## Solution Method

The steady-state solution is found by implicitly time-stepping the unsteady ft>w equations (Eqn. 1) until the steady-state residual has been reduced several orders of magnitude. Local time-stepping is used to increase the rate of convergence. The traditional method [2, 3] of solving the time-linearised ft>w equations (Eqn. 9) is to add pseudo-time derivatives to the left side of the equations. The equations then have a similar form to the unsteady ft>w equations and hence the same method can be employed to solve them. Instead of the traditional pseudo-time-stepping, the current scheme uses GMRES with preconditioning to solve the linearised flow equations.

The Generalized Minimum RESidual (GMRES) [13] can be used to solve large, sparse and non-symmetric linear systems. GMRES belongs to the class of Krylov based iterative methods. Consider the linear system

Ax = b (12)

where A is a square non-singular n x n complex matrix, and b is a complex vector of length n. Let xo Є Cn be an initial guess for this linear system and ro = b — Ax0 be its corresponding residual. The GMRES algorithm builds an approximation of the solution of Equation 12 under the form

xi = xo + Vi y (13)

where Vi is an orthonormal basis for the Krylov space of dimension l

where y belongs to C1. The vector y is determined so that 2-norm of the residual ||r ||2 is minimal over Ki. The 2-norm is defined as

(15)

For more information about GMRES see Saad [13].

The convergence rate of Krylov subspace methods is highly dependent on the condition number of the matrix A. The smaller the condition number, the better the convergence. Preconditioning can be used to reduce the condition number of the linear system. The current scheme uses ILU(p) [13] to right precondition the linear system. The preconditioner is calculated by performing an incomplete Gaussian elimination. The method is incomplete because the non-zero pattern on the preconditioner is determined by the non-zero pattern of the original matrix and the level of fill p. For inviscid cases, a level of fill p =1 gave the best performance and for viscous cases p = 3. In order to construct the preconditioner, it is necessary to explicitly determine and store the governing matrix of the linearised fbw equation (Eqn. 9).

2. Test Cases

In order to verify that a computational method is accurately solving the given equations for real profiles where no analytical solution exists, it is necessary to compare results with previous numerical work. Most of the test cases shown are of Standard Configuration 10 mainly due to the quality, quantity and availability of previous numerical solutions. Unfortunately this is an Euler test case, but many features of the code (correct Euler formulation, correct far-field boundary treatment, and correct treatment of shocks) can be examined by this case. In order to verify the correct formulation of the linearised Navier-Stokes equations and the turbulence model, Standard Configuration 11 is examined.