Multiple-pulse PIV: numerical evaluation and experimental validation
Liuyang Ding1, Stefano Discetti2, R. J. Adrian1, S. Gogineni31School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, Arizona, 85287
Liuyang.Ding@asu.edu
2Department of Aerospace Engineering (DIAS), University of Naples Federico II, P.le, Tecchio 80,80125, Naples, Italy
3Spectral Energies LLC. 5100 Springfield Street, Suite 301 Dayton, OH 45431 ABSTRACT
Many conventional experimental methods in fluid mechanics, such as LDV and HWA, are not able to measure simultaneous acceleration fields directly. Double-pulse PIV is extended to the multiple-pulse PIV enabling people to directly extract acceleration fields as well as to achieve better accuracy in the velocity measurement [13]. We evaluate the performances of triple-pulse PIV and quadruple-pulse PIV based on PTV simulations. By investigating various flow conditions,
optimizations are globally achieved to minimize the error in velocity and acceleration measurements for both triple-pulse and quadruple-pulse PIV. Compared to that with uneven time spacing, triple-pulse PIV with even time spacing yields better spatial resolution in velocity measurement. Moreover, quadruple-pulse PIV significantly reduces the error in both velocity and acceleration measurement relative to triple-pulse PIV. Experimental validation of the acceleration measurement from triple-pulse PIV is conducted by measuring the mean centerline acceleration profile of a round impinging air jet. A custom 8-pulse laser system in combination with an ultra high-speed 4-channel camera are utilized to well satisfy the requirements of multiple-pulse PIV. Conventional double-pulse PIV, together with the polynomial curve fitting, are used to obtain the ground truth for the mean centerline acceleration profile. The comparison between the ground truth and acceleration
measurements by triple-pulse PIV shows a good agreement with the maximum error being 10% of the full scale acceleration. Several potential error sources are also discussed to provide the direction of making improvements in the following stage. 1. INTRODUCTION
Acceleration is of great interest in fluid dynamics because it is associated with the net forces from surrounding fluid materials. In the incompressible form of Navier-Stokes equation,
𝜕𝜕𝑢𝑢𝑖𝑖 𝜕𝜕𝜕𝜕 + 𝑢𝑢𝑗𝑗 𝜕𝜕𝑢𝑢𝑖𝑖 𝜕𝜕𝑥𝑥𝑗𝑗 = − 1 𝜌𝜌 𝜕𝜕𝜕𝜕 𝜕𝜕𝑥𝑥𝑖𝑖+ 𝜈𝜈 𝜕𝜕2𝑢𝑢𝑖𝑖 𝜕𝜕𝑥𝑥𝑗𝑗𝜕𝜕𝑥𝑥𝑗𝑗 , (1) the first term on the left hand side is referred to the local rate of change or temporal derivative, and the second term on the left hand side is referred to the convective change or convective acceleration. The sum of these two terms is defined as the substantial derivative, 𝐷𝐷𝑢𝑢𝑖𝑖 𝐷𝐷𝜕𝜕 = 𝜕𝜕𝑢𝑢𝑖𝑖 𝜕𝜕𝜕𝜕 + 𝑢𝑢𝑗𝑗 𝜕𝜕𝑢𝑢𝑖𝑖 𝜕𝜕𝑥𝑥𝑗𝑗 , (2) which can be interpreted as the derivative in a reference frame following the fluid material. The measurement of the substantial derivative provides a method to investigate the net pressure and viscous forces on the right hand side of (1). In a fluid/structure interaction, such measurement in the region close to the wall can also offer a way to calculate forces acting on the interface. In turbulent flow, the statistics of acceleration also provide the tool for establishing better stochastic modeling for turbulence structure [14].
Acceleration was studied from both Lagrangian and Eulerian point of view. At early times, the Lagrangian acceleration and the acceleration correlation function in turbulence were mostly evaluated by theoretical and numerical methods [5,9,14].
Experimental measurements of fluid acceleration have been increasingly reported. From the Eulerian point of view, Gylfason et al. [4] used hot-wire anemometry (HWA) together with Taylor’s frozen field hypothesis to convert time series to spatial series, measuring the mean squared pressure gradient and the acceleration variance in active-grid wind-tunnel turbulence. From the Lagrangian point of view, Volk et al. [11] measured the particle acceleration in a turbulent Von Karman swirling flow using extended laser-Doppler velocimetry (LDV) with wide laser beams. Particle tracking velocimetry (PTV) was also applied by people to reconstruct particle trajectories and thus compute particle Lagrangian accelerations [2,8,12].
Particle image velocimetry (PIV) is a powerful and widely-used experimental method for simultaneous velocity field measurements in fluid mechanics. With a Eulerian standpoint, it has the advantage of directly offering spatial derivatives and spatial correlation over HWA and LDV, which were invented to measure point-wise velocity [13]. And also PIV is capable to offer a higher data yield than PTV as it is based on the statistical analysis of particles and thus allows higher image density then PTV [1]. Since its invention, a number of robust technologies based on conventional double-pulse PIV have been well developed to measure 2D as well as 3D flow. However, conventional PIV only captures particle positions at two times ( 𝜕𝜕 and 𝜕𝜕 + ∆𝜕𝜕 ) and approximates the Eulerian velocity using first order finite difference,
𝒖𝒖�𝒙𝒙𝒑𝒑(𝜕𝜕), 𝜕𝜕� ≅𝒙𝒙𝒑𝒑(𝜕𝜕+∆𝜕𝜕)−𝒙𝒙∆𝜕𝜕 𝒑𝒑(𝜕𝜕) , (3)
where 𝒙𝒙𝒑𝒑(𝜕𝜕) denotes the Lagrangian particle position at time 𝜕𝜕. Thus, this method is not capable to measure temporal
acceleration, and the accuracy of the convective acceleration measurement is limited by the spatial resolution achieved in PIV cross-correlation analysis.
In order to extend conventional PIV to the measurement of acceleration, particle positions at additional time instants are required. Christensen and Adrian [3] developed a particle image accelerometer (PIA) measuring Eulerian acceleration fields by time differencing two continuous measurements of the Eulerian velocity fields. Two standard PIV systems, each on composed by a dual-cavity Nd:YAG laser and a double frame camera, are used to guarantee a short enough time separation between two measured velocity fields ( two pairs of pulse, two pulses per camera). More recently, time-resolved PIV (TR-PIV) has been well developed thanks to the availability of high-speed CMOS cameras and Nd:YLF laser with high repetition rate. TR-PIV is suited for acceleration field measurements in the sense that it combines the beauty of high frame rate and high data yields. Thus it offers the possibility of measuring other quantities derived from Navier-Stokes equation, such as the pressure field [10]. However, the acquisition frequency of TR-PIV can be typically up to 5 kHz with
1024×1024-pixel images [13], which is still not well suited for high-speed air flow measurements, where the measurement of Lagrangian quantities is of great interest.
Multiple-pulse PIV has been increasingly attractive to people in recent years [6,13]. With additional particle positions at very short delayed times (enough for high speed air flow measurements), a polynomial interpolation can be performed to estimate the particle trajectory. Furthermore, velocity and acceleration can be easily estimated by differentiating the interpolated trajectory. This method is intended to achieve greater velocity accuracy and enable people to measure Eulerian acceleration fields directly. Haranandani [6] quantitatively analyzed the performance of triple-pulse PIV with the second pulse located at 1/3 of the time separation between the first and the third pulse. He found that the velocity error is markedly reduced by triple-pulse PIV compared to double-pulse PIV, except for the case with fairly small path length and
acceleration. The accuracy of locating particle is always significantly improved by triple-pulse PIV.
