×

A hodograph-based method for the design of shock-free cascades. (English) Zbl 0617.76064

A hodograph-based method, originally developed by the first author [Commun. Appl. Numer. Methods 2, 37-45 (1986; Zbl 0604.76043)] for the design of shock-free aerofoils, has been modified and extended to allow for the design of shock-free compressor blades. In the present procedure, the subsonic and supersonic regions of the flow are decoupled, allowing the solution of either an elliptic or a hyperbolic-type partial differential equation for the stream function. The coupling of both regions of the flow is carried out along the sonic line which adjoins both regions. For the subcritical portion of the flow considered here, the pressure distribution is prescribed in addition to the upstream and downstream flow conditions.
For the supercritical portion of the flow, the stream function on the sonic line is given instead of the supercritical pressure distribution which is found as part of the solution. In the special hodograph variables used, the equation for the stream function is solved iteratively using a second-order accurate line relaxation procedure for the subsonic portion of the flow. For the supercritical portion of the flow, a characteristic marching procedure in the hodograph plane is used to solve for the supersonic flow. The results are then mapped back to the physical plane to determine the blade shape and the supercritical pressures. Examples of shock-free compressor blade designs are presented. They show good agreement with the direct computation of the flow past the designed blade.

MSC:

76H05 Transonic flows
76J20 Supersonic flows
76M99 Basic methods in fluid mechanics

Citations:

Zbl 0604.76043

Software:

CAS22
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] The Dynamics and Thermodynamics of Compressible Fluid Flow, The Ronald Press Company, New York, 1954.
[2] and , Elements of Gas Dynamics, Wiley, New York, 1957.
[3] and , ’Analysis of transonic cascade flow using conformal mapping and relaxation techniques’, AIAA Paper No. 76-370, July 1976.
[4] and , ’The calculation of potential flow in cascades using finite area techniques’, AIAA Paper No. 79-0077, January 1979.
[5] and , ’Full potential solution of transonic quasi-3-D flow through a cascade using artificial compressibility’, NASA TM81637, March 1981.
[6] ’A time marching method for two-and three-dimensional blade to blade flows’, Aeronautical Research Council, R&M No. 3775, London, October 1974.
[7] Partial Differential Equations, Wiley, New York, 1964.
[8] Swenson, Communications on Pure and Applied Mathematics XXI pp 175– (1968)
[9] ’Numerical design of transonic cascades’, Courant Institute of Mathematical Sciences, ERDA Research and Development Report C00-3077-72, January 1975.
[10] ’Entwurf uberkritischer profile mit hilfe der rheoelektrischen analogie’, DFVLR-FB 75-43, 1975.
[11] ’Related analytical, analog and numerical methods in transonic airfoil design’, AIAA Paper No. 79-1556, July 1979.
[12] ’Transonic fluid dynamics-lecture notes’, The University of Arizona, TFD 77-01, October 1977.
[13] and , ’CAS22-FORTRAN program for fast design and analysis of shock-free airfoil cascades using fictitious gas concept’, NASA CR 3507, January 1982.
[14] and , ’Design of shock-free compressor cascades including viscous boundary layer effects’, ASME Paper No. 83-GT-134, October 1983.
[15] ’A numerical tool for the design of shock-free transonic cascades’, M. S. Thesis, The University of Arizona, Tucson, Arizona, August 1981.
[16] and , ’Shock-free redesign using finite elements’, Proc. Int. Conf. on Inverse Design Concepts, Austin, Texas, pp. 283-295, October 1984.
[17] and , ’Shockless design and analysis of transonic blade shapes’, AIAA Paper No. 81-1237, June 1981.
[18] ’An efficient transonic shock-free redesign procedure using a fictitious gas method’, AIAA Paper No. 79-0075, June 1979.
[19] ’Subcritical and supercritical airfoils for given pressure distributions’, Ph.D. Dissertation, The University of Arizona, Tuscon, Arizona, August 1981.
[20] Hassan, AIAA Journal 22 pp 1185– (1984)
[21] and , ’The role of constraints in the inverse design problem for transonic airfoils’, AIAA Paper No. 81-1233, June 1981.
[22] Lighthill, Quart. J. of Mech. and Appl. Math. 1 pp 442– (1948)
[23] Thompson, J. Comp. Phys. 15 pp 299– (1974)
[24] The Hodograph Equations, Oliver and Boyd, Edinburgh, 1971.
[25] Hassan, Commun. Appl. Numer. Methods 2 pp 37– (1986)
[26] ’A viscous-inviscid coupling method for the design of low Reynolds number airfoil sections’, Proc. Conf. Low Reynolds Number Airfoil Aerodynamics, Notre Dame, Indiana, pp. 53-63, June 1985.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.