Reynolds stresses derivation

Where the Reynolds stress formula (2) and the universal law of the theory of isotropic turbulence apply to the turbulent velocity fluctuations (4), the relationship (20) for the description of the maximum energy dissipation can be derived from the correlation of the particle diameter (see Fig. 9). It includes the geometrical function F and thus provides a detailed description of the stirrer geometry in the investigated range of impeller and reactor geometry 0.225derived from many turbulence measurements, correlation (9). 

Of course, the role of the artificially introduced stochastics for mimicking the effect of all eddies in a RANS-based particle tracking is much more pronounced than that for mimicking the effect of just the SGS eddies in a LES-based tracking procedure. In addition, the random variations may suffer from lacking the spatial or temporal correlations the turbulent fluctuations exhibit in real life. In RANS-based simulations, these correlations are not contained in the steady spatial distributions of k and e and (if applicable) the Reynolds stresses from which a typical turbulent time scale such as k/s may be derived. One may try and cure the problem of missing the temporal coherence in the velocity fluctuations by picking a new random value for the fluid s velocity only after a certain period of time has lapsed. 

In summary, the mean velocity field (U) could be found by solving (2.93) and (2.98) if a closure were available for the Reynolds stresses. Thus, we next derive the transport equation for lutu ) starting from the momentum equation. 

Like the Reynolds stresses, the scalar flux obeys a transport equation that can be derived from the Navier-Stokes and scalar transport equations. We will first derive the transport equation for the scalar flux of an inert scalar from (2.99), p. 48, and the governing equation for inert-scalar fluctuations. The latter is found by subtracting (3.89) from (1.28) (p. 16), and is given by... 

The derivation of the scalar-flux transport equation proceeds in exactly the same manner as with the Reynolds stresses. We first multiply (2.99), p. 48, by

[Pg.101]

As discussed in Chapter 5, the complexity of the chemical source term restricts the applicability of closures based on second- and higher-order moments of the scalars. Nevertheless, it is instructive to derive the scalar covariance equation for two scalars molecular-diffusion coefficients ra and I, respectively. Starting from (1.28), p. 16, the transport equation for ((,) can be found following the same steps that were used for the Reynolds stresses. This process yields34... 

This type of model is usually referred to as an algebraic scalar-flux model. Similarmodels for the Reynolds-stress tensor are referred to as algebraic second-moment (ASM) closures. They can be derived from the scalar-flux transport equation by ignoring time-dependent and spatial-transport terms. 

Like the Reynolds stresses, the scalar flux obeys a transport equation that was derived in Section 3.3 ... 

In transported PDF methods (Pope 2000), the closure model for A, V, ip) will be a known function26 ofV. Thus, (U,Aj) will be closed and will depend on the moments of U and their spatial derivatives.27 Moreover, Reynolds-stress models derived from the PDF transport equation are guaranteed to be realizable (Pope 1994b), and the corresponding consistent scalar flux model can easily be found. We shall return to this subject after looking at typical conditional acceleration and conditional diffusion models. 

The closed PDF transport equation given above can be employed to derive a transport equation for the Reynolds stresses. The velocity-pressure gradient and the dissipation terms in the corresponding Reynolds-stress model result from... 

Since the mean velocity and Reynolds-stress fields are known given the joint velocity PDF /u(V x, t), the right-hand side of this expression is closed. Thus, in theory, a standard Poisson solver could be employed to find (p)(x, t). However, in practice, (U)(x, t) and (u,Uj)(x, t) must be estimated from a finite-sample Lagrangian particle simulation (Pope 2000), and therefore are subject to considerable statistical noise. The spatial derivatives on the right-hand side of (6.61) are consequently even noisier, and therefore are of no practical use when solving for the mean pressure field. The development of numerical methods to overcome this difficulty has been one of the key areas of research in the development of stand-alone transported PDF codes.38... 

Such a generic derivation was first effected by Landau and Lifshitz (1959) in the absence of any complicating factors. Included in these complicating factors are inertia, which necessitates the introduction of Reynolds stresses, as well as interfacial tension, present when the suspension is composed of droplets rather than rigid particles. Batchelor s (1970) analysis incorporates such factors. 

In the most systematic application of this approach, Harlow and co-workers at Los Alamos have derived a transport equation for the full Reynolds stress tensor pu u j. They have coupled this equation with a scalar dissipation transport equation and have utilized with various semi-empirical approximations to evaluate the numerous unknown velocity, velocity-pressure, and velocity-temperature correlations which appear in the formulation. While this treatment is fairly vigorous, extensive compu-... 

Chou [23] was the first to derive and publish the generalized transport equation for the Reynolds stresses. The exact transport equation for the Reynolds stresses was established by use of the momentum equation, the continuity equation and a moderate amount of algebra. 

Normally, as discussed earlier, in reactor modeling we are interested in the energy associated with the velocity fluctuations v only. The appropriate turbulent kinetic energy balance equation, the k equation, has therefore been derived via an equation for the Reynolds stress tensor. 

The general transport equation for the specific turbulent fluxes of scalar variables is derived in analogy to the corresponding momentum flux equations, i.e., the Reynolds stress equations. The derivation combines two equations for the fluctuations to produce a flux equation. For the first equation we start with the momentum fluctuation equation (1.389), multiply it by the scalar quantity perturbation ip, and Reynolds average ... 

The situation is analogous to momentum flux, where the relative Importance of turbulent shear to viscous shear follows the same general pattern. Under certain ideal conditions, the correspondence between heat flow and momentum flow is exact, and at any specific value of rjr the ratio of heat transfer by conduction to that by turbulence equals the ratio of momentum flux by viscous forces to that by Reynolds stresses. In the general case, however, the correspondence is only approximate and may be greatly in error. The study of the relationship between heat and momentum flux for the entire spectrum of fluids leads to the so-called analogy theory, and the equations so derived are called analogy equations. A detailed treatment of the theory is beyond the scope of this book, but some of the more elementary relationships are considered. 

Equations (16.42) and (16.46) have, as dependent variables, ut,p, 0, u u -, and u -Q. We have thus 14 dependent variables (note that njwj = m-mJ, so there are six Reynolds stresses plus three u i) variables, three velocities, pressure, and temperature), but we have only five equations from (16.46). We need eight additional equations to close the system. We could attempt to write conservation equations for the new dependent variables. For example, we can derive such an equation for the variables w-wj by first subtracting (16.43) from the second equation in (16.39), leaving an equation for u r We then multiply by u - and average all terms. Although we can arrive at an equation for m m this way, we unfortunately have at the same time generated still more dependent variables This problem,... 

Let us assume a steady turbulent shear flow in which u = u i (xj) and 2 = 3 = 0. We first consider turbulent momentum transport, that is, the Reynolds stresses. The mean flux of xi momentum in the X2 direction due to turbulence is pu. Let us see if we can derive an estimate for this flux. 

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