Euler Analysis Research Articles

Euler Analysis of a Machine for Raising Waters Proposed by Mr. de Mour

We examine a publication by Euler, Sur une nouvelle manière d’élever de l’eau propose par M. de Mour (On a new method to raise water proposed by Mr. de Mour), published in 1753, addressing a type of water sprinkler, consisting of an inclined tube with its lower end immersed into water, and discharging at its top by turning a vertical axis to which the inclined tube is attached. In this publication, Euler uses pioneering approaches by applying the concept of mechanical power to estimate the power needed to drive the machine by the human force, and by measuring time by the length of a pendulum necessary to achieve one oscillation. Several design formulas were derived for the determination of the output of the machine, giving also measures for its construction, and supported at the end by a numerical example to show the feasibility of the proposition. This is, perhaps, one of the earliest attempts to design a rotating machine to raise waters.

Large displacement behavior of double parallelogram flexure mechanisms with underconstraint eliminators

This paper presents a new analytical method for predicting the large displacement behavior of flexural double parallelogram (DP) bearings with underconstraint eliminator (UE) linkages. This closed-form perturbative Euler analysis method is able to – for the first time – directly incorporate the elastomechanics of a discrete UE linkage, which is a hybrid flexure element that is linked to ground as well as both stages on the bearing. The models are used to understand a nested linkage UE design, however the method is extensible to other UE linkages. Design rules and figures-of-merit are extracted from the analysis models, which provide powerful tools for accelerating the design process. The models, rules and figures-of-merit enable the rapid design of a UE for a desired large displacement behavior, as well as providing a means for determining the limits of UE and DP structures. This will aid in the adoption of UE linkages into DP bearings for precision mechanisms. Models are generated for a nested linkage UE design, and the performance of this DP with UE structure is compared to a DP-only bearing. The perturbative Euler analysis is shown to match existing theories for DP-only bearings with distributed compliance within ≈2%, and Finite Element Analysis for the DP with UE bearings within an average 10%.

Geometrical statistics of fluid deformation: Restricted Euler approximation and the effects of pressure

The geometrical statistics of fluid deformation are analyzed theoretically within the framework of the restricted Euler approximation, and numerically using direct numerical simulations. The restricted Euler analysis predicts that asymptotically a material line element becomes an eigenvector of the velocity gradient regardless its initial orientation. The asymptotic stretching rate equals the intermediate eigenvalue of the strain rate tensor. Analyses of numerical data show that the pressure Hessian is the leading cause to destroy the alignment between the longest axis of the material element and the strongest stretching eigendirection of the strain rate. It also facilitates the alignment between the longest axis of the element and the intermediate eigendirection of the strain rate during initial evolution, but tends to oppose the alignment later.

Compression Behavior of Single-Layer Graphenes

Central to most applications involving monolayer graphenes is its mechanical response under various stress states. To date most of the work reported is of theoretical nature and refers to tension and compression loading of model graphenes. Most of the experimental work is indeed limited to the bending of single flakes in air and the stretching of flakes up to typically approximately 1% using plastic substrates. Recently we have shown that by employing a cantilever beam we can subject single graphenes to various degrees of axial compression. Here we extend this work much further by measuring in detail both stress uptake and compression buckling strain in single flakes of different geometries. In all cases the mechanical response is monitored by simultaneous Raman measurements through the shift of either the G or 2D phonons of graphene. Despite the infinitely small thickness of the monolayers, the results show that graphenes embedded in plastic beams exhibit remarkable compression buckling strains. For large length (l)-to-width (w) ratios (> or =0.2) the buckling strain is of the order of -0.5% to -0.6%. However, for l/w < 0.2 no failure is observed for strains even higher than -1%. Calculations based on classical Euler analysis show that the buckling strain enhancement provided by the polymer lateral support is more than 6 orders of magnitude compared to that of suspended graphene in air.

