Computational Fluid Dynamics Review 1998

The following sections are included:

• PREFACE

• CONTENTS OF VOLUME I

• CONTENTS OF VOLUME II

After a short interruption due to the defeated war, old Aeronautical Research Institute, Imperial University of Tokyo revived as Aeronautical Research Institute, University of Tokyo at 1953, which has been growing with the aerospace science and technology. In 1964, it renamed as Institute of Space and Aeronautical Science, University of Tokyo to expanding its activity into space. In 1981, it still expands its activity into astronautics under new name of Institute of Space and Astronautical Science. Here is an overview concerning to the history of fluiddynamics and numerical fluiddynamics and their interaction with digital technologies taking place in this Institute, and, in some extent, in Japan and Asia from 1950 to present. It is discussed, how fluid dynamics as a modern science has been introduced and developed, and how digital technology as modern engineering tools has grown up to modern industry and how it has influenced to fluiddynamics in Japan.

The following sections are included:

• INTRODUCTION

• 3D POTENTIAL CASCADE FLOWS

• FUNDAMENTAL EQUATIONS OF COMPRESSIBLE FLOW

• NUMERICAL METHODS FOR COMPRESSIBLE FLOW

• COMPUTATIONAL RESULTS OF COMPRESSIBLE CASCADE FLOWS

• INCOMPRESSIBLE CASCADE FLOWS

• UNSTEADY CASCADE FLOW PROBLEMS

• Bibliography

The paper presents a summary of some of the major activity in the area of Computational Fluid Dynamics within Australia. The summary is divided into various sections - Industrial CFD, DSMC, High Speed Flow, Multi Phase Flow, Application of CFD to study fundamental fluid flow processes and Algorithms. In every section the important research work undertaken is reviewed from the point of view of its aim and achievements.

The appeal of the solution-adaptive Cartesian Grid approach to CFD is the degree to which the grid-generation procedure can be automated. In most grid-generation approaches , the starting point is a discretized surface geometry, and the quality of the volume grid that is ultimately generated is highly dependent on the discretization of the surface geometry. In the Cartesian approach, the quality of the volume grid is much less dependent on the original surface geometry description; in fact, the two are almost entirely decoupled. The Cartesian approach to grid generation introduces difficulties in the development of a solver, however. For proper resolution, adaptive gridding is a virtual necessity, and, even with adaptation, the number of cells necessary for a high-fidelity Cartesian flow solution is higher than that of a solution on a body-fitted grid, whether structured or unstructured. The Cartesian solver must also be able to handle the small cut cells that occur in the Cartesian grid-generation procedure. This paper describes approaches for Cartesian grid generation, and for flow solvers that work well on adaptive Cartesian grids.

An anisotropic Cartesian grid (ACG) method is developed for viscous flow computations. This method is more compatible with the anisotropic feature of flows near the body than the traditional isotropic Cartesian grid (ICG) method and is therefore grid saving. An efficient algorithm for ACG generation is presented and analyzed. A bilinear interpolation is applied to construct an easily implementable second-order accurate solid wall condition. The stability of this solid wall treatment is established using the GKS-stability theory. The ACG method along with the solid wall condition is finally validated by computing viscous flows around airfoils. The ACG method is compared with the ICG method. It is found that the ACG method can significantly save the number of grid points without jeopadizing the accuracy. Such a gain can be expected to be substantially more important in three-dimensional flow computations.

The aim of this survey is to discuss some of the difficulties one can encouner both when solving Navier-Stokes equations for compressible flows by an obstacle and analyzing the approximate solutions. Far to be exhaustive, some main aspects of the numerical simulation are deliberately pointed out, in addition to the way the obstacle is taken into account and to the far field boundary conditions. Then, using one of the robust methods it is possible to simulate the transition to turbulence for increasing Reynolds numbers. That means to compute transient solutions which need to be analyzed and here is the second topic of this paper. Indeed, the classical tools like Fourier analysis are very efficient as long as the solutions is periodic but useless when the solution is more complex. Despite the development of wavelets and new algorithms it seems still difficult to distinguish quasi-periodic and chaotic solutions.

Adaptive grid methods of control function approach in elliptic grid generation developed by the author are described in this article. A use of a modified solution instead of actual solution is shown to be effective for estimation of a weight function related to the control function. In the development of the methods, a use of upwind formulation and a locally-variable relaxation parameter method have been introduced for the elliptic equation, which are described in detail in this article. An orthogonality of the grid and grid adaption cannot be strictly applied simultaneously. The adaptive method of elliptic type produces badly-skewed grid in many cases. To overcome this problem, an overset adaptive-grid method is introduced, which combines the overset grid method with adaptive-grid method. In this article, the developments of the adaptive grid methods are presented with numerical examples.

