An optical measurement of vortex shape at a free surface

Optics & Laser Technology 34 (2002) 107 – 113

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An optical measurement of vortex shape at a free surface Qi-Can Zhang ∗ , Xian-Yu Su Opto-Electronics Department, Sichuan University, Chengdu 610064, China Received 25 November 2000; accepted 17 October 2001

Abstract We have proposed an optical method of vortex shape measurement based on Fourier transform pro5lometry (FTP) and veri5ed it by experiment. The results of our experiment proposed in this paper show that FTP can e7ciently reconstruct the vortex shape at a free surface and this method is suitable for wide use in studying such problems as liquid shear 9ow, wake of an object, 9ow behind a blu< body, and wetting angle. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Fourier transform pro5lometry; Vortex shape measurement; Dynamic liquid surface

1. Introduction

2. Fundamental concept

Understanding the characteristics of vortex shedding is very important in many industrial problems and its physical aspects are important to improve 9ow control methods. Much pioneering works has been undertaken to experimentally analyse or numerically simulate the deformation of a liquid surface; some optical methods among them are based on the refraction and di
A Ronchi grating or sinusoidal grating is projected onto a vortex surface. A sequence of dynamic deformed fringe images can be grabbed by CCD camera and rapidly saved on disk. Firstly, by using Fourier transform, we obtain their spectra, which are isolated in the Fourier plane when the sampling theorem is satis5ed. Secondly, by adopting a suitable bandpass 5ltering (for example, a 2-D Hanning window) in the spatial frequency domain, all frequency components are eliminated except the fundamental component. Hence by calculating the inverse Fourier transform of fundamental component, a sequence of phase-maps can be obtained. Thirdly, by applying a phase unwrapping algorithm in 3-D phase space, we can obtain continuous phase distribution and the shape of the vortex at di
∗

Corresponding author. E-mail addresses: [email protected] (Q-C. Zhang), xianyusu@ mail.sc.cninfo.net (X-Y. Su).

0030-3992/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 0 1 ) 0 0 0 9 7 - 4

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by CCD at di
(3)

The same operations are applied to the fringe pattern on the reference plane to obtain gˆ0 (x; y) = A1 r0 (x; y) exp{i[2f0 x + 0 (x; y)]}:

(4)

Noting that PEp HEc ∼ PCHD in Fig. 1, we can write CD =

In order to eliminate the mirror re9ection of liquid surface as much as possible, we add two polaroids (p1 ; p2 in Fig. 1) in front of the projector and camera systems, respectively. Projecting a Ronchi grating fringe onto the reference plane, the grating image (with period p0 ) on the reference plane observed through a CCD can be expressed by +∞


(1)

where r0 (x; y) is a nonuniform distribution of re9ectivity on the reference plane, An is the weighting factors of Fourier series and f0 (=1=p0 ) is the fundamental frequency of the observed grating image. 0 (x; y) is the original phase on the reference plane R, (i.e. h(x; y) = 0).The coordinate axis is chosen as in Fig. 1. Let us consider a dynamic 3-D object placed in the optical 5eld giving a sequence of deformed fringe patterns grabbed by CCD camera and stored rapidly on disk. The intensity distributions of these fringe patterns in di
P(x; y; t) = (x; y; t) − 0 (x; y) = 2f0 (BD − BC) = 2f0 CD:

(6)

Substituting Eq. (5) into Eq. (6) and solving it for h(x; y; t), we obtain the formula for height distribution h(x; y; t) =

l0 P(x; y; t) l0 P(x; y; t) ≈− : P(x; y; t) − 2f0 d 2f0 d

(7)

An r0 (x; y) exp{i[2nf0 x + n0 (x; y)]};

n=−∞

g(x; y; t) =

(5)

The phase shift resulting from the object height distribution is

Fig. 1. Optical geometry of a FTP system.

g0 (x; y) =

−dh(x; y; t) : [l0 − h(x; y; t)]

An r(x; y; t) exp{i[2nf0 x + n(x; y; t)]}

n=−∞

(t = 1; 2; : : : ; s);

