Topics in vortex motion

Six topics in incompressible, inviscid fluid flow involving vortex motion are presented. The stability of the unsteady flow field due to the vortex filament expanding under the influence of an axial compression is examined in the first chapter as a possible model of the vortex bursting observed in aircraft contrails. The filament with a stagnant core is found to be unstable to axisymmetric disturbances. For initial disturbances with the form of axisymmetric Kelvin waves, the filament with a uniformly rotating core is neutrally stable, but the compression causes the disturbance to undergo a rapid increase in amplitude. The time at which the increase occurs is, however, later than the observed bursting times, indicating the bursting phenomenon is not caused by this type of instability.

In the second and third chapters the stability of a steady vortex filament deformed by two-dimensional strain and shear flows, respectively, is examined. The steady deformations are in the plane of the vortex cross-section. Disturbances which deform the filament centerline into a wave which does not propagate along the filament are shown to be unstable and a method is described to calculate the wave number and corresponding growth rate of the amplified waves for a general distribution of vorticity in the vortex core.

In Chapter Four exact solutions are constructed for two-dimensional potential flow over a wing with a free ideal vortex standing over the wing. The loci of positions of the free vortex are found and the lift is calculated. It is found that the lift on the wing can be significantly increased by the free vortex.

The two-dimensional trajectories of an ideal vortex pair near an orifice are calculated in Chapter Five. Three geometries are examined, and the criteria for the vortices to travel away from the orifice are determined.

Finally, Chapter Six reproduces completely the paper, "Structure of a linear array of hollow vortices of finite cross-section," co-authored with G. R. Baker and P. G. Saffman. Free streamline theory is employed to construct an exact steady solution for a linear array of hollow, or stagnant cored vortices. If each vortex has area A and the separation is L, then there are two possible shapes if A^(1/2)/L is less than 0.38 and none if it is larger. The stability of the shapes to two-dimensional, periodic and symmetric disturbances is considered for hollow vortices. The more deformed of the two possible shapes is found to be unstable, while the less deformed shape is stable.

Some problems of edge waves and standing waves on beaches

Some problems of edge waves and standing waves on beaches are examined.

The nonlinear interaction of a wave normally incident on a sloping beach with a subharmonic edge wave is studied. A two-timing expansion is used in the full nonlinear theory to obtain the modulation equations which describe the evolution of the waves. It is shown how large amplitude edge waves are produced; and the results of the theory are compared with some recent laboratory experiments.

Traveling edge waves are considered in two situations. First, the full linear theory is examined to find the finite depth effect on the edge waves produced by a moving pressure disturbance. In the second situation, a Stokes' expansion is used to discuss the nonlinear effects in shallow water edge waves traveling over a bottom of arbitrary shape. The results are compared with the ones of the full theory for a uniformly sloping bottom.

The finite amplitude effects for waves incident on a sloping beach, with perfect reflection, are considered. A Stokes' expansion is used in the full nonlinear theory to find the corrections to the dispersion relation for the cases of normal and oblique incidence.

Finally, an abstract formulation of the linear water waves problem is given in terms of a self adjoint but nonlocal operator. The appropriate spectral representations are developed for two particular cases.

Earthquakes of the Nepal Himalaya : towards a physical model of the seismic cycle

