In turbomachinery powerful numerical methods have been developed and used successfully to simulate the flow. However, these classical first- and second-order accurate algorithms for spatial discretization are inefficient for flow problems with complex physics and geometry. In particular, such applications as computational aeroacoustics (CAA) or turbulent combustion problems require a more accurate prediction of turbulent phenomena than what is attainable with second-order RANS simulations.
High-Order Spatial Discretization Methods / Discontinous Galerkin
For complex configurations the generation of structured grids can be very time consuming. On the other hand, the generation of unstructured grids can be automated, although automatic generation of high quality grids is still a challenging task.
As a consequence, it is preferred to use special high-order methods, which are better suited to unstructured grids and which are able to attain sufficient solution resolution on a very coarse grids.
One of the most popular high-order methods is the Discontinuous Galerkin (DG) finite element method, which is an extension of the Finite Volume (FV) method. Here the solution is approximated by means of piece-wise polynomials inside the element, where continuity between the elements is not required. Moreover, the method delivers no computational difficulties for unstructured grids.
We focus on the application and development of DG methods for the Euler and Navier-Stokes equation in turbomachinery flows. The main features in this field of research are the application of the methods to turbomachinery physics and the maintenance of computational efficiency.
Implicit Runge-Kutta Schemes
For the temporal integration of the governing equations in TRACE typically numerical algorithms of 2nd-order accucary are used. To increase the temporal accuracy the suitability of several Implicit Runge-Kutta (IRK) methods in the context of blade-row interaction within compressor and turbine configurations is investigated. For this purpose three diagonal implicit Runge-Kutta schemes with an explicit first stage (ESDIRK) of second-, third- and fourth-order accuracy are considered. The results show that these schemes can be efficiently used with high robustness and that they are able to simulate aeroacoustical phenomena obtaining high accuracy.
Real component of the pressure at the first harmonic of BPF on suction (solid lines) and pressure side (dashed lines) of the rotor blade of a low pressure turbine stage, using second order BDF (left) and third order ESDIRK (right) with different time-step sizes.
Real component of the pressure at the 2nd Harmonic of the BPF in the DLR Ultra High Bypass Ratio Fan (UHBR) for Crank-Nicolson with 256 time-steps (left) and ESDIRK 3rd order with 64 time-steps (right).