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COMBUSTION HEAT RELEASE EFFECTS ON THE DYNAMICS OF BLUFF BODYSTABILIZED PREMIXED REACTING FLOWS

Prashant G. Mehta* and Marios C. Soteriou†

United Technologies Research CenterEast Hartford, CT 06108

* Research Engineer† Research Engineer, Senior Member AIAA

ABSTRACT

The physics of bluff body flameholder stabilized premixed combustion is investigated using the results of time dependent two-dimensional Lagrangian simulations. The Vortex Element Method is used to simulate the low Mach number compressible flow field while the reacting field is simulated using a kinematical flamesheet model. The numerical model is validated against non-reacting experimental data for shear layers –simulated by considering a thin bluff body with a velocity difference across it - and traditional bluff body flows. Results for the reacting bluff body flow indicate a shift in the solution from the Von Karman asymmetric shedding of coherent vortices at a characteristic frequency witnessed in the non-reacting flow, to a rather symmetric shedding that is not dominated by any single frequency.Analysis indicates that this shift is mainly due to the dilatation that accompanies the combustion heat release while baroclinic vorticity plays a supporting but secondary role. The dynamics in the near field of the flameholder (4-5 bluff body thickness downstream) is dominated by the vorticity generated at the bluff body walls. Further downstream vorticity generated by the baroclinic torque dominates the dynamics. The amount of the downstream baroclinic vorticity is strongly dependent on the presence of wall generated vorticity. The latter excites the flame in the near field thereby enhancing the conditions for baroclinic vorticity generation.

1. INTRODUCTION

Flameholder stabilized premixed combustion has been a subject of extensive research in the past due to its relevance to a number of power generation devices such as gas turbines and rockets. Much of the initial work in this area focused on flame stabilization issues: assessing flow conditions under which robust combustion can be sustained [1], [2]. More recent efforts [3], [4] have attempted to scrutinize the role played by

the reacting flow dynamics on the onset and sustenance of thermoacoustic instabilities. These instabilities arise due to a positive feedback coupling of the duct acoustic modes with the unsteady heat released due to combustion [3]. Their suppression/control is a major objective of the latter work and is also the long term objective of our research For both flame stabilization and thermoacoustic instability control to be successful, however, an in-depth understanding of the physics of the unsteady reacting flow is necessary. In particular, this includes questions related to the flow structure and dominant dynamics in the flameholder wake, the impact of combustion heat release there and the receptivity of the dynamics to external manipulation.

In the absence of combustion and for the Reynolds number of interest, bluff body flameholders naturally shed vortices in the Von Karman vortex street regime – i.e. asymmetric (about the bluff body centerline) coherent vortex shedding at a well defined frequency that scales with the bluff body width and the free stream flow velocity [5]. In the presence of combustion, however, little evidence of such vortex shedding has been found in experiments [6], [7] (see also review [8]) and some recent numerical simulations [4], [9]. Rather, the bluff body wake is characterized by a more symmetric but less coherent instantaneous flow field. The physical mechanism behind the transition from the asymmetric Von Karman vortex solution of the non-reacting flow to the more symmetric flow regime in the presence of burning, is far from being clear. A variety of explanations have been proposed in the literature, the majority of them pointing to the vorticity generated by the flame via the baroclinic mechanism as the primary reason behind the exhibited shift in flow behavior. For example, Coats [8] attributes the shift to the combination of two possible mechanisms: (i) the dampening of the vorticity due to the increased kinematic viscosity of the reacted fluid and, (ii) the generation of baroclinic vorticity that is of opposite sign to the flameholder generated vorticity and

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AIAA 2003-835

Copyright © 2003 by United Technologies Corporation. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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tends to nullify the effects of the latter. Menon and co-workers also point to the baroclinic vorticity generation as the main mechanism that leads to the shift [9].

A related question, assuming that the wake region is characterized by two shear layers in the presence of combustion, concerns the effect of combustion on the shear layer evolution. Premixed combustion in a shear layer has been studied by Ghoniem and Krishnan [10], who found baroclinic effects to be important in determining the long term shear layer evolution. As a consequence of these and other studies, a number of more recent attempts to model flameholder stabilized premixed combustion have ignored the impact of the flameholder wall generated vorticity altogether, focusing, instead, on vorticity generated by the baroclinic torque [11], [12].

