- Email: [email protected]
Beam configuration for light sheet application
Measurement 38 (2005) 42–52 www.elsevier.com/locate/measurement
Beam configuration for light sheet application W. Baetz a, W. Holzapfel
a,*
, H. Dietrich b, B. Nath
c
a
University of Kassel, Institute for Measurement and Automation, D-34109 Kassel, Germany b Elster Produktion GmbH, Steinernstrasse 19, D-55252 Mainz-Kastel, Germany c AKG Thermotechnik GmbH & Co. KG, Am Hohlen Weg 31, 34369 Hofgeismar, Germany Received 25 January 2003; accepted 21 February 2005 Available online 11 May 2005
Abstract To visualise laminar and turbulent flow patterns of gaseous or liquid fluids, the laser-light-sheet technique can be applied. The optically recorded flow images are useful for research and development of flow measuring and controlling devices. Instead of single high-power gas lasers or solid state lasers six monomode laser diodes emitting in the visible spectral range are applied. They are mounted in parallel and the individual radiation fields overlap to generate a more homogeneous intensity distribution with less coherence than with a single laser source. Therefore, disturbances due to speckle effects are reduced. The resulting light sheet has a total optical output power of 0.3 W and a thickness of 0.7 mm. The working distance amounts from 60 mm up to 160 mm with an operating area of 100 · 100 mm2. Application examples of the internal flow characteristics of a gas–pressure regulator and high-speed gas flow diagnosis in a sonic Venturi nozzle are presented and compared with computational fluid dynamics (CFD) simulations. 2005 Elsevier Ltd. All rights reserved. Keywords: Gas flow; Laser diode; Light sheet
1. Introduction The investigation of gaseous or liquid flow fields often requires only qualitative results about
*
Corresponding author. Tel.: +49 561 804 2758; fax: +49 561 804 2847. E-mail addresses: [email protected] (W. Baetz), [email protected] (W. Holzapfel), [email protected] (H. Dietrich), [email protected] (B. Nath).
the flow pattern to identify unwanted vortex patterns or dead zones in the flow field of interest. For this purpose, the laser light sheet technique (Fig. 1) is applied to a variety of different research fields like vortex pattern visualisation about aircraft models or the experimental investigation of the characteristics of diesel spray heads [1]. It can also be adapted to visualise gas flow pattern in gas pipes, regulators [2,3], furnaces [4,5] and volume-measuring devices [6]. The technique (Fig. 1) applies three basic procedures: (1) illuminating a
0263-2241/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2005.02.004
W. Baetz et al. / Measurement 38 (2005) 42–52
43
This contribution presents an inexpensive modular laser diode light sheet unit, generating a more homogeneous intensity profile by superposition of the nearly Gaussian radiation fields of six monomode laser diodes. Another advantage of this superposition is the lower coherence of the resultant radiation field, because the individual laser diodes are not frequency-stabilized and their mean wavelengths are different over several nanometres. We present the theoretical relations and the layout of the applied laser diode light sheet technique, as well as its application to gas flow diagnosis in a gas pressure regulator and high-speed gas flow in a sonic Venturi nozzle. Fig. 1. Typical experimental set-up for flow field visualisation with a laser light sheet.
2. Laser diode light sheet 2.1. Theory of light sheet generation To generate a thin light sheet with high luminous intensity, in general, gas or solid state lasers are employed. The emission aperture of these lasers is usually circular and they are operating in their fundamental transverse mode (TEM00) so that the output emission corresponds to a circular Gaussian beam. The Gaussian beam is uniquely determined by its wavelength k and its smallest beam radius w0 in the beam waist at z = 0 (Fig. 2).
3
Relative Beam Radius
w w0
2
r
w = w0 -1
zR
1
w= 2 w0 I
thin plane cross-section of the flow field with a light sheet, (2) seeding the flow with diffusely reflecting tracer particles and (3) recording the resulting images of the tracer streaks with a camera. Smoke, vapour or tiny oil droplets in air or bubbles in water can be used as tracer particles. To follow the real flow field without tracking errors, the tracer particles have to be very small and with low mass. For the observation of gas flow, the particle diameter is approximately 1–10 lm and the scattered optical power is very small. Therefore, in general a high-power laser like an argon-ion laser or an Nd:YAG laser must be used to provide high luminous intensities in the light sheet to visualise the gas flow. These laser technologies are very expensive, high power consuming and difficult to handle due to their geometrical size. Therefore, additional light-guiding devices like light arms or fibre optics are needed to illuminate the flow field region to be observed. Other disadvantages of these single laser sources are the inhomogeneous illumination by the Gaussian intensity distribution of the single laser beam and the great coherence length of the single laser radiation. The Gaussian intensity profile often results in an overexposure in the centre and an underexposure in the peripheral regions of the light sheet and the great coherence leads to an irregular illumination by the speckle effects.
1
-1 -2
2
3
Θ1/e
-3
Relative Beam Coordinate z/zR Fig. 2. Propagation characteristics of a circular Gaussian beam with the beam waist w0, the Rayleigh length zR and the divergence angle H1/e.
