is developed in terms of the flow governing specific total enthalpy, specific entropy, and circulation functions. Solutions of the resulting nonlinear ordinary differential equation for the columnar case together with a newly derived flow-force condition describe the outlet state of the reactive flow in the pipe. These solutions can be used to form the bifurcation diagrams of steady reacting flows with swirl as the swirl level is increased and provide theoretical predictions of the critical conditions for the first appearance of reacting flows with vortex breakdown states. The paper sheds light on the dynamics of compressible, reacting flows with swirl and vortex breakdown.

1. Introduction

Swirl is of practical importance in gas-turbine combustion as a means of enhancing the performance of certain lean, premixed, subsonic combustion systems (Syred & Beer 1974, Gupta et al. 1984, Paschereit et al. 1998a,b). When the degree of swirl in the flow is above a certain critical level, a large and nearly stagnant axisymmetric separation (breakdown) region suddenly appears along the vortex axis. In the case of an incoming premixed lean flux of air and fuel, the burned particles are trapped in the vortex breakdown zone where they create a region of high temperature. Heat transfer from this hot region to the surrounding swirling flow helps to stabilize the flame and burn more of the reactants, thereby improving combustion effectiveness (Sivasegaram & Whitelaw 1991). Once appearing in the reacting flow, the breakdown zone serves as a natural fluid dynamic flame holder, specifically in combustion with high upstream speeds, where the flow is slowed down to near stagnation ahead of the breakdown zone and a stable chemical reaction can take place. However, a major problem in inducing axisymmetric vortex breakdown in combustors is the appearance of flow instabilities inside the large separation zone and in its wake (see, for example, the experimental results of Bruecker & Althaus 1995, Sivasegaram & Whitelaw 1991, Paschereit et al. 1998a,b). At certain critical situations, the breakdown zone itself may become unstable and disappear because of changes in the distribution of swirl to the incoming flow, variations in the temperature field, or large perturbations in the back-pressure at the downstream

1 Professor, AIAA Associate Fellow 2 Research Associate

46th AIAA Aerospace Sciences Meeting and Exhibit7 - 10 January 2008, Reno, Nevada

AIAA 2008-1052

Copyright © 2008 by Z. Rusak and J.J. Choi. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


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of the combustor. The appearance and disappearance of breakdown zones at near critical swirl levels can lead to significant flame oscillations in the chamber or flame blowout (see the theoretical predictions of Rusak et al. 2002, the numerical simulations of Biagioli, 2006 and Umeh & Rusak, 2006, and the experimental studies of Zhang et al. 2007 and Lieuwen et al. 2007). On the other hand, at higher swirl levels the breakdown zone moves upstream, forcing the flame position into the fuel mixing region and causing flame flashback (Umeh & Rusak 2006). Extensive literature is available on the fluid mechanics of vortex stability and breakdown, including the reviews by Hall (1972), Leibovich (1978, 1984), Escudier (1988), Sarpkaya (1995), Althaus et al. (1995) and Rusak (2000). The flow’s significant expansion around a central recirculation (breakdown) zone is observed together with fine scale three-dimensional, vortical structures, including a helical vortex tube. The breakdown zone size and location strongly depend on the swirl number, chamber geometry and axial pressure gradient. However, a consistent description of the phenomenon has proved elusive, although several possible explanations were given for certain features of the problem. Until recently, the relationship between the various theoretical and numerical solutions had not been fully clarified, nor were unambiguous criteria given for the occurrence, stability, and dynamics of vortex breakdown.

Over the last ten years, Rusak and co-authors have developed a novel theoretical framework for explaining and predicting the axisymmetric vortex breakdown process in non-reacting vortex flows (Wang & Rusak 1996a,b,c, 1997a,b, Rusak & Wang 1996a,b, Malkiel et al 1996, Rusak et al. 1998a,b, Rusak & Lamb 1999, Judd et al. 2000, and Rusak 2000, Rusak & Lee 2002, 2004, and Rusak et al. 2007). The approach, employing the Euler and Navier-Stokes formulations, examines the global dynamics of incompressible, high Reynolds number, axisymmetric swirling flows in a cylindrical chamber of finite length. The model considers the interaction of a vortex flow downstream of a vortex generator with vorticity waves that propagate in the flow. The model allows the inlet state a degree of freedom to develop a radial velocity due to the effect of upstream influence of propagating disturbances in the flow. The results, established through a rigorous, nonlinear, global analysis, provide a fundamental and nearly complete mathematical description of the dynamics and stability of single-phase axisymmetric swirling flows. This theory (Rusak 2000) gives a global description of the problem and unifies the major previous theoretical and numerical approaches to axisymmetric vortex breakdown, including the works of Benjamin (1962), Hall (1972), Randall & Leibovich (1972), Leibovich & Kribus (1990), Keller et al. (1985), Buntine & Saffman (1995), Brown & Lopez (1990), Beran & Culick (1992), Beran (1994) and Lopez (1994).

The theory also provides a consistent explanation of the physical mechanism leading to axisymmetric vortex breakdown, and the conditions for its occurrence in high Re-flows. The

Fig. 1. The bifurcation and stability diagram of non-reacting vortex flows.