There are reasons, however, to question the conclusions noted above. For example, it is well established that the generation of baroclinic vorticity leads to the streamwise acceleration of the flow behind the flameholder [11]. This would exclude the existence of a re-circulation zone, since re-circulation can only be caused by the flameholder generated vorticity. Experimental evidence is, however, quite clear on the fact that a re-circulation region does exist behind the flameholder and in fact, it is actually enlarged in the presence of burning [2]. This inconsistency suggests that the region immediately behind the flameholder must be dominated by the flameholder generated vorticity, and that baroclinic vorticity must play a secondary role there. Is this sufficientto cause the shift in the flow behavior? If not, what then is the mechanism that causes the shift? Resolving this issue is important because the region immediately behind the flameholder determines flame stability and can have a significant impact on the flow receptivity to external (acoustic) excitation. In this paper, an attempt is made to address these questions using results from two-dimensional non-averaged numerical simulations in which different flow dynamic mechanisms are selectively eliminated and their impact is assessed. In particular, the impact of baroclinic vorticity generation and of dilatation due to combustion heat release is investigated. Studies of non-premixed combustion in shear layers have shown the latter mechanism to have a dominant impact on the flow dynamics [13]. It appears credible that it should have an important role in this case as well. The paper is organized as follows. In Section 2, the numerical model for the simulation of reacting bluff body flow is described. Section 3 presents numerical results. A brief

validation study is given first. The bulk of the Section is then dedicated to understanding the physics of flameholder stabilized premixed combustion and in particular of addressing the impact of combustion on the flow dynamics behind the flameholder. Finally, the major findings of the study are summarized in Section 4.

2. NUMERICAL MODEL

Formulation

The physical problem of premixed combustion stabilized by a single rectangular bluff body flameholder of height h in a channel of height H is considered (see Fig.2.1 (a) for a schematic).

(a)

(b)Figure 2.1 (a) Schematic of the physical scenario under consideration. (b) Detail on the flamesheet representation of the flame.

The size of both the bluff body and the channel in the third dimension (z) is large so that two-dimensionality is assumed to apply locally. The Mach number is low and both reactants and products behave as ideal gases. The combustion time scale is much faster than that of the flow and the reacting field is assumed to be approximated by a flamesheet.

The governing flow equations are solved in the non-primitive variable form. The Helmholtz decomposition is used to decompose the velocity field ),( vuu =

rinto a vorticity induced solenoidal

part, ωur

and two irrotational parts: eur

, due to the

combustion related volumetric expansion and bur

,

due to the inflow boundary conditions:

be uuuurrrr

++= ω (2.1)

nSTˆ

nSuu Tfˆ)(κ+=

rr

fur

Premixed reactants

Bluff bodyflameholder

x

ythin flame - “flamesheet”

magnifyxmin xmax

Shearregions

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The vortical velocity ωur

is obtained by introducing

the streamfunction, ψ ,

,),ˆ( 2 ωψψω −=∇×∇= kur

(2.2)

where k̂ is the unit normal vector normal to the plane of motion, ω is the vorticity and ∇ is the gradient operator. The irrotational components eu

r

and bur

are obtained from the continuity equation

by introducing two velocity potentials φe and φb, respectively, such that

,1

, 2

Dt

Du eee

ρρφφ −=∇∇= (2.3)

and

,0, 2 =∇∇= bbbu φφ (2.4)

where t is the time, D/Dt is the material (substantial) derivative and ρ is the density. The

evolution of the vorticity vector k̂ωω =r

is described by the 2D vorticity transport equation

ωρρρωω rrr

2

Re

1)( ∇+

∇×

∇=⋅∇+

pu

Dt

D, (2.5)

where Re is the Reynolds number and p is the pressure. The second term on the left hand side of Eq.(2.5) is the divergence term and arises because of the dilatation experienced by the flow due to burning. The first term on the right hand side of the Eq.(2.5) is the baroclinic term and accounts for the baroclinic vorticity generation at the flame surface (as density gradients exist only at the flame surface). Finally, the second term on the right hand side is the viscous term and accounts for the diffusion of vorticity. The vorticity equation is solved in the Lagrangian frame of reference requiring the solution of

udt

xd rr= , (2.6)

where xr

is the position vector of a vortex element.The computational domain exists between

[xmin , xmax], defining the start and the end of the domain (see Fig.2.1). Within the computational domain, no-slip and impermeability boundary conditions are specified at the bluff body walls. The channel walls, on the other hand, are modeled

as slip impermeable planes in effect imposing a symmetry boundary condition there. At the entrance of each of the half channels upstream (on either side) of the bluff body, a uniform velocity profile is specified. This, in effect, provides a constant influx of reactants there. An exit condition is imposed by removing any elements that leave the bounds of the computational domain.

