1

Effects of Acoustic Excitation on Bluff-body Stabilized

Premixed Reacting Flows

Vaidyanathan Sankaran,* Robert R. Erickson,

† and Marios C. Soteriou

‡

United Technologies Research Center, 411 Silver Lane, East Hartford, CT, 06040

The response of low Mach number, bluff-body stabilized, premixed flames to acoustic

excitation is investigated numerically. Two dimensional simulations are performed using the

Lagrangian Transport Element Method which employs a variable density version of the

Vortex Method to simulate the vorticity field and a kinematical flame-sheet model to

simulate the flame. The method facilitates direct addition of one-way coupled acoustic

solutions to the flow. Longitudinal excitation in the bulk mode (no phase difference) was

considered at a Strouhal number near unity (based on the bluff-body height) and at

amplitude of 15%. Results indicate that the flame unsteadiness is amplified by the acoustic

excitation. Direct acoustic-flame interaction appears to have small effect in most of the

domain and the flame response is closely linked to the vorticity field response. In the bluff-

body near field, both enhanced shedding of wall-generated vorticity and amplification of

flame-generated vorticity leads to enhanced flame response. In the far field, flame-

generated vorticity dominates the response. Results also point to the critical importance of

the flame anchoring point, which is also the vorticity separation point. When the area

around this point is not forced, i.e. when the forcing is applied in the far-field, flame exhibits

minimal response. The multi-dimensionality of the acoustic field near the anchoring point is

also shown to be important. Simulations where the acoustic field is constrained to be one-

dimensional show substantially reduced response. These results are consistent with earlier

experimental studies which showed that the acoustic velocity in the near-field is multi-

dimensional and that the response of the flame is significantly altered by the local

characteristics at the anchor point.

Nomenclature

H = Channel width

h = bluff-body width

Lm = Markstein length

P = Pressure

Re = Reynolds Number

SL,0 = Nominal flame speed

ST = Turbulent flame speed

T = Temperature

t = time

u = Axial velocity

z = Position vector

σ = Transport element core size

= Local flame curvature

= Velocity potential

= Density

= Vorticity

* Research Engineer, AIAA Member

† Research Engineer, AIAA Member

‡ UTRC Fellow, Senior Member AIAA

48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition4 - 7 January 2010, Orlando, Florida

AIAA 2010-1333

Copyright © 2010 by United Technologies Corporation. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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Sub-scripts:

u = Unburned

b = Burned

e = Expansion

b = Potential

I. Introduction

Bluff bodies are often being used as flame stabilization devices in practical applications such as power generation

devices, ramjets, and rockets. The rich and complex behavior of bluff-body stabilized reacting flows has attracted

substantial attention in the past1-14

. Despite these efforts, the behavior of these flows is not completely understood,

especially in terms of unsteady characteristics. One primary consideration when dealing with bluff-body stabilized

flames is their receptivity to combustion instabilities that result from unstable feedback mechanisms between system

acoustic modes and unsteady heat release rates. As a result of the large amplitude flow/flame oscillations, overall

system performance can be degraded and under certain circumstances, it can even be destructive to hardware. These

instabilities are often the result of acoustic-vortex-flame interactions and a number of different mechanisms have

been proposed in the literature by which they manifest themselves. Some of them are: (i) acoustic perturbations may

change the flame surface area and hence the heat release (ii) acoustic pressure waves may modulate the inlet fuel

flow rate causing equivalence ratio fluctuations (iii) acoustic/sound waves may change the burning rate directly (iv)

acoustic perturbations may excite Kelvin-Helmholtz instability modes and cause large scale coherent structures

which eventually break-up into small scale turbulence to affect unsteady heat release. In addition, for bluff-body

stabilized flames, acoustic forcing can modulate the vortex shedding off a flame holder to cause large scale

fluctuations in the flame area and heat release.

Most of the studies conducted in the past to understand these interactions included some but not all of these

effects at the same time. For instance, several experimental studies1,2

were limited to laminar flames subjected to

longitudinal forcing and characterized the flame response in terms of flame transfer functions. But flows in most

realistic systems are turbulent and hence neglecting the multi-scale vortex-flame interaction may not yield a realistic

picture of the dynamics of these flames. Some of the notable exceptions are studies shown in Refs. 3 and 4 in which

results are reported for the response of turbulent flames to external forcing. On the other hand, theoretical

investigations6-11

of bluff-body stabilized flames used (linear or non-linear) G-equation based models with imposed

velocity fluctuations, ignoring vortex-flame interactions. However, on many occasions of practical interest the

unsteadiness in bluff-body stabilized flames may be the result of shed vortices inducing a flapping motion on the

flame, leading to a time varying heat release. Therefore the hydrodynamics can be as important as the flame itself, if

one tries to understand the acoustic-flame interactions on a fundamental basis.

In this paper, we perform a study that includes this latter piece of physics with particular emphasis on the impact

of acoustic excitation on vortex shedding and hence on the flame response. In particular we investigate the effect of

acoustic forcing on the flame and vorticity fields and we ignore the reverse coupling between the reacting flow and

acoustics.

To achieve the aforementioned objective we numerically investigate the problem of premixed combustion

stabilized by a two dimensional, triangular bluff-body using Lagrangian Transport Element Method (LTEM) 19, 20

.

This formulation has been used in previous work22-23

to investigate the fundamental differences between non-

reacting and reacting flow fields past bluff-bodies. It was shown in Ref. 22 that the non-reacting flow manifests the

Von Karman (V-K) asymmetric vortex shedding behavior while the reacting flow suppresses the asymmetric

shedding mode due to dilatational effects resulting in a more symmetric shedding from the two „shear layers‟

anchored at the bluff body. This model has also been used to show that the transition from the shear layer to the V-K

shedding mode happens as the combustion conditions change 23

. Specifically, the transition back to the V-K mode

(in reacting flows) is witnessed as the temperature ratio across the flame is reduced.

