Characteristics of Helicoid

Anemometers

a/ f

J. M. McMichael and W. G. Cleveland

Fluid Engineering Division

Center for Mechanical Engineering

and Process Technology

National Engineering Laboratory

National Bureau of Standards

Washington, D.C. 20234

August 1 978

Final

Prepared for Federal Highway Administration

Office of Research

Structures and Applied Mechanics Division

qq ____ Washington, D.C. 20590

100

. U56

78-1505
• .
NBSIR 78-1505

CHARACTERISTICS OF HELICOID

ANEMOMETERS

f?s??8nal Bureau of Standards APR 17 197*

n oi face* *

r\o.

J. M. McMichael and W. G. Cleveland

Fluid Engineering Division

Center for Mechanical Engineering

and Process Technology

National Engineering Laboratory

National Bureau of Standards

Washington, D.C. 20234

August 1978

Final

Prepared for Federal Highway Administration

Office of Research

Structures and Applied Mechanics Division

Washington, D.C. 20590

U S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Dr. Sidney Harman. Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

LIST OF SYMBOLS

V“)

fc K

L

S

t

U

U

U

o

u

c u(t)

u

i (t)

e

O)

Q

TO

Power spectral density of true velocity fluctuation.

Power spectral density of indicated velocity fluctuation.

Frequency (Hz)

.

Characteristic frequency.

Slope of steady flow calibration curve.

Response length or distance constant of helicoid anemometer.

Estimate of alongwind integral length scale.

Rotation rate of anemometer rotor.

Time.

True instantaneous velocity.

True mean velocity.

Velocity intercept of steady flow calibration curve.

Velocity indicated by helicoid anemometer.

Mean indicated velocity.

Characteristic amplitude of true velocity fluctuation.

True velocity fluctuation, normalized.

Indicated velocity fluctuation, normalized.

Normalized amplitude of sinusoidal velocity fluctuation.

Circular frequency.

Reduced frequency.

Time constant of helicoid anemometer.

ii

6

LIST OF FIGURES

Figure 1. Variation of Overspeeding-Error with

Amplitude and Frequency.

Figure 2. Variation of Overspeeding-Error with

Amplitude and Frequency.

Figure 3. Variation of Overspeeding-Error with

Amplitude and Frequency.

Figure 4. Variation of Overspeeding-Error with

Amplitude and Frequency.

Figure 5. Comparison of Measured and Predicted

Overspeeding-Error for efi< 0.4.

iii
CONTENTS

Page

ABSTRACT 1

1. INTRODUCTION 1

2. BASIC LIMITATIONS ON THE RANGE

OF APPLICABILITY — 2

3. EXPERIMENTAL PROCEDURES AND RESULTS 3

4. APPLICABILITY TO TYPICAL ATMOSPHERIC CONDITIONS 4

5. PROCEDURES FOR CORRECTING MEASUREMENTS OF

ATMOSPHERIC SPECTRA 5

6. CONCLUSIONS 6

7. REFERENCES 6
CHARACTERISTICS OF HELICOID ANEMOMETERS

J. M. McMichael and William G. Cleveland

ABSTRACT

An experimental study of the overspeeding

error for helicoid anemometers in periodic air

flows is described. The ranges of amplitude and

frequency for which a simple nonlinear model for

the dynamic response of such instruments remains

valid are presented. It is shown that the model

is valid for typical atmospheric applications of

such instruments. A simple method is presented

whereby the effects of inertial lag and nonlinearity may be taken into account in obtaining

measurements of alongwind power spectra in the

atmosphere

Key Words: Air; anemometer; dynamic response;

experimental; lag; unsteady flow.

1 . INTRODUCTION

In November of 1975 a National Bureau of Standards Interagency

Report, NBSIR 75-772, entitled "The Dynamic Response of Helicoid

Anemometers" was prepared for the Federal Highway Administration,

In that report certain questions were raised as to the effect that

nonlinearities inherent in the response characteristics of helicoid

anemometers might have upon the validity and accuracy of atmospheric

turbulence spectra measured with these instruments. In particular,

if the mathematical model presented in the above mentioned report

were valid under typical atmospheric conditions, the linear transfer

function commonly used to correct for the inertial averaging characteristics of these instruments would be inadequate in principle, and

an improved method of correction based on the nonlinear model would

be required to refine spectral data. The present study addresses

this question in two parts. First, over what range of the basic parameters is the nonlinear model valid? And second, if the criterion for

validity were satisfied for typical atmospheric conditions, how then

should one apply corrections to the measured spectral data?