44
W. Baetz et al. / Measurement 38 (2005) 42–52
The intensity distribution I(r, z) of the radiation with the total optical power P in the direction of propagation z, and at the distance r from the beam axis (Fig. 3), is defined by [7]: " # 2P 2 r2 Iðr; zÞ ¼ exp ; ð1Þ wðzÞ2 p wðzÞ2 with the beam radius w(z) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 z ; w0 ¼ wð0Þ wðzÞ ¼ w0 1 þ zR
ð2Þ
and the Rayleigh length zR zR ¼
p w20 . k
ð3Þ
The beam radius w(z) is defined as the distance from the z-axis, where the electric field amplitude has fallen to the 1/e-fold of the maximum amplitude. Here the beam intensity has decreased to the 1/e2-fold of the maximum intensity I(0, z). The Rayleigh length zR characterizes one half of the collimated range on one side pffiffiffiof the beam waist, with the beam radius wðzÞ 6 2w0 . According to Eqs. (1)–(3), in the far field (z zR) the beam radius increases linearly with the distance z from the beam waist. The beam angle H1/e of the radiation cone for r = w(z) only depends on the wavelength k and the beam waist w0 wðzÞ H1=e ¼ lim 2 arctan z!1 z k ¼ 2 arctan . ð4Þ p w0
I(r) I(0)
0.8
Relative Intensity
1.0
0.6
-1.5
r/w = 0.5
0.4 0.2
ln 2 2
r/w=1
The emission aperture of the laser-active zone of a semiconductor laser is not circular, but instead rectangular in contrast to the gas or solid-state lasers. The emission aperture of index-guided monomode laser diodes is typically 3 lm wide and 1 lm high, with the long side of the emission aperture orientated parallel to the semiconductor junction. In the far field, perpendicular to the beam-propagation direction, the intensity distribution also corresponds in good approximation to a twodimensional Gaussian distribution. Because of the rectangular emission aperture, the Gaussian distribution, however, is no longer circular, but elliptic. Here the semiaxes of the elliptic distribution are orientated parallel (index p) and perpendicular (index r) to the semiconductor junction. Because of the very small emission aperture, the output radiation is strongly divergent due to diffraction effects. In the far field, the propagation is characterized by the divergence angles Hp and Hr. These FWHM-angles (FWHM: full width at half maximum) mark the full cone angles in the r- and p-planes of the semiconductor junction, within which the radiant intensity is greater, or equal to, one half—not 1/e2—of the maximum intensity I(0, z). The half-intensity beam radius w1/2 results from the 1/e2-intensity radius w according to Eq. (1) with I(r, z) = 0.5I(0, z) (Fig. 2) rffiffiffiffiffiffiffiffi ln 2 w. ð5Þ w1=2 ¼ 2 Assuming a Gaussian beam, the equivalent waist radii w0p and w0r, in the semiconductor laser resonator, can be calculated from the divergence angles (with Hp < Hr) by Eq. (4) rffiffiffiffiffiffiffiffi ln 2 k w0p ¼ ð6aÞ 2 p tan H2p and
2
1/e
w0r -1
-0.5 0.5 1 0 Relative Beam Radius r/w
1.5
Fig. 3. Relative intensity distribution of a Gaussian beam and definition of the beam radius.
rffiffiffiffiffiffiffiffi ln 2 k . ¼ 2 p tan H2r
ð6bÞ
Applying Eqs. (2) and (3) the elliptic intensity distribution I(x, y, z) of a semiconductor laser is now given by
W. Baetz et al. / Measurement 38 (2005) 42–52
Iðx; y; zÞ ¼
2P p wp ðzÞ wr ðzÞ ( " exp 2
!
x2
wp ðzÞ2
þ
y2 wr ðzÞ2
.
The image location x00 determines the centre of the working area of the light sheet and the Rayleigh length x0R is equal to the depth of focus where the thickness of the light sheet is nearly constant.
ð7Þ
2.2. Laser diode light sheet unit
!#)
Here, x and y are the Cartesian coordinates parallel and perpendicular, respectively, to the semiconductor junction and perpendicular to the propagation direction z. In order to achieve a light sheet as thin and extensive as possible from this divergent intensity distribution, it is necessary to collimate the radiation in the plane with the smaller divergence angle Hp. Then the angle Hr determines the spread of the light fan in the plane of the light sheet. That means that the very small beam waist w0p must be projected into a larger one with the radius w00p using an appropriate cylindrical lens (focus length f). Assuming a thin lens, the image of the Gaussian beam waist is given by (Fig. 4) [7] f k w0p w00p ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . k2 x20 þ p2 w40p
ð8Þ
The image is located at the distance x00 from the focus point F 0 in the image space f 2 x0 x20 þ z2R
x00 ¼
45
The realized light-sheet unit consists of two laser diode modules. Each module consists of three monomode laser diodes with an individual optical output power of 50 mW at a wavelength of k = 685 ± 5 nm. Thus, the total optical output power amounts to 0.3 W. Their FWHM divergence angles in the far field amounts to Hp = 9 and Hr = 22. The equivalent beam waist radii in the laser-active zone are with Eq. (6) w0p = 1.64 lm and w0r = 0.66 lm. To achieve an intensity distribution as even as possible at a higher total intensity the diodes are arranged in parallel with a spacing of 20 mm. Each laser diode is collimated with a cylindrical lens in the plane parallel to the semiconductor junction (Fig. 5). Positioning the beam waist at the focal point F (i.e. x0 = 0) of the cylindrical lens with f = 5 mm, the beam waist w0p of the laser diode is imaged into a waist with the radius w00p ¼ 0.6 mm located
ð9Þ
with the new Rayleigh length z0R ¼
p f 2 w20 k . k2 x20 þ p2 w40
ð10Þ
z r'
zR 2 w0
w0
•
w0'
2 w0' •
F'
F
x0
f
δ
x0'
f
δ'
Fig. 4. Image of the Gaussian beam waist with a thin lens.
Fig. 5. Design of the light sheet unit with six monomode laser diodes.
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W. Baetz et al. / Measurement 38 (2005) 42–52
at the focal point F 0 (i.e. x00 = 0). With Eq. (10) the Rayleigh length in the image space is approximately z0R ¼ 2 m and is therefore very large compared with the focal length f of the cylindrical lens. Therefore the resultant light sheet has a thickness of approximately 0.7 mm (FWHM) within the collimated range of approximately 2 m. The orthogonal divergence angle Hr = 22 (FWHM) defines the fan angle of the individual light sheet. At a greater distance in the propagation direction the Gaussian intensity distributions
z = 60 mm Rel. Intensity
overlap providing a widely constant intensity (Fig. 6). Distributing the laser diodes into two modules (Fig. 7) at a total power of 0.3 W allows a flexible arrangement of the radiation sources so that intensity and dimensions of the light sheet can be adapted to the requirements of the individual measurement task. With the two modules arranged in parallel, the useable illuminated range of the light sheet is about 100 · 100 mm2 at an operating distance of 60–160 mm. The laser diode light sheet can be operated with the control unit in continuous-wave mode, or in pulse mode with a pulse duration of 100 ls to 999 s.
0.06
3. Test set-up
0.04
100
0.02
50
0 -50 0 x/mm
-50 50
0 y/mm
-100
Fig. 6. Relative intensity distribution of the laser diode light sheet at the distance of z = 60 mm.
A test set-up has been built to record flow images of different flow models by means of laser diode light sheet technique (Fig. 8). To conduct the test, a housing is constructed to simulate the environment from which the gas flow is to be studied. To insert the laser light sheet into the housing, two slits are cut into it from directly above and below. Next, a window is created on the front of the housing to give the camera its own optical access. These windows and slits are made with acrylic plastic and are sealed to ensure an airtight enclosure. For the first experiment, the Karman vortex street, a cylinder is placed in the housing at the far left of the cameraÕs inspection window. After the laser light sheet has been directed into this
Viewing Window Laser Light Sheet
Image Area of the Video Camera
Spray Tube
Aerosol Generator
Fig. 7. Laser module with three laser diodes.
