Fluid-structure interaction with mean flow: Over-scattering and unstable resonance growth

It is known theoretically [1-3] that infinitely long fluid loaded plates in mean flow exhibit a range of unusual phenomena in the 'long time' limit. These include convective instability, absolute instability and negative energy waves which are destabilized by dissipation. However, structures are necessarily of finite length and may have discontinuities. Moreover, linear instability waves can only grow over a limited number of cycles before non-linear effects become dominant. We have undertaken an analytical and computational study to investigate the response of finite, discontinuous plates to ascertain if these unusual effects might be realized in practice. Analytically, we take a "wave scattering" [2,4] - as opposed to a "modal superposition" [5] - view of the fluttering plate problem. First, we solve for the scattering coefficients of localized plate discontinuities and identify a range of parameter space, well outside the convective instability regime, where over-scattering or amplified reflection/transmission occurs. These are scattering processes that draw energy from the mean flow into the plate. Next, we use the Wiener-Hopf technique to solve for the scattering coefficients from the leading and trailing edges of a baffled plate. Finally, we construct the response of a finite, baffled plate by a superposition of infinite plate propagating waves continuously scattering off the plate ends and solve for the unstable resonance frequencies and temporal growth rates for long plates. We present a comparison between our computational results and the infinite plate theory. In particular, the resonance response of a moderately sized plate is shown to be in excellent agreement with our long plate analytical predictions. Copyright © 2010 by ASME.

Robust airfoil design with respect to boundary layer transition

In order to disign an airfoil of which maximum lift coefficient (CL max) is not sensitive to location of forced top boundary layer transition. Taking maximizing mean value of CL max and minimizing standard deviation as biobjective, leading edge radius, manximum thickness and its location, maximum camber and its location as deterministic design variables, location of forced top boundary layer transition as stochastic variable, XFOIL as deterministic CFD solver, non-intrusive polynomial chaos as substitute of Monte Carlo method, we completed a robust airfoil design problem. Results demonstrate performance of initial airfoil is enhanced through reducing standard deviation of CL max. Besides, we also know maximum thickness has the most dominating effect on mean value of CL max, location of maximum thickness has the most dominating effect on standard deviation of CL max, maximum camber has a little effect on both mean value and standard deviation, and maximum camber is the only element of which increase can lead increase of mean value and standard deviation at the same time. Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc.

Re-examination of the coefficient of determination (r2) using road materials engineering case studies

The analysis of scientific data is integral to materials engineering and science. The correlation between measured variables is often quantified by estimating the coefficient of determination or the r2 value. This is the recognised procedure for determining linear relationships. The authors review the derivation of the r2 value and derive an associated quantity, termed the relative deviation (RD), which is the ratio of the root mean square of the deviations about the fitted line to the root mean square of the deviations about the y bar line expressed as a percentage. The relative deviation has an advantage over the coefficient of determination in that it has greater numerical sensitivity to changes in the spread of data about the fitted line, especially when the scatter is small. In addition, the relative deviation is able to define, in percentage terms, the reduction in scatter when different independent variables are correlated with a common dependent variable. Four case studies in the materials field (aggregate crushing value, Atterberg limits, permeability and creep of asphalt) from work carried out at the Queensland Main Roads Department are presented to show the use of the new parameter RD.

Blade loading and its application in the mean-line design of low pressure turbines

In order to minimize the number of iterations to a turbine design, reasonable choices of the key parameters must be made at the earliest possible opportunity. The choice of blade loading is of particular concern in the low pressure (LP) turbine of civil aero engines, where the use of high-lift blades is widespread. This paper presents an analytical mean-line design study for a repeating-stage, axial-flow Low Pressure (LP) turbine. The problem of how to measure blade loading is first addressed. The analysis demonstrates that the Zweifel coefficient [1] is not a reasonable gauge of blade loading because it inherently depends on the flow angles. A more appropriate coefficient based on blade circulation is proposed. Without a large set of turbine test data it is not possible to directly evaluate the accuracy of a particular loss correlation. The analysis therefore focuses on the efficiency trends with respect to flow coefficient, stage loading, lift coefficient and Reynolds number. Of the various loss correlations examined, those based on Ainley and Mathieson ([2], [3], [4]) do not produce realistic trends. The profile loss model of Coull and Hodson [5] and the secondary loss models of Craig and Cox [6] and Traupel [7] gave the most reasonable results. The analysis suggests that designs with the highest flow turning are the least sensitive to increases in blade loading. The increase in Reynolds number lapse with loading is also captured, achieving reasonable agreement with experiments. Copyright © 2011 by ASME.

The foundations of a new approach to additive manufacturing: Characteristics of free space metal deposition

In this paper, we report on the realisation of a free space deposition process (FSD). For the first time the use of a moving support structure to deposit tracks of metal starting from a substrate and extending into free space is characterised. The ability to write metal shapes in free space has wide ranging applications in additive manufacturing and rapid prototyping where the tracks can be layered to build overhanging features without the use of fixed support structures (such as is used in selective laser melting (SLM) and stereo lithography (SLA)). We demonstrate and perform a preliminary characterisation of the process in which a soldering iron was used to deposit lead free solder tracks. The factors affecting the stability of tracks and the effect of operating parameters, temperature, velocity, initial track starting diameter and starting volume were measured. A series of 10 tracks at each setting were compared with a control group of tracks; the track width, taper and variation between tracks were compared. Notable results in free space track deposition were that the initial track diameter and volume affected the repeatability and quality of tracks. The standard deviation of mean track width of tracks from the constrained initial diameter group were half that of the unconstrained group. The amount of material fed to the soldering iron before commencing deposition affected the taper of tracks. At an initial volume of 7 mm3 and an initial track diameter of 0.8 mm, none of the ten tracks deposited broke or showed taper > ∼1°. The maximum deposition velocity for free space track deposition using lead-free solder was limited to 1.5 mm s-1. © 2011 Elsevier B.V. All rights reserved.

