CHAPTER XIII FLOW PAST FINITE BODIES

Transcription

1 HAPTER XIII LOW PAST INITE BODIES. The formation of shock waves in supersonic fow past bodies Simpe arguments show that, in supersonic fow past an arbitrar bod, a shock wave must be formed in front of the bod. or the disturbances in the supersonic fow caused b the presence of the bod are propagated on downstream. Hence a uniform supersonic stream incident on the bod woud be unperturbed as far as the eading end of the bod. The norma component of the gas veocit woud then be non-zero at the surface there, in contradiction to the necessar boundar condition. The resoution of this difficut can on be the occurrence of a shock wave, as a resut of which the gas fow between it and the eading end of the bod becomes subsonic. Thus a shock wave is formed in front of the bod when the incident fow is supersonic; it is caed the bow wave. When the eading end of the bod is bunt, the bow wave does not touch the bod. In front of the shock wave, the fow is uniform; behind it, the fow is modified and bends round the bod (ig. 7a). The surface of the shock wave etends to infinit, and at great distances from the bod, where the shock is weak, it intersects the incident streamines at an ange approaching the ach ange. A characteristic feature of fow past a bunt-ended bod is the eistence of a subsonic fow region behind the shock wave at the most forward part of its surface; this region etends to the bod itsef, and thus ies between the discontinuit surface, the bod, and a atera sonic surface (the broken curves in ig. 7a). The shock wave can touch the bod on when the eading end of the atter is pointed. The surface of discontinuit then has a point at the same pace (ig. 7b); in asmmetric fow, part of this surface ma be a weak discontinuit. or a bod with a given shape, however, this tpe of fow pattern is possibe on for veocities eceeding a certain imit; at ower veocities, the shock wave is detached from the eading end of the bod (see 3), even if the atter is pointed. Let us consider aia smmetrica supersonic fow past a soid of revoution and determine the gas pressure at the rounded eading end of the bod (the stagnation point O in ig. 7a). It is evident from smmetr that the streamine which terminates at O intersects the shock wave at right anges, so that the veocit component at A norma to the surface of discontinuit is the same as the tota veocit. The vaues of quantities in the incident stream wi be denoted, as usua, b the suffi, and the vaues behind the shock wave at the point A b the suffi. The atter are determined from formuae (89.6) and (89.7): γ ( γ ) p p, γ +

2 v c + ( γ ) ( γ + ) ( γ + ) ρ ρ. + ( γ ) The pressure p at the point O (where the gas veocit v ) can now be obtained b means of the formuae which give the variation of quantities aong a streamine. We have (see 83, Probem) and a simpe cacuation gives, γ /( γ ) γ v p p +, c ( γ+ ) /( γ ) /( γ ) γ + p p. (.) γ γ This determines the pressure at the eading end for a supersonic incident fow ( ). or comparison, we give the formua for the pressure at the stagnation point obtained for a continuous adiabatic retardation of the gas, with no shock wave (as woud be true for a subsonic incident fow): γ /( γ ) γ p p +. (.) or, the two formuae give the same vaue of p, but for > the pressure given b formua (.) is awas greater than the true pressure p given b formua (.). In the imit of ver arge veocities ( >> ), formua (.) gives p γ + p ( γ+ ) /( γ ) γ /( γ ) >, (.3) i.e., the pressure p is proportiona to the square of the incident veocit. rom this resut we can concude that the tota drag force on the bod at veocities arge compared with that of sound is proportiona to the square of the veocit. It shoud be noticed that this is the same as the aw governing the drag force at veocities sma compared with that of sound but et so arge that the Renods number is arge (see 45). Besides the fact that shock waves must be formed, we can aso sa that in supersonic fow past a finite bod there must be two successive shock waves at arge distances from the bod (L. Landau 945). or the disturbances caused b the bod at arge distances are sma, and can therefore be regarded as a cindrica sound wave outgoing from the -ais (which passes through the bod parae to the direction of fow); considering the fow, as usua, in a coordinate sstem where the bod is at rest, we have a wave in which the time is represented b / v, and the rate of propagation b v (see 3). We can therefore app immediate the resuts obtained in for a cindrica wave at arge distances from the This statement is true genera, and does not depend on the assumption of a po tropic or even a perfect gas in (.), (.). or, when a shock wave is present, the entrop s of the gas at O is greater than s, whereas if the shock wave were absent s woud be equa to s. The heat function is in either case w w + ( / ) v, since the quantit w+ ( / ) v is unchanged when a streamine intersects a norma compression discontinuit. rom the thermodnamic identit dw Tds+ dp / ρ, it foows that the derivative p ρ T <, i.e., an increase in s w entrop when w remains constant invoves a decrease in pressure, whence the resut foows.

