Flow visualization using a Sanderson prism

Transcription

1 J Vis REGULAR PAPER Jennifer Schulz Beric Skews Alessandro Filippi Flow visualization using a Sanderson prism Received: 11 January 2018 / Revised: 31 July 2018 / Accepted: 16 August 2018 Ó The Visualization Society of Japan 2018 Abstract The use of a Sanderson prism for the visualization of density gradients in a compressible flow is shown to be an inexpensive and versatile substitute for Wollaston prisms in the application of shearing interferometry. Experimentation using the Sanderson prism in a schlieren optical setup was performed to examine the effectiveness of the prism s use for flow visualization. A range of polycarbonate prisms were tested for a range of prism heights and different physical deflections to produce a range of divergence angles. For each prism height and deflection, tests were done using a hair dryer, helium jet, a soldering iron and a moving shock wave interaction to generate flow density gradients. The light which passes through the prism was also focused at different distances to determine the effects that different fringe arrangements have on the visibility of the flow. Tests were also done comparing horizontal and vertical fringe arrangements. It was found that the colour of the infinite fringe selected had an influence on the results because the larger the difference in colour between that of the adjacent fringes, the better the contrast in the final result. It was also found that infinite fringe use showed better results due to flow features such as shocks and vortices being more easily identified against a uniform background. It is shown that the technique produces good visualization of the density gradients that form in the flow. Keywords Compressible flow Schlieren interferometry Density gradients 1 Introduction Information on a wide variety of flow visualization techniques have been given in Merzkirch (2012) and Smits and Lim (2012). For transparent flows with density changes, which result in changes in refractive index, the two most common techniques are schlieren and shadowgraph imaging, and various forms of interferometry, as detailed in these works. For schlieren imaging parts of a collimated beam of light is deflected when passing through a density gradient. If deflected downward over a knife edge that portion of the image is darkened and vice-versa. Removal of the knife-edge results in a shadowgraph image. In Mach Zehnder interferometry, a collimated beam is split into two beams which are then subsequently combined. If one beam passes through a phase object, such as a variable density field, interference fringes result giving information on the density variation. In differential interferometry, also referred to as shearing interferometry, only a single beam passes through the area of interest and is the basic technique used in the current paper. This is done using a Wollaston prism ( which splits the incident beam into two orthogonal polarised beams which then interfere on recombination. This has been J. Schulz B. Skews (&) A. Filippi Flow Research Unit, School of Mechanical, Industrial and Aeronautical Engineering, University of the Witwatersrand, PO WITS 2050, Johannesburg, South Africa beric.skews@wits.ac.za Tel.:?

2 J. Schulz et al. shown to be useful technique as shown, for example, in Carlomagno (1986) and Desse and Pegneaux (1993). However, it is somewhat limited due to the high cost of the prisms and having a fixed divergence angle, requiring different prisms for different applications. An inexpensive alternative method was reported by Sanderson (2005). This prism is a polycarbonate material that when stressed forms stress induced birefringence. It is made from standard polycarbonate sheeting and has an added advantage over a Wollaston prism that a single prism can be used for a range of divergence angles. The colour fringes that form in the image when the prism is stressed represent the interference between the two polarised beams when combined. Application and design of a system, including the theoretical background, has been treated by Biss et al. (2008). 2 Experiments In the modified schlieren optical setup for interferometry, the Sanderson prism fixture is used to replace the knife edge in a standard z-format schlieren setup, as shown in Fig. 1. For a white light setup, two Sanderson prism fixtures are required, one on either side of the test section. The light leaves the light source where it is shone through a lens of short focal length and is then focused through a slit in order to create an approximated point light source. The light is then shone through a second lens, through a polariser and is focused though the fringes in the prism. The light is then reflected off of the first parabolic mirror and is collimated over the test section. The light then reflects off of the second parabolic mirror where it is decollimated. It is then focused through the fringes in the second prism fixture and through the second polariser. The light is finally shone through the last lens which is used to focus the light into the camera. It is important to note that the first prism fixture must be placed at the focal length of the parabolic mirror in order to ensure that minimal light is lost (Biss et al. 2008). The prisms used were made from 6-mm-thick polycarbonate sheet (Makrolon) with heights of 6, 8, 10, 12, and 14 mm height, although not all were used for all test cases. These were placed in the fixture shown in Fig. 2. This was manufactured to the same sizes as those quoted in Biss et al. (2008). The upper cylindrical supports were 50 mm apart and the lower ones 76 mm. The pure bending stress was generated using a four point bending method; this allows the fringes to form uniformly across the centre of the prism as shown in Fig. 3. The light source used was a 175W Xenon short arc lamp with a colour of 5700K (Megaray), and the camera, a Photron Fastcam SA5, was used at a variety of frame rates and exposure times depending on the application. The measurement of prism deflection was done with a finger gauge with a resolution of 0.01 mm. 2.1 Calibration During initial testing, the polycarbonate material used revealed that when different sets of prisms where placed in position and stressed, the fringes that formed, formed on opposite sides. Some developed concentrated fringes at the top of the prism, whereas others developed them in the bottom part of the prism as shown in Fig. 4 under same loading conditions, i.e., one on the side of tensile stresses and the other on the Fig. 1 Standard z-layout schlieren system including the Sanderson prisms and polarisers

