The mean value of the fluctuations in pressure and pressure gradient in a turbulent fluid

1938 ◽  
Vol 34 (4) ◽  
pp. 534-539
Author(s):  
A. E. Green
Keyword(s):  
Unit Volume ◽  
Isotropic Turbulence ◽  
Fluid Motion ◽  
Mean Value ◽  
Mean Square ◽  
Turbulent Fluid ◽  
Dissipation Of Energy ◽  
The Mean ◽  
Rate Of Dissipation ◽  
Image Position

1. Taylor has shown (1) that two characteristic lengthsλ and λη may be defined for turbulent fluid motion. The length λ, which is connected with the dissipation of energy, is, for isotropic turbulence, given bywhere is the mean rate of dissipation of energy per unit volume and represents the mean square value of any component of velocity. The length λη can be defined in terms of thuswhere For isotropic turbulence Taylor assumed thatwhere B is a constant. Since the turbulence is isotropic,and so, from (1), (2), (3) and (4) we have

scholarly journals Mechanism of the production of small eddies from large ones

1937 ◽  
Vol 158 (895) ◽  
pp. 499-521 ◽  
Keyword(s):  
Wind Tunnel ◽  
Unit Volume ◽  
Isotropic Turbulence ◽  
Hot Wire ◽  
Statistical Representation ◽  
Dissipation Of Energy ◽  
Linear Dimensions ◽  
The Mean ◽  
Hot Wires ◽  
Rate Of Dissipation

The connexion between the statistical representation of turbulence and dissipation of energy has been discussed in relation to the decay of the isotropic turbulence which is produced in a wind tunnel by means of regular grids. It was shown that a length λ can be defined which may be taken as a measure of the scale of the small eddies which are responsible for dissipation. This λ can be found by measuring the correlation R y between the indications of two hot wire anemometers set at a distance y apart on a line perpendicular to the axis of the tunnel. Then 1/ λ 2 = Lt y→0 1 - R y / y 2 , and the mean rate of dissipation of energy per unit volume is W ¯ = 15 μ u 2 ¯ / λ 2 , (1) where u 2 ¯ is the mean of the square of one component of velocity. When turbulence is generated in a wind stream by a grid of regularly spaced bars it may be expected to possess a definite scale proportional to the linear dimensions of the grid. In any complete statistical description of turbulence this scale must be implicitly or explicitly involved. One way in which the scale can be defined is to measure the distance y apart by which the two hot wires must be separated before the correlation between the indications disappears. Another way is to define the scale as l 2 = ∫ 0 y R y d y . (2)

The mean-square value of the divisor function

2016 ◽  
Vol 100 (548) ◽  
pp. 203-212
Author(s):  
Peter Shiu
Keyword(s):  
Counting Function ◽  
Mean Value ◽  
Mean Square ◽  
Divisor Function ◽  
Dirichlet’S Theorem ◽  
Fixed Power ◽  
The Mean ◽  
Dirichlet's Theorem ◽  
Image Position ◽  
Prime Counting Function

The behaviour of the divisor function d (n) is rather tricky. For a prime p, we have d(p) = 2, but if n is the product of the first k primes then, by Chebyshev's estimate for the prime counting function [1, Theorem 414], we have so thatfor such n then, d (n) is ‘unusually large’ — it can exceed any fixed power of log n, for example.In [2] Jameson gives, amongst other things, a derivation of Dirichlet's theorem, which shows that the mean-value of the divisor function in an interval containing n is log n. However, the result is somewhat deceptive because, for most n, the value of d (n) is substantially smaller than log n.

The mean value of the fluctuations in pressure and pressure gradient in a turbulent fluid

1936 ◽  
Vol 32 (3) ◽  
pp. 380-384 ◽  
Author(s):  
G. I. Taylor
Keyword(s):  
Pressure Gradient ◽  
Mean Value ◽  
Turbulent Fluid ◽  
General Systems ◽  
The Mean ◽  
Image Position

In the general systems of vortices represented by (1) the mean variation in pressureiswhereKis a number which varies between 1 and √2. When the vortices are confined to cubical partitions, the case most nearly analogous to that of free turbulence,K= 1·06, so that the conjecture which I made some years ago, thatwould be equal tois probably nearly correct.

The mean square of the Dedekind zeta function in quadratic number fields

1989 ◽  
Vol 106 (3) ◽  
pp. 403-417 ◽  
Author(s):  
Wolfgang Müller
Keyword(s):  
Zeta Function ◽  
Number Fields ◽  
Mean Value ◽  
Fourth Power ◽  
Mean Value Theorem ◽  
Mean Square ◽  
Quadratic Number ◽  
Dedekind Zeta Function ◽  
The Mean ◽  
Image Position

Let K be a quadratic number field with discriminant D. The aim of this paper is to study the mean square of the Dedekind zeta function ζK on the critical line, i.e.It was proved by Chandrasekharan and Narasimhan[1] that (1) is at most of order O(T(log T)2). As they noted at the end of their paper, it ‘would seem likely’ that (1) behaves asymptotically like a2T(log T)2, with some constant a2 depending on K. Applying a general mean value theorem for Dirichlet polynomials, one can actually proveThis may be done in just the same way as this general mean value theorem can be used to prove Ingham's classical result on the fourth power moment of the Riemann zeta function (cf. [3], chapter 5). In 1979 Heath-Brown [2] improved substantially on Ingham's result. Adapting his method to the above situation a much better result than (2) can be obtained. The following Theorem deals with a slightly more general situation. Note that ζK(s) = ζ(s)L(s, XD) where XD is a real primitive Dirichlet character modulo |D|. There is no additional difficulty in allowing x to be complex.

