The Real-Time Wall Interference Correction System of the NASA Ames 12-Foot Pressure Wind Tunnel

TABLE OF CONTENTS CHAPTER PAGE vii

CHAPTER PAGE

ooo Vlll LIST OF SYMBOLS A cross-sectioni area of the wind tunnel a speed of sound b wing span C wil pressure port calibration C interpolated wall pressure port calibration C wing chord c t mean aerodynamic chord (2/S. f:12 c2 dy) mean geometric chord (S/b) drag coefficient CD uncorrected drag coefficient ACD drag coefficient correction due to inclination of lift and drag force cl rolling moment coefficient corrected rolling moment coefficient Cl,c uncorrected rolling moment coefficient Cl,un, c /X.c z rolling moment coefficient correction (non-symmetric lift distribution) AW rolling moment coefficient correction (inclination of moment vectors) lift coefficient CL local lift coefficient

CL(y)

uncorrected lift coefficient Acz lift coefficient correction due to inclination of lift and drag force CM pitching moment coefficient corrected pitching moment coefficient CM,c uncorrected pitching moment coefficient CM,unc ACM1 pitching moment coefficient correction due to difference between mean and loci wil interference corrections along the 3/4-chord of the wing and tail ACM2 pitching moment coefficient correction due to streamline curvature Cn yawing moment coefficient corrected yawing moment coefficient Cn,c uncorrected yawing moment coefficient Cr$,_r_c ix A-_-d.

yawing moment coemcient correction (inclination of moment vectors) specific heat ; pressure coefficient cp D drag force D' uncorrected drag force camber of circularly cambered airfoil k singularity index L lift force L' uncorrected lift force lift force of a line doublet of the tail L, L_ lift force of a line doublet of the wing l length of a Rankine body M Mach number or number of reference points calibrated Mach number test section reference Mach number M_!

free-stream Mach number Moo m number of wall pressure orifices N number of reference points n number of singularities number of line doublets of the tail nt nw number of line doublets of the wing P pitching moment static pressure P static pressure at a wall orifice ; empty tannel calibration P_mp Pst_p static pressure at a wall orifice ; support system calibration static pressure at a wall orifice ; real-time wind tunnel test Ptun PT total pressure in the settling chamber dynamic pressure q calibrated dynamic pressure qe test section reference dynamic pressure q,'e!

free-stream dynamic pressure q_ R gas constant or rolling moment X R !

uncorrected rolling moment polar coordinate rl _ r2 ro radius of a hMfbody S area of the propeller disc of a propulsion simulator S reference area of test article $ span of a rectangular wing As span of a horseshoe vortex Ast discrete span of a line doublet of the tail Asw discrete span of a line doublet of the wing T temperature

Tp propulsion simulator thrust

TT

total temperature in the settling chamber U velocity in the x-direction U_ calibrated axial velocity; empty tunnel or image plane calculated value of u_ u; test section reference velocity U_ free--stream velocity II perturbation velocity in the x-direction Ui streamwise velocity correction Ui normalized perturbation velocity of the wM1 interference flow field ; x-component "Urn axial perturbation velocity; model in free-air calculated value of u,_ _m i axial perturbation velocity; support system in free-air _s i _t normalized perturbation velocity of the wind tunnel flow field ; x-axis Ut axial perturbation velocity; wind tunnel flow field calculated value of ut _t axial perturbation velocity of a model in the wind tunnel flow field axial perturbation velocity of a support system in the wind tunnel flow field _ts x'-axis component of the velocity vector U flow velocity in the y-direction Vi velocity correction in the y-direction xi Vi normalized perturbation velocity of the wall interference flow field ; y-axis _t y-component of the velocity; empty tunLel or image plane calibration D e ¢s y-component of the perturbation velocity; model in free-air D m _s y-component of the perturbation velocity; support system in free-air D s J y-component of the perturbation velocity; model in wind tunnel flow field Dtm t* y-component of the perturbation velocity; support system in wind tunnel flow field D t yr-axis component of the velocity vector W flow velocity in the z-direction ; reference point weight velocity correction in the z-direction Wi normalized perturbation velocity of the wall interference flow field ; z-axis z-component of the velocity; empty tunnel or image plane calibration W e z-component of the perturbation velocity; model in free--air W m W 8 z-component of the perturbation velocity; support system in free-air z-component of the perturbation velocity; model in wind tunnel flow field W_ z-component of the perturbation velocity; support system in wind tunnel flow field W_ weight of a line doublet of the tail weight of a singularity wj Ww weight of a line doublet of the wing (syn'_metric lift distribution) Ww weight of a line doublet of the wing (non-symmetric lift distribution) W t z'-axis component of the velocity vector X value of first calibration variable during calibration X" value of first calibration variable during real-time test X x-coordinate; roll axis; line doublet starting point _z pitching moment arm incompressible x-coordinate X_r x-coordinate of the pitching moment reference axis x-coordinate ZS of a point source x-coordinate of a line doublet of the tail X_ x-coordinate Xw of a line doublet of the wing x-coordinate of a point source or line doublet starting point x-coordinate X_ of a point sink xii x-coordinate of axis of rotation Xs Xo stagnation point distance ; initial x-coordinate of singularity X / x-coordinate of the reference coordinate system Y value of second calibration variable during calibration or yawing moment yt uncorrected yawing moment y* value of second calibration variable during real-time test y-coordinate; pitch axis Y incompressible y-coordinate y-coordinate of a line doublet starting point yz y-coordinate of a point source ys yt y-coordinate of the reference coordinate system Z z-coordinate; yaw axis incompressible z-coordinate z-coordinate of a line doublet starting point Zl ZS z-coordinate of a point source Z t z-coordinate of the reference coordinate system angle of attack of test article angle of attack correction mean angle of attack correction along 1/4-chord line of wing mean angle of attack correction along 3/4-chord line of wing angle of attack correction caused by the model O_rn angle of attack correction caused by the support system O_ s geometric angle of attack O_t pitch angle of High Angle of Attack Sting _oo free--stream angle of attack

Z x/1 - M 2 or sideslip angle of test article

yaw angle of High Angle of Attack Sting 1-' circulation of a horseshoe vortex in [m 2/sec] or [ft 2/sec] F_ circulation constant of the tail in [m 2/sec] or [ft 2/sec] circulation constant of the wing in [m 2/sec] or [ft 2/sec] ooo Xlll ## circulation of symmetric lift distributionin [m2/sec] or [ft2/sec] r** circulation of non-symmetric lift distributionin [m 2/sec] or [ft2/sec] isentropic exponent wall pressure orifice index blockage factor mean blockage factor minimum of blockage factor _r_in blockage factor caused by a model _rn _s blockage factor caused by a support system number of source / sink pairs ; solidvolume blockage _7 0 test section reference point A aspect ratio of wing model reference point index !/ number of sources and sinks density P test section reference density Prey total density in the settling chamber PT free-stream density P_ O" singularity strength in [m3/sec] or [fta/sec] singularity strength divided by reference velocity ; [m 2] or [ft 2] Or k sink strength of the propulsion simulator divided by reference velocity o" I, sink strength of the propulsion simulator in [ma/sec] or [fta/sec] strength of a line doublet of tail divided by reference velocity O"t strength of a line doublet of the tail in [mZ/sec] or [ftZ/sec] wake source strength or strength of a line doublet divided by reference velocity or w wake source strength or strength of a line doublet in [m a/sec] or [ftZ/sec] strength of a line doublet in [ma/sec] or [fta/sec] singularity strength of a point source or sink ffl _0"2 line doublet orientation angle ; identical with _, T roll angle of test article _ sweep angle of wing sweep angle of 1/4-chord line of wing _oo.2s xiv _0.50 sweep angle of 1/2-chord line of wing line doublet orientation angle ; identical with r T* polar coordinate

eL

line doublet potential CPD point doublet potential

Cs

point source potential wall / support system potential ¢, wind tunnel potential singularity potential

Coo free-stream potential

¢1, ¢2 length scale of wind tunnel model

n normal vector at a panel centroid uoo free-stream velocity vector Woo unit wind vector in model coordinate system XV xvi LIST OF WALL INTERFERENCE CORRECTIONS 1. FIRST ORDER CORRECTIONS Correction Reference Symbol Equation

Blockage Factor Chapter 2 (4),(ga),(15a),(17)

Mach Number Appendix 1 -- (1.8) Dynamic Pressure Appendix 1 -- (1.15) Angle of Attack Chapter 2 (7),(9b),(15b),(18b) O_i _ O_rn j _s 2. SECOND ORDER CORRECTIONS Correction Reference Symbol Equation Lift Coefficient Appendix 17 ACL (17.7a) (Inclination of Force Vectors) Drag Coefficient Appendix 17 ACD (17.7b) (Inclination of Force Vectors) Pitching Moment Coefficient Appendix 18 ACM1 (18.4) (Lateral Shift of Center of Pressure) Pitching Moment Coefficient Appendix 18 ACM2 (18.21a) (Streamline Curvature) (18.215) Rolling Moment Coefficient Appendix 17 A_ (17.13a) (Inclination of Moment Vectors) Rolling Moment Coefficient Appendix 19 Act (19.27) (Non-Symmetric Lift Distribution) Yawing Moment Coefficient Appendix 17 AF_ (17.13b) (Inclination of Moment Vectors) xvii

xviii

CHAPTER1 INTRODUCTION Wind tunnel tests have always played an important role in the development of mod- ern aircraft. These tests are used to simulate atmospheric conditions experienced by an aircraft or spacecraft in free-flight. Aerodynamic forces and moments are measured and related to corresponding free flight values using Mach and Reynolds numbers. These mea- surements provide valuable information about expected performance, stability, and control characteristics of a new aircraft design.

Large wind tunnel models, i.e. wing span on the order of 80% of the wind tunnel width, are often preferred in order to achieve a good simulation of viscous phenomena of the flow field. In this case, however, the presence of the wind tunnel wall and model support system change the free-air flow field experienced by the aircraft model. These flow field interference effects have to be considered to allow a reasonable comparison between wind tunnel test and free flight condition. Therefore, interference corrections to Mach number, dynamic pressure, and angle of attack have to be determined to improve test data quality.

In the 1970s and 1980s, techniques were developed that use boundary measurements during a wind tunnel test to predict wall interference corrections. The Wall Signature Method introduced by Hackelt et al. [1],[2],[3] and the Two-Variable Method introduced by Ashill [4],[5] were used extensively in 3-dimensional wind tunnel testing. Ashill [5] gives a detailed discussion and comparison of these techniques.

The Wall Signature Method and Two-Variable Method are both based on potential flow theory. Computed wall interference corrections agree if each method is applied cor- rectly. However, a few differences exist between these two methods. Each method has its advantages and disadvantages. Table 1 compares important features of the Wall Signature Method and the Two-Variable Method.

The Two-Variable Method does not require a singularity representation of the wind tunnel model to determine wall interference corrections. However, the wall interference correction calculation depends on an integration of the measured and interpolated surface pressure distribution on the wind tunnel wall.

The Wall Signature Method uses a singularity representation of test article, wall pres- sure measurements, precomputed perturbation velocities, and the principle of superpo- sition to compute wall interference correction. No integration and interpolation of the surface pressure distribution is required. Therefore, the Wall Signature Method is more flexible in the selection of wall pressure port measurements required for the wall interfer- ence calculation. Computed corrections are relatively insensitive to the number and the location of wall pressure ports used for the least squares fit.

Table 1 : Comparison Wall Signature Method / Two-Variable Method Wall Signature Two-Variable Method Method (Ita,:kett et al.) (Ashill et at.)

Singularity Representation YES NO of the Wind Tunnel Model Measurement of the Flow Velocity YES YES at the Wind Tunnel Wall Number of Wall Pressure ___ 2 >_ 100 Port Measurements Wall Interference Correction YES NO Calculation based on a Perturbation Velocity Flow Field and the Principle of Superposition Wall Interference Correction NO YES Calculation based on an Integration of the Surface Pressure Distribution at the Wind Tunnel Wall LOW HIGH Sensitivity of Computed Corrections to the Number and the Location of Wall Pressure Port Measurements The Wall Signature Method was selected for use in the real-time Wall Interference Correction System (WICS) of the NASA Ames 12ft Pressure Wind Tunnel, because it is fast and does not depend on an integration of a measured surface pressure distribution.

The Wall Signature method will still provide corrections even if a large number of wall pressure ports do not provide a useful measurement.

In general, the Wall Signature Method computes wind tunnel wall interference cor-

rections by introducing a simplified representation of the test article expressedin terms

of singularities. Sourcesand sinks represent the fuselagevolume and viscous separation

wake blockageeffectsand horseshoevortices or line doublets representthe lifting effects.

In addition, power simulator blockageeffects can be representedby a sink [6] . This sin-

gula_ity representation is combinedwith a least squaresfit of wall pressuremeasurements, data from calibration tests, and solutions of the subsonicpotential equation, in the form of

normalized perturbation velocities, to predict Mach number, dynamic pressure,and angle

of attack corrections.

During the past decadesignificant advancesin the development of low-order panel

method codes and computer hardware have made a fast calculation of complex three-

dimensional internal flow field problems on workstation type computers possible. Panel

method codesallow application of the Laplace Equation to realistic three--dimensionalwind

tunnel geometrieswhich is important if the methodology of the Wall Signature Method is

applied to the quasi-octogonal cross-sectionof the 12ft PressureWind Tunnel (PWT) at

NASA Ames Research Center. It was shown by Ulbrich and S_einle [7],[8] that normalized panel method code solutions of the wind tunnel flow field combined with the Wall Signature Method can be used to predict subsonic wall interference corrections close to real-time.

The revised formulation of the Wall Signature Method developed for the 12ft PWT is described in detail in this report. Figure 1 shows principle elements of the modified Wall Signature Method. Improvements of the Wall Signature Method were introduced to allow an application of the Wall Signature Method in real-time and to deal efficiently with a wide range of model and support system geometries.

Originally, Hacke_t et al. [1] based their formulation of the Wall Signature Method on a "local" least squares fit procedure. They introduced a piecewise approximation of the wall signature using a parabola for its maximum and a tanh - function for its downstream asymptote. The location of singularities was found by matching the location of the maxi- mum of the parabola with the inflection point of the tanh - function. Unfortunately, this feature of the original formulation of the Wall Signature Method is difficult to use in a real-time correction system, as it requires the selection of wall pressure ports used for the "local" least squares fit of the maximum of the real-time wall signature.

Ulbrich [9] introduced improvements to the Wall Signature Method to overcome the

limitations of a "local" least squares fit of the wall pressure signature. He suggested a

"global" least squares fit procedure which matches the wall signature on all wall pressure ports using panel method code solutions of singularities placed inside the wind tunnel test section. In his approach, a "best" singularity location is found by minimizing the standard deviation of the least squares fit of the wall signature as a function of the singularity location.

Support system wall interference corrections for fullspan model tests can also be found by applying the Wall Signature Method. In this case the Wall Signature Method has to be applied to the difference between the support system and the empty tunnel calibration at the wall pressure ports. Support system wall interference effects can be computed off-line and stored in a database.

In the first part of this report, basic relationships of the proposed Wall Signature Method are derived for a fullspan and a semispan model.

The second part of this report discusses the integration of the method into a wind tun- nel facility. Experimental data, obtained during tests of two different size semispan models mounted on an image plane in the NASA Ames 12ft Pressure Wind Tunnel (PWT), are applied to the modified Wall Signature Method to verify computed corrections. Exper- imental data recorded during the calibration of the Ames Bipod are also applied to the method.

CHAPTER 2 WALL INTERFERENCE CORRECTION PREDICTION 2.1 Definition of Interference Correction Wind tunnel tests allow the prediction of aerodynamic forces and moments acting on an aircraft model in atmospheric free-flight. Unfortunately, the wind tunnel wall and the model support system change the flow field experienced by the aircraft. Many of these changes can be ignored if the aircraft model is small compared to the wind tunnel height and width. However, if the span of the test article is large or if substanial flow separation occurs, wall and model support system interference effects cannot be neglected.

Then, reliable estimates of interference corrections to Mach number, dynamic pressure, and angle of attack are necessary so that wind tunnel test data may be compared with free-flight conditions.

In general, wall and support system interference corrections are defined as the dif- ference between the wind tunnel flow field and the free-air flow field experienced by the model (see Fig.2a) . Corrections are described in terms of a blockage factor e and an angle of attack correction _i . Mach number and dynamic pressure corrections are related to the blockage factor computed at some reference point in the wind tunnel. For more detail on classical subsonic wall interference corrections, see AGARDograph 109, [10] . The block- age correction relates the free--stream velocity Uoo to a calibrated empty tunnel velocity Ue at a model reference point v (see Fig. 2b) . The calibrated empty tunnel velocity Ue captures the effects of the wind tunnel wall boundary layer growth, wall divergence, and orifice error. It is still necessary to correct for the wall interference effect of the test article, its separation wake, and the influence of the support system.

The ratio between free-stream velocity Uoo and the calibrated empty tunnel velocity U, is expressed as a function of the blockage factor e, [10] : Uoo(u) = 1 + ui(u) = 1 + e(u) (1) where ui is the axial velocity correction at the model reference point u caused by the model and support system interference effects relative to the calibrated empty tunnel velocity Ue •

For small changes in velocity, second order approximations of Mach number and dy- namic pressure correction can be expressed as a function of the blockage factor e(v) using a Taylor series expansion. These second order approximations are used if a large blockage factor is expected, e.g. during high angle of attack tests of aircraft models. We get (see Appendix 1) : Me(v) = 1 -{- 2 4 " e(v)2 (2a) qoo - q_(v) =q,(v) [2- M2(v)].e(v)+ [1- -_-M_(v)+ 2-TM4(v)2 ] "e(v)2 (2b) The calculation of wall interference corrections based on the ideas of the Wall Signature Method requires the formulation of a blockage factor e and an angle of attack correction c_i such that a direct connection between experiment and the panel method code computation is possible [7],[8].

Studies by the author have shown that differences of the panel method code solu- tions can be used in combination with the Wall Signature Method to predict interference corrections. Figure 2b shows the relationship between the empty tunnel calibration, sup- port system calibration, wind tunnel test, free-air flow field, and the corresponding panel method code calculations using a simplified representation of the test article and support system in terms of singularities. This representation is uniquely defined if type, location, and strength of the singularities is known. The type and location of singularities must be specified by a test engineer. The Wall Signature Method is used to compute the strength of the singularities. The Principle of Superposition arid panel method code solutions of the wall interference flow field are used to determine wall interference corrections.

It is assumed that singularities of the test article an,_ support system are located inside a tunnel of constant cross-sectional area as effects of wall divergence and boundary layer growth are already included in the calibrated empty tlmnel velocity Us . The calibrated velocity U_ corresponds to a constant reference velocity U_ of a constant cross-section wind tunnel.

Comparing the flow fields depicted in Fig. 2b we caz_ make the following approximation of the velocity ratio U_(v)/U_(v) defined in Eq. (1):

u_(_) u; + [( u; + u;(.) ) - ( u; + u;,(.) )]

v_O,) u; (3a)

= 1 + r/i,;(,,,)

- u=(_) ]

L

u;'

The total perturbation velocity u_ of the wind tunnel flow field caused by the test article and support system is expressed as the sum of the perturbation velocity contribution u_, n of the test article and the perturbation velocity contribution u_*, of the support system (see Fig. 2c) . The perturbation velocity component u_s of the support system perturbation velocity flow field can further be represented as the sum of the perturbation velocity component u_s - u_ due to the wall interference of the support system and the perturbation velocity component u_ due to the direct influence of the support system (see Fig. 2d).

Finally we can write :

u_(_)

r - ,-,*(,,)+ ,-,;.(,-,)- ,,:,:,,.,,)+ ,-,:(,-,)

+

L J

u_(_) u; (3b) = 1+ [';"'(')-07 _';'(') ]+ [ ,.,7,,(,,) 57- ,,i(,..')]+ ";(')U; It is difficult to predict the direct influence of the support system on the model flow field in terms of a perturbation velocity component us* . The Wall Signature Method cannot be used to determine the direct influence of the support system on the model as this technique has been developed for wind tunnel wall interference studies. Experimental or CFD studies have to be used to estimate the direct influence u* .

For the present study it is assumed that the direct interference between support system and wind tunnel model is small, i.e. u; _ 0. Then, comparing Eqs. (1) and (3b), we get for _(_) : = ern(_') + es(v) (4)

U; U2

Interference velocity components in pitch and yaw axis direction can be expressed in forms similar to Eq. (4) if direct influence of the support system is neglected, i.e., v_ _ 0 andw s _0. We get :

(5a)

u.(_,) u: u; 1 r 1 Wi(V) W _m (12 )

[

j + - j = +

u,(.) L u: u; The velocity ratios a,,_ and a, are angle of attack corrections due to the model and support system wall interference effects.

The interference velocity component in the z-axis direction is related to the free- stream angle of attack aoo experienced by the test article at the model reference point.

Therefore we get :

aoo(_) = a,(_) + _(.) (6)

where a_ is the geometric angle of attack measured relative to the wind tunnel centerline and _i is the angle of attack correction due to lift interference of the test article and due to the change of the flow field angle caused by the interference flow field of the model support system. The total angle of attack correction of a fullspan model is then :

[ ]

_(_) = w_(_) = v_(_) w_(_) -_- u_(_)

+

(7)

uoo( ,) L

where U_(_)/U_(_) and_,(_)/U_(_) are given byEqs. (3b), (Sb). Theangle ofattack

correction for the semispan model is obtained by replacing w_(v) by v_(v) in Eq. (7).

The calculation of the Mach number, dynamic pressure, and angle of attack correction using Eqs. (2a), (2b), (4), (7) is reduced to finding blockage corrections c,_ and e, and angle of attack corrections am and a_ .

Model corrections c,_ and am, i.e. velocity differences Jut* m - u,.n]/Ve, [v_m - v,_]/Ve or [wt*m - w_n]/U* , and support system corrections es and as, i.e. velocity differences [u;, - u_]/U_, [v_, - v;]/U_ or [w;, - w_]/U*, can be computed by using the modified Wall Signature Method. This is possible because the wall interference flow field caused by the wind tunnel model and support system can be treated as a far field effect. This will be explained in detail in the following sections.

2.2 Panel Method Code Solution In the previous section, it was demonstrated that wall interference effects of a test arti- cle and support system, i.e. e,_, e,, a,_, and as, can be found by calculating dimensionless velocities [u_,_ - u*]/U*, [v_,_ - v_,]/U_, [w_ - w*]/U_, [u;_ - u*_]/U_, [v_s - v*]/U*, and [w;, - w*]/U_ at a selected test article reference point v. These dimensionless veloc- ities are computed by superimposing panel method code solutions and applying the Wall Signature Method. In real-time operation, the Wall Signature Method uses a singularity representation of the test article in combination with the measurement of wall pressure, lift force, propulsion simulator thrust force, pitching moment, mad precalculated normalized perturbation velocities to predict model wall interference corrections. The Wall Signature Method may also be used to predict support system wall interference corrections by taking the difference between the wall pressure port calibration of the support system and empty tunnel. However, the application of the Wall Signature Method is only possible, if precal- culated solutions of the subsonic potential equation in the form of normalized perturbation velocities are linear with respect to singularity strength.

