1、NASA TECHNICAL NOTE NASA TN D-2970 - 4. / . - A DESCRIPTION OF NUMERICAL METHODS AND COMPUTER PROGRAMS FOR TWO-DIMENSIONAL AND AXISYMMETRIC SUPERSONIC FLOW OVER BLUNT-NOSED AND FLARED BODIES by MLZOYU Iaonye, John V. Rkkh, dad Harvmd LOUX Ames Reseurch Center M o ffett Field, Cu Zz? NATIONAL AERONAU
2、TICS AND SPACE ADMINISTRATION 0 WASHINGTON, D. C. AUGUST 1965 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TECH LIBRARY KAFB. NY 0079963 A DESCRIPTION OF NUMERICAL METHODS AND COMPUTER PROGRAMS FOR TWO-DIMENSIONAL AND AXISYMMETWC SUPERSONIC FLOW O
3、VER BLUNT-NOSED AND FLARED BODIES By Mamoru Inouye, John V. Rakich, and Harvard Lomax Ames Research Center Moffett Field, Calif. NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - Price $2.00 Prov
4、ided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-A DESCRIPTION OF NUMERICAL METHODS AND COMPUTER PROGRAMS FOR TWO-DIMENSIONAL AND AXISYMMETRIC SUPERSONIC FLOW OVER BLUNT-NOSED AND F!LABD BODIES By Mamoru Inouye, John V. Rakich, and Harvard Lomax Ames Rese
5、arch Center SUMMARY The computer programs developed at Ames Research Center for calculating the inviscid flow field around blunt-nosed bodies are described briefly and their application to specific shapes is demonstrated. The programs solve numerically the exact equations of motion for plane or axis
6、ymmetric bodies at zero angle of attack and for a perfect gas or a real gas in thermodynamic equilibrium. An inverse method is used for the subsonic-transonic region, and the method of characteristics is used for the supersonic region. Results are shown for several body shapes in both perfect and re
7、al gas flow, including a comparison between air and a C02-N2 mixture. Presented are shock-wave shapes and distributions of pressure and other flow variables along the body and across the shock layer. INTRODUCTION Aircraft and spacecraft designers are faced with the problem of determin- ing the invis
8、cid flow field over blunt-nosed bodies for supersonic flight at speeds encompassing those attained in planetary entry. In addition to the blunt nose, a typical body shape may have a flared afterbody which further complicates the problem. The dominant features of such a flow field are indi- cated in
9、sketch (a). There occurs a detached bow wave that is normal at the Supersonic region Typical flow field Sketch (a) Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-axis of symmetry and decays in strength as it approaches a Mach wave at large distances
10、 from the body. The flow behind the shock wave is subsonic in the nose region bounded by the sonic line and becomes supersonic over the after- body. Expansion waves and embedded shocks may occur as a result of corners. In addition, an embedded shock may arise from coalescence of compression waves fr
11、om the surface or from separation of the boundary layer, which often occurs on this type of body. Analysis of the viscous region is beyond the scope of the present study; however, its study depends on a knowledge of the external inviscid flow. A nuniber of exact and approximate techniques for determ
12、ining the flow Some of the more recent con- field depicted in sketch (a) have been reported. tributions are references 1 through 4 for blunt-body flows, and references 5 and 6 for the supersonic region downstream of the nose. For flared bodies, exact numerical results have been reported in reference
13、 7 while approximate methods may be found in references 8 through 10. present a more complete discussion of the entire flow field. Hayes and Probstein (ref. 11) The computer programs that are described in the present report solve numerically the exact equations of motion for plane and axisymmetric f
14、low at zero angle of attack and provide the complete inviscid flow field between the body and the shock wave. The fluid may be a perfect gas or a real gas in thermodynamic equilibrium. An inverse method (ref. 3) is used for the subsonic-transonic region (referred to as the blunt-body program), and t
15、he method of characteristics (ref. 5) is used to extend the calculations down- stream in the supersonic region. These computer programs were written in FORTRAN I1 for use on an IBM 7094 at Ames Research Center, but have been made available to a number of other organizations. The distribution of thes
16、e pro- grams has created a need for a more complete description and documentation than is presently available. The present report is intended to partially fulfill this need. The purpose of the present report is to provide a general description of the Ames flow-field computer programs and to present
17、results of calculations that demonstrate the range of applicability. The governing equations of motion are introduced briefly at the start. Then the methods used to solve the equations are presented. of all the subroutines and flow charts. Instead, detailed descriptions are provided only for selecte
