Calibration of a three-wire probe for measurements in nonisothermal flow

Calibration of a Three-Wire Probe for Measurements in Nonisothermal Flow L. Meyer

IIA calibration method for a hot-wire probe with an x-wire and a third cold wire has been developed for measurement in nonisothermal flow. The method is an extension of a multiple-angle x-probe calibration technique using a lookup table to the case of moderately heated low-speed flow for the simultaneous measurement of two instantaneous velocity components and the instantaneous temperature. The main quality of the method is its speed of evaluation. The evaluation of the velocity components and temperature is fast enough to determine all turbulent quantities, such as higher order correlations, frequency power spectra, and probability density functions on a microcomputer directly during or immediately after the data acquisition.

Kernforschungszentrum Karlsruhe, INR Karlsruhe, Germany

K e y w o r d s : hot-wire calibration, non&othermal flow, temperature fluctuation measurements, turbulent heat flux, higher order correlations, lookup table method

INTRODUCTION For the study of axial flow through heated rod bundles, the measurement of turbulent quantities at several hundred points in the flow cross section is necessary. The airflow is moderately heated with wall temperatures up to 80°C and mean velocities between 10 and 35 m/s; the hydraulic diameter of the flow cross section is on the order of 50 mm. Reynolds stresses, eddy diffusivities of momentum and heat, and frequency power spectra are determined. The measuring method must be capable of running fully computer controlled and delivering processed data that need only a small storage capacity. Therefore the evaluation of all turbulent quantities must be done during the measuring process. The most practical technique presently available for this task is hot-wire anemometry using a multiple-sensor probe. A widely used hot-wire probe for the simultaneous measurement of two velocity components is the x-wire probe, although it has some inherent error sources. The most severe is the error due to neglect of the velocity component that is normal to the plane of the x-probe; see, for example, Tutu and Chevray [1]. To overcome this, extra wires can be used [2] while maintaining the x configuration, or three- and four-wire probes can be employed that are capable of measuring all three velocity components [eg, 3-5]. There are basically two different methods for measuring the instantaneous temperature by using hot wires. By using two parallel identical wires close to each other at different overheat ratios, it is possible to determine the temperature fluctuations [eg, 6-8]. Since both wires are run in the constant-temperature mode (CTA), the frequency response is very good with relatively thick wires of 5 /zm diameter. Whereas Blair and Bennett [6] obtained good results in a

heated boundary layer, Lienhard and Helland [7] warned of using this technique in high-intensity flows with small temperature signals. Sakao [8] found this technique qualified only for moderate frequencies. This corresponds with my own experience with tests at moderately heated and isothermal jets, where I have found generally excessive temperature fluctuations. This may be the result of the finite spatial resolution of the two-wire probe. Any difference between the two signals of the individual wires is interpreted by this method as a temperature fluctuation. Thus, even in isothermal flow, temperature fluctuations were indicated at higher frequencies. The second method is the well-known cold-wire probe, which is run at very small currents in the constant-current mode (CCA). It has the drawback that very thin wires (diameter _< 1 #m) need to be employed to achieve a sufficiently high natural frequency response. Digital or analog techniques can be used to improve the frequency response and to eliminate phase-shift errors between the velocity and temperature signals. The choice of technique to apply to our situation was governed by the following factors. (1) Only commercially available anemometer systems should be used; (2) the size of the probe must be small but should consist basically of available probes; (3) evaluation of all quantities must be done online; (4) calibration and measuring must be performed fully automatically. Keeping in mind the above-mentioned shortcomings of x-probes for future improvements, I chose a probe with an x-wire/cold-wire arrangement that has been used by many other researchers [9-14]. This paper describes the calibration method for an x-wire probe with an additional cold wire perpendicular to the

Address correspondence to Dr. L. Meyer, Kernforschungszentrum Karlsruhe, INR, Postfach 3640, D-7500 Karlsruhe, Germany.