In general, the perturbation flow field of a singularity placed inside a wind tunnel of constant cross-sectional area is a linear function of the singularity strength. Figure 3 depicts a singularity of strength _o located inside a selected wind tunnel configuration of constant cross-sectional area A. The corresponding flow field solution is obtained by using a modified panel method code as a boundary value problem solver (see Appendix 2 for a detailed description of these modifications). Figure 3 also shows a singularity of the same type and location but with different singularity strength _rl . The strength g0 and al of these two singularities are related as follows : or1 -- _.cr0 (8a) We know for the corresponding panel method code solutions in terms of the perturbation velocities : * = _ * (8b) It 1 • U o Equation (Sb) is valid as long as the cross-section of the selected panel geometry of the wind tunnel test section is constant. This relationship allows the user to find all wind tunnel flow field solutions of or1 # 1.0 [ft3/sec] by simply multiplying the flow field solution of a0 = 1.0 [ft31sec] with the factor A = allao • In the following section, details will be developed as to how the Wall Signature Method can be combined with the linear property of panel method solutions defined by Eqs. (Sa),(Sb) to obtain test article and support systemwall interference corrections for a a fullspan or semispan model.

2.3 Fullspan Model 2.3.1 Test Article Wall Interference Correction The Wall Signature Method can be applied to compute the blockage factor e_n and angle of attack correction _,n caused by test article wall interference effects at a reference point "v" The Wall Signature Method uses a simplified representation of the test article in terms of point sources, point sinks, and line doublets in combination with measurement of wall pressure, lift force, propulsion simulator thrust, pitching moment, and precalculated normalized perturbation velocities to obtain the blockage factor and angle of attack cor- rection. The blockage factor, era, and angle of attack correction, am, are linear functions of the singularity representation of the test article (see previous section). The principle of superposition allows em and am to be expressed as the sum of contributions of "n" singu- larities that represent the test article. Figure 4 shows, as an example, how the principle of superposition can be applied to a simplified representation of a test article using three singularities. Knowing that e,_ and a,_ are a linear function of the singularity strength _, we get for "n" singularities : ?'4

= (ga)

k--1

= (gb)

k=l where _¥(v, k) and _7(v, k) are normalized perturbation velocities of the wall interference flow field. These normalized velocities are dimensionles_ perturbation velocities divided by unit singularity strength per unit velocity. Perturbation velocity _-(v, k) is the normalized axial perturbation velocity component of the wall inte_erence flow field and _'_(v, k) is the normalized perturbation velocity component perpendicular to the wing plane of the test article. Normalized perturbation velocities have the unit [1/rn 2] or [1/ft 2] . Singularity strength _k has the unit [m 2] or [ft 2] and is compatible with these normalized perturbation

velocities. Singularity strength 0-k is defined as singularity strength 0- in [rn3/sec] or

[fta/sec] (see also App. 3,4) divided by reference velocity U* or U_, I : 0- O"

0-k = = (10)

u;' u,- s

The strength of the singularities is computed similar to the procedures outlined by Vlbrich and S_eiule [7],[8], and Kouing [6], using the measurement of lift force, thrust force, pitching moment, and a least squares fit of wall pressure signature. Figure 5a summarizes the basic steps in the application of the Wall Signature Method to a fullspan model configuration. Rectangular boxes in Fig. 5a symbolize real-time measurements, rounded boxes symbolize information stored in database files, and elliptical boxes symbolize computational procedures.

Fuselage volume blockage effects are represented by point sources and point sinks. The location of these sources and sinks has been selected by the user such that pairs of sources and sinks are related to Rankine bodies describing the fuselage volume of the test article.

Strengths trl,..., 0-*?of point sources and strengths 0-*?+1, ..., 0-2*?of point sinks are reduced to a single variable or. if weighting factors wl, ..., w2*? are introduced. These weighting factors must be defined by the user. We then obtain : _j = wj ; 1 <j < 21/ (lla) 0-, where wj = -wj_*? ; 7/+1 _< j _< 27/ (llb) Separation wake blockage effects are represented by a set of sources, o"2,?+1, ..., 0-_¢ .

Again, weighting factors are introduced to reduce the number of independent variables.

Assuming that the strength a** is a common reference strength of sources related to the separation wake we get : = wj ; 2r/+l <j < _ (12) 0-** The weighting factors w2*?+l,..., w_ are, by definition, greater than zero since separation wake blockage effects are modeled as sources. The calculation of the strength of a total number of "_" point sources and sinks representing fuselage volume and wake blockage effects is then reduced to finding the values of a, and a** using the Wall Signature Method.

A least squares fit of velocities derived from wall pressure measurements is used to calculate a, and or,, (see Fig. 5b) . It is assumed that a total number of "n" singularities has been selected to represent the test article. Axial velocities are derived from pres- sure measurements at "m" wall pressure orifices. The measured real-time wall signature, U_(_) + ut(6) , is corrected for orifice error, wall divergence, and wall boundary layer growth by subtracting the measured velocity, Ue(d_) -_- uts(_) , of the support system calibration at each wall orifice location "_". The computational equivalent, U_, to the calibrated velocity, U_ or Ur_f, at the wall pressure orifice "6" is constant everywhere inside of the test section as it is computed using a constant cross-section wind tunnel.

Normalized perturbation velocities of the wind tunnel flow field at wall pressure orifice locations, "_", have to be introduced to compute the strength of singularities representing the test article. They relate the strength of point sources, sinks, and line doublets to the perturbation velocity components at the wall pressure orifices. The strengths, _+1, ..., _r_, of line doublets representing lifting effects of the test article are estimated by combining lift force, pitching moment, and rolling moment measuxements with the Kutia/Joukowski formula (see Appendix 5 and Appendix 19) . The strengths, _,+1, ..., _r,_, of point sinks representing blockage effects of propulsion simulators axe estimated using thrust measure- ments and Koning's formula (see Appendix 10) . The normal equation of the modified linear least squares problem depicted in Fig. 5b is giver., as, [11] : [AT2 xrn Amx2] X2xl = AT2×,_ B,_×I (13a) X2xl -" O'** Arnx2 "- " " (13c) al,1 al 2 ) \ arn,1 am,2 2t7

a6,1 = w,(6, k) (13 )

k=l a6,2 :- wk. _'7(,5, k) (13e) k=2,_+l (b,).

(13D

b_ = [U,(6) -+ u,(6)] - [Ue(6) + uts(6)] _ -h-7(6, k) (139) - z._., o'k U,- S k=_+1 The vector X contains the strength of singularities modeling the fuselage volume and wake blockage effects. The matrix A contains normalized perturbation velocities of the wind tunnel flow field. The vector B contains residual perturbation velocity components caused by the fuselage volume and separation wake. Line doublet and propulsion simulator contributions, k = _ + 1, ..., n, are subtracted from the measured wall signature difference between the wind tunnel flow field and the support system calibration at the wall pressure ports "6" (see Fig. 5b).

Perturbation velocity, _(6, k), is the normalized perturbation velocity of the wind tunnel flow field of singularity "k" at wall pressure orifice "6" It is defined as the dimensionless perturbation velocity divided by unit singularity strength per unit velocity at wall pressure orifice "£' due to a singularity "k" located inside the test section.

In general, it is required that the measured velocity, U_(6) + ut(6), at wall pressure port "6" can be approximated by its component in the streamwise direction. In practical applications, however, a least squares fit can tolerate a few wall pressure measurements that do not fulfill this condition as the normal equation of the least squares fit, Eq. (13a), assigns equal weight to all wall pressure port measurements.

The solution of the two variable linear least squares problem defined in Eq. (13a) can be written in explicit form as :

= [A A]-'

2×2 [AT " B]2×l The solution vector X is computed by using the Singular Value Decomposition tech- nique [12], which is the numerical method of choice for linear least squares problems.

It is now possible to compute the blockage correction, c,_(v), and the angle of attack correction, c_,_(v), at reference point "v" as the strength and location of all singularities representing the test article and wake are known: 2.3.2 Support System Wall Interference Correction The Wall Signature Method can also be used to find the blockage factor, es, and the angle of attack correction, as, caused by support system wall interference effects. The application of the Wall Signature Method to the support system wall interference problem closely follows procedures discussed in the previous section. Figure 6a summarizes basic elements in the application of the Wall Signature Method to the support system wall interference problem.

Support system wall interference effects have been defined in Eqs. (4) and (5b) .

Similar to Eqs. (9a),(9b), e_ and as are a linear function of the singularity strength. It is assumed that a total number of "_" sources and sinks of unknown strength are used to represent blockage effects of the support system. We then get :

= (15a)

k=l k=l where _'7(v, k) and _'(_,, k) are normalized perturbation velocities of the wall interference flow field.

Again, positions of sources and sinks modeling solid volume blockage effects are se- lected such that pairs of sources and sinks are related to Rankine bodies describing the volume of the support system (see Eq. (lla),(llb)). Sources related to wake blockage effects of the support system are placed where a wake separation on the support system is expected (see Eq. (12)). Introducing weighting factors it is possible to reduce the number of unknown singularity strength values to two, i.e.a. _md a** .

Assuming that "m" wall pressure measurements a_e taken during the support system calibration and that "_" singularities are used to represent the support system, we get the following normal equation of the least squares problem, [11] :

[AT ×m Am×s] = (16 )

where bl b6 = [U,(6) + u,,(_)] - U,(6) (16c) U,._.f The vector X contains the strength of singularities describing the support system volume and wake blockage effects (see also Eq. 135). The matrix A is given by Eqs. (13c), (13d), and (13e) using the perturbation velocities _-t(6, k) of the singularities 1 < k _ _ represent- ing the support system. The vector B contains perturbation velocity components caused by support system volume and separation wake. These perturbation velocity components are the measured wall signature difference between the support system calibration and the empty tunnel calibration at wall pressure ports "_" (see Fig. 6b).

2.4 Semispan Model In general, blockage and angle of attack corrections are computed using Eqs. (4) and (5b) if the modified Wall Signature Method is applied to a semispan model. Studies by the author have shown that a semispan model mounted on a finite length image plane may be treated similar to the fullspan model configuration. It is only necessary to select the proper geometry of the wind tunnel, i.e. the cross-section of the wind tunnel channel above the image plane surface plus its reflected image, for the calculation of normalized perturbation velocities (see Fig. 7) . However, a new calibration of the empty tunnel velocities, U,(_) and U,.,I, must be conducted because the installation of the image plane changes the empty tunnel geometry.

Semispan models normally consist of half of the fuselage mounted on the image plane.

It is therefore necessary to place singularities representing the fuselage volume on the surface of the image plane. This requires further modification of a panel method code to compute normalized perturbation velocities (for more detail see Appendix 2) .

No support system is present in the test section, i.e. ut = utm, uts = 0.0, and us = 0.0 (see boundary value problems depicted in Fig. 8). Thus Eq. (4), (55), (7) are replaced by the following expressions :

-

= (17)

~ = (18 )

=

Normalized perturbation velocities, ui, vi, and _-t, of the interference and wind tunnel flow field are computed using a panel method code. These perturbation velocities are required for the least squares fit of the wall signature and the calculation of wall interference corrections. The user has to make sure that the orientation angle of the line doublets is changed for semispan tests (see Appendix 4).

Figure 9a summarizes the basic elements of the Wall Signature Method for semispan model tests. The least squares fit of the wall signature, defined in Eqs. (13a) to (13f), still applies. Again, it is necessary to subtract the contribution of the wing line doublets and the contribution of the propulsion simulator point sinks from the wall signature difference.

However, the image plane is calibrated as a part of the wind tunnel wall and no support system is present in the tunnel. Therefore, the wall signature difference is defined as the difference between the wind tunnel measurement, Lr(6) + u_(6), and the image plane calibration, U_(6), (see Fig. 95). Equation (13g) is replaced by

urns - (19)

k=_+l For a semispan model configuration the lift force measurement and the Kutta/Joukowski formula are needed to determine the strength of the line doublets of the wing (see Ap- pendix 5).

2.5 Compressibility Effects Compressibility effects have to be taken into account if the modified formulation of the Wall Signature Method is applied to a subsonic wind tunnel flow field.

In general, the Prandtl/Glauert transformation may be used to approximate the effects of compressibility in a flow field. This transformation requires a coordinate stretching of the wind tunnel geometry, singularity location, singulexity strength, and reference point location when normalized perturbation velocities are being computed as outlined in the previous sections (for more detail on the transformation see Ref. [13] and [14]).

The application of the Prandtl/Glauert rule to the flow field of a singularity located inside of a wind tunnel requires several steps. First, coordinates (x,y,z) of the tunnel

geometry and of the singulaz-ity position are transformed to corresponding incompressible

coordinates (_,_,_) using the Prandtl/Glauert transformation:

= • (20a)

= y.v/1 - M 2 - y-;3 (20b) 5" = z.X/1 - M 2 = z.fl

(20c)

It is necessary to stretch the strength of the singularity as well. The strength of a source is related to the cross-sectional area of a corresponding halfbody, [13], as

= .R2.Uoo = + z2].Uoo (21)

where R equals the radius of a halfbody far downstream of the source location. Singularity strength cr has the units [m3/sec] or [ft3/sec]. Comparing Eqs. (20b),(20c),(21) we see that the singularity strength is proportional to f12 . The strength of a line doublet is related to a corresponding wing span increment, i.e. Ay, and angle of attack, i.e. w/Uoo or d z/d z. Considering Eqs.(20b) and (20c) we see that the line doublet strength has to be proportional to f12 . Finally, we get for the incompressible singularity strength : = a.[1 - M 21 = cr.t32 (22) A Mach number, M, derived from the calibrated velocity, U_el , at a test section reference point should be used for the coordinate and singularity strength stretching.

Next, the incompressible flow field is obtained by using the modified version of the panel method code PMARC,[15] as a boundary value problem solver. Finally, perturbation velocities calculated relative to the inlet velocity are transformed from the incompressible flow field, (fi, _, _), to the compressible flow field, (u, v, w). We then get : fi u = _-_ (23a) fl (23b) W "-" fl (23c)

A quantitative checkof compressibility effects is possible if we place a point source

of unit strength inside the geometry of the 12ft PWT and use a modified panel method code and the Prandtl/Glauert transformation described above to solve the flow field. Far downstream of the point source the flow field is essentially one-dimensional and asymptotes of perturbation velocities for different Mach numbers can be calculated. The results of the panel method code calculations on Row 1 (see Fig. 17b for location of Row 1) for Mach numbers M - 0.0, 0.3, 0.6 are plotted in Fig. 10 . Asymptotic values of perturbation velocities can be found in Table 2 .

Independent of the panel method code calculation it is also possible to apply the Area-Velocity-Relation of compressible flow, [16] . We can write : du 1 -dA Uoo = 1- M 2 A (24) The 12ft PWT has a cross-sectional area A = 109.74 [ft 2] . A point source of strength a/Uoo = 1.0 [ft 2] corresponds to a change of cross-sectional area of -dA = 1.0 [ft 2] (see also Eq. (21)). Equation (24) is applied to Mach numbers M - 0.0, 0.3, 0.6. Results are shown in the table below.

Table 2 : Calculated Asymptotes of Perturbation Velocity Mach Number Panel Method Code Equation (24) ; du/Uoo 0.0 0.0091 0.0091 0.3 0.0100 0.0100 0.6 0.0140 0.0142 Comparing the results of the panel method code calculation and the Area-Velocity- Relation, we get excellent agreement to verify application of the Prandtl/Glauert trans- formation.

2.6 Normalized Perturbation Velocity Definition In general, normalized perturbation velocities can be defined as dimensionless pertur- bation velocities divided by unit singularity strength per unit velocity caused by some sin- gularity. For example, the normalized perturbation velocity, _-/(_, k), at a point, (x_, y$, z6), in the free-air flow field of a unit strength source located at (zk, Yk, z_) can be defined as (seealsoEq. (3.2a)in Appenaix 3):

_(6, k) = u(6, k) Vro: = _(_,k___2).i

U,._: o" U,._ I o'2 1 x_ - zk (25)

4 _ [ [x, - _]2 + [_ _ _]2 + [z_ - z_]_]3/2

Two different sets of normalized perturbation velocities for a given singularity type, location, and specified model reference point location have to be computed if the Wall Signature Method is applied. The first set is related to the wind tunnel flow field of a singularity. The second set is related to the wall interference flow field of a singularity.

Normalized perturbation velocities, h-7(6, k), of the wind tunnel flow field of a singu- larity are required for the least squares fitting of the wall signature on wall pressure orifices "5" (see Eqs. (13d), (13e), (13g), (19)) . Each normalized perturbation velocity can be interpreted as a dimensionless streamwise perturbation velocity divided by unit singularity strength per unit velocity at wall pressure orifice "6" caused by the wind tunnel flow field of a singularity located at position "k" Normalized perturbation velocities, _-(v, k) and _'(v, k), of the wall interference flow field are required to determine wall interference corrections at reference point "v" (see Eqs. (9a), (95), (15a), (155)) . Each of these perturbation velocities can be interpreted as a dimensionless perturbation velocity component divided by unit singularity strength per unit velocity at flow field reference point "L," caused by the wall interference flow field of a singularity located at position "k" . Normalized perturbation velocity _--_(v, k) is the streamwise perturbation velocity component and normalized perturbation velocity _-(v, k) is the perturbation velocity component perpendicular to the wing plane.

Explicit equations of these normalized perturbation velocities can be derived in the case of a wind tunnel with rectangular cross-section using the Method of Images. However, normalized perturbation velocities of a wind tunnel with non-rectangular cross-section have to be computed using a panel method code as a boundary value problem solver (see aef. [15],[17], mad Appendix 2 for more detail).

Exact Bessel Function solutions of the angle of attack correction of a line doublet located in plane z = 0.0 inside of a wind tunnel with circular cross-section are available.

NACA TN 2454, [18] lists these corrections in the form of upwash factor tables. Tables can be compared with the normalized perturbation velocities of the wall interference flow field that were computed using a modified panel method code [15],[17].

At first, an upwash factor table used for fullspan model tests (Table I on p.32 of NACA TN 2454, [18]) is compared with the corresponding panel method code solution. Figs. lla and 1 lb compare upwash factor F as a function of the dimensionless streamwise coordinate and the dimensionless line doublet location (lateral coordinate !? = 0.7). The exact Bessel Function solution and the numerical panel method code solution of the upwash factor F show reasonable agreement verifying the normalized perturbation velocity definition of the fullspan configuration.

Table 3 compares the input and accuracy characteristics of NACA TN 2454 and the panel method code solution of the normalized perturbation velocities.

Table 3 : Comparison NACA TN 2454 / Panel Method Code NACA TN 2454 Panel Method Code Tunnel Circular Tunnel Any Tunnel Geometry Geometry One Bipolar Tunnel Solution Numerical Exact (Circular) Solution Type Approximation (Bipolar) Singularity Line Doublet Point Source, Point Doublet, Line Doublet Type Minimum distance from wall Singularity Z=0.0 (Circular) Location Y=0.0 (Bipolar) panels _ 0.2 × tunnel radius Minimum distance from wall Reference Point Z=0.0 (Circular) Location Y=0.0 (Bipolar) panels .._ 0.2 × tunnel radius Unfortunately, no rigorous solution of the upwash factor F, i.e. angle of attack cor- rection, of a line doublet located inside of a bipolar wind tunnel (image plane / semispan configuration) is available. However, NACA TN 2454 provides an approximation of the upwash factor F for a bipolar wind tunnel that can be compared with the results obtained by applying a panel method code. Fig. 11c compares tl:.is approximation of upwash factor F (Fig. 5(c) on p.56 of NACA TN 2454, [18]) with the panel method code solution. Both approximations show reasonable agreement verifying the normalized perturbation velocity 2O definition of the semispanmodel configuration.

A panelmethod codeallows the userto computenormalized perturbation velocities for

any type of constant cross-sectionwind tunnel geometrythat canbe paneled. Singularities

and referencepoints can alsobe placed anywhereinside of the wind tunnel as long as the

minimum distance of the singularity or referencepoint from the paneled wind tunnel wall

(60 panelsused to representtunnel cross-section)is greater than 0.2 x tunnel radius.

CHAPTER3

APPLICATION OF THE METHOD TO WIND TUNNEL TESTS

3.1 Real-Time Wall Interference Calculation The revised and improved version of the Wall Signature Method presented in this report can be used to predict the Mach number, dynamic pressure, and angle of attack correction at a test article reference point due to the subsonic wind tunnel wall interference effects. Post-test analysis of subsonic wall interference effects is also possible as long as the exact position of the test article in the wind tunnel is known for a specific angle of attack setting. The location of singularities representing the test article is directly related to this position. Perturbation velocities of the wind tunnel and interference flow field can be computed as outlined in the previous chapters.

The real-time calculation of wall interference corrections is fast because the Wall Signature Method only requires superposition of the perturbation velocities, application of the Kutla/Jonkowski formula (Appendix 5), Koning's formula (Appendix 10), and the solution of a 2 x 2 linear system of equations related to the least squares fit of the wall signature.

The precalculation of normalized perturbation velocities used for the real-time least squares fit of the wall signatures and for the calculation of corrections has to be done on a mainfraxne computer or fast workstation since a realistic implementation of the proposed Wall Signature Method requires the calculation of perturbation velocities for many different singularity types, locations, and Math numbers. Figures 12a,12b depict geometries of the NASA 12ft Pressure Wind Tunnel test section that are selected for the calculation of the perturbation velocities of the wind tunnel and interference flow field using a panel method code as a boundary value problem solver. Preeomputed normalized perturbation velocities have to be stored in a database that is accessed during a wind tunnel test.

A singularity and reference point grid has to be used for the calculation of the per- turbation velocity database. These two grids should be selected such that they allow for a real-time interpolation of all conceivable singularity and reference point locations in the wind tunnel test section. Perturbation velocities required for singularities representing the

test article change as a function of test article geomel;ry and position in the wind tun-

nel test section. Size, complexity, and accuracy requirements of the perturbation velocity database have to be balanced to guarantee best real-time performance. Thus, perturbation velocities of the wind tunnel and interference flow field for a given test article location are found in real-time by applying a tri-linear interpolation (real-time singularity position; Appendix 6) and parabolic interpolation (real-time Mach number; Appendix 7) to the precomputed perturbation velocity database.

Blockage effects of the support system can be computed off-line by applying the Wall Signature Method to the difference between the support system and the empty tunnel calibration. Computed support system wall interference corrections on the reference point grid have to be stored in a database as a function of the support system calibration variables. The support system wall interference corrections are added in real-time to the wall interference corrections caused by the test article.

Post-test analysis of the interference effects is based on minimizing the standard deviation of the least squares fit of the wall signature as a function of the location of the test article singularities (see Appendix 8). This procedure provides an optimal singularity representation of the test article.

Studies of the author have shown that the real-t:ime speed of the wall interference calculation is governed by the efficiency of the interpolation of the perturbation velocities using the precomputed perturbation velocity database.

Figures 13a,13b depict basic elements of the real-time Wall Signature Method for a fullspan and a semispan model configuration. The .empty tunnel calibration, support system calibration, real-time wall pressures , lift force, propulsion simulator thrust, and pitching moment measurements are the critical link between experiment and the panel method code calculation. Matching conditions between the wind tunnel test and the panel method code calculation will be discussed in detail in the following section.

3.2 Matching Conditions The least squares fit proposed for a fullspan and semispan model configuration relates experimental data, i.e. wall pressure measurements, to precomputed normalized perturba-

tion velocities. Therefore suitable equationshave to be found to convert the wall pressure

measurements to perturbation velocities.

