scholarly journals Statistical theory of turbulence III-Distribution of dissipation of energy in a pipe over its cross-section

1935 ◽  
Vol 151 (873) ◽  
pp. 455-464 ◽  
Author(s):  
Geoffrey Ingram Taylor
Keyword(s):  
Statistical Theory ◽  
Turbulent Flow ◽  
Cross Section ◽  
Fundamental Question ◽  
Point Of View ◽  
Turbulent Motion ◽  
Correlation Curve ◽  
Dissipation Of Energy ◽  
Rate Of Dissipation ◽  
Statistical Theory Of Turbulence

The success of the theory of turbulence given in Part I, in predicting the connection between the rate of dissipation of energy in turbulent motion and the shape of the Ky correlation curve, suggests that it may be possible to use measurements of correlation in order to find out how the dissipation of energy is distributed over a field of turbulent flow. This is a fundamental question in any theory of turbulent motion which attempts to penetrate beyond the stage of empirical assumptions. By a fortunate coincidence the necessary observations already exist for the analysis of one case of flow from this point of view. The rate o f dissipation of energy of turbulent flow is W = 7·5 μ ¯¯(∂ u /∂ y ) 2 = 15 μ ¯ u 2 /λ 2 .

Heisenberg’s Statistical Theory of Turbulence and the Equations of Motion for a Turbulent Flow

1968 ◽  
Vol 23 (10) ◽  
pp. 1471-1477
Author(s):  
F. Winterberg
Keyword(s):  
Statistical Theory ◽  
Energy Spectrum ◽  
Turbulent Flow ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Turbulent Convection ◽  
Turbulent Motion ◽  
Navier Stokes ◽  
Statistical Theory Of Turbulence

It is shown that consistent with Heisenberg’s statistical theory of turbulence, equations of motion for a turbulent flow can be derived. These equations are linear integro-differential equations expressing the non-local interaction of eddies with different wave numbers on the basis of Heisenberg’s statistical theory.The nonlocal terms in these equations of motion for turbulent flow have to be determined from the energy spectrum of the turbulent motion, which is obtained from a nonlinear integro-differential equation. A general solution of the linear equations of motion is obtained by an arbitrary superposition of solutions. However, only those linear superpositions are permitted which are selfconsistent with the energy spectrum of turbulent motion.In contrast to the Navier-Stokes equations, the non-linearity occurs here only in the equation for the energy spectrum and not in the equations of motion itself. This fact greatly facilitates the integration of these equations.Our analysis is extended to include turbulent convection. In the spirit of Heisenberg’s hypothesis, equations of motion and energy are formulated which are consistent with the equations of the energy spectrum for free turbulent convection derived by Ledeoux, Schwarzschild and Spiegel. With this method, one may treat turbulent convection problems which arise in stellar and planetary atmospheres where the classical solution of laminar free convection cannot be applied.

Experimental study of particle-driven secondary flow in turbulent pipe flows

2012 ◽  
Vol 709 ◽  
pp. 1-36 ◽  
Author(s):  
R. J. Belt ◽  
A. C. L. M. Daalmans ◽  
L. M. Portela
Keyword(s):  
Turbulent Flow ◽  
Secondary Flow ◽  
Stress Tensor ◽  
Reynolds Stress ◽  
Cross Section ◽  
Circular Pipe ◽  
Horizontal Pipe ◽  
Pipe Cross Section ◽  
Reynolds Stress Tensor ◽  
Pipe Flows

AbstractIn fully developed single-phase turbulent flow in straight pipes, it is known that mean motions can occur in the plane of the pipe cross-section, when the cross-section is non-circular, or when the wall roughness is non-uniform around the circumference of a circular pipe. This phenomenon is known as secondary flow of the second kind and is associated with the anisotropy in the Reynolds stress tensor in the pipe cross-section. In this work, we show, using careful laser Doppler anemometry experiments, that secondary flow of the second kind can also be promoted by a non-uniform non-axisymmetric particle-forcing, in a fully developed turbulent flow in a smooth circular pipe. In order to isolate the particle-forcing from other phenomena, and to prevent the occurrence of mean particle-forcing in the pipe cross-section, which could promote a different type of secondary flow (secondary flow of the first kind), we consider a simplified well-defined situation: a non-uniform distribution of particles, kept at fixed positions in the ‘bottom’ part of the pipe, mimicking, in a way, the particle or droplet distribution in horizontal pipe flows. Our results show that the particles modify the turbulence through ‘direct’ effects (associated with the wake of the particles) and ‘indirect’ effects (associated with the global balance of momentum and the turbulence dynamics). The resulting anisotropy in the Reynolds stress tensor is shown to promote four secondary flow cells in the pipe cross-section. We show that the secondary flow is determined by the projection of the Reynolds stress tensor onto the pipe cross-section. In particular, we show that the direction of the secondary flow is dictated by the gradients of the normal Reynolds stresses in the pipe cross-section, $\partial {\tau }_{rr} / \partial r$ and $\partial {\tau }_{\theta \theta } / \partial \theta $. Finally, a scaling law is proposed, showing that the particle-driven secondary flow scales with the root of the mean particle-forcing in the axial direction, allowing us to estimate the magnitude of the secondary flow.

A mixing length model for turbulent flow in constant cross section ducts

1977 ◽  
Vol 10 (2) ◽  
pp. 125-129 ◽  
Author(s):  
R. P. Hornby ◽  
J. Mistry ◽  
H. Barrow
Keyword(s):  
Turbulent Flow ◽  
Cross Section ◽  
Constant Cross Section ◽  
Mixing Length

CONTRIBUTIONS TO THE THEORY OF WAVE GUIDES

1949 ◽  
Vol 27a (4) ◽  
pp. 69-69 ◽  
Author(s):  
L. Infeld ◽  
J. R. Pounder ◽  
A. F. Stevenson ◽  
W. Z. Chien ◽  
J. L. Synge
Keyword(s):  
Cross Section ◽  
General Problem ◽  
Radiation Resistance ◽  
Arbitrary Cross Section ◽  
Point Of View ◽  
Wave Guide ◽  
Linear Antenna ◽  
Rectangular Wave ◽  
Point Of Entry ◽  
Wave Guides

Part I deals with the problem of determining the field due to a source of radiation inside a semi-infinite rectangular wave guide closed at one end by a plug, the current distribution in the source being regarded as known. Both the walls of the guide and the plug are treated as being perfectly conducting. Three different methods of solving the problem are given. The radiation resistance is then deduced from energy considerations. In particular, an expression for the radiation resistance of a linear antenna perpendicular to the wider face of the plug, fed at the point of entry, is derived, it being assumed that the antenna current is sinusoidal and that only the fundamental H-wave is transmitted by the guide.In Part II, one of the methods of paper I is extended to the case of a guide of arbitrary cross section, and the general problem of the calculation of radiation resistance and reactance is discussed.In Part III, a number of formulae for the radiation resistance of antennae of various shapes, with various assumed current distributions, in rectangular and circular guides, are given.In Part IV, explicit calculations for the impedance of a linear antenna in a rectangular wave guide are given. Further numerical calculations relating to the same problem, from the point of view of matching and sensitivity, have been made by Messrs. Chien and Pounder, but are not reproduced here.

Sign in / Sign up

Export Citation Format

Share Document