Physical and Technical Background

Transcription

1 Physical and Technical Background Last Class: 1. Historical Background 2. Introduction of Particle Image Velocimetry 3. Principle of PIV 4. Major Technologies and Milestones 5. Applications Today s Contents: 1. Streamlines, Streaklines, and Pathlines 2. Tracer Particles 3. Particle Scattering Theories 4. Light Sources 5. Imaging of Small Particles 6. Cameras for PIV 1

2 Flow Visualization Three common concepts are used in visualizing fluid flow : streamlines, streaklines, and pathlines. A Streamline 2

3 Streamlines 3

4 Streamlines (Cont.) 4

5 Streaklines A Streakline 5

6 Pathlines A Pathline 6

7 Pathlines (Cont.) 7

8 Lagrangian and Eulerian Perspectives There are two common ways to study a moving fluid: Look at a particular location and observe how all the fluid passing that location behaves. This is called the Eulerian point of view or control volume point of view. Look at a particular piece of fluid and observe how it behaves as it moves from location location to location. This is called the Lagrangian point of view or system point of view. 8

9 Lagrangian Derivative 9

10 Lagrangian Derivative (Cont.) 10

11 Tracer Particles PIV is a more direct measure of the velocity of a fluid because it depends directly on determining how far something travels during what time period. Thus it is a more direct determination of velocity than pitot probes (measure pressure) and hot wire probes (measure heat loss). However, it is indirect insofar as it is a measure of the velocity of the tracer particles and not the fluid itself. Consequently, the particles following the flow is of paramount importance. 11

12 Inertia Let the particle-to fluid density ratio be ρ = ρ p ρf where H(ω) is frequency response function of the particle. The energy transfer function at for arbitrary e with small Re or asymptotically large e with finite Re is where ε is the Stokes number. i) For a neutrally buoyant particle, ρ = 1, and H(ε) 2, which implies perfect response of the tracer particle. ii) For ρ < 1, H(ε) 2 > 1 so that buoyant particles tend to over-respond. iii) Fo r ρ>>l, H(ε) 2 -> 18ρ -2 ε -4 /16 at intermediate ε. This implies a low-pass filtering behavior of heavy particles; at large ε, H(ε) 2 -> 18ρ -2 /4 which implies practically negligible frequency response. 12

13 Sedimentation The analytical solution for the creeping flow (Re<<1) around a sphere was first given by Stokes in From Stokes' solution, the drag force acting on the sphere can be obtained as g u z u t u F D = 3u t πμd p d p Unlike the macro-scale, in the mciro-scale, the shape is not very determinant. 13

14 Brownian Motion 1D Einstein relation : s 2 = 2D t The error for a single particle: Brownian motion is the foundation of diffusion. Fick's second law predicts how diffusion causes the concentration to change with time: 14

15 Boundary Effect Stokes Flow around a particle U U r Boundary Layer Near wall u r = 1 r u θ = r ψ = UUUUUθ 1 3 R r u θ, u r 1 r R r It implies wakes primarily decay as r Wakes are felt very far from a particle if at least 10 radii from boundaries or other particles. 15

16 Mie Scattering by Particles Side scattering Forward scattering Back scattering 16

17 Mie Scattering by Particles It can be seen from all Mie scattering diagrams, that the light is not blocked by the small particles but spread in all directions. Therefore, for a large number of particles inside the light sheet massive multiscattering occurs. Then the light which is imaged by the recording lens is not only due to direct illumination but also due to fractions of light, which have been scattered by more than one particle. In the case of heavily seeded flows this considerably increases the intensity of individual particle images, because the intensity of directly recorded light at 90 to the incident illumination is orders of magnitude smaller than that scattered in the forward scatter range. 17

18 Mie Scattering by Particles 18

19 Rayleigh Scattering (withfriendship.com/user/mithunss/rayleigh-scattering.php) 19

20 Particle Generation and Supply Requirements: In the Rayleigh scattering regime, where the particle diameter dp is much smaller than the wavelength of light, dp <<λ, the amount of light scattered by a particle varies as d 6. 20

21 Seeding Materials for Liquid Flow 21

22 Why NO Fluprescently-labeled Particles in Gas Flow? First, commercially available fluorescently-labeled particles are generally available as aqueous suspensions. A few manufacturers do have dry fluorescent particles available, but only in larger sizes, > 7 μm. Second, the particle-laden aqueous suspensions can be dried, and the particles subsequently suspended in a gas flow but this often proves problematic because the electrical surface charge that the particles easily acquire allows them to stick to the flow boundaries and to each other. Finally, the emission decay time of many fluorescent molecules is on the order of several nanoseconds, which may cause streaking of the particle images for highspeed flows. 22

23 Seeding Materials for Gas Flow 23

24 Particle Generation and Supply Oil droplet seeding of air flows Powder-based seeding of air flows. Soap bubble seeding for air flows 24

25 Light Sources Light Amplification by Stimulated Emission of Radiation (Laser): The laser material consists of an atomic or molecular gas, semiconductor or solid material. The pump source excites the laser material by the introduction of electromagnetic or chemical energy. The mirror arrangement, i.e., the resonator allows an oscillation within the laser material. 25

26 Laser Models Continuous Wave (CW) Lasers: He-Ne Laser (λ=633 nm) Argon Ion Laser (514 nm, 488 nm) Semiconductor Laser (depends on materials) Ruby Laser (λ=694 nm) Pulsed Lasers: Nd:YAG Laser (λ=1064 nm, 532 nm) Nd:YLF Laser (λ=1053 nm, 526 nm) 26