3 Visualization using a Sanderson prism Fig. 2 Fixture used to apply the bending stress Fig. 3 Uniform fringes created by four point bending Fig. 4 Reversal of fringe formation compression side. For this reason, two sets of 12-mm prisms were made and tested, each set forming the stress fringes on opposite sides. The 8 mm, 10 mm and one 12 mm sets of prisms developed the fringes on the bottom part of the prism. The 6 mm, 14 mm and the other 12 mm sets of prisms developed the fringes on the top of the prism. The reasons for this have not been explored further. A calibration process was used to determine the relationships between the deflection of the prism and the number and thickness of the fringes produced as a result. The relationship between the deflection and the divergence angle produced is also examined. Camera settings were at a frame rate of 8400 fps, shutter speed of 1/91,000 s and a resolution of pixels. Images were imported into a software package (GIMP) which allows a specific colour to be selected and then used to locate colours of a similar RGB value, within a specified threshold, which was set within 15.6% of the selected value. This is done in order to distinguish where one fringe ends and another begins. The main colours represented by the fringes were selected and the size of the fringe was measured in pixel length. An example is given in Fig. 5. It is then necessary to convert the pixel length to millimetres. This is done using the pixel height of the image of the entire prism compared to the actual prism height. Once this ratio is known the individual fringe heights could be found. There clearly are uncertainties in this method as the threshold in which the colours are located is variable, and therefore, it is important to use a small threshold in order to get colours with a small enough range in RGB values. There are also uncertainties in the measurement of the fringes as a single pixel represented

4 J. Schulz et al. Fig. 5 Fringe colour selection and measurement Fig. 6 Average thickness of prominent fringes that form for different deflections and different prism heights mm. The thickness of the fringe was taken at three different points along the fringe and averaged; this was then done for each of the fringes in the image and the overall average fringe thickness recorded. The deflection of the prism and the average fringe thickness is given in Fig. 6. The number of prominent fringes also increases as the deflection is increased as shown in Fig. 7. The tendency is for the number to increase and then flatten out as the deflection increases. An example is given in Fig. 8 for the 10 mm high prism. One of the benefits of using Sanderson prisms compared to Wollaston prisms is that the divergence angle of the two beams is controlled by the amount of prism deflection. For calibration, a thin wire is placed horizontally across the test section to determine the spacing between the two beams and their spacing measured. Figure 9 shows typical results of the double image of the wire as the prism is deflected. The smudge in the region of the wire image is a defect in one of the mirrors and should be ignored. A similar technique was used to determine the distance between the lines on the double image of the wire. In this case the calibration block pixel height and actual length were used to give a scaling factor. The distance between the lines was measured at three locations and averaged, which based on the scaling gives an estimated error of mm. The results are given in Fig. 10. It was found that, within the accuracy of measurement, the wire image spacing closely corresponded for all prisms, for the same degree of deflection. A full theoretical analysis relating the bending moment of the prism, its elastic and birefringent properties, and the geometric arrangement of the loading frame has been given by Biss et al. (2008). In addition, it is shown from geometric optics that the beam divergence angle,, is related to the beam separation distance, d, and the focal length of the second schlieren mirror, f, through ¼ d=f. This is a linear relationship as

5 Visualization using a Sanderson prism Fig. 7 Change in number of prominent fringes as deflection is changed Fig. 8 Images showing fringe variation for the 10 mm high prism at deflections of 0.3, 0.5, 0.7 and 1.0 mm suggested in Fig. 10 with larger errors at small deflections due to measurement of smaller spacings. In the present case for a prism deflection of 1 mm, the divergence angle for all prism heights is 2.75 arcminutes, with a mirror focal length of m.

