Recently, dense gases have been investigated for many engineering applications such as for turbomachinery and wind tunnels. Supersonic nozzle design for these gases is complicated by their nonclassical behavior in the transonic flow regime. In this paper a method of characteristics (MOC) is developed for two-dimensional (planar) and, primarily, axisymmetric flow of a van der Waals gas. Using a straight aortic line assumption, a centered expansion is used to generate an inviscid wall contour of minimum length. The van der Waals results are compared to previous perfect gas results to show the real gas effects on the flow properties and inviscid wall contours

Aldo, Andrew C.

Argrow, Brian M.

English

NASA Technical Reports Server

SUPERSONIC MINIMUM LENGTH NOZZLE DESIGN
FOR DENSE GASES
N94-23656
Andrew C. Aldo" and Brian M. Argrow**
Department of Aerospace Engineering Sciences
University of Colorado
Boulder, Colorado80309
ABSTRACT
Recently, dense gases have been investigated for many engineering applications such as for turbo-
machinery and wind tunnels. Supersonic nozzle design for these gases is complicated by their nonclassical
behavior in the transonic flow regime.. In this paperamethod of characteristics (MOC) is developed for
two-dimensional(planar) and, primarily, axi,symmetric flow of a van der Waals gas. Using a straight sonic
line assumption, a centered expansion is used to generate an invisdd waLlcontour of minimum length_ The
van der Waals resulti are compared to previous perfect gas results to show the real gas effects on the flow
properties and inviscid wall contours.
INTRODUCTION
A minimum length nozzle _ produces a tmiform supersonic flow with a minimum ratio of
throat height or radius to total nozzle length. Argrow and Ematmd (mfs. 1, 2) present detailed discussion
of MLN designs, comparisons of different types, computational flow field analysis, and engineering ap-
plications. Until this paper, all MLN analyses, that we are aware of, have focused only on perfect gas re-
suits. Several authors (rds. 3-10) have recently investigated nozzle flows for nonideal gases. References
3-5 and 8-11 discuss steady flows of dense gases. Cramer (ref. 11) refers to these gases as BZT (Bethe-
Zel'dovich-Thompson)torecognizetheindividualsthatfasttheorizedtheir nonclassical behavior.Such
behavior includeslocala minimum of theMach number duringsteadyisentropicexpansion,expansion
shocks, increase in the critical Mach number, and other nonclassical behaviors. BZT fluids typically have
large polyatomic m01ecules with comparatively largo specific heats.
All the previously mentioned references that iave.,_dgam BZT fluids discuss the thermodynamic
condition that governs the classical or nonclassical behavior. The governing thermodynamic parameter is
where
is the speed of sound, @_p, and _ are the density, pressure, and entropy, respectively. The overbar indicates
dimensionalquantities.The parameter/"isreferredtoby Thompson (ref.13)asthefundamentalderivative
ofgasdynamics.
For most fluidsF > 0 undernormalconditions,butforBZT fluidsmay haveF < 0. The F < 0
region of a BZT gas (dense gas) occurs in the dense gas region near the saturated vapor curve in the pv
plane. Practical uses for the nonclassical behavior of Bz'r fluids include turbomachinery and heavy-gas
*'Aaaimat Pr_
329
https://ntrs.nasa.gov/search.jsp?R=19940019183 2020-06-16T15:24:11+00:00Z
wind tunnels. Discussion of the various applications that may capitalize on the nonclassical behavior of
BZT fluids can be found in refs. 3, 5, and 8-12.
The most apparent application of a MLN designed for dense-gas flow is for heavy-gas wind run-
nels. Anderson (ref. 9) shows Navier-Stokes calculations for the flow of sulfur hexaflouride (HF6) over a
NACA 0012 airfoil.Thisstudyinvestigatedthefeasibilityofusingthislarge-moleculegasforwind tunnel
applications.While most ofthepreviousreferencespeculatethatnozzlescan be designedtoproducesu-
personicflowofa BZT fluid,thecalculationshavebeenLimitedtoone-dimensionalcases(refs.3-5,8,10,
Ii).
THERMODYNAMIC MODEL
The van der Waals equationof state is
_a
P=___ _,
where _ is the specific gas constant, ; is the specific volume, and
- anda"= 27
The c subscriptreferstoconditionsatthecriticalpoint.The followingthermodynamicdevelopment and
nondimensionalization scheme follows that of ref. 19. The ¢nthalpy _ and speed of sound a are given by
_= g_+_ l+d -_"iT,
[( )'___ _(1÷6)-
Here erisan arbitraryreferencenergyand
Because_ there is minimal temperature variation for the flows investigated, we assume a constant _ = _ ®,
where cv ®is the ideal gas specific heat. The temperature variation for isentropic flow of a van der Waals
gas is given by
where the 0 subscript refers to stagnation conditions.
