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A Multi-Objective Factorial Design Methodology for Aerodynamic Off-Takes and Ducts

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Fluid off-takes and complex delivery ducts are common in many engineering systems but designing them can be a challenging task. At the conceptual design phase many system parameters are open to consideration and preliminary design studies are necessary to instruct the conceptualisation process in an iterative development of design ideas. This paper presents a simple methodology to parametrically design, explore and optimise such systems at low cost. The method is then applied to an aerodynamic system including an off-take followed by a complex delivery duct. A selection of nine input variables is explored via a fractional factorial design approach that consists of three individual seven-level cubic factorial designs. Numerical predictions are characterised based on multiple aerodynamic objectives. A scaled representation of these objectives allows for a scalarisation technique to be employed in the form of a global criterion which indicates a set of trade-off geometries. This leads to the selection of a set of nominal designs and the determination of their robustness which will eventually instruct the next conceptual design iteration. The results are presented and discussed based on criterion space, design variable space and contours of several flow quantities on a selection of optimal geometries.
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Citation: Spanelis, A.; Walker, A.D.
A Multi-Objective Factorial Design
Methodology for Aerodynamic
Off-Takes and Ducts. Aerospace 2022,
9, 130. https://doi.org/10.3390/
aerospace9030130
Academic Editor: Fernando Lau
Received: 8 December 2021
Accepted: 27 February 2022
Published: 2 March 2022
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
published maps and institutional affil-
iations.
Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
aerospace
Article
A Multi-Objective Factorial Design Methodology for
Aerodynamic Off-Takes and Ducts
Apostolos Spanelis * and Alastair Duncan Walker
Department of Aeronautical and Automotive Engineering, Loughborough University,
Loughborough LE11 3TU, UK; a.d.walker@lboro.ac.uk
*Correspondence: a.spanelis@lboro.ac.uk; Tel.: +44-(0)150-922-7156
Abstract:
Fluid off-takes and complex delivery ducts are common in many engineering systems but
designing them can be a challenging task. At the conceptual design phase many system parameters are
open to consideration and preliminary design studies are necessary to instruct the conceptualisation
process in an iterative development of design ideas. This paper presents a simple methodology to
parametrically design, explore and optimise such systems at low cost. The method is then applied
to an aerodynamic system including an off-take followed by a complex delivery duct. A selection
of nine input variables is explored via a fractional factorial design approach that consists of three
individual seven-level cubic factorial designs. Numerical predictions are characterised based on
multiple aerodynamic objectives. A scaled representation of these objectives allows for a scalarisation
technique to be employed in the form of a global criterion which indicates a set of trade-off geometries.
This leads to the selection of a set of nominal designs and the determination of their robustness which
will eventually instruct the next conceptual design iteration. The results are presented and discussed
based on criterion space, design variable space and contours of several flow quantities on a selection
of optimal geometries.
Keywords: aerodynamic; off-take; duct; conceptual; preliminary; multi-objective; factorial; design
1. Introduction
Fluid off-takes coupled to complex duct systems are common in many engineering
applications. These include, for example, engine air intakes for both aero-engines and
ground vehicles, secondary air bleeds to provide cooling flows, tidal turbine intakes,
distributary water channels etc. A generic description of a fluid off-take is the diversion
of a portion of the mainstream flow, of the given system, via a discrete opening on a solid
boundary as illustrated in Figure 1.
Aerospace 2022, 9, x. https://doi.org/10.3390/xxxxx www.mdpi.com/journal/aerospace
Article
A Multi-Objective Factorial Design Methodology for
Aerodynamic Off-Takes and Ducts
Apostolos Spanelis * and Alastair Duncan Walker
Department of Aeronautical and Automotive Engineering, Loughborough University,
Loughborough LE129JD, UK; a.d.walker@lboro.ac.uk
* Correspondence: a.spanelis@lboro.ac.uk; Tel.: +44-(0)1509-227156
Abstract: Fluid off-takes and complex delivery ducts are common in many engineering systems but
designing them can be a challenging task. At the conceptual design phase many system parameters
are open to consideration and preliminary design studies are necessary to instruct the conceptuali-
sation process in an iterative development of design ideas. This paper presents a simple methodol-
ogy to parametrically design, explore and optimise such systems at low cost. The method is then
applied to an aerodynamic system including an off-take followed by a complex delivery duct. A
selection of nine input variables is explored via a fractional factorial design approach that consists
of three individual seven-level cubic factorial designs. Numerical predictions are characterised
based on multiple aerodynamic objectives. A scaled representation of these objectives allows for a
scalarisation technique to be employed in the form of a global criterion which indicates a set of
trade-off geometries. This leads to the selection of a set of nominal designs and the determination
of their robustness which will eventually instruct the next conceptual design iteration. The results
are presented and discussed based on criterion space, design variable space and contours of several
flow quantities on a selection of optimal geometries.
Keywords: aerodynamic; off-take; duct; conceptual; preliminary; multi-objective; factorial; design
1. Introduction
Fluid off-takes coupled to complex duct systems are common in many engineering
applications. These include, for example, engine air intakes for both aero-engines and
ground vehicles, secondary air bleeds to provide cooling flows, tidal turbine intakes, dis-
tributary water channels etc. A generic description of a fluid off-take is the diversion of a
portion of the mainstream flow, of the given system, via a discrete opening on a solid
boundary as illustrated in Figure 1.
Figure 1. Generic off-take and delivery duct.
Citation: Spanelis, A.; Walker, A.D.
A Multi-Objective Factorial Design
Methodology for Aerodynamic
Off-Takes and Ducts. Aerospace 2022,
9, x. https://doi.org/10.3390/xxxxx
Academic Editor: Fernando Lau
Received: 8 December 2021
Accepted: 27 February 2022
Published: 2 March 2022
Publisher’s Note: MDPI stays neu-
tral with regard to jurisdictional
claims in published maps and insti-
tutional affiliations.
Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and con-
ditions of the Creative Commons At-
tribution (CC BY) license (https://cre-
ativecommons.org/licenses/by/4.0/).
Figure 1. Generic off-take and delivery duct.
Downstream of the off-take, depending on the application, the flow may be suitable
for use directly or it may be necessary to accelerate or decelerate before delivered to the
Aerospace 2022,9, 130. https://doi.org/10.3390/aerospace9030130 https://www.mdpi.com/journal/aerospace
Aerospace 2022,9, 130 2 of 32
target zone. Therefore, the delivery duct can be of constant or variable cross-sectional
area distribution, i.e., a nozzle or a diffuser. The target zone itself represents the region
where the diverted flow is utilised. This zone may simply represent a reference section
for aerodynamic evaluation or more likely the inlet to an engine, some turbomachinery,
a heat exchanger, a manifold, or some other engineering component. The aerodynamic
performance of the off-take and delivery duct will determine the overall performance of
the application they serve. Inappropriate designs may cause flow phenomena such as flow
separation, high levels of aerodynamic blockage, unacceptable total pressure loss and high
levels of non-uniformity, hence careful aerodynamic design is needed.
This generic concept is studied on a specific application relating to a cooled cooling air
(CCA) system on an aero gas turbine [
1
]. Briefly, a CCA system uses part of the bypass duct
airflow to cool, via an array of heat exchangers (HX), a portion of the core engine airflow,
which is then used to cool hot components in the engine core [
1
3
]. The current work
considers the low pressure (LP) side of this system. This comprises an aerodynamic off-take
that captures the “cooling” air from the bypass duct and a delivery duct that transfers it to
the HX.
Fluid off-takes can be sub-divided in two broad categories: total off-takes and flush off-
takes. In general, a total off-take will produce the best pressure recovery and aerodynamic
performance [
2
4
]. This is mainly because a total off-take is operated by the total pressure
of the mainstream flow as opposed to a flush off-take that in principle is operated solely by
the static pressure. However, in the bypass duct of a gas turbine, the preferred option is
flush or submerged off-takes as they can minimise the aerodynamic effect on the bypass
stream and the associated impact in efficiency and SFC. Examples of work employing flush
off-takes in the bypass duct can be found at Walker et al. [
1
,
5
] and Spanelis and Walker [
6
].
A parameter of importance to the current work is the location of the off-take within
the bypass duct. Traditionally the preferred location is on the inner cowl surface which
offers a wide surface area as well as easy access to the engine’s core. The drawback is
that the quality of the ingested flow is relatively low, due to the thick boundary layers
coexisting with the fan outlet guide vane (OGV) wakes. An alternative location would
be the upper and lower bifurcations (the large struts in the bypass duct—Figure 2). The
trend for increased bypass ratio, to enhance propulsive efficiency, leads to larger bypass
ducts and consequently larger bifurcations struts. See, for instance, the work of Clemen
et al. [
7
] who describe the design and optimisation of the bypass duct system on a large
civil turbofan engine. This enlargement creates additional volume inside the struts for
possible placement of air delivery ducts, hence a submerged off-take on a bifurcation strut
becomes an attractive option.
Aerospace 2022, 9, x 2 of 32
Downstream of the off-take, depending on the application, the flow may be suitable
for use directly or it may be necessary to accelerate or decelerate before delivered to the
target zone. Therefore, the delivery duct can be of constant or variable cross-sectional area
distribution, i.e., a nozzle or a diffuser. The target zone itself represents the region where
the diverted flow is utilised. This zone may simply represent a reference section for aero-
dynamic evaluation or more likely the inlet to an engine, some turbomachinery, a heat
exchanger, a manifold, or some other engineering component. The aerodynamic perfor-
mance of the off-take and delivery duct will determine the overall performance of the
application they serve. Inappropriate designs may cause flow phenomena such as flow
separation, high levels of aerodynamic blockage, unacceptable total pressure loss and
high levels of non-uniformity, hence careful aerodynamic design is needed.
This generic concept is studied on a specific application relating to a cooled cooling
air (CCA) system on an aero gas turbine [1]. Briefly, a CCA system uses part of the bypass
duct airflow to cool, via an array of heat exchangers (HX), a portion of the core engine
airflow, which is then used to cool hot components in the engine core [1–3]. The current
work considers the low pressure (LP) side of this system. This comprises an aerodynamic
off-take that captures the “cooling” air from the bypass duct and a delivery duct that
transfers it to the HX.
Fluid off-takes can be sub-divided in two broad categories: total off-takes and flush
off-takes. In general, a total off-take will produce the best pressure recovery and aerody-
namic performance [2–4]. This is mainly because a total off-take is operated by the total
pressure of the mainstream flow as opposed to a flush off-take that in principle is operated
solely by the static pressure. However, in the bypass duct of a gas turbine, the preferred
option is flush or submerged off-takes as they can minimise the aerodynamic effect on the
bypass stream and the associated impact in efficiency and SFC. Examples of work em-
ploying flush off-takes in the bypass duct can be found at Walker et al. [1,5] and Spanelis
and Walker [6].
A parameter of importance to the current work is the location of the off-take within
the bypass duct. Traditionally the preferred location is on the inner cowl surface which
offers a wide surface area as well as easy access to the engine’s core. The drawback is that
the quality of the ingested flow is relatively low, due to the thick boundary layers coexist-
ing with the fan outlet guide vane (OGV) wakes. An alternative location would be the
upper and lower bifurcations (the large struts in the bypass duct—Figure 2). The trend for
increased bypass ratio, to enhance propulsive efficiency, leads to larger bypass ducts and
consequently larger bifurcations struts. See, for instance, the work of Clemen et al. [7] who
describe the design and optimisation of the bypass duct system on a large civil turbofan
engine. This enlargement creates additional volume inside the struts for possible place-
ment of air delivery ducts, hence a submerged off-take on a bifurcation strut becomes an
attractive option.
Figure 2. Bifurcation off-take concept–sketch.
Figure 2. Bifurcation off-take concept–sketch.
Depending on the position of the aerodynamic off-take on the bifurcation strut, the
type of flow ingestion can vary between a total off-take and a flush off-take as shown
in Figure 3.
Aerospace 2022,9, 130 3 of 32
Aerospace 2022, 9, x 3 of 32
Depending on the position of the aerodynamic off-take on the bifurcation strut, the
type of flow ingestion can vary between a total off-take and a flush off-take as shown in
Figure 3.
Figure 3. Submerged off-take types on a round leading-edge surface.
For a bifurcation off-take the flow must be ducted away from the strut to the target
zone and the aerodynamic and mechanical constraints add to the design challenge. In this
study, the target zone for a CCA system is an array of heat exchangers located at discrete
positions around the turbine casing. The velocity in the bypass duct is typically M0.5–0.6
hence, to avoid excessive total pressure losses in the HX (see Shah and Sekulic [8]) the
flow captured by the off-take must be diffused. It is desirable to spread this diffusion
throughout the duct system. Due to spatial constraints, some diffusion can be achieved
via pre-diffusion of the captured stream-tube and some can be achieved in diffuser 1 (see
Figures 2 and 4). However, approximately 90% of the diffusion will need to be achieved
between the strut exit and the HX. In previous work of Spanelis et al. [9] this region was
comprised of two s-ducts separated by a vane. Their results suggested that flow uni-
formity at duct exit is maximised for designs which concentrate the aerodynamic loading
(both diffusion and curvature) close to the HX. They attributed this to the beneficial effects
that the HX blockage exerts on the flow immediately upstream of the HX. Therefore, as
shown in Figure 4, this region is sub-divided in two components: (1) a plane diffuser re-
ferred to as “diffuser 2” which should be moderately loaded, and (2) an s-shaped manifold
connected to the HX which can then be more aggressively loaded by taking advantage of
the above-mentioned local pressure field near the HX. For the CCA system examined
herein the target diffusion equates to an area ratio of 𝐴𝐴
= 16.7 within an overall non-
dimensional length of 𝐿
=28.7. Where “s” denotes the stream-tube captured by
the off-take and “T” is the target zone or, here, the HX inlet plane. Note that a single off-
take feeds two HXs therefore, 𝐴 represents the combined inlet area of both.
