NUMERICAL MODELING OF TURBULENCE
In volume two of the Annual Review of Fluid Mechanics (1970), Professor Howard Emmons reviewed the possibilities for numerical modeling of fluid dynamics and concluded that, ''The problem of turbulent flows is still the big holdout. This straight-forward calculation of turbulent flows—necessarily three-dimensional and non-steady—requires a number of numerical operations too great for the foreseeable future.'' However, within a year of the article's publication, the field of direct numerical simulation of turbulence began with the achievement of wind tunnel flow simulations at moderate Reynolds numbers. In the past 20 years the field has developed remarkably.
The importance of computational turbulence study is its position at the crossroads of advanced technological applications and cutting-edge science. It is an enabling technology in applications ranging from engine design, weather forecasting, and the dynamics of the universe. (Nearly all terrestrial, atmospheric, and marine fluid dynamical processes involve turbulence). Owing to the current knowledge limitations about turbulence problems, research progress is limited for the entire range of flow problems, from many thousands of kilometers to centimeters or smaller scales.
In all fluids (including air, water, and solar gases) the most important physical consequence of turbulence is its enhancement of transport momentum, energy, and particles. A related turbulence feature, which is indeed the fundamental characteristic that makes it so theoretically and computationally difficult, is that it exhibits far more small-scale structure than its nonturbulent counterparts. This small-scale structure is responsible for the enhanced turbulent transport phenomena and is itself evidence of enhanced transport in the sense that small scales develop from the degradation of large-scale excitations that are maintained by energy transport from one scale to another. In the cascade from large to small scales, the nonlinearity of the underlying dynamical equations, the Navier-Stokes equations, plays a pivotal role in mediating interactions among these scales.
One way to measure the effective nonlinearity of the Navier-Stokes equations is by a nondimensional quantity R, called the Reynolds number, defined as R = UL/v, where U is
a typical velocity change across a typical large length scale L and v is the kinematic viscosity of the fluid. It can easily be shown that the Reynolds number is a typical measure of the ratio of the nonlinear terms in the dynamical equations to the linear (dissipative) term. For typical flows, R is usually quite large. For example, since v ≈ 0.01 cm2/s for water, even U = 1 m/s (2 mph) and L = 1 m gives R = 106.
One of the main results of turbulence theory is that the range of significantly excited scales of motion in both space and time is of the order R3/4 Therefore, to calculate a high Reynolds number flow in three space dimensions plus time, it is necessary to perform order (R3/4)4 = R3 computational work. This means that calculating a flow at Reynolds number 2R requires roughly 10 times more work than calculating at Reynolds number R!
Another important characteristic of turbulent flows is their apparent randomness and instability in the face of small perturbations, a feature readily noticeable in these complex flows. Two turbulent flows that are at some time nearly identical in detail do not remain so on the time scales of dynamical interest. Instability of turbulent motion is related to the limited predictability of, say, atmospheric motions. The onset character of this randomness (i.e., the transition to turbulence) is a subject of much current interest and involves the study of such interesting dynamical phenomena as those of "strange attractors."
Over the past two decades it has become clear that substantial progress in understanding turbulent flows will require the largest and most powerful computer resources available. Several kinds of computer studies are important.
Full numerical solution of the Navier-Stokes equations for turbulent flows to answer fundamental fluid dynamical questions.
Numerical tests of theories of turbulence.
Numerical tests of turbulent transport approximations for use in large-scale computer models of engineering systems, the ocean, atmosphere, etc. (Here a transport approximation means that only the largest scales of motion are calculated dynamically; all smaller scales are modeled, usually in terms of eddy transport coefficients, like eddy viscosity.)
Numerical studies of turbulence dynamics using large-eddy simulations. (In a large-eddy simulation all fluid motions on scales larger than the grid scale in a numerical simulation are calculated in detail, while motions on scales smaller than the grid scale are modeled, often by a turbulence transport approximation.)
Studies of the origin of turbulence, including investigations of possible routes leading to chaos (apparent random behavior).