Perturbation velocity differences [U, + ut] - [U, + ut,] , [U, + ut,] - U, , and [U, + ut] - U, defined in Eqs. (13g), (16c), and (19) can be related to the wall pressure measurements taken during calibration and real-time wind tunnel test by applying the energy equation and the isentropic flow assumption. Assuming that the total temperature TT and total pressure PT in the tunnel settling chamber, and the static pressure p at wall pressure orifice are known, we get for the flow velocity (see Fig. 14a) : Dimensionless perturbation velocities can then be written as : [U_(6) + ut(6)]- [ U_(cS) + ut,(_5)] = V(pt,,,_(tS))- U(p_,_v(_5)) (27a) U,._ I U_<f

[ + - u0(6) - u(v mp(6))

= (27b) u_s u_f

[ uo(6) + - u (6) -

= (27c) U,._ I U,._S The flow velocity U(p(6)) in Eq. (26) is written as a function of the pressure difference PT -- P(_) as it is easier to measure a pressure difference at a wall pressure port. Real-time static pressure pt_,,-,, support system static pressure ps=p, and empty tunnel calibration static pressure Pemp are recorded at each wall pressure port "_" . Measured velocity Ue(_) = U(pe,-r,p($)) at a wall pressure port "_" is the first matching condition. It is required to obtain the perturbation velocities defined in Eqs. (27b),(27c).

The velocity Ur_f in Eqs. (13g), (16c), and (19) is the second matching condition between the wind tunnel flow field and the corresponding dimensionless panel method code calculation. It is required to non-dimensionalize the perturbation velocities defined in Eqs. (27a),(27b),(27c) . It can be considered as the constant flow velocity inside of a hypothetical constant cross-sectional test section. This velocity should be measured during the empty tunnel calibration at a specific model reference point 0 (see Fig. 14b; this point has the coordinates X=120.71 [ft], Y=0.0 [ft], Z=0.0 [ft] in the 12ft PWT). The

velocity Urel can be measured by using, e.g., a static pipe installed in the wind tunnel

during the calibration. We then get :

: [

The real-time lift force _nd pitching moment measurements are related to the strength of the corresponding line doublets using the Kn_ta/Joukowski formula. Its application re- quires knowledge of velocity U,-ef and density p,..! (0) at a model reference point 0 measured during the calibration of the wind tunnel. This velocity and fluid density connect lift force and pitching moment measured in [lbf],[ft • lbf] or [NI,[N • m] to the definition of the line doublet strength in [ft 2] or [m 2] (see Appendix 5). The fluid density is also required to relate the propulsion simulator thrust measurement to the sink strength if Koniug's formula is applied (see Appendix 10). The density P,-e! (9) is the third matching condition.

It is found by applying the ideal gas and the isentropic flow relationship at a test section reference station/9 . We get :

e s(0) = R-- r " -- (29)

where p_mp(0) is the static pressure measured at the model reference point/9 .

It is interesting to note that the perturbation velocity differences [Ue(6) + ut(6)] - [V_(6) + ut_(6)], [g_(6) + u,,(6)] - U_(d_), and [U_(6) + u,(6)] - U_(6) remove the influence of the wall boundary layer growth, orifice error, image plane, and wall divergence from the least squares fit. Therefore it is possible to use the geometry of an equivalent wind tunnel with constant cross-sectional area for the calculation of normalized perturbation velocities of the wind tunnel and interference flow field.

Three matching conditions, i.e. U_(6), U_S, and .o,-_I, establish a link between the measurement of wall pressures, forces, moments and the panel code solutions of the wind tunnel and interference flow field expressed as normali2ed perturbation velocities. Figure 14c summarizes the importance of these matching conditions.

3.3 Application of the Method to Semlspan Models In the summer of 1996, two different sized semispem models were tested in the NASA Ames 12ft Pressure Wind Tunnel (PWT). Both models, i.e. the 8 % and 14 % scale 7J7

semispanmodels,were provided by the Boeing Corporation. Each model wasmounted on

the image plane in the 12ft PWT. Figure 15 shows a similar test configuration.

Both models were tested over a wide range of angle of attack, total pressure, and Mach number settings. For the present study, two runs were selected. During Run No. 154 the 8 % scale model was tested from -20.280 to 19.820 uncorrected angle of attack at a total pressure of 2.0 [atm] and a Mach number of 0.25 . During Run No. 219 the 14 % scale model was tested from -4.030 to 9.980 uncorrected angle of attack at a total pressure of 2.0 [atm] and a Mach number of 0.30 .

The application of the present wall interference correction method was done in several steps. At first, type and initial location of the singularities representing each model were specified. Rules of thumb given in Appendix 9 were used to select type, location, and weighting factors for these singularities. A total of 11 singularities were selected for each model. A source and a sink were selected to represent fuselage blockage, two sources were selected to model the separation wake blockage effects. Seven line doublets, located along the 1/4 chord hne of the wing, were chosen to represent lifting effects. Weighting factors for the line doublets were selected to model an elliptic lift distribution for the wing.

Figures 16a,16b give the singularity representation of each semispan model for 0 ° angle of attack. The real-time coordinates of these singularities as a function of the pitch angle were computed using the known kinematics of a semispan model mounted on the image plane (for more detail see Eqs. (14.19a),(14.19b) in Appendix 14).

In the next step, measured lift force in combination with the Kulta-Joukowski formula was used to determine the strength of line doublets for the wing for each data point.

The strength of the remaining singularities was computed using a least squares fit of the wall pressure measurements on 180 wall pressure ports that were arranged in six rows above the image plane. The least squares fit used wall pressure port rows 1,2,3,6,7,8 depicted in Fig. 17a,17b. For more detail on the least squares fit procedure see Section 2.4.

The standard deviation of the least squares fit of the wall signature was computed for each data point of Runs 154 and 219 (see Figs. 18a,18b) . The standard deviation of the 8 % scale model was on the order of 0.002 in units of the dimensionless perturbation velocity. This agrees with the standard deviation of a wall signature obtained by Rueger et al., [19] who reported a value of 0.005 in units of pressure coefficient, i.e. 0.0025 in units of

the dimensionlessperturbation velocity. The standard deviation of the 14 _ scalemodel

was on the order of 0.003 to 0.006in units of the dimensionlessperturbation velocity.

Figures 19ato 19f showthe result of the least squaresfit of the wall signature for the

8 % semispanmodel at 19.82 o uncorrected angle of attack. Figures 20a to 20f show the

result of the least squares fit of the wall signature for the 14 % semispan model at 9.98 o uncorrected angle of attack. Figures 21a, 21b depict the wall signature for both models at approximately 0.0 ° angle of attack at wall pressure port Row 6 . The large difference in solid volume blockage of both models can clearly be detected in the wall signature. The measured wall signature difference "u" depicted in Figs. (19a) to (21b), i.e. the velocity difference [Ue + ut] - Ue in Fig. 9b, shows excellent agreement with its least squares fit.

The present method (WICS), the two--variable method, and the classical method were used to compute wall interference corrections. Two-variable method results were provided by Mat Rueger of Boeing St. Louis. Classical corrections were provided by Alan Boone of NASA ARC who used NACA Rep. No. 995 (solid volume blockage), [20], R.A.E. Rep.

No. 3400, [21] (separation wake blockage), and NACA TN 2454, [18], to determine wall interference corrections. Mean wall interference corrections for each model were computed using flow field reference points located along the 3/4 chord line of the wing. Corresponding results are compared below.

As expected, wall interference corrections computed by WICS and the two-variable method show excellent agreement because both methods are based on potential flow theory and boundary flow measurements. Angle of attack corrections agree well in all three cases (see Figs. 22a,22b) . The solid volume blockage factor contribution depicted in Fig. 23a agrees well for the 8 % scale model in all three cases. A comparison of the solid volume blockage factor contribution of the 14 % scale model depicted in Fig. 23b shows larger differences between classical corrections and WICS. This can be explained by the fact that the calculation of the solid volume blockage using the classical method (NACA Rep.

No. 995) assumes that a wind tunnel of constant cro,;s-section extends to far upstream and downstream of the semispan model. This assumption, however, cannot be justified anymore in the case of the 14 % scale model as the fllselage length is 16.69 [ft] and the length of the image plane is _ 20.0 [ft] . The classica] method will therefore overpredict the solid volume blockage effect for the 14 % scale model. The separation wake blockage factor contribution for the 8 % scale model determined based on the classical method [21] is larger than the blockage factor computed using WICS or the two-variable method (see Figs. 23a). This agrees with observations reported in the literature,[5],[19].

The 8 % and 14 % scale model have identicalgeometry. Therefore it is possible to compare the minimum of the blockage factor of both models by using a scale factor law (for more detail see Appendix 20). Results discussed in Appendix 20 demonstrate that blockage corrections computed with the present method (WICS) satisfythis scale factor law. Thus, wall pressure measurement accuracy and the solid volume description used by the present method are sufficiently accurate for computing blockage effects.

In general, the ratio between measurements and unknowns of a least squares fithas to be large to take full advantage of its smoothing characteristics. In our application the number of unknowns of the least squares fitis two (see Eqs. (13a),(16a)). Figures 23c, 23d compare the computed dynamic pressure correction for the 8 % scale model with 180 or 30 wall pressure ports used for the least squares fit of the wall signature.

The differencesin the computed corrections depicted in Figs. 23c, 23d are small. This demonstrates a key operational advantage of the Wall Signature Method : the calculation of the corrections is relatively insensitiveto the number and location of the wall pressure ports (see also Table 1 in Chapter 1) . Comparison of the data scatter in the computed dynamic pressure correction depicted in Figs. 23c,23d shows that an increase in the number of wall pressure measurements used in the least squares fitreduces the data scatter of the computed blockage corrections.

The local dynamic pressure correction for the 8 % model at 19.82° angle of attack and for the 14 % model at 9.980 angle of attack are computed in the plane Y=0.0 [ft] (cut through test section parallelto side wall). The dynamic pressure correction for the 8 % model at 19.82° increases gradually as an observer moves from upstream to downstream of the model (see Fig. 24a) . The separation wake blockage effectsdominate the dynamic pressure correction downstream of the model at an angle of attack of 19.82° . The contour lines are nearly parallelto the z-axis. We conclude that the dynamic pressure correction for the 8 % model is almost exclusivelya function of the streamwise coordinate.

The dynamic pressure correction for the 14 % model at 9.98° has a saddle point at the streamwise coordinate 120.0 [ft] (see Fig. 24b). The correction decreases as an observer

moves downstream of the model. Therefore, it can be concluded that the solid volume

blockage effects dominate the correction at an angle of attack of 9.98 ° .

In both contour plots, it can be seen that the minimum of the dynamic pressure correction for a constant streamwise coordinate is about 4.0 [ft] above the image plane.

This is caused by the fact that the tunnel width increases as an observer moves from the image plane surface to the tunnel centerline (see also Fig. 15).

The local angle of attack corrections for 19.82 o and 9.98 o for each model are computed in the plane Y=0.0 [ft] (cut through test section parallel to side wall). The contour plots of the results are depicted in Figs. 24c,24d. The angle of attack correction variation along the 3/4 chord line of the 8 % model at 19.82 °, i.e. from wing root to tip, is on the order of 0.050 . The aerodynamic twist is insignificant in this case. However, the angle of attack correction variation along the 3/4 chord line of the 14 % model at 9.98 ° is on the order of 1.0 ° and cannot be ignored anymore.

3.4 Application of the Method to the Ames Bipod In 1995 the Ames Bipod, a floor-mounted support system, was tested in the NASA Ames 12ft Pressure Wind Tunnel (PWT). Blockage corrections due to support system wall interference effects were computed using the Wall Signature Method (WICS) as outlined in Section 2.3.2 .

The application of the Wall Signature Method to t:ae Ames Bipod was done in several steps. At first, type, initial location, and weight of singularities representing the Ames Bipod were specified. Figure 25a shows the Ames Bipod geometry and lists type, initial location, and weight of these singularities. A total number of 27 singularities was selected for the support system. Singularities No. 1 to No. 18, i.e. 9 source/sink pairs, were chosen to represent solid volume blockage effects of the support system. Singularities No. 19 to No. 27 were chosen to model separation wake blockage effects.

In a second step, strength values of these singularities were computed using a least squares fit of the wall pressure measurements. Wall pressures on Row 1 to Row 8 (see Figs. 17a,17b) were measured during the support system calibration as a function of the total pressure in the settling chamber and the Mach aumber at a test section reference 3O point. Wall pressure measurements obtained during an empty tunnel calibration were also available. The least squares fit was applied to the difference between the wall signature measured during the support system calibration and the wall signature measured during the empty tunnel calibration. Figures 25b,25c show the result of the least squares fit on Row 2 and Row 4. Measured wall signature difference and least squares fit show reasonable agreement. Larger differences between the measured wall signature and the least squares fit on Row 4 are caused by the fact that singularities cannot be placed too close to the floor for numerical reasons (see also Chapter 2.6). Finally, the blockage factor was computed on planes Y=0.0 [ft] (parallel to test section side wall), X=120.71 [ft] (parallel to test section inlet), and Z=0.0 [ft] (parallel to test section floor).

Figure 25d shows the blockage factor on plane Y=0.0 [ft] that was computed using the initial singularity location given in Fig. 25a . Figure 25e shows the blockage factor on plane Y=0.0 [ft] that was computed after the standard deviation of the least squares fit was minimized as a function of the singularity location (see Appendix 8). Comparing both contour plots it can be recognized that differences in the computed blockage factor are small, i.e. computed corrections are not very sensitive to the location of singularities as long as they are placed at the location of the support system. Comparing singularity locations before and after the minimization it can be noticed that sources No. 1 to No. 9 and sources No. 19 to No. 27 have moved closer together. The minimization procedure has correctly deduced from the wall signature difference that the front post of the Ames Bipod has a significantly larger diameter than the pitch strut, i.e. most of the separtion wake of the Ames Bipod is caused by the front post.

Figure 25f shows the blockage factor on plane X=120.71 [ft] . In this contour plot it can be seen that the blockage factor increases in any direction if an observer moves closer to the test section wall. A minimum of the blockage factor is located _ 1.0 [ft] above the tunnel centerline. The blockage factor increases significantly if an observer moves closer to the test section floor.

Figure 25g shows the blockage factor on plane Z=0.0 [ft] . As expected, the blockage factor increases if an observer moves from an upstream position to a downstream position.

This observation is caused by the fact that the separtion wake blockage dominates blockage corrections downstream of the Ames Bipod.

CHAPTER 4 CONCLUSION AND REMARKS A revised version of the Wall Signature Method was developed which allows the user to predict Math number, dynamic pressure, and angle of attack correction due to wall inter- ference effects in three-dimensional subsonic wind tunnel testing of aircraft models. This improved formulation of the Wall Signature Method uses lift force, propulsion simulator thrust force, pitching moment, wall pressure measurements, empty tunnel calibration, sup- port system calibration, a simplified representation of the test article and support system in terms of singularities, and precalculated normalized solutions of the subsonic poten- tial equation expressed as normalized perturbation velocities to predict wall interference corrections at a model reference point in real-time.

The method is applicable to complex wind tunnel and support system configurations.

Wall interference corrections are found by taking the difference between a simplified rep- resentation of the wind tunnel flow field in terms of singularities and the corresponding free-air solution. Computational procedures were developed to predict solid body blockage, separation wake blockage, propulsion simulator blockage, and lift interference correction if a fullspan model or a semispan model is tested.

The definition of normalized perturbation velocities used by the method has been improved to allow the user to take full advantage of the geometry modeling capabilities of a three-dimensional panel method code. A "global" least squares fit procedure of the wall signature was also introduced to improve the application of the Wall Signature Method in real-time. Optimal locations of singularites are defined by minimizing the standard deviation of the least squares fit of the wall signature.

Experimental data obtained during tests of two semispan models mounted on an image plane in the NASA 12ft Pressure Wind Tunnel were applied to the modified Wall Signa- ture Method. Blockage and angle of attack corrections were computed for different angle of attack settings. In all cases, computed angle of attack corrections show good agree- ment with corresponding classical corrections. Computed blockage corrections are smaller than corresponding classical corrections. This result agrees with observations reported in literature, [5],[19] .

Experimental data recorded during the calibration of the Ames Bipod was also suc- cessfuUy applied to the method.

Further experimental studies have to be conducted in the future to gain confidence in the method.

LIST OF REFERENCES [1] Hackett, J. E., Wilsden, D. J. and Lilley, D. E., "Estimation of Tunnel Blockage from Wall Pressure Signatures: a Review and Data Correlation," NASA CR-152241, Lockheed-Georgia Company, Marietta, Georgia, March 1979.

[2] Hackett, J. E., Wilsden, D. J. and Stevens, W. A. , "A Review of the Wall Pressure Signature and other Tunnel Constraint Correction Methods for High Angle-of-Attack Tests," AGARD-R-692, Munich, May 1980.

[3] Hackett, J. E., "Living with Solid-Walled Wind Tunnels," AIAA 82-0583, presented at the AIAA 12th Aerodynamic Testing Conference, March 22-24, 1982/Williamsburg, Virginia.

[4] Ashill, P. R., Keating, R. F. A., "Calculation of Tunnel Wall Interference from Wall Pressure Measurements," The Aeronautical Journal, Vol. 92, No. 911, January 1988, p.36 to p.53.

[5] AshiU, P. R., "Boundaxy-Flow Measurement Methods for Wall Interference Assess- ment and Correction; Classification and Review," Proceedings, 73rd AGARD Fluid Dynamics Panel Meeting and Symposium on Wall Interference, Support Interference and Flowtleld Measurements, Oct. 4-7, 1993, p.12-1 to p.12-21 .

[6] Koning, C., "Influence of the Propeller on other Parts of the Airplane Structure," in "Aerodynamic Theory," edited by Durand, W. F., Springer Verlag, Berlin, 1935, Volume IV, Division M, Section 5, p.367/368 .

[7] Ulbrich, N. and Steinle, F. W., "Real-Time Wall Interference Calculation in Three- Dimensional Subsonic Wind Tunnel Testing, " AIAA 94-0771, presented at the 32nd Aerospace Sciences Meeting, Reno, Nevada, January 10-13, 1994.

[8] Ulbrich, N. and Stein]e, F. W., "Semispan Model Wall Interference Prediction Based

on the Wall Signature Method, " AIAA 95-0793, presented at the 33rd Aerospace Sciences Meeting, Reno, Nevada, January 9-12, 1995.

[9] Ulbrich, N., "Real-Time Wind Tunnel Wall Interference Prediction Based on Lin- earized Panel Method Code Solutions," final report of National Research Council tenure, presented to the Aeronautical Test and Simulation Division at NASA Ames Research Center, Moffett Field, California, December 1994.

[10] Garner, H. C., Rogers, E. W. E., Acum, W. E. A. and Maskell, E. C., "Subsonic Wind Tunnel Wall Corrections," AGARDograph 109, North Atlantic Treaty Organization, October 1966.

[11] Strang, G., "Introduction to Applied Mathematics, " Wellesley-Cambridge Press, Wellesley, Massachusetts, 1986, p.35-39 .

[12] Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P., "Numerical Recipes in FORTRAN 77," Cambridge University Press, reprint of second edition with cor- rections, 1996, p.51-63, p.670-673.

[13] Schlichting, H. and Truckenbroth, E., "Aerodynamik des Flugzeuges," Vol. 2, 2nd Ed., Springer-Verlag, Berlin, Heidelberg, New York, 19q37, p.159-164.

[14] Kuethe, A. M. and Chow, C., "Foundations of Aerodynamics," 4th Ed., John Wiley & Sons, New York, 1986, p.271-272.

[15]

Ashby, D. L., Dudley, M. R., Iguchi, S. K., Browne, L., Katz, J.,"Potential Flow Theory and Operation Guide for the Panel Code PMARC," NASA TM 102851, NASA Ames Research Center, Moffett Field, California, January 1991 .

[16] Liepmann, H. W. and Roshko,A., "Elements of Gasdynamics," 1st Ed., John Wiley

& SonsInc., New York, 1957,p.52 .

[17] Lo, C. F., Xian, X. and Ulbrieh, N., "Wind Tunnel InterferenceCalculated By a Panel

Method, " AIAA 95-0794, presentedat the 33rd AerospaceSciencesMeeting, Reno, Nevada, January 9-12, 1995.

[18] Sivells,J. C. and Salmi, R. M., "Jet-Boundary Correctionsfor Complete and Semispan

Swept Wings in Closed Circular Wind Tunnels," NACA TN 2454, 1951, p.8, p.17, p.18.

[19] Rueger, M., Crites, R., Weirich, R., "Comparison of Conventional and Emerging

(Measured Variable) Wall Correction Techniquesfor Tactical Aircraft in Subsonic

Wind Tunnels," AIAA 95-0108, presented at the 33rd AerospaceSciencesMeeting

and Exhibit, January 9-12, 1995,Reno, Nevada.

[20] Herriot, J. G.,"Blockage Correctionsfor Three-Dimensional-Flow Closed-Throat Wind

Tunnels, with Considerationof the Effect of Compressibility," NACA Report No. 995, Ames Aeronautical Laboratory, Moffett Field, California, 1951.

[21] Maskell, E. C.,"A Theory of the Blockage Effects on Bluff Bodies and Stalled Wings in a Closed Wind Tunnel," R.A.E. Report No. 3400, November 1963 .

[22] Katz, J. and Plotkin, A., "Low-Speed Aerodynamics: From Wing Theory to Panel Methods," McGraw-Hill Series in Aeronautical and Aerospace Engineering, New York, 1991, p.60, p.63, p.378-420 .

[23] Schlichting, H. and Truckenbroth, E., "Aerodynamik des Flugzeuges," Vol. 1, 2nd Ed., Springer-Verlag, Berlin, Heidelberg, New York, 1967, p.62-63 .

[24] Ashley, H. and Landahl, M., "Aerodynamics of Wings and Bodies," Addison-Wesley

Pubhshing Company,Inc., Reading,Massachusetts,1965,p.87.

[25] DeYoung, J. and Harper, C. W., "Theoretical Symmetric Span Loading at Subsonic Speeds for Wings having Arbitrary Plan Form," NACA Report No. 921, Ames Aero- nautical Laboratory, Moffett Field, California, 1948 .

[26] Burden, R. L. and Faires, J. D., "Numerical Analysis," 3rd Ed., PWS-Kent Publishing Company, Boston, Massachusetts, 1985, p.514.

[27] Heyson, H. H., "Jet-Boundary Corrections for Lifting Rotors Centered in Rectangular Wind Tunnels," NASA Technical Report R-71, Langley Research Center, Langley Field, VA, 1960.

[28] Pope, A., "Basic Wing and Airfoil Theory," 1st Ed., McGraw-Hill Book Company, Inc., 1951, p.91.

[29] Schlichting, H. and Truckenbroth, E., "Aerodynamics of the Airplane," translated by H.J. Ramm, McGraw-Hill International Book Company, 1979, p.162/163 .

APPENDIX 1 HIGHER ORDER CORRECTION FORMULAE The testing of an aircraft model at a high angle of attack in a three-dimensional subsonic wind tunnel can create large separation wake blockage effects. In this case, first order approximations of the Mach number and dynamic pressure correction as a function of the blockage factor e are no longer sufficient. Second order approximations have to be derived.

A second order approximation of the Mach number correction can be found using a Taylor series expansion. The Mach number is expressed as a function of a small change in the fluid velocity. Assuming that this velocity change AU is related to the flow velocity U and blockage factor e as :

Au = (1.1)

we get the following Taylor series expansion of the Mach number : M(U + e U) = M(U) + d M(U) (e U) d2M(U) (e U) 2 d U 1! + d U 2 2l + --. (1.2) It is necessary to express the Mach number as a function of the fluid velocity U. Applying the energy equation we know : V 2 % TT = % T + 2 (1.3) Combining the energy equation with the definition of the Mach number, i.e.

