Acknowledgments:

1. Acknowledgments: • T. Craft & D. Laurence at The University of Manchester • CTR Summer Program 2006 • DESider FP6 EU Project • EDF

2. Introduction • Description of Wingtip vortices • Testcase details (Chow et al., 1997) • Development of Curvature Corrected v2-f model • Validation • Results with structured grid • Development of Stress-Strain Lag model • Validation • Results with unstructured grid • Conclusions

3. Wingtip vortices • A wingtip vortex is set up as a result of high pressure flow under the foil escaping to the lower pressure regions above it, around the wingtip. • Result is an anti-clockwise Trailing vortex in wake of wingtip (as viewed from front) • Wingtip Vortices can reduce efficiency of the wing and increase drag. • Accurate prediction of the downstream extent of the highly rotational motion is crucial for safe aircraft separation distances. • Complex separation is a challenge for turbulence models and grid requirements are stringent.

4. Geometry matches the experimental work by Chow et al. (1997) • NACA 0012 wing section at angle of attack of 10degrees. • Reynolds number of 4.35x106 • Transition forced at 4% chord NLEVM EXPT EVM RSM (TCL) Craft et al, IJHFF 27 684-695 (2006) • Results from Linear and Non-linear EVM are found to exhibit a far too rapid decay of the vortex core. • A Reynolds stress transport model (RSM) reproduces the principal features found in the experimental measurements. Testcase details • Previous work established that standard RANS models over-predict turbulence levels in a vortex, and consequently over-predict the decay rate of the vortex downstream of the wing. (Craft et al 2006)

5. To account for frame-rotation (FR) effects in the v2-f model, a strain sensitive version of the eddy viscosity coefficient, C, was proposed (Petterson-Reif et al., 1999). • Based on equilibrium solution of homogenous plane shear under orthogonal rotation • Functional form dependent upon invariants of Strain and Vorticity. • Analytic case of a simple line vortex (Lamb-Oseen). • Reduction of turbulent viscosity at vortex core. • However, viscosity levels are higher outside the vortex core. Modified v2-f model • A range of studies have investigated the inclusion of rotational/curvature effects into a turbulence model. • emphasis on 2 or 3 equation models to conserve cost economy versus full RSM.

6. Where the anti-symmetric tensor is computed according to Wallin & Johansson (2002) as: • The rate of time variation of the strain rate tensor (DS/Dt) is seen to be important.(Kozolv et al 2003). • Curvature sensitivity added to the v2-f model by Duraisamy et al. (2005) using an objective vorticity tensor similar to Gatski (2000) and Hellsten (2002), as: • The extra curvature sensitive terms cause a further reduction of viscosity over a wider radial region. Modified v2-f model

7. v2f (FR) Baseline model v2f (FR + CC) Modified v2-f model: Results • Similar trend in results obtained for full wingtip case on a structured grid of ~9M cells. • Plots of normalized turbulent viscosity (Duraisamy et al. 2005) • Plane located downstream of trailing edge (at x/c = 0.246)

8. Axial and vertical velocity over wing Axial and vertical velocity downstream of wing • x/c =-0.114 • x/c = 0.456 v2-f v2-f (FR) v2-f (FR+CC) EXPT (Chow et al) Modified v2-f model: Results • The v2-f Curvature Corrected model gives a good comparison to experimental data. • Influence of Curvature Correction observed downstream of trailing edge.

9. A transport equation for a parameter that provides a measure of the misalignment of the tensors of stress anisotropy, and strain, . • The misalignment parameter is defined as: Which will be zero when and are mutually perpendicular • The implemented form of the transport equation is derived from a pressure strain model: • Can be incorporated into the two equation SST model by modifying the turbulent viscosity: DS/Dt term Leading to the 3-equation model, SST-Cas The Stress-Strain Lag model • Another method is applied, initially developed for unsteady mean flows (Revell et al. 2005)

10. Model implemented into Code_Saturne • - 3D unstructured finite volume code (open source) • Applied to a range of unsteady flows: • - homogenous cyclic strain, oscillating channel, airfoil at high incidence, around circular cylinder • Third equation found to add a 10-15% cost compared to standard 2 equation EVM model. • standard RSM is around 80% more expensive than a 2-equation model • Reproduces similar results to Reynolds Stress Model: (7-eqns) (3-eqns) (2-eqns) • 2D Flowfield around NACA0012 at 20o, Re= 105 Development of model

11. SST • Black lines are DNS data at fixed time intervals after initialisation. • The 2 equation SST model shows a over-predicted rate of decay as expected. • The RSM and Stress-Strain Lag model (SST-Cas) are able to capture correctly the slower decay rate of the vortex. • Turbulence slightly underpredicted at core with SST-Cas model. Uq Ux SST-Cas Validation of SST-Cas model • Applied to an isolated decaying vortex, initialised from DNS data of Duraisamy et al. (2006) RSM

12. Unstructured mesh using ‘Hanging Nodes’: ~ 1.2 Million cells, using in-house mesh generator from Stanford. • Run at Manchester using Code_Saturne Unstructured Grid • Wingtip case requires fine grid resolution of both the very thin boundary layer over the wing, and the core of the vortex in the wake. • Both regions are crucial in order to correctly model the flow. • Structured meshes for this case quite large: • RANS: ~ 9.3 MillionDuraisamy (2005) • LES at 10x Lower Re: ~ 26 Million Uzun (2006)

13. Axial velocity Standard SST RSM SST-Cas EXPT (Chow et al) • Under-prediction could be a result of under-refinement at the vortex core or other grid issues. Results with the SST-Cas • Results from Unstructured Grid: plane location x/c = 0.456 downstream of trailing edge. EXPT RSM SST SST-Cas

14. Turbulent kinetic energy Tangential velocity EXPT RSM EXPT RSM SST SST-Cas SST-Cas SST • Turbulence is under-predicted. • Misalignment effects are too strong. • Similar findings in earlier results. (Craft et al., 2006) Unstructured Grid

15. Conclusions • Vortical flows have been examined: in particular, the case of the trailing vortex in the wake of a wingtip • Standard RANS models (both EVM and NLEVM) over-predict turbulence at the vortex core, and therefore under predict the downstream extent of the vortex. • The Curvature Corrected v2-f model has been shown to significantly improve the prediction of the downstream vortex. • The Stress-Strain Lag model (SST-Cas) has been shown to give substantial improvement over the baseline two equation model. • Originally developed for unsteady mean flows • First validated for the decaying isolated vortex: results similar to Reynolds Stress Model. • These findings suggest that the inclusion of the advection of the strain is important in these flows. • An unstructured mesh has been used, enabling grid economies of around 80-90% • Optimisation of the unstructured mesh • Further work will focus on the generalisation of the modelling work for application to other models.