Monochromatic heterodyne fiber-optic profile sensor ... - OSA Publishing

Monochromatic heterodyne fiber-optic profile sensor for spatially resolved velocity measurements with frequency division multiplexing Thorsten Pfister, Lars Büttner, Katsuaki Shirai, and Jürgen Czarske

Investigating shear flows is important in technical applications as well as in fundamental research. Velocity measurements with high spatial resolution are necessary. Laser Doppler anemometry allows nonintrusive precise measurements, but the spatial resolution is limited by the size of the measurement volume to ⬃ 50 ␮m. A new laser Doppler profile sensor is proposed, enabling determination of the velocity profile inside the measurement volume. Two fringe systems with contrary fringe spacing gradients are generated to determine the position as well as the velocity of passing tracer particles. Physically discriminating between the two measuring channels is done by a frequency-division-multiplexing technique with acousto-optic modulators. A frequency-doubled Nd:YAG laser and a fiber-optic measuring head were employed, resulting in a portable and flexible sensor. In the center of the measurement volume of ⬃1-mm length, a spatial resolution of ⬃5 ␮m was obtained. Spatially resolved measurements of the Blasius velocity profile are presented. Small velocities as low as 3 cm兾s are measured. The sensor is applied in a wind tunnel to determine the wall shear stress of a boundary layer flow. All measurement results show good agreement with the theoretical prediction. © 2005 Optical Society of America OCIS codes: 060.2370, 120.7250.

1. Introduction

There is great demand for a way to determine flow structure with high spatial resolution down to the micrometer range. In the vicinity of the surface of a solid body, e.g., an aircraft wing, shear flows occur: As the tangential velocity drops to zero at the wall, a large velocity gradient is encountered in the boundary layer. The thickness of the boundary layer scales down with increasing free-stream velocity1 and is often in the submillimeter range. Thus the velocity profile has to be determined with a high spatial resolution. In turbulent flows the scale of vortices becomes smaller and the thickness of the boundary layer becomes thinner with the increasing Reynolds number.2 The Reynolds number of an industrial flow reaches typically to the order of 106 and higher. The smallest vortex scale of the Kolmogorov theory can be

The authors are with the Institut für Grundlagen der Elektrotechnik und Elektronik, Technische Universität Dresden, Helmholtzstrasse 18, D-01062 Dresden, Germany. T. Pfister’s e-mail address is [email protected]. Received 25 August 2004; revised manuscript received 30 November 2004; accepted 1 December 2004. 0003-6935/05/132501-10$15.00/0 © 2005 Optical Society of America

in the micrometer range. To resolve structures in such a flow, a measurement technique with a micrometer resolution is desired. Another demand is for the recently growing microfluidic applications, e.g., microflow channels of lab-on-the-chip devices. To know the structure inside a microfluidic flow,3 a spatial resolution in the micrometer range is necessary. In the past few years powerful direct numerical simulations of velocity fields have become available, and they are successfully applied to the clarification of turbulent flows.4 However, experimental verification is necessary, and at high Reynolds numbers the direct numerical simulation is not available. For example, in small channels of microfluidic elements the flow motion depends on surface interactions, which are difficult to simulate numerically.3 Furthermore the flow may be non-Newtonian, and complicated interactions between macromolecules, cells, and walls can occur. For measurement of such a flow, nonintrusive techniques with high spatial resolution should be applied. Conventional laser Doppler anemometry (LDA) is not capable of obtaining high spatial resolution in the micrometer range because of the spatial averaging effect inside the measurement volume. When the dimensions of the measurement volume are reduced, the measurement error of the velocity generally in1 May 2005 兾 Vol. 44, No. 13 兾 APPLIED OPTICS

