Second-order Mean Research Articles

Design of a Dynamic Positioning System Using Model-Based Control

A dynamic positioning (DP) system includes different control functions for automatic positioning and guidance of marine vessels by means of thruster and propeller actions. This paper describes the control functions which provide station-keeping and tracking. The DP controller is designed using model-based control, where a new modified LQG feedback controller and a model reference feedforward controller are applied. A reference model calculates appropriate reference trajectories. Since it is not desirable nor even possible to counteract the wave-frequency movement caused by first-order wave loads, the control action of the propulsion system should be produced by the low-frequency part of the vessel movement caused by current, wind and second-order mean and slowly varying wave loads. A Kalman filter based state estimator and a Luenberger observer are used to compute the low-frequency feedback and feedforward control signals. Full-scale experiments with a supply vessel demonstrate the performance of the proposed controller.

Tidal-shock relaxation: A reexamination of tidal shocks in stellar systems

view Abstract Citations (61) References (13) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Tidal-Shock Relaxation: A Reexamination of Tidal Shocks in Stellar Systems Kundic, Tomislav ; Ostriker, Jeremiah P. Abstract The phenomenon of 'tidal-shock relaxation' is defined and quantitatively estimated. We show that the second-order term mean value of ((delta E)2) which has usually been neglected in the treatment of tidal shocks, is typically far more important than the first-order term mean value of (delta E). The latter has been found by Aguilar, Ostriker & Hut (1988) to be the dominant physical process driving the evolution of the Galactic system of globular clusters. The reason is simply that the absolute value of (v dot delta v), which contributes to the second-order term, is usually much larger than the absolute value of (delta v)2, the basis of the first-order term. Near the tidal radius the tidal-shock relaxation term mean value of ((delta E)2)ts will accelerate mass loss, and near the half-mass radius it competes with the two-body relaxation mean value of ((delta E)2)rel in driving the evolution of the internal structure in the cluster. Formulae for the evaluation of the second-order term are computed for the idealized case treated by Spitzer (1987) of stars in harmonic potential. For typical parameters of global clusters we find that even at the half-mass point, tidal-shock relaxation may be competitive with two-body relaxation. Publication: The Astrophysical Journal Pub Date: January 1995 DOI: 10.1086/175114 Bibcode: 1995ApJ...438..702K Keywords: Celestial Mechanics; Globular Clusters; Shock Waves; Stars; Stellar Mass Ejection; Stellar Physics; Tides; Approximation; Harmonics; Milky Way Galaxy; Two Body Problem; Astrophysics; CELESTIAL MECHANICS; STELLAR DYNAMICS; GALAXY: GLOBULAR CLUSTERS: GENERAL; SHOCK WAVES full text sources ADS |

Mass transport under partially reflected waves in a rectangular channel

Mass transport under partially reflected waves in a rectangular channel is studied. The effects of sidewalls on the mass transport velocity pattern are the focus of this paper. The mass transport velocity is governed by a nonlinear transport equation for the second-order mean vorticity and the continuity equation of the Eulerian mean velocity. The wave slope, ka, and the Stokes boundary-layer thickness, k (ν/σ)½, are assumed to be of the same order of magnitude. Therefore convection and diffusion are equally important. For the three-dimensional problem, the generation of second-order vorticity due to stretching and rotation of a vorticity line is also included. With appropriate boundary conditions derived from the Stokes boundary layers adjacent to the free surface, the sidewalls and the bottom, the boundary value problem is solved by a vorticity-vector potential formulation; the mass transport is, in gneral, represented by the sum of the gradient of a scalar potential and the curl of a vector potential. In the present case, however, the scalar potential is trivial and is set equal to zero. Because the physical problem is periodic in the streamwise direction (the direction of wave propagation), a Fourier spectral method is used to solve for the vorticity, the scalar potential and the vector potential. Numerical solutions are obtained for different reflection coefficients, wave slopes, and channel cross-sectional geometry.

Journal of Ship Research First and Second-Order Wave Effects on a Submerged Spheroid

Computations are presented for the linearized force and moment acting on a submerged slender spheroid in regular waves, the resulting pitch and heave motions, and the second-order mean force and moment. These numerical results, which are based on the use of a three-dimensional panel code, are compared with the approximations based on slender-body theory. The accuracy of the slender-body approximation is relatively good for the first-order forces and body motions, but substantial errors are revealed for the second-order mean drift force and pitch moment. Unlike the approximate result, the more correct numerical prediction of the mean pitch moment is non-zero, and generally acts in the bow down direction in head seas. To explain this result it is shown that the wave elevation directly above the spheroid increases in amplitude from bow to stern, thus causing a greater upward force on the afterbody relative to the forebody.