330
The nondimensional (reduced variable) form of the van der Waals equation is written as
ffi 8T 3
P 3v_-I - v2'
and the critical compressibility is
(")Z_ = _--_c =
with the reducedvariables
P ffiP"_' -_c V fv"_"
The nondimensional form of the other thermodynamics relations are
--'_0'
2a 11/2
g = ,/2(% - h)
a
where
METHOD OF CHARACTERISTICS FOR REAL GASES
The method of characteristics (MOC) used by Argrow and Emanuel (ref. 1) assumes an isentropic,
ixrotational flow of a perfect gas. For this case, the governing two-dimensional partial differential equation
reduces to a set of four algebraic equations, two characteristic and two compatibility equations. For'the
axisymmeu--ic case, the characteristic and compatibility equations form a set of four ordinary differential
equations that are' solved simultaneously. Derails of the solution procedures can be found in ref. 1.
A MOC for the isentropic two-dimensional or axisymmetric flow of a real gas (ref. 14) is used for
the present study. The method is completely general and we use the van der Waals thermodynamic model
more for simplicity than for accuracy. The governing partial differential equations (PDEs) are given by the
gas dynamic equation, the irrotationality condition, and the speed of sound relation,
(v_ - _2)v_+ (v2 - ,,,2)% + 2v_vy_-f-_ ,,2vy-a--T-=O ,
331
avx
ay ax =0'
a --a(V) = a(Vx,Vy),
whexe o --0 fortwo--dimensionalflow oro --0 foraxisymmetricflow,x and y axetheaxialand radial
(transverse)coordinatesnondimensionalizedwithrespectothroatradiusorhalf-height,Vx and W arethe
correspondingvelocitycomponents,and V isthevelocitymagnitude,allvelocitiesaxenondimensionalized
inthesame manner asthespeedofsound_. Along thecharacteristicl nes,thatcorrespondtoMach linesin
theflow,this ystemofPDEs reducestotwo ordinarydifferentialequationscalledcompatibilityequations.
Fig. 1 is a schematic of the supersonic flow field showing the MLN geometry and the flow geometry
associated with an arbitrary point on a streamline. The angle 0* is the initial inclination of the supersonic
contour. The left and right running characteristics designated as C÷ and C_ are Mach lines inclined at the
Mach angle _t with respect to the velocity vector V. The corresponding chaxacteristic and compatibility
equations are
C± characteristic equation:
C± compatibilityequation:
4.
a2Vv
- a2)avx,+ [2v._vy- (v_ - a2)A±ldVy+ --¢7-.-]._--,'I,c±= O. (Ib)
The gas is assumed to enter the supersonic portion of the nozzle along the straight sonic line OA
uniform and parallel to the axis. It is then expanded and accelerated (or possibly decelerated in the F < 0
region) through the nozzle and exits the nozzle with a uniform flow crossing the terminating characteristic
BC at the exit Mach number Mf. For the two-dimensional nozzle, the centered expansion generated by the
sharp throat is a Prandtl-Meyer expansion. In the axisymmetric case, the flow at the wall is locally two-di-
mensional, thus at the throat the expansion is locally Prandt1-Meyer. Construction of the flow field begins
by discretizing the centered expansion into equally'spaced velocity increments. Each velocity increment
A V has an associated isentropic flow turn angle increment A O. The angle increment A 0 is computed from
the relation
f V202 -O_ := M__-- l dv
J Vt V "
(2)
For a perfect gas, the AM associated with an angular deflection .40 can be easily determined from the
Prandtl-Meyer function. For a real gas, the Prandtl-Meyer computation requires the solution of a system of
ordinary differential equations as shown by Cramer (ref. I0). We avoid the Prandtl-Meyer computation by
using the angle-velocity relation, equation (2).
With the position of the throat specified and the velocity components (Vx,Vy) computed from V and
0, the necessary independent variables are determined at the throat. Equations (1) are solved simultaneous-
ly using the second-order average-property Euler predictor-corrector scheme described in ref. 14. The
characteristic net of the kernel region OAB and the transition region ABC is constructed using the unit pro-
cesses described in ref. 1. The wall contour corresponds to the streamline that passes through points A and
C. This streamline is determined by also using the average-property Euler predictor-eorrector scheme to
integrate the equation
dy....__- tan0w
dz
from the initial condition, 0w = 0* at x = 0 to the exit condition 0w = 0 at xf = 0.