3 source. We thus arrive at the foowing pattern of shock waves far from the bod: in the first shock, the pressure increases discontinuous, so that behind it there is a condensation; then foows a region where the pressure gradua decreases into a rarefaction, after which the pressure again increases discontinuous in the second shock. The intensit of the eading 3/ 4 shock decreases as r with increasing distance from the -ais, and the distance between / 4 the two shocks increases as r. Let us now eamine the appearance and deveopment of the shock waves as the number gradua increases. A supersonic region first appears for some vaue of ess than unit, as a region adjoining the surface of the bod. At east one shock wave occurs in this region, usua at the edge of the supersonic region. As increases, the supersonic region epands, and the ength of the shock wave increases. This is the shock wave whose eistence for has been demonstrated (for the two-dimensiona case) in ; it foows aso that the shock wave must first appear for <. As soon as eceeds unit, another shock wave appears, the bow wave, which intersects the whoe of the infinite wide incident stream of gas. or eact unit, the fow in front of the bod is entire subsonic. or but arbitrari cose to unit, therefore, the supersonic part of the incident stream, and consequent the bow wave, are arbitrari far in front of the bod. As increases further, the bow wave gradua approaches the bod. The shock wave in the oca supersonic region must intersect the sonic ine in some wa. We sha discuss the two-dimensiona case. The nature of this intersection is not et fu understood. If the shock terminates at the intersection, its strength fas to zero there, and the fow is transonic everwhere in the pane near the intersection point. The fow pattern in such a case must be given b the appropriate soution of the Euer-Tricomi equation. In addition to the usua conditions that the soution be singe-vaued in the phsica pane and the boundar conditions at the shock wave, the foowing conditions must aso be satisfied: () if the fow is supersonic on both sides of the shock (as when on the shock terminates at the intersection, being "supported" b the sonic ine), then the shock wave must be one that reaches the intersection; () characteristics in the supersonic region which reach the intersection cannot have an fow singuarities, since these coud arise on from the intersection itsef and woud therefore have to be carried awa from the intersection point. The eistence of a soution of > or shock waves formed in aia smmetrica fow past narrow pointed bodies, the quantitative coefficients in these reationships can be determined; see the second footnote to 3

