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Decaying grid turbulence in a strongly stratified fluid
- OLIVIER PRAUD, ADAM M. FINCHAM, JOEL SOMMERIA
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- 13 January 2005, pp. 1-33
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Grid turbulence experiments have been carried out in a stably stratified fluid at moderately large Reynolds numbers (160 based on the Taylor microscale). A scanning particle image velocimetry technique is used to provide time-resolved velocity fields in a relatively large volume. For late times, in the low-Froude-number regime, the flow consists of quasi-horizontal motion in a sea of weak internal gravity waves. In this regime the dynamics of the flow is found to be independent of the ambient stratification. Fundamental differences with two-dimensional turbulence, due to the strong vertical shearing of horizontal velocity, are observed. In this regime, a self-similar scaling law for the energy decay and the length-scale evolution are observed. This behaviour reflects a process of adjustment of the eddy aspect ratio based on a balance between the horizontal advective motion which tends to vertically decorrelate the flow and the dissipation due to the strong vertical shear. The characteristic vertical size of the eddies grows according to a diffusion law and is found to be independent of the turbulence generation. The organization of the flow into horizontal layers of eddies separated by intense shear leads to a strong anisotropy of the dissipation: this has been checked by direct measurement of the different tensorial components of the viscous dissipation.
Rotating gravity currents. Part 1. Energy loss theory
- J. R. MARTIN, G. F. LANE-SERFF
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- 13 January 2005, pp. 35-62
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A comprehensive energy loss theory for gravity currents in rotating rectangular channels is presented. The model is an extension of the non-rotating energy loss theory of Benjamin (J. Fluid Mech. vol. 31, 1968, p. 209) and the steady-state dissipationless theory of rotating gravity currents of Hacker (PhD thesis, 1996). The theory assumes the fluid is inviscid, there is no shear within the current, and the Boussinesq approximation is made. Dissipation is introduced using a simple method. A head loss term is introduced into the Bernoulli equation and it is assumed that the energy loss is uniform across the stream. Conservation of momentum, volume flux and potential vorticity between upstream and downstream locations is then considered. By allowing for energy dissipation, results are obtained for channels of arbitrary depth and width (relative to the current). The results match those from earlier workers in the two limits of (i) zero rotation (but including dissipation) and (ii) zero dissipation (but including rotation). Three types of flow are identified as the effect of rotation increases, characterized in terms of the location of the outcropping interface between the gravity current and the ambient fluid on the channel boundaries. The parameters for transitions between these cases are quantified, as is the detailed behaviour of the flow in all cases. In particular, the speed of the current can be predicted for any given channel depth and width. As the channel depth increases, the predicted Froude number tends to $\surd 2$, as for non-rotating flows.
Rotating gravity currents. Part 2. Potential vorticity theory
- J. R. MARTIN, D. A. SMEED, G. F. LANE-SERFF
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- 13 January 2005, pp. 63-89
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An extension to the energy-conserving theory of gravity currents in rectangular rotating channels is presented, in which an upstream potential vorticity boundary condition in the current is applied. It is assumed that the fluid is inviscid; that the Boussinesq approximation applies; that the fundamental properties of momentum, energy, volume flux and potential vorticity are conserved between upstream and downstream locations; and that the flow is dissipationless. The upstream potential vorticity in the current is set through the introduction of a new parameter $\delta$, that defines the ratio of the reference depth of the current to the ambient fluid. Flow types are established as a function $\delta$ and the rotation rate, and a fourth flow geometry is identified in addition to the three previously identified for rotating gravity currents. Detailed solutions are obtained for three cases $\delta\,{=}\,$0.5, 1.0 and 1.5, where $\delta\,{<}\,1$ is relevant to currents originating from a shallow source and $\delta\,{>}\,1$ to currents where the source region is deeper than the downstream depth, for example where a deep ocean flow encounters a plateau. The governing equations and solutions for each case are derived, quantifying the flow in terms of the depth, width and front speed. Cross-stream velocity profiles are provided for both the ambient fluid and the current. These predict the evolution of a complex circulation within the current as the rotation rate is varied. The ambient fluid exhibits similar trends to those predicted by the energy-conserving theory, with the Froude number tending to $\surd 2$ at the right-hand wall at high rotation rates. The introduction of the potential vorticity boundary condition into the energy-conserving theory does not appear to have a substantial effect on the main flow parameters (such as current speed and width); however it does provide an insight into the complex dynamics of the flow within the current.