18、d portions of the programs that warrant special con- sideration. The information contained in this report should acquaint the reader with the general logic followed in the programs and be helpful in diag- nosing small difficulties or in making minor modifications. No attempt is made to provide a com
19、plete listing Sample results are presented for shock-wave shape, surface-pressure distribution, and shock-layer profiles of total pressure, static pressure, density, and velocity for various free-stream conditions and body shapes. The first examples demonstrate how a simple modification improves the
20、 accuracy of the calculations in regions with large entropy (or vorticity) gradients. Then comparisons are made with flow-field results obtained by means of an integral method for the blunt-body solution. Comparisons are also made with experimental results obtained for a body with a flare. Finally,
21、examples of calculations for real gases in thermodynamic equilibrium are presented. 2 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-SYMBOLS a h ht M n P Pt R S X XYY Y n 6 E speed of sound ellipsoid bluntness, (b/c)2 semiaxes of ellipsoid enthalpy
22、total enthalpy Mach number coordinate normal to a streamline static pressure total pressure nose or cylinder radius entropy sheared coordinates (see eqs. (8) velocity components in xyy directions velocity shock-wave shape cylindrical coordinates with origin at body nose ratio of specific heats shock
23、 standoff distance angle of corner on the body index for number of degrees of symmetry; E = 0 for plane symmetric flow, and E = 1 for axisymmetric flow flow angle cone angle Mach angle 3 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-p density $ str
24、eam function Subscripts b body S shock free-stream conditions EQUATIONS The partial differential equations that must be satisfied for steady, inviscid flow are as follows: Continuity of mass a a E - (puy 1 +- (PVY 1 = 0 ax dY where E = 0 for plane symmetric flow and E = 1 for axisymmetric flow. Mome
25、ntum equations x direction dU du dP - dX dy dx pu - + pv - + - - 0 y direction a, a, dX dY SY pu - + pv - + 32 = 0 Energy equation where a is the isentropic speed of sound defined by (3) (4) a2 = ()s 4 (5) Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-
26、,-,-To solve these equations for a given set of boundary conditions, the thermodynamic properties of the gas are required; for example, a2 = f(P,P) For a perfect gas these relationships are merely functions of the gas constant and ratio of specific heats. For example, equation (6) becomes For a real
27、 gas the equilibrium composition and thermodynamic properties must be obtained by means of statistical mechanics. These calculations can be done independently of the flow-field equations, and the results can be tabulated for later use. Dr. Harry E. Bailey of Ames Research Center has recently per- fo
28、rmed these calculations for various gas mixtures of current interest follow- ing the assumptions and approximations made by Marrone (ref. 12). cover temperatures to 25,OOOO K in 250 increments and densities from lo3 times a reference density, po, which is the density of the mixture for a temperature
29、 of 273.16 K and a pressure of 0.101325 MN/m2 (1 atmosphere). For example, the properties for carbon dioxide are reported in reference 13. The data to The thermodynamic properties in this form are not suitable for optimal use in a computer program. Some approximations are necessary to minimize the c
30、omputing time and storage requirements. For use in the present programs, the calculated values of the properties have been spline fitted with cubics by the method of reference 14, and the coefficients of the cubics have been stored on magnetic tape. A special subroutine reads the tape, searches for
31、the proper coefficients, and evaluates the desired properties. This approximate tech- nique,in general, yields results within 1 percent of the original data. At present the thermodynamic properties for air and the twelve mixtures of nitro- gen, carbon dioxide, and argon listed in table I are availab
32、le on tape. For moderate temperatures, for example, below about 2000 K for air, dissociation and ionization can be neglected, and the imperfect gas effects are due to the excitation of the vibrational states. The thermodynamic prop- erties for such thermally perfect gases have been calculated in ref
33、erence 15 and have also been stored on tape for use in the present programs. The system of equations is now complete. In general, the four partial differential equations (eqs. (1) through (4) must be solved simultaneously for the four dependent variables p,p,u, and v. In the following sections the m