2501

creases for the following reasons: First, the reduced number of fringes in the generated fringe system causes a systematic frequency measurement error in the fast-Fourier-transform (FFT) signal-processing technique.5,6 Second, sharp focusing of the laser beams involves increasing variation in the fringe spacing within the measurement volume, resulting in virtual turbulence.7 Furthermore a conventional LDA sensor is a pointwise measurement technique, meaning that the velocity profile would have to be scanned point by point, which is time-consuming and requires a stationary flow. Laser Doppler profile sensors have been proposed8 –11 that allow velocity measurements with high spatial resolution inside the measurement volume. A differential profile sensor was developed that is based on the generation of two different fringe systems with diverging and converging fringes inside the measurement volume.10,11 Two different laser wavelengths were employed to physically discriminate between the two measuring channels. Calculation of the two Doppler frequencies has been carried out by a FFT signalprocessing technique after removal of the dc pedestals of the detected burst signals with a high-pass filter. This sensor has already been applied successfully to velocity profile measurements in a boundary layer flow.11 However, one drawback of this setup is that depending on the filter settings and on the velocity range, very low particle velocities of ⬍0.5 m兾s could not be detected,11 since Doppler frequencies close to zero were eliminated together with the dc pedestals by the high-pass filter of a certain cutoff frequency. In this paper a laser Doppler profile sensor is presented that involves the heterodyne technique. Since the dc pedestals of the burst signals are suppressed, small Doppler frequencies can be measured. The physical discrimination between the two measurement channels, i.e., fringe systems, is done by frequency division multiplexing (FDM) with acoustooptic modulators (AOMs) creating two carrier frequencies of 20 and 120 MHz. Optical fibers are employed for the profile sensor. Components, e.g., the laser, the AOM’s, photodetectors, or signal processing, can be located in one basic unit that is connected to an optical measuring head through optical fibers. The advantage of such a setup is that the measuring head can be built up to be very compact and without electrical components, giving great flexibility to the whole measurement system.7,12–14 The concept of the profile sensor allows high spatial resolution in the micrometer range. A low virtual turbulence intensity results, and traversal of the sensor is not necessary, since the velocity profile is determined inside the measurement volume. The monochromatic heterodyne laser Doppler profile sensor presented has further advantages: (1) The heterodyne technique involved suppresses the dc pedestals of the burst signals, allowing measurement of small velocities. Flows close to the wall can be measured. 2502

APPLIED OPTICS 兾 Vol. 44, No. 13 兾 1 May 2005

(2) The FDM is used to discriminate between two measurement channels. Only one single-wavelength laser and one photodetector are sufficient for evaluating two Doppler frequencies. Furthermore dispersion effects at the lens optics as well as in the fluid do not occur. An axial shift between the fringe systems can be suppressed. (3) A fiber-optic measuring head is used. Thus the sensor exhibits a compact and modular setup; components, e.g., the measuring head, can be easily exchanged, and the measurement system is flexible and robust. This paper is structured as follows: Section 2 contains a short recapitulation of the principle of the laser Doppler profile sensor. In Section 3 the experimental setup, alignment, and the system calibration are described. Furthermore the measurement accuracy of the sensor is determined and compared with theoretical estimations in Section 4. The sensor is applied in a wind tunnel experiment. Velocity profile measurements in a laminar boundary layer flow are carried out, and the measurement results are compared with the Blasius theory in Section 5. Finally the conclusions are in Section 6.

2. Principle of the Laser Doppler Profile Sensor

Velocity measurements by means of LDA are based on evaluation of light from scattering inside a measurement volume, which is formed by interference fringes in the cross section of two coherent laser beams. Taking into account the spacing d of the fringes, the velocity v of a scattering particle crossing the measurement volume is calculated according to v ⫽ fd, where f denotes the measured Doppler frequency. Conventionally an average of the velocity distribution over the entire measurement volume is obtained. In contrast the profile sensor allows the velocity measurement to be performed with spatial resolution inside the measurement volume. The position z as well as the velocity v of scattering particles moving through the measurement volume can be determined.8 –11 The principle of the profile sensor is based on generation of two fringe systems with different fringespacing gradients, ⭸di共z兲兾⭸z, i ⫽ 1, 2, inside the same measurement volume. By two different laser wavelengths (wavelength division multiplexing),10,11 FDM techniques,15,16 or time division multiplexing,17 a separate measurement of two Doppler frequencies f1 and f2, corresponding to the two measurement channels, is assured. The z position of the scattering object can be determined by the quotient of the two Doppler frequencies:

q(z) ⫽

f2(v, z) v(z)兾d2(z) d1(z) ⫽ ⫽ , f1(v, z) v(z)兾d1(z) d2(z)

(1)

where the Doppler frequency is given by fi ⫽ v兾di,

Fig. 1. Experimental setup.

i ⫽ 1, 2, with v being the transverse component of the velocity perpendicular to the fringes. The quotient q共z兲 is independent of the velocity v, see Eq. (1). Thus the position of the scattering object can be calculated from the two Doppler frequencies, f1 and f2, through the calibration function q共z兲. An unambiguous measurement of the position requires a monotonic calibration function, which can be achieved by opposite fringe-spacing slopes for the two measurement channels. Therefore the beam-waist positions have to be shifted longitudinally against each other before and behind the measurement volume to obtain one convergent and one divergent fringe system.10,11 The calculated z position allows determination of the actual fringe spacings d1 and d2 from the known fringe-spacing curves, di共z兲, i ⫽ 1, 2, measured beforehand for calibration of the sensor. As a result the velocity can be calculated precisely according to v(z) ⫽ f1(v, z)d1(z) ⫽ f2(v, z)d2(z).

