Scalar mixing and large–scale coherent structures in a turbulent swirling jet
J. Fr¨ohlich , M. Garc´ıa-Villalba , W. Rodi
Institute for Technical Chemistry and Polymer Chemistry, University of Karlsruhe, 76128 Karlsruhe, Germany. Institute for Hydromechanics, University of Karlsruhe, 76128 Karlsruhe, Germany corresponding author:
[email protected]
ABSTRACT The paper presents large eddy simulations of co-annular swirling jets into an open domain. In each of the annuli a passive scalar is introduced and its transport is computed. If the exit of the pilot jet is retracted strong coherent flow structures are generated which substantially impact on the transport and mixing of the scalars. Average and instantaneous fields are discussed to address this issue. A conditional averaging technique is devised and applied to velocity and scalars. This allows to quantify the impact of the coherent structures on the mixing process.
1
I NTRODUCTION
Swirling annular and co–annular jets are widely used in combustion devices such as gas turbine burners to stabilize the flame by means of a swirl–induced recirculation zone [16]. Previous experimental and numerical studies have demonstrated, however, that such flows are often prone to fluid mechanical instabilities generating large–scale coherent vortical structures [16,3,22]. These have a substantial impact on the mixing of scalar quantities such as fuel and oxidizer or hot and cold gas. Hence, they can substantially influence the combustion process and can trigger pronounced unsteadiness as experienced in [20]. Modern gas turbines frequently run in the LPP mode, i.e. in lean premixed conditions, which renders them more sensitive to combustion instabilities than with richer fuels. Syred [27] recently provided an exhaustive review on the role of coherent structures in non-reacting and reacting swirling flows relevant to industrial combustion so that here only some pertinent individual work on coherent structures in swirling jets is recalled. Froud et al. [8] applied phase averaging to experimental data and demonstrated the existence of a precessing vortex core (PVC) in a round jet with swirl number 1.5 emanating from a circular pipe into a confined environment. More recently, Cala et al. [2] provided phase-averaged experimental data for an unconfined round swirling jet issuing from a circular pipe at a Reynolds number of
and swirl number . Under these conditions, the authors identified two further types of vortices, additional to the PVC, an inner and an outer co-spiralling vortex. A low-swirl jet with around 0.22 was numerically investigated in [15] and [18]. In this regime, no recirculation zone exists, but the entrainment of the jet is larger than
in the non-swirling case. It could be shown that this is due to the generation of axial braids resulting from the addition of swirl. Large eddy simulations (LES) of swirling jets have been performed first by Pierce and Moin [23] for a confined situation. Since then, LES of this type of flow has been conducted by several other groups such as [30,28,21,13]. Freitag and Klein [5] presented DNS for an unconfined annular swirling jet with and identified a PVC in the center of the flow. While lately also reactive premixed flows have been computed with LES ([24] and references therein, [17]), papers investigating the scalar transport in cold swirling jets are scarce [29,6]. Mostly, statistical data concerning mean fields and rms-fluctuations are provided. The present paper focusses on the pure mixing process. We investigate the role of the coherent structures present in swirl flows on the large-scale mixing by means of LES for the non-reactive constantdensity flow. To this end, a case with pronounced structures was selected, previously computed in [14,11]. The entire simulation model, consisting of numerical discretization scheme, choice of grid, subgrid–scale modelling and boundary conditions, has been validated in previous work [13,9]. In these studies the generation of coherent structures was investigated together with their dependence on various geometrical and flow conditions. The strongest structures were observed in simulations with the pilot jet retracted into a surrounding tube, so that this case is considered here. In the case without retraction which is taken for comparison, the coherent structures are destroyed by a pilot jet [10]. Both cases are illustrated by snapshots of vortical structures in Fig.1. In the case with retraction a system of two dominant spiralling vortices develops, an inner one with a small angle to the axis and an outer one at a larger angle. In the absence of the pilot jet, this structure is also observed in the non-retracted case [10,13].
