The Cameron—Martin—Wiener method in turbulence and in Burgers’ model: general formulae, and application to late decay

1970 ◽  
Vol 41 (3) ◽  
pp. 593-618 ◽  
Author(s):  
W-H. Kahng ◽  
A. Siegel
Keyword(s):  
Energy Spectrum ◽  
Velocity Field ◽  
Equations Of Motion ◽  
Navier Stokes Equation ◽  
Hermite Expansion ◽  
Burgers Model ◽  
Non Linear ◽  
Gaussian Case ◽  
Non Gaussian ◽  
Flatness Factor

We apply the Cameron—Martin—Wiener (formerly ‘Wiener—Hermite’) expansion of a random velocity field to the analytical study of turbulence. The kernels of this expansion contain all statistical information about the ensemble. Complete expressions are derived for constructing statistical quantities in terms of the kernels, and for the equations of motion of the kernels. We rigorously prove the Gaussian trend of the velocity field of the Navier—Stokes equation in the very late stage when the non-linear term is neglected. Then-dependence (nis the order of derivative) of the flatness factor, minus three for derivatives of the velocity field, shows a rapid increase withnin this stage.The late decay problem of the Burgers model of turbulence is studied analytically with a view to obtaining suggestive guidelines for fitting the non-linear aspects of the model turbulence. We can divide the energy spectrum density into two parts, the larger of which is a kind of steady solution, which we call the ‘equilibrium state’, which remains self-similar in time in terms of an appropriate variable. The deviation from this ‘equilibrium solution’ satisfies the Kármán—Howarth equation. As initial velocity field, we take two particular cases: (a) a pure Gaussian, and (b) a non-Gaussian velocity field. With these two cases a detailed spectral analysis has been obtained. The energy spectrum deviation from ‘equilibrium’ declines exponentially to zero for all wave-numbers. The Gaussian case shows that the flatness factor minus three increases rapidly withn, while the non-Gaussian case does not show any marked dependence onn.

The forced flow of a rotating viscous liquid which is heated from below

1953 ◽  
Vol 246 (907) ◽  
pp. 81-112 ◽  
Keyword(s):  
Velocity Field ◽  
Angular Velocity ◽  
Viscous Liquid ◽  
Opposite Direction ◽  
Equations Of Motion ◽  
Cylindrical Vessel ◽  
Central Axis ◽  
Forced Flow ◽  
Outer Portion ◽  
Non Linear

A liquid is contained in a cylindrical vessel and is subject to heating on the horizontal base of the vessel. The problem of the forced flow arising from the heating has been investigated in the case when the heating function is symmetrically arranged about the central axis. It is found that the relative forced flow tends to become zonal in character when the vessel rotates at a sufficiently high angular velocity. This relative zonal motion is principally in the direction of the rotation except near the outer portion of the fluid where it is in the opposite direction, the former being ‘westerlies’, the latter ‘easterlies’. The easterlies are due to the non-linear inertia terms in the equations of motion. This description of the velocity field is used because the experiment described above has considerable meteorological significance.

Equilibrium characteristics of nearly normal turbulence

1970 ◽  
Vol 41 (1) ◽  
pp. 179-188 ◽  
Author(s):  
W. C. Meecham
Keyword(s):  
Fluid Flow ◽  
Incompressible Fluid ◽  
Linear Model ◽  
Gaussian Processes ◽  
Incompressible Fluid Flow ◽  
Hermite Expansion ◽  
Burgers Model ◽  
Flow Problems ◽  
Non Linear ◽  
Model Equations

We discuss some consequences of assuming that two different non-linear model equations, and real turbulence are nearly Gaussian. It is supposed when necessary that the process is driven and it is supposed that the processes have become statistically stationary. These problems are discussed from the viewpoint of the Wiener–Hermite expansion for non-linear, nearly Gaussian processes. Expected equilibria forms are related to corresponding expressions obtained from the zero-fourth-cumulant assumption. The spectrum for Burgers’ model and for incompressible fluid flow problems is found from this viewpoint to beE∼k−2. The kinematical properties leading to such spectra are discussed. It is noted, as has been remarked earlier, that this spectrum is characteristic of flows with near discontinuities. A conjecture is offered concerning how these discontinuities are related to Gaussianity.

Modeling the Climber Fall Arrest Dynamics

2005 ◽  
Author(s):  
Lionel Manin ◽  
Jarir Mahfoudh ◽  
Matthieu Richard ◽  
David Jauffres
Keyword(s):  
Dynamic Characteristics ◽  
Equations Of Motion ◽  
Close Approximation ◽  
Non Linear ◽  
The Future ◽  
Safety Recommendations ◽  
Improved Design ◽  
Fall Arrest

Sports and mountaineering activities are becoming more and more popular. Equipment constructors seek to develop products and devices that are easy to use and that take into account all safety recommendations. PETZL and INSA have collaborated to develop a model for the simulation of displacements and efforts involved during the fall of a climber in the “safety chain”. The model is based on the classical equations of motion, in which climber and belayer are considered as rigid masses, while the rope is considered as a series of non-linear stiffness passing through several devices as brakes and runners. The main goal is to predict the forces in the rope and on the return anchor at the first rebound of the fall. Experiments were first performed in order to observe and determine the dynamic characteristics of the rope, and then to validate results stemming from simulations. Several fall configurations are simulated, and the model performs satisfactorily. It also provides a close approximation of the phenomena observed experimentally. The model enables the assessment of the existing equipments and the improved design of the future one.