(2)

Based on Eqs. (1) and (2), the position as well as the velocity of scattering particles can be determined without the influence of fringe-spacing variations. 3. Experimental Setup and Calibration A.

Setup

Figure 1 illustrates the setup of the monochromatic laser Doppler profile sensor. The beam of a frequencydoubled Nd:YAG laser 共532 nm兲 with approximately 100᎑mW output power was divided into four partial beams by a beam-splitter cube and two subsequent AOMs (from Pegasus Optik GmbH) with 60- and

80᎑MHz driver frequency. Only the 0th and the ⫹1st diffraction orders of these AOMs were used; all other orders were blocked by beam stops. One of the 0thorder output beams passed another AOM, impressing a frequency shift of 120 MHz to the optical wave. The resulting four partial beams, which had experienced frequency shifts of 0, 60, 80, and 120 MHz, were coupled into single-mode fibers, SMF’s, and guided to the fiber-optic measuring head. Three AOMs are necessary to suppress cross talk between the different measurement channels. Inside the fiber-optical measuring head the beams were collimated by four aspheric lenses and brought to an intersection by a front lens with a 300᎑mm focal length. The sensor head and the fibers were aligned in such a way that two parallel fringe systems were obtained, exhibiting carrier frequencies of 20 and 120 MHz. The AOM shift frequencies were chosen so that all other difference frequencies between the individual beams disagree with the two used carrier frequencies of 20 and 120 MHz (see Fig. 2). Thus parasitic signals at these frequencies are eliminated by the low-pass filters and do not disturb the measuring signals. The scattered light of particles crossing the measurement volume was coupled into a multimode fiber, MMF, and guided to a photodetector. Subsequent to a power splitter the electrical output signal of the photodetector was mixed down into the baseband with two carrier frequencies of 20 and 120 MHz that were created by using the reference outputs of the AOM drivers. The 20᎑MHz reference signal was generated by mixing the 60- and 80᎑MHz driver signals and applying afterward a bandpass with a center frequency of 20 MHz. To prevent aliasing, the resulting 1 May 2005 兾 Vol. 44, No. 13 兾 APPLIED OPTICS

2503

Fig. 2. Difference frequencies between the four frequency shifted laser beams. The carrier frequencies are in boldface.

baseband signals for the two measurement channels were filtered with low-pass filters of 5᎑MHz cutoff frequency. Data acquisition and further signal processing were done with a standard PC containing a 12-bit analog兾digital converter card. Both channels were sampled simultaneously. A LabVIEW program controlled the measurements and read the burst signals represented by ⬃212 samples. The Doppler frequencies for both channels were calculated by means of FFT. A Gaussian function was fitted to the Doppler peaks, which yields the center frequencies. Because of the quadratic arrangement of the beams at the sensor head, the two fringe systems obtained were tilted by ⬃3.6° with respect to the optical axis, i.e., by 7.2° against each other (see Fig. 3). Therefore the effective size of the measurement volume, which is built up by the intersection area of both fringe systems, was explicitly smaller than the size of the individual fringe systems. To analyze this effect in more detail, a scheme with two fringe systems tilted 7.2° against each other was investigated numerically; the results are in Fig. 4. The length of the outline of the fringe systems was 1.5 mm, which is approxi-

Fig. 3. Fringe systems tilted against each other because of the quadratic beam arrangement at the sensor head. 2504


Fig. 4. Top, right half plane of the outline of two elliptical fringe systems of 1.5᎑mm length tilted by 7.2° against each other. Bottom, coincidence probability along the z axis for different tilt angles.

