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

A linearized Euler analysis of unsteady flows in turbomachinery

A method for calculating unsteady flows in cascades is presented. The model, which is based on the linearized unsteady Euler equations, accounts for blade loading shock motion, wake motion, and blade geometry. The mean flow through the cascade is determined by solving the full nonlinear Euler equations. Assuming the unsteadiness in the flow is small, then the Euler equations are linearized about the mean flow to obtain a set of linear variable coefficient equations which describe the small amplitude, harmonic motion of the flow. These equations are discretized on a computational grid via a finite volume operator and solved directly subject to an appropriate set of linearized boundary conditions. The steady flow, which is calculated prior to the unsteady flow, is found via a Newton iteration procedure. An important feature of the analysis is the use of shock fitting to model steady and unsteady shocks. Use of the Euler equations with the unsteady Rankine-Hugoniot shock jump conditions correctly models the generation of steady and unsteady entropy and vorticity at shocks. In particular, the low frequency shock displacement is correctly predicted. Results of this method are presented for a variety of test cases. Predicted unsteady transonic flows in channels are compared to full nonlinear Euler solutions obtained using time-accurate, time-marching methods. The agreement between the two methods is excellent for small to moderate levels of flow unsteadiness. The method is also used to predict unsteady flows in cascades due to blade motion (flutter problem) and incoming disturbances (gust response problem)

Aeroacoustic sensitivity analysis and optimal aeroacoustic design of turbomachinery blades

During the first year of the project, we have developed a theoretical analysis - and wrote a computer code based on this analysis - to compute the sensitivity of unsteady aerodynamic loads acting on airfoils in cascades due to small changes in airfoil geometry. The steady and unsteady flow though a cascade of airfoils is computed using the full potential equation. Once the nominal solutions have been computed, one computes the sensitivity. The analysis takes advantage of the fact that LU decomposition is used to compute the nominal steady and unsteady flow fields. If the LU factors are saved, then the computer time required to compute the sensitivity of both the steady and unsteady flows to changes in airfoil geometry is quite small. The results to date are quite encouraging, and may be summarized as follows: (1) The sensitivity procedure has been validated by comparing the results obtained by 'finite difference' techniques, that is, computing the flow using the nominal flow solver for two slightly different airfoils and differencing the results. The 'analytic' solution computed using the method developed under this grant and the finite difference results are found to be in almost perfect agreement. (2) The present sensitivity analysis is computationally much more efficient than finite difference techniques. We found that using a 129 by 33 node computational grid, the present sensitivity analysis can compute the steady flow sensitivity about ten times more efficiently that the finite difference approach. For the unsteady flow problem, the present sensitivity analysis is about two and one-half times as fast as the finite difference approach. We expect that the relative efficiencies will be even larger for the finer grids which will be used to compute high frequency aeroacoustic solutions. Computational results show that the sensitivity analysis is valid for small to moderate sized design perturbations. (3) We found that the sensitivity analysis provided important insight into how airfoils should be modified to improve their aeroelastic stability. Using this insight, we redesigned an aeroelastically unstable cascade to produce a stable cascade

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

Sensitivity analysis for aeroacoustic and aeroelastic design of turbomachinery blades

A new method for computing the effect that small changes in the airfoil shape and cascade geometry have on the aeroacoustic and aeroelastic behavior of turbomachinery cascades is presented. The nonlinear unsteady flow is assumed to be composed of a nonlinear steady flow plus a small perturbation unsteady flow that is harmonic in time. First, the full potential equation is used to describe the behavior of the nonlinear mean (steady) flow through a two-dimensional cascade. The small disturbance unsteady flow through the cascade is described by the linearized Euler equations. Using rapid distortion theory, the unsteady velocity is split into a rotational part that contains the vorticity and an irrotational part described by a scalar potential. The unsteady vorticity transport is described analytically in terms of the drift and stream functions computed from the steady flow. Hence, the solution of the linearized Euler equations may be reduced to a single inhomogeneous equation for the unsteady potential. The steady flow and small disturbance unsteady flow equations are discretized using bilinear quadrilateral isoparametric finite elements. The nonlinear mean flow solution and streamline computational grid are computed simultaneously using Newton iteration. At each step of the Newton iteration, LU decomposition is used to solve the resulting set of linear equations. The unsteady flow problem is linear, and is also solved using LU decomposition. Next, a sensitivity analysis is performed to determine the effect small changes in cascade and airfoil geometry have on the mean and unsteady flow fields. The sensitivity analysis makes use of the nominal steady and unsteady flow LU decompositions so that no additional matrices need to be factored. Hence, the present method is computationally very efficient. To demonstrate how the sensitivity analysis may be used to redesign cascades, a compressor is redesigned for improved aeroelastic stability and two different fan exit guide vanes are redesigned for reduced downstream radiated noise. In addition, a framework detailing how the two-dimensional version of the method may be used to redesign three-dimensional geometries is presented

