Numerical simulations of turbulent flow around a distributed

propeller wing

António Salreta

afonso.salreta@tecnico.ulisboa.pt

Instituto Superior Técnico, Universidade Técnica de Lisboa, Portugal

November 2016

Abstract: New advances in electric propulsion have been encouraging new studies in aeronautic propulsion.

A new field of study that shows great promise is distributed electric propulsion (DEP). The present work aims

to study the effects of changes of the propellers in a DEP aircraft. This study looks at the effects in lift and

drag coefficients of the aircraft and is divided in three parts: changes made to the geometry and power of

the propellers, changes made to the inflow velocity and changes in the number of propellers working. The

CFD program used for this work is STAR-CCM+, where RANS simulations were run using SST K-Omega

turbulence model. The mesh generation and mesh refinement are also included in this work, as well as the

creation of a performance curve for the propellers from the program JavaProp. Leading edge position

changes are not big enough to influence differences in lift and drag, while differences in diameters and power

lead to some interesting conclusions. Inflow velocity is a big influential factor for DEP aircrafts. Propellers

close to the tip of the wing show more influence on lift.

Keywords: Distributed electric propulsion, lift and drag coefficients, STAR-CCM+, RANS, SST K-Omega,

JavaProp

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Introduction

Electric engines are a good

alternative to combustion engines. These

have many benefits including good

efficiency, low noise and high reliability.

However the most interesting characteristic

of electric engines is the possibility of

producing smaller engines and locating

them in strategic places of the airplane. This

way aircraft designers and engineers have

more design freedom, creating vastly

different designs without being restrained by

the weight and volume of the actual

combustion propellers. Some early concepts

of turboelectric distributed propulsion

vehicles with subsonic fixed wing show

promise for better efficiency and noise

reduction. N3-X is a NASA’s project that predicts the combination of a blended wing

body configuration and distributed

propulsion system produces a 70% fuel-burn

reduction relative to a B777-200LR

reference aircraft [1].

Distributed electric propulsion (DEP)

has shown great promise especially in light

weight aircrafts. In this design many small

propellers are distributed along the span of

the wing increasing dynamic pressure, and

facilitating of flight at lower velocities

excluding the need for multi-element high lift

devices. While being a promising design,

this brings higher structural complexity.

Variable pitch loading can be achieved by

controlling individual propellers, making it

easier to optimally match the needs of the

aircraft. It is also a redundant and robust

system that can be advantageous in case of

engine failure.

This work aims to study the

influence of inflow conditions as well as

parametric changes made to the electric

propellers for a DEP small man-controlled

airplane. This plane is entitled of Silent Air

Taxi (SAT).

Configuration Description

SAT’s configuration was inspired by

NASA’s SCEPTOR configuration, following

the same idea of distributed propellers. In

this case the airplane geometry is presented

in figure 1 and the configurations in table 1,

showing five smaller identical propellers

distributed along the span of the wing, and

one bigger propeller at the tip (this refers

only to one wing). The number of propellers

was already chosen and quantity of

propellers was not tested. The reason for the

location of the propellers in front of the

leading edge compared to locating them

behind the wing is for favourable pitching

moment, acoustic, cruise drag and structural

complexity [2].

Comparisons were made to Cirrus SR22, Cessna 172R and Tecnam P2006T airplanes because these represent the current state of the art for this class of aircraft and share a similar wing span (table 2). Propeller configurations are displayed in table 1. These propeller were chosen to have the same characteristics as pre-existing propellers. Simulations were done only to one wing taking advantage of symmetry and excluding the fuselage to be less computationally demanding. The geometry was tested without flap.

Figure 1 - SAT's geometry

Objectives

To study changes in lift a reference

configuration was chosen as the starting and

comparison point for the other simulations,

while the whole geometry remained

constant. Only numerical simulations were

done. Different speeds were tested, for

cruise at 110.6 m/s and take-off/landing

31.38 m/s. Parametric changes to the

propellers were made to the diameter,

distance to the leading edge and power of

the engine. These were tested for higher

velocity (110.61 m/s) to produce higher

differences. While to study the influences of

each inner propeller, the simulations were

made for an inflow velocity of 31.38 m/s. To

do these last simulations these propellers

were switched off one by one.

