AXIAL SHEAR VIBRATIONS OF STAR-SHAPED BARS - AN APPLICATION OF CONFORMAL TRANSFORMATION
Conformal transformation method applied to natural frequency of axial shear vibration of prismatic bars
CONFORMAL TRANSFORMATION↗
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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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The conformal transformation of an airfoil into a straight line and its application to the inverse problem of airfoil theory
A method of conformal transformation is developed that maps an airfoil into a straight line, the line being chosen as the extended chord line of the airfoil. The mapping is accomplished by operating directly with the airfoil ordinates. The absence of any preliminary transformation is found to shorten the work substantially over that of previous methods. Use is made of the superposition of solutions to obtain a rigorous counterpart of the approximate methods of thin-airfoils theory. The method is applied to the solution of the direct and inverse problems for arbitrary airfoils and pressure distributions. Numerical examples are given. Applications to more general types of regions, in particular to biplanes and to cascades of airfoils, are indicated. (author)
Mutterperl, William↗
Conformal geometry from entanglement
In a physical system with conformal symmetry, observables depend on cross-ratios, measures of distance invariant under global conformal transformations (conformal geometry for short). We identify a quantum information-theoretic mechanism by which the conformal geometry emerges at the gapless edge of a 2+1D quantum many-body system with a bulk energy gap. We introduce a novel pair of information-theoretic quantities (\mathfrak{c}_{\textrm{tot}}, \eta) ( 𝔠 tot , η ) that can be defined locally on the edge from the wavefunction of the many-body system, without prior knowledge of any distance measure. We posit that, for a topological groundstate, the quantity \mathfrak{c}_{\textrm{tot}} 𝔠 tot is stationary under arbitrary variations of the quantum state, and study the logical consequences. We show that stationarity, modulo an entanglement-based assumption about the bulk, implies (i) \mathfrak{c}_{\textrm{tot}} 𝔠 tot is a non-negative constant that can be interpreted as the total central charge of the edge theory. (ii) \eta η is a cross-ratio, obeying the full set of mathematical consistency rules, which further indicates the existence of a distance measure of the edge with global conformal invariance. Thus, the conformal geometry emerges from a simple assumption on groundstate entanglement. We show that stationarity of \mathfrak{c}_{\textrm{tot}} 𝔠 tot is equivalent to a vector fixed-point equation involving \eta η , making our assumption locally checkable. We also derive similar results for 1+1D systems under a suitable set of assumptions.
Kim, Isaac H.↗
Manipulating the Assembly and Architecture of Fibrillar Silk
Silk is a unique and exceptionally strong biological material. However, no synthetic method has yet come close to replicating the properties of natural silk. This shortfall is attributed to an insufficient understanding of both silk nanofibril structure and the mechanism of formation. Here in situ atomic force microscopy (AFM) and photo-induced force microscopy (PiFM) is utilized to investigate the formation process and define the basic structural paradigm of individual silk nanofibrils. By visualizing the multistage process of silk nanofibril formation, the importance of conformational transformations along the assembly pathway is revealed. Unfolded silk structures initially accumulate into amorphous clusters, which then evolve into crystal nuclei via conformational transformation into β-crystallites. Nanofibril elongation then occurs through the attachment of silk molecules at a single end of the nanofibril tip; this is facilitated through the formation of a new amorphous cluster that then repeats the aforementioned conformational transformation. However, enzymatic digestion of the amorphous regions leads to direct, rapid elongation of β-crystalline fibers. These findings imply that the energy landscape is characterized by shallow minima associated with intermediate states, which can be eliminated by introducing β-crystallites, and motivate research into the directed modification of the silk assembly pathway to select for features beneficial to specific applications.
36 MATERIALS SCIENCE↗
Image remapping strategies applied as protheses for the visually impaired
Maculopathy and retinitis pigmentosa (rp) are two vision defects which render the afflicted person with impaired ability to read and recognize visual patterns. For some time there has been interest and work on the use of image remapping techniques to provide a visual aid for individuals with these impairments. The basic concept is to remap an image according to some mathematical transformation such that the image is warped around a maculopathic defect (scotoma) or within the rp foveal region of retinal sensitivity. NASA/JSC has been pursuing this research using angle invariant transformations with testing of the resulting remapping using subjects and facilities of the University of Houston, College of Optometry. Testing is facilitated by use of a hardware device, the Programmable Remapper, to provide the remapping of video images. This report presents the results of studies of alternative remapping transformations with the objective of improving subject reading rates and pattern recognition. In particular a form of conformal transformation was developed which provides for a smooth warping of an image around a scotoma. In such a case it is shown that distortion of characters and lines of characters is minimized which should lead to enhanced character recognition. In addition studies were made of alternative transformations which, although not conformal, provide for similar low character distortion remapping. A second, non-conformal transformation was studied for remapping of images to aid rp impairments. In this case a transformation was investigated which allows remapping of a vision field into a circular area representing the foveal retina region. The size and spatial representation of the image are selectable. It is shown that parametric adjustments allow for a wide variation of how a visual field is presented to the sensitive retina. This study also presents some preliminary considerations of how a prosthetic device could be implemented in a practical sense, vis-a-vis, size, weight and portability.