Experimental Thermal and FluM Science 1992; 5:260-267 © 1992 by Elsevier Science Publishing Co., Inc., 655 Avenue of the Americas, New York, NY 10010 260

0894-1777/92/$5.00

Calibration of a Three-Wire Probe x-wire plane for simultaneous measurement of two components of instantaneous velocity (u and v) and temperature (0) in low-speed ( 5 - 3 0 m/s) airflow at moderate temperatures (10-80"C). The calibration method uses lookup tables (LUTs) as described by Lueptow et al [15], extended by the temperature dimension. The performance of the method was evaluated by measuring the turbulent quantities in a heated turbulent pipe flow. Although my coworkers and I tested this method only with our x-wire probes, it should be suited to any design of x-probes with an additional wire to obtain the instantaneous temperature, as well as to probes in which the two wires are mounted like a V. THE CALIBRATION TECHNIQUE Probe D e s i g n The design of the probe was keyed to the determination of the flowfield in small channels, such as the axial flow through rod bundles. The x-wires have a length of 1.2 ram, a diameter of 5 #m, and a spacing of 0.40 mm. The cold wire has a diameter of 1 #m and a length of 0.9 mm and is positioned 0.1 mm upstream of the x-wire prong tips, perpendicular to the x-wire plane. The measuring volume is approximately 1 mm 3. A different configuration was tested with the cold wire parallel to the x-wire plane, but it turned out to be inaccurate in temperature gradients. The probe was fabricated in our laboratory using the Dantec probe 55P61. Small grooves were cut into the ceramic body among its entire length at both sides. Nickel prongs for the cold wire were glued into each of the grooves. With the spacing of the prong tips of 0.9 mm, the x-wires should not be affected by the cold-wire prongs if the flow vector perpendicular to the x-wire plane is within 16". Great care was taken to ensure that the connections between the two nickel prongs and the two copper wire leads lay close together and were thermally insulated, in order to avoid errors by inducing a thermoelectric emf. The Calibration Unit The Dantec nozzle calibrator 55D90 was modified to achieve airflow at different constant temperatures. The air from the laboratory's compressed air supply system was used and regulated to a constant pressure of approximately 3 bar. The air was led through a thin, 3 m long, stainless steel pipe, which was rolled into a coil to fit into a small well-insulated box. The pipe was heated directly by electrical resistance heating, using low-voltage high current supplied by a controllable transformer. The mass flow rate of the heated air was controlled by two high-precision needle valves that were interlocked by gears and driven by a step motor. When one valve opened, the other one closed. The flow through one was led to the calibration nozzle while the flow through the other was led into the open air. In this way the mass flow through the heating section was kept almost constant. This simplified the control of both velocity and temperature because they were almost independent of each other. The valves and the nozzle unit were thermally well insulated. The nozzle diameter was 12 mm. The measuring position was 6 mm above the outlet. The temperature was determined by a thermocouple close to the measuring position of the hot-wire probe with an uncertainty of 0.1 *C. The flow

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velocity was determined from the static pressure measured in the nozzle settling chamber. Both temperature and velocity were computer controlled and were held constant to within +_0.15"C and 0.05 m/s, respectively. The pitch of the hot-wire probe was achieved by a computer-controlled step motor in steps of 0.05*.

Data Acquisition The data acquisition, mass flow rate, and temperature at the calibration unit and the movement of the probe are controlled by a microcomputer HP-Vectra RS25C. The triple-wire probe is run by two CTA bridges and one CCA bridge from the Dantec 55M system. A TSI-IFA-100 with three channels was used for signal conditioning, with filter, offset, and amplification. The signals were digitized by a Data Translation DT2828 card, which provided sample-and-hold digitization of up to four channels with 12-bit resolution and a maximum input signal of 10 V. Signals were sampled at different rates, ranging from 500 Hz to 20 kHz per channel. The total number of samples taken in a continuous stream was 96,000 per channel, loaded by DMA into extended memory of the computer. For measurements without digital compensation of the temperature signal, a sampling rate of 2 kHz was chosen with measurement times of 48 s. The Calibration Method Conventional calibration methods use empirical relationships to describe the heat transfer from the wire as a function of flow velocity and the angle between the velocity vector and the wire. Three or four parameters have to be determined for each wire in isothermal flow as well as the precise angles of the x-wires. For nonisothermal flow the dependency of each parameter on the temperature has to be established; or other schemes can be employed, generally requiring more than 10 parameters for a triple-wire probe. An entirely different approach to the problem of calibrating x-probes is a lookup table method that does not require any assumptions regarding the nature of the sensor cooling. Nor is it necessary to know the precise angles of the x-wires. However, the number of calibration points necessary is higher than for most of the other methods, as the x-probe has to be exposed to at least four different flow velocities and be oriented through a range of yaw angles at each of them. Several methods have been proposed [15-17] that use a multiple-angle x-probe calibration technique in conjunction with a lookup table. The method described by Lueptow et al [15] seems to be the easiest in setting up the LUT and the fastest in evaluation of the velocity components. It has been stated that LUT methods cannot easily be extended to thermally varying flows. I will show that this is not the case, at least for the temperature and flow velocity range we are dealing with. Since the method has been well described by Lueptow et al for the application to isothermal flow, I will concentrate on the problems arising with nonisothermal flow. The procedure at a constant calibration temperature T is the following. The x-probe is inclined through a certain range of yaw angles ~b, for instance, from - 30 ° to + 30* in steps of 6", at constant flow velocity Q, This is repeated at five different velocities. For each calibration point we obtain the voltage pairs E I and E 2 of the two wires (Fig. 1). As

