MCA-/.-/3 NATIONAL ADVISORy COMMITTEE FOR AERONAUTICS WARTIME REPORT u ORIGINALLY ISSUED January 19^5 as Advance Restricted Report IAL19 CHARTS OF PRESSURE, DENSITY, AND TEMPERATURE CHANGES AT AN ABRUPT INCREASE IN CROSS -SECTIONAL AREA OF FIOW OF COMPRESSIBLE AIR By Upshur T. Joyner Langley Memorial Aeronautical Laboratory Langley Field, Va. 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 war 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. DOCUMENTS DEPARTMENT L-13 Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/chartsofpressureOOIang 3<*l NACA ARR No. liiL19 RESTRICTED NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ADVANCE RESTRICTED REPORT CHARTS OF PRESSURE, DENSITY, AND TEMPERATURE CHANGES AT AN ABRUPT INCREASE IN CROSS-SECTIONAL AREA OF FLOW OF COMPRESSIBLE AIR By Upshur T. Joyner SUMMARY Equations have been derived for the change in the quantities that define the thermodynamic state of air - pressure, density, and temperature - at an abrupt increase in cross-sectional area of flow of compressible air. Results calculated from these equations are given in a table and are plotted as curves showing the variation of the calculated quantities with the area expansion ratio in terras of the initial Mach number as parameter. Only the subsonic region of flow is considered. INTRODUCTION The well-known Borda-Carnot expression for change in o res sure when an incompressible fluid passes an abrupt area expansion has long been used for estimating the pressure changes in compressible air flow at an abrupt expansion. This method is simple but not exact. The expressions for pressure, density, and temperature ratios given herein are for subsonic flow and are in precise agreement with the exact expression for the velocity ratio in a compressible flow at an abrupt area expansion developed In reference 1. In the present report, Mach number, or the ratio of air- flow velocity to existing sound velocity, is used as a parameter; whereas in reference 1 the parameter was the ratio of existing air-flow velocity to the velocity that the air would possess if accelerated isentropically until Its velocity was equal to the then existing local RESTRICTED NACA ARR No. LI4.LI9 sound velocity. This difference in parameters must be considered when equations from the two papers are compared. By use of the same three fundamental equations used herein, a somewhat similar equation showing pressure and density changes across a shock loss in a pipe of uniform cross section was developed by Hugoniot and is given in reference 2. The expressions obtained herein for pressure, density, and temperature changes in a compressible flow at an abrupt expansion are too involved to be of practical use. The present computations have therefore been made and are presented in tabular and graphical form. SYMBOLS A cross-sectional area of flow, square