zbMATH — the first resource for mathematics

Adjoint wall functions: a new concept for use in aerodynamic shape optimization. (English) Zbl 1346.76059
Summary: The continuous adjoint method for the computation of sensitivity derivatives in aerodynamic optimization problems of steady incompressible flows, modeled through the \(k-\varepsilon \) turbulence model with wall functions, is presented. The proposed formulation leads to the adjoint equations along with their boundary conditions by introducing the adjoint to the friction velocity. Based on the latter, an adjoint law of the wall that bridges the gap between the solid wall and the first grid node off the wall is proposed and used during the solution of the system of adjoint (to both the mean flow and turbulence) equations. Any high Reynolds turbulence model, other than the \(k-\varepsilon \) one used in this paper, could also profit from the proposed adjoint wall function technique. In the examined duct flow problems, where the total pressure loss due to viscous effects is used as objective function, emphasis is laid on the accuracy of the computed sensitivity derivatives, rather than the optimization itself. The latter might rely on any descent method, once the objective function gradient has accurately been computed.

76G25 General aerodynamics and subsonic flows
76N25 Flow control and optimization for compressible fluids and gas dynamics
76D05 Navier-Stokes equations for incompressible viscous fluids
76F65 Direct numerical and large eddy simulation of turbulence
49J99 Existence theories in calculus of variations and optimal control
65K10 Numerical optimization and variational techniques
Full Text: DOI
[1] R. Martins, I.M. Kroo, J. Alonso, An automated method for sensitivity analysis using complex variables, AIAA Paper 2000-0689, 2000.
[2] J.C. Newman, W.K. Anderson, D.L. Whitfield, Multidisciplinary sensitivity derivatives using complex variables, Tech. Rep. MSSU-COE-ERC-98-08.
[3] Hazra, S.; Schulz, V.; Brezillon, J.; Gauger, N., Aerodynamic shape optimization using simultaneous pseudo-timestepping, Journal of computational physics, 204, 1, 46-64, (2005) · Zbl 1143.76564
[4] Pironneau, O., On optimum design in fluid mechanics, Journal of fluid mechanics, 64, 97-110, (1974) · Zbl 0281.76020
[5] Jameson, A., Aerodynamic design via control theory, Journal of scientific computing, 3, 233-260, (1988) · Zbl 0676.76055
[6] W.K. Anderson, V. Venkatakrishnan, Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation, AIAA Paper 97-0643, 1997.
[7] Papadimitriou, D.I.; Giannakoglou, K.C., A continuous adjoint method with objective function derivatives based on boundary integrals for inviscid and viscous flows, Computers and fluids, 36, 325-341, (2007) · Zbl 1177.76369
[8] G.R. Shubin, P.D. Frank, 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 journal, 34, 9, 1761-1770, (1996) · Zbl 0909.76082
[10] J. Elliot, J. Peraire, 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, International journal for numerical methods in fluids, 40, 3-4, 323-332, (2002) · Zbl 1036.76056
[12] Othmer, C., A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows, International journal for numerical methods in fluids, 58, 8, 861-877, (2008) · Zbl 1152.76025
[13] Jameson, A.; Pierce, N.; Martinelli, L., Optimum aerodynamic design using the navier – stokes equations, Theoretical and computational fluid dynamics, 10, 213-237, (1998) · Zbl 0912.76067
[14] Zymaris, A.S.; Papadimitriou, D.I.; Giannakoglou, K.C.; Othmer, C., Continuous adjoint approach to the spalart – allmaras turbulence model for incompressible flows, Computers and fluids, (2008) · Zbl 1242.76064
[15] P. Spalart, S. Allmaras, A one-equation turbulence model for aerodynamic flows, AIAA Paper 92-0439, 1992.
[16] Nielsen, E.J.; Lu, J.; Park, M.A.; Darmofal, D.L., An implicit exact dual adjoint solution method for turbulent flows on unstructured grids, Computers and fluids, 33, 1131-1155, (2004) · Zbl 1103.76346
[17] D.W. Zingg, T.M. Leung, L. Diosady, A.H. Truong, S. Elias, M. Nemec, Improvements to a Newton-Krylov adjoint algorithm for aerodynamic optimization, AIAA Paper 2005-4857, 2005.