Combustion is assumed to occur at the infinitely fast limit and is modeled by using a Lagrangian flamesheet model (see Fig.2.1 (b) for a schematic). In this model, the flame is modeled to move normal to itself in the reference flame of the flow so as to burn into the reactants with a prescribed burning velocity ST, i.e.

nSuudt

xdTf

f ˆ)(κ+==rrr

, (2.7)

where fxr

is the flame position vector, n̂ is the

unit normal to the flame oriented into the reactants (see Fig.2.1 (b)), fu

r is the flame velocity and

)1()( 0

MTT LSS κκ −= . (2.8)

Here, κ is the local flame curvature, LM is the Markstein length and 0

TS is the nominal value of

flame speed. In the numerical implementation of the flamesheet model, additional mechanisms for modifying this flame speed are implemented. These are discussed in the Numerical Method part of this Section.

The dilatation effects due to combustion are determined from the mass swept by the motion of the flamesheet, i.e.

fT

b

u

e

e

e

dlSADt

DA

ADt

D)()1(

111 κρρρ

ρ −==− , (2.9)

where ρ is the density and Ae is the area created as a result of burning. The first part of the equality arises due to mass conservation (as fluid of density ρu burns in to a fluid of density ρb across the flame) and the second part of the equality is an equivalent representation of the second term assuming a flamesheet of length dlf burns with flame speed ST

normal to itself into the reactants.

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Numerical Method

The Lagrangian Vortex Element Method (VEM) in the form that accommodates the presence of reaction in the low Mach number limit [14] is used to reproduce the unsteady flow. The Lagrangian implementation of the flamesheet evolution is performed using numerical techniques consistent with the VEM. Specifically, the vorticity field and the flamesheet are discretized over a number of elements. Chorin’s boundary sheets method [15] is used to impose the no-slip boundary condition at the bluff body walls. Using this method, the boundary layer vorticity within a thin flow region (of thickness δ) near the wall is discretised as one-dimensional vorticity sheets, while in the remainder of the (interior flow) domain, the vorticity is discretised as two-dimensional Gaussian blobs (of initial core radius δ). In addition to the wall generated vorticity, baroclinic vorticity is generated at the flame surface to account for the baroclinic term in eq.(2.5). This baroclinic vorticity is also discretised as overlapping two-dimensional Gaussian blobs.

The numerical integration of the governing equations is achieved in two fractional steps. The first step includes all processes other than diffusion: vortex and flame elements are advected with the local velocity vector while, at the same time, their properties are updated by numerically integrating the non-diffusive transport equations. All numerical integrations are achieved using predictor-corrector schemes. The vorticity equation is solved in its circulation form – this eliminates the need to explicitly determine the divergence term. Divergence does impact the element core, however, expanding it as elements cross the flame according to local mass conservation and the condition that the element area to core ratio remains constant during this process. The baroclinic generation term is determined by substituting the pressure gradient with the material acceleration using the inviscid momentum equation. The second integration step accounts for the effects of diffusion. This is accomplished using the core expansion scheme [16].

The velocity field at a given time instant is constructed from the potential, the vortical and the expansion components. The potential component of the velocity field results from the inlet boundary condition which imposes a uniform flow at x=-∞. The potential solution is computed using the Schwarz-Christoffel transformation that maps the infinte domain onto upper half complex plane. The conformal mapping is in the form of a

transcendental equation that is solved using Newton iteration [17]. The conformal mapping techniques are also employed to obtain the field solutions for vortical and expansion velocity using Green’s functions in half-space. This greatly facilitates the implementation of boundary conditions by the use of images.

In addition to the above, there are other miscellaneous algorithms in the numerical implementation and are briefly discussed below. To help with the computational burden, a vortex removal algorithm for vortices of the same sign that get too close (as assessed relative to their core sizes) is implemented. The removal of vortices is carried out according to local conservation laws. Flame smoothing such as flame remeshing using cubic splines helps to suppress numerical and physical instabilities associated with the flame detailed structure. An algorithm is also implemented to prevent a flame from propagating into itself or in to the walls by decaying the flame speed for any part of the flame that gets close to either the wall or to other flame. All these features lead to a grid free and adaptive flow simulation.

3. RESULTS AND DISCUSSION

In this section, numerical results are presented and discussed with two major objectives, namely, to provide validation of the numerical model and to investigate the physics of the wake region of the bluff body.

