an inclined jet in cross-flow studied by means of particle

16th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 09-12 July, 2012

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An Inclined Jet in Cross-Flow Studied by Means of

Particle Image Velocimetry and Magnetic Resonance Imaging

Filippo Coletti1,*, Chris Elkins1, John Eaton1

1: Department of Mechanical Engineering, Stanford University, Stanford (CA), United States

* correspondent author: [email protected] Abstract A jet-in-crossflow configuration relevant to film cooling is studied by means of Particle

Image Velocimetry (PIV), Magnetic Resonance Velocimetry (MRV) and Magnetic Resonance Concentration measurement (MRC). The focus is on the transport of momentum and injected scalar due to turbulence. The experiments are conducted in water, the molecular properties being immaterial in the considered turbulent regimes. Mean velocity and concentration fields are obtained via MRV and MRC, while turbulence statistics in the vicinity of the hole are obtained by PIV. The Reynolds wall-normal and shear stresses are found to correlate well with concentration and magnitude of concentration gradient respectively, highlighting the correspondence between turbulent momentum transport and turbulent scalar transport. The mean flow is dominated by the counter-rotating vortex pair aligned in streamwise direction, which leaves its footprint in the scalar distribution in the vicinity of injection. Few hole diameters downstream turbulent transport takes over, and the mean flow trajectory diverge from the scalar distribution. The jet entrainment is enhanced by the formation of the counter-rotating vortex pair between 2 > s/D > 12; further downstream the proximity to the wall inhibits the process, and the film cooling jet entrains less fluid than an axisymmetric jet.

1. Introduction The jet in cross-flow is one of the most extensively studied cases in fluid mechanics due to its

innumerable applications, from pollutant plumes to film cooling. The latter is arguably the most effective and most widely employed cooling technique in the thermal design of gas turbines: internal cooling air bleeds through small holes along the exterior skin of the blade, forming a protective layer.

Film cooling flows have been analyzed in numerous numerical and experimental studies, as reviewed for example in Bogard and Thole (2006). The problem is a complex one, governed by numerous parameters including: the hole shape and angle with the free-stream, jet Reynolds number, the jet-to free stream blowing ratio and density ratio, the free stream turbulence intensity and the Mach number. Current numerical and analytical film cooling models are unable to satisfactorily predict the mixing of scalars (i.e. temperature or coolant concentration), especially near the injection location (Kholi and Bogard, 2005).

The standard CFD modelling approach for optimizing a cooling design is to employ Reynolds-Ageraged Navier–Stokes (RANS) turbulence closures, where Reynolds-stresses can be determined via a turbulent viscosity model (Luander and Sharma, 1974), or via solution of modeled Reynolds-stress transport equations (Speziale, Sarkar, and Gatski, 1991). On the other hand, the turbulent heat fluxes are typically determined via a gradient-diffusion assumption with a scalar turbulent diffusivity and constant turbulent Prandtl number Prt = νt/αt, where νt and αt are the turbulent viscosity and diffusivity, respectively. While the turbulence closures for the momentum transfer can deliver acceptable results even for practical flows, the Prt model can be used to compute only relatively simple boundary-layer, channel or pipe flows (Kays, 1994). There exists a wide range of more sophisticated alternatives (Abe and Suga, 2001; Younis et al., 2005), but the development and validation of such models in three-dimensional flows is however limited by the scarcity of detailed experimental data. In most technical reports available in literature, the flow quantities are presented in form of one-dimensional profiles, or at best as two-dimensional distributions. Such data cannot


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be used to effectively assess turbulent mixing in complex three-dimensional flows. The present study focuses on a single-hole film-cooling configuration, where a non-buoyant

contaminant is injected in a channel flow. The full three-dimensional mean velocity field is obtained by means of Magnetic Resonance Velocimetry (MRV, Elkins et al., 2003), allowing detailed exploration of the velocity and vorticity fields. The three-dimensional scalar concentration field is obtained by Magnetic Resonance Concentration measurements (MRC, Benson et al., 2010). From the velocity and concentration fields a streamwise distribution of turbulent diffusivity is derived from the advection-diffusion equation. Particle Image Velocimetry (PIV) is used to measure the Reynolds stresses and hence the turbulent viscosity along the jet symmetry plane. The results allow the validity of both the gradient-diffusion hypothesis and on the Reynolds analogy to be tested.

