1、U3-J.FOR AERONAUTICSTECHNICAL NOTE 2744PRACTICAL CALCULATION OF SECOND-ORDER SUPERSONICFLOW PAST NONLIFT13JG BODIES OFBy Milton D. Van DykeREVOLUTION- .,Ames AeronauticalMoffettField,Laborar yCalif.:.)-$r at r = R(x) (lls).Smooth bodies.- For bodies without corners,the choice of tangencycondition ha
2、s no consistent effect upon the error in surface velocity.Greater accuracy in the second-order solution results from using the.exact tangency condition in some cases, but the approximate condition.+ lThe magnitude of this effect was brought to the authorrs attention by. John Huth and E. P. Willisms
3、of the Rand Corporation.Y. . .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-z8 NACA TN 2744.,in others.2 For example, the exact condition leads to eater accuracyfor cones, as shown in sketch (b). This superiority, of course, arisesat the tiP of any
4、 potited body and persists for sQme distance down-stream. On the other hand, the approximate tangency condition leadsgreater accuracy for the boattail following a long cylinder shown insketch.875,900.9?5$.950.975Low(c), for-which the exact solutionh/,YSecond-order sohticw:dExact tangency0 .5 1.0 1.5
5、 2.0 2.5tod“a71-e.x, semicalJbersaSketch (c)!Ibus the conclusion,based upon estimatesneither tangency condition is consistentlyempirically for smooth bodies. .Bodies with corners.- In plane flow, the approximate tangency “. condition insatiablyleads to more accurate first- and second-order uvelociti
6、es than the exact condition. The superiority of the approxi-mate tangency condition is most pronounced for expansions, and becomesgreater as the Mach number falls toward unityAt a corner on a body of revolution the flow is locally hro-dimensional. Therefore the approximate tangency condition is, at
7、least.locally, consistently superior to the exact condition for both them21n the first-order solution,however, the approximate ency condi- .tion eeems invariably to yield greater accuracy. .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2;.NACA TN 9
8、first- and second-order solutions. This is shown.in sketch (d) forthe velocity just behind thecorner of a conical boattailwhich follows a very longcircular cylinder. (The .exact solution is, of course,given by a plane Prandtl-Meyer expansion.) At moderate *Mach numbers, tbe superiorityL ,.of the app
9、roximate tangencycondition is of considerableTractical importance in thesecond-order solution. Thesuperiority is not confinedto the immediatevicinity ofthe corner, but persists fardownstream. This is illus-trated in sketch (e)by comp-arison with the solution fora conical boattail calculatedby the me
10、thod of characteris-tics. (For clarity, the first-order solutions are only par-tially shown.)1 Sketch (d) suggests thatthe large discrepancy associatedwith the choice of tsmgegcy con-dition is in some sense a tran-sonic phenomenon. This isconfirmed by examination of theexpressions for the streamwise
11、velocity just behind the corner.For expansion through an anglewhose tangent is , the second-order solution using the exacttangency condition is(12a)f.31.2ur./I.oExact sofufionISecond-wder/1.35i. 30f.25/.20qz1.51./01.05/.00r? 3 4InSketch (d)First- order soution, _Approx. tangency.fia , second-order S
12、OI.t*enequal to the length of the cone. Otherwise, the meridian curve willordinarilybegti with finite curvature. For a specified limit ofnumerical error, the maximum permissible length of the first intervalmust be proportional to the initial radius of curvature,which is theprimary length in the prob
13、lem. The factor of proportionalitywill, ofcourse, depend won the shape of the body. If the meridian cwve isanalytic, dimensional analysis combinedwith the supersonic sitilitYrule indicates that the first interval is given by an expression of theform5 i(30)Here ?, Rot, Roirt are the first three deriv
14、atives bf R(x) eval-uated at the vertex, and the dots indicate that no appreciable depend-ence upon higher derivatives is to be expected. Tndeed, for slendersmooth bodies even the second.variable Ro/(f2)f2) is no-lly verYsmall comparedwith the first. Hence it may be assumed that the func-tion Go doe
15、s not depend GignifLcantlyupon its second variable, sothat the length of the first interval is given by180= Go(wO)MIRo” ) (31)It is now clear that the body shape need not be analytic throughout thqfirst interval; it.is sufficientthat no violent changes in curvatureoccur.%hat the denominator shouldbe
16、 taken a MRo rather than o isSuggested by the result of equation 32).,.,* “,-*a71 “Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NAC!ATN 2744 21!lheform of the function Go can be determined by analysis,because the second-order solution at the end o
