Computational Efficiency and Robustness

The calculations were performed on a personal computer with two Intel Pentium IV processors (jobs run in serial mode) with a clock speed of 1.7GHz. The computational effort required for some of the test cases presented in this paper are shown in Table 1. In terms of computational time, it can be seen that the current scheme is very efficient. The memory requirements are large but not unreasonable for a modern personal computer. However, if the current scheme is to be extended to three-dimensions, a supercomputer or a cluster of PCs would be necessary to deliver the required memory.

Table 1. Computational effort required for various test cases Problem Equations Flux cells CPU (seconds) Memory (Mb) SC10 Case 17 Euler EFM 3029 4.8 52.6 SC10 Case 17 Euler EFM 12116 32.2 209 SC10 Case 17 Euler EFM 27261 113 678 SC11 RANS AUSMDV 17518 75 559 SC11 RANS AUSMDV 70072 793 3000

The linearised solutions to the Standard Configuration 11 test case described above, using various flux schemes but all based on the same steady-state solu­tion (AUSMDV) are shown in Fig. 6. The fact that it was possible to obtain accurate solutions when the flux scheme used to calculate the steady-state so­lution is different to the scheme used by the unsteady linearised solution is a good demonstration of the robustness of GMRES with preconditioning. Hall and Clark [17] stated that for transonic fbws with shock capturing, the discre – tised perturbation equations should be a faithful linearisation of the discretised unsteady nonlinear equations used to compute the steady flow. Moreover, Hall & Clark could not obtain converged linearised solutions in the vicinity of a res-

Figure 6. Solutions to Standard Configuration 11 using various upwind schemes

onant interblade phase angle when pseudo-time-stepping was employed. The author also encountered similar difficulties when using pseudo-time-stepping. However when GMRES was employed it was possible to find solutions for all interblade phase angles. Note the aerodynamic dampings shown in Fig. 3 are calculated at one degree intervals.

3. Summary

A linearised Navier-Stokes fbw solver which includes the Spalart & All – maras turbulence model that is suitable for flutter investigations has been pre­sented. Various flux schemes were tested and EFM gave the best results for Euler calculations and AUSMDV gave the best results for viscous calculations. GMRES with preconditioning was used to solve the unsteady linearised flow equations in a robust and efficient manner. A three-dimensional version of this method is currently being developed.

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