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Fluid dynamic turbulence is one of the most challenging computational physics problems because of the extremely wide range of time and space scales involved, the strong nonlinearity of the governing equations, and the many practical and important applications. While most linear fluid instabilities are well understood, the nonlinear interactions among them makes even the relatively simple limit of homogeneous isotropic turbulence difficult to treat physically, mathematically, and computationally. Turbulence is modeled computationally by a two-stage bootstrap process. The first stage, direct numerical simulation, attempts to resolve the relevant physical time and space scales but its application is limited to diffusive flows with a relatively small Reynolds number (Re). Using direct numerical simulation to provide a database, in turn, allows calibration of phenomenological turbulence models for engineering applications. Large eddy simulation incorporates a form of turbulence modeling applicable when the large-scale flows of interest are intrinsically time dependent, thus throwing common statistical models into question. A promising approach to large eddy simulation involves the use of high-resolution monotone computational fluid dynamics algorithms such as flux-corrected transport or the piecewise parabolic method which have intrinsic subgrid turbulence models coupled naturally to the resolved scales in the computed flow. The physical considerations underlying and evidence supporting this monotone integrated large eddy simulation approach are discussed.

We propose to use new orthonormal wavelet packet bases, more efficient than the Fourier basis, to compress two-dimensional turbulent flows. We define the "best basis" of wavelet packets as the one which, for a given enstrophy density, condenses the L² norm into a minimum number of non-negligible wavelet packet coefficients. Coefficients below a threshold are discarded, reducing the number of degrees of freedom. We then compare the predictability of the original flow evolution with several such reductions, varying the number of retained coefficients, either from a Fourier basis, or from the best-basis of wavelet packets. We show that for a compression ratio of 1/2, we still have a deterministic predictability using the wavelet packet best-basis, while it is lost when using the Fourier basis. Likewise, for compression ratios of 1/20 and 1/200 we still have statistical predictability using the wavelet packet best-basis, while it is lost when using the Fourier basis. In fact, the significant wavelet packet coefficients in the best-basis appear to correspond to coherent structures. The weak coefficients correspond to vorticity filaments, which are only passively advected by the coherent structures. In conclusion, the wavelet packet best-basis seems to distinguish the low-dimensional dynamically active part of the flow from the high-dimensional passive components. It gives us some hope of drastically reducing the number of degrees of freedom necessary to the computation of two-dimensional turbulent flows.

Results are described of recent research and model developments for performing large-scale multidimensional compressible reacting-flow computations on the Connection Machine, a very fine-grained, parallel computer capable of multigigaflop performance. We are interested both in having general-purpose computer programs for routine production computations and in evaluating the architecture of the computer for a wide range of computational fluid dynamics and reacting-flow applications. We describe the hurdles involved in rethinking the structure and the algorithms to best suit this kind of computer, provide some relative timings for different programs, and describe ways of dealing with special constraints (such as periodic boundary conditions) imposed by the architecture. Finally, representative results are presented for several reacting and nonreacting computations, including the development and propagation of a spark in a hydrogen-oxygen mixture, an imploding detonation, and the generation of beam-channel turbulence.

Direct numerical simulations were made to examine the local structure of the reaction zone for a moderately fast reaction between unmixed species in decaying, homogeneous turbulence and in a homogeneous turbulent shear flow. Pseudospectral techniques were used in domains of 64³ and higher wavenumbers. A finite-rate, single step reaction between non-premixed reactants was considered, and in one case temperature-dependent Arrhenius kinetics was assumed. Locally intense reaction rates that tend to persist throughout the simulations occur in locations where the reactant concentration gradients are large and are amplified by the local rate of strain. The reaction zones are more organized in the case of a uniform mean shear than in isotropic turbulence, and regions of intense reaction rate appear to be associated with vortex structures such as horseshoe vortices and fingers seen in mixing layers. Concentration gradients tend to align with the direction of the most compressive principal strain rate, more so in the isotropic case.

Physical features of turbulence and vortex-flame interaction in engines are investigated by performing large eddy simulation and direct numerical simulation of compressible flows. A BI-SCALES (Boundary-Inner Smoothly Coupled, Alternating multi-Level Equations System) is proposed, which is a mathematical formulation of the system of governing equations suitable for computations of compressible flows. A numerical method based on the proposed formulation is developed. Using the method, the transition to turbulence in the compression process and the cyclic variations are examined. Then, the vortex-flame interaction is studied, mainly on the relation between the flame structure and the Kalvoritz- and turbulent Reynolds number effect. Finally, the large wrinkle simulation (LWS) of engine combusting flow is performed.

