Two-dimensional Stability Analysis Research Articles

Three-Dimensional and Two-Dimensional Stability Analysis of Bentonite Slurry Trenches Using a Shear Strength Reduction Technique and Limit Equilibrium Methods

Bentonite slurry trenches are becoming increasingly popular in the excavation of trenches, especially for diaphragm wall construction. The problem that needs to be addressed is the stability of bentonite slurry trenches. This paper presents a stability analysis of trenches with temporary support from bentonite slurry, with unit weights ranging from 10.5 to 12.0 kN/m3 in the realistic stratum, and C3 in the Hue city area. Our analysis employs the Shear Strength Reduction Technique (SSR) with Mohr–Coulomb materials to numerically evaluate the factor of safety (FS). The finite element method (FEM) software program (RS2 v. 121 and RS3 v. 4.0) and the finite difference method (FDM) FLAC v. 7.0 software were used. Additionally, the limit equilibrium method (LEM) of Bell–Washbourne and three-dimensional (3D) Bishop were used to calculate trench stability. The results of the analysis show a good agreement between RS2 and FLAC2D, and between RS3 and FLAC3D. Secondly, upon comparison, it was noted that the factor of safety of the 3D software programs (RS3 v. 4.0, FLAC v. 7.0) was higher than that of the 2D software programs (RS2 v. 11.0, FLAC v. 7.0), ranging from 52.3 to 63.0% for trench lengths of 6 m. However, for trench lengths of 54 m, the factor of safety values in 2D and 3D configurations were nearly equal. Thirdly, the factor of safety of the Bell–Washbourne method (LEM) was lower than that of the numerical analysis methods (FLAC and RS programs). Using the three-dimensional numerical method appears to be effective for estimating stability.

Case study: 3D mobilized strength of compacted fill

Water infiltration can cause softening of compacted structural fill and a reduction of the shear strength from the peak compacted strength to the fully softened strength (FSS) with an accompanying reduction in drained factor of safety (FoS). This study presents two-dimensional (2D) and 3D stability analyses of a compacted fill slope failure that occurred 6 years after construction due to water leaking from a connection between the main and lateral water pipes in the water supply system. The compacted fill material primarily consists of high plasticity fine-grained soil. The 3D FoS at the end of construction is 2.44 using the peak compacted strength envelope. However, the 3D FoS is close to unity (1.0) when the FSS is assigned to the compacted fill material with the appropriate piezometric surface, which means the 2 H:1 V compacted fill slope softened to the FSS within 6 years. This is an interesting FSS case because the failure surface is 4 m deep and semi-circular, which differs from infiltration cases that exhibit a shallower and more planar surface.

Rimming flow in a rotating horizontal cylinder with phase change at the interface

The two-dimensional solutions and stability analysis are presented for an evaporating thin viscous liquid film flowing inside a uniformly heated rotating horizontal cylinder. A non-linear, fourth-order, partial differential evolution equation is obtained by simplifying mass, momentum, and energy conservation equations within the lubrication approximation. The effect of evaporation, gravity, viscous drag, surface tension, thermocapillary stress, and intermolecular forces has been taken into account. The numerical solutions of the model are validated against the existing experimental as well as the numerical results, along with the analytical result for the limiting cases of the present model. The film thickness model is solved to elucidate two-dimensional spatiotemporal solutions and their stability for a wide range of thermal and other parameters. The evaporative mass flux at the liquid–air interface results in unsteady solutions which are oscillatory in nature, and the amplitude of the oscillations increases with an increase in the evaporative flux. The film ruptures after some time and the rupture time is found to be inversely proportional to the evaporation number, a non-dimensional number quantifying the rate of evaporation. The linear stability analysis shows that the thermocapillary stresses as well as the long-range molecular forces destabilize the film. A negative disjoining pressure is shown to reduce the rupture time and vice versa. Evaporation (condensation) plays a destabilizing (stabilizing) role in the thin film flow. Non-linear computations are carried out for the steady profiles, validating the growth rates obtained from the linear stability analysis.

