European Congress on Computational Methods in Applied Sciences and Engineering
ECCOMAS 2004
P. Neittaanmäki, T. Rossi, K. Majava, and O. Pironneau (eds.)
W. Rodi and P. Le Quéré (assoc. eds.)
Jyväskylä, 24—28 July 2004
INVESTIGATION OF BZT TRANSONIC FLOWS PAST AN AIRFOIL
USING A 5TH POWER VIRIAL EQUATION OF STATE
Paola Cinnella, Pietro M. Congedo and D. Laforgia
Department of Engineering for Innovation
University of Lecce, via per Monteroni, 73100, Lecce, Italy
e-mails: [email protected], [email protected], [email protected]
Key words: Dense gas/BZT fluid/equation of state/transonic flow/airfoil
Abstract. Dense Gasdynamics studies the flow of gases in a thermodynamic region close to
the liquid-vapour critical point. In recent years, great attention has been paid to certain
substances, known as the Bethe—Zel'dovich—Thompson (BZT) fluids, that exhibit in the
vapour phase, for a whole range of temperatures and pressures above the upper saturation
curve, negative values of the Fundamental Derivative of Gasdynamics. This can lead to
nonclassical gasdynamic behaviors, such as rarefaction shock waves, mixed shock/fan waves,
shock splitting and other. The uncommon behavior of BZT fluids can find application in
technology, in particular to reduce losses due to wave drag and shock/boundary layer
interaction in turbines and nozzles. The present work presents a detailed numerical study of
transonic BZT fluid flows past a NACA0012 airfoil. The compressible Euler equations are
solved using a third-order accurate centred scheme, and the fifth virial power equation of
Martin and Hou is used to model the thermodynamic behavior of the fluid. A parametric study
of the influence of free-stream thermodynamic conditions has been performed for a typical
BZT gas flowing past a NACA0012 in transonic regime, at fixed Mach number and angle of
attack. The objective is to investigate the influence of BZT effects on the airfoil performance
and to demonstrate the possible gains with respect to a classical perfect gas flow past the
same configuration.
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P. Cinnella, P.M. Congedo and D. Laforgia.
1 INTRODUCTION
Dense Gasdynamics studies the dynamic behavior of gases in the dense regime, i.e. for a
range of temperatures and pressures close to the thermodynamic critical point, where the
perfect gas law is invalid. Since about a decade, there has been increased interest in dense gas
flows, motivated by the critical role they play in many engineering applications. Examples
include: high Reynolds wind tunnels, chemical transport and processing and Rankine power
cycles. Particular attention has been paid to some heavy polyatomic fluids, referred-to as the
Bethe—Zel’dovich—Thompson (BZT) fluids, for which the well-known compression shocks
of the perfect gas theory violate the entropy inequality, over a certain range of temperatures
and pressures in the vapour phase, and are therefore inadmissible [1, 2]. In fact, such fluids
exhibit a region of negative values of the Fundamental Derivative of Gasdynamics Γ [1], i.e.
reversed isentrope concavity in the p-v plane, which, in the transonic and supersonic regime,
leads to nonclassical gasdynamic behaviors, such as rarefaction shock waves, mixed
shock/fan waves, shock splitting and other. The thermodynamic region where this phenomena
occur is often called the inversion zone. BZT properties are generally encountered in fluids
characterized by large heat capacities and complex, heavy molecules, such as some
commercially available heat transfer fluids. The nonclassical phenomena typical of BZT
fluids could find application in technology: particularly attractive seems the possibility of
reducing losses due to wave drag and shock/boundary layer interactions in turbomachines and
nozzles [3-6], with particular application to Organic Rankine Cycles (ORCs). ORCs’ working
fluids are in fact heavy organic compounds with large heat capacities that possess BZT
properties. Now, one major source of losses in ORC turbines is due to wave drag, because
they usually operate in the transonic/supersonic regime: the use of a BZT fluid could avoid
shock formation and, ideally, allow isentropic turbine expansion.
