Aerodynamic Design of a Propeller for High-Altitude 

Balloon Trajectory Control

NASA/CR-2012-215893

March 2012

Richard Eppler and Dan M. Somers

National Aeronautics and 

Space Administration 

Goddard Space Flight Center

Greenbelt, Maryland 20771
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Aerodynamic Design of a Propeller for High-Altitude 

Balloon Trajectory Control

NASA/CR-2012-215893

March 2012

Richard Eppler 

University of Stuttgart, Stuttgart, Germany 

Dan M. Somers 

Airfoils, Incorporated, Port Matilda, PA

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14 December 2004 to 31 December 2005.

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Goddard Space Flight Center 

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AERODYNAMIC DESIGN OF A

PROPELLER FOR HIGH-ALTITUDE

BALLOON TRAJECTORY CONTROL

RICHARD EPPLER

UNIVERSITY OF STUTTGART

DAN M. SOMERS

AIRFOILS, INCORPORATED

AUGUST 2005
ABSTRACT

The aerodynamic design of a propeller for the trajectory control of a high-altitude, scientific balloon has been performed using theoretical methods developed especially for such

applications. The methods are described. Optimum, nonlinear chord and twist distributions

have been developed in conjunction with the design of a family of airfoils, the SE403, SE404,

and SE405, for the propeller. The very low Reynolds numbers along the propeller blade fall

in a range that has yet to be rigorously investigated, either experimentally or theoretically.

INTRODUCTION

Scientific balloons, including future Ultra Long Duration Balloons (ULDB), require a

system for the control of flight trajectory. One such trajectory-control system employs a propeller. The design of the propeller represents a challenge, however, because of the very low

Reynolds numbers (< 50,000) characteristic of the high operational altitudes. Accordingly,

the aerodynamic design of the propeller has been performed using theoretical methods developed especially for such applications. The propeller design includes not only the optimum

planform of the blades (i.e., chord and twist distributions) but also the airfoil shapes along the

radius of the blade.

Propeller theory, as it has commonly been applied to date, is based on a method developed in 1919 (refs. 1 and 2). This theory assumes a low, propeller loading and neglects viscous drag. It has been successfully applied to many lowly loaded propeller designs, including

those for man-powered and solar-powered aircraft. A theory for more highly loaded, optimum propellers, still ignoring viscous drag, was developed recently (“Propellers and Fans,”

seminar by R. Eppler at Siegen University, Germany, 15 April 1999). One result from the

application of this theory is that the lift is shifted considerably outboard if the propeller is

more highly loaded, which is understandable because the lift outboard produces less torque.

Lately, it has been concluded empirically that viscous drag too influences the optimum lift distribution along the propeller blade because overcoming the drag outboard requires much more

power than inboard.

The design of an optimum, solar-powered propeller for a high-altitude balloon can not

be performed adequately without accounting for the viscous drag because the Reynolds numbers are so low. Accordingly, the recent theory has been extended to include viscous drag.

The first application of the extended theory is presented. The effect of the viscous drag is

much greater than previously suspected. The lift is shifted back inboard but in a different way

than it is shifted outboard in the case of high loading.

Because the airfoil performance is the key to the propeller performance and because

current theoretical airfoil methods (e.g., refs. 3–5) have not been validated for such low Reynolds numbers, it is imperative that an appropriate airfoil from the propeller design (e.g., 75-

percent radius) be experimentally verified.
2

SYMBOLS

B number of blades

Cp power coefficient; pressure coefficient

CT thrust coefficient

c chord, m

cd section profile-drag coefficient

cl section lift coefficient

cm section pitching-moment coefficient about quarter-chord point

D drag, N

H function

L lift, N

M Mach number

P shaft power, W

p static pressure, Pa

R radius to blade element, m; Reynolds number based on free-stream conditions

and chord

r nondimensional radius to blade element, 

T thrust, N

t airfoil thickness, m

tinfl time after which influence of vortex elements on blade forces is neglected, s

tmax maximum lifetime of vortex elements, s

U induced velocity parallel to rotational direction, m/s; blade rotational velocity,

ωR, m/s

u nondimensional induced velocity parallel to rotational direction, 

R

Rt

----

U

ωRt

---------
3

forward velocity of propeller, m/s

W induced velocity parallel to propeller axis, m/s

w nondimensional induced velocity parallel to propeller axis, 

x airfoil abscissa, m

y blade abscissa, perpendicular to propeller axis and propeller radius, m; airfoil

ordinate, m

z blade ordinate, parallel to propeller axis, m

α angle of attack relative to x-axis, deg

Γ circulation, m2

/s

γ velocity jump across vorticity layer, ΔU, m/s; lift distribution

Δ incremental change in quantity

δ

ε

η efficiency

θ rotational angle, deg

Λ Lagrange multiplier

λ nondimensional forward velocity of propeller, 

ρ air density, kg/m3

Φ Lagrange function

φ angle between local flow velocity and rotational direction, deg; twist angle,

deg

Ψ function for u and w

ω angular velocity, rad/s

∞ infinity

V∞

W

ωRt

---------

D

L = atan---

D

L

---=

V∞

ωRt

---------
4

Subscripts:

Betz Betz solution (see ref. 1)

L at blade

max maximum

min minimum

mn near minimum

n normalized

S separation

t tip

0 far upstream of propeller; index of function H for and ; zero

lift

1 upstream of cylinder; at propeller; index of partial function Φ

2 downstream of cylinder; far downstream of propeller; index of partial function

Φ

∞ free-stream conditions

Abbreviations:

L. lower surface

S. boundary-layer separation location, xS /c

U. upper surface

PLANFORM

THEORY

Blade-Element Theory

The blade elements of the propeller are located on concentric cylinders around the propeller axis at a radius R and have a radial length of dR. One such cylinder, unrolled into a

plane, is shown in figure 1. The pressure upstream of the cylinder is p1 and downstream, p2.

u 0= w 0=
5

The velocities induced by the propeller are U1 and W1 upstream of the cylinder, U2 and

W2 downstream, and UL and WL at the propeller blade. The velocity at a blade section is

composed of the downstream velocity and the rotational velocity ,

where is the forward velocity of the propeller (i.e., the velocity of advance) and ω is the

angular velocity of the blade. The velocities U1, W1, U2, and W2 and the pressure difference depend on the radius R of the cylinder and the rotational angle θ. The

propeller has B blades. The direction of the local velocity is determined by the angle φ for

which

(1)

The lift L of the section is perpendicular to the local velocity and the drag D is perpendicular to the lift. The resulting force forms an angle to the lift vector defined by

.

Blade-element theory works with averaged velocities such as

and the corresponding averaged pressure differences . This introduces several difficulties. For example, for continuity reasons, must equal . This is not true for and

, which are difficult to evaluate. Also, the contraction of the propeller jet over the finite

height of the cylinder is neglected.

The only simple and clear averaging that can be performed is to replace the finite number of propeller blades by an infinite number of infinitesimal blades, as illustrated in figure 2.