Figure 4. Schematic representation of the LP-CCA system.
The presented system is a conceptual design. It can be shown via the elementary
timeline of a design process in Figure 5, that in an industrial environment there is a sig-
nificant overlap between different design stages. The purpose of this overlap is to improve
efficiency of the workforce. See for instance the description of the aircraft design process
as explained by Gudmundsson [10]. More specifically, as the conceptual design stage
phases out, the preliminary investigation is already in progress, which provides the op-
portunity to the conceptualisation team to further develop the concept. A new preliminary
Figure 3. Submerged off-take types on a round leading-edge surface.
For a bifurcation off-take the flow must be ducted away from the strut to the target zone
and the aerodynamic and mechanical constraints add to the design challenge. In this study,
the target zone for a CCA system is an array of heat exchangers located at discrete positions
around the turbine casing. The velocity in the bypass duct is typically
M0.5–0.6 hence
, to
avoid excessive total pressure losses in the HX (see Shah and Sekulic [
8
]) the flow captured
by the off-take must be diffused. It is desirable to spread this diffusion throughout the
duct system. Due to spatial constraints, some diffusion can be achieved via pre-diffusion
of the captured stream-tube and some can be achieved in diffuser 1 (
see Figures 2and 4
).
However, approximately 90% of the diffusion will need to be achieved between the strut
exit and the HX. In previous work of Spanelis et al. [
9
] this region was comprised of two
s-ducts separated by a vane. Their results suggested that flow uniformity at duct exit is
maximised for designs which concentrate the aerodynamic loading (both diffusion and
curvature) close to the HX. They attributed this to the beneficial effects that the HX blockage
exerts on the flow immediately upstream of the HX. Therefore, as shown in Figure 4, this
region is sub-divided in two components: (1) a plane diffuser referred to as “diffuser 2”
which should be moderately loaded, and (2) an s-shaped manifold connected to the HX
which can then be more aggressively loaded by taking advantage of the above-mentioned
local pressure field near the HX. For the CCA system examined herein the target diffusion
equates to an area ratio of
AT/As=
16.7 within an overall non-dimensional length of
LsT/hs=
28.7. Where “s” denotes the stream-tube captured by the off-take and “T” is the
target zone or, here, the HX inlet plane. Note that a single off-take feeds two HXs therefore,
ATrepresents the combined inlet area of both.
Aerospace 2022, 9, x 3 of 32
Depending on the position of the aerodynamic off-take on the bifurcation strut, the
type of flow ingestion can vary between a total off-take and a flush off-take as shown in
Figure 3.
Figure 3. Submerged off-take types on a round leading-edge surface.
For a bifurcation off-take the flow must be ducted away from the strut to the target
zone and the aerodynamic and mechanical constraints add to the design challenge. In this
study, the target zone for a CCA system is an array of heat exchangers located at discrete
positions around the turbine casing. The velocity in the bypass duct is typically M0.5–0.6
hence, to avoid excessive total pressure losses in the HX (see Shah and Sekulic [8]) the
flow captured by the off-take must be diffused. It is desirable to spread this diffusion
throughout the duct system. Due to spatial constraints, some diffusion can be achieved
via pre-diffusion of the captured stream-tube and some can be achieved in diffuser 1 (see
Figures 2 and 4). However, approximately 90% of the diffusion will need to be achieved
between the strut exit and the HX. In previous work of Spanelis et al. [9] this region was
comprised of two s-ducts separated by a vane. Their results suggested that flow uni-
formity at duct exit is maximised for designs which concentrate the aerodynamic loading
(both diffusion and curvature) close to the HX. They attributed this to the beneficial effects
that the HX blockage exerts on the flow immediately upstream of the HX. Therefore, as
shown in Figure 4, this region is sub-divided in two components: (1) a plane diffuser re-
ferred to as “diffuser 2” which should be moderately loaded, and (2) an s-shaped manifold
connected to the HX which can then be more aggressively loaded by taking advantage of
the above-mentioned local pressure field near the HX. For the CCA system examined
herein the target diffusion equates to an area ratio of 𝐴𝐴
= 16.7 within an overall non-
dimensional length of 𝐿
=28.7. Where “s” denotes the stream-tube captured by
the off-take and “T” is the target zone or, here, the HX inlet plane. Note that a single off-
take feeds two HXs therefore, 𝐴 represents the combined inlet area of both.
Figure 4. Schematic representation of the LP-CCA system.
The presented system is a conceptual design. It can be shown via the elementary
timeline of a design process in Figure 5, that in an industrial environment there is a sig-
nificant overlap between different design stages. The purpose of this overlap is to improve
efficiency of the workforce. See for instance the description of the aircraft design process
as explained by Gudmundsson [10]. More specifically, as the conceptual design stage
phases out, the preliminary investigation is already in progress, which provides the op-
portunity to the conceptualisation team to further develop the concept. A new preliminary
Figure 4. Schematic representation of the LP-CCA system.
The presented system is a conceptual design. It can be shown via the elementary
timeline of a design process in Figure 5, that in an industrial environment there is a
significant overlap between different design stages. The purpose of this overlap is to
improve efficiency of the workforce. See for instance the description of the aircraft design
process as explained by Gudmundsson [
10
]. More specifically, as the conceptual design
stage phases out, the preliminary investigation is already in progress, which provides
the opportunity to the conceptualisation team to further develop the concept. A new
preliminary design iteration will then follow on the updated concept. This feedback loop
continues until the design is mature enough to proceed to the next design stages. In line
with the above, the general scope of this paper is to present a single step in the evolution of
the above conceptual design, generalised into a suitable method to support the iterative
process between conceptualisation and preliminary design space exploration.
Aerospace 2022,9, 130 4 of 32
Aerospace 2022, 9, x 4 of 32
design iteration will then follow on the updated concept. This feedback loop continues
until the design is mature enough to proceed to the next design stages. In line with the
above, the general scope of this paper is to present a single step in the evolution of the
above conceptual design, generalised into a suitable method to support the iterative pro-
cess between conceptualisation and preliminary design space exploration.
Figure 5. Generic timeline of a design process.
At this early design phase, the engineer must seek simplifications that will not affect
the leading order properties of the system. This task is far from trivial due to component
interactions upstream-to-downstream and vice versa. See for example A’Barrow et al. [11]
who illustrates that the presence of a diffusing off-take downstream of a compressor can
have an upstream effect on the compressor and similarly the compressor’s exit conditions
will also affect the performance of the diffusing off-take. Additionally, Spanelis and
Walker [6] who investigated the effects of an annular bleed between the stator and the
rotor of a low-pressure compressor, showed that the position of the bleed is responsible
for severe flow bias in the radial direction which can reduce the OGV stall margin and
alter the OGV stall topology itself. The current system can be described by a large set of
closely coupled design variables which define the aerodynamic shape. Relevant examples
are given by Jirásek [12] and Hamstra et al. [13], who investigated s-shaped intake ducts
that employed design of experiment (DoE) techniques, a.k.a. factorial design, to obtain an
optimum shape. Furthermore, the work of Yurko and Bondarenko [14], employed facto-
rial design to examine a single annular s-duct by means of four design variables. On a
previous iteration of the present conceptual design evolution, Spanelis et. al. [9] examined
a double s-duct by sequentially exploring eight primary and five secondary design varia-
bles. Their findings have fed back to evolve the conceptual design to the current state by
introducing new features, such as a significant elongation of the vane that separates the
two heat exchangers and an independent diffusing duct segment upstream of the mani-
fold (i.e., diffuser 2).
Further to the general scope of this paper that was stated above, the following specific
objectives are set:
(A) Simplify and parametrically model the conceptual design.
(B) Select and evaluate a suitable approach to sample the design space.
(C) Define a set of quality evaluation criteria (design objectives).
(D) Develop a multi-objective characterisation methodology.
(E) Characterise the design space.
(F) Select and evaluate a set of nominal geometries to advance to the next stage of the
design process.
In the remaining part of this paper, Section 2 provides description and justification
of modelling choices, assumptions and simplifications that render this method capable to
meet the above targets and objectives. Then, specific details such as the description and
quantitative definition of input and output variables, as well as the numerical simulation
parameters, are presented in Section 3. The results are then presented in Sections 4 and 5
followed by a brief conclusion of this paper in Section 6.
Figure 5. Generic timeline of a design process.
At this early design phase, the engineer must seek simplifications that will not affect
the leading order properties of the system. This task is far from trivial due to component
interactions upstream-to-downstream and vice versa. See for example A’Barrow et al. [
11
]
who illustrates that the presence of a diffusing off-take downstream of a compressor can
have an upstream effect on the compressor and similarly the compressor’s exit conditions
will also affect the performance of the diffusing off-take. Additionally, Spanelis and
Walker [
6
] who investigated the effects of an annular bleed between the stator and the rotor
of a low-pressure compressor, showed that the position of the bleed is responsible for severe
flow bias in the radial direction which can reduce the OGV stall margin and alter the OGV
stall topology itself. The current system can be described by a large set of closely coupled
design variables which define the aerodynamic shape. Relevant examples are given by
Jirásek [
12
] and Hamstra et al. [
13
], who investigated s-shaped intake ducts that employed
design of experiment (DoE) techniques, a.k.a. factorial design, to obtain an optimum
shape. Furthermore, the work of Yurko and Bondarenko [
14
], employed factorial design to
examine a single annular s-duct by means of four design variables. On a previous iteration
of the present conceptual design evolution, Spanelis et. al. [
9
] examined a double s-duct by
sequentially exploring eight primary and five secondary design variables. Their findings
have fed back to evolve the conceptual design to the current state by introducing new
features, such as a significant elongation of the vane that separates the two heat exchangers
and an independent diffusing duct segment upstream of the manifold (i.e., diffuser 2).
Further to the general scope of this paper that was stated above, the following specific
objectives are set:
(A)
Simplify and parametrically model the conceptual design.
(B)
Select and evaluate a suitable approach to sample the design space.
(C)
Define a set of quality evaluation criteria (design objectives).
(D)
Develop a multi-objective characterisation methodology.
(E)
Characterise the design space.
(F)
Select and evaluate a set of nominal geometries to advance to the next stage of the
design process.
In the remaining part of this paper, Section 2provides description and justification
of modelling choices, assumptions and simplifications that render this method capable to
meet the above targets and objectives. Then, specific details such as the description and
quantitative definition of input and output variables, as well as the numerical simulation
parameters, are presented in Section 3. The results are then presented in Sections 4and 5
followed by a brief conclusion of this paper in Section 6.
2. Methodology
As mentioned in the introduction, this study concerns the iterative process between
the conceptual and preliminary design phase of the CCA system. In this phase the avail-
ability of computational resources is limited as funding is not fully committed yet. For
instance, the numerical predictions presented in this paper have been acquired as part of
a single iteration of this design cycle and have only utilised a single high-performance
desktop computer for less than two days. To achieve this low computational cost, several
Aerospace 2022,9, 130 5 of 32
simplifications have been implemented both in the examined model and in the sampling
approach. This paragraph discusses and justifies the various design choices and then
quantitatively presents the selected model parameters.
2.1. Modelling Assumptions and Limitations
Typically, at this early phase in the design process it is not viable to employ physical
experiments. A numerical approach is more suitable in terms of both time and cost
considerations. The computational cost of a single design iteration depends on the leading
order accuracy of the model. More specifically, high fidelity, three-dimensional simulations
are not attractive for the exploration of a conceptual design. In this way, to maintain a
balance between numerical accuracy and computational cost, a simplified two-dimensional
version of the problem is considered, which is a valid approximation given that the primary
flow features can be resolved in two dimensions. In terms of turbulence modelling, there is
a vast range of models available in the literature, for which accuracy and computational
cost are in general directly linked to one another. For this study, RANS modelling has been
selected as it is an economic yet sufficiently accurate approach, that has been extensively
validated for duct flows.
Furthermore, during this phase it is recommended to maximise the input variable
ranges in order to identify all valuable design regions. Constraints can be set based on
limits known to exist by theory or by empirical models. In this study, diffuser loading charts
have been employed to limit the expansion rate of various duct components. Additionally,
constraints are often set by physical objects in the environment of the system itself. For
instance, here the maximum limit of the off-take opening and diffuser 1 expansion rate are
set by the available space within the strut, while the width of diffuser 2 is constrained by
the turbofan gearbox which is located near the strut exit.
Maximising the input variable ranges brings about an additional computational chal-
lenge. As the variable ranges increase, the probability for the observed effects to exhibit
linear trends across the boundaries of the design space decreases. In fact, the results of
this study revealed the existence of multiple peaks and valleys of the objective functions
within the design space. Often, classical DoE engineering research uses between two
and three levels per input variable. Such a low number of levels requires the creation
of empirical models to correlate the limited responses observed, i.e., response surface
modelling. With the increased non-linearity of the observed effects in the large design
spaces considered here, a larger number of levels per variable is required (further discussed
in the next sub-section).