In present work, the performances of triple-pulse and quadruple-pulse PIV with even time spacing are assessed. Similarly to the work by Haranandani [6], optimizations regarding minimum error in position and velocity are evaluated and discussed on the basis of PTV simulations. For quadruple-pulse PIV, the optimization regarding acceleration error is also investigated and discussed. Experimental validation for triple-pulse PIV is performed by measuring the mean centerline acceleration profile of the well-known case of a round impinging air jet, and comparing the results with the ground truth obtained from the theoretical formula.
2. THEORETICAL BACKGROUND
Suppose particle images are captured at pulse times 𝜕𝜕1, 𝜕𝜕2, 𝜕𝜕3, …, which can be either a single particle image or particles
clusters for cross-correlation-based PIV. Following the convention of Haranandani [6], also to distinguish triple-pulse and quadruple-pulse PIV, the locations of image centers here are denoted by
𝑿𝑿
𝑁𝑁
𝜕𝜕𝑝𝑝 = 𝑿𝑿𝑁𝑁 𝜕𝜕(𝜕𝜕𝑝𝑝), 𝑝𝑝 = {1,2, … , 𝑁𝑁}, (4)
where the left superscripts denote the total number of pulses with which particle images are sampled. For simplicity, all temporal and spatial points are referred to the first pulse, 𝜕𝜕𝑝𝑝′ = 𝜕𝜕𝑝𝑝− 𝜕𝜕1, 𝑿𝑿𝑁𝑁 𝜕𝜕𝑝𝑝′ = 𝑿𝑿𝑁𝑁 𝜕𝜕𝑝𝑝 − 𝑿𝑿𝑁𝑁 𝜕𝜕1, ∆𝜕𝜕 = 𝜕𝜕𝑁𝑁− 𝜕𝜕1; for the
triple-pulse, 𝛼𝛼∆𝜕𝜕 = 𝜕𝜕2− 𝜕𝜕1; for the quadruple-pulse, 𝜆𝜆∆𝜕𝜕 = 𝜕𝜕2− 𝜕𝜕1, 𝜇𝜇∆𝜕𝜕 = 𝜕𝜕3− 𝜕𝜕1. Then the interpolation for the particle position
is given by 𝑿𝑿� 3 𝜕𝜕 ′(𝜕𝜕′) = 3𝑿𝑿𝜕𝜕2′ −𝛼𝛼23𝑿𝑿𝜕𝜕3′ 𝛼𝛼(1−𝛼𝛼)∆𝜕𝜕 𝜕𝜕′− 𝑿𝑿 3 𝜕𝜕2 ′ −𝛼𝛼 𝑿𝑿3 𝜕𝜕3 ′ 𝛼𝛼(1−𝛼𝛼)∆𝜕𝜕2 𝜕𝜕′2 (5) for the triple-pulse, and
𝑿𝑿�
4 𝜕𝜕
′(𝜕𝜕′) = 𝒂𝒂
1𝜕𝜕′+ 𝒂𝒂2𝜕𝜕′2+ 𝒂𝒂3𝜕𝜕′3 (6)
for the quadruple-pulse, where
𝒂𝒂1= 𝜇𝜇 𝑿𝑿 4 𝜕𝜕2 ′ 𝜆𝜆(1−𝜆𝜆)(𝜇𝜇 −𝜆𝜆)∆𝜕𝜕− 𝜆𝜆 𝑿𝑿4 𝜕𝜕3′ 𝜇𝜇(1−𝜇𝜇 )(𝜇𝜇−𝜆𝜆)∆𝜕𝜕+ 𝜆𝜆𝜇𝜇 𝑿𝑿4 𝜕𝜕4′ (1−𝜆𝜆)(1−𝜇𝜇)∆𝜕𝜕 , (6a) 𝒂𝒂2= − (𝜇𝜇+1) 𝑿𝑿 4 𝜕𝜕2 ′ 𝜆𝜆(1−𝜆𝜆)(𝜇𝜇 −𝜆𝜆)∆𝜕𝜕2+ (𝜆𝜆+1) 𝑿𝑿4 𝜕𝜕3′ 𝜇𝜇(1−𝜇𝜇 )(𝜇𝜇−𝜆𝜆)∆𝜕𝜕2− (𝜆𝜆+𝜇𝜇 ) 𝑿𝑿4 𝜕𝜕4′ (1−𝜆𝜆)(1−𝜇𝜇)∆𝜕𝜕2 , (6b) 𝒂𝒂3= 𝑿𝑿 4 𝜕𝜕2 ′ 𝜆𝜆(1−𝜆𝜆)(𝜇𝜇 −𝜆𝜆)∆𝜕𝜕3− 𝑿𝑿 4 𝜕𝜕3 ′ 𝜇𝜇 (1−𝜇𝜇 )(𝜇𝜇−𝜆𝜆)∆𝜕𝜕3+ 𝑿𝑿 4 𝜕𝜕4 ′ (1−𝜆𝜆)(1−𝜇𝜇)∆𝜕𝜕3 , (6c) and the tilde on the top indicates estimated values.
By differentiating (5) and (6), one can estimate velocity and acceleration along the particle trajectory. The triple-pulse PIV provides linear estimation for velocity and constant estimation for acceleration, while the quadruple-pulse PIV can have quadratic estimation for velocity and linear estimation for acceleration:
𝑿𝑿�̇ 3 𝜕𝜕 ′(𝜕𝜕′) = 3𝑿𝑿𝜕𝜕2′ −𝛼𝛼23𝑿𝑿𝜕𝜕3′ 𝛼𝛼(1−𝛼𝛼)∆𝜕𝜕 − 2 𝑿𝑿 3 𝜕𝜕2 ′ −𝛼𝛼 𝑿𝑿3 𝜕𝜕3 ′ 𝛼𝛼(1−𝛼𝛼)∆𝜕𝜕2 𝜕𝜕′, (7) 𝑿𝑿�̈ 3 𝜕𝜕 ′(𝜕𝜕′) = 2𝛼𝛼 𝑿𝑿3 𝜕𝜕3′ − 𝑿𝑿3 𝜕𝜕2′ 𝛼𝛼(1−𝛼𝛼)∆𝜕𝜕2 (8) 𝑿𝑿�̇4 𝜕𝜕 ′(𝜕𝜕′) = 𝒂𝒂 1+ 2𝒂𝒂2𝜕𝜕′+ 3𝒂𝒂3𝜕𝜕′2 (9) 𝑿𝑿�̈ 4 𝜕𝜕 ′(𝜕𝜕′) = 2𝒂𝒂 2+ 6𝒂𝒂3𝜕𝜕′ (10)
where 𝒂𝒂1, 𝒂𝒂2, 𝒂𝒂3 are given in (6a) ~ (6c).