2. Apparatus and procedures 2.1. Set-up and operating conditions

The experimental model is sketched in Fig. 1. It consists of several sections fabricated using stereolitography and assembled together in a channel model 1350 mm in length. The upstream part transforms the circular inlet with a diameter of 25.4 mm to a square section of 100 by 100 mm2. Seven equally spaced grids in the diffusing sections prevent separation, as verified by dedicated MRV measurements, not shown here for sake of brevity. The central part of the model consists of a flow conditioning section and a contraction. The flow conditioner comprises a honeycomb followed by a grid. The cells of the honeycomb are square (12 by 12 mm2) and have a length of 6 cell sizes, to remove large scale secondary flows. The contraction has an area ratio of 4:1 and smoothly connects to the 50 by 50 mm2 test section. A 2 by 2 mm2 riblet that runs around the channel perimeter 25 mm downstream of the contraction trips the boundary layer in order to give an engine-representative momentum thickness at the point of coolant injection, which is 250 mm downstream. The film hole is circular with a diameter D = 5.8 mm, inclined at 30 degrees with respect to the main flow direction, and a length L = 24 mm (L/D = 4.1). The boundary layer thickness upstream of injection is about 1.9D. The coolant flow is fed from a plenum of 40 by 35 by 25 mm3, where velocities were verified to be negligibly small except in the vicinity of the hole inlet. The outlet section connects to a circular pipe. For the PIV measurements an identical test section is used, but with transparent Plexiglas windows opened on one lateral wall and on the wall opposite to the film cooling hole.

The closed-loop flow system consists of a holding tank from which a Berkeley BPDH10-L electric pump extracts the main flow and a Little Giant TE-6-MD-HC electric pump extracts the jet flow. Both flow rates are monitored by paddlewheel flow meters. The main flow circulates through the channel model, while the jet flow feeds the plenum. The fluid recirculates back to the holding plenum. The elements are connected by 25.4 mm diameter plastic hoses.

For the present study, which is focused on the fundamental aspects of the inclined jet-in-crossflow, a single-hole configuration was preferred over a row of holes, which is more representative of the engine conditions. The mutual interaction of adjacent jets was not the object of the investigation. Moreover, given the limited size of the test section, placing more holes in the spanwise direction would have caused a significant blockage effect of the lateral walls.

The dimensions of the water tunnel are limited by the need of inserting the model in the MRI coil (see following subsection). Because the hydraulic diameter of the channel is only about 8D, the evolution of the jet far downstream of the injection is somewhat influenced by the presence of the walls. However the attention here is focused on the inception of the vortical structures characteristic


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Figure 1. Left: water tunnel (top) and test section (bottom),with dimensions in mm. Right: the channel model about to be inserted in the medical MRI scanner.

. of the film cooling flow and their development up to a distance of Z=10-15D downstream of the injection. In this early phase the wall-confinement effect does not affect the fluid dynamics in a critical manner.

The working fluid is water for both the main flow and the jet. The different viscosity with respect to gas-turbine applications does not diminish the relevance of the experiments, as the flow is fully turbulent and the eddy viscosity dominates against its molecular counterpart. In this regard, Benson (2011) obtained film cooling effectiveness distributions on a pressure side cutback trailing edge both in air and in water, finding satisfactory agreement. The Mach number cannot be matched to practical applications due to the incompressibility of the working fluid. However in most reported studies Mach nymber effects do not appear to be significant (Bogard and Thole, 2006).

The main flow Reynolds number, based on the channel hydraulic diameter and the bulk velocity (Ubulk = 0.5 m/s), is 25000. The velocity ratio is one (Ujet = Ubulk) which results in a jet Reynolds number of 2900. Since the density ratio (ρjet/ ρbulk) is one, the blowing ratio (ρjetUjet/ ρbulkUbulk) and the momentum flux ratio (ρjetUjet

2/ ρbulkUbulk2) are also one.

