Schlieren ''PIV'' for turbulent flows
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Optics and Lasers in Engineering 44 (2006) 190–207
Schlieren ‘‘PIV’’ for turbulent flows Dennis R. Jonassen, Gary S. Settles, Michael D. Tronosky Gas Dynamics Lab, Mechanical and Nuclear Engineering Department, Penn State University, University Park, PA 16802, USA Available online 31 May 2005
Abstract The possibility of using commercial PIV equipment combined with schlieren optics to measure the velocity fields of turbulent flows is explored. Given a sufficiently high Reynolds number and adequate refractive flow differences, turbulent eddies can serve as the PIV ‘‘particles’’ in a schlieren image or shadowgram. The PIV software analyzes motion between consecutive schlieren or shadowgraph frames to obtain velocity fields. Velocimetry examples of an axisymmetric sonic helium jet in air and a 2D turbulent boundary layer at Mach 3 are shown. Due to optical path integration, axisymmetric flows require the inverse Abel transform to extract center-plane velocity data. Conditions for optimum schlieren sensitivity are examined. In its present embodiment, ‘‘schlieren PIV’’ is not useful for laminar flows nor for fully 3D flows. Otherwise it functions much like standard PIV under conditions where individual particles are not resolved and velocimetry is instead based on correlation of the motion of turbulent structures. ‘‘Schlieren PIV’’ shows significant promise for general refractive turbulent flow velocimetry if its integrative nature can be overcome through sharpfocusing optics. r 2005 Elsevier Ltd. All rights reserved. Keywords: Schlieren imaging; PIV; Velocimetry; Turbulent flows; Jets; Boundary layers
1. Introduction and literature review Particle image velocimetry (PIV) has recently become the most important new tool of experimental fluid dynamics. Two consecutive images of a particle-laden flow are Corresponding author. Tel.: +1 814 863 1504; fax: +1 814 865 0118.
E-mail address: [email protected] (G.S. Settles). 0143-8166/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2005.04.004
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interrogated for small particle displacements, from which a slice through a velocity field is measured. This depends upon modern digital camera equipment and desktop computing capability previously beyond the reach of the individual investigator, but now commercial PIV systems with convenient user interfaces are available to anyone who can afford them. One of the main issues in PIV is seeding the flowfield with particles. These particles should match the density of the fluid in order to avoid gravitational forces, and at the same time must not change the flow dynamics [1]. However, if the flowfield is sufficiently turbulent and this turbulence involves refractive-index changes, then turbulent eddies themselves might act as ‘‘PIV particles’’ when the flow is viewed using either schlieren or shadowgraph methods. In such a case the flowfield is selfseeding. An underlying assumption here is that the evolutionary time scale of the eddies is much longer than the time separation of the images in a PIV pair, otherwise no PIV correlation can occur. Likewise some flow symmetry, e.g. planar or axisymmetric, is required in order that integrative optical methods may present an interpretable view of a flow. The aim of the present approach is to explore such velocimetry, not the measurement of the refractive-index field per se. The use of the schlieren method and PIV software for the related purpose of refractive-index or temperature measurement is described by others, e.g. [2,3]. Schlieren velocimetry was first tried in 1936 by Townend [4], but was not pursued in that pre-computer era. In 1989, Papamoschou [5] used a pattern-matching algorithm to compute the convective velocity of a supersonic shear layer from two consecutive schlieren images. Since then, Tokumaru and Dimotakis [6] proposed an ambitious method of image correlation velocimetry for fluid flows, while Fu and Wu applied image analysis to extract velocity fields from sequences of schlieren images [7–9]. Kegerise and Settles also performed schlieren velocimetry on a convective plume [10]. All the cited velocimetry examples require specific ‘‘home-grown’’ algorithms to extract the desired velocity data from schlieren image sequences. In contrast, the present goal is to explore the validity of using a commercially available PIV system and software to measure velocity fields from schlieren images and shadowgrams.
2. Experimental equipment The PIV system used here is a single-camera system made by IDT Inc. [11]. Its software offers two different algorithms for extracting displacements from an image pair: standard and adaptive interrogation modes. The former is the cross-correlation approach most often used in PIV, but is prone to errors such as loss of pairing, image truncation, and spatial averaging of velocity gradients. The adaptive interrogation mode [12] is designed to reduce or avoid such errors. Although both algorithms produce similar results in the present test cases, only the adaptive interrogation mode has been used in obtaining the results presented here. Insofar as schlieren ‘‘PIV’’ measurements are concerned, comparable equipment and software by other manufacturers is expected to function similarly.
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Three different schlieren optical instruments were used in this study, one lensbased and two mirror-based systems [13]. The lens-based system had two 152 mm diameter f =5:67 telescope objective lenses as field elements. The mirror-based systems had z-type layouts with twin f/8 parabolic mirrors of either 108 or 152 mm diameter as field elements. Special schlieren light sources are required in order to provide pulsed illumination for PIV. For low- to moderate-speed flows, inexpensive microsecond white-light xenon strobe lamps suffice, e.g. [14]. Two such lamps are mounted perpendicularly to one another as shown in Fig. 1. A 50/50 beamsplitter combines the beams from these lamps and directs them, via a condenser lens and slit, along the optical axis of the present 108 mm-aperture z-type schlieren system. Neutral density filters are required after each bulb to balance the illumination from the two flashes. One flashbulb is triggered by PIV trigger signal A from the controlling computer and timing box, the second by trigger signal B. In our case, since the PIV system is designed for pulsed lasers, proper strobe timing required an external delay circuit between the PIV timing box and the first strobe lamp [15]. For high-speed flows requiring time separations less than 5 ms between images A and B of a PIV pair, strobe illumination fails and pulsed laser illumination is required. This is provided by a New Wave Gemini 200 dual-head Nd:YAG laser. This powerful laser (200 mJ/pulse) needs strong attenuation for use as a schlieren illuminator, here accomplished by two beam reflections from the planar sides of fused silica plano-concave lenses. Finally, a beam expander overfills the first mirror of the 152 mm-diameter z-type schlieren system to yield approximately uniform laser illumination. Dual pulsed-laser illumination has the distinct advantage of a virtually unlimited range of pulse-separation timing. However, coherent laser light produces schlieren images that are inferior to those produced by non-coherent white light for reasons given in [13] and [16]. Briefly, the geometric-optical approach to schlieren system performance breaks down and a very small focal spot occurs in the cutoff plane. The first issue produces coherent artifact noise and a reduced signal-to-noise ratio compared to the usual evenly illuminated white-light schlieren image. The second
Fig. 1. Diagram of the twin strobe arrangement for schlieren PIV.
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issue rules out the traditional knife-edge cutoff for laser illumination, requiring instead one of several milder cutoff options (graded filter, partially transmitting cutoff, etc.). Here, a simple half-plane sooted microscope slide cutoff [13] is used. The schlieren sensitivity using this cutoff is determined by the optical transmission of the sooted section, which is measured by transmission densitometry. Within limits, greater optical transmission of the cutoff yields less sensitivity in a laser-illuminated schlieren system and vice versa. By comparison, schlieren sensitivity with a traditional extended white-light source and a knife-edge cutoff is determined by the amount of cutoff and the width of the light source image. All of the present schlieren instruments are easily converted to ‘‘focused’’ shadowgraphy [13] by the removal of the cutoff, image focus adjustment, and reduction of the source slit size to an effective ‘‘point’’ in the case of white-light illumination. Of course, these optical methods present somewhat different views of a flowfield, but both reveal refractive turbulence, thus both are included in principle under the general name ‘‘schlieren PIV.’’ Two different turbulent flows, both 2D in the mean, are used here to test the validity of the schlieren PIV concept: a planar turbulent boundary layer and an axisymmetric turbulent jet. The boundary layer is formed on the test section floor of the Penn State Supersonic Wind Tunnel, a cold-flow blowdown facility with a 15 16.3 60 cm test section, up to 2 min test duration, and a Mach number range of 1.5–4.0. Present tests are conducted at Mach 3, boundary-layer thickness d ¼ 25 mm, and momentum-thickness Reynolds number Rey ¼ 98000. The lens-type schlieren system described above was used in these experiments. A converging nozzle of exit diameter 0.787 mm, discharging helium into ambient air, produces the axisymmetric jet tested here. The nozzle stagnation pressure of 208 kPa yields a choked helium jet with a theoretical exit velocity of 890 m/s and a Reynolds number of about 7300. The small nozzle scale allows a large nondimensional jet length to be studied (up to about 200 nozzle diameters downstream) within the fields-of-view of the z-type schlieren instruments employed here.
