Interpretation of data and response of probes in unsteady flow

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Author's Reprint - Presented at the ASME Hydraulic Division Conference, May 21-23, 1962 and published in Symposium on Measurement in Unsteady Flow by The American Society öf Mechanical Engineers, 345 East 47th Street, New York 17, New York.

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INTERPRETATION OF DATA AND RESPONSE

OF PROBES IN UNSTEADY FLOW

Philip G. Hubbard

a radius of pressure probe

C1, C2 constants characterizing transducer system f frequency of sinusoidal pressure fluctuation

g standard acceleration of gravity

H average head indicated, by a stagnation probe

L size of a typical eddy

n number of radii between nose and pressure openings

p instantaneous true pressure at a point in the

flow

instantaneous pressure indicated by a transducer

p average true pressure

error in pressure indicated by a probe

t time

average velocity at a point

u' instantaneous fluctuation of velocity about the

mean

(Vr,) radial component of velocity in absence of

probe

V transverse components of velocity

z elevation of a point in the flow y specific weight of fluid

angle of skew for a probe

p specific mass of fluid

Unsteadiness in a flow field creates- two general types of problems in measuring flow variables: The accuracy of instruments which detect mean properties may be affected adversely, and mean values alone may be in-adequate for a satisfactory analysis. With regard to mean quantities, it is necessary to select instruments which are unaffected by the unsteadiness, or else to evaluate the errors so that corrections can be made. When information about the unsteady components is required, drastic modifications to the system are often necessary, or perhaps an entirely new principle of measurement must be adopted. In addition to the instrumentation problems caused by unsteadiness, it is sometimes difficult to define the mean value. More specifically, the period of time over which the variable must be averaged may require considerable- thought.

Before proceeding to the methods involved in solving

these problems, it will be informative to consider the types of data ordinarily desired, and the types of sensing and readout instruments involved. Some systems can be studied satisfactorily in purely kinematic terms,

requiring measurement of fluid velocities and

accelerations. Analyses of a different nature may re-quire additional information on pressure, density, and mass flow rate. Many mechanical systems cannot be -analyzed completely without a knowledge of temperatures at different points, and the presence of a free surface means that sometimes its elevation must be determined. Any or all of these quantities may vary in an unsteady system, so that the acquisition and interpretation of data frequently require careful consideration of a number of interrelated characteristics.

Instruments used to measure these variables include pitots, current meters, and hot-wire and hot-film

anemometers; accelerometers and pressure cells employing strain gages, crystals, reluctance elements, and capacitors; thermocouples and resistance

thermometers; and, for wave heights, capacitance wires. Indicating or readout instruments include various graphic

and photographic recorders, d'Arsonval-type meters, and cathode-ray oscillographs.

Although maximum or minimum values are some times important, temporal (and sometimes spatial) means are far more frequently required for analyses. When the un-steadiness is periodic, as in flow affected by pumps, vibrating surfaces, vortex trails, and acoustic or hydraulic resonances, information on the period, amplitude, relative phase, and possibly harmonic com-ponents can be used in a systematic analysis. Still another class of flow includes phenomena which are unsteady even in a statistical.sense; shock waves,

surges, waterhammer, and explosions are typical examples. Analyses of these systems involve delay periods, damping constants, energy storage, and the concomitant resonance or "ringing" which may exist in the flow and possibly in the measuring instruments. The final type of unsteadiness to be considered is the con-tinuous random or stochastic motion encountered in turbulent flow, which can be analyzed only in terms of statistical quantities: mean products of. velocity corn -ponents or velocities and pressure, mean scalès, and velocity gradients.

With regard to the time-dependent errors, perhaps the most revealing test of an instrument for use in unsteady flow is its response to a step function. The behavior of most probes can be described in terms of a typical time constant plus a damping coefficient. If the- response to a step function approaches the final value monotonically, reaching about 65% of the final value in time t., then it will be satisfactory for periodic flow with periods down to about -5ç, and its response will decrease- monotonical-ly as the period decreases. If the system has less than

critical damping, then- the response to a step functicri will exhibit an overshoot or even a ringing,, axid the instrument will respond abnormally to periodic stimuli within a certain range. It is generally assumed that the response in a system of random unsteadiness, where the spectrum of the variable is continuous over a wide band of frequencies, will be abnormal in the range where the response to single frequencies is excessive or deficient It thus appears that as far as time-dependent distórtion is concerned, satisfactory response in unsteady flow requires an instrument with high natural frequency and, for optimum conditions, well-controlled damping.