3.1 Non-reacting shear layer flow

The shear layer flow, resulting from the amplification of the Kelvin-Helmholtz instability between two fluids moving at different velocities, has been a subject of extensive experimental study (see [18] for a review). As such, it is an excellent reference flow for validating numerical models that aim to capture the evolution of shear flows. The model described in Section 2 can be made to simulate a shear layer by considering the limit where the bluffbody is thin compared to the boundary layer thickness and by imposing a velocity difference across the bluff body. The first condition is imposed by specifying h/δ<<1 since δis, by construction, of the same order as the boundary layer thickness. Specifically, in the simulations h/δ=0.1 and blockage ratio h/H=10-3 (recall, h is the bluff body width, H is the channel width and δ is the initial core radius of the vortex elements and the thickness of the numerical boundary layer). The velocity ratio r=U2/U1, -where U1 and U2 are the upper and lower inlet

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stream velocities is implemented via the inlet boundary condition. Three velocity ratios are considered, namely r=0.35, 0.5, and 0.75. The Reynolds number is 4000 so as to be in agreement with the experimental results of Spencer and Jones [19].

Figure 3.1 displays a numerical flow visualization for r=0.5. The composite figure presents the instantaneous flow in terms of the vortex elements (points denote centers of on the elements) and streamlines (lines), superimposed mean streamwise velocity field (shades). The streamlines - exhibiting closed loop, “cat’s eye” like patterns [20] - are obtained in a reference frame moving with the average velocity (U1+U2)/2.

Qualitative agreement of the numerical solution with well-established features of the shear layer flow is evident in Fig.3.1. The instantaneous flow solution is dominated by the evolution and interactions of coherent vortices. At small streamwise locations, the shear layer grows by the generation of vortices due to the fundamental mode of the Kelvin-Helmholtz instability. Further downstream, shear layer growth is achieved via the subharmonic pairing interaction of neighboring vortices. Finally, it is noted that the vortical structures disappear in the mean which is characterized by essentially linear, self-similar growth features.

Figure 3.1. Shear layer flow visualization with r =0.5, Re=4000. Black /white points denote the instantaneous locations of vortex elements with -/+ vorticity, respectively. Instantaneous streamlines obtained in a reference frame moving with the average flow velocity (U1+U2)/2 are depicted by the white lines. Color shades indicate the time-averaged streamwise velocity.

Figure 3.2. Mean and turbulent profiles computed at different downstream x-locations plotted against the scaled similarity variable βy*, where β is the Roshko’s growth parameter [20] (chosen here to be 20). The mean profile is compared against Gorteler error function (red) and the turbulent profiles are compared against experimental data (red) from [19].

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-3

-2

-1

0

1

2

3

UU ∆>< /

*yβ

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-3

-2

-1

0

1

2

3Uuu ∆>< /

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-3

-2

-1

0

1

2

3 Uvv ∆>< /

*yβ

-5 0 5 10 15 20

x 10-3

-3

-2

-1

0

1

2

3

2/ Uuv ∆><−

U1

U2

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Quantitative comparison of the numerical solution mean and turbulent velocity profiles to experimental evidence from Spencer and Jones [19] is presented in Fig.3.2. The profiles of the figure are plotted against the similarity variable defined as

0

0*xx

yyy −

−= , (3.1)

where (x0,y0) is the shear layer’s virtual origin [21]. When plotted against this variable, the experimental results collapse onto a single, self-similar profile. The numerical mean velocity profile <U>/∆U of Fig.3.2 too collapses onto a self-similar profile. It, however, exhibits two notable differences, that are, subject to explanation: (i) A small remnant of a wake persists in the numerical profile near the slow stream side. We believe this to be a consequence of the finite width of the thin bluff body – note that the experimental results were obtained with a splitter plate – combined with the impact of relatively low Reynolds number. Indeed, simulations at higher Reynolds numbers indicate a smaller wake in the mean velocity, thus supporting this view. (ii) The numerical profiles appear to be slightly accelerated with respect to the experimental ones. This is a result of the inlet condition specified in the simulation (flat velocity profiles at the inlet) coupled with the effects of confinement – in effect more mass is pushed through the channel than in an equivalent duct in the experiment.

The turbulent stress profiles of Fig.3.2 are in reasonable agreement with experiments, particularly when contrasted with earlier numerical work in this area. The stress Uuu ∆>< / profiles - known to be related to the large flow structure – accurately predict the experimentally observed peak magnitude, indicative of the model’s ability to capture thisstructure well. The experimental profiles are thinner than those predicted numerically – we believe the experimental profiles to be suspect in this case, however, given the relative thickness of the mean velocity and of the stress

2/ Uuv ∆><− profiles. Previous simulations of this flow have also seen the trend of wider profiles in the Uuu ∆>< / profile [21]. The

severe over-prediction of the profile Uvv ∆>< / , seen in Fig.3.2, has been documented in previous numerical work [10], also [22] and is linked to the 2D nature of the simulation. The

stress 2/ Uuv ∆><− , indicative of the momentum exchange between the streams is reasonably well predicted..