332
The accuracy of the characteristic computations and the wall contour are affected by the spacing of
the characteristic nodes. The C_ characteristics that emanate from the comer at point A reflect from the axis
as C+ characteristics and bend away from the center of the expansion. This causes relatively large spacing
between characteristic node points and the subsequent computed wall points just downstream of the throat
where the gradients are largest. To alleviate this situation, the characteristic net is compressed toward the
sonic line by decreasing the step size of the speed increments in the centered expansion discretization. The
Prandtl-Meyer and turn angles in ref. 1 are discretized and compressed in a similar manner.
RESULTS
For the nominal case of TO=1.01, c5= 0.02, and v0 = 0.70, dV a c_V = 10 .6 for equation (2). The
fL,'stcharacteristic is chosen at 6V, then the step size is increased until a characteristic is generated at every
206 V. This allows characteristics to be compressed towards the sonic line for improved accuracy. For the
nominal case, the limits of integration for equation (2) are set at the sonic speed VI = 0.26 to II2 = 0.9Vmax
with Vmax = 9.93. Characteristic spacing in the transition region is controlled by an aspect ratio that keeps
the shape of the characteristic cells as uniform as needed. For the relatively short nozzles produced in this
study, the aspect ratio was set at 0.5. Reference 1 gives a complete description of how the characteristic
compression and transition region aspect ratio affect the overaU computational accuracy.
Figures 2 show the waLl contour and the variation of M, 0, and F along the axis and along the wall
for a two-dimensional nozzle with To =1.01, 6 =0.02, and vo =0.70, 0.85. Figure 2(a) indicates the increase
in nozzle length required for the more dense gas. Note that 0 ° is fixed at 2.5 ° for the comparisons in Figs.
2. This is near the maximum value of 0 ° where the C_ characteristic, near AB of Fig. 1, begin to overlap
other C_ characteristics in the kernel, or there is overlapping from C+ characteristics near BC with both
instances producing an oblique expansion shock. Once a shock occurs (compression or expansion), the
flow is no longer isentropic and the MOC cannot be applied without some special procedure to account for
the placement of the shock (We do not to incorporate such a procedure in the MOC used for this study.)
For Figs. 2(b-d), the axis curves are terminated at the end of the kernel because the uniform flow
region begins at point B as shown in Fig. 1. The wail curves extend to the end of the contour at point C as
also shown in Fig. 1. The density 0 decreases smoothly along the axis and wall for both cases as shown in
Fig. 2(b). Figure 2c shows that the gas is expanded into a F < 0 region that extends through the nozzle exit
for the nominal v0 =0.70 case. This produces a uniform supersonic flow with F < 0. For this case, the
Mach number reaches a maximum of about 1.88 along the axis and along the wall, as shown in fig. 2d.
This agrees with the quasi one-dimensional results obtained by Cramer and Best (ref. 12).
A comparison of the waLl contours at the maximum Mf for the two-dimensional and axisymmetric
nominal cases is shown in Fig. 3. Note that in the contour plots, the y-axis is not to scale. Although diffi-
cur to see, the axisymmetric contour contains an inflection point just downstream of the throat that is not
present in the two-dimensionalcontour. Also, 0* is larger for the two-dimensional case as will be shown in
more detail in the foLlowing figures. The results of this plot agree with the perfect gas results of Argrow and
Emanuel (ref. 1).
O . ,
Figures 4(a) and 4(b) show the 0 vs. M/vanauon for the axisymmetric and two-dimensional cases,
respectively. These figures show the nominal case, a perfect gas with nominal conditions, and nominal
conditions with v0 -- 0.85. Varying stagnation conditions for the perfect gas does not effect the nondimen-
sional results. The dense gas case, v0 = 0.7, shows BZT gas behavior, reaching a maximum Mf of about
1.88 for 0*near 1° for the axisymmetric case shown in Fig 4(a). Then M/decreases until reaching a maxi-
mum 0 *value of about 2.5 °. The maximum Mf for the two-dimensional case is at 0 * B 2.5 °. The decrease
of M/ for 0 * > 2.5 ° is not shown in Fig. 4(b), although the maximum 0 *for this case is about 3.5 °.
Figures 5 and 6 also show 0-vs. M/for the for the nominal axisymmetric case. Figure 5 shows the
effect of varying To, keeping other nominal conditions fixed, compared to a perfect gas. This shows that
slightly increasing TO moves the flow away from the dense gas region. Figure 6 shows the effect of varying
333
cv/R,keepingtheothernominalconditionsfixed,compared toa perfectgas.Thisshows a tendencytoap-
proachperfectgas behaviorascv/Risdecreased.