On the left side of figure 2, several small vortices are shown along with their directions of

rotation. If weaker and weaker vortices are supposed, they approach a vorticity surface,

which is represented on the right side of figure 2 by a broken line that moves to the right with

the velocity ωR. Such a vorticity layer is merely a discontinuity in the velocity field. The

velocity jumps across the vorticity layer by , which is the strength of the vorticity

layer. The direction of ΔU is perpendicular to the direction of the vector , which, in the

present case, is the radial direction. The height of the previously considered cylinder becomes

infinitesimal and the induced velocities become

If the flow upstream of the propeller is irrotational, . Then, as shown in figure 2,

can be set. Upstream of the propeller, the Bernoulli equation

(2)

V∞ WL + ωR UL –

V∞

Δp p2 p1 –=

tanφ

V∞ WL +

ωR UL – --------------------=

tanδ = = D/L ε

U1( ) R 1

2π

------ U1( ) R, θ dθ

0

2π

∫ =

Δp R( )

W1 W2 W1

2

W2

2

ΔU = γ

γ

U2 U1 – = γ UL U1 U2 = ( ) + /2 W1 W2 WL = =

U1 0=

W1 === W2 WL W U2 = = U γ UL = U/2

p∞

ρ

2

--V∞

2 + p1

ρ

2

--( ) V∞ + W 2 +=
6

is valid. Downstream of the propeller, the jet rotates, contracts, and becomes turbulent. It is

assumed that the Bernoulli equation along the streamlines is valid until the pressure has

reached free-stream static . Thus,

(3)

where due to the conservation of momentum. Thus, the pressure difference across

the propeller disk is

(4)

An infinitesimal blade element of the rotating propeller extends over a sector of the

propeller disk having the angle dθ. The circulation on that blade element is γ(R)Rdθ. The

direction of the flow at this element is given by

(5)

The lift on this blade element is perpendicular to the direction of the flow and its magnitude is

(6)

The corresponding thrust element is

and the thrust element for the entire “ring” is, with ,

or, using equation (5) and ,

(7)

The same thrust element results from the pressure difference Δp, which yields, using

equation (4),

(8)

Another formula follows from the total flow through the ring. The flow is contained in

a tube having variable radius R and radial length dR. Its cross section is shown in figure 3.

The radius far upstream of the propeller is R0; far downstream, R2; and at the propeller, R1.

The corresponding widths of the tube are dR0, dR1, and dR2, which are determined by the

continuity condition

(9)

p∞

p2

ρ

2

-- ( ) V∞ + W 2 U2 + ( ) + p∞

ρ

2

-- V∞ W∞ ( ) + 2 U∞

2 += ( ) +

U∞ = U

Δp p2 p1 – ρ

2

-- 2V∞W∞ W∞

2 = = ( ) +

tanφ

V∞ + W

ωR U– /2 = -----------------------

ddL ργRdRdθ ω( ) R U– /2 2 ( ) V∞ + W 2 + ργRdRdθωR U– /2

cosφ = = -----------------------

ddT ddL

cosδ = ----------- cos( ) φ δ +

γ = U

dT 2πρURdRωR U– /2

cosφ = -----------------------( ) cosφ δφ – tan sin

tanδ = ε

dT = 2πρU( ) ωR U– /2 – ε( ) V∞ + W RdR

dT Δp2πRdR πρ 2V∞W∞ W∞

2 = = ( ) + RdR

R0dR0V∞ R1dR1( ) V∞ + W R2dR2 V∞ W∞ = = ( ) +
7

where the factor 2π has been omitted. The pressure integral around this tube vanishes due to

the properties of the flow about an infinite half body. The external force Δp2πR1dR1 acts at

the propeller disk. The momentum equation in the direction of the propeller axis yields then

again without the factor 2π. Together with equation (9), it follows that

Comparing this equation with equation (4) yields immediately

This simple equation results in momentum theory as well. Half the downstream velocity produced by the propeller is already present at the propeller. With this equation, it follows

from equation (8) that

and, by comparing this equation with equation (7),

(10)

The total thrust can thus be evaluated by two formulas

(11)

(12)

where Rt

 is the radius of the propeller. The power required to move a ring element is

with ddL from equation (6). This yields the total power required

(13)

Nondimensional Variables

To nondimensionalize the variables, all lengths are divided by the propeller radius Rt

and all velocities, by the rotational velocity at the propeller tip Ut ( ). The pressures are

ρV∞

2 R0dR0 ΔpR1dR1 ρ V∞ W∞ ( ) + 2

R2dR2 + 0 – =

Δp ρ( ) V∞ + W W∞ =

W∞ = 2W

dT 4πρ V∞W W2 = ( ) + RdR

2W V( ) ∞ + W = U( ) ωR U– /2 – ε( ) V∞ + W

T 4πρ W V( ) ∞ + W R Rd

0

Rt

∫ =

T 2πρ U( ) ωR U– /2 – ε( ) V∞ + W R Rd

0

Rt

∫ =

ddP ddL

cosδ = -----------sin( )ω φ δ + R

P 2πρω U V( ) ∞ + + W ε ω( ) R U– /2 R2 Rd

0

Rt

∫ =

ωRt =
8

divided by and the forces, by . In this report, to the extent possible, the dimensional variables are designated by upper-case letters and the nondimensional variables, by the

corresponding lower-case letters. For example,

The nondimensional forward velocity of the propeller is

which is proportional to the advance ratio.

The nondimensionalized ωR is merely r. Accordingly, equation (5) becomes

(14)

and equation (10),

(15)

The thrust T and the power required P are given in terms of their corresponding

coefficients

(16)

(17)

Optimum Propeller Design

Formulation of problem.- The thrust and power coefficients depend, according to

equations (16) and (17), on the given parameters λ and ε, r, and the functions w(r) and

u(r). The two functions are not independent, however. Using equation (15), one function can

be evaluated if the other is known. Therefore, only one of the functions is free. This means

that CT and CP depend on one of these functions.

The requirements for an optimum propeller can be specified in different ways.

(1) Minimum power for a given thrust

(2) Maximum thrust for a given power

(3) Maximum thrust-to-power ratio

ρUt

2

ρUt

2

Rt

2

u

U

ωRt

= --------- w

W

ωRt

= --------- r

R

Rt

----=

λ

V∞

ωRt

= ---------

tanφ

V∞ + W

ωR U– /2 ----------------------- λ + wr u – /2 = = ----------------

Ψ( ) u w, = = 2w( ) λ + w – 0 ur u ( ) – /2 – ε λ( ) + w

CT

T

2πρRt

2

ωRt ( )2 --------------------------------- T

2πρRt

2

Ut

2 ----------------------- 2w( ) λ + w r rd

0

1

∫ = ==

CP

P

2πρωRt

3

ωRt ( )2 -------------------------------------- P2πρRt

2

Ut

3 ----------------------- u( ) λ + + w ε( ) r u – /2 r

2 rd

0

1

∫ = ==
9

The only function that is free for achieving the optimum is either u(r) or w(r). Such

optimization problems are typical for variational calculus.

From the different formulations of the variational problem, the following has been

selected: for the specified thrust T, the power required P must be minimum. The free function is w(r); is given, with w(r) according to equation (15). Mathematically, this is formulated with a Lagrange multiplier Λ. The Lagrange function

(18)

is defined with

(19)

(20)

and equation (15). The necessary condition for the defined optimum is

(21)

where Φ1 depends on w and u(w) and, therefore,

(22)

where

Using Ψ(u,w) from equation (15),

(23)

The central optimization condition represented by equation (21) is a simple, special

case of the Euler equation of the variational problem because dw/dr does not occur in equation (18). Equation (21) concerns only the integrand of equation (18) and must be satisfied for

each r.