Finally, the shear layer spatial growth rate obtained in the simulations is compared with experimental evidence, summarized by Dimotakis [18] into the empirical formula:

.21

21

UU

UUC

+−

=∆′ ∆ (3.2)

where the growth rate is defined as ∆′=∆(x)/(x-x0) with ∆ being the cross-stream thickness of the shear layer and x0 being the virtual origin. C∆is an experimentally determined constant with the value in the range [0.25,0.45], dependent on initial conditions. Figure 3.3 compares the numerical results obtained for the three velocity ratios investigated. In the simulations the shear layer thickness is defined by considering the point where the flow velocity reaches 0.99 of the nearby free stream velocity. The figure makes clear that the numerical model is effective at capturing the effect of velocity ratio variation on the shear layer growth rate.

∆′

Figure 3.3. Comparison of shear layer growth rate obtained using the model (blue o) against the experimental results summarized in eq.(3.2) (red -) [18].

3.2. Non-reacting bluff body flow

We now consider a flow where the bluff body thickness is much larger than that of the boundary layer, i.e. h/δ>>1, and the velocity ratio r=1. In this configuration, and for moderate Reynolds number, the bluff body flow is dominated by the characteristic Von Karman shedding solution: an asymmetric (about the bluff body centerline) street of counter-rotating vortices shedding at a characteristic Strouhal number St=fUin/h~0.2, where Uin=U1=U2, and f is the shedding frequency [5].

Velocity ratio – U2/U1

Shear layer growthrate

experiment Dimotakis (91)

model

0.35 0.50 0.75

0.04

0.08

0.12

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Simulation results are obtained for blockage ratio h/H=0.16, h/δ=20 and Re=20000 based upon the bluff body thickness h and compared against the experimental results of Raffoul et al. [23]. Figure 3.4, which displays a representative visualization of the numerical results, shows that the model captures the time-dependent asymmetric Von Karman street. The Strouhal number St is found to be 0.2, close to the expected result (see Fig.3.8 (b) for a spectrum). The Lagrangian representation of the vorticity field via the vortex elements (points) clarifies that the vorticity at the core of each vortex comes from both the top and bottom bluff body boundary layers. Also, despite the asymmetry in the instantaneous solution, the mean solution (shades) is symmetric. A re-circulation zone

(approximately defined as the region where the mean velocity is negative) is found to be about half a bluff body thickness - consistent with the results presented in [23].

Figures 3.5 (a) and (b) contrast the numerically obtained mean velocity and turbulent stress profiles, respectively, at two downstream locations with the experimental results of [23]. It is evident from the figure that satisfactory agreement is exhibited for most profiles indicating that the model is capable of quantitatively capturing the flow features. It is worth noting that the rather large difference seen in the <V> profile at x/H=1 has also been encountered in simulations of this flow presented in [23].

Figure 3.4. Flow visualization of the non-reacting bluff body flow with h/H=0.16, h/δ=20, and Re=20000. Black/white points denote the instantaneous centers of vortex elements with -/+ vorticity, respectively. Color shades indicated the mean streamwise velocity.

x/H=.5

x/H=1

(a)

<V>/Uin<U>/Uin

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x/H=.5

x/H=1

(b)Figure 3.5. Comparison of (a) mean and (b) turbulent velocity profiles obtained from the model (blue -) with the experimental results (red o) of Raffoul et al. [23].

3.3 Reacting bluff body flow without wall generated vorticity

As an intermediate step to the simulation of the reacting bluff body flow, we focus on the case where the flow is reacting but vorticity generation on the bluff body walls is artificially turned off, i.e. a slip condition is imposed there. This is a considerably simpler problem to simulate and yields results that can readily be used for the validation of the model. More importantly, this flow has been considered to be relevant to the physics of bluff body stabilized flames in the numerical studies of Pindera et al. [11] and Rhee et al. [12]. These studies assume that the evolution of the vorticity field is dominated by baroclinic vorticity and neglect the impact of the wall generated vorticity as secondary. By simulating this flow, we can obtain data for comparison with the aforementioned studies and also, through comparison with the comprehensive flow simulation (presented in Section 3.4), we can attempt to shed light on the validity of its key assumptions.