Prediction of Undsteady Flows in Turbomachinery 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. The model accounts for blade loading, blade geometry, shock motion, and wake motion. Assuming that the unsteadiness in the flow is small relative to the nonlinear mean solution, the unsteady Euler equations can be linearized about the mean flow. This yields a set of linear variable coefficient equations that describe the small amplitude harmonic motion of the fluid. These linear equations are then discretized on a computational grid and solved using standard numerical techniques. For transonic flows, however, one must use a linear discretization which is a conservative linearization of the non-linear discretized Euler equations to ensure that shock impulse loads are accurately captured. Other important features of this analysis include a continuously deforming grid which eliminates extrapolation errors and hence, increases accuracy, and a new numerically exact, nonreflecting far-field boundary condition treatment based on an eigenanalysis of the discretized equations. Computational results are presented which demonstrate the computational accuracy and efficiency of the method and demonstrate the effectiveness of the deforming grid, far-field nonreflecting boundary conditions, and shock capturing techniques. A comparison of the present unsteady flow predictions to other numerical, semi-analytical, and experimental methods shows excellent agreement. In addition, the linearized Euler method presented requires one or two orders-of-magnitude less computational time than traditional time marching techniques making the present method a viable design tool for aeroelastic analyses

Aeroelastic Studies of a Rectangular Wing with a Hole: Correlation of Theory and Experiment

Two rectangular wing models with a hole have been designed and tested in the Duke University wind tunnel to better understand the effects of damage. A rectangular hole is used to simulate damage. The wing with a hole is modeled structurally as a thin elastic plate using the finite element method. The unsteady aerodynamics of the plate-like wing with a hole is modeled using the doublet lattice method. The aeroelastic equations of motion are derived using Lagrange's equation. The flutter boundary is found using the V-g method. The hole's location effects the wing's mass, stiffness, aerodynamics and therefore the aeroelastic behavior. Linear theoretical models were shown to be capable of predicting the critical flutter velocity and frequency as verified by wind tunnel tests

Design and assembly sequence analysis of option 3 for CETF reference space station

A design and assembly sequence was conducted on one option of the Dual Keel Space Station examined by a NASA Critical Evaluation Task Force to establish viability of several variations of that option. A goal of the study was to produce and analyze technical data to support Task Force decisions to either examine particular Option 3 variations in more depth or eliminate them from further consideration. An analysis of the phasing assembly showed that use of an Expendable Launch Vehicle in conjunction with the Space Transportation System (STS) can accelerate the buildup of the Station and ease the STS launch rate constraints. The study also showed that use of an Orbital Maneuvering Vehicle on the first flight can significantly benefit Station assembly and, by performing Station subsystem functions, can alleviate the need for operational control and reboost systems during the early flights. In addition to launch and assembly sequencing, the study assessed stability and control, and analyzed node-packaging options and the effects of keel removal on the structural dynamics of the Station. Results of these analyses are presented and discussed

Why Terror Networks are Dissimilar: How Structure Relates to Function

Abstract Theories on international terrorist networks are wrought with contradiction. On the one hand, networks that support or facilitate politically motivated violent extremism are thought to pose a threat because they are centralized and hierarchical. On the other hand, the same networks are thought to pose a threat because they are decentralized and operate autonomously. Social networks analysis (SNA) makes it possible to resolve this apparent contradiction by controlling across countries for characteristics and structure of networks linked to the same terrorist organization relative to different functions that such networks perform. One terrorist organization for which sufficient open-source data exist to mount a systematic comparison is Al-Shabaab (AS). Comparing traits such as brokers, centrality characteristics of nodes, international linkages, and use of funds, the chapter compares AS networks as they relate to recruitment, fundraising and attacks across the United States and Australia with corroborating evidence from Canada, the United Kingdom, the Netherlands and Denmark. Although networks differ markedly across these attributes, unrelated networks performing similar functions are consistent in their nature and structure. These findings suggest that networks are functionally differentiated insofar as they serve as strategic repertoires. This is a significant finding. Knowing how a network's function is related strategically to its structure means being able to infer a network's function if only its structure is known and, conversely, being able to infer a network's structure if only its function is known. Not only does SNA thereby facilitate detection and dismantling of networks, it also suggests that recruitment, fundraising and attack networks require differentiated approaches by defence and security agencies insofar as SNA shows them to be distinct phenomena

A Time-Linearized Navier–Stokes Analysis of Stall Flutter,”

ABSTRACT 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 NavierStokes 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 LaxWendroff 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