UNIVERSITY OF BELGRADE FACULTY OF MECHANICAL

UNIVERSITY OF BELGRADE

FACULTY OF MECHANICAL ENGINEERING

Taha A. Abdullah

TWO-DIMENSIONAL WIND TUNNEL MEASUREMENT

CORRECTIONS BY THE SINGULARITY METHOD

Doctoral Dissertation

Belgrade, 2014

UNIVERZITET U BEOGRADU

MAŠINSKI FAKULTET

Taha A. Abdullah

ODREĐIVANJE KOREKCIJA U DVODIMENZIONALNIM

AEROTUNELSKIM MERENJIMA METODOM

SINGULARITETA

doktorska disertacija

Beograd, 2014

Dedicated to my parents, wife and daughters

EXAMINATION COMMITTEE

Advisor: Prof. Dr. Zlatko Petrovic

Full Professor

University of Belgrade, Faculty of Mechanical

Engineering

Co-Advisor: Prof. Dr. Ivan Kostic

Associate Professor


Engineering

Members: Prof. Dr. Zoran Stefanović

Full Professor, retired


Engineering

Date of defence

Komisija za ocenu i odbranu disertacije:

Mentor: Prof. dr. Zlatko Petrovic

redovni profesor

Mašinski fakultet Univerziteta u Beogradu

Komentor: Prof. dr. Ivan Kostic

vanredni profesor


Members: Prof. dr. Zoran Stefanović

redovni profesor u penziji


Datum odbrane:

ACKNOWLEDGEMENTS

I would like to express my sincere thanks and gratitude to my advisor professor

Zlatko Petrovic for his valuable and continuous advice, thoughtfulness and assistance

throughout the duration of this work.

I would like to express my sincere thanks and gratitude to my co-advisor

professor Ivan Kostic. This research would not have been accomplished without their

support and patience in every phase of this thesis from the initial to the final level and

enlightened the work with their vast knowledge on the subject.

I would also like to extend my sincere gratitude to Professor Zoran Stefanovic

for his contributions, guidance and advices.

I also would like to take this opportunity to thank my mother and my brothers

who are surely proud of me on this day.

Special thanks to the staffs, students and friends that I have met during my

research work especially those in the Mechanical Engineering Department.

Lastly but most importantly, I want to express my gratitude to my wife and my

lovely daughters each of whom gave me support, encouragement and love in my life

and made this thesis possible.

Above all, I am very much grateful to almighty Allah for giving me courage

and good health for completing the venture.

Mr. Taha Ahmed

I

TWO-DIMENSIONAL WIND TUNNEL MEASUREMENT

CORRECTIONS BY THE SINGULARITY METHOD

Abstract

The novel approach to two-dimensional wind tunnel measurement corrections for the

airfoils has been established and applied in this thesis. Flow about the airfoil is

simulated by approximating the actual airfoil shape by linearly varying vorticity

elements distributed along a finite number of panels, positioned along its contour

(panel method), both for free flow conditions, and for flow conditions in wind tunnel

test section. The difference in calculated pressure coefficient distributions about the

airfoil in free flow and in wind tunnel is either applied directly as a correction to the

measured pressure distributions, or after its integration, to the measured aerodynamic

lift and moment coefficients. Solid wind tunnel walls are simulated by repeated

mirroring of the paneled airfoil shape with respect to position of the test section walls.

Porous walls are simulated similarly as solid walls, while transpiration is simulated by

singularities of sources/sinks type, distributed along test section walls. Intensity of

sources/sinks is determined to closely approximate results of measurements in wind

tunnel with the calculated aerodynamic parameters (pressure distribution and/or

aerodynamic coefficients). Calculated wind tunnel parameters and corrections have

been compared, and have shown good agreements both with classical wind tunnel

corrections, and with experimental data obtained from two relevant wind tunnel

facilities.

Keywords: wind tunnel corrections, singularity method, solid and porous walls, wall

interference, pressure coefficient distribution.

Scientific field:

Technical Sciences, Mechanical Engineering

Narrow scientific field:

Aeronautical Engineering

UDC number:

II

ODREĐIVANJE KOREKCIJA U DVODIMENZIONALNIM

AEROTUNELSKIM MERENJIMA METODOM SINGULARITETA

Sažetak

U okviru ove disertacije formiran je i primenjen novi proračunski model, namenjen

korekcijama u dvodimenzionalnim aerotunelskom ispitivanjima. Strujanje oko

aeroprofila modelira se aproksimiranjem realnog oblika aeroprofila vrtložnihm

elemenatima linearno promenljivog intenziteta raspoređenih po konačnom broju panela

na njegovoj konturi (panel metod), kako za slučaj slobodnog strujanja, tako i za slučaj

strujanja u radnom delu aerotunela. Razlika u proračunskoj raspodeli koeficijenta

pritiska oko aeroprofila u slobodnoj struji i u aerotunelu primenjuje se ili kao

neposredna korekcija superponiranjem sa izmerenim vrednostima koeficijenta pritiska

u aerotunelu, ili nakon integraljenja kao korekcija izmerenim vrednostima

aerodinamičkih koeficijenata uzgona i momenta. Čvrsti zidovi aerotunela simulirani su

serijom paneliranih kontura konkretnog aeroprofila, preslikanih po principu likova u

ogledalu u odnosu na zidove radnog dela. Porozni zidovi simulirani su na isti način, pri

čemu se prostrujavanje kroz njih simulira singularitetima tipa izvor/ponor postavljenim

po zidovima radnog dela. Intenziteti izvora/ponora određuju se tako da sračinatim

aerodinamičkim parametrima adekvatno aproksimiraju rezultate merenja u aerotunelu

(raspodelama pritiska i/ili aerodinaičkim koeficijentima). Sračunati aerotunelski

parametri i korekcije upoređeni su, i pokazali su dobra poklapanja kako sa korekcijama

dobijenim klasičnim metodama, tako i sa rezultatima merenja obavljenim u dve

renomirane institucije u oblasti eksperimentalne aerodinamike.

Ključne reči: aerotunelske korekcije, metod singulariteta, čvrsti i porozni zidovi,

uticaj zidova, raspodela koeficijenta pritiska.

Naučna oblast:

Tehničke nauke, Mašinstvo,

Uža naučna oblast:

Vazduhoplovstvo

UDK broj:

III

Table of Contents

TWO-DIMENSIONAL WIND TUNNEL MEASUREMENT CORRECTIONS BY

THE SINGULARITY METHOD ........................................................................................... I

Abstract I

CHAPTER ONE ..................................................................................................................... 1

1 INTRODUCTION ................................................................................................... 1

1.1 Background .............................................................................................................. 1

1.2 Literature review ...................................................................................................... 3

1.3 Wall interference corrections from boundary measurements .................................. 9

1.3.1 Early blockage corrections for solid walls ............................................................... 9

1.3.2 Method of Capelier, Chevauier and Bouniol ......................................................... 10

1.3.3 Method of Mokry and Ohman ............................................................................... 11

1.3.4 Method of Paquet ................................................................................................... 12

1.3.5 Method of Ashill and Weeks ................................................................................. 12

1.3.6 Methods of Kemp and Murman ............................................................................. 13

CHAPTER TWO .................................................................................................................. 14

2 THE METHOD OF SINGULARITIES AND ITS APPLICATION IN

SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS .................................. 14

2.1 Two-dimensional point singularity elements ......................................................... 14

2.1.1 Two-dimensional point source ............................................................................... 14

2.1.2 Two-Dimensional Point Doublet ........................................................................... 15

2.1.3 Two-Dimensional Point Vortex ............................................................................. 15

2.2 Two-dimensional constant-strength singularity elements ..................................... 16

2.2.1 Constant-strength source distribution .................................................................... 17

2.2.2 Constant-Strength Doublet Distribution ................................................................ 20

2.2.3 Constant-strength vortex distribution .................................................................... 22

2.3 Two-dimensional linear-strength singularity elements .......................................... 24

IV

2.3.1 Linear Source Distribution ..................................................................................... 25

2.3.2 Linear doublet distribution ..................................................................................... 27

CHAPTER THREE .............................................................................................................. 31

3 CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST SECTIONS

WITH SOLID WALLS ........................................................................................................ 31

3.1 Classical wall corrections assumption ................................................................... 31

3.1.1 Coordinate System and Governing Equations ....................................................... 32

3.1.2 Model Representation ............................................................................................ 34

3.1.3 Tunnel Wall ........................................................................................................... 35

3.2 Application of the correction method .................................................................... 37

3.2.1 Classical correction for Lift Interference ............................................................... 38

3.2.1.1 2D Lift interference ............................................................................................ 38

3.3 Classical correction for blockage interference ....................................................... 41

3.3.1 2D solid blockage for small models ...................................................................... 41

3.4 Wake blockage ....................................................................................................... 44

CHAPTER FOUR ................................................................................................................ 46

4 CLASSICAL CORRECTIONS FOR VENTILATED TEST SECTIONS ........... 46

4.1 Background, assumptions, and definitions ............................................................ 49

4.2 Wall boundary conditions ...................................................................................... 53

4.2.1 Ideal ventilated wall boundary conditions ............................................................. 55

4.3 Interference in 2d testing ....................................................................................... 57

4.3.1 Interference of small models, uniform walls ......................................................... 57

CHAPTER FIVE .................................................................................................................. 62

5 NEW APPROACH IN NUMERICAL MODELING OF WIND TUNNEL

CORRECTIONS .................................................................................................................. 62

5.1 Motivation for the 2D wind tunnel wall corrections .............................................. 62

V

5.1.1 Fundamental ideas of classical 2D wind tunnel wall corrections .......................... 62

5.1.2 Fundamental ideas of this thesis for 2D wind tunnel wall corrections for solid

test sections ........................................................................................................................... 63

5.1.3 Fundamental ideas of this thesis for 2D wind tunnel wall corrections for

ventilated wall test sections .................................................................................................. 65

5.2 Numerical modeling for solid wall wind tunnel .................................................... 66

5.2.1 Fundamental assumptions ...................................................................................... 66

5.2.2 Governing equations .............................................................................................. 66

5.2.3 Boundary conditions .............................................................................................. 67

5.3 Induced velocities .................................................................................................. 68

5.3.1 Two-dimensional point vortex ............................................................................... 68

5.3.2 General linear vortex distribution .......................................................................... 69

5.3.3 Linear vortex distribution with image ................................................................... 70

5.3.4 Numerical solution of the flow about the airfoil .................................................... 72

5.4 Numerical modeling for ventilated wall wind tunnel ............................................ 80

5.4.1 Fundamental assumptions ...................................................................................... 80

5.4.2 Application of Bernoulli equation ......................................................................... 80

5.4.3 Boundary Condition for a ventilated Wall ............................................................. 81

5.4.4 The effect of constant strength sources in the wind tunnel walls panels ............... 82

5.4.5 The effect of sources panels on the vortex panel control points on the airfoil ...... 84

CHAPTER SIX .................................................................................................................... 86

6 RESULTS AND DISCUSSION ............................................................................ 86

6.1 Correction for solid wall test section ..................................................................... 87

6.2 Sources of experimental data for calculations of test sections with ventilated

walls ... ……………………………………………………………………………….92

6.2.1 T-38 wind tunnel (VTI Žarkovo, Belgrade) .......................................................... 93

6.2.2 Transonic cryogenic tunnel (0.3-m NASA Langley TCT) .................................... 93

6.2.3 Models ................................................................................................................... 93

VI

6.2.3.1 Model from T-38 wind tunnel ............................................................................ 93

6.2.3.2 Model from transonic cryogenic tunnel ............................................................. 95

6.3 Calculation of corrections for test sections with ventilated walls ......................... 96

6.3.1 Wind tunnel T-38 ................................................................................................... 97

6.3.2 NASA transonic cryogenic wind tunnel .............................................................. 104

6.3.3 Comparison between T-38 and NASA wind tunnels ........................................... 110

CHAPTER SEVEN ............................................................................................................ 115

7 CONCLUSION .................................................................................................... 115

7.1 Correction procedure for solid wind tunnels walls .............................................. 115

7.2 Correction procedure for ventilated wind tunnel walls ....................................... 118

CHAPTER EIGHT ............................................................................................................. 121

8 Bibliography ........................................................................................................ 121

List of figures

Figure 2-1 Schematic description of a generic panel influence coefficient calculation. ...... 14

Figure 2-2 The influence of a point singularity element at point P. .................................... 15

Figure 2-3 Transformation from panel to global coordinate system. ................................... 16

Figure 2-4 A generic surface distribution element ............................................................... 17

Figure 2-5 Constant-strength source distribution along the x axis ....................................... 18

Figure 2-6 Nomenclature for the panel influence coefficient derivation ............................. 19

Figure 2-7 Constant-strength doublet distribution along the x axis ..................................... 21

Figure 2-8 Equivalence between a constant-strength doublet panel and two point

vortices at the edge of the panel ........................................................................................... 22

Figure 2-9 Constant-strength vortex distribution along the x axis ....................................... 23

Figure 2-10 Decomposition of a generic linear strength element to constant-strength

and linearly varying strength elements ............................................................................... 25

VII

Figure 2-11Nomenclature for calculating the influence of linearly varying strength

source .................................................................................................................................... 26

Figure 2-12 Linearly varying strength doublet element ...................................................... 29

Figure 3-1 Coordinate System and Geometry (2D test section) ........................................... 32

Figure 3-2 Elementary singularities used for model representation in a uniform stream .... 34

Figure 3-3 Method of images for a planar solid wall .......................................................... 36

Figure 3-4 Image systems for a singularity at the center of a 2d tunnel with solid walls .... 37

Figure 3-5 Up-wash interference of a 2d vortex in a solid-wall tunnel ................................ 39

Figure 3-6 Stream-wise interference of a 2d vortex in a solid-wall tunnel .......................... 40

Figure 3-7 Stream-wise interference of a 2d source doublet in a solid wall tunnel ............. 42

Figure 3-8 Up-wash Interference of a 2D Source Doublet in a Solid-Wall Tunnel ............. 43

Figure 3-9 Stream-wise interference of a 2D source in a solid-wall tunnel ......................... 44

Figure 4-1 Ventilated wall wind tunnel, general arrangement ............................................ 47

Figure 4-2 Potential flow in an ideal wind tunnel with ventilated walls .............................. 50

Figure 4-3 Slotted Tunnel Geometry .................................................................................... 56

Figure 4-4 2D Interference in ideal slotted and porous tunnels ........................................... 59

Figure 4-5 Longitudinal variation of blockage interference in 2d slotted and porous

tunnels ................................................................................................................................... 60

Figure 4-6 Longitudinal variation of up-wash interference in 2d slotted and porous

tunnels ................................................................................................................................... 61

Figure 5-1 Wind tunnel solid wall correction approach ...................................................... 62

Figure 5-2 Mokry approach .................................................................................................. 63

Figure 5-3 New approach to 2D wind tunnel correction procedure for solid walls ............. 64

Figure 5-4 New approach to 2D wind tunnel correction procedure for ventilated walls .... 65

Figure 5-5 Airfoil paneling ................................................................................................... 66

Figure 5-6 Point vortex ......................................................................................................... 68

Figure 5-7 Linear strength vortex variation ......................................................................... 69

Figure 5-8 system of image for linear strength vortex ........................................................ 71

Figure 5-9 Nomenclature for a linear-strength vortex element ............................................ 73

Figure 5-10 Constant strength source panels on wind tunnel walls ..................................... 82

VIII

Figure 5-11 Control point and source panel in the same location ...................................... 83

Figure 5-12 Source panels induced control points in linear vortex panels ........................... 85

Figure 6-1 Experimental and numerical lift coefficient ....................................................... 87

Figure 6-2 Numerical Cp distribution for free stream and with wind tunnel wall effect

Alfa=2 ................................................................................................................................... 89

Figure 6-3 Numerical Cp distribution for free stream and with wind tunnel wall effect

Alfa=2 ................................................................................................................................... 90

Figure 6-4 2D test calibration model NACA 0012 ............................................................ 94

Figure 6-5 Cp for airfoil NACA 0012 measured in T-38 wind tunnel at M = 0.3 and α

= 2° ....................................................................................................................................... 95

Figure 6-6 Cp for airfoil NACA 0012 measured in NASA wind tunnel at M = 0.3 and

α = 2° .................................................................................................................................... 96

Figure 6-7 Measured and calculation pressure distribution in T-38 wind tunnel ................. 98



Figure 6-10 Numerical Cp for free stream and wind tunnel wall effect for T-38 .............. 100

Figure 6-11 Numerical Cp for free stream and wind tunnel wall effect for T-38 .............. 100

Figure 6-12 Numerical Cp for free stream and wind tunnel wall effect for T-38 ............... 101

Figure 6-13 Measured Cp after correction in T-38 wind tunnel for T-38 ........................... 102

Figure 6-14 Measured Cp after correction in T-38 wind tunnel ......................................... 103

Figure 6-15 Measured Cp after correction in T-38 wind tunnel......................................... 103

Figure 6-16 Measured and calculation pressure distribution in NASA wind tunnel.......... 105



Figure 6-19 Numerical Cp for free stream and wind tunnel wall effect ............................. 107

Figure 6-20 Numerical Cp for free stream and wind tunnel wall effect ............................. 107

Figure 6-21 Numerical Cp for free stream and wind tunnel wall effect in NASA wind

tunnel .................................................................................................................................. 108

Figure 6-22 Measured Cp after correction in NASA wind tunnel ...................................... 108

Figure 6-23 Measured Cp after correction in NASA wind tunnel ...................................... 109

IX

Figure 6-24 Measured Cp after correction in NASA wind tunnel ..................................... 109

Figure 6-25 Measured Cp in T-38 and NASA wind tunnels for Alfa=2 ............................ 111

Figure 6-26 Measured Cp after numerical correction in both wind tunnels ....................... 111





List of symbols

A effective cross-sectional area of 2D model = Ao + added-mass area

A rectangular tunnel aspect ratio = B/H

a body radius

a slot width

A0 dimensional cross-sectional area of 2D model

Am maximum transverse cross-section of model

B tunnel breadth

b tunnel half-breadth

C cross-sectional area of test section

c airfoil chord

CD drag coefficient

Cd Cd = drag coefficient for 2D model

CL lift coefficient

Cl lift coefficient for 2D model

CLw lift coefficient of wing

CM pitching moment coefficient

Cp pressure coefficient

cpicorr

corrected experimental pressure coefficient distribution

cpimeas

measured pressure coefficient distribution about the airfoil

cpiN

numerical solution for pressure coefficient distribution with walls presence

cpN∞i

numerical solution for pressure coefficient distribution for free stream

d distance of 2D vortex from the floor

f body fineness ratio

F slotted wall parameter

H tunnel height

X

h tunnel half-height

KCD drag correction factor

KCL lift correction factor

KCM Moment correction factor

Kα angle of attack correction factor

L length; wing lift

M Mach number

m source strength

n spatial co-ordinate normal to the test section wall

p static pressure

Q porous wall parameter = 1 /(1 +βR)

q dynamic pressure

R porous wall resistance factor

Re Reynolds number

Rmax maximum body radius

S wing reference area

s wing or vortex semi-span

s source-sink separation distance for Rankin ovals and bodies

T static temperature

t maximum thickness

t slot depth (= wall thickness)

U mean aerodynamic chord

U Stream-wise velocity

u perturbation x-velocity

U∞ upstream reference velocity

V velocity magnitude

v perturbation y-velocity

V∞ Velocity for free stream

w perturbation z-velocity

wk downwash correction at tail position

x Stream-wise spatial co-ordinate

xci, zci Global coordinate of control point

XiJn,

ZiJn Local coordinate of control point xj, zjn global coordinates of the first point of the n-th segment

y Span-wise (or lateral) spatial co-ordinate

Greek Symbols

α angle of attack

β Prandtl-Glauert compressibility factor = (1 – M2)0.5

γ vortex strength in 2D = 1/2 U∞ c CL

δ lift interference parameter

XI

δ0 lift interference parameter evaluated at the model center

δ1 Stream-wise curvature interference parameter

δε Up-wash interference due to blockage

δΩ Up-wash interference due to solid blockage

ε blockage interference ratio = ui/ U∞

εδ blockage factor for bluff-body flow

ζ Non-dimensional vertical co-ordinate = z/Lref

η Non-dimensional span-wise co-ordinate = y/Lref

θ blockage factor for bluff-body flow

Λ wing sweep angle

λ body shape factor

Λ is the form factor and its value function of airfoil thickness t/c.