As an xathple of a system with a time-depéndent response, consider a pressure cell with excellent chaìacteristics up to a frequency beyond the range expected in the flow. If the size of this céll is

exces-sive foE use as a probe or for flush mounting in a boundary, cómmon practice is to connect it by a tube to a piezorneter hole in the probe or boundary. The result is a system whjch is usually underdamped, with natural frequency considerably below tht of the unmodified transducer. All of the undesirable symptoms already described are present, and the performance may be quite unsatisfactory. For the purpose of illustrating this type of behavior, the system illustrated in Figs. 1 and 2 was constructed. Two identical pressure cells pos sessing uniform response characteristics up to 19,000 cycles per second are mounted in a heavy brass block, rith an adjustablé length of tubing connecting them The chamber in front of the upper transducer can be equipped with a rupture diaphragm and connected to a high-pressure source so that a sudden drop in high-pressure can be induced (Fig. 1), or it can be connected to an electromagnetic driver (Fig. 2) which creates an oscillating pressure in the chamber. The flui4-filled space in front of the lower transdücer (referred to as re-cessed) was made as small as practicable in order to avoid cavity resonance, and the entire system, except for the driver or the rupture chamber, could be filled

with either air or liquid. Figures 3 throuh 5 are

oscillograms of the response of this airfilled system to sudden reductions in pressure via the bursting

diaphragm, and three characteristics should be noted: (1) the time delay between the two signals, (2) the much

slower decay of the signal from the lower transducer, and (3) the attenuation or magnification of the fluctuations which follow the burst. Three different lengths of tubing have been used in order to emphasize the desirability of keeping the length at a minimum..

The frequency-rêsponse curves plotted in Figs. 6 and 7 illustrate the performance of the same system when filled with air (Fig. 6) or water (Fig. 7) and subjected to sinusoidal fluctuations in pressure. The resonant peaks are quite pronounced with either fluid and with either short or long connecting tubes. This type of behavior is quite well understood, of course, and a detailed analysis would be superfluous here [il. However, the point of the present discussion is to suggest a method of

improving the performance of the system by compensating

the signal. It is assumed thatall reasonable means

have already been taken to improve the mechanical system, such as minimizing the length of tubing, maximizing its diameter, and increasing the damping. If these efforts still fall short of the desired result, then compensation of the signal is in çrder and an analytical reduction of the system to a simpler components can be

very helpful.

/

Figure 1. Recessed-Transducer System Equipped for Sudden PressUre Reduction by Rupture Diaphragm Attachment

Assuming that the flow in the connecting rube is laminar and that the fluid and second transducer behave in an elastic manner, it is readily shown that the characteristic equation is

dt

+ C1 - + C2 Pt = C2p = f(t)

where p is the pressure jùst outside the piezorneter to which the tubing is connected (indicated by the first transducer), Pt is the pressure detected by the

transducer at the end of the tube, tis time, and C and

C2 are constants. This is the well-known equation for an elastic system with a single degree of freedom, where p is the driving function. The constant C2 deter-mines the natural frequency (ø =C2), and C1 is the damping coefficient. Equation (1) is obviously well-suited to solution by an electronic diffèrential analyzer [21 (analog computer), and a suitable circuit for this

purpose is illustrated in Fig. 8. The first two triangles

Figure 2. Recessed-Transducer System Equipped for Sinusoidal Excitation over an Extended Frequency

'g

Figure 3. Response of o Recessed Transducer(Sloping Line) at End of 75 ft of Tubing, with Sudden Drop in Pressure (Horizontal Line); System Is Air-Filled, and Time Scale