6 J. Schulz et al. Fig. 9 The double images that form for the 6 mm prism at deflections 0.4, 0.6, 0.8 and 1.0 mm Fig. 10 Distance between wire images for different deflections and different prism heights 3 Results Four test items were selected for evaluation. These were a commercial hair dryer, a helium jet, a hot soldering iron and a moving shock wave interaction. There are a significant number of variables which were examined for each of the test items. These are: choice of finite or infinite fringe setting, and for the latter a choice of which colour fringe to select, as well as the choice of prism height and the degree of imposed deflection. Only 6, 12 and 14 mm prism data will be presented. Most assessments will be done for the nominally steady flow cases, with the transient shock wave case, using high-speed imaging, treated subsequently because of the rapid evolution in time. 3.1 Fringe selection A comparison of different infinite fringe colours shows how the selection of a specific colour can influence the visibility of the density gradients in the test section. Figures following show the different visibilities. It is important to note that for each prism height shown, the deflection for that prism remains constant and only the fringe colour is changed. Figure 11 shows images of the test done using a hair dryer, helium, and the soldering iron.

7 Visualization using a Sanderson prism Fig. 11 Infinite fringe background colours for: top: air jet using 14 mm prism deflected by 0.6 mm; middle: helium jet using 12 mm prism deflected by 0.8 mm; bottom: soldering iron using 6 mm prism deflected by 0.8 mm 3.2 Influence of deflection As the prism height and deflection increase the number of fringes and the thickness of the fringes varies. A comparison is done by comparing the visibility of the density gradients with some selected prism heights and increasing deflection for that height. Figure 12 shows typical comparisons for the testing done on the hair dryer, helium and soldering iron respectively. An indication of the sensitivity is given in Fig. 13 which shows the colour fringes and an enlarged view of the top centre image in Fig. 12. The fringe selected for infinite fringe mode is near the interface between the blue and red fringes in the centre of the fringe image. Moving upward in the jet the light passes through yellow/green then through blue/black and finally through orange/yellow. For downward deflection the fringes are wider and changes less marked, but passes through green/yellow and in a few places near the nozzle exit into orange/red. 3.3 Finite and Infinite Fringes In some cases, it is useful to visualize a flow in finite fringe mode rather than infinite fringe, such as those given above. Images with the highest colour contrast for infinite fringes are used in order to see the density gradients clearly. Once again it is important to note that for a single prism the deflection is kept constant. The results were generally found to be unsatisfactory for the jet flows where the jet axis is aligned with the fringes and the fringe width is of the same order as the jet, as shown in Fig. 14 for the three test cases. For flows where the fringes are vertical some cases are shown (Fig. 15). These are all with higher prism deflections in order to increase the number of fringes and keep them narrow.

8 J. Schulz et al. Fig. 12 Images at fixed prism height and different prism deflections. Top row: 6 mm prism height with hot jet, middle row: 14 mm prism height with helium jet, bottom row: 12 mm prism height with soldering iron. Left column: 4 mm deflection, middle column: 6 mm deflection, right column: 8 mm deflection Fig. 13 Correlation of colour fringes and the image for a hot jet

9 Visualization using a Sanderson prism Fig. 14 Horizontal finite fringe images for air jet, helium jet and soldering iron Fig. 15 Vertical finite fringe images for air jet, helium jet and soldering iron 3.4 Shock wave interaction Testing was done, using a shock tube with a 76 mm square cross-section, to study the wave interaction over the two test pieces shown in Fig. 16 which are attached to the bottom wall of the shock tube. These results exhibit both reflection and diffraction processes. The high-speed camera was operated at a frame rate of 75,000 fps and exposure time of 1 l s. All tests were done with the 12 mm prism and 0.4 mm deflection. Density gradients in the gas are significantly higher than those in the tests above and the flows are twodimensional. Figure 17 shows the evolution of the flow over the 15 test piece, after the wave has propagated over the apex, using infinite fringe mode. This test shows a typical shock wave diffraction pattern, the details of Fig. 16 Schematic of the test pieces for shock interaction study