4 the Euer-Tricomi equation satisfing these requirements seems to be not et proved. 3 Another possibe configuration of the shock wave and the sonic ine in the oca supersonic region is for on the sonic ine to terminate at the intersection (ig. 8b); the shock wave strength need not be zero there, and so the fow near it is transonic on on one side of the shock. The shock wave itsef ma have one end "supported" b a soid surface, and the other end (or both ends) starting within the supersonic stream (cf. the end of 5). 3. Supersonic fow past a pointed bod The shape which a bod must have in order to be streamined in supersonic fow, i.e., to be subject to as sma a drag force as possibe, is quite different from the corresponding shape for subsonic fow. We ma reca that, in the subsonic case, streamined bodies are those which are eongated, rounded in front, and pointed behind. In supersonic fow past such a bod, however, a strong shock wave woud be formed in front of it, eading to a considerabe increase in the drag. In the supersonic case, therefore, a ong streamined bod must be pointed at both ends, and the ange of the point must be sma; if the bod is incined to the direction of fow, the ange between them (ange of attack) must aso be sma. In stead supersonic fow past a bod of this shape, the gas veocit is nowhere ver different in magnitude or direction from the incident veocit, even near the bod, and the shock waves formed are weak; the intensit of the bow wave decreases with the ange at the front of the bod. ar from the bod, the gas fow consists of outgoing sound waves. The main part of the drag can be regarded as due to the conversion of kinetic energ of the moving bod into the energ of the sound waves which it emits. This drag, which occurs on in supersonic fow, is caed wave drag; 4 it can be cacuated in a genera form vaid for an cross-section of the bod (T. von Karman and N. B. oore 93). The nature of the fow just described makes it possibe to use the inearized equation (4.4) for the potentia: φ φ + β z where we have introduced for brevit the positive constant φ, (3.) v c β ; (3.) c the -ais is in the direction of the fow, the suffi denotes quantities pertaining to the incident stream, and / β is just the tangent of the ach ange. Equation (3.) is forma identica with the two-dimensiona wave equation with / v representing the time and v β the veocit of the waves. This is no accident; the phsica / significance is that the gas fow far from the bod consists, as aread mentioned, of outgoing sound waves emitted b the bod. If the gas at infinit is regarded as being at rest, and the bod as being in motion, the cross-section of the bod at a given point in space wi var with time, and the distance to which a disturbance is propagated at time t (i.e., the distance to the ach cone) wi increase as v / β. Thus we sha have a two-dimensiona emission of sound (propagated with veocit v / t β ) b the variabe profie. Using this "sonic anaog" as a guide, we can immediate write down the required epression for the veocit potentia of the gas, using formua (74.5) for the potentia of cindrica sound waves emitted from a source (at distances arge compared with the dimension of the source) and repacing ct b / β. 3 Germain has found severa tpes of soution of the Euer-Tricomi equation which might represent the intersection of a shock wave with the sonic ine, but the have not been at a fu investigated. Some of them do not satisf condition () above. igure 8a shows a case which might correspond to the termination of a shock wave forming the boundar of the oca supersonic region: at the point of intersection, the shock wave and the sonic ine both terminate and have a common tangent, and ie on opposite sides of it (the gas moves from eft to right). The fufiment of condition () has not been tested, however. On the possibe range of k has been determined (3/4<k</), but it is not known whether one can satisf the condition for the coordinates to be continuous at the shock wave in the phsica pane. See P. Germain, Progress in Aeronautica Sciences 5, 43, The tota drag is obtained b adding to the wave drag the forces due to friction and to separation at the traiing end of the bod. 4

5 Let S() be the area of the cross-section of the bod in a pane perpendicuar to the direction of fow (the -ais), and the ength of the bod in that direction; we take the origin at the eading end of the bod. Then βr v ( ξ) dξ φ(, r) ; (3.3) π ( ξ) β r the ower imit is taken as zero, since S () for < (and for > ). Thus we have compete determined the gas fow at distances r from the ais which are arge compared with the thickness of the bod. 5 Disturbances eaving the bod in a supersonic fow are, of course, propagated on into the region behind the cone β r, whose verte is at the eading end of the bod; in front of this cone we have simp φ (uniform fow). Between the cones β r and β r, the potentia is determined b formua (3.3); behind the atter cone (whose verte is at the traiing end of the bod) the upper imit of the integra in (3.3) is the constant. Both these cones are weak discontinuities, in the approimation considered; in reait, the are weak shock waves. The drag force acting on the bod is just the -component of the momentum carried awa b the sound waves per unit time. We take a cindrica surface with arge radius r and ais aong the -ais. The -component of the momentum fu densit through this surface is φ φ Π r ρvr ( v + v ) ρ v + r. On integration over the whoe surface, the first term gives zero, since the integra of ρ vr is the tota mass fu through the surface, which is zero. Thus φ φ πr Π r d πrρ d. (3.4) r At arge distances (in the wave region), the derivatives of the potentia can be cacuated as in 74 (see formua (74.7)), and we have φ φ v β ξ ) dξ β r π r. ξ βr This epression is substituted in (3.4), and the squared integra is written as a doube integra; putting for brevit β r X, we obtain ρ X X v ξ) ξ ) dξdξ dx. 4π ( X ξ )( X ξ ) The integration over X can be effected; after changing the order of integration, the integra is from the greater of ξ and ξ to infinit. We first take as the upper imit a arge but finite quantit L, which ater tends to infinit. Thus ξ ξ) ξ )[ og( ξ ξ) og 4L] dξdξ. ρv π The integra of the term containing the constant factor og 4L is zero, since not on the area S () but aso its derivative S) vanishes at the pointed ends of the bod. We therefore have or ξ ξ) ξ ) og( ξ ξ) dξdξ, ρv π 5 or aia smmetrica fow past a soid of revoution, (3.3) is vaid for a r up to the surface of the bod. In particuar, it again gives (3.6) for fow past a narrow cone. On the other hand, if this inear-approimation soution is considered at a great distance from the bod, a correction can be appied to it for the non-inear distortion of the profie, as was done in for a cindrica sound wave. This gives the strength of the shock wave at arge distances from a narrow pointed soid of revoution, incuding the dependence on, i.e., the coefficient in the aw of 3/ 4 damping ( r ) described in. See G. B. Whitham, Linear and Noninear Waves, New York 974,

