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IACA ~rv\~1\~;9
NATIONAL ADVISOrY COMMITTEE FOR AERONAUTICS TECHNICAL IBiORA3PDU4M r:0. 1189 THEORETICAL ANALYSIS OF STATIONARY POTENTIAL FLOWS AID BOUNDARY LAYERS AT HIGH SPEED* Br K. Oswatitach and K; Wieghardt The present report consists of two parts. The first part deals with the twodimonsional stationer,. flow in the presence of local supersonic zones. A numerical method of integration of the equation of gas dynamics is developed. Proceeding from solutions at great distance from the bod:, the flow pattern io calculated step by step. Accordingly the related body form is obtained at the end of th. calculation. The second part treats the relationshipo between the dis placement thickness of laminar and turbulent boundary layers and the pressure distribution at high speeds. Tha stability of the boundary layer is investigated, rezulting in basic differences in the behavior of subsonic an., supersonic flows. Lastly, the decisive importance of the boundary layer for the pressure distribution, particularly for thin profiles, is demon strated. PART I ITOTATION p pressure P density T absolute temperature K ratio of specific heats n= *"Theoretische Untarsuchun6sn 'ber a tationare Potentialstr'dmungen und Grenzschichten bei hohen GeschwindicL.iten." Lilienthal Gesellscheft fur Lftfahrtforschung Bericht S 13/1.Teil, pp. 724. NACA IM No 11U89 p w w U, v c c* Ma = w/c F = 1 Ma2 coefficient of friction velocity vector magnitude of velocity velocity components velocity potential velocity of sound critical velocity of sound Mach number 0 = v/p stream density Re* Reynolds number of the displacement thickness 6 boundarylayer thickness 5* displacement thickness a momentum thickness f stream filament section r radius of curvature of the stream line N normal to the stream line H maximum bump elevation R related radius of curvature Subscript n refers to the conditions in the freestream region, subscript a to the outer flcw, subscript v to the wall. Subscript m refers to the chamber or drum values in the phase quantities and in the velocity to the highest obtainable value. The quantities of state in part I are made dinensionless by the chamber quantities and the velocities by the highest velocity obtainable. TIACA TM No. 1189 PART I 1. NOTES OI THE CHARACTERISTICS OF COiPRECS DL.Ei POTErNTIAL FLOWS The equation of gas dynamics is derived by means of the energy equation rather than ti6 adiabatic equation js customary. A simple formula is obtained rcr the stream density which is valid in a guide range about the critical velocity of sound. By applying this formula a simplified equation of gas dyiemics is derived which in the tran sition zone frcm subosonic to eupei'scnic, for smll velocity compclients v, describes the prccecses very acc.urateiy. Lastly, the problem of flow around a crl.inidricsl bod:., rLrEtLi'j'cal in two diirctions, is analyzed. It is found tnat, from 1 certain flow velocity on, located above the critical velocity, no nia.:maim velocity can occur at the point of moxiimum bouy thicknoss. In the de:ription of a gas flow the mozt generall cae involves six unrlnon functions, namelY', the three com.pon.snts of the velocity and the three pihse quanti.tins of the gac, the preaissure p, the density p, and the absc'luto tanIp .ra'ture T. The cluation of state of the gas permits the teiippeiat.,re to be exprezo'e. in terms of the preessre and aensity, th s lcav'r:.3 five unkinoxin f..nctlons fLr the calculation of which the three Euler equations .nI tlhe ccntlnuit,:, equation :re avai l.blu, For ti'e mi.ss5ng eqtiat'cn it is customary to use the auiauatic curve to eliminate the pr'. ls.re ani density from the equation ansi so arrive at an eyiration brt\ween the v;olocty com ponents, that is, the socalled eo'ut'tion cf gas .jnamics. IHowever, it appears to be uaknmr..n that for th? d:lriation of the equ'.tion of gas dynamics tf. asu:lpt'on of thie alabatic is not necessary at all, but that the use of the cne;,rgy th.3nrr.l itself i, sufficient. This derivation Is brieily carried out in the follo.win:, while havIng recourse to the vector sa lthod. Pressure, den..:ity, .ind tenpe'atu.ire are iialo nconaimensicnal by the corresponding "cnmbr qua,,tituie" p pm, Tm; that is, the quantities of state at verloc'ty O, and all o.curring .'locitie. and velocity components by the maximrir: obtalnaile velocity, that is, the velocity at pressure O. With iencting the ve aity vector, c the aseci'fic heat at constLnt .irS i.ce, ana r: the ratio of the specific her;t:. this ian:.nii v.locitly S= 2,' Tm 1 pa r) 4 NACA TM No. 1189 The equation of energy of an ideal gas In stationary flow has expressed norimenslcnally the following simple form p 2 =1 t(1.1) and the corresponding continuity condition reads div w + grad 0 = 0 (1.2) The oft repeated quantity 2 SI n = x 1 has a physical significance; it Indicates the degrees of freedom of a molecule. For air n = 5 is very exact. The Euler equation is then written as ad rot xn = (1.3) Grad w / (1.3) The quantity xn enters the equation through the nondimensional notation. Pressure and density are eliminated by forming the gradient of the energy equation,thus obtaining 0 = grad w2 + 1grad p + (w2 1) 1 grad o 0 0 Scalar multiplication by w and application of (1.2) and (1.3) gives the equation of gas dynamics 2 (1 w2) div v + nw grid 0 (1. ) This equation is written here in a form where the velocity of sounf is already eliminated and only the flow velocity itself is present. We will use this equation in the next section. For an insight into the Dotential flow properties in the speed range of Ma = 1 the just derived equation is much too complicated. So the processes in a flow filament are analyzed, unsteady variations IJACA TM No. 1189 and friction processes excluded so that adiabatic changes of state can be assumed. The enerr y equation (1.1) then affords a connection between velocity arnd density, and the stream density wp can be represented as a function of the velocity w: 0() = pw = w(l 2)n/2 (1.5) It is known that this function reaches a maximum at the point where the velocity is exactly equal to the velocity of sound. This particular point is generally denoted as the critical speed of sound c*. "ith f as the section of a flow filament the continuity equation reads f9 = ccnst, The speed c* is there ore characterized by the fact that a fl.o filament for Jhi. valuo w reaches a smallest pcsciblc crosz section. For w> c*  for w< c* the flow filament section is .:reater. Less familiar Is th. ~rm ellnes. of the otram density changes over a very subs.tantial peed ranse. To indicate it ,. is represented for x = 1.40 In the ran6e of 0.5c* < w < 1.gc* in figure 1. Q.ELuatity '"* denotes 'he vai.e of for w = :*, the same applies to the derv:c'tive: ci .'. Trhi characteristic of the stream density .s of decisive significance for th: effect of the boundary layer ,n the flow, av3 vil]. be shl u elsewhere. Consider the fiction C in thr vicinity of the maximum developed and signify iti drivative vith e*W, *, ., etc. Nov it is found that the parabola S c2 i(1.6 is already sufficientl seccu.rate for a wide seed ran.e. This approximation is indicated by dashes in figure 1. The calculation for the coefficient of the quadrstic term livess the simp e result 1 i+2 1w + ] n+1 (1. 2 = .. The equation (1.6) serves in good stead for the derivation of a simplified dynamic ges equeti.on for twcdimesi.nal flo.ws on the limiting assumptions that the y component of veloc.t w, signified by 7, is small comLpared tc the vel.ocit, of sound i'. that u, the x component of the velocity, does not differ too much from the velocity of esund. The rtrcar1 ieSi.i.tj up can be NACA TM No. 1189 replaced by up = e(u) and the equation of continuity (1.2) on applying the same omissions as effected with respect to the terms with the factor v for the derivation of the Prandtl law, can be written as (up) + 'U + = 0 The coefficient of depends only on u; it is simply a dif ferent method of expressing the wel..known quantity 1 .Ma. Assuming, aside from the smallness of v, that Pu can be 8 regarded as constant results in the PrandtlGL.u"'t 'vnaeloy. If this coefficient were plotted against u in the vicinity of the sonic velocity,it would show that it can asume negative as well as positive values and at u = c* is equal to zero. So the premise of constancy of this quantity can no longer be main tained, especially since the derivative u changes signs at sonic velocity, as Jeen from figure 1. The variations of u on the other hand are no more weighty than the variations of the entire coefficient anywhere in the range of not too. high subsonic speeds. Thus in support of Prandtl's law u can very well be put equal to this quantity in the freestream region, but not for eu. This quantity is computed by (1.6) and gives . ei. + .. )  i =O So c ^ ox e' / y ck The subscript o denotes the quantities in the freestream region. Using the notation U V (e + 1) V y 6 (o 9 V) (1.7) c i) c )' v 0 c gives for 1 and 1 the simplified equation of gas dynamics Iu ,v U = 0 (1.7) ix Cy ' U can be positive and negative. Here also the introduction of a velocity potential is accompanied, although in simplified form, by the undersirable change of the equation from the elliptical to the hyperbolic type. To secure solutions which have supersonic zones by an analytical method it is advisable to find solutions of (1.7), because it combines the simplifying assumption of small v NACA T' No. 1109 with a very accurate description of the processes in the critical sonic speed range. This wa the reason foi the brief derivation of the equation. The fact that the flow filament section has a minimum at the critical speed may, under certain circumstances, have very characteristic consequences for the velocity distribution at the appearance of supersonic zones on bodies, as will be demonstrated for the case of twodimensional flow pa;t a body that is symmetrical about two mutually