Unsteady aerodynamics of blade rows

The requirements placed on an unsteady aerodynamic theory intended for turbomachinery aeroelastic or aeroacoustic applications are discussed along with a brief description of the various theoretical models that are available to address these requirements. The major emphasis is placed on the description of a linearized inviscid theory which fully accounts for the affects of a nonuniform mean or steady flow on unsteady aerodynamic response. Although this linearization was developed primarily for blade flutter prediction, more general equations are presented which account for unsteady excitations due to incident external aerodynamic disturbances as well as those due to prescribed blade motions. The motivation for this linearized unsteady aerodynamic theory is focused on, its physical and mathematical formulation is outlined and examples are presented to illustrate the status of numerical solution procedures and several effects of mean flow nonuniformity on unsteady aerodynamic response

Development of an unsteady aerodynamic analysis for finite-deflection subsonic cascades

An unsteady potential flow analysis, which accounts for the effects of blade geometry and steady turning, was developed to predict aerodynamic forces and moments associated with free vibration or flutter phenomena in the fan, compressor, or turbine stages of modern jet engines. Based on the assumption of small amplitude blade motions, the unsteady flow is governed by linear equations with variable coefficients which depend on the underlying steady low. These equations were approximated using difference expressions determined from an implicit least squares development and applicable on arbitrary grids. The resulting linear system of algebraic equations is block tridiagonal, which permits an efficient, direct (i.e., noniterative) solution. The solution procedure was extended to treat blades with rounded or blunt edges at incidence relative to the inlet flow

Development of a steady potential solver for use with linearized, unsteady aerodynamic analyses

A full potential steady flow solver (SFLOW) developed explicitly for use with an inviscid unsteady aerodynamic analysis (LINFLO) is described. The steady solver uses the nonconservative form of the nonlinear potential flow equations together with an implicit, least squares, finite difference approximation to solve for the steady flow field. The difference equations were developed on a composite mesh which consists of a C grid embedded in a rectilinear (H grid) cascade mesh. The composite mesh is capable of resolving blade to blade and far field phenomena on the H grid, while accurately resolving local phenomena on the C grid. The resulting system of algebraic equations is arranged in matrix form using a sparse matrix package and solved by Newton's method. Steady and unsteady results are presented for two cascade configurations: a high speed compressor and a turbine with high exit Mach number

Development of a linearized unsteady aerodynamic analysis for cascade gust response predictions

A method for predicting the unsteady aerodynamic response of a cascade of airfoils to entropic, vortical, and acoustic gust excitations is being developed. Here, the unsteady flow is regarded as a small perturbation of a nonuniform isentropic and irrotational steady background flow. A splitting technique is used to decompose the linearized unsteady velocity into rotational and irrotational parts leading to equations for the complex amplitudes of the linearized unsteady entropy, rotational velocity, and velocity potential that are coupled only sequentially. The entropic and rotational velocity fluctuations are described by transport equations for which closed-form solutions in terms of the mean-flow drift and stream functions can be determined. The potential fluctuation is described by an inhomogeneous convected wave equation in which the source term depends on the rotational velocity field, and is determined using finite-difference procedures. The analytical and numerical techniques used to determine the linearized unsteady flow are outlined. Results are presented to indicate the status of the solution procedure and to demonstrate the impact of blade geometry and mean blade loading on the aerodynamic response of cascades to vortical gust excitations. The analysis described herein leads to very efficient predictions of cascade unsteady aerodynamic response phenomena making it useful for turbomachinery aeroelastic and aeroacoustic design applications

Application of a linearized unsteady aerodynamic analysis to standard cascade configurations

A linearized potential flow analysis, which accounts for the effects of nonuniform steady flow phenomena on the linearized unsteady aerodynamic response to prescribed blade motions, has been applied to five cascade configurations. These include the first, fifth, eighth and ninth standard configurations proposed as a result of the Second International Symposium on Aeroelasticity in Turbomachines and a NASA Lewis flutter cascade. Selected results from this study, including comparisons between analytical predictions and the experimental measurements submitted for three of the foregoing configurations, are described. The correlation between theory and experiment for the first standard configuration (a compressor cascade operating at low Mach number and frequency) is quite good. Moreover, the predictions and measurements for the NASA Lewis cascade of symmetric biconvex airfoils show good qualitative agreement. However, wide discrepancies exist between the theoretical predictions and the experimental measurements for the fifth standard configuration (a subsonic transonic fan tip cascade). These can be partially attributed to conditions being imposed in the experiment which differ from those commonly used in unsteady aerodynamic analyses

Development of unsteady aerodynamic analyses for turbomachinery aeroelastic and aeroacoustic applications

