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as the three-dimensionality.

For our simplified transport calculation, we assume the same large-scale perturbation (with an energy 10-4 times the mean-flow kinetic energy) and then add a further 10 per cent (or 10-5 times the mean-flow energy) in small-scale energy. Our results (the second column of Fig. 14) show that accounting for small-scale perturbational energy causes the growth rate of the turbulence to be much larger than in the first method.

We should also note that even though turbulence is inherently three-dimensional, both the first and second methods deal with mean flow in two dimensions. In the second case, however, we are able to account for the third dimension in an average sense via turbulence kinetic energy and its transport.

A third approach is to treat all length scales as turbulence-even the large-scale perturbation, which, so far, has been treated as part of the mean flow. We can do this because the exact definition of turbulence is a relative one that depends on the observer's point of view. If we adopt this scheme, the flow becomes one-dimensional, that is, only vertical changes occur in ū, K, and €.

In our calculations with this method, we assume the same total initial perturbational energy as in the second example, but with the large- and small-scale energies lumped together. Once again, all length scales of turbulence are incorporated, and our results (the third column of Fig. 14) show that the growth rate of the layer matches that in the second example very closely. On the other hand, turbulence on the largest length scale, which corresponds to mean flow in the earlier examples, is not resolved in detail.

Thus, to effectively use turbulence modeling, one must decide which length scales will be considered mean flow and which will be considered turbulence. Once this has been decided, the power of the method allows us to describe the flow accurately without having to dedicate excessive computer resources to resolving minute flow structures in detail.

Current Research

So far we have concentrated on turbulence in a single incompressible fluid with density perfectly constant in position and time (the downward branches of Fig. 11). Recently, our research has included additional features that are of interest to many of the new scientific and engineering directions at Los Alamos and other laboratories. These features are

• Two-phase flow interactions: the sources, sinks, and effects of turbulence in a fluid containing particles, droplets, or bubbles of another material.

• Density gradients: turbulence in an incompressible fluid for which variations of temperature or the presence of some dissolved substance cause large variations in density.

• Supersonic turbulence: the effects of high-speed processes on turbulence.

In all cases, we continue to use the basic philosophy of transport modeling, which, despite some obvious difficulties, seems at present to be by far the most promising approach for the solution of practical problems.

Two-Phase Flow. Particles, drops, or bubbles suspended in a fluid-whether that fluid is a liquid or a gas can significantly alter the turbulence and its effects. Intuitively,

TWO-PHASE INTERPENETRATION

Fig. 15. Our transport modeling techniques are able to handle both ordered interpenetration of two phases, such as occurs in the laminar-flow transport of blood cells, and disordered interpenetration, such as occurs in a rapidly moving gas that contains suspended particles.

Ordered

Disordered

we expect that when distinct entities interpenetrate a surrounding fluid the creation of turbulence is enhanced; on the other hand, we also expect the inertial properties of heavy entities to dampen turbulent fluctuations. How can we describe these effects quantitatively?

From considerations similar to those for incompressible flow of a single fluid, we know that extra turbulence is generated by pressure gradients producing differences in the accelerations of the particles and of the surrounding fluid. Such differential acceleration induces distortions of the fluid around the particles, thereby creating disturbances in the velocity field that would be absent if there were no particles.

For example, consider the flow field of a shock wave moving horizontally and passing a rigid particle suspended in the fluid. If no particle was present, the flow would remain completely horizontal. However, as the shock wave passes the particle, local velocity fluctuations appear, including changes in the horizontal velocity and the generation of vertical velocity. As soon as there is a velocity difference between the velocity fields of the particle and the fluid, viscous drag forces, competing with differential acceleration, begin to diminish any velocity perturbations.

In a manner analogous to that for single-phase flow, the relative contributions of acceleration and viscous drag can be compared through a particle Reynolds number

where Dp is the particle diameter, up and uf are the local velocities of the particle and fluid, respectively, and m is the molecular kinematic viscosity of the fluid.

