![Reference delta wing (Adapted from [4])](https://www.researchgate.net/profile/Miguel-Angel-Aguirre/publication/359857477/figure/fig2/AS:1143272013606912@1649588697132/Reference-delta-wing-Adapted-from-4_Q320.jpg)
Content uploaded by Miguel Angel Aguirre
Author content
All content in this area was uploaded by Miguel Angel Aguirre on Apr 10, 2022
Content may be subject to copyright.
1
56th 3AF International Conference FP46-AERO2022-Aguirre2
on Applied Aerodynamics
28 – 30 March 2022, Toulouse - France
EXERGY ANALYSIS OF A DELTA WING: A CHALLENGING TEST
CASE
Miguel Ángel Aguirre(1) and Sébastien Duplaa(2)
ISAE SUPAERO-Université de Toulouse, 10 avenue Edouard Belin, 31055, Toulouse, France
(1) Miguel-angel.AGUIRRE@isae-supaero.fr
(2) Sebastien.DUPLAA@isae-supaero.fr
Pablo Alfredo Caron(3)
Instituto de Tecnología, Universidad Argentina de la Empresa – UADE, Lima 717, Buenos Aires, Argentina
(3) pcaron@uade.edu.ar
ABSTRACT
The aerodynamic analysis by the exergy method is a
powerful approach which allows establishing the
theoretical limits of the aerodynamic performance of a
given body, based on an aero-thermodynamic approach.
The most used formulation is the Arntz’s method [1],
which is well suited for the analysis of CFD data. This
formulation was widely applied and validated for plenty
of test cases but all of them were high-aspect ratio
configurations. Then, the objective of this work is to
study a test case with low aspect ratio: a delta wing with
74° of sweep angle, which is characterized by extreme
flow features like strong vortices. It was found that the
exergy formulation behaves well even under such
challenging conditions. However, the related extreme
flowfield allows identifying some hard points like the
wake-reduced exergy analysis of the vortex breakdown,
the impact of the solver setting on the final assessment,
the viscous anergy generation, among other topics.
1. NOMENCLATURE
= total anergy, W
α = angle of attack, Degrees
CD = drag coefficient
= transverse kinetic exergy, W
= exergy, W
= effective dissipation, W.m-3
i, j ,k = unit vectors along the x-, y- and z-axes
ρ = air density, kg.m-3
S = surface, m2
= non-isentropic component of u, m.s-1
= isentropic component of u, m.s-1
V = ui, vj, wk, local velocity vector, m.s-1
ξ = axial vorticity, s-1
ω = vorticity, s-1
Subscripts
out = outlet survey plane of infinite size
w = wake region
ref = reference
Acronyms
CFD = Computational Fluid Dynamics
WTT = Wind Tunnel Testing
RANS = Reynolds Averaged Navier-Stokes
2. REVIEW OF EXERGY METHODS
The exergy method is a novel aerodynamic analysis
approach based on the laws of Thermodynamics which
is suited for the aero-propulsive assessment of any
aircraft configuration. The interest behind this method is
that it delivers a deeper drag breakdown analysis than
the far-field method and, at the same time, it establishes
the design limits based on the 2nd law of
Thermodynamics. In particular, it quantifies the room
for improvement for a given design and it allows
identifying the zones were energy can be recovered [2]
and it also provides the clues of how to recover that
wasted energy. There are currently two well-developed
formulations, namely, full survey plane method and
wake-reduced method.
2.1. Full survey plane method
The formulation developed by Arntz [1] combines the
mass conservation equation with the momentum
conservation as well as the 1st and 2nd laws of
Thermodynamics to provide a general exergy equation.
2
For the case of unpowered bodies, it expresses the drag
coefficient “” as the sum of the exergy “” and
anergy “
” of the system:
(1)
This formulation is known as the “Full survey plane
method” since exergy is given by several integral
equations requiring a survey plane of infinite size. For
example, one of its components, the transverse exergy
“Ev” is given by:
(2)
This represents the flux of transverse kinetic energy at
each point of an infinite survey plane placed
downstream of the body. The same applies for the other
exergy components (see details in [1]), and then, the
drag coefficient also requires an infinite survey plane as
shown in Fig. 1. That is why this formulation is only
well suited for the analysis of CFD data.