Application of computational fluid mechanics in two areas of metallurgy is considered: solidification of liquid alloys and MHD turbulence. In the first class of problems, where the technical issue is to obtain a reasonably homogeneous composition of the cast, one has to consider not only the completely molten and solidified regions but also the "mushy zone" that is made up of small-scale dendrites, which appear between the first two regions. In the completely molten region, the composition is practically constant and the fluid is set into motion due to the inhomogeneous temperature field. In the mushy zone, on the other hand, solutal convection often dominates strongly over thermal convection. It is shown that laminar convection is of prime importance for the composition of the solidified alloy. In the second class of problems, two cases of turbulent MHD flows in cylindrical containers are considered: an electromagnetic furnace and an electromagnetic stirrer. In the electromagnetic furnace, the mean flow consists of two toroidal vortices. The mean motion in the electromagnetic stirrer is a swirling motion that is accompanied by a weak meridional circulation, which is reminiscent of that occuring in spin down phenomena. The MHD flows are computed by using large eddy simulation methodology with a new subgrid model of the Smagorinsky type that accounts for a variable mesh. Predictions from the all model computations are compared with experimental observations. In general, the agreement between theory and experiments is satisfactory.

The unsteady Navier-Stokes (NS) analysis of Osswald, Ghia and Ghia in velocity-vorticity variables is modified to study the dynamic stall phenomenon for a NACA 0015 airfoil undergoing constant Ω0 pitch-up maneuvers at Reynolds number Re = 10 000 and 45000. The use of third-order accurate biased upwind differencing for the nonlinear convective terms in the vorticity transport equation removes the spurious oscillations observed in the earlier studies by the authors for these values of Re. The fully implicit and vectorized ADI-BGE method of the authors is used to solve the unsteady NS equations. Instantaneous inertial surface vorticity, which is an invariant of the choice of reference frame selected, is employed to determine the location of separation of the boundary-layer flow on the suction surface; also a separation bubble embedded within the boundary layer is observed for both cases somewhere between the leading edge and the quarter-chord point. Primary, secondary, tertiary and quarternary vortices have been observed before the dynamic-stall vortex evolves and gathers its maximum strength.

The objective of this study is to efficiently simulate vortex-dominated highly unsteady flows. In such flows, the locations as well as the extent of the regions requiring fine-mesh resolution vary with time. A technique has been developed to simulate these flows on a temporally adapting grid in which the adaption is based on the evolving flow solution. The flow in an axisymmetric constriction has been selected as an illustrative problem. The multiple and disparate length scales inherent in this complex flow make this problem ideally suited for evaluating the adaptive-grid technique. Adaption is based on the equidistribution of a weight function, through the use of forcing functions. The significance of this is that the method can be implemented into existing flow-analysis systems with minimal changes. The grid-generation equations developed are viewed as grid-transport equations. The time-dependent control functions perform the role of the convective speed in this transport mechanism. The equations provide the efficiency and flow tracking capability of parabolic equations, while maintaining the smoothness of computationally expensive elliptic equations. The efficiency and flow tracking capability of the approach is demonstrated for both steady and unsteady flows.

An efficient finite-difference scheme solving for the three-dimensional incompressible Navier-Stokes equations is described. Numerical simulations of vortex breakdown are then carried out for a sharp-edged delta wing and a round-edged double-delta wing at high Reynolds numbers. Computed results show that several major features of vortex breakdown are qualitatively in agreement with observations made in experiments.

Steady and unsteady asymmetric vortical flows around slender bodies at high angles of attack are solved using the unsteady, compressible, this-layer Navier-Stokes equations. An implicit, upwind-biased, flux-difference splitting, finite-volume scheme is used for the numerical computations. For supersonic flows past point cones, the locally conical flow assumption has been used for efficient computational studies of this phenomenon. Asymmetric flows past a 5° semiapex-angle circular cone at different angles of attack, free-stream Mach numbers, and Reynolds numbers has been studied in responses to different sources of disturbances. The effects of grid fineness and computational domain size have also been investigated. Next, the responses of three-dimensional supersonic asymmetric flow around a 5° circular cone at different angles of attack and Reynolds numbers to short-duration sideslip disturbances are presented. The results show that flow asymmetry becomes stronger as the Reynolds number and angles of attack are increased. The asymmetric solutions show spatial vortex shedding which is qualitatively similar to the temporal vortex shedding of the unsteady locally conical flow. A cylindrical afterbody is also added to the same cone to study the effect of a cylindrical part on the flow asymmetry. One of the cases of flow over a cone-cylinder configuration is validated fairly well by experimental data.

