mixing and entrainment of transitional non circular plumes

Flow, Turbulence and Combustion 67: 5779, 2001. 2001 Kluwer Academic Publishers. Printed in the Netherlands. 57

Mixing and Entrainment of TransitionalNon-Circular Buoyant Reactive Plumes

X. JIANG and K.H. LUODepartment of Engineering, Queen Mary and Westfield College, University of London,London E1 4NS, U.K.; E-mail: [email protected]

Received 17 April 2001; accepted in revised form 8 October 2001

Abstract. Three-dimensional spatial direct numerical simulation is used to investigate the evolutionof reactive plumes established on non-circular sources. Simulations are performed for three cases:a rectangular plume with an aspect ratio of 2:1, a square plume, and the square plume in a cornerconfiguration. Buoyancy-induced large scale vortical structures evolve spatially in the flow field. Astronger tendency of transition to turbulence is observed for the free rectangular plume than the freesquare case due to the aspect ratio effect. Dynamics of the corner square plume differs significantlyfrom the corresponding free case due to the enhanced mixing by the side-wall effects. A turbulentinertial subrange has been observed for the free rectangular and corner square plumes. Mean flowproperties are also calculated. The study shows significant effects of source geometry and side-wallboundary on the flow transition and entrainment of reactive plumes.

Key words: buoyancy, DNS, entrainment, mixing, non-premixed flame, reaction, transition.

1. Introduction

Reactive jets and plumes are encountered in many industrial and environmentalapplications. Understanding their mixing and entrainment properties is of bothpractical and fundamental importance. The spatial development of a jet dependson its initial momentum and the surrounding environment. For a plume, buoyancyeffects due to density inhomogeneity are also of importance. Major sources ofdensity inhomogeneity include temperature inhomogeneity due to heat release,differences in density of chemical species and phase changes. All of these mayoccur in one reactive flow.

Despite the numerous experimental and computational studies [5, 7, 12], dy-namics of jets and plumes is far from being well understood. For instance, effectsof the geometry of jet nozzle or plume source have not been fully investigated. Onthe other hand, optimizing the nozzle geometry is potentially an efficient techniqueof passive flow control that can improve combustion efficiency, enhance heat andmass transfer, and reduce undesired emissions at a relatively low cost. Non-circularjet nozzles, for example, have been used to improve the mixing and entrainment ofreactive and non-reactive systems [12]. For reactive plumes, another outstanding

58 X. JIANG AND K.H. LUO

issue is the side-wall effects, which is important in applications such as fire controlbut has not been fully investigated.

Dynamics of reactive plumes is a difficult problem to tackle. The complexitiesinclude the effects of plume source geometry and side-walls, together with the cou-pling between fluid dynamics and combustion through buoyancy. Buoyancy effectsare important to low speed reactive flows. In experiments, it is difficult to separatethe individual effects. Accurate measurements of the instantaneous quantities andflow field data are not easy either. In analytical approaches, it is very difficult to takeinto account the non-parallel effects due to strong entrainment and the effects ofsource geometry and surroundings. Numerical approaches based on the Reynolds-Averaged NavierStokes equations (RANS) have been used in the modelling ofreactive plumes for many years [5]. However, the RANS approach is inappropri-ate to investigate unsteady reactive plumes involving transition, intermittency andturbulence.

Direct numerical simulation (DNS) which directly resolves all the relevant timeand length scales described in the NavierStokes equations provides a possibilityto study the complex phenomena occurring in reactive plumes. Recently, a directand large-eddy simulation of the transition of plane plumes in a confined enclosurewas reported for a non-reacting flow with Boussinesq approximation for the effectof buoyancy by Bastiaans et al. [1], in which no clear turbulent inertial subrangewas present. The present authors [17] performed a spatial DNS of the near fielddynamics of a rectangular reactive plume. The spatially developing reactive plumeshowed a tendency of transition to turbulence under the effects of combustion-induced buoyancy. However, the effects of plume source geometry and side-wallswere not fully investigated.

The main objective of this study is to examine the plume source geometryand side-wall effects on the near field dynamics of a buoyant reactive plume,which belongs to diffusion-controlled non-premixed flames. Chemical reactionsproduce heat sources heterogeneously distributed. Under the buoyancy effects dueto temperature inhomogeneity, the reactive plume evolves spatially with transitionoccurring downstream. A comparative study of three cases has been conducted.The first case performed is a rectangular reactive plume with an aspect ratio of 2:1in an open boundary domain, while the second case performed is a square reactiveplume with the same cross-sectional area as the rectangular case. The third caseconsidered is the square reactive plume in a corner configuration. The simulationsdescribe details of the reactive plume evolution through transition to turbulence.Results are discussed in terms of instantaneous quantities, history of the streamwisevelocities and energy spectra, vorticity transport, and time-averaged statistics.

