1、#m /,1TECHNICAL NOTE 2200A STUDY OF SECOND-ORDER SUPERSONIC-FLOW THEORYBy Milton D. Van DykeCaliforniaInstituteof TechnologyI11WashingtonJanuary 1951.-.-. - . . . -. .-. .- . .-. .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TECHLIBRARYKAFB,NM Ill
2、lllllllllllllllllllllluluCIDL505L,NATIONAL ADVISORY COMMITTEEFOR AERONAUTICSCEU?ICAL NOTE 2200A STUDY OF SECOND-ORDER SUPERSONIC-FLOWTREORYBy Milton D. Van Dyke .An attempt is made to develop a secondof problems of supersonicflow which can beorder theory. The method of attack adoptedthe linearized s
3、olution as the first step.approximationto the solutionsolvedby existing first-is an iterationprocess ustigFor plane flow it is found that a particular solution of the iter-ation equation can be written down at once in terms of the first-ordersolution. The second-orderproblem is thereby reduced to an
4、 equivalentfirst-orderproblem and can be readily solved. At the surface of asinglebody, the solution reduces,tothe well-known result of Busemann.The plane :ase is consid the extension to third and fourth order ischiefly of academic interest.The aim of the present study is, therefore, to ftid a secon
5、dapproxhation, analogous to Busemanns result, for supersonicflow pastbodies which can be treatedby existing ftist-order theory. The naturalmethod of attack, and apparently the only practical one, is by means ofan iterationprocess, taking the usual.linearized result as the ftiststep. Severalwriters h
6、ave applied this procedure to subsonic ?1ow. supersonicflow, as usual, the solution is simpler, so that more generalproblems canbe solved.This paper is a revised version Of a thesis in aeronautics for thedegree of doctor of philosophy written at the California Institute ofTechnology under a National
7、 Research Council predoctoral fellowship. Ithas been made available to the NACA for publication because of itsgeneral interest.ITERATION PROCEDUREBasic assumptions.-The problem to he considered is that of steadythree-dhensional supersonicflow of a polytropic gas past one or moreslenderbodies. As ind
8、icated in the fol.lowin The procedure is clearlydescribedby Sauer (reference1, p. 11 for the case of plane flow.-. . - . .-, - - - - .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 2200 5.1), mibject to proper boundary conditions,The lineari
9、zed solution ii taken as the first approximation. Substitutingthis lnmwn solutioninto the right-hand side of equation (4) gives($n(2) (OZZ(2)-:213=(2)=Fl(x,y,z) (6)where Fiis a known function of the independentvariables. This isagain a inear eqYJ ion, the nonhomogeneouswave equation. A second-order
10、solution. Q 2 , stijectto proper boundary con?litions,can besought by standardmethods. The procedure canbe repeatedby mibsti- ,tuting (2) into the right-hand side of equation (kand solvingagain.Continuingthis process yields a sequence of solutions (n) ch,under proper conditions,presumably convergest
11、o the e=ct sdlution.This procedure bears a superficialresemblance to the Picard processfor hyperbolic equations in two independentvariables (reference12,P. 317). There is, however, an essential difference. Iu the Picardprocess, the characteristiclines of the ”x$ %(2)-=u 42)-.w_z(2)- 96(2). ,. -T - -
12、 - -,.(25).w-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TNin bothassumed2200 11Cartesian and cylindrical coordinates. itserves-asa guide in more complicatedproblems. In p g(x . BY) 62 g(x - PY)”g(x - BY) +TMthesame result can be found by so
13、lving equationimpulse method (reference1.2,p. 164).is therefore(45)M%Yp(x - 13Y)-J2+(46)(42)directly,usingOn the surface of the wall, the streamwisevelocity perturbationis given byu-=u -; g(x) - M,; 2 zg(xz (47)The pressure coefficientat the wall can now be calculatedfrcm equa-tion (27) which, upon
14、replacing N by its value from equation (lJ.),givesCp=; “(y+l)M4-42Gg(x)+2P4 g%c)z(48)This is the well-known result of Busemann (references18and 19). Tosecond order, the surfacepressure coefficient depends only upon thelocal slope.-. - . . . . . , . . . .Provided by IHSNot for ResaleNo reproduction o
15、r networking permitted without license from IHS-,-,-3 NACA TN 2200 17.Role of characteristics.-It was pointedbecause of the underlying significanceof theout previously that,characteristicsurfacesfor solutions of hyperbolic equations, it might be expected that thecharacteristicswould have to be revis
16、ed successivelyat each stage of the iteration. However, an iterationprocess was chosen which permitsno such revision. It is therefore pertinent to inquire in this smleexample what role has been played bycharacteristics.Only one of the two families ofThe original characteristicsof thisw=dxthe origina