There are several advantages in using a LTEM for this study. First and foremost is the fact that the artificial

diffusion due to numerical discretization of the governing equations is nearly zero in this method. This is especially

important here because the initial amplitude of the disturbances that grow into large scale oscillations is very small

and in the presence of overwhelming numerical diffusion these initial disturbances may be completely erased from

the system. Dispersion errors of the numerical schemes on the other hand, may introduce frequencies which were

not originally present in the problem and hence can corrupt the predicted frequency response of the system. Both

these errors are minimal in LTEM and this makes it an ideal candidate for this study.

The objective of this paper is to explore the impact of acoustic excitation on the dynamics of premixed flames

stabilized by bluff-bodies. Only longitudinal modes are considered in this study and the acoustic excitation is

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achieved through the use of velocity perturbations motivated by the justification provided in Ref.7. Specifically,

experimental data obtained by Bloxsidge7 et al in bluff-body stabilized flames at low Mach numbers showed that the

fractional change in velocity overwhelms the fractional change in pressure and temperature. Therefore, they

supposed that the unsteady heat release is determined principally by the velocity fluctuations near the flame holder

and it was confirmed using their measurements. Use of a low Mach number formulation in the current work allows

us to employ a similar assumption regarding the nature of the imposed acoustic perturbations and hence, the acoustic

excitation of the flame is realized through the use of velocity perturbations

The paper is organized as follows. In section II, the complete formulation and models for the LTEM will be

introduced. Section III will be used for elaborating on the numerical methods, boundary conditions and the

algorithm of the total approach. Results and discussion will be provided in section IV and finally section V will

summarize the conclusions of this investigation.

II. Formulation

This investigation considers premixed combustion stabilized by a triangular bluff-body flame holder of height h

in a channel of height H, as shown in Fig. 1. A low Mach number formulation which includes variable density

effects of the reacting flow field is used in this study.

Figure 1: Schematic diagram of the bluff-body stabilized reacting flow-field showing the geometry of the

bluff-body, flame anchor point and shear layers

A. Governing Equations

The formulation begins with the Conservation of mass equation expressed as

0 VDt

D

(1)

Here D/Dt is the material derivative and represents density and V

the two-dimensional velocity vector,

respectively. Divergence of velocity, V in Eq. (1) is nonzero only across the flame front in this low Mach

number formulation. The conservation of momentum is expressed as

) ( VpDt

VD

(2)

Here P and μ represent the pressure and dynamic viscosity, respectively. Viscosity is assumed to be constant in both

the reactants and products, but varies across the flame.

The Navier-Stokes equations (Eqs. (1) and [2]) are solved using a non-primitive formulation. The corresponding

equation in terms of non-primitive variable, vorticity (

) can be obtained by taking the curl of Eq. (2), using Eq. (1)

and non-dimensionalizing the resulting equation to get:

4

VVEVp

VDt

D..

Re

1)( 2

(3)

Here Re is the Reynolds number defined as U0L/ with U0 equal to the inlet velocity and L representing some

characteristic length, such as the channel width or the bluff-body width. The term on the left hand side of the

equation accounts for vorticity convection. The first term on the right hand side (RHS) of the Eq. (3) is the dilatation

term due to combustion, second term is the baroclinic term and accounts for the vorticity generation due to the

interaction of density and pressure gradients. The third term in the RHS is the viscous term which accounts for the

diffusion of vorticity, fourth term is vortex-stretching term which is zero in the two-dimensional case (considered

here) and the last term E.V.V stands for extra viscous terms due to spatial viscosity variations (neglected here).

The unsteady flow-field governed by Eq. (3) is solved using the Lagrangian Transport Element Method19, 20, 22, 23

.

By conceiving turbulent eddies or vortical structures as a “fundamental” element of the flow, one can directly

discretize the vorticity into computational elements. Furthermore, by tracking the interaction of these vortical

elements, essential features of the flow-field governed by Eq. (3) can be reproduced. In the case of reacting flows,

flame surface can also be discretized into flame elements as shown later. In the above equation, terms involving

convection and reaction are solved simultaneously whereas diffusion is solved subsequently to obtain the complete

solution. In the next few sections, we will introduce the models for convection, flame/combustion and diffusion

terms.

B. Models for Convection

First, consider the convection step. For a constant density, two-dimensional, inviscid flow, Eq. (3) reduces to the

following form 0/ DtD

. This equation implies that the vorticity associated with a fluid particle is constant as

it moves through the fluid. In other words, if one follows the fluid particle, the associated motion of the vorticity

(vorticity convection) is also known. Therefore, it is natural to use a Lagrangian description of the flow, where one

follows the motion of material points in the fluid. In Lagrangian coordinates, we can discretize the fluid into discrete

vorticity carrying elements and track their motion to model vortex convection as follows:

)0,());,((

)0,(

));,((),(

ttz

z

ttzVdt

tzd

(4)

Here, z

is the position vector of a discretized vortex element and is the initial location of the particle (at time t =

0). By adopting a Lagrangian method, one can eliminate the effects of numerical diffusion introduced by the use of

a fixed grid in the Eulerian view point.

To obtain the local fluid velocity we employ the Helmholtz decomposition15

to split velocity field into curl-free

(irrotational) and divergent-free (solenoidal) components as follows:

VVVV ep

Here pV

and eV

are the curl-free (irrotational) components due to the potential flow and combustion induced

volumetric expansion, respectively. V

is the divergent-free (solenoidal) component due to the vorticity field. The

methodology adopted for computing the three components of velocity would be discussed below.

