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turbulent fluctuations. However, as the scale decreases, the characteristic length of the eddies decreases, and the velocity gradients in the eddies become steeper and steeper. In other words, dH/dt eventually wins, and, at the smallest of turbulence scales, energy goes directly to heat.

Thus, cascading of turbulence is consistent with nature's universal law dictating that ordered motion must become progressively more disordered until the energy in a flow degrades to heat. The direction and magnitude of energy flow within the cascade guides us in mathematically describing the decay of turbulence, not only into heat from very small-scale eddies but also from large scales to smaller scales. Because the transfer of energy through the cascade is, in some sense, equal at all steps, we can easily describe the energy decay rate in a manner independent of molecular processes. We will describe this approach more extensively when we consider detailed modeling

in the next section.

In an idealized steady-state approximation of turbulence, exactly as much energy enters the fluctuational spectrum of motion at the largest scale as leaves it to become heat at the smallest scale. More accurately, there is some loss of energy to heat at every scale, but the loss at the smallest scale is dominant (Fig. 9). Although these ideas have been exploited to derive interesting properties of the small-scale spectrum of turbulence energy, our principal concern here is with the largest scales. It is these scales that contain most of the energy and thus exert the dominant effects on mean-flow dynamics.

Transport Modeling of Turbulence

There are numerous theoretical approaches to turbulence: some reach to the conceptual heart of the matter, others are directed toward the solution of practical problems, and a few attempt to cover the entire range. Despite its present shortcomings, turbulence transport theory, which fits into the last category, already shows promise of considerable success in both illuminating the fundamental dynamical processes and serving as a vehicle for the solution of practical problems.

TURBULENCE CASCADE

Fig. 8. With each reduction in scale, turbulent motion of the larger scale becomes mean-flow motion of the smaller scale (arrows). Because each reduction in scale has approximately the same change in meanflow velocity occurring over a much smaller distance, velocity gradients become steeper, and a larger fraction of the turbulence energy goes directly into heat.

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Rate of
Change

Smallest Eddy

Even for a single fluid with constant, uniform density the relevant mathematical formulations are lengthy, and there are significant difficulties yet to be resolved. Nevertheless, we can capture in a relatively simple manner much of the flavor of turbulence modeling by starting with the Navier-Stokes fluid-dynamics equations for an incompressible fluid that is, a fluid of constant density and viscosity everywhere and for all time. One of our fundamental assumptions is that these familiar and deceptively simple equations describe everything we need to understand about the turbulence of such a fluid, including every "microscopic" detail in every fluctuating part of the turbulent flow.

The Navier-Stokes equations describe the variations of pressure and velocity in the fluid. Using Cartesian index notation with the summation convention, we can write the first equation, which is an expression of the conservation of mass, as

Jui
Əxi

and the second, which is an expression of the conservation of momentum, as

Ju¡ uj

მx;

Advection

=

=

0,

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Əxi
Driving

Force

+

2ui Vm ex

Diffusion

(9)

(10)

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So, there it is in a nutshell: the entire, mysterious world of turbulent fluid flow described by two short lines of mathematical symbols. Well, not quite the entire real world, because many fluids are not the idealized incompressible materials of constant viscosity and density considered here, but we shall return to that point below.

One obvious approach to modeling turbulence is to solve the Navier-Stokes equations directly (the left path in Fig. 11). However, certain difficulties limit the success of this approach. For example, even when the mean flow is one-dimensional, the equations must be solved numerically in three dimensions because turbulence is inherently three-dimensional. Only recently have computers had enough computational capability to begin meeting the task of solving three-dimensional fluid-flow problems. To describe the full spectrum of eddies, the computational mesh would have to be fine enough for the smallest eddies, yet cover a domain large enough to include the mean flow and the largest eddies. Another complication occurs if the system includes a solid boundary. Because the turbulent flow depends on the minute details of the boundary conditions (even stochastic quantitites depend on minute perturbations in the initial and boundary conditions, such as wall roughness), these details must be specified. Furthermore, because a particular set of minute perturbations describe only one possible representation of the boundary conditions, repeated calculations must be made with various boundary conditions and the results of the calculations averaged to give a complete description of the turbulent flow. The memory and speed requirements for the calculations would

MOMENTUM TRANSPORT

Fig. 10. The diffusion, driving-force, and advection terms of the Navier-Stokes momentum equation represent the ways in which momentum is locally added to or taken away from a region in the fluid.

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APPROACHES TO TURBULENCE MODELING

Fig. 11. Typically, modeling of turbulence makes two simplifying assumptions with respect to the full Navier-Stokes equations: an incompressible fluid (V. u = 0) and the density p constant everywhere. Using just the first of these assumptions, one could, in principle, solve the equations directly (left path)—a difficult task. However, if one uses both simplifying assumptions together with ensemble averaging, the result is two sets of equations: the meanflow equations, which include the Reynolds stress tensor Rij, and the Reynolds stress transport equation. Turbulence transport theory (right) uses input from both sets, whereas point-functional turbulence models (middle) deal only with the mean-flow equations (by postulating that R is a function of mean-flow variables). Later in this article we describe work on multiphase flow in which the assumption of constant density has been dropped. Current research is just beginning to approach the full NavierStokes equations for compressible, multiphase flow.

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