U 2 U 2 M 2 = = (1.4) a2 7 RT and with the relationship between specific heat at constant pressure, isentropic exponent, and Gas constant, i.e. %(7 - 1) = 7R , we get : U[ U2 ] -I/2 M(U) = v"-7- 1 " cv TT 2 (1.5) After some algebra we get for the first and second derivative : d2M(U) 3 M3(U) [ 7-1

(1.6b)

_. M2(U) ] dU 2 = --_--'(7-1)" U2 • [1 + 2 Combining Eqs. (1.2), (1.6a), (1.6b) and rearranging terms we get: M(U -t- e U) - M(U) = [1 + 7-1"M2(U)] "e2 M(U) + "(7-- 1).M2(U) • 1 +

[

7-2 I . M2(U )] .e2 (1.7) "_ °'° Using the nomenclature introduced in Chapter 2.1 and assuming that M(U + e U) = Mm and M(U) = Me, we get the second order approximation : Me _ 1 + 7_____2.M_ • e + -_.(7-1).M_.4

[ 1[

Similar to the Mach number correction it is necessary to derive a second order ap- proximation of the dynamic pressure correction using a Taylor series expansion.

Assuming that the velocity change AU is related to the flow velocity U and blockage factor e according to Eq. (1.1) we get the following Taylor series expansion for the dynamic pressure :

q(u+ _ u) = q(u)+ d q(u) (_ U) a_q(u) (_ u) _

dU 1! + dU 2 2! + ... (1.9) The dynamic pressure has to be expressed as a function of the fluid velocity U. Applying the energy equation (Eq. (1.3)), the Mach number defin:ition (Eq. (1.4)), the ideal gas law, i.e° : P = R. T (1.10) P and assuming isentropic flow, i.e. : P PT = (1.11) we get : II(_-i)

Rp) U 2 ] I/(-y-I) US

- % TT 2 2 (1.12)

q(U) = % PT

4O After some algebra the first and second derivative can be obtained as : (1.13a) dq(U)du = q(U---'_)'[ 2 -U M2(U)]

d_q(U) _ q(u) [

(1.13b) - 5M2(U) + (2-7)-M4(U)] d U s -- U-- T- • [ 2 Combining Eqs. (1.9), (1.13a), (1.13b) and rearranging terms, we get:

q(u +_ u) - q(u)

= [2 - M2(U)].e

q(u)

(1.14) + 1 5 M2(U ) + . M4(U ) 2 2 ,°.

Again, using the nomenclature introduced in Chapter 2.1 and assuming that q(U + e U) = qoo, q(U) = q_, and M(U) = M_(U), we get the approximation:

APPENDIX 2 PANEL METHOD CODE MODIFICATIONS Introduction In general, it is necessary to use a three-dimensional panel method code to calculate normalized perturbation velocities caused by singularities if the Wall Signature Method is applied to a wind tunnel with a non-rectangular but constant cross-section. Unfortunately, commercially available panel method codes do not allow the calculation of the flow field of a singularity placed inside of a wind tunnel. However, a few modifications to a panel method code can be introduced which make it possible to solve this type of internal flow field problem and to compute perturbation velocities.

A panel method code may be used to find the velocity potential of a given internal flow problem if the corresponding internal flow geometry, e.g. the geometry of the wind tunnel test section of constant cross-sectional area, is paneled. The specification of normal velocities at the wind tunnel inlet is also required (see Ashby et al. [15] for more detail on the application of a panel method code to internal flow problems; see Katz and Plotkin [22] for a more detailed description of three-dimensional panel method codes) .

The boundary value problem of a singularity placed inside a wind tunnel flow field is depicted in Fig. 26. The velocity potential has to fulfill the Laplace equation :

+ ,, ] = 0 (2.1)

where _c¢ is the free--stream potential and ¢_ is the wind tunnel potential due to the wind Zero normal flow has to be satisfied across the wind tunnel walls and the singularity.

tunnel wall surface and so we get:

an + = 0 (2.2a)

The normal velocity vector uo_ has to be specified by the user at the test section inlet: a--n ¢oo + _, -- ucc (2.2b) Based on the principle of superposition (see Fig. 26) it is possible to express the wind tunnel potential Ct as the sum of the singularity potential _ba and the wind tunnel wall potential dw : ¢_ = _b_ + ¢_ (2.3) Combining Eqs. (2.1),(2.3) and knowing that the singularity potential itself fullfills Laplace's equation we get: V2[¢oo + ¢,_ ] = 0 (2.4) Combining Eqs. (2.2a), (2.3) and rearranging terms we get the boundary condition across the wind tunnel wall surface:

]

cg---n ¢°° + ¢,_ = On ¢ * (2.5a) Combining Eqs. (2.2b),(2.3) and rearranging terms we get the boundary condition at the test section inlet:

0[ ] 0[] cg----n ¢oo + ¢,_ = uoo - O"_ ¢_ (2.5b)

The velocity vector uoo and the singularity potential Ca are known and so the boundary value problem given by Eqs. (2.4),(2.5a),(2.5b) can be solved. This requires the modi- fication of the original panel method code such that 'the boundary conditions given by Eqs. (2.2a),(2.2b) are replaced by the boundary conditions given by Eqs. (2.5a),(2.5b) .

Only the normal velocity component due to the singularity, i.e.

O has to be added to prescribed normal velocities at panel centroids.

Finally, the flow field solution of a singularity placed inside a wind tunnel test section can be found by superimposing the solution of the boundary value problem given by Eqs. (2.4),(2.5a),(2.5b), i.e. ¢_ + _,o, computed using the modified panel method code with the known analytic solution of the singularity potential ¢_, .

The modification of the boundary conditions ba._ed on Eqs.(2.5a),(2.5b) does not cause numerical difficulties if a singularity like a point source, point sink, or point doublet is selected. In these cases, the singularity potential ¢., vanishes if the distance between singularity and panel centroid is large and therefore normalized perturbation velocities can

be compute with no restrictions. However,the boundary conditions have to be modified

carefully if a horseshoevortex or a semi-infinite line doublet is selectedas a singularity.

Panel method code PMARC, [15] models the internal flow geometry as a closedbox and

a horseshoe vortex or a semi-infinite line doublet will intersect panels defining the outflow

conditions. Therefore, the flow field of a line doublet has to be computed by solving the

flow field of a point doublet and this solution has to be made perfectly symmetric or anti-

symmetric relative to the streamwisecoordinate of the point doublet. Finally, a numerical

integration is applied in the streamwisedirection to obtain the flow field of a line doublet (for more detail seeAppendix 4).

Modifications of a panel method codefor semispantest are similar to the procedures

described above. The user only has to make sure that normal velocities induced on the

image plane surfaceby a singularity located on the image plane surface,i.e. point sources

and sinks representing the semispanmodel fuselagevolume, are zero.

Numerical Verification

A slender Rankine body (l/2ro _ 10) is selected to verify the proposed panel method

code modification if a source or sink is selected as a singularity (see Fig. 27a for a detailed desciption of the Rankine body geometry) . A point source of strength +1.0 [ft 2] is located at X=116.0 [ft], Y=2.0 [ft], Z=-3.0 [ft] and a point sink of strength -1.0 [ft 2] is located at X=126.0 [ftl, Y=2.0 [ft], Z=-3.0 [ft]. The shape and surface pressure coefi_cient distribution of the corresponding Rankine body can be described in analytic form using polar coordinates, [23] .

Assuming Zl - zo < z < zl + (z2 - zl)/2, we obtain : sin _1/2 rl = r0" (2.6a) sin_ol =:1 --=: cos T1 - (2.6b) _'1 (2.6c) cp = 1 - 4sin 2 + 3sin 4-_- Assuming zl + (=:2 - Zl)/2 < x < x2 + zo, we obtain : sin T2/2 (2.7a) r2 = r0. sin_2 Z --Z2 cos _o2 = -- (2.7bI r2 cp = 1 - 4 sin 2 _ + 3 sin 4 _o__22 (2.7c) where the stagnation point distance zo and the halfbody radius r0 for a point source strength a of +1.0 are given as: z0 -- (2.8a) ro = V_ (2.8b) In a first step it is necessary to panel the selected Rankine body using the known description of its shape in polar coordinates (see Fig. 27b) . The free-air solution of the surface pressure coefficient distribution of this geometry is calculated using panel method code PMARC, [15] . The result of this panel method code calculation is compared with the corresponding analytic solution given by Eqs. (2.6c),(2.7c). Figure 28 shows excellent agreement between the panel method and corresponding analytic solution of the surface pressure coefficient distribution of a Rankine body.

In the second step the Rankine body is placed inside a wind tunnel test section. The chosen wind tunnel geometry is similar to a test section configuration of the NASA Ames 12ft Pressure Wind Tunnel which will be used for semispan model tests. Figure 29 shows the corresponding wind tunnel wall, support system and Rankine body paneling. Velocities are computed along eight rows on the wind tunnel wall using the original version of panel method code PMARC, [15] .

Then, panel method code PMARC, [15] was modified to solve the boundary value problem given by Eqs. (2.4),(2.5a),(2.5b) . The velocity field due to a point source and point sink is used to represent the Rankine body (see also Appendix 3). The Rankine body is replaced by corresponding point source and sink of strength +1.0 [ft 2] at X=116.0 [ft], Y=2.0 [ft], Z=-3.0 [ft] and -1.0 [ft 2] located at X=126.0 [ft], Y=2.0 [ft], Z=-3.0 [ft] and velocities are again computed on eight rows using the modified version of the panel method code.

Figure 30 compares dimensionless velocities of both calculations on Row 4. Velocities show excellent agreement to verify the proposed panel method code modifications if point sources or point sinks are used to represent volume effects of a test article.

A rectangular wing (s/c = 3.56) with a NACA 0012 airfoil section at an angle of attack

of 50 is selected to verify the proposed panel method code modification if a semi-infinite line doublet is selected as a singularity (see Fig. 31 for a detailed desciption of the wing geometry) .

The rectangular wing is placed inside a wind tunnel test section. Figure 32 shows the corresponding wind tunnel wall, support system and wing paneling. Lift coefficient of the wing and velocities Mong eight rows on the wind tunnel wall are computed using the original version of panel method code PMARC, [15] . Panel method code PMARC was also used to compute the wall signature at angle of attack 0 ° This wall signature is due to the thickness of the wing and was therefore subtracted from the wall signature calculated for 5 o to obtain the wall signature due to lift only.

Then, panel method code PMARC, [15] was modified to solve the boundary value problem given by Eqs. (2.4),(2.5a),(2.5b). The velocity field of four line doublets is used to represent lifting effects of the wing at 5 o angle of attack (see also Appendix 4) . The location of these line doublets is depicted in Fig. 33 . The computed lift coefficient of the wing placed inside the wind tunnel test section (see Fig. 32) has to be related to the strength of the semi-infinite line doublets. Using the definition of the lift coefficient and the Kutta/Joukowski formula we get for the lift force : U 2 L = eL • fl • _.s.c = p.U.F-s (2.9) So we get for the circulation F : U F = cL • --. c (2.10) The line doublet strength _ of each of the four line doublets is then given as (see Ap- pendix 4) : s U s cr = C. --_ = CL" --_'c" T (2.11a) or 6 r c s = eL" --. -- (2.115) U 2 4 Knowing that CL = 0.3556, c = 1.5 [ft], and s/4 = 1.3334 [ft] for the selected wing we get for the singularity strength per unit velocity of each of the four line doublets a /U = 0.3556 [ft 2] .

The wing can now be replaced by four line doublets of strength 0.3556 [ft 2] and

velocities are again computed on eight rows using the modified version of the panel method code.

Figure 34 compares dimensionless velocities on the wind tunnel wall of both calcula- tions at Row 8. Velocities due to lift effects show excellent agreement to verify the proposed panel method code modifications if semi-infinite line doublets are used to represent lift of a wing.

Summary A three-dimensional panel method code was modified to calculate the flow field of a singularity placed inside a wind tunnel test section. The modification was verified by replacing a Rankine body by a point source and point sink and by replacing a rectangular wing at 50 angle of attack by four line doublets. Corresponding flow field solutions compare favorably in both cases. The panel method code modification will work with no restrictions if a point source or point sink is selected as a singularity. However, boundary conditions have to be modified carefully if a horseshoe vortex or semi-infinite line doublet is selected as this type of singularity intersects outflow panels of the internal flow geometry and therefore requires a numerical integration of the corresponding point doublet solution.

APPENDIX 3 POINT SOURCE VELOCITY VECTOR The velocity vector of a three-dimensional point source is required if a panel method code is modified to compute the flow field of a source or sink placed inside a wind tunnel.

The potential of a point source at location (zs,Vs,zs) is given as, [22] :

Cs(x, y, z) = - o"

4. _ [ [=- xs]2 + [y- ys]2 + [z- zs]2 ],/2 (3.1)

The velocity components of this point source are : _S Or X -- XS u(z,y,z) = 69x = 4 7r [ [z-- xS] 2 + [y -- ys] 2 + [z-- zs] 2 ]3/2 (3.2a) a&s a y - ys

v(x_ y_z) (3.2b)

ay- 47r [[x-zs] 2 + [y-ys] 2 + [z-zs] 213/2 _¢S _ z -- zS

w(=,y,z) = az = 4_ [[=_x_]2 + [y_ys]2 + [z-z_]213/2 (3.2c)

The singularity strength cr has the unit [ft3/sec] or [mZ/sec] . The perturbation velocity field given in Eqs. (3.2a),(3.2b),(3.2c) could be used to change a panel method code as outlined in Appendix 2 .

APPENDIX 4 LINE DOUBLET VELOCITY VECTOR The velocity vector of a three-dimensional semi-infinite line doublet is required if a panel method code is modified to compute the flow field of a line doublet placed inside a wind tunnel.

In general, a semi-inflnite line doublet can be considered as an elementary horseshoe vortex ([22], see also Fig. 35). The potential of a line doublet can be obtained by integrating the solution for a point doublet in the x-direction, [24] . A point doublet does not have a radial symmetry as in case of a point source and therefore the line doublet potential will be a function of the orientation of the point doublet. For lift force acting in the positive z- direction (see Fig. 35) it is necessary that point doublets point in the negative z-direction.

The potential of a finite length line doublet with starting point at (zl, Yl, zl ) and end point at (ZN, Yl, zl) is then given as, [22] :

[z- zl]

(4.1)

¢(z, y, z) = 4-"7" Jz, [ [X--_12 2y [y__yl]2 2V [Z-- Z II 2 ]3/2 or --0" Z -- Z 1 _b(z,y,z) . = 4 _r [y- yl] 2 + [z- zl] 2 A (4.2a) X -- XN A

[ 2 + [y- 12 + [z-z112

(4.2b) X -- X 1 [ [X-- Xl] 2 "4- [y-- yl] 2 "4- [Z--Zl] 2 ]1/2 Taking the limit zN _ o0 we get the potential for the semi-infinite line doublet : ¢L(z,y,z) = lim ¢(z,y,z) -- a z-- zl _N--.oo 4----'_ " [y- yl] 2 + [z -- zl] 2 B (4.3a) B = 1 + z- zl (4.3b) [ [X-- Xl] 2 "3 t- [y-- yl] 2 _t. [Z-- Zl] 2 ]1/2 This equation agrees with equation (5-35) in Ref. [24], if the starting point of the line doublet is located at (zl = 0, yl = 0, zl = 0) . The velocity components can now be obtained by taking derivatives of the velocity potential qgL .

In the x-direction we get : U(X y, Z) "-- _¢L -- dr Z -- Z 1 ' az -- 4_" [[Z-Zl] 2 + [y-yl] 2 + [z-zl] 213/2 (4.4) In the y-direction we get :

O_L _ -dr [y- yl]. [z- zl]

v(x,y,z) = oy - 4_ [y_y,]2 + [z-_,]2 c (4.5a)

2 • B z-z1 C = [y-yl] 2 + [z-zl] 2 + [[z-z1] 2 + [y-y1] 2 + [Z-Zl] 2]a/2 (4.5b) In the z-direction we get : aCL a 1 w(z,y,z) = Oz = 4 r [y-- yl] 2 + [z -- zl] 2 D (4.6a) D = [Y- yl]2 - [z- Zl] 2 . B

[y- _,]_ + [z- Zl] 2

[z- zl]" [z - Zl] 2 (4.6b)

[ [_- _,]_ + [_- y,]_ + [z- z,]_ ]_z_

The calculation of the flow field of a semi-infinite line doublet located inside a wind tunnel of constant cross-sectional area is difficult if a modified version of panel method code PMARC is used (see Appendix 2) . The line doublet intersects the exit plane of the paneled wind tunnel geometry and can cause convergence and accuracy problems.

Fortunately, it is possible to calculate the flow fieht of a line doublet in a wind tunnel of constant cross-sectional area by integrating the flow field of the panel method code solution of a point doublet in the streamwise direction. The line doublet flow field solu- tion is obtained by shifting and superimposing corresponding point doublet solutions (see Fig. 36a).

In general, the potential of a semi-infinite line doublet (Eq. (4.1) ; ZN >> zl) can be approximated using numerical integration. We get then : N CL(z,y,z) _ _ _pD(z,y,z; :rk )" /X_r (4.7a) t;=1 dr Z -- Zl CpD(;r,y,Z; T, k )

(4.7b)

4 11"

[ [_- _]_ + [_- y,]_ + [_- Zl]_ ]_/_

zk = zl + (k - 1).Az ; zN >> zl where Az is the step size of the integration. The perturbation velocities of the line doublet can now be approximated as : N 0 u(x,y,z) ,._ Az. E _ dZpD(x,y,z; xk ) (4.Sa) k--1 g 0 k=l iv c9 w(x,y,z) _, Ax. E "_z dppD(x,y,z; xk ) (4.8c) k=l where rr (-3). [x -- xk]-[z-- Zl]

a--_ 6pD(.,y,z; *_ )

4_r [[z-rk] 2 + [y--y1] 2 + [z--zl] 215/2 (4.9a)

0 a (-3). [y- Yl]" [z- zl] (4.9b)

o_ 4".(,,_,z; _ ) = 4 _ [[_-_,]_ + b-y1] 2 + [z-zl]2 ]_/_

o ¢ [_ - _]_ + [y- y_]_- 2. [z - z_] _

0---_ C_pD(x, y,z; xk ) = 4---'_" [ [z- zk] 2 + [y- yl] 2 + [z-- zl] 2 ]5/_ (4.9c)

For a wind tunnel of constant cross-sectional area a streamwise shift of the flow field solution of a point doublet located at (zl, y_, z_) is possible (see Fig. 36b). Therefore Eqs. (4.8a), (4.8b), (4.8c)only require the calculation of dppD(X,y,z; X 1 ) aS the following relationships apply : 0 0

0--; Ce.(.,y,z; _k ) = 0--; ¢Po(*-[k-1]a_,_,z; .] ) (4.10_)

0 0 _n dflPD(_,Y,Z; Xk ) "-" c,_ dPPD(T' -- []¢-- ].]Ax,y,z; Xl ) (4.10b) yo yu 0 0 a---z ¢pD(x,y,z; xk ) = Cg-'-'z CpD(x--[k--1]Ax,y,z; a:l ) (4.10c) The numerical integration defined by Eqs. (4.8a), (4.8b), (4.8c) has to be done care- fully. Integration error due to the fact that the numerical solution of a point doublet flow field is not perfectly antisymmetric (see Eq. (4.9a)) or symmetric (see Eqs. (4.9b),(4.9c)) relative to its x-coordinate xk has to be avoided. This can be done by considering, e.g., the numerical solution upstream (x < xk) of the point doublet as exact and imposing the following conditions on the downstream part (x > x_) of the perturbation velocities : u(x,y,z) - - u(xk-Ix- xk],y,z) ; x > x_ (4.11a)

v(x,y,z) = ; • > (4.11b)

w(x,y,z) = w(zk -[x- zkl,y,z) ; z > xk (4.11c) Unfortunately, the point doublet and therefore also the semi-infinite line doublet does not have a radial symmetry. However, the velocity vector of a point or line doublet not pointing in the negative z-axis direction can easily be found by applying a coordinate system rotation. A rotation angle 7" related to the directional property of the line doublet is introduced as depicted in Fig. 37.

The relationship of coordinates and velocities between the line doublet fixed coordinate system (x,y,z) and reference coordinate system (x',y',z') is depicted in Fig. 37 . The rotation angle direction is defined such that the lift force caused by the semi-infinite line doublet points in the positive y'-axis if the line doublet is rotated by +900 . In this case we get for the transformation of coordinates from r,he reference coordinate sytem, i.e.

(z',y',z'), to coordinates in the line doublet fixed coordinate system (x,y,z) : x - x' (4.12a) y = y'.cos_" - z'.sinr (4.12b) z = y'.sin_" + z'.cosr (4.12c) The velocities components (u,v,w) are computed i:_ the line doublet fixed coordinate system. Finally it is necessary to back-transform thes_ velocity components to the refer- ence coordinate system. We get then : u I = u (4.13a) v' - v- cos r + w. :_in r (4.13b) w' = - v.sinr + w.cosv (4.13c) The singularity strength _r has the unit [ft3/sec] or [m3/sec] . The perturbation velocity field given in Eqs. (4.8a),(4.8b),(4.Sc) could be used to change a panel method code as outlined in Appendix 2 .

APPENDIX 5 CALCULATION OF THE LINE DOUBLET STRENGTH In general, it is possible to compute the strength of line doublets representing lifting effects of art aircraft by using lift and pitching moment measurements recorded during a wind tunnel test (see Ulbrich and Steinle,[7] ).

Assuming that the lifting surfaces of the wing and tail of an aircraft model are dis- cretized by using equally spaced line doublets along the 1/4-chord line of the wing and tail we get for the total lift and pitching moment (see also Fig. 38) : L - ELw(i) + y_L,(j) (5.1) i=I j=l

P = -- + -- l (5.2)

"= ]=1 where L_(i) and L,(j) are the lift contributions of the line doublets of the wing and tail.

Similar to Eqs. (2.9) and (2.11), the Kutta/Joukowstd formula may be used to connect a lift force to a line doublet strength. The application of the Kutta/Joukowski formula in a wind tunnel requires the calculation of the product poo • Uoo where poo is the free-stream density and U_o is the free--stream velocity at the location of the aircraft model. A first order approximation of the product p_- Uoo may be obtained by applying wall interference corrections to the measured reference density prel and reference velocity Uref at a model reference station (see Eqs. (5.78) and (5.82) in Ref. [10]). We then get: poo'Uoo _ prel'Ure," [1 -t- ¢'(1 --M_2el )] (5.3a) The Kutta/Joukowski formula may now be written as : L,_(i) = poo " Uoo " cr_o(i) ,_ P, ef " Vr,l " [ 1 + _ . ( 1 - M_ l ) ].a_(i) (5.3b) L,(j) = poo " Voo " _;(j) ,_ Pref " U, ef " [ I + e" ( 1 - M;2el ) ]. g;(j) (5.3c) Introducing singularity weights w_ (i), w,(j), and discrete line doublet span As_, As, on the wing and tail we get for the singularity strength : _(i) = F_ . As,_ . w_(i) (5.4a)

_;(j) = rt. As,. wt(j) (5.45)

The circulation F_ and F_ are the only unknowns in Eqs. (5.1) and (5.2) if line doublets weights are specified by the user. The weights may represent, e.g., elliptic lift distribution along the wing and tail of an aircraft model. Experimental or CFD solutions of the true lift distribution may also be used to determine weights (see also NACA Report No. 921, [25]). Combining Eqs. (5.1) to (5.45) we get: L - Fto • oq -t- Ft- or2 (5.5a) P = Fto • 171 -t- Ft- 132 (5.5b) where 1"I w

_1 = Asto. _wto(i) (5.6a)

i--1 "1%t

,_ = A,,t._ w,(j) (5.6b)

j=l 131 = Asto • ww(i).[zm,. - zto(i)] (5.7a) *_.