1. Potential flow

The contribution to the velocity vector from the potential flow depends only on the geometry of the system and its

boundary conditions15

. The governing equation for the potential flow is the Laplace equation: 0p2 subject to

the prescribed inflow and boundary conditions. It is solved using a conformal mapping technique based on the

Schwartz-Christoffel transformation15, 21

which projects the exterior region of a body (in the physical complex plane)

to an upper half plane (transformed complex plane). In the transformed coordinates, the physical boundary of the

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body appears along the real axis of the transformed complex plane. The usefulness of the conformal mapping

technique is that a complicated geometry & its flow-field (along with its boundary conditions) may be mapped into a

simple domain with a simple flow pattern (such as a rectangular domain with uniform flow and potential sources) in

the transformed plane. For this simple flow, exact solutions are available which could be inverted to obtain solutions

in the physical plane15

.

2. Vortical flow

The velocity field induced by the vortex elements is obtained by introducing the stream function. Vorticity and

stream function are related by the Poisson‟s equation:

2 (5)

This is related to the vortical velocity by

V (6)

One of the standard approaches for solving the Poisson equation is by using the Green‟s function method15, 16

which

yields the following exact solution.

''),'()',(),( dydxtzzzGtz

(7)

The Green‟s function )',( zzG

satisfies )',()',(2 zzzzG

, where )',( zz

is the Dirac-delta function

and the actual form of the Green‟s function is

|)'ln(|2

1)',( zzzzG

(8)

After substituting Eq. (8) in Eq. (7), V

is obtained from Eq. (6) as

''),'()',( dydxtzzzKV

(9)

Here, K is given by

)'(|'|2

1)',()',(

2zz

zzzzGzzK

(10)

Equations (9) and (10) are called the Biot-Savart law15

. The velocity field given by Eq. (9) is only applicable for free

space flows (without any boundaries). For flows with solid boundaries (in the physical plane), this velocity field has

to be corrected by applying the boundary conditions 19

which will be discussed in the section on numerical methods.

3. Expansion flow

Finally, the velocity field induced by combustion/heat release eV

is computed as follows. From mass conservation

equation (Eq. (1)), we have

Dt

DVe

1

Total velocity in Eq. (1) is replaced by expansion velocity in the above equation, because we are only considering

the expansion induced velocity at the flame front. Since this velocity is irrotational, a velocity potential can be

defined such that eeV

. Substituting this in the above equation, we get

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Dt

De

12 (11)

Evaluation of material derivative of density in the above expression requires the specification of a combustion

model which will be elaborated in the next section. Until then, let us call this material derivative . Equation (11) is

a Poisson equation similar to Eq. (5) which can also be solved using the Green‟s function method explained earlier

to get the following expression for the expansion velocity

''),'()',( dydxtzzzGVe

(12)

Once all three component of velocity VVV ep

and ,, are known, the total velocity at the location of the fluid

particles is known. Therefore, Eq. (4) can be solved numerically to complete the model for vorticity convection. In

the following sections models for flame/combustion (1st and 2

nd term on the RHS of Eq. (3)) and diffusion (3

rd term

in the same equation) will be described.

C. Models for flame/combustion

Premixed combustion is modeled using a flame-sheet approximation which assumes that the reaction zone is

confined to a thin interface between the reactants and products. Furthermore by assuming that chemical kinetic time-

scales are much smaller than the flow time scales one can reduce the combustion problem to that of a flame

propagation problem. In this scenario, flame position is determined by the balance between flame-speed and local

fluid velocity. Flame position in turn determines the location of heat release and the hence the complete flow-flame

interactions. Therefore, we need only three basic “ingredients” to model premixed combustion using flame-sheet

approximation and they are 1) a model for representing the flame surface, 2) a model for flame propagation and 3) a

model for flow-flame interactions. In the following section we will introduce the equations and the models for each

of these three “ingredients” used in flame-sheet approximation of premixed combustion.

The first ingredient, flame surface, is modeled by discretizing the flame sheet into discrete flame surface

elements. These flame surface elements are tracked using the Lagrangian equations of motion

nSVdt

zdT

fˆ

(13)

Here fz

is the position vector of the flame surface elements, n̂ is the unit normal to the flame oriented into the

reactants, V

is the fluid velocity at the flame element location and TS is the local flame propagation speed. This

equation merely states that flame surface elements are convected by the local fluid velocity augmented by the flame

propagation in a direction normal to the local flame element at a prescribed burning velocity TS .

The second ingredient, flame propagation is modeled as follows in this work.

U

u

U

S

U

S LT '1 (14)

Here 'u and U are the turbulent intensity and mean inflow velocity. SL is the laminar flame speed which

includes the effects of local flame curvature16

as follows

) 1(0

MLL LSS (15)

0

LS is the unstrained laminar flame speed, is the local flame curvature, and LM is the Markstein length16

. In

Eq.(14), the term (1+ 'u /U ) represents the enhancement of the laminar burning velocity due to turbulence that

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wrinkles the flame at length scales smaller than what is resolved in the simulation. The ratio 'u /U is an input

parameter to the model that is used only for coarse-grain simulations.

The third ingredient, flow-flame interaction was introduced partially in the previous section, where a model was

described to compute the combustion induced expansion velocity (See Eq. (12)). The only element of the model that

was not elaborated earlier is the model for computing the material derivative of density that appears in Eq. (11).

