II. Determination of constitutive coefficients and illustrative examples

1989 ◽  
Vol 327 (1595) ◽  
pp. 449-475 ◽  
Keyword(s):  
Turbulent Flow ◽  
Constitutive Equations ◽  
Theoretical Calculations ◽  
Equations Of Motion ◽  
Plane Channel ◽  
General System ◽  
Turbulent Fluctuation ◽  
System Of Equations ◽  
Constitutive Response

This paper is a continuation of Part I under the same title and is concerned mainly with the determination of the constitutive response coefficients, as well as some simple illustrative examples. First, a system of simplified constitutive equations for incompressible viscous turbulent flow is obtained from the more general system of equations in Part I through a judicial choice of retaining only those terms which appear to represent major features of the turbulent flow. Even for this simplified system of equations, the identification of some of the constitutive coefficients presents a formidable task; and this is especially true in the case of those coefficients that are associated with the presence of the additional independent variables of the theory due to the manifestation of the alignment of eddies (on the microscopic scale), turbulent fluctuation and eddy density. Because of this difficulty, the present effort for identification of the various constitutive coefficients must be regarded partly as tentative, pending future availability of suitable relevant experimental data and/or pertinent numerical simulation results. Keeping this background in mind, most of the relevant coefficients in the constitutive equations are determined, or the nature of their functional forms are estimated, through consideration of‘cartoon-like’ models on the microscopic level and these results are then used in conjunction with the macroscopic equations of motion to examine a number of simple solutions. These include the possibility of a flow possessing a constant uniform velocity gradient and solutions pertaining to decay of flow anisotropy and plane turbulent channel flows. The predicted theoretical calculations are in general accord with experimental observations. In addition, for plane channel flow, plots of variation along the width of the channel for the turbulent temperature and the macroscopic velocity compare favourably with corresponding known experimental results.

scholarly journals IV. On the dynamical theory of incompressible viscous fluids and the determination of the criterion

1895 ◽  
Vol 186 ◽  
pp. 123-164 ◽  
Keyword(s):  
Singular Solution ◽  
Physical State ◽  
Theoretical Calculations ◽  
Equations Of Motion ◽  
Dynamical Theory ◽  
Viscous Fluids ◽  
Linear Functions ◽  
Incompressible Viscous Fluids ◽  
Theoretical Results

1. The equations of motion of viscous fluid (obtained by grafting on certain terms to the abstract equations of the Eulerian form so as to adapt these equations to the case of fluids subject to stresses depending in some hypothetical manner on the rates of distortion, which equations Navier seems to have first introduced in 1822, and which were much studied by Cauchy and Poisson) were finally shown by St. Venant and Sir Gabriel Stokes, in 1845, to involve no other assumption than that the stresses, other than that of pressure uniform in all directions, are linear functions of the rates of distortion, with a co-efficient depending on the physical state of the fluid. By obtaining a singular solution of these equations as applied to the case of pendulums in steady periodic motion, Sir G. Stokes was able to compare the theoretical results with the numerous experiments that had been recorded, with the result that the theoretical calculations agreed so closely with the experimental determinations as seemingly to prove the truth of the assumption involved. This was also the result of comparing the flow of water through uniform tubes with the flow calculated from a singular solution of the equations so long as the tubes were small and the velocities slow. On the other hand, these results, both theoretical and practical, were directly at variance with common experience as to the resistance encountered by larger bodies moving with higher velocities through water, or by water moving with greater velocities through larger tubes. This discrepancy Sir G. Stokes considered as probably resulting from eddies which rendered the actual motion other than that to which the singular solution referred and not as disproving the assumption.

scholarly journals On the dynamical theory of incompressible viscous fluids and the determination of the criterion

1995 ◽  
Vol 451 (1941) ◽  
pp. 5-47 ◽  
Keyword(s):  
Singular Solution ◽  
Physical State ◽  
Theoretical Calculations ◽  
Equations Of Motion ◽  
Dynamical Theory ◽  
Viscous Fluids ◽  
Linear Functions ◽  
Incompressible Viscous Fluids ◽  
Theoretical Results

1. The equations of motion of viscous fluid (obtained by grafting on certain terms to the abstract equations of the Eulerian form, so as to adapt these equations to the case of fluids subject to stresses depending in some hypothetical manner on the rates of distortion, which equations Navier seems to have first introduced in 1822, and which were much studied by Cauchy and Poisson) were finally shown by St Venant and Sir Gabriel Stokes, in 1845, to involve no other assumption than that the stresses, other than that of pressure uniform in all directions, are linear functions of the rates of distortion, with a coefficient depending on the physical state of the fluid. By obtaining a singular solution of these equations as applied to the case of pendulums in steady periodic motion, Sir G. Stokes was able to compare the theoretical results with the numerous experiments that had been recorded, with the result that the theoretical calculations agreed so closely with the experimental determinations as seemingly to prove the truth of the assumption involved. This was also the result of comparing the flow of water through uniform tubes with the flow calculated from a singular solution of the equations, so long as the tubes were small and the velocities slow. On the other hand, these results, both theoretical and practical, were directly at variance with common experience as to the resistance encountered by larger bodies moving with higher velocities through water, or by water moving with greater velocities through larger tubes. This discrepancy Sir G. Stokes considered as probably resulting from eddies, which rendered the actual motion other than that to which the singular solution referred, and not as disproving the assumption.