feet a velocity of sound, fee b per second f area ratio (A1/A2) M I T ach number ( V/a ) p static pressure, pounds ver square foot V velocity of flow, feet per second Y ratio of specific heat at constant pressure to specific heat at constant volume, dimensionless R universal gas constant, Btu per slug per °F T absolute temperature, b r 60 + °P p density of air, slug per cubic foot Subscripts : 1 before abrupt expansion 2 after abrupt expansion NACA ARR No. 1J.1LI9 ANALYSIS From the fundamental equations for the conservation of energy, of continuity, and for che conservation of momentum, equations are obtained that give the variation of the pressure, density, and temperature ratios with the area expansion ratio f in terms of the initial Mach number as parameter. Figure 1 shows the conditions of flow assumed for the present calculations. The static pressure at a (fig. 1) is taken to be the same as at b Tor subsonic flow, as has been proved experimentally by Nusselt (reference 5). Uniform velocity distribution before and after the expansion is assumed. The ratio of specific heats y is taken as I.I4.O5. The fundamental equations are the equation for the conservation of energy V l 2 , Y gl = V 2 2 , Y P2 (1) 2 y - 1 p 1 2 y ~ ! P 2 the equation of continuity, P 1 A 1 V 1 = P2 A 2 V 2 (2) and the equation for the conservation of momentum p 2 A 2 V 2 2 - P 1 A 1 V 1 2 = -A 2 (p 2 - Pl ) From equation (1) or YPp (3) Y ' \ 2 . . _ Y " VVf,. 2 + P 2 p l ,1, 4 7kCk ARR No. li|L19 Prom equation (3) = 1 + Y fM 2/i - ?if) (5) 1 V p 2 / Prom equation (2) V 2 P x A-[_ p ] _ v l P2 A 2 P2 When equations (5) and (6) are substituted in equation ()].), Y " 1 p Y - i/PA 2 ? ? Pi Pn ? -5 ?/ p l\ 2 1— ~ F X 2 + 1 = 1__(-1) f2 M 2 + Lk + lll Y fM 1 2 _ Y f2 M 2/_J.\ 2 2 \P2/ P2 P2 ~ \ p 2/ or 'Pl\ 2 p 2 Y + 1 Pi / ?\ y^t 2 - M n 2 + 2 — ) f^Mi* 1 — — (1 + Y fM-i )+ -L-i ± =0 (7) When equation (7) is solved for Pt/p 2 an d the resulting equation is inverted, P 2 f2:,! 1 2( Y + i) Pl 1 + Y f --1 2 -v/2YfM 1 2 (l - f ) + 1 - 2f 2 M-, 2 + f 2 !,;^ In order to obtain an expression for the -ores sure ratio, equation (3) is substituted in equation (5) and the following equation results: (8) Vp 1 + Y f % 2 + YV 2 Y fM-| 2 (l - f) + 1 - 2f 2 ;,: 1 2 + f 2 Mi^ — i 1 i ( q ) Pi Y + 1 MCA ARR No. liiL19 5 By differentiating P2/P1 with respect to f, it can be shown that the maximum static-pressure recovery for any value of Mi is obtained when / Y - 3 Y +\j -J~ ¥ l 2 + 1 f = 5— (10) 2( Y + 1) - M x 2 The locus of maximum pressure ratio is shown in figure 2 In equations (8), (9), and (10), the sign of the radical has been chosen so that the results obtained are in the region where the assumptions are valid. The temperature ratio is obtained "oy use of the general gas law and the computed values of pressure and densit;/- ratio as £1 p l RT, - RT P2 P2 T2 P2/P