The range of applicability of the dynamic response equation developed in Reference (1) is ascertained by examining experimentally the

overspeeding error encountered in a periodic flow field produced in the

NBS Unsteady Wind Tunnel. A parametric study of the effect of mean speed,

fluctuation amplitude, and frequency on the overspeeding error was con-
ducted. The overspeeding error was selected to validate the nonlinear

response equation because it is relatively easily measured, and its

existence is due solely to the instrument nonlinearity. Furthmore,

in the previous study the measured overspeeding error, as presented,

exhibited a large amount of apparent scatter, and the overall trend of

the data indicated a departure from the predicted values. Only a careful parametric study can reveal the reasons for this behavior.

Under the conditions of validity established experimentally, a correction procedure based on the model is presented in the present

report

2. BASIC LIMITATIONS ON THE RANGE OF APPLICABILITY

The basic dynamic response equation in (1) may be written

L |f- + (U + U )S - KU2

, (1) dt o where L is the response length (distance constant) characteristic of

the instrument, S is the anemometer rotation rate, U is the instantaneous flow velocity, and K and U are empirically determined from the

asymptotic steady flow calibration curve:

S - K (U - U

q

) . (2)

Equation (1) is based on the assumption that under dynamical

conditions the instantaneous departure from equilibrium is not too

large. For helicoid anemometers this means that instantaneous local

blade angles of attack must remain small. As the departure from

equilibrium increases additional terms may be required in the Taylor

expansion used to represent the driving torque in the derivation of

Equation (1). In extreme cases local flow separation may occur on the

propeller blades radically altering the value of torque coefficients

in the expansion.

Intuitively, at low frequencies the anemometer is expected to

follow the velocity fluctuations quite closely with departures from

equilibrium remaining small. This is so even for amplitudes approaching 100% of the mean speed because at any instant anemometer lag

is small, and the instantaneous rotation rate is never far from its

equilibrium value where S(t) = KU(t). However, as the frequency in- creases the increasing inertial lag of the instrument causes departures

from equilibrium which can be kept small only by correspondingly re- ducing the amplitude of the flow fluctuations. Hence, a reasonable

criterion under which Equation (1) should remain applicable is that the

product of frequency and amplitude (suitably normalized) must remain

less than some critical value. In fact, a rigorous perturbation analysis

of Equation (1) and its associated torque expansion shows that this is

precisely the case. Physically, this condition amounts to restricting

flow accelerations to sufficiently modest values that the anemometer speed

of rotation can change at a comparable rate.

2
The restriction to small departures from equilibrium may be rigorously stated that uc

ft must remain small compared to unity, where u

c is a measure o_f the velocity fluctuation amplitude relative to the mean velocity U, and ft is a reduced characteristic frequency given by

2-nf ch

ft * ——

U

(3)

where f is a frequency in Hz which is characteristic of the velocity

fluctuation. Just how small u ft must remain may be determined experimentally by conducting a parametric study of the overspeeding error at

various mean speeds, frequencies, and amplitudes, and comparing the re- sults to values predicted from the mathematical model.

3. EXPERIMENTAL PROCEDURES AND RESULTS

The experimental apparatus and test procedures for the measurement

of overspeeding errors were substantially the same as those presented

in Reference (1), with the following refinements to reduce measurement

uncertainty. On a day to day basis, steady flow calibrations were performed to determine K and U

q

for each set of overspeeding error measurements.

Small changes in U affected the mean indicated velocity. In addition,

changes in the colci resistance of the hot-wire used to measure the velocity

fluctuations were monitored closely. These changes were due to temperature

drift in the ambient air stream. The hot resistance of the wire was adjusted at frequent intervals to compensate for this drift in ambient temperature. With these refinements uncertainty in the true mean air speed

determined from the hot-wire anemometer was reduced to ± 1%.

