REPORT 1135 - NASA

1 REPORT 1135 . EQUATIONS, TABLES, AND CHARTS FOR. COMPRESSIBLE FLOW. By AMES RESEARCH STAFF. Ames Aeronautical Laboratory Moffett Field, Calif. REPORT 1135 . EQUATIONS, TABLES, AND CHARTS FOR COMPRESSIBLE FLOW 1. By AMEs RESEARCH STAFF. SUMMARY normal force C_ normal-force coefficient for cones, q_S_. This REPORT , which is a revision and extension of NACA TN C_, specific heat at constant pressure 1_28, presents a compilation of equations, tables, and charts Ce specific heat at constant volume useful in the analysis of high-speed flow of a compressible fluid. h enthalpy per unit mass, u-_pv The equations provide relations for continuous o_e-dimensional l characteristic reference length flow, normal and oblique shock waves, and Prandtl-Meyer M Mach number, V.

2 Expansions for both perfect and imperfect gases. The tables a present useful dimensionless ratios for continuous one-dimen- pressure 2. P. sional flow and for normal shock waves as functions of Mach dynamic pressure, p V2/2. q number for air considered as a perfect gas. One series of charts heat added per unit mass q presents the characteristics of the flow of air (considered a perfect R gas constant gas) for oblique shock waves and for cones in a supersonic air R Reynolds number, pVl stream. A second series shows the effects of caloric imperfec- tions on continuous one-dimensional flow and on the flow Sb base area of cone through normal and oblique shock waves. S entropy per unit mass T absolute temperature 2.

3 INTRODUCTION U internal energy per unit mass The practical analysis of compressible flow involves fre- specific volume, 1. p quent application of a few basic results. A convenient compilation of equations, tables, and charts embodying these _tV velocity components parallel and perpendicular results is therefore of great assistance in both research and respectively, to free-stream flow direction design. The present REPORT makes one of the first such velocity components normal and tangential, compilations (ref. 1) more readily available in a revised and respectively, to oblique shock wave extended form. The revisions include a complete rewriting V speed of flow of the lists of equations, as well as the correction of certain maximum speed obtainable by expanding to typographical errors which appeared in the earlier work.

4 Zero absolute temperature The extensions are primarily in the directions dictated by W external work performed per unit mass increasing flight speeds, that is, to higher Mach numbers and a angle of attack to higher temperatures with the accompanying gaseous imperfections. 5' ratio of specific heats, Compilations similar to those of reference 1 have been given in other publications, as, for example, references 2 angle of flow deflection across an oblique shock wave through 6. These references have been utilized in extending the tables and charts to higher values of the Mach number. shock-wave angle measured from upstream flow direction The extension to imperfect gases is based on the relations molecular vibrational-energy constant presented in references 7 and 8.

5 1. Mach angle, sin-l_. SYMBOLS AND NOTATION. absolute viscosity PRIMARY SYMBOLS. p Prandtl-Meyer angle (angle through which a supersonic stream is turned to expand from a speed of sound A cross-sectional area of stream tube or channel M=I to M_I). Supersedes NACA TN 1428, _'Notes and Tables for Use in the Analysis of Supersonic Flow" by the Staff of the Ames 1- by 3-foot Supersonic Wind-Tunnel Section, 1947. When used without subscripts, p, p, and T denote static pressure, static density, and static temperature, respectively. 613. _, - ." .. _'" _ _ _. _ _'_r k_ _ 614 REPORT l135--NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS. FUNDAMENTAL RELATIONS. pressure ratio across a shock wave, PJ.

6 Pl THERMODYNAMICS. p mass density _. a semivertex angle of cone THERMAL EQUATIONS OF STATE. SUBSCRIPTS A thermal equation of state is an equation of the"form free-stream conditions p----p(v, T) (1). 1 conditions just upstream of a shock wave Several of the more commonly used thermal equations of 2 conditions just downstream of a shock wave state are the following: t total conditions (i. e., conditions that would Equation for thermally perfect gas exist if the gas were brought to rest isen- tropically) RT. * critical conditions (i. e., conditions where the p_-_-= pRT [therm per/] (2). local speed is equal to the local speed of sound) or c conditions on the surface of acone dp dp dT.

7 R reference (or datum) values P p _-----0 [therm peril (3). perf quantity evaluated for a gas which is both ther- mally and calorically perfect Eqdations for thermally imperfect gas therm perf quantity evaluated for a gas which is thermally Van der Waals' equation (ref. 9). perfect but calorically imperfect ( )p derivative evaluated at constant pressure RT a (), derivative evaluated at constant entropy P=V----b-_ (4). ( )r derivative evaluated at constant temperature where a is the intermoleeular-force constant and b is (). derivative evaluated at constant specific volume ( ),, quantity evaluated over a reversible path the molecular-size constant (see ref. 9, pp. 390 et seq. for numerical values).]

8 NOTATION. Berthelot's equation (ref. 7). The notation in brackets [ ] after many of the equations signifies that the equation is valid only within certain RT c limitations. )'or example : P=v--b v2T (5). [pelf] means that the equation is restricted to a gas which is both thermally and calorically where b is the molecular-size constant and c is the perfect. (By "thermally perfect" it is intermolecular-force constant (see ref. 7 for numerical meant that the gas obeys the thermal values). equation of state p=pRT. By "calorically Beattie-Bridgeman equation (ref. 10). perfect" it is meant that the specific heats cp and c, are constant.). [therm perf] means that the only restriction on the gas is p-= (l-- C_']Fv+Bo(l--b')7--_-i(1--a') (6).

9 V \ '/l_ \ via v k v/. that it must be thermally perfect. Equa- tions so marked may be used for calorically where a, Ao, b, B0, and c are constants for a given gas imperfect gases. (They are, of course, also (see ref. 10, p. 270 for numerical values). valid for completely perfect gases.). CALORIC EQUATION OF STATE. [isen] means that the flow process must take place isentropically. Equations so marked may A caloric equation of state is an equation of the form not be applied to the flow across a shock wave. u=u(v, T) (7). [adiab] means that the only restriction on the flow It can be shown that process is that it must take place adiabati- cally-that is, without heat transfer. (Such a flow process may or may not be isen- tropic depending on whether it is or is not reversible.)

10 Equations so maxked may be du=c, dT [therm peril (8b). applied to the flow across a shock wave. If the gas is calorically perfect--that is, the specific heats An equation without notation has no restrictions beyond are constant--equation (8b) can be integrated to obtain those basic tb the study of thermodynamics and/or inviscid compressible flow. u=c,T+u, [peril (9). 2 When used without subscripts, p, o, and T danote static pressure, static density, and static temperature, respectively. 615. EQUATIONS, TABLES, AND CHARTS FOR COMPRESSIBLE FLOW. ENERGY. RELATIONS ENTROPY. The law of conservation of energy gives The entropy is defined by dq=du_dw (first law of thermodynamics)_ (21).]]