Mathematical models

RANS simulations were performed using the

commercial program STAR-CCM+. Both

mesh generation and simulations were

accomplished using this program. SST

(Menter) K-Omega was the turbulence

model because it behaves well in adverse

pressure gradient, and solves the flow inside

and outside the boundary layer with

relatively accuracy.

SST (Menter) K-Omega model is a hybrid of

two other models using K-Omega in the

inner boundary layer and K-Epsilon in the far

field. Using the following equation to

calculate the turbulent viscosity:

𝜇𝑡 = 𝜌𝑎1𝑘

𝑚𝑎𝑥(𝑎1𝜔,𝛺𝐹2) ( 1)

where 𝑎1 is a constant, Ω is the absolute value

of the vorticity, and 𝐹2 is a function that is one

for boundary-layer flows and zero for free

shear layers. 𝑘 and 𝜔 are calculated by two

transport equations. [3]

Wall laws for this wall treatment are chosen

automatically based on the behaviour of the

turbulence model. For this case, blended wall

laws are used, which include a buffer region

that smoothly blends the laminar and the

turbulent profiles together.

The axial force and normal force on a surface

are computed by the Star-CCM+ as:

𝑓 = ∑ (𝑓𝑆𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 + 𝑓𝑆

𝑆ℎ𝑒𝑎𝑟). 𝑛𝑓𝑓 ( 2)

𝑓𝑆𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 = (𝑝𝑓 − 𝑝𝑟𝑒𝑓)𝑎𝑓 ( 3)

𝑓𝑆𝑆ℎ𝑒𝑎𝑟 = −𝑇𝑓𝑎𝑓 ( 4)

where 𝑓𝑆𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 and 𝑓𝑆

𝑆ℎ𝑒𝑎𝑟 are the pressure

and shear force vectors on the surface face S,

𝑛𝑓 is the direction defined by the user, 𝑎𝑓 the

face area vector, 𝑝𝑓 if the face static pressure,

Inner Tip

Number of Blades 5 3 Design RPM 4000 3000 Outer diameter 0.75 m 1.5 m Inner diameter 0.056 m 0.292 m Airspeed (m/s) 110.6 Power (kW) 20 50 Airfoil MH 114 13% Re =

500 000

Table 1 - Propellers' configurations

𝑝𝑟𝑒𝑓 is the reference pressure, and 𝑇𝑓 is the

stress tensor at face 𝑓.

Table 2 - Comparison of reference configuration with similar plane

Numerical approximations

Segregated flow model was chosen,

with second order convection and diffusion.

This solves the flow equations (one for each

component of velocity, and one for pressure)

in a segregated or uncoupled manner. The

linkage between the momentum and

continuity equations is achieved with a

predictor corrector approach. This model also

solves the total energy equation with the

temperature as the solved variable. Enthalpy

is then computed from temperature according

to the equation of state. In this case the

equation of state is given by the ideal gas

theory.

Propeller model

For simulating the propellers, a

virtual disk was used to induce force

distributions instead of resolving the

propellers blade geometry. This is a faster way

of simulating a large number of propellers. The

rotor is simulated through source terms in the

momentum equations. The method employs a

uniform volume force distribution over the

cylindrical virtual disk. The volume force

varies with the radial direction. The radial

distribution varies with the Goldstein optimum

and is given by: [4]

𝑓𝑏𝑥 = 𝐴𝑥𝑟∗√(1 − 𝑟∗) ( 5)

𝑓𝑏𝜃 = 𝐴𝜃 .𝑟∗√1−𝑟∗

𝑟∗(1−𝑟ℎ′)+𝑟ℎ

′ ( 6)

𝑟∗ =𝑟′−𝑟ℎ

′

1−𝑟ℎ′ ( 7)

𝑟ℎ′ =

𝑅𝐻

𝑅𝑃, and 𝑟′ =

𝑟

𝑅𝑃 ( 8)

where 𝐴𝑥 and 𝐴𝜃 are functions of the propeller

hub radius 𝑅𝐻 and tip radius 𝑅𝑃.