2.2. Magnetic resonance velocimetry

Velocity data are obtained using the method of magnetic resonance velocimetry (MRV) described by Elkins et al. (2003). MRV makes use of a technique very similar to that used in conventional medical magnetic resonance imaging (MRI). MRI generates spatially resolved images inside an object utilizing magnetic fields and radio frequency pulses. Hydrogen nuclei in a constant magnetic field tend to align their magnetic moments with the field. As this alignment takes place, the net spin vector of each hydrogen atom precesses around the direction of the magnetic field, at a frequency which is proportional to the external field strength. By applying a constant magnetic field and then applying radio frequency pulses for a short time, it is possible to perturb the spins of hydrogen atoms away from their initial alignment with the external magnetic field and then detect the radiation these atoms emit as the spins relax back into their original alignment. Because the precession frequency is proportional to the magnitude of the external magnetic field, the locations of spins can be encoded in their frequencies by spatially varying the magnetic field across an object. Quantitative assessment of flow can be obtained due to the sensitivity of the phase of the MR signal to motion. This can be used to measure the local velocities of moving spins. The procedure for this data acquisition technique is described by Pelc et al. (1994).


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Experiments are performed at the Richard M. Lucas Center for Imaging (Stanford, CA). Figure 2 shows the test section about to be inserted in a the scanner. A 3 Tesla General Electric scanner is utilized, with a standard transmit-and-receive radio frequency coil commonly used for imaging human heads. Three-component velocity measurements are obtained on a uniform Cartesian mesh following the phase contrast MRV (Magnetic Resonance Velocimetry) technique described by Elkins et al. (2003). To enhance imaging, copper sulfate is used as contrast agent at a concentration of 0.06 mol/L in deaerated water. The velocity data are measured on a uniform grid with 384 points in the streamwise direction, 110 points in the wall-normal direction, and 66 points in spanwise direction, at a resolution of 0.6 mm in all three directions. Each scan is about 6 minutes in duration and the measured velocity field represents a time-average. Six scans are performed and averaged to reduce random errors.

The procedure is similar to the one followed by Benson et al. (2011): a flow-off scan (i.e. scan during which the flow rate is set to zero and the fluid is at rest) precedes and follows groups of three flow-on scans (during which both main flow rate and coolant flow rate are at the test point), for a total of six flow-off scans and five series of three flow-on scans. To account for MR system temperature drift, the flow-off runs conducted before and after each set of three flow-on runs are averaged and then subtracted from the average of the three flow-on runs. Subtracting the flow-off data ensures that phase changes due to temperature drift and/or induced by eddy currents were eliminated. The five resulting velocity data sets are averaged to obtain the final mean velocity data.

The MRI scan covers a rectangular volume which includes both the flow and the solid walls of the test section, The location of the flow boundaries is performed using a simple thresholding based on the signal magnitude: when the signal magnitude is less than 10 times the value of the magnitude of the average noise, the point is identified as solid material. Changing the value of the threshold by +/-20% produces no appreciable difference in the identification of the solid walls.

The velocity encoding (Venc) values control the maximum measurable velocity that is free of aliasing. Venc values are 0.8 m/s for the streamwise and wall-normal direction and 0.5 m/s for the spanwise direction. The expected uncertainty in the MRV measurements can be calculated from the formula (Pelc et al., 1994):

SNRVenc2

V πδ = (1)

where signal to noise ratio SNR is given by the ratio of the signal in the flow region over the signal in the solid wall. For the streamwise velocity, this yields an uncertainty of about 0.03 of the bulk flow velocity. The bulk flow rate calculated from the MRV data is within 5% of the nominal flow rate. This includes both velocity measurement uncertainty as well as inevitable inaccuracy in locating the walls.