To evaluate the performance of the interpolated estimation, one needs to know the real particle trajectory for comparison. Unfortunately, in most complex flows, an analytical form of the trajectory is unknown. Therefore a reasonable model is needed here for comparison and validation. Considering a curved particle trajectory, it can be approximately regarded as a circular path line with radius equal to the local radius of curvature 𝑅𝑅. This approximation is accurate enough providing that the path length is sufficiently small relative to 𝑅𝑅 . In fact, this condition can usually be well satisfied in a PIV measurement. Suppose the displacement on the image plane is 10 pixel, with the pixel pitch 9 𝜇𝜇𝜇𝜇 and lateral magnification 0.1. Then the displacement in the physical space is 0.9 mm, which means the arc angle is ~0.1 radians if the radius of curvature 𝑅𝑅 = 10𝜇𝜇𝜇𝜇. Therefore, it is reasonable to assume the real trajectory to be a local circular path line,
𝑿𝑿𝜕𝜕′(𝜕𝜕′) = (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅[𝜃𝜃(𝜕𝜕′)] − 𝑅𝑅)𝒆𝒆𝑥𝑥+ 𝑅𝑅𝑅𝑅𝑖𝑖𝑝𝑝[𝜃𝜃(𝜕𝜕′)]𝒆𝒆𝑦𝑦 . (11)
Taking the acceleration factor into account, 𝜃𝜃(𝜕𝜕′) can be modeled as the sum of displacements due to constant angular
Figure 1 Contours of errors in position and velocity by triple-pulse PIV for γ∆t2ω = 0.3 and δXp
dτ = 0.2. (a) the error in position
σx
R ; (b) the error in velocity σv
Rω. (For visibility, different color scales are used)
𝜃𝜃(𝜕𝜕′) = 𝜔𝜔𝜕𝜕′+1
2𝛾𝛾𝜕𝜕′2 , (12)
where 𝜔𝜔 denotes the angular speed and 𝛾𝛾 denotes the angular acceleration. Thus the location of image centers are modeled as 𝑿𝑿 𝑁𝑁 𝜕𝜕𝑝𝑝 ′ = 𝑿𝑿 𝜕𝜕 ′(𝜕𝜕 𝑝𝑝) = (𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅[𝜃𝜃(𝜕𝜕𝑝𝑝)] − 𝑅𝑅)𝒆𝒆𝑥𝑥+ 𝑅𝑅𝑅𝑅𝑖𝑖𝑝𝑝[𝜃𝜃(𝜕𝜕𝑝𝑝)]𝒆𝒆𝑦𝑦 . (13)
On the other hand, in a real PIV experiment, there are always errors in locating particle centers due to electric noise, lens aberration and so on. To simulate this realistic condition, we artificially add a noise 𝛿𝛿𝑿𝑿𝜕𝜕 to the image center location:
𝑿𝑿 𝑁𝑁 𝜕𝜕𝑝𝑝𝜇𝜇 ′ = 𝑿𝑿𝑁𝑁 𝜕𝜕𝑝𝑝 ′ + 𝛿𝛿𝑋𝑋 𝜕𝜕𝑥𝑥𝒆𝒆𝑥𝑥+ 𝛿𝛿𝑋𝑋𝜕𝜕𝑦𝑦𝒆𝒆𝑦𝑦 , (14)
where the subscript 𝜇𝜇 denotes a measured location perturbed by noise. Plugging (12) and (13) into (14) and substituting all 𝑿𝑿
𝑁𝑁 𝜕𝜕𝑝𝑝
′ in (5) ~ (10) with (14), one can obtain estimations for position, velocity and acceleration. Comparing these to the
modeled “real trajectory” (11), we are able to express errors in location, 𝜎𝜎𝑁𝑁
𝑥𝑥, errors in velocity, 𝜎𝜎𝑁𝑁 𝑣𝑣, and errors in
acceleration, 𝜎𝜎𝑁𝑁
𝑎𝑎, as a function of 𝜕𝜕′,
𝜎𝜎
𝑁𝑁
𝑥𝑥= �𝑿𝑿𝜕𝜕′(𝜕𝜕′) − 𝑿𝑿�𝑁𝑁 𝜕𝜕′(𝜕𝜕′)�, 𝜎𝜎𝑁𝑁 𝑣𝑣 = �𝑿𝑿̇𝜕𝜕′(𝜕𝜕′) − 𝑿𝑿�̇𝑁𝑁 𝜕𝜕′(𝜕𝜕′)� , 𝜎𝜎𝑁𝑁 𝑎𝑎 = �𝑿𝑿̈′𝜕𝜕(𝜕𝜕′) − 𝑿𝑿�̈𝑁𝑁 𝜕𝜕′(𝜕𝜕′)�. (15)
For generalization, we further normalize 𝜎𝜎𝑥𝑥 with the radius of curvature 𝑅𝑅, 𝜎𝜎𝑣𝑣 with the full scale velocity 𝑅𝑅𝜔𝜔 and 𝜎𝜎𝑎𝑎 with
full scale acceleration 𝑅𝑅�𝜔𝜔4+ 𝛾𝛾2. Moreover, a cost function is defined as
𝐾𝐾(𝜁𝜁, 𝜂𝜂) = 𝜁𝜁𝜎𝜎𝑥𝑥 𝑅𝑅 + 𝜂𝜂 𝜎𝜎𝑣𝑣 𝑅𝑅𝜔𝜔+ (1 − 𝜁𝜁 − 𝜂𝜂) 𝜎𝜎𝑎𝑎 𝑅𝑅�𝜔𝜔4+𝛾𝛾2 , (16) to represent the combined error in position, velocity and acceleration.
The optimal 𝜕𝜕′ is determined in the sense that it minimizes the cost function (16). Several parameters are considered to find
a globally optimized 𝜕𝜕′, including normalized path length 𝑅𝑅𝜔𝜔∆𝜕𝜕/𝑅𝑅 = 𝜔𝜔∆𝜕𝜕, normalized acceleration factor 1
2𝛾𝛾∆𝜕𝜕2/𝜔𝜔∆𝜕𝜕 =
𝛾𝛾∆𝜕𝜕/2𝜔𝜔 and normalized noise level in locating the image center 𝛿𝛿𝑋𝑋𝜕𝜕/𝑅𝑅, where we assume 𝛿𝛿𝑋𝑋𝜕𝜕𝑥𝑥 = 𝛿𝛿𝑋𝑋𝜕𝜕𝑦𝑦 = 𝛿𝛿𝑋𝑋𝜕𝜕.
3. NUMERICAL SIMULATION
Haranandani [6] performed numerical simulations of triple-pulse PIV with 𝛼𝛼 = 1/3. He found the combined error in position and velocity is minimized by setting 𝜕𝜕′/Δ𝜕𝜕 = 0.74 over a wide range of varying path length, acceleration and noise
level. The origination of choosing 𝛼𝛼 = 1/3 is to generate unequal spacing between exposures on the image, so that one can easily distinguish the direction of the flow. This is valid for the single-frame triple-exposure recording strategy, however, PIV images are usually recorded today using double-frame single-exposure strategy. For triple-pulse PIV, one can either record the first pulse on the first frame and the second and third pulses on the second frame, or record three pulses on individual frames respectively ( triple-frame single-exposure). For either one the flow direction is always known without distinguishing unequal spacing on images. Therefore it is more natural to consider even time spacing, which can also simplify timings in PIV experiments. The performance of triple-pulse PIV with 𝛼𝛼 = 1/2 are evaluated and compared with the case of 𝛼𝛼 = 1/3 in terms of error in position and velocity. Quadruple–pulse PIV is also analyzed with 𝜆𝜆 = 1/3, 𝜇𝜇 = 2/3, and comparisons between the triple-pulse and quadruple-pulse are shown and discussed.
