Local and Global Instability of Buoyant Jets and Plumes

XXIV ICTAM, 21-26 August 2016, Montreal, Canada

LOCAL AND GLOBAL INSTABILITY OF BUOYANT JETS AND PLUMES

Patrick Huerre1a), R.V.K. Chakravarthy1 & Lutz Lesshafft 1 1Laboratoire d’Hydrodynamique (LadHyX), Ecole Polytechnique, Paris, France

Summary The local and global linear stability of buoyant jets and plumes has been studied as a function of the Richardson number Ri and density ratio S in the low Mach number approximation. Only the m = 0 axisymmetric mode is shown to become globally unstable, provided that the local absolute instability is strong enough. The helical mode of azimuthal wavenumber m = 1 is always globally stable. A sensitivity analysis indicates that in buoyant jets (low Ri), shear is the dominant contributor to the growth rate, while, for plumes (large Ri), it is the buoyancy. A theoretical prediction of the Strouhal number of the self-sustained oscillations in helium jets is obtained that is in good agreement with experimental observations over seven decades of Richardson numbers.

INTRODUCTION

Buoyant jets and plumes occur in a wide variety of environmental and industrial contexts, for instance fires, accidental gas releases, ventilation flows, geothermal vents, and volcanic eruptions. Understanding the onset of instabilities leading to turbulence is a research challenge of great practical and fundamental interest. Somewhat surprisingly, there have been relatively few studies of the linear stability properties of buoyant jets and plumes, in contrast to the related purely momentum driven classical jet. In the present study, local and global stability analyses are conducted to account for the self-sustained oscillations experimentally observed in buoyant jets of helium and helium-air mixtures [1].

FORMULATION AND METHODOLOGY

The flow is assumed to be governed by the compressible Navier-Stokes equations in the low Mach number approximation, a feature that allows for large density variations but filters out acoustic waves. In the limit of small density variations, this system reduces to the Boussinesq approximation. The steady axisymmetric base flow is obtained by resorting to the Newton-Raphson technique for given hyperbolic tangent axial velocity and density profiles at the inlet. Let R denote the inlet radius, ρj and Uj the inlet density and axial velocity and ρ∞ the outer density. The most important parameters of the 2 problem are then the Richardson number Ri = gR(ρ∞ - ρj)/(ρjUj ) and the density ratio S = ρ∞ /ρj with Ri and S in the range 10-4 < Ri < 103 and 1 < S < 7. At low Richardson numbers, buoyancy forces are much smaller than inertia forces, in which case the base flow is referred to as a buoyant jet, while at large Richardson numbers, buoyancy is dominant and the base flow is said to be a plume. The pure jet and pure plume are reached in the limit Ri = 0 and Ri = ∞ respectively. The linearized system of partial differential equations associated with the non-parallel base flow then gives rise to a two- dimensional eigenvalue problem in the axial and radial directions, where the eigenvalue is the complex circular frequency

ω = ωr + iωi of frequency ωr and temporal growth rate ωi. A suitable finite-element discretization of the system followed by the implementation of a shift-invert scheme then leads to the determination of the eigenvalues and associated two- dimensional eigenfunctions. The same methodology is applied to the locally parallel base state at each axial station to compute the eigenvalues ω of the local instability problem as a function of axial wavenumber k and azimuthal wavenumber m. The convectively/absolutely unstable nature of the base flow is then readily determined as shown in [2] and [3].

LOCAL AND GLOBAL INSTABILITY CHARACTERISTICS

For buoyant jets (low Ri), the axisymmetric m = 0 global spectrum typically displays a single unstable discrete frequency, whereas for plumes (large Ri), it is composed of several unstable discrete modes with a maximum growth rate and corresponding frequency that are larger by two or three orders of magnitude. The local and global instability results may be summarized in the Ri – S state diagram displayed in Figure 1. As the density ratio S and the Richardson number Ri increase along a diagonal line, the base flow changes from locally convectively unstable (white region) to locally absolutely unstable (blue and red regions) as the zero absolute growth rate neutral curve (thin line) is crossed. As the streamwise extent of the absolutely unstable domain reaches a critical size, the base flow becomes globally unstable (red region) as the neutral global stability curve (thick line) is crossed. Buoyant jets (low Ri) successively experience all three qualitative states as S increases while plumes are much more unstable and exhibit a rapid transition to global instability. The dip in the global neutral curve at intermediate Ri’s is due to a shift from an inertia driven instability mechanism for buoyant jets to a buoyancy dominated mechanism for plumes. No globally unstable discrete frequency is obtained for the m = 1 helical mode, but the downstream sponge region needs to be carefully tuned in order to stabilize the continuous spectrum, as in classical jets. A local stability analysis [2] in the Boussinesq approximation framework (S close to unity), recently reported that the m = 1 mode is the only

a) Corresponding author. Email: [email protected] one to become absolutely unstable. It may be concluded that the magnitude of the absolute growth rate is then too weak to drive an unstable global mode.

Figure 1: Instability regions and associated neutral curves in Ri – S plane (log-log scale) for the axisymmetric mode m = 0 at Reynolds number 200. White region: locally convectively unstable; Blue region: locally absolutely unstable; Red region: globally unstable (and absolutely unstable).

A sensitivity analysis on the relative contributions to the global growth rate and frequency of shear, buoyancy and other terms in the linear dynamical operator, reveals that shear is the dominant factor for the global instability of buoyant jets while buoyancy becomes dominant for plumes.

GLOBAL FREQUENCY AND SELF-SUSTAINED OSCILLATIONS

Light jets have been known for some time to give rise to axisymmetric self-sustained oscillations. Prior theoretical investigations have relied on local stability approaches, thereby implicitly assuming that the base flow is weakly non-parallel. The predictions of the global analysis are compared in Figure 2 with the experimental observations of Cetegen & Kasper [1]. The blue line represents the empirical formula resulting from experiments and the red line the theoretical prediction of the present study.

Theory: St = 0.24 Ri0.43 Exp.: St = 0.26 Ri0.38 Cetegen & Kasper [1]

-4 3 Figure 2: Strouhal number St = ωr/2π versus Richardson number (log-log scale) in the range 10 < Ri < 10 : Comparison between prediction of global stability theory (blue line) and experimental observations of Cetegen & Kasper [1].

The linear global mode theory is seen to closely agree with the observations over 7 decades of Richardson numbers. For buoyant jets (low Ri), the Strouhal number is almost constant with Ri, while for plumes (large Ri), it continuously increases.

References

[1] Cetegen B.M. & Kasper K.D.: Experiments on the Oscillatory Behavior of Buoyant Plumes of Helium and Helium-Air Mixtures Phys. Fluids 8(11):2074- 2984, 1996. [2] Chakravarthy R.V. K., Lesshafft, L. & Huerre, P.: Local Linear Stability of Laminar Axisymmetric Plumes. J. Fluid Mech 580:344-369, 2015. [3] Chakravarthy R.V. K.: Local and Global Instabilities in Buoyant Jets and Plumes. PhD Thesis, Ecole Polytechnique, LadHyX, Paris, France, 2015.