A note on the stability of inviscid zonal jet flows on a rotating sphere

2012 ◽  
Vol 710 ◽  
pp. 154-165 ◽  
Author(s):  
Eiichi Sasaki ◽  
Shin-ichi Takehiro ◽  
Michio Yamada
Keyword(s):  
Linear Stability ◽  
Shooting Method ◽  
Jet Flows ◽  
Zonal Flows ◽  
Rotating Sphere ◽  
Rotation Rates ◽  
Critical Rotation ◽  
Critical Layers ◽  
The Stability ◽  
Linear Stability Problem

AbstractThe linear stability of inviscid zonal jet flows on a rotating sphere is re-examined. A semi-circle theorem for inviscid zonal flows on a rotating sphere is proved. It is also shown that numerically obtained eigenvalues of the linear stability problem do not converge well with a spectral method which was adopted in previous studies, due to an emergence of critical layers near the poles. By using a shooting method where the integral path bypasses the critical layers in the complex plane, the eigenvalues are successfully obtained with ${\ensuremath{\sim} }10\hspace{0.167em} \% $ correction of the critical rotation rates compared to those obtained in Baines (J. Fluid Mech., vol. 73, 1976, pp. 193–213).

Linear Stability of Viscous Zonal Jet Flows on a Rotating Sphere

2013 ◽  
Vol 82 (9) ◽  
pp. 094402 ◽  
Author(s):  
Eiichi Sasaki ◽  
Shin-ichi Takehiro ◽  
Michio Yamada
Keyword(s):  
Linear Stability ◽  
Jet Flows ◽  
Rotating Sphere

Stability of columnar convection in a porous medium

2013 ◽  
Vol 737 ◽  
pp. 205-231 ◽  
Author(s):  
Duncan R. Hewitt ◽  
Jerome A. Neufeld ◽  
John R. Lister
Keyword(s):  
Porous Medium ◽  
Linear Stability ◽  
Stability Problem ◽  
Columnar Structure ◽  
Marginal Stability ◽  
Vertically Propagating ◽  
The Stability ◽  
Rayleigh Benard ◽  
Linear Stability Problem ◽  
Dimensional Domain

AbstractConvection in a porous medium at high Rayleigh number $\mathit{Ra}$ exhibits a striking quasisteady columnar structure with a well-defined and $\mathit{Ra}$-dependent horizontal scale. The mechanism that controls this scale is not currently understood. Motivated by this problem, the stability of a density-driven ‘heat-exchanger’ flow in a porous medium is investigated. The dimensionless flow comprises interleaving columns of horizontal wavenumber $k$ and amplitude $\widehat{A}$ that are driven by a steady balance between vertical advection of a background linear density stratification and horizontal diffusion between the columns. Stability is governed by the parameter $A= \widehat{A}\mathit{Ra}/ k$. A Floquet analysis of the linear-stability problem in an unbounded two-dimensional domain shows that the flow is always unstable, and that the marginal-stability curve is independent of $A$. The growth rate of the most unstable mode scales with ${A}^{4/ 9} $ for $A\gg 1$, and the corresponding perturbation takes the form of vertically propagating pulses on the background columns. The physical mechanism behind the instability is investigated by an asymptotic analysis of the linear-stability problem. Direct numerical simulations show that nonlinear evolution of the instability ultimately results in a reduction of the horizontal wavenumber of the background flow. The results of the stability analysis are applied to the columnar flow in a porous Rayleigh–Bénard (Rayleigh–Darcy) cell at high $\mathit{Ra}$, and a balance of the time scales for growth and propagation suggests that the flow is unstable for horizontal wavenumbers $k$ greater than $k\sim {\mathit{Ra}}^{5/ 14} $ as $\mathit{Ra}\rightarrow \infty $. This stability criterion is consistent with hitherto unexplained numerical measurements of $k$ in a Rayleigh–Darcy cell.

scholarly journals On the stability of a falling liquid curtain

2002 ◽  
Vol 463 ◽  
pp. 163-171 ◽  
Author(s):  
PETER J. SCHMID ◽  
DAN S. HENNINGSON
Keyword(s):  
Linear Stability ◽  
Low Frequency ◽  
Frequency Range ◽  
Linear Superposition ◽  
Air Cushion ◽  
Global Modes ◽  
The Stability ◽  
Linear Stability Problem ◽  
Stability Results ◽  
Good Agreement

