1. Comparative study of implicit and subgrid-scale model large-eddy simulation techniques for low-Reynolds number airfoil applications Daniel J. Garmann, Miguel R. Visbal, and Paul D. Orkwis, 1) Air Force Research Laboratory, Wright-Patterson AFB, Dayton, OH 45433, USA 2) University of Cincinnati, Cincinnati, OH 45221, USA INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2013; 71:1546–1565 Published online 23 August 2012 in Wiley Online Library (wileyonlinelibrary.com/journal/nmf). DOI: 10.1002/fld.3725

2. Introduction & Background The purpose of this research was to evaluate ILES by comparison to a LES - dynamic Smagorinsky SGS model using a SD 7003 airfoil with various grid resolutions at 2 Reynolds numbers (60,000 and 120,000) ILES has been applied to a number of canonical turbulence flows over the past decade, with encouraging results Alternative to SGS-based technique, e.g. MILES and ILES Motivation is computational cost gains and feasibility, especially for fast, complicated flows and geometries Consider,

3. Solution Methodology – 3D, compressible Navier-Stokes equations

4. LES – Dynamic Smagorinsky SGS Model Recast into Favre-filtered(mass/density weighted) forms Consequence is the SGS stress and heat-flux components which are appended to the filtered stress tenor and heat flux vector as: For the compressible dynamic SGS model And, the test filter to grid filter width ratio is

5. ILES - Implicit LES method ILES approach is to divide the residual stress, so The spatial truncation error/’numerical’ stress is utilized to remove energy from the resolved scale For each computational step a ‘numerical’ stress is generated and addressed, i.e. implicit The relationship between the ‘numerical’ stress and spatial discretization is direct and crucial – a pth-order accurate results in order h^p for this stress

6. ILES Approaches Many strategies for contributions & numeric's of these 2 stresses (refer to Grinstein et al. (2007)): – MILES (Monotone Integrated LES) uses residual stress = 0, Boris et al (1992); – Flux-Corrected Transport (FCT), Boris and Brook (1973) – Monotonic Upstream Centred Scheme for Conservation Laws (MUSCL) – Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) of Smolarkiewicz (1986) – Essentially non-Oscillatory (ENO) schemes (see Harten et al. (1987) – Spectral Vanishing Viscosity (SVV) method of Tadmor (1989) – Etc. etc. – Details for above references can be found in the last reference listed in this presentation

7. ILES Technique Applied - Spatial FDL3DI solver (Developed at Air Force Research Lab) Finite difference method discretizes governing equations Spatial derivatives are obtained with higher-order compact differencing schemes, in this case 6 th order Primes are derivatives and φ is some metric Utilizes computational gains of a tri-diagonal system

8. ILES Technique Applied - Filter To ‘eliminate spurious elements of the solution’, a Padé type, low-pass spatial filtering technique is applied sequentially in each direction by incorporation in a sub-iteration ‘free parameter’ – Implicit filter transfer function is designed to construct filters on uniform meshes that: – Are non-dispersive – Do not amplify any waves – Preserve constant functions – Completely eliminate the odd-even mode

9. ILES Technique Applied – Temporal, etc. Time marching utilizes iterative, implicit approximately factored integration method with 4 th -order accuracy Simplified with diagonalization Supplemented with Newton-like sub-iterations to achieve 2 nd order accuracy 4 th order, non-linear dissipative terms are appended to augment stability Special attention to boundary numerical techniques

10. Application - Computational Mesh 304 points are projected about 100 chords from the airfoil shape to a circular far-field boundary Note that the mesh resolution is purposefully coarser than typical in ILES to extenuate differences with SGS

11. Boundary Conditions Boundary on airfoil is no-slip, adiabatic with 4 th -order, extrapolated zero normal pressure gradient Free-stream prescribed on far-field boundary where meshes are stretched rapidly On branch cut and span-wise boundaries spatial periodicity is imposed

12. Case 1 - Re = 60 000, Incidence angle = 8% These conditions produce separation at about 2% chord length and reattachment at just after 25% chord Initially determined the filter coefficient which controls the degree of filtering by the spatial filtering operator Plot is effect of coefficient on filter transfer function at frequencies

13. Case 1 - Results Evaluation of stream-wise mesh resolution For the fine mesh there is distinct agreement in the values found for: – Stream-wise velocity and time-averaged u-velocity contours – Turbulent kinetic energy contours – Force coefficients Next are the time mean stream-wise velocity and squared fluctuations of u-velocity profiles with various mesh resolutions

14. Results ILES upper profiles (a), SGS lower profiles (b)

15. Mean lift, drag, and moment coefficients Skin friction distributions

16. Effect of stream-wise mesh resolution on eddy viscosity coefficient

17. Temporal energy spectra (spatial energy spectra were also analyzed)

18. Case 1b – Coarse span-wise mesh Evaluation of span-wise mesh resolution The pevious analysis showed that stream-wise grid resolution is ‘insignificant’ between the 2 techniques, with or without SGS Revealed most spatial energy in first 20 wave numbers across span and thus a mesh was generated with reduced resolution to investigate if the SGS model could compensate for the lack of higher wave number content. Filter coefficient in the span-wise direction was set at 0.475

19. Effect of SGS on spatial and temporal energy spectra

20. Case 2 - Re = 120 000 & Incidence angle = 8% This investigation was aimed at examining the upper extreme of low-Reynolds number, airfoil applications The fine mesh with reduced span-wise resolution was used The flow re-attachment is sooner in this case, about 15% chord length Smaller and higher concentration of turbulent kinetic energy All four metrics of comparison showed little or no difference

21. Application - Conclusion No significant differences in flow characterization was found between the computations conducted with or without the inclusion of the dynamic Smagorinsky model for the various configurations investigated. This indicates that ILES may be viewed as a robust and computationally efficient modeling strategy, particularly for low- Reynolds number flow around airfoils. The SGS model was found to increase the computational cost by a factor of 2 Limiting or ramping the start-up eddy viscosity coefficient was sometimes needed to stabilize the SGS model ILES was endorsed to provide adequate dissipation using the filter operator applied More studies at high Reynolds numbers and greater range of spatial resolutions are recommended

22. ILES Summary Directly coupled to spatial discretization High order numerical accuracy is important Implicit methods can provide better stability, convergence, ‘compactness’, etc. so various tactics are used to reduce the associated computational cost Facilitates simulations of more complex flows within the bounds of contemporary computational limits

23. References Comparative study of implicit and SGS model large-eddy simulation techniques for low Reynolds number airfoil applications, Garmann, Visbal & Orkwis, 2012, Int. J. Numer. Meth. Fluids 2013; 71:1546-1565 Turbulent Flows, Pope, Cambridge University Press, 2000, ISBN 978-0-521- 59886-6 Elements of direct and large-eddy simulations, Geurts, Edwards, 2004, ISBN 1- 930217-07-2 Computational Methods for Fluid Dynamics, Ferziger & Peric, Springer, 2002, ISBN 978-3-540-42074-3 AAIA 99-0557 Further Development of a Navier-Stokes Solution Procedure Based on Higher-Order Fomulas, Gaitonde & Visbal, Numerical Techniques for Direct and Large-Eddy Simulations, Jiang & Lai, CRC Press, 2009, ISBN 978-1-4200-7578-6 Direct and Large-eddy Simulation VI, ERCOFTAC SERIES, 2006, ISBN 10-1-4020- 4909-9 Implicit Turbulence Modeling for Large-Eddy Simulation, Hickel, Technische Universitat Munchen, http://www.academia.edu/836591/Implicit_Turbulence_Modeling_for_Large- Eddy_Simulation

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