C.P. No. 397 (19.547)

A R C Technul Report

C.P. No. 397 (19.547)

A R C. Technical Report

MINISTRY OF SUPPLY

AERONAUTICAL RESEARCH COUNCIL

CURRENT PAPERS

Some Methods of Evaluating Imperfect Gas Effects

in Aerodynamic Problems

G. A. Bird

LONDON HER MAJESTY’S STATIONERY OFFICE

1958

PRICE 4s. 6d NET

C.P. RL 397

U.D.C. No. 533.6.011 : 533.6.011.72

Teohnical. Note No. Aero 24.68

ROYAL AIRCRAFT ESTAXJXHMENT -- ---

Some methods of evaluating imperfect gas effects in aerodynamic problems

G. A. Bird

Simple numerical and graphmal procedures are described for the calculation of the imperfect gas effects on the properties of steady and unsteady one-32.msnsions.l isentropic flows, the Prandtl-Meyer expansion round a corner and normal end oblique shock waves. The fun&mental equatmns of each type of flow have been put into a form in which they may be solve2 usmi: the published tables of the equilibrium properties of gases. Both thermal Land caloric imperfections have been t&en into account but relaxation time effects have been neglected.

Numerical examples are given for each type of flow although the main emphasis has been plaaed on the methods rather than on the results. These basic methods have been used to calculate the magnitude of the imperfect gas effects on a number of specific aerd.ynsn3.c problems which have been ccnsidered in detail.

LIST OF GONTEh'l!S

1 Introduction 2 Methods of Imlysis of Basic Flows

2.1 Isentropic processes 2.11 Steady one-di.mens~onal isentropio flow 2.12 Irandtl-i~~eyer expsnslon round corner 2.13 Prcpagation of one-dlmens~onal isentropic wava

2.2 Shock waves 2.21 Ncmal shock waves 2.22 Oblique shock waves

3 Application cf %thcds to Typical Problems

::: Shock tube performance Lift of flnt plate

::z Ncszle of hypersonic impulse tunnel Reflection of normal shook wave from rigid wall

3.5 Two-dmensionel supersonic intake List of Symbols References

LIST OF I.LLUSTR.4TIONS

Steady one-dimensional isentrcpic expansien Steady one-dlmensionsl isentropic expansion Prnndtl-Meyer expCansion round corner Propagatzon of one-dimensional isentropio wave (unsteady expansion) Normal shock wares Oblique shook waves Oblique shock waves Imperfect gas effects w shock tube performance Shock tube performance L5ft of flat plate Nozzle of hmerscnic impulse tunnel Reflection cf normal shock wave from rigid wall Two-dimensional supersonic intake

14 15 15

:z

?6 18

Fiwe 1 2

3 4

5 6

7 8

9 10 11 12

13

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1 Introduction

In the last few years, the high temperatures that have been encountered Zen aerodynamic problems have been such that it has become necessary to consider the effects produced by the difference in behaviour of a real gas compared with that of s 'perfect' gas.

A perfect gas is here deflned as one whxoh has constant specific heats (caloric perfection) ad which obeys the equation of state P R - =- P

m T (thermal perfection). At low temperatures or high pressures,

a real gas departs from the perfect gas equation of state as a result of molecular force and molecular size effects becoming apparent. At considerably higher temperatures, appreciable energy is imparted to additional degrees of freedom and then speczfx heats then cease to be constant. At still higher temperatures, the composition of the gas may change as a result of molecular dissociation ad cause variation in the speolfic heats and in the molecular weight. Molecular ionisation does not commenoe to produce appreciable effects until even higher terriperatul-es have been reached,

A large number of attempts have been made to modify the conventional gas dynamic equations so that they apply to real as well as perfect gases. Some aspects of the problem have been treated by the use of one of the more nearly exact equations of state such as van der Waai's in reference 1, Bertheld's in reference 2 and the Benttxa-Brdgeman equation in refer- ence 3, and allowance was made in several of these reports for changes in the specific heat ratlo also. Some other n.ethcds have been suggested which allow for changes in the specjfic heat ratlo only. However, thus type of treatment does not take into account all types of tiz&?erfectlon ad, in general, it is restrxted to flows in which dissociation does not occur. The expresslcns which are cbtalned by these methds are invariably extremely involved and it is not easy to apply them to the spenific problems which are encountered in practice.

A number of sets of tables have recently been issued which list the equilibrium thermal properties of gases (such aa entropy, enthalpy, density etc.) as functions of temperature ad pressure. For Instance, reference 4 lists them for air up to temperatures of 50004( and pressures of 100 atmospheres and, in reference 5, data is tabulated for a number of gases. Both these sets of tables are based on the now obsolete value of 7.373 e.v. per molecule for the dissociation energy of nitrogen, but reference 6 gives a preliminary form of revised tables based on the currently accepted value of 9.758 e.v. per molecule*. These tables take into account (as far as is possible) all types of imperfection.

In references 7, 8 and 9, the normal shock wave equations are solved for a variety of initial osrditions, by iterative methods which make direot use of these tables. However, 1.t 1s assumed In all these methods that the gas is thermally perfect, i.e. the perfect gas equation of state still applies, This assumption is valid for ccmhinations of high temperatures and low pressures even if the gas is dissociating, but it is not justified at low temperatures or high pressures. In section 2.21 of this report, a simple graphical method 1s presented for the solution of the shock wave equations for any initial ooditions; the

* Unfortunately these tables were not available to the author at the time of preparation of this report and the older sets of tables have necessarily been used in the examples.