μ doublet strength

ξ Non-dimensional stream-wise co-ordinate = x/ Lref

ρ fluid density

σ non-dimensional wing or vortex semi-span

σ Source strength

τ Tunnel shape factor

Φ total velocity potential

φ perturbation potential

φm perturbation potential due to the model

φw, φi perturbation potential due to the walls (= interference potential)

Ωd Up-wash interference parameter due to solid blockage

Ωs Stream-wise interference parameter due to solid blockage

Ωw wake blockage interference ratio

Subscripts

b base

c corrected

corr corrected

i interference

L Lower wall

m model

n normal

p plenum (corresponding to plenum pressure)

ref reference

U Upper wall

unc uncorrected

w walls

CHAPTER ONE INTRODUCTION

1

CHAPTER ONE

1 INTRODUCTION

1.1 Background

Airfoil characteristics are usually determined in wind tunnels, or at least

confirmed by wind tunnels. Results obtained in the wind tunnels are not identical to

flight test data, or free-stream data, not only because it is hard to maintain the same

Reynolds and Mach number but also it is hard to maintain free-stream turbulence level,

roughness characteristics and also because the wind tunnel test section is of limited

size and has a boundary layer attached slightly destroying the two-dimensional flow

field.

The fundamental problem of wall corrections concerns the difference between

the flow fields around a body immersed in a uniform oncoming stream of infinite

lateral, upstream, and downstream extent, and around the same body in a stream

confined or modified by wind tunnel walls. The streamlines around a body in a

uniform subsonic onset flow depend on the shape of the body and on the aerodynamic

forces acting on the body (which may be considered a result of its shape). In the free

stream case, as distance increases laterally from the body, the streamlines approach the

straight and parallel flow of the onset stream. If the wind tunnel's boundaries (the

"walls") are far enough away from a model being tested so that the flow perturbation

due to the model is negligible, the same uniform parallel flow condition is obtained at

the boundary and the flow around the model is therefore not affected by the tunnel

boundaries. However, to the extent that the model's influence is perceptible at the

boundary, the flow within the tunnel (i.e., around the model) is different from that

which would be obtained in a free stream. Classical wall correction theory attempts to

account for this difference under a set of simplifying assumptions and corresponding

restrictions on the theory's range of applicability.


2

One of the typical problems associated with a wind tunnel test is the error

introduced into the measurements by the presence of the wind tunnel walls. Since the

flow in a wind tunnel is constrained by the walls, it must accelerate around the model

in order to satisfy the continuity equation. As a result, the model behaves inside the

wind tunnel as if it were at a slightly greater speed than the nominal wind tunnel

velocity. The increase of velocity or dynamic pressure, caused by the solid blockage of

the model, is results in an increase in all the forces and moments acting on the model.

Because the velocity in the viscous wake is slower than the velocity in the free stream,

an additional blockage, known as wake blockage, is created. As the wake grows, the

free-stream velocity must increase, again as defined by the continuity equation. The

increase in velocity around the model and its wakes causes a pressure gradient to

develop (according to the Bernoulli equation) which creates an apparent increase in

drag on the model. A blockage correction should be able to determine the incremental

velocity that, when added to the free stream, accounts for the extra forces and

moments. Once the velocity increment is found, the aerodynamic data can be

"corrected" to obtain the desired free-air results. The angle of attack of the model is

also affected by the wind tunnel boundaries. The presence of the wind tunnel walls

alters the normal curvature of the flow around the test body, creating an apparent

increase in the angle of attack. To complete the wind tunnel wall corrections, the

geometric angle of attack needs to be corrected for this apparent increase. The most

important corrections are:

– Buoyancy: Wind tunnel buoyancy is caused by the fact that the boundary layer grows

on the walls of the test section. Boundary layer growth is equivalent to a contraction of

the test section area, the flow is accelerated, causing a drop in static pressure.

Therefore, models with a big frontal area are pushed backwards. Buoyancy artificially

increases the drag.

– Solid blockage: The presence of a model in the test section reduces the area through

which the air can flow. The air velocity is increased over the model. This effect is

called the solid blockage. The effect can be corrected by increasing the effective wind

tunnel airspeed.


3

– Wake blockage: The airspeed in the wake is lower than flow field velocity. In a

closed duct this means that the airspeed outside the wake must be larger than flow field

velocity for a constant mass flow rate. The wake blockage effect can also be corrected

using an increment in the effective airspeed.

– Streamline curvature: The wind tunnel ceiling and floor artificially straighten the

curvature of the flow streamlines around the model. The model appears to have more

camber than it really has, i.e. it has too much lift. This effect requires corrections to

angle of attack, lift coefficient and moment coefficient.

1.2 Literature review

In the Ganzers paper (Ganzer, 1980) computational solution of the flow about

airfoil is used to calculate the necessary curvature of the adaptive wall which will

result in the same pressure distribution over the wall as the distribution obtained from

calculations. Sawada in his paper (Sawada, 1980) used horse-shoe vortex distribution

over a wing to calculate interference effects of ventilated wind tunnel walls. Measured

pressure distribution over walls is used as a boundary condition for the potential flow

solution within test section.

In (Mokry M. and Ohman L., 1980), fast Fourier transformation to solve

Laplace equation in two-dimensional wind tunnel test section is used. By using

experimental wind tunnel wall pressure distributions combined with a dipoles - vortex

approximation of the airfoil shape as boundary conditions to solve Laplace’s equation.

The intensity of the vortex is adjusted to the measured lift coefficient while corrections

are made on angle of attack and airspeed due to buoyancy effect. Correction is taken

from the results obtained at the position of the vortex doublet singularity.

The wall correction method of (Kupper A., July 1994) based on measured

pressure distribution on the tunnel walls to solve Laplace’s equation; this method

combines theoretical calculated boundary conditions with experimental test data. The

results of method are compared with the measured and corrected data and the data of

free flight. The calculation of wind tunnel wall interference is based on the solution of

Greens integral. In (Mokry M. D. J., 1987) the first order doublet-panel method to


4

correct Mach number and angle of attack is used obtained by measurement in the test

section of the wind tunnel with ventilated walls. The measured static pressure over the

walls and measured model forces are applied as boundary conditions.

The procedure by (Beutner T. J. Celik, July 1994) utilizes measurements of the

wall pressure distribution to develop a flow field solution based on the method of

singularities. This flow field solution is then imposed as a pressure boundary condition

in a CFD simulation of the internal flow field. The singularity method is applied in two

and three dimensional wind tunnel tests with porous walls.

In (Holt D.R and Hunt B., May 1982) the Greens theorem is used to solve

Laplace’s equation to represent a potential flow field. The wall interferences are

calculated for two and three dimensional model by measuring static pressure as

boundary conditions on the walls.

In (Ashill P.R. and Week D.J, May 1982) subsonic wall interference effects

evaluate in both two and three dimensional model by paneling the roof and ceiling with

linear distribution of vorticity. From reference (Moses D.F., December 1983) wall

interference corrections calculate by an iterative method. The method is applied to low-

speed solid-wall wind tunnels, where the only measurements required are wall static

pressures as a boundary condition for the mathematical description of flow on the walls

(outer region) and then compute the corresponds velocity. This velocity can now be

used to modify the outer boundary conditions of the inner region, and thus to obtain a

mathematical description of an improved flow in this region. The inner boundary

conditions of this region are those imposed by the model and its wake. Specifying

these boundary conditions defines the improved inner flow, from which the

corresponding static pressure can now be calculated. Using these new values as

boundary conditions for the next approximation to the outer flow, another set of

velocity are obtained, and so the iteration goes. The iteration process has converged to

unconfined flow when this error has been reduced to a suitably small value.

In reference (Antonio F. and Paolo B., January 1973), a method for the

determination of wind-tunnel corrections at transonic speed is presented. The method

consists of measuring pressure and streamlines deflection at the walls of the tunnel and


5

analytically determining the streamline deflection corresponding to the measured

pressure and the pressure corresponding to the measured streamline deflection for

external uniform free-stream conditions at the same Mach number as the test. The

comparison between measured and computed pressures and measured and computed

streamline deflections is then utilized to calculate the wall corrections to be applied to

the experimental results. The determination of wall interference for either porous or

slotted walls, are based on linear theory and the general concept of approximating the

model by dipoles and vortices and representing the perturbation velocity potential as

the sum of a free-air potential and a wall interference potential. Application of the

proper wall boundary conditions provides the interference potential and thus the

blockage and lift interference of the model.

In reference (Kraft E.M. and LO C.F., April 1977), two analytical methods for

determining the interference effects of a ventilated wind-tunnel wall on the flow past a

two-dimensional non-lifting airfoil at transonic speeds are represented. The first

method approximates the flow-field with the linearized transonic small disturbance

equation and the interference velocity is determined directly by Fourier transform

techniques. This method is readily extended to axisymmetric flows. The second

method solves the nonlinear transonic small disturbance equation including shock

waves by an integral equation method which is shown to be an order of magnitude

more rapid than the numerical relaxation techniques. It is demonstrated by the integral

equation solution, where the correct shock location as compared to the free-air solution

can be obtained by the proper selection of porosity. However, this optimum porosity is

shown to be dependent on the Mach number and the airfoil configuration.

In (Salvetti M.V. and Morrelli M., 2000) a procedure for the correction of wind

tunnel blockage effects on the experimental measurement of aerodynamic coefficients

is proposed. The correction is obtained as the difference between the values obtained in

two different numerical simulations: in the first one the flow over the model in free-

stream conditions is simulated, while, in the second one, the measured pressure values

over the wind tunnel walls are used as boundary conditions. A necessary preliminary

step is the choice of the number, location and accuracy of the pressure measurements.


6

This strategy is applied to the subsonic flow around a complete aircraft configuration

by means of a potential flow solver. Preliminary results for transonic flow around a

wing section are obtained through a Navier-Stokes solver.

In (Chann Y.Y., May 1982) the boundary-layer developments on the ventilated

walls and the sidewalls of a transonic two-dimensional wind tunnel is studied

experimentally and computationally. For the upper and lower walls, the wall

characteristics are strongly affected by the boundary layer and a correlation depending

explicitly on the displacement thickness is obtained. A method of calculating the

boundary-layer displacement effect is derived, providing the boundary condition for

the calculation of the interference flow in the tunnel. For the sidewalls, the three-

dimensional boundary-layer developments at the vicinity of the model mount have

been calculated and its displacement effect analyzed. The effectiveness of controlling

the adverse effects by moderate surface suction is demonstrated.

In reference (Horsten B.J.C and Veldhuis L.L.M., 2009), a method based on

uncorrected wind tunnel measurements and fast calculation techniques (it is a hybrid

method) to calculate wall interference, support interference and residual interference

for any type of wind tunnel and support configuration is presented. The method applies

a simple formula for the calculation of the interference gradient. This gradient is based

on the uncorrected measurements and a successive calculation of the slopes of the

interference-free aerodynamic coefficients. For the latter purpose a new vortex-lattice

routine is developed that corrects the slopes for viscous effects.

Reference (Mokry M., May 1982), is evaluated subsonic wall interference

corrections using the Fourier solution for the Dirichlet problem in a circular cylinder,

interior to the three-dimensional test section. The required boundary values of the

stream-wise component of wall interference velocity are obtained from pressure

measurements by a few static pressure tubes (pipes) located on the cylinder surface.

The coefficients of the resultant Fourier-Bessel series are obtained in closed form and

the coefficients of the Fourier sine series are calculated by the fast Fourier transform.

The estimation of the far field disturbance due to the model by singularities allows

extracting the axial component of wall interference velocity on the test section


7

boundary from the measured wall static pressures. The velocity correction at the model

position is obtained by solving the Dirichlet problem for the axial velocity in the test

section interior. The normal components of interference velocity are derived from the

zero vorticity condition. However, since it is impractical to measure the pressures over

the whole wall surfaces, a simpler solution, based on the circular cylinder interior to

the test section. The pressures are measured only by two or four static pressure tubes

(pipes) on the surface of the control cylinder. Using the periodicity condition, the

surface distribution of the axial component of the wall interference velocity is

approximated by a Fourier expansion of axisymmetric functions. The values of Fourier

components on the upstream and downstream ends of the cylinder are obtained by a

"tailored" interpolation that allows a closed-form solution for the coefficients of the

resultant Fourier-Bessel series.

In (Fernkrans Lars, October 1993), wind tunnel wall interference correction

based on Greens theorem is predicted. The method gives the interference velocity

potential field in the control volume from the velocities on a control surface around the

model of interest without the need to model the flow field. The boundary velocities

around separated wake flows are measured with static pressure pipes. This is done with

both solid and partially open test section walls. The results are used for validation of

the tool and to evaluate the possibilities to use static pressure pipes in low speed flows

as a means to get the perturbation velocities needed to calculate blockage effects in

nonsolid walls cases. It requires measurements of velocities on the tunnel boundaries.

Only axial velocity is needed for solid wall tunnels, while in tunnels with ventilated or

partially open walls it is necessary to measure both axial and cross flow velocity

components to solve the problem.

In (Allmaras S.R., March 1986) the wall-pressure signature method for

correcting low speed wind tunnel data to free-air conditions has been revised and

improved for two-dimensional tests of bluff bodies. The method uses experimentally

measured tunnel wall pressures to approximately reconstruct the flow field about the

body with potential sources and sinks. With the use of these sources and sinks, the

measured drag and tunnel dynamic pressure are corrected for blockage effects. In the


8

wall-pressure signature method the flow field about the body is approximated using the

superposition of flows associated with a set of sources and sinks. The strengths and

positions of these sources and sinks are determined so as to reconstruct the measured

velocity distribution on the tunnel walls. Once determined the effect of the tunnel walls

on the measured drag and dynamic pressure at the model is estimated, and appropriate

blockage corrections are made.

In (Everhart Venkit Iyer and Joel, 2001) the free-stream corrections to the

measured parameters and aerodynamic coefficients for full span and semi-span models

are calculated, for the tunnels in the solid-wall configuration. These corrections remove

predictable bias errors in the measurement due to the presence of the tunnel walls. At

the NTF (National Transonic Facility), the method is operational in the off-line and on-

line modes, with three tests already computed for wall corrections. At the 14x22-ft

tunnel, initial implementation has been done based on a test on a full span wing.

In (Holst, May 1982) correction factors (angle of incidence and flow curvature)

for ventilated wind tunnels by the vortex lattice method is calculated. The vortex lattice

method is then used to calculate wall pressures in closed and ventilated test sections.

Measurements in a 1.3m closed square test section were made using circular discs for

blockage and a rectangular wing as a lift generator. The above mentioned vortex lattice

method was used for the calculations of interference factors. The tunnel boundaries are

subdivided into panels, the singularities used are vortex squares and the boundary

conditions are fulfilled at a set of control points. The model is represented by

singularities, i.e. horseshoe vortices for lift, and doublets, sources and sinks for

blockage interference.

In (Blackwell James A) an empirical method for correcting two-dimensional

transonic flow results for wind-tunnel wall blockage effects has been developed. The

empirical method utilizes velocity calculations based on linear theory with free-air

boundaries evaluated at vertical positions representative of the wind-tunnel walls and

experimental velocity data obtained near the tunnel walls above and below the model.

The experimental verification also indicated that the empirical method does provide

blockage corrections to the free-stream Mach numbers that are of the right order of


9

magnitude, needed to correct the experimental airfoil data obtained for various wind-

tunnel wall conditions.

In (Holst H., 1983) the wall interference correction method for closed

rectangular test sections which uses measured wall pressures is developed.

Measurements with circular discs for blockage and a rectangular wing as a lift

generator in a square closed test section validate this method. The measurements are

intended to be a basis of comparison for measurements in the same tunnel using

ventilated (in this case, slotted) walls. Using the vortex lattice method and

homogeneous boundary conditions, calculations have been performed which show

sufficiently high pressure levels at the walls for correction purposes in test sections

with porous walls. An adaptive test section (which is a deformable rubber tube of 800

mm diameter) has been built and a computer program has been developed which is

able to find the necessary wall adaptation for interference-free measurements in a

single step. To check the program prior to the first run, the vortex lattice method has

been used to calculate wall pressure distributions in the non-adapted test section as

input data for the "one-step method." Comparison of the pressure distribution in the

adapted test section with "free-flight" data shows nearly perfect agreement.

1.3 Wall interference corrections from boundary measurements

1.3.1 Early blockage corrections for solid walls

The evaluation of wall interference corrections from wall pressures was

proposed by (Franke A. and Weinig, April 1946), (Goethert, Feb. 1952), (Thom A.,

Nov. 1943), (Mair W.A. and Gamble H.E., Dec 1944) and possibly by others. The

development of the method was motivated by observations that the determination of

solid wall corrections from the classical solid wall theory became unreliable at high

speeds and incidences, mainly due to uncertainties in the determination of singularity

strengths, representing the far field of the model. The use of measured wall pressure

data made the estimation of singularity strengths unnecessary.


10

1.3.2 Method of Capelier, Chevauier and Bouniol

This method utilizes the measured boundary pressures differently from that of

Section 1.3.1. In what we have seen so far, it was always the wall boundary condition

that was supposed to be known; the novelty of the approach of (Capelier C. Chevallier

J. and Bouniol F., Jan.-Feb. 1978) is that the measured pressures are directly taken as

the boundary values so that the cross-flow properties of the walls do not enter the

picture at all. This makes the method particularly suited for the evaluation of wall

corrections in test sections with ventilated walls, whose cross-flow properties are

extremely difficult to model mathematically. However, as in the classical wall

interference concept, the far field representation of the model by singularities is still

required.

The idea of the method is very simple, resting again upon the existence of the

linearized flow at the walls and the concept of splitting the disturbance velocity

potential into the free air and wall interference parts. The flow is investigated in the

infinite strip, where the wall interference potential is supposed to satisfy Laplace’s

equation. The lines along which the static pressures are measured and which bound the

analyzed tunnel flow region, are sometimes called the interfaces. Usually, they are

placed some distance from the walls (inside the test section), in order to avoid wall

viscous effects and smooth out discrete disturbances caused by the open and closed

portions of the walls.

The boundary value problem can be solved numerically, for example by the

panel method (Smith J. A., Jan.1981), finite difference or finite element techniques.