6 ρv π ξ ) ξ ) og ξ ξ dξ dξ. (3.5) This is the required formua for the wave drag on a narrow pointed bod. 6 magnitude of the integra is v S The order of, where S is some mean cross-sectiona area of the bod. ρ S Hence ~. The drag coefficient for an eongated bod ma be conventiona defined, in terms of the square of the ength, as. (3.6) ρv Then, in this case, S ~ ; (3.7) 4 it is proportiona to the square of the cross-sectiona area. We ma point out the forma anaog between formua (3.5) and formua (47.4) for the induced drag on a thin wing; the function Γ (z) in (47.4) is here repaced b v ( ). On account of this anaog we can use, to cacuate the integra in (3.5), the method described at the end of 47. It shoud aso be noticed that the wave drag given b formua (3.5) is unchanged if the direction of fow is reversed: the integra is independent of the direction in which the bod etends. This propert of the drag force is characteristic of the inearized theor. 7 ina, et us brief discuss the range of appicabiit of this formua. This subject ma be approached as foows. The ampitude of osciation of the gas partices in the sound waves emitted b the bod is of the order of magnitude of the thickness of the bod, which we denote b δ. The veocit of the osciations is according of the order of the ratio δ : ( / v) of the ampitude δ to the period / v of the wave. The inear approimation for the propagation of sound waves (i.e., the inearized equation for the potentia), however, awas requires that the gas veocit be sma compared with the veocit of sound, i.e., we must have v β >> v /, or, what is in practice the same, / δ <<. (3.8) δ Thus the theor given above becomes inappicabe for vaues of comparabe with the ratio of ength to thickness of the bod. It is aso inappicabe, of course, in the opposite imiting case where is cose to unit and the inearization of the equations is invaid. PROBLE Determine the form of the eongated soid of revoution which eperiences the smaest drag for a given voume V and ength. Soution. On account of the anaog mentioned in the tet, we introduce a variabe θ such that ( cosθ ) ( θ π ; the origin of is at the eading end of the bod); and write 6 The ift (for a bod not aia smmetrica or a non-zero ange of attack) is zero in the approimation here considered. 7 It aso hods in the theor of the wave drag on thin wings given in 5. 6

7 the function f ( ) ( ) as n f A n sin nθ ; the condition S for and means that on terms with n can appear in the sum. The drag coefficient is then n 4 n π na The area S () and the tota voume V of the bod are cacuated from the function f () as S f ( ) d, V S( ) d. A simpe cacuation gives V π 3 A, i.e., the voume is determined b the coefficient A 6 aone. The minimum is therefore reached if A for n 3. The resut is 8 V 9π Sma,min. π 3 3 The cross-sectiona area of the bod is S A sin θ, and the radius as a function of is 3 therefore n / 8 V R( ) π 4 3 The bod is smmetrica about the pane /. 8. 3/ 4 [ ( )]. 4. Subsonic fow past a thin wing Let us consider subsonic fow of a gas past a thin streamined wing. As for an incompressibe fuid, a wing which is streamined for subsonic fow must be thin, pointed at the traiing edge, and rounded at the eading edge, and the ange of attack must be sma. We take the direction of fow as the -ais and the direction of the span as the z-ais. The gas veocit nowhere 9 differs great from the veocit v of the incident stream, so that we can use the inearized equation (4.4) for the potentia: φ φ φ ( ) + + z. (4) At the surface of the wing (which we ca ), the veocit must be tangentia; introducing a unit vector n aong the norma to the surface, we can write this condition as φ φ φ v + n + n + nz. z Since the wing is fattened and the ange of attack is sma, the norma n is amost parae to the -ais, so that n is amost unit, whie n and n z are sma. We can therefore φ φ negect the second-order terms n and n z, and repace n b ± (+ on the z upper surface of the wing and on the ower surface). Thus the boundar condition on equation (4.) is ± φ v n. (4.) 8 Athough R () vanishes at the ends of the bod, R' () becomes infinite, i.e., the bod is not pointed; the approimation undering the method is therefore not strict appicabe near the ends. 9 Ecept in a sma region near the eading edge of the wing, where there is a stagnation ine. 7