perpendicular axes. The flow direction is to be aong one body axis, that is, the aigle of attack equal to zero. The flow is to be adiabatic and iriotational, the latter characteristic being expr?3s2d by w w w 7N =7 where N is the nomal to the streamlinee and r its radiuz of curvature. The sign for N is so cho .n that it is positive when the normal points out from the radius cf curvature. Equation (i.8) holds exactly for a.l twodimenrional potential flow. By the continuity condition in the form, f = Ccnstnnt and the freedom from rotation (1.e) the flow is completely defined. The origin of the coordirnte system x and y i: placed in the center of the body, axis x iL made ccincident with the flow direction (fig. 2). and the r' or C ts't~v ;, 7vaue naL:'rzed. The cylindrical body is visualized as b..ing exposed to a flow velocity which Jeads to the formation oL a supersonic zone near the thickest part of the body and it 1 as.oumed tnat in every stream filament the maximum velocity 1i reached at the point x = 0, an assumption which certainly should be fulfilled for subsoric flows. A point on the yaxi. with zupersonic opened must have a maximum btroam filyment width, a point with su.bsonic speed, a minimum of stream filament width. In the sLper.ornic region the curvature cf the streamlines on the positive portion of the y axis must deciense leso rapia.i' than on concentric circles, in the subsonic zone the stream line curvature must decrease more rapidly than for; concentric circle.. Hence no great error is introduced when in the vicinit; of the point on the y axis where donic velocity is reached, the etreamlineu are replaced by concentric circles, and it will not lead us far astray when this is assumed up to a value of y equal to twice the distance from the cylinder of the point with the critical sonic NACA T4 No. 1189 velocity c*. After the streamline curvatures are approximately known the velocity distribution on the axis in this zone is completely defined by (1.8). If the piece which the body cuts off from the yaxis is denoted by H and the radius of curvature of the profile on the yaxis by R, its velocity distribution is u R UY__ F H+ Y (1.9) According to (1.6) It may be stated that the volume of flow through a section of the y axis is then greater than on an identical section of the free stream, if at a a:t; 'ul ~' oint the inequality u < u < 2c* uo is fulfilled. Since the velocity distribution for x = 0 is defined by (1.9) up to the constant uy_, it also is the difference in through flow vnlurne for > H T.1, the freestream reaon and on the yaxis. It may now be asked at what value of the constants the absolute amount of this through "ow d..'rferencL reaches its highest possible value and the answer is found in the fairly accurate equation u=g = 2c* u (1.9a) that is, that the stream density on the .axis must nowhere be less than in the free stream. For a simple picture it is imagined that (1.9) with the constant (l.9a) is applicable up to the attain ment of speed uo and that from this y value on, the constant flow velocity prSvails. This break may occur at the value y = y for which the equation reads u = u for o R  As near the body more can flow by than on a strip of equal width in the freestream region, because of th3 increase in density. we must proceed from the cylinder only as tar as the free stream is displaced. The result is therefore a highest possible value of H, denoted by Hmax, which is given by the equation oHmax = (e 0) dy (1.10) Y=Hmax NACA TM IHo. 1189 The integrand is given by (1.6), (1.9), (l.9a), H is to be replaced by Hmax, since u is equal to w on the yax.s. The evaluation of (1.10) gives the following relation between flow velocity and Xmas R TABLE I  0.70 0.75 0.80 0.85 0.90 O.Q5 1.00 H e 0.053 0.026 0.013 0.009 0.0020 0.0004 0 The extension of the speed by pieces at y y unquestionably introduces an error; but it can only callse a shift in Hmax, while not changing the existence of such a va'te. In the subsonic zone a streamline may be re.7?rded a.s bnimp and the residual rise in through flow volume iup to inrreass in ..locity corputqd by in approxi mation process that applies in the subcsnic range. The result then is a finite variation o' the inte&rl in (1.10) and a corre.spcndi'ingly different Hmx. The possibility cf a a3red increase in ; direction in the subsonic region must be rejected, as it would invalidate the present considerations. Hence it is seen that thhe assumption cannot be applied to all bodies and therefore .he fo'.lo'in,. principle: To each flow velocity u,, there corresponds a definite ratio H /I. If the ratio H/'R exceeds this l:iit for a body symmet rical in two mutually perpendicular directions and lying along the flow direction, there is no flow for whlic: velocity ma: im'.ums can be reached on the entire yaxis. It must be expected that. the maxyimn, speeds on the yaxis disappear ory in chA supersonic rnnge. But since it cannot be assumed that velocity maximums in the sicnrsoni' range disaspear on a part of the yaxis while a velocity maximum appears on the body, we are led to the following principle. From a definite value of H/P. on, for bodies and flow directions of the described type, there is no flow at which a speed maxim'n with local supersonic zone is reached at the noint of maximum thickness of the body. A boundary point for these specific values of H/R is given in table 1. In the subsonic zone this principle has no analogy. NACA TM No. 1189 2. METHOD FOR THE NUMERICAL INTEGRATION OF THE EQUATION OF GAS DYnIAMICS A numerical graphical method is indicated for finding solutions of the equation of gas dynamics with supersonic zones, by progressive calculation of the entire flow, starting from an exact solution at great distances from the body. The exact body form follows at the end from the shape of the streamlines. Exceeding the sonic velocity causes no special difficulties rr peculairities. Limited to twodimensional, irrotational flows with w = grad past a cylindrical body, equation (1.4) L:iv3s for the velocity potential cp a nonlinear differential equation of the second order D = (n +1) ' S (n (2.1) x y ox y 1)% The zero joint of the coordinate system is placed in the body, its dimensions are of the order of magnitude of unity, and the flow strikes the bod, along the positive xa:is. The boundary conditions for then read: = 0 at the body itself, IN denoting the normal, and at infinity for z x + y ja rO and  u = dimensionlesss) flow yx ox 0 velocity. On passing thriourh the local velocity of sound , n + equation (2.1) changes from the elliptic to the hyperbolical type. For this case the xact integration has b.on successfully secured for single specific examples only. For the subsonic range several general approximate solutions are available, the simlest and best known of which is the solution I. obtained by the Prandtl rule. This satisfies equation (2.1) better as the body becomes more slender and the distance from the body becomes greater. NACA 1T No. 1189 The following method is therefore indicated. Compute the Prandtl solution 0p, for the entire flow and attempt to secure the correction cp in arch a way that ) + c = 0 becomes a solution of the complete equation (2.1). As the analytical calculation of q is too complicated,a numerical method is advisable, starting from the outside (z > 1) where = z 0, and progressively continuing Inwardly toward the body. The exact body shape follows at the end of the calculation from the streamline distribution; however it is to be suspected that it ossontially remains similar to the form of the Prandti solution. For this purpose the differential equation for p = : _ is set up; Q is an exact solution of the considerably simplified equation (2.1: o 2 2 !1 Mno) &2 + 0 with 2 Ma, = = n uo (2.2) o 1 Uo The subscript c refers to the condition at Infinity, (p fulfills the complete equation (2.1) up to an error ep which can be computed by metns of (2.2): D[J "=2(Ma2 +n L n) \ ^y/ + (n + 1) Ma2n (2 Ma2 p 2, S bx (2.3) ox Cvy Bx ,V Putting = 0 + 9c in (2.1) and regarding it as a differential equation for cp, p follows as term of zero degree in p. As <'P ip in the entire range, it is assumed that it applies to the derivatives as well. Merely the terms of the zero and first degree NACA TM No. 1189 and the greatest term of the second degree in p need to be Included; There results  (S (n + 1) ) . 0y) 6y2  (n + 1) ()2 \ox/Y7 2 ox dX 1) Maoa  '2 x2 0xL2 2,,.  n ' ox ,y y n 7p   d)7 cl + n 32 cIx cy 3x Idy = P L + 2 KMea2 + n) 0 / S2 (n + 2n o "y ex 6y + 2 (a + ) Ox 6p ?cp c'7 (2.4) The better satisfies the equation of gas dynamics, the easier is the determination of 'p. Cc, at great distances from the body the equation can be substantially simplified. For z >> 1, especially on slender bodies, ~ ; hence we can put cy x and = 0 in 2.4 but in contrast to the Prandtl rule .jy consider the variation of Since p 0 for z>*m, the term of the second degree is emitted also. And equation (2.4) is simplified c,y with (F  Fo + F' F oFx2. (2.5) 1 (n 1)  ,,1 ( i ,^ .ll^.   Z w 1 y 0 ay c .p . 1 ).) Zp NACA TM Ho. 1189 dF Fd  Tc This equation could also be u . vlation for u 2n I , derived from (1.7); F is an abbre The boundary condition for z > is p = 0 where the Prandtl rule applies exactly. On the other hand, however, the disturbance of the flow by the body is very extended when the flow velocity approaches sonic velocity. It is thercfo.'a necessary to determine an initial approximation for p analyti. flyr so as not to be com pelled to start at unduly great distance'. For this purpose (2.5) is transformed further. While Ma = Mao, by the Prandtl rule, hence F = Fo, the more exact term F = Fo + F'. u so that Fo 2? 32? F  SX o* = F' I 4 \6ox8 14 (H~  u + 1 / CX .AJ (2.7) o2 There  can be ignored with respect to and it is assuim c x2 that x x u at great distances; for example, in the cal culated case: For a parallel flow and dIpole, uo dies dowi 1/z2, but e as 1/z4. Hence finally: 12 ox F C 2 4) (F 1. F)  6x 1 / 3d i as (2. 