Instantaneous turbulent velocity

A more realistic approach to quantify the pressure field is to consider the effect of turbulence [6]. For a pipe flow, the turbulent pressure fluctuations are due to velocity perturbations as a result of the formation of eddies. The instantaneous turbulent velocity can be calculated by assuming a sinusoidal velocity variation in... 

Delichatsios (D4) and Delichatsios and Probstein (D5) also derived a corresponding expression forg(a )- Following Levich (L7), they proposed that a breakup will occur whenever the instantaneous turbulent velocity differences across a drop diameter exceed b. the velocity necessary to break the drop. From similarity arguments. 

The instantaneous turbulent velocity at a certain position was calculated by assuming a sinusoidal velocity variation in the instantaneous local velocity (V ) with a frequency of the velocity perturbation (similar to the case of acoustic cavitation) and is given by ... 

Second, the cross-shore variation of the degree of sediment suspension is estimated using the experimental finding of Kobayashi et aZ. who showed that the turbulent velocities measured in the vicinity of the bottom were related to the energy dissipation rate due to bottom friction. Representing the magnitude of the instantaneous turbulent velocity by with D f = O.Sp/ftt/f in light of... 

In RANS-based simulations, the focus is on the average fluid flow as the complete spectrum of turbulent eddies is modeled and remains unresolved. When nevertheless the turbulent motion of the particles is of interest, this can only be estimated by invoking a stochastic tracking method mimicking the instantaneous turbulent velocity fluctuations. Various particle dispersion models are available, such as discrete random walk models (among which the eddy lifetime or eddy interaction model) and continuous random walk models usually based on the Langevin equation (see, e.g.. Decker and... 

Measurement of the velocity of a large particle. The investigation of the turbulence characteristics in the liquid phase of a bubbly flow has generated detailed studies on the use of thermal anemometry and optical anemometry in gas-liquid two-phase flows. These techniques have been proved to be accurate and reliable for the measurement of the instantaneous liquid velocity in bubble flow. However, the velocity of the gas bubbles—or, more precisely, the speed of displacement of the gas-liquid interfaces—is still an active research area. Three techniques that have been proposed to achieve such measurement were reviewed by Delhaye (1986), as discussed in the following paragraphs. 

In the joint velocity, composition PDF description, the user must supply an external model for the turbulence time scale r . Alternatively, one can develop a higher-order PDF model wherein the turbulence frequency > is treated as a random variable (Pope 2000). In these models, the instantaneous turbulence frequency is defined as... 

Typical Lagrangian approaches include the deterministic trajectory method and the stochastic trajectory method. The deterministic trajectory method neglects all the turbulent transport processes of the particle phase, while the stochastic trajectory method takes into account the effect of gas turbulence on the particle motion by considering the instantaneous gas velocity in the formulation of the equation of motion of particles. To obtain the statistical... 

In trajectory models, the particle turbulent diffusion can be considered by calculating the instantaneous motion of particles in the turbulent flow field. In order to simulate the stochastic characteristics of the instantaneous gas velocity in a turbulent flow, it is required to generate random numbers in the calculation process. 

If one observes the instantaneous macroscopic velocity in a turbulent-flow system, as measured with a laser anemometer or other sensitive device, significant fluctuations about the mean flow velocity are observed as indicated in Fig. 5-11, where u is designated as the mean velocity and is the fluctuation from the mean. The instantaneous velocity is therefore... 

For a unit area of the plane P-P, the instantaneous turbulent mass-transport rate across the plane is pv. Associated with this mass transport is a change in the x component of velocity u. The net momentum flux per unit area, in the x direction, represents the turbulent-shear stress at the plane P-P, or pv u When a turbulent lump moves upward (v > 0), it enters a region of higher u and is therefore likely to effect a slowing-down fluctuation in u , that is, u < 0. A similar argument can be made for v < 0, so that the average turbulent-shear stress will be given as... 

It may be noted that the discussion so far has not considered turbulent flow. When the continuous phase flow field is turbulent, its influence on particle trajectories needs to be represented in the model. The situation becomes quite complex in the case of two-way coupling between continuous phase and dispersed phase, since the presence of dispersed phase can affect turbulence in the continuous phase. The Eulerian framework may be more suitable to model such cases. Even when dispersed phase particles are assumed to have no influence on the continuous phase flow field, the trajectories of the particles will be affected by the presence of turbulence in the continuous phase. For such cases, it is necessary to calculate the trajectories of a sufficiently large number of particles using the instantaneous local velocity to represent the random effects of turbulence on particle dispersion. 