The helicoid anemometer was tested in the NBS Unsteady Wind Tunnel

where essentially sinusoidal longitudinal velocity fluctuations of varying

amplitude and frequency were superimposed on mean air speeds of 15, 20,

30, and 40, feet per second (4.6, 6.1, 9.1, and 12.2 m/s).

Neglecting bearing friction, the overspeeding error for sinusoidal

velocity fluctuations, as derived in Reference (1), is given by

U 2 1+ft 2

where Ui is the mean indicated velocity defined by

l + u K o and e is the peak amplitude of the velocity fluctuation.

(4)

Data for the overspeeding error were acquired parametrically. For

each mean velocity, the fundamental amplitude e was varied while frequency

3
was held constant. Thus for each set of data ft was held constant.

Frequencies of 0.5, 1.0, 2.0, 3.0, and 4.0 Hz were used, and values

of e ranged as high as 0.8. Equation (4) shows that for constant

ft the overspeeding error should vary linearly with e

2 /2.

Figures 1, 2, 3, and 4 present measured values of the overspeeding

error as functions of e

2 /2 for four values of U. The mean velocity is

indicated in the legend to each figure. For each figure several sets of

data are shown corresponding to the different frequencies for which data were acquired at each velocity. Data for successive frequencies are offset vertically for clarity of presentation. The solid straight line for

each frequency represents the theoretical overspeeding error from Equation

(4), as calculated from measured values of e and ft.

It is evident from these figures that at the lowest frequency (0.5

Hz) the theoretical curves based on Equation (4) are in good agreement

with the measured overspeeding error. At higher frequencies it is apparent that the model breaks down for the reasons already discussed, and

agreement between theory and experiment is limited to successively smaller

values of e

2 /2 as the frequency increases. Thus, the inference drawn

earlier, that the range of applicability for the model equation may be

simply stated in terms of a maximum permissable value of eft, appears

reasonable. An appropriate criterion, based on the data, is

eft _< 0.4 . (5)

On each straight line in each of the four figures an asterisk is placed at

the maximum value e

2 /2 for which the above condition is satisfied. In

Figure 5 all the measured overspeeding error data satisfying this criterion

are plotted as a function of the predicted values from Equation (4)

.

It

is inferred from these results that Equation (1) is valid for the helicoid

anemometer within the experimental limits given by Expression (5).

4. APPLICABILITY TO TYPICAL ATMOSPHERIC CONDITIONS

An estimate of the frequency of the energy containing eddies present

in the atmosphere may be obtained from the longitudinal scale and the mean

wind speed at any given elevation. The scale may be conservatively taken

to be of the same order as the elevation. Hence, at an elevation of 52

feet (15.8 meters), the approximate height of the deck of the Sitka, Alaska

cable-stayed bridge (Reference 2) , the longitudinal scale L-^ is on order

of 15 meters. The characteristic frequency is then approximately

and the reduced frequency is simply

ft = 2irL

4
A measure of the amplitude of alongwind velocity fluctuations may be

obtained from typical values of the variance reported in Reference (2).

Thus

,

u -0.2

c An estimate of the numerical value of the dimensionless group, eft,

is

u ft * 0.2 (2tv ) . c L

1

For Gill helicoid anemometers of the type used in the present study,

L = 1.07 m, and one finds

u ft - 0.09 . c This estimate is nearly an order of magnitude smaller than the

maximum value given in Expression (5). It is evident, therefore, that

Equation (1) is indeed valid for typical atmospheric conditions.

5. PROCEDURES FOR CORRECTING MEASUREMENTS OF ATMOSPHERIC SPECTRA

The usual correction procedure for spectral measurements is based

on a linear approximation to Equation (1). For illustrative purposes

the effect of friction (U ) shall be neglected here. Defining U. =

to be the indicated velocity, Equation (1) may be written. ^

dU.

L ~~ + UU. - U

2

. (6) dt l

The velocity field is of the form

U = U (1 + u ( t ) ) , (7)

where u(t) is a stationary random function of time with zero mean. The indicated velocity may be represented by

U. = U. (1 + u. (t)), (8)

where from Equation (6), u^(t) satisfies the following equation to first

order:

du

,

T oF + u

i * u(t) (9)

where the time constant x^is given by x Q = L/U. This linear approximation

necessarily implies that U^ = U . The magnitude of the associated transfer

function is simply

1

V 1 + (wx 0 ) 2

5
where u)T o

= 0, and oj is the circular frequency.

