Slender-body analysis of the motion and stability of a vortex filament containing an axial flow

1971 ◽  
Vol 50 (2) ◽  
pp. 335-353 ◽  
Author(s):  
Sheila E. Widnall ◽  
Donald B. Bliss
Keyword(s):  
Travelling Waves ◽  
Stability Boundary ◽  
Axial Flow ◽  
Vortex Pair ◽  
Force Balance ◽  
Slender Body ◽  
Vortex Filament ◽  
Core Radius ◽  
Single Filament ◽  
The Stability

Previous results concerning the effects of axial velocity on the motion of vortex filaments are reviewed. These results suggest that a slender-body force balance between the Kutta–Joukowski lift on the vortex cross-section and the momentum flux within the curved filament will give some insight into the behaviour of the filament. These simple ideas are exploited for both a single vortex filament and a vortex pair, both containing axial flow. The stability of a straight vortex filament containing an axial flow to long wave sinusoidal displacements of its centre-line is investigated and the stability boundary obtained. The effect of axial flow on the stability of a vortex pair is explored. It is shown that to lowest order (in the ratio of vortex core radius to distance between the vortices) the effect of axial flow is to reduce the self-induced rotation of a single filament and that this effect can be considered as a change in effective core radius. To the next order, travelling waves appear in the instability, the instability mode for the vortex pair becomes non-planar but the amplification rate of the instability is not affected.

Nonaxisymmetric Waves of a Stratified Vertical Vortex

1992 ◽  
Vol 59 (2) ◽  
pp. 445-449 ◽  
Author(s):  
Y. T. Fung
Keyword(s):  
Vortex Breakdown ◽  
Fluid Layer ◽  
Stability Boundary ◽  
Vortex Sheet ◽  
Boussinesq Approximation ◽  
Force Balance ◽  
Flow Profile ◽  
Vortex Sheets ◽  
Interfacial Conditions ◽  
The Stability

The interfacial conditions for a cylindrical and an axial vortex sheet or thin fluid layer are obtained for a general class of vortex flows in a radius and gravity-stratified environment. The flow is assumed to be inviscid and incompressible. No Boussinesq approximation is required. In addition to the kinematic and dynamic conditions that the flow has to satisfy in the centrifugal and gravitational directions, a third condition, which restrains the interaction of the centrifugal and gravitational force fields, has to be imposed on those vortex sheets. This is consistent with the previous derived criteria for this type of vortex motions, in which a third condition based on pressure and force balance must be satisfied. Nonaxisymmetric instability for a special flow profile is examined and the stability boundary is obtained to show the behavior of this type of stratified vertical vortex. The results provide us with some information on the instability mechanism for the generation of the horizontal vortices in the ocean and for the spiral type of vortex breakdown in tornadoes and waterspouts in the atmosphere.

An Experimental Investigation of System Effects in Axial Flow Compressors

1998 ◽  
Author(s):  
Peter O. Silkowski ◽  
Hyoun-Woo Shin
Keyword(s):  
Stability Boundary ◽  
Axial Flow ◽  
Local Environment ◽  
Small Scale ◽  
Length Scale ◽  
Compression System ◽  
Stable Component ◽  
And Performance ◽  
Flow Compressor ◽  
The Stability

In this paper, experimental results are presented which demonstrate the potential importance of component coupling and system effects on the pumping stability and performance of a rotor in an axial flow compressor. Three different configurations are presented: (1) series coupling, (2) parallel coupling, and (3) combined series/parallel coupling. In all three configurations the more stable component or region successfully stabilizes the entire compression system beyond the stability boundary of the less stable component or region. A novel method for assessing the relative stability of the different regions is employed. This method utilizes the small scale pre-stall disturbances of the compressor to probe the stability of the different regions. The role of length scales or domains of dependence in determining stability is consistently demonstrated. Specifically, the stability of small length scale disturbances is shown to be governed by the local environment while being relatively insensitive to other components. However, the stability of larger length scale phenomena is affected by the global environment created by considering all of the parts of the overall system. Consideration and knowledge of these results is important when modeling, analyzing, or designing a compression system. Furthermore, these results serve as a cautionary note when interpreting solutions from a simplified analytical/computational analysis.