/32 = Ast •

_,(j). [=_,. - =t(j) ] (5.7b)

j=l Equations (5.5a) and (5.5b) are a 2 x 2 linear system of equations which can be solved easily for Fto and Ft : Fto = 1 L-132 - P'a2 (5.8a) Ft = 1 (-L)-131 q- P'_I (5.85) Finally, the singularity strength is obtained by applying Eqs. (5.4a) and (5.4b).

The line doublet strength has to be compatible with non-dimensionalized perturba- tion velocities used in the panel method code computation of the wind tunnel and wall interference flow field (see Eq. (25) and Appendix 2). Correct units for velocity, density, force, and pitching moment have to be selected if Eqs. (5.4a), (5.4b) are used. Table 4 lists units : Table 4: Units for Singularity Strength Calculation Quantity SI unit non-SI unit Velocity Tt_ / $eC fg / $CC Density Slug / ft 3 Force N - kg. m / sec 2 lbf -- Slug. ft / sec _ Moment N.m ft . lb f The singularity strength as defined in Eqs. (5.3b),(5.3c) has the unit [rn3/sec] or [ft3/sec] . Similar to Eq. (25), it is necessary to divide the strength by the reference velocity Urey • Therefore we get for the normalized strength in units [rn 2] or [ft 2] : a_(i) = a*(i) (5.9a) at(j) = a;(j) (5.9b) U_f Semispan model tests are often conducted using only the wing and fuselage of an aircraft model. In this case the circulation Ft is zero and only the lift force measurement will be used to calculate the strength of line doublets representing rifting effects of the wing. Using Eqs. (5.5a) we get for F_ : L r_ = (5.10) p_1"Ur_f" [1 + e.(1 -M_ 1 )] "al In general, the application of Eqs. (5.8a), (5.8b), (5.10) to the Wall Signature Method requires at least a one step iteration as blockage factor e can only be estimated after a successful least squares fit of the wall signature. Zero may be selected as an initial approximation of blockage factor e in Eqs. (5.8a), (5.8b), and (5.10).

APPENDIX 6 TRI-LINEAR INTERP OLATION In general, tri-linear interpolation can be used to ffud the value of a scalar function f ( xp , yp , zp ) using known values f ( z¢ , yj , zk ) , f ( zi + l , yj , z_ ) , ... , at corresponding eight grid cell corner points (see Fig. 39) .

Tri-linear interpolation requires several steps. At first we have to find grid cell indices (i,j,k) such that z_ < zp _< zi+l ; yj <_ yp <_ Yj+I ; zk <_ zp <_ zk+l • Fora constant grid cell size Az, Ay, Az, indices (i,j, k) can be computed as : i = INT zp - zl ] Ax + 1 (6.1a)

J

INT yp - y_ 1

+ 1 (6.1b)

]

k = INT[ ZpAz- zl ] + 1 (6.1c) where Xl, Yl, Zl are the starting coordinates of the entire grid.

In the next step, weighting factors of each grid cell corner point have to be found.

The line connecting a corner point with its opposite corner point is assumed to be a spacial diagonal of the grid cell. Then, the weighting factor of a grid cell corner point is computed by dividing the volume of the rectangular prism defined by point (xv, yp, zp) and the opposite corner point by the total volume of the grid cell. For example, the weighting factor of grid cell corner point (zi+l, yj, zk) is : TVi'J+l'k +1 (6.2a) Wi+ l,j,k = Ycell where

(6.2b)

V/,j+l,k+l = ABS[ [T,p -- Zi] " [yp -- Yj+I]" [Zp " Zk+l] ] V_n = ABS[ [xi+l - xi] . [yj-t-1 - yj] " [Zk-t-1 -- Zk] ] : AT. • Ay " AZ (6.2C) Finally, the functional value f(xp, yp, zp) can be interpolated as : i+1 j+l k+l f(z,,yp,zp) = _ _ _ f(xc,,y_,z._).w,_,_,._ (6.3) a=i fl=j "y=k Table 5 lists grid cell comer points and corresponding opposite comer points.

Table 5: Grid Cell Indices No. Corner Point Opposite Corner Point i, j, k i+l,j+l,k+l i,j + l,k i + l,j,k + l i+ 1,j,k i,j+l,k+l i + l,j + l,k i, j, k + 1 i,j,k+l i+l,j+l,k i,j+l,k+l i+ 1,j,k i + l,j,k + l i,j+l,k i + l,j + l,k + l i, j, k Tri-linear interpolation is applied to calculate normalized perturbation velocities of a singularity located at a point (xp, yp, zp) assuming that perturbation velocities are known for singularities located on eight comer points of a singularity grid cell.

APPENDIX 7 PARABOLIC INTERPOLATION A parabola isused to interpolate normalized perturbation velocities y(M) as a function of the real-time Mach number M . The parabola has the following form : y(M) = a.M 2 -{- b.M + c (7.1) Three coefficients a, b, c of the parabola have to be calculated. Therefore it is necessary to compute the perturbation velocity database for three discrete Mach numbers.

Three discrete Mach numbers M1 = 0.0, M2 = 0.3, M3 = 0.6 are selected and corre- sponding normalized perturbation velocities y(M1), y(M2), y(M3) are known. Then we get for the coefficients of the parabola : y(M3) - y(M1) y(M2)- y(M1) a --- (7.2a) M3- M2 M3 - M1 M2 - M1 b -" y(M2)- y(M1) - a.(M1 -b M2) (7.2b) M2 - M1 c = y(M1) - a. M_ - b. M1 (7.2c) Coefficients a, b, c have to be computed for nl • (n2 + n3) parabolas in real-time if, e.g., a test article and support system are represented by nl singularities, wall signatures are measured at a total number of n2 wall pressure ports, and a reference point grid of n3 points is selected.

APPENDIX 8 OPTIMIZATION OF THE SINGULARITY LOCATION The wall signature method calculates wall interference corrections based on a singu- larity representation of the test article and the support system. The wind tunnel test engineer has to specify the initial location of these singularities using simple rules. For example, sources and sinks representing the fuselage volume should be located along the fuselage axis such that they represent a Rankine body approximation of fuselage volume effects, line doublets should be distributed along the 1/4-chord line of lifting surfaces, and sources related to separation wake effects should be located at points where separation is likely to occur during a test.

The initial singularity representation is not unique as the test engineer has to specify the singularity location. However, it is possible to make the singularity location unique by minimizing the standard deviation of the least squares fit as a function of the singularity location.

An efficient optimization can only be achieved if the total number of independent variables of the minimization, i.e. the coordinates of each singularity, is reduced to a reasonable limit. It is also necessary to restrict the direction of change of the singularity coordinates.

During the support system calibration three groups of singularities exist, i.e., sources of the support system volume, sinks of the support system volume, and sources of the support system wake. Therefore it is possible to reduce the number of independent variables down to three, if the relative distances of the singularities of each group is kept constant during the optimization. The direction of change of the optimization is defined by the line connecting the first source and the first sink of singularities representing the support system volume.

Similarly, during the wind tunnel test of a test article five groups of singularities exist, i.e., sources of the fuselage volume, sinks of the fuselage volume_ and sources of the wing separation wake, line doublets of the wing, and line doublets of the tail. Therefore the number of independent variables can be reduced to five, if the relative distances of the singularities of each group is kept constant during the optimization. In this case the direction of change of the optimization is defined by the line connecting the first source and the first sink of singularities representing the fuselage vohune. Assuming that the 1/4-chord line of the wing and tail are ideal locations for line doublets of the wing and tail it is even possible to reduce the ntunber of independent variables to three.

An optimization algorithm based on the Method of Steepest Decent has been included in the real-time software package of WICS (see also Ref. [261).

APPENDIX 9 SELECTION OF THE SINGULARITY / REFERENCE POINT LOCATION The Wall Signature Method uses a singularity representation of the wind tunnel model to predict wall interference corrections. The singularity representation of the model is uniquely defined if TYPE, LOCATION, and STRENGTH of each singularity are known.

The STRENGTH of each singularity is derived from real-time measurements of wall signature, lift force, and pitching moment. If a propulsion simulator is used during a test it is also necessary to measure the propulsion simulator thrust and the propeller disk area.

The TYPE of each singularity is chosen by the wind tunnel test engineer. A singularity TYPE should be selected such that it represents a blockage or lifting effect of the wind tunnel model. Sources and sinks are used to represent volume and wake blockage effects of the wind tunnel model. Line doublets are used to represent lifting effects of the wind tunnel model. A sink is used to model blockage effects of a propulsion simulator in wind tunnel testing.

The LOCATION of each singularity has to be specified by wind tunnel test engineer.

LOCATIONS should be selected based on geometry of wind tunnel model. All singularity coordinates have to be provided in tunnel coordinates.

The following empirical rules will help the test engineer to make reasonable selections of the singularity TYPE and LOCATION : (1) FULLSPAN WIND TUNNEL MODEL : (1.1) Fuselage Volume : A source of weighting factor "+1.0" has to be placed on the fuselage axis approximately one mean fuselage radius downstream of the nose of the fuselage. A source of weighting factor "-1.0" , i.e. a sink, has to be placed on the fuselage axis approximately one mean fuselage radius upstream of the tail end of the fuselage.

(1.2) Wake Separation : Sources with positive weighting factors have to be placed on the wind tunnel model at locations where flow separation occurs. If a wind tunnel model is tested, e.g., in landing configuration sources of equal strength should be placed at the location of the wing flaps.

(1.3) Wing : The wing spanis divided into equal sizewing span increments. One line doublet is assignedto each wing spanincrement. Line doublet starting points are placed where the 1/4 chordline of the wing and the middle of eachwing span increment intersect.

(1.4) Tail : The tail span is divided into equal size tail span increments. One line doublet is assignedto each tail span increment. Line doublet starting points are placed where the 1/4 chord line of the tail and the middle of eachtail span increment intersect.

(1.5) Propulsion Simulator : A sourceof weighting factor "-1.0" , i.e. a sink, is placed at the center of the propeller if a turboprop engine is simulated; a sink is placed halfway between the compressorand turbine if a turbojet or turbofan engine is simulated.

(2) SEMISPAN WIND TUNNEL MODEL : (2.1) FuselageVolume : A sourceof weighting factor "+1.0" has to be placed on the fuselageaxis approximately one mean fuselageradius downstream of the noseof the fuselage. A sourceof weighting factor "-1.0" , i.e. a sink, has to be placed on the fuselage axis approximately one mean fuselageradius upstream of the tail end of the fuselage.

The z-coordinate of the sourceand sink of the semispanmodel fuselagedoesnot have to be specified. The Wall Interference Correction System (WICS) of the NASA Ames 12ft PressureWind Tunnel (PWT) implicitly assumesthat their z-coordinate is identical with the z-coordinate of the image plane surface.

(2.2) Wake Separation : Sourceswith positive weighting factors have to be placed on the wind tunnel model at locations where flow separation occurs. If a wind tunnel model is tested, e.g., in landing configuration sourcesof equal strength should be placed at the location of the wing flaps.

(2.3) Wing : The wing semispanis divided into equaisizewing spanincrements. One line doublet is assignedto each wing span increment. Line doublet starting points are placed where the 1/4 chord line of the semispanwing and the middle of each wing span increment intersect.

(2.4) Propulsion Simulator : A sourceof weighting factor "-1.0" _i.e. a sink, is placed at the center of the propeller if a turboprop engine is simulated; a sink is placed halfway betweenthe compressorand turbine if a turbojet or turbofan engine is simulated.

(3) REFERENCE POINTS : WICS computesmean wall interference corrections for sets of referencepoints. The test engineerhas to specify thesesets of referencepoints in the tunnel coordinate system.

WICS allows the test engineer to specify up to 10 independent sets of reference points.

Setsof referencepoints can be specified along the fuselageaxis and along the 3/4 chord line of the wing or tail.

A simple Expert Systemshouldbe designedin the future that assiststhe test engineer in the selectionof the singulal"itylocation and weighting factors.

APPENDIX 10 BLOCKAGE EFFECT OF A POWERED WIND TUNNEL MODEL The correct simulation of flow interference effects between the aircraft fuselage, the wing, and the engine cowling during a wind tunnel test requires the installation of a propul- sion simulator in a wind tunnel model. The blockage correction caused by a propulsion simulator has to be estimated if a significant amount of thrust is produced.

In general, a propulsion simulator is operated as a small propeller. A point doublet pointing in the streamwise direction may be used to estimate propulsion simulator blockage effects during a wind tunnel test, [22] . A more accurate modeling of blockage effects may be obtained if a semi-infinite line doublet pointing in the streamwise direction is used, [27] .

The velocity potential of a semi-infinite line doublet with a starting point at (zl, yl, zl ) is given as, [22]: - a lim ....

CLD(Z,y,z) -- 47r ==--.oo , [[z__] 2 + [y_yl] 2 + [z_zl] 213/2 (10.1) It can be shown that the velocity potential defined in Eq. (10.1) is identical with the velocity potential of a sink located at (xl, yl, zl) (see Eq. 3.1) : O" CLD(X,y,z) = --¢S(X,y,z) = 4 _r [[_-_1] 2 + [y-yl]2 + [z-zip]l/_ (10.2) Location and strength of this sink have to be specified. It is reasonable to place the sink at the center of the propeller or compressor disk of the propulsion simulator. The sink strength a may be related to the propulsion simulator thrust Tp, propeller disk area S, free-stream velocity Uoo, and free-stream density pco by applying an approximate solution of the flow around the ideal propeller given by Koning, [6] . In wind tunnel testing, free-- stream velocity and density have to be estimated by applying blockage corrections to the measured reference velocity U,e! and reference density p, ej', i.e. Uoo _ U-el (1 + e) and poo _ pr_! (1 - c M]_I) (see Eqs. (5.78) and (5.82) in aef.[10]) . Then, we get for the propulsion simulator sink strength :

F/ 1

2.

Tp /J -- _T_f _ ( 1-[- _)/ (10.3)

% = S. LvU f'(1 +e) 2 + p,.e.f.(1-e M_._I).S

J Equations (10.3), i.e. Koning's formula, may also be obtained by considering blockage effects of a lifting rotor in subsonic wind tumad testing, [27] .

Effects of a propulsion simulator on wind tunnd blockage effects can easily be in- eluded in the least squares fit of the wall signature required for the application of the Wall Signature Method. The strength of the propulsion simulator sink is known from the thrust measurement. It is only necessary to subtract the corresponding wall signature contribution from the total wall signature (see also Eqs. (13g) in Chapter 2).

The sink strength _,; as defined in Eq. (10.3) has the unit [m3/sec] or [ft3/sec]. It has to be divided by U,._I if normalized perturbation velocities as defined in Chapter 2 are used for the least squares fit of the wall signature. Finally we get : [_ 2.Tp -(1+ _)] (10.4) _, = S. (1+,) 2 + p,,_._Lj.s.(I_,MLj ) The application of Eq. (10.4) to the Wall Signature Method requires at least a one step iteration as blockage factor e can only be estimated after a successful least squares fit of the wall signature (see also Appendix 5).

APPENDIX 11 QUALITY CHECK OF THE LEAST SQUARES FIT The Wall Signature Method uses a linear least squares fit of the difference between wall signature and corresponding wall pressure port calibration to predict blockage effects of a test article. In practical applications the facility hardware cannot guarantee that good pressure measurements are recorded on all wall pressure ports at all times. However, not all pressure measurements are required to compute a blockage correction of the test article.

Therefore an efficient implementation of a wall interference correction system based on the Wall Signature Method has to check measurements on each wall pressure port.

Studies using experimental data have shown that a four-step quality check of the least squares fit is sufficient to identify unacceptable wall signature measurements without rejecting too many data points. Wall signatures obtained during the empty tunnel or support system calibration have to be inspected separately. They are used to remove orifice error, wall divergence, and wall boundary layer displacement effects.

The proposed quality check of the difference between the real-time and calibrated wall signature is done as follows (see also Fig. 40) : CHECK 1 : WALL SIGNATURE DIFFERENCE MAGNITUDE The absolute value of the wall signature difference is computed for each wall pressure port. A port is rejected, i.e. its port flag is set to zero, if the wall signature difference is larger than a specified upper bound. This bound is a function of the wind tunnel facility configuration and data aquisition hardware. An upper bound of 0.1 has been selected for the NASA Ames 12ft Pressure Wind Tunnel.

CHECK 2 : OUTLIER IDENTIFICATION The absolute value of the difference between the wall signature difference and its least squares fit is computed for each wall pressure port. A port is rejected, i.e. its port flag is set to zero, if this difference is larger than three times the standard deviation of the least squares fit.

CHECK 3 : STANDARD DEVIATION OF EACH ROW The standard deviation of each wall pressure port row is computed. A wall pressure port row is rejected, i.e. its wall pressure port flags are set to zero, if the standard deviation is larger than a specified upper bound. This bound is a function of the wind tunnel facility configuration and data aquisition hardware. An upper bound of 0.01 (perturbation velocity) has been selected for the NASA Ames 12ft Pressure Wind Tunnel.

CHECK 4 : WALL PRESSURE PORT NUMBER The total number of wall pressure ports used for the least squares fit of the wall signature difference is computed after CHECKS 1 TO 3 are applied. No wall interference corrections are computed if the total number of wall pressure ports is smaller than a specified lower limit. The lower limit of wall pressure ports is a function of the wind tunnel facility. A number of 60 wall pressure ports has been selected as the lower limit of the NASA Ames 12ft Pressure Wind Tunnel.

APPENDIX 12

LINEAR INTERPOLATION OF WALL PRESSURE PORT CALIBRATION The Wall Signature Method uses a linear least squares fit of the difference between real-time wall signature and corresponding wall pressure port calibration to determine blockage effects of a test article. A value of the wall pressure port calibration has to be found that matches real-time test conditions.

The wall pressure port calibration is usually a function of several independent calibra- tion variables. It can be a function of the total pressure PT and Mach number M at some wind tunnel reference station. The wall pressure port calibration can also be a function of the support system kinematics. In this case calibration variables describing the position of the support system during the calibration have to be introduced.

An interpolation algorithm has been developed for the WICS software package that uses linear interpolation to determine the wall pressure port calibration as a function of real-time test conditions. This algorithm fulfills reliability, performance, and accuracy requirements of WICS.

The interpolation algorithm is applicable to up to four independent calibration vari- ables. As an example, a detailed description of this interpolation algorithm for two indepen- dent calibration variables will be presented below. A detailed description of the algorithm for three and four independent variables is beyond the scope of the WICS Theory Guide.

In general, a value of the wall pressure port calibration C(X*, Y*) has to be found for real-time test conditions defined by, e.g., two independent variables X*, Y* . It is assumed that wall pressure ports are calibrated for discrete combinations of these variables. A total of "n" discrete values of the first calibration variable Xi were selected. The second calibration variable ]_,j was changed by keeping the first calibration variable Xi constant.

A total of "re(i)" discrete values of the second calibration variable Y_,j were selected.

Therefore we get the following set of discrete wall pressure port calibrations : C(Xi, Y_,.i) forl<i<nandl<j<rn(i).

The interpolation of the calibration is done in two steps. At first, calibrations are interpolated for the second calibration variable Y* . The first calibration variable Xi

is kept constant. Comparing Y* and Y_j only one of three possible cases applies (see

Fig. 41a) : Case i: Y* < _,I

U(X_) = C(X_,_=I) (12.1)

Case 2 : }_,j _< Y* _< _,j+l

c(x_,Y_,j+_) - c(x_,_,_)

Case 3 : Y* > Y_,m(i)

U(x,) = c(x_,Y_,._(_)) (12.3)

Finally, calibrations axe interpolated for the calibration variable X* by using the first calibration variable X_ and interpolated calibrations C(Xi) . Again, comparing X* and Xi , only one of three cases applies (see Fig. 41b) : Case I: X* < X1 C(X*,Y*) = C(X1) (12.4) Case 2 : Xi <_ X* <_ Xi+l

- v(x,) [x" - x, ] (12.5)

C(X',Y') = "C(Xi) + Xi+l - Xi Case 3: X* > Xn C(X*,Y*) - C(X,_) (12.6) Basic ideas and elements of the interpolation algorithm can easily be extended to three and four independent calibration variables.

Support system wall interference corrections at a reference point grid axe required if the Wall Interference Correction System of the 12ft PWT is applied to a fullspan model test configuration (see Fig. 13a) . These corrections are known as a function of calibra- tion variables. A similar linear interpolation algorithm may be used to interpolate the corrections for real-time test conditions.

APPENDIX 13 SINGULAR VALUE DECOMPOSITION The numerical method of choice for solving linear least-squares problems, i.e. Eq. (13a) or Eq. (16a), is the Singular Value Decomposition (SVD) technique, [12]. This robust numerical technique was selected for the WICS software. The application of the SVD technique to Eqs. (13a), (16a) can be summarized in three steps : Step 1 : Rewrite Eq. (13a) or Eq. (16a) as an overdetermined linear system : A_x2 X2×I = B,_xl (13.1) Step 2 : Write matrix A as the product of a column-orthogonal matrix U, a diagonal matrix W with positive or zero elements, and the transpose of an orthogonal matrix V .

Matrices U, W, and V are found using the SVD algorithm. We get : A,-nx 2 = U,-nx 2 W2x2 v'r2x2 (13.2) Step 3 : Determine the solution of the least squares problem by computing the fol- lowing matrix product : X2xl = V2x2 W-12x2 u'r2×,, B,_xl (13.3) It is important to identify elements of the diagonal matrix W, i.e. singular values of W, that are small. These singular values and their reciprocal will be set to zero if the least squares problem is iU-conditioned.

APPENDIX 14 SUPPORT SYSTEM KINEMATICS In general, it is recommended to compute real-time coordinates of singularities that represent the support system or the test article if WICS is used to predict wall interference corrections in the 12ft PWT. This reduces the standard deviation of the least squares fit of the wall pressure signature and improves the accuracy of computed blockage corrections.

The real-time coordinates of singularities are a function of the kinematics, i.e. the movement, of the support system or the test article. Three different types of support systems are presently available for testing in the 12ft PWT, i.e. the Ames Bipod, the High Angle of Attack Sting, and the Image Plane. Equations describing the movement of these support systems and of the test article in the tunnel coordinate system (see Figs. 17a,17b) have to be derived.

Ames Bipod / High Angle of Attack Sting The kinematics of the Ames Bipod and of the High Angle of Attack Sting in the tunnel coordinate system are essentially identical, if the roll angle of the High Angle of Attack Sting is kept at to = 0.0 ° (the more complex case of a non-zero roll angle is discussed in Appendix 15) . The kinematics may be described by one rotation about the axis x, = 120.71 [ft], z = 0.0 [ft] and one rotation about the axis x. = 120.71 [ft] , y = 0.0 [ft] Angles o_ and f? depicted in Fig. 42a are independent variables that describe the motion. For a roll angle _ = 0.0 °, angle a may be approximated by the pitch angle of the wind tunnel model and angle fl may be approximated by the sideslip angle of the wind tunnel model. Figures 42b and 42c show the connection between angle a, angle fl, and singularity coordinates. It is assumed that point PI(zl, yl, zl) describes the initial location of a singularity. After a first rotation about the axis x. = 120.71 [ft] , z = 0.0 [ft] the singularity moves to point P2(z2, y2, z2). After a second rotation about the axis x. = 120.71 [ft] , y = 0.0 [ft] the singularity moves to point P3(z3,y3,z3).