Theoretical analyses and computer codes are being developed for predicting compressible unsteady inviscid and viscous flows through blade rows. Such analyses are needed to determine the impact of unsteady flow phenomena on the structural durability and noise generation characteristics of turbomachinery blading. Emphasis is being placed on developing analyses based on asymptotic representations of unsteady flow phenomena. Thus, flow driven by small-amplitude unsteady excitations in which viscous effects are concentrated in thin layers are being considered. The resulting analyses should apply in many practical situations, lead to a better understanding of the relevent physics, and they will be efficient computationally, and therefore, appropriate for aeroelastic and aeroacoustic design applications. Under the present phase (Task 3), the effort was focused on providing inviscid and viscid prediction capabilities for subsonic unsteady cascade flows

Rapid Aeroelastic Analysis of Blade Flutter in Turbomachines

The LINFLUX-AE computer code predicts flutter and forced responses of blades and vanes in turbomachines under subsonic, transonic, and supersonic flow conditions. The code solves the Euler equations of unsteady flow in a blade passage under the assumption that the blades vibrate harmonically at small amplitudes. The steady-state nonlinear Euler equations are solved by a separate program, then equations for unsteady flow components are obtained through linearization around the steady-state solution. A structural-dynamics analysis (see figure) is performed to determine the frequencies and mode shapes of blade vibrations, a preprocessor interpolates mode shapes from the structural-dynamics mesh onto the LINFLUX computational-fluid-dynamics mesh, and an interface code is used to convert the steady-state flow solution to a form required by LINFLUX. Then LINFLUX solves the linearized equations in the frequency domain to calculate the unsteady aerodynamic pressure distribution for a given vibration mode, frequency, and interblade phase angle. A post-processor uses the unsteady pressures to calculate generalized aerodynamic forces, response amplitudes, and eigenvalues (which determine the flutter frequency and damping). In comparison with the TURBO-AE aeroelastic-analysis code, which solves the equations in the time domain, LINFLUX-AE is 6 to 7 times faster

Different atmospheric moisture divergence responses to extreme and moderate El Niños

On seasonal and inter-annual time scales, vertically integrated moisture divergence provides a useful measure of the tropical atmospheric hydrological cycle. It reflects the combined dynamical and thermodynamical effects, and is not subject to the limitations that afflict observations of evaporation minus precipitation. An empirical orthogonal function (EOF) analysis of the tropical Pacific moisture divergence fields calculated from the ERA-Interim reanalysis reveals the dominant effects of the El Niño-Southern Oscillation (ENSO) on inter-annual time scales. Two EOFs are necessary to capture the ENSO signature, and regression relationships between their Principal Components and indices of equatorial Pacific sea surface temperature (SST) demonstrate that the transition from strong La Niña through to extreme El Niño events is not a linear one. The largest deviation from linearity is for the strongest El Niños, and we interpret that this arises at least partly because the EOF analysis cannot easily separate different patterns of responses that are not orthogonal to each other. To overcome the orthogonality constraints, a self-organizing map (SOM) analysis of the same moisture divergence fields was performed. The SOM analysis captures the range of responses to ENSO, including the distinction between the moderate and strong El Niños identified by the EOF analysis. The work demonstrates the potential for the application of SOM to large scale climatic analysis, by virtue of its easier interpretation, relaxation of orthogonality constraints and its versatility for serving as an alternative classification method. Both the EOF and SOM analyses suggest a classification of “moderate” and “extreme” El Niños by their differences in the magnitudes of the hydrological cycle responses, spatial patterns and evolutionary paths. Classification from the moisture divergence point of view shows consistency with results based on other physical variables such as SST

Effect of temperature anisotropy on various modes and instabilities for a magnetized non-relativistic bi-Maxwellian plasma

Using kinetic theory for homogeneous collisionless magnetized plasmas, we present an extended review of the plasma waves and instabilities and discuss the anisotropic response of generalized relativistic dielectric tensor and Onsager symmetry properties for arbitrary distribution functions. In general, we observe that for such plasmas only those electromagnetic modes whose magnetic field perturbations are perpendicular to the ambient magneticeld, i.e.,B1 \perp B0, are effected by the anisotropy. However, in oblique propagation all modes do show such anisotropic effects. Considering the non-relativistic bi-Maxwellian distribution and studying the relevant components of the general dielectric tensor under appropriate conditions, we derive the dispersion relations for various modes and instabilities. We show that only the electromagnetic R- and L- waves, those derived from them and the O-mode are affected by thermal anisotropies, since they satisfy the required condition B1\perpB0. By contrast, the perpendicularly propagating X-mode and the modes derived from it (the pure transverse X-mode and Bernstein mode) show no such effect. In general, we note that the thermal anisotropy modifies the parallel propagating modes via the parallel acoustic effect, while it modifies the perpendicular propagating modes via the Larmor-radius effect. In oblique propagation for kinetic Alfven waves, the thermal anisotropy affects the kinetic regime more than it affects the inertial regime. The generalized fast mode exhibits two distinct acoustic effects, one in the direction parallel to the ambient magnetic field and the other in the direction perpendicular to it. In the fast-mode instability, the magneto-sonic wave causes suppression of the firehose instability. We discuss all these propagation characteristics and present graphic illustrations