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Shape optimization of corrugated airfoils

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Abstract

The effect of corrugations on the aerodynamic performance of a Mueller C4 airfoil, placed at a \(5^{\circ }\) angle of attack and \(Re=10{,}000\), is investigated. A stabilized finite element method is employed to solve the incompressible flow equations in two dimensions. A novel parameterization scheme is proposed that enables representation of corrugations on the surface of the airfoil, and their spontaneous appearance in the shape optimization loop, if indeed they improve aerodynamic performance. Computations are carried out for different location and number of corrugations, while holding their height fixed. The first corrugation causes an increase in lift and drag. Each of the later corrugations leads to a reduction in drag. Shape optimization of the Mueller C4 airfoil is carried out using various objective functions and optimization strategies, based on controlling airfoil thickness and camber. One of the optimal shapes leads to 50 % increase in lift coefficient and 23 % increase in aerodynamic efficiency compared to the Mueller C4 airfoil.

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References

  1. Mohammadi B, Pironneau O (2004) Shape optimization in fluid mechanics. Annu Rev Fluid Mech 36:255–279

    Article  MathSciNet  Google Scholar 

  2. Giles MB, Pierce NA (2000) An introduction to the adjoint approach to design. Flow Turbul Combust 65(3–4):393–415

    Article  MATH  Google Scholar 

  3. Katamine E, Azegami H, Tsubata T, Itoh S (2005) Solution to shape optimization problems of viscous flow fields. Int J Comput Fluid Dyn 19(1):45–51

    Article  MATH  MathSciNet  Google Scholar 

  4. Anderson WK, Venkatakrishnan V (1999) Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation. Comput Fluids 28(4):443–480

    Article  MATH  Google Scholar 

  5. Okumura H, Kawahara M (2000) Shape optimization of body located in incompressible Navier–Stokes flow based on optimal control theory. Comput Modell Eng Sci (CMES) 1(2):71–77

    MathSciNet  Google Scholar 

  6. Mohammadi B (2004) Optimization of aerodynamic and acoustic performances of supersonic civil transports. Int J Numer Methods Heat Fluid Flow 14(7):893–909

    Article  MATH  Google Scholar 

  7. Reuther J, Jameson A, Farmer J, Martinelli L, Saunders D (1996) Aerodynamic shape optimization of complex aircraft configurations via an adjoint formulation. Research Institute for Advanced Computer Science, NASA Ames Research Center

  8. Jameson A (1989) Computational aerodynamics for aircraft design. Science (Washington) 245(4916):361–371

    Article  Google Scholar 

  9. Nadarajah SK, Kim S, Jameson A, Alonso JJ (2003) Sonic boom reduction using an adjoint method for supersonic transport aircraft configurations. In: IUTAM symposium transsonicum IV. Springer, Berlin, pp 355–362

  10. Soto O, Löhner R, Yang C (2004) An adjoint-based design methodology for CFD problems. Int J Numer Methods Heat Fluid Flow 14(6):734–759

    Article  MATH  Google Scholar 

  11. Abraham F, Behr M, Heinkenschloss M (2005) Shape optimization in steady blood flow: a numerical study of non-newtonian effects. Comput Methods Biomech Biomed Eng 8(2):127–137

    Article  Google Scholar 

  12. Srinath DN, Mittal S (2007) A stabilized finite element method for shape optimization in low Reynolds number flows. Int J Numer Methods Fluids 54(12):1451–1471

    Article  MATH  MathSciNet  Google Scholar 

  13. Srinath DN, Mittal S (2009) Optimal airfoil shapes for low Reynolds number flows. Int J Numer Methods Fluids 61(4):355–381

    Article  MATH  MathSciNet  Google Scholar 

  14. Srinath DN, Mittal S, Manek V (2009) Multi-point shape optimization of airfoils at low Reynolds numbers. Comput Model Eng Sci (CMES) 51(2):169

    MATH  MathSciNet  Google Scholar 

  15. Srinath DN, Mittal S (2010a) An adjoint method for shape optimization in unsteady viscous flows. J Comput Phys 229(6):1994–2008

