Numerical methods commonly used for simulations of turbomachinery flows are generally robust. However, in order to resolve complex flow structures they require rather fine grids. For complex configurations the generation of structured grids can be very time consuming. 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 Discontinuous Galerkin methods, which are better suited for obtaining high-order solutions on coarse unstructured meshes.
So, our focus is the adaptation of a DG method to the requirements of the turbomachinery physics while preserving numerical efficiency. In this method it is mandatory to modify the discretization schemes for high-order elements at curved surfaces. Therefore, for cells adjacent to curved, solid walls the standard linear parameterization is replaced with a polynomial representation, taking into account additional points on the curved boundary. The integration of two-dimensional, non-reflecting boundary conditions into the DG framework is important to simulate the subsonic flow about a stator vane near midspan.
DG solver has been integrated into TRACE and successfully tested. The figure shows a conventional CFD mesh (left) and a DG mesh (right) with comparable prediction quality. Furthermore, the DG solver has been ported to run on Graphical Processing Units (GPUs). A fivefold decrease in computational time could be achieved, although substantial changes to the code were needed.
Time Integration Methods: Implicit Runge-Kutta
The simulation of unsteady flow phenomena in turbo machines demands robust and efficient time integration methods. They should be able to capture transient as well as periodic phenomena. BDF-methods (Backward-Difference-Formula) of second order are commonly used, because of they are easy to implement and robust. They are linear multistep methods and not suitable for high order methods. Such methods are A-stable only up to second order. On the contrary implicit Runge-Kutta (IRK)-methods as a single-step method can be extended to higher orders. In the flow solver TRACE such methods are implemented and tested. From this research work the importance of the L-stability becomes clear. L-stable methods provide a fast damping of oscillations.
The effect of this characteristic is shown with the Linger problem. The results show the time evolution of a solution variable, calculated with two different time integration schemes. The solution with the A-stable Crank-Nicolson (CN)-method show strong overshoots with a very slow decay. In contrast to this, the L-stable IRK-method show a significant better agreement with the analytical solution and have no overshoots.
Error estimation and control of the time step size
Along with the excellent stability attributes of IRK-methods, they offer the possibility to estimate the error. The estimation is carried out with a second scheme with different order of accuracy integrated in the original one. The difference between the two solutions tents to an estimation of the local error of the main scheme. The calculation of the second embedded scheme means no additional evaluation. Only already known sub-stages are used. This way the additional expenses are marginal. The estimated error in the pressure is shown in the figure at an example of the laminar flow past a circular cylinder.
For a further improvement in efficiency of the time integration method an automated adjustment of the time step size is desirable. Algorithms to control the time step size optimize the time step based on the calculated error in the local time discretization. Based the error estimation an adaptive control of the time step size will integrate in TRACE. The use of a fixed number of time steps per period can be replaced with this method.
Application in aeroacoustics
To quantify the efficiency of IRK-methods we investigated the production and transmission of tonal noise in the UHBR fan-rig of the DLR. The interactions between the rotor wakes and the stators lead to a highly complex acoustic field, which is dominated by a couple of modes. A comparison with the simulated and measured amplitudes of the dominant upstream travelling modes is shown in the picture. The numbers next to the method names correspond to used number of time steps per period in the simulation.
Even with 32 time steps per period the IRK method is able the capture the dominant acoustic modes very well. The CN-method is also able to reproduce these modes. But only by the use of twice as many sub-iterations, because otherwise the simulation becomes unstable. In contract to these results the high numerical damping in the BDF-method leads to significant differences.