Figure 2.11: Density matching of dye solution for flow visualisation experiments: a) Hydrometer;
b) Density of working fluid; c) Density of pure dye solution; d) Dye solution with correct density after addition of methanol (Paice, 2005a).
Figure 2.12: Configuration of dye ports in MOPJ facility for flow visualisation experiments shown for the example ofS/h=10.
Figure 2.13: The phase-locked image capturing system used for recording of flow visualisation data utilising a commercial digital SLR, Hall effect sensor and custom made control system.
Figure 2.14: Sequential advance of the camera trigger mechanism and image splicing to achieve a pseudo-video movie.
For many years fluid dynamicists had to rely on intrusive measurement techniques such as the Pitot-static tube or hot-wires for their investigations. With the advent of lasers during the 1960s, so called “non-intrusive techniques”, such as Laser Doppler Anemometry (Bates, 1977, LDA), Particle Dynamics Analysis (Durst et al., 1997, PDA), Laser Induced Fluorescence (LIF) and Particle Image Velocimetry (PIV) became readily available.
Since then, PIV has become a well established technique. The theoretical background will only be touched on briefly, together with an overview of the experimental equipment and set-up used in the present study. For in-depth information on the theory and applications of PIV, the reader should refer to the works of Adrian (1986, 1991); Willert and Gharib (1991); Keane and Adrian (1992) and texts such as Smits and Lim (2000); Raffelet al.(1998); Westerweel (1993). A more detailed explanation of the PIV arrangement for the present study can be found in Appendix A.
2.7.1 Theoretical Background
The technique used for the present investigation is termed single–exposure, multi–frame PIV, where two laser pulses image the movement of particles onto two separate image frames with a known time separation. For this case it is beneficial to use the fundamental definition of velocity (Kalt, 1998) to estimate the local instantaneous velocityu
u(x,t) = lim
Δt→0
Δx(x,t)
Δt (2.11)
whereΔxis the particle displacement andΔtis the time separation between image recordings.
Consider an image frame with 3 tracer particles at locationsx1,x2 & x3, illuminated by the first laser pulse (Figure 2.15) at timet followed by a second image frame at timet with a known temporal separation ofΔt. In the second frame the pixels have moved from their original to new positions atx1,x2&x3. Assuming it is possible to clearly identify where each respective tracer
Figure 2.15: Schematic representation of 3 particles at locations (x1,x2,x3)at time t moving a distance during a known timeΔtto their respective new positions at(x1,x2,x3)at timet. The resultant velocity vectors(V,V2,V3)can hence be found.
particle has moved, one is now able to determine the displacements in x and y direction, respec- tively. Together withΔy, this allows for the following expression in two geometrical dimensions
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎣ V1x
V1y
V2x
V2y
V3x
V3y
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎦
= 1 Δt
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎣ Δ1x Δ1y Δ2x Δ2y Δ3x Δ3y
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎦
(2.12)
whereVix&Viyare velocity components in the x and y direction, respectively and
Δix = Xi−Xi
Δiy = Yi−Yi
⎫⎪
⎬
⎪⎭i=1,2,3 (2.13)
If one was to use a seeding density where the distance between particles is larger than the individ- ual particle movement between images, it would be very easy to extract the relevant information.
However, this would translate into a very low information density that can be extracted from any given image pair. In practice the seeding density is much higher, i.e. the particle spacing is smaller than the particle movement. Hence it is not possible to track individual particles in an image pair and one has to revert to correlations to identify the most likely corresponding particles in each
Figure 2.16: Conceptual arrangement of frame-to-frame image sampling. (Raffelet al., 1998)
image.
2.7.2 Cross-Correlation of Image Pairs
The system used in this investigation is a fully digital system from the image acquisition to the data analysis. As a result, only the relevant image correlation technique utilising Fast Fourier Transforms (FFT) is described here in detail.
If one considers a pair of images (Figure 2.16), where Image 1 is the input and Image 2 the output with corresponding sample regionsI(m,n)andI(m,n)it is possible to find the displace- mentd(m,n)as described in previous section. In this case,d(m,n)is the spatial transfer function for samplesI andI. Expressed in mathematical terms, this equates to
I(t) =⇒D(t) =⇒I(t) (2.14)
whereI(t)is the input function,I(t)is the output function andD(t)is the linear transfer function,
Figure 2.17: Idealised linear signal processing model relating the input (I(m,n)) to the output (I(m,n)) via the transfer function under the addition of random noise. (Raffelet al., 1998)
all of which are dependent on time. While this is an ideal situation, in reality the image samples as well as the transfer functions will contain an amount of random noise as shown in Figure 2.17.
By cross-correlating samples I andI in the spatial dimension, it is possible to find the relevant transfer function.
This requires large amounts of calculations as shown in Raffelet al. (1998) and hence it is easier, and nowadays common practice, to convert the image samples into the frequency domain using FFT to undertake the cross-correlation followed by re-conversion into the time domain. This holds as
I⇐⇒Iˆ I⇐⇒Iˆ
(2.15)
where ˆI and ˆI are the Fourier transforms ofI(m,n)andI(m,n), respectively, whereby the trans- formations are reversibly and hence hold true in both directions.
