Mean and fluctuating parts

A wave equation governing the unsteady motions is then derived by decomposition of all dependent variables as sums of the mean and fluctuation parts. Thus... 

Apply the Reynolds decomposition procedure and expand the dependent variables within the instantaneous equations into mean and fluctuating parts. 

The general equation for the corresponding mean Reynolds averaged variables in a turbulent flow is derived in the following way. We start with the basic transport equation (1.454) and expand -0 into its mean and fluctuating parts (e.g., [153] [167] [66]) ... 

The velocity fluctuation represents the flow that varies with periods shorter than the averaging time period. Recall that turbulence is a 3D phenomena. Therefore, we expect that fluctuations in the x-direction might be accompanied by fluctuations in the y- and z- directions. Turbuience, by definition, is a type of motion. Yet motions frequently cause variations in the temperature and concentration fields as well, if there is some mean gradient of that variable across the turbulent domain. Hence, it is common practice to portion each of these variables into mean and turbulent parts in the same manner as for the velocity. 

Sometimes the quantities z and y will fluctuate about non-zero mean values (z) and (y) Under such circumstances it is typical to consider just the fluctuating part and to defint the correlation function as ... 

As of this writing, the only practical approach to solving turbulent flow problems is to use statistically averaged equations governing mean flow quantities. These equations, which are usually referred to as the Reynolds equations of motion, are derived by Reynold s decomposition of the Navier-Stokes equations (18). The randomly changing variables are represented by a time mean and a fluctuating part ... 

Turbulent eddies larger than the cloud size, as such, tend to move the cloud as a whole and do not influence the internal concentration distribution. The mean concentration distribution is largely determined by turbulent motion of a scale comparable to the cloud size. These eddies tend to break up the cloud into smaller and smaller parts, so as to render turbulent motion on smaller and smaller scales effective in generating fluctuations of ever smaller scales, and so on. On the small-scale side of the spectrum, concentration fluctuations are homogenized by molecular diffusion. 

Interaction ofthe electrons in the framework of the self-consistent field approximation is accounted for by considering the induced density fluctuations as a response of independent particles to Oext + Poissons equation [2], This means, physically, that collective excitations of the electrons can occur, taken into account via a chain of electron-holeexcitations. These collective excitations show up in S(q, ) as a distinct energy loss feature. Figure 2 shows the shape of the real and imaginary parts of the dielectric function in RPA (er(q, ), Si(q, )) and the resulting dielectric response... 

The constrained-junction model was formulated in order to explain the decrease of the elastic moduli of networks upon stretching. It was first introduced by Ronca and Allegra [39], and Flory [40]. The model assumes that the fluctuations of junctions are diminished below those of the phantom network because of the presence of entanglements and that stretching increases the range of fluctuations back to those of the phantom network. As indicated by the second part of Equation (26), the fluctuations in a phantom network are substantial. For a tetrafunctional network, the mean-square fluctuations of junctions amount to as much as half of the mean-square end-to-end vector of the network chains. The strength of the constraints on these fluctuations is measured by a parameter k, defined as... 

An approach that does not suffer from such problems is the ABF method. This method is based on computing the mean force on and then removing this force in order to improve sampling. This leads to uniform sampling along . The dynamics of corresponds to a random walk with zero mean force. Only the fluctuating part of the instantaneous force on remains. This method is quite simple to implement and leads to a very small statistical error and excellent convergence. 

It has been assumed that the flow is incompressible so that there are no fluctuations of the density. Equation 1.91 shows that the momentum flux consists of a part due to the mean flow and a part due to the velocity fluctuation. The extra momentum flux is proportional to the square of the fluctuation because the momentum is the product of the mass flow rate and the velocity, and the velocity fluctuation contributes to both. The extra momentum flux is equivalent to an extra apparent stress perpendicular to the face, ie a normal stress component. As (v x)2 is always positive it produces a compressive stress, which is positive in the negative sign convention for stress. 

Consequently, the choice of the averaging time s determines which eddies appear in the mean advective transport term and which ones appear in the fluctuating part (and thus are interpreted as turbulence). The scale dependence of turbulent diffusivity is relevant mainly in the case of horizontal diffusion where eddies come in very different sizes, basically from the millimeter scale to the size of the ring structures related to ocean currents like the Gulf Stream, which exceed the hundred-kilometer scale. Horizontal diffusion will be further discussed in Section 22.3 here we first discuss vertical diffusivity where the scale problem is less relevant. 

Interactions of the fluctuating part of the particle motion with the mean particle motion through interparticle collisions, which generate pressure and stresses in the particle assembly, consequently yielding apparent viscosity of the particle phase... 

Here the mean photons number of subharmonic modes are represented as the sum n = rid + Sn of the semiclassical and quantum parts. Straightforward, but complicated analytical calculations (see details in [Kryuchkyan 2004]) show that 5n —> —0.125 in the limit E —> 00, which leads to the asymptotic value Kuin = 0.75 < 1. Therefore, as the analysis shows, allowing for quantum fluctuations of arbitrary level, CV entanglement is always achieved in the NOPO... 

The function (151) can be simplified in the long-time limit t> 1, when the average number of created photons, /L n h fa (V +U)/2 exceeds 1. Then the mean-square fluctuation of the photon number has the same order of magnitude as the mean photon number itself, s/2 Jf, and the most significant part of the spectmm corresponds to the values n > 1. Using the Laplace-Heine asymptotical formula for the Legendre polynomial [283]... 

---