Johnson, Curtis D.↗
On a new method for calculating the potential flow past a body of revolution
A new method is presented for obtaining the velocity potential of the flow about a body of revolution moving uniformly in the direction of its axis of symmetry in a fluid otherwise at rest. This method is based essentially on the fact that the form of the differential equation for the velocity potential is invariant with regard to conformal transformation of the meridian plane. By means of the conformal transformation of the meridian profile into a circle a system of orthogonal curvilinear coordinates is obtained, the main feature of which is that one of the coordinate lines is the meridian profile itself. The use of this type of coordinate system yields a simple expression of the boundary condition at the surface of the solid and leads to a rational process of iteration for the solution of the differential equation for the velocity potential. It is shown that the velocity potential for an arbitrary body of revolution may be expressed in terms of universal functions which, although not normal, are obtainable by means of simple quadratures. The general results are applied to a body of revolution obtained by revolving a symmetrical Joukowski profile about its axis of symmetry. A numerical example further serves to illustrate the theory.
Kaplan, Carl↗
Analysis of accelerated motion in the theory of relativity
Conventional treatments of accelerated motion in the theory of relativity have led to certain difficulties of interpretation. Certain reversals in the apparent gravitational field of an accelerated body may be avoided by simpler analysis based on the use of restricted conformal transformations. In the conformal theory the velocity of light remains constant even for experimenters in accelerated motion. The problem considered is that of rectilinear motion with a variable velocity. The motion takes place along the x or x' axis of two coordinate systems.
Jones, R. T.↗
Minimal Joule dissipation models of magnetospheric convection
This paper gives a topical review of theoretical models of magnetospheric convection based on the concept of minimal Joule dissipation. A two-dimensional slab model of the ionosphere featuring an enhanced conductivity auroral oval is used to compute high-latitude electric fields and currents. Mathematical methods used in the modeling include Fourier analysis, fast Fourier transforms, and variational calculus. Also, conformal transformations are introduced in the analysis, which enable the auroral oval to be represented as a nonconcentric, crescent-shaped figure. Convection patterns appropriate to geomagnetic quiet and disturbed conditions are computed, the differentiating variable being the relative amount of power dissipated in the magnetospheric ring current. When ring current dissipation is small, the convection electric field is restricted to high latitudes (shielding regime), and when it is large, a significant penetration of the field to low latitudes occurs, accompanied by an increase in the ratio of the region I current to the region 2 current.
Barbosa, D. D.↗
Prediction of vortex shedding from circular and noncircular bodies in subsonic flow
An engineering prediction method and associated computer code VTXCLD are presented which predict nose vortex shedding from circular and noncircular bodies in subsonic flow at angles of attack and roll. The axisymmetric body is represented by point sources and doublets, and noncircular cross sections are transformed to a circle by either analytical or numerical conformal transformations. The leeward vortices are modeled by discrete vortices in crossflow planes along the body; thus, the three-dimensional steady flow problem is reduced to a two-dimensional, unsteady, separated flow problem for solution. Comparison of measured and predicted surface pressure distributions, flowfield surveys, and aerodynamic characteristics are presented for bodies with circular and noncircular cross sectional shapes.
Mendenhall, Michael R.↗
Instability of jets of arbitrary geometry
This paper describes a calculation technique for determining the stability of jets of arbitrary cross section. In particular, elliptic and rectangular jets are considered. The numerical procedure involves both a conformal transformation between the computational domain and the physical plane and a solution of the transformed stability equation in the computational domain. Modern, efficient, conformal mappings are used for both simply and doubly connected domains. The numerical solution is based on a finite difference/pseudospectral discretization of the stability equation. The technique is verified by comparison with previous calculations for circular and elliptic jets. Calculations are performed for the stability of elliptic and rectangular jets of aspect ratio 2. Growth rates, phase velocities, and pressure eigenfunctions are calculated.
Baty, Roy S.↗
Prediction of subsonic vortex shedding from forebodies with chines
An engineering prediction method and associated computer code VTXCHN to predict nose vortex shedding from circular and noncircular forebodies with sharp chine edges in subsonic flow at angles of attack and roll are presented. Axisymmetric bodies are represented by point sources and doublets, and noncircular cross sections are transformed to a circle by either analytical or numerical conformal transformations. The lee side vortex wake is modeled by discrete vortices in crossflow planes along the body; thus the three-dimensional steady flow problem is reduced to a two-dimensional, unsteady, separated flow problem for solution. Comparison of measured and predicted surface pressure distributions, flow field surveys, and aerodynamic characteristics are presented for noncircular bodies alone and forebodies with sharp chines.