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results are summarized in the bifurcation diagram (Fig. 1, from Wang and Rusak 1997a) where the horizontal axis represents the variation of the flow from columnar state and the vertical axis represents the vortex energy functional, E. Each energy line is for a fixed level of the swirl ratio ω of the incoming vortex flow. The analysis revealed the existence of three branches of steady vortex states, connected by two critical levels of swirl ω0 and ω1, where ω0 < ω1. The critical swirl ω1 is an extension for a finite-length pipe of Benjamin’s (1962) critical swirl concept and ω0 is an extension for a finite-length pipe of Keller et al.’s (1985) special vortex breakdown solution. The branch of columnar vortex states (the vertical bold line) is composed of absolutely stable states when ω< ω0, linearly stable states when ω0 < ω < ω1 and unstable states when ω > ω1. The branch of solitary waves (the dashed line) connecting between the states at ω0 and ω1 is composed of unstable states describing axisymmetric traveling waves along the vortex core that are convected downstream. The branch of breakdown states (the inclined bold line) starts from the swirl ratio ω0 and is globally stable for ω > ω0. The theory shows that the vortex breakdown phenomenon is a necessary evolution from an initially near-columnar concentrated vortex flow to another relatively stable, lower-energy equilibrium state describing a swirling flow around a large nearly-stagnant breakdown zone. This evolution is the result of the interaction between azimuthal vorticity waves propagating upstream and the relatively fixed incoming vortex flow, which leads to an absolute loss of stability of the base columnar state when the swirl ratio ω is near or above a critical level ω1. This result reveals a previously unknown global instability mechanism and sheds new light on the relationship between flow stability and vortex breakdown. The fundamental critical states at ω0 and ω1 that lead to vortex instability and breakdown are calculated from a single, ordinary differential equation that is governed by the inflow axial and circumferential speed profiles.

The theoretical results of Wang & Rusak (1997a) suggested the existence of hysteresis in the range of swirl between ω0 and ω1, a result not previously predicted. We showed experimentally the existence of the hysteresis loop for various Re in nonreacting flows (Malkiel et al. 1996). Limit levels ω0 and ω1 as function of Re were reported. These results also indicate for the possible existence of limit-cycle oscillations between columnar and breakdown states in the range of swirl between ω0 and ω1 due to external acoustic, vortical or thermal perturbations. It was also shown that even large perturbations to the vortex flow just below ω0 would not permanently change it from being near columnar, while a large perturbation to the flow just above ω0 leads to a breakdown state. Using LDV results of the inlet velocity profiles we computed the limit swirl levels and found good correlation with the data. Moreover, guided by the theory, the breakdown criteria of Q-vortices in a pipe were computed (Rusak et al. 1998b) and showed agreement with Leibovich (1984) data. Independently, Mattner et al. (2002) also found nice agreement between their measured breakdown states and Rusak et al. (1998b) predictions. The most remarkable result is the calculation of breakdown along leading-edge vortices above a 70-deg delta wing using flow data from lower angles of attack no sign of breakdown, see Rusak & Lamb (1999). The nice agreement of predictions of breakdown location as function of angle of attack with measured data was demonstrated.

The effects of flow Reynolds number, inlet vorticity perturbation, and pipe divergence or convergence on the critical swirl ω1 were studied in Wang & Rusak (1997b), Rusak (1998), Rusak et al. (1998, 2001) and Rusak & Meder (2004), respectively. The theory was recently extended to describe the dynamics and breakdown of compressible vortex flows (Rusak & Lee 2002, 2004, 2007). The strong interaction between the convection of vorticity and the baroclinic effects due to the coupling between swirl and temperature gradients was demonstrated. Results


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showed that the bifurcation diagram of compressible swirling flows is similar in nature to that of incompressible vortex flows. However, the critical swirl ratio ω1 (which is also a point of bifurcation of non-columnar states as well as a point of exchange of stability) is delayed to higher levels as the flow Mach number is increased. Similar results were found in the numerical simulations by Herrada et al. (2003). Although not proven yet, it is expected that the critical swirl ω0 is also affected by the baroclinic effects and is also delayed to higher values as flow Mach number is increased. The presence of dilatation in the flow significantly alters the vortex dynamics. As revealed by the vorticity equation, positive flow dilatation causes the damping of vorticity magnitude as the vortex tube expands. It also causes augmentation/damping of vorticity through the baroclinic mechanism, depending upon the relative orientations of the pressure and density gradients (Bray et al., 2001). Furthermore, if associated with temperature changes, it is also accompanied by strong variations in viscosity and, therefore, effective Reynolds number. However, there is limited work quantifying how flow dilatation alters the parameter regimes in which vortex breakdown occurs and the structure of the breakdown state. For example, characteristic signals associated with vortex flow under non-burning and premixed combustion conditions reveal well-defined frequencies associated with swirl (Gupta 1979, 1985). Several different modes of combustion were observed in swirl burners, each with its own characteristic frequency (Gupta, 1979). The frequency and amplitude of the instabilities vary with swirl strength, mixture ratio, mode of fuel entry, and burner design and operational parameters.

The effect of weak reaction on near-critical swirling flows was recently studied by Rusak et al. (2002), Choi et al. (2004, 2007) and Sohn et al. (2006). Using asymptotic analyses and numerical simulations we studied the manner in which heat release resulting from a premixed and non-premixed combustion alters the nature of a steady, axisymmetric, near-critical swirling flow in a straight circular pipe. It was shown that weak exothermicity splits the bifurcation portrait of Fig. 1 of a cold flow into two solution branches separated by a gap in the level of swirl; within this gap no steady near-columnar solutions exist and the critical swirl ω1δ for a reacting flow is lower than ω1 for a cold flow. For a certain range of swirl below ω1δ and for sufficiently low heat release, the solutions yield three equilibrium states, one a near-columnar state, the second a solitary wave state, and the third is a breakdown state. For heat release beyond a limit value the branch loses its fold, suggesting a gradual appearance of large-amplitude disturbances with increase of swirl. It is shown that at near critical swirl levels, small amounts of exothermicity can generate significantly larger changes in the velocity and pressure. The mechanism that leads to breakdown is the nonlinear interaction between swirl, baroclinic effects and propagation of azimuthal vorticity waves. The numerical simulations of Choi et al. (2007) agree with the asymptotic results only for low heat release, and as heat release increases, the critical swirl ω1δ also increases. Similarly, the numerical simulations of Umeh and Rusak (2006) of lean premixed combustion in a sudden expansion combustor have demonstrated that the computed ω0 of the speed profiles at the sudden expansion plane and at the inlet section correlate nicely with the swirl levels at which flame blowout and flame flashback occur.