Non-reflective boundary conditions for non-uniform mean velocity inlet and outlet flows

The numerical solution of problems in unbounded physical space requires a truncation of the computational domain to a reasonable size. As a result, the conditions on the artificial boundaries are generally unknown. Assumptions like constant pressure or velocities are only valid in the far field and lead to spurious reflections if applied on the boundaries of the truncated domain. A number of attempts have been made over the past decades to design conditions that prevent such reflections. One approach is based on characteristics. The standard analysis assumes a spatially uniform mean flow field but this is often impractical. In the present paper we show how to extend the formulation to the more general case of a non-uniform mean velocity field. A number of test cases are provided and our results compare favourably with other boundary conditions. In principle the present approach can be extended to include non-uniformities in all variables.

Stiffness of sands through a laboratory test database

Deformations of sandy soils around geotechnical structures generally involve strains in the range small (0·01%) to medium (0·5%). In this strain range the soil exhibits non-linear stress-strain behaviour, which should be incorporated in any deformation analysis. In order to capture the possible variability in the non-linear behaviour of various sands, a database was constructed including the secant shear modulus degradation curves of 454 tests from the literature. By obtaining a unique S-shaped curve of shear modulus degradation, a modified hyperbolic relationship was fitted. The three curve-fitting parameters are: an elastic threshold strain γe, up to which the elastic shear modulus is effectively constant at G0; a reference strain γr, defined as the shear strain at which the secant modulus has reduced to 0·5G0; and a curvature parameter a, which controls the rate of modulus reduction. The two characteristic strains γe and γr were found to vary with sand type (i.e. uniformity coefficient), soil state (i.e. void ratio, relative density) and mean effective stress. The new empirical expression for shear modulus reduction G/G0 is shown to make predictions that are accurate within a factor of 1·13 for one standard deviation of random error, as determined from 3860 data points. The initial elastic shear modulus, G0, should always be measured if possible, but a new empirical relation is shown to provide estimates within a factor of 1·6 for one standard deviation of random error, as determined from 379 tests. The new expressions for non-linear deformation are easy to apply in practice, and should be useful in the analysis of geotechnical structures under static loading.

Blade Loading and Its Application in the Mean-Line Design of Low Pressure Turbines

In order to minimize the number of iterations to a turbine design, reasonable choices of the key parameters must be made at the preliminary design stage. The choice of blade loading is of particular concern in the low pressure (LP) turbine of civil aero engines, where the use of high-lift blades is widespread. This paper considers how blade loading should be measured, compares the performance of various loss correlations, and explores the impact of blade lift on performance and lapse rates. To these ends, an analytical design study is presented for a repeating-stage, axial-flow LP turbine. It is demonstrated that the long-established Zweifel lift coefficient (Zweifel, 1945, "The Spacing of Turbomachine Blading, Especially with Large Angular Deflection" Brown Boveri Rev., 32(1), pp. 436-444) is flawed because it does not account for the blade camber. As a result the Zweifel coefficient is only meaningful for a fixed set of flow angles and cannot be used as an absolute measure of blade loading. A lift coefficient based on circulation is instead proposed that accounts for the blade curvature and is independent of the flow angles. Various existing profile and secondary loss correlations are examined for their suitability to preliminary design. A largely qualitative comparison demonstrates that the loss correlations based on Ainley and Mathieson (Ainley and Mathieson, 1957, "A Method of Performance Estimation for Axial-Flow Turbines," ARC Reports and Memoranda No. 2974; Dunham and Came, 1970, "Improvements to the Ainley-Mathieson Method of Turbine Performance Prediction," Trans. ASME: J. Eng. Gas Turbines Power, July, pp. 252-256; Kacker and Okapuu, 1982, "A Mean Line Performance Method for Axial Flow Turbine Efficiency," J. Eng. Power, 104, pp. 111-119). are not realistic, while the profile loss model of Coull and Hodson (Coull and Hodson, 2011, "Predicting the Profile Loss of High-Lift Low Pressure Turbines," J. Turbomach., 134(2), pp. 021002) and the secondary loss model of (Traupel, W, 1977, Thermische Turbomaschinen, Springer-Verlag, Berlin) are arguably the most reasonable. A quantitative comparison with multistage rig data indicates that, together, these methods over-predict lapse rates by around 30%, highlighting the need for improved loss models and a better understanding of the multistage environment. By examining the influence of blade lift across the Smith efficiency chart, the analysis demonstrates that designs with higher flow turning will tend to be less sensitive to increases in blade loading. © 2013 American Society of Mechanical Engineers.

Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank

This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. The resulting Riemannian space has strong geometrical properties: it is geodesically complete, and the metric is invariant with respect to all transformations that preserve angles (orthogonal transformations, scalings, and pseudoinversion). A meaningful approximation of the associated Riemannian distance is proposed, that can be efficiently numerically computed via a simple algorithm based on SVD. The induced mean preserves the rank, possesses the most desirable characteristics of a geometric mean, and is easy to compute. © 2009 Society for Industrial and Applied Mathematics.