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method uses the tabulated real gas properties which take into account thermal as well as caloric imperfections. shock waves also. (Secticn 2.22).

The method is extended to cover oblique

References 7 and 8 present methods of dealing with imperfect gas effects on the propagaticn of one-dcmensional unsteady isentropic waves. These are based on the replacement of the usual form of the Riemann variables

2 q's* form q-c

i

which is valid only for an ideal gas by the more fundamental fdp which is valued for real gases also. This integral is

not easy to evaluate in the real gas case, and has been tabulated for a few oases only, The method presented in Section 2.13 of the Report makes direct use of the more fundamental eplation $p+"

pa and a simple numerical

procedure is given for the complete solution of the problem for any initial conditions.

Reference 7 considers also steady isentropic expansions and presents a method which is again very complicated and which depends upon the gas being thermally perfect. In Section 2.11 a graphical method is suggested whioh is consistent vnththat outlined above for unsteady flow. Euler' s equation of motion for steady one-dimensional flow may be put into the form

s?L%L$=-; and the real gas tables enable this to be integrated directly along any required isentrope. A simple extension of the method is given in section 2.12 to cover the Prandtl-Meyer expansion round a corner.

It is not necessary to bring enthalpy into the isentropic flow cslcula- tions as has been done in references 7 md 8. No iteration procedures are involve3 m any of the methods presented in this Report and they are all far more straightforward than those which have previously been proposed, even though they allow for thermal as well as cslorio imperfectrons.

The magnitude of the imperfeot gas effects is dependent on the initial temprature and pressure of the gas to such an extent that it is practically impossible to compile a comprehensive set of compressible airflow tables for a real gas. Emphasis has therefore been put, in this Report, on the development and presentation of simple numerical procedures by which tne imperfect gas effects maybe calculated III any specific example. A number of exsmples are given which illustrate the application of these procedures to typical problems.

It has been assumed in all these methods that relaxation time effects may be neglected. These relaxation time effects arise because excitation of the various degrees of freedom of the molecul.es,the s.dJustment of energy in them, dissociation, reassociation, ionisation and reoombmation all occur as a result of collisions between molecules snd atoms. knumber of collisions is required to produce equilibrium and this number varies greatly with the type of process and the gas in which it is taking place. Equilibrium is reached very quickly between the translational and rotational degrees of free&m but a considerable number of collisions are required to reach equilibrium if the vibrational modes are excited and the fluid moves a certain distance v,hilst it is being established. Experiments have indicated that this "relaxation distance" rarely exceeds a few centimetres and is usually very nmoh less and that it decreases as the temperature and pressure increase. Very little is lolown about rates of dissociation, although eqilibriwn is probably reached far more slowly than in the case of the vibrational energy adjustment. Still less is known about rates of reassooia- tion and these will have a very pronounced effect on the flow cn the nozzle of a hypersonic impulse tunnel. The importance of these non-equilibrium effects till depend on the ratio of a typical dimension of the flow to these

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"relaxation distenoes", and much work remains tobe done before the magnitude of these effects can be estimated in practioalcases.

2 Methods of Analysis of Basic Flows

2.1 Isentropic processes

The oonventionsl gas dynamic eqations for one-dxmensionsl isentropic flows are obtained from the eqations of motion by use of the perfect gas isentropx relation between pressure and density (-$ E const.).

In the following sections, the basic equations will be put into a form in which they maybe integrated directly using the tabulated then& properties of real gases.

2.11 Steady one-dimsnsionsl z.entropic flow

Consider the flow in a stream tube. The momentum equation is

sit 29 at + q ax = ,li?E P a*

For steady flaw at k3 = 0 and, as only one dimension is involved, the equation msy be written:

or

(For a perfect 7

as, this msy be integrated to give the eonventionsl energy equation .

The tables list specific entropy of a real gas as a function of temperature and pressure. The isentrope for the real gas msy therefore be constructed by looking up the value of entropy for the znitisl conditions and then finding the values of temperature and. pressure in the required range for which the vslue of the entmp

% is constant. As

an exzunple, the isentmpe for air expanded from 5000 and 100 atm is plotted in Figure 1.1 and 1s compared with that given by the perfeat gas relation -E = constant.

PY Density and speed of sound are slso listed4J5r6

as functions of temperature and pressure and ourves of p and a (see Figs.Z.3 and 1.2) may be plotted against p for these isentropea.

The reciprocal of the density which by equation 2.2 is equal to

ai -9

(

ap 1s then plotted sgainst p in Flg.1.4 and these curves are inte-

grated graphically to give curves of &q' against p (Fig.i.5). In this exsmple the gas was assumea to be expanded from rest, but the initial value of +q2 111 the integration

47 be adjusted for any initial velocity.

Once the relation between ~q (F1 .1.6)

and p has been found, that between q and p follows

7 and this is used in conJunction with the curve of a

against p Fig.l.2) to find the relation between Mach number and pressure (Fig.l.7).

The aocuracy of the numericsl wd. graphical procedures maybe checked if the curves appropriate to the perfect gas are plotted alongside those

-5-

of the real gas in each figure. The two sets of curves were operatea upon in exactly the same msnner and an indication of the accuracy of the calcula- tions was givenby comparing the calculated values of p against M for

the perfect gas with those given by the relation > =

The continuity equation gives

The pressure at which M = I is found fromFig.1.7 and p* and q* are then A

found from Figs.i.3 snd 1.6 respectively. The area ratio 2 may then be found as a function of Each number and the results for the perfeot gas again provided a oheck on the accuracy of the calculations.

