MECN 4016 Aerodynamics

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MECN 4016 Aerodynamics 06 June 2011

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MECN 4016 Aerodynamics

Assignment –

“Methods of Determining the Low Speed Downwash Angle on an Aft Tail ”

Jameson Bentley

Thando Tshabalala

06 June 2011



MECN 4016 University of the Witwatersrand

School of Aeronautical Engineering

1

Abstract

Various methods were applied to estimate the downwash angle of a wing at a very low Mach

number, and the effect this would have on the effective angle of attack of a tail aft of the wing. The

theory for the analysis is based around Prandtl’s Lifting Line Theory, where calculations were

performed using Microsoft Excel and Matlab. The theory was compared to empirical data obtained

from performing wind tunnel tests of the same downwash scenario. Further models were

established using Vortex Panel Method Tornado, run in Matlab, as well as a DATCOM analysis. From

the four methods of analysis it was found that the downwash angle produced by a rectangular

NACA-0015 (with chord 0.08m and span 0.48m at an airspeed of 33.57m/s) was a function of angle

of attack. This function was represented by a linear relationship as anticipated from the theory.

Furthermore it was validated that the influence coefficients were a function of wing geometry and

remained constant with changing angles of attack. Therefore the downwash angle varied with the

value for lift coefficient. It was also found that the spanwise location at a constant distance aft of the

wing changed the downwash angle because it changed both the lift co-efficient and the influence

coefficients. This relationship was rather more complex and required numerical iteration and matrix

methods to solve. The DATCOM analysis revealed a perfectly linear relationship between lift

coefficient and downwash angle and a similarly linear relationship between downwash and angle of

attack.





2

Declaration

Group declaration with joint task submitted for assessment

We, the undersigned, are registered for MECN 4016, Aerodynamics in the year 2011. We

herewith submit the following task,

“Methods of Determining the Low Speed Downwash Angle on an Aft Tail ”

in partial fulfilment of the requirements of the above course.

We hereby declare the following:

We are aware that plagiarism (the use of someone else’s work without their permission

and/or without acknowledging the original source) is wrong. We confirm that the work submitted herewith for assessment in the above course is our

own unaided work except where we have been explicitly indicated otherwise.

This task has not been submitted before, either individually or jointly, for any course

requirement, examination or degree at this or any other tertiary educational institution.

We have followed the required conventions in referencing the thoughts and ideas of others.

We understand that the University of the Witwatersrand may take disciplinary action against

us if it can be shown that this task is not our own unaided work or that we have failed to

acknowledge the sources of the ideas or words in our writing in this task.

Signed this, the ____________ day of ________________ in the year___________.

Student number Student name Signature % contribution

324628 Thando Tshabalala 50.0

0616194H Jameson Bentley 50.0





3

Contents Abstract ............................................................................................................................................ 1

Declaration ........................................................................................................................................ 2

List of Figures .................................................................................................................................... 5

List of Tables ..................................................................................................................................... 6

1. Introduction .............................................................................................................................. 7

1.1. Background Information .................................................................................................... 7

1.2. Literature ........................................................................................................................... 9

The Downwash Aft of an Unswept Wing (3) ............................................................................... 9

2. Objectives ................................................................................................................................ 17

3. Analysis ................................................................................................................................... 18

3.1. Prandtl’s Lifting Line Theory ............................................................................................. 18

3.2. DATCOM .......................................................................................................................... 20

4. Experimentation ...................................................................................................................... 21

4.1. CFD (Tornado) .................................................................................................................. 21

4.2. Wind-tunnel Test ............................................................................................................. 22

4.2.1. Apparatus ................................................................................................................. 22

4.2.2. Procedure................................................................................................................. 25

4.2.3. Observations ............................................................................................................ 25

5. Results ..................................................................................................................................... 26

6. Discussion ................................................................................................................................ 30

6.1. Thando Tshabalala ........................................................................................................... 30

6.2. Jameson Bentley .............................................................................................................. 30

7. Conclusions and Recommendations ......................................................................................... 34

7.1. Thando Tshabalala ........................................................................................................... 34

7.2. Jameson Bentley .............................................................................................................. 34

8. References ............................................................................................................................... 36





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9. Appendix ................................................................................................................................. 37





5

List of Figures

Figure 1: Variation of Downwash with lift coefficient ......................................................................... 7

Figure 2: Variation of maximum downwash angle in the symmetry plane with Cl for a Clark Y aerofoil

.......................................................................................................................................................... 8

Figure 3: Comparison of calculated and experimental downwash at the tail of the midwing-

monoplane model (Silverstein, S and Bullivant 1939) ......................................................................... 9

Figure 4: Prandtl's model for the bound vorticity and the trailing vortex sheet generated by a finite

wing (Phillips, et al. 2002) ................................................................................................................ 10

Figure 5: Schematic of the vorticity rollup behind a finite wing with elliptical planform shapeb