Figure 1. Drag distribution for an infinite survey plane
placed downstream of the body (α=20°)
2.2. Wake-reduced method
Recently, the original method was further developed to
obtain a wake-reduced version [2]. This new
formulation is suited for the analysis of both, CFD and
WTT data but even more importantly, it provides a
further decomposition of the drag sources which is well
suited for practical aircraft design.
The key element behind this wake-reduction is the
velocity decomposition method [3]. It breaks down the
velocity field into its isentropic and non-isentropic
components as follows:
(3)
Where the isentropic component “” is the baseline
inviscid velocity field whereas the non-isentropic “” is
the part of the velocity field related to the viscous and
wave losses. By implementing this decomposition into
the Arntz equation, the total drag can be broken down
into its isentropic and non-isentropic components:
(4)
The isentropic drag “
” is the vortex drag and the
non-isentropic drag “” is the profile drag [2]. Since
“” requires an infinite surface integral, the two
component also requires it. However, by further
manipulation these components can be reduced to the
wake as shown in Fig. 2 for “”:
Figure 2. Wake-reduced profile drag distribution at a
survey plane behind the body (α=20°)
3. REVIEW OF DELTA WING STUDIES
The exergy formulation was widely deployed for the
study of high-aspect ratio configurations but it was
never applied to low aspect ratio configurations. Hence,
this section summarizes the previous studies carried out
for the most iconic type of this kind of geometries: the
delta wing. This review is classified based on the
aerodynamic analysis method used, as follows.
3.1. Near-field method
The delta wing was widely studied by WTT and CFD.
For this kind of geometry, the predominant source of lift
and drag mostly comes from the surface pressure rather
than the wall shear stress. This explains why using only
the drag values obtained by surface pressure data has
given successful results when it is compared against
balance data [4]. However, very little attention was
devoted to the baseline drag at zero angle of attack,
where the friction forces become predominant.
3.2. Far-field method
Rather surprisingly, there are very few published cases
of delta wings analysed by the far-field method [5]. The
Maskell formulation [6] was mostly used in these cases.
The related wake-based lift prediction matches the
balance data with the same level of agreement than the
cases of high-aspect ratio but the range of angles of
attack studied were very limited. On the other hand, the
reported drag values were reasonably similar to the
balance data if the wake refinement issues are taken into
account. However, in some cases the reported error
exceeded 10% which is quite high.
3
3.3. Lamb vector method
Some experimental and numerical studies were carried
out by implementing this aerodynamic analysis method
[7-10]. The reported lift and drag values were relatively
close to the balance data although there were some
significant discrepancies, especially at high-angle of
attack (in the range between 20° to 25°).
4. TEST CASE: DELTA WING
The test case chosen for this work is a delta wing with
74° of leading edge sweep angle, which was tested in a
subsonic wind tunnel [4]. It is a symmetric flat plate
with beveled sharp edges as shown in Fig. 3 (the very
small edge thickness do not affect the nature of the
feeding sheet separation at the leading edge).
Figure 3. Reference delta wing (Adapted from [4])
5. CFD DATA
A collaborative project was established between ISAE
and UADE to carry out a joint computational study of
the test case. The main goal was to assess the
applicability of the exergy analysis for the study of the
delta wing, without employing excessive computational
effort: no grid adaption or advanced solvers were
considered for this work, instead, just RANS solver and
standard industrial meshing techniques were used. This
approach was chosen because the exergy analysis
should be mainly exploited during the early stages of an
aircraft design (where a large room for improvement is
available). At the preliminary design stage, no
sophisticated CFD computations are usually carried out.
5.1. Geometry
The CAD model used for this study is a half-body of the
actual wind tunnel model shown before in Fig. 3. The
fairing of the model support lying underneath the flat
plate was ignored as well as the support itself. Hence,
the sting interactions are not taken into account in our
numerical study. Moreover, a free-air condition was
considered for the CFD analysis, with a domain of ±20
wing chords (±14m). Note that the wind tunnel walls are
not modelled in the numerical case which ignores the
boundary blockage effects. However, the reference
balance data from [4] has already introduced the
classical wall corrections as well as tare and interference
corrections. Hence, the balance data is consistent with
free-air conditions without the presence of the sting.
