Curvature correction and application of the v 2 − f turbulence model to tip vortex flows

Lifting surfaces such as airplane wings and helicopter rotors generate coherent trailing vortices. The persistence of these vortices is known to be a hazard to following aircraft (in airplane wakes) and a significant source of noise and vibration (in helicopters). The physics of the flow is extremely complex in the near-field region of a tip vortex, as the process is largely turbulent, highly three dimensional, and involves multiple cross flow separations as depicted in Fig.1. Streamwise vorticity, mainly in the form of a feeding sheet, is seen to separate from the wing surface and roll-up into the tip vortex along with a variety of minor structures. Downstream of the trailing edge, these structures rapidly evolve into the largely axisymmetric and coherent tip vortex. Many experimental studies on wing tip vortices (Chow et al. (1997)) have reported largely reduced turbulence levels in the vortical core, even in the near-field. This has been attributed to the near-solid body rotation that exists in the inner core. Analytical studies based on linear stability theory of isolated vortices (Jacquin & Pantano (2002) and references therein) have also supported this argument by showing the damping of imposed disturbances in the core. As a result, downstream of the trailing edge, the major diffusion mechanism appears to be laminar rather than turbulent (Zeman (1995)). The high Reynolds numbers, typically O(10), encountered in typical flight conditions renders the cost of Large Eddy Simulations prohibitive. Therefore, one has to resort to robust, high fidelity Reynolds Averaged Navier Stokes (RANS) simulations. Although RANS models cannot be expected to describe accurately the intricate details of the turbulent flow-field, it has been previously demonstrated by Duraisamy (2005) that reliable solutions of the mean flow-field of a tip vortex can be achieved within engineering accuracy. In this work, the v − f turbulence model of Durbin (1991) is used to simulate vortex formation from a wing in a wind tunnel. The key phenomenon of streamline curvature and its effect on the turbulence levels is addressed by means of a correction to the eddy viscosity coefficient. The primary objective of this work is to evaluate the predictive capabilities of baseline and corrected v − f turbulence model as applied to tip vortex flows, and also to compare it with other widely used closures such as the Spalart Allmaras (SA) model and Menter’s SST model.