In this paper we extend the theoretical foundation for the global analysis of the dynamics of vortex flows in pipes to the case of compressible, swirling flows with lean premixed reaction in a finite-length, straight, circular pipe. A novel nonlinear partial differential equation for the solution of the flow stream function is developed in section 2 in terms of the flow governing specific total enthalpy, specific entropy, and circulation functions. Solutions of the resulting nonlinear ordinary differential equation (section 3) together with a newly derived flow-force


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condition (section 4) describe the outlet state of the reactive flow in the pipe. The paper sheds additional light on the complex dynamics of compressible, reacting flows with swirl and vortex breakdown.

2. Mathematical Model

The non-dimensional state, continuity, linear momentum, species, and energy equations describing the development of a steady, inviscid, compressible and chemically-reacting premixed flow are given respectively by (Williams 1985 or Buckmaster & Ludford 1982): p = ρT, ∇⋅ (ρV)=0, ρV⋅ ∇V = - 1/(γ M0

2) ∇p, (1) ρV⋅ ∇Y - L/Pe ∇2Y= - W, ρV⋅ ∇T - (γ - 1)/γ V⋅ ∇p - 1/Pe ∇2T = αW. Here, the number of reactants is limited to one and Y is the mass fraction of the reactant in the combustible gas, T the temperature, ρ the density, p the pressure, and V the velocity vector. The reacting fluid behaves according to the perfect gas equation of state (1) where R is the specific gas constant. The gas thermophysical properties are assumed to be constant. Non-dimensionalization is carried out with reference to the state of the inlet flow. Velocity is normalized with respect to the inlet flow centerline axial speed U0 and lengths are normalized against the pipe inlet radius R0. Temperature, density and pressure are normalized with respect to the inlet flow centerline temperature T0, density ρ0, and pressure p0. Furthermore, L is the Lewis number, Re the Reynolds number, Pe is the Peclet number, α the heat-release parameter, γ the specific heats ratio, and M0 = U0 /a0 is the flow Mach number, where 0 0a RTγ= is the frozen acoustic speed at the reference state. Also, W is the reaction rate. Tractability requires making certain idealizations of the combustion process. As the emphasis is on heat release-flow interaction, kinetic complexities are avoided in favor of a simple, one-step, first-order Arrhenius reaction W = DρYexp(-θ/T) where θ is the dimensionless activation energy and D is the reaction rate pre-exponential factor. We focus on the case of high activation energy where the high-θ asymptotic approach is applicable (Williams 1985, Buckmaster & Ludford 1982) and where the reaction front is a very thin sheet and W is a localized function around the reaction sheet. We also consider the case of high Pe where the diffusion and conduction terms in the species and energy equations may be neglected.

Using the flow linear momentum equation in (1), the Gibbs relationship for a perfect gas

in its non-dimensional form, ( 1) dpTds dhγ γρ

= − − (where s is the non-dimensional specific

entropy, scaled with the specific heat at constant volume Cv, and h T= is the non-dimensional specific enthalpy, scaled with the specific heat at contact pressure Cp and the temperature T0),

and the vector identity, 1( )2

V V V V V ω∇ = ∇ − ×i i , Crocco’s formula is found,

( )20

1( 1)M V H T sγ ωγ

− × = ∇ − ∇ , (2)


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where H is the non-dimensional specific total enthalpy (scaled with the specific heat at contact

pressure Cp and the temperature T0) and given by 2 20 0( 1) ( 1)

2 2M MH h V V T V Vγ γ− −

= + = +i i .

Also, from the species equation in (1) we find that V Y Wρ ∇ = −i , (3a) from the energy and linear momentum equations in (1), we find that

V H Wρ α∇ =i , (3b) and, from the energy equation and Gibbs relationship, we find that

1V s Wαγ

∇ =i . (3c)

In the case of an axisymmetric flow in a cylindrical ( , , )x rθ coordinate system with the corresponding axial speedw , circumferential speedv , and radial speed u , the continuity equation in (1) is:

( ) ( ) 0ru rwr xρ ρ+ = . (4)

A stream function ( , )x yψ is defined where ,2

yxu wy

ψψρρ

= − = and2

2ry = . Also, the

circumferential momentum equation in (1) is: ( ) ( ) 0u rv w rvr xρ ρ+ = or ( ) ( ) 0yrv rvx y xψ ψ− + = , (5) which shows that the circulation function K rv= is given by ( )K K ψ= . Equations (3a), (3b) and (3c) become, respectively: uY wY Wr xρ ρ+ = − or x y yY Y Wxψ ψ− + = − , (6a) uH wH Wr xρ ρ α+ = or x y yH H Wxψ ψ α− + = , (6b) and

us ws Wr xρ ρ γα+ = or x y ys s Wxψ ψ γα− + = . (6c) With the high-θ asymptotic approach we find that

0 0 0( ) ( ), ( ) ( ) and ( ) ( )Y Y Y H H H s s sψ ψ ψ ψ ψ ψ= = = = = = (6d) ahead of the reaction sheet, and, for a complete reaction across the sheet,

0 0 0 00, ( ) ( ) and ( ) ( ) /Y H H Y Q s s Y Q T fψ ψ ψ ψ= = + = + (6e)

behind the reaction sheet. Here, and fQ T are the non-dimensional heat added by the reaction and the flame temperature, respectively.

