Measurements of turbulence inputs for V/Stol approach paths in a simulated planetary boundary layer

•

July,

1973.

MEASUREMENTS OF TURBULENCE INPUTS FOR V/STOL APPROACH PATHS IN A SIMULATED

PLANETARY BOUNDARY LAYER

by

B. Etkin, G. W. Johnston and H. W. Teunissen

r,

SEP. '918

UTIAS Report No.

189

eN ISSN 0082-5255

MEASUREMENTS OF TURBULENCE INPUTS FOR V/STOL APPROACH PATHS IN A SIMULATED

PLANETARY BOUNDARY LAYER

by

B. Etkin, G. W. Johnston ~d H. W. Teunissen

Sub mi tted June 1973

'

.

Acknowledgement

The authors acknowledge the assistance of Mr. W. H. Pinchin, who carried out the bulk of the experimental measurements described in this report. Fin~cial support for this project was receivedfrom the National Research Couneil of Canada under Operating Grant A.1894 and the United states Air Force, under Contract No. F33615-73-C-3013.

.

,

Abstract

A study relating to the prediction of V/STOL flight path perturbations during steep landing descents has been completed in three phases. In the first phase, the possibility of utilizing turbulence correlation data accummulated by means of fixed survey probes in a simulated planetary boundary layer model to predict the aircraft response is discussed. It is shown that this technique although necessarily limited in certain respects, represents an excellent start-ing point for ·more rigorous aircraft studies which may eventua1ly be required. The theory of the stationary probe technique is deve10ped giving the mean square displacement errors of the aircraft f1ight trajectory at the target plane (break out) in terms of the turbulence correlation characteristics of the shear'layer

~d the aircraft gust impulse response functions.

In the second phase of the work a large number of fixed probe hot wire correlation measurements of the turbulent velocity components were comp1eted

with a scaled model of the planetary boundary layer. The majority of these measure-ments were confined to a mean velocity power law variation of n·= 0.16 with limited

turbulence data also taken with n = 0.35. Finally, in the third phase of the work, the feasihility of extending the present techniques with a more

sophisti-cated moving hot wire turbulence probe has been studied. This technique could eliminate some of the inherent restrictions of the"stationary probe technique, incurring, however, a significant increase in measurement difficulties. A specific moving probe design concept has been examined and a first costing estimate developed .

I. 11. lIl. Dl. V. VI.

'

.

Table of Contents Abstract Notation INTRODUCTION

APPROACHES TO SOLillION OF THE PROBLEM THEORY OF THE STATIONARY PROBE METHOD 3.1 Response to Turbulence

3.2 The Input Vector EXPERIMENTAL MEASUREMENTS

4.1

Assumptions

4.2

Parameter Selection and Velocity Sealing

403

Wind Tunnel Facility

4.4

Instrumentation and Measurement Techniques Hot Wire Anemometry

Analog Computer

4

'

.4.1

4.4.2

4.4.3

4.4.4

4.4.5

Correlation and Spectral Analysis System Random Noise Meters

Correlation Matrix Measurement

4.5

Wind Tunnel Flow Characteristics and Roof Tests

4.6

Results and Discussion

4.6.1

Two-Point Cross-Correlation Data

4.6.2

Flight Path Cross.Correlation Results

4.7

Conclusions

MOVING PROBE FEASIBILITY STUDY

5.1

Relative Merits of the Moving Probe Solution

5.2

Veloeity Sealing

5.3

Frozen Flow Assumption

5.4

Probe Requirements

5.5

Probe-Moving Apparatus

5.6

Instrumentation

5.7

Summary and Conelusions CONCLUSIONS REFERENCES FIGURES Page No. i i iv 1 3 5 6 10 11 11 12

15

15

15

15

16

16

16

17

21 21 22

27

28

28

28

29

33

35

38 38 39

40

A B f f FE FN ~(t) H

=

_[hij} H k 1,1 ,1 u v w n P,Q"P _o ,Q, 0 ,~, R

s

T T t 6t u,v,w u'I ,VI, w'

v

v

I EF Notation glide path starting point glide path decision height

time-varying matrices of perturbation system

vector of forces and moments; also refers to a function frequency, Hz

atmosphere-fixed reference frame - convected with local mean wind speed W(z)

vertical earth-fixed reference frame

nominal trajectory reference fralll$ , .. :.'earth-~fixed gust input vector

matrix of impulse response functions wind tunnel height ( = Bil)

reducedfrequency, f/W

integral scales of turbulence components power law index

reference points along flight path (Figs. 4 and 5)

matrix of cross-correlations(dimensional and nen-dimensional) distance from wind tunnel jet exit plane

total flight time along glide path (A to B) matrix of gust transfer functions

time

13 - ex

time-varying turbulent velocity components in FE

J~

RMS values of u,v,w; u'

=

u , etc. vehicle airspeed

vemcle ground speed (VEI for model tunnel flow)

W(z) x,y,z x',y',z'

,.

'

<

>

(

mean wind speed at height z (W'(z) for ·model tunnel flow) gradient wind speed (W

G' for model tunnel flow) coordinates in FE (f~ll scaleflow)

coordinates in FE (model tunnel flow) coordinates in FN

state vector

reference solution gradient height

flight time to reach P ,Q , respectively on glide path starting from A 0 0

power 'spectral density glide path · ~gle in FA glide path angle in FE time-del~y

ensemble average time average Laplace Transform transpose of matrix

10 11~ODUCTION

Safe and reliable lqndings of STOL and VTOL aircraft on steep des cent paths in congested environments, low visibility, and strong wind must be achieved if the all-weather air transport systems envisaged for the future are to become a reality. A central feature of the overall problem is the design of vehicle control and terminal guid~ce systems that will re sult in small enough dispersions of the flight path and vehicle attitude at the decision height. Figure 1.1 illustrates the nominal flight path with which we are concerned, i.e., the path that would be followed through the planetary boundary layer wi th ~ ideal guidance and con-trol system. It consists of an approach that terminates at point A .wi th speed

VA' a straight line descent at ~gle

rE

to point B, where the speed is V_B, poss-ibly different from VA' qnd a subsequent flare and touchdown. Point B is at the decision height, where the c.hoice is made whether to land or to abort. This decision depends, when.flight path perturbations are present, on whether or not the trajectory (more precisely the state vector) falls "rit hin a certain 'windo1-l'. The problem for the ~alyst and designer is to predict, for a givensituatior., the probabilities associated with this decision. It is not our purpose in this report to deal with the required window size, nor with the probability levels needed, important as these questions may be. We are rather concerned here with the methodology of arriving at the probabilities, and more particularly with the treatment of the dispersions produced by the action of the turbulent wind.