The present study has only considered the “max take-off” condition because at this
stage in the flight envelope the engines are at full thrust which is when the CCA system
is mostly needed. While the concept considers the option to set a mechanical valve that
regulates the flow in the system so that it can be “turned off” when not needed, other
engine conditions are subject to investigation at a more matured design stage. Furthermore,
this study assumes a symmetric geometry and neglects any swirl reminder exiting the
bypass duct OGV. However, this is not a leading order characteristic of this system and
alongside other three-dimensional flow features it is subject to optimisation in a subsequent
design stage. Moreover, the configuration of the return ducting downstream of the HX is
not fully decided yet. For this reason, the exit pressure and the overall loss of the cooling
system are unavailable. Therefore, closure of the numerical model has been achieved here
by fixing the system mass flow rate to the nominal amount for the given HX (more details
in Section 3).
2.2. Sampling Approach
This subsection discusses the approach selected to sample the design space.
A review
on design modelling through numerical experiments has been carried out by
Chen et al. [15]
.
Furthermore, Yondo et al. [
16
] presented a review of DoE and surrogate models for aerody-
Aerospace 2022,9, 130 6 of 32
namic design. From these we can see that advantages of the factorial design approach com-
pared to other design exploration and optimisation methodologies include the following:
(a)
It is much more efficient in the estimation of the main effects, i.e., it allows direct
evaluation of the design variable interactions.
(b)
The complete dataset is available a priori which facilitates weighting of the multi-
objective functions.
(c)
It is relatively simple to apply, and it does not require an expertise in advanced
optimisation algorithms.
The main disadvantage, however, is the high computational cost associated with large
number of design iterations, which can be a prohibitive constraint when many design
variables are closely coupled to one another. Gradient-based methods are efficient in
finding the true optimal solution in such problems while they can directly handle non-
linear constraints. However, true optimality is not the primary objective in the conceptual
design phase. The primary objective here is to identify different design families via the
systematic mapping of areas of interest in the design space. Factorial design offers this
attribute and with sufficiently high sampling resolution it also offers a set of perceived
optimal solutions which is adequate at this phase. Considering the above, the factorial
design technique is selected to sample the design space and to optimise the present concept.
2.3. Design Space Reduction
It will be shown in the next sub-section that nine input variables are needed to
geometrically determine this problem. A full factorial of the nine inputs would per-
haps be computationally viable if sampling was limited to two levels per input variable
(
i.e., 29=512 iterations
). However, as mentioned above, more levels are required to di-
rectly observe the highly nonlinear effects in this study. Even as few as three levels are
not computationally affordable (i.e., 3
9'
2
×
10
4
iterations). Preliminary sampling of
the design space showed that not all input variables interact strongly to one another and
suggested that reduction of this full factorial into three fractions is possible without risking
the elimination of high value design families.
The first step of the fractionalisation was to separate the off-take and diffuser 1 from
all downstream components such as the diffuser 2, the manifold and the two HXs. This
subdivision resulted in two models, hereafter referred to as “model 1” and “model 2”
respectively. Results of the factorial analysis of model 1, which are thoroughly discussed
later, demonstrated that the flow profiles delivered at the exit of model 1 are not sensitive to
changes in the off-take geometry for a substantial portion of the design space surrounding
the optimal geometry.
In the following, the inputs of the two s-ducts in model-2 are explored separately.
Both these s-ducts belong to model-2, hence, when the inputs of one duct are varied, the
inputs of the other remain fixed. It can be argued that decoupling the two s-ducts did not
significantly affect the outcome of this optimisation because the mass flow split between
the two s-ducts is conditioned by the high pressure drop across the HX. Therefore, the
streamtube division at the leading edge of the s-shaped vane that separates the two HX
cannot be considerably affected by moderate geometrical variations in the s-ducts, except
for a small subset of extreme high-loss geometries which have very little value in the
analysis (further discussed in the results section).
The above subdivision has resulted in three cubic factorial designs, i.e., 3
×N3
. In
order to increase resolution, the number of levels per variable was maximised within the
given computational resource availability, while constrained by the computational expense
limit of 50 CPU hours per cubic factorial design. With that in mind, the maximum value
achievable was
N
= 7 which results on a total of 3
×
7
3=
1029 design iterations. The
same resolution would require 7
9'
4
×
10
7
design iterations, were it not for the fractional
approach employed.
Aerospace 2022,9, 130 7 of 32
3. Model Setup Details
This section provides description and quantification of all problem parameters includ-
ing input variables, design objectives and the numerical simulation setup details.
3.1. Input Variables
A sketch of model 1 is presented in Figure 6. A significant geometric feature in this
model is the leading edge of the bifurcation strut (curve E–G). This curve can be defined by
an analytical equation that best describes the strut geometry. A sixth order polynomial was
found adequate to describe the geometry of the investigated strut. It should, however, be
noted that in general high order polynomials may be inappropriate when the polynomial
coefficients are used by the input variables (i.e., a non-fixed geometry). Here, the strut
geometry is fixed hence this is not a concern. Furthermore, line A–B represents the diffuser
1 exit plane, hence, curves C–A and D–B represent the walls of diffuser 1. Note that point
A, alongside the bypass duct and the strut wall, is fixed in space. The three design variables
selected for the exploration of model 1 are:
(a) The axial extent of diffuser 1 (A–C) referred to as “Length”. This variable implicitly de-
termines the position of the off-take along the strut. Low values of length correspond
to flush off-takes and high values to total off-takes.
(b)
The “Angle” of the off-take (C–A) relative to the local slope of the strut (E–G). At a
0angle the duct would locally be parallel to the strut surface.
(c)
The “Height” of the off-take which is explicitly defined by the distance C–D.
Aerospace 2022, 9, x 7 of 32
for a small subset of extreme high-loss geometries which have very little value in the anal-
ysis (further discussed in the results section).
The above subdivision has resulted in three cubic factorial designs, i.e., 3×𝑁. In
order to increase resolution, the number of levels per variable was maximised within the
given computational resource availability, while constrained by the computational ex-
pense limit of 50 CPU hours per cubic factorial design. With that in mind, the maximum
value achievable was 𝑁=7 which results on a total of 3×7= 1029 design iterations. The
same resolution would require 7≃4×10
design iterations, were it not for the frac-
tional approach employed.
3. Model Setup Details
This section provides description and quantification of all problem parameters in-
cluding input variables, design objectives and the numerical simulation setup details.
3.1. Input Variables
A sketch of model 1 is presented in Figure 6. A significant geometric feature in this
model is the leading edge of the bifurcation strut (curve E–G). This curve can be defined
by an analytical equation that best describes the strut geometry. A sixth order polynomial
was found adequate to describe the geometry of the investigated strut. It should, however,
be noted that in general high order polynomials may be inappropriate when the polyno-
mial coefficients are used by the input variables (i.e., a non-fixed geometry). Here, the
strut geometry is fixed hence this is not a concern. Furthermore, line A–B represents the
diffuser 1 exit plane, hence, curves C–A and D–B represent the walls of diffuser 1. Note
that point A, alongside the bypass duct and the strut wall, is fixed in space. The three
design variables selected for the exploration of model 1 are:
(a) The axial extent of diffuser 1 (A–C) referred to as “Length”. This variable implicitly
determines the position of the off-take along the strut. Low values of length corre-
spond to flush off-takes and high values to total off-takes.
(b) The “Angle” of the off-take (C–A) relative to the local slope of the strut (E–G). At a
0° angle the duct would locally be parallel to the strut surface.
(c) The “Height” of the off-take which is explicitly defined by the distance C–D.
A set of different instances of model 1 for various values of the three input variables
can be observed in Figure 7.
Figure 6. Model 1 (CCA1) topology.
Figure 6. Model 1 (CCA1) topology.
A set of different instances of model 1 for various values of the three input variables
can be observed in Figure 7.
Figure 7. Examples of different instances of model 1 variables. (a) Length; (b) Angle; (c) Height.
Figure 8illustrates the control points and curves that constitute model 2. A robust
method to design and parametrically control the shape of the manifold, typical for s-duct
design, is the cubic Bezier curve equation, defined as:
B(τ)=P1(1τ)3+3P2τ(1τ)2+3P3τ2(1τ)+P4τ3(1)
where
P1P4
are the four control points of the Bezier curve
B(τ)
and 0
τ
1 is the
dimensionless length of the Bezier curve. Applying
B(τ)
in two dimensions yields two
Aerospace 2022,9, 130 8 of 32
Cartesian coordinates for each point
X(τ)
and
Y(τ)
. Following this approach, the design
of a single Bezier curve requires the definition of four points, which means that a total of
thirty-two coordinates are required for the full definition of the manifold. Conveniently,
many of these parameters can be eliminated using straightforward boundary relationships.
To begin with, P1and P4, which are the start and end points of each Bezier curve, must be
fixed in both the X and Y directions due to the condition that the s-duct must be connected
to the upstream and downstream components. This only leaves the coordinates of points
P2
and
P3
to be defined for each curve. To ensure a smooth blend of the Bezier curves with the
upstream and downstream components, the Y coordinates of points
P2
and
P3
are defined
as a function of their X coordinate and the slope of the neighbouring curve to each point.
With these simplifications, complete control of a Bezier curve can be provided solely by
the axial coordinate of the two intermediate points
X(P2)
and
X(P3)
and a set of thirty-two
variables defining the manifold has now been reduced to just eight. In this way, the value
of a control point expresses the axial distance from start (upstream) to end (downstream)
of a Bezier curve. Selected (optimal) solutions of s-duct 1 and s-duct 2 are later combined
to construct the subsequent optimal manifold designs. Subscripts and superscripts of the
Bezier control points follow the notation of Figure 8.
Aerospace 2022, 9, x 8 of 32
(a) (b) (c)
Figure 7. Examples of different instances of model 1 variables. (a) Length; (b) Angle; (c) Height.
Figure 8 illustrates the control points and curves that constitute model 2. A robust
method to design and parametrically control the shape of the manifold, typical for s-duct
design, is the cubic Bezier curve equation, defined as:
𝐵(𝜏)=𝑃
(1−𝜏)+3𝑃
𝜏(1−𝜏)+3𝑃
𝜏(1−𝜏)+𝑃
𝜏 (1)
where 𝑃−𝑃
are the four control points of the Bezier curve 𝐵(𝜏) and 0≤𝜏≤1 is the
dimensionless length of the Bezier curve. Applying
𝐵(𝜏) in two dimensions yields two
Cartesian coordinates for each point 𝑋(𝜏) and 𝑌(𝜏). Following this approach, the design
of a single Bezier curve requires the definition of four points, which means that a total of
thirty-two coordinates are required for the full definition of the manifold. Conveniently,
many of these parameters can be eliminated using straightforward boundary relation-
ships. To begin with, 𝑃 and 𝑃, which are the start and end points of each Bezier curve,
must be fixed in both the X and Y directions due to the condition that the s-duct must be
connected to the upstream and downstream components. This only leaves the coordinates
of points 𝑃 and 𝑃 to be defined for each curve. To ensure a smooth blend of the Bezier
curves with the upstream and downstream components, the Y coordinates of points 𝑃
and 𝑃 are defined as a function of their X coordinate and the slope of the neighbouring
curve to each point. With these simplifications, complete control of a Bezier curve can be
provided solely by the axial coordinate of the two intermediate points 𝑋() and 𝑋()
and a set of thirty-two variables defining the manifold has now been reduced to just eight.
In this way, the value of a control point expresses the axial distance from start (upstream)
to end (downstream) of a Bezier curve. Selected (optimal) solutions of s-duct 1 and s-duct
2 are later combined to construct the subsequent optimal manifold designs. Subscripts
and superscripts of the Bezier control points follow the notation of Figure 8.
Figure 8. Model 2 (CCA2) topology.
Details of all curve entities of the two models are summarised on Table 1. Further-
more, Table 2 presents a summary of all components of model 1 and model 2, the associ-
ated design variable description, and the selected evaluation methods.
Figure 8. Model 2 (CCA2) topology.
Details of all curve entities of the two models are summarised on Table 1. Furthermore,
Table 2presents a summary of all components of model 1 and model 2, the associated
design variable description, and the selected evaluation methods.
Table 1. Model 1 and model setup details.
Component Boundary Type Sketch
Reference Entity Type
Model 1
Off-take ramp
no-slip wall
D’–D
cubic Bezier curve
Diffuser 1 D–B
C–A
By-pass duct off-take wall
E–G polynomial
By-pass duct virtual
2D annulus slip wall F*–G*
line segment
By-pass duct inlet velocity
profile F–F*
Cooling duct exit outflow A*–B*
By-pass duct exit G–G*
By-pass duct
symmetry plane symmetry F–E
Target zone flow interior A–B
Aerospace 2022,9, 130 9 of 32
Table 1. Cont.
Component Boundary Type Sketch
Reference Entity Type
Model 2
S-duct 1
no-slip wall
1–4 (A)
cubic Bezier curve
1–4 (B)
S-duct 2
1–4 (C)
1–4 (D)
Diffuser 2
0–1 (A)
line segment
0–1 (D)
Duct inlet inlet profile 0 (A–B)
Duct exit outflow 6 (A–D)
Target zone 1 flow interior 4 (A–B)
Target zone 2 4 (C–D)
HX 1 Porous
media
4–5 (A–B) rectangular block
HX 2 4-5 (C–D)
Table 2. Complete parametric model summary and factorial design input.