(Phillips, et al. 2002) ........................................................................................................................ 11

Figure 6; the vortex used for estimating the downwash a few chord lengths or more aft of an

unswept wing (Phillips, et al. 2002).................................................................................................. 12

Figure 7: The wing tip vortex strength factor as predicted from the series solution to Prandtl’s lifting-

line theory (Phillips, et al. 2002) ...................................................................................................... 15

Figure 8: The wingtip vortex span factor as predicted from the series solution to Prandtl's lifting line

theory (Phillips, et al. 2002) ............................................................................................................. 15

Figure 9: The effect of tail position on the downwash angle in the plane of symmetry aft of an

unswept wing (Phillips, et al. 2002).................................................................................................. 16

Figure 10: Plan view of the Continuous Wind Tunnel used for the test............................................. 22

Figure 11: The configuration of the wing and tail as set up in the continuous wind tunnel ............... 23

Figure 12: Bubble inclinometer ........................................................................................................ 24

Figure 13: Downwash angle versus semi-spanwise position, using Prantdl’s LLT .............................. 26

Figure 14: Variation in downwash angle with AoA for both Prantdl’s LLT and the windtunnel test ... 27

Figure 15: Relationship between downwash angle and lift coefficient according to Prantdl’s LLT, and

the DATCOM analysis ...................................................................................................................... 27

Figure 16: Lift coefficient versus AoA for Prantdl’s LLT compared to the theoretical value. .............. 28

Figure 17: Lift coefficient versus AoA for the wind tunnel experiment compared to the theoretical

value. .............................................................................................................................................. 29

Figure 18: Spanwise variation in lift coefficient produced by Vortex Panel Method Tornado. ........... 38

Figure 19: Spanwise variation in normalized force over the surface of the wing, according to panel

methods run in Tornado. ................................................................................................................. 39





6

List of Tables

Table 1: Results as published from DATCOM ................................................................................... 20

Table 2: Geometric characteristics of the wing and tail sections ...................................................... 23

Table 3: Group member responsibility ............................................................................................. 37





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1. Introduction

1.1.

Background InformationIt has long been known that the angle of downwash, as observed at a given point behind the

aerofoil, is directly proportional to the lift of the aerofoil (Br. A.C.A. R. & M. No.196) and inversely

proportional to the aspect ratio (Lanchester “Aerial Flight” Vol. 1, Chapter 8, Br. A.C.A. R. & M.

No.191).

This has been determined experimentally by Hunk and data from five series of downwash

determinations have been plotted below, with angle of downwash as ordinates and lift coefficients

as abscissa. It is evident from inspection of the graph below that the downwash angle varies directly

with lift coefficient. The validity of the results obtained is obviously confined to that range of angle

of attack of lift coefficient in which the flow about the aerofoil in which the flow is not abnormally

turbulent. (Diehl 1921)

Figure 1: Variation of Downwash with lift coefficient





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Similar result where also found by Silverstein whereby if it was assumed that, for a wing without

twist, the span load distributions ( or circulation distribution) are similar at all angles of attack, itfollows that downwash should be proportional to the lift coefficient. This is proportionality is shown

in (L and E 1999).

Silverstein obtained downwash angles experimentally by comparing tail-off pitching moments with

tail-on pitching moments obtained at different stabilizer settings. The stabilizer settings

corresponding to zero load on the tail were found by interpolation or extrapolation. From these

values, the corresponding angles of attack of the airplane, and the jet-boundary corrections, the

downwash angles are derived. The agreement between theory and experiment is shown in (Diehl

1921) to be satisfactory except at higher angles of attack, where the tips are stalled. This was seen as

favourable as favourable as the model used was well streamlined and had a relatively small fuselage.

It is likely that, similar to this example, interference will be small in modern carefully streamlined

airplanes.

The theoretical computations where done by means of the Biot-Savart equation, the theoretical

span load distribution, and the lifting line concept. (Silverstein, S and Bullivant 1939)

Figure 2: Variation of maximum downwash angle in the symmetry plane with Cl for

a Clark Y aerofoil





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Figure 3: Comparison of calculated and experimental downwash at the tail of the midwing-monoplane model (Silverstein, S and Bullivant 1939)

1.2. Literature

The Downwash Aft of an Unswept Wing (Phillips, et al. 2002)

The well-known infinite series solution to Prandtl’s classical line equation applies to a single finite

wing with no sweep or dihedral, having an arbitrary spanwise variation in chord length. This solution

is based on the change of variables

Equation 1

The variation in section circulation along the span of the wing, as predicted by this solution, is

Equation 2





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Historically, the coefficients in this infinite series solution may have been usually evaluated from

collocation methods. Typically the series is truncated to a finite number of terms and the coefficients

in the finite series are evaluated by requiring the lifting-line equation to be satisfied at a number of

spanwise locations equal to the number of terms in the series. A very straightforward method was

first presented by Glauert (Glauert 1959). Most recently, Rasmussen and Smith (L and E 1999) have

presented a more rigorous and rapidly converging method, based on a Fourier series expansion

similar to that first used by Lots and Karamcheti (Karamcheti 1966).

