lki\ I'HI ACR Ho. L6A22 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED Marcli 19^6 as Adyance Confidential Report L6A22 APPE0XIMA3:E FOEMULAS foe the COMPOTATIOn OF TUHBUiiENT BOIMDARY -LAYER MOMEHTDM TEECKMESSES IN COMEBESSIHLE FLOWS By Heal Tetervin Langley Memorial Aeronautical Laboratory Langley Field, Ya. NACA ^ WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the wsir effort. They were pre- viously held under a security status but are now unclassified. Some of these reports were not tech- nically edited. All have been reproduced without change in order to expedite general distribution. L - 119 DOCUMENTS DEPARTMENT Digitized by tlie Internet Arcliive in 2011 witli funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/approximateformuOOIang 70,1 CI i^"^ ■■t.U;A ACR ITo. L6i\22 I-JATION.AL ADVISORY C0?.1''.'riTTE^ PCR AERONAUTICS ADVAvjcs coi;fide],tti7\l report APPROXIMATE] FORMTTT.AS FOR THE COIVPUTATION OP T T.m BTJI.: ENT BO UTTD ARY -L AYER MO ?/!EIJT UM THICr^^JESSES IIT COlVn^RESSIELE FLOWS By Neal Tetervin Approximate formulas for the computation of the moment-ami thicknesses of turbulent boundary layers on two-dimensional bodies, on bodies of revolution at zero angle of attack, and on the iriner surfaces of round channels all in com.pressible flow are given in the form of integrals that can be conveniently computed. The form.ulas involve the assumptions that the mom.entum thickness may be computed by use of a boundary- layer velocity profile which is fixed and that skin-friction form.ulas for flat plates m.ay be used in the computation of boundary-layer tb.icknesses in flov; with pressure gradients. The effect of density changes on the ratio of the displacement thickness to the momentum thickness of the boundary layer is tfsken into account. Use is made of the experimental finding that the skin-friction coefficient for turbulent flow is independent of Mach number. The comioutstions indicate that the effect of density/ changes on the momentumi thickness in flows with pressure gradients is sm.all for subsonic flov.'s. IITTRODUCTIOI A number of methods are available for the com.putation of boundary-layer momientujn thicknesses for inccmipressible flow. The increasing importance of flows at Mach numbers approaching and exceeding 1 has emphasized the need of formulas that would m.ake possible the com.paratively rapid comiputation of boundary-layer momentum thicknesses for com.pressible flows. The purpose of the present work is therefore to provide approximate formxulas for the compu- tation of boundary-layer-thickness parameters for CONFIDENTIAL NACA ACR ITo . L6iA22 compressible flows. The present work ftirnishes no new Information concerning the boundary-layer shape, skin- friction coefficient, position of the transition point, or likelihood of boundary-layer separation. Approximate formulas for the computation of the m.oraentum thicknesses of turbulent boundary layers on two-dimensional bodies, on bodies of revolution at zero angle of attack, and on the inner surfaces of round channels all in compressible flow are given in the form of integrals that can be conveniently computed. The approximate form.ulas contain the as sum.pt ions that the momentumi thickness may be computed, by use of a boundary- layer velocity profile which is fixed during the inte- gration and that skin-friction formulas for flat plates may be used in the computation of boundary- layer momentum thicknesses for flo¥if with pressure gradients. The formulas are applicable to. all unseparated, turbulent boundary layers and in spepial cases to laminar boundary layers. The numerical values of the ratio of the dis- placement thickness to the momentum thickness, a ratio that appears in the m.omientum. equation and that is capable of specifying approximately the velocity distribution through the turbulent boundary layer in incom;;pressible flow, are corrected for density changes in the boundary layer by use of low-speed velocity distributions. Use is made of the experimental finding that the skin-friction coefficient in turbulent flow is independent of Mach number . The problem of computing