scholarly journals The vortex-entrainment sheet in an inviscid fluid: theory and separation at a sharp edge

2019 ◽  
Vol 866 ◽  
pp. 660-688 ◽  
Author(s):  
A. C. DeVoria ◽  
K. Mohseni
Keyword(s):  
Mass Density ◽  
Normal Component ◽  
Vortex Sheet ◽  
Three Dimensions ◽  
Sharp Edge ◽  
Pressure Jump ◽  
Inviscid Fluid ◽  
Fluid Theory ◽  
Viscous Boundary ◽  
Singularity Removal

In this paper a model for viscous boundary and shear layers in three dimensions is introduced and termed a vortex-entrainment sheet. The vorticity in the layer is accounted for by a conventional vortex sheet. The mass and momentum in the layer are represented by a two-dimensional surface having its own internal tangential flow. Namely, the sheet has a mass density per-unit-area making it dynamically distinct from the surrounding outer fluid and allowing the sheet to support a pressure jump. The mechanism of entrainment is represented by a discontinuity in the normal component of the velocity across the sheet. The velocity field induced by the vortex-entrainment sheet is given by a generalized Birkhoff–Rott equation with a complex sheet strength. The model was applied to the case of separation at a sharp edge. No supplementary Kutta condition in the form of a singularity removal is required as the flow remains bounded through an appropriate balance of normal momentum with the pressure jump across the sheet. A pressure jump at the edge results in the generation of new vorticity. The shedding angle is dictated by the normal impulse of the intrinsic flow inside the bound sheets as they merge to form the free sheet. When there is zero entrainment everywhere the model reduces to the conventional vortex sheet with no mass. Consequently, the pressure jump must be zero and the shedding angle must be tangential so that the sheet simply convects off the wedge face. Lastly, the vortex-entrainment sheet model is demonstrated on several example problems.

Propeller Wake Sheet Roll-up Modeling in Three Dimensions

1997 ◽  
Vol 41 (01) ◽  
pp. 81-92
Author(s):  
Sangwoo Pyo ◽  
Spyros A. Kinnas
Keyword(s):  
Three Dimensional ◽  
Vortex Sheet ◽  
Experimental Information ◽  
Higher Order ◽  
Three Dimensions ◽  
Tip Vortex ◽  
Propeller Wake ◽  
Wake Interaction ◽  
Radial Circulation ◽  
Wake Sheet

An algorithm for predicting the complete three-dimensional vortex sheet roll-up is developed. A higher order panel method, which combines a hyperboloidal panel geometry with a bi-quadratic dipole distribution, is used in order to accurately model the highly rolled-up regions. For given radial circulation distributions, the predicted wake shapes are shown to be convergent and consistent with those predicted from other methods. Then, a previously developed flow-adapted grid and the three-dimensional wake sheet roll-up algorithm are combined in order to estimate the propeller loading/trailing wake interaction. The complete wake geometry is determined by the method without the need of any experimental information on the shape of the wake. Predicted forces and tip vortex trajectories are shown to agree well with those measured in experiments.

STEADY STATES OF SELF-GRAVITATING INCOMPRESSIBLE FLUID WITH AXIAL SYMMETRY

2009 ◽  
Vol 24 (23) ◽  
pp. 4287-4303 ◽  
Author(s):  
MAYER HUMI
Keyword(s):  
Gravitational Field ◽  
Steady State ◽  
Incompressible Fluid ◽  
Axial Symmetry ◽  
Numerical Solutions ◽  
Mass Density ◽  
Three Dimensions ◽  
Analytic Model ◽  
Hydrodynamic Equations ◽  
Isothermal Case

In this paper we develop a simple analytic model for the steady state of self-gravitating incompressible nonswirling gas in three dimensions with axial symmetry which is based on the hydrodynamic equations for stratified fluid. These equations are then reduced to a system of two equations for the mass density and the gravitational field. Numerical solutions of these equations under different modeling assumptions (with special attention to the isothermal case) are then used to study the patterns of the resulting steady state of the fluid.