10) All the equations for the process are nc: available. The general process of calculation is as follows: First determine the entire field of flow of the incompressible fluid numerically, and then the compressible approximate solution by the Prandtl rule for a fixed Ma number of the flow velocity. The correction cp on a strip for great y follows from (2.10). Frcm here on q' is computed numerically, step by stop. At great dictz'i.ce from the body we NACA TM No. 1189 therefore first use equation (2.5) and later in the neighborhood of the body the more complete equation (2.4). Having thus determined cp for the entire field, the stream lines and there fore the body contour itself, as well as pressure and velocity distribution,are obtained from $ = p + p. Naturally the process can be built on another approximate solution; however, the formulas probably become simplest when the Prandtl rule is used. For the present the range of application of this method is confined to the flows where the Prandtl rule affords a good approximate solution and the fundamental assumption p << pt is actually fulfilled. p Excluded are accordingly flows around not sufficiently slender bodies, as well as areas in which the velocity of sound is sub stantially exceeded. This also manifested itself in the calcu lated example. Flow around a cylinder (circular in the incompressible case) at Ma = 0.7454. The calculation was perfectly smooth into the supersonic range, where it had to be broken off especially since c and  quickly rose to the order of magnitude of  ox 8x 6y ox ?2p and but exceeding the sonic velocity itself involved no 8x 6y difficulties. In principle we can also free ourselves from the approximation that p << p, when in the formulation of equation 2.4 we consider terms of the third degree in qj; the length of the calculation, however, becomes disproportionately large. In another more appro priate method the assumption p << OP is emitted and the tedious calculation of Sp in the entire field of flow is eliminated. The previously described p method is utilized only for computing the initial values for large 5. The new method is as follows: 0p is evaluated at large distances from the body, for y >> yl; where yl is chosen so large that the error of the Prandtl solution is suf ficiently small; p is evaluated from equation 2.10. This affords the exact solution of the dynamic gas equation D = p + p in an initial strip. From here on $ itself ic calculated step by step. The width of the initial strip from xl to + x21 must extend upstream and downstream from the body so that for all y at x X1 and x > x2 the Prandtl rule is applicable with sufficient accuracy. From yl on, where $ is then known, 620 1  is graphically extrapolated to yl y for certain 6y2 2 'For bodies which are symmetrical relative to the yaxis also, xl is naturally = x2. NACA 24 No. 1189 fixed abscissae x (Ly is the length of the step; it can be assumed quite large at first, and reduced again later in proximity of the body). Next "71y 1 : l' is plotted against x and graphically differentiated, which gives 2 Plotting I against y and integrating gives yC Oy 6 Y'1 y1 y1ly 1 = ~+ y ox) ox J , oO: Y 1. AY Yl Y1 Lastly the variation of f over x fieldsds by graphical Yxl ox differentiation. Uith it , 6 62 and x' Oy ox' x2 are known for y = y Ay. From the equation of gas dynamics (.l), in which the simpli fication 0 can be made so long as it is valid that ay << 6 ). is calculated for the required x values and plotted against y. Then the calculation is reported, 2 extrapolated for yl y and so forth. If the values of computed for ; > yl, the value J extrapolated at 21 2Ay and that comoited for yl Ly do not form a smooth curve, the step must be repeated with a differently extrapolated value for the particular abscissae. In this event it is better to reduce the length of the step. Since the differentiation NACA TM No. 1189 of the curves x and is uncertain at the y=const. y=const. boundary points x = xl and x = x2, it is advisable to compute D = p + q also in two vertical strips x < xI and x > x2 and to join the progressively defined points to these edgp strips. The direction of integration for this step method must be chosen at right angles to the flow for the following reason. At flow around a body exposed to a flow alon3 x. is sure to be greater than almost everywhere, especially in the supersonic zone. Thus at entry in a supersonic zone the coefficient of  ox2 in (2.1) goes through zero. This does rot interfere in the above method since (2.1) is used for computing y. If, however, we integrate in the xdirection and solve equation (2.1) for L then difficulties will result. The coefficient of 2 can, on the other hand, disappear only far above the speed of sound where is not important at the point dy under consideration. We can also prove this state of affairs with the help of characteristics. When discontinuities in the velocity or their derivatives appear we cannot integrate across a character istic. On the other hand the characteristics of our flow become nearly vertical so that again we can not calculate in the x direction in this supersonic region. 3. ILLUSTRATIVE EXAMPLE A flow symmetrical in x and y is computed for a Mach number of flow of 0.7454. The flows on smooth bumps with supersonic zones are obtained exactly, but on the other hand the flow past a closed body is obtained only with errors in the region of the stagnation point. The described method is tried out on a very simple example; we start from the incompressible flow subscriptt i) past a circular cylinder i = xi + (3.1) xi2 + y2 NACA TM Io. 1189 the freestream velocity and the radius are taken as unity. Pr.ndtl's rule is applied to a fixed Mach number of flow Mao to which the dimensionless flow velocity u uo i/ Z4+. I 2 Ma + n = 1,13.0 + n corresponds. The abscissas for this transformation are contracted by \ Mao2 x x= 1 a o2 Xi (3.2) The ordinateh remain the same: y = y = yri so that and 6x2 a0 i . .TMac ,i To compute p by equation (2.10) the coordinates xi are used, so that xc2 y2 dx'2 X y d x (3.3) and y = yi oy' Uo ox M1 2 xi 2 1 + Ma 2J Development of the righthand side for using equation xi2 S= and ^l 1 (n + 1) uo2 (,i/3xi)2 u i 2 i)2 (3.4) la..ge zi xi + Y by (3.1) gives for the first ;?poroximation, when = 2= 1  2 zi c = 3x = uo I*X Ox (1 Ma2) NACA vT No. 1189 = Mao 4 ij nuo 1 Mao2 Z 12Mao Xi nuo %1 Mao2 Z1  6i + 892) 10,o 8 2\ 3 s+ b, A particular integral of this Poisson equation is obtained with the help of the separation formula general solution is written xi S= f ( ). 12Ma4 Xi nu 1   nuo l Ma2 Zi with tApot = 0 and c arbitrary. As boundary condition for p the sile requirement is that it shall be small compared to Op, that Is,decrease more rapidly 1 than q. But for the rest ePot end c can be chosen at random. The physical meaning of this ambiguity is as follows: Owing to the disregarded terms of higher order in ths formulation of (3.5) only the effects of the first order of the bod;. at great distances are taken into account. But these are the saze for different body forms. So the calculation yields different section forms, depending upon the choice of c and ppot. The manner in which c and Q ot affect the body form cannot be evaluated until several examples h6e been worked out. Up to now only one such example has been worked out, owing to lack of time. The Mach number of flow Mao = V5/9 = 0.7454 had been specifically chosen. The dimensionless velocity is then Uo = 7/10 = 0.3162 and x E p = xi. In (3.6) only half 9 ot 0 and c = 1/8 were assumed for simplicity, thus eliminatiRg the linear term in ; Lp(xi, y) (3.5) Thus the (3.6) c  + Qpot NACA TM No. 1189 Sr (3.7) zi with this p the velocities (and their derivatives) of the exact solution 6 = Op + rp for y = 10 and 0< x< 6, as well as to x = 6, 0 < y< 10 were computed and 0 was determined for y < 10 and x <6 by the described stepbystep method. In view of the symmetry of flow relative to x and y the calculation in one quadrant was sufficient. The step length 4y up to y = 1.5 was ay = 0.5; from there on 0.2. While the exceeding of ths sonic velocity (first at y = 1.35) caused no difficulties, the calculation could not be carried out to the body because of another reason but hl.d to be broken at y = 0.6. For at x 0 0.6 the horizontal components of the velocity Changes so rapidly for smaller ordinates y that the graphical differentiations became too uncertain tD compute the next step. As is seen from the contour of constant velocity (':i. 3) a steady but still very sudden rarefect.on occurs and on a point symmetrically situated with reference t x = 0 a compression occurs. This phenomencn would of course not be pl&in at a lower flow velocity, but it is certainly characteristic of the flow in proximity of the stagnation point where the speed increases quickly from subsonic to supersonic. For this point of the flow field another method must therefore be developed. So while unable to obtain the flow around. a finite body with a stagnation point, the data obtained thus far are nevertheless very informative for subsonic flows with supersonic zones. The calculated streamlines and lines of constant velocity are shown in figure 3. Visualizing, the lowest streamline in figure 3 as rigid wall, we get the flow along a smooth buip with a supersonic zone near the highest point. Since this streamline is already very steep for x" 0.55, it can be assumed that the velocity distribution around the finite body (with s;tmmetry axes x = 0 and y = 0.5) indicated in figure 5 is fairly accurately reproduced by the dotted line. Incidentally,it is notad th&t even Prandtl's rule yields considerable errors near the stagnation zint. Figure 4 shows several streamlines maznified five times in elevation, along with the respective velocity distributions. Not withstanding the similarity of the individual peaks the velocities differ considerably at various places. The velocity and