5.2. Meshing techniques
Two of the most used fast meshing types were tested:
Hybrid Multizone:
A hybrid mesh technique based on the ICEM Multizone
method was used [2]. It uses a rapid 3D blocking around
the geometry (O-block type) and the wake block
(structured blocks extruded along the wake) inside of
which an anisotropic mesh is obtained (Fig. 4).
Afterwards, this mesh is surrounded by tetrahedral cells
(Delaunay) to fill the remaining domain volume (Fig.
5). The mesh cell count across the wake was adjusted in
order to satisfy the typical rules-of-thumb (at least 50
points across thin wakes or up to 100 points across
vortices [2]). The surface mesh size is such that Y+=1 is
achieved, as required by the turbulence model (Spalart
Allmaras). The final mesh size was 14 million of cells.
Figure 4. Blocking around geometry and wake
Figure 5. Cross-section of the resulting mesh
SnappyHexMesh:
This type of mesh is widely used by OpenFOAM users
and that is why this was included for this study. It
4
generates a baseline structured grid that is progressively
refined within zones defined by simple geometries
(circles, cylinder, cones, etc.) and/or surfaces (i.e. STL
imported surfaces). The refinement level indicates how
many times the initial cell will be partitioned. The
refinement zones used in this work can be seen in Fig.
6. The final mesh size was 4.5 million cells.
5.3. Solvers and settings
Two different solvers were implemented for this work.
Fluent was used in combination with the hybrid mesh
and OpenFOAM with the SnappyHexMesh.
Three solver settings were used:
• Setting A: density-based solver + energy equation
activated + density given by ideal gas + viscosity given
by the Sutherland relation.
• Setting B: density-based solver + energy equation
activated + constant density + viscosity given by the
Sutherland relation.
• Setting C: density-based solver + energy equation
deactivated + constant density + constant viscosity.
Setting A corresponds to the recommended practice for
computing the exergy/anergy parameters with high-
fidelity. Indeed, even though the test case is
incompressible and adiabatic, it is convenient to use
these settings in order to properly compute the related
exergy terms and to obtain accurate values. On the other
hand, the setting C is the most common setup used in
practice for low subsonic cases and the setting B is an
intermediate case. Hereafter, the setting A will be used
unless the opposite is specified.
Figure 6. Detail of the SnappyHexMesh
5.4. Boundary conditions
The reference test case [4] was studied at a Reynolds
number of 1.7x106 but our study will be made at
1.0x106 (Freestream velocity of 20m/s) since the
gathered CFD data will be used for future experimental
work. On the other hand, a static pressure of 101325 Pa
and a static temperature of 288K were imposed at the
far-field boundary. The turbulence level was 0.1%.
5.5. Convergence criteria
The CFD runs were launched with first-order
interpolation scheme (for flow and turbulence) and then
switched to second-order scheme up to its final
convergence. The simulations were stopped when the
near-field drag coefficient varied less than 0.1dc (1dc =
1 drag count = 0.0001 ) between iterations.
5.6. Postprocessing tool
The posttreatment of the CFD data was carried out with
Epsilon [11]. It is an open source aerodynamic analysis
tool developed by ISAE-Supaero, suited for CFD and
experimental data. It is mainly devoted to the exergy
analysis but it also includes the classical methods as
well (near field, far-field and Lamb vector).
Epsilon is a set of plugins for Paraview created to
provide an easy to use tool for aero-propulsive
assessment of aircraft. It can be freely downloaded from
the following internet site (GNU GPL V3 license):
https://Epsilon-Exergy.isae-supaero.fr
5.7. Data validation
The near-field lift and drag values were compared
against the reference balance data for the two solvers as
shown in Fig.7 and Fig.8.
Figure 7. Validation of lift data
It is observed that the lift and drag values presents a
good agreement with the experimental data for α<20°.
