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Ultrasonic tomography for visualizing the velocity profile of air flow
Ultrasonic tomography for visualizing the velocity profile of air flow KIN-ICHI KOMIYA and SUTHAM TEERAWATANACHAI Department of Control Engineering, Kyushu Institute of Technology, 1-1, Sensuicho, Tobata, Kitakyushu, 804 Japan
The computerized tomography technique (CT) is used to reconstruct the velocity profile of air flow in a pipe from its average velocities measured by an ultrasonic velocimeter over a specific set of paths. The data, which is detected by a fan beam arrangement, is rearranged into the form of interlaced parallel data, which is further interpolated in each view to obtain uniformly spaced data. This is then applied in the convolution algorithm for parallel scan data. This is then applied in the convolution algorithm for parallel scan data. This method is tested with experimental data and the reconstructed velocity is compared with the values measured by a hot wire anemometer. Keywords: CT technique; velocity profile; air flow
Introduction The velocity profile of fluid flow is one of the important characteristics of flow and many studies have been reported on its measurement. The purpose of those studies concerns mainly fluid dynamics, flow measurement and transport phenomena. In regard to flow measurement, the characteristics of flowmeters depend on the velocity profile upstream. For example, the flow rate can be calculated from the velocity profile by taking the area integral over the plane through which the fluid passes. The measurement of a velocity profile is conducted, in general, by point velocity measurement and a scanning apparatus to traverse the probe over the entire space of interest. Examples of velocity measuring probes are LDV systems, hot wire probes, Pitot-static tubes etc. The other method for velocity profile measurement is to apply the flow visualization technique 1. However, this technique is basically a qualitative measurement. For the purpose of a quantitative measurement, we have to analyse the pattern on a hard copy obtained by, for example, the smoke wire method, the hydrogen bubble method etc. On the other hand, the computerized tomography (CT) technique has been developed and is known as a method of reconstructing the interr~al details of a structure from its projections. The most simplified model of these projections is to interpret them as line integrals along straight paths where the beam passes through. Since the measured data is obtained without affecting the original nature of the objects, the method of CT becomes of great interest in non-invasive medical diagnosis, remote sensing, industrial applications and other scientific fields. 0955-5986/93/020061-05 ~) 1993 Butterworth-HeinemannLtd
Our work, reported here, applies this CT technique to reconstruct the velocity profile of air flow in a circular pipe. The data needed for the reconstruction algorithm in this case comprises the line integrals of velocity over many different paths. Those are obtained by use of an ultrasonic velocimeter; the measured velocities are the average velocities over the path. This idea has been applied with liquid flow, where the transducer parts in those installations are mounted external to the pipe 2,3. This is not possible for the measurement of air flow in a pipe, since the acoustic wave is highly attenuated when propagating across the boundary of air and another medium. In our setup, the transducers are installed directly in open apertures in the wall of the pipe. According to the arrangement of the transducers, the scanning method is a special type of diverged beam scan. In our previous work, some experiments were conducted with air flow in a pipe. The reconstruction algorithm used in that work does not take into account the fact that the velocity of flow at the wall of the pipe is zero, and a large deviation from the reference values near the circumference of the pipe was observed 7. Since the source of the beam is located at the boundary of the region under interest, it also prevents us from directly applying the reconstruction algorithm for diverged scan data. Hence we first arrange the data into the form of a parallel scan geometry, where the data is now non-uniformly spaced, and then interpolated for the other sets of uniformly spaced data. These can be applied in a reconstruction algorithm of parallel scan data. We can visualize the velocity profile and compare the velocity with the measured data by a hot wire anemometer.