Gas Flow Channel
to the Fan
Laser Module
Fig. 8. Schematic diagram of the test set-up.
W. Baetz et al. / Measurement 38 (2005) 42–52
housing, the gas flow is added. As the gas flows around the cylinder, the flow is disturbed. It is possible to record these disturbances to the flow because the camera is mounted perpendicular to the laser light sheet. To generate the gas flow, a vacuum pump and an aerosol generator are used. A constant volume flow through the housing is accomplished by using a Venturi nozzle in-line with the pump. The Venturi nozzle is then operated under critical conditions. To visualise the gas flow, an aerosol generator adds DEHS (diethyl sebacate) particles with the size of 0.2–1 lm to the flow. When these particles pass through the light sheet, they scatter the laser light and become visible. Within the light sheet, an image of the flow pattern appears that can be recorded by a CCD camera. The CCD camera, having a sensitivity of approx. 0.1 lux, is used to record the images. This highly sensitive video camera is necessary to capture the images because, unlike the high-output light intensity of gas lasers, the output intensity of the laser diodes is much lower. As the analogue signals of the camera are transmitted to the PC, a frame-grabber board digitises these signals allowing the PC to store them. Next, the PC is used to perform the digital post-processing of these signals into visible images.
47
recorded, this time offset results in a local offset in Fig. 9. The size of this local offset is dependent on the flow rate. By means of the digital separation of the interlaced frames into fields, any frame digitised by the frame grabber board is split into two fields which have only half the height. Thus the image contents gives the impression of being highly compressed. This effect is compensated for by spacing the lines of the resulting image until a blank line emerges between two adjacent lines. The grey levels of the pixels of these blank lines are generated by a linear interpolation of the grey levels of the pixels above and below. Another important result of the digital separation into fields is the seeming duplication of the sampling frequency of the recording system. Any events which are presented blurred within a frame can now be observed through two consecutive fields. Figs. 10 and 11 show the fields created by the odd and even lines as a result of the separation into fields indicated in Fig. 9. In order to eliminate reflections or extraneous light in the flow images, the separation into fields is followed by the subtraction of dark-field images.
4. Image processing In order to provide an optimal presentation of the information contents of the recorded flow images, various digital image-processing techniques are used. First of all, a digital separation of the frames into fields is carried out. This is required because the video camera operates with the interlaced field technique in accordance with the European CCIR standard. The frame which has been digitised by the frame grabber board consists of two consecutive fields with a time offset of Dt = 20 ms. While one field contains the image information of all even lines at a given time, the other field contains the information of the odd lines at the same time plus Dt = 20 ms. With respect to the example where a non-stationary flow against a cylinder is
Fig. 9. Raw flow image recorded by means of the field technique.
Fig. 10. Field created from odd lines.
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W. Baetz et al. / Measurement 38 (2005) 42–52
5. Experiments and simulation
Fig. 11. Field created from even lines.
The laser diode light sheet was applied to investigations of the internal flow characteristic of a gas pressure regulator and the flow detachment in a sonic Venturi nozzle. The recorded light sheet images are compared with numerical flow fields of a computational fluid dynamics (CFD) calculation. 5.1. Gas pressure regulator
For that purpose, a dark-field image is recorded with the camera being unchanged and the laser light switched off. By subtracting the dark-field image from the flow image, only the contents of the image which is caused by the influence of the laser light remains. Since the original image, in general, consists of only a restricted range of grey levels, finally a spread of the grey levels is carried out to fully utilise the existing pixel depth of 8 bits. For that purpose, the smallest grey level and the highest grey level of the existing grey levels are taken from the histogram. This restricted range is now homogeneously spread across the entire range of possible grey values. Due to the increased resolution of grey values, minor differences between grey levels are now better visible.
Primary, the laser diode light sheet was applied to the study of the internal flow characteristic of a gas pressure regulator. Fig. 12 shows the schematic and modified housing of a gas pressure regulator with a nominal diameter of 50 mm. To get optical access to the valve chamber several areas of the original regulator housing are milled away and replaced by acrylic plastic windows to maintain airtight integrity. The laser light sheet enters the valve chamber through a slotted window at the top of the housing and a CCD camera captures the flow images through a front window of the regulator. The tracer particles are fed to the valve through special tubing at the left hand side of the valve disk. A vacuum pump creates the gas
Fig. 12. Schematic (left) and modified housing (right) of a gas pressure regulator.
W. Baetz et al. / Measurement 38 (2005) 42–52
49
Fig. 13. Light sheet image of the valve chamber down-stream of the valve disk showing the vortices at the valve gap exit (top left corner) and the lower left corner of the valve chamber.
flow (air) in direction of the normal exit of the regulator. Fig. 13 represents a recorded image of the gas flow through the valve chamber at a mean velocity of approximately 2 m/s. The image was recorded at an exposure time of 1/250 s and digitally post-processed as described in Section 4. This image shows the valve chamber upside down compared to Fig. 12. It shows one big vortex in the centre of the left half of the valve chamber. Additionally two small vortices in the top left corner near the valve gap and in the lower left corner of the valve chamber can be identified. The smaller swirls are known as recirculation areas. These swirls in the recirculation area maintain this pattern and shape throughout the gas flow. It is important to note that while this is a two-dimensional picture, these recirculation areas have a tubular shape surrounding the valve if observed three-dimensionally. The low intensity at the right half of the valve chamber results from the greater flow output velocity into the normal exit of the regulator. The flow field inside the valve chamber was simulated by computational fluid dynamics (CFD) calculation for different flow rates. Fig. 14 shows the simulation result of the flow velocity distribution around the valve disk for a flow rate of Q = 622 m3/h and a valve gap of 8 mm. It shows the same structures of the flow field like the light sheet image in Fig. 13 for the lower flow rate of Q = 6 m3/h. The CFD simulation results show that the flow structure
Fig. 14. CFD calculation of the flow velocity.