In previous works [7,8,9], the first two authors have investigated the influence of BZT
effects on the flow field properties for a simplified configuration, represented by an isolated
NACA0012 airfoil in transonic regime. Such computations were performed using the van der
Waals equation of state for polytropic gases, the simplest thermodynamic model that allows to
take into account BZT effects. The results showed a dramatic improvement of the
aerodynamic performance when the free-stream value of the Fundamental Derivative was
taken sufficiently high for supercritical flow to be established, although sufficiently small to
allow significant BZT effects yet [9]. In particular, such optimal choice allowed to obtain
increased lift and reduced drag with respect to a perfect gas flow with the same free-stream
Mach number and angle of attack. The airfoil lift was enhanced by the formation of an
expansion shock wave at the airfoil suction surface, close to the leading edge, that drove the
pressure coefficient at levels much lower than those typically obtained for a perfect diatomic
gas flowing at the same conditions; the flow then recompressed through a classical
compression shock wave located at the rear part of the airfoil. On the other hand, losses
introduced by such discountinuities were found to be very low, due to the small entropy
changes associated to weak shocks in the vicinity of the transition line. When operating the
system at thermodynamic conditions closer to the inversion zone, i.e. at lower free-stream Γ,
the flow was found to be entirely subcritical and characterized by zero drag (in the limit of
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P. Cinnella, P.M. Congedo and D. Laforgia.
vanishing mesh size); even though the lift coefficient was lower than in the perfect gas case.
On the contrary, chosing the system operation point quite far from the inversion zone, the
results were qualitatively similar to perfect gas flow, and the benefits due to BZT effects
progressively disappeared.
The van der Waals equation of state has the advantage of simplicity of implementation and
low computational cost, and is capable to describe all qualitative features of BZT flows.
However, it is well-known that such an equation largely over-predicts the extent of the
inversion zone, so that it is legitimous to wander if results presented in [9] are not too much
optimistic.
In the present work, a realistic equation of state involving five virial expansion terms has
been considered. The equation has been implemented within a numerical code for dense gas
flow simulations, based on a simple and efficient third-order accurate centred solver [10]. The
aim of the present investigations is twofold: (i) to provide more quantitative results on how
dense gas effects influence the airfoil performances (lift, drag, lift-to-drag ratio) and at which
extent far from the inversion zone they are still significant; (ii) to verify that the airfoil
performance enhancement mechanisms pointed out in [9] are qualitatively the same,
independently from the equation of state used to model the fluid thermodynamic behavior. A
parametric study of the influence of free-stream thermodynamic conditions on the
aerodynamic performance of a BZT transonic flow past a NACA0012 airfoil has been
undertaken. The results are critically analized and compared to those obtained for a perfect
diatomic gas and for a heavy van der Waals gas flowing past the same configuration.
2 BASICS OF DENSE GASDYNAMICS
Let us consider the key nondimensional parameter:
Γ := 1 +
ρ  ∂a 
v3  ∂ 2 p 
=

 , (1)


a  ∂ρ s 2a 2  ∂v 2  s
where a denotes the speed of sound, ρ the fluid density, s the entropy, v=1/ρ the fluid specific
volume and p the pressure. The above quantity is commonly referred-to as the Fundamental
Derivative of Gasdynamics [1, 2] because it determines the nonlinear behavior of gases. If
Γ <1, the flow would exhibit an uncommon sound speed variation in isentropic perturbations:
a will grow in isentropic expansions and reduce in isentropic compressions, contrarily to what
γ +1
happens in “common” fluids. For perfect gases, Γ is equal to
, where the specific heats
2
ratio γ is always greater than 1 for thermodynamic stability reasons: thus, Γ > 1 as well. For
heavier gases, composed by sufficiently complex molecules and characterized by high cv/R
ratios (with cv the constant volume specific heat and R the gas constant), Γ is smaller than 1,
or even than 0, for extended ranges of densities and pressures, and it tends to recover its
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P. Cinnella, P.M. Congedo and D. Laforgia.