In this case, the linear transfer function can be found by the complex conjugate multiplaction of the Fourier transforms as
RII⇐⇒Iˆ•Iˆ (2.16)
whereRII is the cross-correlation in the frequency domain. This is equivalent to the FFT of the linear transfer function (D(f)). By taking the inverse Fourier transform of D(f), the transfer function is replaced into the time domain (D(t)) and the displacement can hence be estimated.
This is also shown graphically in Figure 2.18.
Figure 2.18: Implementation of cross-correlation in the frequency domain using FFTs. (Raf- felet al., 1998)
To achieve optimum data correlation it is desirable to follow a number of guidelines such as those describing the correct seeding density, particle movement during the recording interval, particle size as well as out of plane particle movement as described in detail in Keaneet al.(1995).
2.7.3 Experimental PIV Set-up – Overview
The PIV system used for data acquisition in the present study consisted of a Qantel Brilliant B Twin Nd:YAG laser, frequency doubled to a wavelength of 532nm. The system was pulsed at a frequency of 10 Hertz and the power output was measured to be 400mJ per pulse. The laser was fed though a series of lenses and a mirror from underneath the working section and culminated in a laser sheet of approximately 1 mm thickness that ran spanwise across the centreline of the facility as shown in Figure 2.19. A more detailed description of the physical arrangement of the PIV set-up is given in Appendix A.
Hollow glass spheres with a mean diameter 20μmwere used as reflective seeding particles and deemed to be following the flow in a satisfactory manner from information given by Raffelet al.
Figure 2.19: PIV set-up used in present investigation.
Figure 2.20: PIV target image with resolution pattern included.
(1998). The data sheet for the seeding particles can be found in Appendix J.
PIV Images were recorded using a Kodak 1.0 Megapixel CCD camera with a resolution of 1008×1018 pixels mounted onto a three-axis traverse. The physical size of the image recorded had a size of 108 mm square, with a region of interest of 100 mm square. This resulted in an overlap of 8 mm (0.9h) on each side of the image. The actual grid for the PIV experiments conducted can be found in Appendix H. From target images such as Figure 2.20, the resulting image and pixel size was found to be 94 pixels per 10 mm, which equals a resolution of 106.4μm per pixel. For a 32×32 pixel correlation window this equates to a physical size of 3.4mm×3.4mm.
PIV image acquisition was synchronised with the position of the oscillating plate by using a custom designed control system. The control system again takes an input from plate crankshaft similar to the CamTrigger II system to drive the laser system and camera. To allow for identifica- tion of images in relation to the crank position, the control system drives a set of LEDs, which are blended into PIV image representing the crankshaft angle in binary code, which can be examined during the data analysis stage. The system set-up and an example of an image containing the LED binary coding are shown in Figures 2.21 & 2.22. Due to the fact that a mirror was used to blend the LEDs into the images, it is important to note that while the highest bit is on the left hand side and the lowest bit is on the right hand side, that actual HIGH-LOW-bit LEDS are mirror images.
This means that the “LOW”-set LEDS are on the top and the “HIGH”-set LEDs are on the bottom.
Figure 2.21: The trigger control box synchronises the Laser pulses with the camshaft and displays the phase angle in correspondence to the camshaft sensor on a LED display in binary form.
Figure 2.22: The LED display array is recorded as part of the PIV image on the CCD camera to allow phase by phase identification. The array can be seen in the bottom right hand corner.
From previous experimental campaigns using the same image acquisition components, it was known that 2 pixels on the camera’s CCD are permanently damaged and record a binary value of 255 resulting in bright white pixels in the digital image. These pixels get mistakenly treated as seeding particles during the cross-correlation process if left untreated. An automated method to interpolate and replace the faulty pixel value from surrounding pixels in each instant has been developed by England (2009) and was applied to all raw PIV images prior to data analysis. The correction method is described in detail in Appendix B.
The actual PIV analysis was undertaken using the PIVview 2.4 software package (PivTec GmbH, G¨ottingen, Germany). The software contains a number of methods for outlier or spurious vector detection for the PIV post-processing phase. Except for the application of a global his- togram filter algorithm1 to the data, no further image correction was undertaken. Phase-locked and flow averaged PIV images were analysed using a 32×32 pixel grid with a grid overlap of 75%.
The measurement uncertainty in the PIV data was calculated using the methods documented by Raffelet al.(1998). The measurement uncertainty was found to be a maximum of 15% in one case ofReh=1800 withUx,y≥0.25U0. For all other cases investigated, the maximum measurement uncertainty was found to be less than 10% forUx,y≥0.25U0.
During the analysis of the PIV data it was noted that a constant overestimation of the jet velocities was obtained along the facility centreline in the streamwise direction at a distance of x/h=3.3±0.2 and atx/h=5±0.5, as shown in Figure 2.23. Upon inspection of acquired target images used for the system set-up some optical distortion was discovered in those locations. It was hence concluded that the distortions are most probably either an artefact of the experimental facility, as non-optical grade PMMA was used for the walls of the working section, or dried water droplets on the outside of the facility.
