Heims, Steve P

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TECHNICAL NOTE 4230 
PRANDTL-MEYER EXPANSION OF CHEMICALLY REACTING GASES 
IN LOCAL CHEMICAL AND THERMODYNAMIC EQUZIBIZUM 
By Steve P. Eeims 
Ames Aeronautical Laboratory 
Moffett Field, Calif. 
Washington 
March 1958 
https://ntrs.nasa.gov/search.jsp?R=19930085178 2020-06-17T15:43:38+00:00Z
TECH LIBRARY KAFB, NM 
1T 
NATIONALADVISORYCOMMITTEE FORAERONAUTICS 
TECHNICAL NOTE 4230 
PRANDTL-MEXEREX3NSIONOFCHFMIC~YRRACTINGGASES 
INLOCALCHESIICALANDTHNRMOD~CEQUILIRRIUM 
By Steve P. Heims 
It is found that Prandtl-Meyer flow, in which chemical reactions = 7 
are occurring and are in equilibrium, can be simply and exactly cslcu- 
lated. The property of air which governs the flow is found to be a 
quantity which depends only on the ratio of enthalpy to the square of 
the speed of sound; the analogous quantity for an inert gas depends only 
on the ratio of specific heats. The maximum angle through which the 
flow may turn is generally larger when chemical reactions are occurring 
than it is in nonreacting air. A numerical example shows that the pres- 
sure variation with angle, as well as temperature and Mach number varia- 
tions,may be considerably affected by the presence of the chemical 
reaction. 
INTRODUCTION 
At the high temperatures encountered in hypersonic flight, the air 
may no longer be regarded as an inert gas. It does not have a constant 
ratio of specific heats, 7, nor does it generally obey the simple equa- 
tion of state, p/p = RT. These thermodynamic features reflect the fact 
that at high temperatures moleculsx vibrations are excited and chemical 
reactions are taking place in air. Because of this, any flow solutions 
depending on constancy of 7 and the perfect gas law are not valid. 
One elementary supersonic flow solution is the Prandtl-Meyer expan- 
sion around a corner. In this paper the theory of the Prandtl-Meyer 
expansion is extended to include high temperature flow in chemical and 
thermal equilibrium. When only molecular vibrations are active, and no 
chemical reactions occur, one can still use the usual flow equations in 
terms of 7, if the appropriate function of temperature is inserted for 
7 (see ref. 1). However, when chemical reactions are occurring, then 7 
is no longer a useful concept in the Prandtl-Meyer flow. 
Instead of working with 7, we shall employ a quantity 9 to des- 
cribe the thermodynamics of the gas, because it is q and not 7 which 
enters into the flow equations at high temperatures. When no reactions 
are taking place, 9 reduces to (7+1)/(7-l). In the present anslysis, 
it will be shown that by introducing also an auxili.ary variable q, the 
2 NACA TN 4230 
Prsndtl-Meyer expansion can be sFmply and exactly calculatec for equi- 
librium flow without employing any iterative procedures or e.utensive 
numerical. integrations. However, a table orla Moldier diagram for the 
thermodynamic properties of air at high temperatures is required. At 
I 
b- - 
high temperatures, the table of reference 2 or the Mollier chart of 
reference 3 is suitable. At temperatures below 3000’ K, reference 4 is 
useful. 