17、f the first interval ofa general.ogive can be calcul-tedexactly as well.as approximately ifthe interval is very short. Although the result is formidable, itsimplifies greatly in the limiting case when o approaches unity(whichcorrespondsphysically to the Mach cone becoming tangent to tHenose). In thi
18、s case, for a relative numerical error x in stream-wise velocity perturbation, the length of the first interval isJ8= 1 (-P% 2)Jm as ,8R0t+l (Y)7+1 MIRo”lNumerical exmples show that this asymptotic form is, with a revisedconstant of proportionality, a good approximation to the functionthroughout the
19、 range of practical application. The relative numericalerror at the end of the first interval will not exceed 1 percent i1It is conceivable that an unusual body shapefor which the curvature would change considerablyso, the above rule would not apply (the variable,.-might be encounteredover,this leng
20、th. IfRn/(Ro“2j ineqktion (30)would not be negligible), 8nd some eri its form is different because R rather than l/Rtt istaken as the primary reference length. (The second variable here has no counterpart in equation (30)because R is zero at the tip.) Fora smooth slender body, the third variable is
21、ordinarily very small, as%his rule ordinarily permits greater first intervals than the ruleZio= 0.025/ times initial radius of curvature which was previouslySuggested in reference 4-. .,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-22 NACA TN 2744a
22、re all subsequentvariables which involve higher derivatives. Thenaccording to the argument used previously, the function G1 dependssignificantlyupon only its first two variables This conclusion isreinforcedby the empirically determined fact that disconttiuitiesincurvaturemust be accounted for separa
23、tely,but not jumps in third and Ihigher derivatives. Hence the nth interval is giveu byAs before, the assumption that the body is analytic can now be replaced .by the requirement that no violent changes in curvature occur.Analytic determinationof the function GI seems impractical.It detailed form co
24、uld be determined experimefitallybycalculating a 1number of solutionsusing intervals of various lengths. However,experience suggests that for the body shapes encountered in practice G1may be taken as a constant. The relative numerical error will appar-ently not exceed 1 percent if internal intervals
25、 for bodies withoutcorners are chosen so that .P12E1. (36) .- =Modification for corner or curvature discontinuityy.- Two pointsmust be chosen at any discontinuity in slope or curvature, one just oneach side, as indicated in sketch (k). A corner so strongly affectsthe subsequent flow field that it ha
26、sbeen found necessary to reduce the nextr“ nte”d” e”ativeemorwi” -apparently not exceed 1 percent if the -interval following a corner i takenSketch (k) where Rc is the radiusThereafter, intervals can be chosen according to the rulebodies (equation (36).(37).at the corner.for smoothLimitations of rul
27、es.- These rules for choosing intervals areintended only as guides and must not be followed blindly. Althoughadequate for most bodies, they may fail for unusual shapes, particu-larly those having rapid changes in curvature._ For example, the rulefor choosing internal.intervals (equation (36) does no
28、t apply to the.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.b.!NACA TN 2744 23comugatA body shows in sketch (L). In this case the variablenot only is the scatterquadrupled,but successive errors then accumulate to such an extentthat the result dep
29、arts progressively farther from the true solutionwith distance downstream.Description of Computing FormsStandard computing forms have.been prepared which largely reducethe second-order solution to routine calculation with a desk machine.Form A is used for bodies having continuous curvature. Form B i
30、s aninsert to be pasted into form A to account for a corner ses. The forms canreadily be extended to handle longer calculations. Copies of theforms suitable for photosensitive reproduction are enclosed. 7Thus if one extra point is required, every row on each_sMe ofom_AI and B which nov extends to co
31、lumn Pa (except rows to , ,0and s of form A) is extended to form an additional column labeled. PTj and below row 6W of form A is inserted a new group of rows iden-0; oticalwith rows 6a to 6W on the left and o “heright, but labeled 7- - and containingblanks only in colum P7.!Provided by IHSNot for Re
32、saleNo reproduction or networking permitted without license from IHS-,-,-24 NACA TN 2744The desired values of Mach number and 7 =e entered at the topf form A, togetherwith values of XjR,Rt, and Rf at points alongthe body chosen accokding to the rules formulated above. Then the formcan be given to a
33、computer together with tables I and II. The solutiofor a typical ogive or boattiil can be calculated in from 5 to 10 hours.As the solutionprogresses along the body, the results are found asdifferences of increasinglylmge numbers. cnsequently, it is advisableto carry all computations to six significa