Figures 7 and 8 show the variation of the kernel length ._ and nozzle length x/ vs. My, respectively.
Note that for the two-dimensional case, the figures show the increase to the maximum lengths but the de-
&.
crease as 0 increases (and Mf decreases from the maximum) is not shown in these plots. The BZT gas
behavior is indicated in the nominal cases. Because the gas is more dense, the lengths are longer and reach
a maximum before decreasing.
CONCLUSIONS
The MOC is applied to the steady, isentropic flow of a dense gas. A method is presented for gener,
ating inviscid _ contours to produce a uniform supersonic flow. The MI_ procedure presented is lira=
i;-,d to cases where there is only one sonic point. Inclusion of more than one sonic point will require the
coupling of the present method with a subsonic contour design procedure. In order to continuously isen=
tropically expand the gas from a stagnation state to M -,- oo, reference 12 shows that for the nominal case
investigated in the present study, M Rrst reaches a local maximum value of about 1.88 before decreasing to
a supersonic minimum of about 1.05. We have shown that it is not possible for a nozzle to use a single
centered expansion to accomplish _S. At best, the single centered-expansion _ can generate an isen=
tropic expansion slighdy beyond the local maximum before an expansion shock is generated. The maxi-
mum allowable 0* appears to occur as the F < 0 region approaches the throat, resulting in a shock. We
speculate that a design that employs two centered expansions separated by a finite converging wall may be
able to accomplish the expansion in a minimum length. This will be investigated in the future.
It was shown that the nozzle designs presented may be used to produce a steady supersonic flow of
a BZT gas in a/" < 0 state. This raises the possibility of producing interesting wind tunnel experiments to
study the supersonic external flow of a dense gas over airfoils and other aerodynamic shapes.
ACKNOWLEDGEMENT
The authors gratefully acknowledge Mark. S. Cramer and George Emanuel for their insightful cor-
respondence and suggestions.
REFERENCES
i. Argrow, B. M. and Emanuel, G., "Comparison of Minimum Length Nozzles," J'. of Fluids Engineering
2. Argrow, B. M.; and Emanuel, G.: Computational Analysis of the Transonic Flow Field of Two-Dimen-
sional Minimum Length Nozzles. J. Fluids Eng., vol. 113, Sept. 1991, pp. 479-488.
3. Cramer, M. S.; and Fry, R. N.: Nozzle Flows of Dense Gases. Phys. Fluids A, vol. 5, May 1993, pp.
1246-1259.
4. Chandrasekar, D.; and Prasad, P. P.: Transonic Flow of a Fluid with Positive and Negative Nonlineari-
ty through a Nozzle. Phys. Fluids A., vol. 3, Mar. 1991, pp. 427-438.
5. Schnerr, G. H.; and Leidner, P.: Diabatic Supersonic Flows of Dense Gases. Phys. Fluids A, vol. 3,
OCt.1991,pp.2445- 2458.
6. Bober,W.; and Chow, W. L.:NonidealIsentropicGas Flow Through Converging-DivergingNozzles.
L FluidsEng.,vol.112,Dec. 1990,pp 455--460.
7. V'mokur,M.: DiscussionofNonidealIsentropicGas Flow Through Converging-DivergingNozzles.L
FluidsEng.,vol.112,Dec. 1990,pp.460--461.
8. Kluwick,A.: TransonicNozzleFlow ofDense Gases. L FluidMech.,vol.247,1993,pp.661-688.
9. Anderson,W. K.: NumericalStudyon Using SulfurHexaflourideasaWind TunnelTestGas. AIAA
J'.,vol.29,Dec. 1991,pp.2179-2180.
10.Cramer,M. S.;and Crickenberger,A. B.: Prandd-Meyer FunctionforDense Gases.ALAA J.,vol.30,
Feb.1992,pp.561-564.
334
i I. Cramer,M.S.: Nonclassical Dynamics of Classical Gases. Nonlinear Waves in Real Huids, Springer-
Verlag, 1991, pp. 91-145.
12. Cramer, M. S.; and Best, L. M.: Steady, Isentropic Flows of Dense Gases. Phys. Fluids A, vol. 3, Jan.
1991,pp.219-226.
13.Thompson, P. A.: A" Fundamental Derivativein Gasdynamics. Phys. F'luids,vol.14, 1971, pp.
1843-1849.
