Three-dimensional stability analysis for a salt-finger convecting layer

2018 ◽  
Vol 841 ◽  
pp. 636-653
Author(s):  
Ting-Yueh Chang ◽  
Falin Chen ◽  
Min-Hsing Chang
Keyword(s):  
Stability Analysis ◽  
Three Dimensional ◽  
Longitudinal Mode ◽  
Transverse Mode ◽  
Solute Diffusion ◽  
Two Dimensional ◽  
Grashof Number ◽  
Significant Difference ◽  
The Stability ◽  
Mode A

A three-dimensional linear stability analysis is carried out for a convecting layer in which both the temperature and solute distributions are linear in the horizontal direction. The three-dimensional results show that, for $Le=3$ and 100, the most unstable mode occurs invariably as the longitudinal mode, a vortex roll with its axis perpendicular to the longitudinal plane, suggesting that the two-dimensional results are sufficient to illustrate the stability characteristics of the convecting layer. Two-dimensional results show that the stability boundaries of the transverse mode (a vortex roll with its axis perpendicular to the transverse plane) and the longitudinal modes are virtually overlapped in the regime dominated by thermal diffusion and the regime dominated by solute diffusion, while these two modes hold a significant difference in the regime the salt-finger instability prevails. More precisely, the instability area in terms of thermal Grashof number $Gr$ and solute Grashof number $Gs$ is larger for the longitudinal mode than the transverse mode, implying that, under any circumstance, the longitudinal mode is always more unstable than the transverse mode.

scholarly journals Linear stability of confined flow around a 180-degree sharp bend

2017 ◽  
Vol 822 ◽  
pp. 813-847 ◽  
Author(s):  
Azan M. Sapardi ◽  
Wisam K. Hussam ◽  
Alban Pothérat ◽  
Gregory J. Sheard
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Two Dimensional ◽  
Two Dimensional Flow ◽  
The Stability

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

scholarly journals Stability Analysis of Axisymmetric Concave Slopes Based on Two-Dimensional Limit Equilibrium Approach considering Additional Shear Resistance

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Xiao-ming Liu ◽  
Rui Zhang ◽  
Jie Han ◽  
Sha Chen
Keyword(s):  
Stability Analysis ◽  
Limit Equilibrium ◽  
Three Dimensional ◽  
Iteration Algorithm ◽  
Engineering Practice ◽  
Two Dimensional ◽  
Spatial Effects ◽  
Practical Applications ◽  
Equilibrium Approach ◽  
The Stability

Axisymmetric concave slopes, one special type of three-dimensional (3D) slopes, may be encountered in mining and civil engineering practice. Analysis of 3D slopes is generally complex and mostly relies on complicated numerical simulations. This paper proposes an elastoplastic solution for determining the additional shear resistances due to spatial effects of axisymmetric concave slopes. By incorporating the extra antislide forces, this paper proposes a simplified two-dimensional (2D) limit equilibrium procedure for the stability analysis of axisymmetric concave slopes. Combined with an iteration algorithm, the procedure can obtain the factors of safety for axisymmetric concave slopes in a simple and efficient way. Comparisons of the results from the proposed method and the numerical software FLAC3D are performed to demonstrate the validity of the proposed method for practical applications. Finally, the effects of several key parameters on the stability of axisymmetric concave slopes are investigated through a parametric study.

scholarly journals Three-dimensional Floquet stability analysis of the wake of a circular cylinder

1996 ◽  
Vol 322 ◽  
pp. 215-241 ◽  
Author(s):  
Dwight Barkley ◽  
Ronald D. Henderson
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Circular Cylinder ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Neutral Stability ◽  
Two Dimensional ◽  
Numerical Stability Analysis ◽  
Mode A

Results are reported from a highly accurate, global numerical stability analysis of the periodic wake of a circular cylinder for Reynolds numbers between 140 and 300. The analysis shows that the two-dimensional wake becomes (absolutely) linearly unstable to three-dimensional perturbations at a critical Reynolds number of 188.5±1.0. The critical spanwise wavelength is 3.96 ± 0.02 diameters and the critical Floquet mode corresponds to a ‘Mode A’ instability. At Reynolds number 259 the two-dimensional wake becomes linearly unstable to a second branch of modes with wavelength 0.822 diameters at onset. Stability spectra and corresponding neutral stability curves are presented for Reynolds numbers up to 300.