Two-dimensional face stability analysis in rock masses governed by the Hoek-Brown strength criterion with a new multi-horn mechanism

The face stability problem is a major concern for tunnels excavated in rock masses governed by the Hoek-Brown strength criterion. To provide an accurate prediction for the theoretical solution of the critical face pressure, this study adopts the piecewise linear method (PLM) to account for the nonlinearity of the strength envelope and proposes a new multi-horn rotational mechanism based on the Hoek-Brown strength criterion and the associative flow rule. The analytical solution of critical support pressure is derived from the energy-work balance equation in the framework of the plastic limit theorem; it is formulated as a multivariable nonlinear optimization problem relying on 2m dependent variables (m is the number of segments). Meanwhile, two classic linearized measures, the generalized tangential technique (GTT) and equivalent Mohr-Coulomb parameters method (EMM), are incorporated into the analysis for comparison. Surprisingly, the parametric study indicates a significant improvement in support pressure by up to 13% compared with the GTT, and as expected, the stability of the tunnel face is greatly influenced by the rock strength parameters. The stress distribution on the rupture surface is calculated to gain an intuitive understanding of the failure at the limit state. Although the limit analysis is incapable of calculating the true stress distribution in rock masses, a rough approximation of the stress vector on the rupture surface is permitted. In the end, sets of normalized face pressure are provided in the form of charts for a quick assessment of face stability in rock masses.

Numerical Analysis of Linear Traveling Wave in Rotating Rayleigh–Bénard Convection with an Adiabatic Sidewall

In rotating Rayleigh–Bénard problems, convection with traveling waves may occur near the sidewalls. The Rayleigh number, Taylor number and Prandtl number are involved in this phenomenon, and the convection mode is determined depending on their values. We focused on the onset of this convection with traveling waves under the assumption that centrifugal force is neglected. By conducting two-dimensional linear stability analyses assuming periodicity of the flow and temperature fields along the sidewall direction, we investigated the effect of the Taylor number and the Prandtl number on the critical Rayleigh number and also attempted to understand the phenomenon qualitatively through three-dimensional visualizations. It was exhibited that as the Taylor number increases, the wave number, the Rayleigh number and the phase speed are found to increase. On the other hand, as the Prandtl number decreases, the wavenumber and the Rayleigh number decrease, but the phase velocity increases. The present analyses suggest that convection modes localized near the sidewalls are unlikely to emerge for low Prandtl number cases, which are comparable to those of liquid metals.

First instability of the flow past two tandem cylinders with different diameters

Two-dimensional direct stability analysis and numerical simulation are used to identify and characterize the primary instability of the flow around two tandem cylinders with different diameters. The ratio of the diameter d of the upstream cylinder to the diameter D of the downstream cylinder is varied from 1.0 to 0.4, while the distance L from the center of the upstream cylinder to the front stagnation point of the downstream cylinder is varied from 1.0D to 8.0D. Three typical spacings are chosen to describe the different scenarios observed, and the results for each configuration are described in detail. Three different unstable modes are identified, and the structural sensitivity and nonlinear characteristics of each mode are investigated. The results show that when d/D = 0.6 and L/D ≥ 3.5, the migration of the eigenvalues corresponding to the most unstable modes can be observed for a range of Reynolds numbers. This is ultimately shown to be caused by the transition of the unstable mode. At the same time, this mode transition changes the core region of the instability mechanism. Moreover, two different mechanisms that lead to vortex shedding in the gap between the two cylinders and a secondary bifurcation behavior of the flow around the downstream cylinder under a specific configuration are observed. To obtain a more comprehensive view of the first instability, a transient growth analysis of the flow is also performed, and the critical Reynolds number in each case is discussed.