scholarly journals The motion of a buoyant vortex filament

2018 ◽  
Vol 857 ◽  
Author(s):  
Ching Chang ◽  
Stefan G. Llewellyn Smith
Keyword(s):  
Analytic Solution ◽  
Axial Flow ◽  
External Pressure ◽  
Transverse Component ◽  
Tangential Component ◽  
Force Balance ◽  
Vortex Filament ◽  
Vortex Rings ◽  
Asymptotic Model ◽  
Vortex Element

We investigate the motion of a thin vortex filament in the presence of buoyancy. The asymptotic model of Moore & Saffman (Phil. Trans. R. Soc. Lond.A, vol. 272, 1972, pp. 403–429) is extended to take account of buoyancy forces in the force balance on a vortex element. The motion of a buoyant vortex is given by the transverse component of force balance, while the tangential component governs the dynamics of the structure in the core. We show that the local acceleration of axial flow is generated by the external pressure gradient due to gravity. The equations are then solved for vortex rings. An analytic solution for a buoyant vortex ring at a small initial inclination is obtained and asymptotically agrees with the literature.

A Method for Evaluating the Aerodynamic Stability of Multi-Stage Axial-Flow Compressors

2009 ◽  
Author(s):  
Tsuguji Nakano ◽  
Andy Breeze-Stringfellow
Keyword(s):  
Steady State ◽  
Flow Field ◽  
Stability Boundary ◽  
Axial Flow ◽  
Pressure Rise ◽  
State Density ◽  
Constant Density ◽  
Multi Stage ◽  
The Stability ◽  
Axial Flow Compressors

A new simple engineering parameter to evaluate the stability of multi-stage axial compressors has been derived. It is based on the stability analysis for a small circumferential disturbance imposed on the steady state flow field. The analytical model assumes that the flow field is two dimensional and incompressible in the ducts between blade rows although the steady state density is permitted to change across the blade rows. The resulting stall parameter contains terms that relate to the slope of the pressure rise characteristic of the blade rows and the inertia effects of the fluid in the blade rows and ducts. The parameter leads to the classical stability criteria based on the slope of the overall total to static pressure rise coefficient in the limit where constant density and constant blade rotational speed are assumed across the compressor. The proposed stall parameter has been calculated for three different multi-stage axial flow compressors and the results indicate that the parameter has a strong correlation with the measured stability of the compressors. The good correlation with the test data demonstrates that the newly derived stall parameter captures much of the fundamental physics of instability inception in multi-stage compressors, and that it can be a good guideline for designers and engineers needing to evaluate the stability boundary of multi-stage machines.

Tertiary solutions and their stability in rotating plane Couette flow

1998 ◽  
Vol 358 ◽  
pp. 357-378 ◽  
Author(s):  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Stability Boundary ◽  
Streamwise Vortex ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Plane Couette Flow ◽  
Streamwise Direction ◽  
The Stability ◽  
Oscillatory Instabilities

The stability of nonlinear tertiary solutions in rotating plane Couette flow is examined numerically. It is found that the tertiary flows, which bifurcate from two-dimensional streamwise vortex flows, are stable within a certain range of the rotation rate when the Reynolds number is relatively small. The stability boundary is determined by perturbations which are subharmonic in the streamwise direction. As the Reynolds number is increased, the rotation range for the stable tertiary motions is destroyed gradually by oscillatory instabilities. We expect that the tertiary flow is overtaken by time-dependent motions for large Reynolds numbers. The results are compared with the recent experimental observation by Tillmark & Alfredsson (1996).