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sufficient accuracy, the functions Ej, E 2 = f ( T , Q, 49) were investigated. The full Q, 49 range was measured at five temperatures. Figure 1 shows the strong effect of the temperature on the sensor signal E. In search for a reduction formula we follow the logic of Bearman [ 18], who developed a correction formula for small temperature drift. He starts with the basic law of heat transfer at long cylinders,

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= 52 C

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(2)

neglecting the temperature correction factor on the Nusselt number found by Collis and Williams [ 19]. Taking advantage of the fact that for air the property group k ( p / # ) ° 5 is only very slightly dependent on the fluid temperature, Bearman derives the relation

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=vc

(3)

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E~ (v) Figure 1. Sensor signals E l and E 2 at five velocities, 11 probe angles, and two temperatures. described by Lueptow et al, a matrix in the E r E 2 plane is formed that contains at each point in specific intervals, for example, A E = 0.01 V, the values for the velocity components U and V. In the actual measurements, arbitrary values of E 1 and E 2 are obtained. For the evaluation of U and V, the positions of the four E I, E z pairs closest to the measured values are determined in the matrix. A simple bilinear interpolation determines the values of the velocity components U and V pertaining to the measured El, E 2 values. A straightforward extension of this method to the third parameter, the temperature would be the determination of a three-dimensional matrix U, V, T = f ( E l, E 2, E3). Since this means measuring the total velocity-yaw angle field at a minimum of four temperatures each, the cost of calibration time is quite high. Therefore the temperature effect on all three wires was investigated in the velocity range of 5 - 3 5 m / s and the yaw angle range of + 35* to find out if this is necessary. The relation between temperature and the signal of the cold wire is linear in the temperature range investigated. For a CCA signal it is e 3 = A 3 + "/3 r

(1)

If the heating current is small enough (eg, 0.1 mA), the effect of the flow velocity on E 3 is negligible. An effect of the yaw angle is possible during the calibration procedure only if parts of the probe are outside of the heated flow. Then either temperature conduction effects or thermoelectric effects due to faulty construction of the probe can lead to errors. During measurement in turbulent flow, such errors should not occur, with the exception of measurements in very strong temperature gradients. Because of the high accuracy of the measured data obeying Eq. (1) for all flow conditions, a lookup table for T = f ( E 3 ) cannot produce any improvement. In order to find out whether the dependence of E l, E 2 = f ( T ) can also be described by an analytical relation with

which he linearizes for small temperature corrections. The index c denotes the value of E at the reference fluid temperature TC and the specific velocity at which E m was measured, T m is the air temperature, and T w is the temperature of the wire. For a better understanding of the relation between the wire signal E and the temperature difference between wire and fluid, Eq. (3) can be written as f2m

Tw-T -

E2c

-

m

-

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(4)

Tc

Since there is no need for linearization, and for hightemperature reductions a iinearized equation would increase the errors, we use Eq. (3) in the form

EL = El

--

1 + 3"(Tc -

Tin)

(5)

to reduce all calibration points to a constant temperature. The reduction parameter 3' = 1 / ( T w - T c) is dependent on the wire temperature and therefore different for each wire. The examples of some calibration points at Q = 12 m / s and 25 m/s (Fig. 2) show that Eq. (5) holds well; the relation is close to linear. However, we have a different parameter 3' for different yaw angles and a small effect of the flow velocity. This relation, that E 2 is related linearly to T, was also found by other authors (eg, Fulachier [10], Champagne [21], Bremhorst [22], and others). Figure 3 shows the temperature reduction parameter 3' for both wires, which had a slightly different overheat ratio, at five yaw angles each, as a function of the calibration velocity. The error bars at zero and +_30* yaw angle show the standard error in the evaluation of 7, that is, the deviation of linearity in Fig. 2, which was obtained by a least squares fit using IMSL routines. Bremhorst's [22] data reduced by Eq. (5) show a similar trend--a very small variation at velocities above 10 m/s, but a large drop of 3' at smaller velocities. Bremhorst discussed this finding in terms of an effective wire temperature Tw, eft, which is 1