On the disturbed motion of a plane vortex sheet

1958 ◽  
Vol 4 (5) ◽  
pp. 538-552 ◽  
Author(s):  
John W. Miles
Keyword(s):  
Eigenvalue Problem ◽  
Asymptotic Approximation ◽  
Formal Solution ◽  
Vortex Sheet ◽  
Inviscid Fluid ◽  
Initial Value ◽  
Spatially Periodic ◽  
The Stability ◽  
Spurious Eigenvalues ◽  
Disturbed Motion

A formal solution to the initial value problem for a plane vortex sheet in an inviscid fluid is obtained by transform methods. The eigenvalue problem is investigated and the stability criterion determined. This criterion is found to be in agreement with that obtained previously by Landau (1944), Hatanaka (1949), and Pai (1954), all of whom had included spurious eigenvalues in their analyses. It is also established that supersonic disturbances may be unstable; related investigations in hydrodynamic stability have conjectured on this possibility, but the vortex sheet appears to afford the first definite example. Finally, an asymptotic approximation is developed for the displacement of a vortex sheet following a suddenly imposed, spatially periodic velocity.

Finite-wavelength scattering of incident vorticity and acoustic waves at a shrouded-jet exit

2008 ◽  
Vol 612 ◽  
pp. 407-438 ◽  
Author(s):  
ARNAB SAMANTA ◽  
JONATHAN B. FREUND
Keyword(s):  
Mach Number ◽  
Acoustic Waves ◽  
Vortex Sheet ◽  
Acoustic Mode ◽  
Uniform Flow ◽  
Sharp Edge ◽  
Potential Mechanism ◽  
Ambient Flow ◽  
Acoustic Modes ◽  
Cylindrical Vortex

As the vortical disturbances of a shrouded jet pass the sharp edge of the shroud exit some of the energy is scattered into acoustic waves. Scattering into upstream-propagating acoustic modes is a potential mechanism for closing the resonance loop in the ‘howling’ resonances that have been observed in various shrouded jet configurations over the years. A model is developed for this interaction at the shroud exit. The jet is represented as a uniform flow separated by a cylindrical vortex sheet from a concentric co-flow within the cylindrical shroud. A second vortex sheet separates the co-flow from an ambient flow outside the shroud, downstream of its exit. The Wiener–Hopf technique is used to compute reflectivities at the shroud exit. For some conditions it appears that the reflection of finite-wavelength hydrodynamic vorticity modes on the vortex sheet defining the jet could be sufficient to reinforce the shroud acoustic modes to facilitate resonance. The analysis also gives the reflectivities for the shroud acoustic modes, which would also be important in establishing resonance conditions. Interestingly, it is also predicted that the shroud exit can be ‘transparent’ for ranges of Mach numbers, with no reflection into any upstream-propagating acoustic mode. This is phenomenologically consistent with observations that indicate a peculiar sensitivity of resonances of this kind to, say, jet Mach number.

scholarly journals Regularized Vortex Sheet Evolution in Three Dimensions

1998 ◽  
Vol 146 (2) ◽  
pp. 520-545 ◽  
Author(s):  
M. Brady ◽  
A. Leonard ◽  
D.I. Pullin
Keyword(s):  
Vortex Sheet ◽  
Three Dimensions

Computational Modeling of Rolling Wave-Energy Converters in a Viscous Fluid1

2015 ◽  
Vol 137 (6) ◽  
Author(s):  
Yichen Jiang ◽  
Ronald W. Yeung
Keyword(s):  
Wave Energy ◽  
Performance Metrics ◽  
Vortex Method ◽  
Inviscid Fluid ◽  
Mathematical Form ◽  
Wave Energy Converters ◽  
Ocean Wave Energy ◽  
Real Fluid ◽  
Fluid Theory ◽  
Energy Absorbing

