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into fluctuations. Moreover, they contribute significantly to the isotropic rearrangement of anisotropic turbulence.
The unstable slip plane of Fig. 6a is excellent for visualizing the effects of these terms. Previously, we observed that on either side of the slip plane a slight increase in velocity over the mean for that side is accompanied by a slight decrease in the pressure, whereas exactly the opposite occurs in the other half of the fluid. Thus, fluctuations in pressure and velocity are strongly correlated. Because u' and Op'/dy have opposite signs (for example, an increase in u in the lower half results in a downward or negative pressure gradient), the ensemble average of the product of these variables is always negative. But the two ux др terms in Eq. 15 are negative, so these terms are a positive source to Rxy (that is, to the anisotropic, or off- diagonal, components of the Reynolds stress tensor).
Once Rxy is created, it ultimately contributes to the turbulence kinetic energy K, which, as we noted earlier, is proportional to R1; (that is, to the sum of diagonal components of the stress tensor). That Rxy contributes to K is easily illustrated by examining the contracted form of the Reynolds stress transport equation for, say, the type of flow illustrated in Fig. 5. In this case, the mean-flow source terms contribute to the rate of change only as follows:
Diffusion and Decay. Of the last two terms in Eq. 15, the first is usually negligible and represents diffusion of turbulence by molecular viscosity, which requires no further modeling. The second involves the tensor Dij, for which the usual procedure has been to derive a horrendously complicated transport equation and attempt to solve this simultaneously with the Reynolds transport equation. Such a procedure introduces a host of additional correlation terms to be modeled, and much appeal to "intuition" is invoked in the process.
Bypassing the fascinating but tedious discussion of these derivations, we can nevertheless describe several interesting properties of this second term. First, its contraction
is positive definite so that the -4vmDij term in Eq. 16 always describes a decay of the turbulence energy.
The second interesting property, deduced by extensive manipulations of the Dij transport equation, is that Dj should vary inversely as the molecular viscosity under almost all circumstances. Therefore VmDij is essentially independent of viscosity, which seems paradoxical. Resolution of this paradox hinges on an important property of turbulence: most of the turbulence effects and energy are associated with the largest of
the eddies, which decay first by cascading to smaller eddies before converting to thermal energy (Fig. 9). Thus, an alternative to the usual modeling of the behavior of Dj has recently emerged. We can get the same results by treating the decay of the large-scale eddies as the energy source of the small-scale eddies. For this purpose the largescale eddies are momentarily thought of as being "mean flow." In some complicating circumstances, such as interpenetration of particles, this alternative modeling technique has proven so far to be the only tractable approach.
in which 1, the turbulence viscosity, is a measure of the increase in viscosity due to turbulence (see "Reynolds Number" and "Reynolds Number Revisited"), and dij is the Kronecker delta function (8;; = 0 if i j, dij = 1 if i = j, and Sii = 3). This approximation is consistent with the definition of turbulence kinetic energy in terms of the Reynolds stress: R¡¡ = 2K. Furthermore, the approximation bears a strong resemblance to the Stokes formulation for laminar-flow stresses pij, in which the stresses are related to molecular viscosity and fluid pressure (rather than turbulence viscosity and kinetic energy):
The chief advantage of using Boussinesq's approximation is that transport relationships for all individual components of R1; are replaced by a single expression involving an effective turbulence, or eddy, viscosity.
How does one describe ? The simplest imaginable description of the turbulence viscosity is that it is a constant that depends on some average mean-flow parameters. Somewhat better is a formulation that relies on a mixing length 1, which is usually an algebraic estimate of the size of the main energy-containing eddies as a function of flow geometry. For example, one approach that has proven quite successful for boundary-layer flow and some other well-defined jet flows is to define v, by modifying Prandtl's mixing-length theory so that
Reynolds Number Revisited
s discussed in the earlier sidebar, the Reynolds number is a convenient and physically sound basis for comparing similar flows under different circumstances. For instance, as flow speed through a pipe increases, the drag on the fluid increases and, consequently, so also does the required applied pressure; these increases are reflected in a corresponding increase in Reynolds number. Total friction experienced by the fluid undergoing laminar flow is usually expressed in terms of the Reynolds number R, allowing easy comparison between widely varying tests.
Once the Reynolds number reaches a critical value, however, laminar flow in the pipe becomes turbulent, and further increases in Reynolds number no longer reflect significant changes in measured drag. At this point, the effective turbulence Reynolds number Reff becomes a more appropriate gauge, reflecting the ratio of inertial to turbulence momentumdissipation effects (rather than inertial to viscous-dissipation effects).
