Velocity field, mean, time-averaged

The velocity field in turbulent flow can be described by a local mean (or time-average) velocity, upon which is superimposed a time-dependent fluctuating component or eddy. Even in one-dimensional flow, in which the overall average velocity has only one directional component (as illustrated in Fig. 6-3), the turbulent eddies have a three-dimensional structure. Thus, for the flow illustrated in Fig. 6-3, the local velocity components are... 

The application of an electric field E to a conducting material results in an average velocity v of free charge carriers parallel to the field superimposed on their random thermal motion. The motion of charge carriers is retarded by scattering events, for example with acoustic phonons or ionized impurities. From the mean time t between such events, the effective mass m of the relevant charge carrier and the elementary charge e, the velocity v can be calculated ... 

Under the conditions of turbulence, the time-averaged velocity field is symmetric with respect to the free stagnation plane, provided the flow rates from the two nozzles are equal. The mean axial velocity profile has a similar shape to the curve of uju ) vs x. The gradient of the time-averaged axial velocity takes the maximum at the stagnation plane, while it approaches zero near the nozzle. 

Partially filled bands of collective-electron states support metallic conductivity. The electrical conductivity is defined as the ratio of current density J = nev to electric field strength, E, where n is the number of carriers of charge e per unit volume and v is their average velocity. Since the average force on a charged particle is eE = m v/r, where r is the mean time between collisions and m is the effective mass, it follows that... 

A new term is introduced, the so-called Reynolds stresses m-m). The overbar denotes a time average. This term is the correlation between the turbulent velocity fluctuations and uj, and it describes the transport of momentum in the mean flow due to turbulence. This term is difficult to model, and over the years a variety of turbulence models have been developed. Turbulence models are necessary for calculating time-averaged flow fields directly, without first having to calculate a fully time-dependent flow field and then doing time averaging. The use of turbulence models is therefore much more computationally efficient. A detailed discussion is beyond the scope of this entry, but it is important to note that not all turbulence models are equally suited for all types of flow. Table 1 summarizes the most common turbulence models and their properties. 

Hd = time-averaged size distribution function V — mean wind field K = eddy diffusion coefficient Cj = terminal settling velocity... 

Special code modules were developed to compute the water content and the liquid velocity field. We obtained the macroscopic mean values of micropore and maeropore water contents, respectively. 0mit.(r.,.0 and 0Mlm.(Zj,/) by averaging over 50 time steps, and over spatial steps ol 5 sites in the "-direction ami overall the mi-... 

Even if we computed the microscopic velocity field in the x, y and z directions, we only considered the macroscopic averages in the y and z directions, as no heterogeneity is considered in the x-direction. The macroscopic fluxes were calculated by averaging the microscopic velocities over 50 time steps, over five sites in the z-direction, and over half-cross sections of the micropore matrix and of the crack, multiplied by the respective mean water contents. 

Assuming that the velocity field is a random function of space and time, the expectation value of the particles final position can be obtained by averaging the above equation over many realizations of the random velocity field. If the mean velocity is zero this gives the simple result (r(t)) = ro- The standard deviation of the distance from the initial point satisfies the equation... 

The time-averaged velocity field downstream of the TARS, including axial and tangential mean and turbulent velocity components, was measured using a Phase Doppler Particle Analyzer (PDPA) system. 

Time-averaged mean and turbulent velocity data without a confined combustion chamber were used to compare the basic characteristics of the flow field for the different multiple swirler combinations listed in Table 10.1. 

In a turbulent flow field, all the dynamic properties (e.g., velocity, pressure, vorticity) are random functions of position and time. One thus looks at the statistical aspects of the flow characteristics (e.g., mean velocity, rms turbulent intensity). These quantities are meaningful if the flow is stochastically random (i.e., its statistics are independent of time) (Nerem and Rumberger, 1976). The time average of any random quantity is given by... 

The quantity F is related to the mean time between electron collisions with lattice vibrations. T (i.e., to the problem of electron-phonon scattering). By considering the motion of electrons able to make collisions with lattice vibrations in an electric field E having radian frequency o), it is straightforward to show that the average velocity is... 

The hybrid algorithm we use in the actual case is a combination of two sub-models Conventional SIMPLE approach together with a, k — e model and elliptic velocity-composition joint PDF scheme [6]. The sub-models interact as follows The CFD model supplies mean velocity fields, V(p and arrays of turbulent kinetic energy and dissipation rates as input for the PDF part. Having obtained these quantities as input, the fractional time step algorithm provides scalar composition and density as final output. The averaged density-field is finally handed back to the CFD sub-model. 

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