On the dynamics of unsteady gravity waves of finite amplitude Part 2. Local properties of a random wave field

1961 ◽  
Vol 11 (1) ◽  
pp. 143-155 ◽  
Author(s):  
O. M. Phillips
Keyword(s):  
Kinetic Energy ◽  
Wave Field ◽  
Root Mean Square ◽  
Surface Displacement ◽  
Higher Order ◽  
Random Wave ◽  
Order Effects ◽  
Mean Square ◽  
Local Properties ◽  
The Mean

Expressions in closed form are derived for a number of local properties of a random, irrotational wave field. They are: (i) the mean potential and kinetic energies per unit projected area; (ii) the energy balance among the processes of energy input from the surface pressure fluctuations, rate of growth of potential and kinetic energy and horizontal energy flux; and (iii) the partition between potential and kinetic energy. These expressions are mainly in terms of quantities measured at the free surface, which are therefore functions of only two spatial variables (x, y) and of time t.Approximations for these expressions can be found simply by subsequent expansion methods; the fourth order being the highest for which the assumption of irrotational motion is appropriate in a real fluid. It is shown that the mean product of any three first-order quantities is of fourth or higher order in the root-mean-square wave slope, and this result is applied in estimating the magnitude of some higher order effects. In particular, the skewness of the surface displacement is of the order of the root-mean-square surface slope, which has been confirmed observationally by Kinsman (1960).

The dispersion relation for a nonlinear random gravity wave field

1976 ◽  
Vol 75 (2) ◽  
pp. 337-345 ◽  
Author(s):  
Norden E. Huang ◽  
Chi-Chao Tung
Keyword(s):  
Dispersion Relation ◽  
Phase Velocity ◽  
Wave Field ◽  
Gravity Wave ◽  
Root Mean Square ◽  
Surface Velocity ◽  
Mean Deviation ◽  
Mean Square ◽  
Local Phase ◽  
The Mean

The dispersion relation for a random gravity wave field is derived using the complete system of nonlinear equations. It is found that the generally accepted dispersion relation is only a first-order approximation to the mean value. The correction to this approximation is expressed in terms of the energy spectral function of the wave field. The non-zero mean deviation is proportional to the ratio of the mean Eulerian velocity at the surface and the local phase velocity. In addition to the mean deviation, there is a random scatter. The root-mean-square value of this scatter is proportional to the ratio of the root-mean-square surface velocity and the local phase velocity. As for the phase velocity, the nonzero mean deviation is equal to the mean Eulerian velocity while the root-mean-square scatter is equal to the root-mean-square surface velocity. Special cases are considered and a comparison with experimental data is also discussed.

Direct numerical simulation of turbulence in injection-driven three-dimensional cylindrical flows

2011 ◽  
Vol 670 ◽  
pp. 176-203 ◽  
Author(s):  
JU ZHANG ◽  
THOMAS L. JACKSON
Keyword(s):  
Reynolds Number ◽  
Root Mean Square ◽  
Streamwise Velocity ◽  
Wall Turbulence ◽  
Wall Friction ◽  
Mean Square ◽  
The Mean ◽  
Wall Shear ◽  
Strong Injection ◽  
Near Wall

Incompressible turbulent flow in a periodic circular pipe with strong injection is studied as a simplified model for the core flow in a solid-propellant rocket motor and other injection-driven internal flows. The model is based on a multi-scale asymptotic approach. The intended application of the current study is erosive burning of solid propellants. Relevant analysis for easily accessible parameters for this application, such as the magnitudes, main frequencies and wavelengths associated with the near-wall shear, and the assessment of near-wall turbulence viscosity is focused on. It is found that, unlike flows with weak or no injection, the near-wall shear is dominated by the root mean square of the streamwise velocity which is a function of the Reynolds number, while the mean streamwise velocity is only weakly dependent on the Reynolds number. As a result, a new wall-friction velocity $\(u_\tau{\,=\,}\sqrt{\tau_w/\rho}\)$, based on the shear stress derived from the sum of the mean and the root mean square, i.e. $\(\tau_{w,inj} {\,=\,} \mu |{\partial (\bar{u}+u_{rms})}/{\partial r}|_w\)$, is proposed for the scaling of turbulent viscosity for turbulent flows with strong injection. We also show that the mean streamwise velocity profile has an inflection point near the injecting surface.