3.1 Triple-pulse simulation with even time spacing
In the simulation, the local radius of curvature is set to 𝑅𝑅 = 10 𝜇𝜇𝜇𝜇 and the particle image diameter is 𝑑𝑑𝜏𝜏 = 22.5 𝜇𝜇𝜇𝜇. The
parameter 𝛼𝛼 is set to be 1/2, i.e. equal time spacing between the three pulses. 𝜎𝜎𝑥𝑥
𝑅𝑅 , 𝜎𝜎𝑣𝑣
𝑅𝑅𝜔𝜔 and 𝜎𝜎𝑎𝑎
𝑅𝑅�𝜔𝜔4+𝛾𝛾2 are calculated for noise level 𝛿𝛿𝑋𝑋𝜕𝜕
𝑑𝑑𝜏𝜏 = 0%, 10%, 20%, 50% and acceleration factor
𝛾𝛾∆𝜕𝜕
2𝜔𝜔 = -0.4, -0.2, 0, 0.3, 0.5, 1.0 under the condition of normalized
displacement 𝜔𝜔Δ𝜕𝜕 from 0.01 to 0.5 and 𝜕𝜕′⁄ from 0 to 1. Cost function 𝐾𝐾(𝜁𝜁, 𝜂𝜂) are evaluated with 𝜁𝜁 = 𝜂𝜂 = 1/2. Δ𝜕𝜕
Figure 1(a) and 1(b) show the normalized error in position and velocity respectively for 𝛾𝛾∆𝜕𝜕2𝜔𝜔 = 0.3 and 𝛿𝛿𝑋𝑋𝜕𝜕
𝑑𝑑𝜏𝜏 = 0.2. For visibility, the color scales are different on the two figures. The vertical axis, 𝜕𝜕′/Δ𝜕𝜕, denotes the time relative to the time
separation between the first and the third pulse. The horizontal axis, 𝜔𝜔Δ𝜕𝜕, is the displacement as a fraction of the local radius of curvature. For 𝜔𝜔Δ𝜕𝜕 less than 0.2, the particle trajectory approaches a straight line. The dominant error in position of this region is due to the noise in locating particle centers, 𝛿𝛿𝑋𝑋𝜕𝜕. With noise level 𝛿𝛿𝑋𝑋𝑑𝑑𝜏𝜏𝜕𝜕=20%, the error of this region is
around 0.05% and it slightly increases with the time. For larger 𝜔𝜔Δ𝜕𝜕, the error due to the quadratic interpolation becomes the dominant error, and it is much greater than the error of the region with smaller 𝜔𝜔Δ𝜕𝜕. The minimum position error
Figure 2 Contours of the cost function (16) with ζ = η = 1/2 by triple-pulse PIV for γ∆t2ω = −0.4 and δXp
dτ = 0.
Figure 3 Contours of the cost function (16) with ζ = η = 1/2 by triple-pulse PIV for γ∆t2ω = 0 and δXp
dτ = 0.2.
Figure 4 Contours of the cost function (16) with ζ = η = 1/2 by triple-pulse PIV for γ∆t2ω = 0.5 and δXp
Figure 5 Contours of errors in position, velocity and acceleration by quadruple-pulse PIV for γ∆t2ω = 0.3 and δXp
dτ = 0.2.(For visibility, different color scales are used). (a) the error in position σx
R ; (b) the error in velocity σv
Rω; (c) the error in
acceleration σa
R�ω4+γ2.
appears at about Δ𝜕𝜕𝜕𝜕′ = 0, 0.5, 1 for approximately 0.25 < 𝜔𝜔Δ𝜕𝜕 < 0.5, which is the expected result because as the second pulse is placed half way between the first and the third pulse, i.e. 𝛼𝛼 = 1/2. In a similar plot with 𝛾𝛾∆𝜕𝜕2𝜔𝜔 = −0.4 and 𝛿𝛿𝑋𝑋𝜕𝜕
𝑑𝑑𝜏𝜏=50%, we found the position error due to noise becomes more dominant, resulting in that it even extends to the region with larger 𝜔𝜔Δ𝜕𝜕. As a consequence, the minimum position error appears at 0 < 𝜕𝜕′⁄ < 0.05 over a wide range 0.01 < 𝜔𝜔Δ𝜕𝜕 < 0.5. As Δ𝜕𝜕
for the error in velocity, it is obviously observed that 𝜕𝜕′/Δ𝜕𝜕 = 0.78 yields error in velocity very close to the minimum for
the range 0.01 < 𝜔𝜔Δ𝜕𝜕 < 0.5. Interestingly, the minimum of velocity error shows up where the position error appears as its maximum. This might be due to the fact that at this location the tangential vector of the interpolated parabolic trajectory is nearly parallel to the one of the circular path line, but they are apart from each other with a relatively large distance. To find out a optimal 𝜕𝜕′, the cost function (16) with 𝜁𝜁 = 𝜂𝜂 = 1/2 is calculated for various conditions stated above. Figure 2
shows the cost function when the particle is experiencing a deceleration factor -0.4 and with zero noise. Two minimum strips show up at 𝜕𝜕′/Δ𝜕𝜕 = 0.78 and 𝜕𝜕′/Δ𝜕𝜕 = 0.20 respectively for 0.1 < 𝜔𝜔Δ𝜕𝜕 < 0.5, while in the region 𝜔𝜔Δ𝜕𝜕 < 0.1 the error
is globally small. Figure 3 shows the case with no acceleration but with 20% noise added. It appears that the triple-pulse interpolation works pretty well over a broad range when there is no acceleration, except that the noise adds some extreme error where 𝜕𝜕′/Δ𝜕𝜕 and 𝜔𝜔Δ𝜕𝜕 are both small. By carefully checking contour levels, the minimum strip is found at
approximately 𝜕𝜕′/Δ𝜕𝜕 = 0.76~0.78. Increasing the acceleration factor to 0.5 and the noise level to 50%, errors in position
and velocity both significantly increase (see Figure 4). Interestingly enough, we can still find the minimum at approximately 𝜕𝜕′⁄ = 0.76~0.78, which is shown as the slender dark blue strip in Figure 4. Combining Figure 2~4 and other cases we Δ𝜕𝜕
explored, it is concluded that with 𝛼𝛼 = 1/2 one can always choose 𝜕𝜕′/Δ𝜕𝜕 = 0.78 to obtain the nearly minimum combined
error in position and velocity, despite different acceleration factors and noise levels. 3.2 Quadruple-pulse simulation with even time spacing
In principle, by adding an additional pulse, the real trajectory is sampled at more points so that a polynomial interpolation can more accurately recover it. Still keeping 𝑅𝑅 = 10 𝜇𝜇𝜇𝜇 and 𝑑𝑑𝜏𝜏= 22.5 𝜇𝜇𝜇𝜇, and setting 𝜆𝜆 = 1/3, 𝜇𝜇 = 2/3 in (6), we
perform simulations of the quadruple-pulse PIV with the same flow conditions as the triple-pulse simulations above. While the acceleration estimation by triple-pulse PIV is only a constant along the trajectory, we are able to have a linear estimation for acceleration using 4 pulses. Therefore it is necessary to consider the error in position, velocity as well as in acceleration. Figure 5(a) ~(c) show the error in position, velocity and acceleration respectively for 𝛾𝛾∆𝜕𝜕2𝜔𝜔 = 0.3 and 𝛿𝛿𝑋𝑋𝜕𝜕
𝑑𝑑𝜏𝜏 = 0.2. For visibility, different color scales are used. By comparing Figure 5(a) and Figure 1(a), we find that the error in position due to noise 𝛿𝛿𝑋𝑋𝜕𝜕
becomes more dominant in a wide range for quadruple-pulse PIV. Only for the region 0.4 < 𝜔𝜔Δ𝜕𝜕 < 0.5 and 0.4 < 𝜕𝜕′/Δ𝜕𝜕 <
(c) (b)
1, the error due to cubic interpolation becomes significant and it shows up as a bimodal shape. In Figure 5(b), the error in velocity has two minimum strips at approximately 𝜕𝜕′/Δ𝜕𝜕 = 0.47~0.50 and 𝜕𝜕′⁄ = 0.85~0.87. In Figure 5(c), the error in Δ𝜕𝜕
acceleration has only one minimum strip at approximately 𝜕𝜕′⁄ = 0.67~0.72. Interestingly, as discussed in the triple-Δ𝜕𝜕
pulse results, the minimum velocity error appears where the location error reaches its maximum. Furthermore, the minimum acceleration error shows where velocity error has maximum and location error has minimum. Besides the physical
interpretation discussed in triple-pulse, one can also generally relate this to a mathematical description: the nth order zero crossings of a continuous function show up where the (n-1)th order extremum appear.
