The transition from the laminar to the turbulent flow state (transition) and flow separation are still to be scrutinized in detail since profound understanding of many of the underlying processes is still lacking. The understanding of the instability mechanisms causing eventually laminar breakdown is essential for an improved technique to hinder transition or to prolongue the laminar flow region, since in many cases transition is unwanted due to its drag increasing effect. At the institute, transition research is done in two numerical groups, described here, and in experimental groups (Laminar Wind Tunnel, Laminar Water Tunnel, and Shock Tube; cf. Laminarwindkanal, Laminarwasserkanal).

The simulation of the different states of the flow through the numerical solution of the complete Navier-Stokes-Equations is undertaken using the so-called spatial model, where the realistic spatial and temporal development of disturbances in the spatially varying base flow is considered like in most experiments. Various two- and three-dimensional, incompressible and compressible wall-bounded and free shear flows are examined under "controlled" conditions for all stages of transition, i.e. disturbance receptivity, primary instability, secondary instability, streak instability, strong nonlinear stages, generation and development of coherent structures, and randomization in early turbulence.

Since those direct numerical simulations (DNS) of unstable waves and moving small-scale structures in space and time require a very accurate numerical integration, a highly accurate method with an adequate number of gridpoints has to be used. In case of the examination of a whole transition region up to turbulence, the total number of grid points drastically increases and the calculations have to be performed on a high-performance computer (see Project LAMTUR ) with sufficient memory size applying a well optimized calculation method. The used methods are based on the experience on this highly specialized research topic for more than a decade, oops: 2 decades already concerning the vectorization and parallelization.

In general, a combined Finite-Difference/Spectral method of high-order accuracy with a fourth-order Runge-Kutta time-integration scheme is used for the equations in vorticity-velocity formulation for incompressible flows, or for the compressible equations in conservative variables. Finite Differences of 4th or 6th order (the latter compact) are employed for the nonperiodic streamwise and wall-normal directions, and the pseudo-spectral Fourier method is used for the spanwise direction. The optimization of the utilized discretization, including the boundary conditions, with respect to the nonlinear wave-transport phenomena yields a integration method of extreme accuracy and numerical efficiency that is long-time stable. In the outflow region a self-developed buffer zone technique with artificial suppression of the disturbances is employed to substantially reduce the disturbance level at the actual outflow. This prevents undue upstream disturbance scattering and makes proper long-time integrations of transition processes also with initially low amplitudes feasible. In case of incompressible flow, the Poisson-type equations for the wall-normal velocity component are solved either directly or with an iterative line relaxation procedure with multigrid acceleration.