Material Derivative Measurements in High-Speed Flows by Four-Pulse
Tomographic PIV
Kyle Lynch1 and Fulvio Scarano1
1 Department of Aerospace Engineering, TUDelft, Delft, Netherlands
k.p.lynch@tudelft.nl ABSTRACT
A tomographic PIV system is introduced for the instantaneous measurement of the material derivative of velocity (VMD). The system is able to operate with very short temporal separation and is therefore suitable for applications in high-speed flows. The method of operation consists of the imaging of a measurement volume using an array of 12 CCD cameras and two double-cavity laser systems. Four independent recordings of particle images are captured by decomposing the system into three separate tomographic PIV systems comprised of four cameras. A discussion is made that compares the present working principle with other methods used to separate the light scattered from multiple pulses, namely by polarization.
Various approaches are compared to determine the optimal utilization of four-pulse data to measure the VMD: the Eulerian and Lagrangian schemes are compared with the recently introduced fluid trajectory correlation (FTC) technique from the authors (Lynch and Scarano, 2013). The comparison focuses on the behavior of the schemes with respect to truncation errors and how the error estimates for four-pulse data are modified from those typically applied to image sequence data from a time-resolved PIV experiment. The analysis of synthetic images of a translating vortex clearly shows the envelope of applicability of the different schemes and the structure of the measurement errors introduced by truncation.
The 12-camera tomographic system in four-pulse configuration is employed to measure the wake of an axisymmetric truncated base with an afterbody at a Reynolds number of 68,000. The system calibration accuracy and the baseline measurement uncertainty of the velocity are evaluated by performing a test with a negligible time delay between the independent tomographic PIV systems. The comparative performance of the material derivative schemes is estimated by appealing to a physical property of the material derivative field. The results indicate that a 12-camera system can be employed for material derivative evaluation using a variety of estimation schemes. Among these schemes, the FTC technique is found to be the least susceptible to the growth of truncation errors and is thus suitable for measurement at large temporal intervals which are necessary to suppress random errors.
1. INTRODUCTION
The acceleration of fluid parcels in turbulent flows plays a fundamental role in determining the local fluctuations of pressure and is often associated to the action of coherent turbulent structures. Being able to describe the behavior of fluid parcels along their trajectories provides a fundamental link between fluid kinematics and the dynamical interactions within the fluid and forces exerted upon aerodynamic bodies immersed in the stream. This connection is formally governed by the momentum equation,
∇𝑝 = −𝜌!𝐮 !"+ 𝜇∇
!𝐮 (1)
relating the pressure gradient field to the force due to fluid inertia and due to viscous interactions. In eqn. 1, 𝑝 is the pressure, 𝜌 is the density, 𝐮 is the velocity, 𝜇 is the coefficient of viscosity, and D/Dt is the material derivative operator. The term D𝐮/Dt represents the material derivative operator applied to the velocity, and will be referred to as the velocity material derivative (VMD) for the remainder of the paper.
It is stated that in turbulent flows at high Reynolds number the viscous term becomes negligible and the inertial force is dominant (e.g., Liu and Katz, 2006, and van Oudheusden et al., 2007). In the former, a cavity flow at a Reynolds number of 335,000 yielded viscous terms three orders of magnitude smaller than the inertial terms. In the latter, flow past a square-section cylinder at a Reynolds number of 20,000 yielded viscous terms two orders of magnitude lower. Moreover, in experiments performed by Ghamei et al. (2012) on a turbulent boundary layer at a momentum-thickness Reynolds number of 2,400, the viscous terms were again found two orders of magnitude lower than the inertial terms. Therefore the determination of the instantaneous pressure gradient translates into the problem of accurately estimating the VMD term.
The VMD physically represents the change in velocity 𝐮! of a fluid particle travelling along its trajectory defined by the position 𝐱! (van Oudheusden, 2013). This Lagrangian viewpoint can be expressed as follows,
!𝐮 !"= !𝐮!(!) !! = !𝐮(𝐱!! ,!) !! (2)
This expression is intuitive, but cannot be directly applied to measurements because the measurement of the particle field and velocity field is performed in an Eulerian sense, i.e., with respect to a fixed laboratory reference frame. Thus, the evaluation of eqn. 2 requires a transformation into a reference frame in motion with the fluid element following its local position 𝐱!. One transformation is through the application of the differential form of the material derivative operator, which decomposes the VMD into components of local and convective acceleration:
!𝐮 !" =
!𝐮
!"+ 𝐮 ∙ ∇ 𝐮 (3)
This equation is entirely equivalent to eqn. 2 at the differential level. Through the application of finite differencing, both terms in eqn. 3 can be estimated from data acquired from a fixed laboratory reference frame (see de Kat and van Oudheusden, 2011). However, this is but one method for estimating eqn. 2. Other methods for the transformation include the use of a bulk convection velocity (Christensen and Adrian, 2002), a local convection velocity (de Kat and Ganapathisubramani, 2013), and pathline-tracing (Lagrangian) approaches (Jensen et al., 2003, Liu and Katz, 2006, de Kat and van Oudheusden, 2011, among others). In particular, the Eulerian approach of eqn. 3 and the pathline-tracing Lagrangian approach will be discussed at length later in this paper.
A complication for the application of these schemes is determining the temporal derivative of velocity. From the standpoint of PIV measurements, this requires more than two snapshots of the particle tracers to evaluate the change in velocity and thus calculate a single instantaneous VMD. These snapshots must be acquired in accordance with the time scales of the flow; for high-speed flow regimes, the temporal separation may need to be as short as a few microseconds, which corresponds to acquisition rates in the order of hundreds of kHz, clearly outside the range of currently available high-speed laser and camera systems. A novel aspect of the present work is the application of burst recording approach (similar to Thurow et al., 2013) obtained using multiple independent systems as will be detailed in section 2.1. A second complication is the three-dimensionality of many relevant flows, precluding the measurement using planar measurement techniques. While schemes based on eqn. 2 only require knowledge of the velocity and its gradient along a plane, the use of pathline-tracing Lagrangian schemes require the velocity to be defined within the entire volume in which the fluid parcel travels. In this work, the volumetric measurement is provided via the tomographic PIV technique (Elsinga et al., 2006, Scarano, 2013), which is extended to multiple pulses as needed. The use of tomographic PIV entails particular details in the experimental setup which deviate from multiple pulse measurements performed using planar systems, which are outlined in sections 2.1-2.2. However, before addressing these issues, it is instructive to detail the evolution of previous low-speed or planar studies that aimed either at determining the acceleration or the VMD in order to provide a perspective on the technical principles of the method.