The x-component of the Crocco’s equation (2) is ( ) 12( 1) 0M u v H Tsx xγ η ςγ

− − = − ,

where the radial and circumferential components of the non-dimensional vorticity

are 1 and 22

v K u w u ywx x x r x yyς η= − = − = − = − , respectively. With 1

2v K

y= we find

that 12 2( 1) ( 2 ) ( 1)0 0 2KKxM u u yw H M Tsx y x xy

γ γγ

− − = − − − or


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'( ) 12 2( 1) ( 2 ) '( ) ( 1) '( )0 0 22KKxM u yw H M Tsx y xyy

ψ ψγ ψ γ ψ ψγρ

⎡ ⎤− − − = − − −⎢ ⎥

⎣ ⎦or

2 20 0

'( ) 1( 1) '( ) ( 1) '( )22

y xw u KKM H M Ts

yyψγ ψ γ ψ

ρ γρ

⎡ ⎤− − = − − −⎢ ⎥

⎢ ⎥⎣ ⎦ or

1 1 '( ) 12 2( 1) '( ) ( 1) '( )0 02 2KKy xM H M Ts

y yxy

ψ ψ ψγ ψ γ ψρ ρ ρ ρ γ

⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥− + = − − −⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦. (7)

Equation (7) is a novel partial differential equation for the solution ofψ . It extends the classical Squire-Long equation for incompressible and inert swirling flows to the case of compressible, reacting, swirling flows. In (7), the density ρ is computed from the specific entropy definition

for a perfect gas: ln 1Ts γρ

⎛ ⎞= ⎜ ⎟−⎜ ⎟

⎝ ⎠, i.e. ( ) 1/( 1)

/exp ( )T sγ

ρ ψ−

= ⎡ ⎤⎣ ⎦ , where T is found from

2 2( 1) ( )2 20( )2 2M KT H u w

yγ ψψ

⎛ ⎞−= − + +⎜ ⎟⎜ ⎟

⎝ ⎠ with ,

2yxu w

y

ψψρρ

= − = . The solution of (7)

assumes that the functions ( ), ( ), and ( )H K sψ ψ ψ are given, for example determined by the inlet conditions. For a flow in a finite-length, straight circular pipe of non-dimensional length 0x we consider the following boundary conditions. At the pipe inlet at x=0, we assume that the profiles of the axial speed, the circumferential speed, the axial derivative of radial speed and the temperature are given by

(0, ) ( ), (0, ) ( ), (0, ) 0, (0, ) ( ), ( ) 0 0 0 0 for 0 1/ 2.

w y w y v y v y u y T y T y Y Y yxy

ω= = = = =

≤ ≤ (8)

Here ω is the swirl ratio of the incoming flow. The inlet conditions describe a compressible swirling flow that was generated by a vortex generator ahead of the pipe at continuous and steady operation. The inlet condition on the radial speed allows the inlet state a degree of freedom to develop a radial velocity to account for the upstream influence of propagating disturbances in the flow. At the pipe outlet at 0x x= , we assume a fully developed, parallel, x -independent (columnar) flow state where ( , ) 0 for 0 1/20u x y y= ≤ ≤ . (9)

Along the pipe centerline at 0y = , we assume the symmetry condition, 0( , 0) 0 for 0x x xψ = ≤ ≤ , and along the pipe wall at 1/ 2y = , we assume constant flux,

1/2

0 0( ,1/ 2) ( ') ( ') ' for 00 00x y w y dy x xψ ρ= Ψ = ≤ ≤∫ .


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3. The Columnar State

A state is defined as columnar when for all x the flow is x -independent, i.e. 0u = , and ( )yψ ψ= . Note that the outlet state of the solution of the partial differential equation (7) is a

columnar state. We consider a base, inert columnar state where ( ), ( ), ( ), ( ), ( )0 0 0 0 0w w y v v y T T y p p y yω ρ ρ= = = = = . Then, from the radial momentum

equation for a columnar state, 22

0vp Mr r

γ ρ= , together with the equation of state we find that

2( )2 20 00 2 ( )0 0

p v yy Mp yT y

γ ω= . With the condition (0) 10p = we find that

2 2 2( ')0 0( ) exp '00 2 ' ( ')0

M v yyp y dyy T y

γ ω⎡ ⎤⎢ ⎥= ∫⎢ ⎥⎣ ⎦

and 2 2 2( ')1 0 0( ) exp '00 ( ) 2 ' ( ')0 0

M v yyy dyT y y T y

γ ωρ

⎡ ⎤⎢ ⎥= ∫⎢ ⎥⎣ ⎦

. (10)

Then, from ywψρ

= and (10) we can compute

( ) ( ') ( ') '0 00

yy y w y dyψ ρ= ∫ . (11)

This function can be inverted such that ( )y y ψ= . Also, we can compute ( )2 0K yv yω= and then,

( )( ) 2 ( ) ( )0K y v yψ ω ψ ψ= , (12a)

2( 1) 2 2 20( ) ( ( ) ( ))0 0 02M

H T y v y w yγ

ω−

= + + and then,

( ) ( ) ( )( )2( 1) 2 2 20( ) ( ) ( ) ( ) ( )0 0 0 02

MH H T y v y w y

γψ ψ ψ ω ψ ψ

−= = + + , (12b)

and ( )

( 1)2 2 2( ) ( ')0 0 0ln ln ( ) exp '001 2 ' ( ')0( )0

T y M v yys T y dyy T yy

γγ ωγ

γρ

− −⎛ ⎞⎛ ⎞ ⎧ ⎫⎡ ⎤⎜ ⎟⎜ ⎟ ⎪ ⎪⎢ ⎥= = ∫⎨ ⎬⎜ ⎟⎜ ⎟− ⎢ ⎥⎪ ⎪⎜ ⎟⎜ ⎟ ⎣ ⎦⎩ ⎭⎝ ⎠ ⎝ ⎠

and then,

( )( )2 2 2( 1) ( ')( )0 0( ) ( ) ln ( ) '00 0 2 ' ( ')0

M v yys s T y dyy T y

γ ω ψψ ψ γ ψ⎡ ⎤−⎢ ⎥= = − ∫⎢ ⎥⎣ ⎦

. (12c)

From Eqs. (12), we can compute the derivatives


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( ) ( ) ( )

( )

