Problems in Transonic Flow Citation Cole, Julian David (1949) Problems in Transonic Flow. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/QS66-WA09. https://resolver.caltech.edu/CaltechETD:etd-04202004-105506 Abstract No Abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Aeronautics and Mathematics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Minor Option: Mathematics Awards: Caltech Distinguished Alumni Award, 1971 Thesis Availability: Public (worldwide access) Research Advisor(s): Lagerstrom, Paco A. Group: GALCIT, Caltech Distinguished Alumni Award Thesis Committee: Lagerstrom, Paco A. (chair) De Prima, Charles R. Liepmann, Hans Wolfgang Defense Date: 1 January 1949 Record Number: CaltechETD:etd-04202004-105506 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-04202004-105506 DOI: 10.7907/QS66-WA09 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 1420 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 23 Apr 2004 Last Modified: 14 Jun 2023 23:08 Thesis Files Preview PDF (Cole_jd_1949.pdf) - Final Version See Usage Policy. 1MB Repository Staff Only: item control page

PROBLEMS IN TRANSONIC FLOW Thesis by Julian D. Cole In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1949 TABLE OF CONTENTS 1. Introduction and Summary 2. Flow Past a Thin Body 2el Bee Bed 2e4 ed 206 eT 208 General Equations Approximations for Flow Past a Thin Body Discussion of Approximate Equations Laws of Similarity Potential Flow Properties of Flow in a Local Supersonic Zone Drag of Transonic Airfoils Determination of Height of Supersonic Zone 3e Flow Through a Shock Wave Sel Set Derivation of Equations Transonic Flow Through a Shock Wave Page ll 15 26 33 45 50 50 1. INTRODUCTION AND SUMMARY The main problems studied in this thesis deal with transonic flow, flow where the local speed of the gas is close to the local speed of sound. Such problems are difficult because non-linearity of the mathematical problems must be considered if the solutions are to resemble reality. Non-linearity enters the problem in different ways and several of these are considered here. The main aim of the research was a description of the flow past an airfoil through that range of Mach numbers where local supersonic zones and shock waves form on the airfoil. This problem cannot be regarded as solved. However, some progress has been made in that direction. When gas streams past a curved surface the flow speed may ex- ceed the local speed of sound. The non-linearity which enters here is in the acceleration terms like a se and in the variation of the sonic speed and this is balanced by the change of area. When gas flows through a normal shock wave the flow speed also passes through the sonic speed. The non-linearity which is important here is the acceleration term 4 aL which tends to steepen the wave front and which, in this case, is balanced by the viscous stresses. When shock waves occur on an airfoil both effects are important, as well as the effect of the viscous layer near the boundary. This boundary layer is omitted in the present work although its effect may, in certain cases, be vital. A proper study of the boundary layer and its interaction with the "outer flow" depends on an understanding of the non-linear phenomena already mentioned. In the first part of the work some general equations for tran- sonic flow are derived. A law of similarity for these equations is presented and some comparisons made with experiments, Then, omit- ting shock waves, the first non-linear effect is studied in the boundary value problem for transonic potential flow past an airfoil. All the problems studied deal only with the local supersonic zone itself and some conclusions are reached. However, not much could be done about the important problem of the global flow picture, es- — pecially the location of these supersonic zones in arbitrary cases. The main mathematical problems which occur in the first part are questions of the existence and uniqueness of solutions for a non- linear equation of changing type. This has seldom been treated in the literature. The question of the height of the supersonic zone and the drag of a transonic airfoil is also treated briefly in the first part. In the second part the non-linear effects in shock waves are studied. It is shown approximately how a shock wave approaches a steady state due to the non-linearity. It was planned then to study the presence of both types of non-linear effects for the flow past an airfoil but not too much progress has been made yet along these lines. I would like to thank Professor P. A. Lagerstrom, who acted as supervisor, and Professors C. de Prima and H. W. Liepmann for their kind assistance. I would also like to thank the Transonic Research Group, especially H. I. Ashkenas, for cooperating in ob- ' taining experimental results. 2 FLOW PAST A THIN BODY In this section some problems in steady flow are considered. The first aim is aderivation of approximate equations suitable for transonic flow and a check on their approximation. Next the pro- perties of these equations are considered in some detail. Some viscous effects are also studied. 2el General Equations The fundamental equations are statements of well-known physical principles. The starting point of the discussion here is the state- ment of the conservation of momentum in the Navier-Stokes equations. Hence, the assumptions in the derivation of these equations are im- plicit in the results. We consider, at first, steady, two-dimensional flow and later some remarks will be made about three-dimensional flow. The momentum equations for the x and y directions ey be written: ge 2 VA “int ae —p Rp a Ve Keg] ts Asie bY (2.11) ye 2 w 2,4 ),2 7, a hae ee glts Lh 5 seep (b) where i] i] velocity component in x-direction d " velocity component in y-direction pressure density viscosity SOS The gas is also assumed to obey the perfect gas law Le gh 7 (2012) where 7 ce , R = gas constant = Cy ~ Cy ar] | absolute temperature Sp specific heat at constant pressure Cc, = specific heat at constant volume tt In addition the conservation of matter expressed by the continuity Dieu) o(av) see + Gg =0 (2413) The first step will be to derive an equation of motion of a com- equation pressible viscous fluid in a form suitable for discussion. Upon multiplying equation (2.lla) by u and (2.11b) by v and adding, one obtains, a? rar (go ae er ped peg) gh 7 58 Ue Fy a) 95 Z1/ «A fld ppl tap $l Ld - fave 6g iP 5 sd) (2.14) The expression involving the pressure gradients may be eliminated from (2.14) by considering the equation for the conservation of energy and using the continuity equation. Assuming that no external heat is added to the system consisting of all the fluid, the energy equation is “ego =- (pr +4] 7 EAE C427) + ag leg l/ (2.15) where E = internal energy = of k = specific thermal conductivity, and x ue [2l&/* (iy) * (3 '[- gp (ee. ° (2.16) Equations (2.12), (2.13), and (2.15) imply, assuming Cy and cy are the dissipation function HT g constant and writing oor ’ ¥ ACB E)= 2 WA, a Ef) spl # a) (2.17) , Hence, writing Qre Oe » the velocity for an adiabatic disturbance, / and using equation (2.17), equation (2.14) may be transformed to bE El: ay +t /+ bf) oy "74 mae 5 zy? Gl a gl] - Sea" Lay bs ie] * gegl) seth PB ES pvipll tide (2.18) The left hand side of the equation above represents the dynamic terms as usually written while the right hand side represents the effect of viscosity and heat conduction. No problems for (2.18) will even be set up but simplifying assumptions will be made immediately. 