The in-house flow solver EllipSys3D is developed in co-operation between the Department of Mechanical Engineering at DTU and The Department of Wind Energy at Risø National Laboratory, see [i], [ii] and [iii].

The EllipSys3D code is a multiblock finite volume discretization of the incompressible Reynolds Averaged Navier-Stokes (RANS) equations in general curvilinear coordinates. The code uses a collocated variable arrangement, and Rhie/Chow interpolation [iv] is used to avoid odd/even pressure decoupling. As the code solves the incompressible flow equations, no equation of state exists for the pressure, and the SIMPLE algorithm of [v] is used to enforce the pressure/velocity coupling. The EllipSys3D code is parallelized with MPI for executions on distributed memory machines, using a non-overlapping domain decomposition technique. For rotor computations, a moving frame attached to the rotor blades is used, and the necessary fictitious forces are added to the governing equations. Polar velocities (Vr,Vt,Va) are used in order to allow simple treatment of periodic boundary conditions in the azimuth direction, [vi] and [vii].

The solution is advanced in time using a 2nd order iterative time-stepping (or dual time-stepping) method. In each global time-step the equations are solved in an iterative manner, using under-relaxation. First, the momentum equations are used as a predictor to advance the solution in time. At this point in the computation the flowfield will not fulfil the continuity equation. The rewritten continuity equation (the so called pressure correction equation) is used as a corrector making the predicted flowfield satisfy the continuity constraint. This two step procedure corresponds to a single sub-iteration, and the process is repeated until a convergent solution is obtained for the timestep. When a convergent solution is obtained, the variables are updated, and we continue with the next timestep.

For steady state computations, the global time-step is set to infinity and dual time stepping is not used, this corresponds to the use of local time stepping. In order to accelerate the overall algorithm, a three level grid sequence is used in the steady state computations. The convective terms are discretized using a second order upwind scheme, implemented using the deferred correction approach first suggested by [viii]. Central differences are used for the viscous terms, in each sub-iteration only the normal terms are treated fully implicit, while the terms from non-orthogonality and the variable viscosity terms are treated explicitly. Thus, when the sub-iteration process is finished all terms are evaluated at the new time level.

In the present work the turbulence in the boundary layer is modelled by the k-w SST eddy viscosity model [ix]. The details of the model will not be given here, we will only state that the model is chosen because of the very promising results for 2D separated flows, [x], [xi]. The equations for the turbulence model are solved after the momentum and pressure correction equations in every sub-iteration/pseudo time step.

The three momentum equations are solved decoupled using a red/black Gauss-Seidel point solver. The solution of the Poisson system arising from the pressure correction equation is accelerated using a multigrid method. In order to accelerate the overall algorithm, a three level grid sequence and local time stepping are used.

[i] Michelsen J.A., ”Basis3D - a Platform for Development of Multiblock PDE Solvers” Technical Report AFM 92-05, Technical University of Denmark, 1992.

[ii] Michelsen J.A., ”Block structured Multigrid solution of 2D and 3D elliptic PDE's”, Technical Report AFM 94-06, Technical University of Denmark, 1994.

[iii] Sørensen N.N., ”General Purpose Flow Solver Applied to Flow over Hills”, Risø-R-827-(EN), Risø National Laboratory, Roskilde, Denmark, June 1995.

[iv] Rhie C.M. ”A numerical study of the flow past an isolated airfoil with separation” Ph.D. thesis, Univ. of Illinois, Urbane-Champaign, 1981.

[v] Patankar S.V. and Spalding D.B. ”A Calculation Procedure for Heat, Mass and Momentum Transfer in Three-Dimensional Parabolic Flows.” Int. J. Heat Mass Transfer, 15:1787,1972

[vi] Michelsen J.A., ”General curvilinear transformation of the Navier-Stokes equations in a 3D polar rotating frame”, Technical Report ET-AFM 98-01, Technical University of Denmark, 1998.

[vii] Sørensen N.N. and Michelsen J.A.,”Aerodynamic Predictions for the Unsteady Aerodynamics Experiment Phase-II Rotor at the National Renewable Energy Laboratory”, AIAA-2000-0037 Paper, 38th Aerospace Sciences Meeting & Exhibit,2000, Reno.

[viii] Khosla P.K. and Rubin S.G., ”A diagonally dominant second-order accurate implicit scheme”, Computers Fluids, 2:207-209, 1974.

[ix] Menter F.R., ”Zonal Two Equation k-w Turbulence Models for Aerodyanmic Flows”. AIAA-paper-932906, 1993.

[x] Wilcox D.C. ”A Half Century Historical Review of the k-w Model”.. AIAA-91-0615, 1991.

[xi] Menter F.R. ”Performance of Popular Turbulence Models for Attached and Separated Adverse Pressure Gradient Flows”. AIAA Journal 30(8):2066-2072, August 1992