For higher angles of attack, there is an increasing
5
discrepancy due to the appearance of the vortex
breakdown phenomenon, which was not expected to be
properly captured by the current CFD solution (in fact, a
more complex CFD setup is required to capture the
related physics [12]).
Figure 8. Validation of drag data
6. FULL-PLANE EXERGY ANALYSIS
A full survey plane exergy analysis was carried out for
the entire angle of attack range as shown in Fig. 9
(unless specified, the survey plane used hereafter for the
exergy analysis is placed at the wing trailing edge).
Figure 9. Total drag: Exergy vs near-field method
It can be shown that the total drag obtained by the
exergy method matches the reference near-field drag
value. This confirms the fact that the exergy method is
also well suited for studying low aspect ratio
geometries.
It must be kept in mind that the CFD solutions are only
physically accurate for α<20°, hence, the comparison of
exergy-based drag vs near-field drag beyond α=20° is
just to assess the fact that both drag values are identical
even in this vortex breakdown flow regime.
Figure 10 shows the decomposition of the total drag into
its exergy and anergy components. This highlights the
fact that, for this low aspect ratio configuration, the
related amount of exergy involved is higher than the
case of high-aspect ratio configurations [2]. Indeed, the
exergy is an amount of energy that can be theoretically
recovered. If an ideal energy recovery system is
implemented in such a way that all this energy is
capitalized, then, the drag of a body can be reduced and
the lower theoretical limit of this drag reduction is given
by the anergy. Under this condition, the aerodynamic
efficiency of a body will increase and reach its
maximum theoretical value, given by “
” [2] as
shown in Fig. 11.
Figure 10. Exergy-based breakdown of the total drag
Figure 11. Maximum theoretical efficiency
6
This highlight the fact that the current low aerodynamic
efficiency peak of the delta wing (
≈7) can be
further increased and reach a reasonable efficiency
compared to high aspect ratio wings. For example, a
rectangular wing of aspect ratio 8 has a
≈20 and
a
≈60 [2].
7. WAKE-REDUCED EXERGY ANALYSIS
Having verified that the Arntz’s full survey plane
formulation performs well for the entire range of angles
of attack analyzed, now we can study the wake-reduced
exergy formulation. Firstly, the total drag “” is
decomposed into its isentropic “
” and non-isentropic
“” components (full survey plane formulation [2]) as
shown in Fig. 12. These components are basically the
vortex drag and profile drag respectively from an exergy
point of view.
Figure 12. Total drag breakdown into its isentropic and
non-isentropic parts (Fluent data only)
Now, the objective is to compute these parameters from
wake data only by using the wake-reduced formulation
[2], which is shown as “
” and “”. For
the case of the profile drag, a good agreement is
observed between the full-plane and wake-reduced
values for the entire angle of attack range, even under
vortex breakdown condition.
For the vortex drag, the wake reduced version performs
well for α<25°. For higher angles of attack, it
underestimates the drag value of the full-plane version
(taken here as reference since no assumptions are made
on its formulation). Hence, when vortex breakdown
arises, there is a discrepancy between both approaches
just for the vortex drag.
It is interesting to note that the wake-reduced curve
follows a law similar to “
”. Such a
behavior was not expected according to the theory since
the wake reduced expression is supposed to be an exact
version of the full plane equation. However, this
theoretical development was carried out for stable
vortical flows (where it performs well). Then, it is clear
that the wake-reduced theory must be improved to take
into account the cases of vortex flow with recirculation
inside the core. This is the case of the “bubble type”
vortex breakdown, shown in Fig. 13.
Figure 13. Vortex breakdown for α=30°
The full-survey plane decomposition into the isentropic
and non-isentropic drag [2] allows determining the axial
distribution of the profile and vortex drag along the
body and the wake as shown in Fig. 14 (This can be
achieved independently of the presence or not of vortex
breakdown). It must be noticed that, downstream of the
body, the vortex drag decreases (due to viscous
dissipation at the core) and then the profile drag
increases accordingly in such a way that the total drag
remains constant along the wake. The interesting thing
about this test case is that the distinction between profile
drag and vortex drag is kept even if the entire wake is
fully rolled-up. Indeed, as shown before in Fig. 13. The
entire wake rolls up very rapidly: in less than half-chord
downstream the viscous wake coming from the trailing
edge has been rolled around the vortex. For high aspect
ratio wings this roll-up process is slower, which allows
identifying the part of the wake related to the “pure”
viscous drag from the “pure” vortex drag. This is not
possible in a low aspect ratio wing, but the formulation
is robust enough to keep the profile/vortex drag
information despite the rolling-up process.
