1、REPORT 1135EQUATIONS, TABLES, AND CHARTS FORCOMPRESSIBLE FLOWBy AMES RESEARCH STAFFAmes Aeronautical LaboratoryMoffett Field, Calif.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REPORT 1135EQUATIONS, TABLES, AND CHARTS FOR COMPRESSIBLE FLOW 1By AME
2、s RESEARCH STAFFSUMMARYThis report, which is a revision and extension of NACA TN1_28, presents a compilation of equations, tables, and chartsuseful in the analysis of high-speed flow of a compressible fluid.The equations provide relations for continuous o_e-dimensionalflow, normal and oblique shock
3、waves, and Prandtl-Meyerexpansions for both perfect and imperfect gases. The tablespresent useful dimensionless ratios for continuous one-dimen-sional flow and for normal shock waves as functions of Machnumber for air considered as a perfect gas. One series of chartspresents the characteristics of t
4、he flow of air (considered a perfectgas) for oblique shock waves and for cones in a supersonic airstream. A second series shows the effects of caloric imperfec-tions on continuous one-dimensional flow and on the flowthrough normal and oblique shock waves.INTRODUCTIONThe practical analysis of compres
5、sible flow involves fre-quent application of a few basic results. A convenientcompilation of equations, tables, and charts embodying theseresults is therefore of great assistance in both research anddesign. The present report makes one of the first suchcompilations (ref. 1) more readily available in
6、 a revised andextended form. The revisions include a complete rewritingof the lists of equations, as well as the correction of certaintypographical errors which appeared in the earlier work.The extensions are primarily in the directions dictated byincreasing flight speeds, that is, to higher Mach nu
7、mbers andto higher temperatures with the accompanying gaseousimperfections.Compilations similar to those of reference 1 have beengiven in other publications, as, for example, references 2through 6. These references have been utilized in extendingthe tables and charts to higher values of the Mach num
8、ber.The extension to imperfect gases is based on the relationspresented in references 7 and 8.SYMBOLS AND NOTATIONPRIMARY SYMBOLSaAspeed of soundcross-sectional area of stream tube or channelC_C_,CehlMPqqRRSbSTU_tVVWa5pnormal forcenormal-force coefficient for cones,q_S_specific heat at constant pres
9、surespecific heat at constant volumeenthalpy per unit mass, u-_pvcharacteristic reference lengthMach number, Vapressure 2dynamic pressure, pV2/2heat added per unit massgas constantReynolds number, pVlbase area of coneentropy per unit massabsolute temperature 2internal energy per unit massspecific vo
10、lume, 1pvelocity components parallel and perpendicularrespectively, to free-stream flow directionvelocity components normal and tangential,respectively, to oblique shock wavespeed of flowmaximum speed obtainable by expanding tozero absolute temperatureexternal work performed per unit massangle of at
11、tackratio of specific heats,angle of flow deflection across an oblique shockwaveshock-wave angle measured from upstream flowdirectionmolecular vibrational-energy constant1Mach angle, sin-l_absolute viscosityPrandtl-Meyer angle (angle through which asupersonic stream is turned to expand fromM=I to M_
12、I)Supersedes NACA TN 1428, _Notes and Tables for Use in the Analysis of Supersonic Flow“ by the Staff of the Ames 1- by 3-foot Supersonic Wind-Tunnel Section, 1947.When used without subscripts, p, p, and T denote static pressure, static density, and static temperature, respectively.613Provided by IH