The performance of an asymmetrical rolling cam as an ocean-wave energy extractor was studied experimentally and theoretically in the 70s. Previous inviscid-fluid theory indicated that energy-absorbing efficiency could approach 100% in the absence of real-fluid effects. The way viscosity alters the performance is examined in this paper for two distinctive rolling-cam shapes: a smooth “Eyeball Cam (EC)” with a simple mathematical form and a “Keeled Cam (KC)” with a single sharp-edged keel. Frequency-domain solutions in an inviscid fluid were first sought for as baseline performance metrics. As expected, without viscosity, both shapes, despite their differences, perform exceedingly well in terms of extraction efficiency. The hydrodynamic properties of the two shapes were then examined in a real fluid, using the solution methodology called the free-surface random-vortex method (FSRVM). The added inertia and radiation damping were changed, especially for the KC. With the power-take-off (PTO) damping present, nonlinear time-domain solutions were developed to predict the rolling motion, the effects of PTO damping, and the effects of the cam shapes. For the EC, the coupled motion of sway, heave and roll in waves was investigated to understand how energy extraction was affected.

Optimal flexibility of a flapping appendage in an inviscid fluid

2008 ◽  
Vol 614 ◽  
pp. 355-380 ◽  
Author(s):  
SILAS ALBEN
Keyword(s):  
Power Law ◽  
Leading Edge ◽  
Vortex Sheet ◽  
Input Power ◽  
The Body ◽  
Flexible Body ◽  
Inviscid Fluid ◽  
Driving Frequency ◽  
Pitch Frequency ◽  
New Formulation

We present a new formulation of the motion of a flexible body with a vortex-sheet wake and use it to study propulsive forces generated by a flexible body pitched periodically at the leading edge in the small-amplitude regime. We find that the thrust power generated by the body has a series of resonant peaks with respect to rigidity, the highest of which corresponds to a body flexed upwards at the trailing edge in an approximately one-quarter-wavelength mode of deflection. The optimal efficiency approaches 1 as rigidity becomes small and decreases to 30–50% (depending on pitch frequency) as rigidity becomes large. The optimal rigidity for thrust power increases from approximately 60 for large pitching frequency to ∞ for pitching frequency 0.27. Subsequent peaks in response have power-law scalings with respect to rigidity and correspond to higher-wavenumber modes of the body. We derive the power-law scalings by analysing the fin as a damped resonant system. In the limit of small driving frequency, solutions are self-similar at the leading edge. In the limit of large driving frequency, we find that the distribution of resonant rigidities ~k−5, corresponding to fin shapes with wavenumber k. The input power and output power are proportional to rigidity (for small-to-moderate rigidity) and to pitching frequency (for moderate-to-large frequency). We compare these results with the range of rigidity and flapping frequency for the hawkmoth forewing and the bluegill sunfish pectoral fin.

scholarly journals The two-dimensional hydrodynamical theory of moving aerofoils-II

1938 ◽  
Vol 164 (918) ◽  
pp. 346-368 ◽  
Author(s):  
Rosa M. Morris ◽  
Louis Napoleon George Filon
Keyword(s):  
Complete Solution ◽  
Mathematical Problem ◽  
Boundary Curve ◽  
Vortex Sheet ◽  
Unsteady Motion ◽  
Sharp Edge ◽  
Small Motion ◽  
Moving Cylinder ◽  
Simplifying Assumptions ◽  
Hydrodynamical Theory

