scholarly journals Generalized two-dimensional Lagally theorem with free vortices and its application to fluid–body interaction problems

2012 ◽  
Vol 698 ◽  
pp. 73-92 ◽  
Author(s):  
C. T. Wu ◽  
F.-L. Yang ◽  
D. L. Young
Keyword(s):  
Hydrodynamic Force ◽  
Density Ratio ◽  
Vortex Pair ◽  
The Body ◽  
Body Motion ◽  
Liquid Density ◽  
Two Dimensional ◽  
Fixed Target ◽  
Two Dimensional Flow ◽  
Neutrally Buoyant

AbstractThe Lagally theorem describes the unsteady hydrodynamic force on a rigid body exhibiting arbitrary motion in an inviscid and incompressible fluid by the properties of the singularities employed to generate the flow and the body motion and to meet the boundary condition. So far, only sources and dipoles have been considered, and the present work extends the theorem to include free vortices in a two-dimensional flow. The present extension is validated by reproducing the system dynamics or the force evolution of three literature problems: (i) a free cylinder interacting with a free vortex; (ii) the moving Föppl problem; and (iii) a cylinder in constant normal approach to a fixed identical cylinder. This work further extends the bifurcation analysis on the moving Föppl problem by including the solid-to-liquid density ratio as a new parameter, in addition to the system total impulse and the vortex strength. We then apply the theorem to the problem where a moving Föppl system is made to approach a fixed or a free neutrally buoyant target cylinder of identical size from far away. The force developed on each cylinder is examined with respect to the vortex pair configuration and the target mobility. When approaching a fixed target, a greater force is developed if the vortex pair has stronger circulation and larger structure. The mobility of the target cylinder, however, can modify the hydrodynamic force by reducing its magnitude and reversing the force ordering with respect to the vortex pair configuration found for the case with fixed target. Possible mechanisms for such a change of force nature are given based on the currently derived equation of motion.

Large-Amplitude Transient Motion of Two-Dimensional Floating Bodies

1979 ◽  
Vol 23 (01) ◽  
pp. 20-31
Author(s):  
R. B. Chapman
Keyword(s):  
Boundary Condition ◽  
Wave Field ◽  
Large Amplitude ◽  
The Body ◽  
Body Motion ◽  
Source Distribution ◽  
Floating Body ◽  
Two Dimensional ◽  
Body Boundary ◽  
Free Mode

A numerical method is presented for solving the transient two-dimensional flow induced by the motion of a floating body. The free-surface equations are linearized, but an exact body boundary condition permits large-amplitude motion of the body. The flow is divided into two parts: the wave field and the impulsive flow required to satisfy the instantaneous body boundary condition. The wave field is represented by a finite sum of harmonics. A nonuniform spacing of the harmonic components gives an efficient representation over specified time and space intervals. The body is represented by a source distribution over the portion of its surface under the static waterline. Two modes of body motion are discussed—a captive mode and a free mode. In the former case, the body motion is specified, and in the latter, it is calculated from the initial conditions and the inertial properties of the body. Two examples are given—water entry of a wedge in the captive mode and motion of a perturbed floating body in the free mode.

The slow transverse motion of a flat plate through a non-diffusive stratified fluid

1971 ◽  
Vol 47 (1) ◽  
pp. 171-181 ◽  
Author(s):  
G. S. Janowitz
Keyword(s):  
Viscous Fluid ◽  
Shear Layer ◽  
Flat Plate ◽  
Kinematic Viscosity ◽  
Vertical Plate ◽  
The Body ◽  
Vertical Shear ◽  
Transverse Motion ◽  
Two Dimensional ◽  
Two Dimensional Flow

