On Benjamin's theory of conjugate vortex flows

1967 ◽  
Vol 28 (1) ◽  
pp. 85-96 ◽  
Author(s):  
L. E. Fraenkel
Keyword(s):  
Small Amplitude ◽  
Swirling Flow ◽  
Standing Waves ◽  
Vortex Flows ◽  
Primary Flow ◽  
Conjugate Flows

Benjamin (1962) introduced the idea that a given ‘primary’ swirling flow, with cylindrical stream surfaces, may have associated with it ‘conjugate flows’, also swirling and cylindrical, which in a certain sense are equivalent to the primary one. He deduced that, in cases where such conjugate flows exist and where the primary flow cannot support standing waves of small amplitude, the conjugate flow nearest the primary one (a) can support such waves, and (b) has a ‘flow force’ greater than that of the primary flow. In the present paper these two results are proved rigorously by a method which differs from Benjamin's.

Lift-Enhancing Vortex Flows Generated by Plunging Rectangular Wings with Small Amplitude

AIAA Journal ◽  
2013 ◽  
Vol 51 (12) ◽  
pp. 2953-2964 ◽  
Author(s):  
D. E. Calderon ◽  
Z. Wang ◽  
I. Gursul
Keyword(s):  
Small Amplitude ◽  
Vortex Flows

Stability of time-periodic flows in a circular pipe

1977 ◽  
Vol 82 (3) ◽  
pp. 497-505 ◽  
Author(s):  
W. H. Yang ◽  
Chia-Shun Yih
Keyword(s):  
Small Amplitude ◽  
Reynolds Numbers ◽  
Circular Pipe ◽  
Axially Symmetric ◽  
Primary Flow ◽  
Periodic Flows ◽  
Governing Differential Equation ◽  
The Stability ◽  
Time Periodic ◽  
Range Of Values

The stability of time-periodic flows in a circular pipe is investigated. The disturbance is assumed to be axially symmetric and to have a small amplitude, so that the governing differential equation is linear. Calculations are carried out for the first ten modes for a range of values of the frequency of the primary motion, of the wavenumber of the disturbance, and of the Reynolds number of the primary flow. In the ranges of the parameters for which the calculations have been carried out, the flows are found to be stable and, as for Stokes flows (von Kerczek & Davis 1974), it is conjectured that the flows under study here are stable for all frequencies and all Reynolds numbers.

An experimental study of standing waves

1953 ◽  
Vol 218 (1132) ◽  
pp. 44-59 ◽  
Keyword(s):  
Small Amplitude ◽  
Wave Length ◽  
Three Dimensional ◽  
Standing Waves ◽  
Close Approximation ◽  
Frequency Curve ◽  
Free Standing ◽  
Experimental Conditions ◽  
The Moment ◽  
Good Agreement

The experiments here described were designed to test experimentally some conclusions about free standing waves recently reached analytically by Penney & Price. A close approximation to free oscillations was produced in a tank by wave makers operating with small amplitude and at frequencies where great amplification occurred, owing to resonance. The amplitude-frequency curve proved to consist of two non-intersecting branches, a result which can be explained theoretically. A striking prediction made by Penney & Price was that when the height of the crests of standing waves reaches about 0·15 wave-length they will become pointed, in the form of a 90° ridge. Higher waves were expected to be unstable because the downward acceleration of the free surface near the crest would exceed that of gravity. The experimental conditions necessary for producing a crest in the form of an angled ridge were found and the wave photographed in this condition. Good agreement was found with the calculated form of the profile of the highest wave, which had an angle very near to 90°. The predicted instability for two-dimensional waves was found to begin at the moment the crest became a sharp ridge. It rapidly assumed a three-dimensional character which was revealed by two photographic techniques. Even when the amplitude of oscillation of the wave makers was only 0·85°, violent types of instability developed which produced effects that are here recorded.

