Transition to time-dependent convection

1974 ◽  
Vol 65 (4) ◽  
pp. 625-645 ◽  
Author(s):  
R. M. Clever ◽  
F. H. Busse
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Horizontal Layer ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Prandtl Numbers ◽  
Steady Solutions ◽  
The Stability ◽  
Convection Rolls ◽  
Good Agreement

Steady solutions in the form of two-dimensional rolls are obtained for convection in a horizontal layer of fluid heated from below as a function of the Rayleigh and Prandtl numbers. Rigid boundaries of infinite heat conductivity are assumed. The stability of the two-dimensional rolls with respect to three-dimensional disturbances is analysed. It is found that convection rolls are unstable for Prandtl numbers less than about 5 with respect to an oscillatory instability investigated earlier by Busse (1972) for the case of free boundaries. Since the instability is caused by the momentum advection terms in the equations of motion the Rayleigh number for the onset of instability increases strongly with Prandtl number. Good agreement with various experimental observations is found.

Stability of convection rolls in a layer with stress-free boundaries

1985 ◽  
Vol 150 ◽  
pp. 487-498 ◽  
Author(s):  
E. W. Bolton ◽  
F. H. Busse
Keyword(s):  
Three Dimensional ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Stability Boundaries ◽  
Prandtl Numbers ◽  
Steady Solutions ◽  
The Stability ◽  
Convection Rolls ◽  
Steady Convection

Steady finite-amplitude solutions for two-dimensional convection in a layer heated from below with stress-free boundaries are obtained numerically by a Galerkin method. The stability of the steady convection rolls with respect to arbitrary three-dimensional infinitesimal disturbances is investigated. Stability is found only in a small fraction of the Rayleigh-number-wavenumber space where steady solutions exist. The cross-roll instability and the oscillatory and monotonic skewed varicose instabilities are most important in limiting the stability of steady convection rolls. The Prandtlnumbers P = 0.71, 7, 104 areemphasized, but the stability boundaries are sufficiently smoothly dependent on the parameters of the problem to permit qualitative extrapolations to other Prandtl numbers.

Instabilities of convection rolls with stress-free boundaries near threshold

1984 ◽  
Vol 146 ◽  
pp. 115-125 ◽  
Author(s):  
F. H. Busse ◽  
E. W. Bolton
Keyword(s):  
Rayleigh Number ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Limited Range ◽  
Finite Domain ◽  
Free Boundaries ◽  
Prandtl Numbers ◽  
The Stability ◽  
Convection Rolls ◽  
Horizontal Fluid Layer

The stability properties of steady two-dimensional solutions describing convection in a horizontal fluid layer heated from below with stress-free boundaries are investigated in the neighbourhood of the critical Rayleigh number. The region of stable convection rolls as a function of the wavenumber α and the Rayleigh number R is bounded towards higher α by the monotonic skewed varicose instability, while towards low wavenumbers stability is limited by the zigzag instability or by the oscillatory skewed varicose instability. Only for a limited range of Prandtl numbers, 0·543 < P < ∞, does a finite domain of stability exist. In particular, convection rolls with the critical wavenumber αc are always unstable.

The oscillatory instability of convection rolls in a low Prandtl number fluid

1972 ◽  
Vol 52 (1) ◽  
pp. 97-112 ◽  
Author(s):  
F. H. Busse
Keyword(s):  
Prandtl Number ◽  
Reasonable Agreement ◽  
Fluid Layer ◽  
Three Dimensional ◽  
Convective Motion ◽  
Critical Value ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Vertical Vorticity ◽  
Convection Rolls

The instability of convection rolls in a fluid layer heated from below is investigated for stress-free boundaries in the limit of small Prandtl number. It is shown that the two-dimensional rolls become unstable to oscillatory three-dimensional disturbances when the amplitude of the convective motion exceeds a finite critical value. The instability corresponds to the generation of vertical vorticity, a mechanism which is likely to operate in the case of a variety of roll-like motions. In all aspects in which the theory can be related to experiments, reasonable agreement with the observations is found.