2.3. Magnetic resonance concentration measurements

The principles and procedures are the same as described in Benson et al. (2011). In this type of experiment the imaged signal is proportional to the concentration of the contrast agent (copper sulfate) in water. The two fluids used for concentration measurements are: (i) plain deaerated water and (ii) a 0.015mol/L solution of copper sulfate in deaerated water. Between the minimum and maximum copper sulfate concentration the MR signal magnitude varies linearly, as verified in dedicated calibration tests. The MRC scan procedure consists of four scan types: background scans, with plain water in the mainstream and jet flow lines; reference scans, with 0.015 mol/L copper sulfate solution in the mainstream and jet flow lines; inverted scans with copper sulfate in the mainstream and plain water in the jet flow; and standard scans, in which the mainstream fluid is water and the copper sulfate solution is the jet fluid. The Cartesian grid is the same as in the MRV


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experiment. Each scan is 119 s in duration and the resulting concentration field represents a time-average. Eighteen scans are performed and averaged to reduce random errors. The procedure for MRC data processing is detailed in Benson et al. (2011). In summary, the concentration C at each point is calculated as:

( )IS

IS

NN

NBRBI1N

BRBS

C+

⎟⎟⎠⎞

⎜⎜⎝⎛

⎟⎠⎞⎜

⎝⎛

−−−+⎟

⎠⎞⎜

⎝⎛

−−

= (2)

where B, I, R and S indicate the signal obtained in background scans, inverted scans, reference scans and standard scan, respectively. NS and NI refer to the number of standard and inverted scans, respectively.

The uncertainty in the point-by-point concentration measurements is estimated from the variance of the signal magnitude over the set of runs for the different types of scans. The uncertainty evaluated at a 95% confidence level is approximately 4% of the maximum concentration. Higher errors are possible within one voxel (0.6 mm) from the wall, due to the uncertainty in locating the wall itself (partial volume effect). However such locations are not considered in the present study.

2.3 Particle Image Velocimetry

Measurements of mean velocity (to compare with those obtained by MRV) and Reynolds stresses are performed along the symmetry XY plane by means of two-dimensional PIV. The light source is a pulsed Spectra-Physics PIV-400 Nd:YAG laser, emitting a 532 nm light with an intensity of 350 mJ/pulse. The laser beam is shaped into a 0.5 mm thick sheet by means of a convergent spherical lens of focal length f = 1000 mm followed by a cylindrical lens of f = -60 mm. The images are acquired by a 12-bit Princeton Instruments CCD camera with a spatial resolution of 1300 by 1030 pixels2. A magnification factor of about 112 pixel/mm is achieved using a 105 mm Canon objective at an aperture f/4. Two slightly overlapping windows (about 12 by 9 mm2 in size) are considered around the injection location. The separation time between laser pulses is 200 µs, resulting in an average particle displacement of 8 to 12 pixels. Laser and camera are coordinated by a Stanford DG535 synchronizer. The sampling frequency is 10 Hz, which guarantees statistically independent realizations.

A background image is constructed by pixel-wise selection of the minimum intensity over the ensemble of the recordings and subtracted from every image, as recommended by Wereley and Meinhart (2002). The processing is realized by means of a cross-correlation based interrogation algorithm with windows offset and deformation. The initial interrogation windows are 128 by 128 pixel2. Two refinement steps and a 50% overlap lead to a vector spacing of 16 by 16 pixel2 or 0.14 by 0.14 mm2 (the actual resolution being 0.28 by 0.28 mm2) The vector validation is based on the signal-to-noise ratio and the local median threshold. Rejected vectors are filled using a linear interpolation of the surrounding vectors. The imaging is challenging because the laser impacts on the curved surface of the film cooling hole. However at least 75 % of the recorded vectors are valid in every location. Mean velocity and rms of the velocity fluctuations are obtained by averaging over 1000 image pairs.

3. Results In the following, the X-axis indicates the streamwise direction, while the Y- and Z-axes are the

wall-normal and spanwise directions, respectively. The origin is at the center of the hole at the film-cooled wall. U, V, W are the mean velocity components in the X, Y and Z direction, respectively. u,


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v and w indicate the fluctuating components. Ensemble-averaging is implied by the operator <·>. The quantities are non-dimensionalized by the film hole diameter D and the bulk main flow velocity (equal to the bulk jet velocity) U0.

3.1. Mean velocity and vorticity

Figure 2 shows a cut of the volume investigated by MRV, with color contours of mean streamwise velocity along the jet symmetry plane. The two PIV windows are also marked. It is evident how the jet is in the detached regime, as typical for momentum flux ratios larger than 0.8 (Thole et al, 1992). The detached flow at the hole inlet, a signature feature of short-hole film cooling configurations, is clearly visible (X/D = -2, Y/D = -1.5).

