/08/2002SMARTER meeting 1 Solution of 2D Navier-Stokes equations in velocity-vorticity formulation using FD Remo Minero Scientific Computing Group – Dep. Mathematics and Computer Science

/08/2002SMARTER meeting 2 2D NAVIER-STOKES EQUATIONS where D is [-1; 1] x [-1; 1] x y O (1,1)(-1,1) (-1,-1)(1,-1) D

/08/2002SMARTER meeting 3 NUMERICAL METHOD Temporal discretization:  Advection term: Adams-Bashforth scheme  Convection term: Crank Nicolson scheme  2nd order Runge-Kutta scheme for the 1st time step Spatial discretization:  Finite differences Influence matrix technique to enforce boundary condition for boundary condition for  2nd order accuracy Accuracy dependent on derivatives’ discretization (e.g. 1st order for upwind, 2nd for centred differences, etc.)

/08/2002SMARTER meeting 4 SELF ORGANIZATION OF VORTICES Random initial condition for u, initial value for  follows consistently. (1 – Re=1000 ) (11 – Re=2500 )

/08/2002SMARTER meeting 5 FUTURE PERSPECTIVES Investigation on time evolution of some physical quantities like E,  and L Steep gradient of  near the walls: LDC Different initial conditions/ boundary conditions Comparing results and performances with already existing codes

/08/2002SMARTER meeting 6 LDC IN TRANSIENT PROBLEMS Coarse grid BC Defect Fine grid tntn t n-1 t n+1 t n+2 t xxx Max  t non to have instabilities

/08/2002SMARTER meeting 7 xxx x L0 LDC WITH SPECTRAL METHODS Coarse grid BC Defect Fine grid ? xxx x L0

/08/2002SMARTER meeting 8 Re=1000

/08/2002SMARTER meeting 9 Re=2500

/08/2002SMARTER meeting 10 2D NAVIER-STOKES EQUATIONS where D is [-1; 1] x [-1; 1] x y O (1,1)(-1,1) (-1,-1)(1,-1) D