3. Schlieren sensitivity to turbulence As in traditional PIV, image quality is a key issue in schlieren PIV. The image scale and resolution are determined by the nature of the CCD sensor in the PIV camera and by the lens employed, and vary with the scale of the flow under study. Aside from these traditional concerns, however, schlieren PIV has additional image quality issues arising from the sensitivity of the optics to the refractive disturbances in the turbulent flows under investigation. Schlieren sensitivity is here analogous to particle size selection and seeding density in traditional PIV. For both schlieren and shadowgraphy an optimum sensitivity range can be defined over which the best PIV results are had. In simple geometric–optical terms, schlieren sensitivity S is given by Eq. (1) (adapted from [13]), where a is the width of the unobstructed light-source image in the schlieren cutoff plane, b is the width of the entire light-source image, and f 2 is the
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focal length of the second schlieren field lens or mirror. Since f 2 and b are typically fixed for a given schlieren instrument, we may conveniently discuss sensitivity in terms of the percent of knife-edge cutoff of the light-source image. For example, 100% cutoff refers to the complete blockage of the source image by the knife-edge (a situation in which Eq. (1) predicts infinite sensitivity, the geometric–optical description fails, and diffraction effects take over [13]). S¼
f2 f2 . ¼ a b 1 %cutoff=100
(1)
For the white-light strobe-illuminated schlieren system described earlier, a razor blade knife-edge oriented perpendicular to the streamwise direction of the flow is used as the cutoff. Fig. 2 shows the results of the range of cutoffs used with the present helium jet. For 10% and 20% cutoff only an underlying very-slight shadowgraph effect is seen. These images contain insufficient information for PIV processing. On the other hand, high-cutoff images result in over-ranging, shown for example by the entirely white regions in the 90%-cutoff image in Fig. 2. When overranging occurs the local flow details are washed out, which is likewise detrimental to schlieren PIV. To avoid over-ranging, especially when high cutoff is required by a need for high schlieren sensitivity, one can increase the overall measuring range by increasing the light-source slit width b. Better solutions, such as replacing the knife-edge by a graded filter [13] are also available. The optimum case for schlieren PIV requires sufficient sensitivity to reveal the smallest turbulent structures without significant over-ranging. This occurs, for example, in the 30–60% cutoff range in Fig. 2. Here, unlike the case in some other schlieren applications, it is the fine-scale turbulence—PIV ‘‘particles’’—that must be revealed. As noted earlier, laser schlieren illumination requires a less-severe cutoff than the knife-edge just described, whose optical transmission f is 0%. In Fig. 3, different halfplane-sooted microscope slides are used as cutoffs to vary the schlieren sensitivity. Cutoffs with fp26%, being almost as opaque as a knife-edge, yield binarized images in which some or all of the fine turbulence scales are washed out. Likewise the f ¼ 93% cutoff is almost transparent, yielding practically no cutoff and a vanishing schlieren effect. Therefore, the present optimum sensitivity range for laser-illuminated schlieren with partially transmitting cutoffs is roughly 50%ofo90%. Coherent artifact noise is quite apparent in Fig. 3. It has the undesirable effect in schlieren PIV of obscuring some of the fine-scale turbulent motion. Moreover, with twin pulsed-laser illumination a pseudo-velocity can result from the coherent artifact noise if the two laser beams are not aligned with perfect coincidence. As already noted, ‘‘focused’’ shadowgram pairs can also be used for PIV analysis. Here, since shadowgrams involve no cutoff, the product of the focus offset distance L and the Laplacian of the refractive-index field dictates the sensitivity [13]. Fig. 4 shows an example of laser-illuminated shadowgrams of the helium jet with L ranging from 0 to 130 cm. As expected, the sharply focused case yields no usable information for PIV analysis. Shadowgraph PIV becomes possible in this example when L
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Fig. 2. Helium-jet white-light schlieren images with different amounts of knife-edge cutoff (b ¼ 0:86 mm source-slit width).
Fig. 3. Helium-jet laser-schlieren results with different optical transmission values f of the half-plane sooted microscope slide cutoff: (a) f ¼ 13%, (b) 26%, (c) 61%, (d) 72%, (e) 88%, and (f) 93%.
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Fig. 4. Helium-jet shadowgrams with increasing sensitivity: (a) L ¼ 0 cm, (b) 12.5 cm, (c) 38 cm, (d) 53 cm, and (e) 130 cm.
exceeds 12 cm, but fails for L4130 cm because the increasing blur eventually obscures the fine-scale turbulence that serves as PIV ‘‘particles.’’ These L limits will be different, of course, for schlieren objects of different strength than the present helium jet. Note especially that a shadowgram is not a focused image, and that coherent artifact noise in the case of laser illumination can pose the same troublesome issue of pseudo-velocity as described above for schlieren PIV. To better understand exactly how PIV algorithms ‘‘perceive’’ schlieren images, 120 120 pixel turbulence samples were taken from each image in Fig. 2 and converted into binarized 2D intensity maps, which were then analyzed for the number and size of the ‘‘particles’’ within a 24 24 pixel interrogation window. For simplicity, no window overlap was used in the determination of the number of particles per window, and the particles found at the edges of the window were included in the particle count. As shown in Fig. 5, the number of apparent ‘‘particles’’ decreases with increasing knife-edge cutoff, while the average ‘‘particle’’ size increases. Keane and Adrian [17] proposed a method to quantify the minimum number of particles required in a PIV interrogation window. It is based on the number of particles N, the out-of-plane displacement F 0 , and the in-plane displacement F i . In order to achieve 95% accuracy in the probability of displacement detection, they claim that the product of these three parameters should be larger than five. However, Raffel et al. [1] claim that a value of three or four for this product is sufficient for practical situations. When applied to schlieren PIV, F 0 equals unity since all ‘‘particles’’ are always in the ‘‘plane.’’ F i can likewise be set essentially to unity when the adaptive interrogation mode is used. Thus the number of schlieren ‘‘particles’’ in the PIV interrogation window should be at least four. Fig. 5 reveals that a knife-edge cutoff of 50% or less thus ensures an adequate number of ‘‘particles’’ in a 24 24 pixel window for the schlieren equipment used here (f 2 ¼ 864 mm, b ¼ 0:86 mm). Cutoffs exceeding 50% may yield interrogation windows without ‘‘particles,’’ thereby requiring software interpolation to compute displacements.
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Number of Apparent "Particles" in a 24 x 24 pixel interrogaiton window
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Percent Cutoff Fig. 5. ‘‘Particle’’ size and concentration vs. percent schlieren cutoff (based on Fig. 2 and using a 108 mm diameter f =8 z-type schlieren system with b ¼ 0:86 mm source-slit width).
