NBSIR 1^-111

The Dynamic Response of

Helicoid Anemometers

J. M. McMichael and P. S. Klebanoff

Aerodynamics Section

Mechanics Division

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

November 1975

Final

Prepared for Federal Highway Administration

Office of Research

Structures and Applied Mechanics Division

Washington, D. C. 20590

NBSIR 75-772

THE DYNAMIC RESPONSE OF

HELICOID ANEMOMETERS

J. M. McMichael and P. S. Klebanoff

Aerodynamics Section

Mechanics Division

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

November 1975

Final

Prepared for Federal Highway Administration

Office of Research

Structures and Applied Mechanics Division

Washington, D. C. 20590

U.S. DEPARTMENT OF COMMERCE, Rogers C.B. Morton, Secretary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler. Acting Director

LIST OF SYMBOLS

a amplitude of velocity square-wave - ^

e base of natural logarithms

exp (subscript) experimental value

f function defined by Equation (13)

fps feet per second

i square-root of minus one

k,l,n summation indices

t time

u dimensionless velocity fluctuation

y,z normalized transverse test-section axes

A constant defined by Equation (7)

B coefficient of bearing friction defined by Equation (2)

C Fourier coefficients of periodic velocity fluctuation, defined

n

n

n

by Equation (23)

C amplitude of C

' n ' n D Fourier coefficients of rotation rate, defined by Equation (34)

n

Id I amplitude of D

' n' n D^^^ Fourier coefficients of the i^^ term in the perturbation expansion

defined by Equation (25)

E instantaneous voltage output of Gill anemometer system

E average value of E

E Fourier coefficients of the anemometer voltage, defined by

Equation (65)

|E I amplitude of n^^ harmonic of the anemometer voltage

retarding torque on anemometer rotor due to bearing friction

G(^) nondiinensional aerodynamic torque defined by Equation (3)

I moment of inertia of anemometer rotor

K meter factor defined by Equation (13)

anemometer voltage sensitivity defined by Equation (64)

L distance-constant

R characteristic radius of anemometer rotor

S rate of rotation of anemometer rotor

steady flow equilibrium rotation rate

S* rotation rate defined by Equation (18)

5 average rate of rotation

T aerodynamic torque on anemometer rotor

U instantaneous air speed

U average air speed

U constant defined by Equation (14) o

g coefficient defined by Equation (11)

6 coefficient defined by Equation (22) o

e fundamental amplitude of velocity fluctuation relative to U

th

phase of n harmonic of velocity fluctuation

(I) angular frequency

density of air stream

a relative fluctuation in rotation rate, Equation (17)

(ii) j-j... J. th . , , . . . a coefficient of n term m the perturbation expanion for a

T time constant

T time constant defined by Equation (21) o

E, dimensionless rotation rate defined by Equation (4)

5^ asymptotic value of ^ as U -> <».

iii
LIST OF FIGURES

1. Characteristic Curve of C/^^ a Function of Flow Speed.

2. Schematic Steady Flow Calibration Curve Defining S* and a,

3. Theoretical Over-Registration of Mean Speed.

4. Theoretical Phase Dependence of Ratio of Second Harmonics.

5. Patio of Second Harmonics with = 0.

6. Ratio of Second Harmonics with = H.

7. Theoretical Square-Wave Response.

8. Configuration of the Instruments in the Test-Section.

9. Horizontal and Vertical Distributions of Free-Stream Velocity U = 35 fps.

10. Horizontal Distribution of Free-Stream Longitudinal Turbulence

Intensity, U = 35 fps.

11. Instrumentation for Measured Dynamic Response Characteristics.

12. Domain of Mean Speed and Frequency Used in the Experiments.

13. Domain of Fundamental Amplitude and Frequency Covered by the Experiments.

14. Domain of First and Second Harmonics of the Velocity Fluctuation.

15. Steady Flow Calibration Curve.

16. Attenuation of the Response Fundamental.

17. Comparison of Theoretical and Experimental Values of Response Fundamental

Amplitude, Neglecting Bearing Friction.

18. Comparison of Theoretical and Experimental Values of Response Fundamental

Amplitude, with Bearing Friction.

19. Comparison of Measured and Predicted Overspeeding Errors.

20. Response of Second Harmonic.
CONTENTS

Page

ABSTRACT 1

1. INTRODUCTION 1

2. ANALYTICAL CONSIDERATIONS 3 2.1 Torque Balance 3

2.2 The Driving Torque - Dimensional Considerations 4

2.3 The Response Equation 6

2.4 The Condition of Steady Flow Equilibrium 7 2.5 The Dynamic Response Equation 7 2.6 The Perturbation Solution for the Dynamic Response 8

2.6.1 The Perturbation Expansion 9

2.6.2 The Steady-State Solution 9

2.6.3 The Over-Registration of Mean Speed 10

2.6.4 The Amplitude of the Response Fundamental 12

2.6.5 The Amplitude of the Response Second Harmonic 13

2.6.6 The Response to a Square-Wave Velocity Fluctuation 14

3. EXPERIMENTAL APPARATUS AND PROCEDURE 17

3.1 The Apparatus • 17

3.2 Range of Parameters 18

4. EXPERIMENTAL RESULTS 19

4.1 The Steady-Flow Response 19

4.2 Determining the Distance-Constant 19

4.3 The Overspeeding Error 21

4.4 The Response Second Harmonic 22

5. CONCLUSIONS 22

6. ACKNOWLEDGEMENTS 23

7. REFERENCES 23

V

THE DYNAMIC RESPONSE OF HELICOID ANEMOMETERS

J. M. McMichael and P. S. Klebanoff

ABSTRACT

The results of an analytical and experimental investigation

of the dynamic response of a helicoid anemometer are presented.

The experimental investigation was conducted using the NBS

Unsteady Flow Facility and data are presented which illustrate

the dynamic behavior in a spatially uniform, fluctuating flow

with varying amplitudes, frequencies, and mean velocities.

An analytical model governing the dynamic response is also

presented and compared with the experimental results

Key Words: Air; analytical; anemometer; dynamic response;

experimental; lag; unsteady flow.

1. INTRODUCTION

The importance of such problems as the effect of wind loading on structures and buildings under the action of a variable wind, turbulent

transport processes in the atmosphere as they relate to pollution, the

gathering of climatological and weather data, the need to further our understanding of the properties of atmospheric turbulence, etc., has

emphasized the necessity of improving the ability of anemometers to

provide accurate measurements of variations in wind movement, and to

reliably reproduce the true wind speed.

One approach to this problem has been to obtain the necessary information by physical modeling in wind tunnels. Although physical modeling

still appears to be a most useful approach for studying under controlled

conditions the variety of wind effects that may occur, it has become

fully evident that there is a need for full-scale studies. This is

particularly true with respect to understanding the effects of wind on structures and buildings, not only to establish the correctness of

correlations between full-scale and model tests, but because it is not

realistic to expect to simulate in the wind tunnel the lower wave numbers

of atmospheric turbulence pertinent to appropriate dynamic modeling. In

this connection it would be preferable, if one were limited to wind tunnel

modeling, to approach the problem by studying the response of a structure

to certain discrete frequencies and stepwise changes in velocity.

It is therefore important that reliable mean velocity and turbulence

measurements be made in the field. Mechanical-type instruments such as
rotary anemometers, are the most widely used, and will continue to be the

principal Instrumentation for gathering field data. Although rotary

anemometers were Initially conceived primarily for the measurement of

mean speeds, the need for atmospheric turbulence data has led increasingly

to the application of these instruments for this purpose. An understanding

of the dynamic response characteristics of anemometers is therefore essential.

In steady flows rotary anemometers adjust to a rate of rotation such

that the aerodynamic forces on the rotor element produce a sufficient

driving torque to balance the retarding torques, the latter arising from

bearing friction, signal generator resistance, and fluid friction acting

on the rotor. In unsteady flows the net torque at any instant produces

an angular acceleration of the rotor which depends upon the inertia of the

rotor. The mechanical inertia of rotary anemometers leads to an effect

termed inertial averaging, where the dynamic response exhibits an attenuation dependent upon frequency, inertia, flow velocity, fluid density and

viscosity, and the physical and aerodynamic characteristics of the rotor.

