Integral scale velocity

Using the relationship between Lu and r) given in Table 2.2, p. 36, the mixing rate at the velocity integral scale Lu found from (3.5) is approximately... 

Like the velocity spatial correlation function discussed in Section 2.1, the scalar spatial correlation function provides length-scale information about the underlying scalar field. For a homogeneous, isotropic scalar field, the spatial correlation function will depend only on r = r, i.e., R,p(r, t). The scalar integral scale L and the scalar Taylor microscale >-,p can then be computed based on the normalized scalar spatial correlation function fp, defined by... 

Figure 3.12. Model scalar energy spectra at Rk = 500 normalized by the integral scales. The velocity energy spectrum is shown as a dotted line for comparison. The Schmidt numbers range from Sc = 10 4 to Sc = 104 in powers of 102.

The last term on the right-hand side is unclosed and represents scalar transport due to velocity fluctuations. The turbulent scalar flux ( , varies on length scales on the order of the turbulence integral scales Lu, and hence is independent of molecular properties (i.e., v and T).17 In a CFD calculation, this implies that the grid size needed to resolve (4.70) must be proportional to the integral scale, and not the Batchelor scale as required in DNS. In this section, we look at two types of models for the scalar flux. The first is an extension of turbulent-viscosity-based models to describe the scalar field, while the second is a second-order model that is used in conjunction with Reynolds-stress models. 

In many reacting flows, the reactants are introduced into the reactor with an integral scale L that is significantly different from the turbulence integral scale Lu. For example, in a CSTR, Lu is determined primarily by the actions of the impeller. However, is fixed by the feed tube diameter and feed flow rate. Thus, near the feed point the scalar energy spectrum will not be in equilibrium with the velocity spectrum. A relaxation period of duration on the order of xu is required before equilibrium is attained. In a reacting flow, because the relaxation period is relatively long, most of the fast chemical reactions can occur before the equilibrium model, (4.93), is applicable. 

Earlier it was stated that the structure of a turbulent velocity field may be presented in terms of two parameters—the scale and the intensity of turbulence. The intensity was defined as the square root of the turbulent kinetic energy, which essentially gives a root-mean-square velocity fluctuation U. Three length scales were defined the integral scale l0, which characterizes... 

FIGURE 10.9. The ratio of the turbulent burning velocity to the laminar burning velocity as a function of the ratio of the integral scale to the laminar-flame thickness, for various turbulence intensities, for stoichiometric propane-air flames at atmospheric pressure, and initially at room temperature [137],... 

Earlier it was stated that the structure of a turbulent velocity field may be presented in terms of two parameters—the scale and the intensity of turbulence. The intensity was defined as the square root of the turbulent kinetic energy, which essentially gives a root-mean-square velocity fluctuation U. Three length scales were defined the integral scale /q, which characterizes the large eddies the Taylor microscale X, which is obtained from the rate of strain and the Kolmogorov microscale 1, which typifies the smallest dissipative eddies. These length scales and the intensity can be combined to form not one, but three turbulent Reynolds numbers Ri = U lo/v, Rx. = U X/v, and / k = U ly/v. From the relationship between Iq, X, and /k previously derived it is found that / ... 

In a turbulent flow, the entire spectrum of the continuous phase eddies is imparted to the droplets or particles of the dispersed phase present. Theory and experiments (K15, K16, L8, S14) indicate that small drops or particles follow the behavior of the fluid eddies very closely. Drops or particles larger than the integral scale of turbulence follow the mean fluid flow. Experiments (K16) with solid particles (0.013-0.20 cm in diameter) in an agitated vessel show that the particle velocity fluctuations are given by the Maxwell distribution. In addition, the micromotion of the particles was determined mainly by eddies of size comparable to particle diameters. 

The Taylor h rpothesis has been further discussed by [144] (pp 5-7) and [59] (pp. 31-81) considering theories of double correlations between turbulence-velocity components, the features of the double longitudinal and lateral correlations in homogeneous turbulence, integral scales of turbulence, and Eulerian, Lagrangian and mixed Eulerian-Lagrangian correlations. 

Thus, in the Re —> oo limit the turbulent velocity field is a rough non-differentiable function Sv(l) l1/3 for l -h. 0. This expression identifies the so-called Holder exponent as a = 1/3. It is easy to see that the energy transfer rate 8v(1)2/t(1) is independent of the length scale l and applying the first of the above formulas to the integral scale (L) one can relate the energy dissipation rate to the large scale properties of the flow as u3/l. 

For fully developed three-dimensional turbulence one can identify different regimes. Below the Kolmogorov scale the velocity field is smooth and chaotic advection dominates, therefore the FSLE is the same as the standard Lyapunov exponent and is independent of 5. When 5 is within the inertial range, we have A(5) 5 2/3. If the size of the domain is much larger than the integral scale this is followed by a diffusive regime where A(5) 5 2 for 6 > L. Thus the FSLE... 

The inertial-convective range of three-dimensional turbulence covers the range of length scales where inertial forces dominate over viscous forces in the dynamics of the velocity field and advection is the dominant transport process with respect to diffusion. This is valid below the integral scale and limited at small scales by the larger of the Kolmogorov scale (rf) or the diffusive scale (Id)- For... 

A and B are empirical positive coefficients larger than unity (of the order of ten), y and z denote the horizontal and vertical coordinates of the two points in the plane normal to the mean speed, and U denotes the mean velocity [12, 13]. According to this expression, for distances r and frequencies n larger than zero the coherence is smaller than unity. This reflects the physical fact that, at any instant, the velocity fluctuations are not quite the same at two distinct points. The smaller the integral scales of turbulence in the y and z directions, the larger are the constants A and B, and the faster is the decay of the coherence function with increasing n and r. 

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