doesnÕt change significantly when the flow rate increases. 5.2. Venturi nozzle Venturi nozzles have an inlet radius that directly turns from the nozzle throat into the outlet diffuser with a smooth contour. This type of nozzle is used, for example, to calibrate gas flow meters and volume-measuring devices [8]. For this purpose the Venturi nozzle is operating under critical conditions. That means, the pressure ratio across the Venturi nozzle is less or approximately equal to 0.7. At this ‘‘critical pressure ratio’’ the flow velocity reaches speed of sound in the nozzle throat. In this case the flow rate depends only on the cross-section of the nozzle throat. If it is ensured, that at least the critical pressure ratio is applied across the Venturi nozzle, the flow rate will be highly constant and can be reproduced with high accuracy. To get a better understanding of the internal flow characteristic of the Venturi nozzle the laser
50
W. Baetz et al. / Measurement 38 (2005) 42–52
diode light sheet unit is applied. For this purpose a planar Venturi nozzle model is constructed with a rectangular cross-section instead of a normally circular cross-section of the original Venturi tube. The nozzle contour is formed by acrylic plastic to allow both the laser light sheet and the camera good optical access to the internal region of the nozzle diffuser. To double the optical intensity the two laser diode modules are mounted above and below the diffuser to superimpose the light sheets in opposite directions (Fig. 15). A CCD camera is mounted perpendicular to the flow and the light sheet to capture the flow images. Fig. 16 shows at the top the captured laser light sheet image of the whole flow field from the nozzle inlet up to the exit of the diffuser. The right half of the upper part of the diffuser section is extracted from that picture and enlarged displayed below the original image. The enlarged view shows clearly the detachment of the flow from the wall, because of boundary layer separation. The vortices form and separate themselves from the wall in a rolling motion as they travel downstream to the exit of the diffuser. The nozzle flow was simulated by CFD calculation. Fig. 16 shows the resultant distribu-
Laser Diode Module 1 Acrylic Plastic Strips for Nozzle Contour
Light Sheet
Diffuser
Flow Input
Laser Diode Module 2
Fig. 15. Schematic of the planar Venturi nozzle model made of acrylic plastic with the laser diode modules mounted above and below to double the light sheet intensity within the observation region of the diffuser.
Fig. 16. Flow characteristics of a Venturi nozzle: (a) light sheet image of the flow field; (b) enlarged view of the upper right diffuser section; (c) CFD calculation of the static pressure distribution; (d) CFD calculation of the total energy density distribution with streamlines and (e) CFD calculation of the velocity distribution.
tions of the static pressure p, the total energy density E and the Mach number Ma in the flow field. The energy density diagram includes some streamlines illustrating the inhomogeneous structure. The vortex structure of the flow detachment is visible in the light sheet image as well as in the
W. Baetz et al. / Measurement 38 (2005) 42–52
pressure and velocity distribution. The light sheet image can not be directly interpreted as the velocity or the pressure distribution. It is a combination of pressure and velocity effects as well as the energy distribution. The Mach number and pressure distribution shows three shock waves with supersonic velocity and corresponding low static pressure. Simulation at different points of time show that these shock waves travel backwards into the nozzle neck and can modulate the velocity in the nozzle neck. This velocity modulation can lead to a modulation of the flow rate, which then is no more constant [9]. This does not affect the accuracy of the Venturi nozzle as a calibration device in test rigs. This effect acts at high frequencies and leads to a slightly smaller value of the mean flow rate. It is recommended to modify the diffuser outlet of the Venturi nozzle to minimize these effects [10].
6. Conclusion Flow images of gas flow fields inside a gas pressure regulator and a Venturi nozzle are recorded. The observed flow structures are in good agreement with the simulation calculations of computational fluid dynamics software. We have demonstrated that instead of highpower gas lasers, commercially available monomode laser diodes can be used to accomplish the laser light sheet technique. We used a simple and inexpensive laser light sheet unit, which allows a good optical access to the different flow fields under test. Due to its compact size no fibres or light arms are needed. The application examples show that the laser light sheet technique in conjunction with laser diodes is a useful tool for the development of gas measurement and controlling units. Despite of the low output power, the laser diode light sheet can even be used to visualise flow structures in high-speed gas flows as demonstrated by the experiments conducted with sonic nozzles. Increasing the packing density of the laser diodes will increase the optical output power without loss in the optical quality of the light sheet. The rapid progress of semiconductor laser technol-
51
ogy provides high-power laser diodes for the visible spectral range, but the optical beam quality is reduced. The light sheet thickness will be increased to approximately 1–2 mm due to the bigger size of the light source. These high-power light sheets will make it possible to carry out investigations of high-speed gas flows similar to the PIV (particle-image velocimetry) procedure.
Acknowledgments The authors thank Prof. Dr.-Ing. Eckard Beese, Fachbereich Maschinenbau, Stro¨mungslehre und Stro¨mungsmaschinen, Fachhochschule Bochum, and Prof. Dr.-Ing. Ernst von Lavante, Institute of Turbomachinery, University of Essen, Germany, who carried out the CFD calculation of the flow fields in the gas pressure regulator and the Venturi nozzle, respectively.
References [1] L. Araneo, A. Coghe, G.E. Cossali, Characterisation of a diesel spray head by different laser based techniques, in: Proceedings 9th International Symposium on Applications of Laser Techniques to Fluid Mechanics, June 13–16, Lisbon, Portugal, 1998, pp. 7.4.1-8. [2] B. Nath, H. Dietrich, W. Baetz, W. Holzapfel, Application of laser diodes to gas flow diagnosis, in: Proceedings of the International Workshop on Flow Diagnosis Techniques, St. Petersburg, Russia, 1998, pp. 157–164. [3] B. Nath, H. Dietrich, W. Baetz, W. Holzapfel, High-Speed Gas Flow Diagnosis with Semiconductor Lasers, Proceedings of IMEKO XV World Congress, June 13-18, 1999, Osaka, Japan, Vol. XI: Photonic Measurements, Optical Measurements and Laser Applications, pp. 25–32, ISBN 4907764-10-3. [4] R.K. Bunnell, B.J. Krass, I.K. Puri, An investigation of methane–air nonpremixed combustion in a rapid-mixing burner, Proceedings of the ASME Heat Transfer Division, HTD 335 (1993) 193–302. [5] P. Astrup, S. Clausen, Laser sheet visualisation of flames in experimental and power station furnaces, Measurement, Science and Technology (1995) 815–825. [6] B. Nath, W. Lo¨ber, W. Baetz, W. Holzapfel, Gas flow visualization by laser diode light sheet imaging, in: Proceedings of the 9th International Symposium on Applications of Laser Techniques to Fluid Mechanics, June 13–16, Lisbon, Portugal, 1998, pp. 6.2.1-7.
