scholarly journals Three-dimensional instability of axisymmetric flow in a rotating lid–cylinder enclosure

2001 ◽  
Vol 438 ◽  
pp. 363-377 ◽  
Author(s):  
A. Yu. GELFGAT ◽  
P. Z. BAR-YOSEPH ◽  
A. SOLAN
Keyword(s):  
Stability Problem ◽  
Axisymmetric Flow ◽  
Three Dimensional ◽  
Phase Velocities ◽  
Discrete Values ◽  
The Stability ◽  
Two Parameters ◽  
Dimensional Instability ◽  
Axisymmetric Instability ◽  
Axisymmetric Perturbations

The axisymmetry-breaking three-dimensional instability of the axisymmetric flow between a rotating lid and a stationary cylinder is analysed. The flow is governed by two parameters – the Reynolds number Re and the aspect ratio γ (=height/radius). Published experimental results indicate that in different ranges of γ axisymmetric or non-axisymmetric instabilities can be observed. Previous analyses considered only axisymmetric instability. The present analysis is devoted to the linear stability of the basic axisymmetric flow with respect to the non-axisymmetric perturbations. After the linearization the stability problem separates into a family of quasi-axisymmetric subproblems for discrete values of the azimuthal wavenumber k. The computations are done using the global Galerkin method. The stability analysis is carried out at various densely distributed values of γ in the range 1 < γ < 3.5. It is shown that the axisymmetric perturbations are dominant in the range 1.63 < γ < 2.76. Outside this range, for γ < 1.63 and for γ > 2.76, the instability is three-dimensional and sets in with k = 2 and k = 3 or 4, respectively. The azimuthal periodicity, patterns, characteristic frequencies and phase velocities of the dominant perturbations are discussed.

scholarly journals A Simple Analytical Model for Estimating the Dissolution-Driven Instability in a Porous Medium

Fluids ◽  
2018 ◽  
Vol 3 (3) ◽  
pp. 60
Author(s):  
Layachi Hadji
Keyword(s):  
Stability Problem ◽  
Three Dimensional ◽  
Step Function ◽  
New Approach ◽  
Engineering Mathematics ◽  
Alternative Approach ◽  
Function Base ◽  
The Stability ◽  
Rayleigh Benard ◽  
Profile Approach

This article deals with the stability problem that arises in the modeling of the geological sequestration of carbon dioxide. It provides a more detailed description of the alternative approach to tackling the stability problem put forth by Vo and Hadji (Physics of Fluids, 2017, 29, 127101) and Wanstall and Hadji (Journal of Engineering Mathematics, 2018, 108, 53–71), and it extends two-dimensional analysis to the three-dimensional case. This new approach, which is based on a step-function base profile, is contrasted with the usual time-evolving base state. While both provide only estimates for the instability threshold values, the step-function base profile approach has one great advantage in the sense that the problem at hand can be viewed as a stationary Rayleigh–Bénard problem, the model of which is physically sound and the stability of which is not only well-defined but can be analyzed by a variety of existing analytical methods using only paper and pencil.

1997 Best Paper Award—Turbomachinery Committee: A Study of Spike and Modal Stall Phenomena in a Low-Speed Axial Compressor

1998 ◽  
Vol 120 (3) ◽  
pp. 393-401 ◽  
Author(s):  
T. R. Camp ◽  
I. J. Day
Keyword(s):  
High Speed ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Short Length ◽  
Operating Conditions ◽  
Length Scale ◽  
Stall Inception ◽  
Low Speed ◽  
The Stability ◽  
Dimensional Instability

This paper presents a study of stall inception mechanisms in a low-speed axial compressor. Previous work has identified two common flow breakdown sequences, the first associated with a short length-scale disturbance known as a “spike,” and the second with a longer length-scale disturbance known as a “modal oscillation.” In this paper the physical differences between these two mechanisms are illustrated with detailed measurements. Experimental results are also presented that relate the occurrence of the two stalling mechanisms to the operating conditions of the compressor. It is shown that the stability criteria for the two disturbances are different: Long length-scale disturbances are related to a two-dimensional instability of the whole compression system, while short length-scale disturbances indicate a three-dimensional breakdown of the flow-field associated with high rotor incidence angles. Based on the experimental measurements, a simple model is proposed that explains the type of stall inception pattern observed in a particular compressor. Measurements from a single-stage low-speed compressor and from a multistage high-speed compressor are presented in support of the model.

