## Abstract

The Karman vortex street generated behind a circular cylinder in water flow field is displayed and analyzed in real time by means of digital holography. Using a modified Mach-Zehnder interferometer, a digital hologram of the flow field in still state and then a video of continuous digital holograms in flowing state are recorded at 14.6 frames per second by a CCD camera, respectively. A series of sequential phase maps of the flow field are numerically reconstructed from the holograms in different states above based on double-exposure holographic interferometry. By seriating these phase maps, the shape and evolution of Karman vortex street can be displayed in real time in the form of a movie. For comparison, numerical simulation of the Karman vortex street under the boundary conditions adopted in the experiment is also presented, and the consistent results indicate that the experimental observation of Karman vortex street by using digital holography is successful and feasible.

© 2009 OSA

## 1. Introduction

In fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic property changes and flow visualization is a very important method in research of the flow field distribution. Optical methods, such as schlieren technique [1] and shadowgraph technique [2], are traditional means to display the dynamic behavior and flow pattern with the nondestructive advantage. However, the schlieren and shadowgraph technique are limited by the available test section size, reduced sensitivity and difficulty in obtaining quantitative results, which restricts their practical application to some extent.

Besides these, there are several velocity methods widely applied in practice, such as laser induced fluorescence, particle image velocimetry, laser-Doppler velocimetry [3, 4] and so on. In these methods, three-dimensional (3D) measurement of velocity fields is achieved by tracking the spatial and temporal data of the particle position and velocity. A common drawback of these methods, however, is that the so-called velocity field is actually the particles velocity and the particles suspended in the fluid maybe do not follow the fluid. Furthermore, these particles could also disturb the turbulence and lead to inaccurate results.

In this paper, we present a digital holographic method to measure the flow field. In recent years, with the advantage of fast, non-contact, nondestructive and full-field measurement [5,6], digital holography has been widely applied in deformation analysis, particles measurement, vibration analysis, three-dimensional image formation, microscopy, refractive index measurement, information encryption, etc [7–19]. Different from traditional optical holography, in digital holography the hologram is recorded by CCD or CMOS and the holographic image is numerically reconstructed by computer, form which we can conveniently obtain the quantitative amplitude and phase information of the object wave. However, due to the aberration of the recording system and the algorithm limitation in the numerical reconstruction, phase information of the reconstructed object wave is often corrupted and unusable to recover shape information.

In the previous studies the flow measurement based on digital holography merely focused on the air flow and some simple liquid flows, such as laminar thermal convection [20–23]. In the present work we extend the study further on a more complex liquid flow, Karman vortex street formed behind a circular cylinder in a water channel, by using digital holography. By recording a series of the holograms of the liquid flow field in steadily flowing state continuously using CCD and numerically reconstructing the holographic images, we can obtain the full-field distribution of the Karman vortex street and its evolution in the form of a phase map video. The experimental results are well consistent with that of the numerical simulation under the experimental boundary conditions.

## 2. Principles

Karman vortex street is one of the important hydrokinetic phenomena with repeating patterns of swirling eddy caused by the unsteady separation over bluff bodies. For incompressible liquid, the Karman vortex street can be theoretically explained with Reynolds-Averaged Navier-Stokes equations described as follows

*u*

_{i}is the Reynolds-Averaged velocity component,

*p*is the pressure, and

*ρ*and

*μ*are the density and viscosity of the liquid, respectively.

To solve Eq. (1), additional equations are required. In the practical applications, the Spalart-Allmaras turbulence model (S-A model) [24] is employed for better accuracy and computing efficiency. Based on Reynolds-Averaged Navier-Stokes equations and S-A model, the Karman vortex street in different situations can be determined exclusively.

In this paper we adopt digital holography to observe and measure the Karman vortex street described above. Considering a plane reference wave ** R**(

*x, y*) and an object wave

**(**

*O**x*,

*y*), which interfere on the plane of the recording medium, e.g. a CCD, the intensity of the interferogram will be given by

**”*denotes the complex conjugate operation, and

*x*,

*y*are the rectangular coordinates on the CCD target plane.

Assuming the reconstructed object waves in two different flowing states are *O*
_{1}(*x*, *y*) and *O*
_{2}(*x*, *y*), we can separately record two digital holograms *H*
_{1} and *H*
_{2} for the two different flowing states. Based on the double-exposure method, the phase change Δ*ϕ*(*x*, *y*) between *O*
_{1}(*x*, *y*) and *O*
_{2}(*x*, *y*) is given by

As we know, a refractive index change in the flow field leads to a change of the optical path length Δ*l*(*x*, *y*) and thereby to an interference phase change between two light waves passing flow field before and after the change. The phase change Δ*ϕ*(*x*, *y*) due to the index change is given by

*λ*is the wavelength,

*L*is the geometrical path length,

*n*

_{0}is the initial refractive index of the liquid under unperturbed state and

*n*(

*x, y*,

*z*) is the final refractive index distribution. The light beam passes the flow field in

*z*direction and the integration is taken along the propagation direction. For a 2D phase object without index change in the

*z*direction, the phase map representing the phase change directly reflects the variation of refractive index of the liquid. So the flow field can be displayed and analyzed in the form of phase maps.