Tw,eff = -- + Tc 3'

(6)

His conclusion is that Tw, etr = Tw for U > 10 m/s, but below this velocity a significant departure between Tw, err and

Calibration of a Three-Wire Probe 0.2

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Figure 4. The temperature reduction parameter 3' versus the Tw occurs. However, he used a 2.5 /~m diameter wire and states that this is a Reynolds number effect, so that the cutoff velocity above which Tw,err (or 3') is practically independent of velocity is inversely proportional to the wire diameter. Thus, a calibration with a 5 /xm diameter wire should not show this effect at 7 m / s , which is at variance with our findings. We also did measurements with a 2.5 /zm diameter x-probe and found the same dependence of 7 on the calibration velocity as for the 5 ttm diameter wires. Nevertheless, the effect of the velocity on 3' is smaller than the effect of the probe angle in our calibration range. The dependence of 3, on the yaw angle is shown in Fig. 4 for both wires: 3" has a maximum at zero probe angle. While 3" changes little for probe angles that decrease the effective yaw angle of the respective wire, 3' decreases more for high

probe angle.

effective yaw angles. Up to now we have tested six different probes, and the general trend is similar to the one shown, yet not identical. Obviously, this effect is totally or partially an effect of the prongs,~and therefore of the geometry of the welding spot. A useful test, which could not be performed in our laboratory, would be to use a larger calibration nozzle, having the whole length of the prongs in the heated flow at all probe angles. Figure 5 shows the reduced values of E l, E 2 pairs for five velocities and four temperatures. All measured values were

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reduced by Eq. (5) with a constant average value of 3, for each wire. Many of the calibration points at different temperatures but the same velocities and yaw angles are reduced to a single point with very small variances; some do not collapse. If reduced by the individual 3,'s, the variance is practically zero for all points (not shown). Now we have the choice of several alternative ways to employ a lookup table method in nonisothermal flow, which basically fall into two categories: whether an average 3, or individual 3,'s depending on u and v are to be used. The choice will be governed by four criteria: 1. The range of velocity, temperature, and turbulence level to be measured, which determines the quality of the reduction by Eq. (5) 2. The permissible variation of the experimental results 3. The available evaluation time 4. The cost of calibration time Three different methods will be described briefly. Method 1. The full Q, ~b range is measured at a minimum of three temperatures. The reduction of the measured El, E2 pairs to a constant temperature by Eq. (5) is performed for each yaw angle and if necessary for each velocity separately. This way they have a very small variance. One single LUT is determined at the reduced temperature with the averaged values of E 1 and E 2. The evaluation of the measured data is performed in five steps. First, the temperature is determined by Eq. (1) from E 3. Second, E I and E 2 are reduced to the constant temperature by an average 3' of the specific wire. Third, approximate values of Q and ~b are found directly from the LUT without interpolation. Fourth, the reduction to Ej and E 2 is repeated using the specific 3,'s pertaining to the Q and ~b values. Fifth, the final values of U and V are determined using a bilinear interpolation scheme between the four nearest points in the LUT. Method 2. The calibration data are the same as in method 1. The reduction of the E l, E 2 pairs by Eq. (5) is performed with an average 3, for each wire. For each calibration temperature a separate LUT is determined with the reduced E 1, E 2 values. The result are two three-dimensional matrices for U and V, the LUTs U, V = f ( T , Et, E2). Here the reduction to a constant temperature serves only to put all two-dimensional matrices into the same E 1, E 2 range. The evaluation of the measured data is a three-step process. The temperature is evaluated as usual with E 3 by Eq. (1). The measured data E 1 and E 2 are reduced by the respective average 3, by Eq. (5). With T, Elc, E2c we go into the three-dimensional LUT and find the eight nearest points. Using a trilinear interpolation, the U and V values are determined. Method 3. The calibration is performed at only one temperature. Additionally, one calibration point each at several different temperatures is taken at one constant velocity and without inclination of the probe. From these measurements the reduction parameters 3'1,2 are determined. The LUT is determined at the calibration temperature of the Q, ~b field. The evaluation of the measured data is similar to that of method 2. The temperature is evaluated as usual from E 3, and the E I , E 2 values are reduced to the calibration tempera-