with it the pressure distribution of thin bodies is therefore at high flow velocities markedly dependent upon the exact shape of the body. NACA TM No. 1189 Noteworthy also is the steep velocity increase at a point where the streamlines themselves are still comparatively flat. The contours of equal velocity in the supersonic region prove the principle set up in section 1 according to which the highest speeds under certain assumptions do not occur at the point o? maximum thickness of the body. Even the equation (1.9) applied for the derivation of this principle is satisfactorily confirmed in figure 6, where the velocities on the yaxiP are plotted along with the hyperbola (dotted) that touches the curve w(x = 0)/c* at w/c* = 1. From the farreaching agreement of the curves it follows that in the vicinity of w = c* the expression u = w(x = 0) = a b +y 13 a good approximation. With this example the accuracy of table I can be checked. In view of the flow velocity of uc/c* = 0.7746 Hp. /R = 0.019 would have to be expected according to this table, bu by the calculated example it is proved that from H,,a/R = 0.031 on, the speed maximum is no longer situated at the greatest ordinate. Thus, it is seen that table I is a good representation of the order of magnitude of hy,/R. The difference is attributable to the fact that the hyperbola used for the approximation gives too low speeds in the subsonic range. PART II .4 INTRODUCTORY NOTES ON BOUNDARY LAYERS AT HIGH SPEEDS Studies of the behavior of supersonic flosv in parallel channels disclose that in the supersonic zone,principal flow and boundary flow are in unstable equilibrium in certain circum stances. An effect of the boundary layer on the principal flow in the zone of the critical speed is to be expected for the reason that here small variations in stream density cause considerable changes in speed. This is particularly plain in the calculation of the flow through a Laval nozzle at high subsonic speed with observance of the boundary layer. In order to gain an insight into the condition of the boundarylayer flow at high speeds, which we will study in the following, consider an example from the sphere of incompressible flows, where the conditions are better controlled. We consider the circulationfree, incompressible and stationary flow around a circular cylinder at a high but still subcritical Reynolds number. Computing the pressure distribution at the body with the NACA TM No. 1189 aid of the potential theory on the assumption that the cylinder has no deadair region behind it and then calculating on the basis of this the boundarylayer conditions, say, with the aid of a refined Pohlhausen method, we find a separation point in the zone of rising pressure. It is found that the omission of the deadair region was wrong. The pressure distribution on the body must therefore be computed with due allowance for the deadair region and then. it can be hoped to attain a result corresponding to reality when the deadair region is so assumed that the related pressure distribution yields separation exactly at the starting point of the free stream line. Thic example shows that potential flow and boundarylayer flow usually depend upon each other. In general, we can say that the potential flow determines the bounda.ylayer flow, also that the boundarylayer flow deterines the potential flow. The former can be stated with great approximation in flow without pressure rise. It is a knonm experimental fact that for large expansions a flow simply does not follow the boundaries of the region; but it should be remembered that for the development of a deadair region not the expansion of the stream filament but the fact of a pressure rise is decisive, which only in subsonic flows goes hand in hand with an increase in stream filament section. In supersonic flow on the other hand a contraction of the stream filament results in a pressure rise. Thus visualizing a parallel channel with a flow of Ma > 1 a too strong growth in boundary layer caused by some disturbance is followed by a pressure rice, which in turn favors a stronger growth in boundary layer. In contrast to subsonic flow, an unstable equilibrium of boundary layer and principal floor is involved in this instance and a very considerable boundary layer growth must be reckoned with in certain circumstances. It may, in a straight channel result in a sudden ctrcng pressure rise at the flat wall and so in the formation of a deadair region (fig. 7(a)). (Compare reference 11.) If the pressure rise is so great that the flow becomes subsonic, the relation of main flow and boundarylayer flow is stable again, the deadair space cannot remain in this part of the channel. If a small pressure rise is involved of, say, a weaker oblique compressibility shock, the principal flow experiences a directional change in the sense of a channel contraction. The deadair space must increase wedgelike, but this holds only over a short distance, otherwise the flow would hava to revert into the subsonic range. It is therefore to be asuxmed that at an oblique compressibility shock, as met with in fiLvre 7(a), the turbulent intermingling imposes a limit on the growth of the deadair space. These qualitative reflections lead to the conclusion that in the range Ma > 1 an unstable state of equilibrium must be reckoned with in certain circumstances between principal and boundarylayer flow, which may promote the formation of deadair regions even at a flat wall. NACA TM No. 1189 A disturbance of the unstable stato of equilibrium of principal and boundarylayer flow in the supersonic range is favored by the fact that any minor disturbance in a supersonic flow is propagated undamped along Mach lines. Thus, a pressure rise in a supersonic tunnel can be dispersed by a small disturbance far upstream; on the other hand, the pressure rise sets in again some distance downstream as we can also infer from our example. The unstable behavior of the boundary layer in the supersonic zone must disappear when the principal flow approaches sonic velocity. In the critical speed range w = c*, which is of particu lar interest in flows past bodies with high speed, the fact stands out that this is the range of maximum flow density. But the proce dure in computing the incompressible flow past an airfoil is such that the pressure distribution is obtained from the potential flow without consideration of the displacement effect of the boundary layer, and then the boundary layer is computed with the aid of this pressure distribution. This is not permissible however in the region of the velocity of sound, because a minor variation in stream density e exerts a very substantial effect on the speed. This is readily apparent in figure 4, where peaks with comparatively minor form changes produce very unlike pressure distributions. This effect increases with increasing flow velocity. The effect of the boundary layer on the flrw in the vicinity of the velocity of sound is illustrated by a simple example, which, although it involves no flow problem, is nevertheless informative for the appraisal of the displacement effect of boundary layers at high speeds. The velocity distribution in the nozzle used by Stanton (reference 1) for his experiments was :omputed by appli cation of the simple flrw filament theory, once without boundary layer, and once on the assumption of a laminar boundary layer. The boundarylayer calculation is made with the help cf a process which will be explained in the following section. The initial value of momentum and displacement thickness at x = 0.20 was estimated. The dimensions of the nozzle are so small that it can be assumed that no turbulent transition takes place. Stanton's test series C is illustrated in figure 7(b). The velocities were determined by measuring the static pressure on the axis of the axially symmetrical nozzle (lower test points) and adjacent to the wall (upper test points). The theoretical curves by Oswatitsch and Rothstein (reference 2) and the flow filament solution with and without nbundary layer allowed for are included for comparison. The former was computed only as far as the separation point. It is seen that the aPymmetry 'is reproduced qualitatively correct by the flow filament solution with boundary layer taken into account. The displacement thickness at the narrowest point of the nozzle is not quite 2 percent f the nozzle radius. Computing the velocity NACA TM No. 1189 distribution for the same nozzle in incompressible flow with and without consideration of the boundary layer, the results in both cases are essentially even lines. Even aL speeds about 15 percent below those of test C,any boundarylayer effect is quite insignificant. This may be taken as proof that the asymmetry in nozzle flows which at the most, manifest local supersonic zones, are caused by boundary layer effect. As to making the computation, only the following is mentioned. That one gets at first the distribution of the displace ment thickness from the stream filament solution and then a new stream filament solution taking into account the calculated displace ment thickness is proof in itself that ouch an iterative procedure is prrmissable at very high subsonic speeds. Displacement thickness and stream filament solution are obtained stepwise at the same time in the downstream direction. The influence of the boundary layer on a submerged body will be handled in section 7. Our example, however, shows that we cannot hope to obtain results that correspond to the real process in some degree, for the flow problem with high velocity, without examining the boundary layer. We then have to remove, in practice or experiment, the influence of the boundary layer, perhaps by suction. 