2 ( ) ( )2 ( ) 0 00 ( )'( ) 2 2 2( ) ( ) ( ')0 0 ( )0 0( ) exp '00 2 ' ( ')0

yv y T yyv y ydK dy y y yKdy d y w y M v yyw y dy

y T y

ω ψω ψψψ ρ γ ω ψψ

⎡ ⎤⎢ ⎥⎣ ⎦ =

= = =⎡ ⎤⎢ ⎥∫⎢ ⎥⎣ ⎦

, (13a)

( )

( )( ) ( ) ( ) ( ) ( )( )

( )

2 2( 1)0 0 0 0 0 0'( )

( ) ( )0 02 2( ) ( 1) ( ) ( ) ( ) ( )0 0 0 0 0 0

( )0 2 2 2( ')( )0 0( ) exp '00 2 ' ( ')0

T M v v w wy y ydH dyHdy d y w y

T y M v y v y w y w yy y yT y

M v yyw y dyy T y

γ ωψ

ψ ρ

ψ γ ω ψ ψ ψ ψψ

γ ω ψψ

+ − += =

+ − +=

⎡ ⎤⎢ ⎥∫⎢ ⎥⎣ ⎦

, (13b)

and

( )( )

( )

2 ( )2 2 0( ) ( 1) 00 2 ( )'( ) 2 2 2( ')( )0 0( ) exp '00 2 ' ( ')0

v yT y Myds dy ys

dy d M v yyw y dyy T y

ψψ γ ω

ψψψ γ ω ψψ

− −= =

⎡ ⎤⎢ ⎥∫⎢ ⎥⎣ ⎦

. (13c)

Equation (7) becomes in the columnar case

1 '( ) 12 2( 1) '( ) ( 1) '( )0 0 2KKyM H M Ts

yy

ψ ψγ ψ γ ψρ ρ γ⎛ ⎞

− = − − −⎜ ⎟⎜ ⎟⎝ ⎠

(14)

where the various functions of ψ are defined in (12) and (13) and

22 2( 1) ( )0( )2 2M KyT H

y

ψγ ψψρ

⎛ ⎞− ⎛ ⎞⎜ ⎟= − +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

, ( )

1/( 1)2 220( 1) ( )( )

2 2exp ( )

M KH wy

s

γγ ψψ

ρψ

−⎛ ⎞⎛ ⎞−

− +⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟= ⎜ ⎟

⎜ ⎟⎜ ⎟⎝ ⎠

. To

simplify the solution of (13) we use again ywψρ

= and then (13) is equivalent to the following

system of two, first-order, nonlinear, ordinary differential equations in terms of and w ψ :

1 ( ) '( ) 12'( ) ( 1) '( )02 2( 1) 0

K Kw H M Tsy yM

wy

ψ ψρ ψ γ ψγγ

ψ ρ

⎛ ⎞= − − −⎜ ⎟

− ⎝ ⎠

=

(15)


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with 22 2( 1) ( )0( )

2 2M KT H w

yγ ψψ

⎛ ⎞−= − +⎜ ⎟⎜ ⎟

⎝ ⎠,

( )

1/( 1)2 220( 1) ( )( )

2 2exp ( )

M KH wy

s

γγ ψψ

ρψ

−⎛ ⎞⎛ ⎞−

− +⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟= ⎜ ⎟

⎜ ⎟⎜ ⎟⎝ ⎠

and with

boundary conditions: 1/2

0(0) 0, (1/ 2) ( ') ( ') '0 00y w y dyψ ψ ρ= = Ψ = ∫ . (16)

A trivial solution of the columnar flow problem (15) and (16) is for the inert compressible case with 0Q = where ( ), ( ), ( ) with ( ), ( )0 0 0 0 0w w y v v y T T y p p y yω ρ ρ= = = = = given by (10), i.e. the base state. The work of Wang and Rusak (1997a) for incompressible swirling flows shows that when the swirl ratio 0ω ω≥ additional solutions of the nonlinear columnar flow problem exist, showing the existence of multiple solutions of the problem (7) with (8) and (9) and the appearance of vortex breakdown states. Moreover, it is clear that when 0Q ≠ , ( ) and ( )H sψ ψ are given by (6d) and (6e) and the solution of (15) and (16) is different from the base state.

In the next section we derive a flow force condition to determine which of the columnar state solutions of the problem (15) and (16) can be an outlet state of the solution of the steady state problem given by (7) and (8), (9). Once such columnar states are found, the bifurcation diagram of steady-state compressible and reacting flows with swirl in a pipe can be determined as the swirl ratio ω is increased at fixed values of 0 , , and fM Q T . From the previous work of Wang and Rusak (1997a) on incompressible inert swirling flows, it is expected that such columnar states will describe flows with a finite size, central, stagnant (breakdown) zone. Then, the critical conditions for the appearance of vortex breakdown states in reacting flows can be predicted.

4. Flow Force Analysis

Analysis of the axial linear momentum equation in (1), ( )20 r x xM uw ww pγ ρ + = − , for an inviscid

flow in a straight circular pipe shows that the flow force

( )1/2 2 2( ) ( , ) ( , ) ( , ) contant0 0F x p x y M x y w x y dyγ ρ= + =∫ for all 00 x x≤ ≤ . (17)

This means that 0(0) ( )F F x= or

( ) ( )1/2 1/22 2 2 2(0, ) (0, ) ( ) ( , ) ( , ) ( , )0 00 0 0 0 0 0p y M y w y dy p x y M x y w x y dyγ ρ γ ρ+ = +∫ ∫ . (18)

Applying the radial linear momentum equation in (1), 22

0vM uu wu pr x rr

γ ρ⎛ ⎞

+ − = −⎜ ⎟⎜ ⎟⎝ ⎠

, to

the inlet state and using the inlet conditions (8), we find that


11

2 220 ( )(0, )2 (0, ) (0, )0 2 2 y

y

v yu yM y p yy

ωγ ρ⎛ ⎞⎛ ⎞⎜ ⎟− = −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

. From the equation of state in (1) and the inlet

conditions (8), 0(0, ) (0, )/ ( )y p y T yρ = , and then, 2 22

0

0 0

2 2 ( ) (0, )(0, )0 0( ) 2 2 ( ) (0, )

y

y

M M v y p yu yT y yT y p yγ γ ω⎛ ⎞

− = −⎜ ⎟⎝ ⎠

.