One should notice that the errors in position, velocity and acceleration have different order of magnitude. Generally speaking, the acceleration error is one order larger than the velocity error, and the velocity error is one order larger than the position error. To provide more useful guidelines, the performances of the quadruple-pulse are assessed in two scenarios: (i) evaluate cost function (16) with 𝜁𝜁 = 𝜂𝜂 = 1/3 for the situation that acceleration and velocity need to be measured
simultaneously or only acceleration is of great interest; (ii) evaluate cost function (16) with 𝜁𝜁 = 𝜂𝜂 = 1/2 for the situation that one performs only velocity measurement using quadruple-pulse PIV. Two optimal 𝜕𝜕′ are given below for scenario (i)
and (ii).
Figure 6 Contours of the cost function (16) with ζ = η = 1/3 by quadruple-pulse PIV for γ∆t2ω = −0.4 and δXp
dτ = 0.
Figure 7 Contours of the cost function (16) with ζ = η = 1/3 by quadruple-pulse PIV for γ∆t2ω = 0 and δXp
dτ = 0.2.
Figure 8 Contours of the cost function (16) with ζ = η = 1/3 by quadruple-pulse PIV for γ∆t2ω = 1.0 and δXp
dτ = 0.5.
Figure 9 Contours of the cost function (16) with ζ = η = 1/2 by quadruple-pulse PIV for γ∆t2ω = 1.0 and δXp
dτ = 0.5.
Figure 10 Contours of the cost function (16) with ζ = η = 1/2 by quadruple-pulse PIV for γ∆t2ω = −0.4 and δXp
Case Displacement Acceleration factor
Image noise
Velocity error (%) Position error (%)
𝝎𝝎∆𝒕𝒕 𝜸𝜸∆𝒕𝒕/𝟐𝟐𝝎𝝎 𝜹𝜹𝑿𝑿𝒑𝒑/𝒅𝒅𝝉𝝉 4-PIV 3-PIV
𝜶𝜶 = 𝟏𝟏/𝟐𝟐
3-PIV
𝜶𝜶 = 𝟏𝟏/𝟑𝟑
2-PIV* 4-PIV 3-PIV
𝜶𝜶 = 𝟏𝟏/𝟐𝟐 3-PIV 𝜶𝜶 = 𝟏𝟏/𝟑𝟑 2-PIV* 1 0.1 0 1% 0.002 0.006 0.014 0.06 0.003 0.003 0.003 0.13 2 0.1 0.1 1% 0.002 0.018 0.017 0.26 0.003 0.008 0.012 0.29 3 0.2 0 10% 0.009 0.028 0.07 0.28 0.029 0.031 0.034 0.5 4 0.2 0.25 20% 0.016 0.13 0.18 1.33 0.059 0.11 0.16 1.47 5 0.2 0 2% 0.002 0.013 0.015 0.17 0.006 0.005 0.007 0.5 6 0.3 0 10% 0.006 0.035 0.05 0.4 0.029 0.022 0.025 1.12 7 0.3 0.25 10% 0.005 0.22 0.18 1.96 0.026 0.16 0.25 2.6 8 0.6 0 20% 0.007 0.11 0.094 1.5 0.057 0.11 0.21 4.47 9 0.6 0.25 20% 0.037 0.61 0.60 4.38 0.017 0.65 1.03 7.83 10 0.8 0.25 20% 0.083 0.99 1.04 6.4 0.073 1.26 1.99 13.1 11 1 0 0% 0.015 0.31 0.33 4.5 0.051 0.79 1.29 13.4 12 1 0.25 0% 0.15 1.49 1.65 9.41 0.27 2.23 3.48 21.5
Table 1. Comparison of quadruple-pulse (4-PIV), triple-pulse (3-PIV) and double-pulse (2-PIV) errors in velocity and position for various flow conditions. Columns with asterisk are cited from Haranandani (2011).
Case Displacement Acceleration factor Image noise Acceleration error (%) 𝝎𝝎∆𝒕𝒕 𝜸𝜸∆𝒕𝒕/𝟐𝟐𝝎𝝎 𝜹𝜹𝑿𝑿𝒑𝒑/𝒅𝒅𝝉𝝉 4-PIV 3-PIV 𝜶𝜶 = 𝟏𝟏/𝟐𝟐 1 0.1 0 0% 0.003 0.021 2 0.1 -0.4 0% 0.075 0.5 3 0.1 0.5 20% 0.58 2.43 4 0.2 0 10% 0.71 3.24 5 0.2 0.5 0% 0.32 1.58 6 0.2 0.5 20% 0.3 2.13 7 0.5 -0.4 10% 0.37 2.06 8 0.5 0.5 10% 1.59 6.61 9 0.5 1 20% 3.84 12.74 10 0.8 0.5 10% 3.88 13.9 11 1 1 0% 16.42 42.46
Table 2. Comparison of quadruple-pulse (4-PIV) and triple-pulse (3-PIV) errors in acceleration for various flow conditions.
Cost function (16) with 𝜁𝜁 = 𝜂𝜂 = 1/3 is evaluated with various flow conditions as stated above. Three of them are shown in Figure 6~8 respectively. With a deceleration factor -0.4 and zero noise, the quadruple-pulse PIV broadly works well as shown in Figure 6. Two minimum strips are symmetrically found at 𝜕𝜕′/Δ𝜕𝜕 = 0.28 and 𝜕𝜕′⁄ = 0.71~0.72. By adding Δ𝜕𝜕
some noise level, the symmetry is lost but the minimum strip still remains except for some extreme large errors as shown in Figure 7 and Figure 8. In Figure 7 where 𝛾𝛾∆𝜕𝜕2𝜔𝜔 = 0 and 𝛿𝛿𝑋𝑋𝜕𝜕
𝑑𝑑𝜏𝜏 = 0.2, the minimum strip is at 𝜕𝜕
′⁄ = 0.67~0.68. However, by Δ𝜕𝜕
carefully examining raw data, the error at 𝜕𝜕′⁄ = 0.70 is no more than 2% at 0.2 < 𝜔𝜔Δ𝜕𝜕 < 0.27 and no more than 1% at Δ𝜕𝜕
0.27 < 𝜔𝜔Δ𝜕𝜕 < 0.5, where it is even down to 0.1%. Figure 8 shows the cost function when 𝛾𝛾∆𝜕𝜕2𝜔𝜔 = 1.0 and 𝛿𝛿𝑋𝑋𝜕𝜕
𝑑𝑑𝜏𝜏 = 0.5, in which the minimum strip is relatively broad and it is located at 0.67 < 𝜕𝜕′⁄ < 0.72. By observing Figure 6~8 and other Δ𝜕𝜕
cases, for quadruple-pulse PIV with even spacing in time, 𝜕𝜕′⁄ = 0.70 is considered as the optimal location where one can Δ𝜕𝜕
If velocity measurement is of great interest in a quadruple-pulse PIV experiment, with 𝜕𝜕′⁄ = 0.70 one can get relatively Δ𝜕𝜕
small errors in velocity, but not be able to minimize them. Therefore the cost function (16) with 𝜁𝜁 = 𝜂𝜂 = 1/2 is also evaluated for quadruple-pulse PIV so that the acceleration error is ignored. Figure 9 & 10 show two groups of the results. Two minimum strips appear in the contour plots, corresponding to 𝜕𝜕′⁄ = 0.47~0.50 and 𝜕𝜕Δ𝜕𝜕 ′⁄ = 0.86~0.88, and the Δ𝜕𝜕
latter one is preferred by carefully comparing the raw data. Simulations with other flow conditions show nearly the same results. Thus it is concluded that 𝜕𝜕′⁄ = 0.87 yields minimum combined error in velocity and position for quadruple-pulse Δ𝜕𝜕
PIV with even spacing in time.