mately the 1兾e2 length of the fringe systems at this sensor (see Subsection 3.C). For z ⬎ 620 ␮m there is no longer an overlap between the two fringe systems (see Fig. 4), and thus the effective length of the measurement volume is reduced by ⬃17.5% to only 1240 ␮m compared with the length of the individual fringe systems of 1500 ␮m. Furthermore the coincidence probability, which is the lateral range where burst signals from both channels can be received with respect to the total range where burst signals can occur at all, is also significantly reduced (see Fig. 4, bottom). For a tilt angle of 7.2° the overall coincidence probability within the effective measuring area of 1240᎑␮m length is only ⬃50%. This value is determined as the ratio of the areas below the curves for the coincidence probability for tilting angles of 7.2° and 0° (see Fig. 4, bottom). Therefore only 50% of the occurring burst signals are double bursts from both channels and can be evaluated for measurement. In the future this drawback can be overcome by creating another fiber-optic measuring head with a more advantageous arrangement of the fibers and consequently of the partial beams. Such changes in the setup can be done easily because of the flexible and modular design of the sensor due to the employment of fiber optics. In addition a compact laser source with a size of only 44 mm ⫻ 60 mm ⫻ 98 mm was used, which has been mounted onto an optical breadboard together with the AOMs, the fiber incoupling units, and the electronics. Thus a compact and transportable system was obtained. One further advantage of the setup presented is that drifts in the driver frequencies for the AOMs and therefore in the AOM-frequency shifts have no effect on the measured baseband signals, since the same oscillators are used for frequency shifting and down mixing. Furthermore no dc pedestals occur (see Fig. 5) because in the down-mixing process the dc parts of

Table 1. Measured Beam Parameters for Four Partial Beams

Beam

w0 共␮m兲

M2

z0 共mm兲

1 2 3 4

50 51 52 38

1.26 1.47 1.3 1.6

⫺11.74 ⫺10.64 9.74 9.84

tion 3.C). For most applications the resulting amount of the measuring range of 0 ⬍ |v| ⱕ 22.5 m兾s is sufficient. In addition only one laser wavelength is necessary. Thus dispersion as well as chromatic aberrations (e.g., of lenses) is irrelevant, and the wavelength can be selected from spectral regions where compact and powerful laser sources are available. B.

Fig. 5. Top, time-domain burst signal for the 20᎑MHz channel after down mixing and the low-pass filter and, bottom, the corresponding power spectrum. No dc pedestal exists.

the rf signals are shifted in the frequency domain toward the corresponding carrier frequencies, and afterward they are eliminated by the 5᎑MHz low-pass filters (see Fig. 1). Therefore very low particle velocities with Doppler frequencies close to zero can also be measured in contrast with homodyne techniques.10,11 Because of the cutoff frequency of the low-pass filters the maximum measurable Doppler frequency is limited to 5 MHz, corresponding to a maximum particle velocity of approximately v ⫽ fd ⫽ 22.5 m兾s, taking into account a fringe spacing of approximately d ⫽ 4.5 ␮m on average (see Subsec-

The fiber-optic sensor head was adjusted in a way that the four partial beams were brought to an intersection at one single point along the optical axis, which is the center of the measurement volume. In addition opposite fringe-spacing slopes for the two measurement channels are required for an unambiguous measurement of the position (see Section 2). Thus the beam waists of the two partial beams forming the first fringe system with a 120᎑MHz carrier frequency had to be located before the intersection point in order to obtain a divergent fringe system. Analogously the beam waists of beams 3 and 4 forming the second fringe system with a 20᎑MHz carrier frequency had to be located behind the intersection point for generating a convergent fringe system. These adjustments were done by varying the positions of the fiber end faces with respect to the collimating lenses at the fiber-optic measuring head in the axial and lateral direction. All four fibers could be adjusted independently. With the help of an optical beam scanner the crossing of the four partial beams at one single point and the location of the beam waists with respect to this crossing point were monitored. Figure 6 depicts the resulting caustic curves for all four partial beams with respect to the intersection point at z ⫽ 0. The exact values for the location of the beam waists z0, the beam quality factors M2, and the waist radii w0 are in Table 1. The beam waists of the two pairs of partial beams are clearly separated; i.e., they are shifted by ⬃10 mm before and behind the intersection point. The M2 values are around 1.4, and the waist radii are ⬃50 ␮m except for beam 4, which has experienced some distortions from a defect in the optics of the fiber-optic measurement head, which has not been repaired. C.

Fig. 6. Caustic curves for all four partial beams.