Flow Meas. Instrum., 1993 Vol 4 No 2
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K.-I. Komiya and S. Teerawatanachai - Ultrasonic tomography for visualizing the velocity profile of air flow Ultrasonic
velocimeter
Computerized
The ultrasonic velocimeters measure the velocity of a flowing medium by monitoring interaction between the flow stream and an ultrasonic sound wave transmitted into or through it. The two most commonly applied techniques are the Doppler and the transmissive. The Doppler meter is normally used for a flow stream with a discontinuity and essentially measures the point velocity. This is not applicable for our purpose. The transmissive meter depends upon the fact that the velocity of a sound wave in a moving fluid is modified. If the wave has a component in the direction of flow, then the magnitude of the velocity is increased and the direction of the wave is altered towards the flow direction. The reverse happens if the wave is travelling against the flow. In our case, the sound waves are not generated in the direction of flow, but at an angle across it, as shown in Figure 1. Sound wave transit times in the downstream direction ta, and upstream direction tu, along a path of length /_ can be expressed as L tu - C - v
L td - C + v
(1)
where C is the velocity of sound in the fluid and v is the fluid velocity. The average velocity Vm along the path is v V m --
(2)
cos 0
1
td
tu
[(C
-{- v m c o s
0) -
(C -
v m cos
0)]
_ 2vm cos 0
(3)
When the velocity changes along the path, equation (3) shows that the difference of the reciprocal of the propagation time is proportional to the mean velocity. Suppose that the velocity distribution is steady over the measuring volume; then, as will be stated later, we can apply the computerized tomography technique for reconstruction of the velocity profile by using these measured velocities.
T r a n s d u c e r
air f,o.
(4)
Next, we also define P~ (t,0) as the line integral of f(x,y) over the line L, that is
Pt(t,O) = f f(x,y) ds
(5)
The variable of integration s is an ordinate in the other coordinate, t-s, which is obtained by rotating the x-y coordinates by an angle 0. An algorithm called the convolution algorithm is widely used to reconstruct f(x,y) from the projection Pf(t,O). This algorithm was developed based on the Fourier slice theorem 9. This states that the Fourier transform of the projection at an angle 0 yields a central cross-section of the Fourier transform of the object function at the same angle. Now, let us define S(O,w) as the Fourier transform on variable t of P~(t,O) and w as its corresponding radial frequency. The Fourier slice theorem can then be expressed as follows: (6)
where F(w,O) is the two-dimensional Fourier transform of fix,y) on polar coordinates. Then theoretically, fix,y), the distributed property of the object, can be reconstructed if we have enough views of projection data. Reconstruction can then be performed in either frequency space or object space. In the former it becomes the direct Fourier transformation method 4, while in the latter it becomes the convolution method s, which is the one used in this work. It is noted that the reconstruction consists of two steps, filtering and back-projection. The procedure of back-projection for a point (x,y) is to sum the values over all of the views at the projected point of (x,y) on
/
---=-\ \c-vcos 0
N,~~
Figure 2 Straight line L(t,O) and geometric meaning of t and 0
figure 1 Ultrasonic flowmeter 62
L(t,0): x cos 0 + y sin 0 = t
u(t,e)
V \ c+v~~~'~ 2
Let us define f(x,y) as a function which presents the distributed property of an object, as shown in Figure 2. Any straight line, L, on the x-y plane can be specified by two variables, t and 0, where t is the distance from the line to the origin and 0 is the angle between a line through the origin perpendicular to L and the x-axis. Then we can derive the following equation:
S(w, 0) = F(w, 0)
Then we obtain 1
tomography
Flow Meas. Instrum.,
1993 Vol 4 No 2
K.-I. Komiya and S. Teerawatanachai - Ultrasonic tomography for visualizing the velocity profile of air flow to the abscissa of the projection function in each view. The filter in the algorithm which is used here is the one proposed by Shepp et al. 6. As mentioned in the previous section, the line integral or mean of the velocity is obtained by the use of an ultrasonic velocimeter. The velocity distribution could then be reconstructed by this method. The scanning method we used was a diverged scan, and this time we tried to use the algorithm of parallel scanning. In order to obtain the data needed for this algorithm, we rearranged the diverged data into parallel data and interpolated for uniformly spaced points by the Lagrangian method. From the data, we can reconstruct the velocity profile, assuming that the flow is steady in time and space over the measuring volume.
