The trailing vorticity field behind a line source in two-dimensional incompressible linear shear flow

2013 ◽  
Vol 720 ◽  
pp. 618-636 ◽  
Author(s):  
Sjoerd W. Rienstra ◽  
Mirela Darau ◽  
Edward J. Brambley
Keyword(s):  
Shear Flow ◽  
Free Field ◽  
Two Dimensional ◽  
Mean Flow ◽  
Mass Source ◽  
Transform Method ◽  
Impedance Boundary ◽  
Duct Acoustics ◽  
Dominant Feature ◽  
Linear Shear

AbstractThe explicit exact analytic solution for harmonic perturbations from a line mass source in an incompressible inviscid two-dimensional linear shear is derived using a Fourier transform method. The two cases of an infinite shear flow and a semi-infinite shear flow over an impedance boundary are considered. For the free-field and hard-wall configurations, the pressure field is (in general) logarithmically diverging and its Fourier representation involves a diverging integral that is interpreted as an integral of generalized functions; this divergent behaviour is not present for a finite impedance boundary or if the frequency equals the mean flow shear rate. The dominant feature of the solution is the hydrodynamic wake caused by the shed vorticity of the source. For linear shear over an impedance boundary, in addition to the wake, (at most) two surface modes along the wall are excited. The implications for duct acoustics with flow over an impedance wall are discussed.

Compressible magnetoconvection in three dimensions: planforms and nonlinear behaviour

1995 ◽  
Vol 305 ◽  
pp. 281-305 ◽  
Author(s):  
P. C. Matthews ◽  
M. R. E. Proctor ◽  
N. O. Weiss
Keyword(s):  
Magnetic Field ◽  
Shear Flow ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Nonlinear Behaviour ◽  
Two Dimensional ◽  
Horizontal Shear ◽  
Mean Flow ◽  
The Magnetic Field

Convection in a compressible fiuid with an imposed vertical magnetic field is studied numerically in a three-dimensional Cartesian geometry with periodic lateral boundary conditions. Attention is restricted to the mildly nonlinear regime, with parameters chosen first so that convection at onset is steady, and then so that it is oscillatory.Steady convection occurs in the form of two-dimensional rolls when the magnetic field is weak. These rolls can become unstable to a mean horizontal shear flow, which in two dimensions leads to a pulsating wave in which the direction of the mean flow reverses. In three dimensions a new pattern is found in which the alignment of the rolls and the shear flow alternates.If the magnetic field is sufficiently strong, squares or hexagons are stable at the onset of convection. Both the squares and the hexagons have an asymmetrical topology, with upflow in plumes and downflow in sheets. For the squares this involves a resonance between rolls aligned with the box and rolls aligned digonally to the box. The preference for three-dimensional flow when the field is strong is a consequence of the compressibility of the layer- for Boussinesq magnetoconvection rolls are always preferred over squares at onset.In the regime where convection is oscillatory, the preferred planform for moderate fields is found to be alternating rolls - standing waves in both horizontal directions which are out of phase. For stronger fields, both alternating rolls and two-dimensional travelling rolls are stable. As the amplitude of convection is increased, either by dcereasing the magnetic field strength or by increasing the temperature contrast, the regular planform structure seen at onset is soon destroyed by secondary instabilities.

scholarly journals Interaction between mountain waves and shear flow in an inertial layer

2017 ◽  
Vol 816 ◽  
pp. 352-380 ◽  
Author(s):  
Jin-Han Xie ◽  
Jacques Vanneste
Keyword(s):  
Shear Flow ◽  
Inviscid Limit ◽  
Mean Flow ◽  
Mountain Waves ◽  
Planetary Rotation ◽  
Thin Region ◽  
Linear Shear ◽  
Propagation Regime ◽  
Finite Distance ◽  
The Mean