34、ntfigures or seven decimalplaces, whichever is the lesser. It is for this reason that tables Iand 11 mmt be so extensive. It is not, of course, necessarY pre-scribe the problem With such accuracy; it is sufficientto give M, 7,and the body shape to three significantfigures.Details of form A.- The lef
35、t half of form A is devoted chiefly tothe calculation of the first-orderpotential q and its required deriva-tives. The particular integral is also found in the last 23 rows ofthe left side. The right half gives a Ywal-lel cctiation of the com-Tbesecond-orderpressure coefficient ie %:u:f:?$? d the co
36、rrespondingfirst-order result,.Following various preliminary calculations in row8to , eachgroup of from 10 to 13 rows bounded by double lines comprises a sepsratebasic solution. The-first such group (rows q:; ,g)=;gs:ra linear source solutionbeginning at the origa pointed tip. It my be noted that a
37、stratagemhas been introduced in calculating its effect at the tip. There both x and R are zero,so that the value of the conical variable t givenby equation (19)would be indeterminate. This difficulty is surmountedby identifyingvalues at the tip with those at the end of a tangent cone whose lengthis
38、arbitrarily chosead unity, as indicated in sketch (m). The requi-site modification of gven values in the first column i= indiSketch (m) second,calculation of the required)strength of.the solution (row -s from the tangency condition; thlr ,calculationof its contributionstoQ)d-P, -, /p, and - (rows -t
39、to -w, at each of the points P. to I=aticoribui;me8:q:i=:%Tcomplete flrst-order results in rowsand (27)pe t the calculation f the remainin two second derivatives,-Qm (row 2 ) and c only the latteris used at a curvature discontinuity. The two groups are similar instructure to those of form A, with th
40、e addition that qm/ is alsocalculated (rows ) for later use.me right half of form B contains the corresponding corner andcurvature solutions for the complementary function. In addition, astep solution is provided.(rcws ) which, as discussed previously,is required in the complementaryfunction to neut
41、ralize a step in theparticular integral at a corner. This step solution is placed adjacentto the first-order corner solution with which it is associated. Simi- r-larly, the corner solution is placed ad3acent to the first-order curva-ture solution, with which it is associated even if the body has noc
42、orner. The curvature solution is not required in the complementaryfunction except at a corner. At a corner the curvature discontinuityis so great that it must be accountid for at least approximately inorder to prese numerical accuracy. Its strength cannot be calculatedexactly in terms of previously
43、determined quantities, but fortunatelycurvature and corner solutio are so intimately related that it%Ctmaybe noted that the coding is mnemonic to the extent that rowsoand -v are proportional to the first-order velocity perturbationsu and v, am s and to the second-order values.Provided by IHSNot for
44、ResaleNo reproduction or networking permitted without license from IHS-,-,-26 L NACA TN 2744suffices to take them in the same ratio in the complementaryfunctionin the first-order solution. .Use for first-order solution alone.- A very accurate first-ordersolution is found in the course of the second-
45、order competition. Theaspresent scheme can therefore be simplified ifonly a first-order solu-tion is desired. Except for rows to, only the left halves offorms A and B are used, and form A car.be terminatedtith row 22 andQformB with row c (becausecurvature discontinuitiesneedno beaccounted for). Pore
46、over, the folloting rows can be deleted from ,form A:and the followingfrom form B:oCe o, and CtThe restrictions on interval lengthand O-w 1s; and o20cab be considerablyrelaxed.An analysis similar to that described previously shows that the firstinterval for a pointed ogive can be taken asA few numer
47、ical examples suggest that subequent intervals can be taken /at least twice as large as for a second-order-solution,so that.2pRD exceq just behind a cornerbn =Rn just behind a corner 1 ( 39)PRACTICAL USE OF METHODThe following instructionsare intended to permit the reader to - . apply the method wit
48、houtreference to the preceding detailed discussion.Applicability./- - _-The method gives both the first- and second-ordervelocities andpressures at the external surface of a body of revolution in supersonicflow provided that:.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.NACA TN 2744 271. The body has a pointed nose, or has a sharp-edged open nosewith purely supersonic external flow at the entrance, or is a boattail.following an infini10WL rules. T