a
Flow
Straightener
i'~ ~c2::Z:::zi
II ~,~,~ ii "-+- ~
b
50~
"~, ;
Experiment To test the idea mentioned above, an experiment was conducted. Its arrangement is shown in Figure 3a. The vinyl pipe size is 0.49 m in diameter and 2 m in length. A ventilator is used as the air source, and at the inlet of the pipe a nest of small diameter tubes of 20 mm diameter and 150 mm in length serves as a flow straightener. The test section is 1.1 m of acrylic pipe. It is located on four castors to make it rotatable. On the wall of this test section, eight ultrasonic transducers are installed, as shown in Figure 3b. One transducer is installed upstream and the other seven are installed at downstream locations. The two planes are 50 cm apart. The transducers used in the ultrasonic velocimeter are a 40 kHz speaker type and the time resolution of the ultrasonic velocimeter is 0.25 I~S. After finishing measuring in each view, the test section is rotated manually through 45 ° for the next measuring view. Then eight sets of data from eight views with seven values in each view are recorded, as shown in Figure 4a. Each of these paths has two values of the average velocity associated with it. By averaging these two values for each path, we obtain one single value for each path of Figure 4a, that is 28 values. For simply applying ordinary reconstruction algorithms, we have to arrange the data into the form of parallel and uniformly spaced scan data. We arrange the 28 values in two groups: four sets of four parallel paths and four sets of three parallel paths, as shown in Figure 4b. The spacing of the paths is non-uniform and it is different between the two groups. Our idea is to interpolate for the data on uniformly spaced points from the data at hand. The interpolation method used in our work is the Lagrangian interpolation. The number of points which are interpolated for is 15 in each view. The velocity profile is assumed to change smoothly from point to point, and also the a priori knowledge that the velocity is zero on the wall is introduced in the interpolation. Moreover, since the view of the even numbers, as shown by the dotted lines in Figure 4b, does not have a sample point at the centre, the average value of data at the centre from the other four points from the odd numbered view was inserted. Without doing this, a valley at the centre was observed in the interpolated results, which it seems inconsistent to use in reconstruc-
/
~m
Transducers
0 C
a
Figure 3 Experimental arrangement: (a) outline of experimental setup; (b) locations of transducers; (c) velocity points measured by hot wire anemometer tion ~. The consistency of the interpolated results was checked by reprojecting the reconstruction results. To check the reconstructed results, the velocities at four sections with seven points in each section were measured at the downstream plane by a hot wire anemometer. Those points are shown in Figure 3c. The mode of measurements used was 10 s, averaged, and data was recorded every 5 s, 10 times for each point. From the rearranged data, we can construct the velocity profile, assuming (1) that the flow is steady, (2) that the flow in the pipe is parallel to the pipe axis, and (3) that the velocity profile does not change between the two planes where the transducers have been installed. The contour and three-dimensional graph of the velocity profile reconstructed from the data are shown in Figures 5a and b, respectively. The dimension of
Flow Meas. Instrum.,
1993 Vol 4 No 2
63
K.-I. Komiya and S. Teerawatanachai - Ultrasonic tomography for visualizing the velocity profile of air flow 1.o
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Discussion From the results, a deviation of the reconstructed profile from the values measured by a hot wire anemometer is observed. This is expected to be due to the following reasons. The length of the pipe used may be too short compared with its diameter, say, about 6 times the diameter. The profile of flow in the volume of measurement, i.e. from the upstream to the downstream plane, may therefore not be constant. It would therefore be more appropriate to consider this as a three-dimensional reconstruction problem. However, this would need a more complex scanning structure to obtain all the data needed. Some fluctuations in the ventilator which was used 64
Flow Meas. Instrum.,
O.Z
0.2
1993 Vol 4 No 2
-0.8
-1.o
i.o
Figure 5 Experimental results: (a) contour lines of reconstructed velocity profile (plotting step 0.1 m s-l); (b) 3D graph of reconstructed velocity profile in this experiment as the air source may arise. As mentioned before, the data was measured manually, and it would then be inconsistent to use the data measured at different times as if it is the same set of data. However, the reconstructed result near the circumference decreases to zero as desired, and there is general improvement over our previous work. This is due to the fact that the profile of the flow in our experiment is comparatively smooth. It then seems reasonable to use polynomials of low degree, which was realized by Lagrangian interpolation in this work, to interpolate for the data needed. It is intuitively noted that more complex profiles may need more measured data to allow good reconstruction results.