Accordingly, in Section 11, various approaches to this problem are outlined, with the description of a.simplified approach using stationary probes being given in Section 111. The details of an experiment to obtain data for application of this approach are given in Section IV. This experiment was carried out in a small pilot model of a ~ew, larger facility (Ref. 13) presently being constructed at UTIAS for further investigation. Section V outlines the results of a feasibility study into a more co~lex moving probe approach and conclusions are summaxized in Section VI.

Ideally, the state variables ,Ni th which we would be concerned in discussing errors at A and B include the position vector, the velocity vector and the atti· tude, nine scalar variables in all. The angular velocity is probably uniIl!Portant. j\~so, in practice, it is probably good enough to define the windmv at B in terms of a severely restricted set of variables; for example z, y, V and

z.

Of the factors that contribute to dispersion at the target plane (Fig. 1.1) we c~~ usefully separate out three (apart from navigation system errors):

(i) the mean "rind profile,

(ii) errors in initial conditions at A, (iii) turbulence.

The way in which these t~xee contribute to state errors at the target plane is illustrated in Fig. 1.2 by the intercepts of an ensemble of trajectories with the target plane - an ensemble related to a given wind field. R is the point at which the reference trajectory associated with correct initial conditions and

zero turbulence pierces the target plane. The reference trajectory accounts for the mean wind profile, which may include cross wind, rotation·of wind direction with height, and local effects of buildings or terrain (the wind. in Fig. 1.1 is shown coplanar with the trajectory.only for convenience). Curve a on Fig. 1.2

represents the dispersion around R associated with initial errors at A that may occur with an assigned probability P. These may be the outcome of a stationary

random process along the nearly-hori±ontal approach path. A given initial error

in position and/or velocity and/or attitude yields a single point in the target

plane. The continuous curve a derives from a continuous set of initial errors.

P is the probability (e.g.,

:995)

that the trajectory falls inside a. Curve b

0*

Fig. 1.2 shows the additional dispersion at probability level P

2

associated-with turbulence ~d a particular set of initial errors. We assume that the

errors resulting from initial conditions are independent of those produced by

the turbulence, and hence that the joint probability is the product of the

sepa-rate probabilities. If, furthermore, as will frequently be the case, the

dis-persions from the two sources c~ be linearly superposed, only one curve like b

need be obtained - for zero initial error·- ~d combined with every point of a-to

yield the envelope c. The probability that the trajectory will lie inside c Is

then P

=

P

1P2• -

-To recapitulate and clarify the different vehicle trajectories between A

and B (Fig. 1.1) to which we will be referring in this report, see Fig. 1.3. The

nominal path is equivalent to the glide slope and is that which the vehicle would

follow if there were no dispersive factors at all or if it had an ideal guidance

and control system. The refer ence trajectory is the path the aircraft would follow

if there were noinitial condition errors and no turbulence. It reflects the

departure from the nominalp.:atn 'iau..e only to the mean velocity and this would be

zero for

an

ideal control system (and thus R

=

B). Finally, the actual path flown

by the aircraft includes the effect of all three of the factors listed above and

its departure from the reference solution represents the effects of initial c

on-dition errors ~d turbulence.

The current generation of STOL airplanes - deflected slipstream, tilt wing,

tilt duct, tilt rotor, augmentor wing - appear to be limited at approach speeds

of

50

knots or more to descent ~gles of less than about 160 (Ref. 1). Future

developrents may be expected both to demand and to lead to even steeper glide

slopes for STOL airplanes, perhaps by the use of reverse thrust combined wi th

powered lift (Ref. 2). Helicopter8 and other VTOL aircraft are of course capable

of des cents at all angles up to

90.

~y past studies of the response of vehic~es

to turbulence, during landing as well as in cruise, have used a statistically

stationary model of the process. This yields as outputs, among others, the

mean-square values of the vehicle response variables • From these, wi th the common

assumption of a Gaussian process, probabilities can be calculated (Refs. 1,3,4).

Al though this may be a reasonable approach for CTOL airplanes wit h glide slopes

of about 30 and high landing speeds, the increased relative importance of

gusti-ness at low speeds and the steeper penetration of the boundary layer màkes ·.the

stationary model suspect for STOL and VTOL. A guide as to·the expected usefulness

of a stationary model is obtained by ex~ning some data.for a representative

STOL landing. FromRef. l(CL

84,

Case 12) in a typical landing at V =

60

knots

and IE

=

13030, the vehicle has the following characteristic oscillatory periods

a,.nd wavelengths with the SAS on:

Period Wavelength

(sec) (ft)

Longi tUdinal, long

26

2640

- - -

-The path length from ~ altitude of 500' to a CAT IIB decision height of 100' ,

which,would inelude most of the intense shear and turbulence, is only1740 ft.

Thus we see that the characteristic distances are not small relative·to the

app-roach'path length; but this is an essential condition that must be mèt if a

stationary model of the response beginning with zero perturbations is to be valid.

It should be noted as weIl that the mean gradient itself has a strong effect on

the characteristic modes of aircraft (Ref.5, Sec.

9.9),

~~ effect not included in

the table above. Both the phugoid period and its time to balf a.ltlPlitude may be increased by as much as 100% in astrong head wind.

11. APPROACHES TO SOLDrION OF TEE PROBLEM

For purposes of design and operation, one wishes to be able to calculate with

good accuracy how any given landing system - vehicle + controls + terminal

guidance • will perform in a statistical sense at

anY

given airport in any given

wind. It is clear that this ~ntails a knowledge of the wind profile and turbulence,

ineluding local effects of terrain a.nd nearby structures, that pertain to the air ...

port in question. Presumably this information might be obtained in the field, but

this would be a costly

and.

time consuming process that would be undertaken only

rarely. The only feasible alternàtive, ~~d one that can provide reasonably good

results for the lower part of the atmosphere, say below 1000 or' 1500 ft., is a

wind tunnel capable of simulating the atmospheric flow in this region - that is, in

the planetary boundary layer. In such a tunnel, one can model the geometry of the

local airport and environs and col1ect data systematically and relatively inexpen~

sivelyo The principal limitation of these facilities at present is their inability

to include the Coriolis effects that result in the mift in wind direction with height, and hence to simulate faithfully the outer region of the boundary layer.

However, the region of most intense shear~nd local effects of terrain and struc~

tures can be fairly well reproduced.

Assuming then that we have a laboratory facility available which C~D provide

reasonable simulations of the atmospheric bOlli~dary layer, we now consider the

possible alternatives for determining the effects of turbulence alone on the air~

craft. We assume the aircraft is beginning its descent through the boundary layer

from some known initial position A and we ~~sh to determine its dispersion from

the·reference trajectory due to the turbulence it encounters - that is, we wish to

determine the valueof P associated with a given 'window'. The reference tr~­

jectory·itself may be

de~ermined

from a knowledge of the mathematical model of

the aircraft/guid~~ce system and the mean wind profile (see Sec. 111).

