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Available online at www.sciencedirect.com Procedia IUTAM 00 (2014) 000–000 www.elsevier.com/locate/procedia IUTAM ABCM Symposium on Laminar Turbulent Transition Global linear stability of axisymmetric coaxial jets Jacopo Canton a , Franco Auteri a,* , Marco Carini a a Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34, 20156 Milano, Italy Abstract In this work, the first instability of the flow produced by two coaxial jets is investigated as a first step to start to shed light on the related laminar-turbulent transition process. The impact of the ratio between the maximum velocities in the two jets on the stability properties of the flow is studied by the tools of linear stability analysis. For a unitary velocity ratio, an oscillatory mode is found responsible for the onset of a von K´ arm´ an vortex ring street originating in the wake of the duct wall separating the two streams. The corresponding direct and adjoint modes are described. When the velocity ratio is increased, the vortex dynamics is more and more similar to the one produced by a Kelvin–Helmholtz instability and some changes in the computed eigenspectrum are observed. c 2014 The Authors. Published by Elsevier B.V. Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering). Keywords: Coaxial jets; NavierStokes equations; Modal stability; Hopf bifurcation. 1. Introduction Coaxial jet flows are often used in industrial applications as an eective way of mixing two dierent fluid streams: important examples are found in the fuel-air mixing process which takes place inside jet engines 1,2 , as well as in the gas assisted spray formation. Furthermore, much aeronautical interest lies in the noise reduction characteristics achievable by coaxial jets 3 . The stability properties of this class of flows have been investigated experimentally by several authors 4,5 . Based on the results of experimental measurements and theoretical investigations of the inviscid limit 6,7 , it has been argued that incompressible, coaxial jets may present an absolutely unstable region immediately downstream of the wall which separates the two incoming flows, depending on the outer to inner velocity ratio r u . In particular, for a wall of relevant thickness, s D i /10 (where D i is the diameter of the inner pipe), and for r u 1 the flow is dominated by the instability of the wake of the separation wall, which displays a von K´ arm´ an vortex street pattern for this value of the parameters. Changing the value of the control parameters the flow behaviour changes, and in particular by increasing the velocity ratio, the vortex dynamics resembles more and more the one produced by a Kelvin–Helmholtz instability. Self-sustained oscillations characterizing the coaxial jets are investigated here by a comprehensive linear stability analysis. A non-swirling and axisymmetric flow is considered, the axisymmetry assumption being justified at least in the near-wake region by experimental observations 4,5 . In our computations the same geometry used in 8 has been * Corresponding author. Tel.: +39 02-2399-8046 ; fax: +0-000-000-0000. E-mail address: [email protected] 2210-9838 c 2014 The Authors. Published by Elsevier B.V. Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering).

IUTAM ABCM Symposium on Laminar Turbulent Transition Global …€¦ · Jacopo Cantona, Franco Auteria,, Marco Carinia aDipartimento di Scienze e Tecnologie Aerospaziali, Politecnico

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Page 1: IUTAM ABCM Symposium on Laminar Turbulent Transition Global …€¦ · Jacopo Cantona, Franco Auteria,, Marco Carinia aDipartimento di Scienze e Tecnologie Aerospaziali, Politecnico

Available online at www.sciencedirect.com

Procedia IUTAM 00 (2014) 000–000www.elsevier.com/locate/procedia

IUTAM ABCM Symposium on Laminar Turbulent Transition

Global linear stability of axisymmetric coaxial jetsJacopo Cantona, Franco Auteria,∗, Marco Carinia

aDipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34, 20156 Milano, Italy

Abstract

In this work, the first instability of the flow produced by two coaxial jets is investigated as a first step to start to shed light on therelated laminar-turbulent transition process. The impact of the ratio between the maximum velocities in the two jets on the stabilityproperties of the flow is studied by the tools of linear stability analysis. For a unitary velocity ratio, an oscillatory mode is foundresponsible for the onset of a von Karman vortex ring street originating in the wake of the duct wall separating the two streams. Thecorresponding direct and adjoint modes are described. When the velocity ratio is increased, the vortex dynamics is more and moresimilar to the one produced by a Kelvin–Helmholtz instability and some changes in the computed eigenspectrum are observed.c© 2014 The Authors. Published by Elsevier B.V.Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering).