Outside STAR-CCM+, JavaPROP

[5] is used to obtain the propeller curve for

both types of propellers. In this program

equations based on the formulas of Adkins

and Liebeck’s work of “Design of Optimum

Propellers” are used to arrive to an optimum

solution.[6] It is based on the theory of optimum

propeller as developed by Betz, Prandtl and

Glauert. Only a small number of parameters

need to be specified. These are:

Number of blades, B

Axial velocity v of the flow (flight speed or boat speed),

Diameter D of the propeller,

Selected distribution of airfoil lift and drag coefficients Cl and Cd along the radius. In this case the distribution is not directly selected but an airfoil is selected at each station (or radius) with the angle of attack.

Desired thrust T or the available shaft power P

Density 𝜌 of the medium

The disk is applied directly to the

original mesh specifying the diameter,

thickness and orientation. The table from the

resulting performance curve consists of the

advance ratio, propeller efficiency, thrust

coefficient 𝐾𝑇, and torque coefficient 𝐾𝑄 given

as: [4][5]

𝐽 =𝑉∞

𝑛 𝐷𝑂 ( 9)

𝑃 = 𝑇 ∙𝑉∞

𝜂 ( 10)

𝐾𝑇 =𝑇

𝜌𝑛2𝐷𝑂4 ( 11)

𝐾𝑄 =𝑄

𝜌𝑛2𝐷𝑂5 ( 12)

Where 𝑉∞ is the flow velocity, 𝐷𝑂 is the

propeller diameter, and 𝑛 the rotation rate.

Since the operation point is given by

the thrust, the advance ratio is calculated by

solving the following equation numerically:

𝑓(𝐽) = 𝐾𝑇 − 𝐾𝑇′ ( 13)

𝐾𝑇′ =

𝐽2𝑇𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 𝑝𝑜𝑖𝑛𝑡

𝜌𝑖𝑛𝑓𝑙𝑜𝑤 𝑝𝑙𝑎𝑛𝑒𝑉∞2 𝐷0

2 ( 14)

Boundary Conditions

The flow inlet is made from a semi-

spherical surface where a velocity magnitude

and direction are defined. (figure 2) This way

the wing is maintained static, and the flow

direction is only controlled by the inlet and the

propellers. In contact with the wing hub a

symmetry plane is placed. The wing and

nacelles have a no-slip condition. And finally,

the rest of the surfaces (semi-cylindrical and

semi-circle) are considered as a pressure

outlet where the flow is extrapolated. The

outlet has to cover also the cylindrical walls

because of the way the angle of attack is

recreated in these simulations. The semi-

cylindrical surfaces cannot be considered

walls because the inlet flow determines the

angle of attack, not the rotation of the body as

it would be the case in an experimental

procedure.

Mesh Refinement

Mesh refinement was not made

systematically but was instead local. At first

the meshing process was made automatically

but was then refined locally. A hexahedral grid

type was chosen for all the mesh because of

its simplicity in defining wake refinement and

its superior speed to mesh. A prism layer was

used near the body with 20 layers, all

amounting to 1% of the base cell size (cell size

that every other parameter is related to). To

compare between meshes, axial and normal

forces coefficients were observed since these

are the values from which the rest of the

results are computed from. Refinement

convergence criteria was decided to be 0.001

(

size of the first cell closer to the surface is

correct for Mesh 6. The difference between

Mesh 7 and Mesh 6 is 0.0002 (0.4%) for the

normal and axial force coefficients so further

refinement was not made. The slight

fluctuation of wall y+ value between Mesh4,

Mesh5 and Mesh6 is due to the unsteady

nature of the flow. These simulations were

made for angle of attack of 8º where the model

starts to become accurate. Mesh 6 is the final

mesh used for this project.

