摘要:

An investigation of lower bounds on the quantitiesWn(t)=∫&OHgr;|&ohgr;|2n dVforn⩾1for the incompressible three-dimensional (3D) Euler equations has led us to consider a set of spatially averaged weighted “eigenvalues,”&lgr;S(n)(t)and&lgr;P(n)(t), of the strain matrixSand the Hessian matrix of the pressureP={p,ij}, respectively. It is shown that these obey the simple inequality,&lgr;˙S(n)+f(&thgr;n)(&lgr;S(n))2+&lgr;P(n)⩾0, wheref(&thgr;n)=1−tan2 &thgr;n.The&thgr;nare spatially averaged weighted angles between the vorticity vector&ohgr;and the vortex stretching vector&sgr;=&ohgr;⋅&bnabla;u. The weighting in the averaging process highlights regions of large vorticity. This is the angle considered by Tsinober, Kit, and Dracos in their analysis of data from turbulent grid flow experiments in which they noted a tendency toward alignment between&ohgr;and&sgr;. The Burgers vortex turns out to be a sharp solution of this inequality with a corresponding angle&thgr;n=0, giving rise to exponential growth inWn.Some special solutions for cases where&thgr;nmoves between&thgr;n=0and&thgr;n=&pgr;/2are displayed. The work of Ohkitani and Kishiba on the alignment in 3-D Euler flows between&ohgr;and the third eigenvector ofPat maximum enstrophy is also particularly relevant and is applied to the modified pressure matrixQ={p,ij−3&dgr;ijp,ii}in the limitn→∞. The finite time blow-up problem is discussed in this context. In an Appendix it is shown that an identical inequality holds for the barotropic compressible Euler equations where&zgr;=&ohgr;/&rgr; andWn(t)=∫&OHgr;&rgr;|&zgr;|2n dVreplace&ohgr;andWn,respectively. ©1997 American Institute of Physics.

ISSN:1070-6631    

DOI:10.1063/1.869186

出版商:AIP    

年代:1997

数据来源: AIP

摘要:

The convective instability boundary of a circular Couette flow in the annular region bounded by two co- or counter-rotating coaxial cylinders with angular velocities&ohgr;1and&ohgr;2, respectively, is studied in the presence of an axial flow due to a constant axial pressure gradient and a radial flow through the permeable walls of the cylinders. A linear stability analysis is carried out for positive and negative radial Reynolds numbers corresponding to outward and inward radial flows, respectively. Axisymmetric and non-axisymmetric disturbances are considered. In the particular case of no axial flow, the Couette flow is stabilized by an inward, or a strong outward, radial flow, but destabilized by a weak outward radial flow. Non-axisymmetric disturbances lead to instability for some negative values of&mgr;=&ohgr;2/&ohgr;1. Bifurcation diagrams for combined radial and axial flows are more complicated. For particular values of the parameters of the problem, the Couette flow has regions of stabilization and destabilization in the parameter space. Computational results are compared with experimental data. ©1997 American Institute of Physics.

ISSN:1070-6631    

DOI:10.1063/1.869187

出版商:AIP    

年代:1997

数据来源: AIP

摘要:

Interfacial waves grow in a cocurrent, stratified gas–liquid flow by extracting energy from the main flow. The most unstable mode typically has a wavelength comparable to or less than the liquid depth. Experiments show that these short waves can saturate at small amplitude with no generation of long-wave or transverse modes. By decomposing the typical Stuart–Landau analysis into three components, it is found that saturation usually occurs by cubic self-interaction of the fundamental mode but quadratic resonant interaction with the first overtone is also possible. Interaction with mean flow modes is usually much less important. Experiments confirm the predictions of weakly nonlinear theory. The measured overtone is found to beO(|A1|2)and is phase-locked with the fundamental except near a 1–2 resonance point where the fundamental and the overtone have comparable speeds. Near this resonance, the amplitudes are of the same order and the phase angle between them is observed to jump irregularly as predicted by modern dynamical systems theory for intermittent chaos near a heteroclinic cycle. The phase and magnitude of the overtone interaction specify the shape, chaotic dynamics and symmetry of the waves across resonance which are analyzed and confirmed experimentally. ©1997 American Institute of Physics.