Methods for generation of unstructured planar, surface, and volume grids using the advancing-front/local-reconnection (AFLR) procedure are presented. AFLR uses an iterative point creation and insertion scheme wherein points are created using advancing-front, -point, or -normal type point placement. Initially, the connectivity for these generated points is obtained by direct subdivision, without regard to connectivity quality. The connectivity is then improved by iteratively using local reconnection subject to a quality criterion. A min-max type (minimize the maximum angle) criterion is used. The overall procedure is applied repetitively until a complete field grid is generated. Field point distribution is controlled by a point distribution function based on the boundary point spacing. This function is propagated through the field by interpolation or specified growth. Procedures for generating edge and surface grids which are fully compatible with the volume grid generation are presented. Results for a variety of configurations are presented. The results demonstrate that the AFLR procedure consistently produces grids of very-high quality with minimal user input. Efficiency is such that standard PCs or workstations can be used to generate three-dimensional unstructured grids for complex configurations.

Several new ideas for anisotropic adaption of un- structured triangular grids are presented with par- ticular emphasis to fluid flows computations in- volving multiscale phenomena.

Numerical solution of infinite-domain boundary-value problems requires some special techniques that would make the problem available for treatment on the computer. Indeed, the problem must be discretized in a way that the computer operates with only finite amount of information. Therefore, the original infinite-domain formulation must be altered and/or augmented so that on one hand the solution is not changed (or changed slightly) and on the other hand the finite discrete formulation becomes available.

One widely used approach to constructing such discretizations consists of truncating the unbounded original domain and then setting the artificial boundary conditions (ABC's) at the newly formed external boundary. The role of the ABC's is to close the truncated problem and at the same time to ensure that the solution found inside the finite computational domain would be maximally close to (in the ideal case, exactly the same as) the corresponding fragment of the original infinite-domain solution.

Let us emphasize that the proper treatment of artificial boundaries may have a profound impact on the overall quality and performance of numerical algorithms. The latter statement is corroborated by the numerous computational experiments and especially concerns the area of CFD, in which external problems present a wide class of practically important formulations.

In this paper, we review some work that has been done over the recent years on constructing highly accurate nonlocal ABC's for calculation of compressible external flows. The approach is based on implementation of the generalized potentials and pseudodifferential boundary projection operators analogous to those proposed first by Calderon. The difference potentials method (DPM) by Ryaben'kii is used for the effective computation of the generalized potentials and projections. The resulting ABC's clearly outperform the existing methods from the standpoints of accuracy and robustness, in many cases noticeably speed up the multigrid convergence, and at the same time are quite comparable to other methods from the standpoints of geometric universality and easiness of implementation.

In this paper, we review the basic principles of the method of space-time conservation element and solution element for solving the conservation laws in one and two spatial dimensions. The present method is developed on the basis of local and global flux conservation in a space-time domain, in which space and time are treated in a unified manner. In contrast to the modern upwind schemes, the approach here does not use the Riemann solver and the reconstruction procedure as the building blocks. Therefore, the logic and rationale are considerably simpler. The present approach has yielded high-resolution shocks, rarefaction waves, acoustic waves, vortices, ZND detonation waves and shock/acoustic waves/vortices interactions. Moreover, since no directional splitting is employed, numerical resolution of two-dimensional calculations is comparable to that of the one-dimensional calculations.

We present a new finite volume version ([1], [2], [3]) of the 1-dimensional Lax-Friedrichs and Nessyahu-Tadmor schemes ([5]) for nonlinear hyperbolic equations on unstructured grids, and compare it to a recent discontinuous finite element method ([6], [23]) in the computation of some typical test problems for compressible flows.

The non-oscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution of Riemann problems at the cell interfaces is by-passed thanks to the use of the staggered Lax-Friedrichs scheme, is extended here to a two-step, two-dimensional non-oscillatory centered scheme in finite volume formulation. The construction of the scheme rests on a finite volume extension of the Lax-Friedrichs scheme, in which the finite volume cells are the barycentric cells constructed around the nodes of an FEM triangulation, for even time steps, and some quadrilateral cells associated with this triangulation, for odd time steps. Piecewise linear cell interpolants using least-squares gradients combined with a van Leer-type slope limiting allow for an oscillation-free second-order resolution.

The discontinuous finite element method consists of two steps. We first perform a finite element computation which includes calculation of the fluxes across the edges of the triangular elements using 1-D Riemann solvers with a modification to satisfy the entropy condition. We then proceed to a truly multidimensional slope limitation performed on the physical variables.

Numerical applications to several test problems show the accuracy and stability of the finite volume method.

The following sections are included:

• Introduction

• The cell vertex method

• Test problems

• Improved artificial dissipation models for feedback control

• Conclusions

• References

In this article we are concerned with difference approximation for conservation law. We mainly analyze the convergence property from the viewpoint of numerical viscosity. In the main theorem we obtain a rather wide class of difference approximations converging to the entropy solution. The main theorem is one of the best possible results of this kind.

In this paper the non-existence of third order accurate MUSCL schemes for genuinely nonlinear conservation laws is proved. A new class of non-MUSCL highly accurate ENO schemes is constructed in an unified way which is useful for parallel computing. A new discontinuity sharpening technique -artificial stiff source term approach is suggested. Numerical results are presented.