Steady and oscillatory bimodal convection

1994 ◽  
Vol 271 ◽  
pp. 103-118 ◽  
Author(s):  
R. M. Clever ◽  
F. H. Busse
Keyword(s):  
Fluid Layer ◽  
Three Dimensional ◽  
Convection Pattern ◽  
Numerical Computations ◽  
Spatial Symmetry ◽  
Prandtl Numbers ◽  
Steady Solutions ◽  
The Stability

Steady three-dimensional convection in the form of bimodal cells in a fluid layer heated from below with rigid boundaries is studied through numerical computations for Prandtl numbers in the range 10 [lsim ] P [lsim ] 100. The stability of the steady solutions with respect to disturbances of various symmetries has been analysed. Typically, the range of stable steady bimodal convection is restricted by the transition to oscillatory bimodal convection. The oscillations preserve the spatial symmetry of the steady bimodal convection pattern in the case of high P and higher wavenumbers, but break it in the case of lower P or lower wavenumbers in the range that has been investigated. Some comparisons are made with experimental observations. The transition from bimodal to knot convection has also been studied.

Nonlinear properties of convection rolls in a horizontal layer rotating about a vertical axis

1979 ◽  
Vol 94 (4) ◽  
pp. 609-627 ◽  
Author(s):  
R. M. Clever ◽  
F. H. Busse
Keyword(s):  
Rotation Rate ◽  
Fluid Layer ◽  
Three Dimensional ◽  
Vertical Axis ◽  
Horizontal Layer ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Nonlinear Properties ◽  
Prandtl Numbers ◽  
The Galerkin Method

Steady finite amplitude two-dimensional solutions are obtained for the problem of convection in a horizontal fluid layer heated from below and rotating about its vertical axis. Rigid boundaries with prescribed constant temperatures are assumed and the solutions are obtained numerically by the Galerkin method. The existence of steady subcritical finite amplitude solutions is demonstrated for Prandtl numbers P < 1. A stability analysis of the finite amplitude solutions is performed by superimposing arbitrary three-dimensional disturbances. A strong reduction in the domain of stable rolls occurs as the rotation rate is increased. The reduction is most pronounced at low Prandtl numbers. The numerical analysis confirms the small amplitude results of Küppers & Lortz (1969) that all two-dimensional solutions become unstable when the dimensionless rotation rate Ω exceeds a value of about 27 at P ≃ ∞. A brief discussion is given of the three-dimensional time-dependent forms of convection which are realized at rotation rates exceeding the critical value.

Heat-flow experiments in liquid 4He with a variable cylindrical geometry

1987 ◽  
Vol 174 ◽  
pp. 209-231 ◽  
Author(s):  
H. Gao ◽  
G. Metcalfe ◽  
T. Jung ◽  
R. P. Behringer
Keyword(s):  
Cylindrical Geometry ◽  
Small Scale ◽  
Onset Of Convection ◽  
Aspect Ratios ◽  
A Value ◽  
Prandtl Numbers ◽  
Convection Rolls ◽  
Flow Experiments ◽  
Increasing Function ◽  
Good Agreement

This paper first describes an apparatus for measuring the Nusselt number N versus the Rayleigh number R of convecting normal liquid 4He layers. The most important feature of the apparatus is its ability to provide layers of different heights d, and hence different aspect ratios [Gcy ]. The horizontal cross-section of each layer is circular, and [Gcy ] is defined by [Gcy ] = D/2d where D is the diameter of the layer. We report results for 2.4 [les ] [Gcy ] [les ] 16 and for Prandtl numbers Pr spanning 0.5 [lsim ] Pr [lsim ] 0.9 These results are presented in terms of the slope N1 = RcdN/dR evaluated just above the onset of convection at Rc. We find that N1 is only a slowly increasing function of [Gcy ] in the range 6 [lsim ] [Gcy ] [lsim ] 16, and that it has a value there which is quite close to 0.72. This value of N1 is in good agreement with variational calcuations by Ahlers et al. (1981) pertinent to parallel convection rolls in cylindrical geometry. Particularly for [Gcy ] [lsim ] 6, we find additional small-scale structure in N1 associated with changes in the number of convection rolls with changing [Gcy ]. An additional test of the linearzied hydrodynamics is given by measurements of Rc. We find good agreement between theory and our data for Rc.