Using Equation 1 and Equation 2and the following equation for bound vorticity

The spanwise variation of shed vorticity is

Equation 3

The downwash that is predicted directly from Equation 3 is not accurate in the region behind the

wing. This is because the development of Equation 3 is based on the assumption that the vortex

filaments trailing downstream from the wing are all straight and parallel to the freestream flow, is

shown in Figure 4. In reality, the vorticity trailing from each side of the wing will roll up around an

axis trailing slightly inboard from the wingtip, as is shown schematically for an elliptic wing in Figure

5.

Figure 4: Prandtl's model for the bound vorticity and the trailing vortex sheet generated by a finite

wing (Phillips, et al. 2002)





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Figure 5: Schematic of the vorticity rollup behind a finite wing with elliptical planform shapeb

(Phillips, et al. 2002)

The rollup of the vortex sheet trailing behind each semi-span of the wing can be viewed as a result of

the vortex lifting law (see Saffman (Saffman 1992)). This vortex lifting law requires that, in any

potential flow containing vortex filaments, a force is exerted on the surroundings that is

proportional to the cross product of the local fluid velocity with the local filament vorticity. Since a

free vortex filament cannot support a force, the cross product of the local fluid vorticity with the

local filament vorticity must always be zero at every point along a free vortex filament. This means

that all free vortex filaments must follow the streamlines of the flow everywhere. Thus, the free

vortex filaments trailing behind each semi-span of the wing will follow the streamlines and rollup

about the centre of vorticity shed from that semi-span. Within a few chord lengths behind the wing,

the vortex sheet becomes completely rolled-up to form wingtip vortices. This rollup has a significant

effect on the downwash.

Each wingtip vortex is generated from the trailing vortex sheet produced by one-half of the wing.

Therefore, a wingtip vortex, a few chord lengths or more behind the wing, can be approximated by a

single vortex of strength, wt, which is given by

Equation 4

Substituting equations Equation 1 and Equation 3 in Equation 4 gives

Equation 5





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Performing the indicated integration, we have

Equation 6

Figure 6; the vortex used for estimating the downwash a few chord lengths or more aft of an

unswept wing (Phillips, et al. 2002)

When comparing the downwash a few chord lengths or more downstream from a finite wing, we

can approximate the rolled-up vortex sheet as a single horseshoe shaped vortex filament of strength

wt, as shown in Figure 6. The distance between the trailing vortices, b’, is less than the wingspan

because the vortex sheet from each side of the wing rolls up around the centre of vorticity, which is

somewhat inboard from the wingtip. The horseshoe filament starts at an infinite distance

downstream from a point slightly inboard of the left wingtip, (∞,0,b’/2) and runs upstream along the

left wingtip vortex to the left wing, (0,0,b’/2). From here it runs across the quarter-chord to a point

slightly inboard of the right wingtip vortex to infinity, (∞,0, -b’/2). From the Biot-Savart law, the y-

velocity component induced at any point (x, y, z) by this entire horseshoe vortex is

Equation 7





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Since the vortex sheet shed from each semi-span of the wing rolls up about the centre of vorticity,

we have

Equation 8

Using Equation 4 in Equation 8 gives

Equation 9

Now, applying Eqs. 5, 7, and 10, this can be rewritten as

Equation 10

The integration with respect to in Eq. 14 is readily carried out to give

Equation 11

Using Eq. 15 in Eq. 14 results in

Equation 12

Because the downwash is small compared to the freestream velocity, the downwash angle, d, can

be approximated as the downwash velocity divided by the freestream velocity. Thus, applying Eq. 10





14

and Eq. 16 to Eq. 11 the downwash angle a few chord lengths or more downstream from an inswept

wing is approximated as

Equation 13

Where

Equation 14

Equation 15

Equation 16

Equation 17

Equation 18





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The dimensionless parameters Kv and Kb depend on the planform shape of the wing. For an elliptical

wing, all the coefficients, Bn, in the infinite series solution, except for the first are zero. Using this fact

with Equation 14, we find that Kv is 1.0 for an elliptic wing. Thus, from Equation 6 and Equation 14,

we see that the vortex strength factor, Kv, is the ratio of the wingtip vortex strength to that

generated by an elliptic wing having the same lift coefficient and aspect ratio. The vortex span

factor, Kb, is defined as the spacing between the wingtip vortices divided by the wingspan. Both Kv

and Kb were determined analytically from the series solution to Prandtl’s lifting-line equation. For an

elliptic wing with no sweep, dihedral, or twist Kv is 1.0 and Kb is π/4. For an unswept wing with no

dihedral or twist, Kv and Kb are related to the aspect ratio and taper ratio as is shown in Figure 7 and

Figure 8.