This may be convenient if it is required to calculate the whole interference velocity

field and not just the corrections at the model position. The numerical methods are

applicable to more complex test section geometries or combinations of pressure and

normal velocity boundary conditions. In the latter case, which is appropriate to the

solid wall wind tunnel with a finite-length ventilated test section (Smith J. A.,

Jan.1981), care must be taken since we are no longer on the safe ground of the

Dirichlet, respectively the Schwarz boundary value problem (the problem of

determining an analytic function inside a domain from its defined real part on the


11

boundary. The real part is determined uniquely, the imaginary part to within an

arbitrary constant.). The mixed boundary value problem of the Keldysh-Sedov type has

a solution only when x, y components of wall interference velocity is permitted to be

unbounded at the solid wall edges. A unique solution exists if Kutta-like conditions are

satisfied at either the upstream or downstream solid wall edges.

1.3.3 Method of Mokry and Ohman

In this method, described in detail in reference (Mokry M. and Ohman L.,

1980), instead of using the infinite strip solution, the problem is formulated for a

rectangle, which is more appropriate to testing in actual, finite-length test sections. The

method is again of the "Schwarz type", indicating that by using the measured wall

pressures the velocity correction is determined uniquely, whereas the flow angle

correction is obtained only to within an arbitrary constant. A procedure based upon

linear theory has been developed for the evaluation of wall interference corrections for

an arbitrary two-dimensional test section whose walls are operated at subcritical flow

conditions. As verified experimentally, local supercritical flow regions may exist on

the tested airfoil.

An important feature of the method is that it utilizes measured boundary

pressure distributions, but does not require knowledge of the cross-flow properties of

the walls. However, if the pressures on the upstream and downstream boundaries are

not available, the wall pressures should be measured as far as the two-dimensional

portion of the wind tunnel permits, allowing the upstream and downstream boundary

values to be obtained by interpolation. The method is relatively insensitive to

experimental scatter or type of smoothing applied. The integration constant, needed for

the evaluation of the angle of attack correction, should be obtained by measuring the

flow angle at a selected reference point, sufficiently distant from the airfoil. The

utilization of the fast Fourier transform makes the method very efficient and suitable

for routine correcting of two-dimensional wind tunnel measurements.


12

1.3.4 Method of Paquet

In this method the wall interference corrections are derived from the boundary

pressure measurements, utilizing the solution of the Schwarz problem for a semi-

infinite strip. It may well be the best combination of the two above methods, since the

flow angle reference point can be put comfortably far upstream and yet the uncertainty

of the downstream extrapolation avoided by performing the measurement (or

interpolation) across the stream at a finite distance behind the model. The acquisition

of boundary values for the three methods, treated collectively in (Paquet J.B., Jun

1979) thesis.

1.3.5 Method of Ashill and Weeks

The specification of the singularity strengths representing the far field of the

airfoil becomes unnecessary if both the pressure and flow angle distributions are

known along the test section boundary. Wall corrections can then be calculated directly

from these wall quantities, without knowing anything about the cross-flow properties

of the walls and the flow in the neighborhood of the model (Rubbert P.E., Nov. 1981).

Near the model the flow can be separated, supercritical, but near the tunnel walls it is

assumed to be attached and subcritical. Ashill and Weeks were among the first

researchers who fully realized the great potential of this approach, deriving the general

correction formula first from Green's theorem (Ashill, 1978) and then, more concisely,

from Cauchy's integral formula (Ashill P.R. and Weeks D.J., Feb. 1980). The idea of

correcting the model data from measured two components of velocity at a control

surface near tunnel walls was independently also pursued by (Lo.C.F., 1978), who

derived the blockage formula for symmetrical flow past an airfoil between solid tunnel

walls by solving the linearized boundary value problem using the Fourier transform

method.

In the case of solid walls, to which the method of Ashill and Weeks is mainly

addressed, the flow angle is essentially defined by the condition of no flow through the

walls, and so only static pressures need to be measured. To the order of accuracy of the

small disturbance theory, the flow angles can be estimated from the wall shape


13

adjustment (adaptive walls) and boundary layer development (Holt D.R and Hunt B.,

May 1982).

For ventilated walls, the technical problem of measuring flow angles is an

obstacle to the routine application of the method. However, there has been a steady

progress in applications of laser Doppler technology (Satyanarayana B. Schairer E.

Davis S., 1981) and developments of flow angle probes (Sawada H. Hagu H. Komatsu

Y. Nakamura M., 1980) and double orifice static pipes (Nenni C.E. J.p. Erickson J.C.

and Wittliff, 1982), which eventually will make this powerful correction technique

applicable to all types of test sections.

1.3.6 Methods of Kemp and Murman

A rather different approach to the correction of transonic two-dimensional wind

tunnel data is the one taken in the method by Kemp (Kemp W.B., 1976), (Kemp W. ,

March 1978), (Kemp W. T., May 1980) and in the related method by (Murman E.M.,

July 1979). The method is attractive and of practical interest, since it does not require

boundary flow angle measurements. It uses experimental pressures at the model and

the walls and transonic computational codes to determine whether the airfoil pressure

data is correctable in the sense that they can be (with a reasonable accuracy)

reproduced computationally by an optimized search of the free air Mach number and

angle of attack.

An inverse problem is solved to determine the values of the normal component

of velocity on the upper and lower surfaces, from which the effective contour of the

tested model can be constructed. The use of measured pressures on the model ensures

that the boundary layer effects are included in the calculation providing the pressure

gradient across the boundary layer is small: the effective contour contains the actual

airfoil (at given geometrical incidence) augmented by the displacement area of the

boundary layer. Boundary conditions used for the inverse problem include the

measured pressure at the tunnel wall, the measured pressure at the model and suitable

upstream and downstream boundary conditions.

CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN

SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS

14

CHAPTER TWO

2 THE METHOD OF SINGULARITIES AND ITS

APPLICATION IN SIMULATION OF FLOW IN WIND

TUNNEL TEST SECTIONS

2.1 Two-dimensional point singularity elements

These elements are the easiest and simplest to use and also the most efficient in

terms of computational effort. The three point elements that will be discussed are

source, doublet and vortex, and their formulation is given in the following sections.

2.1.1 Two-dimensional point source

As shown in Figure 2-2 consider a point source singularity at ( 0 0,x z ) with a

strength . The increment to the velocity potential at a point P is then

2 2

0 0( ) ( )2

ln x x z z

2-1

, , , ,

p p p

p

x y z influenceu v w

Panel geometry coefficient

Singularity strength calculation

Figure 2-1 Schematic description of a generic panel influence coefficient calculation.

and the velocity components after differentiation of the potential, are:

0

2 2

0 02 ( ) ( )

x xu

x x x z z

2-2

0

2 2

0 02 ( ) ( )

z zw

z x x z z

2-3



15

2.1.2 Two-Dimensional Point Doublet

Consider a doublet that is oriented in the z direction (0, ) . If the

doublet is located at the point 0 0,x z then its incremental influence at point P is:

0

2 2

0 0

( , )2 ( ) ( )

z zx z

x x z z

2-4

and the velocity component increments are

0 0

2 2 2

0 0

( )( )

[( ) ( ) ]

x x z zu

x x x z z

2-5

2 2

0 0

2 2 2

0 0

( ) ( )

2 [( ) ( ) ]

z z x xw

z x x z z

2-6

2.1.3 Two-Dimensional Point Vortex

Consider a point vortex with the strength γ as in Figure 2-2, located at ( 0 0,x z ).

Again using the definitions of the points, we find that the

Figure 2-2 The influence of a point singularity element at point P.

increment to the velocity potential at a point is

1 0

02

z ztan

x x

2-7

and the increments in the velocity components are

0

2 2

0 02 ( ) ( )

z zu

x x z z

2-8



16

0

2 2

0 02 ( ) ( )

x xw

x x z z

2-9

Note that all these point elements fulfill the requirements presented in Figure

2-1. That is, the increments of the velocity components and potential at P depend on

the geometry 0 0( , , , )x z x z and the strength of the element.

As shown in (Figure 2-3) the basic singularity element is given in a system (

,x z ) rotated by the angle relative to the ( x*,z* ) system, then by the

transformation the velocity components can be found

*

*

cos sin uu

sin cos ww

2-10

Figure 2-3 Transformation from panel to global coordinate system.

2.2 Two-dimensional constant-strength singularity elements

The discretization of the vortex, source, or doublet distributions in the previous

section led to discrete singularity elements that are clearly not a continuous surface

representation. The refining representation of these singularity element distributions

can be obtained by dividing the solid surface boundary into elements. This element is

shown schematically in Figure 2-4, both the shape of the singularity strength

distribution and the surface shape are approximated by a polynomial. A straight line

will be used in this section, for the surface representation. For the singularity strength,



17

only the linearly, constant varying, and quadratically varying strength cases are

presented, but to higher order elements the methodology of this section can be applied.

Three examples will be presented (source, doublet, and vortex) for evaluating the

influence of the generic panel at an arbitrary point P. For simplicity, the formulation is

derived in a panel-attached coordinate system, and into the global coordinate system of

the problem the results need to be transformed back.

Figure 2-4 A generic surface distribution element

2.2.1 Constant-strength source distribution

As shown in Figure 2-5, consider a source distribution along the x axis. The

assumption is that the source strength per length is constant such that ( ) .x cons

The effect of this distribution at a point P is an integral of the effects of the point

elements along the segment 1 2x x ;

2

1

2 2

0 0( )2

x

x

ln x x z dx

2-11



18

Figure 2-5 Constant-strength source distribution along the x axis

2

1

002 2

02 ( )

x

x

x xu dx

x x z

2-12

2

1

02 2

02 ( )

x

x

zw dx

x x z

2-13

The integral for the velocity potential in terms of the corner points

1, 2( 0),( ,0)x x of a generic panel element Figure 2-6 the distances 1 2,r r and the angles

1 2, it becomes

2 2

1 1 2 2 2 12 ( )4

x x lnr x x lnr z

2-14

where

1 , 1, 2k

k

ztan k

x x

2-15

2 2( ) , 1,2k kr x x z k

2-16

The velocity components are obtained by differentiating the potential, they are:



19

Figure 2-6 Nomenclature for the panel influence coefficient derivation

2

1 1

2

2 22 4

r ru ln ln

r r

2-17

2 1( )2

w

2-18

Returning to x, z variables we obtain

2 22 2

1 1 2 2

1 1

2 1

4 2 ( )

x x ln x x z x x ln x x z

z zz tan tan

x x x x

2-19

2 2

1

2 2

24

x x zu ln

x x z

2-20

1 1

2 1

( )2

z zw tan tan

x x x x

2-21

Of particular interest is the case when the point P is on the element (usually at

the center).



20

In this case z = 0± and the potential becomes;

2 2

1 1 2 2( ,0 )4

x x x ln x x x x ln x x

2-22

and at the center of the element it becomes:

2

1 2 2 12 1( ,0 )

2 4 2

x x x xx x ln

2-23

The x component of the velocity at z = 0 becomes:

1

2

( ,0 )4 ( )

x xu x ln

x x

2-24

which is zero at the panel center and infinite at the panel edges.

It is important to distinguish between the conditions when the panel is

approached from its lower or from its upper side for evaluating the w component of the

velocity,. For the case of P being above the panel, 1 0 while, 2 . These

conditions are reversed on the lower side and therefore

( ,0 )2

w x

2-25

2.2.2 Constant-Strength Doublet Distribution

Consider a doublet distribution along the x axis consisting of elements pointing

in the z direction (0, ) as shown in Figure 2-7. The influence at a point ( , )p x z is

an integral of the influences of the point elements between x1 and x2;

2

1

02 2

0

,2 ( )

x

x

zx z dx

x x z

2-26

and the velocity components are

2

1

002

2 2

0

( ),

2 ( )

x

x

x x zu x z dx

x x z

2-27

2

1

2 2

002

2 2

0

( ),

2 ( )

x

x

x x zw x z dx

x x z

2-28



21

Figure 2-7 Constant-strength doublet distribution along the x axis

Note that the integral for the w component of the source distribution is similar

to the potential integral of the doublet. Therefore, the potential at P (by using equation

2-21) is:

1 1

2 1

( )2

z zw tan tan

x x x x

2-29

Comparison of this expression to the potential of a point vortex (Eq. 2-7)

indicates that this constant doublet distribution is equivalent to two point vortices with

opposite sign at the panel edges such that see Figure 2-8. Consequently, the

velocity components are readily available by using Eqs. (2-8) and (2-9):

2 2 2 2

1 22 ( ) ( )

z zu

x x z x x z

2-30

1 2

2 2 2 2

1 22 ( ) ( )

x x x xw

x x z x x z

2-31

When the point P is on the element 1 2( 0, )z x x x then we have

, 02

x

2-32

and the velocity components become



22

Figure 2-8 Equivalence between a constant-strength doublet panel and two point vortices at the

edge of the panel

( )

,0 0d x

u xdx

2-33

1 2

1 1,0

2w x

x x x x

2-34

and hence the w velocity component is singular at the panel edges.

2.2.3 Constant-strength vortex distribution

Once the influence terms of the constant-strength source element are obtained,

owing to the similarity between the source and the vortex distributions, the formulation

for this element becomes simple. The constant-strength vortex distribution

(x) const. is placed along the x axis as shown in Figure 2-9. The influence of

this distribution at a point P is an integral of the influences of the point elements

between x1 and x2. So we have:

2

1

1

0

02

x

x

ztan dx

x x

2-35



23

Figure 2-9 Constant-strength vortex distribution along the x axis

2

1

1

02 2

02

x

x

zu tan dx

x x z

2-36

2

1

1 002 2

02

x

x

x xw tan dx

x x z

2-37

The solution of integrals in terms of the distances and angles of equations

(2-15) and (2-16) as shown in Figure 2-6 the potential becomes:

2

11 1 2 2 2

22 2

rzx x x x ln

r

2-38

which in terms of the x, z coordinates is

2 2

1

1 2 2 21 2 2

2 2

x x zz z zx x x x ln

x x x x x x z

2-39

Following the formulation used for the constant-source element, and

observing that the u and w velocity components for the vortex distribution are the same

as the corresponding w and u components of the source distribution, we obtain



24

1 1

2 12

z zu tan tan

x x x x

2-40

2 2

2

2 2

14

x x zw ln

x x z

2-41

The influence of the element on itself at z 0 and 21x ( ) x x can be found

by approaching from above the x axis. In this case 1 20, and

1 2 2,0 02 2

x x x x x x x

2-42

2,02

x x x

2-43

Similarly, when the element is approached from below, then the x component

of the velocity can be found by observing equation. (2-24) for the source

,02

u x

2-44

and the w velocity component is similar to the u component of the source equation (

2-23)

2

2

2

1

,04

x xw x ln

x x

2-45

In most situations the influence is sought at the center of the element where

1 2r r and consequently (panel−center, 0±) = 0.

2.3 Two-dimensional linear-strength singularity elements

The representation of a continuous singularity distribution by a series of

constant strength elements results in a discontinuity of the singularity strength at the

panel edges. To overcome this problem, a linearly varying strength singularity element

can be used. The requirement that the strength of the singularity remains the same at

the edge of two neighbor elements results in an additional equation. Therefore with this

type of element, for N collocation points 2N equations will be formed.



25

2.3.1 Linear Source Distribution

Consider a linear source distribution along the x axis 21x x x with source

strength of, 0 1 1 x x x as shown in Figure 2-1. Based on the principle of

superposition, this can be divided into a constant-strength element and a linearly

varying strength element with the strength 1 x x , for the general case (as

shown in the left-hand side of Figure 2-10. The results of this section must be added to

the results of the constant-strength source element.

Figure 2-10 Decomposition of a generic linear strength element to constant-strength and linearly

varying strength elements

The influence of the simplified linear distribution source element where

1 x x at a point P is obtained by integrating the influences of the point elements

between x1 and x2 see Figure 2-10.

2

1

2 2

0 0 02

x

x

x ln x x z dx

2-46

2

1

0 0

02 2

02

x

x

x x xu dx

x x z

2-47

2

1

002 2

02

x

x

x zw dx

x x z

2-48

The results of integration are:



26

Figure 2-11Nomenclature for calculating the influence of linearly varying strength source

2 2 2 2 2 2

2 21 21 2 2 1 2 12

4 2 2

x x z x x zlnr lnr xz x x x

2-49

where r1,r2,θ1 and θ2 are defined by equations (2-15) and (2-16). The velocity

components are obtained by differentiating the velocity potential which gives:

2

1 11 2 2 12

22 2

rxu ln x x z

r

2-50

2

1 22 12

1

24

rw zln x

r

2-51

Substitution of kr and k from equations (2-16) and (2-17) results in

2 2 2 2 2 22 22 21 2

1 2

1 1

2 1

2 1

2 2

42

x x z x x zln x x z ln x x z

z zxz tan tan x x x

x x x x

2-52



27

2 2

1 1 111 22 2

2 122 2

x x zx z zu ln x x z tan tan

x x x xx x z

2-53

2 2

2 1 11

2 22 11

24

x x z z zw zln x tan tan

x x x xx x z

2-54

When the point P lies on the element 1 2 0 , z x x x , then equation (2-52)

reduces to

2 2 2 2

1 1 2 2 2 14

x x ln x x x x ln x x x x x

2-55

At the center of the element this reduces to

2 2 2 11

1

4 2 2

x xx x ln

2-56

Also, on the element

1 11 2

22

x xu xln x x

x x

2-57

1

2w x

2-58

and at the center of the element

11 2

2u x x

2-59

and

12 1

4w x x

2-60

2.3.2 Linear doublet distribution

Consider a doublet distribution along the x axis with a strength

0 1 1 x x x consisting of elements pointing in the direction 0,μ as

shown in Figure 2-11. In this case, too, only the linear term 1( )x x is considered

and the influence at a point , P x z is an integral of the influences of the point

elements between 1x and 2x



28

2

1

0102 2

0

,2

x

x

x zx z dx

x x z

2-61

2

1

0 0102

2 2

0

,

x

x

x x x zu x z dx

x x z

2-62

2

1

2 2

0102

2 2

0

,2

x

x

x x zw x z dx

x x z

2-63

The integral for the velocity potential is similar to the velocity component of

the linear source equation (2-48). Therefore, following equation (2-51), we obtain

2

1 22 12

1

24

rzln x

r

2-64

and in Cartesian coordinates

2 2

2 1 11

2 22 11

24

x x z z zzln x tan tan

x x x xx x z

2-65

To obtain the velocity components we observe the similarity between equation

(2-64) and the potential of a constant-strength vortex distribution equation (2-38).

Replacing with – in equation (2-39) yields:

2 2

2** 1 111 22 2

1 21

2 24

x x z z zzln x x tan x x tan

x x x xx x z

2-66

and therefore the potential of the linear doublet distribution of equation (2-65) is

** 11 1 2 2

2x x

2-67

and the two last terms are potentials of point vortices with strengths 1 1x and 1 2x

see equation (2-7). The velocity components therefore are readily available, either by

differentiation of this velocity potential or by using equations (2-40) and (2-8).



29

Figure 2-12 Linearly varying strength doublet element

1 11 1 2 1 1

2 22 22 1 2 1

2 2 2

x xz z z zu tan tan

x x x x x x z x x z

2-68

and for the component using Eqs. (2-41) and (2-9) we get

2 2

21 1 1 1 1 2 2

2 2 22 2 2

1 1 24 2 2

x x z x x x x x xw ln

x x z x x z x x z

2-69

The values of the potential and the velocity components on the element

1 2( 0, )z x x x are:

1

2x

2-70

1

2u

2-71

2

21 1 2

2

1 21

2 2

4

x x x xw ln

x x x xx x

2-72



30

and the velocity component at the center of the element becomes

1 2 1

2 1

x xw

x x

2-73

and hence the velocity is singular at the panel edges because of the point vortices there.

