The Onset and Development of Thermal Convection in Fully Developed Shear Flows

Kelly, R. E.

doi:10.1016/s0065-2156(08)70255-2

Cited by 100 publications

(50 citation statements)

References 122 publications

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“…When the shear rate is large enough, the viscous torque dominates over the gravitational one. In this case, the swimming direction of the cell becomes unsteady and changes periodically in time (Pedley & Kessler 1987, 1992, similarly to that of a passive particle in a shear flow (Jeffery 1922). Owing to this behaviour, gyrotactic cells in a strong shear flow tumble and exhibit greatly reduced up-swimming velocity on average.…”

Section: Introductionmentioning

confidence: 96%

Bioconvection under uniform shear: linear stability analysis

Hwang

Pedley²

2013

J. Fluid Mech.

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The role of uniform shear in bioconvective instability in a shallow suspension of swimming gyrotactic cells is studied using linear stability analysis. The shear is introduced by applying a plane Couette flow, and it significantly disturbs gravitaxis of the cell. The unstably stratified basic state of the cell concentration is gradually relieved as the shear rate is increased, and it even becomes stably stratified at very large shear rates. Stability of the basic state is significantly changed. The instability at high wavenumbers is drastically damped out with the shear rate, while that at low wavenumbers is destabilised. However, at very large shear rates, the latter is also suppressed. The most unstable mode is found as a pair of streamwise uniform rolls aligned with the shear, analogous to Rayleigh-Bénard convection in plane Couette flow. To understand these findings, the physical mechanism of the bioconvective instability is reexamined with several sets of numerical experiments. It is shown that the bioconvective instability in a shallow suspension originates from three different physical processes: gravitational overturning, gyrotaxis of the cell, and negative cross diffusion flux. The first mechanism is found to rule the behavior of low-wavenumber instability whereas the last two mechanisms are mainly associated with high-wavenumber instability. With the increase of the shear rate, the former is enhanced, thereby leading to destabilisation at low wavenumbers, whereas the latter two mechanisms are significantly suppressed. For streamwise varying perturbations, shear with sufficiently large rates is also found to play a stabilising role as in Rayleigh-Bénard convection. However, at small shear rates, it destabilises these perturbations through the mechanism of overstability discussed by Hill et al. (1989). Finally, the present findings are compared with a recent experiment by Croze et al. (2010) and they are in qualitative agreement.

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Section: Introductionmentioning

confidence: 96%

Bioconvection under uniform shear: linear stability analysis

Hwang

Pedley²

2013

J. Fluid Mech.

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“…Ingersoll 1965;Clever, Busse & Kelly 1977). Various developments of this theme are discussed in the review article of Kelly (1994).…”

Section: Introductionmentioning

confidence: 99%

The onset of thermal convection in EkmanCouette shear flow with oblique rotation

2003

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The onset of convection of a Boussinesq fluid in a horizontal plane layer is studied. The system rotates with constant angular velocity Ω Ω Ω, which is inclined at an angle ϑ to the vertical. The layer is sheared by keeping the upper boundary fixed, while the lower boundary moves parallel to itself with constant velocity U 0 normal to the plane containing the rotation vector and gravity g (i.e. U 0 g × Ω Ω Ω). The system is characterized by five dimensionless parameters: the Rayleigh number Ra, the Taylor number τ 2 , the Reynolds number Re (based on U 0 ), the Prandtl number Pr and the angle ϑ. The basic equilibrium state consists of a linear temperature profile and an Ekman-Couette flow, both dependent only on the vertical coordinate z. Our linear stability study involves determining the critical Rayleigh number Ra c as a function of τ and Re for representative values of ϑ and Pr.Our main results relate to the case of large Reynolds number, for which there is the possibility of hydrodynamic instability. When the rotation is vertical ϑ = 0 and τ 1, so-called type I and type II Ekman layer instabilities are possible. With the inclusion of buoyancy Ra = 0 mode competition occurs. On increasing τ from zero, with fixed large Re, we identify four types of mode: a convective mode stabilized by the strong shear for moderate τ , hydrodynamic type I and II modes either assisted (Ra > 0) or suppressed (Ra < 0) by buoyancy forces at numerically large τ , and a convective mode for very large τ that is largely uninfluenced by the thin Ekman shear layer, except in that it provides a selection mechanism for roll orientation which would otherwise be arbitrary. Significantly, in the case of oblique rotation ϑ = 0, the symmetry associated with U 0 ↔ −U 0 for the vertical rotation is broken and so the cases of positive and negative Re exhibit distinct stability characteristics, which we consider separately. Detailed numerical results were obtained for the representative case ϑ = π/4. Though the overall features of the stability results are broadly similar to the case of vertical rotation, their detailed structure possesses a surprising variety of subtle differences.

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“…The problem of Rayleigh-Bénard convection in an externally sheared flow has numerous geophysical and engineering analogs (Kelly, 1994), for instance, in the formation of cloud rows in a convective atmospheric boundary layer (Brown, 1980;Shirer, 1986;Atkinson and Zhang, 1996). Usually convection rolls with axes parallel to the shear direction are found, e.g.…”

Section: Externally Forced Shearmentioning

confidence: 99%

“…However, transverse rolls, whose axes are perpendicular to the direction of shear, are known to occur in certain situations, such as moderately supercritical convection with very low Reynolds number in air (Graham, 1933;Chandra, 1938;Bénard and Avsec, 1938). For more discussion of experimental work on transverse rolls, see the reviews by Brunt (1951) and Kelly (1994), and the paper by Ingersoll (1966). The LOM to be developed below is proposed as a model of transverse rolls.…”

Section: Externally Forced Shearmentioning

confidence: 99%

Gyrostatic extensions of the Howard-Krishnamurti model of thermal convection with shear

Tong

Gluhovsky

2008

Nonlin. Processes Geophys.

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Abstract. The Howard & Krishnamurti (1986) low-order model (LOM) of Rayleigh-Bénard convection with spontaneous vertical shear can be extended to incorporate various additional physical effects, such as externally forced vertical shear and magnetic field. Designing such extended LOMs so that their mathematical structure is isomorphic to those of systems of coupled gyrostats, with damping and forcing, allows for a modular approach while respecting conservation laws. Energy conservation (in the limit of no damping and forcing) prevents solutions that diverge to infinity, which are present in the original Howard & Krishnamurti LOM. The first LOM developed here (as a candidate model of transverse rolls) involves adding a new Couette mode to represent externally forced vertical shear. The second LOM is a modification of the Lantz (1995) model for magnetoconvection with shear. The modification eliminates an invariant manifold in the original model that leads to potentially unphysical behavior, namely solutions that diverge to infinity, in violation of energy conservation. This paper reports the first extension of the coupled gyrostats modeling framework to incorporate externally forced vertical shear and magnetoconvection with shear. Its aim is to demonstrate better model building techniques that avoid pathologies present in earlier models; consequently we do not focus here on analysis of dynamics or model validation.

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The Onset and Development of Thermal Convection in Fully Developed Shear Flows

Cited by 100 publications

References 122 publications

Bioconvection under uniform shear: linear stability analysis

Bioconvection under uniform shear: linear stability analysis

The onset of thermal convection in EkmanCouette shear flow with oblique rotation

Gyrostatic extensions of the Howard-Krishnamurti model of thermal convection with shear

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