ture by Eq. (5). The values of U and V are found by bilinear interpolation from the two-dimensional LUT. The advantages and disadvantages of the three methods are shown in Table 1. We tested methods 2 and 3. In a typical example of the range of the flow velocity was Q = 5 - 3 0 m/s, the range of the yaw angle was + 30 °, and that of the temperature was T = 20-65 °. A total of 4 × 55 calibration points were taken. A measure of the accuracy of the methods is to evaluate the original calibration data with the LUT and compare the results with the exact data. In this test run the standard deviation in method 2 was a = 0.45 % in U and o = 0.031 m/s in V; the maximum deviation was 1.5% in U and 0.062 m/s in V. In method 3, the standard deviation was tr = 1.2% in U and 0.12 m/s in V; the maximum deviation was 2.5% in U and 0.35 m/s in V. For our measuring task we use a combination of methods 2 and 3. The calibration is performed as in methods 1 and 2 with the full range of velocities and probe angles at four temperatures. Then average 3,'s are determined for each wire. The evaluation is performed as in method 3 with a two-dimensional LUT. This combines a higher accuracy than that of method 3 with its fast evaluation. The standard deviations are typically o = 0.8% in U and cr = 0.05 m/s in V. P e r f o r m a n c e of the C a l i b r a t i o n The three signals from the triple-wire probe are time-averaged and stored together with the pressure, temperature, and inclination data on a file. A calibration of 55 points at a constant temperature takes approximately 120 min of fully automated operation. The program that determines the lookup tables from the calibration data has three parts. In the first part the data are corrected to the nominal calibration temperatures and velocities. These corrections are small and are performed by a first estimate of the respective 3,i. The second part is the actual calculation of the LUT. The size of the matrices was chosen to be between 40 x 40 and 100 × 100, which was sufficient for a high accuracy for the interpolation. There is no limit to size apart of the computer memory. In the third part the LUT is tested with the original calibration data. The details and program listings can be found in Ref. 23. ERROR SOURCES IN MEASUREMENTS WITH X-PROBES Apart from calibration errors, there are several errors typical of multiwire probes in general and x-probes in particular. There are numerous publications on errors of hot-wire measurements, and I do not want to review any of them here. The following paragraphs serve to put the above-mentioned errors Table 1. Qualities of Three Calibration and Evaluation Methods

Method

Range

Errors

Eval. Time

Calibr. Time

1 2 2

Long Medium Small

Small Small Medium

Medium Medium Short

Large Large Short

Calibration of a Three-Wire Probe due to the calibration technique into perspective and give some data actually determined with out probe in our measurement situation. Error Due to Velocity Components Normal to the X-Wire

Plane. This velocity component decreases the effective angle between the wires and the flow, and therefore the signals E l and E 2 increase. Since both signals increase by the same amount, there is no error in II, but the axial component U is evaluated too high. For example, if the probe is pitched by 12", which relates to a normal velocity component of 20% of the axial component, the axial velocity U is evaluated 7% too high. The corresponding values for a pitch angle of 5 ° are 9% normal velocity component and 1.2% error in U. Probes with a normal wire and an inclined wire, which were suggested by some authors, would measure a correct axial component U under similar conditions, but the evaluated I/ component would be strongly affected by velocity components normal to the wire plane. Errors Due to the Finite Length of the Wires and the Distance Between Them. These effects were described by