2e2 Approximations for Transonic Flow Past a Thin Body In this section approximate equations applicable to a special type of transonic flow will be developed. Specifically, the theory will be restricted to the flow past a thin airfoil which may cause a local supersonic region in a basically subsonic flow. In general there will exist shock waves in this region. The most important parameter affecting the nature of the flow field at a point is the local Mach number, or the ratio of the flow speed to the sound speed at the point. For most of the flow field the local sound speed and the flow speed can be simply related by the well-known Bernoulli equation for compressible fluids x2 atte? | a = enol «= & we “2 x-/ 2 xX~/ (say) (2.21) In all cases a* is well defined from conditions given at upstream infinity so that a* = speed at which the local flow speed and sound speed are equal. Equation (2.21), a statement of the conservation of energy per unit mass, is derived by integrating equation (2.15) along a streamline in a region of the flow field where the dissi- pative terms are not important. Hence equation (2.21) can be written in another form arpy? w? + eT ae Zhe Ftg7 (2.218) Transonic flow is defined here as flow in which the local flow speed is everywhere close to a*, The basic flow is uniform, di- rected along the x-axis and the body, close to the x-axis, is assumed to cause small disturbawces. The assumption of transonic flow and a thin body may be introduced by restricting the velocity components so that # My g) = a woe ty) Vey) =. rng) (2.22) u! v’ where aX, at K/ Using equation (2.21) and neglecting terms of higher order the Mach number M can easily be expressed as a’ Mi; = Kiiz (2.23) a The most important terms on the left hand side of equation (2.18) may be isolated by considering the character of flow near M=1 past a thin body. An essential property is that in general # >? iy iy & - This property may be understood by consider- ing the rapid growth of a supersonic zone in the y-direction as the Mach number at infinity is increased. In another way, in the limit as M,—* 1 in supersonic flow, the linearized theory (Ref. 1, pe 148%) indicates that the disturbance is propagated practically undamped to infinity in the y-direction but is restricted to a small width in the x-direction. Furthermore, shock waves in transonic flow are necessarily weak. It is well-known that along any streamline the change in entropy through a normal shock is proportional to (a - 1)° where M, = Mach number ahead of the shock (Ref. 1, pe 41). Hence for transonic flow the change in entropy depends on [gy « This means that difficulties due to changes in entropy can be neglectéd to the second order in A and outside of shock waves and boundary layers, the flow can be considered irrotational. Specifically, Crocco's vortex theorem (Ref. 2) implies au" a As Lf ¥ OY ax dy ty S = entropy = SIREAN FUNETION so that da’ _ a” x - F (2.24) Equation (224) also implies the existence of a perturbation vel- ocity potential PAY! » such that 1. OF “us Ox oy (2025) / Vi= iY The terms on the right hand side of equation (2.18) are of course unimportant in general except in shock waves or boundary layers. As a further simplification only one type of viscous effect will be considered, namely shock waves. Recent research (Ref. 3) has shown that longitudinal waves (shock waves) and transversal waves in a viscous fluic may to a certain extend be uncoupled and treated sep- arately. The boundary layer represents to a large degree a trans- versal wave spreading into the fluid. For this work the boundary layer will be omitted entirely but it should be remembered that 10 both theoretical and experimental work (Ref. 4) has shown a strong local interaction. Now make the following assumption about the structure of the shock. Viscosity and heat conduction effects are considered to be important in a region whose dimension A in the x=direction is much less than its dimension A in the y-direction. (A<e£k ). Since in this region / vm a? — »* &" a fut OS) aa + Ae} 2 a’ + a’ gicats OS) te hoe OCG] Then, in the right hand side of equation (2.18) all the terms of the dissipation fimction v4 and all of the other viscous terms can be seen to be much less than Pipe) ~ £ L (£) ye ax ax In addition, since across the shock the change in T (see Eq. (2.21la) 2 ¥ / . A7~ du ~ da Ak” 4 temperature term is seen to be of the same order of magnitude, x af fea? tw! Qf 27) ~ ee cls) at ox ax fv Then, under all these assumptions and retaining only the dominant terms, equation (2.18) can be written ¥ = —_— —y 2ec6 at 4x A Fata” ax? foe ax* ‘ Summarizing, equation (2.26) may be considered a valid approximation under the following assumptions: il i) transonic flow ii) thin body iii) Navier-Stokes equation valid iv) flow outside boundary layer v) no heat addition vi) shock wave restricted to a narrow region. Heat conduction has qualitatively the same effect as viscosity and that is now also omitted for purposes of simplification. Then omit- ting the primes the basic system, under the assumptions above, is YY x“ ou av _ Py, Die at dx wy ~~ xt (a) (2027) yg ox ~ (pb) where yr —_ reg In the case of shock free flow the system reduces to Y4/ oe ~~ av -o Ax “ox YY (a) ge Weg ay ox (b) (2-28) or if a potential is introduced H/ 9 ID _ oY a by 3x* ay* =0O (229) 2e3 Discussion of Approximate Equations In this and the next section it is shown that the approximate 12 equations (2027 = 2.28) should be adequate for a description of transonic flow phenomena under the assumptions made. This is done in two ways, first by discussing briefly. the properties of these equations and secondly by deriving a law of similarity which is compared with experimental results. The significance of (2.28) can be appreciated better by a com- parison with the equation used in linearized theory of supersonic and subsonic motions (2.31) where za Z the Mach number of infinity, a constant. It should be noted here that usually 2” is taken to vanish at infinity, at least upstream, while sr does not. The comparison between (2.28) and (2.31) can be made more striking by rewriting (2.29) with the aid of (2.23) as z P) _ ; /- HW) §. 2” * ay" = 0 (2.32) where, of course, N= May) Now the Mach lines or characteristics of equation (2.31) are given by the lines whose slope and are real if the basic flow is supersonic. This is typical of a 13 hyperbolic equation; they are imaginary everywhere typical of an elliptic equation for OX Gy « Partially supersonic and partially subsonic flow cannot be described by the linearized theory. It is clear that an equation for transonic flow must be non-linear like (2.29). The characteristics or Mach lines of such an equation must be real or imaginary depending on the magnitude of the velocity, i.e. the equation must be able to change type, depending on the velocity. The characteristics of equation (2.29) or (2.28) have the differ- ential equation GAA Vg 2 - By, (2.33) / WY Fx The characteristics are evidently real if Ho and imaginary if 3? <Q? , as they should be. The non-linear equation (2.32) is thus the simplest potential equation which retains the essential features of transonic flow. Linear theory applies only locally in mixed flow. Of course, the slope of the characteristic at any point is just the Mach angle /3 for lins3 = » as can be seen from (2033). Another interesting contrast with the linearized theory is pro- vided by the fact that any individual stream tube has a throat, as it should at M=1, in the transonic theory, it cannot have such a throat in the linearized theory. This is essentially a consequence of replacing 1 - Me by 1 - ME, in the one-dimensional continuity equation aS Hi; A -. 