William S. Clark

Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique

A harmonic balance technique for modeling unsteady nonlinear e ows in turbomachinery is presented. The analysis exploits the fact that many unsteady e ows of interest in turbomachinery are periodic in time. Thus, the unsteady e ow conservation variables may be represented by a Fourier series in time with spatially varying coefe cients. This assumption leads to a harmonic balance form of the Euler or Navier ‐Stokes equations, which, in turn, can be solved efe ciently as a steady problem using conventional computational e uid dynamic (CFD) methods, including pseudotime time marching with local time stepping and multigrid acceleration. Thus, the method is computationally efe cient, at least one to two orders of magnitude faster than conventional nonlinear time-domain CFD simulations. Computational results for unsteady, transonic, viscous e ow in the front stage rotor of a high-pressure compressor demonstrate that even strongly nonlinear e ows can be modeled to engineering accuracy with a small number of terms retained in the Fourier series representation of the e ow. Furthermore, in some cases, e uid nonlinearities are found to be important for surprisingly small blade vibrations.

A time-linearized Navier-Stokes analysis of stall flutter

A computational method for predicting unsteady viscous flow through two-dimensional cascades accurately and efficiently is presented. The method is intended to predict the onset of the aeroelastic phenomenon of stall flutter. In stall flutter, viscous effects significantly impact the aeroelastic stability of a cascade. In the present effort, the unsteady flow is modeled using a time-linearized Navier-Stokes analysis. Thus, the unsteady flow field is decomposed into a nonlinear spatially varying mean flow plus a small-perturbation harmonically varying unsteady flow. The resulting equations that govern the perturbation flow are linear, variable coefficient partial differential equations. These equations are discretized on a deforming, multiblock, computational mesh and solved using a finite-volume Lax-Wendroff integration scheme. Numerical modeling issues relevant to the development of the unsteady aerodynamic analysis, including turbulence modeling, are discussed. Results from the present method are compared to experimental stall flutter data, and to a nonlinear time-domain Navier-Stokes analysis. The results presented demonstrate the ability of the present time-linearized analysis to model accurately the unsteady aerodynamics associated with turbomachinery stall flutter.

A Linearized Euler Analysis of Unsteady Transonic Flows in Turbomachinery

A computational method for efficiently predicting unsteady transonic flows in two- and three-dimensional cascades is presented. The unsteady flow is modeled using a linearized Euler analysis whereby the unsteady flow field is decomposed into a nonlinear mean flow plus a linear harmonically varying unsteady flow. The equations that govern the perturbation flow, the linearized Euler equations, are linear variable coefficient equations. For transonic flows containing shocks, shock capturing is used to model the shock impulse (the unsteady load due to the harmonic motion of the shock). A conservative Lax-Wendroff scheme is used to obtain a set of linearized finite volume equations that describe the harmonic small disturbance behavior of the flow. Conditions under which such a discretization will correctly predict the shock impulse are investigated. Computational results are presented that demonstrate the accuracy and efficiency of the present method as well as the essential role of unsteady shock impulse loads on the flutter stability of fans.

Nonreflecting boundary conditions for linearized unsteady aerodynamic calculations

The present method for the implementation of nonreflecting boundary conditions in 2D and 3D linearized unsteady flow computations is applied to cases of unsteady flows in turbomachine blade rows. The eigenmodes of a discrete representation of the governing equations are computed and used to construct nonreflecting boundary conditions. In 3D, a mixed numerical method is used; in 2D, the discrete representation of the governing equations is obtained from the discretized equations used by the flow solver itself. Wavenumbers and radial mode shapes are computed.

Unsteady Simulation of a 1.5 Stage Turbine Using an Implicitly Coupled Nonlinear Harmonic Balance Method

The harmonic balance method implemented within STAR-CCM+ is a mixed frequency/time domain computational fluid dynamic technique, which enables the efficient calculation of time-periodic flows. The unsteady solution is stored at a small number of fixed time levels over one temporal period of the unsteady flow in a single blade passage in each blade row; thus the solution is periodic by construction. The individual time levels are coupled to one another through a spectral operator representing the time derivative term in the Navier-Stokes equation, and at the boundaries of the computational domain through the application of periodic and nonreflecting boundary conditions. The blade rows are connected to one another via a small number of fluid dynamic spinning modes characterized by nodal diameter and frequency. This periodic solution is driven to the correct solution using conventional (steady) CFD acceleration techniques, and thus is computationally efficient. Upon convergence, the time level solutions are Fourier transformed to obtain spatially varying Fourier coefficients of the flow variables. We find that a small number of time levels (or, equivalently, Fourier coefficients) are adequate to model even strongly nonlinear flows. Consequently, the method provides an unsteady solution at a computational cost significantly lower than traditional unsteady time marching methods.The implementation of this nonlinear harmonic balance method within STAR-CCM+ allows for the simulation of multiple blade rows. This capability is demonstrated and validated using a 1.5 stage cold flow axial turbine developed by the University of Aachen. Results produced using the harmonic balance method are compared to conventional time domain simulations using STAR-CCM+, and are also compared to published experimental data. It is shown that the harmonic balance method is able to accurately model the unsteady flow structures at a computational cost significantly lower than unsteady time domain simulation.Copyright

