Thermosolutal and binary fluid convection as a 2×2 matrix problem

Tuckerman, Laurette S.

doi:10.1016/s0167-2789(01)00284-6

Cited by 18 publications

(15 citation statements)

References 58 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When Im͑͒ decreases to zero, the pair of complex conjugate eigenvalues splits into two real eigenvalues, which correspond to the transcritical and pitchfork modes before coalescence. This whole coalescence and splitting process is similar to the situation of a 2 ϫ 2 matrix problem studied by Tuckerman 22 and the Marangoni convection in binary mixtures studied by Bergeon et al 23 It is interesting to note that before coalescence the most unstable mode is transcritical, while after splitting the most unstable mode is pitchfork. In the case of a closed cavity, these two steady modes simply cross at a codimension-two point and near this point no new dynamical behavior is observed.…”

Section: Variations Of Leading Eigenvalues With Grsupporting

confidence: 75%

Onset of double-diffusive convection in a rectangular cavity with stress-free upper boundary

Chen

Zhan

et al. 2010

Physics of Fluids

View full text Add to dashboard Cite

Double diffusive convection in a porous layer saturated with viscoelastic fluid using a thermal non-equilibrium model Phys. Fluids 23, 054101 (2011) Influence of vibrations on thermodiffusion in binary mixture: A benchmark of numerical solutions Phys. Fluids 19, 017111 (2007) Inertial effects on the transfer of heat or mass from neutrally buoyant spheres in a steady linear velocity field Phys. Fluids 18, 073302 (2006) Additional information on Phys. Fluids Double-diffusive buoyancy convection in an open-top rectangular cavity with horizontal temperature and concentration gradients is considered. Attention is restricted to the case where the opposing thermal and solutal buoyancy effects are of equal magnitude ͑buoyancy ratio R =−1͒. In this case, a quiescent equilibrium solution exists and can remain stable up to a critical thermal Grashof number Gr c . Linear stability analysis and direct numerical simulation show that depending on the cavity aspect ratio A, the first primary instability can be oscillatory, while that in a closed cavity is always steady. Near a codimension-two point, the two leading real eigenvalues merge into a complex coalescence that later produces a supercritical Hopf bifurcation. As Gr further increases, this complex coalescence splits into two real eigenvalues again. The oscillatory flow consists of counter-rotating vortices traveling from right to left and there exists a critical aspect ratio below which the onset of convection is always oscillatory. Neutral stability curves showing the influences of A, Lewis number Le, and Prandtl number Pr are obtained. While the number of vortices increases as A decreases, the flow structure of the eigenfunction does not change qualitatively when Le or Pr is varied. The supercritical oscillatory flow later undergoes a period-doubling bifurcation and the new oscillatory flow soon becomes unstable at larger Gr. Random initial fields are used to start simulations and many different subcritical steady states are found. These steady states correspond to much stronger flows when compared to the oscillatory regime. The influence of Le on the onset of steady flows and the corresponding heat and mass transfer properties are also investigated.

show abstract

Section: Variations Of Leading Eigenvalues With Grsupporting

confidence: 75%

Onset of double-diffusive convection in a rectangular cavity with stress-free upper boundary

Chen

Zhan

et al. 2010

Physics of Fluids

View full text Add to dashboard Cite

show abstract

“…One of the emerging real eigenvalues keeps increasing with Re while the other one decreases, producing a steady subcritical bifurcation responsible for the existence of subcritical steady flow branches. 30,33 Unfortunately this scenario does not hold here, at least in the range of Re values investigated ͓Fig. 2͑a͔͒.…”

Section: B Steady Flow Regimementioning

confidence: 80%

Double-diffusive Marangoni convection in a rectangular cavity: Onset of convection