scholarly journals Experimental and Numerical Investigation of Preferential Flow in Fractured Network with Clogging Process

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Xiaobing Chen ◽  
Jian Zhao ◽  
Li Chen
Keyword(s):  
Velocity Field ◽  
Cross Sections ◽  
Preferential Flow ◽  
Pressure Head ◽  
Fracture Network ◽  
Navier Stokes ◽  
Single Fracture ◽  
Navier Stokes Equation ◽  
Physical Experiments ◽  
Fractured Network

In this study, physical experiments and numerical simulations are combined to provide a detailed understanding of flow dynamics in fracture network. Hydraulic parameters such as pressure head, velocity field, Reynolds number on certain monitoring cross points, and total flux rate are examined under various clogging conditions. Applying the COMSOL Multiphysics code to solve the Navier-Stokes equation instead of Reynolds equation and using the measured data to validate the model, the fluid flow in the horizontal 2D cross-sections of the fracture network was simulated. Results show that local clogging leads to a significant reshaping of the flow velocity field and a reduction of the transport capacity of the entire system. The flow rate distribution is highly influenced by the fractures connected to the dominant flow channels, although local disturbances in velocity field are unlikely to spread over the whole network. Also, modeling results indicate that water flow in a fracture network, compared with that in a single fracture, is likely to transit into turbulence earlier under the same hydraulic gradient due to the influence of fracture intersections.

scholarly journals Non-Gaussian inference from non-linear and non-Poisson biased distributed data

2014 ◽  
Vol 10 (S306) ◽  
pp. 258-261
Author(s):  
Metin Ata ◽  
Francisco-Shu Kitaura ◽  
Volker Müller
Keyword(s):  
Dark Matter ◽  
Statistical Inference ◽  
Computer Code ◽  
Matter Density ◽  
Density Field ◽  
Distributed Data ◽  
Dark Matter Density ◽  
Posterior Sampling ◽  
Non Linear ◽  
Non Gaussian

AbstractWe study the statistical inference of the cosmological dark matter density field from non-Gaussian, non-linear and non-Poisson biased distributed tracers. We have implemented a Bayesian posterior sampling computer-code solving this problem and tested it with mock data based onN-body simulations.

scholarly journals Ensemble variational assimilation as a probabilistic estimator – Part 1: The linear and weak non-linear case

2018 ◽  
Vol 25 (3) ◽  
pp. 565-587 ◽  
Author(s):  
Mohamed Jardak ◽  
Olivier Talagrand
Keyword(s):  
Probability Distribution ◽  
Simple Procedure ◽  
Linear Case ◽  
Bayesian Probability ◽  
Variational Assimilation ◽  
Non Linear ◽  
Abstract Data ◽  
Gaussian Case ◽  
Low Dimensional ◽  
Lorenz 96

Abstract. Data assimilation is considered as a problem in Bayesian estimation, viz. determine the probability distribution for the state of the observed system, conditioned by the available data. In the linear and additive Gaussian case, a Monte Carlo sample of the Bayesian probability distribution (which is Gaussian and known explicitly) can be obtained by a simple procedure: perturb the data according to the probability distribution of their own errors, and perform an assimilation on the perturbed data. The performance of that approach, called here ensemble variational assimilation (EnsVAR), also known as ensemble of data assimilations (EDA), is studied in this two-part paper on the non-linear low-dimensional Lorenz-96 chaotic system, with the assimilation being performed by the standard variational procedure. In this first part, EnsVAR is implemented first, for reference, in a linear and Gaussian case, and then in a weakly non-linear case (assimilation over 5 days of the system). The performances of the algorithm, considered either as a probabilistic or a deterministic estimator, are very similar in the two cases. Additional comparison shows that the performance of EnsVAR is better, both in the assimilation and forecast phases, than that of standard algorithms for the ensemble Kalman filter (EnKF) and particle filter (PF), although at a higher cost. Globally similar results are obtained with the Kuramoto–Sivashinsky (K–S) equation.

GPR-based novel approach for non-linear aerodynamic modelling from flight data

2018 ◽  
Vol 123 (1259) ◽  
pp. 79-92
Author(s):  
A. Kumar ◽  
A. K. Ghosh
Keyword(s):  
Aerodynamic Force ◽  
Equations Of Motion ◽  
Gaussian Process Regression ◽  
Likelihood Estimation ◽  
Flight Data ◽  
Novel Approach ◽  
Non Linear ◽  
Research Aircraft ◽  
Novel Method ◽  
Aerodynamic Modelling

ABSTRACTIn this paper, a Gaussian process regression (GPR)-based novel method is proposed for non-linear aerodynamic modelling of the aircraft using flight data. This data-driven regression approach uses the kernel-based probabilistic model to predict the non-linearity. The efficacy of this method is examined and validated by estimating force and moment coefficients using research aircraft flight data. Estimated coefficients of aerodynamic force and moment using GPR method are compared with the estimated coefficients using maximum-likelihood estimation (MLE) method. Estimated coefficients from the GPR method are statistically analysed and found to be at par with estimated coefficients from MLE, which is popularly used as a conventional method. GPR approach does not require to solve the complex equations of motion. GPR further can be directed for the generalised applications in the area of aeroelasticity, load estimation, and optimisation.

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