- 
a speed of sound 
C a constant ofmotion with dimensions of velocity, defined by 
equation (3) 
enthalpy per unit mass 
Plsnck's constant divided by 2n 
Bolt- constant 
Mach number 
P 
r 
R 
S 
T 
V 
vr 
% 
7 
9 
8 
V 
vm 
P 
pressure 
radial coordinate 
gas constant for air 
entropy 
absolute temperature 
speed, ,/m 
component of velocity along the radius vector 
component of velocity perpendicular to the radius vectqr 
ratio of specific heats, cv 
parameter defined by l+ 9 
angular coordinate 
angle through which the flow has turned 
theoretical maximum value of v 
density 
SYMBOLS 
c - 
cP 
NACA TN 4230 3 
. Ik an angle defined by equation (7) 
Jr’ an integration variable 
Subscript 
0 value along the radisl line on which the Prandtl-Meyer expansion 
begins 
In the usual derivation of Prandtl-Meyer flow (ref. 5) one seeks a 
solution for supersonic flow for which the pressure, density, and velocity 
ANAIJYsrs 
are constsnt along radiel lines emanating from the corner around which 
the gas is expanding. Such a solution can exist only when no character- 
istic length enters into the problem. Thus when chemicsl relaxation 
occurs in the air, one cannot have such a solution. However, when the 
air is in equilibrium everywhere, there is no length and one may expect 
to find such a solution. More generally, if each of the various reac- 
tions and internal degrees of freedom of the molecules are either frozen 
or in chemical equilibrium, a Prandtl-Meyer type solution is expected to 
exist. We Seek a formal solution of the Prandtl-Meyer type which is 
valid for any of these isentropic flows. 
If we define a2 = 22 
> ap s' 
the variation of pressure with density at 
constant entropy, we obtain from the requirement of continuity and from 
momentum conservation the usual equations: 
I=v 
de 8 (1) 
(2) 
Equation (2) requires that the flow be supersonic. The energy equation 
iS 
1 c2 vg + vr2 
2 =h+ 2 (3) 
Combining equations (1), (2), and (3) gives the differential equation 
for Vy: 
4 NACA TN 4230 
1 
1 + (2h/a2) b" - vr2) (4) 
Let 8, be the angle at which the Prandtl-Meyer expansion begins. Then 
in view of equation (2), the initial radial velocity is (see sketch): 
vr(Q0) = 43 R-=i (5) 
It is seen from equations (1) and (4) that, for given initial condi- 
tions, the flow is completely specified if the quantity q is known, 
where we define 
-. 
L 
Y 
(6) 
For air at moderate temperatures, the ratio of specific heats is a 
constant, and v = (7+1)/(7-l) = 6. For a diatomic gas in which the 
molecular vibrations are in equilibrium but no reaction is occurring, q 
is a function of temperature (but not of pressure) and varies from 6 at 
moderate temperatures to 8 at temperatures that are large compared to 
c 
* 
NACA TN 4230 5 
Hw/k, where w is the characteristic frequency of molecular vibrations. 
In general, when chemical reaction is also occurring, the quantity v 
is a function 
In order 
convenient to 
or 
of local temperature 
to integrate equation (4) in this general case, it is 
introduce a variable q(6) defined by 
with 
q. = sin -e sin eo)= sin-<%+/-) 
Then equations (4) and (5) become 
= (c’ - v$) 
Vr(lko) = &o Jc 
(7) 
(W 
The solution of equation (8) is 
, 
vr =csinq, * p qk, (9) 
The other component of velocity is then in view of equations (1) and (2): 
ve = a = cq -l/2 COB $ (10) 
It is seen from equations (9) and (10) that when the angle Jc is 
SC/~, the flow is purely radial and the Mach number becomes infinite. 
From this it follows that x/2 is the limiting value of $. 
To compute the flow field, we notice from equation (10) that $ 
can be expressed as a function of the state of the gas: 
6 NACA TN 4230 
Now with equations (2), (6), (g), and (ll), we can compute the flow field 
as a function of *. When chemical reaction is important the simplest 
procedure is to use a Mollier chart for air (ref. 3) or a set of tables 
(ref. 2), and regard h as the independent-variable. Knowing the initial 
(9 = q,) state of the gas, and knowing that it will follow an isenixope, 
one takes successive increments of h, looks up corresponding values of 
the sound speed a, readily evaluates the right-hand side of equation (XL), 
and thus finds the value of 9 corresponding to each h. Any of the 
other thermodynamic properties of the air c-responding to a particular 
J# can then also be read off from the chart or table. At the lower tem- 
peratures where the molecular vibrations are active, but where the cheml- 
cal reaction may be neglected, a somewhat simpler procedure is suggested 
by the fact that 7 and a are in this region functions of temperature 
only: Choose temperature as the independent variable (as is done in the 
tables of ref. 3), and look u-p h and a in the tables. Then equation(11) 
- yields the value of $. 
