1、NASA TECHNICAL NOTE NASA TN D-6997CASE FSLCOPYSHOCK WAVES AND DRAGIN THE NUMERICAL CALCULATIONOF ISENTROPIC TRANSONIC FLOWby Joseph L. Steger and Barrett S. BaldwinAmes Research CenterMoffett Field, Calif. 94035NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. OCTOBER 1972Provided by I
2、HSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1. Report No.NASA TN D-69972. Government Accession No. 3. Recipients Catalog No.4. Title and SubtitleSHOCK WAVES AND DRAG IN THE NUMERICAL CALCULATIONOF ISENTROPIC TRANSONIC FLOW5. Report DateOctober 19726. Performi
3、ng Organization Code7. Author(s)Joseph L. Steger and Barrett S. Baldwin8. Performing Organization Report No.A-45199. Performing Organization Name and AddressNASA-Ames Research CenterMoffett Field, Calif., 9403510. Work Unit No.136-13-05-08-00-2111. Contract or Grant No.12. Sponsoring Agency Name and
4、 AddressNational Aeronautics and Space AdministrationWashington, D. C. 2054613. Type of Report and Period CoveredTechnical Note14. Sponsoring Agency Code15. Supplementary Notes16. AbstractProperties of the shock relations for steady, irrotational, transonic flow are discussed and compared for the fu
5、ll andapproximate governing potential equations in common use. Results from numerical experiments are presented to show thatthe use of proper finite difference schemes provide realistic solutions and do not introduce spurious shock waves. Analysisalso shows that realistic drags can be computed from
6、shock waves that occur in isentropic flow. In analogy to theOswatitsch drag equation, which relates the drag to entropy production in shock waves, a formula is derived for isentropicflow that relates drag to the momentum gain through an isentropic shock. A more accurate formula for drag based onentr
7、opy production is also derived, and examples of wave drag evaluation based on these formulas are given and comparisonsare made with experimental results.17. Key Words (Suggested by Author(s)Transonic flowWave dragShock waves18. Distribution StatementUnclassified - Unlimited19. Security Classif. (of
8、this report)Unclassified20. Security Classif. (of thisUnclassified21. No. of Pages4522. Price$3.00 For sale by the National Technical Information Service, Springfield, Virginia 22151“ “ c .-A.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NOMENCLATU
9、REA shock surface areaa speed of soundCp drag coefficientc airfoil chordD drag forceF function defined by equation (19)/ surface of integrationG function defined by equation (29)h fluid enthalpy/ isentropicM Mach numbercrit Mach number at which sonic flow is reachedn normal distanceP function define
10、d by equation (23)p fluid pressureq fluid velocityR gas constantr Mach number function defined by equation (22) or equation (C3)s specific entropyT fluid temperatureastastx coordinateinProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-y y coordinate5 w
11、edge angle or flow deflection angle7 ratio of specific heats6 angle between shock line and x axisp fluid densityT airfoil“thickness ratioSubscripts1 ahead of shock2 behind shock00 free streamcrit sonic velocity condition/ isentropicn normal componentRH Rankine-Hugoniot flowst stagnationx x component
12、y y componentIVProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-SHOCK WAVES AND DRAG IN THE NUMERICAL CALCULATION OFISENTROPIC TRANSONIC FLOWJoseph L. Steger and Barrett S. BaldwinAmes Research CenterSUMMARYProperties of the shock relations for steady
13、, irrotational, transonic flow are discussed and com-pared for the full and approximate governing potential equations in common use. Results fromnumerical experiments are presented to show that the use of proper finite difference schemes pro-vide realistic solutions and do not introduce spurious sho
14、ck waves. Analysis also shows that real-istic drags can be computed from shock waves that occur in isentropic flow. In analogy to theOswatitsch drag equation, which relates the drag to entropy production in shock waves, a formulais derived for isentropic flow that relates drag to the momentum gain t
15、hrough an isentropic shock.A more accurate formula for drag based on entropy production is also derived, and examples ofwave drag evaluation based on these formulas are given and comparisons are made with experimentalresults.INTRODUCTIONFinite difference procedures using both time-dependent formulat
16、ions and relaxation methodshave been developed to compute the steady, inviscid, transonic flow about arbitrary bodies. In mostof these techniques the flow is assumed to be adiabatic and irrotational - that is, isentropic - andshock waves, if they appear at all, are not strong. The assumption that th
17、e flow is isentropic leadsto considerable savings in computer algebra and storage, and for these reasons of efficiency, theisentropic assumption is quite useful in numerical computation. However, the implications of thisassumption in transonic flow are perhaps not fully appreciated. For example, eve