Mountain-generated inertia–gravity waves (IGWs) affect the dynamics of both the atmosphere and the ocean through the mean force they exert as they interact with the flow. A key to this interaction is the presence of critical-level singularities or, when planetary rotation is taken into account, inertial-level singularities, where the Doppler-shifted wave frequency matches the local Coriolis frequency. We examine the role of the latter singularities by studying the steady wavepacket generated by a multiscale mountain in a rotating linear shear flow at low Rossby number. Using a combination of Wentzel–Kramers–Brillouin (WKB) and saddle-point approximations, we provide an explicit description of the form of the wavepacket, of the mean forcing it induces and of the mean-flow response. We identify two distinguished regimes of wave propagation: Regime I applies far enough from a dominant inertial level for the standard ray-tracing approximation to be valid; Regime II applies to a thin region where the wavepacket structure is controlled by the inertial-level singularities. The wave–mean-flow interaction is governed by the change in Eliassen–Palm (or pseudomomentum) flux. This change is localised in a thin inertial layer where the wavepacket takes a limiting form of that found in Regime II. We solve a quasi-geostrophic potential-vorticity equation forced by the divergence of the Eliassen–Palm flux to compute the wave-induced mean flow. Our results, obtained in an inviscid limit, show that the wavepacket reaches a large-but-finite distance downstream of the mountain (specifically, a distance of order$(k_{\ast }\unicode[STIX]{x1D6E5})^{1/2}\unicode[STIX]{x1D6E5}$, where$k_{\ast }^{-1}$and$\unicode[STIX]{x1D6E5}$measure the wave and envelope scales of the mountain) and extends horizontally over a similar scale.

scholarly journals Two-dimensional fluctuating vesicles in linear shear flow

2008 ◽  
Vol 25 (3) ◽  
pp. 309-321 ◽  
Author(s):  
R. Finken ◽  
A. Lamura ◽  
U. Seifert ◽  
G. Gompper
Keyword(s):  
Shear Flow ◽  
Two Dimensional ◽  
Linear Shear

The evolution of thermocline waves from an oscillatory disturbance

1993 ◽  
Vol 254 ◽  
pp. 401-416 ◽  
Author(s):  
D. Nicolaou ◽  
R. Liu ◽  
T. N. Stevenson
Keyword(s):  
Finite Difference ◽  
Shear Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Ray Theory ◽  
Navier Stokes Equations ◽  
Linear Shear ◽  
Wave Shapes ◽  
The Way

The way in which energy propagates away from a two-dimensional oscillatory disturbance in a thermocline is considered theoretically and experimentally. It is shown how the St. Andrew's-cross-wave is modified by reflections and how the cross-wave can develop into thermocline waves. A linear shear flow is then superimposed on the thermocline. Ray theory is used to evaluate the wave shapes and these are compared to finite-difference solutions of the full Navier–Stokes equations.

Notes on the forces acting on a two-dimensional aerofoil in shear flow in the presence of a plane boundary

1949 ◽  
Vol 45 (4) ◽  
pp. 612-620 ◽  
Author(s):  
A. Coombs
Keyword(s):  
Shear Flow ◽  
Real Variable ◽  
Plane Boundary ◽  
Two Dimensional ◽  
Variable Theory ◽  
Linear Shear ◽  
Limiting Case ◽  
Special Case ◽  
Typical Length ◽  
Uniform Fluid

1. The forces acting on the two-dimensional aerofoil in a bounded uniform stream have been found for a variety of cases, and in the present paper an attempt is made to extend the theory to include linear shear-flow. The special case of the symmetrical Joukowsky aerofoil in unbounded shear-flow has been solved by Tsien (6) using real-variable theory. A more satisfactory method is using complex-variable technique indicated in § 3, and is applicable to a more general shaped aerofoil. The effect on the lift and moment when a plane boundary is present is then considered. When the aerofoil is not too near the boundary, the lift and moment can be expanded in powers of the ratio of a typical length in the aerofoil to the height of the aerofoil above the boundary, by following exactly the technique used by Green (1) in a recent paper dealing with the same problem but with a uniform fluid flow. In the last section the limiting case of the flat plate touching the boundary with its trailing edge is discussed.

Approach to some Properties of Parallel to Wall Shear Flow and its Application

2012 ◽  
Vol 516-517 ◽  
pp. 1053-1057
Author(s):  
Zhao Cun Liu ◽  
Wei Jia Fan ◽  
Ping Yi Wang ◽  
Xiang Zhou Xv
Keyword(s):  
Shear Flow ◽  
Open Channel ◽  
Open Channel Flow ◽  
Resonant Vibration ◽  
Two Dimensional ◽  
Mean Flow ◽  
Frequency Locking ◽  
The Mean ◽  
Wall Shear ◽  
Different Levels

Some properties of two dimensional open channel flow were studied. Energy distribution was calculated to show energy exchanging relationship between the mean flow and the fluctuating flow. Viscous dissipative energy of mean flow, the energy of fluctuating flow and the energy of fluctuating flow taking from the mean flow were calculated. The energy spectrum was explored. The mechanism of two dimensional parallel to wall shear flow were probed to show that not only the energy transfer relates different structures corresponding to different levels of turbulent exciting but also the energy dissipating and diffusing closely connect with resonant vibration and frequency-locking. On the basis of flowing structures, the results applying to waterway regulating was discussed.

