th
5 International Symposium on Particle Image Velocimetry Busan, Korea, September 22-24, 2003
PIV’03 Paper 3125
Application of non-isotropic resolution PIV in supersonic and hypersonic flows 1
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F. Scarano and J. Haertig 1 2
Delft University of Technology - Aerospace Engineering Department – 1 Kluyverweg, 2629 HS, Delft, NL Institut Franco-Allemand de Saint-Louis (ISL) – Rue du Général Cassagnou, F-68301 Saint-Louis, FR
Abstract The non-isotropic image interrogation technique is applied to study the compressible supersonic and hypersonic flow regimes. Supersonic flow experiments are performed in a blow-down windtunnel where the wake turbulence downstream of a wedge-plate model configuration is studied in the transonic M=1.35 regime. The hypersonic experiments are conducted in a short duration facility at Mach 6 and the flow over a sphere is studied. The technical challenges to achieve PIV measurements in compressible flows are discussed with respect to seeding techniques, particle imaging and the analysis of the recordings. Particular attention is given to the analysis of flow features such as shock waves and compressible shear layers with the possibility to enhance the spatial resolution by means of an adaptive interrogation scheme. The image interrogation method is based on the adaptation of the interrogation window shape (elliptical) and its orientation with respect to the velocity components spatial curvature. The analysis of the hypersonic flow case is performed with a fixed and free window method respectively showing that shock waves may be better resolved with the non-isotropic approach. In the case of the supersonic wake flow the measurements of the compressible turbulent separated shear layer attempts the characterization of the shear layer spatial growth rate. 1
Introduction
Velocity measurements in compressible flow regimes constitute a technical challenge due to the presence of a wide range of flow time and length scales, which are orders of magnitude wider than that encountered in subsonic flows. The occurrence of compressible flow phenomena such as shock and expansion waves determines the smallest length scale (of the order of the molecular free path) in the flow. Seeded velocimetry techniques such as PIV and Doppler Global Velocimetry (DGV) are often limited in temporal resolution, due to finite time response of seeding particles. When seeding particles cross a shock wave the deceleration to the flow velocity downstream of the shock follows an exponential decay in time (Melling, 1997). The slip velocity is maximum just after the shock location. The error associated to the particle velocity lag can be dominating (Tedeschi et al 1999, Lang 2000), which becomes the limiting physical factor when the hypersonic flow regime is considered. Other limiting factors are introduced in the process of double pulse illumination and recording. Although the pulse duration of current Nd:Yag lasers is sufficiently short (below 10 ns) to freeze the particle images in their position almost instantly, the time elapsing between two pulses is limited by the minimum required to acquire the particle images on separate frames. Current CCD captors based on the frame interline transfer technology allow to record two images within a time interval sufficiently short (about 1 µs) for supersonic flow conditions, however the limit becomes constraining when the flow velocity exceeds 1000 ms-1 and a time delay below the micro-second must be chosen (Haertig et al. 2002). The situation becomes even more challenging when the flow length scales are considered. The PIV technique is based on the interrogation of the particle image recordings by means of correlation windows, which should contain a sufficient number of particles images. The resulting spatial resolution is therefore limited by the spatial density of particle images (Willert and Gharib, 1991). Since the velocity returned from window cross-correlation is an ensemble average of the particles velocity, the result F. Scarano Author, Department of Aerospace Engineering, Delft University of Technology, 2629HS Delft, The Netherlands J. Haertig, French-German research Institute of Saint-Louis, 68301 Saint-Louis, France Correspondence to: F. Scarano Author, Department of Aerospace Engineering, Delft University of Technology, 2629HS Delft, The Netherlands Kluyverweg 1, 2629 HS Delft, The Netherlands, E-mail:
[email protected]
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can therefore be considered as a spatially low-pass filtered version of the instantaneous velocity distribution even in case of ideal particle tracers (instantaneous response to velocity variations). In case of a shock wave the exponential decay of real particle tracers is then smeared and the maximum error occurs close to the shock location. In these conditions poor spatial resolution leads to overestimate the particle tracers’ response and in turn their physical properties (mass and dimension). It is therefore important to achieve the highest possible spatial resolution when measuring the velocity across shock waves. Several methods were proposed in literature, which attempt the improvement of PIV spatial resolution (Keane et al. 1995; Hart 2000 among others). The current study focuses on the application of an adaptive resolution interrogation technique based on the nonisotropic interrogation windows (Scarano 2003). The latter is particularly appealing in compressible flows because shock waves are strongly non-isotropic velocity fluctuations. According to the above study the spatial resolution of PIV interrogation techniques can be enhanced significantly when the velocity fluctuations occur along a preferential direction (e.g. shock waves and shear layers). The present study applies the concept of nonisotropic spatial resolution to the above flow features in relation to supersonic and hypersonic flow experiments. First the detached bow shock produced over a sphere is considered at high value of the Mach number. The investigation focuses on the particle velocity relaxation. The second application is related to the turbulent compressible flow past a two-dimensional base and the focus is on the separated shear layers. 2
The non-isotropic interrogation technique
In several flow situations, the radius of curvature of the measured velocity fluctuations has not the same extent in all directions. We can recall for instance the flat plate laminar boundary layer flow, where the velocity spatial derivatives in the stream wise direction can be neglected with respect to those in the wall-normal direction. In such a case for a given value of the image density NI the measurement volume can be arranged with a preferential direction parallel to the wall (Figure 1). For the present example the choice of the user is straightforward, however for a general velocity field the selection of the optimum interrogation window shape and orientation is not trivial.