Adjustment of the Beams

Calibration

To calibrate the sensor, the measurement volume was scanned with a tungsten wire of 4᎑␮m diameter and approximately 1᎑cm length acting as a scattering object, which was fixed to an optical chopper. The chopper was mounted on a motorized translation ta1 May 2005 兾 Vol. 44, No. 13 兾 APPLIED OPTICS

2505

Fig. 7. Fringe spacings versus position.

ble and rotated with a well-defined and stabilized angular speed. The measurement and the movement of the motor table were controlled by a LabVIEW program on a PC. The measurement volume was scanned with a defined velocity vCh of the wire fixed on the chopper. According to di ⫽ vCh兾fi, i ⫽ 1, 2, the fringe spacings were calculated with the measured Doppler frequencies f1 and f2. At each position 25 samples were taken in order to reduce the statistical uncertainty by averaging. The mean relative uncertainty for the frequency determination was ⬃0.034%. Figure 7 shows the fringe spacings obtained within the measurement volume of ⬃1᎑mm length for both channels. The opposite slopes, which are necessary for an unambiguous position measurement, are clearly visible. The resulting monotonic calibration function q共z兲 exhibiting a steepness of ⬃0.13 mm⫺1 is shown in Fig. 8. According to the measured power densities of the Doppler frequency spectral lines the 1兾e2 length of the fringe systems was ⬃1.5 and 1.8 mm (see Fig. 9). However, the effective length of the measurement volume, which is built up by the intersection area of both fringe systems, was only ⬃1 mm. One reason for this significant reduction is a tilt of ⬃7.2° between the two fringe systems because of the quadratic arrangement of the partial beams at the sensor head (see Subsection 3.A). As shown in Fig. 4 this decreases the effective length of the measurement volume by ⬃17.5% ending up with only a 1.26᎑mm measurement range for fringe systems with 1.5᎑mm length. In addition the two fringe systems are shifted against each other along the optical axis by ⬃150 ␮m

Fig. 9. Power density of the Doppler frequency spectral lines of the burst signals.

because of the nonideal adjustment of the optical measuring head (see Fig. 9). Thus the measurement volume is reduced further to only an ⬃1.1᎑mm length. Besides the signal-to-noise ratio (SNR) decreases significantly toward the borders of the measurement volume because of the smaller number of interference fringes available. This effect is accompanied by the strong decay in the coincidence probability caused by the tilt between the fringe systems (see Fig. 4). Therefore the obtained measuring range of 1 mm is a reasonable and feasible value. 4. Investigation of the Measuring Error A.

Theory

The accuracy of a determined velocity profile depends on the measurement error of the positions as well as the error in the velocities of the tracer particles moving through the measurement volume. Both errors can be derived analytically by using the law of error propagation.11,18 For the center of the measurement volume the measurement uncertainty in the position is given by11

冏共兲冏

␦z ⬇ 冑2

⭸q z ⭸z

冉冊

2506


␦f . f

(3)

Thus ␦z depends on the relative uncertainty of the frequency measurement ␦f兾f as well as on the inverse of the steepness of the calibration function ⭸q兾⭸z. As a result the steepness of the calibration function should be maximized for a high spatial resolution to be obtained. In contrast the relative statistical measurement error of the velocity depends only on the relative uncertainty of the frequency measurement ␦f兾f and can be simplified to11 ␦v 3 ⬇ v 2

Fig. 8. Calibration function q共z兲.

⫺1

1兾2

␦f . f

(4)

The measurement uncertainty in the velocity of

Fig. 10. Left, standard deviation of the position ␦共z兲 and, right, relative measurement uncertainty in the velocity ␦v兾v over the z position.

conventional LDA systems suffers strongly from variation in the fringe spacing along the optical axis, giving rise to virtual turbulence.7 At the laser Doppler profile sensor this influence of the variation of the fringe spacing ⌬d兾d over the measurement volume is eliminated by principle. Because the position where the tracer particle crosses the interference fringe systems is determined exactly, only the local fringe spacings at this defined z position are taken into account in calculating the velocity. After adjustment of the sensor, the calibration function ⭸q兾⭸z is fixed. Therefore for a fixed adjustment the statistical measurement errors in the position and velocity depend only on the relative uncertainty of the frequency measurement ␦f兾f [see Eqs. (3) and (4)]. When a FFT signal-processing technique is used, ␦f兾f mainly depends on the SNR and on the number of signal periods in the measured timedomain signals. The SNR does not change for the varying velocity of the measurement objects, e.g., the tracer particles. Also the number of signal periods contained in the burst signals do not change with particle velocity. Furthermore the accuracy in the FFT frequency estimation does not significantly improve with the increasing number of signal periods if more than 10 signal periods are available. Therefore the measurement accuracy in position and velocity according to Eqs. (3) and (4) is independent of the velocity of the tracer particles, if the width of the recorded time-domain signals is adapted to the actual width of the occurring burst signals in such a way that at least 10 signal periods are available even for very slow particles. Consequently the record length of the time-domain signals limits the minimum measurable velocity, but it can be set individually according to the particular measurement task. With the slope of the calibration function in the center of the measurement volume of ⭸q兾⭸z ⬇ 0.13 mm⫺1 and with ␦f兾f ⫽ 0.034% (see Subsection 3.C) the standard deviation of the position is approximately ␦z ⬇ 3.7 ␮m, and the relative statistical mea-