The rest of the paper is organized as follows. Mathematical description of thephysical problem and numerical method used for the spatial DNS are presented inthe next section, followed by a discussion on the simulation results from a com-parative study of the effects of source geometry and side-walls on the dynamics ofnon-circular buoyant reactive plumes. Finally, conclusions are drawn.

TRANSITIONAL NON-CIRCULAR BUOYANT REACTIVE PLUMES 59

2. Numerical Approach

2.1. MATHEMATICAL DESCRIPTION

Flow transition to turbulence in reactive jets and plumes occurs in the near field,which is the region close to the jet nozzle or plume source. In this study, the nearfield of transitional buoyant reactive plumes is considered, where a fuel jet issuesfrom the non-circular nozzle vertically into an oxidant ambient. Combustion occurswhen fuel/oxidizer mixing takes place and a non-premixed flame is establishedabove the inlet plane.

The governing equations used to describe the reactive flow field above the inletplane are the compressible time-dependent NavierStokes equations with tempera-ture-dependent viscosity. The conservation laws for mass, momentum, energy andchemical species formulated in non-dimensional forms are adopted. Major refer-ence quantities used in the normalization are: gref = 9.81 m/s2, magnitude of thegravitational acceleration; Lref = L0, width of the fuel jet along the major axis ofthe rectangular source; T ref = T a , the ambient temperature; wref = w0, the max-imum velocity of the fuel jet at the inlet; ref = a , fuel viscosity at the ambienttemperature; ref = a , fuel density at the ambient temperature. Accordingly timeis non-dimensionalized with tref = L0/w0. Here, the subscripts 0 and a refer to thefuel jet (source) and ambient respectively and the superscript stands for dimen-sional quantities. The dynamic viscosity is chosen to be temperature-dependentaccording to = a(T /Ta)0.76.

The non-dimensional governing equations can be written in a vector form as

Ut

+ Ex

+ Fy

+ Gz

+ S = 0. (1)

In the Cartesian coordinate system, the z-direction is assumed to be the streamwise(vertical) direction, the x-direction is aligned with the minor axis of the rectangularfuel jet at the inlet, while the y-coordinate is along the major axis. The termsmajor and minor used here refer to the dimensions of the rectangular inlet.For square geometry, this difference vanishes. The velocity components in the x,y, and z directions are represented by u, v, and w respectively. In Equation (1),vectors U, E, F, G and S are defined as U = (, u, v, w, ET , Yf , Yo)Twith the superscript T representing transposition,

E =

u

u2 + p xxuv xyuw xz

(ET + p)u+ qx uxx vxy wxzuYf 1Re Sc

(

Yf

x

)

uYo 1Re Sc(Yo

x

)

,


F =

v

uv xyv2 + p yyvw yz

(ET + p)v + qy uxy vyy wyzvYf 1Re Sc

(

Yf

y

)

vYo 1Re Sc(Yo

y

)

,

G =

w

uw xzvw yz

w2 + p zz(ET + p)w + qz uxz vyz wzz

wYf 1Re Sc(

Yf

z

)

wYo 1Re Sc(Yo

z

)

, and

S =

000

(a)gzFr

(a)wgzFr hfo

.

As shown, the gravitational effect is expressed as buoyancy terms (a )gz/Frand (a )wgz/Fr in the streamwise momentum equation and energy equation,respectively, where Froude number is defined as Fr = w02/(grefL0) and gz = 1is the gravity imposed in the downward vertical direction. The governing equationsalso include the perfect gas law for the mixture.

In Equation (1), ET = [e + (u2 + v2 + w2)/2] is the total energy with e rep-resenting the internal energy per unit mass, q represents the heat flux components,Re and Sc represent the Reynolds and Schmidt numbers, respectively, Y standsfor mass fraction of the chemical species. Currently, a complete three-dimensional(3D) DNS of reactive flows with multi-species transport and complex chemistry isstill out of reach [2], due to the excessive computer resources required. In this study,a one-step global reaction fMf +oMo pMp with finite-rate Arrhenius kinet-ics is presumed for the chemistry, where Mi and i represent the chemical symboland stoichiometric coefficient for species i, respectively. The subscripts f , o, pstand for fuel, oxidizer and product, respectively. The reaction rate takes the form


of T = Da(Yf /Wf t)f (Yo/Wo)o exp[Ze(1/T 1/Tfl)] after normalization[21], where Da, Ze, and Tfl are the Damkhler number, Zeldovich number, andflame temperature, respectively, Wi stands for the molecular weight of species i.The reaction rates for individual species are f = fWfT , o = oWoTand p = pWpT . The heat release rate in the energy equation is given byh = QhT with Qh representing the heat of combustion.