Particle size is likewise a key parameter in PIV analysis. According to Raffel et al. [1], the optimum particle diameter for conventional PIV is two pixels, based on a simulation of particles traveling one pixel per frame in a light sheet. In this controlled case, particle size is directly related to the RMS uncertainty of particle displacement. Above a particle diameter of two pixels, however, the RMS uncertainty increases exponentially, although it decreases as the interrogation window grows [1]. Based on this, ‘‘particle’’ diameters should be minimized for successful schlieren PIV, especially when small interrogation windows are needed for high spatial resolution. Fig. 5 shows that the present ‘‘particle’’ size increases almost linearly over the range of 10–60% cutoff, above which over-ranging of the helium-jet schlieren image begins. We therefore conclude that effective ‘‘schlieren PIV’’ calls for the minimum knifeedge cutoff compatible with ‘‘particle’’ visibility. One may anticipate extreme cases where a combination of low cutoff and a weak turbulent refractive field prevents the success of schlieren PIV. Even for the present strongly refractive helium jet, a cutoff in the ‘‘ideal’’ range of 10% produces insufficient contrast between the turbulence and the background, i.e. a signal-to-noise ratio that is too small to be effective. Thus the optimum schlieren sensitivity for PIV measurements in the present experiment occurs within the 30–60% range of knife-edge cutoff using white-light illumination.
4. PIV image processing Processing schlieren image pairs with commercially available PIV software is no different than in standard PIV. The first consideration is spatial resolution: high
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resolution of the velocity field requires small interrogation windows. But errors can result due to the ‘‘particle’’ size and concentration constraints noted above. Smaller windows also increase the probability of other errors such as loss of pairing and particle truncation. Both errors result in velocity underestimation [12]. The adaptive interrogation window described earlier alleviates most of these problems. To further increase the probability of detection, a PIV image offset may sometimes be required. This literally shifts the second image of the PIV pair with respect to the first by a predetermined integer pixel value. Such an offset helps the correlation algorithm to determine sub-pixel displacements, permitting the use of smaller interrogation windows. Certain situations further require that the PIV field be analyzed in zones, rather than as a whole. For example, high velocity gradients can produce neighboring regions of a flowfield requiring different image offsets (often by just a single pixel). This zonal evaluation technique was required in analyzing the present compressible boundary layer test case.
5. ‘‘Schlieren PIV’’ results 5.1. Helium jet The helium jet studied here has distinct near- and far-field zones. The near-field extends to about 15 nozzle diameters downstream of the nozzle exit. Its defining feature is its laminar core, containing no turbulent structures (particles) for schlieren PIV. The effect is clearly seen in Fig. 6: a double peak appears in the integrated velocity profile at about y ¼ 8 mm from the nozzle exit. The reason for this is the path-integration of the schlieren beam through this axisymmetric flow. A light ray through the edge of the laminar jet core incurs more refraction due to turbulence than one traversing the jet centerline. The former ray thus yields a greater apparent ‘‘particle’’ displacement and a higher velocity is found. No schlieren PIV is possible within y ¼ 1 mm of the nozzle exit, where the accompanying shadowgram reveals an initially laminar jet. Also, due to optical path integration and turbulent mixing, the measured velocities in the 40 m/s range in Fig. 6 are low compared to the theoretical jet nozzle-exit speed of 890 m/s. The far-field jet region begins around 15 diameters (12 mm) from the nozzle exit and is fully turbulent. Fig. 7 shows laser- and white-light-illuminated schlieren images and a laser shadowgram of this region. The white-light schlieren image, Fig. 7b, has a more severe knife-edge cutoff than the laser schlieren image, and both laser frames have coherent artifact noise that can lead to velocity errors. Nonetheless, each of these three varied depictions of the helium jet does in fact yield, with good accuracy, the same convective velocity field upon ‘‘schlieren PIV’’ analysis (150 image pairs at a time delay of 5 ms between paired images). The resulting mean velocity profile normal to the jet axis is given in Fig. 8. For comparison, traditional laser-sheet-illuminated PIV measurements were made using micron-sized oil droplets that were entrained into the helium jet by way of a
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Fig. 6. Laser shadowgram of the helium jet near field (left, L20 cm, nozzle exit diameter ¼ 0.787 mm) and the corresponding schlieren PIV velocity contour plot (right). Velocities are given in m/s.
Fig. 7. (a) Laser schlieren, (b) white-light schlieren, and (c) laser shadowgram of helium jet showing far-field.
slow coaxial seeded jet. Only far-field measurements were possible in this manner. Example flowfield images and PIV velocity results are compared in Fig. 9. It is clear from this comparison that traditional PIV yields higher velocities than schlieren PIV in this helium jet case, as expected. The difference is attributed to the pathintegration of the schlieren optics. Since the jet is axisymmetric, a proper comparison
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Axial Velocity, path-averaged (m/s)
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Fig. 8. Mean helium-jet velocity profiles from ‘‘schlieren PIV’’ analysis of the cases of Fig. 7 (y ¼ 42 mm downstream of nozzle exit). Only half-profiles are shown for the laser schlieren and shadowgraph cases.
can be made by applying the Abel transform to the traditional PIV results, Eq. (2) (see, e.g. [18]). iP edge
V pa ði; jÞ ¼
V pl ði; jÞ
i¼io
N io !iedge
,
(2)
where V pa ði; jÞ is the path-averaged velocity, V pl ði; jÞ is the planar velocity, N io !iedge is the number of discrete points between io and iedge , and the subscripts ‘o’ and ‘edge’ represent the location of V pa ði; jÞ within the jet and the jet edge, respectively. The result is shown in Fig. 10, where the Abel transform applied to the traditional-PIV jet center-plane data recovers approximately the same velocity profile as that measured directly by ‘‘schlieren PIV.’’ Alternatively, the inverse Abel transform can be applied to the schlieren PIV results to yield equivalent jet center-plane data. In most cases this is the preferable procedure, since the equivalent center-plane data are the most useful form of the experimental results. The opposite procedure has been adopted here merely as a matter of convenience, since deconvolution of the data makes the inverse Abel transform approach somewhat more involved. Nonetheless modern digital computing power is more than sufficient for either of these procedures. To further test the schlieren PIV results, one may compare the jet centerline velocity distribution vs. y with traditional PIV results and with the extensive jet data
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Fig. 9. (Above left) white-light helium-jet schlieren image, (above right) particle-seeded image of the same flow with laser sheet illumination through the jet center-plane. Below the images are shown average PIV velocity contour maps for comparison (schlieren PIV left, traditional PIV right).
correlation by Kleinstein and Witze [19]: ! 1 uc ¼ 1 exp , 0:5 kx re Xc
(3)
where k ¼ 0:074, x ¼ x=rj , re ¼ re =rj (e and j refer to the ambient and jet exit conditions, respectively), and X c ¼ 0:7. The centerline velocity is normalized by the jet nozzle-exit velocity, 890 m/s.
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Fig. 10. Velocity half-profiles of the far-field helium jet at y ¼ 65 mm, comparing ‘‘schlieren PIV’’ with traditional particle PIV results with and without the application of the Abel transform to the latter.
The results show that the present helium-jet centerline velocity data, obtained by traditional PIV in two separate experiments, agree well with the Kleinstein–Witze correlation as shown in Fig. 11. Also, applying the Abel transform to the traditional PIV data yields results comparable to the ‘‘schlieren PIV’’ data, demonstrating once again that ‘‘schlieren PIV’’ gives a path-integrated representation of the axisymmetric velocity field. The noise level of the traditional PIV data in Fig. 11 makes it difficult to determine the exact jet edge location, which may account for some error incurred during the Abel transformation. 5.2. Compressible turbulent boundary layer The above helium-jet ‘‘schlieren PIV’’ results reveal the dominant effect of optical path integration in measuring an axisymmetric flowfield. No such effect is expected in a true 2D flow, however, where schlieren PIV and traditional PIV results should be directly comparable even though the former still integrates across the flow. To test this, the compressible turbulent boundary layer on the test-section floor of the Penn State Supersonic Wind Tunnel was measured by ‘‘schlieren PIV.’’ Given a span of 6d and end effects of less than one d, approximately 2D flow is expected. Note, however, that it is not possible with the current schlieren optics to eliminate end effects by focusing on the center-plane of this flow. Double-pulsed laser illumination through the lens-type schlieren system described earlier is the only approach that allows a sufficiently small time delay between PIV frames (0.55 ms) for this Mach 3 flow. To provide adequate schlieren sensitivity a
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Fig. 11. Helium jet centerline velocity decay as measured by traditional PIV, predicted by the KleinsteinWitze correlation, and measured by schlieren PIV with path integration.