[18] Dwight, R.P.; Brezillon, J., Effect of approximations of the discrete adjoint on gradient-based optimization, AIAA journal, 44, 12, 3022-3031, (2006)
[19] Anderson, W.K.; Bonhaus, D.L., Airfoil design on unstructured grids for turbulent flows, AIAA journal, 37, 2, 185-191, (1999)
[20] Lee, B.J.; Kim, C., Automated design methodology of turbulent internal flow using discrete adjoint formulation, Aerospace science and technology, 11, 163-173, (2007) · Zbl 1195.76348
[21] Mavriplis, D.J., Discrete adjoint-based approach for optimization problems on three-dimensional unstructured meshes, AIAA journal, 45, 4, 740-750, (2007)
[22] Kim, C.S.; Kim, C.; Rho, O.H., Feasibility study of constant eddy-viscosity assumption in gradient-based design optimization, Journal of aircraft, 40, 6, 1168-1176, (2003)
[23] C.H. Norris, W.C. Reynolds, Turbulent channel flow with a moving wall boundary, Rept. No. FM-10, Stanford University, Rept. Mech. Eng., 1975.
[24] Rodi, W.; Scheuerer, G., Scrutinizing the k-&z.epsiv; model under adverse pressure gradient conditions, Journal of fluids engineering, 108, 174-179, (1986)
[25] W. Rodi, Experience with two-layer models combining the k-&z.epsiv; model with a one-equation model near the wall, AIAA Paper 91-0216, 1991.
[26] D.L. Sondak, R.H. Pletcher, Application of wall function to generalized nonorthogonal curvilinear coordinate systems, AIAA Paper 93-3107, 1993. · Zbl 0824.76036
[27] Cabot, W.; Moin, P., Approximate wall boundary conditions in the large-eddy simulation of high Reynolds number flow, Flow turbulence and combustion, 63, 269-291, (1999) · Zbl 0981.76045
[28] J.A. Templeton, M. Wang, P. Moin, Towards LES wall models using optimization techniques, Center for Turbulent Research, Annual Research Briefs, 2002.
[29] Pantano, C.; Pullin, D.I.; Dimotakis, P.E.; Matheou, G., LES approach for high Reynolds number wall-bounded flows with application to turbulent channel flow, Journal of computational physics, 227, 9271-9291, (2008) · Zbl 1166.76025
[30] Ng, E.Y.K.; Tan, H.Y.; Lim, H.N.; Choi, D., Near-wall function for turbulence closure models, Computational mechanics, 29, 178-181, (2002) · Zbl 1053.76513
[31] Papadimitriou, D.I.; Giannakoglou, K.C., Aerodynamic shape optimization using adjoint and direct approaches, Archives of computational methods in engineering, 15, 4, 447-488, (2008) · Zbl 1170.76348
[32] S. Nadarajah, A. Jameson, Optimal control of unsteady flows using a time accurate method, AIAA Paper 2002-5436, 2002.
[33] S. Nadarajah, M. McMullen, A. Jameson, Non-linear frequency domain based optimum shape design for unsteady three-dimensional flow, AIAA Paper 2002-2838, 2002.
[34] A.L. Marsden, M. Wang, J.E. Dennis, P. Moin, Suppression of vortex shedding noise via derivative-free shape optimization, Stanford University. · Zbl 1187.76332
[35] Jones, W.; Launder, B., The prediction of laminarization with a two-equation model of turbulence, Int. J. heat mass transfer, 15, 301-314, (1972)
[36] Chorin, A.J., A numerical method for solving incompressible viscous flow problems, Journal of computational physics, 2, 12-26, (1967) · Zbl 0149.44802
[37] Roe, P., Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of computational physics, 43, 357-371, (1981) · Zbl 0474.65066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.