In any case, once * is known, one obtains the 
equations (9) and (lo), and the local Mach number is 
M= Jl + (Vr/Ve)= = Jl + 7 ts.Z12$ 
flow field from 
given by 
& 
(12) l 
It remains to obtain the physical angle 8 corresponding to the parameter 
$ by performing the integration indicated in equation (7). At tempera- 
tures where molecular vibrations and chemical reaction csn be neglected, 
this integral (for air) is ssmply 8 - e. = 4(7+1)/(7-l) ($ - qo). Even 
at high temperatures fi varies only slowly (as fig. 1 illustrates), and 
numerical evaluation of the integral (7) can be done quickly because one 
may take large steps, treating fi as constant during each step. 
This completes the solution of the Prandtl-Meyer problem; the angle 
the flow has turned through is given by: 
v 8 - COB = -1 & M 
(For an alternative expression see ref. 3.) 
For some applications it may be desirable to have &z1 eqlicit analy-kk 
expression for the density as a function of $. Such an expression can be 
obtained by integrating the continuity equation, (dpve/d@) + pvr = 0 (see 
ref. 5). 
and (10). 
Th$ inn;rtion is performed with the help of equations (7), (g), 
: 
NACA TN 4230 7 
(14) 
The usual ttconstant 7" solution is obtained when q = 7, = (7+1)/(7-l). 
A numerical table of h and a permits fitting (q,- q)/qo, the local 
deviation from constancy, by a sinp?le sine series with integer multiples 
of *t-*o as the argument. The integral in equation (14) is then of an 
elementary type. In the special case where M. is unity, the integral 
in equation (14) may be put equal to zero for small angles 9. 
NUMERICAL EXAMPLF: AND Cr’ENERA QU!YGITATIVEFEATURE!S 
A numerical example of Prandtl-Meyer flow is presented graphically 
in figures 1 to 5. A set of initial conditions were chosen at which the 
difference between frozen and equilibrium flow is marked, and at the same 
time is in the region which is of interest for high-altitude flight. It 
corresponds to the stagnation conditions at an altitude of about 130,000 
feet with a flight velocity of 19,000 feet per second. The frozen flow 
was taken as a flow in which only the random translations and molecular 
rotations keep up with equilibrium, but the chemical reaction and the 
energy of the molecular vibrations are frozen at the values they have at 
the point where the Prandtl-Meyer expansion begins. The initial thermo- 
dynamic state (pressure, temperature, density) at a Mach number of unity 
is taken to be the same in the frozen and equilibrium case. This requires 
that the initial sound speed and the flow velocity are different in the 
frozen case than in the equilibrium case. The sound speed, dp 
l-7 dP s' 
is 
different in the two cases because the variation of pressure with density 
is different, even though the initial thermodynamic states are identical. 
This difference in sound speed results in a significant difference in the 
initial vslue of q (see fig. 1). 
Several qualitative features of the relation between equilibrium 
flow (large q) and frozen flow (smsll q) can be seen from the present 
analysis and from the figures. Because of the simple geometry, we 
restrict ourselves in the following comments to the important type of 
flow in which the expansion begins with sonic velocities, even though 
all of the foregoing equations are valid for any initial Mach number not 
less than unity. 
The Maximum Tuning Angle 
The maximum angle through which the flow may turn corresponds to 
*= 3r/2 and is given by 
8 
When ~j is constant, equation (15) reduces to 
Vm= (@j-l) 5 
NAC!A TN 4230 
(15) 
(154 
,- c 
b 
Even when 11 is not constant, equation (15a) is evidently approximately 
equal to equation (15),provided a suitably averaged vslue of lJii is used. 
For frozen undissociated flow q = 6 and vm = 129.3'. However, for flow 
with chemical reaction (see fig. l), the averaged value of 7 may be 
larger than 9 and consequently the maximum angle Vm is larger than 180'. 