Previous Accelerometry and Multi-Pulse Efforts
The first dedicated efforts for acceleration measurement involved the use of specialized camera hardware (Jackobsen et al., 1997), or hybrid cross- and auto-correlation algorithms applied to data acquired from standard PIV systems (for example, Dong et al., 2001). In both cases, substantial drawbacks were noted; the former by the high cost and complexity of specialized hardware, and the latter from the lack of robustness inherent in the auto-correlation technique and its limitation in spatial resolution due to limitations in particle density.
Realizing these limitations, development shifted toward the application of multiple standard CCD cameras and standard cross-correlation algorithms. In summary, this class of techniques involves using two or more independent cameras to capture pulses L! through L! as shown in figure 1. This allows two velocity fields to be obtained. Finite differencing is then applied to estimate the acceleration. The challenge with implementing this system is the isolation of laser pulses into individual camera frames, made difficult due to the long second exposure of interline CCD cameras. This requires filtering the laser pulses between cameras.
One approach by Guibert and Lemoyne (2002) split the incoming light between two cameras and utilized a ferromagnetic liquid filter to block a percentage of light from entering the first camera during the second set of pulses. This allowed for measurement of the acceleration with very short separation times, critical for their measurements of in-cylinder engine turbulence. However, the liquid filter provides a limited extinction ratio, or percentage of light blocked; in Guibert and Lemoyne (2002) this was estimated at around 60%, leaving artifacts in the first camera. A more robust solution was proposed by Kähler and Kompenhans (2000) in their development of dual-plane stereoscopic PIV, which is based on the use of polarization filtering to discriminate light pulses for particular stereo-PIV systems. While the
Figure 1. Schematic of standard particle image accelerometry system.
focus of their work was on the estimation of the velocity gradient tensor, the authors do mention the utility of the system for conducting accelerometry when the dual-plane system is used in a coincident plane configuration.
The polarization approach was followed by Christensen and Adrian (2002) in a dual planar-PIV configuration. Their focus was on the topology of the acceleration field in turbulent boundary layers and in the generation of acceleration statistics. Furthermore, they proposed a novel zero-time-delay measurement as a method for an in-situ measurement of the PIV uncertainty. Another application of the polarization approach was practiced by Perret et al. (2006), who implemented a dual-time stereo-PIV system along coincident planes to measure the acceleration field in a turbulent plane mixing layer. Furthermore, they extended the analysis of Christensen and Adrian (2002) to determine an optimal value of the time delay ∆t! between subsequent velocity measurements based on the frequency response of a finite differencing operator and an estimate of the PIV random error. A final example of the polarization discrimination approach is exhibited by Liu and Katz (2006), who used the technique as part of a system to measure the instantaneous pressure field in a cavity flow.
These modern applications of accelerometry concentrate on the application of dual-PIV systems; besides the use of such systems in measuring accelerations, other quantities can also be measured. For example, Hu et al. (2001) performed measurements of the auto-correlation of velocity and vorticity in a lobed jet flow by means of a dual-plane stereo PIV arrangement as a proof of concept. Souverein et al. (2009) applied a dual planar-PIV system for measurement of the auto-correlation in a shock-wave turbulent boundary layer interaction. By varying the time separation between the two velocity measurements and applying the autocorrelation function between the measurements, an estimation of local flow timescales and convection velocity was performed.
From the present survey it can be concluded that the dual-time PIV technique was applied successfully only for planar measurements and at best for the stereoscopic configuration. A well-known restriction is given by the out-of-plane particle motion that poses an upper limit to the extension of time separation between the two measurements. More importantly, when the velocity variation is to be evaluated along the local trajectory, for three-dimensional flows the planar approach is ill-posed. Thus the focus of the present work is to develop a three-dimensional approach for the determination of fluid flow acceleration and material derivatives for high-speed three-dimensional flows. A relevant application of this technique will be the measurements of the unsteady pressure field in compressible flows.
2. WORKING PRINCIPLES
The extension of tomographic PIV for measurement of the VMD requires attention to a number of details as described herein. Following this, a related assessment of the analysis schemes for the estimation of eqn. 2 is given, which provides guidance on the application of the technique to the experimental case.
Pulse Separation by Independent Systems
A key attribute of composite systems (neglecting approaches based on autocorrelation image analysis) is the distribution of individual pulses to individual image frames. For planar and stereo studies, the preferred method for achieving this is through the use of orthogonally polarized laser light sheets and the filtering of light entering the cameras based on the polarization direction. The first camera captures vertically polarized light scattered from particles illuminated by the first laser, and the second camera captures horizontally polarized light scattered from particles illuminated by a second laser. Christensen and Adrian (2008) achieved a greater than 90% extinction ratio of one pulse on an alternate frame, preventing artifacts from distorting the correlation.
The use of polarization selectivity in this work is complicated by two factors; first, for high-speed flows, solid particles are often seeded using solid tracer particles. The non-spherical shape of solid particles (e.g., Ragni et al., 2010) results in a scattering behavior that does not retain the polarization of the incident light. A second complication is that for tomographic PIV, the sensors may be positioned at angles not always in-line with the laser polarization axes. In this case, the projection of the electric field oscillations is not orthogonal, which should lead to limited separation by linear polarization plates.
Thus, an alternate approach is adopted whereby the separation of the light scattered by each laser pulse is obtained by controlling the exposure time of three independent systems of CCD cameras. An explanation of how the three imaging systems are used is given in the timing diagram of figure 3, which follows that shown in figure 1, but includes the exposure time of each camera system. The first system operates in “single-frame” mode; therefore, the exposure can be interrupted immediately after the first pulse L1. The second system also operates in “single-frame” mode and is
exposed immediately following the first pulse, integrating light until after the second pulse L2. The third system
operates in “double-frame” and is synchronized as a standard PIV system to the laser pulses L3 and L4. For CCD
cameras capable of a short minimum exposure time (less than 5 microseconds), this configuration can operate in high speed flows up to the supersonic regime. Note that the first two systems cannot be used in double-shutter mode because the duration of the second exposure depends on the readout time and typically lasts several milliseconds for CCD cameras that record at the typical rate of 10 Hz. Furthermore, shuttering the image by mechanical systems has not been demonstrated as a feasible option for the switching on the order of microseconds. The long exposure of the third camera
system does not have an adverse effect on the imaging since the system is run at a low repetition rate (around 5 Hz), and therefore the next set of laser pulses falls far beyond the long exposure time.