The multiple scattering of waves in irregular media

1968 ◽  
Vol 262 (1133) ◽  
pp. 609-640 ◽  
Keyword(s):  
Wave Field ◽  
Probability Distributions ◽  
Essential Feature ◽  
Mean Square ◽  
Angular Power Spectrum ◽  
Absorbing Medium ◽  
Phase Fluctuations ◽  
Field Fluctuations ◽  
The Mean ◽  
Wave Normal

When a wave passes through a large thickness of a non-absorbing medium containing weak random irregularities of refractive index, large amplitude and phase fluctuations of the wave field can develop. The probability distributions of these fluctuations are important, since they may be readily observed and from them can be found the mean square amplitudes of the fluctuations. This paper shows how to calculate these distributions and also the ‘ angular power spectrum ’ for an assembly of media which are statistically stationary with respect to variations in time, and in space for directions perpendicular to the wave normal of the incident wave. The scattered field at a given point is resolved into two components in phase and in quadrature with the residual unscattered wave at that point. The assembly averages of the powers in these two components, and of their correlation coefficient are found, and a set of three integro-differential equations is constructed which show how these three quantities vary as the medium is traversed. The probability distributions of amplitude and phase of the wave field at any point in the medium are functions of these three quantities which are found by integrating the equations through the medium. An essential feature of these equations is that they include waves which have been scattered several or m any times (multiple scatter). The equations are solved analytically for some particular cases. Solutions for the general case have been obtained numerically and are presented, together with the corresponding probability distributions of the field fluctuations and their average values.

Uncertainty Quantification of Energy Harvesting Systems Using Method of Quadratures and Maximum Entropy Principle

2015 ◽  
Author(s):  
Aditya Nanda ◽  
M. Amin Karami ◽  
Puneet Singla
Keyword(s):  
Energy Harvesting ◽  
Maximum Entropy ◽  
Density Function ◽  
Root Mean Square ◽  
Maximum Entropy Principle ◽  
Mean Square ◽  
Mean Power ◽  
Harvesting Systems ◽  
Entropy Principle ◽  
The Mean

This paper uses the method of Quadratures in conjunction with the Maximum Entropy principle to investigate the effect of parametric uncertainties on the mean power output and root mean square deflection of piezoelectric vibrational energy harvesting systems. Uncertainty in parameters of harvesters could arise from insufficient manufacturing controls or change in material properties over time. We investigate bimorph based harvesters that transduce ambient vibrations to electricity via the piezoelectric effect. Three varieties of energy harvesters — Linear, Nonlinear monostable and Nonlinear bistable are considered in this research. This analysis quantitatively shows the probability density function for the mean power and root mean square deflection as a function of the probability densities of the excitation frequency, excitation amplitude, initial deflection of the bimorph and magnet gap of the energy harvester. The method of Quadratures is used for numerically integrating functions by propagating weighted points from the domain and evaluating the integral as a weighted sum of the function values. In this paper, the method of Quadratures is used for evaluating central moments of the distributions of rms deflection and mean harvested power and, then, in conjunction with the principle of Maximum Entropy (MaxEnt) an optimal density function is obtained which maximizes the entropy and satisfies the moment constraints. The The computed nonlinear density functions are validated against Monte Carlo simulations thereby demonstrating the efficiency of the approach. Further, the Maximum Entropy principle is widely applicable to uncertainty quantification of a wide range of dynamic systems.

