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Applied Computational Fluid Dynamics Techniques - Wiley Episode 2 Part 8

Tham khảo tài liệu 'applied computational fluid dynamics techniques - wiley episode 2 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 422 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES where dw x is given by the Hermitian polynomial Lohner 2001 dw 1 - 3f2 2f3 19.13 and f is defined in 19.11 . The final semi-discrete scheme takes the form MiP t r ra u v P rs dw w Vd dh P 19.14 where the subscripts a s and d stand for advection source and damping. This system of ODE s is integrated in time using explicit time-marching schemes e.g. a standard five-stage Runge-Kutta scheme. 19.1.2. OVERALL SCHEME One complete timestep consists of the following steps - given the boundary conditions for the pressure update the solution in the 3-D fluid mesh velocities pressures turbulence variables etc. - extract the velocity vector v u v w at the free surface and transfer it to the 2-D free surface module - given the velocity field update the free surface P - transfer back the new free surface P to the 3-D fluid mesh and impose new boundary conditions for the pressure . For steady-state applications the fluid and free surface domains are updated using local timesteps. This allows some room for variants that may converge faster to the final solution e.g. n steps of the fluid followed by m steps of the free surface complete convergence of the free surface between fluid updates etc. Empirical evidence Lohner et al. 1998 1999a c indicates that most of these variants prove unstable or do not accelerate convergence measurably. For steady-state applications it was found that an equivalent time-interval ratio between the fluid and the free surface of 1 8 yielded the fastest convergence e.g. a Courant number of Cf 0.25 for the fluid and Cs 2.0 for the free surface . 19.1.3. MESH UPDATE Schemes that work with structured grids e.g. Hino 1989 1997 Hino et al. 1993 Farmer et al. 1993 Martinelli and Farmer 1994 Cowles and Martinelli 1996 march the solution in time until a steady state is reached. At each timestep a volume update is followed by a free surface update. The repositioning of points at each timestep implies a complete .