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[7] A.E. Siegman, Lasers, University Science Books, Sausalito, CA, 1986, pp. 663–679. [8] H. Dietrich, B. Nath, E.v. Lavante, High-Accuracy Test Rig for Gas Flows from 0.01 m3/h up to 25000 m3/h, IMEKO-XV, June 13–18, Osaka, Japan, 1999, Vol. 7, pp. 49–56. [9] E.v. Lavante, B. Nath, H. Dietrich, Effects of instabilities in small sonic nozzles, in: Proceedings of the 9th
International Symposium on Flow Measurement FLOMEKO Õ98, June 15–17, Lund, Sweden, 1998, pp. 221–226. [10] E.v. Lavante, A. Zachcial, D. Zeitz, B. Nath, H. Dietrich, Effects of various geometric parameters in sonic nozzles, in: Proceedings of the 10th International Symposium on Flow Measurement FLOMEKO 2000, June 4–8, Salvador, Brazil, paper F6.
Beam configuration for light sheet application W. Baetz a, W. Holzapfel
a,*
, H. Dietrich b, B. Nath
c
a
University of Kassel, Institute for Measurement and Automation, D-34109 Kassel, Germany b Elster Produktion GmbH, Steinernstrasse 19, D-55252 Mainz-Kastel, Germany c AKG Thermotechnik GmbH & Co. KG, Am Hohlen Weg 31, 34369 Hofgeismar, Germany Received 25 January 2003; accepted 21 February 2005 Available online 11 May 2005
Abstract To visualise laminar and turbulent flow patterns of gaseous or liquid fluids, the laser-light-sheet technique can be applied. The optically recorded flow images are useful for research and development of flow measuring and controlling devices. Instead of single high-power gas lasers or solid state lasers six monomode laser diodes emitting in the visible spectral range are applied. They are mounted in parallel and the individual radiation fields overlap to generate a more homogeneous intensity distribution with less coherence than with a single laser source. Therefore, disturbances due to speckle effects are reduced. The resulting light sheet has a total optical output power of 0.3 W and a thickness of 0.7 mm. The working distance amounts from 60 mm up to 160 mm with an operating area of 100 · 100 mm2. Application examples of the internal flow characteristics of a gas–pressure regulator and high-speed gas flow diagnosis in a sonic Venturi nozzle are presented and compared with computational fluid dynamics (CFD) simulations. 2005 Elsevier Ltd. All rights reserved. Keywords: Gas flow; Laser diode; Light sheet
1. Introduction The investigation of gaseous or liquid flow fields often requires only qualitative results about
*
Corresponding author. Tel.: +49 561 804 2758; fax: +49 561 804 2847. E-mail addresses: [email protected] (W. Baetz), [email protected] (W. Holzapfel), [email protected] (H. Dietrich), [email protected] (B. Nath).
the flow pattern to identify unwanted vortex patterns or dead zones in the flow field of interest. For this purpose, the laser light sheet technique (Fig. 1) is applied to a variety of different research fields like vortex pattern visualisation about aircraft models or the experimental investigation of the characteristics of diesel spray heads [1]. It can also be adapted to visualise gas flow pattern in gas pipes, regulators [2,3], furnaces [4,5] and volume-measuring devices [6]. The technique (Fig. 1) applies three basic procedures: (1) illuminating a
0263-2241/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2005.02.004
W. Baetz et al. / Measurement 38 (2005) 42–52
43
This contribution presents an inexpensive modular laser diode light sheet unit, generating a more homogeneous intensity profile by superposition of the nearly Gaussian radiation fields of six monomode laser diodes. Another advantage of this superposition is the lower coherence of the resultant radiation field, because the individual laser diodes are not frequency-stabilized and their mean wavelengths are different over several nanometres. We present the theoretical relations and the layout of the applied laser diode light sheet technique, as well as its application to gas flow diagnosis in a gas pressure regulator and high-speed gas flow in a sonic Venturi nozzle. Fig. 1. Typical experimental set-up for flow field visualisation with a laser light sheet.
2. Laser diode light sheet 2.1. Theory of light sheet generation To generate a thin light sheet with high luminous intensity, in general, gas or solid state lasers are employed. The emission aperture of these lasers is usually circular and they are operating in their fundamental transverse mode (TEM00) so that the output emission corresponds to a circular Gaussian beam. The Gaussian beam is uniquely determined by its wavelength k and its smallest beam radius w0 in the beam waist at z = 0 (Fig. 2).
3
Relative Beam Radius
w w0
2
r
w = w0 -1
zR
1
w= 2 w0 I
thin plane cross-section of the flow field with a light sheet, (2) seeding the flow with diffusely reflecting tracer particles and (3) recording the resulting images of the tracer streaks with a camera. Smoke, vapour or tiny oil droplets in air or bubbles in water can be used as tracer particles. To follow the real flow field without tracking errors, the tracer particles have to be very small and with low mass. For the observation of gas flow, the particle diameter is approximately 1–10 lm and the scattered optical power is very small. Therefore, in general a high-power laser like an argon-ion laser or an Nd:YAG laser must be used to provide high luminous intensities in the light sheet to visualise the gas flow. These laser technologies are very expensive, high power consuming and difficult to handle due to their geometrical size. Therefore, additional light-guiding devices like light arms or fibre optics are needed to illuminate the flow field region to be observed. Other disadvantages of these single laser sources are the inhomogeneous illumination by the Gaussian intensity distribution of the single laser beam and the great coherence length of the single laser radiation. The Gaussian intensity profile often results in an overexposure in the centre and an underexposure in the peripheral regions of the light sheet and the great coherence leads to an irregular illumination by the speckle effects.
1
-1 -2
2
3
Θ1/e
-3
Relative Beam Coordinate z/zR Fig. 2. Propagation characteristics of a circular Gaussian beam with the beam waist w0, the Rayleigh length zR and the divergence angle H1/e.