2
C ONFIGURATION
The geometry, shown in Fig. 2, features two annular jets exiting into still ambient. This setting matches an experiment conducted at the University of Karlsruhe [1]. The outer main jet accounts for 90% and the inner jet for 10% of the mass flux. The Reynolds number is , based on the bulk velocity of the main jet and its outer radius . The total swirl number is , determined at the jet exit. The swirl number of the pilot jet alone is 2 at (recall its small flow rate, however). Inflow conditions for the main stream are imposed substantially upstream of the outlet, where steady flow with an appropriate angular component is prescribed. In the entrance duct turbulence develops which yields the appropriate flow in the outer annular pipe as validated in [12,14]. The swirl in the inner annular pipe is generated by an axial swirler in the experiment. In the present computations this is accomplished by a precursor computation of flow in a periodic annular pipe with body forces instantaneously adjusted so as to provide the desired mass flux and swirl. Two simulations have been performed with the inlet geometries depicted in Fig. 2. One with the pilot jet and the center body ending flush with the outlet of the main jet, the other with a retraction of the pilot jet to (also see Fig. 1). Apart from this difference, all conditions are identical for both cases. The origin of the coordinate in streamwise direction is located at the end of the outer tube. The dynamic Smagorinsky model was used for subgrid-scale modelling in the momentum equation. This model was found appropriate for jet flows in [26]. 2
To investigate the mixing processes, two scalars were introduced, in the inner and in the outer jet, both with Schmidt number equal to one. An additional transport equation is solved for each of them using the bounded HLPA scheme [31] for the convection term in this equation. The use of such schemes is supported by the investigations in [4]. An eddy diffusivity model was employed to account for the subgrid-scale contribution with a turbulent Schmidt number of [7]. No measured data are available for the transport of scalars in the corresponding experiment.
3 3.1
I NSTANTANEOUS
AND STATISTICAL RESULTS
Average flow field
Fig. 3 shows two-dimensional streamlines in the centerplane ( plane) for both cases. Details of the flow field are discussed in [11]. In this reference, the results for the flow field were also validated against corresponding experimental data and good agreement was found. With retraction, the average recirculation zone does not reach upstream to the exit of the jets and is shorter and broader. Also, the spreading angle of the jet is larger than in the reference case. In the case with retraction, the pilot jet is not able to destroy the coherent vortices. Rather, these are enhanced by the surrounding pipe after the first expansion at . 3.2
Instantaneous data for the transported scalar
The coherent structures visualized in Fig. 1 substantially impact on the scalar mixing as shown in Fig. 4. Without retraction, the instantaneous scalar field is more symmetric in the centerplane. This results from a relatively equal distribution around the circumference shown in the lower left plot of the figure, taken close to the outlet at . With retraction, large spots of high concentration appear in plots at
reflected by asymmetry in the centerplane plot. The spots are generated by the coherent structures visible in the right plot of Fig. 1. These spots are correlated with a local excess in axial velocity which brings along undiluted scalar from the jet exit as shown in Fig. 5. Additional to the large-scale features pointed out, small-scale pockets of faster fluid are generated around the outer circumference of the jet due to shear layer instabilities (Fig. 5). They have a spatial period of about and lead to corresponding pockets of fluid with higher concentration (Fig. 4, bottom) which are also visible in the centerplane plots (Fig. 4, top). Without retraction similar structures are visible along the inner circumference of the annular jet as well (Fig. 4, bottom left). 3.3
Statistical data for scalar field and correlations
The mean concentration of is reported in Fig. 3 for both cases. Fig. 6a shows corresponding profiles in radial direction. The flow field close to the exit differs in that the recirculation is stronger with retraction and the spreading angle near is larger than in the reference case (Fig. 3). For these reasons, the concentration is distributed spatially more uniformly (Fig. 6a). Also around , the concentration differs appreciably. Both figures mentioned show that the concentration on the axis and the maximum concentration at a given value of are smaller with retraction. Hence, spatial mixing is stronger in this case. A closer quantitative comparison of the 3
spatial mixing between both cases is somewhat delicate since one could argue that in the retracted case mixing actually starts at and that this length effectively needs to be added. This, however, disregards the different shape of the contours in Fig. 3. Temporal mixedness at a given point is characterized by the rms-value of the scalar considered [25] (normalization by the local mean value of the scalar is not appropriate for jets as the latter drops to zero at the edge of the jet). With retraction, Fig. 6b shows substantially larger values for this quantity around . Further downstream the level is similar for both cases but shifted towards larger . The global temporal mixing defficiency is determined by the spatial integral over the fluctuations [25]. The integral is larger with retraction due to the factor occuring with integration. The bottom row of Fig. 6 reports the turbulent fluxes of tracer in the three coordinate directions. The better spatial mixing with retraction is reflected by the larger values of at . Further downstream, the levels are similar. Visual inspection of Fig. 6a and 6d shows that no pronounced region of radial counter-gradient transport occurs. The axial turbulent flux is similar for both cases except around where a pronounced maximum and negative minimum is observed instead of a single bump. Also, substantial values occur near the axis. Tangential correlations are similar in both cases with somewhat larger values with retraction. Note that in a non-swirling jet this quantity vanishes for symmetry reasons. To sum up, the most noticable differences between the two cases occur around , i.e. where the strong coherent structure exist (Fig. 1b). 3.4