Figure 7: The wing tip vortex strength factor as predicted from the series solution to Prandtl’s

lifting-line theory (Phillips, et al. 2002)

Figure 8: The wingtip vortex span factor as predicted from the series solution to Prandtl's lifting

line theory (Phillips, et al. 2002)

The dimensionless parameter, Kp, is a position factor that account for special variations in

downwash. As a first approximation, the variation in downwash along the span of the horizontal tail

is usually neglected. The downwash for the entire tail is typically taken to be that evaluated at the





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aerodynamic centre. For asymmetric airplane, the aerodynamic centre of the tail is in the plane of

symmetry. The change in the downwash with respect to the spanwise coordinate is zero at the

aircraft plane of symmetry. Furthermore, the span of the horizontal tail is usually small compared to

that of the wing. Thus, the downwash is often fairly uniform over this span, and a reasonable first

approximation for the downwash on an aft tail is found by setting the dimensionless spanwise

coordinate, z, equal to zero in Equation 16. This gives the relatively simple relation

Equation 19

The tail position factor, Kp, depends on the planform shape of the wing and the position of the tail

relative to the wing. The variation of Kp with tail position in the plane of symmetry is shown in fig. 6.

The planform shape of the wing affects the value of Kp only through its effect on Kb. Thus, for a main

wing with no sweep or dihedral, the value of Kp in the plane of symmetry is a unique function of x/Kb

and y/Kb, as is shown in Figure 9.

Figure 9: The effect of tail position on the downwash angle in the plane of symmetry aft of anunswept wing (Phillips, et al. 2002)

Notice from Figure 7 and Figure 8 that the planform shape of the main wing has a very significant

effect on the downwash induced on an aft tail. Similar results were observed empirically by Hoak,

but are not accounted for in the model proposed by McCormick.





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2. Objectives

I. Experimentally estimate the low-speed downwash angle on an aft tail.

II. Compare the results obtained experimentally with theoretical computations found by the

following methods:

a) The modified analytical technique founded on Prandtl’s classical lifting line theory,

developed by W. F. Phillips, E.A. Anderson, J.C. Jenkins, and S. Sunouchi.

b) An empirical method.

c) Predictions found by Computational Vortex Panel Methods using Matlab-based code

Tornado.





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3. Analysis

3.1.

Prandtl’s Lifting Line Theory (Neiswander 2008)For the case of an unswept wing the spanwise position is given by the transformation:

= −2

cos

Where z is the spanwise co-ordinate and θ is the angle between the quarter-chord line of the wing

and the line connecting the wingtip to the root of the tail quarter-chord. It is also given that the

downwash velocity is represented by the fundamental equation:

() = − 1

4 Γ( − )

Converting to polar co-ordinates using the transformation equation and applying the definition of

circulation as a function of θ:

Γ() =

2

sin

It can be deduced after substitution that the downwash angle is given by the equation:

(,,) =

Where the coefficients are defined as:

= 1 + sin 2

=

4

+ ∑ ( − 1) cos 2

1 + ∑ sin 2





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=1 ( − ) + ( − ) 1 +

+ + ( − )+

+ − + + ( − )+

+

+ + ( + )+

( + ) + ( + ) 1 + + + ( + )

Where

=2

= 2

=2

are the non-dimensionalized reference position co-ordinates for measuring the downwash.

The value for aspect ratio, AR can be found using:

=

And the lift coefficient can be found using the relationship between itself and angle of attack,

= .

Where the angle of attack is the geometric angle of attack.

It is also worth noting that the kappa-coefficients are purely functions of the wing geometry, not of

the flow field conditions. The result of this is that the downwash angle varies as a linear function of

alpha (the angle of attack). The coefficients in the summation (the Bn’s) are required in order to

perform the calculation for the kappa-coefficients. This was not an easy task. The following

relationship was given for the calculation of the coefficients from the local geometric angle of attack:

()

(

)

⋮()=

⎣

, , ⋯ ,, ⋱ ,

⋮ ⋱ ⋮, , ⋯ ,⎦.

⋮





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The central g-matrix was determined and expanded in Microsoft Excel. However for the calculation

of the Bn’s that requires the inverse of matrix g multiplied by the angles of attack, i.e.

[] = []. [()]

This operation was input into Matlab from Excel and then plotted back into Excel to solve for the

kappa-coefficients. The results of this can be found in the sections that follow.

3.2. DATCOM

DATCOM is an empirical method that has the fundamental purpose is to estimate aerodynamic

stability and control characteristics in preliminary design applications. This computational method was use by first defining flight condition( Mach number, Altitude, and

angle of attacks to be analyzed), followed by synthesis which sets up the CG location as well as the

position of the wing and tail surfaces, also the wing planform is defined by using variables related to

wing type, span, chord, sweep and etc., and similarly the horizontal tail was also defined. This data was the used to find the tail and wing lift coefficients as well as the lift curve slope. The

values found were then used to calculate the downwash angle.