“perfect gas” value in the low density limit. The sign of Γ is univocally determined by the
 ∂2 p 
sign of the second derivative  2  , i.e. the concavity of the isentrope-lines in the p—v
 ∂v  s
plane. It is possible to show [11], [2] that the entropy change across a weak shock can be
written as:
a 2Γ ( ∆v )
4
∆s = − 3
+ O ( ∆v ) ,
v
6T
3
(
)
(2)
where ∆ represents a change in a given fluid property through the shock and T is the absolute
temperature. As a result, in order to satisfy the second law of thermodynamics, a negative
change in the specific volume, i.e. a compression, is required if Γ > 0, while a positive
change, i.e. an expansion, is the only physically admissible solution when Γ < 0. A BZT fluid
is defined as a fluid that exhibits a region of negative Γ in the vapour phase. Such a region is
often mentioned as “the inversion zone”, and the Γ=0 contour is called the transition line [2].
Another important property of BZT fluids is that the shock strength is reduced up to an
order of magnitude from that predicted by equation (2) for thermodynamic conditions where
Γ≈0. Cramer and Kluwick [2] showed in fact that Γ = O ( ∆v ) for small volume changes in the
vicinity of the transition line. Thus, shock waves having jump conditions in the
thermodynamic region near the Γ=0 contour are expected to be much weaker than normal.
3 GOVERNING EQUATIONS AND THERMODYNAMIC MODEL
As long as the thermodynamic states are restricted to the single-phase regime, the flow is
governed by the equations for equilibrium, non-reacting flows. As BZT effects influence
essentially the inviscid behavior of the flow, in the present study we consider the Euler
equations, written in integral form for a control volume Ω with boundary ∂Ω:
d
w d Ω + ∫ f ⋅ n dS = 0 .
∂Ω
dt ∫Ω
(3)
In equation (3), w is the conservative variable vector,
w = ( ρ , ρ v, ρ E ) ,
T
n is the outer normal to ∂Ω, and f is the flux density:
(
f = ρ v, pI + ρ vv, ρ vH
)
T
where v is the velocity vector, E the specific total energy, H=E+p/ρ the specific total
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P. Cinnella, P.M. Congedo and D. Laforgia.
enthalpy, and I the unit tensor. The above equations are completed by a thermal and a caloric
equation of state:
p = p ( ρ ( w), T ( w) ) ,
(4)
e = e ( ρ ( w), T ( w) ) ,
(5)
where e is the specific internal energy. Formally, equations (4) and (5) can be combined in a
single equation of state of the form:
p = p ( ρ ( w), e( w) ) ,
(6)
even if such an equation can not always be written explicitly.
In the present work, the Martin-Hou equation [12] of state is used, that provides a realistic
description of the gas behavior and of the inversion zone size.
Such equation, involving five virial terms and satisfying ten thermodynamic constraints, is
given by:
p=
5
f (T )
RT

+∑ i
v − b i = 2 (v − b )i
 
where R is the gas constant, and the functions fi (T) are of the form:
f i (T ) = Ai + BiT + Ci e
−k
T
Tc

with Tc the critical temperature and k=5.475. The gas-dependent coefficients Ai, Bi, Ci can be
expressed in terms of the critical temperature and pressure, the critical compressibility factor,
the Boyle temperature TB and one point on the vapour pressure curve. The main advantage of
this equation is to ensure high accuracy with a minimum amount of experimental information.
The caloric equation of state is of the form:
5 
T
Tf ' (T ) − f i (T ) 
e = er + ∫ cv∞ dT + ∑  i
(8)
i −1 
Tr
i = 2  ( −i )( v − b )



where a power law is used to model the low-density specific heat variation with temperature:
n
T 
cv∞ (T ) = cv∞ (Tc )   ,
 Tc 
and the critical value for cv∞ and the exponent n are gas-dependent parameters.