Figure 14. Longitudinal evolution of the drag
components for α=20°
Vortex drag
Total drag
Profile drag
7
8. VISCOUS ANERGY GENERATION
The viscous anergy generation is given by:
(5)
Where the dissipation function is related to the velocity
gradients and the stress tensor “”:
(6)
However, according to Koh [13], the effective
dissipation function can also be expressed as a volume
integral of the vorticity as follows:
(7)
Hence, there are two different ways to compute the
viscous anergy: one is by using the classical dissipation
function (which is based on local velocity gradients) and
the other one is by using the local vorticity. These
methods will be referred hereafter as “gradient” and
“vorticity”.
According to Koh, the Eq. 7 holds only for the total
integrated value but not for a point by point basis. So
the two questions here are: a) which of both approaches
is the most pertinent from a physical point of view, and
b) are the resulting anergies identical? Both questions
can be easily answered by using the delta wing test case
since the vorticity and velocity gradients are very large
(compared to a low aspect ratio wing), which facilitates
the analysis and the comparison of both approaches.
The first question can be answered if it is reminded that
the dissipation function gives, at a point, the amount of
kinetic energy that is being converted into heat through
dissipation mechanisms. This can be easily verified as
shown in Fig. 15-right (Fluent data), where the gradient-
based approach is strictly zero at the vortex cores.
Figure 15. Comparison of viscous anergies (α=20°)
Indeed, at the vortex core there is a “rigid body
rotation” of the flow field, then, no velocity gradients
are involved and thus, no local dissipation. On the
contrary, the vorticity-based approach (Fig. 15-left) is
mostly concentrated in the two vortices (main and
secondary vortices) as well as the feeding sheet. Hence,
in a point-by-point basis, both approaches are not
comparable at all and just the gradient approach is the
most pertinent from a physical point of view.
The vorticity approach can be considered as the wake-
reduced version of the gradient approach. Indeed, as any
wake-reduced method, it can be expressed as a function
of the vorticity because the overall flow physics is
driven by vorticity [2]. One can even make a parallelism
between these two approaches and the vortex drag: for
the vortex drag two approaches are available: the full-
plane and the wake-reduced which are not comparable
in a point-by-point basis, but both provides the same
drag value from different points of view. After all, the
vorticity drives the rotational flow field around the
vortex but also it is linked to the velocity gradients [2]
which are accounted for by the gradient approach.
The second question can be answered in Fig. 16, where
the anergy computed by both approaches is shown. The
results are identical provided that the vortex breakdown
do not occurs in front of the survey plane (placed at the
trailing edge). Once the breakdown occurs over the
wing, both methods are no longer consistent. Also note
that, for α<30°, both volume integral approaches
underpredicts the true anergy value computed by
surface approach [1]. However, for α=0° the surface and
volume methods seems to be consistent. Then, the
discrepancy is lift-dependent and requires further work
to identify the origin of this underprediction.
Figure 16. Comparison of viscous anergy generation
9. LIFT PREDICTION
Lift can be computed from wake data based on the
Maskell method [6] as follows:
(8)
8
The existing reference studies were not too conclusive
about the applicability of the wake-reduced lift equation
of Maskell since the range of angles of attack studied
and the related test cases were quite limited. Hence, this
test case will be used to verify its applicability in cases
with extreme flow features, like the delta wing. As a
matter of fact, our objective is to find all the physical
information from wake data only, and this concerns
drag as well as lift.
The curves of the lift coefficient vs angle of attack for
two methods (near-field and Maskell) are shown in Fig.
17. It can be observed that a good agreement is
observed, even at high angles of attack, where vortex
breakdown is present. This confirms the robustness of
the Maskell method for the prediction of lift from wake
data.