Properties of function u(w).- Equation (15) is quadratic in u and w. Because both

terms containing u2

 and w2 have the same sign, an ellipse is represented for each r. It is

possible to solve equation (15) with respect to u by

u r( ) = uwr ( ) ( )

Φ() Φ w r( ) ( ) 1( ) ΛΦ w – 2( ) w rd

0

1

∫ =

Φ1( ) w u( ) λ + + w ε( ) r u – /2 r

2 =

Φ2( ) w = 2w( ) λ + w r

d

dw

------ Φ1 ΛΦ2 ( ) – 0=

dΦ1

dw

--------- Φ1 ∂

∂w--------- Φ1 ∂

∂u --------- du

dw

+= ------

du

dw

------

Ψ∂

∂w

-------

Ψ∂

∂u

-------

–= -------

du

dw

------ 2λ + + 4w εu

r u – – ε λ( ) + w = --------------------------------------

u r – ε λ( ) + w ( ) r – ε λ( ) + w

2 = – – 4w( ) λ + w
10

Only the negative sign of the square root is realistic because the positive sign would yield

u values greater than r. All solutions go through . A series of the resulting

ellipses is shown in figure 4; the largest, for , and the others, for monotonically smaller

r values.

The maximum w ( ) for which a u is defined makes the argument of the

square root zero. This yields

(24)

It follows that the maximum u ( ) is

(25)

Obviously, there is also a minimum r, below which equation (15) has no solution. For

, it follows from equations (24) and (25) immediately that

In equation (23), the denominator is zero for and ; there, the

ellipses have their vertical tangents. More difficult is the other singularity at . For w

and u → 0, it follows that du/dw → – ∞.

Optimization details.- The Lagrange multiplier Λ in equation (21) is normally determined such that the specified CT is achieved. To obtain an overview of the possible solutions, it is simpler to vary Λ, however. Therefore, the range of Λ values that leads to

solutions must be determined. From equation (19), it follows that

and, from equation (20),

Both equations, when introduced into equation (21), yield the optimum condition

(26)

which must be satisfied for all values of r considered. All variables u, w, ε, and r must be

greater than zero. Moreover, only u less than r makes sense. The opposite would mean that

the induced velocity U is greater than ωR and, thus, the induced velocity would be greater

than the rotational velocity. Accordingly, the term within the inner parentheses of the first

term in equation (26) is not negative and can be zero only for . The first

term becomes + ∞ as w → wmax. For any r, a solution exists once there is a w for which the

left side of equation (26) is negative.

u w = = 0

r 1=

wmax =

wmax

2λ ε + ( ) r – ελ

4 ε

2 –

– ----------------------------------- 2λ ε + ( ) r – ελ

4 ε

2 –

----------------------------------- ⎝ ⎠ ⎛ ⎞ 2 ( ) r – ελ 2

4 ε

2 –

= + + ---------------------

umax =

umax r ε λ wmax –= ( ) +

r = ελ

wmax umax = = 0

u umax = w wmax =

r = ελ

dΦ1

dw

--------- u ( ) λ + + w εr – εu

du

dw

+ ------ ⎝ ⎠ ⎛ ⎞r

2 =

dΦ2

dw

--------- 2 = ( ) λ + 4w r

u ( ) λ + + w εr – εu

du

dw

+ ------ ⎝ ⎠ ⎛ ⎞r

2 – 0 Λ( ) 2λ + 4w r =

u w = == ε 0
11

It has been shown that below , no solution can exist. Moreover, for

,

tends toward – ∞ for u and w → 0. The first term in equation (26) thus has very large values

for r → rmin. For these r values, solutions can exist only for very large Λ values. It is,

therefore, necessary to study the possible solutions in more detail. Once an arbitrary w can

be found for a given r and Λ, for which the function

then a solution w exists because also H → ∞ for w → wmax(r), according to equation (24).

In figure 5, H(w) is shown for and and a series of r values. From this

figure (and a common property of H), it can be concluded that H always increases with w.

A unique solution is, therefore, guaranteed once

Also shown in figure 5 is a diagram of H0(r) for and six values of ε. All curves

are hyperbolas with a vertical asymptote at and the second asymptote according to

The Lagrange multiplier Λ is the same for all r. If

solutions exist, according to the function H0(r) shown in figure 5, for all r down to the small

rmn for which

This limit is represented in figure 5, for and , by broken lines connecting the points H0(1) and H0(rmn).

For Λ > H0(1), this range can be slightly extended. In this case, the lower limit of the

solution range is

The range for which no solution exists is very small and, in practical applications, is

covered by the hub.

rmin = ελ

r rmin =

du

dw

------ εu + + 2λ 4w

u + εw –= --------------------------------

Huw ( ) ( ), , w r

u ( ) λ + + w εr – εu

du

dw

+ ------ ⎝ ⎠ ⎛ ⎞r

2λ + 4w = ------------------------------------------------------------------- < Λ

λ 0.2= ε 0.05=

H0 Hru ( ) , , 0= w 0=

( ) λ ε + r r

2λ

r – ελ --------------

2λ ------------------------------------- ( ) λ ε + r rr – ελ = == ---------------------- < Λ

λ 0.2=

r = ελ

H0 λ 1 ε

2 = ( )ε + + r

Λ H0( ) 1 ε λ +

1 – ελ = = --------------

H0 rmn ( ) H0 = ( ) 1

ε 0.01= ε 0.06=

rmn

Λ λ –

2ε ------------- Λ λ –

2ε ------------- ⎝ ⎠ ⎛ ⎞ 2

–= – λΛ
12

If a specified CT is to be achieved, the algorithm is slightly different. Instead of

selecting Λ values, Λ must be determined iteratively such that the desired CT is achieved.

Finite number of blades.- As in all “averaged” propeller theories, the discretization of

the infinite number of blades into B blades must be performed by concentrating the Bth part

of the vorticity surface onto one blade. The total circulation of this blade is

The lift due to this circulation on a blade element of length dR results from

equation (6) if dθ is replaced by 2π/B

where c(R) is the local chord of the blade. This yields, with ,

or, nondimensionalized,

(27)

This equation determines the chord distribution c(r) once the lift coefficient of the blade element is known.

The twist of the blade is determined by the angle φ. It shows the (average) direction

of the local flow. Relative to this direction, the airfoil must have the angle of attack needed to

generate the lift coefficient that was used to determine the chord according to equation (27).

If the chord is large, the velocities induced by the wake may change along the chord. The

induced velocities along the chord should then be calculated using another method (e.g.,

ref. 6).

RESULTS

Inviscid Solutions

The inviscid case is obtained by setting ε to zero in all the formulas. For two values

of λ, 0.026 and 0.2, only Λ was varied; CT and ur depend on Λ. For easier comparison

with the Betz solution, ur is shown in figure 6 normalized by dividing all ur values by

CT /CT,Betz. Thus, the ordinate is

Γ γR2π

B = ------

dL ργ

2π

B

------ ( ) V∞ + W 2 ( ) ωR U– /2 2 + RdR cl

1

2

--ρ ( ) V∞ + W 2 ( ) ωR U– /2 2 = = ( ) + cdR

γ = U

clc

4πUR

B V( ) ∞ + W 2 ( ) ωR U– /2 2 +

= -----------------------------------------------------------------------

cl

c

Rt

---- 4πur

B ( ) λ + w

2 ( ) r u – /2 2 +

= -----------------------------------------------------------

( ) ur n ur

CT Betz ,

CT

= -----------------
13

The solid lines represent the results of the present theory. The broken line represents the result

from the theory of Betz (ref. 1). The first and last values of CT are written next to the first

and last solid lines.