Home to hundreds of millions of souls and land of excessiveness, the Himalaya is also the locus of a unique seismicity whose scope and peculiarities still remain to this day somewhat mysterious. Having claimed the lives of kings, or turned ancient timeworn cities into heaps of rubbles and ruins, earthquakes eerily inhabit Nepalese folk tales with the fatalistic message that nothing lasts forever. From a scientific point of view as much as from a human perspective, solving the mysteries of Himalayan seismicity thus represents a challenge of prime importance. Documenting geodetic strain across the Nepal Himalaya with various GPS and leveling data, we show that unlike other subduction zones that exhibit a heterogeneous and patchy coupling pattern along strike, the last hundred kilometers of the Main Himalayan Thrust fault, or MHT, appear to be uniformly locked, devoid of any of the “creeping barriers” that traditionally ward off the propagation of large events. The approximately 20 mm/yr of reckoned convergence across the Himalaya matching previously established estimates of the secular deformation at the front of the arc, the slip accumulated at depth has to somehow elastically propagate all the way to the surface at some point. And yet, neither large events from the past nor currently recorded microseismicity nearly compensate for the massive moment deficit that quietly builds up under the giant mountains. Along with this large unbalanced moment deficit, the uncommonly homogeneous coupling pattern on the MHT raises the question of whether or not the locked portion of the MHT can rupture all at once in a giant earthquake. Univocally answering this question appears contingent on the still elusive estimate of the magnitude of the largest possible earthquake in the Himalaya, and requires tight constraints on local fault properties. What makes the Himalaya enigmatic also makes it the potential source of an incredible wealth of information, and we exploit some of the oddities of Himalayan seismicity in an effort to improve the understanding of earthquake physics and cipher out the properties of the MHT. Thanks to the Himalaya, the Indo-Gangetic plain is deluged each year under a tremendous amount of water during the annual summer monsoon that collects and bears down on the Indian plate enough to pull it away from the Eurasian plate slightly, temporarily relieving a small portion of the stress mounting on the MHT. As the rainwater evaporates in the dry winter season, the plate rebounds and tension is increased back on the fault. Interestingly, the mild waggle of stress induced by the monsoon rains is about the same size as that from solid-Earth tides which gently tug at the planets solid layers, but whereas changes in earthquake frequency correspond with the annually occurring monsoon, there is no such correlation with Earth tides, which oscillate back-and-forth twice a day. We therefore investigate the general response of the creeping and seismogenic parts of MHT to periodic stresses in order to link these observations to physical parameters. First, the response of the creeping part of the MHT is analyzed with a simple spring-and-slider system bearing rate-strengthening rheology, and we show that at the transition with the locked zone, where the friction becomes near velocity neutral, the response of the slip rate may be amplified at some periods, which values are analytically related to the physical parameters of the problem. Such predictions therefore hold the potential of constraining fault properties on the MHT, but still await observational counterparts to be applied, as nothing indicates that the variations of seismicity rate on the locked part of the MHT are the direct expressions of variations of the slip rate on its creeping part, and no variations of the slip rate have been singled out from the GPS measurements to this day. When shifting to the locked seismogenic part of the MHT, spring-and-slider models with rate-weakening rheology are insufficient to explain the contrasted responses of the seismicity to the periodic loads that tides and monsoon both place on the MHT. Instead, we resort to numerical simulations using the Boundary Integral CYCLes of Earthquakes algorithm and examine the response of a 2D finite fault embedded with a rate-weakening patch to harmonic stress perturbations of various periods. We show that such simulations are able to reproduce results consistent with a gradual amplification of sensitivity as the perturbing period get larger, up to a critical period corresponding to the characteristic time of evolution of the seismicity in response to a step-like perturbation of stress. This increase of sensitivity was not reproduced by simple 1D-spring-slider systems, probably because of the complexity of the nucleation process, reproduced only by 2D-fault models. When the nucleation zone is close to its critical unstable size, its growth becomes highly sensitive to any external perturbations and the timings of produced events may therefore find themselves highly affected. A fully analytical framework has yet to be developed and further work is needed to fully describe the behavior of the fault in terms of physical parameters, which will likely provide the keys to deduce constitutive properties of the MHT from seismological observations.

Large-eddy simulations of fully developed turbulent channel and pipe flows with smooth and rough walls

Studies in turbulence often focus on two flow conditions, both of which occur frequently in real-world flows and are sought-after for their value in advancing turbulence theory. These are the high Reynolds number regime and the effect of wall surface roughness. In this dissertation, a Large-Eddy Simulation (LES) recreates both conditions over a wide range of Reynolds numbers Re_τ = O(10²)-O(10⁸) and accounts for roughness by locally modeling the statistical effects of near-wall anisotropic fine scales in a thin layer immediately above the rough surface. A subgrid, roughness-corrected wall model is introduced to dynamically transmit this modeled information from the wall to the outer LES, which uses a stretched-vortex subgrid-scale model operating in the bulk of the flow. Of primary interest is the Reynolds number and roughness dependence of these flows in terms of first and second order statistics. The LES is first applied to a fully turbulent uniformly-smooth/rough channel flow to capture the flow dynamics over smooth, transitionally rough and fully rough regimes. Results include a Moody-like diagram for the wall averaged friction factor, believed to be the first of its kind obtained from LES. Confirmation is found for experimentally observed logarithmic behavior in the normalized stream-wise turbulent intensities. Tight logarithmic collapse, scaled on the wall friction velocity, is found for smooth-wall flows when Re_τ ≥ O(10⁶) and in fully rough cases. Since the wall model operates locally and dynamically, the framework is used to investigate non-uniform roughness distribution cases in a channel, where the flow adjustments to sudden surface changes are investigated. Recovery of mean quantities and turbulent statistics after transitions are discussed qualitatively and quantitatively at various roughness and Reynolds number levels. The internal boundary layer, which is defined as the border between the flow affected by the new surface condition and the unaffected part, is computed, and a collapse of the profiles on a length scale containing the logarithm of friction Reynolds number is presented. Finally, we turn to the possibility of expanding the present framework to accommodate more general geometries. As a first step, the whole LES framework is modified for use in the curvilinear geometry of a fully-developed turbulent pipe flow, with implementation carried out in a spectral element solver capable of handling complex wall profiles. The friction factors have shown favorable agreement with the superpipe data, and the LES estimates of the Karman constant and additive constant of the log-law closely match values obtained from experiment.