Illuminating the same volume by two laser systems can be obtained either by beam combiners based on polarization or by directing the laser at the same region from different locations. The latter approach is adopted in the current experiments for its simplicity.
Multipulse Tomographic PIV System Calibration
A second key attribute of composite systems is the alignment requirements among the independent systems. Although the field of view does not need to be identical for each system, the mapping function that relates the “image” space to “object” space needs to be accurately aligned among all views. In previous planar studies, the alignment was conducted using precision optical translation stages, with fine tuning by cross-correlation analysis between images of a calibration target (see e.g., Christensen and Adrian, 2002). An extension of this was explored by Liu and Katz (2006) by using the cross-correlation of simultaneously acquired particle images to determine a dense estimate of the relative displacement between the two cameras, thus forming a basis for a local deformation correction.
For tomographic PIV, a multi-plane physical calibration procedure (e.g., Soloff et al., 1997) can yield residual calibration errors of around 1-2 pixels between cameras throughout the entire volume depth. Fine tuning via cross-correlation cannot be applied to tomographic PIV data, since the reconstruction process itself requires the cameras to be in alignment to within 0.2-0.3 pixels. The alternative is the volume self-calibration procedure developed for tomographic PIV by Wieneke (2008). This triangulation-based approach has been extended for the simultaneous calibration of 12 cameras, allowing all cameras to be calibrated with an accuracy of approximately 0.2-0.3 pixels, corresponding in the experiments conducted herein to errors of less than 20 microns in object space.
Material derivative from an Eulerian approach
Assume that four-pulse recordings are used to extract the velocity material derivative. Three approaches will be compared. The first is an Eulerian evaluation based on eqn. 3. The second is a Lagrangian technique similar to that proposed by Jensen et al. (2003). The third is a recently introduced approach based on a correlation-based pathline measurement called Fluid Trajectory Correlation (Lynch and Scarano, 2013). The first two methods share similar requirements on the hardware configuration, specifically the pulse timing. To define the nomenclature for the next two sections, figure 3 shows the timing diagram for these schemes. It consists of two pairs of laser pulses and corresponding images where the spacing within the pair is given as ∆t! and the spacing between the pairs is given as ∆t!. Moreover, the total temporal measurement interval can be defined as ∆T, which will be useful later as normalization parameter to judge the schemes.
The first approach is known as the Eulerian approach,
and is a method for determining the material acceleration as defined by equation 2 based on the transformation given earlier in equation 3. In this formulation the local and convective contributions to the material acceleration are computed separately using finite differencing. For four-pulse data, the former can be calculated using a first-order forward difference applied to the velocity fields acquired at times 𝑡 and 𝑡 + ∆t!,
!𝐮 !" 𝐱, 𝑡 = 𝐮 𝐱,!!∆!!!𝐮 𝐱,! ∆!! − ! ! !!𝐮 !!!∆t!+ ⋯ (4)
Note that this formulation contains a greater truncation error than given by van Oudheusden (2013) since temporal central differencing cannot be applied to four pulse data. The convective contribution to the material derivative is evaluated using a central difference scheme applied to the velocity field at time 𝑡. For a single coordinate direction this can be expressed as,
𝑢!" !" x, 𝑡 = 𝑢 𝑥, 𝑡 ! !!∆!,! !! !!∆!,! !∆! − ! ! !!! !!!∆x!+ ⋯ (5)
where ∆x is the spacing between grid points in the velocity field. This is identical to the expression given by van Oudheusden (2013). The propagation of random error into the measurement is not discussed here, as it follows similar
Figure 3. ‘Standard’ configuration of the four laser pulses L1-L4 and three tomo-PIV systems S1-S3.
trends to those reported by van Oudheusden (2013). However, it is worth noting that the propagation of random error into the measurement is generally reduced as the temporal interval is increased. Therefore a tradeoff exists between the prevention of truncation errors and the suppression of random errors. This tradeoff was explored in detail by Perret et al. (2006) by performing a Fourier analysis of the differencing operators to reframe the tradeoff in terms of measureable frequencies for a given error level. This approach is not taken here; instead, tests using synthetic data will highlight the numerical properties in a following section.
Material derivative from a Lagrangian approach
The Lagrangian approach stems from the concept defined in equation 2. The numerical implementation is given by the first-order forward difference of the local velocity at the location of a fluid particle at two points along its pathline, i.e.,
!𝐮 !! 𝐱! 𝑡 , 𝑡 ≡ !𝐮! !" = 𝐮!𝐱!!!∆!!,!!∆!! !𝐮!𝐱!! ,! ∆!! − ! ! !!𝐮! !!! ∆t!+ ⋯ (6)
where the quantity 𝐮! 𝑡 represents the local velocity of a fluid particle of fixed identity at a specified time t. The implementation of equation 6 requires the reconstruction of the fluid pathline in order to determine the fluid position, 𝐱! 𝑡 . This reconstruction procedure can be thought of as an initial value problem in the context of ordinary differential equation solvers (see texts e.g., Atkinson, 1989); therefore, with two velocity fields, an implicit trapezoidal scheme can be used as a method for tracing the fluid pathline:
𝐱! 𝑡 + ∆t! !!! = 𝐱
! 𝑡 +∆!!! 𝐮! 𝐱! 𝑡 , 𝑡 + 𝐮! 𝐱! 𝑡 + ∆t! !, 𝑡 + ∆t! −!"! !!𝐱!
!!! ∆t!!+ ⋯ (7) where 𝑘 denotes the iteration number. This equation is implicit in that the final location of the fluid particle 𝐱! 𝑡 + ∆t! appears on both sides. An initial guess for 𝑘 = 0 is supplied using a forward Euler method, and two to three iterations of eqn. 7 are required in order to reach convergence better than 0.001 pixel. Values of the velocity at intermediate spatial grid point locations are determined using linear interpolation. Note that the pathline determination scheme given in eqn. 7 contains a reduced truncation error compared to van Oudheusden (2013) due to the use of the implicit ODE propagation scheme. However, even with the application
of a high-order pathline integration scheme, the dominant source of truncation errors in the Lagrangian approach is in the use of first-order forward differences in eqn. 6.
A graphical depiction of this process is shown in figure 4 with a representative trajectory of a fluid parcel denoted by Γ. The iterations of eqn. 7 are shown in light gray. In this example, the method has converged at k = 3, but the position does not correspond exactly with point 𝐱! 𝑡 + ∆t! due to the truncation error in the pathline tracing. Additionally, the resulting difference between the two velocity vectors (i.e., VMD) is shown in red. The light red arrow indicates the placement of the acceleration if a central difference scheme would be used. The alignment of this vector would be accurate with the vector defining the material derivative of velocity on the chosen pathline. However, since the grid points are defined at the time of the first velocity measurement, the VMD is defined at the time of the first
velocity measurement, and is clearly less suitable for describing the VMD at this point.