In principal, the dynamic behavior of any rotary anemometer could be

predicted from full knowledge of all these factors. However, the detailed

aerodynamics of anemometer rotors is imposingly complex, and all theoretical

treatments of the problem to date rely upon selected models based upon

certain simplifying assumptions and requiring the evaluation of certain

empirical constants. Nevertheless, useful information can be extracted

from the various models in the form of explicit functional dependence of

the response upon the primary independent variables.

All of the theoretical models proposed to date (1-7) reveal that the

driving torque is nonlinearly related to the flow velocity. However, the

lack of an adequate understanding of the effect this nonlinear behavior

has on atmospheric turbulence measurements, in particular on atmospheric

turbulence spectra, has limited experimental efforts (8-11) to make such

measurements to utilizing linear systems theory and the system transfer

function for the linearized response equations to correct for inertial

averaging. However, there does appear to be a fundamental reason to

question the adequacy of linear assumptions in as much as it is well

known that mean flow velocities obtained from rotary anemometers in

fluctuating winds characteristically exhibit an over-speeding or over- registration introduced by the nonlinear nature of the dynamic response.

Since the pioneering experimental research of Schubauer and Adams

(1) established that the time constant for rotary anemometers varies

inversely with the flow speed, it has been common practice to characterize

the dynamic response of rotary anemometers in a simple wind tunnel test

by determining the "distance-constant" using the "locked-rotor" technique.

In this method the rotor is released from rest in a steady flow and the

time-constant of the exponential rise in its rotation rate towards its

equilibrium value is determined, which together with the flow velocity

2
determines the distance constant. On the other hand very little has been

done in studying the dynamic response in controlled fluctuating flows, and

what has been done (2,4,12) has been confined to the phenomenon of over- speeding and its dependence upon the velocity amplitude and frequency with

no attention being given to the higher harmonics of the fluctuations in

the anemometer response.

In the present report the results of wind tunnel tests of the dynamic

response of a Gill helicoid anemometer. Model 27002,* conducted in the NBS

Unsteady Flow Facility over a range of known frequencies, amplitudes, and

mean velocities are presented. The primary purpose is to determine to

what extent the linear system transfer function adequately represents the

observed response, and to what extent, if any, the inherent nonlinearities

of a helicoid anemometer may affect the response.

In addition to the experiments, a dynamic response equation which is

believed to be applicable to rotary anemometers of various types is presented. The equation exhibits the essential nonlinearity which is inherent

in all rotary anemometers, but Reynolds number effects are neglected

although the retarding torque due to bearing friction and signal generator

resistance is incorporated into the model presented.

An approximate solution based upon a second-order perturbation

expansion is presented for the nonlinear dynamic response to arbitrary

periodic flows. Expressions for the over-speeding error and the response

harmonics are compared with the experimental results.

Dimensional quantities throughout this report are expressed in

engineering units which are customarily used in this field.

2. ANALYTICAL CONSIDERATIONS

2.1 Torque Balance

The general torque balance for rotary anemometers may be written

where I is the moment of inertia of the rotor, S is the instantaneous

angular velocity, t is time, T is the driving torque, and F is the retarding torque..

It is assumed at the outset that aerodynamic friction may be neglected

The remaining retarding torques arise from bearing friction and signal gen-

*This particular instrument was selected as being representative of a given class of rotary anemometers and this selection does not represent an endorsement
erator resistance, and it is assumed that these may be combined and expressed

as follows:

F = BS , (2)

where B is a constant resistance coefficient. This relation should suffice

for lubricated bearings, including the effective resistance of the signal

generator, and has been employed by Onuma (13). Starting friction is ignored

since attention shall be restricted to air speeds in excess of the starting

speed.

Many expressions for the driving torque have appeared in the literature

(1-7) with varying degrees of sophistication depending upon the particular

model adopted for the aerodynamic forces on the rotor. Some of the models

account for Reynolds number effects while others simply proceed from dimensional arguments and the assumption of frictionless flow. However, all of

the models indicate that the driving torque is nonlinearly related to the

flow velocity, and it is this aspect upon which the following analysis is

focused

2.2 The Driving Torque - Dimensional Considerations

The view has been expressed by Onuma (13) and Jepson (2), among others,

that both cup and propeller-type anemometers obey the same general response

equations. The plausibility of this contention may be established from

dimensional considerations.

For frictionless flow the driving torque must depend only upon the

following variables

T = T(p , U, S, R)

a

where p is the fluid density, R is a length characteristic of the physical

dimensions of the rotor, and U is the instantaneous air speed. The flow

about the rotor is assumed quasi-steady so that T is independent of the

time derivatives of U. In nondimensional terms

Pa

uIrJ = G(0 , (3)

where

^ = ¥ • ^""^

To develop an expression for G(^) , one proceeds from the observation

that the steady flow calibration curve (see for example Figure (15)) for

rotary anemometers approaches a straight line as U increases. This means

that the calibration data, when E, is plotted as a function of U, character4
istically exhibits the form indicated in Figure 1. As U increases ? asymptotically approaches a constant value 5 > which is simply related to the

meter factor K,

In steady flow without friction, T = 0 and C = over the entire range of

U. It follows that G(Cq) = 0. When friction is present or when the flow

fluctuates a driving torque is produced by departures of C from Cq, and a Taylor series expansion may be written for G(C) as follows;

9G

3? ^o^ + (5)

provided that the derivatives are continuous.

To lowest order then, the driving torque may be written

T = A U2 (K - :|) (6)

where

(7)

is a positive constant. Equation (6) contains the inherent nonlinearity

in the flow speed to which the overspeeding known to occur in all rotary

anemometers may be attributed. The effect of this nonlinearity on anemo- meter response in unsteady flows is explored in subsequent sections of

this report.

In support of the differentiability assumed above it may be noted that

the resulting expression for the driving torque (Equation (6)) is identical

in form to the expression obtained empirically by Ower (4) , the expression

obtained from momentum arguments by Rubin, et al. (3), and the expressions

obtained from airfoil analysis by Rubin, et al. (3) and by Jepson (2),

neglecting aerodynamic friction. The advantage of the more elaborate

models, of course, is that the dependence of the aerodynamic driving

torque upon rotor geometry and aerodynamic characteristics is more

explicit. Nevertheless, these models still require the experimental

evaluation of aerodynamic force coefficients, and there is no loss of

generality by adopting Equation (6) and regarding A as an empirical con- stant to be determined.

While retaining only the lowest order term in the above expansion

would suggest that the resulting torque expression is limited to modest

5
departures of E, from 5 » this assumption appears to be no more restrictive

than the assumption of modest angles of attack for the rotor blades which

underlies the models adopted by Jepson (2) and Rubin, et al. (3)

2.3 The Response Equation

Substitution of Equations (2) and (6) into Equation (1) yields the

following response equation

where K is defined by Equation (13)

In the case of a step change in U at time t the response equation may

be written

T II

+ (1 + 6)S = KU , (9)

where the "time-constant" t is defined by :^

and

U is understood to be the constant flow speed for t > t . Neglecting friction,

it is clear that the exponential response in this case is characterized by a time-constant which varies inversely with the flow velocity, and that the

product

L = tU = I (12) A

\

' ' is a constant. L has been termed the "distance-constant" by Schubauer and

Adams (1) who, having obtained Equation (12) empirically, deduced the form

of Equation (8) (without friction) by assuming the driving torque to be

proportional to (KU - S) , the departure of the rotation rate from its

frictionless equilibrium value.

The time constant of the transient response including bearing friction

is given by

6
which implies that L as determined by the locked-rotor technique and Equation

(12) may be slightly in error unless B/A is sufficiently small compared to U.

For example, in the wind tunnel tests described in detail in later sections

of this report it was found that B/A = 1 fps so that if U = 10 fps in a locked rotor test a 10 percent error in L would result unless the results were corrected as indicated by the above equation.