Figure 1 – Non-isotropic measurement volume. Most flow measurement techniques are based on a non-isotropic measurement volume, which is arranged so that the largest dimension is aligned with the direction where the flow is most uniform (e.g. boundary layer Pitot pressure probe, hot wire anemometer, laser Doppler velocimeter). In PIV, the idea of re-arranging or weighting the interrogation windows in order to improve the measurement of the in-plane particle motion is not new in itself. Lecordier et al. (1999) proposed the re-orientation of square windows in the flow direction in order to minimize the effect of the velocity gradient along the window diagonal. Di Florio et al. (2002) proposed a windowing, re-shaping and re-orientation method stretching the window size in the direction of the flow and proportionally to the measured displacement. The adaptive resolution scheme proposed by Scarano (2003) follows that of Di Florio, however with an important difference, which is the criterion used for the window adaptation. It is demonstrated that the spatial resolution is actually improved only if the interrogation windows are adapted on the basis of the velocity spatial fluctuations. 2.1
Velocity spatial fluctuations
In this section a mathematical procedure is briefly described, which returns the spatial distribution of the velocity fluctuations in term of spatial curvature. The procedure is applied to each of the velocity components separately.
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The following derivations are made for the horizontal velocity component U. The Hessian tensor H is composed by the second spatial derivatives of U(x,y):
U xx U xy H = U xy U yy
(2-1)
The maximum and minimum radius of curvature rmax and rmin of the velocity spatial distribution are the reciprocal of the eigen-values of H, λmin and λmax, aligned along perpendicular eigen-directions θmin and θmax respectively. These parameters are necessary to establish the direction of the minimum and maximum curvature radii and their ratio. The operating principle of the non-isotropic interrogation is based on three important criteria: 1) the window aspect ratio AR = lmin / lmax and orientation should be chosen so as to minimize the error associated to the local curvature of the velocity field. The criterion is implemented associating the eccentricity e = (1 - AR) of the elliptical-shaped window to the eigen-values ratio λmin/ λmax. 2) it is necessary to avoid degenerate cases when the eigen-values both tend to zero (e.g. in uniform velocity flow). The criterion is implemented restricting the eccentricity range to the interval [0, ¾]. 3) the window eccentricity must tend to zero when the minimum radius of curvature rmin = 1/λmax of the velocity spatial fluctuations becomes significantly larger (order of magnitude) than the size of the interrogation window. This criterion is implemented applying an exponential decay law to the eccentricity with rmin/l as governing parameter (l is the window linear size). The final expression for the eccentricity reads as:
λ e = 3 4 1 − min λmax
rmin l ⋅ exp − σr
(2-2)
With a fixed value of the coefficient σr = 102. The elliptical window semi axes α and β are obtained respecting the equivalence on the window area (NI is kept constant). The ellipse equation in the image x-y frame of reference is obtained after rotation 2 2 2 2 cos (θ ) sin (θ ) 2 sin (θ ) cos (θ ) 1 1 + + + + − G ( x, y ) = exp − x 2 y xy 2sin θ cos θ ( ) ( ) 2 2 α2 β2 α2 β2 β α c a b
(2-3)
The above equation is applied to shape the non-isotropic interrogation windows, for which different weighting schemes can be applied. In the present study, the elliptical shape of the interrogation windows is achieved by means of a Gaussian weighting function. In a previous study the weighting function has been applied directly to the interrogation windows, which involved a significantly higher computational effort (8 times higher). In the present study a faster implementation of the technique has been adopted. The weighting coefficients are applied to the sub-window correlation maps obtained dividing each correlation window in smaller blocks. As a result the correlation is computed only once for both velocity components and the amount of computation is almost independent of the overlap factor. In conclusion the present implementation requires practically the same computational effort of the WIDIM algorithm (WIndow Deformation Iterative Multi-grid, Scarano and Riethmuller, 2000). The method assessment with synthetic particle images of known velocity distribution returned a maximum relative improvement when a purely non-isotropic velocity distribution was considered, such as in the case of particle motion across a normal shock wave. Figure 2 shows the simulated behaviour of the WIDIM interrogation with and without the non-isotropic approach. The adapted correlation windows are circular in the uniform flow region and they become increasingly eccentric across the shock where the velocity curvature is non-zero along the stream wise direction. The non-isotropic interrogation with windows of 41 pixels diameter compares well with the conventional interrogation performed with a 21 pixels window diameter.