1For more details on Global Histogram Filters and other outlier detection methods refer to Raffelet al.(1998).
Figure 2.23: Normalised velocity magnitude for the mean flow of a sample MOPJ case (Reh=2700, S/h=0.5, f =2Hz) close to the nozzle exit. The blue circled area highlights the area where velocities were found to be overestimated due to an optical distortion.
2.7.5 Steady Jet Velocity Profiles
The PIV set-up was used to acquire velocity data of steady planar jets of the three cases earmarked for further experiments and to verify the exit velocity profiles and centreline velocities close to the nozzle for a further three Reynolds numbers that were investigated as part of the flow visualisation study. For all steady jet cases a minimum of of 440 image pairs were used in each case to obtain the velocity profiles.
Figures 2.24 & 2.25 show the normalised streamwise velocity profiles of the steady jets cases from the nozzle exit to a distance ofx/h=8. All cases show an initial top-hat velocity profile that gradually transforms into a smooth bell-curve profile as the jet shear layer expands.
Inspection of the individual velocity profiles in the two figures show the overestimation of the velocities at x/h=3 & 5, especially for the higher Reynolds number cases as discussed in the previous section (Figure 2.23), presenting further evidence to the claim that the velocity errors are inherent to the experimental facility and not a result of faulty data analysis.
The normalised centreline velocities for six Reynolds numbers (Re=1000, 1800, 2700, 3420,
Figure 2.24: Normalised streamwise velocity probiles. a)Re=1000; b)Re=1800; c)Re=2700.
Figure 2.25: Normalised streamwise velocity probiles. a)Re=3420; b)Re=5850; c)Re=7650.
chosen for further quantitative investigation and the normalised centreline velocities are presented in the bottom panel of Figure 2.26. The centreline velocity decays approach a linear slope at aroundx/h≥12 and hence fulfil the criteria set out by Rajaratnam (1976) for the behaviour of a planar jet in the self-similar farfield as discussed in Chapter 1.
Figure 2.27 shows the centreline velocity decay in the jet farfield for the current study in comparison with other relevant studies, whereby kis the decay rate coefficient. The data shows that a significantly higher velocity decay is present in this experimental facility when compared to other studies. Examination of the experimental conditions used by the cited reference studies shows that the current study is the only one that is closely confined both in the spanwise and the transverse directions (Table 2.2).
While the physical boundaries are at a distance of 50hin the lateral direction, the higher ve- locity decay raises the question as to whether any existing induced recirculation within the facility may change the jet behaviour and hence cause the observed difference. Further inspection of Table 2.2 also shows that, with the exception of Thomas and Goldschmidt (1986), the present study is the only one that has a back plate behind the nozzle and consequently blocks fluid entrainment from the outside of the jet area. For an unconfined jet, the jet momentum in the self-similar farfield is expected to be constant and, assuming that flow-averaged self-similarity is present, when plotted for the three Reh investigated, this is evidently not the case (Figure 2.28). Initially the jet mo- mentum increases downstream from the nozzle as the velocity profile is not self-similar and hence does not fulfill the original condition stated beforehand. After x/h≈8 where the jet momentum can be expected to become self-similar, a loss in jet momentum is evident.
The loss of jet momentum due to an induced recirculation around the jet stemming from close- ness of the physical facility boundaries to the jet nozzle exit has been extensively discussed by Husseinet al.(1994), Deo (2005) and Deoet al.(2008). If the loss of jet momentum is due to backflow, Deo (2005) shows that the decrease in momentum should follow a linear decay in the jet
a)
b)
Figure 2.26: a) Inverted normalised centreline streamwise velocities of the steady jet from PIV measurements. b) Inverted normalised centreline streamwise velocities of the steady jet from PIV measurements for the cases used in more detailed investigation.
Figure 2.27: Comparison of the present centreline velocity decay values in the steady jet far field with other studies on steady planar jets. Results from the current study are shown in red, results from references are shown in black: D08= Deoet al.(2008); BO75= Bashir and Uberoi (1975);
TG86= Thomas and Goldschmidt (1986).
Table2.2:Summaryofreferencedstudiesforplanarsteadyjetinrelationtojetconfinementconditionsandnozzleshape. AuthorsRehw/hkFluidNozzleShapeJetConfinementBack SidewallsTransverse† Plate PresentStudy10001000.36WaterRadialContrYes50hYes PresentStudy18001000.36WaterRadialContrYes50hYes PresentStudy27001000.36WaterRadialContrYes50hYes Deoetal.(2008)1500600.22AirRadialContrNo125hNo Deoetal.(2008)3000600.19AirRadialContrNo125hNo ThomasandGoldschmidt(1986)6000470.22Airn/a∗No291hYes BashirandUberoi(1975)27001440.24AirSmoothContrNon/aNo Notes:n/a=NotAvailable;Contr=Contraction;†=Distancetonearestboundaryinthetransversedirection; ∗=Squareductterminatedbyplatesformingaslot.Noinformationonnozzleprofileavailable.