Several results are significant.

(1) The present theory converges to the Betz solution as CT → 0.

(2) For the lower λ ( ), relatively low values of CT already yield curves that

deviate considerably from the Betz solution. The curve for is the seventh

one from the Betz solution.

(3) For the higher λ ( ), the deviation from the Betz solution is smaller although still

not negligible.

Optimum propellers for finite loadings have lift distributions shifted toward the blade

tip compared to the classical solution. The highest CT shown in figure 6 yields very large

chords, typical of ship propellers.

Effect of Viscosity

The viscosity is embodied by the parameter ε, which is the drag-to-lift ratio of the airfoil. The design of propellers for low Reynolds numbers must generally consider ε values

greater than 0.03. Propellers having rough leading edges may also have an ε value of 0.015

at higher Reynolds numbers. The following examples are representative of a two-bladed propeller for a balloon operating at very high altitudes (see table I). At the specified altitude, the

very low air density leads to Reynolds numbers around 12,000, despite the required large

chords. Therefore, ε values up to 0.05 have been evaluated. The thrust coefficient CT and

the nondimensional forward velocity λ are given in each example.

In figure 7, λ corresponds to the specified, low velocity of the balloon. The optimum

propeller, including viscous effects, has reduced lift near the blade tip and increased lift

inboard of about 75-percent radius. The difference from the inviscid solution is not dramatic

but also not negligible.

The effects of a higher λ value and different CT values are shown in figure 8. For

the lower CT value, the effect of viscosity is even greater than for the low λ value. Increasing CT decreases the effect.

Optimum Propellers

If no parameters are varied, not only clc/Rt but also the twist angle φ and the Reynolds number R are plotted, as shown in figure 9 for the two-bladed propeller at the specified

forward speed.

0.026=

CT 0.009=

0.2=
14

The given tip speed Ut is a driving parameter. The results are:

(1) The power required for all three tip speeds is within the specified limit.

(2) The power required decreases with decreasing tip speed.

(3) The local blade chords increase with decreasing tip speed.

(4) The Reynolds numbers increase with decreasing tip speed.

The planform of the propeller with the lowest power required (fig. 9(a)) exhibits large

chords inboard (e.g., for cl = 1, c > 2 m). The incorporation of these large chords inboard

requires careful consideration. In this region, the blade design is determined by structural

requirements, as is typical. Also, although the Reynolds number is extremely low, because

this region contributes little to the total thrust and power required, no aerodynamic concerns

arise.

Propeller Analysis

The optimum propellers have been developed using a propeller theory that does not

consider the velocities induced by the vortices in the jet downstream of the propeller. The

most and the strongest vortices are shed from the trailing edge of the propeller blades near the

tips. By neglecting these vortices, a ducted fan is considered rather than a free propeller. This

is also true for the fundamental method of reference 1, which was corrected by the supplement

in the same reference. The corrected method is widely used today (e.g., ref. 2) and has been

employed to modify the tip of the optimum propeller blade. Accordingly, the method of reference 6, referred to as the Zerle code in this report, has been used to investigate the development of the vortex wake produced by the propeller.

Vortex lattice and vorticity layer.- The wake downstream of a wing or propeller blade

is a layer of vorticity having finite thickness. It would best be described by the Navier-Stokes

equations. A viscous, computational fluid dynamics (CFD) code has yet to accurately treat

the vortex wake downstream of a propeller or helicopter rotor, however. Alternatively, several

methods employ an inviscid idealization in which the vorticity layer of the wake is represented by a two-dimensional, vortex surface. The movement of this vortex surface is, however, difficult to compute. Accordingly, the vortex surface is usually discretized by a vortex

lattice, as in the Zerle code.

The representation of the vortex surface by a vortex lattice has several shortcomings.

(1) Mathematically, a single vortex is represented by a potential flow, which is irrotational. The (possibly curved) axis of the vortex represents a strong singularity in this flow.

For example, a vortex of finite strength contains an infinite amount of energy. This singularity is normally eliminated by assuming a vortex core of finite diameter, where the

velocity tends toward zero instead of infinity. Only outside this core is the flow irrotational. The radius of the core influences the results, however.
15

(2) A mathematical vortex with no core induces a velocity at any point outside the vortex

according to the well-known Biot-Savart formula. The term “induced velocity” is somewhat misleading, however. This velocity must exist if the flow outside the vortex is irrotational and, therefore, a potential flow. If the vortex is a straight line of infinite length, the

Biot-Savart formula yields the simple, two-dimensional, vortex flow, in which the streamlines are circles around the vortex axis and the velocity is inversely proportional to the

radius of the circle.

(3) A curved vortex induces an infinite velocity at any point on itself. The vortex would

then move with this velocity. This is again prevented by the vortex core but the numerical

results may vary greatly with core radius.

(4) If a nonsingular potential flow is superposed on a vortex flow, a force acts on each element of the vortex proportional to the strength of the vortex and the local velocity of the

superposed flow. This force is also infinite for a curved vortex with no core.

(5) The vortex surface (or layer) downstream of a wing or propeller blade rolls up aft of

the tips and, thereby, no part of the vortex surface can penetrate any other part of the vortex surface. The cross sections through the wake become spirals. The vortices of the vortex lattice also exhibit the typical roll up but they penetrate the vortex lattice representing

the vortex surface. This leads to a chaotic mixing of the vortices that does not model the

roll up of the vortex surface. Moreover, the roll up of the vortex lattice begins earlier and

much faster than the roll up of the vortex surface.

All these shortcomings raise the question whether reliable results can be obtained from

a vortex-lattice method. Many valuable results have been presented, however, for fixed

geometries and fixed wakes without relaxation. In this case, the induced velocities are only

considered at the center of the doublet elements, which never coincide with a flow singularity.

If the velocity normal to the blade surface is required to be zero at these points, the total “leakage” through the vortex lattice is small because the positive and negative contributions tend to

cancel each other. This is still true if a vortex wake is assumed to have a fixed form. Once the

wake is to be adapted to the local induced velocities, however, the shortcomings become

almost overwhelming.

Zerle code.- To determine the differences between the results from blade-element theory and the propeller with a free wake, it is most important to know the induced velocities

from the wake onto the propeller blade. These velocities depend on the shape of the wake. It

is known that the jet behind a propeller contracts and increases its downstream velocity. If the

propeller moves in the direction of its axis, the vorticity vector always has the direction of the

local velocity vector.

The shape of the wake is thus determined by the velocity field downstream of the propeller. This velocity field is composed of the velocities due to the movement of the propeller

and the induced velocities from the propeller and the wake. No velocity component perpendicular to the surface of the vortex wake is permitted. The solution of this problem normally

requires an iterative approach because the velocity field depends on the lift of the propeller
16

blades and the shape of the wake whereas the lift of the blades and, therefore, the vorticity

strength of the wake depend on the induced velocities.