14 where A = area of stream tube w = velocity along stream tube It is also related, in subsonic flow, to the representation of com- pressible flow as an affine transformation of incompressible flow in the linearized theory. . Thus we have shown that qualitatively the approximate equation should be valid. Next we can indicate some quantitative results both theoretical and experimental. First we can develop the relation- ships which are valid for simple waves (Ref. 5, pe 59) in supersonic flow as given by (2.28). These may be found by seeking the solutions of (2.28) for which u is a function of v alone as Srv] (2.35) (2.48) becomes 2 so that wey ff'= / . a and 4 oA ¥-CL2YVR * (2436) This agrees with the hodograph epicycloid for supersonic simple waves (Ref. 5) when the Mach number is close to 1 and the deflection angles are small. Since, in a sense, a shock free local supersonic 15 zone can be represented as a construction with these simple waves we conclude that the approximate equation (2.28) should quantita- tively describe the flow in such a supersonic zone. This will be discussed in more detail later. Next we consider the system (2.28) in the large and make comparisons with experiments based on definite problems. 2.4 Laws of Similarity This is done by the transonic similarity law. Laws of similar- ity are important for several reasons. They contribute to a better understanding of the problem by showing the relation between the various important parameters. They are also needed for comparison of experiments performed under different conditions. Laws of simi- larity were first given by Karman (Ref. 6) and Guderley (Ref. 7), and some aspects of the problem have been discussed by Graham (Ref. 8). The derivation of the law of similarity can be regarded as an attempt to see what flows can be related to an already known flow by a class of linear transformations. Suppose that of % gid is the known velocity potential for flow past a given body whose shape is x Yo? Z £ (Ee) (2.41) where 7 =1/2 thickness; f (0) =1 f, = shape function 1/2 chord. 16 The origin is taken at the 1/2 chord point of the airfoil and the subscript 1 is used to indicate quantities connected with the "known" problem. It is assumed that the body is symmetric about y =0. Then assuning shock free flow AN fi) satisfies the equation (2629) (2.42) and the following boundary conditions may be taken: (i) Linearized boundary conditions of flow tangent to the body af, = _ =O where the prime denotes differentiation, (ii) Uniform flow, parallel to the x-axis, at infinity + 24 _% (/- 45 / The potential cA is assumed to be determined uniquely by these conditions.* Now, let us determine the constants A, B, C which will relate a second potential Y, to the known potential Y in the following way: 2s , poh cA to) = ABO fi (2.45) *This is a touchy point for mixed flows which is discussed later. However, as applied, visual evidence indicates flow of a similar nature. 17 ¥%2 = LX ae Let us see what problem is solved by the new potential % . Corresponding points are defined by We have ” g* If ry (ne ss ox” : A ox” (2.46) and - so that (2.42) becomes 2 2 52 20 C vA Yj) B ARE $h_ Ff Seo In order that % satisfies an equation similar to that satisfied by g¥ (2.42) it is necessary that 4,* @¢? (2.48) g% may be considered to describe flow past a second body related to the first by a transformation of the boundary condition at (yy = Yeo =0)- Then (2.43) transforms to ~ x p) a ae t+ af/ 2) « Cat 2 fh eo=0 Bef Bo Vg tel= FLEE gp or 4. 4 & o AVE A. @ B45 a/ 42) if %,= Z L1F */ gives the new body shape. In addition the flow 18 at infinity must be altered. Equation (2.44) becomes # a* oy FA BONE) =- 2 fr-he) Om = 4A da, (vey) A F+/ y or #* 2 A a, 1g EG G* 4B (2.410) Equations (2.48), (2.49) -and (2.410) give the conditions for simi- larity. They tell how the potential, space coordinate, body shape, Mach number at infinity, and total enthalpy can be altered to pro- duce similar flow; of course comparisons can be made in various ways. These conditions having been satisfied corresponding points (x15 Xx) and (x95 Yo) are defined and the properties of the two flows at these points can be related. For example, Qt 1M %s pr) _ A g* Grp) © (2.411) In general, the two flows will have different directions and magni- tudes at the corresponding points. Now, for simplicity, consider the case of ag =a , B=1. The constants may then be determined as _ Mowe C= of Lhe A= —T2 ? /-14,7 /-Shee 29 and the condition for similar flows then demands J > “Ae te ) a 2 & 4 le/ = Te St LAL J- Ves ¢ 2 4 E/ Fron this, two important points may be noticed. First, since oe (2.412) 19 are constants, comparisons can only be made when the slopes of the bodies at corresponding points are in a constant ratio. This can be done when the slopes depend on the same power of x. Second, a simultaneous variation of Mach number and body shape must occur if any comparison is to be made. Picking the same body necessitates the same condition at infinity and the comparison is trivial, i.e. intentical flow pattern.e . Then a simple set of bodies that can be compared are aifoils of the same shape functions, i.¢. 4VG Big 4 Ue Ui and of the same chord G=O . If the condition for similarity (2.412), is satisfied % 2 Z 1-/hia Za A= 2 = {= (2.413) ‘7 M20 / corresponding points are given by x, =H; /- 1H (2.414) Peo TH 3 and then the Mach numbers at corresponding points are related by 2 12 MGA) /- Nae = we 20415 1M po) lag (26415) Summarizing, equations (2.413), (2.414) and (2.415) can be taken as the law of similarity for a case of (i) flow outside viscous region (ii) a = a0 » same total enthalpy (iii) o c same chord 1 2’ (iv) £(#) = 4 oe) » same shape function The transonic similarity as developed in (2.413 = 2.415) can be checked against experiments in several ways. Experimental meas- urements (Ref. 4) of the Mach number decay above the maximum thick- ness of 6% and 12% circular are airfoils and the points are plotted in Figures 1 and 2. By regarding the flow past the 6% airfoil as imown ( 4, ), decay curves for the 12% airfoil ( @ ) can be com= puted. The details of the computation are given below. The conditions for the similarity law as developed in (2.413- 2.415) are met. ay = a5 9 C7 =S al 1/2" and in accordance with the linearized boundary conditions £ [P) =b [g)--28 for circular arc airfoils. However since Now is not well-defined in the wind tunnel, the two flows are linked by comparing the Mach numbers at the points of maximum thickness. This can be done since the origin remains fixed in the transformation. Then A= - (2) (2) BV 6b « 1 My fo) Che Ml, gi) 4 “Mace The corresponding points are given by az = x, / 1-Shes a\? p22 Vas = iilZ] = 0794 Ay Curves were faired through the experimental points for the 6% air- foil and values of M, and y, Were read off. Values of My and Yo on the decay for the 12% airfoil were then computed. SYN GYOHD JO % -—~ JOvVsNNS WOsyIv WOYS FJONVISIC WOLLYUBA oz! oll ool oe OZ o9 OS Or o€ oz ol 0 T Se‘ 060 ~- 41660 001 1SO'l L ~ | ~ se r ae ee ins ~ . 