This shows that at high temperatures and pressures, equilibrium flow might 
have to make more than a 180° turn before the theoretical limit is reached. 
- 
.- 
The Mach number, pressure, density, and temperature sll take on their 
limiting values (M becomes infinite, T, p, and p become zero, for the 
ideal gas model)" at v = Vmo This fact is reflected in sl.1 the figures 2 
to 5 in that for the larger turning angles the small 11 values of p, T, 
p, and M have changed more from their initial value, thsn the large 7 
curves. 
, 
Density 
An interesting feature of the density curve (fig. 2) is that the 
equilibrium value is not everywhere higher than the frozen value of 
density, but its init= slope is steeper snd the two curves then must 
cross, because from the above comment about vm we 3mow that the density 
for the frozen case has to reach zero much sooner than the equilibrium 
curve. It will ususlly be true that for the equilibrium flow the density 
decreases initielly more rapidly than for the frozen flow. This can be 
proved rigorously from equation (lb), if the equilibrium flow is identi- 
fied with a certain vslue of 7. and a nonpositive value of all 
%=0 > 
and the frozen flow is identified with a relatively smaller and a 
constsnt q. (see appendix). 
%ch e point here is not so much that T = 0 and p = 0 at vm; in fact 
the ideal gas model breaks down completely at temperatures where the air 
begins to liquefy. For example, near T = 0 the specific heats are zero, 
and do not have the constant values associated with ideal gases at room 
temperature. The point is, more precisely, that the v corresponding to 
the state of the air in which the ideal gas laws break down is much larger 
in the large 7 case than in the small q flow. In the text the simpler 
statement is preferred; however, all of the equations in this paper are 
generally valid and do not depend on any specific gas model. 
- 
- 
Q 
c 
2T NACA TN 4230 
. 
9 
Pressure 
Y First of a31 one sees from figure 3 that unlike the density curve 
the equilibrium values for pressure are always higher than the frozen 
ones. Moreover the equilibrium and frozen pressure curves are well sepa- 
rated, mch more so than the density curves. For example at v = 70' 
the equilibrium pressure is nFne times as large as the frozen pressure. 
The corresponding ratio of the densities is 2. 
The point of interest is that in a Prsndtl-Meyer expsnsion the pres- 
sure is quite sensitive to the chemical reaction, much more so than the 
density. This is just the opposite of what o.ccurs In the cconpression by 
a normsl shock, in which the pressure ratio is very insensitive to sny 
chemical reaction, but the density ratio is not. 
Mach Number snd Turning Angle 
. 
L 
The local Mach number variation and its strong dependence on 7 in 
the example are shown in figure 4. The more general qualitative variation 
oz;y number snd turning sngle can be derived from equations (X2), (13), 
It is seen from these equations that for a given angle 8, the 
Mach &ber and v depend on the local value of 7, as well as on its 
value for smaller 8, but they depend on no other parameters. The differ- 
ence between frozen snd equilibrium flow msy be analyzed as arising (a) 
from the fact that 9 is ususlly larger in the equilibrium case and (b) 
that it is not constant in the equilibrium case. The effect (a) is seen 
from equation (12), and msy be stated in words: The larger the (constant) 
vslue of q the sm&Uer the vslue of the local Mach number at sny fixed 
13, and the larger the corresponding turning angle v(6). It follows that 
the larger the value of q, the smaller the Mach number at any given 
turn- single v. As to the effect (b) due to the variation of q(8) 
with 8, it may be stated Fn words as follows: If q(6) is a monotoni- 
csJ.ly decreasing function, then the actual Mach number at a point 0, 
for equilibrium flow is smaller and ~(0,) is larger than it would be 
for a flow which is characterized by the constant value of q = ~(0,). 
It follows that for any fixed v, the Mach number is smsller than for 
the flow characterized by the constant q = ~(6,). 
Ames Aeronautical Laboratory 
National Advisory Committee for Aeronautics 
Moffett Field, Calif., Dec. 2, 1957 
10 NACA !U!? 4230 
APPEXDIX . 