13、SNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- _, - .“ _“ _ _ _. _ _r k_ _ -._614 REPORT l135-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSpressure ratio across a shock wave, PJplp mass density _a semivertex angle of coneSUBSCRIPTSfree-stream conditions1 condition
14、s just upstream of a shock wave2 conditions just downstream of a shock wavet total conditions (i. e., conditions that wouldexist if the gas were brought to rest isen-tropically)* critical conditions (i. e., conditions where thelocal speed is equal to the local speed of sound)c conditions on the surf
15、ace of aconer reference (or datum) valuesperf quantity evaluated for a gas which is both ther-mally and calorically perfecttherm perf quantity evaluated for a gas which is thermallyperfect but calorically imperfect( )p derivative evaluated at constant pressure(), derivative evaluated at constant ent
16、ropy( )r derivative evaluated at constant temperature(). derivative evaluated at constant specific volume( ), quantity evaluated over a reversible pathNOTATIONThe notation in brackets after many of the equationssignifies that the equation is valid only within certainlimitations. )or example :pelf me
17、ans that the equation is restricted to a gaswhich is both thermally and caloricallyperfect. (By “thermally perfect“ it ismeant that the gas obeys the thermalequation of state p=pRT. By “caloricallyperfect“ it is meant that the specific heatscp and c, are constant.)therm perf means that the only rest
18、riction on the gas isthat it must be thermally perfect. Equa-tions so marked may be used for caloricallyimperfect gases. (They are, of course, alsovalid for completely perfect gases.)isen means that the flow process must take placeisentropically. Equations so marked maynot be applied to the flow acr
19、oss a shockwave.adiab means that the only restriction on the flowprocess is that it must take place adiabati-cally-that is, without heat transfer. (Sucha flow process may or may not be isen-tropic depending on whether it is or is notreversible.) Equations so maxked may beapplied to the flow across a
20、 shock wave.An equation without notation has no restrictions beyondthose basic tb the study of thermodynamics and/or inviscidcompressible flow.FUNDAMENTAL RELATIONSTHERMODYNAMICSTHERMAL EQUATIONS OF STATEA thermal equation of state is an equation of the“formp-p(v, T) (1)Several of the more commonly
21、used thermal equations ofstate are the following:Equation for thermally perfect gasRTp_-_-= pRT therm per/ (2)ordp dp dTP p _-0 therm peril (3)Eqdations for thermally imperfect gasVan der Waals equation (ref. 9)RT aP=V-b-_ (4)where a is the intermoleeular-force constant and b isthe molecular-size co
22、nstant (see ref. 9, pp. 390 et seq. fornumerical values).Berthelots equation (ref. 7)RT cP=v-b v2T (5)where b is the molecular-size constant and c is theintermolecular-force constant (see ref. 7 for numericalvalues).Beattie-Bridgeman equation (ref. 10)p-=R_.T_(l- C_Fv+Bo(l-b)7-_-i(1-a) (6)v v.t/l_ v
23、ia v k v/where a, Ao, b, B0, and c are constants for a given gas(see ref. 10, p. 270 for numerical values).CALORIC EQUATION OF STATEA caloric equation of state is an equation of the formu=u(v, T) (7)It can be shown thatdu=c, dT therm peril (8b)If the gas is calorically perfect-that is, the specific
24、heatsare constant-equation (8b) can be integrated to obtainu=c,T+u, peril (9)2 When used without subscripts, p, o, and T danote static pressure, static density, and static temperature, respectively.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ENER