Table 1: Results as published from DATCOM

Cl Cl Wing Cl Tail alpha wing alpha tail ε deg

0.298 0.211 0.087 0.04014231 0.016552 0.023591 1.35165

0.45 0.329 0.121 0.06259156 0.02302 0.039572 2.267283

0.694 0.521 0.173 0.09911915 0.032913 0.066206 3.79334

0.865 0.656 0.209 0.12480262 0.039762 0.085041 4.872479

0.953 0.726 0.227 0.13811997 0.043186 0.094934 5.4393





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4. Experimentation

4.1.

CFD (Tornado)Tornado is a Vortex Panel Method that was run in Matlab in order to theoretically calculate the

effects of the angle of attack on downwash. First the parameters for analysis had to be input into the

program. This began by defining the panels over the surface of the airfoil that would be the basis for

calculation. The wing was analysed as a combination of 20 identical partitions from tip to tip. These

partitions were each one-twentieth of the span in width and had 10 identical panels segmenting the

upper surface and 10 identical panels segmenting the lower surface. The rule-of-thumb with panel

methods is the more panels that are used; the more the method replicates the actual surface. This

however comes at computational cost in the form of time-expenditure. It was defined also that the

partitions would each have zero dihedral, zero twist, zero flap deflection and zero change in airfoil

profile across the span. This was in accordance with the experimental test piece used in the wind

tunnel test. The span, chord length and profile were defined as those used in the wind tunnel and

the details can be found in table 2 in the following section. The origin was defined as the leading

edge at the wing root.

In addition to the geometry being defined, the flow conditions also had to be input. This was mainly

to define the speed of the airstream and what angle of attack the wing was at. Due to the amount of

time taken to perform a single simulation, the angle of attack was fixed at a reasonable 4°. This was

a fairly good point to choose to compare to the other methods because it is in a range where the lift

curve slope of the wing is roughly linear.

After the geometry and flow field conditions were input, the panel lattice was generated. This was

when the surfaces of the wing were discretized into the various panels that would be used for the

analysis. Following the lattice generation, the various simulations were performed. The flight

condition was classified as a steady, level flight path and as such no values were input for sideslip

angle, pitch angle or any of the angular rates. For this reason only the alpha sheet, force, lift and

coefficient distribution were simulated across the span. The results were obtained in the form of

graphs an tables as output by the Tornado code and can be found in the results section that follows.





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4.2. Wind-tunnel Test

4.2.1. Apparatus

Closed-circuit Wind Tunnel

The wind tunnel used for testing was a closed circuit, low speed tunnel driven by a single fan. The

test section was elliptical, fed by a nozzle and vacated through a diffuser as shown in the figure

below. The cross section of the test chamber was a 916 x 616 mm ellipse, with a clear Perspex

removable observation cover. Test holes were drilled through the sides of the wind tunnel fore and

aft the test piece for measuring pressures using a Pitot tube. The drive motor was an English Electric

DC motor with a rated maximum of 50 hp.

Figure 10: Plan view of the Continuous Wind Tunnel used for the test





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Wing and Tail Configuration

Table 2: Geometric characteristics of the wing and tail sections

Wing Tail

Profile NACA-0015 NACA-0018

Planform shape Rectangular Rectangular

Reference area [m2] 0.0384 0.0093

Span [m] 0.48 0.2

Chord Length [m] 0.08 0.0465

Figure 11: The configuration of the wing and tail as set up in the continuous wind tunnel





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Bubble Inclinometer and electr ic trim calibrator

The angle of attack of the test piece was calibrated using a device that electrically adjusts the trim of

the wing profile from a digital scale between -50 and 1500. At regular recorded intervals this number

was referenced against the actual measured angle of attack using the bubble inclinometer, and the

scale number could be calculated to give the estimated angle of attack. From this relationship it was

possible to observe the angle of attack at which the wing profile started to produce positive lift,

maximum lift and where the wing stalled.

Figure 12: Bubble inclinometer





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4.2.2. Procedure

Precautions before testing

1.

Ensure there are not obstacles within the wind tunnel.2. Ensure that all parts of the wind tunnel are secure.

3. Ensure that the test section door is properly closed before taking aerodynamic

measurements.

4. When varying the angle of attack, only move in one direction (do not increase then decrease

the angle of attack before the end of the test run).

5. Allow the air flow to settle before taking measurements.

Testing procedure

In order to determine the value of the downwash angle, that is, the difference between the

aerodynamic, or effective angle of attack, and the geometrical, actual angle between the tail centre

line and the direction of flight the following procedure was used:

1. The angle of attack was measured on the bubble inclinometer and the arm was set to this

value.

2. The door to the tunnel was then closed and bolted.

3. The tunnel was then started and the airspeed for the oncoming air flow was set.

4. Tail and wing lift was then measured separately by means of a strain gauges in terms of

strain.