In a recent review paper [13], triple-pulse and double-pulse errors are compared and discussed, from which we see that the performance of PIV is markedly improved by adding a third pulse. Here, we compare the performance of the triple-pulse with 𝛼𝛼 = 1/2 and the quadruple-pulse to the triple-pulse with 𝛼𝛼 = 1/3 as well as double-pulse PIV. Table 1 shows the error in velocity and position for different cases. The column of 3-PIV with 𝛼𝛼 = 1/3 are calculated using 𝜕𝜕′⁄ = 0.74 [6]; Δ𝜕𝜕
the column of 3-PIV with 𝛼𝛼 = 1/2 are calculated using 𝜕𝜕′⁄ = 0.78 as concluded above; the column of 4-PIV are Δ𝜕𝜕
calculated using 𝜕𝜕′⁄ = 0.87 to obtain minimum combined error in position and velocity. Comparing 3-PIV with 𝛼𝛼 = 1/2 Δ𝜕𝜕
and 𝛼𝛼 = 1/3, it is observed that the accuracy of velocity measurement does not change much. However the error in locating particles is reduced by approximately 20% ~ 40%, especially for the cases (cases 8 ~12) with large displacement relative to the radius of curvature which might corresponds to small diameter vortices. This improvement implies that one can achieve better spatial accuracy when a Lagrangian velocity vector is assigned to a Eulerian point by using equal temporal spacing in triple-pulse PIV. More interestingly, quadruple-pulse PIV works amazingly well in improving the accuracy of velocity estimation. With small normalized displacement and zero acceleration factor (cases 1, 3), the quadruple pulse reduces the velocity error by 2/3. Increasing the tangential acceleration (cases 2,4) or increasing displacement (cases 5,6), the velocity error is reduced to 10% ~ 20% relative to 3-PIV. For even larger displacement (cases 8 ~ 12), the velocity error of 4-PIV is one order of magnitude smaller than the error of 3-PIV. On the other hand, 4-PIV works better in estimating the particle position when the normalized displacement is large (cases 7 ~12).
Table 2 shows the comparison of acceleration errors between the quadruple-pulse and the triple-pulse. The column of 4-PIV are calculated using 𝜕𝜕′⁄ = 0.70 as proposed above. For triple-pulse PIV, there is no globally optimal 𝜕𝜕Δ𝜕𝜕 ′ to minimize the
acceleration error. However, when the particle has a normalized displacement 𝜔𝜔Δ𝜕𝜕 greater than 0.3, minimum error in acceleration can be found at approximately 𝜕𝜕′⁄ = 0.5. Thus we use 𝜕𝜕Δ𝜕𝜕 ′⁄ = 0.5 to roughly represent the acceleration Δ𝜕𝜕
error of triple-pulse PIV, as listed in the column of 3-PIV with 𝛼𝛼 = 1/2. When the particle displacement is small relative to the radius of curvature and the tangential acceleration factor is small (case 1), both of them works well. It is also found that image noise within 20% of the image diameter does not significantly influence the acceleration estimation accuracy (cases 5,6). Increasing the displacement and the acceleration factor, the variation of the tangential and the centrifugal acceleration of a particle becomes more significant. In such case (cases 8~12), a linear (4-PIV) or a constant (3-PIV) estimation of the acceleration becomes less accurate. In addition, quadruple-pulse PIV can reduce the error by approximately 70%~85%
relative to triple-pulse PIV, resulting in that the error of 4-PIV is no more than 1% for cases 1~7 and 4% for extreme cases 8~10. Therefore we can conclude that acceleration estimation is significantly improved by adding a fourth pulse and using a cubic interpolation to reconstruct the particle trajectory.
4. EXPERIMENT 4.1 Experimental setup
In this preliminary stage, the capability of triple-pulse PIV to measure the acceleration is experimentally validated by investigating the mean centerline axial acceleration profile of a round impinging air jet. A schematic of the multiple-pulse PIV system used in our experiment is shown in Figure 11. To satisfy the requirements of multiple-pulse PIV experiments, a custom 8-pulse Nd:YAG laser system, which produces eight independently-triggered, 532 nm laser pulses, is utilized to achieve time separations between pulses as short as needed. All eight laser beams are co-linearly combined by carefully-designed beam-combing optics in front of four dual-cavity laser heads, to achieve same optical path lengths as well as at least 95% overlap of all output laser beams [7]. Two 8-channel BNC 565 Pulse Generators are used for triggering flash lamps and Q-switches respectively, enabling one to precisely and flexibly control the timing for each laser pulse.
Imaging is achieved by a HSFC-Pro ultra high-speed camera, which is composed by four individual imaging channels and four 1280×1024-pixel CCD arrays with pixel pitch 6.7 𝜇𝜇𝜇𝜇 . The input light ray is divided into four independent ones behind the objective by an integrated beam splitter system that ensures equal optical path lengths for each channel and CCD registration errors within 2 pixels. Such registration errors can be further corrected by cross-correlating simultaneous particle images on different CCD arrays. This imaging architecture makes it possible to straddle different frames on different CCDs without affected by significant registration errors. Therefore the sampling frequency would not be limited by the data transfer speed of CCD arrays. Another BNC 565 Pulse Generator together with a SRS DG 535 unit are used to synchronize the laser pulses and the exposures of the camera. In our experiment, laser pulses 5,7,8 are synchronized with CCD arrays 1,2,3 respectively using single-trigger mode. One can also set the HSFC-Pro camera to the double-shutter mode to fully take advantages of the 8-pulse laser system. By combining the flexibility and versatility of the 8-pulse laser system and the 4-channel high-speed imaging system, the sampling frequency of the tested flow can be well satisfied for multiple-pulse PIV experiments.
A round impinging air jet with approximately 𝑅𝑅𝑅𝑅 = 30000 is investigated using triple-pulse PIV method. The centerline of the jet is normal to the impinging plate within experimental tolerance. A schematic of the test section is shown in Figure 12. The expanded laser sheet is adjusted to go through the axial axis so that the object plane can be approximately considered as an azimuthal iso-surface. The exit nozzle has a diameter D=21.59 mm and it is mounted with a distance H=3.82D above the impinging plate. An area of 960 by 672 pixels is cropped out of the original image, which leads to a 3.00D (w) ×2.10D (h) region of interest in the physical space right above the impinging plate. DEHS (Di-Ethyl-Hexyl-Sebacat) aerosol generated by a Laskin nozzle is used as tracer particles. Particle images are captured with the magnification about 0.10 and the effective f-number about 20, resulting in the particle image diameter is approximately 4.3 pixels. The time separation ∆𝜕𝜕 between pulses 1 and 3 is chosen as 80 𝜇𝜇𝑅𝑅. Using 𝛼𝛼 = 0.5, the time separation 𝛼𝛼∆𝜕𝜕 between pulses 1 and 2 is set to be 40 𝜇𝜇𝑅𝑅. Cross-correlation with multigrid interrogation strategy is applied to analyze frames 1,2 and frames 1,3 to obtain the
predefined 𝑿𝑿3 𝜕𝜕2
′ and 𝑿𝑿3 𝜕𝜕3
′ respectively, with final interrogation window size 48×48 pixels corresponding to
3.24mm×3.24mm in the physical space, and with 75% overlap between adjacent windows. A 3×3 Gaussian smoothing kernel with standard deviation H=1.2 is used to smooth the vector field. A total number of 1000 realizations are recorded and averaged. Eventually mean velocity vector fields with the dimension 53 rows by 77 columns are obtained, and the centerline is selected as the 33rd column by examining the symmetry of the mean velocity fields.