Nonstationary Response of a Two-Degrees-of-Freedom Nonlinear Ship Model Under Modulated Excitation

1995 ◽  
Author(s):  
Ruigui Pan ◽  
Huw G. Davies
Keyword(s):  
Degrees Of Freedom ◽  
Stability Boundary ◽  
Period Doubling ◽  
Approximate Solutions ◽  
Loss Of Stability ◽  
Two Degrees Of Freedom ◽  
Ship Model ◽  
Nonstationary Response ◽  
Period 2 ◽  
The Stability

Abstract Nonstationary response of a two-degrees-of-freedom system with quadratic coupling under a time varying modulated amplitude sinusoidal excitation is studied. The nonlinearly coupled pitch and roll ship model is based on Nayfeh, Mook and Marshall’s work for the case of stationary excitation. The ship model has a 2:1 internal resonance and is excited near the resonance of the pitch mode. The modulated excitation (F0 + F1 cos ωt) cosQt is used to model a narrow band sea-wave excitation. The response demonstrates a variety of bifurcations, loss of stability, and chaos phenomena that are not present in the stationary case. We consider here the periodically modulated response. Chaotic response of the system is discussed in a separate paper. Several approximate solutions, under both small and large modulating amplitudes F1, are obtained and compared with the exact one. The stability of an exact solution with one mode having zero amplitude is studied. Loss of stability in this case involves either a rapid transition from one of two stable (in the stationary sense) branches to another, or a period doubling bifurcation. From Floquet theory, various stability boundary diagrams are obtained in F1 and F0 parameter space which can be used to predict the various transition phenomena and the period-2 bifurcations. The study shows that both the modulation parameters F1 and ω (the modulating frequency) have great effect on the stability boundaries. Because of the modulation, the stable area is greatly expanded, and the stationary bifurcation point can be exceeded without loss of stability. Decreasing ω can make the stability boundary very complicated. For very small ω the response can make periodic transitions between the two (pseudo) stable solutions.

Stability of a String in Axial Flow

1985 ◽  
Vol 107 (4) ◽  
pp. 421-425 ◽  
Author(s):  
G. S. Triantafyllou ◽  
C. Chryssostomidis
Keyword(s):  
Stability Analysis ◽  
Frictional Coefficient ◽  
Axial Flow ◽  
Added Mass ◽  
Equation Of Motion ◽  
Bending Stiffness ◽  
Diameter Ratio ◽  
Constant Speed ◽  
Hamilton’S Principle ◽  
The Stability

The equation of motion of a long slender beam submerged in an infinite fluid moving with constant speed is derived using Hamilton’s principle. The upstream end of the beam is pinned and the downstream end is free to move. The resulting equation of motion is then used to perform the stability analysis of a string, i.e., a beam with negligible bending stiffness. It is found that the string is stable if (a) the external tension at the free end exceeds the value of a U2, where a is the “added mass” of the string and U the fluid speed; or (b) the length-over-diameter ratio exceeds the value 2Cf/π, where Cf is the frictional coefficient of the string.

scholarly journals Flexural-Torsional Vibration of Thin-Walled Beams Subjected to Combined Initial Axial Load and End Bending Moment: Application to the Design of Saw Tooth Blades

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Van Binh Phung ◽  
Anh Tuan Nguyen ◽  
Hoang Minh Dang ◽  
Thanh-Phong Dao ◽  
V. N. Duc
Keyword(s):  
Boundary Conditions ◽  
Axial Load ◽  
Stability Boundary ◽  
Bending Moment ◽  
Element Analysis ◽  
The Finite Element Analysis ◽  
Rayleigh Method ◽  
Thin Walled ◽  
The Stability ◽  
Fixed End

The present paper analyzes the vibration issue of thin-walled beams under combined initial axial load and end moment in two cases with different boundary conditions, specifically the simply supported-end and the laterally fixed-end boundary conditions. The analytical expressions for the first natural frequencies of thin-walled beams were derived by two methods that are a method based on the existence of the roots theorem of differential equation systems and the Rayleigh method. In particular, the stability boundary of a beam can be determined directly from its first natural frequency expression. The analytical results are in good agreement with those from the finite element analysis software ANSYS Mechanical APDL. The research results obtained here are useful for those creating tooth blade designs of innovative frame saw machines.

Sign in / Sign up

Export Citation Format

Share Document