Discussion of PDFs
The mixing of the main jet with the ambient fluid has been investigated by means of the probability density function (PDF) of both scalars at various points. The one of the main jet is more important, so that only these PDFs will be shown here. Le Ribault et al. [26] determined PDFs in a plane jet at from LES and compared them with the data from a corresponding DNS. They found that the diffusion introduced by the subgrid-scale model modifies the PDF of the resolved scalar field in that peaks related to unmixed fluid are broadened and reduced in hight. In general, the qualitative match between PDFs from DNS and LES was good. Fig. 7 shows PDFs of the resolved field obtained from the present computations at different positions in the flow. The middle left plot was generated for a point in the inner structure. Without retraction, the concentration is fairly uniform near the center of the jet as reflected by the narrow shape of the PDF. With retraction, the coherent structures transport almost unmixed scalar during some instants while at other instants more diluted flow is present leading to a bimodal shape of the PDF close to the outlet. Similar observations are made at the outer border of the jet in the upper plots of Fig. 7. The points of investigation are the same for PDFs from both simulations but the respective spreading angle of the jet is different so that the jet is somewhat more remote from the point in the case without retraction. Further downstream, i.e. for , the shape of the PDF from both cases is similar, but mean and variance still differ. In [26] the radial variation of the PDF is discussed in terms of the shape being marching, nonmarching or tilted [19]. Marching PDFs with constant shape but shifted position are related to small-scale mixing. Variable shape (nonmarching PDF) and peaks close to the border of the scalar’s range (tilted PDF) result from large-scale coherent structures transporting unmixed fluid. In the present case sub4
stantial differences in the shape of the PDFs are observed near the outlet due to the presence or absence of coherent structures. PDFs at several further points are reported (not all being reproduced here) and show the following features: At the PDF is marching for and tilted for in the reference case, while it is tilted all over with retraction. At the PDF from the reference case is marching for and tilted for larger radii. With retraction, the PDF is marching for and tilted for larger radii. At the point of change between one and the other behaviour is located around and at for the nonretracted and the retracted geometry, respectively. Furthermore, the PDF is substantially narrower in the latter case.
4
4.1
C ONDITIONAL
AVERAGING
Method
The coherent structures observed in Fig. 1 rotate around the axis at a relatively constant rate. They are however not totally stable due to the high Reynolds number and can, for example, disintegrate into several structures or change their shape to a certain extent. It has been shown that the inner structures trigger the outer ones [10] so that the following conditional averaging strategy was devised [11]. In over 180 stored flow fields the position of the strongest inner structure was detected and its center assigned the zero angular coordinate. The resulting rotated fields of velocity and scalars were ensemble averaged and are identified by a tilde in the following. Note that this type of averaging is different from phase averaging as illustrated by the fact that the result is unchanged if a structure rotates at arbitrary non-constant speed. True phase averaging would amount to distinguishing different phases and to averaging over each of them. This can be done, e.g. fixing the frequency from the observed spectrum or, to avoid problems of synchronization, with a trigger signal from the flow itself. The amount of samples in each phase, however, would be small for the same duration of the simulation or, vice versa, a substantially longer and hence more expensive simulation would be needed. Furthermore, all angular positions are statistically equivalent so that the definition of different phases is physically not needed. This is why the above procedure is preferred. It is applied here for the case with retraction only since in the reference case without retraction the structures observed are very weak. 4.2
Average coherent structures
Fig. 8 provides the conditionally averaged vortex structures by means of a pressure-perturbation iso-surface, similarly to Fig. 1. The averaging generates a sort of prototype structure which is very smooth and can hence be analyzed conveniently. The vortex structures in the upper plot, rotating at constant rate around the axis [11] may constitute an idealized model for the coherent structures in this flow. The bottom figures show iso-surfaces of (left) and (right). They demonstrate how the coherent vortices influence the concentration of the scalar quantities. The maximum of is found in the interiour of the inner structure while has small values there. Due to its position of intrusion, essentially depends on the inner structures. The mixing of with the ambient fluid, on the other hand, is determined by the outer structures. The distortion of the iso-surface of its conditionally averaged concentration by the outer structure is very well visible.