scholarly journals On a sum analogous to Dedekind sum and its mean square value formula

2002 ◽  
Vol 32 (1) ◽  
pp. 47-55 ◽  
Author(s):  
Zhang Wenpeng
Keyword(s):  
Asymptotic Property ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Mean Square ◽  
Dedekind Sum ◽  
The Mean

The main purpose of this paper is using the mean value theorem of DirichletL-functions to study the asymptotic property of a sum analogous to Dedekind sum, and give an interesting mean square value formula.

scholarly journals Microstructure of the Galactic Magnetic Field

1958 ◽  
Vol 8 ◽  
pp. 952-954
Author(s):  
K. Serkowski
Keyword(s):  
Magnetic Field ◽  
Position Angle ◽  
Mean Value ◽  
Open Clusters ◽  
Electric Vector ◽  
Stokes Parameters ◽  
Galactic Magnetic Field ◽  
The Mean ◽  
Image Position

The polarization of the stars in open clusters, explained on the basis of the Davis-Greenstein theory, gives some information on the microstructure of the galactic magnetic field.The polarization is most conveniently described by the parameters Q, U, proportional to the Stokes parameters and defined by where p is the amount of polarization, θ is the position angle of the electric vector, and θ̄ is the mean value of θ for the region under consideration.

scholarly journals Extreme events in computational turbulence

2015 ◽  
Vol 112 (41) ◽  
pp. 12633-12638 ◽  
Author(s):  
P. K. Yeung ◽  
X. M. Zhai ◽  
Katepalli R. Sreenivasan
Keyword(s):  
Extreme Events ◽  
Isotropic Turbulence ◽  
Temporal Evolution ◽  
Energy Dissipation Rate ◽  
Mean Value ◽  
Small Scale ◽  
Homogeneous And Isotropic Turbulence ◽  
The Mean ◽  
Grid Points ◽  
Vortex Tubes

We have performed direct numerical simulations of homogeneous and isotropic turbulence in a periodic box with 8,1923grid points. These are the largest simulations performed, to date, aimed at improving our understanding of turbulence small-scale structure. We present some basic statistical results and focus on “extreme” events (whose magnitudes are several tens of thousands the mean value). The structure of these extreme events is quite different from that of moderately large events (of the order of 10 times the mean value). In particular, intense vorticity occurs primarily in the form of tubes for moderately large events whereas it is much more “chunky” for extreme events (though probably overlaid on the traditional vortex tubes). We track the temporal evolution of extreme events and find that they are generally short-lived. Extreme magnitudes of energy dissipation rate and enstrophy occur simultaneously in space and remain nearly colocated during their evolution.

On the dispersion of small particles suspended in an isotropic turbulent fluid

1977 ◽  
Vol 83 (3) ◽  
pp. 529-546 ◽  
Author(s):  
M. W. Reeks
Keyword(s):  
Relaxation Time ◽  
Fluid Velocity ◽  
Fluid Motion ◽  
Particle Diffusion ◽  
Small Particles ◽  
Mean Velocity ◽  
Turbulent Fluid ◽  
A Value ◽  
Long Time ◽  
The Mean

A solution to the dispersion of small particles suspended in a turbulent fluid is presented, based on the approximation proposed by Phythian for the dispersion of fluid points in an incompressible random fluid. Motion is considered in a frame moving with the mean velocity of the fluid, the forces acting on the particle being taken as gravity and a fluid drag assumed linear in the particle velocity relative to that of the fluid. The probability distribution of the fluid velocity field in this frame is taken as Gaussian, homogeneous, isotropic, stationary and of zero mean. It is shown that, in the absence of gravity, the long-time particle diffusion coefficient is in general greater than that of the fluid, approaching with increasing particle relaxation time a value consistent with the particle being in an Eulerian frame of reference. The effect of gravity is consistent with Yudine's effect of crossing trajectories, reducing unequally the particle diffusion in directions normal to and parallel to the direction of the gravitational field. To characterize the effect of flow and gravity on particle diffusion it has been found useful to use a Froude number defined in terms of the turbulent intensity rather than the mean velocity. Depending upon the value of this number, it is found that the particle integral time scale may initially decrease with increasing particle relaxation time though it eventually rises and approaches the particle relaxation time. It is finally shown how this analysis may be extended to include the extra forces generated by the fluid and particle accelerations.

The theory of the mean square variation of a function formed by adding known functions with random phases, and applications to the theories of the shot effect and of light

1936 ◽  
Vol 32 (4) ◽  
pp. 580-597 ◽  
Author(s):  
E. N. Rowland
Keyword(s):  
Total Effect ◽  
Mean Square ◽  
Single Event ◽  
Random Events ◽  
Variation Of A Function ◽  
The Mean ◽  
Image Position

In the study of random events and associated fluctuations such as occur in the shot effect, a theorem first stated and discussed by Dr N. R. Campbell can often be employed. It applies on any occasion when there occur at random a number of events whose effects are additive. Let us suppose that a single event occurring at time tr causes at time t an effect f(t − tr) in some part of the observed system, and that the effects of different events are additive, so that the total effect or output is ϑ(t), given byWe may suppose that the same events cause another set of effects g(t − tr) with output ϑ(t), whereBoth the functions are assumed to be bounded and integrable in the Riemann sense, as are all the functions studied in physics.

scholarly journals On the mean value of the enumeration function for multiplicative partitions

1990 ◽  
Vol 41 (3) ◽  
pp. 407-410 ◽  
Author(s):  
Cao Hui-Zong ◽  
Ku Tung-Hsin
Keyword(s):  
Natural Number ◽  
Mean Value ◽  
The Mean ◽  
Image Position

Let g(n) denote the number of multiplicative partitions of the natural number n. We prove that

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