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Figure 2 – Performances assessment from synthetic PIV images. Tracers velocity profile across a shock wave (Left). Actual velocity distribution (solid line) and cross-correlation analysis with and without non-isotropic interrogation windows. Velocity contours and non-isotropic interrogation window pattern (right). 3
Experimental apparati
The experimental apparati developed to perform the experiments related to the present study are described with details in Havermann et al. (2002) and Scarano and van Oudheusden (2003). 3.1
Short duration hypersonic flow facility
The ISL shock tunnel STA consists of a 100 mm diameter shock tube connected to a convergent-divergent nozzle and a 10 m3 dump tank. The test section has optical access from three sides allowing flexible illumination and recording of the seeded flow. A 2.7 m long high-pressure section (driver tube) is separated by the 18.4 m long low-pressure section (driven tube) by means of a single stainless steel diaphragm. The driver gas is a mixture of hydrogen and nitrogen at a pressure of 50 MPa, whereas pure nitrogen is used in the driven section at a pressure of 0.5 MPa. The shock tunnel is operated in shock reflection mode with tailored interface condition resulting in duration of the flow of 1 to 2 ms. The flow expands along a contoured Laval nozzle to a Mach number of 6. Flow seeding is accomplished introducing Al2O3 particles with a nominal diameter of 0.3 µm. A fluidised-bed seeder is operated with nitrogen and the seeded fluid is introduced in the driven tube before the experiment is performed. The illumination is obtained with a double Nd:YAG laser (Quantel Twins) with 140 mJ pulse energy. The vertical light sheet has a thickness of 0.2 mm and is 300 mm wide. The pulse separation is chosen at 400 ns. Two separate recordings are obtained using a sharpVISION 1300 DE camera (1280×1024 pixels) by IDT. A Nikon zoom objective was used to change the field of view. The particle image recordings are analysed with the non-isotropic technique with a nominal window size of 32×32 pixels. The images are interrogated with a grid spacing of 8 pixels in each direction. Figure 3 shows the particle image recording obtained in the ISL shock tunnel. The two exposures are superposed with different colour channels (red for the first exposure and bluegreen for the second exposure). The clear challenge for the analysis of this type of recordings is not only given by the very sharp velocity profile across the shock but also in the large difference in seeding density due to flow compressibility.