6880=2% © vpi6'O= "NW e@ ° 9e60= Wo WosdIV YW, € © SS3NWOIHL WOWIXYN 3A08¥v AVD30. H3GNNN PS60 = -W HOVA. HOVAR 77307 Two Y3SNNN 2. Bane CRON) 20 -% ~ SWANS "Noss Woad FONVISIO WwoLISA ol OA & oO one) OS ov Of Oz oy) fo) Ean Sectian ARENT Pa pep ; 0@O S890 © eee SO) - ety Key -%% « on Os SSANND OBiVg —- : : ‘ Wars KE ws SSLO=*W ® | KY O3LOAWOD B7ZO="\A @ 7 (eae eVBO=A sh perp ye --4-0 aS8-0 =F ° oe “SBAMND £C— =. GLB OX WN o. ae ia Ses O-"W @ : ae sLnaWod WANBWIURd%S — — \ coe NOSUIW %e Ue . SSaNADIHL WOW SAOSV AwoSd BaEWNN: HON ROWIN Tae ASSWON ia . SUPER SONIC OF HEIGHT ZONE - oT fb 0.5 230 SUPERSONIC ZONE HEIGHT BICONVEX CIRCULAR ARC AIRFOILSs ® 312% AIRFOIL ° 3°* 6% AIRFOIL @ COMPUTES BY SIM- iLARITY FROM 6% AIRFOIL DATA. -a-H--- APPROXIMATE EQUA- ‘TION (SB 2B) “1.00 1.09 tO AAS Lze MACH NUMBER AT MAXIMUM THICKNESS ~ Me FIGURE 3 24 The computed curves are plotted in Figure 2 and show good _ agreement with the experimental decay curves. The height of the supersonic zone over the maximum thickness was checked in the same way and is given in Figure 3. Extension of Transonic Similarity to Flow with Shock Waves The similarity law developed in (2.48 - 2.410) can be extended to flow with shock waves, under the assumptions previously mentioned, by considering (2.27). If the potential is introduced we have yw 9P 2% ay _ = a at ox Fer ay? ~ axe (2.416) Then, following (2.45) we obtain, in addition to (2.48-2.410), the condition that 5.7 A gt 7° (2.417) Thus if the viscosity may be varied comparisons may be made in various ways, as before. However, in the usual case * * A , 4 "29% (2.418) so that it is necessary that A= 1. This implies that 2 2 1- Mw (thw —— = ——2 & 4 b= [Mag of x 7 Mag 7 (26 10) a distortion of the x-direction is necessary for similarity. Also, in the usual case the shape is given by the same function so that the chord must be correspondingly distorted. This is needed since av? y =o at y = 0 ahead and behind of the airfoil. Thus 25 2 o Lhe 1) —-— ee 3 a 7 Mage (2.420) so that Xz ey Then | 2 C = a9 & /-Aag (2.4422) Finally the condition for similarity (2.48) shows De (ts 1 Mes (2.423a) G (Nig ones or Gey = (4 - Ae /- Mane fe, (2.423b) Corresponding points are now defined and properties at these may be compared. The law of similarity (2.423b) is the same as before (2.413), but it should be remembered that the length scale in the flow direction is altered. This distortion is consistent with the possibility of introducing a characteristic length, e.g. x ’ when a real fluid is considered. Some similarity of thé ratio of shock wave thickness to chord is needed. Experiments have not yet been carried out which permit an evaluation of this law. If any comparisons are made, however, the state of the boundary layer will also have to be taken into account (Ref. 4). Further extension of the similarity law to three-dimensional 26 flows and axially symmetric flows can be made and the results are qualitatively the same. They seem to emphasize the essential non- linearity of the problem. They also indicate that the transonic approximation should give quantitatively correct results for a reasonable range of Mach numbers close to M=1, It was also seen that the non-viscous behavior should hold, at least up to the maximum thickness of the airfoil. 2e5 Potential Flow It was shown in the previous section that potential transonic flow should have some significance. Experiments (Ref. 4) have in- dicated a regime in which shock-free transonic flow exists. There- fore it is desirable to find solutions for transonic potential flow, with local supersonic zones past given bodies and to discuss their meaning. This is difficult for two reasons. First, we are dealing with an equation which is partly elliptic (subsonic ) and partly hyperbolic (supersonic) and the proper prescription of conditions to determine a unique solution to the mathematical problem is now known. Also the existence of such a solution is an open question. Secondly, the equations are non-linear so that representations cannot be built up on @ superposition principle. Some solutions have been given however (Ref. 9), but these all determine the body shape after start- ing with a known solution in the hodograph. Hence they can never discover, except perhaps by inspection, what additional conditions (as contrasted to purely subsonic flow) are implicit in the physical 27 problen. An additional difficulty is that it is not known whether potential solutions can depend continuously on the given data. Guderly (Ref. 7) has attempted to show that it is not possible, but the details of his proof are obscure. For these reasons practically no special problems will be considered here. Instead some very general properties of super- sonic gas flows and of gas flows in local supersonic zones will be discussed. The basic concepts used are found in Ref. 5 and some of the results appear in Nikolsky and Tagonoff (Ref. 10). General Properties of Supersonic Flow In the following work the transonic approximation will be used exclusively, for the sake of simplicity. Almost all of the results can be derived on the basis of the exact equations of motion. The starting point is thus (2.28) ZY uy 2K pA =O A* ox ta - (2,28a) ae . IY og a 0% (2.28b) Now consider a pair of functions u(x,y), v(x,y) in a domain @ of the (x,y) plane. Then to eachpoint P(x,y) there corresponds one point P(u,v) in the hodograph or (u,v) plane. We now study the local mapping from x-y plane to u-v plane in the neighborhood of P. If we consider a neighboring point P, (x + dx, y + dy) this will map on to a point P,(u,, v,) defined by a, = alindd, yr dy)= eiiy) lg, x liyp 4, + smaller terns 28 4 - VURER, hy) = Ving! 1 Mi, Fx AU, 4 + smaller terms In the limit the smaller terms vanish and the infinitesimal vector PP} maps to PP . It has been assumed that u,v are differentiable at P in @ . We thus obtain the mapping of infinitesimal vector (dx,dy) into (du,dv) (de) = (help % +p & (2.518) (av) = lz|p Pk + Gp 4 (2.51b) or, the slope of the mapped vector tly CIEL: dy eee (2.52) will be used to discuss the owen of various local vectors under the assumption that: the mapping functions satisfy the equa- tions of motion in Bi) « When the flow is supersonic two distin- guished directions at P are the characteristic directions (Mach lines). ay Y __a®™ Ze 7 (Fo7) 4 (2455) The streamlines at P are in the first approximation the lines pms. y = constant of the undisturbed flow and these bisect the angle be- tween the 2 characteristics (+) given by (2.53). It is a general result (Ref. 5) that the streamlines bisect the angle between the characteristics in the physical plane. According to (2.52) thé characteristic directions map into at® ly be 2B Jegurja ~*~ o Cart) le 29 or using (2628) ay =—_ +¢ (StL da a* (2.54) Thus a characteristic of one family (+) maps into a direction orthogonal to the characteristic of the other family (-). The characteristic direction in the hodograph may be imbedded in a curve integrating (2.54). lay a / As will be shown later (2.55) is the equation of the characteristics of the hodograph equations for the flow. Hence we have shown that locally, characteristic directions in the physical plane map to characteristic directions in the hodograph. To show that one characteristic in the physical plane maps into one characteristic in hodograph, i.e. that the constant c is the same, we need to approximate the characteristic in the physical plane by a series of segments and use a covering theorem. We use here the result that: a characteristic in the physical plane maps into a definite characteristic in the hodograph and the equation (2.55) with a fixed constant is valid along both these characteristics. The mapping of equation (2.55) to the