Note that for the general element, where 0 1 1 x x x the potential

becomes:

** 0 12 1 1 2 2

2 2x x

2-74

and because of the potential jump at the edges of this doublet distribution two

concentrated vortices exist. The vortex at 1x will have a strength of 0 while at 2x

will have a strength of 1 2 1 0 x x as shown schematically in Figure 2-12.

CHAPTER THREE CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST

SECTIONS WITH SOLID WALLS

31

CHAPTER THREE

3 CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST


The wind tunnel wall corrections problem concerns itself with the difference

between the flow fields around a model in a stream constrained by walls of wind tunnel

and the same model in uniform stream with infinite lateral, upstream and downstream

extend. Around the model for a uniform subsonic flow the streamlines depend on the

shape of the model and on the aerodynamic forces acting on the model. With the

increasing of walls distance from the body (interference-free case) the streamlines

approach the straight line and parallel undisturbed flow far enough from the model

onset. In case of the wind tunnel walls far away from the testing model the perturbation

flow is negligible due to the model, then the obtained flow at the boundary is uniform

parallel flow therefore the effect of walls for the model ignored. The difference

between the flow around the model in case of wall presence and without these walls

can be clearly sensible at the walls. Classical wall correction theory tries to calculate

this difference under a set of simplifying assumptions and corresponding restrictions

on the theory's range of applicability.

3.1 Classical wall corrections assumption

The assumptions that the classical wall interference theory include:

1. Perturbation flow at the tunnel boundaries.

2. Linear potential flow.

3. Tunnel of constant cross-sectional area extending far upstream and downstream of

the model, with boundaries parallel to the direction of the flow far upstream of the

model, and whose boundary condition for a given wall is either no flow normal to the

wall or a constant pressure at the wall location.



32

4. Model whose dimensions generally are small relative to the tunnel and whose wakes

(including both the viscous and vortex wakes) extend straight downstream from the

model.

"Classical" is used as a further classification of wall corrections, which

includes the classical. These corrections are based on classical concepts in that the

perturbation flow assumptions are used, but model size, wake position, and tunnel

boundary conditions are not restricted as above. For present purposes, the tunnel walls

are restricted, however, to a fixed geometry with a known pressure-cross flow

characteristic. Classical wall correction methods do not then include specified

boundary condition methods or adaptive wall methods. Much of the work reported in

AGARDograph109 (Garner, 1966) satisfies this definition of "classical", though

specified boundary condition methods and adaptive wall methods have appeared in the

literature since the 1940s, and are included in AGARDograph 109 (Garner, 1966)as

well.

3.1.1 Coordinate System and Governing Equations

The coordinate system is defined for a classical wing body model such that x

is the stream-wise coordinate, z is the vertical coordinate corresponding to the direction

of primary lift, and y is the lateral or span-wise coordinates, Figure 3-1. The origin of

the coordinate system is typically taken to be on the test section centerline, at the

model center. In 2D flow, the flow field is taken to be invariant with y-axis. Far

upstream of the model, the incoming flow is uniform.

Figure 3-1 Coordinate System and Geometry (2D test section)



33

The linearized potential flow assumption between the tunnel boundaries and the

model is the starting point for classical wall interference corrections development.

Streamline flow is assumed with no allowance for separated wakes or shock waves.

The viscosity effect of fluid is ignored in the governing equations. Velocity is the

gradient of the potential function at any point in the tunnel in the usual way:

( , ) ( , )v x z x z 3-1

The key feature of classical wall interference analysis is the principle of

superposition. This principle allows the interference flow field to be considered as an

increasing flow field to the interference free flow around the model. Thus, the potential

∅ is assumed to be expressible as the superposition of a uniform onset stream, the

model potential, and the wall potential,

( , ) ( , ) ( , )m wx z U x x z x z 3-2

The model and wall potentials can be considered perturbation velocity

potentials in those regions of the flow away from the model where the flow

perturbations to the uniform oncoming stream are small. The effect of compressibility

can be linearized in the full potential equation, for small deviations from the nominal

free stream, resulting in the governing equation for the perturbation velocity potentials,

2 0xx zz 3-3

where that part of the flow field due to the walls, the wall interference

velocity field, is the gradient of the wall interference potential,

( , ) w wV x z i jx z

3-4

The equation for the perturbation velocity potential can be reduced to the

Laplace equation 2 0 with the coordinate transformation (as developed by Prandtl

and Glauert for 2D airfoils and extended to three dimensions by Goethert), X x and

Z z . This transformation relates the linearized compressible flow to an equivalent

incompressible flow in stretched coordinates.



34

3.1.2 Model Representation

The small model and perturbation velocities at the boundary of tunnel

assumptions mean that only the far-field flow around the model must be properly

represented. That is, the model details are not important; only the loading are important

to first order and integrated effects at the tunnel boundary of model geometry.

The first order far field influence of the model arises from three independent features

of a model's aerodynamics:

1. Model shape and volume, which causes a displacement or bulging of streamlines

around the model, with the streamlines reconverging to unperturbed parallel flow

downstream of the model.

2. Model lift, which in three dimensions results in a redirection of momentum of the

stream, resulting in a downwash field that persists to downstream infinity.

3. Model parasite drag (i.e., not including induced drag or drag due to separated

wakes), which results in an outward displacement of streamlines around the viscous

wake that also persists downstream of the model.

Figure 3-2 Elementary singularities used for model representation in a uniform stream

For small models, an elementary analytical singularity is placed at the model

location which is representing these three characteristics. The basic singularities derive

from potential flow theory and are summarized in Figure 3-2 (in 2D flow, point) vortex



35

to represent lift, source doublet to represent model volume, and point source to

represent the displacement effect of the wake.

3.1.3 Tunnel Wall

The extending of tunnel walls far upstream and downstream allows the

application of images method with its corresponding analytic results set. For the

evaluation of interference in tunnels with either solid-wall or open jet boundaries, the

method of images is a simple, yet powerful technique.

For a solid wall the boundary condition is no flow normal to the wall, given

exactly in terms of the perturbation potential,

0n

3-5

where m w

The boundary condition for an open wall (or free jet) is a constant pressure

equal to the static pressure far upstream of the model; in linearized form,

0x

3-6

Finally, a tunnel of constant cross section with assumption of extending to

infinity both upstream and downstream of the model provides the simplifications

(symmetries and asymptotic boundary conditions) permitting the analytic techniques

application, such as the method of images. The model located in most wind tunnel tests

on the centerline of the test section, the advantage of this symmetry condition can be

used to simplify the analysis and to allow a suitable decoupling of up-wash

interference from model volume and wake characteristics, and of blockage interference

from model lift.

Consider a planar solid wall to infinity extending in all directions in vicinity to

an isolated point singularity whose velocity potential is given by ( , , )x y z .Figure 3-3

illustrates this situation in two dimensions for the point vortex and source singularities.

The desired boundary condition at the wall is / 0n . If the velocity potential of



36

the singularity is such that / 0n is an odd function of the coordinate n normal to

the wall (i.e., ( is even with respect to n), then by symmetry, the velocity normal to

Figure 3-3 Method of images for a planar solid wall

the wall due to this singularity is identically cancelled by placing a so-called image

singularity of the same magnitude and strength on the other side of the wall, at the

same distance from the wall, on the line normal to the wall and passing through the



37

original singularity. Conversely, if / 0n for the original singularity is an even

function of the coordinate n (i.e., is odd with respect to n), the normal velocity at the

wall due to the original singularity is cancelled by an image singularity of equal

magnitude and opposite strength. Thus for a planar solid wall, the 2D point vortex

requires an image of the opposite sense, while a point source requires an image of the

same sense.

3.2 Application of the correction method

Wind tunnel with solid test section and aerodynamically parallel walls are the

easiest to understand and analyze. The boundary condition for each wall gives way to

treatment by the images method. The presence of more than one wall requires the use

of multiple images. An infinite array of singularities is required even in the simplest

case of two walls.

Figure 3-4 Image systems for a singularity at the center of a 2d tunnel with solid walls

In two dimensions, the solid wall boundary condition can be satisfied on the

upper and lower walls by using a single row of image singularities both above and



38

below the test section. In constructing the image system each wall initially requires an

image outside the test section of the model within the test section. However, the

presence of the first-order singularity for the lower wall violates the parallel flow

boundary condition on the upper wall, thus requiring a second singularity above the

ceiling, and similarly for the floor. For a model placed midway between the floor and

ceiling this results in an infinite set of singularities, all at the same station as the model,

equally spaced in z, aligned above and below the test section as indicated in Figure 3-4.

A single infinite summation expresses the interference in the test section. This image

system is readily generalized to the case of asymmetric model location.

3.2.1 Classical correction for Lift Interference

The part of the wall interference due to circulation is defined as Lift

interference (i.e., corresponding to a force normal to the oncoming stream direction)

generated by the model. When the small model located in the center line of test section,

the model lifts results in primarily up-wash interference in the vicinity of the model.

This change in effective free air flow direction directly modifies the model

aerodynamic angle of attack and requires the resolution of force balance measurements

relative to the corrected wind axis direction.

3.2.1.1 2D Lift interference

In 2D flow, the lifting effect of an airfoil is represented by a point vortex

singularity. The potential for a point vortex located at 0x z is:

arctan( )2

m

z

x

3-7

where , the vortex strength is1/ 2 LU cC , and c is the airfoil chord. Defining non-

dimensional spatial coordinates / , /x H z H , anywhere in tunnel for a model

centrally located between solid upper and lower walls the up-wash interference is

given by:



39

0

2 2

1( , ) ( 1)

2 ( )n

nnw

nL

H

U cC z n

3-8

Throughout the test section the up-wash interference is shown in Figure 3-5. At

the model station the up-wash interference is zero as expected, since due to each image

singularity the velocity is in the stream-wise direction at this station. The up-wash

gradient, however, is not zero. Additional lift the model will experience due to this

induced camber relative to the interference free case. The stream-wise curvature

interference parameter at the model location 0 is

0

1 2

0,0

1 1(0,0) ( 1)

2 24n

nn

n n

3-9

Figure 3-5 Up-wash interference of a 2d vortex in a solid-wall tunnel



40

Since the up-wash gradient is proportional to LC , the uncorrected lift curve will

be steeper. For convenience, a stream-wise interference parameter (due to lift) can be

defined as:

0

2 2

1( , ) ( 1)

2 ( )n

nnw

nL

H n

U cC x n

3-10

By symmetry, along the tunnel axis the stream-wise interference is identically

zero, being positive above the axis and negative below the axis at the model station

(for positive lift), see Figure 3-6. Both the stream-wise and up-wash interference

velocities far upstream and downstream of the model, approach zero.

Figure 3-6 Stream-wise interference of a 2d vortex in a solid-wall tunnel

These results, only to a small model are strictly applicable, the implications of

finite model size are apparent from consideration of the spatial variations of

interference velocities in Figure 3-5 and Figure 3-6. At zero incidences and the model

centered between the walls have a chord length that places leading and trailing edges

beyond the region of constant interference. Rotation of model through incidence



41

angles, the leading and trailing edges move away from the centerline i.e. in the variable

region of up-wash and stream-wise interference. Along the centerline the limits of

linear stream-wise and up-wash are no more than about / 0.4x H , Figure 3-6.

Both up-wash and stream-wise interference deviations from the centerline value are

small for / 0.2z H .

3.3 Classical correction for blockage interference

Wall interference due to the displacement of streamlines is blockage

interference around a body that carries no lift or side force. In the tunnel, the part of the

blockage due to the volume of the model is solid blockage. This is usually taken to be a

solid body, though if the effect of a support sting is sought, under certain

circumstances modeling of its volume might take the form of a semi-infinite body

which can be represented by a source. A source flow is similarly used to represent the

displacement effect of a viscous wake from the model.

3.3.1 2D solid blockage for small models

As discussed by (Glauert H.), around any non-lifting body the flow field may

be represented by a power series in the inverse of the complex spatial coordinate. At a

large distance from the body, the leading term (of the form of a source doublet)

dominates. In 2D flow, the potential of a source doublet is

2 2 2( )

2m

x

x z

3

-11

In a uniform unconstrained stream, the potential of a source doublet aligned

with the oncoming stream represents the flow around a cylinder whose radius (a) is

related to the doublet strength,

22 a U

3

-12

The far field of any non-lifting body is approximated by this first-order term if

is taken as /AU , where A is the effective cross-sectional area of the model. It is

the sum of the model volume (per unit span) and its virtual volume (per unit span) for



42

accelerated flow in the stream-wise direction. Using non-dimensional coordinates

/ , /x H z H , and summing the effect of all the image doublets, the stream-

wise interference anywhere in the tunnel for a model centrally located between solid

upper and lower walls is given by:

0

2 2 2

23 2 2 2

1 ( )

2 ( )n

iu A c n

U c H n

3-13

It should be noted that at any value of the interference is a maximum at the

model location, as shown in Figure 3-7, which increases the effective free-stream

velocity felt by the model. However, due to the stream-wise symmetry of the

interference, there is no pressure buoyancy force on the model.

Figure 3-7 Stream-wise interference of a 2d source doublet in a solid wall tunnel

At the model location, 0 , the interference is given by:



43

0

2

0 3 2 2 3 2

1 1(0,0)

2 6n

A c A

c H n H

3-14

As for the point vortex, interference at the model station is a minimum on

centerline, with interference velocities / 0.2 /z H x H very close to centerline

values.

In a manner analogous to the point vortex, an up-wash interference parameter

for a non-lifting body can be defined:

0

22 2 2 2

1 1 2 ( )( , )

2 ( )n

nwi

n

w A n

U U z H n

3-15

Figure 3-8 Up-wash Interference of a 2D Source Doublet in a Solid-Wall Tunnel

By symmetry, along the axis of the tunnel the interference up-wash due to solid

blockage is zero, Figure 3-8. Off-centerline the interference up-wash has a character

similar to the up-wash interference of a 2D vortex Figure 3-5.



44

3.4 Wake blockage

In 2D flow the potential of a point source located at the origin is:

2 2 2 0.5( )2

m

mx z

3-16

where m, the source strength, is 1/ 2 LU cC . In terms of non-dimensional coordinates

/ , /x H z H , the stream-wise interference anywhere in the tunnel for a

model centrally located between solid upper and lower walls is given by:

0

2 2 22 ( )n

D

n

C c

H n

3-17

The maximum value of stream-wise interference attains far downstream of the

model location, Figure 3-9.

Figure 3-9 Stream-wise interference of a 2D source in a solid-wall tunnel

Its magnitude is consistent with 1D stream-tube considerations: the tunnel

cross-sectional area is decreased downstream of the model, by the equivalent



45

displacement area of the viscous wake plume, so that the flow external to the wake

must increase proportionately. The image sources add additional mass to the oncoming

stream, so that the uniform velocities far upstream and downstream cannot be equal.

An interesting result for this singularity set is the non-zero interference far upstream of

the model. Formally, this physical paradox can be alleviated by providing each source

with a corresponding sink far downstream of the model, thus closing off each "wake

body". This array of sinks produces an equal and opposite interference flow far

upstream that restores the undisturbed onset stream velocity.

The upstream interference to be zero is a practical approach to wake blockage

corrections. Because the setting of tunnel speed commonly relies on a wall static

pressure measurement upstream of the test section, the influence of the model at this

location is automatically included in the definition of uncorrected tunnel speed.

Therefore, the wake blockage interference at the model location should be taken as the

difference between the interference at the static pressure reference location and the

interference at the model location see Figure 3-9. If the upstream asymptote is used as

a reference, the interference at the model is:

0 24

DC c

H

3

-18

At the model location the stream-wise gradient of wake blockage interference is

maximum and results in a buoyancy force on the model. Differentiating the series

expression for due to the source representing the displacement of the wake, the same

series appears as for solid blockage of a source doublet, so that

2

wake Dsolid

C Hc

A

3

-19

At the model location 0 ,

212

wake DC c

H

3

-20

By symmetry, the interference up-wash is zero along the axis of the tunnel and,

in the vicinity of the model; the interference up-wash is directed from the walls toward

the tunnel axis.

CHAPTER FOUR CLASSICAL CORRECTIONS FOR VENTILATED

TEST SECTIONS

46

CHAPTER FOUR

4 CLASSICAL CORRECTIONS FOR VENTILATED TEST

SECTIONS

As described in Chapter 3, the fundamental characteristics of wall interference

of small models in incompressible flow in these types of tunnel were established by the

mid-1930s, e.g. (Glauert H.), (Theodorsen T.). These analyses of lift and blockage

interference in solid-wall and open-jet test sections predicted corrections of opposite

sign. Reasoning that walls of some intermediate geometry would therefore minimize

the interference, testing with walls having a mix of open and solid elements was

undertaken.

In conjunction with these developments in the testing methodology, the

maturation of the applied aeronautical sciences was enabling flight speeds approaching

the speed of sound. In solid-wall tunnels investigation of aerodynamic characteristics

of flight vehicles encounters serious difficulties in this speed range. Extremely small

model sizes are required to avoid sonic choking of the flow around the model in a

solid-wall test section. One-dimensional compressible flow relationships provide the

limiting case of maximum model cross-sectional area for choked flow: for example, a

model with an area blockage ratio of 0.01 permits a maximum upstream Mach number

of only about 0.89. This problem is manifested even in linearized compressible flow,

for which the Prandtl-Glauert compressibility transformation results in blockage

interference velocities increasing like (Goethert B. H., 1961). The theoretical

singularity at Mach = 1.0 (due to linearization of the compressibility effect) is

consistent with experimental difficulties experienced at high-subsonic test Mach

numbers.

With walls comprising both open and solid elements an unexpected

consequence of testing was a substantial increase in verification upstream Mach

number before the onset of sonic choking around the model. This discovery led to the


TEST SECTIONS

47

ventilated wall. Two basic wall geometries have emerged as preferred ventilated wall

types: slotted walls, comprising solid wall areas (slats) alternating with longitudinal

slots, and ventilated walls, which are characterized by a pattern of holes in an

otherwise solid wall surface. The test section is surrounded by a single large open

plenum chamber, assumed to be at a constant static pressure that is usually used as the

tunnel Mach number reference pressure, Figure 4-1.

Figure 4-1 Ventilated wall wind tunnel, general arrangement

At its downstream end this plenum chamber may be vented to the test section

diffuser through a variable-geometry re-entry flap system, or may be actively pumped

by a plenum evacuation system (PES) which typically can remove up to several

percent of the tunnel mass flow from the plenum, usually to be re-injected elsewhere

into the tunnel circuit. In the transonic speed range use of a PES is especially

advantageous to maximize clear tunnel flow uniformity, the upstream flow to assist

expansion to supersonic test Mach numbers, and to help offset the adverse effects of

wake blockage in the downstream part of the test section. Primarily for subsonic

testing experience with slotted walls has led to their use. In the near-sonic and low-

supersonic speed range ventilated walls are preferred, due to their ability to attenuate

shock (and expansion) wave reflections with the right choice of openness ratio


TEST SECTIONS

48

(Estabrooks B. B., June 1959), (Jacocks J. L., August 1969), (Neiland V. M., July-

August 1989). Ventilated walls of one type or the other (or, in some cases, of a hybrid

type), whose geometry remains fixed (or at most varies uniformly with Mach number)

have been the mainstay of aerodynamic testing at Mach numbers from approximately

0.6 to 1.2 since their introduction in the 1940s and early 1950s (Goethert B. H., 1961).