An analysis of inviscid numerical fluxes is presented with emphasis on their capability for accurately capturing shock and contact discontinuities, preserving the positivity for density and pressure, and maintaining shock stability in multidimensions. We investigate in detail the cause of certain catastrophic breakdowns in some numerical flux schemes. In particular, the terms responsible for the odd-even decoupling and the so-called "carbuncle phenomenon" are identified, leading to a proposed conjecture and hence a cure. The validity of this conjecture is confirmed by examining the mass flux of existing upwind schemes. Therefore, the conjecture is very useful to flux function development as to whether the flux scheme will be afflicted with these failings. Study of mass flux is important because the numerical diffusivity introduced in it can be identified and it is the term common to all conservation equations. It also bears a direct consequence to the prediction of contact (stationary and moving) discontinuities which are considered to be the limiting case of the boundary layer. Specifically, the recent AUSM-family schemes are examined and we prove the positivity property of these schemes for both density and pressure. Furthermore, we show the scheme not only exactly captures the 1D shock condition, but in addtion satisfies the entropy condition, thus yielding only the physically admissible solution.

In this paper we present a summary of the splitting technique for both compressible and incompressible flows previously proposed in [22, 23, 7]. Also, we extend it to the case of a fully implicit treatment of the viscous and convective terms of the momentum equations. For incompressible flows, this scheme reduces to classical fractional step methods, except for a non-standard treatment of the boundary conditions. For compressible flows the continuity equation involves two variables which must be related through the equation of state. Convective terms of the conservation equations to be solved are stabilized by means of a Characteristic - Galerkin scheme. Also, in the presence of shocks some additional dissipation is needed. Both numerical techniques are explained here taking the transport of a scalar quantity as a model problem.

The simulation of two dimensional compressible turbulent flow problems is considered. The turbulence is represented via a two–equation eddy viscosity model and both the k - ∊ and the k - ω models are employed. The mass–averaged compressible Navier–Stokes equations, coupled with the turbulence–transport equations, are solved using a Galerkin finite element method. Consistent numerical fluxes are constructed using Roe flux–difference splitting. A MUSCL scheme is employed to achieve higher–order accuracy for both the mean flow and the turbulence variables. A general unstructured triangular grid is employed to represent the computational domain and the solution algorithm is implemented using an edge–based data structure. Several test cases, involving a range of flow conditions, are solved to show the applicability of the proposed procedure.

In the present paper, unsteady natural convection flows of air in a square cavity with constant heated and cooled walls are investigated numerically using the hybrid-GAMAC (generalized and simplified marker and cell) finite-element method. A direct-numerical simulation of two- dimentional turbulence is shown in the present paper without introducing any random forces. The present result at Ra = 1.0 × 10⁸ is compared with the other numerical results. The present result of calculation is in good agreement with that of the others. This study allows us to say that the hybrid-GSMAC method is very stable at high Rayleigh numbers. In addition, selected frames from an animation generated from the computational results at Ra = 1.0 × 10⁹ show very clearly the formation of large-scale eddies via the following sequence: initial instability, proceeding through transition, and eventually the statistical steady state. Using this method, it is possible to investigate natural convection and heat transfer phenomena at high Rayleigh number up to 10¹⁰.

The following sections are included:

• Introduction

• Discontinuities

• Boundary Conditions

• Complex Geometries

• Time Discretizations

• Conclusions

• REFERENCES

A newly developed spectral finite difference scheme ( Fourier expansion, Dini expansion, Legendre expansion ) applicable to numerical analysis of heat and fluid flows is reviewed. Concrete solution procedures applied to different ( mathematical or physical ) types of problems are given for analyses corresponding to two-dimensional external flows, axisymmetric heat and fluid flows, two-dimensional natural convection in an elliptic enclosure, two-dimensional natural convection with a free surface, supplemented with obtained thermal or flow fields.

The solution of the Navier-Stokes equations within the vorticity-streamfunction formulation by means of spectral methods is presented and discussed. By using semi-implicit time-discretization schemes, the Navier-Stokes problem amounts to the solution of a Stokes-type problem. The solution of this problem makes use of the influence matrix technique for prescribing the boundary conditions in the Chebyshev collocation approximation. The method is direct and reduces to matrix products to be performed at each time-cycle. Then, the method is adapted to the domain decomposition approach. Finally Fourier-Chebyshev methods for calculating free-surface flows with the vorticity-streamfunction equations are described.

This paper gives an overview of the advances made in the recent vortex methods for removing three restrictions on the methods, i.e, (1) inability to treat viscosity, (2) inability to treat compressibility, and (3) prohibitive increase of computational costs with the square of the vortex number.

Addressed in this article are only those problems of aerodynamics that can be reduced to linear singular or hypersingular equations and for which the mathematical foundation of the method of discrete vortices or the method of discrete closed vortex frames is given. Such problems include noncirculatory problems for piecewise smooth contours (surfaces), the linear stationary problem for the rectangular finite-span wing and the linear nonstationary problem for a thin airfoil.