This term is determined using mass conservation & area swept by the motion of the flame as follows

bTswept dAdldtSdA f (16a)

Using mass conservation, we get

bbsweptu dAdA (16b)

Therefore, the new flame area created by burning is

sweptbe dAdAdA (16c)

Substituting Eqs. (16a) and (16b) in (16c) and rearranging

fdl 1 T

b

ue Sdt

dA

(17)

By relating the change in the length of a material line element and the velocity difference at its ends, it can be

proved that15

dt

dAAV e )(

By using continuity equation (Eq. (1)) and Eq. (17), we get

fT

b

ue dlSdt

dAA

Dt

D)1(

1

(18)

Here u and b are the unburnt (reactant) and burnt (products) density. Ae is the area created as a result of burning

and dlf is the length of a flame element. By virtue of Eq. (18), flame acts a volumetric source in generating an

irrotational velocity field eV

which influences the flame propagation through V

in Eq. (13) and hence a full coupling

between the flame and flow is established. Note, to model the interaction between flame sources, the concept of

flame-source elements are used analogous to vortex elements. Again, velocity field induced by the flame sources

must satisfy the boundary conditions (for Eq. (11)) and this will be elaborated in the section on numerical methods.

Baroclinic vorticity

Flame generated or Baroclinic vorticity (2nd

term in the RHS of Eq. (3)) is another mode of flame-flow interaction

that has to be modeled in variable-density flows. This is modeled using the momentum equation as follows.

Neglecting the diffusion term in Eq. (2), pressure gradient term can be equated to the material acceleration term as

DtVDp //

. Substituting this in the baroclinic term, we get Dt

VDp

In the above expression, material acceleration of an element is computed by numerically differentiating its velocity

between two time steps.

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D. Models for Diffusion

Particle based methods like the one used here are well suited for modeling convective processes, but have

difficulty in modeling diffusive processes. As a result there are several approaches in literature16, 17

for modeling

diffusion in discrete vortex methods. Some of the methods model diffusion by changing the vortex element

parameters like: their position (Random Walk Method, Diffusion velocity method); their sizes (Core expansion

method) or their circulation (Deterministic particle methods). Two kinds of diffusion models are used here: one for

modeling the vorticity diffusion near the walls and the other for modeling diffusion in the flow-field away from the

walls.

Due to its simplicity, the Random walk method has been used extensively and it is used in the current work as

well. This method was first proposed by Chorin17

and is used to simulate vorticity diffusion near solid walls. In the

proximity of walls, the velocity gradients are high and hence Chorin17

argued that vortex blobs/cores are not a good

representation of the flow and introduced the concept of one dimensional vortex sheet elements. Motion of these

sheets are governed by the same convection equation shown in eq. (4), however, unlike vortex blobs, their domain

of influence is limited to wall normal direction only. Vorticity diffusion away from the wall is modeled in a

probabilistic sense by altering the position of the vortex sheet elements with random displacements17

that have zero

mean and a variance equal to twice the product of the kinematic viscosity and time step. Once the sheet elements

move away from the boundary layer, they are converted into vortex blobs/cores with appropriate circulation.

Vorticity diffusion in the interior domain (away from the walls) is modeled using a procedure called core

expansion or the core-spreading method and was first introduced by Leonard 18

. It is a deterministic method

accounting for diffusion by allowing each vortex element core to grow at a rate proportional to the kinematic

viscosity using an exact solution of the vorticity diffusion equation

Re// 2

t

E. Models for Acoustics

The acoustic model employed is consistent with the low Mach number approximation invoked in the flow

solution and, as such, assumes small (linear range) amplitude waves that have long wavelength compared to the

reacting flow and propagates in an effectively inviscid manner. Moreover, it is assumed that the acoustic field is

one-way coupled to the flow i.e. acoustics influence the flow but not vice versa - which in effect removes effects of

velocity and temperature variations on the acoustic field.

Under these assumptions, the acoustic field is irrotational (potential) and the formulation proposed herein

enables a straightforward way of incorporating it in the overall solution, namely via the addition of acoustic velocity

acV

in the Helmholtz decomposition, i.e. acep VVVVV

In the general approach, the acoustic velocity is determined by solving the wave equation (shown below) with

appropriate boundary conditions

ac

ac ct

22

02

2

Here acac V

and 0c is the speed of sound.

In this paper, we do not pursue the comprehensive solution of the wave equation and, instead, we follow a more

simplified approach in which we construct approximate “acoustic” solutions. Specifically, recognizing the

similarities in the solutions of the Laplace and Helmholtz equations at the limit of low Mach number, and taking

advantage of the fact that the formulation used herein already provides solutions to the Laplace equation via the

Schwartz-Christoffel transformation, we postulate a solution that has a Laplacian spatial variation multiplied by a

harmonic temporal variation, .i.e. )2sin( ftAVV pac

. Here A is a constant setting the amplitude of the

“acoustic” oscillations and f is the frequency of forcing.

It is worth pointing out that the approximations involved in this method of “acoustic” forcing is equivalent to

forcing the inlet flow velocity an approach frequently followed in other numerical studies. Moreover, the

experimental study of Bloxidge 7 et al justifies this approach because of the low Mach number formulation

employed here. However, we repeat that this is an approximation that is valid strictly in the limit of very low Mach

number and is not a rigorous acoustic solution. Figure 2 shows the streamlines of this approximated acoustic

velocity field at an arbitrary instant. It can be observed that the flow-field is two-dimensional in the near-field of the

bluff-body as observed in some of the velocity measurements in the near field of the experiments8.

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Figure 2: Streamline plot of the acoustic velocity field showing two-dimensional field near the bluff-body at

an arbitrary instant

III. Numerical Methods

In this section, details of numerical models used for approximating their continuous counterparts presented will

be provided. Recall from the previous sub-section (II.B.1) that a conformal transformation is used for mapping the

coordinates of the physical plane to a transformed plane. Owing to the advantages related to the application of

boundary conditions (as shown later) and the fact that the potential velocity field can be determined analytically in

the transformed coordinates, all computations are performed in the transformed complex upper-half plane. In what

follows, physical plane is represented using “ z

”and the transformed plane is represented using “ ” variables,

where is a complex number.