18、n though the flowis assumed to be isentropic, wave drag arising from shock “losses“ can be evaluated. This seeminglycontradictory result occurs because the isentropic shock relations - the permissible weak solutions(Lax, ref. 1) to the isentropic flow equations - do not conserve momentum in the dire
19、ction normalto the shock.Current relaxation procedures developed to treat transonic flow also require the isentropicassumption. Both time-dependent, finite-difference techniques and current relaxation proceduresallow isentropic shock waves to evolve naturally without the explicit use of sharp shock
20、conditions.Unlike the time-dependent schemes, the relaxation procedures do not attempt to follow characteris-tics in time in order to automatically maintain the proper domain of dependence. Instead, “proper“hyperbolic or elliptic difference formulas must be used, depending on whether the flow is sub
21、sonicor supersonic. However, while the concept of proper differencing in transonic flow has been exten-sively used since Murman and Coles first successful exploitation of the idea (ref. 2), it has not beenfully explored.Provided by IHSNot for ResaleNo reproduction or networking permitted without lic
22、ense from IHS-,-,-Both the concept of drag in an isentropic flow and the concept of shock formation can bestudied under guidelines suggested by the theory of weak solutions. Consequently, this paper beginswith the study of the isentropic shock relations as predicted by this theory. Several numerical
23、 ex-periments are reported for the relaxation methods which demonstrate that the differencing tech-nique is general and can give all possible solutions. A major portion of this paper is devoted to adetailed analysis of the drag mechanism in isentropic flow. From this analysis, a practical methodis d
24、eveloped for the evaluation of wave drag which does not require integration of surface pressures.Results from this technique are also presented.WEAK SOLUTIONS FOR TRANSONIC FLOW EQUATIONSConsider the equations of irrotational, inviscid, adiabatic flow for a perfect gas in twodimensionsdpq dpq- + = 0
25、 (la)ax ayby dxhsj. = constant5 = constantEquations (la) through (Id) may be combined with the equation of state of a calorically perfect gasto obtain two equations for the two dependent variables, the velocity components u and vj_Tl I , r _, -i 7-1 v= 0 (2)9“ _ iL = o (3)ay dxwhere _ .“ = la v =Pro
26、vided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-According to the theory of a weak solution (Lax, ref. 1, and Lomax, Kutler and Fuller, ref. 3)for hyperbolic systems, solutions of equations (2) and (3) may be discontinuous across a smoothcurve (which may
27、 be a shock wave) and constitute a weak solution if they satisfy the relationstan0 7-1-i 7-11 -7-1-, 7-17-17-11 -7-17-1(4)and( M! - w2 ) = -tan 0 nj - v2) (5)Here d is the angle between the shock wave and the x axis, while the subscripts 1 and 2indicate values before and behind the shock wave. Note
28、that equations (4) and (5) permit a con-tinuous solution, ul = 2 and Vi = v2, as well as the discontinuous solution. Equations (4) and(5) pertain to isentropic flow and admit solutions analogous to the Rankine-Hugoniot relations.The discontinuous solution of the flow conservation equations of mass,
29、momentum, andenergy1 is given by the Prandtl relation, but a corresponding exact closed-form, discontinuoussolution of equations (4) and (5) has not been found. Across a normal shock, equations (4) and (5)reduce to7-1 7-17-1U2 (6)while for Rankine-Hugoniot flow the Prandtl relation for a normal shoc
30、k is7+ 1 (7)Here the flow described by these equations is referred to as Rankine-Hugoniot flow. In Rankine-Hugoniot flow, entropyis not conserved across a shock plane, and the Rankine-Hugoniot equations are satisfied across any arbitrary plane in the field.An alternate flow, for example, could be de
31、scribed by the conservation equations of mass and momentum (the Euler equations)and conserve entropy in place of energy across a shock wave.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-A numerical evaluation of equation (6) is compared to equation
32、 (7) in figure 1 for the reference Machnumber M* = q/a*. Figure 1 shows that throughout much of the transonic range the two relationsagree well. Nevertheless, there are important differences. The possible expansion shock solution canno longer be excluded by the second law of thermodynamics as it is
33、in the case of Rankine-Hugoniotflow. Also, through an isentropic shock wave, mass, energy, and entropy are conserved, but, trans-verse momentum is not conserved. For example, at M = 1.4, the one-dimensional momentumequation has the difference across the shock ofPi = 0.0301P*t Pst Pst PstBecause mome