8 Since the wing is assumed thin,. φ on its surface can be taken as the imiting vaue for The soution of equation (4.) with the condition (4.) can easi be reduced to the soution of a probem of incompressibe fow. To do so, we use instead of the coordinates,, z the variabes ' ' z' z. (4.3) In these variabes, equation (4.) becomes φ φ φ + +, (4.4) ' ' z' i.e., Lapace's equation. The surface of the bod is repaced b another, ', obtained b eaving unchanged the profies of cross-sections b panes parae to the -pane, but reducing in the ratio a dimensions in the direction of the span (the z-direction). The boundar condition (4.) then becomes φ v n ± ' and it can be reduced to the previous form b introducing in pace of φ a new potentia We then have for 8, φ ' : φ ' φ. (4.5) φ ' Lapace's equation with the boundar condition v φ' ± ' n, (4.6) which must be satisfied for '. Equation (4.4) with the boundar condition (4.6), is, however, the equation which must be satisfied b the veocit potentia of an incompressibe fuid fowing past the surface. Thus the probem of determining the veocit distribution in compressibe fow past a wing with surface is equivaent to that of finding the veocit distribution in incompressibe fow past a wing with surface '. Net, et us consider the ift force acting on the wing. irst of a, we note that the derivation of Zhukovskii's formua (38.4) given in 38 is entire vaid for a compressibe fuid, since the variabe densit ρ of the fuid can be repaced in that approimation b a constant ρ. Thus ρ v Γdz, (4.7) where the integration is taken aong the span z of the wing. rom the reation (4.5) and the equait of the transverse profies of the wings and ' it foows that the veocit circuation Γ in compressibe fow past the wing is reated to the circuation Γ ' in incompressibe fow past the wing ' b Γ ' Γ. (4.8) Substituting this in (4.7) and changing to an integration over z', we obtain Γ ' dz' ρ v. The numerator is the ift force on the wing ' in an incompressibe fuid. Denoting it b we have ',

9 ' (4.9) Introducing the ift coefficients, ρ v z ' ' ρ v z (where, z and, z ' z are the engths of the wings and ' in the and z directions), we can rewrite this equation as '. (4.) or wings with arge span (and constant profie), the ift coefficient in an incompressibe fuid is proportiona to the ange of attack, and does not depend on the ength or width of the wing: constant α, (4.) ' where the constant depends on on the shape of the profie (see 46). In this case, therefore, (4.) can be repaced b where and ' (), (4.) () are the ift coefficients for the same wing in compressibe and incompressibe fuids, respective. Thus we have the rue that the ift force acting on a ong wing in a compressibe fuid is times that on the same wing (at the same ange of attack) in an incompressibe fuid (L. Prandt 9, H. Gauert 98). Simiar reations can be obtained for the drag force. Together with Zhukovskii's formua for the ift force, formua (47.4) for the induced drag on a wing is aso entire appicabe to compressibe fow. Effecting the same transformations (4.3) and (4.8), we obtain ', (4.3) where ' is the drag on the wing in an incompressibe fuid. When the span increases, the induced drag tends to a constant imit ( 47). or sufficient ong wings we can therefore repace ' b coefficient is () (the drag in an incompressibe fuid for the wing ). Then the drag () omparing this with (4.), we see that the ratio. (4.4) ' is the same for compressibe and incompressibe fuids. A the resuts given here are, of course, invaid for vaues of cose to unit, since the inearized theor then becomes inappicabe. 5. Supersonic fow past a wing If a wing is streamined in a supersonic stream, it must be pointed at both ends, ike the thin bodies discussed in 3. Here we sha consider on the fow past a thin wing with ver arge span, the profie being constant aong the span. Regarding the span as infinite, we have a two-dimensiona gas 9