ISSN:1070-6631    

DOI:10.1063/1.869188

出版商:AIP    

年代:1997

数据来源: AIP

摘要:

In this paper we are concerned with the wave generation by a singular forcelet in a viscous fluid of finite depth, where the singularity is located far from the bottom and not very near the free surface. In the first part of this work, the image system of an Oseenlet bounded by a no-slip wall, is considered. It is found that the resultant velocity field can be described by a planar distribution of vertical Oseen doublets and a negative Oseenlet located at the mirror point of the singularity with respect to the plane wall. In the second part of the work we deal with the generation of waves by these solutions. By imposing the linearized free-surface conditions on the solutions obtained from the first part, the wave generated is shown to exhibit the Kelvin ship wave pattern that agrees with observation. The effects of water depth and of submergence on the wave amplitude are also investigated. ©1997 American Institute of Physics.

ISSN:1070-6631    

DOI:10.1063/1.869189

出版商:AIP    

年代:1997

数据来源: AIP

摘要:

The Lagrangian transport of “passive” particles advected by inertia-gravity waves is investigated. We consider two classes of waves, namely, vertically trapped, horizontally propagating waves, and those propagating in three dimensions (3D). In the former case, it is shown that the superposition of at least two waves is necessary to produce chaotic particle paths; whereas for the latter case, at least three waves are required to initiate chaotic mixing. Liapounov exponents are used to quantify the predictability of particle trajectories in the chaotic region. Whether the chaotic mixing process is temporally uniform or intermittent is deduced from the local deviation from the Liapounov exponent. Typical estimates of Liapounov exponents give error-doubling times of the order of a few hours which roughly decreases as the amplitude of the perturbing wave (&egr;) increases. For waves propagating only in the horizontal, the chaotic mixing process tends to be more uniform as &egr; increases, while the reverse is the case for waves propagating in 3D with more intermittent mixing for larger values of &egr;. The chaos induced transport process is characterized from a relation of the form&Dgr;X2(t)∼t&agr;, for larget, where&Dgr;X2(t)is the mean square distance traveled by a cloud of particles. For lower values of &egr;, the horizontally propagating case gives values of &agr; greater than 2 and is nearly 2 for a larger value of &egr;. The value of &agr; is nearly 2 for chaotically dispersing particle clouds in the 3D propagating case. Also, correlation dimensions are used to learn about the geometry of the cloud evolution. The results show that clouds originating in the chaotic zone initially spread more than like a filament, subsequently become area filling, and then proceed toward space filling behavior. This sequence of transition has been found to be faster for the 3D propagating waves than for the vertically trapped case. The implications of the results to the wave-induced mixing phenomena in geophysical flows are discussed. ©1997 American Institute of Physics.

ISSN:1070-6631    

DOI:10.1063/1.869190

出版商:AIP    

年代:1997

数据来源: AIP

摘要:

When layers of salt and sugar solution are separated by a “diffusive” interface, interfacial waves are spontaneously generated by the turbulent convection once the system evolves to a critical value of the density–anomaly ratioR&rgr;≡&bgr;&Dgr;S/&agr;&Dgr;T(Stamp &etal;, to appear in J. Fluid Mech). The waves modulate the interfacial fluxes by modifying the interface thickness and thereby organize the otherwise random convective motions into coherent large-scale circulations. In narrow rectangular channels a wide range of conditions give rise to a single wave which propagates back-and-forth, resulting in quasi-periodic reversals of tank-scale circulations. Here it is shown that in annular and equant rectangular cavities this same coupling phenomenon produces turbulent convection cells of a traveling-wave nature, coupled to large-amplitude solitary waves on the interface. ©1997 American Institute of Physics.

