Time-averaged Quantities Research Articles

Energy flux and dissipation of inhomogeneous plane wave forms in Kelvin-Voigt viscoelasticity

Inhomogeneous plane waves of complex frequency ω and slowness S are propagated through a (linear) Kelvin-Voigt viscoelastic solid. The energy density, energy flux and dissipation are quadratic in the small quantities (namely, the displacement gradient, velocity and velocity gradient) and so give rise to (attenuated) constant terms as well as to inhomogeneous plane waves of frequency 2ω. The latter- terms are usually removed by time averaging, but here we retain them as they are necessarily of a magnitude comparable with those of the time-averaged quantities. A relationship is derived that connects the amplitudes of the terms of the energy density, energy flux and dissipation that have frequency 2ω. It is shown that the complex group velocity is related to these amplitudes rather than to the attenuated constant terms as it is for homogeneous waves in conservative materials. The results are specialized to isotropic viscoelasticity and to viscous fluids.

Vector space decomposition of reactive power for periodic nonsinusoidal signals

A vector space is used to represent periodic voltage and current signals in a single port electrical network. This representation is convenient for expressing time average power quantities, such as average power, apparent power, and certain definitions of reactive power. The vector representation for reactive power provides insight into the difficulty of finding a consistent definition for a scalar measure of reactive power when harmonics are present. Instead, a reactive power vector is defined and shown to obey conservation according to Tellegen's theorem. Projections of the reactive power vector are shown to obey conservation and can be used as signed, scalar measures of reactive power.

A spectral model for turbulence and microphysics dynamics in an ice cloud

Abstract. A one-dimensional, nine-mode spectral model for temperature, velocity, and the mixing ratios of suspended and precipitating ice-particle components is shown to be consistent with ice-cloud observations. The observations include Doppler radar time-series measurements of a single winter ice cloud and direct measurements of mean particle size vs. icewater content for a set of ice clouds. Fitting of the model to the Doppler vertical-velocity measurements allows a prediction to be made of the vertical scale and turbulent Prandtl number active in the ice-cloud vertical motions. The model is then used to explore the question of how turbulence and gravity-wave motions affect the microphysical properties of an ice cloud. The model predicts interesting dynamical effects on the mixing ratios due to these motions, but no significant effects on the time-averaged microphysical quantities.

Magneto-Inertial Confinement of Turbulent Plasma

A one-fluid model describing the axisymmetric plasma confinement in a Simple Magnetized Torus (SMT) is formulated, discussed and studied numerically. This configuration does not exhibit a stationary MHD-equilibrium state, but the present work shows that the Reynolds stress-tensor due to the inertial forces from turbulent velocity fluctuations is sufficient to secure an equilibrium of time-averaged quantities. Numerical solutions demonstrates how a cathode generated plasma develops a spiralling vortex that evolves towards increasingly smaller spatial scales. Hence turbulence develops from a driven, non-equilibrium situation, and not from an instability of an equilibrium.

The Statistics and Sensitivity of a Double-Gyre Model: The Reduced-Gravity, Quasigeostrophic Case

Abstract The sensitivity of the time-averaged circulation of an oceanic double-gyre model to variations in the model's parameters and forcing is studied. Unresolved low-frequency variability in the solution leads to statistical uncertainty in the estimates of the time-averaged quantities. The authors utilize bootstrap analyses of a number of multicentury integrations of a reduced-gravity, quasigeostrophic, eddy-resolving ocean model to estimate these statistical uncertainties. An analysis is then presented of the sensitivity of the system to variations in the strength and asymmetry of the wind forcing and to variations in several other physical and numerical parameters of the system. The bootstrap results enable us to estimate error bounds on these sensitivities The physical measures of the system investigated include the means and variances of the total energy, the peak transport, the location of the separation point, and the penetration scale of the free jet.