Turbulence structure in a boundary layer with two-dimensional roughness

2009 ◽  
Vol 635 ◽  
pp. 75-101 ◽  
Author(s):  
R. J. VOLINO ◽  
M. P. SCHULTZ ◽  
K. A. FLACK
Keyword(s):  
Boundary Layer ◽  
Large Scale ◽  
Reynolds Shear Stress ◽  
Spatial Scales ◽  
Three Dimensional ◽  
Turbulence Structure ◽  
Two Dimensional ◽  
Mean Flow ◽  
Outer Flow ◽  
Dominant Feature

Turbulence measurements for a zero pressure gradient boundary layer over a two-dimensional roughness are presented and compared to previous results for a smooth wall and a three-dimensional roughness (Volino, Schultz & Flack, J. Fluid Mech., vol. 592, 2007, p. 263). The present experiments were made on transverse square bars in the fully rough flow regime. The turbulence structure was documented through the fluctuating velocity components, two-point correlations of the fluctuating velocity and swirl strength and linear stochastic estimation conditioned on the swirl and Reynolds shear stress. The two-dimensional bars lead to significant changes in the turbulence in the outer flow. Reynolds stresses, particularly $\overline {{v'}^2} ^ +$ and $ - \overline {{u}'{v}'} ^ + $, increase, although the mean flow is not as significantly affected. Large-scale turbulent motions originating at the wall lead to increased spatial scales in the outer flow. The dominant feature of the outer flow, however, remains hairpin vortex packets which have similar inclination angles for all wall conditions. The differences between boundary layers over two-dimensional and three-dimensional roughness are attributable to the scales of the motion induced by each type of roughness. This study has shown three-dimensional roughness produces turbulence scales of the order of the roughness height k while the motions generated by two-dimensional roughness may be much larger due to the width of the roughness elements. It is also noted that there are fundamental differences in the response of internal and external flows to strong wall perturbations, with internal flows being less sensitive to roughness effects.

scholarly journals Time-evolving bubbles in two-dimensional Stokes flow

1995 ◽  
Vol 301 ◽  
pp. 325-344 ◽  
Author(s):  
Saleh Tanveer ◽  
Giovani L. Vasconcelos
Keyword(s):  
Surface Tension ◽  
Shear Flow ◽  
Exact Solutions ◽  
Stokes Flow ◽  
Mathematical Structure ◽  
The Other ◽  
Two Dimensional ◽  
Slow Viscous Flow ◽  
Expanding Circle ◽  
Linear Shear

A general class of exact solutions is presented for a time-evolving bubble in a two-dimensional slow viscous flow in the presence of surface tension. These solutions can describe a bubble in a linear shear flow as well as an expanding or contracting bubble in an otherwise quiescent flow. In the case of expanding bubbles, the solutions have a simple behaviour in the sense that for essentially arbitrary initial shapes the bubble its asymptote is expanding circle. Contracting bubbles, on the other hand, can develop narrow structures (‘near-cusps’) on the interface and may undergo ‘breakup’ before all the bubble fluid is completely removed. The mathematical structure underlying the existence of these exact solutions is also investigated.

Wave propagation in a stratified shear flow

1975 ◽  
Vol 71 (1) ◽  
pp. 89-104 ◽  
Author(s):  
R. J. Hartman
Keyword(s):  
Wave Propagation ◽  
Shear Flow ◽  
Wave Packet ◽  
Initial Value Problem ◽  
Two Dimensional ◽  
Plane Couette Flow ◽  
Mean Flow ◽  
Initial Value ◽  
Stratified Shear Flow ◽  
The Mean

The linearized initial-value problem for a two-dimensional, unbounded, exponentially stratified, plane Couette flow is considered. The solution is used to evaluate the evolution of wave-packet perturbations to the mean flow for all Richardson numbers J > ¼, demonstrating that a consistent wave-packet approach to wave propagation in these flows is possible for all J > ¼. It is found that the vertical influence of a wave-packet perturbation is limited to a distance of order (J − ¼)½/k0, where k0 is the magnitude of the initial central wave vector. These results are used to clarify the J [gsim ] ¼ conclusions of an earlier treatment by Booker & Bretherton.

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