2.4 The Condition of Steady Flow Equilibirum

When the flow velocity U is held constant, the rotation rate reaches

an equilibrium value S such that dS/dt = 0. Applying this condition to

Equation (8) it follows that

S =

e

KU^

U +

- f(U) (13)

As U increases the calibration curve, S (U) becomes a straight line of

slope K (see Figure (15)). The equation of this straight line asymptote

may be expressed as

S = K(U - U ) e o

(14)

where Uq is the velocity intercept when Sg = 0. For large values of U such

that B/AU << 1, Equation (13) is approximated by

^e

~~ "^"(1 - AU>

•

Comparing this to Equation (14) it follows that

o A

(15)

Thus two of the three coefficients in Equation (8) may be determined for a given anemometer by a simple steady flow calibration. The remaining co- efficient is the distance-constant which must be determined dynamically.

2 . 5 The Dynamic Response Equation

When Equations (12) and (15) are substituted into Equation (8) the

latter may be written

L + (U + U )S = KU2

at o (16)

While Equation (16) is a linear first-order equation with non-constant

coefficients, the response S is nonlinear in the sense that if Si(t) is a solution when U = Ui(t), and S2(t) is a solution when U = U2(t), then

7
Si + S2 is not a solution when U = + U2.

Direct integration of the equation is possible in principal for arbitrary

functions of time, U(t) . The exact integration has been presented by Onuma

(13) in the case of a purely sinusoidal velocity fluctuation about some mean

value.

Solutions for the case of arbitrary periodic velocity fluctuations have

not appeared in the literature heretofore. In order to develop an approximate solution in this case it is convenient to express Equation (16) in

terms of the variable a defined by the following equation:

S = S*(l + a) (17)

where S* is given by

S* = f(U) (18)

and f is defined by Equation (13) . These definitions are illustrated

graphically in Figure 2. The instantaneous velocity may be expressed as

U = U(l + u) , (19)

where U is the temporal mean flow velocity, and u is the relative velocity

fluctuation.

Substitution of Equations (13), (17), (18), and (19) into Equation (16)

yields the following dimensionless dynamic response equation

+ (1 + 3^ + u)a = (1 + 23^)u + (1 + 3^)u2 , (20)

o dt o o o

where

and

= ^ . , (21)

u

i ° u

(22)

2,6 The Perturbation Solution for the Dynamic Response

It is noteworthy that there are only two coefficients, 6 and t , in

the dynamic response equation which require experimental determination.

3 is obtained with the aid of the steady flow calibration, and t derives

from the distance constant obtained in a dynamic calibration. This observation represents a substantially simpler view than that proposed by Wyngaard

(5) wherein as many as four torque derivatives must be measured experimentally in addition to the distance constant.

8
The perturbation solution developed in this section assumes a general

periodic velocity_fluctuation, and the results obtained are subject to the

assumptions that U remains different from zero and the relative amplitude

of the velocity fluctuation remains less than unity.

2.6.1 The Perturbation Expansion

The magnitude of the velocity fluctuation is characterized by e, the

relative amplitude of the fundamental component of the Fourier series

expansion

_ e ~ inwt , . u = -r- L C e (23)

2 n n = -oo

th where |c | is the amplitude of the n harmonic, 03 is the angular frequency,

and by definition

C = 0 and C, = 1 . (24)

o 1

The assumption is made that e is sufficiently small that a may be expanded

in a power series as follows:

a = a + so + e o + . (25)

Substitution of this expression into Equation (20) , recognizing .that u is

of order e, leads to the following hierarchy of equations for a :

(2) QU

T do ^ (1 ^ 3 )^(2) ^ (1 + 3

)(U)2 _ ^(1) (28)

o dt o o e e

and so on. The solution is sought through second order only.

2.6.2 The Steady-State Solution

Equation (26) has the trivial steady-state solution

.(0) = 0 (29)

The Fourier series expansion for a^^^ may be written

^(1) ^ 1 ^ p(l) ^ina)t

2 n n

9
Substituting Equations (23) and (30) into Equation (27), and equating

coefficients for each value of n, one obtains

(1 + 26 )C

D^l) = ° ^ Ml)

o o

(2) Similarly, the Fourier series expansion for a is

z n

n = - 00

The left side of Equation (28) becomes

: I

? ? ^[(1 + 3^)0^ -D^l^e^('^-^^>'^^ . £ = _oo le = _oo

Equating coefficients such that n = k + 8. leads to the result:

(2) 1 CO C [(1 + 3^)C - D^l>]

D = T ^ ^ ^ , , . (33)

Substituting Equations (29) through (33) into Equation (25) yields the

second-order solution for a:

a = f

E D e^^^'^^ , (34)

2 n

n = -00

where

D = /-, ^ o N^u. • {(1 + 26)0+1 Z C-C . n (1 + 6 ) + inojT 1 o n 2 , k n - k

O O I k = -00

(1 + 26 ) (1 + e J

-

o' (1 + 6 ) + ikwT

O o

(35)

In the following paragraphs this solution is examined for the overspeeding

error, the response fundamental and the response second harmonic. These ex- pressions are compared to the experimental results in Section 4 of this report.

The theoretical response to a square-wave velocity fluctuation is also discussed below.

2.6.3 The Over-Registration of Mean Speed

The mean value of the response, denoted a, is obtained quite simply from

the term in Equation (34) corresponding to n = 0,

a=|D^ (36)

10
where, from Equation (35) and (24), is given by

D = ^ Z C, C , o 2 , k -k

k = -°°

r- 1 + 2E 1 - ( 1 + 6 (1 + 3 ) + ikojT o o o

srving that because u must be real = C* the conjugate of C^, and by

ing corresponding positive and negative values of k in the sum above, a Obseparing

may be written

(37)

In the case of a pure sinusoidal velocity fluctuation = 1 and C^^ = 0

for k > 1, reducing Equation (37) to

— e' a = J

1 + 26

1 -

(1 + 6^)-^ + (WT^)' (38)

The exact expression for a obtained by Onuma (13) in this case may be reduced

to Equation (38) by assuming that the amplitude of the velocity fluctuation

is small compared to the mean velocity.

When friction is negligible, 6^ 0, and Equation (38) becomes

a =

2 1 + (a)T )2

• o (39)

To illustrate the dependence of the overspeeding error upon e and wt , a, as given by Equation (39), is plotted as a function of e in Figure 3 wi?h wt

as a parameter. At high frequencies the maximum possible error is

In order to compare measured values of the mean rotation to those given

by this solution, Equations (17) and (18) are averaged and combined to yield

o

By measuring the true mean speed, U, knowing the d.istance constant and the

steady flow calibration intercept, the values of S predicted by Equations

(37) and (40) can be computed and compared to the measured values presented

in Section 4.

11
2.6.4 The Amplitude of the Response Fundamental

From Equation (35) the magnitude of the response fundamental relative

to the input fundamental can be written

c7

(1 + 26^)

/(I + e )^ + ((OT )^

o o

k, = -00

1 +

1 + 23 (1 + I ) + ikwT

o o

(41)

If the disturbance amplitude e is sufficiently small the summation may be

neglected leaving only the simple linear first-order transfer function as given by the factor outside the absolute value symbols . For larger values

of e contributions to the response fundamental amplitude arise from the

second and higher harmonic amplitudes of the velocity fluctuation. However,

these contributions remain small so long as these input higher harmonics

remain small compared to unity.

To estimate the degree to which the linear first-order transfer function

suffices to describe the response fundamental amplitude, consider that

1 +

1 + 23 (1 + 3 ) + ikwT

o o

< 2 for all k.

This is so because 3^ is always positive. It follows that

1 + 23

v'd + 3 )2 + (cot )2

o o

1 + e k = -co IVI - kl

Suppose that, as is the case for the experiments, C^^ ~ 0 for k > 2. The

correction term, i.e., the summation term, may be written

k = -00 k 1 - k ' '2

Hence, as long as the input second harmonic amplitude, e |C2[> remains

small compared to unity the linear first-order transfer function adequately

approximates the attenuation of the response fundamental. Thus,

Dl 1 + 23

/(I + 3 )2 + (WT )^

o o

(42)

Furthermore, when 3 << 1 the first-order transfer function may be

o written

D

/I + (WT )^

o

(43)

12
This curve is shown in Figure (16) along with the measured values of the

transfer function.