2.2. NUMERICAL METHODS

The equations are solved using a sixth-order compact finite difference scheme [19]with spectral-like resolution for evaluation of the spatial derivatives. This schemeallows more flexibility in the specification of boundary conditions with minimalloss of accuracy compared with spectral methods. A third-order accurate fully-explicit compact-storage RungeKutta scheme is used to advance the equations intime. The time step is limited by the CourantFriedrichsLewy (CFL) condition forstability. It is further limited by the reaction rate, an increase in local mass fractionof product of more than 0.01% is prevented for one time step. For DNS of reactiveflows, the time step is usually limited by the chemical restraint.

The specification of the boundary conditions is performed by using the generalformulation of characteristic boundary conditions for DNS of NavierStokes equa-tions by Poinsot and Lele [24]. The 3D computational domain is bounded by theinflow and outflow boundaries in the streamwise direction. At the inflow boundary,temperature is treated as a soft variable while other variables are imposed withtheir initial values. A soft variable means that the temperature is allowed tofluctuate around the prescribed value, which is associated with the characteristicwave variations at this boundary. This treatment guarantees the numerical stabilityof the high-order numerical scheme near the boundary, while the fluctuation oftemperature with time is less than 0.5% of its prescribed value in the simulationsperformed.

The outflow boundary condition needs careful attention since vortices are beingconvected through this boundary and the flow field outside the domain is unknown.At the outflow boundary, non-reflecting characteristic boundary condition [25] isused. In order to completely eliminate the spurious wave reflections, a sponge layer[15] is applied next to the outflow boundary. This is because that the non-reflectingcharacteristic boundary condition is based on a one-dimensional formulation, butthe flow near the outlet is of multi-dimensional nature due to the convection ofvortices. The computational results in the sponge layer are not truly physical andtherefore not used in the data analysis.

In the cross-streamwise directions, the flow field is either open or bounded bythe side-walls. For the open boundary, the entrainment boundary condition [15] isused, which allows entrainment of the ambient fluid into the computational domain.For the side-wall boundary condition, the wall is assumed to be impermeable andnon-slip. The wall temperature is determined from the characteristic form of the


energy equation by applying the local one-dimensional inviscid relations for a wallboundary [24], which allows variations of the wall temperature.

Atop-hat profile is assumed for the streamwise velocity of the fuel jet on theinlet plane, which is given by

w = {1 + tanh[20 (sx |x Lx/2|)]} {1 + tanh[20 (sy |y|)]}[1 + tanh(20 sx)+ tanh(20 sy)+ tanh(20 sx) tanh(20 sy)] ,while the cross-streamwise velocity components are taken as zero on the inletplane. The inlet plane is of the dimension of (0 Lx,Ly/2 Ly/2), where Lxand Ly stand for the inlet domain lengths in the minor and major axis directions,respectively, and s stands for the half-width of the fuel source on the inlet plane.The fuel and oxidizer are unmixed on the inlet plane and the fuel jet is centered inthe inlet domain. In the simulations, the fuel temperature at the source is assumedto be 3, which was chosen to ensure auto-ignition of the mixture [10, 11, 26].Ignition occurs automatically when the fuel and oxidizer mix with each other. Thetemperature and streamwise velocity profiles on the inlet plane are linked with theCroccoBusemann relation.

The flow field inside the domain is initialized with the inlet conditions. Buoyantjets and plumes display an intrinsic, absolutely unstable flow instability [14, 16,20, 22] in which vortices evolve naturally in the flow field due to the gravitationaleffect [16]. For such as an absolute instability, there is no need to apply continuousexternal perturbations at the inflow boundary for the development of flow vortices[15, 16, 18]. In order to isolate the absolute buoyancy instability from the convec-tive shear instability, external perturbations were not used at the inflow boundaryin the simulations performed. The external stimulus to initiate flow vortices inthe numerical simulations can arise from a mismatch between the imposed initialconditions and solutions to the flow field. For the present spatial simulations, initialconditions are of minor importance after the initial transient period.