There are basically four distinct approaches one may t~e toward solving the

above problem, each requiring different assurnptions. These solutions are des cri bed

belowin descending order of faithfulness.

(1) Obviously the most nearly· 'exact' solution would be to build a faithful free-flying model of the aircraft and have it seek to follow the prescribed nominal flight path from A to B (Fig.

1.1) at correctly scaled speed with the correct guid~~ce and

control laws. By making many descents under a given set of conditions, the ensemble statistics of the state vector at the target plane could be collected. To get good statistics would

t~ke many 'flights' in the tunnel in order to define the 811-important tail of the probability curve with adequate accuracy. This technique may not in principle be beyond the state of the art, but it would appearto be inord,inately expensive, qnd would certainly require a very large wind tunnel in order that the . models of the aircraft not be too small. It does have the

ad-vqntage, however, of requiring no basic assumptions other thah that the laboratory simulation of the boundary layer is satis-factory, ~d this is of course required for

anY

wind tunnel solu-tion.

(2) A considerably simpler approach would be to drive arigid model equipped .wi th a force balance down the reference traj ectory 'open loop'. As noted above this trajectory corresponds to zero initial errors qnd zero turbulence and can be calculatedin advance. Using a six-component force balance and by making appropriate corrections for gravity ~d inert ia forces, the transient aerodyn~c inputs to the vehicle could be obtained. These time histories would then be used with a mathematical model of the vehicle to compute errors in the state vector at the target plane. In this approach, we must assume that aerodynamic forces obtained on the reference

tra-jectory are approximately the same as those that would have been found on the true traj ectory, and hence large perturbations from the reference path would introduce significant errors. If desired, we couldfurther assume that the aircraft guidance system has

sufficiently high perfor~ce that the reference path is very close

to the nominal glide slope. In this case, of course, the model could be dri ven along the latter path rather than the former. As

with the previous solution, a large number of repetitions would be needed to obtain adequate statistics.

(3) The third approach would be somewhat like (2), but instead of measuring aerodynamic forces with a proper vehicle model, we would measure 'gust inputs' fromwhich theseforces could in principle be caJ.culated by linear aerodynamics. That is,

! (

t )

=

1;

{~( t ) }

where f is the six component force vector, L is a .linear operator, a,nd ~ is the m component 'gust vector'. When

f.

is not state-dependent, the simple transfer function form results; that is

in which T(s) is a

6

x m matrix containing constant 'gust deriva-tives' (see Ref.

5,

Sec.

13.3).

The use of this approach requires that this matrix cqn be determined either from theory or experi~

ment qnd that the vector ~ can be defined qnd neasured. This measurement could be performed wi th hot-wire or other turbulence-measuring devices, and various levels of approximation are possible by making different assumptions concerning the makeup of ~ (Ref.

5,

Sec.

13.3).

The advantage of this solution, of course, is that only probes rather than actual vehicle models are required. When f

has been generated in this fashion for ~y one trial, it can be used in the same wa as f measured by method (2) to compute the vehicle response. As with methods

(1)

and

(2),

this one would require a large number of trials to obtain adequate statistical reliability of the results. The feasibility of this method is fUlly discussed in Section V.

(4)

Finally, the simplest approach to solving the problem is one which employs neither moving models nor moving probes but instead collects the flow statistics at fixed points in the flow. This method is the one used in the present investigation and is described in detail

in Section 111. It has the principal advantage of using fixed probes

~d not requiring a large number of repeated runs but it does require more assumptions than the previous methods. Unlike

(3),

for which nonlinearities (other than those excluded by the

! -

~ relation hypothesized above) can be included, method

(4)

requires a campletely

. linear system. From the standpoint of validi ty for linear systems, of course, it is equivalent to (3) insofar as it yields approximate mean-square and mean produets of the deviations of the state variables

at the target plane. However, it is less faithful than (3) in that one can only take the next step and proceed to statements about probabilities with the aid of an assumption about the form of the probability distribution function, e.g., that it is Gaussian. In addition, ergodicity of the turbulence signals is a necessary (but weak) assumption.

lIl. THEORY OF TEE STATIONARY-PROBE MErHOD

Suppose we assume now that we have ~ adequate mathematical model of

the beam-following system, comprising the vehicle, its controls (human or automatic) and the guidance loop. This model may be denoted symbolically in state vector form by

i

=

f(Z,t) +~(Z) ~ (3.1)

where

X

is the n x 1 state vector (including all closed-loop control variables),

~ is the m x 1 t gust' or turbulence input vector, and ~ is an n x m matrix of

coefficients that in general are state dependent. The time variable is included in f to allow for the possibility that the descent might have some programmed variables, e.g., flap angle or propeller pitch or wing tilt as functions of time.

Now consider the reference solution for which the initial conditions are

the nominalor ideal ones, and for which the wind 'is laminar and steady with respect to its mean profile. Let the reference solution be denoted y (t) so that from (3.1), wi th ~

=

0 for a lqminar wind, --0

y

=

f (y

,t)

-0 - 0 (3.2)

This reference solution provides the constant position error at the target plane associated with wind shear. It of course also provides the errors in

airspeed, ba.'1k q,ngle, etc., at the target plane, which may be important factors in the decision to be taken as to whether to land or abort.

Now consider a set of perturbation solutions (i.e., small departures from

the reference solution) associated either with small deviations in the initial

conditions or with turbulence. The constraint of 'smallness' is imposed to

yield a linear system for analysis, and the usual confidence in linear

flight-dyngmic solutions leads to the expectation that they will be of practic&l utility

for realistic'levels of disturbance. The perturbation is denoted by 6y, so

that from (3.1)

-Now to the first order in ~"we have

where

A(y ,t)

- 0

= [

~J

oy.

'

J. 1.

=

Zo

(3.4)

which, on substitution into (3.3), serves to eliminate the reference solution. Also since ~ is taken as small, the second term of (3.3) to fîrst order is

B (lo)~. Finally, since

1.0

is a known function of time, the perturbation

equa-~ion can be written as

Equation (3.5) can now provide the second and third contributions to errors at the target plane referred to in Sec. I. The initial condition errors are found by solving (3.5) with ~

=

0 and 6y(0)

f

0; the gust response is treated

with ~y(O)

=

0 and ~

f

O.

3.1 Response to Turbulence

In the approach taken herein, the basic response elements are the impulse=

response functions for gust inputs. Thus lèt· ,there be a,.n impulse in g. at time

T; g.

=

5(t-T). The subsequent response at time t in state-variable Jy . is

J 1

h .. (t,T). For a linear-invariant system, h .. would be a function only of the

lJ lJ

time difference (t-T) and would be given by the inverse Laplace transform of the associated transfer f~~ction (Ref.5, Sec. 3.4). For time-varying systems like (3.5), the h functions 'WOuld ordinarily have to be found by machine compu=

'.'

the case, about 10 values of ,- might be expected to suffice. The matrix of all the impulse response ·functions is

. H = [ho

.1

(3.6)

- l.J

For any state variable y. the response to random turbulence can then be expressed by the superpositioft,theorem (even though the system is time varying, linearity allows this):

y.(t) = \ '

J

t h ..