1⋅1—In the first paper of this series (referred to subsequently as “I”) I developed in some detail the complete solution of the hydrodynamical problem of the motion of an incompressible homogeneous inviscid liquid when a cylinder with a general aerofoil cross-section is moving in any manner perpendicular to its axis. As was mentioned in that paper, the discussion there suffers from two well-known defects. In the first place it fails generally when there is a sharp edge to the moving cylinder—or a singularity on the boundary curve—because the velocity of the liquid, as defined by the potential function, becomes infinite at such an edge. This difficulty is usually surmounted by a proper choice of the circulation, but in the case of the moving cylinder this artifice is unavailing by itself as it only gives one condition, when the complete vanishing of the velocity at the trailing edge involves two. In any case, however, the existence of finite circulation creates difficulties of its own—the energy involved in such circulations is infinite—so that still further complication of the mathematical problem seems required. In practice it is known that the liquid motion is more complicated than that represented in the simpler problem, as it is always accompanied by the development, at such a sharp edge, and particularly in unsteady motion, of a region of turbulent motion in the fluid which trails behind in the wake of the cylinder. To deal with this wake and its effects mathematically we must, of course, make certain simplifying assumptions. The simplest picture is that used by Wagner (1925), who imagined the wake to be a simple surface of discontinuity (vortex sheet) trailing behind in the liquid. Of course, in a perfect fluid such a sheet could not arise, but nevertheless, by assuming its existence, it is possible without any close inquiry into its structure to calculate its effect on the forces acting on the cylinder. This is, in effect, what Wagner did, but under conditions which imply that his results are only approximately true when the cylinder is a flat plate with very small motion (without rotation). The main objects of the present paper are, first, to show that the general theory of the first paper can be extended to include the effects of such surfaces of discontinuity and, second, to discuss the bearing of the general results obtained in the particular problems discussed by Wagner, and others in extension of his work; and thereby to test the validity of the formulae now used in practice, the various terms in which have been obtained by diverse devices, all indirect and often involving contradictory physical assumptions and approximations.

Grid turbulence near a moving wall

1977 ◽  
Vol 82 (3) ◽  
pp. 481-496 ◽  
Author(s):  
N. H. Thomas ◽  
P. E. Hancock
Keyword(s):  
High Reynolds Number ◽  
Previous Experiment ◽  
Normal Component ◽  
Tangential Component ◽  
Grid Turbulence ◽  
Velocity Fluctuations ◽  
Moving Wall ◽  
Theoretical Predictions ◽  
High Reynolds ◽  
Viscous Boundary

Decaying grid turbulence was passed over a wall moving at the stream speed. For the high Reynolds number of the experiment, the field due to the wall constraint on the normal component of the velocity fluctuations is found to extend further into the flow than the influence of the viscous boundary condition on the tangential-component fluctuations. Measurements of the variances, length scales and spectra of the three velocity components of the turbulence are compared with the results of a previous experiment and with the theoretical predictions for an idealization of the flow. A simple model for some departures from the theory is proposed.

Singularity formation in three-dimensional motion of a vortex sheet

1995 ◽  
Vol 300 ◽  
pp. 339-366 ◽  
Author(s):  
Takashi Ishihara ◽  
Yukio Kaneda
Keyword(s):  
Asymptotic Analysis ◽  
Numerical Integration ◽  
Finite Time ◽  
Fourier Coefficients ◽  
Evolution Equations ◽  
Three Dimensional ◽  
Vortex Sheet ◽  
Three Dimensions ◽  
Leading Order ◽  
Singularity Formation

The evolution of a small but finite three-dimensional disturbance on a flat uniform vortex sheet is analysed on the basis of a Lagrangian representation of the motion. The sheet at time t is expanded in a double periodic Fourier series: R(λ1, λ2, t) = (λ1, λ2, 0) + Σn,mAn,m exp[i(nλ1 + δmλ2)], where λ1 and λ2 are Lagrangian parameters in the streamwise and spanwise directions, respectively, and δ is the aspect ratio of the periodic domain of the disturbance. By generalizing Moore's analysis for two-dimensional motion to three dimensions, we derive evolution equations for the Fourier coefficients An,m. The behaviour of An,m is investigated by both numerical integration of a set of truncated equations and a leading-order asymptotic analysis valid at large t. Both the numerical integration and the asymptotic analysis show that a singularity appears at a finite time tc = O(lnε−1) where ε is the amplitude of the initial disturbance. The singularity is such that An,0 = O(tc−1) behaves like n−5/2, while An,±1 = O(εtc) behaves like n−3/2 for large n. The evolution of A0,m(spanwise mode) is also studied by an asymptotic analysis valid at large t. The analysis shows that a singularity appears at a finite time t = O(ε−1) and the singularity is characterized by A0,2k ∝ k−5/2 for large k.

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