the boundary-layer momentum thickness for com.pressible flow has been treated by Young and Winterbottom (reference 1), who integrated the boundary-layer m.omentum equation for lam.inar flow by using the skin-friction relation from the Pohlhauseri theory (reference 2, p. 109), fixing the velocity profile, and correcting the density through the boundary layer for the effects of compressibility. For the turbulent boundary layer, the m.omentum eqiiation was integrated by a step-by- step process in Avhich a fixed velocity profile was used and the effect of density changes through the boundary layer on the ratio of the displacement thickness to the momientum. thickness v/as ignored. The problem of computing the raom.entum thickness over a body of revolution for incompressible flow has been treated by Young (reference 3), who com.outed the mom.entum_ thickness of the lam.inar boundary layer by a step-by-step CONFIDENTIAL KACA AC?, ITo. lSi'22 COIJFID^^TTIAL computation in Y-'hich an eztension of the Pohlhausen method was used. The thickness of the turbulent part of the boundary layer ivss corrputed by a step-by-step process in which a fixed velocity profile and the inoKientum eq\:.atlon for a body of revolution were used. In order to substantiate the assumption that skin- friction formulas for turbulent flow along flat plates may be used in the computation of mom.enti,im thicknesses for flow with pressure gradients, references 5 to 6 are cited. In these references, good agreement between calcu- lated and experimental results was generally obtained although fairly large adverse pressure gradients were present in many of the cases. The assumption that the mom.entum thickness may be computed to a close approximation by fixing the velocity profile curing integration is substantiated by the work in references 5 and l^. and by that in reference 6, v/hich contains a com.parlson between the computed and experi- mental values of the m.omentiim thickness over the entire chord. That the skin-friction coefficient for turbulent flow is independent of Mach number is established by the work in references 7 3.nc, 8. Prossel (reference 7) presents experimental data for turbulent flow in pipes which show that the velocity profiles for subsonic compressible flow and the skin-friction coefficlenbs for subsonic and super- sonic compressible flow do not differ noticeably from those for inccraoressible flow. Thecdorsen and Regier (reference 8), by experimenting with rotating disks, showed that the skin-friction coefficient for turbulent boiondary layers is independent of Mach number. Keenan and Neuman (reference 9)? after performing experim.ents with pipes, reached conclusions that did not contradict those of references 7 and 8. SYMBOLS a constant in equation relating Eg to A, and Hi b sloDe of velocity distribiition C(^ drag coefficient per unit span CCNFIDTCRTIAL. J4. C0NFIDT3NTIAL IT AG A ACIl ¥.0 . l6A22 Cq velocit;)'' cf soimd in free stream C-) specific heat at constant pressure, foot-pounds per pound-mass per degree H ratio of displacement thickness to mo:mentum thiclmess" {&"'/&) K constant k constant in skin-friction formula L length of airfoil, body of revolution, or roiind channel: measured along chord or axis of revolution Mq free-stream Mach number m exponent in formula for boundary- lajj-er velocity d 1 s t r 1 bu 1 1 on m' particular value of m n exponent in skin-friction formula p static pressure R gas constant Ry^ Reynolds number (UoL/uq) Ry Reynolds mimber based on length of plate (Uq^/'^^o) Rg Reynolds num.ber based on momentum thickness (ue/i;) r radial distance of point from axis of body of revolution or round channel rt radius of body of revolution or round channel rt maxim.um radius of bodv of revolution or round ^max , T channel T .'absolute temperature Tq absolute temperature of free stream. T5 absolute tsm_perature at edge of boundary layer KACA ACR Iio. L6i^-2 COin^IDENTIAL U U\ Ul u w X ^0 J a P = Pi V 5-"- 9 On IS /x velocity parallel to x at outer edge of boundary layer free-stream velocity /e value of U at station at which value of I — \ i-j obtained value of Tj at X = value