The Time Averaging Procedure

In this section we assess the single time averaging technique. Pure time averaging has been examined by [229, 112, 45, 43, 47, 53, 48, 24, 54, 116, 157]. These investigations are recommended for complementary studies. 

The conditions under which the time averaging procedure can be applied for multiphase flows coincide with those for which time averaging can be used for single phase turbulence flows, and can be expressed as ]112, 43, 47]  

It is noted that the requirement of proper separation of scales represents the main drawback of the time averaging method. The constitutive equations used generally depend strongly on this assumption which is hardly ever fulfilled performing simulations of turbulent reactive flows. 

In this averaging procedure we imagine that at any point r in a two-phase flow, phase k passes intermittently so a function ipk associated with phase k will be a piecewise continuous function. However, the interfaces are not stationary so they do not occupy a fixed location for finite time intervals. For this reason the average macroscopic variables are expected to be continuous functions (but this hypothesis has been questioned as it can be shown that the first order time derivative might be discontinuous which is not physical, hence an amended double time averaging operator was later proposed as a way of dealing with this problem [43, 47]). Since T is the overall time period over which the time averaging is performed, phase k is observed within a subset of residence time intervals so that T = for all the phases in the 

In the conventional averaging procedure a phase indicator function Xj. is defined by  

The mathematical properties of Xk ensure that the ordinary mathematical operations of vector calculus can be performed on the local instantaneous variables which are discontinuous [62, 118], 

The local instantaneous equations can be time averaged over a defined time interval [t — T/2 t + T/2], The single time averaging operator which can be applied to any scalar, vector or tensor valued property function ipk associated with phase k, is defined by  

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We obtain by expanding the Lagrangian in orders of the perturbation along with the time-averaged procedure [51] the response functions as derivatives of the time-averaged CC quasienergy L(t) r. Finally, we obtain the response equations from the stationary condition. In particular, the linear response function is given by... 

If the conditions prevailing in the phase under consideration are turbulent, then it will be necessary to time average the conservation equations. The time averaging procedure is discussed by, for example. Bird et al. (1960). Time averaging the component material balances (Eqs. 1.3.6) gives... 

Because NMR spectroscopy is such an insensitive technique, spectra are recorded in a time averaging procedure with the FID of each scan being added to the FID from the previous scans stored in the computer memory. The time averaging procedure is initialized by the loop command go =... Because the same rf coil is used for the transmission of the rf pulse and for the acquisition of the induced voltage, the go command also performs various electronic switching tasks. Once the acquisition has started the accumulated data is saved in a temporary file and is saved at the end of the... 

As it was mentioned in Section 9.4.1, 3D structures generated by DG have to be optimized. For this purpose, MD is a well-suited tool. In addition, MD structure calculations can also be performed if no coarse structural model exists. In both cases, pairwise atom distances obtained from NMR measurements are directly used in the MD computations in order to restrain the degrees of motional freedom of defined atoms (rMD Section 9.4.2.4). To make sure that a calculated molecular conformation is rehable, the time-averaged 3D structure must be stable in a free MD run (fMD Sechon 9.4.2.5J where the distance restraints are removed and the molecule is surrounded by expMcit solvent which was also used in the NMR measurement Before both procedures are described in detail the general preparation of an MD run (Section 9.4.2.1), simulations in vacuo (Section 9.4.2.2) and the handling of distance restraints in a MD calculation (Section 9.4.2.3) are treated. Finally, a short overview of the SA technique as a special M D method is given in Sechon 9.4.2.6. 

The time A between readings is thus greater than the reaction half-life, and the numerical averaging procedure leading to equation 3.3.24 could be used for the analysis of these data if an estimate of the reading at infinite time can be obtained. In order to manipulate this equation to a form that can make use of the available data, several points must be noted. 

Figure 6 The block-averaging procedure considers a full range of block sizes. The upper panel shows the time series for the squared cosine of the central dihedral of butane, with two different block sizes annotated. The lower panel shows the block-averaged standard error for that times series, as a function of block size.