Two-dimensional stability analysis of finite-amplitude interfacial gravity waves in a two-layer fluid

We explore basic mechanisms for the instability of finite-amplitude interfacial gravity waves through a two-dimensional linear stability analysis of the periodic and irrotational plane motion of the interface between two unbounded homogeneous fluids of different density in the absence of background currents. The flow domains are conformally mapped into two half-planes, where the time-varying interface is always mapped onto the real axis. This unsteady conformal mapping technique with a suitable representation of the interface reduces the linear stability problem to a generalized eigenvalue problem, and allows us to accurately compute the growth rates of unstable disturbances superimposed on steady waves for a wide range of parameters. Numerical results show that the wave-induced Kelvin–Helmholtz (KH) instability due to the tangential velocity jump across the interface produced by the steady waves is the major instability mechanism. Any disturbances whose dominant wavenumbers are greater than a critical value grow exponentially. This critical wavenumber that depends on the steady wave steepness and the density ratio can be approximated by a local KH stability analysis, where the spatial variation of the wave-induced currents is neglected. It is shown, however, that the growth rates need to be found numerically with care and the successive collisions of eigenvalues result in the wave-induced KH instability. In addition, the present study extends the previous results for the small-wavenumber instability, such as modulational instability, of relatively small-amplitude steady waves to finite-amplitude ones.

Prediction of Cavitation Surge Onset Point by One-Dimensional Stability Analysis

Cavitation instabilities such as rotating cavitation and cavitation surge often occur in high speed turbopumps. It has been shown by a stability analysis that the cause of cavitation instabilities is explained by the positive mass flow gain factor, representing the increase of cavity volume against the decrease (increase) of flow rate (incidence angle). The cavitation compliance, representing the increase of cavity volume in response to the decrease of inlet pressure, determines the frequency of instabilities. However, one-dimensional stability analysis cannot be directly used for the prediction of the onset point of cavitation surge when quasi-steady assumption is applied. In the present study, unsteady characteristics, i.e. the phase lag in the response of cavity volume against the flow rate/inlet pressure fluctuations, are taken into account in a stability analysis of cavitation surge in the form of lag element. The onset criterion considering the time lag is newly proposed for one-dimensional stability analysis, and the criterion is validated by comparisons with a two-dimensional stability analysis based on a singularity method applied to a free streamline theory.

Modelling and dynamic mode analysis of compressor impeller spike-type stall with global stability approach

Flow instability associated with spike-stall restricts the development of advanced aero-engines with a high-speed centrifugal compressor. However, given the irregular structure of centrifugal compressor, few approaches can be applied to stability analysis. Considering the typical features of spike-stall, an implement of stability analysis method on the flow field of impeller inlet was proposed innovatively. Thus, in-depth comprehension of stall perturbation structure and accurate prediction of stall perturbation propagation can be obtained. In this work, an unsteady full annular simulation of a centrifugal compressor with a volute coupled with experimental validations was carried out by using the Reynolds averaged Navier–Stokes technique. Then, an advanced dynamic mode decomposition method named compressed DMD was employed to extract and analyze the dominant stall perturbation structure. Finally, considering the effects of eddy viscosity and molecular viscosity, a two-dimensional global stability analysis method based on the θ-z cylindrical surface at the impeller inlet is established to reveal the propagation behaviors of stall perturbation. The results indicate that the compressed DMD approach can capture low-frequency perturbation caused by stall accurately but not dependent on samplings. The flow structure near stall can be decomposed into the influenced mean flow, flow perturbation associated with rotor frequency and blade passing frequency with the frequencies of 1333 Hz and 9333 Hz, respectively, and stall perturbation with the frequency of 510 Hz. In the view of perturbation propagation features, it is mainly two propagations along the θ-z cylindrical surface. One is the reverse propagation to the upstream axially; the other is the secondary propagation to the blade leading edge with the effect of the main flow during the first propagation process. Both propagation paths deflect to the rotation direction, and the deflection degree of the second propagation paths is larger than that of the first one, thus, a triangular instability region is formed. The researches not only provide a new thought for the application of stability analysis in the centrifugal compressor but also clarify the propagation characteristics of stall disturbance, which lays a foundation for accurate flow control.