Time-Averaged Quantity of a Low-Temperature Semiconductor Experiment Reflects Scaling Behavior of Saddle-Node Bifurcation to Chaos

Abstract Low-temperature impact ionization breakdown in p-type germanium crystals gives rise to spontaneous oscillations of the current flow. We demonstrate experimental evidence of a particularly high-conducting dynamical state that is limited to a finite parameter regime of the current versus magnetic field characteristic. After bifurcation from a coexisting nonoscillatory state to periodicity, one observes a type-I intermittent transition to chaos and, eventually, a jump back to the nonoscil­latory branch upon increasing the magnetic field control parameter. The scaling behavior of the underlying saddle-node bifurcation, already found in time-resolved measurements, also becomes visible in a square-root dependence of the time-averaged current developing both prior to and after the critical point. Our result might be of interest for time-averaged information is accessible.

Simulation of complex turbulent flows: recent advances and prospects in wind engineering

Publisher Summary Turbulent flows are found everywhere and are of interest to engineers and scientists working in many fields. There is a great variation in the physics of the flows and the required predictive capability among the fields. This might lead to difficulties. Researchers in one field might believe that their field is so unique that they need not borrow from others or they might adopt methods and concepts from other fields without sufficient critical thought. Either approach might lead to difficulties. Accuracy requirements vary greatly among the fields. Turbulence is strongly affected by “extra strains” including stratification, buoyancy, rotation, chemical reaction, and compression. To some, these are essential issues; to others, they are of no interest. In some fields, temporal fluctuations are of great importance; others are satisfied with time-averaged quantities. The accuracy demanded in some high technology areas is a few percent; other fields, including the geophysical and astrophysical sciences, might be satisfied with the order of magnitude results.

Turbulent free jet flows issuing from sharp-edged rectangular slots: The influence of slot aspect ratio

An experimental study was made of the influence of aspect ratio on turbulent free jet flows issuing from sharp-edged rectangular slots. Four slot aspect ratios (2, 5, 10, and 20) were used. The measured time-average quantities, acquired with hot-wire anemometry, are the mean streamwise velocity, turbulence kinetic energy, Reynolds shear stress, and some of the triple-velocity products. The results indicate that, for the slot aspect ratios considered, the speed of mixing in the very near flow field (ie, X/ D e ≤ 5) increases as the slot aspect ratio increases. Indeed, in the region X/ D e ≤ 5, the highest shear-layer values of the turbulence kinetic energy, the Reynolds shear stress and the turbulent transport of the Reynolds stresses are found in the aspect ratio 20 jet, which also has the shortest potential core length. The results also show that a simple eddy-viscosity type of turbulence model would fail to simulate the shear-stress distribution in all but the lowest aspect ratio jet flow and that the turbulent eddies in the jet shear layers are highly organized.

水平環状空間内非定常３次元自然対流，（ＩＩＩ） 乱流自然対流の直接数値シミュレーションの妥当性

Direct numerical simulation using the three-dimensional and time-dependent finite difference scheme which have been reported in the previous paper is performed for turbulent natural convection in a horizontal concentric annulus. Numerical results are obtained for the Rayleigh number based on the gap width from 1.0×106 to 1.72×107. To verify the validity of the present scheme, numerical results dealt with statistically are compared with other experimental data and numerical solutions and the effect of grid size on spatial resolution capability is investigated. It is shown that time-averaged quantities deduced from appropriate spatial resolution are in good agreement with experimental results, whereas turbulence quantities are excessively estimated when spatial resolution is insufficient.

Near-wall structure of a turbulent boundary layer with riblets

A detailed wind tunnel study has been carried out on the near-wall turbulence structure over smooth and riblet wall surfaces under zero pressure gradient. Time-average quantities as ‘well as conditionally sampled profiles were obtained using hotwire/film anemometry, along with a simultaneous flow visualization using the smoke-wire technique and a sheet of laser light. The experimental results indicated a significant change of the structure in the turbulent boundary layer near the riblet surface. The change was confined within a small volume of the flow close to the wall surface. A conceptual model for the sequence of the bursts was then proposed based on an extensive study of the flow visualization, and was supported by the results of conditionally sampled velocity fields. A possible mechanism of turbulent drag reduction by riblets is discussed.

Detonation in a relaxing gas and relaxational instability

It is well known that the one-dimensional structure of DW is encountered relatively rarely in practice. As a rule, detonation waves have a three-dimensional pulsating structure (triple configurations [10, 24]). There is no question that the one-dimensional model of detonation studied here cannot and does not pretend to describe the nature and structure of three-dimensional pulsations caused by the thermal or relaxational instability of the wave front. In the case of a pulsating detonation the proposed model, like the classical theory (limit of small α), refers to time-averaged gasdynamic quantities and to the selection rule for the velocity of detonation. In addition, the one-dimensional model can be used (see the section concerning the relaxational instability) as a starting point for the analysis of the instability of DW.