System Component Design Variable DoE MIN MAX Evaluation Method
Model 1
Off-take
streamtube height -0.12 h4fixed
mass flow rate
length 3.62 h45.44 h4
DoE 1
angle 845
height 0.052 h40.155 h4
Diffuser 1 inlet height
exit height -0.21 h4
engine
constraint (strut)
Model 2
Diffuser 2 inlet height
exit height -0.33 h4ESDU chart
(diffuser)
Manifold
(double s-duct)
inlet height
exit heights - 2h4HX specification
Bezier curve
control points
PA
150% 95%
DoE 3
PA
250% 95%
PB
150% 95%
PB
2- 90% Spanelis et al., 2017
PC
170% 95%
DoE 2
PC
280% 95%
PD
130% 95%
PD
2- 90% Spanelis et al., 2017
3.2. Design Objectives
The criteria that best describe the “quality” of an aerodynamic system may vary
from one problem to the next. However, a parameter of paramount importance in most
aerodynamic systems is total pressure loss. More specifically high-pressure loss may affect
the flow capacity of the cooling system and extensively it can impact the overall engine’s
efficiency. It is therefore important that total pressure loss is considered as an objective in
Aerospace 2022,9, 130 10 of 32
the current system. In this way, spatially averaged values of total and static pressure at a
reference section are derived using the mass-weighted technique described by Klein [17]:
e
P=1
.
mZPd .
m=1
ρUA ZPρUdA (2)
e
p=1
.
mZpd .
m=1
ρUA ZpρUdA (3)
subsequently, the first design objective is a mass-weighted total pressure loss coefficient
defined as:
λ=e
Pse
PT
e
Pse
ps
(4)
where index “s” corresponds to the captured stream-tube and index “T” corresponds to
the target zone. The captured stream-tube condition is defined at an adequate distance
upstream of the bifurcation strut and is not significantly affected by changes in the input
variables of model 1. There are four different regions where a loss coefficient is evaluated in
this work including the loss in the bypass duct (
λBPD
), the loss in the CCA duct of model-1
(
λCCA1
) and the loss in the CCA duct of model-2 (
λCCA2
). Note that the loss across the HX
(λHX ) is incorporated in λCCA2.
Another important parameter in cooling systems is the uniformity of the flow deliv-
ered to the HX. Specifically, mal-distributed feeds can significantly affect the aerothermal
performance of a HX, see Kwan et al. [
18
] and Raul et al. [
19
]. For this reason, the second
design objective of this work is defined as the area-weighted non-uniformity index of the
velocity magnitude at the HX inlet face (the target zone), calculated as:
(1γ)=n
i=1UiUaAi
2Uan
i=1Ai
(5)
where
i
is the facet index of a surface with n facets, and
Ua
is the area weighted average of
the velocity at the HX inlet. A perfectly uniform flow would have a value of
(1γ)=
0.
This parameter was used by Spanelis et al. [
9
] on a single objective optimisation of a dual
s-duct configuration similar to the one tested here.
A common issue in highly loaded duct systems is flow separation and recirculation,
not only because of the high associative pressure losses, but also due to induced vibrations
which can become a mechanical concern. In general, flow separation gives rise to an
increased production of turbulence kinetic energy at the shear layer of the separation
bubble (see for instance the highly resolved Large Eddy Simulation data reported by Luiz
Schiavo [20] who investigated the turbulence kinetic energy budgets in turbulent channel
flows with pressure gradients and separation). An increase in the production of turbulence
kinetic energy is a source of total pressure loss, hence, in most duct systems minimising the
total pressure loss coefficient would be sufficient to minimise or, if possible, to eliminate
flow separation. However, as it will be shown later in this paper, there is an additional
source of total pressure loss that takes place at the frontal face of the heat exchanger which
conflicts with the upstream duct losses. This means that minimising the total pressure loss
coefficient in the current problem does not guarantee minimization of the duct losses and
subsequent flow separation. To resolve this issue, a third design objective is introduced
that explicitly evaluates the turbulent activity in the delivery duct. More specifically, the
kinetic energy ratio is defined as the mass-weighted volume integral of turbulence kinetic
energy (TKE) in the delivery duct normalized by the mean kinetic energy (MKE) of the bulk
flow in the same region. This is:
k=TKE
MKE =n
i=1kiρi|Vi|
n
i=1qi|Vi|(6)
Aerospace 2022,9, 130 11 of 32
where nis the number of cells in the examined fluid volume and
ki
,
ρi
,
Vi
and
qi
are the
cell values for turbulence kinetic energy per unit mass, density, volume and dynamic
pressure respectively. To better understand the meaning of the kinetic energy ratio
k
,
assume a finite fluid volume with uniform velocity and turbulence intensity, (U) and (I),
respectively. The turbulence kinetic energy per unit mass in this fluid volume can be
expressed as
TKE =
3
/
2
(UI)2
. In the same region the bulk flow kinetic energy per unit
mass can be expressed as
MKE =
1
/
2
U2
. Combining these two equations in Equation (6)
the kinetic energy ratio relates to turbulence intensity as
k=
3
I2
. For instance, a value
of
k=
0.03 is representative of an average turbulence intensity in the examined fluid
volume of I=10%. Furthermore, it is important that
k
is weighted appropriately so that
it is insensitive to small variations in the turbulence kinetic energy in the flow-field but
still, maintains sensitivity to radical changes, such as a flow separation/recirculation where
turbulence intensity generally increases by a greater factor.
3.3. Multi-Objective Function
An “optimised” solution can be achieved by minimising one or more of the objective
functions presented above. However, an attempt to improve a design objective may
lead to the degradation of another which constitutes a non-trivial optimisation problem.
A workaround is the establishment of a global function, the minimum value of which
corresponds to the “best” compromise between the conflicting objectives, i.e., a scalarisation
method. A multi-objective function is proposed, which can be classified as a weighted
global criterion method, and is defined by:
Ψ=n
i=1wisi[FiF0
i]
n
i=1wi
where si=1
FCr
iF0
i
(7)
More specifically, function
Ψ
calculates the normalised Euclidean distance in the
criterion space
FiF0
i
, where
Fi
is the value of a given design objective and
F0
i
is its utopia
point, and subsequently applies a scaling and a weighting operation. Particularly,
si
is
the vector of scaling factors and consists of two components. The first component of the
denominator,
FCr
i
, represents the highest acceptable (critical) value for a given design-
objective and the second component,
F0
i
, represents its lowest (ideal) value, i.e., the utopia
point. Both these constants must be defined by the decision maker for each design objective
and the range between these two values is referred to as the “acceptable design range”. In
general, the utopia point is unattainable, and depending on the nature of the given design
objective it may be difficult to evaluate. On a factorial design, however, the complete data
sample is acquired before it is analysed, hence the minimum observed value of each design
objective
(Fi)min
is known in advance. Therefore, the utopia point is approximated based
on the minimum observation as:
F0
i=(F0
i)0+α(Fi)min (F0
i0(8)
where
(F0
i)0
represents fixed contributions to a design objective. There are three such
occurrences in the current study including:
(a)
The contribution of the bifurcation strut blockage to the bypass duct loss in model 1.
(b)
The HX loss in the overall CCA duct loss of model 2.
(c)
The turbulence intensity at the inlet of model 2 in the kinetic energy ratio.
If no constant contributions can be identified, i.e., (F0
i)0=0, Equation (8) reduces to:
F0
i=α(Fi)min (9)
The minimum observation of a design objective,
(Fi)min
, may approach but it can
never equal the utopia point
F0
i
, i.e.,
α<
1. In this study, an estimated value of
α=0.8 is
selected, meaning that the utopia points are approximated at a level 20% lower than the
Aerospace 2022,9, 130 12 of 32
minimum observations. Since the utopia point is always lower than the value of the
objective function, the value of
Ψ
must always be positive. Additionally, on a “bad” design,
the objective function
Fi
could exceed the critical value,
FCr
i
, hence,
Ψ
does not have an
upper limit. Finally,
w
is the vector of weighting factors evaluated by the decision maker.
The relative values of these weights determine the influence of each design-objective in
the multi-objective function. It is noteworthy that the vector of scaling factors,
s
, as well
as the one of weighting factors,
w
, in Equation (7) are both means of weighting the design
objectives. The former is explicitly defined by the physical extremities of the problem and
is therefore a more robust way to weight the outputs. However, the later can be very useful
when the decision maker wishes to assign different levels of importance to the outputs. A
review of multi-objective optimisation methods for engineering can be found at Marler
and Arora [21].
3.4. Numerical Simulation Setup
The commercial CFD platform ANSYS Fluent was employed and the numerical simu-
lations have been executed on a Linux machine using four CPU cores per run. This study
has not considered compressibility effects; hence, a pressure-based solver was employed.
Pressure-velocity coupling has been implemented via the SIMPLE algorithm (see Versteeg
and Malalasekera [
22
]). Turbulence closure was achieved using the Reynolds stress model
(RSM). Walker et al. [
23
,
24
] showed that this higher order model is required to capture
the effects of curvature in an annular s-duct. A second order upwind scheme was applied
for spatial discretization of momentum, turbulence kinetic energy, turbulence dissipation
rate, and Reynolds stresses. Spatial discretization of pressure was implemented with the
PREssure STaggering Option (PRESTO!) scheme in model 2 simulations due to lack of
compatibility of the second order scheme with the porous media approach employed
by Fluent.
A grid convergence study was conducted for two different geometries taken from
model 1. Six different mesh densities in the range 3.0
×
10
4
N
2.5
×
10
5
were evaluated
based on the total pressure loss coefficient of the CCA duct in model 1,
λCCA1
. Results
of this study are presented in Figure 9which shows that the total pressure loss changes
significantly with mesh density at 2
×
10
4
, but the solution is effectively invariable at mesh
density of 5
×
10
4
or higher and this is therefore the chosen density for the current work. At
this mesh density, the off-take mouth is resolved by 20–25 cells depending on the off-take
height and the HX inlet face is resolved by 55 cells. Boundary layer inflation is applied
on wall boundaries with a growth rate of 1.2. The resulted non-dimensional wall distance
varies as 30
y+
100, which is compatible with the selected near-wall modelling
approach, i.e., standard wall functions. Note that model 1 and model 2 mesh sizes are
similar in this work and the details mentioned above apply to both models. Examples of
meshing for the two models are shown in Figures 10 and 11 respectively.
Aerospace 2022, 9, x 13 of 32
Figure 9. Model 1 grid convergence study.
Figure 10. Example of Model 1 meshing.
Figure 11. Example of Model 2 meshing.
Furthermore, the inlet boundary of model 1 is fed by a profile of fan OGV exit veloc-
ity data representative of a commercial turbo-fan engine. Note that this is not explicitly
presented here for reasons of confidentiality, but its effect to the bypass duct flow field
can be observed in the velocity contour plots of section 4. The mass flow ratio between the
two outflow boundaries of model 1 is fixed at 𝑚 𝑚
=9.10
 which is the re-
quirement to feed two heat exchangers at max take-off engine conditions as calculated in
the corresponding HX study by Ha et al. [25]. Velocity and turbulence data at the target
zone of the optimal solution of model 1 are fed into model 2. More specifically, the x and
y-velocity components, the turbulence kinetic energy, its dissipation rate and the full set
of Reynolds stresses 𝑢’𝑣’, 𝑢’𝑢’, 𝑣’𝑣’ and 𝑤’𝑤’, are interpolated at the inlet boundary of
model 2. For further details about boundary type selection, refer to Table 1.
Finally, HX blockage is modelled via porous media through the application of a uni-
form inertial porous resistance of 𝜆 𝑃/𝑞
 =87 in all three directions. Here, the dy-
namic head of a perfectly uniform flow at the HX inlet is used in the evaluation of the loss
coefficient, i.e., 𝑞 =0.5𝑚
(𝜌
𝐴
). This loss coefficient was calculated from the data
of Ha et al. [25] in line with a requirement for a 3.25% pressure drop (Δ𝑃) across the HX
with respect to the bypass duct condition of a commercial turbo-fan engine at max take-
off conditions. Turbulence in the porous medium is solved using the standard conserva-
tion equations and microscopic effects to turbulence kinetic energy are not accounted for
(see Pedras et al. [26]).
Figure 9. Model 1 grid convergence study.
Aerospace 2022,9, 130 13 of 32
Aerospace 2022, 9, x 13 of 32
Figure 9. Model 1 grid convergence study.
Figure 10. Example of Model 1 meshing.
Figure 11. Example of Model 2 meshing.
Furthermore, the inlet boundary of model 1 is fed by a profile of fan OGV exit veloc-
ity data representative of a commercial turbo-fan engine. Note that this is not explicitly
presented here for reasons of confidentiality, but its effect to the bypass duct flow field
can be observed in the velocity contour plots of section 4. The mass flow ratio between the
two outflow boundaries of model 1 is fixed at 𝑚 𝑚 
=9.8×10
 which is the re-
quirement to feed two heat exchangers at max take-off engine conditions as calculated in
the corresponding HX study by Ha et al. [25]. Velocity and turbulence data at the target
zone of the optimal solution of model 1 are fed into model 2. More specifically, the x and
y-velocity components, the turbulence kinetic energy, its dissipation rate and the full set
of Reynolds stresses 𝑢’𝑣’, 𝑢’𝑢’, 𝑣’𝑣’ and 𝑤’𝑤’, are interpolated at the inlet boundary of
model 2. For further details about boundary type selection, refer to Table 1.
Finally, HX blockage is modelled via porous media through the application of a uni-
form inertial porous resistance of 𝜆 𝑃/𝑞
 =87 in all three directions. Here, the dy-
namic head of a perfectly uniform flow at the HX inlet is used in the evaluation of the loss
coefficient, i.e., 𝑞 =0.5𝑚
(𝜌
𝐴
). This loss coefficient was calculated from the data
of Ha et al. [25] in line with a requirement for a 3.25% pressure drop (Δ𝑃) across the HX
with respect to the bypass duct condition of a commercial turbo-fan engine at max take-
off conditions. Turbulence in the porous medium is solved using the standard conserva-
tion equations and microscopic effects to turbulence kinetic energy are not accounted for
(see Pedras et al. [26]).
