Proceedings of Czech–Japanese Seminar in Applied Mathematics 2004 August 4-7, 2004, Czech Technical University in Prague http://geraldine.fjfi.cvut.cz pp. 63–72 FLOW OVER A GIVEN PROFILE IN A CHANNEL WITH DYNAMICAL EFFECTS JI ˇ R ´ IF ¨ URST 1 , RADEK HONZ ´ ATKO 2 , JAROM ´ IR HOR ´ A ˇ CEK 3 , AND KAREL KOZEL 1 Abstract. The work deals with a numerical solution of steady and unsteady 2D inviscid in- compressible flow over the profile NACA 0012 in a channel. The flow is described by the system of Euler equations. Cell-centered finite-volume scheme at quadrilateral C-mesh is used. Steady state solutions and also unsteady flows caused by the prescribed oscillation of the profile were computed. The method of artificial compressibility and the time dependent method are used for computation of the steady state solution. Some presented numerical results are compared with experimental data. Key words. aeroelasticity, cell-centered scheme, Euler equations, finite-volume method, method of artificial compressibility, oscillation, time dependent method AMS subject classifications. 74F10, 76B10 1. Introduction. The cell-centered Lax-Wendroff scheme in Richtmyer form of second order accuracy is considered for the numerical solution of flow over the profile NACA 0012 in a channel. The added artificial viscosity is used in the combination with the Lax-Wendroff scheme. 2. Mathematical model. The behaviour of a flow is described by the system of Euler equations for inviscid incompressible flow in conservation form: ˜ RW t + F x + G y =0 , (2.1) where W =   p ρ u v   ,F =   u u 2 + p ρ uv   ,G =   v uv v 2 + p ρ   , ˜ R = diag ‖0, 1, 1‖ . The system written above is used for the numerical solution with ˜ R = diag   1 a 2 , 1, 1   , a ∈ R (method of artificial compressibility). Here ρ is the density (constant), p is the pressure and (u, v) is the velocity vector. Upstream conditions are u = u ∞ ,v = v ∞ , p is extrapolated. Downstream con- dition is only given by p = p 2 . Next values of W 2 are extrapolated. Wall conditions on fixed walls of the channel are non-permeability conditions (u, v) n = 0 (normal component of velocity vector is equal to zero). Two approaches were applied to the wall conditions on the oscillating profile in the channel. Firstly the non-permeability conditions were applied, secondly the velocity vector of the flow field near the profile 1 Department of Technical Mathematics, Faculty of Mechanical Engineering, Czech Technical University in Prague, Karlovo n´ am. 13, 120 00 Prague, Czech Republic. 2 Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Tech- nical University in Prague, Trojanova 13, 120 00 Prague, Czech Republic. 3 Institute of Thermomechanics, Academy of Sciences of the Czech Republic, Dolejˇ skova 5, 182 00 Prague, Czech Republic. 63