The lattice Boltzmann equation for natural convection in a two-dimensional cavity with a partially heated wall
- G. BARRIOS, R. RECHTMAN, J. ROJAS, R. TOVAR
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- 13 January 2005, pp. 91-100
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The lattice Boltzmann equation method in two dimensions was used to analyse natural convective flows. The method was validated with experiments in an open cavity with one of the vertical walls divided into two parts, the lower part conductive, the upper part and all the other walls adiabatic. An upward thermal boundary layer formed near the conductive wall. This layer gave way to a wall plume. The numerical results compared well with experiments in the laminar ($Ra\,{=}\,2.0\,{\times}\,10^9$) and transition ($Ra\,{=}\,4.9\,{\times}\,10^9$) regimes. The behaviour of the starting plume was numerically studied for Rayleigh numbers Ra from $10^6$ to $4.9\times 10^9$. The wall plume grows in three stages: in the first with constant acceleration, in the second with constant ascending velocity and in the third with negative acceleration due to the presence of the top boundary layer. The acceleration of the first stage and the velocity of the second both scale with the Rayleigh number.
Viscous eddies in a circular cone
- V. S. MALYUGA
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- 13 January 2005, pp. 101-116
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The flow of viscous incompressible fluid in a circular cone induced by a non-zero velocity prescribed at the boundary within a ring $0\,{<}\,a_1\,{<}\,r\,{<}\,a_2\,{<}\,\infty$, where $r$ is the distance from the vertex, is considered in the limits of the Stokes approximation. In the spherical coordinate system $(r,\theta,\phi)$ with the origin at the vertex and the axis $\theta\,{=}\,0$ coincident with the axis of the cone the velocity and pressure fields are represented in the form of a Fourier series on the trigonometric system $\cos m \phi$. The solution is constructed for each term by use of the Mellin transform. The contribution of each term of the Fourier expansion to the local velocity field near the vertex is studied. The kinematics of the local flows is illustrated by two examples. The flows are induced by the motion of two and three equally spaced segments, respectively.
Nested toroidal vortices between concentric cones
- CHETAN P. MALHOTRA, PATRICK D. WEIDMAN, ANTHONY M. J. DAVIS
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- 13 January 2005, pp. 117-139
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A few decades ago, the significance of Moffatt vortices was demonstrated by establishing their existence in various flows. Wedge and cusp regions and their axisymmetric counterparts were preferred to conical regions because the associated analyses were simpler. The lowest even and odd modes were always dominant and the streamline patterns of higher modes were assumed to be similarly simple, especially as their minute strength caused computational difficulties. Here, armed with far more computer power, we return to the vortices' canonical structure, with our principal focus on the region exterior to two cones with common axis and vertex. Many interesting features are revealed, the most unexpected being the structure of the third (second odd in a symmetric geometry) mode. The two-cone geometry allows consideration of asymmetric regions, for the first time. Comparisons are made with the well-known wedge and single-cone results and numerical corrections made to the latter. In all cases, eigenvalue plots play a valuable role in guiding the discussion.