44
W. Baetz et al. / Measurement 38 (2005) 42–52
The intensity distribution I(r, z) of the radiation with the total optical power P in the direction of propagation z, and at the distance r from the beam axis (Fig. 3), is defined by [7]: " # 2P 2 r2 Iðr; zÞ ¼ exp ; ð1Þ wðzÞ2 p wðzÞ2 with the beam radius w(z) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 z ; w0 ¼ wð0Þ wðzÞ ¼ w0 1 þ zR
ð2Þ
and the Rayleigh length zR zR ¼
p w20 . k
ð3Þ
The beam radius w(z) is defined as the distance from the z-axis, where the electric field amplitude has fallen to the 1/e-fold of the maximum amplitude. Here the beam intensity has decreased to the 1/e2-fold of the maximum intensity I(0, z). The Rayleigh length zR characterizes one half of the collimated range on one side pffiffiffiof the beam waist, with the beam radius wðzÞ 6 2w0 . According to Eqs. (1)–(3), in the far field (z zR) the beam radius increases linearly with the distance z from the beam waist. The beam angle H1/e of the radiation cone for r = w(z) only depends on the wavelength k and the beam waist w0 wðzÞ H1=e ¼ lim 2 arctan z!1 z k ¼ 2 arctan . ð4Þ p w0
I(r) I(0)
0.8
Relative Intensity
1.0
0.6
-1.5
r/w = 0.5
0.4 0.2
ln 2 2
r/w=1
The emission aperture of the laser-active zone of a semiconductor laser is not circular, but instead rectangular in contrast to the gas or solid-state lasers. The emission aperture of index-guided monomode laser diodes is typically 3 lm wide and 1 lm high, with the long side of the emission aperture orientated parallel to the semiconductor junction. In the far field, perpendicular to the beam-propagation direction, the intensity distribution also corresponds in good approximation to a twodimensional Gaussian distribution. Because of the rectangular emission aperture, the Gaussian distribution, however, is no longer circular, but elliptic. Here the semiaxes of the elliptic distribution are orientated parallel (index p) and perpendicular (index r) to the semiconductor junction. Because of the very small emission aperture, the output radiation is strongly divergent due to diffraction effects. In the far field, the propagation is characterized by the divergence angles Hp and Hr. These FWHM-angles (FWHM: full width at half maximum) mark the full cone angles in the r- and p-planes of the semiconductor junction, within which the radiant intensity is greater, or equal to, one half—not 1/e2—of the maximum intensity I(0, z). The half-intensity beam radius w1/2 results from the 1/e2-intensity radius w according to Eq. (1) with I(r, z) = 0.5I(0, z) (Fig. 2) rffiffiffiffiffiffiffiffi ln 2 w. ð5Þ w1=2 ¼ 2 Assuming a Gaussian beam, the equivalent waist radii w0p and w0r, in the semiconductor laser resonator, can be calculated from the divergence angles (with Hp < Hr) by Eq. (4) rffiffiffiffiffiffiffiffi ln 2 k w0p ¼ ð6aÞ 2 p tan H2p and
2
1/e
w0r -1
-0.5 0.5 1 0 Relative Beam Radius r/w
1.5
Fig. 3. Relative intensity distribution of a Gaussian beam and definition of the beam radius.
rffiffiffiffiffiffiffiffi ln 2 k . ¼ 2 p tan H2r
ð6bÞ
Applying Eqs. (2) and (3) the elliptic intensity distribution I(x, y, z) of a semiconductor laser is now given by
W. Baetz et al. / Measurement 38 (2005) 42–52
Iðx; y; zÞ ¼
2P p wp ðzÞ wr ðzÞ ( " exp 2
!
x2
wp ðzÞ2
þ
y2 wr ðzÞ2
.
The image location x00 determines the centre of the working area of the light sheet and the Rayleigh length x0R is equal to the depth of focus where the thickness of the light sheet is nearly constant.
ð7Þ
2.2. Laser diode light sheet unit
!#)
Here, x and y are the Cartesian coordinates parallel and perpendicular, respectively, to the semiconductor junction and perpendicular to the propagation direction z. In order to achieve a light sheet as thin and extensive as possible from this divergent intensity distribution, it is necessary to collimate the radiation in the plane with the smaller divergence angle Hp. Then the angle Hr determines the spread of the light fan in the plane of the light sheet. That means that the very small beam waist w0p must be projected into a larger one with the radius w00p using an appropriate cylindrical lens (focus length f). Assuming a thin lens, the image of the Gaussian beam waist is given by (Fig. 4) [7] f k w0p w00p ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . k2 x20 þ p2 w40p
ð8Þ
The image is located at the distance x00 from the focus point F 0 in the image space f 2 x0 x20 þ z2R
x00 ¼
45
The realized light-sheet unit consists of two laser diode modules. Each module consists of three monomode laser diodes with an individual optical output power of 50 mW at a wavelength of k = 685 ± 5 nm. Thus, the total optical output power amounts to 0.3 W. Their FWHM divergence angles in the far field amounts to Hp = 9 and Hr = 22. The equivalent beam waist radii in the laser-active zone are with Eq. (6) w0p = 1.64 lm and w0r = 0.66 lm. To achieve an intensity distribution as even as possible at a higher total intensity the diodes are arranged in parallel with a spacing of 20 mm. Each laser diode is collimated with a cylindrical lens in the plane parallel to the semiconductor junction (Fig. 5). Positioning the beam waist at the focal point F (i.e. x0 = 0) of the cylindrical lens with f = 5 mm, the beam waist w0p of the laser diode is imaged into a waist with the radius w00p ¼ 0.6 mm located
ð9Þ
with the new Rayleigh length z0R ¼
p f 2 w20 k . k2 x20 þ p2 w40
ð10Þ
z r'
zR 2 w0
w0
•
w0'
2 w0' •
F'
F
x0
f
δ
x0'
f
δ'
Fig. 4. Image of the Gaussian beam waist with a thin lens.
Fig. 5. Design of the light sheet unit with six monomode laser diodes.
46
W. Baetz et al. / Measurement 38 (2005) 42–52
at the focal point F 0 (i.e. x00 = 0). With Eq. (10) the Rayleigh length in the image space is approximately z0R ¼ 2 m and is therefore very large compared with the focal length f of the cylindrical lens. Therefore the resultant light sheet has a thickness of approximately 0.7 mm (FWHM) within the collimated range of approximately 2 m. The orthogonal divergence angle Hr = 22 (FWHM) defines the fan angle of the individual light sheet. At a greater distance in the propagation direction the Gaussian intensity distributions
z = 60 mm Rel. Intensity
overlap providing a widely constant intensity (Fig. 6). Distributing the laser diodes into two modules (Fig. 7) at a total power of 0.3 W allows a flexible arrangement of the radiation sources so that intensity and dimensions of the light sheet can be adapted to the requirements of the individual measurement task. With the two modules arranged in parallel, the useable illuminated range of the light sheet is about 100 · 100 mm2 at an operating distance of 60–160 mm. The laser diode light sheet can be operated with the control unit in continuous-wave mode, or in pulse mode with a pulse duration of 100 ls to 999 s.