half-plane sooted microscope slide with 72% light transmission was required. Given the 15 cm extent of the boundary layer along the optical axis, converting to shadowgraphy required only the removal of the knife-edge, without any focus adjustment. (In other words, the shadowgraph image is always present and is combined with the schlieren image when a schlieren cutoff is used.) An example of schlieren image and shadowgram are given in Fig. 12. In both cases fine-scale turbulence is observed, though it does not end as expected at the boundary-layer edge, but rather appears to continue into the freestream. This is, in fact, an end effect due to the wind-tunnel sidewall boundary layers on the glass windows. Slight over-ranging is also seen in the schlieren image, but this is not serious enough to affect the overall PIV results. One hundred and fifty schlieren PIV image pairs were processed, as before, using the IDT software’s adaptive mode with a 20 20 pixel interrogation window. However, since the boundary-layer velocity profile extends from 0 at the wall to 613 m/s in the freestream, the required image offset varies across the height of the boundary layer as discussed earlier, starting at 6 pixels near the wall and ending at 8 pixels near the freestream. Since three separate image-offset zones are needed to analyze this flow, three different meshes are also required. The zones with their mesh locations and corresponding image offsets are as follows: 1. 0.0py/dp0.17 6-pixel offset, 2. 0.14py/dp0.30 7-pixel offset, 3. 0.27py/dp1.01 8-pixel offset.
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Fig. 12. Example schlieren image (left) and shadowgram (right) of the Mach 3 turbulent boundary layer on the test section floor of the Penn State Supersonic Wind Tunnel.
Unfortunately, traditional particle PIV is not currently possible in this boundary layer due to particle seeding difficulties. Instead, the schlieren PIV results are evaluated by comparison with mean velocity profiles derived from pitot-pressure surveys by Garg and Settles [20], and with the established wake-wall similarity law for turbulent boundary layer profiles. The Van Driest transformation was used to convert measured compressible-flow velocity values, u, obtained from both the optical ‘‘PIV’’ and the pitot surveys, to equivalent incompressible values, u* (see, e.g. [21]). The results are plotted in Fig. 13 in traditional u+ vs. y+ coordinates along with the incompressible wall-wake law of Coles [22]. Fig. 13 reveals that the ‘‘schlieren PIV’’ data are in substantial agreement with the pitot-survey results for this boundary layer, even though the differences in the two measurement methods are striking. Schlieren PIV measures, in principle, the broadband convective velocity of turbulent structures directly. Pitot pressure surveys are converted, with the assumption of constant stagnation temperature, to mean Mach number profiles from which the mean velocity is extracted by way of an assumed static temperature profile. Agreement between the two indicates that schlieren PIV actually measures, by way of refractive eddy motion, the mean velocity profile of the boundary layer. Fig. 13 also reveals that this boundary layer has an unusually high wake-strength component, P ¼ 1:9 vs. the usual P0:55 for a flat-plate boundary layer [22]. This doubtless results from the pressure-gradient history along the wall of the wind tunnel’s long asymmetric variable-Mach-number sliding-block nozzle. There are two boundary-layer regions in which ‘‘schlieren PIV’’ is unable to measure the velocity satisfactorily: very near the wall and near the boundary-layer edge. Failure near the wall is due to poor spatial resolution caused by wind-tunnel vibrations and by laser diffraction that obscures the true location of the wall. Even so, schlieren PIV—being non-intrusive—measures closer to the wall than does the
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Fig. 13. Mean velocity profiles of the Mach 3 turbulent boundary layer in wall-wake coordinates.
pitot survey method. Near the boundary-layer edge, however, ‘‘schlieren PIV’’ fails by yielding velocities that are too low. The outer 20% of the turbulent boundary layer is quite intermittent, so the eddies serving as PIV ‘‘particles’’ are fewer there. Eddies in the sidewall boundary layers contaminate the integrated schlieren measurement with incorrect end-effect velocities in this region, as shown in Fig. 12. By way of focusing schlieren optics [20,23], it should be possible to eliminate these end effects and make an accurate measurement up to the boundary-layer edge, but that is beyond the present scope.
6. Conclusion ‘‘Schlieren PIV’’ is shown to yield valid velocimetry data, within certain limits, for a 2D compressible turbulent boundary layer and an axisymmetric turbulent helium jet in air. Effective ‘‘schlieren PIV’’ calls for the minimum knife-edge cutoff compatible with the visibility of fine-scale turbulence. Due to optical path integration, axisymmetric flows require the inverse Abel transform to extract center-plane velocity data. Path integration also causes velocimetry errors due to end effects in the boundary-layer experiment. In both flows studied here the evolutionary time scale of turbulent eddies was much longer than the proper time separation between the images in a PIV pair, so eddy evolution did not limit the ‘‘schlieren PIV’’ measurements. Despite its limitations, the method shows promise as a new optical velocimetry tool. ‘‘Schlieren PIV’’ combines commercially available PIV equipment and software
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with a standard schlieren optical instrument. It requires no actual particle seeding of the flow, since it uses fine-scale turbulence as PIV ‘‘particles.’’ Otherwise it functions much the way standard PIV does under conditions where individual particles are not resolved and velocimetry is instead based on correlation of the motion of turbulent structures. In its present embodiment, ‘‘schlieren PIV’’ is not useful for laminar flows or for fully 3D flows, but it shows significant promise for general refractive turbulent flow velocimetry if its integrative nature can be overcome through sharp-focusing schlieren optics [20,23]. A study of that possibility is recommended for future work, as is the potential use of simple means to generate non-coherent schlieren illumination directly from coherent PIV laser pulses [24]. Additional information on the present study is given in [25].