The Zerle code starts with the given shape of the blade, although the blade sections

have no thickness. The blade and the wake are represented by panels having constant doublet

strength and straight edges, which correspond to vortex rings.

Instead of the iterative method previously discussed, the Zerle code starts from the

propeller with no wake and then develops the wake in time steps. After each time step, the

wake is updated. The wake elements from the preceding time step move with the local velocities and a new row of wake elements is introduced at the trailing edge of the blade. The new

elements contain only longitudinal vortices if the circulation at the radius has not changed. If

the circulation has changed, the elements also contain lateral vortices.

In calculating the strength of the doublet panels of the blade, all induced velocities

from the wake and the blade panels are considered. All vortices represent singularities in the

flow because the induced velocity of a vortex element tends toward infinity with decreasing

distance from the vortex element. This is, as previously mentioned, no problem for fixed

blade geometries and fixed wakes. For a free wake, however, the distance between the vortices may become very small or even zero. For these vortices, a so-called damping radius is

therefore introduced, within which the induced velocity tends toward zero instead of infinity.

This does not, however, prevent the chaos that occurs if vortices penetrate the vortex lattice,

unless the damping radius is very large. In this case, the essential effects of the wake movement are lost. For calculating the displacement of the wake elements, a small damping radius

is also used for the blade vortices.

The greatest difficulty for vortex-lattice methods, including the Zerle code, is the computation of the free wake. The shape of the vortex lattice, in which all vortex elements influence the movement of each other, is much more unstable than the shape of a vorticity surface,

which, in turn, is much more unstable than a vorticity layer of finite thickness. Frequently, the

movement of the vortex lattice becomes chaotic and is no longer representative of the real

wake. The instability is greatest at the beginning of the wake development. After less than

one revolution of the propeller, the wake can become completely chaotic. The chaotic portion

of the wake moves downstream, however, and the wake being shed from the trailing edge is

less unstable. The forces on the propeller often converge to asymptotic values. The induced

velocities from the wake onto the blades decrease rapidly with distance from the propeller.

Accordingly, a maximum lifetime tmax of the vortex elements has been introduced, after

which their induced velocities are neglected. In other words, an artificial termination of the

wake has been introduced. To avoid errors near the artificial terminus, tmax must be large

enough that the evaluation of the forces on the propeller will result in very little error when

elements having a lifetime greater than tinfl (< tmax) are neglected. The most important criterion for the validity of the results is the convergence of the forces on the propeller to asymptotic values with increasing time.

Modifications to Zerle code.- The number of panels in the Zerle code is limited to 6

chordwise and 20 spanwise, which is insufficient for the current problem. Therefore, the
17

number of panels has been increased to 16 chordwise and 32 spanwise. Moreover, nondimensional variables have been introduced in the same way as previously described. Simultaneously, the code was converted to double precision from single precision because of the

increased number of panels and, correspondingly, the size of the linear equation systems.

The major difficulty of the Zerle code is the unstable shape of the vortex lattice of the

wake. The instability is increased by the finer panelling. In two regions, the instability is very

persistent: the center of the wake and the roll up at the outer edge of the wake.

The following modifications to the Zerle code improve the behavior of the wake such

that, in most cases, an asymptotic convergence is achieved.

(1) The amount of rotation between two vortex elements is limited, which is effective primarily near the outer edge of the wake where it retards the roll up.

(2) During the development of the wake, the lift on the blade varies. The resulting lateral

vortices in the wake increase the instability of the wake near the propeller axis. This is

reduced by neglecting the shedding of lateral wake vortices for varying lift. This does not

introduce an error in the asymptotic case, where the lift is constant.

(3) A maximum lifetime tmax of the vortex elements of the wake (typically, equivalent to

four revolutions) is introduced, after which all induced velocities from the elements are

neglected. The wake, which extends downstream to infinity, is artificially terminated at

tmax.

(4) An influence time tinfl (< tmax) is introduced, after which the induced velocities from

the elements onto the blades are neglected.

(5) The total time for which the development of the wake is computed is always greater

than tmax. A considerable portion of the wake has then departed the computation domain.

A total time corresponding to six propeller revolutions is usually sufficient.

Results of analysis.- The two-bladed propeller exhibiting the highest performance

(fig. 9(a)) has been evaluated using the modified Zerle code. The tip speed Ut

 is 50 m/s and

the radius Rt is 5 m. The modification of the tip according to reference 2 increases the radius

to 5.2 m and the tip speed to 52 m/s. A circular arc was used for the airfoil shape. An angle of

attack of 3 degrees is necessary for this airfoil shape to reach the desired lift coefficient of 1.

The chord of the propeller blade has been reduced slightly near the axis. This region will be

determined by structural requirements, in any event. The propeller and the computational

panels are shown in figure 10 as an axonometric projection and a plan view.

The development of the wake is shown in figure 11 after one through six revolutions;

24 time steps per revolution have been computed. (Note that the wakes have been terminated

at tinfl, equivalent to three revolutions in this case.) The projection direction is the longitudinal axis of the propeller, which shows primarily the profiles of the propeller. It should be

noted that the airfoils are laid out on cylinders and appear, in this projection, not in their exact

shapes. After one revolution, the wake seems to develop regularly, although its downstream

velocity is too low and some irregularities occur in the center of the wake. After two revolu-
18

tions, the downstream velocity of the wake has increased and the irregularities in the center of

the wake have grown into a chaotic region. After three revolutions, the chaotic region has

increased in extent and moved downstream. After four revolutions, a portion of the chaotic

region has already exceeded the time range of three revolutions shown. It has not yet, however, exceeded the range where it influences the shape of the wake. This is also true after five

and six revolutions, where the wake gradually becomes regular.

After each time step, the forces and moments are evaluated. During the final revolution, the thrust varies between 21.0 N and 21.3 N, which is near the design requirement.

All the results from the inviscid and viscous theories have been compared to those

from the simple formulas of the momentum or jet theory of propellers (e.g., refs. 7 and 8).

The very low nondimensional forward velocity (i.e., advance ratio) of the present propeller is

close to the static case, in which the power P and the thrust T of the propeller are related by

the formula

The thrust from the momentum theory cannot be realized because this theory neglects the

rotational energy of the propeller jet. Accordingly, an efficiency η is introduced into the

above formula

For all results from the inviscid theory, η is slightly less than 1, which demonstrates that the

details of the vortex wake influence the efficiency very little.

The viscous theory yields . Values of are typical of propellers

that are not designed for the static case. Thus, the value for the present propeller appears to be

reasonable.

The local induced velocities from the wake onto the propeller blade can also be evaluated from the asymptotic wake. Qualitatively, they do not differ greatly from the values computed by the present method. The thrust agrees well and both methods exhibit higher

downstream velocities toward the center of the wake.

A decambering effect on the airfoil may be present because the aspect ratio of the

blade is low and the induced velocities from the wake are lower at the leading edge than at the

trailing edge of the blade. The airfoil must then have more camber than in the infinite, straight

flow assumed in theoretical and experimental airfoil evaluations.

Because of the very low Reynolds numbers, the uncertainties in the section characteristics of the airfoil are at least of the same order of magnitude as the uncertainties in the propeller theories. The final propeller shape can be determined once the aerodynamic

performance of the airfoil is known.