Velocity fields in mixing-enhanced compressible shear layers
- SHIGEYA WATANABE, M. G. MUNGAL
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- 13 January 2005, pp. 141-177
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Planar velocity fields of mixing-enhanced compressible planar shear layers are measured via particle image velocimetry (PIV) in order to investigate the mechanism of mixing enhancement by sub-boundary-layer triangular disturbances. The measurements are conducted at convective Mach numbers, $M_{{c}}$, of 0.62 and 0.24 to examine compressibility effects on effectiveness of the mixing enhancement technique. Instantaneous side- and plan-view vector maps of the shear layers are obtained, and turbulence statistical quantities are derived from the instantaneous velocity data. Schlieren and planar laser Mie scattering (PLMS) techniques are also used to measure the shear-layer thickness and growth rate as well as surveying the qualitative flow fields. The velocity fields for several disturbance configurations with different shape, size, or thickness are compared in terms of the shear-layer thickness and growth rate in order to investigate the effects of the configuration variation on the mixing enhancement strategy. Configuration parameters include thickness, the semi-vertex angle of the triangular disturbance, and the streamwise offset of the disturbance from the splitter tip. The measured transverse profile of the mean streamwise velocity shows a characteristic shape with triple inflection points for the effective mixing-enhanced cases at the two different compressibility conditions, while periodic inflection points are observed in the spanwise direction. A pair of stationary counter-rotating streamwise vortices introduced by the subboundary-layer disturbances are also observed, even in the fully developed region of the shear layers. At $M_{{c}}\,{=}\,0.62$, it is found that in successfully enhanced cases, regardless of the disturbance configurations, the present mixing-enhancement strategy has the effect of increasing the turbulence intensity and Reynolds stress, and suppressing the turbulence anisotropy increase with increasing compressibility, i.e. alleviating the compressibility effect which intrinsically reduces pressure–strain-rate redistribution, leading to effective mixing enhancement. Comparison of the results at the two compressibility conditions reveals that the growth rate of the layer is almost constant in the streamwise direction for all cases at $M_{{c}}\,{=}\,0.62$, while for all disturbed cases at $M_{{c}}\,{=}\,0.24$, after an initial layer thickening, growth rate decreases with downstream distance to the value for the undisturbed case, indicating that the present mixing enhancement is less effective at nearly incompressible conditions.
Regular shock refraction at an oblique planar density interface in magnetohydrodynamics
- V. WHEATLEY, D. I. PULLIN, R. SAMTANEY
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- 13 January 2005, pp. 179-214
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We consider the problem of regular refraction (where regular implies all waves meet at a single point) of a shock at an oblique planar contact discontinuity separating conducting fluids of different densities in the presence of a magnetic field aligned with the incident shock velocity. Planar ideal magnetohydrodynamic (MHD) simulations indicate that the presence of a magnetic field inhibits the deposition of vorticity on the shocked contact. We show that the shock refraction process produces a system of five to seven plane waves that may include fast, intermediate, and slow MHD shocks, slow compound waves, $180^\circ$ rotational discontinuities, and slow-mode expansion fans that intersect at a point. In all solutions, the shocked contact is vorticity free and hence stable. These solutions are not unique, but differ in the types of waves that participate. The set of equations governing the structure of these multiple-wave solutions is obtained in which fluid property variation is allowed only in the azimuthal direction about the wave-intersection point. Corresponding solutions are referred to as either quintuple-points, sextuple-points, or septuple-points, depending on the number of participating waves. A numerical method of solution is described and examples are compared to the results of numerical simulations for moderate magnetic field strengths. The limit of vanishing magnetic field at fixed permeability and pressure is studied for two solution types. The relevant solutions correspond to the hydrodynamic triple-point with the shocked contact replaced by a singular structure consisting of a wedge, whose angle scales with the applied field magnitude, bounded by either two slow compound waves or two $180^\circ$ rotational discontinuities, each followed by a slow-mode expansion fan. These bracket the MHD contact which itself cannot support a tangential velocity jump in the presence of a non-parallel magnetic field. The magnetic field within the singular wedge is finite and the shock-induced change in tangential velocity across the wedge is supported by the expansion fans that form part of the compound waves or follow the rotational discontinuities. To verify these findings, an approximate leading-order asymptotic solution appropriate for both flow structures was computed. The full and asymptotic solutions are compared quantitatively.
