This article reviews some of the work carried out on numerical methods in fluid mechanics in Germany during the past four decades: Early investigations in the sixties dealt with the extension of already, existing solutions for two-dimensional flow problems to those of three-dimensional flows. In the following decades international short courses and conferences were established, and cooperation between scientists of various research centers and universities was successfully initiated and built up. The German Research Foundation generously sponsored these activities in two priority programs and in a third cooperative research program co-sponsored by the French Centre National Recherche et Scientific. In the nineties the German Science Council proposed a recommendation to the Federal Government to establish high-performance computing in Germany on an internationally competitive basis, which finally resulted in the foundation of the German Gauss Center for Supercomputing.

The intersection of a longitudinal vortex by a normal shock is studied with a numerical solution of the Euler and Navier-Stokes equations for time-dependent, three-dimensional, laminar flow. The study is concerned with the destruction of the vortex core, usually referred to as vortex breakdown or bursting, by letting the shock intersect the vortex normal to its axis. Results are presented for a Mach number Ma∞ = 1.6. The calculations were performed on a Cartesian mesh with approximately 2 millions grid points. The computations show noticeable differences between the two solutions, and viscous forces become important in the burst part of the vortex. Visualization studies clearly reveal the time-dependent, three-dimensional nature of the flow after breakdown.

Breakdown of slender vortices' caused by normal (NSVI) and oblique (OSVI) shocks is studied using numerical solutions of the Navier-Stokes equations for unsteady three-dimensional, supersonic flow. A Burgers vortex with a given circulation and axial velocity distribution is prescribed at the inflow boundary. The calculations show that breakdown is primarily controlled by pressure forces and therefore is easily initiated when the vortex interacts with a normal shock. Oblique shocks are deformed into an 's'-shaped part near the vortex core where the shock becomes normal. The results indicate that initiation of breakdown is more sensitive to variations in the axial velocity than in the circulation, and that the flow structure is clearly time-dependent.

Breakdown of a slender vortex caused by a normal shock is studied using a numerical solution of the Navier-Stokes equations for unsteady, three-dimensional, supersonic flow at a free stream Mach number of 1.6. The numerical results clearly reveal the time-dependent flow structure for both the axial and the radial direction. The results compare well with recent experimental findings.

Results of an experimental study of a supersonic flow around the leeward side of a delta wing are presented. The experiments are performed on three delta wings with leading–edge sweep angles χ = 68°, 73°, and 78° for Mach numbers M =2—4 and angles of attack α = 0—22°. Data on the structure and position of internal shock waves are obtained; the size and location of primary and secondary vortices are found. New regimes of the flow around a delta wing are identified. The chart of flow regimes around delta wings is refined and extended.

The leeside vortex structures on delta wings with sharp leading edges were studied for supersonic flow at the Institute of Theoretical and Applied Mechanics of the Russian Academy of Sciences in Novosibirsk. The experiments were carried out with three wings with sweep angles of χ=68°, 73°, and 78° and parabolic profiles in the 0.6 × 0.6 m2 test section of the blow-down wind tunnel T-313 of the institute. The test conditions were varied from Mach numbers M=2 to 4, unit Reynolds numbers from Rel=26 × 106 to 56 × 106 m−1, and angles of attack from α=0° to 22°. The results of the investigations revealed that for certain flow conditions shocks are formed above, below, and between the primary vortices. The experimental data were accurate enough to detect the onset of secondary and tertiary separation as well as other boundaries. The various flow regimes discussed in the literature were extended in several cases. The major findings are reported.

Supersonic flight of aerospace planes is of marked interest since several flow regimes characterized by different local flow structures have to be flown through. This problem was investigated experimentally for the hypersonic research configuration ELAC 1. The aim of the study was to detect the influence of the rounded leading edge, of the thickness distribution prescribed, and of the Reynolds number, especially on the flow on the leeward side of the configuration. The experiments were carried out in the transonic wind tunnel of Aerodynamisches Institut of RWTH Aachen, at a freestream Mach number Ma ∞ =2, a unit Reynolds number of Re ∞ =13×10 6 , angles of attack between −3°?α?10°, and in a wind tunnel of the Institute for Theoretical and Applied Mechanics of the Russian Academy of Sciences in Novosibirsk. The freestream Mach numbers covered in these experiments were varied between 2? Ma ∞ ?4, freestream Reynolds numbers per unit length between 25×10 6 ? Re ∞ ?56×10 6 and angles of attack between −3°?α?10°. Flow visualization studies, measurements of surface pressure distributions and of aerodynamic forces were used to analyze the flow. The results, which will also be compared with numerical data, clearly indicate marked differences in the location of the separation and reattachment lines, and the formation of the primary, secondary and tertiary vortices, for the flow regimes investigated.