Sources of along-strike variation in magnetic anomalies related to intrasedimentary faults: A case study from the Rio Grande Rift, USA

Normal faults within sedimentary basins are commonly associated with subtle linear features in high-resolution, total magnetic intensity (TMI) data. Many of these anomalies arise from the tectonic juxtaposition of sedimentary units of differing magnetic properties. In detail, the anomalies can be quite variable in character, even along the strike of individual faults. To understand this variability, we examine the well-exposed San Ysidro Fault in the central Rio Grande Rift, USA, using detailed magnetic-property measurements, geophysical models based on geology, and Euler analysis. We find that along-strike anomaly variability arises mainly from (1) multi-levelled magnetic contrasts at the fault that are variably sampled by uneven levels of erosion, and to a lesser extent from (2) magnetic susceptibilities that vary along strike within individual units, and (3) variable throw and dip of the fault that produces differences in the extents to which contrasting units are in contact. The multi-levelled magnetic contrasts arise from the juxtaposition of different strata across the fault at discrete depths. Locations of magnetic sources along the fault estimated from Euler analysis of the TMI data reflect the variations in depths to the shallowest sources along strike. Variations in clustering of the Euler solutions suggest that the sources have variable geometry (structural index). The results at the San Ysidro Fault demonstrate the important and complex role of multi-level led magnetic sources in understanding anomalies associated with faulted geologic layers in general. The potential for multiple sources suggests that the use of simple model geometries to represent faults may not always be appropriate.

Toward silent supersonic transport—A fundamental study of supersonic biplane

In this research, aerodynamic design of biplane airfoils in supersonic flight is discussed based on computational fluid dynamics (CFD). In supersonic flight, airfoils generate strong sonic booms and wave drags accompanied by shock waves. New airfoil geometries which significantly reduce shock waves using a biplane concept will be proposed. The background of this concept originates from Busemann biplane and Licher type biplane concepts. In order to focus on the shock‐wave characteristics around biplane configuration, inviscid flow (Euler) analyses are performed (which are particularly suitable for wave drag analyses). For the evaluation of the reduction level of shock waves, the wave drag coefficient is used. The design Mach number is 1.7. The aerodynamic design is conducted using an iterative inverse design method that is newly implemented. A biplane configuration with a desired performance has been obtained. Having total maximum thickness ratio of 0.10, it has the lift‐to‐wave‐drag ratio of 21.7 at a desired lift condition for supersonic flight. At the range of lift coefficient more than 0.14 this designed biplane has lower wave drag than that of a (zero‐thickness) single flat plate airfoil. This result will lead to the realization of silent supersonic transport.

Robust Structural Design of an Active Aeroelastic Wing with Maneuver Load Inaccuracies

A multidisciplinary design methodology for aircraft structures subject to inaccuracies in aerodynamic load predictions is presented. A statistical load-correction model is developed that is based on typical differences between loads as predicted by nonlinear and linear aerodynamic theories. Load corrections are propagated to a response function that represents the magnitude of structural redesign that would be necessary when the structure is subject to variation in the loads to which it was designed. The method is applied to an active aeroelastic wing concept, in which both the structural and control law designs are highly sensitive to variations in the aerodynamic loads. Computational fluid dynamics Euler analysis is used to construct the load-correction model. The most critical loading condition is then identified and used to redesign the structure and control-surface gear ratios. The redesigned structure was found to be 11% heavier and significantly more robust than the structure that was optimized with the linear loads only. The present methodology provides a way to account for load inaccuracies in the early phases of the design process, thereby, reducing the need for redesign late in the design process as load predictions become more accurate.