We find that:

( )

( )

'0

'0

22 2 2 2 (0, ')( ')0 0 0(0, ) ( ) (0, ) exp ' exp '0 02 ' ( ') 2 ( ')0 0

22 (0, ')0 ( )exp '02 ( ')0

y

y

u yM v y My yp y T y y dy dyy T y T y

u yM yp y dyT y

γ ω γρ

γ

⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥= = −∫ ∫⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ ⎤⎢ ⎥= − ∫⎢ ⎥⎢ ⎥⎣ ⎦

.(19)

Here (10) is used. From (18) and the equation of state for the outlet state we have,

2

2 2(0, ')2 ( )' 2 00( )exp ' 10 00 ( ')0 ( )0

2( , )2 0( , ) 10 0 ( , )0

1/20

1/20

u y w yM yyp y dy M dyT y T y

w x yp x y M dy

T x y

γγ

γ

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟

⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥− +∫⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥= +⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

∫

∫

. (20)

Since ( ) ( )

2 2

0 0

2(0, ')(0, ') (0, ')' '0 ( ')0max ( ) min ( )

u yu y u yy y dyT yT y T y

⎛ ⎞⎜ ⎟⎝ ⎠

≤ ≤∫ for 0 1/2y≤ ≤ , we find that

( )2

20

0

2(0, ')2 '00 exp ' exp (0, ) 102 ( ')0 2 max ( )

u yM Myy dy u yT y T yγ γ

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟

⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

⎛ ⎞< − ≤ − ≤⎜ ⎟∫ ⎜ ⎟

⎝ ⎠ for 0 1/ 2y≤ ≤ . Then,

2

2( )2 0( ) 10 0 ( )0

2 2(0, ')2 ( )' 2 00( )exp ' 10 00 ( ')0 ( )0

2( , )2 0( , ) 10 0 ( , )0

1/20

1/20

1/20

w yp y M dy

T y

u y w yM yyp y dy M dyT y T y

w x yp x y M dy

T x y

γ

γγ

γ

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟

⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥+⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥− +∫⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥= +⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

∫

∫

∫

≥ (21)


12

We define 2( , )2( ) ( , ) 1 0 ( , )

1/20

w x yE x p x y M dyT x y

γ⎛ ⎞⎡ ⎤⎜ ⎟= +⎢ ⎥⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

∫ the columnar flow force at cross

section x (the flow force at x if the flow state there was columnar). From (21) we see that a solution of the steady state and axisymmetric problem (1) must satisfy that

0

2( )2 0( ) (0) ( ) 10 0 ( )0

1/20

w yE x E p y M dy

T yγ

⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥≤ = +⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

∫ , (22)

i.e. the columnar flow force at the outlet is equal of less than the columnar flow force at the inlet. This means that solutions of the columnar problem (15) and (16) that satisfy the condition (22) are outlet states of the solution of the axisymmetric problem (1). It should be noted that the columnar problem (15) and (16) is nonlinear and may have more than one solution. The state that gives an equal sign in (22) may be a bifurcation state for the appearance of multiple solutions of the axisymmetric problem (1).

5. Example Let the inlet flow describe a solid body rotation with a uniform axial speed, uniform temperature, and uniform reactant distribution, i.e.

( ) 2 , ( ) ( ) 1, ( )0 0 0 0v y y w y T y Y y δ= = = = for 0 1/2y≤ ≤ . (23)

Then,

( )

( )

( )

( )

2 2( ) ( ) exp , 0 0 02 21 12 2 0( ) exp 1 , (1/ 2) exp 1 ,0 02 2 2 2 20 0

1 2 2( ) ln 1 ,02 20

2 22 2( ) 2 ( ) ln 1 , '( ) ,02 2 2 10 02( 1) 20( ) 1 1

2 0

p y y M y

My M y

M M

y MM

K y M KM M

MH

M

ρ γ ω

γ ωψ γ ω ψ

γ ω γ ω

ψ γ ω ψγ ω

ωψ ω ψ γ ω ψ ψγ ω γ ω ψ

γψ

γ

= =

⎡ ⎤⎛ ⎞⎡ ⎤ ⎢ ⎥⎜ ⎟= − Ψ = = −⎢ ⎥⎣ ⎦ ⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

= +

= = + =+

−= + + ( ) ( )

( ) ( )

2 2( 1)2 2 0ln 1 , '( ) ,02 2 2 102 2( 1)2 2 0( ) ( 1) ln 1 , '( ) .0 2 2 10f

MM Q H

M

MQs M sT M

γ ωγ ω ψ δ ψ

γ ω ψ

γ γ ωψ γ γ ω ψ δ ψ

γ ω ψ

⎛ ⎞ −⎜ ⎟+ + =⎜ ⎟ +⎝ ⎠

−= − − + + = −

+

(24)

Then (15) becomes,


13

( ) ( )2 2 2 21 ln 102 22 2 1 00

w T MyM yM

wy

ωρ γ ω ψγ ωγ ω ψ

ψ ρ

⎛ ⎞⎜ ⎟= + − +⎜ ⎟+ ⎝ ⎠

=

(25)

with

( ) ( )2

22 2( 1) 2 2 12 2 2 201 1 ln 1 ln 10 02 2 22 0 0

MT M w M Q

yM M

γ ωγ ω ψ γ ω ψ δγ γ ω

⎛ ⎞⎛ ⎞− ⎜ ⎟⎜ ⎟= + + + − − + +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

,

( )1/( 1) 2 2 1 exp0 ( 1) f

QT MT

γ δρ γ ω ψγ

−⎛ ⎞

= + −⎜ ⎟⎜ ⎟−⎝ ⎠.