25、GYRELATIONSThe law of conservation of energy givesdq=du_dw (first law of thermodynamics)_ (10a)t=du+p dv=dh-v dpdq=c, dT+p therm pert (lOb)=cp dT-v dpSPECIFIC HEATSspecific heats at constant pressure and constantEQUATIONS, TABLES, AND CHARTS FOR COMPRESSIBLE FLOWENTROPYThe entropy is defined byIt fo
26、llows thatThevolume are defined by/bq “_ bh_c _k_r =k_), (11)It can be shown that( pY5u _T_/.c_c,=(_)r+p _(Op,_ (13a)bv/rc_-co=R therm peril (13b)The ratio of specific heats is defined asCp (14)C_According to the kinetic theory of gases, for many gases overa moderate range of temperature,n+2 (15)nwh
27、ere n is the number of effective degrees of freedom of thegas molecule. Useful relations for thermally perfect gasesaredh _ n .yRc_=_T=C,_=_- 1 therm perf (16)du R therm peril (17)c_=_=c_,-R=,y_ 1ENTHALPYThe enthalpy of a gas is defined byh-uWpv (18)It follows thatdh-du+p dv+v dp=dq+v dpi_p _pdh=(c,
28、+R)dT=c,dT therm perf (19b)h=(c,-t-R)T-t-u,=c,T+u, peril (20)615(21)ds _du+dw_ iZdu+pdv_ dT. Sp ,=. _ ./,= _- -),=c.-_-i.-_a_,) av (22a)dT. _ dr“ds=C, -T-+ _ -ff-dT R dp=c_- -;therm peril (221)_ dT o dp-c_?-cp dp7-.Ts-s,-c, In _/,_, R In -Pp,-c_ In T-Rln -p peg (23a)Ir p_=C, In P-c_.ln -PP,Pr.T/T_s-
29、s,=c, In (p/p,)_-c_ In T/T, perf (23b)(p/p_) (_-_)_=c, ln P/P“(p/p,)_P-P-= P“ e(-, )/, peril (24)p_ pr _The second law of thermodynamics requires thats-s,_O adiab (25)CONTINUOUS ONE-DIMENSIONAL FLOWBASIC EQUATIONS AND DEFINITIONSThe basic equations for the continuous flow of an inviscidnon-heat-cond
30、ucting gas along a streamline are as follows:Thermal equation of stateP=RT therm perfl (26)pDynamic equation1 dp-_VdV=O (27)pProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-:_ ;:7 , , -_,_ _ * “_: (; _;_*_, _- _“ ._ _. _ “ , “ “ !. “2, _:, “:_ “*.7 7
31、_616 REPORT l135-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSEnergy equationdh+VdV=O )cpdT+ VdV=O t3,._d(P)+vdV:O_ adiab, thermperf (28b)Additional useful variables are defined as follows:Speed of soundI-=._,lv P=_ therm pert (29b)_49.0 n/_ ft/sec for airif T is in degrees Rankine(=degrees Fahrenheit
32、 +459.6) (29c)Mach numberDynamic pressure1q -_ o V -_VM-=- (30)a(31 a)therm perf (31b)INTEGRATED FORMS OF ENERGY EQUATIONTile energy equation (28) can be integrated at once toobtainV 2h +_-:constant-ht adiab (32a)V 23“ /p.V 2 3“ (P)3“-1(,-;) +a 2 y _ at 23“- 1 _“ 2 3“- 1 adiab, perf (32b)a 2 . V 2 1
33、 3“-+- 1_3“zi+_-=_ _1 a*_a 2 I7 2 Vm 27-1_2 2The three reference speeds at, a, and V, are related bya, 3“+1“(V_ * _+1-_-, = _ adiab, pert (33)(v.y= zat 3“-1PRESSURE-DENSITYRELATIONFrom equations (27) and (28b) it follows that_P -constant- p isen, perf (34)p_ pt _from whichp (py“ (Tv-1 (a_ “_-p_ p,-/
34、-T-t/ -_/ isen, perf (35)BERNOULLI*SEQUATIONCombination of equations (32b) and (35) gives Bernoullisequation for compressible flow in the form“y-1 ()3“ Pt P _, = 3“ Pt isen, perf (36)RELATIONSBETWEENLOCALANDFREE-STREAMCONDITIONSWith the aid of the foregoing equations it can be shown thatT , _-1 V 2M
35、: adiab, perf (37)!V 2 _-1P-_I-_-I M2(_._)-l isen, perf I (38)p_ ( 2Ip V 2_-_=1-“/2 -1 M*2(.V_)-I v-t isenperf (39)In small-disturbance theory, where it is assumed that(V-V.)_0 (87)It follows immediately from the energy relation (86) thattotal enthalpy, total temperature, and total speed of soundare
36、 constant across the shock and hence (from the previousrelations“ (33) for adiabatic flow) also the critical speed ofsound and limiting speed:htl:htT,= T, 2 1ah=at26*1=6*2 /V,.l=V,.2 jCombining equationsrelationadiab (881)adiab, perf (88b)(84) to (86) leads to Prandtlsuau2-_a,2=P2-P_ adiab, per/ (89