In the example shown in Fig.?, there is considerable dissociation of the gas under stagnation conditions and the very large departures from perfect gas behaviour are a result of the subseqent reassociation. The energy whhlch is released in the reassociation process causes the temperature, at a given pressure, to be much higher than in the case of a perfect gas expsnded to the same pressure. The speed of sound 1s therefore much higher and the density is lower. The reciprocal of the density is higher and so the velocity pmduoea by a given pressure drop 1s higher for the red. gas. However, the effect on speed of sound. is greater than that on velocity and the Mach number for a given pressure dmp is less for the real gas than for the perfect gas. Thercis also a very pronounced real gas effect on srea ratio, the real gds requiring a far greater area ratio to reach a given Mach number.

Figure 2 shows the corresponding results when the gas is Ylitially at a temperature such that no dissociation or reassociation effeots are encountered. There are differenoes of sever&l percent between real and perfect gas cases in the pressure versus temperature, speed of sound, density end velocity curves but these are in directions such that they ten6 to cancel out when the Mach number versus pressure snd area ratio ourves am calculated from them,

2.12 F'rsndtl-Meyer expansion round corner

The equations of momentum for the steady flow of an inviscid oompress- able fluid may be written in polar coordinates as,

and

(2.4)

(2.5)

The equation of continuity is:

(2.6)

-6-

In the case of supersmic flow, It maybe assumed that p, p, u and v are functions of 6 only and the equations then reduce to,

O?.-

and

From (2.91,

But fmm (2.8),

vau v2 r=--F =

0

au v=z5

av vz+u = ( > ,192 P ae

av & pu +px+vao = 0

,122 v ae

Comparing (2.10) with (2.11),

v = a

HEGlC.3 UdgT=@X

and fmm (2.7) and (2.12),

(2.7)

(2.8)

(2.9)

(2.10)

(2.11)

(2.12)

(2.13)

Equation (2.2) applies along a stream tube and the methods of section 2.11 are used to construct curves of a against p, q against p and M against p. Corresponding values of u ardimny then be found for any value of Mach number and these are plotted against one another as shown

mFlgure3.1. As :=g (equation 2.14), these curves msy be integrated

graphically to give ourves of 8 against u (Fig.3.2) and the curves of 6 agamst M (Fig.3.3) follow.

The flow deflection angle is given by,

= e - tan -’ &5 (2.15)

-7-

The curves of Y against N are then plotted as shown in Fig.3.4 snd the accuracy of the calculations maybe checked by ccanparing the perfect gas values with those given by the conventional gas dyntics equation,

The exsmple shown in Figs.3.1 to 3.4 is for air expanded from a temperature of 500Q°K and a pressure of 100 atmospheres, and reassociation again prcduces large imperfeot gas effects. Figure 3.5 shows the corres- ponding effect on Prendtl-Meyer angle when the stagnation temperature is much lower end, in the absence of initial dissociation, the departures from the perfect gas values are of the order of one or two peroent only.

2.13 Fropanation of one-dimensional isentropic wave

The ccd5nuity equation may be wrxtten,

end the momentum equation may again be written,

2s &L+121.1E, at + q ax P ax Treating p as a function of p and S,

ap = ($j)p as + ($)s a~

i.e.

end, as the flow is isentropio,

as as z+qz = 0

Multiplying (2.16) by % and (2.18) by 1 2 0 ap as p

and adding,

+ a ax = O

Now, from (2.1 s.113 (2.191,

(2.16)

(2.1)

(2.17)

(2.18)

(2.19)

(2.20)

-8"

Therefore, if a wave satisfies the direction conditions,

then

r4r. at = q-ta (2.21)

(2.22)

(For a perfect gas this maybe integrated to give the well known relation q&a = constant).

Equation (2.22) may be integrated for a real gas by a sivnlar process described for the integration of equation (2.2) in Section 2.11. An example is shown in Fig.4 for the unsteady expansion of air from 300% end 100 atmospheres. It aen be seen from Figure 4.5 that for a given pressure ratio, the real. gas attains a higher flow velocity thsn the perfect gas.

In Figure 4.6, the flow velocity (s) is plotted against speed of sound (a). A check on the eocuracy of the calculations is provided by these curves as the perfect gas should obey the condition of invariance of the Riemann variable (5 a + q) . It may be seen fromFigure 4.6 that

the real gas departs widely fmm this condition and, as most practioal probleme are solved by using Xiemann's variables, large errors would be introduced by the assumption of a perfect gas in this exsmple, in which relatively moderate temperature and pressures are considered.

2.2 Shock waves

The flow through a normal shock wave is desoribed by the equations of oontinuity, momentum and energy. The conventional normal shock wave relations for a perfect gas are obtained by replacing the enthelpy term

y 2 in the energy equation by G p which enables the pressure, temperature, density and velocity ratios across the shock front to be found as functions of M, end y. In the ease of a real gas, however, enthslpy must be left in the energy equations, as such, and the equations are then solved by making use of the real gas values of specific enthelpy which ere tabulated as functions of temperature and pressure in references 4 and 5.