Reynolds [127] provided the fundamental ideas about averaging and was the first to accomplish the formulation of the governing equations for turbulent flows in terms of mean and fluctuating flow quantities rather than instantaneous quantities. Reynolds stated the mathematical rules for forming mean values. That is, he suggested splitting a turbulent velocity field into its mean and fluctuating components, and wrote down the equations of motion for these two velocity quantities. 

By averaging our velocity measurements over a certain time period, we can eliminate or average out the positive and negative deviations of the turbulent velocities about the mean. Once we have the mean velocity, v, for any time period, we can subtract it from the actual instantaneous velocity v, to calculate the turbulent part, v ... 

Local instantaneous liquid velocity measurements in bioreactors that can quantify turbulence statistics are challenging using conventional laser-based techniques because optical access is critical for effective signal acquisition. Laser Doppler anemometry (LDA) and PIV have been used to determine local liquid velocities within multiphase flows. Reviews of LDA and PIV with applications to multiphase flows have appeared in the literature (Boyer et al., 2002 Chaouki et al., 1997 Cheremisinoff, 1986). 

By comparing these two time-smoothed equations with Eqs. (3.6-24) and (3.10-16) we see that the time-smoothed values everywhere replace the instantaneous values. However, in Eq. (3.10-20) new terms arise in the set of brackets which are related to turbulent velocity fluctuations. For convenience we use the notation... 

The averaging notions introduced above can be directly applied to define the average value of a velocity component, its instantaneous turbulent fluctuation, and the rms turbulent velocity in the ensemble-average sense ... 

An illustrative example is a snapshot in Fig. 19 from a periodic-box DNS of a Hquid—solid suspension with monodisperse spherical particles coUiding as a result of the turbulence level imposed via a random forcing technique it was performed by Ten Cate et al (2004) by means of an LB technique. The snapshot shows both the instantaneous Hquid velocity field (denoted by the arrows) and the spatial distribution of the rate of energy dissipation in the liquid (denoted by the color code) along with the instantaneous position of the particles in some cross-sectional area through the periodic box. In addition, Derksen and Van den Akker (2007) demonstrated the usefiilness of the periodic-box DNS implementation for investigating the effect of... 

With LES we get much more information than with traditional time-averaged turbulence models, since we are resolving most of the turbulence. In Fig. T1.15 the computed u velocity is shown as a function of time in two cells one cell is located in the wall jet (Fig.. 15a), and the other cell is in the middle of the room (Fig. ll.lSh). It is found the instantaneous fluctuations are very large. For example, in the region of the wall jet below the ceiling where the time-averaged velocity u)/l] ) is typically 0.5, the instantaneous velocity fluctuations are between 0.2 and 0.9. In the middle of the room, which is a low-velocity region, the variation of u is much slower, i.e., the frequency is lower. 

The population balance in equation 2.86 employs the local instantaneous values of the velocity and concentration. In turbulent flow, there are fluctuations of the particle velocity as well as fluctuations of species and concentrations (Pope, 1979, 1985, 2000). Baldyga and Orciuch (1997, 2001) provide the appropriate generalization of the moment transformation equation 2.93 for the case of homogeneous and non-homogeneous turbulent particle flow by Reynolds averaging... 

In the present discussion only the problem of steady flow will be considered in which the time average velocity in the main stream direction X is constant and equal to ux. in laminar flow, the instantaneous velocity at any point then has a steady value of ux and does not fluctuate. In turbulent flow the instantaneous velocity at a point will vary about the mean value of ux. It is convenient to consider the components of the eddy velocities in two directions—one along the main stream direction X and the other at right angles to the stream flow Y. Since the net flow in the X-direction is steady, the instantaneous velocity w, may be imagined as being made up of a steady velocity ux and a fluctuating velocity ut, . so that ... 

In turbulent flow there is a complex interconnected series of circulating or eddy currents in the fluid, generally increasing in scale and intensity with increase of distance from any boundary surface. If, for steady-state turbulent flow, the velocity is measured at any fixed point in the fluid, both its magnitude and direction will be found to vary in a random manner with time. This is because a random velocity component, attributable to the circulation of the fluid in the eddies, is superimposed on the steady state mean velocity. No net motion arises from the eddies and therefore their time average in any direction must be zero. The instantaneous magnitude and direction of velocity at any point is therefore the vector sum of the steady and fluctuating components. 

An idea of the scale of turbulence can be obtained by measuring instantaneous values of velocities at two different points within the fluid and examining how the correlation coefficient for the two sets of values changes as the distance between the points is increased. 

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