The Change of Pitot across Oblique Shock Perfect Gas. Pressure. C.P. No W.J. Graham and Miss BM. Davis AERONAUTICAL RESEARCH COUNCIL

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1 C.P. No. 783 MINtSTRY OF AVIATION AERONAUTICAL RESEARCH COUNCIL CURRENT PAPERS The Change of Pitot across Oblique Shock Perfect Gas Pressure 1 Waves in a BY W.J. Graham and Miss BM. Davis LONDON: HER MAJESTY S STATIONERY OFFICE 1965 PRICE 1s. 6d. NET

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3 AERONAUTICAL RESEARCH COUNCIL C.P. No.783 FLUID MOTION SUB-COMMITTEE The Change of Pitot Pressure across Oblique Shock Waves in a Perfect Gas - By - W. J. Graham and Miss B. M. Davis December, In this note the vtiiation of pitot pressure aaross an oblique shock wave in the two-dimensional flow of a perfect gas is examined by means of explicit formulae. Results are presented over the full range of shock angles for Mach numbers from 1'0 to m. This information is not readily available from standard references on shock wave relationships, and is given here to assist in interpreting pitot tube measurements in flows involving oblique shocks. 2. By manipulation of the usual oblique shock relations and the Rayleigh pitot formula (see for example Ref. 1) the ratio of pitot pressure aorom such a shock is given by The pressure, p, is that indicated by a pitot tube: in aupersonia flow this is the total pressuxm behind 8 normal shocki in subsonio flow it is just the stream total pressure. Suffices '1' and '2' denote ocnditions ahead of and behind the oblique shock respectively, M 1s the Mach number, e isthe shock angle, and y the ratio of the speoifio heats (assumed constant). & is related to & and sin 0 by the equation (Raf. 1) 42 = (y+l)a & stie - 4(q ailide - l)(ylq sinpe + I) [2yQ sir?e - (y-l)l[(y-i)@ siride + 21 *... (2) For the case in which h& < 1, the pitot pressure behind the shock is identically equal to the total pressure, and the pitot pressure ratio becomes FA siipe~(y-i)@ + 2j 1 * c 2ybq - (y-l) 1 * -= PI [ (Y-I)I@.5tie + 2 ~42.dSe - (y-~) *... (3) Replaces N.P.L. Aero Note No A.R.C P&fished with the permissxon of the Director, National PhyslcalLaboratory.

4 -2- If ab, is sufficiently large these equations become, respectively, and for M, < 1 pa h - = (,P[ 5142 IY(ym,) IT ",":"," 1 '.. (la) hg = (Y+l)" - Il-y sirl e zy(y-1) sir? e '... (24 P?d - -= (coseo e)yz1.... (34 3. Using the above relations the ratio of the pitot pressure across an oblique shock wave in air (y = T/53 has been calculated over the full range of shook angles for various Mach numbers from unity to infinity and is shown in Fig. 1. The curves to the right of the dotted line correspond to subsonic flow downstream of the shook (I& < 1). 4. For the case in which aa, > 1, Pi is the total pressure behind the normal shock (at &) occurring at the mouth of the pitot tube; whereas Ps is the total pressure behind both an oblique shock (at &) and. a normal shock (at ag, < ML) again at the mouth of the pitot tube. The two-shock system leading to Pa has less losses than the single shock giving Pi, SO that Pe is greater than Pi. As the oblique shock wave angle is decreased, far a given upstream Mach number, the strength of the oblique shock decreases and in the limit, when the shook angle equals the Mach angle, I& = I& and it follows, therefore, that P,/P, is unity. For the case in which h$ < 1, Pi is again the total pressure behind a normal shook at &, but Pa is the total pressure behind a 'strong' oblique shock at the same Hach number. The latter shock has the smaller losses and so Pa is greater than Pii. As the oblique shock angle approaches 9o", the oblique shock becomes a normal shock and Pe/P, is again unity. Retween the two limits (Fig. I), for which P2 = Pi, the ratio P,/PI increases to a maximum value for a given value of upstream Mach number, 4. With increasing & this peak value of Pa/Pi increases, whilst the shock angle at which it ooours decreases. A limiting value of P,/P, equal to 6 oocurs at infinite la, and small a. In general, the rise of pitot pressure across an oblique shock increases with increasing Maoh number for a given shook angle. I. Equations, Tables and Charts for Compressible Flow, N.A.C.A. Rpt PB. D 3il?,Q/l/Wt.00 K4 l/c6 XL/CL

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7 A.B.C. C.P. No. 783 December, 1963 Graham, W. J. and Davis, MISS 9. Y. THE CXANGE OF PITOT PRESSURE ACROSS OBLIQUE SHOCKWAVES INAPEBFGCT GAS The pitot pressure behind an oblique shock wave m a perfect gas has been calculated and compared alth the pitot ressure ahead of the shock. Results, for air (y = v/5) are presented graphically for all shock angles a;dvarious Mach numbers from 1 0 to m. A.B.C. C.P. No. 783 December, 1963 Graham, W. J. and Davis, Miss B. Y. THE CHANGE OF PITOT PRESSURE ACFQSS OBLIQUE SHOCK WA-S IN A PKRFECT MS The pltot pressure be*hind an oblique shock wave m a perfect gas has been calculated and compared mth the pitot ressure ahead of the shock. Results, for air (y =?/5) are presented graphically for all shock angles &d various Mach numbers frcm 1-O to co. A.R.C. C.P. No. 783 December, 2963 Graham, W. J. and Davis, Miss B. 11. THE CHANGE OF PITOT PRESSURE ACROSS OBLIQm SHOCKWAVZS INAPERFECP GAS The pitot pressure behind an oblique shock vave m a perfect gas has been calculated and comparedwith the pitot ressure ahead of the shock. Results, for air (Y = 55) are presented graphically for all shock angles aid various Mach numbers Roan 1'0 to co.

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