References [1] Raffel M, Willert C, Kompenhans J. Particle image velocimetry. Berlin: Springer; 1998. [2] Raffel M, Richard H, Meier GEA. On the applicability of background oriented optical tomography for large scale aerodynamic investigations. Exp.Fluids 2000;28:477–81. [3] Papadopoulos G. Novel shadow image velocimetry technique for inferring temperature. J Thermophys Heat Transfer 2000;14(4):593–600. [4] Townend HCH. A method of airflow cinematography capable of quantitative analysis. J Aero Sci 1936;3(10):343–52. [5] Papamoschou DI. A two-spark schlieren system for very-high velocity measurement. Exp Fluids 1989;7(5):354–6. [6] Tokumaru PT, Dimotakis PE. Image correlation velocimetry. Exp Fluids 1995;19:1–15. [7] Fu S, Wu Y. Detection of velocity distribution of a flow field using sequences of Schlieren images. Opt Eng 2001;40(8):1661–6. [8] Fu S, Wu Y. Quantitative analysis of velocity distribution from schlieren images. In: Carlomagno GM, editor. Proceedings of the eighth international symposium on flow visualization, Sorrento. Paper 233, 1998. [9] Fu S, Wu Y, Kothari RD, Xing H. Flow visualization using the negative-positive grid schlieren system and its image analysis. In: Grant I, Carlomagno GM, editors. Proceedings of the ninth international symposium on flow visualization, Edinburgh. Paper 324, 2000. [10] Kegerise MA, Settles GS. Schlieren image-correlation velocimetry and its application to freeconvection flows. In: Grant I, Carlomagno GM, editors. Proceedings of the ninth international symposium on flow visualization, Edinburgh. Paper 380, 2000. [11] IDT Inc., Tallahassee, FL, USA, http://www.idtpiv.com. [12] Lourenco LM, Krothapalli A. TRUE resolution PIV: a mesh-free second order accurate algorithm. In: Proceedings of the international conference on applications of laser fluid mechanics, Lisbon, 2000. [13] Settles GS. Schlieren and shadowgraph techniques, 1st ed. Berlin: Springer; 2001. [14] Model 1531-AB Genrad Strobotac, IET Labs, Inc., (516) 334-5959, [email protected], http:// www.ietlabs.com/. [15] Tronosky MD. Schlieren PIV. MS thesis, Department of Mechanical and Nuclear Engineering, Penn State University, 2003. [16] Oppenheim AK, Urtiew PA, Weinberg FJ. On the use of laser light sources in schliereninterferometer systems. Proc Roy Soc A 1966;291:279–90. [17] Keane RD, Adrian RJ. Optimization of particle image velocimeters. Part I: double pulsed systems. Meas Sci Technol 1992;1:963–74. [18] Watt DW, Donker Duyvis FJ, van Oudeusden BW, Bannink WJ. Calibrated schlieren and incomplete Abel inversion for the study of axisymmetric wind tunnel flows. In: Grant I, Carlomagno
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GM, editors. Proceedings of the ninth international symposium on flow visualization, Edinburgh. Paper 363, 2000. Witze PO. Centerline velocity decay of compressible free jets. AIAA J 1974;12(4):417–8. Garg S, Settles GS. Measurements of a supersonic turbulent boundary layer by focusing schlieren deflectometry. Exp Fluids 1998;25:254–64. White FM. Viscous fluid flow, 2nd ed. Boston: McGraw-Hill; 1997. Coles D. The young person’s guide to the data. In: Proceedings of AFOSR-IFP-Stanford Conference Computation Turb BLs, vol. 2. Stanford University Press; 1968. p. 1–45. Alvi FS, Settles GS, Weinstein LM. A sharp-focusing schlieren optical deflectometer. AIAA paper 93-0629. Murphy MJ, Adrian RJ, Elliott GS, Thomas KA, Kennedy JE. Visualization in extreme environments. In: Mueller TJ, Grant I, editors. Proceedings of the 11th international symposium on flow visualization, Notre Dame University paper 155, 2000. Jonassen DR. Schlieren PIV and its application to two-dimensional and axisymmetric turbulent flows. MS thesis, Department of Mechanical and Nuclear Engineering, Penn State University, 2004.
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Schlieren ‘‘PIV’’ for turbulent flows Dennis R. Jonassen, Gary S. Settles, Michael D. Tronosky Gas Dynamics Lab, Mechanical and Nuclear Engineering Department, Penn State University, University Park, PA 16802, USA Available online 31 May 2005
Abstract The possibility of using commercial PIV equipment combined with schlieren optics to measure the velocity fields of turbulent flows is explored. Given a sufficiently high Reynolds number and adequate refractive flow differences, turbulent eddies can serve as the PIV ‘‘particles’’ in a schlieren image or shadowgram. The PIV software analyzes motion between consecutive schlieren or shadowgraph frames to obtain velocity fields. Velocimetry examples of an axisymmetric sonic helium jet in air and a 2D turbulent boundary layer at Mach 3 are shown. Due to optical path integration, axisymmetric flows require the inverse Abel transform to extract center-plane velocity data. Conditions for optimum schlieren sensitivity are examined. In its present embodiment, ‘‘schlieren PIV’’ is not useful for laminar flows nor for fully 3D flows. Otherwise it functions much like standard PIV under conditions where individual particles are not resolved and velocimetry is instead based on correlation of the motion of turbulent structures. ‘‘Schlieren PIV’’ shows significant promise for general refractive turbulent flow velocimetry if its integrative nature can be overcome through sharpfocusing optics. r 2005 Elsevier Ltd. All rights reserved. Keywords: Schlieren imaging; PIV; Velocimetry; Turbulent flows; Jets; Boundary layers
1. Introduction and literature review Particle image velocimetry (PIV) has recently become the most important new tool of experimental fluid dynamics. Two consecutive images of a particle-laden flow are Corresponding author. Tel.: +1 814 863 1504; fax: +1 814 865 0118.
E-mail address: [email protected] (G.S. Settles). 0143-8166/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2005.04.004
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interrogated for small particle displacements, from which a slice through a velocity field is measured. This depends upon modern digital camera equipment and desktop computing capability previously beyond the reach of the individual investigator, but now commercial PIV systems with convenient user interfaces are available to anyone who can afford them. One of the main issues in PIV is seeding the flowfield with particles. These particles should match the density of the fluid in order to avoid gravitational forces, and at the same time must not change the flow dynamics [1]. However, if the flowfield is sufficiently turbulent and this turbulence involves refractive-index changes, then turbulent eddies themselves might act as ‘‘PIV particles’’ when the flow is viewed using either schlieren or shadowgraph methods. In such a case the flowfield is selfseeding. An underlying assumption here is that the evolutionary time scale of the eddies is much longer than the time separation of the images in a PIV pair, otherwise no PIV correlation can occur. Likewise some flow symmetry, e.g. planar or axisymmetric, is required in order that integrative optical methods may present an interpretable view of a flow. The aim of the present approach is to explore such velocimetry, not the measurement of the refractive-index field per se. The use of the schlieren method and PIV software for the related purpose of refractive-index or temperature measurement is described by others, e.g. [2,3]. Schlieren velocimetry was first tried in 1936 by Townend [4], but was not pursued in that pre-computer era. In 1989, Papamoschou [5] used a pattern-matching algorithm to compute the convective velocity of a supersonic shear layer from two consecutive schlieren images. Since then, Tokumaru and Dimotakis [6] proposed an ambitious method of image correlation velocimetry for fluid flows, while Fu and Wu applied image analysis to extract velocity fields from sequences of schlieren images [7–9]. Kegerise and Settles also performed schlieren velocimetry on a convective plume [10]. All the cited velocimetry examples require specific ‘‘home-grown’’ algorithms to extract the desired velocity data from schlieren image sequences. In contrast, the present goal is to explore the validity of using a commercially available PIV system and software to measure velocity fields from schlieren images and shadowgrams.
2. Experimental equipment The PIV system used here is a single-camera system made by IDT Inc. [11]. Its software offers two different algorithms for extracting displacements from an image pair: standard and adaptive interrogation modes. The former is the cross-correlation approach most often used in PIV, but is prone to errors such as loss of pairing, image truncation, and spatial averaging of velocity gradients. The adaptive interrogation mode [12] is designed to reduce or avoid such errors. Although both algorithms produce similar results in the present test cases, only the adaptive interrogation mode has been used in obtaining the results presented here. Insofar as schlieren ‘‘PIV’’ measurements are concerned, comparable equipment and software by other manufacturers is expected to function similarly.
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Three different schlieren optical instruments were used in this study, one lensbased and two mirror-based systems [13]. The lens-based system had two 152 mm diameter f =5:67 telescope objective lenses as field elements. The mirror-based systems had z-type layouts with twin f/8 parabolic mirrors of either 108 or 152 mm diameter as field elements. Special schlieren light sources are required in order to provide pulsed illumination for PIV. For low- to moderate-speed flows, inexpensive microsecond white-light xenon strobe lamps suffice, e.g. [14]. Two such lamps are mounted perpendicularly to one another as shown in Fig. 1. A 50/50 beamsplitter combines the beams from these lamps and directs them, via a condenser lens and slit, along the optical axis of the present 108 mm-aperture z-type schlieren system. Neutral density filters are required after each bulb to balance the illumination from the two flashes. One flashbulb is triggered by PIV trigger signal A from the controlling computer and timing box, the second by trigger signal B. In our case, since the PIV system is designed for pulsed lasers, proper strobe timing required an external delay circuit between the PIV timing box and the first strobe lamp [15]. For high-speed flows requiring time separations less than 5 ms between images A and B of a PIV pair, strobe illumination fails and pulsed laser illumination is required. This is provided by a New Wave Gemini 200 dual-head Nd:YAG laser. This powerful laser (200 mJ/pulse) needs strong attenuation for use as a schlieren illuminator, here accomplished by two beam reflections from the planar sides of fused silica plano-concave lenses. Finally, a beam expander overfills the first mirror of the 152 mm-diameter z-type schlieren system to yield approximately uniform laser illumination. Dual pulsed-laser illumination has the distinct advantage of a virtually unlimited range of pulse-separation timing. However, coherent laser light produces schlieren images that are inferior to those produced by non-coherent white light for reasons given in [13] and [16]. Briefly, the geometric-optical approach to schlieren system performance breaks down and a very small focal spot occurs in the cutoff plane. The first issue produces coherent artifact noise and a reduced signal-to-noise ratio compared to the usual evenly illuminated white-light schlieren image. The second
Fig. 1. Diagram of the twin strobe arrangement for schlieren PIV.
