SOLUTION OF EQUILIBRIUM PRANDTL-MEYER FLOW OF A REAL GAS BY THE METHOD OF EQUIVALENT IDEAL FLOW Fang Toh Sun Gas Dynamics Laboratories Department of Aerospace Engineering The University of Michigan Ann Arbor- Michigan June 1966

TABLE OF CONTENTS Page ABSTRACT iv ACKNOWLEDGMENT v LIST OF ILLUSTRATIONS vi NOMENCLATURE vii Io INTRODUCTION 1 II, ANALYSIS 2 III CORRELATION BETWEEN THE REAL FLOW AND THE EQUIVALENT IDEAL FLOW 5 IV. THE PRANDTL-MEYER FLOW OF EQUILIBRIUM AIR 9 V. DISCUSSION 11 REFERENCES 28 APPENDICES A. ANALYTICAL SOLUTION FOR AN IDEAL PRANDTL-MEYER FLOW: SUMMARY OF FORMULAS 29 B. A SHORT TABLE FOR THE PRANDTL-MEYER FLOW OF AN IDEAL DIATOMIC GAS 30 C. AN EMPIRICAL FORMULA FOR THE & vs 1 RELATION 31 Do NUMERICAL EXAMPLES 32 iii

ABSTRACT A practical method for the solution of Prandtl-Meyer flow of a real gas in equilibrium is presented. The concept of an equivalent ideal flow is introduced and the exact solution of the Prandtl-Meyer flow of an ideal gas is utilized to reduce the bulk of graphical work needed for the solution of a real flow. A few typical key-charts for this method are developed for equilibrium air and from these charts the behavior of equilibrium air in a Prandtl-Meyer flow is briefly reviewed. Several numerical examples illustrating the application of this method are included in the appendix. iv

ACKNOWLEDGMENT The author is grateful to The University of Michigan Institute of Science and Technology for its financial support of the work presented in this report. Special thanks are due to Profo JO Ao Nicholls, whose encouragement and support made this work possible. Thanks are also due to Messrso Harry Ho Watanabe and James Howe for their help in the tedious calculations and plottingo v

LIST OF ILLUSTRATIONS Page Figure 1o The Geometry of a Prandtl-Meyer Flow 3 Figure 2-1, Correlation Chart AK vs. &6 (ht/RT 203) 14 Figure 2-2. Correlation Chart: Aj vs.? (hlh /RT - 253) 15 Figure 2-3, Correlation Chart: A v s.o,. (ht/RT - 303) 16 Figure 3-1. Correlation Chart B5 6 vs. 6 (h/RT - 203) 17 Figure 3-2. Correlation Chart B 6 vs. 6, (h /RT: 253) 18 I t. 0 Figure 3-3. Correlation Chart B~ 6 vs. 6, (h /RT - 303) 19 Figure 4-1. Correlation Chart C: I vs. & (h h/RT - 203) 20 i t 0 Figure 4-2. Correlation Chart C > vs. s, (h /RT 253) 21 0 Figure 4-3. Correlation Chart C., vso.. (hh/RT - 303) 22 Figure 5o Temperature Variatio-n f-he Prandtl-Meyer Flow of Equilibrium Air 23 Figure 60 Pressure Variation in the Prandti-Meyer Flow of Equdlibrium Air 24 Figure 7. Density Variation in the5 Prandtl-Meyer Flow of Equilibrium Air 25 Figure 80 Mach Number Variat:orn in the PFaldtfl-Meyer Flow of Equilibrium Air 26 Figure 9. Variation of,. with Local Enthalpy ard Entropy 27 Figure D-1o Example 1. 32 Figure D-2. Example 2. 26 Figure D-3e Isentropic Variation of Mach Number with Local Enthalpy 37 v.