With the maturation of aerodynamic testing technology, data accuracy needs

have become more stringent (Steinle, November 1982), with parallel accuracy

requirements with regard to interference corrections. The continuing expansion of high

Reynolds number testing (Goldhammer, September 1990) has stimulated an increased

appreciation of Reynolds number effects, which in turn has increased the pressure on

model size in order to simulate flight Reynolds numbers more closely. Model size

(relative to test section dimensions) thus continues to play a key role in interference

calculations. Similarly, there is a continuing demand for more comprehensive

predictions of flight characteristics, including increased emphasis on flight regimes

where the effects of compressibility are strong (both on the flight characteristics

themselves and on the wall interference as well). For subsonic flight vehicles whose

design point is close to drag rise or beyond, this includes flight conditions at Mach

numbers approaching 1.0, with substantial regions of supersonic flow, and possibly

with large areas of separated flow. Supersonic flight vehicles require testing through

their entire flight envelope, typically including Mach numbers as close to 1.0 as

possible. Each of these factors increases the magnitude of the wall interference,

consequently maintaining pressure on improving wall interference methods for

ventilated wall tunnels.

Even though the theory of ventilated-wall wind tunnels is less soundly based

than for solid-wall tunnels, classical ventilated-wall tunnels offer several practical

advantages: demonstrated small interference effects in subsonic flow (compared to

solid-wall tunnels), the ability to operate at high-subsonic Mach numbers and through

the sonic and low-supersonic speed range, and the operational simplicity of fixed

geometry ventilated walls. These advantages, coupled with both a substantial capital

investment in existing test facilities and continuing competitive pressure to improve


TEST SECTIONS

49

wind tunnel data accuracy, provide the motivation to understand ventilated wall

behavior.

Perhaps the greatest difficulty in the application of the methodology and results

of ventilated-wall interference theory is the approximate nature of the ideal ventilated-

wall boundary conditions and the unknown relationship between physical wall

geometry and wall cross-flow parameters. This weakness has motivated investigations

of cross-flow characteristics of particular wall geometries, the use of measured

boundary conditions to determine wall characteristics (Mokry M. Peake, February

1974), development of alternate wall cross-flow models, and finally, the direct use of

measurements near the wall as boundary conditions in the computation of interference.

The application of boundary measurement techniques for interference estimation of

ventilated walls appears to be a viable approach, particularly for ventilated walls (e.g.,

in 2D, on (Mokry M. and Ohman L., 1980) in 3D, (Mokry M. D. J., 1987), (Beutner T.

J. Celik, July 1994), and even for slotted walls (Freestone, July 1994). Nonetheless,

because of the additional instrumentation, measurement, and computational

requirements of such methods, testing with passive, non-adaptive, ventilated walls and

the use of classically based corrections predominates in practice, especially for 3D

tunnels.

The impact of improvements in high-speed computing cannot be

overemphasized. The CFD codes and techniques developed over the past three decades

for analysis of flight vehicles in an unconstrained flow are now being applied to the

analysis of models within wind tunnels. More complex and larger test configurations,

asymmetric installations in the test section, general tunnel cross sections, and a variety

of wall boundary conditions can now readily be analyzed. The influences of finite test

section length and model supports can also be evaluated.

4.1 Background, assumptions, and definitions

"Classical" wall corrections are taken to be those that apply to tunnel flows

where the influence of the walls is approximated as an incremental flow field in the

vicinity of the model that is calculable using linearized potential flow theory, and


TEST SECTIONS

50

where the walls are basically of fixed geometry with known cross-flow characteristics.

Thus it is assumed that the flow around the model in the wind tunnel is governed by

Figure 3-3. The potential at any point in the tunnel is expressed as the superposition of

the separate potentials representing a uniform onset free stream, the model, and the

walls:

( , ) ( , , ) ( , , )m wx z U x x y z x y z 4-1

Compressibility is taken into account through the Prandtl-Glauert

compressibility factor . Simply the interference flow field is due to the wall potential.

Throughout its length the test section is usually taken to be of constant cross- section,

with flow through the walls satisfying a boundary condition relating the pressure

difference across the walls and the cross flow velocity, Figure 4-2. The tunnel is

typically taken to be doubly infinite in length for analytic solutions. Tunnel length is

necessarily finite when computational approaches such as panel methods are used.

Model flows with substantial embedded supersonic regions, at high lift coefficients so

that wake position or separated wake effects become important, and in the transonic,

near-sonic, and low-supersonic speed regimes are beyond the scope of this chapter.

Figure 4-2 Potential flow in an ideal wind tunnel with ventilated walls

"Classical" ventilated walls are taken to be either longitudinally slotted walls,

ventilated walls, or a combination of these two wall types, whose behavior is described

locally by a simple pressure-cross-flow relationship and whose geometry remains fixed


TEST SECTIONS

51

over a given range of test conditions. It is assumed that these walls are vented to a

single large plenum chamber, whose pressure is constant and is taken to be the

reference static pressure for the calculation of the onset Mach number in the tunnel.

Note that for a plenum of finite longitudinal extent, the Mach number far upstream

does not necessarily correspond to this plenum reference Mach number. The wall

interference corrections in AGARDograph 109 for steady flows are discussed in terms

of interference velocity components: longitudinal (or stream-wise, iu ) and cross-stream

(typically up-wash, iw ). Because of their one-to-one correspondence to simple

representations of model volume and lift for a model at the center of a tunnel with

uniform walls, these interferences are commonly referred to as blockage and lift

interference, respectively. The separate interference velocity components are assumed

to be independent and superposable. Independence can be obtained by suitable

symmetry restrictions: a small model located at the center of a tunnel of symmetric

cross section and having uniform walls.

Cross-coupling of interference velocity components and model characteristics

(blockage interference due to lift, for example) will occur for models asymmetrically

located relative to the walls and for non-linear wall cross-flow characteristics. Non-

linear wall ventilation can be the result of actual geometric differences among the

walls, but is usually attributed to the action of viscosity at the walls. Superposition is

valid provided the magnitudes of the corrections remain small and the Mach number is

not too close to 1.0.

Interference corrections for ventilated walls are further classified in

AGARDograph 109 according to wall type and test section cross section. The wall

type refers to the boundary condition to be satisfied at the wall, mainly: solid-wall,

open-jet, ideal slotted or ideal porous, though there is some discussion of the hybrid

slotted wall (slots with cross-flow resistance). The test sections considered are the 2D

tunnel (planar flow), circular (or by coordinate transformation, elliptical), rectangular

and, less comprehensively, octagonal (or rectangular with corner fillets). Most of the


TEST SECTIONS

52

results given are for walls whose geometry does not vary stream-wise and that extend

far upstream and downstream of the model.

As suggested in Chapter 3, the interference results for small models in 2D and

rectangular test sections are considered suitably representative of many interference

situations encountered in practice (the major exclusions include sidewall interference

in 2D testing, "large" models, and models "too close" to the walls). Rectangular

sections with corner fillets or elliptical cross sections may be approximated by

rectangular tunnels of equal cross-sectional area and equivalent aspect ratio (width to

height ratio). This approximation is supported by the close correspondence of

interference characteristics of square and circular ventilated test sections.

The interference flow field is commonly described in non-dimensional terms as

defined in Equations (3-6) and 3-8) for stream-wise and cross-stream (up-wash)

interference velocity perturbations.

iu

U

4-2

i

L

w c

U sC

4-3

Solid blockage interference for small models in ventilated-wall tunnels is

conveniently expressed in terms of the blockage parameter S , the ratio of solid

blockage in the ventilated test section to that in a solid wall test section of the same

cross section:

ventilated

S

closed

4-4

Thus S =1 for a solid-wall test section.

The stream-wise gradient of , / x results in a pressure force on the model

(buoyancy drag), whose magnitude is proportional to the effective volume of the model

(for small models in linear gradients). The stream-wise gradient of up-wash, or flow


TEST SECTIONS

53

curvature, characterized by results in additional angle-of-attack and pitching moment

corrections for even small models.

1

( )x

L

4-5

For models of large size, applying only primary corrections to the free stream is

at best approximate. Residual corrections may be adequate for many cases but large

variations of blockage and/or up-wash interference over the region occupied by the

model may ultimately not be correctable. That is, there is no equivalent unconstrained

flow (with a uniform onset velocity) for the model geometry being tested. This

situation is particularly acute in transonic flow fields because of their extreme

sensitivity to small variations in onset flow conditions. The adequacy of corrections

can be tested by careful comparison of computed model aerodynamic characteristics

from in-tunnel and unconstrained-stream solutions (at flight conditions that include

primary interference corrections). Such a test requires a higher degree of sophistication

of model representation than for the calculation of simple linearized corrections.

Paneling or gridding requirements for this type of analysis are the same as for typical

high-resolution free-air analyses.

4.2 Wall boundary conditions

The wall boundary condition distinguishes ventilated walls from solid-wall or

free-jet boundaries. A useful simplification of the actual wall boundary condition is to

treat the walls as homogeneous, wherein the open- and solid-wall areas are not

represented separately, but as an equivalent permeable surface (Davis D. D., June

1953), (Goethert B. H., 1961). The normal velocity through the walls thus is a local

average, varying smoothly and in a continuous manner as a function of the (similarly

spatially averaged) pressure distribution on the walls. Walls with perforations are thus

idealized as permeable porous surfaces with infinitesimally small holes. Slotted tunnels

are idealized as having an infinite number of very small slots distributed around the

tunnel boundaries.


TEST SECTIONS

54

The validity of the assumption of homogeneous walls depends on the length

scale of the wall openness and the Mach number. It is expected that the effect of wall

"graininess" will be felt out into the tunnel stream a distance on the order of / ,

where l is the length scale associated with the wall openings. As long as / is small

compared to the tunnel dimension (or more directly, to the distance from the wall to

the closest model part, such as a wing tip), the interference felt by the model will be the

same for homogeneous walls, as for discretely ventilated walls having equivalent

cross-flow properties. There are often two distinct geometric length scales associated

with a given ventilated wall: the typical size of the discrete openings and their spacing.

A third length scale may also be involved: the wall boundary layer thickness, whose

properties have been found to influence the wall cross-flow characteristics.

For ventilated walls, the openness length scales are the holes diameter and

spacing. For slotted walls, they are the slot width and circumferential slot spacing.

Consideration of typical ventilated wall arrangements suggests that treating ventilated

walls as homogeneous (for wall interference purposes) is a valid assumption given the

typical small scale of perforations. Slotted-wall openness length scales, on the other

hand, are often at least an order of magnitude larger. For some tunnels, the slot spacing

approaches a substantial fraction of a test section dimension. The assumption of

homogeneous walls is more tenuous in this case, especially for models whose

components are on the order of an openness length from a wall surface (e.g., wing tips

of large-span models, body tail or nose for long models at high angles of attack).

For cases where the walls cannot be treated as homogeneous, the alternating

open- and solid-wall areas (slots and slats) can be modeled separately, for example, by

an appropriate mix of solid-wall and open-jet boundary conditions. In such situations,

simplicity and computational efficiency are sacrificed for higher fidelity of the

simulation.

Measured boundary conditions methods with ventilated walls may be strongly

influenced by wall inhomogeneities (solid and open elements). The resulting local flow

gradients are not representative of the far-field homogeneous boundary condition.


TEST SECTIONS

55

Correction methods for individual measurements, alternate measurement strategies, or

explicit computational modeling of wall elements may be required.

4.2.1 Ideal ventilated wall boundary conditions

The boundary conditions of ventilated walls are motivated by physical

considerations (see, for example (Davis D. D., June 1953), (Baldwin, May 1954),

(Goethert B. H., 1961). The so-called ideal porous wall boundary condition can be

derived by consideration of porous walls as a lattice of lifting elements. The pressure

difference across the wall is then proportional to the flow inclination ( ) at the wall,

2 2wall

wall plenum normalp

p pC

q R U R

4-6

In linearized perturbation form with the plenum pressure taken to be the same

as the pressure far upstream,

n xR 4-7

where R, is an experimentally determined constant of proportionality. Note that the

limits R=0 and R correspond to the standard solid-wall and free-jet boundary

conditions, respectively. It is convenient to define an alternate ventilated wall

parameter,

1

(1 )

Q

R

4-8

so that 0Q corresponds to a solid wall, and 1Q to a free jet.

The ideal homogeneous slotted-wall boundary condition is developed by

consideration of the balance of pressure difference across the slots and stream-wise

flow curvature in the vicinity of the slots,

0nx xnK

R

4-9

where the third term represents a viscous pressure drop across the slot and , the slot

parameter, is related to slot geometry, including the approximate effect of slot depth

/t a according to


TEST SECTIONS

56

e

1 πa tK d log cosec

π 2d a

4-10

Slotted-wall geometry definitions are summarized in Figure 4-3. For an ideal

inviscid slotted wall (i.e., R ), solid-wall and free-jet boundary conditions

correspond to K and 0K , respectively.

Figure 4-3 Slotted Tunnel Geometry

As for the ideal porous wall, a convenient alternate slot parameter is defined,

1

1P

F

4-11

where is proportional to according to

2 /F K H for a 2D test section.


TEST SECTIONS

57

0 /F K r for a circular test section.

/ F K H for a rectangular test section.

0P and 1 P correspond to solid-wall and free jet boundary conditions respectively.

The boundary conditions for walls with discrete slots comprise

0n on the slats (i.e., the solid-wall segments between slots).

0n

xR

for slots with cross-flow resistance.

xφ 0 for open slots.

The ideal ventilated-wall boundary conditions may be viewed as first-order

approximations to ventilated wall cross-flow characteristics. These simple analytic

expressions are intended to capture the dominant flow physics at the wall, as perceived

at some distance from the wall (i.e., at the model location). Improvements in ventilated

wall modeling have focused on more accurate descriptions of the flow near the wall,

including:

1) Effect of boundary layer thickness on the wall cross-flow characteristics.

2) Non-linear pressure-drop terms (e.g. proportional to square of cross-flow velocity).

3) Entry of stagnant plenum air into the test section.

4.3 Interference in 2d testing

Some of the principal results given in AGARDograph 109 and (Pindzola, May

1969) are repeated here as benchmarks for small models. Using a Fourier transform

method these results were calculated.

Reference (Data Unit Engineering Sciences, October 1995) has published

comprehensive summary carpet plots of lift and blockage interference and gradient

factors for 2D point singularities in ideal porous and slotted test sections.

4.3.1 Interference of small models, uniform walls

In the (Figure 4-4) interference parameters with (homogeneous) slotted and

porous walls for a small model in the center of a 2D test section are shown as functions

of porous wall parameter Q, and slotted wall parameter P, respectively. As the

superposition of a point source doublet whose strength is proportional to the model


TEST SECTIONS

58

effective cross-sectional area and by a point vortex whose strength is proportional to

lift, the model is represented. In a 2D solid-wall test section the blockage of a small

model is given by:

3 26closed

A

H

4-12

where H, is the height of the test section, and A is the effective cross-sectional area of

the model.

Although the solid-wall and open-jet limits of P and Q (0 and 1, respectively)

are the same for these two types of walls, at intermediate values of P and Q the

interference characteristics are fundamentally distinct (except when consideration is

given to slots with cross-flow resistance). It is not possible to obtain zero blockages

and zero up-wash interference simultaneously with any uniform inviscid slot geometry

or uniform porous wall as shown in Figure 4-4.

The blockage interference distribution longitudinal midway between the walls

is shown in Figure 4-5. The interference velocity along the tunnel centerline is

symmetric fore and aft of the model for ideal slotted walls with no viscous pressure-

drop term Q =0. Consequently, on the model there is no interference buoyancy force.

In contrast, porous walls (except for the limiting cases of solid and open jets) offer

longitudinal interference gradient, producing a buoyancy force on the model. The

gradient is very nearly a maximum for the value of porosity for zero blockage

interference (Pindzola, May 1969). Similar it can be expected interference distributions

for slots with non-zeroQ .

The up-wash interference longitudinal variation is shown in Figure 4-6 for

ideal slotted and porous walls (Pindzola, May 1969). For solid walls only zero up-wash

at the model location is obtained. The gradient of zero up-wash is obtained for

intermediate values of Q and P (for porous and slotted walls, respectively), but the up-

wash is non-zero for these cases.


TEST SECTIONS

59

Figure 4-4 2D Interference in ideal slotted and porous tunnels


TEST SECTIONS

60

Figure 4-5 Longitudinal variation of blockage interference in 2d slotted and porous tunnels

(a)Slotted walls Q = 0

(b) porous walls


TEST SECTIONS

61

Figure 4-6 Longitudinal variation of up-wash interference in 2d slotted and porous tunnels

CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND

TUNNEL CORRECTIONS

62

CHAPTER FIVE

5 NEW APPROACH IN NUMERICAL MODELING OF

WIND TUNNEL CORRECTIONS

5.1 Motivation for the 2D wind tunnel wall corrections

Motivation for research in 2D wind tunnel corrections is fact that most classical

contemporary methods represent airfoil with combined vortex-doublet singularity

which together with approaching parallel flow builds circle with circulation around it.

Intensity of the circulation is related to the measured lift coefficient, while circle radius

is generated by the doublet strength selected in such a way to obtain circle area equal

to the frontal airfoil area.

5.1.1 Fundamental ideas of classical 2D wind tunnel wall corrections

Solid wall 2D wind tunnel corrections are schematically illustrated by the

Figure 5-1.

Figure 5-1 Wind tunnel solid wall correction approach

In the wind tunnel measured parameters are angle of attack α, free-stream

velocity V, and aerodynamic coefficients CL, CM, and CD. For wind tunnel correction

airfoil is substituted by cylinder with circulation Γ which corresponds to measured lift

coefficient. Solid walls are modeled by mirroring of cylinder vortex images with

respect to upper and lower solid wall. When cylinder-vortex combination is removed

from test section, remains the influence of the tunnel walls to the measurement.


TUNNEL CORRECTIONS

63

Remaining cylinder vortex system induces velocity components Δu and Δv at the

control point position of the airfoil. Measured free-stream velocity and angularity is

corrected by these induced velocities.

For ventilated (porous) walls Mokry approach is standard. In the wind-tunnel

test section measured free-stream velocity V, angle of attack α, pressure distribution

along test section walls Cp, and aerodynamic coefficients CL, CD and CM. Correction

procedure is schematically illustrated in Figure 5-2.

Figure 5-2 Mokry approach

Again airfoil is substituted by cylinder-vortex combination. Intensity of the

vortex and intensity of the doublet is determined the same way as in the solid wall

case. Pressure distribution along walls from vortex-cylinder combination is determined

next. Difference between measured pressure coefficient along the walls and the one

calculated for cylinder-vortex combination is suitably applied as the boundary

condition on the rectangle sides which represent empty test section. Laplace equation

for disturbance potential is solved within numerical test section to determine

disturbance flow field. Correction velocities calculated at airfoil mid-chord point are

taken to correct angle of attack α and free stream velocity V or Mach number M.


solid test sections

Correction procedure developed in this thesis is applicable for both cases when

either aerodynamics coefficients are measured directly in the wind tunnel or when

pressure distribution about airfoil is measured. In the wind tunnel measured quantities

are angle of attack α, free-stream velocity V, and either aerodynamic coefficients CL,


TUNNEL CORRECTIONS

64

CD and CM, or measured pressure distribution given by Cp. Correction procedure is

illustrated in Figure 5-3.