Figure 17. Comparison of lift curves (Fluent data)
Now we can tackle one the problematic physical
features associated to its formulation: the streamline
curvature. Indeed, the lift is computed based on the
axial vorticity at the survey plane. As shown in Fig. 18,
the vortex core is well inclined near the trailing edge but
almost horizontal just downstream. Hence, the local
axial vorticity will change even if the vortex strength is
constant along the vortex core (we ignore here for a
moment the dissipation at the vortex core).
Figure 18. vortex curvature near trailing edge (α=20°)
This problem was first identified in [2] and in this work
we will analyze it in detail since the delta wing is a test
case that, unlike high-aspect ratio wings, it presents a
strong streamline curvature.
Fig. 19 shows the lift distribution along the chord and
the wake. It is observed that lift reaches a constant value
behind the trailing edge, which confirms the fact that the
Maskell formulation is robust enough to deal with the
streamline curvature feature.
Figure 19. Longitudinal evolution of the lift coefficient
for α=20° (Fluent data)
10. IMPACT OF SOLVER SETTINGS
Another interest behind this test case is that the related
flow perturbations are very high, which facilitates the
identifications of some numerical issues. Indeed, there
is a certain influence of the solver settings on the result
of the exergy analysis and this do not comes from the
exergy formulation itself, but from the lack of physical
fidelity of the input data. In order to highlight this fact,
several solver settings were used (A, B and C), where
the setting A is the recommended for high fidelity
whereas the setting C is widely used for incompressible
and adiabatic flows (Setting B is somewhere in between
both cases). The exergy and anergy values for these
three cases are shown in Table 1, where the total exergy
“ ” was decomposed into its mechanical “ ” and
thermal components “ ” [1]:
Setting
A
0,0351
0,2542
0,0006
B
0,0995
0,2541
0,3218
C
4043,09
0,2543
1,1297
B corrected
0,0344
0,2541
0,0004
C corrected
0,0341
0,2543
0,0004
Table 1. Exergy and anergy values for several solver
settings (α=20°/Fluent data)
Setting A is taken here as the reference since it is the
setup previously used for the validation of the exergy
Lift
9
formulation and thus, it gives accurate values. It is
observed that the settings B and C provide the same
mechanical exergy but the thermal exergy and the
anergy are very different.
This problem was found to be related to the density and
temperature field from the input data. For example, by
assuming a constant density field, the related density
perturbations will no longer exist. Hence, the static
temperature field “” will be driven by the static
pressure field. This means that the “” field will follow
the “
” field. This is shown in Fig. 20, where the “”
field for Setting C is radically different from the
reference Setting A. Setting B also considers constant
density, but it provides the enthalpy as an input, then it
gives an intermediate “” field.
Figure 20. Influence of solver settings and correction
on the static temperature (α=20°/Fluent data)
The consequence of this erroneous input data is that the
related anergy field will be also erroneous as shown in
Fig. 21. Hence, a method is required to avoid this error.
Since the problem comes from the density field, which
was assumed to be constant, then, the solution is to
recompute the density field before computing the
exergy parameters. The simplest approach is to assume
an isoenergetic flow, which will provide the true static
temperature as follows:
(9)
Then, by using the input static pressure and the equation
of state, the density can now be determined. Hence, with
this new “” and “ρ” fields (designed hereafter as
“corrected”), the exergy quantities can be recomputed.
The result of such “correction” is shown in Table 1,
where the “Setting B corrected” and “Setting C
corrected” are now identical to the reference Setting A.
This can also be observed in Fig. 20 and Fig. 21, where
the corrected fields are identical to the reference Setting
A. This confirms that the input data correction
procedure was successful. However, such a
methodology is only valid for isoenergetic cases
(because the energy conservation equation was used),
i.e., this will not be valid inside jet plumes.
Figure 21. Influence of solver settings and correction
on the total anergy (α=20°/ Fluent data)
All the OpenFOAM characteristic curves shown in this
paper have used the correction method presented in this
section. However, all the Fluent curves have used the
solver setting A, thus, no correction was required.