The stability of a falling liquid curtain is investigated. The sheet of liquid is assumed two-dimensional, driven by gravity and influenced by a compressible cushion of air enclosed on one side of the curtain. The linear stability problem is formulated in the form of an integro-differential eigenvalue problem. Although experimental efforts have consistently reported a peak in the low-frequency range of the spectrum, the linear stability results do not show instabilities at these frequencies. However, a multi-modal approach combined with a projection onto low-frequency modes reveals a dominant and robust instability feature that is in good agreement with experimental measurements. This instability manifests itself as a wave packet, consisting of a linear superposition of linear global modes, that travels down the curtain and causes a strong pressure signal in the enclosed air cushion.

scholarly journals On the linear stability of compressible plane Couette flow

1994 ◽  
Vol 258 ◽  
pp. 131-165 ◽  
Author(s):  
Peter W. Duck ◽  
Gordon Erlebacher ◽  
M. Yousuff Hussaini
Keyword(s):  
Linear Stability ◽  
Couette Flow ◽  
Small Amplitude ◽  
Reynolds Numbers ◽  
Inviscid Limit ◽  
Plane Couette Flow ◽  
Moving Wall ◽  
Type Boundary ◽  
The Stability ◽  
Radiation Type

The linear stability of compressible plane Couette flow is investigated. The appropriate basic velocity and temperature distributions are perturbed by a small-amplitude normal-mode disturbance. The full small-amplitude disturbance equations are solved numerically at finite Reynolds numbers, and the inviscid limit of these equations is then investigated in some detail. It is found that instabilities can occur, although the corresponding growth rates are often quite small; the stability characteristics of the flow are quite different from unbounded flows. The effects of viscosity are also calculated, asymptotically, and shown to have a stabilizing role in all the cases investigated. Exceptional regimes to the problem occur when the wave speed of the disturbances approaches the velocity of either of the walls, and these regimes are also analysed in some detail. Finally, the effect of imposing radiation-type boundary conditions on the upper (moving) wall (in place of impermeability) is investigated, and shown to yield results common to both bounded and unbounded flows.

A Study of the Lateral Stability of Railroad Vehicles Using a Nonlinear Constrained Multibody Formulation

2001 ◽  
Author(s):  
Davide Valtorta ◽  
Khaled E. Zaazaa ◽  
Ahmed A. Shabana ◽  
Jalil R. Sany
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Theory ◽  
Linear Stability Analysis ◽  
Numerical Study ◽  
Operating Conditions ◽  
Lateral Stability ◽  
Railroad Vehicles ◽  
Contact Formulation ◽  
The Stability

Abstract The lateral stability of railroad vehicles travelling on tangent tracks is one of the important problems that has been the subject of extensive research since the nineteenth century. Early detailed studies of this problem in the twentieth century are the work of Carter and Rocard on the stability of locomotives. The linear theory for the lateral stability analysis has been extensively used in the past and can give good results under certain operating conditions. In this paper, the results obtained using a linear stability analysis are compared with the results obtained using a general nonlinear multibody methodology. In the linear stability analysis, the sources of the instability are investigated using Liapunov’s linear theory and the eigenvalue analysis for a simple wheelset model on a tangent track. The effects of the stiffness of the primary and secondary suspensions on the stability results are investigated. The results obtained for the simple model using the linear approach are compared with the results obtained using a new nonlinear multibody based constrained wheel/rail contact formulation. This comparative numerical study can be used to validate the use of the constrained wheel/rail contact formulation in the study of lateral stability. Similar studies can be used in the future to define the limitations of the linear theory under general operating conditions.

Non-linear stability analysis of micropolar fluid lubricated journal bearings with turbulent effect

2019 ◽  
Vol 71 (1) ◽  
pp. 31-39
Author(s):  
Subrata Das ◽  
Sisir Kumar Guha
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Micropolar Fluid ◽  
Journal Bearing ◽  
Journal Bearings ◽  
Turbulent Regime ◽  
Content Type ◽  
Non Linear ◽  
The Stability ◽  
Bearing System