Figure 10. Example of Model 1 meshing.
Aerospace 2022, 9, x 13 of 32
Figure 9. Model 1 grid convergence study.
Figure 10. Example of Model 1 meshing.
Figure 11. Example of Model 2 meshing.
Furthermore, the inlet boundary of model 1 is fed by a profile of fan OGV exit veloc-
ity data representative of a commercial turbo-fan engine. Note that this is not explicitly
presented here for reasons of confidentiality, but its effect to the bypass duct flow field
can be observed in the velocity contour plots of section 4. The mass flow ratio between the
two outflow boundaries of model 1 is fixed at 𝑚 𝑚 
=9.8×10
 which is the re-
quirement to feed two heat exchangers at max take-off engine conditions as calculated in
the corresponding HX study by Ha et al. [25]. Velocity and turbulence data at the target
zone of the optimal solution of model 1 are fed into model 2. More specifically, the x and
y-velocity components, the turbulence kinetic energy, its dissipation rate and the full set
of Reynolds stresses 𝑢’𝑣’, 𝑢’𝑢’, 𝑣’𝑣’ and 𝑤’𝑤’, are interpolated at the inlet boundary of
model 2. For further details about boundary type selection, refer to Table 1.
Finally, HX blockage is modelled via porous media through the application of a uni-
form inertial porous resistance of 𝜆 𝑃/𝑞
 =87 in all three directions. Here, the dy-
namic head of a perfectly uniform flow at the HX inlet is used in the evaluation of the loss
coefficient, i.e., 𝑞 =0.5𝑚
(𝜌
𝐴
). This loss coefficient was calculated from the data
of Ha et al. [25] in line with a requirement for a 3.25% pressure drop (Δ𝑃) across the HX
with respect to the bypass duct condition of a commercial turbo-fan engine at max take-
off conditions. Turbulence in the porous medium is solved using the standard conserva-
tion equations and microscopic effects to turbulence kinetic energy are not accounted for
(see Pedras et al. [26]).
Figure 11. Example of Model 2 meshing.
Furthermore, the inlet boundary of model 1 is fed by a profile of fan OGV exit velocity
data representative of a commercial turbo-fan engine. Note that this is not explicitly
presented here for reasons of confidentiality, but its effect to the bypass duct flow field
can be observed in the velocity contour plots of Section 4. The mass flow ratio between
the two outflow boundaries of model 1 is fixed at
.
mCCA/.
mBPD =
9.8
×
10
3
which is the
requirement to feed two heat exchangers at max take-off engine conditions as calculated in
the corresponding HX study by Ha et al. [
25
]. Velocity and turbulence data at the target
zone of the optimal solution of model 1 are fed into model 2. More specifically, the x and
y-velocity components, the turbulence kinetic energy, its dissipation rate and the full set of
Reynolds stresses
u0v0
,
u0u0
,
v0v0
and
w0w0
, are interpolated at the inlet boundary of model
2. For further details about boundary type selection, refer to Table 1.
Finally, HX blockage is modelled via porous media through the application of a
uniform inertial porous resistance of λHX =P/qin =87 in all three directions. Here, the
dynamic head of a perfectly uniform flow at the HX inlet is used in the evaluation of the loss
coefficient, i.e.,
qin =
0.5
.
m2
HX /(ρA2
HX )
. This loss coefficient was calculated from the data
of Ha et al. [
25
] in line with a requirement for a 3.25% pressure drop (
P)
across the HX
with respect to the bypass duct condition of a commercial turbo-fan engine at max take-off
conditions. Turbulence in the porous medium is solved using the standard conservation
equations and microscopic effects to turbulence kinetic energy are not accounted for (see
Pedras et al. [26]).
4. Model 1 Results and Discussion
4.1. DoE 1 Data Sampling and Objective Function Conditioning
The first step in the execution of a factorial design is the selection of the sampling
method. Here a domain-based approach is employed, aiming to uniformly distribute
the sampling points in the design space. A subset of the sample, including the extreme
and mid-values for each of the three input variables, is illustrated in Figure 7. The actual
variable ranges are available on Table 2.
The results of DoE 1 are initially plotted in a three-dimensional criterion space defined
by the three selected design-objectives. In this type of plot, each individual level of the fac-
torial design assigns a single point, i.e., the points that appear in Figure 12 are the imprints
of the three-dimensional cloud projected on the three mutually perpendicular planes. More
specifically, the three two-dimensional criterion spaces that result this decomposition are
shown in Figure 12 in sub-figures:
Aerospace 2022,9, 130 14 of 32
(a)
total pressure loss in the delivery duct against diffuser 1 exit non-uniformity,
(b)
total pressure loss in the bypass duct against diffuser 1 exit non-uniformity and
(c)
total pressure loss in the delivery duct against the one in the bypass duct.
Aerospace 2022, 9, x 15 of 32
(Figure 14) for every 1% increase of the bypass duct pressure loss (Δ𝑃/𝑃) on a fan at mod-
erate pressure ratio (i.e., 𝑃𝑅~1.5) the equivalent polytropic efficiency drops by approxi-
mately 2%. The current study attempts to limit the equivalent polytropic efficiency drop
induced by the off-take at Δ𝜂. ≤ 0.05%. Based on the plot of Figure 14 this corre-
sponds to a critical increase in by-pass duct loss of order Δ𝑃/𝑃 ~ 0.025%. On a typical
turbofan engine in max take-off conditions this coefficient can be expressed in terms of
inlet dynamic head as Δ𝑃/𝑞 ~ 0.25%. Further to the off-take spillage loss, inherent in the
bypass duct region is part of the aerodynamic loss of the bifurcation geometry itself. This
has been quantified here via a pre-cursor simulation of the same bypass duct section but
without an off-take and it was predicted as Δ𝑃/𝑞 = 0.22%. This value can be considered
as a constant offset of the critical value, i.e., 𝐹
 = 0.0025 +0.0022 = 0.0047. For the same
reason this value also appears as a constant offset in the evaluation of the utopia point in
Equation (8), i.e., (𝐹
)′ = 0.0022,. The utopia points of the other two objectives do not
involve any known fixed contributions and are therefore calculated from Equation (9). A
summary of the scaling parameters of DoE 1 is available in Table 3.
(a) (b) (c)
Figure 12. DoE 1 results—Projected 3D criterion space into three individual 2D components (ac).
Figure 13. Non-dimensional velocity profiles at the exit of diffuser 1 corresponding to a range of
non-uniformities (1 γ).
Figure 12. DoE 1 results—Projected 3D criterion space into three individual 2D components (ac).
In Figure 12a,b it can be observed that there is no design point for which both objectives
are simultaneously minimised. Instead, a frontier of optimal solutions tends to form at
the bottom left of each of these two plots, a.k.a. a Pareto frontier (see Pareto [
27
]). On the
contrary, the two objectives of Figure 12c exert a different behaviour. More specifically, it is
indicated that the losses of the CCA duct and the ones of the bypass duct are minimised
simultaneously, which constitutes a trivial multi-objective optimisation problem. It should,
however, be born in mind that by nature the current factorial design methodology cannot
guarantee an absolute optimal solution nor an actual Pareto frontier; this would effectively
require an infinite number of DoE training points. There is a wide range of intelligent
optimisation techniques, capable to optimise nonlinear multimodal functions, such as
genetic algorithms, tabu search, simulated annealing and neural networks, see for instance
the recent review of Pham and Karaboga [
28
]. However, this endeavour is beyond the
scope of the current work which, as part of a preliminary design exploration, primarily
aims to characterise the design space without necessarily finding the true optimal solution.
Additional information that can be drawn by Figure 12 is the effectiveness of the
selected parametrisation approach. In an optimisation study it is preferable that the
training points appear at an increased concertation in the direction of the utopia point. In
this paper the design variables (inputs) of the DoE study are distributed uniformly in the
design space. The output data distribution in criterion space indicates the effectiveness of
both the input variable definition and to the selected input variable ranges. In this respect
Figure 12 indicates that the parametrisation of DoE 1 is effective in the exploration of the
by-pass duct loss and the CCA duct loss but it appears that the density of the sampling
points near the optimality of non-uniformity is low. To better understand this behaviour
the results must be viewed in design variable space which is discussed in the next section.
The highlighted (grey) region in the plots of Figure 12 is the projection of the three-
dimensional window upon which the objective functions are scaled. The boundaries of
this window are defined by the values of
F0
i
and
FCr
i
applicable in Equation (7). More
specifically, in this example the critical value above which the pressure loss in model 1
(
λCCA1)
is unacceptable has been empirically set to
FCr
1=
0.66. This value is strongly
application-specific; here it was set based on the available pressure differential (in the
engine) which drives the bleed system. Furthermore, the critical value for non-uniformity
(1
γ)
has been set to
FCr
2=
0.08 after a qualitative examination of different non-uniformity
values on a representative selection of velocity profiles as shown in Figure 13. A critical
value for the bypass duct loss (
λBPD)
can be estimated based on a fan equivalent polytropic
Aerospace 2022,9, 130 15 of 32
efficiency of the bypass duct. More specifically, as shown by Kyprianidis [
29
] (Figure 14) for
every 1% increase of the bypass duct pressure loss (
P/P
) on a fan at moderate pressure
ratio (i.e.,
PR
1.5) the equivalent polytropic efficiency drops by approximately 2%. The
current study attempts to limit the equivalent polytropic efficiency drop induced by the
off-take at
ηeq.pol
0.05%. Based on the plot of Figure 14 this corresponds to a critical
increase in by-pass duct loss of order
P/P
0.025%. On a typical turbofan engine in
max take-off conditions this coefficient can be expressed in terms of inlet dynamic head as
P/q
0.25%. Further to the off-take spillage loss, inherent in the bypass duct region is
part of the aerodynamic loss of the bifurcation geometry itself. This has been quantified
here via a pre-cursor simulation of the same bypass duct section but without an off-take
and it was predicted as
P/q=
0.22%. This value can be considered as a constant offset
of the critical value, i.e.,
FCr
3=
0.0025
+
0.0022
=
0.0047. For the same reason this value
also appears as a constant offset in the evaluation of the utopia point in Equation (8), i.e.,
(F0
3)0=
0.0022. The utopia points of the other two objectives do not involve any known
fixed contributions and are therefore calculated from Equation (9). A summary of the
scaling parameters of DoE 1 is available in Table 3.
Aerospace 2022, 9, x 15 of 32
(Figure 14) for every 1% increase of the bypass duct pressure loss (Δ𝑃/𝑃) on a fan at mod-
erate pressure ratio (i.e., 𝑃𝑅~1.5) the equivalent polytropic efficiency drops by approxi-
mately 2%. The current study attempts to limit the equivalent polytropic efficiency drop
induced by the off-take at Δ𝜂. ≤ 0.05%. Based on the plot of Figure 14 this corre-
sponds to a critical increase in by-pass duct loss of order Δ𝑃/𝑃 ~ 0.025%. On a typical
turbofan engine in max take-off conditions this coefficient can be expressed in terms of
inlet dynamic head as Δ𝑃/𝑞 ~ 0.25%. Further to the off-take spillage loss, inherent in the
bypass duct region is part of the aerodynamic loss of the bifurcation geometry itself. This
has been quantified here via a pre-cursor simulation of the same bypass duct section but
without an off-take and it was predicted as Δ𝑃/𝑞 = 0.22%. This value can be considered
as a constant offset of the critical value, i.e., 𝐹
 = 0.0025 +0.0022 = 0.0047. For the same
reason this value also appears as a constant offset in the evaluation of the utopia point in
Equation (8), i.e., (𝐹
)′ = 0.0022,. The utopia points of the other two objectives do not
involve any known fixed contributions and are therefore calculated from Equation (9). A
summary of the scaling parameters of DoE 1 is available in Table 3.
(a) (b) (c)
Figure 12. DoE 1 results—Projected 3D criterion space into three individual 2D components (ac).
Figure 13. Non-dimensional velocity profiles at the exit of diffuser 1 corresponding to a range of
non-uniformities (1 γ).
Figure 13.
Non-dimensional velocity profiles at the exit of diffuser 1 corresponding to a range of
non-uniformities (1 γ).
Aerospace 2022, 9, x 16 of 32
Figure 14. Effect of fan tip pressure ratio and bypass duct pressure losses on fan equivalent poly-
tropic efficiency, (from Kyprianidis [29]).
Table 3. Scaling parameters for DoE 1.
Design Objective ID 𝑭𝒊
𝟎 𝑭𝒊
𝑪𝒓
delivery duct loss, λ
CCA1
1 0.0928 0.66
non-uniformity, 1 γ 2 0.0193 0.07
bypass duct loss, λ
BPD
3 0.00333 0.0047
4.2. DoE 1 System Response Analysis and Optimisation
DoE 1 results are illustrated in Figure 15 on a design variable space representation.
