1、m NASA NASA TR R-171 - “- e, f L z THE CORRELATION OF OBLIQUE SHOCK PARAMETERS FOR RATIOS OF SPECIFIC HEATS FROM 1 ,TO 5 /3 WITH . APPLICATION TO REAL GAS ,FLOWS NATIONAL AERONAUTICS AND SPACE ADMINtSTRATION WASHINGTON, D. C. DECEMBER 1963 Provided by IHSNot for ResaleNo reproduction or networking p
2、ermitted without license from IHS-,-,-TECH LIBRARY KAFB, NM 0067958 THE CORRELATION OF OBLIQUE SHOCK PARAMETERS FOR MTIOS OF SPECIFIC HEATS FROM 1 TO 5/3 WITH APPLICATION TO REAL GAS FLOWS By Mitchel H. Bertram and Barbara S. Cook Langley Research Center Langley Station, Hampton, Va. NATIONAL AERONA
3、UTICS AND SPACE ADMINISTRATION For sale by the Office of Technical Services, Department of Commerce, Washington, D.C. 20230 - Price $1.75 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THE CORRELATION OF OBLIQUE SHOCK PARAMETERS FOR RATIOS OF SPECIF
4、IC HEATS FROM 1 TO 5/3 WITH AFPLICATION TO REAL GAS FLOWS By Mitchel H. Bertram and Barbara S. Cook An analytic investigation was made to show the extent of correlation of the exact oblique shock parameters that may be accomplished by means of similarity parameters suggested by approximate theory. T
5、he exact theory for the inviscid flow of a perfect gas was calculated for Mach numbers of 1.1 to 40, ratios of specific heats from 1 to 5/3, and angles of attack from 0 to shock detachment. From consideration of the approximate theory and the concept of effective ratio of specific heats, the oblique
6、 shock correlations were found to be useful for the rapid calculation of many flow parameters for an equilibrium real gas. Some use- ful correlations are also given for the case of isentropic expansion around a sharp corner. INTRODUCTION Two-dimensional oblique shock theory is one of the basic tools
7、 in aerodynamic work. Tables and charts are now available for results from two-dimensional oblique shock theory for the two gases most widely used in wind-tunnel work, air and helium, over extensive ranges of Mach number. (See refs. 1 to 8, for example.) In reference 9, the desire for air-helium tra
8、nsformations prompted an examination of the oblique shock similarity laws. This examination was for the first-order case with Mach number approaching infinity and some higher order approximations with the air-helium comparison paramount. There are cases where oblique shock parameters are desired for
9、 gases other than air or helium or where existing tables for a gas are inadequate. In these cases, the possibility of using existing tables or charts for gases other than the one of interest to obtain the desired information is suggested by similarity considerations. This analytic investigation was
10、undertaken to determine the correlating powers of certain similarity parameters suggested by approximate theory. Some comparisons with approximate theories were also made. In order to have as wide a comparison as possible, Mach numbers from the low supersonic to the hypersonic range have been includ
11、ed, deflection angles from 0 to shock detachment, and ratios of specific heats from unity to 5/3 in inviscid flow. From consideration of the approximate theory and the effective ratio of specific heats concept, the oblique shock correlations were found useful for the rapid calculation of many flow p
12、arameters for an equilibrium real gas. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-SYMBOLS CP pressure coefficient h static enthalpy K flow deflectton similarity parameter (see eq. (5) Kp (7 + 1) ( 1 -2) Kpu 2 (7 + 1) (l - -) PmUm p22 M Mach numb
13、er P pressure T temperature U velocity 7 isentropic exponent ; equal to ratio of specific heats for a perfect gas 7e 6 * flow deflection angle effective value of 7 for flow across shock waves (eq. (16) e angle between free-stt-eam flow direction and shock wave ,P ,. density Subscripts : 2 condition
14、behind shock wake co free-stream conditions N normal to shock wave S at constant entropy 2 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-t total pressure Superscripts in parentheses refer to series. PRESENTATION the order of an equation developed i
