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Modal Instability Analysis of Light Jets
Available online at www.sciencedirect.com
ScienceDirect Procedia IUTAM 14 (2015) 137 – 140
Modal instability analysis of light jets L. Lesshaffta,∗, W. Coenenb , X. Garnauda , A. Sevillab b Ingenier´ıa
a LadHyX, CNRS – Ecole ´ Polytechnique, 91128 Palaiseau, France T´ermica y de Fluidos, Universidad Carlos III de Madrid, 28911 Legan´es, Spain
Abstract Linear global modes are computed for round helium jets in air at low Reynolds number. The governing equations are formulated in the limit of zero Mach number, and all parameters are chosen such as to model the experiments of Hallberg & Strykowski 1 . A single axisymmetric mode is found to become unstable beyond a critical Reynolds number value that depends on the initial shear layer thickness. The neutral curve for linear global instability is traced as a function of these two parameters, and this result compares favorably with the experimental reference measurements. c 2015 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license © (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering). Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering) Keywords: Global mode analysis, jet instability
1. Introduction Light jets, i.e. jets of a light fluid issuing into a denser ambient fluid, are known to undergo a bifurcation to a periodic flow state, which represents a nonlinear global mode in the sense of Huerre & Monkewitz 2 . This behaviour has clearly been observed experimentally in hot air jets 3 as well as in helium-in-air jets 4,1 . The nonlinear global instability in these configurations is known to be closely associated with the occurrence of a linear absolute instability of the steady flow state 5,6,1,7 . More recently, linear global instability studies of jet flows have been conducted for supersonic 8 and for subsonic constant-density settings 9,10 , but an investigation of linear global instability in the parameter regimes of the above-cited experiments is still wanting. One unstable linear global mode of a light jet at zero Mach number is discussed in the PhD thesis of Qadri 11 . The present paper provides a rigorous comparison between critical parameters for nonlinear bifurcation, as measured by Hallberg & Strykowski 1 , and the onset of linear global instability.
2. Problem statement We consider laminar jets of pure helium in air, with flow parameters corresponding to the experiments by Hallberg & Strykowski 1 . Effects of compressibility and buoyancy are supposed to be negligible. We therefore employ the lowMach-number approximation that has been formulated by McMurtry 12 , and which has been used in several recent jet instability studies 13,14,11 , in the following form: Dρ = −ρ∇ · u , Dt 1 1 Du = −∇p + ∇2 u + ∇(∇ · u) , ρ Dt Re 3 DY 1 ∇ · (ρ∇Y). = ρ Dt Re S c ∗
Corresponding author. Tel.: +33-1-69335256 ; fax: +33-1-69335292. E-mail address: lesshaff[email protected]
2210-9838 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering) doi:10.1016/j.piutam.2015.03.033
(1) (2) (3)
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L. Lesshafft et al. / Procedia IUTAM 14 (2015) 137 – 140
0
growth rate
−0.1 −0.2 −0.3 −0.4 −0.5
0
0.1
0.2
0.3 Strouhal number
0.4
0.5
Fig. 1. Eigenvalue spectra of the Re = 360 and D/θ0 = 25 jet, obtained with two different numerical box lengths xmax = 63 (◦) and xmax = 103 (+).