Prediction of unsteady aerodynamic loads in cascades using the linearized Euler equations on deforming grids

A linearized Euler solver for calculating unsteady flows in turbomachinery blade rows due to both incident gusts and blade motion is presented. Using the linearized Euler technique, one decomposes the flow into a mean (or steady) flow plus an unsteady, harmonically varying, small disturbance flow. Linear variable coefficient equations describe the small disturbance behavior of the flow, and are solved using a pseudo-time marching Lax-Wendroff scheme. For the blade motion problem, a harmonically deforming computational rid that conforms to the motion of vibrating blades eliminates large error producing mean flow gradient terms that would otherwise appear in the unsteady flow tangency boundary condition. The paper also presents a new, numerically exact, nonreflecting far-field boundary condition based on an eigenanalysis of the discretized equations. Computed flow solutions demonstrate the computational accuracy and efficiency of the present method. The solution of the linearized Euler equations requires one to two orders of magnitude less computer time than solution of the nonlinear Euler equations using traditional time-accurate time-marching techniques. In addition, the deformable grid significantly improves the accuracy of the solution.

Submarine Maneuvering Simulations of ONR Body 1

The application of computational fluid dynamics method to the submarine maneuvering simulations of ONR Body 1 is presented. ONR Body 1 is an unclassified submarine radio controlled model with propeller and control surfaces. Unsteady Reynolds-averaged Naviers-Stokes equations of fluid flow is coupled to the six degrees-of-freedom equations of motion of a rigid body via user coding to predict the instantaneous position and body orientation. Propeller and control surface motions are accounted for by using the moving mesh feature integrated into the solution procedure which allows sliding interfaces between different mesh blocks of the computational domain (for propeller rotation), as well as mesh distortion (for control surface deflection). This offers the flexibility of using a single computational grid for the entire simulation period. The maneuvers simulated include a constant depth and heading run as well as a horizontal overshoot maneuver using conditions consistent with the experiment. Predicted results show favorable agreement with experimental measurements.Copyright

A Numerical Model of the Onset of Stall Flutter in Cascades

In this paper, we present a computational fluid dynamic model of the unsteady flow associated with the onset of stall flutter in turbomachinery cascades. The unsteady flow is modeled using the laminar Navier-Stokes equations. We assume that the unsteadiness in the flow is a small harmonic disturbance about the mean or steady flow. Therefore, the unsteady flow is governed by a small-disturbance form of the Navier-Stokes equations. These linear variable coefficient equations are discretized on a deforming computational grid and solved efficiently using a multiple-grid Lax-Wendroff scheme. A number of numerical examples are presented which demonstrate the destabilizing influence of viscosity on the aeroelastic stability of airfoils in cascade, especially for torsional modes of blade vibration.Copyright

Calculation of unsteady linearized Euler flows in cascades using harmonically deforming grids

A method for calculating unsteady, inviscid, compressible flows in cascades is presented. Using the linearized Euler technique, the flow is decomposed into a steady or mean flow plus a harmonically varying small disturbance flow. The equations that describe the small disturbance flow are linear variable coefficient equations, and are solved using a pseudo-time time marching Lax-Wendroff technique. Unlike previous linearized methods, however, the solution is computed on a harmonically deforming computational grid that conforms to the motion of the vibrating airfoils. The mean flow and perturbation flow solutions are defined in the deforming coordinate system rather than in a coordinate system fixed in space. Hence, no extrapolation terms are required to implement the upwash boundary conditions at the airfoil surfaces significantly improving the accuracy of the method. For transonic flow calculations, unsteady shock motions are modelled using shock capturing. The unsteady loads due to the shock motion are then seen as pressure impulses. Representative computational results are presented for transonic channel flows and subsonic and transonic cascade flows.

A Time-Linearized Navier-Stokes Analysis of Stall Flutter

A computational method for accurately and efficiently predicting unsteady viscous flow through two-dimensional cascades is presented. The method is intended to predict the onset of the aeroelastic phenomenon of stall flutter. In stall flutter, viscous effects significantly impact the aeroelastic stability of a cascade. In the present effort, the unsteady flow is modeled using a time-linearized Navier-Stokes analysis. Thus, the unsteady flow field is decomposed into a nonlinear spatially varying mean flow plus a small-perturbation harmonically varying unsteady flow. The resulting equations that govern the perturbation flow are linear, variable coefficient partial differential equations. These equations are discretized on a deforming, multi-block, computational mesh and solved using a finite-volume Lax-Wendroff integration scheme. Numerical modelling issues relevant to the development of the unsteady aerodynamic analysis, including turbulence modelling, are discussed. Results from the present method are compared to experimental stall flutter data, and to a nonlinear time-domain Navier-Stoke analysis. The results presented demonstrate the ability of the present time-linearized analysis to model accurately the unsteady aerodynamics associated with turbomachinery stall flutter.Copyright