2.6.5 The Amplitude of the Response Second Harmonic

The transfer function for the response second harmonic is quite different

from that of the fundamental. In the general situation where the velocity

fluctuation contains a second harmonic, the response second harmonic is not

given by the linear first-order transfer function which suffices for the

fundamental. Rather, the response second harmonic is quite sensitive to

the input fundamental as well, and even in the absence of an input second

harmonic the nonlinear nature of the response is such that a response second

harmonic is generated by the input fundamental. This can be demonstrated

by the solution given by Equations (34) and (35) in the following way. First,

one may write for the magnitude of the second harmonic

1 + 23

V o o

£ 00 ^2 2 , ^ V2 - k

1 +

1 + 26 (1 + I ) + ikojT o o

Again, the linear first-order transfer function has been factored outside the

absolute value sjmibols. The summation term is not generally negligible.

Assuming as before that Cy^ = 0 for k > 2, the summation may be reduced to

"l + 6

1 + 23 (1 +

1

5 ) + ioJT o o

since C =0 and = 1 by definition. It follows that

o 1

1 + 2(

/(I + 3 )^ + (2a)T )^

o o

+ 2

1 +

1 + 23 (1 + 3 ) + iwT

o o (45)

The Fourier coefficient is complex so that the phase of the second

harmonic of the velocity fluctuation may be important in determining the

amplitude and phase of the response second harmonic. To illustrate the

effect of this phase angle consider Equation (45) in the case where 3 vanishes

iwT

1 I c D

/I + (2a)T )^

o

C +^ 2_

2 2 1 + iuT (46)

To define the phase angle, the velocity fluctuation may be represented by a Fourier sine series:

u = e n = 1

C sin (nwt + (j) ) .

' n' (47)

13
The phase of the fundamental, ^, is zero by definition. Comparison of

Equations (23) and (A7) shows that

C^ = IC2I (sin <t>2 - i cos (^^) .

Substitution of this expression into Equation (46) leads to the following:

(48)

A + (2a)T )2

o 'SI' 1 + (0)1 )2<'^'^o *2 - *2> + ¥'

1 + (0)1 )^

(49)

In the special case where the velocity fluctuation is purely sinusoidal

this reduces to

21 2 A + (2a)T )^ /I + (wT )^

o o

(50)

In the more general case, as in the experiments reported in Section 4,

C2 does not vanish, and the transfer function becomes

/I + (2wT )^

o

COT ^ rTT^^'^'^o *2 - *2>-'t^Tb'>'l + (°T )^

(51)

The factor outside the braces is the linear first-order transfer function.

Clearly, this suffices to describe the attenuation of the second harmonic

only when either cot^ is sufficiently small or the parameter e/|C2l is sufficiently small. In either case the transfer function depends only upon the

single parameter cot . Otherwise, Equation (51) shows that the transfer function

is dependent upon tfiree parameters, wt^, e/lc^l, and the phase angle This

observation raises considerable doubt as to tne adequacy of the use of the

linear first-order transfer function for the correction of atmospheric turbulence spectra as is currently widespread practice.

The transfer function given by Equation (51) is plotted in Figure (4)

with e/|C2| = 10 for phase angles of 0, ± n/2, and IT. For any given value of

COT phase angles can be found which produce a maximum and a minimum value of

the transfer function. Envelopes of these maxima and minima are also shown.

It is interesting that the response second harmonic is suppressed strongly

for a particular combination of phase and frequency for a given value of

e/|C2|. Furthermore, for values of 2cot greater than a certain value (dependent upon e/|c^|) the response secon3 harmonic is amplified above that

predicted by the linear transfer function for all phase angles.

Figures (5) and (6) show the dependence of the transfer function upon

the parameter e/|C-| for phase angles of 0 and H, respectively.

14
Because such significant harmonic distortions as indicated in Figures

(4), (5), and (6) are predicted by the perturbation solution, a comparison

of this solution with an exact solution for the case of a square-wave

velocity fluctuation, where the harmonics are large, is warranted in order

to lend credence to the predictions of the theory.

2.6.6 The Response to a Square-Wave Velocity Fluctuation

The velocity fluctuation is represented by a Fourier series as expressed

in Equation (23). For a square-wave the Fourier coefficients are given by:

- — , for n odd

n

C = ( } (52) n J 0 , for n even

The amplitude of the square wave is z. Setting S equal to zero for

sinqjlicity , the Fourier coefficients for the response are obtained by substituting Equation (52) into Equation (35) with the result

— -—;—T

J for n odd

n 1 + xncJT D =< °

} (53) n

o n even

2 1 + in:.' k(n - k) 1 + ik^T

o , 0 k = -= odd

Interestingly, the odd harmonics of the response are given by the linear

transfer function while the even harmonics are produced by the nonlinear nature

of the response even though the velocity fluctuation in this case contains no even harmonics

The overspeeding error in this case can be sho^ra to be

- 1 + (k^T )^ • ''^^ k = -= o odd

To determine the exact steady-state solution to Equation (20) in this case consider Figure (7) which shows the response c in comparison with the

velocity fluctuation in the steady scate. The velocity fluctuation may be

written

u =

£ ^, for 0 < u.(t - t^) <

-£ ^, for < uj(t - t^) < 2..

(55)

where t^ is arbitrary. Considering each half-cycle separately Equation (20)

may be written

15
T 77 + (1 + a)a = (1 + a)a , 0 < a)(t - tj < n

o at 1

. (56)

^

• T 4r + (1 + = - (1 - a)a , n < u(t - t,) < 2n

o at 1

where a = e n/4. The corresponding initial conditions are

, a)(t ~ t^) = 0

a = ( , w(t - t^) = n ] (57)

, a)(t - t^) = 2n

The condition that the steady-state exists requires that

^3 = c^l • (58)

The solution for each half-cycle may be simply written as follows:

( (t - tp

a + (a^ - a)e T , 0 < a)(t - t^) < n

o = I

^ - 7 I (59)

[t - (t —'

'

^ -(1 - a)

- a + (02 + a)e , n < w(t - t^) < 2n / .

To complete the solution, values for and must be found. From the above

equations one may evaluate 02 and 03, respectively:

" d + a)

a„ = a + (o- - a)e wt (60)

z i o

and

n

(1 - a) = - a + (02 + a)e wt^ . (61)

From Equations (58) , (60) , and (61) one obtains

e^^V^oiT ' - cosh (—

n

° '^^'^o Oi=e4 ^ — > (62)

sinh ( ) o
and o ^ may be obtained from Equation (60)

The exact solution is plotted relative to the square-wave in Figure (7)

along with the perturbation solution for the case where the velocity fluctuation is twenty-five percent of the mean speed e = 1/4, and the frequency is

such that urr =1. The perturbation solution as shown represents a close

. o , -, . approximation to the exact solution.

By taking the average value of the exact solution over a complete cycle

the ove^rspeeding error can be obtained. In the case shown in Figure (7) one finds o/e = 0.196. From Equation (54) one finds that o/e = 0.180, in reasonably good agreement.

3. EXPERIMENTAL APPARATUS AND PROCEDURE

3.1 The Apparatus

The experimental measurements were conducted in air in the NBS Unsteady

Flow Facility. The instrument studied was a Gill helicoid anemometer, Model

27002 equipped with four-bladed polystyrene propellers, six inches in diameter. The anemometer was mounted centrally in the wind-tunnel test-section

with the U sensor aligned to the flow as shown in Figure (8) , the V sensor was directed vertically upward, and the W sensor was removed because of

spatial limitations in the test section.

The test-section was 4.5 ft square and 16 ft in length. Profiles of

mean velocity as measured with a Pitot-static tube in the horizontal and

vertical directions are shown in Figure (9) for a centerline mean speed of

35 feet per second. Figure (10) shows the distribution of free-stream

turbulent intensity in the horizontal direction. The centerline turbulent

intensity varies with mean speed, ranging from 0.2 percent at 10 fps down

to 0.08 percent at 40 fps. These data correspond to nominally steady flow.

