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At least 19 records

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↗

Applications of Laplace transform methods to airfoil motion and stability calculations

This paper reviews the development of generalized unsteady aerodynamic theory and presents a derivation of the generalized Possio integral equation. Numerical calculations resolve questions concerning subsonic indicial lift functions and demonstrate the generation of Kutta waves at high values of reduced frequency, subsonic Mach number, or both. The use of rational function approximations of unsteady aerodynamic loads in aeroelastic stability calculations is reviewed, and a reformulation of the matrix Pade approximation technique is given. Numerical examples of flutter boundary calculations for a wing which is to be flight tested are given. Finally, a simplified aerodynamic model of transonic flow is used to study the stability of an airfoil exposed to supersonic and subsonic flow regions.

Edwards, J. W.↗

Design of the Hybrid Wing Body with Nacelle: N3-X Propulsion-Airframe Configuration

The Hybrid Wing Body (HWB) aircraft is of great interest for future transport concepts due to itspromises of reduced aircraft noise, nitrous-oxide emissions, and fuel consumption. A design parameterizationmethod for HWB configurations with mail slot nacelle has been developed for a fast exploration of designspace in conceptual and preliminary design phases of a HWB configuration. A HWB planform model byLaughlin [11] was implemented, and the Class Shape Transformation (CST) airfoil generation method byKulfan [10] was utilized to construct the needed geometry for computational high fidelity aerodynamicsimulations. Geometric constraints for the parameterization such as internal cabin and cargo hold layoutswere imposed on the geometry generation. A CFD simulation was performed for a HWB configurationgenerated by the current geometric modeler, clearly showing a significant effect of the installed nacelle on theflowfield.

Nacelle Installation↗

Isolated and cascade airfoils with prescribed velocity distribution

An exact solution of the problem of designing an airfoil with a prescribed velocity distribution on the suction surface in a given uniform flow of an incompressible perfect fluid is obtained by replacing the boundary of the airfoil by vortices. By this device, a method of solution is developed that is applicable both to isolated airfoils and to airfoils in cascade. The conformal transformation of the designed airfoil into a circle can then be obtained and the velocity distribution at any angle of attack computed. Numerical illustrations of the method are given for the airfoil in cascade.

Goldstein, Arthur W↗

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.↗

Airfoil gust response and the sound produced by airifoil-vortex interaction

This paper contributes to the understanding of the noise generation process of an airfoil encountering an unsteady upwash. By using a fast Fourier transform together with accurate airfoil response functions, the lift-time waveform for an airfoil encountering a delta function gust (the indicial function) is calculated for a flat plate airfoil in a compressible flow. This shows the interesting property that the lift is constant until the generated acoustic wave reaches the trailing edge. Expressions are given for the magnitude of this constant and for the pressure distribution on the airfoil during this time interval. The case of an airfoil cutting through a line vortex is also analyzed. The pressure-time waveform in the far field is closely related to the left-time waveform for the above problem of an airfoil entering a delta function gust. The effects of varying the relevant parameters in the problem are studied, including the observed position, the core diameter of the vortex, the vortex orientation and the airfoil span. The far field sound varies significantly with observer position, illustrating the importance of non-compactness effects. Increasing the viscous core diameter tends to smooth the pressure-time waveform. For small viscous core radius and infinite span, changing the vortex orientation changes only the amplitude of the pressure-time waveform, and not the shape.

Amiet, R. K.↗

Airfoil Smoothing and Scaling Programs

Two programs smooth and scale arbitrary airfoil coordinates. Airfoil smoothing program (AFSMO) utilizes both least-squares polynomial and leastsquares cubic-spline techniques to smooth iteratively second derivatives of y-axis airfoil coordinates with respect to transformed x-axis system that unwraps airfoil and stretches nose and trailing-edge regions.

Morgan, Harry L., Jr.↗

Derivation of an integral equation for transonic flows

It is noted that the nonlinear partial differential equation for the perturbation velocity potential and boundary conditions describing steady inviscid compressible transonic flow past a thin two-dimensional airfoil can be transformed into a singular integrodifferential equation and that differentiation of the latter yields an integral equation. Two forms of this integral equation currently exist: one for the singularity that is enclosed in an infinitely long strip of vanishing thickness and the other for the singularity that is enclosed in a vanishing circle. In the present article, a more general integral equation is derived by enclosing the singularity in a vanishing rectangular cavity of arbitrary aspect ratio. The two existing forms of this equation are deduced as special cases distinguished by the respective values for the aspect ratio (infinity for the first form and unity for the second).