Figure 5-3 New approach to 2D wind tunnel correction procedure for solid walls

Pressure distribution around airfoil is numerically determined for the isolated

airfoil (in the free-stream), as well as the pressure distribution around airfoil in the

wind tunnel. Airfoil is approximated by strait linearly varying vortex elements

(panels), wind tunnel walls are exactly simulated by multiple mirror imaging of the

airfoils. Calculated pressure distribution around airfoil in the wind tunnel is determined

by taking into account all images. Difference between calculated pressure distribution

for isolated airfoil and for airfoil in the wind tunnel represents corrections by this

method. Lift coefficient correction is determined as:

pLc dxc 5-1

while moment coefficient is corrected according to the formula:

4 pmlx c dxc 5-2


TUNNEL CORRECTIONS

65


ventilated wall test sections

Again correction procedure is applicable for both cases when distribution of

pressure is measured around airfoil or when only aerodynamic coefficients CL and CM

are measured. Airfoil is again represented by a system of straight elements, with

linearly varying vortices strength. Nodal intensities of the vortices are determined by

requiring that cross-flow through control point does not exists. Wind tunnel walls are

ensured by multiple mirroring of complete airfoil resulting in exact solid wall

boundary condition. Additionally, source/sink singularity panels are distributed along

wind tunnel walls which simulate wall porosity. Intensity of the sources/sinks is

proportional to the pressure difference between wind tunnel test section and plenum

chamber. Coefficient of the proportionality is determined by comparison of pressure

distributions obtained by numerical calculations and that obtained by measurement. If

only aerodynamic coefficients are measured, then comparison is done between

measured lift coefficient and calculated lift coefficient. Proportionality coefficient is

determined properly if numerical calculation agrees well with measurement.

Corrections to lift coefficient and to moment coefficient are determined the same was

as for the solid wall case. Figure 5-4 illustrates correction procedure for ventilated 2D

wind tunnel test section.

Figure 5-4 New approach to 2D wind tunnel correction procedure for ventilated walls


TUNNEL CORRECTIONS

66

5.2 Numerical modeling for solid wall wind tunnel

5.2.1 Fundamental assumptions

The following assumptions are adopted:

1. The flow about a two-dimensional airfoil is inviscid and irrotational.

2. The airfoil is represented by a sufficiently large number of linear vortex panels

(Figure 5-5).

3. The air flow is subsonic.

4. Corrections are small, and can be applied linearly.

5. Three-dimensional effects are negligible.

Figure 5-5 Airfoil paneling

5.2.2 Governing equations

The first assumption replaces Navier-Stokes equations by potential flow

equation:

2 22

2 20

x z

5-3

Compressibility correction parameter is defined as:

5-4

where M is the free stream Mach number, and the small disturbance perturbation

velocity potential has been defined as follows:

U x V y 5-5

Transforming and ( , )x z by equation:

21 M


TUNNEL CORRECTIONS

67

( , ) ( , )x z x z 5-6

we obtain 0 , where Laplace operator is:

2 2

2 2

x z

5-7

This equation is solved for free-stream conditions, and for airfoil in the wind

tunnel test section, using the superposition principle of singular solutions, since

Laplace equation is linear. Any combination of singular solutions is also the solution of

the Laplace's equation. Our task here is to select arbitrary constants for singularity

solutions that, besides satisfying the Laplace's equation, also satisfy boundary

conditions.

5.2.3 Boundary conditions

On both airfoil and wind tunnel walls, the normal component of the velocity at

any point of the solid surface must be equal to zero. This requirement is achieved by:

1. Establishing an imaging system of the airfoil, represented by linear vortex segments,

with the respect to the floor and ceiling of the wind tunnel test section. This imaging

system ensures simulation of the real flow-field streamlines, which are parallel to the

floor and the ceiling of the test section.

2. Posting the condition that the normal component of the velocities over the solid

surface of the airfoil (i.e. on the control points of the panels) satisfies the following

condition:

. 0i i V n 5-8

Subscript i indicates a control point whose coordinates are defined by:

1 1,2 2

x x z zi i i ix zc c

i i

5-9

where 1 1

( , ), ,i ii i

x zx z

are the coordinates of endpoints of the segments by which

airfoil is specified, ordered in counter-clockwise direction starting from the trailing

edge. Unit normal at arbitrary control point i is calculated as:


TUNNEL CORRECTIONS

68

1 1

2 2

1 1

.i i

i i i i

i x y

i i i i

z z i x x jn i n j

x x z z

n 5-10

1

2 2

1 1

i

i ix

i i i i

z zn

x x z z

5-11

1

2 2

1 1

i

i iy

i i i i

x xn

x x z z

5-12

3. To ensure that velocity at the trailing edge is finite, the Kutta condition must be

satisfied at the trailing edge.

5.3 Induced velocities

5.3.1 Two-dimensional point vortex

Consider a point vortex with strength located at 0 0( , )x z as shown in Figure

5-6.

Figure 5-6 Point vortex

The induced velocity components by this vortex at point P (x, y) are:


TUNNEL CORRECTIONS

69

0

2 2

0 0

( )

2 ( ) ( )

z zu

x x x z z

5-13

0

2 2

0 0

( )

2 ( ) ( )

x xv

z x x z z

5-14

5.3.2 General linear vortex distribution

Velocity induced at some arbitrary point ( , x z ) by vorticity with linear

strength variation along the segment (see Figure 5-7), is calculated by applying the

superposition principle. By this principle contribution of all vortices 0 0

( )x dx along

vortex segment placed between 1

x and 2

x is added to obtain:

2

00 02 2

01

1( )

2 ( )

x

x

xv x dx

x x z

z

5-15

Figure 5-7 Linear strength vortex variation


TUNNEL CORRECTIONS

70

2

0 00 02 2

01

1 ( )( )

2 ( )

x

x

x x xv x dx

x x z

5-16

These velocity components are expressed in local coordinate system. To use

these expressions for arbitrary position of the vortex segment, it is necessary to

transform the coordinates of the end points of the vortex and coordinates of the

arbitrary point (x, z) to the coordinate system fixed to segment. Vorticity distribution

0( )x is determined by vorticity strengths

1 and

2 at segment’s end points. Their

magnitude is determined from boundary conditions (see equation 5-8).

0 1 0 1( ) ( )x x x 5-17

2 1

2 1x x

5-18

Integration of expressions for u and v gives induced velocity by vortex segment at

any point ( , x z ) in the segment-fixed coordinate system:

1 12 1 2 1

2

Δ

2 2

ru yln x

r

5-19

1 1 11 2 2 1

2 2

Δ

2 2

r rv ln xln x x z

r r

5-20

1 1

2 1

2 1

; z z

tan tanx x x x

5-21

2 2 2 2

1 1 2 2( ) ; ( ) r x x z r x x z 5-22

Angles 1 and

2 , as well as

1r and

2r , are shown in Figure 5-7.

5.3.3 Linear vortex distribution with image

In two dimensions, the solid wall boundary condition can be satisfied on the

upper and lower walls by generating a column of airfoil images, represented by their

vortex segments, mirrored both above and below the test section. Theoretically, the

number of images is infinite. In Figure 5-8 the segment of linear vortex strength


TUNNEL CORRECTIONS

71

distribution in the test section is mirrored by an infinite number of its symmetric

images with respect to the ceiling and the floor of the test section. All images as well

as an original segment on the airfoil contribute to the induced velocities.

Figure 5-8 system of image for linear strength vortex

The n-th image of the vortex segment is placed between points1

x and2

x in

global coordinate system, as well as a segment in the test section. The local coordinate

system is fixed to this segment image, with the x-axis passing through1

x and2

x .

Velocity components induced at point ( ,x z ) in the local coordinate system are

calculated by:

112 1

2

Δ( 1) ( )

2 2n

n n

n

n

n

ru yln x

r

5-23


TUNNEL CORRECTIONS

72

111 2 2 1

2

Δ( 1)

2 2n

n n

n

n

n

rv xln x x z

r

5-24

where n

u and n

v represent components of induced velocity at an arbitrary point (x,y)

in the local coordinate system of the n–th segment image.

5.3.4 Numerical solution of the flow about the airfoil

In the equations (5-23) and (5-24) the subscripts 1 and 2 refer to first and last

point of a panel, globally defined by points numerated as j and j+1 respectively. The

airfoil NACA 0012 in this work is given with 1N pairs of ( , x z ) coordinates

ordered counterclockwise, starting from the trailing edge of the airfoil. The shape of

the airfoil is approximated by N panels connecting these 1N point coordinates of the

airfoil. In expressions for induced velocities, 1 and

2 are local parameters, unique

for each panel. These coefficients are used to model vortex strength variation over the

panel. If the strength of γ at the beginning of each panel is set equal to the strength of

the vortex at the end point of the previous panel, the continuous vortex distribution is

obtained. The numerical procedure should determine all vortices (1

, , .j j

)

at the

end points of the panels, see Figure 5-9. If the airfoil shape is approximated by N

distributed vortex panels, then the number of unknown parameters is equal to the

number of points which define vortex segments, i.e. N+1, one greater than number of

panels. To apply expressions (5-19) to (5-24), subscripts 1 and 2 should be replaced by

j and 1j respectively.

The induced velocity components in local coordinate system of the panel at i-th

control point by n-th image is expressed in terms of the panel-edge vorticity strengths

and . This way, equations (5-23) and (5-24) become: j 1j


TUNNEL CORRECTIONS

73

Figure 5-9 Nomenclature for a linear-strength vortex element

1

( j 1) j

j 1 j

j

( j 1) j

( j 1)

( )( 1)

2 2

.

( )

( )

n n

n

n n

n

j j jn

ij n

i i

ux x

rz ln x

r

5-25

j 1

( j 1) j 1 j

j

j j 1 ( j 1) j

( j 1)

( )( 1)

2 2 (

.

)n

n

n

n n

n

j j jn

ij n

i i

rv ln

r x x

rx ln x x z

r

5-26

Control point coordinates with respect to vorticity segment J are transformed in

segment fixed coordinate system by the equations (5-27) and (5-28):

( z )n i i niJ c j jn c j jnX x x cos z sin 5-27

Z z cos ( )sinn i n iiJ c j jn c j jnz x x 5-28


TUNNEL CORRECTIONS

74

where ( ,c ci i

x z ) are global coordinates of the control point, ( ,j jn

x z ) are global

coordinates of the first point of the n-th segment J image, while ( iJnX , Z )iJn

are local

coordinates of the control point, as viewed from the local coordinate system fixed to

segment J (see Figure 5-9). Local coordinates of the starting points are (0, 0), while the

local coordinates of the end points of the segment are given as shown in equations

(5-29) and (5-30).

( 1) 1 1cos (z z )sinn n nj j j jn j j jnX x x 5-29

( 1) ( 1) 1Z (z z )cos (x )sinn n nj j j jn j j jnx 5-30

Slope of the segment with respect to global x-axis is:

( 1)1

1

z

xn nj j

jn

j j

zatan

x

5-31

The distances between the control point and the end points of the segment are:

2 2

n n nij iJ iJR X Z 5-32

2 2

( 1) ( 1)( )n n nij iJ j n iJR X X Z 5-33

Angles between segments and the lines connecting end of the segment with

control point are given as:

1

n

Zθ n

n

iJ

ij

iJ

atanX

5-34

1

1n

1

Zθ

Xn

n

iJ

ij

iJ ij n

atanX

5-35

It is necessary to separate contributions to the induced velocity at control points

into parts influenced only by end segment vorticities. Equations (5-25) and (5-26) can

be divided into a portion of velocity influenced by j

and a portion of velocity

influenced by1j

. The superscripts ( ) j

and 1( ) j

represent the contribution of the

beginning and the contribution of the ending vorticity strength.


TUNNEL CORRECTIONS

75

( j 1) j ( j 1) j

( 1) ( 1) ( 1)

Z1( 1) ( )

2n n n

n n n n n

n n n

iJ j iJj n

i j

j j j

R Xu ln

X R X

5-36

1

1 ( j 1) j

( 1) ( 1) ( 1)

Z1( 1)

2n n n

n n n

n n n

iJ iJ jj n

i j

j j j

X Ru ln

X X R

5-37

( j 1) j

( 1) ( 1) ( 1)

Z1( 1) 1 1

2n n n

n n n

n n n

j iJ iJj n

i j

j j j

R Xv ln

R X X

5-38

1

1 ( j 1) j

( 1) ( 1) ( 1)

Z1( 1) 1

2n n n

n n n

n n n

iJ j iJj n

i j

j j j

X Rv ln

X R X

5-39

where j

inu ,

1j

inu

j

inv and

1j

inv

represent the induced velocity components influenced

by the vorticity strengths at the beginning and at the end of each segment. The

calculations of the equations (5-36), (5-37), (5-38) and (5-39) are based on the

assumption that 1j

and1

0j

. The induced velocity at any point in the flow

field in local (segment fixed) coordinate system is:

1

n n n

j j

iJ i iu u u 5-40

1

n n n

j j

iJ i iv v v 5-41

The equations (5-36), (5-37), (5-38) and (5-39) can be arranged in the form:

.n n

j

i ij ju c 5-42

1

1.n n

j

i ij ju e

5-43

.n n

j

i ij jv w 5-44


TUNNEL CORRECTIONS

76

1 .n n

j

i ij jv z 5-45

where the ijn

c , ijn

e , ijn

w and ijn

z represent the coefficients defined as:

( j 1) j ( j 1) j

( 1) ( 1) ( 1)

Z1( 1) ( )

2n n n

n n n n n

n n n

iJ j iJn

ij

j j j

R Xc ln

X R X

5-46

( j 1) j

( 1) ( 1) ( 1)

Z1( 1)

2n n n

n n n

n n n

iJ iJ jn

ij

j j j

X Re ln

X X R

5-47

( j 1) j

( 1) ( 1) ( 1)

Z1( 1) 1 1

2n n n

n n n

n n n

j iJ iJn

ij

j j j

R Xw ln

R X X

5-48

( j 1) j

( 1) ( 1) ( 1)

Z1( 1) 1

2n n n

n n n

n n n

iJ j iJn

ij

j j j

X Rz ln

X R X

5-49

Induced velocity components, calculated in segment fixed coordinate system,

have to be transformed back into airfoil coordinate system, and summed up to

determine induced velocity at a control point (xci ,zci) by the vorticity segment J:

n n

n k n k

iJ iJ jn iJ jn

n k n k

u u cos v sin

5-50

n n

n k n k

iJ iJ jn iJ jn

n k n k

v u sin v cos

5-51

where k determines the number of images used in the calculation, both with respect to

the upper and lower wall. Number k is determined in such way that the contribution of

the first neglected image, which is too far to generate any practical influence to the

relative velocity, is less than the specified small number ε, defined as:

0

nv

v


TUNNEL CORRECTIONS

77

The vn is velocity induced by n-th image of the vorticity segment, while 0

v is

velocity induced by vorticity segment in the wind tunnel test section. Components of

induced velocity in global coordinate system at ( , )x z point in the flow field are

obtained by transforming local induced velocity components, due to vortex segment

between point’s j and 1j , according to:

1 1( . . ) ( . . )n n n n

n k n k

ij ij j ij j jn ij j ij j jn

n k n k

u c e cos w z sin

5-52

1 1( . . ) ( . . )n n n n

n k n k

ij ij j ij j jn ij j ij j jn

n k n k

v c e sin w z cos

5-53

Equation (5-52) after rearrangement can be written in the form:

1. .ij ij j ij ju a b 5-54

where the coefficients ij

a and ij

b are given by following equations:

( )n n

n k

ij ij jn ij jn

n k

a c cos w sin

5-55

( )n n

n k

ij ij jn ij jn

n k

b e cos z sin

5-56

Similarly for equation (5-44):

1. .ij ij j ij jv k s 5-57

where the coefficients ij

k and ij

s are defined as:

( )n n

n k

ij ij jn ij jn

n k

k c sin w cos

5-58

Since vorticity strength is shared by two neighboring segments and ,

it is necessary to group contributions of each end vorticity. Only first and last points

are not shared by two vorticity segments. Components of induced velocity due to

vorticity strength are given by the equations (5-59) and (5-61):

j 1J J

j


TUNNEL CORRECTIONS

78

, 1

, 1( ) .j j

i ij i j j ij ju a b A

5-59

1

, 1

1

2 .

1

i

ij i jij

ij

a j

a bA j N

b j N

5-60

where the coefficients 1i

a and ij

b represent the coefficients of first and last segment.

, 1

, 1( ) .j j

i ij i j j ij jv k s B

5-61

1

, 1

1

2 .

1

i

ij i jij

ij

k j

k sB j N

s j N

5-62

Total velocity at a control point ( , )c ci i

x z is obtained when all contributions

are summed up:

1

1

.N

i ij j

j

u A U

5-63

1

1

.N

i ij j

j

v B V

5-64

Boundary conditions require that the normal velocity component to the airfoil

surface at arbitrary control point i is equal to zero:

i i iu v V i j 5-65

. 0i ii i i x i zu n v n V n 5-66

When expressions (5-63) and (5-64) are replaced into equation (5-66) the

following is obtained:

1 1

1 1

. . . . 0i i i i

N N

ij x j x ij z j z

j j

A U B V

n n n n 5-67

The far field speed on the right-hand side is obtained as:


TUNNEL CORRECTIONS

79

1

1

. . . .i i i i

N

ij x ij z j x z

j

A B U V

n n n n 5-68

1

1

. . , 1,2, ,i i

N

ij j x z

j

D U V RHS i N

n n 5-69

Since the control point is defined in the middle of the segment, there are N

segments and thus N conditions given by equation (5-69).

Additional necessary condition is obtained from Kutta condition:

1 1 0N 5-70

The system of equations is then solved to determine the coefficients of linear

vortex strength panels.

111 12 1, 1 1

21 22 2, 1

31 32 3, 1

,1 ,2 ,

1

2

1

2

3 3

.... .. .. ..

1 0 1 0

N

N

N

N N N N N N

N

a a a RHS

a a a RHS

a a a RHS

a a a RHS

5-71

Now, as the vorticity strengths j

are known for all panels, then the

induced velocity at each control point can be easily calculated by equations (5-40) and

(5-41).

The pressure coefficient can be calculated as:

2 2

2 21

i

i ip

u vC

U V

5-72

The lift coefficient can be calculated from equation (5-73):

L p

dxC C

c∮ 5-73


TUNNEL CORRECTIONS

80

5.4 Numerical modeling for ventilated wall wind tunnel

5.4.1 Fundamental assumptions

The following assumptions are adopted:

1. The flow is incompressible (compressibility is taken over the parameter

21 M ).

2. The effects of porous walls can be superimposed the solution with the

solid walls.

3. The porosity of one wall does not affect the porosity of the second, in the

calculation sense.

5.4.2 Application of Bernoulli equation

The plenum chamber around the model in wind tunnel is usually at the same

pressure as the flow pressure in front of the model (far enough). If the pressure and

velocity of flow far enough from the model front are denoted as p ,V , then the

Bernoulli equation will be;

2 21 1

2 2p V pV

5-74

and after rearranging

22

2

1(1 )

2

Vp p V

V

5-75

If we assume that the velocity at the walls wind tunnel can be written as:

V V u 5-76

then substituting it in the previous equation we get

2 22

2

211

2

V u up p V

VV

5-77

Also after neglecting u2

in comparison with other terms:

p p V u 5-78


TUNNEL CORRECTIONS

81

5.4.3 Boundary Condition for a ventilated Wall

It is desirable to formulate a boundary condition for the slotted-ventilated wall

in order to generalize the approach adopted in the experiments. It is also desirable to

find a basis for comparison between interference due to an 'ideal' slotted-ventilated

wall (i.e., with inviscid flow) and interference when viscous slot flow is important. In

both cases the porosity of the boundary influences the interference up-wash in the

tunnel, but the porosity of an ideal ventilated wall relates the mass flow through the

wall to the pressure drop across it in inviscid flow, whilst the effective porosity of a

slotted wall is significant only when viscous flow at the boundary is predominant, e.g.,

when slot width is less than the boundary-layer displacement thickness. The porosity

due to viscous slot flow can be associated with that of a truly porous wall, and one

might expect a similar porosity effect for a ventilated wall if the perforation size is less

than the boundary-layer displacement thickness, (A.W. Moore and K. C. Wight, 1969).