The following sections are included:

• Introduction

• Incompressible Equations

• Algorithm I

• Algorithm II

• Symmetric Preconditioner

• Convergence to the Incompressible Equations

• Artificial Viscosity

• Boundary Conditions

• Implementation

• Jacobi Preconditioning

• Preconditioning-squared

• Time Dependent Problems

• Differential Preconditioners

• Canonical Decomposition

• The Two dimensional Case

• Three Dimensions

• Comparison of Preconditioners

• Computational Results

• References

The robustness and effectiveness if iterative solution methods in CFD habe improved significantly in recent years thanks to progress made in both accelerators and preconditioners. This paper highlights some of the idears which have been exploited to obtain these improvements.

An overview of multilevel methods on unstructured grids for elliptic problems will be given. The advantages which make such grids suitable for practical implementations are flexible approximation of the boundaries of complicated physical domains and the ability to adapt the mesh to resolve fine-scaled structures in the solution. Multilevel methods, which include multigrid methods and overlapping and non-overlapping domain decomposition methods, depend on proper splittings of appropriate finite element spaces: either by dividing the original problem into subproblems defined on smaller subdomains, or by generating a hierarchy of coarse spaces. The standard splittings used in structured grid case cannot be directly extended for unstructured grids because they require a hierarchical grid structure, which is not readily available in unstructured grids.

We will discuss some of the issues which arise when applying multilevel methods on unstructured grids, such as how the coarse spaces and transfer operators are defined, and how different types of boundary conditions are treated. An obvious way to generate a coarse mesh is to re-grid the physical domain several times. We will propose and discuss different and possibly better alternatives: node nested coarse spaces and agglomerated coarse spaces.

A recent development in application of multigrid and multilevel adaptive methods for fluid flow by the numerical simulation group at Louisiana Tech University is reviewed. The multigrid method has been successfully applied for a number of flow cases including both steady and time-dependent flows. The steady cases include laminar flow, turbulent flow, and turbulent combustion. The time-dependent cases include flow transition around airfoils at all speeds including incompressible, subsonic, and supersonic flows. The multigrid method has been found very efficient for both laminar and turbulent flows and also useful for complex geometries.

In this paper airfoil shape optimization is described. The shape is defined by a spline interpolating curve passing through data points, which are the design variables. An objective function, which depends on the design varibles, is minimized by using the sequential quadratic programming procedure. At each iteration of the sequential quadratic programming procedure, the gradient of the objective function with respect to the design vaiables is calculated by using the implicit function theorem. CPU time requirecd for the gradient calculation is greatly reduced by this method. An optimization example for a compressor cascade is presented.

A method is presented for performing lift-constrained drag minimization based on the 3D Euler equations and the 2D laminar Navier-Stokes equations. For the 3D Euler equations, the optimization system is based on a flow analysis scheme which uses gradients calculated on an elemental basis. For the 2D Navier-Stokes system, a change was made in the flow analysis scheme to one based on gradients calculated using a finite volume approach due to the fact that this scheme has drastically reduced memory costs associated with the storage of the residual jacobian, and allows direct extension to 3D Navier Stokes. Optimization exercises are presented which demonstrate the effectiveness of the optimization system in finding credible optimal geometries.

This work presents a complete methodology for the use of an advanced CFD code in aerodynamic shape-design optimization of realistic aircraft configurations. The work consists of two parts. In the first part, the well-known, general-purpose, flow-analysis code CFL3D is accurately differentiated with respect to general geometric shape variations using an automatic-differentiation software tool ADIFOR. The result is a new flow-sensitivity-derivative code, CFL3D.ADII, which retains all of the capabilities of the original code (at least for steady flows), including the ability to model complex geometries with multiblock grids. Similar previous studies with advanced flow codes have yielded accurate sensitivity derivatives; however, computational efficiency was unacceptable. In the present study, however, computational efficiency of the differentiated flow code has been greatly improved, primarily through automatic differentiation in incremental iterative form and by a thorough restoration of the original code's vectorization for efficiency on Cray-type computers. In one example, a milestone result was achieved, in that the CPU-timing of the sensitivity-derivative code CFL3D.ADII was found to be more efficient than the original CFL3D code (on the basis of microseconds of CPU-time per multigrid-cycle per grid-point per design-variable). In the second part of the work, attention was focused on the use of CFL3D.ADII in the shape optimization studies of realistic geometries on parallel computers. A coarse-grain parallel implementation of CFL3D.ADII together with an alternative parallel 1-D line search scheme are used in shape improvement design studies of a realistic High-Speed Civil Transport wing/body geometry on an IBM-SP2 parallel computer. The geometry is represented by 60 design variables and over 200,000 grid points. The flow is supersonic and inviscid, represented by the 3-D Euler equations. In addition to making the handling of such a large problem possible, the use of parallel computation provided significantly reduced overall execution time and turnaround time.