MRV and PIV measurements are compared in Fig. 3. The mean streamwise velocity agree within 3-8%. Beside the inherent uncertainty associated to both techniques, possible sources of discrepancy include errors on the blowing ratio (possible errors of about 2% in the flow meters calibration) and in the location of the laser sheet (the estimated possible misplacement of about 0.5 mm correspond to about 0.09D).

Figure 4 shows perspective views that illustrate the signature feature of every jet in crossflow: the counter-rotating vortex pair oriented streamwise. This is evident both from the isosurfaces of streamwise vorticity ωZ (levels at ωZ = 0.3U0/D and ωZ = -0.3U0/D are shown) and from the in-plane velocity vectors along various cross-sections. This feature is in general considered responsible for most of the scalar mixing, especially close to injection.

Figure 2. Contours of mean streamwise velocity along the jet symmetry.

Figure 3. Left: color contours of mean streamwise velocity as measured by PIV and MRV. Right: velocity profiles measured by PIV and MRV at locations X/D = 1 and X/D =2.


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Figure 4. Left: isosurfaces of positive (red) and negative (blue) streamwise vorticity. Right: cross-sections showing in-plane velocity vectors and streamwise velocity. 3.2. Turbulent momentum and scalar transport

The Reynolds stresses measured by PIV and the mean concentration measured by MRC are combined in Figure 5 to illustrate the coupling of turbulent momentum and scalar transport. On the left panel, levels of mean concentration C (in fractional unit, C=1 indicating full concentration of injected scalar) are superimposed on color contours of wall-normal stresses <vv>. High levels of vertical velocity fluctuations correspond to regions of high concentration: the wall-normal component of turbulence carried by the jetting fluid is transported into the mainstream in much the same way as the passive scalar, at least in the vicinity of the film cooling hole. On the right panel levels of in-plane concentration gradient ((∂C/∂X)2 + (∂C/∂X)2)0.5 are superimposed onto colour contours of Reynolds shear stresses <uv>. The correspondence is evident, suggesting that (in the limit of the validity of the gradient-diffusion hypothesis) the turbulent momentum transport correlates well with the turbulent scalar transport.

To illustrate the three-dimensional structure of the coolant jet, iso-surfaces of mean concentration are shown in Fig. 6, for nine levels between 80% (a) and 20% (i). It appears that the counter-rotating vortex pair influences significantly the scalar distribution in the vicinity of the hole, up to about 3 hole diameters downstream of the injection (a–d): the footprint of the two vortices is recognizable in the forked shape of the iso-concentration surfaces. However, further downstream (and therefore at lower concentration levels), the concentration isosurface is more oval, indicating that the turbulent transport is dominating over the advection. The concept of a streamtube is useful to illustrate and quantify the impact of turbulent dispersion on the scalar distribution. Figure 7 shows the streamtube issued by the film cooling hole, formed by the streamlines passing through the ellipsoidal intersection of the hole with the main channel. Its convoluted shape is the result of the counter-rotating vortex pair. In the right panel contours of mean streamwise concentration flux CU (C being the fractional mean concentration are plotted along various cross-sections. The concentration flux within the streamtube drops quickly moving away from the injection due to turbulent dispersion. If the flow was laminar the concentration flux would be nearly conserved at any given section, with any loss of flux due to molecular diffusion.

The decoupling between velocity and concentration field due to turbulent dispersion is


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Figure 5. Left: levels of mean concentration and color contours of wall-normal Reynolds stresses. Right: levels of mean concentration gradient and color contours of Reynolds shear stresses.

Figure 6. Mean scalar structure of the jet, illustrated by iso-surfaces of mean concentration, from 80% (a) to 20% (i).

highlighted in Fig. 8, where color contours of the mean concentration are displayed, together with the velocity centerline, in black. The latter is defined as the streamline originating from the center of


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Figure 7. Left: streamtube originating from the film cooling hole. Right: cross-sections showing contours of mean streamwise concentration flux.