NOMENCLATURE a speed of sound a* critical speed of sound h enthalpy per unit mass M Mach number, V/a M* dimensionless velocity, V/a* p pressure r radial distance R gas constant s entropy per unit mass T absolute temperature V speed Vr radial component of velocity V transversal component of velocity a Mach angle ratio of specific heats, cp/c 6 deflection angle, defined in Fig. 1 d polar angle, defined in Figc 1 p density X flow parameter /1 +a A. average value of X, defined by Eq. (C-l) vi

Subscripts i ideal flow o reference condition s isentropic t stagnation condition vl ~i

I INTRODUCTION The problem of supersonic flow through a centered expansion wave, known as Prandtl-Meyer flow, has been solved analytically for the case of 1 2 an ideal gas by L. Prandtl and Th. Meyer in the early nineteen hundreds, 3, 4) 6) 7 and hodographs and tables have been extensively developed since then " However, when the real gas effects are to be taken into account (such as the variation of specific heats with temperature, dissociation, ionizatlorn and so on), analytical solutions have not been available, and the solution of such problems has to rely heavily on numerical approximations and graphical means. As a great variety of real gases and flow conditions are encountered in present-day engineering, a tremendous amount of numerical and graphical work will be expected. The purpose of this report is to solve the problem of Prandtl-Meyer flow of a real gas by correlating it with an equivalent ideal flow, a concept which will be introduced below. Through such a correlation it: will be shown that the solution of the real flow problems may be obtained from that of an ideal flow, so that the existing analytical solution for the ideal case may be utilized, and the graphical and numerical work necessary for the real flow problems will be greatly reduced. 1

Iio ANALYSIS With the gas assumed in loal) thar;Wmcldynamic ~-quilibriLm throughout its expansion around a convex sharp corner, it is expected that all physical properties of the gas will be constant along the radial lines emanating from the corner since the boundary conditions define no charactenr:sli4c length in 1 5 such a problem according to Prar-,dtlvs argument and the resulting flow is therefore of the Prandtl-Meyer type. Under such assumptions,the flow is isentropic and the following equat ions hold not only for an ideal gas but; for a real gas as well~ dV -, V. a (2) 2 r S, - since they are based on the conservat'."r of mass momentum; and e>nergy only, hence independent of The equatilon of state of the particular gas under considerationo The geometry of such a flow is shown ia Figo lo Eliminating V from Eqs. (1) and (2) results in Heims' equaio. n^ for the radial velocity V r *First introduced by Helms) Steve Po in Ref. 10 with a slightly different notationo 2

x o~~-A II 2 /n 0 0r Figure I. Geometry of the Prandtl - Meyer Flow 3

IdV i der2- 2 1 aht -VV ~l (3) d L)'lA 2 rtJ' S where the parameter X is defined by A 1 + 2 (4) a A comparison of Eqo (3) with the corresponding equation for an ideal gas, /:dV 2 _ ri- - -2ht V- (3a) with.2 2h. y.+ 1 = 1 +2 —-= (4a) a. i suggests the following change of variable, d 0. dO _ 1d (5) Upon substituting the new variable 0. for 0, Eq, (3) reduces to dvC iZ f 2 ht V j di ^h -^ i(6) r 4

which is exactly of the same form as Eq, (3a). Thus, as a consequence of such a transformation, we arrive at V V (7) r ri if we take hti - ht. That is, the radial velocity along the radial line of polar angle 9 in the real flow is the same in magnitude as that in the flow of an ideal gas of the same stagnation enthalpy along the corresponding radial line of polar angle 0. defined by Eq, (5). The fictitious flow of an ideal gas associated with a real gas flow in the sense of Eqo (7) will be referred to as the equivalent ideal Prandtl-Meyer flow (or ideal flow for short) wherein the polar angles of the corresponding radial lines along which the radial componecn:s of the gas velocities in the two flows are equal (in magnitude) are related through Eq. (5), IIL CORRELATION BETWEEN THE REAL FLOW AND THE IDEAL FLOW With the basic correlation between the real Prandtl-Meyer flow and its equivalent ideal flow established in Eqs. (5) and (7) it is a simple matter to deduce the relations between other gas properties along the corresponding radial lines. For example, the corresponding transversal components of the flow velocities are related by dV dV dO, X. rr r- V (8) 0V d d. dO do A i (8) 5

The correlation between other properties may be deduced in the same mannerO A summary of such formulas is given below: Table 1. Correlation Between the Real and Ideal Prandtl-Meyer Flows Flow Parameter Correlation Formula Velocity, radial V - Vi (7) Velocity, transversal V. = -) Vi (8) Local speed of sound a= - ( a (9) Mach angle tan a =- tan a. (10) ~A~ 1 Mach number M2 1 + ( (Mi2 1) (11) Local enthalpy h = 1 h (12) Thus once the correspondence between 0 and 0. is established, all principal flow parameters in the real flow can be obtained from the corresponding quantities in the equivalent ideal flow by simple calculations through these correlation formulas. (A summary of the analytical solution and a short table for the ideal flow will be found in Appendices A and Bo ) 6