The effect of compressibility on the critical swirl of vortex flows in a pipe

2002 ◽  
Vol 461 ◽  
pp. 301-319 ◽  
Author(s):  
Z. RUSAK ◽  
J. H. LEE
Keyword(s):  
Mach Number ◽  
Eigenvalue Problem ◽  
Swirling Flow ◽  
Base Flow ◽  
Vortex Flows ◽  
Flow State ◽  
Flow Perturbation ◽  
Limit Value ◽  
High Reynolds ◽  
Pressure Perturbations

The effect of compressibility on the critical swirl level for breakdown of subsonic vortex flows in a straight circular pipe of finite length is studied. This work extends the critical-state concept of Benjamin (1962) to include the influence of Mach number on the flow behaviour. The analysis is based on a linearized version of the equations for the motion of a steady, axisymmetric, inviscid and compressible swirling flow of a perfect gas. The relationship between the velocity, density, temperature and pressure perturbations to a base columnar flow state are derived. An eigenvalue problem is formulated to determine the first critical level of swirl at which a special mode of a non-columnar small disturbance may appear on the base flow. It is found that when the characteristic Mach number of the base flow tends to zero the eigenvalue problem and the critical swirl are the same as defined by Wang & Rusak (1996a, 1997a) in their study of incompressible swirling flows in pipes. As the characteristic Mach number is increased, the critical swirl level increases and the flow perturbation expands in the radial direction. As the Mach number is increased toward a certain limit value related to the core size of the vortex, the critical swirl reaches very large values and becomes singular. The present results indicate that the axisymmetric breakdown of high-Reynolds-number compressible vortex flows may be delayed with the increase of the flow Mach number.

Seiches over parabolic bottoms

1967 ◽  
Vol 63 (4) ◽  
pp. 1167-1175
Author(s):  
E. V. Laitone
Keyword(s):  
Shallow Water ◽  
Small Amplitude ◽  
Standing Waves ◽  
Explicit Solutions ◽  
Ring Type ◽  
Vertical Walls ◽  
Circular Island

AbstractThe solutions are derived for the shallow water standing waves of small amplitude that can form in channels or lakes of varying breadth with a concave parabolic bottom. In addition explicit solutions are given for standing waves in a ring-type lake with a parabolic bottom and a central circular island that has vertical walls.

Snakes and corkscrews in core–annular down-flow of two fluids

1997 ◽  
Vol 340 ◽  
pp. 297-317 ◽  
Author(s):  
YURIKO Y. RENARDY
Keyword(s):  
Small Amplitude ◽  
Annular Flow ◽  
Travelling Waves ◽  
Standing Waves ◽  
Axial Direction ◽  
Nonlinear Interactions ◽  
Azimuthal Direction ◽  
Weakly Nonlinear ◽  
Two Fluids ◽  
Axisymmetric Instability

Core–annular flow of two fluids is examined at the onset of a non-axisymmetric instability. This is a pattern selection problem: the bifurcating solutions are travelling waves and standing waves. The former travel in the azimuthal direction as well as the axial direction and would be observed as corkscrew waves. The standing waves travel in the axial direction but not in the azimuthal direction and appear as snakes. Weakly nonlinear interactions are studied to see whether one of these waves will be stable to small-amplitude perturbations. Sample situations for down-flow are discussed. The corkscrews tend to be preferred when the annulus is narrow, while snakes are more likely when the annulus is wide.

On the global nonlinear stability of a near-critical swirling flow in a long finite-length pipe and the path to vortex breakdown

2012 ◽  
Vol 712 ◽  
pp. 295-326 ◽  
Author(s):  
Z. Rusak ◽  
S. Wang ◽  
L. Xu ◽  
S. Taylor
Keyword(s):  
Finite Length ◽  
Control Method ◽  
Vortex Breakdown ◽  
Swirling Flow ◽  
Model Problem ◽  
Flow Dynamics ◽  
Axial Distance ◽  
Vortex Flows ◽  
Wide Range

