The nonlinear evolution of centrifugal instabilities in curved, compressible mixing layers

Abstract

The nonlinear development of Görtler vortices in a curved compressible mixing layer is studied. It has been shown both experimentally and theoretically that the curved mixing layer can support a centrifugal mode, which is believed to be similar to the Görtler vortex. This study follows the corresponding incompressible study of Seddougui & Otto [Proceedings of the IUTAM Symposium on Nonlinear Instability and Transition in Three-Dimensional Boundary Layers, edited by P. W. Duck and P. Hall (Kluwer Academic, Amsterdam, 1995)] and attempts to demonstrate the effect compressibility has on the growth of such modes. The formulation follows that of Hall & Lakin [Proc. R. Soc. London, Ser. A 415, 421 (1988)] for the corresponding incompressible boundary-layer problem, and utilises a stream function formulation to solve the governing equations. We will present the nonlinear structure of the so-called thermal modes, whose linear counterparts were first discovered in Owen, Seddougui & Otto [Phys. Fluids 9, 2506 (1997)]. A brief discussion will be made concerning the effect of the nonlinear modes on the inherent Kelvin–Helmholtz modes of instability, which may have significant ramifications in flows dominated by this form of instability.

Centrifugal instabilities in curved compressible wakes

“…The problem can be extended to suit a nonlinear investigation on the effect of the curvature for both the stably curved and the unstably curved wake-dominated curved compressible mixing layers. This has been considered in the absence of a wake by Owen et al 8 Having studied the behavior of the Görtler vortex, it would be interesting to conduct a similar investigation on the Kelvin-Helmholtz and Rayleigh-Taylor instabilities within the wake-dominated curved compressible system to study how the modes of these instabilities behave when both stably and unstably curved wake-dominated mixing layers are considered.…”

“…IV and verified using the alternative method provided in OSO. 8 The variation in the growth rate of the first mode, ␤ = ␤ 1 , for a particular wave number, a = 5, for both the stably and the unstably curved compressible mixing-layer system under increasing wake effect is demonstrated in Fig. 12͑a͒.…”

“…The nonlinear evolution of Görtler vortices in curved mixing layers was investigated by Seddougui and Otto,7 and they determined that the Görtler vortex has a stabilizing effect on the KelvinHelmholtz instability with increasing Görtler number within the incompressible system. Owen et al 8 later investigated the nonlinear evolution of centrifugal instabilities within curved compressible mixing layers, and they found that the increase in compressibility effect would destabilize the thermal mode and thus render the flow more unstable. Owen et al also investigated the effect of a nonlinear vortex upon the linear development of a Kelvin-Helmholtz mode.…”

Görtler vortices in compressible mixing layers

“…In Owen, Seddougui & Otto (1998), which follows the incompressible study of Seddougui & Otto (1995), the nonlinear structure of the centrifugal instabilities was investigated. The calculation was affected in the parameter régime corresponding to the neighbourhood of the right-hand branch of the neutral curve.…”

“…It was found in both papers that the predominantly velocity-driven mode had the effect of reducing the temporal growth rates of the inviscid Rayleigh modes. However, Owen et al (1998) found that the presence of the thermal modes produced a destabilization of the inviscid modes, see Otto, Sarkies & Denier (2000).…”