Material derivative evaluation by direct pattern correlation
It is well known that enlarging the time interval ∆t! allows random errors in the VMD to be reduced due to the reduction of the relative error (e.g., de Kat and van Oudheusden, 2011). However, enlarging ∆t! leads to an increase in truncation error caused by the use of first-order differencing operators in time (as per eqns. 4 and 6). Therefore, the allowable enlargement of ∆t! is limited. In previous studies making use of four-pulse data, extending ∆t! was not explored. For instance, in Liu and Katz (2006) ∆t! was actually shorter than ∆t! to limit the effect of truncation error. Additionally, this and other studies such as Pereria et al. (2006) considered planar measurements, which impose a further restriction on extending ∆t! due to out-of-plane motion. Tomographic PIV alleviates this constraint by allowing fluid pathlines to be constructed in all three dimensions (e.g., Violato et al. 2010, de Kat and van Oudheusden 2011).
The availability of four realizations of the particle field allows a higher-than-linear representation of the fluid parcel pathline to be constructed, allowing the time interval ∆t! to be extended with a less dramatic rise in truncation error.
Figure 4. Schematic diagram of Lagrangian method.
However, in order to achieve this, the fluid parcel pathline should not be estimated by velocity integration (eqn. 7) or differencing (eqn. 4); instead, the pathline should be directly determined from the reconstructed particle fields.
This concept has been explored recently by the authors (Lynch and Scarano, 2013: fluid trajectory correlation or FTC). In summary, this technique consists of four steps: first, the fluid parcel pathline 𝐱! 𝑡 is estimated from available velocity fields. Second, the estimated pathline is corrected using cross-correlation analysis. Third, the pathline is parameterized using a suitable basis function such as polynomial functions. Finally, provided that the chosen basis function is differentiable, the properties of the pathline such as instantaneous velocity and its time rate of change (VMD) can be extracted.
According to the method of operation of FTC, one particle field is designated as the reference pattern and the other fields are cross-correlated with it. The associated timing diagram differs from that of the standard approach because pulse L2 is
designated as the central image. Pulses L1 and L4 are separated
by an equal amount ∆t! from pulse L2. The pulse L3 is
positioned at ∆t! from pulse L2. This shorter time separation allows for a robust predictor to be generated from the
correlation of L2 and L3.
The parameterization of the pathline by polynomial functions allows for a connection to the discussion of truncation errors of the Eulerian and Lagrangian schemes. N is defined as the number of measured samples of the fluid pathline (in this paper, N = 4). A polynomial of degree P is to be fitted to these pathline samples. For the case of a single coordinate direction, this can be expressed as,
𝑥′ 𝑡 = !!!!𝑎! 𝑡 − 𝑡! !+ 𝑂 ∆T !!! (8)
where there are P+1 independent parameters 𝑎!, 𝑎!, … , 𝑎!, requiring at least P + 1 samples to yield a unique solution. Thus, the polynomial order must satisfy P ≤ N − 1. When N = 4, the
problem becomes linear regression analysis with polynomials of first or second degree. The second derivative of these polynomials yields the VMD. Note that a first degree polynomial yields a second derivative of zero, and is thus excluded. A second degree polynomial yields the following expression for the VMD,
!!
!! 𝑡 = 2𝑎! (9)
This is a constant expression, but the use of centered data in the polynomial fitting procedure maintains a reduced truncation error of 𝑂(∆T)!. This can be shown by deriving central-difference operators via polynomial fitting, for example by Tannehill et al. (1997). Note that the value of 𝑂(∆T)! is a reduced truncation error compared to the Eulerian and Lagrangian schemes.
A schematic depiction of the FTC procedure for P = 2 is shown in figure 6, for comparison to the Lagrangian method in figure 4. The trajectory predicted by a linear function is shown in light grey, and is generated from the predictor shown as the black arrow. The blue vectors are the corrections to the trajectory as returned by the cross-correlation analysis. The red dashed curve 𝑥′ 𝑡 is the polynomial fitted to the trajectory points. The acceleration is returned from the second derivative of 𝑥′ 𝑡 , and is shown as the red vector. Some characteristics of the process can be seen: first, for a polynomial of degree 2, the polynomial does not pass exactly through the points and is instead the product of a least squares minimization to the trajectory positions. Second, no truncation error is formed from integration because the corrector is determined from cross-correlation analysis. Finally, the method is time-centered/symmetric compared to the Lagrangian scheme, which reduces the growth of truncation errors compared to the situation in figure 4.
2. NUMERICAL ANALYSIS OF EVALUATION SCHEMES
The three schemes proposed above vary widely in their method of operation, which results in unique characteristics related to their truncation errors and sensitivity to random errors. Additionally, the FTC method has not to this point been evaluated from the perspective of acceleration measurement. The equations and graphical depictions provided in earlier sections provide a guideline to the behavior of the truncation error, but are difficult for providing a direct Figure 5. ‘FTC’ timing configuration of the two laser systems and three tomographic PIV systems.
Figure 6. Schematic diagram of FTC prediction, correction, and fitting procedure.
comparison between methods. In this section, a comparison is provided using synthetic image data to evaluate the performance of the schemes for rotational and convective motion.
Synthetic Data Generation
The reference field used for comparison is that of a translating Gaussian vortex, as used before by de Kat and van Oudheusden (2011) for a parametric study of instantaneous pressure evaluation from PIV data. The tangential linear velocity is given by the equation,
𝑉!"# =!!"! 1 − 𝑒 !!!
!! (11)
where Γ is the circulation, 𝑟! is the vortex core radius, 𝑐!= 𝑟!!/𝛾, and 𝛾 = 1.256 to place the peak tangential linear velocity at 𝑟!. This is converted into corresponding Cartesian components and convection is applied by adding a component 𝑉!"#$ along the x-direction and to the vortex core position. The selection of 𝑟!, Γ, and 𝑉!"#$ are derived from the window size used in the interrogation; for this study, 𝑟!= 5𝑊𝑆, 𝑉!"#$= 0.5𝑊𝑆/∆𝑡!, and Γ is calculated from equation 11 to yield 𝑉!"# = 0.5𝑊𝑆/∆𝑡! at 𝑟!. A set of synthetic images are generated with a size of 15𝑟! x 10𝑟! pixels at a particle density chosen to yield approximately 10 particles within an interrogation window. The particles are modeled at 2 px diameter, with no noise or out-of-plane motion simulated. This is to maintain the focus of the discussion on the differences in the acceleration schemes used instead of the errors in the velocity fields. The particle positions are integrated in time using a fourth-order ODE solver to the corresponding time instants needed for the acceleration schemes.