5
In a further step it was investigated to which extent the coherent structures contribute to the fluctuations scalar. This is demonstrated in Fig. 9. The left graph shows the total fluctuations of the
. Fluctuations generated by highly ordered regular structures are subsequently deter mined as . The right plot in Fig. 9 shows the ratio of these two quantities. In almost the entire region with , more than 40% of the fluctuations result from organized motion. The total fluctuations themselves are not very strong, though, as visible in the left plot of the figure. Another maximum of the ratio is observed in the outer shear layer with 30-40% around and due to the outer structures. Here, also the fluctuations attain large values. 5
C ONCLUSIONS
The analysis of instantaneous and statistical data demonstrates the strong impact of the coherent structures generated by retraction of the pilot jet on the mixing process. In a reacting flow this would alter considerably the entire combustion process. The technique devised for conditional averaging yields smooth and reliable data which characterize the regular part of large-scale features in the velocity and scalar field. Note that this procedure can also be applied to non-swirling flows of round jets since the same arguments given in the text hold for such configurations as well. Acknowledgments: This work was supported by the German Research Foundation through SFB 606 (www.sfb606.uni-karlsruhe.de). Calculation time on an HP XC6000 Cluster was kindly provided by the Computer Center of the University of Karlsruhe.
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[9] M. Garc´ıa-Villalba. Large eddy simulation of turbulent swirling jets. PhD thesis, University of Karlsruhe, 2006. http://www.uvka.de/univerlag/volltexte/2006/113/pdf/Garcia-Villalba Manuel.pdf. [10] M. Garc´ıa-Villalba and J. Fr¨ohlich. LES of a free annular swirling jet – dependence of coherent structures on a pilot jet and the level of swirl. Int. J. Heat Fluid Flow, in press. [11] M. Garc´ıa-Villalba, J. Fr¨ohlich, and W. Rodi. Numerical simulations of isothermal flow in a swirl burner. Paper GT2006-90764 at ASME Turbo Expo 2006, May 8–11, Barcelona, Spain, to appear in Journal of Engineering for Gas Turbines and Power. [12] M. Garc´ıa-Villalba, J. Fr¨ohlich, and W. Rodi. On inflow boundary conditions for large eddy simulation of turbulent swirling jets. In Proc. 21st Int. Congress of Theoretical and Applied Mechanics. Warsaw. Poland, 2004. [13] M. Garc´ıa-Villalba, J. Fr¨ohlich, and W. Rodi. Identification and analysis of coherent structures in the near field of a turbulent unconfined annular swirling jet using large eddy simulation. Phys. Fluids, 18:055103, 2006. [14] M. Garc´ıa-Villalba, J. Fr¨ohlich, W. Rodi, O. Petsch, and H. B¨uchner. Large eddy simulation of flow instabilities in co–annular swirling jets. In E. Lamballais, B.J. Geurts, O. M´etais, and R. Friedrich, editors, Direct and Large-Eddy Simulation VI. Kluwer Academic, 2006. [15] B. Guo, T.A.G. Langrish, and D.F. Fletcher. Simulation of turbulent swirl flow in an axisymmetric sudden expansion. AIAA J., 39:96–102, 2001. [16] A.K. Gupta, D.G. Lilley, and N. Syred. Swirl Flows. Abacus Press, 1984. [17] Y. Huang, S.W. Wang, and V. Yang. Systematic analysis of combustion dynamics in a lean-premixed swirl-stabilized combustor. AIAA J., 44:724–740, 2006. [18] S. Mc Illwain and A. Pollard. Large eddy simulation of the effects of mild swirl on the near field of a round free jet. Phys. Fluids, 14:653–661, 2002. [19] P.S. Karasso and M.G. Mungal. Scalar mixing and reaction in plane liquid shear layers. J. Fluid Mech., 323:23–63, 1996. [20] C. K¨ulsheimer and H. B¨uchner. Combustion dynamics of turbulent swirling flames. Combustion