RECENT HISTORY OF TURBULENCE COMPUTATIONS
The first numerical simulations of two-dimensional turbulence were performed on a CDC 6600 computer by Lilly in 1969; the first numerical simulations of three-dimensional turbulence were performed by Orszag and Patterson in 1971 using the CDC 6600 and 7600; the first large-eddy simulation of shear and convective turbulence was performed by Deardorff in 1970; and the first large-eddy simulation of turbulence in a stratified shear flow was performed by Deardorff in 1972. Since then the field has developed so rapidly that for many flows with simple geometry and a moderate Reynolds number, data can now be gathered more cost effectively, more quickly, and more comprehensively by computation than by laboratory experimentation. This feature is especially true in the field of transition to turbulence, where computation has proven to be an exceptionally useful tool. It is expected that this trend will continue and may, indeed, accelerate over the next decade, especially with the advent of new cost-effective parallel computers.
In the following paragraphs, we review the progress on certain specific key problems in turbulence—theory and application—and note the current unsolved issues that numerical simulations have raised. We start with a brief statement on the state of the art circa 1975, we shall point out how, for these particular problems, the next generation of supercomputers, with the reservations and uncertainties noted above, will yield new insights and results. We will then be in a better position to discuss the kinds of development to expect in the 1990s.
In 1975, before the introduction of the Cray-1, state-of-the-art turbulence simulations involved full numerical solutions of the Navier-Stokes equations with up to 32 × 32 × 32 degrees of freedom in three space dimensions and 128 × 128 degrees of freedom in two dimensions. Low Reynolds number inertial-range (small-scale) dynamics was already studied in two dimensions; however, the study of three-dimensional inertial-range dynamics seemed well beyond the power of existing computers. Numerical studies of thermal convection in two dimensions had been done, as well as some isolated, low-resolution, three-dimensional convection studies. However, there were no systematic studies of the origin of chaotic time
dependence in flows, nor was it clear then that numerical methods would prove definitive in understanding the transition to turbulence. Large-eddy simulations were well on their way, given the pioneering work of Deardorff, but there were no attempts at that time to use the computer to understand the basic fluid dynamics of wall layers in turbulent shear flows. Thus, in 1975, turbulence computations that accounted for all relevant scales of motions were only in their infancy, and scientists were just beginning to perceive the usefulness of such methods to provide answers to fundamental fluid dynamics questions.
With the availability of Cray computers in the late 1970s, studies could be done with computer codes using several orders of magnitude and more resolution than was possible with the CDC 7600. Generally, the CRAY-1 brought a range of three-dimensional turbulence problems within the realm of numerical scrutiny, and it permitted the rudiments of inertial ranges to be defined for simple geometries. For two-dimensional turbulence, considerable progress was made in understanding detailed inertial range effects.
Let us now review some direct numerical simulation studies that have yielded new insights. As yet, most of these problems are not completely solved; they need CRAY 3/MP-level, or higher, computing capabilities before high Reynolds number asymptotics can be ascertained. The survey given here is very limited, but it does highlight some of the range of problems involved.
TAYLOR-GREEN VORTEX, VORTEX RECONNECTION, AND 3-DIMENSIONAL INERTIAL-RANGE DYNAMICS
There are certain simple initial value problems that have yielded much insight into the way in which the nonlinearities in the Navier-Stokes equations lead to ever smaller scales of motion and the rapidity with which this happens. Two problems come immediately to mind in this connection: the Taylor-Green problem, which is a regular system of three-dimensional vortices, and the evolution of a pair of closely spaced, well-defined vortices. Improved resolution on Cray 2-level computers together with new algorithms has allowed an increase in resolution up to 8003 spatial gridpoints for numerical simulation of the Taylor-Green problem. While the Taylor-Green vortex is a special flow, it appears that the special symmetries it invokes do not inhibit small-scale dynamics in the inertial range. Indeed, in the early 1980s it was for this particular problem that the first direct calculation
of a three-dimensional inertial range was possible. In addition, recent studies have permitted the investigation of intermittency effects of the dissipation-fluctuation spectrum. These computations address very fundamental issues concerning the nature of the effects of nonlinearities in Navier-Stokes dynamics; for example, does frictionless (inviscid) flow stretch vorticity an infinite amount in a finite time? Resolution of the existence of this singularity for the inviscid Euler equations in three-dimensional flow is still undecided, but recent calculations suggest that there may be no singularity. If this is the case, numerical simulations call into question conventional assumptions about the universal nature of high Reynolds number turbulence.
The vortex reconnection problem consists of studying the time development of a pair of vortices that evolve by reconnecting, so that their flow topology changes. This change is induced by the action of viscosity that allows vortex lines to be broken and reconnect. The process has the potential not only to build an understanding of boundary phenomena but to be a mechanism in inertial-range dynamics. So far, resolution 2563 calculations have demonstrated cleanly the dynamics of such processes and have displayed the subtleties of dissipation (and acoustic noise generation) during the collision process.