T3 2πρRt

2

P2 =

T η 2πρRt

2

P

2 3 =

η = 0.753 η ≈ 0.7
19

AIRFOILS

DESIGN

Objectives and Constraints

The airfoil design specifications, which are derived from the propeller performance

requirements, are contained in table II. Two primary objectives are evident. The first objective is to achieve a high lift-to-drag ratio for a Reynolds number of 10,000. The second objective is to provide a reasonable range of lift coefficients over which no significant separation

occurs for the same Reynolds number. This range is intended to provide a margin against

such contingencies as manufacturing tolerances, operational deviations, three-dimensional

effects, and inaccuracies in the theoretical methods. Note that, because the free-stream Mach

number for all relevant operating conditions remains below 0.2, the flow is considered incompressible (i.e., M = 0).

Two major constraints were placed on the airfoil design. First, the zero-lift pitchingmoment coefficient must be no more negative than –0.20. Second, the airfoil thickness must

be at least 3-percent chord for structural reasons.

In addition to the airfoil specified by these requirements, designated the primary airfoil, two thicker airfoils are desirable for the root region of the blade because of structural considerations.

Philosophy

Given the above objectives and constraints, certain characteristics of the design are

apparent. To achieve a high lift-to-drag ratio (i.e., a low ε), it is more efficacious to pursue

higher lift coefficients than lower profile-drag coefficients, especially because the very low

Reynolds numbers will likely result in laminar flow over the entire airfoil. Thus, at the maximum lift coefficient, the entire upper surface should be used for the pressure recovery. The

same is true for the lower surface at the minimum lift coefficient. To achieve a reasonable

range of lift coefficients without significant separation, the airfoil thickness should be

increased, although the thickness is constrained by the goal of higher lift coefficients.

From the preceding discussion, the pressure distributions along the drag polar can be

deduced. The pressure distribution near the maximum lift coefficient for the primary airfoil
20

should look something like sketch 1. (The pressure distributions for the root airfoils should be

qualitatively similar.)

Sketch 1

To achieve a high maximum lift coefficient without significant separation, essentially the

entire upper surface is used for the pressure recovery. Because the very low Reynolds number

will result in laminar flow over the entire upper surface, the adverse pressure gradient must be

shallow to avoid laminar separation. The steep, adverse pressure gradient aft of about 90-

percent chord is a “separation ramp,” originally proposed by F. X. Wortmann,1

 which confines

separation to a small region near the trailing edge. By constraining the movement of the separation point at high angles of attack, high lift coefficients can be achieved with little drag

penalty. This feature has the added benefit of initiating docile stall characteristics. (See

ref. 9.)

The amounts of pressure recovery on the upper and lower surfaces are determined by

the airfoil-thickness and pitching-moment constraints and the objective of a reasonable range

of lift coefficients without significant separation.

1Director, Institute for Aerodynamics and Gas Dynamics, University of Stuttgart, Germany.
21

The pressure distribution along the lower surface near the minimum lift coefficient

should be similar in nature to that along the upper surface near the maximum lift coefficient,

as illustrated in sketch 2.

Sketch 2

Execution

Given the pressure distributions previously discussed, the design of the airfoils is

reduced to the inverse problem of transforming the pressure distributions into airfoil shapes.

The Eppler Airfoil Design and Analysis Code (refs. 3 and 4) was used because of its unique

capability for multipoint design and because of confidence gained during the design, analysis,

and experimental verification of many other airfoils. (See refs. 10–12, for example.)

The primary airfoil, which corresponds to the radii outboard of that for the maximum

local blade chord (i.e., ), is designated the SE403. The root airfoils, the SE404 and

SE405, were derived from the SE403 airfoil to increase the aerodynamic and geometric compatibilities of the three airfoils. The airfoil shapes are shown in figure 12 and the coordinates

are contained in tables III, IV, and V. The SE403 airfoil thickness is 4.66-percent chord; the

SE404, 7.01-percent chord; and the SE405, 9.32-percent chord, which satisfy the design constraints.

THEORETICAL PROCEDURE

The pressure distributions and section characteristics are predicted using the method

of references 3 and 4 for Reynolds numbers of 7,000, 10,000, and 15,000. Because the freestream Mach number for all relevant operating conditions remains below 0.2, all results are

incompressible. Because of the very low Reynolds number, it is unlikely that leading-edge

r ≥ 0.17
22

roughness can force transition and, therefore, all results are computed with transition free (i.e.,

smooth) using a critical amplification factor of 11.

For comparison, the section characteristics of the SE403 airfoil are also predicted

using the method of reference 5. The computations are performed with transition free using a

critical amplification factor of 9.

RESULTS

Pressure Distributions

The inviscid pressure distributions for the SE403 airfoil at various angles of attack are

shown in figure 13.

Section Characteristics

The section characteristics of the SE403 airfoil predicted using the method of references 3 and 4 are shown in figure 14, along with the separation locations. At 75-percent

radius, for which the Reynolds number is approximately 12,800, the lift-to-drag ratio at a lift

coefficient of 1 is about 27. Thus, the drag-to-lift ratio is about 0.036. The airfoil exhibits little separation over the range of lift coefficients from 0.75 to 1.03, which meets the design

objective. (The small, trailing-edge separation predicted on the upper surface is caused by the

separation ramp; see fig. 13.) The zero-lift pitching-moment coefficient is predicted to be

–0.22, which exceeds the design constraint. Because of boundary-layer displacement effects

not accounted for in the analysis, the pitching-moment coefficient is probably overpredicted

by at least 20 percent. Therefore, the actual zero-lift pitching-moment coefficient should less

negative than –0.18, which satisfies the design constraint.

The effect of Reynolds number on the section characteristics of the SE403 airfoil is

summarized in figure 15. Because the flow is essentially completely laminar due to the very

low Reynolds numbers, laminar rather than turbulent separation affects the lift and pitchingmoment coefficients. Because laminar separation is independent of Reynolds number, the

zero-lift angle of attack, the lift-curve slope, and the pitching-moment coefficient as well as

the minimum and maximum lift coefficients are unaffected by Reynolds number. Only the

profile-drag coefficient is affected.

Because it is unlikely that leading-edge roughness can force transition, the effect of

roughness on the section characteristics is probably negligible.

The effect of Reynolds number on the section characteristics of the SE404 airfoil is

summarized in figure 16. The characteristics are similar to those of the SE403 airfoil, except

for the extensive separation on the lower surface. This separation becomes so massive on the

lower surface of the SE405 airfoil that the predicted section characteristics are not reliable.
23

Comparison with Other Theoretical Results

The section characteristics of the SE403 airfoil predicted using the method of references 3 and 4 (PROFIL05) are compared with those predicted using the method of reference 5

(MSES) in figure 14. Although the results do not agree well, for such very low Reynolds

numbers, it is remarkable that they are at least similar.

INTEGRATION

The SE403 airfoil is applied to the propeller blade from the radius corresponding to

the maximum local chord ( ) outboard to the blade tip ( ). The SE404 and

SE405 airfoils may be applied inboard of this radius as dictated by structural requirements.

The airfoils should be laid out at a given radius along the curve defined by the intersection of

a line inclined at the corresponding twist angle and a cylinder having the given radius. Note

that the twist angle is relative to the zero-lift angle of the airfoil, not the x-axis.

