Drag coefficient turbulence

Between about Rop = 350,000 and 1 X 10 , the drag coefficient drops dramatically in a drag crisis owing to the transition to turbulent flow in the boundary layer around the particle, which delays aft separation, resulting in a smaller wake and less drag. Beyond Re = 1 X 10 , the drag coefficient may be estimated from (Clift, Grace, and Weber) ... 

Drag coefficient The coefficient relating to the influence of drag over a surface in either laminar or turbulent flow. 

As seen in Fig. 11-2, the drag coefficient for the sphere exhibits a sudden drop from 0.45 to about 0.15 (almost 70%) at a Reynolds number of about 2.5 x 105. For the cylinder, the drop is from about 1.1 to about 0.35. This drop is a consequence of the transition of the boundary layer from laminar to turbulent flow and can be explained as follows. 

In some way, introducing an increased particle drag by means of Eq. (17) resembles the earlier proposal raised by Bakker and Van den Akker (1994b) to increase viscosity in the particle Reynolds number due to turbulence (in agreement with the very old conclusion due to Boussinesq, see Frisch, 1995) with the view of increasing the particle drag coefficient and eventually the bubble holdup in the vessel. Lane et al. (2000) compared the two approaches for an aerated stirred vessel and found neither proposal to yield a correct spatial gas distribution. 

In addition, it is dubious whether this new correlation due to Brucato et al. (1998) should be used in any Euler-Lagrangian approach and in LES which take at least part of the effect of the turbulence on the particle motion into account in a different way. So far, the LES due to Derksen (2003, 2006a) did not need a modified particle drag coefficient to attain agreement with experimental data. Anyhow, the need of modifying particle drag coefficient in some way illustrates the shortcomings of the current RANS-based two-fluid approach of two-phase flow in stirred vessels. 

When the Reynolds number Rep reaches a value of about 300000, transition from a laminar to a turbulent boundary layer occurs and the point of separation moves towards the rear of the sphere as discussed above. As a result, the drag coefficient suddenly falls to a value of 0.10 and remains constant at this value at higher values of Rep. 

Henderson 575 presented a set of new correlations for drag coefficient of a single sphere in continuum and rarefied flows (Table 5.1). These correlations simplify in the limit to certain equations derived from theory and offer significantly improved agreement with experimental data. The flow regimes covered include continuum, slip, transition, and molecular flows at Mach numbers up to 6 and at Reynolds numbers up to the laminar-turbulent transition. The effect on drag of temperature difference between a sphere and gas is also incorporated. 

When the fluid is in turbulent flow, or where turbulence is generated by some external agent such as an agitator, the drag coefficient may be substantially increased. Brucato et al.(24> have shown that the increase in drag coefficient may be expressed in terms of the Kolmogoroff scale of the eddies (kE) given by ... 

The increase in the drag coefficient CD over that in the absence of turbulence CD0 is given by ... 

At still greater Reynolds numbers the boundary layer itself becomes turbulent and separation occurs at the rear of the sphere and closer to the particle. In this fully turbulent region, beyond Re = 2x 10 the drag coefficient falls further to a value of about 0.10. 

The range between these small and large particles is less well understood although some experimental studies have been reported (K9, Ul). Similar problems arise in interpretation as with accelerated motion (see Chapter 11). Measurements are commonly correlated by a turbulence-dependent drag coefficient, which contains a number of possible acceleration-dependent components. With fundamental understanding so poorly advanced, it is impossible to say to what extent results are specific to the experimental conditions employed. 

There has been relatively little work on the motion of bubbles and drops in well-characterized turbulent flow fields. There is some evidence (B3, K7) that mean drag coefficients are not greatly altered by turbulence, although marked fluctuations in velocity (B3) and shape (K7) can occur relative to flows which are free of turbulence. The effect of turbulence on splitting of bubbles and drops is discussed in Chapter 12. 

Data predictions for droplets moving freely in turbulent gas streams are confounded by the problem of ballistics of droplets. Until the droplet is essentially accelerated or decelerated to the gas stream velocity, Reynolds number, thus Nusselt number, and thus X are changing constantly, and precise calculations require very small steps. The drag coefficient is of considerable importance. El Wakil, Uyehara, and Myers (117) em-... 

The vaporization rates and drag coefficients for 2,2,4-trimethylpentane (iso-octane) sprays in turbulent air streams were determined experimentally by Ingebo (40), who reported that the effect of relative velocity on the evaporation rate was represented by the 0.6 power of the Reynolds number and that the drag coefficient varied inversely with the relative velocity of the drops in the spray. By assuming that the evaporation rate was independent of velocity and the drag coefficient for droplets obeyed Stokes s law, the present author derived a mathematical theory for the ballistics of droplets injected into an air stream for which the velocity varied linearly with distance (57) and... 

Drag coefficients may be affected by turbulence in the free-stream flow the drag crisis occurs at lower Reynolds numbers when the free... 

G. Laminar and turbulent, flat plate, forced flow >=7h = = 0.037 Chilton-Colbum analogies, Nsc = 1-0, (gases), /= drag coefficient. Corresponds to item 5-21-F and refers to same conditions. 8000 < < 300,000. Can apply analogy,=/ 2, to entire plate (including laminar portion) if average values are used. [100] p. 193 [109] p. 112 [146] p. 201 [151] p.271... 

In case 3 the relative size of the particles (with respect to the computational cells) is large enough that they contain many hundreds or even thousands of computational cells. It should be noted that the geometry of the particles is not exactly represented by the computational mesh and special, approximate techniques (i.e., body force methods) have to be used to satisfy the appropriate boundary conditions for the continuous phase at the particle surface (see Pan and Banerjee, 1996b). Despite this approximate method, the empirically known dependence of the drag coefficient versus Reynolds number for an isolated sphere could be correctly reproduced using the body force method. Although these computations are at present limited to a relatively low number of particles they clearly have their utility because they can provide detailed information on fluid-particle interaction phenomena (i.e., wake interactions) in turbulent flows. 

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