5 results
Compressibility effects on the secondary instabilities of the circular cylinder wake
- Laura Victoria Rolandi, Jérôme Fontane, Thierry Jardin, Jérémie Gressier, Laurent Joly
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- Journal:
- Journal of Fluid Mechanics / Volume 966 / 10 July 2023
- Published online by Cambridge University Press:
- 05 July 2023, A36
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With a growing interest in low Reynolds number compressible flows, compressibility effects on the secondary instabilities developing on the circular cylinder periodic wake are investigated. The unsteady and time-averaged two-dimensional flows are characterised for Reynolds numbers ${{Re}} \in [200;350]$ and Mach numbers up to ${{M}}_{\infty }=0.5$, revealing different flow structures which influence the characteristics of the secondary unstable modes. The two-dimensional time-periodic solution is used as the base state for a global linear stability analysis performed by means of Floquet theory coupled with a time-stepping finite-difference approach of the nonlinear operator. The influence of compressibility on mode A and mode B secondary instabilities which are responsible for the three-dimensionalisation of the two-dimensional periodic wake is analysed. A stabilising or a destabilising effect of compressibility is observed on mode A, depending on the Reynolds number and the spanwise wavelength of the mode, while mode B is stabilised by the increase of the Mach number. Compressibility is indeed found to decrease the mode kinetic energy production due to base flow shear conversion, which drives the growth of mode B. This results in a delay of the three-dimensionalisation process of the wake due to compressibility.
Optimal growth over a time-evolving variable-density jet at Atwood number $\vert \textit {At} \vert = 0.25$
- Gabriele Nastro, Jérôme Fontane, Laurent Joly
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- Journal:
- Journal of Fluid Mechanics / Volume 936 / 10 April 2022
- Published online by Cambridge University Press:
- 11 February 2022, A15
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Secondary instabilities growing over a time-evolving variable-density round jet subject to the primary Kelvin–Helmholtz (KH) instability at Atwood number $\vert \textit {At} \vert = 0.25$ are investigated with a non-modal linear stability analysis. Despite local modifications of the base flow vorticity induced by the baroclinic torque, these disturbances experience a short-term universal growth due to a combination of the Orr and lift-up mechanisms, whatever the azimuthal wavenumber $m$. At $\textit {Re}=1000$, the secondary energy growth stems from the development of elliptical and hyperbolic instabilities, with an E-type-to-H-type transition as $m$ and $\textit {Re}$ increase, as in the homogeneous case (Nastro et al., J. Fluid Mech., vol. 900, 2020, A13). In the light jet at $\textit {Re} = 1000$, after the KH mode saturation, the high-$m$ H-type instability is replaced by a perturbation organised as counter-rotating streamwise vortices located in the base flow region of promoted strain rate. Increasing the Reynolds number up to $\textit {Re} = 10\,000$ yields larger energy growths and a strong anisotropy among energy and enstrophy components with a preferential increase of axial velocity and azimuthal vorticity. Both come from the linearised baroclinic source that drives the optimal response towards folded sheets of axial velocity that differ from those observed in the variable-density plane shear layers. When the perturbation is injected around the KH saturation time for $\textit {Re}=10\,000$, the response to optimal perturbation takes the form of fast growing secondary KH instabilities whatever $m$. We find these three-dimensional secondary KH instabilities to be good candidates for the transition to turbulence in variable-density jet flows.