Analysis of loess landslide mechanism and numerical simulation stabilization on the Loess Plateau in Central China

Loess landslides have complicated deformation mechanisms. Accurately describing the internal failure deformation of loess landslides and establishing a theoretical method of landslide instability evaluation for the prevention of subsequent landslides have become important topics in western development project construction in China. This paper presents a case study of the Zhonglou Mountain landslide in Shaanxi Province, China. Based on field investigation results, a two-dimensional stability analysis model was constructed using the finite element method. Taking the deformation characteristics of the landslide as the research basis, the distribution laws of the displacement, stress, and shear strain of this landslide were identified with the strength reduction finite element numerical simulation method. Additionally, the safety factor was evaluated under normal and storm conditions. The numerical simulation results show that the horizontal tensile stress of the landslide was mainly distributed in the middle and upper parts of the landslide under normal conditions, while the vertical tensile stress was distributed near the sliding surface. Under heavy rainfall, the sliding force increased, and the anti-sliding force and anti-sliding section decreased; the location of the maximum shear strain shifted down from the middle and upper parts of the landslide body to the area with a shear crack, and the plastic shear strain area expanded along nearly the entire the sliding surface, leading to the occurrence of a landslide. Thus, the use of anti-slide piles to stabilize the landslide was proposed and tested. Monitoring points were arranged along the sliding surface to evaluate the displacement, stress, and strain responses. The on-site observation results agreed with the modeling results. The use of anti-slide piles was demonstrated to be an effective stabilization method for the Zhonglou Mountain landslide.

Coexisting Multiple Attractors in a Fourth Order Chua’s Circuit With Experimental Verifications by Analog and Digital Circuits

Exploring coexistence of multiple attractors brought by the multistability for circuits and systems has a significant meaning in the theoretical researches and practical applications for chaos. In this article, a succinct fourth order Chua’s circuit is proposed by replacing the negative resistance with an ordinary positive resistance in a traditional fourth order one. The two-dimensional stability analysis for equilibrium points shows that this circuit possesses one unstable saddle-focus point with index 1 and two stable node-focus points. Coexisting bifurcation models, multiple attractors and the corresponding attraction basins are revealed by a series of numerical simulations. The clear crisis scenario of the coexisting limit cycles of period-3 bridging the coexisting single-scroll attractors of chaos and the double-scroll one is observed by the bifurcation analyses. The dual-mode experimental verifications by the analog and digital circuits are carried out on the self-made printed circuit boards, which validate the simulated dynamical behaviors with the combination of physics and engineering.

The Effect of Rigging Angle on Longitudinal Direction Motion of Parafoil-Type Vehicle: Basic Stability Analysis and Wind Tunnel Test

The paraglider, a flexible flying vehicle, consists of a parafoil with flexible wings, suspension lines, and a suspended payload. At this time, the suspension lines have several parameters to be designed. Above all, a parameter called Rigging Angle (RA) is sensitive to the aerodynamic characteristics of a paraglider during flight. In this study, the effect of RA is clarified using the two-dimensional stability analysis and a wind tunnel test. The mechanisms about the parafoil-type vehicle stability are clarified through the experimental and analytical approaches as follows. The RA has an allowable range for a stable flight. When the RA is set out of the range, the parafoil cannot fly stably. Furthermore, the behavior of the parafoil wing in the case of lower RA than the allowable range is different from the case of higher RA. The parafoil collapses from the leading edge of the canopy and cannot glide in the case of lower RA.

