Transient growth is quantitatively examined in two prototype separated flows using Direct Numerical Simulation (DNS). Separated flows typically exhibit regions of convective instability due to the inflectional velocity profiles inherent in the shear flow. This can lead to the transient growth of small disturbances by many orders of magnitude. After reviewing the mathematical tools and numerical techniques required, we present an analysis of transient growth in an axisymmetric pipe with a 1:2 diametral expansion. A direct method is used to calculate the optimal transient energy growth for specified time horizons and Reynolds numbers up to Re=1200, and low-order azimuthal wavenumber m. At each Re the maximum growth is in azimuthal mode m=1 and this maximum is found to increase exponentially with Re. The time evolution of optimal perturbations is presented and shown to correspond to sinuous oscillations of the shear layer. Finally, full three-dimensional DNS with the inflow perturbed with Gaussian white-noise confirms the presence of the structures determined by the transient growth analysis.
The second prototype flow considered is the cylinder wake in the subcritical regime. Large energy growth is observed at Reynolds numbers close to the onset of global instability and the optimal perturbations which lead to this growth are determined. Three-dimensional spanwise perturbations are also examined and it is found that, except for short time horizons, the zero wavenumber is dominant. Furthermore, performing accurate linear and transient growth analysis is found to be highly dependent on the size of the computational domain. Adjoint eigenmodes extend far upstream of the cylinder necessitating a long inflow. More importantly, constrictions in the cross-stream direction
are found to distort the basic flow, which has a substantial effect on the accuracy of the
Transition in pipe flow is a topic for which there is still relatively little understanding. Puffs are small regions of turbulence observed close to the transitional Reynolds number. A gradually expanding pipe is proposed as a means to effectively produce
turbulent puffs and study their creation and decay.