university of belgrade faculty of mechanical
TRANSCRIPT
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
University of Belgrade, Faculty of Mechanical
Engineering
Members: Prof. Dr. Zoran Stefanović
Full Professor, retired
University of Belgrade, Faculty of Mechanical
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
Mašinski fakultet Univerziteta u Beogradu
Members: Prof. dr. Zoran Stefanović
redovni profesor u penziji
Mašinski fakultet Univerziteta u Beogradu
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 6-8 Measured and calculation pressure distribution in T-38 wind tunnel ................. 98
Figure 6-9 Measured and calculation pressure distribution in T-38 wind tunnel ................. 99
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 6-17 Measured and calculation pressure distribution in NASA wind tunnel.......... 106
Figure 6-18 Measured and calculation pressure distribution in NASA wind tunnel.......... 106
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
Figure 6-27 Measured Cp in T-38 and NASA wind tunnels for Alfa=4 ............................ 112
Figure 6-28 Measured Cp after numerical correction in both wind tunnels ....................... 112
Figure 6-29 Measured Cp in T-38 and NASA wind tunnels for Alfa=6 ............................ 113
Figure 6-30 Measured Cp after numerical correction in both wind tunnels ....................... 113
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.
CHAPTER ONE INTRODUCTION
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.
CHAPTER ONE INTRODUCTION
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
CHAPTER ONE INTRODUCTION
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
CHAPTER ONE INTRODUCTION
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.
CHAPTER ONE INTRODUCTION
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
CHAPTER ONE INTRODUCTION
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
CHAPTER ONE INTRODUCTION
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
CHAPTER ONE INTRODUCTION
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.
CHAPTER ONE INTRODUCTION
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
CHAPTER ONE INTRODUCTION
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.
CHAPTER ONE INTRODUCTION
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
CHAPTER ONE INTRODUCTION
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
CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN
SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS
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
CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN
SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS
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,
CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN
SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS
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
CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN
SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS
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:
CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN
SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS
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).
CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN
SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS
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
CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN
SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS
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
CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN
SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS
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
CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN
SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS
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
CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN
SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS
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.
CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN
SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS
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:
CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN
SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS
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
CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN
SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS
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
CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN
SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS
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).
CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN
SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS
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
CHAPTER TWO THE METHOD OF SINGULARITIES AND ITS APPLICATION IN
SIMULATION OF FLOW IN WIND TUNNEL TEST SECTIONS
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
SECTIONS WITH SOLID WALLS
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.
CHAPTER THREE CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST
SECTIONS WITH SOLID WALLS
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)
CHAPTER THREE CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST
SECTIONS WITH SOLID WALLS
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.
CHAPTER THREE CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST
SECTIONS WITH SOLID WALLS
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
CHAPTER THREE CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST
SECTIONS WITH SOLID WALLS
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
CHAPTER THREE CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST
SECTIONS WITH SOLID WALLS
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
CHAPTER THREE CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST
SECTIONS WITH SOLID WALLS
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
CHAPTER THREE CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST
SECTIONS WITH SOLID WALLS
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:
CHAPTER THREE CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST
SECTIONS WITH SOLID WALLS
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
CHAPTER THREE CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST
SECTIONS WITH SOLID WALLS
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
CHAPTER THREE CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST
SECTIONS WITH SOLID WALLS
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
CHAPTER THREE CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST
SECTIONS WITH SOLID WALLS
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:
CHAPTER THREE CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST
SECTIONS WITH SOLID WALLS
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.
CHAPTER THREE CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST
SECTIONS WITH SOLID WALLS
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
CHAPTER THREE CLASSICAL WIND TUNNEL CORRECTIONS FOR TEST
SECTIONS WITH SOLID WALLS
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
CHAPTER FOUR CLASSICAL CORRECTIONS FOR VENTILATED
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
CHAPTER FOUR CLASSICAL CORRECTIONS FOR VENTILATED
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
CHAPTER FOUR CLASSICAL CORRECTIONS FOR VENTILATED
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
CHAPTER FOUR CLASSICAL CORRECTIONS FOR VENTILATED
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
CHAPTER FOUR CLASSICAL CORRECTIONS FOR VENTILATED
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
CHAPTER FOUR CLASSICAL CORRECTIONS FOR VENTILATED
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
CHAPTER FOUR CLASSICAL CORRECTIONS FOR VENTILATED
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.
CHAPTER FOUR CLASSICAL CORRECTIONS FOR VENTILATED
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.
CHAPTER FOUR CLASSICAL CORRECTIONS FOR VENTILATED
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
CHAPTER FOUR CLASSICAL CORRECTIONS FOR VENTILATED
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.