A Study of Spike and Modal Stall Phenomena in a Low-Speed Axial Compressor

1997 ◽  
Author(s):  
T. R. Camp ◽  
I. J. Day
Keyword(s):  
High Speed ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Operating Conditions ◽  
Two Dimensional ◽  
Stability Criteria ◽  
Stall Inception ◽  
Low Speed ◽  
The Stability ◽  
Dimensional Instability

This paper presents a study of stall inception mechanisms a in low-speed axial compressor. Previous work has identified two common flow breakdown sequences, the first associated with a short lengthscale disturbance known as a ‘spike’, and the second with a longer lengthscale disturbance known as a ‘modal oscillation’. In this paper the physical differences between these two mechanisms are illustrated with detailed measurements. Experimental results are also presented which relate the occurrence of the two stalling mechanisms to the operating conditions of the compressor. It is shown that the stability criteria for the two disturbances are different: long lengthscale disturbances are related to a two-dimensional instability of the whole compression system, while short lengthscale disturbances indicate a three-dimensional breakdown of the flow-field associated with high rotor incidence angles. Based on the experimental measurements, a simple model is proposed which explains the type of stall inception pattern observed in a particular compressor. Measurements from a single stage low-speed compressor and from a multistage high-speed compressor are presented in support of the model.

Three-dimensional destabilization of Stuart vortices: the influence of rotation and ellipticity

1999 ◽  
Vol 387 ◽  
pp. 205-226 ◽  
Author(s):  
P. G. POTYLITSIN ◽  
W. R. PELTIER
Keyword(s):  
Cross Sections ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Steady State Solution ◽  
Vorticity Distribution ◽  
Columnar Vortex ◽  
Elliptical Instability ◽  
The Stability ◽  
Dimensional Instability ◽  
Stuart Vortices

We investigate the influence of the ellipticity of a columnar vortex in a rotating environment on its linear stability to three-dimensional perturbations. As a model of the basic-state vorticity distribution, we employ the Stuart steady-state solution of the Euler equations. In the presence of background rotation, an anticyclonic vortex column is shown to be strongly destabilized to three-dimensional perturbations when background rotation is weak, while rapid rotation strongly stabilizes both anticyclonic and cyclonic columns, as might be expected on the basis of the Taylor–Proudman theorem. We demonstrate that there exist three distinct forms of three-dimensional instability to which strong anticyclonic vortices are subject. One form consists of a Coriolis force modified form of the ‘elliptical’ instability, which is dominant for vortex columns whose cross-sections are strongly elliptical. This mode was recently discussed by Potylitsin & Peltier (1998) and Leblanc & Cambon (1998). The second form of instability may be understood to constitute a three-dimensional inertial (centrifugal) mode, which becomes the dominant mechanism of instability as the ellipticity of the vortex column decreases. Also evident in the Stuart model of the vorticity distribution is a third ‘hyperbolic’ mode of instability that is focused on the stagnation point that exists between adjacent vortex cores. Although this short-wavelength cross-stream mode is much less important in the spectrum of the Stuart model than it is in the case of a true homogeneous mixing layer, it nevertheless does exist even though its presence has remained undetected in most previous analyses of the stability of the Stuart solution.