So far no attempt has been made to correlate directly the corresponding local flow deflections 6 and 6i as it is more involved. However, with the polar angle 0 and the local Mach angle a determined, the local deflection in the real flow may be simply calculated from 6 - + a - (13) which is evident from the geometry of the flow (see Fig, 1). Now it remains to integrate the differential relation (5) between 0 and Odl In the absence of suitable analytical formulas the integration can only be effected graphically. Before proceeding with such an integration we recall from the analytic solution for an ideal Prandtl-Meyer flow, a. -V =I V2 cos (0/i) (14) i 0i -. t i which when,combined with the correlation formula (9), gives 0. -X. cos -b1 t (15) 1 1 A2h This equation provides a principal linkage between the condition in the real flow and the corresponding polar angle in the ideal flow, and enables one to effect the integration of Eq. (5) according to the procedure suggested below. 7

For a specified stagnation condition defined by h and s, 1) Assume a series of values of local enthalpy, h < ht along the constant entropy line, and find the corresponding local speed of sound from the proper chart (eo g, the Mollier chart) or table for the real gas under consideration; 2) Calculate X according to Eq. (4) and then 0. according to Eq. (15): 3) Plot X versus 0. and carry out the graphical integration according to 0. i 0 | - dO. (5a) 0 It is to be noted that both 0 and 6. are measured from the Mach line M = 1 by definition; and in principle, any ideal gas, monatomic or polyatomic, may be taken as the reference gas in the ideal flow irrespective of the nature of the real gas under considerationo As an illustration, some typical correlation charts for 0 and 6 i 6 and 6. have been worked out for equilibrium air by the present method with a diatomic gas as the reference gas. The results together with the X vs. 6. curves from which these correlation charts were obtained are 1 shown in Figso 2 to 4o Some typical examples showing the use of these charts for the solution of practical problems are given in Appendix Do 8

As further illustrations, the variation of the local properties of equilibrium air (temperature ratio, pressure ratio, density ratio, and Mach number) through the Prandtl-Meyer expansion as obtained from these basic charts are presented in Figs. 5 to 8. The corresponding quantities for an ideal diatomic gas given by the analytical solution are also shown for comparison purpose. IV. THE PRANDTL-MEYER FLOW OF EQUILIBRIUM AIR Based on the graphs presented in Figs. 2 to 8, a few observations on the Prandtl-Meyer flow of equilibrium air may now be made. The relation between the corresponding real and ideal polar angles is nearly linear at a constant stagnation condition. The slope of such a line (do/do ) changes in going from one isentropic line to another, the higher the entropy the higher the slope; but it remains practically constant for different stagnation enthalpy. This is due to the fact that the parameter X of the real gas changes considerably with entropy but very little with enthalpyo With the help of the Mollier chart the variation of X with enthalpy and entropy for equilibrium air is shown in Figo 9. An empirical formula for the average slope of the 0 - 0. curve for equilibrium air is given in Appendix C. A comparison of the equilibrium properties of air in a real flow with an ideal flow shows that, within the range of the present plots, the temperature ratio (T/Tt) in a real flow is higher thIan that in an ideal flow at the 9

same angle of deflection (both measured from the initial direction M = 1) and the same is true for the pressure ratio (p/po)o The deviation of air temperature in the real flow from that in an ideal flow of the same stagnation temperature first increases and then decreases as the flow deflection increases, with the maximum deviation occurring at some intermediate deflection angle (Fig. 5). However, the pressure deviation between a real flow and an ideal flow of the same stagnation pressure is small, and is practically negligible at high flow deflections (Fig. 6). In the case of density ratio, as Figo 7 shows, the difference in the real and ideal flows is very little throughout the range of plotting. For small flow deflection the air density in the real flow is slightly lower than that in the ideal flow of the same stagnation density while for large deflection it is slightly higher, The two become identical at an intermediate flow deflection of approximately 38~ under the assumed stagnation condition. Finally the Mach number in the real flow is lower than that in the ideal flow at the same turning angle, and this deviation increases rapidly as the air turns (Fig. 8). As we know, the maximum angle of turning in an ideal flow has a theoretical limit of [ (f(y + 1)/(y - 1) - 1]'i/2, the present plotting indicates that the equilibrium air may turn through a much higher angle than an ideal diatomic gas. 10