Mendenhall, Michael R.↗
Prediction of vortex shedding from forebodies with chines
An engineering prediction method and associated computer code VTXCHN to predict nose vortex shedding from circular and noncircular forebodies with sharp chines edges in subsonic flow at angles of attack and roll are presented. Axisymmetric bodies are represented by point sources and doublets, and noncircular cross sections are transformed to a circle by either analytical or numerical conformal transformations. The lee side vortex wake is modeled by discrete vortices in crossflow planes along the body; thus the three-dimensional steady flow problem is reduced to a two-dimensional, unsteady, separated flow problem for solution. Comparison of measured and predicted pressure distributions, flow field surveys, and aerodynamic characteristics are presented for noncircular bodies alone and forebodies with sharp chines.
Mendenhall, Michael R.↗
Potential flow about arbitrary biplane wing sections
A rigorous treatment is given of the problem of determining the two-dimensional potential flow around arbitrary biplane cellules. The analysis involves the use of elliptic functions and is sufficiently general to include the effects of such elements as the section shapes, the chord ratio, gap, stagger, and decalage, which elements may be specified arbitrarily. The flow problem is resolved by making use of the methods of conformal representation. Thus the solution of the problem of transforming conformally two arbitrary contours into two circles is expressed by a pair of simultaneous integral equations, for which a method of numerical solution is outlined. As an example of the numerical process, the pressure distribution over certain arrangements of the NACA 4412 airfoil in biplane combinations is presented and compared with the monoplane pressure distribution.
Garrick, I E↗
Improved design of subcritical and supercritical cascades using complex characteristics and boundary layer correction
The method of complex characteristics and hodograph transformation for the design of shockless airfoils was extended to design supercritical cascades with high solidities and large inlet angles. This capability was achieved by introducing a conformal mapping of the hodograph domain onto an ellipse and expanding the solution in terms of Tchebycheff polynomials. A computer code was developd based on this idea. A number of airfoils designed with the code are presented. Various supercritical and subcritical compressor, turbine and propeller sections are shown. The lag-entrainment method for the calculation of a turbulent boundary layer was incorporated to the inviscid design code. The results of this calculation are shown for the airfoils described. The elliptic conformal transformation developed to map the hodograph domain onto an ellipse can be used to generate a conformal grid in the physical domain of a cascade of airfoils with open trailing edges with a single transformation. A grid generated with this transformation is shown for the Korn airfoil.
Sanz, J. M.↗
Improved design of subcritical and supercritical cascades using complex characteristics and boundary-layer correction
The method of complex characteristics and hodograph transformation for the design of shockless airfoils was extended to design supercritical cascades with high solidities and large inlet angles. This capability was achieved by introducing a conformal mapping of the hodograph domain onto an ellipse and expanding the solution in terms of Tchebycheff polynomials. A computer code was developed based on this idea. A number of airfoils designed with the code are presented. Various supercritical and subcritical compressor, turbine and propeller sections are shown. The lag-entrainment method for the calculation of a turbulent boundary layer was incorporated to the inviscid design code. The results of this calculation are shown for the airfoils described. The elliptic conformal transformation developed to map the hodograph domain onto an ellipse can be used to generate a conformal grid in the physical domain of a cascade of airfoils with open trailing edges with a single transformation. A grid generated with this transformation is shown for the Korn airfoil. Previously announced in STAR as N83-24474
Sanz, J. M.↗
Shuttle computational grid generation
The well known Karman-Trefftz conformal transformation, consisting of repeated applications of the same basic formula, were found to be quite successful to body, wing, and wing-body cross sections. This grid generation technique is extended to cross sections of more complex forms, and also more automatic. Computer programs were written for the selection of hinge points on cross section with angular shapes, the Karman-Trefftz tranformation of arbitrary shapes, and the special transform of hinge point on the imaginary axis. A feasibility study is performed for the future application of conformal mapping grid generation to complex three dimensional configurations. Examples such as Orbiter vehicle section and a few others were used.
Ing, Chang↗
Extending the Lorentz transformation by characteristic coordinates
The problem considered is that of rectilinear motion with variable velocity. The paper gives, by an elementary construction, a system of coordinates which is conformal in a restricted region near the axis of the motion. In such coordinates the velocity of light remains invariant even for observers moving with variable velocity. By a particular choice of the scale relation the restricted conformal transformations can be made to reduce to the Lorentz transformation everywhere in the case of constant velocity and locally in the case of variable velocity.
Jones, R. T.↗
Extending the Lorentz Transformation to Motion with Variable Velocity
The problem considered is that of rectilinear motion with variable velocity. The paper gives, by an elementary construction, a system of coordinates which is conformal in the vicinity of the axis of motion. By a particular choice of the scale relation, such restricted conformal transformations can be made to reduce to the Lorentz transformation everywhere in the case of uniform velocity and locally in the case of variable velocity.
Jones, Robert T.↗