Proceedings of GPPS Forum 18 Global Power and Propulsion Society

Montreal, 7th-9

th May 2018

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This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License CC-BY-NC 4.0

GPPS-2018-0143

AEROTHERMODYNAMIC ANALYSIS OF HEAT RECUPERATION BY A NOZZLE-INTEGRATED THERMOELECTRIC GENERATOR

IN A TURBOFAN

Dragan Kožulović Department Automotive and

Aerospace Hamburg University of

Applied Sciences dragan.kozulovic@haw-

hamburg.de Hamburg, Germany

Christoph Bode Institute of Jet Propulsion and

Turbomachinery Technische Universität

Braunschweig chr.bode@tu-braunschweig.de

Braunschweig, Germany

Jens Friedrichs Institute of Jet Propulsion and

Turbomachinery Technische Universität

Braunschweig j.friedrichs@ifas.tu-braunschweig.de

Braunschweig, Germany

ABSTRACT

To harvest the waste energy from the engine exhaust, a

thermoelectric generator is applied to the core nozzle of a

turbofan engine. In this way, a heat flux is generated from

the inner to the outer side of the core nozzle. To assess the

corresponding effect to the boundary layers at both sides and

to the engine performance, RANS-simulations with

prescribed heat flux at both core nozzle surfaces are

conducted. For the cruise and top of climb conditions, heat

flux of 30 kW and for the takeoff of 45 kW were applied.

Only 10% of the heat flux is converted into electrical power.

These simulations have been compared to simulations with

adiabatic walls. The velocity profiles of both boundary layers

(inner and outer side of the core nozzle) and in the shear

layer downstream the nozzle remain almost constant,

whereas density and temperature profiles change

considerably. The overall impact at thrust and propulsive

efficiency is negligible.

INTRODUCTION

The exhaust jet of a modern turbofan engine still has a

very large temperature (500 − 800 𝐾), providing an

opportunity for energy harvesting. In the present approach,

this is done by a TEG (thermoelectric generator), which is

placed in the core nozzle, cf. Figs. 1 and 2. A sketch of the

TEG and its installation in the nozzle is given in Fig. 3. The

TEG redistributes the heat from the core to the bypass flow

and directly converts a part of the corresponding heat flux

into electrical energy. Due to large velocities and large

temperature difference, remarkable amounts of heat flux can

be achieved, as demonstrated in (Bode et al. 2017a). To

assess the TEG performance in future engines, a successor of

current mid-size engines (as used for Airbus A320neo and

Boeing 737 MAX) with an entry into service in 2030 has

been selected as reference. Reasonable assumptions for the

corresponding technology improvements have been made for

both the engine and TEG parameters. The assessment of

TEG has been done on engine cycle and whole-aircraft level

in previous work, cf. (Bode et al. 2017a). Further, some

detailed study on TEG design has been conducted in

Ziolkowski et al. 2016) and (Bode et al. 2017b). Now, the

effect of heat transfer at the nozzle boundary layers and

engine performance is investigated in more detail.

Figure 1: Velocity contours in the meridional plane

As a key parameter of previous investigations, the power

density of the TEG system has to be larger than 149 𝑊/𝑘𝑔

in a favourable scenario and larger than 379 𝑊/𝑘𝑔 in an

unfavourable case, in order to provide fuel savings for a

typical flight mission. In the present work, maximum electric

power output is 4.5 𝑘𝑊 at take-off, which translates into

2

maximum TEG system weight of 30.2 𝑘𝑔 or 11.9 𝑘𝑔 per

engine, respectively. Commercially available TEGs have

power densities up to 500 𝑊/𝑘𝑔 at module level. This value

is reduced at system level, but it still fits well into the

aforementioned range, making the TEG a viable solution for

engine exergy harvesting.

Figure 2: Pressure contours and core nozzle

Figure 3: TEG sketch and installation in the nozzle

As an example for redistribution of heat from the core

exhaust to the bypass flow, the intercooled-recuperated

engine can be mentioned. The beneficial effect at the engine

efficiency is well known, however, large amounts of heat

flux (order of magnitude: 1 − 10 𝑀𝑊) are required, leading

to large and heavy heat exchangers. In contrast, the present

study is focused at much smaller amounts of heat flux

(< 50 𝑘𝑊) in combination with thermoelectric generation,

hence avoiding the excessive heat exchanger weight. On the

one hand, much smaller impact at the engine performance is

expected, due to the small amount of heat flux. On the other

hand, the heat flux is applied to the outer boundary layer of

the core nozzle, which is exposed to adverse pressure

gradients and large thickening. Heating this region, even with

small amounts of energy, may have significantly different

effects than an intercooler, which is heating the freestream

region of the bypass flow.