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P. Cinnella, P.M. Congedo and D. Laforgia.
Figure 1: Amagat diagram for PP10.
Figure 1 shows the p-v diagram and the inversion zone for a heavy fluorocarbon, the pfperhydrofluorene (commercial name PP10). Such a fluid, used in all computations presented
in the sequel, is characterized by the following thermochemical properties:
Chemical formula Tc (K) pc (atm)
Zc
n
TB(K) cv∞(Tc)/R
C13F22
632.2
16
0.283 463.2
78.4
0.5255
Table 1: Thermochemical properties of PP10.
For completeness, we recall here the thermodynamic relations used to compute some
quantities mentioned in the sequel.
1/ 2
•
•
Speed of sound:
Entropy:
 T  ∂p 2 ∂p 
a = v 
 − 
 cv  ∂T  ∂v 
s = sr + ∫
T
Tr
(9)
5
f i ' (T )
 v−b 
cv∞
dT + R ln 
 +∑
i −1
T
 vr − b  i = 2 ( −i )( v − b )
(10)
4. NUMERICAL METHOD
The governing equations are dicretized using a cell-centred finite volume scheme of thirdorder accuracy [13], extended to the computation of flows with an arbitrary equation of state
[10]. The scheme is constructed by correcting the dispersive error term of the second-orderaccurate Jameson’s scheme [14]. The use of a scalar dissipation term simplifies the scheme
implementation with highly complex equations of state and greatly reduces computational
costs [10]. In order to preserve the high accuracy of the scheme on non cartesian grids, the
6
P. Cinnella, P.M. Congedo and D. Laforgia.
numerical fluxes are evaluated using suitably weighted discretisation formulas, which take
into account the stretching and the skewness of the mesh: this allows to ensure a truly thirdorder accuracy on moderately deformed meshes and at least second-order accuracy on highly
distorted meshes (see [13, 15] for details). The governing equations are integrated in time
using a four-stage Runge--Kutta scheme [14]. Local time-stepping, implicit residual
smoothing and multigrid are used to efficiently drive the solution to the steady state. The
method has been successfully validated for a variety of perfect and real gas flows [13, 16, 10].
5. RESULTS
The above numerical method has been used to investigate the inviscid transonic flow past a
NACA0012 airfoil, characterized by a free-stream Mach number equal to 0.85 and an angle of
attack equal to 1°.
The computations have been performed using three C-grids of increasing density, formed by
136×20, 272×40 and 544×80 cells, respectively. The finer and the coarser grids have been
generated by doubling or halving, respectively, the number of cells of the medium one in each
direction. The outer boundary is about 20 chords away from the airfoil and the mean height of
the first cell closest to the wall is about 5×10-2 chords on the medium grid. A view of the
coarse grid is provided in Fig. 2. For most of the computations presented in the sequel, grid
convergence for the wall pressure and Mach number was obtained on the medium grid.
However, the results presented above have been obtained on the fine grid.
5.1 Perfect gas reference solution
Firstly, the solution has been computed for a perfect diatomic gas (specific heat ratio γ ≡1.4):
such results are used in the sequel as a reference for comparisons with BZT flows. The overall
solution (pressure coefficient contours) is presented in Fig. 3. The flow is characterized by
two shocks at about 85% of the chord at the suction side, and 63% at the pressure side (see
also Fig. 4). The computed lift coefficient, drag coefficient, and lift-to-drag ratio are:
CL=0.373;
CD=5.74x10-2;
CL/CD=6.51
Figure 2: View of the coarse grid.
7
P. Cinnella, P.M. Congedo and D. Laforgia.
Figure 3: Perfect gas flow past a NACA0012, M∞=0.85, α=1°. Pressure coefficient contours, ∆CP=0.05.