3. Results and Discussion

In the plume near field, buoyancy effects such as flow acceleration on the resolutionset a limit on the minimum value of the Froude number that can be prescribed ina DNS under a certain number of grid points [6]. In this study, the Froude numberFr = 1.5 used has been chosen so that the buoyant reactive flow field can be fullyresolved. The nominal flow Reynolds number used in the simulations is Re =1000, which is based on the inlet reference quantities. The species are assumed tobe of equal diffusivity. The ratio of specific heats, Prandtl and Schmidt numbersused in the simulations are chosen to be constants: = 1.4, Pr = 1 and Sc = 1.The parameters used for the chemical reaction are: Da = 6, Ze = 12, Tfl = 6, andQh = 1650. These values are chosen to mimic the behavior of a relatively low heatrelease combustion within the computational resources available.

For the free rectangular case, a computational domain of the size of 3 6 8 is used, which has been chosen to minimize the effects of boundaries on the


simulation results. The area of the fuel jet is 0.51.0 on the inlet plane. For the freesquare case, the domain size used is 4.254.258.50 which has approximately thesame cross-sectional area as the rectangular case, while the fuel jet source area isapproximately 0.710.71. The results presented next are obtained from a uniformgrid system with 108216288 nodes for the free rectangular case and a uniformgrid system with 152 152 304 nodes for the free square case. The cornersquare case performed is of the same size as the free square case with side-wallboundaries located at x = 0 and y = Ly/2. For this case, more grid points areneeded to fully resolve the flow field between the wall and plume. A grid systemwith 188 188 376 nodes is used for this corner square case. For a sufficientlysmall reference length scale (e.g. Lref 1 cm), the grid points used in these threecases are sufficient to resolve the energy spectra of the relevant physical problems.In the simulations, the sponge layer used to prevent the spurious wave reflectionsfrom the outflow boundary is located between z = 7.0 and the end of the domainin the streamwise direction. For clarity, some results shown in the figures are for abox which is smaller than the computational domain used.

Simulations have been performed on the massively parallel computer Cray T3E-1200E in Manchester by using 72 processors for the free rectangular case, 76processors for the free square case and 94 processors for the corner square case. Inthis study, grid independence and time-step independence tests were performed. Aspatial resolution study was performed by adding 20% more grid points in each di-rection for the corner square case, while a temporal resolution study was performedby reducing the time step by half. In both cases, history of the vorticity extrema inthe computational domain and typical instantaneous velocity profiles showed noappreciable changes. Therefore the results presented next are considered as gridand time-step independent. In the following, instantaneous quantities are shownfirst to discuss the dynamics and structure of the transitional flow fields, followedby the time-averaged flow statistics of the reactive plumes.

3.1. INSTANTANEOUS FLOW STRUCTURE AND MIXING CHARACTERISTICS

Buoyancy-driven large scale vortical structures are the characteristics of both re-active and non-reactive plumes in the transitional regime, and the breakdown ofthese vortical structures leads to turbulence in the flow field [3, 4]. Figure 1 showsthe instantaneous 3D visualizations of the vorticity magnitude at t = 23.0 andt = 24.0 of the corner square reactive plume. A diagram showing the inlet planeand the Cartesian coordinates is also included. The three components of the vortic-ity are x = w/y v/z, y = u/z w/x, and z = v/x u/y,respectively. The quantity shown in Figure 1 is (2x+2y+2z)1/2. It can be observedthat large vortical structures evolve spatially in the flow field.

Figures 24 show the contour plots on the center plane x = Lx/2 of thecorner square plume at the two different times, corresponding to the 3D plots inFigure 1. In the near field, buoyant jets and plumes display a puffing or flick-


Figure 1. Instantaneous 3D visualizations of the vorticity magnitude at t = 23.0 and t = 24.0of the corner square reactive plume.

Figure 2. Fuel mass fraction contours on the x = Lx/2 plane at t = 23.0 and t = 24.0 of thecorner square reactive plume (15 contours between the minimum and maximum as indicated).


Figure 3. Reaction rate contours on the x = Lx/2 plane at t = 23.0 and t = 24.0 of thecorner square reactive plume (15 contours between the minimum and maximum as indicated).

Figure 4. Temperature contours on the x = Lx/2 plane at t = 23.0 and t = 24.0 of the cornersquare reactive plume (15 contours between the minimum and maximum as indicated).


Figure 5. Temperature contours on the z = 1.0 and z = 5.0 planes at t = 24.0 of the cornersquare reactive plume (15 contours between the minimum and maximum as indicated).

ering phenomenon [3, 4, 1518, 20, 22], which is associated with the formationand convection of buoyancy-induced large vortical structures. The flow exhibitsapproximately a periodic behavior near the plume source due to the puffing orflickering. The times t = 23.0 and t = 24.0 shown in these figures are within onepulsation period of the plume. The necking phenomenon close to the inlet planecan be seen clearly. One prominent feature in Figures 24 is that the flow field isless coherent downstream due to the breakdown of the large vortical structures,which indicates the emergence of small scale turbulence in the flow field.

