Instabilities of convection rolls in a fluid of moderate Prandtl number

Busse, Friedrich H.; Clever, Richard M.

doi:10.1017/s002211207900015x

Cited by 305 publications

(219 citation statements)

References 9 publications

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“…This structure is quite similar to the so-called « skewed-varicose » observed in Rayleigh-B6nard convection [4,6]. However, it is a stationary state in our system, contrary to Rayleigh-B6nard convection.…”

Section: The Defects In the Zig-zag Structure -supporting

confidence: 86%

“…Around these points the vertical velocity decreases to Vz 0. Taking yi as the axis of the tilted roll in the zig-zag phase, the pinching perturbation may be represented as the sum of two elementary modes of perturbation : one of wave vector kyl is an undulation along yi, the second one kxl is a compression-dilation of the roll diameter [6].…”

Section: The Defects In the Zig-zag Structure -mentioning

confidence: 99%

“…Physique 47 (1986) 595 [1,2]. On the other hand spatial structures of decreasing symmetry are observed, depending on both the Rayleigh number (the constraint or control parameter related to the temperature gradient) and the Prandtl number [3][4][5][6]. However [7,8].…”

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confidence: 99%

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Hydrodynamic transitions to chaos in the convection of an anisotropic fluid

1986

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Abstract. -We present the first spontaneous sequence of hydrodynamic instabilities due to the evolution of the convective flow from the rest state toward chaos in an anisotropic fluid : a nematic liquid crystal subjected to an AC electric field. The resulting structures are of decreasing symmetry as the control parameter is gradually increased. The study of the streamlines allows the interpretation of the entire sequence by the evolution of the convective flow up to the onset of chaos. We show that the first important step in the flow evolution is the build up of a second mode of rotation which is associated with a pinching instability of the vortex lines. A simple picture is then proposed in order to describe the essential steps of a possible route to chaos, and which could be extended to other convective systems.

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Section: The Defects In the Zig-zag Structure -supporting

confidence: 86%

Section: The Defects In the Zig-zag Structure -mentioning

confidence: 99%

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Hydrodynamic transitions to chaos in the convection of an anisotropic fluid

1986

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“…A series of theoretical studies [5][6][7][8][9][10] indicate that the parameter regime where two-dimensional rolls remain steady is rather small, in particular for low Pr fluids such as liquid metals. This stable region in the wave number-Rayleigh number space (Busse balloon) is limited by the Eckhaus instability on the smaller wave number side, by the skewedvaricose instability on the larger wave number side, and by the oscillatory instability at Ra ∼ 1900 [7].…”

Section: Introductionmentioning

confidence: 99%

Convection patterns in a liquid metal under an imposed horizontal magnetic field

et al. 2013

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We performed laboratory experiments of Rayleigh-Bénard convection with liquid gallium under various intensities of a uniform imposed horizontal magnetic field. An ultrasonic velocity profiling method was used to visualize the spatiotemporal structure of the flows with simultaneous monitoring of the temperature fluctuations in the liquid gallium layer. The explored Rayleigh numbers Ra range from the critical value for onset of convection to 10 5 ; the Chandrasekhar number Q covers values up to 1100. A regime diagram of the convection patterns was established in relation to the Ra and Q values for a square vessel with aspect ratio 5. We identified five flow regimes: (I) a fluctuating large-scale pattern without rolls, (II) weakly constrained rolls with fluctuations, (III) a continuous oscillation of rolls, (IV) repeated roll number transitions with random reversals of the flow direction, and (V) steady two-dimensional (2D) rolls. These flow regimes are classified by the Ra/Q values, the ratio of the buoyancy to the Lorentz force. Power spectra from the temperature time series indicate that regimes I and II have the features of developed turbulence, while the other regimes do not. The region of steady 2D rolls (Busse balloon) extends to high Ra values in the present setting by a horizontal magnetic field and regime V is located inside the Busse balloon. Concerning the instabilities of the steady 2D rolls, regime III is the traveling wave convection developed from the oscillatory instability. Regime IV can be regarded as a state of phase turbulence, which is induced by intermittent occurrences of the skewed-varicose instability.

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“…The rolls on the left form just above the onset of convection and the ones on the right well (20%) above the onset 1 . Busse and Clever [11,14] established the stability of straight parallel convection rolls and used them to explain many experimental observations, for large Prandtl numbers σ = ν κ (ν is the kinematic viscosity and κ is the thermal diffusivity). For low Prandtl numbers, the situation is much more complicated and it has long been recognized [15,17,41,18] that (complex) mean flows are crucial in understanding the complex pattern dynamics observed in experiments.…”

Section: Where L Is the Width (Radius) And H Is The Height Of ω;mentioning

confidence: 99%

Existence theory and strong attractors for the Rayleigh-Bénard problem with a large aspect ratio

Svanstedt¹,

Birnir²

2003

DCDS-A

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Abstract. The Navier-Stokes equation driven by heat conduction is studied. It is proven that if the driving force is small then the solutions of the Navier-Stokes equation are ultimately regular. As a prototype we consider Rayleigh-Bénard convection, in the Boussinesq approximation. Under a large aspect ratio assumption, which is the case in Rayleigh-Bénard experiments with Prandtl numer close to one, we prove the ultimate existence and regularity of a global strong solution to the 3D Navier-Stokes equation coupled with a heat equation, and the existence of a maximal B-attractor. Examples of simple B-attractors from pattern formation are given and a method to study their instabilities proposed.1. Introduction. In this paper we study the Navier-Stokes equation driven by heat conduction. The existence of strong solutions to the Navier-Stokes equation is still open in three dimensions. On the other hand it has long been known that weak (L 2 ) solutions exist, [32], in three dimensions and if the initial data and the forcing is small then strong solutions exist, see [33,34] [52]. In this paper we show that it suffices to assume the latter of the above conditions, namely the smallness of the forcing, to get ultimate regularity of solutions and the existence of global attractors in three dimensions consisting of smooth solutions. The proof of this results which is stated as Theorem 2.2 is very simple. The existence of an attracting set for weak solutions of the three-dimensional Navier-Stokes equations was proven by Foias and Temam,[20], and Sell [50] proved the existence of a global attractor of weak solutions, without assuming that the forcing is small. Thus our hypothesis is stronger than Sell's but our result is also stronger because it is not clear whether his solutions are ultimately smooth, whereas our attractor consists of strong solutions and has all the nice properties that follow. Many authors have proven the existence of strong solutions and attractors in thin three-dimensional domains, see Raugel [47] and the references therein, and [46,27], etc. None of these results apply to our study of the Rayleigh-Bénard problem. The reason is that we will not assume that the domain is thin unless the driving is very large and even in the latter case the results in thin domains do not apply because we do not restrict the initial data. Instead we will

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Instabilities of convection rolls in a fluid of moderate Prandtl number

Cited by 305 publications

References 9 publications

Hydrodynamic transitions to chaos in the convection of an anisotropic fluid

Hydrodynamic transitions to chaos in the convection of an anisotropic fluid

Convection patterns in a liquid metal under an imposed horizontal magnetic field

Existence theory and strong attractors for the Rayleigh-Bénard problem with a large aspect ratio

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