Mesh Nº of cells (in millions)

𝑪𝑨𝑭 𝑪𝑵𝑭 Wall y+

Mesh 1 0.88 -0.0034 0.5127 4.01 Mesh 2 2.38 0.0287 0.6636 2.02 Mesh 3 2.57 0.0294 0.6532 2.05 Mesh 4 12.16 0.0479 0.7418 0.95 Mesh 5 15.00 0.0486 0.7363 0.98 Mesh 6 15.05 0.0489 0.739 0.99 Mesh 7 18.29 0.0487 0.7388 0.99

Table 3 – Mesh refinement process

Results

Configs 𝑷𝒊𝒏𝒏𝒆𝒓(kW)

𝑷𝑻𝒊𝒑

(kW)

𝒙𝑳𝑬(m)

𝑫𝒊𝒏𝒏𝒆𝒓(m)

𝑫𝒕𝒊𝒑(m)

Ref 20 kw 50 kw 0 0.75 1.5 X1 20 kw 50 kw 0.1 0.75 1.5 X2 20 kw 50 kw 0.2 0.75 1.5 DS 20 kw 50 kw 0 0.50 0.75 DL 20 kw 50 kw 0 m 0.9 1.5

Po

we

r I30 30 kw 50 kw 0 m 0.75 1.5 T70 20 kw 70 kw 0 m 0.75 1.5 T20 20 kw 20 kw 0 m 0.75 1.5

Table 4 - Table of configurations

Table 4 summarizes the parametric

changes made to the propellers in power and

geometry in every configuration. Only for the

reference solution (solution to which every

other result is compared) the variation of angle

of attack has been from 0° to 12°. For the rest

of the cases only the point of maximum lift was

interesting studying. For almost all of the

different propellers configurations, 𝐶𝐿 reaches

a maximum value at the angle of attack of 8°.

This is the reason why the results for other

configurations are only presented for angles

7° to 9°. Unfortunately this model is no longer

accurate after separation is reached because

it ceases to be a steady flow and it does not

take into account possible vortex shedding,

producing results that do not correspond with

reality. That is why it shows a slight increase

of 𝐶𝐿 between the angles of 9° and 10° even

though the values of both simulations have

stabilized.

2D simulations for NACA 2412 wing

profile without nacelles were made using the

program Xfoil [7] where the maximum angle of

attack was found to be at 13.8° with a 𝐶𝐿 𝑀𝑎𝑥

of 1.52 (presented in the annex). In figure 4

the 𝐶𝐿 𝑀𝑎𝑥 for the Reference solution is equal

to 0.85 at 8°. Not only the finite wing has a

decrease of lift but the presence of propellers

causes it to stall sooner at 8º. The results of

the rest of the parametric changes are

presented in phases, depending on the design

modifications made.

Figure 3 shows the wall shear stress

in the x-direction for the reference solution for

9°, after the maximum angle of attack. There is clearly a separation happening more in one

side on the left of the propeller. This is due to

the rotation of the propeller that increases the

local angle of attack on the left side of the

propeller forcing it to stall first. This means the

chosen wing airfoil cannot stall abruptly for

DEP systems.

In figure 4 every case for geometry

and power change are presented and the first

thing noticeable is that all cases have very

close values, stalling near the same angle of

attack. For X1 configurations 𝐶𝐿 𝑀𝑎𝑥 value

decreases to 0.84 at an angle of 8 degrees.

The variations of the lift coefficient are seen to

vary 0.004 (0.47%) from the Reference

configuration, while the drag coefficient varies

0.00026 (0.5%) with 𝐶𝐷 being 0.051 (figure 4).

The case of X2 has no decrease of 𝐶𝐿 𝑀𝑎𝑥,

remaining at a value of 0.85 to an angle of also

8 degrees, and the drag coefficient is

unaltered. This is a very low and not relevant change to conclude such differences.

Smaller propellers’ diameters are also tested (DS table 4). The value of 𝐶𝐿 𝑀𝑎𝑥 seems

to increase to 0.88 (3.7 %), for the larger

diameter configuration, this value drops

slightly to 0.84 (0.59 %). The drag coefficient

doesn’t change much for both cases in relation

to the reference solution, which leads to

conclude that this difference in lift is due to

different rotation rates of the propellers. For

simulations where larger propeller diameters

were tested (DL), the rotation rate is around

2400 rpm, while in the case of the smaller

diameters, propellers rotation is around the

speed of 5500 rpm. This also explains why the

drag for the larger diameter case is the same

as the reference solution even though the

affected area is bigger. In the case of the

smaller diameters, the higher rotation rate

increases the drag even though less area is

affected by the propellers. Looking at the lift

the Small Diameter case has higher values,

since the flow over the wing is less disturbed.