The power spectrum of U, E^(w), may be obtained quite simply from

a measured spectrum of U, , E (to) by using the relation

1 u

i

EyCco) = [ 1 + (wt 0

) 2

] Ey (w). (10)

i

An equivalent linear correction to the spectrum E^Cw) can be com- puted from a record of u^(t) either digitally or by analog methods by

simply performing the linear operations indicated in Equation (9) to obtain a record of u(t) from which E^Cw) may then be obtained.

The latter technique may also be used to obtain an improved record of

U(t) from U^(t) which takes into account the nonlinear response characteristics by performing the nonlinear operations indicated in Equation (6)

Solving this equation for U(t),

U(t)

U.(t)

~2

dU.

l

dt (ID

These operations on the measured signal U^t) can be performed readily

by analog or digital methods, and the spectrum E (w) can be computed

directly from the record of U(t). In this manner both the inertial lag

and the instrument nonlinearity are taken into account. This approach

has the added advantage that an estimate of x is not required a priori

since it is the distance constant L which appears explicitly in Equation

( 11 ).

6. CONCLUSIONS

A suitable mathematical model for helicoid anemometers has been

validated experimentally in a periodic air flow. The model has been

shown to be applicable to such instruments in a typical environment of

atmospheric turbulence. A simple procedure to correct measurements of

atmospheric alongwind spectra for both inertial lag and nonlinearity has

been suggested. These results cannot be expected to apply to indefinitely

high frequencies since turbulence scales on the order of rotor characteristic dimensions will be spatially averaged and the model presented does

not take this into consideration. However, instruments of the type investigated in the course of this study may be used over the range of frequencies

and scales of primary interest in the study of wind loading of such structures as cable-stayed suspension bridges.

7 . REFERENCES

1. J. M. McMichael and P. S. Klebanoff, The Dynamic Response of Helicoid

Anemometers, NBSIR-75-772, November 1975.

2. C. L. Gerhardt and D. D. Nelson, Monitoring and Recording Wind

Velocities, Directions, and Effects at Long-Span Bridge Sites,

Report No. FHWA-RD- 76-10, August 1975.

6
Figure

1.

Variation

of

Overspeeding

-

Error

with

Amplitude

and

Frequency.
Figure

2.

Variation

of

Overspeeding

-

Error

with

Amplitude

and

Frequency.
0.16

0.16

IE3

I l£3 •H

l£3

Figure

3.

Variation

of

Overspeeding

-

Error

with

Amplitude

and

Frequency.
11=)

I li=>

l!=>
0.08

|

'i

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

- 0.01

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

e£ ft?_

2 1+ft 2

Figure 5. Comparison of Measured and Predicted

Overspeeding - Error for eft <_ 0.4
U S. DEPT. OF COMM. BIBLIOGRAPHIC OATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 78-1505

2. Gov’t Accession

No.

4. TITLE AND SUBTITLE CHARACTERISTICS OF HELICOID ANEMOMETERS

7. AUTHOR(S) J. M. McMichael and W. G. Cleveland

*. PERFORMING ORGANIZATION NAME AND ADDRESS NATIONAL BUREAU OF STANDARDS DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

3. Recipient’s Accession No.

5. Publication Date

6. Performing Organization Code

'(MJHfc °-e¥5i5S

port No.

10. Project/Task/Work Unit No. 2130482

11. Contract /Grant No. Purchase Order No 5-3-0200

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

IS. SUPPLEMENTARY NOTES

14. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.) An experimental study of the overspeeding error for helicoid anemometers

in periodic air flows is described. The ranges of amplitude and frequency for which a simple nonlinear model for the dynamic response of

such instruments remains valid are presented. It is shown that the

model is valid for typical atmospheric applications of such instruments.

A simple method is presented whereby the effects of inertial lag and

nonlinearity may be taken into account in obtaining measurements of

alongwind power spectra in the atmosphere.

17. KEY WORDS (mix to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons) Air; anemometer; helicoid anemometer; lag; overspeeding error; rotary anemometer; unsteady flow.

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