5. This procedure was repeated for various angles of attack.

4.2.3. Observations

For low angles of attack there was excessive vibration of the airfoils, causing vast

fluctuations in the strain readings.

Past a certain angle of attack the wing became fully effective and the vibration slowed down

and taking readings was much easier.

The apparatus fairing could have played a role in affecting the airflow towards the tail, or

otherwise changing the pressure variation below the wing.





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5. Results

Figure 13 below shows the variation of downwash angle versus semi-spanwise position according to

Prantdl’s lifting line theory. In this figure the downwash angle is shown as positive downwash and

therefore any upwash is indicated as negative. For both angles of attack the zero up/down wash

locations are slightly inboard of the wingtip (wingtip @ 0.24m). This indicates the position where

airflow trying to move to the upper surface due to differential pressures cancels out the effect of the

air being forced to angle downwards due to the angle of attack. This point slightly inboard of the

wingtips indicates the centreline of the trailing vortex where zero wash conditions are expected.

Importantly the aft position of this up/down wash behaviour was input as the quarter chord position

of the tail. It is clear that with a tail semi span of 0.1m that the trailing vortex was never trailing over

the tail and therefore the tail was experiencing the downwash of the wing.

Figure 13: Downwash angle versus semi-spanwise position, using Prantdl’s LLT

Figure 14 shows the variation in downwash angle for changing angles of attack. According to the

theory the change in angle of attack should cause a linear response in downwash. This however was

assuming that the change in lift coefficient with changing angle of attack was linear, which it was

not. The downwash angle however did show a linear correlation to the lift coefficient as anticipated.

This can be seen in figure 15.

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.05 0.1 0.15 0.2 0.25 0.3 D o w n w

a s h [ d e g ]

Semi-span [m]

AoA=4

AoA=8





27

Figure 14: Variation in downwash angle with AoA for both Prantdl’s LLT and the windtunnel test

Figure 15 below shows the correlation between the downwash angle and lift coefficient. As

anticipated by the theory the relationship is perfectly linear for both the lifting line theory and the

DATCOM analysis. The difference in gradient is of concern, however the differences between the

experimentation and the theoretical analysis have shown to have discrepancies throughout. The

conclusion can only be drawn that the error between results is caused by a fundamental issue

relating to experimental error or inaccurate analysis.

Figure 15: Relationship between downwash angle and lift coefficient according to Prantdl’s LLT,

and the DATCOM analysis

0

0.02

0.04

0.06

0.08

0.1

0.120.14

0.16

0.18

0.2

0 5 10 15

D o w n w a s h [ d e g ]

AoA [deg]

LLT Downwash

EXP Downwash

0

2

4

6

8

10

12

0 1 2 3

D o

w n w a s h a n g l e [ d e g ]

CL

Downwash vs CL

DATCOM





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Figure 16 shows the variation in lift coefficient with angle of attack for Prantdl’s LLT compared to the

theoretical angle of attack. As is clear the LLT follows the theory until about 6.5 degrees after which

it deviates drastically. The reasons behind this could have stemmed from errors in the calculation of

the first series coefficient for varying angles of attack. This error could have come from the process

of performing the matrix inversion, where small errors become amplified. Also aspects of the theory

may have unknowingly incorporated small angle approximations that went unnoticed during lengthy

calculation. This may explain the cohesion for smaller angles of attack and deviation thereafter.

Figure 16: Lift coefficient versus AoA for Prantdl’s LLT compared to the theoretical value.

The experimental lift coefficient versus angle of attack is shown in figure 17 below. Unlike the LLT

the shape of the graph approximates the theoretical line far better despite having some local

random data points. These discrepancies can be put down to slightly inaccurate measurement of the

apparatus during the experiment and elastic movement of the rig during operation. Also the

calibration of the strain gauges was questionable, all of which could have led to the inaccurate

relationship arrived at in figure 17.

0

0.5

1

1.5

2

2.5

3

0 5 10 15

C L

AoA [deg]

CL vs alpha LLT

Theory





29

Figure 17: Lift coefficient versus AoA for the wind tunnel experiment compared to the theoretical

value.

Figure 18 and figure 19 show the spanwise lift distribution on the wing as simulated in Tornado. The

shape of the distribution correlates to that expected by theory of lift coefficient. The irregular shape

of the distribution is due to the fact that the program uses panel methods in order to iterate a

solution. However no results were released form Tornado regarding downwash due a defect with

the program. Also the value for air density could not be changed so it was run as standard air density

of 1.225 instead of the actual test condition of 1.02. These discrepancies and inadequacies with the

program rule the results obtained as inaccurate and unreliable, and as a result they have been

moved to the appendix.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15

C L

AoA [deg]

CL vs alpha EXP

Theory





30

6. Discussion

6.1.