The research on boundary layer heating has attracted

much attention, especially for high speed flows, where

compressibility leads to considerable increase of

temperature. In general, heating leads to a reduction of skin

friction, which may be a rather small effect, but also a very

large one, depending on many parameters: Mach number,

Reynolds number, pressure gradient, turbulence intensity,

beginning of the thermal boundary layer (with respect to the

velocity boundary layer), etc. Following authors give a

comprehensive overview of corresponding methods, where

the last one by White seems to address the most elaborated

approach:

Shapiro (1954, pp. 1107-1117) only flat plates

Hoerner (1965, chapter 17, pp. 3-7)

Schlichting and Gersten (2006, pp. 616-623)

Kays and Crawford (1987, pp. 286, 305-310)

White (1999, pp. 539-562)

Boundary layer heating has been successfully applied to

different objects by (Kramer et al. 1999): a wind tunnel

model, a flying testbad and the fuselage of a business jet.

Although a considerable drag reduction was achieved, the

heating equipment was heavy and it required an extra source

of energy. This disadvantage is not present for the case of jet

engines, since the core flow already provides a large amount

of (wasted) heat.

METHODOLOGY

CFD solver

The parallel CFD-solver TRACE of DLR Cologne has

been applied, cf. Nürnberger (2004), Kügeler (2005) and

Becker et al. (2010). In this solver, the three-dimensional

Reynolds-averaged Navier-Stokes equations are numerically

solved on multi-block meshes by a finite volume technique.

Only structured meshes are used in this study. In addition to

the Navier-Stokes equations, the equation of state for ideal

gases and Fourier’s law for heat conduction are applied.

Here, air with constant heat capacity is used in the complete

computational domain.

The convective fluxes are discretized by the 2nd

order

TVD upwind scheme of Roe (1981) and the diffusive fluxes

by a central differencing scheme. The cross derivatives,

which appear when converting from curvilinear to Cartesian

frame of reference, are taken into account (full Navier-Stokes

approach). Only steady simulations have been performed,

using an implicit predictor-corrector time integration scheme.

The turbulence is modelled by the two-equation SST

model of Menter et al. (2003), where the excessive

turbulence production is prevented by limiting the production

term of the turbulent kinetic energy equation to be less than

10 times the corresponding destruction term. This model is

3

based on the Boussinesq approximation, hence the turbulent

shear stress and turbulent heat flux are calculated by eddy

viscosity and turbulent Prandtl number, the latter being

assumed constant. Due to the high Reynolds number and

high turbulence intensity in both internal engine flows, the

boundary layers have been assumed to be fully turbulent.

Hence, transition has not been modelled.

Non-reflecting boundary conditions by Saxer and Giles

(1993) have been applied to the inlet and outlet boundaries of

following types:

1) Farfield with constant stagnation pressure and

stagnation temperature as well as flight Mach number

according to the investigated operating points, together with

turbulence intensity of 0.1% and turbulent length scale of

0.1 𝑚. Inlet distortion is not present and the angle of attack

was set to zero. Besides the inlet plane above the nacelle (as

shown Fig. 1), the same farfield condition is also prescribed

to the lateral and outlet surfaces (not visible in the Figure).

2) Bypass and core inlet are prescribed as one-

dimensional radial distributions of stagnation pressure,

stagnation temperature, turbulent kinetic energy and

dissipation rate, taking into account both the hub and the

casing boundary layers. These distributions are taken from

previous RANS simulations, where the fan and the low-

pressure turbine have been investigated in more detail. The

integral values are adapted to the previous engine cycle

calculations at corresponding stations. Further, the radial and

circumferential angles are set to zero.