surement error of the velocity can be estimated as ␦v兾v ⬇ 0.042% by using Eqs. (3) and (4). B.

Experimental Results

To investigate the measurement error of the monochromatic laser Doppler profile sensor experimentally, a subsequent measurement with the 4᎑␮m wire was carried out in analogy to the calibration procedure (see Subsection 3.C). The measured calibration curve and the curves for the fringe spacings (see Figs. 7 and 8) were fit by fifth-order polynomial functions and inserted into the LabVIEW program for the measurement. At each position an averaging over 20 samples was carried out. The results for the standard deviation of the position ␦共z兲 and for the relative measurement uncertainty in the velocity ␦v兾v are shown in Fig. 10. The minimum values of ␦共z兲 ⫽ 4.08 ␮m and ␦v兾v ⫽ 0.036% in the middle of the measurement volume are in good agreement with the theoretical estimations above. At the boundaries of the measurement volume, both errors increase because of the smaller SNR, a smaller number of signal periods, and distortions in the signal shape. Over the whole measurement volume the average standard deviation of the position was 8.08 ␮m, and a mean relative uncertainty in the velocity, 具␦v兾v典 ⫽ 0.066%, was achieved. 5. Velocity Profile Measurements of a Flat-Plate Boundary Layer Flow A.

Wind Tunnel Test Facility

All measurements described in Section 5 were performed in a Göttingen-type closed-loop wind tunnel at the Lehrstuhl für Strömungsmechanik (LSTM) of the University of Erlangen. Liquid particles of diethylhexylsebacat (DEHS) and of a size of around 1.5 ␮m were used as tracers. The intrinsic turbulence of the wind channel was determined to be smaller than 0.75% by hot-wire measurements. Near-wall velocity measurements for a well-known flow field, a 1 May 2005 兾 Vol. 44, No. 13 兾 APPLIED OPTICS

2507

Fig. 11. Blasius profile measured in 10 steps by traversing the sensor head, left, depicted in absolute units and, right, in comparison with theory in normalized units. The wall is located at z ⫽ ␩ ⫽ 0.

Blasius boundary layer,19 were carried out to verify the performance of the monochromatic laser Doppler profile sensor. For this reason a flat glass plate ⬃5 mm thick was installed downstream to the nozzle of the wind tunnel in the test section. Thus a two-dimensional laminar boundary layer flow results. The fiber-optic sensor head was arranged in a way that the measurement volume lies at the front side of the glass plate, x ⫽ 160 mm away from the leading edge. A forwardscattering regime of the receiving optics was used. Data acquisition by a 12-bit analog兾digital converter card and signal processing with a LabVIEW program was done on a standard PC. Every time the sensor was traversed the number of samples corresponding to the width of the measurement window in the time domain was adapted to the actual width of the burst signals according to the present flow velocities. Note that no averaging and no software filters were applied. The center frequencies of the FFT spectra were determined by means of peak detection and a seven-

point interpolation algorithm. To validate the burst signals, three procedures were used: (1) verification that the Doppler lines in the FFT spectra exceed a certain threshold, (2) a check that the SNR of the burst signals exceeds a certain threshold, and (3) verification that the determined quotient of the Doppler frequencies is within the default rage of the calibration curve. B.

