scholarly journals On the theory of advective effects on biological dynamics in the sea. III. The role of turbulence in biological–physical interactions

2007 ◽  
Vol 464 (2091) ◽  
pp. 555-572 ◽  
Author(s):  
Louis Goodman ◽  
Allan R Robinson
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Euphotic Zone ◽  
Biological Interaction ◽  
State Variables ◽  
Turbulent Layer ◽  
Wide Range ◽  
The Mean ◽  
Interaction Term

A nonlinear model for biological and physical dynamical interactions in a laminar upwelling flow field in parts I and II of this study is extended to turbulent flow. In the previous studies, a prescription for obtaining quadrature solutions to the fundamental biodynamical equations was developed. In this study, we use a probability density function approach on these solutions to obtain statistics of the biodynamical state variables and their self-interaction for the case of turbulent advection. To illustrate the theory, a simple nutrient ( N ), phytoplankton ( P ) problem is considered, that of upwelling into a surface turbulent layer. Biological interaction is modelled as bilinear, representing the uptake of N by P in a uniform light euphotic zone. A random walk model is used to obtain the appropriate probability density function for the advective turbulent field. The mean quantities, , , as well as the biological interaction term are calculated. The term has two contributions, , and the turbulence-induced interaction term, . It is shown that the often neglected turbulence-induced coupling term is of the order and opposite in sign. This results in, over a wide range of Peclet numbers, the mean interaction term being significantly smaller than either of its constituent terms, and .

Effects of the Fractional Laplacian Order on the Nonlocal Elastic Rod Response

2017 ◽  
Vol 3 (3) ◽  
Author(s):  
Giuseppina Autuori ◽  
Federico Cluni ◽  
Vittorio Gusella ◽  
Patrizia Pucci
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Coefficient Of Variation ◽  
Fractional Laplacian ◽  
Numerical Procedure ◽  
Elastic Rod ◽  
The Mean ◽  
Distributed Forces ◽  
Nonlinear Fashion

In this paper, we yield with a nonlocal elastic rod problem, widely studied in the last decades. The main purpose of the paper is to investigate the effects of the statistic variability of the fractional operator order s on the displacements u of the rod. The rod is supposed to be subjected to external distributed forces, and the displacement field u is obtained by means of numerical procedure. The attention is particularly focused on the parameter s, which influences the response in a nonlinear fashion. The effects of the uncertainty of s on the response at different locations of the rod are investigated by the Monte Carlo simulations. The results obtained highlight the importance of s in the probabilistic feature of the response. In particular, it is found that for a small coefficient of variation of s, the probability density function of the response has a unique well-identifiable mode. On the other hand, for a high coefficient of variation of s, the probability density function of the response decreases monotonically. Finally, the coefficient of variation and, to a small extent, the mean of the response tend to increase as the coefficient of variation of s increases.

A New Microcontact Model Developed for Variable Fractal Dimension, Topothesy, Density of Asperity, and Probability Density Function of Asperity Heights

2006 ◽  
Vol 74 (4) ◽  
pp. 603-613 ◽  
Author(s):  
Jeng Luen Liou ◽  
Jen Fin Lin
Keyword(s):  
Fractal Dimension ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Gaussian Distribution ◽  
Fractal Theory ◽  
Radius Of Curvature ◽  
The Mean ◽  
Contact Surfaces ◽  
Non Gaussian

In the present study, the fractal theory is applied to modify the conventional model (the Greenwood and Williamson model) established in the statistical form for the microcontacts of two contact surfaces. The mean radius of curvature (R) and the density of asperities (η) are no longer taken as constants, but taken as variables as functions of the related parameters including the fractal dimension (D), the topothesy (G), and the mean separation of two contact surfaces. The fractal dimension and the topothesy varied by differing the mean separation of two contact surfaces are completely obtained from the theoretical model. Then the mean radius of curvature and the density of asperities are also varied by differing the mean separation. A numerical scheme is thus developed to determine the convergent values of the fractal dimension and topothesy corresponding to a given mean separation. The topographies of a surface obtained from the theoretical prediction of different separations show the probability density function of asperity heights to be no longer the Gaussian distribution. Both the fractal dimension and the topothesy are elevated by increasing the mean separation. The density of asperities is reduced by decreasing the mean separation. The contact load and the total contact area results predicted by variable D, G*, and η as well as non-Gaussian distribution are always higher than those forecast with constant D, G*, η, and Gaussian distribution.

