Published online by Cambridge University Press: 15 August 2013
A methodology to achieve extremely-low-dimensional models for temporally developing shear layers is extended from incompressible flows to weakly compressible flows. The key idea is to first remove the slow variation (i.e. viscous growth of shear layers) through symmetry reduction, so that the model reduction using proper orthogonal decomposition (POD)-Galerkin projection in the symmetry-reduced space becomes more efficient. However, for the approach to work for compressible flows, thermodynamic variables need to be retained. We choose the isentropic Navier–Stokes equations for the simplicity and the availability of a well-defined inner product for total energy. To capture basic dynamics, the compressible low-dimensional model requires only two POD modes for each frequency. Thus, a two-mode model is capable of representing single-frequency dynamics such as vortex roll-up, and a four-mode model is capable of representing the nonlinear dynamics involving a fundamental frequency and its subharmonic, such as vortex pairing and merging. The compressible model shows similar behaviour and accuracy as the incompressible model. However, because of the consistency of the inner product defined for POD and for projection in the current compressible model, the orthogonality is kept and it results in simple formulation. More importantly, the inclusion of compressibility opens an entirely new avenue for the discussion of compressibility effect and possible description of aeroacoustics and thermodynamics. Finally, the model is extended to different flow parameters without additional numerical simulation. The extension of the compressible four-mode model includes different Mach numbers and Reynolds numbers. We can clearly observe the change in the nonlinear interaction of modes at two frequencies and the associated promotion or delay of vortex pairing by varying compressibility and viscosity. The dynamic response of the low-dimensional model to different flow parameters is consistent with the vortex dynamics observed in experiments and numerical simulation.