Shear velocity 98

Scale-up calculation is based on constant shear forces, where shear is directly related to impeller tip velocity. Shear forces are defined as... 

Fig. 4.3.7 The left figure shows the radius of Figure 4.3.6, at the free surface of the Fano the Fano column as a function of the column column (r = Rr), one half radius of the column height (z), which was enlarged towards the (r= 0.5 R) and the center of the column (r= 0). column base. The right figure shows the local The figure clearly shows that there was a large velocity values at three different locations (z) of velocity shear near the fluid entrance, and the the Fano column as a function of the column velocity at the free surface was almost con-height. The three sets of velocity values were stant. measured from the velocity profiles shown in...

The die-swell (extrudate swell) effect describes the significant expansion of the diameter of the fluid column after exiting from a small pipe (Figure 4.3.8(b)). Some polymer fluids can have a swelling of up to two or three times the exit diameter. A simple proposition for the mechanism of the die-swell phenomenon is that while the fluid is inside the exit pipe, it is subject to a velocity shear, similar to the pipe flow with a maximum shear stress at the wall [18]. This velocity shear stretches... 

Dispersion will be discussed in the subsequent section. At this point we only mention that dispersion always occurs in fluids with a distinct direction of advective flow. It originates from the velocity difference between adjacent streamlines. This effect is called velocity shear. 

The spreading results from a combination of two processes, (1) Fickian horizontal diffusion with scale-independent diffusivity Eb, and (2) dispersion by velocity shear in the direction of the mean flow. The process of dispersion is discussed in Section 22.4. It is related to the flow velocity difference (called velocity shear ) between adjacent streamlines. Since water parcels traveling on different streamlines (e.g., at different depth) have different velocities, a tracer cloud is elongated along the direction of the mean flow (see figure below). 

Although dispersion can be described by the same law as diffusion, its nature is different. Dispersion is the result of the velocity shear, that is, of the velocity difference between adjacent streamlines in an advective flow. Due to turbulent exchange perpendicular to the direction of flow, water parcels continuously change the streamline along which they move. Since these streamlines move at different speeds, each water parcel has its own individual history of speed and thus its individual mean velocity. 

Once the mathematical description of dispersion has been clarified, we are left with the task of quantifying the dispersion coefficient, Eiis. Obviously, Edh depends on the characteristics of the flow field, particularly on the velocity shear, dvx/dy and dvx /dz. As it turns out, the shear is directly related to the mean flow velocity vx. In addition, the probability that the water parcels change between different streamlines must also influence dispersion. This probability must be related to the turbulent diffusivity perpendicular to the flow, that is, to vertical and lateral diffusion. At this point it is essential to know whether the lateral and vertical extension of the system is finite or whether the flow is virtually unlimited. For the former (a situation typical for river flow), the dispersion coefficient is proportional to (vx )2 ... 

Fig. E2.5b Schematic representation velocity shear rate and shear stress profiles of a Newtonian fluid between parallel plates.

These equations are solved numerically under the assumptions of velocity, shear stress, and temperature continuity at all interfaces. They use the Sabia 4-parameter viscosity model (69), because of its ability to include the Newtonian plateau viscosity, which is important for multilayer extrusion, because of the existence of low shear-rate viscosities at the interfaces. 

In addition to the results presented above, we should also note the studies of the climatic BSGC [56] based on the basic Russian prognostic model [57]. The distinctive features of [56] were related to the dependence of the coefficients of horizontal turbulence on lateral velocity shears and to the specifying of the monthly climatic temperature and salinity field at the surface [29] instead of the heat and moisture fluxes. Despite the relatively coarse horizontal calculation grid (about 22 km), this allowed the authors to reproduce [56] a relatively distinct MRC jet and the known NSAEs off the Turkish and Caucasian coasts and off the Danube River mouth. The results of the tuning in [56] of the Munk-Anderson s formula for the coefficient of the vertical turbulent exchange from the point of view of reproduction of the actual CIL were used in [53,54]. 

In addition to deep-sea anticyclones and large cyclonic eddies interacting with them (such as vortical pair Al-Cl in Fig. 4A), an important contribution to the water exchange between the coastal zone and the open sea is made by small cyclones with a diameter of 30-40 km and jets. Two cyclones of this type probably formed as a result of a velocity shear at the seaward boundary of the RC may be seen in Fig. 5. Moving to the northwest at a rate of approximately 10 cm/s and interacting with the RC and the anticyclone separating from the coast, they provided the transport of warm coastal waters over a distance more than 100 km away from the coast [23]. Small cyclonic eddies are also formed in the cyclonic meanders of the RC [8]. 

In the above equations, co and Q are the angular velocity of liquid and cylinder, respectively, p is the sUp coefficient (defined as = slip velocity/shearing stress),

[Pg.68]

As previously discussed, turbulence is caused in part by velocity shear due to a nonuniform velocity profile. In a river, a shear velocity, which is related to the shear force per unit area exerted by the water flow on the river channel, can be estimated (Fischer et al., 1979) as... 

Note that none of the equations in Table 2-5 include wind speed, because it is assumed that the turbulence generated within the stream due to velocity shear and turbulent diffusion primarily controls gas exchange. Unfortunately, the predictions of these equations are often not in good agreement with one another, and it is difficult to know which one is best in a given situation (see Fig. 2-13). 

The sub-script h is used to denote the dispersion associated with velocity shear at the canopy scale, as distinct from velocity shear at the wake scale. Recall that here h = //, as the canopy is emergent. Equation (6.19) can be non-dimensionalized as... 

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