Bremhorst [24], Vagt [25], and others. I will present some experimental results for the measurements in velocity and temperature gradients. When the x-wire plane lies parallel to a wall, the signals E 1 and E2 will be different because of the velocity gradients, which will be interpreted as a component parallel to the wall ( W ) . In addition to the difference in the time mean values, the fluctuating part of the signal is also different. This was investigated with a parallel-wire probe in pipe flow (pipe diameter 100 mm). The wires had a spacing of 0.4 mm and a length of 1 mm. Irrespective of the position of the wire plane with respect to the wall, the correlation of the two signals decreases with decreasing wall distance. The correlation coefficient was 0.97 at a wall distance of y = 20 mm and decreased to 0.87 at y = 2 mm. Thus, an x-probe will also determine erroneous 02, w 2, and u~. With heated walls, the effects are intensified by the temperature gradient. An example for pipe flow with a maximum velocity of 22 m / s will demonstrate this. Due to the velocity gradient normal to the wire plane, the x-probe determines an erroneous velocity parallel to the wall of W = 0.20 m/s. Along the spacing of the wires of 0.4 mm there is a temperature gradient of 0.8"C, which leads to an additional error in W of 0.08 m/s; the total error adds up to 0.28 m/s. At a wall distance of y = 4 mm the total error is still 0.15 m/s, and at y = 20 mm it is 0.05 m/s. Errors Due to the Third Wire. Flow disturbances from the cold wire, which is upstream of the x-wires, should be negligible, because the distance is on the order of 500 wire diameters. The cold wire prongs should not have any effect for flow angles smaller than 16". The time lag between the cold-wire signal and the x-wire signals is approximately 5 × 10 -5 s at U = 10 m/s; this should be negligible for frequencies below 2 kHz. Of greater concern is the poor frequency response of the cold wire compared to that of a CTA system. The time constant can be estimated by the relation

M

d2pc 4Nu k

(7)

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where d is the diameter of the wire, p the density, c the specific heat, k the thermal conductivity of air, and Nu the Nusselt number given by Eq. (2). For a platinum wire with a diameter of 1 /~m, the time constant is M = 3 × 10 -5 s for u = 25 m / s and M = 4.3 x 10 -5 s for 8 m/s. With the assumption that the cold-wire system behaves like a low pass, the attenuation is given by

H(~) = (1 + ~2 M2)-'/2

(8)

and the phase shift is • (w) = a r c t a n ( - w M )

(9)

with o~ = 2 r f , f being the frequency of fluctuation. With an average time constant of 3.5 × 10 -5 s, the attenuation factor is 0.976 and the phase shift is 12" at a frequency of 1 kHz. The respective values for 2 kHz are A = 0.915 and ,I~ = 24*. The corner frequency is 7.9 kHz. These data can give only a rough estimate of the possible errors due to the frequency response of the cold wire because the time constant depends on other parameters also, such as the length of the wire, its connection to the prongs, and the length of time of operation [26]. To estimate the effect of the attenuation and phase shift of the temperature signal on the correlations of u, v, and 0, measurements were performed setting the low-pass filter for the temperature signal at certain corner frequencies. The measurements were taken in heated pipe flow at a wall distance of y = 4 mm (see [23]). According to these measurements, we can estimate the errors for a l #m wire of 02, uO, vO, u 2, v2, and u--~to be the_following: 02 is 2%, - u--0is 4%, and v-'0 is 6% too small; u 2 and - u v are 4% too large; and there is a negligible error in 02. These are rough estimates that will help to decide whether a compensation of the temperature signal should be performed, which requires that the time constant of the cold wire be known. VERIFICATION OF THE METHOD To verify the calibration method, the turbulent velocity and temperature fluctuations in fully developed pipe flow were measured. The test section was a vertical round tube with an inner diameter of 100 mm and a total length of 7 m with a length of 5 m heated by electric heaters. The measurements were taken 25 mm upstream of the outlet. Average velocity and temperature profiles were measured by Pitot tube and thermocouple, respectively, and the wall shear stress was determined by Preston tube. Measurements were taken at two Reynolds numbers of Re = 8 × 10 4 and Re = 13 x 104 , and at three temperature levels--unheated and wall temperatures of 50"C and 80°C. For demonstration of the quality of the calibration method I present only some results. Figure 6 shows the intensities of the velocity and temperature fluctuation normalized by the friction velocity u~ and the friction temperature T, as a function of the relative wall distance y/r. The results for V / o 2 / u T and V / w 2 / u ~ do not show any effect of the heating; the curves for the isothermal case and the heated case collapse on a single curve each. The difference in

~ u 2 / u ~ between the heated and the unheated case might be due to the poor frequency response of the cold wire, which

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tion of higher order correlations, frequency power spectra, and joint probability density functions. This technique is especially well suited for experiments where measurements at a large number of locations have to be made, preferably fully computer controlled, and the evaluation of all turbulence data is done online. It is very simple to apply, because the only mathematics involved to build a lookup table are smoothing and interpolation procedures. It is not necessary to know the exact geometry of the three-wire probe, and no specially built probes or anemometers are needed.