On the Accuracy of Direct Noise Calculations Based on the Euler Model

The paper reviews a class of Direct Noise Calculation (DNC) models based on the compressible Euler equations and analyses their computational accuracy considering the combination of model completeness, scheme accuracy and stability to round off errors. It is shown that a form of the Nonlinear Disturbance Equations (NLDE) is sensitive to the round off errors. An NLDE formulation with Roe average matrix (NLDE/Roe) for minimizing the round off errors is proposed. (The NLDE/Roe is identical to Direct Numerical Simulation (DNS) at rather large solutions and converges to the Linearised Euler Equations (LEE) at the small solutions NLDE/Roe thus accurately treats the signals including both large and small scales of disturbances). A version of NLDE in the case of disturbance decomposition is given in the general form. While studying the DNC model efficiency, a new measure of the total model accuracy is introduced. It is proposed to test all the models and schemes available by comparing the output numerical result with the exact solution of the complete model, that is, with the Euler equations.

Navier-Stokes Analysis of Helicopter Rotor Aerodynamics in Hover and Forward Flight

Reynolds-averaged compressible Navier ‐Stokes computations are presented for the hovering 7A model rotor and a low aspect ratio NACA0012 rotor in nonlifting forward e ight. For enhanced hover performance prediction, aeroelastic effects are taken into account by tightly coupling the e ow solver with a e nite element model of the blade based on Timoshenko beam theory. Good agreement between computed and measured rotor performance is achieved for three collective pitch settings on structured periodic grids featuring noncongruent cell faces along the periodicity planes. Results of a hybrid Navier ‐Stokes/Euler chimera hover computation are presented, in addition to the periodic grid analysis demonstrating the superiority of the overset grid approach with respect to tip vortex conservation. The unsteady Navier ‐Stokes computations for the nonlifting forward e ight test case show more sensitivity relative to time step size than comparable Euler analyses. Overall correlation of computed and experimental near-tip pressure distributions over blade azimuth is considered to be fair, and the strong transonic effects on the advancing blade are adequately captured by the numerical analysis.

A Three-Dimensional Linearized Euler Analysis of Classical Wake/Stator Interactions: Validation and Unsteady Response Predictions

A comprehensive validation of the linearized Euler analysis, LINFLUX, for wake/blade row interaction is carried out. The flow configuration is that of the benchmark problem for rotor-stator interaction proposed at the Third Computational Aeroacoustics Workshop. It consists of an unstaggered, annular, flat-plate blade row excited by the vortical gusts associated with the wakes shed from an upstream rotor. The numerical results for the unsteady pressure responses of the stator are compared with semi-analytic lifting surface and lifting line solutions. The validation is first conducted for narrow-annulus flows, where the numerical results are shown to agree well with classical two-dimensional solutions over a range of frequencies. We then carry out a detailed comparison of the three-dimensional LINFLUX results with the lifting surface results of Namba and Schulten for a blade row with a hub-to-tip ratio of 0.5. This study encompasses gust excitation frequencies for which the stator responses vary from cut off to propagating, as well as gusts with varying degrees of spanwise variation. The numerical and semi-analytical analyses yield results for the stator pressure response, including the complex amplitudes of the propagating and least attenuated, evanescent, pressure modes that are in very good agreement. The effect of increasing the spanwise phase variation of the gust is generally, but not necessarily, to reduce the power associated with the acoustic response of the blade row. A comparison of the present numerical results with those obtained from a stripwise application of classical linear theory reveals that the latter approach can be erroneous and, therefore, of questionable applicability to realistic turbomachinery unsteady flows.

Higher Trigonometry, Hyperreal Numbers, and Euler's Analysis of Infinities

Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use. Although analysis does not require an exhaustive knowledge of algebra,