We consider the two-dimensional flow produced by the slow horizontal motion of a vertical plate of height 2b through a vertically stratified (ρ = ρ0(1 - βz)) non-diffusive viscous fluid. Our results are valid when U2 [Lt ] Ub/ν [Lt ] 1, where U is the speed of the plate and ν the kinematic viscosity of the fluid. Upstream of the body we find a blocking column of length 10−2b4/(Uν/βg. This column is composed of cells of closed streamlines. The convergence of these cells near the tips of the plate leads to alternate jets. The plate itself is embedded in a vertical shear layer of thickness (Uν/βg)1/3. In the upstream portion of this layer the vertical velocities are of order U and in the downstream portion of order Ub/(Uν/βg)1/3 ([Gt ] U). The flow is uniform and undisturbed downstream of this layer.

scholarly journals The steady oblique path of buoyancy-driven disks and spheres

2012 ◽  
Vol 707 ◽  
pp. 24-36 ◽  
Author(s):  
David Fabre ◽  
Joël Tchoufag ◽  
Jacques Magnaudet
Keyword(s):  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Density Ratio ◽  
Steady Motion ◽  
The Body ◽  
Body Motion ◽  
Angle Of Incidence ◽  
Weakly Nonlinear ◽  
Numerical Studies ◽  
Flow Past

AbstractWe consider the steady motion of disks of various thicknesses in a weakly viscous flow, in the case where the angle of incidence $\ensuremath{\alpha} $ (defined as that between the disk axis and its velocity) is small. We derive the structure of the steady flow past the body and the associated hydrodynamic force and torque through a weakly nonlinear expansion of the flow with respect to $\ensuremath{\alpha} $. When buoyancy drives the body motion, we obtain a solution corresponding to an oblique path with a non-zero incidence by requiring the torque to vanish and the hydrodynamic and net buoyancy forces to balance each other. This oblique solution is shown to arise through a bifurcation at a critical Reynolds number ${\mathit{Re}}^{\mathit{SO}} $ which does not depend upon the body-to-fluid density ratio and is distinct from the critical Reynolds number ${\mathit{Re}}^{\mathit{SS}} $ corresponding to the steady bifurcation of the flow past the body held fixed with $\ensuremath{\alpha} = 0$. We then apply the same approach to the related problem of a sphere that weakly rotates about an axis perpendicular to its path and show that an oblique path sets in at a critical Reynolds number ${\mathit{Re}}^{\mathit{SO}} $ slightly lower than ${\mathit{Re}}^{\mathit{SS}} $, in agreement with available numerical studies.

On boundary layers in two-dimensional flow with vorticity

1963 ◽  
Vol 17 (1) ◽  
pp. 141-153 ◽  
Author(s):  
J. F. Harper
Keyword(s):  
Bluff Body ◽  
Steady Motion ◽  
The Body ◽  
Two Dimensional ◽  
Viscous Layer ◽  
Starting Point ◽  
Two Dimensional Flow ◽  
High Reynolds ◽  
Very High ◽  
Rear Stagnation Point

The starting-point for this paper is the suggestion (Batchelor 1956b) that the wake behind a bluff body in a uniform stream may consist principally of two eddies rotating in opposite directions. The fluid is assumed to be incompressible and in two-dimensional steady motion at a very high Reynolds number. Along the boundary between the eddies, a viscous layer must form. This layer is unusual in that merely the vorticity, and not the velocity itself, varies appreciably across it. It will be shown that such layers can be treated theoretically much more simply than the general case, because it is possible to linearize the equation of motion. They may, of course, exist in flows other than that past a bluff body.A discussion is also given of the flow near the rear stagnation point, where this boundary layer meets the body. It had been suggested that a large number of small eddies would have to exist there, but this seems not to be so.