17、l and the revised - .characteristicswill be considered.family are the lines of slope1“F (w)These are the Mach lines of the undisturbed flow which run downstreamfrom the wall (see the preceding -gram). They qre also characteristicsof equation (32) in the mathematical sense (reference12, ch. 5; refer-
18、ence 13, ch. II).It can readilybe shown that, if the r7 st-order streetwiseperturba- tion velocityat any point in a flow is, u 1 , then the revised localvalues of Mach nuuiber-and areM(l)=M1+given to ftist order by“ I= PF+M2(N- 1)$(5)I(50b)By using this result together with the first-order solution
19、(equa-tion (41), the revised downstreamMach lines are found”to have the slope(5U These are not the mathematical characteristicsof the iteration equa-tion (equation (42) for the reason that fractions of the highest-orderderivativeshave there been transferred to the right-hand side andregarded as know
20、n. Mathematically,the characteristicscontinue to begiven by equation (w)., .Physically,the characteristicsare lines along which discontinu- ities in velocity derivativesare propagated, and this definition is._ ._. . - . . . . . - -. -. .- .- - - - . . - . - -.Provided by IHSNot for ResaleNo reproduc
21、tion or networking permitted without license from IHS-,-,-. 18 lWK!ATN 2200completely equivalentto the mathematical one (reference12, P. 297).Therefore, in the second-ordersolution derived above, discontinuitiesin accelerationmust occur along the original characteristics.Suppose,however, that no suc
22、h discontinuitiesoccur. For flowpast a singlebody the downstream characteristicsare also lines alongwhich the velocity iS constantprovided tmt shock waves do not aPPe.Settingit is seen that the velocity is constant if.For the secondalong lines ofw (2) %(2)=-p=-pax(52)(53)approximation (equation (46)
23、 the velocity iS consttslope 1* ll+!$.,(X-J3Y)=.dx Pwhich, according to equation (50b), are the revisedConsequently,although the characteristicshave notmathematical sense, the solutionbehaves physicallylong as discontinuitiesdo not occur. The questionwill be considered in the next section.(54)charac
24、teristics.been revised in theas if they had, soof discontinuitiesThe connectionbetmen the original and revised characteristicscan be interpretedphysically. The right-hand side of the iterationeqtion may be regarded.as representingthe effect of a known distri-bution of supersonic sources throughout t
25、he flow field. The influenceof this source distribution spreads.downstresmalong both families oforiginal characteristics. The resulting velocity changes are just suchthat the second-ordervelocities become constant along the revisedrather than the original characteristics.Finally, it is interestingto
26、 note that the second-orderpotentialis constant-n lines whichistics. For, settingd0(2)bis=ct the original and revised character-= 0X(2) dx+4y) dy=o (55).,. . .+- . . . . .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 22(XI 19r0(2) is found
27、to be constant along lines of slope.dy_l+M%dx P 1 Gg,(x - py) (56)Flow past a corner and a parabolic bend.- A simple case in whichdiscontinuitiesmay occur is that of flow past a sharp corner. The exactsolution is known to involve an oblique shock wave with attendant veloc-ity discontinuitiesfor comp
28、ressionand a continuousPrandtl-Meyerfanfor expansion.Denoting the tangent of the deflectionangle by G, positive forcompression (see the following figure), the function g(x) appearingYtIIII/X=PY “/._L_.-jjx .Flow past a corner.in equation (39) isog(x) =x(57). From equation () the second-orderperturba
29、tion potential is found tobe(x,y) =.: (X- PY).+$(X-PY) -$2X (58).- - - . . - -. - - - - - .-. .- -., ,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-20 NACA TN 2200 ,to the right of the,line x = y and zero to the left. Consequently,in either compres
30、sion (c 0) or eansion (G (77)(referece 21),and.(78),Thediscrepancymeans that the. - - . - -. .- - .-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-,.NACA TM 2200second-ordersolutionfails near the Mch cone. It seems remsrblethat the result is in erro
31、r only to the extent of a constant factor.The entropy increasethrough a weak oblique shock wave is propor-tional to the cube of its inclination away from the Mach lines. Conse-. quently, the entropy rise through the shock wave from a cone is o(e),as noted by Lighthill (reference21).Particular soluti
32、onfor axially symetric flow.- Consider flow pasta body of revolutionwhich is either a slenderpointed body with nose atthe origin or one which extends indefinitelyupstresm with-constanto (79a) .,. . . .- . . . - - - - - - - - -.-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