CONCLUDING REMARKS

The present propeller optimization theory is, in principle, based on blade-element theory. Unlike other blade-element theories, however, the velocities and pressures are not averaged over the blade-element cylinders. Instead, the blades themselves are “averaged” by

distributing them into an infinite number of infinitesimal blades, which yields a vorticity surface that can be treated exactly.

The theory allows optimum propellers to be designed for given thrust coefficients and

advance ratios. The local airfoil drag is considered by a specified drag-to-lift ratio of the

blade sections. For the case of no drag and the thrust coefficient going to zero, the present

theory yields the classical and commonly used results of Betz. A finite thrust coefficient

shifts the lift of the blades toward the blade tip, which is understandable because the advance

angle is lower there. For a propeller of a high-altitude balloon, the difference is not negligible.

The airfoil drag has the opposite effect, which is also understandable because overcoming the

blade drag requires more power as the distance from the propeller axis increases. The effects

do not compensate each other, however, because they have different distributions along the

radius.

The resulting propellers have a finite chord at the blade tip. This has been corrected as

described in the supplement by Prandtl to Betz’s paper. The modification increases the diameter of the propeller slightly but does not cause any further problems.

The shortcoming of any blade-element theory is that the induced velocities from the

vortex wake onto the blades are not explicitly regarded, but they are also not completely

neglected. The theory assumes only that the Bernoulli equation is valid along the streamlines

of the wake until the static pressure returns to free-stream. This is not guaranteed if the wakes

of the propeller blades roll up and form regions with strong vorticity and low pressure. The

r 0.17= r 1.00=
24

assumed flow satisfies the Euler equations for all streamlines that reach free-stream static

pressure before joining a low-pressure region. Therefore, the flow contains the induced

velocities from these streamlines. This conclusion is corroborated by the fact that the inviscid

solution from the present theory converges to the Betz solution, which is derived from an

assumed form of the wake flow.

The first generalization of the present theory should contain an option for specifying

section drag-to-lift ratio along the blade radius. When experimental results for the airfoil performance are available, it should be possible to evaluate the drag-to-lift ratio as a function of

the local Reynolds number.

In summary, the aerodynamic design of a propeller for the trajectory control of a highaltitude, scientific balloon has been performed using theoretical methods developed especially

for such applications. Optimum, nonlinear chord and twist distributions have been developed

in conjunction with the design of a family of airfoils, the SE403, SE404, and SE405, for the

propeller. The very low Reynolds numbers along the propeller blade fall in a range that has

yet to be rigorously investigated, either experimentally or theoretically.

Because the airfoil performance is the key to the propeller performance and because

current theoretical airfoil methods have not been validated for such low Reynolds numbers, it

is imperative that the SE403 airfoil be experimentally verified.

ACKNOWLEDGMENTS

This effort was sponsored by NASA Goddard Space Flight Center, Wallops Flight

Facility. Philip R. Ward served as the technical monitor.
25

REFERENCES

1. Betz, A.: Schraubenpropeller mit geringstem Energieverlust. Mit einem Zusatz von L.

Prandtl (Screw Propeller with Least Energy Loss. With a Supplement by L. Prandtl).

Nachrichten der K. Gesellschaft der Wissenschaften zu Göttingen, Mathematischphysikalische Klasse, 1919, pp. 193–217.

2. Larrabee, E. E.: Practical Design of Minimum Induced Loss Propellers. 790585, Soc.

Automot. Eng., May 1979.

3. Eppler, Richard: Airfoil Design and Data. Springer-Verlag (Berlin), 1990.

4. Eppler, Richard: Airfoil Program System “PROFIL00.” User’s Guide. Richard Eppler,

c.2001.

5. Drela, M.: Design and Optimization Method for Multi-Element Airfoils. AIAA Paper

93-0969, Feb. 1993.

6. Zerle, Lorenz: Aerodynamische und aeroakustische Rotorberechnung unter Anwendung

frei entwickelter Nachlaufwirbelschichten und retardierter Potentiale (Aerodynamic and

Aeroacoustic Rotor Calculation Using Freely Developing Wake-Vortex Sheets and

Retarded Potential). Fortschritt-Berichte VDI, Reihe 7, Nr. 399, 2000. (Ph.D. Dissertation, Univ. Stuttgart, 1999)

7. McCormick, Barnes W.: Aerodynamics, Aeronautics, and Flight Mechanics. Second ed.

John Wiley & Sons, Inc., 1995.

8. Eppler, R.: Strömungsmechanik. Akademische Verlagsgesellschaft (Wiesbaden), 1975.

9. Maughmer, Mark D.; and Somers, Dan M.: Design and Experimental Results for a HighAltitude, Long-Endurance Airfoil. J. Aircr., vol. 26, no. 2, Feb. 1989, pp. 148–153.

10. Somers, Dan M.: Subsonic Natural-Laminar-Flow Airfoils. Natural Laminar Flow and

Laminar Flow Control, R. W. Barnwell and M. Y. Hussaini, eds., Springer-Verlag New

York, Inc., 1992, pp. 143–176.

11. Eppler, Richard; and Somers, Dan M.: Low Speed Airfoil Design and Analysis.

Advanced Technology Airfoil Research – Volume I, NASA CP-2045, Part 1, 1979,

pp. 73–99.

12. Eppler, Richard; and Somers, Dan M.: Airfoil Design for Reynolds Numbers Between

50,000 and 500,000. Proceedings of the Conference on Low Reynolds Number Airfoil

Aerodynamics, UNDAS-CP-77B123, Univ. of Notre Dame, June 1985, pp. 1–14.
26

TABLE I.- PROPELLER OPERATIONAL SPECIFICATIONS

Parameter Value

Propeller radius ≤ 5.334 m (17.50 ft)

Shaft power P < 500 W

Forward velocity 1.29 m/s (4.22 ft/s)

Thrust (at forward velocity) T ≥ 22.7 N (5.10 lbf)

Altitude 36,576 m (120,000 ft)

Air density ρ 6.6486 × 10–3 kg/m3 (1.2900 × 10–5 slug/ft3)

Viscosity 1.5516 × 10–5 N·s/m2

 (3.2406 × 10–7 lbf·s/ft2)

Speed of sound 311.1 m/s (1021 ft/s)

Pressure 459.7 Pa (9.601 lbf/ft2)

Temperature 240.9 K (433.6° R)

Effective gravitational acceleration 9.696 m/s2 (31.81 ft/s2)