Stability of the wakes of cylinders with triangular cross-sections

2018 ◽  
Vol 844 ◽  
pp. 721-745 ◽  
Author(s):  
Zhi Y. Ng ◽  
Tony Vo ◽  
Gregory J. Sheard
Keyword(s):  
Stability Analysis ◽  
Cross Sections ◽  
Reynolds Numbers ◽  
Base Flow ◽  
Half Period ◽  
Two Dimensional ◽  
Instability Modes ◽  
The Stability ◽  
Time Periodic ◽  
Mode A

The stability of the wakes of cylinders with triangular cross-sections at incidence is investigated using Floquet stability analysis to elucidate the effects of cylinder inclination on the dominant flow instability. The upper limit of the Reynolds numbers (scaled by the height projected by the cylinder in this study) at which the wake of the two-dimensional base flow is time periodic is$Re\approx 140$for most cylinder inclinations, exceeding which the flow becomes aperiodic, restricting the range of Reynolds numbers permitted for the stability analysis. Two different instability modes are predicted to manifest as the first-occurring mode at various cylinder inclinations – a regular mode possessing perturbation structures consistent with mode A dominates the wakes of cylinders at inclinations$\unicode[STIX]{x1D6FC}\lesssim 34.6^{\circ }$and$\unicode[STIX]{x1D6FC}\gtrsim 55.4^{\circ }$, with a subharmonic mode consistent with mode C emerging as the primary mode in the wakes of the cylinder at the intermediate range of inclinations. For all inclinations, the mode B branch is not detected within the range of Reynolds numbers examined. The peak instability growth rates corresponding to mode A for all cylinder inclinations describe a linear variation with$(Re-Re_{A})/Re_{A}$, where$Re_{A}$is the mode A transition Reynolds number, while those corresponding to mode C vary only approximately linearly. The generalized trend most pertinently shows mode C to develop more rapidly than mode A at inclinations which permit it. Examination of the near wake of the two-dimensional time-periodic base flow demonstrates the dependence of the development and intensity of mode C on imbalances in the flow solution over each shedding period, directly implying that the two-dimensional base flow solutions deviate from the half-period-flip map as the cylinder inclination is increased. The degree of asymmetry of the two-dimensional base flow relative to the ideal half-period-flip map is quantified using several measures. The results show distinctly different trends in these asymmetry measures between inclinations where modes A or C are dominant, agreeing with results from the stability analysis. The nature of the predicted instability modes at transition are also investigated by applying the Stuart–Landau equation, showing the transitions to be supercritical for all cylinder inclinations, with mode C being consistently more strongly supercritical than mode A.

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.

scholarly journals Directional solidification of a binary alloy into a cellular convective flow: localized morphologies

1999 ◽  
Vol 395 ◽  
pp. 253-270 ◽  
Author(s):  
Y.-J. CHEN ◽  
S. H. DAVIS
Keyword(s):  
Directional Solidification ◽  
Binary Alloy ◽  
Temporal Dynamics ◽  
Three Dimensional ◽  
Multiple Scale ◽  
Solute Diffusion ◽  
Two Dimensional ◽  
Morphological Instability ◽  
Flow Strength ◽  
Weakly Nonlinear