0.06
3. Test set-up
0.04
100
0.02
50
0 -50 0 x/mm
-50 50
0 y/mm
-100
Fig. 6. Relative intensity distribution of the laser diode light sheet at the distance of z = 60 mm.
A test set-up has been built to record flow images of different flow models by means of laser diode light sheet technique (Fig. 8). To conduct the test, a housing is constructed to simulate the environment from which the gas flow is to be studied. To insert the laser light sheet into the housing, two slits are cut into it from directly above and below. Next, a window is created on the front of the housing to give the camera its own optical access. These windows and slits are made with acrylic plastic and are sealed to ensure an airtight enclosure. For the first experiment, the Karman vortex street, a cylinder is placed in the housing at the far left of the cameraÕs inspection window. After the laser light sheet has been directed into this
Viewing Window Laser Light Sheet
Image Area of the Video Camera
Spray Tube
Aerosol Generator
Fig. 7. Laser module with three laser diodes.
Gas Flow Channel
to the Fan
Laser Module
Fig. 8. Schematic diagram of the test set-up.
W. Baetz et al. / Measurement 38 (2005) 42–52
housing, the gas flow is added. As the gas flows around the cylinder, the flow is disturbed. It is possible to record these disturbances to the flow because the camera is mounted perpendicular to the laser light sheet. To generate the gas flow, a vacuum pump and an aerosol generator are used. A constant volume flow through the housing is accomplished by using a Venturi nozzle in-line with the pump. The Venturi nozzle is then operated under critical conditions. To visualise the gas flow, an aerosol generator adds DEHS (diethyl sebacate) particles with the size of 0.2–1 lm to the flow. When these particles pass through the light sheet, they scatter the laser light and become visible. Within the light sheet, an image of the flow pattern appears that can be recorded by a CCD camera. The CCD camera, having a sensitivity of approx. 0.1 lux, is used to record the images. This highly sensitive video camera is necessary to capture the images because, unlike the high-output light intensity of gas lasers, the output intensity of the laser diodes is much lower. As the analogue signals of the camera are transmitted to the PC, a frame-grabber board digitises these signals allowing the PC to store them. Next, the PC is used to perform the digital post-processing of these signals into visible images.
47
recorded, this time offset results in a local offset in Fig. 9. The size of this local offset is dependent on the flow rate. By means of the digital separation of the interlaced frames into fields, any frame digitised by the frame grabber board is split into two fields which have only half the height. Thus the image contents gives the impression of being highly compressed. This effect is compensated for by spacing the lines of the resulting image until a blank line emerges between two adjacent lines. The grey levels of the pixels of these blank lines are generated by a linear interpolation of the grey levels of the pixels above and below. Another important result of the digital separation into fields is the seeming duplication of the sampling frequency of the recording system. Any events which are presented blurred within a frame can now be observed through two consecutive fields. Figs. 10 and 11 show the fields created by the odd and even lines as a result of the separation into fields indicated in Fig. 9. In order to eliminate reflections or extraneous light in the flow images, the separation into fields is followed by the subtraction of dark-field images.
4. Image processing In order to provide an optimal presentation of the information contents of the recorded flow images, various digital image-processing techniques are used. First of all, a digital separation of the frames into fields is carried out. This is required because the video camera operates with the interlaced field technique in accordance with the European CCIR standard. The frame which has been digitised by the frame grabber board consists of two consecutive fields with a time offset of Dt = 20 ms. While one field contains the image information of all even lines at a given time, the other field contains the information of the odd lines at the same time plus Dt = 20 ms. With respect to the example where a non-stationary flow against a cylinder is
Fig. 9. Raw flow image recorded by means of the field technique.
Fig. 10. Field created from odd lines.
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W. Baetz et al. / Measurement 38 (2005) 42–52
5. Experiments and simulation
Fig. 11. Field created from even lines.
The laser diode light sheet was applied to investigations of the internal flow characteristic of a gas pressure regulator and the flow detachment in a sonic Venturi nozzle. The recorded light sheet images are compared with numerical flow fields of a computational fluid dynamics (CFD) calculation. 5.1. Gas pressure regulator
For that purpose, a dark-field image is recorded with the camera being unchanged and the laser light switched off. By subtracting the dark-field image from the flow image, only the contents of the image which is caused by the influence of the laser light remains. Since the original image, in general, consists of only a restricted range of grey levels, finally a spread of the grey levels is carried out to fully utilise the existing pixel depth of 8 bits. For that purpose, the smallest grey level and the highest grey level of the existing grey levels are taken from the histogram. This restricted range is now homogeneously spread across the entire range of possible grey values. Due to the increased resolution of grey values, minor differences between grey levels are now better visible.
Primary, the laser diode light sheet was applied to the study of the internal flow characteristic of a gas pressure regulator. Fig. 12 shows the schematic and modified housing of a gas pressure regulator with a nominal diameter of 50 mm. To get optical access to the valve chamber several areas of the original regulator housing are milled away and replaced by acrylic plastic windows to maintain airtight integrity. The laser light sheet enters the valve chamber through a slotted window at the top of the housing and a CCD camera captures the flow images through a front window of the regulator. The tracer particles are fed to the valve through special tubing at the left hand side of the valve disk. A vacuum pump creates the gas
Fig. 12. Schematic (left) and modified housing (right) of a gas pressure regulator.
W. Baetz et al. / Measurement 38 (2005) 42–52
49
Fig. 13. Light sheet image of the valve chamber down-stream of the valve disk showing the vortices at the valve gap exit (top left corner) and the lower left corner of the valve chamber.
flow (air) in direction of the normal exit of the regulator. Fig. 13 represents a recorded image of the gas flow through the valve chamber at a mean velocity of approximately 2 m/s. The image was recorded at an exposure time of 1/250 s and digitally post-processed as described in Section 4. This image shows the valve chamber upside down compared to Fig. 12. It shows one big vortex in the centre of the left half of the valve chamber. Additionally two small vortices in the top left corner near the valve gap and in the lower left corner of the valve chamber can be identified. The smaller swirls are known as recirculation areas. These swirls in the recirculation area maintain this pattern and shape throughout the gas flow. It is important to note that while this is a two-dimensional picture, these recirculation areas have a tubular shape surrounding the valve if observed three-dimensionally. The low intensity at the right half of the valve chamber results from the greater flow output velocity into the normal exit of the regulator. The flow field inside the valve chamber was simulated by computational fluid dynamics (CFD) calculation for different flow rates. Fig. 14 shows the simulation result of the flow velocity distribution around the valve disk for a flow rate of Q = 622 m3/h and a valve gap of 8 mm. It shows the same structures of the flow field like the light sheet image in Fig. 13 for the lower flow rate of Q = 6 m3/h. The CFD simulation results show that the flow structure
Fig. 14. CFD calculation of the flow velocity.