34、ethods used to solve these equations numerically in the subsonic- transonic region and in the supersonic region are discussed. METHOD OF SOLUTION FOR SUBSONIC-TRANSONIC REGION In the nose region of blunt bodies, equations (1) through (4) exhibit different character; namely, the equations are ellipti
35、c in the subsonic region, 5 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-parabolic on the sonic line, and hyperbolic in the supersonic region. these complications, an inverse method (see, e.g., ref. 1) has been found effectual for solving such flo
36、w fields. shock shape is assumed and the equations are integrated numerically by a finite-difference method to determine the corresponding body shape. ticular version used in this report is reported in detail in reference 3; hence, only a brief description will follow. Despite In the application of
37、this method, a The par- Since the initial boundary conditions are specified along the shock, a sheared, nonorthogonal coordinate system with one axis coincident with the shock is useful (see sketch (b). The new coordinates are defined as follows: Sketch (b) I s = x - x(y) t=y Equations (1) through (
38、4) are then transformed and expressed in the following form: 2Q = F,(t,P,P,U,V, -, aP - dP, a, 9 I 3, at at at a (9) 22- = (P3(t?P?P?U?V? - aP, a, a, *) at at at at as For a given set of free-stream conditions and shock shape, the values of p,p,u, and v/t Hugoniot Telations, and the derivatives with
39、 respect to t are found by numerical differentiation. step toward the body. A flow chart for the computer program is shown in figure 1, and a list of subroutines is provided in table 11. To illustrate the predictor-corrector integration technique, suppose the flow properties are known for the (i - 1
40、)th and ith steps and are to be calculated for the (i + 1)th step (see sketch (b). A second-order predictor and a modified Eulerian second-order corrector are used as follows, where p is a typical flow variable. just behind the shock wave are calculated from the Rankine- Then equations (9) are used
41、to march in step-by- 1. Differentiate numerically data for ith step to obtain (a,/ and Inouye, Mamoru: Numerical Analysis of Flow Properties About Blunt Bodies Moving at Supersonic Speeds in an Equilibrium Gas. NASA TR R-204, 1964. 4. Belotserkovskii, 0. M.: The Calculation of Flow Over Axisymmetric
42、 Bodies With a Detached Shock Wave. Computation Center, Acad. Sci., Moscow, USSR, 1961. AVCO Corp., 1962. Translated: and edited by J. F. Springfield, RAD-TM-62-64, 5. Inouye, Mamoru; and Lomax, Harvard: Comparison of Experimental and Numerical Results for the Flow of a Perfect Gas About Blunt-Nosed
43、 Bodies. NASA TN D-1426, 1962. 6. Chushkin, P. I.; and Shulishnina, N. P.: Tables of Supersonic Flow About Blunted Cones. Computation Center Monograph, Acad. Sci., Moscow, USSR, 1961. Translated and edited by J. F. Springfield, RAD-TM-62-63, AVCO Corp., 1962. 7. Eastman, D. W.; and Radke, L. P.: Eff
44、ect of Nose Bluntness on the Flow AIAA J., vol. 1, no. 10, Oct. 1963, Around a Typical Ballistic Shape. pp . 2401-2402. 8. Palermo, D. A.: Equations for the Hypersonic Flow Field of the Polaris Re-Entry Body. Aircraft Corp., Oct. 1960. LMSD-480934, Lockheed Missiles and Space Div., Lockheed 9. Seiff
45、, A.: Secondary Flow Fields Embedded in Hypersonic Shock Layers. NASA TN D-1304, 1962. 10. Jorgensen, Leland H.; and Graham, Lawrence A.: Predicted and Measured Aerodynamic Characteristics for Two Types of Atmosphere-Entry Vehicles. NASA TM X-1103, 1963. 11. Hayes, Wallace D.; and Probstein, Ronald
46、F.: Hypersonic Flow Theory. Academic Press, New York, 1959. 12. Marrone, Paul V.: Inviscid, Nonequilibrium Flow Behind Bow and Normal Shock Waves, Part I. General Analysis and Numerical Examples. CAL Rep. w-1626-12 (I), May 1963. 13. Bailey, Harry E .: Equilibrium Thermodynamic Properties of Carbon
47、Dioxide. NASA SP-3014, 1965. 16 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-14. Walsh, J. L.; Ahlberg, J. H.; and Nilson, E. N.: Best Approximation Properties of the Spline Fit. J. Math. Mech., vol. 11, no. 2, March 1962, pp . 225-234. 15. Hilsen
48、rath, Joseph, et al.: Tables of Thermal Properties of Gases . Cir . 564, U.S. National Bureau of Standards, Nov. 1, 1955. 16. Ferri, Antonio: The Method of Characteristics, Section G. Supersonic Flows With Shock Waves, Section H. General Theory of High Speed Aerodynamics, William R. Sears, ed., Prin
49、ceton University Press, Princeton, New Jersey, 1954, pp. 583-747. 17. Powers, S. A.; and ONeill, J. B.: Determination of Hypersonic Flow Fields by the Method of Characteristics. AUlA J., vol. 1, no. 7, July 1963, pp . 1693-1694. 18. Inouye, Mamoru; and Sisk, John B.: Wind-Tunnel Measurements at Mach Numbers From 3 to 5 of Pressure and Turbule