Effects of density differences on lateral mixing in open-channel flows

This study investigates lateral mixing of tracer fluids in turbulent open-channel flows when the tracer and ambient fluids have different densities. Longitudinal dispersion in flows with longitudinal density gradients is investigated also.

Lateral mixing was studied in a laboratory flume by introducing fluid tracers at the ambient flow velocity continuously and uniformly across a fraction of the flume width and over the entire depth of the ambient flow. Fluid samples were taken to obtain concentration distributions in cross-sections at various distances, x, downstream from the tracer source. The data were used to calculate variances of the lateral distributions of the depth-averaged concentration. When there was a difference in density between the tracer and the ambient fluids, lateral mixing close to the source was enhanced by density-induced secondary flows; however, far downstream where the density gradients were small, lateral mixing rates were independent of the initial density difference. A dimensional analysis of the problem and the data show that the normalized variance is a function of only three dimensionless numbers, which represent: (1) the x-coordinate, (2) the source width, and (3) the buoyancy flux from the source.

A simplified set of equations of motion for a fluid with a horizontal density gradient was integrated to give an expression for the density-induced velocity distribution. The dispersion coefficient due to this velocity distribution was also obtained. Using this dispersion coefficient in an analysis for predicting lateral mixing rates in the experiments of this investigation gave only qualitative agreement with the data. However, predicted longitudinal salinity distributions in an idealized laboratory estuary agree well with published data.

The instability of fluids with time dependent heating

The stability of a fluid having a non-uniform temperature stratification is examined analytically for the response of infinitesimal disturbances. The growth rates of disturbances have been established for a semi-infinite fluid for Rayleigh numbers of 10³, 10⁴, and 10⁵ and for Prandtl numbers of 7.0 and 0.7.

The critical Rayleigh number for a semi-infinite fluid, based on the effective fluid depth, is found to be 32, while it is shown that for a finite fluid layer the critical Rayleigh number depends on the rate of heating. The minimum critical Rayleigh number, based on the depth of a fluid layer, is found to be 1340.

The stability of a finite fluid layer is examined for two special forms of heating. The first is constant flux heating, while in the second, the temperature of the lower surface is increased uniformly in time. In both cases, it is shown that for moderate rates of heating the critical Rayleigh number is reduced, over the value for very slow heating, while for very rapid heating the critical Rayleigh number is greatly increased. These results agree with published experimental observations.

The question of steady, non-cellular convection is given qualitative consideration. It is concluded that, although the motion may originate from infinitesimal disturbances during non-uniform heating, the final flow field is intrinsically non-linear.

An investigation of the stresses and deflections of swept plates

The problem in this investigation was to determine the stress and deflection patterns of a thick cantilever plate at various angles of sweepback.

The plate was tested at angles of sweepback of zero, twenty, forty, and sixty degrees under uniform shear load at the tip, uniformly distributed load and torsional loading.

For all angles of sweep and for all types of loading the area of critical stress is near the intersection of the root and trailing edge. Stresses near the leading edge at the root decreased rapidly with increase in angle of sweep for all types of loading. In the outer portion of the plate near the trailing edge the stresses due to the uniform shear and the uniformly distributed load did not vary for angles of sweep up to forty degrees. For the uniform shear and the uniformly distributed loads for all angles of sweep the area in which end effect is pronounced extends from the root to approximately three quarters of a chord length outboard of a line perpendicular to the axis of the plate through the trailing edge root. In case of uniform shear and uniformly distributed loads the deflections near the edge at seventy-five per cent semi-span decreased with increase in angle of sweep. Deflections near the trailing edge under the same loading conditions increased with increase in angle of sweep for small angles and then decreased at the higher angles of sweep. The maximum deflection due to torsional loading increased with increase in angle of sweep.

The global optimization of phase-incoherent signals

The problem of global optimization of M phase-incoherent signals in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are established for N = 2 and the optimal signal sets are determined for M = 2, 3, 4, 6, and 12.