11. CONCLUSIONS
Recent developments of the exergy method [2] have
shown that the vortex drag is the main source of
recoverable energy in the airflow of a lifting body.
Hence, in order to better understand the nature behind
this energy recovery possibility, an exergy analysis was
SETTING B - corrected
SETTING A
SETTING B
SETTING C - corrected
SETTING C
SETTING B - corrected
SETTING A
SETTING B
SETTING C - corrected
SETTING C
10
carried out in a delta wing since vortex drag is
predominant for this case.
Near-field CFD data has shown a good agreement with
balance data except where vortex breakdown exists.
Moreover, the Arntz exergy method has proven again its
robustness since its related drag value matches the near-
field reference value for any angle of attack. On the
other hand, the wake-reduced formulation performed
well provided no vortex breakdown exists at the survey
plane. Hence, the wake-reduced formulation can be
implemented for performing exergy-based aerodynamic
assessment from wind tunnel data even for low aspect
ratio geometries (e.g., fighter aircrafts or missiles).
12. FUTURE WORK
The discrepancies found in the wake-reduced vortex
drag prediction method when the survey plane is placed
inside a vortex breakdown region will be deeply studied
in order to solve the related limitation. Moreover, the
increasing underprediction of the volume-based viscous
anergy (respect to the surface-base anergy) with the
increase in lift will also be investigated. Finally, the
development of a correction of the input data for non-
isoenergetic cases will be considered.
13. REFERENCES
1. Arntz, A. (2014). Civil Aircraft Aero-thermo-
propulsive Performance Assessment by an Exergy
Analysis of High-fidelity CFD-RANS Flow
Solutions, Fluids mechanics, Université de Lille 1.
2. Aguirre, M.A. (2022). Exergy analysis of innovative
aircraft with aero-propulsive coupling, PhD thesis,
ISAE-Supaero, Toulouse University, France.
3. Aguirre, M.A., Duplaa, S., Carbonneau, X. and
Turnbull, A. (2020) “Velocity Decomposition
Method for Exergy-Based Drag Prediction,” AIAA
Journal, vol. 58, no. 11, pp. 4686–4701, doi:
10.2514/1.J059414.
4. Wentz, W. (1972) “Effects of leading-edge camber
on low-speed characteristics of slender delta
wings”, NASA CR 2002.
5. Dogar, M. (2012) “An Experimental Investigation of
Aerodynamics and Flow Characteristics of Slender
and Non-Slender Delta Wings”, MSc Thesis,
McGill University, Canada.
6. Maskell, E. (1973). Progress Towards a Method of
Measurement of the Components of the Drag of a
Wing of Finite Span, Royal Aircraft Establishment
TR 72232.
7. Ondrusek, B. (1996) “Force Diagnostics of
Aerodynamic Bodies Based on Wake-Plane Data,”
Master Thesis, University of Tennessee.
8. Wu, J., Ondrusek, B. and Wu, J (1996) “Exact force
diagnostics of vehicles based on wake-plane data,”
in 34th Aerospace Sciences Meeting and Exhibit,
American Institute of Aeronautics and
Astronautics.
9. Yang, Y., Zhang, R., An, Y. and Wu, J.Z. (2007)
“Steady vortex force theory and slender-wing flow
diagnosis,” Acta Mech. Sin., vol. 23, no. 6, pp.
609–619.
10. Zou, S.-F., Wu, J. Z., Gao, A.-K., Liu, L., Kang,L.
and Shi,Y (2019) “On the concept and theory of
induced drag for viscous and incompressible steady
flow,” Physics of Fluids, vol. 31, no. 6, p. 065106.
11. Aguirre, M.A., Duplaa, S. (2020). Epsilon: An Open
Source Tool for Exergy-Based Aerodynamic
analysis, Aerospace Europe conference 2020,
Bordeaux, France.
12. NATO/RTO/AVT Task Group AVT-080 (2009),
“Vortex Breakdown over Slender Wings”. ISBN
978-92-837-0078-4
13. Koh, Y.-M. (1994) “Vorticity and viscous
dissipation in an incompressible flow,” KSME
Journal, vol. 8, no. 1, pp. 35–42, doi:
10.1007/BF02953241.