This report reviews recent progress in incompressible Navier-Stokes solution methods and their application to problems of engineering interest. Discussions are focused on the methods designed for complex geometry applications in three dimensions, and thus are limited to primitive variable formulation. A summary of our recent progress in flow solver development is given followed by numerical studies of a few example problems of our current interest. Both steady and unsteady solution algorithms and their salient features are discussed. Solvers discussed here are based on a structured-grid approach using either a finite-difference or a finite-volume frame work.

In this review, it is presented to survey of the recent development of the finite element method (FEM) which can be applied to the time-dependent problems of Navier-Stokes. This paper focuses on the fractional step method applied to the areas of unsteady incompressible fluid flow. Several frames about time discretization and the corresponding algorithms are analysed. With respect to practical problems in the simulation of incompressible fluid flow, we discuss some results of both steady and unsteady flow. The attention of application to free surface flows also is taken. Thus, the descriptions of fluid motion, i.e., Lagrangian method, Eulerian method, and Arbitrary Lagrangian-Eulerian method (ALE), are presented in this review.

The following sections are included:

• Introduction

• Theory

• NUMERICS

• NUMERICAL RESULTS FOR STARTUPS

• NUMERICAL RESULTS FOR SHUTDOWNS

• CONCLUDING COMMENTS

• ACKNOWLEDGMENTS

• REFERENCES

We provide an overview our finite element methods for parallel computation of unsteady flow problems with interfaces such as two-fluid and free-surface flows. The methods we discuss are the Deformable-Spatial-Domain/Stabilized Space-Time (DSD/SST) formulation, which is an interface-tracking technique, and the Enhanced-Discretization Interface-Capturing Technique (EDICT). Both methods are based on the stabilized finite element formulations which possess good stability and accuracy properties.

In the DSD/SST method, the finite element formulation of the problem is written over its associated space-time domain. This automatically takes into account the motion of the interfaces. The mesh update is achieved with strategies that focus on moving the mesh in an effective way and minimizing the frequency of remeshings. Although this interface-tracking method results in accurate representation of the interfaces, in cases with complex, rapidly-changing interfaces, especially in 3D simulations, reducing the frequency of remeshing becomes difficult, and sometimes not feasible.

With the EDICT, we solve, over a non-moving mesh, the Navier-Stokes equations together with an advection equation governing the evolution of an interface function with two distinct values identifying the two fluids. To increase the accuracy in modeling the interfaces, we use finite element functions corresponding to enhanced discretization at and near the interface. These functions are designed to have multiple components, with each component coming from a different level of mesh refinement over the same computational domain.

With the test problems presented here, we demonstrate that the EDICT can be used very effectively to increase the accuracy of the base finite element formulations. With parallel implementations of the DSD/SST method and EDICT, we have brought our simulation capability to a point where we can now address a wide-range of unsteady flow problems with interfaces.

The purpose of research on this subject is the fact that worldwide most of the plants of the electrical power industry have been in service for more than 25-30 years and recent advances in the development of CFD codes have made possible the detailed calculation of complex single phase flows in turbine stages. However no commercially available code contains an adequate representation of the steam phase transition mechanism, nor the corresponding condensation loss [?]. Because about 10% of the total electrical net power output of plants up to 1500 MW is produced in the last stages of the low pressure steam turbines it is obvious, that understanding and accurate modelling of steady and the control of unsteady phenomena in condensing steam flows are of exceptional great importance for the redesign of modern steam turbine bladings.

Performing experiments and numerical calculations of condensing transonic flows in nozzles with varying shapes we detected new instabilities and bifurcations with sudden changes of the flow patterns, together with discontinuous increase or decrease of the oscillation frequency (Schnerr et al. [?]). The amplitude of the pressure waves increases locally for more than 100 % and the dispersion of the liquid phase changes completely. The first aspect is certainly relevant for flutter excitation of large blades in low pressure stages and the second affects losses and erosion by the droplet impact on the blade surfaces.

In this paper we first investigate the formation of flow instabilities with bifurcations in arbitrary channel geometries with unsymmetric boundaries and we confirm the existence of bifurcation dynamics in axial cascades using vapor/carrier gas mixtures, which is the operating test fluid of the atmospheric indraft wind tunnel where these new instabilities were detected. We then present numerical results of steady and unsteady nucleating vapor flows using actual blading, operating at low pressure conditions, typical for observing condensation.

First-principles based techniques for the prediction of fixed and rotary wing wake geometry are described. It is demonstrated that fifth order accuracy schemes do substantially better than third order spatial accuracy schemes in capturing the details of the vortex core structure. It is demonstrated that the use of embedded grids can further enhance the resolution of the tip vortex, particularly if the boundary conditions and the order of interpolation accuracy are carefully maintained to be fifth order. A hybrid approach where the costly Navier-Stokes analysis is confined to small viscous regions near the blade surface is also described. Sample applications of these methods to the vortex wake behind a fixed wing, and the lift and surface pressure distributions over various rotors are presented.