Thando TshabalalaThe downwash was theoretically expected to be directly proportional to the lift of the airfoil, by

referring to figure 15 one can see that this has been confirmed by theoretical computational

methods used in DATCOM as well as Prandtl’s Lifting line theory as both graphs illustrate linear

behaviour between the downwash and lift coefficient.

The relation between downwash and angle of attack was also proposed to be linear both

experimentally and theoretically by means of figure 3. If one compare this to figure 14 one notes

that this is not what was found by the model used and the results obtained experimentally.

The variation in lift coefficient with angle of attack for Prantdl’s LLT compared to the theoretical

angle of attack may be viewed in Figure 16 .Prandtl’s model follows the theory until about 6.5

degrees after which the values for Prandtl become considerably higher . The reasons behind this

could have stemmed from errors in the calculation of the first series coefficient for varying angles of

attack in addition to the process of performing the matrix inversion, in which small errors become

amplified and rounding off errors also become more prominent. Also aspects of the theory may have

unknowingly incorporated small angle approximations that went unnoticed during lengthy

calculation. This may explain the cohesion for smaller angles of attack and deviation thereafter. It

also should be noted that stall is not taken into account at higher angles of attack (this occurs due to

separation) which is what would happen in reality.

The experimental lift coefficient versus angle of attack is shown in figure 17 below. Unlike the LLT

the shape of the graph approximates the theoretical line far better despite having some local

random data points. These inaccuracies can be attributed to slightly inaccurate measurement of the

apparatus during the experiment and elastic movement of the rig during operation. This also could

be due to the fact that not all tunnel interference effects could be eliminated as no correction was

introduced for the model being in a closed wind tunnel which results in the reduction of the wake

behind the model. This was not taken into account during theoretical analysis.

The calibration of the strain gauges was questionable, which could have led to the inaccurate

relationship arrived at in figure 17. This could also be due to the fact that the experiment was

performed a higher angles of attack in the region where aerodynamic theory may break down due to

separation. Another reason for these discrepancies may be due to the fact that effects of turbulence





31

were also not taken into account (there is always sound level of turbulence in real flows) which

causes more random behaviour in air flows. The approximate manner in which the lifting –line

theory deals with bound voracity may also causes data variations.





32

6.2. Jameson Bentley

The results obtained from the experimentation neither completely validated or completely

invalidated the results obtained from Prantdl’s Lifting Line Theory. However based upon the

research conducted and the generally understood theory, the lifting line method seemed to show

strong correlation to real-world practical examples regarding the airflow over airfoils. This

particularly true when observing figure 15, the relationship between downwash and lift coefficient.

Predictably the graph is a straight line, in accordance with the theoretical relationship, and despite a

contrasting slope to that of the DATCOM analysis, this still lends credibility and confidence to the

data surrounding the lifting line calculations (because both are perfectly linear). In figure 13 the

spanwise downwash is plotted, where it is observed that the downwash increases for increasing

angles of attack. This makes logical sense because lift increases with increasing angles of attack and

the downwash is a function of the lift at any particular spanwise station. This lends itself to another

well known relationship regarding lift, in that it is commonly understood that lift cannot be

produced without the presence of circulation (the Kutta-Joukowsky theorem). Considering that the

downwash angles are calculated based on the generation of circulation, this all correlates

appropriately. The matter of circulation however is not confined to a mere mathematical formality;

it is evident on figure 13. Near the wing tip there is a presence of negative downwash, or upwash.

Practically this represents the tendency of the high pressure air on the lower surface to move

towards the lower pressure air on the upper surface. This potential and resulting separation off the

trailing edge manifests itself as upwash at the tip. A conundrum now exists for the span in-between

the upwash at the tip and the downwash from approximately mid-span. The result is for the flow to

separate off the trailing edge a small spanwise distance inboard of the wingtip with a high-energy

rotational flow due to the difference in upwash and downwash meeting at a point. This forms the

trailing vortex. Importantly the theory suggests that if the wing had infinite aspect ratio and

therefore infinite span, there would never exist the point where the air would try to move from the

high pressure of the lower surface to the low pressure of the upper surface and therefore the trailing

vortex would never form. Practically however, we know this never to be the case.

Figure 14 shows the variation in downwash angle with increasing angles of attack for both the lifting

line theory and the experimental wind tunnel test. The result of this shows a close correlation

between the curve shape of each approach, only with a slight offset in the x-axis, meaning that in

the real case a higher angle of attack is required to induce the same downwash angle. This

discrepancy could be put down to a mistake regarding the calculated air density, a difference in the

ideal and actual airfoil surface roughness, or general experimental error.





33

The variation of lift coefficient with angle of attack is shown by figures 16 and 17, for the lifting line

theory and the experimental data respectively. The general trend is followed for small angles of

attack for the lifting line case; however the graph begins to deviate above approximately 6 degrees.