10 J. Schulz et al. Fig. 17 Infinite fringe shock wave diffraction ms between frames which are fully described by Skews (1967), although slightly modified due to the wave initially reflecting in Mach reflection mode over the ramp, the remains of which are visible above it. The choice of fringe is given in the first frame with the colours of the adjacent frames enabling interpretation of the magnitude and direction of the density gradients. The reference fringe is dark blue. The diffracting shock is shown in frame a. It is clear that the light has been deflected downward through the narrow brown fringe, past the wider blue fringe, and into the narrow yellow and wider orange/red fringe. The flow around the corner is complex and the gradients at the vortex are so high that the light deflection exceeds the range and shows as white. The shear layer becomes visible in frame b and later frames. In frame b gradients above the layer pass from the top right corner through the yellow and orange fringes through the dark blue and green up to the purple fringe and the high gradients of the shear layer. Frame c shows how the expansive regions above the shear layer become separated as the layer s instabilities develop. The strong expansion wave at the corner grows and then reduces as the reflected wave from the top of the test section, shown in frame d, passes through it. Of particular interest in these later frames is the clarity of the contact surface and the way it raps around the vortex and terminates on the underside of the shear layer. Figures 18 and 19 are results for the same test arrangement as in Fig. 17 but in finite fringe and vertical infinite fringe modes. For the horizontal finite fringe mode, the vertical displacement of the fringes as they pass through the shock are apparent in the second frame. The third frames shows an indication of the contact surface in a similar position to that in frame b of Fig. 17, but is less well defined. The significant distortion of the flow in the vicinity of the shear layer and vortex are evident in the next frame with the final frame showing this effect extending to the full height of the image. Details of these flow features in shock wave diffraction are given in Skews (1967). In the vertical fringe infinite fringe mode, Fig. 19, wider fringe spacing is used compared to the horizontal fringe case of Fig. 17. This then more clearly shows up the gradients around the vortex. The centre of the main vortex and those of the small vortices on the shear layer are clearly defined as are the diffracted incident shock and reflected expansion wave. The background fringe is set to the thin blue fringe in the first image, which shows the finite fringe setup. From the right-hand side of the images the light then passes into the reddish fringe, then into the orange fringe and then the next blue fringe at the shear layer, where again the deflection changes due to the associated density change. The opposite occurs on the left hand side of the vortex with the light passing through the red/light blue fringe. These effects are also particularly clear for the 45 test piece shown in Fig. 20. The spiral nature of the flow around the vortex and its trajectory are evident. The features are weakened somewhat due to the reflected wave from the top of the test section. The unevenness in the shear layer due to small vortices on it is particularly noticeable. The extent of the light deflection due to the density gradients is evident in

11 Visualization using a Sanderson prism Fig. 18 Finite fringe shock wave diffraction ms between frames Fig. 19 Vertical fringe, infinite fringe of shock wave diffraction ms between frames comparing the range of colours with the no-flow finite fringe image in the first frame. The background is set in the blue fringe just below the apex of the wedge with light deflection upward going through the yellow, orange, dark blue and then light blue fringes with an indication just entering the narrow white fringe. For deflection on the other side of the vortex the light passes through the corresponding lower bands. In summary, Fig. 21 gives larger images of selected results for the four cases tested in order to illustrate the capability.

12 J. Schulz et al. Fig. 20 Horizontal fringe, infinite fringe of shock wave diffraction over a 45 degree fence ms between frames Fig. 21 Representative differential schlieren-interferometer images of a hot air jet, a Helium jet, a hot soldering iron, and a shock wave diffraction, using a Sanderson prism 4 Conclusion Application of a birefringent polycarbonate bar, known as a Sanderson prism, is used as a differential schlieren-interferometer for the study of a variety of gas flows ranging from gas plumes to high-speed

13 Visualization using a Sanderson prism imaging of shock wave diffraction. It is shown to be a simple and versatile technique exceeding the capability of a Wollaston prism as an interferometer, in that the divergence angle of the interfering beams is adjustable, thus providing an inexpensive alternative for a wide range of applications. It was operated in both finite and infinite fringe modes for visualization of hot turbulent jets, helium jets, plumes from a soldering iron, and shock wave propagation over obstacles. The nature and important details of the flows such as turbulent jet geometry, buoyant plumes, shock waves and compressible vortices are identified, giving clear indication of the density gradients. Acknowledgements We wish to acknowledge the support of the South African National Research Foundation. References Biss MM, Settles GS, Staymates ME, Sanderson SR (2008) Differential schlieren-interferometry with a simple adjustable Wollaston-like prism. Appl Opt 47(3): Carlomagno GM (1986) A Wollaston prism interferometer used as a reference beam interferometer. In: Veret C (ed) Flow visualization IV. Hemisphere Press, Washington, pp Desse JM, Pegneaux JC (1993) Direct measurement of the density field using high speed differential interferometry. Exp Fluids 15: Merzkirch W (2012) Flow visualization, 2nd edn. Academic Press, London Sanderson SR (2005) Simple, adjustable beam splitting element for differential interferometers based on photoelastic birefringence of a prismatic bar. Rev Sci Instrum 76: Skews BW (1967) The perturbed region behind a diffracting shock wave. J Fluid Mech 29: Smits AJ, Lim TT (2012) Flow visualization, 2nd edn. Imperial College Press, London