Blasius Boundary Layer Measurements

The known Blasius velocity profile of a laminar boundary layer1,19,20 allows one to investigate the performance of the spatially resolving monochromatic profile sensor. The experiment was carried out with a free-stream velocity, v⬁ ⬇ 4.85 m⁄s, resulting in a Reynolds number of 5.2 ⫻ 104, which is smaller than the critical range for a laminar-toturbulent transition of 3.5 ⫻ 105–1 ⫻ 106. By traversing the sensor head in steps of several hundred

Fig. 12. Measurements, left, at the slope, at ⬃1᎑mm distance from the wall and, right, in the free-stream region of the Blasius profile for different speeds of the wind tunnel. 2508


micrometers, the whole velocity profile from free stream down to the wall was measured in 10 steps. Afterward the measured profile was normalized according to v→

v , v⬁

冉冊

z→␩⫽z

v⬁ 2␯x

1兾2

(5)

for comparison with a theoretical Blasius profile obtained by numerically solving the Blasius equation.1 Here ␯ denotes the kinematic viscosity of air, 1.5 ⫻ 10⫺5 m2兾s, at a temperature of 20 °C. The results are in Fig. 11 with the wall located at z ⫽ ␩ ⫽ 0. The measured profile clearly exhibits the shape of a Blasius boundary layer profile and agrees very well with theoretical predictions. Because no dc pedestals occur, no digital low-pass filters are required; thus very low velocities, as low as 0.03 m兾s, could be measured close to the wall. Note that no averaging or digital filtering of the measured data has been applied. However, the variance of the measurement points is relatively high, and there are several explanations for this. First, the wind tunnel exhibits some amount of residual turbulence, which has been identified as lower than 0.75% by hot-wire measurements. Besides, the spatial resolution of the sensor of ⬃5 ␮m corresponding to the steepness of the calibration curve of 0.13 mm⫺1 was lower than in former systems. This was mainly determined by the fixed optical arrangement of the fiber-optic sensor head. In Ref. 11, for example, a spatial resolution as low as 1.6 ␮m was obtained with a different optical setup. Therefore for future measurements the spatial resolution can be increased by optimizing the optical setup of the fiber-optic sensor head. C. Measurement of Wall Shear Stress for Different Free-Stream Velocities

The wall shear stress is of great importance in practical applications (e.g., aircraft wings), because the frictional drag of a body within a flow is equal to the integral of the wall shear stress over the whole surface. Thus the free-stream velocities and the corresponding slopes of the Blasius profiles have been measured to determine the wall shear stress at different flow speeds of the wind tunnel. The raw data are in Fig. 12. The free-stream velocities v⬁ have been calculated as the mean values of the right-hand curves. By applying a linear fit to the data points in Fig. 12, left, the corresponding slopes ⭸v兾⭸v were determined. From the viscosity of air at a temperature of 20 °C, ␮ ⫽ 1.82 ⫻ 10⫺5 Ns兾m2, and the velocity gradient ⭸v兾⭸z at the wall 共z ⫽ 0兲, the wall shear stress ␶ can be computed according to20 ␶⫽␮

⭸␯ ⭸z

冏

.

(6)

z⫽0

With the free-stream velocities v⬁ the measured values for the wall shear stress were compared with theory (see Fig. 13).

Fig. 13. Measured wall shear stress for different flow speeds of the wind channel in comparison with theory.

The results for the wall shear stress are in excellent agreement with theory as well as for the Blasius profile measurement. Thus we have proved that the monochromatic laser Doppler profile sensor presented in this paper is capable of measuring spatially resolved velocity profiles with high accuracy, even close to a wall, where very low velocities occur. 6. Conclusions

A monochromatic laser Doppler profile sensor for highly spatially resolved velocity measurements with a fiber-optic measuring head and a frequency division multiplexing technique has been presented. The system features a flexible and modular setup containing only standard components, and it is capable of measuring very low velocities, down to 0.03 m⁄s. In the center of the measurement volume a spatial resolution of ⬃5 ␮m has been obtained. Measurements of a Blasius boundary layer profile and of the wall shear stress in a flat plate laminar flow were in excellent agreement with theory, underlining the capability of this sensor. Therefore this sensor is predestined for highly spatially resolved measurements of wallbounded flows, e.g., turbulent boundary layer flows2,21 and microfluidic flows.3 We thank P. Pfeiffer for sophisticated work on the fiber-optic system and help with the wind tunnel experiments at LSTM. The authors thank F. Durst, H. Lienhart, and S. Becker (LSTM, Erlangen, Germany) for valuable advice and for providing the wind tunnel. We thank especially the Deutsche Forschungsgemeinschaft (DFG) for funding this project (funding code Cz55兾15-1). References 1. H. Schlichting, Boundary Layer Theory (McGraw-Hill, New York, 1987). 2. M. Fischer, J. Jovanovic, and F. Durst, “Reynolds number effects in the near-wall region of turbulent channel flows,” Phys. Fluids 13, 1755–1767 (2001). 3. C. D. Meinhart, S. T. Wereley, and J. G. Santiago, “Micronresolution velocimetry techniques,” in Laser Techniques Applied to Fluid Mechanics, Selected Papers from the Ninth International Symposium, Lisbon, Portugal, July 1998 (Springer-Verlag, Berlin, 2000), pp. 57–70, paper I.4. 1 May 2005 兾 Vol. 44, No. 13 兾 APPLIED OPTICS