V. DISCUSSION 1, As shown throughout the previous sections the parameter A. plays a vital role in the behavior of the Prandtl-Meyer type of gas flow. By writing its defining Eq. (4) in the alternate form I 1 2 2 a2 12 it is seen that the square of A represents the ratio of the total energy to the kinetic energy of the gas if the flow is critical, that is, V = a, under the local condition defined by its enthalpy and entropy. Heim's equation then shows that even though a great variety of real gas effects are present in the real flow, they enter the governing equation only through this parameter as long as the assumption of equilibrium flow is valid. Consequently, the introduction of this parameter provides a simple approach to the analysis of Prandtl-Meyer flow despite the presence of apparently many diversified real gas effects: and it also makes it possible to link the real gas flow with an ideal gas flow which forms the basis of the present method of solution. A few typical curves showing the variation of A with the local enthalpy and entropy have been presented in Figo 9 for equilibrium airo In fact lines of constant X may be added to a regular Mollier chart since it is a local property of the gaso Such Mollier charts, if available, would be of special value in assisting the solution of Prandtl-Meyer flow problems of real gases, 11

2. As Section III shows, by linking the real flow with its equivalent ideal flow through the parameter A the problem of finding the solution for a real flow reduces to that for an ideal flow if the correspondence between the real and ideal polar angles 0 and 0. is known. Thus the graphical work necessary for such a solution is narrowed down to the graphical integration of the single relation X.d0 Xd d. ifrom which the 0 vs. 0. charts may be prepared. Once such charts are available9 the method of solution consists of merely finding the ideal solution according to 0i and converting it to the real solution through the simple correlation formulas in the table of Section III. Basically the 0 vs. 0. charts together with the X vs, 0. charts, constitute the key charts needed in the present method. However, in view of the fact that in many practical problems it is usually the local deflection angle that is specified instead of the polar angle, correlation charts of 6 vso 65 such as those shown in Figs. 3-1 to 3-3 are often desired. Such charts may be easily obtained from the 0 vso 6. chart through formula (13)o Thus, by using the equivalent ideal flow method, at most three types of correlation charts are required; the 0 vs. o 6 vs. 6i V and X vso 0; and to prepare these charts only one graphical integration is needed. In case of equilibrium air the empirical formula in Appendix C may be used in the absence of the 0 vs. 0. charts. 12

Finally it is to be noted that the graphical integration of the differential relation XidO = X i on which the present method hinges can be effected only when the equilibrium properties of the real gas under consideration are known (either in the form of charts, tables, or empirical formulas)o Investigations of such properties of various real gases are beyond the scope of the present study. 3. Although this report deals with the equilibrium flow, same method applies equally to the case of frozen flowo In such a case the value of X is frozen at its initial value and the real gas behaves as an ideal gas throughout the flow. Consequently, the 0 vso 0. relation is exactly linear, and a direct application of ideal flow formulas is permitted with a proper choice of the value of X, 13

180 h1/RT, 203 / 140 120 __ ___ _ _ __c _ ___'__ _ _ _ I coo 80 60 40 20 20 40 60 80 100 120 140 0i (degrees) Figure 2-1. Correlotion Chart A: & vs. ( ht / RTo =203) 14

180 - 160 -- - ht/RTo 253 t/ 140 120 I 100 80 60 20 0 20 40 60 80 100 120 140 e (degrees) Figure 2-2. Correlotion Chort A' e vs. &i (h1/RTo = 253) 15

180 160 ht /RTo: 303 140 120 (100 80 07// 40 20; 20 40 s0 80 100 120 140 oi degrees) 16 16

LI (~OZ = ~lj8/4)' SA s^ 8 PD'Jq D uo!iDolJJOD i 1-~ aJn6!i (saajap)!g 09 0~ O7 0_2 OZ 01 0 09 - ~-^ —w| 4 X i 1~~~09 _ __ _ _ __ -, //I/-0'

90 80 0) h,/RTo=253 y 70____ 60 50 30 20 10 0 10 20 30 40 50 60 18 ~MJ~~~ ~~ ^ —//,/....... — ~4o~ ~~~ 18