Dynamics of axisymmetric bodies rising along a zigzag path

2008 ◽  
Vol 606 ◽  
pp. 209-223 ◽  
Author(s):  
PEDRO C. FERNANDES ◽  
PATRICIA ERN ◽  
FRÉDÉRIC RISSO ◽  
JACQUES MAGNAUDET
Keyword(s):  
Aspect Ratio ◽  
Transverse Direction ◽  
Transverse Velocity ◽  
Density Ratio ◽  
Transverse Component ◽  
The Body ◽  
Force Balance ◽  
Body Motion ◽  
Major Influence ◽  
Inertia Force

The forces and torques governing the planar zigzag motion of thick, slightly buoyant disks rising freely in a liquid at rest are determined by applying the generalized Kirchhoff equations to experimental measurements of the body motion performed for a single body-to-fluid density ratio ρs/ρf ≈ 1. The evolution of the amplitude and phase of the various contributions is discussed as a function of the two control parameters, i.e. the body aspect ratio (the diameter-to-thickness ratio χ = d/h ranges from 2 to 10) and the Reynolds number (100 < Re < 330), Re being based on the rise velocity and diameter of the body. The body oscillatory behaviour is found to be governed by the force balance along the transverse direction and the torque balance. In the transverse direction, the wake-induced force is mainly balanced by two forces that depend on the body inclination, i.e. the inertia force generated by the body rotation and the transverse component of the buoyancy force. The torque balance is dominated by the wake-induced torque and the restoring added-mass torque generated by the transverse velocity component. The results show a major influence of the aspect ratio on the relative magnitude and phase of the various contributions to the hydrodynamic loads. The vortical transverse force scales as fo = (ρf − ρs)ghπd2 whereas the vortical torque involves two contributions, one scaling as fod and the other as f1d with f1 = χfo. Using this normalization, the amplitudes and phases of the vortical loads are made independent of the aspect ratio, the amplitudes evolving as (Re/Rec1 − 1)1/2, where Rec1 is the threshold of the first instability of the wake behind the corresponding body held fixed in a uniform stream.

Added Mass and Damping of Rectangular Bodies Close to the Free Surface

1984 ◽  
Vol 28 (04) ◽  
pp. 219-225
Author(s):  
John Nicholas Newman ◽  
Bjørn Sortland ◽  
Tor Vinje
Keyword(s):  
Hydrodynamic Force ◽  
Standing Waves ◽  
Present Theory ◽  
Added Mass ◽  
The Body ◽  
Simple Explanation ◽  
Two Dimensional ◽  
Infinite Depth ◽  
Calm Water ◽  
Heave Motion

A submerged two-dimensional rectangle in calm water with infinite depth is studied. The rectangle is oscillating in a heave motion. Negative added mass and sharp peaks in the damping and added-mass coefficients have been found when the submergence is small and the width of the shallow region on top of the rectangle is large. Resonant standing waves will occur in this area. A linear theory is developed to provide a relatively simple explanation of the occurrence of negative added mass for submerged bodies. The vertical hydrodynamic force is associated only with the flow in the shallow region, and the resulting pressure which acts on the top face of the rectangle. The results from this theory are compared with numerical results from the Frank method. The importance of the interaction effect between the top and the bottom of the body, which is neglected in the present theory, is discussed.

Two-dimensional vortex motion in the cross-flow of a wing-body configuration

1995 ◽  
Vol 305 ◽  
pp. 93-109 ◽  
Author(s):  
T. W. G. De Laat ◽  
R. Coene
Keyword(s):  
Circular Cylinder ◽  
Initial Conditions ◽  
Vortex Motion ◽  
Cross Flow ◽  
Vortex Pair ◽  
The Body ◽  
Two Dimensional ◽  
Oncoming Flow ◽  
Stagnation Flow ◽  
Closed Orbits

For a two-dimensional potential flow, Föppl obtained the equilibrium positions for a symmetric vortex pair behind a circular cylinder in a uniform oncoming flow. In this article it is shown that such an equilibrium is in general possible for a vortex in a stagnation flow (e. g. in a corner). Furthermore it is found that a vortex near such an equilibrium position will rotate with a definite frequency around this equilibrium. Expressions are derived for the frequencies associated with the closed orbits of the vortices in the case of equilibrium of a vortex in a stagnation flow and for the equilibrium of the symmetric vortex pair behind a circular cylinder in oncoming flow. For the large-amplitude case the vortex trajectories are claculated using a fifth-order Runge-Kutta integration method. The analysis is then extended to the case of a simple wing-body combination in a cross-flow such as arises for a slender aircraft at an angle of attack with vortices generated by strakes or at the front part of the body. At the wint-body junctions the motions of the vortices may be periodic, quasi-periodic or the vortices may be swept away, depending on the initial conditions.