Conclusion A velocity profile of pipe flow visualizing method was proposed. The technique of CT was used to reconstruct the velocity profile of air flow in a pipe from average velocities over a specific set of paths measured by an ultrasonic velocimeter. This set of data is used to
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Figure 6 Velocity distribution for four sections. (a), (b), (c) and (d) correspond to the sections indicated in Figure 3c interpolate for the value at the uniformly spaced points. The interpolated data is then applied in a convolution algorithm to reconstruct the velocity profile of flow. An experiment with a ventilator was conducted and the result was compared with the values measured by a hot wire anemometer. The i m p r o v e m e n t in the result from our previous w o r k was observed. References 1 W.-J. Yang (Ed.) 'Handbook of Flow Visualization', Hemisphere Publishing Corporation, New York (1989) 2 Johnson, S. A., Greenleaf, J. F., Tanaka, T. and Flandro, G. Reconstruction 3D temperature and fluid velocity vector fields from acoustic transmission measurements ISA Trans. 16 (1970) 3-15 3 Hauck, A. Ultrasonic tomography for the non-intrusive measurements of flow-velocity-fields In 'Proc. 5th Int. IMEKO-CON, on Flow Measurement', FLOMEKO, D~isseldorf (1989) 361-369
4 Stark, H., Woods, J. W., Paul, I. P. and Hingorani, R. An investigation of computerized tomography by direct Fourier inversion and optimum interpolation IEEE Trans. Biomed. Eng. 28 (1981) 496-505 5 Ramachandran, G. N. and Lakshminarayanan, A. V. Threedimensional reconstruction from radiographs and electron micrographs: application of convolution instead of Fourier transforms
Proc. Natl. Acad. Sci. USA 68-9 (1971) 2236-2240 6 Shepp, L. A. and Logan, B. F. The Fourier reconstruction of a head section IEEE Trans. Nucl. Sci. 21 (1974) 21~13 7 Teerawalanachai, S., Komiya, K. and Sasamoto, H. Estimating the velocity profile of air flow by means of the ultrasonic translation time computed tomography method In 'Proc. IECON '91, (1991) 2379-2384 8 Teerawalanachi, S., Komiya, K., Sasamoto, H. and Ogata, N. An ultrasonic tomography for reconstructing the velocity profile of air flow from nonuniform spaced projection data Trans. Soc. Instr. Control Eng., Japan (in press) 9 Kak, A. C. and Slaney, M. 'Principles of computerized tomographic imaging', IEEE Press, New York (1988)
Flow Meas. Instrum.,
1993 Vol 4 No 2
65
The computerized tomography technique (CT) is used to reconstruct the velocity profile of air flow in a pipe from its average velocities measured by an ultrasonic velocimeter over a specific set of paths. The data, which is detected by a fan beam arrangement, is rearranged into the form of interlaced parallel data, which is further interpolated in each view to obtain uniformly spaced data. This is then applied in the convolution algorithm for parallel scan data. This is then applied in the convolution algorithm for parallel scan data. This method is tested with experimental data and the reconstructed velocity is compared with the values measured by a hot wire anemometer. Keywords: CT technique; velocity profile; air flow
Introduction The velocity profile of fluid flow is one of the important characteristics of flow and many studies have been reported on its measurement. The purpose of those studies concerns mainly fluid dynamics, flow measurement and transport phenomena. In regard to flow measurement, the characteristics of flowmeters depend on the velocity profile upstream. For example, the flow rate can be calculated from the velocity profile by taking the area integral over the plane through which the fluid passes. The measurement of a velocity profile is conducted, in general, by point velocity measurement and a scanning apparatus to traverse the probe over the entire space of interest. Examples of velocity measuring probes are LDV systems, hot wire probes, Pitot-static tubes etc. The other method for velocity profile measurement is to apply the flow visualization technique 1. However, this technique is basically a qualitative measurement. For the purpose of a quantitative measurement, we have to analyse the pattern on a hard copy obtained by, for example, the smoke wire method, the hydrogen bubble method etc. On the other hand, the computerized tomography (CT) technique has been developed and is known as a method of reconstructing the interr~al details of a structure from its projections. The most simplified model of these projections is to interpret them as line integrals along straight paths where the beam passes through. Since the measured data is obtained without affecting the original nature of the objects, the method of CT becomes of great interest in non-invasive medical diagnosis, remote sensing, industrial applications and other scientific fields. 