Density ρ and helium mass fraction Y are related as Y=
ρ−1 − S , 1−S
(4)
where S = ρ jet /ρ∞ = 0.1429 is the density ratio between helium and air. The helium mass fraction is then unity inside the jet at the nozzle, and zero in the pure ambient air. The Reynolds number is defined as Re = U j R/ν, with the jet volumetric velocity U j , the jet radius R and kinematic viscosity ν. The Schmidt number is kept at S c = 1.69 for helium. The nonlinear equations (1–3) are discretized using the finite-element software FreeFEM++ on a domain that spans over 0 ≤ r ≤ rmax and 0 ≤ x ≤ xmax in the radial and axial directions, respectively, in cylindrical coordinates (similar as figure 1 in Garnaud et al. 10 ). The domain represents half the cross-section of a cylinder. An axisymmetric steady solution is obtained by Newton iteration. An inflow profile is prescribed at x = 0, from where the flow first develops inside a pipe over a streamwise distance of 3R, before it exits into the free ambient. At the end of the pipe, the jet has an initial momentum thickness θ0 . Consistent with the experimental reference 1 , this characteristic parameter is reported as D/θ0 in the following, where D = 2R is the jet diameter. The Reynolds number and the profile parameter D/θ0 are the only parameters that are varied in this study. The equations are then linearized around the base flow, and perturbations q = (ρ , u , p)T are sought in the form of ˆ r)e−iωt . The complex angular frequency ω is obtained as an eigenvalue of the temporal Fourier modes q (x, r, t) = q(x, linear operator 9 . The imaginary part ωi is the temporal growth rate of the eigenmode, and the real part ωr is related to the Strouhal number S t = f D/U j as S t = ωr /π. The eigenvalue problem is solved using the implicitly restarted Arnoldi method as implemented in Matlab. Full details on the numerical procedure for obtaining the steady base flow and the perturbation eigenmodes are given in Garnaud et al. 9 3. Onset of linear global instability Spectra obtained for a configuration with Re = 360 and D/θ0 = 25 are shown in figure 1 for two different numerical domains of streamwise extent xmax = 63 (blue circles) and xmax = 103 (red crosses). One marginally stable mode with S t = 0.293 is found in both cases. At higher Reynolds numbers and higher values of D/θ0 , this mode crosses over into the upper half plane and thereby becomes unstable. All other modes in figure 1 clearly are not converged with respect to the size of the numerical domain. Only the converged eigenmode, which determines the global stability of the flow, will therefore be considered. Its spatial structure is displayed in figure 2; the colour contours show a snapshot of the axial velocity perturbation. The perturbation amplitude attains its maximum near x = 24, and it decays by several orders of magnitude before the downstream boundary (xmax = 63 or 103) is reached. Similar spectra are computed for configurations with Reynolds numbers between 300 and 850, and for inflow profiles with D/θ0 between 15 and 35. The neutral curve that separates the stable from the unstable regime is shown in figure 3. Comparison with the experimentally measured critical parameters demonstrates that the present linear analysis predicts the onset of nonlinear global instability with reasonable accuracy. Bifurcation in the experiments
L. Lesshafft et al. / Procedia IUTAM 14 (2015) 137 – 140
Fig. 2. Spatial structure of the marginally stable eigenmode: real part of axial velocity perturbation. Top frame: full distribution in r and z; bottom frame: absolute values in logarithmic scale, extracted at r = 0.5.
Fig. 3. Marginally unstable parameter settings for a pure Helium jet. Experimental measurements by Hallberg & Strykowski 1 () and present results of the linear analysis (•).
occurs systematically in configurations that are determined to be linearly slightly stable. This observation can be interpreted as the result of low-level excitation of a slightly damped global mode by ambient noise in the experiments (the “slightly damped oscillator” scenario proposed by Huerre & Monkewitz 2 ).
4. Conclusions A linear analysis of axisymmetric perturbation modes, resolved in both the axial and the radial direction, allows to identify a global instability in pure helium jets. The critical values of Reynolds number and initial shear layer thickness match the experimental results of Hallberg & Strykowski 1 with adequate accuracy, considering that the studied jets are very potent amplifiers of external noise. An important finding is that all modes, except the one responsible for the onset of instability, are highly unconverged with respect to the numerical box length.
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The effects of finite Mach and Richardson numbers (compressibility and buoyancy) have not been considered in the present study. In the reference experiments, Mach numbers were below 0.1, and Richardson numbers below 0.005. Compressibility has been found to have a stabilizing effect in a local sense 5 ; the influence of buoyancy will be assessed in future work.