V∞
27

TABLE II.- AIRFOIL DESIGN SPECIFICATIONS

Parameter Objective/

Constraint

Reynolds 

Number R

Mach 

Number M

Lift-to-drag ratio cl /cd

As high as 

possible

Range of lift coefficients without significant 10,000 0

separation 0.2

Zero-lift pitching-moment coefficient cm,0 ≥ –0.20

Airfoil thickness t/c ≥ 3%
28

TABLE III.- SE403 AIRFOIL COORDINATES

Upper Surface Lower Surface

x/c y/c x/c y/c

0.000002 0.000090 0.000021 –0.000292

.000172 .001009 .000059 –.000459

.001019 .002848 .000121 –.000603

.005958 .008441 .000219 –.000728

.015025 .014890 .000273 –.000781

.028398 .021831 .000354 –.000849

.045973 .028997 .000710 –.001082

.067685 .036173 .001726 –.001508

.093345 .043172 .005559 –.002267

.122764 .049803 .016464 –.002433

.155726 .055925 .033284 –.001225

.191944 .061431 .055861 .001035

.231097 .066219 .084072 .004153

.272832 .070203 .117577 .007908

.316767 .073317 .156077 .012137

.362498 .075516 .199147 .016599

.409595 .076773 .246279 .021092

.457613 .077081 .296907 .025429

.506090 .076451 .350413 .029432

.554557 .074911 .406133 .032935

.602537 .072506 .463362 .035787

.649556 .069293 .521357 .037854

.695143 .065343 .579351 .039026

.738838 .060735 .636560 .039221

.780197 .055558 .692188 .038388

.818795 .049904 .745444 .036510

.854235 .043866 .795552 .033608

.886143 .037528 .841757 .029739

.914172 .030933 .883339 .024971

.938138 .023896 .919462 .019319

.958469 .016557 .949091 .013290

.975344 .009714 .971783 .007783

.988447 .004295 .987622 .003523

.996991 .001014 .996930 .000892

1.000000 .000000 1.000000 .000000
29

TABLE IV.- SE404 AIRFOIL COORDINATES

Upper Surface Lower Surface

x/c y/c x/c y/c

0.000018 0.000320 0.000008 –0.000197

.000278 .001495 .000099 –.000645

.000499 .002093 .000324 –.001054

.004860 .008157 .000680 –.001467

.013552 .015150 .001177 –.001914

.026667 .022656 .001697 –.002301

.044065 .030383 .008105 –.005147

.065663 .038103 .020239 –.007704

.091261 .045617 .037628 –.009399

.120661 .052729 .060001 –.010236

.153641 .059294 .087149 –.010136

.189909 .065200 .118764 –.008901

.229139 .070340 .154932 –.006448

.270972 .074625 .195443 –.003028

.315023 .077987 .239965 .001167

.360884 .080379 .288102 .005929

.408121 .081772 .339392 .011021

.456284 .082159 .393303 .016182

.504909 .081550 .449231 .021141

.553522 .079975 .506505 .025623

.601644 .077479 .564387 .029364

.648797 .074121 .622083 .032127

.694506 .069973 .678750 .033711

.738309 .065120 .733514 .033968

.779758 .059652 .785485 .032811

.818426 .053665 .833771 .030218

.853912 .047257 .877499 .026206

.885838 .040514 .915637 .020754

.913850 .033481 .946887 .014484

.937764 .025949 .970698 .008546

.958064 .018048 .987202 .003878

.974982 .010628 .996837 .000982

.988213 .004714 1.000000 .000000

.996915 .001115

1.000000 .000000
30

TABLE V.- SE405 AIRFOIL COORDINATES

Upper Surface Lower Surface

x/c y/c x/c y/c

0.000003 0.000137 0.000035 –0.000487

.000084 .000816 .000209 –.001091

.000513 .002292 .000510 –.001717

.003668 .007390 .001409 –.003020

.011900 .014966 .002638 –.004348

.024710 .023076 .010760 –.009901

.041905 .031397 .023554 –.015171

.063375 .039688 .040919 –.019836

.088906 .047743 .062544 –.023648

.118289 .055358 .088177 –.026292

.151294 .062384 .117577 –.027182

.187623 .068705 .151261 –.026072

.226942 .074209 .189190 –.023287

.268890 .078807 .231172 –.019027

.313077 .082427 .276953 –.013532

.359088 .085020 .326206 –.007100

.406486 .086556 .378515 –.000080

.454817 .087028 .433369 .007139

.503613 .086447 .490156 .014145

.552395 .084842 .548164 .020524

.600682 .082259 .606583 .025885

.647991 .078761 .664519 .029881

.693844 .074423 .721008 .032233

.737775 .069330 .775039 .032748

.779334 .063579 .825577 .031327

.818089 .057269 .871589 .027946

.853636 .050502 .911837 .022552

.885595 .043370 .944748 .015900

.913603 .035916 .969674 .009420

.937472 .027914 .986818 .004274

.957738 .019482 .996754 .001079

.974682 .011518 1.000000 .000000

.988014 .005128

.996849 .001216

1.000000 .000000
31Figure 1.- Fundamental parameters of blade-element theory of propellers.
32

Figure 2.- Vorticity surface for infinite number of propeller blades.
33Figure 3.- Cross section of flow tube.
34

Figure 4.- Ellipses from equation (15) for λ = 0.2 and ε = 0.05.
35

Figure 5.- Function 

H(w) for and and a series of equidistant 

r values; function 

H0(r) for 

values.

λ 0.2= ε 0.05=
36

Figure 6.- Normalized functions (ur)n of optimum propellers for and , with ; broke

solution (ref. 1).

λ 0.026= λ 0.2= ε 0=
37

Figure 7.- Effect of drag-to-lift ratio on lift-chord distribution of high-altitude propeller.
38

Figure 8.- Effect of drag-to-lift ratio on lift-chord distributions of optimum propellers with higher nondimensiona

different thrust coefficients.
39

(a) .

Figure 9.- Optimum propellers for high-altitude balloon.

Ut = 50 m/s
40

(b) .

Figure 9.- Continued.

Ut = 75 m/s
41

(c) .

Figure 9.- Concluded.

Ut = 100 m/s
42

Figure 10.- Axonometric projection and plan view of optimum propeller for high-altitude balloon.
43

Figure 11.- Development of free vortex wake of optimum propeller for high-altitude balloon.
44

Figure 12.- Airfoil shapes.
45Figure 13.- Inviscid pressure distributions for SE403 airfoil.
46

(a) R = 7,000.

Figure 14.- Section characteristics of SE403 airfoil.
47

(b) 

R = 10,000.

Figure 14.- Continued.
48

(c) R = 15,000.

Figure 14.- Concluded.
49Figure 15.- Effect of Reynolds number on section characteristics of SE403 airfoil.
50

Figure 16.- Effect of Reynolds number on section characteristics of SE404 airfoil.

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Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std. Z39-18 

03-01-2012 Contractor Report 14 December 2004 to 31 December 2005.

AERODYNAMIC DESIGN OF A PROPELLER FOR HIGH-ALTITUDE

BALLOON TRAJECTORY CONTROL

NNG05WC00G

 Eppler, Richard and Somers, Dan M.

Wallops Flight Facility, Wallops Island, VA 23337

Airfoils, Incorporated

122 Rose Drive

Port Matilda, PA 16870-7535

National Aeronautics and Space Administration

Washington, DC 20546-0001

NASA

CR-2012-215893

Unclassified-Unlimited, Subject Category: 02

Report available from the NASA Center for Aerospace Information, 7115 Standard Drive, Hanover, MD 21076. (443)757-5802

The aerodynamic design of a propeller for the trajectory control of a high-altitude, scientific balloon has been performed using

theoretical methods developed especially for such applications. The methods are described. Optimum, nonlinear chord and twist

distributions have been developed in conjunction with the design of a family of airfoils, the SE403, SE404, and SE405, for the

propeller. The very low Reynolds numbers along the propeller blade fall in a range that has yet to be rigorously investigated, either

experimentally or theoretically.

Propeller; SE403, SE404, SE405; airfoils; low speed; laminar flow; low Reynolds number

 Unclassified Unclassified Unclassified

Unclassified

51

 Magdi Said

(757) 824-1386