0955-5986/93/020061-05 ~) 1993 Butterworth-HeinemannLtd
Our work, reported here, applies this CT technique to reconstruct the velocity profile of air flow in a circular pipe. The data needed for the reconstruction algorithm in this case comprises the line integrals of velocity over many different paths. Those are obtained by use of an ultrasonic velocimeter; the measured velocities are the average velocities over the path. This idea has been applied with liquid flow, where the transducer parts in those installations are mounted external to the pipe 2,3. This is not possible for the measurement of air flow in a pipe, since the acoustic wave is highly attenuated when propagating across the boundary of air and another medium. In our setup, the transducers are installed directly in open apertures in the wall of the pipe. According to the arrangement of the transducers, the scanning method is a special type of diverged beam scan. In our previous work, some experiments were conducted with air flow in a pipe. The reconstruction algorithm used in that work does not take into account the fact that the velocity of flow at the wall of the pipe is zero, and a large deviation from the reference values near the circumference of the pipe was observed 7. Since the source of the beam is located at the boundary of the region under interest, it also prevents us from directly applying the reconstruction algorithm for diverged scan data. Hence we first arrange the data into the form of a parallel scan geometry, where the data is now non-uniformly spaced, and then interpolated for the other sets of uniformly spaced data. These can be applied in a reconstruction algorithm of parallel scan data. We can visualize the velocity profile and compare the velocity with the measured data by a hot wire anemometer.
Flow Meas. Instrum., 1993 Vol 4 No 2
61
K.-I. Komiya and S. Teerawatanachai - Ultrasonic tomography for visualizing the velocity profile of air flow Ultrasonic
velocimeter
Computerized
The ultrasonic velocimeters measure the velocity of a flowing medium by monitoring interaction between the flow stream and an ultrasonic sound wave transmitted into or through it. The two most commonly applied techniques are the Doppler and the transmissive. The Doppler meter is normally used for a flow stream with a discontinuity and essentially measures the point velocity. This is not applicable for our purpose. The transmissive meter depends upon the fact that the velocity of a sound wave in a moving fluid is modified. If the wave has a component in the direction of flow, then the magnitude of the velocity is increased and the direction of the wave is altered towards the flow direction. The reverse happens if the wave is travelling against the flow. In our case, the sound waves are not generated in the direction of flow, but at an angle across it, as shown in Figure 1. Sound wave transit times in the downstream direction ta, and upstream direction tu, along a path of length /_ can be expressed as L tu - C - v
L td - C + v
(1)
where C is the velocity of sound in the fluid and v is the fluid velocity. The average velocity Vm along the path is v V m --
(2)
cos 0
1
td
tu
[(C
-{- v m c o s
0) -
(C -
v m cos
0)]
_ 2vm cos 0
(3)
When the velocity changes along the path, equation (3) shows that the difference of the reciprocal of the propagation time is proportional to the mean velocity. Suppose that the velocity distribution is steady over the measuring volume; then, as will be stated later, we can apply the computerized tomography technique for reconstruction of the velocity profile by using these measured velocities.
T r a n s d u c e r
air f,o.