Ogana, W.↗

Generation of Performance Model for the Aeolian Wind Tunnel (AWT) Rotor at Reduced Pressure

The NASA Jet Propulsion Laboratory (JPL) designed the Mars Helicopter (MH) in collaboration with AeroVironment Inc., NASA Ames Research Center, and NASA Langley Research Center to explore the possibility of a vertical takeoff and landing (VTOL) Unmanned Aerial Vehicle (UAV) for flight on Mars. A 40-inch-diameter Aeolian Wind Tunnel (AWT) rotor, roughly approximating the proposed MH design by JPL, was tested in forward flight at Mars atmospheric pressure at the NASA Ames Planetary Aeolian Laboratory (PAL) in support of MH research efforts. This report describes the generation of the rotor model used to correlate with that experimental effort as reported by Ament and Koning. The 40-inch-diameter rotor was 3D-scanned and transformed into an airfoil deck. The scanned rotor airfoil sections are analyzed using C81 Generator (C81Gen) to generate the sectional aerodynamic coefficients for comprehensive analyses. A mid-fidelity computational fluid dynamics (CFD) simulation using Rotorcraft CFD (RotCFD) is pursued to efficiently estimate rotor hover and forward flight performance. Simulations at two pressures, 7 mbar (approximate Martian atmospheric pressure) and 1018 mbar (1 atmosphere), are performed to gain an understanding of the performance differences and Reynolds number effects observed. Experimental 1-atmosphere thrust for single- and dual-rotor isolated hover cases correlate well with the modeled rotor. Performance results at reduced pressure (7 mbar) show a drastic decrease in lift for equivalent RPMs tested at 1 atmosphere. Although this is primarily due to pressure reduction, Reynolds number effects also contribute to this decrease, as airfoil lift and drag coefficients are affected when compared with 1-atmosphere results. Further, simulated rotor power coefficient shows drastic increases at reduced pressures, attributed to laminar boundary layer separation, as described in Koning et al. for the MH rotor analysis. PAL experimental Martian Surface Wind Tunnel (MARSWIT) results are presented in the paper by Ament and Koning. The very low Reynolds number range is currently not well understood and presents various challenges for both experimentation and simulation.

Koning, Witold J. F.↗

Computer programs for smoothing and scaling airfoil coordinates

Detailed descriptions are given of the theoretical methods and associated computer codes of a program to smooth and a program to scale arbitrary airfoil coordinates. The smoothing program utilizes both least-squares polynomial and least-squares cubic spline techniques to smooth interatively the second derivatives of the y-axis airfoil coordinates with respect to a transformed x-axis system which unwraps the airfoil and stretches the nose and trailing-edge regions. The corresponding smooth airfoil coordinates are then determined by solving a tridiagonal matrix of simultaneous cubic-spline equations relating the y-axis coordinates and their corresponding second derivatives. A technique for computing the camber and thickness distribution of the smoothed airfoil is also discussed. The scaling program can then be used to scale the thickness distribution generated by the smoothing program to a specific maximum thickness which is then combined with the camber distribution to obtain the final scaled airfoil contour. Computer listings of the smoothing and scaling programs are included.

Morgan, H. L., Jr.↗

Conformal mapping for potential flow about airfoils with attached flap.

The conformal mapping sequence presented transforms the potential flow about a circle into that about an airfoil with an attached flap or spoiler. It is found that adequate versatility of the flap shape for a given airfoil can usually be obtained with the indicated functions, although other transformations would expand the variety of possible flap shapes.