scholarly journals Reproducibility of Peripheral Quantitative Computed Tomography Measurements at the Radius and Tibia in Healthy Pre- and Postmenopausal Women

2011 ◽  
Vol 62 (3) ◽  
pp. 183-189 ◽  
Author(s):  
Kristina A. Szabo ◽  
Colin E. Webber ◽  
Christopher Gordon ◽  
Jonathan D. Adachi ◽  
Richard Tozer ◽  
...  
Keyword(s):  
Postmenopausal Women ◽  
Root Mean Square ◽  
Subcutaneous Fat ◽  
Quantitative Computed Tomography ◽  
Distal Tibia ◽  
Mean Square ◽  
Cross Sectional ◽  
Mean Values ◽  
The Mean

Purpose The objectives of this study were to utilise the XCT-2000 pQCT scanner to determine the mean values and the reproducibility of in vivo total, trabecular, and cortical volumetric bone measurements at distal and diaphyseal sites of the radius and the tibia, as well as calf muscle and subcutaneous fat areas, in healthy pre- and postmenopausal women. Methods Twenty-nine women (14 premenopausal and 15 postmenopausal) were recruited to participate in this study. Distal and diaphyseal sites of the radius (at 4% and 20% of the length of the radius) and tibia (at 4%, 38%, and 66% of the length of the tibia) were examined. Results The root mean square coefficient of variation for measurements at the distal tibia gave the most favorable reproducibility values for total (1.5%) and trabecular (1.6%) density, whereas the diaphyseal tibia showed the most favorable reproducibility value for cortical density (0.3%). The root mean square coefficients of variation for measurements of muscle and fat cross-sectional areas at the calf were 0.6% and 0.7%, respectively. At the distal tibia, the mean values for total ( P < .05) and trabecular ( P < .01) density were significantly lower in postmenopausal women than in premenopausal women. Conclusions The data presented here indicate that XCT-2000 pQCT scans at the tibia provide highly reproducible measurements of total, cortical, and trabecular bone as well as muscle and fat cross-sectional areas. Furthermore, significant differences in volumetric bone measurements between healthy pre- and postmenopausal women were evident only at the distal tibia, suggesting that this site warrants further study.

scholarly journals Changes of Conformation in Albumin with Temperature by Molecular Dynamics Simulations

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 405
Author(s):  
Piotr Weber ◽  
Piotr Bełdowski ◽  
Krzysztof Domino ◽  
Damian Ledziński ◽  
Adam Gadomski
Keyword(s):  
Molecular Dynamics ◽  
Molecular Dynamics Simulations ◽  
Root Mean Square ◽  
Mean Square Displacement ◽  
Conformational Space ◽  
Global Dynamics ◽  
Mean Square ◽  
Conformational Entropy ◽  
The Mean ◽  
Dynamics Simulations

This work presents the analysis of the conformation of albumin in the temperature range of 300 K – 312 K , i.e., in the physiological range. Using molecular dynamics simulations, we calculate values of the backbone and dihedral angles for this molecule. We analyze the global dynamic properties of albumin treated as a chain. In this range of temperature, we study parameters of the molecule and the conformational entropy derived from two angles that reflect global dynamics in the conformational space. A thorough rationalization, based on the scaling theory, for the subdiffusion Flory–De Gennes type exponent of 0 . 4 unfolds in conjunction with picking up the most appreciable fluctuations of the corresponding statistical-test parameter. These fluctuations coincide adequately with entropy fluctuations, namely the oscillations out of thermodynamic equilibrium. Using Fisher’s test, we investigate the conformational entropy over time and suggest its oscillatory properties in the corresponding time domain. Using the Kruscal–Wallis test, we also analyze differences between the mean root mean square displacement of a molecule at various temperatures. Here we show that its values in the range of 306 K – 309 K are different than in another temperature. Using the Kullback–Leibler theory, we investigate differences between the distribution of the mean root mean square displacement for each temperature and time window.

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