For a non-premixed flame, the flow must satisfy two criteria for a significantreaction to occur: both the fuel and oxidizer must be well mixed at a given pointin the field and the temperature must be high enough at that point. The chemicalreactivity of the corner square plume is evident from Figures 2 and 3, which showthe fuel mass fraction and reaction rate T contours, respectively. The fuel is beingconsumed by the chemical reaction when the fuel/oxidizer mixing takes place. Theunburnt fuel being convected out of the computational box for this corner squarecase is less than 20%. The chemical reaction is rather weak at the downstreamlocation z = 6.0 near the outlet as shown in the reaction rate contours. Chemicalheat release also increases the local temperature in the reaction zone, as shown inFigure 4. From the temperature contours, a disorganized flow regime downstreamof the reactive plume characterized by small scales due to the breakdown of largescale structures is also evident.

Figure 5 shows temperature contours on the z = 1.0 and z = 5.0 planes att = 24.0 of the corner square plume. The complex 3D structures in the non-circular reactive plume are evident. It is noticed that the corner square reactiveplume remains symmetric about the bisector of the domain. Unlike the simula-tions of non-buoyant reactive jets performed by Grinstein and Kailasanath [10,


Figure 6. Temperature contours on the z = 1.0 and z = 5.0 planes at t = 24.0 of the freesquare reactive plume (15 contours between the minimum and maximum as indicated).

Figure 7. Temperature contours on the z = 1.0 and z = 5.0 planes at t = 24.0 of the freerectangular reactive plume (15 contours between the minimum and maximum as indicated).

11], external perturbations were not used at the inflow boundary in the simulationsperformed, in order to isolate the absolutely unstable buoyancy instability [14, 16,20, 22] from the shear instability. Therefore the flow remains symmetric due tothe geometric symmetry of the physical problem and absence of external distur-bances. Nevertheless, transition to turbulence still occurs in the flow field due tothe buoyancy and 3D vortex stretching effects.

For comparison, Figures 6 and 7 show the temperature contours of the freesquare and free rectangular cases at the two different vertical locations at t = 24.0,corresponding to those shown in Figure 5 for the corner square case. At the locationz = 1.0 which is close to the fuel source, both the free square and free rectangularreactive plumes resemble their original shapes specified on the inlet plane. At the


downstream location z = 5.0, complex structures are developed with the spatialdevelopment of the flow field, which are characteristics of the non-circular plumes.

For the comparison between the two square cases shown in Figures 5 and 6, it isnoticed that the reactive plume in a corner configuration spreads more in the lateraldirections than the free case at the downstream location. This is mainly because ofthe recirculation zones formed between the wall and plume [1]. With the presenceof impermeable and non-slip side-wall boundaries, mixing between the reactiveplume and its ambient is enhanced by the strong recirculation, therefore the cornersquare plume has larger spreading.

For the comparison between the two free cases shown in Figures 6 and 7,it is also noticed that the free rectangular reactive plume spreads more than thefree square case at the downstream location. This can be explained by the self-induced BiotSavart vortex-ring deformation associated with non-circular nozzlegeometries [9, 12], which is responsible for the complex 3D structures in the flowfield. The differences between the free rectangular and free square cases originatefrom the aspect ratio effect of the rectangular source. Gutmark and Grinstein [12]summarized the application of stability theory to non-circular jets, which iden-tified the aspect ratio effects on the vortex flow evolution. For the BiotSavartdeformation of vortex-ring dynamics, the self-induced velocity responsible for thevortex-ring deformation is proportional to the local curvature Vd C log(1/ )bin a thin, incompressible, inviscid vortex tube, where C is the local curvature ofthe tube, is the local cross-section of the vortex tube, and b is the binormal tothe plane containing the tube. The self-induced Biot-Savart deformation is strongerin the free rectangular case than that in the free square case with the same cross-sectional area, due to the differences in curvature. A stronger vortex deformationin the rectangular case therefore leads to a more vortical flow field and in turn astronger mixing and spreading at the plume downstream.

In the near field, a plume entrains its ambient fluids mainly through the largescale vortical structures. Figure 8 shows a comparison of the instantaneous cross-streamwise velocities on the x = Lx/2 plane at z = 5.0. The three different timesshown in the figure correspond approximately to one pulsation period of the freerectangular and free square cases, which is about +t = 2.0. Therefore the curvesat t = 22.0 and t = 24.0 are almost overlapping for these two cases. For the cornersquare case, the pulsation period at this downstream location differs from the twofree cases. From this figure, it is observed that the cross-streamwise velocities ofthe free rectangular and corner square cases are much higher than that of the freesquare case, which indicate stronger entrainment in these two cases.