of Tj chosen to make value of 1^=-- ) maxireui:n value of IT at xq -la V'xy velocity inside boundary layer and parallel to surface exponent in forraula for viscosity distance measured along surfs ce from forward stagnation point position on surface at beginning of Integration distance me?sured normal to x angle bet¥/een tangent to surface of body of revolution or round charinel and axis of revolution r+-8 + Q G( a value of ^0 ratio of soecific heat at constant pressure to specific heat at constant volujne nominal thickness of boundary layer I A 5 / pu\ cisolacem.ent thickness \ L il - dy mo me nt uin th i c kne s s i ,^5 _pu 1 - u dy value of at Xr CCNFIDENTIAL 6 CONFIDENTIAL i:ACA ACR lie l6-^22 (/ V(it/lTo)" [J, coefficient of viscosity p free-stream viscosity u kinematic viscosity at outer edge of boundary layer u„ kinematic visccsitv In free stream, o p densit^r Pq free -stream density po density at edge of boundary layer T^ surface shearing stress o o = 1 + r - i)Mc 2 ^ = r^e - Q cos a ii-^ value of i]/ at Xq Subscripts ; c compressible flow 1' incomioressible flow ANALYSIS Momentum equation for corrpressible flow about a two- dirrLens 1 oral bo ay . - The boundary-leyer ;riomentum equation for two-dimensional compressible flow (reference 10, p. 132] CClIFIDENTlJil,- nag;. ACR Fo. -l£i^Z COi^IDEi\^TIAL 7 is given, when the static-pressure variation across the boundary layer is negligible, as T^ / pu^ dy - TJ^r— / pu dy = - Tn - 5 -^ From the equation of motion for compressible, inviscid flow the relation between the velocity and. pressure derivatives outside the boundary layer may be written for convenience as '^PSU- 6p5lj _ 6p 6 X bx ox Then by use of the equation for ■^, the definition fo: the momentum thickness rO and the definition for the displaceviient thickness no (1 - P5'J/ -^ = X U-S-x equation (1) can be written in the form given in reference 1 as , q ■?i / p-c + 2 cU 1 6p5\ _ To + e ( ii^i — - _ + _ rr^ )■ = _i^ (2) \ U ox pg 6x/ pgU^ where 5^ The principles used in the derivation of equation (2) were the conservation of mass and llewton's law of motion. Equation (1) is therefore aoplicable to both subsonic and supersonic flow. This equation is not, however, to be used for flovir through a shock wave because in this case the ass-^imptions of the boundary-layer theory may not be applicable , CONPID'^NTIAI; 8 COKFID^NTIAL IT AC A ACR l-Jo . ]^G^ZZ ycmentiim equat?Lon for compressible flow about a body of revolution.- The boundary-layer momentum equation for cOiT.nr e s s 1 bl e f 1 ow about a body of revolution (reference 10, p. 133) can bs Y/ritten, when the static-pressure variation across the boundary layer is negligible j, as 6 .^'S 2 d r5 (A) 6p T'- — / pu r dy - U — / pur dy = _ j^r^ ^- / ex ./o dx Jo ^y- Jr Since r dy (3) ^0 r=r^+ycosa where r+; and cos a are deoendent on x only (fig. 1), equation (3) ^"7/ be rewritten as * r 2 — / DU-^^r + ^x Jo '0 ^y- Jo -.0 t ^y ~ '' TZ / p^-^^'t ^y 6p /"^ 6.p r 6^ Jo ^t ^7 - 6;^ J, (^OEa}ydy - r^r^ If 9 and Q are defined by the relations t2 _ t' ^t^9b^ = ,/^ pu(U - u)rt dy and. C(cos a) pgTj2 ^ J^ pu (IT - u) (ccsa)ydy and the equation of motion for cC'iipressible , inviscid flow cp _ ^ „ 6lt ox 6x CONFIDENTl.-iL IT AG A AGR Ko . Loi^S COJJFIDENTIAL 9 is used, then the morrientum equation becomes / nd / r p.Ur^ dy 6x Vn ^0 t - / pur^ dy) - — (r^eP5TJ-j ^'0 J uo Pc:^lT (cos a) 7 dy - / ' ~ pu (cos a) j dy i -T.2 6x L jQ (cosalpgr^i = - r^To J Then 6"' and C'"-' nay be defined >^y the relations r,5 V^"P5^ = Jo (Pe'' - P"j^t dy and ,-.5 Q'"' (cos o.)p5U = j/q (pf^lT - pu) (cos a) y ay so that the momentum equation becomes — (]^ f "^ + i^ CCS a , + 2; \1 7J + (r+.6 + G cos a) = ^ PfiU^ ;i^) In order to permit inte "ration of the momentum equation for comjoresslble flow about a bod;/ of revolution, equation (Lj.) should be written in the ssjne form as that for t^'-'o-dlmensional ccr.prsssible flow (equation (2)). The approximation is therefore m.ade that for flow on the body of revolution r^e(Pc + 2} + Q(cos a)i^- + 2) Q / = (Hq + 2)(r^9 + C cos a)K CO^TFTDEIITIAI. 