hodograph depends on the possibility of interchanging dependent and independent variables 30 and leads to the linear equations Ye/ — X= ae “Ye “ (2.56a) % - Ju =? (2.56b) This is always possible so long as the Jacobian . Mu, vy) a yeh ty = a e does not vanish, since (Ref. 5) ty = } , a= Ae 5 a Am (2.58) 2058 Y= Jw Vi hae j represents a directed ratio of the local areas in the hodograph to the local area in the physical plane. This may be seen by con- nid 7 sidering the map of vector product of du and dv at P. DP xa? = [ty Ray GB )X (EI GG) —_> . = = 2g (Bi x&y) Thus if j <0 the mapping is order reversing j70O the mapping is order preserving Similarly in hodograph Bex tp J A TE where ~ ala, v/ es sl J represents ratio. of local area in physical plane to local area in hodograph. It can be shown that J, j <0 for M <1 (except for constant flow) but that they may change sign if M71. When J = 0 for a solution of (2,56) along a curve in the hodograph the mapping of (u,v) plene may form a fold in (x,y) plane and the edge of the fold.is called a limiting line. Limiting lines have been studied in much detail (Ref. 1, 5). All we need to know is that solutions, starting in the hodograph, which show this represent physically impossible flow patterns and that: the critical curve in (u,v) plane which maps into a limiting line can be described by the con- dition that the images of the streamlines in the hodograph pass through it in a characteristic direction. The characteristic directions in the hodograph can be found from (2.56) and are known in advance since the equations are linear. According to the usual methods the characteristic differential equation for the system (2.54) and the characteristics are given by the semi-cubical parabolas of (2.55). | The direction (of flow) along the streamline in the physical plane is taken as a time-like direction and the direction of the two characteristics at each point in the physical plane is pre- scribed as downstream. These characteristics bound the region of influence in the initial instants, of a small disturbance intro- duced into the stationary flow. A time-like direction is thus fixed for the image of the streamline in the hodograph and then the direc- tion of the characteristics in the hodograph is knowm. Hence, locally, 32 the characteristic. in the hodograph can be identified as expansion (B) ise. increasing velocity or compression (C). This fixes the characteristics in the physical plane by the mapping. Three types of flow may now be distinguished locally (of course flow of the same type may fill a region): (a) Source-like Flow: The characteristics leaving a stream- line (8) are here both of.same type, either both compression (CC) or both expansion (EE). A typical example is shown below. | v £ = srs Se < a 2 x u (bo) Vortex-like Flow: The characteristics leaving a stream- line are of different types; one carries compression and the other expansion (CE or EC) V ~ c a (c) Characteristic-like Flow: In the hodograph the stream- line S lies on a characteristic; thus S crosses only one set of characteristics, either E or C. This is Prandtl-Meyer Flow of the simple waves of Ref. 5. Exact solutions of these three types of flow exist showing that each type represents a possibility. In all the discussion so Ss) far, except (c}, a local bi-uniqueness of the mapping from hodograph _to physical plane was assumed. There is in general no bi-unique- ness in the large. For a physical problem the flow in the hodograph may have to be represented on two or an infinite number of sheets. The solutions may have to have certain singularities and folds in the hodograph may alg be needed. The problems of the solution in the large will not be studied here and the main attention will be to potential flow in a local supersonic zone near an airfoil. 2e6 Properties of Potential Flow in a Local Supersonic Zone In this section it is assumed that there exists a region of shock=free local supersonic flow and the properties of this flow are studied. In certain cases it will be shown that the assumption of the existence of such a supersonic zone leads to contradictions. To be specific, assume there are two continuous and piece-wise differentiable functions u(x,y), v(x,y) satisfying (2.28) and the conditions of flow tangent to a given body. Assume also there exists a bounded region & » Whose boundary is a simple closed curve, in which u 7 0, and another region in which u< 0. The curve K: wu = 0 which forms part wall of the boundary of R is then rectifiable and simple. (The case where u = 0 over a region is. ruled out.) The body B is assumed to be a thin airfoil lying close to the x-axis. For purposes of the mathematical boundary value problem the airfoil coincides with a segment of the x-axis. I. V(x,y) is a monotonic decreasing function of the arc length on the sonic line u = 0 going in a clockwise direction about the 54 region of higher velocity. x= xXto) v4” Consider the sonic line K represented parametrically by where g- is the arc-length along the line. Consider Cal * AT AT = ty ae dy s (2.61) by using (2.28). Hence Se O on KO (2.62) In the figure, “4 <O end AZO . It is also possible to have ay 7° but then 2%, <0O since the direction of increas- ing & is clockwise about the region of higher velocity. If the equality in (2.62) holds at any point then either # = 0 or uy = O at that point. If qs gand wt=0 a a segment “y | point. aA ty long a seg of K and if we assume the solution is analytic in a neighborhood of K it follows easily that Us Ves Vy and all higher derivatives vanish in that neighborhood. Under those assumptions yy = 0 leads to a con- tradiction. However, the assumption of analyticity is hot always plausible so that it is assumed here that J # 0 on the sonic line. A* W/ vt 2 = “- = au? oO. xk aH JS= X%& GG by ZO, om Ja - Uy Hence Yy #0. Now if z# =o wehave Wy Lh = # SO Yy = 0. Hence ty =z 7 is also ruled out by the assumption J # 0 ou and the sonic line cannot be vertical. Thus, in the following work 35 we use tt <o oo (2.62a) It is interesting also to consider a curve K,, on which u =u, = constant, which is assumed to separate a region of higher velocity from a region of lower velocity. Then along Ky ay 4 Si #Y 4 = & fz am IH, te dr * ar ¢ dle ¢ & (a “)eZp 2 /x%/ 2 Ye XZ {lz 2 (2 o B) I un k (2.63) Hence if 4 So subsonic or sonic the same conclusion prevails 7 Pp that 4’ 20 abng Ks H=4 (2.84) dco If the flow is supersonic (2.64) persists only if the flow is vortex- like. In this case the line Kj: u = uy and the streamline S both lie in the same quadrant between the characteristics leaving a point when the flow is vortex like (see Fig. ). Hence, the slope of Ky ie a | 36 may be bounded in the physical plane as Zz / / )-/81 <a This inequality combined with (2.63) implies (2.64). II. The boundary of the region R: u > O mst contain part of body B. The characteristics (Mach lines) are real at a point where u>0; each real characteristic in the physical plane maps in to a definite characteristic in the hodograph (p27). Now assume that the boundary of R does not contain part of the boundary B. Then each characteristic from K must return to K and thus intersect it in two points Pi Poe These two points are mapped on the same hodograph characteristic and actually on the same point since u only takes the value zero once on a given hodograph characteristic. This follows from the extension theorem for the solution of ordinary differential equations since the characteristics are defined as the solution