Consider a shock moving with a high velocity through a collection of particles that are initially at rest, such that Rp > 1. At first, the effects of differential acceleration dominate and turbulence kinetic energy is created. Then, as viscous drag causes the particles to be swept along with the fluid, the velocity difference and the particle Reynolds number decrease, corresponding to a dampening of turbulent fluctuations. Since the amount of drag depends on the volume fraction of the particles, the turbulence level that is induced will also depend on this parameter.

These effects, however, address only a small fraction of the rich spectrum of dynamic processes that can occur in multifield turbulent flows. In our approach we discard the more conventional procedure of decomposing velocities and volume fractions and, instead, consider momentum and volume fractions as the primary variables to be conserved, decomposing these into their mean and fluctuating components. Such an approach allows us to derive two limiting fluid behaviors: diffusion (in the limit of strong momentum coupling between the particle and fluid fields), and wave-like interpenetration (in the weak-coupling limit). Our model is thus strongly analogous to the interpenetration of two different molecular species: diffusive when the mean free path is short, and wave-like when little or no coupling is present and the species transport as if each were expanding into a vacuum.

In addition, our model handles both ordered and disordered interpenetration of two phases as illustrated in Fig. 15. Other technical accomplishments include the resolution of mathematical ill-posedness of the multiphase flow equations, the emergence of a new closure principle (based on the constraint, with generalized Reynolds-stress expressions,

of exactly neutral stability for the mean-flow equations), and the development of practical modeling equations.

The modeling of turbulent flow with dispersed particles, droplets, or bubbles is of interest to a wide variety of scientific projects at the Laboratory. For example, to model the transport of dust and debris by volcanic eruptions, one must concentrate on the interactions between particulate and hot-gas flows. To improve the design of internal combustion engines, one needs an accurate prediction of both the combustion efficiency and the spatial distribution of heat generation, which, in turn, requires knowing the details of the mixing of fuel droplets and air. Although flow within the body's circulatory system is normally not turbulent, the transport of blood cells can be analyzed by using the equations for ordered two-field interpenetration. Other applications include modeling of the flow within nuclear reactors and the analysis of shock-wave motion in a gas that contains suspended particles.

Density Gradients. The second area we are currently striving to understand with transport modeling is turbulent mixing generated by strong density gradients that are sustained by large variations in thermal or material composition. Coupled with pressure gradients, such density gradients can lead to strongly contorted flow with intense vorticity near the steepest density variations. Again, the proper basis for deriving a generalized Reynolds stress lies in decomposing the momentum rather than the velocity. Among the most important configurations to be studied are those for which adjacent materials initially quiescent and of very different densities-are rapidly accelerated by a strong pressure gradient or heated by a sudden influx of radiation. The ensuing fluid instability (Richtmyer-Meshkov if the shock is going from heavy to light material, Rayleigh-Taylor for the opposite case (Fig. 7)) can act as a strong source for the turbulent mixing of the two materials.

For example, consider an experiment in which a plane shock wave progresses down a closed cylindrical tube divided into two sections by a permeable membrane with air in the first section and helium in the second. As the shock passes from the dense to the less-dense gas, the air-helium interface is accelerated. Later, the interface is repeatedly decelerated by reflections from the rigid wall at the end of the tube. Interface instabilities lead to turbulent mixing of the two gases, and the initially sharp plane separating the gases becomes smeared and indistinct. Our work allows prediction of the average concentration across any strip of fluid taken normal to the nominal streaming direction and calculation of velocity and density profiles within the turbulent mixing zone.

Instabilities driven by density gradients are important to the study of the implosion dynamics of pellets used in inertial confinement fusion (Fig. 16). Radiation from a highpower laser initiates the implosion of an outer spherical capsule, creating a strong shock wave. This shock passes over the interface between the inner surface of the capsule and the enclosed gas, is reflected from the core, and returns to the interface where it induces Rayleigh-Taylor instability. The resultant mixing of gas and capsule in the central region of the pellet can, in many cases, reduce neutron yield.

Another area of interest is the dynamics of fire plumes in the postulated circumstances of "nuclear winter." Extreme heating of the ambient atmosphere produces up to four-fold expansions, resulting in a powerful updraft with intense turbulence.

