Two Dimensional Transonic Flow Past Airfoils

The problem of constructing solutions for transonic flow over symmetric airfoils is treated for simplification of the mapping of the incompressible flow is emphasized. In the case of a symmetric Joukowski airfoil without circulation the mapping is relatively simple, but the coefficients in the power series are difficult to evaluate. As a means of simplification an approximate flow is used which differs only slightly from the exact incompressible one when the thickness is small. Flow with circulation is also considered by the same method.

Interferometer measurements are given of the flow fields near two-dimensional wedge and circular-arc sections at zero angle of attack. Pressure distributions and drag coefficients as functions of Mach number were obtained and the wedge data are compared with theory. It is shown that the local Mach number at any point on the surface of a finite three dimensional body or an unswept two-dimensional body, moving through an infinite fluid, has a stationary value at Mach number 1 and, in fact, remains nearly constant for a range of speeds below and above Mach number 1. On the basis of this concept and the experimental data, pressure distributions and drag coefficients for the wedge and circular-arc sections are presented throughout the entire transonic range of velocities.

Summary: An experimental and theoretical study has been made of the two-dimensional transonic flow past a 12° wedge airfoil with lower surface inclined 13° to the air stream. The experimental portion of the study consists of surveys of the flow field about the leading edge in a closed throat tunnel near choking Mach number by means of an X-ray densitometer. From these surveys various features have been determined; namely, the location of the stagnation point, the location of the sonic line, the size of the separation bubble on the upper surface, and the value of the Mach number at the bubble edge. These features were relatively insensitive to changes in Reynolds number in the range from 1X106 to 3X106 based on chord. The theoretical portion of the study consists of the calculation by relaxation methods of an unbounded, inviscid, two-dimensional flow about a portion of the same body at a free-stream Mach number of exactly unity. The calculations yield a separation bubble on the upper surface which has a shape shown to be compatible with known theoretical conditions for viscous attachment to the upper surface of the body. Agreement between theoretically calculated and the experimentally determined features of the flow was good in regions where the local Mach number was 2 or less. In regions of higher local Mach number the agreement was fair. The theoretically determined Mach number on the bubble edge was 2.80 whereas the experimentally determined value was 2.30. The theoretically determined length of the separation bubble was 1.54-percent chord whereas the experimentally determined length was about 5-percent chord. The theoretically determined location of the stagnation point was 0.16-percent chord from the leading edge whereas the experimentally determined location was at 0.56-percent chord. The experience gained in studying the flow suggests the possibility that the solution obtained is not unique and that another solution in better agreement with experiment might exist.

Written to teach students the nature of transonic flow and its mathematical foundation, this book offers a much-needed introduction to transonic aerodynamics. The authors present a quantitative and qualitative assessment of subsonic, supersonic and transonic flow around bodies in two and three dimensions. The book reviews the governing equations and explores their applications and limitations as employed in modeling and computational fluid dynamics. Some concepts, such as shock and expansion theory, are examined from a numerical perspective. Others, including shock-boundary-layer interaction, are discussed from a qualitative point of view. The book includes 60 examples and more than 200 practice problems. The authors also offer analytical methods such as Method of Characteristics (MOC) that allow readers to practice with the subject matter. The result is a wealth of insight into transonic flow phenomena and their impact on aircraft design, including compressibility effects, shock and expansion waves, shock-boundary-layer interaction and aeroelasticity.

With the aim of developing a fast and accurate computer code for predicting the aerodynamic forces needed for a flutter analysis we review some basic concepts in computational transonics. The unsteady transonic flow past airfoils in rigid body motion is adequately described by the potential flow equation as long as the boundary layer remains attached. The two dimensional unsteady transonic potential flow equation in quasilinear form with first order radiation boundary conditions is solved by an alternating direction implicit scheme in an airfoil attached sheared parabolic coordinate system. Numerical experiments show that the scheme is very stable and is able to resolve the highly nonlinear transonic effects for flutter analysis within the context of an inviscid theory. (Author).

Includes the Committee's Technical reports no. 1-1058, reprinted in v. 1-37.

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