Modeling the Response from a Cascade to an Upstream Acoustic Disturbance

Time-accurate Euler flow field simulations for the flow through a two-dimensional cascade subjected to an upstream acoustic disturbance were used as the basis for a small disturbance model to predict the reflected response upstream of the cascade. The small disturbance model results in a linear system of algebraic equations for the properties of the reflected and transmitted disturbances. The model predicts the reflected and transmitted responses as a function of the cascade blade geometry, the disturbance strength, and the initial flow properties prior to the upstream disturbance. The predicted results from the small disturbance model were then compared with the Euler analysis results for a two-dimensional cascade. Agreement between the model and the Euler data indicated that the model was potentially useful as a basis for an outflow boundary condition for time-accurate Euler/Navier-Stokes (ENS) simulations of supersonic mixed compression inlet flows needed to determine the stability margin of an inlet that encounters an atmospheric disturbance. This boundary condition must provide an approximation of the response from the compressor by the inlet flow at the face of the compressor when a disturbance from upstream passes through the inlet and into the compressor. A new characteristic boundary condition based on the small disturbance response model was formulated and demonstrated independently in two one-dimensional Euler codes. The one-dimensional Euler codes with the new boundary condition and with existing boundary condition formulations were used to predict the reflection response for an axial compressor experiment. The new boundary condition was found to provide a significant improvement in accuracy for the reflection response of an acoustic disturbance from a compressor relative to existing outflow boundary condition models. For a supersonic mixed compression inlet, a one-dimensional Euler code was also used to demonstrate the dependence of the inlet normal-shock response and unstart tolerance on the outflow boundary condition.

A Design Method to Prevent Low Pressure Turbine Blade Flutter

A design approach to avoid flutter of low pressure turbine blades in aircraft engines is described. A linearized Euler analysis, previously validated using experimental data, is used for a series of parameter studies. The influence of mode shape and reduced frequency are investigated. Mode shape is identified as the most important contributor to determining the stability of a blade design. A new stability parameter is introduced to gain additional insight into the key contributors to flutter. This stability parameter is derived from the influence coefficient representation of the cascade, and includes only contributions from the reference blade and its immediate neighbors. This has the effect of retaining the most important contributions to aerodynamic damping while filtering out terms of less significance. This parameter is utilized to develop a stability map, which provides the critical reduced frequency as a function of torsion axis location. Rules for preliminary design and procedures for detailed design analysis are defined. [S0742-4795(00)01401-0]

ROTARY INERTIA, AXIAL AND SHEAR DEFORMATION EFFECTS ON THE IN-PLANE NATURAL FREQUENCIES OF SYMMETRIC CROSS-PLY LAMINATED CIRCULAR ARCHES

The in-plane free vibrational analysis of symmetric cross-ply laminated circular arches is studied using both the Timoshenko and Bernoulli–Euler beam theories. The free vibration equations are derived based on the distributed parameter model. The transfer matrix method is used in the analysis. The numerical algorithm available in the literature is adopted to compute the exact overall dynamic transfer matrix of the curved beam. The rotary inertia, axial and shear deformation effects are considered in the Timoshenko analysis by the first-order shear deformation theory. All these effects are neglected in the Bernoulli–Euler analysis. Radius of the arch/thickness ratios from 5 to 25, fixed–fixed, fixed–simple and fixed–free boundary conditions, and two values of the opening angles (10° and 90°) are taken into consideration in the parametric study. The effects of the ratio of the extensional modulus to the transverse modulus on the natural frequencies are examined. Non-dimensional numerical results are also presented in terms of relative errors.

Nonlinear Separated Inviscid-Viscous Analysis of Oscillating Cascade Aerodynamics Using an Inverse Integral Method

A mathematical model is developed to analyze the unsteady flow through an harmonically oscillating cascade of airfoils, including separated flow. The model incorporates an inverse integral boundary layer solution with the time-marching Euler analysis NPHASE. An embedded composite grid formulation is incorporated, specifically a deforming C-grid embedded in a Cartesian H-grid, thereby simplifying grid generation. To reduce computational requirements, Fourier series unsteady periodic boundary conditions are implemented. The integral turbulent boundary layer model is closed with steady correlations adopted in a quasi-steady manner. To couple the inviscid and viscous solutions, the viscous effect is modeled in the unsteady Euler solution in a quasi-steady manner by a transpiration boundary condition. An isolated airfoil is used to compare the steady interaction model with experimental data. Then a flat plate cascade is used to verify the unsteady flow solver with linear theory predictions. An experimental unsteady aerodynamics data set of a loaded cascade with separated meanflow executing torsional oscillations compared favorably with the analysis. The code is then utilized to study the effect of flow separation on the unsteady aerodynamics.