Keywords:Coaxial jets; NavierStokes equations; Modal stability; Hopf bifurcation.

1. Introduction

Coaxial jet flows are often used in industrial applications as an effective way of mixing two different fluid streams:important examples are found in the fuel-air mixing process which takes place inside jet engines1,2, as well as inthe gas assisted spray formation. Furthermore, much aeronautical interest lies in the noise reduction characteristicsachievable by coaxial jets3.

The stability properties of this class of flows have been investigated experimentally by several authors4,5. Based onthe results of experimental measurements and theoretical investigations of the inviscid limit6,7, it has been argued thatincompressible, coaxial jets may present an absolutely unstable region immediately downstream of the wall whichseparates the two incoming flows, depending on the outer to inner velocity ratio ru. In particular, for a wall of relevantthickness, s ≈ Di/10 (where Di is the diameter of the inner pipe), and for ru ≈ 1 the flow is dominated by theinstability of the wake of the separation wall, which displays a von Karman vortex street pattern for this value of theparameters. Changing the value of the control parameters the flow behaviour changes, and in particular by increasingthe velocity ratio, the vortex dynamics resembles more and more the one produced by a Kelvin–Helmholtz instability.

Self-sustained oscillations characterizing the coaxial jets are investigated here by a comprehensive linear stabilityanalysis. A non-swirling and axisymmetric flow is considered, the axisymmetry assumption being justified at leastin the near-wake region by experimental observations4,5. In our computations the same geometry used in8 has been

∗ Corresponding author. Tel.: +39 02-2399-8046 ; fax: +0-000-000-0000.E-mail address: [email protected]

2210-9838 c© 2014 The Authors. Published by Elsevier B.V.Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering).

Page 2: IUTAM ABCM Symposium on Laminar Turbulent Transition Global …€¦ · Jacopo Cantona, Franco Auteria,, Marco Carinia aDipartimento di Scienze e Tecnologie Aerospaziali, Politecnico

2 J. Canton et al. / Procedia IUTAM 00 (2014) 000–000

employed, with the introduction of a wall perpendicular to the incoming flow. A relevant portion of the pipes hasbeen modelled, since their exclusion from the computational domain may lead to the incorrect computation of theeigenmodes, in particular of the adjoint eigenmodes. The shape of the inlet velocity profile is determined as a functionof the thickness of the boundary layer inside the pipes and has been chosen, following9, according to the velocityprofiles measured in the experiments reported in5,8. On the borders of the pipes and on the wall a no-slip boundarycondition has been assigned while a quasi stress-free boundary condition has been enforced on the outflow and radialborders.

2. Modal analysis

The computed eigenspectrum for Rei = 1420 and ru = 1 is reported in Figure 1a. A pair of marginally stablecomplex-conjugate eigenvalues is found, with imaginary part ωc = 5.73. In addition, the eigenspectrum presentsthree main groups of eigenvalues, labelled with bi in the figure. A first branch, starting at the origin and indicated byb1, corresponds to vortical structures located in the region surrounding the jets. Eigenvalues of this kind are typicalof nearly-parallel flows10, they usually represent a discrete approximation of a continuous branch and are stable forany value of the parameters describing the flow field. A second branch (b2) represents modes localized within thejets. This family of eigenvalues is highly sensitive to domain truncation effects. Branches b1 and b2 are also presentin the spectrum of a single jet and their eigenvalues are always fully resolved9. The third branch (b3) contains poorlyconverged eigenvalues which change with the shift employed in the eigenvalue computation. The direct eigenmodeassociated with the marginally stable eigenvalue is depicted in Figure 2a. This mode displays an array of counter-rotating vortex rings which develop in the wake of the separation wall. The corresponding adjoint mode, whichallows to identify the region of maximum global mode receptivity, is shown in Figure 2c: values of highest sensitivityare attained near the separating wall and at the end of the inlet pipes. For the considered values of the parameters,the transition to a limit cycle through a Hopf bifurcation has been confirmed by DNS. Good agreement is obtainedbetween the global mode frequency and the shedding frequency extracted from the DNS (ωDNS = 5.84 Hz).