CHAPTER FOUR CLASSICAL CORRECTIONS FOR VENTILATED
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
CHAPTER FOUR CLASSICAL CORRECTIONS FOR VENTILATED
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.
CHAPTER FOUR CLASSICAL CORRECTIONS FOR VENTILATED
TEST SECTIONS
59
Figure 4-4 2D Interference in ideal slotted and porous tunnels
CHAPTER FOUR CLASSICAL CORRECTIONS FOR VENTILATED
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
CHAPTER FOUR CLASSICAL CORRECTIONS FOR VENTILATED
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.
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
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.
5.1.2 Fundamental ideas of this thesis for 2D wind tunnel wall corrections for
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,
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
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
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
TUNNEL CORRECTIONS
65
5.1.3 Fundamental ideas of this thesis for 2D wind tunnel wall corrections for
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
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
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
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
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:
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
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:
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
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
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
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
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
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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
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
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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
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
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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
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
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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.
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
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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
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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
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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
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
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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:
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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
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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
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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:
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
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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:
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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
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
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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).
CHAPTER FIVE NEW APPROACH IN NUMERICAL MODELING OF WIND
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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
CHAPTER SIX RESULT AND DISCUSSIONS
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
CHAPTER SIX RESULT AND DISCUSSIONS
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
CHAPTER SIX RESULT AND DISCUSSIONS
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
CHAPTER SIX RESULT AND DISCUSSIONS
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.
CHAPTER SIX RESULT AND DISCUSSIONS
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.
CHAPTER SIX RESULT AND DISCUSSIONS
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
CHAPTER SIX RESULT AND DISCUSSIONS
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).
CHAPTER SIX RESULT AND DISCUSSIONS
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).
CHAPTER SIX RESULT AND DISCUSSIONS
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.
CHAPTER SIX RESULT AND DISCUSSIONS
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
CHAPTER SIX RESULT AND DISCUSSIONS
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.
CHAPTER SIX RESULT AND DISCUSSIONS
98
Figure 6-7 Measured and calculation pressure distribution in T-38 wind tunnel
Figure 6-8 Measured and calculation pressure distribution in T-38 wind tunnel
CHAPTER SIX RESULT AND DISCUSSIONS
99
Figure 6-9 Measured and calculation pressure distribution in T-38 wind tunnel
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.
CHAPTER SIX RESULT AND DISCUSSIONS
100
Figure 6-10 Numerical Cp for free stream and wind tunnel wall effect for T-38
Figure 6-11 Numerical Cp for free stream and wind tunnel wall effect for T-38
CHAPTER SIX RESULT AND DISCUSSIONS
101
Figure 6-12 Numerical Cp for free stream and wind tunnel wall effect for T-38
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
CHAPTER SIX RESULT AND DISCUSSIONS
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
CHAPTER SIX RESULT AND DISCUSSIONS
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
CHAPTER SIX RESULT AND DISCUSSIONS
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.
CHAPTER SIX RESULT AND DISCUSSIONS
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
CHAPTER SIX RESULT AND DISCUSSIONS
106
Figure 6-17 Measured and calculation pressure distribution in NASA wind tunnel
Figure 6-18 Measured and calculation pressure distribution in NASA wind tunnel
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.
CHAPTER SIX RESULT AND DISCUSSIONS
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
CHAPTER SIX RESULT AND DISCUSSIONS
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
CHAPTER SIX RESULT AND DISCUSSIONS
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.
Figure 6-23 Measured Cp after correction in NASA wind tunnel
Figure 6-24 Measured Cp after correction in NASA wind tunnel
CHAPTER SIX RESULT AND DISCUSSIONS
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.
CHAPTER SIX RESULT AND DISCUSSIONS
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°
CHAPTER SIX RESULT AND DISCUSSIONS
112
Figure 6-27 Measured Cp in T-38 and NASA wind tunnels for α = 4°
Figure 6-28 Measured Cp after numerical correction in both wind tunnels α = 4°
CHAPTER SIX RESULT AND DISCUSSIONS
113
Figure 6-29 Measured Cp in T-38 and NASA wind tunnels for α = 6°
Figure 6-30 Measured Cp after numerical correction in both wind tunnels, α = 6°
CHAPTER SIX RESULT AND DISCUSSIONS
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.
CHAPTER SEVEN CONCLUSION
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.
CHAPTER SEVEN CONCLUSION
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
CHAPTER SEVEN CONCLUSION
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
CHAPTER SEVEN CONCLUSION
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
CHAPTER SEVEN CONCLUSION
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
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