The final position of the singularity at point P3 is known if coordinates x3, Y3, z3 are given as a function of initial coordinates xl, Yt, zl, the pitch angle a, and the sideslip angle/? .

Coordinates of point P3 can be derived by considering triangles depicted in Fig. 42b : Xl -- X, cos 7 = (14.1) R1 Zl

8i. 7 = n---_- (14.2)

co8 (7- _) = 2:2 - 2:, (14.3)

R1 sin (7 - c0 = z2 = z__3_3 (14.4) R1 R1 The following trigonometric fomulas are known : cos (7 - a) = cos 7"cos ot + sin 7" sin ot (14.5) sin (7-a) = sin 7" cosa - cos 7" sina

(14.6)

Combining Eqs. (14.1),(14.2),(14.3),(14.5) we get: 2:2 = 2:. + (2:1 - 2:.).cos a + zl.sin a (14.7a) Combining Eqs. (14.1),(14.2),(14.4),(14.6) we get: z2 = z3 = zl.cos o_ - (zl - 2:.)'sin ot (14.7b) Considering triangles depicted in Fig. 42c we get : 2:2 -- 2:* cos 7 = (14.8) R2 Yl

sin 7 y2 (14.9)

R2 R2 2:3 -- 2:,

co_ (7 + 8)

R_ (14.10)

Y3 (14.11)

_i. (7 + 8)

R2 The following trigonometric formulas are known : cos (7+fl) = cos 7" cos fl - sin 7" sin fl

(14.12)

sin (7+8) = sin 7" cos fl + cos 7" sin fl (14.13) 8O Combining Eqs. (14.8),(14.9),(14.10),(14.12) we get: x3 = x. + (x2 - x,).cos 13 - Yl" sin ]3 (14.14a) Combining Eqs. (14.8),(14.9),(14.11),(14.13) we get: Y3 = Yl .cos 13 + (x2 -- x.).sin 13 (14.14b) The calculation of coordinates z3, y3, z3 can be summarized as follows : Calculate z3 using Eqs. (14.7a) and (14.14a). Calculate Y3 using Eqs. (14.7a) and (14.14b). Calculate z3 using Eq. (14.7b) .

Image Plane The movement of a semispan model mounted on the Image Plane can be described by a single rotation about the semispan model pitch axis (x. = 120.71 [ft], y = 0.0 [ft] ).

The pitch angle a is the independent variable that describes the motion (see Fig. 42d).

Figures 42e and 42f show the connection between pitch angle and singularity coordinates for a left and right wing semispan model. It is assumed that point Pl(xl, yl, zl) describes the initial location of a singularity. After a rotation about the pitch axis the singularity moves to the final position at point P2(x2, y2, z2). The coordinate zl of the singularity does not change in this case, i.e. z2 = zl . The final position of the singularity at point P2 is known if coordinates x2, y2 are given as a function of initial coordinates xl, yl and the pitch angle c_.

Coordinates of point P2 can be derived for a left wing semispan model by considering triangles depicted in Fig. 42e : xl - z. (14.15) cos 7 = R Y' (14.16) sin 7 = R cos ('7 -c_) = z2 - x, (14.17) R sin (7 - a) = Y_ (14.18) R Combining Eqs. (14.5),(14.15),(14.16),(14.17)we get:

x2 = x, + (xl - z,).cos a + yl"sin a

(14.19a) Combining Eqs. (14.6),(14.15),(14.16),(14.18) we get: Figure 42f shows initial and final coordinates for a right wing semispan model. In this case coordinates of point P2 can be derived by simply replacing pitch angle a by -a in Eqs.(14.19a),(14.19b) and noting that cos(-a) = cos(a) and sin(-a) = -sin(a) . We then get :

x, Jr (Xl -- x,).cos a -- yl'sin a (14.20a)

= yl. cos a + (_ -- _,). sin a (14.20b)

APPENDIX 15 HIGH ANGLE OF ATTACK STING KINEMATICS The kinematics of the High Angle of Attack Sting (HAA) for a roll angle of to = 0.0 ° is described in detail in Appendix 14. In Appendix 14 the reasonable assumption is made that the pitch angle a, of the HAA may be approximated by the angle of attack c_ of the test article and that the yaw angle fl, of the HAA may be approximated by the sideslip angle fl of the test article. The line doublet orientation angle _, of the test article, defined as the angle between the positive z-direction of the test section fixed coordinate system (see Fig. 42a) and the lift force vector, is also equal to 0.0 ° for a roll angle _ = 0.0 ° . These assumptions, however, are invalid if the roll angle _ of the HAA is not equal to 0.0 ° .

Equations are derived in this Appendix that use the angle of attack c_, sideslip angle/_, and roll angle _ of the test article to (i) compute the pitch angle a, and the yaw angle fl, of the HAA, to (ii) compute the location of test article singularities and reference points, and to (iii) compute the orientation angle of test article line doublets.

(i) PITCH ANGLE c_, AND YAW ANGLE fl, OF THE HAA : The pitch and yaw angle of the HAA may be computed in several steps assuming that angle of attack a, sideslip angle fl, and roll angle _ of the test article are known. In a first step, the unit -----4 -:+ wind vector Woo is expressed in the model coordinate system defined by unit vectors z , j , and k . Using Fig. 43a it can easily be seen that

= | - (15.1)

cos ol o cos ,_)_ t \ sin _. cos In the second step, it is necessary to reverse the roll angle rotation. Assuming that the roll axis of the test article is identical with unit vector z (see Fig. 43b), we get for the rotated unit vectors I , J, and K : I = z = 0 (15.2a) J = cos (15.2b)

(0)

\ sin = - sin _ (15.2c)

(0)

cos Now it is possible to compute the yaw angle fl, of the HAA by using a scalar product and a right-angled triangle depicted in Fig. 43c. The yaw angle fi, is defined as the angle between unit wind vector Woo and the plane spanned by unit vectors -_ and _ . Using the projection [ Woo o J ] J of the unit wind vector Woo in the direction of unit vector J and the sign convention that fi, is positive if the unit wind vector is coming from the right wing, we get : -----} --4 sin fl, = - Woo o ,1 (15.3a) or fl, = arcsin -- Woo o-'f ; 2 - - "_

] < /3. < (15.3b)

where the scalar product _ o 7 is computed by using Eqs. (15.1) and (15.2b) : -----4 --.4 Woo o J = - sin fl. cos _ + sin a . cos fi. sin _ (15.3c) Similarly, the pitch angle a, of the HAA may be computed by using a scalar product and a right-angled triangle depicted in Fig. 43c. The pitch angle a. is defined as the angle ......+ _.___+ between unit vector I and the projection of the wind vector Woo on the plane spanned by unit vectors I and K . Using the projection of vector Woo - [ Woo o J ] J in the direction of unit vector _ we get :

(woo - [Wooo j I-Y) oY

sin c_, = (15.4a) 1 - Woo o J Knowing that J 2_ K,i.e. (Woo - [Wooo J ]"#*) o]_ _ ---+ -'_ "* ----* _ "* = Woo o K , we finally get 7r 7?

_. = arcsin ; 2 < a_ < -_ (15.4b) Scalar products in Eq. (15.4b) are computed by using Eqs. (15.1), (15.2b), and (15.2c) .

(ii) SINGULARITY AND REFERENCE POINT COORDINATES : It is now possible

to computethe location of singularities and referencepoints of a test article that is mounted

on the HAA by using Eqs.(15.35), (15.45) and Eqs. (14.7a), (14.75), (14.14a),and (14.145)

of Appendix 14 . This is done as follows : (1.) Apply roll angle _ to the initial singularity or referencepoint location given by point P0 (xo, yo, z0) • Using Fig. 43d it is possible to compute the new location of the singularity or reference point at point Pl(xl, yl, zl) as : xl = z0 (15.5a)

vl = Ro . cOS [ l_ - _ ] (15.55)

zl = Ro . sin [ _ - _ ] (15.5c)

= yo/Ro and Combining Eqs. (15.55), (15.5c), (14.5), (14.6) and knowning that cos I_ sin !_ = zo/ Ro we finally get :

yl = yo " cos _ + zo . sin (15.5d)

Zl = zo'cos _ - yo'sin _ (15.5e) (2.) Rewrite Eqs. (14.7a), (14.75), (14.14a), (14.145) by replacing a, fl with c_,, fl, :

x2 = x, + (xl - ,).cos + zl.sin (15.6a)

x3 = z, + (z2 - x,).cos fl, - yl"sin fl, (15.6b) ya = yl . cos _. + (a:2 - z.).sin _. (15.6c) z3 = zl.cos _, - (_,_ - z,).sina, (15.6d) The calculation of coordinates z3, y3, za may be summarized as follows : Calculate z3 using Eqs. (15.6a) and (15.6b) . Calculate y3 using Eqs. (15.6a) and (15.6c) . Calculate z3 using Eq. (15.6d) .

(iii) LINE DOUBLET ORIENTATION ANGLE _, : In general, the line doublet orientation angle _, of the test article(identical with angle r in Appendix 4) is defined as the angle between the positivez-direction of the test section fixed coordinate system (see Fig. 42a) and the lift force vector. Angle _0, can easily be computed using a unit vector that is parallel to the positive z-direction of the test section coordinate system and a unit vector that is parallel to the lift force vector.

A unit vector Z that is parallel to the positive z-direction of the test section co- •----4 --_ ordinate system may be obtained by rotating unit vector K in the I - K plane (see Figs. 43c, 43e) . The selected rotation angle is equal to the pitch angle c_, of the HAA given in Eq. (15.4b) . Using Fig. 43e we then get : K - tan _, • I (15.7a) K - tan _, . I where K - tana, • I = - sin_o (15.7b) cos _o - tan _, I = V / 1 + tan 2 _, (15.7c) A unit vector W that is parallel to the lift force vector may be obtained by computing the cross-product Woo × 3 using Eqs. (15.1) and knowing that -*"

= (0; 1;0) (see

also Fig. 43a) . We then get : ---* Woox 3 W = = (15.8)

_

Woo x 3 \ cos a Finally, the line doublet orientation angle _, may be obtained by computing the scalar ----4 ---4 product of unit vectors W and Z We get : o -_ = cos _, (15.9) Combining Eqs. (15.7a), (15.7b), (15.7c), (15.8), and (1.5.9) we obtain: _o, -- arccos W o Z ; _o < _" (15.10a) _, = 2 _ - arccos o Z ; _o > r (15.10b) where ---4 sin _ • tan _, + cos _ • costa WoZ = (15.10c) _/1 + tan 2 _,

APPENDIX 16 PITCHING MOMENT LEVER ARM In Appendix 5 equations are derived that compute the singularity strength of a line doublet of a wind tunnel model by using the measured lift force and pitching moment.

The pitching moment lever arm is approximated in Eq. (.5.2) as the difference between the x-coordinate of the pitching moment reference axis and the x-coordinate of the line doublet starting point (see Fig.38). This approximation, however, is not valid anymore if a wind tunnel model is tested at a high angle of attack. More general equations are derived in this Appendix that use the pitching moment axis and the lift force axis to compute the pitching moment lever arm of a line doublet.

In general, the pitching moment lever arm of a line doublet is defined as the distance between the pitching moment axis and the lift force axis of a line doublet (see Fig. 44) .

Therefore a mathematical description of each axis has to be found in order to compute the pitching moment lever arm.

(i) PITCHING MOMENT AXIS EQUATION : The pitching moment axis equation may be found by using the initial coordinates of the balance center in the tunnel coordinate system (see Fig. 17a, 17b). Assuming that X,nr, Yrn,', Zm,- are the initial coordinates of the bMance center (i.e. angle of attack, sideslip angle, and roll angle of the test article are equal to zero), it is possible to introduce an initial unit pitching moment axis vector -_ as follows : "_ = _22- _ (16.1a) where

A1 = / 06.1b)

\ z._r / For a fullspan model point vector A2 is defined as : 2 = y,,_ + 1.0 Z vn r For a left wing semispan model point vector A2 is defined as :

(16.1d)

A2- |

f Xrrlr / \ z,_,. - 1.0 For a right wing semispan model point vector A9 is defined as :

= l (16.1e)

\ z_,- + 1.0 In the next step, equations describing the kinematics, i.e. the movement of the model support system, have to be applied to the points described by vector _ and _ (see Appendix 14 and 15). Point vector A1 moves to point vector -_ and point vector X22 moves to point vector C, . After points A1, _ are moved to their new location, it is possible to describe the unit pitching moment axis vector D as : .=.=4 D = C, - _ (16.2) ...=@ A point P on the pitching moment axis may now be described by using vectors and an independent variable A : ----4 -----4 ---=4 P(_) = C + A. D (16.3) (ii) LIFT FORCE AXIS EQUATION : The lift force axis equation of a line doublet may be found by using the tunnel coordinates of the line doublet starting point in com- bination with the line doublet orientation angle _, (see also Appendix 4 and 15) . After the application of equations describing the support system kinematics, the line doublet starting point has moved to a location described by point vector E . The line doublet orientation is given by a unit lift vector F in the tunnel coordinate system : F = sin _o, (16.4) \ cos 9, For a left wing semispan model 9, = rr/2, for a fight wing semispan model _, = -r/2 .

Similar to Eq. (16.3), a point _ on the lift force axis may be descibed by using vectors E, F, and an independent variable/_ : _( --'* .._, #) = E + ju. F (16.5)

(iii) PITCHING MOMENT LEVER ARM : The pitching moment lever arm of the

selectedline doublet is found by minimizing the squareof the distance of points located on

eachaxis (seeFig. 44). The square of the distance of two selectedpoints may be computed

using a scalar product : D2(A,/_) -- ['_(A) - _(/_)] o ['_(_) - _(/_)] (16.6) Using Eqs. (16.3), (16.5) and applying the product rule to Eq. (16.6) we get : = 2-[ "_(A) - -Q(/_) ] o _ (16.7a) a_ c9 D2(A,/_)

= 2.[ _'(_) - _(,)] o [- _'] (16.7b)

The variables A,_i, and #,hi,, of the two closest points are found by setting Eqs. (16.7a), (16.7b) to zero. We get : (16.8a)

(16.8b)

Using Eqs. (16.3), (16.5), it is possible to write Eqs. (16.Sa), (16.Sb) as a 2 × 2 system of linear equations : In general, D _L F or D o F = 0 . Therefore, we get for A,'ni,_ and #mi,, of the two closest points : Ami, ___ [ E - -_] o D (16.10a) -----4 DoD

[E - C]o r

(16.10b) -FoP Finally, using Eq. (16.6), we get for the pitching moment lever arm : D(A,_,/_,_,_) = _/[-_(A,_,,,) - _(/_,_,,_)] o [-P(A,n,,_) - -_(/_,n,,,)] (16.11) The following two conditions may be used to determine the correct sign of the pitching moment caused by a line doublet :

[_ × V]o[_(_..,.)-_(_,_,_)] > o _ "+"

[ D x _] o ['_(_,,_i,,) - Q(p_I,_)] < 0 ===_ "-" These conditions ensure that the pitching moment is positive if the lift force axis of a line doublet is upstream of the pitching moment axis (see also Eq.(5.2) and Fig. 38).

9O

APPENDIX 17

INCLINATION OF FORCE AND MOMENT VECTORS Corrections to the lift, drag, rolling moment, and yawing moment coefficient are caused by the wall interference induced inclination of corresponding force and moment vectors.

These four corrections may be estimated by describing the inclination of the force and moment vectors using the induced mean angle of attack correction &] at the 1/4-chord line of the wing (see also Ref. [181, p.17).

LIFT AND DRAG COEFFICIENT CORRECTION The lift and drag coefficient correction caused by the inclination of the lift and drag force may be estimated by using the resultant of the uncorrected lift and drag force vectors.

Using Fig. 45a we get for the resultant vector in the x-z coordinate system : = D' + - L' where Dr is the magnitude of the uncorrected drag force and L I is the magnitude of the uncorrected lift force experienced by the wind tunnel model.

Unit vectors i , 3 in the free-stream direction and perpendicular to the free-stream direction of the wind tunnel flow field have to be specified as they are used to define the lift and drag force. These unit vectors are a function of the mean angle of attack correction _ at the 1/4-chord of the wing. They may be written in the x-z coordinate system as : -.-.':*._ ( cos _ ) (17.2a) z -- ksin a%

_ (17.2b)

3 - \ cos at } The mean angle of attack correction a"i may be computed by calculating the local angle of attack correction at equally spaced reference points along the 1/4-chord line of the wing. We then get : N 1/4-chord line ==_ _i = .N (17.2c) Ej=I w(j) where N is the number of reference points, w is the weighting factor of a reference point (a function of the lift distribution of the wing), and cq is the local angle of attack correction at a reference point. Now it is possible to compute the lift and drag force vectors as : -- "- 3 (17.3a)

[ ]

D - D , _o-_ --+" - = , (17.3b) where ""* "-*" D' sin _ + L' cos o_ L - Ro 3 = - • • (17.4a) -*" D' L' D = -'_o , = "cosffi + .sinffi (17.4b) The lift and drag coefficient correction may now be written as : L - L' ACL "-- CL -- C'L = qoo"-S (17.5a) D - D' ACD = CD -- c_9 = qoo" S (17.5b) where qoo is the free-stream dynamic pressure and S is the reference area of the wind tunnel model. A first order approximation of the free-stream dynamic pressure may be obtained by using Eq. (1.15) in Appendix 1. Using the test section reference density, velocity, and Mach number we get : qoo = qrey [ 1 + (2- MLy).

(17.6a) -- Prey 2 [1 + (2 - Me%y).

-- 2 U¢'eY The mean blockage factor _ has to be computed as a weighted average of local blockage factors e that are computed at a set of user selected reference points. We get : = F_,_=I w(j). e(j)

!,, (17.6b)

Ej: o(j)

where N is the number of reference points, w is the weighting factor of a reference point, and e is the local blockage factor at a reference point.

Important remark: The reference points used to compute the mean blockage factor

in Eq. (17.6b) axe not necessarily identical with the reference points used to compute the mean angle of attack correction in Eq. (17.2c) I Combining Eqs. (17.4a), (17.4b), (17.5a), and (17.5b) we get for the lift and drag coefficient correction : - D'.sina_ + L'.[cos_ - 1] ACL = -- (17.7a) qo_ • S D' . [ cos _ - 1] + L'. sin ACD -- -- (17.7b) qoo " S Finally, corrections have to be added to the uncorrected lift and drag coefficients.

Then we get : CL,c "- eL,uric "_- ACL (17.8a) CD,c = CD,ur, c _L ACD (17.8b) ROLLING AND YAWING MOMENT COEFFICIENT CORRECTION The rolling and yawing moment coefficient correction caused by the inclination of the rolling and yawing moment vector may be computed by using the resultant of the uncorrected rolling and yawing moment vectors. Using Fig. 45b we get for the resultant vector in the x-z coordinate system :

- r' = (:v.9)

-- + _y, where R' is the magnitude of the uncorrected rolling moment and Y' is the magnitude of the uncorrected yawing moment experienced by the wind tunnel model.

Similar to Eqs. (17.3a) and (17.3b), it is possible to combine Eqs. (17.2a),(17.2b) with Eq. (17.9) . Then, we get for the rolling and yawing moment vectors : R -R "-_" "* -_" = , = o i , (17.10a)

Y = -y = (17.10b)

[

where R -_ -_" R' ^ Y' ^ (17.11a) - o _ = ° cos oq -t- ° sin oq --> - R' • Y'.

Y = --_o 3 = sin _ + cos_i (17.11b) The rolling moment and yawing moment coefficient correction may now be written as : R-R' A_ = ct - c_ = qoo.'S.b/2 (17.12a) y _ yt ' -- (17.12b) A'O"j, = c,_ - c,, q_ • S. b/2 where qoo is the free-stream dynamic pressure, S is the reference area of the wind tunnel model, and b is the wing span.

Combining Eqs. (17.11a), (17.lib), (17.12a), and (17.12b) we get for the rolling and yawing moment coefficient correction : R' . [ cos a'_ - 1] + Y' . sin _ AW = _ (17.13a) qoo" S. b/2 - R'.sin_ + Y'.[cos&_ - 1] A_-_ = _ (17.13b) q_o. S. b/2 Finally, corrections have to be added to the uncorrected rolling and yawing moment coefficients. Then we get : ct,¢ = ct._nc + A'_ (17.14a) c,_,_ = c,_,_,,¢ + A_--_ (17.14b)

APPENDIX 18 PITCHING MOMENT COEFFICIENT CORRECTION Wall interference induced angle of attack corrections are not constant in the test section of a wind tunnel (see Fig. 24c). Therefore, it is necessary to compute a pitching moment coefficient correction that will account for the differencebetween the mean and local angle of attack correction in the spanwise and chordwise direction of the wing of the test article.

In general,the corrected pitching moment coefficient CM, c of the complete test article configuration may be written as : (18.1) CM,c --" CM,unc "Jr- ACM1 -Jr- ACM2 where CM,,_,_, is the uncorrected pitching moment coefficient, ACM: is the pitching moment coefficient correction due to the difference between the mean and local angle of attack cor- rection in the spanwise direction of the wing, and ACM2 is the pitching moment coefficient correction due to wall interference induced streamline curvature in the chordwise direction of the wing.

PITCHING MOMENT CORRECTION Acm: : A pitching moment coefficient correction ACM: is due to the difference between the mean mad local angle of attack correction in the spanwise direction of the wing. It may Mso be interpreted as a correction resulting from the lateral shift of the spanwise center of lift caused by wall interference effects.

The pitching moment coefficient correction ACM1 may be computed by using Eq. (3.58) given in Ref. [10], which is essentially identical with Eq. (34) in Ref. [18] . Using the no- tation introduced in the WICS Theory Guide we get : 2 r 3 Y / _d0.75,y) - W / '_,,,(Y) • ACM: = F . 180 b I. J F = (_r/2). A 2. (OcL/ao O. tan _o.2_ (18.2b) r.h + 2.(C%L/OO 0 where A is the aspect ratio of the wing, COCL/Cgo_ is the lift curve slope in [1/rad], _o0.2s is the sweep angle of the 1/4-chord line of the wing, ai(0.75, y) is the local angle of attack correction in [deg] along the 3/4-chord lineof the wing, _'_ is the angle of attack correction in [deg] of the wing, w_ is a weighting factor derived from the lift distribution (see also Eq. (19.2c) in Appendix 19), y isthe spanwise coordinate, and b is the span of the wing.

In general,Simpson's Rule may be applied to Eq. (18.2a) ifACMI has to be determined for a given wing. However, itis also possibleto simply discretize Eq. (18.2a) as long as the local angle of attack correction ai(0.75, y) is computed at a large number of equally spaced points along the 3/4-chord lineof the wing. Assuming that a total number of N equally spaced reference points were selected along the semispan of the wing (i.e. b/2 = N. Ay), we get the approximation : d _ T "Ay =: N Using the abbreviation ww(y(j)) =- w_(j), Eq. (18.2a):may be discretized as: N 180 b N j=l After rearranging terms we get : F. r 1 N

b. 9o W- - • w.(j) y(j) (18.3 1

j=.l Sometimes it is necessary to distribute a total nualber of M equally spaced reference points along the complete span of the wing (i.e. b = 1J • Ay). Then, using the absolute value of y(j), Eq. (18.3c) becomes: F • rr 1 M

ACM1 _ b 90 " _ _ [_,(0.75,j)--<] w,(j), lY(J)I (18.4)

j--1 WICS uses Eq. (18.4) to determine the pitching moment coefficient correction ACM1 that is caused by the wall interference induced shift of the center of pressure of the wing.