of certain differential equations. It can also follow from arguments on the boundedness of the slope of the character- istics. This implies that "P, = "po" But by I. this is possible 7, BS only if K is vertical at all points, which ( reduce R to a line. Hence R occurs next to \ Q the boundary B. The arguments in I and II RB apply just as well if there is more than one supersonic region. The monotonicity properties 37 still hold on the sonic line of each region and each must be ad- _ jacient to the boundary. III. The characteristics in R run from the sonic line K to the boundary B. It has already been proved that the characteristics cannot run from the sonic line to the sonic line. Now it is also shown that characteristics cannot run from the boundary B back to the boundary. The slope of the characteristics which is the Mach angle is a con- tinuous function over R since the velocity u is assumed continuous. Hence the characteristics running from P, to Pp must somewhere ¢ attain the horizontal direction. This 4 implies an infinite velocity which is con- /. a tradiction. Hence III is true. In the ' above reasoning it is assumed that the body 2 x is $o thin that the data are given on the X~axise It is interesting to note that the shape of the sonic line K may be fairly well specified. The lines x = sonstant must run from B through R to the sonic line. If they do not, a contradiction follows. The characteristic leaving P on K must run to the boundary B. If y is not monotone on this character- istic it must assume the horizontal direc- tion, which is forbidden. The supersonic zone must then have the following general appearance. 38 TYPICAL CHAMACTEAIS TICS Iv. (u,v) are monotone functions of the distance from body B along a characteristic in a local supersonic zone of potential flow. Consider a segment AC of a characteristic which hits the sonic line at D and the characteristic of the other family extending from AC to the sonic line K. First, at any point At / Yr = a She a4 (2.65) This follows from the symmetry of the hodograph characteristics onto which aA‘a* x. and A'D are mapped or from their equation Jf, w 2 /xy4/ z Now along A*c*, VY is decreasing. Hence by (2.65) along AC, VY is decreasing. But (2.66) holds along AC with a fixed constant and sigh. Thus u is monotonic along AC. It should be noted that this proof depended on the fact that the characteristics from AC extended to the sonic line K and that 39 A* to c* was clockwise. For the other family of characteristics . the reverse is true when traveling away from B. Vv. ‘Pransonic potential flow around a corner formed by straight sides is not possible; more generally, a flat segment inside a local supersonic zone is not possible. - Consider a body with a corner formed by straight sides and consider a local supersonic zone about that corner. Consider a characteristic running from P on the sonic line K to a point P pefore the corner on the body. From IV PS fore, for the assumed flow, Vp 7 v Vp, 7 Vp. and from I vi > Yea" There- P2 which contradicts the assumption of straight sides: (v = constant). There- fore the sonic velocity can be reached on the body only at the corner. From symmetry and from discussion of the shape of the super- sonic zone under III it then follows the supersonic zone consists of only the corner point. This solution makes sense for a body with infinitely long sides and with the flow at rest at infinity. This problem is solved for incompressible flow. However the infinite velocity which occurs at the corner in the incompressible flow is now replaced by the sonic velocity. When the body dimehsion is in- finite there is no characteristic length in terms of which the height of the supersonic zone can be expressed. However, when the body has a finite size, as a double wedge airfoil, shock waves are needed to satisfy the boundary conditions as soon as a sonic region develops. 40 The proof that potential flow cannot exist when there is a _ flat segment is analogous and depends on monotonicity of along the characteristic. Consider a flat segment joined to the body with continuous slope and the characteristic going downstream from its start. We have va > vp and vp'7 Vg which implies wV7 Vo a contradictions. (vy = Vp! = Vo is ruled out as this implies the existence of a simple wave: see next paragraph.) Another interesting result is that the local supersonic zone cannot enclose a body with a corner (v discontinuous). In turning a corner the supersonic potential flow mst use a simple wave (Ref. 5). This simple wave is a region which has a degenerate hodograph and which has one family of characteristics which are straight lines. If these straight characteristics exténd to the sonic line they must be vertical and hence the velocity is sonic everywhere, a contradiction. The occurrence of a simple wave or Prandtl-Meyer fan in a local supersonic zone always implies the existence of a shock wave. VI. The flow along a convex boundary next to a local super- sonic zone is vortex-like. The convex boundary is defined by Es <o along the boundary 41 in the flow direction. The proof of VI is indicated from an in- spection of the flow picture and its mapping in the hodographe In the physical plane one expansion characteristic of E runs from the sonic line into the streamline and one compression character- istic must run from the streamline to the sonic line. In the hodograph a vortex-like streamline is the only one for which this is possible. The result. may be demonstrated analytically as follows. Consider a point P on S and the relationship which holds along an E characteristic b=he- FVxn % (2.67) and differentiate (2.67) along the streamline. This may be written —- of dy _ dlp — fot) 4? hap (206 yr Ta i a ~68) AY, But by I and since a%<9 along S, Joe >o . Therefore | ? (2.68) gives | 42 M6 ey Flug (2469) Mp av . fii!” The slope of E at P in the hodograph is just —//7%@ ° Similarly by considering C and the relationships that hold on it Ip 74% y, (26610) . Mp | = Hence the flow is vortex-like. By examining the hodograph picture it is apparent the flow must be vortex-like in the entire region R. Hence the extension of I applies and Y must attain its maximum and minimum in the boundary. The lines of constant velocity run from the boundary to the boundary. If for any reason the inequalities (2.610) must be violated and the image of the streamline in the hodograph is made to appear source-like we have a contradiction and the assumed potential flow cannot exist. For example, it is always possible to do this by introducing a sufficient jump in the curvature of the surface if the velocity is assumed fixed at a point. Discussion. The main question so far was the existence of a solution and the consequence of it. It was shown first of all that for certain bodies potential flow could not exist and that shock waves were needed to satisfy the boundary conditions. The necessity of shock waves for this reason is familiar from purely supersonic flow. Also it was shown that although one body may permit a transonic zone, a body close to it with only a small flat segment cannot. Also assuming a solution fora given body and Mach number at infinity the supersonic potential zone may be made impossible by a change in its curvature. However, it is not so clear why, for a given smooth body, the potential flow cannot exist as M is increased to one. Friedrichs (Ref. 11) has shovm that under these circumstances, starting with the solution in the hodograph, the flow cannot breakdown because