The role of computational simulation in the design of advanced aerospace and commercial products is becoming prominent with advances in software technology (geometry modeling and grid generation, solution algorithms and visualization tools), as well as in computer hardware with increasingly powerful workstations and parallel platforms. While the computational technology has advanced to allow detailed simulations of very complex physical processes employing millions of grid cells running on parallel computing architectures, one of the pacing issues in the advancement of this technology for routine use as an engineering design tool is to make it cost-effective and quick-turnaround to meet the demands of shorter design cycle time with reduced budgets. This paper describes some of the recent advances at Rockwell in the development and application of a quick-turnaround, unstructured grid-based parallel computational technology addressing the bottlenecks in various stages of simulation.

Large-scale simulation results for industrial aerospace applications obtained with the structured-mesh solver NSMB and with the unstructured-mesh solver AVBP are reviewed. A brief overview of the basic theory and a comparison of methodologies is given for these two types of solvers: the physical models for compressible turbulent flow are first described, and the different numerical formulations used to solve the governing equations are then examined, with a discussion of their implementation on vector and parallel computers. Finally, three large-scale simulation results are presented: two on transonic aircraft, aerodynamics, and one on hypersonic spacecraft aerothermo-dynamics.

This paper reports a subset of results from a study carried out under the auspices of The Technical Cooperation Program (TTCP) with participants from Canada, the UK, and the US. The purpose of this study was to apply Navier-Stokes computational techniques to a complex flow field with highly separated flow for a missile shape to evaluate the predictive technology. The portion of the study reported here includes results for three Navier-Stokes computations for transonic and supersonic velocities at 8° and 14° angles of attack. The computational results were compared to experimental measurements for surface pressure, pitot surveys of the outer flow field, and strain gage force measurements. Computational results are reported for seven turbulence models and laminar flow. For the conditions of this study, no "best" turbulence model could be identified in the comparisons to experiment. Predictions of surface pressure and outer vortical flow in regions of highly separated flow indicate further development of the predictive technology is required.

This paper describes computational efforts to develop an accurate computational tool for the purpose of establishing a better estimation method of the safety distance for the blast wave. Since this type of simulations require high level of grid resolution, two approaches are considered. One is an overset zonal method and the other is an unstructured solution adaptive method. Two-dimensional computations are carried out and the computed results indicate that the former approach is adequate for this problem. Based on the techniques developed for the two-dimensional simulations, three-dimensional simulations are carried out and the effect of the ground surface geometry is discussed. Sufficient grid resolutions are obtained within the available computer memory and speed. The computed result clearly shows that the ground surface has an important effect on the strength of the blast wave even far away from the point of explosion. To make the result useful for other practical problems, assessments of the computational aspect are given for the required resolution of the computational grid for this type of problems.

Advances in numerical methods and rapid increases in memory capacity and computational speeds of available computers have made computational fluid dynamics (CFD) an essential tool for the development of hypersonic vehicles. This paper presents our recent work on developing high-order numerical methods for transient nonequilibrium hypersonic flow simulations and for the direct numerical simulation (DNS) of transitional and turbulent hypersonic boundary layers. The applications of these numerical techniques to unsteady hypersonic flow computations are demonstrated through several examples of results. A brief overview of general CFD methods as they are applied to hypersonic flow simulations in the continuum regime is also given.

We propose the approach for generalizing the classical model of a continuum medium, whose local kinematics is not described by a diffeomorphic mapping. This approach is based on using the thermodynamics scheme in which distortions are the thermodynamics variables. We derived the complete system of equations for description of the medium. A special attention is payed to the consideration of the dissipative processes. We analyzed the Yang-Mills gauge approach from the thermodynamics point of view. Finally, the linear approximation of the gauge theory is considered.

The following sections are included:

• Introduction

• FIRST-ORDER DISTRIBUTION FUNCTION AND N-S EQUATIONS

• SECOND-ORDER DISTRIBUTION FUNCTIONS AND BGK-BURNETT EQUATIONS

• LINEARIZED STABILITY ANALYSIS AND BOLTZMANN'S H-THEOREM

• NUMERICAL ALGORITHM TO INTEGRATE THE BGK-BURNETT EQUATIONS AND COMPUTATIONAL TEST CASES

• ACKNOWLEDGMENT

• REFERENCES

The following sections are included:

• Introduction

• Kinetic theories

• The Navier-Stokes limit

• Coupling Strategy

• The Local Solvers

• Numerical Tests

• Conclusion

• References

Transition analysis in the engineering environment serves to determine the border between the regions of laminar and turbulent flow in boundary layers. Knowledge of this border permits improved estimates of drag and heat transfer, and is key to the design of low-drag components and flow-control systems. We discuss problems of transition analysis, improvements to the current approach based on linear stability characteristics, and the more advanced nonlinear techniques enabled by the parabolized stability equations. Besides the issues of efficient computation crucial in engineering practice, we address improvements in software design and usability, data quality, and physical models of the transition process.