Vortex-induced vibrations of a pivoted cylinder
- F. FLEMMING, C. H. K. WILLIAMSON
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- 13 January 2005, pp. 215-252
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Much of the research into vortex-induced vibrations has been dedicated to the problem of a cylinder vibrating transverse to a fluid flow ($Y$-motion). There are very few papers studying the more practical case of vibration in two degrees of freedom ($XY$-motion), or the case where there is variation of amplitude along the span of a body. The present two-degree-of-freedom pivoted cylinder apparatus represents the simplest configuration having a spanwise variation of amplitude. A central question concerns how well the results from $Y$-motion studies carry over to the case of a body in two degrees of freedom, and also how effective the quasi-uniform assumption is when there is spanwise amplitude variation.
In a manner comparable with the $Y$-motion cylinder, the principal dynamics of the pivoted body are transverse to the flow. For moderate values of the product: inertia–damping or $(I^*\zeta)$, the system exhibits two amplitude response branches, and for sufficiently low $(I^*\zeta)$, three response branches appear, in strong analogy with previous results for $Y$-motion bodies. The response branches for the bodies with low $(I^*\zeta)$ correspond with both the 2S mode (two single vortices per cycle) and 2P mode (two vortex pairs per cycle) of vortex formation along the span. We also observe a clear 2S-2P hybrid mode, similar to that found for vibrating tapered cylinders by Techet et al. (J. Fluid Mech. vol. 363, 1998, p. 79). These different modes correspond well with the Williamson & Roshko (J. Fluids Struct. vol. 2, 1988, p. 355) map of modes in the plane of amplitude and frequency, so long as the streamwise vibration is small. However, when the inertia of the body is sufficiently small, the correspondence with the map of modes for the upper branch is not close. The response branches cross over each other in this map, and one has to introduce a third dimension to represent streamwise amplitude. This three–dimensional plot shows that the two response branches exist in quite different parameter spaces. The upper branch with the higher streamwise motion corresponds to a new vortex formation mode, which comprises two co-rotating vortices each half-cycle, defined as a ‘2C’ mode. We present the principal three-dimensional vorticity structures corresponding to each vortex wake mode. Vortex dislocations and vortex merging are characteristics of these complex three–dimensional structures. We introduce equations of motion for the case of the pivoted cylinder with two degrees of freedom, and thereby deduce that a critical inertia, $I_{\hbox{\scriptsize\it crit}}$ exists analogous to the ‘critical mass’ of Govardhan & Williamson (J. Fluid Mech. vol. 420, 2000, p. 85; vol. 473, 2002, p. 147), below which the pivoted body is predicted to have an infinitely wide regime of flow velocities where resonant oscillations will occur.
Turbulent rotating disk flow with inward throughflow
- S. PONCET, M. P. CHAUVE, P. LE GAL
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- 13 January 2005, pp. 253-262
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The evolution of the entrainment coefficient $K$ of the rotating fluid in a rotor–stator cavity with an inward throughflow and pre-rotation is studied according to the flow parameters. Measurements are obtained in water for a turbulent Batchelor type of flow with two separated boundary layers on the rotating and stationary disks by means of a laser Doppler anemometer, and the results are compared to those performed using pressure transducers. We show that the entrainment coefficient $K$ depends on a local flow rate coefficient $Cq_r$ according to a $5/7$ power law whose coefficients depend on the boundary conditions. A theoretical analysis confirms this behaviour of $K$.