Computational work in continuum hypersonics and, in particular, the difficulties encountered during the development of the methods at the DLR Institute for Theoretical Fluid Mechanics are presented. Finite-difference schemes with implicit-explicit central discretisation and shock fitting and with upwind-TVD formulation and shock capturing are discussed, e.g., for strong shock-shock interactions. 3D flows of perfect gas at large angle of attack are considered, including those with radiation-adiabatic surface boundary conditions. 2D or axisymmetric as well as 3D flows in thermochemical equilibrium are simulated using either the Tannehill curve-fitting approach for air or the minimization procedure for Gibbs' free energy which is more general at the expense of larger cost. Nonequilibrium flow simulations are discussed for axisymmetric nitrogen and air model flows. With the help of an appropriate model equation for finite-rate chemistry it is shown that the usual Harten-Yee approach does not allow to propagate shock waves in a time-accurate fashion if large source terms are present.

Symmetry preserving difference schemes which are found on the basis of the Lie-group theory for partial differential equations applied to their first differential approximations are used to simulate the evolution and reflection of shock waves in a cubical cavity with rigid walls and walls with outflow boundary conditions. The explicit difference scheme was used for a shock capturing technique to find and trace the discontinuities in the flow field. The numerical model is based on the strong conservative form of the three-dimensional time-dependent Euler equations with an equation of state for an ideal gas. The three-dimensional time-dependent shock waves which are created by a centered gas volume under high pressure interact with the plane walls of the cavity and lead to focusing effects after the explosion. To follow the propagation of strong plane or spherical shock waves the shock capturing method was proved to be sufficient concerning the spatial resolution of the discontinuities. When interacting with the rigid walls of the cavity the shock waves are assumed to be reflected specularly and to interact with shock waves and contact discontinuities moving from the center to the edges and corners of the cavity. The symmetry preserving character of the invariant difference scheme under use was proved numerically by calculations over long time. No significant deviation of the symmetry character of the imposed initial distribution was found. Additionally, a numerical analysis of the conservation of energy in the cube was performed to find the behavior of the energy in time.

A mesoscale model, which is suitable to simulate the distribution of meteorological variables with high resolution in space and time in a region of typically 50 km × 50 km is described. The model's applicability in complex terrain is demonstrated by several examples, namely local weather prediction, calculation of complex wind fields and dispersion, and local climate study.

Energy transfer processes in decaying, three-dimensional, isotropic turbulence are investigated using numerical results from local energy transfer (LET) theory. The study covers a wide range of evolved, microscale Reynolds numbers (5 < R_λ < 250). It is found that the energy transfer is mainly local (between scales of similar size), but there are also some signs of nonlocal transfer at higher Reynolds number. The nature of the underlying triad-wavenumber interactions, on the other hand, seems to depend on both the Reynolds number and the wavenumber range of interest. In the energy containing and dissipation ranges, both local (all three scales of the triad interaction are of comparable size) and nonlocal (one scale is much larger than the remaining two) interactions are important, with the latter becoming more dominant as the Reynolds number increases. But our nonlocal interactions tend to be less severe than those observed by Domaradzki and Rogallo. More significantly, in the inertial range of high Reynolds number flows, the LET theory predicts dominance of local and near-local interactions. While this is contrary to the result from eddy damped quasi-normal Markovian theory that the important triad interactions are mainly nonlocal, it is closer to the Kolmogorov picture of turbulence. Another interesting result is that, despite their inherent differences, the LET theory and the full simulation of Ohkitani and Kida predict inertial-range values for the energy transfer locality function in fairly good agreement, not only with each other, but also with the analytical closure theory result for infinite Reynolds number, stationary turbulence by Kraichnan. The calculated values reveal that the contributions to the net energy transfer are predominantly from near-local interactions (scale ratios ≈ 4), which is indicative of cancellation of large numbers of highly nonlocal interactions.