Nonlinear Asymmetric Excitation of Edge Waves

The resonant excitation of two weakly nonlinear oppositely travelling edge waves on a gently sloping beach by a perfectly reflected obliquely incident gravity wave is calculated. The four first-order nonlinear differential equations that govern the slowly varying amplitudes and phases of the edge waves are associated with a Hamiltonian supplemented by a dissipation function that provides for both viscous and radiation damping. This system is reduced, through successive canonical transformations in action-angle variables, to two evolution equations that are isomorphic to those for the symmetric problem of normal incidence. Asymmetric edge waves of finite amplitude, like their symmetric counterparts, occur in a narrow band of frequency if and only if the dissipation is inferior to a certain critical value, in which case they appear as a slowly modulated (in both space and time) standing wave and induce a second-order mean flow along the shore. The effects of imperfect reflexion are described by reference to the symmetric problem

Evaluation of wave drift forces on vertical cylinders of arbitrary geometry, with application to tension leg platforms (TLPs)

Abstract The “horizontal plane method” of Hwang and Tuck (1970, J. Fluid Mech.42, 447–464) is used to evaluate the first-order wave potentials on vertical cylinders of arbitrary geometry, and together with the near-field approach of Pinkster (1979, Ocean Engng6, 593–615) the second-order mean wave drift forces are calculated. Based on this numerical technique an approximate method is developed to estimate the horizontal mean drift forces on tension leg platforms (TLPs).

Free convection in an infinite porous medium induced by a heated sphere

Natural Convection induced by a heated sphere embedded in an unbounded fluid-saturated porous medium is studied analytically. The temperature of the sphere is assumed to be fluctuating over a non-zero mean value. Solutions for both the velocity and temperature fields are obtained in the form of series expansions in the Rayleigh number based on the permeability of the porous medium and the mean heat flux from the sphere. All discussions are based on the assumption that the flow is governed by Darcy's law and that the thermal Rayleigh number is small. Fluctuating part shows the evolution of different wave patterns. The interaction of the first harmonic fields induce second-order mean components in the velocity and temperature fields, for which analytical expressions are obtained.

A note on long waves induced by short-wave groups over a shelf

The generation of long waves due to the refraction of wave groups over a shelf is reexamined. The appropriate boundary conditions for calculating the second-order mean free-surface displacement are discussed. Numerical results for a plane shelf are presented, which are different from those given by Mei & Benmoussa (1984).

Measuring meaning in life: An examination of three scales

Meaning in life is an important construct for psychological theory which has received little empirical investigation, partly because of uncertainty about measurement scales. This paper examines the factor structure of three scales to measure meaning in life, the purpose in life (PIL) test, the life regard index (LRI) and the sense of coherence (SOC) scale. Results suggest that meaning in life can be regarded as a multidimensional construct, with meaning able to be attained in several different ways. Oblique factor solutions were accepted, and higher-order analyses conducted for all three scales. A general second-order meaning in life dimension was identified only for PIL and it is suggested that this scale may be the best general measure of the construct. It is concluded that further work should be undertaken to explore the specific dimensions of meaning in life.

The identification of metaphor

A number of philosophers, linguists and psychologists have made the dual claim that metaphor is cognitively significant and that metaphorical utterances have a meaning not reducible to literal paraphrase. Such a position requires support from an account of metaphorical meaning that can render metaphors cognitively meaningful without the reduction to literal statement. It therefore requires a theory of meaning that can integrate metaphor within its sematics, yet specify why it is not reducible to literal paraphrase. I introduce the idea of a “second-order meaning”, of which metaphor is but one instance, that is a function on literal-conventional, i.e., first-order meaning, and outline a linguistic framework designed to provide a representation of linguistic meaning for both. This framework is designed to represent linguistic units ranging from a single word to an entire text since I argue that the by-now familiar position that the sentence is the appropriate unit for metaphor has mislead us into asking the wrong questions about metaphorical meaning. With this apparatus, we can specify the conditions under which an utterance may transcend the constraints on first-order meaning (transgressions not always apparent on the sentential level), without thereby being “meaningless”. Conversely, we can specify the conditions that may render apparently odd utterances first-order meaningful rather than metaphorical. In this way we see how metaphorical language differs both from deviant language and from specialized language such as technical language, fanciful and fantastical language (in fairy tales, science fiction, etc.).