Three-dimensional stability analysis for slurry-filled trench wall in cohesionless soil

1996 ◽  
Vol 33 (5) ◽  
pp. 798-808 ◽  
Author(s):  
Jiin-Song Tsai ◽  
Jia-Chyi Chang
Keyword(s):  
Stability Analysis ◽  
Stability Problem ◽  
Slip Surface ◽  
Three Dimensional ◽  
Vertical Direction ◽  
Soil Mass ◽  
Cohesionless Soil ◽  
Cohesionless Soils ◽  
Slurry Trench ◽  
The Stability

On the basis of the limiting equilibrium and arching theory, a three-dimensional analysis is proposed for slurry-supported trenches in cohesionless soils. This analytical approach is developed by considering the trench stability problem as a vertical soil cut within a fictitious half-silo with a rough wall surronding. Arching effects are considered not only in the vertical direction but also in the horizontal direction. A shell-shaped slip surface of the sliding soil mass is defined by Mohr-Coulomb criterion. The factor of safety is defines as the ratio of the resisting force induced by slurry pressure to the horizontal force required to maintain the stability of the trench wall. Results of the proposed method have been compared with those of two existing analytical methods for a typical trench stability problem. Key words: stability analysis, slurry trench wall, cohesionless soil.

Experimental Study on the Stability of the Flow Behind an Accelerating Circular Cylinder

2007 ◽  
Author(s):  
A. Inasawa ◽  
K. Toda ◽  
M. Asai
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Cylinder Wake ◽  
Global Instability ◽  
The Stability ◽  
Dimensional Instability ◽  
Wake Oscillation

Disturbance growth in the wake of a circular cylinder moving at a constant acceleration is examined experimentally. The cylinder is installed on a carriage moving in the still air. The results show that the critical Reynolds number for the onset of the global instability leading to a self-sustained wake oscillation increases with the magnitude of acceleration, while the Strouhal number of the growing disturbance at the critical Reynolds number is not strongly dependent on the magnitude of acceleration. It is also found that with increasing the acceleration, the Ka´rma´n vortex street remains two-dimensional even at the Reynolds numbers around 200 where the three-dimensional instability occurs to lead to the vortex dislocation in the case of cylinder moving at constant velocity or in the case of cylinder wake in the steady oncoming flow.

The equilibrium and stability of axisymmetric pendent drops

1976 ◽  
Vol 351 (1664) ◽  
pp. 117-127 ◽  
Keyword(s):  
Horizontal Plane ◽  
Constant Pressure ◽  
Pressure Head ◽  
Plane Interface ◽  
Wave Number ◽  
Capillary Length ◽  
Asymmetric Mode ◽  
The Stability ◽  
Axisymmetric Instability ◽  
Axisymmetric Perturbations

The stability of the equilibrium of axisymmetric drops suspended from a horizontal circular orifice is studied mathematically in this paper. For axisymmetric perturbations it has been shown by Pitts (1976) that limit point instability occurs at the positions of maximum volume, or of maxi­mum internal pressure at the point of support, depending on whether the drop is held at constant volume, or at constant pressure head respectively. Here a criterion is given for asymmetric instabilities. It is shown that bifur­cation of the equilibrium into an asymmetric mode, with azimuth wave-number m = 1, will occur when the profile of the drop becomes horizontal at the point of support. For a drop grown from an initial horizontal plane interface calculations show that when the orifice radius a is greater than 3.219... in units of the capillary length, onset of this instability will precede the axisymmetric instability. When a reaches the value 3.812... the m = 1 instability sets in at the plane interface, and we recover the earlier results of Plateau (1873) and Maxwell (1875) on the instability of horizontal plane interfaces. Higher order instabilities are briefly discussed, and it is shown that modes m = 2, 3, 4,... will not precede the m = 1 mode for a drop suspended in this way.

Magneto-gravitational convection in a vertical layer of ferrofluid in a uniform oblique magnetic field

2016 ◽  
Vol 795 ◽  
pp. 847-875 ◽  
Author(s):  
Habibur Rahman ◽  
Sergey A. Suslov
Keyword(s):  
Magnetic Field ◽  
Computational Cost ◽  
Three Dimensional ◽  
Vertical Layer ◽  
Gravitational Convection ◽  
Magnetic Parameters ◽  
Instability Modes ◽  
The Stability ◽  
Dimensional Instability ◽  
Oblique Magnetic Field

The stability of base gravitational convection in a layer of ferrofluid confined between two vertical wide and tall non-magnetic plates, heated from one side, cooled from the other and placed in a uniform oblique external magnetic field is studied. Two distinct mechanisms, thermo-gravitational and thermo-magnetic, are found to be responsible for the appearance of various stationary and wave-like instability modes. The characteristics of all instability modes are investigated as functions of the orientation angles of the applied magnetic field and its magnitude for various values of magnetic parameters when both the thermo-magnetic and gravitational buoyancy mechanisms are active. The original three-dimensional problem is cast in an equivalent two-dimensional form using generalised Squire’s transformations, which significantly reduces a computational cost. Subsequently, full three-dimensional instability patterns are recovered using the inverse Squire’s transformation, and the optimal field and pattern orientations are determined.

