Fifth International Symposium on Turbulence and Shear Flow Phenomena Advanced Turbulence Modelling of Wingtip Vortices A. Revell, The University of Manchester K. Duraisamy, The University of Glasgow G. Iaccarino, Stanford University Acknowledgments: • T. Craft & D. Laurence at The University of Manchester • CTR Summer Program 2006 • DESider FP6 EU Project • EDF 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 Fifth International Symposium on Turbulence and Shear Flow Phenomena, TUM, Garching, Germany. Aug 27-29 2007 A. Revell, K. Duraisamy & G. Iaccarino 2 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. Fifth International Symposium on Turbulence and Shear Flow Phenomena, TUM, Garching, Germany. Aug 27-29 2007 A. Revell, K. Duraisamy & G. Iaccarino 3 Testcase details • Geometry matches the experimental work by Chow et al. (1997) • NACA 0012 wing section at angle of attack of 10 degrees. • Reynolds number of 4.35x106 • Transition forced at 4% chord • 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) EXPT EVM NLEVM 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. Fifth International Symposium on Turbulence and Shear Flow Phenomena, TUM, Garching, Germany. Aug 27-29 2007 A. Revell, K. Duraisamy & G. Iaccarino 4 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. • 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. Fifth International Symposium on Turbulence and Shear Flow Phenomena, TUM, Garching, Germany. Aug 27-29 2007 A. Revell, K. Duraisamy & G. Iaccarino 5 Modified v2-f model • 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: • 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). • The extra curvature sensitive terms cause a further reduction of viscosity over a wider radial region. Fifth International Symposium on Turbulence and Shear Flow Phenomena, TUM, Garching, Germany. Aug 27-29 2007 A. Revell, K. Duraisamy & G. Iaccarino 6 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) Baseline model v2f (FR) Fifth International Symposium on Turbulence and Shear Flow Phenomena, TUM, Garching, Germany. Aug 27-29 2007 v2f (FR + CC) A. Revell, K. Duraisamy & G. Iaccarino 7 Modified v2-f model: Results • The v2-f Curvature Corrected model gives a good comparison to experimental data. Axial and vertical velocity over wing • x/c =-0.114 Axial and vertical velocity downstream of wing • x/c = 0.456 v2-f v2-f (FR) v2-f (FR+CC) EXPT (Chow et al) • Influence of Curvature Correction observed downstream of trailing edge. Fifth International Symposium on Turbulence and Shear Flow Phenomena, TUM, Garching, Germany. Aug 27-29 2007 A. Revell, K. Duraisamy & G. Iaccarino 8 The Stress-Strain Lag model • Another method is applied, initially developed for unsteady mean flows (Revell et al. 2005) • 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: DS/Dt term • Can be incorporated into the two equation SST model by modifying the turbulent viscosity: Leading to the 3-equation model, SST-Cas Fifth International Symposium on Turbulence and Shear Flow Phenomena, TUM, Garching, Germany. Aug 27-29 2007 A. Revell, K. Duraisamy & G. Iaccarino 9 Development of model • 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: (2-eqns) – (3-eqns) (7-eqns) 2D Flowfield around NACA0012 at 20o, Re= 105 Fifth International Symposium on Turbulence and Shear Flow Phenomena, TUM, Garching, Germany. Aug 27-29 2007 A. Revell, K. Duraisamy & G. Iaccarino 10 Validation of SST-Cas model • Applied to an isolated decaying vortex, initialised from DNS data of Duraisamy et al. (2006) RSM Uq SST Ux • 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. SST-Cas • 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. Fifth International Symposium on Turbulence and Shear Flow Phenomena, TUM, Garching, Germany. Aug 27-29 2007 A. Revell, K. Duraisamy & G. Iaccarino 11 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 Million Duraisamy (2005) • LES at 10x Lower Re: ~ 26 Million Uzun (2006) • Unstructured mesh using ‘Hanging Nodes’: ~ 1.2 Million cells, using in-house mesh generator from Stanford. • Run at Manchester using Code_Saturne Fifth International Symposium on Turbulence and Shear Flow Phenomena, TUM, Garching, Germany. Aug 27-29 2007 A. Revell, K. Duraisamy & G. Iaccarino 12 Results with the SST-Cas • Results from Unstructured Grid: plane location x/c = 0.456 downstream of trailing edge. Axial velocity EXPT SST RSM SST-Cas 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. Fifth International Symposium on Turbulence and Shear Flow Phenomena, TUM, Garching, Germany. Aug 27-29 2007 A. Revell, K. Duraisamy & G. Iaccarino 13 Unstructured Grid Tangential velocity EXPT RSM SST SST-Cas Turbulent kinetic energy EXPT SST RSM SST-Cas • Turbulence is under-predicted. • Misalignment effects are too strong. • Similar findings in earlier results. (Craft et al., 2006) Fifth International Symposium on Turbulence and Shear Flow Phenomena, TUM, Garching, Germany. Aug 27-29 2007 A. Revell, K. Duraisamy & G. Iaccarino 14 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. Fifth International Symposium on Turbulence and Shear Flow Phenomena, TUM, Garching, Germany. Aug 27-29 2007 A. Revell, K. Duraisamy & G. Iaccarino 15

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