61 (~0~ = ~l18/ )'.sA:E loJD43 uo4lolaJJo3*~-~ )ain6iI (saaj6ap)!8 09 0__ t 02 OZ 01 0 01 0 I ot c 09 ________ _0UA_ A C- X X rr S | WCV~tf >;-09 - --------- -yb,//- o, ---- /7Z^ —I -- -- -- -- - 7 -, (r-/ - -- -- -- ~ 9,i.III1 O' — ^S^ZK —"~~~~~~~~~~~~~~~~ __ __ __ /ill/to / 1 _ /' _ __ _ __ -/^-~/ ~ _ —-0

4.1 4.0 3.7 F. ht / RT = 203 3.69 3.5 3.4 3.2 0 20 40 60 80 100 120 140 0i ( degrees ) Figure 4-1. Correlation Chart C' \ vs. ~i ( ht /RTO = 203) 20

4.2 4.1 ht /RT= 253 4.0 3.9 3.8 _ 3.9 - - - - -- - - -- - - 3.51 34 3.3 3.2 34 0 20 40 60 80 100 120 140 160 8 i (degrees) Figure 4-2. Correlation Chart C: X vs. 8i (ht/RTo = 253) 21

4.2 4.0-_- __ — 3.9-9 3.8 —-- 3.7 ---- - l mml __ — l_3.6 _____ __A 3.5 -9 L 34- 3. - Figure 4-3. Correlation Chart C: X vs. 9e ( ht /RTo = 303) 22

~~~~~ — - t —-- -- ----- --- -- --- ~ — --- — ~~= —— +-.==:::: — 0.4 0.9 h,/RT-= 253 s/R=39 0.8'-0 0.7 G - — ^os5 ^____ 0.60 -__ _ __ ___ 0.5 ~- 04 <: 0.3 S__ 0.2 - 0.l 00 10 20 30 40 50 60 70 80 90 FLOW DEFLECTION, a (degrees) Figure 5. Temperature Variation in the Prandtl- Meyer Flow of Equilibrium Air.

0,, ------------— +0.10 0.6 ---—' —-—' —-------, — -— _... + -— 0.05 a. 0.5 ht/RT:=253 s/R=39 0.4 0.3 -N -_. 0 0.2 0.1 0 0__20 30 40 50 60 70 80 90 FLOW DEFLECTION, (degrees) Figure 6. Pressure Variation in the Prandtl-Meyer Flow of Equilibrium Air.

___ ____ ___ _~ +o.05 Ap/p I 0.65.-, —-_..,-,. — - -..- --—' — -- 0.0~~~~.5 ~~~-0.05 0.5 \ -0.1 ht/RTo=253 s/R=39 0.4 Ideal Gas.. /;()-1{ 1.405). 0.3 CA' 0.1 0 10 20 30 40 50 60 70 80 90 FLOW DEFLECTION, S (degrees) Figure 7 Density Variation in the Prandtl- Meyer Flow of Equibrium Air.

/ 4.0 - - - - - - - - - - ---- ---- + 3. 0.., hRT253, / 0\ toe s/R=39 // / 3.0 - ---- -- -- --- -- + 2. 0 / /.//V i I / j i 1 / 1 i I I / / 1.0 __ __......... 0 10 20 30 40 50 60 70 80 90 FLOW DEFLECTION, < (degrees) Figure 8. Mach Number Variation in the Prandtl -Meyer Flow of Equilibrium Air.

4.2 4.0 80 100 120 140 160 h/RTo 3.8 V 3.6 3.4 3.2 3.0. 70 80 100 120 140 160 180 200 220 240 260 h /RTo Figure 9. Variation of X with Local Enthalpy and Entropy.

REFERENCES 1. Prandtl, L., "Neue Untersuchungen uber die stromende Bewegung der Gase und Dampfe, " Phys. Zeits, 8, 23, 1907. 2. Meyer, Th., Diss. Gottingen 1908. 3. Shapiro, B, "The Dynamics and Thermodynamics of Compressible Fluid Flow," Vol. 1, 1953, pp. 474-477, 632-633. 4. Ames Research Staff, "Equations, Tables, and Charts for Compressible Flow," NACA Report 1135, 1953. 5. Lighthill, M. J., "Higher Approximations, High Speed Aerodynamics and Jet Propulsion," Vol. VI, Part E, Princton, 1954, pp. 375-377. 6. Ferri, A., "The Method of Characteristics, High Speed Aerodynamics and Jet Propulsion," Vol. VI, Part G, Princeton, 1954, pp. 591-605. 7. Sauer, Robert, "Einfuhrung in die theoretische Gasdynamik, " SpringerVerlag, 1960, pp. 99-103. 8. Hilsenrath, J. and Beckett, C. W., "Tables of Thermodynamic Properties of Argon-Free Air to 15, 000K, " AEDC TN 56-12, National Bureau of Standards, Sept. 1956. 9. AVCO Mollier Diagram for Equilibrium Air, AVCO Research Lab, Everett,Mass., Jan, 1957. 10. Heims, S. P., "Prandtl-Meyer Expansion of Chemically Reacting Gases in Local Chemical and Thermodynamic Equilibrium, " NACA TN 4230, 1958. 11. Glass, I. I. and Kawada, "Prandtl-Meyer Flow of Dissociated and Ionized Gases, " Institute of Aerophysics, Univ. of Toronto, Report No. 85, 1962. 28