A. Convection and Reaction

The models for convection of vortex (Eq. (4)) and flame elements (Eq. (13)) are solved using the following

predictor-corrector scheme which is second order accurate in time.

), )()(2

1)(

), )()(

**1

*

tW(tttt

tW(ttt

j

n

jj

n

j

n

j

n

jj

Here is the location of the elements in the transformed plane and W represents the complex velocity of “j”-th

element in the transformed plane and is given by

),(),(),(), tWtWtWtW( jjejpj

The analytical solution of the potential flow in the transformed plane is given by 15

NP

m

mj

mj

mo

jp

hUtW

1

,02

,0

,)(

||),(

(19)

)( jpW is the potential velocity at the “j”-th element due to NP potential sources in the flow-field. m,0

is the

location of the “m”-th potential source in the transformed plane. Here, each potential source represents each inflow

boundary present in the physical plane. U0,m is the “m”-th inflow velocity and h is height of that inflow boundary.

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In LTEM, continuous vorticity field is represented as the sum of large number of discrete vortex elements as

N

j

j

n

j

N

j

jj

n

j

n

jj ftfAttztz1

,

1

, )()(),(),(

j is the circulation of the “j”-th vortex element. jf , is a regularization function

16 and represents a

characteristic size of the vortex core. A second order radially symmetric Gaussian core function of the form

2

,

2

2

,

,

)(exp

1

jz

j

jz

j

zzf

is used in this work 21, 22

. This function satisfies the normalization condition 120

rdrf . By substituting the

above equation in Eq. (11) and followed by Eq. (11) in Eq. (9), we can get the regularized, discrete equivalent of the

vortical velocity at any location z

as follows

NV

j jz

j

j

jn

j

nzz

zz

zzttzV

12

,

2

2

)(exp1

||

)()(

2

1),(

(21)

Here ivuV

gives the x and y component of the vortical velocity in the physical plane. NV is the total number

of vortex elements in the computational domain. Therefore, vortical velocity in the transformed coordinates is given

by

NV

j j

j

j

jn

j

n ti

tW1

2

,

2

2

)(exp1

||

)()(

2),(

(22)

Note, jz , and

j, are the vortex element core sizes in the physical and transformed planes respectively and

1i is the imaginary number.

Finally, expansion velocity due to combustion is obtained analogous to the vortical velocity by using the concept

of flame-source elements as follows

NS

j j

j

j

jn

j

n

e ttW1

2

,

2

2

)(exp1

||

)()(

2

1),(

(23)

NS is the total number of flame-source elements in the computational domain.

Numerical models for flame

The numerical solution for the motion of the flame-sheet requires several algorithms and rules to remain

physically consistent throughout the simulation. First, since the flame-sheet is described by discrete transport

elements, it must be re-discretized (equivalent to re-meshing in Eulerian CFD) during every time step in order to

maintain the desired resolution despite changes in flame length due to stretching and curvature. One final note on the

assumptions of the present combustion model: This work does not address the flame holding capability of the

system, and therefore it is assumed that the two flame-sheets are continuously anchored to the upper and lower

corners of the flame holder wedge. Effect of local flame lift-off on the flame response to imposed acoustic

oscillations is not addressed here.

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B. Diffusion

Numerical implementation of the random-walk method is accomplished as follows. At the end of each corrector

step, two random numbers are sampled from a Gaussian distribution such that they have a zero mean and a

standard deviation of Re/2 t . These two random numbers represent the random displacements of a vortex

sheet element in two mutually perpendicular directions and hence are added to x and y components of the

displacements in the transformed plane. Diffusion of the vorticity in the interior domain (away from the walls) is

modeled using core expansion method and it is implemented numerically as follows. At the end of the convective

step, the core radius of each vortex element is expanded as

Re

4)()(

2

,

12

,

ttt n

j

n

j

(24)

C. Boundary conditions

The following boundary conditions are needed to complete the numerical implementation of LTEM.

Potential velocity field must satisfy zero normal velocity or impermeability boundary conditions at the

walls. Potential solution of the Laplace equation in the transformed plane automatically satisfies this

boundary condition. However, for the vortex elements, impermeability of the walls is imposed using the

method of images. In this approach, an image element with equal vorticity magnitude and opposite sign is

created behind the wall to cancel the wall normal velocity induced by the original element. In the

transformed domain, all the physical boundaries appear along the real axis. This implies that only a single

image is required for each element to enforce the wall normal boundary conditions. On the other hand,

presence of multiple walls in the physical z domain requires an infinite series of images for each element to

implement the wall normal boundary conditions.

Expansion velocity field must also satisfy zero normal velocity at the walls and is implemented using the

same method of images. However, the image element in the case of flame-source elements has equal

magnitude and sign as that of the original element.

Vortical velocity fields must satisfy no-slip velocity at the walls. Chorin‟s boundary sheets method16

is used

to impose the no-slip boundary condition at the walls. Using this method, the tangential velocity produced

by the flow-field (potential, expansion & vortical) at a wall is cancelled by introducing one-dimensional

boundary layer sheets with proper strength. The influence of these sheets is restricted locally to a domain

normal to the wall. These sheets convect and diffuse away from the wall into the interior region just like

any other element in the computational domain. Once the sheets move out of the numerical boundary

layers, they are converted into two-dimensional vortex Gaussian cores as explained in the previous section.

In addition, the channel duct walls are modeled as slip walls for the sake of simplicity. This assumption

introduces confinement effects while not modeling boundary layers on the duct walls. With this

assumption, there is no vorticity generation on the duct walls and therefore no boundary sheets are needed there.