of a limiting line. This is of course connected with the fact that a limiting line demands a corner of the stream- line and this cannot appear on a given body. The reason for the impossibility of a potential sobution might be connected with the violation of equations (2.69) and (2.610) and the impossible flow pattern in the hodograph. However this criterion is not very useful and further research is needed along these lines, 2e7 Drag of Transonic Airfoils Consider a non-lifting airfoil moving at M <1 through air ) at rest, with & local supersonic zone which contains shock waves. In these coordinates it is easy to concentrate the attention on a definite mass of gas,bounded by planes normal to the flight direction at #0. Then the First Law of Thermodynamics applied: to this gas states that: Heat added to gas = Gain in internal energy + work done by gas (2671) If the assumption is made, as before, that no external heat is added, all the heat added to the gas comes from the dissipative effects. These effects are mainly in shock waves and the boundary layer. If the flow pattern is steady the temperature distribution is the same so that there is no gain in internal energy. The work done by the gas is all in retarding the airfoil. Thus in a mit time, if ZX = dissipation function D drag per unit span (two-dimensional flow) Vv velocity of airfoil For transonic flow V = a* and Vis approximated assuming a single normal shock of length / . For convenience assume the airfoil is symmetric. The expression for the dissipation, as worked out in - J ¢ 9 Z ¥/. 2 - 2LVkA “u [ HF f Lee = FS (2478) —_ ~~ OO replacing a’, of ( #2) by ug = velocity ahead of shock. Hence, if A = height of shock ne, + teaiye" /° “ty = Ua or: #/ 41) D7 3 toy EE. “a bar denotes mean value 45 2. 6. Hy 4 (OP Lf+1) (2.74) GS i “| \ C = chord (2.74) gives the "wave drag" of a transonic airfoil. It is normally quite small but increases rapidly as M—* 1 for £ be- comes large rapidly (see Fig. 3). This drag is, of course, felt in the pressure distribution in the airfoil. 28 Determinatién of the Height of the Supersonic Zone from Data on the Airfoil In this section it is assumed that we have transonic potential flow adjacent to a convex boundary and that the velocities are known on the boundary. Theoretically it is possible to determine the height and shape of the supersonic zone by the computational methods of characteristics. Practically this is impossible because the slope of the characteristics become vertical near Me=i1. It is shown here that a certain functional relationship must exist along the streamline and the determination of the height of the supersonic zone is reduced to the solution of an integral equation. A first approximation is given for a special case. A solution to the equation of motion must satisfy the follow- ing equation in the hodograph / -¢Y. = a Hy “fue = 0 (2681) 46 derived from (2.56). DarbYoux (Ref. 12) gives the general solu- tion for (2.81) of an initial value problem, with y, yy given on u =@ (sonic line), sf, - 5, hoe St VF We Ardy Jt O-£) WE -# “Z ane ford Beata] 2 7 (2.82) which is valid at any point P whose domain of dependence lies on the initial line. If v, and vg are the velocities at the entrance and exit of Vv the sonic line the representation (2.82) is valid in the triangle Vy» Uys Vo formed by the character- istics through v, and Vos In par- 1 ticular, assuming a convex body, the bounding streamline SB is vortex-like and hence lies within the tréangle so that (2282) holds in the local super- sonic zone. The functions f and g are related to the initial data since at u=0 -f5 _ * vy) on fer p-t) dé ~ Lyf, tiv) -5f, PUGI where Wi - sere - fr th gt) “dt ey) (2.83) 47 = A fe 0 LY pr (2484) Now approximate the airfoil by y = 0 and assume that all data are given on the airfoil. We may express the data as: On J Se te A, di) (2.85) x Since x, = y,, = At) on Sg we obtain, using (2.85), the follow- ing pair of integral equations for f, g which must noice ‘ -S, ey = -4 - | Wokariy tes be ol My bo 9b hatte Wt) Ye (2.86) / Lx Lo -3f, -5f = JHE] baffle datifety 204 4 ’ ~ -~ » [aloe heobugd 4-0) ee o / J -zhinf 9 Yor katy) Jatyt Gt) 0 92) The general solution of (2.86) and (2.87) has not been worked out. However an approximate answer may be found by putting h = O under the integral sign. This gives ee ee a 48 ely Lt " Lt Mal $9 Le (oe gil H (2.89) a fet 0) 4, fet bi-usae For. a symmetric solution we have zg. (0) = f' (0) so that where / Ju = fe = 4 « Z (2.810) ‘p “4 4 $ 24 / Five hn | Bey A Ys / ox. Now A Ws jv ~¢ along S,- Hence the height of the super- (2.811) sonic zone, on the line v = 0, is approximately (see (2.83)) mage ( 2) = — 2281 Y- AK/o Example: Consider a circular are airfoil so that approximately v= -2a*ex on. Jy 3; @ = thickness ratio ot £# 2 ~2a & Then: 2 pe! fg ip ra f° — Zh) C))0OUAL (2.813) — For @ = 0.06, 0.12 (2.813) gives straight lines which are drawn in Fig. 3. In comparison with the one experimental point which is 49 really available the values seem slightly high but the agreement is fair for M, ¢ 1.10. 50 3- .FLON THROUGH A SHOCK WAVE A simple example of a non-linear longitudinal wave is a shock wave, which in a real fluid is of course not a discontinuity but a “continuous steep variation. The effect of the non-linearity in | maintaining the steepness and preventing viscous dispersion of the wave is vital, and is lost in any lihearized theory. | The actual non-linear problems are very difficult and, at present, not capable of solution. However, the main effects can be shown by considering simplified problems. In the following sections an approximate treatment of weak non-stationary waves will be given. 3el Derivation of Equations Basic Equations for Transonic Flow. Consider one-dimensional longitudinal flow with the velocity u(x,t). The basic assumption is that at x = = % the flow is uni- form and steady, slightly faster than the speed of sound; a ee (3.11) where the constant a* is the speed of sound at M=1. Under the conditions assumed the flow is isentropic at — % and the Bernoulli equation is a2 ue co oe / = ¢ 4 = > y-/ = constant (3.12) The equations to be satisfied by the flow are continuity, momentum energy, and the perfect gas law 4% + atl, = 0 | (30138) 51 fY + ple = a a Aang (3413b) | ¢ 2 fe someny-op Ltt J Fan (3.130) P apt? (3.13d) The assumptions of no heat conduction, no heat addition to the system, and a perfect gas have been made. The first step is the derivation of suitable equations from (3.13) Derivation of a Bernoulli Equation, Including Viscosity. For convenience introduce a velocity potential Ws t/ such that LX, t) = Z (3.14) Then integration of the monenvun equation from —o to x yields ors ad, ~ My flee $/ 2 a3 YH yy “Ly (3615) “Z x The right hand side of (3.15) can be integrated under certain per- turbation assumptions. For transonic flow assume that all the quantities differ slightly from their values at M=1. Thus x 7 ae pes P (/4 p / where @ tT. ~ yeT= % —x is defined from the - *€ Z #) condition at — 0 =-x aii 71+ O) = wr gira *) / ve where Vga (77 52 ; «a and where all the quantities ?