4.2 Experimental results and discussions
Figure 13 (a) and (b) illustrate the mean velocity vector fields together with the contours of the velocity magnitude, from interrogation results of pulses 1,2 and pulses 1,3 respectively. The figures are presented in cylindrical coordinate system (𝑟𝑟, 𝜃𝜃, 𝑥𝑥) and both radial and axial coordinates are normalized by the exit nozzle diameter D. Symmetry of the jet about the axial axis r=0 is obviously observed. For the region above approximately x/D=1.0, the mean flow is nearly a unidirectional constant flow towards the negative axial direction, and the magnitude of the axial velocity symmetrically decays with the radius r increasing. As approaching the impinging plate, the flow starts to separate into radial direction and it experiences a
Figure 13 Mean velocity vector fields and contours of the velocity magnitude (a) measured using pulses 1&2 (b) measured using pulses 1&3
strong deceleration in axial direction. For given x, the maximum radial velocity magnitude shows up at approximately r/D=0.5, where the flow has a strong curvature corresponding to a small radius of curvature.
The r.m.s. fluctuation contour plot, Figure 14, reveals more interesting physical characters of the impinging jet. The potential core, inside which the flow is nearly laminar, is shown as the light blue region along the centerline. Beyond the core , the flow strongly interacts with the ambient still air forming a shear layer surrounding it. It is also noticed that the shear layer grows as it approaches the impinging plate, gradually invading the potential core. Moreover, in the near wall region ( approximately x/D<0.25), the solid boundary has significant influences on fluids leading to relatively intense fluctuations.
By comparing Figure 13 (a) and (b), one can hardly tell the differences between them. However, in principle, using a shorter time separation would yield greater accuracy when the Eulerian quantities are approximated by Lagrangian quantities, especially at the location where fluid materials have strong accelerations. A more quantitative comparison between the measurements by analyzing pulses 1&2 and pulses 1&3 is present in Figure 15, showing the differences of the centerline axial velocity profile. In our experiment, the flow is considered as statistically stationary and ergodic, so the mean acceleration equals to the mean convective acceleration, i.e.
𝐷𝐷
𝐷𝐷𝜕𝜕 < 𝒖𝒖 >=< 𝒖𝒖 ∙ ∇𝒖𝒖 > (17)
where <∙> denotes ensemble averages. Therefore, for approximately x/D>1 in Figure 15, it is considered as no acceleration since the velocity profile does not vary with x. After this, the flow starts to decelerate appeared as the decreasing of the velocity in space. In this region, the measured velocities using pulses 1&3 are smaller than those using pulses 1&2, which is due to the deceleration during the three sampled time instants 𝜕𝜕1, 𝜕𝜕2, 𝜕𝜕3. The amount of the difference in velocity
measurements can be used to quantitatively estimate the acceleration. This is because one can think the two velocity measurements using pulses 1&2 and pulses 1&3 as the velocity estimations for two times, 12(𝜕𝜕1+ 𝜕𝜕2) and 12(𝜕𝜕1+ 𝜕𝜕3), and
then naturally estimate the acceleration by time differencing them. In fact, this is equivalent to what the triple-pulse PIV does in acceleration estimation, as previously shown in equation (8).
Figure 14 Contours of r.m.s fluctuations, measured using pulses 1&3.
Moreover, the mean centerline axial acceleration profile can also be obtained by taking the advantage of (17). Adopting the form in cylindrical coordinate system and simplifying it with symmetry, we can express the mean axial acceleration along the centerline as : 𝐷𝐷 𝐷𝐷𝜕𝜕 < 𝑢𝑢𝑅𝑅>= 1 2 𝜕𝜕 𝜕𝜕𝑥𝑥(< 𝑢𝑢𝑅𝑅>2+< 𝑢𝑢𝑅𝑅′𝑢𝑢𝑅𝑅′ >) (18)
where the centerline axial velocity 𝑢𝑢𝑅𝑅 is decomposed into the mean < 𝑢𝑢𝑅𝑅> and the fluctuation 𝑢𝑢𝑅𝑅′, i.e. 𝑢𝑢𝑅𝑅=< 𝑢𝑢𝑅𝑅 > +𝑢𝑢𝑅𝑅′.
Conventional PIV is applied to pulses 1 and 2 to measure the centerline velocity. A degree 9 polynomial is used to fit a curve to the quantity (< 𝑢𝑢𝑅𝑅>2+< 𝑢𝑢𝑅𝑅′𝑢𝑢𝑅𝑅′ >). Then the acceleration profile can be obtained by analytically differentiating
the fitted polynomial. We set this to be the ground truth and compare it to the acceleration profile measured by the triple-pulse PIV, as shown in Figure 16.
As shown by the ground truth in Figure 16, the acceleration maintains zero beyond x/D=1. As approaching the impinging plate, its magnitude sharply increases to the maximum at approximately x/D=0.2, and then even more sharply decreases within a very thin layer above the plate. Data of acceleration measurements by triple-pulse PIV are plotted as the red circles. Generally speaking, a good agreement is shown in the comparison especially in the decelerating region 0.1<x/D<0.7. Except for extreme errors of the two data points close to the plate (discussed below), the maximum error is about 2000 m/s2 showing up at approximately x/D=0.8 and x/D=1.7, which is about 10% of the full scale acceleration at x/D=0.2. However, in the region of 0.8<x/D<0.2, the error exhibits a particular pattern which may not be considered caused by random errors in the experiment. The reason of this bias error is not fully understood thus far. Several potential sources of the error are discussed here.
(a) Errors of conventional PIV measurements. The errors in PIV measurements can be caused by many factors, such as electrical random noise of cameras, aberrations of lenses, nonuniform distribution of particles, algorithms used to process images, etc. Regardless of source, there is an error 𝜎𝜎 in determining the displacement between different frames on the image plane. Considering the estimation (8) for acceleration by triple-pulse PIV, with pixel pitch 6.7 𝜇𝜇𝜇𝜇, magnification 0.1 and ∆𝜕𝜕 = 80 𝜇𝜇𝑅𝑅 in our experiment, the maximum error in Figure 16, 2000 m/s2, corresponds to an error of 0.024 pixels in the term 𝛼𝛼 𝑿𝑿3
𝜕𝜕3
′ − 𝑿𝑿3
𝜕𝜕2
′ . This means 𝜎𝜎=0.016 pixels could cause an error up to 2000 m/s2 in the acceleration measurement with 𝛼𝛼 = 0.5, which indicates that the accuracy of the acceleration measurement by triple-pulse PIV is quite sensitive to the error in determining the displacement.
(b) Errors due to quadratic interpolation. With three sampled points, the best one can do is to reconstruct the particle trajectory using quadratic interpolation, resulting in a constant estimation for the acceleration. Errors are introduced in this process since the acceleration is estimated using first-order finite difference. As shown in Table 2, this kind of error can be significantly reduced by adding an additional pulse, i.e. the quadruple-pulse PIV, to achieve a linear estimation for the acceleration.