Simple example of the ergodic hypothesis

The kinetic-theory derivation of the classical second virial coefficient of real gases requires replacing a time-averaged quantity by its equivalent spatial average. A rigorous justification of this step is given for repulsive interactions which do not support bound states. This new derivation provides an explicit illustration of the ergodic hypothesis.

Evaluation of the basic systems of equations for turbulence measurements using the Monte Carlo technique

A numerical experiment has been carried out to evaluate two of the methods available for finding the time-averaged mean velocity and the Reynolds stresses of a turbulent flow field using hot wires. The conventional method is based on the series expansion of the response equation, subsequent truncation of the series and time averaging. The improved method is based on squaring and time averaging without neglecting any terms. The method adopted to evaluate these two methods is based on the Monte Carlo simulation of a pseudo turbulent flow field using random-number generators and the corresponding hot-wire response, for a prescribed set of conditions, by assuming an appropriate model for the hot-wire response. The simulated hot-wire response and the calibration constants are then perturbed about their mean values to study the effects of errors in these quantities. The perturbed response is used to compute the time-averaged flow field by the two methods. The deviation of these values from the generated pseudo values, averaged over large number of trials, is used as the criterion to evaluate the methods. This procedure is also used to estimate the errors due to truncation in the conventional method, to study the effect of turbulence-intensity levels and to study the effects of measurement errors. The results indicate that the choice of the method for determining the time-averaged quantities should be based on the turbulence-intensity level and the measurement errors likely to be encountered. The conventional method yields reliable mean-velocity results for turbulence intensities as high as 50% with second-order turbulence correction. If measurement errors are within reasonable limits and the turbulence level is below 20%, the conventional method yields reliable results for Reynolds stresses. The improved method should be used to determine the time-averaged flow field for turbulence intensity above 40–50%. The error in the yaw sensitivity parameter k has an insignificant effect on the mean velocity and Reynolds stresses computed by both methods. By accurately determining the sensitivity s of the hot wire, the accuracy of the measured mean velocity and Reynolds stresses can be improved significantly. An improved method of carrying out the uncertainty analysis for measurements, based on the Monte Carlo technique, has also been outlined.

Relating actual and effective ventilation in determining indoor air quality

Ventilation is both a mechanism for removing indoor air pollutants, and a potential energy load on the heating or cooling system of a building. Quantitative estimates of the ventilation rates, important for both of these applications, necessitate determining time-averaged quantities. The time-averaged ventilation rate appropriate for indoor air pollution, however, is different from that associated with energy load. We derive ventilation efficiencies for well-mixed, homogeneous, time-varying concentrations and corroborate findings with field data from a test house in Edmonton, Alberta, which indicate that monthly ventilation efficiency ranges from 79% to 92% with an annual average of 80%, and that hourly temporal ventilation efficiencies vary over a much larger range than time-averaged quantities.

Sonically produced heat in a fluid with bulk viscosity and shear viscosity.

In a viscous fluid, sound produces heat in a spatial pattern which, in general, depends on the relative magnitudes of the shear viscosity coefficient eta and the bulk viscosity coefficient B'. It is well known that when the particle velocity components ui relative to Cartesian coordinates xi are given for an arbitrary sound field, or any field of flow, the volume rate of heat production qv can be determined from a dissipation function in the form B'T1 + eta T2. Here, T1 and T2 are quadratic functions involving derivatives of the type delta ui/delta xj. In this paper, examples are discussed for continuous monofrequency sound fields, including crossed plane waves, as well as focused and unfocused fields. In these examples, spatial distributions of the time-averaged quantity [qv] for media in which the loss mechanism is primarily bulk viscosity are compared to those for media in which shear viscosity dominates.