Figure 4 : Perfect gas flow, wall Mach number and pressure coefficient.
5.2 Choice of the operation points
Then, a parametric study of the influence of the thermodynamic free-stream conditions on the
airfoil performance has been performed for the heavy fluorocarbon PP10. Twenty-six
operation points were chosen, according to the following considerations:
• in the present study, inviscid steady flows with uniform free-stream are considered,
where the only source of entropy gradients is given by shock waves;
• for flows past isolated airfoils in transonic regime, shock waves are quite weak and
produce relatively small entropy changes;
• for BZT fluids, shocks having jump conditions in the vicinity of the transition line,
8
P. Cinnella, P.M. Congedo and D. Laforgia.
produce entropy changes one order of magnitude lower than shocks in perfect gases
having the same pressure jump (see Section 2).
It is then reasonable, to a first approximation, to neglect entropy changes in the flow. Thus,
the point representing the thermodynamic state of a fluid particle moving across the flow, will
approximately move along an isentrope of the p-v plane, that remains fixed once the
thermodynamic state of the free-stream has been chosen. Moreover, the possible
thermodynamic states of a flow field will all lay on the isentrope arch included between the
minimum and maximum pressures attained in the flow. In order for the flow field to exhibit a
region of BZT effects, the locus of the flow thermodynamic states in the p-v diagram has to
cross the inversion zone. The extent of such a region depends on the extent of the flow locus
portion falling within the inversion zone.
The operation points chosen for the present study were picked on four different isentropes of
the p-v plane crossing the inversion zone. More precisely, six/seven operation points were
chosen on each isentrope, characterized by values of the Fundamental Derivative Γ ranging
from about 0 to O(1). The numerical values for the free-stream pressure, density and Γ are
listed in Table 2. Fig. 6 shows the location of such points in the p-v plane. BZT effects are
expected to be more significant for flows having higher free-stream entropy and, at given
entropy, lower free-stream values of Γ. In particular, the range of the considered operation
points includes: points close to the transition line; points characterized by Γ=O(1), but falling
within or close to the Γ=1 curve; and points laying far from both the transition line and the
Γ=1 curve. Isentrope S4, in particular, has been taken approximately tangent to the transition
line: thus, for flows evolving along S4, dense gas effects are expected to be related primarily
to the presence of flow regions with Γ<1, more than to BZT effects.
Figure 5: Location of the operation points in the p-v diagram
9
P. Cinnella, P.M. Congedo and D. Laforgia.
Isentrope S1
Case
S11
1.00
p∞/pc
0.752
ρ∞/ρc
0.416
Γ∞
CL
0.244
CD
4.23e-4
CL/CD
577
5.79
Λ∞
Mc
0.985
Isentrope S2
Case
S21
0.992
p∞/pc
0.694
ρ∞/ρc
0.118
Γ∞
CL
0.234
CD
3.77e-4
CL/CD
621
4.10
Λ∞
Mc
0.998
Isentrope S3
Case
S31
0.986
p∞/pc
0.658
ρ∞/ρc
0.0165
Γ∞
CL
0.232
CD
3.67e-4
CL/CD
632
2.84
Λ∞
Mc
∼1
Isentrope S4
Case
S41
0.960
p∞/pc
0.571
ρ∞/ρc
-0.0170
Γ∞
CL
0.236
CD
3.81e-4
CL/CD
619
--Λ∞
Mc
---
S12
1.02
0.813
0.886
0.276
4.47e-4
617
5.58
0.930
S13
1.03
0.877
1.33
0.393
3.22e-3
122
3.86
0.773
S14
1.05
0.944
1.62
0.513
6.81e-2
7.53
2.02
0.350
S15
1.07
1.00
1.74
0.344
8.28e-2
4.15
1.91
0.207
S16
1.10
1.09
1.91
0.161
9.36e-2
1.72
-----
S22