(t,,-)g.(,-)d,-1.

L

l.J J

j

T=O

or, for ,the whole state vector,

~(t)

=

J

t .!!(t,'-).s.('-) d,- •

(3.8)

,- = 0

We now proceed to findthe mean products of the values of y.. The squares and products are gi ven by the matrix ' l.

2

T Yl

Y1Y2· •••

1.1.

=

₂ (3.9)

Y2Yl Y2 ••...

of which the diagonal gives'the squares. On substituting (3.8) into (3.9) we get

T

J

t J t T T

1. 1. (t)

=

.!!(t,a).s.(a).s. (~) . .!! (t,~)~~ (3.10)

a=o

~=O

where

a,

~ are dummy variables of integration. time ,t is then

t t

T

The ensemble mean of 1. 1. at

<'1.1.T(t)

> "'"

J

J

.!!(t,a)<

.s.(a).s.T(~» .!!T(t,~)~~

(3.11) o 0

The ensemble mean within the integral is the matrix of cross correlations of the g., which we denote

T

,~ (a,~) =

<

~(a) ~ (~)

>

(3.12)

so that the desired matrix of mean squares and products of the state variables is finally

t t

<'l..XT(t)

>

=

J

J

.!!(t,a)

~(a,~) .!:!T(t,~) da~

• (3.13)

o 0

Equation (3.13)yields the means corresponding to a given transit time t. For results at the target plane, t should be taken as the nominal time T of arrival there. Actually, of course, the ensemble of vehicle positions corresponding to thisfixed time forms a set of points scattered about the target plane, but the means should be qui te accurate, since the errors are of both signs and tend to cancel out. Equation (3.13)Jis the main result of-this section, and provides the basic formula for computing results. Note that it contains one ingredient"R, that depends on the turbulent field, and another, ~, that is purely a system property.

To assist in understanding what (3.13) means, consider the simple case in which there is only one input, say vertical gust w, and one output, say normal displacement error zN; i.e.,

~ = [wJ

Then (3.13) gives the mean square of zN directly as

h (t,a)R _{zw ,ww} (a,~)h (t,~)da~. _zw (3.14)

Here h (t,a) is the zN error at time t that results from a unit impulse in zw

w at time

a,

and the cross-corrèlation

<

w(a) w(~)

>

is the ensemble mean of the products of w at the two different but fixed times

a

and~. Since the vehicle moves between the times

a

and ~ these values of w also pertain to different points in space (see Fig. 3.1). Thus

whereP is a point fixed by

a

and Q by~. We now assume, as is usual in other treatments of response to turbulence, that the turbulence statistics can with sufficient accuracy be taken as those associated with the reference solution. That is, at time

a

thereference position is P and at time·~ it is Q. Thus the values used in forming the mean product arg w{p

,a)

and w(Q ,~.), ~nstead

o 0

of w(p,a) and w(Q,~) (subscript '0' refers to reference solution). Since the error in the product for any.one trajectory of the ensemble is as likely to be positive as negative, the mean should be near enough to the true value -especially when the usual scat ter of correlation measurements is taken into account. Thus the correlation is taken to be

R (a,~) =

<

w(p ,a)w(Q ,~)

> •

ww 0 0

. The two fixed points P and Q , at which the measurements are made for times

o 0

a

and ~, are actually on the reference trajectory, which departs from the glide slope centerline by an amount that depends on the intensity of the wind shear and on the system response characteristics. It would probably be

~atisfactory in most cases to make measurements at the orthogonal projections

o~ P and Q on the nominal straight-line trajeetory, since the differences

o .0

would be expected to be small for a good landing system. Finally, the ergodie

praperty of the turbulence permits us to replace the ensemble mean of

(3.16)

with the time average

R _ww (a,~)

=

w(p,t)w(Q ,t+ ~t), ~t

=

~-a

.

0 (3.17)

The important point in the above derivation is that it leads to a measurement made with stationary probes, instead of moving ones. The result in (3.17) is

si~ly the cross-correlation of w measured at the points P , Q with time _o

0

delay ~-a. .This kind of turbulence measurement is the sort commonly and routinely made in many aerodynamics laboratories and which has been made in the experiments described.in the following section of this report. The general result, referring back to

(3.12)

is

(3.18)

To recapitulate,

(3.13)

is the basic equation from which the mean squares and products of the state variables of interest can be computed at the decision window. To carry out thecomputation one needs three things:

(i)

(ii) (iii)

The computed reference solutiopj

The computed matrix of impulse response functions of interest, . H;

The turbulence input matrix.R, either measured or otherwise approximatecl.

squares and mean produets of

YN'

zN' etc. at the target p1ane, and fina11y,these mean va1ues can be used with the assumption of a Gaussian process to compute probabi1ities P (see Fig. 1.2) associated with a given dispersion. In computing

these

probabi1i~ies

for a general point of the window in state space,one uses

the normal multivariate probabi1ity function (Ref.

5,

Eq. 2.6,31)), which makes

use of the mean products (the off-diagona1 components of (3.13)) as we11 as of the mean squares.

3.2 .The Input Vector

The degree of approximation-in the fina1 cam,puted resu1ts of Qispersion

at the decision window, and the difficulty of measurement, both depend very much

on the choice of gust vector ~ (see Ref.

5,

Sec. 13.3). The simplest case occurs

when the variation of the turbu1ence over the vehic1e is neg1ected, and the three turbulent veloeities at the e.G. are taken as the inputs. Then

For a morefaithful representation of the turbulence, one may add gradients in

the veloeities, such as àwjdy, which producesrolling moments and

dwjdx,

which

produces pitching moments. Which additional terms wi11 be needed in any parti-cu1ar app1ication depends on the vehic1e, the turbulence, and the accuracy desired. Although adding gradient terms to g comp1icates the measurements, there is an exception with respect to streamwise gradients. Assuming that the turbulence components are being measured in the FE coordinate system, which is most convenient, the corre1ations of derivatives with respect to x can be derived

from Tay1or's hypothesis of frozen turbu1ence. That assumption re1ates spatia1 a.nd temporal gradients by

W

₌

-where Wis the loca1 mean wind speed (in the x direction). Thus for ex~le,

w

x

1

- w

.

and the cross-corre1ation of this derivative at

ex

and ~ is

R (a,~)

=

ww 1 x x (3.20) (3.21) where W_p ,W

Q are the wind va1ues at the two reference points. But the

corre-o 0

lation of the time derivative of a variab1e is re1ated to that of the variab1e itse1f, Le.,

(3.22)

so that the correlation of w can be derived from that of w by

x

R _ww (0:,13)

=

xx

1

(3.23)

and similarly for ot her x gradients. The double 'differentiation required by

(3.23) can in principle be performed with sufficient accuracy if the R data

ww

is good enough. With respect to y gradients, no comparable general method is

available, and recourse would be needed to measurements with specia,l probes (for example a small airfoil that could measure rolling moment) or to theory

(for example Ref.