Is 0.1 sec/Cm

represent differentiating amplifiers, the third is a simple phase inverter, and the fourth is a summing amplifier. The circles represent coefficient potentiometers (or amplifiers) which determine the constants C1 and C2. The use of an approximate differentiating circuit based on integrating amplifiers only [3] should improve the noise level of this circuit, but will involve many more components. If the distorted signal from the transducer is fed into this network, the output p(t) will correspond to the undistorted input to the system, provided that the coefficients C1 and C2 are correctly set. In practice, this operation involves supplying a known input pres-sure at the piezometer and adjusting the constants until

a

Figure 4. Response to Sudden Pressure Decrease with 55 Inches of Tubing. Time Scale 2 msec/cm

Figure 5. Response to Sudden Pressure Decrease with 5.5 Inches of Tubing. Time Scale 2 rnsec/cm

the output correctly represents this known input, or the constants must be determined from the characteristic

equation. This latter techñique is not recommended, because of the very poor accuracy with which the equivalent length and diameter of tubing and the

equivalent spring constant can be determined. The systems illustrated in Figs. i and 2 would be ideal, of course, but simpler arrangements may be acceptable. If values of damping constant and natural frequency are to

be obtained from plots similar to those of Figs. 6 and 7, it should be noted that the maximum sensitivity always occurs at a frequency less than the natural frequency if damping is present.

It should be realized that the system just described is purely illustrative, and that much more complicated response relationships representing square-law damping,

multiple degrees of freedom, dead zones, backlash, hysteresis loops, and other types of nonlinear behavior are readily reproducible using standard analog-computer techniques. The amount of complication to be ac-comodated in this way is primarily a matter of economics. Quite obviously, the method is also well suited to compensating the output of transducers such as hot-wire and hot-film anemometers, resistance thermometers, thermocouples, and heavily overdamped systems in which the second-order term is unimportant or missing altogether.

Although phase lag, attenuation, and resonance can be compensated in these analog circuits, time delays due to transmission through an elastic fluid cannot be eliminated in any circuit. This would require the circuit to produce an output before receiving an input, which is not possible. Usually, of course, the existence of a constant delay is unimportant, as in any analysis of a single signal. When correlations between two or more signals are required, however, such delays can be serious sources of error. If all variables are similar, e.g., the stagnation and local pressures from a pitot, then maintaining similarity in the transmission lines can make the delay times equal and hence harmless. How-ever, if it is required to correlate pressure and

tempe-5 4 3

2

Figure 6. Response of a Recessed Transducer to Sinusoidal Pressure Fluctuations at Various Frequencies.

System Is Air.FiIled

o

/80

270

/0

/00

f,cps

Figure 7. Frequency Response of a Water-Filled Transducer

System

Figure 8. Analog Type Compensating Circtiit rature fluctuations, for example, then such similarity cannot be maintained. It is entirely feasible, however, to pass the signal from a thermocouple through a delay line, before comparing it with the signal from a recessed pressure transdúcer. Lines for delays from a few micro-seconds to a few millimicro-seconds are obtainable as

standard items, and longer delays are easily arranged with the aid of a magnetic-tape recorde.

Turning attention now from time-dependent errors which are so important in measuring unsteady components of a flow system, consider the effect of unsteadiness on the performance of instruments which should indicate true mean values only. Much attention has been given to evaluating erEoEs from this source through the years, and from their titles it appears that some of the subsequent papers in the present symposium will cover this subject. With this in mind, the discussion herein will be limited to pointing out sources of error and reviewing some of the significant works in the field. Additional limitations will be imposed by omitting references to tempature and wave-height measurements, simply because the writer believes that temperature-measuring probes and wave.height transducers exist in which errors peculiar to unsteady flow are due either to time-dependent characteristics alxeady discussed, or to dynamic effects which are easily determined and corrected.

Instrumental errors in the presence of unsteadiness are attributable to asymmetrical response of the probe to increases and decreases in the variable, to changes in angle of attack which accompany unsteadiness, to conversion of velocity head of turbulence to pressure head, or to asymmetric transitions in networks between the probe and the indicator. The latter case may be compared with a rectifier in a series electrical circuit, and frequently is related to burrs, wear, or damage in piezometers and transmission tubes. It is also

neces-sary to avoid the counterpart of series resistances coupled with parallel capacitance which make an effective low-pass filter but cause the average output voltage (or pressure) to build up.