2509

4. H. Abe, H. Kawamura, and Y. Matsuo, “Direct numerical simulation of a fully developed turbulent channel flow with respect to the Reynolds number dependence,” Trans. ASME J. Fluids Eng. 123, 382–393 (2001). 5. D. Matovic and C. Tropea, “Spectral peak interpolation with application to LDA signal processing,” Meas. Sci. Technol. 2, 1100 –1106 (1991). 6. J. Czarske and O. Dölle, “Quadrature demodulation technique used in laser Doppler anemometry,” Electron. Lett. 43, 547– 549 (1998). 7. L. Büttner and J. Czarske, “A multimode-fiber laser-Doppler anemometer for highly spatially resolved velocity measurements using low-coherence light,” Meas. Sci. Technol. 12, 1891–1903 (2001). 8. V. Strunck, G. Grosche, and D. Dopheide, “New laser Doppler sensors for spatial velocity information,” in Proceedings of the International Congress on Instrumentation in Aerospace Simulation Facilities ICIAF’93, Saint-Louis, France, 1993 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1993), pp. 36.1–36.5. 9. V. Strunck, T. Sodomann, H. Müller, and D. Dopheide, “How to get spatial resolution inside probe volumes of commercial 3D LDA systems,” Exp. Fluids 36, 141–145 (2004). 10. J. Czarske, “Laser Doppler velocity profile sensor using a chromatic coding,” Meas. Sci. Technol. 12, 52–57 (2001). 11. J. Czarske, L. Büttner, T. Razik, and H. Müller, “Boundary layer velocity measurements by a laser Doppler profile sensor with micrometer spatial resolution,” Meas. Sci. Technol. 13, 1979 –1989 (2002). 12. D. A. Jackson, J. D. C. Jones, and R. K. Y. Chan, “A high-power fiber-optic laser Doppler velocimeter,” J. Phys. E 17, 977–980 (1984). 13. S. L. Kaufmann and L. M. Fingerson, “Fiber optics in LDV applications,” in Proceedings, International Conference on La-

2510


14. 15.

16.

17.

18.

19.

20.

21.

ser Anemometry—Advances and Applications, Manchester, UK, 16 –18 September 1985 (Springer-Verlag, Berlin, 1985), pp. 53– 65. M. Stieglmeier and C. Tropea, “Mobile fiber-optic laser Doppler anemometer,” Appl. Opt. 31, 4096 – 4105 (1992). J. Czarske and H. Müller, “Two-dimensional directional fiberoptic laser Doppler anemometer based on heterodyning by means of a chirp frequency modulated Nd:YAG miniature ring laser,” Opt. Commun. 132, 421– 426 (1996). H. Müller, H. Wang, and D. Dopheide, “Fiber optical multicomponent LDA-system using the optical frequency difference of powerful DBR-laser diodes,” in Developments in Laser Techniques and Fluid Mechanics, Selected Papers from the Eighth International Symposium, Lisbon, Portugal, 8 –11 July 1996 (Springer-Verlag, Berlin, 1996), pp. 11–21, paper I.2. D. Dopheide, V. Strunck, and H. J. Pfeifer, “Miniaturized multicomponent laser Doppler anemometers using high frequency pulsed diode lasers and new electronic signal acquisition systems,” Exp. Fluids 9, 309 –316 (1990). L. Büttner, “Untersuchung neuartiger Laser-DopplerVerfahren zur hochauflösenden Geschwindigkeitsmessung,” Ph.D. dissertation (Cuvillier Verlag, Göttingen, Germany, 2004). H. Blasius, “Grenzschichten in Flüssigkeiten mit kleiner Reibung,” Z. Math. Phys. 56, 1–37 (1908), NACA Tech. Memo 1256. H-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, Laser Doppler and Phase Doppler Measurement Techniques (Springer-Verlag, Berlin, 2003). F. Durst, H. Kikura, I. Lekakis, J. Jovanovic, and Q. Ye, “Wall shear stress determination from near-wall mean velocity data in turbulent pipe and channel flows,” Exp. Fluids 20, 417– 428 (1996).