6).

Dl. EXPERIMENTAL MEASUREMENTS

In this section we describe the results of laboratory measurements

of some of the correlation data required to define the matrix R(o:,13) as given

in Eq. (3.18). With an adequate knowledge of this matrix and the appropriate

mathematical model of the particular aircraft under consideration, the theory of Sec. lIl, and in particular Eq.(3.13), may be used to obtain the desired mean squares and produets of the state variables • The measurements described

here were made in a small (8" x 8") pilot model of the modified UTIAS subsonic

wind tunnel described in Ref. 13.

4

01 Ass~tions

We are considering here the flight of the aircraft from A to B

(Fig. 1.1) in a laboratory simulation of the planetary boundary layer.

Be-cause of the small size of the tunnel used, only the simplest gust input will be considered; that is,

~

= [

so that, from Eq. (3.18),

-

~u2 (0:,13)

-R = V' lu2

-

V?' lu2

where the subscripts 1 and 2 are u(t) v(t) w(t)

]

-

~v2 v lv2

-

_"?'lv2 such that

,\w

₂

-

v lw2 w lw2

and

~= u( Q ,t "+ D.t),

o

In addition, we assume that the control system of the aircraft is such that the reference trajectory is approximately the same as the glide slope. Together with

the ass~tion of small perturbations of· the a,ircraft due to the turb:ulence,

this means we are asswning that the actual path of the vehicle is along the glide slope and it is here that the turbulence will be measured. Thus the points Po

and Q are the points on the glide slope defined by a and ~, respectively, and

the aîrcraft ground speed V

E (see Fig.

4.1).

That is,

and

(4.1)

and for any pair of points P and Q , we ma.y obtain the values of flight time

o 0

reCluired to reach these points if the ground speed is knawn.

In the present experiment, reasurements of only the diagonal components

of the R (a,~) matrix have been made, for various glide slope angles and vehicle

speeds.- These diagonal components are referred to as

R (a,~)

= u(p ,t) u(Q ,t +

D.t)

uu 0 0

Rvv(a,~)

=

v(po,t) v(Qo,t ~ D.t), D.t = ~-a

Rww(a,~)

=

w(p ,t) w(Q ,t + D.t)

o 0

or· in non-dimensional terms,

R

(a,~)

=

R

(a;~)

/ (

J

u2(P )

uu uu· 0

The denominator is of course simply the RMS turbulence level at each of the

two points, and may be expressed in terms of a and ~ using ECl.

(4.1).

4.2

Parameter Selection and Velocity Sealing

In Figs. 1.1 and

4.1

the velocity W(z) represents the time-averaged

mean velocity of the wind in the frame FE" If we restriet ourselves to the

ca.se of a .neutrally stable atmosphere and further neglect Coriolis effects , then

the variation of this velocity with height can be reasonably well represented by

(4.2)

where W

G is the gradient or free-stream velo city above the boundary layer, zG is the gradient height or boundary layer thickness, and n is the power 'law . index. Both zG and n depend on the roughness of the earth's surface, with typical values as suggested by Davenport (Ref. 7) given in the table below. While these values are by no means 'exact, they do give a .reasonable estimate of the mean wind speed if this speed is not too low. Further characteristics of the planetary flow are described in detail in Refs.

8

and

9.

SURF ACE TYPE n _{zG' FT.} Flat, open.

0.16 1000

country

Woodland forest 0.28 1300

Urban 0.35 1600

Consider now the motion of the aircraft along the glide slope as shown in Fig.

4.1.

It flies at airspeed V at same angle

r

relative to the mean wind W

so that its motion relative to the ground is at speed V

E and angle rE" Thus at any particular height z, we may obtain the vector diagram shown below:

W(z)

If we let W, V and V

E represent magnitudes of the respective velocity vectors, then the vector sum

Y

E

=

.

!!

+

;

Y

yields

V_E cos rE

=

-W + V cosy

(4.3)

and

(4.4)

where V

E' W, V and

r

are functions of z and

rE

.

is constant for a given flight (we will also choose V to be constant with height). SOlving, we get

(4.5) a..nd

I

=

sin-1 [VEsin/E/V] (4.6)

where Eq.(4.5) corresponds to the case of IE ~ 900 (V/W >1). Thus if IE' W and V are specified, the above equations may be solved for V

E and I'

As discussed in Sec. I, present maximum values of IE for STOL vehicles are ~ 160• Since in the present case we are concerned with both STOL and VTOL

vehicles, we select values of IE of 150,450 a..nd 900•

As

for W(z), its value is adequately specified by the value chosen for the gradient wind WGo This wind may range from zero to values of 120';".140 f'ps. Higher values are not of interest for the present investigation since this represents an upper'limit on STOL

landing airspeeds and we require that V/W> 1 for all W. Also, the lowerrange of W is not pertinent since shear-generated turbulence will be very small in this range and only thermal instability effects will be of prime concern. Finally,

we will assume constant values of the airspeed V for any given flight. Typical values are determined by vehic1e landing a,nd stall speeds and by maximum allowable

descent rates. Thus if we use the parameter V/W

G to represent airspeed values, the considerations outlined above lead to the following criteria:

(i) ~0-20 f'ps.

<

W

G

<

~ 120 f'ps.

(ii) Vma:x.'" 120 f'ps'~V/WG)ma:x. ~ _120/WG (iii) V /W

G >

1-These criteria are displayed graphical1y in Fig. 4.2. In the present case, we select a value of W

G

=

68 f'ps. as a typical strong wind speed. This yields a value of W ~ 39 f'ps at z

=

33 ft., whichis typical weather station monitoring height, and represents a fairly strong wind.

We may now use Eq. (4.2) to determine

W(z)

for a particular flow simulation.

Equations (4.5) and (4.6) then yield VE(z) and I(z) for any value of z with V/W

G and IE as parameters. The time t to reach any point on the glide path defined by height z is found as in Eq.(4.l); that is

t(z)

=

J

z ZA

dz

a..nd

a:

and ~ may be obtained from this function by selecting zp or zQ' Determina=

tion of the variables as functions of z was performed in this way for V/W

G

=

1.0, 1.2, 1.5 a,nd 1.9, except for the case of IE

=

90

0 (vertical descent) where the 1arger velocity ratios were excluded because they yield unreasonably large descent

rates. The results were tabulated as shown tYJ?ically in Table land, some tYJ?ical

plots are shown in Fig. 4.3. While these curves apply for a .full scale planetary

boundary layer, the appropriate values for a simulated flow may be easily determined

from a knowledge of the length ~d velocity scale factors of the simulation (see

Sec. 4.5).