Numerical investigation of turbulent supersonic axisymmetric wakes

2012 ◽  
Vol 702 ◽  
pp. 488-520 ◽  
Author(s):  
Richard D. Sandberg
Keyword(s):  
Reynolds Number ◽  
Axisymmetric Flow ◽  
Three Dimensional ◽  
Recirculation Region ◽  
Stream Velocity ◽  
Near Wake ◽  
Wake Structure ◽  
Axisymmetric Mode ◽  
Turbulent Inflow ◽  
The Stability

AbstractNumerical experiments are conducted of turbulent supersonic axisymmetric wakes at Mach number $M= 2. 46$ and Reynolds number, based on free-stream velocity and base diameter, ${\mathit{Re}}_{D} = 1\ensuremath{\times} 1{0}^{5} $. Direct numerical simulations (DNS) are used to study the effect of approach flow conditions, and of specific azimuthal modes, on the near-wake behaviour. To that end, DNS are performed with laminar and turbulent approach boundary layers, and additional turbulent approach flow DNS with reduced circumferential size are conducted to deliberately eliminate certain azimuthal/helical modes. DNS with turbulent approach flow show an increased turning angle and increased growth of the separating shear layer, leading to a shorter recirculation region, a stronger recompression shock system, and ultimately good agreement with experimental data at considerably higher Reynolds number. A similar wake structure is found for laminar and turbulent inflow conditions, giving further evidence of the wake structure being a consequence of the global near-wake instabilities and not a result of upstream conditions. Stability analyses of two-dimensional basic states are carried out by computing the temporal pulse response using forced Navier–Stokes simulations to investigate which azimuthal modes are dominant for fully turbulent wakes and how the stability behaviour is influenced by the choice of basic state. Using the time- and azimuthally averaged data from three-dimensional DNS with turbulent inflow as basic state, an absolute instability of the axisymmetric mode was found and helical modes $m= 4, 5, 6$ were found to be linearly most unstable, in contrast to results obtained earlier using an axisymmetric flow solution as the basic state. The addition of a turbulence viscosity in the forced DNS retains most of the stability characteristics but reduces the wavenumber of the linearly most-amplified modes.

Inviscid spatial stability of a three-dimensional compressible mixing layer

1991 ◽  
Vol 231 ◽  
pp. 35-50 ◽  
Author(s):  
C. E. Grosch ◽  
T. L. Jackson
Keyword(s):  
Dimensional Stability ◽  
Stability Problem ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Mean Flow ◽  
Compressible Mixing Layer ◽  
Spatial Stability ◽  
Family Of Curves ◽  
The Stability

We present the results of a study of the inviscid spatial stability of a parallel three-dimensional compressible mixing layer. The parameters of this study are the Mach number of the fast stream, the ratio of the speed of the slow stream to that of the fast stream, the ratio of the temperature of the slow stream to that of the fast stream, the direction of the crossflow in the fast stream, the frequency, and the direction of propagation of the disturbance wave. Stability characteristics of the flow as a function of these parameters are given. Certain theoretical results are presented which show the interrelations between these parameters and their effects on the stability characteristics. In particular, the three-dimensional stability problem for a three-dimensional mixing layer at Mach zero can be transformed to a two-dimensional stability problem for an equivalent two-dimensional mean flow. There exists a one-parameter family of curves such that for any given direction of mean flow and of wave propagation one can apply this transformation and obtain the growth rate from the universal curves. For supersonic couvective Mach numbers, certain combinations of crossflow angle and propagation angle of the disturbance can increase the growth rates by a factor of about two. and thus enhance mixing.

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