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CONCLUSION t~ 0.5

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Figure

shows that a compensation should be applied. The measurements of Hishida and Nagano [27] show a similar effect, except for being more pronounced near the wall and less near the pipe center. The data on intensities of 0 compare well with their data. The cross-correlation coefficient Ruo = ~ / V/~u2 V / ~ and the corresponding coefficients Ruv and Roo are shown in Fig. 7. The cross correlation between w and 0 was close to zero. PRACTICAL SIGNIFICANCE / USEFULNESS The examples presented show the good performance of the calibration technique and its potential use for the determina0.8

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I wish to thank Mr. Mensingerfor his cooperation in manufacturingthe hot-wire probes and in performingthe experiments.

NOMENCLATURE

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0.6

]

A calibration method for a hot-wire probe with an x-wire and a third cold wire has been developed for simultaneous measurement of two components of instantaneous velocity and temperature in low-speed airflow at moderate temperatures. The calibration method uses lookup tables for the velocity signals and a linear relation for the temperature. The velocity signals are reduced to a constant temperature by a simple relation between the squared anemometer signal and the instantaneous temperature. The reduction parameter is different for each wire but varies very little with flow velocity in the range above 10 m/s. The variation with the flow angle is greater. Depending on the range of flow parameters to be measured, the permissible error, and the time available for calibration and evaluation, either a constant reduction parameter can be used with a single lookup table or methods with specific reduction parameters or more than one lookup table can be applied. It was found that for our limited range of velocity and temperature and our specific probe the simple method was accurate enough. The errors of the calibration method were smaller than those due to the specific errors related to x-wire measurements.

R,o, R,,o, and Ruv in

A, B

calibration coefficients; see Eqs. (1) and (2), dimensionless E voltage signals of CTA and CCA, V H attenuation of a low pass, dimensionless M time constant of the cold wire, s Nu Nusselt number of hot wire, dimensionless Q flow velocity at calibration nozzle, m/s Re Reynolds number, dimensionless T i n s t a n t a n e o u s t e m p e r a t u r e , K(T~ friction temperature, K) U instantaneous velocity parallel to probe axis, m/s V instantaneous velocity normal to probe axis and parallel to x-wire plane during calibration or velocity vector normal to wall in measurement, m/s W instantaneous velocity normal to probe axis and normal to x-wire plane during calibration or velocity vector parallel to wall in measurement, m/s c specific heat capacity of wire material, kJ/(kg • K) d diameter of hot wire, /~m

Calibration of a Three-Wire Probe

f k r u

frequency, s - i conductivity of heat, W / ( m . k) radius of pipe, mm fluctuating velocity corresponding to U, m / s [u~ friction velocity, m/s] o fluctuating velocity corresponding to V, m / s w fluctuating velocity corresponding to W, m / s y distance from wall, mm Greek S y m b o l s

q,

phase shift of a low pass probe angle 3' coefficient in Eqs. (1) and (5) 0 fluctuating temperature, K p density of air, k g / m 3 a standard deviation 60 ( = 2 r f ) , s - l Subscripts

1,2,3 c

m w

pertaining to the x-wires 1 and 2 and the cold wire 3 constant or reference temperature measured value hot-wire sensor refers to time average of any quantity REFERENCES

1. Tutu, N. K., and Chevray, R., Cross Wire Anemometry in High Intensity Turbulence, J. Fluid Mech., 71,785-800, 1975. 2. Legg. B. J., Coppin, P. A., and Raupach, M. R., A Three-Hot-Wire Anemometer for Measuring Two Velocity Components in High Intensity Turbulent Boundary Layers, J. Phys. E: Sci. Instrum., 17, 970-976, 1984. 3. Fabris, G., Probe, and Method for Simultaneous Measurements of "True" Instantaneous Temperature and Three Velocity Components in Turbulent Flow, Rev. Sci. Instrum., 49(5), 654-664, 1978. 4. Samet, M., and Einav, S., A Hot-Wire Technique for Simultaneous Measurement of Instantaneous Velocities in 3D Flows, J. Phys. E: Sci. lnstrum., 20, 683-690, 1987. 5. Russ, S., and Simon, T. W., Signal Processing Using the Orthogonal Triple-Wire Equations, Flow lines, TS! Magazine, 3-9, Winter 1990. 6. Blair, M. F., and Bennett, J. C., Hot-Wire Measurements of Velocity and Temperature Fluctuations in a Heated Turbulent Boundary Layer, J. Phys. E: Sci. lnstrum., 20, 209-216, 1987. 7. Lienhard, J. H., and Helland, K. N., An Experimental Analysis of Fluctuating Temperature Measurements Using Hot-Wires at Different Overheats, Exp. Fluids, 7,265-270, 1989. 8. Sakao, F., Constant Temperature Hot Wires for Determining Velocity Fluctuations in an Air Flow Accompanied by Temperature Fluctuations, J. Phys. E: Sci. Instrum., 6(7), 588-676, 1973. 9. Bremhorst, K., and Bullock, K. J., Spectral Measurements of Temperature and Longitudinal Velocity Fluctuations in Fully Developed Pipe Flow, Int. J. Heat Mass Transfer, 13, 1313-1329, 1970.

