Averaged equations

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 ... 

When the continmty equation and the Navier-Stokes equations for incompressible flow are time averaged, equations for the time-averaged velocities and pressures are obtained which appear identical to the original equations (6-18 through 6-28), except for the appearance of additional terms in the Navier-Stokes equations. Called Reynolds stress terms, they result from the nonlinear effects of momentum transport by the velocity fluctuations. In each i-component (i = X, y, z) Navier-Stokes equation, the following additional terms appear on the right-hand side ... 

Although direct numerical simulations under limited circumstances have been carried out to determine (unaveraged) fluctuating velocity fields, in general the solution of the equations of motion for turbulent flow is based on the time-averaged equations. This requires semi-... 

Anderson and Jackson (1967, 1968, 1969) and Ishii (1975) have separately derived the governing equations for TFMs from first principles. Although the details of constructing the averaged equations are different, the final equations are essentially the same. The TFMs differ significantly from each other as different closures for the solid stress tensor are used. 

To update the composition of each segregated mineral, we average the composition (e.g., <5180yfc/) of the mass that precipitated over the reaction step and the composition (e.g., S18f i)) of whatever mass was present at the onset. The averaging equation for oxygen, for example, is... 

Due to its smaller uncertainties, the first measurement is dominant. The final results for the weighted average, equation (5.4.5), its standard errors, equation (5.4.6), and the correlation coefficient are reported in the last row of Table 5.21. o... 

Assuming now that the incident wave interacts with centers whose vectors are randomly oriented with respect to Eo, we can average Equation (5.13) over all possible orientations. Taking into account that (cos 6>) = 1/3 (considering that all... 

A variety of specific mathematical formulations of the CTRW approach have been considered to date, and network models have also been applied (Bijeljic and Blunt 2006). A key result in development of the CTRW approach is a transport equation that represents a strong generalization of the advection-dispersion equation. As shown by Berkowitz et al. (2006), an extremely broad range of transport patterns can be described with the (ensemble-averaged) equation... 

A structure of the obtained set of equations derived by us in [81, 86] is very close to the famous BBGKI set of equations widely used in the statistical physics of dense gases and liquids [76]. Therefore, we presented the master equation of the Markov process in a form of the infinite set of deterministic coupled equations for averages (equation (2.3.34)). Practical use of these equations requires us to reduce them, retaining the joint correlation functions only. 

An extended medium comprising the s-region can be described statistically with the probability distribution Pa(bo) of initial boson amplitudes. The stochastic assumption means that only averages A = J dbQ Ps(b0)A(b0) of properties A over this distribution are needed. With this in mind, we look for statistical averages p — of the p-region density operator, satisfying an averaged equation of motion,... 

When the continuity equation and the Navier-Stokes equations for incompressible flow are time averaged, equations for the time-averaged... 

Thus the standard enthalpy of formation or an isomer group is the mole fraction weighted average. Equations 3.5-11 to 3.5-14 will be especially useful in the next chapter. 

Equation (3.5-18) has been written in terms of molar heat capacities Cpm(i), rather than heat capacities of formation, because the heat capacities of the elements are on both sides and cancel. The second term of this equation is always positive because the weighted average of the squares is always greater than the square of the average. Equation 3.5-18 is in accord with LeChatelier s principle As the temperature is raised, the equilibrium shifts in the direction that causes the absorption of heat. Equation 3.5-18 can also be derived using CP = — T(d2G/dT2)P (equation 2.5-25). 

The most widely adopted method for the turbulent flow analysis is based on time-averaged equations using the Reynolds decomposition concept. In the following, we discuss the Reynolds decomposition and time-averaging method. There are other methods such as direct numerical simulation (DNS), large-eddy simulation (LES), and discrete-vortex simulation (DVS) that are being developed and are not included here. 

The volume-averaged equation of energy conservation in terms of internal energy is given by averaging Eq. (5.18) as... 

The volume-averaged products in the preceding volume-averaged equations can be further expressed by the products of the volume averages. From Eq. (5.100), the volume-averaged mass flux of phase k is given by... 

The volume-time-averaged equations can be summarized as follows ... 

For convenience, we remove the () signs and time-averaging bars from the preceding equations. Thus, the volume-time-averaged equations can be simplified to the form... 

The time averaging equations are straightforward and will not be shown here. 

In the small particle size regime, two equivalent formulations lead to the interpretation of the data in terms of ratios of moments of the particle size distribution or in terms of powers of the D32 average (equations 15 and 20). It is clear that in either case a sufficient number of terms in the series has to be included in order to account for the behavior of the extinction as function of a. The number of terms required cannot be decided a priori, rather the data itself has to dictate how many terms in the power series approximation the measurements can detect. 

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