The Streaming Instability in Two-dimensional Protoplanetary Disks

Abstract We conduct a two-dimensional (2D) stability analysis for the aerodynamically coupled system of gas and dust in protoplanetary disks using the WKB approximation. The 2D analysis may apply to a vertically integrated disk or a thin dust layer at the disk midplane. We show that the streaming instability exists in 2D and examine the growth rate and its dependence on the mean dust-to-gas ratio, ϵ ¯ , and the Stokes number, St. For small ϵ ¯ , the dust back reaction may be viewed as a perturbation to the wave system, and analytical results are successfully derived by the method of perturbation with respect to ϵ ¯ . At a given azimuthal wavenumber, m, the growth rate as a function of the radial wavenumber, kr, shows a peak corresponding to the resonance between the relative drift velocity and the Rossby/trivial waves in the gas. For small ϵ ¯ , the peak growth rate at resonant modes increases as m 1/2 when the Rossby wave effect is negligible and the gas wave becomes trivial. With increasing St, the peak growth rate increases as St 1/2 at St ≪ 1, peaks at St ∼ 1, and finally decreases as St −1/2 for St ≫ 1. We find that the growth rate scales with ϵ ¯ as ϵ ¯ 1 / 2 at ϵ ¯ ≪ 1 and peaks at ϵ ¯ ∼ 1 . The instability is physically interpreted as a runaway process of dust trapping in relative drift-induced anticyclonic vortices. The effects of the compressibility and viscosity of the gas, the applicability of the WKB application, and the effects of the Keplerian shear are discussed.

A flow on the verge of turbulent breakdown

Stably stratified sheared flows are ubiquitous in geophysical flows from the ocean to the stars, and the route to turbulence in these flows remains an open question. The article by Lefauve et al. (J. Fluid Mech., vol. 848, 2018, pp. 508–544) is an invitation to this journey. With impressive experimental precision mastered by few teams in the world, the nature of the coherent structure that dominates the flow on the verge of turbulent breakdown is revealed and analysed through one- or two-dimensional modern stability analysis of an experimentally obtained base flow. The effect of confinement is surprisingly strong, advocating for leaving the textbook flows, inhomogeneous in only one direction, for the more complex shores of real flows, now accessible to analysis of multidimensional stability problems. The route explored by Lefauve et al. (2018) renews with the long tradition of the supercritical bifurcation scenario, it revisits the linear stability theory with possibility of resonances, critical layers and more to be imagined, since complex base flows are now available to explore both experimentally and analytically.

Investigation of landslide failure mechanisms adjacent to lignite mining operations in North Bohemia (Czech Republic) through a limit equilibrium/finite element modelling approach

Understanding the impact of data uncertainty is a fundamental part of ensuring safe design of manmade excavations. Although good levels of knowledge are achievable from field investigations and experience, a natural geological environment is subject to intrinsic variability that may compromise the correct prediction of the system response to the perturbations caused by mining, with direct consequences for the stability and safety of the operations.Different types of geoscientific evidence, including geological, geomorphic, geotechnical, geomatics, and geophysical data have been used to develop and perform two-dimensional Limit Equilibrium and Finite Element Method stability analyses of a lignite open-pit mine in North Bohemia (Czech Republic) affected by recent landslides. A deterministic-probabilistic approach was adopted to investigate the effect of uncertainty of the input parameters on model response. The key factors affecting the system response were identified by specific Limit Equilibrium sensitivity analyses and studied in further detail by Finite Element probabilistic analyses and the results were compared. The work highlights that complementary use of both approaches can be recommended for routine checks of model response and interpretation of the associated results. Such an approach allows a reduction of system uncertainty and provides an improved understanding of the landslides under study. Importantly, two separate failure mechanisms have been identified from the analyses performed and verified through comparisons with inclinometer data and field observations. The results confirm that the water table level and material input parameters have the greatest influence on the stability of the slope.

Successive bifurcations in a fully three-dimensional open cavity flow

The transition to unsteadiness of a three-dimensional open cavity flow is investigated using the joint application of direct numerical simulations and fully three-dimensional linear stability analyses, providing a clear understanding of the first two bifurcations occurring in the flow. The first bifurcation is characterized by the emergence of Taylor–Görtler-like vortices resulting from a centrifugal instability of the primary vortex core. Further increasing the Reynolds number eventually triggers self-sustained periodic oscillations of the flow in the vicinity of the spanwise end walls of the cavity. This secondary instability causes the emergence of a new set of Taylor–Görtler vortices experiencing a spanwise drift directed toward the spanwise end walls of the cavity. While a two-dimensional stability analysis would fail to capture this secondary instability due to the neglect of the lateral walls, it is the first time to our knowledge that this drifting of the vortices can be entirely characterized by a three-dimensional linear stability analysis of the flow. Good agreements with experimental observations and measurements strongly support our claim that the initial stages of the transition to turbulence of three-dimensional open cavity flows are solely governed by modal instabilities.

