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The flow about an airfoil in free air can be described approximately by a boundary-layer flow near the surface of the airfoil and by a potential flow everywhere else. Boundary-layer theory can be applied to the flow about an airfoil in two ways. First, the boundary-layer development can be determined for a given potential flow velocity distribution. This is the direct or analysis problem. Second, the potential-flow field, or at least some of its properties, can be determined for a given boundary-layer development. This is the inverse or design problem. This second application of boundary-layer theory requires the solution of the inverse potential-flow problem where the potential-flow velocity distribution is specified and the airfoil shape is computed. Thus, the viscous airfoil design (inverse) problem can be described as the computation of a shape from a potential flow velocity distribution which is consistent with a desired boundary-layer development. The first application of this inverse procedure has been described in references 1 and 2. Because Tollmien, Schlichting,Ulrich, Pretsch, and others had shown that favorable pressure gradients delay the transition from laminar to turbulent flow, airfoils were designed with aft pressure recoveries. The experimental results for these airfoils confirmed the theoretical predictions. This breakthrough led to the laminar flow airfoil series. Since that time, boundary-layer and potential flow theories have been steadily improved.

Different computer programs have been developed for low-speed (incompressible)airfoils. (For example, see ref. 4.) The present paper describes one of these programs. The potential flow inverse problem still plays a major role in airfoil design. This problem has been solved exactly by means of conformal mapping as shown in reference 5. The method is similar to that of Lighthill (ref. 6), is direct, and solves most multipoint design problems in a very simple manner. A potential-flow analysis method is also required for comparison with wind tunnel tests of given airfoils and for analyses of airfoils generated by the design method and then modified by a flap deflection. The airfoil analysis problem is solved using a distributed surface singularity method similar to those described in references 4 and 7. Some of the details of this method are new and previously unpublished. The boundary-layer method, which uses integral momentum and energy equations, is described in reference 8. The present method does not contain a boundary-layer displacement iteration. The program has been successfully applied at Reynolds numbers from 20 thousand to 100 million. (See ref. 9 for example.)

REFERENCES

  1. Abbott, Ira H.; Von Doenhoff, Albert E.; and Stivers, Louis S., Jr.: Summary of Airfoil Data. NACA Rep. 824, 1945.
  2. Abbott, Ira H.; and Von Doenhoff, Albert E.: Theory of Wing Sections. Dover Publications, Inc., c.1959.
  3. Schlichting, Hermann (J. Kestin, transl.): Boundary Layer Theory. McGraw-Hill Book Co., Inc., 1955.
  4. Smetana, Frederick O.; Summey, Delbert C.; Smith, Neill S.; and Carden, Ronald K.: Light Aircraft Lift, Drag, and Moment Prediction - A Review and Analysis. NASA CR-2523, 1975.
  5. Eppler, Richard: Direct Calculation of Airfoils From Pressure Distribution. NASA TT F-15,417, 1974. (Translated from Ingenieur-Archiv, vol. 25, no. 1, 1957, pp. 32-57.)
  6. Lighthill, M. J.: A New Method of Two-Dimensional Aerodynamic Design. R. & M. No. 2112, British A.R.C., 1945.
  7. Hess, John L.: The Use of Higher-Order Surface Singularity Distributions To Obtain Improved Potential Flow Solutions for Two-Dimensional Lifting Airfoils. Comput. Methods Appl. Mech. and Eng., vol. 5, no. 1, Jan. 1975, pp. 11-35.
  8. Eppler, Richard: Practical Calculation of Laminar and Turbulent Bled-Off Boundary Layers. NASA TM-75328, 1978. (Translated from Ingenieur-Archiv, vol. 32, 1963, pp. 221-245.)
  9. Eppler, Richard; and Somers, Dan M.: Low Speed Airfoil Design and Analysis. Advanced Technology Airfoil Research - Volume I, NASA CP-2045, Part 1, 1979, pp. 73-99.
  10. Kowalik, J.; and Osborne, M. R.: Methods for Unconstrained Optimization Problems. American Elsevier Pub. Co., Inc., 1968.
  11. Eppler, R. (Francesca Neffgen, transl.): Laminar Airfoils for Reynolds Numbers Greater Than 4 Million. B-819-35, Apr. 1969. (Available from NTIS as N69-28178.) (Translated from Ingenieur-Archiv, vol. 38, 1969, pp. 232-240.)
  12. Eppler, Richard: Turbulent Airfoils for General Aviation. Journal of Aircraft, vol. 15, no. 2, Feb. 1978, pp. 93-99.
  13. Squire, H. B.; and Young, A. D.: The Calculation of the Profile Drag of Aerofoils. R. & M. No. 1838, British A.R.C., 1938.
  14. Woods, L. C.: The Theory of Subsonic Plane Flow. Cambridge Univ. Press, 1961.

John Roncz has written an revised version of Eppler that includes a correction for compressibility. You may download a copy from this web site. (ZIP, 864KB)

Although this program is of great historical importance and one still finds current papers that refer to calculations made with PROFILE, it is not the program of choice for someone learning about airfoil plus boundary layer calculations. I would recommend Xfoil for today's students. Xfoil is an interactive program for the design and analysis of subsonic isolated airfoils written by Mark Drela and Harold Youngren. Check the Virginia Tech site for valuable notes on running Xfoil.

PDAS home > Contents > Eppler
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