5. CALCULATION OF DISPLACEMENT THICKNESS OF LAMINAR AND TURBULENT COMPRESSIBLE BOUNDARY LAYERS For more accurate calculations on boundarylayer effect in flows at high speed, formulas for the variation of the displacement thickness are necessary. This is done in this section, first for the laminar and then for the turbulent boundary layers. In the derivation of the formulas for laminar boundary layer a refined Pohlhausen method is given for computing laminar compressible boundary laSers for given pressure distribution. Inasmuch as the pressures in the cci:pressible zone transverse to the flow direction in the boundary layer is also to be regarded as constant, the relative density variations within the boundary layer are in amount equal to the relative temperature variations. Thus, if sonic velocity prevails in the outer flow the relative density variation inside the boundary layer amounts to about 20 percent, since stagnation point temperature can be approximately assumed at the wall. So, at not too high supersonic speeds a qualitatively identical behavior in the boundary layer and in the incompressible range is to be expected. NACA TM No. 1189 So as not to exclude the possibility of compressibility shocks beforehand, the flow outside of the boundary layer is called outer orprincipal flow instead of potential flow. Being primarily interested in the behavior of the displacement thickness, the behavior of other quantities is studied only to the extent that it appears in the result witliout loss of time. This is the case in the ctudy of laminar boundary layers, where the momentum thickness is comparatively easily obtained and the displacement thick. .ess dezivrd. from it. The process is based upon an improved Pohlhausen method in conjunction with the reports by Bohlen referencee 3) and Walz (reference 4). Boundarylayer equation and continuity condition for the stationary case are written as follows u u dp 8 / u pu F + pv y = d+ 7y x (5.1) (pu) + T (pv) = 0 The coordinate system is chosen in the ucual manner so that x Is tangential and y normal to the contour of the body. Thus, in the general case, x, y ply Jj i 'artesian coordinates in the following; A is the friction coefficient dependent on temperature. The quantities p, p, u, and v are not made dimensionless. It further is assumed that the boundarylayer thickness is small relative to the radius of curvature of the wall, so that curvature effects can be disregarded. The second NavierStokes equation gives then exactly as for incompressible flow the result that p is merely dependent on x, but not on y, which in the boundarylayer equation was already evidenced by formation of the ordinary derivative of p. After integration of the boundarylayer equation over y, the application of the continuity condition gives the von Karman momentum equation which is written in the form d (au + PaU 5jd ( (5.2) (.2) s. s Lc iLeTTr an G M'PF 'S A,j TM..11, "t (10.3.) s" a Sp>ii cs. E: (CCA NACA TM No. 1189 The subscript w indicates the values at the wall, subscript a those in the outer flow. In the momentum equation displ.coment thickness and momentum thickness are defined, by *5 i( s 7;u \ u u where 5 is the socalled boundarylayer thickness which is now chosen so great that 5' and :* can be regarded as independent of 5. The displacement thickness has the hiysicsll significance that the through flow volume in the bouindiry la:er is reduced by an amount that, on the assumption of pure yp.tentisla flow. is equivalent to a shift of the wall by piece 5' in positi'r3 y direction. Equation (5.2) .can also be given the form u 2 2 du + 2 2 /a Aside from the usual assunrtions with the aid of which the boundary layer equations are derived, no restrictions of anR kind have been made so far. Only the problem without heat transfer cn the wall, the socalled thermometer problem, is treted in the ollciwing.. Further we will make one .approximation, b: which we will specify the form of the bomudarylayer profile b; only one parameter in addition to the Mach number of Tne oute.ie floe. Next It is necessary to make an assumption concernin the configuration of the density profile. Having se3rn that in the vicinity of the critical velocity the velocity profile especially ,ioht be decisive, while the density variation is unimportant, the case of laminar boundary layers is limited Go Lhe assuimption that the tempsrotur.) at the wall always attains the tank temperature Tm and eas.sfies the energy theorem within the boundary layer. This ties in .lso the assumption that temperature and velocity boundary layers are of equal thickness. Accurste calculations on flat plates indicate that this assumption also holds in a considerable supersonic range (reference 5). The pressure in the y direction being constant, the density variation follows from the temperature variation as NACA TM No. 1189 1 / 0 Ta Q'Imi P T pt, c 2 _U (5.4) The parameter for the boundarylayer characterization is derived from the known boundary conditions which for u = v = 0 in the boundary layer equation leads to + P  du = P aUa Since the internal friction is only dependent on T and no heat transfer takes place, = 0, hence with application of 4 t s a, Y * = L /'^ " D3 O' dua Pw d4 We have avoided the introduction of the boundarylayer thickness itself in the equations other than at the unimportant place as upper limit of the integral. The version of (5.5) was largely taken from Walz's report (reference 4). The parameter k* differs from the conventional Pohlhauson parameter by the factor 2 the density refers to that outside, the internal friction to that at the wall. For a class of velocity profiles, such a for M = 0, for instance, the individual profiles which are represented by the magnitude of parameter ~* the quantities and can be taken as S t function of the parameter %* from the class of profiles, hence d Pae2 obtain us by (5.2a) simply as function Ix II, of %*. Choosing the Pohlhausen profiles as profile class 3Ives the curve of Bohlen Holstein (reference 3) in figure 1, vhile the Hartree or Howarth pro files result in Walze' curves of figure 1. Moreover, it is not necessary at all tc have an analytical representation of the profile class, it can equally be given as family of experimental curves. If the outer flow is dependent on a Mach number, one profile class is used for each Mach number. In order not to come to grief because T Tm (5.5) 2 OL ) ; w,' 6_y (Ry~ NACA T1 No. 109 of nur ignorance in the sphere of comcressible velocity prcfiles two known facts are tak.n advantage of: Firzt, we Jnow that the velocity profiles on the plate aL constant pressure are not verya closely related with the 'Mach n'rmbrs of the outer flo'r (reference 5). Therefore, this is assumed to be the case in retarded or accelerated flow also; secondly, we know that the single paametric riathed in the incompressible, which utilizes the Pohlhausen profiles, leads to fairly practical results, althcugJ the ?ohlhausen profiles themselves do not represent the actual profiless very well. An e:xct represen tation of the velocity profiles thernelves is not needed, the main point is the displacement thickness for which the integration over the profile form is already accomplished. On the basis cf these arcuirLents we therefore select, independent of the Mach number, the Hartroe profiles f'nr the velocity distribution, which, as e rule, p:obably r.irzsent the in7omiressible boundary layer, best of all. The density variation is then (iven by (5.!,) as function of the .elocity distribution and owing to th' presence of 1'a quantity  as function of the Mruch nLrnuber. w1 By a numerical m sthod th3 quantities  and L are u 17 u./ i j /w then obtained as functiorn.i of and Ika, as exemplified for the 'ollowinr Ma which correspond to the  values: si 0 1 I 1 l. .O0 \ o.,'. 0 .hTB O.57 07.'5 07 L I I  We obtain the c:rve system. cf fiur'e S. The curves are shown at the left as tar as the separation point, at the riCht they proceed to the point .p to which the Hartree profiles are calculated. Several ci.rvs 'ere extra .clatd beyond it, and indicated by dashes. The CLoav? la = 0 is identical with the Hartree curve from Walz. Thus with the velocity distributica. the variation of the Mach number of the outer flow, and an initial value of the ouraieter A* can be formed; with it and observirng Mi the quantity NACA TH No. 1189 d apg2 Da P u d a can be taken from figure 8 and the variation of a a CL 1w Pw computed. From it we obtain again this quantity at a point shifted by one step and the calculation can then be repeated (reference 3). Although b* is wanted, it was preferred to compute La because the equation for this quantity is very much simpler. Know ing the outer flow, 32 can now be specified. To determine 5* thus further requires which is a function of X* and Ma, represented in figure 9. Since the quantity in the dx be defined as accurately as possible advantage 1 di* B ,i 5* di* b: dx 6* ,"* d boundarylayer equation is to the following formula is of .0 6 S* v w Wm ( 5 d Ua d Pa dx pw (5.6) This formula contains only quantities dependent either on Ma and *, or which can be taken from the previous boundarylayer calcu lation. The derivative with respect to u which is a function Wm of Ma, was preferred over that with respect to Ma for reasons of simplicity. The first term at the richthand side in (5.6) is generally the principle term. Ii *t If it is desired to eliminate V. Ex so as to secure then because of AS! merely in relation dx and a to i d ua dx dx2 in (5.6) (which * enters in the equation ) and coefficients dx solely dependent on A*, Ma, and Re, the expression becomes fairly long, but since this relationship is used later it is given here, the Re number bsing suitably referred to the out side density, the coefficient of friction at the wall and the displacement thickness: Re* = UaPa* 2 a2 a ( (597) JACA TM No. 1189 The formula reads xd6* 1 R1 *\ 2 a dx Re* 3 (4 W + *dua i u yix fu a o 2* 8* " 2 + Ma2 * 3S1?*  F ua + I R5 " + _ I i m u , f a.  (2 ( 4 ivia) (ua dx a2 l 56* dua &*2 d ua /E dua\ . _ a. aR* aRs* Re* ua dx ct3 u c d.x \aa /x (m.8) It contains the first and second derivative of ua made dimension less with the displaccment thickness and th3 outside rlorcity, and also Re* and coefficients that are dependent cn A* and Ma. The coefficients al, a, and a? for 0* = 0 are given in the following table: TALE II LAMINAR BOUIiDFY LAYER; N:a 0 Ua wM a1 1.4  a0 5.31 rL 21,T I V~~r o.4oS 2.02 4.89 .So o..? u 0.43 2.30 4.6o0 0. 2 '.* = 0 . 0.55 2.76 3.87 J, . 2 C. 2.0 0.667 3.82 2.37 r. 1.1 Later on the equation is to be applied to the case where the velocity distribution differs little and monotonically from Uat d a 5*2 d2ua us = Const, so that and are also regarded as Ua dx ua d:2. 0.21y U...)(+ .. I 30 NACA TM No. 1189 being small. Restricted to the linear terms in the derivatives of u1a, the term with na can be stecs: out and the coefficients a2 and a3 taken at the point X* = 0 for the specified Mach number. The dependence of al on the derivatives of ua proves to be so small that the sane can be done for this quantity too. So on the assumption of smaall derivatives of ua simply (5.8) is taken vith the constants of table II for the corresponding d&* Ma, for of a laminar boundary layer. dx To obtain a formula for the variation of the displacement thickness of a turbulent boundary layer a different procedure is required. On analyzing the cause of the variation of an incom pressible turbulent velocity profile at a specific pressure variation it is found that the pressure forces are primarily responsible. The shearing stresses introduces by the turbulent intermingling play, however, a subordinate part. It is true that the difference of the two effects is not so far reaching that a second profile could be computed accurately eroiug from the specified velocity profile when the ehearirng tr.sses are discounted, because the shearing stresses are able to substantially modify the character of the profile; but for the calculation of the variation of displacement thickness, which essentially involves an integral over the velocity variation in the incompressible case, the shearing stresses can be disregarded. The result at Ma = 0 is the following ap'ro:'imation formula for the turbulent boundary layer: d. a / u (5.9) da ua dx u a As the intoerand is always positive, it can be tarzon from this formula that a speed increase is accompanied by a decrease in displacement thickness and a speed decrease by an increase in displacement thickness. At constant outside velocity the displace ment thickness ra.malns constant, according to (5.9). This result is naturally wrong, as indicated by experiments on the plate at constant pressure. For in this case the variation in displacement thickness is contingent upon the turbulent shearing stresses, so no correct result is to be expected. The formula could be improved by the addition of the conventional formula for the variation of NACA TM No. 1189 the displacement thickness, but it would serve no useful purpose, as will be seen. It is of greater significance that in contrast to equation (5.3) for the laminar boundary layer the second derivative of ua is lacking in (5.9). Since (5,9) was obtained by several rougher omissions its practicability is illustrated in figure 10. The experimental values of dua an3i are shown plotted ua dx dx against the arc length x of Gruschwit's (reference 6) test series 3, along with the variation in displacement thickness calcu dua lated by (5 9), the integral being forced at = 0. It is found that the formula reproduces the actual conditions s adequately, as far as the area of greater accelerations, whore errors begin to be intro duced. This is, of course, due to the fact that 8* = 0 imposes a limit on the decrease in displacement t.lcihness. These experiences in the incompressible zone can now be interpreted to the effect that the turbilnt shearing stresses for the calculation of i* can also be carcelled in the compressible dx zone. But even this assumtion is insufficient to develop a law for the variation in displacement thicknes; additional data on the density distribution in the boundary layer are needed. In the case of turbulent boundary layers the energy theorem is not directly applicable, because the densityboundary layer. i probablytwice as great as the velocityboundary layer; hence, the density varies in an area in which the velocity is already practically considered constant (fig. 11). The result of it is that the varia tion in density plays the same role in the calculation of 6* as the variation in speed within the boundary layer. Unfortunately only one measurement of a turbulent supersonic profile is available, and naturally there is little sense in developing a theory without further basis. However, in order to reach a tolerably correct numerical value, the part of the boundary layer in which the density alone varies is disregarded for the present, since it involves only about 10 percent of the displacement thicknsss, and, in the remaining portion, putting the stream de.isity as a unction of the velocity as follows: pu ' (5.10) Paa u. H to be taken from the experiment. Now the derivatives of p can be expressed by derivatives of u, u., and pa with the aid of (5.10). This enables us to derive a fornila for the variation of displacement thickness, neglecting the turbulent shearing stresses. NACA TM No. 1189 d* = du I Ma2 dx iu dx Jo P pu2 TF  1) e * / &* 5* dua = a dx 2t 6id (5.11) E' is the derivative of the function H according to the argument uu and 5 the place where  can be put equal to unity Ua Ua  = 1), while 6* represents the corrnc.t displacement thickness hence, integrate up to a point where 2 itself is equal to unity Pa L = j). This means, we state that the 10 percent of the dis \Pa placement thickness between the point =1 and = 1 u Pa contributes to the variation of the displacement thickness an amount which corresponds to its portion of the disolacement thickness. For Ma = 0, equation ( 11l) naturally changes to (5.9). If the density and speed in the boundary layer are specified, the integral can be evaluated also. We have calculated the expression in paren du theses for a profile by Gruschwitz, for which d = 0, and for the velocity and density profile represented in figure 11; thus we obtain the constant a2 for two values of the Ilach number. TABLE III TURBULENT BOUNDIIRY LAYER !Ma 0 1.7 a2 5.1 2.2 i The close agreement of coefficient a2 i'or the turbulent and the laminar velocity profile is noteworthy. 6. STABILITY STUDY ON THE FLAT PLATE A study of the equilibrium of boundary layer and supersonic flow on the flat plate indicates that an unstable state is involved. The growth of a small disturbance in a laminar boundary layer differs somewhat from that in a turbulent layer and is, especially in the last I.ACA I M No. 1189 case, very rapid. In incompressible flow a stable equilibrium exists between principal flow and boundary layer. Having secured the variation of displacement thickness 8* in relation to the velocity variation of the outer flow, the reciprocal effect of principal and boundary layer flow is now analyzed in the simplest case, naiely, in the flow at the plate without specified pressure distribution. Since the effect of small disturbances is to be involved, the Mach number of the outer flow Ha is regarded as constant and the v component of the velocity considered small relative to the velocity of sound. Pfter introductionn of a velocity potential the simple equation (2.2) is.involved, and written in the form S ( 1) (6.1) The xaxis is to be in plate direction, the :yIxis normal to it. Now it is necessary to represent the effect of the boundary layer on the potential flow in form of a boundary condition. The boundary layer is thersfcre visualized as being replaced by an elastic layer superimposed on the plate, which has the property of always attaining the thickness equivalent to the displacement thickness of the Doundary layer .,t the particular place for the prevailing velocity distribution. That is, the equation v = u1 (6.2) d. must be satisfied for y = 5*. This condition is incc.venient to the *xt:nt tht the boundary for which it is to be fulfilled 'i not specified beforehand. But, inasmuch as the disturbances are to be small, hence the outerflow is to differ very little from a flow ua = Const., the boundary condition for displacement thickness 5* in undisturted flow is assumed. By assumption the departure of 3* from the valie of the displacement thiclneas for the undisturbed flow must be small. Hence it seems immaterial whether v is specified at y = 5* cr at y = 5* + dS* in the lincarized problem. Besides, the study is do* to be restricted to such a small area that  itself can be dx regarded as constant at ua = Const. NACA TM No. 1189 Furthermore the boundary condition (6.2) has the property of giving the same v component of the velocity at y = 5* as a dS* boundary layer with equal , on the assumption of potential dx' flow in the entire space. In (6.2), v and u are none other than the components of the outer velocity, hence in the notation of the proceeding section equal to va and ua. Applying (5.8) or (5.11) to dX gives then as boundary dx condition of the problem,linearized in the derivatives of u, the following equation for y = 8*: v = u al a 25. aRe* 2 5*2 (6.2a) Re x x2 If a laminar flow is involved the corresponding constants must be taken from table II; if, turbulent flow, table III; in the latter case, a3 must be put = 0. In view of the linearization 5* and Re* must also be regarded as constant, although the variation of u in the first term is not important, it is considered nevertheless, because the solution then is reduced to the treatment of a homogeneous linear differential equation, which means some simplification. Now it is attempted to find the solution for the cabe that the plate is exposed to a flow with the velocity u(x,y) = u = Const and at a point x = 0 at the plate the velocity is artificially varied by an amount U <, uo. The coordinate system is turned through a small angle so that the xaxis in point x = 0 is exactly in flow direction and the y*axis normal to it. The tangent of