4.3 Wind Tunnel Facility

The laboratory facility in which experiment al measurements were carried out

is a small, open~circuit wind tunnel which is driven on the ejector principle by

~ array of jets located across its cross-section. The veloeities of these jets

may be individually ~~d independently controlled so that by appropriate adjustment₉

flows possessing virtually any desired mean velocity profile can be produced in

the test section. In addition, the use of barriers and simulated surface rough~

ness allows turbulence ,levels characteristic of planetary bOQ~dary layer flows

to be obtained. The tunnel has an 8-inch-square cross-section (Le., H

=

8t!)

which allows boundary layer simulations up to about 7" thick. Primary !:j.ir is

supplied to the jets by a centrifugal blower via individuallines, in each of

which a variable area rotameter' is located to measure the mass flow. A schematic

layout of the turmel is shown in Fig. 4.4 and Fig. 4.5 is a photograph of the facility. In Fig. 4.6 the location of the test section and a tYJ?ical tunnel con-figuration used for producing a desired flow are shown. For complete details of the characteristics and capabilities of this 'multiple-jet' tunnel, the reader

is referred to Ref.

9.

4.4 Instrumentation alld Measurements Techniques

The basic in st rument at ion used in the present experiment is identical to

that used and described in Ref.

9.

Complete details can be foundin the

appen-dices of that report and thus only a brief r~sumé of this equipment is included

here.

4.4.1 Hot Wire P~emometry

All turbulent flow velocity measurements made in this experiment were ob-tained with four channels of DISA type 55DOl constant temperature hot wire

anemo-meters ~~d type 55D10 linearizers. The hot wire probes used are DISA tYJ?e 55E30

single wire probes for the longitudinal component and tYJ?e 55A38 miniature

cross-wire probes for both the longitudinal and lateral components. Temperature

com-pensation was used throughout'the experiment and it is estimated that the

experi-ment al error is only about 2-3% in the u-con:g;lOnent measurements and about 5%

in the v- and w-canponent measurements.

4.4.2 ,Analog COnputer

~ipulation of the linearized hot wire anemometer signals to 6btain desired outputs was performed on a PACE model 22lR analog computer. This computer is a

100 volt system in which a digital voltmeter having a ,10 mV resolution is used

for readout. It was also used to obtain time averages of all velocity signals from the anemometers by passing the signals through simple first-order low-pass

filter circuits. These circuits could also be used as high~pass filters and thus

removal of the m::;an values of the fluctuating signals was effected. A time

constant of 5 seconds was used throughout the experiment, resulting in a cut-off

DC removal in this marmer was applied to all velocity signals prior to corre-lation.

The only elements of the a,nalog computer that were used are potentiometers

and ~lifiers. The output of a typical ~lifier with groQ~ded input is

negligibly small, and the fre~uency response of two amplifiers in series with a

potentiometer was found to be flat to at least 10 KHz. Conse~uently, it is

concluded th at any error or distortion introduced by the analog computer is negligible.

4.4.3

Correlation and Spectral Analysis System

Autocorrelations and cross-correlations of the velocity components in this experiment were obtained using a Princeton Applied Research Model 100 Signal Gorrelator. This instrument produces time-delay correlation curves on-line for

~y two input signals for a maximum delay time T

max which may be selected between

0.1 msec. and 10 seconds. One hundred points on the correlation curve are

available in the form of analog voltages stored at the output of simple RC filter

networks. A Cimron Model 6480 Data Logging System is interfaced with the

corre-lator to monitor these volt ages, which are then punched on data cards by an IBM

526

Summary Punch. An IBM 1130 digital computer was then used to

non-dimension-alize,print a,nd plot the correlations so obtained.

Power spectral densities of the velocity components were obtained by Fourier

tr~~sformationof correlation curves using the digital computer, which

subse-quently was also used to plot the results • A schematic diagram of the con:q;:>lete

system, including the hot-wire instrumentation, is shown in Fig.

4.7.

4.4.4

Random Noise Meters

Root-mean-s~uare values of the velocity components were obtained using two

Bruel and Kjaer model

2417

r~dom noise voltmeters. These instruments provide

RMS values to an accuracy of 1% of full scale deflection for any input signal

over the range of frequencies from 0.8 Hz

(-3

dB) to 20 KHz. The time constant

of each meter may be varied from 0.3 to 100 seconds, a,nd a value of at least

10 seconds was used in all cases in this experiment. This corresponds to ~~

effective statistical record length of 20 seconds and from Ref.

9,

this is

sufficiently long that the statistical variability of the rms estimates is negligibly smalle

4.4.5

Correlation Matrix Measurements

Values of the correlation matrix,~ (a,t3 ) ",for various values of a and t3

(0 ~ a,t3 ~ T) were·obtained as follows. For R

Uu(a,t3), for example, a single

wire probe was placed at some fixed point P on the glidepath corresponding to

o

height zp from the floor of the tunnel. Another probe was placed at a point Q below it on the glide path,at height z . The fluctuating velocity signals

f~om

these probes were then

cross-correlat~d

using the P.A.R. correlation system as described above, yielding the correlation between the signals for 100 values

of delay time T of u(Q ) with respect to u(p ), 0

<

T

<

T (see, for example,

o 0 max

r---~~-~~-~---~---~--- -- ---

---•

P I:llld Q . for any particular airspeed V could then be found from the time tables

o 0

described in Sec.

4.2,

with the necessary scale factors for the simulated.flow

tl:lken into account. Consequently the time differe!lce 6t = ~-a ~ould be determined

~nd by selecting the value of the cross-correlation at T

=

6t, Ruu (a,~) was

ob-tainedo For the same pair of points P ~d Q but different airspeed, the time

o 0 ~

tables yield different values of

a,

~ and 6t such that the value of R _uu (a,~) was

obtained from the same cross-correlation curve but at a different value of T.

By moving only the ~ower probe to a different value of

Zo

&~d obtaining a new

cross-correlation, R (a,~) could be found forthe same

a out

differing~. Ob-Xiously( then, by.selecting appro~riate values of P and Q along the glide path,

~llU(a,~) could be determined for the entire r~ge o~ intergst of a ~d ~.

Based on the above procedure for findipg the values of

R

(a,~), it is estimated

that the maximum experiment al error ineach point should be no greater than about

12-14%.

