Turbulent diffusivity fluctuations

The physics and modeling of turbulent flows are affected by combustion through the production of density variations, buoyancy effects, dilation due to heat release, molecular transport, and instabiUty (1,2,3,5,8). Consequently, the conservation equations need to be modified to take these effects into account. This modification is achieved by the use of statistical quantities in the conservation equations. For example, because of the variations and fluctuations in the density that occur in turbulent combustion flows, density weighted mean values, or Favre mean values, are used for velocity components, mass fractions, enthalpy, and temperature. The turbulent diffusion flame can also be treated in terms of a probabiUty distribution function (pdf), the shape of which is assumed to be known a priori (1). 

In the so-called "wrinkled flame regime," the "turbulent flame speed" was expected to be controlled by a characteristic value of the turbulent fluctuations of velocity u rather than by chemistry and molecular diffusivities. Shchelkin [2] was the first to propose the law St/Sl= (1 + A u /Si) ), where A is a universal constant and Sl the laminar flame velocity of propagation. For the other limiting regime, called "distributed combustion," Summerfield [4] inferred that if the turbulent diffusivity simply replaces the molecular one, then the turbulent flame speed is proportional to the laminar flame speed but multiplied by the square root of the turbulence Reynolds number Re. ... 

The term numerical diffusion describes the effect of artificial diffusive fluxes which are induced by discretization errors. This effect becomes visible when the transport of quantities with small diffusivities [with the exact meaning of small yet to be specified in Eq. (42)] is considered. In macroscopic systems such small diffusivities are rarely found, at least when being looked at from a phenomenological point of view. The reason for the reduced importance of numerical diffusion in many macroscopic systems lies in the turbulent nature of most macro flows. The turbulent velocity fluctuations induce an effective diffusivity of comparatively large magnitude which includes transport effects due to turbulent eddies [1]. The effective diffusivity often dominates the numerical diffusivity. In contrast, micro flows are often laminar, and especially for liquid flows numerical diffusion can become the major effect limiting the accuracy of the model predictions. 

Note that the correction terms are proportional to fT and result from turbulent velocity fluctuations (represented by a gradient-diffusion model). For the multi-environment model the composition vector is defined by... 

In order to understand the physical basis for turbulent-diffusivity-based models for the scalar flux, we first consider a homogeneous turbulent flow with zero mean velocity gradient18 and a uniform mean scalar gradient (Taylor 1921). In this flow, velocity fluctuations of characteristic size... 

Coppalle, A., and D. Joyeux. 1994. Temperature and soot volume fraction in turbulent diffusion flames Measurements of mean and fluctuating values. Combustion Flame 96 275-85. 

Fluctuations due to different flow -y velocities and due to molecular / and turbulent diffusion /... 

A temporal mean is the mean of a fluctuating quantity over a time period, T. If the time period is sufficiently long, the temporal mean values are constant over time. Temporal means are often used in analyzing turbulent diffusion. For example, if u is the x-component of velocity and is a function of space and time, u = u x, y, z, t), in cartesian coordinates. Then the temporal mean velocity, u, would only be a function of X, y, and z ... 

Therefore, let us consider the following thought process if the end result of turbulence, when visualized from sufficient distance, looks like diffusion with seemingly random fluctuations, then we should be able to identify the terms causing these fluctuations in equation (5.18). Once we have identified them, we will relate them to a turbulent diffusion coefficient that describes the diffusion caused by turbulent eddies. Looking over the terms in equation (5.18) from left to right, we see an unsteady term, three mean convective terms, the three unknown terms, the diffusive terms, and the source/sink rate terms. It is not hard to figure out which terms should be used to describe our turbulent diffusion. The unknown terms are the only possibility. 

Consequently, the choice of the averaging time s determines which eddies appear in the mean advective transport term and which ones appear in the fluctuating part (and thus are interpreted as turbulence). The scale dependence of turbulent diffusivity is relevant mainly in the case of horizontal diffusion where eddies come in very different sizes, basically from the millimeter scale to the size of the ring structures related to ocean currents like the Gulf Stream, which exceed the hundred-kilometer scale. Horizontal diffusion will be further discussed in Section 22.3 here we first discuss vertical diffusivity where the scale problem is less relevant. 

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... 

Two basic trajectory models, i.e., the deterministic trajectory model [Crowe etal., 1977] and the stochastic trajectory model [Crowe, 1991], are introduced in this section. The deterministic approach, which neglects the turbulent fluctuation of particles, specifically, the turbulent diffusion of the mass, momentum, and energy of particles, is considered the most... 

Interactions of the particles with the fluctuating component of the gas velocity, which lead to particle turbulent diffusion and induce an exchange of the kinetic energy between the fluctuating components of the velocity of the two phases, which results in either damping the fluctuations of the gas velocity and enhancing fluctuations in particle velocity, or vice versa... 

This relationship between the degree of diffusion by the turbulent velocity fluctuation and time is identical to the traditional general knowledge. 

Distributions like those in Figure 10.4, for example, indicate that Yp or T differs from Yp(Z) or T(Z), respectively. If mixing were complete in the sense that all probability-density functions were delta functions and fluctuations vanished, then differences like T — T Z) would be zero. That this situation is not achieved in turbulent diffusion flames has been described qualitatively by the term unmixedness [7]. Although different quantitative definitions of unmixedness have been employed by different authors, in one way or another they all are measures of quantities such as Yp — Yp(Z) or T — T(Z). The unmixedness is readily calculable from P(Z), given any specific definition (see Bilger s contribution to [27]). 

At first sight, this simple model appears to have the capability of accounting only for axial mixing effects. It will be shown, however, that this approach can compensate not only for problems caused by axial mixing, but also for those caused by radial mixing- and other nonflat velocity profiles These fluctuations in concentration can result from different flow velocities and pathways and from moleeular and turbulent diffusion. 

As a result of time averaging, several new terms appear in the GDE. The fourth term on the left-hand side, the tiiictuating growth term, depends on the correlation between the fluctuating size distribution function n and the local concentrations of the gaseous species converted to aerosol. It result.s in a tendency for spread to occur in the particle size range—-a turbulent diffusion through v space (Levin and Sedunov, 1968). 

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... 

In cool skin conditions, an eddy bringing sub-surface water to the surface will create a warm anomaly on the sea surface. Temperature fluctuations on the sea surface will be highly sensitive to the character of surface renewal. The turbulence-diffusion paradigm implies that the vertical scale of eddies diminishes asymptotically to molecular scales close to the surface,... 

If V becomes too large, then the ratio of inertial to viscous forces (characterized by a Reynolds number. Re = pva/fi, where // is a representative average value of the coefficient of viscosity) becomes large enough to cause the flow to be turbulent. Turbulence occurs in most practical burners and introduces qualitative differences from the present predictions the flame fluctuates rapidly and on the average is thicker, with hja approximately independent of i or a [12]. This flame-height behavior may be obtained from the present analysis by replacing D by a turbulent diffusivity that is assumed to be proportional to the product va. However, there are many properties of turbulent diffusion flames that cannot be predicted well by a simple approach of this kind (see Section 10.2). 

In case of turbulent diffusion, the situation is somewhat different. Motion of particles under action of turbulent pulsations is not connected to thermal fluctuations. Therefore B = const and the factor of turbulent diffusion is inversely proportional to the second power of factor of hydrodynamic resistance. 

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