The in-cylinder flow of a reciprocating piston engine is simulated numerically for the intake and compression stroke. A block structured moving grid system is applied to resolve the piston and valve movements. The grid is refined and coarsened during the opening and closing of the valve and the piston up- and downward motion. In addition to the conservation equations for mass, momentum and energy, a conservation equation for the cell volume is solved numerically. No turbulence model is applied, because large eddy simulations of a turbulent channel and plane jet flow show that the influence of subgrid scale models is small for second order schemes. Results are presented for one and four valve engines also in comparison with experimental data.

Depending on volume flux, flow visualizations in a water tunnel showed bubble-, spiral-type breakdown, and periodic transition between both. The initiation and development of bubble-type breakdown can be explained by a nonlinear feedback model. A growing asymmetry of the circumferential vorticity distribution leads to the transition to spiral-type. These conjectures are supported by experiments in which an artificially generated vortex ring induced initiation of bubble- and transition to spiral-type breakdown. To simulate vortex breakdown Navier-Stokes equations for three-dimensional, unsteady, and incompressible flows were solved. A comparison between experimental and numerical flow visualizations showed a good agreement.

Without Abstract

Developments and advances in numerical fluid dynamics are being reviewed with emphasis on physical aspects in preference to methodical questions. The governing equations of fluid dynamics, describing the conservation of mass, momentum, and energy are discussed first. Recent work on predictions of inviscid, and of boundary-layer flows is then described in the following two sections. Thereafter, computations of fully viscous flows by numerical solutions of the Navier-Stokes equations are elucidated with several examples of the recent literature.

The evaluation of the circulation from numerical solutions of the momentum and energy equations is discussed for incompressible and compressible flows. It is shown how artificial damping directly influences the time rate of change of the circulation.

Associated with the breakdown process is the formation of a stagnation point on the axis of the vortex. This requires the deceleration of the axial velocity component, which must be enforced by a positive axial pressure gradient. The analysis presented here shows, how the pressure gradient along the axis of the vortex is influenced by the radial and azimuthal velocity components. An explicit expression for ∂p/∂x (x, 0) can be obtained by integration of the momentum equation for the radial velocity component with respect to the radial and subsequent differentiation of the integral with respect to the axial direction. In an order of magnitude analysis it is then demonstrated that for large Reynolds numbers one component of the frictional force in the azimuthal direction cannot be neglected. In order to obtain an estimate for the pressure gradient rigid body rotation is assumed for the vortex core, and a distribution similar to that of a potential vortex w = kr −n , for the outer portion. The estimate shows that a positive axial pressure gradient can exist only, if the radial velocity component is positive and if the exponent n is less than unity. It is also verified that a potential vortex cannot support an axial pressure gradient, that the pressure gradient in magnitude is directly proportional to the square of the maximum of the azimuthal velocity, referenced to the freestream velocity.

Summary The flow of a Newtonian fluid in cone-plate viscometers is determined by solving the Navier-Stokes equations. It is shown in an order-of-magnitude analysis that the governing equations can be simplified and that the flow field can be split into two parts, if the angle between cone and plate is small. A closedform solution for the inner flow field is established through an iteration procedure. The zeroth-order solution in which the convective terms are neglected is inserted into the governing equations which then can be integrated again. The outer flow field is also described analytically. The solution satisfies the boundary condition of the vanishing radial velocity component at the free surface. The flow is described throughout the whole gap. Velocity components, pressure distributions, torque and normal force due to inertia are calculated. Pressure distributions and the maximum radial velocity are measured. The free surface boundary condition is experimentally examined.