Also, in this case,

( ) ( ) ( ) ( )2 2 21 01/22 2 2 20(0) 1 exp exp 1 100 0 0 02 2 20

M ME M M y dy M

M

γ γ ωγ γ ω γ

γ ω

⎡ ⎤⎛ ⎞+⎢ ⎥⎜ ⎟= + = − = + Ψ∫⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦.(26)

6. Summary The theoretical foundation for the global analysis of vortex flows is extended to the case of compressible, swirling flows with lean premixed reaction in a finite-length, straight, circular pipe. A novel nonlinear partial differential equation for the solution of the compressible, reactive flow stream function is developed in terms of the flow governing specific total enthalpy, specific entropy, and circulation functions. Solutions of the resulting nonlinear ordinary differential equation for the columnar case together with a newly derived flow-force condition describe the outlet state of the reactive flow in the pipe. The paper sheds light on the complex dynamics of compressible, reacting flows with swirl and vortex breakdown.

References

Althaus, W., Bruecker, Ch. and Weimer, M. 1995 “Breakdown of Slender Vortices”, Fluid Vortices, Kluwer Aca., 373-426. Biagioli, F. 2006 “Stabilization Mechanism of Turbulent Premixed Flames in Strongly Swirled Flows,” Combustion Theory & Modeling, 10, 3, 389-412.

Benjamin, T.B. 1962 Journal of Fluid Mechanics, 14, 593, 629. Beran, P.S. 1994 Computers Fluids, 23, 913-937.

Beran, P.S. & Culick, F.E.C. 1992 “The role of non uniqueness in the development of vortex breakdown in tubes,” Journal of Fluid Mechanics, 242, 419-527.

Bray, K. & Louch, D., 2001 Vorticity in Unsteady Premixed Flames: Vortex pair-Premixed Flame Interactions Under Imposed Body Forces”, Combustion and Flame, 125, 1279-1309.


14

Brown, G.L. & Lopez, J.M. 1990 “Axisymmetric vortex breakdown. Part 2. Physical mechanisms,” 221, 1990, 553 – 576. Bruecker, Ch. and Althaus, W. 1995, “Study of vortex breakdown by particle tracking velocimetry (PTV), Part 3: time-dependent structure and development of breakdown modes”, Experiments in Fluids, 18, 174-186.

Buckmaster, J.D. & Ludford, G.S.S 1982 Theory of Laminar Flames (Cambridge: Cambridge University Press). Buntine J. D. and Saffman, P.G. 1995 “Inviscid swirling flows and vortex breakdown,” Proc. R. Soc. A, 449, 139-153. Choi, J.J., Rusak, Z. & Kapila, A.K. 2004 "Numerical Simulations of Premixed Combustion with Swirl," AIAA Paper 2004-0977, 42nd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 7.

Choi, J.J., Rusak, Z. & Kapila, A.K. 2007 "Numerical Simulations of Premixed Chemical Reaction with Swirl," Combustion Theory and Modeling, 11 (in print). Escudier, M. 1988, “Vortex Breakdown: Observations and Explanations”, Progress in Aero. Sci., 25, 189-229. Gupta, A.K, Lilley, D.G. and Syred, N., 1984 Swirl Flows, Abacus Press, UK.

Gupta, A.K., Lewis, M.J., & Qi, S. 1998 “Effect of Swirl on Combustion Characteristics in Premixed Flames,” Journal of Engineering and Gas Turbine and Power, Trans, ASME, 120, July, 488-494. Hall, M.G., 1972, “Vortex breakdown”, Annual Reviews of Fluid Mechanics, 4, 195-217. Herrada, M.A. Prez-Sabroid, M. and Barrero, A. 2003 “Vortex breakdown in compressible flows in pipes,” Physics of Fluids 15, 2208-2215. Judd, K., Rusak, Z., and Hirsa, A., 2000, “Theoretical and Experimental Studies of Swirling Flows in Diverging Streamtubes”, in "Advances in Fluid Mechanics III”, WIT Press, 491-502. Keller, J.J., Egli, W. & Exley,. W. 1985 “Force-and loss-free transitions between flow states,” Z. Angew. Math. Phys. 36, 854-889. Khorrami, M.R. 1995 “Stability of a Compressible Axisymmetric Swirling Jet,” AIAA J., 33, 4, 650-658. Leibovich, S. 1984,“Vortex Stability & Breakdown: Survey and Extension” AIAA J, 22, 1192.

Leibovich, S. and Kribus, A. 1990 Journal of Fluid Mechanics, 216, 459-504. Li, G. and Gutmark, E. 2006 “Experimental Study of Boundary Conditions Effects on

Non-reacting and Reacting Flows in a Multi-Swirl Gas Turbine Combustor,” AIAA J., 44, 3, 444-456.

Lieuwen, T., McDonell, V., Petersen, E., & Santavicca, D. 2007 “Fuel Flexibility Influences on Premixed Combustor Blowout, Flashback, Autoignition, and Stability”, J. Engineering Gas Turbines and Power (in print).

Lopez, J.M. 1994 Physics of Fluids, 6, 3683-3693. Malkiel, E., Cohen, J., Rusak, Z. and Wang, S. 1996, “Axisymmetric vortex breakdown in a pipe–theoretical and experimental studies”, Proc. 36th Israel Ann. Conf. Aerospace Sci.

Mattner, T.W., Joubert, P.N. & Chong, M.S. 2002 “Vortical flow. Part 1. Flow through a constant-diameter pipe,” Journal of Fluid Mechanics, 463, 259 – 291.