It can easily be seen from momentumoonsiderations that the tangential component of the flow velooity is the same on both sides of en oblique shock wave. The equations of continuity, momentum end energy, when written in terms of the normal velocity components, are then identical with those for a normal shock wave. Therefore, for any flow geometry, the oblique shock wave problem reduces to that of finding the strength of the "equivelent normal shook wave" and this may be done by trigonometrioal considerations.

2.21 Nod shook waves

The continuity, momentum and energy eptations msy be written,

and

PI 91 = P2%

PI + PI 91 2 = p2+p2q22

(2.23)

(2.24)

$+?I 1

= 2L.E 2 2 (2.25)

-9-

From, (2.23) end (2.24)~

2 2 L-1 p2 - PI = PI s, p, p2 ( >

also

From (2.25) end (2.27),

Hz-H, =

(2.26)

(2.27)

(2.28)

The following procedure is adopted to solvethlsequation when the gas in front of the shock wave is at a known temperature and pressure.

(i) Find H,, a, and p, from the thermdynemic tables of resl gases.

(ii) Choose a value of T2,

(iii) Use the real gas tables to find the value of H2 at this temperature for a number of values of p2 and plot H2 - H, against p2 as shorn? in Fig.5.2.

(iv) Find p2 from and evaluate & (p, - p,)

s for the seme range of values of p2

in each case. The curve of

against p2 may be plotted on the SW axes as

that of H2 - H, against p2 and the intersection of the two curves gives the value of p2 which corresponds to the chosenvelue of T2. (See Fig.5.2).

the $L, The corresponding values of ~2 and a2 may then be found from .

The procedure is repeated for other values of T2 and consistent T2 sets of values of the ratlos- p2 2 2 and "2 are thus found. The 'JJ, ' 5' p, ' q, al

next step is to find the vslue of shock Mach number to which each of these sets corresponds. This is easily done as, by (2,26),

i.e. (2.29)

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It is often required to find the flow velocity behind a normsl shock wave moving into a gas at rest. This is given by,

and using (2.23)

i.e.

3 %(I-3

PI

u2 = al% '"P ( J (2.30)

The method of solution is shown in Fig.5.2 for three values of T2 for air initially at 3OO'K and 0.1 atm. The calculations were also carried out for a number of other values of T2 and also for air initially at 0.01 atm snd 3009(. The shook Mach number to which each of these solutions corresponds was found from equation 2.29 snd Fig-es 5.3 to 5.6 show the departures fmmthe perfect gas values of pressure, temperature, density and velocity ratios across the shock and of the flow Mach number behind a shock moving into a gas at rest.

The pressure ratio (Figure 5.3) across the shock is greater for the real gas than for the perfect gas end there is no signifioant dependence on pressure in the exemple which has been considered. The temperature ratio (Figure 5.4) for the real ges is far lower then that for the perfect gas and at high shock Mach numbers, when dissociation is the dominant factor, the effect is greater for the gas initially at the lower pressure. However, there is still an appreciable effect at shock Mach numbers around five and, at these speeds, thermal imperfections cause the effect to be slightly greater for the gas initially at the higher pressure. Fw-= 5.5 shows that the density is higher behind a shock in the real gas end the steady flow velocity is less. According to the perfect gas theory, the nwxinium flow Maoh number behind an unsteady normal shock wave in a gas of y = 1.4 is 1.89 but Figure 5.6 shows that the flow Maoh number in a real gas maybe very much higher.

The above results are merely given to illustrate the method and the variation of these and other quantities for different initial conditions may be easily calculated.

2.22 Oblique shock waves

The equation of continuity gives

P, u, = P2 u2

The equation of momentum msy be applied to flow normally end tsngentislly to the wave front to give

2 2 P, - P2 = P2 u2 " PI u, (2.32)

- 11 -

0 = p2 “2 v2 - pi Y vl

comp~ing (2.31) with (2.33) it is seen t&t

vi 5 v 2 = v

The energy eqxd,ion my therefore be m-itten

2 2 Y -U2

2 =%2-%2 =H-H 2 2 I

(2.33)

(2.34)

(2.35)

Hence the equations in u and u for the oblique shock wave are identical to those in q, end 42 fo& the &rmal shook wsve.

Let s = ;;- be the "equivalent nod shock" Mach nuniber. I

Now

But

:.

u2 tanP=y ui end te3a =';-

5 u2 v- ten6 -= V

I + 5 ~tanb

Therefore, 1

"2 = u1 I+

tan6 (2.36)

Equation (2.36) enables the "equivalent normal shook" Mach number to be plotted as a function of "2 for any given flow Mach number and deflection

u1 u angle. (Note that the value of < depends only on the ratio of M, to MH

and not on their individual values so that, after one cme has been

- 12 -

.

calculated, the others for the same value of 6 are simply scaled from it). These curves are based solely on geonletricsl considerations and hold for red or perfect gases.

u2 The shock wave equations must also be satisfied and curves of - Y

against MS which do this maybe obtained using the methods of section

2.21. (As q for the oblique shock is e-al to Q2

nonml shock). Q1

for the equlvslent

Exsmples of the two sets of curves sre plotted in Figure 6.1 and their points of intersection give the appropriate values of the equiva- lent normsl shock Mach number for the given flow Mach number deflection angle and lnitisl gas conditions. There are, 111 general., two solutions for each case, the first corresponds to the "weak" shock which is the one normally obsenrea and the seoond corresponds to the "strong* shock. When the curves do not intersect the shock wave becomes detaohed.