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issue rules out the traditional knife-edge cutoff for laser illumination, requiring instead one of several milder cutoff options (graded filter, partially transmitting cutoff, etc.). Here, a simple half-plane sooted microscope slide cutoff [13] is used. The schlieren sensitivity using this cutoff is determined by the optical transmission of the sooted section, which is measured by transmission densitometry. Within limits, greater optical transmission of the cutoff yields less sensitivity in a laser-illuminated schlieren system and vice versa. By comparison, schlieren sensitivity with a traditional extended white-light source and a knife-edge cutoff is determined by the amount of cutoff and the width of the light source image. All of the present schlieren instruments are easily converted to ‘‘focused’’ shadowgraphy [13] by the removal of the cutoff, image focus adjustment, and reduction of the source slit size to an effective ‘‘point’’ in the case of white-light illumination. Of course, these optical methods present somewhat different views of a flowfield, but both reveal refractive turbulence, thus both are included in principle under the general name ‘‘schlieren PIV.’’ Two different turbulent flows, both 2D in the mean, are used here to test the validity of the schlieren PIV concept: a planar turbulent boundary layer and an axisymmetric turbulent jet. The boundary layer is formed on the test section floor of the Penn State Supersonic Wind Tunnel, a cold-flow blowdown facility with a 15 16.3 60 cm test section, up to 2 min test duration, and a Mach number range of 1.5–4.0. Present tests are conducted at Mach 3, boundary-layer thickness d ¼ 25 mm, and momentum-thickness Reynolds number Rey ¼ 98000. The lens-type schlieren system described above was used in these experiments. A converging nozzle of exit diameter 0.787 mm, discharging helium into ambient air, produces the axisymmetric jet tested here. The nozzle stagnation pressure of 208 kPa yields a choked helium jet with a theoretical exit velocity of 890 m/s and a Reynolds number of about 7300. The small nozzle scale allows a large nondimensional jet length to be studied (up to about 200 nozzle diameters downstream) within the fields-of-view of the z-type schlieren instruments employed here.
3. Schlieren sensitivity to turbulence As in traditional PIV, image quality is a key issue in schlieren PIV. The image scale and resolution are determined by the nature of the CCD sensor in the PIV camera and by the lens employed, and vary with the scale of the flow under study. Aside from these traditional concerns, however, schlieren PIV has additional image quality issues arising from the sensitivity of the optics to the refractive disturbances in the turbulent flows under investigation. Schlieren sensitivity is here analogous to particle size selection and seeding density in traditional PIV. For both schlieren and shadowgraphy an optimum sensitivity range can be defined over which the best PIV results are had. In simple geometric–optical terms, schlieren sensitivity S is given by Eq. (1) (adapted from [13]), where a is the width of the unobstructed light-source image in the schlieren cutoff plane, b is the width of the entire light-source image, and f 2 is the
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focal length of the second schlieren field lens or mirror. Since f 2 and b are typically fixed for a given schlieren instrument, we may conveniently discuss sensitivity in terms of the percent of knife-edge cutoff of the light-source image. For example, 100% cutoff refers to the complete blockage of the source image by the knife-edge (a situation in which Eq. (1) predicts infinite sensitivity, the geometric–optical description fails, and diffraction effects take over [13]). S¼
f2 f2 . ¼ a b 1 %cutoff=100
(1)
For the white-light strobe-illuminated schlieren system described earlier, a razor blade knife-edge oriented perpendicular to the streamwise direction of the flow is used as the cutoff. Fig. 2 shows the results of the range of cutoffs used with the present helium jet. For 10% and 20% cutoff only an underlying very-slight shadowgraph effect is seen. These images contain insufficient information for PIV processing. On the other hand, high-cutoff images result in over-ranging, shown for example by the entirely white regions in the 90%-cutoff image in Fig. 2. When overranging occurs the local flow details are washed out, which is likewise detrimental to schlieren PIV. To avoid over-ranging, especially when high cutoff is required by a need for high schlieren sensitivity, one can increase the overall measuring range by increasing the light-source slit width b. Better solutions, such as replacing the knife-edge by a graded filter [13] are also available. The optimum case for schlieren PIV requires sufficient sensitivity to reveal the smallest turbulent structures without significant over-ranging. This occurs, for example, in the 30–60% cutoff range in Fig. 2. Here, unlike the case in some other schlieren applications, it is the fine-scale turbulence—PIV ‘‘particles’’—that must be revealed. As noted earlier, laser schlieren illumination requires a less-severe cutoff than the knife-edge just described, whose optical transmission f is 0%. In Fig. 3, different halfplane-sooted microscope slides are used as cutoffs to vary the schlieren sensitivity. Cutoffs with fp26%, being almost as opaque as a knife-edge, yield binarized images in which some or all of the fine turbulence scales are washed out. Likewise the f ¼ 93% cutoff is almost transparent, yielding practically no cutoff and a vanishing schlieren effect. Therefore, the present optimum sensitivity range for laser-illuminated schlieren with partially transmitting cutoffs is roughly 50%ofo90%. Coherent artifact noise is quite apparent in Fig. 3. It has the undesirable effect in schlieren PIV of obscuring some of the fine-scale turbulent motion. Moreover, with twin pulsed-laser illumination a pseudo-velocity can result from the coherent artifact noise if the two laser beams are not aligned with perfect coincidence. As already noted, ‘‘focused’’ shadowgram pairs can also be used for PIV analysis. Here, since shadowgrams involve no cutoff, the product of the focus offset distance L and the Laplacian of the refractive-index field dictates the sensitivity [13]. Fig. 4 shows an example of laser-illuminated shadowgrams of the helium jet with L ranging from 0 to 130 cm. As expected, the sharply focused case yields no usable information for PIV analysis. Shadowgraph PIV becomes possible in this example when L
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Fig. 2. Helium-jet white-light schlieren images with different amounts of knife-edge cutoff (b ¼ 0:86 mm source-slit width).
Fig. 3. Helium-jet laser-schlieren results with different optical transmission values f of the half-plane sooted microscope slide cutoff: (a) f ¼ 13%, (b) 26%, (c) 61%, (d) 72%, (e) 88%, and (f) 93%.