Tt (11) P2/Pl Figures 3 and Ij. show the variation of density ratio and temperature ratio with area expansion ratio. In order to make the results shown in figures 2 to ij. usable in cases for which only the conditions after the expansion are known, the value of Mp in terms of M]_ and f is given in figure 5. If M 2 and f are known, Mj can be determined from this figure. The relation plotted in figure 5 I s developed as follows: NACA ARR No. li|L19 '^2\ 2 . (v 2 A 2 ) 2 YP2/P2 V yPq/p 1 ■d) v 2\ 2p l p 2 P 2 Px By use of equation (6), \M Ox 1 M; P l P l fM RESULTS AND DISCUSSION The calculated values of pressure ratio, density ratio, and temperature ratio are given in table I. The values have been computed to 8 decimal places because of the form of the equations, in which small differences in large quantities are involved. In the region where M and f were both small, that is, 0.1 or 0.2, it was necessary to carry some of the calculations to 12 decimal places in order to obtain smooth curves for the quantities calculated. NACA ARR No. li|L19 7 As in the case of incompressible flow, the calcu- lated changes occur gradually after the abrupt increase in cross-sectional area of flow, and the calculated and measured results are in best agreement at a distance the order of 6 to 10 diameters of the large cross section downstream from the abrupt area increase. The comparison of pressure ratios for compressible and incompressible flow is shown in figure 2, in which a long-dash line gives the pressure ratio calculated on the basis of incompressible flow for the same initial conditions that are assumed for compressible flow at an initial Mach number of unity. It is evident that the effect of compressibility is vanishingly small for values of the area ratio of expansion below about 0.25 The short-dash line in figure 2 shows the pressure ratio to be obtained with isentropic expansion and an initial Mach mimber of unity. The experimental points from reference 3 shown in figure 2 were obtained from the only experiments known to the author in which pressure ratio has been measured at an abruot expansion with compressible gas flow at high Mach number. These data were obtained for an area expansion ratio of 0.21+6, however, for which the difference between compressible and incompressible flow is insignificant. These experimental results agree well with the calculated results but are by no means conclusive. Agreement of experimental with calculated values at an area expansion ratio of 0.7 or 0.3 would be conclusive evidence of the difference in pressure ratio obtained with compressible flow from that calculated by the Eorda-Carnot formula for incompressible flow. An experimental investigation of the changes in pressure, density, and temperature at an abrupt increase in cross-sectional area with compressible flow would serve to determine corrections for the effect of nonuniform velocity distribution and friction on the idealized results obtained from the present calculations. Langley Memorial Aeronautical Laboratory National Advisory Committee for Aeronautics Langley Field, Va. 8 MCA ARR No. LUL19 REFERENCES 1. Buseraann, A.: Gasdynamik. I-Iandb. d. Experimentalphys . , Ed. IV, 1. Teil, Akad. Verlags^esellschaft m. b. H. (Leipzig), 1931, pp. liO3-.l4.O7. 