AbstractThe dynamics of a perturbed incompressible, inviscid, axisymmetric, near-critical swirling flow in a long, finite-length, straight, circular pipe is studied through a weakly nonlinear analysis. The flow is subjected to non-periodic inlet and outlet conditions. The long-wave approach involves a rescaling of the axial distance and time. It results in a separation of the perturbation’s structure into a critical standing wave in the radial direction and an evolving wave in the axial direction, that is described by a nonlinear model problem. The approach is first validated by establishing the bifurcation of non-columnar states from the critical swirl and the linear stability modes of these states. Examples of the flow dynamics at various near-critical swirl levels in response to different initial perturbations demonstrate the important role of the nonlinear steepening terms in perturbation dynamics. The computed dynamics shows quantitative agreement with results from numerical simulations that are based on the axisymmetric Euler equations for various swirl levels and as long as perturbations are small, thereby verifying the accuracy of each computation and capturing the essence of flow dynamics. Results demonstrate the various stages of the flow dynamics, specifically during the transition to vortex breakdown states. They reveal the evolution of faster-than-exponential and shape-changing modes as perturbations grow into the vortex breakdown process. These explosive modes provide the sudden and abrupt nature of the vortex breakdown phenomenon. Further analysis of the model problem shows the important role of the nonlinear evolution of perturbations and its relevance to the transfer of the perturbation’s kinetic energy between the boundaries and flow bulk, the evolution of perturbations in practical concentrated vortex flows, and the design of control methods of vortex flows. A robust feedback control method to stabilize a solid-body rotation flow in a pipe at a wide range of swirl levels above critical is developed. The applicability of this method to stabilizing medium and small core-size vortices is also discussed.

Conjugate-flow theory for heterogeneous compressible fluids, with application to non-uniform suspensions of gas bubbles in liquids

1972 ◽  
Vol 54 (3) ◽  
pp. 545-563 ◽  
Author(s):  
T. Brooke Benjamin
Keyword(s):  
Cross Section ◽  
Gas Bubbles ◽  
Arbitrary Cross Section ◽  
Compressible Fluids ◽  
Lagrangian Description ◽  
Pressure Force ◽  
Primary Flow ◽  
Cross Sectional ◽  
Dissipative Flow ◽  
Conjugate Flows

Conjugate flows have been defined generally as flows uniform in the direction of streaming that separately satisfy the relevant hydrodynamical equations, so allowing a transition from one flow to its conjugate to be consistent with mass and energy conservation. In previous studies of various examples, certain general principles have been found to apply to conjugate flows: in particular, one in a pair of such flows is subcritical (subsonic) and the other supercritical (supersonic), the former having greater flow force (i.e. momentum flux plus pressure force). In this paper these principles are confirmed in another field of application, for which the theory of conjugate flows takes a novel course.The theoretical model defined in § 2 consists of a straight duct of arbitrary cross-section filled with a perfect fluid whose constitutive properties vary with cross-sectional position, and whose primary, prescribed flow is axial with a velocity distribution that may be non-uniform. In § 3 the possibility of a conjugate flow in the same duct is investigated, and its principal properties relative to those of the primary flow are deduced from certain simple inequalities between integrals over the cross-section. A Lagrangian description of the conjugate flow is essential, but the properties in question are established without the necessity of determining this flow explicitly. At the end of § 3, a modification of the model is discussed accounting for dissipative, flow-force conserving transitions (shocks). The application of the theory to flows of non-uniform suspensions of gas bubbles is considered in § 4.

The Relationship Between Standing Waves, Pressure Pulse Propagation, and Critical Flow Rate in Two-Phase Mixtures

1989 ◽  
Vol 111 (2) ◽  
pp. 467-473 ◽  
Author(s):  
A. E. Ruggles ◽  
R. T. Lahey ◽  
D. A. Drew ◽  
H. A. Scarton
Keyword(s):  
Flow Rate ◽  
Small Amplitude ◽  
Fluid Model ◽  
Flow Analysis ◽  
Standing Waves ◽  
Critical Flow ◽  
Two Phase ◽  
Fourier Decomposition ◽  
Critical Flow Rate ◽  
Pressure Perturbations

A two-fluid model is presented that can be used to predict the celerity and attenuation of small-amplitude harmonic disturbances in bubbly two-phase flow. This frequency-dependent relationship is then used to predict the propagation of small-amplitude pressure perturbations through the use of Fourier decomposition techniques. Predictions of both standing waves and propagating pressure perturbations agree well with existing data. The low and high-frequency limits of the celerities predicted by the model are examined and their relationship to critical flow rate is demonstrated. Some limitations of the interfacial pressure model employed in conventional critical flow analysis are exposed and the implications to the prediction of critical flow rate are discussed.

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