On high order finite-difference metric discretizations satisfying GCL on moving and deforming grids

In this note we generalize our previous treatment of the discretizations of geometric conservation laws on steady grids (Vinokur and Yee, 2000) to general time dependent grids. The commutative property of mixed difference operators is generalized to apply to time metrics and Jacobians. Our treatment uses half the number of terms as those used in a recent paper by Abe et al. (2012). We also derive the proper temporal discretizations of both Runge–Kutta and linear multistep methods to satisfy the commutativity property for higher than first order

Development of an Unsteady Aeroelastic Solver for the Analysis of Modern Turbomachinery Designs

Developers of aircraft gas turbine engines continually strive for greater efficiency and higher thrust-to-weight ratio designs. To meet these goals, advanced designs generally feature thin, low aspect airfoils, which offer increased performance but are highly susceptible to flow-induced vibrations. As a result, High Cycle Fatigue (HCF) has become a universal problem throughout the gas turbine industry and unsteady aeroelastic computational models are needed to predict and prevent these problems in modern turbomachinery designs. This research presents the development of a 3D unsteady aeroelastic solver for turbomachinery applications. To accomplish this, a well established turbomachinery Computational Fluid Dynamics (CFD) code called Corsair is loosely coupled to the commercial Computational Structural Solver (CSD) Ansys® through the use of a Fluid Structure Interaction (FSI) module. Significant modifications are made to Corsair to handle the integration of the FSI module and improve overall performance. To properly account for fluid grid deformations dictated by the FSI module, temporal based coordinate transformation metrics are incorporated into Corsair. Wall functions with user specified surface roughness are also added to reduce fluid grid density requirements near solid surfaces. To increase overall performance and ease of future modifications to the source code, Corsair is rewritten in Fortran 90 with an emphasis on reducing memory usage and improving source code readability and structure. As part of this effort, the shared memory data structure of Corsair is replaced with a distributed model. Domain decomposition of individual grids in the radial direction is also incorporated into Corsair for additional parallelization, along with a utility to automate this process in an optimal manner based on user input. This additional parallelization helps offset the inability to use the fine grain mp-threads parallelization in the original code on non-distributed memory architectures such as the PC Beowulf cluster used for this research. Conversion routines and utilities are created to handle differences in grid formats between Corsair and the FSI module. The resulting aeroelastic solver is tested using two simplified configurations. First, the well understood case of a flexible cylinder in cross flow is studied with the natural frequency of the cylinder set to the shedding frequency of the Von Karman streets. The cylinder is self excited and thus demonstrates the correct exchange of energy between the fluid and structural models. The second test case is based on the fourth standard configuration and demonstrates the ability of the solver to predict the dominant vibrational modes of an aeroelastic turbomachinery blade. For this case, a single blade from the fourth standard configuration is subjected to a step function from zero loading to the converged flow solution loading in order to excite the structural modes of the blade. These modes are then compared to those obtained from an in vacuo Ansys® analysis with good agreement between the two

High-Order Stabilized Finite Elements on Dynamic Meshes

The development of dynamic mesh capability for turbulent flow simulations using the Streamlined Upwind Petrov-Galerkin (SUPG) discretization is described. The current work extends previous research to include high-order spatial accuracy, including the satisfaction of the discrete geometric conservation law (GCL) on curved elements. Two closely-related schemes are described and the ability of these schemes to satisfy the GCL, while also maintaining temporal accuracy and conservation is assessed. Studies indicate that although one scheme discretizes the time derivative in conservative form, both schemes exhibit temporal conservation errors that decrease according to the expected design order of accuracy. The source of the temporal conservation errors is examined, and it is demonstrated that many finite-volume and finite-element schemes can also be expected to have difficulty strictly satisfying conservation in time. The effects on conservation are examined and, while present in the simulations, are seen to be negligible for the problems considered

Effect of Mesh Quality on Flux Reconstruction in Multi-dimensions

Abstract: Theoretical methods are developed to understand the effect of non-uniform grids on Flux Reconstruction (FR) in multi-dimensions. A better theoretical understanding of the effect of wave angle and grid deformation is established. FR is shown to have a smaller variation in properties than some finite difference counterparts. Subsequent numerical experiments on the Taylor–Green Vortex with jittered elements show the effect of localised regions of expansion and contraction. The effect this had on Nodal DG-like schemes was to increase the dissipation, whereas for more typical FR schemes the effect was to increase the dispersion. Some comparison is made between second-order FR and a second-order finite volume (FV) scheme. FR is found to be more resilient to mesh deformation, however, FV is found to be more resolved when operated at second order on the same mesh