Pulsating flow in a pipe

Turbulent and laminar pulsating flows in a straight smooth pipe are compared at identical frequencies and Reynolds numbers. Most measurements were made at a mean Reynolds number of 4000, but the influence of Re was checked for 2900 < Re < 7500. The period of forcing ranged from 0.5 to 5 s, with corresponding change in the non-dimensional frequency parameter α = R √(ω/ν) from 4.5 to 15. The amplitude of the imposed oscillations did not exceed 35% of the mean in order to avoid flow reversal or relaminarization. Velocities at the exit plane of the pipe and pressure drop along the pipe were measured simultaneously; velocity measurements were made with arrays of normal hot wires. The introduction of the periodic surging had no significant effect on the time-averaged quantities, regardless of the flow regime (i.e. in both laminar and turbulent flows). The time-dependent components at the forcing frequency, represented by a radial distribution of amplitudes and phases, are qualitatively different in laminar and turbulent flows. The ensemble-averaged turbulent quantities may also be represented by an amplitude and a phase; however, the non-harmonic content of these intensities increases with increasing amplitude of the imposed oscillations. A normalization procedure is proposed which relates phase-locked turbulent flow parameters in unsteady flow to similar time-averaged quantities. An integral momentum equation in a time-dependent flow requires that a triad of forces (pressure, inertia and shear) will be in equilibrium at any instant of time. All the terms in the force-balance equation were measured independently, providing a good check of data. The analysis of the experimental results suggests that turbulence adjusts rather slowly to the local mean-flow conditions. A simple eddy-viscosity model described by a complex function can account for ‘memory’ of turbulence and explain the different phase distribution in laminar and turbulent flows.

Rate equation for locally time-averaged dynamics in weakly damped systems

We consider the density matrix for weakly damped quantum oscillators. It is proved that the locally time-averaged density matrix derived from the full density matrix satisfies a closed set of equations, which reduce under specified conditions to population rate equations. Rate coefficients in these equations include contributions from collision-induced transitions and internal collisionless transitions explicitly arranged as sequential processes. We compare solutions of the new rate equations with the full density matrix for a four-level driven system. It is shown that despite some notable success in matching locally time-averaged quantities by extending the Wilcox-Lamb formalism to the weak-damping limit, the new equations are more naturally suited for this limit. In contrast with the Wilcox-Lamb formalism, we show that the new equations give a representation of the local time average whose accuracy is limited only by the degree of time-scale separation between collisions and internal dynamics.

Dynamics and time-averaged chemical potential of proteins: importance in oligomer association.

The chemical potential of a protein is a time-averaged quantity whose value depends upon the fractions of time spent in the different conformations and therefore upon the protein dynamics. If the monomer involved in an association equilibrium undergoes unfolding limited by its lifetime, its chemical potential as well as that of the associated form will not be constant and the free energy of association will be a diminishing function of the degree of dissociation. For a unique free energy of association, the logarithm of the protein concentration must change by 2.86 units to increase the degree of dissociation from 0.1 to 0.9. The dissociation of enolase appears to take place over a significantly smaller range (1.7 units), and dansyl conjugates of enolase show an even narrower range (0.9 unit). A simple descriptive theory is developed, and this shows that the values observed are explained by a difference in free energy of 1-3 kcal/mol (1 cal = 4.18 J) between the conformations present at negligible and almost complete dissociation.

On anisotropy of the pressure tensor and the force density for a polarizable fluid in an electromagnetic field

By means of a statistical derivation it is shown that the momentum-balance equation of a polarized fluid contains a force term that is the divergence of an anisotropic tensor (frsol|1/5)PP + ( 1 10 )P 2 U , where P is the polarization. However, for a fluid in equilibrium this force term is compensated by an anisotropic part in the pressure tensor. The same is true for a fluid in an optical field if time-averaged quantities are considered.

Numerical Simulation of Gross Structure of Turbulent Plane Mixing Layer by Discrete-vortex Model

A discrete-vortex model is used to simulate a turbulent plane mixing layer. The vorticity in the mixing layer is partitioned into a number of vorticity blods which are again replaced by inviscid discrete vortices with an inner structure. It is found that random walks of appropriate variance should be superposed on the motion of the vortices in order to obtain the time-averaged velocity and turbulent quantities consistent with experiments. The random walks are assumed to simulate very roughly the diffusion of vorticity by turbulent transport. The potential field of application of the present model and its limitations are also discussed.