PITCHING MOMENT CORRECTION ACM2 : A pitching moment coefficient correction ACM_ due to the streamline curvature along the chord of the wing has to be computed. Classical theory has shown that the magnitude of this correction may be approximated by the camber of a two--dimensional circularly cambered airfoil (see Ref. [18], p.17/18 and Ref. [28], p.91).

In general, the angle of attack correction _ is not constant along the chord of a wing.

This phenomena may be described as a wall interference induced streamline curvature that is experienced by the wing. A constant streamline curvature may be found if we assume a linear change of the angle of attack correction along the chord (see Fig. 46a) . Then, a linear function 6cq may be defined as (see Fig. 46b) : 7r

_(_) = [ _(_) - _(0.50)] • _ [raa] (18.5)

The linear function 6(_i may be related to a two-dimensional circularly cambered airfoil,[28], because its derivative with respect to _ is the streamline curvature. The pitching moment coefficient about the 1/4-chord point of a circularly cambered airfoil is given in Ref. [28] as:

c.(0.25) = _ _. _L (18.6)

C where f/c is the camber of the airfoil (see Fig. 47). The camber is related to the slope of the circular arc at the 1/4 and 3/4-chord point. The slope is related to the upwash along the chord. Then, using Fig. 47, we get : tan _ _ fl = -- •

[ ( )o75]

(18.7a) -- °

[ _o,(o.25) - ,_,_(o.75) ]

Using triangle ABC in Fig. 47 we also know that

2 f

tan_ _ Z = (18.7b)

C Combining Eqs. (18.7a), (18.7b) we get for the camber as a function of the streamline curvature :

f 1

-_- = -_- • [ 6ai(0.25) - ecq(0.75)] (18.8) Using Eq. (18.5) we also know : 7V 6ai(0.25) -- [cq(0.25) -- ai(0.50)] 180 [rad] (18.9a) 7V 6oL/(0.75) = [o_i(0.75) - cq(0.50) l 180 [rad] (18.9b)

Combining Eqs. (18.6), (18.8), (18.9a), (18.9b) we finally get for the pitching moment

coefficient of the two-dimensional circularly cambered airfoil as a function of the difference between the angle of attack correction on the 1/4 and 3/4-chord point :

cu(0.25) = _ _ (18.10)

-7- ' [_(0.75) - _,(0.25)] • 18--6

In the next step, it is necessary to integrate the pitching moment coefficient in the spanwise direction of the wing. Assuming that the sweep angle to of the wing is 0.0 ° (see Fig. 48), the local pitching moment coefficient of a wing segment of width dy about the 1/4-chord line may be written as

d P(y) (lS.11)

cM(0.25,y) = qo¢ • c(y) d _(y)

where d P(y) is the infinitesimal pitching moment acting on the wing segment, qoo is the free-stream dynamic pressure seen by the wing segment, c(y) is the local chord, and d S(y) is the infinitesimal area of the wing segment. The infinitesimal area d S(y) of the wing segment is a trapezoid and may therefore be expressed as : d S(y) = c(y) d y (18.12) Introducing the mean geometric chord as a function of the wing reference area S and the wing span b we know : S = T (18.13) In the next step we combine Eqs. (18.11), (18.12), divide both sides of the resulting equation by the mean geometric chord -5 and the mean aerodynamic chord c', introduce Eq: (18.13) on the right hand side of the equation, and rearrange terms. Then, we get : cM(O.25, y) c2(Y) 1 d P(y) d._ -_- • dy = - (18.14) qoo .d.S The pitching moment coefficient correction ACM_ for 0.0 ° sweep angle may be obtained by integrating both sides of Eq. (18.14) over the whole wing span : ACM_ = cM(O.25, y) y := d P(y) (18.15)

[b/2 c2(y) d fo P

J-b/2 d • "5 b q oo d • "S

The pitching moment coefficient of a swept wing may be approximated (see Ref. [29])

by using the product of the pitching moment coefficient of the unswept wing and the cosine of an average sweep angle _. For a circular cambered airfoil the sweep angle at the 1/2-chord is selected. We get :

hcM,(_ # 0.0 °) = cos Vo.50 Ac_(_ = 0.0°) (18.16)

Finally, combining Eqs. (18.10), (18.15), (18.16) we get for the pitching moment coefficient correction ACM2 due to streamline curvature : ACM2 ----" 7r. cos to0s0 ./b/2 7r c2(y) d y (18.17)

4 J-bi_ [ _(0.75, v) - _(0.25, y) ]- 1--g6 e. _ b

The variable transformation # = 2y/b is introduced in an effort to compare Eq. (18.17) with an equation reported in the literature. Assuming that the lift distribution is symmet- ric, Eq. (18.17) may then be written as :

z . cos _o so [1 _ c2(_) d

ACM2 (18.18) 4 "Jo [a_(0.75,/_) - cq(0.25,#)]- 1-_ " d.e Equation (18.18) agrees with Eq. (35) in Ref. [18].

Assuming that local angle of attack corrections are computed on a total of N equally spaced reference points along the 1/4-chord line of the wing and N equally spaced reference points along the 3/4-chord line of the wing (i.e. b/2 = N. Ay), we get the approximation : d_ _ A/_ = 2Ay = 1 (18.19) b N Combining Eqs. (18.18), (18.19) we get an approximation of the pitching moment coeffi- cient correction due to streamline curvature : _r. cos _0.50 _" N [ ] c2(j) 1 (18.20a) ACM2 _ 4 ._-_ ai(0.75,j)--a,(0.25,j) "1--8-0" c'- _ g j=l Sometimes it is required to distribute a total number of M equally spaced reference points along the 1/4- and 3/4-chord line of the complete span of the wing (i.e. b = M. Ay).

Then Eq. (18.20a) becomes : M _CM2

.cos 0 0 Z 1 c2(j) 1 (1820b)

4 "1--8-6" d • _ M j=l Finally, introducing compressibility effects in Eq. (18.20b) (see Ref. [10], Eq. (3.59)), we get : M ACM2 _ G • M Z [c_/(0.75,j)- a/(0.25,j)] c2(j) (18.21a) j--1 where G = 7r. cos _oo.so . _r 1 (18.21b) 4. V/1 - M 2 • cos2_oo.5o 180 d Equation (18.21a) and (18.21b) are used by WICS to determine a pitching moment correction due to streamline curvature.

CALCULATION OF PITCHING MOMENT COEFFICIENT CORRECTIONS : WICS uses Eq. (18.4) and Eqs. (18.21a,b) to compute pitching moment coeitlcient corrections due to the difference between the mean and local wall interference corrections in the spanwise and chordwise direction of the wing of a wind tunnel model. Input variables required for the application of these equations axe classified as follows : (1) variables that depend on the geometry of the test article (A, c, c', 5, too.25, to0.50), (2) variables that depend on the lift of the wing (w,,(j), OCL/Oa ), m_d (3) variables that are a result of the calculation of mean and local wall interference corrections (_'_, cq(j); ai(0.25,j), c_(0.75, j)). For best results it is recommended to determine the lift curve slope OCL/(9o_ by using the experimental cL -- o_ curve.

APPENDIX 19 ROLLING MOMENT COEFFICIENT CORRECTION An additional rolling moment coefficientcorrection Act has to be found whenever wall interferenceeffects are the resultof a non-synunetric lift distributionon the wing. A non-symmetric lift distributionmay be caused by deflected ailerons or non-zero sideslip angle of the wind tunnel model.

In general, the corrected rollingmoment coefficient ct,c of the complete test article configuration may be written as : ct,¢ = cz,,.,,-,¢ + Act (19.1) where ct,,.,,.,_ is the uncorrected rolling moment coefficient and Act is the rolling moment coefficient correction due to wall interference effects.

The calculation of the rolling moment coefficient correction is done in three steps. At first, it is necessary to find a reasonable non-symmetric lift distribution and line doublet representation of the wing that satisfies the measured lift force, pitching moment, and rolling moment of the test article. Then, angle of attack corrections along the 3/4-chord of the wing are computed using the non-symmetric line doublet representation of the wing and the wall signature method. Finally, the rolling moment coefficient correction Act is determined by integrating the angle of attack correction distribution.

LIFT DISTRIBUTION AND LINE DOUBLET REPRESENTATION : The calculation of a rolling moment coefficient correction is only possible if a non- symmetric lift distribution of the wing is found that matches the measured lift force, pitching moment, and rolling moment of the test article. This lift distribution may be obtained in two steps by using results presented in Appendix 5.

At first, a symmetric lift distribution of the wing is assumed. The corresponding circulation F* of the wing is written as (see Appendix 5 for more detail) : r*(y) = r_ w_(y) (19.2a) where rw is only a function of the lift force and pitching moment measurement on the test article and w_ (y) is the weighting factor of the selected symmetric lift distribution. The weighting factor w,_ (y) has to fulfill the condition : jff/2 w,_(y) d y = b (19.2b) b/2 The weighting factor ww (y) of an elliptic lift distribution (b - wing span) is given as : The total lift caused by the symmetric lift distribution may be computed by integrating the circulation along the wing span. Using Eqs. (19.2a), (19.2b) we get : bl_

L(r') = poo .uoo. r*(y) d y = poo -uoo. F,_. b (19.3a)

J-b 2 Similarly, we get for the total pitching moment caused by the symmetric lift distribu- tion : b12

P(r*) = pop-up. A_(y) r'(y) d

J -b/2 (19.3b) b/2 = poo .uoo.r_. A_.(y) • w,_(y) d y ,_-b12 where Ax(y) is the pitching moment arm.

In a second step, a weighting factor Aw is applied to the symmetric lift distribution such that the total rolling moment caused by the resulting non-symmetric lift distribution equals the measured rolling moment R. The weighting factor Aw must be introduced such that the total lift force and pitching moment of the nort-symmetric lift distribution equals the lift force and pitching moment of the symmetric lift distribution. This weighting factor Aw may be defined as follows (see Figure 49) :

r"(_) = r'(_) [1 + Aw ] ; y < 0 (19.4a)

r**(y) = r*(y) ; y = 0

(19.4b)

r*'(y) = r'(_) [1 - _w] ; _ > 0

(19.4c) where F** is the circulation of the non-symmetric lift distribution of the wing.

Combining Eqs. (19.2a), (19.4a), (19.4b), and (19.4c) it is possible to express the

non-symmetric lift distribution as a function of the symmetric lift distribution. Similar to Eq. (19.2a), we get for the circulation of the non-symmetric lift distribution :

r'*(y) = r_ • D"g'_(y) (19.5a)

where

W'j(y) = w_,(y) ; y = 0 (19.5c)

W-_(y) = w,,,(y) [1 - Aw ] ; y > 0 (19.5d) In the next step it is necessary to show that the lift and pitching moment caused by the non-symmetric lift distribution F** is identical with values obtained from the symmetric lift distribution F*. Integrating the circulation F** of the non-symmetric lift distribution along the wing span and introducing Eqs. (19.5a) we get : bl2

L(r*') = poo-uoo. F'*(y) d y

J-_/2

(19.6a)

bl2 = poo • uoo. r_ • _'7_(y) d y J-hi2 Using Eqs. (19.2b),(19.5b),(19.5c),(19.5d) and knowing that w_(y) is symmetric to y = 0 we get :

b/2 _--J(y) d y

b/2 - _-_(y) d y q- W-_(y) d y

lj ¢"

b/2 Jo (19.6b)

w_(_) • [ 1 - A_ ] d

= _,,,(_) o [ i + Aw ] d _, +

fj

_ /2 .,o

_(y) d y - w_(y) d y = w,,,(y) d y + Aw •

b/2 ] J0 J-b 2 b/2 = b Thus, combining Eqs. (19.6a), (19.6b) and comparing the result with Eq. (19.3a) we con- clude that : L(F**) = poo .uoo.r_. b = L(F*) (19.6c) Similarly, we get for the pitching moment of the non-symmetric lift distribution : b/2

e(r") = poo- _oo- _(_) • r"(y) d y

J-b/2 (19.7a) hi2 = poo "uoo" r=. _z(y)._-;_(y) d y J-b/2 Using Eqs. (19.5b),(19.5c),(19.5d) and knowing that Az(y). w,_(y) is symmetric to y -- 0 we get :

A_:(y) . _--¢_(y) d y +

[1 - Aw] dy (19.7b) b/2

w,_(y) d _, - /',x(y)

JO Thus, combining Eqs. (19.7a), (19.7b) and comparing the result with Eq. (19.3b) we con- clude that : b/2

P(r**) = poo.uoo-r=. Az(y) • w=(y) dy = P(r*) (19.7c)

,_-b/2 In Eqs. (19.6c) and (19.7c) it was shown that the introduction of the weighting factor Aw does not change the lift force and pitching moment. It remains to determine Aw as a function of the rolling moment measurement. This may be done as follows : The calculation of the angle of attack correction based on the wall signature method requires that the non-symmetric lift distribution of the wing has to be translated to a corresponding non-symmetric line doublet representation of the wing. We assume that a

total number of nw equally spaced line doublets are selected along the 1/4-chord line of

the wing. Then, the strength o'U(y(i)) of a line doublet at the spanwise location y(i) is defined as (see also Eq. (5.4a) in Appendix 5) :

_,(y(i)) = A_ r_ • _-_(u(i)) (19.8_)

where W"_(y(i)) is given by Eqs. (19.5b), (19.5c), (19.5d) and b As_ = (19.Sb) nw

b [ 1] 1o8c, y(i) = 2 + As_o i-- -_-

The total rolling moment caused by the non-symmetric line doublet representation of the wing may now be written as : -R = E L,_(i) . y(i) (19.9a) i=1 where L,_(i) is the lift force caused by a single line doublet at the spanwise station y(i).

Equation (19.9a) may also be written as the sum of the contributions from the left and right wing. For an even number nt0 of line doublets we then get : -R = E L,_(i) y(i) + E L,,,(i) . y(i) (19.9b) i=1 i--n_/2+l Applying the Kutta/Joukowski formula to the wing span increment As,_, introducing Eq. (19.5a), and using the abbreviations F*'(y(i)) -- F'*(i) and _-'_(y(i)) - W'_(i) we get: L_,(i) = p_ .u_. Asw. r**(i) = p_ .u_. As_.F,_._g_(i) (19.10) Combining Eqs. (19.5b), (19.5c), (19.5d), (19.9b) and (19.10) we get : n.12 -R = [I+A_].

Z w_(i). _(0

poo • uoo • Asw • Fw i=1 (19.11) _"b tl;

+[l-Awl. ww(i) . y(i)

E

i=n,_/2+l The weighting factor w,_(i) of the symmetric lift distribution is symmetric to the root of the wing (y = 0). Therefore, we know :

v(i) = - (19.12)

i=rtw[2+l i=it Combining Eqs. (19.11) and (19.12) we get: n./2 -R = 2 • Aw • E Ww(i). y(i) (19.13) poo • uoo • Asw • I'w i=1 After rearranging terms in Eq. (19.13) we get for the weighting factor Aw : (19.14a) V'n,,/2 2 • poo "uoo" Asw. r_o z-..,i=l ww(i) y(i) where the number nw of line doublets of the wing is even. Assuming that the number nw of line doublets of the wing is odd, we get : Aw = -R (19.14b) 2 • j0oo "t too" Asw, I'_w £-_i=1 ww(i ) y(i) Finally, non-symmetric line doublet strength values trw*(y(i)) that satisfy the mea- sured lift, pitching moment, and rolling moment may be determined by using Eqs. (19.5b), (19.5c), (19.5d), (19.8a), (19.14a) and (19.14b).

The angle of attack correction tr_ along the 3/4-chord line of the wing may now be computed by applying the wall signature method as the line doublet representation of the wing is known (for more detail see Chapter 2) .

ROLLING MOMENT COEFFICIENT CORRECTION : It is now possible to compute a rolling moment coefficient correction due to wind tunnel wall interference effects as the angle of attack correction along the 3/4-chord line of the wing is known. Figure 50 shows the angle of attack correction distribution ai(0.75, y) as a function of the spanwise coordinate y of the wirg. Using Fig. 50, we get for the infinitesimal rolling moment of a wing span increment d y : dcL(y) . _(0.75, y) • q:_ d S (19.15) dR = (-1) Y " dcr The infinitesimal area d S of the wing span element may be written as : d S = c(y) • d Y (19.16) where c(y) is the local chord of the wing. Combining Eqs. (19.15) and (19.16) and integrat- ing the corresponding equation in the spanwise direction, we get for the rolling moment due to wall interference effects : [_,/2 d cL(y) . 0_i(0.75, y) • c(y) • d y (19.17) A R = qoo a-b 2 (-y) dot The rolling moment coefficient correction Act has to be added to the uncorrected rolling moment coefficient (see Eq. (19.1)). Therefore we get: AR Ac, -- (-1) • qo _ "S. b/2 (19.18) Finally, combining Eqs. (19.17), (19.18), the rolling moment coefficient correction is ob- tained as : 2 [b/2 d cL(y) . ai(0.75, y) c(y) . d y (19.19a) Act = "S • b a-b/2 y " d o_ In general, Simpson's Rule may be applied to Eq. (19.19a) if Act has to be determined for a given wing. However, it is also possible to simply discretize Eq. (19.19a) as long as the local angle of attack correction cq is computed at a large number of equally spaced points along the 3/4-chord line of the wing. Assuming that a total number of M equally spaced reference points were selected along the complete span of the wing (i.e. b = M. Ay and dy ,_ Ay = b/M), we get the approximation : Ac, _ -_ " M2 1 EM Y(J) d d o_cL(j) . ai(0.75,j) c(j) (19.19b) j=l where -b b [ 1 (19.19c) Y(J) = 2 + -M" [ j 2 The application of Eq. (19.19b) requires an estimate of the local lift curve slope dcL(j)/do_ at the spanwise station of a reference point. This estimate may be obtained by

using the Kutta/Joukowski formula. For an infinitesimal wing span increment dy we get

for the lift dL using the non-symmetric lift distribution :

d L = poo uoo r"(y) • d y (19.20a)

The local lift coefficient is defined as : dL cz(y) = poo/2" u L c(y) . d y (19.20b) Combining Eqs. (19.20a), (19.20b) we then get :

eL(y) = 2 r'*(y) (19.21)

uoo c(_)

Combining Eq. (19.5a) with Eq. (19.21) and differentiating both sides of the resulting equation with respect to angle of attack c_ we get : d CL(y) 2 • _ww(Y) d r w

(19.22)

d a uoo e(y) d a

Introducing the abbreviation y(j) - j, we get at the di,'_crete spanwise position j : d eL(j) 2 _-'_(j) d Fw (19.23)

d _ _ • _(j) d

Assuming that the-total lift of the model is charged to the wing, we get from Eq. (19.6c) : L poo u,_ F_ b cL poo 2 . -- . Uoo 2 (19.24) Differentiating both sides of Eq. (19.21) with respect to the angle of attack a and rearranging terms we get : d F,_ _ uoo • S d cL -- (19.25) dot 2 • b da Thus, combining Eqs. (19.23), (19.25) we get the following approximation of the local lift curve slope :

d _L(j) _-_(j) _ d _L

"_ (1926) d a c(j) b d oL

Finally, combining Eqs. (19.19b), (19.26) we get for the rolling moment coefficient

correction due to wall interference effects : 2 dcL 1 M Act ,_, b do_ U E Y(J) " "w-'_(J) _i(0.75,j) (19.27) j=l where dcL/da is the lift curve slope of the wing in [1/deg], b is the wing span, M is the number of reference points in the spanwise direction of the wing, y(j) is the spanwise coordinate of reference point "j", _'_-_(j) is the weighting factor of the non-symmetric lift distribution of the wing at reference point "j", and ai(0.75,j) is the angle of attack correction at reference point "j" in [deg]. An interesting interpretation of Eq. (19.27) is possible if we introduce the absolute value of y(j) and assume that y(j) # O. Then Eq. (19.27) becomes : Acl "" M E Y(J) ly(J)l _j(9). dcL . _d0.75,j) (19.28) "" lY(J)[ b / 2 do_ j--1 Equation (19.28) may be interpreted as the arithmetic mean value of weighted rolling moment coefficient contributions along the 3/4-chord of the wing where

y(J)

---- sign of the local rolling moment correction

I (J)l

ly(J)l -- dimensionless rolling moment arm b 2 dcL _-_-w(j)" dc_ .ai(O.75,j) = local lift coefficient Equation (19.27) is used by WICS to determine a rolling moment coefficient correction due to wind tunnel wall interference effects.

APPENDIX 20

SCALE FACTOR LAW

The minimum of the blockage factor of a wind tunnel model tested in a clean configu-

ration (i.e. flaps, slats not extended and no landing gear) is more or less identical with the

solid volume blockage contribution. This approximation is valid if such a model is tested

in the vicinity of 0.0 ° angle of attack where solid volume blockage effects are significantly larger that corresponding separation wake blockage effects.

Solid volume blockage effects are a function of the volume displacement of the wind

tunnel model. The solid volume blockage factor is proportional to the volume of the test

article. Therefore, it is possible to develop a scale factor law, that relates the length scale of two models of different size but identical geometry to the observed minimum of the solid volume blockage. The scale factor law may be written as :

¢2 (20.1)

where e,n_ is the minimum of the blockage factor as a function of the length scale of the

model, ¢1 is the length scale of the first model, and ¢2 is the length scale of the second

model.

The following example shows how this scale factor law may be applied to a test of

two different sized Boeing 7J7 semispan models (see Chapter 3.3 for more detail). Both

semispan models had identical geometry. The first model was a 8 % scale model, the second

model was a 14 % scale model. Both models were tested in a cruise configuration. From

Fig. 23a we estimate that the minimum of the blockage factor computed by WICS (wall

signature method) and the two-variable method for the 8 % scale model is approximately

0.0030 . Now it is possible to predict the solid volume blockage factor of the 14 _ model

by applying the scale factor law. Using the blockage factor minimum of the 8 % model

and the cube of the scale factor ratio we get the prediction :

! 8% ] = 0.0030- 5.36 -- 0.0160 (20.9-)

Comparing the predicted solid volume blockage factor, i.e. 0.0160, with the estimate

of the solid volume blockage from Fig. 23b, i.e. 0.0155, we see that both values agree

fairly well. Therefore, it can be concluded that the wall pressure measurements used by

WICS to determine blockage corrections and the corresponding description of the solid

volume blockage effect by a source/sink pair is sufficiently accurate. The scale factor

law also provides an additional absolute magnitude check of blockage corrections that are

computed by using wall pressure measurements.

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lmliilim_mmiim_ ilWmilmmiiimmm_ _AN MODEL SINGULAJ:h'rYREPRESIDI_'rATION

I I I I

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r *

I I

I , I

I I

I , , I

I I

I , I

i U_+ _ i

I I

I .__ . I

I I I I

I I

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Fig. 8 Singularity Representation of Semispan Model.

"U O a.

co

E

O O m C m m m a o i

/\

M U"

i

/

I

I

I

z LL

I

I

Point Source at x = 118.0 [ft], y = 3.0 [it], z = 2.0 [It]

0.031 .....