Purpose The purpose of this paper is to investigate the effect of turbulence on the stability characteristics of finite hydrodynamic journal bearing lubricated with micropolar fluid. Design/methodology/approach The non-dimensional transient Reynolds equation has been solved to obtain the non-dimensional pressure field which in turn used to obtain the load carrying capacity of the bearing. The second-order equations of motion applicable for journal bearing system have been solved using fourth-order Runge–Kutta method to obtain the stability characteristics. Findings It has been observed that turbulence has adverse effect on stability and the whirl ratio at laminar flow condition has the lowest value. Practical implications The paper provides the stability characteristics of the finite journal bearing lubricated with micropolar fluid operating in turbulent regime which is very common in practical applications. Originality/value Non-linear stability analysis of micropolar fluid lubricated journal bearing operating in turbulent regime has not been reported in literatures so far. This paper is an effort to address the problem of non-linear stability of journal bearings under micropolar lubrication with turbulent effect. The results obtained provide useful information for designing the journal bearing system for high speed applications.

Stability of a tokamak plasma with diffuse toroidal rotation

2020 ◽  
Vol 86 (5) ◽  
Author(s):  
O. E. López ◽  
L. Guazzotto
Keyword(s):  
Shooting Method ◽  
Tokamak Plasma ◽  
Solution Technique ◽  
Large Aspect Ratio ◽  
Internal Resonances ◽  
Flow Shear ◽  
Magnetic Shear ◽  
The Stability ◽  
External Kink ◽  
Selection Of

The present work considers the stability of a high- $\beta$ , large aspect ratio, circular plasma with diffuse profiles for the safety factor and the angular toroidal frequency (López & Guazzotto, Phys. Plasmas, vol. 24, 032501). An application of the Frieman–Rotenberg formalism results in a system of scalar eigenmode equations whose coupling is retained at the plasma–vacuum transition but is disregarded across the plasma column, which is a standard practice. The solution technique consists of a multidimensional shooting method for the poloidal harmonics; robust initial guesses are constructed by solving the dispersion relation in the static scenario with vanishing magnetic shear. Flow shear appears as a high- $\beta$ toroidal contribution, and we illustrate its destabilizing influence on $n=1$ external kink modes in the presence of ideal and resistive walls. Internal resonances are avoided by means of the selection of appropriate equilibrium parameters. The stabilizing influence of a finite positive average magnetic shear is also exemplified.

scholarly journals LOCATION AND STABILITY OF THE TRIANGULAR POINTS IN THE TRIAXIAL ELLIPTIC RESTRICTED THREE-BODY PROBLEM

2021 ◽  
Vol 57 (2) ◽  
pp. 311-319
Author(s):  
M. Radwan ◽  
Nihad S. Abd El Motelp
Keyword(s):  
Linear Stability ◽  
Periodic Orbits ◽  
Body Problem ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Stability Regions ◽  
Triangular Points ◽  
Problem Frame ◽  
The Stability ◽  
Three Body

The main goal of the present paper is to evaluate the perturbed locations and investigate the linear stability of the triangular points. We studied the problem in the elliptic restricted three body problem frame of work. The problem is generalized in the sense that the two primaries are considered as triaxial bodies. It was found that the locations of these points are affected by the triaxiality coefficients of the primaries and the eccentricity of orbits. Also, the stability regions depend on the involved perturbations. We also studied the periodic orbits in the vicinity of the triangular points.

Stability of Heat Pipes in Vapor-Dominated Systems

1999 ◽  
Author(s):  
Pouya Amili ◽  
Yanis C. Yortsos
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Oil Recovery ◽  
Critical Rayleigh Number ◽  
Stability Limit ◽  
Wave Number ◽  
Nuclear Waste Repository ◽  
Geothermal Systems ◽  
Dimensionless Numbers ◽  
The Stability

Abstract We study the linear stability of a two-phase heat pipe zone (vapor-liquid counterflow) in a porous medium, overlying a superheated vapor zone. The competing effects of gravity, condensation and heat transfer on the stability of a planar base state are analyzed in the linear stability limit. The rate of growth of unstable disturbances is expressed in terms of the wave number of the disturbance, and dimensionless numbers, such as the Rayleigh number, a dimensionless heat flux and other parameters. A critical Rayleigh number is identified and shown to be different than in natural convection under single phase conditions. The results find applications to geothermal systems, to enhanced oil recovery using steam injection, as well as to the conditions of the proposed Yucca Mountain nuclear waste repository. This study complements recent work of the stability of boiling by Ramesh and Torrance (1993).

Sign in / Sign up

Export Citation Format

Share Document