(2)

where r(x; y; t) and (x; y; t) represent a nonuniform distribution of re9ectivity on the object surface and phase modulation resulting from the object height variation at di
3. Experimental Accuracy Several factors, such as dynamic and frequency ranges of phase signal, back-ground luminance and re9ection coe7cient of the measured object, grating period, 5lter pass band, and time drift of image data, may exert an in9uence on the phase measurement accuracy of FTP. The measurement accuracy is mainly related to the 5ltering and the noise. If the spectrum is 5ltered with a suitable 5lter window, the high frequency will be restrained, but at the same time the details of the surface of the object will also be smoothed. Usually, we use the equivalent wavelength to estimate the measurement accuracy, and the maximum measurement accuracy of FTP is e =30. An equivalent wavelength of the projected grating pattern at the investigated surface could be de5ned as e = p0 =tan 

(8)

where p0 is the mean fringe pattern period at the reference plane, and tan  = d=l0 . Increase of e , for example by increasing the period (p0 ) or=and decreasing  leads to slowing down of the fringe phase variation, or=and decrease in the in9uence of the local shadows. In this way the phase unwrapping reliability will be enhanced, but this is an “empty” improvement because at the same time, it inevitably leads to lower measurement accuracy.

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4. Phase Unwrapping

The 5nite sum calculated from Eq. (14) to Eq. (16) is expressed as t
Pn (x; y) = (x; y; t) − 0 (x; y): (17)

4.1. 2-D Phase unwrapping There are two algorithmic methods, which may be used to obtain the phase P(x; y; t) = (x; y; t) − 0 (x; y) shift resulting from variation in height over a di
= arctg

(10)

where Im and Re represent the imaginary part and real part of g(x; ˆ y; t)gˆ∗0 (x; y), respectively. By this method, we can obtain a sequence of phase distributions contained in each deformed fringe, which includes the 9uctuation information of a dynamic object. (2) Phase di2erence algorithm. By calculating the product of the gˆ∗0 (x; y) with g(x; ˆ y; 1) and that of gˆ ∗ (x; y; t − 1) with g(x; ˆ y; t) after an increment in the time, we obtain g(x; ˆ y; 1)gˆ∗0 (x; y) =|A1 |2 r0 (x; y; 1)r(x; y; 1) exp[iP1 (x; y)];

(11)

g(x; ˆ y; 2)gˆ ∗ (x; y; 1) .. .

=|A1 |2 r(x; y; 1)r(x; y; 2) exp[iP2 (x; y)]

(12)

g(x; ˆ y; t)gˆ ∗ (x; y; t − 1) = |A1 |2 r(x; y; t − 1)r(x; y; t) exp[iPt (x; y)];

(13)

where P1 (x; y) = (x; y; 1) − 0 (x; y) = arctg

Im[g(x; ˆ y; 1)gˆ∗0 (x; y)] ; Re[g(x; ˆ y; 1)gˆ∗0 (x; y)]

(14)

P2 (x; y) = (x; y; 2) − (x; y; 1) .. .

= arctg

Im[g(x; ˆ y; 2)gˆ ∗ (x; y; 1)] Re[g(x; ˆ y; 2)gˆ ∗ (x; y; 1)]

(15)

Pt (x; y) = (x; y; t) − (x; y; t − 1) = arctg

Im[g(x; ˆ y; t)gˆ ∗ (x; y; t − 1)] : Re[g(x; ˆ y; t)gˆ ∗ (x; y; t − 1)]

109

(16)

n=1

Comparing Eq. (17) with Eq. (10), we obtain t
P(x; y; t) = Pn (x; y):