For I30 configuration, 𝐶𝐿 increases to

0.87 at 8° of angle of attack. It’s an increase of

0.021 (2.5 %) in 𝐶𝐿 for 50 kW of total power

increase. The reason for this increase is due

to the higher induced velocity over the wing

that ultimately generates higher lift. Obviously

that also means higher drag which in this case

is a 0.001 (1.5 %) of difference of 𝐶𝐷 for the

same angles, in this case 8°. There is a

significant change in drag but the lift is not

changing enough, leading to conclude this

increase in power is not worth it. The overall

change in 𝐶𝐿 is not big enough.

Others studied effects were the ones

created by the increase and decrease of

power in the tip propellers. Both higher and

lower power changes were tested. For the

case of T20, the lift not only gets bigger to a

value of 0.872, a change of 0.02 (2.4 %), but

also happens in a higher angle of attack of 9°.

However for the T70 case the change in lift is

not noticeable, being 0.85 for the angle of 8°.

The drag difference is clearly more noticeable

with changes of 0.016 (32%) for the T20 case,

and 0.0003 (0.6%) for the T70 case. Either

one of these parametric changes bring

benefits in terms of lift, but only T20 presents

also a decrease in drag, concluding that the tip

(or “cruise”) propeller does not bring benefits

when at a higher power than the “high-lift”

propellers.

Finally a change in inflow velocity is studied

and a clear change in results happens. Two

different cases of 77.2 m/s (150 knots) and

31.38 m/s (61 knots) were tested. The

maximum angle still seems to be at 8° for the

77.2 m/s. There is an increase of 0.042 (4.95

%) for the lift coefficient and decrease of

0.0027 (5.45 %) for the drag coefficient. This

shows that lower cruise velocity produces

higher L/D ratios, 16.86 for 110.6 m/s and

18.71 for 31.38 m/s, both at 8°.

Shown in figure 9, there are the lift

coefficients for different speeds. First the

maximum angle of attack is at 12° for 31.38

m/s, and 𝐶𝐿 and 𝐶𝐷 have respectively, an

increase of 0.409 (48.12%) and an increase of

0.098 (194.10%). Not only stall happens later,

as the overall lift coefficient of the wing is

higher. Clearly this type of “high-lift” devices

behave better and bring more efficiency at

lower speeds.

Figure 3 - Wall Shear Stress in the positive x-direction for Reference solution case at 9°

Figure 4 - Lift coefficient for all of the propeller change cases

Figure 5 - Cl/Clref vs angle of attack for all of the propeller change cases

Figure 6 – Drag coefficient for all of the propeller change case

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Lift

Co

effi

cien

t

Angle of attack (°)

DEFAULTX1X2DSDLT20I30T70

0,92

0,94

0,96

0,98

1,00

1,02

1,04

1,06

6 7 8 9 10 11

CL

/ C

Lref

Angle of attack (°)

X1X2DSDLT20I30

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,10

6 7 8 9 10 11

Dra

g C

oef

fici

ent

Angle of attack (°)

DEFAULT

X1

X2

DS

DL

T20

I30

T70

Figure 7 – Lift coefficient for all the cases with one propeller turned off

Figure 8 - Cl/Clref for all the cases with one propeller turned off

1,05

1,10

1,15

1,20

1,25

1,30

6 7 8 9 10 11 12 13 14 15

Lift

ceo

ffic

ien

t

Angle of attack (°)

61knots

1prop

2prop

3prop

4prop

5prop

0,92

0,93

0,94

0,95

0,96

0,97

0,98

0,99

1,00

6 7 8 9 10 11 12 13 14 15

Cl/

Clr

ef

Angle of attack (°)

1prop

2prop

3prop

4prop

5prop

Figure 9 - Lift coefficient for all the cases with one propeller turned off

Figure 10 – Lift coefficient change for velocity change, and without propellers

----------------

Conclusion

A series of different cases have

converged successfully showing both

expected and unexpected results.