APPENDIX A Analytical Solution for an Ideal Prandtl-Meyer Flow Summary of Formulas* Flow velocity, radial V = v2h sin - r t N Flow velocity, transverse V= cos - V'2ht 2 Flow velocity, total V - + ( - 1) sinl A' A,*2 _ 2. Flow velocity, dimensionless M* + (x ) il 2 A2 A Mach number M = \+ A2 tan Mach angle a = tan cot - N N -1 ( Deflection angle 5 = ( - tan A tanlA Temperature T T - cos 2 2 Enthalpy h ht 1 - c os - - (A ) - 1 Pressure p t 2 cos I (1 2 Flow parameter A - / +29 29 as~ ~ t X

APPENDIX B A Short Table for the Prandtl-Meyer Flow of an Ideal Diatomic Gas (1-= 1. 405)* [~J01 6[01 [OJ p/pt M M* T/T 0 0,00 90,00 0,527 1,000 1,000 0,832 1 23,72 67,28 0,477 1,084 1,068 0,808 2 30,04 61,96 0,449 1,133 1,107 0,794 3 34,82 58, 18 0, 424 1, 178 1,141 0,781 4 38,88 55, 12 0, 401 1,220 1,173 0,768 5 42,34 52,66 0,381 1,258 1,201 0,757 6 45,42 50,58 0,363 1,295 1,227 0,747 7 48,30 48,70 0,345 1,332 1,253 0,736 8 50,93 47,07 0,329 1,366 1,276 0,726 9 53,46 45,54 0,313 1,401 1,299 0,716 10 55,84 44,16 0,298 1,435 1,322 0,706 11 58,16 42,84 0,284 1,470 1,344 0,696 12 60,38 41,62 0,270 1,505 1,366 0,686 13 62,49 40,51 0, 2 57 1,539 1,387 0,676 14 64,52 39,48 0,245 1,572 1,407 0,667 15 66,53 38,47 0,233 1,608 1,428 0,657 16 68,47 37,53 0,221 1,641 1,448 0,647 17 70,33 36,67 0,210 1,675 1,467 0,638 18 72,18 35,82 0,199 1,710 1,486 0,628 19 73,98 35,02 0,189 1,744 1, 504 0,619 20 75,74 34,26 0,179 1,779 1,523 0,609 21 77,49 33,51 0,170 1,815 1,541 0,600 22 79,20 32,80 0,161 1,850 1,559 0,591 23 80,90 32,10 0, 1 53 1,884 1,576 0,582 24 82, 55 31, 45 0, 145 1,918 1, 592 0, 573 25 84,20 30,80 0,137 1,954 1,609 0,564 26 85,81 30,19 0,130 1,989 1,625 0,555 27 87,42 29,58 0,123 2,025 1,641 0,546 28 89,02 28,98 0,116 2,062 1,657 0,537 29 90,58 28,42 0,110 2,098 1,673 0,529 30 92,12 27,88 0,104 2,135 1,688 0,520 31 93,66 27,34 0,097 2,174 1,704 0, 511 32 95,18 26,82 0,092 2,214 1,720 0,502 33 96,68 26,32 0,086 2,251 1,735 0,493 34 98,20 25,80 0,080 2,296 1,752 0,483 35 99,67 25,33 0,075 2,339 1, 7 67 0,474 36 101,13 24,87 0,071 2,378 1,781 0,466 37 102,58 24,42 0,066 2,422 1,795 0,457 38 104,02 23,98 0,062 2,466 1,810 0,448 39 105, 46 23, 54 0, 058 2, 508 1, 824 0, 440 40 106,88 23, 12 0, 054 2, 550 1,837 0, 432 41 108,30 22,70 0,051 2, 595 1,851 0,423 42 109,71 22,29 0, 047 2, 640 1,864 0, 415 43 111,11 21,89 0,044 2,689 1,878 0,406 44 112,51 21,49 0,041 2,734 1,891 0,398 45 113,89 21,11 0,038 2,778 1,903 0,390 46 115,27 20,73 0,036 2,826 1,917 0,382 47 116,63 20,37 0,033 2,873 1,928 0,374 48 118,00 20,00 0,031 2,920 1,939 0,367 49 119,36 19,64 0,029 2,968 1,951 0,359 50 120,71 19,29 0,027 3,021 1,963 0, 351 51 122,07 18,93 0,025 3,074 1,975 0,343 52 123,41 18,59 0,023 3,131 1,987 0,335 53 124,74 18,26 0,021 3,188 1,999 0,327 54 126,03 17,97 0,019 3,350 2,012 0,319 129,32 219,32 0,00 0,000 xC 2, 437 0,000 (Taken from Ref. 7, p. 102) *For simplicity, all subscripts (i) for ideal flow have been omitted. 30