The timescales present in the flow include a convective time scale, defined as 𝜏!= 2𝑟!/𝑣!, and a rotational time scale, defined as 𝜏!= 2𝜋𝑟!/𝑣!"# where 𝑣!"# is the rotational velocity at 𝑟!. For brevity, these two timescales are not investigated separately. Rather, a global timescale of the flow 𝜏! is defined as the minimum between 𝜏! and 𝜏!.
The velocity fields are estimated from the synthetic images using a multi-pass, multi-grid FFT-based correlation with Gaussian weighted windows and image deformation of a linearly interpolated predictor with an 8 x 8 sinc kernel. A final interrogation window size of 16 px2 is chosen with vector spacing of 4 pixels to yield a 75% overlap factor.
Results
The tradeoff of truncation errors and random errors in the acceleration schemes is primarily a function of the temporal measurement interval, and this is the variable to be investigated in this section. There is a secondary effect of spatial modulation based on the velocity field gradient; this is for the moment not considered since for all simulations the window size is much smaller than the vortex core radius.
A visual comparison of the schemes is provided in figure 7, which shows contours of the horizontal component of the VMD as analyzed by all three schemes (columns) for increasing temporal intervals (rows). The values of the material acceleration are normalized by multiplication by ∆𝑡!! and a division by the window size; thus, the value represents the change in pathline velocity within a window as a fraction of the window size. The reference value is shown at the top of the figure. Inspecting the Eulerian and Lagrangian methods, both show a distortion in the VMD which increases with an increasing temporal interval. It has been confirmed through inspection of the propagated trajectories (Lagrangian approach) that the cause of this distortion is not due to the trajectory estimation procedure (eqn. 7), but rather by the limited-order finite difference used with the endpoints of the trajectory (eqn. 6). This causes the acceleration to be defined at a non-centered time (recall figure 4). Furthermore this is consistent with the differences in the order of the truncation errors for both stages of the Lagrangian method. Regarding random errors, the Eulerian method qualitatively appears more susceptible to the random noise in the measurement (this is particularly pronounced as the overlap factor is increased), while the Lagrangian method, since velocity measurements at the endpoints of a long trajectory are differenced, is relatively immune to the small scale fluctuations caused by random error in the velocity.
In contrast, the FTC scheme does not exhibit the distortion affecting the first two schemes. This is firstly due to the correlation-based approach that reduces the truncation error of pathline estimation. Secondly, the use of a higher-order polynomial reduces the truncation compared to velocity differencing. Finally, the time-centered nature of the technique prevents a preferential distortion due to convection compared to the Eulerian and Lagrangian schemes. The occurrence of a few outliers for FTC is observed in the case with largest temporal separation. This is ascribed to the combined effect of the low accuracy of the predictor and the limited search range for the corrector.
Figure 7. Comparison of material acceleration estimators for a translating Gaussian vortex. Actual value of material acceleration and color scaling given at top. Columns represent methods: Eulerian (left), Lagrangian (center), and FTC (right). Rows show the effect of increasing the total measurement interval ∆T.
It can be concluded from the above results that the increase of temporal interval ∆T can be very detrimental to both the Eulerian and Lagrangian schemes. A more specific comparison is offered in Figure 8, which shows the relative errors of the schemes as evaluated at points P1, P2, P3, and averaged over the region Pint as indicated in the actual data in
figure 7. The error grows about linearly for the Eulerian and Lagrangian schemes, the former with approximately three times larger slope. This change in slope can be explained through the differences in the truncation error expressions given in equations 4 and 6. Specifically, the truncation error of the Eulerian scheme is dependent on the magnitude of the second derivative of the velocity, and the Lagrangian scheme is dependent on the magnitude of the second derivative of the pathline velocity. As detailed in Christensen and Adrian (2002), the former can be much larger due to the convection of coherent structures which contributes to the derivative. If convection is accounted for, the magnitude of the derivative is reduced and represents primarily the effects of rotation.
In the synthetic flow used in this test, the ratio of rotational timescales to convection timescales (𝜏!/𝜏!) was calculated and found to be approximately 3.1, i.e., the vortex turnover time is three times larger than the convection timescale. This ratio is in agreement with the different slopes observed in the error behavior between the Eulerian and Lagrangian schemes. Furthermore, at points P1 and P2 where the rotation levels are near their maximum, the values
agree closely with the estimate. Point P3 lies outside of the vortex core, therefore the rotational timescale is longer in
this region of the flow, and accordingly, the Lagrangian method shows lower estimates of the error as would be expected in a region of low rotational motion.
Regarding the FTC method, it shows very little dependence on the temporal separation up to values of the time ratio of 0.7 and above. Beyond this value, the occurrence of outliers causes a rapid rise in the error. Due to the time-central nature of the FTC scheme, the truncation error does not follow the same linear behaviour as the Eulerian and Lagrangian schemes, but behaves as a higher-order scheme.
Figure 8. Relative error magnitude of the VMD for Eulerian, Lagrangian, and FTC schemes for points P1, P2, P3, and the region Pint, as a function of the temporal measurement interval.
Dashed lines indicate scaling guidelines for the truncation errors per equations 4, 6, and 9.
These results, although not completely representative of the situation encountered in true measurements, provide fundamental insights on the application of these techniques. It is clear that the use of four-pulse data leads to constraints on time accuracy of the Eulerian and Lagrangian approaches, and the truncation error becomes substantial as the temporal measurement interval is increased relative to the flow timescales. For example, with the Eulerian method the truncation error becomes greater than 10% even when the temporal measurement interval is only 10% of the flow timescale. The FTC technique retains the time accuracy, circumventing truncation errors over a much larger temporal measurement interval compared to the other schemes. However, for large temporal measurement intervals (> 60% of the flow timescale), the FTC approach may become unable to correct the pathline since the correction may lie outside of the correlation search radius, resulting in a rapid growth in the error. A simple outlier detection or SNR threshold may be considered to detect this limit in practice and prevent the solution from degrading.
