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Continuous adjoint approach to the Spalart-Allmaras turbulence model for incompressible flows. (English) Zbl 1242.76064
Summary: A continuous adjoint formulation for the computation of the sensitivities of integral functions used in steady-flow, incompressible aerodynamics is presented. Unlike earlier continuous adjoint methods, this paper computes the adjoint to both the mean-flow and turbulence equations by overcoming the frequently made assumption that the variation in turbulent viscosity can be neglected. The development is based on the Spalart-Allmaras turbulence model, using the adjoint to the corresponding differential equation and boundary conditions. The proposed formulation is general and can be used with any other integral function. Here, the continuous adjoint method yielding the sensitivities of the total pressure loss functional for duct flows with respect to the normal displacements of the solid wall nodes is presented. Using three duct flow problems, it is demonstrated that the adjoint to the turbulence equations should be taken into account to compute the sensitivity derivatives of this functional with high accuracy. The so-computed derivatives almost coincide with “reference” sensitivities resulting from the computationally expensive direct differentiation. This is not, however, the case of the sensitivities computed without solving the turbulence adjoint equation, which deviates from the reference values. The role of all newly appearing terms in the adjoint equations, their boundary conditions and the gradient expression is investigated, significant and insignificant terms are identified and a study on the Reynolds number effect is included.

MSC:
76F10 Shear flows and turbulence
76M12 Finite volume methods applied to problems in fluid mechanics
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[1] Pironneau, O., On optimum design in fluid mechanics, J fluid mech, 64, 97-110, (1974) · Zbl 0281.76020
[2] Jameson, A., Aerodynamic design via control theory, J sci comput, 3, 233-260, (1988) · Zbl 0676.76055
[3] Anderson WK, Venkatakrishnan V. Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation. AIAA Paper, 97-0643; 1997.
[4] Arian E, Salas MD. Admitting the inadmissible: adjoint formulation for incomplete cost functionals in aerodynamic optimization. NASA/CR-97-206269, ICASE Report No. 97-69; 1997.
[5] Hazra, S.; Schulz, V.; Brezillon, J.; Gauger, N., Aerodynamic shape optimization using simultaneous pseudo-timestepping, J comput phys, 204, 1, 46-64, (2005) · Zbl 1143.76564
[6] Papadimitriou, D.; Giannakoglou, K., A continuous adjoint method with objective function derivatives based on boundary integrals for inviscid and viscous flows, Comput fluids, 36, 325-341, (2007) · Zbl 1177.76369
[7] Othmer, C., A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows, Int J numer methods fluids, 58, 8, 861-877, (2008) · Zbl 1152.76025
[8] Shubin GR, Frank PD. A comparison of the implicit gradient approach and the variational approach to aerodynamic design optimization. Boeing computer services report AMS-TR-163; 1991.
[9] Burgreen, G.W.; Baysal, O., Three-dimensional aerodynamic shape optimization using discrete sensitivity analysis, Aiaa j, 34, 9, 1761-1770, (1996) · Zbl 0909.76082
[10] Elliot J, Peraire J. Aerodynamic design using unstructured meshes. AIAA Paper, 96-1941; 1996.
[11] Duta, M.C.; Giles, M.B.; Campobasso, M.S., The harmonic adjoint approach to unsteady turbomachinery design, Int J numer methods fluids, 40, 3-4, 323-332, (2002) · Zbl 1036.76056
[12] Jameson, A.; Pierce, N.; Martinelli, L., Optimum aerodynamic design using the navier – stokes equations, Theor comput fluid dyn, 10, 213-237, (1998) · Zbl 0912.76067
[13] Mohammadi, B.; Pironneau, O., Applied shape optimization for fluids, (2001), Clarendon Press Oxford · Zbl 0970.76003
[14] Jameson A, Kim S. Reduction of the adjoint gradient formula in the continous limit. AIAA Paper, 2003-0040; 2003.
[15] Soto O, Lohner R. On the computation of flow sensitivities from boundary integrals. AIAA Paper, 2004-0112; 2004.
[16] Papadimitriou, D.I.; Giannakoglou, K.C., Total pressure losses minimization in turbomachinery cascades using a new continuous adjoint formulation, Proc inst mech eng A, 222, 6, 865-872, (2007)
[17] Papadimitriou DI, Giannakoglou KC. A continuous adjoint method for the minimization of losses in cascade viscous flows. AIAA Paper, 2006-0049; 2006.
[18] Papadimitriou DI, Giannakoglou KC. Compressor blade optimization using a continuous adjoint formulation. ASME TURBO EXPO, GT2006/90466, Barcelona; 2006.
[19] Spalart P, Allmaras S. A one-equation turbulence model for aerodynamic flows. AIAA Paper, 92-0439; 1992.
[20] Giles MB, Pierce NA. Adjoint equations in CFD: duality, boundary conditions and solution behaviour. AIAA Paper, 97-1850; 1997.
[21] Giles MB, Pierce NA. An introduction to the adjoint approach to design. In: ERCOFTAC workshop on adjoint methods, Toulouse, 21-23 June 1999.
[22] Anderson, W.K.; Bonhaus, D.L., Airfoil design on unstructured grids for turbulent flows, Aiaa j, 37, 2, 185-191, (1999)
[23] Nielsen, E.J.; Lu, J.; Park, M.A.; Darmofal, D.L., An implicit exact dual adjoint solution method for turbulent flows on unstructured grids, Comput fluids, 33, 1131-1155, (2004) · Zbl 1103.76346
[24] Lee, B.J.; Kim, C., Automated design methodology of turbulent internal flow using discrete adjoint formulation, Aerosp sci technol, 11, 163-173, (2007) · Zbl 1195.76348
[25] Zingg DW, Leung TM, Diosady L, Truong AH, Elias S, Nemec M. Improvements to a Newton-Krylov adjoint algorithm for aerodynamic optimization. AIAA Paper, 2005-4857, 2005.
[26] Dwight, R.P.; Brezillon, J., Effect of approximations of the discrete adjoint on gradient-based optimization, Aiaa j, 44, 12, 3022-3031, (2006)
[27] Mavriplis, D.J., Discrete adjoint-based approach for optimization problems on three-dimensional unstructured meshes, Aiaa j, 45, 4, 740-750, (2007)
[28] Kim, C.S.; Kim, C.; Rho, O.H., Feasibility study of constant eddy-viscosity assumption in gradient-based design optimization, J aircraft, 40, 6, 1168-1176, (2003)
[29] Caretto LS, Gisman AD, Patankar SV, Spalding DB. Two calculation procedures for steady three-dimensional flows with recirculation. In: Proceedings of the third international conference on numerical methods in fluid dynamics, Paris; 1972.
[30] Saad Y. Iterative methods for sparse linear systems. Electronic Edition (copyright by Y. Saad); 2000.
[31] Nadarajah SK, Jameson A. Studies of the continuous and discrete adjoint approaches to viscous automatic aerodynamic shape optimization. AIAA Paper, 2001-2530; 2001.
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