In the absence of wall generated vorticity, the fluid dynamics is a consequence of the steady potential solution due to inlet velocity, dilatation due to burning and baroclinic vorticity. As a first step in the simulation of this flow we consider two simpler but instructive scenaria: In the first, the flow field is decoupled from the reacting field, i.e., both the volumetric expansion

and the baroclinic vorticity generation are turned off, while in the second only the baroclinic vorticity is eliminated. Given the stated assumptions, both flows are irrotational but the first corresponds to non-exothermic (cold) combustion while in the second, exothermicity-related dilatation effects are accounted for (hot).

Using the model, both these simplified flows were found to reach a steady state solution. Table I compares the flame angles from these steady solutions for different values of the nominal flame speeds for the cold and hot (dilatational) combustion cases.

in

T

U

S θUinSin(θ)=ST

θCOLD

θHOT

0.1 5.75 5.7 5.350.25 14.5 14 110.50 30 32 13.5

Table I. Flame angle (in degrees) as a function of the flame speed (ST) for vorticity free bluff body stabilized premixed reacting flows. Cold and hot cases are distinguished by the fact that in the latter, dilatation effects are accounted for -density ratio, 5/ =bu ρρ .

The expected solution for the cold flow is a steady flame that lies at an angle, θ, to the streamwise direction as determined by the flow and flame speed: )(Sin θinT US = . The small

<vv>/Uin2 <uv>/Uin

2<uu>/Uin2

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discrepancy between the exact and computed angles in the co lumns 2 and 3 (cold cases) of Table I is due to the bluff body geometry which causes the potential flow to turn inward very close to the trailing edge. For the hot case, on the other hand, the volume generated due to burning, together with the confinement effects due to the presence of symmetry channel walls cause the flame angles to become shallower – as seen in column 4 - with increasing streamwise coordinate.

The impact of baroclinic vorticity generation is considered next. Figure 3.6 (a) and (b) display flow visualizations, including instantaneous flame (lines) and vortex element (points) locations and time averaged streamwise velocity (shades), of the reacting flow without and with baroclinic vorticity generation, respectively.

Figure 3.6. Flow visualization without (a) and with (b) baroclinicity. Flames (lines) and vortex elements (magenta/green points denoting +/-signed vorticity) together with the mean streamwise velocity field (shades).

The acceleration of the flow and the reduction in the flame angle with streamwise coordinate in the presence of dilatation noted in the previous paragraph is evident in Fig.3.6(a). Figure 3.6(b), on the other hand, indicates that the net effect of the baroclinic vorticity is to accelerate the fluid in the burnt region and decelerate it in the unburnt region. Moreover, each flame sheet tends to generate single sign vorticity but opposite to that of the other sheet. This is because of the orientation of the density gradient vector that is opposite in direction on each sheet and because the distortion of the sheet due to unsteadiness is minimal. Unlike the non-

vortical solution in Fig.3.6 (a), which reaches steady state, the one in Fig.3.6(b) is unsteady, albeit only slightly so. This and all the above observations are consistent with the results in [11], [12].

3.4. Reacting bluff body flow

Simulations for the reacting bluff body flow problem (described in section 2 and illustrated in Fig.2.1) are performed for the same configuration as the non-reacting flow of Section 3.2, namely h/H=0.16, h/δ=20, and Re=20000 based upon the bluff body width. The density ratio 5.2/

bu=ρρ and the nominal flame speed

1.0/ =in

o

T US . The Markstein length, LM, is chosen

to be 0.04 consistent with [12]. Characteristic simulation results are shown

in Fig.3.7. Part (a) of the figure contrasts the instantaneous solution in terms of the vortex elements (points) and flame surface (line) to the mean streamwise velocity field (color shades) while part (b) presents the mean flame location (lines) and the vorticity field (shades). Comparison of the reacting solution in Fig. 3.7(a) with its non-reacting counterpart in Fig. 3.4 makes evident that the flow is drastically altered in the presence of combustion. A shift from the asymmetric Von Karman shedding of coherent vortices in the non-reacting case to a more symmetric but less coherent shedding in the reacting case can be seen. This shift is consistent with the experimental observation of [6], [7] (see also review paper [8]) and recent numerical results of [4], [9]. It is noted that the reacting solution is symmetric only in a coarsesense – it is distinguished from the Von Karman solution in the absence of any single frequency asymmetric oscillation of the fluid dynamics.