The angle of inclination 01 of the shock wave is given by,

sina = zlEMs

41 5 (2.37)

Ml As this relation 1s a function only of the ratio K , the farmly of curves

which It gives are easily constructed (Figure 6.2):

Figure 6.3 has been constructed fromFigures 6.1 and 6.2 end gives the shock wave angle as a function of M, for various values of deflection angle for the resl and perfect gas cases. It is seen that the sngle of the "weak" shock is less for the red. gas than for the perfect gas. Also, at any given Mach number, the shock wave remains attached until higher values of defleotion angle have been attained in the real gas than in the perfeot gas. For mstsnce, there is no solution at al.1 at a deflection angle of SO0 for the perfect gas but the shock 1s attached above M, = 8 for the real gas.

Once the value of the "equivalent non& shook" Mach number has been found fromFigure 6.1, the normal shock wave cdadations (Seotion 2.21) give the ratios of pressure, density etc. across the oblique shock wave. The pressure ratio across the shock is plotted zn Figure 7.1 and, for a given initial Mach number and flow deflection angle, the pressure rise is less for the real gas than for the perfect gas.

The flow Mach number behind the shock 1s often required and is given by,

i.e. (2.38)

- 13 ”

u2 For a givenvalue of M, and &,Mg end-arefoundfmmFigure 6.1 a, Y

end the corresponding velue of 1 a2

is given by the normel shock wave oelcula-

tions. The Mach number behind the shock is greater for the reel gas than for the perfect gas. (Figure 7.2).

3 kpplication of Methods to Typical Problems

3.1 Shock tube perfo-ce .

The general principles of operation of shock tubes are well known, A tube is divided by a diaphragm into two sections, one containing gas at high pressure and the other at low oressure. The diaphragm is mptured and the gas in the high pressure section expands isentropicelly into the low press- section. The gas in the law pressure section is compressed by a shockwave which travels into it. The gas which was originally in the high pressure section is divided fmm that which was originally in the low pressure section by a contact surface which also travels into the low pressure section.

The values of pressure and velocity are the same on eech side of the contact surface end this provides the basis of a simple graphical method (see Figure 8) for calculating the performsnce of a shock tube for both the perfect and reel gas oases. The methods of seotion 2.13 enable curves to be constructed of flow velocity (q) against pressure (p) in the expended high pressure gas. These ere shorn in Fig.8 for both air and hydrogen expanded from 100 atmospheres ma 3CQ"iL The methods of se&ion 2.21 may then be used to oonstruot -es of flow velocity behind an unsteady normel. shock wave (U,) against pressure behind the shock (p,). These are shown in Figure 8 for air compressed fmm 0.1 atmospheres and. 3CPK. All these curves are drawn for both the reel and perfect gas cases.

As flow velocity and pressure are continuous aoross the oontact surface, the intersections of these two families of curves give solutions of the shock tube problem. For instance, points (A) end (B) in Figure 8 give respectively the perfect end real gas solution for the flow velocity and pressure at the contaot surface of a shock tube having hydrogen at 100 atmospheres snd 300°K in the high pressure end end sir at 0.1 atmospheres and 300 %-C in the low pressure end. In order to find the strength of the shock wave which is produced, the methods of section 2.21 ere again used to find the relation between shock Mach number (Mg) and the flow velocity (U,) behind it. This is different for the real end perfeat gas oases and a scale of M8, for the two cases, hasbeen added to Figure 8. It mey be seen fmm this that point (A) corresponds to a shock Mach nunher of 7.39 and point (B) corresponds to one of 7.06.

Figures 9.1 and 9.2 have been constructed solely from the information given directly by curves of the type shown in Figure 8 and show the reduction in shock Mach number which is produced by the reel gas effects. The magnitude of the effects varies with the pressure of the high pressure gas as a result of thermal imperfeotions beooming apparent at the higher pressures. One of the advantages of this graphical method is that it mey be seen, in eny particular ease, to what extent the reduction in shock Mach number is due to effeots in the expansion of the high pressure gas (mainly thermal imperfeo- tions) and what is due to effects in the campresaion of the low pressure gas (meinly caloric imperfeations). In particular, it is seen that considerable effects are produced with eir as the high pressure gas at 103 atmospheres even when the shock Mach number is only two or three (see Figure 9.2). Also, if the high presare gas is such that the imperfections in its expensicm are negligible, it mey be seen fmm Figure 8 that the velocity of the contact surface may be increased even though the shock Mach number is reduced. In eny case, when the shock velocity is high, the velocity of the contact surface

- II+ -

is not reduced as much as the shock front and so, at any point down the tube, the interval between the arrival of the shock wave and contact surface is reduced. As the gas between the shook wave and contact surface is used for testing purposes in impulse tunnels, the real gas effects cause a reduction in the running tiao. The time interval between the shook wave end contact surface per unit length of tube is given by:

: = (+&)I (3.1)

and Figure 9.3 ohows the reduction Ln running time due to imperfect gas effeots in a typical case.

3.2 Lift of flat plate

An exact solution for the lift of a two-dimensional flat plate at incidence in a supersonic streemmaybe found by ce&ulating the drop in pressure over the upper surface through the Prandtl-Meyer expansion and the increase in pressure over the lower surface through the oblique shook WEiVB. The lift coefficient of a flat plate is shown in Figure IO for incidences of up to 400 and at Mach numbers of 5, IO and 15 in air at 300°K snd. 0.01 atmospheres.