Bifurcation Analysis of an Energy Harvesting System with Fractional Order Damping Driven by Colored Noise

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Yong-Ge Yang ◽  
Ya-Hui Sun ◽  
Wei Xu
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Fractional Order ◽  
Colored Noise ◽  
Energy Harvester ◽  
Advanced Materials ◽  
Vibration Energy ◽  
Vibration Energy Harvester ◽  
The Mean

Vibration energy harvester, which can convert mechanical energy to electrical energy so as to achieve self-powered micro-electromechanical systems (MEMS), has received extensive attention. In order to improve the efficiency of vibration energy harvesters, many approaches, including the use of advanced materials and stochastic loading, have been adopted. As the viscoelastic property of advanced materials can be well described by fractional calculus, it is necessary to further discuss the dynamical behavior of the fractional-order vibration energy harvester. In this paper, the stochastic P-bifurcation of a fractional-order vibration energy harvester subjected to colored noise is investigated. Variable transformation is utilized to obtain the approximate equivalent system. Probability density function for the amplitude of the system response is derived via the stochastic averaging method. Numerical results are presented to verify the proposed method. Critical conditions for stochastic P-bifurcation are provided according to the change of the peak number for the probability density function. Then bifurcation diagrams in the parameter planes are analyzed. The influences of parameters in the system on the mean harvested power are discussed. It is found that the mean harvested power increases with the enhancement of the noise intensity, while it decreases with the increase of the fractional order and the correlation time.

The Probabilistic Solutions to Nonlinear Random Vibrations of Multi-Degree-of-Freedom Systems

1999 ◽  
Vol 67 (2) ◽  
pp. 355-359 ◽  
Author(s):  
G.-K. Er
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Degree Of Freedom ◽  
State Variables ◽  
Algebraic Equations ◽  
Nonlinear Random Vibration ◽  
Highly Nonlinear ◽  
Fpk Equation ◽  
Two Degree Of Freedom

The probability density function of the responses of nonlinear random vibration of a multi-degree-of-freedom system is formulated in the defined domain as an exponential function of polynomials in state variables. The probability density function is assumed to be governed by Fokker-Planck-Kolmogorov (FPK) equation. Special measure is taken to satisfy the FPK equation in the average sense of integration with the assumed function and quadratic algebraic equations are obtained for determining the unknown probability density function. Two-degree-of-freedom systems are analyzed with the proposed method to validate the method for nonlinear multi-degree-of-freedom systems. The probability density functions obtained with the proposed method are compared with the obtainable exact and simulated ones. Numerical results showed that the probability density function solutions obtained with the presented method are much closer to the exact and simulated solutions even for highly nonlinear systems with both external and parametric excitations. [S0021-8936(00)01602-0]

scholarly journals A numerical study of entropy and residual entropy estimators based on smooth density estimators for non-negative random variables

2021 ◽  
Vol 54 (2) ◽  
pp. 99-121
Author(s):  
Yogendra P. Chaubey ◽  
Nhat Linh Vu
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Numerical Study ◽  
Random Variables ◽  
Random Variable ◽  
Residual Entropy ◽  
Density Estimator ◽  
Wide Range ◽  
Smooth Density

In this paper, we are interested in estimating the entropy of a non-negative random variable. Since the underlying probability density function is unknown, we propose the use of the Poisson smoothed histogram density estimator to estimate the entropy. To study the per- formance of our estimator, we run simulations on a wide range of densities and compare our entropy estimators with the existing estimators based on different approaches such as spacing estimators. Furthermore, we extend our study to residual entropy estimators which is the entropy of a random variable given that it has been survived up to time $t$.

scholarly journals Mechanical Integrity of 3D Rough Surfaces during Contact

Coatings ◽  
2019 ◽  
Vol 10 (1) ◽  
pp. 15
Author(s):  
Maxence Bigerelle ◽  
Franck Plouraboue ◽  
Frederic Robache ◽  
Abdeljalil Jourani ◽  
Agnes Fabre
Keyword(s):  
Plastic Deformation ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Contact Pressure ◽  
Rough Surfaces ◽  
Morphological Structure ◽  
Random Variable ◽  
Wide Range ◽  
Weierstrass Functions