2. Ackeret, J.: Gasdynamik. Handb. d. Fhys., 3d. VII, Eap. 5, Julius Springer (Berlin), 1§27j P- 3 2 5 • 3. Nusrselt, W.: Der Druck im Ringquerschnitt von Rohren rait plotzlicher Erweiterung lDeiirx Durchfluss von Luft mit hoher Geschv;indigkeit . Forschung a. d. Geo. d. Ingenieurwesens , Ausg. B, Bd. 11, Heft 5, Sept. -Oct. 19i;.0, pp. 25O-255. NACA ARR No. L4L19 e J <* > m P o H ce < > c O fc o w H << X o M W H 1 << F- ■* Z K o W M Cm tn S 2! e • X pa u D < .J o I VI < IT': < o ^ « <« ST <: ce a o i-i g t* z 5 S3 h ^ (J o CO o o «: E w DC t> CO to M M ►J m o -3 O rC\ o O OJ CO A-UNO OJ ONnOnO OJ_rf U"\nO ONONCO CAA-O »-H »-4 _H-OJ KNH rH CO -d^O UNON a-oj_h>o o Ovaircoso oj oj oj rn rH__+uN.-H o nn. r-i unoj •aonojno c^-rvj o OOhoj nnno o\_h> rvj n-\ OOOOOOOrt(\|ift .-! ON__+rH OJVO_-tO a-a- A-rH^n a-j-oj o »h ihoo rH__tO O O OJ A-NO UNON no rnco o_hvoj_3co A-O fVJ OJ NNA-nO NNvO C\J NN£0 oj CNr-t c^i/NrH as_^-oyr) o o oj nnno 04H ono OOOOOHHIM OJ_H> N"\ A-Onk\oj Onnncq hv£ a-co ono nnno no o oxo IO_^H*0 OCO ONONUNrH no ih_^co A-rHNO a-onco OnOnOnO n-\nn.nO OJC0__t OJ OJ A-O OJ_H-CTst*-OJ A~ O -hoj ircprg t> ~z±*\tc\ O O O O O nl rH OJ N"V_t oj rHCO rH oj a-ojoj o un _H> rH UNNO A-NO NO N"\nO OJ NO UNnOCO OnO A-nNunCO OO'AOHJ' ONnO NN.CO ON SnO ITNO O CNr-t O rH OJ rAO_)NO A-OJNO O ONCO O O O r-t OJ_3vO 0_d-rH OOOOOOOrH^-1 oj rM A~SO ^OvO NN.NO ON-J- O A-rAA-NO OJvO-J-ONO A-lArH UNuNA- rANNCOCO OJ r-* ONNN.NN.NNnO rH NN.CO UNOJ rAsOCO OJ_3\_H;ONOJ rH NO .^NO^N, A-CO ONHf- o o rH oj jvp onnnonun OOOOOOOHH(\J O NN.UNOJ ONrH OVd-_H;f\J UN rHNO r-OJ_4"r-l\OVO i-t OJnO C^rr\NS_-TLTV-3-OJ ON CO lf\CN_JvO LfSCO rHCQCO rTNNO rHNOOO ONrH ON.d'O rf\ tC\t-4 NO O.J-0 f^ONNO O iHrT\ir\ONrr\OM^^Nrf\ O OOOO rH cHOirA^ CO OJ^H-lTNrAM>f*-0 O »A _d-iOfOJ rH OOOOOrHrHOJ K\JH; 1/NOnO O^rH KNirNO^lfNt^- r-r-^o^ONO o oj_^kd IT« C^r-CO _^rH rr\fAf— rH t^-j-J-o3ior-co_^o k>u-*o cNirvo>_H/_j-KVjH- rf\H"\Q_-^NQ LT«N^OV^CO O f~t fAt/^CO OJ r-OJ 0^^0 OOOOOrHrHOJNrA Ofti rH t-U^ rH CO »AK% ON ONrAONO f-rAfANO t— UN f-i -d" UN UN_^NOfAO^ NO ON unN^o unonO rvj ONf^ t- ONCONO UNONONSOCOCO UN OJ rH NO t*-J-t— f-rANONO O rH OJ J-f-O^tON-^O OOOOOrlHHfyo r^-NO rH co aj r- uno no rvj rH rANO_J-OJ LAO LAlAUN CO NO lAONO lAOO t^-OJ O _J-UN-^rH_^hAOJ ONNO_d- OJ O OJ O f\J rH LA_j+0 OJ OJ y\ONC>NO i-* 0_3-NN.NO O O OJ fALTOD h3aD OJ OqOOOOhHH(\| ONnO rA_H;rH rAOJ„_+rAuN •ArArHCONOOJ OjIB-^ t— -ZfNO OJ UNONrA__t75 O t*- ^0 LAt^O__trAl^ rA O r^ OJ OlAOJ UN_^C*-|_NCOUN rH LA rH O r-i UNrH O H IT O O rH OJrA O O O OO r-t OJ rA_H^LANO f— CO OnO a f^r\j>0 O CTvi/NCNvD O i/\ COvptNJCO O t~-CO OV_tv£) or-oovoso f-tx-N--tco rH t^f-r\j njco <-i f\i a] o o OHPoySr-Hi^ovo ooooooHHnjftj KVj*33 rT\fT\hT\0 CT^vO ON -J-OJ COVOO_JTH e-OJ J-OJ O ►f\O__+t~-C0 0\LT\1\) O r\4C0 OS-Z}-itC5 H ONKNrfN O O <-(«> OOOO 6^^S^ UNOCO OJCO r-lArH__tf^ OCp OJNO CAUNrH_H;HN,UN >-i 0__trHNO •AONC^-nO ft ono onco o 'Aco__tt^-oj ONO O f— ONCO r-i ONCTnO rHCOCO OJ O i*\rH OJ CO ON O O rH NN.UNt-^0 rANO O OOOOOOrHrHrHOJ eg >-i ono_^oj uaOcono CO 0_J-ONf-t^-_ALAOJ i-i O O^no WNONlAt^-LAr— CO t— C-- rH »ACO O O «N C^NO NO NO r-t 'AuNf-C^OcO i-* rH NO lAC— rH CAO LArH rH O O rH OJ_^uNCD O hANO O O O 0^» O O r4r4r4 OJkACO ** OJCQOJ-^-OCO t^rAOJ ~ J-N-^nc A-^O V« OJ NO -3CONO K\ rA UNNO O rACO UNrArHJ _AO_j+OJ rH _4co_^-o un r-h-co fA *-* _-T ON A- A-CO rH NO OJ O O O O rH OJ HN.UNNO CO O OOOOOOOOOrH CO 0__*rH CAA-ONONOJ ON lArAT3/UN_H^UNSO OJ ONOJ OJ CO CO NO HN U"V_+CO NO NO o o c»N__t rH no 13/ on r—o NNOJNO A-rArAflONOCO NN. O r-t OJ_H^t— O hA A- rH NO OOOOOrHrHrHOJOJ OOOOOOOOOO UNO i-i ONrHCO_^NO O ON OCO KMorfiAH OJno__* rHNO_zKO OnO A-rH ONnO NO CO fy ONrH UNrH NO OJ A- OJ 03_JH:__+0CQ A-O A- ONA-rACO (V UNNO A-CQ A- ON ON ONCO CO A-*^> UN__i"rA QN ON ON On ON ON ON ON CA ON &CO rH OJ O rHNO OJQO *A O O OJOO-Cf'^NCO i-i OCO ON UN rHNO -3/ KNrH ONCO O A-NO OHH UNrA.NO CO O ONONrHNOCOOA-OJ C5nO A^ rH OJCO rH OJ ON UN ONrH On ONCO NO UNrA O CO UN rA ONO ON ON ON ON ONCO CO CO i-i OJ fA_^UNsO A-CO CAO H \ ^ f-i 0_H*rH_J-sO UNUNO OS COCOCOnOUnOnO UNODnO A-rH UNrH -AOJ O A- UN ON COCO O A---»'rH r-t NN.CAOO rAUNA-ONA-^O UNCO_H^ON O fi rANO i-i CO CO OJ rA rM O O O O r-t r-i OJ__JnO ON OOOOOOOOOO A-CANN.A-rAO_HKO rArH UNCO NO O l-i no _-JUNNO CO KSONONnO OnS-UNi-H UTV_H; ►A A-CO OJCO fAC^rH ^ArH A- ONCO t>_H>NO OJ UN UNNO -d-UNrHCO O r-i uNrArA_J- CO rANO UNNNOD rH OJ _=J_3" rH A-NO OnjvOnO r-i ONrH A- O O r-t OJ__H>0 ONrH UNCO OOOOOOOrHrHrH O r-i UNA-UNUNCO OCO r-i OO ONOJ_H;CFNO rHCO UN UN A- C^NO A- NN NN__H-_H; CT __± A- ON r-i A-__jNO V NN O CT r-_J- rH UN O rA UN A-CO rH A- A- rHCO O UVtA_H OOH rA__*A-CTOJ i/^CT OOOOOOOrHrHrH rA_d-0JC0 OJ A-NO nn.NO O COCO^ rM t-i r^NNUNN© KN, J-OJNONO ONNO NN__tQCO O ONNO A-rH A-NOCD O On O ONO O O rH rH rH OJ rH OJ A-OO OJO OJCO 00 OJ O OO rHrAuN A- ON OJNO O OOOOOOOrHrHOJ OO O O O OO O OO UNO UNO lA ojo oj o oj Oh oj_hno OJCO CO OJO OO i-i NNUN OO O O O O O O O O O O O O O O UNO UNO O OJ O OJ O CAOJnO O un OJ CNON.HJ-OJ A-ONOJnO O O O r-i r-t OJ i-i OJ K\_tm\0 A-CO O^O 0) o M ccw Q CO O gi § ■* -J p o DO c o 3 o .-H cd u S-< O V-i a y^s'?y. ssM 7^^ 2» jd --— — H •*" -** -' ---"3 ^Jffy'-fi. -'- if? ! - _--'" i> -^ r~l m p-*pT ,^, ^ t \/ i °^ s;*> ^ /, a d> , • / JL ■H / / J f a > ^ c -~ V- z / c * ■ ■A t \ «-* s !/ r- : i J X w4 - / / I y, |j:v / fU J n w r-r \ \