10 COMFIDENTIAL ITAGA ACPJ 1 -o . l6A22 / li_ + 2 \ Q cos a ; Q ^ \ 1 + '. / r^e \H. + 2 / Q cos a 1 + In order to detsr^:iine the Kagnit-ade of the ratio % + 2 -^i: -, the assijonptlon is made that the velocity distri- He + 2' ' butions through the bcundarv layer may be approximated by pov:er curves of the type u / Y \ l^ - = 1^1 U \ / "Mien the flov/ is incompressible, the definitions of Q''' , Q, and E^ may be used to obtain Q* 1 + m and Therefore m 2 + m E^ = m ^'i _ 3m + 1 Hi + 2 3m + 2 for inccmnressible flow, prom L'F(,^/?)ydy=l/%(yl'^ ,^ UO ^ uO it can be sho:vn that got>ifid?;ktiai. IT AG A AGR ¥o . l6 ^2Z CCNFIDZ'KTIAL 11 for corapressible as well as for inco?npre3si"ble flow. 3""" use of and fig\;re 2_, v>'hlch is discussed later in connection with the effect of coi-ZDressibillt" en H^ , the ratio -^ K, 4- 2 • can be e'/alMated for co^ioressilole flow. For all cases of + 2 imsenarated flew, the valu.e of the ratio — differs H., + 2 fron uniity oy less than 20 percent and, for raost cases, by O CO 3 Cl rou^vhly 10 percent. Thus, even when — is not a small fraction, the apprcximation that K =1 is not far froiii true. alien, in addition, it is noted that ^ cos g . _c_ ^t9 ' ~t and is therefore small over most of the body, the approxi- mation that K = 1 is permissible. If PI =1 is used in equation (Ij.), it becomes ^!rj-9 + Q cos a] + — '■ r-— (j'-l-Q + ^ °''s a) + r — [vi-B + Q cos a) = rtTo or bv substitution of 2 for t3 + ^ cos a •^ t 63 /iic + 2 dU 1 6p^ rtT^ ox y ij ox p^ c-y pj^u- This equation for Incompressible flow has been given in reference p. The principles used in the derivation of equation (5) •A'ere the conservation of mass and ITewton's law of motion. The approximation v/as made that for flov/ on the body K = 1. CONFIDEI-ITIAL 12 CONFIDSTJTIAL NACA ACR No. l6a22 Morientu'n equation for coi:ipres3i"ble flov; ovsr iiiner surface of a j-cjiid channel .- ',,li9n che flov/ in a round channel is siich that the flovi throughout the houndary la^-er is approxiniately parallel to the "-vail and a region exists outside the hoij^dary layer in vvhich viscosity has no effect, the ho"jndar3r-la77er momentiim equation for this flow is the same as equation (5). In the moinent'uni equa- tion for flow over the inner su.rface of a ro^ond channel, the pressure change across t]ie bouJidary layer is assumed to be negligible. All of the S3nnbols in the momentum equation for compressible flow over the irjier surface of a roujid channel have their- meanings ujichanged except for the distance y, which is positive when measured inward from the wall so that r =: r^ - y cos a The derivation of the monentuin equation for flow over the inner surface of a round channel follows the same pro- cedure and invol-\7e3 the same ass-'umptions as the dei'ivatlon of equation (5). The deri\'-ed equation is _ '^To + '4/, + 1 _ (o) bx \ U 6x Pq bx y P^^ where ij/ = r^O - ^ cos c Effect of compressibility on lig .- Since tiie ratio of the dis-olacement thiclmess to the momentum thickness H^ occurs in the momentimi eauation, the effect of com- pressibilit:/ upon this ratio should be considered. The equation for Hq is XT --C C do \ ^5 7 CONFIDSrTTIAI CCFFIDEIITIAL 15 The density variation through the ooundary lajT'er can be obtained from the velocity distribution through the boujndary layer hj restricting the treatment to the case in v;hlch there is no heat flov/ through the surface and the effective Frandtl number is equal to Vuiity. The equation c„T + ^^ P 2 constant is then applicable to flow in the boundary layer. W'p.eii the additional restriction is made that the static-pressure variation throv.gh the boLindary layer is negligible, the expression Po becomes _P_ P6 -[--& Y d. "o r - [- m. 1.;q anc ;he express ion for De C O"!?- 3 e 5 pi 1 + 1 _ 2 - 1 2 -m _ 1 2 T.. 2 7 - 1 - - d U/ b Yvhich can be reduced to G Y - 1 o 1 ^ ^ -" _ 2 . 1 5 h y V - 1 - p w 2 ■'° -(1 - -)^l Y - 1 2 ^"O ^ Y 1-.^ , 2 2 ;ONP ID3NTIA] ill CONFIDENTIAL IT AC A ACR No. l6A22 Let 1 + X^-J: Mo^ ^. ~ . X2 (}lY ^-^-^ M.2 V-^o/ P or where ^= 1 then /'"L u /' 11 \ I . (la - i) / LAilii c| (7) *'o ^' - lu/ By application of the same procedure to the definition of 5'''-/o , then f = . - (x2 - X) r' ^^X^ a| ,8) •Jo ■■ - Vu7 Large values of \^ mean siaall values of Mq • The assumption v/as made that u ./y\1A and val.ues were chosen for X^; B'-'/S and 6/5 vvere then calculated for a range of \ bet?/een I.5 and lli..0 v/ith m - 3 J h, 5 J 9-^c! '] . The curves of 1/Hc ^^^ given in figure 2. Since - - 2 + ^ -^ ' mi CONFIDENT I AT. HACA ACR No. l6/\22 COJJFID'^NTI AL 15 and sines H appears in the moii'ent.ixm equation, the curves of figure 2 are designated by values of Ki rather than m. PoT;er curves are usea. inerely for convenience in computing the effect of ccmpresslbillty on H« The value of He for the Blasius flat-plate orofile foz" laminar boundar^^ layers (reference 2, p. 38) has been computed for various values of \ and a Prancltl number of unit3:^ by use of equations (7) and (8). The computations were repeated with the velocity distribution for laminar • flow over a flat plate at Mq ~ 2 (reference 11). ?.Taen He ^"ss plotted against \, the results of both computa- tions were practically identical for small values of \. On'' J the results obtained for the Blasius profile are therefore presented in figure J. Eqi.iations (7) and (8) show that although the velocity distribution through the boundary la^^er is assumed to be independent of position along the surface. He n^-^-y vary with surface position because of its d.ependence on A,. For integration of the momentum equations, the dependence of He on \ may be taken into account by approxim.ating the curves of figures 2 and 3 by the equation He = X - 1 ^^ ) \Uo/ + Hi (9) - 1 The values of a chosen to fit the curves of figures 2 and 5 '■''"ith sufficient accuracy over miost of the range of \ are plotted against H^ in figure .'4. Integration of m.om.entum. equa-tion for two- d imens 1 onal f low . - Before equation ( 2 ) can oe integrated, (Hc+ 2), 1 ^•■0?) Tq — -^ and 7- should be re-Tlaced by functions of ths Po ox Po^- velocity dlstribvition over the body, the free-stream. Mach num.ber, and the mom.entijmi thickness. The term Hq + 2 is replaced by its ecuivalent - — _. . + (H-; + 2 "i . B^r use ^ - i:M\ .1 ^ " u / COI^TFIDENTIAI, ..Uo./ 16 CONFIDENTIAL NAG A ACR No. 1.6/22 of the .!?ss law P5 _ _P_ RTg the eauaticn of moticn 6p _ dx P5' 6x ths Bernoulli equation for compressible flow P CpTo + -J 'D-^O Uo^ and the equation for the velocity of sound Co = (y - 1) °i: 1 '^Pfi the term -^ — - can be written as pJ-o Po 6x 1 ^p^ p'lL') ^' Wo / \-'q/ dx P5 ^x (y - 1) r - Oif \Uo/ J (10) The local skin-friction coefficient ^ is expressed P6U2 as a pov'er function of the local Reynolds number based on the boundary-layer momentum thickness by '^o _ k _ , fu\n -n 11) By analogy v/ith the work of reference 11, the viscosity and density used to calculate Rg are those at the outer edge of the boundary layer. The viscosity at the outer edge of the boundary lawyer is assumed to be given by '•o \To/ (12) where w = O.768 for air (reference 11). COl^IPIDl^NTIAI, IT AC A -ICn no. l6A22 CONFIDENTIAT. 17 The dePaSity at the outer edge of the boundary layer is given by ^-11 £0 Po ^ -iff ^ - 1 ■Y (15) in -A'hich the flow outside the boundary layer is as seamed to be adiabatic. Equations (12} end (15)"'are used to give the kinematic viscosity u In equation (11) as a function cf the velocity distribution over the bod;/ and the free- stream Mach nunber. Equation (2) may then be v;ritten as 6G + 9< Wo/ 'TT "^ "Utq/ ku' ,n in ox r (y- 1) IP' I J This equation is a differential equation of the Bernoulli type. 'Ciien it is made linesr and integrated by standard methods, the result is ^ -7 v^ - fe)-. ,. - PL- k(l + n) n 1) ^ Y-1 U y-0/ /L ^ ^^■^a(l.n) v d- L Uo/ , ^1+n , (Ei+2)(l+n) ! 2 ^ (111 r-^^1^ i G( ^ J ■ l+n ,^-T J-, la(l+n) Ur lU) ;C^TFIDENTIAL l8 CONFID'En'TTIAL IT AC A ACR He "l6a22 where 9i/L and Ui/Uo ^^'^ the values of g/L and u/Uo at — = — ^. The value of j? is always greater than 1. By use of the Bernoulli equation for corcpressible flow, -^^i can be shown to be always less than ^. VJhen Mq \Uo/ approaches zero, equation (114.) becoraes equation (1) of reference 6, that is, the integrated moir^entuin equation for incompressible flov/. Integration of ncmentum equation for flow over a body of revolution.