3. EXPERIMENTAL APPARATUS
The model used for testing is an axisymmetric backward facing step manufactured from black anodized aluminium. The model consists of a 100 mm long contoured nose transitioning to a 160 mm long and 50 mm diameter cylindrical body as shown in figure 9. Protruding from the base of the model is a 20 mm diameter cylindrical afterbody (corresponding to a step height of 15 mm, or h/D = 0.3). The length of the afterbody is set to 90 mm, corresponding to a ratio l/D = 1.8 for a fully solid reattachment (see e.g., Deck et al., 2009). The model is supported by a thin sting with a symmetric airfoil profile of 100 mm chord length and maximum thickness of 8 mm. Separate measurements have confirmed a velocity deficit of less than 5% at the separation point due to the sting. To ensure uniform transition to turbulent flow, a
strand of 0.3 mm thick zig-zag tape is placed near the nose. Within the wake, a field of view is chosen from x/D of -0.1 to 1.25 (-5 to 62.5 mm, total 67.5 mm), and y/D of -0.4 to 0.3 (-20 to 15 mm, total 35 mm). Furthermore, the laser volume thickness is set to 10 mm, corresponding to +/- 0.1 z/D.
Testing is performed in the W-tunnel at the Aerodynamics Laboratory of TUDelft. This is an circuit, open-test-section wind tunnel capable of operation in the range of 5-35 m/s; for all experiments conducted herein, the freestream tunnel velocity is set to 20 m/s, corresponding to a Reynolds number based on body diameter of 62,000. Particle seeding is provided by a PIVtec Laskin Nozzle aerosol generator operating with DEHS particles of a nominal 1 micron diameter, injected upstream of the turbine inlet.
In order to acquire images in accordance with the four-pulse timing diagrams of figures 3 and 4, and without the use of polarization filtering, a large array of independent cameras are used. The camera array consists of 12 Lavision Imager LX 2MP cameras in a configuration of 2 rails, with 6 cameras on each rail, as shown in figure 10. The cameras are connected to a single acquisition computer using three four-port Ethernet cards. These cameras have 1628 x 1236 pixel resolution, with square pixels of 4.4 micron size. Each camera is equipped with a 75 mm Tamron C-mount lens attached to custom-manufactured Schiempflug mounts. In accordance with the field of view specified above, the digital resolution of the system is 24 px/mm, corresponding to a magnification of 0.1. Using an f-number of 8, the depth of field of the optical system is approximately 18.5 mm and the resulting particle image diameter in pixel units (𝑑!∗) is 2.6 px.
Figure 10. Hardware configuration of 12 camera system (left) and custom low-cost Schiempflug adapters (right).
Illumination is provided by two lasers: pulses L1 and L2 from a Spectra-Physics PIV-400 laser (400 mJ/pulse,
reduced to 200 mJ/pulse), and L3 and L4 from a Big Sky PIV-200 laser (200 mJ/pulse). The beams are conditioned with
optical irises to ensure comparable beam diameters within the measurement volume. The two output beams are passed through independent cylindrical lens arrangements to form volumetric laser slabs of 35 mm height and 10 mm depth that are combined in the measurement volume. The time separation ∆t! between laser pulses was set to 26 microseconds to correspond to an average freestream displacement of 12 px. The separation ∆t! is adaptable and will be explicitly denoted in the following results.
The synchronization of all components and acquisition of image data was performed using Lavision Davis 8.1 software and an external timing unit. The acquired images were pre-processed using time-series statistics, including a time-series pixelwise minimum subtraction, time-series pixelwise average normalization, and a subtraction of a constant background intensity. The tomographic reconstruction is performed using the FastMART algorithm in the Davis
Figure 9. Schematic of the generic axisymmetric backward facing step model (left). Photograph of model mounted in wind tunnel (right).
software using 10 iterations and a constant initial condition. A sample of the reconstruction is presented in figure 11, showing the laser sheet projections within the volume (summed along the x-direction). Note that the first three pulses are well-aligned; due to an oversight in the experimental setup, pulse L4 is not completely overlapping the existing
pulses. This deficiency has been taken into account in the remainder of the processing, and only overlapping regions of the volume are used for analysis.
Figure 11. Laser sheet projections summed along the x-direction (top row); corresponding intensity profiles and SNR values (bottom row).
The overlapping reconstruction volumes are analysed with a multi-pass, multi-grid iterative volume deformation algorithm using symmetric block direct correlations over Gaussian weighted windows, with a final window size of 48 px3 and a vector spacing of 12 pixels, yielding volumes of 133 x 60 x 21 vectors, or a total of approximately 168,000
vectors. Trilinear interpolation is used for both velocity field interpolation and volume deformation. Within the algorithm, a three-dimensional mask defining the main body and the afterbody is applied to the volumes and to the velocity fields to prevent interferences from the walls from introducing errors in the correlation and to make the solution respect the no-slip condition at solid interfaces. The FTC results are provided using an identical algorithm, but with modifications as outlined in Lynch and Scarano (2013) and modified to handle three-dimensional flows, nonuniform pulse spacing, and a single initial predictor.
3. VELOCITY FIELD CHARACTERISTICS AND ERROR ESTIMATION
Figure 12 presents two sample instantaneous velocity fields, from the perspective of a centered streamwise slice and multiple cross-stream planes. Clearly, the standard features of a separating-reattaching flow are present. At the separation point a thin shear layer forms which grows with downstream distance, eventually being characterized by the existence of multiple vortical structures. These structures travel downstream and towards the afterbody. Beneath the shear layer, a recirculation region is found which can exhibit reverse velocities on the order of 30% of the freestream velocity, in accordance with previous findings by Hudy et al. (2007). As evidenced in the two snapshots given below, the location of the recirculation region is not fixed, and meanders along the afterbody in time resulting in a highly unsteady recirculation bubble. These are all classic features of an unsteady separated flow, and indicate the ability of the measurement system to capture the primary dynamics of interest.
Figure 12. Streamwise (left column) and cross-stream (right column) visualizations of the instantaneous velocity fields. Streamwise component of velocity shown. Stripes along afterbody indicate positions in increments of 0.25 x/D.
To qualify the accuracy of the data, an analysis of the system performance is made using a zero-time delay measurement as inspired by Christensen and Adrian (2002), Perret et al. (2006), among others. The determination of 𝜀! is critical to ensure the independent tomographic systems are calibrated to a sufficient accuracy. Briefly, a zero-time delay measurement is performed by setting ∆t! to 3 microseconds (the minimum achievable exposure time of the cameras), and ∆t! to a variable setting, typically much longer than ∆t!. In this configuration, the flow can be assumed ‘frozen’ within the time interval ∆t!, but with substantial motion over ∆t!. The image analysis of data acquired in this manner involves the correlation of images from pulses L1 and L3 (yielding u1) for comparison to the correlation result
from L2 and L4 (yielding u2). The resulting error in the velocity estimate at any point within these velocity fields can be
defined as a difference in the magnitude of the two results, which encompasses both the fluctuating random and bias errors inherent in the measurement (see e.g., Christensen and Adrian, 2002),
𝜀!≈ !! (|𝐮!| − |𝐮!|)! (12)
Furthermore, the measurement error is often made dimensionless to provide insight into the dynamic range of the measurement. For this a relative error 𝜀!,! can be defined as 𝜀!,!= 𝜀!/|𝐮!"#|, where 𝐮!"# is the velocity at a chosen reference point, or in this case, the freestream velocity of 20 m/s. It follows then, as per Adrian (1997), that the dynamic velocity range is the inverse of 𝜀!,!.