The transport equations for laminar motion can be formulated, in general, easily and difficulties may lie only in their solution. On the other hand, for turbulent motion the formulation of the basic equations for the time-averaged local quantities constitutes a major physical difficulty. In recent developments, one considers that turbulence (chaos) is predictable from the time-dependent transport equations. However, this point of view is beyond the scope of the present treatment. For the present, some simple procedures based on physical models and scaling will be employed to obtain useful results concerning turbulent heat or mass transfer. 

Two approaches can be used for the analysis of turbulent mass transfer near a liquid-fluid interface. One has the time-averaged convective diffusion equation as the starting point. For obtaining in that procedure an equation for... 

Equation (5.40) reveals the essence of the diffusion coefficient. Since d( 2)/d/ = 2 , then = 2 >/, The same conclusion was reached in Section4.3 where we used an ensemble averaging procedure instead of introducing F°, the time average of the stochastic force, F(t), acting upon a single particle. 

Another possibility is to smooth the seasonal fluctuations using the moving average procedure, but this leads to lack of sharpness because a certain number of values in the lagged time window is included. Seasonally adjusted series achieved by seasonal decomposition are also good starting points for trend searching. 

For ice Ih, at the melting temperature, Ty =10 s and tj, = 10" s. In contrast, in water, owing to the disorder of the liquid phase the two times are Xy = 2-3x10 s and Td = 1.8x10" s, respectively. The mean lifetime of the hydrogen bond tiq assumes values interme ate between the characteristic time of structure V and structure D (thb = 9x10" s). We aim to describe the dynamic properties of water in structure D, by means of a theory that can account for diffusional phenomena in the different molecular environments to do this we shall have to apply suitable time-average procedures to processes the time scale of which is shorter than Thb-... 

In WP analysis the time evolution of an initial wavefunction (or wave-packet) is obtained by the solution of the appropriate time-dependent Schrbdinger equation. The initial wavefunction is determined by the conditions of the collision. The Schrbdinger equation is then integrated, which given the complexity of the potentials usually has to be performed numerically. Information about the crossections can be obtained from this technique and again canonical rate coefficients obtained by the above averaging procedure. 

A fundamental hypothesis of statistical mechanics is the ergodic theorem. Basically it says the system evolves so quickly in the phase space that it visits all of the possible phase points during the time considered. If the system is eigodic, the ensemble average is equivalent to the time average over the trajectory for the time period. The ergodidty of a system depends on the search procedure, force... 

The first attempts to model flow and transport in plant canopies that accommodated (i) the distinct microclimates of different stands of vegetation (ii) the separation of soil surface and layers of canopy as distinct sources and sinks of heat and mass and (iii) the influence of atmospheric stability or advection effects, applied gradient transfer to diffusion within the canopy space ([493]). In this procedure, a flux density is expressed as the product of a diffusion coefficient (turbuient or eddy diffusivity) and the gradient of the time average of the quantity of interest, as in the following examples ... 

Note that the turbulent viscosity parameter has an empirical origin. In connection with a qualitative analysis of pressure drop measurements Boussinesq [19] introduced some apparent internal friction forces, which were assumed to be proportional to the strain rate ([20], p 8), to fit the data. To explain these observations Boussinesq proceeded to derive the same basic equations of motion as had others before him, but he specifically considered the molecular viscosity coefficient to be a function of the state of flow and not only on the system properties [135]. It follows that the turbulent viscosity concept (frequently referred to as the Boussinesq hypothesis in the CFD literature) represents an empirical first attempt to account for turbulence effects by increasing the viscosity coefficient in an empirical manner fitting experimental data. Moreover, at the time Boussinesq [19] [20] was apparently not aware of the Reynolds averaging procedure that was published 18 years after the first report by Boussinesq [19] on the apparent viscosity parameter. 

It has been shown that the averaging procedure is valid for statistically stationary variables only, thus the time average should be independent of the origin t. Apparently, the conventional Reynolds averaging procedure requires that the transients in the governing equations should be negligibly small (e.g., [66], p 6 [167], p. 28). 

The time after volume averaging procedure can be applied under a unified set of conditions denoting the sum of the two sub-sets of requirements formulated in sects 3.4.1 and 3.4.2 for the pure volume and time averaging procedures to handle the scale disparity in a proper manner. 

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