- The assurrotions concerning Hq are the same as those for the two-dimensional case. The equations for -^, — , and — — ~ are also the same ^-■o P5 P5 Ax as for the two-dirnensional case. The expression for the skin friction, however, involves the aoprcxirnation that e 9 can be replaced bv -±—. Tnis approximation is equivalent to nesrlecting the term — cos a in the - ° v^ equation e = -t^ - -— cos a ^t ^'t Q 5 Since the vali-xe of — cos a is ali'/ays less than rr"9, rt -^ ^t it follows that the aoprcxlrnatlon is justified only in cases in v'hich - — is a small fraction of unity. In r-h -^ regions near the tail, therefore, the accuracy of the approximation for the skin friction m-ay be expected to decrease. The momentum equation for flow over a body of revolution (equation (5)) indicates, hov;ever, thst the contribution of the skin friction to the boundary-layer thickness beccm^es less imiportant as the tail is approached. The approximation that fl can be replaced by -^ is r-HTo therefore allowable and the term %- in equation (5) may be v/ritten l^.^Et__^ (15) P6l> pn,jn GCNFIDENTIAL FACA AGR NO. LdA22 C0N"PIDENTT AL 19 and T- are replaced by Waen the terms Hp , ^ P5 6x ' P5U' equations (9)> (10), and (15'> then equation (51 becomes ox ■ n TT Vs^ - ^Uo ^ Hi ^ 2 Up \Uo/ dx 6x U o VJo/ \2"i .' ^ VUo/ j ^z' VUo This equation is a differential equation of the Bernoulli type. 'Tnen it is made linear and integrated by standard methods, the result is ^ ^■ / X ^ r-, ^tt\ "ja \^ o / ,. A J t 2' r/n\ ]CHi+2) Wx A:/L 1 — 1 ^ ^•J VJo/v/r P ■'^O-'x/l ! k(l+n) / I. \ 1+n (^-l)"fw--i-) r.-/t , [( H i +1 ) ( 1 +n ) +1 1 r ^ , n \ 2I (nw+'^ ) / rt \ 1 i u V "^-rj i ^ ' n^T^ I \iJoy v^o/ J \ 1-max/ +n dxo/L i^^-(^)| ~{ a(l+n) 4 L ^_^^(H-n),^^(Hi^2)(l 2 I max / ^t 1+n) i^ Un\2i I 1+n iY- •1 ■'O/ J 1 1+n y-® a (1+n) where and Ui/Uq are the values of (16) and ^max ^t max COIJFID'illoTI.AI, 20 CONFIBENTIAI, NAG A ACR lAO . l6a22 Int8,;5raticn of iricrr-entum equation for flow over irai er su.rface or~""ar roimd TTninne l . -~The equation resulting from the "integration of equation (6), when the same procedure and assuiTiptions made to integrate equation (5) s.re used, may be obtained from equ5.tion (l6) by replacing 5 oj \[;. In the equation for \1/, the q'^.iantities denoting reference conditions. Uq i ^tynn -• ^^'^- ^"^o^ ^^® ^^"® condi- tions at a convenient reference "section of the channel and the length L is any convenient length. The -definite integrals occurring in equations (ill-) and (l6) and the equation for u,' ma^'- be evaluated by either an analytical or a rraphical method, whichever is more convenient. DISCUSSION Pefore equation (Llj.), equation (l6), or the integrated equation containing -d/ can be used^ it is necessary to know the velocity distribution over the surface, the con- stants in the skin-friction formula, and the value to choose for E±. The velocity/' distribution must be t'nat for the Mach number and Reynolds number for vvhich the com- putation is being made. Skin-friction formula s.- Tor the turbulent boundary layer, a n c w e r f un c t loh of Rg is used I'or the skin- friction formula. because references 7 ^--ci 3 indicate no noticeable effect of Mach number on skin friction, a skin- friction formula for low speeds may be chosen and approximated by To _ k P5TJ2 Rg^- as outlined in reference 6. One forrrula of the required form is that of Palkner (reference 12) If the skin-friction data available are riven in the form, of C(5 ~ R^j they may be converted to — ^ ~ Ra by use PoU^ COIJFID^FTIAL :ACA ACR ilo. hGi^Z GOiTFIDENTIAL 21 of the relations and L ami ne r bo undary layer.