(4)
Next, we also define P~ (t,0) as the line integral of f(x,y) over the line L, that is
Pt(t,O) = f f(x,y) ds
(5)
The variable of integration s is an ordinate in the other coordinate, t-s, which is obtained by rotating the x-y coordinates by an angle 0. An algorithm called the convolution algorithm is widely used to reconstruct f(x,y) from the projection Pf(t,O). This algorithm was developed based on the Fourier slice theorem 9. This states that the Fourier transform of the projection at an angle 0 yields a central cross-section of the Fourier transform of the object function at the same angle. Now, let us define S(O,w) as the Fourier transform on variable t of P~(t,O) and w as its corresponding radial frequency. The Fourier slice theorem can then be expressed as follows: (6)
where F(w,O) is the two-dimensional Fourier transform of fix,y) on polar coordinates. Then theoretically, fix,y), the distributed property of the object, can be reconstructed if we have enough views of projection data. Reconstruction can then be performed in either frequency space or object space. In the former it becomes the direct Fourier transformation method 4, while in the latter it becomes the convolution method s, which is the one used in this work. It is noted that the reconstruction consists of two steps, filtering and back-projection. The procedure of back-projection for a point (x,y) is to sum the values over all of the views at the projected point of (x,y) on
/
---=-\ \c-vcos 0
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Figure 2 Straight line L(t,O) and geometric meaning of t and 0
figure 1 Ultrasonic flowmeter 62
L(t,0): x cos 0 + y sin 0 = t
u(t,e)
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Let us define f(x,y) as a function which presents the distributed property of an object, as shown in Figure 2. Any straight line, L, on the x-y plane can be specified by two variables, t and 0, where t is the distance from the line to the origin and 0 is the angle between a line through the origin perpendicular to L and the x-axis. Then we can derive the following equation:
S(w, 0) = F(w, 0)
Then we obtain 1
tomography
Flow Meas. Instrum.,
1993 Vol 4 No 2
K.-I. Komiya and S. Teerawatanachai - Ultrasonic tomography for visualizing the velocity profile of air flow to the abscissa of the projection function in each view. The filter in the algorithm which is used here is the one proposed by Shepp et al. 6. As mentioned in the previous section, the line integral or mean of the velocity is obtained by the use of an ultrasonic velocimeter. The velocity distribution could then be reconstructed by this method. The scanning method we used was a diverged scan, and this time we tried to use the algorithm of parallel scanning. In order to obtain the data needed for this algorithm, we rearranged the diverged data into parallel data and interpolated for uniformly spaced points by the Lagrangian method. From the data, we can reconstruct the velocity profile, assuming that the flow is steady in time and space over the measuring volume.
a
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Straightener
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Experiment To test the idea mentioned above, an experiment was conducted. Its arrangement is shown in Figure 3a. The vinyl pipe size is 0.49 m in diameter and 2 m in length. A ventilator is used as the air source, and at the inlet of the pipe a nest of small diameter tubes of 20 mm diameter and 150 mm in length serves as a flow straightener. The test section is 1.1 m of acrylic pipe. It is located on four castors to make it rotatable. On the wall of this test section, eight ultrasonic transducers are installed, as shown in Figure 3b. One transducer is installed upstream and the other seven are installed at downstream locations. The two planes are 50 cm apart. The transducers used in the ultrasonic velocimeter are a 40 kHz speaker type and the time resolution of the ultrasonic velocimeter is 0.25 I~S. After finishing measuring in each view, the test section is rotated manually through 45 ° for the next measuring view. Then eight sets of data from eight views with seven values in each view are recorded, as shown in Figure 4a. Each of these paths has two values of the average velocity associated with it. By averaging these two values for each path, we obtain one single value for each path of Figure 4a, that is 28 values. For simply applying ordinary reconstruction algorithms, we have to arrange the data into the form of parallel and uniformly spaced scan data. We arrange the 28 values in two groups: four sets of four parallel paths and four sets of three parallel paths, as shown in Figure 4b. The spacing of the paths is non-uniform and it is different between the two groups. Our idea is to interpolate for the data on uniformly spaced points from the data at hand. The interpolation method used in our work is the Lagrangian interpolation. The number of points which are interpolated for is 15 in each view. The velocity profile is assumed to change smoothly from point to point, and also the a priori knowledge that the velocity is zero on the wall is introduced in the interpolation. Moreover, since the view of the even numbers, as shown by the dotted lines in Figure 4b, does not have a sample point at the centre, the average value of data at the centre from the other four points from the odd numbered view was inserted. Without doing this, a valley at the centre was observed in the interpolated results, which it seems inconsistent to use in reconstruc-