3.2. MIXING TRANSITION AND VORTEX DYNAMICS

In the present non-circular buoyant reactive plumes, the 3D vortex deformationassociated with the non-circular geometry and side-wall effects leads to the break-down of large vortical structures into small scales. The emergence of turbulence in


Figure 8. Comparison of the instantaneous cross-streamwise velocities on the x = Lx/2plane at z = 5.0 of the free rectangular, free square, and corner square reactive plumes (t = 22.0; t = 23.0; t = 24.0).


the non-circular buoyant reactive plumes can be testified by spectral analysis of theflow field.

Figure 9 shows the comparison of the history of the centerline streamwise ve-locities at z = 5.0 of the three cases. The velocity fluctuations downstream of theplume shown in Figure 9 are associated with the spatial development of the flowfield since there is no velocity fluctuations at the inlet. At the later stage (approxi-mately after t = 12.0) when the flow is more developed, it can be observed that thedownstream centerline velocities of the free rectangular and corner square plumesfluctuate much more than the centerline velocity of the free square plume. Thedownstream centerline velocities of the free rectangular and corner square cases arealso lower than that of the free square case. This is mainly because of the strongermixing and larger spreading associated with the stronger vortex deformation inthe free rectangular case and the side-wall effects in the corner square case. Withstronger mixing, the density inhomogeneity and buoyancy in the free rectangularand corner square cases decay faster, therefore the buoyancy acceleration decaysfaster in these two cases. The important feature in Figure 9 is that the variations ofthe downstream centerline velocity of the free rectangular and corner square casesare less coherent than that of the free square case, which indicates the emergenceof small scale turbulence in the flow field.

The energy spectra of the three cases determined from the history of the center-line streamwise velocities at z = 5.0 using Fourier analysis are shown in Figure 10,which is expressed in logarithmic scales (to the base 10) of the non-dimensionalfrequency (Strouhal number St = f L0/w0) and kinetic energy. The spectralanalysis used velocity data of the developed reactive plumes after the initial stageof the simulation. It is observed that the most energetic mode for each case occursat a low frequency, which is the puffing or flickering frequency of the buoyantreactive plume associated with the convection of large vortical structures. For thefree rectangular and corner square cases, the flows are more energetic than thefree square case and high frequency harmonics are developed in the flow field. Aprominent feature in Figure 10 is the development of these high frequency harmon-ics, which is associated with the emergence of high frequency small scales in theflow field at the downstream location. The flow turbulence can be measured by theKolmogorov cascade theory (cf. [9]), which states a power law correlation betweenthe energy and frequency: E(St) St5/3. In Figure 10, the Kolmogorov powerlaw is plotted together with the energy spectra. It can be seen that the behavior ofthe free rectangular and corner square reactive plumes at the downstream locationapproximately follows the 5/3 power law. The spectra shown reflect the energyfluctuations of the high frequency harmonics in the free rectangular and cornersquare cases.

Flow transition to turbulence is the direct consequence of the breakdown oflarge scale vortical structures due to strong vortex interactions, especially the inter-actions between the streamwise vorticity and cross-streamwise vorticity [9]. Theoccurrence of flow transition to turbulence downstream of the free rectangular


Figure 9. Comparison of the history of the centerline streamwise velocities at z = 5.0 of thefree rectangular, free square, and corner square reactive plumes.


Figure 10. Comparison of the energy spectra of the centerline streamwise velocity variationsat z = 5.0 of the free rectangular, free square, and corner square reactive plumes (Kolmogorov cascade theory E(St) St5/3).


and corner square cases is due to the much stronger vortex interactions in thesetwo cases in comparison with the free square case. For the flow configurationinvestigated, there are only vortex rings very close to the plume source since thestreamwise vorticity z is zero on the inlet plane. The spatial development ofstreamwise vorticity above the inlet plane and the strong vortex interactions furtherdownstream involving all the three vorticity components are essential to the flowtransition to turbulence [9, 17].

In buoyant flows with gravitational effect, the governing equation for vorticitytransport can be written in a vector form as

D

Dt= ()V (V)+ 1

2(p)

+ 12

a

Fr(g)+

(1

). (2)

The terms on the right-hand side of Equation (2) are the vortex stretching term,dilatation term, baroclinic torque, gravitational term, and viscous term, respec-tively. Among these five terms, it is known that the dilatation term and viscousterm mainly attenuate flow vorticity [8], therefore chemical heat release reduces themixing and entrainment in non-buoyant reactive mixing layers [13, 23]. However,their importance in the vorticity transport is relatively less than the other terms forbuoyant plumes with moderately high Reynolds numbers and low Froude numbers[15, 16]. Therefore, these two terms are not discussed further in the following. Inthis vorticity transport equation, a transport term on the right-hand side promotesthe flow vortical level if this term and the vorticity are of the same sign, while theterm attenuates flow vorticity if they are of opposite signs.