For a cubic factorial design, the subsequent design space is three-dimensional. For sim-
plicity the three principal axes of this space are labelled by the level number (1–7) of a
variable instead of its actual value. The latter can be computed from the input data avail-
able in Table 2 with “min” corresponding to level 1 and “max” to level 7. Data are plotted
on a selection of planes in this volume. More specifically, Figure 15a–d illustrates data on
four intermediate planes of constant “height”. Each plot also includes an iso-surface of the
design objective at a slightly higher value than the observed minimum. This iso-surface
encloses a broader region of optimality. It is observed in Figure 15a that by increasing the
length of diffuser 1 the cooling system experiences a reduction in delivery duct loss. The
direct cause for this observation is the increase in the non-dimensional length of the dif-
fuser as shown in the loading chart of Figure 16. The observed trend is dominant in the
plot, i.e., there is a low level of interaction with the other two input variables (off-take
angle and height). Such a strong behaviour is also attributed to the following conditions:
(a) There is a positive velocity gradient in the streamwise direction due to the strut cur-
vature, i.e., the velocity near the leading edge is lower. Therefore, the pre-diffusion
requirement of the captured stream-tube is reduced near the leading edge.
(b) At the leading edge of the strut the off-take can fully exploit the dynamic pressure of
the bypass stream, a general advantage of total off-takes over flush off-takes.
It is also observed that the losses in the delivery duct are inversely proportional to
the off-take height. The observed output remains acceptable between planes 7 and 3, but
near plane 1, it reaches prohibitively high levels, i.e., λ 𝐹
 (red colour). This can
be explained by the increasing flow speed for a decreasing off-take throat area. The flow
velocity at the off-take throat is inversely proportional to its height (i.e., 𝑈∝1/𝐻) and the
pressure loss is proportional to the square of the velocity (Δ𝑃 ∝ 𝑈). Accordingly, the pres-
sure loss is inversely proportional to the cube of the off-take height (Δ𝑃 ∝ 1/𝐻).
Figure 15b indicates that the non-uniformity of the velocity profile at the exit of dif-
fuser 1, for most of the design space is maintained at (𝐹−𝐹
)𝑠<0.6 or 60% of the ac-
ceptable range. This corresponds to (1 −𝛾) < 0.05 which according to Figure 13 trans-
lates to highly uniform velocity profiles. Nonetheless, there is a region at low length, low
Figure 14.
Effect of fan tip pressure ratio and bypass duct pressure losses on fan equivalent polytropic
efficiency, (from Kyprianidis [29]).
Table 3. Scaling parameters for DoE 1.
Design Objective ID F0
iFCr
i
delivery duct loss, λCCA1 1 0.0928 0.66
non-uniformity, 1 γ2 0.0193 0.07
bypass duct loss, λBPD 3 0.00333 0.0047
Aerospace 2022,9, 130 16 of 32
4.2. DoE 1 System Response Analysis and Optimisation
DoE 1 results are illustrated in Figure 15 on a design variable space representation. For
a cubic factorial design, the subsequent design space is three-dimensional. For simplicity
the three principal axes of this space are labelled by the level number (1–7) of a variable
instead of its actual value. The latter can be computed from the input data available in
Table 2with “min” corresponding to level 1 and “max” to level 7. Data are plotted on a
selection of planes in this volume. More specifically, Figure 15a–d illustrates data on four
intermediate planes of constant “height”. Each plot also includes an iso-surface of the
design objective at a slightly higher value than the observed minimum. This iso-surface
encloses a broader region of optimality. It is observed in Figure 15a that by increasing the
length of diffuser 1 the cooling system experiences a reduction in delivery duct loss. The
direct cause for this observation is the increase in the non-dimensional length of the diffuser
as shown in the loading chart of Figure 16. The observed trend is dominant in the plot, i.e.,
there is a low level of interaction with the other two input variables (off-take angle and
height). Such a strong behaviour is also attributed to the following conditions:
(a)
There is a positive velocity gradient in the streamwise direction due to the strut
curvature, i.e., the velocity near the leading edge is lower. Therefore, the pre-diffusion
requirement of the captured stream-tube is reduced near the leading edge.
(b)
At the leading edge of the strut the off-take can fully exploit the dynamic pressure of
the bypass stream, a general advantage of total off-takes over flush off-takes.
Aerospace 2022, 9, x 17 of 32
off-take angle and intermediate off-take height, where non-uniformity increases beyond
the acceptable limit, i.e., (𝐹−𝐹
)𝑠1.0, which translates to (1− 𝛾)  0.07. There is a
narrow region of optimality for this design objective at low length (flush off-take), high
off-take angle and large off-take height (open throat). This is a highly loaded off-take de-
sign as is well reflected by the increased losses of the captured flow in Figure 15a. A rep-
resentative design from that region is illustrated in Figure 17 which shows that the in-
creased CCA duct loss is caused by a separated off-take. The long duct section between
the off-take and the target zone allows for the flow to reattach, while the convoluted flow
path created by this separation coincidentally leads to a symmetric velocity profile near
the target zone. The narrowness of this optimal region in the design space is product of
this unstable flowfield and hence is a non-attractive option in a subsequent design selec-
tion.
Furthermore, the effect of the three input variables to the bypass duct loss ) is
illustrated in Figure 15c. The plot indicates that both a flush off-take and a total off-take
may offer design options with minimal effect to the bypass duct flow (recall that one of
the main reasons for the use of flash off-takes in the bypass duct of commercial turbofan
engines is to protect the bypass duct flow from aerodynamic losses). Figure 15g and k
suggest that, in a bifurcation installation a submerged off-take near the leading edge (i.e.,
a total off-take) may in fact impact the bypass duct even less than a flash off-take. Even
more encouraging in the selection of a total off-take is the lack of observed interactions
with the other two input variables, i.e., the off-take’s angle and height as shown Figure
15g. The robust behaviour of total off-takes is also demonstrated by a considerably flat
distribution of non-uniformity across the angle-height design space (Figure 15f).
Figure 15. Model 1 (DoE 1) system responses. Full design variable space (ad), total offtakes (eh),
and flush offtakes (il).
Figure 15.
Model 1 (DoE 1) system responses. Full design variable space (
a
d
), total offtakes (
e
h
),
and flush offtakes (il).
Aerospace 2022,9, 130 17 of 32
Aerospace 2022, 9, x 18 of 32
Figure 16. Flow regimes for plane-walled, single-plane expansion diffusers [30].
Figure 17. Design of optimal non-uniformity in DoE 1, a1b6c7.
An optimal off-take design can now be selected from DoE 1. This is achieved via the
multi-objective function, 𝛹, as defined by Equation (7). The vector of scaling factors
𝑠(𝐹,𝐹) is constructed from the values of Table 3 while the vector of weighting factors
is set to 𝑤=1 (i.e., equal significance between all three design objectives). The resulted
contours are illustrated in Figure 15d,h,l. Red colour, i.e., 𝛹 1 indicates regions of low
interest, where one or more of the objective functions exceed their critical value, i.e., 𝐹
𝐹, and green-blue colour indicates regions of increased interest, where the multi-objec-
tive function tends to minimise. The iso-surface of 𝛹 = 0.35 highlights the existence of a
minimal value plateau at maximised length which confirms earlier indications of the su-
periority of total off-takes compared to flush off-takes. This plateau can be better observed
in the contours of 𝛹 in Figure 15h.
Table 4 presents a summary of the input variable settings and the corresponding ob-
jective function values (output) for the three single-objective optimisations as well as the
three-objective optimisation. The design with the minimum value of the multi-objective
function is length = 7, angle = 1 and height = 7, or simply design “a7b1c7”. This geometry
incurs a CCA duct loss of 𝜆 = 0.145, which can be expressed as (𝐹−𝐹
)𝑠=0.09 in
a scaled representation, or else 9% of the acceptable output range. The non-uniformity
of the selected optimal solution is (1 − 𝛾) = 0.0414 and corresponds to a value of 44%
of the acceptable output range. The reason why the scaled value of non-uniformity of the
selected design is relatively high is the existence of a group of aerodynamically unstable
designs of low uniformity in the family of the one shown in Figure 17. The imprints of
such designs can be distinctly observed in the criterion space of Figure 12a,b where a very
low number of design points spreads through an extremely low non-uniformity region,
i.e., 0.025 ≲ (1 − 𝛾) ≲ 0.05. Lastly, the value of the bypass duct loss coefficient of the op-
timal geometry is 𝜆 = 0.00377 corresponding to 32% of the acceptable output range.
The reason for this relatively high scaled value is that the constant offset of the bypass
Figure 16. Flow regimes for plane-walled, single-plane expansion diffusers [30].
It is also observed that the losses in the delivery duct are inversely proportional to
the off-take height. The observed output remains acceptable between planes 7 and 3, but
near plane 1, it reaches prohibitively high levels, i.e.,
λCCA1>FCr
1
(red colour). This can
be explained by the increasing flow speed for a decreasing off-take throat area. The flow
velocity at the off-take throat is inversely proportional to its height (i.e.,
U
1
/H
) and
the pressure loss is proportional to the square of the velocity (
PU2
). Accordingly, the
pressure loss is inversely proportional to the cube of the off-take height (P1/H3).
Figure 15b indicates that the non-uniformity of the velocity profile at the exit of diffuser
1, for most of the design space is maintained at
(F2F0
2)s2<
0.6 or 60% of the acceptable
range. This corresponds to
(1γ)<
0.05 which according to Figure 13 translates to highly
uniform velocity profiles. Nonetheless, there is a region at low length, low off-take angle
and intermediate off-take height, where non-uniformity increases beyond the acceptable
limit, i.e.,
(F2F0
2)s2
1.0, which translates to
(1γ)
0.07. There is a narrow region
of optimality for this design objective at low length (flush off-take), high off-take angle
and large off-take height (open throat). This is a highly loaded off-take design as is well
reflected by the increased losses of the captured flow in Figure 15a. A representative design
from that region is illustrated in Figure 17 which shows that the increased CCA duct loss
is caused by a separated off-take. The long duct section between the off-take and the
target zone allows for the flow to reattach, while the convoluted flow path created by this
separation coincidentally leads to a symmetric velocity profile near the target zone. The
narrowness of this optimal region in the design space is product of this unstable flowfield
and hence is a non-attractive option in a subsequent design selection.
Aerospace 2022, 9, x 18 of 32
Figure 16. Flow regimes for plane-walled, single-plane expansion diffusers [30].
Figure 17. Design of optimal non-uniformity in DoE 1, a1b6c7.
An optimal off-take design can now be selected from DoE 1. This is achieved via the
multi-objective function, 𝛹, as defined by Equation (7). The vector of scaling factors
𝑠(𝐹,𝐹) is constructed from the values of Table 3 while the vector of weighting factors
is set to 𝑤=1 (i.e., equal significance between all three design objectives). The resulted
contours are illustrated in Figure 15d,h,l. Red colour, i.e., 𝛹 1 indicates regions of low
interest, where one or more of the objective functions exceed their critical value, i.e., 𝐹
𝐹, and green-blue colour indicates regions of increased interest, where the multi-objec-
tive function tends to minimise. The iso-surface of 𝛹 = 0.35 highlights the existence of a
minimal value plateau at maximised length which confirms earlier indications of the su-
periority of total off-takes compared to flush off-takes. This plateau can be better observed
in the contours of 𝛹 in Figure 15h.
Table 4 presents a summary of the input variable settings and the corresponding ob-
jective function values (output) for the three single-objective optimisations as well as the
three-objective optimisation. The design with the minimum value of the multi-objective
function is length = 7, angle = 1 and height = 7, or simply design “a7b1c7”. This geometry
incurs a CCA duct loss of 𝜆 = 0.145, which can be expressed as (𝐹−𝐹
)𝑠=0.09 in
a scaled representation, or else 9% of the acceptable output range. The non-uniformity
of the selected optimal solution is (1 − 𝛾) = 0.0414 and corresponds to a value of 44%
of the acceptable output range. The reason why the scaled value of non-uniformity of the
selected design is relatively high is the existence of a group of aerodynamically unstable
designs of low uniformity in the family of the one shown in Figure 17. The imprints of
such designs can be distinctly observed in the criterion space of Figure 12a,b where a very
low number of design points spreads through an extremely low non-uniformity region,
i.e., 0.025 ≲ (1 − 𝛾) ≲ 0.05. Lastly, the value of the bypass duct loss coefficient of the op-
timal geometry is 𝜆 = 0.00377 corresponding to 32% of the acceptable output range.
The reason for this relatively high scaled value is that the constant offset of the bypass
Figure 17. Design of optimal non-uniformity in DoE 1, a1b6c7.
Furthermore, the effect of the three input variables to the bypass duct loss
(λBPD)
is
illustrated in Figure 15c. The plot indicates that both a flush off-take and a total off-take
may offer design options with minimal effect to the bypass duct flow (recall that one of
the main reasons for the use of flash off-takes in the bypass duct of commercial turbofan
Aerospace 2022,9, 130 18 of 32
engines is to protect the bypass duct flow from aerodynamic losses). Figure 15g and k
suggest that, in a bifurcation installation a submerged off-take near the leading edge (i.e., a
total off-take) may in fact impact the bypass duct even less than a flash off-take. Even more
encouraging in the selection of a total off-take is the lack of observed interactions with the
other two input variables, i.e., the off-take’s angle and height as shown Figure 15g. The
robust behaviour of total off-takes is also demonstrated by a considerably flat distribution
of non-uniformity across the angle-height design space (Figure 15f).
An optimal off-take design can now be selected from DoE 1. This is achieved via
the multi-objective function,
Ψ
, as defined by Equation (7). The vector of scaling factors
siF0
i,FCr
i
is constructed from the values of Table 3while the vector of weighting factors
is set to
w=
1 (i.e., equal significance between all three design objectives). The resulted
contours are illustrated in Figure 15d,h,l. Red colour, i.e.,
Ψ
1 indicates regions of
low interest, where one or more of the objective functions exceed their critical value, i.e.,
Fi>FCr
i
, and green-blue colour indicates regions of increased interest, where the multi-
objective function tends to minimise. The iso-surface of
Ψ=
0.35 highlights the existence
of a minimal value plateau at maximised length which confirms earlier indications of
the superiority of total off-takes compared to flush off-takes. This plateau can be better
observed in the contours of Ψin Figure 15h.
