NASA NACA-TN-3527-1956 A Second-Order Shock-Expansion Method Applicable to Bodies of Revolution near Zero Lift《在接近零升力时 回转体适用的二阶冲波膨胀法》.pdf

1、NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL NOTE 3527 A SECOND-ORDER SHOCK-EXPANSION METHOD APPLICABLE TO BODIES OF REVOLUTION NEAR ZERO LIFT By Clarence A. Syvertson and David H. Dennis Ames Aeronautical LaboratoryMoffett Field, Calif. CASE FILE copy Washington January 1956LkProvided by I

2、HSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- -+ - * - -“- a .- - - - - - : I2Q:. 7jY.T:LI :i:t: iJe - - - - -. - - -+ -+ -f- -.-+- -+ -.- -.- -f-.-JOScl -1- -?-4-4- (2) the flow appears locally two-dimensional; (3) surface streamlines may be taken as meridian

3、 lines. In the intermediate range of supersonic speeds of interest here, the first approximation is particularly well justified (see, ref. 9), and it will not be considered further. As a consequence of the second approximation, a solution given by the generalized method satisfies the continuity equa

4、tion only approximately. Although the con-tinuity equation does not appear explicitly in the following analysis, it is this approximation that is refined by the present method. The third approximation is one for bodies of revolution only when they are inclined. In the present investigation, only bod

5、ies near zero lift will be consid-ered. Under this restriction to infinitesimal angles of attack, an anal-ysis has shown that the deviation of true streamlines from meridian lines has negligible effect on surface pressures. In the following development, therefore, the use of meridian lines as stream

6、lines will be retained. Nonlifting Bodies The generalized generalized shock-expansion method was developed for nonhifting bodies ofrevoIbion from the method of characteristics (ref. 2). This development may be summarized with the aid of the equation for the stream-wise pressure gradient.2ap 2yp 1 P

7、(1) s sin 2iis cos tCi In the generalized method the pressure is considered constant along first-family MTies (rf s. 1 and )4). As a consequence, the right-hand member of equation (1) is approximated by zero, and the equation can be integrated to yield the well-known Prandtl-Meyer relation. The obje

8、ctive of the present analysis is to refine this approximation to the right-hand member of equation (1). To this end, consider the flow about a body of revolution which has a pointed nose and over which the flow is everywhere supersonic. The problem will be simplified by approximating the profile In

9、the treatment of two-dimensional flows, the first approximation is used but continuity is exactly satisfied. 2This equation can be derived directly from the continuity, momentum, energy, and state equations with the aid of characteristics theory (see, e.g., refs. 2 and 9). In this form, the equation

10、 applies equally well for rotational and irrotational flows, requiring only that dE/ds not dE/dn be zero.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 NACATN3727 of the body with a series of tangents to the original contour (see sketch (a). It mi

11、ght be noted that Ferrari (ref. 10) suggested a similar scheme with a body whose profile was made up of chord lines joining points Sketch (a) on the original contour. While either approximation is permissible, the tangent body was selected here so that the conical flow at the vertex will be correct

12、regardless of the degree of approximation used downstream of the vertex. The generalized method gives the exact change in surface pressure around the corners of the tangent body but predicts no change along the reduces hdtermination othe_pressure VH1on aloiEe straight-line elements (teh). I _L. ?_ S

13、ketch (b)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3527 7 For simplification, the derivative, /c1, will be approximated with a difference equation; thus, along the straight-line element, equation (1) may be written (since /s = 0)AP 2 .f

14、 tCicos where ip is the net change in pressure along Mach lines emanating fom the surface and AC, is the corresponding length. This equation will be solved by an iteration procedure based on the solution given by the gen-eralized method. As previously noted, with the generliedmthod the flow is consi

15、dered to-dimensional and, consequently, nopressure change is redicted along streamlines between the expansion fans at either end of the straight section. While this approximation may be appreciably in error for the surface streamline, it is apparent that the real flow will appear more nearly two-dim

16、ensional at large distances from the body axis. It is reasoned, therefore, that a streamline, well removed from the axis (line AB in sketch (b), can be found along which the pressure will also be constant to the accuracy required here. 3 For all Mach lines (such as CD) emanating from the straight su

