The Third Two-Dimensional Problem of Three-Dimensional Blade Systems of Hydraulic Machines—Part 2: Analytical Results

1981 ◽  
Vol 103 (1) ◽  
pp. 42-51
Author(s):  
P. K. Agarwal ◽  
G. V. Viktorov
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Blade Profile ◽  
Flow Conditions ◽  
Numerical Examples ◽  
The Third ◽  
Hydraulic Machines ◽  
Method Of Solution ◽  
Axisymmetric Stream

This is the second part of a study of the “third” two-dimensional problem of three-dimensional blade systems of hydraulic machines. Part I described the formulation of the problem and the proposed method of solution to determine the velocity field on surfaces orthogonal to mean axisymmetric stream surfaces. Part II presents the numerical method of solving the integral equations; a few numerical examples for actual impellers/runners are also given. The results are presented in a series of figures and tables showing the distribution of the velocity component c2 along the blade profile on the surface q1 = const. The purpose of these numerical examples is to demonstrate the method and to help create a general understanding and awareness of the flow conditions existing in the runner passage.

The Third Two-Dimensional Problem of Three-Dimensional Blade Systems of Hydraulic Machines—Part 1: Theoretical Analysis

1981 ◽  
Vol 103 (1) ◽  
pp. 33-41
Author(s):  
P. K. Agarwal ◽  
G. V. Viktorov
Keyword(s):  
Three Dimensional ◽  
Tangential Velocity ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Analysis Method ◽  
Flow Conditions ◽  
Three Dimensional Flow ◽  
Hydraulic Machines ◽  
The Mean ◽  
Axisymmetric Stream

This is the first part of a study of “third” two-dimensional problem of three-dimensional blade systems of hydraulic machines. Proposed herein is a method of obtaining and evaluating the three-dimensional effect on a system of hydraulic machine blades with arbitrary geometry. An analysis method indicating the velocity distributions on surfaces perpendicular to the mean axisymmetric stream surfaces has been formulated to help create a general understanding and awareness of the flow conditions in the runner passage. An application of generalized analytic functions for inviscid, incompressible flow has been made to find out the general solution on an auxiliary plane, transformed conformally from the physical plane. Integral equation systems for the tangential velocity and the velocity potential function have been deduced. Thus, the solution of the three-dimensional flow problem is supplied by two-dimensional computation methods.

Compressor Performance 2D Modelling at Reverse Flow Conditions

2010 ◽  
Author(s):  
L. Gallar ◽  
I. Tzagarakis ◽  
V. Pachidis ◽  
R. Singh
Keyword(s):  
Experimental Data ◽  
Reverse Flow ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Engine Performance ◽  
Two Dimensional ◽  
Flow Conditions ◽  
Compressor Surge ◽  
Compressor Performance ◽  
Shaft Failure

After a shaft failure the compression system of a gas turbine is likely to surge due to the heavy vibrations induced on the engine after the breakage. Unlike at any other conditions of operation, compressor surge during a shaft over-speed event is regarded as desirable as it limits the air flow across the engine and hence the power available to accelerate the free turbine. It is for this reason that the proper prediction of the engine performance during a shaft over-speed event claims for an accurate modelling of the compressor operation at reverse flow conditions. The present study investigates the ability of the existent two dimensional algorithms to simulate the compressor performance in backflow conditions. Results for a three stage axial compressor at reverse flow were produced and compared against stage by stage experimental data published by Gamache. The research shows that due to the strong radial fluxes present over the blades, two dimensional approaches are inadequate to provide satisfactory results. Three dimensional effects and inaccuracies are accounted for by the introduction of a correction parameter that is a measure of the pressure loss across the blades. Such parameter is tailored for rotors and stators and enables the satisfactory agreement between calculations and experiments in a stage by stage basis. The paper concludes with the comparison of the numerical results with the experimental data supplied by Day on a four stage axial compressor.

scholarly journals New insights into the suitability of the third dimension for visualizing multivariate/multidimensional data: A study based on loss of quality quantification

2014 ◽  
Vol 15 (1) ◽  
pp. 3-30 ◽  
Author(s):  
Antonio Gracia ◽  
Santiago González ◽  
Víctor Robles ◽  
Ernestina Menasalvas ◽  
Tatiana von Landesberger
Keyword(s):  
Analytical Approach ◽  
Distance Perception ◽  
Statistical Tests ◽  
Three Dimensional ◽  
Multidimensional Data ◽  
Two Dimensional ◽  
Dimensional Representation ◽  
The Third ◽  
Quality Degradation ◽  
Third Dimension

Most visualization techniques have traditionally used two-dimensional, instead of three-dimensional representations to visualize multidimensional and multivariate data. In this article, a way to demonstrate the underlying superiority of three-dimensional, with respect to two-dimensional, representation is proposed. Specifically, it is based on the inevitable quality degradation produced when reducing the data dimensionality. The problem is tackled from two different approaches: a visual and an analytical approach. First, a set of statistical tests (point classification, distance perception, and outlier identification) using the two-dimensional and three-dimensional visualization are carried out on a group of 40 users. The results indicate that there is an improvement in the accuracy introduced by the inclusion of a third dimension; however, these results do not allow to obtain definitive conclusions on the superiority of three-dimensional representation. Therefore, in order to draw further conclusions, a deeper study based on an analytical approach is proposed. The aim is to quantify the real loss of quality produced when the data are visualized in two-dimensional and three-dimensional spaces, in relation to the original data dimensionality, to analyze the difference between them. To achieve this, a recently proposed methodology is used. The results obtained by the analytical approach reported that the loss of quality reaches significantly high values only when switching from three-dimensional to two-dimensional representation. The considerable quality degradation suffered in the two-dimensional visualization strongly suggests the suitability of the third dimension to visualize data.