-* fF 7 KS / - Further intro- duce a perturbation potential PY Z/) such that Zits Wea * 4 4e) (3.17) where z ge Koa Then using (3611), (3.16) and (3.17) and neglecting higher order terms we approximate (3.15) by YD /¢.,* # yf printi= fa * alt’ che (Co Ilf (5,18) (3.18) is the required relationship and expresses the pressure at any point in terms of the derivatives of the perturbation potential Derivation of an Equation of Motion in Terms of the Perturbation Potential P . Multiplying the momentum equation (3.13b) by u yields Z Ad, +u4, ce * Go 4 Votan (3.19) The continuity, energy and state equations of (3.13) can be used to eliminate Be since # & = FOYRG (Y-4£4 (3.110) Then (3.19) and (3.1W are combined as | 2 _ £6 wy + (4 Big = 2 + ZF ip he -3 hie (3.111) Introducing the perturbation assumptions (3.16) and (3.17) in (3.111) 53 and neglecting smaller terms, we have 2 ee L# ¥ + #.9/f = & a* by + (xa -4 Op y ax 7s (3.112) In order to obtain an equation for fp » we eliminate Pe from (3.113) by using the Bernoulli equation (3.18) and we relate p* and s* by using the energy equation (3.13c). Introducing the perturbation assumptions in the energy equations 2 2 * *] £2 Gy [oF * -J = 72 (2? “ aNF as ax" (341152) or integrating + a + _ = - aa z ¥ ix 4 Z) Mt) (3.113b) where J = integral of the dissipation, and J(- ~,t) =0. The flow is isentropic at x = - © so that f = 0 and x BY: et/ p-2 -S,=- Ss A x | (3.114) Hence s/f Z _* ?- pam oe SO" teh ag (3.115) Then using (3.18) and (3.115), we obtain from (3.112) the following basic equation for Y (x,t). Bey 2A “Ye + JO a % 49-2 -q" The f° Ja ee" hae} (3.116) Non-linear terms which allow for both the steepening of the wave v4 ee and variations in the local speed of sound are present in this equation. It should be remembered that the equation is ex- _ pressed in a system of coordinates where the flow is slightly super- sonic at x = - co. Notice also that the dissipation integral does not occur in the right hand side but only in the lower order terms. 3e2 Transonic Flow Through a Shock Wave Steady-State Solution. For a shock wave which is already developed pe Yer) and (3.116) becomes a C4) Fe Fen aX S/H = #” Faxx (3.21a) where x a _ 2% 41) / 22 be SL ¢/ 7 at KF (3.21b) It is now assumed that the dissipation in (3.2la) is small enough to be neglected and (3.2la) is then integrated from —© to x, to give Pu® ap?) Ww _ —% | (3.22) may also be integrated and if x = 0 when w = 0 we obtain So AX —— ~ Bh (7) (3.23) a Also we have the potential / ( x PA a — 4 43 : a (3 24) it 9 @/= 0 55 (3.24) represents a velocity distribution which is supersonic at -X = - o and subsonic at x = +”, with a continuous variation which is most rapid in the neighborhood of the origin x = 0. For smaller values of wv "this region of rapid change becomes increasingly narrower and approaches zero as Yree . The important effect of the non-linearity in allowing a tran- sition from w 70, (supersonic flow), to w< 0, (subsonic flow) is thus shown. This transition is missing in any linearized theory. As a check on the approximations the order of magnitude of the dissipation integral (3.2lb) can be found using the solution (323). We obtain prhd “aX 1 of “Wx 7 3 SLX) = (PL) ye & 3 so that the total dissipation is SA) = FON SF Ye ~ if. 1) (3024) This indicates that the dissipation can be neglected if the Mach number ahead of the wave is sufficiently close to 1. Actually the steady state problem including the viscosity and dissipation can be solved for waves of arbitrary strength (see Ref. 13). However the viscosity is assumed to be constant in that derivation. For a strong wave the variations of viscosity and heat conduction with temperature, and the dissipation all become important. The weak wave assumption is the only one consistent with assuming constant viscosity. 56 An Approximate Non-Steady Solution. In this approximation the integral of the dissipation is again neglected. Further, in order to simplify the problem the assump- tion is made that AK % (3.25 so that certain terms in (3.116) may be neglected. This is an approximation which becomes better for large t when the solution will be shown to approach the steady state. (3.116) is thus approxi- mated by - £y* “xu + (+) IS Bey = 7% Foxx (326) Integration of (3626) from — © to x gives t1/ 49°. rn a we H+ <a 3° Tex (3.27) since Yor uf, » Ry Y> 0 at x=- 0 , (3.27) isa non-linear equation for which the general solution of the initial value problem for the domain (~"*<* <<“ | t > © ) can be given. Only a special example is treated here which is the diffusion by viscosity of an initially sharp wave front. The initial choice is the solution for a weak wave without viscosity and then at t = 0 the viscosity is introduced. The initial conditions are YX o)= Wax x<o =-a,,% 2X>o (3228) The solution is found by assuming 57 ~liyt) = Ff a4 (3.29) where & (x,t) is to satisfy a linear equation and the function F must be determined. Such an assumption about the solution is sug- gested by similarity solutions of (3.27). Under the assumption (3.29) (3227) becomes | Ml p-42 92 22 _ wv FtO) G+ F 4 Fie! Q* uf7/ => VE (3)0,> Fa) 8, (3.210) (3.210) is satisfied if av* (3.211) and 4 F/,,2 av” cA -fg) F Wo a (3.212) The general solution of (3.211) is o#/ The constants Cos Cy may be chosen by having pf Oo as - aeo , and having F = 0 where @= oc . then =- fous ty) F(@) A fog fi- 4 (30213) ) / —_ a F (6) = 1 fOlGt) (3.214) and the linear equations to be satisfied by @ is a modified heat equation 58 uw The boundary conditions for (3.215) are found from the formulae re- lating oO, P through the known function F: gpnt)= - AAag f/- 7 ox, ti} (3,216a) Ott, t) = 4 / /- apf — #7 rst] J (3.216d) Thus using (3.28) the initial conditions for (3.215) are _ tar HX, 0) = a fi-e€ a v<o (3.2178) “Seg X = Afl- © 7 %> 0 (3.217) The solution to (3,215) satisfying the boundary conditions (3.217) may be obtained by Laplace tiansformation or other methods Dest oc B= 7 Axt)= - = -ct cea Zax) “t 2 o a (8+) Lg — 2 corh ey X>o (3.218) It may be verified that (3.218) satisfies the boundary condition (3.217b) by virtue of the formula oo af = wx C) /- Coo zur dt = e ° ” ye Elle, ~) W>o (3.219) av*/ 2) 59 Thus the potential and velocity distribution for x 7 0 are obtained , ond Pyl(-%, tla - AK t) P(x, t) = Bix, t) = the t d gpa rh 2 wok ver a el (3.220a) : oo - tots et ain [YE x fa ard “2%, he? ° (BSH aus, 7 a7 =~s/ ° Pur Me ore) a (3.220b) aur _ faa t (ST cook = —4 e 22 a or Z. [Grbide SF + 7 sui x? — “Ra tA ft) ? a" ¢ =-8f ia” (3 e220c) e* fant CNS EE tool, > wa 4 ert ae dt F Crd u,, 2 “a c “ec av 2 > ” This solution (3.220) approaches the steady-state solution very rapidly because of the exponential damping with time. The ratio ¥é ag ‘may be computed from the solution (3.220) and it can be x seen that for t 2? 0 this term is very small except when x = 0, where he Oo . For any fixed x = 0 the ratio becomes very small after some time. Thus one simple example has been given where the steady state of the shock wave is the limit of special non-steady solution. The 60 problem had to be simplified greatly in order to make an analytical treatment possible. However, the essential features of non-linearity and viscosity were retained. 1. 26 Se Te 8. 10. ll. 12. 13. 61 REFERENCES Liepmann, H. W. and Puckett, A. B.: “Introduction to Aero- dynamics of a Compressible Fluid". John Wiley and Sons, 1947. Sauer, Ro: "Theoretical Gas Dynamics". J. W. Edwards, 1947. 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I.: "Flow of a Gas ina Local Supersonic Zone and Some Conditions for the Breakdown of Potential Flow". Graduate Division of Applied Mathematics, Brown University, Translation A9-T-17. Friedrichs, K. 0.: "On the Non-Occurrence of a Limiting Line in Transonic Flow". I.M.li., New York University, No. 165. Darboux, Ge: “Lecons sur la Theorie des Surfaces". Gauthier- Villars, 1915. Lamb, H.: “Hydrodynamics". Dover, 1945.