Scale-resolving simulations, such as large eddy simulations, have become affordable tools to investigate the flow in turbomachinery components. The resulting time-resolved flow field is typically analyzed using first- and second-order statistical moments. However, two sources of uncertainty are present when statistical moments from scale-resolving simulations are computed: the influence of initial transients and statistical errors due to the finite number of samples. In this paper, both are systematically analyzed for several quantities of engineering interest using time series from a long-time large eddy simulation of the low-pressure turbine cascade T106C. A set of statistical tools to either remove or quantify these sources of uncertainty is assessed. First, the Marginal Standard Error Rule is used to detect the end of the initial transient. The method is validated for integral and local quantities and guidelines on how to handle spatially varying initial transients are formulated. With the initial transient reliably removed, the statistical error is estimated based on standard error relations considering correlations in the time series. The resulting confidence intervals are carefully verified for quantities of engineering interest utilizing cumulative and simple moving averages. Furthermore, the influence of periodic content from large scale vortex shedding on the error estimation is studied. Based on the confidence intervals, the required averaging interval to reduce the statistical uncertainty to a specific level is indicated for each considered quantity.

In this first paper of a three-part series, we present the extension and validation of the high-order discontinuous Galerkin scheme in DLR’s CFD-solver trace for scale-resolving simulations of unsteady row interactions. The translational movement of rows in linear cascade experiments is represented in the numerical model by solving the equations in the relative frame of reference. To couple rows in different frames of reference, a sliding interface approach based on the mortar technique for non-conforming meshes has been developed. The verification of the approach is exemplified by three canonical test cases. First, the experimental order of convergence is verified for the isentropic vortex convection. Subsequently, the suitability of the sliding interface approach for scale-resolving simulations is tested on the Taylor–Green vortex flow and a turbulent cylinder flow. Finally, the LES solver is applied to the T106D cascade with upstream moving bars at an exit Reynolds number of 200,000 and exit Mach number of 0.4. The flow physics with and without bars is discussed in terms of the instantaneous flow field, and time- and phase-averaged quantities. The comparison with experimental data shows overall a good agreement, especially for the total pressure losses in the wake, but also reveals uncertainties related to the reproduction of an experiment in the numerical model.

In the second paper of this three-part series, we focus on the simulation of transonic test cases for turbomachinery applications using a high-order discontinuous Galerkin spectral element method (DGSEM). High-fidelity simulations of transonic compressors and turbines are particularly challenging, as they typically occur at high Reynolds numbers and require additional treatment to reliably capture the shock waves characterizing such flows. A recently developed finite-volume subcell shock capturing scheme tailored for the DGSEM is applied and evaluated with regard to the shock sensor. To this end, we conduct implicit large eddy simulations of a high-pressure turbine cascade from the public literature and a transonic compressor cascade measured at the German Aerospace Center, both at a high Reynolds number above 106. Based on the results, we examine modal-energy and flow-feature based shock indicator functions, compare the simulation data to experimental and numerical studies, and present an analysis of the unsteady features of the flows.

In this final paper of a three-part series, we apply the numerical test rig based on a high-order discontinuous Galerkin scheme to the MTU T161 low-pressure turbine with diverging end walls at off-design Reynolds number of 90,000, Mach number of 0.6, and inflow angle of 41 deg. The inflow end wall boundary layers are prescribed in accordance with the experiment. Validation of the setup is shown against recent numerical references and the corresponding experimental data. Additionally, we propose and conduct a purely numerical experiment with upstream bar wake generators at a Strouhal number of 1.25, which is well above what was possible in the experiment. We discuss the flow physics at midspan and in the end wall region and highlight the influence of the wakes from the upstream row on the complex secondary flow system using instantaneous flow visualization, phase averages, and modal decomposition techniques.

In this paper, we quantify the turbulence modeling uncertainty for the transonic Technische Universität Darmstadt (TUDa) compressor. The present work applies the Eigenspace Perturbation Framework (EPF) as it is the only published physics-based framework capable of addressing the model-form uncertainty in turbulence closure modeling. To sample from the possible solution space and obtain the modeling uncertainty, we perform simulations perturbing the eigenvalues of the Reynolds stress tensor in addition to simulations using an unperturbed turbulence model. We show that the shape of the Reynolds stress tensor ellipsoid has significant impact on the evolution of turbulence, flow separation, vortex systems, shock-boundary layer interaction, and finally the overall performance of the compressor. We compare the estimated uncertainties with available measurements and transitional Delayed Detached-Eddy Simulations (DDES). This allows us to assess the confidence of the chosen turbulence model and to evaluate the sharpness and coverage of the resulting uncertainty bounds. Thus, the EPF is comprehensively validated and suggestions for its future applicability with respect to turbomachinery components are made.