Estimation of population pharmacokinetic parameters using destructively obtained experimental data: a simulation study of the one-compartment open model.

A simulation study of the one-compartment open pharmacokinetic model has been made. The population pharmacokinetic parameters which characterize the population of drug residues over time are assumed to be stochastic. A general theoretical model framework for parameter estimation via the method of extended least squares is presented. Formulas approximating the required mean and variance time functions are developed and subsequently used in the simulation study. The effects of four different designs in four different animal populations are presented. The simulated data are those of the single observation per animal per time point type. The characterizing population pharmacokinetic parameters have been analyzed for bias and reliability in both a naive and second-order mean model. Recommendations for choosing an appropriate sampling design are included.

Mean flows induced by internal gravity wave packets propagating in a shear flow

An inviscid, incompressible, stably stratified fluid occupies a horizontal channel, along which an internal gravity wave packet is propagating in the presence of a basic shear flow. By using a generalized Lagrangian mean formulation, the equation for wave action conservation is derived to describe the manner in which the basic flow affects the waves. Equations describing the second-order (in amplitude) wave-induced Lagrangian mean flows are obtained. Two kinds of applications are discussed: (i) steady mean flows, due to waves encountering an inhomogeneity in their environment, such as a varying channel depth; (ii) mean flows induced by modulations in the wave amplitude.

Comparison between standard waveguide wattmeters of mean and high power levels with a state standard

1. Noninclusive systematic errors of the wattmeters that were compared were within the limits determined by independent certification at a low power level. 2. Transmitted-power, bolometric, waveguide wattmeters and ponderomotive wattmeters can be recommended for use as standard, second-order measurement means at a mean power level (0.1–100 W) with error of 1.5–2.0%.

Mean motions and impulse of a guided internal gravity wave packet

Second-order mean fields of motion and density are calculated for the two-dimensional problem of an internal gravity wave packet (the waves are predominantly of a single frequency o and wavenumberk) propagating as a wave-guide mode in an inviscid, diffusionless Boussinesq fluid of constant buoyancy frequencyN, confined between horizontal boundaries. (The same mathematical analysis applies to the formally identical problem for inertia waves in a homogeneous rotating fluid.)To leading order the mean motions turn out to be zero outside the wave packet, which consequently possesses a well-defined fluid impulse [Iscr ]. This is directed horizontally, and is given in magnitude and sense by\[ {\cal I} = \alpha{\cal M};\quad\alpha = \frac{2c_{\rm g}(c-c_{\rm g})(c+2c_{\rm g})}{c^3-4c^3_{\rm g}}. \]Here [Mscr ] is the so-called ‘wave momentum’, defined as wave energy divided by horizontal phase velocity c ≡ ω/k, andcg=c(N2–ω2)/N2, the group velocity.If the wave packet is supposed generated by a horizontally towed obstacle, [Mscr ] appears as the total fluid impulse, but of this a portion [Mscr ]-[Iscr ] in general propagates independently away from the wave packet in the form of long waves. When the wave packet itself is totally reflected by a vertical barrier immersed in the fluid, the time-integrated horizontal force on the barrier equals 2 [Iscr ] (and not 2 [Mscr ] as might have been expected from a naive analogy with the radiation pressure of electromagnetic waves.)

Analysis of Unsteady Laminar Boundary Layer Flow by an Integral Method

An approximate integral method of analysis is developed for unsteady laminar boundary layer flows. The case of a flat plate in a free stream with small harmonic velocity oscillations about a steady mean is used to formulate the method. Results are compared with the available experimental data. The essence of the method is to: First obtain asymptotic solutions for limiting cases of the flow under consideration. Then, assume velocity profiles with sufficient generality to include the asymptotic solutions. The profile form functions are determined by applying integral relations (velocity weighted averages of the momentum equation) and compatibility conditions (normal derivatives of the momentum equation evaluated at the boundary). As an example of the method, the second-order mean flow correction is determined for oscillating flow over a flat plate.