Figure 14 shows an instantaneous (left) and average (right) error map. From an instantaneous perspective, errors seem to be distributed randomly in space, but concentrated slightly more within the recirculation region. On average, the largest errors occur near the wall, where large shear levels as well as reflections and limited viewing from the cameras contribute to an increase in the uncertainty. This is accentuated at the back wall of the model where the reflections are strongest, which is believed to cause the inability to visualize the secondary corner vortex structure which are seen in other planar studies such as Hudy et al. (2007).
Figure 14. Contours of the instantaneous (left) and average (right) relative error, for the case of ∆t! = 3 microseconds, ∆t! = 30 microseconds.
The adaptability of the system to various values of ∆t! allows for an estimation of an optimal time delay for velocity measurements, which will set the value of ∆t! for the acceleration measurement. Three points are extracted from figure 14 corresponding to topological features of the flow, including the freestream region, the separated shear layer, and the recirculation region near the wall. Additionally, an integral is taken over the region Pint which represents
the global relative error of the measurement.
In the freestream region, the relative error continually decreases as the time separation is increased, due to the lack of constraining factors such as shear or out-of-plane motion to affect the measurement. In contrast, point P2 shows a
increase in the relative error after a time separation between 20 and 30 microseconds as the shear level grows within an interrogation volume. These represent the two most extreme cases,
and the remaining points show intermediate behavior. Based on these observations, 26 microseconds is taken as the choice of the time separation ∆t! for the following measurements of acceleration. 4. VMD MEASUREMENT
The attention now turns to the measurement of the VMD. The previous two sections provided guidelines on the time separations to be used for the acceleration measurement (refer to figures 3 and 5): ∆t! is set based on a minimization of the global error of the velocity field as determined from zero time delay measurements, and ∆t! is set based on the timescale of the underlying flow structures. This flow timescale is calculated based on the Strouhal number of the smallest structures estimated to occur in the flow, the small-scale vortices generated in the shear layer, which occur at a StD ≈ 1.6 (see
Berger et al., 1990). By using an estimate of the convection velocity equal to half the freestream velocity, the timescale t!"#$ associated with this Strouhal number is 1.56 msec. However, from figure 8 it is
Figure 15. Relative error at multiple points in the flow as shown in figure 14, as a function of the time separation between snapshots.
seen that for a Lagrangian method, the growth of errors greatly increases as the time interval is extended, while FTC remains relatively constant. For a proper comparison, the measurement interval is chosen where the errors of the two methods are largely equivalent, i.e., ∆T = 10t!"#$. Note that this selection of the measurement interval is in agreement with previous works related to pressure evaluation which suggest a oversampling of the flow frequency by a factor of 10 (see de Kat and van Oudheusden [18]. For this experiment, this corresponds to a measurement interval of ∆T = 156 microseconds. For the Lagrangian approach this corresponds to setting ∆𝑡!= 4∆𝑡! and for an equivalent FTC timing, ∆𝑡!= 3∆𝑡!.
Figure 16 presents an instantaneous snapshot of the acceleration field using the Lagrangian (left) and FTC (right) schemes. Keep in mind that due to the different timing schemes used by the two techniques, they are unable to be applied on identical snapshots and thus the flow field will be different. As expected, the variation in the VMD is largest in the shear layer region, attributed to the formation of vortical structures. In particular, the signature of an azimuthally aligned vortex tube in the u-component of the VMD consists of alternating negative and positive values as shown earlier in the synthetic testing section. This pattern can be loosely identified in the visualizations of figure 16, particularly for the Lagrangian case (keep in mind these are separate snapshots). A qualitative comparison of the two indicates the FTC method to return greater values within the recirculation region, but also a greater number of outliers particularly near the walls and at some locations within the flow that affect the smoothness of the result. The cause of the outliers in the latter case is still under investigation, but is conjectured to be due to a limited spatial resolution of the measurement. This limited spatial resolution affects the correlations performed at extended temporal separations in the FTC algorithm.
Figure 16. Streamwise (left column) and cross-stream (right column) visualizations of the instantaneous velocity fields. Streamwise component of velocity shown.
Since a reference measurement of the material acceleration is unavailable by other physical probes, a comparison of accuracy between the two schemes must call upon the adherence to the physical properties of the flow. For incompressible flows, a common test used for comparisons of velocity is appealing to the solenoidal nature of the flow, i.e., the divergence must be zero across the vector field. For the material acceleration, imposing the zero divergence condition on equation 3 does not yield a considerable simplification due to the nonlinear convective terms. However, it is possible to use the connection between the material acceleration and the pressure gradient to enforce a physical condition. If the viscous terms of equation 1 are neglected (valid in most regions of the flow except shear layers) and the density is held constant, taking the curl of both sides yields equation 13. Note that the curl of the gradient of a scalar quantity is zero; therefore, the curl of the material acceleration field must also be zero. The physical interpretation is thus that neglecting viscosity, the material acceleration field is irrotational.
∇×!𝐮!"= ∇×∇𝑝 = 0 (13)
An example of this operation is shown in figure 17, which shows the Laplacian of the velocity field (left), and the curl of the VMD (right). The Laplacian provides an indication of regions of the flow where viscous behaviors are present, and as expected, these are concentrated in the shear layer and reattachment region of the flow. A white rectangle is shown in the figure, which indicates the portion of the flow from which statistics will be computed.
Figure 17. Sample Laplacian of the velocity field (left), and magnitude of the curl of the material derivative (right).
A probability distribution function is compiled for the curl of the VMD within the indicated region. The results are shown in figure 18 for all VMD methods. Note that per equation 13, the PDF should collapse to a peak centered at 0 for the physically correct measurement. A clear trend is observed in the PDFs for all coordinate directions: the Lagrangian method provides a noticeable improvement over the Eulerian method, and furthermore the FTC method provides an additional improvement even more substantial than the shift from Eulerian to Lagrangian methods. This indicates that, at least for this region of the flow, the FTC method is superior to the Eulerian and Lagrangian methods for VMD estimation.
a) x-component
b) y-component
c) z-component
Figure 17. Streamwise (left column) and cross-stream (right column) visualizations of the instantaneous velocity fields. Streamwise component of velocity shown.
