In order to study the dynamical effects in turbomachinery it is necessary to simulate the unsteady flow in a complete turbine including the dynamical behaviour caused by the stator - rotor interaction. In this chapter the two algorithms which are implemented in FENFLOSS are described. In addition some results are presented.


Dynamic boundary conditions

At the interface from the stator to the rotor non-matching grids have to be allowed. Otherwise the grid generation would be very complicated and the time steps would be severely restricted. Overlapping meshes are used, schematically shown in figure1. The rotor and stator domains are calculated independently of each other. The information exchange from one domain to the other is organised in form of dynamic boundaries conditions, which were updated in the global iteration, figure 2. The node values (velocities and turbulence quantities) are interpolated in the elements of the upstream component to the nodes of the downstream part. The pressure , the momentum fluxes and the turbulence fluxes at the interface are integrated in the downstream elements and transferred to the upstream where they are applied as Neumann boundary conditions. In order to obtain an accurate interpolation and integration slightly over lapping meshes are applied. This guarantees that all nodes of the downstream part are always inside the upstream mesh and the entire upstream boundary is completely inside the downstream mesh.

Figure 1: Dynamic boundary

Restriction node approach

The restriction node approach is shown in figure 2. In this method the two parts are discretised independently of each other. The coupling effect is introduced by additional compulsory conditions, which are obtained by expressing the restriction nodes by the nodes of the other domain. This results in additional restriction matrices, which lead to the following linear equation system:



With the system matrices A₁₁ and A₂₂ for the two domains, the unknown x₁ and x₂, the Lagrangian multiplier λ, the restriction operators B₁ and B₂ and the vectors of the right hand side b₁ and b₂. In case of matching grids, when the restriction nodes are located in the same position than the nodes of the other domain, the restriction operators only express the equality of these nodes. In order to obtain a robust behaviour and accurate results the restriction nodes should belong to the finer grid. The nodes of the coarse grid should be treated as degrees of freedom.

Figure 2: Restriction node approach

Rotor-stator interface

A rotor-stator problem is shown schematically in figure.3. The computational grid is distributed to different processors (different colours represent different processors). As it can be seen during the calculation the connectivity of the runner nodes to the stator nodes changes. This requires dynamic communication tables. In the case of the using restriction operators this would lead to a different matrix structure in each time step. In the case of the dynamic boundary conditions the necessary information is sent to each processor of the other domain. From this data each processor can interpolate its own boundary conditions.

Figure 3: Sliding interface, overlapping non-matching grids

Results

In figure 4 the pressure distribution of a complete Francis turbine to a certain time step is shown. Dynamic boundary conditions were used. A non-uniform over the heigth of the stay vanes and the runner blades can be noticed. Figure 5 shows a sample of the restriction node approach.

Figure 4: Pressure distribution with dynamic boundary

Figure 5: Pressure distribution with restriction node approach