The method is the following: The signals are assumed to be equally probable and to have equal energy, and thus are represented by points ṡ_i, i = 1, 2, …, M, on the unit sphere S₁ in C^N. If W_ik is the halfspace determined by ṡ_i and ṡ_k and containing ṡ_i, i.e. W_ik = {ṙϵC^N:| ≥ | ˂ṙ, ṡ_k˃|}, then the Ʀ_i = ∩/k≠i W_ik, i = 1, 2, …, M, the maximum likelihood decision regions, partition S₁. For additive complex Gaussian noise ṅ and a received signal ṙ = ṡ_ie^jϴ + ṅ, where ϴ is uniformly distributed over [0, 2π], the probability of correct decoding is P_C = 1/π^N ∞/ʃ/0 r^2N-1e^-(r2+1)U(r)dr, where U(r) = 1/M M/Ʃ/i=1 Ʀ_i ʃ/∩ S₁ I₀(2r | ˂ṡ, ṡ_i˃|)dσ(ṡ), and r = ǁṙǁ.

For N = 2, it is proved that U(r) ≤ ʃ/C_α I₀(2r|˂ṡ, ṡ_i˃|)dσ(ṡ) – 2K/M. h(1/2K [Mσ(C_α)-σ(S₁)]), where C_α = {ṡϵS₁:|˂ṡ, ṡ_i˃| ≥ α}, K is the total number of boundaries of the net on S₁ determined by the decision regions, and h is the strictly increasing strictly convex function of σ(C_α∩W), (where W is a halfspace not containing ṡ_i), given by h = ʃ/C_α∩W I₀ (2r|˂ṡ, ṡ_i˃|)dσ(ṡ). Conditions for equality are established and these give rise to the globally optimal signal sets for M = 2, 3, 4, 6, and 12.

Carbon-13 magnetic resonance spectroscopy

High-resolution, natural-abundance ¹³C spectra have been obtained from a wide variety of organic compounds; ¹³C chemical shifts and coupling constants have been correlated with other molecular properties.

Geminal and vicinal, carbon-proton couplings in benzene and the five- and six-membered aromatic heterocycles have been related to the corresponding proton-proton couplings in substituted ethylenes. The carbon-proton coupling constants in benzene are J_CCH = + 1.0, J_CCCH = +7.4 and J_CCCH = -1.1 Hz. Extended Hückel wavefunctions are uniformly poor in explaining the long-range, carbon-proton couplings in aromatic systems.

Couplings between carbon and elements other than hydrogen have been observed in proton decoupled ¹³C spectra. All of the carbons in fluorobenzene and 1-fluoronaphthalene, but only six of the carbons in 2-fluoronaphthalene are coupled to the fluorine. One-bond, carbon-phosphorus coupling in trialkylphosphines is negative, while one-bond, carbon-phosphorus coupling in tetra-alkylphosphonium ions is positive. Atoms which do not use hybrid orbitals to form bonds to carbon (F, P(III), Se, Te) may have negative, one-bond coupling constants because of the failure of the average energy approximation. One-bond couplings between carbon and carbon, silicon, tin, lead and mercury appear to be explainable in terms of an effective nuclear charge and the s-bond order of the metal. Couplings between carbon and nitrogen and phosphorus (IV) have significant negative contributions to the Fermi contact coupling expression, though, within one series, correlations with s-bond order may be valid. Carbon-carbon coupling in cyclopropane derivatives (10-15 Hz) is consistent with a high degree of p character in the interior orbitals. Some two- and three-bond carbon-carbon coupling constants have also been observed.

Substituent effects of hydroxyl groups on the ¹³C chemical shifts of continuous-chain alkanes depend both on steric and electronic factors. The hydroxyl substituent effects in the long-chain, primary alcohols are α = -48.3, β = -10.2, and γ = +6.0 ppm. The upfield γ effect is attributed to steric crowding in the gauche conformations. Additivity of the hydroxyl and carbonyl and alkyl substituent effects in alkyl-substituted cyclohexanols and cyclohexanones has been demonstrated.

A singular perturbation method of calculating the behavior of supercavitating hydrofoils with rounded noses

A simple, direct and accurate method to predict the pressure distribution on supercavitating hydrofoils with rounded noses is presented. The thickness of body and cavity is assumed to be small. The method adopted in the present work is that of singular perturbation theory. Far from the leading edge linearized free streamline theory is applied. Near the leading edge, however, where singularities of the linearized theory occur, a non-linear local solution is employed. The two unknown parameters which characterize this local solution are determined by a matching procedure. A uniformly valid solution is then constructed with the aid of the singular perturbation approach.

The present work is divided into two parts. In Part I isolated supercavitating hydrofoils of arbitrary profile shape with parabolic noses are investigated by the present method and its results are compared with the new computational results made with Wu and Wang's exact "functional iterative" method. The agreement is very good. In Part II this method is applied to a linear cascade of such hydrofoils with elliptic noses. A number of cases are worked out over a range of cascade parameters from which a good idea of the behavior of this type of important flow configuration is obtained.

Some of the computational aspects of Wu and Wang's functional iterative method heretofore not successfully applied to this type of problem are described in an appendix.