Rough surfaces are in contact locally by the peaks of roughness. At this local scale, the pressure of contact can be sharply superior to the macroscopic pressure. If the roughness is assumed to be a random morphology, a well-established observation in many practical cases, mechanical indicators built from the contact zone are then also random variables. Consequently, the probability density function (PDF) of any mechanical random variable obviously depends upon the morphological structure of the surface. The contact pressure PDF, or the probability of damage of this surface can be determined for example when plastic deformation occurs. In this study, the contact pressure PDF is modeled using a particular probability density function, the generalized Lambda distributions (GLD). The GLD are generic and polymorphic. They approach a large number of known distributions (Weibull, Normal, and Lognormal). The later were successfully used to model damage in materials. A semi-analytical model of elastic contact which takes into account the morphology of real surfaces is used to compute the contact pressure. In a first step, surfaces are simulated by Weierstrass functions which have been previously used to model a wide range of surfaces met in tribology. The Lambda distributions adequacy is qualified to model contact pressure. Using these functions, a statistical analysis allows us to extract the probability density of the maximal pressure. It turns out that this density can be described by a GLD. It is then possible to determine the probability that the contact pressure generates plastic deformation.

scholarly journals Mixing and combustion at low heat release in large eddy simulations of a reacting shear layer

2021 ◽  
Author(s):  
J. X. Huang ◽  
W. A. McMullan
Keyword(s):  
Probability Density Function ◽  
Heat Release ◽  
Shear Layer ◽  
Probability Density ◽  
Density Function ◽  
Temperature Rise ◽  
Mixing Layer ◽  
Large Eddy ◽  
Inflow Condition ◽  
The Mean

AbstractIn this paper, the mixing and combustion at low-heat release in a turbulent mixing layer are studied numerically using large eddy simulation. The primary aim of this paper is to successfully replicate the flow physics observed in experiments of low-heat release reacting mixing layers, where a duty cycle of hot structures and cool braid regions was observed. The nature of the imposed inflow condition shows a dramatic influence on the mechanisms governing entrainment, and mixing, in the shear layer. An inflow condition perturbed by Gaussian white noise produces a shear layer which entrains fluid through a nibbling mechanism, which has a marching scalar probability density function where the most probable scalar value varies across the layer, and where the mean-temperature rise is substantially over-predicted. A more sophisticated inflow condition produced by a recycling and rescaling method results in a shear layer which entrains fluid through an engulfment mechanism, which has a non-marching scalar probability density function where a preferred scalar concentration is present across the thickness of the layer, and where the mean-temperature rise is predicted to a good degree of accuracy. The latter simulation type replicates all of the flow physics observed in the experiment. Extensive testing of subgrid-scale models, and simple combustion models, shows that the WALE model coupled with the Steady Laminar Flamelet model produces reliable predictions of mixing layer diffusion flames undergoing with fast chemistry.

Stochastic estimation of organized turbulent structure: homogeneous shear flow

1988 ◽  
Vol 190 ◽  
pp. 531-559 ◽  
Author(s):  
Ronald J. Adrian ◽  
Parviz Moin
Keyword(s):  
Probability Density Function ◽  
Shear Flow ◽  
Probability Density ◽  
Density Function ◽  
Deformation Tensor ◽  
Stochastic Estimation ◽  
Correlation Tensor ◽  
Homogeneous Shear ◽  
Wide Range ◽  
Homogeneous Shear Flow

The large-scale organized structures of turbulent flow can be characterized quantitatively by a conditional eddy, given the local kinematic state of the flow as specified by the conditional average of u(x’, t) given the velocity and the deformation tensor at a point x: 〈u(x’, t)|u(x, t), d(x, t)〉. By means of linear mean-square stochastic estimation, 〈u’|u, d〉 is approximated in terms of the two-point spatial correlation tensor, and the conditional eddy is evaluated for arbitrary values of u(x, t) and d(x, t), permitting study of the turbulent field for a wide range of local kinematic states. The linear estimate is applied to homogeneous turbulent shear flow data generated by direct numerical simulation. The joint velocity-deformation probability density function is used to obtain conditions corresponding to those events that contribute most to the Reynolds shear stress. The primary contributions to the second-quadrant and fourth-quadrant Reynolds-stress events in homogeneous shear flow come from flow induced through the ‘legs’ and close to the ‘heads’ of upright and inverted ‘hairpins’, respectively.The equation governing the joint probability density function of fu,d (u, d) is derived. It is shown that this equation contains 〈u’/u, d〉 and that the equations for second-order closure can be derived from it. Closure requires approximation of 〈u’/u, d〉.

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