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Fig. 4. Helium-jet shadowgrams with increasing sensitivity: (a) L ¼ 0 cm, (b) 12.5 cm, (c) 38 cm, (d) 53 cm, and (e) 130 cm.
exceeds 12 cm, but fails for L4130 cm because the increasing blur eventually obscures the fine-scale turbulence that serves as PIV ‘‘particles.’’ These L limits will be different, of course, for schlieren objects of different strength than the present helium jet. Note especially that a shadowgram is not a focused image, and that coherent artifact noise in the case of laser illumination can pose the same troublesome issue of pseudo-velocity as described above for schlieren PIV. To better understand exactly how PIV algorithms ‘‘perceive’’ schlieren images, 120 120 pixel turbulence samples were taken from each image in Fig. 2 and converted into binarized 2D intensity maps, which were then analyzed for the number and size of the ‘‘particles’’ within a 24 24 pixel interrogation window. For simplicity, no window overlap was used in the determination of the number of particles per window, and the particles found at the edges of the window were included in the particle count. As shown in Fig. 5, the number of apparent ‘‘particles’’ decreases with increasing knife-edge cutoff, while the average ‘‘particle’’ size increases. Keane and Adrian [17] proposed a method to quantify the minimum number of particles required in a PIV interrogation window. It is based on the number of particles N, the out-of-plane displacement F 0 , and the in-plane displacement F i . In order to achieve 95% accuracy in the probability of displacement detection, they claim that the product of these three parameters should be larger than five. However, Raffel et al. [1] claim that a value of three or four for this product is sufficient for practical situations. When applied to schlieren PIV, F 0 equals unity since all ‘‘particles’’ are always in the ‘‘plane.’’ F i can likewise be set essentially to unity when the adaptive interrogation mode is used. Thus the number of schlieren ‘‘particles’’ in the PIV interrogation window should be at least four. Fig. 5 reveals that a knife-edge cutoff of 50% or less thus ensures an adequate number of ‘‘particles’’ in a 24 24 pixel window for the schlieren equipment used here (f 2 ¼ 864 mm, b ¼ 0:86 mm). Cutoffs exceeding 50% may yield interrogation windows without ‘‘particles,’’ thereby requiring software interpolation to compute displacements.
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Particle size is likewise a key parameter in PIV analysis. According to Raffel et al. [1], the optimum particle diameter for conventional PIV is two pixels, based on a simulation of particles traveling one pixel per frame in a light sheet. In this controlled case, particle size is directly related to the RMS uncertainty of particle displacement. Above a particle diameter of two pixels, however, the RMS uncertainty increases exponentially, although it decreases as the interrogation window grows [1]. Based on this, ‘‘particle’’ diameters should be minimized for successful schlieren PIV, especially when small interrogation windows are needed for high spatial resolution. Fig. 5 shows that the present ‘‘particle’’ size increases almost linearly over the range of 10–60% cutoff, above which over-ranging of the helium-jet schlieren image begins. We therefore conclude that effective ‘‘schlieren PIV’’ calls for the minimum knifeedge cutoff compatible with ‘‘particle’’ visibility. One may anticipate extreme cases where a combination of low cutoff and a weak turbulent refractive field prevents the success of schlieren PIV. Even for the present strongly refractive helium jet, a cutoff in the ‘‘ideal’’ range of 10% produces insufficient contrast between the turbulence and the background, i.e. a signal-to-noise ratio that is too small to be effective. Thus the optimum schlieren sensitivity for PIV measurements in the present experiment occurs within the 30–60% range of knife-edge cutoff using white-light illumination.
4. PIV image processing Processing schlieren image pairs with commercially available PIV software is no different than in standard PIV. The first consideration is spatial resolution: high
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resolution of the velocity field requires small interrogation windows. But errors can result due to the ‘‘particle’’ size and concentration constraints noted above. Smaller windows also increase the probability of other errors such as loss of pairing and particle truncation. Both errors result in velocity underestimation [12]. The adaptive interrogation window described earlier alleviates most of these problems. To further increase the probability of detection, a PIV image offset may sometimes be required. This literally shifts the second image of the PIV pair with respect to the first by a predetermined integer pixel value. Such an offset helps the correlation algorithm to determine sub-pixel displacements, permitting the use of smaller interrogation windows. Certain situations further require that the PIV field be analyzed in zones, rather than as a whole. For example, high velocity gradients can produce neighboring regions of a flowfield requiring different image offsets (often by just a single pixel). This zonal evaluation technique was required in analyzing the present compressible boundary layer test case.
5. ‘‘Schlieren PIV’’ results 5.1. Helium jet The helium jet studied here has distinct near- and far-field zones. The near-field extends to about 15 nozzle diameters downstream of the nozzle exit. Its defining feature is its laminar core, containing no turbulent structures (particles) for schlieren PIV. The effect is clearly seen in Fig. 6: a double peak appears in the integrated velocity profile at about y ¼ 8 mm from the nozzle exit. The reason for this is the path-integration of the schlieren beam through this axisymmetric flow. A light ray through the edge of the laminar jet core incurs more refraction due to turbulence than one traversing the jet centerline. The former ray thus yields a greater apparent ‘‘particle’’ displacement and a higher velocity is found. No schlieren PIV is possible within y ¼ 1 mm of the nozzle exit, where the accompanying shadowgram reveals an initially laminar jet. Also, due to optical path integration and turbulent mixing, the measured velocities in the 40 m/s range in Fig. 6 are low compared to the theoretical jet nozzle-exit speed of 890 m/s. The far-field jet region begins around 15 diameters (12 mm) from the nozzle exit and is fully turbulent. Fig. 7 shows laser- and white-light-illuminated schlieren images and a laser shadowgram of this region. The white-light schlieren image, Fig. 7b, has a more severe knife-edge cutoff than the laser schlieren image, and both laser frames have coherent artifact noise that can lead to velocity errors. Nonetheless, each of these three varied depictions of the helium jet does in fact yield, with good accuracy, the same convective velocity field upon ‘‘schlieren PIV’’ analysis (150 image pairs at a time delay of 5 ms between paired images). The resulting mean velocity profile normal to the jet axis is given in Fig. 8. For comparison, traditional laser-sheet-illuminated PIV measurements were made using micron-sized oil droplets that were entrained into the helium jet by way of a
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Fig. 6. Laser shadowgram of the helium jet near field (left, L20 cm, nozzle exit diameter ¼ 0.787 mm) and the corresponding schlieren PIV velocity contour plot (right). Velocities are given in m/s.
Fig. 7. (a) Laser schlieren, (b) white-light schlieren, and (c) laser shadowgram of helium jet showing far-field.
slow coaxial seeded jet. Only far-field measurements were possible in this manner. Example flowfield images and PIV velocity results are compared in Fig. 9. It is clear from this comparison that traditional PIV yields higher velocities than schlieren PIV in this helium jet case, as expected. The difference is attributed to the pathintegration of the schlieren optics. Since the jet is axisymmetric, a proper comparison
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can be made by applying the Abel transform to the traditional PIV results, Eq. (2) (see, e.g. [18]). iP edge
V pa ði; jÞ ¼
V pl ði; jÞ
i¼io
N io !iedge
,
(2)
where V pa ði; jÞ is the path-averaged velocity, V pl ði; jÞ is the planar velocity, N io !iedge is the number of discrete points between io and iedge , and the subscripts ‘o’ and ‘edge’ represent the location of V pa ði; jÞ within the jet and the jet edge, respectively. The result is shown in Fig. 10, where the Abel transform applied to the traditional-PIV jet center-plane data recovers approximately the same velocity profile as that measured directly by ‘‘schlieren PIV.’’ Alternatively, the inverse Abel transform can be applied to the schlieren PIV results to yield equivalent jet center-plane data. In most cases this is the preferable procedure, since the equivalent center-plane data are the most useful form of the experimental results. The opposite procedure has been adopted here merely as a matter of convenience, since deconvolution of the data makes the inverse Abel transform approach somewhat more involved. Nonetheless modern digital computing power is more than sufficient for either of these procedures. To further test the schlieren PIV results, one may compare the jet centerline velocity distribution vs. y with traditional PIV results and with the extensive jet data
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Fig. 9. (Above left) white-light helium-jet schlieren image, (above right) particle-seeded image of the same flow with laser sheet illumination through the jet center-plane. Below the images are shown average PIV velocity contour maps for comparison (schlieren PIV left, traditional PIV right).
correlation by Kleinstein and Witze [19]: ! 1 uc ¼ 1 exp , 0:5 kx re Xc
(3)
where k ¼ 0:074, x ¼ x=rj , re ¼ re =rj (e and j refer to the ambient and jet exit conditions, respectively), and X c ¼ 0:7. The centerline velocity is normalized by the jet nozzle-exit velocity, 890 m/s.