    Article  MATH  MathSciNet  Google Scholar 

  16. Srinath DN, Mittal S (2010b) Optimal aerodynamic design of airfoils in unsteady viscous flows. Comput Methods Appl Mech Eng 199(29):1976–1991

    Article  MATH  MathSciNet  Google Scholar 

  17. Diwakar A, Srinath DN, Mittal S (2010) Aerodynamic shape optimization of airfoils in unsteady flow. Comput Model Eng Sci 69(1):61

    Google Scholar 

  18. Kumar N, Diwakar A, Attree SK, Mittal S (2013) A method to carry out shape optimization with a large number of design variables. Int J Numer Methods Fluids 71(12):1494–1508

    Article  Google Scholar 

  19. Mittal S, Bhatt V, Srinath DN (2015) Aerodynamic shape optimization using stabilized finite element method. Math Models Methods Appl Sci 25:1540010

    Article  MathSciNet  Google Scholar 

  20. Pelletier A, Mueller TJ (2000) Low Reynolds number aerodynamics of low-aspect-ratio, thin/flat/cambered-plate wings. J Aircr 37(5):825–832

    Article  Google Scholar 

  21. Shyy W, Lian Y, Tang J, Viieru D, Liu H (2008) Aerodynamics of low Reynolds number flyers. Cambridge University Press, New York

    Google Scholar 

  22. Shyy W, Berg M, Ljungqvist D (1999) Flapping and flexible wings for biological and micro air vehicles. Prog Aerosp Sci 35(5):455–505

    Article  Google Scholar 

  23. Harmon RL (2008) Aerodynamic modeling of a flapping membrane wing using motion tracking experiments. ProQuest

  24. Takizawa K, Henicke B, Puntel A, Kostov N, Tezduyar TE (2012) Space-time techniques for computational aerodynamics modeling of flapping wings of an actual locust. Comput Mech 50(6):743– 760

    Article  MATH  Google Scholar 

  25. Takizawa K, Tezduyar TE, Kostov N (2014) Sequentially-coupled space-time FSI analysis of bio-inspired flapping-wing aerodynamics of an MAV. Comput Mech 54(2):213–233

    Article  MATH  MathSciNet  Google Scholar 

  26. Takizawa K, Tezduyar TE, Buscher A (2015) Space-time computational analysis of MAV flapping-wing aerodynamics with wing clapping. Comput Mech 55(6):1131–1141

    Article  Google Scholar 

  27. Rees CJ (1975) Aerodynamic properties of an insect wing section and a smooth aerofoil compared. Nature 258:141–142

    Article  Google Scholar 

  28. Okamoto M, Yasuda K, Azuma A (1996) Aerodynamic characteristics of the wings and body of a dragonfly. J Exp Biol 199(2):281–294

    Google Scholar 

  29. Levy DE, Seifert A (2009) Simplified dragonfly airfoil aerodynamics at Reynolds numbers below 8000. Phys Fluids 21(7):071,901

    Article  Google Scholar 

  30. Levy DE, Seifert A (2010) Parameter study of simplified dragonfly airfoil geometry at Reynolds number of 6000. J Theor Biol 266(4):691–702

    Article  Google Scholar 

  31. Brooks AN, Hughes TJR (1982) Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 32(1):199–259

    Article  MATH  MathSciNet  Google Scholar 

  32. Hughes TJR, Tezduyar TE (1984) Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations. Comput Methods Appl Mech Eng 45(1):217–284

    Article  MATH  MathSciNet  Google Scholar 

  33. Hughes TJR, Franca L, Balestra M (1986) A new finite element formulation for computational fluid dynamics V. Circumventing the Babuska–Brezzi condition: a stable Petrov–Galerkin formulation for the Stokes problem accommodating equal-order interpolations. Comput Methods Appl Mech Eng 59:85–99

    Article  MATH  MathSciNet  Google Scholar 

  34. Tezduyar TE, Mittal S, Ray S, Shih R (1992) Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements. Comput Methods Appl Mech Eng 95(2):221–242