IV. Results and Discussion

In this section, results are presented from the numerical simulations of a flow past a triangular bluff-body in a

rectangular channel/duct. All quantities are non-dimensionalized as follows. All length scales are normalized by the

channel/duct height, H and velocity scales by the inflow velocity U0. Reynolds number, defined as Re = U0 H/ν is set

at 31,000. Here is the kinematic viscosity of the reactants. Non-dimensional flame speed, Markstein‟s length scale

and temperature ratio Tb/Tu are set at 0.1, 0.04 and 6 respectively.

A non-dimensional acoustic frequency defined based on the bluff-body height h, St = fh/U0 = 1.11 is used here.

Forcing amplitude is set at 15% of the inflow velocity. These acoustic parameters were chosen based on the

experiments12, 13

that showed good flame response at these conditions. In the current study, longitudinal forcing was

imposed using the approach described in sub-section II. E and only bulk mode (with no phase difference) is

considered here.

12

A. Fluid-dynamics of unforced flames

Figure 3a shows the contours of temperature at an arbitrary instant from unforced simulations. Earlier studies on

non-reacting flows have shown that these flow-fields are known to exhibit asymmetric vortex shedding behavior

known as Von-Karman shedding15

. On the other hand, studies on reacting flows past bluff-bodies 22, 23

have shown

that in the presence of heat release, the asymmetric V-K mode is suppressed and a more symmetric shedding from

the shear-layers anchored at the bluff-body is observed. This is also clearly exhibited by the flames seen in Fig. 3(a).

Figure 3(b) shows an instantaneous image of the distribution of the Lagrangian (vortical & flame) elements

along with the contours of time averaged vorticity for the unforced case. The vortical elements are colored black and

the black lines on the periphery of the flow represent the flame surface elements. An important aspect of the variable

density reacting flows can be seen in the contours of mean vorticity. The sign of the mean vorticity changes as we

move from the near-field of the bluff-body to the far-field. Specifically, in the top-half of the flow-field, near the

bluff-body (left-most edge) mean vorticity is negative (blue color) indicating that the vorticity generated near the

walls have a clock-wise rotation. As we move downstream, the sign of the vorticity contours changes to positive

(red color) indicating that the vorticity generated along the flame (baroclinic vorticity) has a counter-clockwise

rotation. This was also observed in the earlier work reported in Ref. 22. It is well known that in reacting flows,

density stratification occurs due to the presence of heavier reactant and lighter product fluid which creates a density

gradient. Flow acceleration on the other hand creates sets up pressure gradient. When the pressure and density

gradient vectors are not in alignment, the fluid particle is subjected to a local shear and hence vorticity is generated.

This is called the baroclinic vorticity or the flame-generated vorticity and a schematic diagram shown in Fig. 4

explains this behavior.

B. Physics of unforced vs. forced flames

Figure 5 compares the contours of temperature at an arbitrary instant from unforced and forced simulations. As seen

in Fig. 5(b), in the presence of acoustic forcing, more symmetric, periodic, coherent structures begin to evolve on the

flame and persist far downstream. Period and spacing of these coherent structures seem to be correlated with the

acoustic forcing. In order to quantify frequency response of these flames, power spectral density (PSD) of the

transverse oscillations of unforced and forced flames were computed. Following Ref. 12 and 13, the vertical distance

from the center line to the flame edge is tracked as a measure of transverse oscillations of the flames. Figure 6(a)

and 6(b) shows the PSD of the transverse oscillations of the unforced and forced flames respectively. Whereas the

unforced simulations do not show any peak at the forcing frequency, forced simulations show a clear peak at the

non-dimensional forcing frequency. Note the non-dimensional frequency shown in Fig. 6(a) and 6(b) is based on the

bluff-body height, h. Presence of higher harmonics (at f = 2.22) clearly indicates the non-linearity of the flame

response.

Effect of wall-generated & baroclinic vorticity on flame response

To better understand the reason for the increased flame response in the presence of forcing, as seen in Fig. 5(b)

we analyzed the effect of the individual mechanisms by which the acoustic forcing influences the flame. To this

effect, two forced simulations (using 2D bulk mode) were performed, the first of which included only wall-

generated vorticity and the second included only flame-generated vorticity. This kind of analysis showcases one of

the unique features of LTEM which is not straight-forward by other means of investigation (such as experiments or

Eulerian CFD). Since each of the processes: acoustics, vortex and flame are modeled explicitly in this approach, it

enables us to turn-on or off the individual interactions selectively. This kind of analysis provides valuable insight

into the fundamental mechanisms involved in the dynamics of acoustically forced flames.

Results from these simulations are shown in Figs. 7(a) and 7(b). It is evident that there is a striking difference in

flame response between the two cases and the quantitative evidence of this difference will be provided later. Figure

7(a) which only included wall-generated vorticity, shows that initially the flame wraps around the vortical structures

that form at the shear layers near the bluff-body. The orientation of the flow structures slanting in the direction of the

flow strongly suggests that the flame is indeed wrapped around the wall-generated vorticity which has a clock-wise

rotation (in the top-half and vice-versa in the bottom-half). This disturbance/response begins to grow as we move

downstream due to the growth of the coherent vortical structures. In about six bluff body heights, the response

seems to saturate and the flame flattens itself. It can be observed that the depth of the flame cusps reduce

continuously as we move downstream due to adjacent flames propagating into each other (kinematic restoration).

This picture is in qualitative agreement with recent studies12, 13

on bluff-body stabilized flames where it was

suggested that the kinematic restoration diminishes and saturates the response. But it should be kept in mind that the

explanation offered here only considers the role of wall-generated vorticity and ignores the effect of flame-generated

vorticity, which is continuously generated in real flames.