A good example of asymmetry in the probe itself is the stagnation tube aligned with flow, which indicates the total head of the mean flow plus a head due to turbulent velocity fluctuations. For incompressible

flow,

If the tube is designed to respond only to the longitu-dinal component of velócity, then u' represents the deviation from the mean of that component only. If the rube is insensitive to attack angle over aconsiderable range or actually increases its indication when yawed [5], then u' represents the resultant deviation.

The effect of turbulence pon the mean pressure transmitted by the "static" openings of a probe is typical of errors due to angularity and coñversion of energy. In a recent paper, Toornre [61presents an

analysis of this effect, and shows that the error may be either positive or negative. The sign of the error depends on the "typical eddy size" of the turbulence in comparison with the probe radius, being positive for small-scale turbulence in which the fluctuations at the various holes are not in phase, and negative for larger scale turbulence in which the tendency of the probe to underregister when severely yawed is the dominant effect. Still another variable is introduced when he shows that the longitudinal location of the pressure openings relative to nose is also important, a positive error being produced with a magnitude

p (V)

+2L/a

wherein is the error, (Vr) is the radial component of turbulence which would exist if the tube were absent,

a is the radius of the tube, L is the eddy size, and n is

the number of radii between the nose and the pressure openings. The. magnitude of the resultant error due to this nose effect and the aforementioned «sutface» effect is shown to vary between the limits

i

'nP

<p(V+Ì7) <

where the subscripts y and z indicatethe.transverse components. The error appears to be zero for eddy

sizes equal to or slightly larger than the tube radius. Kronauer and Grant [71 considered errors due to

asymmetry in internal passages of a probe, compres-sibility effects resulting in higher density during inflow, and rurbúlent flow in the piezometer opening. They referred to other works in which the first two effects were found to be small, and proceeded to concentrate on the effect of asymmetrical fluctuations coupled with turbulent flow internally. Their conclusion was that the error depends on the magnitude of the fluctuation, of course, and also upon the frequency of the prçssure changes rélative to a typical probe frequency. No error exists if the fluctuation frequency is much higher thañ the typical probe frequency, but this condition is not met for many conventional probes (pitot-static, certain cylinder probes, etc;); easily-made modifications can make these acceptable, however. Inasmuch as the nonlinear effects are due to the existence of turbulent flow internally, there is also a critical amplitude of pres-sure fluctuation below which no error is introduced. Where the error exists, it can be either positive or negative, and as much as 15% of the amplitude of the fluctuation.

The effect of various types of unsteadiness on the indications of a Conrad probe, a 5-tube probe, an axial probe, a cowled pitot tube, a rectangular pitot tube, and a disk-static probe was studied in low-speed flow by Waishe and Garner at N.P.L. [41. They subjected the

probes to several types of angular and linear

oscil-H

r-2

-

1i2

+

zI-j

làtions in steady flow with and without a super-imposed. steady yaw angle, and then used them to explore the turbulence in the wake of an airfoil. The amplitude of the angular oscillations was varied up to15°, correspond-. ing to equivalent intensities of turlulence as high as

44%. Hot-wire measurements of the turbulence in the wake showed intensities of approximately 35% in the

test region. The general impression of the authors was

. that the greatestcause of rot in pressure nasurement

is associated with- the behavior of the flow within the probes themselves, so that future work would probably

involve probes of various sizes and ratios of wall

thick-.

ness to diameter. Perhaps the best summazy of their results is their own list of conclusions, as follows:

«(1) In fluctuating flow, the measured value of pres-sure will give the time-average value if the fluctuations are in flow directioñ only. When velocity fluctuations occur, the measured value of pressure will be less than the time-average value.

«(2) The cowled pitot tube will give the most ac-curate measure of. total head. In a highly turbulent. wake, typical discrepancies betwen this and other probes amounted to 7 percent of the free-stream dynamic pressure.

"(3) When the mean angle of flow.is required as well as the total pressure and static pressure from one setof

readings, the 5-tube probe is preferred to the axial prçbe.

«(4) A comprehensive assessment of the usefulness of probes in fluctuating flow wouldrequire more extensive data on the effects of Reynolds number and mode, frequency and amplitude of oscillation. More fundamental studiès of internal and external flows re needed to explain the behavior of any probe.