Gaussian-Wave-Packet Dynamics in Uniform Magnetic and Quadratic Potential Fields

For the case of uniform magnetic field and a potential quadratic in its coordinates an electron state originally represented by a Gaussian wave packet remains Gaussian in its subsequent evolution. Under these conditions the state is completely characterized by the first-and second-order quantum means, that is, quantities such as $〈{q}_{i}〉$, $〈{p}_{i}〉$ and $〈{q}_{i}{p}_{j}〉$, $〈{p}_{i}{p}_{j}〉$, respectively. A tensor-product relationship is shown to exist between the problem of the determination of the second- and first-order means which facilitates the solution. Two problems are treated in detail: (i) an isotropic wave packet in an isotropic potential well and (ii) the dynamics of a wave packet on a quadratic saddle point. In the former case, the wave packet "spins" and its radius pulsates as its mean follows the classical trajectory. In the second the magnetic field effect on tunneling through such barriers is studied and found to be small at attainable field strengths unless the effective electronic mass is very small. A method for the numerical solution for the time-dependent Schr\"odinger equation, termed the particle method and based on the hydrodynamic analogy to that equation, is here extended to include magnetic fields. It is applied to the two problems just described and found to agree well with the analytical solutions.

On large-scale solar convection

We examine the effects of rotation about a vertical axis on thermal convection with a simple model in which an inviscid, incompressible fluid of zero thermal conductivity and electrical resistivity is contained in a thin annulus of rectangular cross-section. The initial steady state assumed is one of no motion relative to the rotating frame with constant (unstable) vertical temperature gradient and uniform toroidal magnetic field. Small periodic disturbances are then introduced and the linearized perturbation equations solved. We also determine the second-order mean circulations and magnetic fields that are forced by non-zero Reynolds and thermal stresses and magnetic field transports.

Currents induced by tides and gravity waves

Two-dimensional oscillatory potential flow was shown by Rayleigh and Schlichting to give rise to a steady second-order streaming at the outer edge of a laminar boundary layer. Their method is here extended to a three-dimensional oscillatory flow which may be represented at a plane boundary by two harmonic oscillations ( U, V ) of frequency s?/2π in perpendicular horizontal directions. The resulting second-order mean velocity components are interpreted in terms of bottom currents induced by tidal motion in shallow seas, or by doubly modulated gravity waves. These currents depend critically on the relative magnitudes of U, V and f/s? , where f is the Coriolis parameter. It is suggested that on the tidal scale, such currents may contribute to the circulation within the North Sea, while on the gravity wave scale they contribute to the formation of offshore sandbars around the coasts. DOI: 10.1111/j.2153-3490.1963.tb01397.x

A consequence of the zero-fourth-cumulant approximation in the decay of isotropic turbulence

This paper is a continuation of previous work (Ogura 1962 a, b ) on the dynamical consequence of the hypothesis that fourth-order mean values of the fluctuating velocity components are related to second-order mean values as they would be for a normal joint-probability distribution. The equations derived by Tatsumi (1957) for isotropic turbulence on the basis of this hypothesis are integrated numerically for specific intitial conditions. The initial values of the Reynolds number, $R_ \lambda = (u^{\overline{2}})^{\frac {1}{2}} \lambda|v$ , assigned in this investigation are 28·8, 14·4, 7·2 and 1·8, where $(u^{\overline{2}})^{\frac {1}{2}}$ is the root-mean-square turbulent velocity, λ the dissipation length and v the kinematic viscosity coefficient. The result of such computations is that the energy spectrum does develop negative values for R λ = 28·8 and 14·4. This first occurs at a time approximately 2·8 for R λ = 28·8 and 4·2 for R λ = 14·4 The time-scale here is $(E_0 k^3_0)^-{\frac{1}{2}}$ , where k 0 is a wave-number scale typical of the energy-containing velocity component and E1 0 , a typical value of the energy spectrum, is given by $4 \pi ^-{\frac{1}{2}}k_0^{-1}\overline{u^2}$ There is no evidence of the energy distribution tending to become negative for R λ = 7·2 and 1·8. It is observed that inertial effects are relatively weak at R λ = 7·2 and the decay process is largely controlled by viscous effects. For R λ = 1·8 a purely viscous calculation is found to be adequate to account for the numerically integrated results.

Currents induced by tides and gravity waves

Two-dimensional oscillatory potential flow was shown by Rayleigh and Schlichting to give rise to a steady second-order streaming at the outer edge of a laminar boundary layer. Their method is here extended to a three-dimensional Oscillatory flow which may be represented at a plane boundary by two harmonic oscillations (U, V) of frequency σ/2π in perpendicular horizontal directions.
 The resulting second-order mean velocity components are interpreted in terms of bottom currents induced by tidal motion in shallow sew, or by doubly modulated gravity waves. These currents depend critically on the relative magnitudes of U, V and f/σ, where f is the Coriolis parameter.
 It is suggested that on the tidal scale, such currents may contribute to the circulation within the North Sea, while on the gravity wave scale they contribute to the formation of offshore sandbars around the coasts.