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Supersonic Turbulence. Mach-number effects often can be ignored, but, in some cases (such as the high-Mach-number mitigation of a Kelvin-Helmholtz instability), such effects are significant. Thus, a third feature of our recent work has been to include the principal phenomena resulting from supersonic flow speeds. These effects arise across shock waves, in the shear layers behind Mach-reflection triple-shock intersections, and in the shear layers behind shock waves normal to a deformable wall.

An unexpected result of our work is the discovery that laminar instability theory (as sketched out in the section entitled "Turbulence Energy: Sources and Sinks") is applicable to the study of supersonic turbulence. Despite the seeming inconsistency, this theory is providing highly relevant guidance to our early modeling efforts.

Concluding Remarks

A pertinent question is: What good is all this? Not only has our discussion illustrated several ways in which turbulence transport theory is heuristic or empirical, but the current large inventory of undetermined “universal" dimensionless parameters in its formulation is disturbing. Moreover, full expression of the theory is long and complicated, involving numerous coupled nonlinear partial differential equations. As a result, a transport calculation requires either costly numerical solutions or questionable approximations, or both.

What are the alternatives? There is no way to resolve turbulence in sufficient detail for numerical calculations based on turbulence transport theory to represent the effects of any but the simplest circumstances. Mixing-length theories and other point-functional approaches are hopelessly limited in their applicability. Fundamental approaches purporting to describe turbulence without empiricism are, in general, also restricted to highly idealized circumstances. Yet we are faced with the task of solving an endless variety of fluid-flow problems, a large fraction of which include significant turbulence effects. We need to supply answers to old questions and guidance for new developments in a meaningful way. At present, there seems to be no better approach to these challenging analytical tasks than that provided by turbulence transport theory. Despite the shadows cast by these comments, the situation is actually far from gloomy. Turbulence transport theory seems to be functioning far better than we have any right to expect. There are at least four reasons for this good performance.

First, complex processes of nature often display a near universality in the collective effects that are of most interest. Just as gas molecules almost always have a nearly Maxwell-Boltzmann velocity distribution, it appears that turbulence tends toward a similar universality in its stochastic structure. The success of the few-variable (or collective, or moment) approach to turbulence modeling relies strongly on the validity of this contention. Although the extent to which universal behavior underlies most of the random processes of nature is currently a matter of intense scientific and philosophical discussion, much evidence supports the ubiquitous nature of this property. Perhaps, eventually, such universalities will help to successfully model such diverse instances as thoughts in a brain, activities of groups of organisms (such as mobs of people), and the dynamics of galaxies.

Next, turbulence transport modeling pays close attention to the binding constraints of real physics: conservation of mass, momentum, and energy, as well as rotational and

translational invariance. Such modeling also accounts for history-dependent variations lacking in many other turbulence theories.

We have also paid great care to physically meaningful closure modeling. Auxiliary derivations (like those of laminar instability analysis) combine with new formulations of mathematical restrictions (like that of precisely neutral mean-flow stability in the presence of generalized Reynolds-stress terms) to constrain our modeling procedures in the most physically meaningful manner possible at each stage of the development.

Finally, investigators throughout the world have made numerous comparisons with experiments, leading to corrections, improvements, and ultimately to considerable confidence in the broad applicability of the results.

Future research will concentrate on several significant aspects of the theory. Closure modeling, of course, continually needs strengthening, especially by first-principle techniques that decrease our reliance on empiricism. The numerical techniques need greater stability, accuracy, and efficiency for a host of larger and more complicated problems.

But the most intriguing challenge is how to incorporate new and different physical processes into our theories. For example, with dispersed-entity flow, we have scarcely begun to understand the effects of a spectrum of entity sizes or the deformation of individual entities (including their fragmentation and coalescence) or the modifications that arise when the entities become close-packed (as they do, for example, during deposition and scouring of river-bed sand). The dispersal of turbulence energy through acoustic or electromagnetic radiation is another interesting topic that needs considerable development. Deriving, testing, and applying the appropriate models will keep many investigators busy for a long time.■

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