On probability density function equations for particle dispersion in a uniform shear flow
- MICHAEL W. REEKS
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- 13 January 2005, pp. 263-302
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The paper examines a fundamental discrepancy between two probability density function (PDF) models, the kinetic model (KM) and generalized Langevin model (GLM), currently used to model the dispersion of particles in turbulent flows. This discrepancy is manifest in particle dispersion in an unbounded simple shear flow where model predictions for the values of the streamwise fluid-particle diffusion coefficients are not only different but are of opposite sign. It is shown that this discrepancy arises through a neglect of the inertial convection term in the GLM equation for the mean carrier flow velocity local to a particle which eventually leads to algebraic forms for the particle-fluid diffusion coefficients. Evaluating this term for a Gaussian process leads to identical results for both PDF formulations. This also resolves a fundamental long-standing discrepancy in previous forms reported for the passive scalar diffusion coefficients in a simple shear flow where similar assumptions were made. Avoiding this assumption, the exact solutions are given for the dispersion of particles in this simple shear flow case derived from the solution of the GLM PDF equation which show explicitly the dependence on the particle response time and the strain rate, both normalized on the integral timescale of the turbulence. The analysis shows that the particle diffusion coefficient in the streamwise direction is negative when the strain rate $\,{\geq}\,$ a certain value. The origin of negative diffusion coefficients is explained and their influence is shown in the way in which the mean concentration and mean velocity flow fields of the particle and carrier flow (seen by the particle) evolve with time for particles released from the centre of the shear.
Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows
- T. WEI, P. FIFE, J. KLEWICKI, P. McMURTRY
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- 13 January 2005, pp. 303-327
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The properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows are explored both experimentally and theoretically. Available high-quality data reveal a dynamically relevant four-layer description that is a departure from the mean profile four-layer description traditionally and nearly universally ascribed to turbulent wall flows. Each of the four layers is characterized by a predominance of two of the three terms in the governing equations, and thus the mean dynamics of these four layers are unambiguously defined. The inner normalized physical extent of three of the layers exhibits significant Reynolds-number dependence. The scaling properties of these layer thicknesses are determined. Particular significance is attached to the viscous/Reynolds-stress-gradient balance layer since its thickness defines a required length scale. Multiscale analysis (necessarily incomplete) substantiates the four-layer structure in developed turbulent channel flow. In particular, the analysis verifies the existence of at least one intermediate layer, with its own characteristic scaling, between the traditional inner and outer layers. Other information is obtained, such as (i) the widths (in order of magnitude) of the four layers, (ii) a flattening of the Reynolds stress profile near its maximum, and (iii) the asymptotic increase rate of the peak value of the Reynolds stress as the Reynolds number approaches infinity. Finally, on the basis of the experimental observation that the velocity increments over two of the four layers are unbounded with increasing Reynolds number and have the same order of magnitude, there is additional theoretical evidence (outside traditional arguments) for the asymptotically logarithmic character of the mean velocity profile in two of the layers; and (in order of magnitude) the mean velocity increments across each of the four layers are determined. All of these results follow from a systematic train of reasoning, using the averaged momentum balance equation together with other minimal assumptions, such as that the mean velocity increases monotonically from the wall.
Linear and nonlinear dynamics of cylindrically and spherically expanding detonation waves
- SIMON D. WATT, GARY J. SHARPE
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- 13 January 2005, pp. 329-356
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The nonlinear stability of cylindrically and spherically expanding detonation waves is investigated using numerical simulations for both directly (blast) initiated detonations and cases where the simulations are initialized by placing quasi-steady solutions corresponding to different initial shock radii onto the grid. First, high-resolution one-dimensional (axially or radially symmetric) simulations of pulsating detonations are performed. Emphasis is on comparing with the predictions of a recent one-dimensional linear stability analysis of weakly curved detonation waves. The simulations show that, in agreement with the linear analysis, increasing curvature has a rapid destabilizing effect on detonation waves. The initial size and growth rate of the pulsation amplitude decreases as the radius where the detonation first forms increases. The pulsations may reach a saturated nonlinear behaviour as the amplitude grows, such that the subsequent evolution is independent of the initial conditions. As the wave expands outwards towards higher (and hence more stable) radii, the nature of the saturated nonlinear dynamics evolves to that of more stable behaviour (e.g. the amplitude of the saturated nonlinear oscillation decreases, or for sufficiently unstable cases, the oscillations evolve from multi-mode to period-doubled to limit-cycle-type behaviour). For parameter regimes where the planar detonation is stable, the linear stability prediction of the neutrally stable curvature gives a good prediction of the location of the maximum amplitude (provided the stability boundary is reached before the oscillations saturate) and of the critical radius of formation above which no oscillations are seen. The linear analysis also predicts very well the dependence of the period on the radius, even in the saturated nonlinear regimes. Secondly, preliminary two-dimensional numerical simulations of expanding cellular detonations are performed, but it is shown that resolved and accurate calculations of the cellular dynamics are currently computationally prohibitive, even with a dynamically adaptive numerical scheme.