Rossow, V. J.↗

Unsteady aerodynamic modeling and active aeroelastic control

Unsteady aerodynamic modeling techniques are developed and applied to the study of active control of elastic vehicles. The problem of active control of a supercritical flutter mode poses a definite design goal stability, and is treated in detail. The transfer functions relating the arbitrary airfoil motions to the airloads are derived from the Laplace transforms of the linearized airload expressions for incompressible two dimensional flow. The transfer function relating the motions to the circulatory part of these loads is recognized as the Theodorsen function extended to complex values of reduced frequency, and is termed the generalized Theodorsen function. Inversion of the Laplace transforms yields exact transient airloads and airfoil motions. Exact root loci of aeroelastic modes are calculated, providing quantitative information regarding subcritical and supercritical flutter conditions.

Edwards, J. W.↗

A theory of supercritical wing sections, with computer programs and examples.

Mathematical methods for the design of supercritical wings, which depend on the numerical solution of the partial differential equations of two-dimensional gas dynamics, are developed. The main contribution is a computer program for the design of shockless transonic airfoils using the hodograph transformation and analytic continuation into the complex domain. The mathematical theory is described, and a manual for users of the programs is provided. Numerical examples are given and computational results are discussed, and the computer programs themselves are listed. The analysis routine can be used to ascertain whether the profiles behave well at off-design conditions, or to smooth coordinates and obtain a desirable shape more quickly when perfectly shockless flow is not essential.

Bauer, F.↗

Flow of Gas Through Turbine Lattices

This report is concerned with fluid mechanics of two-dimensional cascades, particularly turbine cascades. Methods of solving the incompressible ideal flow in cascades are presented. The causes and the order of magnitude of the two-dimensional losses at subsonic velocities are discussed. Methods are presented for estimating the flow and losses at high subsonic velocities. Transonic and supersonic flows in lattices are then analyzed. Some three-dimensional features of the flow in turbines are noted.

BOUNDARY LAYER - AIRFOILS - CASCADES↗

An alternating-direction-implicit algorithm for the unsteady potential equation in conservation form

An implicit finite difference scheme for an efficient computation of unsteady potential flow about airfoils is presented. The formulation uses density and velocity potential as dependent variables, and is cast in conservation form to assure the theoretically correct determination of shockwave location and speed. To enable boundary conditions to be imposed directly on the airfoil surface, a time varying sheared rectilinear coordinate transformation is employed. Calculated time history solutions on a pulsating airfoil are compared with the results of another unsteady transonic code. It is concluded that the method has excellent numerical stability and gives accurate solutions with sharply resolved shocks.

Chipman, R. R.↗

On the Theory of the Unsteady Motion of an Airfoil

The paper presents a systematical analysis of the problem of the determination of the unsteady motion about an airfoil moving in an infinite fluid that contains a system of vortices and the determination of the hydrodynamical forces acting on the airfoil. The hydrodynamical problem is reduced to the determination of the function f (xi) which transforms conformally the external region of the airfoil into the interior of a circle. The proposed methods of determining the irrotational motion of a fluid that is produced by any motion of the airfoil are especially simple and effective if the function f (xi) is rational. As an example the flow is determined for the case of an arbitrary motion of an airfoil of the Joukowsky type. The formulas obtained for the determination of the hydrodynamical forces by means of contour integration are similar to those given by S. Chaplygin. These formulas are used to determine the force acting on the airfoil in the cases where the unsteady motion is potential throughout and the circulation about the airfoil is constant and also when the fluid contains a system of vortices. A full discussion is given of the concept of virtual masses together with practical formulas for computing the virtual mass coefficients.

Sedov, L. I.↗

An exploratory study of finite difference grids for transonic unsteady aerodynamics

Unsteady aerodynamic forces are calculated by the XTRAN2L finite difference program which solves the complete two dimensional unsteady transonic small perturbation equation. The unsteady forces are obtained using a pulse transfer function technique which assumes the flow field behaves in a locally linear fashion about a mean condition. Forces are calculated for a linear flat plate using the default grids from the LTRAN2-NLR, LTRAN2-HI, and XTRAN3S programs. The forces are compared to the exact theoretical values for flat plate, and grid generated boundary and internal numerical reflections are observed to cause significant errors in the unsteady airloads. Grids are presented that alleviate the reflections while reducing computational time up to fifty-three percent and program size up to twenty-eight percent. Forces are presented for a six percent thick parabolic arc airfoil which demonstrate that the transform technique may be successfully applied to nonlinear transonic flows.

Seidel, D. A.↗