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Figure 3 – PIV recording of the Mach-6 flow around a 120 mm diameter sphere. Top-left corner: schematic of the ISL shock-tunnel facility STA. 3.2
Blow-down supersonic wind tunnel
Experiments are performed in the transonic-supersonic wind tunnel (TST-27) of the High-Speed Aerodynamics Laboratories at Delft University of Technology. The facility generates flows in a Mach range from 0.5 to 4.2 in the test section (300-W × 270-H mm2). The tunnel operates at values of the unit Reynolds number ranging from 38 × 106 to 130 × 106 m-1. The 14500 kg of stored dry air is sufficient for the blow-down use of the wind-tunnel for 300 s. The two-dimensional model producing the base flow consists of a symmetrical double wedge with sharp leading edge imposing a flow deflection of 11.31º, followed by a thick plate 50 mm long and of constant thickness h = 20 mm. The plate terminates sharply with a vertical base. The model spans the entire section width. In the present experiments the wind-tunnel is operated at a stagnation pressure P0 = 204 kPa. The Mach number over the thick plate is then M∞ = 1.35 as measured by the PIV measurements. 1
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Figure 4 – Schematic of the seeding distribution device and light sheet delivery in the TST wind-tunnel. 1) Air+seeds inlet from seeding generator; 2) multi-orifices distribution pipe in the settling chamber; 3) seeded air stream; 4) camera optical window and model; 5) laser sheet; 6) reflecting prism; 7) laser light optical window. The colour Schlieren photograph in Figure 5 illustrates the Mach-2 flow around the wedge-plate model. The density gradient is visualized in red, green and blue for vertical lower, vertical upper and horizontal light beam deflection respectively. A second expansion occurs when the flow separates at the model base. The free shear layers developing downstream are visualized as blue streaks.
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Figure 5 – Color Schlieren photograph of the supersonic flow around the wedge-plate model (left). Particle response across the oblique shock wave (right). Labels are actual measurements and lines are curve fits. A fraction of the flow is seeded in the settling chamber by means of a two-dimensional rake seeding distributor producing a seeded stream-tube of flow of approximately 6-H × 3-W cm2 in the test section (see Figure 4). Under operating conditions, the seeded flow exhibits a mean particle concentration of about 10 particles/mm3, as estimated from single exposure PIV recordings. The particle-laden flow is produced by entraining Titanium dioxide TiO2 particles (dp = 0.27 µm, ρp = 4.0 × 103 kg m-3) with a high-pressure cyclone separator operated at a pressure of 10 bars. The particle relaxation time τp is evaluated from the measurement of the relaxation length/time and knowledge of the flow velocity behind the shock wave. The particle velocity across the OSW is measured by means of PIV (Figure 5-right). The particle relaxation length ξ is obtained with the aid of an exponential curve fit returning ξ = 0.76 mm. The effects of the finite spatial and temporal resolution are evident mostly around the shock location (s = 0). In fact, at about half a window size upstream (s = -0.28 mm) the measured velocity starts decreasing due to the averaging effect intrinsic to the PIV interrogation method. The exponential curve fit of the particle velocity versus time yields τp = 2.4 µs (fp = 417 kHz). The seeded air flow is illuminated by a double-cavity pulsed Nd:Yag laser with 400 mJ pulse energy and 6 ns pulse duration at a wavelength of 532 nm. A sheet 500 µm thick illuminates an area of about 300-L × 200-H mm2. The laser pulse separation ∆t is set to 1.0 µs yielding a reference particle displacement of about 0.5 mm in the free stream flow. The light is inserted in the flow from the bottom side of the tunnel and reflected towards the model with a prism (Figure 4). The light scattered by the particle tracers is imaged with a 60 mm focal length lens (f# = 5.6) on a 12 bit, Peltier-cooled charge coupled device (1280-W × 1024-H pixels CCD) digital camera. The measurement of the overall wake is performed with a field-of-view of 64-W × 50-H mm2 (50 µm/pixel). A data set of 200 images is acquired during a 60 s duration experiment. 4 4.1
Results Hypersonic flow
The measurement of the hypersonic flow (M = 6) over a sphere (Haertig et al. 2002) is proposed here as an example of application of the non-isotropic interrogation method. The flow velocity is uniform (1740 m/s) upstream of the curved shock formed in front of the sphere. Figure 6 top-left shows the velocity distribution as obtained from the non-isotropic interrogation technique. On the symmetry axis the flow is decelerated through a locally normal shock wave and then decelerates further in the subsonic flow region up to the stagnation point. The flow then accelerates along the sphere circumference where a boundary layer develops at the wall. The highest resolution is desirable across the shock wave and close to the surface of the sphere. Figure 6 top-right presents the result obtained with the isotropic interrogation technique. Square windows of 64×64 pixels are used with a Gaussian weighting function. The spatial resolution is equivalent to that of a top-hat square window of 32×32 pixels. The displacement distribution is qualitatively very similar to that obtained with the non-isotropic method, nevertheless the shock wave representation by the contours is slightly more smeared. The flow between the shock and the object surface does not show any visible difference. At the bottom of Figure 6 the eccentricity of the interrogation windows related to the two velocity components is displayed respectively. The eccentricity
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of the horizontal component peaks on the symmetry axis with two stripes of high eccentricity located at the shock and slightly downstream of it. Increasing the distance from the axis of symmetry, the shock strength weakens as well as the difference of the U-component across the shock. The eccentricity associated to the Vcomponent is zero on the symmetry axis and increases with the distance from the axis until the maximum flow deflection is reached. Shock wave theory predicts the maximum deflection at a wave angle β = 66 deg for M = 5. A slight window eccentricity is also found at the wall of the sphere where a strong curvature is expected normal to the wall for the tangential velocity component. However the result is less clear due to correlation noise ascribed to light scattering from the wall.