In (T.R. Goodman, November, 1950) the pressure drop across the wall to the outflow

is related. The pressure drop across a porous wall is given by Darcy's law:

p w 5-79

Vp w

P

5-80

where P is the porosity factor.

Goodman obtains the boundary condition at the porous wall from equations

(5-76) and (5-77):

( )w Pu P V V 5-81

where the w and u is the vertical and axial component of induced velocity, V is the

free velocity of flow far from the model.

To determine a porosity parameter for the slotted-ventilated walls in inviscid

flow, consider first the boundary condition for a ventilated wall. Assuming that an

idealized ventilated wall consists of an infinite number of traverse slots, reference (P.F.

Maeder, May, 1953) shows that in incompressible flow the relation between the

perturbation velocity in the stream direction and the velocity normal to the wall is:


TUNNEL CORRECTIONS

82

Iw P u 5-82

where PI is a constant of porosity for a given wall configuration.

From equation (5-80) we start to calculate the interference of wind tunnel with

porosity walls. The axial velocity on the walls u is calculated from the numerical

solution of the effects of linear vortex panels and their images inside wind tunnel on

the solid walls. This velocity is regarded as initial value to solve the equations of

constant strength source which are taken in case of ventilated wind tunnel walls.

5.4.4 The effect of constant strength sources in the wind tunnel walls panels

In the previous modeling we considered the solid walls interference in the

testing model inside test section of wind tunnel. In this modeling: the wind tunnel walls

are divided with n panels and each panel is approximated with constant strength source

to model the porosity effects through the wind tunnel walls as shown in Figure 5-10.

Figure 5-10 Constant strength source panels on wind tunnel walls

For each panel the induced velocities are calculated in the control point for the

same panel without contribution of the other panels of constant strength sources,

because the panels in the same straight line cannot induce velocity in the point which

the line of velocity normal to the same straight line. The effect of constant strength

source panels on upper wall does not affect porosity lower wall panels, and vice versa.

The induced velocities for constant strength source can be calculated from the

equations below:


TUNNEL CORRECTIONS

83

2 2

1

2 2

24

x x zu ln

x x z

5-83

1 1

2 1

( )2

z zw tan tan

x x x x

5-84

where is the source strength. The control points of the constant strength source

panels are defined by coordinates:

2 1 2 11 1,

2 2i ic c

x x z zx x z z

5-85

If the induced velocities at control points are calculated from the constant

strength sources, positioned on the same height, then the local vertical axis coordinate

for all of them is z = 0 (see Figure 5-11).This means that the normal component of

induced velocity at a panel control point equation (5-84) is affected only by its own

panel, and takes the form:

(x)(x)

2w

5-86

Figure 5-11 Control point and source panel in the same location

The sign plus or minus in the previous equation is important to distinguish

between the conditions when the panel is approached from its upper or from its lower

side.

The source strength can be calculated from the equations (5-80) and (5-85) as:

2 2 ( )Pu P V V 5-87

In the calculation of source strength (sigma), the value of the axial velocity on

the wall V is taken from the numerical calculation of the influence of the airfoil’s linear


TUNNEL CORRECTIONS

84

vortex panels on the upper and lower solid wall control points of wind tunnel. Each

wall control point on the wind tunnel walls is affected by all airfoil’s panels of linear

strength vortices. The value of porosity parameter P must be estimated depending on

the procedure correction of the measured data in the wind tunnel i.e. iterate the value of

porosity factor in the numerical calculation until the results be equivalent to the

measured wind tunnel data.

After calculation the strength source panel, the induced velocity in the control

points of the panels in the upper and lower walls can be easily calculated from the

equations (5-82) and (5-84).

5.4.5 The effect of sources panels on the vortex panel control points on the

airfoil

As already mentioned, the induced velocity in control points of constant

sources panels on the walls are calculated only as self-influence, without contribution

of the other panels in the calculation.

Also, the airfoil inside test section of the wind tunnel was approximated with n

panels of linear vortex strength segments and the numerical solution is carried out for

free stream (without walls effect i.e. without image system calculation ), and with wind

tunnel walls effect (in this case with image system calculation). As a result, the

pressure distributions and lift coefficients are obtained.

Now the effects between the linear strength vortices on the approximated airfoil

and constant strength sources on the upper and lower wind tunnel walls must be

considered.

All the source panels for the upper and lower walls induce velocity at each

control point of the airfoil’s the vortex panel; therefor the velocity in the control point

of the linear strength vortex panel is a summation of two velocities, induced from

linear vortex panels and constant source panels see Figure 5-12. The velocity induced

from vortex panel was numerically calculated from equations (5-29) and (5-30) after

solving the system of equations see equation (5-71). The additional velocity in the

control point of vortex panel as a result of the constant source panel’s effects was

calculated from equations (5-83) and (5-84).


TUNNEL CORRECTIONS

85

Figure 5-12 Source panels induced control points in linear vortex panels

The system of equations which were solved in case of solid walls to calculate

the vortex strength parameter, now takes another form therefore it must be solved for

the second time to calculate the induced velocities, i.e. the additional influence of

source panels at test section walls. As a result, the pressure distribution and lift

coefficients are obtained.

CHAPTER SIX RESULT AND DISCUSSIONS

86

CHAPTER SIX

6 RESULTS AND DISCUSSION

Analyses performed for the verification of here presented calculation model

were done both in the sense of calculations and corrections of global parameters -

aerodynamic coefficients (for cases when only they are measured in wind tunnels), and

detailed analyses, i.e. determination of pressure coefficient distributions. It should be

emphasized that classical methods, based on airfoil representation by a singular point,

are inherently unable to determine local pressure coefficient distributions over the

airfoil contour.

Initial numerical calculations were conducted for free stream case, with an aim

to evaluate the lift coefficient variations versus angle of attack (global analysis case)

for airfoil NACA 0012 the standard calibration airfoil, which is the standard calibration

airfoil both for wind tunnel calibration purposes, and for numerical models and

software verifications. This airfoil was approximated by linear vortex strength panels,

as described in previous chapters, and calculations were performed for angles of attack

α = -2º, 0º, 2º, 4º, 6º and 8º. The results numerically calculated lift coefficient values

were compared with the existing relevant experimental data, carefully corrected to

represent free stream values (Abbott I. H. Von Doenhoff).

Figure 6-1 shows that agreements are very good at low and moderate angles of

attack. At higher angles of attack, the viscosity effects start to influence the

experimental lift curve more remarkably, and results of here applied potential model

begin to slowly diverge. This is the "natural" behavior of all correction methods based

on potential flow simulations, which neglect viscosity influence (and most present

correction methods are of this category). Since moment coefficient about aerodynamic

center for all symmetrical airfoils, such as NACA 0012, is equal to zero (except at

around critical angles of attack, which cannot be treated for potential models), it was

not considered in this sense. Also, the experiments used for this verification were


87

conducted in low-speed wind tunnel, so Mach number M = 0.15 was used for

numerical calculations.

This initial verification gives advantages to this numerical method to calculate

the corrections needed both for solid wall and ventilated test section experimental

results.

Figure 6-1 Experimental and numerical lift coefficient

6.1 Correction for solid wall test section

Results of the here presented calculation model will be compared with some

standard correction procedures, widely used in many experimental facilities involving

solid wall test sections. Since they were initially derived for low speed tests, the Mach

number M = 0.15 is applied at this time as well, and numerical corrections are

calculated for angles of attack of = 2o and = 6

o.

The lift coefficient correction factor CL

k represents the ratio between the lift

coefficient in free stream, and the lift coefficient in the tunnel test section:

tunL

freeL

CC

CK

L 6-1


88

The angle of attack correction in here applied numerical model is zero (defined

by the concept of the calculation model itself), meaning that the corrected lift

coefficient applies for the same angle of attack as in the wind tunnel. On the other

hand, in classical methods, angle of attack correction is also applied, where the

correction factor K for angle of attack is:

free

tun

K

6-2

Because of that, comparisons between the numerical methods and classical

methods must be done using the lift curve slope correction factor:

K

KK LC

a 6-3

which is the ratio between the lift coefficient and angle of attack correction factors.

For the purpose of verification, the relative test section heights h = 3, 4, 5 and 6

have been applied, where h represents the ratio between the test section height and

model chord length. Numerically obtained results and corrections are presented in

Table 6-1, Table 6-2, Figure 6-2 and Figure 6-3. From (Robert E. Sheldahl) and (I. H.

Abbott), the free stream lift coefficients for NACA 0012 airfoil, for angles of attack of

= 2o and = 6

o, are 22.0LC and 66.0LC , respectively.

Table 6-1: Numerically obtained correction parameters, = 2º, M = 0.15

h = 3 h = 4 h = 5 h = 6

freeLC 0.2390 0.2390 0.2390 0.2390

0.2539 0.2477 0.2445 0.2426

0.9413 0.9649 0.9776 0.9847

free o 2 2 2 2

tuno 2 2 2 2

K 1 1 1 1

aK 0.9413 0.9649 0.9776 0.9847

tunLC

LCK


89

Table 6-2 Numerically obtained correction parameters, = 6º, M = 0.15

h = 3 h = 4 h = 5 h = 6

freeLC 0.7145 0.7145 0.7145 0.7145

tunLC 0.7565 0.7385 0.7302 0.7245

LCK 0.9445 0.9675 0.9785 0.9862

free o 6 6 6 6

tuno 6 6 6 6

K 1 1 1 1

aK 0.9445 0.9675 0.9785 0.9862

Figure 6-2 Numerical Cp distribution for free stream and with wind tunnel wall effect =2


90

Figure 6-3 Numerical Cp distribution for free stream and with wind tunnel wall effect =2

Numerically obtained free stream values are slightly larger, because the applied

calculations are based on the inviscid flow model which, as previously mentioned,

inherently overestimates lift with the increase of angle of attack, due to the lack of

boundary layer influence. On the other hand, it must be emphasized that the same

model is applied both for the free stream and wind tunnel calculations; since pressure

coefficients are subtracted, this shortcoming of inviscid calculation model vanishes and

does not affect the calculated corrections, which are applied to the "raw" experimental

measurement data.

Pressure distributions calculated by here presented method, shown in Figure

6-2 and Figure 6-3, clearly indicate that pressure coefficient (especially on upper airfoil

surface - upper curves on diagrams) is noticeably affected by low relative test section

height h = 3, while for large h = 6 it is very small.


91

The same trend of the influence of relative test section height on the amount of

required corrections is clearly seen in Table 6-1and Table 6-2, where required

corrections progressively increase with the decrease of the test section relative height,

while on the other hand, the angle of attack influence is practically negligible on

numerically obtained results.

The classical lift coefficient and angle of attack corrections, based on (I. H.

Abbott), and (Pope A. and Harper), have been calculated for the purpose of the

comparisons (see Table 6-3, Table 6-4 and Table 6-5 Lift curve slope correction factor

by different methods). In case of classical methods, the same values of correction

parameters apply for both considered angles of attack.

Table 6-3 Analytical corrections: Abbot, Doenhoff and Stivers

h = 3 h = 4 h = 5 h = 6

LCK 0.9657 0.9807 0.9876 0.9914

K 1.0228 1.0128 1.0082 1.0057

aK 0.9441 0.9682 0.9796 0.9858

Table 6-4 Analytical corrections: Pope & Harper

h = 3 h = 4 h = 5 h = 6

LCK 0.9635 0.9790 0.9863 0.9903

K 1.0229 1.0128 1.0082 1.0057

aK 0.9419 0.9667 0.9782 0.9846

Numerically obtained values are compared with these methods in Table 6-5.


92

Table 6-5 Lift curve slope correction factor by different methods

aK h = 3 h = 4 h = 5 h = 6

Numerical

α = 2o

0.9413 0.9649 0.9776 0.9847

Numerical

α = 6o

0.9445 0.9675 0.9785 0.9862

Abbott, Doen. &

Stivers 0.9441 0.9682 0.9796 0.9858

Pope &

Harper 0.9419 0.9667 0.9782 0.9846

Numerically obtained values of the lift curve slope correction factors show very

good agreements with those obtained analytically by standard methods. Small

differences, considering results of the two analyzed angles of attack by here applied

numerical model, are of the same order as differences between the two relevant

analytical methods, and are irrelevant for practical engineering purposes. By here

presented method, the moment coefficient corrections can readily be obtained as well,

but were not considered for analyzed symmetrical airfoil, for already mentioned

reasons. Generally speaking, since moment coefficient is obtained by multiplying

upper and lower surface pressure differences (the same used in lift coefficient

calculations) by local distance from the reference point, calculation errors in moment

determination are practically of the same order as for the lift coefficient.

Once again, it should be noted that these corrections apply for solid walls case

only. Here applied model can also be used for the corrections of measurements at

higher subsonic Mach numbers because compressibility influence factor has been

applied within the calculation model.

6.2 Sources of experimental data for calculations of test sections

with ventilated walls

Verifications of the calculation model for the case of ventilated walls will be

verified referencing experimental data from two relevant experimental facilities, the


93

T-38 Transonic wind tunnel VTI Žarkovo - Belgrade, and NASA Transonic cryogenic

tunnel 0.3-m Langley TCT.

6.2.1 T-38 wind tunnel (VTI Žarkovo, Belgrade)

Wind tunnel tests were performed in the test section 0.38 x 1.5 m using the

NACA 0012 calibration model (see Figure 6-4), in the subsonic and transonic Mach

number range. Pressures about the airfoil were measured with two Scanivalves. Wing

pressure holes 1 to 40 inclusive were connected to the Scanivalve 1, while ports 41 to

80 inclusive were connected to the Scanivalve 2. The two additional Scanivalves were

used to measure pressures on upper and lower wind tunnel walls (Aleksandar Vitić).

6.2.2 Transonic cryogenic tunnel (0.3-m NASA Langley TCT)

The test sections (various sizes of test sections can be used) are rectangular,

have solid sidewalls, and slotted top and bottom walls. Two slots are located in each of

these walls with a spacing of 4.0 in (10.16 cm). All model, surface and tunnel floor,

and ceiling pressures were measured using 48-port Scanivalves, connected to high

precision variable capacitance type pressure transducers. The test program considered

in this paper was conducted in the 8 in by 24 in (20.32 cm by 60.96 cm) two

dimensional test section of the Langley 0.3-meter transonic cryogenic tunnel to obtain

the aerodynamic characteristics of a series of 2D airfoils (including NACA 0012), at

subsonic and transonic speeds and flight-equivalent Reynolds numbers (C. L. Ladson

A. S. Hill).

6.2.3 Models

6.2.3.1 Model from T-38 wind tunnel

The 254 mm chord NACA 0012 model is constructed of steel. It is fixed to

double-ended 2D balance in the 2D inset of the wind tunnel T-38 (see Figure 6-4).


94

Figure 6-4 2D test calibration model NACA 0012

The model is instrumented with a total of 80 static pressure holes located on the

medial upper and lower surfaces. There were 50 holes on the upper surface, 28 on the

lower surface and one at both the leading and trailing edges. The model is used for

measuring the pressure distribution, forces and moments, and losses of total pressure

downstream of the model see reference (Aleksandar Vitić).


95

The example of experimentally obtained pressure coefficient distribution about

this airfoil for Mach number M = 0.3, angle of attack α = 2° and Reynolds number Re

= 4.4x106 is shown in Figure 6-5.

Figure 6-5 Cp for airfoil NACA 0012 measured in T-38 wind tunnel at M = 0.3 and α = 2°

6.2.3.2 Model from transonic cryogenic tunnel

For here considered analyses, test data were used from a two dimensional

model of the NACA 0012 airfoil with a chord of 6.00 in (15.24 cm) and a span of 8.00

in (20.32 cm). The model was constructed of A286 stainless steel which is an

acceptable material for cryogenic test conditions. To locate all pressure instrumentation

tubes internally in the model, it was constructed in two halves, the tubing installed, and

then these two halves were bonded together. By locating the tubes internally, the model

surface should be maintained in a very smooth condition, see reference (C. L. Ladson

A. S. Hill). The example of experimentally obtained pressure coefficient distribution

about this airfoil for Mach number M = 0.3, angle of attack α = 2° and Reynolds

number Re= 6.0795x106, is shown in Figure 6-6.


96

Figure 6-6 Cp for airfoil NACA 0012 measured in NASA wind tunnel at M = 0.3 and α = 2°

6.3 Calculation of corrections for test sections with ventilated

walls

In this chapter numerical solutions of flow about NACA0012 airfoil are used to

apply corrections to the measured pressure distributions around an airfoil due to wall

effects with porosity factor, in the test sections of the two considered wind tunnels.

Numerical solutions for the free stream case and for the flow around airfoil in the test

section are calculated, and pressure distributions for both cases are determined. The

pressure coefficient difference between solutions for the flow in the test section of the

wind tunnel and the free stream solution is superimposed to the measured pressure

coefficient distribution, at the corresponding points, used in measurements:

i i i

N N

p p pc c c

6-4


97

i i i

corr meas

p p pc c c

6-5

where ipC is pressure coefficient difference between solution for the flow in test

section of the wind tunnel and free stream solution, and:

Npi

c is the numerical solution for coefficient pressure distribution about airfoil in the

test section;

Np i

c is the numerical solution for coefficient pressure distribution about airfoil in free

stream;

corrpi

c is the measured pressure coefficient distribution about the airfoil after correction;

measpi

c is the measured coefficient pressure distribution about airfoil.

The new value of the lift coefficient is calculated from correcting pressure

distribution by numerical integration.

6.3.1 Wind tunnel T-38

The numerical calculations were carried out for the NACA0012 airfoil at α =

2°, 4°, 6° and M = 0.3, corresponding to one of the test cases in T-38 wind tunnel.

The pressure coefficient distributions measured on the NACA0012 airfoil in the

T-38 test section and the numerically calculated values of pressure coefficient for same

airfoil (approximated by the linear vortices strength panels, and with the effects of

wind tunnel walls simulated with constant strength sources) are shown in Figure 6-7,

Figure 6-8 and Figure 6-9 for the angles of attack α = 2°, 4° and 6° respectively.

The differences between measured and numerically calculated distributions of

pressure coefficient are small, and they can be contributed partially to the neglected

viscous effects within the calculation method, and partially to inevitable small

measurement errors. Those results verify the introduced and applied method of

ventilated walls modeling, and their influence on the calculated pressure distributions

around the airfoil in the simulated test section.


98

Figure 6-7 Measured and calculation pressure distribution in T-38 wind tunnel



99


The next step was numerical determination of required corrections, quantified

as difference between pressure coefficient distributions between previously calculated

values in ventilated test section, and the free stream distributions. Pressure coefficient

distribution about the NACA0012 airfoil, approximated by linear vortices strength

panels, is numerically calculated this time with number of mirrored images set to zero

(walls excluded), and with zero strength of sources/sinks (ventilation effects excluded).