M-0.0 M = 0.3 -+---

0.021 M - o.s .0-

," "Q, B-D. g : "_'Q'B-B O O fOD :' _-_-,. ....

0.01

3=

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0.00_

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-o.011

!

I I I

105.0

134.0

x [ft]

Fig. 10 Wall Signature of Point Source as Function of Mach Number.

! i | ! | i i NA(_A TN 2454 ; sigma, -0.9 NACA TN 2454 ; s_ma = -0.8 NACA TN 2454 ;sigma = -0.7 NACA TN 2454 ; sigma = -0.5 0.8 NACA TN 2454 ; sigma = -0.2 NACA TN 2454 ;sigma = 0.0 WICS ; sigma, -0.8 WICS ; sigma, -0.9 i WlCS ; sigma = -0.7 WlCS sigma=-0.5 x 0.6 WIC$ ; sigma = -0.2 A WICS ; sigma = 0.0 no 0.4 0.2 -2 Fig. 11a Comparison Bessel Function Solution / Panel Code Solution (WlCS).

i !

NACA TN 2454 ;sl ]ma = 0.2 NAGA TN 2454 ;si |ma - 0.5 .....

NACA TN 2454 ;sl jma - 0.7 ......

NACA TN 2454 ; si ]ma - 0.8 NACA TN 2454 ; sl Bma - 0.9 ....

WlCS ; si _ma - 0.2 * WlCS ; sl 0ma = 0.5 + WlCS ; si oma - 0.7 o WICS ; si gma = 0.8 x WICS ; si gma - 0.9 • ./ u. 3

i/ f

_/ .x" ;_/.e.oS..(a--_"O"B"GP"

i i ..""

0 -- ==i.= -- --.,I I I -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 3d Fig. 11b Comparison Bessel Function Solution ! Panel Code Solution (WlCS).

1.0 I i i i NAC& TN 2454 ; sigma - 0.2 -- NACA TN 2454 ; sigma. 0.6 ....

NACA TN 2454 ; sigma - 1.2 ......

WICS ; sigma - 0.2 • WICS ; sigma. 0.6 + 0.8 WlCS ; sigma - 1 2. o 0.6

...p....q "..-"

o. + 4" U.

..-_] + ._--o --'e 0.4

• . ¢"._-'_/

I ../ i °.-°" [ I ! !

0.0 -2.0 2.0 -1.5 -1.0 -0.5 0.0 0.!; 1.0 1.5 xt Fig. 11 c Comparison NACA TN 2454 Approximation / Panel Code Solution (WlCS).

Fig. 12a Panel Model of Wind Tunnel Geometry/Fullspan Model Configuration.

Fig. 12b Panel Modal of Wind Tunnel Geometry / Semispan Model Configuration.

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ROW 8

ROW 1

ROW 7

ROW 6

ROW 4

ROW S

LOOKING UPSTREAM

Fig. 15 Testsection, Semispan Model, and Image Plane Geometry.

7J7 8% SEMI-SPAN SINGULARITY

AND REFERENCE LOCATIONS

• SINGU.AFgTY LOCATION

O RB=B_ENCE L(T_ON

TEST SECTION CB, ffEFU_

I

IMAGE PLANE IMAGE PLANE

No.

Type z [ft] y [f_,] z _t] Weight

I SOURCE 116.45 0.000 -3.920 1.000

SINK 124.73 0.000 -3.920 -1.000

3 SOURCE 121.47 0.000 -2.930 1.000

4 SOURCE 121.69 O.CO0 -1.430 1.000

5 LINE DOUBLET 119.89 0.000 -3.587 0.997

-2.901

6 LINE DOUBLET 120.25 0.000 0.977

7 LINE DOUBLET 120.60 0.000 -2.216 0.934

8 LINE DOUBLET 120.96 0.000 -1.530 0.866

9 LINE DOUBLET 121.32 0 200 -0.844 0.766

LINE DOUBLET

10 121.68 0.000 -0.159 0.619

LINE DOUBLET 0.000

11 122.03 0.527 0.371

Fig. 16a Singularity Representation of Boeing 7J7 8% Scale Semispan Model.

7J7 14% SEMI-SPAN SINGULARITY

AND REFERENCE LOCATIONS

• SINGULARITY LOCATION O REFERENCE LOCATION ... TEST SECTIONCENTERLINE IMAGE PLANE No.

Type z [ft] y [ft] z [ft] Weight 1 SOURCE 113.26 0.000 -3.920 1.000 2 SINK 127.75 0.000 -3.920 -I.000 3 SOURCE 122.04 0.000 -1.930 1.000 4 SOURCE 122.31 0.000 0.070 1.000 -3.330 0.997 5 LINE DOUBLET 119.28 0.000 6 LINE DOUBLET -2.130 0.977 119.90 0.000 7 LINE DOUBLET 0.000 -0.930 0.934 120.53 8 LINE DOUBLET 121.16 0.000 0.270 0.866 9 LINE DOUBLET 121.78 0.000 1.470 0.766 10 LINE DOUBLET 122.41 0.000 2.670 0.619 LINE DOUBLET 123.03 0.000 3.870 0.371 Fig. 16b Singularity Representation of Boeing 7J7 14% Scale Semispan Model.

,lira v,- 04 04 ROWS I & $ ....

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ROWS 2 & 7 ....

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.... , , , ., . ° , , ° , o , , o , i , . '-.-. °_° .

ROWS 4& 5 .... I---I--- F--I---;-t-t--I- 4-t-I-I-4--I-4. H 4--t-I-&F-I- 4-,l-.l--.-l.--.-l---._-4---

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I /

Tunnel Stations of Wall Pressure Ports in [ft] Fig. 17a Wall Pressure Port Location in 12ft Pressure Wind Tunnel.

ROW 8 ROW 1

ROW 7

., .. ROW 2

S ,'" ." _- Radius = 6.0 [ft] • ', j y L /

ROW 6

ROW 3

_,/ /

ROW 5 ROW 4

Test Section Cross Section at x = 120.71 [ft] (Looking Downstream) Fig. 17b Wall Pressure Port Row Definition in l!:2ft Pressure Wind Tunnel• ......... I ......... I ......... I ......... I ......... I ........

O.OtO 0.008

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I I 0.006

o =

o -£.

0.004 o Lf) 0,002 0.000 3O -20 -t0 0 tO 20 -30 Uncorrected Angle of Attack [deg] Fig. 18a Standard Deviation of Least Squares Fit of Wall Signature 18% Scale Model.

I ! i 0.010 0.008

'T"

I I 0.006 _ 0.004 c o 0.002 I 0.000 / 0 5 10 15 -5 Uncorrected Angle of Attack [deg] Fig. 18b Standard Deviation of Least Squares Fit of Wall Signature/14% Scale Model.

0_0 0.05

....-.--- -_-_ ;_-.._-..-:1111-_

v -0.05 Wolf Signature ; Row 1 -- - Least Squares Fit ..... Blockage Contribution .... Lift Contribution , , I , , ] I , , ; I -0 tc 5 3O 10 15 20 25 Wotl Pressure Port Number Fig. 19a Wall Signature at Row I /8% Scale Model at 19.82[deg] Angle of Attack.

0.10 t ' i I r ' _ 1 r ' i 0.05 F

r

0oo ._-.-=--:z_ ; _ ...................

v

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G"_q Walt Signature ; Row 2 -0 05 I - -- Least Squares _'it r - - • 81ockage Contribution -- " L;ft Contnbution I , , _ , r L _ , , 35 40 60 45 !!,0 55 Wail Pressure Port Number Fig. 19b Wall Signature at Row 2 / 8% Scale Model at 19.82[deg] Angle of Attack.

r I .... I ' 0 I0 I _ ' ' I .... r-" ' ' ' l

E

0.05 I F

L- --_-_-_ -__ ............................... - -

o.oo __._ _- ..... _- _,_-=- ....- ............

-0.05 ES----E2 Wali Signolure ; Row 3 - - - Least Squares Fit ..... 81ockoge Contribution ..... Lift Contribution , , , I , , _ , I , , , t 1 L _ _ T ! , L , r I _ , _ , -0.10 9O 65 70 75 80 85 Wall Pressure Port Number Fig. 19c Wall Signature at Row 3 / 8% Scale Model at 19.82[deg] Angle of Attack.

' ' ' I ' ' ' I " I .... I " ' I 0.10 0.05 0.00 rr --005 _ Wall Signature : Row 6 L - - - Least Squares Fit I .... Blockage Contribution P .... Lift ContributiOn -0101 J , , F , L , , I , , , J I , , , , I , , , , L , , , , , 155 160 165 170 ! 75 180 Woll Pressure Port Number Fig. 19d Wall Signature at Row 6 / 8% Scale Model at 19.82[deg] Angle of Attack.

' ' ' I ' ' ' r I 0.05 o 0 O0 D _] WoII Signature ; Row 7 J - -- Leost Squores Fit J ..... 810ckrge ContributiOn ..... Lift Colnt riDution -005. I ,,I,,,,I K i I i i i i I i i _ i -0.10 , , , I 185 190 195 2:)0 205 210 WOII Pressure Port Numb,;r Fig. 19e Wall Signature at Row 718% Scale Model at 19.82[deg] Angle of Attack.

I I ' ' ' 1 ' I I 0.05

ooo

-0 05 [_'-----g WOH Sicjrcture ; Row 8 - -- Leos! Squores rlt .... Blockage Cont _;bution .... Lift Contribut,On -0 10 2 _5 220 225 : 30 235 240 WoIt Pressure Port Number Fig. 19f Wall Signature at Row 8 / 8% Scale Model at 19.82[deg] Angle of Attack.

0_0 0.05 o 000 re / Least Squares Fit ..... Blockoge Contr_'bution -0'05 I _ WEHI Signature : Row 1 ..... Lift Contributior_ -0,101 , , , I , , , , ! , , , , I , , , , I , , , , I , , , , 5 10 i5 20 25 30 Wall Pressure Port Number Fig. 20a Wall Signature at Row I / 14% Scale Model at 9.98[deg] Angle of Attack.

I ' ' I ' ' ' I ' I I 0.05 o_ 0.00 v i -005 WoH Signature ; Row 2 - - - Least Squares Fit .... Blockage Contr_ution .... Lift Contribution -010 55 40 45 50 55 Wall Pressure Port Number Fig. 20b Wall Signature at Row 2 / 14% Scale Model at 9.98[deg] Angle of Attack.

010 ' ' I .... I [ ' ' 0.05 / o 0.00 n- --0.05 [3-_--E] Wall Signature ; Row 3 --- Least Squares Fit ..... Blockage Contribution ..... Lift Contribution -0._0 i i l i J r l I I l = l I I l i i , i = i i P = = = = , 65 70 75 _0 85 90 WoII Pressure Port Number Fig. 20c Wall Signature at Row 31 14% Scale Model at 9.98[deg] Angle of Attack.

O. tO I I ' ' I I .... I ' ' --/ _. o.oo -0 05 C_-{] Wall Signature ; Row 6 ..... Blockage Contr;but;On -- - LJt Contribution I Least Squares Fit -0 10 , , , I , , r , r , , , b I , T , , , , , , I , , , , 155 160 165 T '0 175 8O Wall Pressure Port Number Fig. 20d Wall Signature at Row 6 / 14% Scale Model at 9.98[deg] Angle of Attack.

• ' ' I ' ] .... f ' ' i .... i 0.10 OO5 m o 0.0C e,- D -005 Wail Signoture ; Row 7 --- Leost Squores Fit ..... Blockage Cor_tribution ..... Lift Contributior_ i l J I h i J i I , l i _ I i i i , I l l I _ l , i _ l -0.10 !1o 185 190 195 200 205 W011 Pressure Port Number Fig. 20e Wall Signature at Row 7 / 14% Scale Model at 9.98[deg] Angle of Attack.

0.10 0.05

"o o oo

t_ - - - Leost £_uores Fit - - - BPockoge Cor_tribution .... Lift Contributior_ -005 I _ WOII Signoture ; Row 8 -0 10/ , _ , ; , , , , T _ , _ , I _ , , , I , I , , _ , 215 220 225 250 255 240 WO_ Pressure Port Numbe¢ Fig. 201'Wall Signature at Row 8 / 14% Scale Model at 9.98[deg] Angle of Attack.

005 i -005 {_ WoII Signature ; Row 6 - -- Least Squares Fit ..... Blockage Contribution ..... Lift Contribution -0.10 , , , 1 , , , , I , , , A i , L i I .... 1 , , , 155 160 165 170 175 Woll Pressure Port Number Fig. 21a Wall Signature at Row 6 / 8% Scale Model at -0.18 [deg] Angle of Attack.

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_o o.oo

D --005 _3"_"=-_] Woll Signature ; ROW 6 - - - Least Squares rlt ..... Blockage Contribution .... Lift ContribuUon 155 160 165 I 70 175 WOIP Pressure Port Numb,tr Fig. 21 b Wall Signature at Row 6 1 14% Scale Model at -0.02 [deg] Angle of Attack.

E] Classical Method (Courtesy of Alan Boone, NASA ARC) O TwO Variable Method (Courtesy of Mot Rueger, Boeing St.Louis) Wall Signature Method (WlCS)

g

o _D 0.5 ___0000 O0

liil m_il !

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-0.5 -20 -tO 0 10 20 Uncorrected Angle of Attack [deg] Fig. 22a Comparison of Angle of Attack Correction / 8% Scale Model.

.... I ......... I ' ' ' '" ' ' ' ' I .... ' ' ' ' ' I ' ' ' ' ' ' ' ' ' I ' ' ' _.5 O Classical Method (Courtesy of Alan Boone, NASA ARC) O Two Variable Method (Courtesy of Mot Rueger. Boeing St.Louis) WoN Signature Method (WICS) 1,0

g

O 0.5 o -05 -20 -10 0 lO 20 Uncorrected Angle of Attack [de9] Fig. 22b Comparison of Angle of Attack Correction / 14% Scale Model.

--T ' ' ' I ......... I ......... I ......... I ......... I ....

0.050 O Clossicol Method (Courtesy of Alon Boone, NASA ARC) O Two Vorioble Method (Courtesy of Mot Rueger, Boeln 9 St.Louis) Woll Signoture Method (WlCS) _" 0020.

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' ' ' ' I ......... I ......... I ......... I ' ........ I ' ' ' 0.030 [] ClossIcol Method (Courtesy of Alon Boone, NASA ARC) O Two Voriobte Method (Courtesy of Mot Rueger, Boeing St.Louis) 0,025 Woll Signoture Method (WlCS) _ O 0 0 '_' 0 020 o o o

s

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0005 I p- 0.0001 .... i ......... _ ......... t ......... i ......... i,.....,.__ -20 -I0 0 10 20 Uncorrected Angle of Atto:k [deg] Fig. 23b Comparison of Blockage Factor/14% Scale Model.

......... 1 ......... i ......... i ......... i ......... } ........

q

180 PORTS / ROWS 1,2,3.6,7,8 mow 8 & A A A A // ,/ o IK_7 0J AA & n E ROWG 3O -30 -20 -I0 0 10 20 Uncorrected AngLe of Attack [deg] Fig. 23c Dynamic Pressure Correction of 8% Scale Model using 180 Ports.

......... I ......... r''' ...... I ......... I .... _ .... I .........

.30 PORTS / ROW 1 3 A A CL & A O A AA A ¢U Az_A 1 5, &

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T T "I' T T T Analytic Solution / r._nel Method Code Solution ....

1.0

e-

_o

_u

(J

m

0.0

n .... _ ............................... _nk,ne_ ............................. _.....

116.0 126.0

x [.]

Fig. 28 Comparison of Surface Pressure Distribution of Rankine Body.

Fig. 29 Panel Model of Rankine Body inside Test Section.

0.03 t i !

Panel modet of Rankine body Source/sink mode4 of Rankine body 0.025 0.02 0.015 o 0.01 0.005 -0.005 -0.01 95 100 105 110 115 120 wall pressure port number

Fig. 30 Comparison of Wall Signature of Rankine Body with equivalent

Source-Sink Representation.

m

I

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5.33 [ft]

/

Tunnel Station 120.0 [ft]

1.50 [ft]

v Fig. 31 Geometry of Rectangular Wing.

Fig. 32 Panel Model of Wing inside Test Section.

Z

A

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I SEMHNRNrrE UNE DOUBLET J_ w t_

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I

I

I

i Fig. 33 Singularity Representation of Rectangular Wing.

0.02 i i i !

Panel Model of Wing Panel Model of Wing 5.0[DEGI - -

Panel Model of Wing ; No Thickness 0.0 [DEG] s.o [DEG]

Line Doublet Model ol Wing S.0 [DEG]

0.015 0,01 J= O 0.005 (]C -0.005 -0,01 t i I I , I 215 220 225 230 235 240 wall pressure po_ number

Fig. 34 Comparison of Wall Signature of Wing with

equivalent Line Doublet Representation.

I- .i I.U .J U m n-

_w

O IL o I=* .i C3 IL Z o --I m

c

,=.J -I 4- x o Q o J_ cff o w L_ o 3_

"6

c o M m D.

E

X .i I- Iii 31:: O*J iii

.3

IZ: 3C O O O O Ii I I I

Y

Fig. 37 Line Doublet Flow Field as a Function of Orientation Angle.

LIFT FORCE 114 chord of the wing PITCHING MOMENT LINE DOUBLET 114 chord of the elevstor Z Fig. 38 Calculation of Line Doublet Strength using Lift Force and Pitching Moment.

k

J_ i,j+l,k+l i+l,j÷l,k+l i,j,k+l

J

I

i÷l,j÷l ,k Fig. 40 Quality Check of Least Squares Fit of Wall Signature.

.t:2 o_ k_ "0 c u u Ul

II

r- °_ 4.0 L_ t_ e,- C c ..1

I I

!

-/

i1 _p ..0 o_ 5=.

>

I

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I I

4-I L_

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a.

I : I

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.=l 0'} a..

/i

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¢::

I

4=_

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C:)

I

4=,' ¢.- c o_ ..J ¢....

L IJ.

\

-4- ¢.==..-¢ m.._J s===l

II II

c: 4.J t,n (J

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r.,-_q ira.,..1 (U

\

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\

"6

\

o)

Q

e..

<c

r,.

\ I

II II

\ I 2::

\ I

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"0 I=

\ I

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\ I

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I:1.

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r,n In

E

,\

I \

I \ !-

I

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I \

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I \

o

I

o_ X

I

I

.__

I

I

v

I

!

I

_/_- --'T-II I' P_(=_' _' _)

I I

Xl X2 Fig. 4Zb Rotation No.1 (x=120.71 [ft]; z=O.OO[ft]).

..

Y

P3(=3,y3, z3)

I I z

X3 X2 Fig. 42c Rotation No.Z (x=120.71 [ft]; y=O.OO[ft]).

/

/

/

/

\

\

o c _o .m

E

"6

o Ix o X ,< LL

/

/

/

/

/

/

#

Y

_ Pl(zl,ul,zl)

Z _1 :1:2 Fig. 42e Left Wing Semispan Model.

Y

• 2 Z1 Fig. 42t: Right Wing Semispan Model.

sin

k

cos

Fig. 43a Unit Wind Vector in Model Coordinate System.

k

J

/

Fig. 43b Reversal of Roll Axis Rotation.

_ wA m_

\ I

\ I

o

I

\

0_

\ I

C

\ I

e- im

\

"l"

\

qCB G_ m e..

\

\

e- 11Dram

el

c_

6_

IL

Yo Yl

Fig. 43d Singularity or Reference Point Location as

a Function of Test Article Roll Angle.

K

I

Fig. 43e Pitch Rotation of Unit Vector K .

LIFT FORCE AXIS

I

UNE DOUBLET I STARTING !

PITCHING MOMENT AXIS Fig. 44 Pitching Moment Lever Arm Calculation.

c.I 1._ U.

l l 1__ l l "0 C w l b i .4

.._._._-_---]]]'_

C: I l I l m I l e- ra I l m I l r" m I l I l I m I¢.

l I C

E

O c m c l m l m m l O l mr l O l l O l tm l II m m u i l l l g_

_(_)

LOCAL CHORD Fig. 46a Angle of Attack Correction at Wing Chord.

LOCAL CHORD

J

Fig. 46b Change of Angle of Attack Correction at Wing Chord.

2;

T

B

b _ o! A

Fig. 47 Geometry of Circular Arc Airfoil.

Local Chord c(_) 1/4 - Chord Line

Y

r Fig. 48 Geometry of Wing with 0.0 [deg] Sweep Angle.

r"(v)

v(;)

r'(v)

=, f r'(v)..

LEFT WING RIGHTW1NG "__ _ Y

R

v Line Doublet Fig. 49 Non-symmetric Lilt Distribution.

_(o.75,v)

LEFT WING

/

-_/2

Reference Point Fig. 50 Angle of Attack Correction Distribution.

Form Approved

REPORT DOCUMENTATION PAGE No.ozo4-o18o

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwo=_ Reduction Project (0704-0188). Washington, DC 20503.

i 1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

July 1998 Contractor Report

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

The Real-Time Wall Interference Correction System of the NASA Ames

12-Foot Pressure Wind Tunnel

NAS2-13605

6. AUTHOR(S)

Norbert Ulbrich

8. PERFORMING ORGANIZATION

7. PERFORMING ORGANIZATION NAME(S) ANDADDRESS(ES)

REPORT NUMBER

VERIDIAN/Calspan Operations

NASA Ames Research Center

A-98-11989

P.O. Box 7

Moffett Field, California 94035-0007

10. SPONSORING/MONITORING

9. SPONSORING/MONITORING AGENCY NAMEtS) AND ADDRESS(ES)

AGENCY REPORT NUMBER

National Aeronautics and Space Administration

NASA/CR--1998-208537

Washington, DC 20546-0001

11. SUPPLEMENTARY NOTES Point of Contact: Norbert Ulbrich, Ames Research Center, MS 227-3, Moffett Field, CA 94035-1000;

(415) 604-6893

12b. DISTRIBUTION CODE 12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified -- Unlimited

Subject Category 09

13. ABSTRACT (Maximum 200 Words) An improved version of the Wall Signature Method was developed to compute wall interference effects in three-dimensional subsonic wind tunnel testing of aircraft models i]1 real-time. The method may be applied to a fullspan or a semispan model.

A simplified singularity representation of the aircraft model is used. Fuselage, support system, propulsion simulator, and separation wake volume blockage effects are represented by point sources and sinks.

Lifting effects are represented by semi--infinite line doublets. The singularity representation of the test article is combined with the measurement of wind tunnel test reference conditions, wall pressure, lift force, thrust force, pitching moment, rolling moment, and precomputed solutions of the subsonic potential equation to determine first order wall interference corrections.

Second order wall interference corrections for pitching and rolling _aaoment coefficient are also determined. A new procedure is presented that estimates a rolling moment coefficient correction for wings with non-symmetric lift distribution.

Experimental data obtained during the calibration of the Ames Bipod model support system and during tests of two semispan models mounted on an image plane in the NASA Araes 12ft Pressure Wind Tunnel are used to demonstrate the application of the wall interference correction method.

15. NUMBER OF PAGES 14. SUBJECT TERMS

Subsonic flow, Wind tunnel wall interference, Wall pressure measurements

16. PRICE CODE

A10

19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION OF ABSTRACT OF REPORT OF THIS PAGE

Unclassified Unclassified

Standard Form 298 (Rev. 2-89) NSN 7540-01-280-5500 Prescribed by ANSI Std. Z39-1e