Table 4presents a summary of the input variable settings and the corresponding
objective function values (output) for the three single-objective optimisations as well as the
three-objective optimisation. The design with the minimum value of the multi-objective
function is length = 7, angle = 1 and height = 7, or simply design “a7b1c7”. This geometry
incurs a CCA duct loss of
λCCA1
= 0.145, which can be expressed as
(F1F0
1)s1=
0.09 in
a scaled representation, or else 9% of the acceptable output range. The non-uniformity
of the selected optimal solution is
(1γ)=
0.0414 and corresponds to a value of 44% of
the acceptable output range. The reason why the scaled value of non-uniformity of the
selected design is relatively high is the existence of a group of aerodynamically unstable
designs of low uniformity in the family of the one shown in Figure 17. The imprints of
such designs can be distinctly observed in the criterion space of Figure 12a,b where a very
low number of design points spreads through an extremely low non-uniformity region, i.e.,
0.025
.(1γ).
0.05. Lastly, the value of the bypass duct loss coefficient of the optimal
geometry is
λBPD
= 0.00377 corresponding to 32% of the acceptable output range. The
reason for this relatively high scaled value is that the constant offset of the bypass duct loss,
i.e.,
(F0
3)0=
0.0022, is relatively small compared to the minimum observation
(F3)min
, i.e.,
(F0
3)0/(F3)min
0.5. Nevertheless, this is only 11% higher than the minimum observation
as shown on Table 4.
Table 4. Characteristics of the optimal geometries (inputs and outputs).
Input: Output: (FiF0
i)si
Optimised for: (Design ID) λCCA1 1γ λBPD
CCA loss, λCCA1 (F1)a7b7c7 4% 65% 24%
Non-uniformity, 1 γ(F2)a1b6c7 88% 10% 61%
BPD loss, λBPD (F3)a7b3c5 14% 69% 21%
All three (F13)a7b1c7 9% 44% 32%
Figure 18 illustrates contours of normalised velocity magnitude and total pressure on
the selected optimal geometry as well as a plot of the profile of velocity magnitude at the
target zone. This profile, alongside the detailed description of turbulence, including the
Reynolds stresses for the RSM model, are fed at the inlet boundary of model 2 to support
DoE 2 and DoE 3 that follow.
Aerospace 2022,9, 130 19 of 32
Aerospace 2022, 9, x 19 of 32
duct loss, i.e., (𝐹
)′ = 0.0022, is relatively small compared to the minimum observation
(𝐹), i.e., (𝐹
)′/(𝐹) ~ 0.5. Nevertheless, this is only 11% higher than the mini-
mum observation as shown on Table 4.
Table 4. Characteristics of the optimal geometries (inputs and outputs).
Input: Output: (𝑭𝒊−𝑭𝒊
𝟎)𝒔𝒊
Optimised for: (Design ID) λ
CCA1
1 γ λ
BPD
CCA loss, λ
CCA1
(𝐹) a7b7c7 4% 65% 24%
Non-uniformity, 1
γ
(𝐹) a1b6c7 88% 10% 61%
BPD loss, λ
BPD
(𝐹) a7b3c5 14% 69% 21%
All three (𝐹) a7b1c7 9% 44% 32%
Figure 18 illustrates contours of normalised velocity magnitude and total pressure
on the selected optimal geometry as well as a plot of the profile of velocity magnitude at
the target zone. This profile, alongside the detailed description of turbulence, including
the Reynolds stresses for the RSM model, are fed at the inlet boundary of model 2 to sup-
port DoE 2 and DoE 3 that follow.
Figure 18. Selected design of DoE 1, a7b1c7.
Figure 19 shows that for a broad range in the design space near the optimal region
(low to moderate loss geometries) the flow profile delivered at the exit of diffuser 1 is
virtually unchanged. Significant changes in the profile are observed in regions of high loss
which can be understood by the mechanisms described in Figure 17. Such designs are of
low value, hence, it can be claimed that the presented profile is representative of model 1
and indicates that fractionalisation of the problem to model 1 and model 2 should have a
small effect to the outcome of this optimisation.
Figure 19. Relative variation of the axial velocity profile at diffuser 1 exit in DoE 1.
Figure 18. Selected design of DoE 1, a7b1c7.
Figure 19 shows that for a broad range in the design space near the optimal region
(low to moderate loss geometries) the flow profile delivered at the exit of diffuser 1 is
virtually unchanged. Significant changes in the profile are observed in regions of high loss
which can be understood by the mechanisms described in Figure 17. Such designs are of
low value, hence, it can be claimed that the presented profile is representative of model 1
and indicates that fractionalisation of the problem to model 1 and model 2 should have a
small effect to the outcome of this optimisation.
Aerospace 2022, 9, x 19 of 32
duct loss, i.e., (𝐹
)′ = 0.0022, is relatively small compared to the minimum observation
(𝐹), i.e., (𝐹
)′/(𝐹) ~ 0.5. Nevertheless, this is only 11% higher than the mini-
mum observation as shown on Table 4.
Table 4. Characteristics of the optimal geometries (inputs and outputs).
Input: Output: (𝑭𝒊−𝑭𝒊
𝟎)𝒔𝒊
Optimised for: (Design ID) λ
CCA1
1 γ λ
BPD
CCA loss, λ
CCA1
(𝐹) a7b7c7 4% 65% 24%
Non-uniformity, 1
γ
(𝐹) a1b6c7 88% 10% 61%
BPD loss, λ
BPD
(𝐹) a7b3c5 14% 69% 21%
All three (𝐹) a7b1c7 9% 44% 32%
Figure 18 illustrates contours of normalised velocity magnitude and total pressure
on the selected optimal geometry as well as a plot of the profile of velocity magnitude at
the target zone. This profile, alongside the detailed description of turbulence, including
the Reynolds stresses for the RSM model, are fed at the inlet boundary of model 2 to sup-
port DoE 2 and DoE 3 that follow.
Figure 18. Selected design of DoE 1, a7b1c7.
Figure 19 shows that for a broad range in the design space near the optimal region
(low to moderate loss geometries) the flow profile delivered at the exit of diffuser 1 is
virtually unchanged. Significant changes in the profile are observed in regions of high loss
which can be understood by the mechanisms described in Figure 17. Such designs are of
low value, hence, it can be claimed that the presented profile is representative of model 1
and indicates that fractionalisation of the problem to model 1 and model 2 should have a
small effect to the outcome of this optimisation.
Figure 19. Relative variation of the axial velocity profile at diffuser 1 exit in DoE 1.
Figure 19. Relative variation of the axial velocity profile at diffuser 1 exit in DoE 1.
5. Model 2 Results and Discussion
5.1. DoE 2 and DoE 3 Data Sampling and Objective Function Conditioning
As mentioned in the introduction, model 2 is a variation of the s-duct that was previ-
ously studied by Spanelis et al. [
9
]. In their work they mentioned that the duct losses are
insignificant compared to the HX loss and accordingly they based their optimisation on the
uniformity criterion alone. However, the present work considers a different parametrisa-
tion method which allows for even higher concentration of the aerodynamic loading near
the HX, via an aggressive manifold. The mission of this aggressive manifold is to diffuse
the flow at an area ratio of
A4/A1=h4/h1=
6 within a non-dimensional duct length of
L14/h1=
7.9 for s-duct 1 and
L14/h1=
5.5 for s-duct 2. According to the loading chart
of Figure 16 both these s-ducts are severely loaded and would strongly separate were it not
for the beneficial upstream influence of the HX.
Below it will be shown that this separation can in fact be avoided even in this extreme
loading scenario, however, this is only achievable at the cost of an increased aerodynamic
loss. For this reason, the pressure loss can no longer be omitted but it becomes an essential
design objective.
To better understand the physical importance of the selected parametrisation method
and subsequent selection of the design objectives, Figure 20 presents an aerodynamic
Aerospace 2022,9, 130 20 of 32
comparison of two extreme manifold designs. Example A (left) is representative of a
uniform loading distribution in the manifold and Example B (right) represents geometries
that concentrate the loading at the exit, i.e., the front face of the HX. The figure shows
contours of velocity
U
, turbulence kinetic energy
(k)
, production of k(
Aerospace 2022, 9, x 20 of 32
5. Model 2 Results and Discussion
5.1. DoE 2 and DoE 3 Data Sampling and Objective Function Conditioning
As mentioned in the introduction, model 2 is a variation of the s-duct that was pre-
viously studied by Spanelis et al. [9]. In their work they mentioned that the duct losses are
insignificant compared to the HX loss and accordingly they based their optimisation on
the uniformity criterion alone. However, the present work considers a different parametri-
sation method which allows for even higher concentration of the aerodynamic loading
near the HX, via an aggressive manifold. The mission of this aggressive manifold is to
diffuse the flow at an area ratio of 𝐴𝐴=
=
6 within a non-dimensional duct
length of 𝐿 =
7.9 for s-duct 1 and 𝐿 =
5.5 for s-duct 2. According to the
loading chart of Figure 16 both these s-ducts are severely loaded and would strongly sep-
arate were it not for the beneficial upstream influence of the HX.
Below it will be shown that this separation can in fact be avoided even in this extreme
loading scenario, however, this is only achievable at the cost of an increased aerodynamic
loss. For this reason, the pressure loss can no longer be omitted but it becomes an essential
design objective.
To better understand the physical importance of the selected parametrisation method
and subsequent selection of the design objectives, Figure 20 presents an aerodynamic
comparison of two extreme manifold designs. Example A (left) is representative of a uni-
form loading distribution in the manifold and Example B (right) represents geometries
that concentrate the loading at the exit, i.e., the front face of the HX. The figure shows
contours of velocity 𝑈, turbulence kinetic energy (𝑘), production of 𝑘 (
P
) and total
pressure (𝑃), all normalised by suitable reference values to facilitate the comparison.
Figure 20. Aerodynamic comparison of two extreme manifold designs in terms of velocity (a,b),
TKE (cd), production of TKE (ef), and total pressure (gh).
It can be observed that Example A is not an ideal approach and as suggested by the
loading chart of Figure 16 a large flow separation forms in the s-duct. This separation is
reflected on both the contours and streamlines of velocity magnitude of Figure 20a as well
as the contours of turbulence kinetic energy of Figure 20c. Note that the latter increases
considerably along the mixing layer of the separation wake of Example A. On the other
hand, the flow in Example B remains attached throughout the s-duct as there is no signif-
icant cross-sectional area variation. This leaves almost all the diffusion and turning to take
place within an extremely short distance immediately in the front of the HX. Nonetheless,
this task is made possible due to the high blockage imposed by the HX, but as indicated
) and total pressure
(P), all normalised by suitable reference values to facilitate the comparison.
Aerospace 2022, 9, x 20 of 32
5. Model 2 Results and Discussion
5.1. DoE 2 and DoE 3 Data Sampling and Objective Function Conditioning
As mentioned in the introduction, model 2 is a variation of the s-duct that was pre-
viously studied by Spanelis et al. [9]. In their work they mentioned that the duct losses are
insignificant compared to the HX loss and accordingly they based their optimisation on
the uniformity criterion alone. However, the present work considers a different parametri-
sation method which allows for even higher concentration of the aerodynamic loading
near the HX, via an aggressive manifold. The mission of this aggressive manifold is to
diffuse the flow at an area ratio of 𝐴𝐴=
=
6 within a non-dimensional duct
length of 𝐿 =
7.9 for s-duct 1 and 𝐿 =
5.5 for s-duct 2. According to the
loading chart of Figure 16 both these s-ducts are severely loaded and would strongly sep-
arate were it not for the beneficial upstream influence of the HX.
Below it will be shown that this separation can in fact be avoided even in this extreme
loading scenario, however, this is only achievable at the cost of an increased aerodynamic
loss. For this reason, the pressure loss can no longer be omitted but it becomes an essential
design objective.
To better understand the physical importance of the selected parametrisation method
and subsequent selection of the design objectives, Figure 20 presents an aerodynamic
comparison of two extreme manifold designs. Example A (left) is representative of a uni-
form loading distribution in the manifold and Example B (right) represents geometries
that concentrate the loading at the exit, i.e., the front face of the HX. The figure shows
contours of velocity 𝑈, turbulence kinetic energy (𝑘), production of 𝑘 (
P
) and total
pressure (𝑃), all normalised by suitable reference values to facilitate the comparison.
Figure 20. Aerodynamic comparison of two extreme manifold designs in terms of velocity (a,b),
TKE (cd), production of TKE (ef), and total pressure (gh).
It can be observed that Example A is not an ideal approach and as suggested by the
loading chart of Figure 16 a large flow separation forms in the s-duct. This separation is
reflected on both the contours and streamlines of velocity magnitude of Figure 20a as well
as the contours of turbulence kinetic energy of Figure 20c. Note that the latter increases
considerably along the mixing layer of the separation wake of Example A. On the other
hand, the flow in Example B remains attached throughout the s-duct as there is no signif-
icant cross-sectional area variation. This leaves almost all the diffusion and turning to take
place within an extremely short distance immediately in the front of the HX. Nonetheless,
this task is made possible due to the high blockage imposed by the HX, but as indicated
Figure 20.