- Although equation (llj.), equation fib), 'and the~equation for \i,' have been derived, for turbulent boundary layers, these equations may be used for the approximate ccrnputation of 6, j3 , and \|; in laminar boundary layers by the appropriate choice of a value for Yi± and of a sl'in-fric'tlon formula. Although by use of the skin-friction relation for the Blasius flat- plate profile the effect of pressure gradient on the skin- friction coefficient is neglected, the error introdiiced is small so long as the average pressure gradient over the extent of the laminar boundary layer is small. The skin- friction relation then is To 0.220 P^t/ R9 where k = 0.220 and n :^ 1 . Although a sm.all decrease occrrs in k as the Mach number increases (reference 11" the value of k for incom.pressible flow (0.220) may be used in view of the aporcximation already made concerning the skin friction. Choice of T fi.- In references 1, 5j &ncl U-, the value of the boundary- layer shape parameter E was restricted to l.I|_, If equation (1I4-) is used, Hj_ Is given an arbitrary increm.ent AH^ , and — ~ 0, then the change / e ■■ T. m ( — I may be given by V I- /x \\ — = '_ii ) { - \'Ax/ Vl/ - 1 X where U>^ lies betv.een the mxaximum axid minimum velocity in the interval x, x^, . The term^ — I has been GCKFIDZni'TTIAL CONFID^NTIiil i-lAGA ACR No. l6:i22 evali::at8d for incompressible flow and a straight-line velocit/ distribution TT — TT. L - L.XO b>: The ex"orej-sion for / L, » \ - r becomes r^.^. W i i 1 - Aixo\KKi+l ) ( n+l ) +2+^Hi ( n+1 )} ' " r ; 1 J ■■p^ 4. ] >! /-ri + 1 ^ +2 1+1 Alth expression for (Hj_ + 1) (n + 1) + 2 + iHj_(n + 1) (1?) I - '^\ >r incompressible flow and the ~~-'^ h-'^JS is only approxlmstely correct for bi ^y/b cases m v/mcn -— ? L excessive error. c in e/L quatlcn (17) indicates that are possible if a in equation (9) is made to equal zero and Hi is replaced by He and rescricted to a constant low-speed value v/hen the actual variation of Hf, is large. Increasing the 0007" thickness, the lift coefficient, or the Mach number increases the variation of Ec over the surface. In the integrated equations the value of Ec ^''is.y vavj over the sxirface but that of H-' is assumed to be constant. Because Hj_ can usually be estlir-ated to within 0.2 for ci-ses in which flo\" separation does not cccTor, the error m e/f na'' is unlikei^ to be more than 10 percent. be caused b37' a poor choice of Hj_ When the velocitv increases in the direction of flow, the value of Hi may be taken ss 1.2. When the velocity decreases_ in the direction of flov/, the value of Hj_ should be increased from^ l.h-, for cases in Vv'hich the total change of velocitjr is abouo 20 percent of the initial velocity, to values near 1^7? i'c's? cases in which the total change in velocity exceeds JO percent of the initial velocity. Because the velocity profile for the laminar C0T«]FIDE1TTIAL KACA ACR NO. l6A22 COl'IFIDllNTIAI. 25 boundary layer has been restricted tc the Blaslus flat- plate profile. Hi is equal to 2.6. For the larninap boundary layer, a then has the value 1.2. Full thickness of boundary layer .- The full thickness of the tvirbulent boundary lay/er in the case of tv/o- dlrner.sional flew may be obtained by use of the relation 5 ^ /9_V/5N L VlAg/ For the body of r evolutior, the full thickness of the turbuleni; boundery layer iriay be obtsined, after ^— 2 -■max is comouted, bv use of the relation Vt M \/^^ / •-'/ i^t 2)\52, •. ^max/ \' \ -mar/ ' \ '^max / cos a r-H /(-) ^™^^^ -'--■ cos a Ve-/ (18) Curves of 6/5 and O/o against A, are given in figures 5 ^^id 6 for four values of Tij_. The variation of G/5 for the Blaslus flat-plate profile Is given in figure 7 f'-'^^ use in computing the full thickness of the laminar boundary layer. For flow about a body of revolution, Q/52 may be neglected since the lairanar boundary layers are usually thin. The expression for the full thickness of the laminar boundary layer on a body of revolution therefore becomes '^max \ V^mxax/ \ / For flov/ over the inner surface of a round channel und are replaced by v/hen the boundary layer is thin, equations (l6) and (19) r-v -max COTjFIDENTIAL COITPID^^ITIAL WACA ACR No. l6/i22 and ^t max (21) In order to obtain a general statement regarding the effect of density changes on the momentum thickness, com- outatlons of Q were made for a linear velocity distri- bution and a range of Mach number. The velocity distri- -■ bu.tion for ~ — := 1 was defined by T, il- . 