1.01
0.752
0.460
0.248
4.44e-4
559
5.55
0.981
S23
1.03
0.812
0.910
0.282
4.65e-4
606
5.07
0.918
S24
1.05
0.877
1.31
0.418
7.70e-3
54.3
3.58
0.760
S25
1.07
0.950
1.61
0.487
6.98e-2
6.98
1.78
0.272
S26
1.09
1.01
1.73
0.317
8.34e-2
3.80
-----
S32
1.01
0.714
0.258
0.240
4.05e-4
593
4.55
0.993
S33
1.02
0.773
0.635
0.259
4.86e-4
533
5.31
0.962
S34
1.04
0.835
1.06
0.312
8.79e-4
355
4.84
0.884
S35
1.05
0.877
1.31
0.411
1.61e-2
25.5
3.78
0.775
S36
1.07
0.920
1.49
0.504
5.46e-2
9.23
2.65
0.581
S37
1.08
0.969
1.63
0.442
7.48e-2
5.91
-----
S42
0.985
0.622
0.0199
0.235
3.69e-4
637
0.450
∼1
S43
1.01
0.676
0.168
0.239
3.99e-4
599
1.85
0.992
S44
1.03
0.733
0.445
0.250
4.65e-4
538
3.56
0.972
S45
1.05
0.794
0.815
0.281
4.55e-4
618
4.82
0.931
S46
1.07
0.850
1.15
0.353
5.65e-3
62.5
5.13
0.870
S47
1.08
0.882
1.31
0.432
3.01e-2
14.4
4.38
0.804
Table 2: Operation points and corresponding aerodynamic performance.
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P. Cinnella, P.M. Congedo and D. Laforgia.
5.3 Dense gas results
Aerodynamic performances (lift, drag, lift-to-drag ratio) for dense gas PP10 are also presented
in Tab. 2. Based on such results, three typical flow regimes can be identified.
A. Subcritical flows
For flows characterized by relatively low free-stream pressures and small values of the freestream Fundamental Derivative (Γ∞ less than about 1), an extremely high computed CL/CD
(O(102)) is found. Such high ratios are due to the low drag coefficient, although the lift
coefficients takes lower values than in the perfect gas case.
Such flows are entirely subsonic: moreover, the free-stream being uniform and steady and
viscous effects having been neglected, they are also isentropic. In such conditions, the drag
coefficient should be exactly equal to zero. In practice, small entropy gradients are generated
close to the wall, because of numerical errors introduced by the solution scheme and boundary
conditions, that lead to small nonzero values, O(10-4), for the computed drag.
An estimate for the critical Mach number, Mc, in BZT transonic flows has been provided in
[4], using the transonic small disturbance theory:
Γ ∞2
M c 1−
,
(11)
2Λ ∞
ρ ∞2 ∂Γ
where Λ ∞ =
( ρ∞ , s∞ ) is the second nonlinearity parameter [2], representing the rate of
a∞ ∂ρ
change of Γ along an isentrope. The predicted critical Mach numbers for the considered cases
are presented in Tab. 2. The corresponding values of Λ ∞ , evaluated numerically using first
order finite differences, are also reported. There is an excellent agreement between such
estimates and the present numerical results.
Fig. 6: Pressure contours for subcritical case S31, ∆CP=0.05.
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P. Cinnella, P.M. Congedo and D. Laforgia.
A typical pressure contour plot for subcritical flow cases is presented in Fig. 6, and typical
distributions of Mach, pressure coefficient, Γ, and sound speed at the wall are presented in
Figs 7-10. The flow begins at the free-stream conditions and undergoes a compression as the
airfoil leading edge is approached. This involves a rise in the local Γ value, that takes a
maximum at the stagnation point, where Γmax≈1.5÷2. Then, Γ suddenly drops when the flow
begins to expand over the top of the airfoil. If Γ∞ is sufficiently small, the local Fundamental
Derivative becomes smaller than 1 or even negative within less than 1% of the chord from the
leading edge: consequently, the flow speed of sound quickly grows, thus preventing the
appearence of supersonic flow.