A steady, two-dimensional cellular convection modifies the morphological instability of a binary alloy that undergoes directional solidification. When the convection wavelength is far longer than that of the morphological cells, the behaviour of the moving front is described by a slow, spatial–temporal dynamics obtained through a multiple-scale analysis. The resulting system has a parametric-excitation structure in space, with complex parameters characterizing the interactions between flow, solute diffusion, and rejection. The convection in general stabilizes two-dimensional disturbances, but destabilizes three-dimensional disturbances. When the flow is weak, the morphological instability is incommensurate with the flow wavelength, but as the flow gets stronger, the instability becomes quantized and forced to fit into the flow box. At large flow strength the instability is localized, confined in narrow envelopes. In this case the solutions are discrete eigenstates in an unbounded space. Their stability boundaries and asymptotics are obtained by a WKB analysis. The weakly nonlinear interaction is delivered through the Lyapunov–Schmidt method.

scholarly journals The Biomechanics of Erections: Modeling the Penis as a One-Comparment Pressurized Vessel vs. Modeling it as a Two-Compartments Pressurized Vessel

2009 ◽  
Vol 3 (2) ◽  
Author(s):  
A. Mohamed ◽  
A. Erdman ◽  
G. Timm
Keyword(s):  
Structural Integrity ◽  
Tunica Albuginea ◽  
Three Dimensional ◽  
The Body ◽  
Physical Models ◽  
Von Mises Stress ◽  
Two Dimensional ◽  
Biomechanical Models ◽  
Significant Difference ◽  
The One

Previous biomechanical models of the penis that have attempted to simulate penile erections have either been limited to two-dimensional geometry, simplified three-dimensional geometry or made inaccurate assumptions altogether. Most models designed the shaft of the penis as a one-compartment pressurized vessel fixed at one end, when in reality it is a two-compartments pressurized vessel, in which the compartments diverge as they enter the body and are fixed at two separate points. This study began by designing simplified two-dimensional and three-dimensional models of the erect penis using Finite Element Analysis (FEA) methods with varying anatomical considerations for analyzing structural stresses, axial buckling and lateral deformation. The study then validated the results by building physical models replicating the computer models. Finally a more complex and anatomically accurate model of the penis was designed and analyzed. There was a significant difference in the peak von-Mises stress distribution between the one-compartment pressurized vessel and the more anatomically correct two-compartments pressurized vessel. Furthermore, the two-compartments diverging pressurized vessel was found to have more structural integrity when subject to external lateral forces than the one-compartment pressurized vessel. This study suggests that Mother Nature has favored an anatomy of two corporal cavernosal bodies separated by a perforated septum as opposed to one corporal body, due to better structural integrity of the tunica albuginea when subject to external forces.

On the stability of two-dimensional stagnation flow

1970 ◽  
Vol 44 (3) ◽  
pp. 461-479 ◽  
Author(s):  
J. Kestin ◽  
R. T. Wood
Keyword(s):  
Blunt Body ◽  
Theoretical Estimate ◽  
Three Dimensional ◽  
Liouville Problem ◽  
Radius Of Curvature ◽  
Two Dimensional ◽  
Stagnation Flow ◽  
Stagnation Region ◽  
Free Stream Turbulence ◽  
The Stability

The paper examines the stability of the uniform flow which approaches a two-dimensional stagnation region formed when a cylinder or a two-dimensional blunt body of finite curvature is immersed in a crossflow. It is shown that such a flow is unstable with respect to three-dimensional disturbances. This conclusion is reached on the basis of a mathematical analysis of a simplified form of the disturbance equation for the stream-wise component of the vorticity vector. The ultimate, or stable, flow pattern is governed by a singular Sturm–Liouville problem whose solution possesses a single eigenvalue. The resulting flow is one in which a regularly distributed system of counter-rotating vortices is super-imposed on the basic, Hiemenz-like pattern of streamlines. The spacing of the vortices is a unique function of the characteristics of the flow, and a theoretical estimate for it agrees well with experimental results. The analysis is extended heuristically to include the effect of free-stream turbulence on the spacing.The problem is similar to the classical Görtler–Hämmerlin study of the stability of stagnation flow against an infinite flat plate, which revealed the existence of a spectrum of eigenvalues for the disturbance equation. The present analysis yields the same result when an infinite radius of curvature is assumed for the blunt body.

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