Abstract
In the paper, a new inverse method for viscous 2D laminar flows is developed. The method is based on incompressible Navier–Stokes equations transformed to the stream-function coordinate system (von Mises coordinates). The flow design problem with appropriate boundary conditions is formulated and solved numerically. The geometrical shape of the boundary is obtained through the integration along streamlines. The method may be coupled with a flow analysis solver to model the influence of known parts of geometry. Results for two analytically solvable cases (the Poiseuille and the Jeffery–Hamel flows) are presented. Then, the foil design problem is considered as an example. Potential applications and developments towards axisymmetric and 3D flows are discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Dulikravich, G.S.: Shape inverse design and optimization for three dimensional aerodynamics. AIAA Paper 95–0695 (1995)
Takahashi, S., Obayashi, S., Nakahashi, K.: Inverse optimization of transonic wing shape for mid-size regional aircraft. AIAA Paper 98–0601 (1998)
Puzyrewski, R.: 14 Lectures on Turbine Stages Theory–Two Dimensional Model [in Polish]. Wydawnictwo PG, Gdańsk (1998)
Rosa Taddei, S., Larocca, F., Bertini, F., Spano, E.: Euler inverse throughflow model based on an implicit upwind time marching technique. 26th International Congress of the Aeronautical Science (2008)
Ferlauto M., Marsilio R.: A viscous inverse method for aerodynamic design. Comput. Fluids 35, 304–325 (2006)
Ramamurthy, R., Roidl, B., Ghaly, W.: A viscous inverse design method for internal and external flow over airfoils using CFD techniques. V ECCOMAS CFD (2010)
von Mises R.: Bemerkungen zur Hydrodynamik. ZAMM 7, 425–431 (1927)
Stanitz, J.D.: Design of two-dimensional channels with prescribed velocity distribution along the channel walls. NACA TN, 2595 (1952)
Martin M.H.: The flow of a viscous fluid. Arch. Rat. Mech. Anal. 41, 266–286 (1971)
Huang C.Y., Dulikravich G.S.: Stream function and stream function coordinate (SFC) formulation for inviscid flow field calculation. Comput. Meth. Appl. Mech. Eng. 59, 155–177 (1986)
Latypov, A.M.: Streamline-aligned orthogonal coordinates. IMA Prepr. Ser. 1182 (1993)
Hamdan, M.H.: Recent developments in the von Mises transformation and its applications in the computational sciences. 11th WSEAS International Conference on Mathematics Methods, Computational Techniques and Intelligence System (2009)
Shen M., Liu Q., Zhang Z.: Calculation of three dimensional transonic flow in turbomachinery with generalized von Mises coordinate system. Sci. China (Series A) 39, 1084–1095 (1996)
Dulikravich, G.S.: Aerodynamic shape design using stream-function-coordinate (SFC) formulation. AIAA Paper 91–0189 (1991)
Keller J.J.: Inverse Euler equations. ZAMP 49, 363–383 (1998)
Keller J.J.: Inverse equations. Phys. Fluids 11, 513–520 (1999)
Scascighini A., Troxler A., Jeltsch R.: A numerical method for inverse design based on the inverse Euler equations. Int. J. Numer. Methods Fluids 41, 339–355 (2003)
Zannetti L.: A natural formulation for the solution of two-dimensional or axis-symmetric inverse problems. Int. J. Numer. Methods Eng. 22, 451–463 (1986)
Giles M.B., Drela M.: Two dimensional transonic aerodynamic design method. AIAA J. 25, 1199–1205 (1987)
Mangler, W.: Die Berechunung eines Tragflügelprofiles mit vorgeschriebener Druckverteilung. Jahrbuch der Deutschen Luftfahrtforschung (1938)
Lighthill, M.J.: A new method of two-dimensional aerodynamic design. Aeronautical Research Council Reports and Memoranda, 2112 (1945)
Alderson T.L., Allan F.M., Hamdan M.H.: On the universality of the von Mises transformation. Int. J. Appl. Math. 20, 109–121 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Butterweck, M., Pozorski, J. Inverse method for viscous flow design using stream-function coordinates. Acta Mech 224, 1801–1812 (2013). https://doi.org/10.1007/s00707-013-0841-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-013-0841-2