/
~m
Transducers
0 C
a
Figure 3 Experimental arrangement: (a) outline of experimental setup; (b) locations of transducers; (c) velocity points measured by hot wire anemometer tion ~. The consistency of the interpolated results was checked by reprojecting the reconstruction results. To check the reconstructed results, the velocities at four sections with seven points in each section were measured at the downstream plane by a hot wire anemometer. Those points are shown in Figure 3c. The mode of measurements used was 10 s, averaged, and data was recorded every 5 s, 10 times for each point. From the rearranged data, we can construct the velocity profile, assuming (1) that the flow is steady, (2) that the flow in the pipe is parallel to the pipe axis, and (3) that the velocity profile does not change between the two planes where the transducers have been installed. The contour and three-dimensional graph of the velocity profile reconstructed from the data are shown in Figures 5a and b, respectively. The dimension of
Flow Meas. Instrum.,
1993 Vol 4 No 2
63
K.-I. Komiya and S. Teerawatanachai - Ultrasonic tomography for visualizing the velocity profile of air flow 1.o
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Discussion From the results, a deviation of the reconstructed profile from the values measured by a hot wire anemometer is observed. This is expected to be due to the following reasons. The length of the pipe used may be too short compared with its diameter, say, about 6 times the diameter. The profile of flow in the volume of measurement, i.e. from the upstream to the downstream plane, may therefore not be constant. It would therefore be more appropriate to consider this as a three-dimensional reconstruction problem. However, this would need a more complex scanning structure to obtain all the data needed. Some fluctuations in the ventilator which was used 64
Flow Meas. Instrum.,
O.Z
0.2
1993 Vol 4 No 2
-0.8
-1.o
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Figure 5 Experimental results: (a) contour lines of reconstructed velocity profile (plotting step 0.1 m s-l); (b) 3D graph of reconstructed velocity profile in this experiment as the air source may arise. As mentioned before, the data was measured manually, and it would then be inconsistent to use the data measured at different times as if it is the same set of data. However, the reconstructed result near the circumference decreases to zero as desired, and there is general improvement over our previous work. This is due to the fact that the profile of the flow in our experiment is comparatively smooth. It then seems reasonable to use polynomials of low degree, which was realized by Lagrangian interpolation in this work, to interpolate for the data needed. It is intuitively noted that more complex profiles may need more measured data to allow good reconstruction results.
Conclusion A velocity profile of pipe flow visualizing method was proposed. The technique of CT was used to reconstruct the velocity profile of air flow in a pipe from average velocities over a specific set of paths measured by an ultrasonic velocimeter. This set of data is used to
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Figure 6 Velocity distribution for four sections. (a), (b), (c) and (d) correspond to the sections indicated in Figure 3c interpolate for the value at the uniformly spaced points. The interpolated data is then applied in a convolution algorithm to reconstruct the velocity profile of flow. An experiment with a ventilator was conducted and the result was compared with the values measured by a hot wire anemometer. The i m p r o v e m e n t in the result from our previous w o r k was observed. References 1 W.-J. Yang (Ed.) 'Handbook of Flow Visualization', Hemisphere Publishing Corporation, New York (1989) 2 Johnson, S. A., Greenleaf, J. F., Tanaka, T. and Flandro, G. Reconstruction 3D temperature and fluid velocity vector fields from acoustic transmission measurements ISA Trans. 16 (1970) 3-15 3 Hauck, A. Ultrasonic tomography for the non-intrusive measurements of flow-velocity-fields In 'Proc. 5th Int. IMEKO-CON, on Flow Measurement', FLOMEKO, D~isseldorf (1989) 361-369
4 Stark, H., Woods, J. W., Paul, I. P. and Hingorani, R. An investigation of computerized tomography by direct Fourier inversion and optimum interpolation IEEE Trans. Biomed. Eng. 28 (1981) 496-505 5 Ramachandran, G. N. and Lakshminarayanan, A. V. Threedimensional reconstruction from radiographs and electron micrographs: application of convolution instead of Fourier transforms
Proc. Natl. Acad. Sci. USA 68-9 (1971) 2236-2240 6 Shepp, L. A. and Logan, B. F. The Fourier reconstruction of a head section IEEE Trans. Nucl. Sci. 21 (1974) 21~13 7 Teerawalanachai, S., Komiya, K. and Sasamoto, H. Estimating the velocity profile of air flow by means of the ultrasonic translation time computed tomography method In 'Proc. IECON '91, (1991) 2379-2384 8 Teerawalanachi, S., Komiya, K., Sasamoto, H. and Ogata, N. An ultrasonic tomography for reconstructing the velocity profile of air flow from nonuniform spaced projection data Trans. Soc. Instr. Control Eng., Japan (in press) 9 Kak, A. C. and Slaney, M. 'Principles of computerized tomographic imaging', IEEE Press, New York (1988)
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