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10. Subramanian, C. S., and Antonia, R. A., Effect of Reynolds Number on a Slightly Heated Turbulent Boundary Layer, Int. J. Heat Mass Transfer, 24(11), 1833-1846, 1981. 11. Abdel-Rahman, A. A., Hitchman, G. J., Slawson, P. R., and Strong, A. B., An X-Array Hot-Wire Technique for Heated Turbulent Flows of Low Velocity, J. Phys. E: Sci. Instrum., 22, 638-644, 1989. 12. Smits, A. J., and Perry, A. E., A Note on Hot-Wire Anemometer Measurement of Turbulence in the Presence of Temperature Fluctuations, J. Phys. E: Sci. Instrum., 14, 311-312, 1981. 13. Chambers, A. J., Antonia, R. A., and Fulachier, L., Turbulent Prandtl Number and Spectral Characteristics of a Turbulent Mixing Layer, Int. J. Heat Mass Transfer, 28(8), 1461-1468, 1985. 14. Browne, L. W, Antonia, R. A., and Chua, L. P., Calibration of X-Probes for Turbulent Flow Measurements, Exp. Fluids, 7, 201208, 1989. 15. Lueptow, R. M., Breuer, K. S., and Haritonidis, J. H., ComputerAided Calibration of X-Probes Using a Look-up Table, Exp. Fluids, 6, 115-118, 1988. 16. Johnson, F. D., and Eckelmann, H., A Variable Angle Method of Calibration for X-Probes Applied to Wall-Bounded Turbulent Shear Flow, Exp. Fluids, 2, 121-130, 1984. 17. Willmarth, W. W., and Bogar, T. J., Survey and New Measurements of Turbulent Structure Near the Wall, Phys. Fluids, 20, $9-$21, 1977. 18. Bearman, P. W., Corrections for the Effect of Ambient Temperature Drift on Hot-Wire Measurements in Incompressible Flow, DISA Inf., 11, 25-30, 1971. 19. Collis, D. C., and Williams, M. J., Two-Dimensional Convection from Heated Wires at Low Reynolds Numbers, J. Fluid Mech., 6, 357-384, 1959,. 20. Fulachier, L., Hot-Wire Measurements in Low Speed Heated Flow, Proc. Dynamic Flow Conference, Skovlunde, Denmark, 1978. 21. Champagne, F. H., The Temperature Sensitivity of Hot Wires, Proc. Dynamic Flow Conference, Skovlunde, Denmark, 1978. 22. Bremhorst, K., Effect of Fluid Temperature on Hot-Wire Anemometers and an Improved Method of Temperature Compensation and Linearisation Without Use of Small Signal Sensitivities, J. Phys. E: Sci. Instrum., 18, 44-49, 1985. 23. Meyer, L., Kalibration einer 3-Draht Sonde fiir Messungen in n i c h t - i s o t h e r m e r L u f t s t r 6 m u n g , Report KfK 4707, Kernforschungszentrum Karlsruhe, 1990. 24. Bremhorst, K., The Effect of Wire Length and Separation on x-Array Hot Wire Anemometer Measurements, IEEE Trans. Instr. Measurement, IM-21(3), 244-248, 1972. 25. Vagt, J.-D., Hot-Wire Probes in Low Speed Flow, Progr. Aerosp. Sci., lg, 271-323, 1979. 26. Smits, A. J., Perry, A. E., and Hoffmann, P. H., The Response to Temperature Fluctuations of a Constant-Current Hot-Wire Anemometer, J. Phys. E: Sci. lnstrum., 11,909-914, 1978. 27. Hishida, M., and Nagano, Y., Structure of Turbulent Velocity and Temperature Fluctuations in Fully Developed Pipe Flow, Trans. ASME, J. Heat Transfer, 101, 15-22, 1979.

Received April 3, 1990; revised November 18, 1991