(c) Errors due to experimental setup. Because of the non-ideal geometry of the exit nozzle and the mounting errors, the jet impinges onto the plate with a small angel of about 1.1°, leading to a approximately 24-pixel distance on the image between the stagnation point and the plumb point. Therefore when the centerline velocity is considered along a single column of the velocity vector field, what one actually does is considering the project of the centerline velocity onto the vertical direction. In addition, due to the illumination scheme, strong reflections occur on the plate surface. The region of interest is selected right above the plate, so interrogation spots near the surface are highly polluted by the noise. This may explain the reason that the two data points close to the pate in Figure 16 are bad.
5. SUMMARY
Acceleration is a critical quantity in fluid mechanics as it is associated with forces in fluids. In turbulence, its statistical properties also help to better interpret turbulent structures. Conventional quantitative experimental methods, such as LDV and HWA, are not able to measure simultaneous acceleration fields directly. Thus the double-pulse PIV is extended to the multiple-pulse PIV enabling people to directly extract the acceleration fields as well as to achieve better accuracy and better spatial resolution in velocity measurements. With multiple particle positions, the particle trajectory can be reconstructed by fitting a polynomial to them. The velocity and acceleration are estimated by differentiating the fitted polynomial with respect to time.
In present study, the performances of triple-pulse PIV and quadruple-pulse PIV are assessed on the basis of PTV simulations. By modeling the real trajectory as a local circular path, errors in position, velocity and acceleration are
considered for the optimization. Conclusions on the optimized 𝜕𝜕′ are given through simulations for different flow conditions.
As for the triple-pulse PIV with even time spacing, i.e. 𝛼𝛼 = 1/2, the optimized 𝜕𝜕′ is globally determined at 𝜕𝜕′/Δ𝜕𝜕 = 0.78,
where the combined error in position and velocity is minimized. From the comparison between two different timing Figure 15 Comparison of mean centerline axial
velocity profiles. Blue squares: measured using pulses 1&2; Red diamonds: measured using pulses 1&3.
Figure 16 Comparison of mean centerline axial acceleration profiles. Black solid line: ground truth; Red circles: acceleration measurement by triple-pulse PIV.
configurations (see Table 1), 𝛼𝛼 = 1/2 and 𝛼𝛼 = 1/3, the accuracy of velocity measurement doesn’t exhibit obvious
improvements, whereas greater accuracy in locating the particle position is achieved by setting 𝛼𝛼 = 1/2. This indicates that triple-pulse PIV yields better spatial resolution in velocity measurement when using even time spacing, as it assigns Lagrangian velocities to Eulerian points in the flow.
The optimizations for the quadruple-pulse PIV are performed in two scenarios, aiming at the velocity measurement and the acceleration measurement, respectively. 𝜕𝜕′/Δ𝜕𝜕 = 0.70 is concluded as the optimized time that minimizes the combined error
in position, velocity and acceleration, while 𝜕𝜕′/Δ𝜕𝜕 = 0.87 should be used to obtain the best accuracy when only the velocity
measurement is of great interest in a quadruple-pulse PIV experiment. The performance of quadruple-pulse PIV is compared to that of triple-pulse PIV in terms of the position error, velocity error and acceleration error (see Table 1,2). It turns out that quadruple-pulse PIV significantly, up to an order of magnitude, increases the accuracy in velocity and acceleration measurements, especially in the cases with large normalized displacements and strong variations occurring in velocity and acceleration
The capability of triple-pulse PIV to measure the acceleration is experimentally validated by measuring the mean centerline acceleration profile of a round impinging air jet. A custom 8-pulse laser system in combination with an ultra high-speed 4-channel camera are utilized to well satisfy the requirements of multiple-pulse PIV. Because the flow is statistical stationary and ergodic, the mean centerline acceleration can be expressed as the mean convective acceleration involving the evolution of centerline velocities ( see (17), (18) ). Conventional PIV is applied to measure velocities along the centerline. A
continuous centerline acceleration profile is obtained by fitting a polynomial to those centerline velocities and
differentiating it analytically. We set this to be the ground truth and compare our triple-pulse acceleration measurements with it. Despite the two data points close to the wall which may be affected by the experimental setup, the comparison in Figure 16 shows a good agreement with the maximum error being about 10% of the full scale acceleration. The result is very promising in the decelerating region, while there may be some bias errors in the region of constant flow which is not yet fully understood. Three potential sources of errors are discussed, including errors of conventional PIV method, errors due to quadratic interpolation applied in the triple-pulse PIV ( i.e. due to the finite number of sampled positions) and errors introduced by the experimental setup. Interestingly, the error of acceleration measurement by triple-pulse PIV is found to be highly sensitive to the error in determining the displacement between different pulses. For instance, a 0.016-pixel error in our experiment could cause a 2000 m/s2 error in the acceleration measurement, which is the maximum error showing up in the comparison of the centerline acceleration profile (Figure 16).
Considering the sensitivity of the error in acceleration measurements by triple-pulse PIV, the quadruple-pulse PIV may be more suited to extract the instantaneous Eulerian acceleration fields. With our 8-pulse laser system and 4-channel camera, quadruple-pulse PIV would be experimentally investigated in the following stage. Methods of compensating for the errors will also be studied. On the other hand, the higher accuracy in velocity measurements by multiple-pulse PIV needs to be validated both numerically and experimentally in the future.
REFERENCES
[1] Adrian RJ and Westerweel J “Particle Image Velocimetry” Cambridge University Press (2010)
[2] Ayyalasomayajula S, Gylfason A, Collins LR, Bodenschatz E, Warhaft Z “Lagrangian measurements of inertial particle accelerations in grid generated wind tunnel turbulence” Physical Review Letters 97 (2006) 144507
[3] Christensen KT and Adrian RJ “Measurement of instantaneous Eulerian acceleration fields by particle image accelerometry: method and accuracy” Experiments in Fluids 33 (2002) pp. 759-769
[4] Gylfason A, Ayyalasomayajula S, Warhaft Z “Intermittency, pressure and acceleration statistics from hot-wire measurements in wind-tunnel turbulence” J. Fluid Mech. 501 (2004) pp. 213-229
[5] Hill RJ and Thoroddsen ST “Experimental evaluation of acceleration correlations for locally isotropic turbulence” Physical Review E 55 (1997) pp. 1600-1606
[7] Murphy MJ “Development of an ultra-high speed dynamic witness-plate particle image velocimeter for micro-detonator studies” PhD thesis, Arizona State Univ.
[8] Ni R, Huang SD, Xia KQ “Lagrangian acceleration measurements in convective thermal turbulence” J. Fluid Mech. 692 (2012) pp. 395-419 [9] Pinsky M, Khain A, Tsinober A “Accelerations in isotropic and homogeneous turbulence and Taylor’s hypothesis” Phys. Fluids 12 (2000) pp. 3195-3204
[10] Violato D, Moore P, Scarano F “Lagrangian and Eulerian pressure field evaluation of rod-airfoil flow from time-resolved tomographic PIV” Exp. Fluids 50 (2011) pp.1057-1070
[11] Volk R, Mordant N, Verhille G, Pinton JF “Laser Doppler measurement of inertial particle and bubble accelerations in turbulence” EPL 81 (2008) 34002
[12] Voth GA, Porta AL, Crawford AM, Alexander J, Bodenschatz E, “Measurement of particle accelerations in fully developed turbulence” J. Fluid Mech. 469 (2002) pp. 121-160
[13] Westerweel J, Elsinga GE, Adrian RJ “Particle image velocimetry for complex and turbulent flow” Annu. Rev. Fluid Mech. 45 (2013) pp.409-436
[14] Yeung PK “One- and two-particle Lagrangian acceleration correlation in numerically simulated homogenous turbulence” Phys. Fluids 9 (1997) pp. 2981-2990