Figure (8) indicates the location of the Pitot-static tube used to

calibrate the hot-wire anemometer, the location of which in proximity to

the helicoid anemometer is also indicated. All three instruments were

mounted essentially in a horizontal plane as shown. The location of the

three instruments with respect to the velocity and intensity profiles is

also indicated in Figures (9) and (10)

The electronic instrumentation was arranged as shown in Figure (11)

The signals from both the hot-wire system and the Gill anemometer were

processed in the same manner with the same instruments, a DC voltmeter,

and a Spectral Dynamics Model SD301D Real Time Spectral Analyser* for

the measurement of the amplitudes of the Fourier components. The analyser

was D.C. coupled to permit measurements at fundamental frequencies down

*Brand names of equipment are used solely to provide a reference for

performance characteristics and do not represent an endorsement.

17
to 0.2 Hz and this necessitated the use of a DC voltage bucking circuit as shovm in the figure. All measurements were performed in real time. The

frequency resolution was ± 0.01 Hz.

The hot-wire system consisted of a platinum wire 0.040 inches in length

with a diameter of 0.0001 inches, operated in a linearized constant- temperature mode. Thermal rise in the air stream of approximately 1.0 °C per hour

presented some problems with changes in the hot-wire sensitivity during the

experimental runs. This problem was minimized by operating the hot-wire at

an overheat of approximately 100 percent, restricting run time to less than

two hours, and calibrating before and after each run, using an average of

the calibrations in the data reduction. Errors introduced by this thermal

drift are estimated at ± three percent for the mean velocities and ± one percent for the measured harmonic amplitudes.

3.2 Range of Parameters

The dynamic characteristics of the wind tunnel are presented in detail

in a subsequent report. The facility was used to generate periodic axial

flow pulsations over a range of mean speeds and fundamental frequencies.

The relative fluctuation amplitude was also varied. The harmonic content

of the velocity fluctuations varied as well. The primary independent ex- perimental parameters are U, a)/2II, and e. The harmonic amplitudes and phases

of the velocity fluctuation may be regarded as secondary parameters. The

range of mean speeds and fundamental frequencies studied is shown in Figure

(12). This information is also presented in tabular form in Table 1.

In addition, as revealed in Section 2 of this report, two particular

instrument properties (not varied in this study) also determine the

response. These are the distance constant, L, and the velocity intercept

U . Consequently, the response is dependent upon the following three

primary dimensionless groups

- ' = L

3 = — , WT = w — , e . (63)

. ; ° u ° u

Of secondary importance are the dimensionless Fourier coefficients, C , where

n = 2, 3, ... which characterize the amplitudes and phases of the harmonics

of the velocity fluctuation.

It should be noted at this point that both the intercept of the linearly

extrapolated steady-flow calibration curve, U , which reflects the effect of

bearing friction in the anemometer, and the distance constant, L, are inversely proportional to the density of the air stream through the quantity

A (Equation (7)), and a simple correction can be made to both L and U when

the anemometer is used in streams of different densities. The density of

the air stream in the present study was held to a value of p = .0704 Ibm/ft^

with a maximum variation of ± one percent. This variation arises from

variations in stream temperature and pressure.

18
In terms of nondimensional quantities the range of fluctuation amplitudes

and frequencies is shown in Figure (13) and tabulated in Table 1. Fluctuation

levels up to approximately 75 percent of the mean speed were studied.

While the harmonics are secondary in terms of their effects on the

value determined for the distance constant, they are important in determining

the higher harmonics of the response. The range of the second harmonic

amplitude relative to the fundamental amplitude is tabulated in Table 1 and

shown in Figure (14) as a function of the fundamental amplitude. Values

range up to about 40 percent. The third harmonic in all cases was less than

10 percent of the second harmonic and is neglected throughout along with all

higher harmonics.

4. EXPERIMENTAL RESULTS

4.1 The Steady-Flow Response

The voltage, E, obtained from the helicoid anemometer under steady conditions is plotted as a function of the speed, U, in Figure (15). The data

shown were accumulated under steady flow conditions during the calibrations

of the hot-wire anemometer before and after each set of runs on the dynamic

response characteristics. The spread of the data in this figure is due

primarily to variations in air stream density as discussed in Section 5.

Were there no bearing friction, no sensitivity to density would be present

and the steady-flow response would be purely kinematic. A least squares

straight-line fit of the data produces a velocity intercept of Uq = 0.851

fps

4.2 Determining the Distance-Constant

According to the model developed in Section 2, all that remains to

"calibrate" the anemometer for its dynamic response characteristics is to

determine the distance-constant. The model indicates that a sinusoidal

frequency response test where the velocity fluctuation is free of harmonics

should produce a linear first-order transfer function for the response fundamental as represented by Equation (42) with friction present, or by Equation

(43) in the absence of friction. It was argued in Section 2 that the effect

of the second harmonic of the velocity fluctuation upon this transfer function

should be small as long as the second harmonic amplitude is small relative

to the mean velocity. Values of the ratio of the amplitude of the second

harmonic to the mean velocity are tabulated in Table 1 along with all of

the experimental data presented in this section. This ratio is less than

23 percent for all of the data, and less than 10 percent for the most part.

Accordingly, the distance-constant was determined by neglecting the contribution of the harmonics.

The Gill anemometer produces a voltage proportional to the rotation

rate:

S = K^E , (64)

19
where contains the amplifier gain and the signal generator sensitivity.

The spectrum of the signal from the anemometer under steady flow conditions

was observed to contain sharp peaks at frequencies corresponding to five

and ten times the rotation rate. In performing the experiments care was

taken so that frequency components of the velocity fluctuation were not

confused with these two frequencies generated by the rotating coil system

in the anemometer signal generator.

The Fourier series representation of the anemometer signal may be

written

. E = i Z E e

i"'^^

(65) 2 n n = -«>

To compare the harmonic amplitudes, |e |, with the mathematical model.

Equations (64), (65), (34), and (17) may be combined with the result:

K

^

? E e^^'^^ = S*

2 n

n = -00

1 , e ~ mwt

1 + E D e

2 n

n = -oo (66)

It follows that

exp

eS* '1' • (67)

From this and Equations (13) , (18) , and (22) one may write

^1

a (1 + e^) |Ej .

r-r.. i : ; 1 exp KeU

where the subscript "exp" refers to "experimental."

(68)

To determine the distance-constant the procedure is as follows. First,

3 is assumed to be zero as a first approximation. Equation (43) gives the

expected transfer function. The experimental values obtained from the measured, values of | E^ | and Equation (68) are then plotted as a function of

w/2nu which is proportional to wt as indicated in Equation (63), and a least squares fit of the data to the expression in Equation (43) yields

the best value for the distance-constant. For the 89 data points the

distance-constant is found to be L = 3.35 ft. Using this value all of the

data are replotted as a function of cot as shown in Figure (16) . The linear

o transfer function without friction is represented by the solid line in the

figure. From the quality of the fit of the experimental data to the predicted

linear transfer function the approximations made in arriving at Equation (43)

appear reasonable.

20
However, there is a significant effect of the bearing friction upon the

value obtained for the distance-constant. When the experimental data are reduced according to Equation (68) with 3^ ^ 0, the values of io^l^ plotted

in Figure (16) must be increased by an amount (1 + 3 ) which variel'^rom

point to point as 3 varies. As shown in Table 1, 3 varies from about 0.02

to about 0.09. Similarly, the theoretical expression for the transferfunction, Equation (42), which includes the effect of 3 must be used rather

than Equation (43). Thus the theoretical values of [D^f^ from Equation (42)

will be somewhat larger than those obtained assuming 3 =0. The overall

effect of 3 would be to shift the data in Figure (16) upward and to the

right. The result is that when a least squares fit is applied to the values

of jD^ I obtained from Equation (68) using Equation (42) for the expected

transr effunction , the distance-constant so obtained is L = 3.52 ft — some

5.1 percent higher than the value obtained assuming 3 =0. This latter value

should be more realistic, and such frequency response tests in general should

be analyzed so as to account for frictional effects.