15、n OF THEORY 8, First let us examine some sample results from exact two-dimensional oblique shock theory for a perfect gas with a constant value for the ratio of specific heats. (See ref. 2, for example. ) Two Mach numbers have been chosen; one is (KPY %) much better than given by the uncorrelated fo
16、rm examples in figures 1(a) 5 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-one can readily note that at a given Mach number the first set of computations to deviate from correlation is that for 7 = 5/3, the next, the 7 = 1.4 results, and so on. On
17、e finds by comparing the results between the various parts of figure 2 that for therefore, the total-pressure ratio which is plotted in figure 11 does not contain the values for 7 = 1. This ratio is plotted in figure 11 as a function of the wedge deflection correlation param- eter. Mach numbers from
18、 1.1 to 40 are shown for a constant value of 7. One finds that for values of the deflection parameter below roughly 2, the ratio of total pressures is virtually independent of the value of 7 assumed. However, for values of the deflection parameter greater than about 2, the total-pressure ratio becom
19、es a strong function of the value of 7 assumed. 7 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-APPLICATION TO A REAL GAS In a relatively simple manner, a method can be derived which allows the preceding correlations to be used for evaluation of ma
20、ny of the oblique shock parameters in a real gas. From mass flow and geometrical considerations uN 2 tan 0 ” uN,2 p, - (15) As given by Trimpi and Jones (ref. 8), an effective value of the isentropic exponent ye is used to describe the density change across the oblique shock applied to a real gas in
21、 equilibrium. By definition 03 wave so that the computations may be ( 1 K)y-l - ,-= 1+- P, In this development there is a limitation imposed by certain series approxima- tions used that also, equation (27) has no meaning unless (7 - l)K -2. The similarity parameter K was given by Linnell as the hype
22、r- sonic small angle result K = however, this correlation becomes progressively poorer as the angle of attack for shock detachment is approached. However, even at the higher deflection angles, improvements in reducing the effect of 7 are found so that accurate interpolation of the results can be mad
23、e. The same similarity param- eters also correlate the effect of Mach number, in the same range where the cor- relation of the effect of y is good, for Mach numbers greater than about . No simple parameter was found for correlating the higher order effects of 7 on static temperature ratio across the
24、 oblique shock. However, first-order correlations, although not as successful as those previously referred to, gave a major improvement in reducing the effect of a change in 7 so that accurate interpolations of the results are possible. The correlation of the effect of Mach number on temperature rat
25、io was, however, apparently as good as that for the previously described flow parameters. From ccmsideration of the approximate theory and the concept of effective ratio of specific heats, the oblique shock correlations have been found to be useful for the rapid calculation of many flow parameters f
26、or an equilibrium real gas. 12 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Some useful correlations are also given for the case of isentropic expan- sion around a sharp corner. Langley Research Center, National Aeronautics and Space Administratio
27、n, n Langley Station, Hampton, Va., March 22, 1963. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REFERENCES 2. Ames Research Staff: Equations, Tables, and Charts for Compressible Flow. NACA Rep. 1135, 1953. (Supersedes NACA TN 1428.) 3. Dailey, C.
28、 L., and Wood, F. C. : Computation Curves for Compressible Fluid Problems. John Wiley .214674 +1 .112591 +2 .500866 +2 .242520 +2 .319276 +1 ,759594 +1 .450811 +2 .243262 +2, .415072 +li .536071 +1 I .218082 +1 .490711 +l .240753 +1 .A80802 +1 .263489 +1 .472381 +1 .309121 +1 .458823 +1 .370225 +1 .