14. Zucrow, M. J.; and Hoffman Gas Dynamics. Vol. 1. W'dey, 1976, vol. 2 Krieger, 1977.
335
CA
streamline
O+
0 B
Fig. 1 MLN supersonic flow field geometry, upper half-plane.
336
y1.15
' I ' t
1.0 2.0
1.10
1.O5
1.00
0.0
)
v, = 0.70,
....... v, = 0.85
i
3.0
(a)
1.20
' I ' I
p 0.60
0.00
0.0
L I I .. I
1.0 2.0
% = 0.70. wail
% = 0,70. axis
..... % : 0.85. wall
...... % ,, 0.85. a,_is
3.0
(b)
6.0 I ' I
r 2.0
-2.0
0.0
_; = 0.70, wall
v, • 0.70, axis
....... v, = 0.8.5, walt
....... % = 0.85, axis
2.0 3.0
(c)
M
2.0
1.5
1.0
0.0
_ v, ,, 0.70, wall
v, - 0.70, aXiS
...... v, - 0.85, wall
....... = O B5,aXis
i _ i I i I I
1.0 2.0
X
3.0
(d)
Fig. 2 (a) Wail contour, (b) density, O, distributions (c) fundamental derivative, F, distribu-
tions; (d)Mach number,M, distributions,alongthewalland theaxis,allfortheaxisymmetric
case,nominaland forvo = 0.85.
337
1.12
Y
1.10
1.08
1.06
1.04
1.02
1.00
............2D,MI= 1.88
AX,M,= _.8S
°°
•°
o°
,,
,°° .
o°
°°
o,
,°
°°
,o
°,
°o
,o•
. o..° ....
,°°
°.o°
o
•o•°
o°
oO
o•
I i 1 i
0.0 1.0 2.0
X
1
3.0
Fig. 3 Comparison of two-dimensional and axisymmetric contours for maximum
M:,nomi_ case.
4.0
338
3.0 i i I I
..........perfect
- . --- Vo = 0.85
.," -- vo = 0.70
2.0 ' ""
I I .,:"
Ii ,,,:"
ii ,."
1.0 ' ""
I I .."
f ."°
I I .."
/:'"""
O,O , I ......... 1 , I , I ,
1.0 1.2 1.4 1.6 1.8
M,
Fig. 4 (a) Izlidal wall turn angle 0*vs. M/, nomJnal conditions, variable v0,
axisymmetric case.
2.0
3.0 I ' I ' I I
A
@
_D
2.0
1.0
0.0
I /
:' ......... - perfect gas /
/ .. ---  o=O.SS /
/ ': vo=0"70 /
] ::
I "
I .;
I ";
# °°A"
; I , I , I , I ,
1.0 1.2 1.4 1.6 1.8
M,
Fig.4 Co) Initialwallmm angte0*vs.M f,nominalconditions,variablev0,
two-dimensionalcase.
2.0
339
3_0 I I I I
2.0
1.0
0.0
1.0
..........perfect
/°°o /°/°/0/°°//"° "° _ _ .
.°
I j I , ,I, _ I
1.2 1.4 1.6 1.8
Fig.5 Initialwallturnangle0"vs.Mr,nominalconditions,variableTo,axi-
symmetric case.
2.0
,,0 I _ I I I
.,. .......... perfect gas
"__ "'"'"-.::' ".........6 = 0.0400
2.0 _'_"'-... _ 6=0.0133
......t_.. - - - 8 --0.0200
o" jl I* i
.." o°oop °° _, _'
°. .jo.-" _ _
1.0 // • oo-°°_ _,/ ° o.O°'J,_ ,_
0.0 , , , I , I , I ,
1.0 1.2 1.4 1.6 1.8
M:
Fig.6 Initialwallmm angle0 "vs.Mr, nominalconditions,variablecv/R,
axisymmetriccase.
2.0
34O
1.0
xs
.......... perfect, 2D
--- vo=0.85,2D
vo = 0.70, 2D
.......... perfect, AX
--- vo= 0.85, AX
%= 0.70, AX
, ] I
1.8
, I
1.2 1.4 1.6
M,
Fig. 7 Kernel lengthxa vs. Mf, nominal two-dimensional and axisymmetric
case, s with variable v0.
2.0
3.0
2.0
xl
1.0
I I I r, I
m
AX
_# vo 0 70, AX
0,0 , I = I , I , I
1.0 1.2 1.4 1.6 1.8
Fig. 8 Nozzle length xf vs. My,nominal two-dimensional and axisymmet-
tic cases with variable vo.
2.0
341



Supersonic minimum length nozzle design for dense gases

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