Laminar-turbulent transition on aerodynamic surfaces occurs between often widely apart limits, namely the upper limit relevant in extremely low-disturbance environments and the lower limit given by the bypass transition. The e^N method only provides the upper bound on the transition location, while experimental information or direct Navier-Stokes simulations are required to determine the lower bound. Computational tools (receptivity theory/linear Navier-Stokes, nonlinear parabolized stability equations, etc.) are now available to estimate the departure from the upper bound due to factors such as surface roughness and acoustic disturbances but await validation against experiments. In this paper, progress made in boundary-layer stability and transition is reviewed and description is given for the computational tools developed to ease the determination of the upper limit via N-factor calculations. This includes a suction optimization code for the efficient design of laminar flow control wings. A brief discussion of some of the physical issues associated with boundary-layer transition is given and some key results, including those pertaining to the question of absolute instability, are presented.

Progress in the development of the hierarchy of turbulence models for Reynolds-averaged Navier-Stokes codes used in aerodynamic applications is reviewed. Steady progress is demonstrated, but transfer of the modeling technology has not kept pace with the development and demands of the CFD tools. An examination of the process of model development leads to recommendations for a mid-course correction involving close coordination between modelers, CFD developers and application engineers. In instances where the old process is changed and cooperation enhanced timely transfer is realized. A turbulence modeling information data base is proposed to refine the process and open it to greater participation among modeling and CFD practitioners.

A brief overview of the recent progress achieved by the author's research group on turbulence control for drag reduction in turbulent boundary layers is given. Control schemes are developed based on the premise that the most effective way to control turbulent boundary layers is through proper manipulation of the near-wall streamwise vortices. Two different methods — neural networks and suboptimal control theory — are utilized to achieve this goal. The resulting feedback laws are applied to a low Reynolds-number turbulent channel flow. Numerical experiments indicate that both approaches yield substantial drag reduction. The application of a system theory approach to the feedback stabilization to delay the transition in a laminar channel flow is also discussed. It is shown that the system theory approach is a valuable tool both for designing control systems to stabilize the flow as well as for understanding the physics of controlled transitional flows.

The technique of large eddy simulation (LES) has proven very useful for the prediction of turbulent flows. Its extension to turbulent, reacting flows has been rather slow, however, due to difficulties in modeling the chemistry in the subgrid scales. In this paper, the general technique of large eddy simulation is briefly described, as well as several recent approaches to the simulation of turbulent, reacting flows. One particular method, based upon the concept of flamelets, is discussed in detail. In this approach, finite-rate chemistry is accounted for by invoking the flamelet approximation, and an assumed form is employed for the subgrid-scale or 'large eddy' probability density function. The method requires the simulation of the filtered mixture fraction, its second moment, and its dissipation rate. Some a priori tests of the model are presented, giving optimism that large eddy simulation can be successfully applied to turbulent combustion.

Interaction between the stellar wind, including the solar wind, and the interstellar medium has long been the subject of investigation by both astrophysicists and fluid dynamicists. This is, first, due to the possibility of comparison of physical models for such interaction with the measurements performed by Voyager, Pioneer, and Ulysses spacecrafts. On the other hand, a complicated structure of the flow containing several discontinuities makes it a challenging problem for the application of modern numerical methods both in gasdynamic and magnetogasdynamic (MHD) cases.

In the solar wind case, the problem becomes even more complicated, since the charge-exchange processes between ions and neutral particles must be taken into account. The continuum equations are not applicable to the description of the neutral particle motion, for their mean free path is much larger than the characteristic length scale of the problem. In this case, either approximate coupling models or direct Monte-Carlo simulation are required. The spatial nonuniformity of the solar wind and its perturbations and periodicity make the problem three-dimensional and non-stationary. From a mechanical viewpoint the problem represents the interaction of the uniform interstellar medium and the spherically-symmetric (or asymmetric) solar wind flow. We consider various approaches used by different authors to solve this problem numerically.

The presence of the contact surface dividing the two flows rises the question of its stability. We discuss the reasons of such instabilities and parameters which influence it.

The presence of the interstellar magnetic field necessitates solution of the MHD equations for proper analysis of the obtained data. Although the system of governing equations remains hyperbolic in this case, the multivariance of the exact solution to the MHD Riemann problem makes inefficient its application for regular calculations. On the other hand, the solution to the linearized Riemann problem is nonunique. We discuss the possible ways of applying the Roe-type methods and some simplified approaches for numerical solution of the ideal MHD equations. One of the difficulties in the solution of the MHD system is the satisfaction of the magnetic field divergence-free condition. Different ways to solve this task are discussed. If the magnetic field vector in the uniform interstellar medium flow is not parallel to the velocity vector, the problem becomes three-dimensional. Both approximate and exact numerical solutions are considered which were applied in this case. Far-field numerical boundary conditions play an essential role in astrophysical applications owing to very large length scales usual for these problems. We discuss several approaches that may be useful to solve problems similar to the stellar wind and interstellar medium interaction.

In the article a number of approximate methods of the solution of ocean dynamics is offered. Evaluation of rate of convergence of the approximate solutions to the exact one, is carried out numerically.

In this paper we review the state of art for shallow water equations using various approches by finite element methods. Such methods are very effective and powerfull and can be still more efficient by the use of parallel computers. Numerical results are given of an implementation made for the Venice Lagoon.