Chen

Zhan

2010

Physics of Fluids

View full text Add to dashboard Cite

Double-diffusive Marangoni convection in a rectangular cavity with horizontal temperature and concentration gradients is considered. Attention is restricted to the case where the opposing thermal and solutal Marangoni effects are of equal magnitude ͑solutal to thermal Marangoni number ratio R =−1͒. In this case a no-flow equilibrium solution exists and can remain stable up to a critical thermal Marangoni number. Linear stability analysis and direct numerical simulation show that this critical value corresponds to a supercritical Hopf bifurcation point, which leads the quiescent fluid directly into the oscillatory flow regime. Influences of the Lewis number Le, Prandtl number Pr, and the cavity aspect ratio A ͑height/length͒ on the onset of instability are systematically investigated and different modes of oscillation are obtained. The first mode is first destabilized and then stabilized. Sometimes it never gets onset. A physical illustration is provided to demonstrate the instability mechanism and to explain why the oscillatory flow after the onset of instability corresponds to countersense rotating vortices traveling from right to left in the present configuration, as obtained by direct numerical simulation. Finally the simultaneous existence of both steady and oscillatory flow regimes is shown. While the oscillatory flow arises from small disturbances, the steady flow, which has been described in the literature, is induced by finite amplitude disturbances.

show abstract

“…Swirling has a dramatic destabilizing effect, since the critical Reynolds number decreases by approximately 85 %, from Re = 300 at S = 0.845 to Re = 40.5 at S = 1.8. Note the distortions of the m = −1 neutral curve close to the value S = 1.05, caused by the avoided crossing of two eigenvalues (Tuckerman 2001). This phenomenon is best seen in figure 4(b) showing the evolution of the two largest m = −1 growth rates when increasing the swirl and keeping the Reynolds number constant (Re = 164, i.e.…”

Section: Mode Selection At the Threshold Of Helical Instabilitymentioning

confidence: 96%

A weakly nonlinear mechanism for mode selection in swirling jets

2012

View full text Add to dashboard Cite

Global linear and nonlinear bifurcation analysis is used to revisit the spiral vortex breakdown of nominally axisymmetric swirling jets. For the parameters considered herein, stability analyses single out two unstable linear modes of azimuthal wavenumber $m= \ensuremath{-} 1$ and $m= \ensuremath{-} 2$, bifurcating from the axisymmetric breakdown solution. These modes are interpreted in terms of spiral perturbations wrapped around and behind the axisymmetric bubble, rotating in time in the same direction as the swirling flow but winding in space in the opposite direction. Issues are addressed regarding the role of these modes with respect to the existence, mode selection and internal structure of vortex breakdown, as assessed from the three-dimensional direct numerical simulations of Ruith et al. (J. Fluid Mech., vol. 486, 2003, pp. 331–378). The normal form describing the leading-order nonlinear interaction between modes is computed and analysed. It admits two stable solutions corresponding to pure single and double helices. At large swirl, the axisymmetric solution bifurcates to the double helix which remains the only stable solution. At low and moderate swirl, it bifurcates first to the single helix, and subsequently to the double helix through a series of subcritical bifurcations yielding hysteresis over a finite range of Reynolds numbers, the estimated bifurcation threshold being in good agreement with that observed in the direct numerical simulations. Evidence is provided that this selection is not to be ascribed to classical mean flow corrections induced by the existence of the unstable modes, but to a non-trivial competition between harmonics. Because the frequencies of the leading modes approach a strong $2$:$1$ resonance, an alternative normal form allowing interactions between the $m= \ensuremath{-} 2$ mode and the first harmonics of the $m= \ensuremath{-} 1$ mode is computed and analysed. It admits two stable solutions, the double helix already identified in the non-resonant case, and a single helix differing from that observed in the non-resonant case only by the presence of a slaved, phase-locked harmonic deformation. On behalf of the finite departure from the $2$:$1$ resonance, the amplitude of the slaved harmonic is however low, and the effect of the resonance on the bifurcation structure is merely limited to a reduction of the hysteresis range.

show abstract

Thermosolutal and binary fluid convection as a 2×2 matrix problem

Cited by 18 publications

References 58 publications

Onset of double-diffusive convection in a rectangular cavity with stress-free upper boundary

Onset of double-diffusive convection in a rectangular cavity with stress-free upper boundary

Double-diffusive Marangoni convection in a rectangular cavity: Onset of convection

A weakly nonlinear mechanism for mode selection in swirling jets

Contact Info

Product

Resources

About