(18)

n=1

In practice, the phase di
where uwK stands for the unwrapped phase map of t = k, and Pi (x; y) represents the phase di
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Fig. 2. (a) Temporal phase unwrapping scheme at a discontinuity point. (b) Scheme of 3-D direct phase unwrapping. (c) Scheme of 3-D phase unwrapping based on phase di
this condition is satis5ed. So precise phase values over the whole 3-D phase 5eld can be obtained by calculating the sum along the t direction. 5. Experiment and results A schematic diagram of the experimental set-up used for the purpose of veri5cation of the current method is given in Fig. 3. The projecting image is focused to generate a Ronchi optical 5eld onto the poster paint surface from a Ronchi grating of 3 lines=mm. An electromagnetic stirrer revolves the poster paint. The deformed grating image is

observed by a low-distortion TV CCD camera (TM560) with a 12 mm focal length lens, via a special video-frame grabber digitizing the image in 320 × 240 picture elements, and scaled down to 128 × 128 pixels. The paint is located at a distance l0 = 890 mm from the projection system exit pupil, and d = 290 mm. In practice, we captured a fringe pattern as the reference plane while the paint is still. Then, we started to capture the deformed fringe images at the instant of commencement of stirring. From quiescence to stable rotation, 205 frame images of the paint were obtained over an interval of 10:4 s and one of them is shown in Fig. 4. All data are rapidly saved on disk.

Q-C. Zhang, X-Y. Su / Optics & Laser Technology 34 (2002) 107–113

Fig. 3. Schematic diagram of the experimental set-up.

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Fig. 5. The pro5le chart of reconstructed vortices (The mark beside each curve is time.)

6. Discussion and conclusions In this paper, we have described and demonstrated a method based on FTP for vortex shape measurement. It has the obvious advantage of real-time data acquisition and measurement of surface shape change for the vortex. In the 5eld of measurement of dynamic object, Chuncai Wu has measured the variation of human chest at inspiration and expiration [13,20]. The process of data collection costs about 4 s. The time interval is 0:2 s. The result of the measurement of the breathing chest is wonderful and it took a short time than PMP, which needs 20 s to do it at least. In the liquid research 5eld, there is no one who used this method up to now. The results of the basic theoretical analysis and of the experiment indicate that: Fig. 4. The liquid vortex shape for measurement (The area inside the dashed frame is to be reconstructed.)

If the re9ected light of the liquid surface is so strong as to submerge and interrupt the fringe, the phases behind the re9ected point will be unreliable. We adjusted the polarization direction of p2 to eliminate as much as possible both the mirror re9ection of the liquid surface and the re9ection of the curved surface of the vortex while the paint is undergoing a dynamic change. While the former can be eliminated completely, the latter cannot, but nevertheless, it has proved possible to obtain a satisfactory result. Fig. 5 shows the pro5les of the reconstructed vortices when t =0:05; 2:84; 3:25; 3:65; 4:06; 4:87 s (their corresponding frame numbers are 1, 56, 64, 72, 80, 96). Fig. 6 gives the grid charts for the corresponding frames in Fig. 5 and illuminate the formation and evolution of the vortex clearly. The inverse shape (−h) is given for convenience of observation.

1. Our method can e7ciently deal with vortex shape measurement. It is not only able to reconstruct the vortex, but also can provide some characteristic parameters for the generation of the vortex. With the development of a higher resolution CCD camera and a higher frame rate frame grabber, the method, as proposed here, should be a promising one in studying such 9uid problems as liquid shear 9ows, wakes of object, 9ows behind a blu< body, wetting angle and surface tension. 2. Since FTP is based on 5ltering for selecting only a single spectrum of the fundamental frequency component, the carrier frequency f0 must separate this spectrum from all other spectra. This condition limits the maximum range measurable by FTP, to:    @h(x; y)  L0   (20)  @x  ¡ 3d ; max where |@h=@x|max denotes the maximum absolute value which is a large value of |@h=@x|max and |@h=@x|min . Eq. (20)

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Fig. 6. The reconstructed shape of vortices (drawing inverse shape (−h) for convenient observing).

states that the maximum range of measurement is limited by the derivative of h(x; y) in the direction normal to the line of the grating. The maximum range can be extended by employing a geometry large enough to prevent the phase from being over-modulated. 3. The optical axis of the projector system and that of the camera system must be in the same plane in traditional FTP. If we persist in following this rule in liquid measurement, the CCD will obtain so much mirror re9ection that results in false phase e
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