scholarly journals I. The flow of air and of an inviscid fluid around an elliptic cylinder and an aerofoil of infinite span, especially in the region of the forward stagnation point

1928 ◽  
Vol 227 (647-658) ◽  
pp. 1-19 ◽  
Keyword(s):  
Wind Tunnel ◽  
Flow Pattern ◽  
Stagnation Point ◽  
Elliptic Cylinder ◽  
Inviscid Flow ◽  
The Body ◽  
Inviscid Fluid ◽  
Two Dimensional ◽  
Two Dimensional Flow ◽  
The Relationship

(1) A study of some of the characteristics of the two-dimensional flow around an aerofoil mounted in a wind tunnel has been made by L. W. Bryant and D. H. Williams. This work included measurements of velocity in the neighbourhood of the aerofoil, and showed that under certain conditions the theoretical law of Kutta and Joukowski can be applied in practice to an aerofoil. The flow pattern measured in the wind tunnel was also compared with that for an inviscid flow having an equal circulation. The purpose of the present paper is to examine, in detail, the relationship between the inviscid and the wind-tunnel flows at the nose of an elliptic cylinder and also of an aerofoil of infinite span. Throughout the paper the term circulation is used in the usual hydrodynamic sense, it being understood that for a wind-tunnel flow the contour is not taken too close to the boundary of the body. Attention has been focussed on the forward stagnation point, because it is a well-defined point on the surface where the pressure is a maximum.

Pressure-driven flow in a two-dimensional channel with porous walls

2009 ◽  
Vol 631 ◽  
pp. 1-21 ◽  
Author(s):  
QUAN ZHANG ◽  
ANDREA PROSPERETTI
Keyword(s):  
Reynolds Number ◽  
Porous Medium ◽  
Hydrodynamic Force ◽  
Reynolds Numbers ◽  
Two Dimensional ◽  
Slip Coefficient ◽  
Staggered Arrangement ◽  
Two Dimensional Flow ◽  
Porous Walls ◽  
Pressure Driven Flow

The finite-Reynolds-number two-dimensional flow in a channel bounded by a porous medium is studied numerically. The medium is modelled by aligned cylinders in a square or staggered arrangement. Detailed results on the flow structure and slip coefficient are reported. The hydrodynamic force and couple acting on the cylinder layer bounding the porous medium are also evaluated as a function of the Reynolds number. In particular, it is shown that, at finite Reynolds numbers, a lift force acts on the bodies, which may be significant for the mobilization of bottom sediments.

The Effect of the Axial Velocity Density Ratio on the Aerodynamic Coefficients of Compressor Cascades

1981 ◽  
Vol 103 (1) ◽  
pp. 210-219 ◽  
Author(s):  
J. Starke
Keyword(s):  
Axial Velocity ◽  
Density Ratio ◽  
Compressor Blade ◽  
Momentum Thickness ◽  
Two Dimensional ◽  
Aerodynamic Coefficients ◽  
Minimum Loss ◽  
Turning Angle ◽  
Two Dimensional Flow ◽  
Inlet Angle

The aerodynamic coefficients of compressor blade sections in two-dimensional flow can easily and very accurately be determined by use of the well-known Lieblein correlations. The flow across the compressor blade sections is often quasi-two-dimensional with the axial velocity density ratio (AVDR) differing from unity. To establish simple correlations for this type of flow as well, the AVDR effect on the aerodynamic coefficients of compressor cascades is theoretically and experimentally investigated. This results in simple but accurate formulae for the calculation of the AVDR effect on the turning angle, the reference minimum-loss inlet angle, and the losses in terms of the wake momentum thickness and the diffusion ratio.

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