ISSN:1070-6631    

DOI:10.1063/1.869191

出版商:AIP    

年代:1997

数据来源: AIP

摘要:

Several approaches for slender vortex motion (the local induction equation, the Klein–Majda equation, and the Klein–Knio equation) are compared on a specific example of sideband instability of Kelvin waves on a vortex. Numerical experiments on this model problem indicate that all these equations yield qualitatively similar behavior, and this behavior is different from the behavior of a nonslender vortex with variable cross-section. It is found that the boundaries between stable, recurrent, and chaotic regimes in the parameter space of the model problem depend on the equation used. The boundaries of these domains in the parameter space for the Klein–Majda equation and for the Klein–Knio equation are closely related to the core size. When the core size is large enough, the Klein–Majda equation always exhibits stable solutions for our model problem. Various conclusions are drawn; in particular, the behavior of turbulent vortices cannot be captured by these approximations, and probably cannot be captured by any slender vortex model with constant vortex cross-section. Speculations about the differences between classical and superfluid hydrodynamics are also offered. ©1997 American Institute of Physics.

ISSN:1070-6631    

DOI:10.1063/1.869192

出版商:AIP    

年代:1997

数据来源: AIP

摘要:

The formation and dynamics of dipolar vortex structures in two-dimensional flows are studied. Localized initial structures possessing a finite linear momentum are found to develop into dipoles by direct numerical solutions of the two-dimensional Navier-Stokes equations. The detailed structure of the evolving dipoles depend on the initial condition. However, the gross properties of their evolution are only weakly dependent on the detailed structure and can be well-described by the so-called Lamb-dipole solution. The viscous decay of the Lamb-dipole, leading to an expansion and a decreasing velocity, is well described by an adiabatic theory. During the expansion the dipole is found to trap fluid as it evolves. ©1997 American Institute of Physics.

ISSN:1070-6631    

DOI:10.1063/1.869193

出版商:AIP    

年代:1997

数据来源: AIP

摘要:

The three-dimensional, compressible Navier–Stokes equations in primitive variables are solved numerically to simulate vortex breakdown in a constricted tube. Time integration is performed with an implicit Beam-Warming algorithm using fourth-order compact operators to discretize spatial derivatives. Initial conditions are obtained by solving the steady, compressible, and axisymmetric form of the Navier–Stokes equations with Newton’s method. The effects of three-dimensionality on flows that are initially axisymmetric and stable to 2-D disturbances are examined. Stability of the axisymmetric base flow is assessed through 3-D time integration. Axisymmetric solutions at a Mach number of 0.3 and a Reynolds number of 1000 contain a region of nonuniqueness. Within this region, 3-D time integration reveals only unique solutions, with nonunique axisymmetric initial conditions converging to a unique solution that is steady and axisymmetric. Past the primary limit point, which approximately identifies the appearance of critical flow (a flow that can support an axisymmetric standing wave), the solutions bifurcate into 3-D time-periodic flows. Thus this numerical study shows that the vortex strength associated with the loss of stability to 3-D disturbances and that of the primary limit point are in close proximity. Additional numerical and theoretical studies of 3-D swirling flows are needed to determine the impact of various parameters on dynamic behavior. For example, it is possible that a different flow behavior, leading to a nearly axisymmetric vortex breakdown state, may develop with other inlet profiles and tube geometries.

ISSN:1070-6631    

DOI:10.1063/1.869194

出版商:AIP    

年代:1997

数据来源: AIP

摘要:

The stability of “top-hat” and fully developed jet profiles is investigated by an inviscid linear stability theory for compressible flow. The study covers a wide range of the Mach number and the temperature ratio. Two types of instabilities are found: vortical and acoustic, each of which can be subdivided into non-radiating (subsonic) and radiating (supersonic) modes. The vortical mode is the continuation of the Kelvin-Helmholtz instability from incompressible flow. The acoustic mode is a compressible flow phenomenon, which becomes important at large Mach numbers. Temperature-ratio effects can be destabilizing or stabilizing, depending on the Mach number and mode of instability. A spectrum of unstable acoustic modes, including axisymmetric ones, are found to exist in the fully developed jet. For this jet, acoustic axisymmetric waves become more unstable than both vortical and acoustic helical waves at Mach numbers over about 3. Strong evidence of a resonance mechanism for acoustic modes is seen in the growth rate curves at high Mach numbers, where a spectrum of local peaks and valleys appears at regularly distributed frequencies. ©1997 American Institute of Physics.

ISSN:1070-6631    

DOI:10.1063/1.869196

出版商:AIP    

年代:1997

数据来源: AIP