Figure 3.8 contrasts the time-series and frequency spectra, based on the v-velocity signal taken on the centerline half bluff body width downstream of the bluff body trailing edge, for the reacting and non-reacting cases. The reduction in unsteadiness (with respect to the non-reacting case) seen in Fig.3.8 is consistent with the experimentally obtained spectral plots, comparing non-reacting and reacting spectra, presented in [7].

Returning to Fig.3.7, the presence of a re-circulation region - note negative mean streamwise velocities near the bluff body– can be seen. This re-circulation region is found to be larger than that of the non-reacting case

(a)

(b)

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Figure 3.7 Flow visualization of the reacting flow with h/H=0.16, h/δ=20, Re=20000, and 5.2/bu

=ρρ .

(a) Instantaneous flame location (lines), center of vortex elements (black/white points denoting -/+ signed vorticity) together with the mean streamwise velocity field (shades). (b). Mean flame location (lines)together with mean vorticity (shades) - the color axis range for vorticity plot is chopped to better view the weaker baroclinic vorticity.

- (a) (b)Figure 3.8. (a) Time series and (b) spectral plots of the non-reacting (blue) and reacting (red) v-velocity signal on the centerline (y=0) at half a bluff body width downstream from trailing edge.

(Fig.3.4). The re-circulation region arises because of the opposite signed (negative clockwise for the top and positive, anticlockwise for the bottom bluff body wall) vorticity generated at the bluff body horizontal walls. Baroclinic vorticity generation, on the other hand, creates vorticity near each flame sheet that

is – in net - of an opposite sign with respect to the closest bluffbody horizontal wall (i.e. positive, anticlockwise for top sheet and negative, clockwise for bottom sheet – see Fig.3.6(b)). As such, it induces a streamwise acceleration of the fluid between the two sheets and thus is not directly responsible for the

(a)

(b)

0 2 4 6 8 10 12 14 16-1.5

-1

-0.5

0

0.5

1

1.5Time-series – v/Uin

Time (non-dimn)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Strouhal Frequency

Spectrum

11American Institute of Aeronautics and Astronautics

presence of re-circulating flow. Indeed, the baroclinic vorticity is actually detrimental to the size of the re-circulation region and causes this region to shrink.

In order to interpret the averaged vorticity field in Fig.3.7(b), it is important to further scrutinize the role played by baroclinic vorticity on shaping the flow dynamics. Figure 3.9 (which follows the format of Fig.3.7) displays results from a flow simulation with same conditions as those of Fig.3.7 but with baroclinic vorticity generation artificially turned off. The two opposite signed regions of mean vorticity seen in Fig.3.9 (b) arise due to the no-slip boundary condition at the two horizontal bluff body walls.

Figure 3.9 Flow visualization for the reacting case with baroclinic vorticity generation turned off. The figure follows the format of Fig.3.7.

On comparing the mean vorticity fields of Figs.3.7 (b) and 3.9 (b), it is noted that the bluff body near field (4-5 bluff body widths downstream of the flameholder) is dominated by the wall generated vorticity. This conclusion follows because the mean baroclinic vorticity is of an opposite sign with respect to the wall vorticity. However, the figures also make evident that the baroclinic vorticity is important in the far field, where it overwhelms the wall vorticity. As a result, the baroclinic vorticity dictates the vortical dynamics in the far field of the bluff body, while in the near field, the wall generated vorticity is dominant.

Another conclusion that can be obtained from comparing Figs.3.7 and 3.9 is that the flow

in either case is symmetric (in a coarse sense), i.e., turning baroclinicity off does not cause the flow to transition back to a Von Karman shedding regime. This conclusion is not surprising given the discussion above; nevertheless it is important because baroclinicity has been postulated as the mechanism that causes the transition to a symmetric regime in some of the earlier studies [9]. If not baroclinicity, what then causes the transition from asymmetric to symmetric shedding behavior? It has already been concluded that the near field bluff body wake (for both the reacting and non-reacting flow) is dominated by the opposite signed wall generated vorticity. For the reacting flow, this vorticity is affected by the dilatation due to burning via two mechanisms discussed below: (i) The dilatation affects the vorticity evolution due to the presence of the term ω)( u⋅∇ in Eq.(2.5).

As a result of this term, the vortical fluid elements expand when they cross the flame into the product region, thereby causing their vorticity magnitude to reduce (their circulation, however, is preserved [13]). This weakens the ability of opposite signed vortical elements to interact and cause Von Karman shedding. (ii) The dilatation affects the incoming boundary layer vorticity - as it comes off the upper (and lower) bluff body corner – by partially lifting it from the wall and causing some of it to evolve in the reactant region. This vorticity crosses into the products further downstream - see Fig.3.7(a). As the density ratio increases, less and less of the wall generated vorticity in the boundary layer crosses in to the product region immediately near the bluff body edge, where it may interact with the vorticity of opposite sign, to produce the Von Karman vortex street.