Boundary layers of all no-slip boundaries are resolved

with a dimensionless wall distance of the wall adjacent cells

of 𝑦+ ≈ 3, which is, for the aim of the current study, well

enough for the low-Reynolds formulation. Usually, adiabatic

condition is used for the Stokes-walls, except for the both

sides of the core nozzle (cf. Fig. 2), where constant area-

specific heat flux �̇�𝑎 is prescribed to mimic the heat

redistribution by the TEG. Due to the TEG efficiency of

10%, the �̇�𝑎 values are set to extract 100% of the required

heat flux �̇� from the inner side and to add only 90% of it to

the outer side:

�̇�𝑜 = −0.9 �̇�𝑖 (1)

with: �̇�𝑖 = �̇�𝑎,𝑖 𝑎𝑖 and: �̇�𝑜 = �̇�𝑎,𝑜 𝑎𝑜

The remaining 10% of the heat flux are converted into

electric power and are removed from the computational

domain. Following the standard convention, the extracted

heat flux is negative and the added heat flux is positive,

hence the minus sign in Eq. 1. Further, the slightly larger

outer nozzle area 𝑎𝑜 (as compared to the inner one 𝑎𝑖), is

also taken into account when setting the �̇�𝑎,𝑜 and �̇�𝑎,𝑖 values.

a) 3D view

b) Meridional plane of nozzle region

Figure 4: Mesh of the test case

Mesh

The 3D-mesh was generated by the commercial tool

ANSYS ICEM CFD 17.0, and it consists of about 0.9 million

nodes, cf. Fig. 4. The boundary layers are resolved by ca. 30

cells. Since the flow is axisymmetric, only a 45° segment

with 10 equidistant cells in circumferential direction was

meshed. The extension of the computational domain is 10

diameters in both directions (axial and radial). Previous

investigations of grid resolution show that further increase of

node number has only negligible effect at the results, in

particular at the thrust and propulsive efficiency, cf. Bode et

al. (2017b).

Simulation Quality

Depending on the test case, the convergence of steady

simulations was achieved after less than 10 000 iterations,

and it was characterized by a density residual drop of at least

three orders of magnitude (see Fig. 5) and a relative

difference of in- and outlet massflow ≤ 10−3. A typical

simulation is conducted in ca. 4 hours on a commodity

cluster with 8 CPUs (Intel 𝑖7 − 3770, each with 4 cores).

𝑈∞

nozzle region

see below

4

Figure 5: Density residual, cruise condition with heat transfer

Besides the aforementioned grid sensitivity study, two

additional measures have been done to increase the

simulation quality and credibility. First, the turbulence model

has been changed from SST to k− of Wilcox (2006),

together with the Kato-Launder (1993) fix for the stagnation

point anomaly. Second, instead of constant turbulent Prandtl

number, it has been correlated to the Peclet number,

according to Kays and Crawford (1987, p. 228). Results from

both studies show some significant deviations in boundary

layer prediction when compared to the SST model and a

constant Prandtl number. However, the difference between

adiabatic and heat transfer simulations, which is of major

interest in the present investigation, was very similar to

original solver settings.

Post-processing

As part of the post processing, the integral boundary

layer parameters are determined by integration of the flow

field perpendicular to the blade surface up to a point where

99% of the total pressure deficit is reached, cf. Kozulovic

(2007) for more details. Further, several planes have been

selected for the result analysis, cf. Figs. 1 and 2:

IS and IE: start and end of inner core nozzle side

OS and OE: start and end of outer core nozzle side

OPP: peak pressure at outer core nozzle side

W1, W2 and W3: wake planes downstream the core

nozzle

RESULTS AND DISCUSSION

Scope of Investigation

Three different operating points have been investigated:

cruise – CRZ, top of climb – TOC and take-off – TAKO.

Corresponding engine and flight conditions are known from

previous investigations. For each case, both adiabatic and

heat flux simulations have been conducted. The applied heat

fluxes are given in Tab. 1, together with the flow

temperatures in Tab. 2. As an additional study, the heat flux

has been varied for the cruise condition.

Heat flux �̇� in [kW] CRZ TOC TAKO

Inner side -30 -30 -45

Outer side 27 27 40.5

Difference

= electric power

3 3 4.5

Tab. 1: Heat flux of investigated configurations

Temperature 𝑇 [K] CRZ TOC TAKO

Core flow 605 623 693

Bypass flow 238 238 290

Tab. 2: Free-stream temperatures of investigated configurations

Due to the hollow structure of the clean nozzle (cf. Fig.

3), which consists of two thin plates with air in between, a

very small amount of heat flux will establish between the

core and bypass flow. Preliminary calculations provide a heat

flux which is less than 10% of the value with TEG’s installed

in the nozzle. For this reason, the heat flux is neglected for

the configuration without TEG, that means adiabatic walls

have been applied.

Most analysis work is done with the cruise condition:

boundary layer analysis, shear layer development and

integral values. The boundary and shear layer analysis has

also been done for the other operating points, but is not

presented here, due to very similar results as cruise

condition. For this reason, only integral values are provided

for TOC and TAKO.