Deflection of Turbofan Exhaust Streams for Enhanced Engine/Nacelle Integration

An analytical and experimental description of how the fan exhaust of a modern turbofan can be dee ected into an annular cascade using only core bleed e ow is presented. Thrust reversing, emphasized herein, and/or vectoring is achieved without the need for fan duct blockers or other hardware devices to turn the e ow, allowing lighter, less complex, and lower loss exhaust systems for more efe cient up-andaway e ight. Two approaches for dee ecting fan e ow into an annular cascade, using two different models, are described, and test results are discussed. Results of a Euler analysis and its critical relationship with the testing are also described. Both methods use injected core e ow to turn the fan e ow. One method uses an annular air-curtain emanating from an annular injection slot located opposite the cascade and the other uses a similar slot in the nozzle throat to force upstream e ow into the cascade. Both approaches are impacted by engine core bleed limits. Injector jet arrangements to meet this requirement while providing adequate reverse thrust are described; analyses and test results are reviewed.

Bowed Stators: An Example of CFD Applied to Improve Multistage Compressor Efficiency

Analysis of multistage compressor stator surface static pressure data reveals that the radial growth of suction surface corner separation prematurely separates core flow stator sections, limiting their pressure rise capability and generating endwall loss. Modeling of the stator flowfield, using a three-dimensional Euler analysis, has led to the development of “bowed” stator shapes, which generate radial forces that reduce diffusion rates in the suction surface corners, in order to delay the onset of corner separation. Experimental testing of the bowed stator concept in a three-stage research compressor has confirmed the elimination of suction surface corner separation, the resulting reduction of the endwall loss, and the increase in pressure rise capability of the stator core sections. This results in more robust pressure rise characteristics and substantially improved efficiency over the entire flow range of the compressor. The strong interaction effects of the bowed stator with the viscous endwall flowfield are shown to be predictable using a three-dimensional multistage Navier–Stokes analysis. This permits matching of the rotors to the altered stator exit profiles, in order to avoid potential stability limiting interactions. Application of bowed stators to a high bypass ratio engine eleven-stage high-pressure compressor has resulted in substantial improvement in efficiency, with no stability penalty.

A Numerical Method for the Prediction of Bladed Disk Forced Response

A numerical method has been developed to predict the forced response of bladed disks due to a wake excitation from upstream blade rows. The structure is modeled by a three-dimensional finite element mesh of a bladed disk segment. Using cyclic symmetry, this model provides a modal base for the rotating structure. The aerodynamic damping of the vibratory modes and the excitation pressures on the blades due to the propagation of upstream flow defects are computed separately using the same three-dimensional unsteady Euler analysis software. A modal response solution of the aeromechanical system is then performed. This analytical methodology has been used to study the forced response of an experimental high pressure compressor blisk. The results are analyzed and compared with actual rig tests.

Euler Analysis of Oscillating Cascade Aerodynamics Using Embeded Composite Grids

Article Euler Analysis of Oscillating Cascade Aerodynamics Using Embeded Composite Grids was published on September 1, 1996 in the journal International Journal of Turbo & Jet-Engines (volume 13, issue 3).

Time-marching Euler analysis of ducted-propellers

A time-accurate solution procedure and its application to ducted-propeller configuration is described in this article. Three-dimensional unsteady Euler equations are solved using a hybrid scheme to reduce the computational time and memory requirements. The solution procedure is applied to a ducted-propeller obtained by enclosing an eight-bladed SR7 advanced propeller within a duct with NACA 0003 airfoil cross section, operating in a steady axisymmetric flowfield. The performance characteristics are compared with other published numerical results. The effect of tip-gap is found to be significant on performance and flow characteristics.