Figure 3.10, which follows the same format as Figs.3.7 and 3.9, displays a flow visualization from a simulation where only the dilatation effects of the burning are turned off. On account of the presence of baroclinic vorticity, the solution does not quite revert back to the Von Karman vortex solution even though the solution exhibits substantial similarities to it – e.g. asymmetric large scale features. The similarity is particularly pronounced in the bluff body near field where a strong asymmetric unsteadiness is witnessed. In the far field of the bluff body, the incipient Von Karman vortex street is over-whelmed by the baroclinic vorticity much as discussed before and the flow evolves accordingly.

(a)

(b)

12American Institute of Aeronautics and Astronautics

Figure 3.10. Flow visualization for the reacting case with dilatation effects turned off. The figure follows the format of Fig.3.7.

Numerical results also indicate that the flame motion is dictated by the spatio-temporal evolution of the vorticity field. For the flame attached to the upper corner of the bluff body, the clockwise spinning of the flow due to the wall generated vorticity causes the flame to curl up in a clockwise sense. In the far field downstream, on the other hand, where the baroclinic effects are important, the flame can be seen to curl up in the counter-clockwise sense due to the counter-clockwise spinning baroclinic vorticity (see Fig.3.7). A similar (mirror) argument can be made for the flame attached to the bottom bluff body corner. Here, we also contrast the unsteady flame evolution seen in the present study with the nearly steady flame of Fig.3.6(b), the case where the wall vorticity generation was turned off but baroclinic vorticity generation was present. Such a comparison suggests that the wall generated vorticity has a strong impact on the far field baroclinic vorticity generation. In effect, the unsteadiness due to the wall generated vorticity forces/excites the flame thereby causing large undulations of this flame downstream. These undulations set-up conditions for enhanced generation of baroclinic vorticity. Without the presence of excitation due to the wall generated vorticity, a much less interesting picture emerges downstream of the bluffbody characterized by much less unsteadiness and diminished baroclinic vorticity generation as seen in Fig.3.6(b). This suggests that simulations of flameholder stabilized

premixed flames in which wall generated vorticity is neglected may lead to misleading results.

4. CONCLUSIONS

A Lagrangian two dimensional numerical model of premixed combustion that uses the Vortex Element Method for the simulation of the flow and a kinematical flamesheet model for the simulation of the reacting field is presented and implemented in the simulation of bluff body flameholder stabilized combustion. The model quantitatively captures the non-reacting flow behavior for both the shear layer (thin bluff body with velocity difference) and bluff body flows. Reacting flow results are in qualitative agreement with experimental and other numerical evidence.

The presence of burning changes the dynamics of the wake in a manner that leads to a shift from a Von Karman, single frequency dominated, limit cycling, asymmetric solution to a ‘symmetric’ solution that is not characterized by a well defined shedding frequency. The reacting flow solution is unsteady and thus symmetric (at any given time instant) only in a coarse sense. In addition, it exhibits substantially diminished unsteadiness in the wake region.

Vorticity is introduced into the reacting flow wake via generation at the bluff body walls and by baroclinic vorticity generation at the flame front. The wall generated vorticity is shown to be dominant in the bluff body near field (4-5 widths downstream of trailing edge), while further downstream, the baroclinic vorticity is important. Results indicate that the shift from the asymmetric Von-Karman shedding to the symmetric one in the presence of burning is primarily a consequence of dilatation due to the combustion heat release. Baroclinic vorticity augments the impact of dilatation but its effect is secondary. The dilatation weakens the wall generated vorticity as it goes through the flame and delays the entrainment of some of this vorticity into the products region of the wake. Both effects tend to diminish the interaction of the opposite signed vorticity emerging from the boundary layers on the two horizontal bluff body walls, thereby diminishing the possibility of a Von Karman street.

The baroclinic vorticity generation is found to be strongly linked to the presence of wall generated vorticity. The latter tends to force the flame in the near field thus setting up the

(a)

(b)

13American Institute of Aeronautics and Astronautics

conditions for the baroclinic vorticity generation. Neglecting the wall generated vorticity results in a much diminished reduction in the unsteadiness of the flow.

ACNOWLEDGEMENT

The work presented in this paper was supported by the United Technologies Corporation and in part by AFOSR (under contract F49620-01-C-0021).

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