CONCLUSIONS
A system for determining the material derivative of velocity that suitable for high-speed flows has been described. Through the use of a 12-camera tomographic PIV configuration, four independent tomograms of the particle field can be acquired, which allows for the temporal rate of change of the velocity to be determined. It was found that through the volume self-calibration procedure the independent camera systems could be calibrated to the same physical coordinates, eliminating ambiguities in the positional matching of the reconstructed volumes. This was subsequently verified through the use of a zero-time-delay test, which demonstrated errors below 0.1 pixel displacement in the freestream between independent camera systems.
Multiple methods for analyzing pulse data were investigated. In particular, it was noted that the use of four-pulse data results in larger temporal truncation behavior for Eulerian and Lagrangian techniques, the latter even if advanced pathline tracing schemes are employed. In contrast, the authors propose the use of the fluid trajectory correlation (FTC) technique to retain a time-centered approach for evaluating the VMD (due to the timing of the laser pulses) while simultaneously reducing the influence random errors (due to the least-squares trajectory fit). Synthetic tests of a translating vortex over increasing temporal intervals confirmed these hypotheses, showing a truncation error growth for the Eulerian and Lagrangian schemes, and a slower growth of truncation error for the FTC technique.
The analysis was extended to application on actual experimental data of a highly unsteady separated flow. Without knowledge of the actual VMD field, an appeal was proposed to use the irrotationality of the VMD field in inviscid flow regions. Within these regions, it was clearly seen that the FTC technique provided an improvement in the measurement compared to the Eulerian and Lagrangian schemes in the sense that the solution more clearly respected the irrotationality of the VMD field.
Based on these findings, accelerometry in high-speed flows using tomographic PIV has been demonstrated to be feasible using a 12-camera system. The timing configuration and temporal measurement interval however are heavily dependent on the post-processing techniques to be used to extract the acceleration. The proposed FTC method provides the ability for truncation errors to be avoided and is thus highly recommended in order to extend the measurement time interval and thus the range of the measurement. However, the authors note that the FTC method currently exhibits sensitivity to outliers in the VMD field. This must be addressed in order to ensure the robustness of the technique compared to the Eulerian and Lagrangian methods.
REFERENCES
Adrian RJ (1997) Dynamic ranges of velocity and spatial resolution of particle image velocimetry. Meas Sci Technol 8:1393-1398. Atkinson KE (1989) An introduction to numerical analysis. Wiley.
Berger E, Scholz D, Schumm M (1990) Coherent vortex structures in the wake of a sphere and a circular disk at rest and under forced vibrations. J Fluids Structs 4:231-257.
Christensen KT, Adrian RJ (2002) Measurement of instantaneous Eulerian acceleration fields by particle image accelerometry: method and accuracy. Exp Fluids 33:759-769.
Dong P, Hsu TY, Atsavapranee P, Wei T (2001) Digital particle image accelerometry. Exp Fluids 30:626-632.
Elsinga GE, Wieneke B, Scarano F, van Oudheusden BW (2006) Tomographic particle image velocimetry. Exp Fluids 41:933-947.
Ghaemi S, Ragni D, Scarano F (2012) PIV-based pressure fluctuations in the turbulent boundary layer. Exp. Fluids 53:1823-1840. Guibert P, Lemoyne L (2002) Dual particle image velocimetry for transient flow field measurements. Exp Fluids 33:355-367.
Hu H, Saga T, Kobayashi T, Taniguchi N, Yasuki M (2001) Dual-plane stereoscopic particle image velocimetry: system set-up and its application on a lobed jet mixing flow. Exp Fluids 31:277-293.
Hudy LM, Naguib A, Humphreys WM (2007) Stochastic estimation of a separated-flow field using wall-pressure-array measurements. Phys Fluids 19:024103.
Jakobsen ML, Dewhirst TP, Greated CA (1997) Particle image velocimetry for predictions of acceleration fields and force within fluid flows. Meas Sci Technol 8:1502-1516.
Kähler CJ, Kompenhans J (2000) Fundamentals of multiple plane stereo particle image velocimetry. Exp Fluids S70-S77. de Kat R, van Oudheusden BW (2011) Instantaneous planar pressure determination from PIV in turbulent flow. Exp Fluids.
de Kat R, Ganapathisubramani B (2013) Pressure from particle image velocimetry for convective flows: a Taylor’s hypothesis approach. Meas Sci Technol 24:024002.
Lynch K, Scarano F (2013) A high-order time-accurate interrogation method for time-resolved PIV. Meas Sci Technol 24:035305.
Liu X, Katz J (2006) Instantaneous pressure and material acceleration measurements using a four-exposure PIV system. Exp Fluids 41:227-240.
van Oudheusden BW, Scarano F, Roosenboom EWM, Casimiri EWF, Souverein LJ (2007) Evaluation of integral forces and pressure fields from planar velocimetry data for incompressible and compressible flows. Exp. Fluids 42:153-62.
van Oudheusden BW (2013) PIV-based pressure measurement. Meas Sci. Technol. 24:032001.
Perret L, Braud P, Fourment C, David L, Delville J (2006) 3-component acceleration field measurement by dual-time stereoscopic particle image velocimetry. Exp Fluids 40:813-824.
Scarano F (2013) Tomographic PIV: principles and practice. Meas Sci Technol 24:012001.
Soloff SM, Adrian RJ, Liu ZC (1997) Distortion compensation for generalized stereoscopic particle image velocimetry. Meas Sci Technol 8:1441-1454.
Souverein LJ, van Oudheusden BW, Scarano F, Dupont P (2009) Application of a dual-plane particle image velocimetry (dual-PIV) technique for the unsteadiness characterization of a shock wave turbulent boundary layer interaction. Meas Sci Technol 20:074003. Thurow B, Jiang N, Lempert W (2013) Review of ultra-high repetition rate laser diagnostics for fluid dynamic measurements. Meas. Sci
Technol. 24:012002.
Violato D, Moore P, Scarano F (2010) Lagrangian and Eulerian pressure field evaluation of rod-airfoil flow from time-resolved tomographic PIV. Exp fluids 50:1057-1070.
Wieneke B (2008) Volume self-calibration for 3D particle image velocimetry. Exp Fluids 45:549-556.