10 fow (in the -pane). Instead of equation (3.), we now have for the potentia the equation with the boundar condition φ β φ ± φ, (5.) mv n, (5.) where the signs m on the right reate to the upper and ower surfaces of the wing, respective. Equation (5.) is a one-dimensiona wave equation, and its genera soution has the form φ f β) + f ( + β ). The fact that disturbances which affect the fow ( start from the bod means that above the wing ( > ) we must have f, so that φ f( β), and beow the wing ( <) φ f ( + β). or definiteness, we sha consider the region above the wing, where φ f ( β). The function f is determined from the condition (5.) b putting n ζ ), where ζ ( ) is the equation of the φ upper part of the wing profie (ig. 9a). We have βf ) vζ ), whence vζ ( ) f. Thus the veocit distribution for > is given b the potentia β v φ(, ) ζ ( β). (5.3) β + v Simiar we obtain, for <, φ (, ) ζ ( + β), where ( ) β ζ is the equation of the ower part of the profie. It shoud be noticed that the potentia, and therefore the other quantities, are constant aong the straight ines ± β constant (the characteristics), in accordance with the resuts of 5, of which the soution just obtained is a particuar case. The fow pattern is quaitative as foows. Weak discontinuities (aaa' and bbb' in ig. 9b) eave the pointed eading and traiing edges. In the regions in front of the This statement is vaid on in the approimation used here. In reait we have not weak discontinuities but weak shock waves or narrow centred rarefaction waves, depending on the direction in which the veocit is turned b them.

11 discontinuit aaa' and behind bbb' the fow is uniform, but between them it is turned so as to go round the surface of the wing; the fow here is a simpe wave, and in the present inearized approimation the characteristics are a parae and incined at the ach ange of the incident stream. φ The pressure distribution is given b the formua p p ρ v ; the term in v in the genera formua (4.5) can here be omitted, since v and v are of the same order of magnitude. Substituting (5.3) and introducing the pressure coefficient p, we obtain in p p ζ β) the upper haf-pane p. In particuar, the pressure coefficient on β ρv the upper surface of the wing is ζ ) p. (5.4) β Simiar, we find for the ower surface ζ ) p. (5.5) β It shoud be noted that the pressure at an point on the wing profie depends on on the sope of the profie contour at that point. Since the ange between the profie contour and the -ais is awas sma, the vertica component of the pressure force can be taken, with sufficient accurac, as the pressure itsef. The resutant ift force on the wing is equa to the difference of the pressures on the ower and upper surfaces. The ift coefficient is therefore see ig. 9a for the definition of 4 ) p p ) d ; β,. We define the ange of attack α as the ange between the chord AB through the ends of the profie (ig. 9a) and the -ais: obtain the foowing simpe formua: 4α α, and (5.6) (J. Ackeret 95). We see that the ift force is determined b the ange of attack, and does not depend on the form of the wing cross-section, unike what happens for subsonic fow (see formua (48.7)). Let us net determine the drag force on the wing (i.e., the wave drag, which is of the same nature as that on thin bodies; see 3). To do so, we must take the -component of the pressure force and integrate over the profie contour. The drag coefficient is then found to be β ( ζ ' ζ ) + ' d. (5.7) We put ζ ' θ α, ζ ' θ α, where θ ( ) and θ ( ) are the anges between the upper and ower parts of the contour and the chord AB. The integras of θ and θ are evident zero, and the resut is therefore or the profie shown in ig. 9b, for eampe, Aa and Bb' are rarefaction waves, whie Aa and Bb are shock waves. The streamine eaving the traiing edge (B in ig. 9b) is actua a tangentia discontinuit of the veocit (which in practice becomes a narrow turbuent wake).

12 4α + ( θ + θ ) ; (5.8) the bar denotes an average with respect to. or a given ange of attack, the drag coefficient is seen to be east for a wing in the form of a fat pate (for which θ θ ). In this case α. If we app formua (5.8) to a rough surface, we find that the roughness ma resut in a considerabe increase in the drag, even if the height of the irreguarities is sma. or the drag is independent of the height of the irreguarities if the mean sope of the surface, i.e., the mean ratio of the height of the irreguarities to the distance between them, remains constant. ina, we ma make the foowing remark. Here, as everwhere, when we speak of a wing we imp that its edges are perpendicuar to the fow. The generaization to the case of an ange γ between the direction of fow and the edge (the ange of aw) is quite obvious. It is cear that the forces on an infinite wing with constant cross-section depend on on the component of the incident veocit norma to its edges; in an idea fuid, the veocit component parae to the edges does not resut in a force. The forces acting on a wing at an ange of aw other than π / in a stream with ach number are therefore the same as those on the same wing for γ π / in a stream with ach number sinγ. In particuar, if but sin γ, the wave drag, which is pecuiar to supersonic fow, wi not occur. > < But nevertheess greater than the thickness of the boundar aer.