Low-dimensional models of a temporally evolving free shear layer

Abstract

We develop low-dimensional models for the evolution of a free shear layer in a periodic domain. The goal is to obtain models simple enough to be analysed using standard tools from dynamical systems theory, yet including enough of the physics to model nonlinear saturation and energy transfer between modes (e.g. pairing). In the present paper, two-dimensional direct numerical simulations of a spatially periodic, temporally developing shear layer are performed. Low-dimensional models for the dynamics are obtained using a modified version of proper orthogonal decomposition (POD)/Galerkin projection, in which the basis functions can scale in space as the shear layer spreads. Equations are obtained for the rate of change of the shear-layer thickness. A model with two complex modes can describe certain single-wavenumber features of the system, such as vortex roll-up, nonlinear saturation, and viscous damping. A model with four complex modes can describe interactions between two wavenumbers (vortex pairing) as well. At least two POD modes are required for each wavenumber in space to sufficiently describe the dynamics, though, for each wavenumber, more than 90% energy is captured by the first POD mode in the scaled space. The comparison of POD modes to stability eigenfunction modes seems to give a plausible explanation. We have also observed a relation between the phase difference of the first and second POD modes of the same wavenumber and the sudden turning point for shear-layer dynamics in both direct numerical simulations and model computations.

Low-dimensional models for compressible temporally developing shear layers

“…The simulation code has been extensively validated and used in our previous work in both SD shear layers (Wei & Freund 2006;Wei et al 2012) and TD shear layers (Wei & Rowley 2009). As also shown in figure 1, a two-dimensional shear layer is simulated in a rectangular computational domain of 0 < x < 5π and −50 < y < 50 with 128 × 800 mesh points along x and y directions.…”

“…Rowley & Marsden (2000) applied the idea of symmetry reduction from geometric mechanics to factor out the slow travelling solution, which was generalized to other symmetry groups later (Rowley et al 2003). For shear flows, it has been shown in our previous work on incompressible flows (Wei & Rowley 2009;Wei et al 2012) that more efficient models can be achieved in a new space where the viscous growth of shear layers is removed through symmetry reduction. Here, a similar idea of constructing POD modes on a symmetry-reduced space is applied on compressible flows.…”

“…It is worth noting that the current definition of scaled variable is simpler than that used in the incompressible model (Wei & Rowley 2009), where a special coefficient matrix was added to re-enforce the divergence free condition for incompressibility. Choosing the initial velocity profile q 0 = (u 0 , v 0 , a 0 ) as the template, we define g(t) by…”

“…We need to point out that the inner product used here to compute POD modes is the same as that used later in Galerkin projection, unlike the case of the incompressible flow where two different inner products were used (Wei & Rowley 2009). So, the orthogonality of POD modes is now preserved when flow equations are projected onto these bases.…”

“…Proper orthogonal decomposition (POD)-Galerkin projection, as a classic approach, has been successful in many cases (Noack & Eckelmann 1994;Holmes, Lumley & Berkooz 1996;Noack et al 2003;Rowley, Colonius & Murray 2004). Recently, Wei & Rowley (2009) used a modified POD-Galerkin approach to model an incompressible TD shear layer by only two modes for each characteristic frequency. The key idea was to factor out the thickness growth first and then apply POD-Galerkin projection more efficiently in the new symmetry-reduced space, which is related to the technique used for travelling solutions by Rowley & Marsden (2000), and self-similar solutions by Rowley et al (2003).…”