THREE‐DIMENSIONAL GRAVITY INSTABILITY DERIVED FROM TWO‐DIMENSIONAL SOLUTIONS

Geophysics ◽  
1966 ◽  
Vol 31 (1) ◽  
pp. 153-166 ◽  
Author(s):  
M. A. Biot
Keyword(s):  
Three Dimensional ◽  
Time History ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Periodic Patterns ◽  
Numerical Work ◽  
Characteristic Distance ◽  
General Validity ◽  
Dimensional Gravity ◽  
History Of

The theory of three‐dimensional gravity instability of multilayers is developed with particular application to salt structures. It is shown that three‐dimensional solutions are immediately obtained without further numerical work from the solution of the corresponding two‐dimensional problem. Application to a number of typical three‐dimensional structures yields the characteristic distance between peaks and crests and shows that this distance does not differ significantly from the wavelength of the two‐dimensional solution. Various periodic patterns are examined corresponding to rectangular and hexagonal cells. The time history of nonperiodic structures corresponding to initial deviations from perfect horizontality is also derived. The method is applied to the three‐dimensional problem of generation of salt structures when the time‐history of sedimentation is taken into account with variable thickness and compaction of the overburden and establishes the general validity of the geological conclusions derived from the previous two‐dimensional treatment of the same problem (Biot and Odé, 1965). The present method of deriving three‐dimensional solutions, which is developed here in the special context of gravity instability, is valid for a wide variety of problems in theoretical physics.

Sufficient Symmetry Conditions for Isotropy of the Elastic Moduli Tensor

1987 ◽  
Vol 54 (4) ◽  
pp. 772-777 ◽  
Author(s):  
R. M. Christensen
Keyword(s):  
Elastic Moduli ◽  
Fiber Composite ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Axial Deformation ◽  
Space Network ◽  
Plane Normal ◽  
Rank Tensor ◽  
Three Space

Symmetry conditions are found that assure isotropy of the fourth rank tensor of elastic moduli. Crystallography provides the answer to this problem in the two-dimensional context, namely one axis of three-fold symmetry assures the isotropy of properties in the plane normal to the axis. The present work provides the answer in the three-dimensional problem: 6 axes of five-fold symmetry are sufficient to give isotropy of the elastic moduli. An important restriction must accompany the present result. The derivation is given in the special form appropriate to low density materials which have a microstructure that transmits load according to the axial deformation of a space network of material distributed into micro-struts. The corresponding fiber composite idealization is that of a fiber dominated system, it therefore follows that if the fibers take the 6 specific orientations in three-space then isotropy is obtained.

A contribution to the application of photoelasticity to the micromechanics of composite materials

1969 ◽  
Vol 4 (2) ◽  
pp. 88-94 ◽  
Author(s):  
D E W Stone
Keyword(s):  
Composite Materials ◽  
Three Dimensional ◽  
Maximum Amount ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Casting Technique ◽  
Intensity Measurements ◽  
Photoelastic Model ◽  
Composite Models

Photoelastic-model methods can prove advantageous for the investigation of microstresses in composite materials. Some two-dimensional investigations of this type are discussed and the extension of this work into three dimensions is considered. It is suggested that more than one approach to the three-dimensional problem may be practicable, and special attention is paid to obtaining the maximum amount of information from a sandwiched polariscope by means of light-intensity measurements. A cold-casting technique for the fabrication of composite models is also described.

3D structure–2D plate interaction model

2019 ◽  
Vol 24 (10) ◽  
pp. 3354-3377 ◽  
Author(s):  
Matko Ljulj ◽  
Josip Tambača
Keyword(s):  
Elastic Layer ◽  
3D Structure ◽  
Three Dimensional ◽  
Interaction Model ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Dimensional Model ◽  
Limit Model ◽  
Two Dimensional Model ◽  
Thin Elastic Layer

In this paper, we derive models for the interaction of a linearized three-dimensional elastic structure with a thin elastic layer of possibly different material attached to it. Rigorous derivation is performed by considering a thin three-dimensional layer and the asymptotics of the solution of the full remaining three-dimensional problem when the thickness [Formula: see text] of the thin layer tends to zero. Furthermore, the attached thin material is assumed to have the elasticity coefficients which are of order [Formula: see text], for [Formula: see text] with respect to the coefficients of the three-dimensional body. In the limit, five different models are obtained with respect to different choices of p, namely [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. Furthermore a three-dimensional–two-dimensional model is proposed that has the same asymptotics as the original three-dimensional problem. This is convenient for applications because one does not have to decide in advance which limit model to use.

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