Optimal perturbations in viscous round jets subject to Kelvin–Helmholtz instability
- Gabriele Nastro, Jérôme Fontane, Laurent Joly
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- Journal:
- Journal of Fluid Mechanics / Volume 900 / 10 October 2020
- Published online by Cambridge University Press:
- 07 August 2020, A13
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We investigate the development of three-dimensional instabilities on a time-dependent round jet undergoing the axisymmetric Kelvin–Helmholtz (KH) instability. A non-modal linear stability analysis of the resulting unsteady roll-up into a vortex ring is performed based on a direct-adjoint approach. Varying the azimuthal wavenumber $m$, the Reynolds number ${Re}$ and the aspect ratio $\alpha$ of the jet base flow, we explore the potential for secondary energy growth beyond the initial phase when the base flow is still quasi-parallel and universal shear-induced transient growth occurs. For ${Re}=1000$ and $\alpha = 10$, the helical $m=1$ and double-helix $m=2$ perturbations stand as global optimals with larger growth rates in the post roll-up phase. The secondary energy growth stems from the development of elliptical (E-type) and hyperbolic (H-type) instabilities. For $m>2$, the maximum of the kinetic energy of the optimal perturbation moves from the large scale vortex core towards the thin vorticity braid. With a Reynolds number one order of magnitude larger, the kinetic energy of the optimal perturbations exhibits sustained growth well after the saturation time of the base flow KH wave and the underlying length scale selection favours higher azimuthal wavenumbers associated with H-type instability in the less diffused vorticity braid. Doubling the jet aspect ratio yields initially thinner shear layers only slightly affected by axisymmetry. The resulting unsteady base flow loses scale selectivity and is prone to a common path of initial transient growth followed by the optimal secondary growth of a wide range of wavenumbers. Increasing both the aspect ratio and the Reynolds number thus yields an even larger secondary growth and a lower wavenumber selectivity. At a lower aspect ratio of $\alpha =5$, the base flow is smooth and a genuine round jet affected by the axisymmetry condition. The axisymmetric modal perturbation of the base flow parallel jet only weakly affects the first common phase of transient growth and the optimal helical perturbation $m=1$ dominates with energy gains considerably larger than those of larger azimuthal wavenumbers whatever the horizon time.
Optimal perturbations in time-dependent variable-density Kelvin–Helmholtz billows
- Adriana Lopez-Zazueta, Jérôme Fontane, Laurent Joly
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- Journal:
- Journal of Fluid Mechanics / Volume 803 / 25 September 2016
- Published online by Cambridge University Press:
- 30 August 2016, pp. 466-501
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We analyse the influence of the specific features of time-dependent variable-density Kelvin–Helmholtz (VDKH) roll-ups on the development of three-dimensional 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 direct-adjoint non-modal linear approach to determine the fastest growing perturbations over a single period of the time-evolving two-dimensional base flow during a given time interval $[t_{0},T]$. When perturbations are seeded at the initial time of the primary KH mode growth, i.e. $t_{0}=0$, it is found that additional mechanisms of energy growth are onset around a bifurcation time $t_{b}$, a little before the saturation of the primary two-dimensional instability. The evolution of optimal perturbations is thus observed to develop in two distinct stages. Whatever the Atwood number, the first period $[t_{0},t_{b}]$ is characterised by a unique route for optimal energy growth resulting from a combination of the Orr and lift-up transient mechanisms. In the second period $[t_{b},T]$, 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 variable-density 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 three-dimensional 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 variable-density 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 counter-rotating 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 two-dimensional 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 two-dimensional fractal cascade.
The stability of the variable-density Kelvin–Helmholtz billow
- JÉRÔME FONTANE, LAURENT JOLY
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- Journal:
- Journal of Fluid Mechanics / Volume 612 / 10 October 2008
- Published online by Cambridge University Press:
- 10 October 2008, pp. 237-260
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We perform a three-dimensional stability analysis of the Kelvin–Helmholtz (KH) billow, developing in a shear layer between two fluids with different density. We begin with two-dimensional simulations of the temporally evolving mixing layer, yielding the unsteady base flow fields. The Reynolds number is 1500 while the Schmidt and Froude numbers are infinite. Then exponentially unstable modes are extracted from a linear stability analysis performed at the saturation of the primary mode kinetic energy. The spectrum of the least stable modes exhibits two main classes. The first class comprises three-dimensional core-centred and braid-centred modes already present in the homogeneous case. The baroclinic vorticity concentration in the braid lying on the light side of the KH billow turns the flow into a sharp vorticity ridge holding high shear levels. The hyperbolic modes benefit from the enhanced level of shear in the braid whereas elliptic modes remain quite insensitive to the modifications of the base flow. In the second class, we found typical two-dimensional modes resulting from a shear instability of the curved vorticity-enhanced braid. For a density contrast of 0.5, the wavelength of the two-dimensional instability is about ten times shorter than that of the primary wave. Its amplification rate competes well against those of the hyperbolic three-dimensional modes. The vorticity-enhanced braid thus becomes the preferred location for the development of secondary instabilities. This stands as the key feature of the transition of the variable-density mixing layer. We carry out a fully resolved numerical continuation of the nonlinear development of the two-dimensional braid-mode. Secondary roll-ups due to a small-scale Kelvin–Helmholtz mechanism are promoted by the underlying strain field and develop rapidly in the compression part of the braid. Originally analysed by Reinoud et al. (Phys. Fluids, vol. 12, 2000, p. 2489) from two-dimensional non-viscous numerical simulations, this instability is shown to substantially increase the mixing.