APPENDIX C An Empirical Formula for the 0 vSo. Relation for Equilibrium Air As shown in Figs. 2-1 to 2-3 the 0 vs. 0. relations for equilibrium air are nearly linear under the given stagnation conditions. Such a relation may be approximated by 0 o. (c-l) i _ where A is the average A over the range of plotting. The value of. may be obtained either from the X vs. 0. curve, or by directly measuring the average slope of the 0 vso 0i curve, By so doing we find that A is strongly dependent on s/R, and only weakly dependent on ht/RT, either dependence being also nearly linear. An empirical formula is found to be -4 -5 0-= o 633 +.0172 (s/R) - 2. 00 (10) (ht/RT ) + 1 20 (10)5 (ht/RT ) (s/R) A. t o t o (C -2) which holds approximately for equilibrium air within the following range of stagnation conditions ht/RT = 203 to 303 s/R 34 to 59 31

APPENDIX D Numerical Examples 1. Equilibrium air is approaching a convex sharp corner at a Mach number of 1 under the stagnation condition: ht s RT =253, = 39 o RT R 0 Find the Mach number, temperature, and pressure of the gas along the radial line inclined at 10 from the initial flow direction as shown in Fig. D-1. Find also the local flow deflection. MO=l I FgrD-/// / Figure D-. Example I. Solution: The polar angle of the radial line is 0 = 90 - 10 = 80~ From the correlation chart A (Fig. 2-2) we find 0 = 58.5~ 32

The analytical solution of the ideal flow at this polar angle is found from Appendix B (or A) to be h. T. 1 1 M.= 1.475, - - 06945, 6. 1L 15 ~1 ~ht Tth The local value of X as found from Fig. 4-2 is X = 3, 372 or X2 11. 37 The value of X for an ideal gas (diatomic) is 2 k. - N6 or X. - 6 1 1 The local Mach number in the real flow, according to formula (11) is given by M =1+ 11 37 (1. 475 - 1) 3. 228 M = 1. 796 The ideal local enthalpy is given by h. h. hi RT h- h 1 k =R (0o 6945) (253)- 175. 7 0 \ t' 01 The local enthalpy in the real flow is then calculated from correlation formula (12)o 33

1- 1 h 11 37 1.094 h. 1 1 h hhi RT h RT i (1. 094) (175 7) 192. 2 o 1 o0 The corresponding temperature and pressure are then found from the Mollier chart: T =67250k p 13.0atm To find the local flow deflection we first calculate 1 1 sin a= — 0. 5568 a= 33 50v M 1.796 Then by Eq. (13) we find 6 = 0 + a -- = 80 + 33~ 50 - 90~ 23 50' The value of 6 could also be found from the 6 vs. 6, chart (Fig. 3-2). 2. For the flow of example 1 find the Mach number, temperature, and pressure of the gas when the local flow deflection is 60~. Solution: The ideal flow deflection corresponding to 5 - 60~ is found from correlation chart B (Fig. 3-2). S6 310 341 34