No single curve such as the solid line of Figure (16) can be drawn in

the case where 3 7^ 0 since the transfer function depends upon 5 as v/ell

o o as ojT 5 and 3 is not constant for all of the data. However, the correlation

between theoretical and experimental values of jD^| can be shown in both

cases as indicated in Figures (17) and (18)

4.3 The Overspeeding Error

From Equations (40) and (64) , it follows that _

(1 + a) = — (1 + 3 ) . (69

The experimental data obtained from the measured mean anemometer voltage

according to this equation are plotted as against the values predicted from

the mathematical model, Equation (38), in Figure (19). The general agreement

is fair. The scatter in the data i^s due to two factors: 1) The sensitivity

of the_theoretical expression for a to uncertainties in the measured values

of e, U and oj, and 2) uncertainties in the measured mean voltages from both

instruments. A trend is evident however, whereby the measured overspeeding

error tends to be somewhat lovrer than the predicted values. This departure

increases as the overspeeding error increases. On the average the largest

overspeeding error produced in the tests was about 10 percent. The expected

values were about a factor of two larger than this.

An examination of the effects of viscous friction on the propeller

blades should be conducted. In general this friction will retard the

rotation rate and may account for some of the average departure of the

data in the figure from the expected values

21
4. A The Response Second Harmonic

From Equation (66) it follows that

exp

K,

KeU

(1 + 3 ) o (70)

The corresponding theoretical expression for the ratio of second harmonics

is given by Equation (51) . Instrumentation for the measurement of the phase

angle was not available during the course of the experiments. Consequently

a direct comparison between theory and experiment is not available.

However, it can be shown that the linear transfer function is inadequate

to represent the second harmonics. Figure (20) has been prepared by neglecting 3 in Equation (70) . The experimental values from the resultant equation

are plotted as a function of 2cot . The solid curve represents the linear

transfer function. The departure of the data from this curve is not simply

experimental scatter. Departures of as much as an order of magnitude are evident. These can be qualitatively explained by the sensitivity of the

second harmonic to the input fundamental amplitude, e, and to the phase

angle indicated in Figures (4), (5) and (6).

Because such large departures from the linear transfer function are found experimentally, considerable doubt arises as to the adequacy of the

linear transfer function for the correction of atmospheric turbulence data.

This point was raised earlier in this report. Further studies are needed

to quantify the spectral characteristics of helicoid anemometers.

5. CONCLUSIONS

Although perhaps heuristically , the case has been made that all rotary

anemometers will exhibit the same basic dynamic response equation (as well

as the same basic steady-flow response equation) , and that only two experimental constants, Uq and L are required to characterize the dynamic response.

Furthermore, the sinusoidal frequency response test, even with modest harmonic

content present in the velocity fluctuation, provides a simple technique for

determining the distance-constant. The effects of bearing friction can be

accounted for as well.

The mathematical model developed accounts reasonably well for the

observed overspeeding phenomenon, but probably needs refinement to account

for the viscous friction on the blades of the helicoid propeller.

Sufficient doubt has been raised as to the use of a linear transfer

function for the correction of measured atmospheric turbulence data to

warrant further study of the effect of the inherent nonlinearities of

rotary anemometers upon their spectral transfer characteristics.

The model presented can be modified to account for angles of yaw,

and the basic equations derived in the text can be examined to determine

22
the instrument response when the aean velocity component vanishes as in the case of vertical sensors in three-component arrays.

6 . ACKNOI^^EDGMENTS

The authors are indebted to William G. Cleveland for his valuable

assistance in performing the experiments, and to Karl E. Lofquist for his

helpful discussions concerning the basic equations.

7, REFERENCES

1. G. B. Schubauer and G. H. Adams, Lag of Anemometers, NBS Report 3245,

April 16, 1954.

2. P. Jepson, Currentmeter Errors Under Pulsating Flow Conditions, J.

Mechanical Engineering Science 9_, 1 (1967), pp. 45-54.

3. M. Rubin, R. W. Miller and W. G. Fox, Driving Torques in a Theoretical

Model of a Turbine Meter, J. Basic Engineering, ASME, June 19 65, pp.

413-420.

4. E. Over, On the Response of a Vane Anemometer to An Air-Stream of

Pulsating Speed, Phil. Mag. S.7. 23_, 157, May 1937, pp. 992-1004.

5. J. C. Wyngaard, J. T. Bauman and R. A. Lynch, Cup Anemometer iK'namics,

in FLOW, Its Measurement and Control in Science and Industry 1^, Rodger B, Dowdell, Editor-in-Chief, Part 2, Flow Measuring Devices,

edited by R. E. Wendt , Jr. (1972).

6. S. Ramachandran, A Theoretical Study of Cup and Vane Anemometers,

Quarterly J. of the Royal Meteorological Soc. 95_ (1969), pp. 163-180.

7. Rikiya Takeda, Theoretical Research on Propeller Type Current Meters,

ASMZ Paper No. 74-WA/FM-6

8. J. P. Bennett and R. S. McQuivey, Comparison of a Propeller Flowmeter

With a Hot-Film Anemometer in Measuring Turbulence in Movable-Boundary

Open-Channel Flows, U. S. Geol . Survey Prof. Paper 700-B, pp. B254-

B262.

9. Thomas W^. Horst, Corrections for Response Errors in a Three-Component

Propeller Anemometer, J. of Applied Meteorology 1^, 4, June 1973,

pp. 716-725.

10. Gordon A. McBean, Instrument Requirements for Eddy Correlation

Measurements, J. of Applied Meteorology 11, pp. 1078-1084.

11. B. B. Hicks, Propeller Anemometers as Sensors of Atmospheric Turbulence,

Boundary-Layer Meteorology 3_ (1972), pp. 214-228.

23
12. E. L. Deacon, The Over-Estimation Error of Cup Anemometers In Fluctuating

Winds, J. of Scientific Instruments 2^, August 1951, pp. 231-234.

13. Keisuke Onuma, Over-Estimation Error of Rotating Type Anemometers in

Sinusoidally Fluctuating Wind Speed, Proc. 7th Japan National Congress

for Applied Mechanics (1957), 11-23, pp. 269-272.

USCOMM-NBS-DC 24
I I I L HoO 00 OO

<3- O

O

W US-



csi
•e- H

COuMIoW

UiOoMHH I
COo

CO

EE] ><<o3

l-l

d oO

d oo
GILL ANEMOMETER

HOT-WIRE

PROBE

I"

PITOTSTATIC

TUBE

TOP VIEW

54

ALL DIMENSIONS EXPRESSED IN INCHES,

FRONT VIEW

FIGURE 8. CONFIGURATION OF THE INSTRUMENTS IN THE TEST-SECTION.
- VERTICAL, z O - HORIZONTAL, y

- QG>© ® ® ®^ ©if ©S) ©I3> ©C^

©

©

-1.0 0 1.0

y , z

FIGURE 9. HORIZONTAL AND VERTICAL DISTRIBUTIONS OF FREE-STREAM VELOCITY,

U =35 fps

0

1 1

j—1 1 1j

1 1 I"

'

©

©

©

© ©- ^^©©©© ©

© ©

1 i {—1 1 1 ^ ' 1 1 •- • -1.0 0 1.0

y

FIGURE 10. HORIZONTAL DISTRIBUTION OF FREE-STREAM LONGITUDINAL TURBULENCE

INTENSITY, U = 35 fps.
o

•Hu

w

•H

l-l <U 4JO

cv)

!-J CO J2O

a;

COCo

p.1 cn

OJ

a

•H

Q

13

0)