2D simulations show that 𝐶𝐿 increases

with higher velocity (110.61 m/s) but stall first

than for lower velocities (31.38 m/s). The

same can’t be said for the 3D cases. With

propellers at lower velocities stall seems to

also be occurring later as it would be

expected, but because of the propellers 𝐶𝐿 is

much higher than for lower velocities. There is

also a slight increase comparing with the

geometry without propellers. Here the

comparison is difficult since the virtual disk are

not present anymore in NoProp case (figure 9)

producing much less drag. This means DEP is

producing higher 𝐶𝐿 overall but stalls first. Also

DEP systems seem to work much better at

lower velocities, and should not be used at

higher velocities.

Parametric changes to diameter and

distance to the leading edge seem to have

produce no conclusive data, even for higher

velocities. However power changes hints that

0,05

0,07

0,09

0,11

0,13

0,15

0,17

0,19

0,21

0,23

7 8 9 10 11 12 13 14 15

Dra

g co

effi

cien

t

Angle of attack (°)

61knots

1prop

2prop

3prop

4prop

5prop

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Lift

Co

effi

cien

t

Angle of attack (°)

DEFAULT

150 knts

61knots

the tip propeller should not have relatively

more power than the smaller ones. Vortices

produced by the propellers seem to be

interfering with one another, producing more

harm than good.

Finally the propellers that seemed to

influence more the lift are the ones positioned

more to the tip of the wing. However without

the fifth propeller (counting from the hub to the

tip of the wing) 𝐶𝐿 is slightly higher than without

the fourth propeller. As explained before, one

possible explanation is the influence of

different vortices produced. One possible

explanation is the influence of different

vortices produced from the fifth and the tip

propeller on one another. This may cause an

increase of dynamic pressure over the wing

due to not contra-rotating propellers.

Acknowledgement

The author would like to thank first

and foremost Prof. Dr. -Ing. Andreas Henze

and Prof. Dr. Luís Eça for all the guidance and

support in the making of this work. This project

was possible thanks to the opportunity given

by Prof. Dr. –Ing. Wolfgang Schröder who

made it possible to work at the Aerodynamic

Institute of RWTH Aachen, and also to

Michael Kreimeier for the opportunity to work

on this project. This project belongs to the

Aeronautics and Aerospace Department of

RWTH Aachen that was done in collaboration

with the Aerodynamic Institute of the same

university.

References

[1] Felder J.L., Brown G.V., DaeKim H., and

Chu J., (2011) Turboelectric Distributed

Propulsion in a Hybrid Wing Body Aircraft.

NASA Glenn Research Center, USA.

[2] Moore, M. D. and Fredericks, W. J., (2014) Misconceptions of Electric Aircraft and their Emerging Aviation Markets," AIAA SciTech, American Institute of Aeronautics

and Astronautics.

[3] Menter F.R., (1994) Two-Equation Eddy-

Viscosity Turbulence Models for Engineering

Applications. AIAA Journal, 32(8):1598-1605

[4] STAR-CCM+ User Guide. Star-CD version

4.02 (2006)

[5] Hepperle M.. (2013). Java PROP – Design

and Analysis of Propellers. Accessed in 15-

03-2016 at: http://www.mh-

aerotools.de/airfoils/javaprop.htm

[6] Adkins C. N.. Liebeck R.H.. (1994) Design

of Optimum Propellers. Douglas Aircraft

Company. Long Beach. California. Journal of

Propulsion and Power. Vol 10. No. 5.

[7] XFoil Subsonic Development System

Website, Drela M.. Youngren H., Accessed on

06-04-2016 at:

http://web.mit.edu/drela/Public/web/xfoil/

http://www.mh-aerotools.de/airfoils/javaprop.htmhttp://www.mh-aerotools.de/airfoils/javaprop.htm