## 3. Experiment setup

Figure 1
shows the experimental setup for recording the digital hologram of the flow field. A thin beam from a He-Ne laser with *λ*=632.8nm is expanded by microscope objective MO and then collimated by lens L_{1}. The pinhole between MO and L_{1} is used as a spatial filter. Then the plane wave is divided into two parts by beam splitter BS_{1}. The transmitted beam is reflected by a pair of mirrors M_{1}, M_{2} for exalting beam height. The exalted beam illuminates the sample by transmitting along the vertical direction to form the object beam, which is collected by telecentric lens set TL attached to CCD target, performing a minified image of the sample on the CCD target plane. The beam reflected by BS_{1} is transmitted by beam splitter BS_{2} and TL to form the reference beam. The mirrors M_{6}, M_{7} and M_{8} are located to equalize the optical paths of the reference and object beams. Both beams interfere with each other on the CCD target plane. To avoid the twin images problem and to eliminate the zero order diffraction, off-axis holograms are recorded [25]. The CCD is a white-black type with 1626×1326 pixels and pixel size of 4μm×4μm. The magnification of TL is 1/6.25 and the distance between TL and the flow field is 150mm. Besides the capacity of imaging a large field of view, owing to the constant transverse magnification in a certain variable range of the object distance, the TL can image clearly without distortion.

The geometry and the relevant dimensions of the channel sample are schematically shown in Fig. 2 . The specially designed rectangular channel with cross section of 5cm×7cm and length of 1m is the main part of the module. The channel is made of uniformly transparent organic glass, which allows vertical incident beam with diameter of 4cm propagating through the measured section. U-type design at the end of the channel ensures a steady flow velocity profile in the measured section. The channel is connected to a large tank with constant water level which is much higher than the channel. The water flow is driven by pressure difference, and the flow velocity is adjustable by the valve on the tank. A smooth metal circular cylinder with diameter of 7mm was vertically inserted between the top and bottom walls of the channel in the measured section for generating Karman vortex street.

The measured section in *xy* plane shown in the inset of Fig. 2 is the object plane to be recorded by CCD. The flow velocity in the measured section is kept approximately at 2cm/s. Due to the viscidity of the channel wall, the flow velocity of the water close to the right-and-left walls of the channel is slower. Thus equal distance between the circular cylinder and the right-and-left walls is required to avoid asymmetrical distortion of the Karman vortex street. In addition, the circular cylinder should be perpendicular to the top and bottom walls of the channel to make sure that the flow field is consistent along with the *z* direction.

## 4. Experimental results and discussions

Firstly when the water in the channel is in still state, we record a hologram of the measured section. Then we keep the flow field in steady state with a velocity of 2cm/s and continuously record a series of the holograms as uncompressed digital video at 14.6 frames per second. Figure 3(a) and (b) show the holograms in still state and steady flow state, respectively. Figure 3(c) shows the spatial spectra of the hologram shown in Fig. 3(a), where the rectangle region represents a filter window used for filtering operation.

Based on double-exposure holographic interferometry, each individual hologram is separately numerically reconstructed first, and the calculating results can be obtained according to Eq. (4) which is a series of sequential phase maps of the object wave in steady flow state relative to that in still state. After the hologram in still state is recorded, the computer can start the calculation in real time along with the experimental process of recording holograms in flow state. By seriating these phase maps, the shape and evolution of the Karman vortex street can be displayed in the form of a movie (Media 1) at 14.6 frames per second. By comparing these phase maps and calculating the recording time, we obtain that the same distribution of the Karman vortex street reoccurs in every 1.8s. Figure 4 shows a sequence of the reconstructed two-dimensional wrapped phase distributions, where each picture separately present a frame of the phase distributions in one period. Since we mainly concern the flow field behind the cylinder rather than the cylinder itself, and the recorded field of view is limited by the size of CCD, the cylinder is posited out of the field of view but just near to the middle of the left edge. It is also clearly shown that the periodic shedding of the vortices occurs first from one side of the cylinder and then from the other, and the alternating vortices present a double row of vortices. This evolution characteristic conforms to the natural evolution law of the flow around a circular cylinder. Moreover, from Fig. 4 we can also see that in the region close to the channel wall, the refractive index of the water presents an evident gradient and random fluctuation. This can be explained by the pressure interaction between the wall and the vortices.

Most flows in the nature contain numerous vortices with different sizes and periodicities. These vortices sometimes induce serious vibration and drag problems when designing a wide range of structures, from submarine periscopes [26] to industrial chimneys [27]. Because the turbulent Karman vortex street past circular cylinders involves most of the characteristic features of technical applications, it is an ideal test case for basically analyzing complex turbulent flow.

To verify the experimental results obtained by digital holography, the flow field is numerically simulated with the computational fluid dynamics code FLUENT by solving the equations discussed in Sec. 2. The boundary conditions used in the simulation process are the same as in the experiment. Figure 5 shows the numerical simulation result of the Karman vortex streets (Media 2), where the gray level represents the contour of the vorticity magnitude. Comparing the rectangle region shown in Fig. 5 with Fig. 4(b), the corresponding vortices have the uniform size, the shape and the same distance from the cylinder. In addition, the period of the Karman vortex street by numerical calculation is also 1.8s, which is in agreement with the period measured by digital holography.

In general, the numerical results agreed fairly well with the experimental data, especially in the near wake. However, the simplified numerical model, such as the coarse resolution in the far wake, will result in large deviations, whereas the digital holography in the far wake remains applicable.

## 5. Conclusions

The Karman vortex street generated behind a circular cylinder in flowing water and its dynamic evolution are displayed and analyzed in real time using digital holography. The numerical simulation results under the experimental boundary conditions with the computational fluid dynamics code FLUENT conform to the experiment results. It is shown that the real-time visualization idea of a complex liquid turbulent flow by digital holography is feasible. It is noticed that the experimental setup can be also extended to other complex fluid flow fields distributed in two dimensions. Future experiments will include the liquid flow field in microflow.

## Acknowledgements

This work is supported by the Science Foundation of Aeronautics of China under Grants No 2006ZD53042. We are grateful to Mr. Peng Song from Xi’an Jiaotong University for the numerical simulation of Karman vortex street.

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