Fig. 7: Case S31, wall Mach number.
Fig. 8: Case S31, wall pressure.
Fig. 9: Case S31, Fundamental Derivative at the wall
Fig. 10: Case S31. Speed of sound at the wall
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P. Cinnella, P.M. Congedo and D. Laforgia.
The smaller Γ∞ is, the lower the wall Fundamental Derivative, and the quicker the increase in
the sound speed: this also implies that, flows with lower Γ∞ are also less compressible, and
characterized by higher values of the pressure coefficient. In summary, the previous reasoning
allows to state that, for “sufficiently low” Γ∞:
1. the flow past the airfoil is subcrical;
2. the drag coefficient vanishes;
3. lift is lower than in the perfect gas case, because of reduced flow compressibility;
4. lift tends to increase with Γ∞.
The last remark is in agreement with results of the small transonic disturbance theory [4]. In
fact, writing the classical transonic similarity parameter in terms of Γ [17]:
1 − M ∞2
,
κ=
2/3
( Γ ∞ε )
with ε the airfoil thickness, and using the Prandtl-Glauert similarity:
CPinc
CP = 1/ 2 ,
εκ
inc
where CP is the pressure coefficient for the same airfoil, albeit in incompressible flow, we
deduce that:
CPinc Γ1/∞ 3
CP =
,
ε 2 / 3 1 − M ∞2
that is, the pressure coefficient approximately grows as Γ1/∞ 3 , and a similar behavior can be
expected for the lift. Pressure coefficient distributions for different values of Γ∞ (at fixed
entropy) are shown in Fig.11.
Fig. 11: Wall pressure distribution for different Γ∞, subcritical cases.
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P. Cinnella, P.M. Congedo and D. Laforgia.
B. Low-pressure transonic BZT flow
A different flow regime is found when Γ∞ is approximately in the range 1÷1.3. In this range, a
significant growth in both lift and drag is observed with respect to the previous case.
Nevertheless, the lift-to-drag ratio is about one order of magnitude greater than in the perfect
gas case due, on the one hand, to the higher values obtained for the lift and, on the other, to
the lower drag coefficient. Such flows are found to be supercritical, in agreement with the
estimate (11) (see Tab. 3), except for case S47, for which the critical Mach number has been
slightly over-estimated. Significant BZT effects are present, which are responsible for the
high aerodynamic performance. A typical view of the pressure contours for this flow regime
is presented in Fig. 12, and Figs 13-16 show the wall distributions of the Mach number,
pressure coefficient, Fundamental Derivative, and sound speed.
For this kind of flows, characterized by a higher free-stream Fundamental Derivative, the
local value of Γ drops more slowly, and the reversed speed of sound behavior, associated to
flow regions with Γ<1, is delayed. Thus, when the flow expands past the leading edge, the
speed of sound is allowed to reach sufficiently low values for the flow to become supersonic.
Finally, Γ becomes negative, downstream of the leading edge, where the pressure is still
quickly falling, and an expansion shock is generated. Downstream of the shock, the pressure
drops to values much lower than in the perfect gas case. The expansion shock is followed first
by a continuous expansion and, then, by a gradual compression, that terminates in a
compression shock as soon as the flow exits the inversion zone. At the pressure side, only a
weak compression shock forms. Both the expansion and the compression shocks have jump
conditions in the vicinity of the transition line: the entropy jump across such shocks
(normalized with the free-stream entropy) is found to be O(10-5), against a jump O(10-2) found
for the perfect gas flow.
In summary, flows in the transonic BZT regime are supercritical and characterized by high lift
and low wave drag. The mechanism responsible for the high aerodynamic performance is
qualitatively the same already observed in [9], i.e. the formation of an expansion shock close
to the leading edge, that strongly enhances the suction peak at the airfoil upper surface.
