HYDRA is a suite of CFD programs being developed in conjunction with Rolls-Royce. The initial research and development was carried out in our group over the period 1998-2002, but the further development now involves Rolls-Royce funded university research groups at Cambridge, Imperial College, Loughborough, Surrey and Sussex.

HYDRA consists of four main codes:

**hyd**to solve the nonlinear, steady or unsteady, flow equations.**hydlin**to solve the harmonic flow equations corresponding to linearised unsteady flow at a given frequency.**hydadj**to solve the steady adjoint flow equations for steady-state design optimisation.**hydadjh**to solve the adjoint harmonic flow equations, for certain flutter and forced response design applications.

The linear code **hydlin** is obtained by linearising the discrete flow
equations in the nonlinear code **hyd**, and the adjoint codes are obtained
by transposing these equations appropriately. This corresponds to what is sometimes
referred to as the "fully-discrete" approach, as opposed to the "continuous" approach
in which the linearisation and adjoint formulation is performed at the PDE level.

Because of this approach, all four codes share a number of common features:

- choice of Euler equations or Navier-Stokes equations with Spalart-Allmaras turbulence model
- use of hybrid unstructured grids comprising a collection of tetrahedra, pyramids, prisms and hexahedra in 3D, or triangles and quadrilaterals in 2D
- multigrid solution with pseudo-time-marching and a Jacobi preconditioner
- distributed-memory parallel computation using the OPlus parallel framework
- use of Visual3 and pV3 for visualisation

In the references below, [1] covers the research on the multigrid and preconditioning and [2,8] explain the formulation and use of the harmonic adjoint solution. Paper [3] contains our main contributions on the formulation and solution of the adjoint equations, in particular the "fiully-discrete" approach, the complexities introduced by the imposition of strong boundary conditions, and the adjoint iterative approach to solving the adjoint equations.

Papers [4,5,7] address an important problem that arose with both the linear and adjoint solvers. In situations where the nonlinear calculation had failed to converge properly, and had instead settled into a low level limit cycle oscillation, it was observed that the linear and adjoint calculatiosn would be unstable, with the unstable eigenmode being localised in the region affected by the nonlinear limit cycle. Sergio Campobasso found that a good way to stabilise such calculations was to wrap a GMRES or RPM iteration around the standard multigrid solver, using the multigrid solver as a preconditioner for the outer GMRES or RPM solver. This has been very effective in solving what was otherwise becoming an increasingly troublesome problem in 3D viscous applications.

- P. Moinier, J.-D. Muller and M.B. Giles.
`Edge-based multigrid and preconditioning for hybrid grids'.
*AIAA Journal*, 40(10):1954-1960, 2002. (PDF). - M.C. Duta, M.B. Giles and M.S. Campobasso.
`The harmonic adjoint approach to unsteady turbomachinery design'.
*International Journal for Numerical Methods in Fluids*, 40(3-4):323-332, 2002. (PDF file: 108kb). - M.B. Giles, M.C. Duta, J.-D. Muller and N.A. Pierce.
`Algorithm developments for discrete adjoint methods'.
*AIAA Journal*, 41(2), 2003. (PDF). - M.S. Campobasso and M.B. Giles.
`Stabilization of a linearized Navier-Stokes solver for turbomachinery
aeroelasticity' in
*Computational Fluid Dynamics 2002*. Springer-Verlag, 2003. (PDF). - M.S. Campobasso and M.B. Giles.
`Effect of flow instabilities on the linear analysis of
turbomachinery aeroelasticity',
*AIAA Journal of Propulsion and Power*, 19(2), 2003 (PDF). - M.S. Campobasso, M.C. Duta and M.B. Giles.
`Adjoint calculation of sensitivities of turbomachinery objective
functions',
*AIAA Journal of Propulsion and Power*, 19(4), 2003. (PDF). - M.S. Campobasso and M.B. Giles.
`Stabilization of a linear flow solver for turbomachinery aeroelasticity
by means of the recursive projection method',
*AIAA Journal*, 42(9) 1765-1774, 2004. (PDF). - M.C. Duta, M.S. Campobasso, M.B. Giles and L.B. Lapworth.
`Adjoint harmonic sensitivities for forced response minimisation',
to appear in the
*ASME Journal of Turbomachinery*, 2005. - P. Moinier and M.B. Giles.
`Eigenmode analysis for turbomachinery applications',
to appear in
*AIAA Journal of Propulsion and Power*, 2005. (PDF).