34、ntum is not conserved across the shock, equation (1) contains a mechanism for dragproduction.It is also a matter of interest to examine the shock relations for the transonic small perturba-tion equations. Consider, as a representative example, the small perturbation equation of Guderley,(ref. 4):3/2
35、 _ 2/7+l(9bwherea* -Across a normal shock wave the jump relation for the Guderley equation is(u, a*)2 = (u2 - 0*)2 (10)Note that this relation, illustrated in figure 2 in terms of M*, has a closed-form solution and closelyapproximates equation (6). The Guderley equation is not a valid approximation
36、for subsonic, low-speed flow.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-As a final example, consider the small perturbation equation of Spreiter (ref. 5)(l - _ _qox dv _ T -r U (11)du _ dvdy dx (12)In this instance, the shock relation is a funct
37、ion of the free-stream Mach number and is given for anormal shock by2(1(7- (13)Figure 3 illustrates this relation for various choices of the free-stream Mach number. Clearly, thisjump relation should not be used at lower free-stream Mach numbers if one intends to approximateRankine-Hugoniot flow.The
38、 transonic small disturbance equations conserve mass, momentum, and energy only tolowest order. Consequently, unlike Rankine-Hugoniot or isentropic flow, it is not clear that only asingle mechanism produces wave drag. The normal shock relation for the Guderley equation (10) andthe Spreiter equation
39、(13) can also be obtained by an expansion of the Rankine-Hugoniot relations,and this approach is taken in references 4 and 5.PROPER DIFFERENCING SCHEMESThe theory of a weak solution shows that the isentropic equations permit a possible discon-tinuity, which might be an expansion or a compression, an
40、d it is necessary that the numericalmethod be able to give these solutions. Here we are concerned with only the relaxation schemes andsurvey results, which demonstrate that proper difference schemes do in fact permit all possiblesolutions.Murman and Cole (ref. 2) first demonstrated that shock waves
41、can be established in the relaxa-tion schemes if upwind (i.e., backwards) differencing formulas are used in the supersonic regions.This is the correct hyperbolic differencing scheme in the sense that it marches away from an initialdata plane, and downstream influences cannot propagate upstream. Shoc
42、k waves, when they form,appear where characteristics of the same family begin to coalesce in the supersonic flow. In a sub-sonic flow region, central difference schemes are used and these correctly bring in information fromall directions this is proper for elliptic equations.Provided by IHSNot for R
43、esaleNo reproduction or networking permitted without license from IHS-,-,-Results obtained by using “proper differencing“ are in good agreement with the jump pre-dicted by the weak solution. The supersonic flow about a wedge as computed by relaxing theGuderley equations illustrates the capturing of
44、the proper jump in figure 4. In this example, theequations were relaxed by the interchange algorithm of reference 6. In the more complex cases oftransonic flow about airfoils, the numerical results predicted by proper mixed differencing areconsidered to agree well with experiment (see e.g., ref. 7).
45、 However, these solutions usually do notshow a shock jump of the proper strength because a rapid expansion persists in the subsonic flowimmediately behind the shock that is not resolved in a relatively coarse finite difference grid. Amore detailed discussion of this flow phenomenon is given in refer
46、ence 8.The differencing schemes also permit multiple shocks to appear. An example of this flow isshown in figure 5, and again these results are plausible since similar results are found experimentally(see, e.g., ref. 9). The locations of these shocks have also been numerically tested and have beenfo
47、und to be fixed and independent of the path along which the solution was relaxed.Numerical experiments with shock-free profiles show that the discontinuous solution is notspuriously forced into the flow field by the finite-difference procedure. Figure 6 illustrates the con-tinuous surface pressure d
48、istribution about a thin “sine wave profile“ in supersonic flow, which wasfound by means of the Guderley equations, and figure 7 illustrates the transonic flow found about ashock-free Nieuwland profile (ref. 10) using the method of reference 7. The very weak shocks thatdo appear in this latter solution are not attributed to the finite-difference procedure but to nu-merical truncation error and imperfection in describing the profile in the finite-difference network.The theory of a weak solution also predicts the existence of an expansion shock, and yet thismathematically correct solution is ne