13

Figure 7(b) which only included the effect of flame-generated vorticity shows that the flame responds differently

in this case. The orientation of the flame structures show that the flame wraps around the flame-generated vorticity

in the counter-clockwise direction (in top-half) as explained in the previous section. Even though the flame

response seems to saturate, flattening of the flames is not observed in this case as in Fig. 7(a). Based on this

qualitative analysis, it looks like the saturation in flame response is a result of combination of kinematic restoration

of the flame and the competing actions of the wall-generated and flame-generated vorticity.

Further understanding of this physics can be obtained by looking at the image of the Lagrangian elements and

the vorticity field. Figures 8(a) and 8(b) shows the location of vorticity (and flame) elements and the contours of

mean vorticity. Figure 8(a) is from the simulation which only included wall-generated vorticity and shows that the

vorticity formed in the shear layers near the anchor point undergoes Kelvin-Helmholtz (K-H) instability and begins

to roll-up. But this is quickly suppressed by the dilatation and the extensive strain of the potential flow and flow

acceleration due to combustion in the wake. As a result elongated vortical structures begin to appear downstream

and the transverse growth rate of the shear layer is reduced. This in turn diminishes the flame response towards the

tail end of the channel/duct.

Figure 8(b) shows the distribution of vorticity elements from the simulation which only included flame-

generated vorticity. In this case, flame-generated vorticity is generated continuously along the flame and hence the

influence of extensive strain is not as effective in suppressing the growth of the shear layer as in the previous case

where the source of disturbance (wall-generated vorticity) is confined to the bluff-body. As a result of this

continuous presence of the disturbance (flame-generated vorticity) and external forcing, these elements seem to

organize into coherent structures around which the flame wraps. Further downstream, this growth begins to saturate

and hence the flame response is reduced. However, in the presence of both wall-generated and flame-generated

vorticity, flame response to the K-H mode at the shear layers is amplified further by the flame-generated vorticity

and hence the collective response is much higher than the individual responses as shown in Fig. 8(c).

To quantify this assertion, we looked at the amplitude of transverse flame oscillations at the forcing frequency at

every point in the axial direction. Figure 9(a) compares this amplitude for three cases: first case included only wall-

generated vorticity; the second case included only flame-generated vorticity and a third case (called full-case)

including both the mechanisms. For x/h < 0.5, flame response is nearly identical for the full case (green line with

circles) and the case with wall-generated vorticity only (blue line with square), indicating that the influence of

flame-generated vorticity on the flame response is minimal in the near field. In addition, the difference in the slope

of the profiles (blue and black lines) strongly suggests that the wall-generated vorticity contributes significantly to

the initial growth rate of the disturbance. Flame response to wall-generated (blue line with squares) and flame-

generated (black line with triangles) vorticity saturates between 2 ≤ x/h ≤ 3 and begins to decline further

downstream. Therefore, the final picture that emerges is that of a flame response that is dominated by wall-generated

vorticity in the near-field and by flame-generated vorticity downstream. But the flame-generated vorticity plays the

dual role of amplifying the response for a short distance (1 ≤ x/h ≤ 3), before beginning to diminish the response in

the far-field. Moreover, it is under the collective action of the mechanisms that the flame shows maximum response

(green line with circles).

Figure 9(b) shows the coherence of the flame response to the input forcing signal for three cases described in the

previous paragraph. Again, in the presence of wall-generated vorticity alone, the coherence persists longer (x/h > 6)

compared to the case that included only flame-generated vorticity (x/h ≤ 3), which also suggests that the flame-

generated vorticity tends to diminish the response in the far-field corroborating the assertions made from

instantaneous plots shown in Figs. 5(a) and 5(b). In the presence of both the mechanisms an intermediate level of

coherence is seen.

C. Flame response to far-field versus near-field forcing

From the analysis presented in the previous sub-section, it is evident that the first observable flame

response/disturbance begins at the tip of the bluff-body where the flame is anchored. This indicates that the presence

of the bluff-body influences the nature of the acoustic-flame coupling and to verify this, we conducted a simulation

where the forcing was limited to far-field only. If the acoustic-interaction with the flame is independent of the

geometrical effects of the bluff-body, no change in response should be observed. A tan-hyperbolic function of the

form

14

)()(

tanh1),(' 0 tFzz

tzFs

is used to restrict the forcing to the far-field. F(t) is the longitudinal, bulk mode forcing. In the above expression

z0 = 4, δs = 0.1 and z is the axial location.

Figure 10 shows the results from the case which was forced in the far-field only. It is very clear from this figure

that the flame does not respond to the forcing and it appears qualitatively similar to the unforced case shown in Fig

3(a). This clearly shows that the geometry of the bluff-body significantly influences the nature of the imposed

perturbations and hence the acoustic-flame interactions. In the absence of near-field forcing, the initial amplification

of the disturbance by the wall-generated vorticity is not present. In other words, K-H instability mode is not

triggered. Further downstream, where the forcing is applied, the flame-generated vorticity diminishes the response.

As a result the overall response of the flame is low. This is also evident from Fig. 12 (a) which shows the amplitude

of the transverse oscillations at the forcing frequency. It can be observed from this figure that the flame response

(black line with triangles) is significantly lower than the other modes of forcing. The lack of coherence of the flame

response to the imposed excitations exhibited in Fig. 12(b) (black line with triangles) also indicates that forcing only

in the far-field has minimal influence on the flame.