The critical merger distance between two co-rotating quasi-geostrophic vortices
- JEAN N. REINAUD, DAVID G. DRITSCHEL
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- 13 January 2005, pp. 357-381
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This paper examines the critical merger or strong interaction distance between two equal-potential-vorticity quasi-geostrophic vortices. The interaction between the two vortices depends on five parameters: their volume ratio, their height-to-width aspect ratios and their vertical and horizontal offsets. Due to the size of the parameter space, a direct investigation solving the full quasi-geostrophic equations is impossible. We instead determine the critical merger distance approximately using an asymptotic approach. We associate the merger distance with the margin of stability for a family of equilibrium states having prescribed aspect and volume ratios, and vertical offset. The equilibrium states are obtained using an asymptotic solution method which models vortices by ellipsoids. The margin itself is determined by a linear stability analysis. We focus on the interaction between oblate to moderately prolate vortices, the shapes most commonly found in turbulence. Here, a new unexpected instability is found and discussed for prolate vortices which is manifested by the tilting of vortices toward each other. It implies than tall vortices may merge starting from greater separation distances than previously thought.
Pressure corrections for potential flow analysis of capillary instability of viscous fluids
- J. WANG, D. D. JOSEPH, T. FUNADA
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- 13 January 2005, pp. 383-394
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Funada & Joseph (Intl J. Multiphase Flow, vol. 28, 2002, p. 1459) analysed capillary instability assuming that the flow is irrotational but the fluids are viscous (viscous potential flow, VPF). They compared their results with the exact normal-mode solution of the linearized Navier–Stokes equations (fully viscous flow, FVF) and with the irrotational flow of inviscid fluids (inviscid potential flow, IPF). They showed that the growth rates computed by VPF are close to the exact solution when Reynolds number is larger than $O(10)$ and are always more accurate than those computed using IPF. Recently, Joseph & Wang (J. Fluid Mech., vol. 505, 2004, p. 365) presented a method for computing a viscous correction of the irrotational pressure induced by the discrepancy between non-zero irrotational shear stress and the zero-shear-stress boundary condition at a free surface. The irrotational flow with a corrected pressure is called the viscous correction of VPF (VCVPF). Here we compute the pressure correction for capillary instability in cases in which one fluid is viscous and the other fluid is a gas of negligible density and viscosity. The growth rates computed using VCVPF are in remarkably good agreement with the exact solution FVF.
Symmetry breaking of two-dimensional time-periodic wakes
- H. M. BLACKBURN, F. MARQUES, J. M. LOPEZ
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- 13 January 2005, pp. 395-411
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A number of two-dimensional time-periodic flows, for example the Kármán street wake of a symmetrical bluff body such as a circular cylinder, possess a spatio-temporal symmetry: a combination of evolution by half a period in time and a spatial reflection leaves the flow invariant. Floquet analyses for the stability of these flows to three-dimensional perturbations have in the past been based on the Poincaré map, without attempting to exploit the spatio-temporal symmetry. Here, Floquet analysis based on the half-period-flip map provides a comprehensive interpretation of the symmetry-breaking bifurcations.