Cruise – Outer Side of Core Nozzle

The development of different parameters along the outer

side is shown in Figs. 6 and 7. According to the pressure

coefficient 𝑐𝑝, this side is exposed to an adverse pressure

gradient between the TEG start position (OS) and peak

pressure position (OPP). Only a small region downstream the

OPP position shows flow acceleration. Both the adiabatic

and heat transfer simulation show almost identical pressure

distribution. In other words, the heat transfer effect is only

restricted to a thin viscous region near the wall, without

influencing the potential flow.

All other parameters show a significant influence of the

heat transfer. As expected, the skin friction coefficient 𝑐𝑓 and

momentum thickness 𝛿2 are reduced, which is in agreement

to the literature on heated walls. As a different behaviour

between 𝑐𝑓 and 𝛿2, the skin friction is reduced very quickly

from the beginning of the heated region, whereas the

momentum thickness is diverging only slowly from the

adiabatic solution. At the end of the nozzle, the momentum

thickness is reduced by 4%. In contrast, the displacement

thickness 𝛿1 is increased by 11%. The wall temperature 𝑇𝑊

and wall density 𝜌𝑊 are considerably changed by heating,

both in a very rapid way at the beginning and keeping the

difference further downstream.

0 2000 4000 6000 8000 1000010

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Residual L1

Residual max.

| |

5

Fig 6: Development of pressure, skin friction and temperature along the outer side of core nozzle

To study the effect of heating at turbulence, the

maximum turbulence intensity across the boundary layer

𝑇𝑢𝑚𝑎𝑥 is also displayed. Obviously, the turbulence is very

quickly reduced, but after reaching the minimum value of

12%, it remains almost constant and ends up with the same

value as adiabatic result. In other words, the very rapid

decrease, which is induced by heating, is suddenly stopped at

this value. In boundary layers, turbulence balance is

dominated by the source terms (production and destruction)

of the 𝑘- and -equation. These terms find a balance at the

above mentioned turbulence value, leading to the sudden

change of the trend. The reasons therefore are not understood

yet.

Fig 7: Development of different values along the outer side of core nozzle

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6-1.10

-1.05

-1.00

-0.95

-0.90

-0.85

adiabatic

heat transfer

OS

x [m]

cp

[-]

OE

OPP

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.000

0.002

0.004

0.006

0.008

0.010

adiabatic

heat transfer

cf[-]

x [m]

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6200

250

300

350

400

450

adiabatic

heat transfer

TW

[K]

x [m]

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.0

0.1

0.2

0.3

0.4

0.5

adiabatic

heat transfer

W

[kg/m3]

x [m]

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.000

0.001

0.002

0.003

0.004

0.005

1

[m]

2[m]

x [m]

2

1

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.0

0.1

0.2

0.3

0.4

0.5

adiabatic

heat transfer

Tumax

[-]

x [m]

6

Cruise – Inner Side of Core Nozzle

The opposite effects occur at the inner side, where the

heat is extracted from the flow, cf. Fig. 8. Only the pressure

distribution remains almost constant. Contrary to the outer

side, the inner boundary layer is accelerated, which results in

a much smaller momentum thickness. However, the increase

of momentum thickness due to heat transfer is roughly the

Fig 8: Development of different values along the inner side of core nozzle

same as the reduction at the outer side (+0.095 𝑚𝑚 vs.

−0.102 𝑚𝑚). This indicates that the beneficial and

detrimental effects at viscous losses are nearly compensated,

giving a very small overall effect at the engine performance.

Cruise – Boundary Layer Profiles

The profiles of last position OE are displayed in Fig. 9,

since the complete upstream influence is agglomerated there.

The effect of heat transfer at the velocity is almost negligible.

Hence, the change of the momentum thickness must be a

result of density change, solely. Indeed, both the density and

temperature profile are considerably altered by the heat

addition. Further, the turbulent kinetic energy 𝑡𝑘𝑒 is only

slightly reduced.

Fig 9: Boundary layer profiles, outer side

Looking at the boundary layer profiles of the inner

nozzle side (position IE), opposite observations are found, cf.

Fig. 10: density is increased, temperature is decreased and

turbulent kinetic energy is slightly increased. Only the

velocity profile is almost unchanged by the cooling, which is

consistent to the case of the heating of the outer side.