D. Sensitivity of flame response on the flame anchor point

Results from the far-field forcing study demonstrated that the flame does not respond to forcing away from the

anchor point and that it may be sensitive to the characteristics of the acoustic field at the anchor point. Specifically,

it was noticed that the main difference in the acoustic field between the 2D bulk mode forcing and the far-field

forcing is the dimensionality of the acoustic velocity field. In the former case, the acoustic field is 2D near the

anchor point, whereas in the latter case, it is 1D in the far-field. This raises the question, what is the influence of the

dimensionality of the acoustic field on the flame response? Other researchers have looked into this issue, for

instance, it has been shown in Ref. 10 that in the vicinity of the anchor point the acoustic velocity field is two-

dimensional and the pressure field is nearly one-dimensional. In addition, it was also reported that the two-

dimensionality of the acoustic velocity field increases with increasing temperature ratio. On the other hand, other

studies 9-11

in the past have used simplified one-dimensional acoustic models to predict flame response that are in

agreement with experimental observations. Therefore, to evaluate the accuracy of the simplified 1D representation

of the acoustic field and to understand the influence of multi-dimensionality of the acoustic field on flame response,

we performed another simulation where a 1D bulk mode forcing was applied to the flame and the response of the

flame is compared with the 2D bulk mode forcing.

Figure 11 shows the contours of temperature from a simulation that was forced with a 1D bulk mode at an

arbitrary instant. In comparison with Fig. 5(b) we see that there are substantial differences in flame response,

pointing to the sensitivity of the response to the characteristics of acoustic field at the anchor point. To compare the

response quantitatively, we looked at the amplitude of transverse flame oscillations at the forcing frequency and this

is shown in Fig. 12(a). In this figure, green line (with circles) is from the simulation with 2D bulk mode forcing, and

blue line (with squares) is from the 1D bulk mode forcing. Clearly the response for the 1D bulk mode forcing is not

as strong as that of the 2D bulk mode forcing, indicating that the characteristics of acoustic fluctuations near the

anchor point are not the same. In other words, constraining the acoustic velocity perturbations to one dimension only

(in the axial direction) interferes with the growth of the K-H instability modes and diminishes the flame response

significantly. This indicates that the presence of the bluff-body changes the characteristics of the acoustic

fluctuations strongly near the bluff-body and a simple 1D mode exhibits significantly lower flame response

compared to the 2D bulk mode forcing. Further analysis is required to understand whether the multi-dimensionality

of the acoustic mode acts as a means of amplification or if it introduces additional physical processes to the flame

response.

Figure 12(b) shows the coherence of the flame oscillations to the input forcing signal and compares the flame

response to 1D and 2D bulk mode forcing. Again, it is clear that the 2D bulk mode forcing exhibits a much longer

coherence compared to the 1D bulk mode forcing. This clearly suggests that the response and the persistence are of

the response are both strong functions of the characteristics of the acoustic fluctuations at the anchor point.

15

V. Conclusions

The response of bluff-body stabilized flames to external forcing was studied using a numerical formulation based on

Lagrangian transport element method in the limit of low Mach number. The acoustic-vortex-flame interactions

present in these flow-fields were modeled explicitly using Lagrangian elements that discretize the vorticity and

flame surface. Several simulations were performed to study the response of the flame to longitudinal forcing and to

understand the individual mechanisms responsible for this response. Longitudinal forcing in the bulk mode was

applied in two ways: a 2D bulk mode which felt the presence of the bluff-body and a 1D bulk mode using spatially

uniform axial velocity fluctuations. Results from the 2D bulk mode forcing showed that the flame dynamics are

controlled mainly by the wall-generated vorticity in the near-field and flame-generated vorticity in the far-field.

Flame-generated vorticity plays the dual role of augmenting the flame response for a short distance before reducing

it in the far-field. By using a simulation that only forced the flame in the far-field, away from the flame anchor-point

we demonstrated that the flame does not respond to far-field disturbance as much as it does to the near-field

disturbances. It also clearly illustrated that the initial response is generated by the wall-generated vorticity at the

anchor point and that in the far-field flame-generated vorticity only acts to diminish this response. This strongly

suggests that the presence of the bluff-body changes the characteristics of the acoustic fluctuations near the bluff-

body. The study employing a 1D bulk mode showed a significantly lower flame response compared to the forcing

which felt the effect of the bluff-body (2D bulk mode). This confirms the earlier studies on forcing which showed

that the acoustic velocity in the near-field is multi-dimensional and that the response of the flame is significantly

altered by the local characteristics at the anchor point.

Acknowledgements

The work presented in this paper was supported by the United Technologies Corporation through collaboration

of the United Technologies Research Center with the Pratt and Whitney aircraft engine division.

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17

a)

(b)

Figure 3: (a) Instantaneous contours of temperature and (b) instantaneous location of the Lagrangian

elements and mean contours of vorticity

Figure 4: Schematic diagram explaining the physics of flame-generated vorticity

18

(a)

(b)

Figure 5: Instantaneous contours of non-dimensional temperature in (a) unforced and (b) forced simulations

using 2D bulk mode

19

(a)

(b)

Figure 6: PSD of transverse flame oscillations for (a) unforced and (b) forced simulations using 2D bulk mode

20

(a)

(b)

Figure 7: Instantaneous contours of temperature in forced simulations (2d bulk mode) considering (a) only

wall generated vorticity (b) only flame generated vorticity

21

(a)

(b)

(c)

Figure 8: Instantaneous contours of mean vorticity and distribution of Lagrangian elements in forced

simulations (2d bulk mode) including (a) only wall generated vorticity (b) only flame generated vorticity (c)

both the mechanisms

22

(a)

(b)

Figure 9: (a) Amplitude of transverse flame oscillations at the forcing frequency (b) Coherence of the

transverse flame oscillations to the forcing signal

23

Figure 10: Instantaneous contours of non-dimensional flame temperature in forced simulations due to far

field forcing.

Figure 11: Instantaneous contours of non-dimensional flame temperature in forced simulations using 1D bulk

mode forcing.

24

(a)

(b)

Figure 12: (a) Amplitude of transverse flame oscillations at the forcing frequency (b) Coherence of the

transverse flame oscillations to the forcing signal.