Although the Reynolds number can, in theory, be increased without bound, the turbulence Reynolds number cannot. The value of Reff is not directly and uniquely set by readily measured properties and flow geometry but rather depends on eddy generation and the resulting eddy sizes within the flow field. A limiting value of Reff is observed in turbulent-flow experiments.
To demonstrate this behavior quantita
that is, Reff is proportional to the ratio of the length scales. As turbulence gains in intensity, its average length scale usually decreases slightly, but not without limit. In fact, the largest eddies, those that contain the major fraction of the turbulence kinetic energy, will be some portion of the mean-flow length scale (such as pipe diameter). Therefore, since C, is usually about 10 and an upper bound on L/s is typically 20, Reff will seldom exceed several hundred, even in the most intensely turbulent flows. On the other hand, R can be several million or more. ■
For more complex flows the simplifications introduced through Eq. 26 are not justified, and a transport model that details evolution of 1, or quantities related to it, is needed. Although this approach is viable only at the expense of much added complexity, it has recently been favored by investigators working with the complicated flow patterns of high-speed jets, shock-boundary-layer interactions, and two-phase flows.
To produce a simplified transport description of turbulence, we rely on flow properties already introduced to develop a dimensionally correct form of 1. For example,
where C, is a model constant, hopefully universal in its applicability, and € = 2mDii. The last parameter, e, is related to the mean rate of dissipation of turbulence kinetic energy and, as discussed in the earlier section entitled "Diffusion and Decay," is independent of molecular viscosity.
From this definition of turbulence viscosity, we can generate transport equations for € and K. For instance, after performing the tensor contraction of the Reynolds transport equation (Eq. 15) and introducing appropriate closure expressions, we obtain the following simplified transport equation for K:
Next, a treatment similar to that used earlier to split the flow variables into mean and fluctuating parts is applied to the Navier-Stokes momentum equation to create a transport equation for Dij. Again, after contraction and closure modeling, we get the following transport equation for €:
To show how we apply this model, we return to the problem of turbulence in a slip plane (Fig. 6). Our goal is to demonstrate numerically that turbulence is indeed. generated by such a configuration and that we can follow its development throughout a two-dimensional flow field as a function of time and position. We use three different methods of accounting for the turbulence and discuss the pros and cons of each.
The first method involves direct solution of the Navier-Stokes equations by a finitedifference method as an approximation to the left path in Fig. 11. Our calculations use a two-dimensional velocity field, and turbulence below the scale of the computational grid is thus ignored. We assume that a slight, sinusoidal vertical velocity is imparted to the interface separating the oppositely flowing fluids. The maximum speed of this perturbation is only 1 per cent of the mean translational speed (and thus the kinetic energy associated with the perturbation is, at most, 10-4 times the mean-flow kinetic energy).
Fig. 14. The evolution in time (from top to bottom) of the turbulence in the slip-plane problem of Fig. 6, as determined by three different types of simplified calculations. In (a), a large-scale sinusoidal perturbation in the vertical direction is calculated with full equations without modeling the unresolved turbulence. The marker-particle plot (corresponding to mean-flow streaklines) in the third panel shows that a slip-plane instability is a strong source of perturbation in the velocity field. In (b), the perturbational energy of (a) has been increased 10 per cent with the addition of small-scale fluctuations. These fluctuations are accounted for with a turbulence kinetic energy K and its transport equation, and the panels show contour plots of K. This more realistic approach reveals a faster growth in the turbulence. Finally, in (c), all the perturbational energy (both small- and large-scale motion in the vertical direction) is accounted for as turbulence kinetic energy. From this perspective, mean flow can only be horizontal and thus varies in only one (vertical) direction. The contour plots of turbulence kinetic energy show the same growth rate as in (b) for mixing between the layers of undisturbed flow.
The results of our calculations are shown in the left column of Fig. 14 as markerparticle plots in which the lines correspond to mean-flow streaklines (representing what you would see if you had introduced a stream of smoke). As time progresses (from top to bottom in the figure), we see that the width of the mixing layer spreads and displays wave-like structures characteristic of the Kelvin-Helmholtz instability. Thus, our calculations show that a slip-layer instability is indeed a strong source of turbulent mixing.
A more interesting and realistic approach incorporates simplified transport of turbulence in the calculations. Consider the same flow, only with additional smallscale sinusoidal perturbations superimposed on the initial large-scale perturbation. If we were to use the first method and treat these minute fluctuations as part of the resolved flow, we would need a much finer computational grid to resolve the details of the velocity field. Rather than do this, we account for the microscopic perturbations through a turbulence kinetic energy K and its corresponding transport equation, then plot the results of our calculations as contour plots of K. This model is more realistic because the kinetic-energy variable incorporates all length scales of turbulence, as well