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6-0.4

-0.2

0.0

0.2

0.4

0.6

adiabatic

heat transfer

cp

[-]

x [m]

IS IE

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.000

0.002

0.004

0.006

0.008

0.010

cf[-]

x [m]

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.000

0.001

0.002

0.003

0.004

0.005

1

[m]

2[m]

x [m]

2

1

0 100 200 300 400-0.01

0.00

0.01

0.02

0.03

0.04

0.05

adiabatic

heat transfer

d[m]

u [m/s]0.25 0.30 0.35 0.40 0.45

-0.01

0.00

0.01

0.02

0.03

0.04

0.05d

[m]

[kg/m3]

200 250 300 350 400-0.01

0.00

0.01

0.02

0.03

0.04

0.05d

[m]

T [K]0 100 200 300 400

-0.01

0.00

0.01

0.02

0.03

0.04

0.05d

[m]

tke [m2/s

2]

7

Fig 10: Boundary layer profiles, inner side

Cruise – Downstream development

Since the velocity profiles are not affected by the heat

transfer, the same can be observed in the downstream

development, planes W1, W2 and W3, cf. Fig. 11. Only

density and temperature show smaller gradients, according to

the corresponding boundary layer profiles. This is clearly

visible in W1, but, to a smaller degree, also in W2 and W3.

From this point of view, the heat redistribution starts the jet

mixing procedure (of some parameters) prior to the nozzle

exit.

Fig 11: Downstream development

0 100 200 300 400-0.01

0.00

0.01

0.02

0.03

0.04

0.05

adiabatic

heat transfer

d[m]

u [m/s]0.12 0.14 0.16 0.18 0.20

-0.01

0.00

0.01

0.02

0.03

0.04

0.05d

[m]

[kg/m3]

450 500 550 600 650-0.01

0.00

0.01

0.02

0.03

0.04

0.05d

[m]

T [K]0 200 400 600 800

-0.01

0.00

0.01

0.02

0.03

0.04

0.05d

[m]

tke [m2/s

2]

-100 0 100 200 300 4000.20

0.25

0.30

0.35

0.40

adiabatic

heat transfer

ux

[m/s]

r [m]

W1

W2

W3

0 0.1 0.2 0.3 0.4 0.50.20

0.25

0.30

0.35

0.40

[kg/m3]

r [m]

W1

W2

W3

200 300 400 500 600 7000.20

0.25

0.30

0.35

0.40

T [K]

r [m]

W1

W2

W3

8

Cruise – Variation of heat flux

In principal, the abovementioned phenomena are

attenuated or intensified if the heat flux is lowered or

increased, respectively. However, this mechanism is not

linear, as indicated by the outer momentum thickness in Fig.

12. The heat flux increase seems to have less impact than a

decrease of the same size. Probably a saturation takes place

at large heat transfer rates.

Fig 12: Impact of heat flux variation at momentum thickness

Integral Values As a final investigation, the engine thrust 𝐹 and the

propulsive efficiency 𝑝𝑟𝑜𝑝

are determined, cf. (Kožulović

2010). Here, the radial variation of different values is taken

into account, cf. Fig. 13. In this way, there is a contribution

of force 𝑑𝐹(𝑟) and of rate of change of kinetic energy 𝑑�̇�(𝑟)

at each radial position 𝑟. When integrated in radial direction,

they provide 𝐹 and 𝑝𝑟𝑜𝑝

:

𝐹 = ∫ 𝑑𝐹(𝑟) 𝑅

0 (2)

𝑑𝐹(𝑟) = 𝑑�̇�(𝑟)(𝑢𝑒(𝑟) − 𝑢0) + 𝑑𝐴 (𝑝𝑒(𝑟) − 𝑝0) (3)

𝑑�̇�(𝑟) = 𝜌𝑒(𝑟) 𝑢𝑒(𝑟) 𝑑𝐴 (4)

𝑑𝐴 = 2 𝜋 𝑑𝑟 (5)

𝑝𝑟𝑜𝑝

=𝑡ℎ𝑟𝑢𝑠𝑡 𝑝𝑜𝑤𝑒𝑟

𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦=

𝐹 𝑢0

�̇� (6)

�̇� = ∫ 𝑑�̇�(𝑟) 𝑅

0 (7)

𝑑�̇�(𝑟) =1

2 𝑑�̇�(𝑟) [𝑢𝑒

2(𝑟) − 𝑢02] (8)

Fig 13: Engine control surface

In this derivation, the fuel mass flow �̇�𝑓 has been

neglected. Further, the integration includes only the engine

mass flow. As integration plane, a distance of roughly one

nacelle diameter downstream the engine has been selected,

since the lateral velocities were almost diminished at this

position. Further downstream positions yield very similar

results and almost the same trends.