Figure 8. Velocity centerline (black) and maximum concentration line (purple) superimposed to color contours of mean concentration along the jet symmetry plane. the hole at its intersection with the film-cooled wall. The maximum concentration line, defined as the loci of highest scalar concentration at each streamwise location, is plotted in purple. The velocity centerline penetrates further into the flow than the maximum concentration line: in fact the former almost coincides with the upper edge of the streamtube in Fig. 7, whereas the latter is closer to its centroid. Again, the divergence of the two trajectories is a consequence of turbulent diffusion.

Mass entrainment brings jet fluid in contact with crossflow fluid, creating regions of high scalar gradient and mixing. The entrainment rate can be determined by integrating the jet volume flux through a section normal to the jet trajectory. Here we consider the maximum concentration line (which describes the location of the jet fluid more accurately than the velocity centerline) and define for it an abscissa s. The origin of s is taken right below the film-cooled wall, so that the cross-section at s=0 is all inside the hole. To distinguish between jet fluid and crossflow fluid, a threshold value of scalar concentration has to be selected. This is chosen as C=0.05, which was shown by Yuan and Street (1998) to deliver acceptable results for a perpendicular jet in crossflow. Significantly smaller values would increase errors due to measurement noise. Figure 9 (left panel)


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Figure 9: Left: isosurface at C=0.05 and sections perpendicular to the jet trajectory colored by levels of mean velocity. Right: evolution of jet volume flux along the abscissa of the jet trajectory.

depicts the C=0.05 isosurface, along with sections normal to the jet, color-coded by the mean velocity projected along the jet trajectory, Us. The evolution of the jet volume flux Vs (normalized by the injection value V0) is presented on the right panel of Fig. 9. The correlation of Ridou and Spalding (1961) for an axisymmetric jet into quiescent flow and the DNS of Muppidi and Mahesh (2008) for a perpendicular jet in crossflow (blowing ratio = 5.7) are also included for comparison. The insert shows that up to s/D=2 the regular jet has a higher entrainment than the jets in crossflow. At larger distances the jets in crossflow entrain significantly more fluid, presumably due to the formation of the counter-rotating vortex pair. However, while the high blowing ratio jet of Muppidi and Mahesh (2008) displays an increasingly larger entrainment at further s/D, the present inclined jet appears to engulf less and less fluid moving away from injection: for s/D > 12 it entrains even less crossflow fluid than a regular jet. In every jet in cross-flow the counter-rotating vortex pair causes more mixing to happen on the leeward side than on the windward side of the jet. In this case the leeward side is closer to the wall than in Muppidi and Mahesh (2008), due to the shallow injection angle and the relatively low blowing ratio. This is likely the reason for the reduced levels of entrainment and mixing.

4. Conclusions An inclined jet in cross-flow has been investigated in a water tunnel by means of Particle Image

Velocimetry, Magnetic Resonance Velocimetry and Magnetic Resonance Concentration measurement. The flow configuration is relevant to film cooling technology, the molecular properties of the fluid being irrelevant in the considered turbulent regime. The full three-dimensional structure of the mean velocity and concentration fields is accessed via the MRI-based techniques, while turbulence statistics in the vicinity of the hole are obtained by PIV.

The Reynolds wall-normal and shear stresses are found to correlate well with the concentration and the magnitude of the concentration gradient, respectively, highlighting the correspondence between turbulent momentum and scalar transport. The mean flow is dominated by the counter-rotating vortex pair aligned in streamwise direction, which results in a convoluted streamtube issued by the film hole and shapes the scalar distribution in the vicinity of injection. However, few hole diameters downstream, the turbulent transport takes over causing a decoupling of mean flow


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trajectory and scalar concentration distribution. The jet entrainment is enhanced by the formation of the counter-rotating vortex pair between 2 > s/D > 12, but further downstream the proximity to the wall inhibits the process, and the film cooling jet entrains less fluid than an axisymmetric jet.

Acknowledgments

The authors would like to thank Honeywell Inc. for the financial support. Use of the facilities at the Richard M. Lucas Center for Magnetic Resonance Spectroscopy and Imaging (Stanford, CA) is gratefully acknowledged.

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an inclined jet in cross-flow studied by means of particle

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