1 -^ b^ IJo L The only variable \v9s Mq . The. curves of 9 against Mq are given in figure 8 for b = and ±0.2. The results indicate that the effect of density variation becom.es important only at Mach numbers exceeding unity. CCNCU'SIONS A comparatively rapid m.ethod is presented for the computation of bound a.ry-layer momentum thicknesses for flow over two-dimensional bodies, over bodies of revolu- tion at zero angle of attack, and over the inner surface of round charjiels all in compressible flow. The computations indicate that the effect of density changes on the momientum thickness in flows with pressure gradients is small for subsonic flows, Langley Memorial Aeronautical Labora-t-ory Rational Advisory Gommiittee i or aci-onautics Langley Field, 7a. CONFIDENT I AL. IT AC A ACR No. l6/^2 CClMTniDENTIAL 25 RliPT^RENCES 1. Young, A. D. , and •.'"'inter bet torn, N. E. : Note on the Effect of Compressibility on the Profile Drag of Aerofoils in. the Absence of Shock Waves. Reo . NO, B. A. 1595, R.A.E. (British), May I9IJ.O . 2. Prandtl, L.: The Mechanics of Viscous Fluids. A/ol.m of Aerodyna.-mic Theory, div. G, sees. iJ^ and 17, "iV. F. Durand, ed., ji.d.ius Springer (Berlin), 1955? pp. 8)4.-90 and 102-112. 3. Young, ii. D. : The Calculation of the Total and Skin Friction Drags of Bodies of Revolution at Zero Incidence. R. & M. No. iGy.lj., British A.R.C., 1939. ii. Squire, H. 3., and Young, A. D. : The Calculation of the Profile Dr?g of Aerofoils. R. S- ¥.. No. 1858, . British ;\.R.C. , '1938. 5. von Doenhoff, Albert E. , ar\d Tetervin, Neal : Determi- nation of General Relations for the Behavior of Turbulent Boundary Layers. NA.GA ACF No. 3-G15, 19<'4-3. 6. Tetervin, Neel: A Method for the Rapid Estimation of Turbulent Boundary-! ajrer Thicknesses .for Calculating profile Drag. KACA ACR No. LLLGli4., I9I1I4-. 7. Frossel, "' . : Flew in Smooth Straight Pipes at A/elocities above and below Sound Velocity. NACA TM No. P'-'iii IQ^B fa. Theor'orsen, Theodore, and Regier, Arthur: Experiments on Drag of .Revolving Disks, Cylinders, and Stream- line Rods at High Speeds. NAC^ ACR No. EU.F16, I9I1J+. 9. Eeenan, Joseph H., and Neumann, Ernest P.: Friction in pipes at Supersonic and Subsonic Velocities. NACA TN NO. 963, 19li-5 . 10. Fluid Motion Panel of the Aeronautical Research Committee and Others: Modern Developments in Fluid Dynamics. Vol. I., 3. Goldstein, ed.. The Clarendon Press (Oxford), 1938. / N? I DENT I AT. CCNFIDEKTIAI MAC A ACR No. l6A2.2 11. ■Rrainerd, J. C'., and '^jnons, H. "J.: "Effect of Vari- able Viscosity en Bcundsry LP-yers, v/i tli a Li^^- cussion of Drag Measurements. Jour. April. Mech., vol. 9» no. 1, March 19^2, pp. A-1 - A-6. 12. Falkner, V. ¥,.: A Ne"; Taw for Calculating Drag. The Resistance of a Smooth Flat "oiate with Turbulent Boiindary Layer. Aircraft Enj^lneering, vol. XV, no. 169, March 19^5 ^ ??■> 65-59. GOJTS'IDT^NTIAL NACA ACR No. L6A22 Fig. 1 z UJ o o o UJ o o o •H 4) F4 o ^*. O d o n a n •< > O < UJ < -1 < s 1^ •2 Z O -i < -J H < Z LI H 9 UJ Z , 8 2 o 1 o , \ \ \ \ \ ^ A i v \ V V \ \ \ \ s\ \ \, X ;:< ^ ^ ^ •^ ■^ _ •o b cr» < lO »0 o n ?( iH cd > a 7i o o -p o o a o •H -P at ft h u to •H 00 *o ^< o X NACA ACR No. L6A22 Fig. 3 >- -a o > a -< _i -< z (/) i- § Ul. z E 8- -1 < -1 < - ■ z Z UJ a ' i UJ ' Q o Li. 2 u l5 ■ _ i \ \ \ k \ V <>^ V ,^ ■^ -~- — !n -- ^ o> -^ •r, »0 ^ N "k^ a o •H ■P :i •H -P n ■P •H o O O •P at iH P« I ■P m a cd « h O •P •H O g iH (d *^ cd I •H Fig. 4 NACA ACR No. L6A22 -I < UJ O U a O W)— 5- S Ul \ < \ < z o 2 \ uj- \ i \ 8 \ \ _i . \ < \ FIDENT \ \ \ Z 8 \ \ 1 1 1 Oi «0 C5 VO ^ ^ :ir •H O o •H +» (U •H I ^ '\) NACA ACR No. L6A22 Fig. 5 • II X »o - ~\ .J X 1 1 . .\i - ■X 1 1 1 1 1 1 NATIONAL ADVISORY :OHMITTEE FOR AERONAUTK 1 1 1 1 1 1 I -1 < -1 < H Z z UJ o ' u. z o o 7 o U 1 \ 1 1 \ \ 1 V \ \ \ \ s. V 1 s, V \ ^^: ■V, \ •v "^■^ : ^ =i =*- — — - b Ol < k w o n PS rH o 0() > O Si o d o -P cd I 05 CO Fig. 6 NACA ACR No. L5A22 < LiJ a o > H -or ^2 r— II - ^ II ~ II . -1 — II -J L ■ X z o ui 1— !5 t -,z 8 -J- < 1- z 9 Li. Z o u \ \ \ V \ \ \ \ N, \ \. \ ^ ^^ V ^l""^ — ^^__ ^ b o^ N ^ W a -o (/> i _< -1 -< X Ui < I z o "!< -2 UI UJ H X~ O -1 < < 1- Z UJ Z UJ a 9 z z 8 8 I \ \ V \, V ^ ^ — - _ b 0^ N •o ^ ^ 55 <0 M •o O Qi