    Article  MATH  Google Scholar 

  35. Hughes TJR (1995) Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput Methods Appl Mech Eng 127:387–401

  36. Hughes TJR, Feijoo G, Mazzei L, Quincy J (1998) The variational multiscale method: a paradigm for computational mechanics. Comput Methods Appl Mech Eng 166:3–24

    Article  MATH  MathSciNet  Google Scholar 

  37. Takizawa K, Montes D, McIntyre S, Tezduyar TE (2013) Space-time VMS methods for modeling of incompressible flows at high Reynolds numbers. Math Models Methods Appl Sci 23(02):223–248

    Article  MATH  MathSciNet  Google Scholar 

  38. Bazilevs Y, Takizawa K, Tezduyar TE (2015) New directions and challenging computations in fluid dynamics modeling with stabilized and multiscale methods. Math Models Methods Appl Sci 25:1502002

    MathSciNet  Google Scholar 

  39. Byrd RH, Lu P, Nocedal J, Zhu C (1995) A limited memory algorithm for bound constrained optimization. SIAM J Sci Comput 16(5):1190–1208

    Article  MATH  MathSciNet  Google Scholar 

  40. Savaliya SB, Kumar SP, Mittal S (2010) Laminar separation bubble on an Eppler 61 airfoil. Int J Numer Methods Fluids 64(6):627–652

    MATH  MathSciNet  Google Scholar 

  41. Mittal S (2000) On the performance of high aspect ratio elements for incompressible flows. Comput Methods Appl Mech Eng 188(1):269–287

    Article  MATH  Google Scholar 

  42. Tezduyar TE, Osawa Y (2000) Finite element stabilization parameters computed from element matrices and vectors. Comput Methods Appl Mech Eng 190:411–430

    Article  MATH  Google Scholar 

  43. Tezduyar TE (2003) Computation of moving boundaries and interfaces and stabilization parameters. Int J Numer Methods Fluids 43(5):555–575

    Article  MATH  MathSciNet  Google Scholar 

  44. Akin J, Tezduyar TE, Ungor M, Mittal S (2003) Stabilization parameters and Smagorinsky turbulence model. J Appl Mech 70(1):2–9

    Article  MATH  Google Scholar 

  45. Rispoli F, Corsini A, Tezduyar TE (2007) Finite element computation of turbulent flows with the discontinuity-capturing directional dissipation (DCDD). Comput Fluids 36(1):121–126

    Article  MATH  Google Scholar 

  46. Hsu MC, Bazilevs Y, Calo VM, Tezduyar TE, Hughes TJR (2010) Improving stability of stabilized and multiscale formulations in flow simulations at small time steps. Comput Methods Appl Mech Eng 199(13):828–840

    Article  MATH  MathSciNet  Google Scholar 

  47. Bazilevs Y, Takizawa K, Tezduyar TE (2013) Computational fluid-structure interaction: methods and applications. Wiley, Hoboken

    Book  Google Scholar 

  48. Tezduyar TE, Behr M, Mittal S, Johnson A (1992) Computation of unsteady incompressible flows with the stabilized finite element methods: space-time formulations, iterative strategies and massively parallel implementations. ASME Press Vessels Pip Div Publ PVP 246:7–24

    Google Scholar 

  49. Tezduyar T, Aliabadi S, Behr M, Johnson A, Mittal S (1993) Parallel finite-element computation of 3D flows. Computer 26(10):27–36

    Article  Google Scholar 

  50. Johnson AA, Tezduyar TE (1994) Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces. Comput Methods Appl Mech Eng 119(1):73–94

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, UP, 208 016, India

    Sambhav Jain, Varun Dhananjay Bhatt & Sanjay Mittal

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  1. Sambhav Jain
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  2. Varun Dhananjay Bhatt
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  3. Sanjay Mittal
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Correspondence to Sanjay Mittal.

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Jain, S., Bhatt, V.D. & Mittal, S. Shape optimization of corrugated airfoils. Comput Mech 56, 917–930 (2015). https://doi.org/10.1007/s00466-015-1210-x

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