17、rface then, the pressures at the points of intersection with this streamline will be equal. The term, L.p, in equation (2) therefore can be written as k1 -p, where k1 is a constant and, of course, p is the y ryjng urac sure. The generalized method also prescribes that the length (from the surface to

18、 streamline ADB) and inclinations of all Mach lines will not change when the surface is straight. The term, LC1cos p., therefore canbe represented by a second constant, 1/k2. Equation (2) thus may be written = kk1 = p) which can be integrated to yieldV -k2s = 3 - (3) where k3 is the constant of inte

19、gration. This analysis serves to establish the form of the equation representing the pressure distribution on an element of the tangent body.4 It remains now to evaluate the three unknown constants in equation (3). Three known conditions can be employed 3Examination of characteristic solutions for t

20、he flow about cone-cylinders indicates that the pressure variation along streamlines, a moder-ate distance from the surface, is markedly less, than that along the surface. 41t might be noted that Ebret, Rossow, and Stevens (ref. ii) found that pressures on ogives correlated according to the hyperson

21、ic similarity law could be represented approximately by an exponential function of distance.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3527 for this purpose. First the pressure, just downstream of the corner, P2, can be calculated exactl

22、y from the Prandtl-Meyer equations if the pres-sure, P1, and the Mach number, M1, upstream of the corner are known. Second, the pressure gradient just downstream of the corner may be cal-culated from the results given in Appendix B. The expression defining this gradient is - sir + B2 ill); Ai() + A2

23、(1i) where (o j =sin 2(5) B- ypM2 - 2(42 - 1) and a is the one-dimensional area ratio or.7+1 r - 2(7-1) 1, + (_2 M2 OL 6VO M Y + 2 For the third condition, it is apparent that the pressure on the element shown in sketch (b) would approach some limiting value if, rather than ending at point 3, the el

24、ement were extended as indicated by the dashed line. If the element were considered to be infinitely long, so as to form an extended conical surface, then the only effect the region ahead of point 2 will have on the flow at infinity is to form an infinitesimally thin layer near the surface across wh

25、ich the entropy varies. It can be demonstrated, however, that there is no pressure change through this layer and that the flow outside the layer is conical. Consequently, the limiting pressure is simply, PC the pressure on a cone tangent to the original body at the same point as the element shon in

26、sketch (b) (and, of course, traveling at the same free-stream Mach number). With these three condi-tions, the three unknowns in equation (3) may be evaluated and there is obtainedPc - (p - p2)e (8)(6) (7)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-

27、,-NACA TN 3527 9 where- (p - X2 (9) - (Pc - p2) however, fur-ther simplification can be obtained by the use of ao_rdp“ tangent This body consists of a cone tangent to the original body at the vertex and a conical surface tangent to the body at the station where the pressure is to be calculated. With

28、 this two-step body, the second surface is a variable depending on the station in question on the original body. For this approximation, equation (8) becomes1f P = Pc - (Pc - p5)e (U)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-10 NACA TN 3727 whe

29、rexsin - rcos(12) r Cos B-x sin 8 B sin =- (13) ( - PS) k95 sin, IlvThe subscript, 5 denotes quantities at the station on the body as eval-uated by the With equation (ii) it is possible to obtain, very rapidly, a first approximation to the pressure distribution. The second-order shock-expansion meth

30、od has been developed toPredict the pressures on a noninclined body of revolution. In the following sc-tion this method will be extended to lifting bodies. Lifting Bodies For inclined bodies of revolution, a second-order shock-expansion method would involve not only a revised expression for the pres

31、sures, but, in addition, a revised approximation to the shape of the surface stream-lines. It is recalled from the results of Eggers (ref. 1) that, according to the generalized method, surface streamlines may be approximated by goedesics. For bodies of revolution, Savin (ref. 3) noted that the per-t

32、inent geodesics are simply meridian lines. While this result is exact for noninclined bodies of revolution, it is only an approximation in the case of inclined bodies. A refined approximation corresponding to a second-order method undoubtedly could be obtained by graphical integration of the momentu

33、m equations employing the pressure distribution given by the generalized method. However, it seems at present that this procedure would involve extensive calculations. If attention is restricted to bodies near a. = 0, it can be demonstrated that the deviation of the true streamlines from the meridia