When the velocity ratio is increased, the vortex dynamics observed in DNSs resembles more and more the oneproduced by a Kelvin–Helmholtz instability mechanism. For ru = 1.5 the complex-conjugate eigenpair crosses theimaginary axis for Rei = 1386 at ωc = 7.57. Corresponding direct and adjoint eigenmodes are depicted in Figures 2band 2d, respectively. The computed eigenspectrum is reported in Figure 1b. It can be noticed that for this value of ru,branch b2 has moved towards the imaginary axis.

−0.6 −0.4 −0.2 00

2

4

6

8

10

b1

b2

b3

σ

ω

(a) Rei = 1420, ru = 1.

−0.6 −0.4 −0.2 00

2

4

6

8

10

σ

ω

(b) Rei = 1386, ru = 1.5.

Fig. 1: Eigenspectra. A marginally stable eigenvalue (highlighted by the dotted lines) is found for ωc = 5.73 (a), ωc = 7.57 (b).

Page 3: IUTAM ABCM Symposium on Laminar Turbulent Transition Global …€¦ · Jacopo Cantona, Franco Auteria,, Marco Carinia aDipartimento di Scienze e Tecnologie Aerospaziali, Politecnico

J. Canton et al. / Procedia IUTAM 00 (2014) 000–000 3

r

z

0 10 20

0

1

−0.79

0

0.79

(a) Axial velocity component of the direct critical eigenvector for Rei = 1420, ru = 1.

r

z

0 10 20

0

1

−0.34

0

0.34

(b) Axial velocity component of the direct critical eigenvector for Rei = 1386, ru = 1.5.

r

z

−4 −3 −2 −1 0

0

1

0

2.9

5.8

(c) Velocity magnitude of the adjoint critical eigenvectorfor Rei = 1420, ru = 1.

r

z

−4 −3 −2 −1 0

0

1

0

4.08

8.15

(d) Velocity magnitude of the adjoint critical eigenvectorfor Rei = 1386, ru = 1.5.

Fig. 2: Real part of the direct and adjoint eigenvectors associated with the marginally stable eigenvalue. Rei = 1420, ru = 1 in (a) and (c),Rei = 1386, ru = 1.5 in (b) and (d).

References

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2003;482:257–269.3. Williams, T.J., Ali, M.R.M.H., Anderson, J.S.. Noise and flow characteristics of coaxial jets. J Mech Eng Sci 1969;11:133–142.4. Dahm, W.J.A., Clifford, E.F., Tryggvanson, G.. Vortex structure and dynamics in the near field of a coaxial jet. J Fluid Mech 1992;

241:371–402.5. Rehab, H., Villermaux, E., Hopfinger, E.J.. Flow regimes of large-velocity-ratio coaxial jets. J Fluid Mech 1997;345:357–381.6. Michalke, A.. On the influence of a wake on the inviscid instability of a circular jet with external flow. Eur J Mech 1993;12:579–595.7. Talamelli, A., Gavarini, I.. Linear stability characteristics of incompressible coaxial jets. Flow Turb Combust 2006;76:221–240.8. Segalini, A.. Experimental analysis of coaxial jets: instability, flow and mixing characterization. Ph.D. thesis; Universita di Bologna; 2010.9. Garnaud, X., Lesshafft, L., Schmid, P.J., Huerre, P.. Modal and transient dynamics of jet flows. Phys Fluids 2013;25.

10. Schmid, P.J., Henningson, D.S.. Stability and Transition in Shear Flows. Springer; 2001.