5-1 dm

c«

<u

»-l o

flo

•H

4J

CO U

c

(i) I

>-< u

CO

60

•H
©

®

Q) 00 0000

fiB© 00 5^0 0 0 0 09

^<2X^ 0 © 0

CDC3>Q0 0 0 - oo X3 CM

COH

ia3 wMHQWH

faQZ

<dQW<ooo o o oCO oCM I o 3
H

525

® ® ®

0

0

0 0

0®

0

0

00 0

©0 ® 0 © 0

0 0

0

0

0

0

0

0

^ 0

0

0

0 0

0

0

0 0 0 0

0

0

0

0 00

©

(50 00

^ 0

09 0 ©0

0 . © ©QB 0

®0

©

©

0

0

0 0

0 ©0

0

o

0 0 0 I

I o

o

u

W

CM M

H

O

o

Q

M

FZ4 00 CM

0)
0 ®

© e 0 i ® 0 0 ©©®

©0©

© ^® 0 . ® © ($) ® ^ ©© © 0© 0 ^ © © 0 © 0 0 © 0 ©© ©

3® i ^ ® © 0

00oOoo

00d

d

Sto o

CM
oO

in ooCO o

CM O

00O o o

CMo . to 4J iHO>

00ddo

CMo

CM COd

d

CMo oo oo o

p-l oM

H

PiW

U ^ s Q O

2i 2 U PQ M M O H W O X wo O !Z o

Oil oooH

0)

1.0 1.1 1.2 1.3

(1 + o)

FIGURE 19. COMPARISON OF MEASURED AND PREDICTED OVERSPEEDING

ERRORS

.
CM O
1 D O

+ -n- rH O

ca

•>-

i.po 1.02 1.02 1.03 1.02 1.03 1.03 1.02 1.03 0.99 1.00 1.00 1.01 1.00 1.01 1.01 1.02 1.04 1.04 1.03 1.03 0.98 1.00 1.01 1.02

+

o

o

CMO

CMoCOo

CMO

fOoCOo

CMO

COo

ONON o

o

o

o

rHO

o

o

rHOrHO

CMO o

si- o

COo

COo00On o

o

rHo

CMO

rH rH H rH rn rH _ J rn rH 1 1 r*1 1 ri rn

* rH O * rH H rH

O

CM Ik

P

O

ca

sr

00o

CJv CMO

.044 CO

CMO

00

rHO

O

o

m

rHO

CMO m

o

o

a\

rHo

00

CMOCMO

SOsoOrHo

CMrH

SO

CM

rH

00H

cy\ CMH

CMsr

rH O

o

rH

CO

OnO

ino

m

rHO .015 m

rHO

o o o o O o O O o o O o O o o o O o O o O o o o O

CM Q

rH

00o

00

CMo

sro

CM

CMO

''4' "'

rH o

o no*

CMo

ino

o

00

rHo

1^

CM CO

CMo

CO

vOo

00vOo

CM

CM oCM

t

rH

•<r

1

CO

CM

- 1

SO

CO

- t so

o>

C^

00o

CMm

o

St

rHo

rHo

srH

O

o o o o o o o o o o O o o o O o o o o o O o o o o

/-\ o

iH Ik ^

O

ca

VO CO

so om CN rH

CMiH oes

rHo

COm

r>. o

ONo

soO

CO

rH o

o

CMO

ino

m

o

m

00

sr

00 o

ON O m

On 00

On

00 COo

o O o O O O o rH o rH rH rH rH rH rH rH rH o o o rH O o O H

rH Q

CMOv oso

00 SO

rsi SOrH

CMrH o>

rH SO0^

rHin

COo

m

o

CMO

ONOso

CJN CO. ON o

o

O

O

rH

00 o

00

so

00 SO

<JN rH m00 m

00 ON

On O O o O O O o O o rH rH rH H o o H rH O o o o o o o O

O

ca

.0397 <

O

sro

.0407 .0413 .0415 .0416 .0415 .0417 .0421 .0412 .0413 .0418 .0422 .0435

<•

•<r o

.0467 .0484 .0460 .0457 .0438 .0432

soo

Sto

.0411 .0414 .0408 o o o o o o o o o o o o o o o o o o o o o o o o o

CM O,0093 ,0139 .0164 .0170 .0138 .0089 .0100 .0007 .0003 .0015 .0020 .0019 .0121 .0218 .0321 .0424 .0535

so

COsDO

.0526 .0321 .0167

CO

CM

rHo

,0036 .0015 .0007

CO o o o o o o o o o o o o o o o o o O o o o o o o o

\ u

VO

CM

«cr asH .111 St

00o

o

soo

CO

soo

CO

r*. o

ON

CMO

CO

rHO

ONm

o

si- orH

CMSO

rH

00

CM

ino<

ON

sr

sr

rHSO

CO

(JN VO

sr

so

p^

CJN m

sr

insf o

rH

CO

p>. sO

rH

CM

COo

so

sro

in

CMo

P o o o o o o o O o O O o o o o o O o o O C o o o

<r rH ON so in in sr Csl SO CM in CM CM so m so <r P«. o r*. m ON

rH

CM HCM o

CM OCM oCM OCM oCM oCM OCM oCM OCM oCM oCM c^H

C3N rH 00

rH rH

00

rH

ODH

ON

rH

CTN rH

rH

CM ocs oCM cCM

^ ll—

I

CMm rH o CM

CM O

r>. o

o

CMP- SO

<f

00

CM

CM

CM CM

CM

CM

CM

CM

CM

csCM

CM

CM

CM

CM

CMCM in

somSOm

SO

in ovO oso oso oso o rH rH CO in r- St O rH O o o O o O o O o o O o O o O O
o o> I—I p- m

O o o o

pH 0^ O ON o or>* c m o o

O -H rH O O

a^ < O O «N ON O iH C O

CN rH m CM o o o o o

iH O iH iH iH iH O O iH iH iHiHrHiH OfHiHiHr-l iH

o ON iH in C ON o o o

CM O ON < m O 0^ C O

r>» o NO o

O tH iH O O

ON »3- C O CM 0> O iH O O

CM r-i in tM O O C O O

O iH .H iH iH iH O >H iH O »H iH iH

in <r o> ON o

o nC m vD sO c c o c o

v£> CN 00 r-l OC NO in n o o

o c o c c

CC CO fM vO

CN CM rH O >—( >—I o c

O ON C iH <• O fNi o m

o o o o c

in in in m

»T vO <J- CN o c c o o

o o o o o o o o o o o o o o o o o o o o o o o o o

m CO NO o NO m o in c c c c c

CN ON vo <•

NO m o

o o o o c

ON CN iH m

H rsl r-l C H rH C C

O r-i iH O tM ri O

o c c c c

«3- H iH CN n

nO sj- m CM c c c c c

o c o o o o o o c c o c o o o o c o o o c c c o o

rH nc O O ON in r«. «a" CM CO n en vo NO ro oc in c C

\0 p«» 1—I iH c CO o^ C r<* iH i-l rH iH iH

CD o o c o

f-v oc r>. NO CM

iH iH CM CM CM O iH O O O o o o o o o o o o o o o o o o

NO a\ 'O >3' CO O ON On C^ CO CM ^a- CM C CO

cr m o m

0^ C ON On

in c o

ON CO c NO O "H iH iH iH

nC in <• c

1-i rH CM CM CM o o o o o o o o o o o o c o o o c o c c c o c o o

NO NO

NO C O CM CM CM CM p^

c c c o c

CO o NO

ON vo m ON NO nO nO ^O o o o o c

in tn CM r>« vo

CNj cN m c C

p^ r>. CO r*- c c c o o

ON P^ X rH CM <• CM CO CM O r>. \0 o

o c c o c

CM CO nc CO va- in nC rH <}• o NO CO X r» C C C 3 O

c c c o c c o o c o c c c o o o o c o c c o o c c

ON P^ ON X

CM PO ON X nD

rH c C X O C O rH C

C^ ON O NO CM m O rH ON CM rH CO o o o o o

CM CM vo X ra

CM CM X in rH m in «j- rH p^ O C C CM C

c X

m c CM

rH P>. C C

C O rH C O

tn r-» CM CM X

in <• St <o vj- C rH X m CM c o c c o

o o c o o o o c o c o c c c o c c o c c c c c c c

r>» fo r«. CM o NO NO O^ ON O

CO CM rH Ln un

•3" CM CM O X

nO ON CM On ^

rH r-i r-i C< o X NO NO m

CM CM O C^ C ^ L-) L-^ n CM

rH < O rH NO

in p^ en p^ rH C CM L-l O O

«^ <• CM X NO n p^ < CM m

C O vff CO rH c c c o o o o o o o o c o o o c o c c c o o c c o

rH X r-. X p^ CM p*. o o n t>» r« o Ln o rH X m o m « S3- H o rH rH CM CM PO CM CM rH rH rH o CM n rH O cn m m CM o> c rH

