D. Graham Holmes

Three Dimensional Linearized Navier-Stokes Calculations for Flutter and Forced Response

This paper describes a new computational tool developed for the analysis of two or three dimensional, viscous or inviscid turbomachinery flutter and forced response problems. Time harmonic results are obtained by linearizing the underlying steady flow numerical algorithm, including the k-ω turbulence model. Solutions are obtained via explicit time marching, with multigrid, on multiblock grids. We present results that focus on the role of the linearization of the viscous terms and the turbulence model in the accurate computation of linearized unsteady flows.

OPTIMIZED DUAL-TIME STEPPING TECHNIQUE FOR TIME-ACCURATE NAVIER-STOKES CALCULATIONS

This paper presents an optimized discretization of the time derivative term for the dual-time stepping method. The proposed discretization is second order accurate and has a lower level of dissipation and dispersion errors than the conventional non-optimized second order discretization. Sample calculations demonstrate that the optimized scheme requires approximately 45-50% less time steps per unsteady cycle compared to the standard non-optimized scheme to resolve an unsteady flow within a certain margin of amplitude error. The number of time steps per cycle can be reduced by 10-15% to keep the phase error less than a certain level when the optimized scheme is used. Since time-accurate calculations are expensive, the proposed approach leads to significant savings of computational time and resources.

Mixing Planes Revisited: A Steady Mixing Plane Approach Designed to Combine High Levels of Conservation and Robustness

The use of “mixing planes” for steady multistage turbomachinery calculations has been common for some time. The basic idea is simple: in order to perform quasi-steady multistage turbomachinery calculations, the exchange of information at an interface between stationary and rotating blade rows must involve a pitchwise averaging process. This pitchwise averaging implies a mixing process of some sort, which may or may not have a physical analog. This paper revisits the mixing plane idea, and describes an attempt to produce a novel mixing plane algorithm that achieves several key goals, including complete flux conservation at the interface, robustness, indifference to local flow direction and non-reflectivity. The key elements of the algorithm are described, along with some examples of its application.Copyright

Adaptive Unstructured 2D Navier-Stokes Solutions on Mixed Quadrilateral/Triangular Meshes

This paper presents a solution adaptive scheme for solving the Navier-Stokes equations on an unstructured mixed grid of triangles and quadrilaterals. The solution procedure uses an explicit Runge-Kutta finite volume time marching scheme with an adaptive blend of second and fourth order smoothing. The governing equations are solved in a 2D, axisymmetric or quasi-3D form.In viscous regions quadrilateral elements are used to facilitate the one dimensional refinement required for the efficient resolution of boundary layers and wakes. The effect of turbulence is incorporated through using either a Baldwin-Lomax or k-e turbulence model.Solutions are presented for several examples that illustrate the capability of the algorithm to predict viscous phenomena accurately. The examples are a transonic turbine, a nozzle and a combustor diffuser.© 1993 ASME

Development of Adjoint Solvers for Engineering Gradient-Based Turbomachinery Design Applications

High-fidelity computational fluid dynamics (CFD) are common practice in turbomachinery design. Typically, several cases are run with manually modified parameters based on designer expertise to fine-tune a machine. Although successful, a more efficient process is desired. Choosing a gradient-based optimization approach, the gradients of the functions of interest need to be estimated. When the number of variables greatly exceeds the number of functions, the adjoint method is the best-suited approach to efficiently estimate gradients. Until recently, the development of CFD adjoint solvers was regarded as complex and difficult, which limited their use mostly to academia. This paper focuses on the problem of developing adjoint solvers for legacy industrial CFD solvers. A discrete adjoint solver is derived with the aid of an automatic differentiation tool that is selectively applied to the CFD code that handles the residual and function evaluations. The adjoint-based gradients are validated against finite-difference and complex-step derivative approximations.Copyright

Gas Turbine Temperature Prediction Using Unsteady CFD and Realistic Non-Uniform 2D Combustor Exit Properties

Typical Computational Fluid Dynamics (CFD) studies performed on High Pressure Turbines (HPT) do not include the combustor domain in their analyses. Boundary conditions from the combustor exit have to be prescribed at the inlet of the computational domain for the first HPT nozzle. It is desirable to include the effect of combustor non-uniformities and flow gradients in order to enhance the accuracy of the aerodynamics and heat transfer predictions on the nozzle guide vanes and downstream turbine blades. The present work is the continuation of steady and quasi-unsteady studies performed previously by the authors. A fully unsteady nonlinear approach, also referred to as sliding mesh, is now used to investigate a first HPT stage and the impact of realistic non-uniformities and flow gradients found along the exit plane of a gas turbine combustor. Two Turbine Inlet Boundary Conditions (TIBC) are investigated. Simulations using a two-dimensional TIBC dependant on both the radial and circumferential directions are performed and compared to baseline analyses, where the previous two-dimensional TIBC is circumferentially averaged in order to generate inlet boundary conditions dependant only on the radial direction. The two elements included in the present work, combustor pitchwise non-uniformities and full unsteady blade row interactions are shown to: (1) alter the gas temperature profile predictions up to ±5%; (2) modify the surface temperature predictions by ±8% near the trailing edge of the vane suction side; (3) increase the overall pressure losses by roughly 1%, and (4) modified the ingestion behavior of the purge cavity flow. In addition, keeping in mind the tradeoff between improved predictions and computational cost, the use of an unsteady sliding mesh formulation, instead of a quasiunsteady frozen gust, reveals the importance of the two-way unsteady coupling between adjacent blade rows for temperature and pressure predictions.Copyright

Quasi-3D Solutions for Transonic, Inviscid Flows by Adapative Triangulation

This paper describes an algorithm for computing two-dimensional transonic, inviscid flows. The solution procedure uses an explicit Runge-Kutta time marching, finite volume scheme. The computational grid is an irregular triangulation. The algorithm can be applied to arbitrary two-dimensional geometries. When used for analyzing flows in blade rows, terms representing the effects of changes in streamsheet thickness and radius, and the effects of rotation, are included. The solution is begun on a coarse grid, and grid points are added adaptively during the solution process, using criteria such as pressure and velocity gradients.Advantages claimed for this approach are (a) the capability of handling arbitrary geometries (e.g., multiple, dissimilar blades), (b) the ability to resolve small-scale features (e.g., flows around leading edges, shocks) with arbitrary precision, and (c) freedom from the necessity of generating “good” grids (the algorithm generates its own grid, given an initial coarse grid).Solutions are presented for several examples that illustrate the usefulness of the algorithm.Copyright