Numerical simulations of multiphase flows are generally classified into two categories. One is direct numerical simulation(DNS) of all phases and the other is simulation with the multiphase flow models like two fluid model. Recent progress of computer performance and numerical algorithms enable us to conduct DNS of relatively complicated multiphase flows. In the present paper, some of the techniques recently developed for tracking the interfaces are reviewed and examples of numerical simulations are shown. These results show quantitatively good agreements with experimental ones. It is also shown that macro-scale model of bubbly flows, which is constructed precisely taking micro scale phenomena like the motion of bubbles into account, can predict the transient phenomena and instabilities of the multiphase flows in a good accuracy.

We have succeeded for the first time to simulate dynamic phase transition from metal to vapor. This success is due to the CIP (Cubic-Interporated Pseudoparticle/Propagation) method that can treat solid, liquid and gas together and can trace a sharp interface with almost one grid. The strategy of the CIP method and its various modifications are reviewed including application to laser- induced evaporation, slamming of ship and formation of milk crown in three dimensions.

A methodology for the simulation of strongly unsteady flows with hundreds of moving bodies has been developed. An unstructured-grid, high-order, monotonicity preserving, ALF solver with automatic refinement and remeshing capabilities was enhanced by adding: equations of state for high explosives, deactivation techniques and optimal data structures to minimize CPU overheads, two new remeshing options, and a number of visualization tools for the pre- processing phase of large runs. The combination of these improvements has enabled the simulation of strongly unsteady flows with hundreds of moving bodies. Several examples demonstrate the effectiveness of the proposed methodology.

A new method is presented for developing finite difference algorithms that are high order in both space and time, and several example algorithms are compared. Differences in efficiency of several orders of magnitude are shown, even for propagation through a few periods. Accurate and efficient propagation to O[10⁵] periods is shown to be possible with high order and high resolution finite difference algorithms. High resolution examples have spectral like properties. Two dimensional examples correctly propagate information along characteristic surfaces. Well posed, stable, and convergent artificial boundary conditions are reported.

The following sections are included:

• Introduction

• Nomenclature

• Governing Equations

• Maxwell Equations on Curvilinear Frame

• Eigenvalues and Eigenvectors

• Flux-Vector Splitting

• Spatial Discretization

• Temporal Discretization

• Numerical Results

• Summary

• Acknowledgment

• References

The following sections are included:

• Introduction

• Mathematical Equations for Compressible Flows

• Mathematical Analysis of Interface Conditions

• Numerical Approximation

• Applications

• Acknowledgement

• REFERENCES

With teraflops-scale computational modeling expected to be routine by 2003-04, under the terms of the Accelerated Strategic Computing Initiative (ASCI) of the U.S. Department of Energy, and with teraflops-capable platforms already available to a small group of users, attention naturally focuses on the next symbolically important milestone, computing at rates of 10 15 floating point operations per second, or "petaflop/s". For architectural designs that are in any sense extrapolations of today's, petaflops-scale computing will require approximately one-million-fold instruction-level concurrency. Given that cost-effective one-thousand-fold concurrency is challenging in practical computational fluid dynamics simulations today, algorithms are among the many possible bottlenecks to CFD on petaflops systems. After a general outline of the problems and prospects of petaflops computing, we examine the issue of algorithms for PDE computations in particular. A back-of-the-envelope parallel complexity analysis focuses on the latency of global synchronization steps in the implicit algorithm. We argue that the latency of synchronization steps is a fundamental, but addressable, challenge for PDE computations with static data structures, which are primarily determined by grids. We provide recent results with encouraging scalability for parallel implicit Euler simulations using the Newton-Krylov-Schwarz solver in the PETSc software library. The prospects for PDE simulations with dynamically evolving data structures are far less clear.

The use of parallel computers and in particular distributed memory architectures imposes a number of requirements on the visualization systems for Computational Fluid Dynamics (CFD). We describe the design and capabilities of our distributed visualization software which can handle parallel multidisciplinary applications, can connect to a running solver for on-line display and steering, supports collaborative visualization and can be used as a tool for the automatic making of movies and animations. This discussion serves as an overview of the current trends and developments in visualization as applied to CFD and other computational sciences.

This paper describes the virtual windtunnel, a virtual reality-based, near-real-time interactive system for CFD visualization. The virtual windtunnel supports several visualization techniques and meets several requirements, including extensibility, guaranteed performance while being very computationally intensive, time management including the maintenance of simultaneity in the virtual environment, and interface independence. Creating a framework which meets all of these requirements presented a major challenge. We describe the interaction techniques, object-oriented structure and process architecture, including interprocess communications and control. Subtle issues regarding the flow of time in the visualization environment are described and addressed. The problem of time-critical computation is discussed, and a solution is presented. Device independence of both the command and display structures are developed, providing the ability to use a wide variety of interface hardware options. The resulting framework supports a high-performance visualization environment which can be easily extended to new capabilities as desired.