H -a- sr rH rH rH rH rH rH rH rH rH rH rH rH H iH rH rH rH rH rH rH rH

O C CM C O p>. CN4 m in O O O CM CM

vO NO NO X X

rH ^ O C> P^ P^ C On O O O C O NO O X P^ C C CM CO

in en ON rH sr r^OOrHrH rH r-i r-i O O COOmm NOmcMNOn COfOrHCMCM
Ocncaoro oc^ococm oisj-cmco ocNr^e^^c^ cnvocMcocM HrHrHOO OOOOO OOOOO OOOOO iHOOOO

rH iH iH »H iH rH iH iH iH iH »H iH iH r-l

O CS O cn

rH rH iH O O

O es iH CO CM OOOOO O St CN O O OOOOO O CM iH CM cr> OOOOO fO vO CSI CO CM

iH O O O O

rH iH rH iH tH rH rH rH rH

00 iH O

00 O 0^ 00 v£) O iH O O O

m CM m S3- rH m en T-\ r-\ 1-i OOOOO

o fs CM m

vo m CO CM OOOOO

m sj- m o m CS CM O rH O O O rH rH o 00 rH in

a* in sr CO <s OOOOO

O O O O O OOOOO OOOOO OOOOO OOOOO

^ vO so CM

00 0^ 00 00 vo OOOOO

si- rH St CO rH

CO CO rH rH rH OOOOO

O O St vO m in St CO CM OOOOO

St CO m rH 00

CM CM O CO O

O O rH rH

St St <T> CO

00 m St CM CM OOOOO

OOOOO OOOOO OOOOO OOOOO OOOOO

o r- St m v£> in m m in in

vo r*» O o^ m

m in \o m vo

Oi 00 rH rH rH

CM CM CO CO CO m m o 00

CO CO CO o

cyi in CO m o in CO CM rH rH o o o c o OOOOO OOOOO O O O rH O OOOOO

00 St CM CM St

St in m in in St m \o CO

in in m m so

00 r>. o o o

CM CM CO CO CO St St VO St CO

CO CO CO 0^ in CO CM St o\ m CO CM rH O

OOOOO OOOOO OOOOO OOOOO OOOOO

St 1*^ m o^ vO

CO 00 1*^ m CM m St -d- St St OOOOO

O rH St a. 00

rH rH O O rH St St St St OOOOO

\0 Cr\ rH CO CO

CM 00 CO rH rH in St St St St OOOOO

in rH O CM rH

rH rH CM m 00 St St St ^ VD OOOOO

>£) m CO 00 O

rH rH O CM 00

r«. r». r-^ OOOOO

OOOOO OOOOO OOOOO OOOOO OOOOO

vD O O O

CO0^ On St OOOOO

CO o 00 cy> St

<t 1^ rH O O

rH O O O O OOOOO

00 CM St rH CO rH VO CO 00

CM CM m CM o

CM rH O O O

»3- VO ON VO St

CO rH O CO CM o o o CO m OOOOO

vD m VO rH VD

00 St 00 CM r>. VO o m o

O rH rH rH CM OOOOO OOOOO OOOOO OOOOO OOOOO

St St VO CO 00

00 CO rH CM vo 1^ r»» vo CO

St m in CM CM O O St CM rH

CM H O O O

CM CM cy> r-^ 00 o r«. m CM m vO St CM rH O

in rH 00 St O

CM rH O 0^ O O O vD r>.

vO 00 CJV rH CTi in vo m 00 St

vo in St sr St OOOOO OOOOO OOOOO OOOOO OOOOO

On m o% m o 00 rH 00 St CM St t>» vi> VD m CO o m On Ov rH ON

in r>- CO O M r-i O

rH rH rH H CM O O rH O O CM CM CM CM CM vD r-^ CJN O O

rH rH rH CM CM C O O CO CM CM CM CM rH rH

O O 00 rH H

CO CO CO ^ St

CM CM CM CM CM St >* "d- -3- St

vO St St CM CM

CO 00 00 00 00

CM CM CM o m

00 00 00 CM m

o 00 vD St in

cjN in St m CO

r-i T-i t-i t-\ rH rH T-i CM CM CM CM CM CM CM CM O O O rH CM CO m
m sr o o o o o

o

o

CO f>> o o o

CM vO CM o o o o

^ •* -a- m

o o o o o

CO o r«. <r O fH O O O

CM tM so m

o o o o

iH iH iH

iH in \o m c^ 00 iH CM CO CO

CM «H T-i r-l CO CO m CM iH o O o O O o O o o O o O o o o • • • • • • • • • • • • • • O o o o o o o o o o o o o o

ON m m CO 00 o o m CO CO

CM CM H fH CO iH CO CO CM iH o O o o o O o O o o o o o O o • • • • • • • • • • • • • • o o o o o o o o o o o o O o

in CO CO CO <£) iH CM CO CM vO

CM CM CM CM CM CM CO m CO o • • • • • • • • • • • • • • o o O o O O O o o o o o O o

CO iH tH CM CM CM o CM rH in

CM CM CM CM CM CM CO in ON iH CM iH o • • • • • • • • • • • • • • o O O O O O o o O o O O O o

CM CnI 1^ CM 00 o <J- m o tH 00 m iH CO O vO r» CO CO O

r-» •<r sr 00 o O o o o tH O o o o o O O o • • • • • • • • • • • • • • o o O o o O o o o o o o O o

tH tH m CO o vO m vO CO CO

CM m m in ON St ON CM CM CJ> vo ON

CM o iH o r>. CM in in m CO CM O o o o o o H o o o O o tH o • • • • • • • • • • • • • • o c o o o o O o o o o o o o

CO CO O Cs| CO 00 CO Cvl 00 ON 00 vO

so vO CM m CM tH CO vO CO CM o CM iH o iH o CO m m m S3- CO m •a-

• • • • • • • • • • • • • • O o O O o o O o o o O O o o

00 CO ON so rH CO 00 tH ON CN sr r>. 00 m in iH vo O H iH iH CM tH tH tH O iH rH tH tH rH

CO CM sj- 00 vO St

•H CO m m ON m • • • • • • • CM CM CM CM CM O CM

vo St vo o o in o CO 00 CO CM CO O ON

• • • • • • • tH O O CO H m CTi
NSS-1 14A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR RKI^ORT NO. NBSIR 75-7722. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

THE DYNAMIC RESPONSE OF HELICOID ANEMOMETERS

5. Publication Date November 1975

6. Performing Organization Code

7. AUTHOR(S)

James M. McMichael and Philip S. Klebanoff

8. Performing Organ. Report No. NBSIR 75-772

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2130482

11. Contract/Grant No. 1-1-1377

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Federal Highway Administration (DoT)

Office of Research

Structures and Applied Mechanics Division

Washington, D. C. 20590

13. Type of Report & Period Covered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most si^ificant information. If document includes a significant

bibliography or literature survey, mention it here.)

The results of an analytical and experimental investigation of the dynamic response

of a helicoid anemometer are presented. The experimental investigation was conducted

using the NBS Unsteady Flow Facility and data are presented which illustrate the

dynamic behavior in a spatially uniform, fluctuating flow with varying amplitudes,

frequencies, and mean velocities. An analytical model governing the dynamic response

is also presented and compared with the experimental results.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Air; analytical; anemometer; dynamic response; experimental; lag; unsteady flow.

18. AVAILABILITY [X^ Unlimited 19. SECURITY CLASS

(THIS REPORT)

21. NO. OF PAGES

1 For Official Distribution. Do Not Release to NTIS

UNCL ASSIFIED 53-

1

' Order From Sup. of Doc, U.S. Government Printing

Wasliineton, D.C. 20402. SD Cat. No. C13

Office 20. SECURITY CLASS

(THIS PAGE)

22. Price

t^torder From National Technical Information Service

Springfield, Virginia 22151 (NTIS)

UNCLASSIFIED $ 4.25

USCOMM-DC 29042-P74