scholarly journals II. On the dynamical theory of incompressible viscous fluids and the determination of the criterion

1894 ◽  
Vol 56 (336-339) ◽  
pp. 40-45 ◽  
Keyword(s):  
Singular Solution ◽  
Physical State ◽  
Theoretical Calculations ◽  
Equations Of Motion ◽  
Dynamical Theory ◽  
Viscous Fluids ◽  
Linear Functions ◽  
Incompressible Viscous Fluids ◽  
Theoretical Results

The equations of motion of viscous fluid (obtained by grafting on certain terms to the abstract equations of the Eulerian form so as to adapt these equations to the case of fluids subject to stresses depending in some hypothetical manner on the rates of distortion, which equations Navier seems to have first introduced in 1822, and which were much studied by Cauchy and Poisson) were finally shown by St. Venant and Sir Gabriel Stokes, in 1845, to involve no other assumption than that the stresses, other than that of pressure uniform in all directions, are linear functions of the rates of distortion with a co-efficient depending on the physical state of the fluid. By obtaining a singular solution of these equations as applied to the case of pendulums in steady periodic motion Sir G. Stokes was able to compare the theoretical results with the numerous experiments that had been recorded, with the result that the theoretical calculations agreed so closely with the experimental determinations as seemingly to prove the truth of the assumption involved. This was also the result of comparing the flow of water through uniform tubes with the flow calculated from a singular solution of the equations so long as the tubes were small and the velocities slow.

Error determination of hidden flows by hydrostatic weighing in parts obtained by casting

2020 ◽  
pp. 339-342
Author(s):  
M.S. Razumov ◽  
P.V. Glazkov ◽  
V.S. Kochergin ◽  
A.S. Byshkin
Keyword(s):  
Theoretical Calculations ◽  
Casting Defects ◽  
Casting Method ◽  
Error Determination ◽  
Hydrostatic Weighing ◽  
Weighing Method ◽  
Hydrostatic Weighing Method

The existing methods for detecting of casting defects and their disadvantages are considered. Hydrostatic weighing method is proposed for detecting of hidden fl aws by comparing the reference and calculated densities of the casting, method for calculating of the error is recommended. The adequacy of theoretical calculations is verifi ed experimentally

scholarly journals Shallow water waves in a rotating rectangular basin

2000 ◽  
Vol 24 (10) ◽  
pp. 649-661 ◽  
Author(s):  
Mohamed Atef Helal
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Equations Of Motion ◽  
System Of Nonlinear Equations ◽  
System Of Equations ◽  
Order Of Approximation ◽  
Shallow Water Waves ◽  
Nonlinear System Of Equations ◽  
Rectangular Basin ◽  
Shallow Water Approximation

This paper is mainly concerned with the motion of an incompressible fluid in a slowly rotating rectangular basin. The equations of motion of such a problem with its boundary conditions are reduced to a system of nonlinear equations, which is to be solved by applying the shallow water approximation theory. Each unknown of the problem is expanded asymptotically in terms of the small parameterϵwhich generally depends on some intrinsic quantities of the problem of study. For each order of approximation, the nonlinear system of equations is presented successively. It is worthy to note that such a study has useful applications in the oceanography.

Numerical simulation of reciprocating turbulent flow in a plane channel

2009 ◽  
Vol 21 (9) ◽  
pp. 095106 ◽  
Author(s):  
Massimiliano Di Liberto ◽  
Michele Ciofalo
Keyword(s):  
Numerical Simulation ◽  
Turbulent Flow ◽  
Plane Channel

Investigation of Train Dynamics in Passing Through Curves Using a Full Model

Joint Rail ◽  
2004 ◽  
Author(s):  
Mohammad Durali ◽  
Mohammad Mehdi Jalili Bahabadi
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Full Model ◽  
Bogie Frame ◽  
Nonlinear Characteristics ◽  
Train Derailment ◽  
Train Dynamics

In this article a train model is developed for studying train derailment in passing through bends. The model is three dimensional, nonlinear, and considers 43 degrees of freedom for each wagon. All nonlinear characteristics of suspension elements as well as flexibilities of wagon body and bogie frame, and the effect of coupler forces are included in the model. The equations of motion for the train are solved numerically for different train conditions. A neural network was constructed as an element in solution loop for determination of wheel-rail contact geometry. Derailment factor was calculated for each case. The results are presented and show the major role of coupler forces on possible train derailment.

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