27 White Light (Broad band): Light Sources (Vernier.com) Energy Time duration, Δt 27

28 Simple Lenses Biconvex Biconcave Lensmaker's equation Thin lens (d->0) 28

29 A Little Geometric Optics Note that if S 1 < f, S 2 becomes negative In the special case that S 1 =, then S 2 = f and M = f / = 0. This corresponds to a collimated beam being focused to a single spot at the focal point. 29

30 Light Sheet Optics Light sheet optics using three cylindrical lenses (one of them with negative focal length). Light sheet optics using two spherical lenses (one of them with negative focal length) and one cylindrical lens. Light sheet optics using three cylindrical lenses. 30

31 Sample of Light Sheet Calculation 31

32 32

33 33

34 Imaging of Small Particles: Diffraction Limit 1 st dark ring If plane light waves impinge on an opaque screen containing a circular aperture they generate a far-field diffraction pattern on a distant observing screen. Using an approximation (the Fraunhofer approximation) for the far field it can be shown that the intensity of the Airy pattern represents the Fourier transform of the aperture s transmissivity distribution. The Airy function is equivalent to the square of the first order Bessel function. Therefore, the first dark ring, which defines the extension of the Airy disk, corresponds to the first zero of the first order Bessel function. 34

35 Lens Aberrations It that occurs due to the increased refraction of light rays when they strike a lens in comparison with those that strike nearer the center. Spherical aberration can be minimized by careful choice of the curvature of the surfaces for a particular application. Coma occurs when an object off the optical axis of the lens is imaged, where rays pass through the lens at an angle to the axis θ. coma can be minimized by choosing the curvature of the two lens surfaces to match the application. If an optical system is not axisymmetric, either due to an error in the shape of the optical surfaces or due to misalignment of the components, astigmatism can occur. Chromatic aberration is caused by the dispersion of the lens material the variation of its refractive index, n, with the wavelength of light 35

36 Perspective Projection 36

37 Digital Imaging Recording : CCD In general a CCD (Charge Coupled Device) is an electronic sensor that can convert light (i.e. photons) into electric charge (i.e. electrons). Another characteristic of a pixel is its fill factor or aperture which is defined as the ratio of its optically sensitive area and its entire area. This value can reach 100% for special, scientific-grade, back-illuminated sensors or may be as low as 15% for complex interline-transfer sensors 37

38 Digital Imaging Recording : CMOS In most of the CMOS sensors the underlying electro-optical principle of each pixel is the photodiode. Their main advantage compared to other techniques like photogates or phototransistors is their high sensitivity and relatively low noise. But in contrast to CCD pixels, the photodiodes in CMOS sensors can be controlled separately by MOS-FET transistors. 38

39 Feature CCD CMOS Signal out of pixel Electron packet Voltage Signal out of chip Voltage (analog) Bits (digital) Signal out of camera Bits (digital) Bits (digital) Fill factor High Moderate Amplifier mismatch N/A Moderate System Noise Low Moderate System Complexity High Low Sensor Complexity Low High Camera components Sensor + multiple support chips + lens Sensor + lens possible, but additional support chips common Relative R&D cost Lower Higher Relative system cost Depends on Application Depends on Application Performance CCD CMOS Responsivity Moderate Slightly better Dynamic Range High Moderate Uniformity High Low to Moderate Uniform Shuttering Fast, common Poor Uniformity High Low to Moderate Speed Moderate to High Higher Windowing Limited Extensive Antiblooming High to none High Biasing and Clocking Multiple, higher voltage Single, low-voltage 39

40 Spectral Characteristics A pixel s sensitivity or quantum efficiency, QE, is defined as the ratio between the number of collected photoelectrons and the number of incident photons per pixel and is measured in collected charge over light intensity Cb/( J cm2). Due to the width and position of the frequency-dependent band-gap of silicon, the sensors substrate material, photons of different frequencies will penetrate the sensor differently resulting in a wavelength dependent quantum efficiency of the sensor. 40

41 Linearity and Dynamic Range Linearity is of importance in PIV recording when small particle images are to be located with accuracies below half a pixel. Any nonlinear behavior during recording jeopardizes the capability of measuring the particle image displacement in the subpixel regime. Since each electron captured in the potential well adds linearly to the cumulative collected charge, the output signal voltage for the individual pixel is practically directly proportional to the collected charge. Nonlinearities of CCD images are usually due to overexposure or poorly designed output amplifiers. The sensors dynamic range is defined as the ratio between the full-well capacity and the dark current noise. Since the dark current noise is temperature dependent, the dynamic range of a CCD increases as the temperature is lowered. Standard video sensors operating at room temperature typically have a dynamic range of gray levels which exceeds that of human perception. Once digitized the useful signal is 7 8 bits in depth. With additional cooling and careful camera design a dynamic range exceeding gray levels (16 bits/pixel) is possible. 41

42 Progressive Scan CCD Conventional CCDs use interlaced scanning across the chip. The chip is divided into two fields: the odd field (rows 1, 3, 5..., etc.) and the even field (rows 2, 4, 6..., etc.). These fields are then integrated to produce a full frame. For example, with a frame rate of 30 frames per second, each field takes 1/60 of a second to read. For most applications, interlaced scanning will not cause a problem. Some trouble can develop in high-speed applications because by the time the second field is scanned, the object has already moved. This causes ghosting or blurring effects in the resulting image. 42

43 Interlaced Scan CCD The progressive scan CCD solves this problem by scanning the lines sequentially (rows 1, 2, 3, 4..., etc.). The progressive scan output has not been standardized so care should be taken when choosing hardware. Some progressive scan cameras offer an analog output signal, but few monitors are able to display the image. For this reason, capture boards are recommended to digitize the image for display. 43