34、n lines will not influence surface pressures. The approx-imation of meridian lines as streamlines can, in effect, be retained and relatively simple expressions can be obtained, therefore, for the initial slopes of the normal-force and pitching-moment curves. To this end, the expression for the norma

35、l-force derivative can be written5 dCN 2v-=_-j Ardx (iii.) dm AB 5The subscript, a = 0, has been omitted for simplicity of notation. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3527 11 where A is the nodimensional loading on a thin disk n

36、ormal to the body axis and having unit radius. This loading A is given by the equation A 2 p d(p/po) cos dcp (is) - 72 f 0 The problem then is to evaluate d(p/p0)/da The development given pre-viously which led to equation (3) also applies to bodies at infinitesimal angles of attack. Equation (8) als

37、o applies; however, the variables in this equation must be idds dependent on angle of attack. By differentiation of equation (8), there is obtained d(p/p0) - (1 - e) d(p/p0) + eT d(p2/p0) + (p - - dr p)e - (16) dct - dct dm C 2 dm This equation must satisfy the condition d(p/p0) d(p2/p0)d = at T = 0

38、 (i.e., x = x2 ). By the application of this condition to equation (16), the last term (involving dT/d.1) is eliminated. 6 The term, d(p2/po)./dm may be evaluated with the aid of the Prandtl-Meyer equation d(p2/p0) - 72 d(p i/P0 ) Pi 1 d(Hi)1 p2 1 d(H2) d.ct dm (11) Fern (ref. 13) has shown that the

39、 entropy (and hence the total pressure, H) on the surface of an inclined cone is constant (independent of cp). When equations (15), (16), and (17) are combined, then, the integrals of. the terms involving . d111/dm and d112/dcL will be zero (since fcos p dcp=0). Equation (15) may therefore be writte

40、n e) A d(P/p0) + d(p.1/p) cos p dcp (18) = yMO2 Jr (1 - dm i da. 0 The only terms in equation (18) that are functions of p are d(pc/po)/da. and d(pj./p0)/da These two terms may be evaluated in terms of the normal-dCN force derivative of the tangent cone, -tcx and in terms of A1 . After performance o

41、f the necessary manipulations, there is obtained SThis result indicates that the lifting pressures at small angles of attack vary in a manner analogous to that of the pressures at a. = 0.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-12 NACA TN 3527

42、 dCNI eA1 (i9) da A = (1 - el)(tan b) + It is apparent from equation (19) that dCNjdm for cones must be known before the loading A can be evaluated. Fortunately, results for and have been plotted for convenience in figure 2. The loading, A, may thus be calculated in the same manner as the zero-lift

43、pressures. In this case, A1, for the first corner is simply (tan v) dCNltcv*As before, a first approximation to A can be obtained with the two-step body. This approximation gives A = (1- eV)(tan + !.eta dCNIdct Itcx 7v v (20) In Appendix C, it is shown that equation (20) leads to very simple results

44、 With the loading, A, known, the normal-force derivative may be eval-uated by integration of equation (lii). In like manner, the pitching-moment derivative can be determined from the equation7 dCm-2it da Arxdx (21) Bad 0 A second-order shock-expansion method for bodies of revolution has been develop

45、ed to predict the pressure distribution and the normal-force and pitching-moment derivatives at a = 0. The results are relatively simple in form and may be applied to a given body with only a moderate amount of computations required. Simplified expressions based on an addi-.tional approximation have

46、 also been presented which further reduce the amount of work required. It should be noted, however, that open-nosed bodies and pointed bodies which produce shock waves other than the one at the vertex require special forms of the method. 8 The necessary equations 7The contribution to the pitching mo

47、ment of the variation in local axial forces with angle of attack is small for slender bodies (see ref. 15) and will be neglected throughout the present analysis. 81t may also be noted that boattailed bodies present a special problem dCN since neither Pc nortcx is defined in this case. In practice, h

48、ow-ever, it has been found by comparison with results given in reference 16 dCN that the use of = p0 and - = 2 gives reasonable results for bodies having moderate amounts of boattail.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3527 13 for these cases are contained in Appendix B. In addition, there are several restrictions on the present method which should be mentioned. First, it is apparent that if the exponential variation of the