The Prandtl-Meyer expansion is treated in Se&ion 2.12 but, for the initiel oonditions of this example, the imperfect gas effects on the expansion will be negligible. The main factor will be the pressure rise through the oblique shock wave and this is shown in Figure 7.1 for the real end perfect gas cases. The pressure rise 1s less for the real gas so that the lift coeffioient will be reduoed from that given by perfect gas theory.

3.3 Nozzle of hypersonic impulse tunnel

The Maoh number of the steedyflowbehind en unsteady normal shock wave is limited to quite low values snd, in order to produce higher Hach numbers, the flow 1s often expanded 3n a supersonic nozzle. Figure 11 shows the nozzl.e profile which would be required to produce a steady flow af Mach 6 at the end of a shock tube in tiich a shock of alaoh nuder 12 is produced. in air initially at 300% and 0.5 atmospheres. The inlet Mach number is higher for the resl gas than for the perfect gas (see Fwe 5.6).

The Prandtl-Meyer angle was found for the real gas by the ssme methods as those used to produce Figure 5.4. B sinusoidal distribution of Mach number we.8 chosen along the centre line and the methods of characteristics was use& to construct the flow field. The nozzle profile was found as a streamline of this flow. It would be expeoted fromFi* 1.8 that the overall erea ratlo would be mob higher for the real gas than for the perfect gas end this has been borne out in this exsmple. As this example inmlves reassociation of the initially dissociated gas, the neglect of non-equilibrium effeots may be serious.

3.4 Reflection of norm&l shock wave from rinid wall

Consider an unsteady normal shock wave moving in a one-dimensional channel which has a closed end. When the shock wave meets the end it is reflected and the strength of the reflected wave is deternuned by the oondition of zero flow adjaoent to the well. The across each wave must therefore be the ssme (i.e.

I*41 a perfeot gas, - is a function of %I only snd

~A~=i~AJ$Vt~or Aq R -is afunction of

a2 62

- 15 -

% only and they may be plotted as shown in Figure12.2. In the aase of a

l*s& real gas, - a2

depends on MS, as well as MSS snd has been plotted in

Figure 12.2. For several v$LLy of hf.+. To find the vslue of MsR for my

value of Ms,, the value

Ms is that which gives

of I"qII - is found for Y &2

SI and the required value of

I*& I*41 - = - * The process is illustrated in a7 a3

Figure 12.2 for Ms1 z 8.0 for the real and perfect gas cases, giving

MQ = 2.89 and 2.55 respectively. The real gas ourves have all been

calculated by the methods described in Section 2.21.

The Mach number of the reflected shock is plotted as a function of the ;Y;ach nwiber of the incident shock in Figure 12.3 aud it is seen that, whereas the Mach number of the refleoted shock is asymptotic to a value of 2.63 for a perfect gas, it may be much higher for a real gas. Figure 12.4 shows that the temperature ratio across the refleoted shock is muoh less for the real gas than for the perfect gas. The ratio of the temperature adjacent to the wall before and after the arrival of the shock is shown in Fig.12.5 snd is again considerably less for the real gas. However, it is seen fmmFigure 12.6 that the pressure ratio is greater for the real gas and, for incident shock Mach numbers above about 8, the difference is very sigzifY.csnt.

3.5 Two-dimensional supersonic intake

A two-dimensional intake maybe designed in which a uniform super- sonic stream is compressed by two oblique shock waves to form another uniform stream in the same direction at a much lower l&oh number. The general configuration is shown in Figure '13.1 for a perfect gas of Y = 1.4 and for air initially at jW?K and 0.01 atmospheres at intake Mach numbers of 6, IO and I&. An oblique shock wave is formed when the stream is turned through an angle of 30° end the stream is returned to its original direction by the reflected oblique shock wave. At an intake Mach number of 6, a solution is possible with regular reflection for the real gas but Mach reflection occurs in the perfeat gss ease.

The properties of the incident oblique shock may be found for the perfect snd real gas cases fmm the graphs in Figures 6 end 7. These also enable the reflected oblique ehwk properties toba found for the perfect gas but, to obtain the solution for the real gas, similar curves must be constructed for initial temperatures and pressures corresponding to those behind the incident shock.

Figures 13.2 to 13.5 show the variation with intake Mach number of the temperature and area ratios and the exit Maoh number and total pressure. It is seen that the real gas intake provides a narrower stream of air at a lower temperature but at a higher Maoh nuuioer and totel pressure than that for the perfeot gas. List of Symbols

P pressure

P density

T temperature

R universal gas constant

- 46 -

s

H

m

C P

cv

5 Y=c v

9

x

r

e

t

u

v

a

M = :

%

u2 A

Y (I

P 6

entropy

enthalpy

moleoular weight

specific heat at constent pressure

specific heat at oonstant volume

speoific heat ratio

flow velocity

linear coordinate

radial coodnate

angular coorainate

time

radial flow velocity component 3.n Prandtl-Meyer expansion or normal flow velocity component through oblique shock wave

tangential velooity component

local velocity of sound

flow Mach number

normal shock Mach number

absolute flow velocity behind unsteady nod shook wave

cross-sectional area

flow defleotion engle (Prandtl-Meyer angle) angle between cblique shook wave and original flow direction angle between oblique shock wave and-final flow direction flow deflection angle through oblique shook vrave

Superscript

$ sonic contitions

Subscripts

0 stagnation conditions

1 conditions in front of shock wave

2 conditions behind shock wave

3 conditions behind reflected shoolc wave

- 17 -

REFERENCES

No. /,uthor

1 Tsien, H.S.

Eggers, A.J.