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Fig. 10. Velocity half-profiles of the far-field helium jet at y ¼ 65 mm, comparing ‘‘schlieren PIV’’ with traditional particle PIV results with and without the application of the Abel transform to the latter.
The results show that the present helium-jet centerline velocity data, obtained by traditional PIV in two separate experiments, agree well with the Kleinstein–Witze correlation as shown in Fig. 11. Also, applying the Abel transform to the traditional PIV data yields results comparable to the ‘‘schlieren PIV’’ data, demonstrating once again that ‘‘schlieren PIV’’ gives a path-integrated representation of the axisymmetric velocity field. The noise level of the traditional PIV data in Fig. 11 makes it difficult to determine the exact jet edge location, which may account for some error incurred during the Abel transformation. 5.2. Compressible turbulent boundary layer The above helium-jet ‘‘schlieren PIV’’ results reveal the dominant effect of optical path integration in measuring an axisymmetric flowfield. No such effect is expected in a true 2D flow, however, where schlieren PIV and traditional PIV results should be directly comparable even though the former still integrates across the flow. To test this, the compressible turbulent boundary layer on the test-section floor of the Penn State Supersonic Wind Tunnel was measured by ‘‘schlieren PIV.’’ Given a span of 6d and end effects of less than one d, approximately 2D flow is expected. Note, however, that it is not possible with the current schlieren optics to eliminate end effects by focusing on the center-plane of this flow. Double-pulsed laser illumination through the lens-type schlieren system described earlier is the only approach that allows a sufficiently small time delay between PIV frames (0.55 ms) for this Mach 3 flow. To provide adequate schlieren sensitivity a
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Fig. 11. Helium jet centerline velocity decay as measured by traditional PIV, predicted by the KleinsteinWitze correlation, and measured by schlieren PIV with path integration.
half-plane sooted microscope slide with 72% light transmission was required. Given the 15 cm extent of the boundary layer along the optical axis, converting to shadowgraphy required only the removal of the knife-edge, without any focus adjustment. (In other words, the shadowgraph image is always present and is combined with the schlieren image when a schlieren cutoff is used.) An example of schlieren image and shadowgram are given in Fig. 12. In both cases fine-scale turbulence is observed, though it does not end as expected at the boundary-layer edge, but rather appears to continue into the freestream. This is, in fact, an end effect due to the wind-tunnel sidewall boundary layers on the glass windows. Slight over-ranging is also seen in the schlieren image, but this is not serious enough to affect the overall PIV results. One hundred and fifty schlieren PIV image pairs were processed, as before, using the IDT software’s adaptive mode with a 20 20 pixel interrogation window. However, since the boundary-layer velocity profile extends from 0 at the wall to 613 m/s in the freestream, the required image offset varies across the height of the boundary layer as discussed earlier, starting at 6 pixels near the wall and ending at 8 pixels near the freestream. Since three separate image-offset zones are needed to analyze this flow, three different meshes are also required. The zones with their mesh locations and corresponding image offsets are as follows: 1. 0.0py/dp0.17 6-pixel offset, 2. 0.14py/dp0.30 7-pixel offset, 3. 0.27py/dp1.01 8-pixel offset.
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Fig. 12. Example schlieren image (left) and shadowgram (right) of the Mach 3 turbulent boundary layer on the test section floor of the Penn State Supersonic Wind Tunnel.
Unfortunately, traditional particle PIV is not currently possible in this boundary layer due to particle seeding difficulties. Instead, the schlieren PIV results are evaluated by comparison with mean velocity profiles derived from pitot-pressure surveys by Garg and Settles [20], and with the established wake-wall similarity law for turbulent boundary layer profiles. The Van Driest transformation was used to convert measured compressible-flow velocity values, u, obtained from both the optical ‘‘PIV’’ and the pitot surveys, to equivalent incompressible values, u* (see, e.g. [21]). The results are plotted in Fig. 13 in traditional u+ vs. y+ coordinates along with the incompressible wall-wake law of Coles [22]. Fig. 13 reveals that the ‘‘schlieren PIV’’ data are in substantial agreement with the pitot-survey results for this boundary layer, even though the differences in the two measurement methods are striking. Schlieren PIV measures, in principle, the broadband convective velocity of turbulent structures directly. Pitot pressure surveys are converted, with the assumption of constant stagnation temperature, to mean Mach number profiles from which the mean velocity is extracted by way of an assumed static temperature profile. Agreement between the two indicates that schlieren PIV actually measures, by way of refractive eddy motion, the mean velocity profile of the boundary layer. Fig. 13 also reveals that this boundary layer has an unusually high wake-strength component, P ¼ 1:9 vs. the usual P0:55 for a flat-plate boundary layer [22]. This doubtless results from the pressure-gradient history along the wall of the wind tunnel’s long asymmetric variable-Mach-number sliding-block nozzle. There are two boundary-layer regions in which ‘‘schlieren PIV’’ is unable to measure the velocity satisfactorily: very near the wall and near the boundary-layer edge. Failure near the wall is due to poor spatial resolution caused by wind-tunnel vibrations and by laser diffraction that obscures the true location of the wall. Even so, schlieren PIV—being non-intrusive—measures closer to the wall than does the
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40 Pitot Pressure Survey Wall Law Shadowgraph PIV Schlieren PIV Wall-Wake Law
U+ = u*/uτ
35
30
M=3 Tw = 260 K δ = 25 mm cf = 8.98 x 10-4 uτ = 18.5 m/s νw = 2.52 x 10-5 m2/s Reθ = 98,000 Π = 1.9
25
20
15 102
103
104
105
Y+ = y(uτ/νw)
Fig. 13. Mean velocity profiles of the Mach 3 turbulent boundary layer in wall-wake coordinates.
pitot survey method. Near the boundary-layer edge, however, ‘‘schlieren PIV’’ fails by yielding velocities that are too low. The outer 20% of the turbulent boundary layer is quite intermittent, so the eddies serving as PIV ‘‘particles’’ are fewer there. Eddies in the sidewall boundary layers contaminate the integrated schlieren measurement with incorrect end-effect velocities in this region, as shown in Fig. 12. By way of focusing schlieren optics [20,23], it should be possible to eliminate these end effects and make an accurate measurement up to the boundary-layer edge, but that is beyond the present scope.
6. Conclusion ‘‘Schlieren PIV’’ is shown to yield valid velocimetry data, within certain limits, for a 2D compressible turbulent boundary layer and an axisymmetric turbulent helium jet in air. Effective ‘‘schlieren PIV’’ calls for the minimum knife-edge cutoff compatible with the visibility of fine-scale turbulence. Due to optical path integration, axisymmetric flows require the inverse Abel transform to extract center-plane velocity data. Path integration also causes velocimetry errors due to end effects in the boundary-layer experiment. In both flows studied here the evolutionary time scale of turbulent eddies was much longer than the proper time separation between the images in a PIV pair, so eddy evolution did not limit the ‘‘schlieren PIV’’ measurements. Despite its limitations, the method shows promise as a new optical velocimetry tool. ‘‘Schlieren PIV’’ combines commercially available PIV equipment and software
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with a standard schlieren optical instrument. It requires no actual particle seeding of the flow, since it uses fine-scale turbulence as PIV ‘‘particles.’’ Otherwise it functions much the way standard PIV does under conditions where individual particles are not resolved and velocimetry is instead based on correlation of the motion of turbulent structures. In its present embodiment, ‘‘schlieren PIV’’ is not useful for laminar flows or for fully 3D flows, but it shows significant promise for general refractive turbulent flow velocimetry if its integrative nature can be overcome through sharp-focusing schlieren optics [20,23]. A study of that possibility is recommended for future work, as is the potential use of simple means to generate non-coherent schlieren illumination directly from coherent PIV laser pulses [24]. Additional information on the present study is given in [25].
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