Experimental and Theoretical Investigation of the Supersonic Boundary Layer Stability on the Swept Wing

2008 ◽  
Vol 3 (3) ◽  
pp. 34-38
Author(s):  
Sergey A. Gaponov ◽  
Yuri G. Yermolaev ◽  
Aleksandr D. Kosinov ◽  
Nikolay V. Semionov ◽  
Boris V. Smorodsky
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Leading Edge ◽  
Supersonic Boundary Layer ◽  
Mean Flow ◽  
Swept Wing ◽  
Wing Model ◽  
Dimensional Boundary ◽  
The Stability ◽  
Good Agreement

Theoretical and an experimental research results of the disturbances development in a swept wing boundary layer are presented at Mach number М = 2. In experiments development of natural and small amplitude controllable disturbances downstream was studied. Experiments were carried out on a swept wing model with a lenticular profile at a zero attack angle. The swept angle of a leading edge was 40°. Wave parameters of moving disturbances were determined. In frames of the linear theory and an approach of the local self-similar mean flow the stability of a compressible three-dimensional boundary layer is studied. Good agreement of the theory with experimental results for transversal scales of unstable vertices of the secondary flow was obtained. However the calculated amplification rates differ from measured values considerably. This disagreement is explained by the nonlinear processes observed in experiment

scholarly journals Different Benzendicarboxylate-Directed Structural Variations and Properties of Four New Porous Cd(II)-Pyridyl-Triazole Coordination Polymers

2020 ◽  
Vol 8 ◽  
Author(s):  
Ying Zhao ◽  
Jin Jing ◽  
Ning Yan ◽  
Min-Le Han ◽  
Guo-Ping Yang ◽  
...  
Keyword(s):  
Solid State ◽  
Coordination Polymers ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Gas Sorption ◽  
Structural Variations ◽  
Benzenedicarboxylic Acid ◽  
Topological Framework ◽  
The Stability

Four new different porous crystalline Cd(II)-based coordination polymers (CPs), i. e., [Cd(mdpt)2]·2H2O (1), [Cd2(mdpt)2(m-bdc)(H2O)2] (2), [Cd(Hmdpt)(p-bdc)]·2H2O (3), and [Cd3(mdpt)2(bpdc)2]·2.5NMP (4), were obtained successfully by the assembly of Cd(II) ions and bitopic 3-(3-methyl-2-pyridyl)-5-(4-pyridyl)-1,2,4-triazole (Hmdpt) in the presence of various benzendicarboxylate ligands, i.e., 1,3/1,4-benzenedicarboxylic acid (m-H2bdc, p-H2bdc) and biphenyl-4,4′-bicarboxylate (H2bpdc). Herein, complex 1 is a porous 2-fold interpenetrated four-connected 3D NbO topological framework based on the mdpt− ligand; 2 reveals a two-dimensional (2D) hcb network. Interestingly, 3 presents a three-dimensional (3D) rare interpenetrated double-insertion supramolecular net via 2D ···ABAB··· layers and can be viewed as an fsh topological net, while complex 4 displays a 3D sqc117 framework. Then, the different gas sorption performances were carried out carefully for complexes 1 and 4, the results of which showed 4 has preferable sorption than that of 1 and can be the potential CO2 storage and separation material. Furthermore, the stability and luminescence of four complexes were performed carefully in the solid state.

Resistive Ballooning Modes in Three-Dimensional Configurations

1982 ◽  
Vol 37 (8) ◽  
pp. 848-858 ◽  
Author(s):  
D. Correa-Restrepo
Keyword(s):  
Growth Rates ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Coupled System ◽  
Small Constant ◽  
Ballooning Mode ◽  
The Stability ◽  
The Magnetic Field ◽  
Symmetric Systems ◽  
The Ideal

Resistive ballooning modes in general three-dimensional configurations are studied on the basis of the equations of motion of resistive MHD. Assuming small, constant resistivity and perturbations localized transversally to the magnetic field, a stability criterion is derived in the form of a coupled system of two second-order differential equations. This criterion contains several limiting cases, in particular the ideal ballooning mode criterion and criteria for the stability of symmetric systems. Assuming small growth rates, analytical results are derived by multiple-length-scale expansion techniques. Instabilities are found, their growth rates scaling as fractional powers of the resistivity

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