FLOW: Bifurcation and stability analysis of a jet in crossflow
Caption: Visualization of the lambda_2 vortex identification criterion using volume rendering for the limit cycle at R=0.675 (top), R=1 (middle) and R=2 (bottom). Bottom: the `wavemaker’ for R=0.675 (grey), shown with contour of zero streamwise velocity (blue), spanwise vorticity (green) and wall-normal velocity (red).
This project builds on the work of Refs.  and , where the Direct Numerical Simulation (DNS) of a jet in crossflow was studied in detail and a stability analysis was performed. Here, we study the development of instabilities using the tools of stability analysis as the ratio of the jet velocity to the crossflow velocity, denoted R, is increased.
Practical applications of jets in crossflow are numerous, and an overview of research over the past few decades can be found in Ref. . One application is film cooling of blades or combustor walls in turbines and jet engines, which is achieved by injecting jets of coolant air from holes in the blade or combustor surface to form a cooling film. It is desirable to achieve efficient cooling at a low inflow velocity ratio R, as this would allow a low flow rate of coolant air, thereby improving turbine efficiency, since the coolant air is usually taken out of the compressed air at turbine inflow. It is therefore important to study the basic physics of this highly complex flow.
The jet is simulated using our in-house code SIMSON, in which the inflow is imposed as an inhomogeneous Dirichlet boundary condition on the wall. While this setup is simplified compared to practical configurations, the DNS simulations reproduce the main physical mechanisms of the jet in crossflow at R=3 , as well as the shedding of hairpin vortices observed by  at low R. Above a critical value of R=0.675, self-sustained oscillation arises immediately downstream of the jet inflow. This oscillation is amplified, resulting in the periodic shedding of hairpin vortices further downstream. As R is increased (see figure below), the structures that are shed become more complex.
We use the extensive set of tools for stability analysis implemented in SIMSON to characterize the instability described above. Using the eigenvectors of the linearized Navier-Stokes operator and its adjoint, we locate a region in the flow known as a `wavemaker’ , which may be interpreted as the core of the instability. This region is important for flow control, since forcing applied there can stabilize or destabilize the flow. The `wavemaker’ is found to be in the shear layer just above the backflow region (see figure below), as predicted by careful studies of the DNS data .
In addition, a video was produced  as part of the Gallery of Fluid Motion for the American Physical Society Division of Fluid Dynamics meeting, held in Long Beach, California, November 21-23. The video was awarded one of the three Milton Van Dyke prizes as an outstanding entry based on artistic value, scientific content, and originality.
A number of additions to the SIMSON code have been made as part of the project: the 3D adjoint linearized Navier-Stokes were added, and the tools for stability analysis have been extended to the 2D-parallelization. The simulations were performed on the Ekman and Lindgren systems at PDC. Further work involves more detailed stability analysis, an investigation of mixing properties using a passive scalar, and a study of passive control of the jet at low R.
 Bagheri, S. , Schlatter, P., Schmid, P. and Henninsgon, D.S., J. Fluid Mech., 624, 33-44, 2009
 Schlatter, P., Bagheri, S. and Henningson, D.S., Theor. Comput. Fluid Dyn., 2010
 Karagozian, A.R., Progress in Energy and Combustion Science, 30, 1-23, 2010
 Ziefle, J., Large-eddy simulation of complex massively-separated turbulent flows, Phd Thesis, ETH Zurich, 2007
 Giannetti, F. and Lucchini, P., J. Fluid Mech., 581, 167-197, 2007
 Ilak, M., Schlatter, P., Bagheri, S., Chevalier, M. and Henningson, D.S., arxXiv:1010.3766v1
 Ilak, M., Schlatter, P., Bagheri, S. and Henningson, D.S., in preparation.