References 1. Hallberg, M. P. and Strykowski, P. J., “On the universality of global modes in low-density axisymmetric jets,” Journal of Fluid Mechanics, Vol. 569, 2006, pp. 493–507. 2. Huerre, P. and Monkewitz, P., “Local and global instabilities in spatially developing flows,” Annual Review of Fluid Mechanics, Vol. 22, 1990, pp. 473–537. 3. Monkewitz, P., Bechert, D., Barsikow, B., and Lehmann, B., “Self-excited oscillations and mixing in a heated round jet,” Journal of Fluid Mechanics, Vol. 213, 1990, pp. 611–639. 4. Kyle, D. and Sreenivasan, K., “The instability and breakdown of a round variable-density jet,” Journal of Fluid Mechanics, Vol. 249, 1993, pp. 619–664. 5. Monkewitz, P. and Sohn, K., “Absolute instability in hot jets,” AIAA J., Vol. 26, 1988, pp. 911–916. 6. Lesshafft, L., Huerre, P., Sagaut, P., and Terracol, M., “Nonlinear global modes in hot jets,” Journal of Fluid Mechanics, Vol. 554, 2006, pp. 393–409. 7. Coenen, W. and Sevilla, A., “The structure of the absolutely unstable regions in the near field of low-density jets,” Journal of Fluid Mechanics, Vol. 713, 2012, pp. 123–149. 8. Nichols, J. and Lele, S., “Global modes and transient response of a cold supersonic jet,” Journal of Fluid Mechanics, Vol. 669, 2011, pp. 225–241. 9. Garnaud, X., Lesshafft, L., Schmid, P., and Huerre, P., “Modal and transient dynamics of jet flows,” Physics of Fluids, Vol. 25, 2013, pp. 044103. 10. Garnaud, X., Lesshafft, L., Schmid, P., and Huerre, P., “The preferred mode of incompressible jets: linear frequency response analysis,” Journal of Fluid Mechanics, Vol. 716, 2013, pp. 189–202. 11. Qadri, U., Global stability and control of swirling jets and flames, Ph.D. thesis, Department of Engineering, University of Cambridge, UK, 2013. 12. McMurtry, P., Riley, J., and Metcalfe, R., “Effects of heat release on the large-scale structure in turbulent mixing layers,” Journal of Fluid Mechanics, Vol. 199, 1989, pp. 297–332. 13. Nichols, J. W., Schmid, P. J., and Riley, J. J., “Self-sustained oscillations in variable-density round jets,” Journal of Fluid Mechanics, Vol. 582, 2007, pp. 341–376. 14. Chandler, G. J., Juniper, M. P., Nichols, J. W., and Schmid, P. J., “Adjoint algorithms for the Navier–Stokes equations in the Low Mach Number limit,” Journal of Computational Physics, Vol. 231, No. 4, 2012, pp. 1900–1916.
ScienceDirect Procedia IUTAM 14 (2015) 137 – 140
Modal instability analysis of light jets L. Lesshaffta,∗, W. Coenenb , X. Garnauda , A. Sevillab b Ingenier´ıa
a LadHyX, CNRS – Ecole ´ Polytechnique, 91128 Palaiseau, France T´ermica y de Fluidos, Universidad Carlos III de Madrid, 28911 Legan´es, Spain
Abstract Linear global modes are computed for round helium jets in air at low Reynolds number. The governing equations are formulated in the limit of zero Mach number, and all parameters are chosen such as to model the experiments of Hallberg & Strykowski 1 . A single axisymmetric mode is found to become unstable beyond a critical Reynolds number value that depends on the initial shear layer thickness. The neutral curve for linear global instability is traced as a function of these two parameters, and this result compares favorably with the experimental reference measurements. c 2015 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license © (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering). Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering) Keywords: Global mode analysis, jet instability
1. Introduction Light jets, i.e. jets of a light fluid issuing into a denser ambient fluid, are known to undergo a bifurcation to a periodic flow state, which represents a nonlinear global mode in the sense of Huerre & Monkewitz 2 . This behaviour has clearly been observed experimentally in hot air jets 3 as well as in helium-in-air jets 4,1 . The nonlinear global instability in these configurations is known to be closely associated with the occurrence of a linear absolute instability of the steady flow state 5,6,1,7 . More recently, linear global instability studies of jet flows have been conducted for supersonic 8 and for subsonic constant-density settings 9,10 , but an investigation of linear global instability in the parameter regimes of the above-cited experiments is still wanting. One unstable linear global mode of a light jet at zero Mach number is discussed in the PhD thesis of Qadri 11 . The present paper provides a rigorous comparison between critical parameters for nonlinear bifurcation, as measured by Hallberg & Strykowski 1 , and the onset of linear global instability.