doesnÕt change significantly when the flow rate increases. 5.2. Venturi nozzle Venturi nozzles have an inlet radius that directly turns from the nozzle throat into the outlet diffuser with a smooth contour. This type of nozzle is used, for example, to calibrate gas flow meters and volume-measuring devices [8]. For this purpose the Venturi nozzle is operating under critical conditions. That means, the pressure ratio across the Venturi nozzle is less or approximately equal to 0.7. At this ‘‘critical pressure ratio’’ the flow velocity reaches speed of sound in the nozzle throat. In this case the flow rate depends only on the cross-section of the nozzle throat. If it is ensured, that at least the critical pressure ratio is applied across the Venturi nozzle, the flow rate will be highly constant and can be reproduced with high accuracy. To get a better understanding of the internal flow characteristic of the Venturi nozzle the laser
50
W. Baetz et al. / Measurement 38 (2005) 42–52
diode light sheet unit is applied. For this purpose a planar Venturi nozzle model is constructed with a rectangular cross-section instead of a normally circular cross-section of the original Venturi tube. The nozzle contour is formed by acrylic plastic to allow both the laser light sheet and the camera good optical access to the internal region of the nozzle diffuser. To double the optical intensity the two laser diode modules are mounted above and below the diffuser to superimpose the light sheets in opposite directions (Fig. 15). A CCD camera is mounted perpendicular to the flow and the light sheet to capture the flow images. Fig. 16 shows at the top the captured laser light sheet image of the whole flow field from the nozzle inlet up to the exit of the diffuser. The right half of the upper part of the diffuser section is extracted from that picture and enlarged displayed below the original image. The enlarged view shows clearly the detachment of the flow from the wall, because of boundary layer separation. The vortices form and separate themselves from the wall in a rolling motion as they travel downstream to the exit of the diffuser. The nozzle flow was simulated by CFD calculation. Fig. 16 shows the resultant distribu-
Laser Diode Module 1 Acrylic Plastic Strips for Nozzle Contour
Light Sheet
Diffuser
Flow Input
Laser Diode Module 2
Fig. 15. Schematic of the planar Venturi nozzle model made of acrylic plastic with the laser diode modules mounted above and below to double the light sheet intensity within the observation region of the diffuser.
Fig. 16. Flow characteristics of a Venturi nozzle: (a) light sheet image of the flow field; (b) enlarged view of the upper right diffuser section; (c) CFD calculation of the static pressure distribution; (d) CFD calculation of the total energy density distribution with streamlines and (e) CFD calculation of the velocity distribution.
tions of the static pressure p, the total energy density E and the Mach number Ma in the flow field. The energy density diagram includes some streamlines illustrating the inhomogeneous structure. The vortex structure of the flow detachment is visible in the light sheet image as well as in the
W. Baetz et al. / Measurement 38 (2005) 42–52
pressure and velocity distribution. The light sheet image can not be directly interpreted as the velocity or the pressure distribution. It is a combination of pressure and velocity effects as well as the energy distribution. The Mach number and pressure distribution shows three shock waves with supersonic velocity and corresponding low static pressure. Simulation at different points of time show that these shock waves travel backwards into the nozzle neck and can modulate the velocity in the nozzle neck. This velocity modulation can lead to a modulation of the flow rate, which then is no more constant [9]. This does not affect the accuracy of the Venturi nozzle as a calibration device in test rigs. This effect acts at high frequencies and leads to a slightly smaller value of the mean flow rate. It is recommended to modify the diffuser outlet of the Venturi nozzle to minimize these effects [10].
6. Conclusion Flow images of gas flow fields inside a gas pressure regulator and a Venturi nozzle are recorded. The observed flow structures are in good agreement with the simulation calculations of computational fluid dynamics software. We have demonstrated that instead of highpower gas lasers, commercially available monomode laser diodes can be used to accomplish the laser light sheet technique. We used a simple and inexpensive laser light sheet unit, which allows a good optical access to the different flow fields under test. Due to its compact size no fibres or light arms are needed. The application examples show that the laser light sheet technique in conjunction with laser diodes is a useful tool for the development of gas measurement and controlling units. Despite of the low output power, the laser diode light sheet can even be used to visualise flow structures in high-speed gas flows as demonstrated by the experiments conducted with sonic nozzles. Increasing the packing density of the laser diodes will increase the optical output power without loss in the optical quality of the light sheet. The rapid progress of semiconductor laser technol-
51
ogy provides high-power laser diodes for the visible spectral range, but the optical beam quality is reduced. The light sheet thickness will be increased to approximately 1–2 mm due to the bigger size of the light source. These high-power light sheets will make it possible to carry out investigations of high-speed gas flows similar to the PIV (particle-image velocimetry) procedure.
Acknowledgments The authors thank Prof. Dr.-Ing. Eckard Beese, Fachbereich Maschinenbau, Stro¨mungslehre und Stro¨mungsmaschinen, Fachhochschule Bochum, and Prof. Dr.-Ing. Ernst von Lavante, Institute of Turbomachinery, University of Essen, Germany, who carried out the CFD calculation of the flow fields in the gas pressure regulator and the Venturi nozzle, respectively.
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[7] A.E. Siegman, Lasers, University Science Books, Sausalito, CA, 1986, pp. 663–679. [8] H. Dietrich, B. Nath, E.v. Lavante, High-Accuracy Test Rig for Gas Flows from 0.01 m3/h up to 25000 m3/h, IMEKO-XV, June 13–18, Osaka, Japan, 1999, Vol. 7, pp. 49–56. [9] E.v. Lavante, B. Nath, H. Dietrich, Effects of instabilities in small sonic nozzles, in: Proceedings of the 9th
International Symposium on Flow Measurement FLOMEKO Õ98, June 15–17, Lund, Sweden, 1998, pp. 221–226. [10] E.v. Lavante, A. Zachcial, D. Zeitz, B. Nath, H. Dietrich, Effects of various geometric parameters in sonic nozzles, in: Proceedings of the 10th International Symposium on Flow Measurement FLOMEKO 2000, June 4–8, Salvador, Brazil, paper F6.