The stretching term, baroclinic torque and gravitational term are the majorterms in the vorticity transport budget of a buoyancy-driven flow. Axisymmetricand planar simulations [15, 16] identified that the effect of gravitational term is topromote the flow vortical level. This term is responsible for the absolute instabilityof buoyant jets and plumes that can initiate flow vorticity. For the flow config-uration studied here, the three components of this term are a/y /(2 Fr),a /x /(

2 Fr), and 0, in x, y, and z directions, respectively. This implies thatthe gravitational term does not contribute to the streamwise vorticity directly. Inthis study, it was found that the gravitational term mainly promotes the flow vortic-ity in the cross-streamwise directions. The stretching term and baroclinic torquecan either promote or destroy the cross-streamwise vorticity depending on thelocal flow structure. These trends are consistent with the previous observationson axisymmetric buoyant reactive plumes [15, 16], but the vortex stretching termfor 3D cases studied here is very significant. The vortex stretching is completelyabsent in 2D planar simulations, while the stretching in axisymmetric simulationsis less significant.

Figure 11 shows the streamwise vorticity contours on the z = 1.0 and z = 5.0planes at t = 24.0 of the corner square reactive plume. The streamwise vorticity


Figure 11. Streamwise vorticity contours on the z = 1.0 and z = 5.0 planes at t = 24.0 of thecorner square reactive plume (15 contours between the minimum and maximum as indicated;solid line: positive; dotted line: negative).

Figure 12. Contours of the major transport terms of the streamwise vorticity on the z = 1.0plane at t = 24.0 of the corner square reactive plume (15 contours between the minimum andmaximum as indicated; solid line: positive; dotted line: negative).

shows a distribution which is symmetric but with a sign change across the sym-metry plane of the corner configuration. For the streamwise vorticity, the majortransport terms are the 3D vortex stretching and the baroclinic torque. Figure 12shows contours of the major transport terms of the streamwise vorticity shown inFigure 11a.

From Figure 12, it is observed that the vortex stretching x w/x+y w/y+z w/z in z-direction is very significant in the transport of the streamwise vor-


Figure 13. Comparison of the time-averaged centerline temperature and streamwise velocitiesof the free rectangular, free square, and corner square reactive plumes ( free rectangular; free square; corner square).

ticity. A careful examination on the vortex stretching term distribution in Fig-ure 12a shows that this term predominantly takes the same sign of the corre-sponding streamwise vorticity shown in Figure 11a. This indicates that the vortexstretching term is a major source for the streamwise vorticity. For the baroclinictorque shown in Figure 12b, it can either promote or destroy streamwise vorticity,which is of the same nature as its contribution to the cross-streamwise vorticity. It isalso noticed that this term is less significant than the stretching term in the transportbudget. Analysis on streamwise vorticity transport at different times and differentlocations and for other cases follow the same trend. Therefore it can be concludedthat the streamwise vorticity in the non-circular buoyant reactive plumes is mainlycaused by the 3D vortex stretching. The development of streamwise vorticity due tovortex stretching leads to the coexistence of vortex rings and spirals in the reactiveplume, and consequently, strong vortex interactions occur downstream which areresponsible for the flow transition.

3.3. FLOW STATISTICS

Flow statistics were obtained by averaging the flow quantities over six pulsationperiods after the initial transients had been convected out of the computationaldomain. Figure 13 shows the comparison of the time-averaged centerline temper-ature and streamwise velocities of the free rectangular, free square, and cornersquare reactive plumes, while Figure 14 shows the comparison of the entrainmentproperties of these three cases. It is worth noting that the bumps in the profilesin Figures 13 and 14 are mainly because of the existence of buoyancy-induced


Figure 14. Comparison of the entrainment properties of the free rectangular, free square, andcorner square reactive plumes ( free rectangular; free square; cornersquare).

large scale vortical structures in the near field. Although the large scale vortices arebeing convected by the mean flow, the time-averaged quantities still indicate theirexistence due to the spatial nature of the simulation.