THE DENSITY AT VERY SMAII, ANGLES z 
We wish to consider two flows initis3.l.y at M = 1 and at the same 
density (which we choose as our unit of density), but with different 
initial values of q, 'lo, and qo2- Also generally the derivatives 
d log tlL 
> 
d lo@; 
QP de 60 
s.nct g2 = 
r12 
de > 
will be different. We ask the 
60 
question: Under what conditions will the density p, of flow 1 be 
smaller than that of flow 2 for small angles? - 
Expansion of equation (14) for small angles 6 yields: 
[ 
0 k, + g2M2 
P/P,= F” 8 1 (g, - g2) + g Cg”3+ o(P) 
From this expression it is seen that the condition that p, be less than 
p, for sufficiently small 8 is either 
Q, < Q2 
or 
. 
In the example give-n here, where p1 represents equilibrium flow and p2 
frozen flow, one has g, < g2 = 0, as well as 'lo,> vo2. 
NELL TN 4230 ll 
REFxmmcEs 
1. Ames Research St&f: Equations, Ta;bles, and Charts for Ccmpressible 
Flow. NACA Rep. n35, 1953. 
2. Hilsenrath, Joseph, and Beckett, Charles W.: Tables of Thermodynamic 
Properties of Argon-Free Air to 15,OOOO K. AJDC TN 56-12, NBS, 
Sept. 1956. 
3. Feldman, Saul: Rypersonic Gas Dynamics Chart for Equilibrium Air. 
AK!0 Research Lab., Jan. 1957. 
4. Hilsenrath, Joseph, et al.: Tables of Thermal Properties of Gases. 
NBS Circular 564, Nov. 1955. 
5. Taylor, G. I., and Maccoll, J. W.: The Mechanics of Compressible 
Fluids. Vol. III of Aerodynsmfc Theory, div. H, ch. IV, sec. 5, 
W. F. Durand, ed., Julius Springer (Berlin), 1943, pp. 243-246. 
. 
. 
NACA l!N 4230 
NACA TN 4230 13 
1 t 
- Equilibrium 
---- Frozen 
7 
6 
\ 
5 
I 
O0 
\ 
100 2cP 300 400 5o” 6o” 70* 800 
Flow deflection, u 
2h 
Figure l,- Comparison of q = l-t 2 audfi for frozen and equilibrium 
Prskdtl-Meyer flow; M. = 1.00, To = 614-0~ K, p. = 1.2 atiosppleres. 
14 IW!A TN 4230 
_-. 
O0 IO0 20* 4o" 60' 80° 
Flow deflection, Y 
Fi .gure 2.- Comparison of density for frozen and equilibrium Prandtl-Meyer 
flow; M, = 1.00, To = 6140’ K, p. = 1.2. atmospheres. .-. 
NACA TN 4230 
O0 100 2o" 40* 60' 80* 
Flow deflection, v 
3 
c 
Figure 3.- Comparison of pressure for frozen and equilibrium Prmdtl-Meyer 
flow; M, = 1.00, To = 6140~ K, p. = 1.2 atmospheres. 
14 
12 I I I 
j 
I I I I I 
I 
M 
0 
- Equilibrium 
---- Frozen 
0 lo” 28 30’ 40° 50° SO” 70’ 80’ 90” 
Flow deflection, Y 
Figure 4.- Comparison of Msch number for frozen and eqtilibrium Prandtl-Meyer flow; M, = 1.00, !s 
T, = 6140~ K, p. = 1.2 atmospheres. 
;G 
% 
c . I c 
rl I P L 2 
E 1.4 
:D 
F I 
f 
I 
I.2 
.8 
.6 
- Equilibrium 
---- Frozen 
0 lo” 20° 30° 40° 50” 60’ 70” 80° 9C 
Flow deflection,u 
)0° 
Figure 5.- Ccmrparj.son of tmperature for frozen and equilibrium F?rmdtl-Player flow; M. = 1.00, 
To = 6140~ K, p. = 1.2 atispheres. 


Prandtl-Meyer expansion of chemically reacting gases in local chemical and thermodynamic equilibrium

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