and Flame, 131:70–84, 2002. [21] X.Y. Lu, S.W. Wang, H.G. Song, S.Y. Hsieh, and V. Yang. Large-eddy simulations of turbulent swirling flows injected into a dump chamber. J. Fluid Mech., 527:171–195, 2005. [22] O. Lucca-Negro and T. O’Doherty. Vortex breakdown: a review. Prog. Energy and Comb. Sci., 27:431– 481, 2001. [23] C.D. Pierce and P. Moin. Large eddy simulation of a confined coaxial jet with swirl and heat release. AIAA-Paper 98-2892, 1998. [24] T. Poinsot and D. Veynante. Theoretical and Numerical Combustion. R.T. Edwards, 2005. [25] C. Pri`ere, L.Y.M. Gicquell, P. Kaufmann, W. Krebs, and T. Poinsot. Large eddy simulation predictions of mixing enhancement for jets in cross flows. J. Turbulence, 5:5, 2005. [26] C. Le Ribault, S. Sarkar, and S.A. Stanley. Large eddy simulation of evolution of a passive scalar in plane jet. AIAA J., 39:1505–1516, 2001. [27] N. Syred. A review of oscillation mechanisms and the role of the precessing vortex core (PVC) in swirl combustion systems. Progr. Energ. Combust. Sci., 32:93–161, 2006.
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[28] P. Wang, X.S. Bai, M. Wessman, and J. Klingmann. Large eddy simulation and experimental studies of a confined turbulent swirling flow. Phys. Fluids, 16:3306–3324, 2004. [29] B. Wegner, B. Janus, A. Sadiki, A. Dreizler, and J. Janicka. Study of flow and mixing in a generic GT combustor using LES. In W. Rodi and M. Mulas, editors, Engineering Turbulence Modelling and Experiments, volume 6, pages 731–740. Elsevier, 2005. [30] B. Wegner, A. Kempf, C. Schneider, A. Sadiki, A. Dreizler, M. Sch¨afer, and J. Janicka. Large eddy simulation of combustion processes under gas turbine conditions. Prog. Comp. Fluid Dyn, 4:257–263, 2004. [31] J. Zhu. Low diffusive and oscillation–free convection scheme. Comm. Appl. Num. Meth., 7:225–232, 1991.
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Fig. 1. Visualization of coherent vortex structures by means of an iso-surface of instantaneous pressure perturbation, in both cases. Left: no retraction, right: with retraction. Light colour: structures in the axis, darker colour: structures in the outer shear layer, inner shear layer exhibiting a small angle with the exhibiting a larger angle with the axis.
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in the centerplane. Left: no retraction, right:
Fig. 4. Snapshots of instantaneous concentration. Left: reference case without retraction, right: case with re
. The thin white traction. The upper picture shows the centerplane, the lower one a cut at circle is located at . The centerplane cut is oriented horizontally in the bottom figure. The lower irght in order to identify the position of the dominant vortex. figure contains iso-lines of
Fig. 5. Instantaneous streamwise velocity in the plane data set shown in Fig. 4, right. Left: , right: .
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for the case with retraction resulting from the
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Fig. 6. Radial profiles of mean scalar and rms-fluctuations (top) together with the turbulent fluxes of this scalar, , , in the bottom row (left to right). Note the different range for . Data in each graph are shown for , , and , respectively (bottom to top). – – – without , retraction, ———- with retraction.
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Fig. 8. Conditional averages of flow field and scalars for the retracted case. Top: iso-surface showing the inner and the outer vortex structure. The former extends behind the outer one and is connected to it. Bottom left: iso-surface which has been rotated by 180 degrees to visualize its deformation. Bottom right: iso-surface of .
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