In the near field of a buoyant reactive plume, the temperature depends on thereaction and mixing. Combustion heat release increases the local temperature. Inthe meantime, large spreading and strong mixing with the ambient reduces thelocal temperature through the convective heat transfer. For the time-averaged cen-terline temperature shown in Figure 13a, it maintains the initial temperature of thefuel near the inlet plane (z < 1.0 in the streamwise direction) since no reactiontakes place due to the unmixedness of the mixture. The temperature then increasesdue to the combustion heat release. Downstream, the time-averaged centerlinetemperature behaves differently for the three cases due to the different mixingcharacteristics. For the free rectangular and corner square cases, the temperaturedecreases due to the large spreading and the fuel consumption. For the free squarecase, however, the downstream temperature still increases due to the small spread-ing and the combustion heat release. It is worth noting that the plume centerlinedoes not correspond to the intense reaction zone, therefore the temperature shownis lower than the highest temperature within the non-premixed flame.

For the reactive plume, the mean streamwise velocity strongly depends on thebuoyancy effects. The time-averaged centerline streamwise velocity increases atlocations close to the plume source due to the buoyancy acceleration. The buoyancyacceleration decays downstream because of the plume spreading and mixing withthe ambient. The free rectangular and corner square cases have larger spreading


than the free square case as shown before, therefore the downstream streamwisevelocities in these two cases decay faster.

Figure 14a shows the comparison of the streamwise mass fluxesLy/2

Ly/2

Lx0

w dx dy

of the free rectangular, free square, and corner square reactive plumes, while Fig-ure 14b shows the comparison of the entrainment rates which are the gradientsof the streamwise mass fluxes in the vertical direction of the three cases. Com-pared with the entrainment rates around 0.3 of square non-buoyant jets reported byGrinstein and Kailasanath [10], the entrainment rate of the square buoyant plumestudied here is about three times of this value due to the strong buoyancy effects.This trend is consistent with the comparison between the near field entrainmentproperties of circular buoyant plumes and non-buoyant jets [7]. It is worth notingthat there is no comparable experimental data available for the non-circular buoyantreactive plumes in the near field, and thus, a direct comparison with experimentsis not feasible. However, the mean properties qualitatively follow the trend ofbuoyancy enhanced entrainment in the near field [7].

In Figure 14, the higher entrainment rate of the free rectangular case comparedwith the free square case is due to the aspect ratio effects [12]. Although thecorner square reactive plume is half-confined in the cross-streamwise directions,the entrainment of the corner square case is significantly higher than that of thefree square case. This can be explained by the side-wall enhanced mixing in thecorner case. From a macro-scale mixing point of view, the corner square case hasmore length scales compared to the one obvious length scale (the fuel jet width) inthe free square case. Also, there is only one symmetry plane in the corner squarecase (the bisector of the domain) compared to the many in the free square case.For the corner case, the plume spreading is greatly enhanced by the recirculationzones formed between the wall and plume as discussed in Section 3.1. This leads toenhanced mixing at large scales in the corner square case. For the mixing at smallscales, the corner square case is much more turbulent than its free counterpart asdiscussed in Section 3.2, which leads to enhanced small scale mixing in the cornercase. The side-wall enhanced mixing is also evident in the sample instantaneousmixing characteristics shown in Figure 8.

4. Conclusions

In this paper, spatial direct numerical simulations of non-circular reactive plumeswith buoyancy effects have been performed for three cases. A one-step chemicalreaction with Arrhenius kinetics is used. The simulations are devoted to a betterunderstanding of the effects of plume source geometry and side-wall confinementon the flow structure and dynamics of reactive plumes. Particular attention has been


given to the effects on the flow transition to turbulence and mixing and entrainmentin non-circular reactive plumes.

Complex 3D structures are observed for the buoyant reactive plumes establishedon non-circular sources. Large scale vortical structures develop naturally in theflow field due to the gravitational effect. Analysis on the vorticity transport showsthat 3D vortex stretching leads to the development of streamwise vorticity in theflow field. Vortex interactions and the consequent breakdown lead to the emergenceof turbulence in the buoyant reactive plumes. A turbulent inertial subrange has beenobserved downstream of the free rectangular and corner square reactive plumes.

The geometry and side-wall effects on the mixing and entrainment of non-circular reactive plumes are significant. Compared with the square geometry, therectangular geometry with an aspect ratio of 2:1 promotes the plume entrainmentof the ambient fluids due to the aspect ratio effects. By creating recirculation zonesbetween the wall and plume and promoting transition to turbulence, the side-wallsin the corner square case enhance the mixing and entrainment of the reactive plumeat both large and small scales.

Acknowledgment

This work was funded by the UK Engineering and Physical Sciences ResearchCouncil under Grant No. GR/L67271.

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mixing and entrainment of transitional non circular plumes

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