2. Problem statement We consider laminar jets of pure helium in air, with flow parameters corresponding to the experiments by Hallberg & Strykowski 1 . Effects of compressibility and buoyancy are supposed to be negligible. We therefore employ the lowMach-number approximation that has been formulated by McMurtry 12 , and which has been used in several recent jet instability studies 13,14,11 , in the following form: Dρ = −ρ∇ · u , Dt 1 1 Du = −∇p + ∇2 u + ∇(∇ · u) , ρ Dt Re 3 DY 1 ∇ · (ρ∇Y). = ρ Dt Re S c ∗
Corresponding author. Tel.: +33-1-69335256 ; fax: +33-1-69335292. E-mail address: lesshaff[email protected]
2210-9838 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering) doi:10.1016/j.piutam.2015.03.033
(1) (2) (3)
138
L. Lesshafft et al. / Procedia IUTAM 14 (2015) 137 – 140
0
growth rate
−0.1 −0.2 −0.3 −0.4 −0.5
0
0.1
0.2
0.3 Strouhal number
0.4
0.5
Fig. 1. Eigenvalue spectra of the Re = 360 and D/θ0 = 25 jet, obtained with two different numerical box lengths xmax = 63 (◦) and xmax = 103 (+).
Density ρ and helium mass fraction Y are related as Y=
ρ−1 − S , 1−S
(4)
where S = ρ jet /ρ∞ = 0.1429 is the density ratio between helium and air. The helium mass fraction is then unity inside the jet at the nozzle, and zero in the pure ambient air. The Reynolds number is defined as Re = U j R/ν, with the jet volumetric velocity U j , the jet radius R and kinematic viscosity ν. The Schmidt number is kept at S c = 1.69 for helium. The nonlinear equations (1–3) are discretized using the finite-element software FreeFEM++ on a domain that spans over 0 ≤ r ≤ rmax and 0 ≤ x ≤ xmax in the radial and axial directions, respectively, in cylindrical coordinates (similar as figure 1 in Garnaud et al. 10 ). The domain represents half the cross-section of a cylinder. An axisymmetric steady solution is obtained by Newton iteration. An inflow profile is prescribed at x = 0, from where the flow first develops inside a pipe over a streamwise distance of 3R, before it exits into the free ambient. At the end of the pipe, the jet has an initial momentum thickness θ0 . Consistent with the experimental reference 1 , this characteristic parameter is reported as D/θ0 in the following, where D = 2R is the jet diameter. The Reynolds number and the profile parameter D/θ0 are the only parameters that are varied in this study. The equations are then linearized around the base flow, and perturbations q = (ρ , u , p)T are sought in the form of ˆ r)e−iωt . The complex angular frequency ω is obtained as an eigenvalue of the temporal Fourier modes q (x, r, t) = q(x, linear operator 9 . The imaginary part ωi is the temporal growth rate of the eigenmode, and the real part ωr is related to the Strouhal number S t = f D/U j as S t = ωr /π. The eigenvalue problem is solved using the implicitly restarted Arnoldi method as implemented in Matlab. Full details on the numerical procedure for obtaining the steady base flow and the perturbation eigenmodes are given in Garnaud et al. 9 3. Onset of linear global instability Spectra obtained for a configuration with Re = 360 and D/θ0 = 25 are shown in figure 1 for two different numerical domains of streamwise extent xmax = 63 (blue circles) and xmax = 103 (red crosses). One marginally stable mode with S t = 0.293 is found in both cases. At higher Reynolds numbers and higher values of D/θ0 , this mode crosses over into the upper half plane and thereby becomes unstable. All other modes in figure 1 clearly are not converged with respect to the size of the numerical domain. Only the converged eigenmode, which determines the global stability of the flow, will therefore be considered. Its spatial structure is displayed in figure 2; the colour contours show a snapshot of the axial velocity perturbation. The perturbation amplitude attains its maximum near x = 24, and it decays by several orders of magnitude before the downstream boundary (xmax = 63 or 103) is reached. Similar spectra are computed for configurations with Reynolds numbers between 300 and 850, and for inflow profiles with D/θ0 between 15 and 35. The neutral curve that separates the stable from the unstable regime is shown in figure 3. Comparison with the experimentally measured critical parameters demonstrates that the present linear analysis predicts the onset of nonlinear global instability with reasonable accuracy. Bifurcation in the experiments
L. Lesshafft et al. / Procedia IUTAM 14 (2015) 137 – 140
Fig. 2. Spatial structure of the marginally stable eigenmode: real part of axial velocity perturbation. Top frame: full distribution in r and z; bottom frame: absolute values in logarithmic scale, extracted at r = 0.5.