For all investigated cases, the thrust is only slightly

increased and the propulsive efficiency is only slightly

decreased by heat transfer, cf. Tab. 3. Both effects are mainly

due to density change, which is also accounted for in the

integration. As already mentioned, the velocity distributions

are almost unchanged, hence their impact is negligible.

Operating point 𝐹 [𝑁] 𝑝𝑟𝑜𝑝

[−]

CRZ-adiab.

CRZ-heat

21910.4

21922.4

0.87207

0.87199

TOC-adiab.

TOC-heat

22257.5

22268.4

0.86422

0.86412

TAKO-adiab.

TAKO-heat

54585.8

54578.1

0.50328

0.50322

Tab. 3: Thrust and propulsive efficiency

CONCLUSIONS

By installation of TEG in a core nozzle of a turbofan, a

heat flux is generated from the core to the bypass flow. The

corresponding effects at the inner and outer boundary layers

of the core nozzle have been investigated by RANS

simulations. Following key conclusions can be drawn:

By heating the outer boundary layer, the momentum

thickness is reduced by 4%.

By extracting the heat from the inner boundary

layer, its momentum thickness is increased by

roughly the same amount as the reduction in the

outer boundary layer.

Velocity profiles remain almost constant, whereas

the density and temperature profiles are

considerably changed. This holds true for both

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.0008

0.0012

0.0016

0.0020

0.0024

0.0028

Q'=2 kW

Q'=3 kW

Q'=4 kW

2

[m]

x [m]

𝑟

9

nozzle boundary layers as well as for the shear layer

downstream the nozzle.

Turbulent kinetic energy is slightly reduced by heat

addition and slightly increased by heat extraction.

Thrust is slightly increased and propulsive

efficiency is slightly decreased by the heat transfer.

Both parameters show a change of the 4th

significant

figure, indicating an almost negligible influence.

NOMENCLATURE

Latin Symbols

𝐴 area

𝑐𝑓 =𝜏𝑊𝜌

2𝑢0

2 skin friction coefficient

𝑐𝑝 =𝑝−𝑝0𝜌

2𝑢0

2 pressure coefficient

𝐹 thrust

𝑓 fuel

�̇� rate of change of kinetic energy

�̇� massflow

𝑝 pressure

�̇� heat flux

�̇� =�̇�

𝑎 area-specific heat flux

𝑅 radius

𝑟 radial coordinate

𝑇 temperature

𝑇𝑢 turbulence intensity

𝑡𝑘𝑒 turbulent kinetic energy

𝑡𝑠𝑓𝑐 thrust-specific fuel consumption

𝑢 velocity

𝑊 wall

𝑥, 𝑦, 𝑧 Cartesian coordinates

Greek Symbols

𝛿1 = ∫ (1 −𝜌(𝑦) 𝑢(𝑦)

𝜌0 𝑢0)

∞

0𝑑𝑦 displacement thickness

𝛿2 = ∫𝜌(𝑦) 𝑢(𝑦)

𝜌0 𝑢0(1 −

𝑢(𝑦)

𝑢0)

∞

0𝑑𝑦 momentum thickness

𝑝𝑟𝑜𝑝

propulsive efficiency

𝜌 density

Abbreviations

CRZ cruise

RANS Reynolds-averaged Navier-Stokes (equations)

TSFC thrust-specific fuel consumption

TAKO take-off

TEG thermoelectric generator

TERA Thermoelectric Energy Recuperation for Aviation

TOC top-of-climb

TRACE Turbomachinery Research Aerodynamic

Computational Environment

ACKNOWLEDGMENTS

Financial support from the German Federal Ministry of

Economic Affairs and Energy is gratefully acknowledged for

funding the TERA-project (grant: 20E1303). We thank the

colleagues from this project for their support and fruitful

discussions: Dr. J. Sieber from MTU Aero Engines, F.

Ahrendts from TU Braunschweig, Dr. P. Ziolkowski from

DLR Cologne and Dr. K.-D. Büchter from Bauhaus

Luftfahrt. Further, the TRACE software package was

provided by the DLR-Institute of Propulsion Technology,

Cologne. We highly appreciate the outstanding TRACE-

support by Dr. E. Kügeler and his colleagues.

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