Fig. 12: Pressure contours for case S31, ∆CP=0.1.
14
P. Cinnella, P.M. Congedo and D. Laforgia.
Fig. 13: Case S24, wall Mach number.
Fig. 14: Case S24, wall pressure.
Fig. 15: Case S24 Fundamental Derivative at the wall
Fig. 16: Case S24. Speed of sound at the wall
C. High-pressure transonic BZT flow.
When the free-stream pressure and Γ∞ are even higher, the flow becomes qualitalively similar
to that of a perfect gas. The flow accelerates from the stagnation point to supersonic velocities
and then recompresses, at the rear part of the airfoil, by means of compression shocks. As the
free-stream Fundamental Derivative is increased, the region of flow characterized by Γ<0
becomes smaller and finally disappears. At the same time, the lift coefficient decreases, and
the drag increases, due to the stronger entropy gradients generated across the shocks. This
progressively reduces the airfoil aerodynamic performance, that becomes finally very poor.
Figure 17 shows typical pressure contours for such flows. Figs 18-21 present the wall Mach
number, pressure coefficient, Fundamental Derivative and sound speed.
15
P. Cinnella, P.M. Congedo and D. Laforgia.
Fig. 17: Pressure contours for case S35, ∆CP=0.1.
Fig. 18: Case S35, wall Mach number.
Fig. 19: Case S35, wall pressure.
16
P. Cinnella, P.M. Congedo and D. Laforgia.
Fig. 21: Case S35. Speed of sound at the wall
Fig. 20: Case S35 Fundamental Derivative at the wall
FINAL REMARKS
In the present work, the influence of the free-stream thermodynamic conditions on the
aerodynamic performance of a NACA0012 airfoil in transonic regime has been investigated
using a realistic 5th power virial equation of state. Results obtained for lift, drag, and lift-todrag ratio for different choices of the free-stream Fundamental Derivative and free-stream
entropy are summarized in Figs 22-24. The three flow regimes described in the previous
section can be clearly identified from the sudden changes in slope of the plots. While the drag
monotonically increases with Γ∞, the lift coefficient presents an optimum value at the
beginning of the third regime, referred-to as high pressure transonic in the present study, and
then drops dramatically. The lift-to-drag ratio is very poor for high Γ∞ flows, but tends to
infinity as the free-stream value of the Fundamental Derivative approaches unity. The best
compromise solution between high lift and low drag is obtained in the low pressure transonic
regime, for Γ∞ approximately in the range 1÷1.3: in such conditions, the flow presents higher
lift values and significantly reduced wave drag compared to perfect gas results. The main
mechanisms that govern the three regimes, pointed out in the present work, are qualitatively
simular to those decribed in [9], where the aerodynamic performance of transonic BZT flows
past a NACA0012 was investigated using the very simple van der Waals equation of state.
In general, the computed aerodynamic performances for BZT flows have been found to be
significantly higher with respect to a perfect gas flow past the same configuration. Moreover,
it is noteworthy that very good results are obtained for operation points relatively far from the
inversion zone, that is, for flows with Γ∞ of the order of unity. Such results are of noticeable
importance for practical applications: in particular, it seems possible to overcome one of the
major difficulties for the development of BZT Organic Rankine Cycles: the necessity of
operating the turbine in the very vicinity of the inversion zone, that has a very limited extent.
Present results indicate that in practice, significant performance enhancement can be achieved
17
P. Cinnella, P.M. Congedo and D. Laforgia.
operating the system within the Γ<1 region, whose extent is about ten times greater.
The present numerical model does not take into account viscous effects. However, it is
expected that the airfoil viscous performance should also be improved, as adverse pressure
gradients due to compression shocks are greatly reduced. Inclusion of viscous effects in the
model is planned as future work.
Fig. 22: Lift coefficient versus Γ∞
Fig. 23: Drag coefficient versus Γ∞
Fig. 24: Lift-to-drag versus Γ∞
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