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Figure 6 – PIV measurement of the velocity distribution around a sphere in hypersonic (M = 6) flow. Top-left: velocity vectors and horizontal component contours (in pixel units) from the non-isotropic interrogation. Topright: conventional interrogation. Bottom-left: interrogation window eccentricity for the horizontal component. Bottom-right: eccentricity for the vertical component. A quantitative comparison is made extracting several velocity profiles across the shock at locations indicated by the profiles in Figure 6 and the result is shown in Figure 7. The normalised velocity profiles obtained with the isotropic analysis are given with solid lines, whereas the non-isotropic resolution data are plotted with dashed lines. Profile 1 corresponds to the stagnation streamline crossing the shock wave, where the difference between the two methods can be appreciated more clearly. Moving upward, the shock wave weakens and the difference
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between the dataset vanishes. The eccentricity peaks twice along each profile with a minimum located in between and corresponding to the inflexion point of the velocity distribution. Based on the peak eccentricity value of 0.6, a window of 20×50 pixels is expected in the vicinity of the shock. Given the relatively small difference between the different analysis methods, it may be concluded that for the present case the measurement accuracy is limited by the particle tracers relaxation length, which was already estimated by Havermann et al. (2002) as 2.1 mm (based on the 1/e2 criterion).
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Figure 7 – Normalised velocity profiles (horizontal component) across the bow shock. Solid line: isotropic resolution; dashed line non-isotropic resolution. Purple line: normal shock wave theory. Dash-dot line: profiles of eccentricity relative to the horizontal component. 4.2
Supersonic flow
The mean base flow is described in Figure 8 as obtained from an average of 200 instantaneous velocity fields. The Mach number before the expansion around the corner is M∞=1.35. The flow upstream of the base expansion is parallel and uniform turning and accelerating along the Prandtl-Meyer expansion fans emanating from the base corners. The flow deflection is about 12.6 degrees in the region surrounding the separated wake and the velocity increases with 21% with respect to the pre-expansion value. The mean flow in the separated region exhibits a nearly symmetrical recirculation pattern of two counter-rotating vortices. This supposedly dead-air region exhibits a maximum reversal velocity of 0.28 U∞ at x/h = 1.12. Spurious reflections were caused from direct laser light impingement on the object surface, therefore the particle images could be distinguished from a minimum distance of about 2 mm from the base. The velocity vector pattern reveals a roughly symmetrical base flow with the reattachment location downstream of the base at x/h = 1.62. Downstream reattachment the recompression pattern is formed. Figure 8-right shows the interrogation window eccentricity for the horizontal velocity component. Two stripes of high eccentricity follow the shear layers, corresponding to the convex and concave part of the velocity profile. The maximum is attained close to the separation region and decreases progressively downstream with the shear layer spreading.
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Figure 8 – Mean velocity over a compressible base flow. Left: horizontal velocity component (in pixel units) and vector field. Right: interrogation window eccentricity for the horizontal component. The application of the non-isotropic interrogation method focuses on the measurement of the separated shear layer properties such as thickness and spatial growth rate. In the present discussion, the reference shear layer length-scale is taken as the 10%-90% velocity thickness b and the growth rate is evaluated in the ξ-direction that is aligned with the shear layer (inclined at 12.6 degrees with respect to the x-direction). The analysis of the mean velocity distribution across the shear layers is shown in Figure 9. In the upper part the solid and dashed lines describe the spatial development of the shear layer axis and its boundary respectively. The shear layer thickness (circle-labels) is determined as the distance in the direction η normal to ξ between the high-speed and low-speed boundaries. The extracted values for the shear layer thickness are b = 1.6 mm at x/h = 0.25 and b = 2.55 mm at x/h = 0.9. The resulting growth rate is ∂b/∂ξ = 0.065, which compares satisfactorily with the “Langley curve” (Kline et al. 1981). Downstream of x/h = 1 the reversal flow in the recirculation region increases the velocity difference across the shear layer of about 30%, and downstream of x/h = 1.5 a recompression of the flow occurs, therefore the data reduction based on the velocity difference does not apply beyond x/h = 1.