Aerodynamic comparison of two extreme manifold designs in terms of velocity (
a
,
b
),
TKE (c,d), production of TKE (e,f), and total pressure (g,h).
It can be observed that Example A is not an ideal approach and as suggested by the
loading chart of Figure 16 a large flow separation forms in the s-duct. This separation
is reflected on both the contours and streamlines of velocity magnitude of Figure 20a
as well as the contours of turbulence kinetic energy of Figure 20c. Note that the latter
increases considerably along the mixing layer of the separation wake of Example A. On
the other hand, the flow in Example B remains attached throughout the s-duct as there
is no significant cross-sectional area variation. This leaves almost all the diffusion and
turning to take place within an extremely short distance immediately in the front of the HX.
Nonetheless, this task is made possible due to the high blockage imposed by the HX, but as
indicated by Figure 20f this highly concentrated loading brings about a large increase in
the local shear stress, which is the source of an increased production of turbulence kinetic
energy (Figure 20d) and consequently of an increase in total pressure loss (Figure 20h),
hereinafter referred to as “HX entry loss”. While most of this loss takes place inside the HX,
a significant proportion of it occurs inside the manifold (see Figure 21).
Aerospace 2022, 9, x 21 of 32
by Figure 20f this highly concentrated loading brings about a large increase in the local
shear stress, which is the source of an increased production of turbulence kinetic energy
(Figure 20d) and consequently of an increase in total pressure loss (Figure 20h), hereinaf-
ter referred to as “HX entry loss”. While most of this loss takes place inside the HX, a
significant proportion of it occurs inside the manifold (see Figure 21).
Figure 21. Extreme production of turbulence kinetic energy around the manifold-HX interface.
The dominant loss in the current system takes place at the core of the HX and is re-
ferred to as “HX core loss”. Nonetheless, the HX entry loss may grow significantly, even
to a comparable level to the HX core loss depending on the specific values of the input
variables.
With the above in mind, the evaluation of the pressure loss of model 2 is not imple-
mented directly at the target zone, which is at the HX inlet, but the evaluation is trans-
ferred at the HX exit plane which ensures that the HX entry loss is fully accounted for.
More specifically, the total pressure loss coefficient in model 2 (DoE 2 and DoE 3) is cal-
culated as:
𝜆 =𝑃
−𝑃
𝑃
−𝑝+𝑃
−𝑃
𝑃
−𝑝
(10)
Furthermore, flow separation and recirculation are common threats in duct systems,
not only because they contribute strongly to the total pressure loss, but also due to vibra-
tion, noise and component fatigue that it may also lead to. Conversely, reduction of this
type of loss in the current system, hereafter referred to as “delivery duct loss”, is predom-
inantly achievable at the expense of an increased HX entry loss. As shown above the de-
livery duct loss and the HX entry loss are in strong conflict with one another; therefore,
minimising their sum does not necessarily guarantee the minimisation of both individual
contributions. Consequently, the pressure loss coefficient of Equation (10) is not suitable
to indicate flow separation in the manifold. For this reason, the kinetic energy ratio of
Equation (6) is employed as a separate design objective. This parameter indicates varia-
tions in the overall turbulence level in the delivery duct, therefore, it can identify the pres-
ence of significant flow separation. In summary, the design exploration of model 2 con-
siders the following objectives:
(1) Total pressure loss across the system.
(2) Non-uniformity of the velocity magnitude at the inlet of the HX.
(3) Kinetic energy ratio in the delivery duct.
Additionally, the multi-objective function of Equation (7) is evaluated based on the
vector of scaling factors 𝑠(𝐹,𝐹) available on Table 5. More specifically, the utopia
point for non-uniformity, 𝐹
, and turbulence kinetic energy, 𝐹
, are calculated based on
the observed minima from Equation (9). On the other hand, the HX loss coefficient is a
fixed boundary condition and given that the HX core flow is highly uniform in the ex-
plored design space, the HX core loss is not likely to vary significantly. In this way, the
utopia point for the total pressure loss, 𝐹
is calculated from equation (8) by setting the
HX loss as a fixed contribution, i.e., (𝐹
)′ = 𝜆 =


=0.32. Furthermore, the critical
value for total pressure loss has been set to 𝐹
 =0.5 for both DoE 2 and DoE 3 ensuring
that the HX core loss will remain as the dominant contribution in the acceptable design
Figure 21. Extreme production of turbulence kinetic energy around the manifold-HX interface.
The dominant loss in the current system takes place at the core of the HX and is
referred to as “HX core loss”. Nonetheless, the HX entry loss may grow significantly,
even to a comparable level to the HX core loss depending on the specific values of the
input variables.
Aerospace 2022,9, 130 21 of 32
With the above in mind, the evaluation of the pressure loss of model 2 is not imple-
mented directly at the target zone, which is at the HX inlet, but the evaluation is transferred
at the HX exit plane which ensures that the HX entry loss is fully accounted for. More
specifically, the total pressure loss coefficient in model 2 (DoE 2 and DoE 3) is calculated as:
λCCA2=e
P0e
P4
e
Pse
ps
+e
P4e
P5
e
Pse
ps
(10)
Furthermore, flow separation and recirculation are common threats in duct systems,
not only because they contribute strongly to the total pressure loss, but also due to vibration,
noise and component fatigue that it may also lead to. Conversely, reduction of this type of
loss in the current system, hereafter referred to as “delivery duct loss”, is predominantly
achievable at the expense of an increased HX entry loss. As shown above the delivery duct
loss and the HX entry loss are in strong conflict with one another; therefore, minimising
their sum does not necessarily guarantee the minimisation of both individual contributions.
Consequently, the pressure loss coefficient of Equation (10) is not suitable to indicate
flow separation in the manifold. For this reason, the kinetic energy ratio of Equation (6)
is employed as a separate design objective. This parameter indicates variations in the
overall turbulence level in the delivery duct, therefore, it can identify the presence of
significant flow separation. In summary, the design exploration of model 2 considers the
following objectives:
(1)
Total pressure loss across the system.
(2)
Non-uniformity of the velocity magnitude at the inlet of the HX.
(3)
Kinetic energy ratio in the delivery duct.
Additionally, the multi-objective function of Equation (7) is evaluated based on the
vector of scaling factors
siF0
i,FCr
i
available on Table 5. More specifically, the utopia point
for non-uniformity,
F0
2
, and turbulence kinetic energy,
F0
3
, are calculated based on the
observed minima from Equation (9). On the other hand, the HX loss coefficient is a fixed
boundary condition and given that the HX core flow is highly uniform in the explored
design space, the HX core loss is not likely to vary significantly. In this way, the utopia
point for the total pressure loss,
F0
1
is calculated from equation (8) by setting the HX loss
as a fixed contribution, i.e.,
(F0
1)0=λHX =e
P4e
P5
e
Pse
ps=
0.32. Furthermore, the critical value
for total pressure loss has been set to
FCr
1=
0.5 for both DoE 2 and DoE 3 ensuring that
the HX core loss will remain as the dominant contribution in the acceptable design range.
Moreover, the critical values of non-uniformity and kinetic energy ratio are empirically set
to 0.2 and 0.1 respectively based on the observed ranges of DoE 2 and DoE 3. A complete
summary of the scaling coefficients of DoE 2 and DoE 3 is available on Table 5.
Table 5. Scaling parameters for DoE 2 and DoE 3.
Design Objective ID F0
iFCr
i
DoE 2
total pressure loss, λCCA2 1 0.383 0.5
non-uniformity, 1 γ2 0.046 0.2
kinetic energy ratio, k* 3 0.036 0.1
DoE 3
total pressure loss, λCCA2 1 0.387 0.5
non-uniformity, 1 γ2 0.061 0.2
kinetic energy ratio, k* 3 0.030 0.1
Finally, Figures 22 and 23 illustrate the results of DoE 2 and DoE 3, respectively, on
a projected criterion space using a similar format to that of Figure 12. It can be observed
that the training points are well distributed in the criterion space except for the non-
uniformity which presents a significant drop in density in the direction of the utopia point.
Aerospace 2022,9, 130 22 of 32
Furthermore, the ranges of the observed output criteria of DoE 2 are relatively narrow which
indicates weak response level to the input variables compared to DoE 3. This is attributed to
the relatively low vertical to axial displacement ratio of s-duct 2
(Y/X= 0.77)
compared
to s-duct 1 (
Y/
X= 1.63). More specifically, according to the current parametrisation of
the Bezier curves (i.e., varying only the x-coordinates of the intermediate control points) a
curve with elevation X/X = 0 would be unresponsive to the input variables. However,
as the value of
Y/
Xincreases the interactions between the two curves also increase,
leading to an increase in the degrees of freedom of the aerodynamic definition of the
duct geometry.
Aerospace 2022, 9, x 22 of 32
range. Moreover, the critical values of non-uniformity and kinetic energy ratio are empir-
ically set to 0.2 and 0.1 respectively based on the observed ranges of DoE 2 and DoE 3. A
complete summary of the scaling coefficients of DoE 2 and DoE 3 is available on Table 5.
Table 5. Scaling parameters for DoE 2 and DoE 3.
Design Objective ID 𝑭𝒊
𝟎 𝑭𝒊
𝑪𝒓
DoE 2
total pressure loss, λ
CCA2
1 0.383 0.5
non-uniformity, 1 γ 2 0.046 0.2
kinetic energy ratio, k* 3 0.036 0.1
DoE 3
total pressure loss, λ
CCA2
1 0.387 0.5
non-uniformity, 1 γ 2 0.061 0.2
kinetic energy ratio, k* 3 0.030 0.1
Finally, Figures 22 and 23 illustrate the results of DoE 2 and DoE 3, respectively, on
a projected criterion space using a similar format to that of Figure 12. It can be observed
that the training points are well distributed in the criterion space except for the non-uni-
formity which presents a significant drop in density in the direction of the utopia point.
Furthermore, the ranges of the observed output criteria of DoE 2 are relatively narrow
which indicates weak response level to the input variables compared to DoE 3. This is
attributed to the relatively low vertical to axial displacement ratio of s-duct 2 (∆𝑌/∆𝛸 =
0.77) compared to s-duct 1 (∆𝑌/∆𝛸 = 1.63). More specifically, according to the current par-
ametrisation of the Bezier curves (i.e., varying only the x-coordinates of the intermediate
control points) a curve with elevation ∆𝑌/∆𝛸 = 0 would be unresponsive to the input var-
iables. However, as the value of ∆𝑌/∆𝛸 increases the interactions between the two curves
also increase, leading to an increase in the degrees of freedom of the aerodynamic defini-
tion of the duct geometry.
To construct a more efficient parametric model and overcome the issues mentioned
above, one must first understand which regions of the explored design space offer im-
proved aerodynamic properties. The following analysis is dedicated to this goal.
(a) (b) (c)
Figure 22. DoE 2 results—Projected 3D criterion space into three individual 2D components (ac).
Figure 22. DoE 2 results—Projected 3D criterion space into three individual 2D components (ac).
Aerospace 2022, 9, x 23 of 32
(a) (b) (c)
Figure 23. DoE 3 results—Projected 3D criterion space into three individual 2D components (ac).
5.2. DoE 2 and DoE 3 System Response Analysis
Contour plots of the scaled objective functions in the design-variable spaces of DoE
2 and DoE 3 are illustrated in Figure 24 following the same format as previously described
for Figure 15a–d. It is observed that all three objective functions conflict with one another
and each design objective is minimised at a different region. The loss coefficient minimises
at coordinates 𝑃
= 1, 𝑃
= 1 and 𝑃
= 4 in DoE 2. For simplicity, this design is referred
to as “a1b1c4” (Figure 15a). Similarly, the minimum loss coefficient in DoE 3 is observed
at design “a5b5c6” (Figure 15b). Furthermore, the non-uniformity minimises near the op-
posite side of the cubic volume, i.e., design a7b5c6 in DoE 2 (Figure 15c) and design a6b7c6
in DoE 3 (Figure 15d). Additionally, the kinetic energy ratio minimises at design a4b7c1
in DoE 2 (Figure 15e) and design a5b7c1 in DoE 3 (Figure 15f). Finally, the three design
objectives are combined into a multi-objective function which minimises at design a4b7c3
in DoE 2 (Figure 15g) and a5b7c5 in DoE 3 (Figure 15h).
Note that there are certain regions in both the DoE 2 and DoE 3 design spaces that
have been blanked out. The reason is that CFD data are unavailable at those regions due
to some failure in the geometry/mesh generation or the numerical simulation. The most
typical reason for failure of an instance execution is that the input variables lead to an
unphysical shape such as a self-intersecting geometry. A less frequent scenario is when
the geometry and mesh were in fact successfully generated but the numerical simulation
did not achieve convergence. Usually, such designs are of low aerodynamic interest and
failing to execute them is in fact beneficial as it saves computational resources.
Figure 23. DoE 3 results—Projected 3D criterion space into three individual 2D components (ac).
To construct a more efficient parametric model and overcome the issues mentioned
above, one must first understand which regions of the explored design space offer improved
aerodynamic properties. The following analysis is dedicated to this goal.
5.2. DoE 2 and DoE 3 System Response Analysis
Contour plots of the scaled objective functions in the design-variable spaces of DoE 2
and DoE 3 are illustrated in Figure 24 following the same format as previously described
for Figure 15a–d. It is observed that all three objective functions conflict with one another
and each design objective is minimised at a different region. The loss coefficient minimises
at coordinates
PC
1
= 1,
PC
2
= 1 and
PD
1
= 4 in DoE 2. For simplicity, this design is referred