Conditionally averaged velocity field

The relaxation lengthscales, non-dimensionalised using the bubble diameter d, for up and downstream distances from the bubble surface are denoted by L and L d respectively and are plotted in the Fig. 7.7. These distances are estimated by exponential fitting of the up and downstream sections of the filtered conditional average. The upstream distance L is weakly dependent on void fraction, and larger than the inviscid estimate of 0.2d. The downstream distance L d extends further than the upstream distance, and is smaller than the wake disappearance length, estimated from equation (7.29), Ld = 2d/3aCd. 

---

Figure 7.6 Relaxation lengthscales of the unfiltered conditional average velocity field normalised by the bubble diameter d as a function of void fraction. (The difference from estimates based on the filtered conditionally averaged velocity field, are negligible). L and L d correspond to decay distance for up and downstream velocity. The theoretical prediction, from (7.29), of the relaxation lengthscale is Ld/d = 2d/aCd, for Ret = 10.

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. 

As in imaging, aliasing can occur if velocities outside the FOF exist. For measurements of average velocities, the same condition holds if the total phase shift due to velocity exceeds 2 n. For velocity fields of simple structure, this problem can be accounted for by so-called unwrapping algorithms (see e.g. Chapters 2.9 and 4.2). 

The mean velocity field is obtained by averaging 60 instantaneous velocity fields. For these conditions the flame is anchored to the collar lip. Superimposed on... 

The stability of most colloidal solutions depends critically on the magnitude of the electrostatic potential ( /o) at the surface of the colloidal particle. One of the most important tasks in colloid science is therefore to obtain an estimate of /o under a wide range of electrolyte conditions. In practice, the most convenient method of obtaining /q uses the fact that a charged particle will move at some constant, limiting velocity under the influence of an applied electric field. Even quite small particles (i.e. <1 xm) can be observed using a dark-field illumination microscope and their (average) velocity directly measured. The technique is called microelectrophoresis . 

The traditional parabolic model with Danckwerts boundary conditions is also used in the literature to describe dispersion effects in packed beds (and porous media). However, unlike the case of capillaries and straight tubes, the flow field in packed beds is more complex and is three-dimensional. However, for many cases of interest, the average velocity in the transverse directions is zero. In such cases, dispersion in the flow direction can be described by the... 

In the absence of stirring, the front propagation speed reaches an asymptotic value, vo = 2 jD/x, and the thickness of the reaction zone is E, = 8VDx [9,10]. While in the presence of a velocity field the front propagates usually with an average speed v/ greater than vo [13-15], Moreover, if/(0) is not convex, under special conditions, the flow may stop ( quench ) the reaction [16]. 

Stockman et al. (1997) provide details on the practical problems and limitations of lattice gas and lattice Boltzmann methods in flow and transport simulation. In particular, they focus on errors associated with boundary conditions, the accuracy required for useful comparison with experimental data, programming, and problem size and run-time issues. For lattice gas methods, they find that averaging over a large number of time steps is sometimes needed to resolve the flow velocity field. This limits the applicability of lattice gas methods to flow simulation under steady state or slowly varying conditions. In contrast, dispersive processes alone can be adequately simulated with lattice gasses by averaging a much smaller number of time steps. Lattice Boltzmann methods do not require averaging. 

We see that the ratio between tube wall concentration and cup mix concentration is highest for undeveloped flow and lowest for a fully developed velocity profile. It further follows that under our experimental conditions (no 1, 2, 3 Table I) for all three flow conditions the average concentration is roughly two times as hi as the wall concentration. This implies that also the rate constant roughly spoken has to be two times as high as the experimental value. In the preceeding paragraph we saw that for the fully developed velocity field this indeed is the case. 

Thus when an electric field is appHed to a soHd material the mobile charge carriers are accelerated to an average drift velocity v, which, under steady-state conditions, is proportional to the field strength. The proportionality factor is defined as the mobility, = v/E. An absolute mobility defined as the velocity pet unit driving force acting on the particle, is given as ... 

---