the angle of rotation is defined by the variation of the displacement thickness at x = y = 0, which is equal to dB' J dx Re* strictly speaking a transformation of the coordinates in the equations themselves should be effected. But since the boundary layer itself makes no difference between these two directions, and so a rotation merely involves more paper work without any physical significance, it is disregarded and the equations applied to the new coordinates. The coordinates are in addition visualized as being made dimensionless by the displacement thickness and the origin shifted to the point x = 0, y = 5*. NACA TM No. 1189 These new coordinates are denoted with 5 x =y 8* and after introducing the velocity potential in (6.2a) give the following differential equation with the respective boundary condition at 'y'y = "x'x' (Ma' 1) (6.la) y' = 0; y' = 2t'x, a RetO'x'xy (6.2b) Assuming a very general solution of (6.3a), and writing the potential as sum of a potential of a principal flow u and a small disturbance 0= uo*X' 4 u* f(x' 4:.Ia2 ) + ( + \W2 1 y') f and g are arbitrary functions of which it is merely required that their sum at x' = y' = 0 be equal to unity. It is seen that g gives Mech lines which point toward the boundary layer, hence stem from a disturbance from the outside. This function is thus put identically zero since such disturbances are to be disregarded. Introduction of' the thus obtained solution in the boundary condition gives the functional form of f. Denoting the derivative with respect to the argument T = x JMa2 1 y' with subscript Tr, we get x' = u 0* + U6*fT ; xx'W = ~8*f.i x'x'x' = b6*f ri; 1 y' = lb* Ma2 If which inserted in (6.2b) gives an ordinary differential equation of the form Re*f + f Ma2 = (6.3) 3 1Tfrj 71 1 30 NACA 4TM Io. 1189 This equation is easily solved. First postulatine a laminar boundary lawyer, hence a3 / 0. Then from the renuireLent for x' = y' = 0: f = 1, a requirement for the second derivative f can be satisfied, because the upper equation can be regarded as differential equation of the second order of f,. Since at this point only the consequences of small velocity disturbances, not the consequences of disturbances of the velocity difference are to be studied, the aaded requirement for x' = y' = 0 is fq = 0, which gives the solution t t f 1 e t 2 e t tl f ti t tl t2 and t2 are abbreviations for the expressions 1 t1,2 2 M2 Ma3e*  1 t 2 a2 _1 i/ 41 2 1e q e, /,Re* 3 . (6.4) As Re* in general term under the root appraisals at high has the order of nmanitude of 103, the last is a term of greatest influence. Thus for Be* we can put (6.4a) t l+2 Z VNa 1 aRe* V1 J It is to be noted however that the critical Re* which corresponds to a value of abnut 1.4 x 103 must not be exceeded as will be shown later. By use of (6.4) the velocity distribution in a laminar boundary layer on the plate is obtained as: 1 x' = + t2 t2let2 x Ma ly t2e tl ' a 2ly (6.5) tl t2 I'C\ which by (6.4a) is reduced to the simple form u + u coh\(x a 1 y where tl (6.5a) NACA TM 1o. 1189 If a turbulent boundary layer were involved, hence a. = 0. the first summand in (6.3) cancels out and only one boundary condition can be satisfied. Again requiring f = 1 for x' = y' = 0 gives fMa2.. 1 ft =e cu? The velocity field under the assumption of a turbulent boundary layer at the plate is u = u + ue a2 (6.6) From (6.5) and (6.6) it is seen that the boundary disturbance along Mach lines is propagated into the flow. The interference velocity U is always accompanied by a function which grows considerably with rising value of the argument, while in the case of the laminar boundary layers the coefficient a3 plays the principal part. In turbulent boundary layers the coefilcient a2 is essentially involved. Thus the boundary layer of a flat plate in flow with constant velocity is in both instances in an unstable state of equilibrium with the principal flow, which with observance of the terms of the first order only, lets a sm.ll disturbance grow infinitely. The type of growth is, of course, quite dissimilar on the two boundary layers. To secure a measure for the instability x of the state, we may ask for which value of x' = g at y' = 0 the disturbance has grown to twice the amount and call this quantity the length of growth A. It is not made dimensionless by the displacement thickness. The length of growth in a laminar boundary layer A is assessed by (6.5a). The hyperbolic cosine grows for a value of the argument of around 1.3 to the amount 2. Accordingly a Re* A 1.3 ,* (6.7) 1. Ma2 1 The length of growth of a turbulent boundary layer At is At = 0.70 ,2 5* (6.8) MaS 1 NACA TM No. 1189 Postulating a laminar boundary layer at Re* = 1000, tables II and III give the following length of growth TABLE IV Noteworthy is the unusually short length of growth in the turbulent boundary layer; but even that in the lam'nTr lyer is still very small when bearing in mind that the displacement thickness in supersonic flows is of the order of magnitude of 103 to 102 centimeters. The investigation was restricted to small disturbances. The extent of growth once they have reached greater amounts remains to be proved. One thing is certain that the outerflow cannot increase to great velocities, because the boundary layer cannot drop below the amount 5* = 0. Thus no limit in velocity decrease is imposed. It may be presumed that the velocity decreases until the boundary layer breaks away. In general, the instability of the discussed equilibrium condition will became evident in a pressure rise, probably an oblique compressibility shock. It would not be sur prisirg if oblique compressibility shock occurred in the center on a flat wall (fig. 7(a)). The example cited here could be multiplied by many others, perhaps cven by flow around conical tips. It should be kept in mind that a pressure rise can cause transition of the boundary layer. In the example adduced here the boundary layer is already certainly turbulent. This study of plate flow can be regarded as first result in this sphere of instability of supersonic boundary layers. It would be desirable to get away from the assumption of small disturbances and constant flow velocity. This seems altogether possible by a combination of characteristics method and boundary layer computation. For the turbulent boundary layer, of course, the laws of variation b* would have to be analyzed first. One unusual fact is that in the measured pressure distribution on a wing, such as those by Gbthert (roference 7), for instance, pressure increases were almost over observed in the supersonic zone, except in form of compressibility shock or occasionally at small Reynolds numbers, where laminar boundary layers must be assumed. Ma 1.2 1.5 1.7 2.0 S 25 18 12 At 1.1 & , IIACA TM No. 1189 It appears entirely possible that this fact might be explainable by the cited properties of the supersonic boundary layer. The corresponding behavior of a laminar boundary layer in incompressible flow (Ma = 0) is briefly indicated. The disturbance at great distances from the wall, that is, for great values of y, must disappear. On these premises, (6.la), (6.2b) by the same method of calculation give u = uo + ue B '2'cos (02x' Ply) (6.9) with the abbreviations 1 C 2)+ ( 3Re ")2 + 0l am3BRe*j g + 3 2 3Re*][( (ct ) ( The decisive term at high Re* numbers is again a Re*. For Re* = 500 it approximately is Il 2 m Re 0.053 that is, a strongly damped oscillation is involved. The analyzed equilibrium of laminar boundary layer and outer flow in the sub sonic zone is extremely stable according to it. This method of analyzing offers the further possibility of exploring the stability of laminar subsonic boundary layer relative to nonstatic.nary dis turbances and comparing the results with Tollmien's calculations (reference 8). For nonstationary velocity variations Pohlhausen's method is, of course, not practical in general, in the form given here. Incidentally, the requirement of damping of the disturbance for great y is not fulfillable in subsonic flow on the assumption of a turbulent boundary layer at the plate. This result may have its cause in the fact that (5.9) does not meet all requirements. NACA TM No. 1189 7. SIGNIFICANCE OF BOUNDARY LAYER IN THE PRESSURE DISTRIBUTION ON A BODY Appraisals indicate that the flow in the critical range of sonic velocity is very substantially affected by the boundary layer. Without its inclusion a correct calculation of the pressure distribution therefore seems, in general, not very promising. In many instances the behavior of the boundary layer actually governs the pressure distribution. On examining the press'ire distribution at a bump computed in section 3, (Cig. U), a symmetrical velocity distribution is also found on a body symmetrical about the yaxis. This is, however, in great contrast to the experience in tests (compare, fig. 12). where symmetrical peaks were Invariably accompanied by asymmetrical velocity distributions. Naturally the question ia whether there is only one solution for each bump but it will be shown that, owing to the boundarylayer effect, symmetrical solutions can be expected as little as in the example of the velocity distribution in a nozzle (fig. 7(b)). By (5.8) the displacement thickness of a laminar boundary layer for constant outer speed is 5* = 2a1 S1 u Pa what is the possible extent of the bump in order that the boundary layer remain laminar? Figuring with tests in a lowpressure tunnel, the values at critical velocity are ua = 3 x l04cm/sec; na = 0.8 x 103g/cm3; 1. = 1.8 x 10 CGSE It is to be presumed that the critical Reynolds number at sonic velocity does not differ substantially from that In incompressible flow. Taking the critical Reynolds number formed with the plate length at Rcrit. = 5 x 105 NACA TI1 No. 1189 gives the critical Reynolds number (5.7) formed with the displacement thickness at Re*crit = 1.4 x 103 with the previous values of ua, Oa, and Pw the critical.values of plate length and