LopezZazueta, Adriana and Fontane, Jérôme and Joly, Laurent Optimal perturbations in timedependent variabledensity Kelvin–Helmholtz billows. (2016) Journal of Fluid Mechanics, 803. 466501. ISSN 00221120

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Official URL: http://dx.doi.org/10.1017/jfm.2016.509
Abstract
We analyse the influence of the specific features of timedependent variabledensity Kelvin–Helmholtz (VDKH) rollups on the development of threedimensional secondary instabilities. Due to inertial (high Froude number) baroclinic sources of spanwise vorticity at high Atwood number (up to 0.5 here), temporally evolving mixing layers exhibit a layered structure associated with a strain field radically different from their homogeneous counterpart. We use a directadjoint nonmodal linear approach to determine the fastest growing perturbations over a single period of the timeevolving twodimensional base flow during a given time interval [t0,T]. When perturbations are seeded at the initial time of the primary KH mode growth, i.e. t0=0, it is found that additional mechanisms of energy growth are onset around a bifurcation time tb, a little before the saturation of the primary twodimensional instability. The evolution of optimal perturbations is thus observed to develop in two distinct stages. Whatever the Atwood number, the first period [t0,tb] is characterised by a unique route for optimal energy growth resulting from a combination of the Orr and liftup transient mechanisms. In the second period [tb,T], the growing influence of mass inhomogeneities raises the energy gain over the whole range of spanwise wavenumbers. As the Atwood number increases, the short spanwise wavelength perturbations tend to benefit more from the onset of variabledensity effects than large wavelength ones. The extra energy gain due to increasing Atwood numbers relies on contributions from spanwise baroclinic sources. The resulting vorticity field is structured into two elongated dipoles localised along the braid on either side of the saddle point. In return they yield two longitudinal velocity streaks of opposite sign which account for most of the energy growth. This transition towards threedimensional motions is in marked contrast with the classic streamwise rib vortices, so far accepted as the paradigm for the transition of free shear flows, either homogeneous or not. It is argued that the emergence of these longitudinal velocity streaks is generic of the transition in variabledensity shear flows. Among them, the light round jet is known to display striking side ejections as a result of the loss of axisymmetry. The present analysis helps to renew the question of the underlying flow structure behind side jets, otherwise based on radial induction between pairs of counterrotating longitudinal vortices (Monkewitz & Pfizenmaier, Phys. Fluids A, vol. 3 (5), 1991, pp. 1356–1361). Instead, it is more likely that side ejections would result from the convergence of the longitudinal velocity streaks near the braid saddle point. When the injection time is delayed so as to suppress the initial stage of energy growth, a new class of perturbations arises at low wavenumber with energy gains far larger than those observed so far. They correspond to the twodimensional Kelvin–Helmholtz secondary instability of the baroclinically enhanced vorticity braid discovered by Reinaud et al. (Phys. Fluids, vol. 12 (10), pp. 2489–2505), leading potentially to another route to turbulence through a twodimensional fractal cascade.
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Audience (journal):  International peerreviewed journal 
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Institution:  Université de Toulouse > Institut Supérieur de l'Aéronautique et de l'Espace  ISAESUPAERO (FRANCE) 
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Deposited On:  31 Aug 2016 12:18 
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