Fig. 3. Marginally unstable parameter settings for a pure Helium jet. Experimental measurements by Hallberg & Strykowski 1 () and present results of the linear analysis (•).
occurs systematically in configurations that are determined to be linearly slightly stable. This observation can be interpreted as the result of low-level excitation of a slightly damped global mode by ambient noise in the experiments (the “slightly damped oscillator” scenario proposed by Huerre & Monkewitz 2 ).
4. Conclusions A linear analysis of axisymmetric perturbation modes, resolved in both the axial and the radial direction, allows to identify a global instability in pure helium jets. The critical values of Reynolds number and initial shear layer thickness match the experimental results of Hallberg & Strykowski 1 with adequate accuracy, considering that the studied jets are very potent amplifiers of external noise. An important finding is that all modes, except the one responsible for the onset of instability, are highly unconverged with respect to the numerical box length.
139
140
L. Lesshafft et al. / Procedia IUTAM 14 (2015) 137 – 140
The effects of finite Mach and Richardson numbers (compressibility and buoyancy) have not been considered in the present study. In the reference experiments, Mach numbers were below 0.1, and Richardson numbers below 0.005. Compressibility has been found to have a stabilizing effect in a local sense 5 ; the influence of buoyancy will be assessed in future work.
References 1. Hallberg, M. P. and Strykowski, P. J., “On the universality of global modes in low-density axisymmetric jets,” Journal of Fluid Mechanics, Vol. 569, 2006, pp. 493–507. 2. Huerre, P. and Monkewitz, P., “Local and global instabilities in spatially developing flows,” Annual Review of Fluid Mechanics, Vol. 22, 1990, pp. 473–537. 3. Monkewitz, P., Bechert, D., Barsikow, B., and Lehmann, B., “Self-excited oscillations and mixing in a heated round jet,” Journal of Fluid Mechanics, Vol. 213, 1990, pp. 611–639. 4. Kyle, D. and Sreenivasan, K., “The instability and breakdown of a round variable-density jet,” Journal of Fluid Mechanics, Vol. 249, 1993, pp. 619–664. 5. Monkewitz, P. and Sohn, K., “Absolute instability in hot jets,” AIAA J., Vol. 26, 1988, pp. 911–916. 6. Lesshafft, L., Huerre, P., Sagaut, P., and Terracol, M., “Nonlinear global modes in hot jets,” Journal of Fluid Mechanics, Vol. 554, 2006, pp. 393–409. 7. Coenen, W. and Sevilla, A., “The structure of the absolutely unstable regions in the near field of low-density jets,” Journal of Fluid Mechanics, Vol. 713, 2012, pp. 123–149. 8. Nichols, J. and Lele, S., “Global modes and transient response of a cold supersonic jet,” Journal of Fluid Mechanics, Vol. 669, 2011, pp. 225–241. 9. Garnaud, X., Lesshafft, L., Schmid, P., and Huerre, P., “Modal and transient dynamics of jet flows,” Physics of Fluids, Vol. 25, 2013, pp. 044103. 10. Garnaud, X., Lesshafft, L., Schmid, P., and Huerre, P., “The preferred mode of incompressible jets: linear frequency response analysis,” Journal of Fluid Mechanics, Vol. 716, 2013, pp. 189–202. 11. Qadri, U., Global stability and control of swirling jets and flames, Ph.D. thesis, Department of Engineering, University of Cambridge, UK, 2013. 12. McMurtry, P., Riley, J., and Metcalfe, R., “Effects of heat release on the large-scale structure in turbulent mixing layers,” Journal of Fluid Mechanics, Vol. 199, 1989, pp. 297–332. 13. Nichols, J. W., Schmid, P. J., and Riley, J. J., “Self-sustained oscillations in variable-density round jets,” Journal of Fluid Mechanics, Vol. 582, 2007, pp. 341–376. 14. Chandler, G. J., Juniper, M. P., Nichols, J. W., and Schmid, P. J., “Adjoint algorithms for the Navier–Stokes equations in the Low Mach Number limit,” Journal of Computational Physics, Vol. 231, No. 4, 2012, pp. 1900–1916.