Tao, L.N.

Staff of App. Phys. Lab., J.H.U.

Hilsenrath, J. et al.

Hilsemath, 3, end Beckett, C.R.

Zienkiewice, H.R.

Squire, W., Hertzberg, A., end Smth, I7.E.

Romig, M.F.

Title, etc.

One-dmensional flows of a gas charac- terised by van der Waal's equatlon of state Vol. 25, No.6, pp.301-324, 1947 Journ. of Xnthematics and Physics

One-dimensional flow of an imperfect diatomio gas NACA Teoh Note No. 1861, 1949

Gas dynamic behaviour of resl gases Vol. 22, No. Ii, @.763-771+, Nov. 1955 Journ. of keronautloel Sciences

Handbook of supersonic aerodynamics (NAVORD Report No. 1488). Vol. 5, Sect. 15. Properties of gases

Tables of thermal properties of gases National Bureau of Standards, Circular 566. November, 1955

~hermdynsm~c propertles of argon - free air (0.78847 N2, 0.21153 02) to 15OOQ% 'National Bureau of Standards, Report No. 3991, April, 1955

On some problems m one-&imensional flow of imperfect gases English Electric Co. Ltd., Report No. L.A.~. 082. March 1955

Real gas effects in a hypersanic shook tunnel Arnold Engmeenng Development Centre Report No. AEDC-TN-55-14. ?lfsroh 1955

The normel shock properties for air in dissociation eqilibrium Vol. 22, No. 4, April 1955 Journ. of ~,eronautical Sciences

- Ia - W.20?8.C.P.307.K3 - Printed in Oreat Britam

FROM PO = IOOatm, To = 5000 OK (AIR.) -----_--_ PERFECT GAS (Y = I. 4)

REAL GAS (THERMAL EQIJILII~RIUMJ

REAL GAS VALUES FROM TABLES REAL GAS VALUE FROM TABLES

PERFECT GAS VALUES GNEN BY ; =@’ PERFECT GAS VALUES GIVEN BY cl-40fio I I

7- 7- IA. 01 IA. 01

J J = = & 0 01 & 0 01 0. 0.

0001 0001

0.0001 0.0001 100 100 IO IO 01 01 0.01 0.01

REAL GAS VALUES FROM TABLES. REAL GAS VALUES FROM TABLES.

PERFECT GAS VALUES GIVEN BY P= g

30 30 I5 I5

25 25 : :

20 20 /’ /’ 1’ 1’

I5 I5 /’ /’

IO IO

lIszT!l lIszT!l

A’ A’

5 5 0 0

100 30 60 70 60 50 40 30 20 IO 0 100 30 60 70 60 50 40 30 20 IO 0 P <aw P <aw

FROM I 4 BY GRAPHICAL INTEGRATION FROM I 4 BY GRAPHICAL INTEGRATION B I7 7 /’

6 1’

5 ,I’ 4

/ ,’

3

lzElq C’

/- 2

0 100 IO

p&m) O’ 0 01

FROM l6& I.2 AS M=g

PERFECT GAS VALUES CHECKED BY

PWm) FROM I 3 ONE GRAPH DRAWN FOR EACH SECTION OF LOGARITHMIC SCALE OF p

P @tm) FROM 1.5.

8

7 6

s

4 =3

2

I 0

PERFECT GAS VALUiS CHECKED BY +e!* =i (F)

f G.I. STEADY ONE-DIMENSIONAL ISENTROPIC EXPANSION.

FROM PO = 100 atm. To = 1000°K(AIR~

---------- ~ELE;-rGAS (Y = I 4)

200 .

100 - 01 I I I I

I00 IO I 01 0 01 P (atm>

IO

r ’

5 2 d

0.1

4

0.01

0.001 100 IO I 01 0 01

P(atm)

IO B

9-

0 -

7 .

6-

5 -

100 IO 0.01

P (atm)

100 IO I 01 0 01

P catm>

IO

3

8

7

6

5

= 4

3

2

I

0

FIG.2. STEADY ONE- DIMENSIONAL ISENTROPIC EXPANSION.

FROM PO = 100 atm , To = 5000 OK. (AIR)

--------_ PERFECT GAS (Y = ,. 4) REAL GAS (THERMAL EQUILIBRIUM)

I CONSTRUCT CURVES FOR STEAOY ISENTROPIC EXPANSION. (FIG I.) 2 TABULATE RESULTS AS FOLLOWS (ONE TABLE FOR REAL AN0 ONE FOR PERFECT GAS)

3 CARRY OUT FOLLOWING GRAPHICAL PROCEDURE

FROM TASLE SH‘OiN A&E

FROM 3 2 AND +kLE SHOWN ABOVE

FROM 3 I BY GRAPHICAL INTEGRATION (NOTE - e-o WHEN M=l)

FROM 3 3 AS V-e! tan-‘m PERFECT GAS VALUES CHECKED BY, v-8 (tm-’ @, -tan-‘Jai-l

60

Y 2 60

z e, 40

z 20

FROM Pgp 100 atm , T’=lOOO°K (AIR) 100

5

I

FIG.3. PRANDTL -MEYER EXPANSION ROUND CORNER.