15

Paschereit, C.O., Gutmark,E., Weisenstein, W. 1998a, “Control of Thermo-acoustic instabilities and Emissions in an Industrial type Gas-Turbine Combustor”, 27, Publisher: The Combustion Inst. Paschereit, C.O., Gutmark, E., and Weisenstein, W., 1998b, “Structure and Control of Thermoacoustic Instabilities in a Gas-Turbine Combustor,” Combust. Sci. Technol., 138, 213–232. Paschereit, C.O., Gutmark,E., Weisenstein,W., 1999a, “Control of Premixed Combustion Instabilities by Fuel Modulations”, AIAA 99-0711, 37th AIAA Aero. Sciences Conf., Reno, NV. Paschereit, C. O., Gutmark, E., and Weisenstein, W., 1999b, “Coherent Structures in Swirling Flows and Their Role in Acoustic Combustion Control,” Phys. Fluids, 9, pp. 2667–2678. Paschereit, C.O., Flohr, P., and Gutmark, E. 2006, "Combustion Control By Vortex Breakdown Stabilization" Journal of Turbomachinery, Vol. 128, No. 4, 679-688. Randall, J.D. & Leibovich, S. 1973 “The critical state: a trapped wave model of vortex breakdown,” Journal of Fluid Mechanics, 53, 481-493.

Rusak, Z. 1998 Physics of Fluids, 10, 7, 1672-1684. Rusak, Z., 2000, “Review of Recent Studies on the Axisymmetric Vortex Breakdown Phenomenon”, AIAA Paper 2000-2529, AIAA Fluids 2000 Conference, Denver, Colorado. Rusak, Z. 2007 “A novel approach for the analysis of compressible vortex flows and vortex breakdown,” RPI Aero. Eng. Report. Rusak, Z., Judd K.P. and Wang, S. 1997 Physics of Fluids, 9, 8, 2273-2285. Rusak, Z. & Judd, K.P. 2001 Physics of Fluids, 13, 10, 2835-2844.

Rusak, Z., Kapila, A.K. and Choi, J.J. 2002 Combustion Theory and Modeling, 6, 625-645. Rusak, Z. and Lamb, D. 1999, “Prediction of Vortex Breakdown in Leading Edge Vortices Above Slender Delta Wings” Journal of Aircraft, 36, 4, 659-667.

Rusak, Z. and Lee, J.-H. 2002 "The Effect of Compressibility on the Critical Swirl of Vortex Flows in a Pipe," Journal of Fluid Mechanics, 461, 301 – 319.

Rusak, Z. and Lee J.-H. 2004 "On the Stability of a Compressible Axisymmetric Rotating Flow in a Pipe," Journal of Fluid Mechanics, 501, 25-42.

Rusak, Z. & Lee, J.-H. 2007 “On the bifurcation and global stability of near-columnar compressible axisymmetric vortex flows in a pipe,” Submitted, Physics of Fluids.

Rusak, Z. and Meder, C.C. 2004 AIAA Journal, 11, 2284-2293. Rusak, Z. and Wang, S. 1996a, “Review of Theoretical Approaches to The Vortex Breakdown Phenomenon”, AIAA paper 96-2126. Rusak, Z. and Wang, S. 1996b, “Theoretical Study of The Axisymmetric Vortex Breakdown Phenomenon”, SIAM Proc. App. Math., 84, 65-75. Rusak, Z. and Wang, C.-W., 1997 "Transonic Flow of Dense Gases Around an Airfoil with a Parabolic Nose," Journal of Fluid Mechanics, 346, 1-21. Rusak, Z., Wang, S. and Whiting C. 1998a, “The evolution of a perturbed vortex in a pipe to axisymmetric vortex breakdown”, JFM, 366, 211-237. Rusak, Z., Whiting, C., and Wang, S., 1998b, “Axisymmetric Breakdown of a Q-Vortex in a Pipe”, AIAA J., 36, 10, 1848-1853. Sarpkaya T. 1995, “Vortex Breakdown and Turbulence”, AIAA paper 95-0433. Sivasegaram, S. and Whitelaw, J.H. 1991, Combustion and Flame, 85, 195-205.


16

Sohn, K.H., Rusak, Z. & Kapila, A.K. 2006 “Effect of near-critical swirl on the Burke-Schumann reaction sheet,” Journal of Eng. Mathematics, 54 (2), 181-196. Syred, N. and Beer, J.M. 1974 “Combustion in Swirling Flows: A Review,” Comb. and Flame, 23, 143-201.

Umeh, C. and Rusak, Z. 2006 "Swirling Flows in an Axisymmetric Lean Premixed Combustor," AIAA-2006-3732, 36th AIAA Fluid Dynamics Conference and Exhibit, San Francisco, California, June 7.

Wang, S. and Rusak, Z. 1995 Applied Mathematics: Methods and Applications, 118-127. Wang, S. and Rusak, Z. 1996a, “On the Stability of an Axisymmetric Rotating Flow”, Physics of Fluids, 8, 4, 1007-1016. Wang, S. and Rusak, Z. 1996b, “On the Stability of Non-Columnar Swirling Flows”, Physics of Fluids, 8, 4, 1017-1023. Wang, S. and Rusak, Z. 1996c ESAIM Journal, 1, 101-112. Wang, S. and Rusak, Z. 1997a, “The dynamics of a swirling flow in a pipe and transition to axisymmetric vortex breakdown”, JFM, 340, 177-223. Wang, S. and Rusak, Z. 1997b, “The Effect of Slight Viscosity on Near Critical Swirling Flows”, Physics of Fluids, 9, 7, 1914-1927.

Williams F A 1985 Combustion Theory (New York: Benjamin Cummings). Zhang, Q., Noble, D., Shanbogue, S. & Lieuwen, T. 2007 “PIV Measurements in H2/CH4 Swirling Flames under near Blowoff Conditions”, 3rd Meeting of the US Sections of the Combustion Institute, March 26-28.

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