«(5) It is coñflrmed that in'steady flow the Conrad probe will give accurate results within a boundary layer if the. angle of i cal flow direçtion in pitch does

not exceed 6 deg." .

-Passing reference. at least should be made to the fact that practically all probes - pitot, static, hot-wire, etc.) depend on unidirectional flow for correct operation. If the instantaneous, velocity becomes negative with respect tothe orientation of the probe, then serious errors are intròduced and several successive corrections ma be required to yield reasonably accurate results.. The general method to be employed is to plot streamlines from data corrected in accordance with equation (1), ánd to adjust' them according to continuity requirements.. A similar' correction for angularity is illustrated in [81....

In view of the fairly recent revival of rotating nemometers as very small probçs for«point" measüre-, ments [91,:it. might be well to call attention to a study, made on larger instruments some 33 years ago .110].

Although these tests predated much' of the work in, turbulence, the conclusion that propellçr-type meters

-tend toj under-register in turbulent flów should be. given careful consideratipn. Hydraulic engineers will also be interested in the observation that cup-type metçrs tènd. to over-register in turbulent flow. In view .of the fact that the behavior of a probe when misaligned with the mean flow is important in all cases.where turbulençe is

involved, the development of a "component runner' Ill] in Germany deserves notice. This screw-type runner indicates a velocity V cos O, wheré O is the angle of skew, for all values of O from _900 to +90°, and hence should be free of errórs due to the large-scale

fluctuations in unsteady flow. To point out the obvious, the use of very small meters briñgs more of the tur bulence into the "large-scale" range...

In conclusion, it is hoped that this discussion has sèrved to point out sources of error and design criteria for minimizing error due to unsteadiness in the measurement' of some mean-flow properties: Select probes which yield the correct component when yawed, makè internal flow passages symmetrical, keep the. internal flow laminar and well damped, match the'scale of the probe to the scale of the fluctuations' where feasible, and remember that errors, when present, tend to increase rapidly with increasing unsteadiness of the flow. : When the problem is measurement of the' unsteady properties per se,, very short time constants ib the probç (short, largediameter tubes in pressure probes') and' all associated circuits are of paramount importance., .

Controlled damping can improve, frequency response by a factor of perhaps 2 where.performance is marginal, and judicious use of analog-computer techniques on the signal can extend performance markedly for a particular probe, even though its behavior is quite complex.

References ' . . . .

1. Iberail, A. S., "Attenuation of Oscillatory 'Pres-sures, in Instrument Lines,'. Nat'l. dur.. Stds. Researçh Paper RP 2115,' Vol. 45, 1.950. , ' . . '

2. Korn and Korn, Electronic Analog Computer,. Second Edition, McGraw-Hill, 1.956.

Johnson, Analog Computer. Techniques, McGraw-, Hill, 1956,, p. 97. ..

Walshe, D. E., and Garner, H. C., "Usefulness of

Varus Pressure Probes in Fluctuating Low-Speed

Flow," Aer Res. Çoun. .Rpt. A.R.C. 21,714 (1960). Goldstein, 1odern Develo pmens in Fluid Dynamies, Oxford, 1938, p. 252.

6.. Toonire, A., "The Effect.of Turbulence on Static Pressure Measurements," A.R.C. 22,010 (1960).,.

Kronauer, R. E., and Grant, H. P.,"Pressure

Probe Response' in Fluctuating Flow," Proc. 2nd U.S. Nat'!. Congr. -AppI. Mech., 1954.

R usC, H., Siao, T.' T., and Nagaratnam, S., "Turbulence Characteristics of the Hydraulic Jump,." Trans. ASCE, Vol. 124, 1959, p. 937.

Molyneux, L., and Edington, J. M., "Portable Propellor Flowmeter Determines Water Velocity,' Electronics,'June 23, 1961, p. 60..

"10. Yarnell,,D. L., and Nagler, F. A., "Effectof

Turbulence on'.the Registration of Current Meters," Trans. ASCE, Vol. 95(193.1), p. 766.

11. Kolupaila, S.,"Recent Developments in Current Meter.Design," Trans AGIJ,,Vol. 30,.No. 6,1949.