The pressure coefficient distributions for these two cases are compared and shown in

Figure 6-10, Figure 6-11 and Figure 6-12, for angles of attack α = 2°, 4° and 6°

respectively.

The differences between the two pressure coefficient distributions at control

points i, denoted as ΔCpi, represent the values of calculated corrections.


100

Figure 6-10 Numerical Cp for free stream and wind tunnel wall effect for T-38



101


Finally, the measured pressure coefficient distributions in T-38 wind tunnel are

corrected by superimposing ΔCpi to them. For that purpose, it is necessary to

interpolate the ΔCpi values, calculated at panel control points, to positions which

corresponds pressure measurement. The corrections applied to T-38 measurements are

shown in Figure 6-13, Figure 6-14 and Figure 6-15 for previously defined angles of

attack, while integrated values of corrections, as global parameters, are shown in Table

6-6.

Those results indicate that the influence of ventilated walls on the pressure

coefficient distribution is relatively small, primarily because of the large relative height

of test section h = 6 applied in the T-38 tunnel, and thus also applied in the

calculations. Table 6-6 shows that, with the increase of angle of attack, corrections

have slight increasing tendency, although in Figure 6-13, Figure 6-14 and Figure 6-15

the differences considering angles of attack are visually hardly noticeable, because of

relatively small order. On the other hand, when compared with Figure 6-3 for solid


102

walls case and the same relative height h = 6, corrections for ventilated test section are

obviously larger. Considering global parameter comparisons, Table 6-6 also shows

good agreements of lift coefficients obtained in wind tunnel (CL measured), and by

numerical calculations (denoted as CL with wall effect), verifying the here applied

method of ventilated walls numerical modeling. Since lift coefficient corrections are

obtained by subtracting calculated free stream values from values with wall effect, they

are all of negative sign. Thus when subtracted from "raw" wind tunnel lift coefficients,

they give corrected values which are larger than measured. In this sense, lift coefficient

corrections for T-38 with ventilated walls is of the opposite sign, compared with

previously discussed corrections for test sections with solid walls (in that case, the lift

curve slope corrections were lower than one, giving smaller corrected lift coefficients

than those measured in wind the tunnel).

Figure 6-13 Measured Cp after correction in T-38 wind tunnel for T-38


103

Figure 6-14 Measured Cp after correction in T-38 wind tunnel

Figure 6-15 Measured Cp after correction in T-38 wind tunnel


104

Table 6-6 Lift coefficient correction for T-38 wind tunnel

For T-38 wind tunnel with height h = 6, M = 0.3

Alfa CL with

wall effect

CL for

free

stream

ΔCL CL

measured

CL after

correction

2o 0.1931 0.2475 -0.0544 0.1900 0.2444

4o 0.3970 0.4944 -0.0974 0.3860 0.4834

6o 0.5918 0.7403 -0.1485 0.5650 0.7135

6.3.2 NASA transonic cryogenic wind tunnel

The same procedure of analysis, as described in previous section, has been

applied here as well, also keeping the same airfoil, Mach number and analyzed angles

of attack. Experimental conditions, comparing T-38 and NASA tunnels, on one side -

slightly differ in the sense of Reynolds numbers which, for example for M = 0.3, are

Re = 4.4x106 in T-38 and Re = 6.0795x10

6 in NASA tunnel. These Reynolds numbers

are of the same order (4.4 and 6 million) and cannot introduce any substantial

difference in the sense of flow patterns around the airfoil, comparing measurements in

two different facilities. (It should also be noted that Reynolds number affects only

experimental data, and not numerical results based on potential model, because it is

inherently inviscid). On the other hand, the difference in relative test section heights (h

= 6 for T-38, and h = 4 for NASA tunnel) suggests that for this reason, required

corrections for NASA wind tunnel should generally be slightly larger, than for T-38.

Comparisons between experimental and numerically obtained pressure

coefficient distributions for M = 0.3 and angles of attack of α = 2o, 4

o, 6

o, are shown in

Figure 6-16, Figure 6-17 and Figure 6-18, respectively.


105

For angle of attack α = 4o, agreements between experimental and calculated Cp

are generally very good. For other two angles of attack, slight discrepancies exist in the

aft domain and trailing edge, while the differences in Cp between upper and lower

surface, considering experiment and calculations, are practically of the same order (and

it should be remembered thatCp generates lift and moment). Keeping in mind that

numerical models always "think" in the same way, while experimental data can be

subject to inevitable small measurement errors, associated to the actual test run

(explanation for certain oscillations on all measured Cp curves that can be hardly be

explained otherwise), the comment considering obtained numerical values is the same

as for T-38, and numerical results can be qualified as proper in the sense of the

verification of the numerical model.

Figure 6-16 Measured and calculation pressure distribution in NASA wind tunnel


106



The calculated free flow values for analyzed angles of attack are compared with

numerically obtained tunnel pressure distributions in Figure 6-19, Figure 6-20 and

Figure 6-21. These differences have been integrated and quantified in Table 6-7.


107

It is obvious that for NASA wind tunnel case, with smaller relative test section

height than in T-38, required corrections for all angles of attack are proportionally

larger, also with increasing tendency for higher angles.

Figure 6-19 Numerical Cp for free stream and wind tunnel wall effect

Figure 6-20 Numerical Cp for free stream and wind tunnel wall effect


108

Figure 6-21 Numerical Cp for free stream and wind tunnel wall effect in NASA wind tunnel

Finally, the calculated values of Cp distributions have been superimposed to

the measured pressure coefficient distributions, and the corrected pressure coefficients

for NASA wind tunnel have been obtained, as shown in Figure 6-22, Figure 6-23 and

Figure 6-24.

Figure 6-22 Measured Cp after correction in NASA wind tunnel


109

Experimental lift coefficients after applied corrections (as global parameters)

are shown in Table 6-7. These values, obtained for M=0.3 and angles of attack of α =

2o, 4

o, 6

o, are practically the same as corrected values of lift coefficient for T-38 wind

tunnel under the same nominal flow conditions, which is the expected outcome of the

entire calculation and correction procedure, when established properly.




110

Table 6-7 Lift coefficient correction for NASA wind tunnel

For NASA wind tunnel with height h = 4, M=0.3

Alfa

CL with

wall

effect

CL for

free

stream

ΔCL CL

measured

CL after

correction

2o 0.1740 0.2475 -0.0735 0.1694 0.2429

4o 0.3715 0.4944 -0.1229 0.3544 0.4773

6o 0.5686 0.7403 -0.1717 0.5351 0.7068

6.3.3 Comparison between T-38 and NASA wind tunnels

The measured (uncorrected) pressure coefficient distributions, obtained in T-38

and NASA wind tunnels, for the NACA 0012 airfoil under the same nominal test

conditions defined by Mach number M=0.3 and angles of attack of α = 2o, 4

o, 6

o, are

compared in Figure 6-25, Figure 6-27 and Figure 6-29. Both measurements have been

performed in two highly respectable test facilities. The expected differences in

measured Cp distributions should be primarily the consequence of different wall

porosities, relative test section heights (h = 6 in T-38 and h = 4 in NASA tunnel) and to

a certain extent because of different Reynolds numbers, although they are generally of

the same order (Re = 4.4x106 in T-38 and Re = 6.0795x10

6 in NASA tunnel).

Under the ideal conditions, measured Cp distributions should have quite similar

smooth shapes for each angle of attack, but mutually slightly shifted with respect to

each other, because of the above mentioned differences. Figures, on the other hand,

show that it is not quite so (oscillations on curves, differences in trailing edge domains

for the two tunnels, etc.), because of the inevitable minor errors in measurement

instrumentation and data acquisition, different positions of pressure probes on test

models, effects of viscosity and model smoothness on flow patterns specially in rear

airfoil domains, etc.


111

Figure 6-25 Measured Cp in T-38 and NASA wind tunnels for α = 2°

Figure 6-26 Measured Cp after numerical correction in both wind tunnels, α = 2°


112


Figure 6-28 Measured Cp after numerical correction in both wind tunnels α = 4°


113


Figure 6-30 Measured Cp after numerical correction in both wind tunnels, α = 6°


114

In an ideal case, when numerically obtained ΔCp corrections are locally

superimposed to the measured Cp curves, the corrected Cp distributions should

practically coincide. On the other hand, Figure 6-26, Figure 6-28 and Figure 6-30,

which compare corrected measurement values, show certain differences. The primary

reason for that is the fact that all "imperfections" which occurred during experiments

are built in, and contained within the corrected Cp distributions.

In spite of that, here established and applied calculation method for 2D wind

tunnel corrections, although based on potential flow model, has shown very good

capabilities in applying the required corrections. Final verification is obtained through

the global comparison of lift coefficients from the two wind tunnels under the same

nominal flight conditions, which after the applied corrections have practically the same

values, although measured (uncorrected) lift coefficients were quite different, as shown

in Table 6-8.

Table 6-8. Lift coefficients from T-38 and NASA wind tunnels before and after applied corrections

for M = 0.3

Alfa

T-38

CL

measured

NASA

CL

measured

T-38

CL after

correction

NASA

CL after

correction

2o 0.1900 0.1694 0.2444 0.2429

4o 0.3860 0.3544 0.4834 0.4773

6o 0.5650 0.5351 0.7135 0.7068

CHAPTER SEVEN CONCLUSION

115

CHAPTER SEVEN

7 CONCLUSION

This thesis describes novel approach to subsonic two-dimensional wind tunnel

wall corrections. While classical subsonic, two-dimensional wind tunnel wall

corrections represent airfoil with singular point at which vortex and doublet are placed,

the approach applied in this thesis treats the airfoil as its true 2D shape, approximated

by a finite number of straight, linearly varying vortex singularity segments (panels).

The wind tunnel test sections with solid walls are modeled by mirroring the complete

airfoil shape, with respect to the upper and lower wall with sufficient number of

images. This ensures that solid wall boundary conditions are satisfied in all points of

the solid walls, in contrast to some other methods which satisfy wall boundary

conditions only in selected points. This also ensures that flow at `infinity’ is parallel to

the solid walls, and that the airfoil setup angle is the true angle of attack in the sense of

calculation, the same as in the free stream. Test sections with porous walls are

simulated in the same way, but with additional constant strength panels with

source/sink singularities, whose strength is such that actual test section porosity

characteristics are simulated.

General theoretical background is the same as in classical 2D subsonic wind

tunnel wall corrections. It is assumed that the differences between measured and

calculated flow properties are small, which allows linearization. Also it is assumed that

velocity has potential, what allows superposition of singular solutions.

7.1 Correction procedure for solid wind tunnels walls

Mirroring of the true airfoil shape is applied in 2D wind tunnel subsonic wall

corrections, both for the cases when pressure distribution is measured, or when

aerodynamic coefficients are measured by balances.


116

In validation section of this thesis, and in reference (Taha A. A. Petrovic Z.

Stefanovic Z. Kostic I. Isakovic J.), it is shown that classical wind tunnel 2D wall

corrections agree favorably with here established calculation model.

For solid walls the numerical calculation carried out in the following order.

1. Numerical calculation is executed to calculate the pressure distribution at all

control points of linear strength vortex panels, i.e. around the airfoil model and

the panels are mirrored by a sufficient number of their symmetric images with

the respect to the ceiling and floor of the test section to simulate the effect of

solid walls in this calculation.

2. Numerical calculation is then carried out to calculate pressure coefficient

distribution around airfoil for free stream i.e. without imaging system.

3. The difference between the two numerical calculation of pressure distribution

around the airfoil with and without walls effects is calculated and it represents

the correction needed to be superimposing to the measured data from wind

tunnel.

When pressure distribution is measured in wind tunnel, then measured pressure

coefficient Cp distribution is corrected by superimposing the difference between

numerically calculated pressure coefficient distribution in free stream, and numerically

calculated pressure distribution in the wind tunnel. This Cp distribution is then

integrated to obtain corrected lift and moment coefficient values.

When aerodynamic coefficients are directly measured by balances (without

pressure measurements), then the difference between numerically calculated Cp

distribution in free stream, and numerically calculated Cp distribution in the wind

tunnel is directly integrated, and global correction values for lift and moment are

obtained.

In both cases, angle of attack and speed remain uncorrected, i.e. nominal values

of these parameters from wind tunnel also apply for corrected aerodynamic

coefficients.


117

The verification of here presented calculation method has been performed by

comparing numerically obtained lift curve slope corrections with those obtained by two

well-known classical methods, Abbot, Doenhoff and Stivers, and the method of Pope

and Harper. The analyzed airfoil was NACA 0012, since this airfoil has been used

worldwide as standard airfoil both for wind tunnel calibrations, and for software

development and verification purposes.

The numerical calculations have been performed for relative test section

heights of h = 3, 4, 5, 6, angles of attack of α = 2o, 6

o and Mach number M = 0.15

(since both classical methods correspond to incompressible flow conditions), in order

to calculate lift slope curve correction factors for the all cases, as shown in Table 6-5.

From this table it can be readily calculated that the differences between numerical

calculations of the lift slope correction factors for α = 2o, and h = 3, 4, 5 and 6,

and

classical method of Abbot, Doenhoff and Stivers, are 0.29%, 0.34%, 0.2% and 0.11%

respectively. From the same table for angle of attack α = 6o, it can be determined that

the differences between numerical calculations of lift slope factor, compared with same

standard method are 0.04%, 0.07%, 0.1% and 0.04% .

Another comparison has been made as well, between numerical calculations of

lift slope factor correction, and Pope and Harper method for the same conditions and

relative heights as mentioned before. The differences for angle α = 2o

are 0.06%,

0.18%, 0.06% and 0.01% respectively, and for α = 6o

the differences between the

numerical and analytical methods are 0.27%, 0.08%, 0.03% and 0.02% respectively.

It is obvious that very good agreements have been obtained for all considered

test cases, confirming capability of here presented calculation model to perform

reliable lift coefficient corrections. Since compressibility correction factor is

incorporated in the algorithm, calculations by this model can be spread to subsonic

Mach numbers at which compressibility influences are not negligible.

Also, here presented method can readily calculate the quarter-chord moment

coefficient corrections from the numerically determined solutions as well, since they

are obtained by multiplying pressure coefficient differences Cp between lower and

upper airfoil camber (the same as used for lift coefficient determination) by relative


118

distance from this reference point. On the other hand, since standard symmetrical

calibration airfoil NACA 0012 has been considered in the entire thesis (with near-zero

quarter-chord moment values), the moment corrections were not considered for

verification purposes. Keeping in mind that they are calculated from same Cp values,

the accuracy of moment corrections would be of the same order as for the lift

coefficient.

7.2 Correction procedure for ventilated wind tunnel walls

Again correction procedure depends on what is measured in wind tunnel. If the

aerodynamic coefficients are measured directly, together with pressure distribution on

the wind tunnel walls, it is necessary to numerically calculate pressure distribution

along walls while the airfoil is assumed to be in the free stream. Then it is necessary to

subtract measured and calculated pressure distributions, and determine speed from

these differences along wind tunnel walls. This wall air speed is then applied as a

boundary condition to determine flow in the empty wind tunnel. Calculated speed and

angularity at representative point determine wind tunnel corrections, as it is done by

Mokry’s method.

If the pressure distribution is measured, then procedure requires repetition of

numerical determination for the free stream flow around airfoil, and for the flow about

airfoil within wind tunnel. To simulate ventilation, additional sources are distributed

along wind tunnel walls. Intensities of sources are systematically varied until

numerically calculated pressure distribution for the flow about airfoil in the wind

tunnel and measurement are agreed well. Difference in numerically calculated pressure

coefficient distributions about airfoil in the free stream and in the wind tunnel is added

to measured pressure coefficient distribution in the wind tunnel. Aerodynamic

coefficients are obtained by integration of corrected pressure distribution.

For perforated walls, the numerical calculation is described below.

1. Numerical calculation is executed to calculate the pressure distribution at all

control points of simulated airfoil, as a result of velocities induced by

constant strength source/sink panels which simulate porosity of wind tunnel


119

walls, and of linear strength vortex panels of the approximated airfoil NACA

0012. The linear strength vortex panels are mirrored by a sufficient number

of their symmetric images with the respect to the ceiling and floor of the test

section, to take into account the presence of walls in this calculation. All

linear strength vortex panels, their images and all the constant strength

source panels on the upper and lower wind tunnel walls contribute to the

induced velocity components (i.e. pressure distribution coefficient) at airfoil

control points.

2. The pressure distribution around airfoil is numerically calculated for free

stream, excluding the influence of mirrored images and of source/sink

singularities.

3. The difference ∆Cpi between the two numerically calculated pressure

distributions around the airfoil (with and without ventilated walls effects)

represents the correction needed to superimpose to the measured pressure

coefficients from wind tunnel.

Practically speaking, the same procedure as for solid walls is carried out for

perforated walls correction, where the only difference is the addition of constant

strength sources panels on the upper and lower wind tunnel walls, to simulate the

porosity through the tunnel walls.

For the verification purposes, measurements performed by two relevant

experimental facilities with ventilated test section walls, the T-38 wind tunnel (VTI

Žarkovo, Belgrade) and transonic cryogenic tunnel (0.3-m NASA Langley TCT) were

analyzed, with same airflow conditions defined by Mach number M = 0.3, and angles

of attack α = 2°, 4°, 6°. The differences in experimental environments were defined by

relative test section heights, h = 6 for T-38 and h = 4 for NASA tunnel, and by slight

difference in Reynolds numbers (Re = 4.4x106 in T-38 and Re = 6.0795x10

6 in NASA

tunnel). In both wind tunnels, pressures were measured on NACA 0012 airfoil

contours.

In the first calculation steps, for both wind tunnels, here applied calculation

model has given good agreements, with expected accuracy level, with "raw" wind


120

tunnel measurements, both in the sense of local pressure coefficient distributions, and

global lift coefficient determinations for analyzed angles of attack. The differences

between measured and calculated lift coefficients with wall effect for T-38 wind tunnel

were 1.63%, 2.85% and 5.85% for angles of attack α = 2°, 4°, 6°, while the

corresponding values for NASA tunnel were 2.7%, 4.82% and 6.2% . The obtained

relative errors obviously increase with angle of attack. It must be kept in mind that in

case of potential models including this one, for the case of free stream analyses, the

relative errors do increase exactly in the same manner and amount, because at higher

angles of attack viscosity effects (neglected by calculation method) become more and

more immanent. On the other hand, since corrections are determined as differences

between calculated free stream values and values with wall influence, this problem is

canceled out, and what remains is the "pure" difference, i.e. the required wind tunnel

correction.

The final verification in this sense was obtained after applying calculated

corrections to the measured lift coefficients from the two wind tunnel facilities, which

were different predominantly because of the different relative test section heights. The

application of correction has the role to eliminate such differences, and provide unique

results, that would correspond to undisturbed free flow, regardless of which facility the

measurements were actually made. As can be calculated from Table 6-8, the lift

coefficients corrected by here applied method, mutually differed only by 0.61%, 1.26%

and 0.94% for Mach number M = 0.3 and angles of attack α = 2°, 4°, 6°, respectively,

which is more than satisfactory level of accuracy for operational engineering purposes.

The application of here presented calculation model on contemporary hardware

makes this method very resource and time efficient and suitable for routine corrections

of two-dimensional wind tunnel measurements, both for test sections with solid, and

with ventilated walls.

CHAPTER EIGHT BIBLIOGRAPHY

121

CHAPTER EIGHT

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