37、)p_-Plwhich implies that the flow is supersonic ahead of the shockwave and subsonic behind (the reverse possibility is ruledout by the requirement of nondecreasing entropy), and tothe Rankine-Hugoniot relationsp_2 (v-t-l) p2-(_- 1) p, adiab, perf (90)p, (5“+1) p,-(v-1) p2P2 (5_- 1) P2+(5“- 1) p, adi
38、ab, perf (91)p-_-(_+ 1) p1-_-(5“- 1) p2_ p2+pt adiab, perf (92)p2-P, 5“p2-Pl P2qt-PtUSEFUL RELATIONSMany relations for normal shock waves are convenientlyexpressed in terms of either upstream Mach number M_ orttm static-pressure ratio across the shock _-p2/px. The fol-lowing relations apply to adiab
39、atic flow of a completelyperfect fluid. The last form of each equation holds for_=7/5.Parameter M,.-P_t_-= 27M_2- (7- 1)_ 7 M_ 2- I (93)p, 5“+1 6p_ Ul U, 2 a. 2_ (_-t-1) M_ 2 _ 6M_ 2 (94)_-_a a2_-_-(5“-I) M,2+2-M_%5T2 a22 25“M,2-(5“-1)1 (_-1) M,+2_=_- (_-t- 1) 2 M,=(7M, 2- 1) (M,2+5)36M, _(5-1) M,2+
40、 2 M,2+ 5M22“ 25“ M_ 2- (5“ - 1 )-_7 M, 2- 1!p_ 25“ M12-(7- l)V,=- _-_ (5“_ 7_ -l-1) M_2+21_ 5 _“7M_2-1(M_+5 )6(95)(96)(97)7p2 -45“M,2-2(v-I)-_=5(7M2-1) _=L (_i-)_ _ a 36M,2P_2=P _._2=e-_Pq Pq .17- (5“+ 1) M, * - _ _+! .T -1=L(_ i-)_,-_ 2- L2,_M,-(5“- l)j7 5= U ,-_4_/1.),p,2 F(5“+I)/,27_F 5“+1 i-p-7
41、,=L- 2 j L25“M,2-(5“- 17 |=(6M“_) (7M16-I) (100)(Rayleigh pitot formula)As P2_-s- (_- 1) _: - (5“- 1) In (_-_h)lnF25“M?_(5“_1)l 5“in - (5“+I)M_ 1= L _ J-L(5“- |)M“+ 2J_ln(7M6-1 7, 6Mr (101)- )-_,n LM_)p2-P, 4(M, 2-1) 5(Mt2-1) (102)q, - (-1)M, _=-Numerical values from equations (93), (94), (95), (96)
42、, (99),and (100) (with 5“=7/5) are given in table II.For weak shock waves (M_ only slightly greater than unity)the following series are useful:_= 1)s+ 25“225“ (M2 (M_ _- 1)+ -Ph1 -2_6 2 s 245= (M, -1) -b_-g_ (M,- 1)-t- (103) As 1 As 25“1)2(_2 .,s 25“ 2 t_2_l)*._._-5“-1 c, 3(5“+ _-(5“+1) _135 , -s 24
43、5 (M2_l),_t - .- (104)=2_ (M, - 1)-gg_(98)619EQUATIONS, TABLES, AND CHARTS FOR COMPRESSIBLE FLOWProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-620 REPORT 1135-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSParameter _ _p2/p1.-M,2= (.y+ 1),_+(-r- 1) 6_+1
44、 (lO5)2_ 7O_2_u_2=(-y+1)_+(-y-1) 6_+ 1 (106)pl u2 (_-1)_+(_(+1) _+6T2_a22=_ (_/- 1)_+(_,+ 1) _+6 (107)Tl-al 2 “ (_+l)_+(-y-1) _6_+1M22_ (_- 1)_+(_ + 1)_ _-+6 (108)2-y_ 7_ 7P2=_ P, _ ( 4“y )- J- 35 -iVq p,_-_i(l+ 1)(_,- 1)_ + (5, + 1)1 _ =_J(10 9)“y 7P-_t2- _t2 = (_+ 1)(_-_ 1)+ (7- 1) _ L6(6+ 1)_J_f-
45、 F(,y+ i)_+(,),- i)7,-,L_)J(1 10)AS 1Pt_ pt_ R “r-1Pq Pq5 7 _-_-/ (11 1)AS= (,),- I) _= - (3,- I) In (P“_=In _-ev pq/,_F(.y+l)_+(-r-1)-I ,. 7. 6_+1For weak shock waves (_ only slightly greater than unity)-_ _+1P2 1-2_ (_- 1)3+ (_- 1)+ Pqa 15=1-5(_-1) +_(_-1)+ . (1 13)As_ 1 As _/+1 1)a _+1R -_-1 c,-1
46、2_ (_- -8_-_ (_-1)4+“5 s 15=4-9 (_- 1) -_-_ (_-1)4+ , (114)In unsteady flow a normal shock wave acts at each in-stant as a steady shock. Hence all the above relations arevalid across a moving normal shock wave if instantaneousvelocities are measured relative to the shock.OBLIQUE SHOCK WAVESIn genera
47、l, a three-dimensional shock wave will be curved,and will separate two regions of nonuniform flow. How-ever, the shock transition at each point takes place instan-taxmously, so that it is sufficient to consider an arbitrarilysmall neighborhood of the point. In such a neighborhoodShock wove _.P, PI P_ P2u2_ ?2 o2T_ o_M I S I M2 $ 2Sireomline= _ _ _ _/FIaURE 2.-Notation for oblique shock wave.the shock wave may be regarded as plane to any desireddegree of accuracy, and the flows on either side as uniformand parallel. Moreover, with the proper orientation ofaxes the flow is locally two-dimen