displacement thickness are Xcrit. = 3.8 cm; B*crit. = 1.1 x l02 cm So in order to prevent transition from laminar to turbulent flow in the boundarylayer model, lengths of only a few centimeters may be permitted in the usual test arrangements, provided that no strong accelerations are involved. Conversely, the critical length indicated here gives a measure for when the transition point is to be expected on a plate flow in an exhaustion tunnel at sonic velocity. In a freeair test this length is reduced by about half because of the higher density. In the schlieren photograph of an infantry shell in flight at around sonic velocity (fig. 13) (references 9 and 10) the oblique compressibility shock is evidently released by transition, its effect being probably amplified by the unstable behavior.of the boundary layer. The fact that a missile at small supersonic speed is involved is immaterial; since a straight compressibility shock prevails in front of nose of the missile, it actually flies as if in a subsonic flow. Analyzing the bump in figure 4, which at the point of its greatest height has nearly constant sonic velocity for some distance, and supposing the points of strong velocity rise and velocity decrease (x = 0.6) to be about 2 centimeters apart, the displace ment thickness at the peak Is certainly greater than that of a plate 1 centimeter in length in flow at sonic velocity. Therefore 5*,=0 > 0.56 x 102 cm At the point of substantial speed decrease, separation must be definitely expected. A calculation by the expanded Pohlhausen method shows that the momentum thickness grows with increasing arc length. Much greater is the rise in the ratio of displacement thickness to momentum thickness (fig. 9) which for Ma = 1 increases NACA TM No. 1189 from point X* = 0 to the separation point from value 3.2 to 4.7. Considering the fact that the momentum thickness itself increases up to the separation point, an empirical rule can be established according to which the displacement thickness is doubled between k* = 0 and the point of separation. The difference between the displacement thickness at the separation point and at the highest point of the peak is in the example; therefore 6* =o 0.56 x 102 cm separ. x=o The difference in height of the highest point and at point of separation hsepar is (compare fit. 4) separ. While the variation in h due to the boundarylayer effect amounts to a mere 20 percent, the illustration shows that a change in height of bump by this amount must be followed by an extra ordinarily great chance in velocity distribution, so that there can be no question of attaining symmetrical results in the experiment. The conditions in the presence of a turbulent boundary layer are considerably worse. A little calculation on Grushwttz's test series 3 (reference 6) discloses that the displacement thickness multiplies from the point of transition to the point of turbulent separation by about 25 times. Assuming turbulent separation at the point of severe velocity drop the greatest displacement effect (height of bump + displacement thickness) would also exist on a bump of considerably greater absolute dimensions at the po'nt of separation due to boundarylayer growth. It is supposed that the displacement effect of the body, increased by the displacement effect of the boundary layer, undergoes no substantial increase behind the highest point of the bump. In turbulent boundry layer and thin profiles or low bumps this is possible only to the extent that a compressibility shock occurs at the point of creetly reduced profile thickness; furthermore a compressibility shock would have to occur so much farther downstream as the bump or tha profile is flatter. It also is feasible that the effect of the increase in displacement thickness is raised by stron, return flow behind the point of separation. Th.e qualitative results can be checked against the work cf GCithert (reference .). NACA TM No. 1189 The fact that a compressibility shock can occur when there is enough space available for the increased displacement thickness caused by it is to be regarded as reason for the fact that the separation computed by stream filament theory in figure 7(b) is almost exactly coincident with the start of the compressibility shock in the test. It may be asked how the streamline pattern in a flow problem must look, in order that the compressibility shock be possible. This can be answered to the effect that the compressibility shock on slender bodies is to be expected near the point of vanishing streamline curva ture. Since the streamlines in the zone of critical sonic velocity are approximately parallel, the points of vanishing streamline curvature must lie near a common orthogonal trajectory, hence, a potential line. Along it the velocity changes little accoding to.(1.8). In a flow that differs little from the critical soic velocity, the freestream velocity is therefore to be expected in the vicinity of points with zero streamline curvature. If the curve decreases rapidly at a place with supersonic velocity a decrease to the outer velocity must be counted on. The marked velocity variations in figure 4 coincide with the streamline inflection points. On flat profiles a point of separa tion can be regarded as starting point of a free streamline with very little curvature. The streamline curvature must thus decrease very substantially in the separation point and it is seen that a strong compressibility shock produces through the separation connected with it a streamline pattern that favors the appearance of the compressibility shock. This argument is therefore not suitable for finding the location of a compressibility shock. 8. CONCLUDING REMARKS The proceeding work shows that in a calculation conforming to reality the pressure distribution of a body in a flow at supercritical freestroam velocity may not be given by the potential flow, that the boundary layer plays a decisive role here. In general, the potential flow around the body permits not even an approximate calculation of the boundary layer. This means that in contrast to incompressible flow the pressure distribution on flat bodies can also be much different. It is therefore intended to first improve the process of calcula tion of the potential flow with a supersonic region. With the process we will ascertain the flow around a substitute body. This will have approximately the same displacement effect that is found on an experimentally investigated body including its deadwater region and 4 IIACA TM No. 1189 the displacement offoct of its boundary layer. We can also anticipate from our calculation a strong velocity increase at the body nose and a strong velocity decrease at the point where the curvature of the substitute body disappears. Translated by J. Vanier National Advisory Committee for Aeronautics NACA TM No. 1189 REFERENCES 1. Stanton, T. E.: Velocity in a Wind Channel Throat. Aeron. Res. Comm. R. & M., No. 1388. 2. Oswatitsch, K1., and Rothstein, W.: Das Strbmungsfeld in einer LavaldUoe. Vorabdr. d. Jahrb. 1942 d. deutsch. Luftfahrtforsch. in den Techn. Ber. Heft 5, Oct. 15, 1942. 3. Holstein, H., and Bohlen, T.: Verfahrcn zur Berechnung laminarer Grenzschichten. LilienthalGCs. f. Luftfahrtforschung Bericht S 10 (Preisausschreiben 19'0.), p. 5. 4. Walz, A.: Ein neuer Ansatz fUr das Gezchwindigkeitsprofil der laminaren Reibungsschicht. LilienthalGes. f. Luftfahrtforschung Bericht 141. 5. Hantzsche,W., and Wendt, H.: Zum Kompressibilit&tseinfluss bei der laminaren Grenaschicht der ebenen Platte. Jahrb. 1940 d. deutsch. Luftfahrtforsch. I, p. 517. 6. Gruschwitz, E.: Die turbulente Reiburigeschicht in ebener Strtmung bei Druckabfall und Druckanstieg. Ing.Arch., II. Bd., 1931, p. 321. 7. Gothert, B., and Richter, G.: Messung am Frofil NACA 001564 im Hochgeschwindigkeitskanal der DVL, FB 147. GCthert, B.: DruclckerteilunGs und Impulsverlustschaubilder fur die Profile NACA 00061, 130, usw. bei hohen Unterschallgeschwindigkeiten. FB 1505/15. 8. Tollmien, W.: Ein allgemeines Kriterium der Instabilit&t laminarer Geschwindigkeitsverteiluneen. Nachr. d. Ges. d. Wiss. zu GOttingen, Math.ohys. Kl., Fachgr. I, Math., Neue Folge, Bd. I, Nr. 5, 1935, p. 79. 9. Cranz, C.: Lehrbuch der Ballistik, Bd. II, p. 452, Fig. 22. 10. Ackeret, J.: Gasdynamik. Handb. d. Physik, Bd. VII, p. 338. 11. Busemann, A.: Das Abreissen der Gren.schicht bei Annaherung an die Schallgeschwindigkeit. Jahi'b. 1940 d. deutsch. Luftfahrtforschung I, p. 539. 12. Fr6ssel, W.: Experimentelle Untersuchung der kompressiblen Strboung an und in der Nahe oiner Eewblbten Wand, 1. Tell. UM 6608 (Abb. 12 entstammt einem nicht verbffentlichten Vorversuch). 46 NACA TM No. 1189 * C, u .s 0 C't Zbcd I O rl. a) 0 ci) p41  SO / ir^ ^ / S3 /~~ P7 <" " \ ^ NACA TM No. 1189 .I/u o It P44 / I y s, mb~ B 48 NACA TM No. 1189 I S ii 0 0 >? a) cn C N 0 m Cd 4 4I 0 Cd a OO co )3 F) / i 0 M 'S NACA TM No. 1189 49 Streamline with the asymptote y = 0.6 Streamline with the asymptote y = 0,8 Streamline with the asymptote y = 1.0 Streamline with the asymptote y = 1.2 Streamline with the asymptote y t 1.5 0 1 2 3 4 X 5 Figure 4. Velocity distributions over various bumps.  i .   NACA TM No. 1189 O rC >O 0 oa) ni 4 0 0 ' C a D .q 0 o nI ~ , Kly W*3 NACA TM No. 1189 0) 0 U) ao a to tn II) cd3 cd 1r I PI  ^0 d ^ S 1i U)  ^ 4 U) I c $ ^ .2 ^ ^ 'ti  <= c' a .l o .l Z, I 52 NACA TM No. 1189 Df 0 . 0 Sel C3 a) og a) oo i . 0 0 a a ps a Cdk 0 o S a) a) a) I I C Ij 0 tQ I rJ; rlwcd I NACA TM No. 1189 53  ,s ___ I C o / o " U a /n 0 4z) Co ,o S :O N '? ^ LS CY  /,// F g 54 NACA TM No. 1189 cp 3d S Il I Q rI _U it c I *l cd I I I O . SI I 0 , 0 . 0I I 0 ( 1 ty cI ^ i ^ ^'.r NACA TM No. 1189 0 cd o * I I 4 *0T3 CH * . j0 ..O V oa1 ( o 1r1 r~la 56 NACA TM No. 1189 ,r* Ti 0 * cd 4 r. o s T , 4 04 NACA TM No. 1189 57 S" a) u cd 0l o U0 SIi 1I t' "4 __ o ^r f & r^ A ^ * ^  t r ^. t I  ^ ^ \  ^^, \ rs ^ ^^^ \ NACA TM No. 1189 59 t o'i :4 A]' Figure 13. 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