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Figure 9 – The structure of the mean shear layers. Solid lines: shear layers axes. Dashed lines: 10%-90% velocity thickness. Circles: shear layer thickness.
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Conclusions
PIV recordings obtained during hypersonic and supersonic flow experiments have been analysed with an adaptive interrogation method based on non-isotropic resolution. The analysis of the Mach-6 flow around a sphere showed that spatial resolution is essential in order to accurately measure the particle velocity relaxation past the shock. High resolution is also necessary in order to limit the smearing effect of the interrogation windows, corrupting the exponential decay, therefore making impossible an accurate estimate of the shock location. The high level of eccentricity returned by the non-isotropic method demonstrated that the suitability of the method to this type of flow study. However in hypersonic experiments also the time separation between the two light pulses plays a role limiting the spatio-temporal resolution of the measurement. The measurements of the transonic base flow also revealed that a significant improvement could be achieved in terms of spatial resolution when the non-isotropic technique is applied. In this case the spatial velocity fluctuation is perpendicular to the flow direction and spatial and temporal resolution are de-coupled. A high level of eccentricity is found on the two sides of the shear layers, which seem to be the only features necessitating of adaptive resolution treatment in this flow study. Furthermore, the measurement resolution allowed measuring the shear layer thickness and spatial growth rate, yielding results in agreement with literature data. Considering that the method can be implemented with a minor increase of the computational cost when compared to the WIDIM algorithm, it is concluded that the non-isotropic interrogation technique may constitute a further advance for the performances of PIV in high-speed applications. References Di Florio D, Di Felice F and Romano GP, (2002) Windowing, re-shaping and re-orientation interrogation windows in particle image velocimetry for the investigation of shear flows, Meas. Sci. Technol. 13: 953-962 Kline SJ, Cantwell CBJ, and Lilley GM eds., (1981) The 1980-81 AFOSR-HTTM-Stanford Conference on Complex Turbulent Flows: Comparison of Computation and Experiment”, Thermosciences Division, Mechanical Engineering Department, Stanford University, Stanford-US Keane RD, Adrian RJ, and Zhang Y, (1995) Super-resolution particle image velocimetry, Meas. Sci. Technol. 6: 754-768 Lang N, (2000) Investigation of the supersonic flow field around a delta wing using particle image velocimetry, Proceedings of the 10th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon,PT Lecordier B, Lecordier J C and Trinite M, (1999) Iterative sub-pixel algorithm for the cross-correlation PIV measurements Proceedings of the 3rd International Workshop on PIV, Santa Barbara, US Melling A, (1997) Tracer Particles and Seeding for Particle Image Velocimetry, Meas. Sci. Technol. 8, 1406 Hart DP, (2000) DPIV error correction. Exp. Fluids 29: 13-22 Havermann M, Rey C and Haertig J, (2002) Application de la PIV a l’etude des ecoulements a tres haute vitesse produits par une sufflerie a choc, 8eme Congres Francophone de velocimetrie Laser, Orsay, FR Haertig J, Havermann M, Rey C and George A, (2002) particle image velocimetry in Mach 3.5 and 4.5 shocktunnel flows, AIAA J. 40: 1056 Scarano F and Riethmuller ML, (2000) Advances in iterative multigrid PIV image processing. Exp. Fluids 29: S051-S060 Scarano F and van Oudheusden BW, (2003) Planar velocity measurements of a two-dimensional compressible wake, Exp. Fluids 34: 430-441 Scarano F, (2003) Theory of non-isotropic spatial resolution in PIV, Exp. Fluids DOI: 10.1007/s00348-0030655-4 G. Tedeschi, H. Gouin and M. Elena, (1999) Motion of tracer particles in supersonic flows, Exp. Fluids 26, 288296 Willert CE and Gharib M, (1991) Digital particle image velocimetry. Exp. Fluids 10: 181-193
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