TWO-DIMENSIONAL AND QUASI-GEOSTROPHIC FLOWS
Two-dimensional flows are the prototype of the large scales of the atmosphere in which the rotational constraint is of primary consideration. In the past decade, calculations using improved resolution (up to 10242) have profoundly affected our understanding of these flows; for example, an initial state of random vortices tends, in time, to become a system of isolated vortices whose dynamics are controlled by their distant interactions with an occasional collisional interaction. Their scale-size distribution is set by their initial distribution. The classical picture of a universal scale-size distribution E(k) ~ k-3 has very limited validity at intermediate times. These ideas are being pressed toward the study of quasi-geostrophic flows (fully three-dimensional but still rapidly rotating), and it will be most important to see how much of the strictly two-dimensional picture carries over. Undoubtedly, the next computer generation will do for quasi-geostrophic flows (with various additional effects, such as topography) what the last generation did for strictly two-dimensional turbulence. With respect to meteorology, these
studies assume even more relevance than the two-dimensional studies mentioned above because of the improved realism.
Early studies of turbulent magnetohydrodynamics (MHD) at 643 resolution have been extended to 1283 in the past few years. For the two-dimensional case a considerable inertial range is computable, but for three dimensions the inertial-range effects are unresolved. Currently, calculations have been able to explore issues of anisotropic flows and, in particular, have revealed unsuspected large-scale instabilities of these flows, which indicated that helical flows (flows with spiral structure) are able to organize—under certain conditions—a significant amount of their energy into large-scale structures. Very large computers are necessary here to extend and improve these studies in order to provide the large-scale separation implied by theoretical analysis. Another effect brought under numerical scrutiny is the problem of magnetic field line reconnection (in two dimensions), which is a dissipative process, at extremely small scales. It is analogous to the vortex-reconnection problem but considerably more complicated. New directions in MHD are study of the effects of compressibility and study of three-dimensional field line reconnections. These computations have applications in fusion research, as well as heating of the corona.
Large-eddy simulations (LESs) are of great practical importance. They are calculations in which the effects of scales smaller than the grid scale on those retained in the calculation are statistically modeled. In reality this method is the only hope we have to detail flow models of real-world engineering complexity, and it has wide applications ranging from calculating detailed flows over aircraft wings to global models of the planetary boundary layer. The current state of the art here is 1283 calculations in relatively simple geometries. In the atmospheric sciences, applications include simulation of the full planetary boundary layer, including radiation and condensation. Stratus-topped planetary boundary layers may now be studied via LES simulations, and their input into global climate models will be of great significance in global warming studies. For this latter step, new-generation computers are required. Equally important is the extension of LESs
to boundaries via new algorithms derived from the renormalization group method. Such procedures have the potential to eliminate much uncertainty in the small-scale modeling required for LESs. On the next generation of computers, LESs should be applied to study flows in complex geometries and with complex physics, such as combustion and multiphase flow effects. Other extensions of LESs include such problems as cloud-top entrainment instability mechanisms. Here, fine resolution is essential and, at present, unavailable.
Numerical simulation of thermal convection (and stably stratified flows) has proceeded to resolution sufficiently high (1923) that fully turbulent, high aspect ratio convection may be simulated (at least at moderate Prandtl numbers). Of interest here is not only verification (or repudiation) of traditional scaling laws but also the statistics (distribution function level) of various flow quantities such as vorticity and temperature differences. Here recent theoretical/numerical developments by Sinai and Yakhot are particularly significant. We should note that recent high-precision experiments have shown a sequence of transitions toward progressively "stronger" turbulence through which the flow proceeds as the basic buoyancy driving force is increased. Recent numerical simulations have been able to track these transitions. With increased computing power, it should be possible to explore the asymptotics of this important class of flows.
In summary, John von Neumann's 1949 prediction that computers would prove particularly useful to the study of turbulent flows has come true. Because of still limited theoretical understanding of nonlinear phenomena, engineers, atmospheric scientists, astrophysicists, fluid dynamicists, and others are in great need of computer simulations. To continue recent advances in many aspects of turbulence study, the power and memory of new supercomputers and massively parallel computers are required. In basic research the need for such capabilities will continue until we can achieve resolutions in which asymptotic regimes are manifest, probably not less than 10243 resolution and often much more, depending on the problem.