The statistical variability of ~y value depends on its magnitude and on

the shape of the signal spectrum but should in general be considerably less than

the ab ove figure •

4.5

Wind Tup~el Flow Characteristics - Roof Tests

As mentioned previously, we consider in the present experiment a planetary

flow which is neutrally stabie ~d in which the effects of Coriolis accelerations

may be neglected. On this basis it can be concluded that theoretically correct

simulation of this flow may be achieved in the laboratory provided the lat ter flow

is aerodynamically rough*, relatively free from pressure gradients, ~d the mean

velocity and turbulence characteristics are correctly simulated (Ref.

9).

In

particular, we require that the integral scales of the turbulence be properly scaled. In the present tunnel, the aerodynamic roughness is provided by the base plates ~d cylindricql bricks (when necessary) of a 'LEGO'construction set, and proper simulation requires only that the mean velocity ~d turbulence be correctly represented.

The characteristicsof the laboratory simulation of the flow over flat,

open country (n

=

0.16)

are shown in Figso

4.8 - 4.12.

The mean velocity profiles

produced at the test section entrance are seen in Fig.

4.8

to follow the desired curve ·very weIl. Good lateral uniformity (i.e., two-dimensionality) has been

aehieved ~d little change in this profile was found throughout the test section.

The parameters of the desired curve are W

G'

=

34 fps and zG'

=

7". Thus wi th

respect to the full scale planetary flow

(W

=

68

fps, zG

=

1000

ft.), the velo-city scale factor is

1/2

and the length

sca~e

factor is

1/1715

for the present

simulation. These factors may be used to obtain from the time tables described

in Sec.

4.2

the vehicle flight times appropriate to this flow.

Turbulence intensity profiles for the three fluctuating components are shown

in Fig.

4.9

and agree weIl with corresponding atmospheric data. Typical power

spectral density curves are presented in Fig.

4.10

~d Re~~olds stress results

in Fig.

4.11.

In Fig.

4.12,

the longitudinal component integral scale is seen

to agree reasonably weIl with suggested atmospheric results and also with the results obtained in a boundary layer wind tunnel simulation of this flow. The vertical scale, however, agrees with suggested atmospheric data only in the lower regions of the flow and is considerably smaller than desired in the upper half of the tunnel. Since representing the vertical eddies in the simulated flow on too small a scale is a distinct disadvantage, particularly for aircraft applications such as the present investigation, some effort was made to increase these scale values.

In the planetary boundary layer, large scale velocity fluctuations in the vertical direction tend to be suppressed at lower heights due to the presence of the earth's surface. For this reason vertical component scales in this region are smaller than at higher altitudes and we find a distinct increase in L with

w

height, as indicated by the suggested atmospheric results in Fig. 4.12. The same effect should of course apply in the simulated flow and this accounts for the observed increase of L with height in the lower half of the tunnel. In this

w

flow, however, we also have asolid boundary at the top of the boundary layer and it is suggested that this is the probable cause of the low values of L found in

. w

the upper region of the flow. That is, unlike the full scale planetary flow, we have in this simulation no free stream region above the boundary layer, with the result that the presence of the roof suppresses large scale vertical fluctuations and thus prevents L fram increasing with height as it should. One possible

w

method of overcoming this difficulty would be to raise the roof to allow for a free stream region. This would, however, considerably increase the tunnel power requirements and in the present facility it would necessitate major reconstruction. Another approach would be simply to simulate the planetary flow on a smaller scale

(Le., ze}

<

7"), but in the present case the boundary layer thickness would be unusably small and in addition we would be left wi th too few jets in the boundary layer region of the tunnel to provide adequate control of the mean velocity

profile. It was therefore decided in the present case, due to time limitations, to perform some fairly crude experiments involving variation of the tunnel roof configuration in an attempt to remove its suppressive effect on L .

w

In order to carry outthe desired tests, the last three tunnel sections shown in Fig. 4.4 (s

=

36" to 76") were removed and replaced by a new 40" section with a removable roof. In this way the roof boundary condition could be easily

altered by simply exchanging various roof sections. The configurations investi-gated are shown and identified in Fig. 4.13.

The effect of the different roof types on the vertical component scale was determined by measuring L for each case in the test section entrance plane

(SiR

= 6.75). For the case

0'

the slotted roof· (No.2) several heights were

considered while for the remaining cases, the value at z'

=

6.5"

(z'/z

_G

=

0.93) was used to typify the roof effects. Scales were found using the spectral-fit

approach in which it is assumed that the vertical component power spectrum can be reasonably fi tted by a von Karman model spectrum. Then fram the location of the peak of the spectrwn (k ) along the frequency axis, L is determined using

the model spectrum result p w

L

=

w

0.106

k

This approach yields true integral sca1es on1y insofar as the measured spectra actually fo11ow the model spectrum shape, but in the case of the origina1 solid roof the agreement is seen to be fair1y good (Fig. 4.14). This is in fact

the only reasonable method for estimating sca1ein the present case, particularly when corre1ation curves have a.significant negative region. Further discussion of the various approaches for scale determination can be found in Ref.

9,

Appendix E.

The scale results obtained for the case of a slotted roof (No. 2, Fig. 4.13) are shown in Fig. 4.12 where it is evident that no significant change has resulted in the lower region of the flow. In the upper section, good data could not be obtained. The reason for this stems from the fact that the use of the slots in the roof resu1tèd in a serious 'beating' phenomenon in this region of the flow. This phenomenon cou1d be felt by an observer by placing a hand above the slots, and its presence is c1ear1y seen in the spectral measure-ments as disp1ayed in Fig. 4.14. It is obvious that the spectrum no longer

resernb1es the von Káxmán model, and comparison of the sca1es obtained by the above technique from these two spectra is virtually meaningless. The possible aIternative of using correlations rather than spectra to determine scales is prec1uded in practise by the large negative region of the appropriate

auto-corre1ation curve as shown in Fig. 4.15. In addition to the measurement problem, of course, is the fact that aspectral shape including this low frequency peak is entire1y unacceptable as a simulation of atmospheric turbulence.

Initia1 attempts at removing the beating phenomenon from the flow involved the use of screens placed over the slots in an attempt to break up the organized pattern which was producing it. This procedure in fact had no major effect on the spectra1 peak save for shifting it slightly along the frequency axis (Fig. 4.14). That is, for the slotted roof with no screens (Case NO.2), the frequency of the peak was rough1y 16 Hz, while with the coarse screen (Case No.3) it was about 22 Hz and with the fine screen (Case No.4) about 31 Hz. lts magnitude, however, was relative1y unaffected by the screens. Suspecting that the beati~g might somehow be associated with the size or shape of the slots, we tested a roof with holes rather than slots (Case No.5). This resulted in removal of the beating but only at the expense of returning the flow, including L , to vir-tually·the same condition it was in with the origina1 solid roof. Whe~ the roof was removed altogether (Case No.6) the spectral peak reappeared at ~ 19 Hz, and when a fine screen was added (Case No.7) it was merely shifted again to about 30 Hz. Finally, slots normal to the flow direction were tried (Case No.8) by plaeing 2" wide slats across the open roof at spacings (d) ranging from 1/2" to 3". In this case, no peak was observed when the probe was located at z'

=

6.5" direct1y below a slat, but it reappeared at about 38 Hz if the probe was below the apen space when d was 3". In the former case, however, the scale was again

no larger than that obtained with a .solid roof.