Stable solitons in the 1D and 2D generalized nonlinear Schrödinger equations with the periodic effective mass and [formula omitted]-symmetric potentials

We explore the influence of effective-mass modulation on beam dynamics in the one- and two-dimensional generalized linear and nonlinear Schrödinger equations with parity-time (PT) symmetric periodic potentials. In the linear regime, we successively give the first and second PT threshold curves of optical lattices under the modulation of different effective masses, and analyze the associated band structure and diffraction dynamics of beams. In the Kerr-nonlinear regime, a family of novel optical solitons can exist in the semi-infinite gap. As the effective-mass parameter grows, the existing range of these solitons gradually increases whereas their stable domain gradually diminishes. Two-dimensional (2D) stability analysis reveals that the larger effective-mass parameter, gain-and-loss amplitude, and propagation constant may more easily lead to instable solitons. The 1D and 2D transverse power flows are also examined. Our results may open a new design window for the nonlinear optics in the effective mass and PT-symmetric lattices and other related fields.

Aspect ratio effect on electroconvection in a suspended liquid crystal film with a rectangular boundary

The aspect ratio dependence of the electroconvection phenomenon in a suspended nematic liquid crystal film with a rectangular boundary is investigated. Two-dimensional global stability analysis is carried out on the coupled electrohydrodynamic system to calculate the instability boundary of the phenomenon for different aspect ratios. The calculated critical $R$ number (Rayleigh-like number) shows a rapidly decreasing trend in the low-aspect-ratio region (roughly $\unicode[STIX]{x1D6FE}<1.5$, where $\unicode[STIX]{x1D6FE}$ is defined as the aspect ratio of the film), and then the variation becomes slow until $\unicode[STIX]{x1D6FE}\approx 2.5$, where the critical $R$ number starts to increase slightly. Convective patterns of liquid films with different aspect ratios are also obtained from stability analysis and validated by particle image velocimetry measurement.

Three-dimensional reinforced slopes: Evaluation of required reinforcement strength and embedment length using limit analysis

Virtually all analyses of reinforced earth structures are based on two-dimensional (2D) conditions, ignoring three-dimensional (3D) effects despite adhering to three-dimensional behavior in reality. In this study, the three-dimensional effects of three-dimensional conditions on reinforced earth structure stability are considered, and employed to determine the required strength and length of reinforcement using a Limit Analysis approach. The results of the three-dimensional stability analysis are compared with idealized two-dimensional results and examples from prior literature. As expected, use of two-dimensional stability analyses for reinforced soil slopes yield results that are more conservative than those considering three-dimensional effects. The effects of various parameters on the three-dimensional solution are investigated and design charts are proposed. The solutions demonstrate that for practical scenarios, three-dimensional effects may affect the required length of reinforcement more than the associated required tensile strength.

Two-dimensional stability analysis of a soil slope using the finite element method and the limit equilibrium principle

A slope stability analysis method using the finite element method and the limit equilibrium principle is presented in this paper to determine the critical slip surface and to calculate the minimum safety factor based on the stress field produced by a numerical simulation. This method assumes a cutting export and makes good use of geometric combination to reduce the search range during the calculation process. The proposed method was validated using two classical benchmark slopes and an engineering slope; it was also compared with other conventional limit equilibrium methods. The error between the proposed method and the limit equilibrium method was relatively small. The proposed method resolved several limitations of the traditional methods, and a comparison of the benchmark slopes showed that the proposed method exhibited good accuracy and efficiency. The proposed method can thus analyse both the stability of a natural slope and the stability of a soil slope under seismic loading conditions.