FROM PO = 100 atm., To = 300 OK (AIR3 ---------- PERFECT GAS. (b’ = I 4)

REAL GAS. (THERMAL EQUILIBRIUM> ?

2 A & t

II DO . SATURATION POINT

01 I I I I I I 9 I I 100 90 80 70 60 50 40 30 20 IO

P (atm)

6001~ 100 90 80 70 60 50 40 30 20 10

REAL GAS VAL”E.!@fRzk TABLES REAL GAS VALUES FROM TABLES

PERFECT GAS VALUES GIVEN BY E = (;)“’ PERFECT GASVALUES GIVEN BY a=aoE

3

1 0 OOE

r 2 0005

2 ::

*+ 0.004

~~OOOZ +I II

_,d oooi P

0001

2.

I -

01 3 6 I I I I I I loo 90 a0 70 60 50 40 30 20 IO

P@W REAL GAS VALUES FROM TABLES - PERFECT GAS VALUES GIVEN BY f’ = g

3’ooo I------

d 800 -

700 -

‘3‘ 2,000

-5 !- t

ONE GRAPH DRAWN FOR EACH SECTION OF LOGARITHMIC SCALE OF p.

e!? rj\

1,000

0 600 800 IO00

FROM 0 g q a (i/SEC)

PERFECT GAS VALUES CHECKED BY

OK 100 90 80 70 60 SO 40

P (Qtm) 30 20 IO o

FROM q BY GRAPHICAL INTEGRATION

5a+y = 500

3

FIG.4. PROPOGATION OF ONE-DIMENSIONAL ISENTROPIC WAVE. (UNSTEADY EXPANSION)

FIG. 5 I.

-$- 71 OR u2=(pT2 u,=o.

SHOCK VELOCITY-O

STEADY FLOW.

SHOCK VELOCITY- y,

UNSTEADY FLOW.

AIR AS REAL GAS. --------- AIR AS PERFECT GAS

250 I I , , , , /,>w

zoo p, = 0.1 aim. I

p, = o~olatn ,I1

1.50 - /'

a IIT too - "I

7:

/'

/' /'

50 -

0 I2 4 6 is IO 12 I4 16

12 PI - 0. I atm.,

10 PI 5 0 01 atm.

0

VG

4, 6 e*lc

IA

,/- ___---- __--___-_

4 /’

/

IL I 2 4 6 6 IO ~i2 14 16

25

IS

FIG,,

5

0 I 2 4 6 8 IO I2 I4 I6

MS

01.. 1, 8. 8.8. I. I. I2 4 6

1 8 IO I2 I4 16

MS

FIG. 5. NORMAL SHOCK WAVES.

_--------- AIR AS PERFECT GAS@ = 1.4) AIR AS REAL GAS (T = 300°K.

Pn = O,Ol atm.‘l

= 30’ (EQiJA% 2 36) [FIG 6.

1 M,=2

FROM NORMAL SHOCK WAVE CALCULATIONS

Y I 3 4” s 0 3 4 I5

EQLlI”AL% NO~NAL3SH~CK~rll’AC”l~~M~~RlMS

90

80

/;j\ y 70 lx $ 60 e ox 50

y 40

Y 4 30

y s 20

:: IO

z 0

INITIAL MACH NUMBER M,

FIG.6. OBLIQUE SHOCK WAVES.

--------- PERFECT GAS

WING AIR (300% 1 FIG 9. I 7

.

.

I I

I IO 100 I A,-.,-. I

PRESSURE RATIO ACRI

I I ~~- I,““” .O

355 DIAPHRAGM.

I I

it&SURE PRESSURE RATIO ACROSS DIAPHRAGM. RATIO ACROSS OlAPHRiGM.

SHOCK MACH NUMBER Ms

FIG.9. SHOCK TUBE PERFORMANCE.

/ p ‘RANDTL - MEYER EXPANSION.

CO = C, tan 6.

\

.

OBLIQUE SHOCK WAVE.

O-8 - O-8 -

d d

3 3 ii ii

F F 0.4 - 0.4 - L L ? ?

0 0 IO IO 20 20 30 30 40 40

INCIOENCE 6’

FIG.10. LIFT OF FLAT PLATE.

H

.

4 tA

N

R

DISTANCE.

__----- ---- PERFECT GAS (X = 1 4) REAL GAS f&R, p; = 0 01 atm. T, = 300%)

30

600 - 600 -

600 - 600 -

FlG.12. REFLECTION OF NORMAL SHOCK WAVE FROM RIGID WALL.

REFLECTED SHOCK

MACH REFLECTION IN PERFECT GAS CASE

FIG. 13.1.

_---------- PERFECT GAS (g - 1.4) REAL GAS @AIR, T, = 300’~. p, = 0 01 ah)

5 6 7 8 9 IO II 12 I3 14 15 INLET MACH NUMBER M,

025

ol~~t~~.t.BJ 5 6 7 8 3 IO II I2 13 19 15

INLET MACH NUMBER M,

INLET MACH NUMBER M,

30

2.5

n

=20 B m

z’ =I 5

3 I.0 ___-----

E /-- _---

t s 0.5 )___i

5 6 7 8 9 IO II 12 13 14 15 INLET MACH NUMBER M,

FIG.13. TWO - DIMENSIONAL SUPERSONIC INTAKE. (OBLIQUE SHOCK COMPRESSION)

C.P. No. 397 (19,547)

A.M. Techmcal Report

0 Crown copyrzght 1958

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C.P. No. 397