This could have been caused by amplified errors in the matrix inversion for calculating the series of

lifting line coefficients or just due to some mathematical inconsistency. For figure 17 the lift

coefficient versus angle of attack showed little correlation to the exact slope outlined by the theory.

Unfortunately very few data points were taken and this affected the resolution of the results. The

rough shape of the linear relationship is approximately followed. This discrepancy can be put down

to inaccurate experimentation techniques and questionable calibration data.





34

7. Conclusions and Recommendations

7.1.

Thando TshabalalaConclusions

The lifting line theory provides a sufficient basis for computation on the downwash angle behind

aerofoil without flaps on an aft tail but this concept breaks down at high angles of attack where

separation occurs.

The empirical model (DATCOM) approximately agrees with theoretical expectations but not

necessarily experimental due to approximations, inaccuracies and random behaviour due to

separation followed by turbulence mentioned above.

Abbot was also found to be theoretically very accurate for simulation.

Recommendations

It would be more ideal if the experiment was performed at lower angles of attack for better

correlation with theoretical data. And if it were possible to have performed the experiment at lower

speeds to model more laminar flow.

In addition to this a correction factor could have been introduced to take into affect wake effects in

the tunnel as well as changes in density being accounted for.

7.2. Jameson Bentley

Conclusions

Downwash angle varies linearly with coefficient of lift and spanwise location. It was also

found that the downwash at the point of trailing vortex formation was zero. This point was

slightly inboard of the wingtip.

Experimental wind tunnel testing can validate the theoretical data providing that methods of

experimentation are accurate.

The downwash produced by a wing makes a significant difference to the lift produced by a

tail aft of the main wing.





35

Prantdl’s Lifting Line Theory is an accurate and acceptable method for evaluating the

downwash produced by a wing, where the effect of the downwash can be found at varying

locations aft of the wing.

Recommendations

Perform the experiment knowing the geometry and profile information in greater detail.

Take far more data points for the experimental data.

Use a more sophisticated CFD program compared to Tornado in order to validate/invalidate

the results.

Use a generic program based on Prantdl’s Lifting Line Theory in order to validate/invalidate

the data obtained.

Perform the experiment using a cambered airfoil as well.





36

8. References

Barlow, J. B., W. H. Rae, and A Pope. Low-speed wind tunnel testing. Wiley, 1999.

Diehl, Walter S. “The Determination of Downwash.” National Advisory Comittee for Aeronautics,

1921.

Glauert, H. The Elements of Aerfoil an Airscrew Theory. Cambridge: Cambridge University Press,

1959.

Karamcheti, K. Ideal Fluid Dynamics. New York: Wiley, 1966.

L, Rasmussen M, and Smith D E. “Liftind-Line Theory for Arbitrary Shaped Wings.” Journal of Aircraft,1999.

Neiswander, Brian. Prantdl's Lifting Line Theory and Finite Wings. University of Notre Dame, 2008.

Phillips, W. F, Anderson E.A, J. C. jenkins, and S Sunouchi. “Estimationg the Low-Speed Downwash

Angle on an Aft Tail.” 40th Aerospace Sciences Meeting & Exhibit, 2002.

Saffman, P. G. Vortex Dynamins. Cambridge: Cambridge University Press, 1992.

Silverstein, Abe, Katzoff S, and Kenneth W Bullivant. “Downwah and Wake Behind Plain and Flapped

Airfoils.” National Advisory Committee for Aeronautics, 1939.





37

9. Appendix

Table 3: Group member responsibility

Section Person responsible

Background Thando Tshabalala

Literature survey: Research and write up Thando Tshabalala

Objectives Thando Tshabalala

Apparatus: Write up and diagrams Thando Tshabalala

Procedure for windtunnel Thando Tshabalala

Analysis: Prantdl Lifting Line theory Jameson Bentley

Analysis: Datcom Thando Tshabalala

Experimentation: Windtunnel data processing Jameson Bentley

Experimentation: Tornado Jameson Bentley

Results: Processing and interpretation Jameson Bentley

Discussion Individual

Conclusions and Recommendations Individual

Formatting, final write up, abstract Jameson Bentley

Compilation Jameson Bentley

EQUATIONS TO DESCRIBE THE PROPERTIES OF AIR

Air density as a function of Temp and Relative Humidity

= 0.0034847 ( − 0.003796) /

= (1.7526 × 10)(. / )

Where:

−

−





38

P – Air pressure in Pa

= [ %; 40% ℎ 40 ]

T – Temperature in K

Viscosity of air at a given temp

Used to calculate Reynolds number

=

+ 198.6

+ 198.6

Where: = °

= 518.6°

= 3.74 × 10 − / (Barlow, Rae and Pope 1999)

Figure 18: Spanwise variation in lift coefficient produced by Vortex Panel Method Tornado.





Figure 19: Spanwise variation in normalized force over the surface of the wing, according to panel

methods run in Tornado.

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MECN 4016 Aerodynamics