and theideal gassolution (Appendix A or B) gives h T, M.-2.174 T -O. 511 0. - 93. 66~ t t The value of X is then found from Fig. 4-2. - 3, 287 By using the correlation formulas (11) and (12) we calculate for the real flow, 2 3.2872 2 M 1 + 2 (2. 174 -1) 7. 711 M -= 2 777 6 1- 1 h 3. 2872 1 = 1.089 h. 61 - The local enthalpy in the real flow is thus given by h h _ RT (~h. ) () ( - (1 089) (0. 511) (253) = 140. 8 oT i \R t... From the Mollier chart we find T = 5350 K p = 2 19 atm 3. Same as example 1 except that the air is approaching the sharp corner at Mach number 2. 35

0~2'ta2 Figure'e D2 EaV2 Figure D-2. Example 2. Solution: To start the calculation the local enthalpy of the air at the, initial Mach number (M1 = 2) must be determined. This can be done graphically with the help of the Mollier chart as follows: Assume a series of values of h/RT (< ht/RT ) and find the corresponding value of "a" along the constant entropy line (s/R - 39) in the Mollier chart; calculate the flow velocity V from the energy equation V = 2 (ht - h) and then the Mach number M = V/a; finally plot M vs. h and thus find from the graph the value of h at the given Mach number (M1 = 2). For the present problem the result of such a plotting is shown in Fig. D-3, from which 36

5 M2 3 ht / RTo =253 2 ----- s/R=39 180 190 200 h / RTo Figure D-3. Isentropic Variation of Mach Number with Local Enthalpy. From which we find RT - 182. 5 0 while the corresponding speed of sound is given by the Mollier chart9 a1 = 5. 025 a o The initial value of the parameter X is thus found from Eq. (4) or Fig. 9, X = 3.366 The initial ideal polar angle is then found from Eq. (15), or Fig. 4-2, (O l)i 620 and the corresponding polar angle, found from the correlation chart A (Fig. 2-2), 37

01 85~ which, as we understand, is measured from the fictitious Mach line M lo To find the polar angle of the 10~ radial line we first calculate 1 1 -1 1 30~ a n = sin sin 30 and then from the geometry of the flow (see Fig. D-2) we find 0 0 + a1 - 10 = 85 + 30 - 10~ - 105~ 2 1 1 From now on the procedures are the same as thos in example 1o The results are as follows: Ideal flow: (2)i= 770 (Fig 2-2) (M2)i = 1. 805 (Ideal solution, Appendix A or B) (h2)/ht 0. 603 ditto) Real flow: X2 3. 335 (Fig. 4-2) M2 2. 277 (Eq. (11)) h2/RT = 166. 6 (Eq. (12)) T2 = 6100~K (Mollier chart9) 38

P2 = 5. 96 atm Mollier chart -1 1 06 a2~sin M - 26 3? The flow deflection measured from the initial flow direction at M = 2 is, from the geometry in Fig. D-2, 62 = 26~ 3' - 100 = 16~ 3' 2 4. For the flow of example 3, find the Mach number, temperature, and pressure of the gas when the local flow deflection is 60 from the initial flow direction at M1 = 2. Solution: From example 3 we have 01 850, 1 =30~ The initial flow deflection measured from the fictitious flow M = 1 is, by Eq. (13), 6 =85+ 30~-90~ 25~ The corresponding local flow deflection is 6 225 +60 =858 2 From now on the procedures are the same as those in example 2. The results are as follows: 39

Ideal flow: (6 2)i 46. 7 (Fig. 3-2) (M 2). =.2,859 (Appendix A or B) 2i/ht T2i/Tt =0. 376 (ditto) 0 2i =116.2 (ditto) Real flow: X2 = 3. 291 (Fig. 4-2) M = 3 72 (Iq. (11)) h /RT - 103. 5 (Eq. (12)) TV - 3990~K (Mollier chart9) P2 0. 288 atm (ditto) 40

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