Mean square fluctuation turbulence

Sachs (Si) attempted to establish the extent of the turbulent fluctuations present. He reports root-mean-square fluctuating velocities ranging up to values on the order of 40% of the mean velocities. Although their quantitative significance may be doubtful, these data do represent... 

If a flow in the tank is turbulent, either because of high power levels or low viscosity, then a typical velocity pattern at a point would be illustrated by Fig. 3. The velocity fluctuation i can be changed into a root mean square value (RMS), which has great utility in estimating the intensity of turbulence at a point. So in addition to the definitions above, based on average velocity point, we also have the same quantities based on the root mean square fluctuations at a point. We re interested in this value at various rates of power dissipation, since energy dissipation is one of the major contributors to a particular value of RMS v. ... 

Brownian diffusion is neglected compared with turbulent transport. The left-hand side represents Dpc/Dt, the Stokes or substantive derivative of p--. The first term on the right-hand side is the turbulent diffusion of The second term —2v p j Vp7 is generally positive and represents the generation of p by transfer from the mean How. The third term. 2p//fl, is the contribution of variations in the mte of gas-to-particle conversion by chemical reaction to the rate of production of p . The la.st term is the decrease of mean square fluctuations pj due to the action of small scale diffusion (dissipation). Thus three types of terms appear on the right-hand side of (13,16), the balance equation for Pi (i) turbulent diffusion of p, and tnmsfer from the mean (low to p.. which alTeci... 

Let us consider a shear flow that is steady and homogeneous in the x X2 plane with the only nonzero mean velocity m 1( 3). The kinetic energy of the turbulence is given by jUjU j = j(u u -I- M2 2 + 3 3)- A measure of the effect of the turbulence on temperature fluctuations is the mean-square fluctuation 9 The dynamic equations governing and 9 in this situation reduce to... 

An apropriate Re-number expression characteristic of local isotropic turbulence can be derived using the root-mean-square fluctuating velocity postulated by Batchelor (1951). 

The transport equations describing the instantaneous behavior of turbulent liquid flow are three Navier-Stokes equations (transport of momentum corresponding to the three spatial coordinates r, z, in a cylindrical polar coordinate system) and a continuity equation. The instantaneous velocity components and the pressure can be replaced by the sum of a time-averaged mean component and a root-mean-square fluctuation component according to Reynolds. The resulting Reynolds equations and the continuity equation are summarized below ... 

As Dryden (4) emphasizes, turbulence is a lack of uniformity in the flow conditions, characterized bylin rF gularfluctuation of the fliiid Ioc-ity at any point from instant to instant. Two factors are ordinarily required to express the degree of turbulence in quantitative fashion, the intensity and the scale. Intensity is defined as the root-mean-square fluctuation of velocity at anjTpoint. The scale relates the fluctuations at different points within the fluid at the same instant and has been quantitatively defined by Taylor (28) as the area under a curve of the correlation between the velocity fluctuation at two points taken perpendicular to the line joining the points plotted against the distance between the points. The scale may be taken as the size of an eddy. 

Vfjp is the friction velocity and =/pVV2 is the wall stress. The friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu p/. . The universal velocity profile is vahd in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, = AP/4L where AP is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are vahd only near the pipe wall. 

Even though the Reynolds number gives some measure of turbulent phenomena, flow quantities characteristic of turbulence itself are of more direct relevance to modeling turbulent reacting systems. The turbulent kinetic energy q may be assigned a representative value <7o at a suitable reference point. The relative intensity of the turbulence is then characterized by either q()KH2 U2) or (77(7, where (/ = (2q0)m is a representative root-mean-square velocity fluctuation. Weak turbulence corresponds to U /U < 1 and intense turbulence has (77(7 of the order unity. 

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 4 depicts the variation of the root-mean-square longitudinal fluctuation as a function of position in a floiving stream. These data were taken from the recent experimental investigation by Laufer (L3) and illustrate the complexity of behavior encountered in a steady, uniformly flowing turbulent stream. It is to be expected that fluctuations of temperature and composition are encountered in turbulent streams involving thermal or material transport. 

In complete accord with a simple numerical evaluation, / — Pmaxr/D0 and the magnitude of DQ decreases the turbulent diffusion is independent of the molecular diffusion coefficient. Let us carefully consider the structure of the quantity r = vX. It is obvious that in a turbulent flow we cannot directly determine a quantity which is linear in the fluctuation velocity. It is no accident that the square of the velocity figured in the original equation. It is precisely the mean square of the velocity and its spectral representation that may be determined in a turbulent flow. Therefore, consistently performing all the calculations, we obtain... 

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 / ... 

It is the fluctuating element of the velocity in a turbulent flow that drives the dispersion process. The foundation for determining the rate of dispersion was set out in papers by G. 1. Taylor, who first noted the ability of eddy motion in the atmosphere to diffuse matter in a manner analogous to molecular diffusion (though over much larger length scales) (Taylor 1915), and later identified the existence of a direct relation between the standard deviation in the displacement of a parcel of fluid (and thus any affinely transported particles) and the standard deviation of the velocity (which represents the root-mean-square value of the velocity fluctuations) (Taylor 1923). Roberts (1924) used the molecular diffusion analogy to derive concentration profiles... 

The turbulent velocity scale is often approximated by, root-mean-square of the fluctuating velocity components. 

The mathematical models for simulating turbulent flame acceleration and the onset of detonation in chemically reacting flows were described in detail in [1-3]. The system of equations for the gaseous mixture was obtained by Favre averaging. The standard k e model was modified an equation was added that determined the mean squared deviate of temperature in order to model the temperature fluctuations. 

FIGURE 8-3 Three-dimensional distribution of root-mean-square turbulent velocity fluctuations upstream of a step (Kasagi and Matsunaga 1995). (Reprinted by permission of Butterworth/Heinemann.)... 

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