29、445403 +1 .za6m +x .465131 +I .234287 +1 1 .21341L +1 .411177 +2 1 .243938 +2 i .503126 +1 .484845 +1, .335415 +2( .245480 +2 1 .706505 +I1 347457 +1 i .181276 +1 I .9S1414 J. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- 1.0 1.1 1.2 1.3 1.4 5/3 1
30、.0 1.1 1.3 1.2 1.4 5/3 1.0 1.1 1.2 1.3 1.4 5/3 1.0 1.1 1.2 1.3 $2 1.0 1.1 1.2 1.3 $2 1.0 1.1 1.2 1.3 1.4 5/3 - .785223 +2 .629890 +2 .547542 +2 489265 +2 !: 1: -159900 +4 -9568 -347 .207284 +2 .27764 -17 .lo9255 +2 .162611 -9 .763071 +1 .139546 -6 .597820 +1 .510679 -5 -399065 +I. e497315 -3 .220513
31、 .303029 .LO9192 .500624 -249940 -1 .219512 .302366 .361811 .500351. .362318 .4oa779 .843187 .766320 .x5587 .674544 .696773 -632990 +2 +2 +2 +2 +2 +2 .1.96040 .163665 .178389 . U0498 .151198 .118213 +1 +1 +1 +1 +l +1 .woo9 +2 .997785 .86195 +2 .198230 +I .322581 +1 .997789 ,729200 +2 .165327 +1 .271
32、308 +1 .997790 I .6993u +2 52662 +I .436392 +1 .997787 .772038 +2 .180285 +1 240939 +1 .997792 I .676390 +2 .la805 +1 .197697 +1 .997795 .633727 +2 .119201 +1 .377718 +1 .99875L -,. _ .199255 +2 .998752 .871412 +2 .199002 +1 .LL562L +1 -998751 .774132 +2 .180954 +1 .730504 +2 .165910 +1 .700229 +2 .
33、153177 +1 .633996 +2 .ll9548 +1 .677054 +2 .142262 +1 .299504 +2 .999442 -452531 +1 -999445 .329337 +1 -999445 -275469 +1 -999445 .a3933 +1 .999445 SI99419 +l -999445 .399654 +2 .999765 .455Ol4 +1 -999687 .330350 +1 .999688 .276088 +1 .999688 .244376 +1 .999688 .199672 +l .999688 .880919 +2 .199570
34、+1 .775659 +2 .181435 +l .731 pressure difference ratio. 10 12 14 Figure 1.- Flow parameters across an oblique shock as a function of flow-deflection angle. 18 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-400 300 101 I 20 30 40 50 6, deg Y 0 1.0 0
35、 1.1 0 1.2 A 1.3 h 1.4 5 513 Oblique I shock 4 theory 60 70 80 90 (b) pressure difference ratio. Figure 1. - Continued. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 10 12 14 (e) = c; shock angle. Figure 1. - Continued. Provided by IHSNot for Res
36、aleNo reproduction or networking permitted without license from IHS-,-,-90 80 70 60 50 0, deg 40 30 20 10 “0 10 20 30 40 50 60 70 80 (a) shock angle. Figure 1.- Continued. 21 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-p2 “ 1 prm 0 0 0 n a 2 4 6
37、8 12 (e) m = fi; density difference ratio. Figure 1.- Continued. 22 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- 1 *Lm 11- L . _. LI - - . - c- . (f) M, = 20; density difference ratio. Figure 1.- Continued. Provided by IHSNot for ResaleNo reprodu
38、ction or networking permitted without license from IHS-,-,-I p22 0 2 4 (g) = $?; mass flow difference ratio. Figure 1.- Continued. 12 14 24 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-0 .4 .a 1.2 1.6 2.0 4 deg 2.4 (h) Ez, = 20; mass flow differen
39、ce ratio. Figure 1.- Continued. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-10 8 6 4 2 1 .8 .6 .4 .1 .08 .06 .M .02 .01 (i) m = and 20; temperature difference ratio. Figure 1.- Concluded. Provided by IHSNot for ResaleNo reproduction or networking
40、 permitted without license from IHS-,-,-shock similarity theory /I/ Third-order small- F Ii i 06 .08 .10 .I2 I I I Y“ 1.4 1.2 1.0 .8 .6 .4 .2 L .9 1. u 0 .1 0 I8 .o i/ ii ii I 1 1 I j j .2 I I jz I I L 7 .4 .1 .5 (a) Mach numbers 1.1, 1.2, and Figure 2.- Correlation of pressure ratio across oblique shock. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-X CL 16 14 12 10 a 6 4 2 0 (b) Mach numbers 2, 3, and 5. Figure 2.- Continued. 28 b - . _ . . Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