The exact cause of the beating phenomenon described above remains unknown, a1though we may speculate on its origine The fact that it appears to be associated with those configurations for which no solid boundary exists between the flow

inside the tunnel and the laboratory air outside leads one to consider the inter-mittency found at the edge of shearlayers, and in particu1ar in.the outer regions of a turbulent boundary 1ayer. This intermittency has been discussed by Klebanoff (Ref. 10) and stems from the fact that the boundary between the turbulent and

non-turbulent flow is qui te sharp a,nd has an irregular, constantly changing shape such as that shown in the sketch below. Thus a probe located at a point such asx

Free stream

-...

N~nal

_t

₎

~)

)

" ' - _L~er Boundary\ ( \

I

'"

\ Turbülent

(

<- _Thickness Flow

~

'-

l

~

'"

_'-

)

alternately sees a.turbulent and non-turbulent flow. Klebanoff's results suggest

an average wavelength for this irregular outline to be about twice the boundary Iayer thickness, which in the present case would be about 1.3 ft. If the

outline were then assumed to be frozen ~d carried past the probe with the free

stream velocity ( '" 32 f'ps), the resulting intermittency would have a .. frequency

of roughly 25 Hz, ~d this is in the general region of the frequencies observed

for the spectral peak. As a .further investigation of this intermittency,

oscilloscope traces of the turbulent signal just above the level of the tunnel roof with no roof present were studied. Intermittent bursts of turbulence were indeed observed here with a frequency of '" 15-20 per second, suggesting that

there might in fact be a correlation between this phenomenon and the beating

that was observed. However, it seems extremely unlikely that this effect could remain significant when screens are located between the tunnel flow a,nd the laboratory air as for Case No.7.

In the present experiment, there was unfortunately no further time available to pursue other possible investigations into either the beating phenomenon or the problem of the lew values of L near the roof. These investigations are

strongly recommended for future re~earch. For the present, the solid roof

con-figuration was accepted as the most reasonable a,ndwas'used for all the

4.6 Results ~d Discussion

4.6.1 Two-Point Cross-Corre1ation Data

In Fig. 4.16 we see a typical two-point time-delay cross-correlation

curve for the 10ngitudin~1 velocity component for

rE

=

90

0 • It is c1ear1y seen

here that the maximum corre1ation between the two points occurs at a value of time de1ay other than zero, which is where it wou1d occur if the flow were homo-geneous. Rather, the peak is found at a positive lag time T , which represents

m

a de1ay of the signa1 from Q with respect to that from Po (see sketch). If we

o

assume frozen flow

W(z)

th en we may trans late this time lag into the spatia1 separation

-/::, x

=

W (zQ)' T

111 m

Thus the line Po QM in the sketch may be considered the line of maximum

corre-lation at this 'location in the flow, ~d its slope is given by /::,z .and /::, xm' In

a homogeneous flow, this line would be normal to the floor, of the tunnel but in

a boundary layer flow it has a slope which may be a .function of z.

If we consider va1ues of

rE

1ess than

90

0, we u1timate1y wi11 re ach

a situation in which the1ine P _o 'Q (see sketch) wi1l have a slope smaller than

0

that of the 1ine of maximum corre1ation, Po~' In this case, we wou1d expect

the peak of the time-de1ay cross-corre1ation curve to appear 'at a negative value

of T. This of course corresponds to a de1ay of the signa1 at P , with respect

m 0

to that at Q rather than vice-versa. This is in fact the case, as indicated by

o

the resu1ts shown in Fig. 4.17 for the case of

rE

=

450• In the present

experi-ment, we are interested on1y in de1ays of the signa1 from Q with respect to that

o

from P (i.e.,T

>

0) and thus in most cases other than

rE

= 900 we do not observe

o

-a pe-ak -at -all in our r-ange . of interest. For more resu1ts and discussion of

optimum time-de1ays for maximum two-point corre1ations in a boundary 1ayer flow,

The curves of Figs. 4.16 and 4.17 are typical of all the raw correlation data obtained for the three velocity components and the three values of rE used in this experiment. Curves of this type have been used to obtain the final flight

path correlation data, R(a,~), as discussed below.

4.6.2 Flight Path Cross-Correlation Results

As described above the present experiment al work in the pilot wind tunnel facility involves a large number of two-point space-time turbulent velocity

com-ponent. correlations (approximately 1000) within the simulated planetary boundary

layer. Spatial separations of the probes were determined b~ the desired §lide

path geometry with fixed aircraft descent angles of rE

=

15 , 450, and 90 being

selected for the present study. Cross-correlations covering a rarge of time delays

(positive time delays of lower·measurement with respect to the upper measurement,

generally) were obtained with the correct time delay for each point pair, being

selected later (as ~escribed in Section 111 - Theory) to develop the final flight

path correlations,. R(a,~). Of the basic correlation information, two typical

plots only (for

R

T

are shown, Fig. 4.16 and Fig. 4.17, in non-dimensional form.

uu

The derived flight path correlations,obtained with a mean wind variation

corresponding to a power'law index of 0.16, are plotted beginning at Fig. 4.19

and continuing through to Fig. 4.66. The following cases have been covered for this mean wind speed variation:

~

i)

R , R and R turbulent cross-correlations

uu vv ww

ii) Flight path approach angles, rE

=

150,450,900•

iii) Aircraft velocity ratios, V/Wg = 1.0, 1.2, 1.5, 1.9.

In addition, initial measurements, with a mean wind speed variation corresponding

to a power'law index of 0.35, are also included, Figs. 4.77 through 4.80. For

this wind profile the following data have been obtained:

i)

R turbulent cross-correlations

uu

ii) Flight path approach angle rE 450

iii) Aircraft velocity ratios V/Wg = 1.0,1.2,1.5,1.9.

In all cases the derived flight correlations correspond to simulated aircraft landing approaches flown at constant air speed along straight-in approach paths. These particular flight constraints result in a continuously increasing aircraft ground speed as the target plane is approached (see Fig. 4.3). The turbulenèe characteristics of the shearlayer, as simulated and measured in this facility,are presented in Figs. 4.9 through 4.12.

The desired flight path correlations, ,~ (a,~), form, in general,

a warped surface over the à,~ plane with the final aircraft flight dispersio~s

being obtained by a double (area) integration in this plane (see Eqs. 3.13 or

3.14) over the range 0

<

a

<

T; 0

<

~

<

T. In a homogeneous turbulent field,

the.R

(a,~)

surface

co~d

b; described-in terms of a .single curve if plotted as